Transportation and Traffic Theory

TRANSPORTATION AND TRAFFIC THEORY

Related Pergamon books LESORT Transportation and Traffic Theory: Proceedings of the 13th ISTT BELL Transportation Networks: Recent Methodological Advances DAGANZO Fundamentals of Transportation and Traffic Operations ETTEMA & TIMMERMANS Activity Based Approaches to Travel Analysis GARLING, LAITILA & WESTIN Theoretical Foundations of Travel Choice Modeling GRIFFITHS Mathematics in Transport Planning and Control STOPHER & LEE-GOSSELIN Understanding Travel Behaviour in an Era of Change Related Pergamon journals Transportation Research Part A: Policy and Practice Transportation Research Part B: Methodological Free specimen copies of journals available on request Editor: Frank A. Haight Editor: Frank A. Haight

TRANSPORTATION AND TRAFFIC THEORYProceedings of the 14th International Symposium on Transportation and Traffic Theory Jerusalem, Israel, 20-23 July, 1999edited by AVISHAI CEDERTransportation Research Institute Faculty of Civil Engineering Technion - Israel Institute of Technology Haifa, Israel

PERGAMONAn Imprint of Elsevier Science Amsterdam - Lausanne - New York - Oxford - Shannon - Singapore - Tokyo

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CONTENTSPreface and Overview International Advisory Committee In Memoriam Contributors ix xiii xv xix

Chapter 1 - Traffic Flow Models MACROSCOPIC TRAFFIC FLOW MODELS: A QUESTION OF ORDER J.P Lebacque, J.B. Lesort MACROSCOPIC MULTIPLE USER-CLASS TRAFFIC FLOW MODELLING: A MULTILANE GENERALISATION USING GAS-KINETIC THEORY S.P. Hoogendoorn, P.H.L. Bovy THE CHAPMAN-ENSKOG EXPANSION: A NOVEL APPROACH TO HIERARCHICAL EXTENSION OF LIGHTHILL-WHITHAM MODELS P. Nelson, A. Sopasakis THE LAGGED CELL-TRANSMISSION MODEL C.F. Daganzo

1 -104

3

27

51 81

Chapter 2 - Traffic Flow Behaviour OBSERVATIONS AT A FREEWAY BOTTLENECK M.J. Cassidy, R.L. Bertini . TLOWS UPSTREAM OF A HIGHWAY BOTTLENECK G.F. Newell THEORY OF CONGESTED TRAFFIC FLOW: SELF-ORGANIZATION WITHOUT BOTTLENECKS B.S.Kerner A MERGING-GIVEWAY BEHAVIOR MODEL CONSIDERING INTERACTIONS AT EXPRESSWAY ON-RAMPS H. Kita,K. Fukuyama

105-188 107 725

147

173

Transportation and Traffic

Theory

Chapter 3 - Road Safety and Pedestrians COMPARISON OF RESULTS OF METHODS OF THE IDENTIFICATION OF HIGH RISK ROAD SECTIONS M. Tracz, M. Nowakowska BEHAVIOURAL ADAPTATION AND SEAT-BELT USE: A HYPOTHESIS INVOKING LOOMING AS A NEGATIVE REINFORCER A.M. Reinhardt-Rutland BI-DIRECTIONAL EMERGENT FUNDAMENTAL PEDESTRIAN FLOWS FROM CELLULAR AUTOMATA MICROSIMULATION V. J. Blue, J.L. Adler

189 - 254

191

213

235

Chapter 4 - Flow Evaluation on Road Networks FLOW MODEL AND PERFORMABILITY OF A ROAD NETWORK UNDER DEGRADED CONDITIONS Y. Asakura, M. Kashiwadani, E. Hato

255 - 324

257 283

A SENSITIVITY BASED APPROACH TO NETWORK RELIABILITY ASSESSMENT M.G.H. Bell, C. Cassir, Y. lida, W.H.K. Lam A CAPACITY INCREASING PARADOX FOR A DYNAMIC TRAFFIC ASSIGNMENT WITH DEPARTURE TIME CHOICE T. Akamatsu, M. Kuwahara

301

Chapter 5 - Traffic Assignment A DYNAMIC TRAFFIC ASSIGNMENT FORMULATION THAT ENCAPSULATES THE CELL-TRANSMISSION MODEL H.K.Lo FORMULATIONS OF EXTENDED LOGIT STOCHASTIC USER EQUILIBRIUM ASSIGNMENTS S. Bekhor, J.N. Prashker A DOUBLY DYNAMIC TRAFFIC ASSIGNMENT MODEL FOR PLANNING APPLICATIONS V. Astarita, V. Adamo, G.E. Cantarella, E. Cascetta ROUTE FLOW ENTROPY MAXIMIZATION IN ORIGIN-BASED TRAFFIC ASSIGNMENT H. Bar-Gera, D. Boyce

325- 416

327 '-x 351

373

397

Contents

Chapter 6 - Traffic Demand, Forecasting and Decision Tools THE USE OF NEURAL NETWORKS FOR SHORT-TERM PREDICTION OF TRAFFIC DEMAND J. Barcelo,J. Casas ALGORITHMS FOR THE SOLUTION OF THE CONGESTED TRIP MATRIX ESTIMATION PROBLEM M. Maker, X. Zhang

417 - 514

419

445 471

COMBINING PREDICTIVE SCHEMES IN SHORT-TERM TRAFFIC FORECASTING N.-E. ElFaouzi A THEORETICAL BASIS FOR IMPLEMENTATION OF A QUANTITATIVE DECISION SUPPORT SYSTEM - USING BILEVEL OPTIMISATION A. CluneM. Smith, Y. Xiang

489

Chapter 7 - Traffic Simulation MACROSCOPIC MODELLING OF TRAFFIC FLOW BY AN APPROACH OF MOVING SEGMENTS M. Cremer, D. Staecker, P. Unbehaun MICROSCOPIC ONLINE SIMULATIONS OF URBAN TRAFFIC J. Esser, L. Neubert, J. Wahle, M. Schreckenberg

515-574

517 535

MODELLING THE SPILL-BACK OF CONGESTION IN LINK BASED DYNAMIC NETWORK LOADING MODELS: A SIMULATION MODEL WITH APPLICATION V. Adamo, V. Astanta,M. Florian, M. Mahut, J.H. Wu

555

Chapter 8 - Traffic Information and Control INVESTIGATION OF ROUTE GUIDANCE GENERATION ISSUES BY SIMULATION WITH DynaMIT J. Bottom, M. Ben-Akiva, M. Bierlaire, I. Chabini, H. Koutsopoulos, Q. Yang A NEW FEED-BACK PROCESS BY MEANS OF DYNAMIC REFERENCE VALUES IN REROUTING CONTROL A. Poschinger, M. Cremer, H. Keller OPTIMAL CO-ORDINATED AND INTEGRATED MOTORWAY NETWORK TRAFFIC CONTROL A. Kotsialos, M. Papageorgiou, A. Messmer PROGRESSION OPTIMIZATION IN LARGE SCALE URBAN NETWORKS. A HEURISTIC DECOMPOSITION APPROACH C. Stamatiadis, N.H. Gartner

575-662

577

601

627

645


Theory

Chapter 9 - Road Tolling and Parking Balance TOLLING AT A FRONTIER: A GAME THEORETIC ANALYSIS D.M. Levinson CARPOOLING AND PRICING IN A MULTILANE HIGHWAY WITH HIGH-OCCUPANCY-VEHICLE LANES AND BOTTLENECK CONGESTION H.-J. Huang, H. Yang BALANCE OF DEMAND AND SUPPLY OF PARKING SPACES W.H.K. Lam, M.L. Tarn, H. Yang, S.C. Wong

663 - 732 665

685 707

Chapter 10 - Traveller Survey and Transit Planning THE ROLE OF LIFESTYLE AND ATTITUDINAL CHARACTERISTICS IN RESIDENTIAL NEIGHBORHOOD CHOICE M.N. Bagley, P.L. Mokhtarian PLANNING OF SUBWAY TRANSIT SYSTEMS S.C. Wirasinghe, U. Vandebona

733 - 796

735 759 779

SCHEDULING RAIL TRACK MAINTENANCE TO MINIMISE OVERALL DELAYS A. Higgins, L. Ferreira, M. Lake

Index

797

PREFACE AND OVERVIEWThese proceedings represent a further step forward in the understanding and solution of transportation and traffic problems. This is the 14th publication in the series of Symposia on Transportation and Traffic Theory. We are endeavouring to continue in the tradition of Professor Robert Herman and a group of scientists who gathered at the General Motors Research Laboratory in Michigan and initiated this undertaking in 1959. We have all been inspired by their leadership. It is with great sadness that we learned that Professor Robert Herman (Honourary Member) and Professor Michael Cremer (International Advisory Committee) have passed away since our last symposium. We have included a tribute to them in this book. This publication goes a long way towards providing innovative, advanced knowledge regarding traffic and transpiration problems and the analytical tools required to achieve their solutions. Transportation issues about: Safety, Mobility, Efficiency, Productivity, Planning and Environmental elements are of direct interest to a growing number of professionals in the fields of communication, data processing, electronics, environmental quality, policy makers and others. Every advanced transportation or traffic system, existing or under development, absorbs its "know how" from models, methods and analyses represented by papers like the ones in this publication. To name but a few: ATMS (Advanced Traffic Management System), ATIS (Advanced Traveller-Information System), APTS (Advanced Public Transportation System), ACVO (Advanced Commercial Vehicle Operations) and ETC (Electronic Toll Collection), all of which are known as ITS (Intelligent Transportation Systems). ITS can be seen as a link between technology and the driver-vehicle-road system and models. There is a saying: "A man's real worth is determined by what he does, when he has nothing to do." The contributors to this publication are among a group of scientists who even when they have nothing to do, think how to resolve and deal with transportation and traffic problems. They are following George Bernard Shaw's remark that people look at existing things that do not work (on transportation and traffic problems) and ask: why? But dream about things that do not exist, but work and ask: why not? This book has been compiled to follow the same session order as the Symposium and each of the ten chapters has been prefaced by three sayings. An overview of these chapters has been created in the following table. The thirty-five papers in this publication represent contributions from sixteen different countries.

