Project report on a study of trip distribution characteristics of central zone.surat.
Descripción: highway engineering
final exam uthm
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hey all!! i was given an assignment on ethical issues in transportation industry.. it was then dat i found dat very little material was available on this topic.. on net as well as in boks.. so.. af...
SYLLABUS
Authors of the Book used: Atty. Timoteo B. Aquino and Assoc. Justice Ramon Paul L. Hernando Professor: Atty. Jose Glenn C. Capanas University of San Carlos ~ This is sourced from the notes …Full description
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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 THEORY Proceedings of the 14th International Symposium on Transportation and Traffic Theory Jerusalem, Israel, 20-23 July, 1999 edited by AVISHAI CEDER Transportation Research Institute Faculty of Civil Engineering Technion - Israel Institute of Technology Haifa, Israel
PERGAMON An Imprint of Elsevier Science Amsterdam - Lausanne - New York - Oxford - Shannon - Singapore - Tokyo
First edition 1999 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for. British Library Cataloguing in Publication Data A catalogue record from the British Library has been applied for.
CONTENTS Preface and Overview International Advisory Committee
ix xiii
In Memoriam
xv
Contributors
xix
Chapter 1 - Traffic Flow Models MACROSCOPIC TRAFFIC FLOW MODELS: A QUESTION OF ORDER J.P Lebacque, J.B. Lesort
1 -104
3
MACROSCOPIC MULTIPLE USER-CLASS TRAFFIC FLOW MODELLING: A MULTILANE GENERALISATION USING GAS-KINETIC THEORY S.P. Hoogendoorn, P.H.L. Bovy
27
THE CHAPMAN-ENSKOG EXPANSION: A NOVEL APPROACH TO HIERARCHICAL EXTENSION OF LIGHTHILL-WHITHAM MODELS P. Nelson, A. Sopasakis
51
THE LAGGED CELL-TRANSMISSION MODEL C.F. Daganzo
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
105-188 107 725
THEORY OF CONGESTED TRAFFIC FLOW: SELF-ORGANIZATION WITHOUT BOTTLENECKS B.S.Kerner
147
A MERGING-GIVEWAY BEHAVIOR MODEL CONSIDERING INTERACTIONS AT EXPRESSWAY ON-RAMPS H. Kita,K. Fukuyama
173
Transportation and Traffic
Theory
Chapter 3 - Road Safety and Pedestrians
189 - 254
COMPARISON OF RESULTS OF METHODS OF THE IDENTIFICATION OF HIGH RISK ROAD SECTIONS M. Tracz, M. Nowakowska
191
BEHAVIOURAL ADAPTATION AND SEAT-BELT USE: A HYPOTHESIS INVOKING LOOMING AS A NEGATIVE REINFORCER A.M. Reinhardt-Rutland
213
BI-DIRECTIONAL EMERGENT FUNDAMENTAL PEDESTRIAN FLOWS FROM CELLULAR AUTOMATA MICROSIMULATION V. J. Blue, J.L. Adler
235
Chapter 4 - Flow Evaluation on Road Networks
255 - 324
FLOW MODEL AND PERFORMABILITY OF A ROAD NETWORK UNDER DEGRADED CONDITIONS Y. Asakura, M. Kashiwadani, E. Hato
257
A SENSITIVITY BASED APPROACH TO NETWORK RELIABILITY ASSESSMENT M.G.H. Bell, C. Cassir, Y. lida, W.H.K. Lam
283
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
325- 416
327 '-x 351
A DOUBLY DYNAMIC TRAFFIC ASSIGNMENT MODEL FOR PLANNING APPLICATIONS V. Astarita, V. Adamo, G.E. Cantarella, E. Cascetta
373
ROUTE FLOW ENTROPY MAXIMIZATION IN ORIGIN-BASED TRAFFIC ASSIGNMENT H. Bar-Gera, D. Boyce
397
Contents
Chapter 6 - Traffic Demand, Forecasting and Decision Tools
417 - 514
THE USE OF NEURAL NETWORKS FOR SHORT-TERM PREDICTION OF TRAFFIC DEMAND J. Barcelo,J. Casas
419
ALGORITHMS FOR THE SOLUTION OF THE CONGESTED TRIP MATRIX ESTIMATION PROBLEM M. Maker, X. Zhang
445
COMBINING PREDICTIVE SCHEMES IN SHORT-TERM TRAFFIC FORECASTING N.-E. ElFaouzi
471
A THEORETICAL BASIS FOR IMPLEMENTATION OF A QUANTITATIVE DECISION SUPPORT SYSTEM - USING BILEVEL OPTIMISATION A. CluneM. Smith, Y. Xiang
489
Chapter 7 - Traffic Simulation
515-574
MACROSCOPIC MODELLING OF TRAFFIC FLOW BY AN APPROACH OF MOVING SEGMENTS M. Cremer, D. Staecker, P. Unbehaun
517
MICROSCOPIC ONLINE SIMULATIONS OF URBAN TRAFFIC J. Esser, L. Neubert, J. Wahle, M. Schreckenberg
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
575-662
INVESTIGATION OF ROUTE GUIDANCE GENERATION ISSUES BY SIMULATION WITH DynaMIT J. Bottom, M. Ben-Akiva, M. Bierlaire, I. Chabini, H. Koutsopoulos, Q. Yang
577
A NEW FEED-BACK PROCESS BY MEANS OF DYNAMIC REFERENCE VALUES IN REROUTING CONTROL A. Poschinger, M. Cremer, H. Keller
601
OPTIMAL CO-ORDINATED AND INTEGRATED MOTORWAY NETWORK TRAFFIC CONTROL A. Kotsialos, M. Papageorgiou, A. Messmer
627
PROGRESSION OPTIMIZATION IN LARGE SCALE URBAN NETWORKS. A HEURISTIC DECOMPOSITION APPROACH C. Stamatiadis, N.H. Gartner
645
Transportation and Traffic
Theory
Chapter 9 - Road Tolling and Parking Balance
663 - 732
TOLLING AT A FRONTIER: A GAME THEORETIC ANALYSIS D.M. Levinson
665
CARPOOLING AND PRICING IN A MULTILANE HIGHWAY WITH HIGH-OCCUPANCY-VEHICLE LANES AND BOTTLENECK CONGESTION H.-J. Huang, H. Yang
685
BALANCE OF DEMAND AND SUPPLY OF PARKING SPACES W.H.K. Lam, M.L. Tarn, H. Yang, S.C. Wong
707
Chapter 10 - Traveller Survey and Transit Planning
733 - 796
THE ROLE OF LIFESTYLE AND ATTITUDINAL CHARACTERISTICS IN RESIDENTIAL NEIGHBORHOOD CHOICE M.N. Bagley, P.L. Mokhtarian
735
PLANNING OF SUBWAY TRANSIT SYSTEMS S.C. Wirasinghe, U. Vandebona
759
SCHEDULING RAIL TRACK MAINTENANCE TO MINIMISE OVERALL DELAYS A. Higgins, L. Ferreira, M. Lake
779
Index
797
PREFACE AND OVERVIEW These 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.
Methods used for identifying high risk road sections, general belief in seat belts and pedestrian flow models.
3. ROAD SAFETY AND PEDESTRIANS
Travel behaviour, origin-destination and capacity models on road networks.
4. FLOW EVALUATION ON ROAD NETWORKS
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.
Advance in Knowledge New formula that allow more accurate and better understanding of the various traffic flow scenarios.
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.
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.
Means for road financing, toll collection ideas, road tolling along with high-occupancy-vehicle lanes and attributes of public car parks.
Connection between residential location and density and urban travel patterns, methods for subway network plan and procedures for rail track maintenance.
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]
9. ROAD TOLLING AND PARKING BALANCE
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.
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.
Transportation and Traffic
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 COMMITTEE Prof. 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 MEMBERS Prof. 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 5°K, thus anticipating later work on the primordial cosmic fireball radiation at 2.8°K 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
Transportation and Traffic
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
IN MEMORIAM
xvii
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
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CONTRIBUTORS Adler, 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.
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.
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
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.
Transportation and Traffic Theory
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
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
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CHAPTER 1 TRAFFIC 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)
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MACROSCOPIC TRAFFIC FLOW MODELS: A QUESTION OF ORDER J.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 INTRODUCTION Since 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)
+ 0 lr 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
Transportation and Traffic Theory
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):
u(x,
(4)
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 ISSUES In 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:
dw dt
, (duf\
dk
\ ok I dx
(
\ *
, due\ dup * dk iI ox
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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: dw ~JT at
(9
)
=
(due\2 dk ~k HjT \dk J ^~ 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 dx
dqe dx
It can be seen that: • 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.
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Theory
2.3 Models solutions 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]: 2
du , c dk =1-MAO -u)-j^ 0
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)
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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
11
• It can be identified using real data, • 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
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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 CASES This 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
Macroscopic Traffic Flow Models
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:
du
(14)
1 k 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
Transportation and Traffic Theory
It follows from (14) that: (15)
du
Q u
I
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:
du (u2 - v/T]
(16)
+ 1vQ
dx T
with boundary conditions given at infinity by the equilibrium conditions:
_ vf u 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
Macroscopic Traffic Flow Models
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: dx -^(t + T) =A(z n _ 1 (*)-z f l (t) J z n _ 1 (*))
(18) or:
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:
~ x K, u(x,t) w l/k(x + l / ( 2 k ( x , t ) ) , t ) w x + l/(2k(x,t))
xn xn xn-i-xn (xn.l-xn)/2 it follows: ..... <20)
du
1 I
.,
1 due/,
.
introducing a supplementary term
.dk\
I
due
1 due
accounting for a specific acceleration due to the variation of the physical environment. The same remark applies to the more recent model of Zhang [1998]. Indeed, in keeping with the derivation of the model, it would be necessary to start from the following macro-micro traffic model: (21) dx^ +T^Ug (k(Xn + A, *),*„ + A) at (replacing equation (16) of the above-mentionned paper). In the above equation (21), we expressed the awareness of the driver of the physical layout ahead of him (at a distance A, i.e. the anticipatory nature of the driver's behavior, in keeping with the ideas of the paper [Zhang 1998]. Now, after setting x = xn+i, xn+i = u, and developing at first order in T and A, it follows: .„. (22)
du 1 . ., . = (u (k x} dt ±
. A 9 u e / , . dk u] H -(k x)
J. OK
ox
Since the characteristics speed being still
Zhang's argument still applies to yield: due aA; \
A
1
~\7
J
/
dqe OT a/c
Finally, Zhang's model becomes: (23)
du 1 , ^ =I
,,
,
,
(du,.,,
C\ dk
, dup
16
Transportation and Traffic
Theory
It is worth mentionning that this last model (23) does not admit the LWR model as its limiting case when ue(k, x) = u since the acceleration in this last model is given by (8) when the equilibrium speed-density depends on the position. As a conclusion, some further research should go into adapting second order models to the possible variability of physical characteristics of the links. Some ideas in that direction have been developed in a purely numerical setting, for instance in [Papageorgiou et al. 1989] (the a coefficients mentionned previously (13)). 3.2 Multiclass-multibehavior/multilane traffic modelling. Both problems, which are related, have been treated in the literature. The necessity of these problems (and the related intersection modelling problem) stems from the fact that traffic models should actually describe flow on networks, and not simply on the real line. We should distinguish between multibehavior flow, in which the the equilibrium relationships applying to different classes of users are different (for instance multilane flow), and multiclass flow, in which users may belong to different classes, as for instance in the dynamic assignment context, but whose equilibrium relationships are the same. Multiclass models are an inescapable ingredient of any traffic flow model aiming at representing dynamic assignment situations. Considering the case of first order models, multiclass flow description relies on the partial flowsdensities concept. For each user class d, partial flow Qd, density Kd and speed Vd are introduced; and the partial flow and density are related by the usual conservation equation and speed definition equation Qd = KdVd. An equation for Vd yields a complete traffic flow model for class d. The simplest possible model is the FIFO model Vd = V, Vd (all users have the same speed, irrespective of their class). Characteristic of the FIFO partial flow model is the fundamental fact that the composition of density and flow are the same. Such a FIFO model was introduced in a discretized version in [Daganzo, 1994]; a non-FIFO extension was proposed (still in a discretized setting) in the STRADA model [Buisson et a/., 1996]. A continuous version of this model is described in [Lebacque and Khoshyaran, 1998] the partial speed Vd results from the partial flow Qd, estimated directly at each point as the minimum of the downstream partial supply and upstream partial demand. The behavioral part of the model is contained in the derivation of partial supplies and demands from their global counterparts. Second order models have a much older history of multiclass models. We might cite for instance the Payne-derived METANET [Messner and Papageorgiou, 1990] in which the partial flow model is approximately FIFO the discretized partial flows are proportional to the composition of the discretized density. The METANET and SSMT derived METACOR [Elloumi et al., 1994] follows similar principles in its motorway part. Hilliges' model [Hilliges, 1995] is approximately FIFO too (same speed irrespective of class d). Basically, models differ on the issue of FIFO behavior of partial flows. FIFO behavior constrains the model drastically, notably in respect to intersection modelling. Indeed, let [a, b] be a link, T(t) the travel time of the link for vehicles entering it at time t, and xd(xi t) me composition of the traffic at location x and time t, then x d ( G > 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
Macroscopic Traffic Flow Models
17
the density in lane i (given by Ki = £) kf) being less than the maximum density Kjti of lane i, d
the following constraints apply to the unknowns kf:
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
de f
with 7j = Kjj/Kj the relative width of lane i (system optimal, entropy maximizing-like model), or the following criterion <-Kihi
ue(s) ds
(user-optimal, similar to the Beckmann transform), yielding [Daganzo 1997] as a particular case. An older model can be recalled here the model of [Michalopoulos et a/., 1984a), in which the interaction between lanes is assumed to follow a simple linear relaxation behavior, of the form: 9kh
for instance in the case of two lanes. Second order models accomodate also multilane models (see again the paper by Michalopoulos et al. [Michalopoulos et al., 1984a], and Hoogendoorn [1996] (combining multiclass and multilane). More recent developments favor the derivation from microscopic lane interaction models, aggregated through the agency of kinetical models, as in Helbing [1997]. 3.3 Platoons and their dispersion. Platoon dispersion is an experimental fact, incorporated as a fundamental feature in the TRANSYT model [Robertson, 1969]. The platoon dispersion problem can be expressed as the solution of the problem presenting the following initial conditions, with kQ an undercritical concentration value:
to which we should add eventually an initial condition for speed.
Transportation and Traffic
18
3.3.1
Theory
First order models
The standard solution of the first order model corresponds to a shock-wave occurring at the end of the initial platoon, and a fan-like characteristic scheme at the front as the following example shows, with Greenshields relationship (the straight lines represent as usual the characteristics and the curves the Shockwaves):
This means that the evolution of the platoon is as follows: kA
The dispersion as predicted by first order models occurs always at the head of the platoon and concerns the faster vehicles (this is the consequence of the entropy condition). The situation is slightly improved by considering discretized first order models, but as we have seen previously, numerical viscosity is not really a reliable modelling tool. It can be noticed that, when computing the solution numerically, the platoon aspect is more realistic, because of the numerical viscosity introduced by discretization. An illustration of the emulation of platoon dispersion by numerical viscosity effects can be given by a derivation similar to the one presented above. Let the road be divided into cells of uniform size 6x. If a concentration k0 is observed in cell i = 0 at time t = 0, with a null concentration everywhere else, with the simplification that ^ is constant between 0 and ko, the concentration at time-step p in the cell i is fc i (p) = Cj^(l-^- i fc 0 with Q = H^f. /? is necessarily less than 1 due to the Courant-Friedrich-Lewy (CFL) condition, and the distribution of concentration inside the platoon is thus binomial, which is quite similar to the platoon dispersion model proposed by (Robertson, 1969) for the TRANSYT programm. Again, this is a purely numerical effect depending on the discretization parameters. If || = ^, which minimizes the numerical viscosity, then /? = 1 and there is no dispersion at all. 3.3.2
Higher order models.
Higher order models on the other hand, thanks to their diffusion term, (or the || term in the speed equation) yield a much better emulation of platoon diffusion. For instance, the platoon diffusion model based on a normal speed distribution proposed by Pacey [1956] has been shown [Grace and Potts, 1964] to correspond to a diffusion equation, a result hardly surprising in view of the relation between stochastic diffusion processes and parabolic partial differential equations.
Macroscopic Traffic Flow Models 3.4
19
traffic signals - interrupted flows
The presence of a traffic signal on the road can be considered in a simplified way as a boundary condition q(xo, t) = 0 when the signal is red. The problem is to compute traffic flow evolution when the signal turns red and when it turns green again. We can use as an archetypical initial conditions that the signal has been green for an infinite time, and that a uniform undercritical flow q0 is present. Boundary conditions are a constant demand q0 upstream and an infinite supply downstream. 3.4.1
First order models
The solution of the traffic signal problem is well known and presented in many books and papers, for one or several intersections (see for instance [Michalopoulos et al., 1980]). For an uncongested signal it has the following form:
What is interesting to notice again is that the accelerations of vehicles are not bounded, neither when negative (vehicles stop instantaneously when they join the queue, nor when positive The first vehicle starting when the signal turns green reaches instantaneously its free speed, and the next ones may present unphysical acceleration values too. For instance, using a Greenshield equilibrium relationship ue = umax (l - jr—\ it is easy to calculate the acceleration of a vehicle moving with the flow at position x and time t in the fan area du 1dt 4t The maximum value of this acceleration is obtained for x — —umaxt which corresponds to the place where vehicles start. It is equal to:
dt 2t It is clear that, whatever the value of a physical maximum acceleration, it is possible to find a time t when this value is exceeded. However, bounded acceleration solutions can be constructed, using for the solution of the above example the trajectory of the first vehicle of the platoon as a boundary condition. These solutions must nevertheless be defined in a proper functional setting [Lebacque, 1997] and still need to be investigated in more detail. 3.4.2
Higher order models
It is clear that higher order models present a better representation of the accelerations downstream the stopline when the signal turns green. On the other hand, as pointed out by Daganzo [1995], the presence of a stopped queue at a red traffic signal may result, if the upstream demand is unsufficient, in negative speeds. As explained in the first part of this paper, some models designed to avoid negative speeds may present other flaws, as the one proposed by [Liu et al., 1998].
20
Transportation and Traffic
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3.5 Intersections. Modelling intersections is a very difficult task in any macroscopic model, considering that the interactions inside an intersection are very complex, generally microscopic in nature, and below the scale-level of the validity of the continuum hypothesis. From a geometrical point of view, two possibilities coexist describe the intersection as a point, or as an extended object. The latter approach has been used mainly in the discretized context and first order models, with SSMT [Lebacque 1984], in which each movement inside the intersection is discretized as a single cell, and the intersection cells overlap, with the model proposed by [Michalopoulos, 1988], also based on overlapping cells, but each movement being discretized into several cells (the discretization was a three cell discretization), or with METACOR [Elloumi et a/., 1994], whose urban part is partially SSMT-derived. A different concept, the concept of non-overlapping exchange zones that accomodate several movements the interaction of which is summarized in the zone dynamics, has been introduced with STRADA [Buisson et al, 1996]. Modelling interactions as points requires to combine quantities describing the traffic state upstream and downstream of the intersection. Particularly illuminating in this respect, notably as far as difficulties occuring in higher order models go, is Hilliges' model [1995], which exists only in semi-discretized form: dNj
__
(24)
with /j the length of cell i. It is obvious that the necessary downstream information for any upstream cell u is an equivalent speed of downstream cells d, whereas a downstream cell d requires an equivalent speed of upstream cells: u
(formulas (6.12) to (6.18) of [Hilliges, 1995]. Furthermore, the FIFO or non-FIFO character of the flow on the upstream links has an impact on the intersection models. A few solutions have been proposed in the first order case. Daganzo [1994] considered merges and diverges, and proposed a priority rule for merges, and a maximum flow rule for diverges (with a FIFO upstream flow). Maximum entropy-like models, constrained by upstream demand and downstream supply, were proposed in [Lebacque and Khoshyaran, 1998]. The impact of the nature of the upstream flow is particular clear in the case of a diverge if the flow is non FIFO, the partial demands are proportional to the traffic composition coefficients upstream of the diverge, but if the flow is FIFO, it is the actual partial flows (throughflows through the intersection) that are proportional to the traffic composition coefficients. To illustrate this point, let us consider for instance a pointwise diverge say a, with upstream link u, downstream links i, upstream demand 5u(a, t), downstream suplies <7j(a, t) resulting from the up- and downstream densities and equilibrium functions (11) by: crj(a, t) = £ e (fcj(a+, £), a; i) Su(a,t) = A e (fc u (a-,t),a;u) . Let qi(a, t),q(a, t) be the partial and total flow through the intersection, and 8i(a, t) and 5(a, t) the partial and total demands upstream. Finally, let Xi(a~i t) be the composition of traffic entering
Macroscopic Traffic Flow Models
21
the intersection. Then, if the model is non-FIFO, the partial demands are proportional to the compositions, yielding:
q(a,t)
= Eft (a,*) • z
On the other hand, if the upstream flow is FIFO, the partial flows are proportional to the compositions, yielding: q(a,t) = Mini[ai(a,t)/Xi(a-,t),6(a,t)] qi(a,t) = Xi(o-,*)g(M) In the case of second order models, friction coefficients in the speed equation have ben used to account for the effect of off- and on-ramps (i.e. diverges and merges). The basic idea is to assume that the say outflow (on the off -ramp) has a given speed related to that of traffic upstream of the intersection (generally smaller), as in [Papageorgiou et al. 1989]. Similar models with friction terms were proposed in [Michalopoulos et al., 1992], generalizing Papageorgiou's model:
du $ - =-(Uf(x)
dk
with G = nkfg and g the the inflow (on-ramp). 3.6 Highway traffic jams. [Kerner, 1997], exhibited a beautifull experimental example of two traffic jams of constant size, persisting on a german highway for a very long period (of the order of an hour), surrounded by fluid traffic. Such a structure cannot be explained by first oder models, except by recoursing to non entropic solutions as illustrated by the following characteristics chart:
But such a solution is highly unstable in the sense that the solution is not unique, and that infinitely small divergences from the right initial conditions may produce an (unique) solution totally different from the above solution, but very close to the entropy solution with its acceleration fan. Let us recall the entropy solution satisfies to continuity relatively to initial conditions. Some second order models of the Kerner-Konhaiiser-Helbing type might be able to explain the persistence of such structures. The above example definitely constitutes a challenge for future traffic flow models. 3.7 Boundary conditions. Boundary conditions are an important feature they determine both intersection models and network entry and exit point models. No traffic equation should be writen only on an infinite track! In the case of first order models, the supply and demand concepts yield the boundary conditions for any link. For instance the link inflow and outflow Q(a, t) and Q(b, t) of the following link:
22
are given by:
Transportation and Traffic
Theory
Q(a,t) = Min Q(M) = Min
with the boundary conditions A u (t) and S
A(M) = In the case of second order models, to the authors knowledge, no simple boundary conditions exist. Let us consider again Hilliges' model (eq. 24) to show the difficulties involved. For a network entry point, corresponding to an entry cell say i, the inflow ki-\Ui must be given, i.e. the upstream density fcj_i must be given. entry point
The same applies to the speed equation, which requires an upstream speed. Basically, upstream boundary conditions will comprise both an upstream density and an upstream speed. For exit points, only the downstream speed is required (the analogy with Papageorgiou's exit ramp model is obvious). There is no simple way to accommodate supply and demand concepts (for example a constraint on the demand at an exit point would imply a constraint on the downstream speed Ui+i, dependent on the exit link density fcj).
4 CONCLUSION The analysis presented above has obvious limits, since it is clear that the only real validation of a traffic model is its confrontation to real data. However, it is also true that real data always present a high proportion of noise that may mask some inconsistencies of a tested model, and that several and possibly conflicting interpretations may often be given of the same observed situations. The validation of a model against real data is thus not necessarily a guarantee that model will behave properly in all situations. This is the reason why many papers presenting the validation of some traffic models are of limited interest. It thus seems interesting to have a basic framework to analyse the behaviour of a model on a set of well-defined cases, to point out eventual inconsistencies or side effects and to compare a particular model (or class of models) to the existing ones. From that point of view, the various cases presented in this paper may contribute to constitute such a framework.
REFERENCES Astarita,V. (1996). A continuous time link model for dynamic network loading based on travel time function, in: Transportation and Traffic Theory, proceeding of the 13th ISTTT (J.B. Lesort ed.). 79-102, Pergamon, Oxford. Bui D.D., P. Nelson, S.L. Narasimhan, (1992). Computational realizations of the entropy condition in modelling congested traffic flow. Report 1232-7. Texas Transportation Institute, USA.
Macroscopic Traffic Flow Models
23
Buisson C. , J.P. Lebacque, J.B. Lesort, H. Mongeot, (1996). The STRADA model for dynamic assignment. Proc. of the 1996 ITS Conference. Orlando, USA. Cassidy M. J.,(1998). Bivariate relation in nearly stationary highway traffic, Trsp. Res. 32B, pp 49-60. Cremer M., M. Papageorgiou, (1981). Parameter identification for a traffic flow model, Automatica 17 pp 837-843 Daganzo C.F., (1994). The cell transmission model 1: a dynamic representation of highway traffic consistent with the hydrodynamic theory. Daganzo C.F., (1995a). The cell transmission model 2: network traffic simulation.Trsp. Res. 28B. 4 269-287. Transportation Research 29B. 2 79-93.1995. Daganzo C.F, (1995b). A finite difference approximation of the kinematic wave model. Trsp. Res. 29Bpp 261-276. Daganzo C.F., (1995c). Requiem for second-order fluid approximation of traffic flow.Trsp. Res., 29B pp 277-286. Daganzo C.F., Cassidy M. J. and Bertini R. L., (1997).Causes and effects of phase transitions in highway traffic, report UCB-ITS-RR-97-8. Daganzo C.F., (1997) A continuum theory of traffic dynamics for freeways with special lanes. Transportation Research 31 B. p 83-102. Del Castillo J.M., Pintado P. and Benitez F. G., (1993). A formulation for the reaction time of traffic flow models. Twelfth International Symposium on Transportation and Traffic Theory, Berkeley, California. Elloumi, E.,H.Hadj Salem, M. Papageorgiou, (1994). METACOR, a macroscopic modelling tool for urban corridors. TRISTAN II Int. Conf., Capri. Friesz T.L., J. Luque, R.L. Tobin, B.W. Wie, (1989). Dynamic network traffic assignment considered as a continuous time optimal control problem. Op. Res. 37, 6 893-901. Grace M.J., R.B. Potts, (1964). A theory of the diffusion of traffic platoons, Op. Res. 12-2 pp 529-533. Greenshields B.D. (1935). A study of traffic capacity, Proceedings of the Highway Research Board, vol 14, pp 448-477 Helbing D., (1997). Verkehrsdynamik, Springer, . Heydecker B.G, (1994). Incidents and interventions on freeways. PATH Research Report UCBITS-PRR 94-5. Hilliges M., (1995). Bin phanomenologisches Modell des dynamischen Verkehrsflusses in schnellstralknnetzen. PHD dissertation. Institut fiir theoretische Physik der Universitdt Stuttgart. Shaker Verlag. Kerner B. S., (1997). Experimental characteristics of traffic flow for evaluation of traffic modelling. IFAC/IFIP/IFORS symposium, Chania, Greece. Kerner B. S. and Rehborn H., (1997). Experimental properties of phase transitions in traffic flow. Phys. Rev. Let.19, pp 4030-4033.
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Kerner B. S., Konhaiiser P. and Shike M., (1996). A new approach to problems of traffic flow theory.in: Transportation and Traffic Theory, proceeding of the 13th ISTTT (J.B. Lesort ed.). 79-102, Pergamon, Oxford. Kiihne R. D., (1984). Macroscopic freeway model for dense traffic. Stop-start waves and incident detection. Ninth Int. Symposium on transportation and traffic theory. VNU Science Press pp 2142. Lebacque J.P., J.B. Lesort, F. Giorgi, (1998). Introducing buses into first order macroscopic traffic flow models. Trsp. Res. Rec 1664, pp 70-79. Lebacque J.P., (1984). Semimacroscopic simulation of urban traffic. Int. 84 Minneapolis Summer Conference. AMSE. Lebacque J.P., (1995). Le modele de trafic de P. Ross Solutions analytiques, Actes INRETS 45, Arcueil,France Lebacque J.P. , (1996a). Instantaneous travel times for macroscopic traffic flow models. CERMICS Report 59-96. Lebacque J.P., (1996b). The Godunov scheme and what it means for first order traffic flow models. in:Transportation and Traffic Theory, proceeding of the 13th 75/77 (J.B. Lesort ed.),pp 647677, Pergamon, Oxford. Lebacque J.-P, (1997). A finite acceleration scheme for first order macroscopic traffic flow models. The 8th IFAC symposium on transportation systems, Chania, Greece. Lebacque J.P. , M.M. Khoshyaran, (1998). First order macroscopic traffic flow models for networks in the context of dynamic assignment. CERMICS Report. To be Published. Presented at the 6th Meeting of the EURO Working Group On Transportation. Goteborg. Leo C.J. , R.L. Pretty, (1992). Numerical simulation of macroscopic traffic models. Transportation Research 26B. 3 207-220, Lighthill M.H., G.B. Whitham, (1955). On kinematic waves II A theory of traffic flow on long crowded roads. Proc. Royal Soc. (Lond.) A 229 317-345. Liu G., Lyrint/is A. and Michalopoulos P., (1998). Improved high Order model for freeway traffic flow. TRB 77th meeting, Washington, DC. Merchant D.K., G.L. Nemhauser, (1978). A model and an algorithm for the dynamic traffic assignment problem.Transportation Science 12 183-(199 & 200-207. Messner A., M. Papageorgiou, (1990). METANET, a macroscopic modelling simulation for motorway networks. Technische Universitat Miinchen. Michalopoulos P.G, G. Stephanopoulos, V.B. Pisharody (1980). Modelling of traffic flow at signalized links, Trsp. Sc. Vol 14-1, pp 9-41 Michalopoulos P.G., D.E. Beskos, Y. Yamauchi, (1984a). Multilane traffic flow dynamics some macroscopic considerations. Michalopoulos P.G. , D.E. Beskos, J.K. Lin, (1984b). Analysis of interrupted flow by finite difference methods. Trsp. Res. B 18B pp 409-421. Michalopoulos P.G., (1988). Analysis of traffic flow at complex congested arterials. Trsp. Res. Rec 1194 pp 77-86. Trsp. res. 18B, 377-395.
Macroscopic Traffic Flow Models
25
Michalopoulos P. G., Yi P. and Lyrintzis A. S., (1992). Developement of an improved high-order continuum traffic flow model.Trap. Res. Rec., Vol. 1365, pp 125-132. Mongeot H., (1997). Traffic incident modelling in mixed urban network, Traff. Eng. + Ctrl, vol 38-1 l.pp 584-592 Newell G. R, (1989). Comments on traffic dynamics. Trsp. Res. 23B, 386-389. Newell G. R, (1993). A moving bottleneck, UCB ITS report UCB-ITS-RR-93-3. Newell G. R, (1998). A moving bottleneck. Trsp. Res., 32B, pp 531-538. Pacey G.M., (1956). The progress of a bunch of vehicles released from a traffic signal, Research note RN/2665/GMP, RRL, London Papageorgiou M., Posch B. and Schmidt G., (1983). Comparison of macroscopic models for control of freeway traffic.Trsp. Res., 17B, pp 107-116. Papageorgiou M., Blosseville J.-M. and Hadj-Salem H., (1989). Macroscopic modelling of traffic flow on the boulevard peripherique in Paris. Trsp. Res., Vol. 23B, pp 29-47. Papageorgiou M., (1998). Some remarks on macroscopic traffic flow modelling.7>.sp. Res. 32A 5 p 323-330. Payne H. J., (1971). Models of freeway traffic and control. Simulation Council proceedings, 1, ch6. Phillips W. R, (1979). A kinetic model for traffic flow with continuum implications. Transportation Planning and Technology, Vol. 5-3, pp 131-138. Prigogine I. and Herman R., (1971). Kinetic theory of vehicular traffic, American Elsevier, New York. Richards P.I. ,(1956). Shock-waves on the highway. Op. Res. 4 42-51. Robertson, D.I, (1969) TRANS YT, a traffic network study tool, RRL Report 153, Crowthorne Ross, P., (1988). Traffic dynamics, Trsp. Res. 22B pp 421-434. Ross, P., (1989). Response to Newell. Trsp. Res. 23B, pp 390-391. Schochet S., (1988). The instant response limit in Witham's non linear traffic model uniform well-posedness and global existence. Asymptotic Analysis 1, pp 263-282. Zhang H. M. and Wu T., (1997). Numerical simulation and analysis of trafic flow. TRB 76th meeting, Washington, DC. Zhang H. M., (1998). Theoretical inquiry into transient speed-concentration relationship in traffic flow. TRB 77th meeting, Washington, DC.
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MULTICLASS MACROSCOPIC TRAFFIC FLOW MODELLING: A MULTILANE GENERALISATION USING GAS-KINETIC THEORY Serge P. Hoogendoorn and Piet H.L. Bovy, Delft University of Technology, Faculty of Civil Engineering and Geosciences, Transportation and Traffic Engineering Section, Delft, The Netherlands
ABSTRACT In contrast to microscopic traffic flow models, macroscopic models describe traffic in terms of aggregate variables such as traffic density, flow-rate, and velocity. The implied mean traffic behavior depends on the traffic conditions in the direct environment of the vehicles in the traffic stream. Using the analogy between the vehicular flow and flow in fluids encouraged deriving these models (e.g. Lighthill and Whitham (1955), Payne (1979)). The advantages of macroscopic models are among others the insight gained into traffic flow operations (e.g. shock-wave analysis), the applicability in model based control, the relatively small number of parameters simplifying model calibration, and the applicability to large traffic networks. Generally, macroscopic models consider the behavior of the aggregate traffic flow. That is, neither a distinction of user-classes, such as traveler types (commuters, freight, recreational, etc.), vehicles types (person-cars, trucks, busses, vans), paying and non-paying traffic, and various types of guided vehicles, nor a distinction of roadway lanes is made. However, we envisage that a generalization of macroscopic traffic flow models to both user-classes and lanes is advantageous. On the one hand, this generalization increases the applicability of macroscopic models to the synthesis and analysis of multilane multiclass (MLMC) traffic flow. As a result, more insight is gained into the response-behavior of the heterogeneous multilane flow, such as effective capacity, velocity distribution, and the distribution of vehicles over the roadway lanes. On the other hand, from the traffic control perspective, contemporary policies pursue a more efficient use of the available infrastructure (e.g dynamic allocation of roadway lanes to
28
Transportation and Traffic
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classes, class-selective ramp-metering). The heterogeneous multilane network-wide traffic control problem is characterized by multiple objectives (efficiency, safety, etc.), multiple target groups (the user-classes), and a high complexity. The latter is caused by the interaction between the user-classes, the interplay between the available control instruments, and the interaction between the different parts of the network. This complexity requires a model-based approach, demanding the availability of operational models providing deterministic conditional predictions of the multilane heterogeneous traffic flow, given some specific control configuration. Only very recently, attempts to generalize the classical macroscopic models emerged. Hoogendoorn (1997), and Hoogendoorn and Bovy (1998a) present a multiclass generalization of the model of Helbing (1996) based on gas-kinetic principles. Research on the multilane generalization of macroscopic flow models is reported by Daganzo (1997), Helbing (1997), and Klar et al. (1998). Helbing (1997) briefly discusses the multiclass generalization of the gaskinetic multilane equations. In this paper, we present a macroscopic model describing the dynamics of heterogeneous multilane traffic flow, based on gas-kinetic multiclass multilane traffic dynamics. In contrast to the aforementioned models, the MLMC model describes the traffic flow by considering the conservative variables density, momentum, and energy, rather than the primitive variables density, velocity, and velocity variance. Using these so-called conservatives simplifies the derivation approach and enables improved mathematical and numerical analysis (cf. Hoogendoorn and Bovy (1998d,1999)). Since the acceleration and lane-changing behavior differs significantly between free-flowing and constrained drivers, the macroscopic flow model considers both driver's states. Other novelties are the derived expressions of the MLMC equilibrium momentum and energy, quantifying the asymmetric user-class and lane interaction. From these expressions, the MLMC equilibrium velocity and velocity variance can be determined. The equilibrium relations result from competitive acceleration and deceleration processes: on the one hand, vehicles accelerate towards their desired velocity, while on the other hand, vehicles that interact with slower vehicles from different user-classes - without being able to immediately overtake to an adjacent lane - decelerate. Also, the equilibrium lane-distribution of the classes as a result of overtaking can be determined. On the input-side, the model allows the specification of the class specific desired velocity, acceleration time, and overtaking probabilities. The paper is organized as follows. First we present the MLMC generalization of the gaskinetic flow equations of Paveri-Fontana (1975) for constrained and free-flowing vehicles. Secondly, we present the macroscopic flow model using conservative variables, while subsequently discussing the equilibrium speed-density relations and the density lane distribution. After discussing the numerical solution approach, we present results from application of the macroscopic model to two test cases. Finally, in the closing section we summarize our research findings.
Macroscopic Traffic Flow Modelling
29
DERIVATION OF THE MLMC GAS-KINETIC EQUATIONS Gas-kinetic models describe traffic using the reduced phase-space density (PSD) p(*,v,/), where p(jc,v,/)dxdv equals the expected vehicle number in [jt,jc+djt) driving with velocity [v,v+dv) at instant t. This concept is borrowed from statistical physics and can be considered as a mesoscopic generalization of the traffic density r(x,t). Equations describing dynamic changes in the implied velocity distributions are based on the work of Prigogine and Herman (1971), who assumed that changes in the reduced PSD are caused by acceleration, deceleration, and convection. The latter simply describes changes due to the movement of the traffic. Their deliberations yielded the following equation: dp dp [dpi — + v— = —
dt
dx
[dpi + —
The acceleration term [dp/dt]^cc describes relaxation of drivers' speed towards a
CD
traffic-
condition dependent velocity. Prigogine and Herman (1971) proposed: [dp/dt]ACC =(Q°(jt,v,f)-p(JC,v,f))/T
(2)
where T denotes the acceleration time and Q°(x,v,0 reflects the distribution of desired velocities, that is, of the expected desired velocity of vehicles driving with velocity v. PaveriFontana (1975) improved the relaxation process by considering the Phase-Space-Density (PSD), which can be considered as a generalization of the reduced PSD extended with an independent variable describing the desired velocity v°, that is p(jc,v,v°,0The interaction term [3p/3?]iNT reflects fast vehicles catching up with slower vehicles, considering the immediate overtaking probability n. The interaction term is composed of contributions of active and passive interactions. An active interaction occurs when a vehicle driving with velocity v interacts with a slower vehicle driving with velocity wv). The assumption of vehicular chaos (cf. Prigogine and Herman (1971)) yields the following expressions for the contributions of active and passive interactions on the dynamics to the reduced PSD: [dp I &CTIVE = (l - n)p(x, v, 0 J ( w - v)pU, w, f)dw
Hoogendoorn and Bovy (1998a,b) propose a multiclass generalization of the gas-kinetic equations of Paveri-Fontana (1975) by considering the Multiclass Phase-Space Density (MUC-PSD). Their model is characterized by asymmetric interactions, caused by the distinction of slow and fast classes. Helbing (1998) proposes similar models for the multilane case.
Transportation and Traffic Theory
30
In this section we will establish the first step in the derivation approach which is the determination of gas-kinetic equations for multiclass multilane traffic flow operations. These equations describe the dynamics of the multilane multiclass Phase-Space Density (abbreviated as MLMC-PSD), which is dis-aggregated into contributions of platooning and free-flowing vehicles respectively due to differences in driving-characteristics. Similar to other gas-kinetic models, several class-specific and lane-specific processes govern the dynamics of the MLMC-PSD (acceleration towards the desired velocity, deceleration caused by vehicle interactions, immediate, postponed and spontaneous lane changing, and state-transitions). Typical class-specific parameters are the desired velocity, the acceleration time, the reaction time, the vehicle length, and the within-user-class velocity variance (cf. Hoogendoorn and Bovy (1998a,b)). The traffic conditions on the roadway lanes differ, due to among others the classdependent overtaking behavior and lane preferences.
The phase-space density for constrained and free-flowing vehicles In order to accommodate the multilane description of heterogeneous traffic, we dis-aggregate the PSD p(jt,v,vV) (cf. Paveri-Fontana (1975)) by distinguishing classes and lanes. The MLMC-PSD f>uj(x,v,v°,t) denotes the expected number of vehicles per unit road-length of user-class u at x on lane j at instant t which are currently driving at a velocity equal to v while aiming to traverse along the road at a desired velocity v°, where j=l,...,M and wsU. By definition, y'=l denotes the rightmost lane while j=M denotes the leftmost lane. By notational convention, dropping the respective index from the notation indicates lane- and/or classaggregation (e.g. p«=2/pujF). The reduced MLMC-PSD and the MLMC traffic density are respectively defined by: pj(;t,v,0 = Jp,!U,v,v 0 ,Odv 0
(5)
and: rJ(x,t)
def . -
=
.
pJu(x,v,v0,t)dv°dv = pJu(x,v,t)dv
(6)
Constrained and free-flowing vehicles In the sequel we will show that the state of a driver - that is, whether he is constrained or freely flowing - to a large extent determines both the lane changing behavior and the acceleration behavior. A constrained or platooning driver refers to any driver who is impeded by a slower vehicle in front without being able to immediately change to an adjacent lane. Conversely, we will refer to any driver not impeded by any slow vehicle or who is able to immediately change lanes as a free-flowing driver. The MLMC-PSD is then dis-aggregated as follows:
Macroscopic Traffic Flow Modelling
,v,v,r)
31
(7)
where auj(x,v,v°,t) and £,u\x,v,v°,t) respectively denote the contribution of the constrained and the free-flowing MLMC-PSD. Correspondingly to the mixed-state reduced MLMC-PSD, we define the separate reduced MLMC-PSD 's, for constrained and free-flowing vehicles.
The gas-kinetic equations for free-flowing and constrained vehicles The gas-kinetic equations that describe the dynamics of £,,/ are governed by both continuum and non-continuum processes. The continuum processes reflect the smooth changes in the free-flowing MLMC-PSD due to balancing inflow and outflow in the phase-space (x,v,v ,t). We can show that the gas-kinetic equations describing the dynamics of the free-flowing MLMC-PSD equal (cf. Hoogendoorn and Bovy (1998c)): —— + V——H
dt
dx
— = 1—— I
dt dv
I a* I
fS"i (8)
where [d£j/dt]uc reflects dynamic changes caused by non-continuum processes. The derivative dv/dt in (8) reflects the acceleration of vehicles. We assume that the freeflowing drivers of class w accelerate to their desired velocity v° in an exponential fashion, i.e.: dv
where TM° denotes the class-specific relaxation constant, reflecting the acceleration capabilities. The continuum terms describing the changes in aj can be derived similarly. However, in opposition to free-flowing drivers, constrained drivers are by definition unable to accelerate towards their (own) desired velocity. Assuming a,/(v0-v)/Tu°=0, the gas-kinetic equations for the constrained MLMC-PSD become: dt
dx
dt
(10)
where [dGuj/dt]wc reflects dynamic changes caused by non-continuum processes. For both free-flowing and constrained traffic, the non-continuum processes reflect the influences of deceleration and lane changing. Let us now discuss these processes in some detail. Braking and immediate lane-changing. Given the assumption of vehicular chaos, the expected number of actively interacting free-flowing vehicles of user-class u at (x,t) on laney driving with a velocity v while having a desired velocity v° equals:
Uo 0
Transportation and Traffic Theory
32
where xF;(v)=2uxFH;(v)>0 equals the expected number of active interactions per unit time of a vehicle driving with velocity v with slower vehicles of any user-class on laney. When a free-flowing vehicle interacts, it either immediately changes lanes, or it decelerates to the (exact) velocity of the preceding vehicle, while becoming constrained (cf. Hoogendoorn and Bovy (1998c)). Consequently, the contribution of active interactions to the dynamics of £j equals:
L a^/ar]™
= -(i-^
(12)
where pj denotes the immediate lane-changing probability on lane j, When the interacting free-flowing vehicles are able to change lanes, the £,J±l on the target lane j±l increases, yielding:
where p,/^1 denotes the immediate overtaking probability to the left-lane 0+1)
an
d right-lane
(/-I) respectively. Similarly, actively interacting free-flowing vehicles on lane y'±l able to immediately change lanes to lane j cause an increase in ^,/. In Europe, overtaking regulations during non-congested traffic conditions follow the 'drive on the right - overtake on the leff principle. That is,pJ^~l=Q, when traffic is not congested. Similarly to the immediate overtaking processes of free-flowing vehicles, a constrained vehicle may be able to change to either of the adjacent lanes when it has actively interacted with a slower vehicle. An active interaction of a constrained vehicle is defined by the corresponding active interaction of the platoon-leading vehicle. Since at this stage, we assume that vehicular particles have no physical length, following (constrained) vehicles interact with a slower vehicle at the same instant and location as the leading vehicle does. As a consequence, we propose expressions similar to (12) and (13) for constrained traffic. Postponed lane changing. Constrained vehicles previously not able to change lanes may change lanes if the opportunity arises, after which the driver is able to accelerate towards his desired velocity until it is again impeded. We will model the postponed lane changes using the postponed lane-change rates X,rj±\ with Xul"°=XuM"M+l=0, and Xj= X^1 +X,Tj+l . Postponed lane changing causes a migration of constrained vehicles a,/ to free-flowing vehicles and
t^f / WPLC~' = ^r 7± V,(v, v°)
(14)
Spontaneous lane changing. Free-flowing drivers may choose to change to either of the adjacent lanes depending on the driver's preferences. This yields a flow from the current lane to an adjacent lane in correspondence to the class-specific driver's preference. Since this process is comparable to the postponed lane-changing process it yields comparable expressions. For European legislation , this spontaneous lane changing process to a large extent results from
Macroscopic Traffic Flow Modelling
33
vehicles which have overtaken slower vehicles using the left lane returning to their origin lane. In this case, spontaneous lane changes to the left lane are rare. Thus, the spontaneous lane changing intensities satisfy yJ~>i'+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]£J±l=-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 v-v)c j u (w,v°)dw
(16)
due to passive interactions with respectively free-flowing and constrained vehicles respectively (cf. Hoogendoorn and Bovy (1998c)), where pj=pj(x,v,i) and quj=quj(x,v,t) denote the immediate lane-changing probabilities of interacting free-flowing and constrained vehicles. Let us define the passive interaction rates by: def ™
def ~
^'(v,v°) = j(w-v)^(w,v°)dw and ^'(v,v°) = j(w-v)a^(w,v°)dw Thus, the total contribution of passive interactions equals: [dai/dt]^IVE=pJ(vm-pJuW(v,v°)
+ (l-qi)®i(v,v0))
(18)
Relaxation due to vanishing impeding vehicles. When the constrained vehicles on lane j are able to changes lanes, vehicles that were previously constrained by the lane-changing vehicles are 'free-flowing'. This event can in itself yield a relaxation of other vehicles. Since this process is very complex, we will assume that the process can be modeled by: [a^'/a/] REL =d( v F ; (v))a^(v,v°) and [3^/3r] REL =-d(^(v)X(v,v°)
(19)
where $ is a monotonic increasing function of the mean number of active interactions ^(v) of vehicles driving at a velocity v.
Transportation and Traffic Theory
34
The resulting gas-kinetic equations Combining the gas-kinetic equation for free-flowing traffic (8) with the specifications for the non-continuum processes derived in the previous paragraphs, yields: ^ + v ^ - + ^-(^(v 0 -v)/<) = -(l- / 7,;)T^(v)^+ 1 3(T ; (v))a^ at dx ov
-Z/=,±1(^
/XI/;
i f
M^' -p^ v M&-Zf_j±lw;"'tt -yr^' -^rx)
(20)
for all «eU andy'=l,...,M. Similarly, the gas-kinetic dynamics of o,/become:
a? -.
a* -\
r
i u / —u
V
(v))-V ,
^
/'
/
(g
\
jWx
J ;'— :4-i ^ l U
/
ur
\
/ \\
r u / i ( \ i
/
\*
" H ^
u V
'
^/
(21)
]
DERIVATION OF THE MACROSCOPIC EQUATIONS This section presents the macroscopic equations describing the dynamics of the MLMC macroscopic traffic equations. To this end, aggregation operators are applied to the gas-dynamic equations (20) and (21). These operators aggregate the contributions of free-flowing and constrained drivers to the respective conservative variables density, momentum, and energy. Hoogendoorn and Bovy (1998c) show that using these conservative variables yields a simplified derivation approach and improved numerical analysis of the resulting macroscopic models. Moreover, the model can be easily recast in its primitive form (density, expected velocity, velocity variance). Aggregation operators. Let us define the operator H on any function a(v,v°) by: 2^[a(v,v°)] = J|a(w,H' 0 )^(jc,w,w 0 ,Odw 0 dw
(22)
For instance, if we consider the momentum v of a free-flowing vehicle of class u on lane j driving at velocity v, EuJ[v] determines the total free-flowing traffic momentum of class u on lane/ Equivalently, we can define the operator Z by: Z}u [a(v, v°)] = J J a(w, w° X (x, w, w°, t)dw°dw (23) To derive the macroscopic flow equations, the gas-kinetic equations (20) and (21) are multiplied by V A , with £=0,1,2. Subsequently, the resulting equations are integrated with respect to the velocity v and the desired velocity v°. Since in effect these aggregation operators total the contributions of the vehicles driving at various velocities v to the respective conservative variable. For instance, v^(v) reflects the contribution to the traffic momentum EuJ[v] of freeflowing vehicles driving at a velocity v. In multiplying the reduced Paveri-Fontana equations vk for £=0,1, and 2, the dynamics of Ej[vk] and £,/[v*] are established. Hoogendoorn and Bovy (1998c) show that these respectively equal to traffic density (£=0), the traffic momentum or
Macroscopic Traffic Flow Modelling
35
flow (&=1), and two times the traffic energy (k=2) of class u on lane j of free-flowing and constrained vehicles respectively.
State-specific macroscopic traffic flow equations State-specific MLMC conservation-of-vehicles. We establish the conservation of free-flowing vehicles equation by assessing the aforementioned equations for £=0. Hoogendoorn and Bovy (1998c) show that this yields: deceleration after active interaction
immediate lane-changing
a* . "(v))]' *
spontaneous / postponed lane-changing
Compared to the regular conservation-of-vehicle equation, the generalized conservative-offree-flowing vehicles of class u on lane j changes both due to lane changing and transitions between constrained to unconstrained driving. Equivalently, we can determine the conservation of constrained vehicles equations. Moreover, we can show that the effect of active and passive interactions of constrained vehicles without immediate overtaking cancel each other out. That is, the constrained traffic density is not affected by interactions of constrained vehicles that cannot immediately changes to another lane. We find:
dt
dx
(25)
state-changing
postponed lane-changing
where the operator P,/ is defined by: P,/[«(v,v°)] = JJa(w,w 0 )p^(x,w,w°,Odw ( ) dw-E:^[a(v,v ( ) )] + Z^[a(v,v 0 )]
(26)
and thus: 0
def
)] =
P>(v,v 0 )]
(27)
State-specific MLMC momentum dynamics. Similar to the derivation of the vehicle conservation equations, we can establish the free-flowing momentum dynamics equation by assessing the aforementioned equations for k-\ (cf. Hoogendoorn and Bovy (1998c)). state-changing
dt
Transportation and Traffic
36
Theory
where vu denotes the mean desired velocity of class u. Equation (28) shows that among others the mean momentum of free-flowing traffic of user-class u on lane j changes due to the inflow and outflow of momentum, reflected by the spatial derivative of the traffic energy. Alternatively stated, the arithmetic mean velocity vj-mj/rj changes due to the balance of inflow and outflow of vehicles with different velocities. We can also derive the momentum dynamics equation for the constrained vehicles:
dt
dx state-transitions
immediate overtaking
postp.lane changing
Let us remark the subtle difference between the reduction in the traffic momentum due to active interactions for free-flowing and constrained vehicles. On the one hand, when a freeflowing driver interacts, he either changes lanes or joins the platoon. As a result, a density flux from the free-flowing spatial density to the constrained traffic density causes the momentum to decrease. However, this does not necessarily imply that the mean velocity of freeflowing vehicles decreases. On the other hand, when a constrained vehicle is impeded, it will reduce its velocity. This causes the mean velocity of the constrained vehicles to decrease. As a consequence, the traffic momentum of the constrained vehicles is decreased. Note that this does not necessarily imply a reduction in the spatial density of constrained vehicles. State-specific MLMC energy dynamics. Finally, the multilane free-flowing energy dynamics can be determined for k-2. These equations are similar to the dynamic equations for the unconstrained and constrained momentum presented in the previous section, and will not be explicitly presented in this paper. Their formulation can be found in Hoogendoorn and Bovy (1998c).
Mixed-state macroscopic traffic flow equations To study the differences between aggregate-lane conservative models (cf. Hoogendoorn and Bovy (1998d)), and the presented MLMC model, it is worthwhile to determine the mixedstate traffic dynamics. To this end, we define the fraction of constrained vehicles: a]u (x, t) = ruj (x,t)/ ru} (x, t)
(30)
the mixed- state immediate overtaking probability from lane j to laney'il:
and the mixed-state overtaking rates from laney to lane y'+l: A;-;±i
tf (i - «
i±l
+ <*•->j±l
(32)
Macroscopic Traffic Flow Modelling
37
MLMC conservation-of-vehicle equations. By adding the equations (24) and (25), the multiclass multilane conservation-of-vehicles equation results: ^ + ^ = -£/=;±1(^/pu'^^
(33)
By state-aggregation, both the state-changing term and the within-lane influence of vehicles interacting vanish, due to the fact that neither of these processes cause changes in the number of vehicles of class u on laney. In the sequel of this section, we will see that these processes do change the momentum and energy. Compared to the aggregate-lane multiclass model of Hoogendoorn and Bovy (1998d), additional density fluxes between the motorway lanes are present due to the different types of lane-changing. MLMC momentum dynamics. By defining the equilibrium momentum: Mju t' rujv° - T " ( 1 ~^" ) ^ s (P,/ [vV/ (v)] + P/ [vO;! (v)]) \
~
(34)
u '
we can establish the mixed-state momentum dynamics by adding equation (28) and (29): dt
• + 2-
dx
Ml -ml ^/(1-aO /p
(35) j
- ^.._.+l (rcr ,/ [v¥ (v)] - n;f ^'P/ [v*F > (v)]) - £ _ +i (A^'mj - A{-*X ) Compared to the aggregate-lane multiclass model of Hoogendoorn and Bovy (1998d), the distinction of lanes introduces momentum flows between the motorway lanes caused by the different types of lane-changing. The expression for the equilibrium momentum M,/ is equal to the expression derived by Hoogendoorn and Bovy (1998d) for aggregate-lane multiclass traffic flow. The equilibrium momentum (34) describes changes in the traffic momentum of class u on lane j due to vehicles interacting with other vehicles, without having the ability to immediately overtake to either adjacent lane. Hoogendoorn and Bovy (1998d) show that this interaction is asymmetric. That is, the presence of relatively slow vehicles, such as trucks, have a more profound impact on the momentum of faster vehicles than vice versa. Bliemer (1998) has used this quantification of the asymmetric interaction to determine multiclass travel time functions for multiclass dynamic traffic assignment. Alternatively, we can determine the equilibrium velocity V,/. This velocity equals the equilibrium momentum MJ divided by the density rj, i.e.: V> = v° - T " (1 ~ 7l " ) V (P,/ [v*F/ (v)] + P/ [v®Ju (v)]) -
(36)
MLMC energy dynamics. Using a similar approach, we can determine the mixed-state energy dynamics:
Transportation and Traffic Theory
38 dej
Fj - cj
r)
g ™>L ". +°-(mJHJ+jJ,2) = 2 0 " dt fa T°/(l-a^) - y - (ni~"'Vs[± v 2x F y (v)] -TC/'^P/R-v 2x F r (v)]) - T , A—0 =j±l
M
M
/
M
H
Z
^^ j
=j
(37) ±
(A^V - A^V") \
M
M
J
where then equilibrium energy EU of class u on lane 7' is defined by: s ^
P/ [v2T/ (v)] + P/ [v 2 (frf (v)]
(38)
M '
where Hl/=3euj/rl/-(muj/rl/)2 and 7^ respectively depict the traffic enthalpy (convective energy flux) and the flux of velocity variance (non-convective energy flux). By noticing that the traffic energy, the traffic velocity and the variance relate as follows: eJu=jruJ((vJJ2+QJu) where
9M;
(39)
denotes the velocity variance, we can establish (Hoogendoorn and Bovy (1998a)): El = \ rj (vJuVuj + QJu )
(40)
where Qj denotes the equilibrium velocity variance: u
s
(41)
In the sequel, we specify relations for the equilibrium velocity and velocity variance. Using Muj=rujVuj and (40), we can determine respectively the equilibrium momentum and energy.
MLMC-MODEL FORMULATIONS We can summarize the model equations by defining the vector \vj=(rj,mj,ej). Hoogendoorn and Bovy (1998c) show that the model equations (33), (35) and (37) can be recast as follows: dt dx dt dx where A.J is the conservative flux-Jacobian. It describes how small spatial variations in the conservative variables influence the other conservative variables over time. The vector x«; summarizes the right-hand sides of the equations (33), (35) and (37).
Using this formulation, the model can be recast into among others its primitive form, and its characteristic or Riemann form. The former describes the dynamics of the MLMC density, velocity and velocity variance and is consequently well suited for comparing the MLMC model equations with other macroscopic flow models. The characteristic form describes the dynamics of the characteristic variables. Although these variables lack intuitive appeal, they are of dominant importance when mathematically analyzing the properties of the flow equations. For instance, they reveal the way in which small perturbations are transported in the flow along the so-called characteristic curves. It can be shown (cf. Hoogendoorn (1998c)) that when the traffic conditions are free-flow, disturbances are transported downstream. In oppo-
Macroscopic Traffic Flow Modelling
39
sition, when traffic conditions are congested, perturbations are transported in both upstream and downstream directions. Figure 1 shows the relations between the various formulations and their respective uses. For a detailed account on the different model formulations, we refer to Hoogendoorn (1998c,1999). CONSERVATIVE
PRIMITIVE
RIEMANN
density r
path-line variable z
momentum m
mach-line var. z
kinetic energy e
mach-line var. z
\v=(r,m,e)
z=(z,z,z)
conservation of vehicles
characteristic equations
momentum dynamics
decoupled system describing dyn. Riemann variables
Figure 1: Different forms of traffic flow models, the relevant variables, and the applicable numerical solution methods.
