/00faý, THE HIGHWAY DESIGN AND MAINTENANCE STANDARDS SERIES Vehicle Speeds 10082 and Operating Costs o Models for Road Planning and Management Thawat Watanatada, Ashok M. Dhareshwar, and Paulo Roberto S. Rezende Lima Report No. 10082 FILE COPY A World Bank Publication THE HIGHWAY DESIGN AND MAINTENANCE STANDARDS SERIES Vehicle Speeds and Operating Costs Models for Road Planning and Management Thawat Watanatada Ashok M. Dhareshwar Paulo Roberto S. Rezende Lima in collaboration with Patrick M. O'Keefe Per E. Fossberg Published for The World Bank The Johns Hopkins University Press Baltimore and London C 1987 The International Bank for Reconstruction and Development / The World Bank 1818 H Street, N.W., Washington, D.C. 20433, U.S.A. All rights reserved Manufactured in the United States of America The Johns Hopkins University Press Baltimore, Maryland 21211 First printing December 1987 The findings, interpretations, and conclusions expressed in this study are the results of research supported by the World Bank, but they are entirely those of the authors and should not be attributed in any manner to the World Bank, to its affiliated organizations, or to members of its Board of Executive Directors or the countries they represent. Library of Congress Cataloging-in-Publication Data Watanatada, Thawat Vehicle speeds and operating costs: models for road planning and management / by Thawat Watanatada, Ashok M. Dhareshwar, Paulo Roberto S. Rezende Lima; in collaboration with Patrick M. O'Keefe, Per E. Fossberg. p. cm. - (The Highway design and maintenance standards series) Bibliography: p. 1. Motor vehicles-Cost of operation-Mathematical models. 2. Motor vehicles-Speed-Mathematical models. I. Dhareshwar, Ashok M., 1949- . II. Lima, Paulo Roberto S. Rezende, 1944- III. Title. IV. Series. TL151.5.T48 1987 388.3'144-cl9 87-22176 ISBN 0-8018-3589-5 Foreword An effective road transportation network is an important factor in economic and social development. It is also costly. Road constructian and maintenance consume a large proportion of the national budget, while the costs borne by the road-using public for vehicle operation and depreciation are even greater. It is therefore vitally important that policies be pursed which, within financial and other constraints, minimize total transport costs for the individual road links and for the road network as a whole. To do this meaningfully, particularly when dealing with large and diverse road networks, alternatives must be compared and the tradeoffs between them carefully assessed. This in turn requires the ability to quantify and predict performance and cost functions for the desired period of analysis. Because of the need for such quantitative functions, the World Bank initiated a study in 1969 that later became a large-scale program of collaborative research with leading research institutions and road agencies in several countries. This Highway Design and Maintenance Standards Study (HDM) has focused both on the rigorous empirical quantification of the tradeoffs between the costs of road construction, road maintenance, and vehicle operation and on the development of planning models incorporating total life- cycle cost simulation as a basis for highways decisionmaking. This volume is one in a series that documents the results of the HDM study. The other volumes are: Vehicle Operating Costs Evidence from Developing Countries Road Deterioration and Maintenance Effects Models for Planning and Management The Highway Design and Maintenance Standards Model Volume 1. Description of the HDM-III Model The Highway Design and Maintenance Standards Model Volume 2. Users' Manual for the HDM-III Model Road-user costs are by far the largest cost elements in road transport. Improvements in road conditions, although costly, can yet pay substantial dividends by reducing vehicle operating costs and hence generate large net benefits to the national economy as a whole. Thus, expressing vehicle operating costs in relation to road characteristics-geometry and pavement condition-is the logical approach. For certain cost components, especially fuel consumption, the required data can be obtained through controlled experiments, whereas for others, especially vehicle maintenance costs, extensive road-user surveys are needed. Both approaches were used in the HDM studies in Kenya, Brazil, and India and in the study in the Caribbean sponsored by the British Transport and Road Research Laboratory. The resulting body of knowledge on road-user costs is enormous. It covers conditions on three continents with diverse highway conditions and in radically different economic environments. This volume takes an aggregate-mechanistic view of vehicle speed and operating costs under free flow conditions. Basing their analysis on the mechanistic principles of propulsion and motion as well as on postulated assumptions of driver behavior, the authors arrive at predictions at three levels of detail, ranging from a simulation method for use in detailed geometric design at the link level to an aggregate method for use in investment planning at the sectoral level. The models were estimated using the comprehensive data base collected in the Brazil-UNDP-World Bank highway research project and were validated along with data sets from India. One of the significant contributions to highway economics research made in this study is the probabilistic limiting velocity approach to steady-state speed prediction, which, combined with the aggregate-mechanistic methodology, will provide a possible basis for future research on the far more complex problem of operating costs under congested conditions. This volume is to some extent a companion to Vehicle Operating Costs: Evidence from Developing iii Countries, which is based on an aggregate-correlative methodology that considers vehicle operating costs equations in an economic context. These two approaches are complementary and elucidate different aspects and different components of the road-user cost complex. Although the relationships described in this volume form part of the HDM-III model, they can also be used on their own. Chapter 14 provides guidelines for implementing the relationships on either mainframe or microcomputers, as well as a set of resource component tables. With the specific road characteristics provided by the user, the models predict travel speed and resource consumption in physical units for the typical vehicle classes and, when combined with prices or unit costs of the relevant resources, predict the user costs. A guide to the approach and methodology may be obtained by reading the chapters and sections shown in the diagram below. Also, readers interested only in prediction formulas and policy applications may continue with Part III (Chapters 13 through 16) after Chapter 1. Clell G. Harral Per E. Fossberg Principal Transport Economist Highways Adviser 1I 2.1, 2.2 3.1, 3.2, 3.3, 3.5 9.1, 9.3, 941. 112.1, 12.2,9 2. 10.3, 10.8 Fuel1 Tire Maintenance, depreciation, interest I E E El EApplications iv Contents Chapter 1. Introduction 1.1 Study Objectives 1 1.2 Model Criteria and Research Approach 3 1.2.1 General modelling framework 3 1.2.2 Criteria for vehicle operating cost prediction models 4 1.3 Outline of Aggregate-Mechanistic Approach 7 1.3.1 Scope 7 1.3.2 Basicapproach 9 1.3.3 Steady-state models 10 1.3.4 Micro transitional and non-transitional models 11 1.4 Adaptive Behavior in Response to Changes in Road Characteristics 12 1.5 Organization of the Book 13 Appendix 1A. Population and Test Vehicles 14 Part I. Vehicle Speed Prediction Models 17 Chapter 2. The Force Balance Relationship 19 2.1 Roadway Attributes and Their Measures 19 2.1.1 Vertical alignment 19 2.1.2 Horizontal alignment 21 2.1.3 Surface characteristics 22 2.1.4 Cross section 22 2.2 Basic Relationships 23 2.2.1 Effective vehicle mass 23 2.2.2 Vehicle drive force 25 2.2.3 Gravitational force 25 2.2.4 Air resistance 25 2.2.5 Rolling resistance 26 2.3 Applications 29 Appendix 2A. Development of Relationship between Coefficient of Rolling Resistance and Road Roughness 31 Chapter 3. Steady-State Speed Model: Formulation 43 3.1 Problem Statement and Approach 43 3.2 Limiting Speed Model: Approach and Basic Formulation 46 3.3 Limiting Speed Model: Derivation of Constraining Speeds 51 3.3.1 Driving power-limited speed, VDRIVE 51 3.3.2 Braking power-limited speed, VBRAKE 52 3.3.3 Curvature-limited speed constraint, VCURVE 53 3.3.4 Roughness-limited speed constraint, VROUGH 56 v 3.3.5 Desired speed, VDESIR 58 3.3.6 Summary and discussion 59 3.4 Limiting Speed Model: Probabilistic Formulation 60 3.5 Summary of Steady-State Speed Model 74 Appendix 3A. Extreme Value Distributions 76 Appendix 3B. General Solution to Steady-State Force-Balance Equation 78 Chapter 4. Steady-State Speed Model: Parameter Estimation, Adaptation and Transferability 81 4.1 Data 81 4.2 Estimation Results 83 4.3 Adaptation of Parameter Estimates for Prediction 91 4.3.1 Redefinition of vehicle classes 92 4.3.2 Determination of HPDRIVE and HPBRAKE parameters 93 4.3.3 Parameterization of perceived friction ratio for partially loaded trucks 97 4.4 Transferability of the Limiting Speed Model: A Case Study of India 98 4.4.1 Data 99 4.4.2 Approach 100 4.4.3 Results 104 Appendix 4A. Steady-State Speed Model: Estimation Procedure 109 Appendix 4B. The Test for Normality of Residuals of the Final Steady-State Speed Model 112 Appendix 4C. Estimation Results for Separate Regressions by Load Level and by Paved and Unpaved Grouping 113 Appendix 4D. Estimation Results Based on Speed-Sensitive Average Rectified Slope 120 Chapter 5. Micro Non-Transitional Speed Prediction Model 123 5.1 Prediction Procedure 123 5.2 Steady-State versus Transitional Speed Behavior 124 Chapter 6. Micro Transitional Speed Prediction Model 131 6.1 Overview of Speed Profile Simulation 131 6.2 Determination of Maximum Allowable Speed Profile: Backward Recursion 134 6.2.1 Maximum allowable entry speed 134 6.2.2 Deceleration 137 6.2.3 Downhill momentum acceleration 141 6.2.4 Iteration procedure 141 6.3 Simulation of Actual Speed Profile: Forward Recursion 143 6.3.1 Exitspeed 143 6.3.2 Powerusage 146 vi 6.4 Verification of Acceleration, Deceleration and Power Usage Logic 150 Chapter 7. Aggregate Prediction Model 153 7.1 Review of Alternative Aggregation Methods 153 7.2 Aggregate Speed Prediction Based on Integration Approach 156 7.3 Aggregate Speed Prediction Model 160 Appendix 7A. Procedure for Computing Average Roadway Characteristics from Detailed Geometric Profile 167 Chapter 8. Validation of Speed Prediction Models 175 8.1 Sources of Prediction Errors 175 8.2 Validation Data 178 8.3 Validation of Micro Models 181 8.4 Validation of Aggregate Model Based on Test Sections 186 8.5 Validation of Aggregate Model Based On User Cost Survey 193 Part II. Vehicle Operating Cost Prediction Models 197 Chapter 9. Unit Fuel Consumption Function 199 9.1 Model Formulation 199 9.2 Estimation Data 200 9.3 Statistical Estimation 201 9.4 Transferability of the UFC Function 208 Appendix 9A. Determination of the Form of Unit Fuel Consumption Function 211 Chapter 10. Fuel Consumption Prediction Models 215 10.1 Micro Transitional Model 215 10.2 Micro Non-Transitional Model 216 10.3 Aggregate Model 217 10.4 Data for Calibration and Validation 218 10.5 Adaptation of Steady-State Speed Model Parameters for Test Vehicles 218 10.6 Calibration of Nominal Engine Speeds 218 10.7 Validation 222 10.7.1 Micro transitional model 222 10.7.2 Micro non-transitional model 225 10.7.3 Aggregate model 225 10.8 Adjustment Factors for Actual Operating Conditions 225 10.8.1 Sources of discrepancy in fuel consumption 238 10.8.2 Data 238 10.8.3 Results and discussion 241 vii Chapter 11. Aggregate Tire Wear Prediction Model 247 11.1 Model Formulation 247 11.1.1 Tire wear cost model 247 11.1.2 Carcass life model 249 11.1.3 Micro tread wear model 250 11.1.4 Aggregate tread wear model 252 11.2 Data for Estimation 253 11.3 Estimation results 256 11.3.1 Carcass life model 256 11.3.2 Treadwearmodel 258 11.4 Recommended Tire Wear Prediction Model and Comparison with Other Studies 259 11.4.1 Busesandtrucks 259 11.4.2 Cars and utilities 262 11.4.3 Comparison of tire wear prediction model with other studies 263 Appendix 1 LA. Derivation of Micro Tread Wear Model 268 Appendix 11 B. Review of Previous Empirical Tread Wear Research 273 Appendix 1 1C. Aggregate Tread Wear Model: Formulation and Numerical Testing 280 Appendix l1D. Average Rubber Volume per Tire 288 Chapter 12. Vehicle Maintenance, Depreciation, Interest and Utilization 291 12.1 Vehicle Maintenance Parts and Labor 291 12.1.1 Maintenance parts model 292 12.1.2 Maintenance labor hours model 294 12.2 Depreciation and Interest Charges 295 12.3 Model Formulation 296 12.3.1 Relationship of vehicle depreciation and interest costs to vehicle utilization 296 12.3.2 Vehicle utilization-simple model 297 12.4 Recommended Vehicle Utilization Prediction Model 303 Appendix 12A. Estimation of Model Parameters 307 Appendix 12B. Calibration of Recommended Vehicle Utilization Prediction Model 312 Appendix 12C. Lubricants Consumption 313 Part III. Applications 315 Chapter 13. Guidelines for Local Adaptation 317 13.1 Use of Guidelines 317 13.2 Coefficient of Rolling Resistance 321 13.3 Speed Prediction Models 321 13.3.1 Desired speed 322 13.3.2 Used driving power 323 13.3.3 Used braking power 325 13.3.4 Perceived friction ratio 326 13.3.5 Average rectified velocity 327 13.3.6 Width effect parameter 328 viii 13.3.7 Weibull shape parameter 328 13.4 Fuel Consumption Prediction Model 329 13.5 Tire Wear Prediction Model 330 13.6 Vehicle Utilization Prediction Model 332 13.7 Maintenance Parts and Labor Prediction Models 333 13.7.1 Maintenance parts model 333 13.7.2 Maintenance labor hours model 334 Chapter 14. User's Guide to Aggregate Prediction 337 14.1 Outline of Methodology 337 14.2 Instructions for Providing Information on the Roadway 339 14.3 Instructions for Providing Information on the Vehicle 342 14.4 Instructions for Providing Information Needed for Predicting Various Components of Vehicle Operating Cost 344 14.5 Steps for Computing Predicted Speed 347 14.6 Steps for Computing Predicted Fuel Consumption, Tire Wear and Vehicle Utilization 352 14.7 Steps for Computing Predicted Maintenance Resources and Lubricant Consumption 355 Appendix 14A. Illustrative Example 362 Chapter 15. Model Predictions and Policy Sensitivity 369 15.1 Vehicle Running Costs versus Road Attributes 369 15.2 Breakdown into Components of Vehicle Running Costs 374 15.3 Total Running Cost versus Vehicle Weight 375 15.4 Detailed Prediction Tables 376 Appendix 15A. Sensitivity of Vehicle Operating Costs to Road Characteristics for Heavy Trucks on Paved Roads 378 Appendix 15B. Breakdown of Operating Costs 394 Appendix 15C. Effect of Vehicle Weight on Operating Costs 397 Appendix 15D. Detailed Prediction Tables 399 Chapter 16. Conclusions 431 16.1 PlanningTools 431 16.2 Recommendations for Future Research 432 Units 435 Glossary of Symbols 437 References 453 ix Acknowledgments Several organizations and numerous individuals have contributed either directly or indirectly to this book. First of all, we want to thank the Government of Brazil and the United Nations Development Programme (UNDP) for their sponsorship of the highway research project entitled "Research on the Interrelationships between Costs of Highway Construction, Maintenance and Utilization" (PICR) from which this book originated. The massive field data were collected and assembled by Empresa Brasileira de Planejamento de Transportes (GEIPOT), the project's responsible executing agency, in collaboration with the Texas Research and Development Foundation (TRDF) and the World Bank. Of the many organizations that assisted the PICR project, particular mention should be made of the British Transport and Road Research Laboratory, the Western Australia Main Roads Department, the Massachusetts Institute of Technology, the Swedish National Road and Traffic Research Institute (VTI), the South African National Transport and Road Research Institute, and the Australian Road Research Board (ARRB). Among the many individuals at GEIPOT, we would like to make special mention of Jose Teixeira and Teodoro Lustosa of the PICR project; Francisco Magalhaes, GEIPOT Director; and Jose Menezes Senna, GEIPOT President, for their guidance, encouragement, and provision of resources. The results here achieved would not have been possible without a solid and realistic experimental design, followed by painstaking data collection in the field, thorough experimentation and surveys. In this regard particular appreciation is due to William R. Hudson, Bertell Butler, Robert Harrison, Richard Wyatt, Russel Kaesehagen, Hugo Orellana, Stephen Linder, Stanley Buller, Pedro Moraes, John Zaniewski, Barry Moser, Leonard Moser, Douglas Plautz, and Joffre Swait of the PICR study, as well as Virgil Anderson of Purdue University and Paul Irick of the U.S. Transportation Research Board. We further want to thank the several individuals who carefully read either the early reports on the subject or the manuscript of this book and made many useful comments and suggestions-particularly, in addition to the anonymous formal reviewers, Andrew Chesher of Bristol University, Gosta Gynnerstedt of the VTI, John Mclean and Chris Hoban of the ARRB, Thomas Gillespie of the Transportation Research Institute at the University of Michigan, Leonard Della-Moretta of the U.S. Forest Service, and Edward Sullivan of the Institute of Traffic Studies at the University of California, Berkeley. At the World Bank, we appreciate the continued support and encouragement given by Christopher Willoughby and Louis Pouliquen, former directors, Transportation Department. Esra Bennathan and Alan Walters are acknowledged for their helpful reviews and advice and also for their intellectual challenge regarding the economic interpretations of our engineering-oriented road-user cost prediction models. Over the years many World Bank transport engineers and economists, too numerous to mention, have made suggestions about the requirements of road-user cost prediction, and we have attempted to build several features into the prediction models in response to those suggestions. We want to thank our colleagues Anil Bhandari, Bill Paterson, E. Viswanathan, and Sujiv Vurgese for their help in numerous ways, including a thorough review of the manuscript at the draft stage. On the production side, we want to acknowledge the outstanding work of Sabine Shive as the publication coordinator. The editorial help given by Ann Petty was invaluable. Special thanks are due to Emiliana Danowski and Roberta Bensky for their great skill and unfailing patience in producing the innumerable drafts of the manuscript. Finally, we owe our deepest gratitude to Clell Harral, who has been the single most influential and motivating force behind all these efforts. Without his total dedication, the book would not have come into being. x CHAPTER 1 Introduction 1.1 STUDY OBJECTIVES A consensus has been growing among highway administrators, economists and engineers, in developed and developing countries alike, on using the principle of total transport cost minimization as a basis for determining road construction and maintenance policies. It has long been known that of the three main components of total life-cycle costs of a roadway, which also include those of road construction and maintenance, the road user costs of vehicle operation (including ownership) and travel time are by far the largest and can amount to more than 90 percent for two-lane highways serving a few thousand vehicles per day or more. Therefore, it is important to quantify empirically, as accu- rately as practicable, how road user costs are affected by road charac- teristics, including geometric standards, which reflect the amount of capital investment in the road, and surfacing standards, which reflect both initial capital and subsequent maintenance expenditures. Moreover, it is important for the form of quantification, or prediction models, to be as convenient to use as possible. These issues are the main concern of this book, which, specifically, has two principal objectives: 1. To present the theory and empirical quantification of models for predicting, under free-flowing conditions, the speed, fuel consumption, tire wear and time-related compo- nents of running costs (primarily depreciation and interest charges and crew costs) of cars, utilities, buses, and a range of trucks from 6 to 40 ton gross weights. These components constitute approximately 75 percent of the total operating cost1 of typical truck operations in Brazil (GEIPOT, 1982 Volume 5). Appendix 1A shows the classifica- tion and specifications of the vehicles employed in the Brazil study. The models are generally based on theories of vehicle mech- anics and behavior of drivers and operators. They have been quantified almost exclusively using extensive data collected in the Brazil-UN.DP-World Bank highway research 1 The other components of running costs are maintenance parts and labor and lubricants; they are reported in detail in a companion volume in the series (Chesher and Harrison, forthcoming) and, for the sake of completeness, summarized in Chapter 12. 2 INTRODUCTION project.2 The exceptions are the effect of narrow road widths and lubricant consumption, which are based on the data from the India Road User Cost Study (CRRI, 1982). The models are available in easy-to-use aggregate form of algebraic functions which produce predictions sensitive to major road design and maintenance variables, can reasonably be extrapolated over the likely range of policy variables and also can, in general, be adapted to suit local condi- tions. Although these models are useful for link-level feasibility and prefeasibility studies, they are particu- larly valuable for highway sector planning in which the planner searches the combination of road construction and maintenance standards that minimizes the discounted total transport cost of the road network under budget constraints. 2. To provide detailed guidelines on: first, how to use these models, along with other models reported in the companion volume (Chesher and Harrison, forthcoming) for predicting maintenance parts and labor requirements and lubricants consumption, in the prediction of total vehicle operating costs; and, second, how to adapt the model parameters to suit the conditions of a new region or country. These models have been implemented in the new version of the Highway Design and Maintenance Standards Model (HDM-III) for total transport cost prediction (Watanatada et al., forthcoming). They can also be used for predicting vehicle operating costs alone and are easily implemented on personal computers either as independent application soft- ware or via a spreadsheet package. The important limitations of the models described in this volume may be identified as follows: 1. The models predict vehicle operating costs under the regime of unimpeded flow. Thus, the impact of vehicle interaction and congestion on vehicle operating costs is not modelled. Extension of the current results to the regime of impeded flow would be an important item in any agenda for future research on vehicle operating costs. 2. While the models predict speeds, and hence travel times, they do not address the valuation of travel time savings. The latter must be provided by the user. 2 Financed by the Government of Brazil and the United Nations Development Programme (UNDP) and executed by the Empresa Brasileira de Planejamento de Transportes (GEIPOT) with the assistance of the University of Texas, Austin and the Texas Research and Development Foundation. The World Bank acted as the Executing Agency for the UNDP. INTRODUCTION 3 3. The models do not encompass the impact of road design and maintenance choices on road user costs through safety- related factors. For quantitative evidence of such user benefits based on the data from a study conducted in India, see CRRI (1982). 1.2 MDDEL CRITERIA AND RESEARCH APPROACH 1.2.1 General Modelling Framework The variables that affect the cost of operating a vehicle on a given route may be divided into three broad groups: 1. Road attributes, which comprise the relevant geometric and surfacing characteristics of the route, e.g., vertical and horizontal alignments, road width, and surface profile irregularity or 'roughness'; 2. Vehicle attributes, which comprise the relevant physical and operating characteristics of the vehicle, e.g., the weight, payload, engine power, suspension design, and number of hours operated per year; and 3. Regional factors, which comprise the relevant economic, social, technological and institutional characteristics of the region, e.g., the region-wide speed limit, fuel prices, relative prices of new vehicles, parts and labor, stage of technological development, driver training and driving attitudes, such as lane discipline, and general attitude of vehicle drivers toward safety. Generally speaking, vehicle operators have a tendency to adjust the operating and physical characteristics of the vehicle so as to mini- mize operating and travel time costs, subject to constraints imposed by the road (e.g., through vertical gradient and horizontal curvature) and the regional environment (e.g., relative resource costs, the regional speed limit and other traffic regulations). Chesher and Harrison (forth- coming) have employed a relatively simple mathematical construct to illustrate the principle of operators' cost minimization through trade- offs among major cost components. From this broad ideal, it is in prin- ciple possible to postulate a generalized mathematical model of total cost minimization which is expressed as a function of the relative prices of vehicle operating cost components, as well as the road characteristics and regional factors. Such a model would, at least in theory, be generally transferable across countries and also over time within a given country. Local adaptation would be a simple matter of providing the values of the necessary regional factors. While the above approach has intellectual appeal, it would be extremely difficult to implement. First, a realistic mathematical form of the model would have to be very elaborate in order to capture properly all the major causal relationsips between the above three groups of variables. At the present time, our collective knowledge of vehicle 4 INTRODUCTION operators' adaptive behavior does not seem to be sufficient to provide the necessary model specifications. Furthermore, even if such specifica- tions were available, the task of calibrating the generalized model would still be immense. The data collection would have to be based on very large factorial designs covering multiple dimensions of relative prices, and vehicle, road and regional variables. This means that standardized data would have to be obtained from several countries of disparate characteristics. Cultural differences among countries would also have to be accounted for, which is by no means a small task. For these reasons, it is not surprising that we have yet to see a serious pursuit of the above approach. What we have seen instead is a different, but feasible approach adopted by many previous researchers: modelling on a country- or region-specific basis physical components of vehicle operating cost, including principally speed (or its inverse, travel time per unit distance), fuel consumption, tire wear and mainte- nance parts and labor (Winfrey, 1963; Claffey, 1966; de Weille, 1966; Hide et al., 1975; Morosiuk and Abaynayaka, 1982; Hide, 1982). Vehicle operating costs are computed simply by multiplying the predicted quanti- ties of physical resources consumed with their relative prices. Compared to the approach of modelling vehicle operating costs directly, the physical-component approach yields models which, if correctly specified in form, are more easily transferable to new countries, since the most direct effects of prices have been removed. However, it is important to emphasize that the calibrated coefficients of physical-component models represent a result of opera- tors' long-term adaptation to the regional environment and, strictly speaking, cannot be transferred to another region or country without re- calibration. In fact, our general view is that, unless there is empiri- cal evidence or theory to indicate the contrary, the coefficients of physical-component models should be subject to at least some re-calibra- tion when they are applied elsewhere. We return to the issue of adaptive behavior in response to changes in road characteristics below, and the issue of transferability and regional adaptation is dealt with in Chapter 14. 1.2.2 Criteria for Vehicle Operating Cost Prediction Models Within the physical-component modelling framework stated above, the basic purpose of this study was to develop vehicle operating cost prediction models that: 1. Can be expressed in aggregate form, i.e., as algebraic functions of aggregate descriptors of road attributes, viz, average gradient, horizontal curvature and roughness, so that detailed information on road alignment is not needed (this requirement is essential for sector-level planning); 2. Are policy-sensitive, i.e., possess discriminatory power to distinguish subtle differences in vehicle physical and operating characteristics as well as among investment INTRODUCTION 5 alternatives involving tradeoffs between vertical and horizontal alignments and surfacing standards; 3. Have an adequate extrapolative capability, i.e., can be extrapolated over the range of vehicle and road character- istics which must be considered in an incremental economic analysis of alternatives; and 4. Have an adequate local adaptability, i.e., the model para- meters can be easily adapted to new regions or countries without having to repeat a full-scale data collection exercise. Over the past 10 to 15 years, research efforts in developing vehicle operating cost prediction models may be categorized into two broad approaches: aggregate-correlative and micro-mechanistic. As will be elaborated below, both approaches have their own merits, and our main task was to combine them into a new approach which took advantage of the strengths of each. For discussion purposes, this new approach is called aggregate-mechanistic. Aggregate-correlative approach This approach, which is more extensively quantified than the micro-mechanistic one, rests on generally large data bases obtained from vehicle operator surveys as well as field experiments (e.g., measurements of fuel consumption) using specially-instrumented vehicles.3 The models produced are expressed in relatively simple algebraic form. Although a priori reasoning has played a significant role in the development of the model form, these models still tend to rely heavily on trends indicated by the data, as opposed to mathematical forms (which are generally non-linear) that result from a more rigorous theoretical postulation. Aggregate-correlative models have been developed in the major field studies conducted in Kenya (Hide et al., 1975), the Caribbean (Hide, 1982; Morosiuk and Abaynayaka, 1982), India (CRRI, 1982) and Brazil (GEIPOT, 1982), and the earlier ones have enjoyed a wide-spread use in highway planning. The pioneering Kenya models have been implemented in the Massachusetts Institute of Technology's Road Investment Analysis Model (RIAM - Brademeyer et al., 1977), the first version of the British Transport and Road Research Laboratory's Road Transport Investment Model (RTIM - Robinson et al., 1975) and an early version of the World Bank's HDM model (Watanatada et al., 1981). The Caribbean models have been incorporated in the latest version of the RTIM model (Parsley and Robinson, 1982), and the Kenya, India and some of the Brazil models have been included in the new version of the HDM model (Watanatada et al., forthcoming). 3 It proved impossible to find an appropriate term for these models. The word "correlative" is intended to imply that the mathematical forms of the models relied more heavily on a combination of observed data trends and common sense than on rigorous theory. It is not, however, intended to imply any lack of causality between the dependent and independent variables. 6 INTRODUCTION Judged against the above four criteria, the overall conclusion is that these aggregate-correlative models are, as a group, generally adequate with respect to the first two, although a large scope exists for possible improvement even there, and are not generally adequate with respect to the latter two criteria, i.e., extrapolation and local adapta- tion. The models enjoy the empirical support of large data bases, which is indeed their greatest strength. The aggregate algebraic functions are easy to use and are mostly expressed in terms of important vehicle and road descriptors. However, the model forms tend to be dictated by trends in the data which were often ambiguous (due to great variability in the data, incomplete coverage of the range of, or correlation among, the independent variables, etc.). As a result, they have a propensity to miss the more subtle interaction of the policy variables (e.g., the speed reduction effect of a steep gradient, sharp curve and rough surface acting simultaneously should be smaller than the sum of the effects of these factors acting separately). Furthermore, although they generally produce reasonable predictions over the range of policy variables covered by the data, they usually do not extrapolate well over the range of the null (or baseline) alternatives which must be postulated for incremental benefit-cost analyses, and can give unreasonable predictions (e.g., excessively low or even negative predicted speed). Finally, the model coefficients are generally difficult to interpret in physical terms (unlike parameters such as "vehicle power" for governing uphill speeds or "friction coefficient" for governing curve speeds) and, consequently, do not lend themselves to local adaptation. The types of adaptation we have found from experience with these models have generally been relatively simple, e.g., applying a multiplicative adjustment factor to bring predictions closer to locally observed values. Micro-mechanistic approach Employing a detailed representation of the road alignment (i.e., using data on the gradient, radius of curvature, superelevation, etc. of the generally small homogeneous subsections into which the road section can be divided), this approach draws upon theories of vehicle mechanics and driver behavior to simulate a detailed speed profile of the vehicle as it traverses the road section (see Chapter 6 for a detailed exposition).4 Once the speed profile is known, fuel consumption and tire wear are predicted in increments at small distance intervals along the road; this is relatively straightforward since fuel consumption and tire wear can be expressed, through theories of vehicle mechanics, as analyti- cal functions of speed. Sullivan (1976) employed this approach in the prediction of the operating costs of logging trucks, while Bester (1981) and Andersen and Gravem (1979) were specifically concerned with fuel consumption. Gynnerstedt et al. (1977), St. John and Kobett (1978) and Hoban (1983) employed mechanistic-behavioral theories to construct free speed profiles primarily as the building block for simulating traffic interaction, which in turn, led to speed prediction under impeded flow conditions. 4 These models also incorporate assumptions of driver/operator behavior but for compactness of terminology they are referred to simply as "mechanistic." INTRODUCTION 7 The micro-mechanistic approach has a general tendency to rely less on large data bases collected in one single field study than the aggregate-correlative approach. Rather, micro-mechanistic models have tended to draw upon the results of previous work; this has been possible since the theories of vehicle mechanics and driver behavior give rise to model parameters that have readily interpretable meanings (e.g., desired speeds, acceleration/deceleration rates, maximum used power, etc.), and therefore can be transferred from one model to another. In addition, the values of unknown parameters can, as a rule, be determined from relative- ly small-scale experiments (e.g., to obtain coefficients of vehicle rolling resistance). Because of their generally strong theoretical basis micro-mechanistic models have an inherent tendency to transfer and extrapolate well. However, since the micro-mechanistic models examined apparently were not developed with all the above criteria in mind, they are not in a form suitable for sector-level planning purposes. Specifically, the re- quirement of detailed information on road geometry is intrinsically unwieldy for quick policy analyses. In terms of empirical quantification the micro-mechanistic models tend to be less extensively developed than the aggregate-correlative counterparts. The models generally have under- gone far less validation by independent data than they should have been. Finally, none of the models examined have incorporated road roughness as a major surface condition variable influencing vehicle operating costs. This has precluded the possibility of using these models at the current state of quantification to examine tradeoffs between surfacing and geometric standards even at the detailed project level. Aggregate-mechanistic approach Having seen the two approaches with their comparative advan- tages, it seemed appealing to combine their virtues, namely, aggregate form, policy-sensitivity, extrapolative ability and local adaptability into new prediction models. This was the basic approach taken in this study. It is briefly outlined in the section below. Table 1.1 provides a summary comparison of the aggregate-corrective, micro-mechanistic and aggregate-mechanistic approaches in the light of the four criteria and the state of model quantification at the completion of this study. 1.3 OUTLINE OF AGGREGATE-NECHANISTIC APPROACH 1.3.1 Scope As mentioned earlier, in this study we are concerned with modelling speed, fuel consumption, tire wear, vehicle depreciation and interest, and crew cost. As shown in Chapter 12, vehicle depreciation and interest are inverse functions of vehicle utilization (defined as the total distance driven per vehicle per year), and the crew cost varies inversely with speed. Therefore, besides speed, fuel and tire wear, vehicle utilization also had to be modelled. The remaining components of the total operating costs--namely, maintenance parts and labor, and lubricants--are supported by well-established mechanistic-behavioral 8 INTRODUCTION Table 1.1: Summary comparison of aggregate-correlative micro-mechanistic and aggregate-mechanistic approaches vis-a-vis research objectives and state of model quantification Modelling approach Aggregate- Micro- Aggregate Research objective correlative mechanistic mechanistic Aggregate formulation Yes No Yes Policy sensitivity Mostly medium1 Mostly high Mostly high Extrapolative ability Mostly low Mostly high Mostly high Local adaptability Mostly low-medium Mostly high Mostly high State of model quantifi- Mostly extensive Lacking Mostly fication at completion validation extensive of this study and surface effects 1 If relevant policy variables are incorporated. Figure 1.1: Logical sequence of mechanistic modelling approach Formulation, calibration Formulation and testing -- and validation of micro ---aof aggregate speed pre--- speed prediction models1 diction model Formulation, calibration Formulation and testing and validation of micro of aggregate fuel fuel consumption consumption prediction prediction models1 model Formulation of micro Formulation and calibra- -e tire wear prediction tion of aggregate tire models1 wear prediction model Formulation and calibra- tion of aggregate vehicle utilization prediction model 1 Including both micro transitional and non-transitional models, as introduced in Section 1.3.4. INTRODUCTION 9 theories, and consequently are estimated on an aggregate-correlative basis and reported elsewhere (GEIPOT, 1982, Volume 5; Chesher and Harrison, 1987); in particular, the maintenance parts and labor models were based on the Brazil data. 1.3.2 Basic Approach As alluded to previously, speed is functionally the most crucial variable to predict: once speed is known, it is fairly simple to predict fuel consumption, tire wear and vehicle utilization. Among these latter three variables, fuel and tires are "consumed" in an instantaneous manner and therefore can be predicted in micro increments as the vehicle traverses the road.5 Vehicle utilization, on the other hand, represents a measure of vehicle productivity which is based on the number of round trips a given vehicle can make over a fixed route in a year, and there- fore could only be modelled as an aggregate function of speed. The basic modelling approach taken for speed and fuel consump- tion was first to construct micro-mechanistic models (Chapters 5, 6, 9 and 10), validate them with independent data (Chapters 8 and 10) and then transform them by means of numerical techniques into aggregate form (Chapters 7, 8 and 10). The main advantages of this approach are not only that deep insights could be gained into the physical-behavioral phenomena at the micro level, but also that once validated to satisfaction, the micro models could be treated as a close approximation of the "truth" and consequently used as a benchmark for developing and testing aggregate- mechanistic models. Furthermore, although not intended as a prime objec- tive of this study, the validated micro models could serve as practical prediction models in their own right for project-level studies in which the detailed response of vehicle speeds and operating costs to specific spot improvements need to be estimated. Another potentially important application of such models would be to serve as a building block for constructing models for simulating the effects of traffic flow inter- action. This approach, however, needs a large store of detailed field data for both model calibration and validation, involving the observation of actual vehicle speeds at a sample of road sites covering a full range of roughness and geometry, as well as the measurement of the speed and fuel consumption of specially instrumented vehicles traversing a sample of test sections. Moreover, since the experimental fuel consumption data were obtained from the test vehicles operated under controlled conditions in favor of fuel economy, it was necessary to adjust the experimentally- based fuel predictions to real-world conditions; this had to be done by comparison with actual fuel consumption records of vehicle operators obtained from a road user survey. 5 The exception is the tire carcass which can survive through several recappings and the number of recappings to which it can be subjected affects the tire cost. This issue is dealt with separately in the "carcass life" model in Chapter 11. 10 INTRODUCTION Although the Brazil speed and fuel consumption data had not been collected with the mechanistic approach in mind, they were broad in scope and deep in detail (Zaniewski et al., 1980; GEIPOT, 1982, Vol. 5), and with few exceptions proved to be virtually ideal for the mechanistic approach. On the other hand, as discussed in Chapter 11, the tire data fell short of the ideal. Because of the similarities in the physical phenomena, tire wear modelling called for the same kind of data as fuel consumption modelling, i.e., a balanced mix of detailed experimental and survey data. Unfortunately, only survey data were collected. Consequently, it was not possible to follow the same approach as taken for speed and fuel modelling, and it became necessary to calibrate the tire wear model at the aggregate level. However, as shown in Chapter 11, it was still possible to formulate a theoretical tire wear model at the micro level and then transform it into an aggregate model for calibration purposes. As mentioned earlier, by its nature the vehicle utilization model had to be formulated at the aggregate level of a fixed route. As derived in Chapter 12, the resulting model form was expressed as a function of the average round trip speed, and aggregate data were available from the road user survey for calibrating this theoretical model. The logical sequence of the mechanistic modelling approach followed in this study is depicted in Figure 1.1. It can be seen that speed modelling always preceded the modelling of fuel consumption, tire wear and vehicle utilization; furthermore, wherever possible micro modelling preceded aggregate modelling. 1.3.3 Steady-state Models The basic building blocks of the mechanistic modelling approach were the models for predicting speed, fuel consumption and tire wear under steady-state conditions. Steady-state conditions arise when a homogeneous subsection is sufficiently long so that the vehicle can reach and maintain a constant speed, at which the forces acting on the vehicle and the resulting fuel consumption and tire wear remain constant. Under steady-state conditions, the vehicle motion and associated fuel consumption and tire wear can be described in a relatively simple manner using closed form algebraic functions derived from mechanistic-behavioral theories. The steady-state models (described in detail in Chapters 3, 9 and 11) are summarized in Table 1.2 along with their underlying theories. Strict steady-state conditions do not occur frequently in most driving situations, which are mostly transient in nature involving continual speed changes in response to variations in road alignment and surface conditions. However, the steady-state models have their appeal in their representation of the asymptotic behavior of vehicle motion. Because of this they were employed not only as the fundamental building blocks for the micro models, but also as an approximation for aggregate INTRODUCTION 11 Table 1.2: Summary of steady-state models for speed, fuel consumption and tire wear Variable to be Steady-state Chapter(s) predicted model described Underlying theory and origin Speed Steady-state 3, 4 Probabilistic limiting speed theory speed model developed in this study as an extension of deterministic limiting speed theory theory using mirror-image of probabi- listic discrete choice theory. Fuel Unit fuel 9 Empirical extension of specific fuel consump- consumption consumption concept to negative vehicle tion function cle power. Principles of internal com- bustion engines. Tire wear Tread wear 11 Generalized form of slip energy theory model Synthesis of tire slip and tread wear wear theories prediction purposes. From the test results (reported in Chapters 8, 10 and 11), it was in fact found that within the range of road severity tested, the aggregate models derived directly from the steady-state models produce satisfactorily accurate predictions relative to the micro models. 1.3.4 Micro Transitional and Non-transitional Models Two types of micro models--transitional and non-transitional- for speed, fuel consumption and tire wear, were constructed on the foun- dation of the steady-state models. As noted above, vehicle motion along the road is for the most part a transient phenomenon. The main objective of the micro transitional models was to simulate the transient phenomena as closely as possible so that they could be used as the next-to-the- truth reference for testing the more simplified micro non-transitional and aggregate models. Employing the same detailed information on road characteristics (i.e., attributes of consecutive homogeneous subsections), the micro non- transitional models differ from the micro transitional counterparts in that they completely ignore the transient or transitional effects and produce predictions only on the basis of the steady-state models applied to the individual homogeneous subsections. The micro non-transitional models were developed with two purposes in mind: first, as building blocks for the micro transitional models; and, second, as an intermediate step between the micro transitional and aggregate models in terms of 12 INTRODUCTION micro transitional and aggregate models in terms of complexity. As elaborated in Chapters 8, 10 and 11, these intermediate models were used to compare with the micro transitional models to ascertain the effects of omitting vehicle transitional behavior, and with the aggregate models to test the accuracy of numerical approximation on the part of the latter. In sum, the development of prediction models at naturally progressive levels of complexity proved to be useful not only in under- standing the physical-behavioral phenomena associated with vehicle motion but also in providing appropriate benchmarks for testing the predictive accuracy of the models themselves. 1.4 ADAPTIVE BEHAVIOR IN RESPONSE TO CHANGES IN ROAD CHARACTERISTICS To facilitate local adaptation and enhance flexibility, the vehicle operating cost models outlined above are expressed as functions of as many relevant vehicle attributes as possible. The user of the models must specify these attributes as inputs to the model, but he must bear in mind that the vehicle attributes could change, at least in the long run, in response to changes in road characteristics. Operators have a proclivity to adapt their vehicles and operating rules to changes in route conditions in order to maximize profit or minimize costs; they can, within constraints, vary a number of variables to minimize transport cost, e.g., the number, type and make of vehicles, the engine size, age, and tires. However, only utilization (number of kilometers driven per year) and, implicitly, fleet size (number of vehicles employed in a company) are endogenous within the vehicle operating cost relationships presented herein. Fleet size is implicitly tied to utilization in the prediction of depreciation and interest per km: when the average route speed is raised by road improvement, a vehicle can make more trips per year, thereby resulting in fewer vehicles needed to haul a given volume of transport--assuming a constant number of hours of operation per year. The treatment of utilization and fleet size as endogenous variables has arisen from their sensitivity to road improvements as well as their con- siderable influence on vehicle operating cost via the sizable components of depreciation and interest. The other vehicle attributes, once exogenously specified, are regarded as constants by the model. Where the user has reason to believe that the changes in road conditions or policies being modelled will lead to other important adaptations of vehicle characteristics, he can incor- porate an estimate of these changes in the input of vehicle characteris- tics and future traffic composition. To do this the user could either employ a subjective assessment of the likely changes or he could system- atically search for the combination of vehicle attributes that yields a minimum cost for the anticipated traffic and road characteristics. To do either is clearly a larger task than the more traditional straightforward comparison of alternatives with the vehicle characteristics held fixed. Whether it is worthwhile undertaking such a task depends on the policy question at hand. For example, the traditional approach should generally suffice for evaluating minor road improvements. On the other hand, to compare alternative schemes involving large-scale changes in the INTRODUCTION 13 road network or such broad policies as axle load regulation and enforce- ment, the user should make a careful judgment on how vehicle operators are likely to respond to these schemes, in terms of the vehicle type and axle configuration, payload, etc. and incorporate these in the projec- tions of future traffic, which can include anticipated changes in the composition of the vehicle fleet. 1.5 ORGANIZATION OF THE BOOK The topics are divided into three parts, the first two dealing with the theoretical development and quantification of the mechanistic- behavioral models and the third with their applications. Part I is devoted to the speed models. It begins with Chapter 2 which presents the basic definitions and relationships concerning the mechanics of vehicle motion. Chapter 2 highlights the vehicle rolling resistance-road roughness model, calibrated with Brazil experimental data. These basic physical relationships are employed in Chapters 3 and 4 which present the theoretical formulation and statistical estimation of the steady-state speed model based on the probabilistic limiting speed theory (mentioned in Table 1.2 above), for cars, utilities, buses, and light/medium, heavy and articulated trucks. The steady-state speed model provides a foundation for the formulation of the micro non-transitional and transitional speed prediction models described in Chapters 5 and 6, respectively. The micro models are then followed by the aggregate speed prediction model formulated in Chapter 7. The three speed prediction models are validated /against independent: data and compared with each other in Chapter 8. As a direct follow-up application of the speed models, Part II presents the formulation and quantification of the aggregate models for predicting fuel consumption, tire wear and vehicle utilization. Chapter 9 deals with the unit fuel consumption function which provides the basic building block for the micro and aggregate fuel consumption prediction models. The formulation, calibration and validation of the latter models are presented in Chapter 10. Chapter 11 deals with tire wear while Chapter 12 describes the vehicle utilization model and also summarizes the vehicle maintenance models (including parts, labor and lubricants) reported in Chesher and Harrison, 1987. Applications of the aggregate-mechanistic models developed above are dealt with in Part III. As several of the model parameters may need to be adapted to suit local conditions, guidelines for local adapta- tion are provided in Chapter 13. Chapter 14 provides a detailed step-by- step user's guide for computing total vehicle operating costs. As a reference for quick policy analysis, based on the Brazil-specific model parameters presented in Chapter 14, Chapter 15 provides tables and graphs of predicted physical resources consumed and the resulting costs of vehicle operation for a full range of road surfacing and geometric stan- dards. These predictions are accompanied by a brief discussion on their investment policy implications. Finally, conclusions and recommendations are given in Chapter 16. 14 INTRODUCTION APPENDIX 1A POPULATION AND TEST VEHICLES Two types of vehicles were employed: those sampled from the actual vehicle population and those procured and instrumented specially for conducting controlled experiments on speed and fuel consumption6(GEIPOT, 1982). The population vehicles were intended to provide actual operating data both at the detailed level through radar and stopwatch speed observations at selected road sites and at the aggregate level through operators' records of route speeds, fuel consumption, tire wear and vehicle utilization. The specially instrumented or test vehicles were to provide accurate detailed speed and fuel consumption information not obtainable from population vehicles. For the purposes of this study, the population vehicles were categorized into ten basic classes, ranging from passenger cars to articulated trucks, as depicted in Table 1A.1 along with their principl attributes, e.g., rated gross weight. Each class is matched by a test vehicle which is considered to be representative of the class. The test vehicles comprised nine vehicles treated as representative vehicles in Table 1A.1 and two replicate vehicles, a utility (VW-Kombi) and a heavy truck (Mercedes Benz 1113 with crane). Their detailed characteristics are assembled in Table 1A.2. 6 No detailed experiments on tire wear were planned in the Brazil study. INTRODUCTION 15 Table 1A.1: Basic classes of population vehicles and principal attributes Engine Approx- Maximum imate SAE rated Weight rated Number gross classi- Number Number power of Vehicle class weight fica- of of Fuel (metric cylin- Representative (tons) tion1 tires axles type2 hp) ders vehicle 1. Passenger cars 1.2 L 4 2 G 49 4 Volkswagen 1300 2. Passenger cars 1.5 L 4 2 G 148 6 Opals (medium) 3. Passenger cars 1.9 L 4 2 G 201 8 Dodge Dart (large) 4. Utilities 2.1 L 4 2 G 61 4 Volkswagen Kombi 5. Large buses 11.5 H 6 2 D 149 6 Mercedes Benz 0362 6. Light trucks 6.1 H 6 2 G 171 8 Ford-400 (gasoline) 7. Light trucks 6.1 H 6 2 D 103 4 Ford-4000 (diesel) 8. Medium trucks 15.0 H 6 2 D 149 6 Mercedes Benz 11133 9. Heavy trucks 18.5 H 10 3 D 149 6 Mercedes Benz 11134 10. Articulated 40.0 H 18 5 D 289 6 Scania 110/39 Note: Figures are those of representative vehicles. 1 L= light vehicle (rated gross weight under 3.5 tons); H= heavy vehicle (rated gross weight of 3.5 tons or more). 2 G= gasoline engine; D= diesel engine. 3 Excludes third rear axle. 4 Includes third rear axle. Source: Classification scheme adapted from Brazil-UNDP-World Bank highway research project data physical characteristics adapted from those of the matched representative vehicle (see table 1A.2). 16 INTRODUCTION Table 1A.2: Test vehicles and physical characteristics Articu- Large lated Class Car Utility Bus Light Truck Heavy Truck Truck Mercedes Mercedes Mercedes Make Volkswagen Chevrolet Chrysler Volkswagen Benz Ford Ford Benz Bens Scania Dodge 1,1132 Model 1,300 Opals Dart Kombi1 0-362 F-400 P-4000 1,1132 with 110/39 crane weight (kg) Tare4 960 1,200 1,650 1,320 8,100 3,120 3,270 6,600 6,570 14,730 Rated gross 1,160 1,466 1,915 2,155 11,500 6,060 6,060 18,500 18,500 40,000 Engine Fuel type Gas Gas Gas Gas Diesel Gas Diesel Diesel Diesel Diesel Number of cylinders 4 6 8 4 6 8 4 6 6 6 Maximum rated torque (o-kgf) 9.1 30.9 41.5 10.3 37 33.5 29.2 37 37 79 Engine speed at maximum 2,600 2,400 2,400 2,600 2,000 2,200 1,600 2,000 2,000 1,200 rated torque (rpm) Maximum SAE rated 49 148 201 61 149 171 103 149 149 289 power (metric hp) Engine speed at 4,600 4,000 4,400 4,600 2,800 4,400 3,000 2,800 2,800 2,200 maximum power (rpm) Drive train Gear 1 3.80 3.07 2.71 3.80 8.02 6.40 5.90 8.02 8.02 13.51 Gear 2 2.06 1.68 1.60 2.06 4.77 3.09 2.85 4.77 4.77 10.07 Gear 3 1.32 1.00 1.00 1.32 2.75 1.69 1.56 2.75 2.75 7.55 Gear 4 0.89 0.89 1.66 1.00 1.00 1.66 1.66 5.66 Gear 5 1.00 1.00 1.00 4.24 Gear 6 3.19 Gear 7 2.38 Gear 8 1.78 Gear 9 1.34 Gear 10 1.00 Differential 4.375 3.08 3.15 4.375 4.875 5.140 4.630 4.875 4.875 4.710 go. of axles 2 2 2 2 2 2 2 34 34 5 Tires Number 4 4 4 4 6 6 10 10 18 Weight (kg) 5 15 17 19 88 43 4 88 88 92 Diameter(m) 0.645 0.626 0.649 0.654 1.016 0.808 0.80 1.016 1.016 1.080 Radius of gyration (a) 0.260 0.235 0.240 0.250 0.380 0.300 0.30 0.380 0.380 0.405 Rotational inertial 39.0 .33.8 37.2 44.4 295.4 142.3 142. 492.4 492.4 931.5 mass of tires (kg) Aerodynamic drag coefficient 0.45 0.50 0.45 0.46 0.65 0.70 0.70 0.85 0.85 0.63 Projected frontal area (m2) 1.80 2.08 2.20 2.72 6.30 3.25 3.25 5.20 5.20 5.75 1 Two identical vehicles were procured for this model. 2 These are virtually identical vehicles except for slight difference in tare weight. 4 Tare weight includes 150 kg weight of two drivers. These models were also available without the rear tandem-axle. In this case they were be classified as medium trucks. Source: Zaniewski et al. (1982) and vehicle manufacturers. PARI탬 I vehicle Speed Predlch·on Models CHAPTER 2 The Force Balance Relationship This chapter is concerned with an exposition of Newton's fundamental law governing motion as it applies in the context of a vehicle traversing a roadway. Section 2.1 briefly describes the roadway attributes and their measures used in the study. Section 2.2 presents the basic force balance equation and derives expressions for the different physical quantities brought together by the relationship. Section 2.3 gives an outline" of various concrete applications of the basic equation made in the study. As part of the study, an empirical relationship between the coefficient of rolling resistance and road roughness was established. The derivation and estimation of the relationship is presented in Appendix 2A. 2.1 ROADWAY ATTRIBUTES AND THEIR IEASURES The attributes of the roadway at the point of contact with the vehicle, which determine vehicle operation on it, are vertical alignment, horizontal alignment, surface characteristics and cross section. These attributes are relevant for short road sections as well as for long homogeneous sections, and the way these attributes are represented in the study is outlined below. Analogous aggregate descriptors for a heterogeneous roadway are discussed in Chapter 7. 2.1.1 Vertical Alignment Generally, the vertical alignment of a roadway consists of tangent grades, and vertical curves which are generally parabolic. In this study, the vertical alignment of a road is approximated by tangent or linear grades. At the point of contact with the vehicle, the vertical alignment of the road is represented by its gradient (also referred to as slope or grade). It is a signed dimensionless quantity defined as the sine of the angle of inclination the roadway makes with the horizontal plane, as shown in Figure 2.1. It is generally approximated by the tangent of the incline angle. Let GR = road gradient, expressed as a fraction; and 0 = angle of incline of the road, in radians. Then, we have GR = tang (2.1) GR is positive for traversal against gravity (uphill travel), and negative for traversal along gravity (downhill travel). Often, for ease of expression, the gradient is stated in hundredths (%), and occasionally, even in thousandths (m/km). 19 20 THE FORCE BALANCE RELATIONSHIP Figure 2.1: Gradient and vehicle motion forces do (kzn) Horizontal curvature, C- Y dges / -180,000 'r ' Figure 2.2:- Equivalent definitions of horizontal curvature THE FORCE BALANCE RELATIONSHIP 21 For modelling purposes it is often convenient to characterize vertical alignment using two non-negative quantities called "rise (RS)" and "fall (FL)." These are defined as follows: GR if GR > 0 (.a RSm={R (2 .2a) RS- 0 othervwTse - GR if GR < 0 FL f otewie (2 .2b) 0 othervilse, These measures are useful for treating positive and negative gradients asymmetrically, and for developing aggregate descriptors of vertical alignment. 2.1.2 Horizontal Alignment Generally, the horizontal alignment consists of tangents, horizontal curves and banking. In this study, horizontal curves are assumed to be circular. At the point of contact with the vehicle the horizontal alignment is represented by its horizontal curvature (also called degree of curvature) and rate of superelevation. The horizontal curvature of a curve is defined as the angle (in degrees) subtended at the center by a unit arc-length of the curve (in km). Referring to Figure 2.2, let C = horizontal curvature, in degrees/km; y = central angle subtended by the curve, in degrees; and L - arc-length of the curve, in km. Then we have, C - y/L (2.3) It may be noted that C is an inverse function of the radius of curvature of the curve: C = 180,000 (2.4) wRC where RC - radius of curvature of the curve, in m. Further, it may be noted that the angle subtended by an arc of a circle at the center is equal to the external angle made by the tangents to the circle at the ends of the arc. Thus C also expresses the absolute angular deviation of the two tangent lines at the end-points of the curve. For a straight road, the value of C is zero and the value of radius of curvature is taken to be infinity. For practical purposes, a road with a radius of curvature of 10,000 m or more may be assumed to be straight. 22 THE FORCE BALANCE RELATIONSHIP The banking of a curve is represented by the rate of superele- vation, defined as the sine of the superelevation angle, that is, the vertical distance between the heights of the inner and outer edges of the road divided by the road width (assuming zero road camber). It is gene- rally approximated by the tangent of the superelevation angle. It is expressed as a fraction, and is denoted by SP. The rate of superelevation is often referred to simply as superelevation. 2.1.3 Surface Characteristics The surface characteristics considered in this study are the road profile irregularity and surface type. The unit of roughness originally used in this study is QI (which is short for QI counts/km). It is a calibrated Maysmeter estimate of a reference Quarter-car Index of profile measured by a dynamic profilometer (GEIPOT, 1982; Paterson, 1987). Since the completion of this study, roughness has come to be defined in the now-standard International Roughness Index (m/km IRI). This measure is a summary index of the irregularity of the road profile in the wheelpath; it is measured as the accumulated axle-body movement made by a simulated passenger car over a unit distance of travel (in m/km IRI). The IRI quantifies the impact of roughness on a moving vehicle in much the same way as vibrations induced by roughness influence vehicle costs, and hence is considered to be the most applicable measure of roughness for economic evaluation purposes (Sayers, Gillespie and Paterson, 1986, and Paterson, 1987). An approximate relationship between the two units is: IRI = QI/13. (2.5) In this study only two types of road surface are distinguished, paved and unpaved. The former class includes primarily asphalt concrete and surface treatment surfacings, and the latter class includes compacted gravel and earth roads. 2.1.4 Cross Section Important cross sectional aspects of rural roads are effective number of lanes or carriageway width, shoulder width, and shoulder condition. Since most of the road sections included in the Brazil study were more than six meters wide and had adequate shoulders, road cross section was not considered to be an important determinant of vehicle operation in the main study. However, the effect of carriageway width on vehicle speeds was investigated as a part of the analysis of transferability of the steady-state speed model based on Indian data (Section 4.4). As a result of the analysis, roads are divided into two width classes based on the effective number of lanes, namely, single-lane roads and intermediate and dual-lane roads. The former class consists of roads which are 3.5 to 4 m wide and on which vehicles in the two directions share both wheel paths. The latter class consists of roads which are at least 5.5 m wide; on these roads the vehicles in the two THE FORCE BALANCE RELATIONSHIP 23 directions either share one wheel path or have distinct wheel paths. It is common practice in India to widen single-lane roads in two stages: initially, to an intermediate width, and then into a double-lane road. It was observed in India that roads with widths between 4 m and 5.5 m were not common. The apparent reason for this is that these widths do not serve useful purposes. For example, a 5 m wide road would be too narrow for two trucks to cross, but too wide for a single-lane road. The effect of shoulder width and condition has not been modeled in the present study. 2.2 BASIC ERLATIONSHIPS The physical relationship that has the single most extensive use in this study is that of vehicle force balance. The force balance relationship has been used by a number of previous researchers in the simulation of vehicle speeds under both free and impeded traffic flows including, among others, Gynnerstedt et al. (1977), St John and Kobett (1978), Sullivan (1976), and Hoban (19837. When a vehicle traverses a straight road that makes an incline angle 4 radians to the horizontal, as illustrated in Figure 2.1, the force balance equation is given by mta = DF - GF - AF - RR (2.6) where m' = the effective mass of the vehicle, in kg; a = the acceleration of the vehicle in the direction of travel, in m/m ; DF = the vehicle drive force delivered at the driving wheels, in newtons; GF = the component of the gravitational force in the direction opposite to the vehicle travel, in newtons; AF = the air resistance, in newtons; and RR = the vehicle rolling resistance, in newtons. 2.2.1 Effective Vehicle Mass A general formula for computing the effective mass of the vehicle is given by (Taborek, 1957): m' =m + mW + me (2.7) where m = the vehicle mass, in kg; mw = the component of the vehicle effective mass attributable to the rotational inertia of the wheels, in kg; and 24 THE FORCE BALANCE RELATIONSHIP me = the component of the vehicle effective mass attributable to the rotational inertia of the parts that rotate at the engine speed, in kg. The above equation ignores the relatively small contributions of the manual transmission parts, gears and shafts. The effective rotational inertia" mass of the wheels, mw, is given by: I m = - (2.8) RRT2 where I, = the moment of inertia of mass of mounted wheels, in kg-m2; and RRT = the rolling radius of each tire, in meters. The term Iw is given by: 1w = Mw RWG2 (2.9) where Mw = the total mass of all mounted wheels; and RWG = the radius of gyration of each wheel, in meters. The term me is expressed as: I SR2 m =e (2.10) e RRT2 where Ie = the total moment of inertia of mass at engine speed of all parts that rotate, in kg-m2; and SR the ratio of the engine speed to the drive axle speed (i.e., the speed reduction ratio). For a given vehicle the term mw is constant and normally smaller than 5 percent of the vehicle mass. An average value of 4 per- cent is recommended by Koffman (1955). The term me is proportional to the gearspeed ratio. For example, when the vehicle is in a low gear (e.g., when accelerating from standstill), me is relatively large, but when the vehicle changes to the top gear me assumes its smallest possible value. Average values of the "effective mass factor," defined as the ratio m'/m, vary from about 1.1 in top gear for both cars and trucks to about 1.4 in first gear for cars and 2.5 in low gears for trucks. THE FORCE BALANCE RELATIONSHIP 25 2.2.2 Vehicle Drive Force The vehicle drive force, DF, may be related to the vehicle power as: DF = 736 HP (2.1) V where HP = the vehicle power delivered at the driving wheels, in metric hp; and V = the vehicle travel speed, in m/s. 2.2.3 Gravitational Force The gravitational resistance force is given by: GF = m g sink (2.12) where 4 - angle of incline in radians (see Figure 2.1); and g = acceleration due to gravity, in m/s2. This can be approximated for small * by: GF = m g GR (2.13) where GR - tan * is the road gradient introduced in Section 2.1. For 15 percent gradient, which is an extreme case, the approximate formula above (Equation 2.13) has an error of 1.1 percent. 2.2.4 Air Resistance The air resistance is given by: AF = -1-- RHO CD AR (V + Vw)2 (2.14) 2 where RHO = the mass density of air, in kg/m3; CD the dimensionless aerodynamic drag coefficient of the vehicle AR = the projected frontal area of the vehicle, in m2; and Vw = the component of wind velocity in the direction opposite to the vehicle travel, in m/s. The mass density of air, RHO, is a function of both altitude and atmospheric temperature. Atmospheric temparature is also a function of altitude but the relationship also depends on latitude. Since RHO is 26 THE FORCE BALANCE RELATIONSHIP relatively insensitive to altitude and atmospheric temparature within the range we normally encounter, the following formula which relates RHO to altitude alone has been suggested (St. John and Kobett, 1978): RHO = 1.225 (1 - 2.26 ALT 10-5 4.225 (2.15) where ALT = elevation of the road above the mean sea level, in meters. The aerodynamic drag coefficient, CD, represents three sources of air resistance (Taborek, 1957; Wong, 1978): 1. Form drag, which is caused by the turbulence in the wake of the vehicle; it is a function of the shape of the vehicle body and accounts for most of air resistance; 2. Skin friction, which is caused by the shear force exerted on the vehicle exterior surfaces by the air stream; skin friction accounts for about 10 percent of total air resistance; and 3. Interior friction, which is caused by the flow of air through the raiator or vehicle interior for cooling and ventilating purposes; interior friction accounts for a small part of total air resistance. Typical values of the aerodynamic drag coefficient for different types of vehicles are shown in Table 2.1. Table 2.1: Aerodynamic drag coefficients Vehicle type CD (dimensionless) Passenger car 0.3 - 0.6 Convertible 0.4 - 0.65 Racing cars 0.25 - 0.3 Bus 0.6 - 0.7 Truck 0.8 - 1.0 Source: Wong (1978). 2.2.5 Rolling Resistance The following discussion is restricted in scope to vehicle rolling resistance on hard surfaces (e.g., paved roads and compacted unpaved roads). The main reason is that almost all the roads in service today fall into this category (with the rare exceptions of fair-weather-only unpaved roads which turn into a very wet, soggy state THE FORCE BALANCE RELATIONSHIP 27 after a flood or heavy rainfall, roads after a major snow storm, unpaved roads during a spring thaw, etc). A second reason is that less is known of the more complicated relationship between vehicle rolling resistance and the deformation characteristics of soft road surfaces. Smooth surfaces were the major focus of earlier research on vehicle rolling resistance. The single most important cause of vehicle rolling resistance was found to be the tire, which, in turn, arose primarily from the internal friction in the rubber and cord of the tire carcass resulting from tire deflection (Taborek, 1957; Wong, 1978; van Eldik Thieme and Dijks, 1981). Internal friction or hysteresis accounts for 90 - 95 percent of total tire rolling resistance. Two other causes provide secondary contributions: 1. Slippage between the surfaces of the rubber tread and the road (about 5-10 percent); and 2. Windage between the rotating tire and the surrounding air (1 - 3 percent). In more recent research, Lu (1983) investigated the effect of rough surfaces on vehicle rolling resistance. In addition to the effect of tire hysteresis when the road is smooth, Lu identified two other important sources which turn increasingly dominant when the road becomes rougher: 1. Additional hysteresis losses in the tire due to road surface irregularities; and 2. Energy dissipation in the suspension system due to the rel- ative motion between the sprung and unsprung masses. Lu simulated these effects on a computer by traversing an idealized vehicle of given tire and suspension mechanical properties over longitudinal road profiles of different roughness levels. The rolling resistance of the vehicle was found to increase by an order of 100 percent from smooth to very rough. While the results from the simulation are specific to the particular characteristics of the idealized vehicle, they highlight the importance of road roughness. Generally, the vehicle rolling resistance, RR, is expressed as: RR = mg cos CR (2.16) where mg coso = the component of the vehicle gravitational force in the direction perpendicular to the road surface, in newtons; and CR = the dimensionless coefficient of rolling resistance. 28 THE FORCE BALANCE RELATIONSHIP Equation 2.15 above may be simplified to: RR = m g CR (2.17) which introduces a 1 percent error for the extreme case of 15 percent gradient. Tire mechanical properties which influence rolling resistance depend on a number of factors including primarily: the inflation pressure, ambient temperature, tire diameter and other geometric characteristics, and tire construction and rubber material. Taborek (1957), Wong (1978), and van Eldik Thieme and Dijks (1981) provide discussions of the effect of these factors. The suspension stiffness and damping characteristics depend primarily on the mechanical design of the suspension system, and the viscosity of the damping fluid used; the latter, in turn, is influenced by the ambient temperature (Gillespie, 1980). Finally, the coefficient of rolling resistance is influenced by the speed of the vehicle because the energy dissipation due to tread deformation and carcass vibration increases with speed (Wong, 1981). The relationship between the coefficient of rolling resistance and speed is non-linear of degree 2.5, with the effect of speed being more significant at lower inflation pressures. For inflation pressures above 28 psi, the relationship could be divided roughly into three regimes: a fairly flat portion up to about 100 km/h, a linear portion between 100 km/h and 130 km/h and a steep non-linear portion above 130 km/h (Taborek, 1957). Owing to the complexity of the relationship between the above factors and the coefficient of rolling resistance, it is necessary to strive for simplification by focussing on policy-relevant factors. In the Brazil study, road roughness was considered to be of primary importance because of its sensitivity to construction and maintenance standards policies. Further, because of the wide range of vehicles involved (from small cars to articulated trucks) with varying tire radii and constructions, it was considered desirable to investigate the effect of the size of the vehicle on the coefficient of rolling resistance. The effects of tire pressure and ambient temperature were minimized by conducting the test runs under standard pressure and temperature conditions. Finally, it was decided to ignore the effect of vehicle speed because the range of speeds of interest in the context of rural roads in the developing countries lies predominantly in the flat regime of the rolling resistance-speed relationship described above. An experiment was conducted to test the roughness effect on vehicle rolling resistance with the other variables held constant. The test vehicles were coasted down in neutral over paved and unpaved road sections of different roughness levels. The time-distance data obtained from the experiment enabled the relationship between the coefficient of rolling resistance and road roughness to be estimated separately for light and heavy vehicles: THE FORCE BALANCE RELATIONSHIP 29 Light vehicles (cars and utilities): CR = 0.0218 + 0.0000467 QI (2.18a) Heavy vehicles (buses and trucks): CR = 0.0139 + 0.0000198 QI (2.18b) where QI is the road roughness, in QI units. The analysis procedure and interpretation of the results are given in Appendix 2A. 2.3 APPLICATIONS As a result of the discussion in the previous section, the force balance may be written as: m'a = 736 HP - mg GR - mg CR - 1/2 RHO CD AR V2 (2.19) V In this study the force balance equation has four principal applications: 1. To derive the coefficients of rolling resistance for the test vehicles. In this application the power term in Equation 2.19 is zero since the test vehicles were coasting down in neutral. By integrating Equation 2.19 over the coast-down distance, an energy balance equation is obtained. The coefficient of rolling resistance can be computed given that the other parameters and variables are known. (See details in Appendix 2A). 2. To derive the used driving power and the used braking power of vehicles under steady-state conditions. In this application, the acceleration term in Equation 2.19 is zero. When a vehicle travels up a steep hill the driver usually utilizes the maximum sustainable level of engine power. This is particularly true for loaded trucks. If the uphill speed of the vehicle is known, the force balance equation 2.19 can be used to compute the used driving power, denoted by HPDRIVE. Similarly, when a vehicle travels down a very steep hill the driver usually utilizes both the engine and the regular brakes to keep the vehicle from accelerating out of control. Again this is particularly evident for heavily loaded trucks. If the downhill speed of the vehicle is known, the magnitude of the used braking power, denoted by HPBRAKE, can be computed. The used driving and braking powers have two main uses: (i) in developing the steady-state speed model (Chapters 3 and 4) and the free-speed profile simulation model (Chapter 6); and (ii) in the adaption of the vehicle operating cost relationships developed in this study to suit local conditions (Chapter 13). 30 THE FORCE BALANCE RELATIONSHIP 3. As a converse to the above, to compute the driving and braking constraining speeds which are the necessary components in the prediction of vehicle travel speeds. Using the force balance relationship these constraining speeds can be computed once the used driving and braking powers are known (see Chapter 3). 4. To simulate the free-speed profile of a vehicle as it traverses a given road stretch. It is necessary to first establish a set of behavioral assumptions regarding how the driver uses the vehicle driving and braking powers in controlling the vehicle's speed and acceleration/deceleration. Given these assumptions, the free-speed profile of the vehicle can be determined by successive application of the force balance relationship at discrete intervals along the road (see Chapter 6). THE FORCE BALANCE RELATIONSHIP 31 APPENDIX 2A DEVELOPMENT OF RELATIONSHIP BETWEEN COEFFICIENTOF ROLLING RESISTANCE AND ROAD ROUGHNESS This appendix presents the results of the analysis to determine the relationship of the coefficient of rolling resistance to road roughness. 2A.1 COAST-DOWN EXPERIMENT The coast-down experiment is one of several methods available for determining vehicle rolling resistance (Van Eldik Thieme and Dijks, 1981). Basically it involves coasting down a vehicle in neutral from a relatively high speed (preferably 120 km/h or more) over a road section of known uniform geometry (usually level-tangent) and surface characteristics. Information on speed and deceleration and/or time and distance during the coast-down is recorded for subsequent calculations to arrive at an estimate of the rolling resistance coefficient. The coast-down experiment in this study involved 9 test vehicles, one car, two replicate utilities (VW-Kombi's), one large bus, two light trucks, two replicate heavy trucks and one articulated truck. The major characteristics of these vehicles are shown in Table 1A.2. Four level-tangent road sections were selected for the experiment, two paved (smooth and rough) and two compacted unpaved (smooth and rough). The characteristics of these sites are summarized in Table 2A.1(a). At each test site, two load levels, empty and loaded, were used on each test vehicle (see the numerical values in Table 2A.1(b)). At each load level the vehicle was coasted down 6 times in each direction. The distance travelled during the coast-down was recorded on a special instrument at 1-second intervals. Also measured during the experiment were the wind speed and direction, the ambient temperature, the rainfall, and the road surface conditions. The tire inflation pressure of each vehicle was maintained at a standard value throughout. More detailed descriptions of the coast-down experiment are given in GEIPOT (1982). 2A.2 ENERGY BALANCE EQUATION Based on the force balance equation different alternative analytical formulas for determining the rolling resistance coefficient can be derived. For a vehicle coasting down in neutral, the drive force equals zero, so the force balance equation can be re-written as: m' -l = - m g GR - m g CR - 1/2 RHO CD AR (V - V )2 (2A.1) where t is elapsed time and V = V(t). 