Transportation and Traffic Theory

Input Relationships between traffic flow, density, speed, time, spacing and headway, including analogy to fluid dynamics and gas-kinetic. Reproducible pattern of traffic congestion around bottlenecks and on-ramps including the basic model of fluid dynamics.

Chapter and Authors

1. TRAFFIC FLOW MODELS[Lebacque, Lesort] [Hoogendoorn, Bovy] [Nelson, Sopasakis] [Daganzo]

Advance in Knowledge New formula that allow more accurate and better understanding of the various traffic flow scenarios.

2. TRAFFIC FLOW BEHAVIOUR[Cassidy, Bertini] [Newell] [Kerner] [Kita, Fukuyama]

Methods used for identifying high risk road sections, general belief in seat belts and pedestrian flow models.

3. ROAD SAFETY AND PEDESTRIANS[Tracz, Nowakowska] [Reinhardt-Rutland] [Blue, Adler]

Travel behaviour, origin-destination and capacity models on road networks.

4. FLOW EVALUATION ON ROAD NETWORKS[Asakura, Kashiwadani, Hato] [Bell, Cassir, lida, Lam] [Aramatsu, Kuwahara] 5. TRAFFIC ASSIGNMENT [Lo] [Bekhor, Prashker] [Astarita, Adamo, Cantarella, Cascetta] [Bar-Gera, Boyce]

Better understanding of traffic evolution around bottlenecks, on-ramps and traffic congestion using vehicle count, occupancy, speed and by analytical flow patterns. Improved methods for identifying high risk road sections, explaining changes in driving behaviour while using seat belts and improved pedestrian dynamics for uniand bi-directional cases. Performance and reliability measures, and origin-destination patterns for short- and long-term deteriorated road networks. New formulations for optimal, dynamic and user equilibrium traffic assignments with route choice, origin and stochastic considerations. New and more accurate methods and algorithms for short-term origin-destination and trip predictions using real-time traffic flows and a decision support model for transportation strategies.

Analytical and simulated behavioural rules for dynamic traffic assignment.

Forecasting procedures and algorithms for short-term origin-destination and trip predictions and basic decision strategies about traffic demand and choices.

6. TRAFFIC DEMAND. FORECASTING AND DECISION TOOLS[Barcelo, Casas] [Maher, Zhang] [El Faouzi] [Clune, Smith, Xiang]

Preface and Overview

Input Traffic as fluid dynamics, fuzzy logic for dynamic route guidance and reproduced route flows on road networks.

Analysis framework for route guidance, control theory with optimal signal setting and advanced software for optimal control on roads with ramp metering.

Chapter and Authors 7. TRAFFIC SIMULATION [Cremer, Stacker, Unbehaun] [Esker, Neubert, Wahle, Schreckenberg] [Adamo, Astarita, Florian, Mahut, Wu] 8. TRAFFIC INFORMATION AND CONTROL [Bottom, Ben-Akiva, Bierlaire, Chabini, Koutsopoulos, Yang] [Poschinger, Cremer, Keller] [Kotsialos, Papageorgiou, Messmer] [Stamatiadis, Gartner]

Advance in Knowledge Simulation of traffic flow, density and travel time with examination of dynamic traffic management and new modelling of congestion spill-back. New algorithms and models for route guidance, optimal traffic signal setting on networks, integration of traffic control strategies and traffic control with a feedback component.

Means for road financing, toll collection ideas, road tolling along with high-occupancy-vehicle lanes and attributes of public car parks.

9. ROAD TOLLING AND PARKING BALANCE

Connection between residential location and density and urban travel patterns, methods for subway network plan and procedures for rail track maintenance.

Better understanding the welfare implications of road [Levinson] tolling, optimal strategies for [Huang, Yang] congestion pricing with [Lam, Tarn, Yang, Wong] high-occupancy-vehicle lanes, and optimization model for balancing demand and supply of parking spaces. 10. TRAVELLER SURVEY AND Definition and understanding TRANSIT PLANNING of the variables to influence [Bagley, Mokhtarian] travel patterns, optimal [Wirasinghe, Vandebona] subway plan with minimum [Higgins, Ferreira, Lake] system cost and optimal model for rail track maintenance crew and projects.


Theory

The review process that allowed the selection of .these 35 papers was particularly difficult, following a two-stage international review process. It is for this reason that, for the first time, 22 additional papers will also be published, in a separate bound volume which will be available at the Symposium. Warmest thanks are due to all the reviewers who completed this arduous task within a very limited time framework. Thanks are also due to Mr. J.-B. Lesort, who organised the last Symposium, to the International Advisory Committee (listed separately) and to the Local Programme Committee (Y. Berechman, Y. Gur, Y. Israeli, T. Lotan, D. Mahalel, A. Mandelbaum, M. Pollatschek, Y. Prashker, D. Shefer, Y. Shiftan, I. Salomon). We are indebted to the following main sponsors: the Transportation Research Institute of the Technion-Israel Institute of Technology, the Israel Ministry of Transport, the General Motors Foundation, and the European Commission (DG Transport).

Avishai Ceder April 1999

INTERNATIONAL ADVISORY COMMITTEEProf. E. Hauer Prof. R.E. Allsop Prof. M.G.H. Bell Prof. P.H.L. Bovy Prof. W. Brilon Prof. A. Ceder Prof. C.F. Daganzo Prof. N. Gartner Prof. H. Keller Prof. M. Kuwahara Mr. J.B. Lesort Prof. H. Mahmassani Prof. Y. Makigami Prof. V.V. Silyanov Prof. M.A.P. Taylor Prof. M. Tracz Prof. S.C. Wirasinghe University of Toronto, Canada (Convenor) University College London, UK University of Newcastle upon Tyne, UK Delft University of Technology, The Netherlands Ruhr University, Bochum, Germany Technion - Israel Institute of Technology, Israel University of California, Berkeley, USA University of Massachusetts at Lowell, USA Technical University of Munich, Germany University of Tokyo, Japan INRETS/ENTPE, Lyon, France University of Texas at Austin, USA Ritsumeikan University, Japan Moscow Automobile and Road Construction Institute, Russia University of South Australia, Australia Cracow Technical University, Poland University of Calgary, Canada

HONOURARY MEMBERSProf. R. Hamerslag Prof. M. Koshi Prof. W. Leutzbach Prof. G.F. Newell Prof. H.G. Retzko Dr. D.I. Robertson Prof. T. Sasaki Prof. S. Yagar Delft University of Technology, The Netherlands Tokyo University, Japan University of Karlsruhe, Germany University of California at Berkeley, USA Technical University Darmstadt, Germany University of Nottingham, UK Kyoto University, Japan University of Waterloo, Canada

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IN MEMORIAM

ROBERT HERMAN

"If you live long enough then even some nice things will happen..." was Robert Herman's characteristically modest response to well-wishers congratulating him upon receiving any of the many prestigious awards that he has received over the years in recognition of his tremendous contribution across a spectrum of fields. Born August 29, 1914, in New York City, he was graduated cum laude and with special honors in physics from the City College of New York in 1935, and was awarded the M.A. and Ph.D. degrees in physics in 1940 from Princeton University. After a distinguished career at Johns Hopkins' Applied Physics Laboratory, he joined the General Motors Research Laboratories in 1956, where he was appointed Head of the Theoretical Physics Department in 1959, and later Head of the Traffic Science Department, a position he held until 1979 when he became General Motors Research Fellow. In September 1979, he joined the faculty of The University of Texas at Austin as Professor of Physics, in the Center for Studies in Statistical Mechanics, and L.P. Gilvin Professor in Civil Engineering. He continued as the L.P. Gilvin Centennial Professor Emeritus in Civil Engineering and sometime Professor of Physics until his death, at home in Austin, on February 13, 1997, after a battle with lung cancer. Dr. Herman is widely acknowledged as a pioneer in the rapid development of the field of vehicular traffic science. He has made significant contributions to the areas of single lane and multiple lane traffic flow. With Ilya Prigogine, he has developed a Boltzmann-like kinetic theory of multi-lane traffic flow which provides what is perhaps the best description up to this time of this complex traffic situation. He has developed a two-fluid model of town traffic which coupled with observation provides a description of vehicular traffic in an overall macroscopic sense. In 1959, Dr. Herman organized a General Motors Research Laboratories symposium on the theory of traffic flow, the first international gathering of its kind on this subject, and remained actively involved in the organization of the subsequent symposia held under various auspices around the world. The Tenth International Symposium on Transportation and Traffic Theory was held in his honor at The Massachusetts Institute of Technology in July 1987. His interest in developing a scientific foundation for the study of traffic phenomena came after a distinguished career as a physicist and cosmologist with far-reaching contributions. In collaboration with Ralph A. Alpher and George Gamow, he initiated a theory of the origin and relative abundance of the chemical elements in a relativistic "Big Bang" expanding universe. In subsequent work, he and Alpher examined the properties of the expanding radiation-matter universe according to general relativity theory, and in 1948 made the prediction that the temperature of the residual black-body radiation, a vestige of the initial "explosion" of the "Big Bang" universe, should be about 5K, thus anticipating later work on the primordial cosmic fireball radiation at 2.8K which pervades the universe homogeneously and isotropically. The theory was confirmed in the 1960s by scientists at Bell Laboratories trying to solve a problem of microwave noise. In recognition of this work, Herman and Alpher were awarded the Henry Draper Medal from the National Academy of Sciences in 1993. They were recognized "for their insight and skill in developing a physical model of the evolution of the universe and in predicting the existence of a microwave background radiation years before this radiation was serendipitously discovered". Herman also received the Magellanic Premium of the American Philosophical Society, the oldest scientific award in the United States (1975), and the eighth quadrennial George Vanderlinden Prix of the Belgian Royal Academy (1975), for the radiation prediction. In 1980, he and Alpher were awarded the John Price Wetherill Gold Medal of The Franklin