THE MLMC-EQUILIBRIUM CONDITIONS In this section we will consider the MLMC equilibrium conditions for mixed-state traffic. To this end, we propose a simple procedure to determine these equilibrium conditions. The discussion focuses on both the distribution of density on the roadway lanes, and the equilibrium velocity. We will consider two user-classes, namely trucks and person-cars.
40
Transportation and Traffic
Theory
Specification of model relations Before presenting the approach to determine the equilibrium traffic conditions for the MLMC-model, the acceleration time iu, the desired velocities vu°, the fraction of constrained vehicles O.J, the immediate lane-changing probabilities nuj, and the lane changing rates A«; are specified. In the scope of this preliminary study, we have neglected the role of the flux of velocity variance, i.e. jJ=Q. Desired velocities. For the desired velocities of person-cars and trucks, the following values have been respectively chosen: v^rson.car =32m/s
vt°ruck = 24m / s
and
(2)
These values agree with average values observed on two-lane motorways in the Netherlands. Note that in the Netherlands, the distinct speed limits on motorways for person-cars and trucks are 32m/s and 22m/s respectively. The acceleration times. The acceleration time reflects the average acceleration capabilities of vehicles of a specific user-class. Since person-cars generally have better acceleration capabilities than trucks, we assume Tperson-car
^
T
truck = ^
(3)
Immediate overtaking probabilities. To specify the immediate overtaking probabilities, we consider the distribution of gaps on the destination lane. Let G7 be a random variate, describing the gap on lane j. We define the gap by the distance between the rear bumper of the leading vehicles and the front bumper of the following vehicle. The mean gap E(GO can be expressed in terms of the lane density r* and the mean vehicle length ll on the lane:
E(Gi) = \lr1 -LJ
(4)
Z/=£v(r/Lv)/r'
(5)
where:
We have assumed that the available gaps can be modeled by a log-normal distribution. Let sj denote the space needed by a vehicle of class u driving on lane j on either of the destination lanes j±\. The space needed is expressed as a function of the mean vehicle length Lu, the average velocity vj and the reaction time Tj. For a vehicle of class u driving with velocity v we assume: ^(v) = 2.5(L u +7»
(6)
Let us assume that the velocities of user-class u on lane j are Gatm/an-distributed random variates with mean vuj and variance Quj. The random variate SUJ describing the distribution of space needed by vehicles of class u on lane j is also Gaussian with mean 2.5(LM+7Yvu) and
Macroscopic Traffic Flaw Modelling
41
standard deviation (2.5Tltr)2Quj. The probability that a vehicle of class u on lane j can change to either of the adjacent lanes equals: X^1 = x^>±! ( V j, QJu, r ;±1 ) = Pr(S/ < GJ±1)
(7)
We assume that the immediate lane-changing probabilities for constrained vehicles are negligible, i.e. qj=0. That is, the probability that an overtaking opportunity occurs precisely when a constrained vehicle actively interacts equals zero. Thus nj equals: ±l
n^J
= Tt^1 ( V ;, 9;, a', r&) = (i - a;jp^'xf"'11
(8)
11
where (V^ models both the preference for either of the adjacent lane, and whether the driver may use that lane for overtaking. For instance, for non-congested traffic operations in Europe we have (3,T;+1=1 and P/^O. Spontaneous and postponed lane changing rates. The spontaneous lane changing rates are also specified by considering the gap distribution on the destination lane. Considering European traffic regulations, traffic must use the rightmost lane if possible. Thus, we assume: Yr ; ' + '=0 and Yf^' = Xf"'"' / T °
(9)
where l/Tu° is the free-flow spontaneous lane changing rate. Tu° can be considered to be the mean time needed for an overtaking maneuver. Also postponed lane changing is a function of the available gap distribution on the destination lanes. We propose:
K^j±l =$Ju-*j±1ti~*j±l/w?
(10)
where Wu° denotes the mean time waiting behind the leading vehicle given free-flow conditions. Velocity variance and fraction of constrained vehicles. We assume that both the velocity variance and the constrained vehicle fraction can be adequately expressed as functions of the mean number of vehicles per unit unoccupied lane-space ^ r =l/E(G / ) (see Figure 2).
Transportation and Traffic
42
Theory
EQUILIBRIUM VELOCITY VARIANCE
0 0.00
CONSTRAINED VEHICLE FRACTION
0.02 0.04 0.06 0.08 effective density [veil / (in lane)]
0.00
0.02 0.04 0.06 0.08 effective density [veh I (in lane)}
Figure 2: Equilibrium velocity variance and constrained vehicle fraction as functions of the mean number of vehicles per unit unoccupied space on a lane.
Determination of the equilibrium conditions We have assumed that the velocities are Gaussian distributed random variates, and can thus be specified by the mean velocity vj and the velocity variance Qj. The assumption of Gaussian distributed velocities, and the specifications of the constrained vehicle fraction, and the immediate overtaking probability enables determining the equilibrium velocity (36) given rj, vj, and QUJ, i.e.:
Vuj=Vus(r,\,Q)
(11)
where r, v, and 0 are vectors of respectively the densities, velocities, and velocity variance of each class on each lane. We define equilibrium traffic conditions by: 1. Equilibrium of inflow and outflow for each of the lanes. That is, the number of vehicles leaving lane j equals the number of vehicle arriving at lane j due to lane changing. 2. The velocities and velocity variances equal the equilibrium velocities and variances. Considering the MLMC conservation of vehicle equation (33), condition 1 yields: 7i/P y 'pF ; (v)]-tiR' = V , U
It
U
U
^^ j =j±\
(Ti^P/L^v)]-A{^'/?j') "
U
(12) ^~
'
where RJ denotes the equilibrium density distribution. Condition 2 yields: y/=\//(R,V,0)
0.02 0.04 0.06 effective density [veh I (in lane)]
Figure 3: The equilibrium velocity on the right-lane and the left-lane of a two-lane motorway, for constant truck densities, and different person-car densities.
0.00
0.02 0.04 0.06 0.08 effective density [veil I (m lane)}
Figure 4: The equilibrium fraction of person-cars and trucks using the left lane for fixed truck density values and increasing person-car densities. Given the aggregate-lane densities ru for the respective user-classes, we can iteratively determine the equilibrium lane distribution, velocity, and velocity variance satisfying (12) and (13). In illustration, Figure 3 and 4 show the equilibrium velocities for fixed truck densities (^truck=0,2 , and 10 truckslkmllane).
44
Transportation and Traffic Theory
NUMERICAL SOLUTION APPROACHES In the past, several numerical approximation schemes have been proposed to determine solutions to a variety of macroscopic models (see Lyritnzis et al. (1994), Lebaque (1996), and Hoogendoorn and Bovy (1998a)). Because of the increased complexity of the developed higher-order traffic flow model there is a need for more efficient numerical approaches to approximate solutions. As with other traffic flow models, the numerical treatment of our multilane multiclass flow model is quite cumbersome. Hoogendoorn (1999) describes a new approach to solve higherorder multilane flow models, based on the flow equations cast in conservative variables. The resulting scheme is an adaptation of the Van Leer Flux-Vector splitting scheme (cf. Van Leer (1982)). It considers the direction in which the perturbations are transported in the MLMC traffic flow, while conserving the density, momentum, and energy in the distinguished roadway segments that follow if only convective processes are considered. The scheme adapts to the prevailing traffic conditions (free-flow/congested). Moreover, the effects of the non-continuum processes are quantified in a multistep approach, in order to prevent vehicles laterally flowing out of an empty lane. A fourth-order Runge-Kutta approach was chosen for the temporal discretization. For details, we refer to Hoogendoorn (1999).
APPLICATION OF THE MLMC-MODEL In this section we discuss results of macroscopic simulation of two test-case examples, based on the specifications proposed in the preceding section. These are: mixing traffic classes, and a lane drop.
Mixing of classes In the first example, we consider a two-lane ringroad of 20km length. At time t=0, three homogeneous regions are present. The first region Xo°=[2km,6km) consists of person-cars only. The second region X0l=[6km,8km) consists of both person-cars and trucks, while the third Xo2=[Skm,l2km) consists of trucks only (Figure 5). We assume that initially, traffic is in equilibrium. That is, the initial velocities, velocity variances, and the distribution of vehicles on the lanes are determined by applying the approach described in the previous section (Figure 6).
When we consider the dynamics of the vehicles in the head of the region Xo2, trucks flow into the empty downstream roadway section at a velocity nearly equal to the desired velocity. For instance, considering instant t=4min, the fastest trucks are located at approximately x=iS.5km. These vehicles have traveled at a velocity of approximately 95km/hr. By studying the char-
Macroscopic Traffic Flow Modelling
45
acteristics of the flow model, we can show that these vehicles have traveled along the socalled Mach-line of the flow, defined by djc/d/=(v1/+(36,/)1/2) (cf. Hoogendoorn and Bovy (1998d)). trucks
x=8km
person-cars
x=2km x=0km Figure 5: Schematics of two-lane ringroad, and initial distribution of person-cars and trucks. Compared to the trucks-only region XQ where trucks use both lanes of the two-lane roadway comparably, in the mixed region XQI the lane-use of trucks is more confined to the right-lane. Fast person-cars flowing from this region into the truck-only region do not affect the velocities of the trucks (e.g. t=4min, x=[\lkm,\lkm)). However, the truck-density lane-distribution is affected, in that trucks only use the right roadway lane. In the upstream region, where person-car densities are higher, the velocities of person-cars are reduced to such an extent, that trucks actively interact with slow person-cars, and consequently need to reduce their velocity. A jam forms between the person-car only region XQ° and the mixed region XQ{ . This area appears due to fast person-cars from the upstream region XQ° flowing into the low-velocity mixed region XQI. These fast person-cars interact with trucks and slow person-cars, and consequently need to slow down. This results in both a decreased velocity and an increased density. The increase in the density results in even more interactions with slow vehicles. This avalanche-like process causes the formation of congestion in the transitional area between the two regions. This phenomenon is comparable to the formation of localized structures leading to the development of phantom-jams (cf. Kerner et al. (1996)). Let us finally consider the tails of the regions XQ° and XQI. Clearly, none of the vehicles flow back into the lower-density regions upstream of these tails. We may therefore conclude that the model satisfies the so-called anisotropy condition (cf. Daganzo (1995)). That is, vehicles mainly react to stimuli in front.
Figure 6: Traffic conditions on a two-lane ringroad for 'mixing of traffic' example.
The lane drop The second test case describes the traffic conditions on the 20km two-lane ringroad at a lane drop. The lane drop is located at X=[lkm,\3km). We will consider a left-lane drop (Figure 6). Upstream of the lane drop, both person-cars and trucks are present. We assume that at ?=0, traffic conditions are in equilibrium (see Figure 8). The lane-drop is modeled by placing motionless virtual vehicles on the left lane. The presence of these virtual vehicles causes an increased number of active interactions, yielding on the one hand an increase in the number of lane-changes to the left lane, and on the other hand, an increase in the density on the left lane.
Macroscopic Traffic Flow Modelling
47
x=\Qkm
x=0km Figure 7: Schematics of left-lane drop on a unidirectional two-lane ringroad. From the simulation results we see that initially, person-cars and trucks flow into the left lane. However, they change to the left lane quickly, resulting in an empty lane. Some of the personcars change to the left roadway lane when arriving at the end of the lane drop at x-\?>km. 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 OUTLOOK This 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
timestamp t = 1 min densities velocities
=5
2 solane drop
person-cars right-lane person-cars left-lane trucks right-lane trucks left-lane timestamp t = 10 min
timestamp t = 4 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.
REFERENCES Bliemer, 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.
51
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.
ABSTRACT Attempts 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.
INTRODUCTION Traffic 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).
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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 TRAFFIC In 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 as 9 = 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 dx The 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 Kiihne and Beckschulte (1993), dv
dv
1 ,Tr, .
,
oldc
d2v
a+'S-^M-O-^-s-i-si.
(2)
Here v — q/c is the mean speed, V(c] = Q(c)/c is the mean speed at concentration c according to some associated static traffic stream model, and r, CQ, v are coefficients that presumably are to be provided from observations. Such higher-order models have been rather widely studied in recent years, especially in the physics literature. An exhaustive list of references, and an excellent systematic overview of this subject (including references to workers who have used somewhat different forms of a dynamic traffic stream model), appear in recent works by Helbing (1995a, 1995b, 1996a). The precise relationship between LWR and higher-order models has been the subject of considerable discussion in the literature (e.g., Kiihne and Beckschulte, 1993; del Castillo, Pintado and 3
The terminology used here is that of Gerlough and Huber (1974), and of May (1990).
The Chapman-Enskog Expansion
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Benitez, 1993; Daganzo, 1995b; Helbing, 1996a). The view of the present work is that, in regard to either of these types (i.e., LWR or higher-order) of continuum models of traffic flow, there are three fundamental issues that arise: Validity: What is the basis, either empirical or theoretical, for the associated (static or dynamic) traffic stream model? Mathematical solvability: What additional conditions (boundary, initial, diverge) are necessary to provide a solution of the traffic flow model that represents, to some reasonable degree of approximation, what one actually observes in a real traffic network? Computational solution: How does one effectively obtain these "real" solutions on a modem high-speed digital computer? A brief overview of the authors' view of the current and historical understanding of each of these issues will be presented in, respectively, the following three subsections.
VALIDITY In the traffic-flow literature it usually is indicated that static traffic stream models should be determined from observations. However, as already mentioned in the Introduction, it is well-known that observations tend to show considerable scatter in observed flows (or speeds), especially in the region of unstable flow. This phenomenon has led some workers (e.g., Ross (1988), p. 422; Kiihne and Beckschulte (1993), p. 367) to doubt existence of traffic stream models, in the sense of a relationship that expresses mean speed (or flow) as a single-valued function of concentration. Others, similarly motivated, have suggested a variety of alternatives to traditional continuum theories (Ceder (1976), Hall (1987), Disbro and Frame (1989)), although none of these as yet seems to have found significant application to the quantitative modeling of traffic flow. Higher-order models seem primarily to be validated on the basis of adjusting the variety of parameters that they contain (e.g., r, CQ, v) to obtain a solution that provides a reasonable fit to observations. The authors are unaware of any efforts to validate dynamic traffic stream models themselves by direct comparison with observations. Any such program would appear to be rather difficult to effect operationally, because (e.g.) each of the four derivatives appearing in Eq. (2) (especially the second derivative) would be rather sensitive to uncertainty in measurements, and then these errors could be subject to even greater magnification from combining them algebraically as in Eq. (2). Additionally, most dynamic traffic stream models invoke an underlying static traffic stream model, and it is unclear how such a dynamic model can be valid without some degree of validity attaching to the associated static model. (The semiviscous and viscous models of Lyrintzis, Liu and Michalopoulos (1994) are exceptions. These dynamic traffic stream
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models were developed on the basis of consideration of a number of car-following models, and seem to give rather good results.) There seems to have been relatively little work taking the form of a comparison of both LWR and higher-order models against some independent standard (e.g., field data). It appears likely that any early efforts (e.g., prior to the 1971 paper of Payne) to compare LWR models against field data were significantly hampered by the primitive state, at that time, of the development of numerical methods known to approximate the entropy solution that is thought (Ansorge, 1990) to be relevant to traffic flow (see the following subsection on computation for more details). It is a hypothesis that is consistent with the authors' reading of the literature that such efforts were not undertaken in the U.S. soon after 1971, because of an "official" commitment to higher-order models, notably as implemented in the well-known FREFLO code (Payne, 1979). The work of Michalopoulos and co-workers is an exception to the apparent general lack of an effort to provide an objective comparison of LWR and higher-order models. This work is particularly noteworthy, because it seems to represent one of the first attempts to use, for LWR models, modern numerical methods that are known to approximate the entropy solution. Michalopoulos, Beskos and Lin (1984, p. 418) concluded, in a comparison against field data of LWR ("simple continuum") models as solved by three distinct numerical methods, that "despite the absence of a momentum equation, the close agreement of the numerical results with field data . . . is notable." Michalopoulos and Beskos (1984) carried out a comparison of LWR and higher-order models, as solved using a variety of numerical methods, against detailed microscopic simulations, and concluded that "existing high order models are only slightly better than the simple continuum model (i.e., LWR) at uncongested flows; however at congested flow conditions the simple continuum model performs better ..." (p. 103). (More recent results by Lyrintzis, Liu and Michalopoulos (1994) seem more favorable to high-order models.) Daganzo (1995b, p. 284) has noted, in regard to high-order models, that "although reasonable fits to specific data-sets can be achieved with adjusted models that include many parameters ... these limited successes do not constitute a validation," and that "given enough degrees of freedom ... a good fit should be expected."
MATHEMATICAL SOLVABILITY Mathematical solvability is best known (and understood) in the context of LWR models. The latter are a class of nonlinear conservation laws that can have multiple solutions (even infinitely many) that satisfy given initial conditions (cf. Ansorge, 1990; LeVeque, 1990). This nonuniqueness phenomenon requires imposition of some additional condition to select the solution that represents actual traffic flow. This is typically done by selecting the solution that satisfies the so-called "entropy condition." But in fact this condition originates from application of the second law of thermodynamics in fluid flow (Ansorge, 1990), and it is unclear why one should expect such a condition to hold sway in traffic theory (the valiant effort of Ansorge (1990) toward an
The Chapman-Enskog Expansion
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interpretation notwithstanding). Lebacque (1997) has recently noted that there are conditions under which the entropy solutions imply a clearly nonphysical infinite acceleration, and suggested that there are circumstances (e.g., entrance ramps) in which this is perhaps so badly wrong that some other criterion should be used. Newell (1993) suggested the geometrically motivated criterion of selecting as the solution of interest the lower envelope of all solutions, when the LWR model is reformulated in terms of A(x, t), the cumulative flow past x at time t, as the dependent variable. A version of this criterion that is suitable for digital computation does not seem to have been developed, nor does its precise relationship to the entropy condition seem to have been clarified. Boundary and intersection conditions for LWR models have been discussed by Lebacque (1996). Uniqueness issues do not seem to have been systematically studied in conjunction with higherorder models. Lebacque (1995) notes that Schochet (1988) has shown: i) Existence and uniqueness of solutions to Payne's (1971) higher-order model, which corresponds to (2) with v = 0. ii) The solutions of Payne's model converge, as T —> 0, to the solution of the continuity equation with an added diffusion-like term, iii) The solution of this "continuity plus diffusion" equation converges, as i>0 —> 0, to the entropy solution of the LWR model. Thus this "continuity plus diffusion" model plays the same role in justifying the entropy condition for the LWR model as the Navier-Stokes equations play for the Euler equations in fluid dynamics. This is rather strongly suggestive that the "correct" higher-order generalization of the LWR model is this "continuity plus diffusion" model, or something very similar. The firstorder form of the continuum approximation corresponding to the Chapman-Enskog asymptotic expansion has exactly that form, as described in the following section on The Chapman-Enskog Expansion.
COMPUTATIONAL SOLUTION Numerous discrete approximations, dating back to the 1960's, have been developed (see LeVeque (1990); Ansorge (1990); Leo and Pretty (1992); and extensive references cited in these works) that converge, in the fine-mesh limit, to the entropy solution of the Euler equations of fluid dynamics. Although the Euler equations are the fluid-flow counterpart of the LWR equation of traffic flow, the first applications of such entropy methods to the LWR equations of fluid flow seem to have come somewhat later (Lebacque, 1984; Michalopoulos, Beskos and Lin, 1984).
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Other workers (Bui, Nelson and Narasimhan, 1992; Leo and Pretty, 1992; Daganzo, 1995a) have subsequently developed and applied such methods within the theory of traffic flow. There is some evidence that this apparent early lack of awareness of the necessity to use, for LWR models, discrete approximations that converge to the entropy solution (and of the existence of such approximations) provided some of the disbelief in LWR models that led to the original motivation for the development of higher-order models. For example, as recently as ten years past (Ross, 1988, p. 422) the LWR model was criticized in terms that make it clear the author was thinking of a discrete approximation that did not predict maximal flow between an upstream region at jam density and a downstream region devoid of traffic, and therefore could not correctly capture the "acceleration wave" that is the entropy solution of this problem (and the traffic flow equivalent of a rarefaction wave in fluid flow). (See also Newell, 1989; Ross, 1989; Nelson, 1995b.) Similarly, the seminal work on higher-order models (Payne, 1971, Eq. (7.10)) approximates the flow between two cells as the average of the two densities times the average of the two velocities. This discrete approximation shares with the upstream method the property that it incorrectly predicts zero flow between an upstream region at jam density and an a downstream region devoid of traffic, and therefore it cannot converge to the desired entropy solution of an LWR model. The discrete approximation apparently employed in the original version of FREFLO (Payne, 1979) is the upwind method, which has this same defect. Because of this, one would expect FREFLO to exhibit some characteristics of poor performance (e.g., slow convergence with mesh refinement) under conditions such that this model approximates the LWR model (see the above discussion of the work of Schochet, 1988). It is a reasonable suspicion that this contributes to the deficiencies of FREFLO and related codes that have been noted by numerous workers (Rathi, Lieberman and Yedlin, 1987; Ross, 1988; Leo and Pretty, 1992; Michalopoulos, Yi and Lyrintzis, 1993; Lyrintzis, Liu and Michalopoulos, 1994; and other works cited in these).
ALTERNATIVE CONTINUUM FORMULATIONS On the one hand it appears almost certain that there exist traffic phenomena (e.g., the "spontaneous traffic jams" of Kerner, Konhauser and Schilke, 1995) that are unlikely to be describable by LWR models. Therefore, one suspects it should be possible to develop macroscopic (hydrodynamic) models of traffic flow that improve upon the LWR model. On the other hand, higher-order models of the type that have been introduced to date provide, at best, marginal improvements on LWR models. (Even the explanations given by Kerner, Konhauser and Schilke (1995) for spontaneous traffic jams seem more based on ad hoc physical models of traffic flow than on the specific mathematical higher-order model that is nominally introduced to provide an explanatory framework.) This raises the question of what alternative type of continuum model might be developed in an effort to improve on LWR models. The overall objective of the present work is to describe initial results for an alternative approach that seems to lead to models more nearly consistent
The Chapman-Enskog Expansion
59
with the improvements suggested in the seminal work of Lighthill and Whitham (1955, pp. 344) than are the higher-order models that employ a dynamic traffic stream model. The motivation for the idea that there should be some alternative treatment arises from the observation that the often repeated analogy between current higher-order models of traffic flow and the Navier-Stokes equations of fluid dynamics is questionable, as follows. The compressible Euler equations of fluid dynamics are a system of five equations in five unknowns, with each equation representing conservation of some quantity (particle number, three components of momentum, and energy) that is conserved in molecular interactions, and each of the five unknowns representing a macroscopic analog (density, fluid velocity, temperature) of one of these microscopically conserved quantities. The LWR model of traffic flow likewise is a single equation, in a single unknown, that represents conservation of the sole quantity (number of vehicles) that is conserved in vehicular interactions, and the single unknown (concentration, or density) is a macroscopic analog of that microscopically conserved quantity. Thus, it seems reasonable to regard LWR models as the traffic flow analog of the compressible Euler equations of fluid dynamics. But the Navier-Stokes equations of fluid dynamics likewise consist of five conservation laws in these same five unknowns as the compressible Euler equations. These are merely of a different form from the Euler equations, in that viscosity and diffusion are now represented, but by terms expressed in the same five dependent variables (and their spatial or temporal derivatives). On the other hand, as described in the introduction, current higher-order models of traffic flow introduce a "dynamic traffic stream model" that has the form of an additional conservation law, and an additional independent variable (e.g., mean speed) that is not a macroscopic analog of some quantity that is microscopically conserved in vehicular interactions. On this basis alone it seems not quite appropriate to regard current higher-order models as traffic-flow analogs of the Navier-Stokes equations.4 Rather they seem to stem from following the/orm of the development of the Navier-Stokes equations of fluid dynamics, as opposed to following the spirit of such development, with due regard for the differences between fluids and traffic streams.5 If one accepts this contention that current higher-order macroscopic models of traffic flow are not the proper analogs of the Navier-Stokes equations of fluid flow, then it follows that the true analog remains to be discovered. How should it be found? The answer proposed here is a development following the lines of the Chapman-Enskog asymptotic expansion of the Boltzmann equation and its solutions that has been shown (e.g., Chapman and Cowling, 1952; Cercignani, 4
In itself this says nothing about the validity or invalidity of such higher order models. In fact it seems quite reasonable to consider them as analogs of the Thirteen Moment method of Grad (1949), which is a quite respectable, although seldom used, description of fluid dynamics. 5 Kerner and Konhauser (1993) even simply directly import the Navier-Stokes equations from fluid flow into traffic flow. Daganzo (1995b) has emphasized the differences between traffic and ordinary fluids, and warned of the dangers of pushing too far the interesting, and sometimes useful, analogy between the two.
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1988; Liboff, 1990) to lead to a hierarchy of macroscopic equations for rarefied gases. The (compressible) Euler equations are the lowest-order (zero-order) member of this hierarchy, the Navier-Stokes equations are the first-order approximation, and the seldom-used Burnett equations are the second-order approximation. The proposed approach thus follows the spirit of the standard theoretical development of macroscopic fluid dynamic equations, as opposed to simply adopting the form of the fluid-dynamic result of that development. In particular, the specific characteristics of vehicular flow, as opposed to fluid flow, presumably will be represented in such an approach, to the extent that these characteristics are incorporated in the underlying kinetic equation that forms the starting point for the development. The foundation of any such development is a suitable kinetic equation for vehicular traffic. Such kinetic equations are the subject of the next section.
KINETIC EQUATIONS FOR VEHICULAR TRAFFIC A kinetic equation is formed by setting the rate of change of the vehicular distribution in velocity and position space equal to the rate of change caused by the changes of vehicle speeds due to vehicular interactions (i.e., slowing-down, speeding up, and passing) according to some (microscopic) mechanical model of driver responses to various situations. An equilibrium solution of such a kinetic equation is a distribution function such that the latter rate of change is identically zero. The equilibrium solutions of a kinetic equation are crucial elements of the connection of that kinetic equation to a continuum model of traffic flow. Indeed, the equilibrium solutions themselves normally lead directly to a traffic stream model. Further, the ChapmanEnskog asymptotic expansion is a formal expansion of the solution of the kinetic equation about an arbitrary equilibrium solution. For these reasons, in order to develop the Chapman-Enskog continuum approximations corresponding to a particular kinetic equation it is essential to have a good mathematical characterization of the equilibrium solutions of that kinetic equation. In this work the venerable kinetic equation of Prigogine and Herman (1971) will be employed. One certainly can raise legitimate questions about the degree of validity of some aspects of the Prigogine-Herman kinetic equation, particularly the somewhat phenomenological "relaxation term." As a result, other kinetic equations for vehicular traffic have been suggested; these are summarized briefly in the concluding subsection of the present section, especially as regards the state of knowledge of their equilibrium solutions and their prior use in developing continuum models. In the first subsection the equilibrium solutions of the Prigogine-Herman kinetic equation are described. This description will be primarily for the modified Prigogine-Herman kinetic model recently considered by Nelson and Sopasakis (1998). Their equilibrium solution contains, as a special case, that originally given by Prigogine, Herman and Anderson (1962).
The Chapman-Enskog Expansion
61
EQUILIBRIUM SOLUTIONS OF THE PRIGOGINE-HERMAN KINETIC MODEL The Prigogine-Herman (1971) kinetic equation of vehicular traffic is
Here / is the density function for the distribution of vehicles in phase space, so that f ( x , v, t) dx dv is the expected number of vehicles at time t that have position between x and x + dx and speed between v and v + dv, c — c(x, t) is the spatial density of vehicles (vehicles per unit length), /0 is the corresponding distribution function for the desired speed of vehicles, v is mean speed, and P, T are respectively the relaxation time and the passing probability. The equilibrium solutions of the P-H model are given by setting the right-hand side of (3) equal to zero, and solving for / = feq. Suppose all desired speeds lie between some positive lower limit w_ and some upper limit w+. Then there is some critical density c^t,6 defined as the root of 1 such that the equilibrium solutions of the Prigogine-Herman model are as follows: Stable flow: For c < c^n,
with C = C(c) defined implicitly by JJ_+ ^iM dv = Tc2(l- P}. Unstable flow: For c > c^n, /„ = /„(«;<:, a) :=
L_-A_ +ao«(« - C).
(5)
Here 8 is the delta function of Dirac, and a is not determined as a function of c, but rather merely required only to satisfy a min (c) = max{0, 1 - re2( ^_ p) /J_+ £^ dv} < a < 1
~ rc3(i-p) fw- ^jr^-dv = amax(c). Further, £ = £(c, a) is defined implicitly, as a function of both a and c, byTc2(}_P) /J_+ ^^ dv = 1 - a. This means that, in the regime of unstable flow, the mean speed (or flow) is a function of two independent parameters. In the above description these parameters were taken as density and the fraction (a) of the traffic in the collective ("highly platooned") mode. It is equally possible to take them as the density and the speed (Q of the collective flow. Notice that c^u is infinite, and therefore the collective flow does not exist, unless /0 is such that / *°^ dv is finite. 6 The critical density is to be interpreted as the smallest density at which equilibrium flow involves some vehicles travelling slower than any driver wishes. This differs from the definition of the 1985 Highway Capacity Manual (Transportation Research Board, 1985, pp. 1-6) as the density at which maximum (i.e., capacity) flow is reached. However, the values of the two are quite close, under reasonable assumptions on /Q.
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In any case a (classical) traffic stream model is obtained in the stable regime, as q(c) = I
vfeq(v;c)dv.
Prigogine, Herman and Anderson (1962) (see also Chap. 4 of Prigogine and Herman, 1971) assumed that w_ = 0. In that case it turns out that a mai (c) = a m in(c) = a(c), so that again a (classical) traffic stream model is obtained, as q(c) — f v f e q ( v ; c, a(c)) dv. In this case £(c, a) = 0, so that the critical density is the smallest density at which some traffic is stopped, at equilibrium flow, and the component (or mode) a6(v) represents a "collective flow" in which some fraction (a) of the vehicles are stopped.7 Prigogine and Herman (1971, Chap. 4) provide graphical presentations of a number of such classical traffic stream models, both with and without collective flows. But, the fact that equilibrium solutions lead to a traffic stream model is closely associated with the fact that the equilibrium solutions are a one-parameter family, and that it is possible to take the density as that parameter. This is the normal expectation. That expectation has its roots in the Boltzmann equation of the kinetic theory of gases, where it is typically found that the parameters required to characterize an equilibrium solution are macroscopic counterparts of the microscopic invariants of molecular interactions, so that the number of parameters is the same as the number of invariants.8 As there is only one invariant in vehicular interactions, the number of vehicles, one therefore expects the equilibrium solutions of a kinetic equation for vehicular traffic to comprise a one-parameter family. That expectation is realized for stable flow, and even for unstable flow, provided one makes the somewhat unrealistic assumption that there exist drivers desiring arbitrarily small speeds. It is therefore perhaps surprising9 to find that this expectation is not met, in the general case of unstable flow. To emphasize the radical distinction between the stable and unstable flow regimes, note that in the unstable flow regime the classical static traffic stream model generally becomes a surface in three-dimensional c/a/v (or c/C/v) space, rather than a two-dimensional curve. (See Nelson and Sopasakis (1998) for figures illustrating this phenomenon.) The projection of this surface on two-dimensional c/v space is, of course, a region rather than a curve. This is qualitatively similar to what is observed in attempted measurements of traffic stream models. 7
Thus, in the unstable flow regime, a is very similar to the "percent time delay" that is used as a primary level
of service indicator for two-lane highways in the 1985 Highway Capacity Manual (Transportation Research Board, 1985). This quantity is defined as "the average percent of time that all vehicles are delayed while traveling in platoons due to the inability to pass." 8 The equilibrium solutions of the Boltzmann equation depend upon five parameters, one for each of the five invariants of molecular interactions. Specifically, these invariants are mass, three components of momentum, and energy. The corresponding parameters can be taken as density, three components of the mean speed, and temperature. These are the parameters that normally are taken as characterizing the well-known Maxwellian distributions, which are the equilibrium solutions of the Boltzmann equations, for some intermolecular force laws. 9 And emphasizes the danger of pushing fluid flow analogies too far in traffic flow.
The Chapman-Enskog Expansion
63
The continuum models to be developed in the following section will be based upon the equilibrium solutions described above. However, in order to obtain a LWR model as the zero-order approximation (i.e., to obtain a static traffic stream model from the equilibrium solution) it appears to be necessary to restrict these developments to the stable-flow regime (i.e., densities below critical). That restriction will be followed in the present work. The development of suitable continuum models for the unstable-flow regime is presently an open problem. Note that the nature of the equilibrium solution in the region of unstable flow appears to cast some doubt on the validity of LWR models per se, in this region.
OTHER APPROACHES TO DERIVING HYDRODYNAMIC MODELS FROM MICROSCOPIC MODELS Approaches to obtaining LWR models from the Prigogine-Herman kinetic theory of vehicular travel have been discussed above, along with a theoretical basis for the limit of their range of validity. The objective of this subsection is to briefly summarize work on the development of continuum models (LWR or higher-order) from other microscopic models. The central focus will be on recent work based on kinetic models other than that of Prigogine and Herman (1971). Somewhat different approaches to this subject seem to have been initiated independently by Nelson (1995a) and by Helbing (1995a). Nelson (1995a) developed a kinetic equation in which the phenomenological relaxation term in the Prigogine-Herman equation is replaced by a more fundamental representation of speeding-up (and passing is neglected). He and co-workers (Bui, Nelson and Sopasakis, 1996; Nelson, Bui and Sopasakis, 1997) subsequently showed that the equilibrium solutions of this kinetic equation lead to a traffic stream model that provides a fit to data that is as close as that of any of several classical such models, but with use of only parameters taken directly from a microscopic model of driver behavior (i.e., no "free" parameters available to provide the best fit to observations). Note that the equilibrium solutions found from this kinetic equation do not display collective flow at nonzero speeds, and thus the newer results of Nelson and Sopasakis as described above, along with the empirical scatter of flow under unstable conditions, bring into doubt the applicability in the unstable regime of LWR models based upon this traffic stream model. On the other hand, Helbing (1995a, 1995b, 1996a, 1996b) has based his approach upon the Paveri-Fontana (1975) extension of the Prigogine-Herman (1971) kinetic equation. This extension includes the desired speed as an additional independent variable, on the same footing with the desired speed, as opposed to being a parameter as in the original Prigogine-Herman model. (Daganzo (1995b) has emphasized the necessity of this, in order that the desired speed be a property of vehicles and drivers, not of roads.) Helbing (1996a) has developed a number of hydrodynamic models, by proceeding in the manner that frequently is used in fluid flow; i.e., by taking
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low-order polynomial moments of the underlying (Paveri-Fontana) kinetic equation, then using ad hoc techniques to close these (i.e., reduce them to determined systems). Note also that Helbing (1995a, pp. 383-384) was unable to obtain the equilibrium solutions of the Paveri-Fontana kinetic equation, but rather assumed them to be Gaussian (Maxwellian) distributions, based upon empirical data that was specifically indicated as being for stable traffic. This certainly brings into question the validity for unstable flow of the continuum equations thus obtained. Wegener and Klar (1996) built upon the approach of Nelson (1995 a) to obtain a more realistic kinetic model that uses first principles to incorporate speeding-up. They studied numerically the equilibrium solutions, and associated traffic stream models, of their class of kinetic equations, but no evidence of a collective flow seems to have been uncovered. These workers (Klar and Wegener, 1997) also developed an "Enskog-like" kinetic model that differs from all previous kinetic models of vehicular traffic in that the effect of nonzero vehicle length is taken into account in the slowing-down and speeding up of vehicles. They derived from this approach a set of macroscopic equations formally similar to the higher-order model of Payne (1979), but with coefficients computable from microscopic considerations, and that clearly showed the importance of including nonzero vehicle lengths. They also obtained computational comparisons between the solution of the kinetic (Enskog) equation and their hydrodynamic model, for a rather realistic lane-drop problem. It is not yet clear how suitable their particular kinetic equation might be for unstable flow (e.g., whether it exhibits collective flow), but it seems likely that ultimately the effect of nonzero vehicle length must be incorporated into any realistic treatment of traffic flow. Finally, Klar and Wagner (to appear-a, to appear-b) are currently pursuing very interesting studies of multilane kinetic models. For completeness, two further lines of development should be mentioned. First, Phillips (1977, 1979, 1981) also has developed continuum models on the basis of an underlying kinetic equation. However, the underlying equilibrium solution does not contain a collective-flow mode, so that it appears unlikely this continuum model will be suitable for modelling unstable flow. Second, there is an extensive older literature on obtaining traffic stream models from car-following models. Most of this is based on the assumption of steady state conditions (constant headway; see Nelson, 1995b), and thus seems unlikely to be helpful in providing insight into the collective dynamics underlying unstable traffic flow. However, a paper of Newell (1962) that is perhaps less well-known than deserved is specifically directed toward adapting car-following models to unstable flow. Before turning to the development of continuum models via asymptotic expansion of the PrigogineHerman kinetic equation, it is appropriate to note a significant difference between this approach and that in the kinetically based developments of continuum descriptions of traffic flow in some of the references cited above. In these previous works the approach is to take the first two polynomial moments of the kinetic equation, and then introduce ad hoc approximations as necessary
The Chapman-Enskog Expansion
65
to close the system (i.e., reduce to two the number of unknowns = dependent variables). By contrast, in the approach via asymptotic expansions the number and nature of the moments of the kinetic equation that are introduced at any order are determined automatically by the requirement that the next higher-order approximation exist. Further, the number and nature of these moments usually are very closely related to the number and nature of the quantities conserved in the interactions described by the particular kinetic equation (invariants). For vehicular kinetic equations one expects only one quantity (the number of vehicles) to be conserved, and therefore the associated continuum models to involve only one moment (as opposed to the two moments of current higher-order models) in any corresponding continuum equation. In this respect a kinetic model of vehicular traffic is perhaps closer to the Lorentz model of the Boltzmann Equation, for which number of particles is also the sole conserved quantity, than to the Bojtzmann equation of the kinetic theory of gases. For this model, it has been shown analytically (Hauge, 1969) that the nth-order Chapman-Enskog approximation has the form of a single partial differential equation of order 2n. One can perhaps reasonably anticipate somewhat similar results for ChapmanEnskog approximations in the kinetic theory of vehicular traffic.
THE CHAPMAN-ENSKOG EXPANSION In this section the Chapman-Enskog expansions, of orders zero and one, will be developed for the Prigogine-Herman kinetic equation, with the probability of passing and the relaxation time taken as constant. The treatment given here is adapted from that of Cercignani (1988, Section V.3). To begin, it is convenient to slightly rewrite the Prigogine-Herman equation (3) as
Here the arguments x, v and t have been omitted, in order to keep the notation as concise as possible. Additionally, w+ is the maximum speed desired by any driver, as in the preceding section, and
(7)
Here / and Kf are respectively the left-hand and right-hand sides of (3), and e is a parameter that is introduced as a formal device to separate (3) into (an infinite number of) more easily solved
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problems. A formal series solution, in the form
is then sought. However, the density (the sole "hydrodynamic" variable, in the case of traffic flow) is specifically not expanded in a power series in e. That means it is required that the densities corresponding to the higher-order corrections to the distribution function are required to be zero, w+ f(n\x,v,t)dv = Q,n = l,2,....
/
(9)
Thus density is carried entirely by the zero-order approximation to the distribution function, f
Jo
f0(0) (x,v,t)dv.
(10)
In the Chapman-Enskog expansion the (partial) time derivative in the Prigogine-Herman equation (7) is further expanded as
fc=0 f\(k\
(In the last equality the expansion (8) has been used, with the ^- treated as linear operators.) Here the -—^— are to be determined so as to ensure existence of the terms of the formal solution (8). That is, the -—^— are to be regarded as unknowns, to be determined by substituting the expansions (8) and (11) into the Prigogine-Herman (7), in exactly the same way as the various terms in the expansion (8) (i.e., the /^) are determined. The result of carrying out these substitutions, and equating coefficients of like powers of e is
K/ (0) = 0,
(12)
at order n = 0. Otherwise it is df(n-V
^ d (fc) f^-*-1) fc=0
Here L(^) is, for any function g, the linear operator defined by 1
w+
/ and S
Jo
The Chapman-Enskog Expansion
67
where, as usual, the dependence on x and t is understood, but is not explicit in the notation. (The assumption that P and T are constants, independent of c, is used in obtaining these equations.) If / (0) , /(1), / (n ~ 1) exist, then (13) has a solution for /^-^ if, and only if, the right-hand side is an element of the range of L(/^), or equivalently it is orthogonal to the null space of the adjoint of L(/<°>), notationally L(/<°>)*. But L(/<°>)* is given by
and it is readily shown from this representation that the null space of L(/(°))* is exactly the onedimensional space of constants. Thus a necessary and sufficient condition for the equations (13) to have a solution for the f^ is the compatibility conditions that the integral over 0 < v < w+ of their respective right-hand sides to be zero. But where Eqs. (9) and (10) have been employed. Similarly, (9) implies that f™+ Sndv = 0, for n > 1. Therefore, the compatibility conditions for Eqs. (13) are the (LWR-like) equations ^("-l)/>
/^(n-l)
ot
ox
= 1,2,...,
(15)
where .- f Jo
+
(16)
If (15) is multiplied by e""1, and the results summed over all n = 1 , 2 . . . , then the result can be written as n=0
Here the definition of the expansion (11) has been employed. This equation, which arises as a consequence of the basic assumptions of the asymptotic expansion rather than from ad hoc assumptions, is the basis for the continuum approximations arising from the Chapman-Enskog approximation, as follows. If the q^ are computed by solving Eqs. (13), then it turns out that q^ depends on x and t only through c(x, t} and its spatial derivatives up to order n. If the sum in (17) is truncated at n = N, and e set to unity, then the resulting partial differential equation of order ./V is the continuum form of the ./Vth-order Chapman-Enskog approximation. The details of these continuum approximations will now be developed, for ./V = 0, 1, in the following two respective subsections.
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THE ZERO-ORDER CHAPMAN-ENSKOG APPROXIMATION From (4) and (8) the zero-order Chapman-Enskog solution of the Prigogine-Herman kinetic equation, for the stable flow regime (c < c^it), is given by
_ (o) _- ^ * 'AM f (v.c] _ /f _ /f M-T(1_p)u_c(c). Here £(c) is the root of
F0(C) = T(l - P)c, +
where Fn(C) := J0™ , J^in+i dv. From (17), the corresponding continuum approximation is
along with the classical traffic stream model given by
(19) Note that the stable flow regime is defined quantitatively by c < c^n := rf°i°L > so that necessarily C(c) is negative in this regime. Eqs. (18) and (19) comprise a LWR model. As a partial differential equation for the zero-order approximation to the concentration, this LWR model reads as dQ0
_
~°'
(20)
Subject to suitable auxiliary (initial and boundary) conditions, this LWR model presumably determines c(°) as a function of position and time. Note that the formal asymptotic procedure gives rise to both the conservation law and the traffic stream model, and that the latter is expressed in terms of the given data in the underlying (Prigogine-Herman) kinetic model. In the following subsection the corresponding problem of determining /^ is considered. This section will be concluded by briefly discussing how the problem of determining /^ is modified in the case of unstable flow (c^ > 0^4). In the unstable case, all of the preceding formal development goes through verbatim, except that QQ no longer depends upon only c^°\ but rather upon both c^0) and a. Therefore, in this case the partial differential equation (20) is a single equation, which cannot be expected to determine the two unknowns, c^ and a. The central problem of extending the Chapman-Enskog expansion to encompass unstable flow is to find a suitable second equation in c^ and a that will, along with (20) and suitable auxiliary conditions, serve to determine c^ and a. This problem is outside the scope of the present work.
The Chapman-Enskog Expansion
69
THE FIRST-ORDER CHAPMAN-ENSKOG APPROXIMATION
Thefirst-orderChapman-Enskog solution of the Prigogine-Herman kinetic equation is / = /(1), where /(°) is the solution of (12) and /(1) is the solution of the instance n = 1 of (13). The resulting zero-order approximation to the vehicular distribution is /(0) = feq(v; c), just as before. However, there is a subtle but important difference, as regards the density parameter appearing in this equilibrium solution, between this equilibrium solution and that arising in the zero-order approximation of the previous subsection. In the equilibrium solutions arising as the zero-order approximation for the Chapman-Enskog expansion truncated at zero order, the appropriate value of the density parameter is determined by the LWR model of the preceding subsection. But in the first-order Chapman-Enskog approximation the appropriate value of the density is the solution of the continuum model corresponding to (17), with the infinite series truncated at N = 1. It will be seen below that this is not a LWR model. In order to determine this first-order Chapman-Enskog continuum model it is necessary to calculate q^\ as defined by the instance n = 1 of (16). This calculation will now be outlined. As already mentioned, the starting point for the computation of /(*), and hence q^l\ is the instance n — 1 of (13). The compatibility condition (15) for this instance is 0(o)c
^(0)
-—- + -^— = 0, n = 1 , 2 , . . . . dt dx The right-hand side of the relevant instance of (13) is then
dt
0fO) dc d(0) f(0) f + v-^— = —^—-7— + v-4r-^c- = —^— < —^— +v— > = dt dc dt dc dx dc { dx dx
Here £' is the derivative of C, and the explicit expression (19) for gO) has been employed. The instance n = 1 of (13) can then be written as (l-P)ct,-C(c)' This is a (degenerate) integral equation of the first kind for /(1). It can be solved explicitly for /M, but it is somewhat easier to solve for q^l\ which is the quantity of actual interest. If this integral equation is integrated over 0 < v < w+J then the result can be solved explicitly for q^ as But it is relatively easy to show that f™+ g-\.(v) dv = 0 and ^(C(C))^
Jo
dx
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Therefore,
where the "diffusion coefficient" is
The first-order continuum approximation associated with the Chapman-Enskog expansion then is defined by the continuity equation (18), but now the flow is given by
Thus this first-order continuum approximation is the advection-diffusion equation
It is an easy application of the Cauchy-Schwartz inequality, along with the equality F 0 (C(c)) = cT(l - P], which defines C(c), to show that Fi(C(c))2 < cT(l - P)F2(C(c)), so that D(c) > 0. Therefore, the initial-value problem for this partial differential equation is well-posed computationally. Some initial computational results for this equation are presented in the following section. The diffusive term in (21) can be interpreted as a correction to the zero-order LWR model, which can be written as
l+flwol-0-
It is interesting to note that 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. (The present approach thus confirms the diffusion term, but not the inertia term.) By the time of his book on nonlinear waves, Whitham (1974) mentioned both this possibility and an approach similar to current dynamic stream models (as an effort to take account of the "time lag in the response of the driver"), but there was no associated citation of the work of Payne (1971). Payne (1971) and Schochet (1988) have observed that the Payne model, as its relaxation time (which does not appear simply related to that in the Prigogine-Herman model) approaches zero, reduces to a form similar to (21). On the basis that the first-order Chapman-Enskog approximation to the Boltzmann equation is the Navier-Stokes equations of fluid dynamics, it appears that the advection-diffusion equation (21) is better considered as the true analog for traffic flow of the Navier-Stokes equations than are current higher-order models. This supposition seems to be rather strongly supported by the result of Schochet (1988) described earlier, which shows that this "continuity plus diffusion" model plays the same role in justifying the entropy condition for
The Chapman-Enskog Expansion
71
the LWR model as the Navier-Stokes equations play in justifying it for the Euler equations of fluid dynamics.
COMPUTATIONAL RESULTS For the example of this section the values P = .67 and T = .0015 hours were used. These correspond to 77: c/Cjam = .3333 and T = .003 hours « 11 seconds, in the notation of Chapter 4 of Prigogine and Herman (1971). The reduced desired speed distribution used is that corresponding to a uniform distribution of desired speeds from 40 miles per hour to 80 miles per hour. The corresponding equilibrium solution, in the stable regime, is f -^T^> /e,(v;c) = < v C(c) I 0,
f°r 40 mph < v < 80 mph . otherwise.
Here C is given explicitly by £(c) = 40(e 02c — 2)/(e 02c — 1) mph, where c is density in vehicles per mile per lane (vpmpl). The critical density is the root of £(c) = 0, which is C^H = 50 In 2 = 34.67 vpmpl. The corresponding traffic stream model, from the equilibrium solution, is q^ (c) = 2000 4- c£(c) vehicles per hour per lane (vphpl). A plot of this traffic stream model, for the stable regime, is shown in Figure 1. The critical flow (i.e., the flow at critical density) is q^a = 1/T(1-P)= 2000 vphpl. The problem considered here is defined by the parameters of the preceding paragraph, and the initial conditions , x , x, f Ccrit = 34.67 vpmpl, x < 0, c(z,0) = c(z,t)| t = 0 = { I 3 vpmpl, x > 0. This corresponds to release, into a relatively vacant region, at t = 0 of a semi-infinite "platoon" of vehicles extending indefinitely to the left from x = 0, and initially at the critical concentration. The corresponding exact solution to the LWR model (18) consists of a shock, with density Q = Ccrit = 34.67 vpmpl on the left and density cv = 3 vpmpl on the right. The shock thus propagates to therightat a speed of (g(0)(cr) - 9(0)(c*)) / (cr - Q) = 57.3 mph. This problem was solved computationally, for both the zero-order LWR approximation and the first-order "advection-diffusion" approximation (21), by means of the generalized Godunov method (Morton, 1996, Section 7.3) This method was selected specifically because it is a shockcapturing method, when applied to pure hyperbolic,advection problems such as the LWR model, but it also is applicable to advection-diffusion equations, such as the first-order Chapman-Enskog approximation (21) obtained above. Thus this method permits comparison of the zero-order and first-order approximations, without extraneous effects stemming from the use of fundamentally different computational methods. Some sense of the capabilities and limits of this method can be obtained from Fig. 2. In this figure, the densities for both the exact solution of the LWR model, as described above, and for its computational approximation by the generalized Godunov
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method, are displayed, for t = 1.03 minutes. Notice that the computational method captures very well the location of the shock. The shock in the computational result displays a nonzero thickness, that is just discernible on the scale of the motion of the front (it is three to five computational spatial cells thick). This "numerical dispersion" phenomenon is practically unavoidable for computational approximations to nonlinear conservation laws (such as LWR models). The LWR computational results also display relatively small (about 10% of the critical density) nonphysical oscillations near either edge of the computed shock. Many (but not all) computational methods for nonlinear conservation laws display such high-frequency oscillatory errors. All in all, these results provide some evidence of credibility for the computational methodology. The computational result for the advection-diffusion model (21) at the same time also are displayed in Fig. 2. The most significant differences from the LWR results are that the shock profile is significantly wider and smoother. (The shock is approximately .3 miles wide for the diffusive approximation, versus .1 miles for the computed LWR results). To understand the basis for this difference, notice that the LWR model effectively approximates the vehicular flow as if all vehicles travelled at the average speed. (The average speed is 57.7 miles per hour unstream of the shock, and 59.7 mph downstream.) However, according to the underlying kinetic model there is in fact a distribution of vehicular speeds extending from v = 40 mph to v = 80 mph. Therefore, on either side of the shock a significant fraction of the vehicles are in fact moving faster than the average speed. Those vehicles in fact penetrate into the low-density downstream region further in a given time than would be predicted on the basis of the assumption that all vehicles move at the average speed corresponding to the local density. It is this physical dispersive effect, arising from the underlying statistical distribution of vehicular speeds, that seems to be represented by the additional flow (i.e., q^) giving rise to the diffusive term in the first-order approximation
(21). Figure 3 is a plot of the densities, as computed from the first-order diffusive model, from t = 0 tot = 1.03 minutes, and over a significant spatial region surrounding the initial location of the leading edge of the platoon. The region to the left of the slowly widening shock is the relatively crowded region initially to the left of x — 0, and that to the right is the freeflow region initially to the right of x = 0. From the perspective of Fig. 3 the widening of the shock as time increases is quite apparent. This is due to dispersion of vehicles at the leading edge of the platoon, because of their different desired speed. By contrast, the shock in the comparable LWR results (not shown) maintain a constant and narrower width.
CONCLUSIONS It has been shown that the Chapman-Enskog expansion applied to the Prigogine-Herman kinetic equation gives rise, in the regime of stable flow, to a LWR model at order zero, and to an
The Chapman-Enskog Expansion
73
Figure 1: Traffic Stream Model
10
5
15 20 25 Concentration (vehicles per mile)
30
Rgure 2: The platoon release problem at t = 1.03 minutes w /\
35 +
+
\ +
30
t +
1 E |25
+
i t20 .2
2
a
i \ + i + 4
+
I
Diffusive
H
V1
- - Cmptd. LWR
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— Exact LWR 10
1
Dettat = .97sec.
+
I
Deltax = 88ft. 5
+
.
+
1
T=.0015hrs
+ +
++
>
P = .666
+
w
n 0.6
,
i
0.7
0.8
0.9
1
Space (miles)
1.1
1.2
1.3
1.4
74
Transportation and Traffic
Theory
Figure 3: The Diffusive Approximation to the Platoon Release Problem
,
1
Space (miles)
advection-diffusion equation, consisting of an LWR model with an additional diffusive term, at order one. It is suggested that the diffusive term represents the tendency of vehicles initially together to disperse, because of the different desired speeds of the drivers. (By contrast, the LWR model proceeds as if all vehicles were traveling at the average speed corresponding to the density of vehicles in their imediate vicinity.) Helbing (1996, p. 2379) has earlier discussed this dispersion phenomenon, by way of answering the objection of Daganzo (1996b) to the fact that higher-order models predict vehicular movement faster than the (average) vehicular speed. Dispersion of vehicles due to differing speeds clearly is a real phenomenon that is not represented in LWR models. But this phenomenon is captured, to some degree, by either existing higher-order models or by the first-order ChapmanEnskog approximation presented here. The choice between these two alternatives, at least in terms of capturing the dispersion effect, may ultimately be a matter of taste. The authors prefer the advection-diffusion model developed here, on the basis of the following two points: A corrective term in the conservation equation, with no additional unknowns, seems simpler than an additional equation, with an additional unknown. (This point invokes the principle of Occam's razor.)
The Chapman-Enskog Expansion
75
• The first-order model obtained here arises from the kinetic theory via standard asymptotic expansions, rather than by ad hoc techniques. In any event, the results presented above clearly provide an alternative to current higher-order models, for those seeking continuum models of traffic flow that are not subject to some of the limitations of LWR methods. The approach presented here also has the advantage, relative to current higher-order models, that it can be systematically extended to give a hierarchy of arbitrarily high-order extensions of the zero-order LWR model. For example, in the theory of rarefied gases the second-order ChapmanEnskog expansion gives rise to the Burnett equations of fluid flow (Cercignani, 1988). These equations are not used in fluid flow as often as their first-order Navier-Stokes counterpart, but they have found application. It remains to be seen whether the traffic-flow counterpart of the Burnett equations has any significant application. Work related to this issue will be presented elsewhere. It would be of some interest to extend the Chapman-Enskog expansions presented here to the case that P and T vary with density (c) in the manner assumed in Chap. 4 of Prigogine and Herman (1971). However, this interest may be mostly academic in nature. This is because the effect of this dependence upon c is most pronounced for larger values of c, which are outside the stable-flow regime for which the present theory applies. Probably a more important issue, as regards the potential application to congested flow, is to determine a counterpart continuum theory that is applicable to unstable flow. The results of this paper, combined with those of Nelson and Sopasakis (1998), suggest that the applicability of LWR models may be essentially limited to stable flow. The asymptotic expansion of Hilbert (e.g., Cercignani, 1988) has been used in the kinetic theory of gases, as yet another approach to obtaining a hierarchy of continuum approximations. When applied to the Prigogine-Herman kinetic model (in the stable flow regime) it gives rise to a hierarchy of continuum models in which the model of order n arises by adding a first-order partial differential equation of LWR type to the previous n first-order partial differential equations. However, the unknowns in these equation are successive corrections to the density, and the equation added to comprise the model of order n contains not only the nth order correction, say c^n\ but also all lower-order contributions c^n~l\ c ^ n ~ 2 ) , . . . , c^. Thus there is a one-directional (triangular) coupling between the equations marking the successive orders. More details of this approach will be presented elsewhere. The first-order diffusive Chapman-Enskog approximation is a differential equation of order two. Thus one would expect to need two boundary conditions. And at each successive further order in the Chapman-Enskog hierarchy one would need one further boundary condition. Thus difficulties with boundary and interface conditions, which already are significant even for the zero-order
Transportation and Traffic
76
Theory
LWR models, are only compounded for succesive entries in the Chapman-Enskog hierarchy. This is to be expected, as boundary conditions for the counterpart expansion in the kinetic theory of gases also are a difficult issue. This issue also arises for current higher-order models, and does not seem to have been addressed in that context. Finally, it would be of great interest to extend the types of results presented here to other, and more realistic, kinetic models of vehicular traffic. Perhaps the most immediate such extension should be to a Prigogine-Herman equation modified so as to incorporate the spatial extent of vehicles in the interaction term.
REFERENCES Ansorge R. (1990). What does the entropy condition mean in traffic flow theory? Transportation Research B 24B, 133-144. Bui, D. D., Nelson, P. and Narasimhan, S. L. (1992). Computational realizations of the entropy condition in modelling congested traffic flow. Texas Transportation Institute report no. 1232-7. Bui, D. D., Nelson, P. and Sopasakis, A. (1996). The generalized bimodal traffic stream model and two-regime theory. Transportation and Traffic Theory, (J.-B. Lesort, Ed.), Elsevier, Oxford, 679-696. del Castillo J. M., Pintado, P. and Benitez, F. G. (1993). A formulation for the reaction time of traffic flow models. Transportation and Traffic Theory, (C. F. Daganzo, Ed.), Elsevier, Amsterdam, 387-406. Ceder A. (1976). A deterministic flow model for the two-regime approach. Transp. Res. Rec. 567, 16-30. Cercignani C. (1988). The Boltzmann equation and its applications. Springer-Verlag, New York. Chapman S. and Cowling, T. G. (1952). The mathematical theory of nonuniform gases. Cambridge University Press, Cambridge. Daganzo C. F. (1995a). A finite difference approximation of the kinematic wave model of traffic flow. Transportation Research B 29B, 261-276. Daganzo C. F. (1995b). Requiem for second-order fluid approximations of traffic flow. Transportation Research B 29B, 277-286. Disbro J. E. and Frame M. (1989). Traffic flow theory and chaotic behavior. Transp. Res. Rec. 1225,109-115. Drake J. S., Schofer, J. L. and May, A. D. (1967). A statistical analysis of speed density hypothesis. Highway Research Record 154,53-87.
The Chapman-Enskog Expansion Gerlough D.L. and Huber M. J. (1975).