32 THE FORCE BALANCE RELATIONSHIP Table 2A.1(a): Summary characteristics of road sections employed in coast-down experiment Section Characteristics Paved Paved Unpaved Unpaved smooth rough smooth rough Double Double Type of surfacing material surface surface Laterite Laterite treated treated gravel gravel Average gradient (percent) 0.2 0.2 1.5 1.0 Average horizontal curvature 14 0 0 0 (degrees/km) Average roughness (QI) 29 80 58 178 Average rut depth (mm) 7 2 6 54 Average depth of loose material (mm) - - 4 9 Average moisture content (percent) - - 1 2 Average ambient temperature (*C) 24.5 23.2 24.2 20.8 Average wind speed (m/s) 3.6 3.0 3.5 3.4 Average rainfall (mm/month) 0 0 0 0 Average spacing of corrugation (m) - - 0 0 Average depth of corrugation (mm) - - 0 0 Source: Brazil-UNDP-World Bank highway research project data. One possible derivation is to re-state the above equation as: Y =B + AX where Y = m' dV dt B = - m g GR - m g CR A = - 1/2 RHO CD AR and X (V - Vw)2 From a vehicle coast-down, a series of data points (X, Y) can in principle be obtained which can then be used in linear regression analysis to estimate the parameters B and A. With knowledge of the road Table gradient (GR), the vehicle weight (m), the air mass density (RHO) and the vehicle projected frontal area (AR), the coefficients of rolling resistance (CR) and aerodynamic resistance (CD) can be determined (St. John and Kobett, 1978). Although relatively simple and analytically ap- pealing, this approach was not possible in the Brazil-UNDP study. Instru- mentation mitations precluded accurate time-distance profile measurement THE FORCE BALANCE RELATIONSHIP 33 Table 2A.1(b): Summary of loads and number of test runs employed in coast-down experiment Load carried Number of runs (kg) by load level By Load level By section Un- Un- Test Un- Un- Paved Paved paved paved Total vehicle loaded Loaded loaded Loaded smooth rough smooth rough Small car 0 280 39 45 22 24 20 18 84 Utility 0 550 40 45 24 19 22 20 85 Replicate 0 550 17 24 20 16 4 1 41 utility Bus 0 2250 41 34 19 19 22 15 75 Light gasoline 150 3510 22 32 9 22 11 12 54 truck Light diesel 0 3325 34 23 9 19 9 29 57 truck Heavy truck 1730 11970 44 47 23 35 23 10 91 Heavy truck 1670 12045 35 36 21 12 21 17 71 with crane Articulated 0 26600 21 31 23 13 5 11 52 truck Total number of runs 293 317 170 179 137 124 610 Source: Brazil-UNDP-World Bank highway research project data. at vehicle speeds greater than 80 km/h. Therefore the test vehicles could not be coasted down from a sufficiently high speed (over 120 km/h) needed to generate a good range of data points (X, Y). However, as aerodynamic drag coefficients of the test vehicles were available from the manufacturers, it was decided to use the energy balance approach for determining the rolling resistance coefficient. The energy balance equation can be derived from the force balance equation as follows: first, rewrite Equation 2A.1 as : m d V + m g GR + m g CR + 1/2 RHO CD AR (V + V )2 = 0 (2A.2) where x = x(t) = the distance travelled by the vehicle during the coast-down. Integrating the above equation between any two distance points xl and X2 during the coast-down, we obtain an energy balance equation: 34 THE FORCE BALANCE RELATIONSHIP AKE12 + APE12 + ARE12 + AAR12 = 0 (2A.3) x2 where AKBl2 f m' V dV = 1/2 m' (V22 - V12) xl APE12 = f m g GR dx = m g GR (x2 - x) xl x2 ARE12 f m g CR dx = m g CR (x2 - xj) X1 X2 and AAR12 f 1/2 RHO CD AR (V - Vw)2 dx xl The terms AKE12 and APE12 are the changes in the kinetic and potential energies, respectively, of the vehicle between points x, and X2 (in J). The remaining terms ARE12 and AAR12 are the energy losses due to rolling and air resistances, respectively, in covering the distance between these points (in J). The air resistance term generally does not have a closed form solution. However, assuming constant acceleration and wind speed, the energy loss due to air resistance can be expressed in the following closed form: ARE12 = 1/2 RHO CD AR [(V22 + V12)/2 + V2 (2A.4) 2 1 w + 2/3 Vw (V2 + V2 V1 + V1)2/(V2 + V1)] (x2 - xd The assumption of constant acceleration is adequate if the distance between x, and x2 is relatively short, or if the vehicle speed is so low that the air resistance loss becomes a small component of the total loss. The adequacy of this assumption will be discussed further subsequently. Strictly speaking, the vehicle rolling reoistance produced in the coast-down experiment does not consist entirely of tire and suspension motion resistances. It also includes small resistances due to friction in the wheel bearings and in the drive-shaft and differential. According to Taborek (1958), the efficiency of bearings is in the range 0.98 - 0.99 and the efficiency of differentials is about 0.95. For modelling purposes in this study the 1-2 percent losses in the wheel bearings were ignored. Although the differential was not so efficient as the wheel bearings, the loss in it would have been a negligible fraction of the total rolling resistance losses since the drive shaft was spinning under virtually no load. Therefore, the losses in the differential were also ignored. To compute the effective mass, m', we set the term me to 0 in Equation 2.7, since the vehicle was in neutral. The term mw is computed based on the following simplifications: THE FORCE BALANCE RELATIONSHIP 35 1. Using the average of the inner and outer radii of the tire,denoted by RMT (in meters), to approximate the radius of gyration of each wheel; and 2. Using the nominal radius of each tire, denoted by RNT (in m), to approximate the effective rolling radius, RRT.1 The resulting formula is given by: m' = m + M = M + M [RMT]2 (2A.5) w w RNT where Mw is the total mass of all mounted wheels, in kg, as defined earlier. The values of mw computed for the test vehicles are shown in Table 1A.2. 2A.3 DETERMENATION OF ROLLING RESISTANCE COEFFICIENTS The computational procedure employed can be described in the following steps. 1. Using linear regression analysis, the following quadratic curve was fitted to the last 15 seconds of each time distance profile: x = x(t) = p0 + p1 t + p2 t2 where po, p1 and P2 are estimated parameters. The above quadratic curve assumes that the vehicle deceleration (equal to -d2x/dt2 or -2P2) was constant. This assumption was considered to be realistic when only the last 15 seconds of the time-distance profile was used. Figure 2A.1 shows a plot of the time-distance data points and the fitted curve, for a typical vehicle run. 2. The coefficient CR for each vehicle run was computed using the energy balance equation (2A.3) with the vehicle speeds at the beginning and end of the 15-second period (denoted by subscripts 1 and 2, respectively) given by: Vi = p1 + 2 p2 tl V2 = P1 + 2 P2 t2 3. For each combination of vehicle, road section, load level and direction an average CR coefficient was computed from the replicate run values obtained in step 2 and compiled in Table 2A.2. Let CRvstd denote this average, where v, From experimental results compiled by van Eldik Thieme and Dijks (1981) the ratio of effective radius to the radius when the tire is unloaded varies in the range 0.95 - 1.00 depending primarily on the tire construction, vehicle speed and load. 36 THE FORCE BALANCE RELATIONSHIP Figure 2A.1: Time distance observations from a typical coast-down run and the fitted curve Distance (a) ++Observation 600 -Pitted 700- 600 500 400 300 200 100 5 5 10 15 20 25 30 35 40 45 50 55 Time (s) Source: Brazil-UNDP-World Bank highway research project data and author's analysis Figure 2A.2: Computed rolling resistance coefficient versus roughness for all combinations of test vehicle and road section Coefficient of rollieg resistance 0.0 A A A Truck or bus * 0 * Car or utility 0.03 0.02 0.01 0.00 ./ka IRI 25 50 75 100 125 150 175 20 QI Roughnes THE FORCE BALANCE RELATIONSHIP 37 Table 2A.2: Computed coefficients of rolling resistance for each test vehicle by road section, load level and direction paved Pvd Unpad Unpved Smoth touzh smoth Rough Vehicle Type LC D 1,.ed llV. qe IV.14 cel c%e Cveeld Cvel C,e vReld CRVe1 CRe 1 0.0242 0.0227 0.0354 0.0207 U - 0.0251 - 0.0219 0.0283 - 0.0353 2 0.0259 0.0210 0.0211 0.0498 Ss11 ar - 0.0249 0.0242 0.0285 0.0319 1 0.0236 0.0261 0.0400 0.0258 L 0.0247 - 0.0266 - 0.0288 - 0.0286 2 0.0259 0.0270 0.0177 0.0314 1 0.0228 0.0221 0.0254 0.0164 U 0.0267 - 0.0230 - - 0.0219 - 0.0293 2 0.0267 0.0240 0.0184 0.0422 Utility 0.0218 0.0232 0.0205 0.0285 1 0.0167 0.0215 0.0238 0.0093 L. 0.0188 0.0233 0.0192 0.0276 2 0.0209 0.0250 0.0166 0.0459 1 0.0217 0.0279 -- ' 0.0221 0.0297 2 *0.0226 0.0315 - Replicate 0.0232 0.0284 0.0241 - utility 1 0.0237 0.0265 0.0370 0.0131 L 0.0243 - 0.0271 - 0.0241 - * 2 0.0249 0.0276 0.0113 - 1 0.0115 0.0161 0.0.[64 0.0013 U 0.0109 - 0.0114 - 0.0136 - 0.0193 2 0.0104 0.0067 0.0109 0.0374 Sue 0.0108 0.0134 0.0165 0.0193 1 0.0080 - 0.0165 0.0269 L. 0.0107 - 0.0153 - - 0.0194 * 2 0.0134 0.0142 0.01.19 0.0361 1 - 0.0184 0.0223 - Light U - 0.0187 - - 0.0183 2 - 0.189 0.0143 touck 0.0145 0.171 0.0183 0.0179 1 0.0131 0.0151 - 0.0013 L. 0.0145 0.0156 -- 0.0179 2 0.0159 0.0160 - 0.0346 I - 0.0239 0.0534 0.0172 U - - 0.0215 * 0.0232 Light 2 - 0.0191 - 0.0292 dieseI 0.0166 0.0215 * 0.0223 1 I 0.0226 - 0.0433 0.0175 L 0.0166 --- * 0.0214 2 0.0106 - - 0.0253 1 0.0172 0.0147 0.0192 0.0034 Neavy U 0.0190 - 0.0177 - - 0.0151 0.0094 track 2 0.0207 0.0206 0.0110 0.0154 0.0164 0.0150 - 0.0146 0.0094 1 0.0126 0.0125 0.0159- L 0.0138 - 0.0124 - 0.0141 2 0.0151 0.0122 0.0126 - 1 0.0198 0.0136 0.0185 0.0007 .avy U 0.0127 - 0.0128 0.0158 - 0.0182 2 0.0056 0.0121 0.0131 0.0358 With 0.0121 0.128 - -- 0.0139 0.0172 1 0.0103 - 0.0133 0.011 L 0.0115 - 0.0121 0.0161 2 0.0127 - 0.0109 0.0162 1 0.0128 - - 0.0020 A.ticu- U 0.0141 *- - - 0.0194 lt.d 2 0.0133 0.0131 0.0126 0.0369 truck 0.0127 0.0128 - * 0.0174 1 0.0107 0.0137 - 0.0157 L- - 0.0114 - 0.0119 - * 0.0154 2 0.0120 0.0119 0.0133 0.0151 LC = Loading condition, U = Unloaded, L = Loaded, D = Direction of travel 1 = Downhill in the case of the paved smooth and unpaved smooth sections and uphill in the case of the unpaved rough section. The paved rough section is flat. 2 = Reverse direction to the above (1). - = Means that no usable run was available for the particular combination of vehicle, Loading condition and direction. * = Signifies that average was not taken because the value for one of the directions was not available. Source: Analysis of Brazil-UNDP-World Bank highway research project data. 38 THE FORCE BALANCE RELATIONSHIP s, j and d stand for vehicle, road section, load level and direction, respectively. 4. For each combination of vehicle, road section and load level, a CR coefficient, denoted by CRvsk, was computed by simple averaging over the two directions: CR 1 (CR + CR ) vst vsi1 vs2 2 where subscripts 1 and 2 denote the two directions. The significance of this step is to cancel out direction- dependent errors, viz., in the measurement of road gradient and wind speed. Korst and White (1973) provide an analyti- cal rationale for the need to average out the coefficient estimates over the two directions. The values of CRvsp are tabulated in Table 2A.2. 5. For each combination of vehicle and road section, a CR coefficient, denoted by CRvs was computed by averaging over the load levels: 1 CR =- (CR + CR ) vs 2 vs1 vs2 2 where subscripts 1 and 2 denote the low and high load levels, respectively. Averaging over load levels was based on previous findings that the rolling resistance coeffi- cient is independent of vehicle load (within the operating range). This finding is clearly supported by the numerical results in Table 2A.2. (For the vehicle section combina- tions that had useful time distance data for only one level no averaging was necessary). The values of CRvs are shown in Table 2A.2. 2A.4 DETEREMINATION OF RELATIONSHIP BETWEEN ROLLING RESISTANCE COEFFICIENT AND ROAD ROUGHNESS Figure 2A.2 shows a plot of rolling resistance coefficient against road roughness for all combinations of test vehicles and road sections obtained from the data points in Table 2A.2. The test vehicles are separated into two groups: 1. Light vehicles - car or utility 2. Heavy vehicles - bus or truck. From the data plot two trends are evident: 1. The rolling resistance of a vehicle generally increases with road roughness. This agrees with the findings by Lu THE FORCE BALANCE RELATIONSHIP 39 Table 2A.3: Computed rolling resistance coefficents by vehicle group and section Paved Paved Unpaved Unpaved Section smooth rough smooth rough Vehicle group Cars and utilities 0.02331 0.02525 0.02440 0.3020 Trucks and buses 0.01386 0.01544 0.01584 0.01727 Source: Analysis of Brazil-UNDP-World Bank highway research project data. Figure 2A.3: Rolling resistance coefficient versus roughness by vehicle group: data points and fitted lines Coefficient of rolling resistance 0.04- 0. 0S 0.02 1 0.01 0.00 0 25 50 75 100 125 150 175 200 Roughness QI A Truck or bus --" Car or utility Source: Analysis of Brazil-UNDP-World Bank highway research project data. 40 THE FORCE BALANCE RELATIONSHIP (1983) cited in the text. Furthermore, according to Limpert (1982), the rolling resistance can increase by as much as 30 percent between a smooth and a badly potholed road. 2. As a group, the rolling resistance of the light vehicles is considerably higher than that of the heavy vehicles. From the discussion in Van Eldik Thieme and Dijks (1981), tire inflation pressures used on commercial vehicles are generally 3 - 4 times those used on passenger cars (the tires used on the VW-Kombi's are of passenger-car size). This results in a smaller hysteresis loss on commercial vehicle tires. Since there are too few data points to estimate coefficients specific for each test vehicle, it was decided to develop separate relationships only for the light and heavy vehicle groups. This was done by first averaging out the CR values of the test vehicles by section and vehicle group, yielding results shown in Table 2A.3. Figure 2A.3 shows a plot of the vehicle-group averge of rolling resistance was then regressedagainst road roughness, resulting in the following relationships: Light vehicle: CR = 0.0218 + 0.0000467 QI Heavy vehicle: CR = 0.0139 + 0.0000198 QI Figure 2A.3 also shows a graph of the fitted regression lines superimposed with the data points. It can be seen that both the intercept and the slope for the light vehicle class are greater than those for the heavy vehicle class. This observation is consistent with the early discussion on the difference between light and heavy vehicle tires. It is also consistent with Taborek's findings (1958) that the coefficient of rolling resistance tends to be more sensitive to road surface conditions as the tire diameter decreases. The coefficient values for light and heavy vehicles increase by 46 and 31 percent, respectively, as roughness increases from very smooth (QI = 20) to very rough (QI = 250). This observation is broadly consistant with the computer simulation results obtained by Lu (1983). Two other studies investigating the relationship between the coefficient of rolling resistance and roughness have been reported in the literature from India (Kadiyali et al., 1982) and South Africa (Bester, 1984). Due to the preliminary nature of the India results, only the South Africa results are discussed below. The South African study also used coast-down technique, and runs were made with cars (8 sections) and trucks (5 sections). The roughness of the sections was between 12 QI and 75 QI. Using the recommended value for the speed-dependent component of the coefficient of rolling resistance, and assuming the roughness effect to be entirely due to rolling resistance, the relationships estimated in the study may be written as follows: THE FORCE BALANCE RELATIONSHIP 41 Car: CR = 0.0152 + 0.0000331 QI + (6.99 + 0.049 QI) 10-6 V2 Truck: CR = 0.0086 + 0.0000826 QI + 6.99 10-6 V2 where V = vehicle speed in m/s. Ignoring the marginal contribution of the speed-dependent part to the coefficient of rolling resistance (e.g., about 0.001 at 12 m/s for trucks), it can be seen that the South Africa relationships are numerically similar to those from Brazil--with the exception of the roughness coefficient for trucks being larger in the former (0.0000826) than in the latter (0.0000198). This discrepancy may be due in part to the relatively small range of roughness used in the South Africa Study. Further, the road sections in the South African study sample were bunched over the very low end of the roughness spectrum with five of the eight sections having roughness values below 24 QI. The relative magnitude of the roughness coefficients in the Brazil relationships seem to be more plausible on a priori grounds, because the extent of pentration is less for larger tires. At any rate, both studies predict a lower value of the coefficient of rolling resistance for heavy vehicles than for light vehicles. A significant finding of the South African study is that, while concrete and asphaltic concrete pavements have a lower rolling resistance than roads with a surface treatment with the same roughness, the differences are negligible. 2A.5 CONCLUSIONS AND RECOMMENDATIONS The main limitation of the Brazil relationships is that, due to the fact that there were only four road sections, two for each pavement type, the relationships are less robust than they should be, nor do they make a distinction between pavement types. For highway investment planning purposes the relationships from the Brazilian study are recommended on an interim basis, with their applications restricted to paved and compacted unpaved surfaces and vehicles with relatively low speeds. Further work is recommended to strengthen these relationships by using field data from more road sections of different surface characteristics, and also by merging experimental data with quarter-car simulation models which can incorporate the effect of suspension characteristics and tire damping. CHAPTER 3 Steady-State Speed Model: Formulation This chapter is concerned with the theory and analytical deri- vation of the steady-state speed model, functionally the most important model developed in this volume. The approach to speed prediction adopted is described first (Section 3.1). It is followed by a discussion of the limiting speed model (Section 3.2) and the detailed analytical derivation of the associated "limiting" speeds (Section 3.3). This provides the basis for the probabilistic version which enables the parameters of the limiting speed model to be estimated (Section 3.4). Section 3.5 provides a summary of the steady-state speed prediction formula. 3.1 PROBLEN STATENENT AND APPROACR The problem of predicting free-flow speeds for a roadway may be stated as follows: given the vertical and horizontal profiles, the sur- face type and condition, and the width of the roadway, predict the speed profile of an unimpeded vehicle of known characteristics along the road- way. The level and nature of the decision for which the speed pre- dictions are required (for example, sectoral policy analysis, appraisal of a new road project, or evaluation of a proposed minor improvement) has implications both for the inputs into and the output from the predic- tion. On the input side, the level of detail available on the roadway may vary from just a few aggregate descriptors of the roadway to detailed information based on geometric design and condition survey. Similarly, on the output side, the accuracy and the level of detail desired of the predicted speed may range from just the space-mean speed for the round- trip travel on the roadway to nearly continuous speed profile across the roadway with statistical confidence statements. In order to cope with the diversity of demands the speed predic- tion model is called upon to cope with, it is convenient, even necessary, to structure the process of speed prediction into two inter-related, but conceptually distinct, components: 1. A model to predict an appropriately defined concept of vehicle speed on a stretch of road over which the charac- teristics of interest do not change appreciably. We refer to such a road stretch as a homogeneous section of the roadway. The concept of speed used is that of a steady- state speed and this part of the speed prediction process may be called a steady-state speed prediction model. 43 44 STEADY-STATE SPEED MODEL: FORMULATION 2. A set of procedures to apply the above for predicting the vehicle speed profile over a heterogeneous roadway, using the available information on the roadway and with the de- sired degree of accuracy for the particular application. We may refer to these procedures as roadway speed predic- tion methods. The current chapter is devoted to a theoretical account of the steady-state speed prediction model. (The next chapter presents the details regarding the econometric implementation of the model.) Chapters 5 through 7 describe alternative procedures for predicting roadway speeds based on the steady-state speed predictions. The steady-state speed of an unimpeded vehicle of known attributes traversing a homogeneous road section of known characteristics, located in a given socioeconomic and traffic environment may be defined as the speed the vehicle would eventually attain and maintain if the homogeneous road section were indefinitely long. Thus steady-state speed in a given environment may be regarded as an inherent property we can associate with a given combination of a homogeneous road section and a vehicle. The distance covered by a vehicle before it reaches its steady-state speed varies considerably. For a heavily loaded truck entering a steep uphill gradient from a long level-tangent section, the distance would be only a few tens of meters long. On the other hand, when the same truck enters a level-tangent section at crawl speed, it would need to travel several hundred meters before reaching its maximum cruising speed. In the model to be described, a homogeneous road section is assumed to be completely defined if its surface type, slope, curvature, superelevation and surface irregularity measure are specified.1 It may variables under the control of the highway planner. We occasionally refer to these as "speed-influencing characteristics" or "road severity factors," collectively denoted by the symbol X. It should be noted that the direction of travel on the homogeneous section is part of the description of the section. In other words, the term section stands for the longer expression "section-direction." The word "vehicle" is also a short-hand for "the operator-vehicle system" and the term operator is used to bring out the fact that the speed decision-maker may be an individual driver or a transport firm. As for the characteristics of the vehicle, the vehicle-class and loading (in the case of a truck) are supposed to be known. Also, a set of technical characteristics of the vehicle, such as 1 Roads are assumed to be at least 5.5 m wide. As estimated initially using roadside speed observations made in Brazil, road width was not one of the independent variables, but the effect of road width on the steady-state speed has been incorporated based on data collected in India, as described in Section 4.4. STEADY-STATE SPEED MODEL: FORMULATION 45 unladen weight, drag coefficient, and so on, are assumed to be known or assignable with reasonable accuracy. We denote the technical characteristics of the vehicle by the symbol Y. Finally, we come to a set of behavioral-technical characteristics of the vehicle, such as used power, perceived friction ratio, desired speed, and so on. These are the estimated parameters of the steady-state speed preciction model and are collectively denoted by the symbol e. For a given application these parameters may be estimated afresh, calibrated on the basis of limited observations or, in some cases, judgmentally determined. The socioeconomic and traffic environment is explicitly mentioned as given because it is not explicitly modelled. In other words, the speed decision has not been modelled as being part of the economic decision problem of a vehicle operating firm or a private driver. Also, features of the general traffic environment within which the particular homogeneous section is located, such as design consistency, have not been related to attained speed. Thus, the model parameters would embody the effect of the environment in which they were estimated, and the degree of transferability of different parameter estimates may vary. The steady-state speed, as defined above, is the appropriate speed concept to work with in the context of homogeneous sections because it represents a state of equilibrium in the driver-vehicle-roadway system so that the process of measurement on the system is well-defined. In fact, the concept of steady-state speed is fundamental for any speed prediction model, whether or not it is explicitly recognized. Since homogeneous road sections represent the most natural way of dividing a roadway, the steady-state speed on the homogeneous section is a theoretical starting point for further analysis. It also represents a "base" level of detail at which speed predictions for the entire roadway could be carried out. Relative to it, one can conceptualize predictions on a more detailed basis as well as on a more aggregate basis. Essentially, the process of incorporating more details consists in modelling transitions betwen successive steady-state speeds on adjacent homogeneous sections. Moreover, as discussed later, the aggregate methods to predict roadway speeds entail representing the entire heterogeneous roadway as a small number of hypothetical "average" homogeneous sections; in fact, as explained in Chapter 7, the procedure of aggregate prediction basically entails treating the entire roadway as two homogeneous sections of positive and negative grades with road characteristics represented by averages. The average journey speed over the roadway is computed on the basis of the predicted steady-state speeds on these idealized road sections. In practical terms also, prediction of speed simply as the space mean speed of the steady-state speeds of homogeneous sections as described in Chapter 5 represents one plausible level of detail which might be adequate for some applications. Indeed we shall study the performance of more detailed level predictions and more aggregate level 46 STEADY-STATE SPEED MODEL: FORMULATION predictions by comparing them with the performance of steady-state speed predictions. 3.2 LIMITING SPEED MODEL: APPROACH AND BASIC FORMULATION Our modelling objective is to develop a functional expression which relates the steady-state speed of a given vehicle class to a set X of descriptors or variables which capture the geometric and surface characteristics of the homogeneous road section. Specifically, these descriptors are: 1. The road gradient, GR, expressed as a fraction (to represent the vertical alignment); 2. The horizontal curvature, C, in degrees/km (or the radius of curvature, RC, in m), and the curve superelevation, SP, expressed as a fraction (to represent the horizontal alignment); and 3. The road roughness, QI, in QI (or m/km IRI) units, and the surface type, ST, which may be paved or unpaved (to represent surface characteristics). Brief descriptions of these variables were given in Section 2.1. One approach which has been employed in several previous studies is to relate the steady-state speed directly to these road descriptors. Because of the difficulties in developing a proper mathematical expression, a linear form has generally been adopted. The most basic of such models may be stylized as: VSS = q0 + q, GR + q2 C + q3 QI (3.1a) where VSS denotes the steady-state speed (in m/s) and q0 through q3 are model parameters. Some of the refinements over the basic model found in the literature follow. First, the model is estimated independently for different surface types. Second, vehicle characteristics, such as power-to-weight ratio and other roadway descriptors, such as rut depth, are introduced as needed as independent variables. Third, a dummy variable formulation is sometimes used to allow the positive and negative grades to have different coefficients. Briefly, this is done by using the rise (RS) and fall (FL) variables defined in Section 2.1, and specifying the model as: VSS = q0 q11 RS + q12 FL + q2 C + q3 QI (3.1b) Linear models of similar forms have been empirically estimated in several studies yielding coefficients of expected signs and reasonable magnitudes (Hide, 1975; Morosiuk and Abaynayaka, 1982; CRRI, 1982; and TRDF, 1980). The values of coefficients other than q0 have generally been found to be negative, indicating the negative influence of road STEADY-STATE SPEED MODEL: FORMULATION 47 severity on vehicle speed, although the coefficient of "fall," q12, is occasionally found to have a small positive value for some vehicle classes in some studies. The constant term, q0, which has generally been found to be positive, represents various subjective (or not easily quantifiable) factors. When road severity variables are at their minimum values, e.g., when the road is flat (GR = 0), straight (C = 0) and smooth (QI approaching zero), the steady-state speed approaches q0; at this point the vehicle speed tends to be constrained by subjective factors. Although the direct approach is evidently workable, its application requires care to circumvent the following inherent problems: 1. It is possible to predict unreasonably low (at times negative) steady-state speeds, especially for low standard roads where the road gradient (GR), curvature (C), and roughness (QI) simultaneously assume large values. In these cases, one has to take recourse to imposing a floor value on the predicted steady-state speeds, but such a procedure distorts benefits from road improvement projects. This is a serious limitation in the light of the fact that, more often than not, it is precisely such roads that are candidates for upgrading. 2. The partial derivative of steady-state speed with respect to each road severity variable is constant (equal to qi); this is unrealistic, particularly when applied to low-standard roads. For example, on a very rough, steep uphill road section, vehicle speeds cannot be raised significantly simply by making the surface smoother while the gradient is still the speed-limiting factor. The issues raised above point to the lack in the linear form of a desirable property, namely, asymptotic consistency; that is, as one road severity variable increases and the others are held constant, the steady-state speed should decrease and should approach zero at a decreasing rate and never become negative. This problem can be mitigated partly by making the mathematical form nonlinear and incorporating interaction terms to make the partial derivatives sensitive to road standard, but the approach would still be ad hoc. That is, the model would not incorporate our prior scientific and behavioral knowledge regarding vehicle dynamics and driver behavior and would consequently be of questionable reliability in extrapolation or transference to a new country. The limiting speed approach is an alternative to the above. Instead of directly associating the steady-state speed with the speed-influencing parameters of the homogeneous section, the approach consists of first associating a set of steady-state "limiting or constraining speeds" with the speed-influencing parameters. Then the vehicle is postulated to be driven at the maximum attainable speed subject to these speed constraints. That is, the observed steady-state speed of the vehicle is regarded as the minimum of the unobservable or latent speed constraints generated by the interaction of road severity 48 STEADY-STATE SPEED MODEL: FORMULATION factors with relevant characteristics of the vehicle. Formally, the model may be written as VSS = min [VDRIVE, VBRAKE, VCURVE, VROUGH, VDESIR] (3.2) where VSS = the attained steady-state speed; VDRIVE = the speed limited by gradient and used driving power, i.e., power applied in the direction of motion; VBRAKE = the speed limited by gradient and used braking power, i.e., power applied against the direction of motion; VCURVE = the speed limited by curvature; VROUGH = the speed limited by roughness; and VDESIR = the desired speed in the absence of road severity factors. All the above quantities are expressed in m/s. As will be seen in the next section, VDRIVE and VBRAKE are primarily functions of the road gradient and the load level and secondarily a function of road roughness. VCURVE is a function of the horizontal curvature and superelevation as well as the surface type and load level. VROUGH is a function of road roughness. Finally, VDESIR, which in general should be a function of several factors, is treated as a constant for a given surface type in the original model estimated with Brazil data. However, in the later extension of the steady-state speed model using data from India, VDESIR also depends on the width class of the homogeneous section. It should be noted that a given road descriptor or vehicle characteristic may be associated with more than one constraining speed. For example, as derived in the next section, the vehicle weight influences both VDRIVE and VBRAKE and road roughness appears as a determinant of VDRIVE, VBRAKE and VROUGH. In other words vehicle and road characteristics affect the steady-state speed through one or more constraining speeds. Figures 3.1-3.3 provide graphical illustrations of constraining speeds and the attained speed for a loaded heavy truck, Mercedes Benz 1113.2 Figure 3.1 plots the steady-state and constraining speeds against the road gradient for a slightly curved, smooth paved section. In this figure three constraining speeds are binding over the + 10 percent range of the road gradient. The maximum possible driving speed (VDRIVE) dominates over slightly negative (-0.2 percent) and 2 The constraining speeds are as derived in Section 3.4, with the parameter values compiled in Chapter 14 for the Mercedes Benz 1113. STEADY-STATE SPEED MODEL: FORMULATION 49 Figure 3.1: Constraining and steady-state speeds versus gradient Speed (m/8) 70- VDRIVE so --VBRAKE VROUGH 0- - - ~ VCURVE 40 30- VDESIR 20 VSS 10 20- -10 -8 -6 -4 -2 0 2 4 6 a 10 Gradient (% Source: Analysis of Brazil-UNDP-World Bank highway research project data. Figure 3.2: Constraining and steady-state speeds versus curvature Speed (m/s) 60- VCUJRVE VROUGH 50 --- -- - - - - 40- VDESIR 20 V VDRIVE 10 0 100 200 300 400 500 800 700 800 900 1000 Curvature (Degrees/km) Source: Analysis of Brazil-UNDP-World Bank highway research project data. 50 STEADY-STATE SPEED MODEL: FORMULATION Figure 3.3: Constraining and steady-state speeds versus curvature Speed (a/s) 100 VROUGH so 70j 601 so -VCURVE 40- 30 - VDESIR 20 0 25 50 75 100 125 150 175 200 Roughness QI Notes for Figures 3.1 - 3.3 1. To interpret the figures, choose any value on the respective road characteristic (horizontal) axis, imagine a vertical line passing through the value, and read off various speed values. 2. The figures are for a loaded heavy truck (Mercedes Benz 1113) on a paved road. The parameter values used are the ones given in Table 14.1 and 14.2 for a heavy truck. 3. The non-varying road characteristic values are GR=O, C=57 degrees/km, and QI-40. The VBRAKE curve is not shown in Figures 3.2 and 3.3 because its value is infinity. 4. These notes also apply to Figures 3.6 - 3.8. Source: Analysis of Brazil-UNDP-World Bank highway research project data. STEADY-STATE SPEED MODEL: FORMULATION 51 positive gradients. For negative gradients of magnitude greater than 7.5 percent the maximum allowable braking speed (VBRAKE) becomes dominant. In the mid-range the steady-state speed is determined by the desired speed (VDESIR). Neither the maximum allowable curve speed (VCURVE) northe maximum ride severity speed (VROUGH) has any influence on the steady-state speed. The plot in Figure 3.2 shows the effect of the horizontal curvature on the steady-state speed for a level smooth paved road. In this example two constraining speeds are binding. The maximum possible driving speed (VDRIVE) prevails over gentle curvature up to 300 degrees/km (approximately corresponding to a radius of curvature of 200 m) beyond which the steady-state speed is dictated by the curve speed constraint (VCURVE). The effect of road roughness on the steady-state speed for a level-tangent paved road is shown in Figure 3.3. Two constraining speeds are binding, with the maximum possible driving speed (VDRIVE) over the smooth range (QI under about 90) and the maximum ride severity speed (VROUGH) over the rough range (QI over 90). It should be noted that the VSS values graphed in these figures refer to a particular instance of vehicle traversal, and not to average VSS values. This point will be further clarified when the probabilistic version of the model is described in the next section. In all three examples the steady-state speed drops monotonically but always remains positive as the road attributes--gradient, curvature and roughness--become severe. Thus the limiting speed approach satisfies the asymptotic consistency requirements discussed above. An even stronger appeal of this approach is that a considerable amount of prior scientific and engineering knowledge can be incorporated in the process of associating the constraining speeds with speed-influencing parameters. This has in fact been done by previous researchers, including Guenther (1969), Moavenzadeh et al. (1971), Sullivan (1977) and Galanis (1980). The various studies differ in the number of limiting speeds used as well as the way the limiting speeds are related to road and vehicle characteristics. 3.3 LIKITING SPEED MDDEL: IERIVATION OF CONSTRAINING SPEEDS In this section, expressions are derived for the five steady-state limiting speeds for a vehicle class in terms of the homogeneous section characteristics X, the vehicle attributes Y, and estimable parameters 6. 3.3.1 Driving Power-limited Speed, VDRIVE The concept of driving speed constraint based on a constant used driving power has been employed in detailed vehicle speed simulation by previous researchers, including Sullivan (1977) and Gynnerstedt et al. (1977), among others. Under steady-state conditions the acceleration is zero, and the force balance introduced in Chapter 2 (Equation 2.19) 52 STEADY-STATE SPEED MODEL: FORMULATION may be written as: 736 HP 736 HP= m g GR + m g CR + 1/2 RHO CD AR V2 (3.3) V The variables above are as defined in Chapter 2. The value of mass density of air, RHO, may be calculated as a function of elevation of the section above the Mean Sea Level (ALT, in m), using Equation 2.15. The wind velocity, Vw, is assumed to be negligible. The speed limited by driving power, VDRIVE, is assumed to be governed by a constant "used driving power," denoted by HPDRIVE. Substituting HPDRIVE for HP in Equation 3.3 above and simplifying, we get 1/2 RHO CD AR V3 + m g (GR + CR) V - 736 HPDRIVE = 0 (3.4) which is a cubic equation with V as the unknown. For all values of GR, the number of sign changes in the coefficients of the equation is unity. Thus, by Descartes' rule of signs, the equation always has exactly one positive root (Dickson, 1957). We define VDRIVE as the unique positive solution to Equation 3.4. Cubic equations are generally solved by iterative methods. However, since the coefficient of the square term in Equation 3.4 is zero, the equation has a closed form solution. The solution procedure is outlined in Appendix 3B. The VDRIVE curve as a function of gradient may be seen in Figure 3.1. The used driving power, HPDRIVE, is the model parameter to be estimated from observed speed data for this speed constraint. The vehicle attributes were obtained from other sources. The average vehicle mass (m) was obtained from actual portable-scale weighing of sample vehicles in the population at various sites (GEIPOT, 1982). The average vehicle aerodynamic drag coefficient and projected frontal area were adapted from values supplied by the manufacturers of the test vehicles (Appendix 1A). 3.3.2 Braking Power-limited Speed, VBRAKE The concept of the braking crawl speed has been previously employed in traffic simulation modelling (Sullivan, 1977; and St. John and Kobett, 1978). When a vehicle descends a long steep grade its maximum descent speed is known to be controlled by the vehicle braking capability, which, according to informal conversations with and observations of truck drivers, results from the use of the vehicle engine retardation power or the regular brakes themselves or both. This braking capability is defined as the "used braking power," a positive quantity denoted by HPBRAKE, which is assumed to govern the braking power-limited speed constraint, VBRAKE. Substituting HPBRAKE for HP in the force balance equation for steady-state speed (Equation 3.3) results in: STEADY-STATE SPEED MODEL: FORMULATION 53 736 HPBRAKE = m g GR + m g CR - 0.5 RHO CD AR V2 (3.5) V Since the braking speed constraint is likely to become binding only on steep negative grades (more than 5 percent) where the steady-state speeds are relatively low, the air resistance term in Equation 3.5 becomes insignificant and may be dropped without causing a large error in the solution. Further, when the "effective gradient" (GR + CR) of a homogeneous section is non-negative, VBRAKE becomes irrelevant since braking is not needed to control the vehicle speed. This is identical to defining the value of VBRAKE to be infinity for such sections. Thus, the braking power-limited speed constraint is obtained as: - 736 HPBRAKE if GR + CR < 0 VBRAKE={ mg (GR + CR ) (3.6) if GR + CR > 0 where HPBRAKE is the model parameter to be estimated. The VBRAKE curve as a function of gradient may be seen in Figure 3.1. 3.3.3 Curvature-limited Speed Constraint, VCURVE The optimal speed at which a vehicle can negotiate a horizontal curve has been a subject of numerous empirical studies (Gynnerstedt et al., 1977, among others). Let RC denote the radius of the curve of the section-direction, in meters, and SP the curve's superelevation, expressed as a fraction (e.g., SP = 0.06 for 6 percent superelevation). For the vehicle travelling at a steady-state speed V, the lateral or side force on the vehicle in the direction parallel to the road surface, LF, in newtons, is given by the following kinematic relationship: LF m V2 cos SP - m g sin SP (3.7) RC The two terms on the right hand side of Equation 3.7 are, respectively, the centrifugal and gravitational forces acting on the vehicle, both in directions parallel to the road surface. The force on the vehicle in the direction perpendicular to the road surface, the normal force as denoted by NF, in newtons, is given by: m V2 NF = m g cos SP + - sin SP (3.8) RC 54 STEADY-STATE SPEED MODEL: FORMULATION Since curve superelevations normally do not exceed 20 percent, we may use the following approximations: cos SP 1 sin SP SP Consequently, Equations 3.7 and 3.8 simplify to: m V2 LF = - - m g SP (3.9) RC m V2 and NF = mg + - SP (3.10) RC The curvature-limited speed constraint is assumed to be governed by the "used perceived friction ratio," denoted by FRATIO and defined as the ratio of the lateral force to the normal force: LF FRATIO = - (3.11) NF Substituting Equations 3.9 and 3.10 in the above equation yields: v2- SP FRATIO - g RC (3.12) 1 + [ 2] SP g RC v2 which can be simplified further by neglecting the small term [- ] SP thereby producing: g RC V2 FRATIO = V - SP (3.13) g RC Solving for V, we have the curvature-limited speed constraint, VCURVE, STEADY-STATE SPEED MODEL: FORMULATION 55 expressed as: VCURVE = / (FRATIO + SP) g RC (3.14) where FRATIO is the parameter to be estimated in the curve speed constraint. VCURVE as a function of horizontal curvature is shown in Figure 3.2. As shall be seen in Section 4.2, the FRATIO parameter for a vehicle class depends on the surface type of the section and, for a truck, the net load. For practical purposes the curvature-constrained speed need be considered only when the radius of curvature (RC) is smaller than 10,000 meters. The adjective "perceived" in "used perceived friction ratio" is used to distinguish this ratio from the actual friction ratio, defined as TF/NF, where TF is the total force on the vehicle in the direction parallel to the road surface; TF is equal to the vectorial sum of the lateral force, LF, as defined above, and the vehicle's drive force, DF, which is longitudinal to the vehicle's axis. TF =/ LF2 + DF2 (3.15) The use of the used actual friction ratio for determining the curvature-limited speed constraint was also attempted in regression analysis. In terms of R-square and t-statistics the regression results were comparable to those based on the used perceived friction ratio. However, the actual friction ratio was found to be inappropriate for steep and highly banked curves where the driving speed constraint dominates but the used actual friction ratios computed are considerably higher than those obtained on flat curves of similar radii. These high friction ratios are in fact heavily dominated by the drive force (DF) as opposed to the lateral force (LF), and are thus associated with the tendency of longitudinal wheel slipping as opposed to lateral wheel skidding. For these reasons the used perceived friction ratio was preferred to the used actual friction ratio. An extension of the model with the used perceived friction ratio as a function of road roughness was also attempted. On a priori grounds, used perceived friction ratio would be related to road roughness because vehicles generally tend to lose contact with the road surface as it becomes rougher. Thus a model form, FRATIO = po + p1 QI (3.16) where QI = the roughness, in QI; and p, p1 = the model parameters 56 STEADY-STATE SPEED MODEL: FORMULATION was attempted. The results seemed to indicate that the coefficient p1 was statistically insignificant. This is probably due to part of the roughness effect being captured by the difference in the FRATIO estimates for paved and unpaved roads. Additional research would be needed to arrive at well determined estimates of p, for each surface type. 3.3.4 Roughness-limited Speed Constraint, VOUGH This speed constraint is derived from the "average rectified velocity" measure (ARV) defined in general for a given vehicle with a rigid rear-axle as the average rate of rear-axle suspension motion, which is defined more specifically as the rate of cumulative absolute displacement of the rear-axle relative to the vehicle body (in mm/s). It is advocated as an adequate measure of ride discomfort, or severity (Gillespie et al., 1980; Gillespie, 1981). ARV is related to the vehicle speed, V, by means of the following identity (Gillespie et al., 1980): ARV - V ARS (3.17) where ARS - the "average rectified slope" measure, defined as the amount of rear-axle suspension motion per unit travel distance (in mm/m or m/km). For modelling purposes the relevant ARV and ARS measures are those of the "calibrated" standard Maysmeter-equipped Opala passenger car used in the Brazil-UNDP study (GEIPOT, 1982; Paterson, 1987). For the Opala-Maysmeter car, the ARS measure was found to be sensitive to the travel speed, through the following general relationship (Paterson and Watanatada, 1985): ARS(V) ARS80 V (rO + r1 in ARS80) (3.18) 22.2 where ARSO - the average rectified slope measured at the travel speed of 22.2 m/s or 80 km/h; and ro, r1 - empirically determined parameters which vary from one surface type to another, as given in the table below: Surface type ro rl Asphaltic concrete 0 0 Surface treated or gravel 1.31 -0.291 Earth or clay 2.27 -0.529 Source: Paterson and Watanatada (1985). STEADY-STATE SPEED MODEL: FORMULATION 57 The ARS80 measure is related to QI, the standard road roughness measure used in the Brazil-UNDP study, through the following relationships: ARS80 = 0.0882 QI (3.19) Substituting the above equation in Eq. 3.18 we have V [ro + r1 in (ko QI)] ARS (V) = ko QI [ ] (3.20) 22.2 where ko = 0.0882. Combining the above equation with Equation 3.17 results in a relationship which expresses the ARV measure for the calibrated standard Opala-Maysmeter vehicle as a function of the vehicle speed and QI roughness measure: V [rO + r1 £n (kO I) ARV (V) = k0 QIVV [ X (3.21) 22.2 For ro and rl equal to zero the above relationship reduces to a simple form in which ARV is proportional to V: ARV (V) = ko QI V (3.22) Gillespie (1981) argues that the ARV statistic is a measure that closely relates to vibration levels on road using vehicles which, in turn, relate to ride discomfort and perception of vehicle and cargo damage. U.S. military research has determined relationships between the vibration levels imposed on the driver, and the maximum travel speed (Lee and Pradko, 1968; Beck, 1978). While the vibration limits found from the military experiments may not correspond directly to the limits for civilian transport which is also associated with the concern for vehicle and cargo damage, these findings have suggested a methodology for formulating the ride severity speed constraint. Applying the basic methodology to a limited data set obtained from the road user cost survey Paterson (1982) found that observed vehicle speeds tended to be dominated by a limiting ARV value for travelling on rough roads. 3 The conversion factor is computed as: 5.08 0.0882 = 0.18 x 320 where 5.08 = the rear-axle suspension motion, in mm, per unit count of the Maysmeter readout; and 0.18 = the average calibration factor for Opala-Maysmeter vehicles, which is used for conversion between the QI roughness unit and Maysmeter counts over 320 meters of standard travel distance. 58 STEADY-STATE SPEED MODEL: FORMULATION In applying these findings to the ride severity speed constraint we assume that the roughness-limited speed is governed by the limiting ARV of the average population vehicle (in a given class). We assume further that the ARV of the average population vehicle is proportional to that of the "calibrated" standard Opala-Maysmeter vehicle. Thus, it is possible to use the limiting ARV for the calibrated standard Opala-Maysmeter vehicle, denoted by ARVMAX, as the surrogate for that of the average population vehicle. By substituting ARVMAX for ARV in Equation 3.21 and solving for V we obtain a closed form solution which expresses the ride severity speed constraint, VROUGH, as a function of ARVMAX: in [ARVMAX/(ko QI)] + kn (22.2)[r0 + r1 in (ko QI)] VROUGH = exp { } (3.23) 1 + ro + rl Zn (ko QI) where ARVMAX is the only parameter to be statistically estimated. When the parameters ro and r1 equal zero the above relationship reduces to: ARVMAX VROUGH = - (3.24) ko QI From a theoretical standpoint, Equation 3.23 is more desirable than Equation 3.24 since it recognizes the speed-sensitive nature of vehicle suspension systems. However, as it is relatively unwiedly to use in practice, Equation 3.24 was also attempted for all surface types in the regression analysis to see whether the results were acceptable. As it turned out the results based on Equation 3.24 were virtually the same in terms of R-square, t-statistics and goodness of fit. Therefore Equation 3.24, which is much simpler, was chosen for the final steady-state speed model. These results are reported in Chapter 4. VROUGH as a function of QI is shown in Figure 3.3. 3.3.5 Desired Speed, VDESIR On a straight, flat and smooth road, although the driving, braking, curve and ride severity speed constraints do not exist, the vehicle still does not normally travel at the speed afforded by its own maximum or even used power. Rather, its speed is usually governed by subjective considerations of such factors as fuel economy, vehicle wear, safety or blanket speed limits (together with the driver's perception of the strictness of enforcement). Since it is not possible to separate these effects in the study data, they are combined in the parameter "desired speed," VDESIR (in m/s). As we shall see in Section 4.2, where estimation based on observed speed data from Brazil is presented, it was found satisfactory to assume that VDESIR for a vehicle class depends only on the surface STEADY-STATE SPEED MODEL: FORMULATION 59 type of the homogeneous section. However, in the extension to the steady-state speed prediction model based on Indian data, VDESIR depends also on the width class of the homogeneous section. The details regarding the extended model are presented in Section 4.4. 3.3.6 Summary and Discussion A stylized summary of the limiting speed specification of the steady-state speed model is as follows: Let X = the set of characteristics of the homogeneous road section = (GR, C, SP, QI, ST, CR, ALT) Y = the set of characteristics of a given class of vehicles = (m, CD, AR) e = the set of parameters for the combination of driver and vehicle class, specific to the environment = (HPDRIVE, HPBRAKE, FRATIO, ARVMAX, VDESIR). Then the predicted steady-state speed VSS of a vehicle of given class with characteristics Y, traversing a homogeneous road section with characteristics X in an environment with parameters e, is VSS = VSS(X,Y:6) = min (VDRIVE, VBRAKE, VCURVE, VROUGH, VDESIR) where VDRIVE, etc., are as described above. We shall briefly discuss the nature of the information requirements X, Y and 6 in turn. Of the information on the homogeneous road section needed to use the model, the important variables are GR, C, R and ST which are precisely the policy variables under the control of the highway planner. As for the other variables, while accurate values would improve the quality of the predictions, approximate values would produce acceptable results. The superelevation values may be derived as a function of curvature based on the average design standards in the region. The following relationships based on average Brazil standards are used as defaults in the HDM-III: SP 0.00012 C for paved sections 0.00017 C for unpaved sections For the coefficient of rolling resistance, the relationship expressing it as a function of section roughness, presented in Appendix 2A may be used.Finally, the value of altitude, if not known, may be taken to be zero. As regards the information on the vehicle needed to use the model, the values of mass of the vehicle, drag coefficient and frontal area may be taken to be the same as those for a representative make prevalent in the region, or average values may be used. In fact, the 60 STEADY-STATE SPEED MODEL: FORMULATION each of the vehicle classes estimated. The mass of the vehicle may be taken to be the sum of tare weight of the representative vehicle and average payload for the vehicle class in the region. The latter are important only for trucks, and are generally available from axle load studies in the region. We turn next to the question of estimation of model parameters 0* 3.4 LIMITING SPEED MDEL: PROBABILISTIC FORMULATION As regards the parameter estimates needed to use the limiting speed model, the approach taken in previous studies has been to calibrate the desired parameters individually (i.e., one at a time) based either on controlled experiments using test fleet vehicles and driven by trained staff, or on a limited set of speed observations. Since the driving environment under experimental conditions is artificial, the parameter values do not reflect aspects of decision-making involved in the driving behaviour of the vehicle population. While it is tempting to interpret parameters such as used driving power as purely technical they would, nevertheless, embody behavioral aspects and, as such, the transferability of different parameter estimates are likely to be different and has to be evaluated with care. Further, and even more important, the data obtained from controlled experiments still contain random variations in the constraining speeds and the resultant attained speeds. Such variations would still arise for a number of reasons even if the hypotheses regarding driver behavior were to hold strictly. Some of the important reasons are measurement errors, omission of characteristics of the road section and vehicle (for example, sight distance, sag curve, age and repair condition of the vehicle, trip purpose, nature of business), deviations of the section and vehicle characteristics from the values actually used, the inability of the driver to determine the binding constraint with certainty, and the inability of the modeller to completely specify the decision procedure of the vehicle operator. In sum, the limiting speeds have to be treated as random variables and the parameters have to be estimated on that basis. The explicit recognition of the stochastic (or probabilistic or random) nature of the constraining speeds is what distinguishes the probabilistic steady-state speed prediction model presented here from those of earlier studies. The recognition of randomness mentioned above provides the link with econometrics. It may be observed that the essential arithmetic operation involved in the specification of the model is that of taking the minimum of a finite number (four, in our case) of quantities. Estimation techniques to handle specifications based on minimization (or maximization) over a set of discrete alternatives have been developed relatively recently by researchers in the field of urban travel demand, based on earlier work in fields such as psychometry, biometry and reliability. One of the well-known formulations, namely, multinomial logit, has been used with an error component structure to derive and STEADY-STATE SPEED MODEL: FORMULATION 61 estimate the probabilistic version of the limiting speed model described above. It was observed that, even if the assumption of constant 6 is true, the limiting speeds vary over different homogeneous sections and over different vehicles of the same class and that these variations can only be partially explained by the variation in the observed characteristics of the section (X) and vehicle (Y). That is, the limiting speeds have a systematic (or deterministic or characteristic) part and a random part (or disturbance or error). We formalize this notion by treating the limiting speeds as random variables (or variates) with means or expected values given by the expressions derived in Section 3.3. For example, denoting the driving power-limited speed variate by Vdr = Vdr(X,Y:8), we may write: Vdr(X,Y:O) = VDRIVE(X,Y:B) Edr(X,Ye) where Cdr = Edr(X,Y:6) is the random part of Vdr* For notational clarity, we occasionally suppress the arguments, and use the traditional expression Vdr for the mean of variate Vdr, instead of VDRIVE. Thus, we may write the above expression as: Vdr = VDRIVE Edr = Vdr Edr* Treating the other limiting speeds analogously, we have, Vbr = VBRAKE Ebr = Vbr Ebr Vc = VCURVE Ec = Vc EC Vr = VROUGH Er Vr Er and Vd = VDESIR Ed = Vd Ed* We may express the above definitions compactly as: Vz= VZEZ, for z = dr, br, c, r and d (3.26) So far, there is no loss of generality in the above, because we have not imposed any restrictions on the error term. Now, we define the observed steady-state speed to be the random variable V given by: V = V(X,Y:e) = min [Vdr, Vbrp Vc Vr, Vd] Just as the speed constraint random variables, V may be expressed as: V(X,Y:e) = VSS(X,Y:6) E(X,Y:O) 62 STEADY-STATE SPEED MODEL: FORMULATION or, V = VSS ( = V 4. (3.27) These random variables may be interpreted as follows. A vehicle which has reached a steady-state speed (say, v m/s) on a homogeneous road section is assumed to give rise to one set of particular values (or realizations) of these random variables. Following the convention of using corresponding lower case letters for realiza- tions, we denote the realizations of random variables Vdr, Vbr, Vc, Vr and Vd, by vdr, vbr, vc, and vd. The realizations are latent or unobservable. The (observable) steady-state speed v and the (unobservable) realizations v dr, v br, v , v r and v are related by: v = min [vdr, vbr, vc, vr, vd] We interpret V = V(X,Y:e) as a random variable whose realizations are the steady-state speeds (such as v) of vehicles characterized by Y on sections characterized by X for fixed e. The VSS curves shown in Figure 3.1-3.3 may be interpreted under the probabilistic framework to be particular realizations of the random variable V. The mean VSS = V(X,Y:e) of the random variable V(X,Y,O) may be seen as the characteristic or average steady-state speed that we can associate with the section X and vehicle Y_for fixed 6. _ Basically, we are interested in V as a function of Vdr br c Vr and Vd* Now we come to the slightly surprising result that, in general, V V, where V = min [Vdr, Vbr, Vc, V, Vdl* In fact, V < V, with the equality holding if and only if the random variables are perfectly postively correlated or they are degenerate, that is, the variations are all zero. In other words, the mean of the minimum is generally less than the minimum of the means. This apparently counter-intuitive result may be illustrated by means of a simple example. Suppose two fair dice are thrown. Let us define variate X, as the number on the first die and variate X as the number on the second die. Finally, define the variate X as the minimum of X1 and X2. Then X1 = X2 = 31/2. The value of T depends on the correlation between the throws. If the throws are independent X = 223 / and, if the throws are perfectly positively correlated, X = 31/23 STEADY-STATE SPEED MODEL: FORMULATION 63 The result is illustrated for two constraining speed variates in Figure 3.4 where the distributions of Ve and Vr with means 30 m/s and 33 m/s are drawn along with the distribution of V = min [Vc,Vr1* By definition, V' is 30 m/s, but the mean of V is seen to be about 26 m/s. This is further illustrated in Figure 3.5 which is the stochastic version of Figure 3.1. For each value of GR, five random realizations are generated for the variate V. The VSS curve is seen to be below the V' curve. The relation between the means of the speed constraints and the mean steady-state speed depends on the assumptions imposed on the joint distribution of the errors of the speed constraint variates. We specify the error structure by making use of two well-known distributions (lognormal and normal), and two distributions from a class known as asymptotic extreme value distributions. These are the Weibull distribution and the Gumbel distribution. A brief account of the distributions is given in Appendix 3A. Further details may be found in Johnson and Kotz (1970) or Benjamin and Cornell (1970). The main property of both the distributions, highlighted in the appendix, is that the minimum or maximum of a set of independent Weibull (Gumbel) variates is also Weibull (respectively, Gumbel) whose mean can be expressed as a closed form function of the means of the individual variates. This property is what makes these distributions suitable for minimization formulation. Just as normal distributions are preserved when the arithmetic operation involved is one of addition, the Weibull and the Gumbel distributions are preserved when the arithmetic operation involved is minimization (or maximization). This enables one to derive the distribution of the minimum variate as a relatively more tractable closed function. The Weibull and the Gumbel distributions are closely related to each other. If W is a Weibull distribution with shape parameter 0 and scale parameter a then the variate jn W has a Gumbel distribution with scale parameter 8 and location parameter in a. The relationship between them is analogous to the relation between lognormal and normal distributions. The distributional assumptions are theoretically equivalent whether they are formulated in terms of a Weibull and lognormal distributions or Gumbel and normal distributions. In the case of steady-state speed model, it is more natural to formulate the model in terms of the Weibull and lognormal distributions, because they deal with speeds rather than logarithms of speeds. However, it is more convenient to estimate the model by working with logarithms, using the Gumbel and normal distributions; this would make errors additive rather than multiplicative, thereby considerably reducing the problem of heteroskedasticity. The speed predictions are then obtained by exponentiating the log-speed predictions. It should be noted that a correction factor is needed because the antilog of the mean is not the same as the mean of the antilog. The error structure for the speed model is presented in detail below, followed by the final form of the equivalent log-speed model. We specify the disturbances pertaining to a particular speed 64 STEADY-STATE SPEED MODEL: FORMULATION Figure 3.4: Probability densities of two constraining speeds and the attained speed Probability density 0.05- ,v v 0.04-t II iV I0.2-k 0.03 o.oo- v,~ vI i- 0 5 10 15 20 25 30 35 410 45 5e 55 so Speed (a/s) Figure 3.5: Ranomly generated spee relzations, and the V' and curves Sp~d (a/s) 40- × 35- × ×× × × 25- x . ×x x × x' × 15- 4× ×XX× × x ×x X- x x -10 -8 - 5 -4 -2 0 2 4 5 5 10 Gradlønt () STEADY-STATE SPEED MODEL: FORMULATION 65 observation and the associated speed constraints using three nested components of error. First, errors pertaining to the homogeneous section itself. These errors encompass factors such as unmeasured characteris- tics of the section and of the vehicles in general at that section, as well as speed measurement errors at that section. Second, errors pertaining to the particular vehicle observed at that section. These errors include unmeasured characterstics of the particular vehicle at the section. Finally, given the above errors, there would be errors specific to the various speed constraint variates for that speed observation. We proceed by imposing fairly standard assumptions (lognormality) regarding the first two components of error. We use the Weibull distribution in respect of the third component to derive the conditional distribution of the observed speed variate. Thus in Equation 3.26, we regard the random part Ez for a given realization vz of a constraining speed variate Vz for vehicle Y on section X to be the product of three mutually independent components of error: first, error e(X) specific to the particular section; second, error C(X,Y) specific to the particular vehicle at the particular section; and, third, the error vz(X,Y) specific to the particular speed constraint for the particular vehicle at the particular section. That is, v = V E = V e(X) (X,Y) v (X,Y) (3.28) The specific distributional assumptions made on the error components are as follows: 1. C(X) are independent and have identical lognormal distributions with mean 1. 2. C(X,Y) are independent and have identical lognormal distributions with mean 1. 3. vz(X,Y) are independent and have identical Weibull distributions with mean 1 and shape parameter . Under the above assumptions, for any particular realizations of the section and the vehicle specific errors, the speed constraint variates Vdr, Vbr, Vc, Vr and Vd have independent Weibull distributions with the same shape parameter 3. Thus, using the main property of the Weibull distribution given in Appendix 3A, we have the following results: 1. Conditional on c(X) and C(X,Y), the attained speed variate V has a Weibull distribution with a shape parameter a. 2. The relationship between the mean of the attained speed and the means of the limiting speed variates is: V = (Vdr-" + Vbr - c + VcL+ Vrl+ VdI/)8 (3.29) 66 STEADY-STATE SPEED MODEL: FORMULATION where Vz = Vz(X,Y:6) and hence, V = V(X,Y:6,3). Thus, observed speed random variable, V, may be written as V = V E = V E(X) (X,Y) v(X,Y) where v(X,Y) have independent Weibull distributions with mean 1 and shape parameter a. The error components C and v are specific to a particular speed observation. It is convenient to introduce a random variable w which is the product of and v. We define W(X,Y) = C(X,Y) v(X,Y). (3.30) The variate w is the product of independent lognormal and Weibull variates. Its distribution is observed to be approximately lognormal. Thus, the distribution of observed speed variate itself is approximately lognormal. Thus, the probabilistic steady-state speed model may be written, in terms of the variables used earlier, as V = VSS (3.31) where VSS = (VDRIVE-/0 + VBRAKE-1/a+ VCURVE -1/a + VROUGH-1/B + VDESIR 1/a)-a (3.32) We next present the equivalent log-speed model. For this, we first define the logarithms of observed speed variate, V, and the constraining speed variates, Vz. Let U = U(X,Y:e) = in V(X,Y:O) (3.33) UZX,Y:O) = in Vz(X,Y:O), for z = dr, br, c, r and d (3.34) Assuming that for a given speed observation, the logarithms of the constraining speeds have independent Gumbel distributions with constant mean and scale parameter a, we can use the main property of the Gumbel distribution given in Appendix 3A to derive the expression for the mean, U, of the log-speed variate U as U Z - En {exp(- dr/8)+ exp(-UFb /a)+ exp(-ZJ /a)+ exp(-Ur /0)+ exp(-Ud/a)1 (3.35) where Uz = Uz(X,Y:e) are the means of the random variables Uz, and U= U(X,Y:6, $) is the mean of U. STEADY-STATE SPEED MODEL: FORMULATION 67 This may be simplified to = -i in(VDRIVE-1/a + VBRAKE-1/8 + VCURVE-1/8 + VROUGH-1/0 + VDESIR-1/8) (3.36) Thus, by analogy with the speed model, we may write the log-speed model as U = U + e + w (3.37) In Equation 3.37, e = e(X) are the logarithms of section-specific errors in the observed speed. They have independent normal distributions with constant mean and variance. The w = w(X,Y) are the logarithms of errors specific to the speed observation. They are independent and identically distributed as convolutions of independent normal and Gumbel distributions with constant means. The latter are approximately normal with constant mean and variance. Thus, the distribution of log-speed variate itself is approximately normal. 0 is an additional parameter introduced by the particular probabilistic approach over and above the set of other parameters e. It has an intuitive interpretation as an indicator of the dispersion of the constraining speed random variables. More rigorously, a is related to the standard deviation of the constraining speed variates and hence that of the observed speeds, that is, the spread in the speeds of a vehicle repeatedly traversing a homogeneous section. This may readily be seen by noting the relation between the shape parameter and the variance of the Weibull distribution. The relationship is even clearer in the case of the Gumbel distribution, for which Variance = 8 7r2/6, or Standard deviation = 8 w/V6 In other words, under the assumptions of the model, the conditional standard deviation of log-speeds is approximately 1.28 8. The value of 8 may be expected to be well below unity. The estimates of the variances of e and w (ae 2 and aw 2 respectively) along with the total variance a2 are useful in deriving the sampling distributions of predicted speeds for different applications. As mentioned earlier, it is the log-speed model defined by Equation 3.36 and 3.37 which is used to estimate the parameters e and 0. Since the means of the error terms in this model are not zero, in order to obtain unbiased steady-state speed predictions an estimation bias correction factor has to be introduced. The choice of the correction factor depends on the application for which the speed predictions are desired. These factors are functions of the variances of the error terms in Equation 3.37. The variances are also useful in calculating the confidence intervals for the speed predictions. These issues are discussed next. 68 STEADY-STATE SPEED MODEL: FORMULATION Various predicted steady-state speeds which may be computed using the estimates of model parameters are themselves random variables. Three important concepts of speed prediction are as follows: 1. Prediction of the mean of a large number (notionally, infinity) of individual steady-state speeds on a given homogeneous section; 2. Prediction of the mean of a large number (notionally, infinity) ofindividual steady-state speeds for a specific vehicle; and 3. Prediction of the steady-state speed of an individual vehicle of giveattributes over a homogeneous section of given characteristics. The first kind of prediction is important for general economic analysis. The second kind of prediction is useful for Monte-Carlo studies of speed behavior on specific vehicles traversing a stretch of road. The third kind of prediction is useful for evaluating the results of statistical estimation. The approximate formulas for unbiased estimates of the mean, the variance and the coetficient of variation of VSS for the above three regimes of prediction may be derived assuming normality of the error terms. When the respective variances are small, say less than 0.05, these formulas may be simplified using the following approximation: exp(x) = (1 + x) Both sets of formulas for the three regimes of speed prediction are given below. Section-specific speed prediction -1/S -1/5 -1/5 -1/8 1/5 -3 E[VSS]=(VDRIVE + VBRAKE + VCURVE + VROUGH + VDESI ) exp(ae/2) Var [VSS] E2[VSS] (exp(a2) - 1) e CV [VSS] = (exp(ae) - 1) As mentioned earlier, the above formulas are relevant for speed prediction in the context of economic analysis and are used in Chapters 13 through 15. We denote the "estimation bias correction factor" for this regime of prediction, exp(a2/2), by E0. e The set of simplified formulas is as follows: STEADY-STATE SPEED MODEL: FORMULATION 69 E[VSS]-(VDRIVi1/+ 8VBRAK 1/+ VCURVE1/0+ VROUGH 1/+ VDESIR O)0(1+ 2/2) Var [VSS] = E2(VSS] a2 e CV [VSS) a e Vehicle-specific speed prediction E[VSSI=(VDRIV01/8+ VBRAK1/0+ VCURVil/o+ VROUGR1/0+ VDES&/) exp(a /2) Var [VSSI E2[VSS] (exp(a ) - 1) CV [VSS] - / (exp(or ) - 1) For this case, the set of simpler formulas is: E[VSSI-(VDRIVl/+ VBRAK 10+ VCURVi'/O+ VROUGf/+ VDESlil/o -0(1+ 2/2) Var [VSS] E2[VSS] a2 CV (VSS] -O Individual speed prediction E[VSS]-(VDRIVi1/0+ VBRAKI1 + VCURVil10+ VROUG1 +VDESI/ )-Sexp(Cr2/2) Var [VSS] = E2[VSS] (exp(a2) - 1) CV IVSS] / (exp(CF2) - 1) The simpler formulas are: E[VSS] -(VDRIVE +VBRAKE-1/$+VCURVE-1/0+VROUGH-1/1+VDESI1/O)-0(1+a2/2) Var[VSS] - E2[VSS] a2 CV[VSS] a a 70 STEADY-STATE SPEED MODEL: FORMULATION The difference between the predicted values of E[VSS] for the three cases is at most 2%, as may be seen in Table 4.3(c) where the estimates of the variances based on Brazil data are presented. Equation 3.32, along with the expressions for VDRIVE, etc., given in the last section, constitutes a multinomial logit model which is non-linear in the parameters 6 and a. It is of interest to note the similarities and differences between the probabilistic steady-state speed model and the probabilistic discrete choice demand models. The latter have been employed to develop models for predicting the market shares of competing alternatives (for example, travel modes, universities) as responses of utility maximizing individuals of known characteristics (Y) to policy variables (X) influencing the utilities of the alternatives (Domencich and MacFadden 1975; Ben-Akiva and Lerman, 1985). In the the speed model, the section characteristics are regarded as policy variables under the control of highway planners which influence the alternative limiting speeds on the sections depending on the characteristics of the vehicles. The vehicle operators are hypothesized as choosing the speed constraint with the highest speed. The differences between the two models is that in probabilistic discrete choice theory, the alternative chosen by the individual is observable but the realized utility is not. In contrast, in the probabilistic limiting speed model, the binding speed constraint is not observable but the realized speed is. Further, the demand models are based on the notion of utillity which is an ordinal measure, and thus the 0 parameter cannot be identified. However, since speed is a cardinal measure, the a parameter in the speed model can be identified. As such, the proper analogue of the speed model in economic theory would be the qualitative response models in the theory of the firm (MacFadden, 1982). Some of the important properties of the VSS function (Equation 3.32) are noted below. The function is homogeneous of degree one in VDRIVE, etc. That is, if each of the average speeds is multiplied by a constant, say, 3.6, the average steady-state speed is also multiplied by 3.6. The partial derivatives aVSS/aVDRIVE, etc., are all positive; thus, as any of the average constraining speeds decreases, so does the -average steady-state speed. As noted before, the average steady-state speed is somewhat less than the minimum of the average constraining speeds. The extent of the difference depends on $ and other constraining speeds. If 6 is zero, that is, if there is no variation in the constraining speeds, then the mean steady-state speed is identical to the arithmetic minimum of the means of the constraining speeds, as in Figures 3.1 - 3.3. As 8 increases from zero, so does the spread around the means of the constraining speeds and, as a result, the mean of the resulting speeds falls below the minimum of the means. This feature is illustrated in Figures 3.6 - 3.8 which show the mean steady-state speed VSS as a function, respectively of gradient, curvature and roughness for a heavy truck, with 8 = 0.27. These figures are the probabilistic counterparts of Figures 3.1 - 3.3. STEADY-STATE SPEED MODEL: FORMULATION 71 Figure 3.6: Constraining and steady-state speeds versus gradient: probabilistic version Speed (W/.) 70 VDRIVE -VBRAKE VDESIR 2061 -10 -8 -B -4 -2 0 2 4 6 8 10 Gradient (%) Source: Analysis of Brazil-UNDP-World Bank highway research project data (See also notes for Figures 3.1 - 3.3). Figure 3.7: Constraining and steady-state speeds versus curvature: probabilistic version Speed (m/s) 607 50- .PCURVE VROUGH -- -- - -- -- D 40 VDESIR 20 VDRIVE 207 01 100 200 300 400 500 600 700 800 900 1000 Curvature (DeRress/kv) Source: Analysis of Brazil-UNDP-World Bank highway research project data (See also notes for Figures 3.1 - 3.3). 72 STEADY-STATE SPEED MODEL: FORMULATION Figure 3.8: Constraining and steady-state speeds versus roughness: probabilistic version spad (%/N) 100 90 70 D0 vS50 7c10 12 50 15 D (see~~~ als noe fo Fiue -. - 3.3).-- 40- 30. 20- VDMIE 10.5 10 0 25 so 75 100 125 15Sr 170 200) Source: Analysis of Brazil-UNDP-World Bank highway research project data (see also notes for Figures 3.1 - 3.3). A better feel for the function may be obtained by examining a series of section diagrams. For these illustrations, it is assumed that VBRAKE is infinity; that is, the effective gradient of the section is non-negative. 1. Holding the means of all the constraining speeds at a constant value V0, and letting only $ vary, we have: VSS = 40VO This function is illustrated for VO=30 m/s in Figure 3.9. 2. Holding VDRIVE, VROUGH, and VDESIR constant at 30 m/s, the curve showing the VSS as a function of VCURVE is given in Figure 3.10, for different values of 8. 3. The contours resulting from varying two of the speed constraints,VCURVE and VROUGH while holding VDRIVE and VDESIR constant at 30 m/s, and 8 constant at 0.27 may be seen in Figure 3.11. Finally, Figure 3.12 demonstrates the sensitivity of VSS to 8. Here the mean steady-state speed versus gradient curves are drawn for values of S ranging from 0 to 0.5. STEADY-STATE SPEED MODEL: FORMJLATION 73 Figure 3.9: Man steady-state speed as a function of g with wa constraining speeds as 30 a/s sped (m/s) 40- 35- Vdr, Ve, -fr, Vd 25- 20 15- vss 0.0 0.1 0.2 0.3 0.4 0.5 0.8 Figure 3.10: Nan steady-state speed as a function of one mean constraining speed for various values of g VSS (si) 40- 35- Vdr. Vr- Vd 30- ---- - -- -- -- --- - -- -- - ------------ 0 25- 20 15-0 10 5- 0 5 10 15 20 25 30 35 40 Nean conatralning apeed Vc (elm) 74 STEADY-STATE SPEED MODEL: FORMULATION Figure 3.11: Mean steady-state speed contours as two mean constraining speeds vary, with p - 0.27 Mean constraining speed V (M/s) r 38- 34- -l 30- 25 22 18 _8 SS S13 1 4 ---.. -4 10 14 18 22 26 30 34 38 Mean constraining speed V (m/s) 3.5 SUMMARY OF STEADY-STATE SPEED MODEL The desired steady-state speed prediction, VSS, for a vehicle class and specific section is given by the following relationship: VSS = EO(VDRIVE-1/0 + VBRAKE-1/8+ VCURVE-1/ 8+ VROUGH-1/8+ VDESIR-1/ )-a (3.38) where E* is the estimation bias correction factor, given by: Eo = exp (a'/2) (3.39) e VDRIVE, VBRAKE, VCURVE, VROUGH and VDESIR are the limiting speeds as derived in Section 3.3. The parameters and statistics to be estimated are HPDRIVE, HPBRAKE, FRATIO, ARVMAX, VDESIR, j and a2. e STEADY-STATE SPEED MODEL: FORMULATION 75 Figure 3.12: Sensitivity of steady-state speed to the Ø parameter Speed (m/s) 25. 0- 22. 5- 20.0O 17. 5 15. C- 10.0- 7.5- 5. 0- 2.5- 0. 0 -10 -8 -6 -4 -2 0 2 4 6 8 10 Gradient (%) 76 STEADY-STATE SPEED MODEL: FORMULATION APPENDIX 3A EXTRE VALUE DISTRIBUTIONS 3A.1 WIBULL DISTRIBUTION The Weibull distribution is also known as Weibull-Gnedenko or type-III extreme value distibution. It is characterized by two positive parameters: a parameter 0 which determines its shape, and a parameter a which determines its scale. A Weibull random variable W can only take on positive values. It is an asymmetric distribution with a right tail and somewhat resembles the lognormal distribution which is often a close alternative assumption. It may also be seen as a flexible form of the better-known exponential distribution. When 8 is 1, the Weibull distribution is the same as exponential distribution with mean a. Its distribution function is: P[W < w] - 1 - exp(-a-1/0 w1/0) Some of its properties are as follows: 1. Mean : W - a r(1+s) where r(.) is the Gamma function, given by r(k) f xk-l e-x dx 0 2. Median : a (Un 2)8 3. Mode : a (1 - 8)0 for 8 < 1, and 0 for 8 > 1 4. Variance : [r(l 2) -( 2] 5. If W has a Weibull distribution with shape parameter 8 and scale parameter a then the variate aW, where a is a positive constant, has a Weibull distribution with the same shape parameter 8 and scale parameter aa. Main property If W1, W2***, Wn have independent Weibull distributions with the same shape parameter 8, and with scale parameters al, a2,.** an, respectively, then the random variable W - min [W1, W2,***, WnI has a Weibull distribution with the same shape parameter 8 and scale parameter a, given by a - (al-1/ + a21/0 + *** + an-1/) -B STEADY-STATE SPEED MODEL: FORMULATION 77 The mean W is related to the means 51, 92#***, n by -1/+ -1/ - -1 - W= (W + 2 + ... + n 3A.2 GUMBEL DISTRIBUTION The Gumbel distribution is also known as type-I extreme value distribution or often simply as the extreme value distribution. It is also characterized by two parameters: a scale parameter X which is positive and a location parameter V which may be positive or negative. The range of a Gumbel variate Z is the entire real line. It is an asymmetric distribution with a right tale. When X is less than 0.5 the distribution begins to resemble a normal distribution and the resemblance is closest when X is about 0.27. Its cumulative distribution function is P[Z < z} = 1 - exp(-exp((z - Some of its properties are: 1. Mean : i + x r'(1) where r'(1) - -y u - 0.5772, which is known as Euler's constant. 2. Median : p + X Ln Xn 2 3. Mode : p 4. Variance : V2/6 5. If Z has a Gumbel distribution with parameters X and U, then the variate aZ + b, where a > 0 and b are constants, has a Gumbel distribution with parameters aX and ap + b. Main property If Z1, 2, *..., Zn have independent Gumbel distributions with the same scale parameter X and location parameters P 2'*** Un, respectively, then the variate Z = min [1, Z2, *..*., Zn] has a Gumbel distribution with the same scale parameter and with location parameter V given by: p = Rn {exp(p/X) + exp(P2/A) + ... + exp(pn/}lAX The mean T is related to the means 71, 72, **** Z by: -1X- -1 /1- -1/A- i= In (exp(21)) + (exp(Z2)) + ... + (exp(Zn 78 STEADY-STATE SPEED MODEL: FORMULATION APPENDIX 3B GENERAL SOLUTION TO STEADY-STATE FORCE-BALANCE EQUATION The formulas to compute the driving power-limited speed constraint, VDRIVE, discussed in Section 3.3.1 are presented here. Recall that VDRIVE is defined as the positive root of the force-balance equation under steady-state conditions (Equation 3.4): 1/2 RHO CD AR V3 + mg (GR + CR) V - 736 HPDRIVE = 0 (3B.1) Equation 3B.1 may be written as: V3 + 3c V - 2b = 0 (3B.2) where c = m g (GR + CR) 3A A = 1/2 RHO CD AR b HPDRIVE 736 2 A The nature of the roots to Equation 3B.2 depends on a quantity D, which may be termed the discriminant of the equation, defined by: D = b2 + c There are two important cases: (1) D > 0 : One real root and two complex roots. (2) D < 0 : Three real roots. Case 1: D > 0 The solution is: VDRIVE - 3V VD + b - 3 VD - b This formula is applicable to all positive values of GR and also when GR is negative but IGRI is small. STEADY-STATE SPEED MODEL: FORMULATION 79 Case 2: D < 0 This case is known in the theory of equations as the "case with three real roots" or "the irreducible case". The algebraic formulas for solving a cubic equation become "circular" in this case and trigonometric formulas are required (Dickson, 1957). The formulas are derived using the following triple angle identity: cos 3z = 4 cos3z - 3 cos z (3B.3) It may be shown that: cos z, cos (z + 2 ) and cos (z + ) 3 3 are the three roots of (3B.3). The method of finding the roots for the case of negative discriminant depends on transforming Equation 3B.2 into the form of Equation 3B.3 through an appropriate change of variable, as follows: Step 1 : Compute r such that: r = 2/rE = 2/- mg (GR + CR) 3 A Step 2 : Find an angle z in radians such that: cos 3z 2b cr That is, z = arc cos [- -- 3 cr Step 3 Determine the three roots of the cubic using vi = r cos (z) v2 = r cos [z + --- 3 v3 = r cos iz + -K 3 In the context of positive power, two of these will be negative and the remaining positive. The positive root is taken as the desired solution. CHAPTER 4 Steady-State Speed Model: Parameter Estimation, Adaptation and Transferability The main thrust of this chapter is on the statistical estimation of the steady-state speed model based on speed data from Brazil. A description of the data is given in Section 4.1 and the estimation results are presented and discussed in Section 4.2. Guidelines for adapting some of the model parameters to reflect specific vehicle characteristics are provided in Section 4.3. Section 4.4 investigates the question of transferability of model parameters to other environments based on speed data from India, and discusses the effect of road width on steady-state speed. The technical details regarding the estimation procedure are given in Appendix 4A, and the results of some auxiliary analyses are presented in Appendices 4B, 4C and 4D. 4.1 DATA The data for the steady-state speed model were obtained from some 100,000 radar speed observations of vehicles at selected homogeneous road sections over a period of a year. Detailed description of the field instrumentation and measurement procedure are given in GEIPOT (1982, Vol. 3). Sections were distinguished by direction of travel. The unpaved sections were further distinguished by roughness intervals of 50 QI, since the roughness of these sections often vary significantly over non-consecutive observation periods. This procedure resulted in a total of 216 homogeneous sections with the gradient ranging from -9 to 11 percent, the radius of curvature of curvy sections from 20 to 2000 m, and the surface condition from paved and smooth (20 QI) to unpaved and very rough (200 QI). The summary of characteristics of the road sections is given in Table 4.1. The observations made for each vehicle sighting were one or more spot speeds on the section, the vehicle type and load condition. The vehicle types observed were categorized into six classes. These classes are shown in Table 4.2(a) along with the adopted average vehicle characteristics. The average gross vehicle masses were obtained from a separate axle load study for Brazil. The values of aerodynamic drag coefficient and frontal area were adapted from those for typical makes and models prevalent in Brazil for each vehicle class. Because of the influence of payload on vehicle performance, a distinction is made in this classification between loaded and unloaded trucks. However, for cars, utilities and buses, no such distinction is made since these vehicles have a small payload range relative to their total weights. For the unloaded and loaded categories of the respective truck classes, the same values of projected frontal area and drag coefficient are used, although these values would be slightly different depending on the nature of loading. It should be noted that the vehicle classes used in the 81 82 STEADY-STATE SPEED MODEL: ESTIMATION Table 4.1: Statistics of test sections Number Characteristics of Mean Min. Max. Std. CV sections dev. (%) Vertical Positivel 111 3.4 0.0 10.8 2.8 80.8 gradient Negativel (%) (absolute values) 105 3.8 0.0 8.7 2.7 71.1 Radius of Tangent sections 10 W - - curvature Curvy: paved 46 298.9 20.0 950.0 258.7 86.6 (m) unpaved 52 251.0 37.0 870.0 217.8 86.8 Super- Tangent sections 10 0 - - - - elevation Curvy: paved 46 5.6 0.0 11.0 3.1 55.3 (%) unpaved 52 9.1 0.0 20.0 4.7 51.4 Roughness Paved 50 51.6 20.0 122.0 25.5 49.4 (QI) Unpaved 58 123.9 56.6 192.5 38.1 30.8 1 Differentiated by section-direction. Source: Brazil-UNDP-World Bank highway research project data. Table 4.2(a): Vehicle classes and characteristics used in steady-state speed model estimation Aerodynamic drag Projected Total Vehicle class coefficient frontal area, vehicle weight (dimensionless)1 AR (m2)1 (kg) 1. Car 0.50 2.00 1,200 2. Utility 0.60 3.00 2,000 3. Bus 0.65 6.30 10,400 4. Light2/medium truck3 0.70 4.5 5,400 unloaded 11,900 loaded 7,900 unloaded 5. Heavy truck 0.85 5.2 1900 loaded 19,200 loaded 6. Articulated truck 0.65 5.8 15,900 unloaded 37,700 loaded 1 Obtained from the vehicle manufacturers. 