xvi


Theory

Institute, and in 1981, they received The New York Academy of Sciences Award in Physical and Mathematical Sciences. Dr. Herman was a member of Phi Beta Kappa, Sigma Xi, the Washington Philosophical Society, and the Royal Institution of Great Britain, and a Fellow of the American Physical Society, the Washington Academy of Sciences, and The Franklin Institute. He was elected to the National Academy of Engineering in 1978 for his contributions to the science of vehicular traffic. In 1979, he was elected a fellow in the mathematical and physical sciences of the American Academy of Arts and Sciences. He has been an associate editor of the Reviews of Modern Physics; was one of the founders of the Transportation Science Section of the Operations Research Society of America (ORS A, now INFORMS), and also became its first chairman as well as the founding editor of its journal, Transportation Science. He has served as the President of ORS A (1980-1981). In 1959, Dr. Herman was co-recipient of the Lanchester Prize in Operations Research for pioneering research on the stability and flow of single-lane traffic. In 1963, he was awarded an honorary medal by the Universite Libre de Bruxelles, and during that same year received the Townsend Harris Medal from the Alumni Association of the City College of New York as a distinguished alumnus and for his scientific contributions. Other prestigious INFORMS awards include the George E. Kimball Medal (1976) and the John Von Neumann Theory Prize (1993). Also in 1993, he received the Roy W. Crum Distinguished Service Award of the Transportation Research Board for "his pioneering contributions to the field of traffic science". In 1984, he was awarded an Honorary Doctorate in Engineering by the University of Karlsruhe in recognition of his outstanding research in the mathematical foundations and development of the theory of traffic flow. He received the first Lifetime Achievement Award of the Operation Research Society's (now INFORMS') Transportation Science Section for his body of work on vehicular traffic science in 1990. The award was subsequently renamed the Robert Herman Lifetime Achievement Award. In October and November 1990, The National Academy of Engineering presented the first public showing of some of Dr. Herman's small abstract sculptures in exotic wood, which he had produced over a period of about 30 years. Notwithstanding all these accomplishments, two activities occupied a very special place in Bob's heart: the ISTTT series and Transportation Science (the journal). In addition to symbolizing the coming of age of traffic science as a legitimate field of human scientific inquiry, and providing it with its scientific pillars, they also defined Bob's special family and community. With us, he shared the love for the work, the ideas, the excitement and the people engaged in pushing the intellectual frontiers of this field, through theoretical development and painstaking measurement. The collective effects of ISTTT gatherings always ensured a far greater impact than the sum of its individual parts. Bob's unbounded human contributions as a colleague, friend, mentor and educator to virtually every generation that has entered the Transportation Science domain will endure. As it showcases recent developments in traffic science and transportation analysis methods, the 14th ISTTT stands as a testimonial to the house that Robert Herman built, and a celebration of the life and intellectual energy that he so freely shared with all of us. Hani S. Mahmassani Austin, Texas, February 1999

In Memoriam

xvii

IN MEMORIAM

MICHAEL CREMER

Dr. Michael Cremer, university professor at the Technical University Hamburg-Harburg, died on September 3, 1998, after a long illness which he bore patiently and with hope to recover. Prof. Cremer studied Electrical Engineering and Control Engineering at the Berlin University of Technology, got his doctorate from the Ruhr University Bochum, and was already involved in both Control and Traffic Engineering during his post doctoral work at the Munich University of Technology. He accepted the position of a university professor at the Hamburg University in 1979, and at the Hamburg-Harburg University of Technology in 1992, where he taught in the departments Industrial Engineering and Control Engineering. It was there where he built up the Automation Engineering Unit. Apart from his research activities in the area of control engineering and its applications to the distribution of pollutants in waterways, his major research interests were control and traffic engineering. There is a series of original contributions of Michael Cremer in this area, such as on the dynamic fundamental diagram, on the application of system dynamics to the estimation of the matrix of traffic flows from traffic counts, or the development of control rules for the optimisation of the traffic flows in motorway networks. Cremer produced a large number of simulation tools which have been used and developed further by his staff and his colleagues. The macroscopic and microscopic traffic flow models which are offsprings of Cremer's institution have been named SIMONE, MAKSIMOS, and MIKROSIM. The deployment of the Kalman filter with its wide variety of applications and expansions to allow an analysis of traffic flow or incident detection is another example of his innovative developments in traffic engineering. The consideration of floating car generated data in macroscopic traffic flow models was the subject of one of his first and of his last scientific achievements. Michael Cremer played a significant role in the European research and development programmes, from the design of system architectures within PROMETHEUS and DRIVE to optimum traffic control strategies for congestion reduction in over-saturated networks within the projects HERMES, COSMOS and OFFENSIVE, to name a few. Michael Cremer was an active representative and monitor of science in his capacity as a member of German and international professional and research associations. The most noteworthy of these being the committee on "Traffic Flow Theory" of the German Forschungsgesellschaft fur StraBen- und Verkehrswesen and the Advisory Committee to the "International Symposion on Traffic and Transportation Theory". Professor Cremer earned significant national and international reputation with his innovative contributions in the field of traffic control and engineering. The research community in general and his colleagues in particular will have to miss his originality and profound competence in research, his witty and sophisticated way and his positive view of life. Hartmut Keller 1998


CONTRIBUTORSAdler, J.L. Adamo, V. Akamatsu, T. Asakura, Y. Astarita, V. Bagley, M.N. Barcelo, J. Bar-Gera, H. Bekhor, S. Bell, M.G.H. Ben-Akiva, M. Bertini, R.L. Rensselaer Polytechnic Institute, Troy, NY, USA Universita della Calabria, Italy Dept. of Knowledge-based Information Engineering, Toyohashi University of Technology, Toyohashi, Japan Dept. of Civil & Environmental Engineering, Ehime University, Matsuyama, Japan Universita della Calabria, Italy University of California, Davis, USA TSS-Transport Simulation Systems, Barcelona, Spain Univ. of Illinois at Chicago, USA Dept. of Civil Engineering, Technion - Israel Institute of Technology, Haifa, Israel Transport Operations Research Group, University of Newcastle, UK Massachusetts Institute of Technology, USA Dept. of Civil & Environmental Engineering & Institute of Transportation Studies, University of California at Berkeley, USA Swiss Federal Institute of Technology, Switzerland New York State Dept. of Transportation, Poughkeepsie, NY, USA Massachusetts Institute of Technology, USA Delft University of Technology, Traffic Engineering Section, The Netherlands University of Illinois at Chicago, USA Universita di Napoli, Naples, Italy Universitat de Vic, Spain Universita di Napoli, Italy Dept. of Civil & Environmental Engineering & Institute of Transportation Studies, University of California at Berkeley, USA Transport Operations Research Group, University of Newcastle, UK Massachusetts Institute of Technology, USA York Network Control Group, Dept. of Mathematics, University of York, UK Technical University of Hamburg-Harburg, Germany Inst. of Transportation Studies, University of California, Berkeley, USA Laboratoire d'Ingenierie Circulation - Transport, Unite Mixte de Recherche INRETS - ENTPE, Bron, France Los Alamos National Laboratory, NM, USA School of Civil Engineering, Queensland University of Technology, Australia Centre for Research on Transportation, University of Montreal, Quebec, Canada

Bierlaire, M. Blue, V.J. Bottom, J. Bovy, P.H.L. Boyce, D. Cantarella, G.E. Casas, J. Cascetta, E. Cassidy, M.J.

Cassir, C. Chabini, I. Clune, A. Cremer, M. Daganzo, C.F. El Faouzi, N.E. Esser, J. Ferreira, L. Florian, M.

XX


Fukuyama, K. Gartner, N.H. Hato, E. Higgins, A. Hoogendoorn, S.P. Huang, H.J. lida, Y. Kashiwadani, M. Keller, H. Kerner, B.S. Kita, H. Kotsialos, A. Koutsopoulos, H. Kuwahara, M. Lake, M. Lam, W.H.K. Lebacque, J.P. Lesort, J.B. Levinson, D.M. Lo, H.K. Maher, M. Mahut, M. Messmer, A. Mokhtarian, P.L. Nelson, P. Neubert, L. Newell, G.F.

Nowakowska, M. Papageorgiou, M. Poschinger, A. Prashker, N.J.