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Traffic Flow Theory. Transportation Research Board
Special Report 165, Washington, D.C. Grad, H. (1949). On the kinetic theory of rarefied gases. Communications on Pure and Applied Mathematics 2, 331-407. Greenshields B. D. (1934). A study of traffic capacity. Procs. Highw. Res. Board 14, 448-477. Hauge E. H. (1969). Exact and Chapman-Enskog solutions of the Boltzmann equation for the Lrentz model. Arkivfor Det Fysiske Seminar i Trondheim, No. 5. Hall F. L. (1987). An interpretation of speed-flow concentration relationships using catastrophe theory. Transp. Res. A 21 A, 191-201. Helbing, D. (1995a). Theoretical foundation of macroscopic traffic modejs. Physica A 219, 375-390. Helbing, D. (1995b). High-fidelity macroscopic traffic equations. Physica A 219, 391-407. Helbing D. (1996a). Gas-kinetic derivation of Navier-Stokes-like traffic equations. Physical Review E 53, 2366-2381. Helbing, D. (1996b). Derivation and empirical validation of a refined traffic flow model. Physica A 233, 253-282. Kerner B. S., Konhauser, P. and Schilke, M. (1995). Deterministic spontaneous appearance of traffic jams in slightly inhomogeneous traffic flow. Physical Review E 51, 6243-6246. Klar A. and Wegener, R. (1997). Enskog-like kinetic models for vehicular traffic. J. Statistical Physics 87, 91-114. Klar A. and Wegener, R. (to appear-a). A hierarchy of models for multilane vehicular traffic I: Modeling. SIAM J. Applied Math. Klar A. and Wegener, R. (to appear-b). A hierarchy of models for multilane vehicular traffic I: Numerical and stochastic investigations. SIAM J. Applied Math. Kiihne R. D. and Beckschulte R. (1993). Non-linearity stochastics of unstable traffic flow. Transportation and Traffic Theory, (C. F. Daganzo, Ed.), Elsevier, Amsterdam. Lebacque, J.-P. (1984). Semimacroscopic simulation of urban traffic. Procs. Int. AMSE Conf "Modelling & Simulation," 4, 273-292. Lebacque, J.-P. (1995). The Godunov scheme and what it means for first order traffic flow models. CERMICS Report No. 95-48, November. Lebacque J.-P. (1996). The Godunov scheme and what it means for first order traffic flow models. Transportation and Traffic Theory, (J.-B. Lesort, Ed.), Elsevier, Oxford, 647-678. Lebacque, J.-P. (1997). A finite acceleration scheme for first order macroscopic traffic flow models. Transportation Systems, Procs. 8th IFAC/IMP/IFORS Symposium, (M. Papageorgiou and A. Pouliezos, Eds.), Tech. Univ. Crete, Chania, 2, 815-820.
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Leo, C. J. and Pretty, R. L. (1992). Numerical simulation of macroscopic continuum traffic models. Transportation Research B, 26B, 207-220. LeVeque, R. L. (1990). Numerical methods for conservation laws. Birkhauser-Verlag, Basel. Liboff R. L. (1990). Kinetic theory: classical, quantum and relativistic descriptions. PrenticeHall, Englewood Cliffs, NJ. Lighthill M. J. and Whitham G. B. (1955). On kinematic waves n - a theory of traffic flow on long crowded roads. Proc. Royal Society, London A229, 317-345. Lyrintzis A. S., Liu, G. and Michalopoulos, P. G. (1994). Development and comparative evaluation of high-order traffic flow models. Transportation Research Record, 1457, 174-183. May, A. D. (1990). Traffic Flow Fundamentals. Prentice-Hall, Englewood Cliffs, NJ. Michalopoulos, P. G., Beskos, D. E. and Lin, J-K (1984). Analysis of interrupted traffic flow by finite difference methods. Transp. Res. B 18(B)(4/5), 409-421. Michalopoulos, P. G. and Beskos, D. E. (1984). Improved continuum models of freeway flow. Ninth International Symposium on Transportation and Traffic Theory, (J. Volmuller and R. Hamerslag, Eds.,) VNU Science Press, Utrecht, 89-111. Michalopoulos, P. G., Yi, P. and Lyrintzis, A. S. (1993). Continuum modelling of traffic dynamics for congested freeways. Transportation Research B, 27B, 315-332. Morton K. W. (1996). Numerical solution of convection-diffusion problems, Chapman & Hall, London. Nelson, P. (1995a). A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions. Transport Theory and Statistical Physics 24(1-3), 383-409. Nelson P. (1995b). On deterministic developments of traffic stream models. Transp. Res. B 29B No 4, 297-302. Nelson, P., Bui, D. D. and Sopasakis, A. (1997). A novel traffic stream model deriving from a bimodal kinetic equilibrium. Transportation Systems, Procs. 8th IFAC/IFIP/IFORS Symposium, (M. Papageorgiou and A. Pouliezos, Eds.), Tech. Univ. Crete, Chania, 2,799804. Nelson P. and Sopasakis, A. (1998). The Prigogine-Herman kinetic model predicts widely scattered traffic flow data at high concentrations. Transportation Research B 32,8, 589-604. Newell G. F. (1962). Nonlinear effects in theory of car following. Operations Res. 9, 209-229. Newell, G. F. (1989). Comments on traffic dynamics. Transportation Research B 23B, 386-389. Newell, G. F. (1993). A simplified theory of kinematic waves in highway traffic, Part I: General theory. Transportation Research B 27B No 4,281-287. 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. Transp. Res. 9, 225-235.
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Payne H. J. (1971). Models of freeway traffic and control. Simulation Council Proc., 1, 51-61. La Jolla, CA; Simulations Council, Inc. Payne, H. J. (1979). FREFLO: A macroscopic simulation model of freeway traffic. Transportation Research Record 722, 68-77. Phillips W. F. (1977). Kinetic model for traffic flow. Report No. DOT/RSPD/DPB/50-77/17, Utah State University for the U.S. Department of Transportation. Phillips W. F. (1979). A kinetic model for traffic flow with continuum implications. Transportation Planning and Technology 5, 131-138. Phillips, W. F. (1981). A new continuum model for traffic flow. Report No. DOT/RSPA/DPA50/81/5, U. S. Department of Transportation, Washington D. C. (Prepared by Utah State University, Logan, Utah). Prigogine I. and Herman R. (1971). Kinetic Theory of Vehicular Traffic. American Elsevier Pub. Co., New York. Prigogine, I. Herman, R. and Anderson, R. L. (1962). On individual and collective flow. Bulletin Acad. Royale de Belgique 48, 792-804. Rathi, A. K., Lieberman, E. B. and Yedlin, M. (1987). Enhanced FREFLO Program: Simulation of congested environments. Transportation Research Record 1112, 61-71. Richards, P. I. (1956). Shockwaves on the highway. Operations Research, 4,42-51. Ross P. (1988). Traffic dynamics. Transp. Res. 228, 421-435. Ross, P. (1989). Response to Newell. Transportation Research B 23B, 390-391. Schochet S. (1988). The instant response limit in Whitham's nonlinear traffic-flow model: Uniform well-posedness and global existence. Asymptotic Analysis, 1, 263-282. Transportation Research Board (1985), Highway Capacity Manual. Special Report 209, National Academy of Sciences, Washington, DC. Wegener, R. and Klar, A. (1996). A kinetic model for vehicular traffic derived from a stochastic microscopic model. Transport Theory and Statistical Physics 25, 785-798. Whitham G. B. (1974). Linear and Nonlinear Waves. Wiley, New York.
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The Lagged Cell-Transmission Model
81
THE LAGGED CELL-TRANSMISSION MODEL Carlos F. Daganzo, Inst. ofTransp. Studies, Univ of California, Berkeley, California, USA
ABSTRACT Cell-transmission models of highway traffic are discrete versions of the simple continuum (kinematic wave) model of traffic flow that are convenient for computer implementation. They are in the Godunov family of finite difference approximation methods for partial differential equations. In a cell-transmission scheme one partitions a highway into small sections (cells) and keeps track of the cell contents (number of vehicles) as time passes. The record is updated at closely spaced instants (clock ticks) by calculating the number of vehicles that cross the boundary separating each pair of adjoining cells during the corresponding clock interval. This average flow is the result of a comparison between the maximum number of vehicles that can be "sent" by the cell directly upstream of the boundary and those that can be "received" by the downstream cell. The sending (receiving) flow is a simple function of the current traffic density in the upstream (downstream) cell. The particular form of the sending and receiving functions depends on the shape of the highway's flow-density relation, the proximity of junctions and on whether the highway has special (e.g., turning) lanes for certain (e.g., exiting) vehicles. Although the discrete and continuum models are equivalent in the limit of vanishingly small cells and clock ticks, the need for practically sized cells and clock intervals generates numerical errors in actual applications. This paper shows that the accuracy of the cell-transmission approach is enhanced if the downstream density that is used to calculate the receiving flow(s) is read { clock intervals earlier
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than the current time, where 0 is a non-negative integer that should be chosen by means of a simple formula. The rationale for the introduction of this lag is explained in the paper. The lagged celltransmission model is related (but not equivalent) to both Godunov's first order method for general flow-density relations and Newell's exact method for concave flow density relations. It is easier to apply and more general than the latter, and more accurate than the former. In fact, if the flowdensity relation is triangular and the lag is chosen optimally, then the lagged cell-tansmission model is a conservative, second order, finite difference scheme. As a result, very accurate results can be obtained with relatively large cells. Accuracy formulae and sample illustrations are presented for both the triangular and the general case.
1. INTRODUCTION This paper describes a new finite difference approximation for the kinematic wave model of traffic flow formulated by Lighthill and Whitham (1955) and Richards (1956), called here the LWR model, and for the generalized continuum model that applies to freeways with special lanes (Daganzo, 1995). The proposed scheme is conservative, in that vehicles are not created or lost during the simulation except at the highway's entrances and exits, like the procedures described in Bui et al (1992), Lebacque (1993) and Daganzo (1993 and 1993a). Unlike its predecessors, however, the proposed scheme is not in Godunov's family of finite difference approximations (see Godunov, 1961, or LeVeque, 1992), and is more accurate than they are. In the new scheme, flows across the boundary between two cells are calculated with a sending/receiving metaphor similar to that introduced in Lebacque (1993) and Daganzo (1993a), using only information from the two neighboring cells.1 Such a metaphor is called here the celltransmission (CT) model. The CT recipe has been modified to model junctions (Daganzo, 1994) and highway links with special lanes, e.g., for turning or high occupancy vehicles (Daganzo, Lin and DelCastillo, 1995). These modifications make it possible to model complex networks. The similarity between the new and the old CT recipes ensures that these modifications can be extended trivially to the new method. Thus, the improvement in accuracy obtained with the new method should apply to network models. The main difference between the new approach and the original CT recipe is that the density of the downstream cell, used to calculate the "receiving" flow, is now taken from an earlier time, with a lag of C simulation clock intervals. Newell's exact solution method for concave flowdensity relations (Newell, 1993) is also based on the introduction of lags. Lags are useful for highway traffic modeling because traffic information travels several times more slowly in the
Lebacque (1993) uses the terms "local demand" and "local supply" to express the metaphor.
The Lagged Cell-Transmission Model
83
upstream than in the downstream direction. In our case, we shall see that if the two wave speeds (forward and backward) are independent of density, then the lagged cell-transmission (LCT) model turns out to be second order accurate. Because this paper is closely related to Daganzo (1993a), the reader is referred to that reference for more extensive introductory remarks. The remainder of this paper is organized as follows. Section 2 presents experimental evidence pertaining to wave speeds and illustrates by means of an example the minor difficulties one encounters with the LWR theory when the wave speed does not decline with density monotonically, i.e., when the flow-density relation is nonconcave; in particular, it is shown that the "stable" LWR solution can have waves emanating from a shock. The new finite difference approximation is then presented in Section 3, together with an analysis of its accuracy and stability. The paper concludes with some examples (Sec. 4) that illustrate the results of Sec. 3.
2. NON-DECLINING WAVE SPEEDS The LWR model is intended to describe traffic on large scales of observation, where it makes sense to define a density function k(t, x) and a flow function q(t, x) in time-space (t, x). It is based on the assumption that q and k are locally related by a flow-density relation q = T(k, t, x). When the highway is homogeneous and its features do not depend on time (e.g. no incidents or moving bottlenecks) then the relation only includes k as an argument: q = T(k).
(1)
This is the case that will receive attention for the most part of this paper. Flow and density are also related by the conservation equation: k, + qx = 0,
(2)
where subscripts have been used to denote partial derivatives.2 Thus, it is possible to eliminate q from (1) and (2). The result is the following simple first order quasi-linear partial differential equation in k: kt + T k k x = 0.
2
(3)
Subscripts t, x and k will be used in this paper to denote partial derivatives of the subscripted variable with respect to time, distance and density. All other subscripts, e.g., i, j, and {, will be indices for the subscripted variables.
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In this equation kt and kx are functions of t and x, and Tk is a known function of k. This is the conventional way of expressing the LWR model for numerical approximations. If Tk is independent of k then the solution to (3) has the form: k(t, x) = g(x-Tkt) ; i.e., it is a translationally symmetric function (wave) with wave velocity T k . The particular form of g depends on the boundary conditions. For example, for the initial value problem where k(0, x) is given, g(x) = k(0, x) and k(t, x) = k(0, x-Tkt).
(4)
The wave velocity Tk also plays an important role in the quasi-linear case. In this case too, the density is constant along wave-lines (characteristics) issued from the boundary, but the wavelines can now focus and cross. The solution can be extended into these regions of the (t, x) plane by introducing curves, called shocks in the LWR theory, where k(t, x) is discontinuous. Conservation of vehicles across such discontinuities results in the following equation for the shock velocity, u : u = Aq/Ak
(5)
where Aq and Ak represent the changes in flow and density across the shock. Equations (3) and (5), however, are not always enough to specify a solution. It turns out that shocks can often be introduced in more than one way to form a mathematically correct solution of a properly formulated problem; i.e., a solution that is consistent with (3) and (5) and with the initial data. If the problem has been properly formulated, however, i.e., if it makes physical sense, then there should be one and only one solution that makes physical sense. This solution can be identified with a standard stability argument; i.e., by making sure that the solution does not come undone if a small perturbation is introduced in it.3 If q = T(k) is represented by a curve as in the top part of Fig. 1, then the wave velocity is the slope of the curve. Note that if the curve is concave, then the wave velocity declines with density, and also with the traffic speed. That is, waves focus (into a shock) when traffic decelerates and fan out (as an expanding wave) when it accelerates. The best evidence available and common sense indicates that the maximum wave speed in the forward direction, obtained for low densities, is comparable with the free-flow traffic speed, e.g., on the order of 60 mi/hr (27 m/s), and that the backward wave speeds are several times slower. This is illustrated on the bottom part of Fig. 1,
3
The physically relevant solution must be one where the shocks are "stable"; i.e., where if (at any given time, t 0 ) one were to replace a shock by a quick but gradual transition in density between the values prevailing on both sides of the shock, then the shock would reform itself and the solution would quickly approach the original (for t > t0).
The Lagged Cell-Transmission Model
85
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Figure 1. Stationary relations between traffic variables. Top: hypothetical non-concave relation between flow and density. Bottom: actual form of a steady-state relationship between flow and occupancy (source: Cassidy, 1998).
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which has been taken from Cassidy (1998). Additional experiments correlating data from several detectors point to the existence of expansive deceleration waves and focused acceleration waves in congested traffic (Windover and Cassidy, 1997). Related anecdotal evidence has also been recorded by this author on Highway US50 West of Placerville, California. This is a heavily traveled two-lane highway with very few intersections, which experienced a sustained capacity-reducing incident on the particular date. This created queues of many miles on both sides of the road. On crossing the incident location and traveling past the queue in the opposite direction, this author noted a period of a minute or two (spanning between 1 and 2 highway miles) where rather dense traffic appeared to be coasting toward the end of the queue(!). This was not a stop-and-go wave, for those were observed within the queue and they had a much shorter period (from the stand point of the moving observer). This coasting effect cannot be explained by the LWR theory with a concave T(k), since in that theory the end of a queue should involve a transition with just a few vehicles. It cannot be explained either by linear car-following models (e.g., as in Herman et al, 1959), but it can be explained by the LWR theory with a non-concave T(k) and a "tail" to the right (such as that in the top part of Fig. 1) and/or by the corresponding non-linear car-following model. We do not wish to speculate further about this phenomenon in this paper, since the goal of this comment is only to establish the desirability of having numerical methods that can treat non-concave T(k) relations. Such methods can be used for prediction if such tail exists, or they may help disprove its existence if it does not. Improved numerical methods are desirable because the exact procedures in Newell (1993) cannot be used with non-concave relations, and because the cell-transmission approaches introduce some error into the calculation. Before the new method is presented, a brief example is introduced which illustrates the stability condition for non-concave T(k).
2.1 An example Let us examine how a line of cars comes to a halt according to the LWR theory, when the flow-density relation, T(k), is as in Fig. 2a. We assume that the corners of our curve have been smoothed very slightly, not enough to be seen in the figure, so that the wave velocity is defined for all densities.4 It is assumed that the leading cars in the line are in state A' (q = 0.6 veh/sec, k = 12 veh/Km) and that heavier traffic (state A with q = 1 veh/sec and k = 20 veh/Km) is found 6 Km upstream of the leading car; see Fig. 2b. The origin of the (t, x) coordinates has been chosen so that
4
Smoothed piece-wise linear q-k curves such as ours, are good for illustration purposes because they lead to solutions that are simple and easy to interpret.
The Lagged Cell-Transmission Model
-200
-100
o
100
200
300
400
87
l(sec)
Figure 2. Solutions of a lead-vehicle problem with kinematic wave theory: (a) T(k) relation; (b) a mathematically valid but physically unacceptable solution; and (c) the stable solution.
the first car stops at x = 0 (Km) when t = 0 (sees). Part b of the figure satisfies (3) and (5) but is unstable. To see this, note that if the shock at time t0 is replaced by a smooth transition in density (e.g., at t0 = 400s) then a fan of waves would emanate from it and the solution would not evolve into the future (t > t0) as originally assumed because the fan would introduce a state "E" into the solution.
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Part c is the stable solution to this problem. It includes a fan of waves (carrying state "E") that are issued tangentially from the shock at the point where it bends. That this solution is stable is seen by noting that if the shock is replaced by a smooth but rapid transition in density anywhere, even at the point where the shock bends, then the wave pattern so generated will match the one in the solution.5 This means that no new states can be introduced into the solution of Fig. 2c by an infinitesimal perturbation and that the solution is the one that would arise physically. That Fig. 2.c is the relevant solution can also be verified from microscopic (car-following) considerations, although this is more tedious. This statement is based on the principle that any asymptotically stable car-following model, in the sense of Herman et al (1959), must be consistent with the corresponding (stable) LWR result; e.g. with the macroscopic result of Fig. 2c in our case, provided of course that the equilibrium speed-spacing relation is consistent with the T(k) of Fig. 2a and that the initial conditions are also consistent with those of Fig. 2. The development of a coasting state, such as state "E" of Fig. 2c, is demonstrated with car-following theory in a longer version of this paper (Daganzo, 1997). Figure 10 of that reference shows that the car-following vehicle trajectories indeed develop a coasting state when a platoon is brought to a halt.
3. THE LAGGED CELL-TRANSMISSION RULE
3.1 Background The cell-transmission model can be applied to unimodal T(k) curves with maximum flow qmax. First one defines two monotonic curves that take values in the interval [0, qmax] , as shown in Fig. 3: a non-increasing receiving curve R(k) and a non-decreasing sending curve S(k). The symbol k° is used to denote one of the densities where the maximum is achieved. A rectangular lattice with spacings e and d is then overlaid on the (t, x)-plane, as shown on Fig. 4. It is understood that traffic flows in the direction of increasing x. The x-coordinates of the lattice points, denoted {Xj} , represent the center of the "cells" into which the highway has been discretized, and the t-coordinates {tj, the times at which the cell densities are evaluated. The numbering scheme is such that xj+1 > ^ and ti+1 > tj so that traffic advances in the direction of increasing j .
5
To check stability at the point where the shock bends, one should treat the transition from A' to A as being rapid but smooth, remembering that the diagram of Fig. 2c is on a large scale. On a resolution scale where the transition from A' to A can be discerned the stable shock would have to bend gradually. Furthermore, waves would peel off from it as it curves. This detailed geometry is consistent with the macroscopic picture painted in Fig. 2c.
The Lagged Cell-Transmission Model
89
If we now let K(tj, Xj) denote the average density estimated for cell j at time t ; , and we write Q(tj + e/2, Xj + d/2) for the average flow that would advance from cell j to cell j+1 (i.e., crossing location Xj + d/2 ) in the time interval [t ; , tj+1] , then we require: K(t+e, x) = K(t, x) - (e/d)[ Q(t+e/2, x+d/2) - Q(t+e/2, x-d/2)]
(6)
by virtue of conservation. The subscripts i and j have been omitted in (6) for simplicity of notation. This will be done from now on when reference to particular cells and/or a time slices is not necessary. In these cases it should be understood that (t, x) is a point on the lattice. The cell transmission model is completed by a formula that gives Q in terms of the sending and receiving functions evaluated at the upstream and downstream cells, Q(t+e/2, x+d/2) = min{ S(K(t, x)) , R(K(t, x+d)) },
(7)
and by specifying that d/|T k | max , where Tk max is the maximum of the absolute wave speed for the given T(k) relation. maximum accuracy, one should choose e=d/|Tk|
Figure 3. Sending and receiving functions of the cell-transmission model.
(8)
For
(9)
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Figure 4. Lattice and stencil for the lagged cell-transmission model. Dots are lattice points; crosses are points where the average flows are evaluated.
Equation (8) ensures that data from outside the two neighboring cells cannot influence the calculated flow, which is a requirement for convergence. Recall now that in Godunov's approach, Eq.(7) would be the flow at the discontinuity in the stable solution of a Riemann problem6 for which the upstream density, ku , is that of the upstream cell, ku = K(t, x ) , and the downstream density, k d , is that of the downstream cell, kd = K(t, x+d); see LeVeque, 1992. The reader can verify, e.g., using the ideas in Sec. 2, that Eq. (7) indeed yields the stable flow at the location of the discontinuity for a Riemann problem with densities K(t, x) and K(t, x+d) — no matter how these two values are chosen -- if the T(k) relation is unimodal. This establishes that the CT model (6-7) is in Godunov's family of finite difference approximations for unimodal T(k)'s.
3.2 The new rule. Let Sk max and | Rk max denote the maximum (absolute) wave speeds in the forward and reverse directions, and T k | max the maximum in any direction. We show below that whenever
6
A Riemann problem is an initial value problem where the initial density is a step function with one step; i.e., k(0, x) = kd if x > x°, and k(0, x) = ku if x < x°.
The Lagged Cell-Transmission Model
91
ax,
as one would expect for most traffic streams, it is advantageous to read the sending and receiving flows from different time slices. That is, one can define a lag, Q. = 0, 1, 2,..., and use: Q(t+e/2, x+d/2) = min{ S(K(t, x)), R(K(t-fie, x+d)) },
(10)
instead of (7). This corresponds to the stencil depicted in Fig. 4, which predicts the flow at point "P" when 0 = 2. The special case with { = 0 reduces to the conventional cell-transmission rule. It will be shown in Sec. 3.3 that rule (10) is most accurate when the velocity of the wave reaching "P" happens to match the slope of one of the arrows in the figure; i.e., if the prediction at "P" is evaluated as close as possible to the source of its wave.7 This will be the basis for our choice for {. It also turns out, for stability reasons discussed in Sees. 3.3 and 3.5, that the backward slope of the arrow in our diagram should not be less than the maximum backward wave speed, R k | m a x . This means that C must satisfy: e > d/[|Rk|max(2C+l)].
(11)
A choice of f where (11) is as close as possible to a pure equality is recommended; i.e., where: 0 = V2[d(e\Rk\maJl - 1].
(12)
The accuracy of (8), (10) and (12) is evaluated below. The steps parallel those in Sec. 4 of Daganzo(1993a).
3.3 Error estimation. Consider a region of the time-space plane where the waves move back (congested traffic). Then, (6) and (10) may be rewritten as: K(t+e, x) = K(t, x) - (e/d)[ T(K(t-«e, x+d) - T(K(t-Ce, x) ]
(13)
since in this region T(k) = R(k).
7
Insofar as cause and effect relations propagate as waves in the LWR model, it should not be surprising to see that accuracy is greatest when we take our data from the lattice point closest to the wave.
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In the exact theory, the solution at time t+e is related to the solution at time t by: k(t+e, x) = k(t, x-Tke) where Tk is evaluated for the density prevailing at (t, x-Tke). Thus, in a region where K > k°, it is convenient to rewrite (13) in the following manner: K(t+e, x) = K(t, x-Tke) + [K(t, x) - K(t, x-Tke)] - (e/d)[T(K(t-0e, x+d)) - T(K(We, x))]. (14) This is useful because a second order power series expansion of the second and third terms in this expression about point (t, x-Tke) yields an estimate of the error in (13) in terms of known quantities. The expansion of the second term is: [K(t, x) - K(t, x-Tke)] - (Tke)Kx + l/2 (Tke)2Kxx , where a double subscript denotes a second derivative. Likewise, the expansion of the third term can be reduced to: -(e/d)[T(K(t-0e, x+d)) - T(K(t-£e, x))] = -(e/d)[ (dTkKx) + V2(d2 + 2dTke)(TkkKx2 + T k K xx ) - (d0e)(TkkKxKt + TkKxt) ], and the sum of these two expressions can be further simplified. The combined result is: - '/2d2Kxx[(Tke/d)2 + (Tke/d)] - >/2(TkkKx2)(de)[l + 2(Tke/d)] - (Ce2)(TkkKxKt + T k K x t ), which allows us to rewrite (14) as follows: K(t+e, x) = K(t, x-Tke) - Vzd'KJCI^e/d)2 + (Tke/d)] - i/2(TkkKx2)(de)[l + 2(Tke/d)] - (fe^T^R, + TkKxt).
(15)
The first three terms of this expression coincide with (9) in Daganzo (1993a). The last term is the contribution to the error caused by the lag. This term becomes more meaningful if the timederivatives Kj and Kxt are eliminated from the solution. This can be done if one notes from (3) that kt = - T k k x , and that the x-derivative of this expression (keeping time constant) is: ktx = - Tkkkx2 Tkkxx . If we use these relationships in the last term of (15) and use p as an abbreviation for | Tke/d , then we obtain the following expression for the error committed in time e (when the system is congested):
The Lagged Cell-Transmission Model 0 = d 2 K xx p 2 [V4(l/p-l) - fi] + T kk K x 2 de[ -Vi + p(l+2«)].
93 (16)
This is the generalization of (10) in Daganzo (1993a). When the system is uncongested, with T(k) = S(k), the LCT recipe does not use a lag. Therefore, the original derivation applies. The result is again (16), but with 0 = 0. Note from (15) that the last three terms of that expression, i.e., what we have called 0 , represent the change in K(t, x) along the characteristic in time e. Therefore, the ratio 0/e is the directional time-derivative of K(t, x) when e - 0 . Thus, in the case of a constant wave speed (i.e., linear T(k)) where the second term of (16) disappears because Tkk = 0, the value of K as seen by an observer moving with the wave satisfies the diffusion equation if e is small. The solution of the diffusion equation is stable only if the diffusion coefficient is non-negative ; i.e., if the bracketed quantity in the first term (16) is non-negative. Thus, one should choose an H that satisfies this condition for the largest possible p. This is the rationale for condition (11). Section 3.5, below, looks at the stability issue in a different way. We have just seen that the second term of (16) vanishes if T kk = 0 . Note as well that if (12) is used to choose C, then the first term will also vanish whenever the prevailing (backward) wave speed is at its maximum value. Since (16) also holds with C = 0 when the highway is uncongested, the first term also vanishes where the wave velocity is at its maximum value in the forward direction (i.e., when p = 1). Since the term also vanishes if p = 0, we see that the LCT model must be second order accurate when the T(k) relation is trapezoidal or triangular, provided that the lag is chosen with (12). We also see from (16) that the (first order) errors arising from the first term of (16) should be proportional to the difference between the lag one would like to use for the given wave speed, which is Vi(l/p - 1) , and the actual lag. This difference is minimized with rule (12), and this is particularly important when T(k) is piecewise linear.
3.4 Variable meshes The LCT method can be applied to highways and networks that have been discretized with cells of different lengths. In this more general case the LCT lag should be cell-dependent so as to ensure that the density is always read from the earliest possible time without violating the stability condition. That is, the lag for cell j , ^ , should satisfy as tightly as possibly the inequality: e>dj/[|Rk|max(2ej+l)],
(17)
where dj is the cell length, instead of inequality (11). To preserve the good properties of the LCT method, one should also introduce a forward
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lag, fj , which should satisfy: e>dJ/[Sk,max(2fJ+l)],
(18)
instead of (8). The forward lag can forestall the deterioration of accuracy where large cells are being used; especially if fj is chosen so that (18) is as close to an equality as possible. The logic for this choice is the same as that in Sec. 3.3, and the outcome is also similar; e.g., in that the resulting method is still second order accurate in the case of a triangular T(k) relation. The LCT rule with a variable discretization is then: K(t+e, Xj) = K(t, Xj) - (e/dj)[ Q(t+e/2, Xj+d/2) - Q(t+e/2, Xj-d/2) ]
3.5. Stability and relationship to Godunov's method Godunov's procedure identifies the stable solution because, as we have seen, its flows are always derived from the stable solution of a Riemann problem. Likewise, it will be shown here that the flows of the LCT method (20) are based on the stable solution of a modified Riemann problem, and therefore that the LCT solution should too approximate the stable solution. Let us consider a modified Riemann problem (MRP) in which constant upstream and downstream densities have been defined on a V-shaped boundary, such as the dark line in Fig. 5. For an MRP problem to be well-posed, waves from both legs of the "V" should point into the solution space. This will happen if the (t, x)-slopes, su and sd , of these legs satisfy su > Sk max and sd < -|R k max ; i.e., if the V-shaped boundary is inside the shaded wedges shown in the figure. Consideration reveals that the stable solution of the MRP for t > 0 is independent of the slopes of the "V" and that, as a result, said solution is also the stable solution of the conventional Riemann problem with the same initial densities. In other words, the stable flows arising at the discontinuity are the same for the conventional and modified Riemann problems. Since (20) has the same form as (7), i.e., it is the minimum of a sending and a receiving value, it expresses the stable flow of a Riemann problem with ku = K(t-fje, Xj) and kd = K(t-f j+1 e, xj+1) . As we have just shown, this is also the solution of a well-posed MRP with the same initial densities. In particular, it is the stable flow for the MRP in which the "V" passes through the lattice points at which ku and kd have been evaluated (points "A" and "B" in Fig. 5). This MRP will be well-posed if the "V" passing through the lattice points is inside the shaded
The Lagged Cell-Transmission Model
95
= 4
A"
«*
max
^"^-^~^
s S*, max
'
f
a, II
\"B* Figure 5. The modified Riemann problem at the core of the LCT. wedges; i.e., if (17) and (18) are satisfied. Thus, when (17) and (18) are satisfied the LCT method is nothing but a recursive solution of well-posed MRP's, just like the Godunov/CT method involves the recursive solution of ordinary Riemann problems. Insofar as the stable solution is always used in both procedures, we can conclude that the LCT method should share the good stability properties of the conventional CT method. The advantage of the former is that it is based on "older" data, which should be less corrupted by numerical errors.8 The following section presents some examples that illustrate both the accuracy and stability properties of the LCT method.
4. EXAMPLES Let us start by examining the accuracy of the LCT model with both smooth and discontinuous initial conditions. 8
The accuracy formulae of Sec. 3.3 confirm that the LCT method is most accurate when f, and <> j+ i are chosen to be as large as possible, i.e., when the "V" forms as acute an angle as possible.
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Tables 1 and 2 present 17 iterations of the conventional and lagged cell transmission models for a triangular flow-density relation of the form: q = min{k, (180-k)/5),
(21)
where k is in veh/mile and q in veh/min. It is assumed that the initial density profile is quadratic and in the congested range k e [30, 180] ; more specifically, that k(0,x) = 50 + Vix2 and x ~ 10 ± 5. Because the (congested) wave velocity is constant (i.e., Rk = -0.2) the exact solution of the problem in this range of x for small t is: k(t, x) = 50 4-VXx-f t/5)2. That is, in the exact solution the density at mile x-1 must equal the density that prevailed at mile x five minutes earlier. Table 1 presents the results of the ordinary CT model when one uses e = 1 and d = 1. In the exact solution the numbers in boldface would be equal, so that the observed discrepancy is the CT error. The discrepancies observed in the table are consistent with what would be expected from (16) with { = 0, and from the more detailed analysis in Daganzo (1993a). If one uses the LCT model with e = 1, d = 1 and f = 2 , i.e., the value recommended by (12), then Table 2 is obtained. In this case the LCT model reproduces the exact results, as one would expect from (16) and the analysis leading to it.9 Of course, the performance of the LCT procedure deteriorates in less favorable cases, e.g., with non-quadratic density profiles, non-linear T(k) functions, and non-ideal lags. This is illustrated by Fig. 6, which contains the initial and final density profiles (at t = 25 min) for a Riemann initial value problem where (21) holds and where the initial density changes suddenly from k = 50 to k = 100 at x = 20 mile. The thin lines are the CT and LCT results obtained with e = 1 min and d = 1 mile. Note the lesser spread of the LCT result, and that the LCT model does not transition monotonically from one density value to the other.10 The relative accuracy of the two models can be evaluated better with the cumulative curves of vehicle count; see Fig. 7. Whereas in the CT model the maximum error in vehicle position is on the order of 2/3 miles (which is not surprising since d = 1 mile), with the LCT the largest error is less than 1/5 mile. In order to illustrate the stability of the method for non-concave T(k) relations, Figs. 8 and 9 depict the LCT solution of the lead vehicle problem in Fig. 2c using cells with d = 100 m and time steps e = 2 sees. This is efficient because with this arrangement (8) is satisfied as an equality — note from Fig. 2a that in our case |Tk max = vf =50 m/s. Since |Rk max = 350/13 m/s , the
9
Note that, for this to be the case, 0+1 time slices of internally consistent initial data had to be specified. 10
This undesirable feature of the LCT model is typical of second order approximations (see, e.g., LeVeque, 1992). As a result, the LCT model may produce densities slightly greater than the theoretical maximum from time to time. Therefore, the receiving function R(k) should be defined for 0 < k < °° , in an implementation.
The Lagged Cell-Transmission Model
97
Table 1. Estimated densities at different positions with the cell-transmission model. Row 3 is the position (x) in Km. Rows 5 and 6 are the maximum flow and the jam density at the given position. Rows 9 to 11 are the initial data, k = 50 + V^x +1/5)2, where t = 0 for row 9, t = 1 for row 10 and t = 2 for row 11.
Table 2. Estimated densities at different positions with the lagged cell-transmission model. Row 3 is the position (x) in Km. Rows 5 and 6 are the maximum flow and the jam density at the given position. Rows 9 to 11 are the initial data, k = 50 + 1A(\ + t/5)2, where t = 0 for row 9, t = 1 for row 10 and t = 2 for row 11.
Figure 6. Evolution of a discontinuous traffic disturbance as predicted by the CT and LCT models for the case of a triangular T(k) curve with p = 0.2: density profile comparison..
optimum lag according to (12) should be f = 3/7 . Because this is not an integer, R(k) was evaluated for a k that was an interpolation of the densities obtained with { = 0 and C = 1. The numerical results should then still be second order accurate for densities less than or equal to that of state "E" in Fig. 2a, and less accurate for more congested states. The density profiles of Fig. 8 confirm this. Note how the curves have sharper steps below the line K = 0.5 than near the top. (In the exact solution these curves would be perfect step functions.) Note as well the good agreement between this figure and 2c; in particular, the development of intermediate state "E", with K = 0.4, after t = 120 sees. Figure 9 displays the N-curves (of vehicle number) at five locations in ¥2 Km increments upstream of the stoppage. They also agree qualitatively with the exact solution of the problem,
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900
750
o 700
650
Figure 7. Evolution of a discontinuous traffic disturbance as predicted by the CT and LCT models for the case of a triangular T(k) curve with p = 0.2: cumulative vehicle count comparison.
which in this case is piecewise linear. Clearly, the curved arcs in the figure, which correspond to episodes of very congested traffic, have some numerical error. However, the straight portions of the curves match the exact solution precisely, and therefore it is easy to ascertain from the figure the magnitude of the numerical errors in the curved sections. This error does not exceed 5 vehicles in any of the curves.
The Lagged Cell-Transmission Model
101
5. CONCLUSION The LCT procedure is also well suited for modeling intersections and inhomogeneous highways, since the only change needed in the existing procedures is reading the traffic density for the downstream conditions with a time lag. This modification is so minor that it can also be applied to highways with special lanes and their junctions; e.g. by modifying the procedure described in Daganzo et al. (1995). It should also be said that lags impose additional memory storage requirements on an LCT simulation, since cell densities must be stored for the past 1+fj time slices. For multi-destination networks, however, most of the storage is consumed by the cell content proportions (by destination and entry time), which are only used as arguments as part of the "sending" functions. Thus, in an LCT implementation the bulk of the information would only have to be kept for 1+fj time slices.
0.1
•t = 0s -1 = 40 s t = 80 s -t=120s -t=160s
0.075
'w c
0.05
0.025
0
10
20
30
40
50
60
70
Distance, x(100 m)
Figure 8. Density profiles for the lead vehicle problem of Fig. 2.
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N-plot for the lead vehicle problem
..-
,•*
,'"
Vehicle number i3 J
/
/ ,--—
B
///
'{/ '^ / /
,.
^
,---—
V//
'/ 0
40
a3
120
160
200
240
280
320
360
Time (sees)
x=OKm x = -.5Km x=-1 Km x = -1.5Km - • • x = -2.0Km
Figure 9. N-curves for the lead vehicle problem of Fig. 2.
REFERENCES Bui, D.D., P. Nelson and S. L. Narasimhan (1992) Computational realizations of the entropy condition in modeling congested traffic flow. FHWA Report TX-92/1232-7. Cassidy, M. (1998), Bivariate relations in highway traffic. Trans. Res. 32B(1), 49-59.
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Daganzo, C.F. (1993) The cell-transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory. Institute of Transportation Studies, Research Report, UCB-ITS-RR-93-7, Univ. of California, Berkeley, CA. Short version in Trans. Res., 28B(4), 269-287. Daganzo, C.F. (1993a) A finite difference approximation for the kinematic wave model. Institute of Transportation Studies, Research Report, UCB-ITS-RR-93-11, Univ. of California, Berkeley, CA. Short version in Trans. Res. 29B(4) 261-276. Daganzo, C.F. (1994) The cell-transmission model. Part II: Network traffic. PATH working paper UCB-ITS-PWP-94-12, Univ. of California, Berkeley, CA. Short version in Trans. Res., 29B(2), 79-94. Daganzo, C.F. (1995) A continuum theory of traffic dynamics for freeways with special lanes. PATH Technical Note UCB-ITS-PTN-95-08, Univ. of California, Berkeley, CA. Short version in Trans Res. 31B (2), 83-102. Daganzo, C.F. (1997) An enhanced cell-transmission rule for traffic simulation. Institute of Transportation Studies Research UCB-ITS-RR-97-06, Univ. of California, Berkeley, CA. Daganzo, C.F., Lin, W.H. and del Castillo, J.M. (1995) A simple physical principle for the simulation of freeways with special lanes and priority vehicles. PATH Technical Note UCB-ITS-PTN-95-09, Univ. of California, Berkeley, CA. Short version in Trans. Res. 31B (2), 105-125. Godunov, S.K. (1961) Bounds on the discrepancy of approximate solutions constructed for the equations of gas dynamics. J. Com. Math, and Math. Phys. 1, 623-637. Herman, R., E.W. Montroll, R.B. Potts, and R.W. Rothery (1959), "Traffic dynamics: Analysis of stability in car following", Opns. Res. 1, 86-106. Lax, P.D. (1973) Hyperbolic systems of conservation laws and the mathematical theory of Shockwaves. Regional Conf. Series in Applied Mathematics. SLAM, Philadelphia, PA. Lebacque, J.P. (1993) Les modeles macroscopiques de trafic. Annales de Fonts, (3rd trim) 67, 2845.
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LeVeque, R.J. (1992) Numerical methods for conservation laws, (2nd edition), Birkhauser-Verlag, Boston, MA. Lighthill, M.J. and G.B. Whitham (1955) On kinematic waves. I flow movement in long rives. n A theory of traffic flow on long crowded roads. Proc. Roy. Soc. A, 229, 281-345. Luke, J.C. (1972) Mathematical models for landform evolution. J.Geophys. Res., 77, 2460-2464. Newell, G.F. (1961) Non-linear effects in the dynamics of car-following. Opns. Res. 9(2), pp. 209-229. Newell, G.F. (1993) A simplified theory of kinematic waves in highway traffic, I general theory, n queuing at freeway bottlenecks, HI multi-destination flows. Trans. Res., 27B, 281-313. Richards P.I. (1956) Shockwaves on the highway. Opns. Res., 4, 42-51. Windover, J. and Cassidy, M.J. (1997) Private communication.
CHAPTER 2 TRAFFIC FLOW BEHAVIOUR
•
Common sense is the least common of all senses.
• Doubt is not a very pleasant status but certainty it is a ridiculous one. (Voltaire) • Every extension of knowledge arises from making the conscious the unconscious. (Friedrich Nietzsche)
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OBSERVATIONS AT A FREEWAY BOTTLENECK Michael J. Cassidy and Robert L. Bertini Department of Civil and Environmental Engineering and Institute of Transportation Studies, University of California at Berkeley
ABSTRACT Traffic was studied upstream and downstream of a bottleneck on a freeway in Toronto, Canada using transformed curves of cumulative vehicle count and cumulative occupancy. The bottleneck was located more than a kilometer downstream of a busy on-ramp. After diagnosing its location and the times that it remained active each day, the study focused on the traffic patterns that arose in each travel lane. It was observed that prior to the bottleneck's activation, the vehicles' lane-changing trends created extraordinarily high flows in the median (i.e., left-most) lane and that these high flows were sustained for extended durations. When a queue eventually formed at the bottleneck, its discharge rates were considerably lower than those flows measured prior to queueing. Within each lane, the queue discharge rates remained nearly constant over the rush and the average rates varied only slightly across days. Finally, vehicles arriving to the bottleneck from the nearby upstream on-ramp entered the freeway at high rates, even after the bottleneck's queue propagated beyond this ramp.
1. INTRODUCTION In an earlier study, the authors examined traffic at two bottlenecks on freeways in and near Toronto, Canada (Cassidy and Bertini, 1999). From this study, certain reproducible patterns were observed. For example, each bottleneck formed more than a kilometer downstream of an on-ramp and their formations occurred at these same locations each day. While the bottlenecks
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were active,1 the vehicles discharged through them at nearly constant rates, although some time dependencies were observed for short periods following the onset of queueing. Furthermore, a bottleneck's average queue discharge rate did not vary much from one day to the next and these average rates were typically 8 to 10 percent lower than the flows measured prior to queueing upstream. These earlier findings came by visually comparing sets of transformed cumulative curves. Each curve was constructed from either the counts or the occupancies collected at one of several neighboring loop detector stations. Of note, these curves described measurements that were taken across all travel lanes, meaning that the above findings came by grouping together the traffic streams in multiple lanes and studying them in the aggregate. In this paper, we add to the previous findings on bottleneck flow by reporting on some observations taken from individual lanes. At each of three neighboring detector stations located upstream and downstream of a bottleneck, curves of cumulative vehicle count and cumulative occupancy were separately constructed for each travel lane. Visual comparisons of these curves revealed certain details of traffic evolution, some of which were unexpected. It was observed, for example, that large numbers of vehicles gradually moved into the median lane as they approached and passed through the bottleneck. This lane-changing pattern even continued at locations more than 2 kilometers downstream of the neighboring on-ramp. As a consequence, flows measured in the median lane were remarkably high; e.g., at a location well downstream of the on-ramp, the median lane flows sometimes exceeded 2,600 vehicles per hour. These high rates persisted each day for durations of up to 40 minutes before queues formed upstream and lower discharge rates ensued. The bottleneck's queue formed at nearly, but not exactly, the same times in each lane; i.e., this formation occurred in the shoulder lane several minutes after it had occurred in the adjacent lanes. Following the queue formations, the flow reductions observed downstream were most pronounced in the median and center lanes and less so in the shoulder lane. The discharge rates remained nearly constant so long as the bottleneck was active and free of any incidents nearby. Although these average rates varied across lanes, each lane's average was reproduced from day to day. It was further observed that vehicles entered the freeway from the upstream on-ramp at very high rates, even after the bottleneck's queue propagated beyond the ramp and obstructed this flow. Thus, the on-ramp vehicles did not share the available capacity with vehicles in the 1
The term "active bottleneck" denotes that the discharge rates measured downstream of a queue were not affected by traffic conditions from further downstream (Daganzo, 1997).
Observations at a Freeway Bottleneck
109
adjacent freeway lane in a strictly alternating or "one-to-one" basis. Rather, motorists from the on-ramp forced themselves into the queue in such a way that the gaps between the freeway vehicles were filled by multiple on-ramp vehicles. The following section provides descriptions of the bottleneck and of the loop detector data used in this study. Section 3 describes the use of cumulative curves to identify the bottleneck's location and the times that it remained active; uncovering these details was a requisite step for studying the evolution of the bottleneck flows. Section 4 presents the bottleneck's traffic patterns that were observed in each lane during a single rush. Some findings from repeating these analyses with data from other days, and comments regarding future research directions, are provided in the fifth and final section.
2. THE DATA The segment of the Gardiner Expressway shown in Figure 1 was the site used in our study. As an aside, this site has been used in several earlier studies of freeway capacity (Persaud, 1986; Persaud and Hurdle, 1991; Persaud, et al., 1998), including the previous one by the authors (Cassidy and Bertini, 1999). It is located in Toronto, Canada and meters are not deployed on its on-ramps (although the Jameson Avenue on-ramp is closed during a portion of each afternoon rush).
Loop Detector Detector Station
Figure 1: Gardiner Expressway, Toronto, Canada
The loop detector stations for measuring traffic data are labeled in Figure 1 as per the numbering strategy adopted by the City of Toronto. These detectors record counts, occupancies and (time) mean speeds in each lane over 20-second intervals. In total,
110
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measurements were made during three weekday afternoons when the local weather bureau reported clear skies and no measurable precipitation. In the next section, data from one of the observation days are used to demonstrate that a bottleneck was activated between detector stations 60 and 70. This bottleneck remained active for more than two hours before a queue spilled over from further downstream and restricted its discharge. We will also show that, some minutes prior to its deactivation, the bottleneck's flow was impeded by an incident that occurred near detector 50.
3. THE ACTIVE BOTTLENECK Figure 2 presents transformed curves of cumulative count, N, versus time, t, for detectors 40 through 80. These were constructed using the vehicle counts taken in all lanes during a 30minute period that spanned the onset of queueing.2 Untransformed JV-curves give the cumulative number of vehicles to have passed (detector) location x by t. By constructing the curves as linear interpolations through the cumulative counts measured every 20 seconds, each curve's slopes would be the flows past x during the 20-second measurement intervals. Moreover, since the counts for each curve in Figure 2 started (N = 0) with the passage of a reference vehicle, the horizontal and vertical separations between the curves would have been the trip times and the vehicle accumulations between detectors, respectively (Newell, 1982; Newell, 1993). In Figure 2, however, each curve, along with its corresponding time axis, was shifted to the right by the average free-flow trip time between the respective detector and downstream detector 80. Consequently, the vertical separations in the curves are the excess vehicle accumulations between detectors due to vehicular delays. These shifts are advantageous because two superimposed curves indicate that traffic in the intervening segment was freelyflowing; every feature of an upstream W-curve is passed to its downstream neighbor a free-flow trip time later. In addition, Figure 2 shows only the differences between each curve of cumulative count and the line N-qo-1', where go is the rate used to re-scale the curves and t' is the elapsed time from the start of each curve. This is important because reducing the curves' cumulative count by a background flow qo magnifies details, such as the time dependencies in the flows, without changing the excess accumulations (Cassidy and Windover, 1995).
The on-ramp counts at Spadina Avenue were also used to construct the curve for detector 40 so that all of the curves in Figure 2 describe the same collection of vehicles. Conversely, a curve for detector 30 was not included in Figure 2, since the vehicles measured at this location were not identical to those measured at the detector stations further downstream.
111
Observations at a Freeway Bottleneck
The (nearly) superimposed curve portions in Figure 2 indicate that traffic was initially in free flow and remained in free flow between detectors 70 and 80. The reader may use a straightedge to confirm that curves 40, 50 and 60 exhibit increased slopes sometime shortly after 15:40, as delimited by the large arrow in Figure 2, and this caused these three curves to
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Transportation and Traffic
112
Theory
diverge from their (two) downstream counterparts. These curve features indicate that a bottleneck was activated between detectors 60 and 70 when increased flows arrived from upstream. The subsequent separation between curves 60 and 50 (at about 15:51:23) reveal when the queue arrived at detector 60. Likewise, the queue's arrival at detector 50 is made evident by the divergence in curves 50 and 40 (at about 15:53:03). In short, the transformed TV-curves in Figure 2 conclusively diagnose the bottleneck's location by showing that excess vehicle accumulations occurred upstream of detector 70 while freeflow conditions prevailed immediately downstream. Especially notable are the pronounced flow reductions (i.e., the reduced slopes of the TV-curves) that followed the queue's formation; further details on this are provided in the next section.
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Figure 3: Re-scaled 7V- and T-curves, Detector 40 Figure 3 reveals the approximate time that the backward-moving queue arrived at the (freeway) detectors at station 40. Presented in this figure is a re-scaled TV-curve for the freeway lanes at this detector station along with a re-scaled curve of cumulative occupancy versus time, or a Tcurve, where cumulative occupancy is the total vehicular trip time over the detectors by time t (Lin and Daganzo, 1997). Again for the purpose of magnifying details, the T-curve shown here is the difference between the cumulative occupancy actually measured in the three freeway lanes at detector 40 and the line T=b0- t', where b0 is the background occupancy rate used to
Observations at a Freeway Bottleneck
113
re-scale the curve and /' is the elapsed time from the curve's start. The two curves show that a sharp reduction in flow was followed closely by an increase in the occupancy rate at about 15:54:23. These features reveal the arrival of the queue at station 40. That this arrival occurred some time around 15:54 will be an important part of later discussion regarding the observed time dependencies in the on-ramp flows from Spadina Avenue. Finally, Figures 4a and 4b reveal the period that the bottleneck remained active. Figure 4a shows transformed TV-curves for detectors 60 and 80; these were constructed in the manner previously described, but for an extended period of over 4 hours. The slope of curve 80 drops noticeably some time around 18:00 and a similar reduction is displayed by curve 60 soon thereafter. Notably, the queue between detectors 60 and 80 persisted, even after these flow reductions; this is evident from the continued displacement between the two curves. These curve features indicate that a queue from further downstream arrived at these detector stations and thereby deactivated our bottleneck. In Figure 4b, the divergence in the re-scaled curves of TV and rreveal that the queue from downstream arrived at station 80 at about 18:12:03. Before concluding this section, it is worth re-emphasizing that the curves in Figures 2 and 4a were instrumental in identifying the bottleneck's location and the period that it remained active. These curves derived their value, in part, by displaying the excess vehicle accumulations that arose between detectors and this required that the curves be constructed from the counts taken over all travel lanes.3 Having now identified these bottleneck details (i.e., its location and the time it was active), traffic patterns could be studied in the individual lanes at locations upstream and downstream of the bottleneck. Some notable findings from this study are presented next.
4. SOME OBSERVATIONS IN INDIVIDUAL LANES Figure 5 presents re-scaled TV-curves in the shoulder lane for detectors 60, 70 and 80. Figures 6 and 7 present the re-scaled curves for these same detector stations in the center and median lanes, respectively. These three detector stations (i.e., 60, 70 and 80) were selected because they were situated immediately upstream and downstream of the bottleneck. In each figure, the curves span a period of more than 3 hours, which includes the time that the bottleneck was active. Before presenting some of the findings obtained by jointly examining Figures 5, 6 and 7, a brief explanation of their annotations is warranted. First, piece-wise linear approximations
If a set of TV-curves do not describe node conservation, their vertical displacements would not be the excess accumulations (Newell, 1982; Newell, 1993; Cassidy and Windover, 1995).
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Transportation and Traffic Theory
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Observations at a Freeway Bottleneck
115
were superimposed on the N to highlight periods of nearly constant flow; the N usually deviated from its corresponding linear approximation by no more than about 10 vehicles. The start and end times for each period of near-constant flow were selected "by eye" and re-scaled 7-curves (not shown here) were also used to aid in these delimitations, since (sizable) changes in flow are accompanied by changes in occupancy. The times marking these flow changes are labeled on each curve.4 Further, the times marking the onset and the termination of queue discharge flows are noted in boldface type for the curves at downstream stations 70 and 80. Also shown are the rates corresponding to each period of near-constant flow; the numbers shown without parentheses are in units of vehicles per hour (vph) and those in parentheses are the corresponding average counts per minute. Finally, dotted lines are used to highlight the average queue discharge rates measured over the rush at downstream detectors 70 and 80. Labels specifying these average discharge flows are shown on the figures in boldface type. We now turn our attention to the traffic patterns revealed by Figures 5 through 7. From even cursory examination of these figures, one observes a trend in vehicle lane changing; namely, that large numbers of vehicles changed lanes to the left while they traveled between detectors 60 and 80. The curves in Figure 5, for example, were each started with the same value of N, but curve 60, N(60, t), lies well above curve 70 for all time t. In similar fashion, curve 70 rises above curve 80, indicating that vehicles continued to exit the shoulder lane at locations well downstream of the Spadina Avenue on-ramp. Figure 6, on the other hand, shows no such obvious trend. Rather, the (net) flows in the center lane remained nearly unchanged as traffic moved through the bottleneck, indicating that, between detectors, the number that moved into the center lane nearly equaled the number that moved out. In fact, the Figure 6 curves have been vertically displaced (by arbitrary distances) because not separating these curves would have made it difficult to view their details. Figure 7 shows that large numbers entered the median lane between detectors 60 and 80 and that this trend gave rise to some extraordinarily high flows. For example, a flow in excess of 2,600 vph was measured in the median lane at detector 80 prior to the bottleneck's activation. Remarkably, this very high rate was observed for over 40 minutes before the queue formed upstream and a lower discharge rate ensued. (The queue's formation is signaled by the onset of its discharge flow at detector 70). The onset of queueing occurred at nearly, but not precisely, the same times in each lane; i.e., the queue appears to have formed in the shoulder lane several minutes after it formed in the adjacent lanes. The curves at detector 60 (in all three lanes) exhibit sustained surges some 4
A few of these times are not labeled in Figure 5 so that the figure would not become cluttered.
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Theory
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17:19:03
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Observations at a Freeway Bottleneck
119
minutes prior to the queue's formation. Conversely, the onset of this queue was accompanied by rather dramatic flow reductions. Figures 5 through 7 show that the periods immediately following the bottleneck's activation were marked by some of the lowest discharge rates observed during the rush, but that these relatively low flows were short-lived relative to the rush. In the center and median lanes (Figures 6 and 7), these so-called flow collapses (Cassidy and Windover, 1995) prevailed for just under 20 minutes before being replaced by higher discharge flows. In the shoulder lane (Figure 5), the collapse persisted for less than 10 minutes. Also of note, Figures 5 through 7 show that sizable reductions in the discharge flows were measured in all lanes some minutes before the bottleneck was deactivated by the arrival of the queue from downstream (recall that this queue arrived at detector 80 at about 18:12). Visual inspection of N- and T-curves measured in individual lanes revealed that these flow drops were caused by an incident (perhaps a vehicle stall or a small collision) that occurred in the shoulder lane near detector station 50. Notably, detector 50 measured near-zero counts and occupancies in the shoulder lane from 18:02:23 to 18:08:23. Figure 8a shows that during this period, the shoulder lane traffic at upstream station 40 exhibited a sharp reduction in the flow coupled with a rise in the occupancy, features which mark the passage of a queue. Conversely, Figure 8b shows that, within this period, shoulder lane traffic at downstream station 60 exhibited sudden reductions in both the flow and the occupancy. These features would be expected to occur downstream of a sudden restriction (i.e., as traffic passed the obstruction and moved into the shoulder lane). A reduction in the discharge flow also occurred in the shoulder lane at about 16:59 (see Figure 5), but this reduction was short-lived and of no real consequence. Despite the flow variations that occurred during the bottleneck's active period, Figures 5, 6 and 7 reveal that the queue discharge rates never deviated much from the linear trends shown with the dotted lines (the reduced discharge flows that accompanied the incident near detector 50 were excluded from consideration here). Thus, for each lane, the discharge rates can be described as being nearly constant over the rush. By comparing the average rates that correspond to each dotted line with the flows that prevailed prior to the bottleneck's activation, it is clear that queueing was accompanied by long-run flow reductions, especially in the median and center lanes. Also by examining these three figures collectively, it is evident that the average queue discharge rates varied across lanes. Finally, Figure 9 shows a re-scaled TV-curve constructed solely from the counts on the Spadina Avenue on-ramp (at detector station 40). The sustained surge in the ramp flow evident at 15:42:43 corresponds to the measured increase in upstream flow previously revealed in Figure
Transportation and Traffic
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Theory
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Observations at a Freeway Bottleneck
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2.5 Likewise, the sharp reduction in this ramp flow at 15:50:23 corresponds closely to the arrival of the queue from our active bottleneck; recall that this queue was shown (in Figure 3) to have arrived to the freeway detectors at station 40 some time shortly after 15:50. Although this queue apparently suppressed the on-ramp flow, vehicles continued to enter the freeway via this ramp at a high rate; i.e., an average ramp flow of 1,650 vph persisted for nearly an hour. This flow dropped at about 16:46:43, perhaps due to a reduction in the on-ramp demand (although additional ramp detectors do not exist to confirm this). In any event, these high ramp flows mean that, just downstream of the merge, more than half of the vehicles traveling in the shoulder lane originated from the on-ramp. Thus, the merging process did not exhibit the socalled "zipper effect" (Newman, 1986) whereby freeway and ramp vehicles share the shoulder lane in a strictly alternating fashion.
Study of the N-curves constructed from freeway counts at detector stations 30 and 40 (not shown here) revealed these increased flows observed in Figure 2 were part of sustained surges in the freeway flows as well as in the flow from the Spadina Avenue on-ramp.