2 Includes both gasoline and diesel light trucks introduced in Table 1A.1. 3 Because of the way in which light and medium trucks were coded in speed observations it was not possible to separate these truck classes for separate model estimation. Source: Brazil-UNDP-World Bank highway research project data. STEADY-STATE SPEED MODEL: ESTIMATION 83 Table 4.2(b): Statistics of observed mean speeds Number individual speed observation Mean speed (km/h) Number of Vehicle Loading mean-speed Std. CV class condition data points Mean Min. Max. Mean Min. Max. dev. (%) Car - 216 164.1 9 825 62.5 21.4 99.6 16.8 27.0 Utility - 216 62.6 2 239 56.1 19.8 91.2 15.0 26.8 Bus - 216 23.7 1 76 52.4 19.0 92.0 18.3 34.9 Light/ Unloaded 216 29.8 1 251 53.2 18.9 83.0 14.5 27.3 medium Loaded 215 34.0 2 223 45.4 17.6 79.9 16.1 35.4 truck Heavy Unloaded 187 14.9 1 102 52.5 20.0 88.3 16.4 31.2 truck Loaded 194 31.9 1 216 42.8 9.0 78.6 18.5 43.3 Artic. Unloaded 112 4.7 1 36 54.8 17.0 89.8 18.8 34.4 truck Loaded 120 11.1 1 79 44.2 9.1 91.7 21.9 49.5 Source: Brazil-UNDP-World Bank highway research project data. estimation are aggregate groupings of the basic vehicle classes introduced earlier in Appendix 1A. Because of this, some of the estimated model parameters would have to be adjusted for the latter classes. This issue is discussed in Section 4.3. The sections had one, three or five observation stations. For sections with more than one station, the spot speeds for each observed vehicle were averaged after eliminating vehicles judged not to have attained a steady-state speed, on the basis of a visual examination of the speed-distance plots. Further, speed observations obtained with the radar visible to the drivers of the observed vehicles were excluded.1 In all, about 80,000 speed observations were included. Finally, for each vehicle class, the logarithms of individual speed observations pertaining to a section were averaged, yielding the dependent variable values of the estimation data set. The summary statistics regarding the range of speeds observed and the number of individual speeds making up various mean-log data points are presented in Table 4.2(b). For ease of reference, mean speeds rather than the logarithmic values are given. 4.2 ESTIMATION RESULTS The estimation was done in two stages.2 In the first stage, 1 This essentially eliminated all observations taken before November 1976. 2 The technical details regarding the estimation procedure are given in Appendix 4A. The tests for normality of residuals are presented in Appendix 4B. 84 STEADY-STATE SPEED MODEL: ESTIMATION for each of the six vehicle classes, regression runs were made separately for paved and unpaved sections, and in the case of trucks, for unladen and laden categories. That is, the model parameters were estimated without any restrictions being placed on them. The objective in this stage was to see whether the parameter estimates were significantly different across these sub-classes. Insignificant variation of some parameters across the sub-classes would permit pooling the observations by imposing restrictions on these parameters leading to better-determined demonstrated in the next section, there is a well defined relationship between these quantities. The magnitudes of the braking power appears to increase with the gross vehicle weight. As would be expected, the estimates. The results of unrestricted estimates indicated that the used driving power (HPDRIVE), the used braking power (HPBRAKE), the limiting average rectified velocity (ARVMAX), and the parameters did not vary significantly by surface type or load level. The friction ratio (FRATIO) and the desired speed (VDESIR) parameters were considerably different between paved and unpaved surfaces. Further, for trucks traversing paved sections the FRATIO parameter differed by loading condition. In the second stage, regression runs were made with restrictions of equality being imposed on the corresponding parameters pertaining to different surface types and load categories. The results from these runs confirmed the indications from the first stage. The complete details regarding these runs with different combinations of pooling the observations, along with the results of X2 tests for model equivalence, are given in Appendix 4C. The final results are presented in Table 4.3(a). These consist of six sets of parameter estimates, for cars, utilities, buses, light/medium trucks, heavy trucks and articulated trucks. Each set includes estimates of the following parameters: 1. , HPDRIVE, HPBRAKE and ARVMAX parameters which are applicable for paved and unpaved surfaces, and unloaded and loaded trucks. 2. VDESIR parameter for unpaved surfaces, and the increment for paved surfaces. For ease of reference, the computed value of VDESIR for paved sections is also included. 3. FRATIO parameter for unpaved surfaces, the increment for paved surfaces, and the further increment for loaded trucks on paved surfaces. Again, for ease of reference, the computed values for the respective cases are included. The asymptotic t-statistics associated with these parameter estimates are given in Table 4.3(b) in a similar format. A goodness-of-fit measure analogous to the R2 value in linear models is also given. This was obtained by regressing the mean observed speeds against predicted speeds. The estimated standard errors, Ce, a,, and C, of the model error terms are given in Table 4.3(c). There were too few observations for articulated trucks to support the determination of all the model parameters and the HPBRAKE and STEADY-STATE SPEED MODEL: ESTIMATION 85 Table 4.3(a): Estimation results of steady-state speed model for six vehicle classes: estimates VDESIR (km/h) FRATIO Unhaved Paved Unpaved Pavel - _ _ unloadel or Unloadel lowed loaded Vehicle 0 FPRIW IBRWE ARWAX Incremt Value Incraent Value Incramet Value class etric setric m/s over over un- over hp hp unpaved paved paved value value unloaded g + h a b c d e f e+f g h g+h I +I Car 0.274 36.4 21.7 259.7 82.2 16.2 98.3 0.124 0.144 0.268 - - Utility 0.306 44.4 32.6 239.7 78.3 16.6 94.9 0.117 0.104 0.221 - - Bus 0.273 112.9 213.6 212.8 69.4 24.0 93.4 0.095 0.138 0.233 - - Tmck: Light/ 0.304 94.7 190.8 194.0 71.9 9.7 81.6 0.099 0.154 0.253 -0.083 0.170 medium Heavy 0.310 108.2 257.1 177.7 72.1 16.8 88.8 0.087 0.205 0.292 -0.107 0.185 Artic. 0.244 200.0 500.01 130.9 49.6 34.5 84.1 0.0401 0.139 0.179 -0.049 0.130 1 These parameters for the articulated truck class were exogenously assigned. Source: Analysis of Brazil-UNDP-World Bank highway research project data as described in Section 4.2 Table 4.3 (b): Some Important statistics associated with the estimation VIESIR FRATI VARTA1ES Vehicle 0 W1PIE HPBRAlE ARMAX Urpaved Incraient Unpaved Incraent for paved a2 a2 R2 Nrmer SSR class for paved e w of surface Unloaded loaded observations a b c 4 e f g h i Car 10.9 15.2 7.9 20.3 22.8 4.9 12.4 10.3 - 0.00654 0.0224 0.92 216 1.36 Utility 10.9 19.3 7.9 17.5 17.4 4.0 9.3 7.0 - 0.00808 0.0355 0.89 216 1.68 Bus 6.8 27.3 4.6 12.0 12.0 3.8 5.3 5.3 - 0.02477 0.0276 0.83 216 5.15 Tmck: Light/ 13.4 44.0 9.5 24.6 18.4 2.8 9.0 7.3 -3.7 0.01574 0.0405 0.87 431 6.64 medium Heavy 9.7 41.6 10.9 19.1 11.6 2.8 3.8 5.9 -3.1 0.02578 0.0369 0.85 381 9.59 Artic. 7.0 33.9 - 19.0 11.6 7.0 - 5.0 -1.5 0.03588 0.0365 0.81 232 8.07 1 The numbers in the parameter columns are the respective asymptotic t-statistics. 2 SSR denotes the sum of squared residuals., 3 For explanation of variances, see Chapter 3. Source: Analysis of Brazil-UNDP-World Bank highway research project data as described in Section 4.2 86 STEADY-STATE SPEED MODEL: ESTIMATION Table 4.3(c): Estimated standard errors of model error terms Estimated standard error Vehicle class ae a a e w Car 0.081 0.150 0.170 Utility 0.090 0.188 0.209 Bus 0.157 0.166 0.229 Light/medium truck 0.125 0.201 0.237 Heavy truck 0.161 0.192 0.250 Articulated truck 0.189 0.191 0.269 Source: Analysis of Brazil-UNDP-World Bank highway research project data as described in Section 4.2. FRATIO parameters were assigned the values of 500 metric hp and 0.40, respectively, when the estimation was carried out. All parameter estimates shown in Table 4.3 are of the expected sign and magnitude, and all but one are significant at the 95 percent confidence level, with values of asymptotic t-statistics as high as 44. The range of the a estimates of 0.244 - 0.310 is well below unity, which agrees with prior expectation discussed in Chapter 3. Except for the mixed class of light/medium trucks, the magnitudes of the used driving power (HPDRIVE) are consistently smaller than the maximum rated power values of the corresponding test vehicles (Table 1A.2). In fact, as greater the weight of a vehicle the more the braking capability needed to render the vehicle operations safe. An approximate relationship between the braking power and the gross vehicle weight is also derived in the next section. The FRATIO estimates, from 0.087 to 0.292, are well below the range of 0.6 - 0.7 found from skid-pad tests of modern high-performance passenger cars. This seems to indicate a large margin of safety within which vehicles are generally operated on public roads. This may also be due to the tendency of the average driver to limit lateral acceleration to an acceptable level of comfort of the driver and the passengers. The role of friction ratio as a dual criterion for skid and comfort has been discussed in Good (1978) and Oglesby and Hicks (1982). Within the total range of FRATIO estimates, the following observations can be made: 1. The range for unpaved roads (0.087 - 0.124) are significantly smaller than that for paved roads (0.130 - 0.292). 2. For paved road operations, loaded trucks have smaller FRATIO values than unloaded trucks (0.130 - 0.185 versus 0.179 - 0.292). 3. The FRATIO tends to vary inversely with the size of STEADY-STATE SPEED MODEL: ESTIMATION 87 the vehicle. For example, the FRATIO estimates for paved roads are the largest for passenger cars (0.268) and smallest for loaded articulated trucks (0.130). The estimates of the average rectified velocity (ARVMAX) show a clear tendency to vary inversely with the vehicle size, starting with the largest value (259.7 mm/s) for cars and ending with the smallest (130.9 mm/s) for articulated trucks. This is somewhat surprising because on purely physical reasoning one would expect the smaller vehicles to be more sensitive to road roughness than larger ones. The reversal of relative magnitudes is probably explained in part by higher tire stiffness of larger vehicles, and in part by the economic response of the driver to the relatively higher cost impact of roughness on larger vehicles. As mentioned in Chapter 3, regression was also carried out for the more complicated ARV formula (speed-sensitive ARS). The results presented in Appendix 4D indicate no improvement in goodness-of-fit offered by the more elaborate formula. The estimates for the desired speed (VDESIR) are, as expected, higher for paved roads (81.6 - 98.3 km/h) than for unpaved roads (49.6 - 82.8 km/h). Moreover, they tend to be larger for smaller vehicles, although, with the exception of the estimate for articulated trucks on unpaved roads (49.6 km/h), they are relatively constant for each surface type. According to the interpretations made in Section 3.4, the estimated values of ae (0.081 - 0.189) indicate that the standard errors of model predictions associated with unmeasured vehicle and road attributes at a given road site are 8.1 - 18.9 percent of the predicted speed. Similarly, the estimated values of a (0.170 - 0.269) imply that the standard errors of model predictions associated with the above measurement error and the random nature of individual speed observations themselves are 17.0 - 26.9 percent of the predicted speeds. To get an idea of the goodness-of-fit of the estimated steady-state speed relationships, a plot of observed against predicted speeds is shown in Figure 4.1 for heavy trucks. Superimposed in the plot are the lines of equality and its plus and minus deviations by the ae and a standard errors (narrow and wide bands, respectively). Also for heavy trucks, Table 4.4 shows average prediction errors for different groupings of the data points based on the road curvature, gradient, roughness and surface type and the vehicle load level. It can be seen that even though the differences between observed and predicted speeds for data points aggregated at the section-direction level (Figure 4.1) are relatively large (standard error of about 1.95 m/s or 7 km/h) they are reduced to a much smaller magnitude when aggregated into groups of section-directions of similar characteristics. This indicates a good fit of the model with the data. Figures 4.2 - 4.4 show, for unloaded and loaded heavy trucks, graphs of predicted steady-state speed plotted against the gradient, radius of curvature and roughness respectively, for both paved and 88 STEADY-STATE SPEED MODEL: ESTIMATION Figure 4.1: Observed speed versus speed predicted by steady-state speed model for heavy trucks Observed speed (m/s) 30- /VSS (1 +o) ,VSS (1 + ce) 25- ++VSS (Line of + equality) 20 ~/ / + VS 1 e + + 20 + + -Y+ +~ +,VSS (1 - e) / ~ ,+VSS (1 -) + , +, + 15- I-' .- 15 4 +- + +4- +1 -W + + 1+0 -. , + 0-. a 5 10s1 20 25 30 Predicted speed (m/s) Source: Analysis of Brazil-UNDP-World Bank highway research project data as described in Section 4.2 STEADY-STATE SPEED MODEL: ESTIMATION 89 Table 4.4: Average speed prediction errors for different groupings of data points (for heavy trucks) Basis for Definition of Number Mean Mean predicted speed grouping group of observed for group (km/h) observa- observations speed for tions in group group Predicted difference (km/h) speed (obs-pred) Gradient (GR) GR<-4 94 54.13 54.21 -0.08 in percent -4 D) due size D-statistic to chance Car 216 0.062 0.04 Utility 216 0.039 0.15 Bus 216 0.060 0.06 Truck: Light/medium 431 0.058 Less than 0.01 Heavy 381 0.075 Less than 0.01 Articulated 232 0.059 0.05 Source: Analysis of Brazil-UNDP-World Bank highway research project data. STEADY-STATE SPEED MODEL: ESTIMATION 113 APPENDIX 4C ESTIMATION RESULTS FOR SEPARATE REGRESSIONS BY LOAD LEVEL AND BY PAVED AND UNPAVED GROUPING This appendix reports on different models estimated during the exploratory phase of the steady-state model estimation process. Basically, different models arise because of the multiplicity of ways in which observations for a given vehicle class may be pooled. In the case of cars, utilities and buses the choice was between independent estimation by surface type and pooled estimation. In the case of the three truck classes, the choice was between four combinations of pooling or not pooling the observations in respect of different surface types and loading conditions. In addition to presenting the estimates under different conditions of pooling the observations, a discussion of the statistical significance of different sets of estimates, based on the log-likelihood ratio test, is included. In making the final choice between different possible models, the following considerations were taken into account. 1. A model which pools observations is to be preferred because of its inherent simplicity and low data requirements for applications in different environments. Further, if the parameters to be estimated may be classified on a priori grounds between those that are sensitive to the basis of pooling (viz. surface type and loading condition) and those that are expected to be invariant, the pooled model would utilize the information more efficiently. 2. The selected model should have significant estimates as judged by the asymptotic t-statistic values. 3. The relative magnitudes and signs of the estimates should be in conformity with theoretical expectations. 4. Ideally, the selected model should not be statistically different from the other models. On a priori grounds, we would expect the desired speed (VDESIR) parameter to be different for unpaved and paved surfaces. In addition, we would expect the perceived friction ratio (FRATIO) parameter to differ by surface traversed as well as by the loading condition for trucks. Since the value of FRATIO would already be low for unpaved surfaces, it was decided to separate the data on the basis of loading condition only in the case of trucks travelling on paved surfaces in intermediate levels of pooling. 114 STEADY-STATE SPEED MODEL: ESTIMATION The following tabulation lists the parameters estimated for each model. Note that the $, HPDRIVE, HPBRAKE, and ARVMAX parameters are estimated for all the models. Only the additional parameters are listed for each model: Vehicle Model name Parameters estimated class Cars, Unpooled VDESIR for the relevant surface Utilities, FRATIO for the relevant surface Buses Pooled VDESIR for unpaved surfaces VDESIR or increment to above for paved surfaces FRATIO for unpaved surfaces VFRATIO or increment to above for paved surfaces Trucks: light, Unpooled VDESIR for the relevant surface and loading medium, FRATIO for the relevant surface and loading heavy and articulated Loads VDESIR for the relevant surface Pooled FRATIO for the relevant surface and empty vehicle VFRATIO or increment to the above for a loaded vehicle (applicable for paved surfaces only) Surfaces VDESIR for unpaved surfaces and relevant loading pooled VDESIR or increment to above for paved surfaces FRATIO for unpaved surfaces and relevant loading VFRATIO or increment to above for paved surfaces Loads VDESIR for unpaved surfaces and either loading and VDESIR or increment to above for paved surfaces surfaces FRATIO for unpaved surfaces and either loading pooled VFRATIO or increment to above for paved surfaces and empty vehicle VVFRATIO or further increment for paved surfaces and loaded vehicle A study of Tables 4.3, 4C.1 and 4C.2 reveals the following: 1. The parameter HPDRIVE changes very little between different models for the same vehicle class. The maximum difference is 14 percent for heavy trucks. The asymptotic t-statistics are uniformly significant. 2. The same is true of the parameter HPBRAKE except for articulated trucks and for the cases where the value was very high. Some t-statistics are poor. 3. ARVMAX and 0 parameters vary between models by about 25-30 percent. The t-statistics are uniformly significant. STEADY-STATE SPEED MODEL: ESTIMATION 115 Table 4C.1(a): Estimation results with surface type unpooled for cars, utilities and bases: estimates Vehicle Model Surface 8 HPDRIVE HPBRAKE ARVMAX VDESIR FRATIO class name Car Unpooled Unpaved 0.317 34.5 18.4 289.7 89.4 0.136 Paved 0.212 34.3 19.2 204.0 93.0 0.250 Utility Unpooled Unpaved 0.346 44.1 28.9 263.8 84.5 0.129 Paved 0.257 43.1 30.1 192.8 89.4 0.210 Bus Unpooled Unpaved 0.266 95.6 W 1 215.7 70.0 0.096 Paved 0.179 121.7 165.6 184.7 82.9 0.214 1 The value was very large, nearly infinity. Source: Analysis of Brazil-UNDP-World Bank highway research project data. Table 4C.1(b): Some important statistics associated with the estimation No. Vehicle Mxel Surface 8 HP HP AWMAX VIESIR FRATIO R2 of SSR D class DRIVE BRAE obs. Car Unpooled Unpaved 7.9 10.0 6.4 12.4 13.1 8.6 0.006812 0.86 112 0.72 0.064 Paved 7.6 15.6 8.8 17.0 32.7 20.8 0.005503 0.92 104 0.54 0.054 Utility Unpooled Unpaved 6.9 11.0 5.6 10.6 9.9 6.1 0.008970 0.79 112 0.95 0.049 Paved 8.6 18.7 7.8 13.0 21.8 17.5 0.006822 0.91 104 0.67 0.075 Bus Unpooled Unpaved 4.5 18.7 -1 9.9 9.7 4.6 0.026826 0.70 112 2.87 0.049 Paved 3.9 31.0 7.3 10.7 20.9 12.6 0.016585 0.87 104 1.63 0.100 1 The value was very large, nearly infinity. Notes: a) The numbers in the parameter columns are the respective asymptotic t-statistics. b) SSR denotes the sum of squared residuals. c) D denotes the Kolmogorov-Smirnov D2 statistic (see Appendix 4B). d) For explanation of the variance, a2, see Chapter 3. Source: Analysis of Brazil-UNDP-World Bank highway reseach project data. 116 STEADY-STATE SPEED MODEL: ESTIMATION Table 4C.2(a): Estimation results at various levels of pooling by surface type and loading condition for trucks: estimates VESIR FRATIO Load- irg Empty loaded Loaded cordi- un- Empty un- pavel Class Model tion Surface 8 HPDRIVE HPBRAW_ AR*%X Unpaved Paved paved paved paved Empty Unpaved 0.314 84.1 131.3 206.0 78.8 0.116 Un- Paved 0.213 82.0 1 169.7 75.9 0.231 pooled I_ I Light Urpaved 0.332 95.9 257.1 192.7 68.9 0.094 and Paved 0.234 97.7 170.9 160.5 75.4 0.158 medium Loads Urpaved 0.350 94.0 174.6 206.9 78.8 0.115 0.115 Pooled Both I Paved 0.262 95.3 180.3 174.3 78.0 0.241 0.162 Sur- Empty 0.265 83.7 1 192.9 72.3 79.2 0.102 0.246 faces Both IIII pooled Loaded 0.298 97.1 203.2 183.3 65.9 80.4 0.086 0.168 Urpaved 0.346 94.9 194.8 186.4 97.5 0.092 Eupty___ Paved 0.288 107.6 297.9 177.6 84.3 0.286 Urr- pooled Unpaved 0.363 107.9 268.7 193.9 64.4 0.166 Paved 0.215 111.2 231.0 147.1 78.7 0.166 Heavy - I I I Loads Urpaved 0.405 106.8 241.2 196.7 87.6 0.152 0.152 pooled Both I Paved 0.256 109.9 244.4 163.0 81.8 0.274 0.173 Sur- Enpty 0.322 104.7 290.8 182.1 87.6 89.7 0.073 0.302 faces - Both pooled Loaded 0.275 108.9 251.1 170.7 58.0 85.2 0.103 0.176 Arti- Un- Enpty Paved 0.207 185.5 265.6 148.9 82.3 0.175 ar- pooled lated Loaded Paved 0.165 210.2 594.2 125.6 74.2 0.106 Loads Both Paved 0.220 203.9 616.3 141.3 80.1 0.169 0.112 pooled Sur- Empty 0.222 184.0 302.2 134.7 75.3 84.9 0.010 0.183 face Both pooled Loaded 0.236 205.4 847.1 123.4 42.9 79.7 0.028 0.116 1 The value was very large, nearly infinity. Source: Analysis of Brazil-UNDP-World Bank highway research project data. STEADY-STATE SPEED MODEL: ESTIMATION 117 Table 4C.2(b) Some Important statistics associated with the estimation VDESIR FR&TIO Variances Load- I.g Empty Loaded Loaded N condi- HP- HP- Un- un- Empty un paved 2 2 R of SSR D Class Model tion Surface 0 DRIVE BRAKE AR.MAX paved Paved paved paved paved % 0e obs. Empty Unpaved 5.6 11.3 1.9 13.5 8.8 5.0 0.01407 0.03659 0.743 112 1.49 0.064 Paved 6.8 28.3 - 1/ 15.5 36.2 - 17.7 - - 0.00721 0.02304 0.893 104 0.71 0.087 Light pooled Unpaved 6.1 20.8 2.0 11.8 8.8 4.3 0.01716 0.06128 0.793 111 1.80 0.058 Lode -.- - - and Paved 4.7 29.1 6.4 8.6 16.4 9.9 0.02184 0.03890 0.869 104 2.14 0.112 medi. Loads Unpaved 8.8 25.4 6.9 16.8 11.1 6.4 0.01613 0.04980 0.774 223 3.50 0.045 Pooled Both Paved 9.4 38.1 8.2 13.8 30.0 13.0 -4.0 0.01423 0.03149 0.881 208 2.86 0.085 So- Empty 9.1 24.3 23.0 17.2 1.9 7.8 7.8 0.01090 0.02969 0.867 216 2.28 0.070 faes - Both - pooled Loaded 8.3 34.7 6.4 15.1 12.0 2.8 5.0 4.1 0.01938 0.04995 0.863 215 4.01 0.083 Unpaved 3.9 10.3 1.7 11.1 3.7 2.1 0.01841 0.04150 0.729 86 1.47 0.089 Em Paved 8.1 27.7 1.1 12.9 21.0 15.1 0.00736 0.01799 0.892 101 0.70 0.058 pooled Unpaved 3.3 17.3 4.2 6.5 5.5 1.4 0.02727 0.05673 0.761 91 2.32 0.067 Paved 3.2 24.2 7.5 6.8 9.8 6.9 0.04535 0.03757 0.835 103 4.40 0.111 Loads Unpaved 5.7 21.4 7.1 11.6 5.7 2.5 0.02314 0.04921 0.761 177 3.96 0.057 poldBth - - - -- pool Both Paved 6.7 35.4 10.2 11.5 18.8 9.8 -3.3 0.02518 0.03285 0.863 204 4.96 0.110 Sur- Empty 8.7 23.9 1.4 19.4 8.2 0.2 4.1 8.8 0.01295 0.02744 0.863 187 2.32 0.070 faold --a-e B .5 32.1 8.0 10.3 9.7 3.3 1.9 1.3 0.03808 0.04100 0.828 194 7.08 0.083 Empty Paved 4.5 26.5 5.9 15.3 19.2 8.8 0.01451 0.02439 0.814 84 1.13 0.08 Un- pooled Loaded Paved 2.3 23.8 2.3 9.7 13.6 4.8 0.04600 0.03748 0.829 85 3.63 0.10 Articu- Loads Both Paved 5.1 31.9 3.9 14.1 20.0 6.8 -2.0 0.03085 0.03424 0.824 169 5.00 0.090 lated pooled Sur- Empty 7.0 31.9 5.1 28.1 5.0 0.7 3.3 10.2 0.01093 0.02896 0.814 112 1.14 0.057 face. Both - 0_ pooled Loaded 3.3 23.9 1.0 9.8 8.0 4.7 1.1 2.6 0.03030 0.03910 0.819 120 5.64 0.073 The value was very large, nearly infinity. Notes: a) the numbers in the parameter columns are the respective asymptotic t-statistics. b) SSR denotes the sum of squared residuals. c) D denotes the Kolmogorov-Smirnov D statistic (see Appendix 4B). d) For explanation of the variances see Chapter 3. Source: Analysis of Brazil-UNDP-World Bank highway research project data. 118 STEADY-STATE SPEED MODEL: ESTIMATION 4. There are two anomalous results for the VDESIR parameter. In the case of light/medium and heavy trucks, the unpaved values are higher than paved values. This is contrary to theoretical expectations. However, three of the t-statistics are very low for the unpooled models. 5. The t-statistics are low for heavy and articulated truck FRATIO parameters in the unpooled cases. 6. As mentioned in the text of Chapter 4, for the pooled models, the signs and relative magnitudes of the estimates are in conformity with theoretical expectations. All the t-statistics are significant with a single exception. Tests were performed to judge the statistical significance of the differences between different models. The appropriate statistical methodology to judge the significance of the difference between the pooled and unpooled runs for each vehicle class is the log-likelihood ratio test. This ratio is defined as follows: SSR of pooled run x = 2 kn (likelihood ratio) = n( -1) Sum of SSRs of unpooled runs where n = total sample size, and SSR = sum of squared residuals for the appropriate regression. The random variable X as defined above is distributed as 2 with degrees of freedom, f, equal to the number of restricted parameters in the more constrained model. The SSR values are shown in Tables 4.3(b) and 4C.1(b) and 4C.2(b), and the critical values for the relevant X2 f distributions are given below: Degrees of freedom, f 4 5 6 7 9 Significance level 1% 9.49 11.07 12.59 14.07 16.92 5% 13.28 15.09 16.81 18.48 21.67 In choosing the level of significance of the test two considerations were taken into account. 1. For practical reasons, the simplicity of the pooled model is strongly preferred. 2. Due to the large sample size at our disposal, the power of the significance test is high even when the level of significance chosen is stronger than the 5 or 1 percent level commonly used for small and intermediate sample sizes. STEADY-STATE SPEED MODEL: ESTIMATION 119 The test results are shown in Table 4C.3. From this table it is evident that the differences between unpooled and intermediate-pooled models are not significant. However, the difference between intermediate-pooled and pooled models are generally significant at traditional critical levels. Nonetheless, on grounds stated in the foregoing discussion, it is reasonable to accept models estimated by pooling observations by surface types as well as loading condition. Table 4C.3: Log-likelihood ratio tests Models compared Vehicle Relatively Number Value of log class unrestricted Restricted Case of likelihood model model restrictions ratio Car Unpooled Pooled 4 17.14 Utility Unpooled Pooled 4 8.00 Bus Unpooled Pooled 4 31.20 Truck: Light/ Unpooled Loads pooled Unpaved 6 14.23 medium Paved 5 0.73 Surfaces pooled Empty 4 7.85 Loaded 4 3.82 Loads pooled Pooled 4 18.97 Surfaces pooled Pooled 7 23.98 Heavy Unpooled Loads pooled Unpaved 6 7.94 Paved 5 0.00 Surfaces pooled Empty 4 12.93 Loaded 4 10.39 Loads pooled Pooled 4 28.62 Surfaces pooled 7 7.70 Artic. Unpooled Loads pooled Paved 5 8.52 Surfaces pooled Pooled 9 44.14 Source: Analysis of Brazil-UNDP-World Bank highway research project data. 120 STEADY-STATE SPEED MODEL: ESTIMATION APPENDIX 4D ESTIMATION RESULTS BASED ON SPEED-SENSITIVE AVERAGE RECTIFIED SLOPE This appendix is devoted to discussion of the results of the estimation of the steady-state speed model which uses the more compli- cated ARV formula (i.e., the speed-sensitive average rectified slope for- mulation). The principles underlying this formulation were discussed in .Section 3.3.4. While computationally trivial, the additional effort needed to use the speed-sensitive version in practical applications is considerable in terms of data requirements. Specifically, the material type of the road surface has to be input which is not required elsewhere in the model. Thus it is of interest to investigate how the simpler ver- sion, of which the parameter estimates are presented in the main text of this chapter, compares with the speed-sensitive version. The comparison consists of three aspects. The first was to find out whether the two formulations of the ARVMAX parameter give rise to markedly different numerical estimates of parameters other than ARVMAX. The second was to compare explanatory power of the two versions in terms of the regression statistics. The final aspect was to compare speed predictions arrived at using the two versions for different groupings of speed observations by various speed-influencing factors for a heavy truck. The estimation results of the speed-sensitive formulation are presented in Table 4D.1. This table is formatted identically as Table 4.3, with parameter estimates in part a of the table and the associated regression statistics in part b. A comparison of Tables 4.3(a) and 4D.1(a) reveals that the estimates of parameters other than ARVMAX do not change significantly in absolute magnitude, and their relative magnitude across the vehicle classes is preserved. In fact, the change in absolute magnitude is five percent or less. Turning next to a comparison of the explanatory power of the two versions of the speed model, it may be seen from Tables 4.3(b) and 4D.1(b) that the R2 values and the standard errors of residuals of the two estimations are almost identical and that the simpler version is marginally better. Finally, Table 4D.2 presents, for a heavy truck, the observed speeds, and the speed predictions using the different ARVMAX formula- tions, averaged over different groupings of the data points based on the road curvature, gradient, roughness and surface type, and vehicle load level. The differences between the two predictions are less than one percent in almost all the groupings. Further, there is no systematic trend in the relative magnitudes of the two predictions. Thus it may be concluded that the speed prediction model using the simple ARV formula- tion is the version to be preferred. STEADY-STATE SPEED MODEL: ESTIMATION 121 Table 4D.1(a): Estimation results of steady-state speed model for six vehicle classes: estimates V DESIR (laU/h) FRATIO Unpaved Paved Paved Unloaded loaded Vehicle 8 HPDRIVE PBRAWE AR*MX Increnent Value Unpaved Increent Value Incraent Value class netric metric nmVs over unloaded over un- over hp hp unpaved or paved paved value loaded value unloaded a b c d e f e+f g h g+b b g ti Car 0.288 37.3 22.8 283.8 80.7 18.5 99.3 0.128 0.143 0.271 - - Utility 0.322 45.2 34.2 269.6 76.3 19.9 96.2 0.122 0.103 0.225 - - Bus 0.290 113.9 222.6 243.1 68.1 26.6 94.7 0.100 0.137 0.237 - - Truck: Light/ 0.321 95.6 196.6 227.8 69.1 13.2 82.4 0.106 0.153 0.259 -0.085 0.174 melium Heavy 0.338 109.3 265.8 214.6 68.7 22.7 91.4 0.103 0.199 0.302 -0.110 0.192 Artic. 0.264 201.9 500.01 154.0 50.2 34.6 84.8 0.0401 0.141 0.181 -0.046 0.135 1 These parameters were exogenously determined. Source: Analysis of Brazil-UNDP-World Bank highway research project data. Table 4D.1(b): Some Important statistics associated with the estimation VIESIR FRATIO VARAlES Vehicle 8 IPDRIE HPBRAE AlRWAX Unpaved Incranent Unpaved Incrament for paved a2 , R2 RIber SS D class for paved e w of surface Unloaded loaded obserwtions a b c d e f g h i Car 11.1 14.3 7.1 25.3 23.7 5.8 11.6 9.5 - 0.00683 0.0224 0.911 216 1.42 0.059 Utility 11.3 18.5 7.3 21.9 18.3 5.0 8.7 6.5 - 0.00838 0.0355 0.885 216 1.74 0.037 Bus 6.9 26.5 4.4 15.2 12.8 4.2 5.3 5.1 - 0.02504 0.0276 0.828 216 5.21 0.048 Truck: Llght/ 13.9 42.3 9.2 31.2 20.8 4.4 8.4 6.9 -3.6 0.01604 0.0405 0.861 431 6.77 0.055 medium Heavy 10.2 39.6 10.5 23.4 13.5 4.0 3.7 5.3 -3.0 0.02635 0.0369 0.846 381 9.80 0.069 Artic. 6.9 33.1 - 24.1 12.2 6.9 - 5.0 -1.4 0.03552 0.0365 0.811 231 7.99 0.070 1 The numbers in the parameter columns are the respective asymptotic t-statistics. 2 SSR denotes the sum of squared residuals. 3 For explanation of variances, see Chapter 3. Source: Analysis of Brazil-UNDP-World Bank highway research project data. 122 STEADY-STATE SPEED MODEL: ESTIMATION Table 4D.2: Average speed predictions for different groupings of data points (for heavy trucks) Mean predicted speed Mean observed for group (In/h) Basis for Niumber of speed for grouping Definition observations group With simple ARV With speed-sensitive observations of group in group (kn/h) fonmlation ARV formulation Gradient (GR) GR < -4 94 54.13 54.21 54.06 in percent -4 0) VENMAX. = min [VCTL ; VDECEL.] (6.3a) Negative effective gradient (GRj + CRj < 0): VENMAX = min [VCTL ; VDECEL.; max (VSS.; VMOMEN.)] (6.3b) j ] J jJ where VDECELj = the maximum allowable speed at the beginning of the simulation interval from which it is possible to decelerate (at a given rate) to the maximum allowable exit speed (VEXMAXj), in m/s; and VMOMENj = the maximum allowable speed at the beginning of the interval from which it is possible to accelerate under control to the maximum allowable exit speed, in m/s. The maximum allowable "deceleration" speed, VDECEL3- is com- puted as a kinematic function of the maximum allowable exit speed, VEXMAXj and a given constant deceleration rate, DECELj (in m/s2): VDECEL = max [(VEXMAX2 + 2 DECEL AL.); 1]0.5 (6.4) where DECELj is defined in Section 6.2.2 as a behavioral function of the vehicle driving power and braking capabilities. In Equation 6.4, the minimum limit of 1 is imposed to insure that VDECELj does not become MICRO TRANSITIONAL PREDICTION MODEL 137 too small. Similarly, the maximum allowable "momentum" speed, VMOMENj is given by: VMOMEN. = max ((VEXMAX? - 2 AMOMEN. AL.); 1 0.5 (6.5) where AMOMEN , the acceleration rate governed by the vehicle braking capability, In m/s, is defined in Section 6.2.3 below. Like DECELj, AMOMENj is assumed to be constant for the simulation interval. It is useful to mention an asymptotic property of the maximum allowable entry speed, VENMAX. For a relatively long subsection which contains a number of identical simulation intervals, repeated application of Equation 6.3 will cause VENMAXj to converge to a constant value. For positive effective gradient (GR- + CR* > 0) VENMAXJ converges to the control speed (VCTLj) whereas for negative effective gradient (GRj + CRj < 0) it converges to the steady-state speed (VSSj). An illustration of the convergence of the maximum allowable speed profile for both positive and negative effective gradients is given in Figure 6.4. For positive effective gradient, it is reasonable to regard the control speed as the "maximum allowable safe speed" rather than the steady-state speed. On a steep positive grade (say, more than 6 per- cent), the steady-state speed can be much lower than the control speed; this occurs when the vehicle does not have the necessary engine power to attain the latter speed. For negative gradient, however, it is reason- able to regard the steady-state speed as the "maximum allowable safe speed." On a subsection of steep negative grade, the braking power- limited speed, VBRAKE, tends to dominate so that the control speed con- verges to the steady-state speed. 6.2.2 Deceleration The deceleration logic basically assumes that the driver em- ploys a "desired" deceleration rate when the vehicle is unencumbered by its driving and braking capabilities; otherwise the deceleration rate is determined on the basis of the used driving or braking power (HPDRIVE or HPBRAKE), whichever is applicable. Graphs of the deceleration rate (DECEL ) and vehicle power (HPj) plotted against the effective gradi- ent (GR + CR ) in Figure 6.5 illustrates the deceleration logic. Mathematically, the deceleration rate is expressed as: DECEL = min [max ( DDESIR ; DDRIVE ); DBRAKEJ] (6.6) where DDESIRj = the "desired" deceleration rate, in m/s2; DDRIVEj = the deceleration rate governed by the used driving power (HPDRIVE), in m/s2; and DBRAKEj = the deceleration rate governed by the used braking power (HPBRAKE), in m/s2. 138 MICRO TRANSITIONAL PREDICTION MODEL Figure 6.4: Convergence of maximum allowable speed profile (a) Subsection with gentle positive gradient(s) followed by a sharp curve (s+1) Subsection s 1 L5 Maximum allowable VCTL, exit speed of Maximum allow- subsec- fable speed tion 8-1 profile of subsection s Maximum allow- able entry speed of sub- section (s+1) (sharp curve) 100 100 100 100 100 100 Distance (W) (b) Subsection with steep negative gradient(s) followed by a tangent positive grade (s+1) Maximum allow- able entry speed of next subsection s+1 (tangent pos- itive grade) Maximum allow- Maximum L able speed profile of exit l ---u subsection s exit I /-- speed of subsec- ion s-i 100 100 100 100 100 100 Distance (m) Travel direction MICRO TRANSITIONAL PREDICTION MODEL 139 Figure 6.5: BelationsЬips betиeen acceleэration and deceleration таtея, power and adjusted grad3ent DECEL� 1 1 1 г 1 1 1 г 1 1 1 г DDRIVE� 1 1 1 DDSIRE j г 1 1 1 1 г DBRAKE� 1 1 г г --�- ск� + ск3 г г г г г г г г г г г г г г г г г г j Regime 3 j Regime 1 j Regime 2 НР� 1 1 1 1 1 1 1 1 i 1 I HPDRIVF. rввиивви�вввввввввввв г г г г г г г г г г г г GRi + CR� 1 1 1 ! 1 1 1 ! 1 1 1 I 1 1 ! 1 1 1 1 1 1 1 1 1 •■ -HpBRAKE 1 1 1 1 1 AMOMENj 1 1 1 1 ! 1 1 ! 1 1 1 1 1 1 1 1 1 1 1 1 �-DBRAKEj1 1 1 г i i i 1 г~ GR j+ CR� + г г г г� г г г г `. г r 1 1 `1 1 1 1 1� 1 1 1 1 1 140 MICRO TRANSITIONAL PREDICTION MODEL The deceleration rates, DDESIRj, DDRIVEj and DBRAKE i , are further elaborated in terms of three regimes of road gradient, as follows. Regime 1: moderate grade (positive and negative) In this regime, the vehicle power employed at the desired de- celeration rate, HPj, falls within the vehicle driving and braking ca- pabilities, i.e., -HPBRAKE < HPj < HPDRIVE For simplicity, the desired deceleration rate, DDESIRj is assumed to be constant for a given vehicle class; values obtained from a field experi- ment in the Brazil study (Zaniewski et al., 1980) are given below (in m/s2): Car 0.6 Utility or bus 0.5 Truck 0.4 Regime 2: steep positive grade The desired deceleration rate, DDESIRj, could not be used, since the required power would exceed the maximum used driving power, HPDRIVE (i.e., HP > HPDRIVE). Therefore, the deceleration rate that can be used is that determined by HPDRIVE, as follows: DDRIVE.= - 736 HPDRIVE + m (GR. + CR + A VAVG2 (6.7) i m' VAVG. mv I ml i where3 VAVG i = 0.5 (VENMAXj + VEXMAXj) i 'NMAXj2 + VEM4AXj2) VAVG2 = 0.5 (VF and the other terms are defined before. For simplicity, the values of the effective vehicle mass (ml) used in determining the rolling resis- tance coefficients (Appendix 2A) are also employed in the simulation log- ic, even though they should be higher when the vehicle is in low gear (as discussed in Chapter 2). The values of the effective vehicle mass are compiled in Table 1A.2. In this regime, we have DDRIVEj > DDESIRj1, meaning that the full driving force of the engine is being exerted but the vehicle is still losing speed at a rate faster than would be desir- able to the driver. 3 Since VENMAX is unknown but needed to compute VAVGj, it is necessary to employ an iteration procedure for determining VENMAXj,- as described in Section 6.2.4. MICRO TRANSITIONAL PREDICTION MODEL 141 Regime 3: steep negative grade Like regime 2, the desired deceleration rate, DDESIRj, could not be used, but for a different reason: the required power would vio- late the vehicle braking capacity constraint (i.e., HP < -HPBRAKE). Consequently, the feasible deceleration rate is determined by HPBRAKE: 736 HPBRAKE mn A ]AV DBRAKE.= + (GR. + CR.) + (6.8) m' VAVG. m I m As shown in Figure 6.5, in this regime the driver is applying the used braking power (HPBRAKE), although the vehicle is still decelerating at a rate slower than desirable. 6.2.3 Downhill Momentum Acceleration As discussed in Chapter 5, when the vehicle approaches the bot- tom of a steep negative grade, the driver can save both fuel and travel time by building up the vehicle momentum to the highest feasible level at the bottom of the hill. In building the momentum, we assume that the driver utilizes the used braking power, HPBRAKE. This permits accelera- tion to begin the earliest.4 A smaller braking power would result in a delayed onset of acceleration, and, hence, longer travel time. The down- hill momentum acceleration, AMOMEN, is given by: 736 HPBRAKE m A jAV2 AMOMEN. = max [0; - 736_HBRAKE- - (GR. + CR.) + A (6.9) m VAVG. m' 3m' where the limit of zero is imposed to indicate that negative AMOMENj is irrelevant to downhill momentum situations. Comparing the above equation with Equation 6.8, it can be seen that DBRAKEj = -AMOMENj when AMOMEN- is positive. This means that whenever a downhill momentum situation occurs, we have DECELj DBRAKE -AMOMEN, and, consequently, the maximum allowable 'deceleration" and "momentum speeds become equal (VDECELj = VMOMENj). 6.2.4 Iteration Procedure As mentioned above, the maximum allowable exit speed of the simulation interval (VENMAXj) is the sought after unknown quantity that is also used in computing the acceleration and deceleration rates. Thus, an iteration procedure is employed in determining VENMAX, as shown sche- matically in Figure 6.6. First, a trial value of VENMAXj is set equal 4 This assumption was verified in comparison between simulated and ob- served profiles of speed and vehicle power, as presented in Section 6.4. 142 MICRO TRANSITIONAL PREDICTION MODEL Figure 6.6: Iteration procedure for determining maximum allowable exit speed, VElMAXJ Set a trial value of maximum allowable entry speed: VENMAX (trial) = VEXMAX Set iteration counter: N = 0 Increase N by one Compute average speed and squared speed VAVC = [0.5 VENMAX + VEXMAX 2 j L (trial) j VAVG2. = [0.5 VENMAX + VEXMAX 1 j L. (trial) j Compute acceleration and deceleration rates AMOMEN, and DECEL as functions of VAVC and VAVG2 -1 Compute maximum allowable "momentum" and "deceleration" rates, VMOMEN and VDECEL as functions of AMOMEN and DECEL Compute new value of maximum allowable entry speed: VENMAX as a function of VMOMENi, VDECELip VCTL. and VSS (new) 3' i 3I Check convergence: VENMAX(new) - VENMAX(trial)< 0.1 YES VENMAX(trial) NO Convergence has been obtained N >E Set VENMAX = VENMAX (new) YES --max: No Reset trial value of maximum allowable entry speed: VEMX(trial) EMX(new) MICRO TRANSITIONAL PREDICTION MODEL 143 to VEXMAX . From this trial value, denoted by VENMAX(trial), a se- ries of speed, acceleration and deceleration quantities are computed leading to a new value of VENMAX. This new value, denoted by VENMAX(new), is compared with the existing trial value. If the differ- ence is less than 10 percent, convergence is considered to be obtained and VENMAX(new) is taken as the solution. Otherwise, the process is repeated with the new trial value of VENMAX set equal to VENMAX(new)* Based on 100-meter simulation intervals, it was found from numerical testing that convergence could be obtained within few iterations. How- ever, a limit of Nmax is imposed to prevent the possibility of loop- ing. Values of 4 or 5 for Nmax was found to be adequate. 6.3 SIMULATION OF ACTUAL SPEED PROFILK: FORWARD RECURSION 6.3.1 Exit Speed In the actual speed profile simulation, the general behavioral assumption is that the driver uses the vehicle power according to a well-defined rule and proceeds as fast as possible provided that the max- imum allowable speed is not exceeded. Again, consider the simulation in- terval j, j = 2 to J, its adjacent intervals, and the corresponding entry and exit speeds as depicted schematically in Figure 6.7. The objective of the forward recursion is to determine the exit speed, denoted by VEXi (in m/s) as a function of the entry speed, denoted by VENj (in m/s , for interval j. The entry speed for the current simulation inter- val, VENj, is equal to the exit speed of the preceding interval Figure 6.7: Simulation intervals and entry and exit speeds Speed Simulated speed profile VEXj+L' VEXj_l VEX VENEN-1 VEN____ _Distance AL 1AL i ALj+l 1 1 I I Simulation j-1 j j+1 intervals Travel direction ON 144 MICRO TRANSITIONAL PREDICTION MODEL upstream, VEXj-1. The forward recursion proceeds from interval J-1 to j by setting: VENj = VEXj_1 (6.10) The entry speed for the first simulation interval, however, must be specified exogenously as a boundary condition.5 The formulas for computing the exit speed for simulation inter- val j (VEXj) are given separately for positive and negative adjusted gradients below. Case I: negative effective gradient (GRj + CRj < 0) In this relatively straightforward case, the formula is: VEXj = min [VEXMAXj; VPOWERj] (6.11) where VEXMAXj = the maximum allowable exit speed for the simula- tion interval (as determined in the backward recursion phase); and VPOWERj = the speed at the end of the interval based on a constant acceleration rate APOWERj, in m/s, as given below: VPOWER = max [(VEN2 + 2 APOWER AL )1 0.5 (6.12) where VENj = the entry speed for the interval, in m/s; and APOWERj = the acceleration rate resulting from vehicle power HPRULEj defined in Section 6.3.2, in m/s2. As before, the limit of 1 m/s is imposed in the above formula to ensure that VPOWER- does not become too low. Examples of various possible speed profiies involving negative effective gradient are illustrated with annotations in Figure 6.8. Case II: positive effective gradient (GRj + CRJ > 0) Separate formulas are provided for subcases of positive effective gradient. 5 In the absence of information on the road conditions upstream, the steady state speed for the interval, VSSI, could be used. MICRO TRANSITIONAL PREDICTION MODEL 145 Figure 6.8: Examples of possible speed profiles for negative effective gradient A Li (a) Gentle grade, high VEXMAX1. The VEXMAXj vehicle accelerates to end of the simulation interval under HPRULEj VEXj VPOWERj unrestricted by VEXMAXj VENj (b) Gentle grade, low VEXMAXj. The VPOWERj vehicle has sufficient power to VENj accelerate, but is forced to decelerate to VEXMAX*. VEXj = VEXMAXJ (c) Steep grade, moderate VEXMAX . The VPOWERj vehicle is accelerating but Its rate of acceleration is restricted VEXj VEXMAXj by VEXMAXj. VENj 146 MICRO TRANSITIONAL PREDICTION MODEL Case II.A: VEXAX < VSSj The formula for this case is the same as for Case I: VEX. = min [VEXMAX.; VPOWER.,] (6.13) Examples of possible speed profiles arising in this case provided in Figure 6.9. Case II.B: VEXAXJ > VSSj This case is more complicated than cases I and II.A since pro- visions must be made to insure convergence toward the steady-state speed, VSS . Formulas are provided separately for convergence from above (VENj > VSSj) and below (VENj < VSS*). For VENj > VSSj, the formula is: VEX = min [VEXMAX.; max (VSS.; VPOWER.)] (6.14) and examples of possible speed profiles are given in Figure 6.10. For VENj < VSSj, we have the following formula: VEX = min [ VSS ; VPOWERJ] (6.15) and examples of possible speed profiles shown in Figure 6.11. 6.3.2 Power Usage This section provides the definition of HPRULEj, the vehicle power used in computing VPOWERj. The rule of vehicle power usage may be described briefly as follows. If the vehicle enters the simulation interval at a speed higher than the steady-state speed (i.e., VENj > VSS) it is assumed to decelerate toward the latter under the power equal to that needed to maintain the steady-state speed itself. Con- versely, if the vehicle enters the interval at a speed lower than the steady-state (i.e., VENj < VSSj) it is assumed to accelerate toward the latter under a power which is dependent on the road gradient but al- ways within the bounds imposed by the maximum driving and braking pow- ers. In the following paragraphs, formulas are provided for the cases with VENj > VSSj and VENj < VSSj. Case of VENj > VSSj This case applies only to positive effective gradient (GR + CRj > 0) since VENj is always smaller than or equal to VSSj when the effective gradient is negative. When VENj is greater than VSSj. MICRO TRANSITIONAL PREDICTION MODEL 147 Figure 6.9: Examples of possible speed profiles for positive effective gradient (with VERMAXj > VSSj) A Li (a) Gentle grade, high VEXMAX1. The - VSSJ vehicle accelerates to end of the simulation interval under HPRULEj - VEXMAXj unrestricted by VEXMAXj VEXj = VPOWERj VENj (b) Gentle grade, low VEXMAXj. The vehicle accelerates at a rate con- - VSSj trolled by VEXMAXj. VPOWERj VENj VEXj = VEXMAXj (c) Steep grade, low VEXMAXj. The VENj vehicle decelerates at a rate controlled by VEXMAX1. VPOWERj - VSSJ VEXj = VEXMAXj 148 MICRO TRANSITIONAL PREDICTION MODEL Figure 6.10: Examples of possible speed profiles for positive effective gradient (with WAIMXj > VSSj and W#j > VSSj) (a) Steep grade, high VEXMAXj. The vehicle has almost reached VENj VSSj at entry but VPOWERj uRdershoots VSSj. Setting VEXj - VSSj insures conver- VEXj - VSSj gence. - VPOWERj - VEXMAXj b) Steep grade, high VEXMAX1. The vehicle is decelerating YE@ toward VSSj under HPRULEj unrestricted by VEXMAXj. VEX1 - VPOUER1 VSS1 (c) Steep grade, low VEXMAXj. The vehicle has not reached VSSj VE N1 and is decelerating at a rate VPOWER controlled by VEXMAXj. VEX1 - VEXMAXj VSSj MICRO TRANSITIONAL PREDICTION MODEL 149 Figure 6.11: Examples of possible speed profiles for positive effective gradient (with VEXNAXj > VSSj and VENJ < VSSj) Lj (a) Gentle grade, high VEXMAXj. VEXMAXj The vehicle has almost con- verged to VSS- at entry but VPOWERj overshoots VSS . Setting VEXj = VSSj insures VEXj = VSSj convergence. VE Ni (b) Gentle grade, high VEXMAXj. - VEXMAXj The vehicle is accelerating under HPRULEj toward VSSj, - VSSj unrestricted by VEXMAXj. VEXj = VPOWERj VENj the vehicle has a tendency to decelerate toward VSSj, and the vehicle power is given by: HPRULE = HPSS (6.16) where HPSSj = the driving power required to sustain the steady- state speed of the simulation interval, given by: HPSS = L mg (GR + CR ) VSS + A VSS (6.17) j 736 -' 1 1 150 MICRO TRANSITIONAL PREDICTION MODEL The above equation provides the largest possible driving power the driver can use while still guaranteeing deceleration toward the steady-state speed (when the subsection is infinitely long). The maximum used driving power, HPDRIVE does not guarantee convergence, since it is greater than HPSSj by virtue of the inequality VDRIVEj > VSSj. Case of VENj < VSSj In this case, the vehicle has a tendency to accelerate toward the steady-state speed. The power usage is defined as follows: HPRULE. = min [HPDRIVE; max (HPGRAD - HPBRAKE)] (6.18) where HPGRADj = the used driving power modified for the effect of the effective gradient, given by: m g (GR. + CR.) VAVG. HPGRAD = HPDRIVE + m 3 ( (6.19) 736 The graphs of HPRULEj plotted against the effective gradient are shown schematically in Figure 6.12 for both VENj > VSSj and VENJ < VSSj cases. In these graphs, the road characteristics other than the gradient (GRj) are held constant. For the case VEN- > VSSj, the vehicle power (HPRULEj = HPSSj) increases with the effec- tive gradient and asymptotically approaches the used driving power, HPDRIVE, when the effective gradient becomes large. For the case VENj < VSSj, the driver is assumed to apply the used driving power (HPRULEj = HPDRIVE) when the effective gradient is positive (GR, + CRj > 0). When the effective gradient is nega- tive, the driver is assumed to apply smaller driving power than HPDRIVE (but not smaller than -HPBRAKE). The power reduction (equal to HPDRIVE - HPGRADj) is such that, for moderately steep negative grades with HPGRADJ > -HPBRAKE, the acceleration rate would equal the rate under full power HPDRIVE when the effective gradient is zero. However, for steep negative grades with HPGRADj < -HPBRAKE, this acceleration rate cannot be sustained and a higher rate must be used as the driver is con- strained by the used braking power, HPBRAKE. 6.4 VERIFICATION OF ACCELERATION, IECELERATION AND POWER USAGE LOGIC The relationships for computing the vehicle deceleration rate (DECELj), downhill momentum acceleration rate (AMOMENj), and power (HPRULEj) were developed broadly in the following manner. First, pre- linary relationships were constructed based on the results of a review of literature (Sullivan, 1977; Gynnerstedt, et al., 1977; St. John and Kobett, 1978; among others), and conversations with truck drivers. Second, profiles of the vehicle speed, acceleration, and power produced MICRO TRANSITIONAL PREDICTION MODEL 151 Figure 6.12: Vehicle power versus effective gradient HPRULE HPDRIVE HPGRAD HPSS )_ GR. + CR. -HPBRAKE and the simulation logic were compared with those derived from observation of the test vehicles traversing several 10-km road sections. Third, modifications were made to the behavioral assumptions after major discrepancies between the simulation and observation were found. A major modification worth mentioning involved the initial assumption on extensive use of coasting in neutral (i.e., HPRULE - 0) in both uphill and downhill travel, as employed by Sullivan (19763. However, the test vehicle data revealed that the drivers generally employed: (i) the used braking power, HPBRAKE, when accelerating in a downhill momentum situation, and (ii) the used driving power, HPDRIVE, when decelerating toward the the steady-state speed on a steep uphill grade. To get an idea of how well the acceleration, deceleration and power usage assumptions approximate reality, Figure 6.13 compares profiles of simulated and observed acceleration and power for one of the test vehicles (Mercedes Benz 1113 heavy truck) traversing a hilly paved road section. The profile of the gradient of the road is also shown in the figure. 152 MICRO TRANSITIONAL PREDICTION MODEL Figure 6.13: Profiles of road gradient and observed and simulated power and acceleration for a loaded heavy truck on a paved hilly road Acceleration (m/s/s) Observed acceleration profile 0.6 Simulated acceleration profile 0.3 -0.6 Power (metric hp) Observed power profile 300- 200 Simulated power profile 100- -100- -200 -300 +5- Road gradient (%) Road gradient I / profile -5. 0 2000 4000 6000 8000 10000 Distance (m) Source: Analysis of Brazil-UNDP-World Bank highway research project data. CHAPTER 7 Aggregate Prediction Model The two micro models for predicting speeds discussed so far need detailed information regarding geometric alignment and surface con- dition, and in turn they can provide a whole array of speed predictions across the length of the section under consideration. While such a rich set of predictions is appropriate for handling minor geometric improve- ments (for example, to eliminate severely sharp bends), it would be not only superfluous but also unwieldy for policy-level decisions where the emphasis is placed on sensitivity of the average speed to broad changes in geometric alignment and surfacing standards. Hence, we sought a sim- pler model for predicting the average speed over a given roadway which requires only aggregate information on road attributes, while at the same time, retains as much as possible the desirable properties of the more elaborate models, namely, predictive accuracy, policy-sensitivity, ex- trapolative ability, and local adaptability. The "aggregation issue" arises from the fact that the micro prediction models represent a non-linear function of road attributes. This means that the required aggregate speed prediction cannot be ob- tained directly from the average attributes of the roadway in question1 but must be derived from a method that properly accounts for the mathe- matical properties of the prediction models themselves. A discussion of some of the important approaches to the aggre- gation issue is taken up in the next section. The discussion motivates the selection, on a priori grounds, of the "integration" approach, and the various aggregation methods under this approach are described in Section 7.2. Finally, a detailed description of the recommended aggrega- tion method is given in Section 7.3. Appendix 7A gives a step-by-step procedure to compute aggregate geometric descriptors starting from the engineering profile of a roadway, along with an illustrative example. Since, as will become apparent, the chosen aggregation method relies on the micro non-transitional prediction model, the discussion of issues re- lating to validation of the aggregation method is incorporated in the overall treatment of validation of all the speed prediction models, which constitutes the subject matter of Chapter 8. 7.1 REVIEW OF ALTERNATIVE AGGREGATION METHODS The importance of aggregation was discussed above in the con- text of speed and travel time prediction. More generally, aggregation is equally important in the context of prediction of various components of vehicle operating costs, especially fuel consumption and tire wear. 1 This is the "naive" method as discussed in Section 7.2. 153 154 AGGREGATE SPEED PREDICTION MODEL Figure 7.1: Schematic display of alternative aggregation approaches and methods Theoretic method Aggregate estimation approach Empirical method Mbnte Method based on --- Carlo simulation approach modelling Aggregation methods based method Integration approach Naive method Method of statistical differentials Because of its pervasive importance, some methodological issues regarding aggregation will be discussed at some length. For excellent discussions of the aggregation issue, see Fisher (1969) and Koppelman (1975). As shown in Figure 7.1, there are three major approaches to ag- gregate prediction. First, one can estimate a prediction equation di- rectly at the level of aggregation desired. This approach may be termed the aggregate estimation approach. Second, one can employ Monte Carlo AGGREGATE SPEED PREDICTION MODEL 155 simulation to generate predictions from the micro transitional model; the predictions can then be used to estimate an aggregate model. Finally, one may use the corresponding estimation equations derived at a more disaggregate level to arrive at disaggregate predictions and aggregate them explicitly. This approach may be referred to as the integration ap- proach. We shall discuss some of the important methods which fall under these three groupings, in turn. The ideal method under the aggregate estimation approach would be an "aggregate theoretical" one. Analogies for such a method from oth- er disciplines are general properties of matter in physics and macro- economics. In the context of speed prediction, such a method would en- tail direct specification of a theoretical model of average speed as a function of average and more detailed information of geometric and sur- face attributes of a roadway. Given the complex and dynamic nature of vehicle motion it is unlikely that such a model can be specified in a manageable aggregate form. A second method under the aggregate estimation approach is a purely empirical procedure where a simple, generally linear, equation re- lating average speed to various speed-influencing factors is estimated. This method has been followed in the TRRL Kenya study (Hide et al., 1975), and in the RUCS India study (CRRI, 1982). The method is easy to use and gives satisfactory predictions within the range of observed vari- ation of selected factors, but is subject to the criticisms made in the context of steady-state speed modelling in Chapter 3. The Monte Carlo approach, which entails a method based on simu- lation modelling, involves three steps. First, one starts with a large sample of detailed road or network profiles; these could be profiles of real roads or artificially generated. For each of these profiles, vari- ous aggregate indices that characterize the average alignments and sur- face conditions would be computed. Second, the micro transitional pre- diction model would be used to obtain detailed speed predictions from which average space-mean speeds would be computed for each road profile. The final step would be to estimate a model specification with the road- way indices as independent variables and the average space-mean speed as the dependent variable. This approach has considerable appeal. In fact, it was the ba- sis of an aggregation procedure developed during the earlier phase of the Brazil-UNDP-World Bank research project. However, it was not adopted be- cause the underlying disaggregate speed prediction models were found to be unsatisfactory. (For details regarding this phase of the research project, see GEIPOT (1982, Volume 4)). The difficulties associated with this approach relate to the sampling and the estimation steps. Since the sampling units are entire road profiles, generating a wide-ranging sample of the population of road profiles is an overwhelming task. More important, in the estimation step, it is not easy to specify a model form that possesses the desirable properties of the micro models mentioned above. Thus, it was decided first to explore the simpler methods belonging to the integration 156 AGGREGATE SPEED PREDICTION MODEL approach. Since, as will be described shortly, a satisfactory method was found in that approach, the Monte Carlo approach was not followed through. However, as part of the overall validation strategy, this ap- proach was used to examine the structure of numerical errors introduced by various integration methods. The third approach to aggregation, the integration approach, conceptually treats road attributes as if they were random variables of known distribution. The required space-mean speed is predicted as the inverse of the average time per distance over the roadway. The latter, in turn, is predicted as the expectation of the time distance function over the distribution of the road attributes. The name of this approach derives from the fact that taking expectation of a function over the dis- tribution of its arguments amounts to mathematical integration. However, as will be seen, "mathematical integration" has only a conceptual conno- tation; this is because exact integration which produces a closed form solution can usually be accomplished only under highly restricted assump- tions and is therefore rarely feasible in practice. The integration methods that will be discussed subsequently are generally approximate methods of taking expectation. 7.2 AGGREGATE SPEED PREDICTION BASED ON INTEGRATION APPROACH Because of certain mathematical restrictions that will be elab- orated below, the integration approach has found its natural applications in the prediction of consumer demand (Theil, 1955; Green, 1964; Gupta, 1969; and Koppelman, 1975). For purposes of aggregate speed prediction, the integration approach has two main advantages. First, alternative methods are available which can be tailored to suit the needs of indi- vidual applications (Koppelman, 1975). And, second, since the structure of the original detailed model can be preserved, the resulting aggregate prediction model tends to retain the desirable properties of predictive accuracy, policy sensitivity, extrapolative ability, and local adapt- ability. The major disadvantage of the integration approach, however, is that it poses certain restrictions on the detailed prediction model. Specifically, the approach requires that: 1. The entity for which the aggregate prediction is made can be divided into a number of mutually exclusive units; and 2. The prediction made for each unit is dependent only on the attributes of the unit. In the case of aggregate consumer demand prediction, these re- quirements are generally satisfied. The population of consumers of in- terest may be divided into different market segments or even into units as small as households. The demand for each unit is normally affected only by the attributes of the consumers within the unit. (This restric- tion is violated when the demand of, say, a consumer is affected by that of another consumer as in the proverbial case of "keeping up with the Joneses.") AGGREGATE SPEED PREDICTION MODEL 157 In the case of aggregate speed prediction, the entity of inter- est, the roadway, may be divided into a number of homogeneous subsec- tions. However, the second of the requirements above can only be satis- fied by the micro non-transitional model, not the micro transitional one. The transitional effects in the latter method make the prediction of speed over a given subsection dependent not only on the attributes of the subsection but also on those of the adjacent subsections. In the mi- cro non-transitional model, with the transitional effects removed, the prediction of speed is dependent only on the attributes of the subsec- tion. Therefore, the micro non-transitional model must be used as the benchmark for testing the accuracy of candidate aggregation methods. This means that the prediction errors involved are entirely due to the numerical approximation in the aggregation methods, and have nothing to do with errors arising from ignoring the transitional effects. The lat- ter errors have to be dealt with separately, as is done in Chapter 8. Koppelman (op. cit.) identified four main alternative integra- tion methods: 1. Exhaustive enumeration method; 2. Classification-based method; 3. Naive method; and 4. Method of statistical differentials. For purposes of aggregate speed prediction, only the first two methods are directly relevant and are dealt with in this section in some detail. The latter two methods are discussed only briefly at the end of the section. Applying the schematic used by Koppelman (op. cit.), each meth- od involves three ingredients, as shown in Figure 7.2: 1. The distribution of road attributes, as represented by an array of vectors X., where X. is the vector of road attributes for homogeneous subsection s (as used in Chapter 5); 2. The steady-state speed model, denoted by VSS; and 3. The aggregation procedure, which operates on the first two ingredients to produce the required space-mean speed prediction. The exhaustive enumeration method uses all information given in the array of vectors Xs and amounts to the micro non-transitional model described in Chapter 5. That is, the time per distance over the roadway (r) is a weighted average by road length of the time per distance predictions for the individual homogeneous subsections, T = 1/VSS (Xs): I s L s L 158 AGGREGATE SPEED PREDICTION MODEL Figure 7.2: Schematic display of integration approach I I I I Distribution IsbuoSteady-state I of road attributes speed model I I I I I I I I I I I I II II Aggregation I procedure Aggregate speed prediction where Ls and L denote the lengths of homogeneous subsection s and the entire roadway, respectively (as given in Chapter 5). In travel demand prediction, Koppelman (op. cit.) found from numerical testing that use of full information on road attributes in ag- gregate prediction, as implied by the exhaustive enumeration method above, was generally not necessary. In fact, he found that use of limit information, if carried out judiciously, could produce aggregate predic- tions almost as accurate as predictions based on full information. The classification-based method was found to be the one that made the most efficient use of limited information. For this reason and also because of its simplicity, the method was chosen as the basis for developing the aggregate speed prediction model (presented in detail in the next section). The method may be briefly summarized by the following steps: AGGREGATE SPEED PREDICTION MODEL 159 1. Classify the subsection into two or more relatively homogeneous classes (e.g., of similar gradients, curvatures, etc.). 2. Compute the time per distance for each class on the basis of the average road attributes, weighted by road length, for the class. 3. Compute the time per distance for the entire roadway as a weighted average by length over all classes. Generally, the more classes employed, the more accurate would be the aggregate prediction but also the more unwieldy. Thus, the choice between the different classification schemes is a tradeoff between a gain in accuracy on the one hand and simplicity of the model and minimality of information requirements on the other. The classification scheme adopted was one that struck a reasonable balance between these opposing criteria. The starting point for exploring alternative classification schemes was to note the following salient properties of the steady-state speed model and their implications on classification: 1. The model is discontinuous with respect to surface type, as evident from Chapter 4, in that the estimated desired speeds and perceived friction ratios are markedly differ- ent between paved and unpaved surfaces. This suggested that separate classes were needed for the surface types. 2. When plotted against gradient (Figure 3.1), the speed curve is not a monotonically increasing or decreasing one, but is bell-shaped with the top of the bell to the left of the zero gradient. This suggested that a relatively large num- ber of classes might be needed for road gradient. 3. When plotted against curvature and roughness (Figures 3.2 and 3.4), speed is monotonically decreasing with respect to these variables. This suggested that a relatively small number of classes might be needed for curvature and rough- ness. (The behavior of speed with respect to supereleva- tion was not examined as the latter was considered to be well correlated with curvature.) The above considerations led to a numerical test of several candidate classification schemes using 10 actual road sections. The test yielded two major findings: 1. For a given surface type and two classes of road gradient, there was little gain in accuracy from classification by curvature and roughness. 2. The optimal number of classes for gradient was two. Very little gain in accuracy was obtained from four classes of gradient. 160 AGGREGATE SPEED PREDICTION MODEL The above numerical results led to the final classification scheme embedded in the aggregate speed prediction model described in the next section. This scheme may be summarized in physical terms as one of converting a heterogeneous roadway of given surface type into two ideal- ized subsections of uniform positive and negative gradients, each having the same curvature, superelevation and roughness which equal average at- tributes of the actual roadway. The other integration methods alluded to earlier, the "naive" method and the method of "statistical differentials," are worth a brief discussion. The naive method is the same as the classification-based method in every respect except that the entire roadway is treated as one class of homogeneous subsections, as opposed to two or more. This method is linear. While it may be useful in applications where the detailed prediction models approximate a linear function, the naive method was found to be unsuited to the structure of the steady-state speed model. Specifically, the naive method involves averaging across positive and negative gradients whereas the steady-state model, as mentioned earlier, treats positive and negative gradients differently. In the case of pre- dicting a round trip speed, the mean gradient would be zero by definition irrespective of the individual gradients of the subsections. Therefore, the naive method was rejected on a priori grounds. The method of statistical differentials could be regarded as an extension of the classification-based method in that, in addition to us- ing the means of road attributes, higher moments including the standard deviation and covariances of the attributes are also employed in aggre- gate prediction through Taylor's series expansion of the steady-state model (Koppelman, op. cit.). Compared to both the naive and classifica- tion-based methods, the method of statistical differentials is much more complicated and requires considerably more information if greater accu- racy is sought. Since classification using first moments was already found to be satisfactory, the method of statistical differentials was not pursued. The aggregation procedure adopted above is also applicable to other resources consumed in vehicle operation than "time per distance," in particular, fuel consumption and tire wear per distance, as described in Chapters 10 and 11, respectively. 7.3 THE AGGREGATE SPEED PREDICTION MDEL It is assumed that the roadway is of a uniform surface type. If necessary, it may be treated as two distinct roadways and predictions of aggregate speed may be obtained separately for the paved and unpaved portions of the roadway. In other words, classification with respect to surface type is taken for granted in the following discussion. A brief note towards the end indicates the procedure to follow if the only infor- mation available regarding the surface type is the proportion of paved road. It should be noted that there is a distinction between a given physical roadway and a traversed roadway. Depending on the direction of AGGREGATE SPEED PREDICTION MODEL 161 travel, three distinct traversed roadways can be identified for a given physical roadway. The distinction is important in defining the aggregate vertical geometric attributes for a traversed roadway. Specifically, given a stretch of road between two points, say, A and B, the three tra- versed roadways are: 1. The roadway traversed in travelling from A to B, 2. The roadway traversed in travelling from B to A, and 3. The roadway traversed in making the round-trip journey be- tween A and B in either order. Homogeneous subsections of the roadway between A and B which would have a positive grade in (1) will have a negative grade in (2), and vice versa. The roadway traversed in (3) is conceptually identical to a roadway which is twice in length and has the homogeneous subsections of both (1) and (2). A level subsection poses a slight problem in classifi- cation as a positive or negative grade and requires a special treatment, as will be dealt with below. The aggregation procedure to be described first is applicable to travel in direction A to B. It will be seen that the aggregated in- formation on road attributes obtained for the A-to-B journey is all that is needed for the B-to-A journey as well as for the round trip. Further, the aggregation procedure for the round trip employs data on road attri- butes at the same level of detail as that used in the HDM model. Consid- erable simplification results when predictions are desired for a round trip journey, as described below. Let L be the length of the traversed roadway; ST be the surface type of the roadway; and The roadway be divided into n homogenous subsections indexed by subscript s; For subsection (s), let GRs be the grade expressed as a fraction; Cs be the curvature, in degrees/km; SPS be the superelevation, expressed as a fraction; QIs be the roughness, in QI counts; and Ls be the length of the subsection, in m. Define the following five attributes for the traversed roadway: 1. Let "PGab" be the weighted average of positive grades. 162 AGGREGATE SPEED PREDICTION MODEL That is: I GR L GRi _ S _ S S PG - _____- ___ ab = L LP sab S where the summation is over all the subsections with posi- tive gradient in the direction of travel (i.e., with GRS > 0), LPab is the proportion of uphill travel, and is = Ls/L. 2. Let "NGab" be the weighted average of the absolute values of negative grades. That is, GRr' Ls GRs s JGRs is NG - _ _ _ _ _ - _ _ _ _ _ ab L LN 1 - LP s ab ab s where the summation is over all the subsections with a negative gradient in the direction of travel (i.e., with GRs < 0) and LNab is the proportion of downhill travel. It should be noted that NGab is a non-negative quantity. 3. Let "C" (average curvature) be the weighted arithmetic mean of the curvatures. That is, CL C C s C __ s Cs a L s where the summation is over all the subsections. Since straight subsections have a curvature value of 0 degree/km, the summation is effectively over the curvy subsections. 4. Let "SP" (average superelevation) be the weighted average of superelevations. That is, I SP L SP= = SP L L s where the summation is, as above, over the curvy subsections. AGGREGATE SPEED PREDICTION MODEL 163 5. Let "QI" be the weighted average of roughnesses. That is, SQI L QI QIQI 9. s s L s where the summation is over all subsections. Now use the micro non-transitional speed prediction model as described in Chapter 5, for a hypothetical roadway consisting of two homogeneous subsections as defined in the following array: A-to-B journey Homogenous Length Surface Gradient Curvature Super- Roughness subsection type elevation "Uphill" LPab ST PGab C SP QI "Downhill" 1-LPab ST -NG ab C SP QI To predict the space-mean speed for the journey in the reverse direction (B-to-A), we adopt a slight change in the way level subsections are treated. For the A-to-B journey, a level subsection (GRr = 0) has been classified as having a positive gradient. For the B-to-A journey we now classify it as having a negative gradient. With this convention adopted it is easy to see that the average positive and negative gra- dients as well as the proportion of uphill travel which have been evalu- ated above for the A-to-B journey are exactly reversed for the opposite travel direction, i.e., A-to-B journey B-to-A journey Average positive gradient PGab PBba ' NGab Average negative gradient NGab NGba = PGab Proportion of uphill travel LIPab LPba 1-LPab 164 AGGREGATE SPEED PREDICTION MODEL and the B-to-A journey speed would be computed based on the array of in- formation shown below: B-to-A journey Homogenous Length Surface Gradient Curvature Super- Roughness subsection type elevation "Uphill" 1-LPb ST NGab C SP QI "Downhill" LPab ST -PGab C SP QI As for the round trip, by way of symmetry it can be easily shown that: 1. The proportion of the uphill travel is always equal to one-half; and 2. The average positive and negative gradients are both equal to one-half of the sum of the corresponding quantities for the A-to-B journey. Letting, PNG = [PGab + NGabI/2 the space-mean speed for the round trip can be computed for the informa- tion array below: Round trip Homogenous Length Surface Gradient Curvature Super- Roughness subsection type elevation "Uphill" 1/2 ST PNG C SP QI "Downhill" 1/2 ST -PNG C SP QI AGGREGATE SPEED PREDICTION MODEL 165 We next discuss the way to handle the case where the roadway is a combination of paved and unpaved surface types and the other attributes are known for the roadway as a whole and not separately for the two portions. Let PP be proportion of paved portion of the roadway. For this case, set up the summary aggregate descriptor array consisting of four imaginary homogeneous subsections as follows: Homogenous Length Surface Gradient Curvature Super- Rough- subsection type elevation ness Paved uphill PP/2 Paved PGN C SP QI Paved downhill PP/2 Paved -PGN C SP QI Unpaved uphill 1-PP)/2 Unpaved PGN C SP QI Unpaved downhill (1-PP)/2 Unpaved -PGN C SP QI Now apply the micro non-transitional prediction model to an hypothetical roadway with the above four homogenous subsections. The aggregate descriptors of vertical and horizontal geometry for the round-trip travel, viz., PNG and C, respectively, are closely related to average rise plus fall, RF, and average horizontal curvature, which are the aggregate descriptors of the roadway used in the HDM-III. The following paragraphs define these concepts and indicate their interrelationship. The average rise plus fall, RF, for a round-trip travel on a roadway is defined as the sum of the absolute values, in meters, of all ascents and all descents along the roadway divided by the length of the roadway, in kilometers. The concept is illustrated in Figure 7.3(a). The average horizontal curvature of a roadway is defined as the sum of the absolute values of angular deviations (in degrees) of succes- sive tangent lines of the horizontal alignment along the roadway, divided by the length of the roadway (in kilometers). This concept is illustrat- ed in Figure 7.3(b). The average rise plus fall is identical to PNG, defined above, but for a factor of 1000. While PNG is a dimensionless quantity, RF is expressed in m/km. Thus, PGN = RF/1000 It was pointed out in Section 3.1 that for a homogeneous curve, the horizontal curve expresses both the angle subtended by it at the center as well as the angular deviation of the tangent lines bounding the curve. By extending the argument to all the curvy subsections of the roadway, it may be seen that the average curvature, C, as defined earlier and the average horizontal curvature as defined above are identical. 166 AGGREGATE SPEED PREDICTION MODEL Figure 7.3: Illustration of road rise plus fall and horizontal curvature a. Vertical profile of the road section AR F1 RF2R3B R1 + R2 + R3+ F + F2 (meters) Average rise plus fall, RF = L (km) b. Horizontal profile of the road section C, C3 A C2 Cl + C2 + C3 + C4 (degrees) Average horizontal curvature, C = L (km) AGGREGATE SPEED PREDICTION MODEL 167 APPENDIX 7A PROCEDURE FOR COMPUTING AVERAGE ROADWAY CHARACTERISTICS FROM DETAILED GEOMETRIC PROFILE It is assumed in the following discussion that the user starts with detailed vertical and horizontal geometric profiles of the roadway as typified by Figures 7A.1(a) and (b), respectively. Step I. Computation of vertical geometric aggregates Divide the roadway into subsections with crests and troughs as boundary points. Determine the lengths and average gradients (with signs retained) of the subsections and form the tabular profile of vertical geometry as shown in columns a, b and c below. The next five columns form a working table: (a) (b) (c) (d) (e) (f) (g) (h) Sub- Length Gradient Positive Negative section (meters) with sign gradient gradient (as a (as a (as a fraction) fraction) fraction) pgs= 0 ngs= 0 Pts= ns= s= 0 or or pgsts ngs s or a ks g pgs= g ngs= g ps= t 2 m Total L PL NL P The working table (defined by column) is used as follows: Column d Determine the "positive gradient" pgs of subsection s: If the gradient of subsection s is positive, i.e., gs > 0, then pg. = gs. If the gradient of the subsection is negative, i.e., g. < 0, then pg, = 0. 