Dept. of Social Systems Engineering, Tottori University, Japan University of Massachusetts at Lowell, MA, USA Dept. of Civil & Environmental Engineering, Ehime University, Matsuyama, Japan CSIRO, Brisbane, Australia Delft University of Technology, Traffic Engineering Section, The Netherlands Beijing University of Aeronautics & Astronautics, P.R. China Transport Operations Research Group, University of Newcastle, UK Dept. of Civil & Environmental Engineering, Ehime University, Matsuyama, Japan Fachgebiet Verkehrstechnik und Verkehrsplanung, Technische Universitat Munchen, Germany Daimler Chrysler AG, Stuttgart, Germany Dept. of Social Systems Engineering, Tottori University, Japan Dynamic Systems & Simulation Laboratory, Technical University of Crete, Chania, Greece Volpe Transportation Systems Center Institute of Industrial Science, University of Tokyo, Japan School of Civil Engineering, Queensland University of Technology, Australia Dept. of Civil & Structural Engineering, Hong Kong University of Science & Technology, P.R. China ENPC-CERMICS, Marnes La Vallee, France INRETS/ENTPE, Lyon, France Institute of Transportation Studies, University of California at Berkeley, USA Dept. of Civil Engineering, Hong Kong University of Science & Technology, Clear Water Bay, .R. China School of Built Environment, Napier University, UK Centre for Research on Transportation, University of Montreal, Quebec, Canada Ingenieurburo A. Messmer, Munich, Germany University of California, Davis, USA Dept. of Mathematics, Texas A&M University, College Station, USA Physik von Transport und Verkehr, Gerhard-Mercator-Universitat, Duisburg, Germany Dept. of Civil & Environmental Engineering & Institute of Transportation Studies, University of California at Berkeley, USA Laboratory of Computer Science, Kielce University of Technology, Kielce, Poland Dynamic Systems & Simulation Laboratory, Technical University of Crete, Chania, Greece Fachgebiet Verkehrstechnik und Verkehrsplanung, Technische Universitat Munchen, Germany Dept. of Civil Engineering, Technion - Israel Institute of Technology, Haifa, Israel

Contributors

XXI

Reinhardt -Rutland, A.H. Schreckenberg, M. Smith, M. Sopasakis, A. Staecker, D. Stamatiadis, C. Tarn, M.L. Tracz. M. Unbehaun, P. Vandebona, U. Wahle, J. Wirasinghe, S.C. Wong, S.C. Wu, J.H. Xiang, Y. Yang, H. Yang, Q. Zhang, X.

Psychology Dept, University of Ulster at Jordanstown, UK Physik von Transport und Verkehr, Gerhard-Mercator-Universitat, Germany York Network Control Group, Dept. of Mathematics, University of York, Heslington, UK Dept. of Mathematics, Texas A&M University, College Station, USA Technical University of Hamburg-Harburg, Germany University of Massachusetts, USA Dept. of Civil & Structural Engineering, The Hong Kong University of Science & Technology, P.R. China Cracow University of Technology, Poland Technical University of Hamburg-Harburg, Germany University of New South Wales, Australia Physik von Transport und Verkehr, Gerhard-Mercator-Universitat, Germany Dept. of Civil Engineering, The University of Calgary, Canada Dept. of Civil Engineering, The University of Hong Kong, P.R. China Centre for Research on Transportation, University of Montreal, Quebec, Canada York Network Control Group, Dept. of Mathematics, University of York, Heslington, UK Dept. of Civil & Structural Engineering, The Hong Kong University of Science & Technology, P.R. China Caliper Corporation, Boston, USA School of Built Environment, Napier University, UK


CHAPTER 1TRAFFIC FLOW MODELS

Imagination is more important than knowledge. (Albert Einstein) We think in generalities, we live in details. (Alfred North Whitehead) Science proceeds more by what it has learned to ignore than what it takes into account. (Galileo)


MACROSCOPIC TRAFFIC FLOW MODELS: A QUESTION OF ORDERJ.PLebacque, ENPC-CERMICS, Marnes La Vallee, France J.BLesort, LICIT INRETS/ENTPE, Lyon, France

Abstract The aim of this paper is to propose a methodology for the comparison of first and second order macroscopic traffic flow models. First, we identify a certain number of difficulties and investigate how various macroscopic models cope with them. Second, we propose a set of problems or situations which could be used as a test workbench for the comparison of models.

i INTRODUCTIONSince the first papers by Lighthill and Whitham [1955] and Richards [1956] and the introduction of higher order models by Payne [1971], a great number of macroscopic traffic flow models have been developed. They are all based on a few similar variables and assumptions the state variables are the flow q(x, t), the density k(x, t) and the mean flow speed u(x, t), defined as:

(1)

= kf

The variables q, k and u are considered as piece-wise differentiable functions of space x and time t, still allowing the existence of some singularities (discontinuities in space-time corresponding to shock waves, discontinuities in time corresponding to incidents and discontinuities in space corresponding to variations in the road layout). The conservation equation constitutes the second basic equation. It can be written as:

(2)

lr + = 0 ox ot

To realise a complete model, a third equation is necessary. It can be derived in different ways. The first one, which results in the basic Lighthill-Whitham-Richards (LWR) model, is to consider an empirical relationship between speed and concentration: (3) u(x,t) ue(k(x,t)}

4


These are first order models. In various extensions, the equilibrium relationship may depend on space (variations in the road layout) and time (occurrence of incidents):

(4)

u(x,

The important point is that this dependence is exogeneous to the model and not a part of it ue is a function of the three variables k, x, t. A second way to obtain a third equation is to consider relation (3) only as an equilibrium relationship, and to describe explicitly transitory states, using for instance a relaxation process expressing the tendency of traffic to tend to an equilibrium. Models of that kind may be derived from microscopic considerations, as made by Payne [1971], resulting in a speed equation:

du

du

1/

v dk

with v an anticipation coefficient and r a reaction time. These are higher order models. More recently, many higher order models have been developed from the kinetic models generalizing the model proposed by Prigogine and Hermann [1971] . Examples are [Phillips, 1979], [Helbing, 1997], [Kerner et al, 1996]. To develop such models, some simplifying assumptions have to be made, such as considering a gaussian repartition of individual speeds [Helbing, 1997]. The resulting equation are somewhat similar to Payne's. For a long time, numerous experiments have been conducted to compare various classes of models. (e.g [Michalopoulos et al., 1992] and [Papageorgiou et al, 1983, 1989]). Most of these experiments have concluded to the superiority of higher order models. However it must be noticed that most of these experiments had as their first aim to validate a higher order model. For these experiments, the LWR model discretization and implementation had not been optimized. On the other hand, the values obtained in the calibration process for parameters as the reaction time are often far from physical, as pointed out by Del Castillo et al. [1993]. More recently, comparisons and analysis have been conducted on a more theoretical basis Schochet [1988] has shown that Payne's model converges towards the simple LWR model when the reaction time tends to zero. Daganzo [1995b] claimed that higher order models included fundamental flaws, such as negative speeds, which made them unsuitable for practical use. In reaction, models have been proposed [Liu et al, 1998], [Zhang, 1998] trying to avoid such flaws. In-depth experimental analysis have been conducted [Kerner and Rehborn, 1997] to understand how theoretical models are able to explain observations. In turn, these experimental results are contested [Daganzo et al., 1997], various explanations being given of the same traffic phenomenons. To date, no definite conclusion is possible on the respective advantages and drawbacks of each kind of model. The purpose of this paper is to propose some clarifications to this debate. In order to achieve this aim, we first identify some specific difficulties, and assess how the various macroscopic models deal with them, thus achieving some measure of comparison. Further, we propose a set of archetypical problems of modelling situations, for which we give some elements of comparison between models, and on which various models could be tested for performance or simply modelling capability. Such a comparison procedure is of course quite theoretical, but the results expected from the models ultimately rely on experience. Actually, some of the proposed archetypes, such as traffic dispersion or congestion formation, are inspired directly by experimental data, whereas others, such as intersection or lane drop modelling, result more from theoretical or logical considerations.

2 THE MAIN MODELING ISSUESIn the literature, a number of theoretical difficulties have been identified for each type of model. They may be grouped into a few classes, which we shall review now.

Macroscopic Traffic Flow Models 2.1 Basic variables

It has been noticed for long that most models have difficulties in keeping some of the basic variables describing the traffic flow dynamics within physically sound limits. This is particularly true for speed and acceleration. 2.1.1 Flow speed

The nature of the flow speed varies depending on the class of the model. For 1st order models, the flow speed is only a derived variable, and not an intrinsic one (the model is based only on flow and concentration). It is one of the reasons why, as noticed by [Buisson et al, 1996], the calculation of speed in discretized versions is not straightforward. In higher order models, the flow speed is one of the basic variables, as one of the equations of the model, say (5), is a speed equation. On the other hand, it has been noticed by [Daganzo, 1995b] and others that this speed equation (5) may lead, under some initial conditions, to unphysical values of speed (negative speeds). In reaction, some authors have recently presented models designed to avoid these negative speeds. However, it seems that these models often present other inconsistencies. For instance, the speed equation of the model proposed by [Liu et al, 1998] is given by:(6)

with T(k] a reaction time depending on k. The other equations of the models are the classical equations (1) and (2). It is then clear that: If an initial condition is u = 0 (locally), then | is positive, which avoids negative speeds, | On the other hand, consider the following initial conditions, which are those of a stopped queue, with k = kjam, u = ue(kjam) = 0 for x < 0 and k = 0, u = Vf = ue(0) for x > 0, with Vf the desired speed (i.e. q = 0 uniformly). Then ^ = 0 (because ue(k) = u for all x), H = 0 (because initially q = 0 uniformly) and the queue will never start again the solution is stationary. It is to date not obvious whether higher order models presenting no risk of negative speed can be developed. In first order models, speeds are always physical for they are bounded by 0 and a maximum equilibrium free speed. 2.1.2 Acceleration