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Transportation and Traffic Theory
5. FINDINGS FROM REPEATED EXPERIMENTS AND FUTURE RESEARCH DIRECTIONS Data from detectors 40 through 80 were extracted during two other weekday afternoons and were examined in the manner previously described. On each of these two additional days, the observed traffic patterns were similar to those presented above. As a means of exemplifying some of these day to day similarities, Table 1 presents certain observations taken each day at detector 80. Row 1 of this table shows that, in the median lane, very high flows were observed each day prior to queueing. The table's second row reveals that these high rates were always sustained for periods of at least 5 minutes, and in two instances, for much longer. Row 3 of Table 1 shows that the flow collapse at the onset of queueing was a reproducible feature in the median lane (note that the rates shown in this row are lower than their corresponding average discharge rates measured over the rush and presented in row 5). However, these collapses persisted for durations that varied across days, as shown in row 4 of the table. The flow collapse was likewise reproduced each day in the center lane, although this information is excluded from the table. Important features also not shown in Table 1 are that the bottleneck always formed at the same location (i.e., between detectors 60 and 70) and that it was always activated by a sustained surge in the flow from upstream. Thus, traffic transitioned from free flow to queued conditions in a predictable way; the queues formed at an inhomogeneity, the bottleneck, due to reproducible, exogenous reasons, i.e., the increased flows. In this instance, one might presume that the freeway's horizontal curve is the inhomogeneity creating the bottleneck (see Figure 1). While this may indeed be the case, it is worth noting that the same analysis methods applied to data from another freeway location found that a bottleneck consistently formed more than a kilometer downstream of an on-ramp, even though there was no obvious inhomogeneity at this location (Cassidy and Bertini, 1999). In any event, our studies to date have revealed no evidence suggesting that traffic can break down and form queues in a spontaneous manner. Also of note, rows 5, 6 and 7 of Table 1 indicate that, while the bottleneck was active, the average discharge flow (in a given lane) exhibited only small variation across days. On each day, these rates can be described as "near-constant" since the cumulative counts never deviated much from a linear trend. Given these predictable features of its discharge rates, it seems reasonable to postulate about how queues might evolve upstream of this bottleneck; e.g., by using a continuum model of highway traffic (Lighthill and Whitham, 1955; Richards, 1956; Newell, 1993).
Observations at a Freeway Bottleneck
123
Table 1 Some Observations Taken From Station 80 D a y l Day2
ROW ~1 2 3 4
Day3
3/5/972/20/97 7/21/97 Maximum Flow (vph) - Median Lane 2630 2630 2400 Measured Duration of Maximum Flow (min: sec)-Median Lane 43:40 5:40 19:40 Flow Collapse (vph) - Median Lane 2280 2110 2300 Measured Duration of Flow Collapse (min: sec)-Median Lane 19:20 7:00 5:20
5 Average Queue Discharge Rate (vph) - Median Lane 6 Average Queue Discharge Rate (vph) - Center Lane 7 Average Queue Discharge Rate (vph) - Shoulder Lane
2340 2290 2330 1920 1910 1950 1720 1690 1690
Note: Data from Day 1 were used in Figures 2 through 9. In closing, we note that the observations reported above bring to light a number of unanswered questions. For example, the reason(s) why the observed lane-changing trends persisted well downstream of the on-ramp, and the extent to which the high flows observed in the median lane might be reproduced at other bottlenecks, are unknown. Also unknown are the causes of the flow reductions that accompanied queueing, especially the relatively large reductions at the onset of queueing. Finally, the potential for using control measures, such as ramp metering, to extend the periods marked by high flows (observed prior to queueing, for example) is uncertain. Answers to the above will only come through additional empirical study. Since (freeway) bottlenecks come in many forms, including merges, diverges, weaves and lane reductions, and since the traffic patterns on each type of bottleneck may exhibit their own peculiarities, the study of bottlenecks in each of their forms seems warranted. Cumulative curves like those described here might be used to conduct these studies since they provide a robust way of diagnosing the details of bottleneck traffic.
ACKNOWLEDGEMENTS The authors are indebted to Mr. David Nesbitt, City of Toronto, for providing the data used in this study, and to G.F. Newell for his helpful comments.
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Transportation and Traffic Theory
REFERENCES Cassidy, M.J. and R.L. Bertini (1999). Some traffic features at freeway bottlenecks. Transpn. Res., 33B, 25-42. Cassidy, M.J. and J.R. Windover (1995). Methodology for assessing dynamics of freeway traffic flow. Transpn Res. Rec., 1484, 73-79. Daganzo, C.F. (1997). Fundamentals of transportation and traffic operations. Elsevier, New York, p. 133. Lighthill, M.J. and G.B. Whitham (1955). On kinematic waves. 7: Flood movement in long rivers. //: A theory of traffic flow on long crowded roads. Proc. Royal Soc., A229, 281345. Lin, W.H. and C.F. Daganzo (1997). A simple detection scheme for delay-inducing freeway incidents. Transpn Res., 31A, 141-155. Newell, G.F. (1982). Applications ofqueueing theory. Chapman Hall, London. Newell, G.F. (1993). A simplified theory of kinematic waves in highway traffic 7: General theory. 77: Queuing at freeway bottlenecks. 777: Multi-destination flows. Transpn Res., 278,281-313. Newman, L. (1986). Freeway operations analysis course notes. Institute of Transportation Studies, University Extension, Univ. of California, Berkeley, U.S.A. Persaud, B.N. (1986). Study of a freeway bottleneck to explore some unresolved traffic flow issues. PhD thesis, Univ. of Toronto, Toronto, Canada. Persaud, B.N. and V.F. Hurdle (1991). Freeway capacity: definition and measurement issues. Proc., International Symposium of Highway Capacity, A.A. Balkema press, Germany, 289-307. Persaud, B., S. Yagar and R. Brownlee (1998). Exploration of the breakdown phenomenon in freeway traffic. Transpn Res. Rec., 1634, 64-69. Richards, P.I. (1956). Shock waves on the highway. Opns. Res., 4, 42-51.
125
FLOWS UPSTREAM OF A HIGHWAY BOTTLENECK
Gordon F. Newell Department of Civil and Environmental Engineering and Institute of Transportation Studies University of California at Berkeley
ABSTRACT Suppose that the local capacity of a highway is a smooth function of location, approximated by a parabolic function with a minimum value at some location (the bottleneck). The flow approaching the bottleneck increases approximately linearly with time as it exceeds the capacity of the bottleneck. We present here an analytic solution for the resulting flow pattern upstream of the bottleneck as predicted by the theory of Lighthill and Whitham (1955) for two different types of analytic forms for the relation between flow and density. Although, in each of the two cases, the formulation of the problem contains seven parameters, it is shown that, by appropriate linear transformation of variables, the flow pattern can be described in terms of a single dimensionless pattern. In each case, a shock first forms at some point upstream of the bottleneck with an amplitude which increases proportional to the square root of the time from its beginning.
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Transportation and Traffic Theory
1. INTRODUCTION Suppose that the capacity at each location along a section of highway is a smoothly varying function of location, as might be the case if the highway curves or changes grade, and that the capacity has a minimum value at some location which we arbitrarily identify as x = 0. Over some distance in the vicinity of the bottleneck, x = 0, we will assume that the capacity Qm(x) can be approximated by a quadratic function of the form
Qm(x) = q0 + Ax2
(1.1)
with q0 the capacity of the bottleneck and A some positive constant describing how rapidly the capacity changes with the distance from the bottleneck. Suppose also that at some location x = - L , well upstream of the bottleneck, there is a flow q(-L, t) approaching the bottleneck. This flow is increasing with time and at some time t = t0 becomes equal to the capacity, q0. Over some interval of time near to we will assume that the flow is increasing (nearly) linearly with time so we can approximate q(-L,t) = q0 + B(t-to) for some positive constant B describing the rate of increase of the flow.
(1.2)
A flow larger than q0 certainly cannot pass the bottleneck nor can it pass any point upstream oi the bottleneck where that flow exceeds the local capacity. As this flow approaches the bottleneck vehicles accumulate behind the bottleneck. The accumulation will eventually create a shock which propagates upstream. Lighthill and Whitham, L-W, (1955, p. 333) gave a qualitative description of how the shock forms and propagates upstream based upon their theory of kinematic waves. In this theory it is i postulated that there is a specified functional relation between the density, k(x, t), and the flo\\
q(x, t), q(x,t) = Q(k(x,t),x).
(1.3)
For any fixed x, Q(k, x), is a concave function of k having a maximum with respect to k, th< capacity Qm(x) at that location. L-W did not specify any specific form for Q(k, x) but it wa; assumed to be a smooth function of k with a locally parabolic maximum with respect to k, and tha Qm(x) had a minimum at the bottleneck.
Flows Upstream of a Highway Bottleneck
127
Our objective here is to give an analytic solution for q(x, t) based on the L-W theory for two hypothetical forms for the function Q(k, x). In the first case it is assumed that Q(k, x) for fixed x has a triangular shape, as illustrated in figure 1 for several values of x. The slope Q(k, x)/k on the left hand side of the triangle is the "free speed" v0 assumed to be independent of x. The slope on the right hand side of the triangle, the (negative) wave velocity, is also assumed to be independent of x. The maximum height of the triangle is Qm(x) as in (1. 1). Thus the family of Q(k, x) curves for various x differ only by a scaling factor Q(x) for both q and k.
Q(x) CT
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ko Density, k
Density, k Fig. 1. Triangular q-k relations
Fig. 2. Parabolic q-k relations In the second case, it is assumed that the function Q(k, x) has a local parabolic maximum with respect to k at some density ko and some specified curvature. The ko and curvature are both independent of x. Thus for k in some vicinity of k, we can, approximate Q(k, x) by Q(k, x) = q0 + Ax2 - C(k - ko )2
(1.4)
for some positive constant C. This form is illustrated in figure 2. We will show for each of the two forms for Q(k, x) that, by appropriate choice of units and coordinates, q(x, t) can be described in terms of a single dimensionless function q*(x*, t*) of a dimensionless length x* and time t*, independent of the parameters q0, A, B, C, L etc. The function q*(x*, t*) can also be evaluated analytically. Similar methods could be applied also to other types of q - k relations.
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2. TRIANGULAR Q - K RELATION For the triangular q - k relation illustrated in figure 1, it is advantageous first to introduce a moving time origin traveling at velocity v0. If we measure time at each location x from the time that some reference vehicle traveling at velocity v0 would pass (Newell 1993II) t' = t-x/v0
then the relation between q' and k' would have the form illustrated in figure 3 in which the "free speed" is, in effect, infinite in the new coordinates. The negative "wave pace" (- w0), the reciprocal of the wave speed, is (- l/v0) plus the corresponding wave pace of figure 1, with w0 independent of
This eliminates the parameter v0, in the sense that q'(x, t') is a solution of the L-W equations for the q' - k' relation of figure 3, which is independent of v0. According to the L-W theory, the flow q(x, t') in the absence of shocks must be a constant q' along "characteristic curves" which travel at the wave velocity associated with the flow q'. For the q' - k' relation of figure 3 there are only two possible values for the wave pace 0 or -w,. Some characteristic curves are illustrated in figure 4. (The numerical values of the flow, distance, and time in figure 4 are the dimensionless values q*, k*, and t* to be specified shortly.) The characteristic curves for flow less than q0 (not shown in figure 4) are vertical lines that extend through the bottleneck. The characteristic curve for the flow q0, labeled as 0 in figure 4, travels vertically until it reaches the bottleneck at x = 0 then travels backward from that point at wave pace -w0. A characteristic curve for some flow larger than q0 travels vertically until it reaches the location where the local capacity is equal to the flow, and then turns back from that point at pace -w0. The "boundary conditions" specify q(-L, t) or equivalently q'(-L, t') = q(-L, t'-L/v0) = q0 + B(t' - L/v0 - t0).
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If q(-L, t) is linearly increasing in t, q'(-L, t') is linearly increasing in f. Since the characteristic curves (before they turn back) are vertical, a linearly increasing flow at one value of L implies a linearly increasing flow at other values of L. Thus the value of L is irrelevant, and we can arbitrarily choose the time origin (i.e. the t0) or the L so that the characteristic for flow qo reaches the bottleneck at t' = 0.
Q(x)-
\
Density, k' Fig. 3. A q'-k' relation in moving coordinates The locus of points where the characteristic curves turn back are points where q'(x, t') = q0 + Bt' = q0 + Ax2 i.e. Bt' = Ax2
(2.4)
shown in figure 4 by a dotted curve. Obviously figure 4 depends on the values of q'(x, t') - q0, but the value of q0 itself is irrelevant. If we increase the flows by some fixed amount and increase the capacities by the same amount the figure would not change. A shock will form at the earliest time when two characteristic curves of different flows intersect. This is obviously at such time t[ , location X], and flow qi , when the slope of the curve (2.4) is equal to the wave velocity -l/w 0 , i.e. dt' _ 2A dx B ' —
x, = - w 0 B / 2 A , t [ =
^1 —
Wn .
°
(2.5)
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0.5
2.0
1.0
Theory
3.0
Flow, q*
Fig. 4. Curves of constant flow We still have the option of choosing units for x, t! and q' - q0. We could, for example, choose a "dimensionless" distance x* x* = -x2A/Bw0
(2.6)
measured upstream of the bottleneck so that x,* = + 1, and a dimensionless time t* t* = t'2/Bw<>
(2.7)
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so that the dimensionless wave velocity or pace is +1 (in the direction of increasing x*) t,* = 1/2 . We can also choose the units of flow so that q*(x*, t*) = [q'(x, t') - q0] 2A/B2w02.
(2.8)
The approaching flow is then q*(x*, t*) = t* and the q,* at t* = 1/2 is 1/2. Thus, starting from a formulation which contained potentially seven parameters, q0, A, L, B, t0, v0, and WG , the problem has been reduced to a dimensionless form with no parameters. The q*(x*, t*) is equivalent to the q(-x, t) for q0 = 0, A = 1/2, L arbitrary, B = 1, t0 = 0, v0 = °° and w0 -1 (or + 1)-
Finally to evaluate the q*(x*, t*) we note that, at any point (x*, t*), it can have only one of three possible values: q*(x*,t*) = t*
(2.9)
if it is determined by the approaching flow, q*(x*,t*) = 0,
(2.10)
if it is determined by the flow which can pass the bottleneck at x = 0, or by a value determined from the characteristic curve passing through (x*, t*) coming from the curve (2.4), t* = (l/2)x*2 where the flow is q* = t*. This last value gives q*(x*, t*) = 1 - (x* -t*) - [1 - 2(x* - t*)]l/2.
(2.11)
This last expression is a function only of x* -1* since the flow is constant along the characteristic curves x* - t* = constant, but it applies only for 0 < x* - t* < 1/2. One can verify that (2.11) is correct by checking that along the curve (2.4), t* = (l/2)x*2, (2.11) gives the approach flow q*(x*, (l/2)x*2) = (l/2)x* 2 = t*. For small values of (x* -1*) an expansion of (2.11) in powers of x* -1* gives q*(x* , t*) = (x* -1*)2/2 + (x* -1*)312 +...
(2.11 a)
thus q*(x*, t*) vanishes quadratically in (x* -1*) as x* -1* —> 0. For x* -1* close to 1/2, however, q*(x*, t*) varies rapidly with a 1/2 power singularly. The function (2.11) is illustrated by the curve of figure 5. For 0 < x* < 1, it is obvious that (2.9) applies for t* < (1/2) x*2, (2.11) for (l/2)x*2 < t* < x*, and (2.10) for x* < t*. For 1 < x*, however, the boundary separating these solutions is a shock path.
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L-W determine the path of the shock by integrating the equations for the velocity of the shock. A simpler and more general method (Newell 19931) is to evaluate the cumulative flow and require that the cumulative flow be continuous across the shock.
c> u_
0.5
-r • Time , tj *-x *
Fig. 5. Flow vs. time at various locations
We can define a dimensionless cumulative flow A*(x*, t*) such that 8A*(x*, t*)/3t* = q*(x*, t*), 3A*(x*, t*)/9x* = k*(x*, t*)
(2.12
with k*(x*, t*) = 0 if the waves at (x*, t*) are going forward and k*(x*, t*) = (l/2)x *2 -q*(x*, t*)
(2.13
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if the waves at (x*, t*) are going backward. Since q*(x*, t*) = 0 along the characteristic curve t* = 0 for all x* > 0 and also at the bottleneck x* = 0 for all t* > 0, we can define A*(x*, t*) = 0 along these lines. The A*(x*, t*) can be evaluated by integrating (2.12) along any path from a point where A* = 0 to (x*, t*). On the upstream side of the shock (2.9) applies. Integration of this with respect to t* for fixed x* gives A*(x*, t*) = (l/2)t* 2 .
(2.14)
On the downstream side of the shock, for t* > x*, (2.10) applies and k*(x*, t*) = (l/2)x*2. Integration of this with respect to x* for fixed t* gives A*(x*, t*) = (l/6)x 3 , t*> x* .
(2.15)
Equating (2.14) and (2.15) we conclude that the path of the shock is (l/2)t*2 = (l/6)x *3 ; x* = 3(t*/3)2/3
(2.16)
if t* > x*, or, equivalently, for t* > 3, x* > 3. The formula for the shock path for 1 < x* < 3 is much more complicated because the flow on the downstream side satisfies (2.11). In the region where (2.11) applies, we can evaluate A*(x*, t*) by integrating the flow (2.11) along a line of constant x* from the point (x*, x*) where A*(x*, x*) = (l/6)x*3. A * (x * , t*) = (l/6)x *3 - Jq * (x*,T)dT x*3 1 3Cx*-t*1 2 7 = — + -{l-3(x*-t*) + — —-[l-2(x*-t*) }. 6 3 2
(2.17)
If one integrates the expansion (2. 11 a), one can verify directly that x
*3
cx*_t*^3
]
— - -(x*-t*) 4 +.... (2.17a) 6 8 Over most of the region where (2.17) applies, the second term of (2.17) is very small. Even at the A*(x*,t*) = 6
—
point where the shock first forms at x* = 1, t* = 1/2 where x* - t* is largest, the second term of (2.17) is only (-1/4) times the first term and the relative size of this term decreases very rapidly as (x* -1*) decreases. The path of the shock for 1 < x* < 3 is obtained by equating (2.17) and (2.14). It is quite easy to evaluate numerically the small deviation of the shock path from the curve defined by (2.16).
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Figure 4 shows the path of the shock and the curves of constant flow q*. Figure 5 shows the flow q*(x*, t*) as a function oft* - x* for various values of x*. To give a more complete description of the solution one might wish also to draw some vehicle trajectories. For any dimensionless solution q*(x*, t*), however, there is a whole family of possible (dimensionless) trajectories depending on the capacity q0 of the bottleneck or its dimensionless form q0* = 2A qo/B2 w02.
To see this one need only observe that the "dimensionless cumulative flow" A*(x*, t*) defined by (2.12) is the dimensionless cumulative flow for a hypothetical capacity qo - 0 or, equivalently, it is the actual dimensionless cumulative flow less a "background" cumulative flow of q0*t*. Thus, for any q0* > 0, the dimensionless cumulative flow is actually q0*t* + A*(x*, t*) and the trajectories are the curves for which this is a constant. In particular, the trajectory which passes x* = 0 at some time t0* is the curve A*(x*, t*) = q0*(to* -1*).
(2.18)
There is just one dimensionless function A*(x*, t*) as described above; the t0* labels the trajectory, but there is still the parameter q0*. There is a different set of trajectories for each qo*. In the region where the characteristics of figure 4 are vertical, A*(x*, t*) is given by (2.14) and the trajectories are defined by the equation t*2/2 = q0*(t0* -1*).
(2.18a)
As expected, the trajectories are also vertical, i.e. t* is independent of x* (the vehicle speed and the wave speed are the same), but the labels of the trajectories (vehicle number) depend on q0*. In the region t* > x*, the trajectories are also simple. From (2.15) x*3/6 = q0*(to* -1*)
(2.18b)
which means that the time displacement of a trajectory from a vertical trajectory is proportional to x*3 (for any qo*). The velocity in the x*, t* coordinate system is proportional to x*"2. In the region where (2.17) applies, the trajectories are more complicated but, for any q0*, thi trajectories are continuous across the shock path.
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3. PARABOLIC Q - K RELATION For the parabolic q - k relation described by ( 1 .4) the formulas are somewhat more complicated because the characteristic curves are not straight lines. In the absence of shocks the formal solution of the L-W equations is that q(x, t) is constant along characteristic curves in the x - t plane, x(t; q), having a slope, for fixed q,
dt
ok
Along the characteristic curve for flow q, the density as given by (1.4) is k - k 0 = ± [ q 0 - q + Ax2]1/2C-1/2. Thus the equation for the characteristic curve of flow q is dx(t; q)/dt = ± 2(CA)I/2 [(q0 - q)/A + x2]1/2.
(3.2)
in which either the + or - sign might apply depending on whether the velocity of the wave is positive or negative. A "formal" solution of (3.2) is r
in which D(q) is some unspecified "integration constant" for each value of q. The D(q) is to be determined from appropriate boundary or initial conditions. In particular, we assume here that "sufficiently far" upstream at x = -L, the flow satisfies (1.2). Sufficiently far upstream is now interpreted to mean at a value of L such that the hyperbolic functions in (3.3) can be approximated by positive exponentials sinh x| ~ cosh x ~ (1/2) exp (|x|). Thus in (3.3), the boundary condition requires that -L = ± | q0 - q(-L, T) | 1/2(l/2)A-'/2 exp(|2(AC)1/2 T + D(q(-L, T)|) for all T . Equivalently if we represent I as a function of q rather than q as a function of T through (1.4), this gives -L = ± | q0 - q|l/2(l/2)A-|/2 exp(| 2(AC)1/2 (t0 + B'1 (q0 - q)) + D(q) |). We would naturally specify that, for some sufficiently large L, the flow would reach q0 at some "large" negative t0, so the argument of exp(| • |) above is negative and the ± sign must be -. Thus exp(D(q)) = | q0 - q | 1/2(2L)-'A-1/2 exp(-2(AC)1/210 - 2(AC)I/2 (q0 -
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This relation depends on t0 and L only through a single factor L"1 exp(-2(AC)1/2 t0) on the righl hand side. Since a change in L is equivalent to a suitable change in t0, this means that if the flow is linearly increasing with t as in (1.2) for some sufficiently large L, it is also linearly increasing with t at other large values of L but with a displaced value of t0. Any change in this factor is equivalent to a translation of the time coordinate and, by suitable choice of the t0, we can assign this factor any value we wish. In particular, we will choose the time origin so that exp(D(q)) = (AC)1/4 [2(q0 - q)/B]1/2 exp(-2(AC)1/2 (q0 - q)/B). The equations for the characteristics (3.3) can now be written in the simple dimensionless form x*(t*; q*) = + e-'*+q* +q*e'*'q*
(3.4)
with x*(t*; q*) = -23/2A 3/4C1/4B-|/2 x(t; q)
(3.5) t* - 2(AC)
I/2
1/2
t, q* = 2(AC)
(q0 - q)/B.
Equation (3.4) is valid either for q < q0 or q > q0 since the q* changes sign when q passes q0 . The x*, t*, and q* are simply rescaled dimensionless versions of the x, t, and q - q0, respectively (but not necessarily related to the corresponding symbols in section 2). Again, as in section 2, the formulation of this problem started with seven parameters q0, A, L, B, t0, ko, and C, the first five of which are the same as before but the parameters ko and C associated with the shape of the parabolic q - k curve replace the parameters v0 and w0 associated with the triangular q - k curve Again the final dimensionless form of the characteristic curves contain none of these parameters. To complete the solution of the dimensionless flow pattern we note that (3.4) can also be writter in the form f-21 q* l l / 2 sinh(t * -q * +l/2^n I q* I) for q* < 0 x*(t*;q) = < [+2q *' / 2 cosh(t * -q * +1 / 2£n q*) for q* > 0 .
(3.6)
Thus the characteristic curves for q* < 0 and those for q* > 0 all have the same shape except for i scaling factor (q*)I/2 and a translation in time by q* - (l/2)£n | q* |. Figure 6 shows a family of characteristic curves evaluated from (3.4) or (3.6). The curves for q*< ( pass through the bottleneck at x* = 0. Note that these are drawn for a geometric sequence of q" values -1, -1/2, -1/4, etc. These characteristic curves will cover the whole space downstream of thf
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137
bottleneck, but clearly the flow at any location downstream rapidly approaches the flow q*= 0 (q = q0) as t* increases.
Time , t '
Fig. 6. Curves of constant flow
The characteristic curve for q* = 0 is simply an exponential e"1* which is asymptotically horizontal for t*—>°°. This characteristic cannot pass the bottleneck. The characteristic curves for q* > 0 move forward in time until the flow q* reaches the location where the local capacity of the highway is q*. At this point the slope of the characteristic curve becomes zero. Unlike the characteristic for q* = 0, however, the slope of the characteristic curve for q* > 0 becomes zero at a finite time after which the characteristic proceeds to move back upstream. Note that this is
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similar to the pattern of figure 4 except that in figure 4 the characteristic curves change direction abruptly. The points at which the characteristic curves become horizontal are identified in figure 6 by the dotted curve. They occur where the argument of the cosh (•) in (3.6) vanishes t* = q* - 1/2 In q* , x* = 2q*/2 or
t* = (x*/2)2 - In (x*/2). As a function of q* or x*, this point starts at t* = oo for q* = 0. It decreases with q* until it reaches a minimum at q* = 1/2 and then increases again. This is the analogue of the dotted line curve of figure 4 except that figure 4 is drawn with a moving time origin. After a characteristic curve turns upstream it will, at some time, intersect a characteristic curve of higher flow moving downstream so as to create a shock. In figure 6 it appears as if all the characteristics for 1 < q* < 2 intersect at (nearly) the same point. The shock actually starts at the first time that 9x*(t*; q*)/3q* vanishes. From (3.4) we see that 3x * (t*;q*)/3q *
= e"'*+q* + e'*~ q * - q * e'*~q*
= -e'* +q *[(q*-l)e~ 2q *-e~ 21 *]. The earliest that this can vanish is at that value of q* for which (q* - 1) exp(-2q*) has a maximum, namely at qi* = 3/2 , t,* = (3/2) + (1/2) ^n2 = 1.846, x,* = 23/2 = 2.83.
(3.7)
To analyse the behavior of the characteristics in the vicinity of the point xi*, t]*, it is advantageous to make a power series expansion of (3.4) in powers of x' = x* - X]*, t1 = t* - t!*, and q' = q*-q,* .
,
„„
,
(l + — - — + -) + ^4^(1 - — + 2 6 3V2 4
The first term on the right hand side of (3.8) is an expansion in powers of t' of the characteristic curve for q' = 0. The second term is a power series expansion in q' for t' = 0. Successive terms give power series expansions in q' of the coefficients of powers oft', t'2, etc (for powers of q' of at least 1).
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Since the q]*, t)*, X|* were chosen so that 3x'/3q' = 0 at t' = 0, q' = 0, we knew that the second term of (3.8) would not contain a term proportional to q'. We would have expected this term to start with a (positive) term proportional to q'2 , but, by some coincidence, this term is also missing. The second term of (3.8) starts with a term proportional to q'3. This is the reason why it appears in figure 6 as if a wide range of characteristic curves are crossing at (nearly) the same point xi*, t]. Also in (3.8) the fourth term is proportional to q' 3 . Equations (3.4), (3.6) or (3.8) define x* or x' as a single-valued function of q* and t*, but, to determine the path of the shock, one must determine the flow (and corresponding densities) on either side of the shock, i.e. q* as a function of x* and t*. From figure 6 it would appear that, in the region where the characteristic curves intersect, there are actually three values of q* at each point x*, t*.
In (3.8) it is not a -priori obvious how many terms one needs to retain in a first approximation because one does not know the relevant relative magnitudes of t' vs q'. If one were to keep only terms linear or quadratic in t' and q', one might approximate (3.8) by x' = A/2 t'(l +1') - A/2 t'q' + ...
This could then be solved for q' as a function of x' and t' q' = x'/2 1 / 2 t' + l + t' + .. This describes what appears to occur in figure 6. Over some range of (small) q', the characteristic curves (nearly) pass through x' = 0, t1 = 0. The value of q' at any point (x', t') depends mostly on the slope x'/t' of the line from x - 0, t' = 0. But this also shows that the relevant magnitudes of t' and q' are not comparable. One is interested in values of q' large compared with t' and should retain at least the term 2l/2q'3/3 in (3.8). Along the characteristic curve x' = A/2 t'(l + t' + t ' 2 / 6 + ...)
which, as an equation for q', is satisfied not only for q1 - 0 but also for q' = ±(3t') 1/2 .
(3.9)
Indeed these represent (to the lowest approximation) the other two values of q' along the characteristic curve for q' = 0.
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Theory
The shock first forms at t' = 0 as an "infinitesimal" shock between the values q' = ±(3t)1/2 and a: such, it travels at essentially the wave velocity for q' = 0, dx*/dt* = (2) l/2 . But even as the jump ii flow increases with t', the shock velocity between the flows q* = q,* + (3t) l/2 and q!* - (3t)1/2 wil stay very close to the wave velocity at the average of these two flows, namely at the wav< velocity for q1 = 0. Thus the shock path will, in turn, stay very close to the characteristic curve fo q' = 0 even though the amplitude of the shock is increasing rapidly with time. Starting with (3.9) as a first approximation one can iteratively obtain higher order approximation by including other terms from (3.8). Figure 7 shows a magnitude view (by a factor of 10) of thi characteristics in the vicinity of X] , t ) . The cross indicates the place where the shock begins. Not that there is a rather large curvature of the shock path near t,*; the shock velocity increases fron (2) l/2 = 1.41 at t,* = 1.85 to about 2 fort,* = 2.0. To determine the path of the shock over long distances it is advantageous to introduce dimensionless cumulative flow as in (2.12), but now the dimensionless density is given by k*2(x*, t*) = (l/4)x *2 - q*(x*, t*)
(3.10
rather than (2.13). Along a characteristic curve of flow q* dA*(x*, t*)/dt = k* dx*/dt* + q* with dx*/dt* = -3q*/ak* = 2k* so
dA*(x*, t*)/dt* = 2k*2 + q* = (1/2) x*2(t*; q*) -q*. Substitution of (3.4) for x* gives dA*(x*, t*)/dt* = (l/2X 2t% + 2q* + (l/2)q*2 e2t* ' 2q * and integration of this with respect to t along the curve of constant q* gives A*(x*, t*) - (l/4)K 2t * + 2q* + q*2 e 2t *' 2q * + D*(q*)]
(3.11)
for some integration constant D*(q*) for each q*. We can arbitrarily specify that for q*= 0 and t*—» + °° , A*(x*, t*) -» 0 so that D*(0) = 0 and A*(x*, t*) = -(l/4X 2t *, for q* = 0, or from (3.4)
(3.12)
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141
A*(x*, t*) = -(l/4)x* 2 , for q* = 0.
.5
1.6
1.7
1.8
1.9
2.0
(3.13)
2.1
2.2
2.3
Fig. 7. Curves of constant flow, a magnified view
The boundary conditions specify that, for any large fixed x*, q*(x*, t*) is linearly increasing in t* on the upstream side of the shock, so A*(x*, t*) = -(l/4)x *2 + (l/2)(t* + In x*)2 for large x*
(3.14)
From this one can show that D*(q*) = -2q* + 2q*2 so that (3.11) determines A*(x*, t*) parametrically as a function of q* along the characteristic curve (3.4) describing x*(t*; q*) also as a parametric function of q*. Where characteristic curves
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Theory
intersect each other giving multiple values of q* for the same x*, t*, there will also be multiple values of A*(x*, t*). The "correct" A*(x*,t*) is the smallest of its multiple values and the shock path is where the A*(x*, t*) becomes multiple-valued (Newell 19931). To determine the path of the shock for large t*, it suffices to note that, for large positive t*, q* is nearly zero for all x* on the downstream of the shock and A*(0, t*) is also nearly zero for large t*. If, as in section 2, we integrate (2.12) along a line of constant t*, but now with the density given by (3.10) for q*(x*, t*) = 0, we have A * (x*,t*) = jk * (z) dz = (1/2) Jz dz = (l/4)x* 2 . 0
(3.15)
0
The path of the shock is now obtained by equating (3.14) and (3.15) or
t = x*-.ftix*.
(3.16)
Actually the formula (3.16) is fairly accurate soon after the shock forms. One can show that to 2 second approximation t = x* - &ix* - x*e -x* The additional term is negligible for x* > 4, only about one distance unit after the shock forms Figure 6 shows the shock path. As with the triangular q - k relation, one might also like to describe the vehicle trajectories. For an> dimensionless q*(x*, t*) as illustrated in figures 6 and 7, however, there is a two-parameter family of possible trajectories. For the triangular q - k relation there was only a one-parameter family ir the transformed coordinate system but in the original coordinate system the trajectories would alsc depend on the free-flow speed v0. The dimensionless cumulative flow defined by (3.11) is a cumulative flow less a dimensionles: background flow of q0* and density ko* so the actual dimensionless cumulative flow is A*(x*, t*) + q0* t* + ko* x*
(3.17)
with q0* - 2(AC)1/2 q0/B,
k0* = 21/2 A1/4 C3/4B'1/2 k0.
Although there is only one dimensionless A*(x*, t*), there are two arbitrary parameters qo* and ko* in (3.17) (regardless of how these may be related to the q0 and ko).
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The exact parametric representation of A*(x*, t*) along characteristic curves is not very convenient for the evaluation of trajectories but, for most trajectories which pass through the shock, (3.15) is valid downstream of the shock. The trajectory which passes x* = 0 at time t0* satisfies the equation qo*(to* -1*) - V x* = x*2/4.
(3.18)
Thus the deviation of a trajectory from a straight line is proportional to x*2 (instead of x*3 as in (2.18b)). Upstream of the shock, according to (3.14), this trajectory would satisfy qo*(to* -1*) - V x* = -x*2/4 +(t* + ^nx *)2 12.
(3.18a)
4. SOME ESTIMATES There is no guarantee that traffic behaves in the manner described here, but, for the theory to be plausible, the various scaling parameter should have values in a reasonable range. In a typical rush hour we might expect the flow per lane to increase at a rate of about 1000 cars/hr. in a period of about 1 hour. Thus a reasonable value of B would be about 103/hour2. For the triangular q - k relation a plausible value of w0 is about 1/10 hr/km (a wave velocity of lOkm/hr). A "typical" value of A in (1.1) is not as well defined but it might be helpful to relate the A to a distance L* required for the capacity (1.1) to change by 100 vehicles/hr. from its value at x = 0, i.e. let
100/hr = AL* 2 , A = 100/L*2hr. With these values of A, B, and w0, (2.5) gives _ L* 2 , _ L *2 hr _ _ 20L* 2 X t 2 q ' ~ 2 k m ' ' ~ 40 (km) ' ' q° ~ km 2 hr Thus, if L* is measured in km, x\ = L*2/2. If the capacity of the road section did vary as in (1.1), a value of L* = 1km would certainly described a very slowly changing capacity for typical roads. The prediction here is that, for L* = 1 km, a queue would first form about 1/2 km upstream of the bottleneck (a rather substantial distance) at a time t] 1 of about 1 minute and qi - q0 = 25 veh./hr.
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The value of \\, being proportional to L*2 , decreases rapidly with decreasing L*. If L* is reduced to 1/2 km, X] becomes only 1/8 km. For the parabolic q - k relation a typical value of C in (1.4) would be such that the flow would drop by about 2000 veh./hr from capacity at a density displacement of about 100 veh./km. Thus, 2000/hr = C(100)2/km2, C ~ (1/5) km2/hr. For the same values of B and A as above, (3.5) and (3.7) predict that the shock forms at x,
This value of x, will typically be considerably larger than for the triangular q - k relation because the parabolic q - k relation provides less room to store excess vehicles near the bottleneck (but more space far from the bottleneck). For L* = 1 km, the value of x, is about 1.5 km. The X] decreases with decreasing L*, proportional to L*(3/2) , but not as rapidly as for the triangular q - k relation. For L* = (1/2) kin, x, is still about (1/2) km.
5. CONCLUSIONS We have described here an analytic solution of the problem posed in the introduction. No claim is made that the theory described here is consistent in detail with how traffic actually behaves bul any attempt to verify the L-W theory should recognize that a shock wave does not necessarily start at the bottleneck itself. It may begin at some point upstream of the bottleneck at a poinl depending on how the capacity varies with location (the A of eq. (1.1)), the rate at which the flow increases (the B of eq. (1.2)), and the shape of the q - k curve near its maximum. Despite all the parameters in the formulation of the problem, the final evaluation of the flow pattern is expressed in terms of a single dimensionless form. All the parameters are absorbed in coordinate transformations. Installations of vehicle detectors at multiple locations along freeway sections at various places throughout the world are presently providing very detailed information about traffic flows or freeways. Data from these system will soon provide a means of testing the validity or deficiencies of various models of traffic flow, including the L-W theory, but to deal with masses of data ai different locations will required special techniques of analysis.
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Many parameters are needed to characterize any particular set of observations and to compare one set of observations with another. Even if the L-W theory should fail to describe correctly some features of the flow pattern, maybe some of the dimensional arguments will still be valid. If one makes observations at one location having a certain set of parameters, such as the A, B, q0, ko, etc. here, and then observations at other locations with different values of these parameters, but one presents the data in terms of the rescaled coordinates, the A*(x*, t*) for example, then maybe one will obtain nearly the same A*(x*, t*) for all locations. This, in itself, would mean that observation at one location can be used to predict behavior at another location having different parameter values. The trick of comparing observations at two or more location by measuring time relative to a coordinate system moving with the free speed has been exploited before. If there are no "delays", an A'(x, t') derived from (2.2) would be independent of x. The trick of rescaling time and/or space variables involves nothing more than relabeling the coordinate scales of x and/or t on a graph or drawing graphs for two different locations on different physical coordinate scales. The trick of subtracting some "background flow" from the actual flow or, equivalently, drawing graphs of
A(x, t) - q0 t+ ko x
(5.1)
for some possibly arbitrary values of q0 and ko is less "well-known". This transformation does not change the wave pattern or the characteristic curves and was exploited in section 3 to eliminate two parameters from the dimensionless wave solution of section 3. This transformation, however, does not eliminate these parameters from the trajectory pattern. Cassidy and Windover (1995) have subtracted a background flow as in (5.1) from experimental data, not to compare observations at different locations, but simply as a tool to "magnify" observed flow variations in space and time. If, by appropriate choice of q0 and ko, one can greatly reduce the magnitude of (5.1), then one can draw the graph on a magnified scale. The effect is quite dramatic.
REFERENCES Cassidy, M. J. and J. R. Windover (1995). A methodology, for assessing the dynamics of freeway traffic flow. Transportation Research Record, (Washington, D.C.) (in press).
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Lighthill, M. J. and G. B. Whitham (1955). Kinematic waves II a theory of traffic flow on long crowded roads. Proc. Royal Soc. (London) A 229, 317-345. Newell, G.F. (1993) A simplified theory of kinematic waves in highway traffic, I general theory, II queueing at freeway bottlenecks, III multi-destination flows. Transpn. Res. 27B, 281-287, 289-303,305-313.
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THEORY OF CONGESTED TRAFFIC FLOW: SELFORGANIZATION WITHOUT BOTTLENECKS Boris S. Kerner, DaimlerChrysler AG, FT1/V, HPC: E224, 70546 Stuttgart, Germany
ABSTRACT Results of experimental observations of phenomena of self-organization in traffic flow on German highways are presented. The observations allow to suggest that there are at least two phenomena of 'self-organization without bottlenecks' in real traffic flow: (i) The spontaneous formation of a local region of synchronized traffic flow in an initially free traffic flow and (ii) The spontaneous formation of a traffic jam in synchronized traffic flow. A theory of congested traffic flow which may qualitatively explain these and other diverse effects of selforganization in real traffic flow is discussed.
1. INTRODUCTION 1.1. Self-Organization Self-organization, which is usually viewed as a spontaneous formation and evolution of different patterns in a non-linear system, is the usual phenomenon in many physical systems, biological systems and chemical reactions (e.g., Nicolis and Prigogine, 1977; Kerner and Osipov, 1994). The ideas of self-organization have also been applied in different theories to explain phenomena observed in traffic flow. Indeed, already in 1958 Chandler, Herman and Montroll (1958) and Komentani and Sasaki (1958), and later other authors (e.g., Prigogine and Herman, 1971; Kiihne, 1991; Schreckenberg et al., 1995, Helbing, 1997) proposed that there is a density range where homogeneous states of traffic flow due either to an instability or to some other kind of phase transition cannot exist. Therefore a sequence of traffic jams has to occur spontaneously - the so-called 'stop and go' phenomenon, which is often observed in real traffic flow (e.g., Treiterer, 1975; Koshi et al., 1983).
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A different scenario of self-organization for the jam's formation has been proposed in 1994 by Kerner and Konhauser (1994), and later in other papers (Bando et al, 1995; KrauB et al., 1997, Barlovic et al., 1998): Before the density range mentioned above is reached, there should be a broad range of lower densities where homogeneous states of free traffic flow are metastable states (see an explanation of 'metastable states' below in Sect. 1.4) and 'the local cluster effect' leading to jam formation can occur (a comparison of the local cluster effect in different traffic models can be found in (Herrmann and Kemer, 1998; Kerner, 1998b). On the contrary, there are alternative theories which claim that there are no self-organization processes in traffic flow and that all phenomena of the formation of spatial-temporal patterns in traffic flow are without exception determined by an influence of on- and off-ramps or other freeway bottlenecks. As the consequence of these theories it is also often claimed that spatialtemporal behavior of traffic flow can be described satisfactorily by the well-known classical Lighthill-Whitham-theory of traffic flow (see Whitham, 1974), which is based on the Lighthill-Whitham-model or by further developments of such 'first order' traffic flow models where no self-organization processes are possible (e.g., Daganzo, 1997; Daganzo et al., 1998).
1.2. Phenomena of Pattern Formation in Traffic Flow Only results of experimental observations of real traffic flow are able to answer the question which traffic flow theories may actually explain related phenomena in real traffic flow. One may mention two well-known phenomena experimentally observed in real traffic flow: A. The occurrence of spatial-temporal patterns consisting of different traffic jams - the 'stop and go' phenomenon (e.g., Treiterer, 1975; Koshi et al., 1983) and B. The breakdown phenomenon in the vicinity of a freeway bottleneck which is often accompanied by the related 'capacity drop' (Fig. l(a)) (e.g., Agyemang-Duah and Hall, 1991; Cassidy and Bertini, 1998; Persaud et al., 1998). Are these phenomena the result of some self-organization processes or are they only the result of initial permanent non-homogeneity caused by freeway bottlenecks !? Recently the nature of the breakdown phenomenon in the vicinity of a freeway bottleneck has been disclosed (Kerner and Rehbom, 1997): This breakdown phenomenon is linked to an occurrence of a first order local phase transition 'Free Flow => Synchronized Flow' (the definition of synchronized traffic flow will be given in Sect. 1.6). Such local phase transitions are one phenomenon of self-organization. Kerner (1998c) discovered the nature of the 'stopand-go' phenomenon: It turned out that this phenomenon is linked to the double (cascade) phase transitions 'Free Flow => Synchronized Flow => Traffic Jams and the 'pinch effect' in synchronized flow which are both very complex phenomena of self-organization.
1.3. About the Role of Non-Homogeneity in Self-Organization Processes A freeway bottleneck usually causes a permanent non-homogeneity in traffic flow (e.g., Koshi et al, 1983; May, 1990; Daganzo, 1997; Cassidy and Bertini, 1998). On the one hand, it is
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well-known from investigations of physical, biological and chemical systems (Kerner and Osipov, 1994) that a permanent non-homogeneity acts as 'a permanent nucleus' for phase transitions: Spontaneous formation of a spatial-temporal pattern occurs considerably more frequently in the vicinity of the permanent non-homogeneity. Such phase transitions can occur in a deterministic way, i.e., even if fluctuations are negligible. On the other hand, this deterministic effect is also one phenomenon of self-organization (Kerner and Osipov, 1994). Indeed, the reason of the spontaneous occurrence and the main non-linear properties of spatial-temporal patterns are nevertheless determined by intrinsic non-linear properties of a system. In other words, phenomena of self-organization can occur in such systems (although usually considerably more seldom) even when no initial permanent non-homogeneity exists. The aim of this paper is to show that self-organization in experimentally observed traffic flow occurs outside any bottlenecks. To show such 'self-organization without bottlenecks' in real traffic flow, one needs to perform experimental observation of the behavior of traffic flow on a long enough highway section where no on- and off-ramps and other bottlenecks exist (Sect. 2). A qualitative theory of congested traffic flow (Kerner, 1998 a, b, 1999) which may explain phenomena of self-organization will be considered in Sect. 3. However, before the new results are considered (Sect. 2), some recent experimental observations which are an 'indirect' proof of'self-organization without bottlenecks' (Sect. 1.4), some additional definitions (Sect. 1.5), and features of synchronized flow which have been found out (Sect. 1.6) should be briefly reviewed.
1.4. Metastable States of Free Traffic Flow and Characteristic Parameters of Wide Jams as 'Indirect' Proof of 'Self-Organization Without Bottlenecks' It should be noted that an 'indirect' proof of the existence of 'self-organization without bottlenecks' in real traffic flow has already been done by Kerner and Rehborn (1996a, 1998a) from their experimental investigation of the propagation of 'wide' jams on highways. A wide traffic jam is a jam whose width, i.e., the longitudinal distance between jam fronts, is considerably higher than the widths of the jam's fronts. A jam front is a region where the flow rate, the density and the average vehicle speed change sharply spatially. Note that the term 'jam front' is used instead of 'shock wave' because the term shock wave in traffic flow theories is usually associated with the classical Lighthill-Whitham theory of shock waves (see Whitham, 1974). However, as it has recently been shown in (Kemer, et al, 1997), the Lighthill-Whitham theory and other 'first order models' cannot explain the existence of characteristic parameters of wide jams observed in experiments (Kerner and Rehborn, 1996a). Indeed, when wide jams propagate outside bottlenecks and free flow is formed in the outflow of the jam, it has been found out (Kerner and Rehborn, 1996a) that there are the characteristic (unique, or coherent) parameters of traffic flow: (i) the velocity v g of the downstream front of a jam, (ii) the jam density p m a x , (iii) and also the flow rate q o u t , the vehicle density p m i n , the average vehicle speed in free flow which is formed by a wide jam downstream. The characteristic parameters do not depend on initial conditions. In particular, it has been found out (Kerner and Rehborn, 1996b) that independently on initial conditions the characteristic parameters were spontaneously self-formed during a development of any jam whose width
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monotonically increased in time. This process of self-organization cannot be explained by any 'first order' models (e.g., Daganzo, 1997; Daganzo, et al, 1998); On the contrary it can be explained by 'second order' models (Kerner and Konhauser, 1994; Kemer et al., 1997). Such a stationary propagation of the downstream front of a wide jam can be represented by a line in the flow-density plane. This characteristic line for the downstream front of the jam, which will be called 'the line J', has the coordinates (p mjn , q out ) and (pmax, 0) in the flowdensity plane; the slope of 'the line J' is equal to the mean value of the velocity of the downstream front v g (Fig. l(b), line J). It should be noted that 'the line J' is not a part of the fundamental diagram: It represents the characteristics of the downstream front of a wide jam. The other result of experimental observations of wide jams found out in (Kemer and Rehborn, 1996a) is that the maximal possible flow rate in free flow qmraexe) can be considerably higher than the flow rate out of a wide jam q out (Fig. l(b)): q (free)
In 4 max /lout
(1)
Therefore, in the range of the flow rate (and in the corresponding range of the vehicle density) (Fig. l(b)) (2) w (free) A fl° rate Qmax
D (free) Kmax
flow rate
(b)
density
Pmax
Fig. 1. A possible shape of the fundamental diagram of traffic flow (a) (e.g., Brannolte, 1991; Ceder, 1976; Hall, 1987; Hall, et al, 1986; Agyemang-Duah and Hall, 1991; Koshi, et al, 1983) and (b) - the concatenation of states of free flow with the characteristic line for the downstream front of a wide traffic jam ('the line J") (Kemer and Rehborn, 1996a; Kemer and Konhauser, 1994). at any chosen average vehicle density there are at least two different states of traffic: (i) free traffic flow and (ii) wide jams. It must be noted that the existence of this range of the density (2) is not linked to a freeway bottleneck. Therefore, a local perturbation of an initially in average homogeneous free traffic flow can force an appearance of a jam, if the amplitude of this local perturbation is high enough. Because there is always a finite probability of a spontaneous occurrence of such a fluctuation in traffic flow outside any bottleneck, a traffic jam can spontaneously occur in traffic flow without any influence from a freeway bottleneck, i.e. 'self-organization without bottlenecks' in traffic flow can really occur.
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As well as in physical, biological and chemical systems, states of free traffic flow in the range (2) may be called metastable states. Recall that a metastable state of a spatial system is stable with respect to any infinitesimal perturbations. However, if the amplitude of a local perturbation exceeds some critical amplitude, this local critical perturbation begins to grow. The same definitions may be applied to traffic flow where local perturbations (fluctuations) of traffic variables (vehicle speed, density and flow rate) usually occur. As it follows from (2), the flow rate and the density q b = q o u t , pb = p min are the boundary (threshold) values which separate stable states of free flow and metastable states of free flow with respect to the jam formation (Kerner and Konhauser, 1994). This means that at q < q b (p < p b ) no jams can exist for a long time or be exited in free flow (Fig. 2). 1
critical amplitude of local perturbation
!\
(a)
^v_ P b °r P s
density (flow rate)
probability of f first order local phase transition 1
!
(b)
/
_J
p b or p
/
density (flow rate)
Fig. 2. Explanation of local first order phase transitions: (a) A qualitative shape of the dependence of the critical amplitude of a local perturbation (see Fig. 6(a) in Kerner and Konhauser, 1994); (b) the dependence of the probability of first order phase transitions which is related to Fig. (a) (see Figs. 3 and 4 in Persaud et at., 1998). Threshold vehicle densities in free flow are designated as p b for the phase transition 'Free Flow => Jam' and as ps for the phase transition 'Free Flow => Synchronized Flow'.
1.5. Nucleation Effect and Local First Order Phase Transitions The effect of the growth of a local perturbation in a metastable state of a system whose amplitude exceeds the critical value, i.e., the growth of a critical local perturbation, is called the nucleation effect. The critical local perturbation plays the role of a 'nucleation center' for local phase transitions in an initial state of the system (e.g., Kerner and Osipov, 1994). A first order local phase transition is a phase transition which occurs in a metastable state of a distributed system and is caused by the nucleation effect. First order local phase transitions are accompanied by a jumping (i.e., breakdown) behavior of variables of a system and hysteresis effects. Some other general properties of first order local phase transitions are shown in Fig. 2: The critical amplitude of the critical local perturbation is maximal at the threshold which separates stable and metastable states of a system. This critical amplitude decreases as the density (or the flow rate) deviates from the threshold value inside the metastable range. Apparently, for traffic flow these properties have first been discovered by Kemer and Konhauser (1994) from their theoretical investigation of the phase transition 'Free Flow => Jam' (Fig. 2(a)). Note that from statistical physics it is known that the probability of the spontaneous occurrence of a fluctuation in a distributed system decreases with the increase in the amplitude of this perturbation. Therefore, from Fig. 2(a) follows the other general property
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of first order local phase transitions: The lower the amplitude of the critical perturbation is, the higher is the probability of the spontaneous occurrence of a first order local phase transition (Fig. 2(b)). Note that the nature of the well-known breakdown phenomenon in the freeway bottleneck, as it has recently been discovered in (Kerner and Rehborn, 1997), is linked to another first order local phase transition 'Free Flow => Synchronized Flow'. Therefore, the latter phase transition should show qualitatively the same behavior as it is shown in Fig. 2 (with a different threshold density p s ). The behavior of the probability of the breakdown phenomenon in a freeway bottleneck shown in Fig. 2(b) has been discovered by Persaud et al. (1998) that confirms the mentioned conclusion by Kerner and Rehborn (1997). Observations (Kemer and Rehbom, 1996b) show that there are three qualitatively different phases of traffic flow: (i) free flow, (ii) synchronized flow and (iii) wide jams. As a result, there may be three qualitatively different types of phase transitions: 1) 'Free flow <=> Jam', 2) 'Free flow <=> Synchronized flow', and 3) 'Synchronized flow <=> Jam(s)'. A spontaneous occurrence of these phase transitions determines the complexity of traffic flow observed in experiments. It is also linked to the result (Kerner and Rehborn, 1997; Kerner, 1998c)) that all these transitions are related to the same class of first order phase transitions, i.e., they are accompanied by similar looking breakdown and hysteresis effects. Besides, for each of these phase transitions the properties shown in Fig. 2 are valid. However, they must be distinguished one from another because the non-linear features of these phase transitions are qualitatively different. This differentiation determines one of the main difficulties in traffic flow theory.
1.6. Properties of Synchronized Traffic Flow In free traffic flow, due to the relatively low densities of vehicles, drivers on a multi-lane road are able to change a lane and to pass. On the contrary, due to the higher density in 'congested' flow, vehicles are almost not able to pass. As a result, a bunching of drivers both on each individual lane and between different lanes of a highway can occur. Therefore, when all lanes of a highway correspond to the same route without on- and off-ramps or other bottlenecks, drivers move with nearly synchronized average speed on the different lanes of the highway. A possibility of such 'synchronized' ('collective') flow which is related to a part of the fundamental diagram (e.g., Fig. l(a)) at higher density has been theoretically predicted by Prigogine and Herman (1971). Synchronized flow has been observed by Koshi et al. (1983). In congested flow a broad and complex spreading of measurement points which cover a twodimensional region on the flow-density plane is observed (e.g., Koshi et al, 1983). This spreading is interpreted either as fluctuations, or as an instability, or else as a jam formation (e.g., Hall, 1987; Helbing, 1997; Koshi et al, 1983; Hall etal., 1986). It is well-known that in traffic flow with high density traffic jams can appear (e.g., Treiterer, 1975). Inside a jam both the vehicle speed and the flow rate are very low or even zero. Recall that in synchronized traffic flow, on the contrary, the vehicle speed is relatively low (but a finite value) and the flow rate can be nearly as high as the flow rate in free flow. This 'quantitative' difference as it follows from results of experimental observations leads to
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qualitatively different non-linear properties of traffic jams and of synchronized flow (Kerner and Rehborn, 1996a, 1996b). Traffic flow with high density, where both traffic jams and synchronized flow may occur, is called congested traffic flow. Kerner and Rehborn (1996b) have found out that synchronized flow has totally different dynamical properties in contrast to free flow. In particular, a multitude of states of free flow may really be described by a curve on the flow-density plane, i.e., by the fundamental diagram (Fig. l(a), curve 'free'). On the contrary, even a multitude of homogeneous states of synchronized flow (i.e., states which are homogeneous spatially and stationary in time; often such states are called 'steady speed' states) cover a two-dimensional region in the flowdensity plane: A given vehicle speed (a steady speed) in a homogeneous state of synchronized flow may be related to an infinity multitude of vehicle densities, and a given density may be related to an infinity multitude of different speeds. In other words, the real dynamics of synchronized flow cannot be described in the frame of the hypothesis about the fundamental diagram: There is no fundamental diagram which is able to describe the properties even of the multitude of homogeneous states of synchronized traffic flow (Fig. 3). Q (free)
4 max
D (free)
pmax
density
density
Fig. 3. Homogeneous states of traffic flow (Kerner and Rehborn, 1996b; Kerner, 1998a): (a) Multitudes of homogeneous states of free (curve F) and of synchronized flow (hatched region) on a multi-lane road, (b) A multitude of homogeneous states of flow on a one-lane road.
2. SELF-ORGANIZATION WITHOUT BOTTLENECKS Between 1995 and 1998 all mentioned above types of phase transitions in traffic flow on the German highways A5, Al, A3 and A44 have been investigated on different days. Since it has been found out that the features of these phenomena are similar in all cases, some general results may be illustrated by a representative data set measured on Monday, March 17, 1997 on a section of the highway A5 (Fig. 4) (Figs. 5 - 9). The section of the highway has three intersections with other highways (II, "Friedberg", 12, "Bad Homburger Kreuz" and 13, "Nordwestkreuz Frankfurt") and is equipped with 24 sets of induction loop detectors (D1,...,D24) (Fig. 4). Each of the sets D4-D6, D12-D15, and D23, D24 consist of four detectors for a left (passing), a middle and a right lane, plus one for the lane related to on-ramps or to off-ramps. The other sets of detectors are situated on the three-lane road without on- and off-ramps, where each of them consist of three detectors only. Each
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traffic flow 1
o
. u>
] ]
p ]
o
] ]
1 L JP
]
1 1
1
o —
o
hff
IQ
en
\ ]
1 ]
1 ] ]
1 ]
1 1 1
] 1
b
i ih i 112 i ]u
a ro D ro a
—• o
o
—'
i i i i \ i i i i i ] j ] ] ] ]
oa o
ro to 10 ro cx> b
ii ii i]
Fig. 4. Schematic configuration of the section of the highway A5-South in Germany. induction loop detector records the crossing of a vehicle and measures its crossing speed. A local road computer calculates the flow rate and the average vehicle speed in one minute intervals. The accuracy of the detectors according to official regulations is: better than 10% for the flow rate higher than 600 veh/h, better than 20% for the flow rate lower than 600 veh/km; better than 3% for the vehicle speed higher than 100 km/h; to within 3 km/h for the vehicle speed lower than 100 km/h. If each vehicle during the interval of the averaging (one minute) has the vehicle speed lower than 20 km/h, then the average speed is set to 10 km/h. Only if no vehicle crosses a detector during the interval of the averaging, the average speed is set to zero.