168 AGGREGATE SPEED PREDICTION MODEL Figure 7A.1: Vertical and horizontal geometric profiles (a) Vertical geometric profile direction of travel A gl4 B 1 12 ' 13 01 -1 14 1 15 1 Length (meters) gl Positive Grade g, Negative Grade (b) Horizontal geometric profile 4 _1 16l s -- - -16 A2aB - a Straight Subsection 7 Sl Curvy Subsection l I Length (meters) r, Radius of Curvature (meters) sl Superelevation AGGREGATE SPEED PREDICTION MODEL 169 Column e Determine the "negative gradient" ngs of the subsection s: If the gradient of subsection s is positive, then ngs = 0. If the gradient of the subsection is negative, then ngs 2 gs I, where gs is the absolute value of gs. Hence, both pgs and ng, are non-negative quantities, and one of them is necessarily zero for each subsection. Column f Multiply columns b and d to get pts: pes = pgs ts Column g Multiply columns b and e to get nts: nts = ng, it Column h This column chooses lengths of subsections with positive gradients. Enter the length t. of the subsection if the subsection has a positive gradient; enter zero if the subsection has a negative gradient. That is, is if gs > 0 ps 0f if gs < 0 Note that this column is not needed for round trip predictions. Finally, form the totals of columns b, f, g and h as L, PL, NL and P, respectively, where L is the length of the roadway in meters. Step II. Computation of horizontal geometric aggregates Divide the roadway into subsections with uniform curvature using the end points of curves as boundary points. Determine the lengths, curvatures and superelevations (if known) of the curvy subsections and form the tabular profile of horizontal geometry as shown in columns i through 1 below. The curvature of curvy subsection is given by (from Section 3.2): c = 180,000 s Tr rs The next two columns form a working table: (i (j) (k) (1) (m (n) Curvy Length Curvature Superelevetion subsection (meters) (deg/km) (as a fraction) s ts c s cts= c I st = s I 1 2 Total K S 170 AGGREGATE SPEED PREDICTION MODEL The working table (defined by column) is used as follows: Column a Multiply columns j and k to get cis: cis = C is Column n Multiply columns j and 1 to get sis: sk, = s is Finally, form the totals of columns m and n as K and S respectively. Step III. Computation of the average geometric characteristics F o r m u 1 a Average geometric Symbol One-way trip Round trip characteristics Forward Reverse direction direction Average positive PL NL PL + NL gradient PG P L - P L Average negative NL PL PL + NL gradient NG L - P P L Proportion of P L - P uphill travel LP L L 0.5 Average K K K curvature C L L L Average S S S superelevation SP L L L Note: The proportion of uphill travel, LP, is exactly 0.5 for a round trip because of symmetry (see Section 7.3). Illustrative example The example roadway is about 3.5 km long and has fairly extreme geometry. Aggregate geometric attributes are desired for a one-way trip, starting from point A and ending at point B. The vertical and horizontal geometric profiles of the roadway are shown on Figure 7A.2(a) and (b), respectively. In terms of vertical geometry, the roadway may be divided into five subsections of which three subsections have negative gradients and two subsections have positive gradients for the specified direction of travel. The lengths and gradients of the subsections are shown in Figure 7A.2(a) and summarized AGGREGATE SPEED PREDICTION MODEL 171 Figure 7A.2: Vertical and horizontal geometric profiles: example (a) Vertical Geometric Profile direction of travel 044 --3 ... -2 044- - - - B 1,300 1 450 Wf -400 P 4 600 01- 670 11 Length (meters) Positive Grade g, Negative Grade (b) Horizontal Geometric Profile 180 4 170 150 150 2B0 300 500 ' -- 04 032 ' 200 350 . B A- - - - - - - - - Straight Subsection 1 S1 Curvy Subsection I1 Length (meters) fl Radius of Curvature (meters) sl Superelevation 172 AGGREGATE SPEED PREDICTION MODEL Table 7A.1: Working table for vertical geometry (a) (b) (c) (d) (e) (f) (g) (h) Sub- Length Gradient Positive Negative section (meters) with sign gradient gradient 8 9 g pgs= 0 ng = 0 pt = nR ps= 0 s s s s s a or or pgs is ngs s or Pg= gs ngs= s s= I 1 1,300 -0.042 0 0.042 0 54.60 0 2 450 0.044 0.044 0 19.80 0 450 3 400 -0.044 0 0.044 0 17.60 0 4 600 0.037 0.037 0 22.20 0 600 5 670 -0.064 0 0.064 0 42.88 0 Total 3,420 42.00 115.08 1,050 in Table 7A.1. In terms of horizontal geometry, the roadway may be divided into 14 subsections of which seven are curved. The lengths, curvatures and superelevations of the curvy subsections are shown in figure 7A.2(b) and summarized in Table 7A.2. The working tables (see Tables 7A.1 and 7A.2) are then formed following the instructions, and the average geometric characteristics, computed. Since predictions for a one-way trip are desired, the computations are as follows: Table 7A.2: Working table for horizontal geometry (i) (j) (k) (1) (m) (n) Curvy Length Curvature Superelevation subsection (meters) (deg/km) (as a fraction) s Ac s c = c . st =s~ t~ s is cs sss s s s s s a is 1 240 254.78 0.037 61,147 8.93 2 280 286.62 0.040 80,254 11.20 3 350 191.08 0.032 66,878 11.08 4 180 382.17 0.048 68,791 8.70 5 150 286.62 0.040 42,993 6.00 6 170 95.54 0.023 16,242 3.97 7 220 458.60 0.055 100,892 12.10 Total 437,197 61.98 AGGREGATE SPEED PREDICTION MODEL 173 Average positive gradient, PG = 42.00 = 0.040 1050 Average negative gradient, NG = 115.08 = 0.049 2370 Proportion of uphill travel, LP 1050 = 0.307 3420 437197 Average curvature, C - = 127.835 3420 Average superelevation, SP 61.98 = 0.018 CHAPTER 8 Validation of Speed Prediction Models The development of a range of speed prediction models at different levels of complexity has the advantage that the prediction errors introduced in these models, defined as the difference between observed and predicted speeds, can be traced through various steps of simplification. This permits the different sources of errors to be examined separately, thereby forming a basis for conducting the validation exercise of the speed prediction model reported herein. Section 8.1 discusses the nature of the sources of errors, followed by Section 8.2 which gives an overview of the experimental and user survey data employed in the validation. The main validation results are presented in Sections 8.3 - 8.5, with Section 8.3 dealing with the micro models, and Sections 8.4 and 8.5 with the aggregate model. 8.1 SOURCES OF PREDICTION ERRORS As depicted in Figure 8.1, the three steps of model simplifi- cation introduce four sources of errors relating to random sampling, model specifications, transitional effects and numerical approximation. 1. Micro transitional model. The most elaborate of the three, the micro transitional model is considered on a priori grounds to be the most accurate or closest to the "truth." Its accuracy can be assessed by comparison with validation data, i.e., speed observations obtained independently from those used in developing or "calibrating" the prediction models. Granted that the validation data were from the same population of vehicles and roads as the calibration data, would expect prediction errors to arise primarily from two sources: random sampling and model mis-specifi- cations. Random sampling of speed observations is the only source of prediction error that has nothing to do with the accuracy of the prediction models themselves, but rather with our inability to observe the "truth." If observable, the "truth" would clearly be the ideal absolute reference for measuring the prediction accuracy. For a given vehicle class and road characteristics, we define the truth," in the sense of classical statistics, as the mean of the means of speed observations at road sites of the same characteristics. According to this definition, the "truth" is not observable, for to do so would take an infinitely large number of sites. Therefore, we must be content with something less than ideal to serve as the 175 176 VALIDATION OF SPEED PREDICTION MODELS Figure 8.1: Sources of speed prediction errors Aggregate model , Numerical approximation errors Micro non-transitional model . Errors due to ignoring 4 transitional effects Micro transitional model . Random sampling errors Errors due to model mis-specifications Validation data (actual observations) reference. The definition of the "truth" stated above suggests that our second-best reference should be the average of averages of individual speeds at as many sites as possible. This argument may be pursued further by examining the results of the estimation steady-state speed model in Chapter 4. As illustrated in Table 4.4, for heavy trucks the average prediction errors for various groups of roads of similar characteristics are extremely small -- in the range of 0.03 - 1.5 km/h in magnitude. If the steady- state speed model were perfect, in the sense that it could exactly replicate the "truth" without introducing any systematic bias, then this magnitude of error would be entirely attributable to random sampling. As the model is not perfect, the magnitude of random sampling error is even smaller than the values computed. The range of 24-192 section-directions in these groups demonstrates that, given a sufficiently large sample size within a practical limit, it is possible to reduce the random sampling error to an acceptable level. The above example is contrasted with Figure 4.1, also for heavy trucks, in which observed and predicted speeds are plotted and the regression between these speeds is shown to have a standard error of 7.0 km/h. The standard error of prediction due to random sampling of sites equasl 16.1 percent of the predicted speed (from as = 0.161). For a typical predicted speed of 50 km/h, this standard error becomes 0.161 x 50 = 8.05 km/h, which, as expected, is similar to the above standard error of regression residuals. If we use say, 25 sites for developing validation data, the random sampling standard error would be reduced to 8.0 + /100 = 0.80 km/h. These VALIDATION OF SPEED PREDICTION MODELS 177 standard errors are similar in magnitude to the average prediction errors for groups of roads shown in Table 4.4. Following the results in Chapters 3 and 4, if an observed speed is taken at random from a heavy truck arriving at a randomly selected site, the standard error of prediction due to this random sampling process is equal to 25.0 percent (from a = 0.250) or 0.250 x 50 = 12.5 km/h for 50 km/h prediction. One hundred random vehicles sampled in this manner would reduce the standard error to 25.0 V 100 - 2.5 percent. The upshot of the above discussion is that, since the "truth" is not observable, our ability to validate the prediction models depends crucially on our ability to obtain a sufficiently large amount of independent data. It is not possible to do validation simply on the basis either of 'individual speed observations at a site or of averages of observations at a few sites. Systematic prediction biases can be gleaned only by comparing the model predictions with observed speeds obtained from a sufficiently large number of vehicles and sites. The second source of prediction erors, model mis-specifi- cations, arise if the micro transitional model has not been properly specified so that the effects of vehicle and road characteristics are not captured fully in the steady- state speed relationship and the transitional speed logic. Although the micro transitional model is quite comprehensive in its attempt to simulate a highly complex real-world phenomenon, like most other behavioral models, it still is to some degree an idealization of reality. Thus, the question is not whether it has been correctly specified but really how close it is in approximating reality. Therefore, a good validation test would be to see how good the predictions are over a wide range of road and vehicle conditions. 2. Micro non-transitional model. The next step of simplifi- cation, from the micro transition to micro non-transition model, introduces a clearly identifiable source of error: the omission of transitional effects. The magnitude of this type of error can be determined by comparing the predictions produced by the two models. 3. Aggregate model. The last step of simplification, from the micro non-transitional to the aggregate model, involves no change in the treatment of transitional effects since neither model makes any attempt to simulate vehicle speed transitions. The only source of discrepancy between these models lies in the degree of numerical approximation of the geometric alignment of the road: the micro non-transitional model employs a large number of short, relatively homogeneous subsections whereas the 178 VALIDATION OF SPEED PREDICTION MODELS aggregate model employs only two "average" subsections. In fact, the aggregate model as defined in Chapter 7 represents only one level of aggregation --- a rather extreme one. Other levels of aggregation are possible. For example, an intermediate one in which four or more "average" sub- sections are used to distinguish between gentle and sharp curves. At any level of aggregation the associated errors can be assessed by comparing aggregate predictions with predictions produced by the micro non-transitional model. 8.2 VALIDATION DATA Two validation exercises were conducted, one primary and one supplementary. The primary exercise employed average speeds by vehicle classes observed at a range of test sections; the supplementary exercise employed speed data obtained from the user cost survey (GEIPOT, 1982, Volume 5). The primary exercise tested all three prediction models and used speed data from two paved and four unpaved sections (of 2-4 km lengths). Detailed information on the vertical and horizontal alignment of the road stretches was obtained from an engineering survey and the road roughness was measured using the standard Opala-Maysmeter vehicle (as alluded to earlier in Chapter 3). The average gradient of these sections varied in the range 1-5 percent and surface characteristics in the range smooth paved to medium-rough unpaved (QI = 26 - 111). The minimum radius of curvature surveyed was 100 meters. Table 8.1 provides summary characteristics of these test sections. The space-mean speeds of vehicles traversing each section were obtained from stopwatch readings at several designated stations along the stretch. Predictions were obtained using the same vehicle characteristics as for the speed model estimation (Table 4.2a), and parameter values as originally estimated (Table 4.3). Because of the large number and variety of vehicles and also the wide range of speeds observed, the data set for the primary validation exercise is considered to be fairly encompassing. However, there are two main drawbacks. First, because the sections are relatively few in number and also short in length, the resulting random sampling errors are considered to be larger than desirable. Second, although some of the sections tested contain steep hills, none can be considered to be very rough or of severe geometric alignment, especially with respect to horizontal curvature. Besides the six test sections listed in Table 8.1, five more actual road sections of 10 km length were selected for testing the micro non-transitional and aggregate models. Summary characteristics of these sections are compiled in Table 8.2. The addition of these sections extended the range of geometric alignment for testing purposes. Although an engineering survey was used to obtain detailed information on road geometry, no speed observations of population vehicles were made. Therefore, these additional sections were used only for comparison VALIDATION OF SPEED PREDICTION MODELS 179 Table 8.1: Summary characteristics of test sections (with speed observa- tions) used in primary validation of speed prediction models SECTI 0 N NUMBER General description 1 2 3 4 5 6 514 558 564 565 566 584 One-way length (km) 2.00 4.20 2.87 3.13 3.30 2.90 Number of homogenous 4 9 22 15 15 6 subsections Gradients (%) Maximum absolute 0.2 1.7 6.0 6.0 6.9 3.6 Average of positive 0.10 0.65 1.55 3.36 2.71 1.21 Average of negative 0.13 0.45 3.04 1.92 2.36 1.63 Length of positive 35.0 50.0 44.4 60.4 50.0 50.3 RF (m/km) 1.2 5.5 24.6 27.9 25.3 14.2 Number of curvy 2 0 10 2 3 0 subsections Minimum radius of 2929.0 101.0 603.0 1130.0 curvature (m) Maximum superele- 3.0 0.0 11.2 3.0 3.0 0.0 vation (%) Radius of curvature 5230.4 m 353.9 2773.3 7488.0 0 (m) Superelevation (%) 16.8 0.0 5.35 0.65 0.45 0.0 Horizontal curvature 11.0 0.0 162.0 20.7 0.0 5.7 (degrees/km) Surface type double- gravel gravel gravel double- gravel treated treated Roughness (QI) 26.0 110.9 103.4 76.0 73.0 80.8 Note: The aggregate descriptors of vertical geometry are for a one-way travel in an arbitrarily chosen direction. Source: The Brazil-UNDP-World Bank highway research project data. between the speed prediction models.1 The supplementary validation exercise used speed data obtained from tachographs of individual journeys of interstate buses. Altogether, tachographs were obtained from 11 companies operating over 41 routes with characteristics as summarized in Table 8.3, along with average observed speeds. Relative to the primary validation data set, this data set has the advantage that the bus routes are relatively long (averaging 116 km one-way), and some are very rough (up to 200 QI roughness). However, as 1 In fact, these sections were used mainly in the fuel consumption experiments for calibration and validation for the fuel prediction methods. This is dealt with in Chapter 10. 180 VALIDATION OF SPEED PREDICTION MODELS Table 8.2: Summary characteristics of test sections (without speed observations) used in primary validation of speed prediction models SECTI 0 N NUMBER General description 1 2 3 4 5 559 560 561 562 563 Paved/ Paved/ Unpaved/ Unpaved/ Unpaved/ rolling/ flat/ flat/ hilly rolling straight straight straight One-way length (km) 10.0 10.0 10.0 10.0 10.0 Number of homogenous 52 27 40 58 72 subsections Gradients (%) Maximum absolute 6.0 2.0 6.6 6.5 6.8 Average of positive 0.79 0.86 0.60 4.86 8.40 Average of negative 2.61 3.00 6.30 0.38 2.42 Length of positive 25.3 68.9 41.0 89.0 38.5 RF (m/km) 22.2 6.5 6.2 42.9 19.1 Number of curvy 7 2 1 23 37 subsections Minimum radius of 491.0 80.0 702.0 72.0 185.0 curvature (m) Maximum superelevation 4.0 3.0 2.0 12.0 4.6 Radius of curvature 3637.0 4311.2 10000.0 535.7 565.0 Superelevation (%) 0.54 0.08 0.0 2.50 2.41 Horizontal curvature 15.8 13.3 3.1 107.0 101.5 (degrees/km) Surface type double- double- gravel gravel gravel treated treated Roughness (QI) 26.0 30.0 86.4 63.7 124.9 Note: The aggregate descriptors of vertical geometry are for a one-way travel in an arbitrarily chosen direction. Source: Brazil-UNDP-World Bank highway research project data. in the primary data set, these routes are only of gentle to moderate geometric alignment. Another shortcoming is that although most bus routes had a number of stops to pick up and drop off passengers, the average stop time was not known and had to be estimated in order to compute the total driving time. This proved to be a major source of discrepancy between the observed and predicted speeds, as reported in Section 8.5 below. VALIDATION OF SPEED PREDICTION MODELS 181 Table 8.3: Summary characteristics of interstate bus routes used in validation of aggregate speed prediction model Standard CV Characteristics Mean Minimum Maximum dev. (%) Rise plus fall (m/km) 26.76 14.00 43.00 6.24 23.32 Horizontal curvature 46.12 9.00 189.00 43.32 93.92 (degrees/km) Roughness (QI) 67.07 24.00 196.00 46.69 69.61 Proportion of the route 67.59 0.00 100.00 44.64 66.04 that is paved (%) Round trip route length 231.88 44.00 705.00 153.83 66.34 (km) Number of stops 7.49 2 57 11.92 68.18 Number of stops per 12.16 0.85 103.64 16.87 138.75 100 km Observed space-mean speed 55.88 33.00 73.00 10.46 18.79 (km/h) Source: Brazil-UNDP-World Bank highway research project data. For road lengths of several hundred kilometers, as in the case of many of the above bus routes, the amount of data preparation and computation involved in using the micro speed prediction models is extremely large. Therefore, these tachograph bus speeds were used to test only the aggregate model. 8.3 VALIDATION OF MICRO MIELS For the micro transitional model the observed speeds were taken as the benchmark for comparison. For each vehicle class, travel direction and, for trucks only, load level, the average observed space-mean speeds were regressed against the predicted space-mean speeds obtained from the speed profile simulation over the test sections presented in Table 8.1. Using ordinary least-squares analysis, performed with and without an intercept, the regressions yielded results as summarized in Table 8.4, along with the results based on all data points pooled together. Figure 8.2 plots the observed against predicted speeds with the data points distinguishable by vehicle class. 182 VALIDATION OF SPEED PREDICTION MODELS Table 8.4: Regression of observed space-mean speeds (a/a) versus those predicted by the micro transitional model Number Standard of error of Vehicle class observations Intercept Slope R2 residuals Car 12 4.98 0.76 0.70 (1.5) (4.9) 1.30 - 0.99 0.64 1.37 (53.1) Utility 12 2.33 0.88 0.68 1.36 (0.6) (4.7) - 1.01 0.67 1.33 (49.9) Bus 11 8.17 0.63 0.47 2.08 (2.1) (2.8) - 1.09 0.22 2.40 (26.5) Light/medium truck 24 1.53 0.89 0.73 1.55 (0.8) (7.7) - 0.98 0.72 1.53 (51.4) Heavy truck 16 0.23 0.96 0.83 1.66 (0.1) (8.3) - 0.98 0.83 1.61 (37.4) Articulated truck 5 4.34 0.76 0.48 3.30 (0.8) (0.46) - 1.11 0.47 3.13 All vehicle 80 1.94 0.899 0.78 1.81 classes combined (2.1) (16.6) - 1.008 0.77 1.84 (84.9) Source: Analysis of Brazil-UNDP-World Bank highway research project data. VALIDATION OF SPEED PREDICTION MODELS 183 The R2 values in Table 8.4 and also in subsequent tables should be interpreted with care. In cases where observations are few and very similar the R2 values could be quite low. In Table 8.4, the with-intercept regression has R2 values above 68 percent for a majority of the vehicle classes. The pooled regression has an R2 value of 78 percent. The R2 values for two of the vehicle classes, namely, buses and articulated trucks, are somewhat lower. For articulated trucks, the low R2 value (48 percent) is explained by there being very few (five) observations. For buses the reasons are unclear but the relatively narrow range of the observed speeds could have been a contributing factor. Because of the relatively small number of data points (5 to 24) for each regression by vehicle class, the with-intercept regressions have relatively large, although generally insignificant, intercepts and slopes which markedly differ from unity, the value that would indicate bias-free predictions. The with-intercept regression in which all vehicle classes were combined into 80 data points, produced a low intercept (1.94 m/s or 7 km/h) and a slope close to one. The standard errors of residuals of these regressions fall within a 4-12 km/h range which is broadly comparable to standard errors of typical speed predictions at the section-direction level using the steady-state speed models estimated in Chapter 4. Following the discussion in Section 8.1 above, we may ascribe the departure of these intercepts and slopes from their ideal (zero and one, respectively), as well as most of the standard errors of regression, to errors resulting from random sampling and model mis-specifications. With the intercept removed, the slopes of all regressions by vehicle class in Table 8.4 are much closer to one (0.98 to 1.11). Most noteworthy is the slope of the pooled regression (1.008) which, according to a t-test, is not significantly different from unity at the 95 percent confidence level. In fact, the relatively tight confidence interval of the estimated slope of [0.984, 1.031] suggests that on average the simulation logic mimics reality very well. For the micro non-transitional model, two comparisons were made employing the simulation-predicted speeds and observed speeds as the benchmarks, respectively. The first comparison was to assess the nature and magnitude of the prediction errors resulting from ignoring the transitional effects and the second to evaluate the overall predictive performance of the model. The procedure used was basically the same as that for the micro transitional model, i.e., regression of the benchmark speeds against speeds predicted by the micro non-transitional model (for the six test sections presented in Table 8.1), with and without the intercept. The results of the first comparison are compiled in Table 8.5 and Figure 8.3. Judging from the regression statistics it can be seen that the predictions produced under the steady-state speed assumption are remarkably close to those produced from full-fledged simulation. All without-intercept regressions yielded slopes slightly greater than one (1.02-1.04). In particular, the slope of 1.03 from the regression with all data points is significantly different from one, suggesting that the 184 VALIDATION OF SPEED PREDICTION MODELS Figure 8.2: The plot of observed space-mean speeds versus those predicted by micro transitional model Observed speed (Wl.) Sl 25 20 + ~x 15 101 0 51 >y 0 5 10 15 20 25 S0 Simulation predicted speed (als) Legend: X X Cr- + + Utility A A 6 Bo z 0 0 Truck -- - egreast on-line Source: Analysis of Brazil-UNDP-World Bank highway research project data. Figure 8.3: The plot of space-mean speeds predicted by micro transitional model versus those by micro non-transitional model Simulation predicted speed (s1a) 30 25 10 Simulation predicted speed (a/e) Leend: Car ++utility Bus OU Truck Regreeston line Source: Analysis of Brazil-UNDP-World Bank highway research project data. VALIDATION OF SPEED PREDICTION MODELS 185 Table 8.5: Regression of simulated space-mean speeds (m/s) computed by by the micro transitional model, versus those by the micro non-transitional model Number Standard of error of Vehicle class observations Intercept Slope R2 residuals Car 26 -0.03 1.02 0.93 0.73 (-0.0) (17.8) 1.02 0.93 0.72 - (155.1) Utility 29 1.47 0.94 0.94 0.58 (1.8) (21.5) 1.02 0.94 0.60 - (173.0) Bus 31 0.78 0.99 0.95 0.76 (1.1) (24.2) 1.04 0.95 0.76 - (127.2) Light/medium truck 40 0.96 0.98 0.96 0.63 (1.9) (32.5) 1.03 0.95 0.64 - (185.6) Heavy truck 54 1.12 0.97 0.96 0.73 (2.7) (35.6) 1.04 0.96 0.77 - (149.2) Articulated truck 41 0.45 0.99 0.98 0.61 (1.6) (48.3) 1.02 0.98 0.62 - (148.8) All vehicle 230 0.73 0.986 0.97 0.68 clases combined (4.1) (92.3) 1.029 0.97 0.70 - (374.7) Source: Analysis of Brazil-UNDP-World Bank highway research project data. 186 VALIDATION OF SPEED PREDICTION MODELS suppression of transitional effects resulted in a slight downward bias of 3 percent on average. Table 8.6 and Figure 8.4 present the results of the second comparison. Except for the steeper regression slopes, these results are very similar to those in Table 8.4 and Figure 8.2 for the observed versus micro transitional comparison. This indicates that barring the small downward bias the non-transitional model predicts almost as well as the transitional counterpart over the range of geometric alignment tested. 8.4 VALIDATION OF AGGREGATE MODEL BASED ON TEST SECTIONS This section presents results of validating the aggregate model against three benchmarks: the micro non-transitional and transitional models and the observed speeds. The comparisons employed the same regression procedure as for the micro models, i.e., benchmark values versus to-be-tested predictions. In this exercise all three models were used to predict speeds for each direction of the 11 test sections shown in Tables 8.1 and 8.2 by vehicle class and, for trucks, load level. As speed observations were obtained from the first six sections (Table 8.1), only these sections were employed in the third comparison. The results of the first comparison, aggregate versus non- transitional, are compiled in Table 8.7 and Figure 8.5. From the regression statistics it can be seen that predictions by the two models are very close. The without-intercept regression slopes of 0.92 - 0.97 indicate the tendency of the aggregate model to produce an upward bias, with an average value of 2 percent based on the slope of 0.982 from the pooled regression. Comparing the standard errors of residuals in Tables 8.7 and 8.5 we can see that, with the exception of cars and utilities, the "random" errors in numerical approximation arising from the aggregate procedure are somewhat larger in magnitude than those resulting from omitting transitional speed effects. It is worth noting that the biases due to these sources of errors are in opposite directions. The second comparison, aggregate vs. micro transitional, yielded results compiled in Table 8.8 and Figure 8.6 The slopes of the regressions in Table 8.8 (0.95-1.05) are higher than those of the regressions in Table 8.7 (0.92-0.97). This is apparently owing to cancellation between the upward bias caused by the aggregation procedure and the downward bias caused by ignoring transitional effects. The net effect is that, within the range of road alignment in the sample of test sections, the aggregate model produces an average underprediction of about 1 percent relative to the micro transitional model (based on the slope of 1.01 from the without-intercept regression including all data points). Another point worth mentioning is that the statistics in Table 8.8 indicate even a better regression fit than those in Table 8.7. This implies that the cancellation effect of the two error sources even extends into the "random" errors. Table 8.9 and Figure 8.7 present results from the third and final comparison: aggregate versus observed. Comparing these results VALIDATION OF SPEED PREDICTION MODELS 187 Table 8.6: Regression of observed space-mean speeds (=/s) versus those computed by the micro non-transitional model Number Standard of error of Vehicle class observations Intercept Slope R2 residuals Car 12 3.49 0.85 0.77 1.15 (1.1) (5.7) - 1.02 0.74 1.17 (62.5) Utility 12 2.79 0.89 0.67 1.39 (0.8) (4.5) - 1.04 0.65 1.37 (48.4) Bus 11 9.30 0.59 0.49 2.04 (2.7) (3.0) - 1.14 0.07 2.62 (24.2) Light/medium truck 24 3.28 0.81 0.71 1.60 (1.9) (7.3) - 1.02 0.66 1.68 (46.8) Heavy truck 16 1.77 0.90 0.79 1.87 (1.0) (7.2) - 1.02 0.77 1.86 (32.3) Articulated truck 5 3.41 0.87 0.52 3.16 (0.6) (1.8) - 1.15 0.46 2.90 (10.4) All vehicle combined 80 2.89 0.875 0.77 1.84 (3.2) (16.2) - 1.042 0.74 1.95 (80.3) Source: Analysis of Brazil-UNDP-World Bank highway research project data. 188 VALIDATION OF SPEED PREDICTION MODELS Table 8.7: Regression of the space-mean speeds (m/s) predicted by the micro non-transitional model versus those by the aggregate model Number Standard of error of Vehicle class observations Intercept Slope R2 residuals Car 26 -0.09 0.98 0.95 0.56 (-0.1) (21.0) 0.98 0.95 0.58 - (188.2) Utility 29 0.63 0.94 0.97 0.44 (1.0 (29.6) 0.97 0.97 0.44 - (232.6) Bus 31 0.85 0.97 0.95 0.78 (1.2) (23.3) 1.02 0.95 0.79 - (119.4) Light/medium truck 49 0.64 0.95 0.89 1.01 (0.8) (19.5) 0.99 0.89 1.00 - (115.0) Heavy truck 54 0.25 0.95 0.81 1.62 (.03) (15.0) 0.97 0.81 1.60 - (68.7) Articulated truck 41 0.10 0.97 0.86 1.78 (0.1) (15.6) 0.98 0.86 1.75 - (51.5) All vehicle classes 230 0.35 0.962 0.91 1.25 combined (1.0) (48.5) 0.982 0.91 1.25 - (203.7) Source: Analysis of Brazil-UNDP-World Bank highway research project data. VALIDATION OF SPEED PREDICTION MODELS 189 Figure 8.4: The plot of observed space-wean speeds versus those predicted by micro wan-transitional model Observed speed (al.) Sl 30- as x xx + 20 x A a~e< ' cP AD ' 15 DIO 100 D 0 5 10 15 20 25 30 Steady-State predicted speed (aWe) Leged: < X XX Car + + + Utility AAA Bus 0 0 Truck -- egression line Source: Analysis of Brazil-UNDP-World Bank highway research project data. Figure 8.5: The plot of space-ean speeds predicted by micro non- transitional model versus those by aggregate model Steady-state predicted speed (e/a) 30 25 c1) X 20 CP+ O 5 15 D 0 CP 0 C 0 5 10 15 20 25 30 Aggregate predicted speed (als) Legentd C XXXCar + + + Utility a 6 A Sn s] 0 0 Truck - RegresLon line Source: Analysis of Brazil-UNDP-World Bank highway research project data. 190 VALIDATION OF SPEED PREDICTION MODELS Table 8.8: Results of regression of space-mean speeds (als) predicted by the micro transitional model versus those by the aggregate model Number Standard of error of Vehicle class observations Intercept Slope R2 residuals Car 26 -0.74 1.03 0.93 0.72 (-0.6) (18.2) 1.00 0.93 0.71 - (157.2) Utility 29 1.90 0.89 0.94 0.62 (2.2) (19.7) 0.99 0.92 0.66 - (156.0) Bus 31 1.55 0.96 0.91 1.04 (1.7) (17.3) 1.05 0.90 1.07 - (90.4) Light/medium truck 49 1.23 0.95 0.89 0.99 (1.5) (19.7) 1.02 0.89 1.01 - (118.3) Heavy truck 54 1.24 0.93 0.79 1.67 (1.2) (14.1) 1.01 0.79 1.68 - (68.2) Articulated truck 41 0.50 0.97 0.86 1.81 (0.6) (15.2) 1.00 0.86 1.80 - (51.3) All vehicle classes 230 0.97 0.955 0.90 1.33 combined (2.7) (45.2) 1.010 0.90 1.35 - (194.2) Source: Analysis of Brazil-UNDP-World Bank highway research project data. VALIDATION OF SPEED PREDICTION MODELS 191 Table 8.9: Regression of observed space-mean speeds (m/s) versus those predicted by the aggregate model Number Standard of error of Vehicle class observations Intercept Slope R2 residuals Small car 12 3.88 0.83 0.72 1.26 (1.2) (5.1) - 1.01 0.69 1.28 (57.2) Utility 12 3.58 0.82 0.67 1.40 (1.0) (4.5) - 1.01 0.63 1.41 (47.1) Large bus 11 8.74 0.66 0.47 2.08 (2.3) (2.8) - 1.20 0.15 2.51 (25.4) Light/medium truck 24 3.05 0.82 0.72 1.57 (1.8) (7.5) - 1.01 0.68 1.64 (48.0) Heavy truck 16 1.65 0.90 0.77 1.92 (0.9) (6.9) - 1.01 0.76 1.91 (31.5) Articulated truck 5 3.80 0.83 0.50 3.24 (0.7) (1.7) - 1.14 0.42 3.01 (10.0) All vehicle classes 80 3.27 0.848 0.73 1.99 combined (3.3) (14.6) - 1.037 0.69 2.12 Source: Analysis of Brazil-UNDP-World Bank highway research project data. 192 VALIDATION OF SPEED PREDICTION MODELS Figure 8.6: The plot of space-sn speeds predicted by micro transitional model versus those by aggregate model Simlation predicted speed as) 25 O 20 X 00 10 0 O S2 5 0 5 Ic is 20 25 30 Aggrgate predicted speed (M/s) Legend: XXX Car + + + Util ty BAABus L] c Truck - -Regression line Source: Analysis of Brazil-UNDP-World Bank highway research project data. Figures 8.7: The plot of observed space-mean speeds versus those predicted by aggregate model Observed speed (.1.) 30 25- x 10< A A 0 A 104 Cl 0 5 10 15 20 25 30 Aggregte, predicted speed (We) Legend: X X X Car + + + U tility AAA Bus O D Truck ---Regresion line Source: Analysis of Brazil-UNDP-World Bank highway research project data. VALIDATION OF SPEED PREDICTION MODELS 193 with those in Table 8.4 we can see that for the road sections tested the much simpler aggregate model performs almost as well as the full-fledged speed profile simulation. The average overprediction of the aggregate model is less than 3 percent (based on the slope of 0.975 for the pooled regression, without the intercept) compared to the virtually bias-free predictions generated by the micro transitional model. The standard errors of residuals for the aggregate model are slightly larger than those for the micro transitional counterpart. 8.5 VALIDATION OF AGGREGATE MDEL BASED ON USER COST SURVEY The observed speed of each of the 41 bus routes was computed as the average of the round trip space-mean speeds obtained from one or more tachographs available for the route. Each tachograph provided a time-distance log of a complete bus tour. A difficulty encountered was that the 5-minute divisions on the tachographs did not provide information on the amount of time buses spent at stops which was generally well below 5 minutes. Therefore, the latter had to be estimated and the value of 40 seconds per stop was adopted. The predicted speeds were computed based on the aggregate procedure described in Chapter 7. The values of the total vehicle weight (10,400 kg), aerodynamic drag coefficient (0.65) and projected frontal area (6.30 m2) are the same as those used in the steady-state speed model for buses (Table 4.1). These values are considered to be fairly representative of the Brazilian interstate bus population. For each route that had both paved and unpaved portions, speeds were first computed separately for those portions, and then the weighted harmonic mean of these speeds was taken as the predicted speed for the entire route. The predicted and observed speeds and prediction errors are presented in Table 8.10 along with the characteristics of the routes. It is seen than the predicted speeds are higher than the observed speeds on 31 of the 41 routes. No effect of road geometry, surface type or roughness was found in the error trends. However, a further examination revealed that the number of stops per unit distance had a strong influence on the prediction errors. Table 8.11 shows mean prediction errors for 6 groupings of the routes by the number of stops per 100 km. It is evident that the magnitude of overprediction increases sharply with the number of stops per unit distance. For the two groups of routes with fewer than 10 stops per 100 km, the average amount of overprediction is smaller than 2 km/h. Figure 8.8 presents a plot of the observed against predicted speeds for these groups of routes. It can be seen that the predicted speeds track the trend of the actual speeds closely. Ordinary least-squares regression of the observed against the predicted speed with the intercept constrained to zero produced a slope of 0.97 as plotted in Figure 8.8. The standard error of residual of 4.8 km/h is about half the standard error of 8.6 km/h for 55 km/h prediction using the steady-state model for buses. This difference could be attributable to the fact that the steady-state model was estimated on the basis of spot speeds measured at short sections, whereas the observed bus speeds were made over long routes, most of which were over 100 km in round trip length (Table 8.10). 194 VALIDATION OF SPEED PREDICTION MODELS Table 8.10: Route characteristics and observed and predicted speeds Speed (km/h) Paved proportion Round-trip No.of Difference RF C Roughness of the No.of route stops observed/ (m/km) (deg/km) (QI) route (%) stops length(km) per 100km Observed Predicted predicted 1 32 24 25 100 5 44 11 52 64.38 -12.38 2 33 13 41 100 27 184 15 63 63.43 -0.43 3 34 22 85 4 30 150 20 47 53.90 -6.90 4 22 21 58 88 57 55 104 56 67.27 -11.27 5 24 11 29 100 16 196 8 68 69.00 -1.00 6 22 10 37 100 6 50 12 52 69.82 -17.82 7 28 10 31 100 8 57 14 69 66.70 2.30 8 24 12 24 100 11 115 10 67 69.11 -2.11 9 27 9 32 100 20 197 10 66 67.25 -1.25 10 30 74 39 100 12 262 5 60 64.39 -4.39 11 34 58 35 100 2 170 1 60 62.51 -2.51 12 25 109 41 100 14 455 3 63 66.14 -3.14 13 18 155 47 100 8 190 4 59 67.63 -8.63 14 14 189 45 100 14 133 11 53 67.90 -14.90 15 29 16 44 88 12 415 3 61 64.55 -3.55 16 29 16 44 88 14 415 3 62 64.55 -2.55 17 31 25 37 92 16 512 3 71 63.95 7.05 18 31 78 40 100 34 444 8 58 63.71 -5.86 19 31 29 42 90 18 348 5 65 63.56 1.27 20 29 28 37 91 20 407 5 58 64.95 -6.75 21 30 27 41 92 19 457 4 66 64.36 1.20 22 43 130 40 100 20 140 14 50 55.86 -5.05 23 29 60 38 100 14 124 11 48 65.27 -17.27 24 34 93 38 100 10 309 3 59 61.75 -2.75 25 24 26 196 0 20 250 8 37 41.20 -4.20 26 31 30 142 0 7 144 5 45 48.19 -3.19 27 18 23 41 100 23 90 26 57 71.52 14.52 28 25 55 104 0 23 192 12 40 54.28 -14.28 29 29 50 127 0 29 95 31 37 50.48 -13.48 30 26 52 116 0 18 140 13 37 52.66 -15.66 31 22 52 148 0 18 128 14 49 48.86 0.14 32 18 11 101 14 30 481 6 54 58.18 -4.18 33 26 9 27 100 6 705 1 73 67.95 5.05 34 27 11 36 100 24 91 26 47 67.12 -20.12 35 21 29 106 11 4 181 2 66 56.29 9.71 36 14 25 136 4 11 142 8 56 52.63 3.37 37 14 14 123 0 50 129 39 40 54.86 -14.86 38 24 23 122 9 7 206 3 53 53.22 -0.22 39 28 50 39 100 33 235 14 71 65.98 5.02 40 34 93 38 100 3 330 1 55 61.75 -6.75 41 33 119 178 0 4 139 3 33 41.84 -8.84 Source: Analysis of Brazil-UNDP-World Bank highway research project data. Source: Analysis of Brazil-UNDP-World Bank highway research project data. VALIDATION OF SPEED PREDICTION MODELS 195 Table 8.11: Effect of number of stops per unit distance on prediction errors Number of stops Number of routes Mean error of per 100 km prediction (km/h) Fewer than 5 16 -1.84 6 - 10 7 -1.77 11 - 15 12 -7.70 16 - 26 2 -10.71 31 - 39 3 -16.15 104 1 -11.23 Source: Analysis of Brazil-UNDP-World Bank highway research project data. Figure 8.8: Observed speeds versus speeds predicted by the aggregate model for bus routes with fewer than 10 stops per 100 km Observed speed (km/h) 70 sop 40 30- 20 10 0 10 20 30 40 50 60 70 Predieted speed (km/h) Legend: # Data points Line of equality Source: Analysis of Brazil-UNDP-World Bank highway research project data. PART II Vehicle Operating Cost Prediction Models CHAPTER 9 Unit Fuel Consumption Function Once the vehicle speed has been predicted using either one of the micro models or the aggregate model, to predict fuel consumption is a relatively straightforward matter. This is done through the "unit fuel consumption function," the formulation of which is first presented (Section 9.1), followed by a short description of the data and the experiment from which the data were obtained (Section 9.2). Finally, the results of model estimation are presented and discussed in Section 9.3. 9.1 MODEL FORMULATION Under ideal environmental conditions, i.e., constant ambient temperature, atmospheric pressure and humidity, the basic principles of internal combustion engines suggest that the rate of fuel consumption per unit time of a vehicle engine may be expressed as a function of two variables, namely the power output and the engine speed (Taylor, 1966, Volume 1). Letting UFC denote the unit fuel consumption (in ml/s) we have: UFC = UFC(HP,RPM) (9.1) where HP = the vehicle power delivered at the driving wheels, in metric hp (as defined in Chapter 2); and RPM = the engine speed, in revolutions per minute (rpm). Although the theory of internal combustion engines is generally concerned with positive values of vehicle power, both positive and negative power must be dealt with in this study. As discussed earlier, when HP is positive, the engine is being used to provide propulsive power for the vehicle. When HP is negative, the engine is being used as a brake, either by itself or in conjunction with the regular brakes. This can occur when the vehicle is decelerating on approaching a curve or travelling downhill since negative power is needed to keep the speed under control. For the purposes of this study, instead of using the actual engine speed, the variable RPM is computed from the following "nominal" formula which ignores tire deflection and slip: RPM = 60 V DRT GRT (9.2) w TD 199 200 UNIT FUEL CONSUMPTION FUNCTION where DRT = the differential speed ratio; GRT = the gear speed ratio; and TD = the nominal tire diameter, in meters; and V = the vehicle travelling speed, in m/s. The values of DRT, GRT and TD for the test vehicles are listed in Table 1A.2. 9.2 ESTIMATION DATA The data for estimating the unit fuel consumption function were obtained from an experiment using all 11 test vehicles. The experiment basically involved running the vehicles over a 1-km length in both directions on 51 selected test sections of constant slope under different loads. Of these sections, 36 were paved and 15 were unpaved. Their characteristics are compiled in Table 9.1. In each run, the vehicle travel and gear speeds were kept constant. Across runs, the vehicle speeds were varied in the range 10-120 km/h by 10 km/h increments. For each vehicle speed, all feasible gear speeds were employed. In each run, the amounts of fuel consumed and time taken to traverse the 1-km stretch were recorded. Generally six replicate runs were made for each combination of test vehicle, load level, section, direction, travel speed and gear. In all, about 60,000 runs were conducted. For each run, the unit fuel consumption (UFC), engine speed (RPM) and vehicle power (HP) were computed. The unit fuel consumption was computed as the amount of fuel consumed over the 1-km stretch divided by the travel time; the engine speed as a function of the measured vehicle speed and gear speed ratio, according to Equation 9.2; and the vehicle power as given by Equation 2.19, with a = 0 for steady-state speed conditions: HP = I [m g(GR + CR) V + 0.5 RHO CD AR V31 (9.3) 736 where GR = the gradient of the test section (positive or negative depending on the direction of travel) as summarized in Table 9.1; CR = the rolling resistance coefficient, computed as a function of the roughness of the test section (summarized in Table 9.1) according to Equation 2.13; and the mass (m), aerodynamic drag coefficient (CD) and frontal area (AR) of each test vehicle are listed in Table 1A.2. For each set of replicate runs for a particular combination of test vehicle, load level, section, direction, travel speed and gear, average values of the variables UFC, HP and RPM were computed and treated UNIT FUEL CONSUMPTION FUNCTION 201 Table 9.1: Characteristics of test sections used in experiments to determine unit fuel consumption function Std. C.V. Mean Min. Max. dev. (%) Absolute value of gradient (%) 3.2 0.0 13.0 2.7 85.8 Curvature (degrees/km) 18.9 0.0 340.0 52.4 277.8 Roughness (QI) 84.6 26.7 212.5 58.5 69.1 Surface type Number of sections Asphaltic concrete 16 Surface treatment 19 Other 1 Total paved 36 Laterite 10 Quartzite 5 Total unpaved 15 Total number of sections 51 Source: Brazil-UNDP-World Bank highway research project data. as one observation in the statistical estimation. Summary statistics of the experiment, including the UFC, HP and RPM variables, are compiled in Table 9.2. 9.3 STATISTICAL ESTIMATION From a preliminary analysis described in Appendix 9A, a general form of the UFC function was obtained for all test vehicles: UFCO + (a3 + a4 RPM) HP + a5 HP2 for 0 < HP UFC = UFCO + a6 HP + a7 HP2 for NHO < HP < 0 (9.4) UFCO + a6 NHQ + a7 NHO for HP < NHO where UFCO = ao + al RPM + a2 RPM2 (9.5) NHO = lower limit on negative power (see Appendix 9A); and a0 - a7 = model parameters to be estimated from regression analysis. 202 UNIT FUEL CONSUMPTION FUNCTION Table 9.2: Summary statistics of experiment to determine unit fuel consumption function No. of mean fuel Pange of observa- speed Pavload Power Engine speed Unit fuel consunption Vehicle tions (kn/h) (1g) (me-tric hp) (rm) (ml/s) Semi- Fully loaded loaded van W4in. Max. Mean tn. #x. W1ean Ifin. max. Srall car 1224 20-120 - 200 5.8 -14.4 34.6 2356.3 959 4096 1.04 0.13 3.30 Redi-n car 398 30-120 - 350 13.7 -10.3 82.5 2384.7 781 4001 3.06 0.33 8.77 Large car 421 20-120 - 350 16.5 -15.5 98.6 2385.4 1028 4525 3.14 0.33 10.36 Utility 1007 10-120 280 550 7.9 -20.1 46.4 2220.3 63C 4040 1.53 0.18 4.73 Large bus 784 10-100 1010 2250 -0.46 -134.7 93.5 1877.6 1016 2796 1.71 0.01 6.62 Tmx: Light gas 1142 10-120 1730 3510 3.8 -110.4 89.3 2412.5 1011 4165 3.40 0.37 13.71 Light & 1020 10-100 1540 3325 0.00 -108.1 61.0 1918.2 865 3035 1.33 0.01 4.97 diesel Heavy 798 8-100 5985 11970 -17.2 -268.6 113.3 1600.0 1016 2541 1.75 0.01 8.45 Articulated 811 10-70 13300 26600 -16.7 -529.9 231.1 1447.1 1155 1649 3.17 0.01 13.61 Replicate 1043 20-120 280 550 7.6 -36.6 46.8 2219.1 935 3784 1.52 0.19 4.76 utility Heavy with 774 8-90 6060 12045 -18.8 -269.9 109.5 1598.4 970 2425 1.60 0.01 8.03 with crane Source: Brazil-UNDP-World Bank highway research project data. UNIT FUEL CONSUMPTION FUNCTION 203 The estimation results based on ordinary least-squares regression, are shown in Table 9.3,1 along with the manually determined value of NHO for each test vehicle. Except for the intercept term,2 a0, all estimated coefficients are significant at the 95 percent confidence level or greater, with t-statistics running from 2.0 to as high as 90.7 and R2 in the range 0.92-0.98. For the Mercedes Benz 1113 as an example, a graph of the estimated UFC function plotted against HP for different values of RPM is shown in Figure 9.1. The estimated UFC function3 is a continuous function of HP and RPM and for a fixed RPM it increases monotonically with HP. This means that when the vehicle is called upon to exerL more power, fuel consumption rises. When HP = 0, we have UFC = UFCO. At idling speed, UFCO approaches the idling fuel consumption. Table 9.4 presents the values of idling fuel consumption for the test vehicles as approximated by UFCO, along with the corresponding idling engine speeds. When HP is smaller than the threshold value NHO, Equation 9.4 above states that the fuel consumption stays constant. While is it true that when the engine is being used as a brake, it does not really need any fuel so that, ideally, the fuel consumption should be zero, the engine, when it is running, always uses some fuel, for instance, the idling fuel consumption. Such relatively small amount of fuel flow is probably due to imperfections in the fuel delivery system. The concave shape of the UFC function as illustrated in Figure 9.1 - constant UFC for HP < NHO and rising UFC with HP for HP > NHO -- has an important implication in studies to determine the effect of speed change cycles on fuel consumption. Contrary to what many may believe, speed change cycles are not themselves the cause of excess fuel consumption. The excess consumption is in fact caused by operating the vehicle in such a way that the power alternates between the positive and negative regimes. While positive power always requires a substantial amount of fuel, negative power dissipates into heat the potential and kinetic energies built up previously by positive power with no compensation. When a vehicle is operated with speed changes but the power still remains within the positive regime, the potential and kinetic energies are not wasted. Another property of the estimated UFC worth mentioning is that given the same HP, fuel consumption increases with the engine speed. This agrees with the general findings in Taylor (1966, Volume 1) and Wong (1978) that for the same power output it is always more economical to operate the internal combustion engine at low speed and high torque than vice versa. 1 For regression purposes, the model form in Equation 9.4 are expressed as shown in Equation 9A.4. 2 There,is no strong a priori reason to expect a0 to be non-zero. 3 Combinations of negative HP and unrealistically low values of RPM can cause the value of UFC computed from the estimated coefficients to be negative. When this occurs, the value should be set to zero. 204 UNIT FUEL CONSUMPTION FUNCTION Table 9.3: Results of regression to estimate unit fuel consumption function (al/s) Nunber Stardard of error of Test vehicle obser- residr- R2 Intercept 1IR 1pM2 PH PH RPM pH2 0 2 vations als ao a, a2 a3 a4 a5 a a7 (0-:) (10-7) (10-2) (0-6) 07 5) (0r (1I 04) 1 Opala 398 0.1473 0.98 0.23453 4.06 1.214 7.775 - - -12 6.552 - (3.2) (6.1) (9.1) (89.0) (11.9) 2 Mercedes Benz 1113 774 0.5635 0.96 -0.41555 10.36 - 3.858 - 16.02 -85 2.764 1.530 with crane (-6.2) (27.1) - (19.9) - (7.1) (15.3) (8.9) 3 Mercedes Bemi 1113 798 0.6746 0.96 -0.22955 9.50 - 3.758 - 19.12 -85 2.394 1.376 (-3.2) (23.0) (19.1) (8.9) (12.6) (7.6) 4 Mercedes &s 0-362 784 0.3261 0.97 -0.07276 6.35 - 4.323 - 8.64 -50 2.479 1.150 (-1.3) (24.7) (24.6) (3.9) (8.3) (2.0) 5 Scania 110/39 810 0.9480 0.98 -0.30559 15.61 - 4.002 - 4.41 -85 4.435 2.608 (-1.8) (13.9) (29.7) (6.3) (15.4) (10.1) 6 Ford 400 1142 2.03% 0.92 -0.48381 12.71 - 5.867 - 43.70 -50 3.843 - (-5.9) (38.6) (17.5) (8.6) (24.3) 7 Fbrd 4000 1020 0.3413 0.94 -0.41803 7.16 - 5.129 - - -30 2.653 - (-10.2) (34.5) (72.8) (30.1) 8 Dodge Dart 421 0.2478 0.98 -0.23705 10.08 - 2.784 9.38 13.91 -15 4.590 - (-4.2) (41.0) (8.4) (8.5) (4.0) (9.2) 9 W-Kmbi 1 1043 0.1915 0.96 0.06014 3.76 - 3.846 13.98 - -12 3.604 - (1.8) (25.4) (13.4) (12.8) (16.4) 10 VW-Kanbi 2 1007 0.2071 0.95 -0.05173 4.69 - 2.963 14.66 - -12 4.867 - (-1.5) (30.2) (9.7) (12.5) (18.9) 11 VW-4300 1224 0.0839 0.96 -0.08201 3.34 - 5.630 - - -10 4.460 - (-5.8) (55.3) (90.7) (22.7) Notes: 1. Parentheses ( ) denote t-statistics. 2. A dash (-) indicates that the corresponding coefficient has been constrained to zero. 3. The variables PH and NHX are defined in Appendix 9A. Source: Analysis of Brazil-UNDP-World Bank highway research project data. UNIT FUEL CONSUMPTION FUNCTION 205 Figure 9.1: Predicted unit fuel consumption versus vehicle power for - different nominal engine speeds - for Mercedes Benz 1113 heavy truck Unft fuel con mption (ml/s) 2400 rp, '400 I 1900 rpm -100 -75 -50 -25 a 25 so 75 100 Power (metric hp) Source: Analysis of Brazil-UNDP-World Bank highway research project data. Table 9.4: Idling fuel consumption for test vehicles based on estimated unit fuel consumption function Idling fuel consumption Typical engine speed Vehicle (ml/s) while idling (rpm) Small car 0.239 960 Medium car 0.625 780 Large car 0.772 1000 Utility 0.323 700 Large bus 0.562 1000 Light gasoline truck 0.787 1000 Light diesel truck 0.205 870 Heavy truck 0.701 980 Articulated truck 1.568 1200 Source: Analysis of Brazil-UNDP-World Bank highway research project data. 206 UNIT FUEL CONSUMPTION FUNCTION By virtue of Equations 9.3 and 9.4 and assuming a constant "normal average" engine speed, the fuel consumption per unit travel distance of each test vehicle can be computed, via the estimated UFC, as a function of the vehicle speed and vehicle and road characteristics. An example of this is presented in Figures 9.2 (a) and (b) for the Mercedes Benz 1113 heavy truck travelling on a level road. These figures show how the vehicle power and fuel consumption per unit distance are affected by the vehicle speed, load level and road roughness and their interaction. At a given load level and roughness, as the speed increases, the power rises, partly due to increased air resistance and partly due to the need to overcome resistance forces at a faster rate; however, the fuel consumption per unit distance drops initially to a minimum before rising. This U-shaped curve is to be expected, since the engine is relatively inefficient at low power. The analogous model developed by Bester (1981) also exhibits this feature. The empirical studies conducted by Hide et al. (1975), Morosiuk and Abaynayaka (1982), and CRRI (1982) developed fuel consumption relationships directly as a U-shape function of speed, without using the unit fuel consumption function as an intermediate step. At given speed and roughness, both vehicle power and fuel consumption increase considerably with the load level (from unloaded to loaded). This is attributable purely to the fact that the rolling resistance is a function of vehicle weight. Similarly, at given speed and load level, increasing roughness (from very smooth paved to very rough unpaved) causes the vehicle power and fuel consumption to go up via an increase in the rolling resistance coefficient. Finally, by the combination of the above factors the effect of roughness on fuel consumption is stronger for the loaded than the unloaded truck. As expected, the amount of the extra fuel goes up with the load level and speed, other things remaining equal. Note that the effect of roughness on fuel consumption is stronger for the loaded than the unloaded truck. A commonly quoted characteristic of internal combustion engines is the specific fuel consumption. For the test vehicles we define the specific fuel consumption, denoted by SFC (in ml/metric hp.s), as the ratio of the fuel consumption rate to the power output: SFC = SFC (HP, RPM) = UFC for HP > 0 (9.6) HP For each test vehicle operating at a "nominal average" engine speed, the values of SFC corresponding to value of HP equal to 25, 50, 75 and 100 percent of the vehicles' maximum used driving power (HPDRIVE) are compiled in Table 9.5. The determination of the "nominal average" engine speeds and the maximum used driving powers of the test vehicles is described in Chapter 10. Figures 9.3 (a) and (b) show specific fuel UNIT FUEL CONSUMPTION FUNCTION 207 Figure 9.2: The effect of vehicle speed, load and road surface on vehicle power and fuel consumption for a heavy truck on a level tangent road Power (metric hp) 150 J a) Vehicle Power 1251 100 ' 75- Unpaved, 0= 200 Paved, QI= 20 25 0 5 10 15 20 25 30 Speed (W/s) Fuel consumption (liters/km) 0. 5 (b) Fuel consumption 0.4- Unpaved, QI= 200 Paved, QI= 20 0. 31 0. 2 0. 11. . . . . . .. 0 5 10 15 20 25 30 Speed (m/s) Legend: Empty, unpaved ------ Empty, paved - Loaded, unpaved -- Loaded, paved Source: Analysis of Brazil-UNDP-World Bank highway research project data. 208 UNIT FUEL CONSUMPTION FUNCTION consumption as a function of power for different values of engine speed for typical gasoline and diesel engines, respectively. Table 9.5: Specific fuel consumption for test vehicles at nominal average engine speed for different values of power Specific fuel consumption (ml/metric hp-s) Nominal average HPDRIVE engine speed (metric hp) 25% of 50% of 75% of Test vehicle (rpm) HPDRIVE HPDRIVE HPDRIVE HPDRIVE VW-1300 3500 30 0.20 0.13 0.10 0.09 Opala 3000 70 0.22 0.15 0.13 0.11 Dodge Dart 3300 85 0.18 0.11 0.09 0.08 VW-Kombi 1 3300 40 0.17 0.10 0.08 0.07 VW-Kombi 2 3300 40 0.18 0.10 0.08 0.07 Ford 400 3300 80 0.25 0.17 0.15 0.14 Ford 4000 2500 60 0.14 0.10 0.08 0.07 Mercedes 1113 2000 100 0.11 0.08 0.07 0.07 Mercedes 1113 2000 100 0.11 0.08 0.07 0.07 Replica Mercedes Bus 1900 100 0.09 0.07 0.06 0.06 0-326 Scania 110/39 1600 210 0.08 0.07 0.06 0.06 9.4 TRANSFERABILITY OF THE UFC FUNCTION The question of transferability of the estimated unit fuel consumption function revolves around two issues. The simpler of the two is the extent to which the estimated function is representative of the particular make and model of the test vehicle used. This issue may be answered in the affirmative based on the evidence of the two replicate vehicles (the VW/Kombi utility and the Mercedes Benz-1113 truck). The coefficients estimated for the two sets of replicate vehicles are close to each other, and the predictions produced by the replicate sets of coefficients are even closer. The second and larger issue is the extent to which the estimated function is representative of the vehicle class. Although we expect that the estimated UFC will be adequately representative of the respective vehicle classes in many cases, it is possible that some makes and models have a different UFC function, owing to improvements or changes in vehicle technology over time (e.g., with respect to engine design (e.g., turbocharged engine with intercooler) or fuel type (e.g., gasohol or alcohol). Thus, it is important to guard against indiscriminate use of the fuel consumption model presented here. The sample of test vehicles for the Brazil study was chosen before the two major oil crises, in the early and the late seventies, stimulated an UNIT FUEL CONSUMPTION FUNCTION 209 Figure 9.3: Specific fuel consumption curves for (a) a typical gasoline engine (Chevrolet Opala) and (b) a typical diesel engine (Mercedes 1113) Specific fuel consumption (ml/metric hp.s) 0. 5 0.14 0. 3 0.2 0.'1- - - - - - - ---- -- 0.01 (a) Chevrolet Opals car o 10 20 30 40 50 60 70 80 90 100 Power (metric hp) Specific fuel consumption (mi/metric hp.s) 0. 14 0. 13 0. 12 0. 11- . 0. 10 0. 09 0.08 0.O 07- ------ 0.06- 0.05. 0.04- 0. 03 0.02- 0. 01 1 (b) Mercedes Benz 1113 truck 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Power (metric hp) Legend: --.-.- 1000 rpm - 1400 rpm -1800 rpm (CRPM) .2200 rpm Source: Analysis of Brazil-UNDP-World Bank highway research project data. 210 UNIT FUEL CONSUMPTION FUNCTION unprecedented change in vehicle technology to improve fuel economy. The problem of representativeness is likely to be particularly serious for cars and the articulated truck. While popular in the sixties and greater part of the seventies, the VW1300 is no longer considered a good representation of the class of small cars. The Chevrolet Opala vehicle in the Brazil study test fleet had six cylinders whereas the most popular configuration for a typical medium car is a four cylinder engine. Similarly, the 8-cylinder Dodge is expected to have relatively few counterparts in most developing countries. Finally, the particular unit of Scania chosen as representative of the class of articulated trucks had a somewhat non-standard transmission which could have resulted in a relatively high fuel consumption. The ideal course of action for obtaining reliable estimates of the UFC function for a particular vehicle would be a re-calibration. However, this would entail a major data collection and analysis effort. A crude but relatively simple alternative would be to employ a multiplicative adjustment factor, a , loosely termed "relative energy-efficiency factor." This factor is specific to a given vehicle class, and has a value of 1 for the particular makes and models used in the estimation of the unit fuel consumption function in the Brazilian study. For a different vehicle of the same class, which is expected to be more efficient than the test vehicle, a value of al, smaller than 1 may be used. Possible ranges of values of al for different vehicle classes are given in Table 9.6 along with recommended values. Table 9.6: Relative energy-efficiency factors Relative energy efficiency factor al Comparable Modern Possible Vehicle class Test vehicle design design range Small car VW-1300 1.00 0.85 0.70-1.00 Medium car Chevrolet Opala 1.00 0.85 0.70-1.00 Large car Dodge Dart 1.00 0.95 0.80-1.00 Utility VW-Kombi 1.00 0.95 0.80-1.00 Bus Mercedes 0-326 1.00 0.95 0.80-1.00 Light gasoline truck Ford 400 1.00 0.95 0.80-1.00 Light diesel truck Ford 4000 1.00 0.95 0.80-1.00 Medium truck Mercedes 1113 1.00 0.95 0.80-1.00 (2 axles) Heavy truck Mercedes 1113 1.00 0.95 0.80-1.00 (3 axles) Articulated truck Scania 110/39 1.00 0.80 0.65-1.00 Source: Author's recommendation. UNIT FUEL CONSUMPTION FUNCTION 211 APPENDIX 9A DETERMINATION OF THE FORM OF UNIT FUEL CONSUMPTION FUNCTION The procedure followed to determine the form of the unit fuel consumption function, UFC(HP, RPM) for the test vehicles is described below, first in general terms, followed by illustration of the procedure for the Mercedes Benz 1113 heavy truck. First, the contours of the observed unit fuel consumption (UFC) values against vehicle power (HP) were obtained for each available nomina engine speed (RPM). Visual examination of the plots led to the following model form relating the UFC to HP for any given value of RPM, say RPM0: b0(RPMO) + bl(RPM0) HP + b2(RPM0) Hp2 for HP > 0 UFC(HP, RPM0) = (9A.1) b0(RPMO) + b3(RPM0) HP + b4(RPM0) HP2 for HP < 0 where bo(RPM0) through b4(RPM0) are parametric functions to be estimated for each tested value of RPM. The notation may be simplified by defining "positive power" (PH) and "negative power" (NH) as follows: PH = Max [0; HP] NH = Min [HP; 0] Further, it was found desirable to impose a lower limit on negative power to simplify the functional form. The limiting value, denoted by NHO, was determined for each test vehicle by inspecting the contour plots. Denoting the bounded negative power by NHX, we may write NHX = Max [NH; NH0 Equation 9A.1 may now be written as: UFC = bo + bl PH + b2 PH2 + b3 NHX + b4 NHX2 (9A.2) where the arguments of UFC and the b's are suppressed for simplicity. In the first stage of the data analysis regression runs were made using Equation 9A.2 for each test vehicle and for each nominal engine speed tested. The next step was to determine the form of the parametric functions bo(RPM) through b4(RPM) for each test vehicle. The most general form used was a quadratic equation: 212 UNIT FUEL CONSUMPTION FUNCTION bi = ci + di RPM + e1 RPM2, i = 0....4 (9A.3) where ci, di and ei are constant parameters. For any particular test vehicle a number of these constant parameters were found not to be significantly different from zero. Combining Equations 9A.2 and 9A.3, and retaining the significant parameters only, the following general form of the UFC function was arrived at: UFC(HP, RPM) = (C0 + do RPM + eo RPM2) + (c + di RPM) PH + c2 PH2 + c3 NHX + c4 NHX2 or, using ao through a7 to denote the parameters for convenience, we may write: UFC= ao + al RPM + a2 RPM2 + a3PH + a4 RPM PH + a5 PH2 + a6 NHX + a7 NHX2 (9A.4) Table 9.3 may be consulted for details regarding the parameters judged to be significant for each test vehicle, noting, as stated above, that for a given test vehicle some of the parameters may be zero. The final step was to derive the "single-stage" estimates of the parameters ao through a7. The results are shown in Table 9.3. Figure 9A.1 shows the contour plot for the Mercedes Benz 1113 heavy truck, using a random sample of the dataset. The value of the lower limit on negative power, NHO, was determined to be -85 metric hp. To illustrate the second step, the plots of b0 and b, against engine speed are shown in Figures 9A.2 (a) and (b), respectively. It may be seen from Figure 9A.2 (a) that bo as a function of RPM has a significant linear trend but only a slight curvature. Thus, for the heavy truck, bo(RPM) = C0 + do RPM. Figure 9A.2 (b) shows that bi is nearly constant over the range of RPM values tested, yielding: bl(RPM) = c1 Similarly, it was found that b2 through b4 were also nearly constant over the RPM values tested. Thus, the UFC as a function of HP and RPM for the heavy truck was: UFC(HP, RPM) = (c0+ do RPM) + cl PH + c2 PH2 + c3 NHX + c4 NHX2 In other words, for the heavy truck the equation 9A.4 was estimated with the parameters a2 and a4 constrained to zero. UNIT FUEL CONSUMPTION FUNCTION 213 Figure 9A.1: Observed unit fuel consumption versus vehicle power with contours for nominal engine speed for heavy truck Unit fuel consumption (ml/s) z 5 A 4 ZO :: * I # 0 Y 0 Z -30 -200 -100 Bn 10 200 Horsepower (metric hp) Legend: N1RPM .+.+ 1000 X X X< 1200 * * * 1300 D F El 1400 0 0 0 1500 A A A 1700 #t #tS 1800 Y Y Y 2000 Z Z Z 2100 2300 A A A 2400 Source: Analysis of Brazil-UNDP-World Bank highway research project data. 214 UNIT FUEL CONSUMPTION FUNCTION Figure 9A.2: bo and b, versus nominal engine speed for heavy truck b0 3. 5 2. 5 2. C 1. 5- 10 Quadratic regression Linear regression 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 Nominal engine speed (rpm) b1 0. 08 0. 06 Mean 0. 04 Linear regression 0. 02 0. 001 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 Nominal engine speed (rpm) Source: Analysis of Brazil-UNDP-World Bank highway research project data. CHAPTER 10 Fuel Consumption Prediction Models Once speed information is known, be it in detailed or aggregate form, fuel consumption can be computed through the unit fuel consumption function presented in Chapter 9. This is the concern of this chapter, which presents specifically: 1. The fuel consumption prediction procedures based on the micro and aggregate models of speed prediction dealt with in Chapters 5-7 (Sections 10.1-10.3); 2. The calibration and validation of these models using independent fuel data (Sections 10.4-10.7); and 3. The development of adjustment factors to bring the fuel consumption predicted by these models closer to real-world operating conditions (Section 10.8). 10.1 NICRO TRANSITIONAL MD)DEL This model is presented first since it is the most general. Let L denote the total distance travelled by a vehicle. Our objective is to predict the amount of fuel consumed by the vehicle per unit distance travelled, denoted by FC (in ml/m, or equivalently, liters/km). The prediction formula is: I AF. FC - (10.1) L where the summation above is over all simulation intervals; and AFj is the predicted amount over fuel consumed over simulation interval j, in ml, given by: AL. AF = UFC (HPj, RPM ) - (10.2) V. J where HPj, RPMj and Vj are, respectively, the average values of vehicle power, engine speed and travel speed over simulation interval j; and ALj is the length of simulation interval j, in meters (as defined in Chapter 6). The vehicle travel speed and power are computed from the following formulas: V 1 [VEN. + VEX (10.3) 215 216 FUEL CONSUMPTION PREDICTION MODELS where VEN and VEXj are the entry and exit speeds determined from the speed proAile simulation described in Chapter 6; and HP 1 {[m' a. + m g (GR + CR.)] V + A V3} (10.4) 1 736 J where aj = the vehicle acceleration over the simulation interval, given by: VEX? - VEN? a = J(10.5) 2 AL. and the other variables are as defined previously, with subscript j denoting those that are specific to the simulation interval. The engine speed, RPM,, is treated as a model parameter to be calibrated, as described in Section 10.6. The general procedure for predicting fuel consumption on the basis of vehicle mechanics described above is not new and has been used in previous studies including Sullivan (1977) and Andersen and Gravem (1979). With respect to the unit fuel consumption function derived herein, simplifying assumptions have been made in these studies. One is that for positive power fuel consumption is taken to be proportional to the power. Another is that for negative power fuel consumption is set constant. 10.2 MICRO NON-TRANSITIONAL MDKL As discussed before, the micro non-transitional model differs from its micro transitional counterpart in two major respects. First, the smaller unit of the road section for which fuel consumption is computed is the subsection (cf. Figure 5.1) as opposed to the simulation interval (cf. Figures 6.1 and 6.3). And second, within each subsection, the vehicle acceleration is zero. By adapting the relationships provided for the micro transitional model, we arrive at the following prediction formula: F FC = s (10.6) L where the summation above is over all homogeneous subsections; and Fs is the predicted amount of fuel consumed over subsection s, in ml, given by: L F = UFC (HP , RPM ) s (10.7) VSS s FUEL CONSUMPTION PREDICTION MODELS 217 where HPs, RPMs and VSSS are, respectively, the vehicle power, engine speed and steady-state speed over subsection s; and Ls is the length of the subsection, in meters (as defined in Chapter 5). For the engine speed RPMs, the calibrated value obtained in Section 10.6 is used. The vehicle power, HPs, is given by: HP = - [m g (GR + CR ) VSS + A VSS (10.8) s 73 s s s s1 736 where the variables on the right-hand side of the above equation are as defined earler, with subscript s denoting those that are specific to the subsection. 10.3 AGGREGATE MODEL As a special case of the micro non-transitional model in which there are only two subsections, uphill and downhill, the prediction formula for the aggregate method is as follows: F + F FC = u d (10.9) L where Fu and Fd are the predicted amounts of fuel consumption over the uphill and downhill subsections, respectively, in ml, given by: L F = UFC (HPU, RPM u) (10.10a) u 'u u VSS u L Fd = UFC (HPd, RPMd) L (10.10b) VSSd where HP 1 [m g (GR + CR) VSS + A VSS3 (10.11a) u 736 u u ul HPd 1 [mg (GRd + CR) VSSd + A VSSd3 (10.11b) 736 and the other variables are as defined previously, with subscripts u and d denoting those that are specific to the uphill and downhill subsections, respectively. 218 FUEL CONSUMPTION PREDICTION MODELS 10.4 DATA FOR CALIBRATION AND VALIDATION The data used in calibrating and validating the fuel consumption prediction models described above were obtained from a field experiment. The experiment involved running nine test vehicles1 over five 10-km test sections in each direction, loaded and unloaded. Two of the sections were paved and three were unpaved. The individual grades vary in steepness and have a maximum value of 9.3 percent. The sharpest curve has a 72-meter radius. Summary statistics of these sections are compiled in Table 8.2.2 During the experiment, the drivers were instructed to drive in a natural manner as if they were in a normal operating situation. During each vehicle run, the amount of fuel consumed was recorded at every 500-meter interval as well as the detailed profile of the elapsed travel time. Using finite differences, the latter was used to derive the profiles of observed speed and acceleration for the run. An illustrative example of profiles of elapsed time, vehicle speed and acceleration is presented in Figure 10.1 for a loaded heavy truck travelling on an unpaved section of moderately severe geometry. Typically, for each combination of the test vehicle, load level, section and direction, six replicate vehicle runs were made. For analysis purposes, profiles of the vehicle speed, acceleration and power for each combination were computed as averages of the profiles obtained from the individual replicate runs. In all, a total of 535 vehicle runs were made. Summary statistics of the runs are compiled in Table 10.1. 10.5 ADAPTATION OF STEADY-STATE SPEED MODEL PARAMETERS FOR TEST VEHICLES In the estimation of the steady-state speed relationships in Chapter 4, the vehicle classes employed generally do not exactly correspond to the test vehicles in terms of weight, load and power. The model parameters considered to be sensitive to the vehicle size, power and load are the used driving and braking powers (HPDRIVE and HPBRAKE, respectively) and the perceived friction ratio (FRATIO) for trucks on paved roads. Therefore, the values of these parameters were revised to make them more specific to the test vehicles following the guidelines given in Chapter 4. The new values of these parameters are compiled in Table 10.2 along with the originally estimated values for the closest vehicle classes. 10.6 CALIBRATION OF NOMINAL ENGINE SPEEDS As mentioned earlier, in order to use the three fuel consumption prediction models described in Sections 10.1-10.3, we must know the engine speed (RPM). If the gear speed is known, RPM can be easily computed (using Equation 9.2). The problem is that for a given 1 Excluding the Dodge and the Opala for logistical reasons. 2 As presented in Chapter 8, these sections were also used in the validation of the micro non-transitional and aggregate speed prediction models. Figure 10.1: Typical profiles of elapsed travel time, speed and acceleration: loaded Mercedes Benz 1113 travelling on unpaved section of relatively severe geometry Elapsed time (min), speed (m/s) 24 Speed profile Elapsed time profile 6 £ UAcceleration (m/s2) 0.4 Acceleration profile 0.2 r / / | 1______I ^ ; I I \ / \ | |lJ' 0.4 j 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Distance (m) Source: Analysis of Brazil-UNDP-World Bank highway research program 220 FUEL CONSUMPTION PREDICTION MODELS Table 10.1: Summary statistics of fuel calibration/validation experiment Number of runs Payload when Vehicle loaded (kg) Unloaded Loaded Small car 280 13 28 Utility (2 vehicles) 550 24 31 Large bus 2,250 37 57 Light gasoline truck 3,510 13 12 Light diesel truck 3,325 36 42 Heavy truck 11,970 30 40 Heavy truck replicate 13,715 48 32 Articulated truck 26,600 50 42 Source: Brazil-UNDP-World Bank highway research project data. combination of vehicle speed and power there can be more than one feasible gear and the choice of gear depends on the behavior of individual drivers. Thus, to predict used engine speed on a rigorous basis, one needs to formulate plausible hypotheses regarding gear change behavior as a function of vehicle speed, acceleration, power used and other factors and to test and calibrate the relationships based on observed data. However, engine-speed profiles were not observed in the fuel consumption validation experiments and recourse had to be taken to simpler approaches. Two approaches were tried out to predict the engine speed. The first was based on an assumption of rational driver behavior: from among the feasible gears the driver chooses the one that minimizes the fuel consumption given the power. The second approach was to make the simple assumption of constant or nominal engine speed and to determine or "calibrate" it using the fuel consumption data. These two approaches yielded comparable accuracy. Since the second approach is much simpler, it was selected for use in all three prediction models. The nominal engine speed or the calibrated RPM value for a test vehicle was obtained simply by varying the value of RPM over the feasible range until the averages of observed fuel consumption and corresponding predictions agreed. The observed speed profiles were all available combinations of test vehicle, load level section and direction were employed as the basis for fuel consumption prediction. The calibrated RPM values for eight test vehicles are shown in Table 10.3 along with the manufacturers' RPM values at the maximum rated power. It is seen that on average, these calibrated values are about 75 percent of the rated values. Thus, for the three test vehicles for which no RPM calibration was actually carried out3, the calibrated RPM values were assumed to equal 75 percent of the rated values (as shown in Table 10.3). To get an idea of the goodness of fit for each test vehicle calibrated, the observed fuel consumption (by load level, section and 3 As mentioned earlier, the validation experiments did not include the Dodge and the Opala. The Ford-400 was included in the experiment, but the fuel measurements were found to be erroneous. Table 10.2: Steady-state speed model parameter adjusted for test vehicles PERCEIVED FRICTION RATIO Used driving power Used braking power Original FRATIO Parameterized HPDRIVE (metric hp) HPBRAKE (metric hp) Test Paved Unpaved Paved Unpaved vehicle Original Adjusted Original Adjusted unload/loaded FRATIO0 FRATIO1 FRATIOO FRATIO1 Small car 36.36 30 21.72 17 0.2674 0.1235 0.2674 0.0 0.1235 0.0 Utility 44.37 40 32.64 30 0.2217 0.1173 0.2217 0.0 0.1173 0.0 Large bus 112.95 100 213.58 160 0.2333 0.0949 0.2333 0.0 0.0949 0.0 Light gas 94.74 80 190.80 100 0.2535 0.0994 0.2535 1.28 x 10-6 0.0994 0.0 truck 0.1705 Light diesel 94.74 60 190.80 100 0.2535 0.0994 0.2535 1.28 x 10- 0.0994 0.0 truck 0.1705 Heavy truck 108.19 100 257.10 250 0.2926 0.0873 0.2926 0.94 x 10-6 0.0873 0.0 0.1858 Articulated 199.98 210 500.00 500 0.1789 0.0400 0.1789 0.23 x 10-6 0.0400 0.0 truck 0.1295 Source: Adapted from analysis of Brazil-UNDP-World Bank Highway research project data. 222 FUEL CONSUMPTION PREDICTION MODELS Table 10.3: Calibrated engine speeds CRPM Calibrated engine MRPM Ratio of CRPM Vehicle speed (rpm) Maximum rated to MRPM Small car 3500 4600 0.7 Medium carl 3000 4000 0.75 Large carl 3300 4400 0.75 Utility1 3300 4600 0.72 Large bus 2300 2800 0.82 Light gasoline truck1 3300 4400 0.75 Light diesel truck 2600 3000 0.87 Heavy truck 1800 2800 0.64 Articulated truck 1700 2200 0.77 1 As no validation data were available for these vehicles their calibrated engine speeds were determined judgementally. Source: Adapted from analysis of Brazil-UNDP-World Bank highway research project data. direction) was regressed against the fuel consumption predicted using observed speeds, with and without the intercept as in the speed model validation presented in Chapter 8. The regression results compiled in Table 10.4 indicate an excellent agreement between the observed and predicted fuel consumption especially in the case of the heavy vehicles which, it may be noted, are all diesel-driven. The fit is somewhat lower for the light gasoline-driven vehicles. The poorest fit is in the case of the small car (VW-1300), with an R2 value of 57 percent. Figure 10.2 shows in graphical form the regression results with the calibrated test vehicles pooled together (into two groups, trucks and non-trucks). Again, an excellent fit is evident. Also shown in Figure 10.2 are regression lines obtained from fuel predictions with RPM varied by + 30 percent of the calibrated values. The small deviations of the slopes of these lines from unity indicate the relative insensitivity of fuel predictions to the engine speed. 10.7 VALIDATION 10.7.1 Micro Transitional Model For each calibrated test vehicle, the micro transitional method was applied to each combination of load level, section and direction for which fuel consumption was recorded. This was accomplished by first constructing a simulated speed profile (following the procedure in Chapter 6) and then applying the relationships described in Section 10.2 (using the calibrated RPM values in Table 10.3) to the speed profile to arrive at the desired prediction. FUEL CONSUMPTION PREDICTION MODELS 223 Table 10.4: Regression results: observed fuel consumption versus fuel consumption predicted with observed speed profiles Standard Number of error of Vehicle observations Intercept Slope R2 residuals Small car 8 -0.04 1.448 0.57 0.017 (-0.4) (38) - 1.010 0.51 0.017 (14.1) Utility 12 0.06 0.588 0.63 0.010 (2.8) (4.2) - 0.987 0.34 0.013 (39.9) Large bus 19 0.00 1.009 0.99 0.015 (0.1) (38.5) - 1.010 0.99 0.015 (88.1) Light diesel 18 0.01 0.933 0.97 0.013 truck (1.7) (24.7) - 0.990 0.97 0.014 (62.6) Heavy truck 15 0.02 0.926 0.96 0.032 (1.1) (16.6) - 0.982 0.96 0.032 (45.3) Articulated 17 0.06 0.930 0.99 0.054 truck (2.6) (33.9) - 0.988 0.98 0.063 (52.2) All vehicles 86 0.01 0.969 0.99 0.031 combined (2.5) (86.3) 0.989 0.99 0.032 (123.8) Source: Analysis of Brazil-UNDP-World Bank highway research project data. 224 FUEL CONSUMPTION PREDICTION MODELS Figure 10.2: Plots of observed fuel consumption versus fuel consumption predicted with observed speed Observed fuel cosumption (al/a) F1 1.0D 0.6 0.4 4 D. 2 X 0. 2 0.0 0.2 0.4 0.6 0.6 1.0 Prediction with observed speed (al/s) Legend: Class X X X Sall car + + +Utility * * * Large bus -- Regression line Observed fuel consuption (al/e) F1 2. 5 2.0 1. 5 1. 0 0-0- 0. 5 0.0 0.5 1.0 1.5 2.0 2.5 Prediction with observed speed (l/.) Legend: Class 0 0 Light ds1 truck A H sevy truck 0 0 Art. truck Regression line Source: Analysis of Brazil-UNDP-World Bank highway research project data. FUEL CONSUMPTION PREDICTION MODELS 225 The results of comparison between the predicted and observed the form of regression statistics similar to those in Tables 10.4 and Figure 10.2. It can be seen that as indicated by the standard errors of regression and regression slopes the prediction accuracy obtained from simulated speed profiles is uniformly high and comparable to the accuracy obtained from the observed profiles. 10.7.2 Micro Eon-transitional Model The procedure used for validating this model paralleled that of the micro transitional counterpart. The main difference was that the "step" steady-state speed profiles were used as the basis for prediction. The predictions produced were first compared with those obtained by the micro transitional model. The results are in the form of regressiostatistics as shown in Table 10.6 and Figure 10.4. The slopes of the without-intercept regression lines virtually equal unity. This implies that the omission of transitional speed behavior introduces virtually no bias in the predictions for the road sections tested. Similarly to the above comparison, the predictions were next compared with the corresponding observed values as shown in Table 10.7 and Figure 10.5. It is evident from the regression slopes and standard errors of regression that the non-transitional model is almost as accurate as thetransitional counterpart. These results indicate that the accuracy of the non- transitional model is adequate for most purposes, especially within the range of road geometry tested. 10.7.3 Aggregate Model This model was tested against the micro models as well as the observed fuel consumption, using the same procedure for comparison as those used for the micro methods, i.e., regression analysis. The results of the comparison are shown in Tables 10.8-10.10 and Figures 10.6-10.8. It can be seen that aggregate predictions closely approximate predictions by both micro models, as indicated by the slopes of the fitted lines being close to one and the small standard errors of regression. 10.8 ADJUSTMENT FACTORS FOR ACTUAL OPERATING CONDITIONS The fuel consumption data employed in the development and validation of the foregoing fuel consumption prediction models were obtained under rather idealized controlled conditions in favor of fuel efficiency. Predictions by these models were found to be generally lower than values experienced by vehicle operators in the same geographic region but under actual conditions. Therefore, adjustment factors, denoted by a2 were developed to bring the predictions closer to vehicle operators' values, as reported in this section. 226 FUEL CONSUMPTION PREDICTION MODELS Table 10.5: Regression results: observed fuel consumption versus fuel consumption predicted using micro transitional model Standard Number of error of Vehicle observations Intercept Slope R2 residuals Small car 8 -0.04 1.436 0.53 0.018 (-0.8) (2.6) - 1.010 0.48 0.018 (13.6) Utility 12 0.05 0.681 0.73 0.001 (2.4) (5.2) - 0.998 0.57 0.011 (49.6) Large bus 19 -0.02 1.017 0.98 0.017 (-1.9) (33.0) - 1.014 0.98 0.019 (68.5) Light diesel 15 0.01 0.953 0.98 0.012 truck (1.3) (25.6) - 0.999 0.98 0.012 (64.5) Heavy truck 15 0.02 0.944 0.95 0.032 (0.9) (16.5) - 0.994 0.95 0.032 (45.5) Articulated 17 0.03 0.961 0.99 0.050 truck (1.3) (38.6) - 0.988 0.99 0.051 (64.5) All vehicles 86 0.01 0.981 0.99 0.028 combined (1.5) (94.6) - 0.992 0.99 0.028 (138.5) Source: Analysis of Brazil-UNDP-World Bank highway research project data. FUEL CONSUMPTION PREDICTION MODELS 227 Table 10.6: Regression results: micro transitional versus micro non- transitional steady-state speed fuel consumption prediction Standard Number of error of Vehicle observations Intercept Slope R2 residuals Small car 8 0.00 0.979 0.99 0.001 (0.8) (27.5) - 1.006 0.99 0.001 (205.4) Utility 12 0.01 0.954 0.99 0.002 (2.1) (35.3) - 1.009 0.99 0.002 (243.2) Large bus 19 0.01 0.976 0.99 0.011 (2.0) (49.8) - 1.012 0.99 0.012 (108.3) Light diesel 15 0.01 0.957 0.99 0.006 truck (2.3) (50.1) - 0.997 0.93 0.007 (109.3) Heavy truck 15 0.03 0.938 0.98 0.023 (1.8) (24.4) - 1.000 0.97 0.024 (59.9) Articulated 17 0.03 0.960 0.99 0.041 truck (1.8) (46.4) - 0.990 0.99 0.044 (75.9) All vehicles 86 0.01 0.977 0.99 0.021 combined (3.1) (126.5) 0.994 0.99 0.022 (176.1) Source: Analysis of Brazil-UNDP-World Bank highway research project data. 228 FUEL CONSUMPTION PREDICTION MODELS Figure 10.3: Plots of observed fuel consumption versus fuel consumption predicted using micro transitional model Observed fuel consumption (mlim) Fl 2.05 1.5 1.0- 0.0 205 1.0 1.5 2.0 2.5 Prediction with simulated speed (ml/m) I egs nd: Claoss O Q Light dsl truck AAA Heavy truck ] 0 0 Art. truck - Regression line Observed fuel cosumption (ml/m) F! 1.0- c. e D. 6 0.2 0.02 0.0 0.2 0.4 0.b 0.8 1.0 Prediction with simulated speed (1l/m) Legende: Cla.ss X X X Sall car + + + Utility & 0 *Large bu ----Regression line Source: Analysis of Brazil-UNDP-World Bank highway research project data. FUEL CONSUMPTION PREDICTION MODELS 229 Figure 10.4: Plots of fuel consumption prediction using micro-transion- al model versus predictions with steady-state speed model Fuel consumption predicted by micro treasitional method (al/.) P2 0. 8 0. 6 0. 4 0. 2 0.0 0.2 0.4 0.6 0.8 1.0 Prediction with steady-state Op (ml/e) Legend: Class XXX Sesll car + +- + Utility * * * Large bus -- Regression line Fuel consuption predicted by micro treasitional ethod (al/) P2 2. 5 2.0 1. 