Acceleration is not an intrinsic variable of any macroscopic model. However, it is interesting to examine whether the acceleration of a vehicle moving with the flow is kept within reasonable bounds. Indeed, models aiming for instance at the description of sound or pollutant emission by traffic flow require reasonable acceleration estimates. If w(t) is the speed of such a vehicle, the acceleration can be written as:

dw _ du dt dt

du dx

In the case of first order models, the possibility of the occurrence of unbounded acceleration values will be illustrated by two examples. First, let us recall that, if the equilibrium speeddensity relationship is dependent on the position x, it follows from the conservation equation and the speed-density relationship that:

dwdt

, (duf\

dk

\ ok I dx

\ *

(

, due\ dupdk i ox I *

6


Theory

If we consider a homogeneous roadway, the equilibrium speed is only a function of density u(x, t) = ue(k(x, t)). Under these conditions the acceleration may be written:(9

)

dw ~JT at

=

(due\2 dk ~k HjT J ^~ \dk ox

Since the value of | cannot be bounded (through a Shockwave, or when a queue discharges), it | is clear that the acceleration may take any value. A second instance which may be mentionned is the effect of a variation in the road layout, i.e ue is a function of x u(x, t) = ue(k(x, t ) , x ) . It is then possible to examine the acceleration of a vehicle under steady-state flow conditions with ff = 0 and |^ = 0, since the analytical solution is straightforward. Under these conditions the acceleration of a vehicle can be written:

dw

u2 due

This formula results from (8) by noting that the steady state conditions | = 0 = f^ imply |

dqe dk dk dxIt can be seen that:

dqe dx

For a given road lay-out profile ( given ^jf), the acceleration sign depends on traffic conditions (fluid or congested) since the derivative ^ is bounded by the free speed, if ^ is very high, there is no physical bound to the acceleration. What this presumably means in reality is that the actual use of the road by traffic (actual capacity)is adapted to the acceleration/deceleration capabilities of vehicles. 2.2 Supply representation The representation of various road layout configurations is one of the problems to be dealt with by traffic flow models. This problem will be investigated more into details in the next section, but two questions are related to some theoretical issues: the notion of capacity, and the occurrence of discontinuities in the road layout. 2.2.1 Capacities

Even if the capacity has a straightforward definition (maximum possible flow), its representation is not that obvious. The main point of controversy is the concept of dynamic capacity for instance, does the maximum flow depend on the presence of a congestion [Papageorgiou, 1998] or not. In first order models, the solutions used for the model are entropy solutions which always maximise the flow. The definition of capacity is thus obvious, since capacity is one of the parameters of the equilibrium relationship. On the other hand, if non-entropy solutions are used, as proposed for instance in [Lebacque, 1997] for bounded acceleration models, the notion of capacity corresponds to that of maximum flow only under steady state traffic conditions. (Actually the flow can never exceed the capacity value, but it is lower under transitory conditions). In higher order models, there is a notion of maximum flow in steady state conditions, when the flow speed is equal to the equilibrium one. On the other hand, outside of these conditions, the

Macroscopic Traffic Flow Models

1

flow may be lower but also higher than this value. Indeed, the speed u and the density k being independent variables, there is no possibility to control the value of their product q = ku. This is particularly apparent in Ross's model [Ross, 1988], ^ = (vj - u) (with v/ the desired speed), in which the capacity constraint ku < qmax must be added to the model in order to recapture such basic features as congestion propagation. For Payne's or similar models, Schochet's results [Schochet, 1988] imply that such excessive values occur only in limited ranges of space and time. This last result is actually only valid if the viscosity z/ and relaxation time T parameters take physical values, i.e. are very small, as proposed in [Del Castillo et a/., 1993]. Otherwise, convergence towards traffic states satisfying to capacity constraints is slow, or takes place over a large space scale only. 2.2.2 Discontinuities

By discontinuities we mean mainly either space or time discontinuities of the equilibrium relationships ue and qe. Spacewise discontinuities would be related to abrupt increases or decreases in capacity. Timewise discontinuities represent accidents or incidents, which imply a more or less extended capacity restriction for some finite duration. Finally, moving capacity restrictions might also be considered, such as buses (in urban areas) [Lebacque, Lesort, Giorgi 1998] or slow moving truck convoys on highways [Newell, 1993 and 1997]. Discontinuities constitute a normal feature of macroscopic models, a necessary consequence of the continuum hypothesis and of the corresponding approximation. Actually, the size of any feature smaller than say 50 to 100 meters should be neglected, a remark that applies also to the unbounded acceleration estimate problem mentionned in subsection 2.1.2. We shall discuss spacewise discontinuities more in detail in subsection 3.1. A few remarks are of order here. First order models accommodate such discontinuities well, since boundary conditions are well defined for these models, see subsection 3.7 the flow at such a discontinuity is continuous (spacewise) and equal to the minimum between upstream demand and downstream supply (entropy solution). Since speed is a function of density, a discontinuity of the speed is generated, implying infinite acceleration and an unrealistic solution, which is flow maximizing, a point often criticized in the literature [Papageorgiou, 1998]. It might be argued that speed discontinuities constitute a normal feature of macroscopic models too, i.e. that the velocity changes abruptly over a range of the same order as the vehicle spacing. Such a velocity gradient is nevertheless not compatible with the finite acceleration of traffic (and the acceleration capabilities of vehicles). Second order models are much better behaved in theory, owing to the damping effect of the diffusion term (term in | in the speed equation (5)) only the acceleration would be discontinuous. The speed should gradually relax towards the equilibrium speed. Nevertheless, it may well be that this favorable situation actually results from the fact that variations of the equilibrium relationships have not been considered in the derivation of higher order models. Indeed, if such variations are considered, as in subsection 3.1.2, supplementary terms in ^ should be introduced into the speed equation, implying a speed-discontinuity at the discontinuity of ue. Timewise discontinuities will not be discussed in detail here, we refer the reader to [Buisson et al., 1996], [Mongeot, 1997], [Heydecker, 1994]. The difficulties involved are the same, with a difference which is that the density, and not the flow, is,conserved at the discontinuity. Moving discontinuities can be treated by considering the moving frame associated to the discontinuity, a programme which has been carried out in the case of first order models (see the references cited above), but not, to the authors knowledge, in the case of second order models. It is actually questionnable whether moving capacity restrictions of small size, such as buses or convoys, can be modelled at all with second order models, considering that the model equations may not be able to enforce a small sized capacity restriction through the mechanism of relaxation towards the equilibrium state.

Transportation and Traffic 2.3 Models solutions

Theory

There are two possible ways of computing solutions to traffic models either to calculate analytical solutions, which is not always possible, or to derive a discretized model from the continuous one and to compute simulation solutions. 2.3.1 Analytical solutions

One of the main advantages of first order models is probably that they make it possible to compute analytical solutions for a wide variety of simple but nontrivial cases. This is due to the simplicity of the model and the existence of characteristic lines (straight in the homogeneous case) carrying constant flows and densities. It is thus possible to calculate the analytical solutions of the Riemann problem for all possible initial and boundary conditions. The necessary initial and boundary conditions are also clearly defined: The knowledge of initial densities k(x, t 0 ) is a sufficient initial condition, and the knowledge of the traffic demands at the entrances of the network and traffic supplies at the exits are sufficient boundary conditions. Let us recall that following [Lebacque, 1996b], in the LWR model, the traffic demand at any point x is the greatest outflow at that point, and the traffic supply is the greatest inflow at that point. These quantities result from the local left repectively right hand side density values at x through the equilibrium demand and supply functions: A e (K,z) = qe(K,x-) if K < kcrit(x-}

(11)

=

qmax(x-)

if K > kcrit(x-}

S e (,z) = qmax(x+) = qe(K,,x+)

if < kcrit(x+) if K > kcrit(x+)

Concerning higher order models, the possibilities to compute analytical solutions are much more restricted. For some specific models such as the model proposed by Ross [Ross, 1988], analytical solutions can be derived in a variety of cases, as shown by [Lebacque, 1995]. This is why Ross's model, although it has been much criticized (see [Newell, 1989] and Ross's response, [Ross, 1989]), constitutes a simplified archetype of second order models (much as Greenshields model or the simplified piecewise linear equilibrium flow-density relationship model in [Daganzo, 1994] do for first order models). An interesting example of analytical calculations for second order models was given by Kuhne in his model [Kuhne, 1984]: du , c dk - =1-MAO -u)-j^0 2

This model results from Phillips' model, in which the anticipation term of Payne's model is replaced by a pressure term _\&P_ k dx (as resulting from a kinetical model) by assuming that the traffic pressure V is a linear function of the density. This model, with addition of a viscosity term ^f^f , and a special choice of the viscosity coefficient v = ^ and of the equilibrium relationship ue(k] = u [(!/ (1 + exp((k/kmax - 0.25)/0.06))) - 3.72,1(T6] became the study object of Kerner and Konhaiiser [Kerner et a/., 1996].