2.1. Experimental Properties of Phase Transitions 'Free Flow => Synchronized Flow' outside Freeway Bottlenecks Let us first consider the dependence of the average vehicle speed (v) and flow rate (q) within the time interval 06:20 - 06:40 (Fig. 5(a)) in the vicinity of the freeway bottleneck which exists on the section of the highway due to the off-ramp inside the intersection 13 (D23, D22, Fig. 4) and in the vicinity of the freeway bottleneck which exists due to the on-ramp inside the intersection 12 (D15, D16, Fig. 4). It can be seen from the dependence of the average vehicle speed (Fig. 5(a), left) and flow rate (Fig. 5(a), right) that in the whole time interval 06:20 06:40 in the vicinity of both freeway bottlenecks free flow is realized. On the contrary, outside from both bottlenecks some transitions between free and synchronized flow occur (D18, at t « 06:27, t « 06:32 and t « 06:36). As it has already been mentioned in Sect. 1.6, in synchronized flow the flow rate may be of the same order of magnitude as in free flow (Fig. 5(a), D18, right), but the vehicle speed is noticeably lower than in free flow and it is approximately the same on different lanes of the highway (Fig. 5(a), D18, right, t>06:36). To show that the transition from free flow to synchronized flow outside freeway bottlenecks, which occurs at t=06:36 in the vicinity of the detectors D18 (Fig. 5(a), D18, left, up arrow), is a local phase transition 'Free Flow => Synchronized Flow' outside freeway bottlenecks, spatialtemporal distributions of vehicle speed and the flow rate both upstream and downstream from the detectors D18 at later times should be studied. The results of such an investigation in the time interval 06:35 - 06:50 are shown in Fig. 5(b). It can be seen from Fig. 5(b) that transitions from free to synchronized flow both upstream (D17, D16, left in Fig. 5(b), up arrows) and downstream (D19, D20, left in Fig. 5(b), up arrows) occur later than at the detectors D18 (up arrow at t=06:36, left in Fig. 5(b)). Besides, the greater the distance from the detectors D18, the
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later the transition from free to synchronized flow occurs. This conclusion is true both upstream (D17, D16, left in Fig. 5(b), up arrows) and downstream (D19, D20, left in Fig. 5(b), up arrows) of the detectors D18. This confirms that the transition from free flow to synchronized Row first occurs only in the vicinity of the detectors D18, i.e., it is really a local phase transition 'Free Flow => Synchronized Flow' outside freeway bottlenecks. (a) A5-South, 17.03.1997, v [km/h] 120 --
Fig. 5(a). See caption to Fig. 5 (a, b) below. Note that the transitions from free flow to synchronized flow, which occur upstream (D17, D16, left in Fig. 5(b), up arrows) of the detectors D18 are linked to the appearance of a wave of induced transitions from free flow to synchronized flow. Respectively, the transitions from free to synchronized flow downstream of the detectors D18 (D19, D20, left in Fig. 5(b), up arrows) are linked to a wave of the propagating synchronized flow. The propagation of synchronized flow supplants free flow downstream. Indeed, as it has already been mentioned, the transitions from free flow to synchronized flow, which occur upstream (D17, D16, left in Fig. 5(b), up arrows) and downstream (D19, D20, left in Fig. 5(b), up arrows) of the detectors D18, occur later than at the detectors D18. Besides, the greater the distance from the detectors D18 is, the
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later they occur. Therefore, the transitions from free flow to synchronized flow which occur upstream (D17, D16, left in Fig. 5(b), up arrows) and downstream (D19, D20, left in Fig. 5(b), up arrows) are not local phase transitions, but induced transitions. The induced transitions upstream and the propagating synchronized flow downstream cause a widening of synchronized flow, which has first spontaneously occurred in the vicinity of the detectors D18: The region of localization of synchronized flow is widening both upstream and downstream over time. It must be noted that the detectors D20-D17 are situated outside bottlenecks. Therefore, the spontaneous occurrence of the local phase transition 'Free Flow => Synchronized Flow' is really a process of 'self-organization without bottlenecks' in traffic flow. 3000 T
1500
D20 0
06:50 06:35 06:40 3000 jq[veh/h]
06:45
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06:45
06:50
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06:50
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1500 --
D19 06:45
(b)
0
06:50 06:35 06:40 3000 T Q[veh/h] 1500 --
D18 06:45
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06:35
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3000 1500 --
06:50
0
D17
06:35
06:40
3000 TV 1500 "
06:35
06:40
06:45
0
D16
06:50 06:35
06:40
Fig. 5(a, b). Results of experimental observations of the phase transition 'Free Flow => Synchronized Flow' outside highway bottlenecks: The dependence of the vehicle speed (left) and the flow rate (right) at the different detectors both downstream and upstream from the location of the phase transition (D18) within time interval 06:20 - 06:40 (a) and 06:35 06:50 (b) for three lanes of the highway (for the detectors D23 the vehicle speed on the offramp is additionally shown, D23-off). Up arrows symbolically show the transitions 'Free Flow => Synchronized Flow' at the related detectors.
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2.2. Experimental Properties of Phase Transitions 'Synchronized Flow => Jam' outside Freeway Bottlenecks A different process of 'self-organization without bottlenecks' - the spontaneous occurrence of a traffic jam inside synchronized flow outside bottlenecks, i.e., the phase transition 'Synchronized Flow => Jam' outside freeway bottlenecks - occurs in the example under consideration a few minutes later (Fig. 6 (a), down arrows). It must be noted that the synchronized flow where the jam spontaneously emerges (D20) is located downstream from the region where the synchronized flow has earlier spontaneously occurred outside bottlenecks in the initially free flow (D16, up arrow, Fig. 5(b)). The jam occurs due to the sequence of the double phase transitions 'Free Flow => Synchronized Flow => Jam' (Kerner, 1998c): (i) First the phase transition 'Free => Synchronized flow' occurs (D18, Figs. 5(b) and 7(a)). (ii) Then the 'pinch effect' in synchronized flow, i.e., a self-compression of synchronized flow is realized, where states of synchronized flow are lying noticeably above the line J in the flow-density plane (D20, Fig. 7(b)). (iii) In the pinch region outside bottlenecks a growing local perturbation appears (down arrows at t = 06:50 in Fig. 6(a), D20). In contrast to the local phase transition 'Free flow => Synchronized flow' (Fig. 5), both the vehicle speed and the flow rate have decreased simultaneously and noticeably (down arrows left and right at t = 06:50 in Fig. 6(a), D20). The further behavior of this local perturbation shows the process of spontaneous self-formation of the traffic jam (down arrows in Fig. 6(a), D19-D15): The local perturbation gradually increases in the amplitude during its propagation upstream (from the detectors D20 to D15, Fig. 6(a)), i.e., both the vehicle speed and the flow rate in the jam decrease over time until they reach the values nearly zero (D16, Fig. 6(a)). It should be noted that the detectors D20-D17, where the processes of self-emergence and selfformation of the jam are realized, are located outside bottlenecks. Therefore, the processes of self-organization: (i) the spontaneous occurrence of a local perturbation (D20, t= 06:50, Fig. 6(a)), (ii) the self-maintaining and self-growth of this perturbation; (iii) the appearance of the traffic jam due to the further self-growth of the local perturbation really occur outside bottlenecks. For an additional verification of the fact that the self-organization of the jam formation has occurred outside bottlenecks, the vehicle speed and the flow rate at the bottleneck in the same time interval as in Fig. 6(a) (from 06:45 to 07:10) have been studied (Fig. 6(b)). It can been seen from Fig. 6(b) that no growing local perturbations have occurred downstream from the detectors D20. Besides, both earlier and during the spontaneous occurrence of the local perturbation (at t= 06:50, D20, Fig. 6(a)) and even later, when the process of the selfformation of the jam continues (D19 - D17, Fig. 6(a)), only free flow exists in the vicinity of the freeway bottleneck downstream (D23 and D22, Fig. 6(b)).
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A5-South, 17. 03.1997, — l e f t lane -- middle lane - . - r i g h t lane v[km/h] 3000 Tq[veh/h] 36:50
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After the jam has been self-formed, it propagates upstream with in average the same velocity of its downstream front independently of the complexity of states of traffic flow and independently of whether there is a freeway bottleneck or not (Fig. 8, down arrows).
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Fig. 6(a, b). Results of experimental observations of the phase transition 'Synchronized Flow => Jam' outside highway bottlenecks: The dependence of the vehicle speed (left) and the flow rate (right) at the different detectors both upstream (a) and downstream (b) from the location of the phase transition (D20). Arrows 'down' symbolically show the location of the jam at the different detectors. Arrows 'up' symbolically show the transitions 'Free Flow => Synchronized Flow' at the detectors D16 and D15 in the vicinity of the freeway bottleneck in the intersection 12 (Fig. 4).
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Fig. 7. Pinch effect in synchronized flow: The concatenation of states of free flow (black quadrates), synchronized flow (circles) and the line J at D18 (a) and D20 (b). Synchronized flow at D20 is related to higher densities than at D18.
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Fig. 8. Propagation of the jam (Fig. 6) through the highway bottleneck and upstream of the bottleneck: The dependence of the vehicle speed (left) and the flow rate (right) at the different detectors. Down arrows symbolically show the location of the jam at the different detectors. Up arrow at D16 symbolically show the phase transition 'Free Flow => Synchronized Flow' at in the vicinity of the freeway bottleneck in the intersection 12 (Fig. 4).
2.3. Comparison of Phase Transitions 'Free Flow the vicinity of Freeway Bottlenecks
Synchronized Flow' outside and in
It is interesting to make a comparison of the phase transition 'Free Flow => Synchronized Flow' which occurs outside freeway bottlenecks (Fig. 5) with the related effects where freeway bottlenecks may play an important role. First note that the wave of induced transitions 'Free Flow •=> Synchronized Flow' upstream of the location of the initial local phase transition (t = 06:36, D18, Fig. 5, left) at time t ~ 06:43 reaches the detectors D17 (up arrow, Fig. 5(b), left) and at time t « 06:44 reaches the detectors
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D16 (up arrows in Figs. 5(b), left). The latter detector is already located in the vicinity of the bottleneck (on-ramp, D15-on). It could be expected that if the wave of induced transitions 'Free Flow => Synchronized Flow' would propagate further upstream, then synchronized flow would be caught in the vicinity of this freeway bottleneck, i.e., it would be self-maintained in the vicinity of the freeway bottleneck for a long time. Such the 'catch-effect', as it follows from experimental observations which have been made during other days, can be really observed on a highway. However, in the case under consideration it does not occur. On the contrary, the reverse phase transition 'Synchronized Flow => Free Flow' is realized at the detectors D16 (up dotted arrow in Fig. 6(a), left). Therefore, the synchronized flow which has spontaneously occurred outside bottlenecks (t = 06:36, D18, Fig. 5, left) has later been also localized outside bottlenecks during the whole time of its existence. The free flow which has occurred due to the mentioned reverse phase transition 'Synchronized Flow => Free Flow' has existed at the detectors D16 only during about 5 min. After this time, the 'usual' phase transition 'Free Flow => Synchronized Flow' in the vicinity of a freeway bottleneck in the intersection 12 at t « 06:57 occurs (up solid arrow in Fig. 6(a), left). It should be noted that at a distance of about 12,6 km from the intersection 12 at time t « 06:37 another local phase transition 'Free Flow => Synchronized Flow' in a freeway bottleneck which is situated in the vicinity of the latter intersection has occurred (up arrow at the detectors D6 in Fig. 9, left). Both phase transitions 'Free Flow => Synchronized Flow' (up solid arrow in Fig. 6(a), left and up arrow at the detectors D6 in Fig. 9, left) show qualitatively the same peculiarities of first order phase transitions which have been considered in (Kerner and Reborn, 1997). A5-South, 17.03.1997,
D6 06:35 06:40 06:45 06:50 06:55 07:00 q[veh/h] 3000 T
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Fig. 9. The phase transition 'Free Flow => Synchronized Flow' in the bottleneck in the intersection II: The dependence of the vehicle speed (left) and the flow rate (right) at the different detectors both downstream and upstream from the location of the phase transition (D6). Up arrows show the transitions to synchronized flow at the related detectors.
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Thus, there are at least two types of transitions 'Free Flow => Synchronized Flow' in experimentally observed traffic flow: (i) The local phase transition outside bottlenecks which is an example of 'selforganization without bottlenecks' in traffic flow (up arrow, D18, Fig. 5, left). (ii) The local phase transition in the vicinity of a freeway bottleneck (the bottleneck located in the intersection 12, up solid arrow at D16, Fig. 6, left and the bottleneck located in the intersection II, up arrow at D6, Fig. 9, left). The first difference between these phase transitions is the different reasons of their occurrence: The phase transition (i) occurs without obvious reason due to the growth of a local perturbation outside bottlenecks and the transition (ii) occurs due to the growth of a local perturbation in the vicinity of a bottleneck. The second difference is that in the cases of the phase transition (ii) synchronized flow can be self-maintained in the vicinity of the bottleneck for several hours after the transition has occurred. On the contrary, in the case (i) synchronized flow usually exists for a relatively short time interval and it may propagate both upstream and downstream from the location where the phase transition has initially occurred.
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Fig. 10. The transitions 'Free Flow => Synchronized Flow' in the flow-density plane: The local first order phase transition outside bottlenecks (a), the local first order phase transitions in the freeway bottlenecks in the intersection 12 (b) and in the intersection II (c), respectively. Free flow is shown by black quadrates and synchronized flow by circles. Arrows symbolically show the transitions at the related detectors (Figs. 5 - 8). However, these two types of transitions 'Free Flow => Synchronized Flow' have important common properties: In the flow-density plane all of them show qualitatively the same breakdown effect in free flow (Fig. 10) which is well-known from numerous observations of freeway bottlenecks (e.g., Agyemang-Duah and Hall, 1991; Brannolte, 1991; Cassidy and Bertini, 1998; Persaud, et al, 1998). However, one could see from the consideration made above that the breakdown effect in free traffic is not exclusively a property of freeway bottlenecks: It can spontaneously occur outside any bottlenecks in free flow (up arrow at D18, Fig. 5). The latter is linked to the fact that these breakdown phenomena have the same nature - they are the local first order phase transitions 'Free Flow => Synchronized Flow' (Kerner andRehborn, 1997). Thus, 'self-organization without bottlenecks' is really observed in traffic flow. Therefore, phase transitions in traffic flow may not be explained by theories (Daganzo, 1997; Daganzo, et al, 1998) where congestion in traffic flow can only occur due to some capacity restriction in freeway bottlenecks rather than due to spontaneous effects of self-organization. On the contrary, experimental observations presented in the paper show that congestion can indeed spontaneously occur in traffic flow outside freeway bottlenecks. In other words, phase
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transitions in traffic flow may really be explained in the frame of the self-organization phenomena, in particular as considered in (Kerner and Rehborn, 1997; Kerner, 1998c).
3. THEORY OF CONGESTED TRAFFIC FLOW The results of experimental observations presented in Sect. 2 show that 'self-organization without bottlenecks' can really occur and play an important role in real traffic flow. 'Selforganization without bottlenecks' can be responsible for all three types of phase transitions in traffic flow: 'Free Flow => Synchronized Flow', 'Free Flow => Jam', and 'Synchronized Flow => Jam(s)'. Naturally, the local phase transition 'Free Flow => Synchronized Flow' occurs considerably more frequently in the vicinity of a freeway bottleneck. It is linked to the fact that traffic flow is usually permanently strongly non-homogeneous in the vicinity of freeway bottlenecks. This permanent non-homogeneity can act as 'a permanent nucleus' for the phase transition, whereas outside bottlenecks there is usually no such non-homogeneity which may 'support' the spontaneous emergence of synchronized traffic flow. On the one hand, the observations confirm that 'self-organization without bottlenecks' can really occur. On the other hand, from these observations it follows that some features of this 'self-organization without bottlenecks' and of self-organization phenomena which occur in the vicinity of a freeway bottleneck are qualitatively the same. Therefore, some intrinsic nonlinear properties of ' homogeneous traffic flow, which do not depend on whether bottlenecks exist on a highway or not, are essentially responsible for these effects of self-organization. A theory of these intrinsic non-linear properties of homogeneous traffic flow has recently been developed in (Kerner, 1998a, 1998c, 1999). Some of the hypotheses of this theory which may explain the observed phenomena of self-organizations presented in this paper and in (Kemer and Rehborn, 1996a, 1996b, 1997; Kerner, 1998c) will be explained below shortly.
3.1. Homogeneous States of Traffic Flow Experimental observations show that real traffic flow on a highway is usually nonhomogeneous. However, to understand features of real traffic flow, the properties of homogeneous states (steady states) of traffic flow should first be understood. The hypotheses presented below are devoted to these steady states of flow. (i)
A hypothesis about the multitude of homogeneous states (steady states) of traffic flow (Kerner and Rehborn, 1996b; Kerner, 1998a): In the flowdensity plane homogeneous states (steady states) of flow on a multi-laneroad are related to a curve (curve F) for free flow and to a two-dimensional region (hatched region in Fig. 3 (a)) for synchronized flow. The multitudes of states of free flow on a multi-lane-road overlap homogeneous states (steady states) of synchronized flow in the density. They are separated by a gap in the flow rate at a given density.
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(ii)
A hypothesis about the behavior of infinitesimal perturbations in initially homogeneous states of traffic flow (Kerner, 199 8 a): Independently of the vehicle density in an initial state of flow infinitesimal perturbations of traffic flow variables (the vehicle speed and/or the density) do not grow in any homogeneous states of either free or synchronized flow or else of homogeneous-in-speed states of synchronized flow: In the whole possible density range homogeneous states of flow can exist. In other words, in the whole possible density range (Fig. 3) there are no unstable homogeneous states of traffic flow with respect to infinitesimal perturbations of any traffic flow variables.
(iii)
A hypothesis about continuous spatial-temporal transitions between different states of synchronized flow (Kerner, 1998a): Local perturbations in synchronized flow can cause continuous spatial-temporal transitions between different states of both homogeneous and homogeneous-in-speed states of synchronized flow (hatched region in Fig. 3(a)).
To explain the hypotheses (i)-(iii), note that in synchronized flow spacing between vehicles is relatively low (i.e., the density is relatively high) in comparison with free flow at the same flow rate. At low spacing a driver is able to recognize a change in the spacing to the vehicle in front of him, even if the difference in speed is negligible. In other words, the driver is able to maintain a time-independent spacing (without taking fluctuations into account) to the vehicle in front of him in an initially homogeneous state of synchronized flow. The ability of drivers to maintain a time-independent spacing should be valid for a finite range of spacing. Therefore, a given vehicle speed may be related to an infinite multitude of homogeneous states with different densities in a limited range (p^^ < p < p^"' in Fig. 11 (a)). For this reason, the multitude of homogeneous states of synchronized flow covers a two-dimensional region in the flow-density plane (hatched region in Figs. 3(a), 11 (a)). States of free flow (curve F in Fig. 11 (a)) and states of synchronized flow (hatched region) overlap in densities. However, in free flow on a multi-lane road due to the possibility to pass the average vehicle speed can be higher than the maximum speed in synchronized flow at the same density. Therefore, there is a gap in the flow rate between states of free flow and synchronized flow at a given density. Small enough perturbations in spacing which are linked to an original fluctuation in the braking of a vehicle do not grow. Indeed, small enough changes in spacing are allowed, therefore drivers should not immediately react on it. For this reason even after a time delay, which is due to a finite reaction time of drivers T reac , the drivers upstream should not brake stronger than drivers in front of them to avoid an accident. As a result, a local perturbation of traffic variables (flow rate, density, or vehicle speed) of small enough amplitude does not grow. An occurrence of this perturbation may cause a spatial-temporal transition to another state of synchronized flow. In other words, local perturbations may cause continuous spatialtemporal transitions between different states of synchronized flow.
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Note that on a one-lane road, independently of the vehicle density, vehicles cannot pass. Therefore, homogeneous states of flow on the one-lane road at higher density are identical to homogeneous states of synchronized flow on a multi-lane road (hatched region in Fig. 3(b)).
3.2. Nucleation Effects and Phase Transitions in Traffic Flow (iv)
A hypothesis about two different kinds of nucleation effects in traffic flow (Kerner, 1999): There are two qualitatively different kinds of first order local phase transitions and, respectively, two qualitatively different kinds of 'nucleation effects' in traffic flow: (1) The 'nucleation effect' which is responsible for the jam's formation (Fig. ll(a,b)) and (2) The 'nucleation effect' which is responsible for the phase transition 'Free flow => Synchronized flow' (Fig. 1 l(c, d)). The nucleation effect which is responsible for the phase transition 'Free flow => Synchronized flow' is linked to a self-decrease in the probability of passing in traffic flow (P), i.e., in the probability that a driver is able to pass. The self-decrease in the probability of passing occurs if, owing to some local perturbation of traffic variables (the vehicle speed or/and the density), the probability of passing in the related local region of traffic flow has decreased below some critical value of the probability of passing Pcr (dotted curve Pcr in Fig. ll(d)). In contrast, the 'nucleation effect' which is responsible for the jam's formation is linked to the growth of a local critical perturbation of the traffic variables (the vehicle speed or/and the density). The local critical perturbation occurs if the amplitude of a local perturbation exceeds some critical amplitude (curves F(pert) and S(pert) in Fig. 1 l(b)).
(v)
A hypothesis about the probability of phase transitions in free flow (Kerner, 1999): The limit density ps (Fig. ll(d)) for the phase transition 'Free flow => Synchronized flow' can differ from the threshold of the phase transition 'Free flow => Jam' pb (Fig. 1 l(a, b)). Because the critical value of the probability of passing is an increasing function on the density (Fig. 1 l(d), dotted curve Pcr), the higher the density is the lower the amplitude of the local perturbation of the traffic variables which can cause the critical value of the probability of passing. Within the density range of free flow where both phase transitions can occur, the probability of an occurrence of the phase transition 'Free flow => Synchronized flow' is considerably higher than of the phase transition 'Free flow => Jam'. It is linked to the assumption that the amplitude of a local perturbation of traffic variables (the vehicle speed or/and the density), which is needed for the occurrence of the critical value of the probability of passing Pcr (Fig. ll(d), curve Pcr), is considerably lower than the critical amplitude of a local perturbation (Fig. ll(b), curve F*611*), which is needed
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for the jam's formation. The latter assumption is related to the result of observations (Kerner, 1998c) where it has been found out that the 'double' phase transitions 'Free flow => Synchronized flow => Jam' occurs considerably more frequently than the phase transition 'Free flow => Jam'.
Flow rate, q (c)
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Fig. 11. Hypotheses about different types of phase transitions in traffic flow: (a, b) Explanation of the phase transitions 'Free flow => Jam' and 'Synchronized flow => Jam(s)', and (c, d) - Explanation of the phase transition 'Free flow => Synchronized flow'. In Fig. a the concatenation of the line J with homogeneous states of free (curve F) and synchronized flow (hatched region) is shown. In Fig. b qualitative dependence of the critical amplitude of the density local perturbation on the density in metastable homogeneous states of free flow (curve F (pert) ) and of synchronized (curve S^pert\ which is related to a constant vehicle speed) flow are shown. In Fig. d a qualitative shape of the dependence of the probability of passing P on the density is shown. Homogeneous states of free (curve F) and synchronized flow (hatched region) in Figs, a, c are the same as in Fig. 3(a).
(vi)
A hypothesis about the nucleation effect which is responsible for the jam's formation (Kerner, 1998a, c): The line J (line J in Fig. ll(a)) determines the threshold of the jam's existence and excitation. In other
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words, all (an infinite number !) homogeneous states of traffic flow which are related to the line J in the flow-density plane are threshold states with respect to the jam's formation. The line J separates all homogeneous states of both free and synchronized flow into two qualitatively different classes: (1) In states which are related to points in the flow-density plane lying below (see axes in Fig. 1 l(a)) the line J no jams either can continue to exist or can be excited, and (2) States which are related to points in the flowdensity plane lying on and above the line J are 'metastable' states with respect to the jam's formation where the related nucleation effect can be realized. The local perturbations of traffic variables whose amplitude exceeds some critical amplitude grow and can lead to the jam's formation (up arrows in Fig. 1 l(b)), otherwise jams do not occur (down arrows in Fig. ll(b)). These critical local perturbations act as 'nucleation centers' (nuclei) for the jam's formation in traffic flow. The critical amplitude of the local perturbations is maximal at the line J and depends both on the density and on the flow rate above the line J. The critical amplitude of the local perturbations is considerably lower in synchronized flow than in free flow (Fig. ll(b)). The lower the vehicle speed in synchronized flow is the lower the critical amplitude of perturbations for states of synchronized flow being at the same distance above the line J in the flow-density plane. To explain the hypotheses (iv), (v), note that in free flow of low enough densities a driver is not hindered to pass, i.e., the related probability of passing is P=l (Fig. ll(d)). The higher the density in free flow is the lower the probability of passing; however in free flow up to the limit point p = p^f' tne probability of passing does not decrease drastically (solid curve PF, Fig. ll(d)). On the contrast to that in synchronized flow of high enough density drivers are not able to pass at all, i.e., P=0. The lower the density of synchronized flow is the higher the probability of passing; however in synchronized flow up to the limit point p = ps the probability of passing cannot increase drastically (solid curve Ps, Fig.
Synchronized and free flows overlap in the density, therefore pj^ > ps . As a result, the dependence P(p) is Z-shaped, i.e., it has a hysteresis loop, and therefore it consists of three branches: (i) The branch PF for free flow, (ii) the branch Ps for synchronized flow, and (iii) the branch Pcr which is related to the critical value of the probability of passing (dotted curve Pcr, Fig. 1 l(d)). The latter means that if in a local region of free flow due to a local perturbation of traffic variables (the speed or/and the density) the probability of passing is decreased below the critical value Pcr, then an avalanche self-decrease in the probability of passing occurs leading to a self-formation of synchronized flow where P=PS (down arrow in Fig. ll(d)). If on the contrary in a local region of synchronized flow due to a local perturbation of traffic variables the probability of passing is increased above the critical value Pcr, then an avalanche self-increase in the probability of passing occurs leading to a self- formation of free flow where P=PF (up arrow in Fig. 1 l(d)).
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It has been mentioned in the Introduction that above a threshold of any first order phase transition, the higher the distance from the threshold is the higher the related probability of the occurrence of the phase transition. In other words, the probability of the phase transition 'Free flow => Synchronized flow' should be an increasing function of the density in free flow, and it should be one at the limit point p = p|^aexe). Such a behavior of the probability of the breakdown phenomenon in a freeway bottleneck has recently been discovered by Persaud, et al (1998). This experimental fact confirms the conclusion made by Kerner and Rehborn in (1997) that the latter phenomenon is caused by the first order local phase transition 'Free flow => Synchronized flow'. An explanation of the hypothesis (vi) has been made in (Kerner, 1998c). (vii)
A hypothesis about the resulting states of traffic where the nucleation effect responsible for the jam's formation occurs. The resulting state of traffic flow, where the nucleation effect responsible for the jam's formation occurs, essentially depends even on small peculiarities of the initial state of flow. In particular, if the initial state is homogeneous, then jams appear caused by the nucleation effect. If, on the contrary, spacing between vehicles in an initial state of flow is essentially non-homogeneous, then the hypothesis (viii) which will be considered below may be valid.
The hypotheses (iv)-(vii) may explain the results of experimental observations where two qualitatively different types of first order phase transitions in traffic flow have been distinguished (Kerner and Rehbom, 1997). Besides, they may explain the following results which have been discovered in (Kerner, 1998c): (1) 'Double' phase transitions 'Free flow => Synchronized flow => Jam(s)' occur considerably more frequently than the phase transition 'Free flow => Jam'; (2) While 'stop-and-go' traffic patterns occur in synchronized flow, only single jams can appear in free flow.
3.3. Nucleation-Interruption Processes in Traffic Flow (viii)
A hypothesis about 'nucleation-interruption' processes in traffic flow (Kemer, 1998a): In traffic flow, where spacing between vehicles are very different one from another, the nucleation effect which is responsible for the jam's formation can be interrupted. A related 'nucleation-interruption' process may cause a sequence, i.e., a cascade, of qualitatively the same nucleation-interruption processes upstream from the initial perturbation, if the dispersion between spacing in an initial flow is large enough. The nucleation-interruption processes may be responsible for phase transitions from free to synchronized flow and for an occurrence of spatial-temporal chaos in traffic flow.
An explanation of this hypothesis has been made in (Kemer, 1998a).
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3.4. Propagation of Jams and 'Three Kinds' of Highway Capacity (ix)
A hypothesis about the process of jam propagation (Kerner, 1998c): The velocity of the downstream front of a wide jam v g does not depend on whether free flow or synchronized flow is formed in the outflow of the jam. Note that the existence of three qualitatively different phases of traffic (Kerner and Rehborn, 1996b): (i) free flow, (ii) synchronized flow and (iii) jams indicates that highway capacity depends on the phase (i, ii, or iii) on which traffic actually is in. The related 'maximal capacity' for free flow is q™raexe), for synchronized flow it is qj^, and downstream of a wide jam it is q out (Fig. 11 (a)).
4. CONCLUSIONS 1. Results of experimental observations allow to conclude the following: • There are at least two phenomena of 'self-organization without bottlenecks' in real traffic flow: (i) the local first order phase transition 'Free Flow => Synchronized Flow' and (ii) the local first order phase transition 'Synchronized Flow => Jam'. • The local phase transition 'Free Flow => Synchronized Flow', which occurs outside freeway bottlenecks, can cause two waves: (i) a wave of induced transitions 'Free Flow => Synchronized Flow' upstream of the initial location of this phase transition and (ii) a wave of the propagating synchronized flow downstream. • These waves, in turn, can cause a widening of the region of synchronized flow (i.e., a widening of congestion) both upstream and downstream. • The well-known breakdown phenomenon in a freeway bottleneck (e.g., Agyemang-Duah and Hall, 1991; Brannolte, 1991; Cassidy and Bertini, 1998; Persaud, et al, 1998), whose nature is linked to the local first order phase transition 'Free Flow => Synchronized Flow' in the vicinity of a freeway bottleneck (Kerner and Rehborn, 1997), has some differences and some common features with the local phase transition 'Free Flow => Synchronized Flow', which occurs outside freeway bottlenecks. A difference is that synchronized flow can be self-maintained for several hours in the case of the phase transition in a freeway bottleneck. On the contrary, in case of the local phase transition 'Free Flow => Synchronized Flow', which occurs outside freeway bottlenecks, synchronized flow usually exist for a relatively short time interval and the region of the location of the synchronized flow may propagate both upstream and downstream from the location where the phase transition has initially occurred. 2. A theory of congested traffic flow (Kerner, 1998a, c, 1999) may qualitatively explain the different local first order phase transitions in traffic flow.
ACKNOWLEDGMENTS I would like to thank H. Rehborn, S. Valkenberg and M. Aleksic' for their help, the Autobahnamt Frankfurt for the support in the preparation of the experimental data and the
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German Ministry of Education and Research for the financial support within the BMBF project ,,SANDY".
REFERENCES Agyemang-Duah, K., and F. Hall (1991) Some issues regarding the numerical value of capacity. In: Proceedings of International Symposium of Highway Capacity (U. Brannolte, ed.), p.1-15. A.A. Balkema, Rotterdam. Bando, M., K. Hasebe, K. Nakanishy, A. Nakayama, A. Shibata, and Y. Sugiyama (1995). Phenomenological study of dynamical model of traffic flow. Phys. I (France) 6, 1389-1399. Barlovic, R., L. Santen, A. Schadschneider, and M. Schreckenberg (1998). Metastable states in cellular automata for traffic flow. Eur. Phys. J. B, 5, 793-800. Cassidy, M.J., and R.L. Bertini. (1998). Some Traffic Features at Freeway Bottlenecks. Trans. Res. B (in press). Ceder, A. (1976). A deterministic flow model for two-regime approach. Trans. Res. Rec. 567, 16-30. Chandler, R. E., R. Herman, and E. W. Montroll (1958). Traffic dynamics: Studies in car following. Oper. Res. 6, 165-184. Daganzo, C. F. (1997). Fundamentals of Transportation and Traffic Operations. Elsevier Science Inc., New York. Daganzo, C. F., M.J. Cassidy and R.L. Bertini (1998). Causes of Phase Transitions in Highway Traffic. Trans. Res. B (in press). Hall, F. L. (1987). An interpretation of speed-flow concentration relationships using catastrophe theory. Transp. Res. A, 21, 191-201. Hall, F.L., B.L. Allen and M.A. Gunter. (1986). Empirical analysis of freeway flow-density relationships. Transp. Res. A, 20, 197-210. Helbing, D. (1997). Verkehrsdynamik. Springer, Berlin. Herrmann, M., and B.S. Kerner (1998). Local cluster effect in different traffic flow models. Physica A, 255, \63-\SS. Kerner, B.S. (1998a) A Theory of Congested Traffic Flow. In: Proceedings of 3rd International Symposium on Highway Capacity (R. Rysgaard, ed.), Vol. 2, pp. 621642. Road Directorate, Ministry of Transport - Denmark. Kerner, B. S. (1998b). Traffic flow: Experiment and Theory. In: Traffic and Granular Flow 97 (M. Schreckenberg and D. E. Wolf, eds.), pp. 239-268. Springer, Singapore. Kerner, B. S. (1998c). Experimental features of self-organization in traffic flow. Phys. Rev. Letters, 81, 3797-3800. Kemer, B.S. (1999). Congested traffic flow: Observations and theory. Preprint No. 990106, TRB, 78th Annual Meeting, January 10-14, Washington D.C. Kerner, B. S., S. L. Klenov, and P. Konhauser. (1997). Asymptotic theory of traffic jams. Phys. Rev. E, 56, 4200-4216. Kerner, B. S. and P. Konhauser. (1994). Structure and parameters of clusters in traffic flow. Phys. Rev. E, 50, 54-83.
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Kerner, B. S. and V.V. Osipov. (1994). Autosolitons: A New Approach to Problems of SelfOrganization and Turbulence. Kluwer Academic Publishers, Dordrecht, Boston, London. Kerner, B. S. and H. Rehborn. (1996a). Experimental features and characteristics of traffic jams. Phys. Rev. E, 53, R4275-R4278. Kerner, B. S. and H. Rehborn. (1996b). Experimental properties of complexity in traffic flow. Phys. Rev. E, 53, R1297-R1300. Kerner, B. S. and H. Rehborn. (1997). Experimental properties of phase transitions in traffic flow. Phys. Rev. Letters, 79, 4030-4033. Kerner, B. S. and H. Rehborn. (1998). Messungen des Verkehrsflusses: Charakteristische Eigenschaften von Staus auf Autobahnen. Internationales Verkehrswesen 50, 5/98, 196-203. Komentani, E., and T. Sasaki (1958). On stability of traffic flow. J. Oper. Res. (Japan) 2, 1126. Koshi, M., M. Iwasaki and I. Ohkura. (1983). Overview on vehicular flow characteristics. In: Transportation and Traffic Theory (V. F. Hurdle, E. Hauer and G. N. Stewart, eds.), pp. 403-426. Proceedings of 8th International Symposium on Transportation and Traffic Theory, University of Toronto Press, Toronto.. KrauB, S., P. Wagner, and C. Gawron, (1997). Metastable states in a microscopic model of traffic flow. Phys. Rev. E, 55, 5597. Ku'hne, R. (1991). Traffic patterns in unstable traffic flow on freeway. In: Highway Capacity and Level of Services (U. Brannolte, ed.), pp. 211-223. A. A. Balkema, Rotterdam. May, A.D. (1990). Traffic Flow Fundamentals. Prentice-Hall, Englewood Cliffs, New York. Nicolis, G., and I. Prigogine. (1977). Self-Organization in Non-equilibrium Systems. Wiley, New York. Persaud, B., S. Yagar and R. Brownlee. (1998). Exploration of the Breakdown Phenomenon in Freeway Traffic, Tranportation Research Board, Preprints of the 77th Annual Meeting, Washington, D.C. Prigogine, I. and R. Herman. (1971). Kinetic Theory of Vehicular Traffic. American Elsevier, New York. Schreckenberg, M., A. Schadschneider, K. Nagel, and N. Ito. (1995). Discrete stochastic models for traffic flow. Phys. Rev. E, 51, 2939-2949. Treiterer, J. (1975). Investigation of traffic dynamics by aerial photogrammetry techniques. Ohio State University, Report No. PB 246 094, Columbus, Ohio. Whitham, G. B. (1974). Linear and Nonlinear Waves. Wiley, New York.
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A MERGING-GlVEWAY BEHAVIOR MODEL CONSIDERING INTERACTIONS AT EXPRESSWAY ON-RAMPS Hideyuki Kita and Kei Fukuyama, Department of Social Systems Engineering. Tottori University, Tottori, Japan
INTRODUCTION As seen in the car following model, traditional traffic flow theory assumes that the influence of a car to the peripheral cars is one-directional. More or less, driving action depends on the actions of surrounding cars with each other. This traditional approach has given us useful information to understand the traffic phenomena. This approach is, however, not necessarily sufficient in describing the traffic behavior when the bi-directional influence plays a dominant role in driving actions (e.g. Troutbeck, 1995). Driving behavior between merging and through cars on an on-ramp merging section of an expressway is a case of this sort. While merging cars usually avoid unsafe situations by controlling the timing of their merge, the through car sometimes makes a "giveway" motion and keeps safer passing by changing its lane to the passing lane next to the merging lane. The behavior of the merging car under the influence of the through traffic also influences the through traffic. That is, both merging and through cars affect each other. Since their influences are not independent from one another, it should be jointly treated in the analysis as an "interaction". Ignoring this interaction may cause an inaccurate description of the traffic phenomena on a merging section.
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While many studies point out the need to analyze the giveway behavior, few studies have explicitly dealt with such behavior. Troutbeck (1995) analyzes the phenomena on a roundabout where the cars with the right of way running on the circle lane sometimes give their way to the merging cars at the entrances. Nielsen and Rysgaard (1995) reported merging and giveway behavior at motorway on-ramps in four EU countries and compared them with several indices. These studies are useful data sources for understanding giveway behavior. There exist many studies of lane changing behavior for traffic capacity analysis (e.g., TRB (1985), Cassidy et al: (1990), Vermijs (1991)), they merely try to clarify the relationship between the macroscopic characteristics such as traffic distribution ratio over lanes or lane changing ratio and road & traffic characteristics. Chang and Cao (1991) model the frequency of lane changing from the viewpoint of microscopic behavior analysis. Their study, however, does not explicitly handle the fundamental mechanism that the traffic behavior as a whole is formed as the result of interactions of driving decisions among cars. Under the recognition mentioned above, we developed a simple model for describing the traffic behavior of a couple of merging and through cars while taking into consideration the interaction between them explicitly (Kita, 1999). The study views the situation, in which each of the drivers chooses their best action by considering his/her forecast of the other driver's action, as a "game" and then, clarifies the mechanism under which the traffic phenomena such as the merging and/or giveway ratio over lanes are determined by the driving environments consisting of road and traffic characteristics and the driving decisions of other surrounding cars. However, the study does not necessarily investigate the existence of multiple equilibria and the transition of equilibria, so that the correspondence between the driving conditions as input and the chosen driving actions as output is not unique. This means that it is difficult to examine the replication capability of the model by using observed data. In this study, we show that a unique correspondence can be found between a certain driving environment and a pair of the resultant driving actions of the drivers as an equilibrium solution of the game, and the possibility to examine the replication ability of the model based on observation data. For this purpose, we refine the earlier version model extensively in order to specify the timing and location of their driving actions corresponding to the initial conditions, by analyzing the equilibria of the game especially their transition along time. According to these results, a test of the model using observation data will be implemented in a practice case study.
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A GAME THEORETIC INTERPRETATION OF MERGING-GIVEWAY INTERACTION Direction of Progress
[Wl
]
'fRil'
74
j
-Yz
-Ys
~Y4
_y
Figure 1: Merging section and the influencing variables Here, we shall analyze a give way motion often seen in the downflow section from the merging gore by the through car encountering a merging car and avoiding facing conflict. The merging section and the cars concerned in this study are depicted in Fig. 1. The speed of the merging cars is assumed to be slower than that of the through cars. Under this situation, if the merging car (Car [1] in Fig. 1) decides to merge behind the through car (Car [2] in Fig. 1), the through car may not give its way. If the through car wishes to give its way to the merging car, the merging car may merge in front of the through car. In this way, both the merging car and the through car attempt to take the best action for themselves by forecasting the other's action, respectively. To describe this situation as a game, we specify the structure of the game as follows.
1. A merging car (Car [1] in Fig. 1, also named 'Merging Car' in this study) and the closest through car approaching from the rear side in the adjacent lane to the acceleration lane (Car [2] and named 'Through Car') are the only players of the game. The other cars depicted in Fig. 1 such as Car [3] (which is also named 'Passing Car') and Car [4] (named 'Following Car') constitute the driving environment of Car [1]
and [2]. 2. The number of plays between two players over a merging section is one. 3. No communication and therefore no coalition can exist between the players. 4. The players have the complete information.
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THE MODEL AND THE EQUILIBRIUM ANALYSIS The Model
Through Car Go with Giveway
Go without Giveway
Merge
Fll, Gn
Fwt Gw
Pass
F01, Goi
F
Merging Car
°°' G°°
Figure 2: Merge-Giveway Game in Normal Form The behavior of the Merging Car and Through Car can be modeled as a two-person nonzero-sum non-cooperative game. The strategy of the Merging Car consists of "Merge (merging in front of the through car)" and "Pass (passing the through car)". The strategy of Through Car consists of "Go with Giveway" and "Go without Giveway" . This game is given in Fig. 2 as the normal form game. In Fig. 2, F^ is the payoff of the Merging Car where i = I means "Merge" and i = 0 means "Pass" , while dj is the payoff of the Through Car where j ' = I means "Go with Giveway" and j = 0 means "Go without Giveway", respectively. We denote the probabilities of merging and giveway choices by x and y, respectively. Then, the expected payoff of the Merging Car, U, and that of the Through Car, V, are given as follow.
U =
(1)
V =
(2)
Since driver behavior can be well described by the model provided in "Time to Collision" (Kita, 1993), assume that each payoff is determined solely by the (TTC) function fa, "Time to Collision (TTC)", t 1? £ 2 , *3, and t4. These are:
FQQ
=
Gn =
Goo
=
0
(3)
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177
where t\, t 2 , is, and £ 4 , represent TTCs between Car [1] and the end of the merging lane, Car [1] and Car [2], Car [2] and Car [3], and Car [1] and Car [4], respectively, and are denned asii = X/vi, t2 = y2/(v$-vi), t3 = (y3 -yz)/(v$ - v$), and U = 2/4/^2- y i)> where u1} uf, 1)3, and u|(= ^2)1 are the velocities of the car [1], [2], [3], and [4], respectively. Functions fij and §ij determine the payoffs of the Merging Car and Through Car, respectively, when the Merging Car and Through Car choose the strategies i(= 0 or 1) and j(= 0 or 1).
Equilibria The game has the following eight possible Nash equilibria, (z*, y*). (For details about how to obtain Nash equilibrium, refer, for example, to Rasmusen (1994)). I. (z*, y*) = (1,1) when Fn - F01 > 0, Fw - F00 > 0, Gn - GIO > 0 II. (z*,y*) = (1,0) when Fn - F01 > 0, Fw - F00 > 0, Gu - Gw < 0 III. (z*, y*) = (1, 0), (0,0), or (F, G) when Fn - Fol > 0, F10 - F00 < 0, Gn - G10 > 0 IV. (z*, y*) - (0, 0) when Fn - F01 > 0, F10 - F00 < 0, Gn - G10 < 0 V. (z*, y*) = (G, F) when Fn - F01 < 0, F10 - F00 > 0, G u - G10 > 0 VI. (z*, y*} = (1,0) when Fu - F01 < 0, F10 - F00 > 0, Gn - G10 < 0 VII. (z*, y*) = (0,0) when Fu - F0i < 0, F10 - F00 < 0, Gn - G10 > 0 VIII. (z*, y*) = (0,0) when Fu - F01 < 0, F10 - F00 < 0, G u - G10 < 0 where F = (F00 - F10)/{(F00 - F10) + (Fn + F01)} and G = -Gol/(Gn - G10 - G 01 ). All the cases above have unique equilibrium except for Case III which has multiple (three possible) equilibria of (1,0), (0,0), and the mixed strategy equilibrium. Case I and its equilibrium show that when the merging car has higher risk at the end of the merging lane and the through car recognizes the low risk on the passing lane, then (Merge, Go with Giveway) occurs. Here, (-, •) indicates ('behavior of merging car (Merging Car)', 'behavior of the through car (Through Car)'). Case II and its equilibrium indicate that when the situation of the merging car is same as in Case I and the through car recognizes the higher risk on the passing lane, then (Merge, Go without Giveway) may be realized. Case III and its equilibria mean that when the two cars are very close in TTC, either (Merge, Go with Giveway), (Pass, Go without Giveway), or their mixed-strategy equilibrium are
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Theory
(vn) (0,0) 1
(0,0) n\H ,
t4
o 33 OQ
cs
2
4.
" (0,0) rr \ .. -, ' (In ) n.n (G,F) i
O
"•h
(E)
(I) change of \.^ or U
Figure 3: Transition of Equilibria realized. Case IV and its equilibrium reveal that when the situation of the through car is same as in Case II and the merging car recognizes higher risk of merging, then (Pass, Go without Giveway) is realized. Case V and its equilibrium show that when the merging car still has a longer merging lane remaining and feels more risk from the following car than the through car, and the through car recognizes a low risk on the passing lane, then the mixed strategy equilibrium (in which the merging car sometimes chooses 'Merge' and the through car sometimes chooses 'Go with Giveway') occurs. Case VI and its equilibrium indicate that when the situation of the merging car is same as in V while the through car recognizes a higher risk on the passing lane, then (Merge, Go without Giveway) occurs. Case VII and its equilibrium show that when the merging car is close to the following car and still has a longer merging lane remaining and the through car recognizes a low risk on the passing lane, then (Pass, Go without Giveway) is realized. Finally, Case VIII and its equilibrium mean the situation of the merging car is the same as in Case VII, and the through car recognizes a higher risk on the passing lane, then (Merge, Go without Giveway) may be realized.
Analyses of Equilibrium The transition among the equilibria according to the change of the environmental parameters, TTCs, are analyzed in this section. Fig. 3 shows the possible transitions of equilibria due to the change of (decrease in) TTCs.
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At the equilibria I and II the merging car chooses Merge and therefore the merging-giveway game ends. At IV, because the merging car does not choose Merge, the game does not end. When the initial environmental situation is at IV, the game ends when the situation moves to either II or I via III. The transition of the equilibrium from IV to II means the change of inequality of FIQ — F0o < 0 into > 0. This change is brought about either by the shortening of the merging lane length remaining for the merging car or brought about by decreasing the TTC to that of the following car [3] for the merging car [1]. This transition of the equilibrium shows that due to the increase in both the risk of staying in the merging lane and the risk of merging to the next gap, the merge to the first gap by the merging car becomes the outcome. Next consider the case where the equilibrium moves from IV to I via III. The equilibrium transition from IV to III occurs when the inequality GH — G\Q < 0 is reversed. The merging car and through car become closer (t^ becomes smaller) and the through car, recognizing the higher risk against the merging car, gives its way to it, resulting in III. In III there exist three equilibria of (0.0), (1,1), and (G,F). We assume that the realizing equilibrium is (0, 0) in III by assuming path dependent decisions by the players; before the transition from IV to III the merging car and through car has chosen 'Pass' and 'Go without Giveway', respectively, at the previous equilibrium of IV. The transition from III to I occurs when the inequality F10 — FQQ < 0 is reversed by the change of the environment, similar to the transition from IV to II. This change of environment is brought about either by the shortening of the merging lane remaining for the merging car [1] or by the decrease in the TTC to the following car (£3) and therefore it means that the merging car chooses Merge due to the increase in risks of merging lane and also due to the next gap to merge. Next, consider when the game starts with III where there exists three equilibria. Without any additional information we cannot designate one equilibrium to realize out of three. By considering the 'pre-game' situation, however, the equilibrium selection can be done. The game has been set to begin when the merging car appearing at the merging section. This means that at 'pre-game situation' the initial situation before the strategy choices by both drivers are 'Pass' and 'Go without Giveway' and therefore (0,0). Employing the path dependence assumption again, we can designate (0, 0) among the three equilibria as the equilibrium to realize outcome III.
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GAME EQUILIBRIA AND TRAFFIC PHENOMENA Specification of Payoff Functions In the previous section, the equilibria are obtained by modeling the merging-giveway behaviors by focusing on the decision making process of the drivers at a certain environment 'at a moment'. Additionally by considering the transitions of conditions under which the equilibria hold, the changes of the merging and giveway decisions with time are explained. In this section, by specifying the payoff functions the traffic phenomena realized at the merging section are explained and the merging and giveway time and location are clarified. Here the relationships between the payoffs and the TTCs given in (3) are specified so that the payoffs are given by the corresponding TTCs themselves. The payoffs are specified by TTCs as follows.
„, + - i^ L Vt
X -vit
_ -TOO —
- 2/2) - (vst - 3/3) t>3 - vl
- 3/2) - (v3t - 3/3) -V20 Gin —
Goo = 0
(4)
where t is the length of time measured from when the merging car first appears at the merging lane, 7/2 > 2/3, and 2/4 are the distances of the car [2], [3], and [4], respectively, from the merging nose when t = 0, and x is the length of the merging lane. By substituting the specification of the payoff functions in (4) into the three inequality conditions, Fn - Fol > (<)0, FIO - FQO > (<)0, and Gn - G10 > (<)0, they can be rewritten as follows. <
V2 - Vi
< Vi
FwGn — GIO . 0 <
V 2 — V i > V3 — V i
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Notice that in these three conditions that determine the equilibrium realized they do not include the time variable t and therefore they are not affected by the progress of time. In other words, when assuming payoffs consisting of TTC themselves the equilibrium transitions never appear. The merging in addition to the giveway times and locations realized are therefore determined right away by the location of all cars when the merging car appears at the merging nose. These three conditions can be interpreted as follows.
Inequality Fn — F0i > (<)0: The sign of this inequality is determined by the TTC of the merging car [1] and the following car [4], and the TTC of the merging car [1] and the end of the merging lane. This corresponds to the comparison of the sizes of t\ and £3, indicating whether the merging car [1] will be passed by the following car [4] before reaching the end of the merging lane. Inequality Fw — F00 > (<)0: The sign of this inequality is determined by the TTC of the merging car [1] and the end of the merging lane, and the TTC of the merging car [1] and the through car [2]. This corresponds to the comparison of the sizes of t\ and t2, indicating whether the merging car [1] will be passed by the through car [2] before reaching the end of the merging lane. Inequality G\\ — G\Q > (<)0: The sign of this inequality is determined by the TTC of the merging car [1] and the through car [2], and the TTC of the merging car [1] and the passing car [3]. This corresponds to the comparison of the sizes of t-2 and £3 + ^4, indicating whether the through car [2] will be passed by the passing car [3] before catching the merging car [1].
Traffic Phenomena and the Game Equilibrium The game is constructed to capture the merging-giveway situation 'at a moment' of the merging section. Accordingly, the equilibria include the ones in which the merging car decides not to merge to the central lane (or chooses 'Pass'; y* = 0). We are interested in analyzing not the drivers' behavior of the moment but the traffic phenomena that occur at the merging section. The merging car should merge to the central lane at some point in time; otherwise it stacks at the end of the merging lane. Using the game analyses given above, we should examine when and where the merging car merges to the central lane and also when and where the through car chooses to giveway.
Transportation and Traffic Theory
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Table 1: Merging and Giveway Behaviors and Equilibrium Conditions
Equilibrium I II III IV
(x*,y*)
(i,i)
(1,0) (1,1)(0,0)(G,F) (0,0)
FH — FQI + +
FIO — FQQ
G\l — GW
+
-
+
+
(1,0)
+ + -
-
-
(i,i)*
-
V + (G,F) VI + (1,0) VII (0,0) VIII (0,0) * Indicates the results different from the game equilibrium
+ +
(z,y) (i.i) (1,0)*
— —
-
(i,i)*
+
(1,0)*
\J"mi t j j
(0,0)
(o,-) *>
Vy—Vl
(
'
V%— Vl'
yz
\v%—vi '
N '
(-M Jte_) ^v* — vi ' v^—vi '
(-M v^f=^7'
>)
According to the interpretation of the three equilibrium conditions of inequality given in the previous section, the game equilibria can be interpreted as the traffic phenomena (merge and giveway behaviors and their timings) under the payoff function specification by the TTCs themselves given above. They are summarized in the sixth and seventh columns in Table 1. In Table 1, (x,y) means whether the merging car [1] ever merges or not (x = I when merging) and whether the through car [2] changes the lane and chooses giveway behavior or not (y = 1 when giveway), respectively. Notice that "—" in Table 1 means the corresponding behavior cannot occur because of the inconsistent conditions exist. tm and tg indicate the time that the merging car [1] merges to the central lane and the time that the through car [2] changes to the far right lane (chooses 'Go with Giveway'), respectively. The traffic phenomenon to realize in the cases that correspond to equilibria I and II are exactly same as the equilibrium strategy ((x*,y*) = (x, y)). In equilibrium I, the merging car merges to the central lane immediately and the through car goes with giveway immediately (tm = tg = 0). In equilibrium II, the merging car merges to the central lane immediately, and the passing car [3] stays at the central lane. Other cases, Case III to VIII, have a different traffic phenomena from the game equilibria (compare the entries in the second and sixth columns in table 1). The reason and the interpretation of these cases are as follows. In equilibrium III, The merging car [1], which will be passed by the through car [2] before the end of the merging lane but not by the following car [4], merges to the central lane right after being passed by the through car (into the gap between the through car [1] and
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183
the following car [4]). Also, the through car [2], faced with risk of catching the merging car [1], chooses to give its way to the merging car and changes the lane to the passing lane because it is never caught by the passing car [3]. Consequently, the traffic phenomenon in this case will be (Merge, Go with Giveway), and the times of merging and giveway that occur simultaneously are given by tm=tg = yij(v\ — t'i). In equilibrium IV, the situation and the behavior of the merging car [1] are same as in the equilibrium III above. The through car [2], which is under risk of catching the merging car [1] before the end of the merging lane, cannot change to the passing lane because it will be caught by the passing car [3]. Consequently, the corresponding traffic phenomenon is (Merge, Go without Giveway). The merging time is given by tm = yil{v\ — v\] because the merging car merges to the central lane just after being passed by the through car, and the giveway time does not exist because no giveway behavior exists. Under the specification of the payoff in this study, the two conditions, Fu — F0i < 0 and FW — -Foo > 0, constituting the equilibrium V and VI, imply that the following car [4] is running ahead of the through car [2]. Consequently, these conditions cannot hold and the traffic phenomenon corresponding to these two equilibria do not exist. In equilibrium VII, the merging car [1] is passed by the through car [2] and also by the following car [4] before reaching the end of the merging lane. It merges to the central lane just after being passed by the following car [4]. The through car [2], which is assumed to always run at the front of the following car [4], already passed the merging car when it merges to that lane. Also, the through car [2] which is never being caught by the passing car [3], changes the lane to the passing one. Consequently, the traffic phenomenon to realize is given by (Merge, Go with Giveway). The corresponding merging and giveway times, that occur simultaneously, are given by tm = tg = y$/(v\ — v\). Finally, In equilibrium VIII, The situation and the behavior of the merging car [1] are exactly same as the ones given in the equilibrium VII above, and it merges to the central lane right after being passed by the following car [4]. The through car [2], which will be caught by the passing car [3] before the merging car reaches the end of the merging lane, does not change the lane to the passing one. Consequently, the realizing traffic phenomenon is (Merge, Go without Giveway). The corresponding giveway time is given as tm = ?/4/(f| — fi) while the giveway time does not exist. Consequently, the merging car merges to the central lane immediately when equilibrium is I or II, merges after being passed by the through car [2] when III and IV, and merges
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Theory
erge i m m e d i a t e l y erge behind Car [2] erge behind Car [4]
0 TTC of Car [1] and Car [4]
a) TTCs of cars and the merging behavior
i 2 TTC of Car [1] and Car [2]
b)TTCs of cars and the giveway behavior
Figure 4: Occurrence of the merging and giveway behaviors after being passed by the following car [4] when VII or VIII. On the other hand, the through car changes to the far right lane when the equilibrium is I, III, or VII, and does not change the lane for II, IV, or VIII. These relationships between the equilibria and the resulting traffic behaviors are depicted in Fig. 4 on the TTC planes of (£2,^4) and (£2^3)5 by using the three inequalities governing the realizing equilibrium. Fig. 4 a) depicts the merging behavior, and the horizontal and vertical lines that distinguish the equilibrium areas indicate £2 = X/v\ and U = X/vi, respectively. Fig. 4 b) shows the equilibrium area for the giveway one, and the line that distinguish the two equilibrium area is given
by £3 = (t>2 —
A CASE STUDY Data and Their Handling To examine the proposed model, a case study is carried out by using a set of video-recorded data observed at an on-ramp merging section of Nagoya I.C., Tohmei Expressway, in Nagoya (data source is Research Group on Design of Intersections (1987)). The number of observation samples of the merging behavior was 74. Among these, observations that have complete sets of explanatory variables were used: there are 10 such complete sets of data in the observation data set. The merging lane has 200m in length, and the average velocities of the vehicles were assumed to be constant and estimated as v\ =82.8km/h for the merging car, v\ = v\ =85.8km/h for the through and following cars, and v3 =95.7km/h
A Merging-Giveway Behavior Model
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for the passing car.
Results The data is plotted on Fig. 5 a) and Fig. 5 b). The solid dots and circles indicate data that "Car [1] merges behind Car [2]" and "Car [1] merges immediately", respectively, in Fig. 5 a), and "Car [3] goes with giveway" and "Car [2] goes without giveway" in Fig. 5 b), respectively. Due to the velocities estimated, the lines, t^ — x/v\ and £4 = x/Vi, that distinguish the equilibrium areas in Fig. 4 a) are both 8.696 sec. The plots of the observed data ('Merge immediately' and 'Merge behind Car [2]') are consistent with the designated areas of the game equilibrium, while no data that a merging car 'merges to the central lane after being passed by the following car' or 'merge behind Car [4]' is available. Next consider Fig. 5 b). According to the velocities of the car set, the slope that distinguishes the two areas is (v\ — v\)/(v^ — v\] = 0.36. The nine data plots for the cars that did not do giveway are consistent with the area designated by the game equilibrium analyses, and that is also true for the one observation plot in which the car went without giveway. Though the sample size is not large enough to fully support the validity of the model, through this data references to the proposed model can be recognized, for instance it gives a good description of the traffic behavior of giveway and merging in on-ramp merging sections. This result verifies the ability of this new approach to understand the traffic phenomena with interactions.
CONCLUSION In this study, the traffic behaviors such as merging and giveway are expressed as game strategy, and the relationships between the traffic phenomena to realize and the driving environment surrounding the drivers are clarified. By understanding the change of driving behaviors (and therefore the traffic phenomena) as the transition from one game equilibrium to another, the mechanism under which the drivers' behaviors and the resulting phenomenon occur and change according to time progress and the change of driving environment. Furthermore, by specifying the payoff functions in the merging-giveway game
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(sec)
White: Merge behind Car [2] Black: Merge immediately
White: Go with Giveway Black: Go without Giveway
40
20 t3=0.36t2
0
8.696
20
60
20 t4
(sec)
a) TTCs of cars and merging behavior
40
t2 (sec)
b) TTCs of cars and the giveway behavior
Figure 5: Observed merging and giveway behaviors model, the conditions under which the merging and giveway behaviors emerge are explicitly induced and the merging and the giveway timing and locations are specified. With this model development a unique correspondence is found between a certain driving environment and a pair of the resultant driving actions of the drivers as an equilibrium solution of the game, and presents the possibility to examine the replication ability of the model based on observation data. Finally, the validity and applicability of the proposed model are checked by applying the observation data. By employing the approach and models employed in this study, the occurrence and change (location and timing) of driver's behaviors such as giveway and merging motions, and the traffic phenomena resulting from the combination of such behaviors can be specified. Accordingly, application of this study contributes to the construction of effective traffic control systems and the design of safer and more comfortable intersections of expressways, explicitly considering the car motions. The game theoretic approach to the traffic behavior modeling has very little research accumulation and several important aspect remain uninvestigated. Among others, most importantly, the development of a new payoff estimation technique is necessary. With this, other important progresses will be also possible: non-linear payoff function applications which enables us to explain the merging behavior at the middle of the gap, and game modeling with incomplete information. Formulation to translate this microscopic model into macroscopic results should be also developed.