5 1.0O O 0.5 0.0 0.5 1.0 1.5 2.0 2.5 Prediction with steady-state ap (.l/e) Legend: Clues o C 0 Light del truck A A A Heavy truck o 0 0 Art. truck -- Regression line Source: Analysis of Brazil-UNDP-World Bank highway research project data. 230 FUEL CONSUMPTION PREDICTION MODELS Table 10.7: Regression results: observed fuel consumption versus fuel consumption predicted using micro non-transitional steady- state speed model Number of Standard Vehicle observations Intercept Slope R2 error of residuals Small car 8 -0.04 0.504 0.59 0.017 (-0.9) (2.9) - 1.018 0.53 0.017 (14.3) Utility 12 0.05 0.657 0.74 0.009 (2.9) (5.3) - 1.007 0.53 0.011 (47.3) Large bus 19 -0.01 1.047 0.98 0.020 (-0.6) (29.4) - 1.027 0.98 0.020 (66.8) Light diesel 15 0.02 0.911 0.97 0.014 truck (2.2) (21.4) - 0.995 0.96 0.016 (49.4) Heavy truck 15 0.05 0.882 0.93 0.041 (1.8) (12.8) - 0.993 0.91 0.043 (33.2) Articulated 17 0.06 0.922 0.98 0.066 truck (2.1) (27.4) - 0.978 0.97 0.073 (44.9) All vehicles 86 0.02 0.958 0.98 0.037 combined (2.9) (71.3) 0.986 0.98 0.039 (100.9) Source: Analysis of Brazil-UNDP-World Bank highway research project data. FUEL CONSUMPTION PREDICTION MODELS 231 Figure 10.5: Plots of observed fuel consumption versus that predicted with steady-state speed model Observed fuel co-sumption (ml/m) FL1 . 8- C. 6 0. 2 D.0 0.2 0.4 D. E 0.8 10 Prediction with steady-state sp (.l/m) Legend: ClJass X XX Small car + + Utility * * Large bus Regression line Observed fuel co nsumPtiOn (ml/m) F1 2. 5 - /A a. O iEx 0.5 i C.0 C. 5 1. 1.5 2.O 2.5 Predicticn with steady-state sp (ml/m) Leger~: Class O Light del tr ck AcaA He avy truck * *0 0 Art truck -- _ Rgre si n line Source: Analysis of Brazil-UNDP-World Bank highway research project data. 232 FUEL CONSUMPTION PREDICTION MODELS Table 10.8: Regression results: steady-state consumption prediction versus fuel consumption predicted using aggregate model Number of Standard Vehicle observations Intercept Slope R2 error of residuals Small car 8 0.00 1.007 1.00 0.0 (-0.2) (118.3) - 1.006 1.00 0.0 (898.2) Utility 0.00 1.015 1.00 0.0 (-2.3) (227.7) 12 - 1.005 1.00 0.0 (1432.1) Large bus 19 0.00 0.993 1.00 0.0 (1.5) (676.6) - 0.995 1.00 0.0 (1487.4) Light diesel 15 0.00 1.020 1.00 0.001 truck (-2.3) (312.7) - 1.013 1.00 0.001 (654.0) Heavy truck 15 0.01 0.960 1.00 0.002 (3.5) (254.4) - 0.972 1.00 0.003 (468.6) Articulated 17 0.00 0.987 1.00 0.002 truck (4.2) (850.0) - 0.991 1.00 0.003 (1011.0) All vehicles 86 0.00 0.986 1.00 0.004 combined (3.1) (686.9) 0.990 1.00 0.004 (939.9) Source: Analysis of Brazil-UNDP-World Bank highway research project data. FUEL CONSUMPTION PREDICTION MODELS 233 Figure 10.6 : Plots of steady-state fuel consumption prediction versus fuel consumption predicted using aggregate model Fuel consumption predicted by micro non-transitional method (mI/.) F3 1.0D 0. 6 0. 4 0. 2 0.0 0.2 0.4 0.8 0.8 1.0 Prediction with aggregate speed (al/u) Logend: Closse XX X Sml car + + utility * * * Large bus -- Regression line Fuel consumption predicted by sicro non-transitional method (al/m) F3 2. 5 2.0 0. 5- 0.0 0.5 1.0 1.5 2.0 2.5 Prediction wLth aggregate speed (./m) Legend: Clse 0 0 Light del truck AAA H.erY truck 0 0 0 Art. truck - Regressios te Source: Analysis of Brazil-UNDP-World Bank highway research project data. 234 FUEL CONSUMPTION PREDICTION MODELS Table 10.9: Regression results: fuel consumption predicted using micro transitional model versus fuel consumption predicted using aggregate model Number of Standard Vehicle observations Intercept Slope R2 error of residuals Small car 8 0.00 0.987 0.99 0.001 (0.8) (31.3) - 1.012 0.99 0.001 (232.3) Utility 12 0.00 0.968 0.99 0.002 - 1.014 0.99 0.002 (260.7) Large bus 19 0.01 0.969 0.99 0.010 (2.2) (52.0) - 1.007 0.99 0.011 (111.0) Light diesel 15 0.01 0.977 1.00 0.006 truck (2.0) (54.9) - 1.010 1.00 0.006 (122.9) Heavy truck 15 0.03 0.900 0.98 0.023 (2.1) (23.6) - 0.972 0.98 0.026 (56.1) Articulated 17 0.03 0.947 0.99 0.041 truck (2.0) (45.7) - 0.981 0.99 0.045 (73.1) All vehicles 86 0.01 0.964 0.99 0.022 combined (3.5) (122.4) 0.984 0.99 0.024 (167.8) Source: Analysis of Brazil-UNDP-World Bank highway research project data. FUEL CONSUMPTION PREDICTION MODELS 235 Figure 10.7: Plots of fuel consumption predicted using micro transition- al model versus fuel consumption predicted using aggregate model Fuel consumption predieted by micrc nontransitional method (ml/m) F2 1.01 0.8- 0. 2 0.G 0.2 D.4 0.6 0.8 1.0 Prediction with aggregate speed (.l/) Legend: Class XXX XSall car + + + Utility * * iarge a Regression line Fuel consumption predicted by micro no-transitional method (ml/m) F2 2. 5 2.0O c.- - - ,T- 1.1 5 0.25 0./ 0.0 2.5 1.0 1.5 2.0 2.5 Predictioc with aggregate speed (ml/m) Legend: Class 2 0 0 Light dal truck A 6 A Heavy truck 0 0 0 Art. cruck -- Regression lice Source: Analysis of Brazil-UNDP-World Bank highway research project data. 236 FUEL CONSUMPTION PREDICTION MODELS Table 10.10: Regression results: observed fuel consumption versus fuel consumption predicted using aggregate model Number of Standard Vehicle observations Intercept Slope R2 error of residuals Small car 8 -0.04 1.475 0.57 0.017 (-0.88) (2.8) - 1.023 0.52 0.017 (14.1) Utility 12 0.05 0.669 0.75 0.009 (2.8) (5.4) - 1.012 0.55 0.011 (48.3) Large bus 19 -0.01 1.040 0.98 0.019 (-0.6) (29.9) - 1.022 0.98 0.019 (67.9) Light diesel 15 0.02 0.930 0.97 0.013 truck (2.1) (22.5) - 1.009 0.97 0.015 (52.6) Heavy truck 15 0.05 0.845 0.92 0.042 (1.9) (12.5) - 0.964 0.90 0.045 (31.8) Articulated 17 0.06 0.910 0.98 0.066 truck (2.2) (27.5) - 0.969 0.97 0.074 (44.3) All vehicles 86 0.02 0.945 0.98 0.038 combined (3.2) (70.4) 0.976 0.98 0.040 (98.6) Source: Analysis of Brazil-UNDP-World Bank highway research project data. FUEL CONSUMPTION PREDICTION MODELS 237 Figure 10.8: Plots of observed fuel consumption versus consumption predicted using aggregate model Observed fuel consumption (al/m) Fi 0 2 1. 0 0-0 / 0.0 D.2 D. 4 D D. 8 1.O Prediction with aggregate speed (mil/a) Lgn:Class X XX Small car + + + Utili ty * * * Large bus - Regression line Observed fuel constraption (al/.) P1 2.5- 0. 1. D 0. 5 0.0 0.0 0.5 0 2.D 2.5 Prediction with aggregate speed (ali) Legent: TClas 0 0 0 Light de truck AAA Heav) truck 0 0 0 Art. truck - Regression line Source: Analysis of Brazil-UNDP-World Bank highway research project data. 238 FUEL CONSUMPTION PREDICTION MODELS 10.8.1 Sources of Discrepancy in Actual Fuel Consumption The conditions under which the test vehicles were operated can be highlighted by the following features: 1. The drivers were well-trained and well-controlled. 2. The vehicles were relatively new and maintained in excellent mechanical condition, particularly the engine. 3. The experiments were conducted generally when the road surface was dry. 4. The experiments were conducted only after the engine had reached a steady-state operating temperature. 5. The experiments were conducted with virtually no interference from other vehicular traffic. According to the international data compiled by the Organization for Economic Cooperation and Development (OECD, 1982) deviations from the above ideals tend to cause significant increases in average fuel consumption. Table 10.11 gives approximate magnitudes of possible increases due to various sources. Though not exhaustive, they should explain most of the discrepancies between the experimental and survey data. 10.8.2 Data For comparison purposes, only the survey vehicles that matched one of the test vehicles in terms of major characteristics such as the engine power, tare weight, rated load carrying capacity and fuel efficiency were included. The resulting correspondence of test and survey vehicles is shown in Table 10.12.4 For each class of survey vehicles in Table 10.12, except for the total weight, the values of vehicle characteristics and model parameters (e.g., the aerodynamic drag coefficient, projected frontal area, tare weight and calibrated engine speed) obtained for the corresponding test vehicles were adopted. While these characteristics and parameters may be slightly different from those of the survey vehicles, the resulting fuel prediction errors should be relatively small. The total vehicle weight, which consists of the tare weight and payload, has a strong influence on predicted fuel consumption. However, as no reliable information was available for individual survey vehicles, estimates of their average values were made for each vehicle class. For each survey vehicle, the characteristics of the routes on which the vehicle was operated were readily available in aggregate form, i.e., average road roughness (in QI), average rise plus fall (in No matches were found for the following experimental vehicles: the Chevrolet Opala, the Dodge Dart and the Ford-400. FUEL CONSUMPTION PREDICTION MODELS 239 Table 10.11: Sources of fuel consumption discrepancy Source Possible variation in fuel consumption1 Driver behavior 20 percent difference between economical and hard driving for gasoline-engined passenger cars. Mechanical condition 20 percent difference or more between well and badly tuned gasoline engines. 2-4 percent difference between correct and incorrect wheel alignment of passenger cars. Road wetness 2 percent difference between dry and wet sur- faces (paved only) for gasoline engined passenger cars. Vehicle preparation 23 percent difference between a 1400 cc gaso- line engine of a passenger car properly warm ed up and at ambiant temperature of 25 degree celsius. Over 4-km distance; 19 percent difference for a 2070 cc diesel engine. For trips longer than 10 km, the percent differ- ence generally reduces to less than 2-3 percent. 1 The percentage differences are very rough estimates. Source: Adapted from OECD, 1982. meters/km), horizontal curvature (in degrees/km), and percentage paved. Summary statistics of the survey routes by vehicle class are compiled in Table 10.13. It can be seen that these routes are generally of modest information was available on the superelevation of the survey routes, the average superelevation was computed as a function of the average horizontal curvature using the relationship given in Chapter 3 (Equation 3.25). The above vehicle and road characteristics were used to predict the average fuel consumption for each survey vehicle. Only the aggregate model was used. This was because, as alluded to earlier, no detailed information on road geometry was readily available. For routes having mixed paved and unpaved portions, predictions were computed separately by surface type and then their average values were obtained using the paved/unpaved percentages as the weights. 240 FUEL CONSUMPTION PREDICTION MODELS Table 10.12: Correspondence between experimental and survey vehicles and estimates of tare weights and payloads S U R V E Y V E H I C L E Vehicle Test Tare weight Average class vehicle Model Number (kg) round trip payload(kg) Car Volkswagen 1300 37 960 0 1300 1600 6 Total 43 Utility Volkswagen Enclosed 2 1320 300 Kombi Passenger 3 Total 5 Bus Mercedes 0 362 32 Benz LPO 321/45/48 12 0 362 LPO 1113/45 113 LP 113/48/51 145 OF 1113/61 8 8100 2300 OH 1313/51 20 0 321 10 0 326 1 0 352 33 1111 2 Total 376 Light Ford 4000 Ford 4000 8 diesel Mercedes L-608 26 3270 2000 truck Total 34 Medium/ Mercedes L/LK/LS 1113 54 heavy Benz L/LK/LS 1313 29 3400/6600 4500/6000 truck L/LK 1513 20 Total 103 Articulated Scania L 110 20 truck L 110 LS 1519 18 L 75 4 L 76 3 14700 13000 LS 36 6 L 111 28 LT 111 7 LK 140 6 Total 92 Source: Brazil-UNDP-World Bank highway research project data. FUEL CONSUMPTION PREDICTION MODELS 241 10.8.3 Results and Discussion The observed and predicted fuel consumption are compared in the form of summary statistics in Table 10.14. Generally, the range predicted is smaller than the range observed. The high variability in the observed fuel consumption relative to the predicted may to be attributed mainfy to the variability in the unmeasured factors discussed in Section 10.8.1, and, in particular, the variability in the total weights of the survey vehicles. Regression of observed fuel consumption against predicted values by means of ordinary least squares5, with and without intercept, yielded results as summarized in Table 10.15. Figure 10.9 shows a plot of observed against predicted fuel consumption distinguished by vehicle class. The regressions with-intercept for utilities and buses yielded intercepts substantially different from zero and slopes substantially different from one. For the other vehicle classes, the regressions with and without the intercept yielded similar results. The slopes of the regressions without-intercept are virtually identical to the simple ratios of mean observed to mean predicted fuel consumption in Table 10.14. For the purposes of adjusting experimentally-based fuel consumption predictions to account for real-world conditions, the intercept is assumed to be zero for each vehicle class and the value of the slope from the regression without the intercept is taken as the required adjustment factor, a2. This procedure is justified by the following reasons: 1. Besides convenience, a single multiplicative adjustment factor provides a desirable asymptotic property of fuel prediction, i.e., when actual fuel consumption approaches zero (for example, when a vehicle is traversing a downhill section under retarding power of the engine) the prediction should do the same. 2. For vehicle classes with a large range of predicted fuel consumption (see the minimum and the maximum in Table 10.14), the results of regression with and without the intercept are very similar. Except for the class of medium/heavy trucks the slopes of the without-intercept regression equations in Table 10.15 fall in a narrow range (1.11 - 1.20). Since there is no fundamental reason why the adjust- 5 The regression should, strictly speaking, employ an error component technique since the observation errors are not independent but consist of components specific to both companies and vehicles (Harrison and Chesher, 1984). The use of ordinary least squares would result in biased t-statistics although the coefficient estimates are still unbiased. In this analysis t-statistics are used only to indicate relative significance. 242 FUEL CONSUMPTION PREDICTION MODELS Table 10.13: Summary statistics of observed and predicted fuel consumption Fuel consumption (liters/1000 km) Vehicle Number Variables Mean Std. Coef. of Minimum Maximum class of vehicles dev. variation (%) Car 43 Observed 100.8 12.4 12.0 80.0 129.0 Predicted 87.8 3.6 4.1 84.3 94.3 Ratio O/P 1.15 Utility 5 Observed 177.2 20.9 11.8 142.0 195.0 Predicted 149.8 2.6 1.7 146.1 151.6 Ratio 0/P 1.18 Bus 376 Observed 305.8 36.5 11.9 211.0 486.0 Predicted 253.5 12.0 4.7 236.4 294.1 Ratio O/P 1.21 Light diesel 34 Observed 193.3 31.9 16.5 157.0 304.0 truck Predicted 174.3 12.2 7.0 164.8 216.3 Ratio 0/P 1.11 Medium/heavy 103 Observed 319.8 45.2 14.1 251.0 424.0 truck Predicted 311.1 40.5 13.0 272.1 392.8 Ratio 0/P 1.03 Articulated 92 Observed 618.9 158.4 6 25. 373.0 1020.0 truck Predicted 539.3 68.5 7 12. 472.9 759.4 Ratio O/P 1.15 Source: Brazil-UNDP-World Bank highway research project data and analysis. FUEL CONSUMPTION PREDICTION MODELS 243 Table 10.14: Sumary of road characteristics Vehicle Statistic Rise plus Horizontal Road Percent class fall curvature roughness paved (m/km) ("/km) (QI) (%) Car Mean 26.8 45.4 76.4 63.1 Std. dev. 7.9 50.1 39.9 35.5 Maximum 39.2 202.3 140.0 100.0 Minimum 12.0 12.0 26.9 10.0 Utility Mean 21.5 27.4 94.0 38.0 Std. dev. 7.1 7.8 31.0 27.4 Maximum 32.3 33.0 115.0 68.0 Minimum 16.6 18.6 46.0 18.0 Bus Mean 24.4 34.7 86.7 48.9 Std. dev. 6.8 39.1 46.4 42.3 Maximum 39.1 188.8 172.6 100.0 Minimum 9.8 6.2 23.0 0.0 Light diesel Mean 25.5 43.9 66.3 68.4 truck Std. dev. 5.6 43.0 40.8 38.5 Maximum 38.7 214.9 166.5 100.0 Minimum 15.9 6.1 28.0 0.0 Medium/heavy Mean 32.5 43.2 42.8 88.9 truck Std. dev. 3.9 30.9 11.7 10.9 Maximum 41.4 108.8 85.0 100.0 Minimum 27.4 5.7 24.0 43.0 Articulated Mean 27.8 40.6 55.2 75.6 truck Std. dev. 10.9 50.9 28.9 40.8 Maximum 48.6 293.7 127.1 100.0 Minimum 10.0 5.7 26.0 1.0 Source: Brazil-UNDP-World Bank highway research project data. 244 FUEL CONSUMPTION PREDICTION MODELS Table 10.15: Regression results observed versus predicted fuel consumption by vehicle class Vehicle Number Intercept Slope R-square Standard error class of vehicles of residuals Car 43 -13.8 1.31 0.14 11.60 (-0.3) (2.6) 1.15 0.14 11.47 (57.7) Utility 5 -897.8 7.17 0.81 10.57 (-3.0) (3.6) 1.18 0.24 18.14 (21.9) Bus 376 272.8 0.13 0.00 36.57 (6.8) (0.8) 1.20 0.00 38.72 (153.0) Light diesel 34 -8.5 1.16 0.20 29.11 truck (-0.1) (2.8) 1.11 0.19 28.67 (39.4) Medium/heavy 103 89.8 0.74 0.44 34.03 truck (3.4) (8.9) 1.02 0.37 35.79 (91.0) Articulated 92 41.4 1.07 0.21 141.14 truck (0.4) (5.0) 1.15 0.21 140.46 (42.6) All vehicle 653 15.3 1.10 0.80 64.83 classes (2.3) (51.7) 1.15 0.80 65.04 (140.1) Notes: Fuel consumption is expressed in liters/1000 km. Values in the parentheses are t-statistics. Source: Analysis of Brazil-UNDP-World Bank highway research project data. FUEL CONSUMPTION PREDICTION MODELS 245 ment factors should be distinctly different, except for fuel type, the vehicles were lumped into light (gasoline) and heavy (diesel) groups, as follows: Light (gasoline) Heavy (diesel) Car Bus Utility All trucks and regressions were run for these groups yielding results as summarized in Table 10.16. Figure 10.10 shows a plot of predicted vs. observed fuel consumption distinguishable by vehicle group; the plot is superimposed with the regression lines of slopes equal to 1.16 and 1.15 for the light and heavy vehicle groups, respectively. As seen in the validation results in Section 10.7, the discrepancies between fuel predictions by the aggregate and micro models are on the order of less than 2 percent, which is considerably smaller than the 15-16 percent discrepancies between experimentally based fuel predictions and fuel consumption observed under actual operating conditions. Therefore, it seems appropriate to recommend the following values of adjustment factor, 42, for use with all three prediction models, aggregate and micro: Cars and utilities: 1.16 Buses and trucks : 1.15 Table 10.16: Regression results of observed versus predicted fuel consumption for heavy and light vehicle groups Vehicle Number Intercept Slope R-square Standard error group of vehicles of residuals Heavy 605 20.5 1.09 0.76 67.18 (2.6) (43.7) 1.15 0.76 67.49 (134.6) Light 48 -8.4 1.24 0.80 12.17 (-1.0) (13.6) 1.16 0.80 12.16 (63.5) Note: Fuel consumption is expressed in liters/1,000 km. Values in the parentheses are t-statistics. Source: Analysis of Brazil-UNDP-World Bank highway research project data. 246 FUEL CONSUMPTION PREDICTION MODELS Figure 10.9: The plot of observed versus predicted fuel consumption for all vehicle classes Obsered~ fuel consuption(m/m i. C / 2 . C 2 0, 4 U,60 Predieted fuel coumption, 'm(/m) Leged: l XX Sall car + 4 tlity * L. Large b 0 0 Light d1l. truck t,AA Md!heavy trk C 0 Ar tru-k ote Only randonly selected data points are shown Note: Only randomly selected data points are shown. Source: Brazil-UNDP-World Bank highway research project data and analysis. Figure 10.10: The plot of observed versus predicted fuel consumption for light and heavy vehicle groups and regression lines Observed fuel osum,ptin( m 22 1. C .1 O . 2 0.0j O 0. 2- 0.0 0.2 0.4 0.6 0.8 1.C PFedicted fu1 tonsumption (ml"m) Clas " - Light vehicl ED F 1eavy" v ticle Source: Brazil-UNDP-World Bank highway research project data and analysis. CHAPTER 11 Aggregate Tire Wear Prediction Model A major component of road user costs, especially for heavy goods vehicles, tire wear accounts for some 23 percent of the average running cost of a typical heavy truck operating on paved roads in rolling terrain in Brazil (GEIPOT, 1982, Volume 5). Current knowledge of tire mechanics indicates a strong dependence of tread wear on road alignment through forces acting on the tire which result from cornering, hill climbing, acceleration and braking. Also, a previous analysis of Brazil tire data (Chesher and Harrison, 1980) found that carcass failures are caused by increased severity of road alignment and surface conditions. This sug- gests that road improvements can have a disproportionate impact on tire costs relative to the other components. Therefore, to optimize highway investments, it is important to quantify properly the effects of geometric and surface characteristics on tire cost. This chapter describes the formulation and estimation of a mathematical model which predicts tire wear as a function of vehicle and road characteristics (Section 11.1). Based largely on theories of tire mechanics, the model is an aggregate form of an algebraic function similar to the aggregate speed and fuel models (Chapters 7 and 10) and was statis- tically estimated using data obtained from the Brazil road user cost survey, for cross-ply tires only (GEIPOT, 1982 Volume 5). The data and estimation results, available for buses and trucks only, are presented in Sections 11.2 and 11.3, respectively. Section 11.4 presents the recom- mended tire wear prediction model based on the above results (for buses and trucks) and compares it with results from other studies. Also pre- sented in Section 11.4 is a relatively simple tire wear prediction model obtained from highly aggregate data which is recommended for cars and utilities. During the period of the study, the tires prevalent in Brazil were almost exclusively cross-ply type of tires and only this type of tires was included in the sample. Thus, the relationships presented here are applicable to cross-ply only. 11.1 MDEL FORMULATION 11.1.1 Tire Wear Cost Model Two principal modes of tire wear contribute to tire cost: 1. Carcass wear; and 2. Tread wear. 247 248 AGGREGATE TIRE WEAR PREDICTION Carcass wear Carcass wear is defined in terms of the number of retreads to which a tire can be subjected before scrappage. Since the number of retreads, r, is an integer, its modelling should be done on a probabilistic basis, by defining: PNR(r) = the probability that the tire will last through r re- treads, r = 0, 1, 2, ..., Nr where Nr is the maximum practical number of retreads (equal to 6 in the Brazil data). Tread wear Treat wear is defined (for a given tire size and model) as the fraction of tread worn per 1,000 tire-km.1 Since the wear rates of new treads and retreads may differ, the symbols TWN and TWR are used to denote the former and the latter, respectively. Tire wear cost per unit travel distance Let CN and CRT denote the cost of a new tire and one retreading, respectively. Then, the cost of tire wear per 1,000 tire-km, CTW, is given by: CTW = CN + CRT NR (11.1) DISTOT where NR = the average number of retreads, given by N NR I r PNR(r) (11.2) r=1 and DISTOT = the total distance of travel (in 1,000 km) provided by the tire carcass through its new tread and retreads, given by: DISTOT = I + NR (11.3) TWN TWR Two assumptions are implicit in Equations 11.1-11.3. First, all the treads completely wear out before any failure (e.g., irreparable damage to the carcass when the tread is only half worn). This assumption is considered to be acceptable since, as demonstrated in Section 11.2, the average distances afforded by terminal treads are greater than 85 percent those of the treads that were returned for retreading. The second assumption is that the wear rate of retreads is constant regardless of the number of retreadings to which the carcass has been subjected. This assumption is substantiated to a large extent in the Brazil data, as shown in Section 11.3. 1 There are other definitions of tread wear, viz., mils of tire tread thickness worn/km travelled, cubic inches/mile travelled, etc. These other definitions will be used on special occasions where appropriate. AGGREGATE TIRE WEAR PREDICTION 249 Equivalent new tire life. By rearranging Equation 11.1, the expression for the distance life of a cost-equivalent or simply equivalent new tire, TLNEW, is obtained: TLNEW = DISTOT (11.4) 1 + NR CRT/CN We can interpret TLNEW as the distance life of CN dollars worth of tires (in 1000 km). The equivalent new tire life is useful for comparing the results from the mechanistic tire wear analysis herein with those from previous analyses and from other studies (Harrison and Chesher, 1984; CRRI, 1982; Hide, 1982; and Hide et al., 1975). The above manner of tire wear model formulation offers several advantages. First, instead of lumping them together, the carcass and tread wear models are considered separately. This gives rise to a more meaningful interpretation of the causal mechanisms affecting the tire wear phenomenon. Second, the definition of tire cost per km of Equation 11.1 is provided as a function of the relative costs of retreads and new tires which appear to vary substantially across countries. While, as noted above, the ratio of the cost of a retreading to the cost of a new tire is about 15 percent in Brazil, it equals 50-60 percent in India (CRRI, 1982) and 40-50 percent in Costa Rica (Harral et al., 1984). Hence, Equation 11.1 provides greater flexibility in model transference to a new country than Equation 11.4 in which the ratio CR/CN is fixed to the originating country. Third, both the carcass life and tread wear models can be easily interpreted in physical terms and the estimates of the model parameters can be compared with data from other sources on a more systematic basis. This provides a greater scope for the model parameters to be adapted to local conditions, for example, with respect to the general recapping policy of vehicle operators, quality of recapping and rubber compound, abrasiveness of surfacing materials, etc. 11.1.2 Carcass Life Model The effect of the carcass life on tire cost may be demonstrated by the following example. From the Brazil data the cost of a retreading is, as noted above, about 15 percent that of a new tire, whereas the average life of a retread is about 75 percent that of a new tread. So, CR = 0.15 CN and TWN = 0.75 TWR. Substituting these in Equations 11.1 and 11.3, we have the tire wear cost per 1,000 km given by: CTW = CN(1 + 0.15 NR) TWR (11.5) 1.33 + NR From Equation 11.5, the ratio of tire wear cost per km for an arbitrary number of retreadings to that for zero retreading is: CTW(NR) _ 1 + 0.15 NR (11.6) CTW(NR=0) I + 0.75 NR 250 AGGREGATE TIRE WEAR PREDICTION This ratio varies from 1 for NR = 0 to 0.45 for NR = 3. Thus, vehicle operators can achieve substantial savings by having the carcass retreaded as many times as practical subject to safety constraints. Although, as shown in the above example, the carcass life is a major variable influencing tire operating cost, there seems to be little work done to date in relating quantitatively the number of retreads to road severity measures.2 The only exception we are aware of is the analysis of Brazil data by Chesher and Harrison (1980) to obtain by multiple regression analysis relationships between the number of retreads to road characteristics, mainly road roughness, for bus and truck tires. On a riori grounds we would expect road roughness to reduce the carcass life in at least two ways: first, through fatigue, and second, through causing irreparable damage to the carcass structure. The former failure mechanism is gradual whereas the latter is sudden. 11.1.3 Micro Tread Wear Model Our literature search revealed a wealth of theoretic-empiric knowledge on tire mechanics which is sufficient to permit a reasonable mathematical formulation of a tread wear model. However, as will be seen below, additional empirical information is still needed, especially with respect to actual driving tests over public roads. Consider a homogeneous road section of uniform characteristics (i.e., constant gradient, curvature, roughness, surface texture) being traversed by a vehicle. At the tire-road contact area, there exists a force in the direction tangential to the road surface. The magnitude and direction of this tangential force depends on the circumferential force used to propel or retard the vehicle and the lateral force used to keep the vehicle from sliding when it is negotiating a bend. According to Schallamach (1981), the tangential force is transmitted through the elastic deformation of the part of the tire next to the road surface (called grip or adhesion). Upon return to its original shape, the tire surface at the contact area slips against the road surface. The abrasive slip action results in some loss of tire material. The amount of wear increases with the amount of tire slip and the magnitude of the accompanying shear force. The physical mechanism of tread wear sketched above is described in detail in mathematical form by Schallamach (1981). Schallamach's comprehensive tread wear model is a synthesis of a series of theoretical and experimental findings dating back to the mid-nineteen fifties. Based on mechanical principles of pneumatic tires designed to handle large deflections and sliding distances, the full mathematical formulas are very complicated. However, for small tire slips these formulas reduce to a simple form. Since normal vehicle operations are associated with small slips, we are mainly concerned with the simplified formulas, the derivation of which is given in Appendix 11A. 2 According to F. C. Brenner and T.E. Gillespie (private communication) and literature search. AGGREGATE TIRE WEAR PREDICTION 251 For purposes of statistical analysis, the micro tread wear, TWT, model so derived can be stated as: TWT = TWT + TWT + TWT (11.7) CFT2 LFT2 or TWT = TWT + CT - + CT -- (11.8) 0 c NFT NFT where TWTc, TWTZ = tire wear resulting from the circumferential and lateral forces, respectively (Equations 11A.8a and 11A.8b); CFT, LFT = the circumferential and lateral forces acting on the tire, in newtons, respectively (as defined earlier); NFT = the load on the tire in the direction normal to the tire-road contact area, in newtons; TWTO = a constant term; CTc = KO b/kc CTk. KO b/kk; b a parameter which depends on the tire dimensions and technical properties; kc, kX = parameters denoting circumferential and lateral lateral stiffness of the tire, respectively; KO = a parameter which depends on the physical properties of the tire and the road surface (as elaborated in Appendix 11A); and the parameters TWTO, CTc and CT9, are to be estimated. According to Bergman and Crum (1973) who employed a similar model form in actual road tests, the constant term TWTO reflects unmeasured forces including the forces due to incorrect wheel alignment, tire construction properties, and road camber. As shown in Appendix 110, the above tread wear model has built into it a dependency of fourth or higher power on the vehicle speed on curves. This is supported by empirical findings given in Appendix 11B. Seeing that circumferential and lateral forces are significant parameters affecting the tire wear, so are the tire properties in regard to circumferential and lateral stiffness. This issue is discussed in Appendix 11B, which draws on earlier empirical tire wear research. Theoretical analysis and practical testing on the basis of slip energy theory indicate that the tire wear rate is relatively independent of the 252 AGGREGATE TIRE WEAR PREDICTION direction of the total tangential force, and that circumferential and lateral tire stiffness can be assumed to be equal; this leads to a considerably simplified tire wear equation (Equation 11B.5). The research results referred to in Appendix 11B also show a significant difference in tire wear: 1. Between radial and bias tires, radials having considerably smaller slip coefficients -- consistent with the general observation that radials provide better grip - and hence also wearing slower than bias tires; 2. Between truck and car tires, the former having wear coefficients as large as 10 times or more those of car tires -- this results from both greater slip and slip energy coefficients for truck tires, which may be attributable to truck tires being operating typically under much higher loads and inflation pressures than passenger car tires; 3. Between dense and open-graded asphalt concrete - the slip energy coefficient for the former being almost three times that of the latter; but 4. While there is conceptual difference in slip between paved and unpaved surfaces, the data available do not demonstrate a difference in tire wear on the basis of road surface type alone. 11.1.4 Aggregate Tread Wear Model As physical phenomena, tire tread wear and fuel consumption share two common features: first, they occur continuously as the vehicle is traversing a road stretch; and, second, they are explained by well-established theories. Therefore, tread wear should ideally be modelled in a similar way to fuel consumption, as briefly outlined in the following manner. Detailed experiments using specially-instrumented test vehicles would be conducted to obtain data to relate tread wear as a function of the normal, circumferential and lateral forces acting on the tires, for different types of rubber compounds, tire construction, and road surfacing aggregates under different loads, inflation pressures, etc. These forces, in turn, would be related to the vehicle speed and characteristics of the vehicle and the road. Once a micro tread wear model is satisfactorily validated, it would be used as the building block for developing an aggregate tread wear model, in a manner similar to the aggregate fuel consumption model described in Chapter 10. Finally, experimentally based tread wear predictions can be adjusted to real-world operating conditions using actual operators records. However, such detailed experimental data on tread wear were not collected in the Brazil study as the original intent was to acquire only survey-based data for modelling tire wear on the basis of empirical trends. The resulting empirical relationships are reported elsewhere (GEIPOT, 1982, Volume 5; Chesher and Harrison, 1985). In order to use AGGREGATE TIRE WEAR PREDICTION 253 this data base, the micro tread wear model formulated in Section 11.1.3 must be converted to aggregate form. Appendix 11C presents a detailed description of the aggregation procedure, along with the results of a numerical test to show that the accuracy of the aggregate model is close to that of the micro model. 11.2 DATA FOR ESTIKATION Detailed data of 2886 tires were compiled for the final analysis; the data were obtained from the road user cost survey, as described in GEIPOT (1982, Volume 5) and Chesher and Harrison (1985). All tires included were of bias construction and were used by buses and medium, heavy and articulated trucks. The nominal dimensions of the tires (rim width/rim diameter, in inches) were 9.00/20 and 10.00/20 for buses and medium and heavy trucks and 9.00/20, 10.00/20 and 11.00/22 for articulated trucks. Each tire had a complete historical record associated with various stages of its life cycle, from the first time it was mounted, through various recaps, and finally to the time it was removed for scrappage. The record of each tire contained not only the distance travelled during each stage, but also the aggregate characteristics of the routes. To get an idea of what typical tires went through during their service lives, we examine the average survival rates and distances travelled by the tires during the various stages of their lives, as given in Table 11.1. It can be seen that of all the tires in the sample, none managed to survive to undergo the seventh recapping. In fact, the survival rate, expressed as a percentage of the total number of new tires, decreases quite rapidly with the recap stage. Only slightly more than 50 percent of the new tires could be recapped and less than 5 percent lasted long enough for the fourth recapping. The distances travelled by new tires were on average greater than those travelled by recaps by more than 20 percent. This may indicate differences in the workmanship and/or the quality of the rubber itself between new tires and recaps. It is interesting to note that: first, the average distance travelled by recapped tires were more or less constant regardless of the recapping stage; and, second, at each stage, new tire or recap, the average distance travelled by non-survivors were smaller than that by survivors but not by more than 16 percent margin. These observations support the assumptions made earlier in the tire wear cost model (Section 11.1.1). The characteristics of the routes on which those tires were operated are summarized on Table 11.2. It can be seen that the tire data set covers a wide range of road roughness, from smooth paved to very rough unpaved. A large percentage of routes apparently have mixed paved/unpaved portions. The range of the vertical alignment is moderately high, with a maximum average gradient approaching 5 percent (rise plus fall = 50 m/km). However, the horizontal curvature suffers from a relatively narrow range of 7 to 294 degrees/km with a standard deviation of only 80 degrees/km. 254 AGGREGATE TIRE WEAR PREDICTION Table 11.1: Survival rates and distances travelled at various stages of tire life cycle Survival rate Distance travelled by Stage of Number Previous New Non- Non-survivors tire life of stage stage Survivors survivors vs. survivors cycle survivors (%) (%) (km) (km) (%) New 2,886 100.0 100.0 26,939 22,612 84 1st recapping 1,549 53.7 53.7 18,628 18,051 97 2nd recapping 780 50.3 27.0 18,707 16,165 86 3rd recapping 360 46.1 12.5 18,495 16,178 87 4th recapping 138 38.3 4.8 18,581 16,211 87 5th recapping 43 31.2 1.5 19,165 16,451 86 6th recapping 11 25.6 0.4 - 13,818 - 7th recapping 0 0 0 - - Source: Brazil-UNDP-World Bank highway research project data. As mentioned before, since the road user survey was not originally designed for the mechanistic approach, the data not only are in highly aggregate form but also lack the following information: 1. Superelevation: As curve superelevation was not measured in the survey of route characteristics, it was not possible to obtain a meaningful estimate of the coefficient for the lateral force component (LFT). Section 11.3 provides a further discussion of this effect. 2. Vehicle characteristics: No detailed information was surveyed except for the make/model and payloads of the vehicles on which the tires were mounted. For simplicity, except for the number of wheels per vehicle, average characteristics including the payload were used for each class of vehicles, as shown in Table 11.3. To predict aggregate vehicle speeds, the closest test vehicle was matched against each of the vehicle classes in Table 11.3, and the steady-state speed model parameters adapted for these test vehicles (as described in Chapter 10) were used. 3. Tire rubber volume: No data were obtained on the make/model and tread type of each tire which were necessary for accurate determination of the tire's wearable rubber volume. A procedure had to be devised to obtain an estimate of the average rubber volume per tire, as detailed in Appendix 11D. AGGREGATE TIRE WEAR PREDICTION 255 Table 11.2: Summary statistics of tire survey routes Route Characteristics Rise plus fall Curvature Roughness Proportion (m/km) (degrees/km) (QI) paved (%) Summary statistic: Mean 30.78 61.78 87.07 57.27 Minimum 9.80 7.50 23.00 0.00 Maximum 48.60 293.70 239.00 100.00 Standard deviation 8.50 80.82 40.30 32.75 Correlation coefficient with: Rise plus fall 1.000 0.800 -0.006 0.405 Curvature 1.000 0.217 0.089 Roughness 1.000 -0.874 Proportion paved 1.000 Source: Brazil-UNDP-World Bank highway research project data. Table 11.3: Vehicle characteristics used in aggregate speed prediction Vehicle Class Characteristics Large Medium Heavy Articulated bus truck truck truck Aerodynamic drag coefficient 0.65 0.85 0.85 0.63 Projected fromtal area (m2) 6.30 5.20 5.20 5.75 Tare weight (kg) 81,00 54,00 66,00 14,700 Payload (kg) 23,00 4500 60,00 13,000 Number of highway tires 6 4 6 14 Number of mud and snow tires 0 2 4 4 Source: Brazil-UNDP-World Bank highway research project data. 256 AGGREGATE TIRE WEAR PREDICTION 11.3 ESTIMATION RESULTS Using the tire data set described above, the carcass life and aggregate tread wear models were estimated on the basis of ordinary least-squares regression, but with an individual intercept estimated for each vehicle operator or company. Company-specific intercepts were intended to capture the differences in tire wear attributable to variations among companies in tire usage, maintenance and recapping policies. The rationale for this procedure is discussed in Harrison and Chesher (1984). 11.3.1 Carcass Life Model The relationship selected relates the average number of recaps (NR) to the road roughness (QI) and horizontal curvature (C): NR = NRO exp[-0.00248 QI - 0.00118 min(300,C)]-1 (-3.32) (-2.20) where: Xo NRO = the natural logarithm of the average of the company-specific intercepts by tire size: 0.66 for 9.00/20 tires (64.8) 1.22 for 10.00/20 tires (52.5) 1.52 for 11.00/22 tires; (58.7) The number of data points is 2,886; The t-statistics are shown in parentheses; The standard error of regression = 0.40; and R2= 0.478 (based on the average intercepts above). In Equation 11.9 above the parameter NRO may be interpreted as equal to the average number of recaps for very smooth tangent roads. The effect of the horizontal curvature (C) is limited to the maximum value of 300 degrees per km which represents the upper end of the data range. It may be noted that the predicted NR can be negative for very rough and curvy roads (but not smaller than an asymptotic minimum of -1), indicating carcass failure before the tire is ready for the first recap. This means the total number of treads (new tread plus retreads) is smaller than one (but never smaller than zero). Graphs of predicted NR plotted against QI and C are shown in Figures 11.1(a) and (b), respectively, for each tire size. NR is depicted to decrease with increasing road severity. An increase in roughness by 150 QI (e.g., from a smooth paved to a rough unpaved surface) causes a 30 percent drop in the number of recaps. The same affect can be achieved by a 300 deg/km rise in the horizontal curvature (e.g., from a tangent to a moderately curved road). AGGREGATE TIRE WEAR PREDICTION 257 Figure 11.1: Number of retreads, MR, for different tire sizes as a function of (a) roughness and (b) curvature (a) Number of retreads 4 3 Curvature = 40 degrees/k. -- - ----- - -- - ----- -- 0 0 2 20/20 -1- 0 25 50 75 100 125 150 175 200 225 250 Roughness (QI) (b) Number of retreads 4 Roughness = 30 QI 3 2 1100/20 1000/20 - 900/20 0 0 50 100 150 200 250 300 350 400 Curvature (Degrees/km) Source: Analysis of Brazil-UNDP-World Bank highway research project data. 258 AGGREGATE TIRE WEAR PREDICTION 11.3.2 Tread Wear Model An earlier analysis (Rezende-Lima, 1984) yielded separate relationships for new treads and retreads, showing that the latter wear out faster than the former. However, due to the approximate nature of the coefficient estimates, it was decided for simplicity to pool the new tires and recaps, survivors and non-survivors, into one regression. This resulted in the following relationship which predicts tire tread wear, TWT, in dm3/1000 tire-km: TWT = 0.164 + 0.01278 CFT2 (11.10) NFT (22.7) (5.12) where the constant term is the average of the company-specific intercepts; the number of data points is 2,886; the t-statistics are shown in parentheses; the standard error of regression = 3.63; and (v) R2 = 0.181 (based on the average intercept above). Conspicuously absent from the above relationship is the term for the lateral force, LFT2/NFT (shown in Equation 11C.4). This does not mean the coefficient CTX for LFT2/NFT is believed to be zero. In fact as discussed this coefficient should be of more or less the same magnitude as the CTc coefficient of CFT2/NFT. However, it was not possible to estimate the CTt coefficient with any degree of confidence. This is because the magnitude of the lateral force is highly dependent on the curve superelevation which, as noted earlier, was not recorded in the road user cost survey. An attempt was made to estimate the lateral force using relationships between superelevation (SP) and horizontal curvature (C) constructed from a sample of Brazilian roads3. Based on these lateral force estimates, the CTx coefficient was found to be insignificant. Two related factors were thought to be possible causes. First, the range of horizontal curvature in the data was too small (0 < C < 300). Second, within this range, superelevation was so well-designed that the actual lateral forces of the surveyed vehicles were indeed small. However, the lateral force is not believed to remain small for values of C greater than 400 degrees/km even with properly designed superelevation. In the high range of C, the curve speed constraint begins to dominate the vehicle speed, thereby raising the lateral force to a significant. level. Therefore, the above tire wear formula is expected to produce conservative predictions for C greater than 400 and for roads without well-designed superelevation. For straight driving, in which the lateral force, LFT, is zero, the estimate of 0.01278 for the circumferential force coefficient (CTc) is directly comparable to the empirically obtained wear coefficients for bias truck tires presented in Table 11B.1. These empirical wear coefficients fall in the range 0.00225 to 0.00570, and the largest value is smaller than one-half of 0.01278. While there are numerous 3 These superelevation-horizontal curvature relationships, obtained separately for paved and unpaved roads, are given in Chapter 3 (Equation 3.25). AGGREGATE TIRE WEAR PREDICTION 259 possibilites for the discrepancy, the following are more plausible: 1. The non-survivors, which amounted to 46.3 percent of the data points used in the model estimation tended to make the estimated tread wear rate higher than should be. 2. The tires surveyed (new and recapped) were likely to be of lesser quality than those used in the controlled experiments (all new), thereby making them more susceptible to wear. For straight driving, graphs of predicted tread wear, expressed in the number of treads per 1000 tire-km, plotted against road roughness (QI) and rise plus fall (RF) for a heavy truck, with and without the payload, are shown in Figures 11.2(a) and (b). The effect of 6,000 kg payload, compared to the 6,600 kg tare weight of the heavy truck, is shown to cause a significant extra tread wear which increases with road severity (roughness or steepness). As expected from the model formulation, tread wear is relatively insensitive to roughness which has only a small effect on the circumferential force through the rolling resistance. In contrast, tread wear is heavily influenced by road steepness. In particular, when rise plus fall exceeds 40 m/km the gravitational component of the circumferential force becomes dominant and causes the tires to wear at an increasing rate. 11.4 RECOMMENIED TIRE WEAR PREDICTION MODEL AND COMPARISON WITH OTFHER STUDIES Separate relationships for tire wear prediction are provided for buses and trucks as one group, and cars and utilities as another group. While the former is based on the mechanistic formulation and estimation results in the preceding section, the latter is based on a relatively simple formulation and highly aggregate data. Like the speed and fuel prediction models, the tire relationships are adapted specifically for each test vehicle. 11.4.1 Buses and Trucks By adapting Equation 11.1, the formula for computing tire cost per 1,000 vehicle-km is given by: CTV = NT (CN + CRT NR) (11.11) DISTOT where NR is the predicted number of retreads, as given in Eq.11.14; NT is the number of tires on the vehicle; and the other variables are as defined before. Specific values of NRO are assigned to the bus and trucks in the test vehicle fleet, as shown in Table 11.4. Since no tires in the light truck classes (nominal dimensions of 7.50/16) were analyzed, NRO = 1.93 which correspond to the 9.00/20 dimensions, were adopted for these truck classes. Recalling that the total travel distance per tire carcass, DISTOT, is given by: 260 AGGREGATE TIRE WEAR PREDICTION Figure 11.2: The plots of predicted umber of equivalent new tires worn versus (a) roughness and (b) road rise plus fall for a heavy truck on a paved road. ,a) Number of equivalent new tires per vehicle per 1000 km 0. 30 Rise plus fall, RF = 20 m/km C. 251 Loaded 0. OC' C 2= 50 C 100 i25 150 Roughness QI (b) Number of equivalent new tires per vehicle per 1000 km 0.71 Roughness, QI 30 0.6 Loaded 0. 5 0.4- Unloaded -* 0.2 1 ----- - - - - - - - - - - - - -I 0. 1 0 10 20 30 40 50 60 70 Road rise plus fall (m/km) Source: Analysis of Brazil-UNDP-World Bank highway research project data. AGGREGATE TIRE WEAR PREDICTION 261 Table 11.4: Basic parameters for tire wear prediction model for buses and trucks NUMBER OF TIRES Number of recaps Average wearable Vehicle type Nominal tire - at zero horizontal rubber volume per Test vehicle dimensions Highway Mud/snow Total curvature NRo per tire VOL (dm3) Large bus 10.00/20 6 0 6 3.39 6.85 Mercedes Benz 0-362 Light diesel truck 7.50/16 6 0 6 1.93 2.52 Ford 4000 Light gasoline truck 7.50/16 6 0 6 1.93 2.52 Ford 400 Medium truck 10.00/20 2 4 6 3.39 7.60 MB 1113 with 2 axles Heavy truck 10.00/20 6 4 10 3.39 7.30 MB 1113 with 3 axles Articulated truck 11.00/22 14 4 18 4.57 8.39 Scania 110/39 Source: Adapted from Brazil-UNDP-World Bank highway research project data. DISTOT = + NR (11.12) TWN TWR and assuming that TWN and TWR, the numbers of treads consumed per 1000 tire-km for new tires and recaps, respectively, are equal, then: TWN = TWR = VOL where TWT is the predicted volume of rubber loss, in dm3/1,000 tire-km, as given in Equation 11.10; and VOL denotes the average volume of rubber per tire, in dm3. Specific values of VOL for the test vehicles were computed on the basis of the vehicles' axle-wheel configurations and are given in Table 11.4. To express tire costs in terms of equivalent new tires, Equation 11.11 may be rewritten as: CTV = CN EQNTV (11.13) where EQNTV = the predicted number of "cost equivalent" or, simply, "equivalent" new tires consumed per 1,000 vehicle-km, given by: 262 AGGREGATE TIRE WEAR PREDICTION EQNTV (1 + RREC NR) (11.14) DISTOT where RREC = the ratio of the cost of one retreading to the cost of one new tire (CRT/CN), or the "recap cost ratio." Eliminating DISTOT, TWR and TWN in Equations 11.3, 11.12 and 11.4 results in: EQNTV NT (1 + RREC NR) TWT (11.15) 1 + NR VOL Substituting Equations 11.9 and 11.10 for NR and TWT in the above equation, and applying bias correction for model nonlinearity, we have: EQNTV = NT (1 + RREC NR) (0.164 + 0.01278 (CFT2/NFT)) + 0.00751 (1 + NR) VOL (11.16) where NR = NRO exp(-0.00248 QI - 0.00118 C) - 1.0 0.0075 is value of the correction term for prediction bias.4 11.4.2 Cars and Utilities As the tire data for cars and utilities obtained in the road user test survey are very aggregated, a relatively crude model was constructed for predicting the number of equivalent new tires per 1000 vehicle-km. The aggregate tire data for utilities (GEIPOT, 1982, Volume 5) come in two groups, paved and unpaved. The paved group has an average road roughness of 40 QI and average equivalent new tire life of 59,100 km. The corresponding averages for the unpaved group are 140 QI and 32,600 km. Assuming that EQNTV is a linear function of QI, the above aggregate data points yield the following relationship: EQNTV = NT (0.0114 + 0.000137 QI) (11.17) 4 Since the above equation is a non-linear transformation of the original linear prediction relationships (Equations 11.9 and 11.11), a prediction bias is produced. This is similar to the prediction bias caused by the exponential transformation of the originally estimated speed model in logarithmic form. However, the non-linear transformation in Eqxation 11.16 is more complicated mathematically. Due to the approximate nature of Equation 11.16, a relatively simple additive bias correction term was applied. To quantify this bias correction term, EQNT was predicted for the survey vehicles in the tire data using Equation 11.16 and also computed based on the observed number of re treadings and total travel distance per carcass. The difference between the averages of the observed and predicted values of EQNT was found to equal 0.0075 and was taken as the bias correction term. AGGREGATE TIRE WEAR PREDICTION 263 for predicting tire wear for cars and utilities with NT = 4; a road roughness ceiling of 200 QI is recommended. 11.4.3 Comparison of Tire Wear Prediction Mbdel with Other Studies Figures 11.3-11.6 compare the number of equivalent new tires predicted by Equations 11.16 and 11.17 with predictions using models obtained from other studies, namely those of TRRL-Kenya (Hide et al., 1975), TRRL-Caribbean (Hide, 1982), India (CRRI, 1982),5 and Brazil in an earlier "aggregate-correlative" analysis (GEIPOT, 1982, Volume 5)6. As the "aggregate-correlative" Brazil models predict the equivalent new tire life (in the form of Equation 11.4) they had to be converted by inversion into the number of equivalent new tires. This non-linear transformation caused these Brazil models to predict generally lower tire consumption, measured in terms of the number of equivalent new tires, than the mechanistic models, even though they were derived from the same data base. For medium/heavy trucks and buses, tire wear predictions by different models are plotted against roughness, curvature and rise plus fall in Figures 11.3-11.57. For.the mechanistic models, different curves are shown separately for paved and unpaved roads. Relative to the others, the Kenya and Caribbean models predict more than twice the influence of roughness (the steeper slopes of the prediction curves in Figure 11.3); however, these models assume no impact of road-geometry on tire wear (the flat slopes of the curves in Figures 11.4 and 11.5). One possible cause of these discrepancies is that the underlying data bases for the Kenya and Caribbean models were considerably smaller than for the India and Brazil models, thereby making the coefficient estimates somewhat less reliable. Among the models derived from the India and Brazil data bases, the predicted influence of roughness is very similar over the entire range, as indicated by almost equal slopes of the curves in Figure 11.3. As to the influence of road geometry, the mechanistic models predict increases in tire consumption over the extreme range of rise plus fall (0-70 m/km) and horizontal curvature (0-1,000 degrees/km) which are 3-4 times as large as the earlier low predictions made by the "aggregate-correlative" Brazil models. The predicted geometric effects by the India models are even lower. 5 The TRRL Towed Fifth Wheel Bump Integrator Scale for road roughness (BI) was used in the TRRL-Kenya, TRRL-Caribbean, and India studies. As recommended by Paterson (1984), the conversion factor of 55 mm/km on the BI scale to one QI count was employed for comparison purposes. The models from these studies are summarized in Chesher and Harrison (forthcoming) and Watanatada, et al. (1987). 6 The model coefficients originally reported in the 1982 GEIPOT report were slightly revised subsequently to make them conform with the revised definition of road roughness. The new coefficients, as reported in Chesher and Harrison (forthcoming), are used in the comparison. Because of variations in design and size among the study areas the prediction models are not strictly comparable. However, efforts were made to match the vehicle classes as closely as possible. 264 AGGREGATE TIRE WEAR PREDICTION Figure 11.3 Tire wear versus roughness for (a) medium/heavy trucks and (b) buses as predicted by the mechanistic model and models from other studies (a) Number of equivalent new tires per vehicle per 1,000 km 0. 4 0.21 jvdia 0. 1 -0 25 50 75 10O 125 150 175 200 225 250 Roughness (QI) (b; Number of equivalent new tires per vehicle per 1,000 km C. 30 C. 20 ---~ ---_ --- --- --- --- - - - - - - - - - Brazil - ndia 0.10 0. 05 0. 00 0 25 50 75 100 125 150 175 200 225 250 Roughness (QI) Source: See the first paragraph of Section 11.4.3. AGGREGATE TIRE WEAR PREDICTION 265 Figure 11.4: Tire wear versus rise plus fall for (a) medium/heavy trucks and (b) buses as predicted by the mechanistic model and models from other studies (a) Number of equivalent new tires per vehicle per 1,000 km 0. 5 ' 0. 54 0.49 3.21 - 0. 1 0 10 20 30 40 50 60 70 Rise plus fall (m/km) (b) Nuvber of equivalent new tires per vehicle per 1,000 km 0. 8- 0. 7- 0. 6- 0. 5 0. 4 V- Brazil (Mech.) - 0. 2 --- Inda Kenya 0. 1 Caribbean 0.0. 0 10 20 30 40 50 60 70 Rise plus fall (m/ke) Source: See the first paragraph of Section 11.4.3. 266 AGGREGATE TIRE WEAR PREDICTION Figure 11.5: Tire wear versus curvature for (a) medium/heavy trucks and (b) buses as predicted by the mechanistic model and models from other studies (a) Number of equivalent new tires per vehicle per 1,000 km 0.2 Brazil 0.201 -a 0. 151 Caribbean C. 10 0. 05 0. OO C 1C 200 3CC 400 500 50C 700 8C 900 1000 Curvature (degrees/km) (b) Number of equivalent new tires per vehicle per 1,000 km 0.301 0. 25 0. 20 paved 0. 15- Brazil n- - - - Keany (-- p. -- - - - - - - - - - 0.10 0.05 r. cO C 1C 200 300 400 5CC 60C 700 800 900 1000 Curvature (degrees/km) Source: See the first paragraph of Section 11.4.3. AGGREGATE TIRE WEAR PREDICTION 267 Figure 11.6: Tire wear versus roughness for cars and utilities as predicted by Equation 11.7 and models from other studies Number of equivalent new tires per vehicle per 1,000 km 0.65 0.5 0.4 0.2 I I 0.2 -- - - - --- - -- Brail 0 25 50 75 100 125 150 Roughness (QI) Source: See the first paragraph of Section 11.4.3. For cars and utilities, tire wear predictions by the simple model of Equation 11.17 and the Kenya, Caribbean and India models are plotted against roughness in Figure 11.6. As before, the Kenya and Caribbean models predict a very strong effect of roughness on tire wear compared to the India and Brazil models which produce very similar predictions up to about 120 QI8. 8 Like the aggregate-empiric Brazil models, the India models predict the equivalent new tire life which can become zero. For cars and utilities, the India model predicts zero tire life when roughness is about 186 QI, at which point the number of equivalent new tires becomes infinite. Therefore, an artificial ceiling must be imposed on the India model for general application. 268 AGGREGATE TIRE WEAR PREDICTION APPENDIX 11A IRIVATION OF KICRO TREAD WEAR MOWML This appendix provides a description of the micro tire tread wear model derived by Schallamach (1981). The physical mechanism of tire tread wear may be divided into two parts: 1. the slip mechanism which deals with the relationship of the tire slip to the tangential and vertical forces on the tire; and 2. the wear mechanism which deals with the relationship of the rate of tread wear to the tangential force and the tire slip. Slip mechanism Tire slip velocity T (in m/s) is defined as the vectorial difference between the travel and circumferential velocity of the wheel, Schallamach (1981): T R - W (11A.1) where R the velocity of the road relative to the wheel axle, in m/s; and W = the velocity of the wheel circumference relative to its 1211e, in m/s. An illustration of the slip velocity definition is given in Figure 11A.1. Tire slip, A, is defined on a relative basis: A - (11A.2) JRI where R L the magnitude of the wheel travel velocity (identically equal to the vehicle speed). Denote the magnitudes of the components of the tire slip in the tire's lateral and circumferential directions by Atand Ac, respectively. When the wheel rolls freely in a direction that makes"slip angle" n (in rad) to the travel direction (see Figure 11A.1), we have zero circumferential slip ( 8 = 0) and the lateral slip is given by: Xt - sin (11A.3) For small slips, A = t , and we may use the slip angle, n, in the place of the lateral slip, At. When the wheel rolls in straight driving, the AGGREGATE TIRE WEAR PREDICTION 269 Figure 11A.1: Illustration of slip velocity (tire-road contact area viewed from below) Travel Direction Wheel Axle (Wheel Axle Velocity) T (Tire Slip Velocity) / W (Tire Circumferential Velocity) slip angle and lateral slip are zero, and the circumferential slip is given by: = 1 - - (11A.4) JR1 where Wj and R denote the magnitudes of the wheel circumferential and travel e! ocities, respectively. The circumferential slip, 1c, may be interpreted as the meters of slip per meter wheel travel. An illustation tire deformation and slip when rolling freely, side slipping, braking and propelling is given in Figure 11A.2. The slip mechanism is mathematically described by means of an elastic toothed wheel (Schallamach and Turner, 1960). The teeth are assumed to deform independently of each other and to obey Hooke's law of linear stress-strain relationships, up to the limit of the available friction coefficient between the tire and the road surface. When a part of, the contact area has its shear stress exceeding the available friction it begins to slide against the road surface and no further grip can be developed. For small slips (Xt below 0.07 radians or 4 degrees and Ae below 5 percent), the elastic toothed wheel model yields the following simplified relationships: 1 = CFT (11A.5a) k NFT c 270 AGGREGATE TIRE WEAR PREDICTION Figure 11A.2: Illustration of tire deformation and slip in contact area model experiment of wheel on transparent track (viewed from below) (a) R-1T=O R60inR (b) Sim Slipping (C) Braking ------------ W (d) Propelng 0 W Tire-Track Marks on Tire Contact Area Circunference ravel Direct o SoLurce Scta'lrmach (1960) Source: Adapted from Schallamach, 1960. and b LFT z X- (11.A5b) k NFT where b = a function of the tire's dimensions and mechanical properties; kc, kt = the circumferential and lateral stiffness of the tire, respectively; CFT, LFT = the circumferential and lateral forces acting on the tire, newtons, respectively; and NFT = the load on the tire in the direction normal to the tire-road contact area, in newtons; Using a bias tire under different loads, Nordeen and Cortese (1964) validated Schallamach and Turner's slip model in full form up to the limit of the available friction coefficient when sliding occurred over the whole contact area. AGGREGATE TIRE WEAR PREDICTION 271 Wear mechanism Based on an earlier experimental finding that the abrasion depth was an increasing function of the normal pressure on the contact area, Schallamach (1981) derived the following wear relationships (for small slips): TWT = CFT n (11A.6a) c NFT(n+1)/2 n LFT TWTZ KO NFT(n+l)/2 (11A.6b) where TWTc, TWTz = tire wear resulting from the circumferential and lateral forces, respectively; K0 = a function of the tire's geometry and mechanical properties as well as mechanical properties which depend on both the tire and the road surface (including available friction coefficient, road surface abrasiveness, and tire abrasion resistance); and n = a constant, equal to 1 or more. By substituting for Xc and Xk from Equations 11A.5 in Equations 11A.6, the tire wear relationships may be expressed in terms of the forces as: K b n+1 0 CFT (1.a TWT - (11A.7a) c k c NFT(n+1)/2 K b n+1 and TWT T- (11A.7b) k (n+1)/2 2. NFT The magnitude of the exponent n has been a subject of controversy. Veith (1974) provided a review of previous work on the exponent n. According to the theory that abrasive wear is proportional to the friction energy dissipated, the value of n should equal 1, and this is borne out in laboratory tests using relatively abrasive surfaces. However, tests using blunt surfaces and very high normal pressures resulted in n values as high as 2 or more. The departure from n=1 has been ascribed to fatigue failure accompanied by heightened temperatures in the contact area (Schallamach, 1981). 272 AGGREGATE TIRE WEAR PREDICTION The experimental results which support the fatigue theory appear to have been based on testing severities considerably beyond those found under normal vehicle operating conditions. For example, from a trailer-cornering test of natural and synthetic rubber tires (Grosch and Schallamach, 1961) the average values of n are 1.4 and 1.5, respectively, for 1-4 degrees range of the slip angle (n). However, for the range 1-2 degrees slip angle, the values equal unity. Except for sharp steering on urban-street corners and sudden braking and other maneuvers to avoid accidents vehicle operators on rural roads are generally expected to use tire slips of no more than 2 degrees (or 0.035 radius) on sharp bends and under one percent on average. Results from tread wear tests using actual vehicles on public roads, carried out by Della-Moretta (1974), Bergman and Crum (1973) and Hodges and Koch (1979) give no indications that n should be greater than 1 for tires under normal usage. Therefore, n=1 was the value adopted for data analysis, and the wear relationships Equations 11A.7 reduce to KO b CFT2 TWT - - K (11A.8a) k NFT c K b LFT2 and TWT (11A.8b) k NFT AGGREGATE TIRE WEAR PREDICTION 273 APPENDIX 11B REVIEW OF PREVIOUS EMPIRICAL TREAD WEAR RESEARCH Speed dependence of wear The tread wear model developed in Appendix 11A has built-in fourth power or higher dependency on the vehicle speed on curves. This is illustrated in Figure 11B.1 using results obtained from race tracks in which measured tread wear is plotted against vehicle speed. In the Schallamach and Turner test (1960), tire wear increased by more than 5 times when the vehicle speed went from 50 to 80 km/h. In the Chiesa and Ghilandi test (1975), an increase in vehicle speed from 45 to 68 km/h resulted in a tire wear increase of more than 12 times. Lateral versus circumferential stiffness An issue of importance in road investment decision is concerned with the relative effects of the horizontal and vertical alignments on tread wear, which, in turn, depend on the relative magnitudes of the lateral and circumferential stiffness of the tire (kk and kc respectively). The results of road tests using actual vehicles conducted by Della-Moretta (1974), Bergman and Crum (1973) and Hodges and Koch (1979) provide useful information on this question, as discussed below. Using a four-wheel drive vehicle and bias tires, Della-Moretta (1974) (also Della-Moretta and Sullivan, 1976) obtained a relationship between tread wear per unit "slip" energy9. The slip energy model is a simplified form of the more general tread wear relationship of Equations 11A.8a and 11A.8b in which the lateral and circumferential stiffness of the tire are assumed to be equal, i.e., kt = kc = k, and the constant term TWTo (in Equation 11.8) is set to zero. So, the tread wear equation reduces to: K 0b TFT2 TWT = -IO b TFT (11B.1) k NFT where TFT = the total tangential force which is the vectorial sum of the (mutually perpendicular) lateral and circumferential components: TFT = VCFT2 + LFT2 (11B.2) Defining the slip energy, denoted by SE, as the product of tire slip 1 and the total tangential force: 9 Although Henry C. Hodges did not produce any publications on slip energy he is acknowledged by Sullivan (1979) as another originator of the slip energy concept. A related report by Hodges and Koch (1979) is referenced later in this appendix. 274 AGGREGATE TIRE WEAR PREDICTION Figure 11B.1: Effect of vehicle curve speed on tread wear (a) Relative tire wear 10 , 80 4 /0 2 1. ,Speed (km/h) 40 50 60 70 80 90 Source: Adapted from Schallamach and Turner, 1960. (b) Specific wear rate (mm/1,000 km) Very high severity I I 2.51, 2.0., Ii I I 1.5 / I I . I High severity / 1.0 I I .5 Medium severity ------------------------ I I Low severity ,.* 8=**** I 1 , 4Maximum speed 0 10 20 30 40 50 60 70 in curves (km/h) Source: Adapted from Chiesa and Ghilardi, 1978. AGGREGATE TIRE WEAR PREDICTION 275 Defining the slip energy, denoted by SE, as the product of tire slip j and the total tangential force: SE = 1XI TFT (11B.3) in joules per meter of travel, where b TFT (11B.4) k NFT The dimensionless term b/k may be called the "slip coefficient." Thus, Equation 11B.1 above converts to the slip energy form: TWT = KO S E (11B.5) where KO can be interpreted as the amount of tire wear per unit slip energy and may be called the "slip energy coefficient." The product 10 b/k may be called the "wear coefficient" for the tangential force, and denoted by CTt. To test the lateral/circumferential tire stiffness assumption, Della-Moretta (1974) conducted experiments with the circumferential force alone (LFT = 0), the lateral force alone (CFT - 0), and both forces combined. The plot of results in Figure 11B.2 shows fairly close agreement among the three test modes. The Della-Moretta experiment was performed in an accelerated manner under conditions which ranged from light to highly Figure 11B.2: Tread wear versus slip energy Tire wear (da3/k.) 0. 10 0 0.09. X 0. 0QB-1o 0.07- 0. 08 0.05 + X 0.04 0.03. + X x Longitudinal force aode O Sideways force mode 0.02- + Combined forces X0 Tire wear poin on gravl 0.01 0 0 200 400 600 BO0 3000 1200 1400 1600 1800 Slip energy per distance (kJ/km) Source: Adapted from Della-Moretta and Sullivan, 1976. Note: The ablative wear results have been omitted from the graph for illustration purposes. 276 AGGREGATE TIRE WEAR PREDICTION severe involving values of the friction ratio TFT/NFT and the slip IX) as high as 0.5 and 0.15, respectively. This may explain why there' is no detectable intercept term TWTo which is associated with the relatively small unmeasured forces. The test conducted by Bergman and Crum (1973) was a close simulation of actual vehicle operations. A 5,100-lb passenger car with bias belted tires was driven over several routes of different surface textures at various average speeds. The lateral and longitudinal forces on the vehicle were measured along the route continuously. Each data point represents route-average tire wear and forces averaged over several runs, with tire rotation among the wheels after each run. The most and least severe test conditions had route-average friction ratios, TFT/NFT, of 0.050 and 0.007, respectively; their respective measured tread wear rates were 12.4 and 1.46 mils/100 miles. From the observed data, Bergman and Crum obtained regression results with R2 = 0.984 and the intercept TWTO equal to 46 and 17 percent of the average and maximum wear rates, respectively. Most important, the lateral force coefficient was found to be greater than the circumferential one by 14 percent. The foregoing results seem to indicate the relative independence of the tire wear rate on the direction of the total tangential force. In a later, larger scale experimental study, Hodges and Koch (1979) used the slip energy theory directly for tire wear prediction without explicitly testing the lateral/circumferential tire stiffness assumption. The study employed two four-wheel drive instrumented vehicles and six 4,100-lb passenger cars to test the wear behavior of bias, bias belted and radial tires. The instrumented vehicles were used to quantify the wear-slip energy relationships for the three types of tire construction and various surface textures. The passenger cars were used to provide independent wear observations for comparison with the wear prediction using these wear-slip energy relationships. Good agreement was obtained although the published results provided no information for evaluating the later/circumferential tire stiffness assumption. Empirical slip, slip energy and wear coefficients In constructing a tire wear prediction model, Zaniewski et al., (1982) compiled a set of slip and slip energy coefficients (b/k and Ko, respectively) obtained from several experimental studies. These coefficients are reproduced in Table 11B.1 (with adapted physical units), along with the corresponding wear coefficients (b K0). The results are for both radial and bias tires tested over bituminous surfaces. In the Hodges and Koch (1979) experiment in which passenger cars were used, the radial tires tended to have considerably smaller slip coefficients than the bias tires. This is consistent with the general tendency of radial tires to provide more grip for the same amount of slip. There is no consistent pattern between the slip energy coefficients of bias and radial tires. Overall, the passenger radial tires tended to wear slower than the bias tires, as indicated by the smaller wear coefficients. AGGREGATE TIRE WEAR PREDICTION 277 Table 11B.1(a): oSumary of available slip energy coefficients Slip coefficient Wear coefficient Vehicle Surface type, tested & aggregate & b 1 Energy2 b source location Tire type --K CT = - K k 0 Passenger Nevada: cars Bituminous Radial 13 0.031 0.0275 0.00086 4,100 lb. concrete of Radial 2 0.038 0.0219 0.00084 (Hodges crushed & sifted Bias 1 0.067 0.0240 0.00160 & Koch, aggregate from Bias 2 0.055 0.0230 0.00128 1979) natural bank Bias 3 0.045 0.0285 0.00129 deposits with Bias 4 0.059 0.0275 0.00161 abrasion 20-25%4 Connecticut: Bituminous Radial 1 0.024 0.0163 0.00028 concrete of Radial 2 0.026 0.0105 0.00027 gravel from Bias 1 0.050 0.0076 0.00038 natural bank Bias 2 0.059 0.0078 0.00046 deposits with Bias 3 0.040 0.0103 0.00041 abrasion 35-37% Bias 4 0.042 0.0105 0.00044 Texas: Bituminous Radial 1 0.029 0.0131 0.00038 concrete of Radial 2 0.023 0.0174 0.00040 crushed lime- Bias 1 0.062 0.0115 0.00072 stone aggregate Bias 2 0.050 0.0111 0.00055 from deposits Bias 3 0.050 0.0106 0.00053 with abrasion Bias 4 0.042 0.0141 0.00059 26-29% Dump truck Nevada: 5-ton Bituminous 10.00-20 (Della- concrete of 12-ply, Moretta crushed basalt, bias Sullivan, 1" nylon- 0.10 0.0570 0.00570 1977) Goodyear Super High Miler (See notes on following page) 278 AGGREGATE TIRE WEAR PREDICTION Table 11B.1(b): Summary of available slip energy coefficients (continued) Slip coefficient Wear coefficient Vehicle Surface type, tested & aggregate & b 1 Energy2 CT b source location Tire type k K t k0 Light Nevada: truck, Bituminous 7.50-16 0.091 0.0616 0.00562 5,000 lb concrete of 10 ply, (Barriere, crushed basalt, Bias et al., 1 Goodyear 1974) Super High Miler 1970 Jeep Nevada: Wagoneer Bituminous G78-14: B 0.045 0.0496 0.00225 (Barriere6 concrete of polyester/ et al. crushed basalt, fiberglass 1974) 1" bias belted 1977 Jeep California: Wagoneer Open graded 1OR15LT 0.077 0.0735 0.00566 modified asphalt Recap 4 wheel concrete logging drive overlay truck tires (Jones radial Della- California: Moretta, Dense graded 0.091 0.0257 0.00252 1979) asphalt concrete road-mixed Footnotes for Table 11B.1(a) and 11B.1(b): Slip coefficient expressed in meters of wheel slip per meter of wheel travel. 2 Slip energy coefficient expressed in dm3/1,000 km of wheel travel per joule per meter of slip energy. 3 Tire types - all mounted on 6.0 x 15.0 rims - passenger car tires: Radial 1 Goodyear steel belted radial - 76.5 in.3 (total volume of available tread rubber) Radial 2 Firestone steel radial 500 - 78.3 in.3 Bias 1 Uniroyal Tiger Paw - polyester bias - 82.0 in.3 Bias 2 Goodyear Power Cushion - polyester/fiberglass bias belted 79.8 in.3 Bias 3 B.F. Goodrich Long Miler - polyester/fiberglass belted 101.5 in,3 Bias 4 Cooper Lifeliner Premium 78 - polyester/fiberglass belted 101.5 in.3 4 ASTM Standard Abrasion Test C535 Source: Adapted from Zaniewski et al., 1982. AGGREGATE TIRE WEAR PREDICTION 279 The data for truck tires show wear coefficients as large as 10 or more times those for passenger car tires. This results from both greater slip and slip energy coefficients for truck tires, which, in turn, may be attributable to the fact that truck tires are typically operated under much higher loads and inflation pressures than passenger car tires. Jones and Della-Moretta (1979) conducted an experiment to compare tire tread wear over dense and open graded asphalt concrete. As shown in Table 11B.1, they found that while the slip coefficients were similar, the slip energy coefficient for the open graded surface were almost three times as large as that for the dense graded surface. The wear coefficients in Table 11B.1 are compared with the estimation results in sub-section 11.3.2. Paved versus unpaved surface Della-Moretta (1974) explained the differences in tire slip behavior between paved and unpaved surfaces. For paved surfaces, slip occurs only between the tire and the road. For unpavedsurfaces, slip may be divided into two parts: the first between the tire and the road surface material immediately adjacent to the tire (termed the "bound slip") and the second between the road surface material adjacent to the tire and the material below it (termed the "loose slip"). Both slips give rise to the total slip observed on the tire but only the first causes tread wear. Since the bound slip on unpaved roads was not immediately observable, Della-Moretta assumed it to equal the slip observed on paved roads of similar aggregates. One test run was made over a gravel road and the bound slip was estimated using the slip-friction ratio relationship obtained from an equivalent paved road with the friction ratio TFT/NFT measured for the run. The observed wear is plotted against the estimated slip energy as shown in Figure 11B.2. The good agreement with the paved road data points may be attributed to similarity in the abrasiveness of the surface aggregate as well as similarity in the bound slip values. In fact, according to the Schallamach-Turner slip model, small interfacial slips should depend only on the mechanical properties of the tire irrespective of the road surface, provided that the latter affords sufficient traction. The simplified slip relationships do not apply to slippery surfaces since their available coefficients of friction are small. Della-Moretta's unpaved road bound slip assumption was used in the Brazil data analysis. 280 AGGREGATE TIRE WEAR PREDICTION APPENDIX 11C AGGREGATE TREAD WEAR MDEL: FORMULATION AND NUMERICAL TESTING 11C.1 FORMULATION OF AGGREGATE TREAD WEAR MODEL Following the same aggregate prediction procedure for speed in Chapter 7, the route is represented by two homogenous segments, one positive and one negative grade of the same length, roughness, horizontal curvature (and superelevation), and absolute gradient. The length of each idealized segment equals that of the actual route. The roughness, horizontal curvature, superelevation and absolute gradient are averages of the values measured over short homogeneous subsections of the actual roadway. The aggregate tire wear model computes tire consumption per unit distance travelled on each of these idealized segments and then averages out these predictions to obtain the tire wear prediction for the round trip. For each idealized segment the following road characteristics are the same: QI = the road roughness, in QI; C = the horizontal curvature, in degrees per km road length; and SP = the superelevation, in fraction. For the quantities that differ between the two segments, subscripts u and d are used to denote the uphill and downhill segments, respectively. Let RF be the average rise plus fall of the actual roadway, in m/km. The gradients for the uphill and downhill segments, expressed as fractions, are thus obtained as: GR - R (11C.1a) 1000 GRd = - RF (11C.1b) 1000 For a given vehicle let TWTU and TWTd denote the volume of rubber worn per 1000 tire-km on the uphill and downhill segments, respectively. From the micro tread wear model of Equation 11.8, TWTu and TWTd are given by: CFT2 LFT2 u u (1.a TWT = TWT + CT + CT (11C.2a) u Ou NFT uNFT LFu, LFd = the vehicle lateral force on the uphill and downhill segments, respectively; and AGGREGATE TIRE WEAR PREDICTION 281 CFT2 LFT2 and TWTd = TWTOd + CTcd d + CT id d (11C.2b) NFT NFT where the terms are as defined earlier but with subscript u and d representing the uphill and downhill segments, respectively. The tire wear predicted for the whole route is computed as the average of the predictions for each road segment: TWT = 0.5 (TWTu + TWT d) (110.3) or TWT = TWT + CT FT2+ CTLFT2 (11C.4) NFT NFT where TWTo = 0.5 (TWTOu + TWTOd); CTc = 0.5 (CTcu + CTcd); CTk = 0.5 (CTzu + CTed); CFT2 = 0.5 (CFT2 + CFT2) u d and LFT2 = 0.5 (LFT2 + LFT2). u d The parameters TWTO, CTc and CTZ are to be estimated. The circumferential and lateral forces per tire are computed as: DF CFT u (11C.5a) u OFT d=ADF. (11C.5b) NT d LF LFT = -u (11C.5c) u LFT - (11C.5d) d where DFu, DFd = the vehicle drive force on the uphill and downhill segments, respectively; LFu, LFd = the vehicle lateral force on the uphill and downhill segments, respectively; 282 AGGREGATE TIRE WEAR PREDICTION and NT = the total number of tires. Similarly, the normal force per tire is given by: NFT = m g (11C.6) NT where m is the vehicle mass and g the gravitational constant as defined in Chapter 2. The vehicle drive forces are computed as in Chapter 2, i.e., DF = mg GR + mg CR + 0.5 RHO CD AR VSS (11C.7a) DFd = mg GRd + m g CR + 0.5 RHO CD AR VSSd (11C.7b) where VSSu and VSSd are the steady-state speeds on the idealized segments, computed according to the procedure in Chapter 7; and all other variables are defined before. The coefficient of rolling resistance, CR, is a function of road roughness, QI, as given in Chapter 2. The vehicle lateral forces are computed as in Chapter 3, i.e., m VSS2 u LF = - mg SP (11C.8a) m VSS2 LFd = ---- - mg SP (11C.8b) RC where RC = the radius of curvature of each road segment, in meters, computed as inverse function of the horizontal curvature, C (as defined in Chapter 3): RC - 18,0000 for C > 0 (11C.9) ir C and SP is the average superelevation, expressed as a fraction. 11C.2 NUMERICAL TESTING The aggregate tread wear prediction model described above was tested for numerical accuracy using the micro transitional and non-transition tread wear prediction models as the benchmarks. The testing procedure employed the tread wear model coefficient estimates in Section 11.4 which were assumed for testing purposes to be the "correct" coefficients. This assumption was considered appropriate because we were mainly interested in relative, not absolute, errors. The micro tread wear model of Equation 11.8 was first quantified into the following: CFT2 LFT2 TWT = 0.164 + 0.01278 [ -T + --2] (11C.10) NFT NFT where TWT is expressed in dm3/1,000 tire-km. The above relationship served as the basis for both the aggregate and micro models. The micro prediction procedures are summarized as follows: AGGREGATE TIRE WEAR PREDICTION 283 Micro transitional model The prediction formula is: ATIRE. TWT = (1C.1 L where the summation is over all simulation intervals; and ATIREj is the predicted amount of rubber lost over simulation interval j, in dm3, given by: CFT? LFT ATIRE. = AL. { 0.164 + 0.01278 [ + -- ] } (11C.12) NFT NFT where CFTj and LFTj denote the average circumferential and lateral forces on the tire, in newtons; and NFT is the normal force on the tire as given in Eq.10.27. CFTj and LFTj are given by: CFT. = 1- Wm' a. + m g (GR. + CR.) + A V?} (11C.13) NT 3 m V2 LFT. = -- - m g sp (11C.14) NT RC. J where aj is the vehicle acceleration as defined in Equation 10.5 and the other variables are as defined previously, but with subscript j denoting those that are specific to simulation interval j. Micro non-transitional model The prediction formula is: I TIRE TWT = s (11C.15) L where the summation is over all homogenous subsections; and TIRES is the predicted amount of rubber lost over subsection s, in dm3, given by: CFT2 LFT2 S S TIRE = L {0.164 + 0.01278 [--- + - ]} (11C.16) NFT NFT where CFTS and LFTS denote the circumferential and lateral forces on the tire over homogeneous subsection s, respectively; and the other variables are as defined before. CFTS and LFTS are given by: 284 AGGREGATE TIRE WEAR PREDICTION CFT = [mg (GR + CR) + A VSS2 (11C.17) s NTs s s] m VSS.18 NT RC s where all variables are as defined previously, with subscript s denoting those that are specific to subsection s. Results The test vehicle characteristics and the steady-state model parameters adapted for the test vehicles (as described in Chapter 10) were used with the above prediction procedures. The detailed road characteristics employed were obtained from five 10-km road sections used in the fuel validation experiment (with summary statistics given in Table 8.2). The numerical results are summarized in Figures 11C.1 and 11C.2, and Tables 11C.1 and 11C.2. Figure 11C.1 plots aggregate tread wear predictions against predictions made by the micro non-transitional models for all test vehicles; each data point represents a 10-km journey of a test vehicle in one direction. The points are superimposed by the fitted ordinary least squares regression line passing through the origin. Figure 11C.2 does the same as Figure 11C.1, but for the micro transitional model. Tables 110.1 and 11C.2 give a summary of regression results by test vehicle for aggregate vs micro non-transitional and aggregate vs micro transitional predictions, respectively. It can be seen from the regression statistics (Table 11C.1) and data plot (Figure 11C.1) that the aggregate and macro transitional models produce almost identical predictions, implying negligible errors due to numerical approximation in the aggregation procedure. As seen in Table 11C.2 and Figure 11C.2, predictions by the aggregate model correlate very well with those obtained from speed profile simulation, with R2 values for the without-intercept regressions falling within a 0.91-0.99 range. The slopes of all without-intercept regressions are smaller than one. The regression with all vehicles combined has a slope equal to 0.965, indicating a relatively small (3.5 percent) average upward bias on the part of the aggregate model. This seems sensible since the driver behavior assumptions underlying the simulated profiles are expected to cause less tire tread wear than that predicted on the basis of steady-state speed profiles. AGGREGATE TIRE WEAR PREDICTION 285 Figure 11C.1 The plot of steady-state tire wear prediction vs. tire wear predicted using aggregate method Tire wear predicted by micro non-transitional method (No. of tires per vehicle per 1000 km) 3. 