Macroscopic Traffic Flow Models 2.3.2 Simulation

Discretizations have been proposed for most models presented in the literature. Concerning first order models, various schemes have been proposed, including [Lebacque 1984] (in which the Godunov flux was introduced in a heuristic way], [Michalopoulos et al., 1984a] (based on a shock-fitting Lax-Wendroff scheme), [Michalopoulos et al., 1984b], [Bui et al., 1992] (using Osher's formula for the Godunov flux function), [Leo and Pretty, 1992] (applying Roe's approximate formula for the Godunov flux function to several models including the LWR model). More recently, dicretizations based on the Godunov scheme and presenting a better consistency with the continuous model have been proposed by [Lebacque, 1996b] and [Daganzo, 1994]. However, few systematical analyses of the discretization question have been made. An older example is provided by [Leo and Pretty, 1992] and a more recent example is [Zhang and Wu, 1997]. On the other hand, some models exist only under a discretized form. It is the case of the model proposed by [Hilliges, 1995], or of some models introducing bounded acceleration (the phenomenological model in [Lebacque, 1997]). The main drawback of this kind of model is that the behaviour of the model is dependent on the discretization parameters (time 8t and space 5x discretization steps), and that the convergence of the model is not guaranteed when 8x and 6t tends towards zero. Considering the equivalent equation is usually not helpfull. Furthermore, the convergences towards zero of 8x and 8t should not be considered independently, as the following cursory analysis of numerical viscosity effects (and also Schochet's convergence results) show. When the discretization scheme is consistent with the continuous model, the only effect of the discretization is to introduce a numerical viscosity into the model behaviour. The effect of this viscosity is visible on a simple example. Let us consider a simple first order model, under fluid traffic conditions. Let ql(t) and q(i) be the flows entering and leaving a discretization segment during time-step

[M + A], k(t) be the mean density at time t. Under fluid conditions, and at first order approximation (for example a perturbation of a stationary state) qr(t) can be considered as given, and q(t + 6t) written as:

= qe(k(f}} + -T-~ (qx(t) - q(t)) by limited development

This is actually a smoothing formula (a fact not incompatible with the existence of shock-waves) which bears some resemblance with the experimental TRANSYT smoothing formula [Robertson, 1969]. Introducing the maximum speed umax, the viscosity factor J|^f can be considered as the product of two terms: A physical and intrinsic term ^ ^ which expresses that the viscosity is a function of density A purely numerical term umaxj^ (The Courant-Friedrichs-Lewy number)

10


A symmetric formula relating qj(t + 6t) to ? 7 (t) and q(t) applies in the congested case. The Courant-Friedrichs-Lewy (CFL) condition 6x > umax5t imposes to this numerical term to be lesser than 1 in order to guarantee the stability of the model. On the other hand, the lesser this term, the higher the numerical viscosity (therefore it should be as close to one as feasable in order to optimize the discretization). An extreme example is given by some flow models used for dynamic assignment, which can be considered as first order models continuous in time, with an imposed space discretization (the link). The basic example is the model proposed by [Merchant and Nemhauser, 1978], from which a time-continuous version has been given by [Friesz et al, 1989]. Indeed, a one segment Godunov discretization of a link yields (assumingthat the downstream supply is sufficient): x(t + St) = x(t) + 6t [q!(t) with q1 the given inflow, x(t) the total number of vehicles at time t, A e the demand function and x ( t ) / l the mean density, I the length of the link. Taking 6t > 0, the following model results:d

ft(t}=qI(t}-g(x(t}} with

As explained by [Astarita, 1996] these models present a high degree of numerical viscosity, resulting in unphysical behaviour (null or infinite travel-times ...). Numerical viscosity can be considered as a positive effect, as it looks like a platoon dispersion effect (this will be developed further on). However, it must be kept in mind that part of this is a purely numerical effect, with little possibility to fit it to physical observations. Second order models in simulation have been analyzed carefully, notably by [Papageorgiou et al, 1989], [Michalopoulos et al., 1992] for instance. For lack of sufficient supportive mathematical results (the theory is not as well developed as that of conservation equations, few analytical solutions or convergence results are known), analyses concentrate on stability, parameter calibration and ability to reproduce satisfactorily observations. The idea of these simulations is to use cells, cell-averages, and finite difference schemes, an approach similar to the one used with first order models. The cell dimension is liable to be much greater than in discretized first order models. Some allowance must be made for special features adding a constant K to the density in the l/k terms for instance, in order to avoid division-by-zero problems. The equation q = ku must be discretized with care too; [Papageorgiou et al., 1989] proposed a linear interpolation formula for the cell outflow of cell (i) during time-step k, qf, relating it to the mean flows of cells (i) and (i + 1): (13) g* = afc*ut* + (1 - a)fc?+1u?+1 Flux vector splitting upwind schemes were proposed too [Michalopoulos et al, 1992], and may yield more rigorous discretizations. At this point, parameter calibration has still an enormous importance for second order models, which means that discretized second models should be considered as phenomenological in that sense, and as having an existence in their own right, irrespective of the continuous model from which they are derived. This is possibly a good thing corrective features can be introduced into discretized second order models, correcting some flaws of the corresponding continuous models, such as pointed out in [Daganzo, 1995c] (negative speeds for instance), possibly introducing capacity bounds too. 2.4 Models vs. reality The physical soundness of a model can only be checked by comparison to real world observations. This means that The model must be able to explain observed physical phenomenons,

Macroscopic Traffic Flow Models It can be identified using real data,

11

The values of the model variables and parameters must be kept within physically sound bounds. 2.4.1 Interpretation of measurements

The basic variables of macroscopic models, flow, speed and density, are readily accessible experimentally. It is thus interesting to analyse whether various kinds of models may explain the values of this parameters as observed on actual traffic. This has been made through many investigations, starting with (Greenshields, 1935], but surprisingly few definite conclusions can be drawn from this literature. For steady-state conditions, an important analysis has been made by [Cassidy, 1998], who, using an original filtering technique, has shown the validity of the speed/flow equilibrium relationships. This is an important result, but it does not concede any kind of superiority to any model, for all models behave quite similarly under steady-state conditions. Concerning dynamic conditions, many phenomenons can occur, resulting in global perturbations which are not easy to explain: The demand/supply mechanism may result, at a measurement point, and for a given value of flow, in various density values: oscillation between both fluid and congested states, mixed into a single observation period, can yield different values of the density, deterministic scatter and hysteresis, even if the flow is constant; The composition of traffic, particularly with respect to the various destinations, which is never measured, may result in the occurrence of unexplained perturbations, particularly at diverge points. This has been observed for instance by [Daganzo et al., 1998]. It is only recently that systematic analysis of observations in relation to traffic models have been conducted. Classical experiments have mainly focused on the calibration/global validation of the model on a statistical basis. One of the first attempts at an in-depth analysis has been made by [Kerner and Rebhorn, 1997]. Considering their paper and the explanations given of the same and similar data by [Daganzo et al., 1997], it is interesting to notice how multiple explanations can be given of the same physical phenomenons, each following its own modelling rationale. Other phenomenons are much more difficult to explain at, such as the constant size perturbations observed by Kerner and Rebhorn, which cannot be explained through classical models. It can be noticed that it would be possible to derive some non-entropy solutions of the LWR model explaining these perturbations. However, these solutions would lead to instability problems. 2.4.2 Parameters values

This topic will be considered again in the sequel. At this point, it can be noted that first order models require few parameters, which all have a physical meaning and are easily identified by network operators (maximum density, speed, flow, critical density and flow). On the other hand, second order models require the same above parameters, which are those of the equilibrium relationships, plus many others whose physical meaning is not always obvious, with resulting values contradicting both theory and common sense (reaction time T of the order of 30 sec. for instance) [Cremer and Papageorgiou 1981], [Papageorgiou et al., 1989], but also [Michalopoulos et al, 1992] parameter to. These parameters require extensive identification. A good example is given by [Papageorgiou et al. 1989], in which the speed equation is given by:

du du 1 / v dk -^7 + C -5- = - ue(k) - u- at ox r \ k + KOX

12


Theory

and the discretized version of the corresponding model requires the parameter values of the equilibrium relationship ue, of the physical parameters T, 5, v, and of the nonphysical parameters K, C and a (this last one, already mentionned, is specific to the discretization equation (13)). Let us recall that in the above, g would be the ramp-inflow for instance. The issue of the reaction time T is also illuminating, since the estimated values vary from 1 to 50 seconds or more depending on the authors and very different functional forms (for r as a function of density k) have been proposed (see [Del Castillo et ai, 1993] and [Michalopoulos et al, 1992]).

3 ANALYSIS OF SOME TYPICAL CASESThis section is devoted to the definition of problems on which to test macroscopic models. Since traffic flow is essentially nonlinear, there exists no set of problems allowing an exhaustive comparison. Therefore, the problems described hereafter represent simply a set of basic problems, aimed at the representation of the difficulties described in the previous section and some other difficulties related to measurements. 3.1 Variation of the road layout

This point has already been mentionned in the paragraph dealing with the basic variables of the model. The basic problem is to represent a discontinuity in the road layout, i.e a sudden increase or decrease in capacity (addition or suppression of one lane for instance). The study of two different situations is of interest: Steady-state traffic conditions, either fluid or congested, A variation of traffic demand upstream (or supply downstream). 3.1.1 First order models

It has been explained in [Lebacque, 1996b] how the supply/demand expression provide complete boundary conditions for first order models. This makes possible a very simple resolution of the Riemann problem when there is a discontinuity relative to variable x in the equilibrium function qe(k, x), for the supply at the discontinuity point is given considering the function qe(k, x+) corresponding to downstream conditions, and the demand is computed using the function qe(k, x~) corresponding to upstream conditions. The boundary condition is given by the continuity of traffic flow through the discontinuity, which can also be expressed as a stationary Shockwave denoting x+ and x~ points immediately downstream and upstream of the discontinuity x, the speed of the shock wave is expressed as: 77 ff\=

s()

9(x+,t) -q(x~,t) k(x+,t)-k(x~,t)

The null speed of the Shockwave implies q(x+, t) = q(x~, t) Vt. The following examples show the solutions in different cases. Solutions for capacity increase and decrease are completely symmetric, depending on the fluid/congested conditions. Ex 1. Capacity restriction, demand below capacity


13

Ex 2. Capacity restriction, increase of the demand from below to above capacity

Ex 3. Capacity expansion, increase of downstream supply from below to above upstream capacity (This case is symetric of the previous one).