A Merging-Giveway Behavior Model
18 7
These various extensions of this study that have not been considered in this study are all important; the existence of many possible future directions of this study indicates that the model and approach can be easily extended to become more robust.
ACKNOWLEDGEMENT The authors would like to express their appreciation to the anonymous referees for their valuable comments. This study was supported financially in part by the Grant-in-Aid for Scientific Research, Ministry of Education, and Sumitomo Marine Welfare Foundation.
REFERENCES Cassidy, M. J. et al. (1990). A proposed analytical technique for the design and analysis of major freeway weaving sections, Inst. of Transp. Studies, Univ. of California at Berkeley, UCB-ITS-RR-90-16. Chang G.-L., and Y.-M. Cao (1991). An empirical investigation of macroscopic lane changing characteristics on uncongested multilane freeways, Trans. Res., 25A, 6, 375-389. Kita, H. (1993). Effects of merging lane length on the merging behavior at expressway on-ramps. In: Transportation and Traffic Theory (C. Daganzo ed.), Elsevier, 37-51. Kita, H. (1999). A merging-giveway interaction model of cars in a merging section: a game theoretic analysis, Transportation Research, Vol.33A, No.3/4, 305-312. Nielsen, M. A., and Rysgaard, R. (1995). Merging Contra Give Way When Entering A Motorway. Road Directorate Report, 27, Danish Road Directorate, Copenhagen. Rasmusen, E. (1994). Games and Information. 2nd edition, Blackwell Publisher, Massachusetts, 67-91. Japan Society of Traffic Engineers (1987). Report on the Design of the Intersections, Research Group on Design of Intersections, 2, Japan Society of Traffic Engineers (in Japanese). Transportation Research Board (1985). Highway Capacity Manual, Special Report, 209. Troutbeck, R. J. (1995). The capacity of a limited priority merge. Physical Infrastructure Centre Research Report, 95-8, Queensland University of Technology, Australia. Vermijs, R. G. M. M. (1991). The use of micro simulation for the design of weaving sections, In: Highway Capacity and Level of Service (U. Brannolte. ed.), 419-227, A. A. Balkema.
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CHAPTER 3 ROAD SAFETY AND PEDESTRIANS
• Nothing in life is to be feared. It is only to be understood. (Marie Curie) •
There is more to life than increasing its speed. (Mahatma Ghandi)
•
A problem adequately stated is a problem well on its way to being solved.
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Comparison of Results of Methods of the Identification of High Risk Road Sections
Marian Tracz, Chair of Highway & Traffic Engineering, Cracow University of Technology, ul. Warszawska 24, 31-155 Cracow, POLAND, e-mail: [email protected] Marzena Nowakowska, Laboratory of Computer Science, Kielce University of Technology, Al. 1000-leciaP.P. 3, 25-314 Kielce, POLAND, e-mail: [email protected]
ABSTRACT In this paper some measures for effectiveness evaluation of the methods of the identification of high risk sections on rural roads have been proposed. Two categories of measures have been taken into account. First concerns the level of accidents' concentration along hazardous road sections as well as the level of concentration of accidents severity. The second category concerns the problem of repeatability of the identification procedures during two consecutive time periods in relation both to the location of dangerous sections along a road and to some features of road accidents. The measures have been applied to three Polish methods and the real data have been used to conduct a comparison. Some lacks which were found in the identification methods in this study have been completed. The results varied for different aspects and accident data from different roads.
INTRODUCTION Accident risk at dangerous road sites can be an important factor while identifying and evaluating them. In practice these dangerous sites can be found in parts of a road network where accident intensity is comparatively large. So, the studies are usually focused on urban
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roads and on the sites that are prone to accidents of various types because of their geometrical characteristics - as urban junctions. In the identification and evaluation of dangerous road sections, the majority of highway authorities in Poland use methods that are based on rather simple criteria such as numbers of accidents or average density of accidents (ace./km/year) that happened on a road section during a certain period of time (usually 3 years). In research it is commonly assumed that the number of accidents at a site over a fixed time period is well modelled by a Poisson distribution. Traffic volume is not taken into account due to the lack of current traffic data. When investigating the safety of rural roads a problem arises how to identify dangerous sections as they can have various lengths and they are not evenly spaced along a road. Polish methods use different algorithms of dividing a road into sections for further investigations. In addition, the time periods for which high risk road sites are identified also differ in various methods. In consequence, the results of accident analysis are frequently not comparable. In this study some measures have been proposed for evaluation of the effectiveness of the methods that are oriented towards rural roads. For those who deal with the problem, these measures should be helpful in comparing results. The paper does not only present a pure methodological study of the analysed methods but also includes some practical aspects.
MEASURES FOR COMPARING RESULTS OF THE IDENTIFICATION OF HIGH RISK ROAD SECTIONS There are several methodologies that are used in analysis of traffic accidents and in the identification of high risk sections on a road (Hauer, 1995). When using different methods, different distributions and numbers of dangerous sections along a road can be a consequence of changing a selection criterion. Then this can affect the effectiveness of road safety measures. All these issues depend to a great extent on the reliability and validity of accident data (Hakkert and Hauer). Consequently, the results of the identification of dangerous road sections are influenced by accident analysis methodology, data collection and data handling. Therefore, an important question arises: which identification procedure should be used to get the best results? A variety of techniques have been applied to answer the question (Maher and Mountain, 1988; Mountain at all, 1992a, 1992b). To choose a proper method reaserchers use statistical tools and indices derived on the basis of probability theory. In the analyses they use real or artificial (generated by a computer) accident data. In this paper some measures for evaluating and comparing the performance of various identification methods have been introduced. The process of comparison was carried out on the basis of real accident data.
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Method Inaccuracy Coefficient Identification procedures applied to rural roads are usually accomplished on the basis of the data that come from time periods of different lengths. Such data contain information of various levels of reference. So the comparion of results of the identification of dangerous road sections obtained from these data should be processed on the basis of indices that can be easily examined. For road sections a method can be considered as a more accurate if there are: • the high proportion of the total number of accidents on dangerous sections in relation to the total number of accidents on the whole analysed road, • the low proportion of the total lengths of these sections in relation to the length of the road. An average accident density (ace. /km/year) is the ratio used very commonly. The greater its value calculated for hazardous road sections the better the results of an identification procedure. In extreme situations, the accident density can reach infinity for zero length sections. There are, however, some disadvantages of using this measure. Comparing the accuracy of results this measure works quite well for one road but not for several roads. Average accident density is an absolute measure and can be as well considered as a suplemantary measure. In order to consider more intuitive measure, a certain relative ratio has been put forward in this work. It has been called the Method Inaccuracy Coefficient (MIC) and is defined as follows:
MIC = [ I l./rl]/[
where: n /, rl nacCj rnacc
I
nacc./rnacc]
0)
- number of high risk sections on a road, - length of the z-th high risk road section, - length of the considered road, - total number of accidents on the j-th high risk road section, - total number of accidents on the road.
The MIC coefficient expresses the proportion of the fraction of the total length of dangerous sections in the road length to the fraction of the total number of accidents occurred on these sections in the total number of accidents on the road. The value of the coefficient can exceed unity. However, if the value really concerns high risk road sections, it is included in the interval <0,1>. The closer to zero the value of the coefficient is, the more accurate or precise the method is and reciprocally. Coefficients describing accidents severity are defined in the same manner and are used here as supporting measures. They are dentoed as MIC(I), MIC(F) and MIC(V) and calculated
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according to the formula (1) in which the number of accidents is respectively replaced by the number of injuries (I), fatalities (F) and vehicles (V). The interpretation of the values of these coefficients is similar to the MIC interpretation.
Space Compatibility Index When considering the process of the identification of hazardous rural road sections, additional potential impact should be taken into account. This is commonly known as "accident migration" or "migration of accident risk" and referred to a phenomenon arising after applying some remedial treatments on a road section or because of an increased percent of time delay (travelling in platoon). These are, for example, accidents typical for dangerous overtaking occurring sometimes a few kilometers from a place where the overtaking demand has appeared. However, during this study it has been noticed that the ends of a high risk road section can change during two consecutive time periods of the identification process, even if no treatment had been applied in relation to the "earlier period" section. In this paper, such period-afterperiod changing of locations of the hazardous sections along a road can be expressed by a certain index. This index has been build using "positive" and "negative" differences between the mutual position of the ends of consecutive (earlier and later) overlapping dangerous road sections: (2)
dr = sgn(r, (1) - r, (2)) • \ r, (1) - r (2) \
where: !,(!)
r,(l)
dlj
drt
(3)
- kilometreage of the left end of the z'-th overlapping section obtained from an identification procedure for the earlier time period; lt(2) is defined likewise for the later time period, - kilometreage of the right end of the /-th overlapping section obtained from an identification procedure for the earlier time period; rt(2) is defined likewise for the later time period, - difference between left ends of the two r'-th overlapping road sections; the sign of the difference informs whether the kilometreage of the end of the later time period section is lower (,,plus" sign) or higher (,,minus" sign) than the kilometreage of the respective end of the earlier time period section, - defined similarity to the dlt difference with respect to the right ends.
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The earlier time period, for which identification procedures have been processed, will also be called the first time period in this paper, and the later time period will be called the second time period. The scheme of overlapping of two identified dangerous road sections is presented in Figure 1. In this scheme, the lower section s(l) represents a road section from the first time period whereas the upper section s(2) represents a road section from the second time period. The notations s(l) and s(2) also represent the respective sections lengths. The remaining notations follow the explanation given for the equations (2) and (3). Subscripts were omitted for simplification.
kilometreage
Figure. 1. Illustration of a mutual position of overlapping dangerous road sections identified for two consecutive time periods. All possible alternatives of mutual position of the two overlapping dangerous road sections s(l) and s(2), obtained for the first and the second time periods in accident analysis, are presented in Figure 2. For each alternative a short description is given, that explains the meaning of signs of the positive and negative differences (formulae (2) and (3)). As can be seen, there is one perfect overlapping (alternative IX) among the nine presented cases. There are also a few other alternatives that can be considered as representing satisfactory space compatibility of overlapping sections - alternatives III- VIII. Only in two cases I and II the space compatibility seems doubtful. Uncertainty is especially justified when the length of the common part of the two sections is shorter than 50% of the length of the total section in the case I or of the total section in the case II. In order to evaluate the degree of repeatability of an identification method - regarding the space distribution of the identified dangerous sections, the Space Compatibility Index (SCI) was introduced as follows:
(4)
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196 where: n/ ri2 n
x,
- number of dangerous sections on a road identified in the first time period, - number of dangerous sections on the same road identified in the second time period, - number of dangerous sections on the road identified in the first time period that have common parts with dangerous sections on the same road identified in the second time period; i.e. the number of pairs of dangerous sections on the road identified in two consecutive time periods. The mutual locations of sections treated as a pair follow one of the schemes peresented in Figure 2. If all dangerous sections from both time periods follow the schemes then ni=ri2=n, - coefficient that specifies the way of overlapping of the pair of dangerous road sections. The coefficient has been defined only for those pairs of sections that have common parts (i=l,...,n) and it distinguishes satisfactory space compatibility (alternatives III-IX in Figure 2) and not satisfactory space compatibility (alternatives I-II): {1.0 x . = \ ,
'
(ssljsjl)
dl -dr. <=0 ' '
dl,-dr>0
i = l...n
(5)
In the formula (5) ssli is the total length of the sum of the two overlapping road sections determined by the very ends of these sections: sslt = maxfafl),
rf(2)} - min{lt(l), lt(2)}
Sj(l) is the length of the /-th overlapping road section identified in the first time period. The idea of the SCI index described by (4) has been taken from the concept of standard logarithm information function (Nicholson, 1995). The SCI index is a complex measure that takes into account the number and the mutual positions of dangerous road sections identified in two consecutive time periods. The minimal value of SCI is equal to 0.3. This value describes the most satisfactory space compatibility, where «/=«2 = « and xt =1.0 for each i=l,...,n. This means that all dangerous road sections identified in the first time period were identified as dangerous in the second time period with the satisfactory period-after-period shifts along a road (schemes III-IX), and any additional sections were not identified in the second time period.
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s(2) I) dl>0, dr>0; the s(2) section is shifted to the left in relation to the s(l) section
II) dl<0, dr<0; the s(2) section is shifted to the right in relation to the s(l) section
s(2) III) dl>0, dr<0; the s(2) section covers the s(l) section
IV) dl<0, dr>0; the s(2) section is covered by the s(l) section
V) dl=0, dr>0; the s(2) section is covered by the s(l) section, left ends are equal
s(2) VI) dl=0, dr<0; the s(2) section covers the s(l) section, left ends are equal
s(2) VII) dl>0, dr=0; the s(2) section covers the s(l) section, right ends are equal
VIII) dl<0, dr=0; the s(2) section is covered by the s( 1) section, right ends are equal
IX) dl=0, dr=0; the perfect overlapping - s(2) and s(l) are the same sections
Figure. 2. Schemes of mutual positions of overlapping dangerous road sections obtained from an identification procedure for two consecutive time periods.
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The value of the SCI higher than 0.3 means that: ni = ri2 but there are ,,unsatisfactory shifts" in mutual positions of overlapping sections such as presented by the schemes I or II (Figure 2), • HI * ri2 but there are satisfactory shifts" in mutual positions of overlapping sections such as presented by the schemes III-IX, • nj * ri2 and there are ,,unsatisfactory shifts" in mutual positions of overlapping sections. The last case is usually the most frequent one - the greater the value of SCI is, the worse the method of high risk road sections identification is with regard to space compatibility. •
Accident Patterns Repeatability Measures The SCI index evaluates the space repeatability of danagerous road sections in two consecutive time periods. In order to support this index, some additional measures are suggested in this chapter to check the repeatability of accident patterns in relation to overlapping road sections. In this work, the accident patterns are expressed by qualitative features of accidents recorded on dangerous road sections such as driver's behaviour and accident type. The classification categories representing these features have to be independent of each other within each of the features (Tracz and Nowakowska, 1998). The measures accessing repeatability of accident patterns are build using commonly known standard error ideas derived from the estimation theory. For each overlapping dangerous section /, the occurence of classification categories of a chosen accident feature was checked for two consecutive time periods t=l,2. Then, if a given accident category was recorded for at least one time period, the percentages of this category occurence for the respective time periods were calculated according to the formula (6):
FCPij(t)=
where: FCPyft) nij(t) kt
"'J
t = l,2
i = l...n
j = L..k,
(6)
- percentage of the occurence of an accident feature category, recorded on the z'-th overlapping dangerous road section in the time period t, - frequency of the occurence of an accident feature category on the z'-th overlapping dangerous road section recorded in the time period /, - number of different categories of the accident feature recorded on the z'-th overlapping pair of road sections.
The introduction of the percentages FCPy allowed to define three complementary measures of accident pattern repeatability:
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- minimum absolute difference MIAD(FCP) in accident feature pattern:
- mean absolute difference MAD in accident feature pattern:
(9)
All notations on the right-hand sides of the definitions (7) - (9) are the same as in the measures described earlier (formulae (1), (4), (6)). The range of the measures of accident pattern repeatability is the interval <0,1>. The value 0 of the measures MAAD and MAD confirms perfect repeatability in accident patterns. The closer to unity the values of the MIAD and the MAD are, the lower the degree of repeatability.
METHODS OF THE IDENTIFIACTION OF HIGH RISK ROAD SECTIONS USED IN POLAND A few methods for the evaluation of traffic safety levels on rural roads have been implemented in Poland. Some of them are attractive for traffic engineers because of their simplicity. These are commonly used methods known as the Warsaw method and the Gdansk method (Datka at all., 1989, 1997). Both are based on the knowledge of accident numbers and average accident densities. Other methods are less popular due to their requirements regarding accident and traffic data or due to probability aspects regarding stochastic character of road accidents. One of such methods has been worked up recently (Tracz and Nowakowska, 1996). Is is based on cluster analysis and Bayesian theory and,takes into account random character of accidents in time and space. The three methods were taken as the subject of a survey in this work. For simplicity they have been named here the WM method, the GM method and the BM method.
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The Warsaw Method A unit road section is used to define the method. It is the one-kilometer-long section determined by a road kilometreage. In the WM method the unit section on which not less than four road accidents have been recorded during a year is classified as a pecularily dangerous unit road section whereas the unit section with the number of road accidents equal to two or three during a year is classified as a dangerous unit road section. In order to characterise the safety of any road section, the numbers of both types of unit sections on this section are considered. However, there is a disadvantage in the definition given above. It has not been expressed how many pecularily dangerous and dangerous unit road sections should be contained in the considered road section in order to identify this section as a high risk one. To make the definition more precise, the performace of the WM method applied to real accident data has been studied. As a result, the following completion has been made in this work: At least three unit road sections with the number of accidents not less than 2 and with the distance between these sections not less than 2 km identify the location of a high risk section on a road.
The Gdansk Method The method classifies a road section on the basis of its accident density D (acc./km/year). A classification criterion uses the accident density of the road Drd that contains the considered section. The Drd value depends on the administrative division of the Polish roads as it has to be calculated for the part of the road placed within the administrative border of a province. According to the GM method a road section can be classified into one of the three categories: •
relatively safe, if Drd
•
being threatened, if 2 • Drd < D <= 3- Drd,
•
dangerous, if D > 3- Drd .
The disadvantage of the GM method lies in lack of the information how the ends of the section should be determined. After the analyses of some possibilities for solving this problem, the following completion has been made in this work: The sections where accidents cluster are determined initially using the single linkage method in the way described in the BM method. Then each section is checked in order to decide to which category it belongs according to the GM method classification.
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The Bayesian Method In order to perform the road safety analysis, road sections where accidents cluster are determined first. This is done by applying the single linkage method - one of the methods of cluster analysis. To identify hazardous sections among the accident cluster sections, an accident-proneness model has been developed (Tracz and Nowakowska, 1996). The model describes the distributions of the following accident variables: number of injuries, number of fatalities and number of vehicles involved in accidents. As the model is derived from the Bayesian approach (Benjamin and Cornell, 1977) the identification method has been called a Bayesian one. The model is based on two sources of information. The first one h(x) is a prior distribution of the accident variable X over the accident cluster sections:
for
x =0
for
x >1
(10)
The second source h(x\z,i) is a posterior distribution of the variable Shaving reference to those accident cluster sections, where the value of z of the accident variable over the i time units have been recorded:
for
x =0
for
x> 1
h(x\z,i) = h(x-l\z,i)-
x • (i + 1 + s)
In the above formulas s>0 is a scale parameter and k>0 is a shape parameter of the distribution. It is obvious that the level of safety varies between different roads and different road sections along a given road. Therefore, to define the change in the value of accident variable X for each road section, the value of the cumulative probability of the h(x z,i) function, calculated for the median m of the h(x) distribution, has been proposed. Thus, a single section can be characterised according to the cumulative posterior probability h (x z, i) of the X value above the median, i.e. by the weight W(X) expressed as follows: (12)
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where m is determined by the equation: v h(x)dx « Y h(x)dx « 0.5 • 0
m
In order to include the influence of a road section length / in the identification process, a weight function for this length has been defined by the following form:
LW(l) u=
max
for
In the BM method the accident cluster sections are subsequently used to find the prior distributions for injuries, fatalities and total vehicles involved in accidents. Then posterior distributions are detremined to calculate accident severity weights from the formula (12). These weights and the road section length weight (13) are the elements of a set S, that characterises the safety of a section: S = { W(X,), W(X2), W(X3), LW(l) }
(14)
The variable X} identifies the number ofy-th accident variable as follows: X\ is the number of injuries, X^ is the number fatalities, Xs is the number of vehicles involved in accidents on a section. The identification is processed on the basis of a classification value calculated from the elements of the set S. This value is called a Safety Weight SW and is defined in the following form: SW = /Z W(X t ) 2 + LW(l)2 V j
(15)
A criterion for high risk road section is determined by a critical set S* obtained from the set S of the elements equal to 0.5, 0.4, 0.6 and 0.75. These numbers reflect the rank of respective elements of the set - the lower value, the higher rank. An accident cluster section is indicated as a high risk road section, if its classification value is greater than the value SW for the critical set S*.
ANALYSIS OF EMPIRICAL DATA In order to present practical application of the described measures and to compare the methods of the identification of high risk road sections, real accident data recorded on four selected rural roads in the period 1991-1996, from two neighbouring provinces of south-central Poland were
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investigated. These roads represent different categories and are characterised by various lengths, traffic volumes and accident densities. High risk road sections were identified for different time periods according to the identification methods. The studied road sections were not treated during the two considered periods. The comparison of results was conducted using calculus evaluation. This enabled to present advantages and disadvantages of the methods.
Comparison of methods in relation to the degree of accident clustering The main differences in the concentration of accidents and in the concentration of accident consequences (injuries, fatalities and vehicles involved in accidents) obtained when using the three methods (WM, GM, BM) are presented in Table 1. The WM method involved one-year time period data to process the identification procedure. The results given in Table 1 for this method were calculated for the year 1991. The other methods (GB and BM) were used for a three-year time period. In this case, the results are given for the period 1991-1993. Table 1. Comparison of the values of the Method Inaccuracy Coefficient and additional measures for high risk road sections obtained for different indentification methods. Accident Time Method period
Accident denisties for dangerous sections are presented in the column 3 of Table 1. Some additional supporting measures have been included in brackets: -
the percentage in a row of the column 4 represents the ratio of the total length of dangerous road sections to the length of a road,
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-
the percentage in a row of the column 5 represents the ratio of the total number of accidents on dangerous road sections to the total number of accidents on a road,
-
the percentages in the columns 6, 7 and 8 are ratios such as in the column 5 but calculated for injuries, fatalities and number of vehicles involved in accidents.
For a road with a high accident density (road No 7) the WM method determines dangerous sections on which the ratios of the numbers of accident features exceed 70%. However, the total length of the sections can be fairly significant - even more than 40% of the road length (see the percentage in the column 5 for the road No 7). The lower the category of a road, the shorter the total length of dangerous sections but also the lower the percentage values of accident features (roads: 44, 74, 728). For all four roads, accident densities on dangerous road sections identified by the WM method are lower than identified by the GM and BM methods. Values of the Method Inaccuracy Coefficient are the highest also for all roads. The situation was similar when using this coefficient for subsequent years (1992, 1993). So, from the point of view of the concentration of accident features the WM is not advisable. The accuracy of the GM method proves to be very good for all roads. Accident densities on high risk road sections obtained from this method have the highest values. The supporting ratios (percentages in the columns 5-8) vary from road to road: they are the lowest for the roads No 7 and No 74, whereas for the roads No 44 and No 728 they are the highest. The MIC measures are very low: they range from 0.17 to 0.24. Taking into account the Method Inaccuracy Coefficient, the performance of this method the best. Nevertheless, it should be pointed out that the results of the GM method depend on the way in which road sections are determined to be the subject of the further identification procedure. If the sections are selected using the cluster analysis the selection criterion can be very weak and only a few accident cluster sections are abandoned. So a sieve role of the GM method (Hauer and Persaud, 1984) is almost none. The values of comparative measures calculated for the BM method place the method, on average, between the WM and the GM methods on a ranking list of the method performace with regard to the degree of accident clustering. In the case of the roads No 7, No 44 and No 74 the MIC values are relatively low (0.20-0.34), but a little higher than for the GM method. In the case of the road No 728 the BM gives the lowest (i.e. the best) values of the MIC coefficient. Considering the values of the measures of the degree of accident clustering it can be said that the best average results were obtained for the GM method, the BM method has occurred quite acceptable and the WM method can be classified as the worst.
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Comparison of methods in relation to the repeatability of locations of high risk road sections The time period for which high risk road sections are identified differs in the considered methods. Consequently, two consecutive time periods taken for calculation of the Space Compatibility Index differ in their lengths. In order to investigate the space repeatability of the WM method the results of identification for two years (1991 and 1992) were taken into account. For the two other methods (GM and BM) trade-off analyses were carried out on the basis of the results from two three-year time intervals 1991-1993 and 1994-1996. The results of analyses are presented in Table 2. Notations in columns 3-7 were taken form the definition of the SCI measure. The different numbers of high risk road sections identified by any of the three methods for two consecutive time intervals can confirm the phenomenon of accidents migration and consequently the migration of accident risk, even if a remedial treatment was not applied. For the three roads (No 7, No 44, No 74) the ratios of the number of overlapping dangerous road sections to the total number of such sections obtained in two consecutive time periods, are for the WM method equal to, or very close to, the same ratios calculated for the BM method. These ratios are better than ratios for the GM method - see column 6. The values of the SCI index are very diversified and rank the WM method on a perfect repeatability list on the first place (road No 74), but also on the second (road No 7) or on the last place (road No 44). The extremally high value of the SCI measure for the WM method for the road No 44 (1.12) results mainly from only one overlapping of high risk road sections, whereas there were together five high risk sections identified by this method in two considered time periods. On the road No 728 the WM method identified one dangerous section in the first time period and also one such section in the second period. However, they do not overlap and there is over fifteen kilometers distance between them. For all considered roads values of the n/(nl+n2) ratio for the GM method are lower than respective ratios for two other methods. The SCI measures never place this method on the first place on the perfect repeatability list. The worst results were obtained for the road No 74. There were seven dangerous road sections identified in the first time period and five sections identified in the second time period using the GM method. From these sections only one pair of road sections overlaps - so the values of both measures of space repeatability (columns 6 and 7) calculated for this method are most unsatisfactory. On average, the GM method gives the worst results.
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Table 2. Comparison of the values of the Space Compatibility Index and suplementary measures obtained for different methods of the identification of high risk road sections. Number of dangerous road sections
For the roads No 7, No 44 and No 728 the BM method gives the lowest values of the SCI measure and for the road No 74 only marginally larger than the SCI calculated for the WM method. Consequently, in qualitative terms, the BM method can be considered as working better than the two other methods. This can be confirmed by the n/(nl+n2) ratios, which are comparatively best of all. Evaluating all values of the space compatibility measures, it can be said that the best results were obtained for the BM method. It means that the precision in the location of dangerous road sections determined by this method was generaly greater than in the case of the two other methods. The WM and GM methods perform more or less on the same level with regard to space compatibility.
Comparison of methods in relation to the repeatability of accident patterns on overlapping high risk road sections In Poland the description of accident details is included in a road police report known as Road Accident Card. This accident information contains nine qualitative features describing accident circumstances (Tracz and Nowakowska, 1997). Two of them have been chosen as subjects of the analysis of the accident pattern repeatability on overlapping high risk road sections. These are: accident type and driver's behaviour. Some categories of these features have been combined in order to obtain more coherent classification (Tracz and Nowakowska, 1998). In such way accident type was classified in eight values and driver's behaviour was classified in
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eleven values. To mark the chosen feature in the measures MIAD, MAAD and MAD, the notation FCP is replaced by ATP for accident type pattern and by DBF for driver's behaviour pattern. The values of the pattern repeatability measures for the analysed methods and for the four considered roads are presented in Table 3. The WM method gives the best results for the road No 7 for both qualitative accident features. The average differences in accident type pattern MAD(ATP) and in driver's behaviour pattern MAD(DBP) do not exceed 10%. The method performs better than the GM method but worse than the BM method for the other considered roads. The results of accident pattern repeatability on overlapping dangerous road sections are the worst for the GM method both in the case of accident type (columns 3-5) and in the case of driver's behaviour (columns 6-8) for all roads. Table 3. Comparison of the measures of repeatability of accident patterns on overlapping high risk road sections obtained for different identification methods. Road number
The BM method performs best of all for the roads No 44, No 74 and No 728 despite the fact that MIAD and MAAD values are not the smallest for some cases. The average differences in pattern repeatability (the MAD measure) are smaller for the BM method than for the other two methods. They range from 9.3% to 13.0%. This range is determined by the results obtained for the road No 44. The MAD values for the roads No 74 and No 728 are included in this interval. The MAD(DBP) value for the WM method is only 0.3% greater than such value for BM method in the case of the road No 44. Considering all values of the measures of accident pattern repeatability it can be said that the best average results were obtained for the BM method. This confirms the general advantage of
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this method over the WM and BM methods with regard to accident pattern repeatability in relation to overlapping dangerous road sections.
Summing up evaluation of the measures used for comparison Accident data such as numbers of: accidents, fatalities, injuries and involved vehicles that were recorded on dangerous road sections as well as total length of these sections determine the degree of accident clustering. The ratios of these numbers on dangerous sections to their totals for a considered road can indicate the degree of clustering; the closer to unity such ratio is, the better the results of identification. However, a comparatively large value of the described percentage can be accompanied by a comparatively large value of the total length of dangerous sections in relation to the length of a road - see the results in Table 1. The Method Inaccuracy Coefficient strikes a balance of these two characteristics, thus becoming a measure which, with regard to accident clustering, plays a fundamental role in the comparison of methods of high risk sections identification. The lower the values of the coefficients MIC, the better the performace of the method. The mentioned earlier ratios can help in the interpretation of MIC values. The repeatability measures can confirm the accuracy of a method with respect to the location of sites with high accident risk. It is obvious that accidents can, but not always, occur in the same sites every year but if the location of sites where these occurences oscillate around two consecutive time intervals is indicated, the method can be considered as proper. Therefore two aspects are important: • the number of dangerous sections that overlap, • the way of overlapping (i.e. the range of overlapping). The more important is the ratio of overlapping dangerous sections to the total number of dangerous sections from two consecutive time intervals. If this ratio is the same for two methods, the range of overlapping indicates a preferable method - see the results for the road No 7 in the columns 6 and 7 in Table 2. Nothwithstanding all these aspects, the SCI values are quite satisfactory to evaluate the repeatability of methods - compare, for example, the results in columns 6 and 7 for all roads in Table 2. Accident patterns described by accident type and driver's behaviour can help in making a remedial treatment decision to be taken in order to improve road safety. So, in relation to overlapping dangerous sections, the perfect repeatability of accident pattern is strongly recommended. As measures of repeatability of accident pattern are related to overlapping dangerous sections they should be treated as supplementary measures to the SCI indexes. The interpretation of these measures is very easy and intuitive - the lower values the better.
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The considerations presented in this paper have shown that the range of the values of MIC, SCI, MAD(ATP) and MAD(DBP) measures depends on a method of high risk road sections identification and also depends on a studied road. Following a suggestion of a referee, the authors have considered the accuracy of these measures (Hauer, 1997). However, in the presented study the sample size is too small (only four roads) to derive general conlusions concerning this issue. Therefore, to look into some results, only the preliminary calculations dealing with variability of the measures were done. The variance of the measures in relation to the three methods is presented by coefficients of variations in the Table 4. It can be seen from this table that coefficients of variance are small for three measures (SCI, MAD(ATP), MAD(DBP)) for the BM method. This method can be expected to have the same efficiency for other roads. For the WM and GM methods the variance is comparatively large. Considering the repeatability of these methods, their efficiency for other roads can be either satisfactory or not. For the MIC measure all coefficients of variance are rather large - so the efficiensy of all considered methods, with respect to this measure, can be difficult to predict. Table 4. Percent coefficients of variation of accuracy measures of high risk road sections identification methods. All roads
MIC
SCI
MAD(ATP)
MAD(DBP)
WM GM BM
22.3%
40.0%
26.4%
21.3%
15.1%
17.8%
33.2%
29.0%
27.6%
4.7%
11.1%
10.2%
These results seem to be interesting but it should be pointed out that more accident data are required in order to confirm conclusions concerning the accuracy of the measures. This is the aim of the authors' future research in this area.
CONCLUDING REMARKS The purpose of this study was to develope a methodology for comparison of the efficiency of different methods of the identification of high risk road sections. The analysis carried out in this work enabled to detect a few deficiences in the dangerous section identification methods (WM and GM). Thus, some suggestions of their improvement were formulated. The suggestions concern: •
the way of defining road sections for further high risk sections classification, and
•
making some definitions of classification criteria more precise.
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In order to generalize the results and to qualify the methods, some main and supplementary measures independent of a methodology have been defined. Two aspects were taken into account. One concerns the level of concentration of accidents along hazardous road sections and the level of accidents severity concentration. The other aspect concerns the problem of repeatability of the identification procedures during two consecutive time periods in relation to the location of dangerous sections along a road and in relation to some descriptive features of road accidents. It has turned out that conclusions from comparison are not explicit. Even simplified methods can give satisfactory identification results provided that their deficiences are made up for. A very simple method such as the GM method performs well in terms of accident feature concentration and badly in terms of repeatability of the method. Comparison measures for the WM method (also a simple one) are the worst for the feature concentration aspect and the most diversified for the other aspects. Only in the case of the more sophisticated BM method the values of all indices are the least controversial. All measures considered in this work are based on the results of the process of the identifcation of high risk road sections. The methods used in such processes varies in different countries despite the common roots in several cases. However, having the results of identification, one can use the proposed measures, which are transferable to other countries, in comparing different methods or in estimating the efficiency of a method.
ACKNOWLEDGEMENTS The authors wishes to acknowledge helpful comments and suggestions of anonymous referees.
REFERENCES Datka S., W. Suchorzewski and Tracz M. (1989, 1997). Traffic engineering. WKiL Press, Warszawa (in Polish). Hakkert A. S. and E. Hauer. Extent and some implications of incomplete accident reporting. In: Methods for evaluating highway improvements. Transporation Research Record 1185, TRB, Washington DC. Hauer E., and B. N. Persaud (1984). Problem of Identifying Hazardous Locations Using Accident Data. Transp. Res. Rec., 975, Transportation Research Board, 131-140. Hauer E. (1995). Identification of'Sites With Promise'. International Conference in Prague on Strategic Highway Research Program and Traffic Safety on Two Continents. The Czech Republic, 20-22 September.
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Hauer E. (1997). Observational before-after studies in road safety. Estimating the effect of highway and traffic engineering measures on road safety. Pergamon (Elsevier), Oxford, New York, Tokyo. Maher M. J. and L. J. Mountain (1988). The identification of accident blackspots: a comparison of current methods. Ace. Anal, and Prev., 20, 143-151. Mountain L., Fawaz B., Sineng L. (1992a). The assessment of changes in accident frequencies at treated intersections: a comparison of four methods. TE&C, 2, 85-87. Mountain L., Fawaz B., Sineng L. (1992b). The assessment of changes in accident frequencies on link segments: a comparison of four methods. TE&Cl, 7-8, 429-431. Nicholson A. (1995). Indices of accident clustering: a re-evaluation. TE&C, 5, 291-294. Tracz M. and M. Nowakowska (1996). Bayesian Theory and Cluster Analysis in the Identification of Road Accident Blackspots. 13-th International Symposium on Transporatation and Traffic Theory, (Editor J-B Lesort), Pergamon (Elsevier), Oxford, New York, Tokyo, 261-276. Tracz M. and M. Nowakowska (1997). Characteristics of some accident circumstances on road blackspot sections. 20-th International Conference on Theories and Concepts in Traffic Safety, Lund, Sweden, 5-7 November. Tracz M. and Nowakowska M. (1998). Using qualitative analysis in road safety research. Third IMA International Conference on Mathematics in Transportat Planning and Control, Cardiff, Great Britain, 1-3 April.
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BEHAVIORAL ADAPTATION AND SEAT-BELT USE: A HYPOTHESIS INVOKING LOOMING AS A NEGATIVE REINFORCER
Anthony H. Reinhardt-Rutland, Psychology Department, University of Ulster at Jordanstown, UK
ABSTRACT The technical performance of seat-belts is not in doubt, but their continuing value is diminished if driving deteriorates following the switch to seat-belt use. Janssen (1994) demonstrated such behavioral adaptation over one year after the switch, but the effects were not sufficient to nullify seat-belt use. Longer experiments are probably impractical because of near-universal seat-belt legislation. However, UK statistics suggest that driving speeds have increased over the years since legislation to the extent of more than nullifying the effectiveness of seat-belts. In the present paper, Fuller's learning model of road behavior is developed in conjunction with the perceptual phenomenon of looming as an alternative to risk formulations of behavioral adaptation. Looming acts as negative reinforcement for unbelted drivers, but not for belted drivers. Because it represents threat to life, negative reinforcement persists in its effectiveness: it will take some years for the loss of looming to affect fully the new seat-belt user's behavior. In addition, behavioral adaptation in motorists inevitably militates against the encouragement of environmentally-sustainable - but vulnerable - modes of travel, such as walking and cycling. If safety-related interventions are to be properly assessed, there must be adequate empirical data obtained over a time-span which permits a full assessment of behavioral adaptation - in conjunction with a plausible theoretical framework in which the data can be interpreted.
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INTRODUCTION For many years, driving had been regarded simply as a perceptual-motor skill subject to straightforward cause-effect relationships between the driver's actions and the road environment. In this context, any technical intervention to reduce the consequences of failure in the performance of any aspect of driving skill would be deemed beneficial; the intervention may be to the road environment - for example, modification of the road-layout at an "accident black-spot" - or to the vehicle - for example, equipping the vehicle with seat-belts, ABS braking and air-bags. However, in the last thirty years or so, it has become clear that such thinking is superficial (Summala, 1996). For example, casualties may simply "migrate" from the former accident black-spot to other parts of the road-system (Davis, 1993). One crucial difficulty is that the technical intervention can be followed over time by changes in driving behavior, a phenomenon labelled behavioral adaptation. Occasionally, behavioral adaptation is in the direction of more cautious driving. More commonly, however, behavioral adaptation entails less cautious driving characteristics. It seems that behavioral adaptation is an automatized process occurring with little, if any, conscious effort on the part of the driver. One well-documented illustration of undesirable behavioral adaptation concerns antilock braking (ABS), which renders a braked vehicle more controllable in slippery conditions. In a study of two matched groups of drivers operating either ABS-fitted or conventionally-braked taxis, collisions over three years were in fact not statistically different for the two groups; the ABS group evinced changes in behavior, particularly increase in abrupt braking, which nullified the expected reduction of casualties (Aschenbrenner and Biehl, 1994). In a contrasting example, the dangers inherent in switching from left- to right-hand driving in Sweden and Iceland were accepted as an unfortunate by-product of legislation required to achieve conformity with neighbouring jurisdictions. In fact, the switch initially elicited reduced casualty rates, presumably because drivers were sensitive to the particular problems and made a conscious effort to avoid them. Unfortunately, casualty rates then increased steadily over the subsequent years to the levels applying before the switch (Naatanen and Summala, 1976; Wilde, 1994). An inference that might be drawn from the above two cases of behavioral adaptation is that all safety-pertinent interventions or situations become void over time, which has profound implications for road-safety policy: there seems little point in devising safety interventions if inevitably there will merely be a return to previous casualty rates. It might be argued that little is lost if the group to whom the safety-pertinent intervention or situation is targeted fails to demonstrate better casualty rates, but this is simplistic. The targeted group does not exist in
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isolation: it is not helpful for vulnerable road-users - pedestrians, cyclists and, arguably, motorcyclists - if the targeted group consists of motorists who now drive more dangerously and generate a more hostile environment for vulnerable road-users.
CURRENT THEORIES: CONCEPTS OF RISK, HABITUAL BEHAVIOR AND BEHAVIORAL ADAPTATION Theories to explain behavioral adaptation generally invoke risk as a mediating variable: it is assumed that subjective risk is adequately manifested in observable behavior. Much has been written about risk theories, so only a brief review will be attempted here. In the best-known example of risk theory, Wilde (1982, 1994) proposes that an individual's performance is determined by a target level of risk, dependent mainly on psychological factors, but including economic and cultural factors. Risk is linked with the individual's "need for stimulation", so the frequently-cited link between high casualty rates and young male drivers can be attributed to the high target risk-levels among this group (Heino et al, 1996). In brief, the target risk-level for the driver is reflected in behavioral factors such as habitual speed, following distance and frequency of overtaking. Following a safety-pertinent intervention or situation, risk homeostasis returns risk to the target level, as reflected in adjustments to habitual speed, for example. Summala and Naatanen (1988) conceptualise risk differently. Their zero-risk theory argues that drivers internalise safety margins; these lead to habitual patterns of behavior which the driver does not consider to be risky. Only when the driver determines that safety margins have become inappropriate - for example, unnecessarily eliciting slow speed - does perceived risk "switch in" to affect behavior. Taking the case of ABS brakes, the differences between Wilde's theory and Summala and Naatanen's theory can be illustrated. For Wilde, the driver's experience after adopting ABS brakes may suggest that the cautious behavior applying in previously hazardous conditions is now unduly low in risk: increase of speed in such conditions will raise risk to tolerable levels. In contrast, for Summala and Naatanen the driver's experience after adopting ABS brakes perhaps obtained by chance - may suggest that cautious behavior in previously hazardous conditions is no longer necessary: there will be no risk if the driver now increases speed in such conditions.
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Criticisms of risk Risk theories have intuitive appeal but several issues prevent full acceptance. One issue concerns definition. In most research, risk is necessarily defined in terms of objective casualty statistics. However, risk has a major subjective component which completely eludes objective statistics (Adams, 1988; Rumar, 1988). For example, flying for the population at large is subjectively riskier than driving, but in terms of casualties per unit distance traveled the reverse is emphatically the case (UK Government Statistical Service, 1997). Given such ambiguities, it is unsurprising that Wilde's theory and the competing theory of Summala and Naatenen can conceptualise risk in such contrasting ways. The proposed mechanisms to explain behavioral adaptation are dubious in their relationship to other evidence. For example, typical cases of homeostasis concerning energy, water and temperature regulation are mediated via specific brain areas, particularly the hypothalamus (Carlson, 1993; Kolb and Wishaw, 1985). In such cases, the time-constant for homeostasis is of the order of hours and a plausible physiological model can be postulated. In contrast, the timeconstant for risk homeostasis is of the order of years, suggesting an entirely different and unfamiliar mechanism. Finally, evidence of complete nullification of a safety-pertinent intervention or situation is not always straightforward. For example, ABS braking is technically less effective than had been expected, since certain collisions are in fact exacerbated by ABS (Farmer et a/, 1997; Kahane, 1994): Aschenbrenner and Biehl's (1994) casualty rates - referred to above - cannot exclude this as a possible contributory factor. Another well-publicised intervention - the airbag - is affected likewise, with reports of injuries to short drivers caused by the inopportune operation of airbags at low speeds (Duma et al, 1996; Mahmud and Alrabady, 1995). The implication is that the determination of a target level of risk can rarely be precise, so it follows that we cannot know for sure that behavioral adaptation elicits changes that fulfil the requirements of a homeostatic conceptualization. Another case suggests little behavioral adaptation: British casualties in poor visibility such as fog and night are invariably higher than in good visibility (Parker and Cross, 1981; White and JefFery, 1980; Smeed, 1977). Since poor visibility is familiar in Britain, risk theories should apply: drivers should evince slower speeds and longer following distances to compensate for the increased incidence of casualties in poor visibility. Although some reduction of speed and increase of following distances may occur, the changes in behavior are far from adequate. Furthermore, the dangers of poor visibility might elicit caution amongst new drivers, given the
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evidence from switching direction of travel in Sweden and Iceland. However, there is little evidence that these inferences apply. The increased casualties in poor visibility are better understood in terms of visual perception: there are several phenomena - reduced vection, anomalous myopia, motion contrast, aerial perspective and so on - which entail serious misperception of speed and distance in conditions of restricted visibility (Cavallo et al, 1997, in press; Leibowitz et al, 1982; Reinhardt-Rutland, 1986, 1992a,b,c). The above reasons suggest that the invoking of risk as a global factor in road-behavior requires caution. As a psychological concept risk is plausible and perhaps even necessary, if only as a short-hand descriptor in regard to behaviour that seems unnecessarily dangerous. However, current risk formulations seem unable to link risk to any established body of psychological knowledge. For example, one might expect to link risk to theories and research concerning motivation and personality (Maslow, 1987) and issues of social interaction (Hogg and Vaughan, 1998; Moghaddam, 1998), but there seems little evidence that this has been or can be realistically attempted. As a starting point in developing theories of road-behavior, it may be better to consider (a) each safety-pertinent intervention or situation in detail on an individual basis, while (b) seeking to relate the intervention or situation closely to empirical research and theory in pertinent areas of psychology.
SEAT-BELT USE AND LEGISLATION Of all safety-pertinent interventions, seat-belts have been regarded as particularly efficacious (e.g., Evans (1991), Wyatt and Richardson (1994) and many governmental agencies). Certainly the potential of seat-belts in reducing casualty rates is difficult to fault. For a given severity of collision, seat-belt use limits motion of the self within the vehicle and ejection from the vehicle. Furthermore - and in contrast with ABS and airbags - the cases in which seat-belt use is disadvantageous are agreed generally to be minor (Rutherford et al, 1985; Salam and Frauenhoffer, 1996). Particularly compelling evidence arises from those cases in which a vehicle involved in a collision contains both belted and unbelted occupants: the belted occupants have a fatality rate which is 40% less than for the unbelted occupants (Evans and Frick 1989; Evans 1990).
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Seat-belt effectiveness modified by driver behavior The limitation of the above evidence is that it can tell us nothing about whether the driver's behavior has altered as a result of wearing a seat-belt. This is an issue that is more difficult to establish. Evans (1991) provides a useful overview of the types of non-intrusive data-collection procedures that may be employed for drawing conclusions about the actual effectiveness of seat-belts among vehicle-occupants. One procedure is to compare casualty rates in jurisdictions matched as for as possible except in regard to seat-belt legislation. Seemingly, this is unsuccessful because the necessary degree of matching is difficult to attain: other legislation and differing patterns of road-use are likely to confound the effects of seat-belt use. Other procedures relate to the time around the introduction of legislation for compulsory seatbelt use in a given jurisdiction. This can be a simple before-and-after count of casualties or a more sophisticated time-series analysis: a discontinuity is expected in an otherwise continuous function for casualty-rates against time. The ideal source of data for such a study is a jurisdiction in which seat-belt use switches from zero before legislation is introduced to 100% after legislation is introduced. Unfortunately, in many jurisdictions seat-belt use has tended to change only gradually over the years, whether legislation is introduced or not: any changes that may result are thus difficult to attribute to seat-belt use. However, the case of the United Kingdom may provide sufficient of a contrast in seat-belt use during pre- and post-legislation times: legislation in 1983 brought a change in seat-belt usage from about 40% to over 90%. Scott and Willis (1985) reported about a 20% reduction in fatalities, which - given the 50% change in seat-belt usage - is in line with Evans and Prick's results from belted and unbelted occupants of the same vehicle. Evans and Prick's results were also broadly confirmed in a more sophisticated time-series analysis (Harvey and Durbin, 1986). Nonetheless, caution is required in the interpretation: drink-driving legislation was also introduced simultaneously in 1983 (Broughton and Stark, 1986) and selective recruitment may apply when a safety-pertinent intervention is not compulsory: seat-belt users prior to legislation were probably individuals who put a premium on safety (Evans, 1985). The 20% reduction in fatalities therefore overestimates the effect of seat-belt use. This indicates that the effectiveness of seat-belts suggested by Evans and Prick diminished over the duration of Scott and Willis' study. Since the study extended one year either side of the introduction of legislation, behavioral adaptation was possible.
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An experimental study Janssen (1994) confirms this view. This important study was unique in the degree of control exercised over nuisance variables. Three groups were investigated - new seat-belt users who up to the beginning of the study had been habitual non-users, habitual seat-belt users and habitual seat-belt non-users - in various driving behaviors on Dutch freeways over one year. Janssen had intended to include a fourth group - new seat-belt non-users who had been habitual users - but could not obtain willing participants. Compared with the other groups, new seat-belt users increased their speeds by 1.6 km/hr - average speeds were 110 km/hr - and reduced their following distances. These changes do not negate the benefit for seat-belt users. Empirical data suggest that the ratio of road fatality rates at two speeds is determined by the fourth power of the ratio of the speeds (Evans, 1991). Hence, the above increase in speed implies a 6% increase in fatalities. The reduced following distances would add to fatalities, although quantification in this case is not easy. Janssen found that behavioral changes had not stabilised: given time, they could have seriously compromised the benefit for seat-belt users.
Trends in British road-traffic This assertion is consistent with trends in British road-traffic speeds since the early 80s. Until very recently officially-available data on speeds and speed trends has been sparse. However, Thomson et al (1985) as part of a study concerning road-use in relation to school-age children reported that on urban and suburban roads for which a 30 mph (48 km/h) limit applies, speeds in 1981 averaged 28 mph (45 km/h), with 38% of vehicles exceeding the limit. Corresponding 1996 figures are 33 mph (53 km/h) and 72% (UK Government Statistical Service, 1997). Following Evans' relationship, the 8 km/h increase in mean speed implies a 92% increase in fatalities - much more than enough to nullify the benefit of seat-belt for their users, given Evans and Prick's 40% reduction in fatalities due to belt-use from their data for belted and unbelted occupants of the same automobiles. Changes other than seat-belt legislation have occurred over the eighties and nineties - for example, in regard to the comfort and ease-of-use of automobiles - which might also have contributed to the elevation of speed. It is unfortunate that much less is known about other trends in driving behavior, such as closefollowing and violation of overtaking rules. With regard to long-term empirical trends in fatalities for British automobile occupants, data over a 20 year period - much longer than was available for the studies of Scott and Willis (1985) and Harvey and Durbin (1986) - show a dip around 1983 and 1984 which was not
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maintained over subsequent years (UK Government Statistical Service, 1991). Adams (1994) reports at length on this and similar long-term trends from other jurisdictions.
The possibility of further empirical studies The problem for empirical research is that controlled seat-belt studies with the time-span of, say, Achenbrenner and Biehl's (1994) ABS study are probably unrealistic, since most jurisdictions now opt for compulsory seat-belt use. Indeed, this applies in the Netherlands, the location of Janssen's study. In the absence of a long-term experimental study, one might appeal to theoretical issues. However, the currently-prevailing risk formulations that are postulated for behavioral adaptation are problematic. Hence, the extent of behavioral adaptation for seat-belt use remains something of an open question. Proponents of seat-belt use can argue that other factors in the ever-changing roadway environment account for the disappointing trends. Of course, proponents need to identify these alternative factors - which they have yet to do in any systematic way. However, the theoretical underpinnings for behavioral adaptation need not be couched in terms of the slippery concepts of risk and homeostasis. Earlier, I argued that each safety-pertinent intervention or situation should be considered individually and in the context of established bodies of psychological research. In fact, there is an alternative model of driver behavior offered by Fuller (1984, 1988, 1992) which can be developed to explain why behavioral adaptation eventually becomes substantial in the case of seat-belt use.
THE BASIS OF SIMPLE LEARNING: CLASSICAL CONDITIONING AND OPERANT CONDITIONING Fuller's model refers to simple learning in humans and non-humans, particularly classical conditioning - associated with Ivan Pavlov - and operant conditioning - associated with B. F. Skinner (Mazur, 1994; Leslie, 1996). The foundation for such learning resides in associations with desired outcomes - rewarding stimuli or events - and associations with undesired outcomes - aversive stimuli or events. In classical conditioning, low-level or autonomic physiological responses become associated with otherwise neutral stimuli. In a typical Pavlovian study with dogs, the normally neutral stimulus of a tone instigates salivation, after the tone and food have been presented
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contemporaneously a number of times. In a contrasting example, electric shock elicits physiological responses associated with fear, undesired outcomes preparing the individual to cope with threat to life; classical conditioning is illustrated if the site at which the electric shock was given evokes fear responses in its own right. Operant conditioning concerns behavior that is under voluntary control at least in the early stages in the acquisition of a given behavior. In a typical Skinnerian study, a food-deprived pigeon learns to peck a key if the pecking is followed by the presentation of a pellet of food. In a contrasting example, an electrician always turns off the mains supply before examination of an appliance, since this has prevented electric shock in the past. As learning becomes established, the learnt behavior is likely to become increasingly automatized, with the learner having little insight or understanding of the learnt behavior; the learning can then take on some of the characteristics of classical conditioning. Indeed, the different types of conditioning can simultaneously affect many cases of learning. Conditioning depends on the reinforcement contingencies with which it is associated: a positive reinforcer elicits learning by its presence - as in the two food-related examples above - while a negative reinforcer elicits learning by its removal - as in the two electricity-related examples above. Learning can be lost if a reinforcer no longer operates, a phenomenon known as extinction. Positive reinforcement for the driver includes early arrival at destination: behaviors associated with fast driving become learnt. Collision is likely to be the main source of negative reinforcement, since it entails tangible damage and injury: slow driving behaviors should be learnt. Learning due to negative reinforcement is often less easily extinguished than learning due to positive reinforcement, since many negative reinforcers are linked with threat to life. On the face of it, this should apply strongly to the roadway. However, despite very high casualty rates in relation to other forms of transport, the roadway is a "forgiving" environment and collisions are rare in relation to distance driven, the average vehicle has only about a 20% probability of being involved in a collision every 15,000 km or so (Fuller, 1992). The unhappy conclusion from Fuller's theory is that positive reinforcement has an overriding effect on driver behavior, because of the rarity of negative reinforcement. Another feature of the rarity of negative reinforcement on the road is that learning may not generalise beyond the specific circumstances of the negative reinforcement. Thus, surviving participants in fatal collisions drive cautiously ever afterwards when faced with the specific circumstances of the collision, particularly the location of the collision - but their driving may be little altered away from these specific circumstances (Foeckler et a/, 1978; Rajalin and Summala, 1997).
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Fuller's Model and Seat-Belt Use A major interest of Fuller concerns the difficulty in getting drivers to anticipate danger. For example, discriminative stimuli signal potential reinforcement. A road-sign or markings before a bend on a narrow winding road is a potential discriminative stimulus leading to slowing, since such signs indicate hidden hazards. However, such road-signs and markings are often ineffective (Shinar et al, 1980): collisions are rare, so the driver fails to slow. Fuller has less to say about behavioral adaptation as such. Indeed one area in which his model and risk theories seem to conflict concerns seat-belts: Fuller (1984, p. 1139) believes that seatbelts remain beneficial for their users. This implies that the effectiveness of a negative reinforcer is independent of its consequences: the reduced injuries for a seat-belt user do not alter the effectiveness of a given collision as a negative reinforcer. Although this seems a dubious proposition, the rarity of collisions argues that such issues would in fact have little effect. Furthermore, the specificity of the learning among fatal-collision survivors was noted above (Foeckler etal, 1978; Rajalin and Summala, 1997).
Near-Misses as Sources of Negative Reinforcement However, the scope of negative reinforcement can be extended in another way. For every collision that a driver may experience, the "forgiving" nature of the roadway suggests that there are likely to be many near-misses. Although, not entailing tangible damage and injury, nearmisses are in fact likely to be negatively reinforcing. It might be retorted that this is simply a matter of classically-conditioned or secondary reinforcement, by which an effect - say, vestibular reaction to sudden deceleration - becomes negatively reinforcing because of its association with the primary reinforcer of collision. Indeed, were this the case little might be added by considering near-misses, given the rarity of collisions and the limited chances of associating deceleration and collision. However, near-misses provide primary negative reinforcement in their own right; the features of near-misses do not need to occur with collisions in order to become effective.
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VISUAL EXPANSION Potential collision with an object - for example, a pedestrian, another vehicle or a static item of road "furniture" - is accompanied by visual expansion at the driver's eye elicited by the object. Features of this expansion, such as angular velocity and acceleration, in conjunction with knowledge of the object's physical size and level of driving experience, provide information from which the driver can determine time-to-collision and take controlled avoiding action (Cavallo & Laurent, 1988; Lee, 1976; Stewart etal, 1993). Near-misses often entail sharp braking, with concomitant visual, vestibular and kinaesthetic changes: the unbelted, unrestrained motorist will be thrown towards interior fittings - the steering-wheel, dashboard, windscreen, windscreen pillars and so on. The visual changes associated with collision between the motorist and the internal fittings are similar to those for collision between the automobile and an object external to the automobile, with the crucial exception that simple mathematical considerations show that visual expansion will be much greater in the former case. This is because collision between motorist and internal fittings concerns much closer distances and directly affects the self: while the motorist is separated from collision with external objects by the automobile's superstructure, the motorist's motion towards the internal fittings is not. Objects close to oneself subtend bigger visual angles than distant objects. When object and self move closer together - as with a motorist being thrown towards the vehicle's internal fittings - the rate of change of the subtended visual angles is of a different order of magnitude, in comparison with the motorist and external objects.
The Looming Phenomenon The unbelted motorist thrown towards the vehicles's interior fittings provides conditions that evoke the looming phenomenon, defensive and fear responses elicited by the threat of immediate collision conveyed visually. For a wide range of non-human species, nothing more elaborate than a rapidly expanding shadow back-projected onto a screen is sufficient to elicit looming responses; neonates are affected similarly to adults - neonates having not encountered collision cannot have learned to link visual expansion with collision (Schiff et al, 1962; Schiff, 1965). Human infants are affected similarly to other species (Ball and Tronic, 1971), while human adults evince looming responses to moving objects that stop short of hitting them (King et al, 1992). Such evidence argues that looming responses are "hardwired": they are the subject of automatic and inborn neural processes.
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Further support for this assertion comes from human psychophysical evidence that centrifugal visual motion - corresponding to visual expansion - elicits special responses in a number of phenomena: motion aftereffects, reaction-times and threshold responses (Ball and Sekuler, 1980; Georgeson and Harris, 1978; Reinhardt-Rutland, 1994). Motion aftereffects in particular provide evidence of low-level neural processing (Reinhardt-Rutland, 1987, 1988). Looming itself can therefore act as a primary negative reinforcer, without any requirement that it be preceded by experience of a collision.
Looming and Seat-Belt Use The negative reinforcement contingencies associated with sharp braking differ according to whether or not a seat-belt is worn. The seat-belt user suffers much less motion within the interior of the automobile than does the seat-belt non-user (Evans, 1991): the former will be protected from looming responses. The greater vestibular and kinaesthetic responses for seatbelt non-users will no doubt also provide other primary negative reinforcers for this group, although there seems to be less in the way of research to which this can be related, probably because such responses are relatively inconvenient and difficult to control in the laboratory. Research in aviation psychology may provide some clues (Hawkins, 1987). Given that near-misses are more frequent than collisions on the road - and therefore likely to be experienced in a variety of situations and locations - greater generalisation of any learning instigated by near-misses is likely. In contrast with the case of fatal-collision survivors noted above (Foeckler et al, 1978; Rajalin and Summala, 1997), slower driving among seat-belt nonusers is not likely to be confined to specific circumstances of near-misses, but is likely to apply to all circumstances.