5- 3. 0- 2. 5- 2. 0- 1.5- 1.0- 0. 5- 0. O - 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Tire wear predicted by aggregate method Legend Class * * * Large bus 0 Light dsl truck A A A Heavy truck 0 0 0 Art. truck - Regression line Source: Road characteristics data from the Brazil-UNDP-World Bank highway research project, and relationships as given in the text. 286 AGGREGATE TIRE WEAR PREDICTION Figure 11C.2: The plot of simulated tire wear prediction vs. tire wear predicted using aggregate method Tire wear predicted by micro non-transitional method (No. of tires per vehicle per 1000 km) N1 3. 5- O 2. 5- 2. 0- 1.5 - 1.0 * 0.5 - 0. 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Tire wear predicted by aggregate method Legend Class * * * Large bus Q Q Light dsl truck AAA Heavy truck 0 0 0 Art. truck Regression line Source: Road characteristics data from the Brazil-UNDP-World Bank highway research project, and relationships as given in the text. AGGREGATE TIRE WEAR PREDICTION 287 Table 11C.1: Regression results: micro non-transitional versus aggregate tread wear predictions Standard Vehicle Number of Intercept Slope R2 error of observations residuals Large bus 19 0.00 0.999 1.00 0.014 (-0.7) (198.0) - 0.996 1.00 0.014 (441.8) Light truck 15 -0.00 0.999 1.00 0.002 (-1.3) (289.5) - 0.995 1.00 0.002 (634.8) Heavy truck 15 -0.01 1.010 1.00 0.005 (-1.8) (151.7) - 0.999 1.00 0.005 (400.8) Articulated truck 17 -0.00 1.001 1.00 0.049 (-1.2) (616.6) - 1.00 1.00 0.050 (1072.0) All vehicles 66 -0.00 0.999 1.00 0.008 combined (-1.2) (625.9) - 0.997 1.00 0.008 (1008.8) Source: Analysis of Brazil-UNDP-World Bank highway research project data. Table 11C.2: Regression results: micro transitional versus aggregate tread wear predictions Standard Vehicle Number of Intercept Slope R2 error of observations residuals Large bus 19 0.07 0.938 0.97 0.115 (1.3) (22.7) - 0.984 0.97 0.117 (51.7) Light truck 15 0.03 0.908 0.98 0.023 (1.9) (23.3) - 0.974 0.98 0.025 (51.5) Heavy truck 15 0.09 0.777 0.92 0.047 (2.7) (12.4) - 0.935 0.91 0.057 (35.8) Articulated truck 17 0.07 0.900 0.98 0.107 (1.5) (25.5) - 0.943 0.97 0.110 (45.5) Source: Analysis of Brazil-UNDP-World Bank highway research project data. 288 AGGREGATE TIRE WEAR PREDICTION APPENDIX 11D AVERACE RUBBER VOLUME PER TIRE For a given tire size and type, the volume of rubber in the tire tread was calculated on the basis of dimensions furnished by Brazilian tire manufacturers. The dimensions considered for this calculation were the tire diameter, tread depth and upper and lower tread widths. The formula for calculating the volume (in dm3) is: Volume = i x Diam x Area (11D.1) 1,000,000 where "Area" = the area of the tread cross section in mm 2, and "Diam" = the tire diameter in mm. The tread cross-section is a trapezoid with the upper and lower bases representing the upper and lower parts of the tread (the lower part being the tread surface in contact with the ground). In general, the two bases differ by 10 mm. The height of the trapezoid (which corresponds to the tread that will eventually be worn away) is equal to the tread depth minus the 1.6 mm remaining tread required by law. The empty parts of the trapezoid (which correspond to the gro-ves) are approximtely 30 percent ofthe total area of the trapezoid. Theiafore, the cross-sectional area of the tread to be worn away is calculated (in mm2) as follows: Area = LL + LU x (DE - 1.6) x 0.70 (11D.2) 2 where LL = the length of the lower trapezoid base, in mm; LU = the length of the upper trapezoid base, in mm; DE = the depth of the grooves, in mm; Considering that LU = LL + 10, we have: Area = (LL + 5) x (DE - 1.6) x 0.70 The estimated tread rubber volumes for different tire sizes and types are shown in Table 11D.1. The tires listed in the data base are of two types: highway (h) and mud and snow (m/s). For a given size, the mud/snow tire has a greater volume of rubber in the tread area than the highway tire and is used only on the driving wheels of trucks. Unfortunately, no information was recorded in the road user survey to indicate the type of each tire. Therefore, it was necessary to estimate the average wearable rubber volume per tire on the basis of the axle-wheel configuration of the vehicles on which the tire were mounted. The axle-wheel configurations of the various vehicle classes are presented in Table 11D.1, along with the corresponding numbers of highway and mud/snow tires. AGGREGATE TIRE WEAR PREDICTION 289 The average wearable rubber volume per tire, VOL, is computed as: N VOL + N VOL VOL = h h N/S rn/S (11D.3) N + N h m/s where VOLh and VOLm/s denote the volumes of one highway and mud/snow tire, respectively; and Nh and Nm/s denote the numbers of highway and mud/snow tires, respectively, for a given vehicle of known axle configuration. Table 11D.l: Wearable tire rubber volumes by tire size, vehicle class, and axle configuration Axle Number Tire size Vehicle class configuration1 of tires 9.00/20 10.00/20 11.00/22 Volume (dn3) tor highway tires: 5.76 6.85 8.00 Buses 1.2 6 Medium trucks 1.2 2 Heavy trucks 1.22 6 Articulated trucks 1.2 - 222 14 Volume (di3) for mud and snow tires: 7.24 7.98 9.77 Buses 1.2 0 Medium trucks 1.2 4 Heavy trucks 1.22 4 Articulated trucks 1.2 - 222 4 Average volume (dm3) used in analysis: Buses 1.2 6 5.76 6.85 8.00 Medium trucks 1.2 6 6.75 7.60 9.18 Heavy trucks 1.22 10 6.35 7.30 8.71 Articulated trucks 1.2 - 222 18 6.09 7.10 8.39 1 This code is the same as that used in the GEIPOT, 1982. Source: Adapted from information provided by Brazilian tire manufac- turers. CHAPTER 12 Vehicle Maintenance, Depreciation, Interest and Utilization Vehicle maintenance, depreciation and interest are important, interrelated components of the costs of vehicle ownership and operation. Maintenance parts and labor typically constitute 15-35 percent, and depre- ciation and interest 15-25 percent, of total costs excluding overhead; together they account for about the same proportion as fuel and tire consumption combined. However, maintenance, and particularly the capital charges of depreciation and interest, are rather different in nature from such direct "running" costs as fuel and tire wear. First, depreciation and interest are less related to actual vehicle kilometers driven than they are to the passage of time, a point to which we return below. Second, vehicle owners normally interrelate expected future maintenance costs and capital costs of a new vehicle in decisions concerning maintenance and replacement, and the owner normally has some greater discretion over when or whether to incur these expenditures and in what form. Some owners may prefer to spend more on maintenance to prolong the life of the vehicle, while others may spend less on maintenance and replace the vehicle more frequently; either course may be optimal, depending on the individual owner's specific circumstances. A formal theoretical model of this type of integrated behavior is provided in Chapter 8 of Chesher and Harrison (1987). We have not attempted to statistically estimate such "integra- ted" models, however, since our focus is not on vehicle maintenance and replacement decisions per se, but rather on the effect of road character- istics on the total costs of vehicle ownership and operation. Our parti- cular objective permits some simplification of the task through separating the estimation of maintenance costs from the estimation of vehicle depre- ciation and interest. Nonetheless, the task is still a difficult one. Because of limitations in both the underlying theory and the empirical data base, the estimation of the relationships of vehicle maintenance, depreciation and interest costs to road characteristics is somewhat cruder, and relies on more restrictive assumptions, than that for fuel consumption, tire wear and travel time. We deal first with vehicle maintenance and then with depreciation and interest costs. 12.1 VEHICLE HAINTENANCE PARTS AND LABOR Vehicle usage-related stresses that cause wear or failure of vehicle parts are mainly of two types: those associated with road rough- ness and those associated with road geometry. The former type of stresses cause, through their repeated applications, wear on the steering and sus- pension systems as well as failure of certain components such as springs 291 , 292 VEHICLE MAINTENANCE AND UTILIZATION and brackets (Gillespie, 1981). The stresses associated with road geo- metry come in the form of forces imposed on the engine, the vehicle drive train (gear box, clutch, rear axle, etc.), and the brakes in order to pro- pel or resist the vehicle against forces acting on the vehicle. Repeated applications of these forces cause wear and failures of the vehicle compo- nents that bear against those forces. In addition, abrasive dust frequently occurring in dry climates can enter the engine and thereby cause extra wear. An attempt was made in Brazil to estimate a mechanistic model for vehicle maintenance parts similar to those given above for fuel and tire wear. A linear model form was postulated to relate parts consumption per 1,000 vehicle-km to vehicle suspension and propulsive energies, per distance unit. However, becausa of limitations in both the existing body of theory and the available data in Brazil, the exercise was not success- ful. As a consequence, for vehicle maintenance costs, resort was made to simpler models correlating spare parts and mechanics' labor, respec- tively, with road characteristics, as reported by Chesher and Harrison (1985). Since these relationships have been adopted for the HDM-III model, we repeat their results here, with only minor changes for differ- ences in units, for the reader's convenience. 12.1.1 Maintenance Parts Model Spare parts consumption was found to depend on the roughness of the routes on which the vehicle ply and the average age of the fleet of vehicles of the same class. The proxy for vehicle age is the average cumulative distance driven by vehicles in the region. The age effect and the roughness effect combine multiplicatively. Holding the age constant, the relationship between parts consumption and roughness is generally non- linear with an exponential model providing the best fit, although for trucks a linear model was found to be more suitable. However, since the exponential relationship overpredicts parts consumption for high values of roughness, a linear form is assumed beyond a certain transitional rough- ness value. The parts consumption cost model may be written as: APART = CP0 exp(CP QI) CKM for QI < QIP 0 (12.1) where APART = parts cost per 1,000 vehicle-km for the given vehicle class expressed as a fraction of the cost of a new vehicle; CPO = constant coefficient in the exponential relationship between spare parts consumption and roughness; CPq = roughness coefficient in the exponential relationships between spare parts consumption and roughness (per QI); VEHICLE MAINTENANCE AND UTILIZATION 293 CKM = average age of the vehicle group in km, defined as the average number of kilometers the vehicles belonging to theparticular vehicle class in the region have been diven since they were built; Kp = age exponent; and QIPO = transitional value of roughness in QI beyond which the relationship between spare parts consumption and roughness is linear. The tangential extension of the above relationship for values of roughness higher than QIPO may be derived analytically as: APART = (P0+ 1QI) CKMKP for QI > QIP0 (12.2) where PO = CPO exp(CPq QIPO)(1 - CPq QIPO) P1 = CPO exp(CPq QIPO) CPq Figure 12.1 depicts the above relationship for a large bus for two values of CKM. The broken lines show the part of the curves replaced by the tangential extension. It should be noted that if the value of QIPO is very low (notionally zero), the relationship between parts consumption and roughness for a given value of age becomes linear. For the sake of uniformity of presentation the completely linear form is treated as a special case of the piece-wise exponential-linear form, with QIPO value being zero. In this case, the coefficients of the equation of the tangential extension become PO = CPO P1 = CPO CPq Using these relationships we can express a linear relationship in the standard" form, as follows. If the linear prediction equation is: APART = (PO + P1 QI) CKMKp then CPO = P0 CPq = P1/P0 and QIPO= 0 The values of the parameters using Brazil data are given in Table 14.4 of Chapter 14. It may be noted that, generally, the age of the vehicle is an intermediate construct determined from the optimal scrapping date decision. However, it has been assumed in this volume that average age and vehicle lifetime are exogenous variables to be specified by the user. This amounts to the assumpticn that the age distribution of the vehicles 294 VEHICLE MAINTENANCE AND UTILIZATION Figure 12.1: Maintenance parts model for a large bus APARTS (%) a. 20o C. 25 0. 291 C. 10 CKM=300,000 0. 05 ass- -- - - - - . -- - D 25 50 75 100 125 150 175 200 225 250 Roughness QI Source: Analysis of Brazil-UNDP-World Bank highway research project data. in the region is stationary, and in any case does not affect the estimated maintenance costs significantly. 12.1.2 Maintenance Labor Hours Model Maintenance labor resources required for operating a vehicle were found to be related to the maintenance parts consumption and, in the case of large buses in Brazil, roughness of the route on which the vehicle plies. The roughness effect, when significant, combines multiplicatively with the parts effect. The model, in its full generality, may be written, ALABOR = CLO APARTCLP exp(CL QI) (12.3) where ALABOR = predicted number of maintenance labor hours per 1,000 vehicle-km; CLO = constant coefficient in the relationship between labor hours and parts consumption; CLp = exponent of parts cost in the relationship between labor hours and parts consumption; CLq = roughness coefficient in the exponential relationships between labor hours and roughness (per QI). A value of zero for CLq would imply that the roughness effect is not significant. VEHICLE MAINTENANCE AND UTILIZATION 295 The constant coefficient CLO captures the result of labor-capital trade-off in the specific economic regime, and, as such, it is of importance to calibrate this parameter for a given application. On the other hand, the parts exponent CLp reflects the economies of vehicle size and is found to be reasonably constant for a similar vehicle class between Brazil and India. Thus, it is not anticipated that the latter parameter needs to be locally calibrated, especially given the richness of vehicle classification in the Brazil study. In the Brazil study, only buses had a significant roughness effect. However, in the Indian study the absence of roughness effect was not decisive in the case of a number of vehicle classes. Guidance for local calibration of the maintenance parts and labor models is provided in Chapter 13. 12.2 ]EPRECIATION AND INTEREST CHARGES A vehicle constitutes a medium-term capital asset; its purchase cost represents an investment which yields services over a period of several years. The market value of the asset declines with both the passage of time (typically rather quickly at first and then more slowly) and, generally to a much smaller degree, with the amount and type of usage. It is this loss of market value (as distinct from some physical or accounting concept) which is here taken to represent "depreciation", measured on an annual basis. Since depreciation occurs gradually, at any given point in time there is a residual (undepreciated) amount of capital tied up in the vehicle, which normally could be invested elsewhere, so that an annual interest charge, in addition to the annual depreciation, is incurred. The interest charge is purely a function of time and the undepreciated value of the vehicle. In order to relate these predominantly time-related costs to other predominantly usage-related costs, it is convenient (and customary) to divide annual depreciation and interest charges by an annual utilization factor, such as average vehicle kilometerage. It should be noted, however, that where annual utilization is much above average, one would expect vehicle depreciation to increase and the vehicle life (in years) to be foreshortened, although probably less than proportionately. Thus, in order to estimate the effects of varying road characteristics on vehicle depreciation and interest costs, one needs to establish the effects of road characteristics on (i) vehicle life and (ii) vehicle utilization. Unfortunately, neither of these relationships has ever been properly quantified empirically; traditionally they have simply been assumed in benefit-cost studies of road investments. In his classic treatise, de Weille (1966) assumed that annual utilization was fully proportional to any change in vehicle speeds due to changes in road characteristics, but the effect on depreciation and interest charges per vehicle kilometer was attenuated by the concomitant assumption that the vehicle life in years was reduced somewhat, so that the annual depreciation charge was increased. 296 VEHICLE MAINTENANCE AND UTILIZATION Studies in Kenya (Hide et al., 1975) India (CRRI, 1982; Chesher, 1983), and Brazil (GEIPOT, 1982, Volume 5) have quantified the relationship of annual vehicle depreciation to the age of the vehicle in years (but not the effect of road characteristics thereon), and the studies in India and Brazil went on to investigate the effect of road characteristics on vehicle utilization. In the Brazil models, as reported below, road characteristics affect depreciation and interest costs per kilometer through utilization; vehicle life in years (and hence annual depreciation) is left to be exogenously specified by the model user as estimated from local evidence. Utilization is modelled through a general set of accounting identities which interrelate route length, total annual operating time and non-driving time (which are estimated as parametric constants), with vehicle speed and driving times (as a function of road characteristics). A change in road characteristics which leads to a change in vehicle speeds results in a less than proportionate change in annual vehicle utilization, and the vehicle fleet size necessary to meet a given level of transport demand changes proportionately. This model is not considered to provide an entirely adequate representation of the actual changes which will occur, particularly over the longer run, in vehicle utilization as a result of changes in the road network, since it is likely that the parameters of the utilization model (as well as the average life of vehicles) will be adjusted in response to any large scale improvements in the road network. Nonetheless, the model does rely less on such restrictive assumptions than do the traditional methods (which, indeed, rely entirely on a priori assumptions) for calculating depreciation and interest in quantifying the benefits of road improvements. When locally calibrated, as discussed below, the model should provide somewhat improved estimates of the impact of road changes on these important cost elements. The remainder of this chapter presents: 1. The mathematical formulation of the Brazil models relating vehicle depreciation and interest to utilization, and utilization to speed (Section 12.3); 2. The results of statistical estimation of the model using the Brazil road user cost survey data (Section 12.3 and Appendix 12A); and 3. A recommended model for predicting vehicle utilization which utilizes local information (Section 12.4 and Appendix 12B). 12.3 MDEL FORMULATION 12.3.1 Relationship of Vehicle Depreciation and Interest Costs to Vehicle Utilization The cost of vehicle depreciation per 1,000 km travelled, DEP, VEHICLE MAINTENANCE AND UTILIZATION 297 expressed as a fraction of the price of a new vehicle of the same class: DEP = 1,000 (12.4) LIFE AKM where LIFE = the average vehicle service life in years; and AKM = the average annual vehicle utilization, in km of travel per year. Similarly, the interest cost per 1,000 km travelled, INT, expressed as a fraction of the price of a new vehicle of the same class: INT = 1,000 AINV (12.5) 200 AKM where AINV is the interest rate in percent, and the annual interest charge is taken as the average of the residual vehicle value, decreasing in a linear fashion from full purchase price at the end of year 0 to zero at the end of year LIFE. The average service life for a given vehicle class may be assumed to be equal to a constant value LIFE0 (in years) for the region. That is, LIFE = LIFEO (12.6a) where LIFEO = the baseline average vehicle life, in years. Alternatively, following de Weille (1966), the average vehicle service life may be assumed to be related to the predicted vehicle operating speed, as follows: LIFE = 1 ( S + 2) LIFEO (12.6b) 3 S where So = the baseline average vehicle speed, in km/h, given by: So = AKMO/HRDO AKMO = the baseline average annual kilometerage, in km/year; HRDO = the baseline number of hours driven per vehicle per year; and S = the predicted round-trip speed, in km/h. 12.3.2 Vehicle Utilization - Simple Model Because of its importance in establishing the impact of changes in road characteristics, vehicle utilization became a major collection item of the road user cost surveys in Brazil and India. From the data obtained, GEIPOT (1982) and CRRI (1982) estimated vehicle utilization relationships as a function of vehicle and road characteristics as well as operating parameters including route length. In the Brazil road user cost survey, speed data were available only for some cars and buses, so in the 298 VEHICLE MAINTENANCE AND UTILIZATION statistical estimation vehicle utilization was not related to road characteristics through speed. This may partially explain why no clear geometric effects on vehicle utilization were found. In the India survey speed data were obtained from timetables and operating schedules. Although the speeds derived in this manner may be quite different from the actual speeds, vehicle utilization was found to be strongly influenced by speed. Subsequently, using a constant elasticity model on India data, Chesher (1983) obtained alternative relationships between vehicle utilization and speed. For bus operations, the elasticity of bus utilization with respect to speed was found to equal approximately 0.7. As will be seen, a subsequent theoretical analysis suggests that the elasticity should lie between zero and one. However, possibly due to unsatisfactory speed data, elasticities of utilization for the other vehicle classes could not be meaningfully quantified. The model formulated below is an adaptation of the "adjusted utilization" method in the earlier HDM-II model (Watanatada, et al., 1981). It assumes that each vehicle operates on a fixed route throughout a given year. The total time spent on making a complete round trip, TT, is given by: TT = TN + TD (12.7) where TN = the amount of time spent on non-driving activities as part of the round trip tour, including loading, unloading, refueling, layovers, etc., in hours per trip; and TD = the amount of time spent on driving over the route, in hours per trip. For a given class of vehicles of similar operating characteristics (nature and size of cargo carried, etc.), the amount of non-driving time per trip, TN, is assumed on average to be constant. Let RL denote the round trip driving distance or route length in km, and S the average round trip speed in km/h. Thus, the driving time, TD, can be written as: TD = RL (12.8) S Let HAV denote the vehicle availability, defined as the total amount of time the vehicle is available for vehicle operation, in hours per year. In general, HAV is equal to the total number of hours per year (8,760 h/y) less the time allowed for crew rest, infeasibility of vehicle operation (e.g., during holidays or hours labor does not normally work), vehicle repairs, etc. Within a given vehicle class, HAV is assumed to be constant and independent of vehicle speed; however, we would expect HAV to be dependent on labor work rules (e.g., scheduling of drivers and other crew members), vehicle age and condition, etc. Vehicle operators are assumed to maximize vehicle productivity by making as many trips possible within the vehicle availability constraint. Under this assumption, the number of trips per year, denoted by NTRIPS is given by: VEHICLE MAINTENANCE AND UTILIZATION 299 NTRIPS = HAV (12.9) TT The annual kilometerage, AKM, may be written as: AKM = NTRIPS RL (12.10) The above assumption is considered appropriate when there is no excess capacity" in the vehicle fleet within the region in question. During a period of economic downturn, it is possible that the total vehicle availability is not fully utilized, i.e., we have: NTRIPS < HAV TT so that the number of trips and the annual utilization are constants inde- pendent of vehicle speed. Thus, in the "excess capacity" case, the assump- tion of constant annual utilization is more appropriate. Private passenger cars and commuting vehicles would also fall into this case, since vehicle utilization is expected to be somewhat independent of speed unless there is significant residential and workplace relocation. As will be seen subse- quently, the "constant annual utilization" method is in fact a special case of the more general model described herein. Where there is "no excess capacity," Equations 12.9 and 12.10 yield a speed-sensitive annual vehicle utilization, AKM, as given by: AM = HAV RL (12.11) TT Substituting Equations 12.7 and 12.8 in Equation 12.11, we have annual vehicle utilization expressed as a function of the number of hours available, the operating speed, the non-driving time per trip, and the route length: AKM = RAV (12.12) TN 1 RL S where HAV and TN are model parameters to be estimated within a given class of vehicles. By differentiation it can be shown that the elasticity of vehicle utilization with respect to speed, denoted by EVU, is equal to the ratio of the driving time to the total trip time: EVU _ AKM S _ TD (12.13) aS AKM TT 300 VEHICLE MAINTENANCE AND UTILIZATION Since by definition the driving time, TD, is never greater than the total trip time, TT, the elasticity is always smaller than one. Denoting the number of hours driven per year by HRD, we have: HRD = TD NTRIPS (12.14) Substituting NTRIPS from Equation 12.9 in Equation 12.14 yields: HRD =TD (12.15) HAV TT Thus, the elasticity of vehicle utilization with respect to speed can be alternatively expressed as equal to the ratio of the annual number of hours driven to the annual number of hours available: EVU = HRD (12.16) HAV The estimation of model parameters HAV and TN is described in Appendix 12A. Using these estimated parameters and average route length in the estimation data, predicted vehicle utilization is graphed against vehicle speed for each vehicle class (except for the "bus without tachograph" and "medium/heavy truck non-tipper" classes) as shown in Figure 12.2. As expected, vehicle utilization for each vehicle class increases with vehicle speed but at a decreasing rate. For cars, utilities, and medium and articulated trucks, Figure 12.3 compares vehicle utilization predicted by the model estimates above with that predicted by the constant annual hourly utilization method. The latter method, which is represented by lines bearing plus symbols in Figure 12.3, employed the following formula: AKM = ----AKM(avg) (12.17) S (avgg) (avg) where S(avg) and AKM(avg) are the mean predicted vehicle speed and observed annual utilization for the survey vehicles, as computed in Table 12A.3. We can see that while predictions produced by these two methods are generally similar for average speeds, they are different for the low and high ends of the speed range. At the low end, the former method predicts a higher utilization than the latter. At the high end, the reverse occurs. As discussed earlier, the elasticity of vehicle utilization with respect to operating speed (EVU) is a useful parameter, and it is desirable to estimate its representative values for the vehicle classes. Two alternative methods were used as described below. Method 1 First, differentiate AKM in Equation 12.12 with respect to S to obtain the elasticity EVU expressed in the following form: VEHICLE MAINTENANCE AND UTILIZATION 301 Figure 12.2: Predicted vehicle utilization versus speed Vehicle utilization (kn/y) 200000- 175000- 150000- 125000 P c 100000 75000- 25000 0-- F 0- 0 10 20 30 40 50 60 70 B0 90 100 Speed (km/h) Note: These plots use the adjusted utilization method. Note: These plots compare the adjusted and the constant hourly utilization method indicated by continuous and broken lines respectively. Source: Analysis of Brazil-UNDP-World Bank highway research project data. 302 VEHICLE MAINTENANCE AND UTILIZATION Figure 12.3: Predicted vebicle utilization versus speed Vehicle utilization (km/y) 200000 - 175000- 150000 - 125000- 1000001- 75000- 50000111 D- lill ' I I O 10 20 30 40 50 60 70 80 90 100 Speed (km/h) Note: These plots use the adjusted utilization method. Source: Analysis of Brazil-UNDP-World Bank highway research project data. VEHICLE MAINTENANCE AND UTILIZATION 303 EVU - 8 AKM S _ 1 (12.18) 3 S AKM S TN + RL Then, for each vehicle class, use the estimated value of TN in Table 12.5 and average values of RL and S to compute an estimate of EVU. Method 2 This method uses the formula in Equation 12.16 (i.e., EVU = HRD/HAV) directly. For each vehicle class, the estimated value of HAV in Table 12A.5 is used. An average value of HRD is computed by averaging over the predicted values of HRD for the survey vehicles. The latter were computed by dividing the annual utilization of each survey vehicle by the corresponding predicted speed. These two methods employ the same source of information in highly aggregate manner but in somewhat different ways. However, if the survey vehicles in the different vehicle classes are not too heterogeneous with respect to their operating behavior, then the resulting elasticities should not be very different. The results of the calculations for all vehicle classes except the "bus without tachograph" and "medium/heavy truck - non-tipper" classes are summarized in Table 12.1. Except for the "articulated truck" class, the elasticities of vehicle utilization based on the two methods are quite similar. The relatively high values for the truck classes indicate relatively low percentages of non-driving time compared to the other vehicle classes. This implies that truck utilization tends to be more responsive to increased operating speeds. In this respect, car utilization is the least responsive, as indicated by the smallest values of elasticities. It should be noted that the average annual utilization value for cars (94,596 km/y) is signiiicantly on the higher side even for commercial usage. It is probably due to the special nature of business in which the cars in the sample were employed, namely, courier work in and around Brasilia. While the Brazil values are treated as defaults, it is recommended that users determine representative values for the region of application. Figure 12.4 illustrates graphically how the relationship between vehicle utilization and speed varies with the elasticity (assuming 10,000 km/y utilization and 54 km/h average route speed at the designated elasticity values). 12.4 RECOMMENDED VEHICLE UTILIZATION PREDICTION MODEL The estimated coefficients (in Table 12A.5) are specific to the Brazilian vehicle operators in the survey. As mentioned previously, the HAV and TN parameters depend on the trip distance, type of the cargo or passengers carried, the pick-up and delivery system, work rules, and other operating characteristics. Since these latter variables can vary from one locale to another, it is anticipated that the coefficients would be 304 VEHICLE MAINTENANCE AND UTILIZATION Table 12.1: Estimation of average elasticities of vehicle utilization Average Estimated Average Annual Route Predicted Predicted Hours Elasticity of vehicle Vehicle utilization length Speed h/year available Non-drive utilization, EVU class AEM RL S driven per year h/trip (k/y) (la) (lk/h) RD HAV TN Method 1 Method 2 Car 94,596 327 79.8 1,162 1,972 2.26 0.64 0.59 Utility 39,283 46 60.2 652 839 0.19 0.80 0.78 Bus with 102,404 328 64.0 1,601 2,302 1.57 0.77 0.70 tachograph Light 120,484 836 63.8 1,863 2,200 1.74 0.88 0.85 Mediun/heavy 98,369 188 54.6 1,825 2,227 0.72 0.83 0.82 truck-tipper Articulated 95,517 1,230 65.6 1,400 2,414 4.86 0.79 0.58 Source: Analysis of Brazil-UNDP-World Bank highway research project data. different in a new country or region. Therefore, it is advisable to develop a more generally applicable method for predicting vehicle utilization. The method should satisfy the following criteria: 1. Reliance on local data; and 2. Simplicity of input data requirements. With the above criteria in mind, we propose a general model which requires three input parameters representing regional or national averages: HRDO = an average annual number of hours driven; AKMO = an average annual utilization, in km per year; and EVU0 = an average elasticity of vehicle utilization. A relatively simple procedure for obtaining values of HRD0, AKMO and EVU0 is described in Chapter 14. Once these parameters are available for a given vehicle class, they can be used to "calibrate" the vehicle utilization model of Equation 12.2, using a simple algebraic procedure described in Appendix 12B. The resulting form of the vehicle utilization prediction model is given by: AKM (1 - EVU)+ EVU_a 1 -1 (12.19) AKMO HRDO S The behavior of predicted vehicle utilization with respect to the elasticity EVUO is of interest. On one extreme, when EVUO equals zero, Equation 12.19 reduces to: VEHICLE MAINTENANCE AND UTILIZATION 305 Figure 12.4: Predicted vehicle utilization versus speed for various EVU Vehicle utilization (km/y) 200000- 175000- 013A 150000 125000- . .--EVU =0 100000- 75000 50000 0- -- I * I l lI I ' 0 10 20 S0 40 50 60 70 80 90 100 Speed (km/h) Note: These plots use the adjusted utilization method. Note: These plots use the adjusted utilization method. Source: Analysis of Brazil-UNDP-World Bank highway research project data. 306 VEHICLE MAINTENANCE AND UTILIZATION AKM = AKMO (12.20) which is recognized as the "constant annual utilization method" mentioned earlier. On the opposite extreme, when EVU0 equals one, Equation 12.19 becomes: AKM = HRDO S (12.21) which is recognized as the "constant annual hourly utilization method" used in many studies. The two extremes of EVU0 = 0 and EVU0 = I are also shown in Figure 12.3 above (as a horizontal line and a positively sloped straight line through the origin, respectively). VEHICLE MAINTENANCE AND UTILIZATION 307 APPENDIE 12A ESTIMATION OF MDEL PARAMETERS 12A.1 DATA FOR ESTIMATION The data used in model estimation were obtained from the road user cost survey as described in detail in GEIPOT (1982, Volume 5) and Chesher and Harrison (1985). For analysis purposes the survey vehicles were divided into the following classes: cars, utilities, buses with and without tachographs, and light, medium/heavy and articulated trucks, covering various models and makes as shown in Table 12A.1. Summary statistics of the route lengths and annual utilization of these vehicle classes are given in Table 12A.2. Table 12A.3 provides summary statistics by vehicle class of the routes' length, rise plus fall, horizontal curvature, roughness, and percentage paved. Since speed data were not generally collected in the survey, the average operating speeds of the survey vehicles were predicted as a function of the above aggregate route characteristics using the aggregate speed prediction model described in Chapter 7. The closest test vehicle was matched against each of the vehicle classes as shown in Table 12A.1, and the steady-state speed model parameters adapted for those test vehicles (as described in Chapter 10) were used with the vehicle characteristics shown in Table 12A.4. The means and standard deviation of the predicted speeds by vehicle class are presented in Table 12A.2. 12A.2 MODEL ESTIMATION RESULTS AND ELASTICITIES OF UTILIZATION The vehicle utilization model of Equation 12.12 is non-linear in the parameters TN and HAV, so a non-linear estimation procedure provided in the Statistical Analysis System (SAS) was employed (SAS Institute, 1982). Ideally HAV and TN should be estimated as a function of operating characteristics such as the type and size of commodities carried, route length, company operating policy, labor work rules, etc. However, because the necessary information was not available in the survey in suitable disaggregate form, it was possible only to obtain representative values of the HAV and TN parameters for broad vehicle classes. Regression runs were carried out separately for the vehicle classes yielding results and coefficient estimates as summarized in Table 12A.5. Except for the "bus without tachograph" and "medium/heavy truck - non-tipper" classes, all coefficients are of the expected sign and reasonable magnitudes. For the vehicle classes with relatively short haul and presumably simple operations, namely, "utility" and "medium/heavy truck-tipper" with average route lengths of 46 and 188 km, respectively, the estimated non-driving time per trip of 0.19 and 0.72 hour, respectively, are smaller than the estimated values for the vehicle classes with relatively long haul operations, notably the "articulated 308 VEHICLE MAINTENANCE AND UTILIZATION Table 12A.1 Classification of vehicle makes and models for vehicle utilization analysis Vehicle class Make/modell Number of vehicles (test vehicle) Car VW-Brasilia 12 (VW-1300) VW-1300 87 VW-1500 5 VW-1600 36 VW-1600 (double carburator) 6 Fiat-147L 51 Total 197 Utility Ford F-75 Pick-up 11 (VW-Kombi) VW-Kombi 6 Total 17 With Without tachograph tachograph Bus MB-LPO 321/45/48 0 13 (MB-0362) MB-LPO 1113/45 36 54 MB-LP 113/48/51 19 121 MB-OF 1113/51 7 1 MB-OH 1313/51 3 7 MB-0 321 7 2 MB-0 326 1 0 MB-0 352 22 8 MB-0 362 0 33 MB-0 355 23 1 MB-1111 (truck chassis) 0 2 Total 118 242 Light truck Ford F 400 12 (Ford 4000) Ford F 4000 6 MB-L 608 14 Total 32 Tipper Non-tipper Medium/heavy truck MB-L/LPK/LPS 312 0 1 (MB-1113) MB-L/LK/LS 1113 40 41 MB-L/LK/LS 1313 23 3 MB-L/LK 1513 8 12 MB-L 2013 0 8 MB-L 2213 1 0 MB-LK/LB 2213 6 0 MB-L/LK 1519 16 0 Total 94 65 Articulated truck MB-LS 1519 35 (Scania 110/39) Scania-L 75 4 Scania-L 76 3 Scania-L 110 24 Scania-LS 36 2 Scania-L 111 45 Scania-LT 111 7 Scania-LK 140 7 Total 127 1 VW = Volkswagen; MB = Mercedes Benz Source: Authors' judgement based on Brazil-UNDP-World Bank highway research project data. VEHICLE MAINTENANCE AND UTILIZATION 309 Table 12A.2: Summary of route length and utilization Vehicle class Round trip Annual Number of Average (Number of Types of route length utilization round trips round trip vehicles) statistics (km) (km) per year speed(km/h) Car Mean 327 94596 374 79.8 (197) Std. dev. 167 43314 256 7.1 Utility Mean 46 39283 975 60.2 (17) Std. dev. 35 10791 281 6.6 Bus: Tachograph Mean 328 102404 411 64.0 (118) Std. dev. 203 34994 214 7.0 W/o tachograph Mean 319 82556 667 56.5 (242) Std. dev. 187 28181 1227 8.8 Truck: Light Mean 836 120484 209 63.8 (32) Std. dev. 364 33731 242 7.1 Medium/heavy Mean 188 98369 661 54.6 tipper (94) Std. dev. 113 15556 319 9.5 Medium/heavy Mean 836 93039 138 54.7 non-tipper (65) Std. dev. 256 29000 97 4.2 Articulated Mean 1230 95547 595 65.6 (127) Std. dev. 1835 56811 820 13.2 Source: Brazil-UNDP-World Bank highway research project data. truck" class with average route length of 1230 km and estimated TN equal to 4.86 hours. (Of course, the number of trips is inversely related to trip distance, so that the total non-driving time per year is much higher for the vehicles with short trip distances.) The estimated non-driving time per round trip, TN, for the "bus without tachograph" and "medium/heavy truck - non-tipper" classes are virtually zero which is clearly unrealistic. An examination of the statistics in Table 12A.2 reveals that the standard deviation of predicted speed for the "medium/heavy truck - non-tipper" class is rather small, only 7.7 percent of the mean, which is seemingly a result of a relatively small range of road geometry and roughness variation. The poor TN estimate for the "bus without tachograph" class does not seem to be caused by the small variation in predicted speed, the standard deviation of which (8.8 km/h) is comparable to those of the other vehicle classes. A plausible explanation is that the speeds were predicted for free-flowing conditions whereas a substantial portion of these bus routes may have been built-up areas in which the actual operating speeds were considerably lower. 310 VEHICLE MAINTENANCE AND UTILIZATION Table 12A.3: Summary of route characteristics Vehicle Types of Rise plus Horizontal class statis- fall curvature Road Percentage tics (m/km) (degrees/km) roughness paved Car Mean 30.0 64.5 61.5 80.8 Std. dev. 6.8 48.1 31.1 29.2 Utility Mean 13.4 17.6 121.3 21.4 Std. dev. 2.1 9.8 32.7 23.9 Bus: Tachograph Mean 22.7 16.3 64.3 65.4 Std. dev. 4.8 10.2 39.5 42.3 W/o tachograph Mean 26.1 42.8 96.0 44.3 Std. dev. 7.8 42.9 48.1 42.2 Truck: Light Mean 27.5 35.4 42.1 88.0 Std. dev. 6.4 49.6 31.2 29.0 Medium/heavy Mean 35.7 86.1 63.8 76.3 tipper Std. dev. 6.5 92.1 39.2 26.4 Medium/heavy Mean 30.0 28.8 61.1 76.8 non-tipper Std. dev. 3.2 8.9 24.3 14.6 Articulated Mean 27.6 40.8 57.1 73.9 Std. dev. 10.8 48.1 28.9 40.3 Source: Brazil-UNDP-World Bank highway research project data. Table 12A.4: Vehicle characteristics used in operating speed prediction Vehicle class Tare weight Load Aerodynamic Frontal (kg) (kg) drag coefficient area (m2) Car 960 0 0.45 1.80 Utility 1,320 300 0.46 2.72 Bus 8,100 2,300 0.65 6.30 (with/without tachograph) Truck: Light 3,270 2,000 0.70 3.25 Medium 5,400 4,500 Heavy 6,600 6,000 Articulated 14,700 13,000 0.63 5.75 Source: Authors' adaptation based on Brazil-UNDP-World Bank highway research project data. VEHICLE MAINTENANCE AND UTILIZATION 311 Table 12A.5: Summary of regression results and estimates of model parameters Parameter estimates Standard Number of error of Vehicle class vehicles HAy (h/y) TN (h/trip) R-square residuals Car 197 1,972 2.26 0.43 32,760 (9.7) (3.4) Utility 17 839 0.19 0.31 9,120 (5.1) (1.1) Bus: Tachograph 118 2,302 1.57 0.35 28,320 (17.1) (4.9) W/o tachograph 242 1,460 0.01 0.12 26,520 (42.0) (0.4) Truck: Light 32 2,200 1.74 0.53 23,520 (10.8) (1.4) Medium/heavy 94 2,227 0.72 0.51 10,900 tipper (26.1) (5.2) Medium/heavy 65 1,734 .0.0 0.24 25,495 non-tipper (13.5) (0.0) Articulated 127 2,414 4.86 0.71 30,600 (24.0) (6.6) Note: Values shown in parentheses are asymptotic t-statistics. Source: Analysis of Brazil-UNDP-World Bank highway research project data. 312 VEHICLE MAINTENANCE AND UTILIZATION APPENDIX 12B CALIBRATION OF RECOMMENDED VERICLE UTILIZATION PREDICTION HDDEL The procedure for calibrating the general vehicle utilization prediction model with parameters HRDO, AKMO and EVU0 defined in Section 12.4 is described as follows. First, from Equation 12.16, we express HAV as the ratio of HRDO to EVU0 HAV = (12B.1) EVU0 Then, express the ratio of the non-driving time per trip to the route length as: TN = HAV - HRD (12B.2) RL AKM0 Substituting Equation 12B.1 for HAV in Equation 12B.2 gives: HRD0 -RD - HRD0 TN = EVU 0 (12B.3) RL AKM0 Finally, substituting Equations 12B.1 and 12B.3 for HAV and TN/RL, respectivelyin Equation 12.12 yields the desired calibrated model: [1 - EVU0] EVU AKM = [+ (12B.4) AKMO S HRDO] VEHICLE MAINTENANCE AND UTILIZATION 313 APPENDIX 12C LUBRICANTS CONSUMPTION Lubricants consumption was not part of the Brazil study. For completeness, the following relationships where lubricants consumption is a function of roughness are reported. These relationships are as modified by Chesher and Harrison (forthcoming) from those obtained from the India study (CRRI, 1982): Passenger cars and utilities: AOIL - 1.55 + 0.011605 QI Light trucks: AOIL - 2.20 + 0.011605 QI Buses and medium and heavy trucks: AOIL - 3.07 + 0.011605 QI Articulated trucks: AOIL - 5.15 + 0.011605 QI where AOIL = the lubricants consumption, in liters/1000 vehicle-km. PART III CHAPTER 13 Guidelines for Local Adaptations As mentioned in Chapter 1, components of transport costs are generally assessed in the HDMS study as the product of a predicted physi- cal quantity and an exogenously determined price or unit cost. This approach is clearly superior to that of directly predicting costs, since physical, as opposed to economic, models are more easily transfered geo- graphically. Even so, transference problems still remain, and the objec- tive of this chapter is to provide the user with some guidelines on how the vehicle operating cost models presented herein may be adapted to a new locality. Essentially the same issues arise as to the effect of large changes over time in the technological and economic circumstances of the countries where the models were originally estimated, and similar proce- dures may be pursued to update the models. This chapter begins with a general discussion on the use of these guidelines (Section 13.1). The remaining sections provide specific guidelines dealing with models for predicting vehicle rolling resistance (13.2), speed (13.3), fuel consumption (13.4), tire wear (13.5), vehicle utilization (13.6) and maintenance parts and labor (13.7). 13.1 USE OF GUIDELINES For economic profitability (or even survival), transport opera- tors are expected to adapt their operations in response to the economic, social, technological and institutional setting of a given region. On the revenue side, the obvious tendency is to provide services where demand has not been satisfied or is growing. On the expenditure side, which is pertinent to this chapter, the operators may adapt by making appropriate choices not only on the vehicle technology (e.g., vehicle type, size, engine power, and suspension system) but also on the operating policy (e.g., vehicle operating hours, speeds, overloading, tire recapping and replacement). For example, in a high-wage economy, the tendency would be to purchase large, expensive trucks which carry large volumes of cargo, require minimum maintenance and provide comfort for high-speed travel over extended periods of time; on the other hand, since shift operations may be costly, the number of hours operated per year may be relatively short. In a low-wage economy, the opposite tendency would be expected. In another example, trucks operating in mountainous regions, e.g., in Nepal or Bolivia, need to employ large engines relative to their size, in order to provide the necessary propulsive power for hill-climbing; in contrast, operators in generally flat countries like Argentina or Australia can afford to use relatively small engines for even very heavy loads. 317 318 GUIDELINES FOR LOCAL ADAPTATION As discussed in Chapter 1, the ideal modelling framework would be to incorporate explicitly operators' adaptive behavior in the specifi- cations of the prediction models. However, since this theoretical ideal was impractical to implement, an alternative, but workable framework of physical-component modelling was adopted. Within the latter framework, the calibrated values of the model parameters generally reflect the adap- tations made by operators within a specific region and, strictly speaking, it would be desirable to re-calibrate them when applied in a new locali- ty. However, attempts were made in two ways to maximize model transfer- ability and thereby facilitate the task of local adaptation. First, the mathematical forms of the models were made as general as possible. The mechanistic-behavioral theories made this possible to a large extent for the speed and fuel models, and to a lesser extent for the tire wear and vehicle utilization models. The "aggregate-correlative" models of maintenance parts and labor reported in Chesher and Harrison (forthcoming) and summarized in Chapter 12, on the other hand, had less theoretical support. However, empirical testing of-the model forms across countries, i.e., Brazil and India, has yielded satisfactory results in certain respects. In particular, the exponents of the cumulative kilo- meterage age in the parts model and the parts variable in the labor model, were found to be reasonably stable. Second, to the extent possible, the models were specified as explicit functions of as many relevant vehicle characteristics as possible (e.g., total weight and engine power), which are routine input require- ments. Since these variables are expected to explain much of the varia- tion across countries, the calibrated parameters in the functions so formulated tend to be less dependent on the peculiarities of a particular region. This was possible to a substantial extent with the speed and fuel models, and, because of data limitations, considerably less so with the tire wear model. The parameters of the vehicle utilization model, owing to their nature, as discussed in Chapter 12, are highly region-specific. As for the maintenance models, Chesher and Harrison, op. cit., found from empirical testing that, except for the exponent parameters mentioned above, the parameters of these models are sensitive to local conditions. From the above discussion, we draw the following conclusions concerning local adaptation of the vehicle operating cost prediction models: 1. The mathematical forms of the models are generally adequate and need not be changed except for special reasons (e.g., to incorporate a new policy variable). 2. The vehicle attributes which appear as explanatory vari- ables in the models should generally be determined for a new region. Table 13.1 compiles the list of the vehicle attributes and indicates possible sources of data needed. 3. Some of the parameters are obviously sensitive to local conditions while others appear less so, as discussed below. GUIDELINES FOR LOCAL ADAPTATION 319 Table 13.1: Summary of vehicle attributes used in aggregate vehicle operating cost prediction models Vehicle attribute Prediction model Chapter Description Symbol Units defined Possible sources of data Rolling resistance roughness equation Contains no vehicle attributes - - 2 Field experiment required Gross vehicle mass a kg 2 Payload LOAD kg 10 User survey/ Speed and field experiment fuel consumption Projected frontal area AR m 2 Vehicle manufacturers Aerodynamic drag coefficient CD dimensionless 2 Above attributes plus below: Tire wear Number of tires per vehicle NT dimensionless 11 User survey/ Wearable rubber volume VOL do i tire manufacturers Maintenance parts Cumulative kilometerage CKM km 12 User survey and labor Depreciation and Life LIFE years 12 User survey interest Utilization AKM km/year 12 Endogenously modelled (see Table 13.2) Lubricants Contains no vehicle - - 13 (Insignificant item, which does not consumption attributes I warrant adjustment) However, location-sensitivity should not be used as the only criterion for deciding whether a given parameter should be re-calibrated. Two other criteria are also important. The first has to do with how much influence the parameter has on the policy issues being addressed. The second is concerned with the amount of effort needed to carry out the re-calibration, involving data collection as well as analysis. Based on these three criteria we offer in Table 13.2 general guidelines for the recalibration of the model parameters. Since circumstances are expected to vary a good deal from one application to another, these recommendations need not be followed too strictly. Depending on the degree of prediction accuracy desired in an application, the user should exercise independent judgment against the three criteria above. To help the user in this respect Table 13.2 also provides broad subjective ratings, "low," "medium," or "high," for each parameter in the three categories, namely "sensitivity to local conditions," "influence on policy-making," and "re-calibration effort." The ratings with respect to "influence on policy-making" were based on subjective assessment of the sensitivity of predictions of changes in total vehicle operating cost (in response to changes in road attributes) to a percentage change in the value of the model parameter. As to the "re-calibration effort," a "high" rating, (e.g., for the maintenance parts, roughness coefficient or for the parameters in the rolling resistance model) means that either a major user survey or a major Table 13.2: Local adaptation of parameters used in aggregate vehicle operating cost prediction models Mdel parameter 1 2 3 4 5 6 7 8 9 10 Prediction Qhapter Sensitivity Influence Re-cali- Re-calibration model Description Symbol Units defined to local on policy- bration generally Possible source(s) of data conditions making effort recomended Rolling resistance- Constant term CR dimensionless 2 low med high no Field experiment roughness equation Roughness coefficient CR QI-1 2 low-med high high no qI Used driving power HPDRIVE metric hp 3 high med-high low yes Vehicle manufacturers/survey Used braking power HPBRAKE metric hp 3 med-high low low yes Speed Desired speed (road width > 5.5 m) VDESIR m/s 3 high low-ned med yes Field experiment Perceived friction ratio FRATIO dimensionless 3 low low-med med no Average rectified velocity ARVMAX mm/s 3 low low-ned med no Width effect parameter BW dimensionless 4 high ned-high med no Weibull shape parameter a dimensionless 3 low-wed ned-high high no Unit fuel consumption parameters ai various 9 low-med ned-high low yes Field experiment Calibrated engine speed CRPM rpm 10 med low low no Vehicle manufacturers Fuel consumption Experimental-to-actual fuel conversion factor a2 dimensionless 10 med low high no Field experiment/survey Ratio of retread to new tire cost RREC dimensionless 11 high high low yes Base number of recaps NR dimensionless 11 high med-high ned yes Tire wenar Other carcass life smodel various various 11 ad-high ned-high high no parameters Survey Constant term in tread war nodel TWT dm3/1000 tire-km 11 ned-high ned-high med yes Wear coefficient for circum- CTc dm3/joule-m/1000 11 ned-high med-high ned yes ferential force tire-km Base number of hours driven HRDO hours/vehicle/year 12 high high med yes Vehicle utilization Base utilization AKMo km/year 12 high high med yes Survey Base elasticity of utilization EVUo0 m/year 12 med-high high med yes Maintenance parts Constant term CPO complicated units 13 high high med yes Roughness coefficient (exponen- CP Q11 13 med high high no Survey tial or linear form) q Exponent of kilometerage age K dimensionless 13 low med high no Maintenance labor Constant term CLo complicated units 13 high high med yes Survey Exponent of maintenance parts CL dimensionless 3 low high med no Roughness coefficient (bus only) CL QI-1 13 high m med high no Lubricants Constant term CO liters/100 veh-km 13 med low ned no consuption Roughness coefficient CO (1ters/ 1000 veh-km) med low high no Survey ,. I GUIDELINES FOR LOCAL ADAPTATION 321 field experiment would be called for; therefore, the parameters with a "high" rating in this category are not recommended for re-calibration, except under exceptional circumstances. Those users who intend to do so should first determine the scope of work involved based on the size of observations sample needed for a given degree of accuracy. Some further discussions on sampling are given in the remaining sections. A "medium" rating, e.g., for the maximum perceived friction ratio (FRATIO) or the the base number of recaps (NRO), indicates that a minor user survey or field experiment would be required. Finally, a "low" re-calibration effort (e.g., for the maximum used driving power (HPDRIVE) or for the ratio of retread cost to new tire cost (RREC)) indicates that the information is either readily available or could be obtained easily from vehicle manufacturers or operators. For those parameters rated "low" or "medium" in re-calibration effort, simple guidelines are provided in the remaining sections (with the exception of the parameters in the lubricants consumption model which is clearly trivial). 13.2 COEFFICIENT OF ROLLING RESISTANCE As discussed in Chapter 2, the rolling resistance coefficient CR, is primarily dependent on the tires and the suspension system. The tires affect the overall magnitude as well as the sensitivity of the coefficient to road roughness; the suspension system affects only the latter. From the current knowledge of vehicle mechanics, it is not possible to provide simple guidelines for adjusting the model for different tire and suspension types. An obvious way to obtain a new CR-roughness relationship is from an experiment similar to the coast-down runs described in Appendix 2A. This would amount to a major effort, however, and is therefore not generally recommended. It would be feasible only where a competent, well-equipped research unit exists. 13.3 SPEED PREDICTION MODELS Of the parameters employed in the speed prediction models, both micro and aggregate, the most influential ones are the speed constraint parameters in the steady-state speed function, VDESIR, HPDRIVE, HPBRAKE, FRATIO and ARVMAX, and the $ parameter.1 As seen in the India case study (Section 4.4), it is not essential to re-calibrate all the speed constraing parameters, for example, FRATIO and ARVMAX. Further, the power parameters HPDRIVE and HPBRAKE can be calculated from rated power and weight, respectively. If warranted, any particular speed constraint parameter can be obtained from a small-scale experiment of spot speed observations at selected road sites. Where different road widths exist for observation, the experiment could also yield data for computing a new value of BW, the width effect parameter. 1 No guidelines are provided herein for adaptation of the desired deceleration rate (DDESIR) used in the micro transitional model (Chapter 6). Although the DDESIR parameter could differ significantly across countries, the effect of the differential is expected to be of a much smaller magnitude than those for the VDESIR, HPDRIVE, HPBRAKE, FRATIO and ARVMAX parameters. 322 GUIDELINES FOR LOCAL ADAPTATION In order to re-calibrate the $ parameter, a somewhat larger-scale experiment could be needed. However, it is expected that the Brazil estimates would, in general, be appropriate except for very specific traffic environments. The estimation bias correction factor, E', need not be changed unless a full-fledged re-estimation of other parameters is attempted. The estimation of each parameter requires a different type of road configuration depending on the type of the associated speed constraint. The general rule is to find sites that are dominated only by the speed constraint to which the parameter belongs. For example, for the VDESIR parameter, the sites selected should be over 6 meters wide, level, straight smooth, and with a slight decline (2-3 percent) so that only the "desired speed" constraint dominates. A slightly negatively sloped section is preferable to a level one since the maximum possible driving speed (VDRIVE) on the latter may become unacceptably low for heavily loaded trucks. Intuitively, the reason for this selection rule is that by allowing only the relevant speed constraint to prevail, we may get a better resolution of the parameter of interest. This point will become more apparent below. For any speed constraint, care should be taken to make sure that the sections are long enough to allow steady-state conditions to be reached. At each section selected, 100 or more speed observations should be taken for each vehicle class. Assuming that individual speed observations have a standard error equal to 18 percent of the average speed at the site2, the average observed speed over 100 observations would reduce the standard error to 1.8 percent (18 & 7M = 1.8). This degree of accuracy should be satisfactory for most purposes. 13.3.1 Desired Speed The objective is to determine the original desired speed parameter (VDESIR) unadjusted for the road width effect for a given surface type (paved/unpaved). As noted earlier, only smooth, tangent roads with a decline of 2-3 percent and wider than 6 meters should be selected3. For each section-direction s, we compute an-estimate of this parameter, denoted by VDESIRs, as 1 1 1 1 VDESIR = Eca VSS - VDRIVE - VROUGH (13.1) where VSSso is the average observed speed at section-direction s; and the other symbols are as defined in Chapter 3 with subscript s denoting 2 From the estimation results of the steady-state speed model in Table 4.3, the range of estimated standard errors due to random sampling of individual speed observations at a given site, aw, is 15.0 to 20.1 percent of the mean predicted speed. 3 Although some sections may have a slightly negative effective gradient (i.e., CR,+GRs QIP0 PO = CPO exp(CPq QIP0)(1 - CP qQI.PO) (13.26) P1 = CPO exp(CPq QIP0) CPq (13.27) where CPO, CP and K are model parameters and QIP0 is the threshold 0 q p0 roughness level. The above formula applies differently to trucks and non-trucks in the following way. For non-trucks, the threshold roughness, QIP0, corresponds to a relatively rough road (roughness exceeding 100 QI). Thus the second part of the above formula represents a straight line extrapolation beyond a high roughness level where the extrapolated line is tangent to the exponential curve represented by the first part of the formula. For trucks, the threshold roughness QIPO is zero by definition as the formula is linear over the entire range of roughness. Thus the above formula reduces to the following form for trucks: PC = CP (1 + CP QI) CKMKP (13.28) It can be seen from both Equations. 13.25 and 13.28 above that for all vehicle classes the constant term CPO is a multiplicative factor in the parts consumption formula. This means that changing the value of CPO affects both the general level and the sensitivity of parts consumption to roughness. The CPO parameter should be re-calibrated in general because it reflects relative prices of vehicle parts and new vehicles which vary significantly across economies. The required data for each vehicle class may be obtained from a few representative companies each operating on a set of routes having similar roughness. For each company average values of the following characteristics should be obtained over all vehicles and over a period of one year or more: parts consumption per 1,000 vehicle-km (PC), expressed as a percentage of new vehicle price; cumulative kilometerage (CKM); and road roughness (QI). Substituting these values and the original estimates for the CPq and Kp in Equations 13.25 or 13.28, whichever is applicable, will yield the value of CPO for each company. The desired value of CPO is computed as the simple average over the individual companies' CPO values. As noted in Section 13.1, the exponent of cumulative 334 GUIDELINES FOR LOCAL ADAPTATION kilometerage, K , is expected to be relatively insensitive to local conditions and therefore need not be re-calibrated. The roughness coefficient, CPq, on the other hand, is believed to reflect durability of the vehicles as well as average maintenance practices in the region or country, and therefore could be sensitive to local conditions, although not as much as the multiplicative parameter, CP0. Again, as discussed in Section 13.1, re-calibration of the roughness coefficient represents a major effort and consequently not recommended in general. However, should the user desire to do so the possibility of employing the small-scale sample design suggested by Chesher (Section 13.3) should be considered if companies that operate on both smooth and rough roads (QI < 40 and QI > 120) can be found in sufficient numbers. 13.7.2 Maintenance Labor Hours Model From Chapter 12, Section 12.1.2, the amount of maintenance labor-hours required (LH) is dependent on the amount of maintenance parts consumption (PC) and, in the case of buses only, roughness (QI) as given by the following general formula: LH = CLO PCCLP exp(CL QI) (13.29) where CLo CL and CL are model parameters. The exponent of parts consumption (CL), which has been found from the Brazil and India studies to lie in the range 0.34 - 0.48 (Chesher and Harrison, 1987), relfects the diminishing amount of labor input per unit parts consumption as parts consumption increases. As noted in Section 14.1 this parameter has been found to be stable across conditions as disparate as Brazil and India and therefore re-calibration is not recommended. The roughness coefficient (CLq) is, of course, zero except for buses. Since re-calibration of this coefficient calls for a relatively major data collection effort, it is not recommended generally. The constant term (CLO) basically represents the amount of labor input resulting from a tradeoff between labor and capital given their relative prices; therefore, the CLO parameter should be re-calibrated. For a given vehicle class, the procedure would be similar to that for calibrating the constant term in the parts model (CPO). This involves first selecting representative companies, which should be the same as those selected for calibrating the parts model constant term (CPO). Then, for each company obtain average values of the following variables over all vehicles and a period of one year or more: labor-hours required per 1,000 vehicle-km (LH); parts consumption (PC), expressed as a percentage of new vehicle price; and, in the case of buses, roughness (QI). Substituting these values and the original CL and CL parameter estimates in Equation 13.29 above will produce the value ol CLO for each company. The desired value of CLO is established as the simple average over the individual companies' CLO values. GUIDELINES FOR LOCAL ADAPTATION 335 percentage of new vehicle price; and, in the case of buses, roughness (QI). Substituting these values and the original CL p and CL parameter estimates in Equation 13.29 above will produce the value of CLo for each company. The desired value of CLO is established as the simple average over the individual companies' CLO values. CHAPTER 14 User's Guide to Aggregate Prediction This chapter is a step-by-step user's guide to predict speed and various components of vehicle operating costs at the aggregate level using information on the roadway (geometry, roughness, surface type and width class) and information on the vehicle (including, at a minimum, class and, for trucks, load). Its main purpose is to bring together in one place material that is covered in different parts of the text and to complement it with prediction equations for the remaining cost components from other sources. The aggregate prediction models are presented in such a way that they can easily be programmed on a personal computer. For convenience, the steps are cross-referenced with equations from earlier chapters from which they are derived. For various model parameters needed in the predictions, the values arrived at in the Brazil study are included as defaults. The user is urged to go over the discussion on local adaptation of model parameters given in Chapter 13 carefully and determine values appropriate for the region of application. 14.1 OUTLINE OF METHODOLOGY Completing the sequence of steps once would yield the following predictions for one combination of roadway and vehicle class: 1. a. Space-mean speed of travel over the roadway in km/h; b. Average passenger time, crew labor and cargo holding in hours/1,000 vehicle-km; 2. Average fuel consumption over the roadway in liters/1,000 vehicle-km; 3. Average tire wear over the roadway in number of equivalent new tires per 1,000 vehicle-km; 4. a. Average yearly utilization of the vehicle in making trips over the roadway in vehicle-km/year; b. Average depreciation and interest as percentages of new vehicle price per 1,000 vehicle-km; 5. a. Average consumption of maintenance parts over the roadway as percentage of new vehicle price per 1,000 vehicle-km; 337 338 USER'S GUIDE TO AGGREGATE PREDICTION b. Average maintenance labor needed for trips over the roadway in hours/1,000 vehicle-km; c. Average consumption of lubricants over the roadway in liters/1,000 vehicle-km. Given a stretch of road between points, say A and B, three distinct types of journey on the road may be identified for a vehicle. These are: 1. One-way journey from A to B; 2. One-way journey from B to A; and 3. Round-trip journey either from A to B and back to A, or from B to A and back to B. Of the three, the first two are basic in the sense that predictions for the round-trip journey may be obtained from the predictions for the two one-way journeys by means of appropriate averages. In order to obtain desired predictions for each journey type, three aggregate attributes of vertical geometry of the roadway are required. However, as pointed out in Section 7.3, the aggregate information on vertical geometry obtained for either journey type (1) or (2) is also sufficient for the other type and also for the round trip journey. For most applications, predictions for a round trip are adequate. The only significant exception is in the case of a truck with very different load levels in the two opposite directions. In this case, it is recommended that predictions be obtained separately for (1) and (2). Further, it may be noted that, while in the cases of speed, fuel consumption and tire wear all the three modes of travel are natural modes for prediction purposes, in the case of vehicle utilization round-trip travel is the natural mode because non-driven time on a trip is not well-defined for a one-way travel. At the aggregate level, the choice among the three modes of travel is governed exclusively by the way the average attributes of vertical geometry are computed for the roadway (Section 7.3). The roadway for which predictions are desired may be of arbitrary length and geometry. However, for one set of predictions, the surface type and width class of the roadway must be the same. If the roadway is made up of both paved and unpaved, or narrow and wide segments, it should be treated as different roadways and the sequence of steps performed separately for each. Two sets of inputs are required from the user for arriving at predicted aggregate speed: information on the roadway and information on the vehicle. In addition, for each operating cost component for which predictions are desired some specific inputs would be needed. Classified lists of information items, as well as instructions for entering the information, are provided. USER'S GUIDE TO AGGREGATE PREDICTION 339 The sequence of steps to arrive at the aggregate speed and fuel consumption predictions is organized as follows: 14.2 Instructions for providing information on the roadway 14.3 Instructions for providing information on the vehicle 14.4 Instructions for providing information needed for predicting fuel consumptioi, tire wear, vehicle utilization and maintenance resources 14.5 Steps for computing predicted speed 14.6 Steps for computing predicted fuel consumption, tire wear and vehicle utilization 14.7 Steps for computing predicted maintenance resources and lubricants consumption These steps are illustrated through a numerical example in Appendix 14A. 14.2 INSTRUCTIONS FOR PROVIDING INFORMATION ON THE ROADWAY The information required on the roadway is as follows: Information item Symbol Units Recommended Value range 1. Surface type - Categorical Paved/unpaved . . 2. Average roughness QI QI counts 15 - 300 . . 3. Average positive gradient PG Fraction 0.0 - 0.12 . . 4. Average negative gradient NG Fraction 0.0 - 0.12 . . 5. Proportion of uphill travel LP Fraction 0.0 - 1.0 . . 6. Average horizontal curvature C deg/km 0 - 1000 . . 7. Average superelevation SP Fraction 0.0 - 0.2 . . 8. Altitude of the terrain ALT m 0.0 - 5000 above Mean Sea Level . . 9. Effective number of lanes - Categorical Single-lane/ . More than one lane In completing the above table, care should be taken that the values are within the recommended range given above, subject to the following qualifications: 1. For prediction of maintenance parts and labor, the data used in model estimation covered the following range of roughness: Cars and utilities 25-120 QI Buses 25-190 QI Trucks 25-120 QI 340 USER'S GUIDE TO AGGREGATE PREDICTION Even though extrapolation is difficult to avoid, the user should be aware of it when roughness values fall outside of the range indicated above. 2. For tire wear prediction, the maximum recommended horizontal curvature is 300 degrees/km, which corresponds to the minimum average radius of curvature of 190 m. Overprediction of tire wear is expected when extrapolated beyond this limit (see Chapter 11 for discussion). The detailed procedure for data input is as follows: 1. Enter surface type (paved or unpaved). 2. Enter average roughness in QI units if, available. If roughness is given as m/km IRI (International Roughness Index), convert into QI units using the formula: QI = 13 IRI If roughness is given in BI units (as measured with the TRRL Towed Fifth Wheel Bump Integrator), convert into QI units using the formula: QI = BI/55 If roughness is measured using any other methodology, transform into QI units using an appropriate calibration method. If a roughness value is not available in any of the above units, the user may translate his subjective assessment of the roughness of the roadway into QI units by using a five-point scale as shown below (since these guidelines can only provide very broad approximations, the user is urged to work with actual roughness measures, if possible): Qualitative QI evaluation Paved roadway Unpaved roadway Smooth 25 50 Reasonably smooth 50 100 Medium rough 75 150 Rough 100 200 Very rough 125 250 3, 4, 5, 6. Enter geometric attributes (defined in Chapter 7). If starting with detailed vertical and horizontal geometric profiles of the roadway, follow the steps in Appendix 7A to determine average geometric attributes. This procedure provides the required aggregate information for both USER'S GUIDE TO AGGREGATE PREDICTION 341 one-way journey directions as well as the round trip. Note the sign convention for gradients: both "average positive gradient" and "average negative gradient," as entered in the table, are non-negative quantities. If starting with the HDM-II or HDM-III input data, the following formulas can be used, but for round trip predictions only: 3. Average positive gradient, PG = RF/1000 4. Average positive gradient, NG = RF/1000 5. Proportion of uphill travel, LP = 0.5 where RF = average rise plus fall, in m/km. The information in the HDM input data is not sufficient to compute predictions for one-way journeys. As pointed out in Section 7.3,' because of symmetry, the average negative gradient for a round trip journey is identical to average positive gradient, and proportion of level and uphill travel is, by definition, 0.5. 7. Enter superelevation. If the superelevation is known on a detailed basis as part of the horizontal profile, the steps in Appendix 7A will yield the average superelevation. If the superelevation is known on an average basis for the roadway, enter it as a fraction. If the superelevation is not known, either on a detailed basis or on an average basis, the following formula obtained from a sample of roads in Brazil may be used to determine superelevation: For a paved roadway: SP = 0.00012 C (3.25) For an unpaved roadway: SP = 0.00017 C 8. Enter the altitude of the terrain above the Mean Sea Level, if known. (This value is not crucial.) 9. Enter the effective number of lanes in the roadway. The only important distinction in the current model is between single-lane roads and other roads. If the carriageway width is less than 4 m the roadway may be classified as single-lane. If the roadway is wider than 5.5 m, it may be classified as having more than one lane. Roadways with width between 4 and 5.5 m, will have to be classified judgementally, based on other factors such as, shoulder width and condition, average daily traffic, and traffic composition. It might be of help to note that, in India, cars seemed to experience the effect of narrow widths more than buses and trucks. 342 USER'S GUIDE TO AGGREGATE PREDICTION 14.3 INSTRUCTIONS FOR PROVIDING INFORMATION ON THE VEHICLE The information required on the vehicle for speed prediction is as follows: Information item Symbol Units Value 1 Vehicle class - Categorical . . 2 (a) Tare weight TARE kg . . (b) Load carried LOAD kg . . 3 Used driving power HPDRIVE metric hp . . 4 Used braking power HPBRAKE metric hp . . 5 Surface type-specific VDESIR m/s desired speed . . 6 Aerodynamic drag coefficient CD Dimensionless . . coefficient number 7 Projected frontal area AR m2 It may be observed that the only required inputs are the vehicle class and, in the case of a truck, the load carried. As for other input items the user has the option of using the "standard" values associated with the particular vehicle class as default values. These values are given in Table 14.1. (Please note that for user convenience, all tables in this chapter are given at the end). Alternatively, the user may retain some of the default values and substitute more suitable values for others, if necessary. Using a value of zero for the load carried by a truck signifies that predictions are desired for an empty (unloaded) truck. In completing the above table, care should be taken that the values of vehicle attributes are within the recommended range given in Table 14.5. The detailed procedure for data input is as follows: 1. The user may choose one of the following vehicle classes (representative makes and models for the respective classes as employed in the Brazil-UNDP Study are given in parentheses): Small car (Volkswagen 1300) Medium car (Chevrolet - Opala) Large car (Chrysler - Dodge Dart) Utility or pick-up (Volkswagen Kombi) Bus (Mercedes Benz 0-362) Light gasoline truck (Ford F-400) Light diesel truck (Ford F-4000) Medium truck (Mercedes Benz 1113 with 2 axles) Heavy truck (Mercedes Benz 1113 with 3 axles) Articulated truck (Scania 110/39) USER'S GUIDE TO AGGREGATE PREDICTION 343 2. a. Look up the tare weight value for the particular vehicle class from Table 14.1 or enter own value. b. If the vehicle is a car, a bus or a utility, look up the default value provided in Table 14.1 or enter own value. Here the load carried represents the weight of the passengers and some light load. If the vehicle is a truck, the user is urged to enter the value of load carried after carefully considering factors such as the nature of the material transported in the region, the loading practice, and the maximum rated tonnage for the vehicle. The following chart is intended to be illustrative of the order of the magnitude involved: Load carried (LOAD) in kg by truck Loading condition Light Medium Heavy Articulated Unloaded 0 0 0 0 Partially loaded 1800 4500 6000 13000 Fully loaded 3600 9000 12000 26000 3, 4, 5. Determine the values of used driving power (HPDRIVE), used braking power (HPBRAKE), and desired speed (VDESIR) values for the particular vehicle class using the procedure given in Section 14.1, or look up the default values given in Table 14.1. Note that the value of VDESIR is different for paved and unpaved surfaces. 6, 7. Look up the standard values of CD and AR for the particular vehicle class from Table 14.1 and enter. If the values are not representative of the vehicles in the region of the roadway, enter appropriate values. In completing the above table in respect of cost components for which predictions are desired, the following procedure should be used for data input: la. Look up the calibrated RPM (CRPM) value for the particular vehicle class from Table 14.1. The table also gives the value of maximum rated RPM (MRPM) for the vehicle class. If the maximum rated RPM value for the typical vehicle in user's region is markedly different from the one given, enter 0.75 times the maximum rated RPM as the value of calibrated RPM. That is, the recommended formula in order to override the default value of CRPM is: 344 USER'S GUIDE TO AGGREGATE PREDICTION 14.4 INSTRUCTIONS FOR PROVIDING INFORMATION NEEDED FOR PREDICTING VARIOUS COMPONENTS OF VEHICLE OPERATING COST The information required for predicting various cost components follows: Prediction desired Information item Symbol Units Value 1. Fuel (a) Calibrated rpm CRPM rpm consumption (b) Relative energy-efficiency al dimensionless . . (Chapters 9 factor and 10) (c) Fuel adjustment factor a2 dimensionless . . 2. Tire wear (a) Number of tires per vehicle NT integer . . (buses and (b) Average wearable volume VOL dm3 . . trucks) of rubber per tire (Chapter 11) (c) Ratio of cost of retreading RREC fraction . . to cost of new tire (d) Maximum number of recaps NRO dimensionless . . (e) Constant term of tire TWTO dm3/m . . consumption model (f) Tire wear coefficient CTc dm3/J-m . . 3. Vehicle (a) Average annual utilization AKMO km . . utilization in a baseline case depreciation (b) Average annual number of HRDO h . . and interest hours driven in a (Chapter 12) baseline case (c) Hourly utilization ratio (or EVU0 fraction . . elasticity of vehicle utili- zation) in a baseline case (d) Average vehicle service life LIFEO year . . 4. Maintenance (a) Constant term of parts CPO dimensionless . . parts and consumption model labor (b) Roughness coefficient of CPq dimensionless . . (Chapter 12) parts consumption model (c) Threshold roughness value QIPO QI * (d) Constant term of maintenance CLO dimensionless . . labor model (e) Parts exponent of maintenanc CLP dimensionless . . labor model (f) Roughness coefficient of CLq dimensionless . . maintenance labor model (g) Average kilometerage of CKM km . . vehicles belonging to this class USER'S GUIDE TO AGGREGATE PREDICTION 345 CRPM = 0.75 MRPM lb. Enter the value of relative energy-efficiency factor, al. If the typical vehicle in user's region is similar to the make and model of the representative vehicle in the Brazilian study, the value of 1.00 may be used. If the typical vehicle in the region mber is more modern, enter a suitable value based on the discussion in Section 9.4. lc. Enter the value of fuel adjustment factor, a2, which eonverts fuel predictions based on controlled experiments to those based on real life operating conditions. If suitable, the following values derived in the Brazil study may be used: Vehicle class a2 Car or utility 1.16 Bus or truck 1.15 For a detailed discussion, the user is referred to Section 10.8. 2a. Enter the number of tires on a typical vehicle of the vehicle class. Table 14.1 gives the values for the representative makes and models. 2b. Enter the average volume of wearable rubber on a new tire of a size suitable for the vehicle class. It should be possible to obtain this information from tire manufacturers in the country. Table 14.1 gives the average volume for tires commonly used in Brazil (in cubic decimeters). 2c. Determine the ratio of the cost of one retreading (or recapping) to the cost of a new tire. Note that the default value of 0.15, based on the Brazil case, is quite low in comparison with many countries. 2d. Enter the hypothetical maximum number of recaps for a typical carcass used on vehicles of this class in the region given that the vehicle would always traverse on extremely smooth and straight road sections. Conceptually, this value can be fractional. For guidelines for calibrating NRO see Section 13.5. 2e,f. Look up the TWTO and CTc values for the particular vehicle class from Table 14.4 or enter own values. See Section 13.5 for discussion on how to determine appropriate values for the region. 346 USER'S GUIDE TO AGGREGATE PREDICTION 3a,b. For a baseline case of a roadway in the region, for a representative vehicle belonging to the vehicle class of interest, and for a typical year of operation, determine: 1. The number of kilometers covered by the vehicle in the year (AKMO); and 2. The number of hours the vehicle was driven to cover that distance (HRDO). In the case of improvement of an existing link, the baseline case could be the present roadway itself. In the case of a new link, the baseline case may be taken to be any similar, or an average, roadway in the region. In either case, these values are relatively easily obtained from vehicle operators. 3c. For the baseline case, determine the average proportion of time the vehicle was in motion relative to the time it was in use: in other words, the hourly utilization ratio, also called the elasticity of vehicle utilization (EVUO). . The method for determining the hourly utilization ratio is given in Section 13.6. Since the operating parameters on which the hourly utilization ratio depends vary considerably from country to country or even from one region in a country to another, the user is urged to provide estimates specific to the country or region. However, if the information is not available the following values based on Brazil data are suggested (provided that operating conditions in the region in question are similar to those in Brazil): 0.60 for a car 0.80 for a utility EVU0 < 0.75 for a bus 0.85 for a truck 3d. Enter the average service life of the vehicles belonging to the particular vehicle class in the region, in years. 