The lines in these examples are either characteristics (straight lines) or the obvious shock-waves. What is important to notice is that: Accelerations are not bounded and are infinite in some cases The traffic flow through the discontinuity is always maximized by the entropy solutions computed by the supply/demand process. Under stationary conditions anyway, the accelerations would be infinite whatever the solution used since the flow is uniform and the concentration is not. 3.1.2 Higher order models

The situation is somewhat less clear for second order models. In principle, the terms | in the | speed equation should smoothen the discontinuities. Let us consider for instance Payne's model:(14)

du

1k dx J

The stationary solution of this model satisfies: q k u Q = = = Q independent of x and t k(x) function of only u(x) function of a; only ku implying f| = -jg

14


It follows from (14) that:(15)du

I

Q u

vdu\ udx J

Let us consider Greenshield's relationship:

(and v = Vf/(2kj)), with parameters Vf and kj having different values on the left- and righthand-side of the origin. The generic solution of (15) (i.e. excluding the particular solution u= (jf + 1vQ\ /Vf) is given by:(16)

du (u2 - v/T]+ 1vQ

dxT

with boundary conditions given at infinity by the equilibrium conditions:

_ vfu is continuous at the origin, but ~ is not, yielding for instance in the diverge case a solution of the following type:

(the exact functional form depending on the exact value of Q). It should be emphasized here that nothing prevents Q from being equal to the maximum throughflow of the system (i.e. the smaller of the maximum equilibrium flow values on the left- and right-hand-side of the origin). This observation suggests that in general, outside transitory (unstationary) situations, second and first order models should yield the same outflows of say capacity restrictions]. This last remark is consistent both with Schochet's [1988] convergence result and with Del Castillo et a/.'s [1993] analysis, which showed that for theoretically consistent values of the parameters v and T of Payne's model:

T(k)

= -1

!/(*) = -(1/2) (t-) the LWR and Payne models yield very close results. There is nevertheless a fundamental difficulty which must be mentionned here the derivation of second order models usually does not take into account the effect of the possible variability of the equilibrium speed-density relationship. Let us consider for example the derivation of Payne's model from a car-following model:(17)fir


15

with xn the position of the n-th vehicle and A(l/fc) == u e ( k ) . Now, if ue depends on the position x, we should rewrite (17) as: (18)or: dx -^(t + T) =A(z n _ 1 (*)-z f l (t) J z n _ 1 (*))

There is not really any argument enabling us to prefer one of the formulas (18), (19) over the other. Both formulas express the fact that the driver anticipates the variation of the physical layout of the track, they differ by the range of this perception of the driver. Let us consider (19), which results in the simplest calculations and is consistent with Payne's original approximation. With the usual approximations:

xn xn xn-i-xn (xn.l-xn)/2it follows: ..... i) Xd(b, t + T ( t } ) . This last identity shows that in FIFO models, the inflow composition of intersections may be completely determined by past upstream conditions. Thus, the modelling of intersections in which this is not the case (intersections with preselection lanes, left-turning stoarage capacity etc...) requires non FIFO links. This is the purpose of multi-lane modelling. First order models do not accomodate these models easily, because of the equilibrium speed-density relationship. Two examples may be cited here. The model of [Daganzo et al. 1997], treats the special case of two lane types and two user classes, with the effective flow resulting from a user optimum. [Daganzo et al. 1997] derived analytical solutions for the corresponding Riemann problem. Lebacque and Khoshyaran [1998] present a general lane assignment model, based on the analogy with classical static assignment problems. In this last model, the users d, of density Kd, have access to the set of lanes i 6 Id, and the density Kd is split between lanes i as Kd = kf. With kd = 0 if i Id, and with Ki


17

the density in lane i (given by Ki = ) kf) being less than the maximum density Kjti of lane i, the following constraints apply to the unknowns kf:d

Kd = kf

\/d

kf = 0 if i g J d , kf > 0 Vi, d

Vi,

The unknowns kf are determined by maximizing the total flow

with 7j = Kjj/Kj the relative width of lane i (system optimal, entropy maximizing-like model), or the following criterioni'+l=Q. If we consider American legislation, the spontaneous lane changes result from drivers having a distinct preference for a specific lane. That is, a driver changes to the left lane if he prefers to be on any of lanes left of his current lane. Assuming that free-flowing drivers only change lanes if they remain free-flowing, then spontaneous lane changing yields the following contribution to the dynamics of ^,/ (cf. Hoogendoorn and Bovy (1998c)): [d^/dt]Jl=-yCJ^Ju(v,vG) and [a^/a;]^=Yf^fV,v 0 ) (15)

Passive interactions. If any vehicle of class u on lane j driving at a velocity w interacts with a vehicle driving at a velocity vkm. Figure 8 shows that congestion occurs on the right roadway lane. That is, although vehicles are able to flow into the right roadway lane without causing congestion upstream of the lane drop, fast vehicles flowing into the right lane cause interacting with slow vehicle downstream on the right lane need to reduce their velocity. This reduction in more profound, since no overtaking possibilities occur. Since each interaction leads to velocity decrease, the right lane is more susceptible to the seemingly spontaneous formation of jams due to vehicle interaction.

CONCLUSIONS AND OUTLOOKThis paper presents a macroscopic multilane multiclass traffic flow model that is founded on gas-kinetic principles. The model dynamics are governed by processes of a convective nature, acceleration towards the desired velocity, deceleration due to interaction, and various types of lane changing. The model distinguishes constrained and free-flowing vehicles. It is cast using the class-specific and lane-specific conservative variables density, momentum, and energy, enabling a simplified model derivation and improved numerical analysis. The model allows the specification of several parameters that define the characteristics of the classes and the roadway lanes. Among these parameters are the vehicle length, the desired velocity, the acceleration time, the reaction time, the constrained vehicle fraction, and the lane-changing probability functions.

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The gas-kinetic approach yields equilibrium expressions for the equilibrium momentum, energy, and the distribution of densities over the roadway lanes. From these relations, expressions describing the equilibrium velocity, and velocity variance have been determined. The equilibrium relations are functions of the user-class specific immediate overtaking probabilities, acceleration times, desired velocities, covariances between the velocity and the desired velocity, and the asymmetric interaction between fast and slow vehicles of the same and different user-classes.

timestamp t = 0 min densities velocities

timestamp t = 1 min

=5

2 solane drop

person-cars right-lane person-cars left-lane trucks right-lane trucks left-lane timestamp t = 4 min timestamp t = 10 min

I

80-

5

10 road position [km]

15

5

10 road position [km]

15

Figure 8: Traffic conditions for the 'left lane drop' example. From the test-case example of a two-lane ringroad we conclude that the preliminary results of macroscopic simulation are plausible. Trucks increasingly use the right lane when the personcar density increases, leaving the left lane for the faster person-cars. The slower trucks are

Macroscopic Traffic Flow Modelling

49

virtually unaffected by faster person-cars. However, when the velocity of person-cars decreases, trucks are affected by these slower person-cars and consequently need to reduce their velocity as well. MLMC traffic operations near a lane drop were simulated. Again, the macroscopic MLMC model was able simulate the traffic operations at the lane drop realistically. Moreover, also incidents and on-ramps can be described. Hoogendoorn (1999) identifies and remedies some of the model's current shortcomings. For instance, he argues that the vehicular chaos assumption only holds for dilute traffic, due to the increased correlation between vehicles for increasing traffic densities. This can be remedied by considering the correlation of the velocities of unconstrained vehicles and platooning vehicles. Moreover, Hoogendoorn (1999) shows that platooning vehicles are able to accelerate (to the desired velocity of the unconstrained platoon leader). Finally, the author incorporates the vehicular space requirements in the MLMC dynamic equations.

REFERENCESBliemer, M. (1998). Multiclass travel time functions. Proceedings of the 6th meeting of the EURO Working Group of Transportation. Daganzo, C.F. (1995). Requiem for second-order fluid approximations of traffic flow. Transportation Research B, 29, 277-286. Daganzo, C.F. (1997). A Continuum Theory of Traffic Dynamics for Freeways with Special Lanes. Transportation Research B, 31, vol. 2, 83-102. Helbing, D. (1996). Gas-kinetic derivation of Navier-Stokes-like traffic equations, Physical Review E, 53, vol. 3, 2266-2381. Helbing, D. (1997). Modelling multilane traffic flow with queuing effects. Physica A, 242, 175-194. Hoogendoorn, S.P. (1997). A Macroscopic Model for Multiple User-Class Traffic Flow. Proceedings of the 3rd TRAIL PhD. Congress, vol. I. Hoogendoorn, S.P. (1999). Multiclass Continuum Modelling of Multilane Traffic Flow. Dissertation Thesis, Delft University Press. Hoogendoorn, S.P. and Bovy, P.H.L. (1998a). Multiple User-Class Traffic Flow Modelling Derivation, Analysis and Numerical Results. Research Report VK2205.328, Delft University of Technology. Hoogendoorn, S.P. and Bovy, P.H.L. (1998c). Continuum Modelling of Multilane Heterogeneous Traffic Flow Operations. Research Report VK 2205.330, Delft University of Technology. Hoogendoorn, S.P. and Bovy, P.H.L. (1998d). Macroscopic Modelling of Multiple User-Class Traffic Flow using Conservative Variables. Proceedings of the 6th meeting of the EURO Working Group of Transportation.