The Time Course of Learning and Extinction Regarding Negative Reinforcement As noted earlier, behaviors learnt during negative reinforcement are not easily lost, because of the threat to life that negative reinforcement represents; hence, the long-term fear of the collision circumstances shown by fatal-collision participants (Foeckler et al, 1978, Rajalin and Summala, 1997). Often the individual can avoid the negative reinforcement altogether: a consequence is that the individual is then unlikely to find out if reinforcement contingencies remain constant. Away from the road, a good example concerns food aversions. An individual who consumes an unfamiliar food that is followed by sickness is likely to avoid that food thereafter: the sickness is a negative reinforcer for avoidance of the food. The individual will
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not find out if the case of sickness was a one-off occurrence, or was truly a consequence of that food (Carlson 1993). A difference between seat-belt non-use and food aversion is that the reinforcement contingencies for the driver are less predictable, since other road-users are ever-present: the driver is likely to experience near-misses, no matter how slowly he/she drives. When nearmisses are found to be no longer associated with looming, extinction of slow driving becomes possible, while positive reinforcement such as early arrival at destination can become more salient. This analysis supports Janssen's assertion that behavioral adaptation requires much time to be complete.
New Licence Holders One factor not easily included in a controlled study of seat-belt use is the increasing proportion of the driver population - new licence holders - who will always have worn seat-belts and will never experience looming: a negative reinforcer leading to safer driving behaviors is permanently unavailable. The performance of these drivers early in their driving careers will therefore be more dangerous than the performance of their predecessors who initially lacked seat-belts. Given that new licence holders include young males - notorious for dangerous driving (Evans, 1991) - this factor will over time come to affect reported statistics substantially.
DISCUSSION In this paper, it is argued that risk models have serious shortcomings in explaining behavioral adaptation: definition of risk, substantiation of risk homeostasis and the fact that behavioral adaptation does not always relate in obvious ways to risk however defined are problems. Instead, given our rudimentary knowledge of psychological factors - particularly as they act over long periods of time - behavioral adaptation should currently be considered on a case-bycase basis. Regarding the case of seat-belts, Fuller's discussion of negative reinforcement on the road can be developed to provide an explanation of behavioral adaptation among seat-belt users. While collisions entail tangible damage and injury - so have obvious reinforcing properties - primary negative reinforcement need not be so restricted. In near-misses, not using a seat-belt leads to vehicle occupants being thrown towards the internal fittings of the automobile and the looming phenomenon - fear and avoidance responses conveyed by visual information for imminent
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collision - can be a primary negative reinforcer in the driver learning slow, careful behaviors. In contrast, the near-misses encountered by seat-belt users are not accompanied by substantial movements within the automobile: looming is eliminated as a negative reinforcer, to the detriment of driving behavior. After switching from not using seat-belts to using seat-belts, the driver may avoid the fast driving behaviors that lead to near-misses, analogous to the avoidance applying in food aversions. However, given that the driver must interact with other road-users, he/she will necessarily encounter near-misses which with belt use no longer engender looming. Only then can the behaviors learned under the influence of looming begin to be extinguished. Nonetheless, the persistence of learning under the influence of primary negative reinforcers - with their link with threat to life - suggests that learning faster driving behaviors will take much time. Another factor as time proceeds after seat-belt legislation, is that there will be increasing numbers of new licence-holders - particularly young males - who have always used seat-belts, for whom looming will never have been a negative reinforcer. The eventually substantial effects envisaged in the present proposal are consistent with other evidence. Janssen's (1994) study demonstrated small behavioral changes, which nonetheless would have been greater had the study been longer, an assertion supported by long-term increases in UK traffic speeds and in casualty rates. In fact, such data suggest that behavioral adaptation may have reversed the benefit for seat-belt users in Great Britain.
The urgent need for an adequate mix of empirical and theoretical knowledge The case of seat-belts is an especially instructive one for our appreciation of interventions intended to reduce casualties. Of all interventions, seat-belts have been regarded as particularly efficacious by many governmental or quasi-governmental authorities. Their belief may seem to be supported by certain short-term empirical investigations. However, it is much more difficult to amass evidence to support such an intervention in the long term, particularly when, as in the case of seat-belts, legislation has made usage mandatory. Furthermore, the effects of other interventions and developments can never be discounted: in Britain, the introduction of drinkdriving legislation simultaneously with seat-belt legislation is a clear case in point. It is not unusual to hear the off-the-cuff remark among the road-safety community that present driving behavior would be careful and sedate if the clock was turned back and the driver was unbelted and surrounded by sharp rigid interior-fittings which would impact with him/her in the event of sudden braking. Yet, it seems highly unlikely that such an assertion would ever be put
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to the test. Once a safety intervention attains compulsory status, it is thereafter highly improbable that that intervention will be the subject of rescinding legislation leading to its abandonment. Furthermore, such an intervention would probably not have societal support: recall Janssen's (1994) inability in recruiting habitual seat-belt users who were prepared to give up seat-belt use for the year's duration of this study.
Non-drivers The main concern of this paper has been with the consequences of a particular safety intervention as it affects drivers. In that regard, it might be argued that it is unfortunate that a safety intervention becomes ineffective for the group to whom it is targeted, but nothing is lost by this: the targeted group at least may not be worse off than before the intervention. However, this is to imply a proposition that the road-system entails more-or-less separate and non-interacting groups of road-users. Unless groups are rigorously segregated, this is simply untrue: how one group performs on the road inevitably affects other groups. As an increasing number of authors have noted, some groups are going to be disadvantaged by any level of behavioral adaptation among drivers resulting from interventions directed at driver safety: it is of absolutely no benefit to vulnerable groups such as pedestrians and cyclists that drivers travel faster than they did twenty-five years ago. As Davis (1993) eloquently shows, the persistent disregard of the needs of vulnerable groups of road-users amounts to an issue of civil rights; it is one that governments - particularly in the UK - have consistently ignored under the pressure of rich and vocal interests representing various strands of "the motor industry". Now this lack of official concern with these civil rights is reaping problems that affect all roadusers, including motorists themselves. Increasingly, congestion is affecting urban and suburban roads. In governmental circles, the answer no longer lies in new road build. Beyond concerns about quality of life, pollution and the like - even if the civil rights of vulnerable road-users are discounted - there is a realization that automobiles are extremely inefficient users of road-space (Highway Code, 1993; Marchetti, 1983): each new stretch of road only accommodates a minimal increase in motoring. Such issues compel governments to campaign for greater dependence on walking and cycling, whatever might be the dangers (Adams, 1988, 1994; Davis, 1993;Hillman
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most expendable form of car-dependency - but little or no parallel effort is directed to diminishing motoring for other purposes.
Comments on possible developments Such observations all point to the need for coherent combinations of empirical and theoretical understanding regarding the psychological factors at play in safety-related interventions. It may be that this understanding always has to be atomistic and specific to individual safety interventions. On the other hand, it may be that more holistic model-building can be undertaken; Fuller's learning model might well form the basis for such an undertaking. Compared with risk formulations, the empirical and theoretical underpinnings of Fuller's learning model are considerably more secure and there is no need to posit intervening variables whose definition and investigation are at the least problematic. Nonetheless, such holistic model-building has limitations. In the particular circumstances of poor visibility, the increased casualties are - as noted in the introduction - best understood in terms of visual perception (Cavallo et al, 1997, in press; Leibowitz et a/, 1982; ReinhardtRutland, 1992a,b). Note that the issue of poor visibility represents another area in which greater attention to adequate empirical and theoretical understanding is needed. The problems of poor visibility have never been adequately addressed in safety-related interventions. For the most part, these interventions have been restricted to issues of basic conspicuity: for example, such interventions have entailed the introduction of high-intensity lamps and retroreflective material, neither of which has been particularly successful. Unfortunately, interventions affecting conspicuity do not address the underlying problems revealed by a deeper analysis based on the psychology of perception: the illusory effects of motion and space - alluded to earlier - cannot be eliminated by reference to conspicuity alone. The consequence is, for example, the continuing occurrence of "motorway madness" cases each winter when fog affects fast through-routes (Reinhardt-Rutland, 1992a). Another and recently-recognized illustration of the need for adequate empirical and theoretical understanding regarding psychological factors concerns the ever-expanding propensity of information technology to automate, direct navigation or otherwise affect the task of driving. As Harris and Smith (1997) and Young and Stanton (1997) show, it is necessary that such developments can be understood and evaluated within a solid framework of psychological knowledge. Such technology will be of limited value for the motorist if it merely displaces congestion from one location to another. More crucially, if the cognitive demands (or lack of them) of the technology affect driver-performance in relation to other road-users, then its value
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really does become contentious - as already remarked, it is at last becoming unacceptable that technical gains for one group of road-users entail losses for vulnerable groups to whom they are not applicable. Judging from the past, if the use of information technology on the road is allowed to develop without restraint, it will become difficult to reverse the process if and when detrimental effects that should have been foreseen become serious.
REFERENCES Adams, J. G. U. (1988). Risk homeostasis and the purpose of safety regulation. Ergonomics, 31, 407-428. Adams, J. G. U. (1994). Seat belt legislation: the evidence revisited. Safety Science, 18, 135152. Aschenbrenner, M. and B. Biehl (1994). Improved safety through improved technical measures? Empirical studies regarding risk compensation processes in relation to antilock brake systems. In: Changes in Accident Prevention: The Issue of Risk Compensation (R. M. Trimpop and G. J. S. Wilde, eds.),. Styx, Groningen Netherlands. Ball, K. and R. Sekuler (1980). Human vision favors centrifugal motion. Perception, 9, 317325. Ball, W. and E. Tronik (1971). Infant responses to impending collision: optical and real. Science, 111, 818-820. Broughton, J. and D. C. Stark (1986). The effect of the 1983 changes to the law relating to drink/driving. Research Report 89. Transport and Road Research Laboratory, Crowthorne UK. Carlson, N. R. (1993). Psychology, the Science of Behavior. Allyn and Bacon, Boston MA. Cavallo, V. and M. Laurent (1988). Visual information and skill level in time-to-collision estimation. Perception, 17, 623-632. Cavallo, V., J. Dore, M. Colomb, and G. Legoueix (1997). Distance perception of vehicles rear in fog. In: Human Factors in Road Traffic II (P. Albuquerque, J. Santos, A. Pires da Costa and C. Rodrigues, eds.). Braga, University of Minho, Braga, Portugal. Cavallo, V., J. Dore, M. Colomb, and G. Legoueix (in press). Distance over-estimation of vehicle rear lights in fog. In: Vision in Vehicles 7 (A. G. Gale, ed.), North-Holland, Amsterdam. Davis, R. (1993). Death on the Streets: Cars and the Mythology of Road Safety. Leading Edge, Hawes UK. Duma, S. M., T. A. Kress, D. J. Porta, C. D. Woods, J. N. Snider, P. M. Fuller, and R. J.
230
Transportation and Traffic
Theory
Simmons, R. J. (1996) Airbag-induced eye injuries - a report of 25 cases. Journal of Trauma-Injury Infection and Critical Care, 41, 114-119. Evans, L. (1985). Human behavior, feedback and traffic safety. Human Factors, 27, 555-576. Evans, L. (1990). Restraint effectiveness, occupant ejection from cars and fatality reductions. Accident Analysis and Prevention, 22, 167-175. Evans, L. (1991). Traffic Safety and the Driver. Van Nostrand Reinhold, New York. Evans, L. and M. C. Frick (1988). Safety belt effectiveness in preventing driver fatalities versus a number of vehicular, accident, roadway and environmental factors. Journal of Safety Research, 17, 143-154. Farmer, C. M., A. K. Lund, R. E. Trempel, and E. R. Braver (1997). Fatal erases of passenger vehicles before and after adding antilock braking systems. Accident Analysis and Prevention, 29, 745-757. Foeckler, M., F. Hutchenson, C. Williams, A. Thomas, and T. Jones (1978). Vehicle drivers and fatal accidents. Suicide and Life Threatening Behavior, 8, 174-182. Fuller, R. (1984). A conceptualization of driving behavior as threat avoidance. Ergonomics, 27, 1139-1155. Fuller, R. (1988). On learning to make risky decisions. Ergonomics, 31, 519-526. Fuller, R. (1992). Learned riskiness. Irish Journal of Psychology, 13, 250-257. Georgeson, M. A. and M. G. Harris (1978). Apparent foveofugal drift of counterphase gratings. Perception, 1. 527-536. Harris, D. and F. J. Smith (1997). What can be done versus what should be done: A critical evaluation of the transfer of human engineering solutions between application domains. In: Engineering Psychology and Cognitive Ergonomics. Vol. 1: Transportation Systems (D. Harris, ed.). Ashgate, Aldershot UK. Harvey, A. C. and J. Durbin (1986). The effects of seat belt legislation on British road casualties: A case stuy in structural time series modelling. Journal of the Royal Statistical Society, A149, 187-227. Hawkins, F. H. (1987). Human Factors in Flight. Gower, Aldershot UK. Heino, A., H. H. van der Molen, and G. J. S. Wilde (1996). Risk perception, risk taking, accident involvement and the need for stimulation. Safety Science, 22, 35-48. Highway Code (1993). Belfast UK: HMSO. Hillman, M., J. Adams, and J. Whitelegg (1991). One False Move: a Study of Children's Independent Mobility. Policy Studies Unit, London. Hogg, M. A., and G. M. Vaughan (1998). Social Psychology: an Introduction. Prentice-Hall, New York. Janssen, W. (1994). Seat-belt wearing and driving behavior: an instrumented-vehicle study. Accident Analysis and Prevention, 26, 249-261. Kahane, C. J. (1994). Preliminary Evaluation of the Effectiveness of Antilock Brake Systems
Behavioural Adaptation and Seat-Belt Use
231
for Passenger Cars. Report DOT-HS-808-206. National Highway Traffic Safety Administration, Washington DC. King, S. M., D. Dykeman, P. Redgrave, and P. Dean (1992). Use of a distracting task to obtain defensive head movements to looming stimuli by human adults in a laboratory setting. Perception, 21, 245-259. Kolb, B., and I. Q. Whishaw (1985). Fundamentals of Human Neuropsychology. Freeman, New York. Lee, D. N. (1976). A theory of visual control of braking based on information about time-tocollision. Perception, 5, 437-459. Leibowitz H. W., R. B. Post, T. Brandt, and J. Dichgans (1982). Implications of recent developments in dynamic spatial orientation and visual resolution for vehicle guidance. In: Tutorials in Motion Perception (A.H. Wertheim, W. A. Wagenaar and H. W. Leibowitz, eds.). Plenum, New York. Leslie, J. C. (1996). Principles of Behavioral Analysis. Harwood, Amsterdam. Mahmud, S. M. and A. I. Alrabady (1995). A new decision-making algorithm for airbag control. IEEE Transactions on Vehicular Technology, 44, 690-697. Marchetti, C. (1983). The automobile in a system context: The past eighty years and the next twenty years. Technological Forecasting and Social Change, 23, 3-23. Maslow, A. H. (1987). Motivation and Personality. Harper and Row, New York. Mazur, J. E. (1994). Learning and Behavior. Prentice-Hall, Englewood Cliffs, NJ. Moghaddam, F. M. (1998). Social Psychology: Exploring Universals across Cultures. Freeman, New York. Naatanen, R. and Summala, H. (1976). Road-User Behavior and Traffic Accidents. North Holland, Amsterdam. Parker, D. B. and I. G. Cross (1981). The effectiveness of motorway matrix signalling - a police view. Police Journal, 54, 266-276. Rajalin, S. and H. Summala (1997). What surviving drivers learn from a fatal road accident. Accident Analysis and Prevention, 29, 277-283. Reinhardt-Rutland, A. H. (1986). Misleading perception and vehicle guidance under poor conditions of visibility. In Vision in Vehicles (A. G. Gale, M. H. Freeman, C. M. Haslegrave, P. Smith and S. P. Taylor, eds.), pp. 413-416. North Holland. Amsterdam. Reinhardt-Rutland, A. H. (1987). Aftereffect of visual movement - the role of relative movement: A review. Current Psychological Research and Reviews, 6, 275-288. Reinhardt-Rutland, A. H. (1988). Induced movement in the visual modality: an overview. Psychological Bulletin, 103, 57-72. Reinhardt-Rutland, A. H. (1992a). On learning, distance overstimation and mist-related motorway accidents. Perceptual and Motor Skills, 126, 130.
232
Transportation and Traffic
Theory
Reinhardt-Rutland, A. H. (1992b). Poor-visibility road accidents: theories invoking "target" risk-level and relative visual motion. Journal of Psychology, 126, 63-71. Reinhardt-Rutland, A. H. (1992c). Some implications of motion perception theory for road accidents. Journal of the International of Traffic and Safety Sciences (IATSS Research), 16, 9-14. Reinhardt-Rutland, A. H. (1994). Perception of motion-in-depth from luminous rotating spirals: Direction asymmetries during and after rotation. Perception, 23, 763-769. Rumar, K. (1988). Collective risk but individual safety. Ergonomics, 31, 507-518. Rutherford, W. H., T. Greenfield, H. R. M. Hayes, and J. K. Nelson (1985). The medical effects of seat belt legislation in the United Kingdom. Research Report 13. Department of Health and Social Security, Office of the Chief Scientist. HMSO, London. Salam, M. M. and E. E. Frauenhoffer (1996). Left arterial appendage rupture caused by a seatbelt - a case-report and review of the literature. Journal of Trauma-Injury Infection and Critical Care, 40, 642-643. Scruff, W. (1965). Perception of impending collision: a study of visually directed avoidant behavior. Psychological Monographs, 79:604, 1-26. Schiff, W., J. A. Caviness, and J. J. Gibson (1962). Persistent fear responses in rhesus to the optical stimulus of "looming". Science, 136, 982-983. Scott, P. P. and P. A. Willis (1985). Road casualties in Great Britain in the first year with seat belt legislation. Report 9. Transport and Road Research Laboratory, Crowthorne UK. Shinar, D., T. H. Rockwell, and J. A. Malecki (1980). The effects of changes in driver perception on rural curve negotiation. Ergonomics, 23, 263-275. Smeed, R. J. (1977). Pedestrian accidents. In Proceedings of the International Conference on Pedestrian Safety II (A. S. Hakkert, ed.), pp. 7-21. Michlol: Haifa. Stewart, D., C. J. Cudworth, and J. R. Lishman (1993). Misperception of time-to-collision by drivers in pedestrian accidents. Perception, 22, 1227-1244. Summala, H. (1996). Accident risk and driver behavior. Safety Science, 22, 103-117. Summala, H. and R. Naatanen (1988). Risk control is not risk adjustment: The zero-risk theory of driver behavior and its implications. Ergonomics, 31, 491-506. Thomson, S. J., E. J. Fraser, and C. I. Howarth (1985). Driver behavior in the presence of child and adult pedestrians. Ergonomics, 28, 1469-1474. UK Government Statistical Service (1991). Transport Statistics Great Britain. HMSO, London. UK Government Statistical Service (1997). Transport Statistics Great Britain. HMSO, London. White, M. E. and D. J. Jeffery (1980). Some aspects of motorway traffic in fog. Report LR958.
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Transport and Road Research Laboratory, Crowthorne UK. Wilde, G. J. S. (1982). The theory of risk-homeostasis: implications for safety and health. Risk Analysis, 2, 209-255. Wilde, G. J. S. (1994). Target Risk. PDE Publications, Toronto. Wyatt, J. P. and J. M. Richardson (1994). The use of seat belts on British motorways. Journal of the Royal Society of Medicine, 87, 206-207. Young, M. S. and N. A. Stanton (1997). Automotive automation: effects, problems and implications for driver mental workload. In Engineering Psychology and Cognitive Ergonomics. Vol. 1: Transportation Systems (D. Harris, ed.). Ashgate, Aldershot UK.
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Bi-Directional Emergent Fundamental Pedestrian Flows From Cellular Automata Microsimulation Victor J. Blue, New York State Department of Transportation, Poughkeepsie, NY, USA Jeffrey L. Adler, Rensselaer Polytechnic Institute, Troy, NY, USA
INTRODUCTION Pedestrian flow is an important component in the design and analysis of transportation facilities and in urban transportation planning. The need to assess the level of service of pedestrian walkways motivates much of the work on modeling pedestrian flows. Statistically derived flow models, expressing the fundamental relationships between density, flow, and speed, are used to generate level of service criteria. The methodology of assessing level of service for pedestrian walkways is similar to that for vehicle roadways. Microscopic modeling of vehicle flows is well studied and most, if not all, of the resulting models have been shown to generate fundamental relationships. In contrast, few attempts have been made to develop microscopic models of pedestrian flows that capture the complex behavioral movements and decisions. Historically, researchers have found the modeling task to be daunting. In recent years, Cellular Automata (CA) has emerged as a technique for modeling complex behavior using a set of simple rules. Cellular Automata (CA), when emulating the complex behaviors of living systems, is characterized as an artificial life approach to simulation modeling (Levy 1997). CA is named after the principle of automata (entities) occupying cells according to localized neighborhood rules of occupancy. The CA local rules prescribe the behavior of automata creating an approximation of actual behavior. Emergent group behavior is an outgrowth of the interaction of the microsimulation rule set. Unlike traditional simulation models that apply equations and not behavioral rules; CA behavior-based cellular changes of state determine the emergent results. Furthermore, CA models function as discrete
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idealizations of the partial differential equations that describe fluid flows and allow simulation of flows and interactions that are otherwise intractable (Wolfram, 1994). The purpose of this paper is to present CA microsimulation models of uni- and bi-directional pedestrian walkways to demonstrate the ability of capturing fundamental properties of pedestrian movements using a fairly simple rule-based approach. The paper begins by characterizing the pedestrian flow problem. A brief literature review of previous modeling efforts is provided. The formulations for the uni- and bi-directional walkway cases are described. This is followed by a discussion of results gathered from numerous simulation experiments. Included is a comparison of emergent flow patterns to the fundamental parameters described in the Highway Capacity Manual (1994).
BACKGROUND Microscopic models of car-following behavior and continuum models of traffic streams are mathematical relationships to describe the complex, and often chaotic, movement of vehicles and the high degree of interaction between vehicles. It is well known that vehicle flow exhibits non-linear speed transitions and self-organized criticality. Lieberman and Rathi (1997) suggest that, while the local behavior of individual vehicles may be reliably represented logically and mathematically with acceptable confidence, the complex simultaneous interactions of the system are difficult to describe and in mathematical forms. In response to the need to study traffic networks, researchers have turned to using simulation. Microscopic simulation attempts to model the behavior and movement of individual vehicles. Simulations are constructed from a set of models that represent a variety of behaviors from changing vehicle speed to lane switching and wayfinding decisions. Traditionally, microscopic simulation of vehicle flow is constructed around car-following models. However, microscopic simulations are difficult to develop and calibrate due to the complex behaviors and large number of parameters that are needed. In addition, chaotic vehicular traffic phenomena that occur at higher densities are difficult to capture with equation-based models and therefore are also difficult to simulate with sufficient realism. Cellular automata microsimulation has been proven to provide a good approximation of complex flow patterns over a range of densities. (Nagel and Rasmussen, 1994; Paczuski and Nagel, 1995; Nagel 1996). Over the past several years researchers have demonstrated the applicability of CA to car-following and vehicular flows including traffic within a single-lane (Nagel and Schreckenberg, 1992), two-lane flow with passing (Rickert, Nagel, Schreckenberg
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and Latour, 1995), and network-level vehicle flows in the TRANSIMS model (Nagel, Barrett and Rickert, 1996). While pedestrian flows are much more complex and chaotic than vehicular flows, the success with which researchers were able to use CA to model traffic flow provides the impetus to explore the use of CA for pedestrian flows. Although fundamental properties for pedestrian flows and level of service methods for assessing capacity are well established, it has been difficult to develop microscopic models of pedestrian flows that generate these fundamental properties. Pedestrian flow is highly chaotic and individualized. Pedestrian corridors may have several openings and support movement in several directions. Pedestrian walkways are not regulated as are roadways. For the most part pedestrian flows are not channeled; pedestrians are free to vary speed and allowed to occupy any part of a walkway. Unlike roadways where vehicle flow is separated by direction, bidirectional walkways are the norm rather than the exception. However, since safety and crash avoidance are less of a concern to pedestrians, slight bumping and nudging is often a part of walking through crowded corridors. Pedestrians are capable of changing speed more quickly when gaps arise and can accelerate to full speed from a standstill. In addition, it is not uncommon for pairs or groups of pedestrians to walk side-by-side or in clusters. Gipps (1986) came closest to developing a realistic microscopic model of pedestrian flow with a cell-based approach using a hexagonal lattice. Individual pedestrians occupy hexagonal cells and over discrete time intervals are moved in relation to one another. Movement is dictated according to reverse gravity-based rules. Pedestrians are repelled from each other as they seek their destinations in buildings. This work was limited in that the formulation is encumbered by sequential floating-point calculations based on the number and proximity of pedestrians who aim to avoid each other. In addition, the approach was tested only on one-directional uncongested flows with inconclusive, though apparently reasonable results for the uncongested speed-volume relationship. Gipps also failed to yield a tenable method for handling bidirectional movements. Fruin discusses several efforts from the 1960's to develop models of pedestrian flow within different scenarios. There are few notable recent efforts by researchers to model pedestrians using more traditional techniques. Lovas (1994) examined pedestrian traffic in building evacuation with a discrete event queuing network. AlGadhi and Mahmassani (1991) simulated crowd behavior using a set of conservation of mass flow simultaneous partial differential equations solved for several classifications of pedestrians over discrete time steps.
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Helbing and Molnar (1995) developed a model in which a pedestrian behaves as if acted upon by external attractive and repulsive forces, termed social fields. Their model displayed the formation of lanes by opposing direction flows and oscillatory changes in walking direction at a doorway. The paper demonstrated a single instance of these abilities and did not show results over a range of densities or calibrate them against established fundamental flows. The CA approach explored in this paper is distinctly different from the social fields model, computationally much simpler and based on maximizing forward progress. The attractiveness of using C A is that the models are based on behavioral rules, rather than performance functions. This paper will demonstrate that emergent group behavior comes from the dynamics of the pedestrians in motion across the defined walkway. The CA behavior-based cellular changes of state determine the emergent results, giving rise to very lifelike phenomena, and to new possibilities of modeling based on behavioral rules. The relevant behaviors become the focus of the modeling and that leads to an understanding of the factors that contribute to the dynamics.
MODEL FORMULATION The objective is to develop an intuitively and empirically appealing CA microsimulation for modeling bi-directional flows on a pedestrian walkway. The model aims to employ the essential minimal rule set needed and avoids the pitfall of trying to capture all possible behaviors of pedestrians which are numerous to identify and problematic to verify. Many pedestrian behaviors are unnecessary distractions from the essential parameters. It has been observed in CA simulations that very simple models are capable of capturing essential system features of extraordinary complexity (Bak 1996). Aiming at the essential and minimal rule set by eliminating anything but critical behavioral factors permits a clear understanding of the underlying fundamental dynamics. There are two fundamental elements of pedestrian movement to capture: forward movement and resolving conflicts. Forward movement refers to the velocity and acceleration of each pedestrian. Desired walking speeds vary over the pedestrian population. Under less crowded conditions, individual pedestrians will strive to accelerate toward their desired maximum walking speed. In congested walkways, pedestrians will adapt their speed to the prevailing flow rates in the immediate walkway neighborhood. To account for differences in desired maximum walking speeds (termed vjmax), a pedestrian population consisting of three walker classes is adopted:
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Table 1. Maximum Walking Speeds Pedestrian Class Fast Standard Slow
\ | j I
Walking Speed (m/s) 1.80 1.30 0.85
Conflict resolution refers to movements that are undertaken to avoid head-on collisions or to enable passing or bypassing. In a bi-directional walkway, pedestrians from opposing directions can vie for the same location. The proposed model will introduce the concept of "place exchange", whereby the positions of opposing pedestrians are swapped to emulate the collision avoidance behavior. Passing movements occur when two pedestrians are moving one in front of the other in the same direction and the trailing pedestrian wishes to accelerate past the lead pedestrian. Bypassing occurs when two pedestrians approach one another from opposite directions and circumvent one another. Termed "lane-changing" or "side-stepping" behavior, the passing pedestrian shifts position to the right or left in order to enable a higher walking speed. Though pedestrians do not follow lanes, lane changing is a helpful term to use when referring to a CA grid of cells where de facto lanes exist. Rule Set The CA microsimulation is based on six rules applied across four parallel updating stages. In the first parallel update stage, a set of lane changing rules is applied to each pedestrian on the lattice to determine the next lane of each pedestrian. All pedestrians are assigned to the new cells during the second stage. In the third stage, a set of forward movement rules is applied to each pedestrian. The allowable speed of each pedestrian is based on the available gap ahead and the pedestrian's desired speed with all the entities in their current positions. Finally, the pedestrians hop forward to new cells in the fourth update. In the first parallel update stage Rules 1-5 determine lane switching. Pedestrians can change lanes only when an adjacent cell is available. Rule 1 determines if the adjacent cells to the immediate right and left are available. Rule 2 resolves possible conflicts if an adjacent cell is available but the cell two lanes over is occupied. A random number is drawn to designate the lane as free to this pedestrian or to the pedestrian two cells away. Rule 3 eliminates a lane change if both adjacent cells are unavailable. If an adjacent lane is free, then lane change is determined by the maximum gap in Rule 4. If the maximum gap is common to two or more lanes, Rule 5 breaks ties for making lane assignments. Rule 5a, an 80/10/10 split for all three
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lanes, assumes pedestrians generally stay in the current lane and drift out of it occasionally. Rule 5b, a 50/50 split between the adjacent lanes, is the most reasonable assumption. Rule 5c, a 50/50 split between the current lane and an adjacent lane, assumes pedestrians wish to have a cell-width of separation from a person in the adjacent lane as much as they want to stay with the current course. These probabilities worked well for the simulation. The second parallel update stage moves all the pedestrians into their new lanes. The third parallel update stage determines the forward movement of the pedestrians. The gap ahead is determined first. If they are opposing pedestrians, Rule 6 guards against deadlocks by emulating what people actually do. Under constrained conditions opposing pedestrians may exchange places. In actuality temporary standoffs may occur when people guess which way to step past one another. Thus, the simulation contains a probability of a temporary standoff between closely opposing walkers. With probability p_exchg closely opposing pedestrians exchange places in the time step. The opposing entities each move the same number of cells, which is 0, 1, or 2 cells. In the fourth parallel update, the pedestrians are moved forward based on the gaps determined in the third parallel update stage. Parallel Update #1: Lane Assignments (Rule 1): Check adjacent cells IF the cell immediately to the left (right) is unavailable THEN assign the cell to be occupied and GOTO Rule 3 ELSE GOTO Rule 2 (Rule 2): Determine if adjacent lanes are free IF the cell two lanes over to the left (right) is occupied by a pedestrian THEN with probability r assign the left (right) lane to be occupied GOTO Rule 3 (Rule 3): Determine if pedestrian must remain in current lane IF the lane immediately to the right is occupied AND the cell immediately to the left is occupied THEN assign pedestrian pn to his current lane ELSE GOTO Rule 4 (Rule 4): Assign to uniquely maximal gaps Compute the available gaps for the current lane and for unoccupied adjacent lanes
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IF a gap is uniquely maximal THEN assign pedestrian pn to the lane having maximum gap ELSE GOTO Rule 5 (Rule 5): Tie-breaking of equal maximum gaps Use the appropriate tie-breaking rule: (a - 3-way tie): Randomly apply 80/10/10 split for current lane and two adjacent lanes. (b - 2-way tie between the adjacent lanes): Randomly apply 50/50 split. (c - 2-way tie between current lane and single adjacent lane): Randomly apply 50/50 split. Parallel Update #2: Lane Movement Move each pedestrian pn, to the lane assigned in the lattice. Parallel Update #3: Assigning Travel Speeds (Rule 6): Update velocity Let v(pn) = available gap IF gap = 0 or 1 and gap = gap2 (cell occupied by an opposing pedestrian) THEN with probability/? exchg v(pn) = gap + 1 ELSE v(pn) = 0 Parallel Update #4: Forward Movement Advance each pedestrian pn, v(pn) cells forward in the lattice. The available gap ahead depends on the direction of flow of the next vehicle downstream. From the gap calculation, if both pedestrians are going in the same direction, the new velocity of the follower is the minimum of the desired velocity of the pedestrian (v_max) and the available gap ahead. If the gap between opposing pedestrians is less than the total distance that both pedestrians could move at maximum speed (i.e., 8 cells is the maximum - 4 in each direction), then the updated velocity is the minimum of v_max and moving halfway forward. Moving halfway forward guards against collisions and hopping over other entities. Taking the minimum gap of the same and opposite directions of movement is shown as a combined task at the end of the gap calculation.
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Computing the Available Gap Look ahead a max of 8 cells (since 2 * largest v_max = 8) IF occupied cell found with same direction THEN set gapl to number of cells between entities ELSE gap 1 = 8 IF occupied cell found with opposite direction THEN set gap2 to INT (0.5 * number of cells between entities) ELSE gap2 = 4 Assign gap = NUN (gapl, gap2, v_max)
SIMULATION EXPERIMENTS A series of simulation experiments were conducted to evaluate the rule set over different unidirectional and bi-directions scenarios. The pedestrian walkway is modeled as a circular lattice of width W and length G. Each cell in the lattice is denoted L(i, j) where 1 < i <, W and 1 ^ j < G. Pedestrian densities are predetermined at the start of the simulation and remain constant throughout each run. At the start of each simulation, for a density d, the proportion of occupied cells, where 0.05 < d < 1.0, N = INT (d*W*G) pedestrians are created and assigned randomly to the lattice. The circular lattice enables the set of pedestrians to interact at constant density while maintaining strict conservation of flow. Cells in the lattice are considered square at 0.457 m per side. This cell size is scaled according to minimal requirements for personal space as described in the Highway Capacity Manual. The scale is also used to generate the speed-flow-density relationships that emerge. Density as the proportion of occupied cells is considered more helpful for discussion of results than units of pedestrians/m2, which certainly can be converted from d/(0.457m)2. To capture variations in walking speeds across the population a distribution of 5 percent 'fast' walkers; 90 percent 'standard' walkers, and 5 percent "slow" walkers (termed 5:90:5 distribution) was used. This distribution had the best realization of the fundamental diagram compared with other distributions in the single-direction case (Blue and Adler 1998). For the circular lattice, the walking speeds are 4, 3, and 2 cells per time step respectively and these values fall within ranges of speed and standard deviations used by others researchers (Lovas 1994).
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Fundamental parameters of pedestrian flows, space, flaw rate, and, average walking speed were generated from the experiments. For the circular lattice, space is the reciprocal of density and is constant for each run. Flow rate is computed as the total number of revolutions made by each pedestrian over the duration of the simulation. Speed is computed as number of total steps divided by the time duration. Each simulation was run for 11,000 time steps of one second each. The first 1000 time steps were used to initiate the simulation and the latter 10,000 were used to generate performance statistics. Each set of experiments included runs at 19 densities ranging from 0.05 to 0.95 percent occupancy in increments of 0.05. For statistical accuracy, twenty repetitions at each density level are run and the fundamental parameters are computed as the average over these replications. The resulting emergent fundamental profile is a map of the relationships between speed and flow over the range of densities.
Uni-Directional Flow The first set of experiments was conducted on a uni-directional pedestrian walkway. This eliminates the need to resolve head-on conflicts as all pedestrians are moving in the same direction. Several model forms were fitted against the resulting fundamental data. Figure 1 depicts three volume vs. density curves: (1) the results of the microsimulation experiment, (2) a fitted May's bell-shaped curve, and (3) a composite two-regime model curve-fitted to the data. May's flow equation was estimated from the simulated data. following speed-density relationship bell curve
S = 83.16e-4 11D2
It was found to have the
(i)
The best fitting model is a two-regime speed-density relationship. Under low densities, the flows appear to follow May's bell-shaped relationship; a linear relationship between flow and density is suggested at higher densities. The best estimated model for the data had the following speed-density relationship:
84.4e~4 75°2 D < 0.45 1 [(26.59/D)-26.39 D>0.45
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As seen on the chart, the maximum flow of this two regime model is approximately 80 ped/min/meter of width. The HCM suggests a capacity around 25 Ped/min/ft-of-width, equivalent to 82 Ped/min/meter-of-width. The results compare favorably with work reported by Virkler and Elayadath (1994) who examined uni-directional flows and fit various curves to field data. Their study found that two functional forms fit the data best: (1) single regime model represented by May's bell-shaped function and (2) a two-regime model with separate linear regimes. A closer look at this study reveals that the estimation of flow equations was based on a data set where density reached a maximum value of 3.12 Ped/m2, corresponding to a density of 0.65 on the circular lattice. Therefore, Virkler and Elayadath's estimation is limited to low and medium densities. The results of the uni-directional CA model for low densities is consistent with these previous findings, combining the best features of May's bell-shaped curved and the two regime linear model. The CA data set continues to high density and provides improvements to the fundamental model. The CA model is a useful tool to extending examination and knowledge of pedestrian behavior and aggregate movement. Figure 1. Volume-Density Curves for Uni-Directional Flow
•CA Simulation •May's Bell Curve 2-regime
0.2
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Density (Cells Occupied / Total Cells)
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Bi-Directional Flow In a bi-directional walkway, pedestrian movements are limited to two opposing directions. According to the HCM (1994), flows along a bi-directional walkway will tend to segregate over time and occupy proportional shares of the walkway width. For example, if sixty percent of a pedestrian population is walking northbound, it can be expected that over time, that sixty percent of the walkway capacity will become dedicated to the northbound pedestrians. For a walkway lattice of ten cells wide, it would be expected that the northbound pedestrians would occupy the six rightmost lanes and the southbound pedestrians would occupy the remaining four lanes. Under this condition, the flow is directionally separated and crossover movements to lanes supporting flow in the opposing direction only occur briefly to accommodate passing. Since cases of directionally separated flow are nearly identical to having two adjoining unidirectional walkways, the emerging flow-density profile should be similar to the unidirectional case. The HCM suggests that depending on the directional split, the flow profile will retain the same shape except for some reduction in capacity. If the directional flows are roughly equal then little reduction in capacity is expected. However when there is a significant imbalance, say, a 90-10 split, capacity reductions of about 15 percent have been observed (HCM 1994). Figure 2 depicts results of microsimulation experiments conducted over a range of directional splits from 100-0 to 50-50. As expected, it was found that the emergent flow-density curves were virtually identical to the results of the unidirectional case. In addition, very little variation in the resulting behaviors between splits was found. There are cases in which flows along bi-directional walkways are not segregated; opposing traffic intermingles and directional lanes do not readily evolve. This usually rather short-lived case of randomly interspersed pedestrians moving in opposing directions may occur at busy crosswalks, in crowded subway stations, and in emergency situations, among others. Historically interspersed bi-directional flow has not been well examined, documented, or modeled. As a result this case is not thoroughly understood. The C A microsimulation was used to investigate the case of bi-directional interspersed flow. When opposing pedestrians converge on the same cell but cannot negotiate sidestepping, the conflict is resolved by exchanging the places of the opposing pedestrians. This is realized through Rule 6 with the use of a variable, p_exchg. To the authors' knowledge, the characteristics of place exchange have not been examined in any field study of pedestrian
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activity, though it is well known that pedestrians will bypass, bump, slip past, and otherwise exchange places with one another when necessary. Figure 2. Volume-Density Curves for Bi-directional Walkway and Varied Splits
— 100/0 — 90/10
— 80/20 — 70/30 — 60/40 — 50/50
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Density (Cells Occupied / Total Cells) Given that the microsimulation model correctly approximates speed-volume-density relationships for the single and lane-based bi-directional cases, a set of runs were conducted to examine the effect on handling conflicts using the place exchange logic. In the absence of empirical data, these findings are considered preliminary, though they are quite instructive regarding the capabilities of the model and, to a lesser extent, actual crowd behavior in the random bi-directional setting. Exchange rates of 100, 75, 50, 25, and 0 percent were applied across the various directional splits. 0 percent is the base case where pedestrians cannot move if an opposing pedestrian creates a direct conflict; 100 percent indicates that all conflicts are resolved. Figure 3 illustrates the resulting speed-density curves for several directional splits based on exchange probabilities of 100 percent success. The speed-density plots of each directional split show a pattern emerging that is orderly and understandable. At a density below 0.45, speeds are reasonably reduced according to the directional split. Above a density of 0.45, the high
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degree of exchange rate allows higher speeds, as splits become more even. This is because pedestrians are not as often held back by opposing persons ahead and because more balanced directional flow provides more opportunities for face-to face conflicts and smooth exchanges of position. However, perfect exchanges of opposing pedestrians allows flows to do what appears to be unreasonably well, allowing evenly split flows to increase at very high density above what single direction flow would allow at its maximum. Thirty or more exchanges per pedestrian per minute occur, a number that appears unlikely in practice. Primarily because of this result, the perfect exchange concept appears more useful as a benchmark of maximum flow. Actual levels of exchange are likely to be hampered by occasional wrong-steps. Figure 3. Speed vs. Density Curves for Bi-directional Walkway, Interspersed Flow, and varied splits for 1.0 exchange probability
100/0 90/10 80/20 70/30 60/40
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Density (Cells Occupied / Total Cells)
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Figure 4 illustrates flow-density curves for 90-10 splits over the range of exchange probabilities. As the rate goes down, significant attenuation of flow is observed. In addition, even for the 100 percent exchange case there is a significant (approximately 15 percent) reduction in capacity as the HCM describes for the case where lane formation does not occur. The 75 percent exchange case has a similar peak volume, appropriately diminished from the uni-directional and lane-separated bi-directional case. Lower exchange rates appear to have peak volumes out of range of what the HCM describes. Figure 4. Flow vs. Density Curves for Bi-directional Walkway, Interspersed Flow, and 90:10 volume split
Lane Changing Behavior Understandably, most of the historical effort that has gone into capturing pedestrian flow characteristics has been directed toward their forward movement. However, an important factor in understanding pedestrian flow is lane changing. Lane changing has been modeled in
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vehicular traffic (see Rickert, Nagel, Schreckenberg & Latour 1995; Ben-Akiva, Koutsopolous, & Yang 1995 for example). Since extensive data gathering and modeling of pedestrian lane changes has not been previously done, the effort here has been to perceive pedestrian lane changing from experience and empirically capture the forward movement with reasonable rules. Attention to modeling of lane changing is also needed due to mode locking of pedestrians, or a marching effect, which may occur in some instances. Mode locking, or synchronization of motions, occurs in many physical processes (Schroeder 1991). In this case pedestrian selforganization effects the ability to synchronize movement. Mode locking is almost certain to occur at about 0.25 density if all the entities are of the same walker class, having the same maximum speed. If rigid mode locking occurs, slowdowns and small jams do not occur and average speed will be unreasonably high. Even with substantial opportunity to pass, the pedestrians achieve flexible self-organization. Self-organization arises naturally in the model, but the lane changing rules should avoid rigid locking into step, or the forward flow characteristics will not emerge correctly. While self-organization is of interest, and the model does display it, it is not the primary focus of attention at this stage of research investigation. Figure 5 illustrates the relationship between lattice density and rate of side stepping. The unidirectional lane change rate (100/0 split case in Figure 5) is effected by density in an unexpected way with a local minimum at 0.25 and a local maximum at 0.35 density. The phenomenon appears to be an outgrowth of self-organization at various stages of interaction among the pedestrians. There are, thus, some spatial configurations of entities that promote lane changing and others that impede it. In fact, spatial use and the uni-directional lane change curve can be used to explain the shape of the S-curve in the speed-density relationship. As the density for the 100/0 split increases above 0.25, the spatial efficiency for forward movement decreases and lane changing increases as entities tend to self-organize into the available space, breaking up any even spacing between pedestrians and making forward movement more difficult. Figure 5 also illustrates the relationship between sidesteps and density at the 1.0 exchange probability for interspersed flow over a range of directional splits. As can be seen from the figure, the spatial advantages of mode locking are entirely lost and the opposite effect even occurs. In fact, the spatial advantage of mode locking at 0.25 density becomes a disadvantage with opposing flows and increases as flows become more evenly divided. Avoiding entities coming from the opposite direction is most effective for reasons similar to why mode locking is so effective in uni-directional flow.
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Figure 5. Sidesteps vs. Density Curves for Bi-directional Walkway, Interspersed Flow, and varied splits for 1.0 exchange probability
100/0 90/10 80/20 70/30 60/40 50/50
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Density (Cells Occupied/Total Cells) When exchanges occur freely, the lane changing is not as impeded. As the exchange probability goes down, the lane changing increases (not shown). One finding from this analysis is that lane changing is sometimes helpful and other times not very helpful in promoting flows. A lot depends on the spatial use and density. Especially as density increases, when a pedestrian changes lane to pass, another pedestrian can change lanes and block the first pedestrian's movement. Pedestrians make their decisions independently and are myopic to the movements others may take. In actual pedestrian traffic pedestrians can "read" the body movements of others to a certain degree, but this can also be deceptive and oncoming pedestrians may still wind up in a face off. In road traffic lane changing, a phenomenon, referred by some as "snaking," can be observed (Resnick 1994). In congested traffic if an adjacent lane moves faster, drivers switch lanes and congest the formerly moving lane. Then, the formerly stopped lane begins to move. The traffic moves like a snake, alternating the lane that is in freer flow. The fact that lane changing can be counterproductive or non-productive for forward flow does not stop lane changing from happening. Pedestrians can be as myopic as
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drivers, as can be observed, when at multiple queues, queue hopping occurs to the seeming eternal frustration of the person who always seems to switch to the wrong queue. It is noteworthy that the CA model has a significant capability in capturing lane changing to the degree needed for apparently any application. At the very least, lane change rules in CA models must prevent excessive mode locking from occurring while permitting reasonable amounts of self-organization of pedestrians.
CONCLUSIONS This paper demonstrated the use of CA microscopic simulation for modeling pedestrian behavior. The basic pedestrian behaviors have been approximated with a minimum rule set that effectively captures the essential pedestrian dynamics for the single- and bi-directional case. The rule set and scaling can be adjusted to situations and conditions that correspond, in some measure, to actual behaviors. The correspondence of the rules to actual behaviors need not be exact. The essential rules are simple enough to program and modify without imposing unnecessary detail, and yet capture complex phenomena. The CA microsimulation exhibits emergent fundamental flows that correspond to published field data and accepted norms. The resultant single-directional flow-density case corresponds to a two-regime curve consisting of (a) May's Bell curve at low-to-mid density and (b) a linear curve at mid-to-high high density. The bi-directional case where lanes are dedicated by directional split appears sufficiently validated in that those flows are not significantly different from single-direction flows. The 90-10 interspersed bi-directional case corresponds well to the 15 percent reduction in capacity noted in the HCM. Over all the simulation tests, the region of density 0.2-0.4, where the maximum flows occur, has the largest differences in speed and volume between tests. These are presumed to be the speed-flow-density combinations that have the most volatile dynamics. This inference agrees with Paczuski and Nagel (1995) that reveals complex dynamics at work, especially in the maximum flow range of the Nagel-Schreckenberg automobile traffic model. This is evidently due in part to spatial sensitivities of lane changing that affects forward movement in this region. Since nonlinear effects occur around maximum flow, the region of maximum flow would then be the area where the greatest concentration of effort in fine-tuning the model would benefit. With further work it would be possible to reveal more about and to better understand the dynamics in the maximum flow range.
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Lane changing effects yield some interesting insights into the speed-density relationship. At lower densities mode locking may occur that is efficient for forward movement, but requires and permits few lane changes. At higher volumes, lane changing may help as well as hinder overall flow and evidently has relatively small impact upon the emergent group flow behavior. However, though not well studied or understood by field research to date, lane change behavior is an important feature of pedestrian movement. Its inclusion in the model is essential and adds realism. Among those issues that would benefit from further examination, lane changing and place exchange, especially, emerge as factors that are critical to the functioning of the model. Field studies should verify the hypothetical lane change and position exchange phenomena. Previously published data were used to model the distribution of walker speeds. However, the use of a CA model as a design tool would require careful study of local pedestrian populations with respect to characteristics of lane changing, position exchange, and the speed distribution of walkers in free flow. The more complex and hypothetical case of interspersed bi-directional flows clearly illustrates the modeling power of the CA method. Its capabilities have been merely touched upon in this investigation. The CA approach yields a viable tool for pedestrian modeling that has daunted the efforts of researchers for years. It captures micro-level pedestrian dynamics and offers an experimental platform for better grasping the important parameters of pedestrian flows. As a new method, its possibilities have barely begun to be theoretically explored and considerable opportunities exist for innovative applications. Among the many possibilities, the authors are investigating a 4-directional pedestrian model and complex walking environments, such as shopping malls, street intersections, and bus/rail stations. Multiple-mode models that combine pedestrians, autos, trucks, buses, bicycles, and so forth (e.g., auto-rickshaws and scooters in Asian traffic) are plausible extensions of the CA pedestrian and auto work done to date. A version of the simulation can be seen online at http://www.ulster.net/~vjblue.
REFERENCES AlGadhi, S.A.H. and H. Mahmassani (1991). Simulation of Crowd Behavior And Movement: Fundamental Relations And Application. Transportation Research Record, 1320, 260-268. Bak, P. (1996). How Nature Works: The Science of Self-Organized Criticality, SpringerVerlag New York, Inc.
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Ben-Akiva, M, H. N. Koutsopoulos, and Q. Yang, Q. (1995). A Simulation Laboratory for Testing Traffic Management Systems. (WCTR). Syndey, Australia.
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Blue, VJ. and J. L. Adler (1998). Emergent Fundamental Pedestrian Flows from Cellular Automata Microsimulation, Transportation Research Record, 1644, 29-36. Fruin, J. J. (1971). Pedestrian Planning and Design. Metropolitan Association of Urban Designers and Environmental Planners, New York, N.Y.. Gipps, P. G. (1986). Simulation of Pedestrian Traffic in Buildings. Schriftenreihe des Instituts fuer Verkehrswesen, 35, Institut fiter Verkehrswesen, Universitaet Karlsruhe, Germany. Helbing D. and P. Molnar (1995). Social Force Model for Pedestrian Dynamics. Physical Review E, 51 (5) 4282-4286. Lieberman E. and A. Rathi (1997). Traffic Simulation, In: Revised Monograph on Traffic Flow Theory, Federal Highway Administration, (N. Gartner, M. Messer, and A. Rathi, eds.) WWW publication http://www.tfhrc.gov/its/tft/tft.htm. Levy, S. (1992). Artificial Life. Vintage Books, New York. Lovas, G. G. (1994). Modeling and Simulation of Pedestrian Traffic Flow. Transportation Research, 28B (6) pp. 429-443 Nagel, K. and M. Schreckenberg (1992). A Cellular Automaton Model for Freeway Traffic. J. Physique I, 2. Nagel, K. and S. Rasmussen. (1994). Traffic at the Edge Of Chaos. Artificial Life IV: Proceedings of the 4th International Workshop on the Synthesis and Simulation of Living Systems, pp. 222-225. Nagel, K. (1996). Particle Hopping Models and Traffic Flow Theory. Physical Review E, 53 (5)4655-4661 Nagel, K., C. Barrett, and M. Rickert (1996). Parallel Traffic Micro-simulation by Cellular Automata and Application for Large Scale Transportation Modeling. Los Alamos Unclassified Report 96:0050, Los Alamos National Laboratory, Los Alamos New Mexico. Paczuski, M. and K. Nagel (1995). Self-Organized Criticality and 1/f Noise in Traffic, Los Alamos Unclassified Report 95:4108 (published in Traffic and Granular Flow, eds. D.E. Wolf, M. Schreckenberg, and A. Bachem, Singapore: World Scientific, 1996, p 41) Los Alamos National Laboratory, Los Alamos New Mexico.
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Resnick, M. (1994). Turtles, Termites, and Traffic Jams: Explorations in Massively Parallel Microworlds. MIT Press, Cambridge, Mass. Rickert, M., Nagel, K., Schreckenberg, M. and Latour, A (1995). Two-Lane Traffic Simulations Using Cellular Automata. Los Alamos Unclassified Report 95:4367 (published in Physica A, Vol. 231, pp. 534, 1996), Los Alamos National Laboratory, Los Alamos New Mexico. Schroeder, M. (1991). Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman and Company, New York. Special Report 209: Highway Capacity Manual (1994). Transportation Research Board, National Research Council, Washington, D.C., Virkler, M. R. and S. Elayadath (1994). Pedestrian Speed-Flow-Density Relationships. Transportation Research Record, 1438, 51-58. Wolfram, S. (1994). Cellular Automata and Complexity. Addison-Wesley Publishing Company
CHAPTER 4 FLOW EVALUATION IN ROAD NETWORKS
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Time is nature's way of keeping everything from happening at once.
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You cannot depend on your eyes when your imagination is out of focus. (Mark Twain)
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Discovery consists of seeing what everybody has seen and thinking what nobody has thought.
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FLOW MODEL AND PERFORMABILITY OF A ROAD NETWORK UNDER DEGRADED CONDITIONS Yasuo Asakura, Masuo Kashiwadani and Eiji Hato Department of Civil and Environmental Engineering, Ehime University Matsuyama, 790-77, JAPAN, FAX. +81-89-927-9843, E-mail, [email protected]
INTRODUCTION Transport network flows are the results of the interaction between travel demand and supply conditions. Flows in a network are not stable due to the fluctuation of travel demand or the occurrence of failure in supply conditions. Even if links and nodes in a network are not physically damaged and the supply condition of the network is normal, flows in the network may not always be stable. Travel demand is variable from time-to-time or day-by-day, and the resulting network flow will be fluctuating. We sometimes experience unusual network flow conditions due to seasonal fluctuations of travel demand On the other hand, traffic accidents, road-works or natural disasters occasionally causes damage on the supply conditions of a network. Some links in the network may be closed to traffic, and flows in the network will become unstable. Almost all of the links are damaged in an extraordinary natural disaster such as the great Hanshin Earthquake. We may suffer from extremely heavy traffic congestion caused by the interaction between fluctuating travel demand and degraded network capacity. Figure 1 shows a conceptual relation of physical conditions and traffic flows in a transport network. As mentioned above, the degree of unusuality of traffic flows in a network is observed even in a physically sound network. The instability of network flows will be magnified as the degree of the physical damage of the supply conditions becomes worse. In the countries often suffering from natural disasters like Japan, it is necessary to describe
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network flows and estimate network performance in deteriorated conditions for the strategic transport network planning under uncertainty.
As well as those countries, the modern
society requires more reliable transport systems. A reliable network means a network which can guarantee an acceptable level of service for traffic even if some links of the network are physically damaged or large amount of travel demand is occasionally generated. Degree of Unusuality of Traffic
Degraded Network
Physically Normal Network Degree of Physical Damage of Network Figure 1 Level of Network Degradation and Corresponding Flow Conditions Network reliability models have been studied for evaluating transport networks in both usual and unusual conditions. Turnquist and Bowman(1980) presented a set of simulation experiences to study the effects of the structure of the urban network on service reliability, lida and Wakabayashi(1989) developed an approximation method of calculating the connectivity between a node pair in a network. These studies are concerned with the reliability analysis of a. pure network. Flows in the network were not explicitly considered in those studies. Asakura and Kashiwadani(1991) proposed a time reliability model of a network considering day-to-day fluctuation of traffic flow. Although they used a traffic assignment model and considered flows in the network, the reliability model was calculated independently from network supply conditions. Du and Nicholson(1993) showed a general framework of the analysis and design of Degradable Transportation Systems. The User Equilibrium assignment was involved in the reliability analysis. By extending the algorithm shown by Du and
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Nicholson, Asakura(1996) presented an approximation algorithm of the distribution function of a performance measure in a deteriorated network. Asakura and Kashiwadani(1997) compared some different reliability models of an origin and destination (OD) pair in a road network. Travel time, travel demand and consumer's surplus between an OD pair were respectively used as performance measures in those reliability models. These studies focus on a flow network, in which the interaction between travel demand and network condition is described. Network flows in degraded conditions might be different from the ones in
ordinary
conditions. As Khattak and Palma (1997) mentioned, however, few studies have been made for describing traveller response in normal and unexpected situations. In particular, network flow models have not been studied for unusual conditions. The uncertainty and inconvenience of travel may bring the reduction of travel demand and it will result in network flow patterns. Deterministic flow models have been used in the previous network reliability analysis. However, this may cause inconsistency between flow description and reliability evaluation process since flow is deterministic while network itself is not. Thus, it is necessary to employ stochastic models for describing flows in a network. The objective of this paper is first to formulate a network flow model for describing flows in degraded conditions. The second objective is to propose some performance measures of a network when some links are closed to traffic. The network flow model is involved in the evaluation process of the performance measures. In the following Chapter 2, the Stochastic User Equilibrium model with variable demand is applied to describe flows in a network with some disconnected links. Chapter 3 shows some performance measures considering uncertain factors of the network flow. Approximation algorithms for estimating the expected value and the distribution functions of performance measures are also described in this chapter. Numerical examples are calculated in Chapter 4 to verify the convergence of the algorithm. A brief conclusion and implication of the models are presented in Chapter 5.
260
Transportation and Traffic Theory
FLOWS IN A NETWORK WITH SOME DISCONNECTED LINKS Network State Vector and State Probability A transport network is represented by a directed graph which consists of a set of numbered nodes and a set of numbered links. We assume that the function of links may be deteriorated by some reasons, while nodes will not be. In order to describe flows in a deteriorated network as simply as possible, we assume that links with failure are completely closed to traffic and such a condition continues for a long time. This assumption is introduced so that the static User Equilibrium conditions could be applied for describing network flows. A degraded road network is identified by the state vector x = {x:,..., xa,... ,x J-. The element of the vector xa denotes whether link a is degraded or not, namely x a is equal to 0 if the link is closed to traffic, or x a is equal to 1 if the link maintains its functions in the ordinary condition. If all links are connected, the state vector is written as xg = {2,
,1} . This
state is referred as the ordinary or normal state in the following part of this paper. The worst state vector is written as x w = {0,.... ,0},ir\ which all links are disconnected. X = {x} denotes the set of the possible state vectors. Since the occurrence of a failure in a link is uncertain, the state of a network
is not
deterministic. We introduce a link connectivity, which is the probability of whether the link is connected and not closed to traffic.
The connectivity for link a is denoted by
pa (a = l,... ,L). We assume the value of link connectivity is exogenously estimated and fixed. When the link connectivity pa is independent each other, the probability of the occurrence of a state x in a network is calculated as; P(x)=TlPaX° (I-PJ1^* a £A
(^
We call p(x) as the state probability.