4a. Enter the value of CPO, the constant term of the parts consumption model, which relates to the relative price of spare parts items typical for vehicles of this class vis-a-vis the average price of a new vehicle of this class. The user is urged to calibrate this parameter following the guidelines in Section 13.7. However, if conditions are similar, the Brazilian values given in Table 14.4 may be used. USER'S GUIDE TO AGGREGATE PREDICTION 347 4b,c. Enter the values of CPq and QIPO for the particular vehicle class from Table 14.4. See Section 4.d for discussion on how to determine appropriate values. 4d. Enter the value of CLO, the constant term of the maintenance labor hours model, which relates to the labor-capital trade-off in the region of interest. The user is urged to calibrate this parameter following the guidelines given in Section 13.7. For countries or regions similar to Brazil, the default values given in Table 14.4 may be used. 4e,f. Enter the values of CLP and CLq for the particular vehicle class from Table 14.4. For reasons explained in Chapter 13, these values are generally recommended for use without re-calibration. However, should the user desire to re-calibrate them, Section 13.7 provides a disucssion on how to determine appropriate values. 4g. Enter the average cumulative kilometerage, that is, the average number of kilometers driven over the lifetime of a vehicle of the particular vehicle class. This value may be arrived at using information from steps 3a and 3d above, subject to the ceiling CKM' given in Table 14.4: CKM = min(O.5 LIFEO AKMO, CKM') 14.5 STEPS FOR COMPUTING PREDICTED SPEED 14.5.1 Compute the Rolling Resistance Coefficient (CR): If the vehicle is a car or a utility: CR = 0.0218 + 0.0000467 QI (2.18a) If the vehicle is a bus or a truck: CR = 0.0139 + 0.0000198 QI (2.18b) 14.5.2 Compute RHO, the Mass Density of Air in the Terrain of the Roadway, in kg/m : If the altitude (ALT) above the Mean Sea Level is known, compute the air density from the following formula: RHO = 1.225 [1 - 2.26 ALT ]4.255 (2.15) 100,000 otherwise use the default value: 348 USER'S GUIDE TO AGGREGATE PREDICTION RHO = 1.225 14.5.3 Compute the -Mass" of the Vehicle, a, in kg: m = TARE + LOAD 14.5.4 Compute the Driving Power-constrained Speed for Uphill Travel, VDRIVE, in m/s (the formulas are from Section 3.3.1 and Appendix 3B): First compute the following intermediate quantities: A = 0.5 RHO CD AR b HPDRIVE 736 2 A C i g (CR + PG) where g = 9.81 m/s2 1 3,A D1= b2 + c d1= yDjl Then, VDRIVE = u '1 Note: HPDRIVE is the used driving power value obtained in Section 14.3. 14.5.5 Compute the Driving Power-constrained Speed for Downhill Travel, VDRIVEd, in m/s (formulas are from Appendix 3B): First compute the following intermediate quantities: m g (CR - NG) c2 3 A D2= b2 + c3 where A and b are as computed in 14.5.4 above. Next, examine the sign of the quantity D2* USER'S GUIDE TO AGGREGATE PREDICTION 349 Case 1: D2 > 0. If D2 is positive, then, d2 = 2 and VDRIVEd 3Vd2 + b - 3dd - b Case 2: D ( 0. If D2 is non-positive, proceed as follows: r = 2 / C2 1 2 b z = -T- arc cos [- c2r where z is in radians. Then compute the following three roots of the cubic equation: v1 = r cos (z) v = r cos (z + v3 = r cos (z + w) Note that exactly one of the three roots is positive. Set VDRIVEd to the positive root. That is: VDRIVEd = max {v1 v2, v3 14.5.6 Compute the Braking Power-constrained Speed, VBRAKE, in m/s: First examine the sign of the quantity (CR - NG). Case 1: CR - NG > 0. If (CR - NG) is non-negative, set VBRAKE to infinity. This is an acceptable value because VBRAKE will appear only as a denominator. That is, 1 = 0 VBRAKE and this factor will then drop out of the denominator of the speed prediction formula. 350 USER'S GUIDE TO AGGREGATE PREDICTION Case 2: CR - NG < 0. If (CR- NG) is negative: VBRAKE = - HPBRAKE 736 (3.6) m g (CR - NG) Note: HPBRAKE is the used braking power value obtained in Section 14.3. 14.5.7 Compute the Maximum Allowable Curve Speed, VCURVE, in m/s: Look up the values of FRATIO0 and FRATIO1 for the particular vehicle class and surface type from Table 14.2. Then, VCURVE = V((FRATIO0 - FRATIO1 LOAD) + SP) g RC (3.14, 4.7) where RC = 180,000/nC If the curvature C is zero or very small, VCURVE may be taken to be infinity, i.e.: 1/VCURVE = 0. 14.5.8 Compute the Roughness-constrained Speed, VROUGH, i-u/a: Look up the value of ARVMAX in Table 14.2 for the particular vehicle class. Then, VROUGH = ARVMAX (3.24) 0.0882 QI 14.5.9 Determine the Desired Speed Adjusted for Width Effect, VUESIR', in m/s: Look up the value of VDESIR for the particular vehicle class and surface type as obtained in Section 14.3. If the roadway is classified as single-lane, look up the value of "width effect parameter," BW, for the particular vehicle class and width class from Table 14.2. If the roadway is classified as having more than one lane, there is no width effect, and the value of BW is 1.00 for all vehicle classes. Then, VDESIR' = BW VDESIR 14.5.10 Look up the Values of 0 and E* for the Particular Vehicle Class from Table 14.2. USER'S GUIDE TO AGGREGATE PREDICTION 351 14.5.11 Compute the Aggregate Uphill Steady-state Speed, VSSu,in m/s: 1 1 1 1 VSS-= E ( 1 1 1 - 1 -+ (3.38) u+ + (1 ) + ( ) ' VDRIVE VCURVE VROUGH VDESIR' u 14.5.12 Compute the Aggregate Downhill Steady-state Speed, VSSd, in a/s: 1 1 1 1 1 VSSd = E0 ( 1 ) 1 Ts )8 1 I 7 1- VDRIVEd VBRAKE VROUGH VCURVE VDESIR' 14.5.13 Final Step for Speed Prediction: Compute the Predicted Average Speed, ASPRED for the Journey (converted from m/s to km/h): ASPEED 3.6 km/h. LP 1 - LP VSS VSSd Note: Some resources which may be straight-forwardly computed using predicted speed are as follows: 1. The number of passenger-hours spent per 1,000 vehicle-km of travel, PHX: PHX = 1,000 PAX ASPEED where PAX is the average number of passengers per vehicle. 2. The number of crew hours required per 1,000 vehicle-km, CRH: C 1,000 CRH1 = 3. The cargo holding cost per 1,000 vehicle-km, CHC: CHC -10 MVC AINV where: MVC is the monetary value of the cargo; AINV is the annual interest rate in percentage; and 8760 is the number of hours per year. 352 USER'S GUIDE TO AGGREGATE PREDICTION 14.6 STEPS FOR COMPUTING PREDICTED FUEL CONSUMPTION, TIRE WEAR AND VEHICLE UTILIZATION Steps for computing predicted fuel consumption: 14.6.1 Compute the Gravitational Resistances in N: Uphill : GF = mg PG (2.13) Downhill : GFd = m g NG 14.6.2 Compute the Rolling Resistance in N: Uphill and downhill: RR = m g CR (2.17) 14.6.3 Compute the Air Resistances in N: Uphill : AF = 0.5 RHO CD AR (VSS )2 A (VSS )2 (2.9) Downhill : AFd = 0.5 RHO CD AR (VSSd)2 A (VSSd )2 14.6.4 Compute the Drive Forces in N: Uphill : DF = (GF + RR + AF ) (2.6) Downhill : DFd = (-GFd + RR + AF d 14.6.5 Compute the Vehicle Power in Metric hp: DF VSS Uphill : HP = u u (2.11) u 736 DFd VSSd Downhill HP = DF_d_____ d 736 14.6.6 Compute the Aggregate Uphill Unit Fuel Consumption, UFC., in Liters/1000 Vehicle-km: Look up the fuel consumption coefficients a, al, a,, a4 and as for the particular vehicle from Table 14.3. Then, UFC = (a0 + al CRPM + a2 CRpM2 + a3HP + aHP CRPM + a5 HP2) x 10-2 R9.4) 5u (CRPM is the calibrated RPM value obtained in Section 14.4.) 14.6.7 Compute the Aggregate Downhill Unit Fuel Consumption, UFCd, in Liters/1,000 Vehicle-ka: USER'S GUIDE TO AGGREGATE PREDICTION 353 First check whether HPd > 0; if so look up fuel consumption coefficients ao, al, a2, a3, a4, a5 from Table 14.3. Then compute: UFCd = (ao + al CRPM + a2 CRPM2 + a3 HPd + a4 HPd CRPM + a5 HPd2) x 10-2 (9.4) On the other hand, if HPd < 0, look up the threshold negative power value, NHO, and the coefficients a0, a1, a2, a., a7 for the particular vehicle from Table 14.3. Compare HPd and NHO. If NHO < HPd < 0, then, UFCd = (ao + al CRPM + a2 CRPM2 + a6 HPd + a7 HPd2) x 10-2 Finally, if HPd < NHO, then, UFCd = (a0 + a1 CRPM + a2 CRPM2 + a6 NHO + a7 NH02) x 10-2 14.6.8 Compute the Predicted Aggregate Experimental Fuel Consumption per Distance Unit, FIELA, in Liters/1,000 Vehicle-ka: LP 1 -LP FUEIA = al (UFC u + UFC d VSS VSSd ud where al is the "relative energy-efficiency factor" obtained in Section 14.4. 14.6.9 Finally, Adjust the Predicted Experimental Fuel Consumption to Account for Real World Operating Conditions Yielding the Predicted Aggregate Operating Fuel Consumption, AFUEL, in Liters/1,000 Vehicle-km.: AFUEL = a2 FUELA where a2 is the "fuel adjustment factor" obtained in Section 14.4. Steps for computing predicted tire wear: 14.6.10 If the Vehicle is a Car or a Utility Go Directly to Step 14.6.17. Otherwise, continue with step 14.6.11. 14.6.11 Compute the Square of Circumferential Force, CFT2, in N2. CFT2 = Lp (DF2) + (1 - LP) (DF2) u d 354 USER'S GUIDE TO AGGREGATE PREDICTION 14.6.12 Compute the Circumferential Energy per Tire, CE, in J: CE = CFT2 m g NT where NT is the number of tires in the vehicle (default value is given in Table 14.1). 14.6.13 Compute the Predicted Volume of Rubber Worn, TWT, in dm3/1,000 ka: TWT = TWT + CT CE (11.8, 11.10) 0 c where TWTo and CTc are given in Table 14.4. 14.6.14 Compute the Predicted Number of Retreadings, NR: NR = (NRO + 1) exp [- 0.00248 QI - 0.00118 min (C.300)] -1 (11.9) where NRO is the maximum number of recaps (default value is given in Table 14.4). 14.6.15 Compute the Total Travel Distance Afforded by One Tire Carcass, DISTOT, in 1,000 ki: DISTOT = (1 + NR) VOL (11.3, 11.2) TWT where VOL is the volume of wearable rubber per tire (default value is given in Table 14.1). 14.6.16 Compute the Predicted Number of Equivalent New Tires Consumed per 1,000 km for Each Tire on the Vehicle (EQNT): Trucks and buses: 1 + RREC NR +007 EQNT = + 0.0075 (11.16) DISTOT where RREC is the percentage ratio of the cost of one retreading to the cost of a new tire (default value from Brazil is 15%). 14.6.17 Compute the Predicted Number of Equivalent New Tires Consumed per 1,000 km of Vehicle Travel (EQMT): USER'S GUIDE TO AGGREGATE PREDICTION 355 1. If the vehicle is-a truck or a bus: EQNTV = NT EQNT (11.16) 2. If the vehicle is a car or a utility: EQNTV = NT min [0.0388, 0.0114 + 0.000137 QII (11.17) Steps for computing predicted utilization, depreciation and interest: 14.6.18 Compute the Predicted Annual Vehicle Utilization, AKK, in k: AKM= 1 - EVUO EVUO --1 (12.9) AKMO ASPEED HRDO 14.6.19 Determine the Average Service Life of the Vehicle, LIFE, in Years, Using one of the Following Formulas: If the vehicle service life is expected to be constant, then: LIFE = LIFE0 (12.6a) If sensitivity of service life to speed is desired, then LIFE = ( -AKH + 2) LIFEO (12.6b) HRDO ASPEED 3 14.6.20 Compute the Vehicle Depreciation per 1,000 Vehicle-km, DEP, in Fraction of the Price of a New Vehicle of the Same Class: DEP = 1,000/(LIFE AKM) 14.6.21 Compute the Amount of Interest Charge per 1,000 Vehicle-ka, INT, in Fraction of the Price of a New Vehicle of the Same Class: INT = 5 AINV/AKM (12.5) 14.7 STEPS FOR COMPUTING PREDICTED MAINTENANCE RESOURCES AND LUBRICANT CONSUMPTION 14.7.1 Compute Standardized Parts Cost per 1000 vehicle-km as Fraction of the Average Price of a New vehicle of the Same Class, APART: Look up Kp from Table 14.4 for the particular vehicle class. Look up the values of QIPO, CPO and CPq as obtained in Table 14.4. Compare QIPO and the value of QI for the road section. If QI < QIPO, then, APART = CP0 exp(CPq QI0) CKMkp (12.1) 356 USER'S GUIDE TO AGGREGATE PREDICTION If QI > QIPO, then, APART = CP0 exp(CP QIPO) CKM Kp((1 - CPq QIPO) + CP QI) 14.7.2 Compute the Predicted Maintenance Labor, ALABOR in Labor-hours per 1,000 vehicle-ka: Look up the values of CLO, CLP and CLq as obtained in Table 14.4. Then, ALABOR = CL APARTCLP exp(CL QI) (12.3) 14.7.3 Compute the Predicted Lubricants Consumption, ADIL, in Liters per 1,000 vehicle-ka: Look up the value Co0 in Table 14.4. Then, AOIL = CO0 + 0.011605 QI (Appendix 12C) USER'S GUIDE TO AGGREGATE PREDICTION 357 Table 14.1: Vehicle classes and their standard characteristics Vehicle class Cars Utility Bus Light truck Mbdium/ Heavy Articulated Characteristics Heavy truck truck Small Ad. Large Gas Diesel truck 1. psI*ve Volks Chey- Chrys Volks arc. Ford Ford Jrc. Arc. Scania vehicle une wagen rolet ler wagen Benz Benz Benz 110/39 model 1300 Opel Dodge Kanbi 0-362 F-400 F-4000 1113 1113 Dart (two (three axles) axles) 2. Wight (fg): Tare weight (TARE) 960 1200 1650 1320 8100 3120 3270 5400 6600 14730 Load carried (LAD) 400 400 400 900 4000 3. Driving power (SAE): Maxima used (HPDRIVE) 30.0 70.0 85.0 40.0 100.0 80.0 60.0 100.0 100.0 210.0 Maxin rated (HPRATED) 48.0 146.0 198.0 60.0 147.0 169.0 102.0 147.0 147.0 285.0 4. Maxim used brdking: power (HPBRAKE) 17.0 21.0 27.0 30.0 160.0 100.0 100.0 250.0 250.0 500.0 5. Desired speed (als) (VIESIR): Paved sections 27.3 27.3 27.3 26.4 25.9 22.7 22.7 24.7 24.7 23.4 Unpaved sections 22.8 22.8 22.8 21.8 19.3 20.0 20.0 20.0 20.0 13.8 6. WK: Calibrated (CRPM) 3500 3000 3300 3300 2300 3300 2600 1800 1800 1700 Maxinun rated (MRPM) 4600 4000 4400 4600 2800 4400 3000 2800 2800 2200 7. Aerodynauic dra: coefficient (CD) 0.45 0.50 0.45 0.46 0.65 0.70 0.70 0.85 0.85 0.63 8. Projected frontal area (m ) (AR) 1.80 2.08 2.20 2.72 6.30 3.25 3.25 5.20 5.20 5.75 9. TLres: Number (NT) 4 4 4 4 6 6 6 6 10 18 Nominal diameter (W) 1000 900 900 1000 1000 1100 Wear rubber volume (dm) 6.85 4.30 4.30 7.60 7.30 8.39 (VOL) To be specified by the user. Source: Adapted from Brazil-UNDP-World Bank highway research project data and analysis. 358 USER'S GUIDE TO AGGREGATE PREDICTION Table 14.2: Parameter values for computing speed predictions Vehicle class Carl Utility Bus Truck Parameter Medium/ Light Heavy3 Articulated FRATIOO Paved roads 0.268 0.221 0.233 0.253 0.292 0.179 Unpaved roads 0.124 0.117 0.095 0.099 0.087 0.040 FRATIO, Paved roads 0 0 0 0.128 0.094 0.023 x 10- x 10-4 x 10 Unpaved roads 0 0 0 0 0 0 ARVMAX 259.7 239.7 212.8 194.0 177.7 130.9 BW 0.74 0.74 0.78 0.73 0.73 0.73 6 0.274 0.306 0.273 0.304 0.310 0.244 E0 1.003 1.004 1.012 1.008 1.013 1.018 1 The parameter values are the same for small, medium and large cars 2 The parameter values are the same for both kinds of light trucks, gasoline and diesel. 3 The parameter values are the same for medium and heavy trucks. Source: Adapted from Brazil-UNDP-World Bank highway research project data and analysis. USER'S GUIDE TO AGGREGATE PREDICTION 359 Table 14.3: Parameter values for computing fuel consumption predictions Vehicle Car Utility Bus Light truck Medium/ Articu- class Heavy lated Parameter Small Medium Large Gas Diesel truck truck ao -8201 23453 -23705 6014 -7276 -48381 -41803 -22955 -30559 a, 33.4 40.6 100.8 37.6 63.5 127.1 71.6 95.0 156.1 a2 0 0.01214 0 0 0 0 0 0 0 a3 5630 7775 2784 3846 4323 5867 5129 3758 4002 a4 0 0 0.938 1.398 0 0 0 0 0 a5 0 0 13.91 0 8.64 43.70 0 19.12 4.41 a6 4460 6552 4590 3604 2479 3843 2653 2394 4435 a7 0 0 0 0 11.50 0 0 13.76 26.08 NHO -10 -12 -15 -12 -50 ~50 -30 -85 -85 Source: Adapted from Brazil-UNDP-World Bank highway research project data and analysis. 360 USER'S GUIDE TO AGGREGATE PREDICTION Table 14.4: Parameters for computing tire wear, maintenance parts, maintenance labor and lubricant consumption Prediction Vehicle class Car Bus Truck model Parameter or Values Utility' Light2 Median Heavy Articulated Tire sar NRO - 2.39 0.93 2.39 2.39 3.57 TWO - 0.164 0.164 0.164 0.164 0.164 CTC - 12.78x1 12.78x1F 12.78xdF- 12.78M(r 12.78x1 3 yin m Kp 0.308 0.483 0.371 0.371 0.371 0.371 parts CPO 32.49xld06 1.77xtO6 1.49x106 1.49x1M6 8.61x106 13.94xt(r6 coamption CPq 13.70x103 3.56x103 251.79xla3 251.79xl(3 35.31x-3 15.65x10-3 QIPO 120 190 0 0 0 0 C1M' 300,000 1,000,000 600,000 600,000 600,000 600,000 CIOenmro CL.j 77.14 293.44 242.03 242.03 301.46 652.51 labor CLP 0.547 0.517 0.519 0.519 0.519 0.519 Ci ~ 0- 0.0055 0 0 0 0 lubricats 1.55 3.07 2.20 3.07 3.07 5.15 coommption 1 The parameter values are the same for cars of all three sizes as well as for utilities. 2 The parameter values are the same for gasoline as well as diesel trucks. Source: Adapted from Brazil-UNDP-World Bank highway research project data and analysis. USER'S GUIDE TO AGGREGATE PREDICTION 361 Table 14.5: Recommended range of vehicle attributes Vehicle attribute Units Recommended range Gross vehicle weight, a kg Cars 800- 2,000 Utilities 1,100- 2,500 Buses 7,500-12,000 Light trucks 3,000- 6,500 Medium trucks 5,000-16,000 Heavy trucks 6,000-22,000 Articulated trucks 13,000-45,000 Payload, LOAD kg Cars 0- 400 Utilities 0- 1,400 Buses 0- 4,500 Light trucks 0- 3,500 Medium trucks 0-11,000 Heavy trucks 0-16,000 Articulated trucks 0-32,000 Projected frontal area, AR m2 Cars 1.5 - 2.4 Utilities 2.3 - 3.2 Buses 6.0 - 7.0 Light trucks 3.0 - 5.0 Medium trucks 5.0 - 8.0 Heavy trucks 5.0 - 8.0 Articulated trucks 5.5 - 10.0 Aerodynamic drag coefficient, CD dimensionless 0.3 - 1.0 Wearable rubber volume per tire, VOL cm3 Buses 5.6 - 8.0 Light trucks 2.0 - 3.5 Medium trucks 6.5 - 9.3 Heavy trucks 6.3 - 8.8 Articulated trucks 6.0 - 8.5 Cumulative kilometerage CEK km Cars and utilities 0- 300,000 Buses 0-1,000,000 Trucks 0- 600,000 Source: Authors' recommendation. 362 USER'S GUIDE TO AGGREGATE PREDICTION APPENDIX 14A ILLUSTRATIVE EXAMPLE In this appendix, the procedure to predict aggregate speeds and various components of vehicle operating costs is illustrated through a numerical example. For this purpose, use is made of the roadway for which the aggregate geometrical attributes were computed in Appendix 7B. It is further assumed that the roadway is paved, quite smooth (QI = 40) and wide (width = 7m); it is 700 meters above the mean sea level. Predictions are desired for a one-way trip, starting from point A and ending at point B (Figure 7B.1). Using the aggregate geometric attributes computed in Appendix 7B, all the information required on the roadway may be furnished and is given in Table 14A.1. The vehicle for which predictions are desired is a heavy truck. It carries a load of 9,900 kg. In this illustrative example the default values provided in Table 14.1 are judged to be adequate, and therefore used to furnish the information required on the vehicle, as shown in Table 14A.2. We desire predictions for fuel consumption, tire wear, vehicle utilization, maintenance resources and oil consumption. We judge that the Brazil default values for the fuel parameters, the tire parameters, and the maintenance resource parameters are appropriate, and we expect a heavy truck in the region to be driven about 80,000 km in a year over about 2,000 hours on an average roadway. Further, we estimate that, in order for a heavy truck to be driven for 2,000 hours, it needs non-driven time of about 350 hours yielding an hourly utilization ratio of approximately 0.85. Finally, an avereage heavy truck in the region lasts for about 8 years. Thus, we estimate the average cumulative kilometerage to be 320,000 km. Now we are ready to furnish the information required to predict the desired cost components (Table 14A.3). The steps for predicting speed and other quantities are shown below. For each step, the final result is given. Note that the number of significant digits shown is more than necessary--they are included to facilitate the user's checking of calculations. Speed Prediction 14.5.1 The coefficient of rolling resistance is CR = 0.01469 using the equation for the truck. USER'S GUIDE TO AGGREGATE PREDICTION 363 Table 14A.1: Roadway information Information item Symbol Units Value 1. Surface type categorical Paved 2. Average roughness QI QI counts 40 3. Average positive gradient PG fraction 0.040 4. Average negative gradient NG fraction 0.049 5. Proportion of uphill travel LP fraction 0.307 6. Average horizontal curvature C deg/km 127.835 7. Average superelevation SP fraction 0.018 8. Altitude of the terrain ALT m 700 above Mean Sea Level 9. Effective number of lanes - categorical More than one Table 14A.2: Vehicle information for heavy truck Information item Symbol Units Value 1. Vehicle class categorical Heavy truck 2.(a) Tare weight TARE kg 6,600 (b) Load carried LOAD kg 9,900 3. Maximum used driving power HPDRIVE metric hp 100.0 4. Maximum used braking power HPBRAKE metric hp 250.0 5. Desired speed on: mZ (a) Paved sections VDESIR(P) m/s 23.8 (b) Unpaved sections VDESIR(U) m/s 19.8 6. Calibrated RPM CRPM rpm 2,000 7. Drag coefficient CD dimensionless 0.85 number 8. Projected frontal area AR mZ 5.20 364 USER'S GUIDE TO AGGREGATE PREDICTION Table 14A.3: Input information for prediction of fuel consumption tire wear, vehicle utilization, depreciation and interest, and maintenance resources Symbol Value 1.(a) CRPM 2,000 (b) al 1.00 (c) a2 1.15 2.(a) NT 10 (b) VOL 7.3 (c) RREC 0.15 (d) NRO 2.39 (e) TWTO 0.164 (f) CT 0.01278 c 3.(a) AKMO 80,000 (b) HRDO 2,000 (c) EVUO 0.85 (d) LIFEO 8 4.(a) CPO 0.0000861 (c) CP 0.03531 q (d) QIPO 0 (e) CLO 0.519 (f) CL 0 q (g) CKM 320,000 14.5.2 The mass density of air is computed using the formula since the altitude is known. RHO - 1.1446. 14.5.3 m = 16500 14.5.4 The value of HPDRIVE is 100. We start with, A = 2.529649 b = 14,547.47 ci = 1,166.53 D1 = 1,799,023,609, and dl - 42,414.93 Thus, VDRIVEu - 8.15865 USER'S GUIDE TO AGGREGATE PREDICTION 365 14.5.5 We start with, C2 = -731.7569, and D2 = -180,203,600. Now, since D2 is non-positive we follow the steps outlined in Case 2: r = 54.10201 z = 0.2484181 The three roots of the cubic equation are: v1 = 52.44123 v2 = -37.74059 v3 = -14.70064 We select the positive root, in this case v1, as the value of VDRIVEd* Thus, VDRIVEd = 52.44123 14.5.6 Since CR - NG is negative, we use the formula in Case 2 to get the value of VBRAKE. VBRAKE = 33.13366 using HPBRAKE = 250 14.5.7 VCURVE = 30.89198 using FRATIO0 0.2926 FRATIO1 - 0.00000945. 14.5.8 VROUGH = 50.37982 using ARVMAX value of 177.74. 14.5.9 Since the effective number of lanes is more than one, BW = 1.00 and VDESIR' = VDESIR = 24.67 14.5.10 For a heavy truck: 8 = 0.3095 and E* = 1.013. 14.5.11 VSSu = 8.15447 14.5.12 VSSd = 19.99359 14.5.13 ASPEED = 49.78622 km/h. 366 USER'S GUIDE TO AGGREGATE PREDICTION Fuel Consumption Prediction 14.6.1 GFu = 6,474.600 GFd = 7,931.386 14.6.2 RR = 2,378.121 14.6.3 AFu = 168.210 AFd = 1,011.050 14.6.4 DFu = 9,020.92 DFd = -4,542.054 14.6.5 HPu = 99.94688 HPd = -123.3859 14.6.6 UFCu= 7,146.423 14.6.7 Since HPd is negative, we compare it with NHO for a heavy truck which is -10. Since it is smaller than NHO we use the last formula for UFCd* UFCd= 439.71 14.6.8 FUELA = 284.2899 14.6.9 AFUEL = 326.9333 Tire Wear Prediction 14.6.10 We proceed with the next step since the vehicle is a truck. 14.6.11 CFT2 = 39280487 14.6.12 CE = 24.26687 14.6.13 TWT = 0.4741306 USER'S GUIDE TO AGGREGATE PREDICTION 367 14.6.14 NR = 1.63963 14.6.15 DISTOT = 40.64727 14.6.16 EQNT = 0.03815402 14.6.17 EQNTV 0.3815402 Predicting Vehicle Utilization Depreciation and Interest 14.6.18 AKM = 96,047.64 14.6.19 Using the speed sensitive service life formula: LIFE = 7.475828 14.6.20 DEP = 0.001392689 14.6.21 Using an investment rate, AINV, of 12%, we have: INT = 0.00062469 Predicting Maintenance Resources and Oil Consumption 14.7.1 Since QI is more than the value of QIPo, which is 0 for a heavy truck,we use the linear equation: APART = 0.002290146 and, 14.7.2 ALABOR = 12.85282 Finally, 14.7.3 AOIL = 3.5342. CHAPTER 15 Model Predictions and Policy Sensitivity As noted before, a desirable feature of the vehicle operating cost prediction models is their power to distinguish subtle differences among investment alternatives. For example, one would expect the time- related savings of paving a gravel road with poor alignment to be some- what lower than that of paving a level-tangent gravel road. This is because even after paving, the poor alignment still prevents speed from increasing above that dictated by geometry. This type of discriminatory power plays an important role in the optimization of highway investments under budget constraints. As evident from practical experience, the more severe the funding constraints the more difficult it becomes to choose from a number of competing alternatives, all appearing to be equally attractive. The discriminatory power afforded by the prediction models helps the policy- maker decide, on the basis of total transport cost minimization, where best to allocate scarce highway funds. Although ideally it is essential to consider other costs also in the context of total transport cost minimization, the scope of this chapter is confined to vehicle operating costs. Specifically, the chapter has two main purposes: 1. To explore, with graphs and tables of model predictions, the effects of road roughness and geometry on vehicle operating costs (Sections '15.1-15.3); and 2. To provide a set of comprehensive and detailed tables of predicted physical resource consumption to serve as an initial reference in project or policy analyses to indicate the sensitivity of costs to alternative standards -- provided that the country or region in question is not too different from Brazil (Section 15.4). Most of the graphs are presented in terms of physical quanti- ties so that use or interpretations can be made in a broader context of other countries. However, to get a better perspective of policy implica- tions, it is necessary to examine model predictions in cost terms based on typical unit costs. 15.1 VEHICLE RUNNING COSTS VERSUS ROAD ATTRIBUTES All predictions presented in this chapter are based on the aggregate models as well as the default vehicle characteristics and model parameters compiled in Chapter 13. The basis for predicting vehicle 369 370 MODEL PREDICTIONS AND POLICY Table 15.1: Basis for calculation of depreciation and interest Vehicle class Car Utility Bus All trucks Base number of hours driven 1,200 1,200 1,600 1,600 HRDO (h/vehicle/y) Base number of km driven 95,0001 95,000 110,000 100,000 AKMO (km/vehicle/y) Elasticity of vehicle 0.6 0.8 0.75 0.85 utilization (EVU0 For all vehiclesl: Vehicle depreciation: "vehicle life" method Vehicle utilization : adjusted utilization method Vehicle service life: 6 years Note: The information in this table should be regarded as indicative only. 1 Note that the average km driven for the car, reflecting commercial usage in the Brazilian case, is much higher than would normally be the case for a passenger car in private usage. Consequently, the depreciation and interest costs per kilometer are much lower. Table 15.2: Economic unit costs for calculation of vehicle operating costs in pesos Vehicle Representative New vehicle price New tire price Crew wages class test vehicle (pesos/vehicle) (pesos/tire) (pesos/crew-h) Small car VW-1300 3,300 19 1.93 Utility VW-Kombi 5,900 27 1.93 Bus MB-D362 40,000 168 4.40 Light truck F-4000 11,700 66 1.93 Medium truck MB-1113/2-axle 18,300 168 2.20 Heavy truck MB-1113/3-axle 22,700 168 2.20 Artic.truck Scania 110/39 56,300 258 4.13 Unit costs applicable to all vehicles: Maintenance labor wages (pesos/labor-h) 2.75 Interest rate (percent/year) 11 Gasoline (pesos/liter) 0.266 Diesel (pesos/liter) 0.282 Lubricants (pesos/liter) gasoline 1.16 diesel 1.07 Note: Adjustments for taxes have been made to approximate economic (as distinct from financial) costs, but the information in this table should be regarded as indicative only. MODEL PREDICTIONS AND POLICY 371 depreciation, interest and passenger value-of-time costs is shown in Table 15.1. Table 15.2 presents unit costs for different vehicle classes expressed in economic terms in arbitrary monetary units, which are called "pesos." While the data in these tables are considered to be fairly typical, they are not intended to be construed as being specific to any particular country. Appendix 15A contains graphs of predictions of the heavy truck's total running cost, speed and individual cost components (per 1000 vehicle-km) plotted against rise plus fall (RF), horizontal curvature (C), and roughness (QI) for paved roads under both loaded and unloaded conditions. The total running cost includes the following components: fuel and lubricants consumption, tire wear, maintenance parts and labor, depreciation and interest, and crew labor. The range of the RF, C and QI variables are: RF 0 - 80 m/km level - steep C 0 - 1,000 degrees/km tangent - curvy QI 25 - 125 smooth - rough The plots in Figures 15A.1-15A.9 are intended to illustrate by means of one vehicle class how the total running cost and its components vary with road and vehicle attributes. Although only plots for paved roads are presented, the patterns are very similar between paved and unpaved roads. While the graphs are largely self-explanatory, the following observations are particularly noteworthy: Speed, depreciation and interest, and crew hours (Figures 15A.1-15A.3) 1. Vehicle load has an adverse effect on speed (and consequently depreciation and interest and crew hours). The effect becomes increasingly pronounced with increasing rise plus fall but attenuates slightly with increasing roughness or curvature or both. This behavior is consistent with the derivation of the constraining speeds in Chapter 3. 2. For all geometric combinations, an increase in roughness causes a reduction in speed, with the greatest reduction occurring when the geometry is the most severe (steep and curvy) and the smallest when the geometry is the least severe (level and tangent), given the same load level. 3. When the road is steep (RF = 80 m/km) the speed of the loaded truck is hardly affected by large changes in roughness and horizontal curvature. This is because to the maximum possible driving and allowable braking speeds are dominant. 4. Horizontal curvature always has an adverse effect on speed. At the same load level, the magnitude of the effect is the largest when the road is smooth and level and the 372 MODEL PREDICTIONS AND POLICY smallest when the road is rough and steep. Fuel consumption (Figure 15A.4) 5. At a given speed, vehicle load affects fuel consumption through vehicle power, which in turn is directly dependent on rolling resistance and gravitational forces. Both of these forces are directly proportional to vehicle load. This explains why the truck consumes more fuel when loaded than when unloaded for any given road geometry. 6. Vehicle load and rise plus fall are the key determinants of fuel consumption and almost completely dominate roughness and horizontal curvature. In Figure 15A.4(b), the effect of rise plus fall on fuel consumption is much stronger when the truck is loaded than when it is unloaded. 7. For both loaded and unloaded conditions, fuel consumption goes up slowly with rise plus fall up to about 20-30 m/km, and then climbs rapidly. This non-linear behavior is explained by the non-linear (upwards concave) shape of the unit fuel consumption function quantified in Chapter 9. The policy implication of this is that improvement in vertical alignment of a road by more than a certain threshold level (say 2-3 percent average gradient) may not be worthwhile from the standpoint of fuel savings. 8. Figure 15A.4(a) shows that when the road is level-tangent the increase in rolling resistance due to increasing roughness over the range of 50-125 QI is more than countervailed by the reduction in air resistance caused by reduced speed on a rougher surface. (However, beyond 125 QI, which is generally applicable to unpaved roads, the rolling resistance effect becomes dominant.) Tire wear (Figure 15A.5) 9. Because vehicle load is a major contributor to vehicle drive force which causes tread wear, tire consumption is larger when the truck is loaded than when it is unloaded, given the same road geometry. The vehicle load effect becomes more prominent with road severity (except when implicit ceilings are imposed on tire consumption as shown in Figure 15A.5(c)). 10. As shown in Figure 15A.5(a), given the same load level, roughness has a small effect on tire wear when the road is level and a large effect when it is steep. 11. The effect of rise plus fall on tire wear is particularly strong. Figure 15A.5(b) shows that tire wear increases with rise plus fall at an increasing rate. At the same load level, the increase is the largest when the road is MODEL PREDICTIONS AND POLICY 373 curvy and rough and the smallest when it is tangent and smooth. 12. Similarly for roughness, as depicted in Figure 15A.5(c), at the same roughness and load level the effect of horizontal curvature on tire wear is relatively small when the road is level and relatively large when it is steep. Maintenance parts and labor and lubricants consumption (Figure 15A.6) 13. Our data show that these three contributors to the total running cost are dependent only on roughness (not on geometry and vehicle load), as predicted by the "aggregate-correlative" models reported in Chesher and Harrison (1985). The graphs show the relationships. Total running cost (Figure 15A.7) 14. For each geometric combination, the total running cost is significantly larger when the truck is loaded than when it is unloaded. 15. For all geometric combinations, roughness has a strong effect on the total running cost. It is shown in Appendix 15B (for heavy truck) that vehicle maintenance (parts and labor) can represent as much as 50 percent of the total running cost. This explains why the trend of the total running cost curves are similar to that of the maintenance parts curve. The effect of roughness on the total running cost is the greatest when the road is level and tangent and the smallest when the road is curvey and steep; however, the differential is small. 16. The total running cost is more sensitive to rise plus fall when the vehicle is loaded than when it is unloaded, other things being equal. At the same roughness and load level, the sensitivity is greater when the road is curvy than when it is tangent. This trend is similar to that of tire wear. 17. Horizontal curvature has a significant effect on the total running cost. At a given roughness and load level, the effect is greater when the road is steep than when it is level (up to the limit imposed by an implicit ceiling). Sensitivity of model predictions to superelevation coefficient (Figures 15A.8 and 15A.9) 18. In these graphs the predicted speed, fuel consumption, tire curvature using the following relationship which relates superelevation (SP) to horizontal curvature (C) for paved roads (from Chapter 13): SP = 0.00012 C x (sensitivity factor) 374 MODEL PREDICTIONS AND POLICY where the "sensitivity factor" were given values of 0.5, 1.0 and 1.5. The value of 1.0 means that the recommended formula was used whereas the values of 0.5 and 1.5 mean that the computed superelevation was +50 and -50 percent of the recommended value, respectively. It can be seen from the graphs that speed predictions are slightly sensitive to the superelevation coefficient, but fuel consumption, tire wear and the total running cost are hardly affected. Speed predictions based on the recommended formula (sensitivity factor = 1) are larger than predictions based on ±50 percent perturbations from the recommended value. This is an indication that the superelevations employed in deriving the SP-C relationship were reasonably well chosen, given the model's sensitivity to superelevation. 15.2 BREAKDOWN INTO COMPONENTS OF VEHICLE RUNNING COSTS Appendix 15B contains in tabular form percentage breakdowns of the running costs of three selected vehicle classes (utility, bus and heavy truck - all loaded) for various combinations of the rise plus fall, horizontal curvature and roughness variables for paved and unpaved surfaces, as laid out below: Vertical Horizontal RF C QI alignment alignment Roughness 1 0 0 25/50 level tangent smooth 2 80 0 25/50 steep tangent smooth 3 0 1,000 25/50 level curvy smooth 4 80 1,000 25/50 steep curvy smooth 5 0 0 125/250 level tangent rough 6 80 0 125/250 steep curvy rough 7 0 1,000 125/250 level tangent rough 8 80 1,000 125/250 steep curvy rough The purpose of this appendix is to illustrate the relative contributions of the various vehicle operating cost components for different vehicle class under different operating conditions. The following observations are noted: 1. The tables show clearly that fuel cost is the largest single cost component. This is brought out most forcefully in the case of utility vehicles, where, for the "smooth" roads, fuel consumption represents some 40 percent of total running costs on paved roads, and slightly lower on unpaved roads. While the absolute total costs (as well as the fuel cost component) go up with increasing roughness, the fuel cost relative to the total is reduced sharply, down to some 32 percent on very rough paved roads, and to some 25 percent on unpaved roads. The same finding holds true for curvature and rise and fall: the more severe the road MODEL PREDICTIONS AND POLICY 375 condition, the smaller the percentage share of fuel costs, i.e., the increase in total running costs are on the account of other cost components. The above findings also apply generally for the cases of buses and heavy trucks, although here the proportions of fuel costs are lower, and thus more in line with what is normally expected. 2. The data show that tire costs are fairly insignificant for utility vehicles, but very large for both buses and heavy trucks and in fact generally assume an ever increasing percentage of an ever increasing total cost, from around 10 percent under the most favorable conditions to around 25 percent under the most unfavorable. Since at the same time total running costs increase by a factor of 2.2-3 this means a six to seven 7 fold increase in absolute tire costs (for paved and unpaved roads, respectively) between the most favorable and the most unfavorable conditions here considered. 3. The crew cost component, which is linearly proportionate to the inverse of speed, varies as expected with road attributes.. The component is very large in the case of utility vehicles and buses, but considerably less for heavy trucks. 4. The maintenance cost component increases sharply with increase in roughness; for a utility vehicle increasing some three-fold on unpaved roads going from QI 50 to QI 250, and slightly less on paved roads, going from QI 25 to QI 125. This effect is equally clear for buses and heavy trucks, but with a factor of 1.5 to 2 between the extreme conditions. In this sense again, the maintenance takes on an increasing proportion of an increasing total running cost; in the case of a utility van on a gravel road, the absolute costs typically increasing by a factor of about 6 due to roughness increase alone. 5. Finally, depreciation and interest account for a substantial proportion of the total running costs for all vehicles on both road types, again with somewhat lower values for the heavy trucks. 15.3 TOTAL RUNNING COST VERSUS VEHICLE WEIGHT In Appendix 15C, Figure 15C.1(a) plots the predicted total running cost per 1,000 vehicle-km against the total vehicle weight for all vehicle classes and a number of load levels. The data points represent two extreme road conditions: the best (level-tangent and smooth paved with RF=O, C=0 and QI=25) and the worst (steep-curvy and rough unpaved with RF=80 m/km, C=1,000 degrees/km and QI=250). Each set of data points, which exhibit a straight-line trend, is accompanied by the line of best fit using ordinary least-squares regression. Both regression lines have a positive slope and positive intercept, and the line corresponding to the 376 MODEL PREDICTIONS AND POLICY worst road condition is steeper and has a larger intercept. This observation suggests an economy of scale with respect to vehicle size under a wide range of operating conditions. As also illustrated in Figure 15C.1(b), the predicted total running cost per 1,000 ton-km of vehicle weight tends to decline as the vehicle weight increases, assuming constant load factors. Even more striking is Figure 15C.1(c) in which the predicted total running cost per 1,000 ton-km of payload is plotted against the vehicle weight (for trucks only), exhibiting an even sharper declining trend. 15.4 DETAIIED PREDICTION TABLES Appendix 15D contains 30 tables (15 for paved and 15 for unpaved roads), each comprising predicted physical quantities consumed per 1,000 vehicle-km of operation on a given surface type (paved or unpaved) under different sets of road conditions for a given vehicle class and load level. The road conditions are represented by all possible combinations of: RF 0 - 40 80 m/km C 0 - 500 1000 degrees/km and QI 25 - 75 125 paved or QI 50 - 150 250 unpaved The tables are identified for the following vehicle classes and load levels: Vehicle class - load level 1. Small car 9. Light diesel truck - loaded 2. Medium car 10. Medium truck - unloaded 3. Large car 11. Medium truck - loaded 4. Utility 12. Heavy truck - unloaded 5. Bus (with passengers) 13. Heavy truck - loaded 6. Light gasoline truck - unloaded 14. Articulated truck - unloaded 7. Light gasoline truck - loaded 15. Articulated truck - loaded. 8. Light diesel truck - unloaded In order to save space the components of crew labor, passenger delay and cargo holding share the same column labelled "time per distance" and expressed in vehicle-hours per 1,000 vehicle-km. This implies that one passenger per vehicle is assumed. However, the user can obtain the desired passenger-hours delayed by simply multiplying the value shown with the actual number of passengers per vehicle. MODEL PREDICTIONS AND POLICY 377 The predictions in each table are for the following vehicle operating cost components: Cost component Units (per 1,000 vehicle-km) 1. Fuel consumption liters 2. Lubricants consumption liters 3. Tire wear equivalent new tires 4. Maintenance parts percent of new vehicle price 5. Maintenance labor labor-hours 6. Depreciation percent of new vehicle price 7. Interest percent of new vehicle price 8. Crew labor crew-hours 9. Passenger delay passenger-hours 10. Cargo holding vehicle-hours 378 MODEL PREDICTIONS AND POLICY APPENDIX 15A SENSITIVITY OF VEHICLE OPERATING COSTS TO ROAD CHARACTERISTICS FOR HEAVY TRUCKS ON PAVED ROADS Figure 15A.1: Speed versus characteristic road parameters for a heavy truck on paved roads: (a) roughness, (b) rise plus fall, and (c) curvature Notes: 1. (U) stands for unloaded, and (L) stands for loaded. 2. Roughness: Smooth means 25 QI, and rough means 125 QI. 3. Rise plus fall: Level means 0 m/km, and steep means 80 m/km 4. Curvature: Tangent means 0 deg/km, and curvy means 1,000 deg/km 5. For other variables default values as given in Chapter 14 are used. Source: Analysis of Brazil-UNDP-World Bank highway research project data (a) Speed (km/h) 80- Tangent-level (U) 70 - Tangent-level (L) 60- Tangent-steep (U) 50 Curvy-level (U) Curvy-level (L) 4--- -- i . . .- Curvy-steep (U) -- 40. Tangent-steep (L) +-------+-------------+ 30- Curvy-steep (L) 20 10- 0 25 50 75 100 125 Roughness (QI) MODEL PREDICTIONS AND POLICY 379 (b) Speed (km/h) 50 oi& v Tangent-snooth (U) 70 - Tang"-amoth 60 - -------------------- --T-n-ent-rou --h-(-) 50-E--=---~--+- ----- - ----- - - - -~ -- --- C 40 - - +- 3 U) 20- 10. 0 10 20 30 40 50 60 70 80 Rise plus fall (m/km) (C) Speed (km/h) eel - 0 oth-e I 20 -.p sel y + --------- - -- --+ -- - _- - .. "A,-e 40ý - -- - -- -- - -- ----------...- ft- t e 20 20 0 100 200 300 400 500 600 700 600 900 1000 Curvature (degrees/km) 380 MODEL PREDICTIONS AND POLICY Figures 15A.2: Depreciation and interest versus characteristic road parameters for a heavy truck on paved roads: (a) roughness, (b) rise plus fall, and (c) curvature Notes: 1. (U) stands for unloaded, and (L) stands for loaded. 2. Roughness: Smooth means 25 QI, and rough means 125 QI. 3. Rise plus fall: Level means 0 m/km, and steep means 80 m/km 4. Curvature: Tangent means 0 deg/km, and curvy means 1,000 deg/km 5. For other variables default values as given in Chapter 14 are used. Source: Analysis of Brazil-UNDP-World Bank highway research project data (a) Depreciation and interest (percent of new vehicle price/1000 vehicle-ka) 0. SO Curvy-steep (L) +---+---+---+- 0. 25- ------- Tangent-steep (L) +- ------- -------+---- Curvy-steep (U) --------- ------- 0.20 Curvy-level (L) ----+-- --4-----------+ Curvy-level (U) --------------------- Tangent-level (L)f 0. 15 Tangent-level (U) 0. 10 0.05 0. 00- 0 25 50 75 100 125 Roughness (QI) MODEL PREDICTIONS AND POLICY 381 (b) Depreciation and intere.t (percent of new vehicle price/1000 vehicle-kim) 0.30 0.25-. --- 0. 25 .....*- -. -- - u curvy-rouz h .- -s -- - - - + - ~ ~Curvy-oug (i) ~ ---- t - - - - - ------ - - ,urvy-szooth ( -=I 0-20-- - -. - - - Tangent-rough ( Ta ent- loth L h()- Tangent-5 0t -s 0. 15 0. 10. 0.05 0.00• 0 10 20 30 40 50 60 70 e0 Rise plus fall (m/km) (C) Depreciation and interest (percent of new vehicle price/1000 vehicle-km) 0.30- 0 , 2 5 sm oo th - s t e e p - ------- + .4.--- - - ---.----------+---+ 0.20 Rou-h-v - .-- - - - -- ---- ou-h- -ee- rU)- S5s,oth-levei (L)- 0. 15 suoo 0. 10 0.05 0.00- 0 100 200 300 400 500 600 700 600 900 1000 Curvature (degrees/km) 382 MODEL PREDICTIONS AND POLICY Figure 15A.3: Crew hours versus characteristic road parameters for a heavy truck on paved roads: (a) roughness, (b) rise plus fall, and (c) curvature Notes: 1. (U) stands for unloaded, and (L) stands for loaded. 2. Roughness: Smooth means 25 QI, and rough means 125 QI. 3. Rise plus fall: Level means 0 m/km, and steep means 80 m/km 4. Curvature: Tangent means 0 deg/km, and curvy means 1,000 deg/k 5. For other variables default values as given in Chapter 14 are used. Source: Analysis of Brazil-UNDP-World Bank highway research project data (a) Crew labor (h/1000 vehicle-km) 40- Curvy-steep (L) Curvy-level-(-------+-------- 0 Tangent-steep (L) ------- -------------- Curvy-steep (U) - - - - - - - - - - - - - - - - - - - - - - - Curvy-level (L) 4 ---- - - + + 4 20 - Curvy-level ()--- -- ---U)- - --- -- - -- ------ Tangent-steep (U)-- - - - - - - - - - - - - - - - - - - - - Tangent-level (L)l Tangent-level (U) 10 0- 0 25 50 75 100 125 Roughness (QI) MODEL PREDICTIONS AND POLICY 383 (b) Crev labor (h/1000 vehicle-km) 40 30 - Curvyrou (L - . 20 -----+•^=- - - - ------ Tangent-rough (U) Tangent-s-0 Tangent-8woth 0 10 20 30 40 50 B0 70 GC Rise plus fall (m/km) Crev labor (h/1000 vehicle-km) 40- 30 --ep-L-____...------..+-------' . 30-... ----+-- Siooth-steep (U) -o --t_e --.-.. - - - - - - - .0ý 'eeve 20 10 0 100 200 300 400 500 So 700 BOD 900 1DD Curvature (degrees/km) 384 MODEL PREDICTIONS AND POLICY Figure 15A.4: Fuel consumption versus characteristic road parameters for a heavy truck on paved roads: (a) roughness, (b) rise plus fall, and (c) curvature Notes: 1. (U) stands for unloaded, and (L) stands for loaded. 2. Roughness: Smooth means 25 QI, and rough means 125 QI. 3. Rise plus fall: Level means 0 m/km, and steep means 80 m/km 4. Curvature: Tangent means 0 deg/km, and curvy means 1,000 deg/km 5. For other variables default values as given in Chapter 14 are used. Source: Analysis of Brazil-UNDP-World Bank highway research project data (a) Fuel consumption (liters/1000 vehicle-km) 70 Curvysteep (L)~ - 4 --- 700 T 500 400 40- Curvy-steep (U) U.enet-s teep (U) lfr *lf lLlC*tf rnw c4 U=l iS tS -S* 30 Tangent-lovel WLl ~.~ Sa00 Curvy-level (L) --- - - -+-- --- Tangent-level (U) Curvy-level (U). ----- ---- 200 100 0 0 25 50 75 200 125 Roughness (QI) MODEL PREDICTIONS AND POLICY 385 (b) Fue1 consu.ption (liters/1000 vehicie-kw) 700 800 Tnetruh(i Saod ''ed 500 400 Tangent-rough (L)C 300 .---j~'hri:5 Cu-- -y-------L-.--- 200 Cuv-moh()Tangent-rough (U) 100 0 10 20 30 40 50 60 70 80 Rise plus fall (m/km) ()Fuel conaumption (liters/1000 vehicle-km) 700 - Rouh-se_L 600, 500 400 Rough-steep (U) 300. -+-- - --L -- Smoth-level (L) Smooth-level (u) 200 100 0 0 100 200 300 400 500 600 700 800 900 1000 Curvature (degrees/km) 386 MODEL PREDICTIONS AND POLICY Figure 15A.5: 'Tire wear versus characteristic road parameters for a heavy truck on paved roads: (a) roughness, (b) rise plus fall, and (c) curvature Notes: 1. (U) stands for unloaded, and (L) stands for loaded. 2. Roughness: Smooth means 25 QI, and rough means 125 QI. 3. Rise plus fall: Level means 0 m/km, and steep means 80 m/km 4. Curvature: Tangent means 0 deg/km, and curvy means 1,000 deg/km 5. For other variables default values as given in Chapter 14 are used. Source: Analysis of Brazil-UNDP-World Bank highway research project data a) Tire wear (number of tires/1000 vehicle-km) 1. 1 0. 9 Curvy-steep (L) 0. 8- --4-- 0.7 Tangent-steep (L)-- 0.8-- Curvy-steep (U) 0.5- 0. 4 - Tangent-steep (11)---- Curvy-level (U) Cu -1e ILe1 *1LL 0.2 i--I-- Tangent-level (U) Tangent-level (L 0. 1- 0. O0 0 25 50 75 100 125 Roughness (QI) MODEL PREDICTIONS AND POLICY 387 (b) Tira wear (number of tires/1000 vehicle-km) 1.91 1. 0 0.c9 V ? 0. 7 0.o6 0. 5- - 0. 4 - -e - 0.1 0 10 20 30 40 50 60 70 80 Rise plus fall (m/km) (C) Tire wear (nurber of tires/1000 vehicle-km) 1. 1 -Rough-steep (L) 1.0O Smoth-steep (L) 0.9- ,.---+-------------+--------+---------+---------+-------+-- - - - - - - - - - -- -- - - - 0.4 0.7- Rough-teep (U) 0.8 Smooth-steep (U) 0.5 - 0. 4- R - h-levl() 0.1. 0 100 200 300 400 500 600 700 800 900 1000 Curvature (degrees/km) 388 MODEL PREDICTIONS AND POLICY Figure 15A.6: Maintenance parts consumption and labor and lubricants consumption versus roughness for a heavy track on paved roads: (a) maintenance parts consumption, (b) maintenance labor, and (c) lubricants consumption Note: These resources depend only on roughness among the road characteristics. Sources: Analysis of Brazil-UNDP-World Bank highway research project data for (a) and (b). (See Chesher and Harrison (1987) and GEIPOT (1982).) Analysis of Indian Road User Costs Study data for (c). (See Chesher and Harrison (1987) and CRRI (1982).) a Maintenance parts consumption (percent of new vehicle price/1000 vehicle-km) 0.8 0. 5 0.4 0.3 0.2 0.1 .. 0 25 50 75 100 125 Roughness (QI) MODEL PREDICTIONS AND POLICY 389 (b) Maintenance labor (h//1000 vehicle-km) 20 15 10 5 0 25 50 75 100 125 Roughness (QI) C)Lubricants consumption (liters/1000 vehicle-km) 5 4 2 0- 0 25 50 75 100 125 Roughness (QI) 390 MODEL PREDICTIONS AND POLICY Figure 15A.7: Total running cost versus characteristic road parameters for a heavy truck on paved roads: (a) roughness, (b) rise plus fall, and (c) curvature Notes: 1. (U) stands for unloaded, and (L) stands for loaded. 2. Roughness: Smooth means 25 QI, and rough means 125 QI. 3. Rise plus fall: Level means 0 m/km, and steep means 80 m/km 4. Curvature: Tangent means 0 deg/km, and curvy means 1,000 deg/km 5. For other variables default values as given in Chapter 14 are used. Source: Analysis of Brazil-UNDP-World Bank highway research project data (a) Total running cost (pesos/1000 vehicle-km) 700 Curvy-steep (L) 4 500 Tangent-steep (L)+ .. 400- Curvy-steep (U) Tangent-steep (U) 300 Curvy-level L4- Tangent-level ( L) Curvy-level (Uj)- Tangent-level (U) 200 100 0 0 25 50 75 100 125 Roughness (QI) (ms�/аааа8ар� ааn3влап0 OOOt 006 008 OOL 009 OOS 004 ООЕ OOZ ООТ 0 0 OOI OOZ (П) tanaj-Ч7oomg ('I) Тала[-q7o 5 ООЕ даа3в-Ч3оот8 ----°°-- - (п)---------------- �_..__.._w...__..._..._............,...,...,.......г..� ���-�--__�-------=----------------- (П) Талаi-48nog �------f------t---'----�--^---1----�--#----�--F------F----�-t -----�--'---- ('[) гаыаi-Ч'$nog 004 •__-----------°----'---�-----------'-----'--"'_ "-_---'_ (А) даа7в-у8по8 OOS "-i---'_'---i�------'- �-°-°-'�-'°'---i-----°-'i---'--'-'}.'---°---�-•--'-'-'-i-------°-1-�4� Saaaa°ц�о®S 009 �Ld-_а7�у8по8 -t-'--i----^�---'-t---�----F^-�' "i'� --F� . QQL (а;-аlогуал OOOI/вовад) �есо 8асиипа Ts7oj � з ) (°�/�I ТТа3 мТд автg ОВ OL 09 DS ОУ ОЕ OZ ОТ 0 1г г!г .( 1 0 ООТ OOZ ---'__-_---"-- � _ 0 аоовв-�оа9иа �_____-� -...__.�_._$` оа8иеl __� _.+--------+------- 006 до��..iТлiл�� 1 �®ат��(1i 4до�В xnsn9i0� Ч8поа_7иа8nву �Oj ц _ _ _ ----------------- -- ��"�"- "' "" ----- ----------�_---- _ - __ ---"-'-.._-_.� __ .�-�((1) ySnoa=Rлт�,_�_____ -��"'._�-+---- 004 _- -' -..---' ����'t��'� ` -ir''`�`+��'� ��' •'♦'' - ---- �'' -.i' -''- /�i'��' OOS i''�' ''�-----�i i .Чg�о х�ая /���� �--- �1j ,Уд ,оа оое � ����у Оос (�Ч-аТоFЧал ОООТ/яоэад) 3воо 8атапnа Тв3оZ /� \ тбs ��тzоа алп� s�отг�та�иа z�ao`т�! 392 MODEL PREDICTIONS AND POLICY Figure 15A.8: Sensitivity of (a) fuel use and (b) tire wear to super- elevation formula for a heavy truck on paved roads (a) Fuel consumption (1i t. c./1000 -hicle-k.) 700- Rough-steep Boo 500 400, 300 5-th-le-1 200 100 C. 0 100 200 300 400 500 600 700 800 Soo 1000 Curvature (d gce../k.) Tire e.r (b) (number of tires/1000 veh'cle-km) 1. Rough-steep 0.8. 0.7 0.6. 0.5- 0.4- 0.3- C.2- Smooth-level 0. 0.0. 0 100 200 300 400 500 Boo 700 Soo 900 1000 Curvature (degrees/ka) Legend: Sensitivity factors: Notes: 1. Smooth mearis 25 QI, and rough means 125 QI. 2. Level means 0 m/km, and steep means 80 m/km. Source: Analysis of Brazil-UNDP-World Bank highway research project data MODEL PREDICTIONS AND POLICY 393 Figure 15A.9: Sensitivity of (a) speed and (b) total running cost to superelevation formula for a heavy truck on paved roads (a) Speed (kIn/h) ac 70 - so- 5C i 40 Rough-s teep I4-~ 10- 0 100 200 300 400 500 600 700 800 900 1000 Curvature (degree/kw) (b) Total running cost (pesos/1000 vehicle-km) 700 600 Rough-steep 500 400 300 Swooth-level 200 100 0 -- -' fl * i l l_ _ - - 0 100 200 300 400 500 600 700 600 900 1000 Curvature (degrees/km) Legend: Sensitivity factors: Notes: 1. Smooth means 25 QI, and rough means 125 QI. 2. Level means 0 m/km, and steep means 80 m/km. Source: Analysis of Brazil-UNDP-World Bank highway research project data 394 MODEL PREDICTIONS AND POLICY APPENDIX 15B BREAKDIWN OF OPERATING COSTS This appendix gives the vehicle cost breakdowns under various operating conditions both as "peso" costs per 1,000 vehicle-km and as percentage of the total cost. The quantities in the columns with labels prefixed by "%" are the percentage breakdowns. The "fuel" column comprises both fuel and lubricants and the "parts" column comprises both parts and labor. MODEL PREDICTIONS AND POLICY 395 Table 15B.1: Cost breakdown by percentage and total running cost for 1,000 vehicle km on paved roads 1 I I i % depr. + Total cost : % fuel % tire i% crew % maint. nt. j(pesos/1000km) IRF |C QI 10 0 25 41.31 1.41 21.3! 13.41 22.61 112.80 11 1125 31.31 1.9" 18.91 30.1: 17.8: 160.18 1000 25 35.7: 1.1: 29.0: 10.7: 23.6; 141.12 11 1125 30.2 1.7: 23.4; 26.2; 18.6; 184.18 80 25 41.4: 1.21 23.7; 11.5 22.1; 130.65 1-+1125 33.4 1.81 19.9; 27.4: 17.5: 175.73 1 h25. 39.0 1.0 28.11 9.7 22.21 155.37 11 1125 i 32.7; 1.6 23.3; 24.5; 17.91 196.60 f% depr. + Total cost % fuel % tire % crew % maint. int. (pesos 1000km) IRF ;C ;QI I - - - - - - - -- -- - -- - - - - - - O o 25 28.0 9.O 19.9 19.2! 23.8! 290.611 1 1 1125 23.0; 7.8; 20.7; 26.8; 21.7; 355.26 11000 25 22.8 7.9 27.1 16.7 25.5 334.41 1- : 1 2 125 21.01 8.01 24.6; 24.2: 22.3; 393.931 180 0 t25 28.7; 19.61 22.5; 10.7; 18.5 520.261 1 1 1125 i 26.0 20.4; 20.8; 16.1 16.7; 592.381 1000 25 26.5; 22.9; 23.0 9.8 17.9 569.851 1125 24.0; 24.3; 20.9: 14.7; 16.0 645.88 '% depr. + Total cost fuel % tire % crew 1% maint. i ( IRF 1C oI 0 0 25 35.5! 11.7! 12.2! 27.1! 13.6! 259.15 ,125 i 24.4; 8.6; 11.5 44.2; 11.3; 376.98 1000 25 i 30.4; 11.6 16.91 25.1 16.0 279.27 ,125 i 23.4; 9.5 13.4; 41.7; 12.1; 399.50 180 '0 '25 i 38.7; 23.4; 12.9; 14.1, io.a 498.131 I I 1--++--++ ,125 31.31 22.i 11.31 26.21 9.11 635.49 ' 25 i 35.6; 27.51 13.41 12.8; 10.7 546.41' 125 i 29.2; 26.4; 11.3; 24.3; 8.8; 685.88! 396 MODEL PREDICTIONS AND POLICY Table 15B.2 Cost breakdown by percentage and running cost for 1,000 vehicle km on unpaved roads 1% depr. + t Total cost : % fuel % tire % crew % maint. int (pesos/1000km) 'RF C QaI O O 5O 38.0 1.61 22.2! 16.21 22.0 123 73 I 250 - - - - - - -- - - - - - - - -- - - - - - - - -- - - - - - - - - 24.:---+- 1-+1250 24.0 1.5j 18.9: 41.81 13.81 272 03 h000 50 34.5 1.3: 28.9, 12.9 22.5: 155.52' - 1 1250 i 24.41 1.5: 20.4, 39.71 14.01 286.26 80 0 50 39.61 1.41 23.61 14.21 21.21 140-901 11--250 25.9 1.51 18.81 40.21 13.51 282.52 1000 50 37.51 1.21 28.11 11.91 2131 168.43 1250 i 26.3j 1.41 20.21 38.41 13.7: 296.08! I %depr. + Total cost : % fuel % tire % crew % maint. int. (pesos/lOOokm) IRF jC ;QI +I - '0 '50 25.1 8. 22.3! 20.5 24.1, 311.151 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - --- -- - - - - - - +250 17.61 5.71 23.51 34.81 18.4: 550.35, h000 50 21.8 7.31 28.71 17.3 24.9 367.94 250 17.41 6.61 24.41 33.2: 18.31 577.29 80 0 50 27.91 19.81 22.41 11.81 18.21 540.31 1 1 1250 20.91 19.91 20,11 24.71 14.41 777.681 111000 so i 25.5 22.9 23.31 10.61 17.61 598.221 250 19.51 24.2: 19.71 22.91 13.7 83777 l i i 1% depr. + Total cost : fuel % tire % crew % maint. int. (pesos/1000km)1 I +-- - - - - - - - - - - - - - +- - - --- -+ +- - - - - - - - - - - -- - - - - - - - - - - - - IRF jC Qai 10 iO 50 30.8 10.5 12.5 33.1 13.2! 288.55 250 20.1: 6.71 13.6 491 10.6 567501 1000 50 27.71 10.31 17.1j 29.71 15.21 321.51 I 1 250 j 20.0 7.91 14.0 47.4: 10.71 587.751 BO O 50 36.41 22.91 12.51 17.8 10.3; 535.001 I-- 2 250 25.2 21.71 11.41 33.61 8.21 828.831 I 25----I-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1000 50 33.3 26.91 13.3 16.21 10.31 588.811 250 23.51 26.3: 11.1: 31.31 7.81 890.13! MODEL PREDICTIONS AND POLICY 397 APPENDIX 15C EFFECT OF VEHICLE WEIGHT ON OPERATING COSTS Figure 15C.1: Total running costs per (a) 1,000 vehicle-ka, (b) 1,000 ton-km of vehicle weight and (c) 1,000 ton-km of payload, versus gross vehicle weight for two extreme road conditions (a) Total running cost (pesos/lOO vehicle-km) 3000 2500-.- 2000 100 + 2000 + + + 0 0 5 10 15 20 25 3 35 40 45 Gross vehicle weight (ton) Best condition +--4+-( Worst condition 398 MODEL PREDICTIONS AND POLICY �Ь � Tota1 running сов[ (ревоа/1000 [on-km of vehicle иeight) 300 + + 200 + + '} ++ + ж + ж + # �+ + + 100 ж + ++ + .{. ж + ++ + ж t + + + + *ж * '� ж*ж * ж ж,� ж* '�` ж ж ж ж * * ж 0 0 5 10 15 20 25 ЭО Э5 40 45 Groas vehicle иeight (ton) ж ж ж Best condi[ion +++ Wогв[ condiCion �С� То[а1 runniг� сов[ (ревов/1000 ton-km of рауlоад) 700 + 600 + 500 400 + 30D + + t + + 200 ж + t * 'Е + t t + 100 ж + t + *ж * * + ж + ж� ж ж* ж* ж ж * ж 0 0 5 1D 15 20 25 ЭО Э5 40 45 Gговв vehicle иeigh[ ([оп) # ж ж Вевt condition +++ Worat condition MODEL PREDICTIONS AND POLICY 399 APPENDIX IND DETAT-1 PREDICTION TABLES The units for the quantities predicted in these tables are as follows: Units Quantities Fuel consumption liters/1,000 vehicle-km Tire wear number of equivalent new tires/1,000 vehicle-km Depreciation and interest percentage of new vehicle price/1,000 vehicle-km Time per distance vehicle hours/1,000 vehicle-km Maintenance and parts percentage of new vehicle price/1,000 vehicle-km Maintenance and labor labor hours/1,000 vehicle-km Lubricant consumption liters/1,000 vehicle-km Table 15D.1: Physical quantities for a small car on a paved road surface Maint.- Maint.- Depre- iTime per Fuel LubricantsiTire wear part labor ciatton Interest distance IRF C 1I '0o 25 97. 1.8 0.0593| 0.1241| 2.0! 0.3056i 11.5 75 99.2! 2.4: 0.0867: 0.2461: 2.9: 1.1111: 0.3056: 12 1125 i 104.2 3.0: 0.11411 0.4872: 4.2: 111111 0.3056: 14.1 100 25 104.8: 1.8: 0.093: 0.1241: 2.0: 1 .i11n o.3o56: 16 0 175 : 107.5: 2.41 0.08671 0.24611 2.91 1.11111 0.3056 16.2 125 112.81 301 0.11411 0.48721 4.2 1.11111 0.30561 17 . 7I 11.10040 +-+-o+-+-2+-4-+11-+-+-e19 1o I 25 1168 1.81 0.05931 0.1241: 2.01 1.1111 0.30561 14 61 IlI 175 i 119,3 2.41 00867 0.24611 2.91 1.11111 0.30561 j9 7 -- -- - -- -- - - -- - I I-4+--- - -- -- -- - -- ------ - --- ----- - -------- ---- - 1125 123.7: 3.01 0.11411 0.48721 4.2 1.11111 0.30561 2041 140 '0o 25 197.7 181 0.05931 0. 12411 2.01 1.1111 0.3056 11 175 99.9 2.41 0.08671 0.24611 2.91 1.1111 0.30561 12.51 125 105 e 3.01 0.11411 0.48721 4.21 1.11111 0.30561 144 10 25105.b1 1.6, 0.05931 0l1241: - 201 1.11111 0.3056! 16-21 1 175 I 1B.11 2.41 0.08671 0.24611 2.9: 1.11111 0.30561 16.41 - - - - - - - - I - - ---+-- -- ---- ---- --- -- -- ---+-- - - ----- - I12I , 113.41 3.0: 0,1141: 0.48721 4.3-1 1.11ii! 0.30561; 17. I- - - - - - - - - 11000- - -- - - - - - - - - - - - - - - '2 '_ -- ---- - - -------- --- 10 i25 i 117.8: 1.8: 0.05931 0.12411 2.01: jl 0.30561 19.7! 1 ~ - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - ---- i -- - - - II75 i 120.11 2.41 0.0667; 0. 2461; 2-ý 11 0. 30i61 19,3 I I ---4-+- -----4- 11235 I 2.1 3.o1 0.ý1141: 0.48721 4.21 111 0.3056! 2ý0- -- - - -- - - - - ------------ ---- -- -V . 2b144ý ý8 0.05931 0.12411 2o1: ill 0.30561 14.5 y /5 106.71 2.4: 0.086Vý 0-24611: 2.9' 1.11111 0.30561 14.1 25112.21) 3.V 0.11411 0.46721 -2' 1.111<ý 0.3056V i i 500 25 511 5'1 009: 0-1 4V i33 ý 1 75 I-.. .8 0.05931 0.24611 02. 1.1111! 0. 30b5i51 ,. - - - -- - - - - --- - - - - - - - i i2%-.6is: 2.41 0.067: 0.2461, 2.9 1.111<1 0.20561 20. 125 120.41 3.0: 0.11411 0.48721 4A'.1211 0.20561: - is2 II.--------4-- --4-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Table 15D.2: Physieal quantities for a medium ear on a paved surface i Fuel LbiatMaint.- Maint.- i rDepre- ' Time per- Fuel iLubricantsTire wear parts labor c1at1on Interest distance RF C :QI o 0 25 1 0.s 1. 0.093 0.1282! 2.o! 1.oo! 0.2750! 10. 75 203.9! 2.41 0.0867! 0.2543! 2.9| i.ooo 0.2750 11.41 1125 217.31 3.01 0.11411 0.50331 4.31 1.oooo1 0.2750 13.7 500 25 224.0: 1.81 0.05931 0.122 2.01 1.0o00 0.27501 15.6 75 | 229.11 2.4: 0.0867 0.25431 2.91 1.oooo 0.27501 15.91 125 1 241.3: 3.01 0.11411 0.50331 4.31 1.oooo 0.27501 17.0 1000 25 1 255.31 1.8% 0.0593% 0.1282% 2.0% 1.0000% 0.2750% 19.4 75 259.91 2.41 0.08671 0.25431 2.91 1.oooo 0.27501 19.5 125 1 269.41 3.01 0.11411 0.50331 4.31 1.0oo 0.2750 20.2 40 0 25 1 199.7: 1.81 0.05931 0.1282; 2.0 1.0000 0.2750: 10.9 75 1 204.21 2.41 0.08671 0.25431 2.91 1.00001 0.27501 11.5 125 1 217.71 3.01 0.1141| 0.50331 4.31 1.00001 0.2750 13.7 I o 25 1 224.3| 1.6: 0.05931 0.12821 2.01 1.00001 0.2750| 15.7 j75 229.41 2.41 0.08671 0.25431 2.91 1.00001 0.27501 15.9 - 125 241.61 3.01 0.1141| 0.50331 4.31 1.00001 0.2750 17 .0 1000 25 256.2: 1.8: 0.0593| 0.1282 2.01 1.00001 0.2750 19.4 75 26o.5: 2.41 0.08671 0.25431 2.91 1.oooo 0.2750 19.61 125 269.8: 3.01 0.1141| 0.50331 4.3: 1.00001 0.2750 20.2 180 o 25 | 210.21 1.81 0.05931 o.1282| 2.01 1.oooo| 0.2750 12.8 1----------------------------------- 1-++------- - - - ---------- 7| 214.31 2.41 0.08671 0.25431 2.9I 1.oooo| 0.27501 13.01 125 1 227.61 3.01 0.1141| 0.5033 A 3 1.0001 0.2750 14.7 15o 25 235.8| 1.81 0.0593| 0.12821 2. 0 1.o00o0 0.2750 16.5 - - -- --0- - 25-- - +-- - -- -- - ----- - -- -- - - - - - - - --- - - - - - - -- - - 75 i 239.s| 2.41 0.08671 0.25431 2.9| 1.00001 0.27501 16.7 - - ---- - ---- --- - - --- ----- - -- -- --4- ---- - - -- .125 250.71 3 C, 0.11411 0.5033 4. 3 1.0000 0.27501 17,6 ------ -- - -- - - -- -- - -- ---~+-- - ---+-- - -- - -- - o25 e265s41 4-4 0.0s71 2. 1232 2.0 i oocoo 0.2750' 2I 0 S.---4-------- ------------ X 715 202 2.41 0.08o71 0.25431 2.91 1.0000 0.27501 20.0 12- 277. 3 -- 54 ------4-3-1.oooo| -o ---o| 20.6 Table 15D.3: Physical quantities for a large car on a paved surface MaInt. - Maint.- ' Depre- ' ' Time per- Fuel LubricantsTire wear partor ciation, Interest distance IRF Ci: QI i i o o 2s 214.9! 1.8! 0.0593! 0.1320! 2.1! 0.9091! 0.2500! 10.7 7i5 | - 221.9| 2.4: 0.0867: 0.2618: 3.01 0.9091: 0.25001 11.3 ,125 243.3: 3.01 0.1141: 0.5182: 4.3: 0.90911 0.2500: 13.6 soo 25 255.1| 1.B: 0.05931 0.1320: 2.1| 0.9091: 0.2500: 15.6 175 261.3: 2.4: 0.08671 0.2618: 3.01 0.9091| 0.25001 15.8 I125 | 277.1: 3.01 0.11411 0.5182: 4.31 0.9091| 0.25001 17.0 100 '25 i 297.0| 1.8| 0.0593 0.1320: 2.11 0.90911 0.25001 19.41 75 | 302.21 2.41 0.0867| 0.2618| 3.01 0.90911 0.2500 19.5 I125 I 313.8| 3.01 0.1141| 0.51821 4.31 o.9091| 0.25001 20.2 140 '0 25 I 218.4: 1.81 0.05931 0.1320 2.11 0.9091| 0.25001 10.8 175 225.31 2.41 0.08671 0.26181 3.ol 0.90911 0.25001 11.41 1125 | 246.4: 3.0| 0.1141: 0.5182: 4.3| 0.9091: 0.2500: 13.7 soo 25 | 258.7: 1.8: 0.0593: 0.13201 2.11 0.90911 0.25001 15.7 175 i 264.5: 2.4: 0.0867| 0.2618: 3.01 0.90911 0.25001 15.9 125 279.9: 3.0 0.11411 0.51821 4.3 0.90911 0.25001 17.0 1000 25 300.61 1.81 0.05931 0.13201 2.11 0.90911 0.25001 19.4 175 I 305.4| 2.41 0.08671 0.2618| 3.01 0.9091| 0.25001 19.5 I125 i 316.61 3.01 0.11411 0.5182| 4.31 0.9091| 0.25001 20.2 o io 25 254.61 1.s; 0.0593; 0.13201 2.1| 0.90911 0.2500 13.1 75 i 257.51 2.41 0.0867| 0.26181 3.01 0.90911 0.2500 13.3! 125 I 272.0| 3.01 0.11411 0.51821 4.3 0.90911 0.2500 14.9 o 25 | 285.1: 1.8: 0.0593! 0.1320: 2.1: 0.9091: 0.2500: 16.71 175 288.11 2.41 0.08671 0.26181 3.01 0.9091| 0.25001 16 -125 299.41 3.01 0.11411 O.SM121 4.31 0.90911 0.25001 17.7 1000 25 i 318.8ý 1.8| 0.05931 0.13201 2.11 0.90911 0.25001 20.1 75 i 322.8: 2.41 0.0867: 0.26181 3.0: 0.9091: 0.2500: 20.11 125 332.8| 3.01 0.1141| 0.S182| 4.31 0.9091| 0.2500 20.7, でbble ISD-4：川WBicd qu肌tltles for a utillty veUcle on a paved surface ・―---------_____---------------------------------------------------------------------- Table 15D.5: Physical quantities for a loaded bus on a paved surface ------------------------------------------------------------------------------------------------------------------------- Maint.- Maint- Depre- Time per-' 1 Fuel iLubricantsiTire wear 1 c i Interest i I I I parts labor iation , I distance ---------------------------------------------------------- -------------------------------------------------------- IRF 1C :01 I ------------------------------- 10 0 25 i 276.31 3.41 0,155131 0.0813l 8.51 0.13591 0.03701 13.2 ---------------------------------------- ------------------------------------------------- 75 i 277.51 3.91 0.16441 0.0971! 12.31 0.13911 0.03831 13.91 ------------------------------------------------------- ------------------------------------------- I 1125 i 272.71 4.51 0.16591 0.11r01 17 71 0.14971 0.04281 16.7 ------------------------------------------------------------------------------------------------------ ---- I 500 25 255.61 3.41 0.1649: 0.08131 8.5: 0.15151 0.04361 17.2 --------------------------------------------------------------------------------------------------- I 1 263.3! 3.9; 0.1795; 0.0971: 12.31 0.15291 0.04431 17.6 75 1 --------------------------------------------------------------------------------------------------- I !125 i 271.51 4,51 OA9241 0.11601 17.71 0.15891 0.0471: 19.3 ----------------------------------------------------------------------------------------------------- ---- 1000 25 i 257.6; 3.4: O iS701 0.08131 8.51 0.16351 0.04931 20.6 ---- ---------------------------------------------------------------------------------------------- 175 265.81 3.91 0.17141 0.09711 12.31 0.16431 0.04971 20.9 --------------------------------------------------------------------------------------------------- '125 276.21 4.51 0.18641 0.11601 17.71 0.16801 0.0516: 22-0 ----------------------------------------------------------------------------------------------------------------------- 140 i, 25 i 319.51 3.41 0.24371 0.08131 8.51 0.15071 0.04331 17.0 i - ------------------------------------------------------------------------------------------------- 175 i 325.01 3.91 0.26581 0.09711 12.31 0-15321 0.04441 17.71 1 --------------------------------------------------------------------------------------------------- 1125 333.21 4.51 0.2939: 0-1160; 17.7' 0.16111 0.04821 i9_9 ---------- I --------------------------------------------------------------------------------------------------- C) 500 25 i 322.41 3.41 0.3067; 0.08131 8.51 0.1618: 0.04851 20.1 4- ---------------------------------------------------------------------------------------------- 75 328-81 3.91 0.3357: 0.09711 12.31 0.16311 0.04921 20.5 ---- ---------------------------------------------------------------------------------------------- 125 337.91 4.51 0.37001 0.11601 17.7: 0.16781 0.05151 21 9 ---------------------------------------------------------------------------------------------------- ... 11000 25 329.11 3-41 0.30891 0.08131 8,51 0.1707: 0.05301 __ 22.9 i - ------------------------------------------------------------------------------------------------- 175 335-61 3.91 0.33771 0.09711 12.3: 0.1716: 0.05351 23.2 --------------------------------------------------------------------------------------------------- 1125 344.31 4-51 0.37091 0.11601 17.7: 0.17471 0.05511 24.2 ----------------------------------------------------------------------------------------------------------------------- [so 10 25 516.71 3.4: 0.6058 0.08131 8.51 0.18181 0.05911 26.6 i - ------------------------------------------------------------------------------------------------- 175 i 522.0: 3.91 0.65701 0.0971' 12.31 0.18261 0.0596: 26.91 1 --------------------------------------------------------------------------------------------------- 1125 528.71 4.51 0.7205: 0.11601 17.71 0.18591 0.06151 28.0 ------------------------------------------------------------------------------------------------------------- Soo 25 i 519.3: 3.41 0.76851 0.08131 8.51 0.18601 0-06161 28.11 ---- ---------------------------------------------------------------------------------------------- 75 i 524.51 3.9: 0.8411: 0.09711 12.31 0.18671 0.06201 28.3 ---- ----------------------------------------------------------------------------------------------- 1125 i 530 81 4.51 0.92731 0.11601 17.71 0.18921 0.06351 29.2 -------------------------------------------------------------------------------------------------------------- 1 11000 25 i 522.31 3.41 0.77551 0.08131 8.51 0.19051 0.0643 29 71 1 1 i ---- --------------------------------------------------------------------------------------------- * - I 175 527.4: 3.9 0.84841 0.09711 12.3: 0.19111 0.0647 30.01 1 --------------------------------------------------------------------------------------------------- 1125 533.41 4.51 0.93321 0.11601 17.71 0.1930: 0.0659 30.71 ------------------------------------------------------------------------------------------------------------------------- Table I5D.6: Physical quantities for an unloaded, light gasoline truck on a paved surface l l i lMaint.- Maint.- Depre- Time per- Fuel Lubricants!Tire wear parts labor , ciation Interest dislance +---- -----------1---- 25 321.6! 2.5! 03994! 0.0968! 6.6! 0.1616 0.04201 1.3 5329.3 1| 0.4203| 0.2639 11 1 0 1G88 0.0444- 14. -- --I-l- - - - - - - - - - -- - ---- -- - ---- - -- --- 12! | 362.8: 3. 7 0.4243: 0.4309| 14.3: 0.1898: 0.0520| 18.0 I -1---- - - - - - - - - - 100 2 343 6; 25 0.4509 G 09651 6.36 .18581 0505: 17.3 --- -- - - --- -- -- -- -- -- ---- - - -- - - - - - - - ------ - -- -- - -- -- - ----- -- -- - 75 -358.9| 3.1| o.49116 0.26391 11.1: 0 18961 0.019|1 18.01 !125 390.21 3.7| 0.53371 0.4309 14.31 0.203i| 0.05711 20.4 i- -- - - - - -+ - - - - - - - - - - - - 1 - - - - - - - - - - - ----------- - - - - - - - - - 4- - ---i-- - - 000 !2 386.3| 2.5: 0.42941 0.0968| 6.6| 0.2043| 0.0576| 20.6 75 395.3| 3.1 0.47491 0.26391 11.1| 0.20671 0.0585| 21.1 .123 421.6: 3.7| 0.5209: 0.4309: 14.31 0.2160: 0.0623 22 9 440 0 25 347-3 2.5| 0.5276| 0.0968: 6.6: 0.16421 0.0429| 13.71 75 356.81 3.11 0.5700 0.26391 11.11 0.17111 0.04521 14.8' 125 390.31 3.il 0.6034| 0.43091 14.31 0.1912; 0.05251 18.3 +o ----- -- - - - -- - ---+-+-- -- - -------4- - - - ---+-- - - - - C500 25 I 376.9: 2.51 0.6449: 0.09681 6.6; 0.1873I .05111 17.6 75 386.7: 3.1: 0.7120: 0.26391 11.11 0.1910 0.05251 18.21 125 416.6| 3.71 0.77841 0.4309| 14.3| 0.20411 0.0575 20.6 1000 25 413.1: 2.51 0.62881 0.09681 6.61 0.20521 0.05801 20.81 i 175 421,51 3.11 0.69611 0.26391 11.11 0.20761 0.0589 21.3 125 I 446.61 3.71 0.76781 0.43091 14.31 0.21681 0.06271 23.0 80 O 25 : 419.7: 2.5: 0.9244| 0.09681 6.61 0.1740 0.0463 15.3 75 | 429.21 3.1: 1.02361 0.26391 11.1: 0.1795 0.0482 16.21 125 8 458.2: 3.71 1.1376| 0.43091 14.31 0.19681 0.0547 19.2 1oo 25 | 445.41 2.51 1.22371 0.09681 6.61 0.19321 0.05331 18.61 75 I 454.4 3.11 1.3574| 0.2639| 11.1| 0.19651 0.05461 19.21 I125 +480.2 3.7| 1.50831 0.4309| 14.31 0.2083| 0.05921 21.4 I1000 25 475.5: 2.5 1.2240| 0.09681 6.61 0.20921 0.05961 21.6 I75 i 483.51 3.11 1.35661 0.26391 11.11 0.2115| 0.06051 22.0' 1125 sOs.8 3.71 1.50571 0.4309| 14.31 0.2200; 0.0640| 23.61 Table 15D.7: Physical quantities for a loaded, light gasoline truck on a paved surface Maint.- Maint.- Depre- Time per i Fuel LubricantsiTire wear a parts labor claton Interest distance ---------------------------- ----------------------------------------------------------------------------------- 'RF |C :01 -- - - - - --- -- -- -- -- -- '0 '25 360.6! 2.5! 0.39411 0.0968i 6.6! 0.1630! 0.04251 13.51 - --------------- --------------------------------------------- ---------- ----------+----------- 75 370.2 3.11 0.42221 0.26391 11.1 0.17001 0.04491 14.6 1 -----------+-------------------------------------------------------------------------------- 4 125 402.7 3.71 0.4381: 0.4309: 14.3: 0.19061 0.0523: 18.1 5 ----------+------------------------------------------------------------------------------------------- 8500 25 387.71 2.51 0.4570 0.09681 6.61 0.1890: 0.0517: 17.91 ----------------------------------------------------------------------------------------- 75 i 400.21 3.11 0.50671 0.26391 11.1| 0.19261 0.0531| 18.5 S ------------------------------------------------------------------------------------------ -125 431.91 3.71 0.5540 0.43091 14.3| 0.20531 0.05801 20.8 ---------+-----------------------------------------+ ----------- -------------------+----------+---------- 11000 25 425.91 2.51 0.44041 0.0968: 6.61 0.2077 0.0590 21.3 S-----------+------------------------------------------------------------------------------- 75 437.0: 3.11 0.4903: 0.26391 11.1 0.2100j 0.0599 21.71 ------------------------------------------------------------------------------------------- I 1125 464.1: 3.71 0.5430 0.4309 14.31 0.21871 0.06351 23.41 S-- --- -- 2+--------- ------------ -----+ ----------+ ---------- -------- ---------- ----------- ------------------- 40 0 25 425.11 2.51 0.5999 0.0968: 6.61 0.1714: 0.04541 14.91 1 ; -----------+----------------------------------------------------------------------------I 1 1 1.75 437.11 3.11 0.66171 0.2639| 11.11 0.17761 0.04751 15.91 S1 -----------+-------------------- ---------- ---------- ---------- ---------- ---------- ---------- 1-+ 1125 i 4692 3.7: 0.72701 0.43091 14.31 0.19561 0.05421 19.0 ----------------------------------------+------------------------------------------------------------ 1500 25 455.31 2.51 0.77211 0.0968| 6.61 0.1940| 0.0536 18.7 I i-----------+-------------------- ---------- ---------- ---------- ---------- ---------- ---------- 1 175 i 466.71 3.11 0.8580| 0.2639| 11.11 0.19731 0.05491 19.31 S-------------------------------------------------------------------------------------------- 1 -+-2 494.9 3.71 0.9520 0.43091 14.31 0.2090: 0.05951 21.5 ------------+---------------------------------------------------------------------- '1000 25 i 488.7| 2.51 0.76581 0.0968 6.6| 0.2110| 0.0603| 21.9 1 -----------+-------------------- ---------- ---------- ---------- ---------- ---------- ---------- 75 498.8| 3.11 0.85111 0.26391 11.11 0.21331 0.0612 22.3 1 ------------------------------------------------------------------------------------------- ---125 523.1| 3.71 0.94611 0.43091 14.31 0.2215 0.0646 24.01 1 ----------------------- ------------------------------------------------------------------------------------- 80 O 25 596.9| 2.51 1.2959: 0.09681 6.6| 0.19871 0.0554 19.6 S-----------+-------------------------------------------------------------------------------- 1I 175 i 605.11 3.11 1.4339: 0.26391 11.1| 0.20221 0.0568 20.21 S1 -----------+------------------------------------------------------------------------ ------ I S--2125 i 622.7 3.7| 1.60321 0.4309 14.31 0.2138 0.06141 22.4 +------------------------------+---------------------------------------------------------------------- 500 25 612.11 2.51 1.7278: 0.09681 6.61 0.21251 0.06091 22.21 1 -----------+-------------------------------------------------------------------------------- 175 619.41 3.11 1.91961 0.26391 11.1| 0.2149; 0.06191 22.6' -----------+----------+----------+----------+----------+----------+----------+----------+----------- 125 i 634.51 3.71 2.14441 0.43091 14.31 0.22341 0.06541 24.3 1 ---------------------------------------------------------------------------------------------------- h00 '25 627.3 2.5 1.73941 0.0968: 6.6 0.2247 0.0660 24.61 ---- ------+-------------------------------------------------------------------------------- 1 I75 634.11 3.11 1.93061 0.26391 11.1| 0.2265 0.0668| 25.01 11-----------+----------+----------+--------------------------------+----------+----------+----------- I I62 -2-+031+033 06+6 115652.21 3.71 2. 1508: 0.43091 14.31 0.23301 0.06971 26.31 1「able 15D.8: Physical guantities f。r an ｀긔끄10aded, light diesel truck on a paved s&lrface ·―-...,.-.-.,-,-―【1！【――【1-,,,,.--1!!.，』―:--&-!-,!1---,:,!1-1-!111--,.-.-4---!.,-1-- Table 15D.9: Physical quantities for a loaded, light diesel truck on a paved surface ------------------------------------------------------------------------------------------------------------------------- 1 Maint.- Maint.- Depre- I ' Time perl Fuel iLubricantsiTire wear p 1 c ! Interest ! I arts abor iation distance_ ---------------------------------------------------------------------------------------------------------------------- IRF 1C 10I I ------------------------------- 10 10 25 187.2! 2.5! 0.3863! 0.0968! 6.6! 0.1669! 0.0438! 14.1 i --------------------------------------------------------------------------------------------------- 175 191.91 3.11 0.41671 0.2639: 11.11 0.1734: 0.0461: 15.21 1 ------------------------------------------------------------------------------------ ---- I 1 1 203.61 3.7: 0.4371: 0.4309: 14.31 0.1927: 0.0531: 18.51 ,125 1 1 I ------------------------------------------------------------------------------------------------------ - 1500 25 194.4: 2.51 0.4558: 0.09681 6.6: 0.1909; 0.0524: 18 21 I ; ---------------------------------------------------------------------------------------- --- 175 i 200.6: 3.1; 0.50591 0.26391 11.11 0.19451 0.0538: 18.81 1 --------------------------------------------------------------------------------------------------- I 1 1125 213.21 3.7: 0.5545: 0.4309: 14.3: 0.2067: 0.0586: 21.11 1 1 ------------------------------------------------------------------------------------------------- ------------ I 1 11000 25 206.8: 2.5: 0.44081 0.09681 6.61 0.20901 0.0595: 21.5 i -------------------------------------------------------------------------------------------------- I 175 212.8: 3.1: 0.4909: 0.2639: 11.11 0.2112l 0.0604: 21.91 1 -------------------------------------------------------------------------------------------------- I 1 1125 i 224.4: 3.7: 0.5442: 0.43091 14.31 0-219s: 0.06391 23.61 1 ------------------------------------------------------------------------------------------------------------ ------------ 140 10 25 208.7: 2.51 0.5858: 0.0968: 6.6: 0.1822: 0.04921 16.7 i -------------------------------------------------------------------------------------------------- 175 214.61 3.1: 0.6497: 0.2639: 11.11 0.1877: 0.0512: 17.61 I --------------------------------------------------------------------------------------------------- I 1 1 1125 228.81 3.7: 0.7238: 0.4309: 14.3: 0.2034: 0.05721 20.51 CD I I -------------------------------------------- I 00 ----------------------------------------------------------------- 1 500 25 220.9: 2.5: 0.7685: 0.09681 6.61 0.2014: 0.0564: 20.11 1 1 i -------------------------------------------------------------------------------------------------- 1 1 175 i 226.71 3.1: 0.8554 0.26391 11.11 0.20451 1 1 1 0.0577: 20.7 ------------------------------------------------------------------------------------ 1 1 1125 i 239.31 3.71 0.9544: 0.4309: 14.3: 0.21501 0.06191 22.7 1 1 -------------------------------------------------------------------------------------------------------------- 1 11000 25 i 234.4; 2.51 0.76881 0.0968: 6.6: 0.2164: 0.0625: 22.91 1 1 i --------------------------------------------------------------------------------------------------- 1 1 175 i 239.7: 3.1: 0.8550: 0.26391 11.11 0.2186: 0.06341 23.41 I I I ---------------------- I ---------------------------------------------------------------------------- I I 1 1125 250.6: . 3.71 0.95271 0.4309 14.31 0.22631 0.06671 24.91 1 ------------------------------------------------------------------------------------------------------------------------- I 180 io 25 315.0: 2.5: 1.30441 0.09681 6.61 0.2194: 0.06371 23.51 1 i -------------------------------------------------------------------------------------------------- I 1 1 175 319.2: 3. Ij 1.44401 0.26391 11.11 0.2224: 0. 06 50: 24.11 1 1 1 --------------------------------------------------------------------------------------------------- I 1 1 1125 i 327.9: 3.7: 1.61821 0.43091 14.3: 0.23181 0.06911 26.11 1 1 -------------------------------------------------------------------------------------------------------------- I 1 1500 25 1 322.01 2.5: 1.74371 0.0968: 6.6: 0.23021 0.06841 25 71 1 1 i ----------- I --------------------------------------------------------------------------------------- 1 1 175 i 325,91 3.1: 1.9383; 0.26391 11.1: 0.2324: 0.0694: 26.2 1 --------------------------------------------------------------------------------------------------- I 1125 i 333.31 3.71 2.1695: 0.43091 14.3: 0.2394: 0.07251 27.71 -------------------------------------------------------------------------------------------------------------- I 11000 25 329.0: 2.5: 1.7605: 0.0968: 6.6: 0.2399: 0.0727' 27.81 1 i --------------------------------------------------------------------------------------- I ----------- I 1 175 332.6: 3.1: 1.95471 0.26391 11.11 0.24161 0.07351 28.11 1 1 --------------------------------------------------------------------------------------------------- I I I 1 025 i 339.2: 3.71 2.18061 0.4309: 14.3: 0.24711 0.0761: 29.3! -------------------------------------------------------------------------------------- ----------------------------------- Table 15D.10: Physical quantities for an unloaded, medium truck on a paved surface Maint. Maint. Depre- Time per-' Fuel iLubricantsiTire wear parts labor a n Interest i d e- part labor ciation Iners distance ---- -- --- -- - ------------ I+~---------+-------------------------------+----------+--------------------+----------- RF C QI 1 1 1 1 1 1 O0 '25 227.9 3.4 0.1492 0.1140 7.2 0.1180 0.0373 13.3 1 ~ -- ---- --+----------+----------+----------+----------+---------+-----------+------------ --- --- - 1175 i 218.21 3.91 0.14831 0.31091 12.11 0.12521 0.04041 14.8 -125 213.81 4.5 0.14111 0.50771 15.61 0.14371 0.04911 19.11 500 25 206.4: 3.4 0.1518: 0.11401 7.2 0.1339 0.0444 16.8 i ------------ +--------7.----- + --------- - --- -- - ------- ------- 175 i 209.4: 3.91 0.1604: 0.31091 12.11 0.13811 0.04641 17.71 1125 i 216.9 4.51 0.1640: 0.50771 15.61 0.15101 0.05291 20.9 I - - - - - - - - -- - - - - - - - -- - - - - - - - -- - - - - - - - -- - - - - - - - -- - - - - - - - '2I-------- 1000 25 207.01 3.41 0.14171 0.11401 7.21 0.14681 0.05071 19.9 ----- ------------- -- ------+- --------- ----------- ---- + ----------- ------ ------ 175 212.2 3.91 0.15171 0.31091 12.11 0.14951 0.05211 20.51 1125 222.91 4.5: 0.1597f 0.50771 15.61 0.1588f 0.05711 23.0 40 0 25 244.01 3.4: 0.18881 0.11401 7.21 0.12211 0.03911 14.11 1 I 175 241.41 3.91 0.1987 0.3109: 12.1 0.12871 0.04201 15.61 +--------+----------+--- - ---------------------- --------------------------------- 4- 1125 i 244.11 4.51 0.20631 0.50771 15.61 0.14581 0.05021 19.61 1 ---------------- - ------------------------------------------------------------------------ ----- '500 25 i 235.81 3.41 0.21821 0.1140 7.21 0.13661 0.04571 17.4 S-------------------- ---------- ----- --------------------+ ---------- ------------------- 175 i239.4 3.91 0z2352! 0.3109! 12.11 0.1405: 0.04761 18.31 1 --+++---------- ---------- ------------------------------------------------------------------ 2125 247.6 4.51 0.2508: 0.5077! 15.61 0.15271 0.0538| 21.41 1 ------- ---------------------------- --------------------------+ -------------------- --------I 1I 11000 25 238.8: 3.41 0.21321 0.1140i 7.21 0.14861 0.0516| 20 3 1 1 +---------- -------------------- -------- --------------------+---------- ----------+--------:- 75 I 243.41 3.91 0.23051 0.31091 12.1: 0.15121 0.05301 21 01 1125 i 253.21 4.51 0.24801 0.50771 15.61 0.16021 0.05781 23.4 180 0 25 317.21 3.41 0.32311 - 0.11401 7.21 0.13331 0.04411 16.61 1I175 320.01 3.91 0.35791 0.3109; 12.11 0.13881 0.04671 17.91 1 - 15125 325.61 4.5: 0.40091 0.50771 15.61 0.15281 0.05381 21.4 - ----- -- - - - ------------ - -- --- --- --- --- --- -- ---- - - -- -- -- -- -- -- -- - -- -- -- -- -- -- - - -- - I- 500 25 317.81 3.41 0.41881 0.11401 7.21 0.1450 0.04981 19.4 ------- - +---------- --------- ---------- ---------- -- ------- + ---------- - -------- ---------- 175 i 321.41 3.91 0.45951 0.31091 12.11 0.14831 0.05151 20.21 +---- -------- --- ++--------+ ---------- -------------------------------- ----- 1-25 327.9 4.5 0.50881 0.5077 15.6 0.1586 0.05701 23.0 11000 25 320.41 3.41 0.42581 0.11401 7.21 0.15491 0.05491 22.01 1 175 324.31 3.91 0.46491 0.31091 12.11 0.15721 0.05621 22,61 1125 i 331.41 4.51 0.51061 0.50771 15.6 0.16511 0.0606: 24.7! Table 15d.11: Physical quantities for a loaded, medium truck on a paved surface Maint.- Maint.- Depre- Time per- Fuel iLubricants;Tire wear I parts labor eiation Interest distance - - - - - - - - - - - ------------------ ---- - -------- - - - --- ... -- - -- - -- - - - - -- - I--- - - RF jC :QI - - -- - - - - - - - - - - - - - - --+- - I I . . . . . I O 0 25 280.6! 3.4! 0.1433! 0.1140! 7.2 0.12101 0.0386! 13.9 0 10 275 1 - ---+---+---+---+---+- - - --- S 1~75 i 275.3: 3.9 0.1478: 0.3109 12.1: 0.1277: 0.0415: 15.4 - - - - - - - - - - - - - - - - - -I - 125 i 271.6; 4.5 0.14781 0.5077 15.6: 0.1451: 0.0498: 19.5 I----- 500 25 256.8; 3.4; 0.15271 0.1140 7.2: 0.13801 0.04631 17.7 175 i 263.81 39 0.1643; 0.31091 12.1: O14181 0.0482; 18 6 1 125 i 273.7; 451 0.17331 0.5077: 15.6; O15361 0.0542 21.61 100 25 2572 3.4 0.1459 0.11401 7.2 0. 