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Hoogendoorn, S.P., and Bovy, P.H.L. (1998b). Modelling Multiple User-Class Traffic. Preprint 980692 of the 1998 TRB annual meeting. Kerner, B.S., Konhauser, P., and Schilke, M. (1996). A new approach to problems of traffic flow theory, Proceedings of the 13th International Symposium of Transportation and Traffic Theory, INRETS, Lyon, 119-145. Klar, A., R.D. Kiihne, and R. Wegener (1998). A hierarchy of models for multilane vehicular traffic I: Modeling. To appear in SIAM J. Appl. Math. Lebaque, J.P. (1996). The Godunov scheme and what it means for first order traffic flow models. Proceedings of the 13th International Symposium of Transportation and Traffic Theory, 647-677. Lyrintzis, A.D., Liu, G., Michalopoulos, P.G. (1994). Development and comparative evaluation of high-order traffic flow models, Transportation Research Record 1547, 174-183. Paveri-Fontana, S.L. (1975). On Boltzmann-Like treatments for traffic flow: a critical review of the basic model and an alternative proposal for dilute traffic analysis. Transportation Research 5,9,225-235. Payne, HJ. (1979). FREFLO: A Macroscopic Simulation Model For Freeway Traffic. Transportation Research Record 772, 68-75. Prigogine, I., and Herman, R. (1971). Kinetic theory of vehicular traffic. American Elsevier Publishing Co., New York. Van Leer (1982). Flux vector splitting for the Euler equations. Proceedings of the 8th International Conference in Numerical Methods in Fluid Dynamics, Berlin, Springer-Verlag.

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THE CHAPMAN-ENSKOG EXPANSION: A NOVEL APPROACH TO HIERARCHICAL EXTENSION OF LIGHTHILL-WHITHAM MODELS

Paul Nelson1 Alexandras Sopasakis Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368, U.S.A.

ABSTRACTAttempts to improve on the basic continuum (hydrodynamic, macroscopic) model of traffic flow, as developed in the seminal 1955 paper of Lighthill and Whitham, have largely followed the 1971 work of Payne in retaining the continuity equation, but replacing the classical traffic stream model by a "dynamic traffic stream model" (or "momentum equation"). In the present work it is suggested that, whatever may be the advantages and disadvantages of the Payne models, they should not properly be regarded as the traffic flow analog of the Navier-Stokes equations of fluid dynamics. Further, the Chapman-Enskog asymptotic expansion in a small parameter is shown to lead to an alternate class of models that seem to have a more legitimate claim to that distinction. Details of this expansion, about the stable-flow equilibria of the Prigogine-Herman kinetic equation and in the case that the passing probability and relaxation time are constant, are presented to orders zero and one. The zero-order and first-order expansions correspond respectively'Also affiliated with the Departments of Computer Science and of Nuclear Engineering at Texas A&M University.

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Theory

to the Lighthill-Whitham (LWR) model, and to the Lighthill-Whitham model with a diffusive correction. These are suggested to be the correct traffic-flow analogs of respectively the Euler and Navier-Stokes equations of fluid dynamics. Results of a numerical simulation for a simple traffic-flow problem suggest that the diffusive term represents a correction to the LWR model that captures, to some extent, effects stemming from the fact that vehicles actually travel at various speeds. (By contrast, Lighthill-Whitham models proceed as if all vehicles travel at the average speed corresponding to the density of vehicles in their immediate vicinity.) Some further related work is suggested.

INTRODUCTIONTraffic stream models (fundamental diagrams, flow/density relations) date back to (at least) the work of Greenshields (1934), and are often considered as the foundation of capacity analysis (Transportation Research Board, 1985). Lighthill and Whitham (1955), and independently Richards (1956), observed that if a traffic stream model is supplemented by the equation of continuity (conservation of vehicles), then the resulting partial differential equation presumably could be solved, subject to suitable initial and boundary conditions, for the concentration (and hence the mean speed and flow) as a function of location and time. This Lighthill-Whitham-Richards model (LWR model) is widely considered the most fundamental continuum (macroscopic, hydrodynamic) model of traffic flow. Unfortunately, observational data (e.g., Drake, Schofer and May, 1965) are at best ambiguous as regards existence of traffic stream models.2 This has led some workers (e.g., Ceder (1976), Hall (1987), Disbro and Frame (1989)) to suggest alternative macroscopic models that differ from LWR models in a revolutionary manner. However, the usual approach to alternative continuum models is more evolutionary, in following the work of Payne (1971) by supplementing the continuity equation with a "dynamic traffic stream model" (or "momentum equation") that is itself a differential equation, as opposed to the classical "static" traffic stream models that do not contain derivatives. This gives rise to a system of two first-order partial differential equations in two unknowns, typically concentration and mean speed. Such systems are customarily known as higher-order models. Higher-order continuum theories of traffic flow frequently (e.g., Helbing, 1996b; Kerner and Konhauser, 1993) are considered as analogs of the Navier-Stokes equations of fluid flow, and correspondingly the LWR theory is viewed as analogous to the Euler equations of fluid flow. The present work is intended as somewhat of a counterpoint to the first of these views. Specifically, the primary objectives of this paper are as follows:2

This difficulty was already noted by Lighthill and Whitham (1955, p. 344).

The Chapman-Enskog Expansion

53

1. It is suggested, in some detail, that there is a viewpoint from which the often repeated supposed analogy between current higher-order models of traffic flow and the Navier-Stokes equations of fluid dynamics is questionable. 2. It is noted that a systematic application of the Chapman-Enskog asymptotic expansion to a kinetic model of vehicular traffic, following the lines of the use of this technique for the Boltzmann kinetic equation and its solutions in the flow of rarefied gases (e.g, Chapman and Cowling, 1952; Cercignani, 1988; Liboff, 1990), has the potential to provide not only continuum models of traffic flow that are true analogs of the Navier-Stokes equations, but even a systematic hierarchy of continuum models for traffic flow. 3. Some initial developments of this approach via the Chapman-Enskog asymptotic expansion are presented. These developments are based on the classical Prigogine-Herman (1971) equation of the kinetic theory of vehicular traffic, The first two of these objectives are met in the following section, in the context of a general critical review of Continuum Models of Vehicular Traffic. The remainder of this work is devoted to the third objective. This is initiated with a section in which the salient features of Kinetic Equations for Vehicular Traffic are collected. The focus in this section is particularly upon the equilibrium solutions of the Prigogine-Herman kinetic equation, and the necessity to limit further considerations to the regime of stable flow, because of the form of these equilibrium solutions. The next section is devoted to the The Chapman-Enskog Expansion, and contains the central new results presented in this work. First the Chapman-Enskog expansion is described generally, in the context of the Prigogine-Herman kinetic equation. Then the zero-order instance of this expansion is shown to give rise to an LWR model, with traffic stream model completely defined in terms of the knowns of the underlying Prigogine-Herman equation. Finally, the first-order Chapman-Enskog expansion is shown to give rise to a continuum approximation that consists of an LWR model with an additional "diffusive" term. The section on Computational Results provides a comparison of numerical results, and a discussion of the interpretation of these differences, for this first-order diffusive approximation and the LWR model that is the corresponding zero-order approximation, in a simple instance of traffic flow. The paper closes with a section of Conclusions, which primarily consists of suggestions for further related work.

CONTINUUM MODELS OF VEHICULAR TRAFFICIn the opening paragraphs of this section the elements of LWR and higher-order models are collected, and three key issues (validity, mathematical solvability and computational solution) affecting each of these types of models are identified. Then follows three subsections in which the current state of affairs of both LWR and higher-order models vis-d.-vis these three issues is briefly reviewed. The concluding subsection is devoted to a discussion of possible continuum

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Theory

models that potentially improve upon LWR models, but in a fundamentally different manner from that underlying current higher-order models. Traffic stream models3 will be written as9 = Q(c),(1)

where q is flow, c is density, and Q is a known function. The equation of continuity (conservation of vehicles) will be expressed as

^ + ^i-n + dt dxThe two can be combined to give a determined system consisting of a single (nonlinear) partial differential equation in one unknown (the density),dc .. . dc

ii+'(c)^ =0'(q1 = dq/dc) that presumably can be solved, subject to suitable initial and boundary conditions, for the density (and hence the mean speed and flow) as a function of location and time. The latter equation is the most concise mathematical form of a LWR model. These are widely used, at least conceptually. The possibility of further improved continuum descriptions of traffic flow already was considered by Lighthill and Whitham (1955, p. 344), who suggested adding "diffusion" (representing adjustments by drivers to the concentration slightly ahead) and "inertia" (representing the nonzero time required for accelerations or decelerations) effects to the continuity equation. However, subsequent development of presumably better models, beginning with the work of Payne (1971), has instead been directed toward so-called higher-order models. In these, the "static" traffic stream model (1) is replaced by a "dynamic" counterpart that constitutes a second differential equation (in addition to the continuity equation). A typical instance of such a "dynamic traffic stream model" (also known as "momentum equation") is that due to Kiih

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