1509: 0.05281 20 9 - 75 4 266.01 3.9: 0.1581: 0.3109; 12.1 0.1534: 0.05411 21 61 1 1125 i 279.1: 4.5; 0.1698 0.50771 15.6 0.16181 0.05881 23-81 40 25 i 3394: 3.41 0.2203 0.11401 7.2: 0.1351: 0.0450 17 01 I 175 344.8 3.91 0.2406: 0.3109 12.11 0.1407: 0.0476 18.41 Ill I 1125 354.4; 4.51 0.26431 0.5077 15.6; 0.1544 0.05471 21.81 1- 1500 25 340.4; 3.41 0.2740 0.1140 7.21 0.14831 0.0515 20.2 175 347.33 3.9; 0.29981 0.3109: 12.11 0.1515 0.0532 21.11 -+- 3 125 8358.0 4.5; 0.3295 0.50771 15.6; 0.16121 0.0584 23.71 ++----+---I- 1000 25 344.7; 3.41 0.27551 0.1140 7.2; 0.1586; 0.0569: 22.9 175 351.9 3.91 0.30071 0.3109; 12.11 0.16081 0.05821 23.61 I------- 125 362.61 4.5; 0.3294; 0.50771 15.61 0.1681: 0.0623 25.61 180 25 533.6 3.4: 0.5095: 0.1140: 7.2 0. 1628; 0.05931 24.1 75 539.9 3.91 0.56121 0.3109 12.11 0.1663 0.06131 25.1 1 - 1 2 1125 548.0: 4.5; 0.6275 0.50771 15.61 0.1756; 0.0669; 27.81 1 +-- -4-+-+--++--------------------------------------------------- 1500 25 i553665. 3.41 0.6609; 0.1140: 7.21 0.1711 0.0642: 26.5 175 542.3 3.91 0.72601 0.3109 12.1; 0.1734 0.06561 27.2' 1125 549.4 4.5; 0.8064 0.50771 15.6 0.1801: 0.0697 29.2 -- ------- I - - - - - - - - - - - - - - - - - - - - - - ----- - --- -------------- -- - -- - -- -- - -- - - I- 1000 25 538.6 3.41 0.6713 0.1140 7.21 0.17801 0.06841 28.6 1I175 544.1: 3.91 0.7352; 0.31091 12.1; 0.1797: 0.06951 29 11 125 550.8: 4.5; 0.8113; 0.5077 15.6 0.1849: 0.0728: 30.81 - Table 15D.12: Physical quantities for an unloaded, heavy truck on a paved surface Maint.- Maint.- Depre- ' Time per-' Fuel LubricantsiTire wear parts labor eiation Interest distance - - - - - - - - - - - - I - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - RF :C :QI '0 '25 1 241.8! 3.4! 0.1842! 0.1740! 11.1 01129 0.0359 13.4 175 233.1! 3.9: 0.18681 0.33711 15.71 0.1196 0.03881 14.9 I125 228.71 4.51 0.18371 0.50021 19.31 0.13691 0.0471: 19.2 500 25 219.8 3.41 0.19511 0.17401 11.11 0.1276 0.0425 16.9 175 223.61 3-91 0.2072: 0.33711 15.71 O13161 0.04441 17.8' ,125 231.41 4.51 0.21501 0.50021 19.31 0.14381 0.05061 21.0 1000 25 219.71 3.41 0.18581 0.17401 11.11 0.13971 0.04851 19.9 175 225.71 3.91 0.19901 0.33711 15.71 0.14231 0.04991 20.61 125 236.91 4.51 0.2109 0.5002 19.31 0.15111 0.0546 23.11 40 '0 25 i 267.01 3.41 0.23491 0.17401 11.11 0.11881 0.03851 1481 175 267.31 3.91 0.25051 0.33711 15.71 0.12481 0.04121 16.2' 125 271.81 4.51 0.26591 0.50021 19.31 0.14021 0.04881 20.11 1500 25 i 262.0 3.4 0.27871 0.17401 11.11 0.13171 0.04451 17.9 75 266.4; 3.91 0.3014 0.33711 15.71 0.13541 0.04631 18-R! ,125 274.91 4.51 0.3246 0.50021 19.31 0.14651 0.05211 21.8 1000 25 264.9 3.41 0.27601 0.17401 11.1 0 1426 0500 20.7 I75 i 270.1: 3.91 0.2986: 0.33711 15.71 0.14511 0.05131 21.41 125 i 279.91 451 0.32261 0.50021 19.31 0.15331 0.0559 23.7 80 O 25 371.2 3.4 0.4145 0.17401 11.11 0.13381 0.04551 18.4 '0 '25 -- - -- - - -- - - - - - -- - -- 0 -- - - --4145- -- - -- - -- - - -- - -- - - -- - -- - - 175 375.21 3.91 0.45821 0.33711 15.71 0.13851 0.04791 19.6 125 381.2: 4.5 0.51351 0.50021 19.31 0.15031 0.0542 22.8I 500 25 372.71 3.4 0.53611 0.17401 11.11 0.14351 0.05051 20.9 I75 376.71 3.9 0.58791 0.33711 15.71 0.14641 0.05211 21.71 1125 i 383.01 4.51 0.6521 0.50021 19.31 0.15521 0.05701 24.3 1000 25 : 374.71 3.41 0.54611 0.17401 11.1J 0.1519: 0.05511 23.3 1 1 75 378.91 3.91 0.59601 0.33711 15.71 0. 15401 0.05621 23.9 125 i 385.71 4.51 0.65551 0.50021 19.31 0.16071 O06021 25.9! Table 15D.13: Physical quantities for a loaded, heavy truck on a paved surface Maint.- Maint.- Depre- Time per-' Fuel LubricantsiTire wear parts labor ciation Interest distance RF 1C QI1 ------------------------------- O0 '0 25 1 313.2! 3.41 0.1809! 0.1740! 1i.i 0.1170! 0.0377! 14.41 175 31.21 3.91 0.18931 0.3371: 15.71 0.1232 0.04051 15.81 I 1 125 308.71 4.5: 0.1941 0.50021 19.31 0.1390: 0.0482 19.7 500 25 i 288.91 3.41 0.1988: 0.17401 11.1 0.13311 0.04521 18.2 1 175 1 298.41 3.91 0.21461 0.3371; 15.71 0.1366 0.0469: 19.11 I ~ - 125 309.81 4.5 0.2290: 0.50021 19.31 0.1473: 0.0525 22.0 ---------+-----------+----------+----------+----------+----------+----------+----------+----------+--------- 11000 25 i 288.71 3.41 0.19281 0.17401 11.11 0.14521 0.05141 21.4 175 299.8: 3.91 0.2090: 0.33711 15.71 0.14751 0.05261 22.01 ,125 i 314.61 4.51 0.22561 0.50021 19.31 0.15521 0.0569 24.2 40 O 25 406.4 3.41 0.28881 0.17401 11.11 0.13741 0.0473 19.3 175 414.41 3.91 0.31571 0.33711 15.71 0. 14221 0.04981 20.6 ,125 426.91 4.5 0.3483 0.50021 19.31 0.1538, 0.0561, 23.8 1500 25 | 410.0 3.41 0.36111 0.17401 11.11 0.14891 0.0534 22.4 175 418.5 3.91 0.39501 0.3371: 15.71 0.15171 0.0550 23.21 S--------- --------- ---------- ---------- ---------- -- -------+--------- ---------- -- --- 125 430.71 4.5 0.43531 0.50021 19.31 0.15971 0.0596: 25.6 1000 25 414.71 3.41 0.36421 0.17401 11.11 0.1575 0.0583 24.91 75 i 423.31 3.91 0.39751 0.3371: 15.71 0.15951 0.05951 25.51 -125 435.11 4.51 0.43631 0.50021 19.31 0.1656 0.06311 27.4 BO O 25 671.3 3.4: 0.6940 0.17401 11.11 0.17131 0.06681 29.3 175 679.0 3.91 0.75741 0.33711 15.71 0.1739 0.0685 30.21 1 --+------------+---- - - --- - -- ..- - - I 125 688.61 4.51 0.83771 0.50021 19.31 0.18061 0.07311 32.5 -----+-+-+-+-+-+-------------- --------- - I 1500 25 674.2: 3.41 0.88611 0.1740: 11.11 0.17741 0.0709 31.4 i ------5-+-+-+-+---------------- I I75 i 681.51 3.91 0.97131 0.33711 15.71 0.17921 0.07211 32.01 1 125 690.21 4.5 1.07421 0.50021 19.31 0.18431 0.0757 33.9 11000 25 676.5 3.41 0.8955 0.1740 11.1 0.18261 0.0744 33.2 1 +--- - - - - - - - --+-- - --+- - - --+- - - --+- - -- - - - - - - - - - - - - - - - - - - - - - - - - 1 175 683.61 3.91 0.9799 0.33711 15.7 0.18401 0.07551 33.81 1 691+8 45I74 5029+180 . 1 125 1 691.81 4.51 1.07941 0.50021 19.31 0.18801 0.07841 35.31 Table 15D.14: Physical quantities for an unloaded, articulated truck on a paved surface Maint.- Maint.- Depre- Time per-' Fuel LubricantsiTire wear parts labor clation Interest distance RF 1C :QI pat ----nterest----distance-- I I III0 0 06 0 O 25 342.01 5.4, 0.29511 0.23061 27.91 0.0846 0.0306 12.31 175 i 348.71 6.01 0.30051 0.36021 35.2: 0.0958: 0.03631 15.5 - 125 402.91 6.6: 0.30751 0.48991 41.31 0.1195: 0.05051 23.4 500 25 346.4: 5.41 0.31841 0.2306: 27.9! 0.10291 0.04031 17.71 1 I 175 363.71 6.01 0.33591 0.36021 35.21 0.1071 0.04271 19.11 I125 412.91 6.6: 0.35271 0.48991 41.31 0.1229 0.05281 24.7 1000 25 370.1 5.4 0.31421 0.2306: 27.91 0.1145; 0.04721 21.6 75 385.5 6.0 0.3326: 0.3602: 35.2: 0.1167: 0.0487: 22. 4 -125 i 426.8: 6.61 0.35171 0.48991 41.31 0.12731 0.05591 26.4 140 'O 25 414.4! 5.41 0.40131 0.23061 27.91 0.09091 0.03371 14 11 i-- --- --+-+-+-+-+-+-- - I 175 423.31 6.01 0.42561 0.36021 35.21 0.09971 0.03841 16.7 I125 457.71 6.61 0.4488 0.48991 41.31 0.12061 0.05131 23.8 500 25 420+4 5-+ 0.46671 0.23061 27.91 0.10551 0.0418 185 75 431.51 6.01 0.49821 0.36021 35.21 0.10931 0.0440 198 I125 464.71 6.61 0.53191 0.4899 41.3 0.1239 0.0535 25.1 11000 25 433.6 5.41 0.46511 0.23061 27.91 0.1158 0.0481 22.1 175 i 444.81 6.01 0.49661 0.36021 3521 0.11801 0.04951 22.81 +125 475.01 6.61 O53121 0.48991 41.31 0.12811 0.0565 26.7 80 O 25 668.81 5.41 0.73931 023061 27.91 0.10531 0.04161 18.5 75 678.41 6.01 0.80171 0.36021 35.21 0.11131 0.0453 20.5 125 694.31 6.61 0.86971 0.48991 41.31 0.12631 0.05521 26.0 500 25 674.21 5.4 0.90811 0.23061 27.91 0.11501 0.0476 21.8 175 682.21 6.0! 0.98081 0.36021 35.21 0.11781 0.04941 22.81 125 i 696.41 6.61 1.06661 0.48991 41.31 .288 0.0570 27.0 1000 25 678.81 5.41 0.91381 0.23061 27.91 0.12241 0.05251 24.51 175 686.31 6.01 0.98491 0.36021 35.21 0.12411 0.05371 25 21 12 699.61 6.61 1.0673 0.48991 41.31 0.1322 0.0596 28.4! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Table 15D.15: Physical quantities for a loaded, articulated truck on a paved surface Maint.- Maint.- Depre- Time per-; Fuel !Lubricants!Tire wear parts labor ciation Interest distance 1 F:1I I I pa t I Ce --------------------------------------------------------------------------- -- ---------- ------------ ---------- RF 1C 10I ---------+---------+------------ 0 '0 25 i 483.4! 5.4! 0.3049 0.2306! 27.9! 0.0869 0.03171 13.0 i ---------------------------------------- + ---------- -----------------------------+----------I 75 497.2: 6.0: 0.3161: 0.3602: 35.21 0.0971: 0.0370 15.9 -----------+----------+----------+----------+----------+-------~-+----------+----------+----------- 125 i 551.61 6.6: 0.3283: 0.4899: 41.3 0.1198 0.0507 23.5 S---- -------------------------------------------------------------------------+------------+---------- 1500 25 484.81 5.41 0.3338 0.2306 27.91 0.10651 0.0423: 18.91 --------------------------------------+------------------------------------------ ---- 1I 175 510.61 6.01 0.35591 0.3602: 35.21 0.11001 0.0444 20.0 1--------------------------------------------------------------+-----------+-------------------- 125 563.01 6.6: 0.37881 0.4899 41.31 0.12411 0.05371 25.2 ---------+-----------00+ ---------- ------- ---------- ----------+------------------------------ ---------- 1000 25 i 507.11 5.41 0.33061 0.2306: 27.9 0.1177 0.0493 22.7 --------------------- ------+----------------------------------------+-------------------- 75 531.31 6.01 0.3532 0.36021 35.2 0.11951 0-0506: 23 41 -----------+----------+----------+----------+----------+----------+----------+----------+---------:-I 125 577.5 6.6: 0.3778 0.4899: 41.31 0.1289| 0.05711 27.1 --------+ ---------+---- ------- -------------------------+------------------------------------------------ 140 'O 25 721.31 5.41 0.5099 0.2306: 27.91 0.1079: 0.0432 19.31 i ---------------- ------ --------------------+----------+-------------------- -------------------- 75 737.01 6.01 0.54871 0.36021 35.21 0.1141: 0.0470 21.5j -----------+----------+----------+----------+----------+----------+----------+----------+ ---------- 1125 : 758.41 6.6: 0.5914: 0.48991 41.3 0.1283: 0.0567 26.9 ---------+ ----------- --------------------------------------------------------+-------------------- 1 1500 25 1 728.01 5.41 0.6103 0.23061 27.9 0.1190 0.05021 23.21 I I i ----------+----------+----------+----------+----------+----------+----------+----------+----------- 75 741.6 6.01 0.65801 0.36021 35.21 0.12161 0.0519; 24.2 1 ------------------ - ------ ---------- ---------------------------------------------------------- 125 : 760.81 6.61 0.71361 0.48991 41.31 0.13131 0.0589 28.11 S-------+ ----------- --------------------------+-------------------------------------------------- 1 11000 25 732.81 5.41 0.61211 0.23061 27.91 0.12611 0.0551: 26.01 1 2 - ------------+ ---------- ------- ---------- ----------+----------+-------------------- ---------- 1 I 175 i 746.01 6.0 0.65931 0.36021 35:21 0.12781 0.05631 26.61 S--------- -------------------- -------------------------------------------------+-----------I -125 764.11 6.6| 0.71371 0.4899 41.3 0.1349 0.06161 29.6 ------------ ----+------------ ----------------------------------------------+-------------------------------- I 80 O 25 1230.3 5.4 1.1877 0.2306| 27.91 0.13631 0.06271 30.21 ------ ----------------------------------------+---------------------------------------- 1 1 I175 1244.9 6.01 1.2779 0.36021 35.2 0.13911 0.06501 31.5 S------------------- ---------------------------------------------------------------------- 1125 1 1264.61 6.61 1.38501 0.4899 41.31 0.1474| 0.07211 35.4 + -------------------- --------+---------+----------------- 500 25 1 1235.01 5.41 1.4556 0.23061 27.91 0.1417| 0.0671: 32.71 1- ------+-------- --------------------------------+---------------------------------------- 1I 175 1 1248.61 6.01 1.5773 0.36021 35.21 0.14331 0.06851 33.41 S----- -----+---------- ---------- -------- ---------- ---------- ----------+-------------------- ,125 | 1266.2: 6.6: 1.7204: 0.4899: 41.3: 0.1492: 0.0737: 36.3 ----------------------------- ---------- + ------------------------------------------------------------ 1000 25 : 1238.6: 5.41 1.4609: 0.2306: 27.9: 0.1458 0.0707: 34.61 1 1 i -----------+----------+----------+----------+----------+----------+----------+----------+----------- 175 1251.9: 6.0: 1.5818 0.3602: 35.2: 0.1469: 0.0717: 35.21 S-----------+--------------------+ ---------- 0--------------------------------------+---------- 1 1 1~125 | 1268.31 6.61 1.72201 0.4899| 41.3| 0.15141 0.07571 37.41 Table 15D.16: Physical quantities for a small car on an unpaved surface Maint.- Maint.- Depre Time per- Fuel iLubricants Tire wear i l tion 1Interest d -I+pat pat labor I ,latdistaerste - - - - - - - - - - - -- - - - -+-- - - - - - - - - - - - - - - -+- -+- - - - - - - - - - - - - - - - - - - - - - - - IRF C Ic I --- -- - - - - - - - - - - - - - '0 50 99.5! 2.1! 0.0730 0.17481 2.4! 1.1111 0.3056 13.2 1 1150 111.41 3.31 0.12781 0.64331 4.91 1.1111 0.30561 16.5 250 i 143.8: 4.5 0.15521 1.26801 7.11 1.1111 0.30561 24.3 1 1500 50 117.41 2.1 0.0730 0.1748: 2.4: 1.11111 0.30561 19.5 I160 I 126.8: 3.3: 0 12781 0.64331 4.91 1.1111: 0.3056 21 01 II 1--+++---++ ,250 152.11 4.51 0.15521 1.26801 7.11 1.11111 0.30561 26.3 1000 50 i 128.81 2.11 0.07301 0.17481 2.41 1.1111 0.30561 22.5 '150 1 136.91 3.31 0.12781 0.64331 4.91 1.11111 0.30561 23.51 I - -+-+-+-+-+-+-+-+- 2250 158.51 4.51 0.15521 1.26801 7.11 1.1111 0.30561 27.8 40 50 100.21 21 0.07301 0.17481 2.41 1.11111 0.30561 13.5 1 5 112.11 3.1 03 12781 0.64331 4.91 1.11111 0.30561 16.71 250 144.11 4.51 O15521 1.26801 7.11 1.1111! 0.30561 24.4 500 50 118.3: 2.1 0.07301 0.17481 2.41 1.11111 0.30561 19.6' 150 127.41 3.31 012781 0.64331 4.91 1.1111f 0.30561 21.1 250 1 152.41 4.51 O 15521 1.26801 7.11 1.1111 0.30561 26.4' f0SO 1 129.81 2.11 0.07301 0.17481 2.41 1.1111 0.30561 22 6 1150 i 137.51 3.31 O12781 0.64331 4-91 1.11111 0.30561 23.61 ,250 1 158.91 4.51 O15521 1.26801 7.11 1.11ti 0.30561 27.9 0 O 50 107.9 2.11 0.07301 0.17481 2.41 1.11il 0.30561 15.51 150 119.01 3.31 O 12781 0.64331 4.91 1.1111 0.30561 17.91 I I---++I--- 1-1 1250 i 149.41 4.51 0.15521 1.26801 7.11 1.1111: 0.30561 24.8 1500 50 125.01 2.11 0.0730 0.17481 2.4 1.1111 0.30561 20.5 150 133.5: 3.31 O12781 0.64331 4.91 1.1111 0.30561 21.71 I I-++4--- 44+ 1250 157.51 4.51 0.15521 1.26801 7.1: 1.1111; 0.30561 26.7 h000 50 135.71 2.11 0.07301 0.17481 2.41 1.11111 0.30561 23.21 150 143.11 3.31 0.12781 0.64331 4.91 1.1111 0.30561 24.11 I 250 1250 i 163.71 4,51 0.15521 1.26801 7A11 1.1111: 0.30561 28.2! Table 15D.17: Physical quantities for a medium car on an unpaved surface SMant.- Maint.- cDepre- ' Time per-I Fuel LubricantsTire wear parts l ciation Interest distance IR - oI I I I ars I abr itI IRF :C 10I O 0 '50 O206.91 2.1! 0.07301 0.18051 2.41 .OOOO 0.27501 12.6 1150 236.61 3.31 0.1278: 0.66461 5.0 1.0000l 0.27501 16.21 250 315.11 4.51 0.15521 1.30981 7.21 1.0000 0.27501 24.1 1500 50 256.1 2.1! 0.0730 0.18051 2.4: 1.00001 0.27501 19.31 1 I 1150 276.41 3.3: 0.12781 0.66461 5.0: 1.0000 0.2750: 20.81 250 335.4 4.5 0.15521 1.30981 7.21 1.0000 0.27501 26.2 - - - - - - - - - - - - - - - - - - - - - - - - --- - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - 4 1000 50 284.31 2.11 0.0730 0.18051 2.41 1.00001 0.27501 22.31 1 1 1150 301.2 3.31 0.12781 0.66461 5.01 1.00001 0.2750 23.4 1 1 25 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 250 1 350.81 4.5I 0.15521 1.30981 7.21 1.0000 0.27501 27.7 140 '0 '50 207.2 2.11 0.07301 0.18051 2.41 1.00001 0.27501 12.71 1 1 1150 236.9 3.31 0.1278: 0.66461 5.0 1.0000 0.27501 16.2 II150-017-.3+-1780-6465.-1OOO-0 270 3- 1- 1250 315.3 4.5 0.15521 1.30981 7.21 1.00001 0.27501 24.2 '500 50 256.81 2.11 0.07301 0.18051 2.41 1.0000 0.2750 19.3 1 -150 276.81 3.31 0.12781 0.66461 5.01 1.0000 0.27501 20.8 11 - - - - - - - - - - - - - --- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -I-- - - - - - 1 I-, 250 33.4: 4.51 0.15521 1.30981 7.21 1.00001 0.27501 26.2 o 50 285.3 2.11 0.0730 0.18051 2.41 1.0000 0.2750: 22. - -----_ -__I 1150 301.7 3.31 0.1278: 0.66461 5.0 1.00001 0.2750: 23.41 250-| 351.7| 4.5: 0.1552| 1.3098| 7.2: 1.000 0.2750 27.7 18 10 5 287 2.11 0.0730: 0. 18051 2.41 1.00001 0,27501 14.01 h5so 245.8: 3.31 0.12781 0.66461 5.0l 1.00001 0.2750: 16.81 250 1 321.41 4.51 0.15521 1.3098: 7.21 1.00001 0.27501 24.31 I~ '01 258 2.1 001 0. 18051 2.41 1.00001 0.27501 19. I 150 284.31 3.31 0.12781 0.66461 5.01 1.00001 0.27501 21.11 I I50 341.41 4.51 0.15521 1.30981 7.21 1 .00001: 0.27501 26.31 1 100 5o0 293.11 2.1: 0.0730: 0.1805: 2.4: i.oooo: 0.2750: 22.71 110308.61 3.31 0.12781 0.66461 5.01 1.00001 0.2750, 23.7 1 1250 1 356.7: 4.51 0.1552: 1.3098: 7.21 1.00001 0.27501 27.9! Table 15D.18: Physical quantities for a large car on an unpaved surface Maint.- Maint.- Depre- I ' Time per-' Fuel iLubricantsiTire wear parts labor cation Interest distance - - - - ---- - - - ---- - - - - - - - - - ---- - - - - - - - - - - -*+- - - - ---+-- - - - IRF :C Q0I I O 0 50 228.81 2.11 0.07301 0.18591 2.51 0.90911 0.25001 I 1 o150 i 270.3 3.31 0.12781 0.6844: 5.01 0.90911 0.25001 16.2 - 250 370.1: 4.51 0.15521 1.34881 7.31 0.90911 0.2500 24.1 500 50 297.61 2.11 0.07301 0.18591 2.51 0.9091: 0.25001 19.3 I 1150 322.51 3.31 0.12781 0.68441 5.0: 0.90911 0.25001 20.81 ,250 395.3 4.5 0.15521 1.34881 7.31 0-90911 0.25001 26.2 1000 50 333.7 2.1 0.07301 0.18591 2.5: 0.90911 0.25001 22.3 - --- ---- I 1 1150 353.81 3.31 0.12781 0.68441 5.01 0.90911 0.25001 23.4 250 1 414.4: 4.51 0.15521 1.34881 7.31 0.90911 0.25001 27.7 140 '0 '5 1 232.21 2. 0.07301 0.18591 2.51 0.90911 0.25001 12.7 I i 0 2.11-- - - - - -- - - - - - - - - -- - - - - - - - - - -- - - - - - - - - -- - - - - - - - - - 7 I 1 150 272.91 3.31 0.12781 0.68441 5.01 0.90911 0.25001 16.21 I250 372.01 4.5 0.15521 1.34881 7.31 0.90911 0.25001 24.1 500 50 301.01 2.11 0.07301 0.18591 2.51 0.90911 0.2500 19.3 I50 325.2! 3.31 0.12781 0.68441 5.01 0.90911 0.2500: 20.81 250 397.21 4.51 0 15521 1.34881 7.31 0.90911 0.25001 26. 2 1000 50 337.21 2.11 0.0730 0.18591 2.51 0.90911 02500 22.4 1150 1 356.51 3.31 0.12781 0.68441 5.01 0 .90911 0.25001 23.41 1250 416.3: 4.51 0.15521 1.34881 7.31 0.9091: 0.2500 27.7 80 O 50 i 264.11 2.11 0.07301 0.18591 2.51 0.90911 0.25001 14.3 I I -+++----- 1150 i 293.11 3.3 0 12781 0.68441 5.01 0.90911 0.2500 17.0 250 i 384.81 4.5 O 15521 1.34881 7.31 0.90911 0.2500: 24.41 1500 50 319.01 2.11 007301 0.18591 2.51 0.9091; 0.2500 19.9 1150 i 340.61 3.3 0.12781 0.68441 5.01 0.90911 0.2500 21.21 250 409.21 4.5: 0.15521 1.3488: 7.31 0.9091; 0.25001 26.4 1000 50 352.4: 2.11 0.0730 0.18591 2.51 0.90911 0.2500 22.81 150 370.21 3.31 O1278: O6844 5.0 0.9091: 0.2500 23.71 I -- 250 i 427.8 4.51 0.15521 1.34881 7.31 0.9091: 0.25001 27.91 Table 15D.19: Physical quantities for a utility vehicle on an unpaved surface i Maint.- Maint.- Depre- Time per-' Fuel LubricantsTire wear parts labor ciation Interest distance I --- --b---a-s-T-r wear---------- 'RF IC QIQI+++++-- 0 O 50 167.3 2.11 0.0730! 0.2145! 2.7! 0.3780 0.0837! 14.3 150 183.81 3.3 0.12781 0.7896: 5.5: 0.4239: 0.09841 18.3 1250 226.51 4.51 0.15521 1.5562: 7.91 0.5069: 0.1290 26.6 1500 50 181.21 2.11 0.07301 0.21451 2.7: 0.44761 0.1066 20.51 I 1 150 1 199.2: 3.3: 0.1278: 0.78961 5.5 0.4692: 0.1144: 22.6 I I 1--+++---+4 1250 236.81 4.51 0.1552: 1.55621 7.91 0.52641 0.1370: 28.8 1000 1 192.21 2.11 0.0730 0.21451 2.71 0.47581 0.11691 23.3 1150 1 209.11 33 0.1278: 0.78961 5.51 0.49161 0.12291 25.01 I------- I 2250 243.61 4.51 0.15521 1.55621 7.9 0.53861 0.14231 30.2 -- - - - -- - - - -- - - - - - - - - -- - - - - - - ------------ ------------------------------- --------------- 140 10 50 168.1: 2.11 0.07301 0.21451 2.7: 0.3842 0.0856 14.8 150 185.0 3.3 0.1278 0.7896 5.51 0.4279 0.0998 18.61 1 1 ---- - +-+-+-4----------------------------- --------------- 250 228.11 4.5 0.15521 1.55621 7.91 0.50861 0.12971 26.8 500 50 t 186.31 2.11 0.0730 0.21451 2.71 0.4504 0.1076 2081 1150 202.5: 3.31 0.12781 0.78961 5.5 0.4715: 0.11531 22.91 I 1--44+---4+ 2250 238.71 4.51 0.15521 1.55621 7.9 0.52781 0.13761 29.0 1000 50 198.41 2.11 0.07301 0.21451 2.71 0.47781 0.1176 23.51 I-- I- 1150 213.1: 3.31 0.12781 0.78961 5.51 0.4935 0.1237; 25.2 2250 245.8 4.5 0.15521 1.55621 7.91 0.5398 0.14281 30.4 80 0 50 200.61 2.11 0.07301 0.2145 2.71 0.4120 0.09451 17.2 150 i 2216.51 3.31 0.12781 0.78961 5.5l 0.44461 0.1055 20.21 I250 i 256.11 4.51 0.15521 1.55621 7.91 0.51561 0.13251 27.6 500 50 218.5: 2.11 0.07301 0.21451 2.71 0.46361 0.11241 22.1 150 231.61 3.31 0.12781 0.78961 551 0.4819 0.1192 23.9 ,250 266.31 4.51 0.15521 1.55621 7.91 0.5335 0.14011 29.6 11000 50 i 228.21 2.11 0.0730 0.2145 2.71 0.48761 0.1214 24.51 1 1 ~ 150 241.6 3.31 0.12781 0.78961 5.51 0.50171 0.1269: 26 01 5I I I 250 1 273.1; 4.51 0.15521 1.55621 7.9: 0.49 0.1450: 31.01 Table 15D.20: Physical quantities for a loaded bus on an unpaved surface Maint. - Maint.- Depre- Time per- Fuel iLubricants!Tire wear parts labor clation I Interest distance RF 1C :or S------ - - - - - - - - II 0 0 soi 262.81 3.71 0.14921 0.08881 10.21 0.14621 0.04131 15.8 1 I 1150 276.1 4.81 0.1646: 0.1268! 21.31 0.16111 0.04811 19.91 1 1 - - - - - - -I-- - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 1250 i321261 6.01 0.18551 0.1775: 43.9: 0.18951 0.06371 29.3 1 150O s0 263.21 3.71 0.1628: 0.0888 10.21 0.16621 0.05071 21.51 150 284.8 4.81 0.19311 0.12681 21.31 0.17331 0.05441 23.71 I I I- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - --+- - - - - - - - I 1I 1 1250 i 328.71 6.01 0.22921 0.1775 43.91 0.19371 0.06631 30.91 1000 SO 270.5 3.71 0.1595 0.08881 10.21 0.17421 0.05491 24.0 150 i 291.91 4.81 0.1905 0.12681 21.31 0.17941 0.05771 25.71 1 250 333.71 6.01 0.22831 0.17751 43.9 0.19641 0.0681: 32.01 -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - 0 -- -- --0- +-+- +- - 40 0 . 323.3 3.71 0.2590 0.08881 10.21 0.15821 0.04681 19.1 I i - - - - - - - - - - - - - - - - - - - --+- - - - - - - - - - - - - - - --+- - - --+- - - --+- - - - - - - - 150 342.21 4.81 0.30921 0.12681 21.31 0.16951 0.05241 22.51 1250 10 382.51 6.01 0.37031 0.1775 43.91 0.19301 0.06591 30.7 500 50 334.1 3.71 0.3230: 0.08881 10.21 0.17291 0.05421 23.61 150 351.91 4.81 0.38921 0.12681 21.3: 0.17901 0.05751 25.6 -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - I - ---+- - 250 i 388.5: 6.01 0.47431 0.17751 43.91 0.19671 0.06831 32.1' 10 -----------------------------------------------4----------- 1000 50 340.91 3.71 0.32311 0.08881 10.21 0.17941 0.05771 25.7 1150 358.11 4, Fi 0.38891 0.1268: 21.31 0.18401 0.06041 27.4 -250 392.7: 6.01 0.47391 0.17751 43.91 0.19911 0.0699: 33.11 180 O SO0 520.7 3.71 0.63541 0.08881 10.21 0.1845: 0.0607 27.51 +--+ _ I---- 150 533.91 4.81 0.7593 0.12681 21.31 0.19021 0.06411 29.61 -250 554.21 6.01 0.92341 0.1775: 43.91 0.20501 0.0740 35.6 5 -------------+---- I 500 50 525.51 3.7 0.81201 0.08881 10.21 0.1918 0.06521 30.2 i -- - - - - - - - - - --1- - - - - - - - - - - - -- - - - - - - - - - --0- - - - - - - - - - - - +-----I--- 150 537.61 4.81 0.98181 0.12681 21.31 0.19571 0.06761 31.7 - 250 556.41 6.01 1.2070: 0.17751 43.91 0.20761 0.07581 36.71 1000 50 528.1 3.71 0.81581 0.08881 10.2: 0.19561 0.06761 31.71 1 1150 1 539.91 4.81 0.98471 0.12681 21.3) 0.19881 0.06971 33 0' 1250 i 557.91 6.01 1.2078 0.17751 43.91 0.20931 0.07711 37.51 T l 1D21:: Physcal q n:ities fir an n ag gasollea on an unpaved aurface M A aint. - Maint.- Depre- Time per- Fuel LubricantsiTire wear parts labor ciation Interest distance RF :c :Q -- - - - - - -- - - - - - +----I---. 10 0 50330.5! 2.8! 0.3928! 0.1803! 9.1! 0.1718! 0.0455 14.9 150 399.5 3.9: 0.4285: 0.5145: 15.7| 0.2064: 0.05841 21.0 1250 i 559.0: 5,1: 0.49781 0.84861 20.41 0.2591: 0.08201 32.1 i500 50 | 397.4| 2.81 0.4490: 0.18031 9.11 0.20831 0.05921 21.41 150 449.71 3.91 0.54281 0.5145| 15.71 0.2259: 0.06651 24.81 I250 i 583.51 5.11 0.65731 0.84861 20.4: 0.26611 0.08551 33.81 1000 50 | 432.3| 2.81 0.43971 0.18031 9.11 0.22191 0.0648 24.0 150 | 477.1| 3.91 0.53691 0.51451 15.71 0.23541 0.07071 26.8j £250 | 598.7| 5.11 0.65571 0.8486| 20.41 0.27031 0.08771 34.81 140 50 | 358.41 2.81 0.53851 0.1803 9.1| 0.17391 0.04621 15.31 I150 | 425.31 3.91 0.62091 0.51451 15.71 0.20731 0.05881 21.21 1250 i 579.4 5.11 0.73791 0.84861 20.41 0.2594 0.0821| 32.21 500 50 423.61 2.81 0.65941 0.18031 9.11 0.2092| 0.05951 21.5 I150 473.61 3.91 0.8042| 0.5145| 15.71 0.22651 0.06681 25.0 I1250 603.4| 5.11 0.98381 0.8486| 20.41 0.2664 0.0856 33.8 11000 50 i 457.11 2.81 0.65161 0.18031 9.11 0.2226 0.06511 24.2 150 500.11 3.91 0.79891 0.51451 15.71 0.2359 0.0710 26.9 £ 250 i 618.31 5.1! 0.98221 0.84861 20.41 0.27061 0.0878 34.9 580 lo 50 430.31 2.81 0.97741 0.18031 9.11 0.1818 0.04911 16.6 150 488.01 3.9| 1.19531 0.51451 15.7| 0.2113| 0.06041 21.91 l-- 250 627.31 5.1| 1.45741 0.84861 20.41 0.2608| 0.08281 32.5 500 50 484.91 2.81 1.2878| 0.1803| 9.11 0.21291 0.0610| 22.3 i +-- - - - -- - - - - - - - - - - --+-+- - - - - - - - --+- - - - --+- - - --+- - - - - -- - - - - - 1150 529.81 3.91 1.58601 0.5145| 15.71 0.22911 0.06791 25.51 I -250 i 649.61 5.11 1.96231 0.8486 20.41 0.2676 0.08631 34.11 1000 50 4 513.91 2.81 1.2854 0.18031 9.11 0.22531 0.0663 24.7 I150 553.41 3.91 1.58331 0.51451 15.71 0.23811 0.0719 27.41 £250 i 663.61 5.1 1.9611| 0.8486| 20.41 0.27177 0.08847 35.1 Table 15D.22: Physical quantities for a loaded, light gasoline truck on an unpaved surface Maint.- Maint.- Depre- Time per-' Fuel LubricantsiTire wear p l Interest distance 1 partsprt labor' ciation I IditneI IRF 1C QI- 1O 1O 50 369.1! 2.8 0.39551 0.18031 9.1! 0.1728 0.0459! 15.11 1150 438.51 3.9: 0.44741 0.51451 15.71 0.20691 0.05861 21.1 250 598.7 5.11 0.53131 0.84861 20.41 0.2593; 0.0821: 32.1 1O SO 431.01 2.81 0.46461 0.18031 9.11 0.20871 0.05931 21 5 150 486.81 3.9 0.57051 0.5145: 15.7: 02262: 0.0667 24A9 1250 622.81 5.1: 0.70301 0.84861 20.41 0.26631 0.08561 33.81 1000 50 464.81 2.81 0.45661 0.18031 9.1 0.22221 0.06491 24 11 1150 513.51 3.91 0.5652: 0.51451 15.71 0.23571 0.0709: 269 - - - - - - - - - - - - - -- - - - - - --+- - --+- - --+- - - --+- - --+- - --+- - - --+- - - - - - I 250 637.81 5.11 0.70t51 0.84861 20.41 0.2705 0.0878 4 8 40 O 50 437.11 2.81 0.62881 0.18031 9.11 0.17991 0.0484f 16 3 1 1 ~ 150 500.9: 3.9: 0.7600 0.51451 15.71 0.21051 0.06011 21 81 1250 646.11 5.1: 0.92461 0.8486 20.41 0.26061 0.08271 32,4' 500 SO 493.3: 2.81 0.8071: 0.18031 9.11 0.2120: 0.0607 22.1 1:50 543.5! 3.9! 0.9973: 0.51451 15.7 0.2286: 0.06771 25.4 + 250 668.68 6 5.1 1.23801 0.84861 20.41 0.26741 0.08621 34.11 h000 50 523.11 2.81 0.80261 0.18031 9.11 0.22471 0.06601 24.61 1150 i 567.51 3.91 0.99381 0.51451 15.71 0.23771 0.0717 27 31 +250 682.71 5.11 1.23681 0.84861 20.41 0.27151 0.08831 35.1 80O S 604.31 2.81 1.36931 0.18031 9.11 0.20351 0.05731 20.51 150 638.31 3.91 1.69651 0.51451 15.71 0.22461 0.06601 24.61 I I1:2 250 i 754.81 5.11 2.10191 0.84861 20.41 0.2664 0.0856 33.81 1 1 0 '50-- - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - I ---+++- 500 50 630.5 2.81 1.83181 0.18031 9.11 0.22551 0.06631 247 150 668.8 3.91 2.27261 0.51451 15.71 0.23881 0.07231 27.51 I I I - - - - - -- - - - - - - --+- - - - - - - - --+- - - --+- - - --+- - --+- - - --+- - - - - - - - - - I 1-+- 50 773.71 5.11 2.83991 0.84861 20.41 0.2725: 0.0888 35.3 1000 50 649.4 2.81 1.83651 0.18031 9.1 0.23541 0.07071 26.8 I I 1150 687.61 3.91 2.27451 0.51451 15.71 0.24631 0.0757: 29 21 2 7 I 1250 1 785.6, 5.ij 2.83961 0.84861 20.41 0.2762: 0.09081 36 3! Table 15D.23: Physical quantities for an unloaded, light diesel truck on an unpaved surface Maint.- Maint.- Depre- Time per- Fuel iLubricantsiTire wear parts i Interest distance -prt abr I iatin Iteet dsac IRFI 0 -- - - - - - - - - - - -----+- I- - - - - I+-+I-I+-±- 1o s 166.. 2.8! 0.38821 0.1S031 9.11 0.1740! 0.04631 15.31 1 1o150 s189.31 3.91 0.42901 0.5145: 15.7: 0.2074: 0.0588: 21.2 - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - I ,250 249.1: 5.11 0.5001: 0.84861 20.4: 0.25941 0.08211 32.2 500 50 185.6: 2.81 0.4494: 0.1803: 9.11 0.2091: 0.0595: 21.51 1150 206.41 3.91 0.54441 0.51451 15.71 0.2265 00668 25.O 250 258.21 5.1 0.6606: 0.8486: 20.41 026641 0.08561 33.81 1 11000 O i 197.41 2.8: 0.44051 0.1803: 9.1 0.22251 0.06511 24.2 1150 216.2: 3.91 0.53871 0.5145: 15.7: 0.23591 0.07101 26.9 1 1250 263. 91 5.11 0.6590: o84861 20.41 0.27061 0.08781 34.91 140 10 '50 1 143 .:1 0 1o4.a 2.s 0.5315 0.1803 9.11 0.17851 0.0479 16.1 150 200.81 3.91 0.62731 0.51451 15.71 O20971 0.05981 21.71 X, -1 12so 261.21 5.11 0.75121 0.84861 20.41 0.2603: 0.08251 3241 1I 1500 O i 198.41 2.81 0.66681 0.18031 9.11 0.2113: 0.06041 22.0 i - - - - - - - - - - - - - - - - - - - -I 4 1150 218.9: 3.91 0.81621 0.51451 15.71 0.22811 0.06751 25.31 1 ----------------- ---+----------------------------- -I 1 1-1250 270.41 5.11 1.00211 0.8486: 20.41 0.2671: 0.08601 34.0 1 1~ 's ---- -+--------------------------------------------------------- 1000 5o 20.91 2.8: 0.66081 0.18031 9.11 0.22411 0.0658 24.5 1Iso 229.0 3.91 0.8119 0.51451 15.71 0.23721 0.0715: 27.21 2250 276.21 5.11 1.00071 0.84861 20.41 0.2713: 0.08821 35.0 so o 5o 217.1 2.81 0.9759: 0.1803 9.1: 0.1930 0.05321 18.6 so0 235.81 3.91 1.21801 0.51451 15.71 0.2183: 0.06331 23.31 2 250 290.51 5.11 1.5023: 0.8486: 20.41 0.26371 0.0842; 33.21 - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ------------ - - ----- I oo500 '50 233.9: 2.81 1.3140 0.1803: 9.11 0.21941 0.06371 23.5 1Iso 250.2: 3.91 1.6268; 0.51451 15.71 0.23421 0.0702: 26.51 --I 2I 1- + 250 299.3 5.11 2.02451 0.84861 20.41 0.2701: 0.0876 34.71 1000 5o 242.31 2.81 1.31731 0.1SO31 9.11 0.2305: 0.06851 25.81 I I150 259.61 3.91 1.62731 0.51451 15.71 0.24231 0.07391 28.31 2 3 1250 304.91 5.11 2.02381 0.84861 20.41 0.27401 0.08961 35.7! Table 15D.24: Physical quantities for a loaded, light diesel truck on an unpaved surface Maint Maint.- Depre- Time per-' Fuel ,Lubricants'Tire wear parts labor ciation Interest distance IRF c QI i l-------------------+----------- i iI o 0 50 190.3 2.8! 0.3915 0.1803 .1 0.1759 0.0469 15.6 l 1 i5 216.6: 3.91 0.44791 0.51451 15.71 0.2084 0.05921 21.4 1250 279.2| 5.11 0.5336 0.84861 20.41 0.25981 0.08231 32.3 Ioo s0 209.6: 2.8: 0.46501 0.18031 9.1| 0.2099: 0.05981 21.7 150 II233.61 3.9: 0.57211 0.51451 15.71 0.2272| 0.0671: 25.11 -- -- -- - -- -- -- - +-- --- --- -- - --- --- - -- ----- - -- - - - - - - - -- - - - - - - - - - - - - - - - - - 250 288.3 5.1 0.70631 0.84861 20.41 0.26671 0.0858 33.91 1000 so 221.4! 2.8; 0.4575 0.1803! 9.1! 0.22321 0.0653 24.3 1i 243.21 3.91 0.56711 0.51451 15.7: 0.23651 0.0712! 27.0 --- 2 - 250 294. 5.11 0.7049| 0.84861 20.41 0.2709 0.0880 34.9 ------ -------+-----------+----------+----------+----------+----------+----------+------------------ ~---+--- ~-------I 40 o 0 214.0: 2.8| 0.6190| 0.18031 9,11 0.1895 0.0519: 17.9l 1i 1 242.31 3.9: 0.76231 0.51451 15,7: 0.21661 0.06261 23 --- --- -- --- --- -- -- - --- - - -- - --- - - -- 3 . 1250 302.61 5.1 0.93671 0,84861 20.41 0.2631| 0.0840| 33.o soo o 0 236.9: 2.81 0.8105 0.18031 9.11 0.21741 0.06291 23.1 150 259.31 3.91 1.o0060 0.51451 15,71 0.23291 0.0696 26.3 250 I 311.5: 5.11 1.2551: 0.84861 20.41 0.26961 0.08731 34.6 11000 o50 248.7| 2.8! 0.8090 0.18031 9.11 0.22891 0.0678; 25.5 I150 | 268.8: 3.91 1.0042| 0.51451 15.7| 0.24131 0.0734| 28.1 I - - - - - - - - - - - - - - - - - - - - - - ~ - - - - - - -- - - - - - - - - - - - - - - - - - - - - -- - - - - - - 0250 i 317.11 5.11 1.2541| 0.84861 20,4| 0.27361 0.08941 35.6 o ,o 0 318.61 2.81 1.37921 0.1803| 9.11 0.22331 o.o654 24.3 150 | 335.3! 3.9! 1.71651 0.5145: 15.7' 0.24051 0.07301 27.9 -- --- -- ---------- ---------- ------------------------- ------ ------ --- -- 250 | 369.7| 5.1: 2.1418 0.84861i 20,41 0.2754| 0.09041 36.11 - - - - - -- - - - - - - -- - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - 1500 o I 330.7! 2.81 1.85431 0.18031 9,11 0.24061 0.0731: 27.9 -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - +---++++--1 150 | 344.2ý 3.91 2.30661 o.5145; 15.7: 0.2519 0.07841 30.41 - - - - - - - - - - - ~ - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1250 374.9: 5.1! 2.8960 0.84861 20,4! 0.28071 0.09331 37.4! 11000 so i 336.9| 2.81 1.86301 0.1803| 9.11 0.2486| 0.0768 29.71 11s | 349.3 3.9! 2.31161 0.51451 15,7| 0.2580| 0.0s14| 31.s 250 1 378.3| 5.11 2.8967| 0.84861 20.41 0.2839| 0.09511 38.3! Table 15D.25: Physical quantities for an unloaded, medium truck on an unpaved surface Maint- Maint.- Depre- 'Time per-' Fuel LubricantsiTire wear parts labor ciation Interest distaIc -++- rt I MI I i stne RF :C jQI 0 '0 so 212.2! 3.7! 0.1389! 0.2125! 9.9! 0.1285! 0.0419 15.5 I 1 1150 223.4: 4.8: 0.1392: 0.60611 17.11 0.1573: 0.05631 22.6 1 -1 2250 286.41 6.O 0.15161 0.99981 22.2: 0.1967: 0.0810 34.81 1I 1500 50 214.51 3.71 0.14271 0.21251 9.9: 0.1556: 0.0553 22.2 1150 237.6 4.81 0.16181 0.6061: 17.11 0.17061 0.06391 26.3' 250 295.11 6.01 0.18491 0.9998 22.21 0.20121 0.08431 36.4 1000 50 223.4: 3.7t 0.1394: 0.2125: 9.91 0.16481 0.0604 24.7 150 245.71 4.81 0.1599 0.6061 17.1 0.1766 0.0675 28.1 +250 300.11 6.01 0.18451 0.99981 22.21 0.2036 0.0862 37.3 40 0 50 238.4: 3.7: 0.1900 0.21251 9.9 0.1316 0.0433 16.2 150 253.51 4.81 0.21011 0.60611 17.11 0.1588 0.05701 23.01 1250 1 311.31 6.01 0.2391 0.99981 22.21 0.19721 0.08141 35.0 500 50 245.9 3.71 0.21921 0.21251 9.91 0.15711 0.05611 22.5 I 1-: 50 266.6: 4.8: 0.2557: 0.60611 17.11 0.17161 0.06441 26.61 250 319.5 6.0 0.3006 0.99981 22.21 0.20161 0.08471 36.6 11000 '50i 1 i - 254.01 3.71 0.21691 0.21251 9.91 0.1659: 0.06111 25.0 1150 273.91 4.81 0.25421 0.60611 17.11 0.1774 0.0680 28.4 1 4..250 324.2: 6.0: 0.3002: 0.9998: 22.2: 0.2040 0.08651 37.51 180 0 SO 318.81 3.71 0.34761 0.21251 9.91 0.1410 0.04781 18.4 150 332.21 4-81 0.42121 0.60611 17.11 0.16391 0.0599 24.41 S----+--- - - --+-+----------- I 1+1 250 375.01 6.01 0.5010: 0.99981 22.21 0.19901 0.0827 35.6 500 50 325.21 3.71 0.44671 0.2125 9.91 0.16221 0.05901 23.9 i- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ---------- ------------------- - I o150 i 340.51 4.81 0.53561 0.60611 17.11 0.17521 0.06661 27.71 1 o------- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - --I --4250 381.6: 6.0: 0.6471: 0.9998: 22.2: 0.2032: 0.0859: 37.21 1I 11000 50 i 330.21 3.71 0.44781 0.21251 9-91 0.1699 0.0634; 26.11 1 1 1150 345.51 4.81 0.53561 0.60611 17.11 0.1805 0.06991 29.31 250 385.4 6.01 O6468 0.99981 22.21 0.20551 0.08771 38.1 Table 1SD.26: Physical quantities for a loaded, medium truck on an unpaved surface Ma int. - Maint.- Depre- Time per-' Fuel iLubricants Tire wear part labor I ciation Interest distance IRF |C -QI --- ~ - +-I---------+---..---..---- 1 0 '50 I 266.9! 3.7! 0.1396! 0.21251 9.9 0.13051 0.0428 16.0 150 i 280.21 4.81 0.1485: 0.6061: 17.11 0.15831 0.05681 22.9 250I| 344.4| 6.01 0.16751 o.9998| 22.21 0.19711 0.08131 34.9 soo 0 1 264.0: 3.71 0.15011 0.21251 9.91 0.15651 0.05581 22.4 iso i 292.11 4.8! 0.1749: 0.60611 17.11 0.17121 0.o642 26.51 I250 352.7| 6.01 0.20611 0.99981 22.21 0.20151 0.08461 36.5 1000 50 i 271.4: 3.71 0.14741 0.2125 9.91 0.1654| 0.0601 24.8 I 1150 i 299.21 4.81 0.17331 0.60611 17.11 0.17711 0.06781 28.3 I22o 1 357.41 6.01 0.20561 0.99981 22.21 0.20391 0.0864 37.4 140 0 o 342.11 3.71 0.23221 0.21251 9.9: o.1427 0.04861 18.9 IIo 363.51 4.81 0.2760 0.60611 17.11 0.16521 0.0607| 24.8 ,250 413.31 6.01 0.32721 0.99981 22.2 0.1997 0.08321 35.9 Io 50 i 350.41 3.71 0.2877 0.2125 9.91 0.1632 0.05961 24.21 S 150 I 372.5! 4.81 0.34491 0.60611 17.11 0.1762| 0.06731 28.0 ,250 1 419.8, 6.0' 0.41741 0.99981 22.21 0.20381 0.08641 37.41 11000 5o i 355.9| 3.71 0.28761 0.21251 9.91 0.17081 0.0639| 26.41 150 377-71 4.81 0.3445 0.60611 17.11 0.18131 0.07051 29.6 250 i 423.6; 6.01 0.41711 0.99981 22.21 0.20611 0.08811 38.3 o 0 so 537. 3.71 0.54291 0.2125! 9.91 0.16761 0.0620 25.51 I -150 i 552.8| 4.81 0.6629 0.60611 17.11 0.18301 0.07161 30.1 5 250 577.91 6.0 0.8041 0.99981 22.2ý 0.2089| 0.09031 39.3 soo so 542.0' 3.71 0.70471 0.2125 9.91 0.18121 0.07041 29.6 150 555.1| 4.81 0.85491 0.60611 17.11 0.1907 0.0768 32.7 250 581.2 6.01 1.0495: 0.9998| 22.2 0.21221 0.09301 40.7 1000 50 543.51 3.71 0.70891 0.21251 9.91 0.1865| 0.0739' 31.31 I- - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - I - - - - - +---+++-- 150 i 556.7 4.81 0.85691 0.60611 17.11 0.19451 0.07941 34.0 250 583.11 6.0 1.04981 0.9998 22.21 0.21401 0.0945 41.41 Table 15D.27: Physical quantities for an unloaded, heavy truck on an unpaved surface Maint.- Maint.- Depre- Time per- Fuel LubricantsiTire wear r ation Interest distance RF CI O0 so 226.4 3.7! 0.1768! 0.2555! 13.6 0.1226 0.0402 15.6 1150 238.01 4.8: 0.18351 0.5818: 20.91 0.1497: 0.0538 22.7 250 301.41 6.0: 0.20301 0.90811 26.31 0.1869: 0.07751 34.8 5- -----+---------------------++ ----------+----------- ---------- - ---- ----------+----------+------ 500 50 227.31 3.71 0.18851 0.25551 13.6: 0.14811 0.0530: 22.21 1--+------ ------------------------------------------------------------------------ ---- 150 251.71 4.8: 0.21511 0.5818j 20.91 0.1622: 0.0611 26,41 ----- ---------------- ------------+- ------------------------------------------------------- 1250 310.01 6.0 0.24811 0.90811 26.31 0.1911: 0.0807 36.4 --------------------------------------+------------------------------------------------------------ 1000 50 1 235.81 3.71 0.18521 0.25551 13.61 0.1567; 0.05781 24.71 I 1150 259.51 4.81 0.21331 0.5818: 20.91 0.16781 0.06461 28.2 - - +---------- ---------- ---------- ---------- ---------- ------- -- ---------- - --- 12O i 315.0 6.0 0.24761 0.9081; 26.31 0.19341 0.08251 37.4 40 i 50 264.51 3.71 0.24111 0.25551 13.61 0.12741 0.04241 16.8 i ------------4----------- ----- --------- ---- ----- 1150 | 280.81 4.81 0.27331 0.58181 20.91 0.1520 0.0551 23.31 +------------------------------------------------------------------------------+------- 250 | 336.21 6.0 0.31431 0.90811 26.31 0.1877 0.07811 35.1 ------------------------------------------------------+ -----------------------------+---------- 500 50 271.71 3.71 0.2853 0.25551 13.61 0.1504 0.05421 22.8 ---------+----------------- -----------------------------------+ --------------------+---------- 150 292.6 4.81 0.33421 0.58181 20.91 0.16381 0.06211 26.9 250 344.0 6.0 0.39521 0.9081 26.31 0.19181 0.08121 36.7 -------- --1-0- ---------- ---------------- 1000 5O 279.0; 3.71 0.28361 0.25551 13.61 0.15841 0.05881 25 21 1 1150 299.31 4.81 0.3330 0.58181 20.91 0.16921 0.0654 28.6 - -- - -- - ----- -- ----- - -- -- --- - -- -- --- -- --- - - - - - -------- ,250 i 348.51 6.01 0.39481 0.9081: 26.31 0.1940| 0.0830| 37.6 --------+----+-------------- ------------------------------ +------+---------- -------+ ------ 80 0 +O373.8| 3.7| 0.44521 0.25551 13.61 0.14031 0.0488| 20.11 S ------+-------------+----------+-----------------+------------------- ---------+-- - - 150 I 386.91 4.81 0.54091 0.5818: 20.91 0.1597| 0.0596| 25.6 -------- ----- -- ---- ---- - 250 424.11 6.01 0.64691 0.90811 26.31 0.1907: 0.0804: 36.3 --------- ---- - ----- - ------ - - ----- - ------- --- - - I-- - - - - - - -- - - - - - - - - - - - - 500 50 379.01 3.71 0.57371 0.25551 13.61 0.1581f 0.05871 25.1 150 393.41 4.81 0.68911 0.58181 20.91 0.16951 0.06561 28.7 - - - - - - - - - - - - -- - - - - - -- - ---- - -- ----- -- - ---- - ---- - - ---+-+-- - - - - - - 250 0 429.71 6.01 0.83541 0.90811 26.31 0.19451 0.08331 37.8 - ---000 -+-+-4--+----+--------------------------+--------- 1000 50 382.71 3.7 0.57631 0.25551 13.61 0.16481 0.06261 27.21 1 1150 397.31 4.81 0.68991 0.5818: 20.91 0.17411 0.06861 30.21 S1 ---------------------------------------------------+--------------------------------------- I 250 | 433.01 6.01 0.83531 0.90811 26.31 0.19661 0.0850 38.6, Table 15D.28: Physical quantities for a loaded, heavy truck on an unpaved surface Maint- Maint-- Depre- Time per-' Fuel LubricantsiTire wear 1 parts Or ciation Interest distance --------------------------------------------------------------------------- RF 1C 01 '0 '50 O301.4! 3.7! 0.1802! 0.2555 13.6! 0.1256! 0.0416! 16.4 S----------+---------- ----------- ------------------- ---------- ---------- + ---------- ----- --- 1150 316.81 4.01 0.1970 0.5818: 20.91 0.15121 0.05471 23.1 +------- ---------- -------------------- ---------- ---------------------------+--------- - 250 381.51 6.01 0.2253 0,90811 26.31 0.18741 0.07791 35.0 +------+------+--------------------- -------------------------+----------+----------------------------- 1500 50 1 295.71 3.71 0.19941 0.25551 13.61 0.14931 0.05361 22.5 S--------------------------------------------------------------------------------------- I I1 327.01 4.8: 0.23361 0.5818: 20.91 0.16321 0.06171 26.71 S+ -----------------------------------------------------------------------+------------------- I ,250 5 389.41 6.o: 0.27761 0.9081; 26.31 0.19161 0.08111 36.6 -- -+-----------+---- ------ ------------------------------+---------------------------- 11000 50 i 302.11 3.71 0.19681 0.25551 13.61 0.15761 0.05831 25.0 i -----+-------------------------------------------------------------------------------- 150 333.51 4.81 0.23211 0.58181 20.91 0.16861 0.06511 28.4 -+--------+-- --------------------------------------------------------+------------------- 5 250 393.91 6.01 0.27711 0.90811 26.31 0.19381 0.08281 37.5 40 0 50 411.41 3.71 0.30511 0.25551 13.61 0.14381 0.05061 21.0 +------- -+------ ---------- -----------------------------------+----------------------------- 150 1 436.6 4.8 0.36501 0.58181 20.91 0.16271 0.06141 26.5 ----+------------------------------- ---------- ---------------------------+---- ----I 250 2 484.11 6.01 0.4363 0.90811 26.31 0.19241 0.08171 37.0I ---------------------------------------------------------+ ---------- ------------------- 500 50 1 420.41 3.71 0.38071 O25551 13.61 0.16071 0.06011 25.9 S-++---------- ---------- ---------- ---------- ---------- ---------- + ---------- - --- ---- I150 i 444.41 4.81 0.45771 0,5818 20.91 0.1719 0.06721 29. -----------------+--------------------+--------------------+---------------------------- - 250 i 489.61 6.01 0.55691 0.90811 26.31 0.19601 0-0845 38.4 -------------+------------ ---------- ------------------ ---------- ---------------------------- 1000 50 1 425.21 3.7 0.3815: 0.25551 13.61 0.16691 0.0640 27.9 ------ ---------------------------------------------------+ ----------------------------- 150 ' 448.71 4.81 0.45781 0.58181 20.91 0.17621 0.07001 31.01 -----------------------------+ ---------- -----------------------------------+-------- 250 1 492.81 6.01 0.55671 0.90811 26.31 0.1980 0.0861l 39.21 +--------+---------------------------------+------------------------------------------+----------+--------- S 1 50 i 676.11 3.71 0.73081 0.25551 13.61 0.17461 0.06901 30.4 i----+-+- --------------- --------------+---------------------------------------+---------- 1150 694.41 4.81 0.88191 0.58181 20.91 0.18631 0.07711 34.61 --------------------+----------+----------+-------------------------------------------------- 1250 I250 716.6I 6.01 1.06881 0.90811 26.31 0.20651 0.09331 42.9 --I -+-+------ ----------------------------- ------ ---------- ---------- ----------------------------I 500 50 i 680.61 3.7 0.93821 0.25551 13.6: 0.18461 0.07591 34.01 1 i----------+----------+----------+----------+----------+----------+----------+----------+----------- I 1 696.91 4.81 1.13591 0.58181 20.91 0.19221 0.08151 36.9 ----- - +----------+----------+----------+----------+----------+----------+----------- --------I I I-+++---+++ 250 718.51 6.01 1.39391 0.90811 26.31 0.20911 0.0955, 44.11 -------------------+------------------------------------------------------------------------------ 11000 50 682.31 3.71 0.9425 0.2555 13.61 0.18861 0.07881 35.51 -+--------- ---------+ ----------------------------------------------------------I I IM 698.01 4.81 1.13821 0.58181 20.91 0.19501 0.08381 38.01 S -------------------------------------------------------------------------------------- 250 1 719.61 6.0: 1.39431 0.90811 26.31 0.21051 0.09681 44.71 實able 1 SD。29:P勿日icd quantities for an uOloaded,artlculated truck恤 a觀”upaved surface ：一〕＿___＿〕一〕一〔〕一〕〕一〕〕一〕〕〕〕一〕一〕〕一〕〕〕一〕〕 實able 15D。30:P勿slcd quautlties for a loaded,artlculated truckou題 觀。p&ed→face ：〕！一〕一〕一一〕〕一〕一〕〕一〕〕一一！〕〕一〕〕〕 CHAPTER 16 Conclusions The major conclusions from this study may be drawn from the perspectives of a planner as well as a researcher, as laid down in the following paragraphs. 16.1 PLANNING TOOLS The study has yielded the following tools useful for a range of highway planning purposes: 1. A set of aggregate models for predicting speed, fuel consumption, tire wear and vehicle utilization of cars, utilities, buses, and light to articulated trucks under free-flowing conditions. These predictions constitute some 75 percent of the total operating cost of a typical Brazilian heavy truck through the component costs of fuel, tires, depreciation and interest and crew (the remaining 25 percent comprise lubricants and vehicle maintenance). Expressed generally as non-linear algebraic functions of vehicle characteristics and road geometry, surface type and roughness, these models were mostly formulated under reasonable assumptions on driver/operator behavior and well-established principles of vehicle mechanics.1 With one exception2, they have been either calibrated (vehicle utilization model) or validated with independent data (speed and fuel models). To a substantial extent, the models are sensitive to vehicle and road characteristics and their interactions, and therefore are capable of discriminating among similar investment alternatives involving total transport cost tradeoffs. They are suit- able for a number of highway planning applications ranging from project to sector level, involving alternative sur- facing and geometric standards, as well as policy issues on road user taxes and vehicle axle load limits. Local adaptation of these models is facilitated by the fact that most of the model parameters can be readily interpreted in Because of the lack of data the tire wear prediction model for cars and utilities was calibrated as a simple linear function of road roughness. 2 As shown in Chapter 11, because of unsuitable data the wear coeffi- cient of the aggregate tread wear model could not be calibrated for the lateral force component. This is expected to cause some underpre- diction for roads with average horizontal curvatures greater than 300 degrees/km. 431 432 CONCLUSIONS physical terms. Now incorporated in the World Bank's HDM-III model for total transport cost prediction, these models can be used for predicting vehicle operating costs alone and can be implemented on hand-held or personal computers (e.g., through a spreadsheet package). 2. Micro prediction models for speed, fuel and tires. Although developed primarily as a basis for formulating and testing the aggregate models, when implemented in a production version3 these micro models can be used to evaluate the economic benefits of specific road link improvements. 16.2 RECOMMENDATIONS FOR FUTURE RESEARCH While the study has yielded planning tools that can be of immediate use, there is still a wide scope for possible future research to improve or extend the results, as discussed below: 1. The probabilistic limiting speed theory, which served as the basis for formulating the steady-state speed model reported herein, provides a generalized framework which can and should be used to incorporate additional policy variables such as, speed limit and curve sight distance. Also, the effect of road width could be further enhanced.4 The unit fuel consumption functions, quantified for normally aspirated gasoline and diesel engines, should be adequate for most applications after appropriate adjust- ment for the progress in engine efficiency since the early seventies. New functions could be calibrated for substan- tially different engine designs (e.g., supercharged engines with an intercooler or gasohol/alcohol powered engines). And further tests could be carried out to improve the adjustment factors. 3. The technique for transforming the micro speed, fuel consumption and tire wear prediction models into 3 For research purposes these models were coded in SAS (Statistical Analysis System) which is generally available only on mainframe IBM computers. Work is currently underway at GEIPOT in Brazil to develop a production version of the micro transitional prediction models for predicting speed, fuel consumption and tire wear. The Brazil data did not include roads narrower than 5 meters. As presented in Chapter 4, based on data from India, the effect of narrow roads on speed has been added to the steady-state speed model in a rather simple way. If possible, a future field study should incorporate other factors related to road width in the data collection effort such as shoulder conditions. CONCLUSIONS 433 convenient aggregate form has been found to produce acceptable accuracy over a moderately wide range of road geometry. However, it should be tested over a more extreme range to ascertain whether an adjustment factor is needed to accommodate more severe transitional driving effects. 4. While a reasonable relationship between vehicle rolling resistance and road roughness has been quantified, alddi- tional field data should be collected to strengthen the existing coefficients. Future research efforts should also investigate the effects of tire properties, suspen- sion characteristics and road roughness pattern on the magnitudes of the coefficients. This should yield a more generalized and easily adaptable relationship. 5. The micro prediction models can be used as a building block of a model for simulating traffic flow interaction and the resulting fuel consumption, and the wear of tires and possibly parts under impeded flow conditions. 6. Conjunctive use of detailed experimental and more aggre- gate survey data has been found to be more effective than either type of data alone. This is clearly evident in the contrast between the fuel consumption and tire wear models; for the former there was an abundance of both experimental and survey data whereas for the latter there were survey data alone. As a result, the estimated coefficients of fuel consumption models are much better determined and refined than those of the tire wear models. Future field studies should include a comprehen- sive program of controlled experiments on tire wear as well as a road user survey covering sufficiently detailed information for mechanistic tire wear modelling purposes. More comprehensive and detailed data should permit the tire wear prediction models to be not only better quanti- fied but also able to distinguish between tire construc- tion types (e.g., bias vs. radial), types of rubber com- pounds and road surface aggregates, etc. Special efforts should be devoted to quantification of the effect of hori- zontal geometry on tire tread wear. This would make the models more policy-responsive as well as enhance their local adaptability. 7. The treatment of vehicle depreciation and interest costs could be improved with incorporation of models for the economic life of the vehicle and with more detailed data on operating characteristics (e.g., actual operating speed, type and size of loads, time spent on driving, loading/unloading, etc.) for calibration of the vehicle utilization model. Emphasis should be given to examining possible effects of road characteristics on vehicle life, non-driving time and vehicle availability. Units Quantity Unit Synbol Note Length Meter in Fundamental Kilometer km 1,000 m Millimeter mm m/1,000 Mass Kilogram kg Fundamental Ton ton 1,000 kg (Also written "tonne") Time Second S Fundamental Hour h 3,600 a Year year or y Auxiliary Angle Radian rad Supplementary- fundamental Degree deg w/180 rad Thermodynamic Kelvin K Fundamental temperature Area Square meter in2 Volume Cubic meter m3 Cubic decimeter din3 In3/1,000 (for tire rubber volume) Liquid volume Liter liter m3/1,000 Milliliter ml liter/1,000 Mass density Kilogram per kg/m3 per cubic meter 435 436 UNITS Quantity Unit Symbol Note Moment of Kilogram-square kg-m2 inertia of meter mass Linear Meter per second m/s - velocity Kilometer per hour km/h 3.6 m/s Millimeter per second mm/s (m/s)/1,000 (for average rectified velocity) Time per Second per meter s/m 1/(m/s) distance Linear Meter per second m/s 2 acceleration square Angular Revolution per minute rpm w/30 rad/s velocity Force Newton N kg-m/s2 Kilogram force kgf 9.81 N Power Watt W N-m/s Metric horsepower metric hp 736 W or or hp 75 kgf-m/s (equal to 0.986 British hp) Energy Joule J N-m or W-s Kilojoule kJ 1,000 J Pressure Pascal Pa N/m2 Kilopascal kPa 1,000 Pa Glossary of Symbols Notes: 1. An additional entry for a subscripted form of a symbol is included for significant instances only. A separate list of subscripts is given following the Greek symbols. 2. For each entry a brief explanation is given. The chapter and section offirst occurrence may be consulted for details. 3. In the column for units: 1 means that the symbol stands for a dimensionless quantity. NVP signifies that the quantity has units of the average price of a new vehicle of the same class, i.e., it is expressed as a fraction of new vehicle price. km refers to vehicle-km unless specifically stated otherwise (e.g., tire-km). $ denotes any monetary unit. -- is used whenever units are inapplicable (e.g., expectation operator) or unhelpful (e.g., logarithm of speed) or various (e.g., a vector of disparate quantities). 437 438 GLOSSARY Symbol Meaning Units Section A Intermediate expression (= 0.5 RHO CD AR) -- 2B a Acceleration m/s2 2.2 a,...,a7 Coefficients of unit fuel consumption function -- 9.2 ADEP Annual depreciation NVP/1,000 km 12.3 AF Air resistance N 2.2 AFUEL Predicted aggregate operating fuel consumption liters/1,000 km 14.6 AINT Annual interest NVP/1,000 km 12.3 AKM Vehicle utilization km/year 12.3 ALABOR Predicted maintenance labor h/1,000 km 12.1 ALT Road elevation above mean sea level m 2.2 AMOMEN Acceleration governed by braking capacity m/s2 6.2 AOIL Lubricants consumption liters/1,000 km 12.1 APART Predicted maintenance parts NVP/1,000 km 12.1 APOWER Acceleration resulting from the use of HPRULE m/s2 6.3 AR Vehicle projected frontal area m2 2.2 ARS (= ARS(V)) Average rectified slope: rear axle suspension motion per distance mm/m 3.3 ARS80 = ARS(80 km/h) mm/m 3.3 ARV (= ARV(V)) Average rectified velocity: rate of cumulative absolute displacement of rear-axle relative to vehicle body mm/s 3.3 ASPEED Pred cted aggregate space-mean km/h 14.5 GLOSSARY 439 Symbol Meaning Units Section B Intermediate expression b...,b4 (= bi(RPM)) Parametric coefficients of intermediate form of UFC -- 9A BI Road roughness measured by Bump Integrator trailer at 32 km/h mm/km 14.2 BP Barometric pressure kPa 13.3 BW Free-flow width effect parameter 1 4.4 C Horizontal curvature degrees/km 2.1 es Curvature of homogeneous sections degrees/km 7A 00..,4 Coefficients of bi(RPM) -- 9A CD Aerodynamic drag coefficient 1 2.2 CFT Circumferential force on a tire N 11.1 CHC Cargo holding cost $/1,000 km 14.5 CKM Average road age of vehicle group km 12.1 CLp Parts exponent in maintenance labor equation 1 12.1 CLq Roughness coefficient in maintenance labor equation (QI)1 12.1 CLO Constant term in the maintenance labor equation 1 12.1 eR = c . 1,000 degrees 7A CN Cost of a new tire $ 11.1 COo Constant term in the lubricants equation liters/1,000 km 14.7 CPq Roughness coefficient in parts-roughness relation (QI)-1 12.1 CPO Constant term in parts-roughness relation 1 12.1 440 GLOSSARY Symbol Meaning Units Section CR Rolling resistance coefficient 1 2.2 CRH Crew hours h/1,000 km 14.5 CRPM "Average" (calibrated) nominal engine speed rpm 10.6 CRT Cost of a retread $ 11.1 CTC Tire tread wear coefficient - circumferential dm3/kJ 11.1 CTk Tire tread wear coefficient - lateral dm3/kJ 11.1 CTV Cost of tire wear (by vehicle) $/1,000 km 11.4 CTW Cost of tire wear (by tire) $/1,000 tire-km 11.1 CV[.] Coefficient of variation of (= standard deviation/mean) -- -- D Discriminant of a cubic 3B D Kolmogorov-Smirnov statistic for test of normality 1 4B D1,D2 Intermediate expressions -- 14.5 d Differential sign -- -- do...,d4 Coefficients of bi(RPM) -- 9A dj,d2 Intermediate expressions -- 14.5 DBRAKE Deceleration goverened by used braking power m/s2 6.2 DDESIR Desired deceleration m/s2 6.2 DDRIVE Decleration governed by used driving power m/s2 6.2 DE Depth of tire tread grooves mm 11D DECEL Predicted deceleration m/s2 6.2 DF Drive force N 2.2 GLOSSARY 441 Symbol Meaning Units Section DISTOT Total distance of travel provided by a tire carcass 1,000 km 11.1 DRT Differential speed ratio 1 9.1 E0 Estimation bias correction factor for section-specific steady-state speed prediction 1 3.4 E[.] Expectation of Elm.] =(El.])2 e Section-specific error in log-speed model - 3.4 e,...e4 Coefficients of bi(RPM) -- 9A EQNT Predicted number of equivalent new tires (by tire) (1,000 tire-km)- 14.6 EQNTV Predicted number of equivalent new tires (by vehicle) (1,000-km)-l 11.4 EVU Elasticity of vehicle utilization 1 12.2 F Predicted amount of fuel consumed ml 10.1 FC Fuel consumption per distance ml/m 10.1 FL Road "fall" 1 2.1 FRATIO Perceived friction ratio 1 3.3 FRATIO0 Intercept term in the relation between FRATIO and payload 1 4.3 FRATIO1 Slope term in the relation betweem FRATIO and payload kg-1 4.3 FUELA Predicted aggregate experimental fuel consumption liters/1,000 km 14.6 g Acceleration due to gravity (= 9.81) m/s2 -- gs Vertical gradient of homogeneous section s I 7A GF Gravitational force N 2.2 442 GLOSSARY Symbol Meaning Units Section GR Road gradient 1 2.2 GRT Gear speed ratio 1 9.1 GVW Rated gross vehicle weight kg 4.3 HAV Vehicle availability h/year 12.3 HP Vehicle power delivered at the driving wheels metric hp 2.2 HPBRAKE Used braking power metric hp 3.3 HPDRIVE Used driving power metrip hp 3.3 HPGRAD Used driving power modified to take into account the effective gradient metric hp 6.3 HPRATED Vehicle maximum rated power metric hp 4.3 HPRULE Driving power used according to the rule derived from assumed transitional driver behavior metric hp 6.3 HPSS Power required to sustain steady-state speed metric hp 6.3 HRD Hours driven h/year 12.2 I Number of sections in steady- state speed model estimation 1 4A Ie Moment of inertia of mass of rotating parts of the vehicle kg-m2 2.2 Iw Moment of inertia of mass of wheels kg-m2 2.2 IRI International Roughness Index m/km 2.1 Ji Number of vehicles of a class observed at section i I 4A K = e cs 1,000 degrees 7A s Ky pAge exponent in parts consumption equation 1 12.1 GLOSSARY 443 Symbol Meaning Units Section KO Tread wear per unit slip energy dm3/kJ 11.1 k Tire stiffness 1 11.1 ko Conversion factor between the ARS80 and the QI measures of roughness (= 0.0882) counts/m 3.3 L Length m Various ks Length of homogeneous section s m 7A LF Lateral force on vehicle N 3.3 LFT Lateral force on tire N 11.1 LL Length of lower trapezoid base of tire tread mm 11D LN Proportional length of downhill travel 1 7.3 LOAD Vehicle payload kg 4.3 LP Proportional length of uphill travel 1 7.3 LU Length of upper trapezoid base of tire tread mm 11D MW Mass of wheels kg 2.2 m Vehicle mass kg 2.2 mr Effective vehicle mass kg 2.2 me Equivalent translatory mass of vehicle rotating parts kg 2.2 MW Equivalent translatory mass of wheels kg 2.2 MRPM Maximum rated engine speed rpm 13.4 MVC Cargo monetary value $ 14.5 N Iteration counter 1 6.2 Nh Number of highway tires on a vehicle 1 11D 444 GLOSSARY Symbol Meaning Units Section Nr Maximum practical number of retreads for a tire 1 11.2.1 Ns Number of section-directions 1 13.3 n Sample size 1 4C n An exponent 1 11A NF Normal force on vehicle N 3.4 NFT Normal force on tire N 11.1 NG Weighted average of absolute values of negative grades 1 7.3 ngs = -Min (0, gs) 1 7A NH Negative power metric hp 9A NHO Lower limit on negative power metric hp 9.2 NHX Bounded negative power metric hp 9A NL I nzt m 71 s nte= ng 2e m 7A NL = nts m 7A NR Average number of retreads for a carcass 1 11.1 NRO Base number of recaps for a carcass 1 11.3 NTRIPS Number of trips a year 1 12.3 P = ps m 7A P[.) Probability of . 1 -- P0 Constant term in tangential exten- sion of parts-roughness relation 1 12.1 P1 Roughness coefficient in tangential extension of parts-roughness relation (QI) 12.1 GLOSSARY 445 Symbol Meaning Units Section p, po, pl, Coefficients in an algebraic etc. expression -- Various p5 Length of a homogeneous section s with a non-negative grade, zero otherwise m 7A PAX Average number of passengers per vehicle 1 14.5 PG Weighted average of positive and level grades 1 7.3 pgs = Max (0, ga) 1 7A PH Positive power metric hp 9A PHX Number of passenger-hours h/1,000 km 14.5 PL - pts m 7A Pz9 " Pgs is m 7A PNG Average of PG and NG 1 7.3 PNR(r) The probability that tire will last through r retreads 1 11.1 PP Proportion of paved part of a roadway 1 7.3 q, q0, q1, Coefficients in an algebraic etc expression -- Various QI Road roughness measured by quarter-car index scale counts/km 2.1 QIPo Exponential-linear transitional value of roughness in parts consumption equation QI 12.1 R Road velocity relative to wheel axle m/s 11A R2 Explained variance/total variance 1 -- r Intermediate expression -- 3B r Variable denoting number of retreads for a tire 1 11.1 446 GLOSSARY Symbol Meaning Units Section r,, rl Surface type-specific parameter in the speed-sensitive ARS formula -- 3.3 RC Radius of curvature m 2.1 RF Roadway rise plus fall m/km 7.3 RHO Mass density of air kg/m3 2.2 RL Route length km 12.3 RMT Average of the inner and outer radii of a tire m 2A.2 RNT Nominal radius of a tire m 2A.2 RPM Engine speed rpm 9.1 RR Rolling resistance N 2.2 RREC Ratio of costs of a retreading and a new tire 1 11.4 RRT Tire rolling radius m 2.2 RS Road "rise" 1 2.1 RWG Wheel radius of gyration m 2.2 S Round-trip speed on a given route km/h 12.3 S I st m 7A 5 ss Superelevation of homogeneous section s 1 7A SE Slip energy J 11B SFC (=SFC(HP, RPM)) Specific fuel consumption ml/metric hp.s 9.3 st s % m 7A SMS Space-mean speed m/s 5.1 SP Superelevation 1 3.2 GLOSSARY 447 Symbol Meaning Units Section SR Speed reduction ratio 1 2.2 SSQ(.) Sum of squares to be minimized with respect to SSR Sum of squared residuals -- -- ST Surface type Qualitative 3.2 STDLOAD Assumed average payload of observed population vehicles kg 4.3 T Tire slip velocity m/s 11A t Student's test statistic 1 -- t Elapsed time s 2A.2 TARE Vehicle tare weight kg 14.3 TF Nominal tire diameter m 9.1 TD Driving time h/trip 12.3 TF Tangential force on vehicle N 3.3 TFT Tangential force on tire N 11.1 TIRE Predicted amount of rubber lost dm3 11C TLNEW Distance life of a cost-equivalent new tire 1,000 km 11.1 TM Temperature K 13.3 TN Non-driving time h/trip 12.3 TT Trip time h 12.3 TWN New tread wear rate (1,000 tire-km)-1 11.1 TWR Retread wear rate (1,000 tire-km)-1 11.1 TWT Tire tread wear - total dm3/1,000 tire-km 11.1 TWTc Tire tread wear - due to circumferential force TWT.t Tire tread wear - due to lateral force 448 GLOSSARY Symbol Meaning Units Section TWTO Tire tread wear - due to unmeasured influences U, U , U, Logarithms of respective V etc. quantities -- 3.4 ui Mean of log-speeds at section i -- 4A UFC (= UFC(HP, RPM)) Unit fuel consumption mi/s 9.1 UFC0 = ao + alRPM + a2RMP2 ml/s 9.2 V Vehicle speed m/s 2.2 Vbr Braking power-limited speed constraint as a random variable m/s 3.4 Vc Curvature-limited speed constraint as a random variable m/s 3.4 Vd Desired speed as a random variable m/s 3.4 Vdr Driving power-limited speed constraint as a random variable m/s 3.4 Vr Roughness-limited speed constraint as a random variable m/s 3.4 Vz General expression for speed con- straint treated as a random variable m/s 3.4 V' Arithmetic minimum of mean speed constraints m/s 3.4 V (=V(X,Y:e,8)) Systematic part of observed speed variate V m/s 3.4 Vz Systematic part of speed constraint variate Vz m/s 3.4 v Realization of observed speed random variable V m/s 3.4 vz Realization of speed constraint variate Vz m/s 3.4 Vw Wind speed m/s 2.2 Var[.] Variance of GLOSSARY 449 Symbol Meaning Units Section VAVG Average of VENMAX and VEXMAX m/s 6.2 VAVG2 Average of squares of VENMAX and VEXHAS -- 6.2 VBRAKE Constraining speed related to braking power m/s 3.2 VCTL Control speed: maximum allowed "safe" speed m/s 6.2 VCURVE Constraining speed related to curvature m/s 3.2 VDECEL Maximum allowed "deceleration" speed m/s 6.2 VDESIR Desired speed: constraining speed related to subjective factors m/s 3.2 VDESIR VDESIR modified to take into account effect of road width m/s 4.4 VDRIVE Constraining speed related to driving power m/s 3.2 VEN Predicted entry speed m/s 6.3 VENTMAX Maximum allowed entry speed m/s 6.2 VEX Predicted exit speed m/s 6.3 VEXMAX Maximum allowed exit speed m/s 6.2 VMOMEN Maximum allowed "momentum" speed m/s 6.2 VOL Tire rubber volume dm3 11.5 VPOWER Exit speed resulting from acceleration APOWER m/s 6.3 VROUGH Constraining speed related to roughness m/s 3.2 VSS Steady-state speed m/s 3.2 VSSa Average observed speed at section-direction s m/s 13.3 W Weibull random variable -- 3A.1 W Roadway width M 4.4 450 GLOSSARY Symbol Meaning Units Section w Error specific to speed observation in log-speed model -- 3.4 x (= x(t)) Distance traversed m 2A.2 x, x2 Arbitrary distance points m 2A.2 Y Intermediate expression -- 2A.2 Y Vector of vehicle characteristics -- 3.2 Z Gumbel random variable -- 3A.2 z Intermediate expression -- 3B a Scale parameter of a Weibull distribution -- 3A.1 01 Relative energy-efficiency factor 1 9.4 a2 Fuel adjustment factor 1 10.8 Shape parameter of Weibull distribution -- 3.4 r Gamma function -- 3A.1 Location parameter of Gumbel distribution -- 3A.2 y Central angle subtended by a curve Degree 2.1 y Euler's constant (~ 0.577) 1 3A.2 Yi Weights in the steady-state speed model estimation -- 4A A Increment sign AAR12 Energy loss due to air resistance between points 1 and 2 J 2A.2 AKE12 Change in kinetic energy between points 1 and 2 J 2A.2 APE12 Change in potential energy between points 1 and 2 J 2A.2 GLOSSARY 451 Symbol Meaning Units Section ARE12 Energy loss due to rolling resis- tance between points 1 and 2 J 2A.2 6ij Weibull model error treated additively -- 4A C (=c(X)) Section-specific error in observed speed -- 3.4 (=;(X,Y)) Vehicle-specific error in observed speed -- 3.4 l Slip angle rad 11A e Vector of steady-state speed model parameters -- 3.2 Scale parameter of Gumbel distribution -- 3A x Wheel slip 1 11A v (=v(X,Y,)) Joint contribution of speed constraint errors to observed speed variate -- 3.4 vz (=Vz(X,Y)) Random part of speed constraint variate Vz -- 3.4 Random part of observed speed variate V -- 3.4 Ez Random part of speed constraint variate Vz -- 3.4 7r Ratio of circumference to diameter (m 3.146) 1 -- Summation sign a Standard error of log-speed -- 3.4 ae Standard error of section-specific error in log-speed model -- 3.4 aw Standard error of error specific to speed observation in log-speed model --3.4 452 GLOSSARY Symbol Meaning Units Section T Time per distance s/m or h/1000 km 5.1 0 Road inclination rad 2.1 x Likelihood ratio 1 4C 2X Chi-square test statistic with f f degrees of freedom -- 4C w (=w(X.Y)) Error specific to a speed observation -- 3.4 References Abaynayaka, S.W. and others. A Study of Factors Affecting Traffic Speeds on Rural Roads in Kenya. TRRL Supplementary Report 28 UC. Department of the Environment, Transport and Road Research Laboratory. Crowthorne, England, 1974. Andersen, S.W. and S. Graven. "Fuel Consumption Model and Preliminary Abstract." Private communication. October 1979. Beck, R. Parametric Analysis of the Off-Road Performance of a Special Tracked Test Vehicle. Proceedings of the 6th International Conference, International Society of Terrain Vehicle Systems, vol. 3, August 22-25. Vienna, Austria, 1978. Ben-Akiva, X. and S.R. Lerman. Discrete Choice Analysis: Theory and Application to Predict Travel Demand. MIT Press, Cambridge, Massachusetts, 1985. Benjamin, J.R. and C.A. Cornel. Probability, Statistics and Decision for Civil Engineers. McGraw-Hill, New York, 1970. Bergman, W. and W.B. Crum. "New Concepts of Tire Wear: Measurement and Analysis." Society of Automotive Engineers, SAE transactions 730615. Warrendale, Pennsylvania, 1973. Bester, C.J. Effect of Pavement Type and Condition of the Fuel Consumption of Vehicles. Transportation Research Record (1,000), pp. 28-32, 1984. . "Fuel Consumption of Highway Traffic." Doctoral dissertation, University of Pretoria. Pretoria, South Africa, 1981. Bhandari, A. and others. An Investigation of Cost Effective Road Maintenance and Improvement Options. Discussion Paper No. 5. Transportation Department, World Bank. Washington, DC (forthcoming). Brademeyer, B. The Road Investment Analysis Model: User Manual Technology Adaptation Program, Massachusetts Institute of Technology, (TAP report 77-10). Cambridge, Massachusetts, 1977. Brodin, A., G. Gynnerstedt and G. Levander. A Program for Monte Carlo Simulation of Vehicle along Two-Lane Rural Roads. An Application of Structured Programming Technique and Simula-67 Language. Meddelande Institute, No. 143, National Road and Traffic Research. Linkoping, Sweden, 1979. 453 454 REFERENCES Central Road Research Institute. Road User Cost Study in India. Final report. New Delhi, India, 1982. Chesher, A. "Vehicle Operating Cost Equations Developed from the Indian Road User Cost Survey Data." Unpublished report prepared for Transportation and Water Department, World Bank. Washington, D.C., May 1983. 'Transferability of User Cost Equations: Methodology for Country Studies." GEIPOT/PICR project memorandum ICR/AC/002/82. March 5, 1982(a). "A Method for Estimating Limiting Velocity Vehicle Speed Models." Unpublished note prepared for Transportation Department, World Bank. Washington, DC, April 1982(b). - . "Transferability of User Cost Equations: An Example Using the Brazil Study Data." GEIPOT/PICR project memorandum ICR/AC/099/82. May 24, 1982(c). Chesher, A. and R. Harrison. Vehicle Operating Costs: Evidence from Developing Countries. Transportation Department, World Bank. Washington, DC (forthcoming). Chesher, A. and R. Harrison. "Tire Wear and the Effect of Highway Characteristics: Preliminary Report." Unpublished internal technical memorandum of GEIPOT/PICR Research. GEIPOT, Brasilia, Brazil, 1980. Chiesa, A. and G. Ghilardi. "Evaluation of Tire Abrasion in Terms of Driving Severity." Society of Automotive Engineers, SAE transactions 750459. Warrendale, Pennsylvania, 1975. Della-Moretta, L. "Relating Operational Variables to Tire Wear." Paper presented at the International Society for Terrain Vehicle Systems, Regional Meeting. Carson City, Nevada, October 1974. Della-Moretta, L. and B.C. Sullivan. "U.S. Forest Service Vehicle Operating Cost Model." Proceedings of the Workshop on Lessening Energy Waste in the Road-Tire-Vehicle-Driven System, sponsored by the U.S. Department of Agriculture, Forest Service and the University of California, Institute of Transportation Studies. Claremont, California, 1976. de Weille, J. Quantification of Road User Savings. World Bank Staff Occasional Paper no. 2, World Bank. Washington, DC, 1966. Dickson, L.E. New First Course in the Theory of Equations. Wiley, New York, 1957. Domencich, T.A. and D. McFadden. Urban Travel Demand: A Behavioral Analysis. North-Holland Publishing Co., Amsterdam, Netherlands, 1975. REFERENCES 455 Fisher, W.D. Clustering and Aggregation in Economics. Johns Hopkins University Press, Baltimore, Maryland, 1969. GEIPOT. Research on the Interrelationships Between Costs of Highway Construction Maintenance and Utilization: Final Report on Brazil-UNDP Highway Research Project. (12 volumes). Brasilia, Brazil, 1982. Galanis, N.H. Users Manual for Program RODES. National Institute for Transport and Road Research. Reports RX/1/80, RX/11/80, RX/12/80. Pretoria, South Africa, 1980. Gillespie, T.D. "Technical Consideration in the Worldwide Standardizatio and Road Roughness Measurement." Highway Safety Research Institute. University of Michigan, Report No. UM-HSRI-81-28. Ann Arbor, Michigan: 1981. Gillespie, T.D., L.W. Sayers and L. Segel. Calibration of Response-Type and Roughness Measuring Systems. NCHRP Report 228, National Cooperative Highway Research Program. Washington, DC, 1980. Good, M.C. Road Curve Geometry and Driver Behavior. ARRB Special Report No. 15, Australian Road Research Board. Victoria, Australia, 1978. Gourieroux, C. and others. Pseudo Maximum Likelihood Methods: Theory. Econometrica (52), pp. 681-700, 1984. Graven, S. "A Microscopic Discrete-Event Simulation Model for Two-Lane Highway Network." Unpublished report. Preliminary report. A Traffic Simulation Model for Two-Lane Roads, Institute of Transport. Economics, Norway, 1978. Green, J.H.A. Aggregation in Economic Analysis: An Introductory Survey. Princeton University Press, Princeton, New Jersey, 1964. Grosch, K.A. and A. Schallamach. "Tire Wear of Controlled Slip." Vol. 4 (356). 1961. Guenther, Karl W. Predictive Models for Vehicle Operating Consequences. M.S. thesis, Civil Engineering Department, Massachusetts Institute of Technology. Cambridge, Massachusetts, 1969. Gupta, K.L. Aggregation in Economics: A Theoretical and Empirical Study Rotterdam University Press, Rotterdam, Netherlands, 1969. Gynnerstedt, G., A. Carlsson and B. Westerlund. A Model for the Monte Carlo Simulation of Traffic Flow along Two-Lane Single-Carriageway Rural Roads, National Road and Traffic Research Institute. Linkoping, Sweden, 1977. Gynnerstedt, G. and R.J. Troutbeck. Computer Programs and Simulation Models with Associated Field Experiments for the Indian Road User Cost Study. Australian Road Research Board Internal Report AIR 838-2, Australian Road Research Board. July 1981. 456 REFERENCES Hide, 0. Vehicle Operating Costs in the Caribbean: Results of a Survey of Vehicle Operators. TRRL Report 1031. Department of the Environment, Transport and Road Research Laboratory. Crowthorne, England, 1982. Vehicle Operating Costs in the Caribbean. Working Paper no. 81. TRRL Overseas Unit. Department of the Environment, Transport and Road Research Laboratory. Crowthorne, England, 1980. Hide, H. and others. The Kenya Road Transport Cost Study: Research on Vehicle Operating Costs. TRRL Report LR 672. Department of the Environment, Transport and Road Research Laboratory. Crowthorne, England, 1975. Hoban, C.J. "Methods of Evaluation used in ARRB Simulation Studies." Paper presented at the Workshop on Rural Traffic Simulation. Australian Road Research Board. Victoria, Australia, 1983. Hodges, H.C. and K.D. Koch. Tire Treadwear Validation, vol 1: Technical Report. National Highway Traffic Safety Administration, U.S. Department of Transportation. Washington, DC, 1979. Johnson, N.L. and S. Kotz. Continuous Univariate Distributions, vol. I. Houghton Mifflin, Boston, 1970. Jones, D. and L. Della-Moretta. Truck Tire Wear Rates on Open vs. Dense-Graded Asphalt Pavement. Report No. 7977-1210. U.S. Department of Agriculture, Forest Service. Washington, DC, 1979. Kadiyali, L.R. and others. Rolling Resistance and Air Resistance Factors for Simulation Study. Road User Costs Study Technical Paper No. 92, Seventeenth Quarterly Report, Central Road Research Institute. New Delhi, India, 1982. Koppelman, F.S. Aggregation in Economic Analysis: An Introductory Survey, Princeton University Press, Princeton, New Jersey, 1964. Korst, H.H. and R.A. White. "Aerodynamic and Rolling Resistances of Vehicles as Obtained from Coast-down Experiments." In the Proceedings of the Second International Conference on Vehicle Mechanics, edited by P. Rapin and H.K. Sachs. Swets and Zeitlinger, Amsterdam, Netherlands, 1973. Lee, R.A. and F. Pradko. Analytical Analysis of Human Vibration. Society of Automotive Engineers No. 680091. Warrendale, Pennsylvania, 1968. Limpert, R. "Tires and Rims." Vehicle System Components. Wiley, New York, 1982. Lind, Robert C. and others. Discounting for Time and Risk in Energy Policy. Resources for the Future, Inc., Washington, DC, 1982. REFERENCES 457 Lu, X.P. "Effects of Road Roughness on Vehicular Rolling Resistance." Paper presented at the ASTM Symposium on Roughness Methodology. Bal Harbor, Florida, 1983. Nalinvaud, E. Statistical Methods of Econometrics. 3d ed. North-Holland Publishing Co.. Amsterdam, Netherlands, 1980. McAdams, H.T. and others. Fuel Economy of Automotive Vehicles: A Compendium of Statistical Techniques. National Technical Information Service, U.S. Department of Commerce. Springfield, Virginia, 1977. McFadden, D. "Qualitative Response Models." In Advances in Econometrics, edited by Werner Hildenbrand, Cambridge University Press. Cambridge, England: 1982. McFadden, D. "Econometric Models for Probabilistic Choice." In Structural Analysis of Discrete Choice Data with Econometric Applications, edited by C.F. Manski and D. McFadden. MIT Press, Cambridge, Massachusetts, 1981. McFadden, D. and F. Reid. "Aggregate Travel Demand Forecasting from Disaggregated Behavioral Models." Paper presented at the 54th Annual Meeting of the Transportation Research Board. Washington, DC, 1975. McLean, J.R. Observed Speed Distribution and Rural Road Traffc Operations.Australian Road Research Board Proceedings, vol. 9 part 5. Vermont-South, Australia, 1978. Hoavenzadeh, F. and others. Highway Design Standards Study Phase I: The Model. Staff Working Paper No. 96, World Bank. Washington, DC. 1971. Morosiuk, G. and S.W. Abaynayaka. Vehicle Operating Costs in the Caribbean: An Experimental Study of Vehicle Performance. TRRL Report 1056. Department of the Environment, Transport and Road Research Laboratory. Crowthorne, England, 1982. Nordeen, D.L. and A.D. Cortese. Force and Moment Characteristics of Rolling Tires." Society of Automotive Engineers. Warrendale, Pennsylvania, 1964. Oglesby, C.H. and R.G. Hicks. Highway Engineering, 4th ed., Wiley, New York, 1982. Organization for Economic Cooperation and Development. Automobile Fuel Consumption in Actual Traffic Conditions. A report prepared by an OECD Research Group, Road Research, 1982. Parsley, L. and R. Robinson. The TRRL Road Investment Model for Developing Countries (RTIM2). TRRL Report 1057. Department of the Environment, Transport and Road Research Laboratory. Crowthorne, England, 1982. 458 REFERENCES Paterson, W.D.O. Road Deterioration and Maintenance Effects: Models for Planning and Management. Transportation Department, World Bank. Washington, DC, 1987. . "Interim Report Reviewing Data Collection and Analysis in the Brazil PICR Project." Unpublished. December 1982. Paterson, W.D.O. and T. Watanatada. "Relationships between vehicle Speed, Ride Quality, and Road Roughness." In T.D. Gillespie and M. Sayers (eds.), Measuring Road Roughness and Its Effect on User Cost and Comfort. ASTM STP 884. American Society for Testing and Materials. Philadelphia, Pennsylvania, 1985. Rezende-Lima, P.R.S. "Analysis of Tire Wear Based on Principles of Pneumatic Tire Mechanics." Unpublished internal memorandum of the Highway Design and Maintenance Standards Study. Transportation Department. World Bank, Washington, DC, January 1984. Robinson, R. and others. A Road Transport Investment Model for Developing Countries. TRRL Report LR 674. Department of the Environment, Transport and Road Research Laboratory. Crowthorne, England, 1975. SAS Institute. SAS User's Guide. Raleigh, North Carolina, 1979. Sayers, M.W., T.D. Gillespie and W.D.O. Paterson. Guidelines for Conducting and Calibrating Road Roughness Measurements. Technical Paper No. 46, World Bank. Washington, DC, 1986. Schallamach, A. "Tire Traction and Wear." In Mechanics of Pneumatic Tires, edited by S.K. Clark. U.S. Department of Transportation, National Highway Traffic Safety Administration. Washington, DC, 1981. Schallamach, A. and D.R. Turner. "The Wear of Slipping Wheels." Wear, vol. 3. 1960. St. John, A.D. and D.R. Kobett. Grade Effects on Traffic Flow Stability and Capacity. National Cooperative Highway Research Program Report 185. Transportation Research Board, National Research Council. Washington, DC, 1978. Sullivan, E.C. The Slip Energy Approach to Vehicle-Road Interactions: Background and Prospects. Institute of Transportation Studies Working Paper No.79-6, University of California. Berkeley, California: January 1979. . Vehicle Operating Cost Model - User's Guide. (2nd ed.) Institute of Transportation Studies Research Report (UCBITS-RR-77-3), University of California. Berkeley, California, 1977. Taborek, J.J. "Mechanics of Vehicles." Machine Design. Towmotor Corporation. Cleveland, Ohio, 1957. REFERENCES 459 Taylor, C.F. The Internal Combustion Engine in Theory and Practice. vol. I, MIT Press, Cambridge, Massachusetts, 1966. Theil, H. Linear Aggregation of Economic Relations. North-Holland Publishing Co., Amsterdam, Netherlands, 1955. Thieme, H. van E. and A. Dijks. "Measurement of Tire Properties." Mechanics of Pneumatic Tires. National Highway Traffic Safety Adminis- tration, U.S. Department of Transportation. Washington, DC, 1981. Veith, A.G. "Tire Wear: Some Knowns and Unknowns." Paper presented at January 25, 1974 meeting of Akron Rubber Group. 1974. Watanatada, T. "Formulation of the Steady-State Speed Model." Unpublished working paper. Transportation and Water Department, World Bank. Washington, DC, April 1982. Highway Design and Maintenance Standards Model (HDM-II): Model Description and User's Manual. Transportation and Water Department, World Bank. Washington, DC, December 1981. Watanatada, T. and P.W. O'Keefe. "Steady-State Speed Model Based on Probalistic Limiting Velocity Concept." Unpublished working paper. Transportation and Water Department, World Bank. Washington, DC, August 1982. Watanatada, T. and others. The Highway Design and Maintenance Standards Model: Volume 4a, HDM-III Model Description and Volume 4b, HDM-III User's Manual. Transportation Department, World Bank. Washington, DC (forthcoming). Williams, J.S. Efficient Analysis of Weibull Survival Data from Experiments on Heterogeneous Patient Populations. Biometrica (34), pp. 209-22, 1978. 1ong, J.Y. Theory of Ground Vehicles. Wiley, New York, 1978. Wu, Shie-Shin and C.L. Heimbach. Simulation of Highway Traffic on Two-lane Two-way Rural Highways. North Carolina Department of Transportation. Raleigh, North Carolina. Zaniewski, J. and others. Vehicle Operating Costs, Fuel Consumption, and Pavement Type and Condition Factors. Technical report prepared for U.S. Department of Transportation, Federal Highway Administration. Washington, DC, 1982. Zanievski, J. and others. Research on the Interrelationships Between Costs of Highway Construction, Maintenance and Utilization Final Report IV: Road User Costs and Traffic Experiments. Texas Research and Development Foundation. Austin, Texas, 1980. 460 REFERENCES List of Abbreviations ARRB - Australian Road Research Board ASTM - American Society for Testing and Measurement CRRI - Central Road Research Institute (India) HDMS - Highway Design and Maintenance Study NCHRP - National Cooperative Highway Research Program GEIPOT - Brazilian Agency for Transportation Planning TRRL - Transport and Road Research Laboratory 6 Å The World Bank The Highway Design and Maintenance Standards Series To provide road design and maintenance standards appropriate to the physical and economic circumstances of developing countries, the World Bank in 1969 instituted the Highway Design and Maintenance Standards Study, which developed into a major collaborative research project with leading research institutions and highway administrations in Australia, Brazil, France, India, Kenya, Sweden, the United Kingdom, and the United States. The aims of the study comprised the rigorous empirical quantification of cost tradeoffs between road construction, maintenance, and vehicle operating costs; and, as a basis for highway decisionmaking, the development of planning models incorporating total life-cycle cost simulation. Controlled experiments and extensive road user surveys were conducted to provide comprehensive data on highway conditions and vehicle operating costs in radically different economic environments on three continents. The five volumes in the series represent the culmination of the 18-year endeavor, along with a computerized highway sector planning and investment model, currently in its third version (HDM-III). The first three volumes in the series provide theoretical foundations and statistical estimation of the underlying physical and economic relationships. The other two discuss the model and its use and are essential references for applying HDM-III. Vehicle Operating Costs: Evidence from Developing Countries Andrew Chesher and Robert Harrison Presents an economic model of firms' management of vehicle fleets, which serves as a framework for the statistical analysis of vehicle operating cost data. Vehicle Speeds and Operating Costs: Models for Road Planning and Management Thawat Watanatada, Ashok M. Dhareshwar, and Paulo Roberto S. Rezende Lima Presents the theory and estimation of a comprehensive set of models to predict speeds and operating costs under free flow conditions for a wide range of vehicles on medium- and low-volume roads as functions of road geometry and condition. Road Deterioration and Maintenance Effects: Models for Planning and Management William D. 0. Paterson Contains an extensive analysis of the physical processes, causes of deterioration, and performance prediction relationships, as well as the effectiveness of maintenance practices on unpaved and paved roads. The Highway Design and Maintenance Standards Model Volume 1. Description of the HDM-III Model Volume 2. User's Manual for the HDM-III Model Thawat Watanatada, Clell G. Harral, William D. 0. Paterson, Ashok M. Dhareshwar, Anil Bhandari, and Koji Tsunokawa Volume 1 organizes relationships described in the first three volumes, as well as a road construction submodel, into interacting sets of costs related to construction, maintenance, and road use. Volume 2 provides guidance on the use of this model- -including input data forms, inference ranges, and default values-and gives numerical examples. ISBN 0-8018-3589-5 ISBN 0-8018-3668-9 (5-volume set)