Formulation Stochastic User Equilibrium Model with Variable Demand
According to the survey by authors, 15% of drivers cancelled their trips in the deteriorated network, and 75% of diverted drivers were obliged to choose the routes with higher risks. We
Road Network Under Degraded Conditions
261
formulate a network flow model in a deteriorated network considering those travel choice behaviour in the network. Travelling in a deteriorated network is more uncertain than in the ordinary network. A stochastic travel choice model is appropriate to describe a driver's travel behaviour. Since some of the drivers may cancel their trips, the travel demand is assumed elastic for network situations. In the previous studies (Asakura(1996), we have applied the Deterministic User Equilibrium with Variable Demand. Although the elasticity of demand was considered in the model, the route choice behaviour was assumed deterministic. In order to represent uncertain route choice behaviour in a deteriorated network, the stochastic route choice modelling is desirable. This will also resolve the inconsistency between route choice behaviour and trip making behaviour. We will apply a Nested Logit based Stochastic User Equilibrium model with Variable Demand (NLSUE-VD) for describing flows in the transport network. Congestion effect is involved in the route travel cost which is the function of the number of travellers. The trip making and route choice behaviour is written by the Nested Logit model. Those choice probability functions between an OD pair are given as follows,
p[k\rs] =
exp(-6,c[s) I
k€K
— exp(-6jc")
p [ r s ] = 2-p[rs]
VkEKrs,rstQ
VrsEQ
(2)
(4)
where p [k\rs] denotes the choice probability of the route k between origin r and destination s, p [rs] presents the choice probability of generating a trip between OD pair rs and p [ rs } is the choice probability of cancelling a trip between the OD pair, c" denotes the cost of travelling the route k between OD pair rs, and S rs represents the expected minimum cost between the OD pair, which is written as Eq.(5).
(5)
262
Transportation and Traffic
Theory
These route cost and the expected minimum cost are the function of flows in a network. Crs and Rrs are the fixed costs of travelling OD pair and of cancelling the travel, respectively. 6}, 02 are the parameters of route choice and travel choice. Q,Krs are the set of OD pairs and the set of routes between OD pair rs. When the number of potential trips of an OD pair is given as 7rs, the number of generated trips qrs and the number of cancelled trips q rs of the OD pair are written as;
exp{-82(Srs
exp{ -
\/rsE Q
exp(-e2RJ
(6)
(7)
The flow of the route k of OD pair is also written as;
exp(-d1c[s)
rs,
rse Q
(8)
An optimization model equivalent to the above conditions is formulated as the following Eqs.
(9) to (13). [NLSUE-VD]
min.Z(f,q, q ) =
J k
(9)
subject to
(10)
Road Network Under Degraded Conditions VI
^.7-5
fks>0
y ,
,
263
—^
jy-
/
\/kEKrs,rsEQ
qrs>0, ~qrs>0
1 1 \
(12)
\/rsEQ
(13)
where ta(w) denotes the link cost function, and xa is the flow of a link. We assume the link cost function is separable and monotonically increasing for its link flow. Let us prove that the first order necessity conditions of the above non-linear optimization model [NLSUE-VD] are equivalent to the Nested Logit model. Introducing the Lagrange multipliers {A ri } and {/j.rs}, the Lagrangianof the [NLSUE-VD] can be written as;
L\f,q, q,*,l*] = Z } ( x ( f ) ) + Z 2 ( q ,
q ) + Z3(q,f)+Z4(q,
q ) (14)
+ I *-r,(Tri-(qr, + ~qJ) + I Hrs(qrs- I ft) rsea
where Z j ( x ( f ) } ,
rs€fi
k€K
Z2(q, q ), Z ^ ( q , f ) and Z4(q, q ) correspond to the 1 st to the 4th terms
of the original objective function, respectively. The Kuhn-Tucker conditions are;
d
qrs
and
fk BL
=0
and
q rs—=— = 0 d q rs
and
>0 VkEKrs,rseQ
8L
> 0 V rs E Q
—-=—>Q V rsEQ d q rs
(15)
(16)
(17)
The constraints of the original problem Eqs.(lO) to ( 1 3 ) are also hold. Since the route flow and the OD flow are positive, the above conditions can be rewritten as;
Transportation and Traffic
264 dL
dL
1
rs
1
In
1
Theory
(18)
In
(19)
In
+1
(20)
Then, the route flow is calculated using Eq.(18) as, fks = qrsexp(- 6jCrks
(21)
When this equation is put into the route flow conservation constraint Eq.(l 1), we obtain the Logit model for route choice. Similarly, the OD flow is calculated using Eqs.(19), (20) and (10). Then, we obtain the Binary Logit equation for trip making. These are the Nested Logit equations which we have assumed for the traveller's choice behaviour in a network. Thus, the optimal conditions of the [NLSUE-VD] is proved equivalent to the stochastic route and trip making choice models. The objective function of [NLSUE-VD] is strictly convex for path flow and OD flow, and feasible region is convex as well. I f w e eliminate cyclic paths, the feasible region of path flow is bounded. Thus, the solution of [NLSUE-VD] is unique for path flow and OD flow. The conventional SUE models assumed probability distribution of perceived travel times of drivers in a deterministic network. This includes both the logit based SUE model by Fisk (1980) and the probit based SUE model by Sheffi and Powell(1981). Although the probit based SUE model overcomes the deficiency of the logit based model, it still remains computational inefficiency for solving probit based SUE model. Mirchandani and Soroush (1987) proposed a general SUE model in a stochastic network. Some limited cases are approximately solved and consistent extension to variable demand seems difficult. The formulated [NLSUE-VD] is categorized into the logit based SUE models, and it is not sufficient to capture the similarity between different routes "Independent from irrelevant alternatives" property of the logit model may cause overloading to the overlapping paths in a network. Nevertheless, the [NLSUE-VD] seems useful since the route choice and the trip
Road Network Under Degraded Conditions
265
generation behaviour in a network are consistently formulated and the model is computationally efficient to solve. Solution Algorithm for NLSUE-VD Applying the Simplicial Decomposition method, the formulated NLSUE-VD can be solved. Two phases are iterated; the phase of solving the Restricted Master problem for given path set and the Column Generation phase of extending the path set. The iterative procedure for the Restricted Master problem and the Column Generation problem can be summarized as follows. The route flow vector and the OD flow vector are referred as /"={/"} and q = (qrs, q rs j, respectively. [Phase 0] Initialization. Set initial path set JA"M. Calculate initial feasible network flows { f ° , q ° } . Set iteration counter m=0. [Phase 1] Restricted Master problem. Solve the Restricted Master problem for the given path set {.K™ ] and obtain network flows \f,q } . [Phase 2] Column Generation problem. Examine the possibility of path set extension. If no additional route is found, close the iteration. Otherwise, up-date the path set (K™*1 } and set iteration counter m=m+l, and go back to [Phase 1] For finding the additional route in the [Phase 2] , the shortest path is calculated for the link cost with loading route flow vector f . If the shortest path is new and not included in the current path set |/C™ ], add the path to the path set.
The Restricted Master problem in the [Phase 1] can be calculated using partial linearization method. The algorithm for the Restricted Master problem is as follows, Initialization. Initial feasible network flows \f°, ^°}were calculated for given path set { K r s } . Set iteration counter n=0. Travel cost up-date. Up-date link flow xa= £ L o"kf^ rs-:Slk€K
and link travel cost
266
Transportation and Traffic Theory
ta=ta(xa) for all links in a network. Then, calculate route cost
c™ = £ S"k ta and the a €A
expected minimum costS^. Direction search. Calculate auxiliary OD flows
min. Z (a) =min. Z (fn + a(g -f"), q" + a(v-q") ) subject to
0 < a <7
Flow up-date. Up-date the route flow vector and the OD flow vector using the optimal value aopt.
Convergence check. If the difference of the flow vectors ||/'!+J -/"|| + is sufficiently small, then the flow vectors are converged and close the Restricted Master problem. Otherwise, set n=n+l and return to .
Road Network Under Degraded Conditions
PERFORMABILITY ALGORITHMS
MEASURES
AND
267
APPROXIMATION
Performance Measures Solving [NLSUE-VD] for the network at state x, we obtain the flow variables as well as travel cost in the network. The equilibrium OD flow {qrs(x)} and the equilibrium route flow \f"(x) } are the representative flow variables, and can be used for evaluating the performance of a deteriorated network. In addition to those flow variables, the equilibrium route travel cost {c"(x) j and the expected minimum travel cost between OD pair
{Srs(x)} are also
available for evaluation. The OD travel demand may decrease when some links in a network become deteriorated. Thus, the simplest performance measure is the degree of reduction of OD travel demand from the normal network state, that is, the ratio of the OD travel demand of a state x to that of the normal state x 0. This is denoted by qrs(x) lqrs(x0). This measure is convenient to compare the performance between different OD pairs. When it is necessary to evaluate the reduction rate for an entire network, the reduction rate of total travel demand £ qrs(x)l £ qrs(xo)can rs€n
rsttt
be used as a performance measure. The other performance measure is the travel cost of an OD pair. Using route travel cost c"(x) and the route choice probability p [ k \ r s ] , the averaged travel cost of an OD pair is represented as,
c[s(x)p(k\rS) = £ c[s(x)
exp(-6,c"(x))
(22)
exp(-
The expected minimum travel cost between an OD pair
Srs(x) will be the alternative
composite measure using route travel cost. As well as the performance measure using flow variables, it is possible to define the ratio of the cost based performance measures, for example, the increase ratio of the expected minimum cost such as S r s ( x ) / S r s ( x 0 ) . We will define the reliability of a flow network using a performance measure. A reliable
Transportation and Traffic
268
Theory
transport network generally means the network in which one can travel from his/her origin to the destination without much uncertainty. The state of a network is probabilistic and the performance measures are also random variables. Therefore, we define the reliability as the probability of whether a performance measure is sustained within an acceptable level. The probability is written as: f Prob.[PM(x) < c] , when PM(x) is increasing. \Prob [PM(X) > c] , when PM(x) is decreasing. PM(x) denotes the value of a performance measure at a network state x . The flow based performance measures are usually decreasing since the ratio will become smaller for worse network state. The cost based performance measure is increasing to the contrary. Parameter c denotes an acceptable level of the performance measure. The value of c is exogenously determined considering the level of service which should be maintained even in deteriorated situations. When the ratio of OD flow, qrs(x) I qrs(x 0 ) , is used as a performance measure,
Rrs(c)
becomes the OD flow reliability. This means the probability of whether the travel between the OD pair is possible within an acceptable reduction rate c (0 < c < 1 ). In other words, the OD pair is regarded connected in condition that (1-c) of the travellers cancel their trips. When the value of c is set equal to 1, any reduction of OD flow is not permitted and the highest level of service is required. The reliability is numerically evaluated using the expected value of the operated/failure function, which determines whether the performance measure is within the given level. Taking a decreasing performance measure like OD flow rate as for example, the operated/failure function is written as, (1 if
PM(x)>c
Z(c,x)= p,,, ,
(24) ^ '
Note that the subscripts rs are dropped in order to avoid complexity. The probability R(c) is the mathematical expectation of Z(c,x) weighted by the state probability written as,
When we evaluate the reliability measure for different criteria c's using the approach above, it is necessary to calculate the operated/failure function for each value of c. Even if we use an approximation algorithm, this is time consuming work. Here, we will show an alternative approach for evaluating reliability. The occurrence of a state is stochastic and performance measures are randomly distributed. If we could estimate the cumulative distribution function of a performance measure Frs [PM ], the probability is easily calculated for any values of c, such as, Rrs(c)=Frs[PM>c]
(26)
The cumulative distribution function is also approximated using similar methods for approximating the expectation of the operated/failure function. This is explained in the next section as well. Approximation Algorithm For a network with L links, the number of possible state amounts to 2L. If [NLSUE-VD] is calculated for each network state, the direct calculation of the expected value of reliability using Eq.(25) requires huge computation cost. This is also true for estimating cumulative distribution function. In this section, we show two algorithms for approximating those equations. The original idea was presented by Li and Silvester (1984). The algorithm defines the lower and upper bounds using the J most probable state vectors. Sorting state vectors in the order of the state probability as Eq.(27). p(x ;)>... > p ( x j ) > p ( x j ^ ! ) > . . .
>p(xN)
(27)
where xj denotes the j-th most probable state vector, p(x.) represents the state probability for the state x f and N is the number of all state vectors. Using the state vectors by the J-th most probable state vector |jc;,
,jc y ) and corresponding the values of the operated/failure
function{Z rs (c,jc y/ );j = 1,... , J } , the upper and lower bounds of the expected value can be defined as follows. The upper bound is obtained through the optimistic expectation of the
Transportation and Traffic
270
Theory
operated/failure function. If we assume that the states jc . for j=J+l to N are equivalent to the normal state xa, the performance measure for those states are also equivalent to that of the normal state, namely
PMrs(Xj)=PMrs(x
0)
for j=J+l to N. This means the values of the
operated/failure function Zrs(c,Xj) are equal to 1 for j=J+l to N. Thus, the upper bound of R^s(c) is obtained as the expected value of Zrs(c,x) of these conditions. That is,
I
p(Xj)Zn(c,x0)
On the other hand, the lower bound is obtained through the pessimistic expectation of the operated/failure function. Assuming that the states X j for j=J+l to N are equivalent to the worst state x w, the performance measure for those states are also equivalent to that of the worst state; namely PMrs(Xj)=PMrs(x
w)
for j=J+l to N. For example, the OD flow of the
worst state qrs(x w) will probably be zero or extremely smaller than that of the normal state qrs(x o ) . Since the ratio of the two qrs(x w)/q rs(x a) is also very small, the value of the operated/failure function for the worst state Zrs(c,xw) is equal to 0 for any criterion c, Accordingly, the values of the operated/failure function Zrs(c, Xj) are equal to 0 for j=J+l to N. Thus, the lower bound of RLrs(c) is obtained as the expected value of Zrs(c,x) of these conditions. That is,
(29)
The expected value of Rrs(c) stays between Rvrs(c) and RLrs(c). Comparing the values of the upper and the lower bounds of the J-th iteration with those of the (J+l)-th iteration, we obtain the folio wings:
RLrs(c,Xj)
Road Network Under Degraded Conditions
271
This means the upper bound and the lower bound converge to the exact expected value of Rrs(c) from the upper side and the lower side, respectively. Figure 2 shows the convergence image of the upper and the lower bounds. We take the next most probable state vectors one after another and update the approximated expected value of the reliability measure Rrs(c) until the difference between the upper and lower bounds becomes small enough.
R(c)
Upper Bound
R (c)
R*(c)
Lower Bound Approx. Value N
1 2
Number of iteration J
Figure 2 An Image of Convergence of Approximation Algorithm
This procedure can be represented as the following algorithm. Step.O Calculate the flow and cost variables by solving the [NLSUE-VD] for the normal state xo .Set iteration counter J=l. Step. 1 Take the J-th most probable state vector x y , where the probability p(xj) is the J-th largest. An efficient algorithm proposed by Lam and Li (1986). Step.2 Calculate the performance measure PMrs(Xj)
by solving [NLSUE-VD] for state Xj.
Step.3 Eevaluatethe operated/failure function. 1 if
PMrs(Xj)>c
0 if
PMrs(Xj)
(31)
Transportation and Traffic
272
Theory
Step. 4 Calculate the upper and lower bounds using Eq.(28) and Eq.(29), respectively. Step. 5 Check the convergence. If the difference of the upper and the lower bounds is small enough, go to Step. 6. Otherwise, set J=J+1 and return to Step. 2. Step. 6 Approximate the expected value as, c)l2
(32)
This algorithm seems efficient for evaluating the reliability between OD pairs rs (rs e £2) for given criteria c. However, one must calculate the number of iterations again if it is necessary to obtain the expected value o f R n ( c ' ) for different criteria c'.
Then, we propose another algorithm to approximate the cumulative distribution function Frs(t) of a decreasing performance measure. The same as the approximation of the expected value of Rrs(c), we can approximate the function Frs(t). Using the J most probable state vectors
{x ,,..., jr y ]
and corresponding
values of the
occurrence probability
\p(x l),...,p( Xj) j, the upper and the lower bounds of the cumulative distribution function are represented as,
where H(t) denotes the subset of [1,...,J], for which performance measure PMrs(Xj)
is greater
than or equal to t. The difference between the upper and the lower bounds is: j
F"(t) -!*,(*) = l-Zp(xj) j-i
(34)
which is kept constant for any range oft. This is convenient for examining the convergence of the iteration. When we take the next most probable state vector xj+1, the upper bound is lowered and the
Road Network Under Degraded Conditions
273
lower bound is raised, respectively. That is,
F",.j(t)*F?,,J+1(t)
FrsJ(t) and FrsJ(t)
denote the upper and the lower bounds for the J-th iteration.
FrSiJ+](t) and FrSiJ+1(t) represent the upper and the lower bounds for the J-th iteration, respectively. This means that the difference of the upper and the lower bounds becomes small enough when a sufficient number of the probable state vectors are included. The cumulative distribution function of performance measure Frs(t) is then approximated as: F
rS (*) = {F",(t) + FLrs (t)} 12
for all t> 0
(36)
Once the distribution function is estimated, we can evaluate the value of Rrs(c) using Eq. (26) for any criteria of c. This is quite useful for analyzing network reliability measures since it is not necessary to consume additional computational cost for different criteria. The algorithm is similar to the one for approximating the expected value of Rrs(c). That is summarized as follows: Step.O Step. 1
Calculate [NLSUE-VD] for the normal network state. Set iteration counter J=l. Take the J-th most probable state vector.
Step. 2 Calculate the performance measure by solving the NLSUE problem for the J-th most probable state vector. Step. 3
Evaluate the upper
and the lower bounds using Eq.(33) for all ranges of the
perfomance measure. Step. 4 Go to Step. 5 if the difference of the upper and lower bounds is small enough. Othewise, set J=J+1 and return to Step. 2.
Step. 5
Approximate the distribution function using Eq.(36).
Transportation and Traffic Theory
274
NUMERICAL EXAMPLES In^ut Conditions Figure 3 presents a small scale network with 4 nodes 5 directed links. Linear link cost functions are also shown in the same figure. A pair of origin and destination flow is assumed from node 1 to node 2. In the following part of the numerical examples, the subscript of origin and destination pair is omitted since only one OD pair is concerned. The maximum number of paths is 3 in the full network., and those paths are identified as shown in Figure 4.
(tj-10+Xj
Figure 3
Network Configuration and Link Cost Function
path-1
path-3
Figure 4
Paths in the Full Network
The upper limit of OD flow T equals to 40. The fixed travel cost C and the cost R for cancelling the travel of the OD pair are assumed 0 and 50, respectively. The parameter values 0j, 02 are 0.1 and 0.2, respectively. Using these conditions for OD demand and route choice, the OD flow q and the path flow/^ (k = l,2 and 3) are written as follows;
exp(-02(C exp(-
- 62R)
-40
exp(-0.2S) exp(-0.2S) + exp(-0.2x 50)
Road Network Under Degraded Conditions exp(-B1ct)
exp(- e,ct)
exp( -O.lck)
275
for k = 1,2,3,
YJexp(-0.1ck)
where S denotes the expected cost between the OD pair and written as;
exp(-61ck) = -10xln lJexp(-0.1ck) The values of link connectivity pa (for a= 1,2,3,4 and5) are assumed 0.9, 0.8, 0.7, 0.9 and 0.8. When the occurrance of link closure is independent each other, the probability for X
network state is given by p(x) = Y[pa °(l-pa)
1 -X
'• F°r example, the state probability for
x = (1,1,1,1,0) is calculated as p(x) = 0.9 x 0.8 x 0. 7 x 0.9 x (1 - 0.8) = 0.09072. SUE Flow for Different Network States Before evaluating network performance measures under degraded network conditions, we will show the results of the network flow analysis. The number of links is 5, and the number of the states is 25=32. These states are aggregated into 8 network patterns with the common set of paths. Table 1 presents the 8 patterns and corresponding state probabilities. When the OD pair 1-2 is not connected, the state is categorized into "disconnected".
[NLSUE-VD] is
calculated for each network state. The results are summarized in the Table 1. It is possible to load 36.67 of OD flow onto the full network (PI) without any disconnected links. This amount of flow is equal to 92% of the given upper limit of the potential OD demand. For this normal state of the network, the largest proportion of the OD flow uses the path-3, and the second largest is the path-1. The path-2 will be used very few, since the cost of the path-2 is the highest among three possible paths. Comparing the network patterns with some disconnected links, it is found that the performance of the network becomes worse when link 5 is not available. If only link 5 is removed (P2) from the full network, the amount of the OD flow loaded onto the network is 20.67. This corresponds to 56% of the OD flow for the full network. On the other hand, the influence of removing link 4 is not so large. Just 1% of OD demand is reduced if link 4 is disconnected (P3). Similarly, the closure of link 1(P5) results in the 10% reduction of OD flow of the full network.
Transportation and Traffic
276 Table
1
Theory
Network State Probability and SUE Flows for Network Patterns
Network Pattern
Network State Probability
OD Flow
Path Flow
Path Cost
(q)
&«)
(Ci,C2,C3)
PI
0.3629
36.67
14.90 1.12 20.65
50.90 68.31 42.18
0.0907
20.67
11.94 8.73
52.64 72.41
0
-
15.01
50.02
O
\T/C-^-'^0 ^-^rx
P2
^-^\
0.0403
36.31
^-^°\
<^^° P4
0.2261
15.90
a^^^^^^o o P5
0.0403
33.35
/^o
P6
0.0101
"N/"
8.33
0
P7
44.26
15.90 0
51.8
-
0 6.51 26.84
60.49 43.30
0 8.33
59.99
0 0.1155
29.52
O
"N^^0 Disconnected
-
21.30
0
CL
cx
0
0.1141
0
0 0
-
29.52
45.90
0
-
When only one path is available (P4,P6 and P7), the performance of the network becomes decreased. In particular, either path-1 or path-2 can carry 20-40% of OD demand of the full network. However, path-3 is still capable to maintain 80% of the OD flows (P7). Although the SUE calculation is executed just for a small scale toy network, it would be useful to find the paths with higher performance under degraded conditions.
Road Network Under Degraded Conditions
111
Convergence of the Approximation Algorithm Figure 5 shows the convergence of the upper and the lower bounds of the expected value of the OD flow. From the previous Table 1, the exact value of the expected value of the OD flow is calculated as; Y J p ( X ) q ( X ) = 25.078, where q(x .) denotes the OD flow from node 1 to 2 for the state vector jc y . Using the J-th most probable state vector, the upper and lower bounds of the expected value can be approximated. It is found that the difference of the upper and the bounds becomes smaller as the number of the probable state vectors increases. This figure proves the convergence of approximation algorithm for the upper and the lower bounds, Eb [q] and EL [q], given by the following equations. j))
J) q(Xj)
q(x ,
OD Flow 40-
\
Lower Bound Upper Bound Expected
V
30" -0-0 "tic-iAr,
20"
10
Figure 5
11
16
21
26 31 N. of iteration
Convergence of the Upper & Lower Bounds of OD Flow and the Expected Value
The values of the upper and the lower bounds approximated using up to the 6-th most probable state are 31.20 and 23.14, respectively. The averaged value of the upper and the lower bounds is 27.17, and the approximation error is 8.3%. When the approximation goes to the 12-th most probable state vector, the upper and the lower bounds become 26.92 and
278
Transportation and Traffic
Theory
24.19, respectively. The averaged value of these two bounds is 25.56, and only 1.9% of the approximation error remains at the 12-th iteration. The upper and the lower bounds of the cumulative distribution function of the OD flow, Fu(q), and FL(q), are approximated as follows; j FU(q)= I pfrJ + V jtH(q)
L
F (q)= £ p(xj) H(q) where denotes the subset of [1,...,J], for which OD flow is less than or equal to q. 1.0
0.8
Lower Bound Upper Bound
O.6-
,-S
0.4-
n—B
J, 0.2
0.0° 0
Figure
6
10
20
30
40
Upper and Lower Bounds of OD Flow Distribution (up-to 6th probable states)
i.o
Lower Bound Upp er B ound
0.2: 0.0
Figure
7
Upper and Lower Bounds of OD Flow Distribution (up-to 12th probable states)
Road Network Under Degraded Conditions
279
Figure 6 shows the upper and the lower bounds of the cumulative distribution function of the OD flow at the 6-th iteration. The difference between the upper and the lower bounds is nearly equal to 0.220 and may not be converged enough. Figure 7 shows the approximated function for the 12-th iteration, in which the difference of the upper and lower bounds reaches 0.074. The difference may not be satisfactory small. However, the distribution function can be well approximated since it is given by the average of the upper and the lower bounds. Once the distribution function is estimated, it is possible to calculate other performance measures as well as the expected value of the OD flow. The shape of the function will give some information for discussing the reliability of the network.
We have just shown the
distribution function of the flow for an OD pair. However, the similar discussion is possible for any other performance measures such as the expected travel cost. It is also possible to compare the distribution of the link flows for different links in the network. The results will be used to find the links with larger fluctuations of flows. Those links should be carefully operated since the influence of the closure of the other links might be magnified.
CONCLUSION Flows in a transport network are not stable even if all links are in service. When several links are deteriorated by some reasons, the fluctuation of network flows will be magnified. The uncertainty and inconvenience of travel may bring the reduction of travel demand. It is necessary to consider those aspects of network flows when we evaluate the performance of the transport network. This paper aims to show a network flow model in degraded conditions, and then to propose performance measures incorporating flows in the network with some links being closed to traffic. Travelling in a deteriorated network is more uncertain than in the ordinary network. A stochastic travel choice model is appropriate to describe a driver's travel behaviour. Some of the drivers may cancel their trips, and the travel demand is assumed elastic for network situations. We have applied Stochastic User Equilibrium model with variable demand to describe flows in a network. Assuming that the trip generation and route choice behaviour are written by the Nested Logit model, we have formulated the Nested Logit based Stochastic User Equilibrium model with Variable Demand (NLSUE-VD). The optimal conditions of the [NLSUE-VD] was proved equivalent to the stochastic route and trip making choice models.
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Formulated [NLSUE-VD] can be solved using the Simplicial Decomposition method. Solving [NLSUE-VD] for a state of the network, we obtain the flow variables as well as travel cost in the network. Those flow and cost variables can be used for evaluating the performance of a deteriorated network. The simplest performance measure is the degree of reduction of OD travel demand from the normal network state, that is, the ratio of the OD travel demand of a state to that of the normal state. When it is necessary to evaluate the reduction rate for an entire network, the reduction rate of total travel demand can be used as a performance measure. Then, the reliability of a flow network was shown using a performance measure. It is defined as the probability of whether a performance measure is sustained within an acceptable level. For a large scale network, the number of possible state is enormous. The direct calculation of the expected value of the reliability requires huge computation cost. We have shown the algorithm for approximating the upper and the lower bounds of the expected value. The algorithm is applied to approximate the probaility distribution function of a performance measure. Through numerical examples in Chapter 4, the convergence of the algorithm was verified. As well as being useful for estimating the expected value, the distribution functions will be widely available to discuss the characteristics of the performance measures. When those performance measures are compared between different OD pairs, it is possible to find less reliable OD pairs. The links reducing network performability will be found and the results will be useful to improve the physical durability of those links. The formulation of the network design problem from the view of reliability is possible.
Although solving the reliability
network design model will be more difficult than usual bilevel network design problems, this will be an attractive field of extension. Of course, we have not yet applied the proposed flow model and the performance measures to actual transport networks. In parallel with this study, the behavioural survey of car drivers is now on going for studying their travel choice behabiour when some links are closed to traffic due to heavy rains. The external validity of the methodology will be examined using those data.
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REFERENCES Asakura, Y. and M. Kashiwadani (1991). Road Network Reliability Caused by Daily Fluctuation of Traffic Flow. Proc. of the 19th PTRC Summer Annual Meeting m Brighton, Seminar G, pp.73-84. Asakura, Y. and M. Kashiwadani (1995). Traffic Assignment in a Road Network with Degraded Links by Natural Disasters. Journal of the Eastern Asia Society for Transport Studies, Vol.1, No.3, pp.1135-1152. Asakura, Y. (1996) Reliability Measures of an Origin and Destination Pair in a Deteriorated Road Network with Variable Flows. Paper presented at the 4th Meeting of the EURO Working Group on Transportation in New Castle upon Tyne. 14 Pages. Asakura, Y.(1997). Comparison of Some Reliability Models in a Deteriorated Road Network. Journal of the Eastern Asia Society for Transport Studies, Vol.2, No.3, pp.705-720. Du, Z. P. and A.J.Nicholson (1993). Degradable Transportation Systems Performance, Sensitivity and Reliability Analysis. Research Report, No.93-8, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand. Fisk, C. (1980). Some Developments in Equilibrium Traffic Assigment Methodology. Transpn. Res.-B, Vol.l4-B, pp.243-255. lida, Y. and H. Wakabayashi (1989). An Approximation Method of Terminal Reliability of Road Network Using Partial Minimal Path and Cut Set. Proceedings of the 5th World Conference on Transport Research, VolJV, Yokohama, Japan, pp.367-380. Khattak, A.J. and A.D. Palma(1997). The Impact of Adverse Weather Conditions on the Propensity to Change Travel Decisions: A Survey of Brussels Commuters. Transpn. /tej.-yi., Vol.31, No.3,pp.l81-203. Li, V.O.K. and J.A. Silvester (1984). Performance Analysis of Networks with Unreliable Components. IEEE Trans, on Communications, Vol.COM-32, No. 10, pp.1105-1110. Mirchandani, P. and H. Soroush (1987). Generalized Traffic Equilibrium with Probabilistic Travel Times and Perceptions. Transportation Science, Vol.21, No.3, pp.133-152. Patnksson, M. (1994) The Traffic Assignment Problem; Models and Methods. VSP Utrecht, The Netherlands. Sheffi, Y. and W. Powell (1981). A Comparison of Stochastic and Deterministic Traffic Assignment Over Congested Networks. Transpn. Res.-B, Vol.l5-B, pp.53-64. Turnquist, M.A. and L.A. Bowman (1980). The Effect of Network Structure on Reliability of Transit Service. Transpn. Res.-B, Vol.l4-B, pp.79-86.
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283
A SENSITIVITY BASED RELIABILITY ASSESSMENT
APPROACH
TO
NETWORK
Michael G H Bell', Chris Cassir1, Yasunori lida2 and William H K Lam3
Abstract This paper presents a methodology for evaluating the reliability of transportation networks. While tools already exist to determine the expected benefits of travel demand management or new infrastructure, tools have yet to be developed which take into account disbenefits arising from randomly occurring disturbances. The paper focuses on the sensitivity of both path travel times and expected minimum origin-destination travel times to normal within-day demand and supply variation, where demand variation takes the form of perturbations to origin-destination flows and supply variation takes the form of perturbations to link saturation flows. Two extreme cases are distinguished; one where route choices fully respond to the perturbations, corresponding to the more major, longer-term incident and the other where route choice does not respond, corresponding to the more minor, shorter-term incident. A logit assignment model, referred to as the Path Flow Estimator (PFE), is linearised with respect to the parameters affected by within-day variation, using sensitivity expressions. As analytically derived sensitivities are sometimes difficult to calculate for large networks, their approximation by finite differencing is considered. Results obtained for a large network (5000 links, 8000 OD pairs) in York are discussed, as well as results obtained for a much smaller network (100 links, 60 OD pairs) in Leicester.
i.
INTRODUCTION
1.1
Background
The potential sources of disruption to transportation networks are numerous, ranging at one extreme from natural or man-made disasters, which tend to occur rather infrequently, to at the other extreme events, which occur on a daily basis. The scale, impact, frequency and predictability of such events will of course vary enormously. While little can be done about their scale, frequency or predictability, particularly where natural disasters are concerned, it
! 1
Transport Operations Research Group, University of Newcastle, Newcastle upon Tyne, NE1 7RU Department of Transportation Engineering, Kyoto University Department of Civil and Structural Engineering, Hong Kong Polytechnic University
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should be possible to design transportation networks so as to minimise the disruption such events can cause. This paper describes the theory behind a software tool being developed for the analysis of network reliability. While many tools exist for studying the impact of new transport infrastructure or travel demand management measures on traffic flows (for example, the widelyused CONTRAM and SATURN programs), there are no tools for assessing the impact of such measures on the reliability of transportation networks. Network reliability has two dimensions. The first relates to the connectivity reliability of a network. When links fail in unfavourable configurations it may no longer be possible to reach a given destination from a given origin, in which case the network becomes disconnected. However, even a connected network may fail to provide an adequate level of service. For example, random events may cause unacceptable variation in origin-to-destination travel times, making it difficult for travellers to arrive at their destinations on schedule. The second dimension of reliability is therefore the performance reliability of a network. Previous work on network reliability (Du and Nicholson, 1993; lida and Wakayabashi, 1989) has focused principally on connectivity in degradable transportation networks. Although Asakura and Kashiwadani (1991, 1995) have looked at the problem of travel time reliability, the field of performance reliability is distinctly under-researched. This paper focuses on the performance reliability of transportation networks in the face of normal within-day variation of origin-destination flows and link properties. It makes use of the Path Flow Estimator (PFE), a Stochastic User Equilibrium (SUE) assignment model, in order to estimate the distributions of the variables of interest (flows, travel times). Reliability with respect to given performance criteria can then be computed on the basis of those distributions. The eventual objective is to develop a software tool that facilitates the design of robust transportation networks.
1.2
Path Flow Estimator
The Path Flow Estimator (PFE) is a flexible traffic assignment tool. It was originally developed in the DRIVE 2 project MARGOT to estimate path travel times for route guidance systems given vehicle detector data, traffic signal times, and possibly also travel time data from probe vehicles. The PFE is based on the notion of a Stochastic User Equilibrium (SUE) between demand and supply. The demand for the paths is determined by a logit path choice model and the link costs, which characterise the supply side, are determined primarily by flow-dependent delay formulae. The delay function currently used is the Kimber and Hollis (1979), time dependent function which allows for over-saturated conditions . The PFE utilises the fact that under a SUE, in contrast to a deterministic user equilibrium, path flows are uniquely defined at the equilibrium. The equilibrium path flows are found by solving an equivalent optimisation problem iteratively. An outer loop generates paths and an inner loop assigns flows to paths according to the logit path choice model. The inputs take the form of traffic counts and/or trip tables and the outputs take the form of estimated path flows and travel times, as well as data which may be derived
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from these, like junction turning proportions, link flows where unmeasured and link delays. Further information about the PFE can be found on http://www.ncl.ac.uk/~nsg5/PFE/html. The structure of the algorithm is illustrated in Fig. 1. Interesting features of the algorithm are; the inclusion in path generation of the "pressure" exerted by active link constraints, and the allowance for measurement errors through user-defined confidence intervals.
Input: Static network data and dynamic flow and signal control data Outer loop Generate paths of "least resistance" taking delay and the pressure exerted by active link constraints into account Inner loop Assign measured flows to paths according to the logit path choice model taking flow-dependent delays into account Output: Path flows and travel times, turning movements, unmeasured link flows Fig. 1 : The Path Flow Estimator The approach to network reliability analysis adopted in this study is based on the sensitivity of the PFE outputs, in particular path travel times and expected minimum origin-destination travel times, to perturbations of the inputs, in particular the origin-destination demands and link saturation flows. Tobin and Friesz (1988) obtained expressions for the sensitivity of deterministic user equilibrium link flows to perturbations in origin-destination demands and link travel times. Bell and lida (1997) extended the method to obtain expressions for the sensitivity of minimum origin-to-destination travel times to perturbations in origin-destination flows and link travel times. Using the same approach, Bell and lida (1997) have obtained comparable sensitivity expressions for the PFE.
2.
NETWORK PERFORMANCE RELIABILITY
2.1
Network performance reliability measures
This paper focuses on normal within-day variations of traffic conditions, like the morning peakhour for instance. The measures of network performance reliability considered are based on travel times, since broadly speaking the main function of a transportation network is to carry travellers to their destinations within acceptable times. Two measures of potential interest to both network users and planners are: • Reliability of path travel times • Reliability of the expected minimum travel time between the origin and destination The first measure is defined as the probability that the travel time on a given path is less than an acceptable threshold. This is of direct interest to the users, since path travel time reliability is a factor likely to influence their route choice. The second measure is taken at a more aggregate level, namely the level of the Origin-Destination (OD) pair. The expected minimum OD travel time is the expected lowest perceived travel time amongst all the relevant paths, allowing for
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random variations across users in the perception of travel time. Thus changes of route undertaken by users to avoid perceived increases in link travel times are taken into account. The second measure may be more useful for planners than for the network users themselves. Consider the logit path choice model with perceived path travel time Ck=g?—j where g™ is the actual travel time on a path k between an OD pair rs, £ is a random, Gumbel distributed, perception error term, and 9 is a scaling factor, then the expected perceived minimum travel time is given by E[min kers '
kers
For both types of reliability measure, an estimation of the distributions of actual (not perceived) path travel times in the face of normal within-day variations of exogenous parameters in the demand or the supply side of the transportation system is required.
2.2
Sources of unreliability
There are basically two types of variation which affect travel times in a transportation network on a daily basis. One relates to the demand, characterised by average OD flows during the period of interest, and the other relates to the supply, for example the saturation flows of the links in the network. Both types of variation can have a range of causes and impacts in terms of their scale and duration. Examples of sources of demand- and supply-side variation are presented in Table 1. Impact Minor/Short-term
Maj or/Long-term
Demand Addition or cancellation of trips; change of destination; change of departure time; etc.
Supply Reduction of capacity due to minor accidents; signal timing changes (where not demand related); etc. Exceptional events (football Major accidents; road matches, fairs, etc.); holiday closures; etc. periods; etc. Table 1: Sources of demand- and supply-side variation
For both the demand- and the supply-side sources of variation, the distinction between minor/short-term and major/long-term impacts serves to indicate whether or not there is likely to be an induced re-routing. For more minor, short-term incidents, there may be insufficient time for information to disseminate to network users, and even where information has disseminated, it may not be worthwhile for users to respond by changing route. These sources of fluctuation can be modelled by the use of appropriate discrete or continuous probability distributions for the OD flows and link saturation flows. To model the impacts the fluctuations will have on the travel times, and hence estimate path travel time distributions,
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linear relationships between the PFE estimates of path travel times and the sources of random variation (the OD flows and the link saturation flows) are established. Two approaches are adopted, depending on the severity and time-scale of the perturbations, reflecting whether or not induced re-routing is expected.
3.
PFE SENSITIVITY
3.1
Linearising the PFE
Linearising the PFE allows approximate distributions for model outputs to be obtained from distributions of model inputs thanks to the conservation of form for certain commonly used types of distribution. For instance, the summation of independent normal variates has a normal distribution whose defining moments are straightforward to compute. The linearisation is valid only for small variations about some point, in this case taken to be the SUE- 'solution obtained for average input values. / Where the fluctuations may induce re-routing, one extreme case arises where full adjustment to the fluctuations has occurred. SUE sensitivity analysis may be used to estimate the distributions of equilibrium path travel times. The other extreme case arises where no re-routing at all occurs. The fluctuations effect only the supply-side, since the route-choice proportions do not respond.
3.2
Notation
Let: h = Vector of path flows v = Vector of link flows c = Vector of link travel times g = Vector of path travel times t = Vector of Origin-Destination (OD) flows s = Vector of link saturation flows A = Link-path incidence matrix B = OD-path incidence matrix P = Matrix of path choice proportions a = Dispersion parameter of the logit model.
The PFE gives an estimation of average flows and travel times in a network for some period of the day. The solution yields flows and travel times consistent with each other by virtue of the equilibrium principle, and the assignment of flows to paths is governed by the logit model as follows
with /7 = exp(-a.gy(h,s))/ for any path flow hj between an OD pair W .
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The relation between path costs and path flows is through the fact that path travel times are the summation of link travel times, and that link travel times depend on link flows in relation to their saturation flows through monotonically increasing functions
where / e j indicates that the summation is over all links on path j , and the relation between link flows and path flows is
where j e / indicates that the summation is over all paths using link i. This SUE solution can be interpreted as the mean value of a network equilibrium where users minimise their perceived travel times. Due to random errors in travel time perception, the model is stochastic and therefore the estimated flows and travel times are also random variables. However the mean values provided by the SUE are a reasonable estimator of average network traffic conditions on a particular period of the day, like the morning peak hour. We shall hereafter refer to the 'base solution' as being the PFE stochastic equilibrium obtained for a given trip matrix t 0 , considered to be the average trip table, and the vector s0 of average link saturation flows. From this base we can then look at the effects of random variations in both demand and saturation flows on path travel times and expected minimum OD travel times necessary for calculating reliability measures.
3.3
PFE sensitivity analysis with full route response
Assuming OD flows are normally distributed with known parameters, Asakura and Kashiwadani (1991) ran a static User Equilibrium (UE) assignment several times, with a demand sampled from these normal distributions, in order to estimate OD travel time distributions. A similar approach would be possible for the PFE. However, for large networks with many OD pairs, this method is computationally prohibitive and also simulation adds an element of non-reproducibility. Another approach would be to estimate variances of travel times from the variances of OD trips by making linear approximations of the relationship between OD flows and the path travel times obtained from the equilibrium assignment model. This can be done by applying sensitivity analysis to the PFE model. Fisk (1980) proposed the following objective function f(h) = S j h J ( l n h J - l ) + aS i I 0vi c i (x)dx When f(h) is minimised subject to t = Bh and h > 0, the Kuhn- Karush -Tucker optimality conditions are Vf(h) + BTu* > 0, h* > 0 and (Vf(h) + BTu*)T h* = 0
Network Reliability Assessment where Vf(h) = In h + ag. As h* > 0 (since all paths are used) In h* = -ag* - BTu* It can be shown that this implies the logit path choice model hj* = tw exp(-agJ*)/EjeP(w) exp(-agj*) Hence at the optimum Vf(h*) + BT u* = 0 where Vf(h*) = In h* + ag*, so at equilibrium V 2 f(h*) Ah + BT Au = -aAg where Ag is the exogenous perturbation of path cost. This leads to r«Agl
~H B T TAhl
[At J ~ B
0 J_AuJ
where H = V2f(h*). Following Bell and lida (1997) TAhlJj,, Au
L J
J
L 2i
J12TaAgl J22iAt J
where J n = H'1 (I - BT (B H-1 BT) ' B H'1) J12 = H ' BT (B H-1 B T )' J21 = (B H'1 B T )' B H ' — /i> u-1 uTVl 72 — ~\ •*-*• Jt> I
Let z = Expected minimum OD costs At optimality In h* + ag* + BTu* = 0 implying hj = exp(- agj -1^). From this logit model exp(-uw) = tw / Sj 6 P(w) exp(-agj) so
-uw = In tw - In Sj e P(w) exp(-agj) Hence -u = In t + az, so dz/dt = a(B H-1 B7)'1 Putting this together
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[dz/ds|s = s,] = -(l/ct)(B H-1 B) !B H-' AT[dc/ds|s = s*] [dz/dt|t=t*] = -(l/cc)[du/dt|t = t,] = (l/a)(B H > B)-1 [dh/ds s= s*] = H-' (I - BT (B H ' B1)'1 B H'1) AT[dc/ds|s = s»] [dh/dt|t = t,] = H-1 BT (B H-1 BT)-' This leads to the following approximate variance and covariance expressions Var(z) = [dz/ds|s = s,] Var(s) [dz/ds|s = s,]T + [dz/dt|t=t»] Var(t) [dz/dt|t = t*]T Var(h) = [dh/ds|s = s,] Var(s) [dh/ds|s = S*]T + [dh/dt|t = t,] Var(t) [dh/dt|t = t*]T assuming that s and t are vectors of independent random variables. Gradients and variances for path travel times rather than path flows may be obtained from the gradients of path travel times to path flows, via the link travel time Jacobian. Calculating those expressions requires extensive matrix operations (like inversion), which for large networks are thought to be unmanageable. Therefore it may be preferable to approximate these gradients by finite differencing
|t = t,]*[Az/At|t = t*] |s = s,]«[Ah/As|s =s,]
To obtain finite differences one needs first to run the PFE program with the trip table t 0 to obtain a base solution in all variables of interest . Then the model has to be run sequentially after applying a small change A/y (say 1 vehicle/hr) to each OD pair j , one at a time. To reduce the number of iterations within the PFE at each run, the initial solution is taken to be the base solution, which is assumed to be not so distant from the perturbed solutions. Thus after each run j , the differences with respect to the base solution in terms of path travel times can be calculated. Those differences divided by the htj provide the finite differences approximations to the partial derivatives needed.
3.3
PFE sensitivity with no route response
If we assume that there is no re-routing induced by the fluctuations of demand and saturation flow, then it is no longer appropriate to consider expected minimum OD travel time. Instead we can look at distributions of path travel times around the equilibrium caused by the demand- and supply-side perturbations. The following gradients reflect the corresponding sensitivities
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[dg/dt|t = t*] = AT [dc/dv|v = v.] A P
4. EXAMPLES 4.1 Example 1: network in York (2323 links) 4.1.1
Network Data.
Network data were provided by York City Council (YCC) in SATURN (a simulation and assignment model) format, as part of a European project AIUTO, which looked at how to model the effects of various travel demand management measures. The SATURN model was based upon one constructed by YCC in 1992 and extensively updated by them in 1996. The node-based data from SATURN had to be translated into a link-based format suitable for the PFE. Also the morning peak hour trip matrix, obtained from a survey with real number entries, was transformed into an integer trip matrix by a rounding operation that sought to maintain approximately the same column and row totals as in the original real valued matrix. The resulting PFE network thus consists of about 2000 links, 8635 Origin-Destination (OD) pairs, and includes such necessary information as link free-flow average speed, link saturation flows, as well as the signal times. A representation of the modelled network is shown in Fig. 2.
Fig. 2: The York network 4.1.2
Base SUE solution.
The base SUE solution was obtained by running the PFE on the York network with the average morning peak hour trip matrix. Fig. 3 shows the network with the congested links highlighted (links with the ratio of flow over saturation flow greater than 0.9 were defined as congested).
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Fig. 3: York network with congested links highlighted 4.1.3 Finite differences and variance computation. It was proposed in this study to obtain morning peak hour variances of both equilibrium and non-equilibrium path travel times. The source of variation was taken to be the trip table (OD flows). For equilibrium path travel times we used sensitivity of the PFE solution about the base solution calculated previously. Since applying analytical sensitivity expressions was not practically possible for such a large network, it was decided to approximate the derivatives of equilibrium path travel times with respect to OD flow variation by finite differences. This was carried out by simply changing each OD flow tt , one at a time, by one unit of flow, and then calculating a new PFE solution trip matrix thus perturbed. The difference in path travel times with respect to the base solution give the approximate derivatives
where g* is the SUE travel time of path j. This meant running the PFE 8635 times (the number of OD pairs) with a differently perturbed OD matrix each time. Eventually the outcome would yield an estimation of the equilibrium path travel time variances, through the linearisation of the relationship between equilibrium path travel times and OD flows discussed in the previous section
i<=OD pair
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What is needed as input are the variances of each OD flow. Due to lack of empirical statistical data, some assumptions about the distributions of OD flows were required. For simplicity, we used a normal distribution for each OD flow centred around tf , the base solution. This implied a normal distribution as output for the path travel times. To have a range of OD flows with negligible negative values, and given that the majority of all OD pairs had flows of the order of 1 veh/hr, it seemed quite reasonable to take a standard deviation a at a third of the mean value // . So for each OD pair i , the variance of flow was thus assumed to be
4.1.4
Analysis of results.
The finite differences method was applied to obtain variances of SUE path travel times. Four days of computational time were necessary in order to compute the required variances, despite the fact that each of the 8635 PFE runs were started from the base solution, expected to be not too distant from the perturbed solutions. We also obtained the variances for non-equilibrium path travel times using finite differences for proper comparison, even though the variances could have been computed analytically as shown in the previous section. With the variances thus calculated, reliability measures could be computed for each path. The performance criteria, or the travel time threshold set for reliability, was arbitrarily taken at 1 1 0% of the mean path travel time. Thus for each path j the travel time reliability Rj , defined as the probability that the travel time will be less than 1 10% of the mean travel time g° , was calculated as follows: /Var[g,]' where O(
0"
) = Fx(x) = —f= v27r
|exp[-—y 2 ]dy J 2
is the cumulative distribution of a normally
-00
distributed variable x with mean tabulation.
//
and variance a1. The values of O are obtained by
Given the huge number of paths in this network (around 23000), a thorough analysis of all the results was not really presentable here. For illustrative purposes, we chose a particular OD pair with two clear cut alternatives to analyse the results and illustrate the differences between the equilibrium and non-equilibrium methodology for path travel time reliability. Fig. 4 shows the two paths (with the arrows) on a zoomed part of the network (in the centre of town), along with the congested links (highlighted). The table below indicates the average path travel times with the values of the standard deviations and reliability for both equilibrium (full route adjustment) and non-equilibrium (no route adjustment) distributions.
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Fig. 4: Two alternative paths path
1 2
Average time (ran)
7.06 4.63
Std deviations (sec) (Equilibrium *) 49* 1*
66 8
Reliability (Equilibrium *)
0.81* 1.00*
0.76 0.99
We can see that for both paths the standard deviation increase quite significantly when readjustment to equilibrium is not considered in the calculation. This suggests that the effects of drivers re-routing is to lessen to impacts of variations. We can also notice that path 2 experiences hardly any significant variations (whether in the equilibrium case or not) compared to path 1. By looking at the map, the explanation is clear; path 2 does not include any congested links whereas path has two of them. So fluctuations of demand on those congested links is bound to produce greater changes in travel times than on the links that are less congested in the base solution. This is a feature that actually appears for all the most unreliable paths in the network; they all contain one or several congested links. While this appears sensible, it still prompts a question about the usefulness of conducting such a time-consuming calculation; the most unreliable paths could indeed have been identified before the calculations of the variances just by finding out the paths containing congested links in the base solution. This limitation comes from the linear approximation used in both methods, which is only valid in the vicinity of the base solution. Therefore it would be useful to account for larger variations,
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which tend to happen in reality anyway. However to model the impacts of these larger variations the non-linear effects need to be included.
4.2 Example 2 : network in Leicester (103 links) To illustrate the points made in the previous example further, the methodology was applied to a much smaller network , thereby allowing a more complete overview of the results to be presented in the scope of this paper. 4.2.1 The network. The network considered here is a small part of an urban network in Leicester , England. It consists of 103 links (including micro-links at junctions), 9 origins and 9 destinations . A representation of the network is shown in Fig. 5. The topological data (including signal timings), along with a trip table for a peak period were made available for a Phd project on OD estimation based on SCOOT traffic counts. We didn't use the link detector data here.
O f
• —•^ .
signalised junction
m
unsignalised junction *.._. 1
1 d«J d«_.
. -"- - +1. _ *-
212i
' !,•-<:
222.
NK.222
^222.
2*1*
C
22«
N00224 )22^
21Jb
iin
»-
(* NM2« )
22.,
Fig 5: Leicester network 4.2.2 Computation of path time variances and reliability measures with respect to OD flow fluctuations. We used the PFE model on the Leicester network with the trip table available, and applied the sensitivity expressions defined in chapter 3, with and without route response. It was here possible to compute the sensitivity expressions in the full (equilibrium) route response case,
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directly without using finite differences, given the small size of the network (typically the number of paths generated was of the order of 100 veh/hr). As in the York case we assumed normal distributions of OD flows centred around the trip table values t°, and again chose a standard deviation of (t° 13), to be consistent with the previous example. However since the order of magnitude of most OD flow was here much larger than in the York case , we put a limit of 30 veh/hr on the standard deviation, which corresponds to the largest deviation in the York case. This was to avoid very large deviations, which given the first order approximation used in the method, would probably lead to unrealistically large variances in the equilibrium travel times. Note that a deviation of 30 veh/hr is not strictly speaking small enough to warrant a valid use of the first order approximation for the relation between the equilibrium solution and the OD flows. However it is conjectured that for moderately congested network, the second-order derivatives of the equilibrium flows will be sufficiently small so as to make the second order terms negligible. The following table shows the time deviations and reliability (defined as in the York example) results obtained for the 20 most 'unreliable' paths, in the equilibrium and non-equilibrium variance evaluation case. Orig. Dest. OR_F DES_B (non-equilibrium)
As for the two paths in the York example, we can see here that, for all the paths, the standard deviation is greater, and thus the reliability lower, when equilibrium route response is not taken into account. We also highlighted two links , 215L and 215K, which are included in all the most unreliable paths, and those two links happen to be the only over-saturated links in the whole network This confirms the relation between congestion in the average equilibrium situation and high variations in travel times and again points to the limitation of the first-order approximation in gaining more useful insights. The next table shows the standard deviation of the expected minimum OD travel time, for the 20 most 'unreliable OD pairs:
Orig
Dest
OR OR OR OR OR OR OR OR OR OR OR OR OR OR OR OR OR OR OR OR
DES_B DES B DES B DES B DES B DES B DES B DES A DES C DES J DES G DES H DES D DES E DES A DES_E DES D DES A DES A DES A
We can see that all OD pairs connecting DES_B have a substantially higher variability in travel time compared to all other pairs. A quick look at the network shows why: To get to DES_B, it is impossible to avoid either link 215L or link 215K. The same could be said for DES_A on the evidence of Fig 5; however in reality link 215K is divided into 2 sub-links, and it is only the right-turn sub-link leading to DES_B that is actually over-saturated.
s.
FUTURE WORK
Including non-linear effects is a problem in the sense that the form of the input distributions (demand, capacity) is no longer conserved in the output (travel time) distributions. We present here briefly the problem of attempting to derive path travel times distributions knowing the distributions of OD flows , and keeping the path-choice proportions fixed (thus restricting the non-linearity to the supply side only). We shall concentrate on the link travel times c,(v,(h),^ ; ). Suppose, without loss of generality, the demand t to be normally distributed around t°, the base trip table.
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Then assuming the path choice proportion matrix fixed at the base solution, P0 , the link flow v, (h) also becomes a normally distributed random variable, centred around v,° , since there is a linear relationship:
v 76i
W 7a
and
Var[v,]= provided the fw are independent, (a condition that is not necessarily satisfied, but which we will assume holds here as we did in the previous calculations). Thus the travel time ct (v,, st) is also a random variable, being a function of a random variable, though it is not normal, since the link cost function is non-linear. However the expected value and the density function can be calculated, using the density function of v^, fv
and the cumulative probability distribution of Cj is given by
where the inverse function c~'(x) exists, since the link cost function is monotonically increasing as a function of link flow v only. The density function fc (x) can then be obtained by taking the derivative of F ( x ) . For reliability studies, we are mainly interested in path travel times, which being summations of random link travel times, are also random variables whose cumulative probability function is just what is needed to obtain the required measures of reliability . The probability of a trip cost on a path j being close enough to the 'normal situation' (base solution) value serves as a good performance measure and will be given by Fg (gj + s). The problem is thus how to compute
V Suppose , for the sake of clarity, that a path X is composed of link 1 and 2. Then we have:
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as the summation of two random variables. The cumulative distribution of gx is then
z+y
If Cj and c, were independent variables, Fgx (jc) would then be quite easy compute, since in that case:
The fc (x) , density functions for the link costs, had been calculated previously. So Fg (x) may be calculated relatively easily ( at least in this case of two variates ).
= z+y
6.
CONCLUSIONS
We have presented a methodology for obtaining some performance reliability measures for a transportation network. The work is essentially centred around estimating typical distributions of path travel times resulting from random fluctuations of exogenous factors. Having identified those factors, a method based on sensitivity (linearisation) of a SUE model, with two different options depending on whether re-routing is considered or not. We showed some results for a large realistic network and also a smaller one to illustrate the differences between the two options and noticed that paths subject to bigger variations in time are those containing links that are congested in the average situation. We concluded by indicating that more useful results might be obtained if non-linear effects were taken into account so as to allow a wider range of fluctuations to be considered.
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REFERENCES Asakura, Y and Kashiwadani, M(1991). Road Network Reliability Caused by Daily Fluctuation of Traffic Flow. Proceedings of the J91 PTRC Summer Annual Meeting in Brighton, Seminar G, 73-84. Asakura, Y and Kashiwadani, M (1992). Road Network Reliability Measures Based on Statistic Estimation of Day-to-Day Fluctuation of Link Traffic. Proceedings of the 6th World Conference on Transport Research, Lyon, France, June. Asakura, Y and Kashiwadani, M (1995). Traffic Assignment in a Road Network with Degraded Links by Natural Disasters. Journal of the Eastern Asia for Transport Studies, Vol.1, No.3, 1135-1152. Asakura, Y (1996). Reliability Measures of an Origin and Destination pair in a Deteriorated Road Network with Variable Flow. Proceea Proceedings of the 4th Meeting of the EURO Working Group, Newcastle-upon-Tyne,UK,September. Bell, M.G.H. , Lam, W and lida, Y (1996), A Time-Dependent Multiclass Path Flow Estimator. Proceedings of the 13th International Symposium on Transportation and Traffic Theory, Lyon, France, July. Bell, M.G.H. and lida, Y (1997), Transportation Network Analysis, Wiley, England. Du, Z.P. and Nicholson, A.J. (1993). Degradable Transportation Systems Performance, Sensitivity and Reliability Analysis. Research Report, No. 93-8, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand. Du, Z.P. and Nicholson, A.J. (1997). Degradable Transportation Systems: Sensitivity and Reliability Analysis. Transportation Research B, 31, No 3, 225-237. Fisk, C (1980). Some developments in equilibrium traffic assignment. Research, 14B, 243-255.
Transportation
lida, Y and Wakayabashi, H (1989). An Approximation Method of Terminal Reliability of Road Network Using Partial Minimal Pat Path and Cut Set. Proceedings of the 5th World Conference, Vol.IV, Yokohama, Japan. 367-380. Kimber, R.M. and Hollis, E.M. (1979). Traffic queues and delays at road junctions. TRRL Laboratory Report 909, Transport and Road Research laboratory, Crowthorne, England. Tobin, S.J. and Friesz T.L. (1988). Transportation Science, 22, 242-250.
Sensitivity Analysis for equilibrium network flow.
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A CAPACITY INCREASING PARADOX FOR A DYNAMIC TRAFFIC ASSIGNMENT WITH DEPARTURE TIME CHOICE Takashi Akamatsu Department of Knowledge-based Information Engineering Toyohashi University of Technology Toyohashi, Aichi 441-8580, Japan Masao Kuwahara Institute of Industrial Science University of Tokyo Minato-ku, Tokyo 106-8558, Japan
ABSTRACT This paper demonstrates that the capacity increasing paradox in a transportation networks as in Braess(1968) does also occur under non-stationary settings, in particular, under dynamic traffic assignment with endogenous time-varying origin-destination (OD) demands. Through the analyses, the analytical formulae for the solutions of the dynamic equilibrium assignment are explicitly derived for two kind of networks: the networks with a one-to-many OD pattern and the reversed networks with a many-to-one OD pattern; the formulae clarify the significant difference in the properties of the two dynamic flow patterns. This also leads us to the findings that one of the crucial conditions that determine whether the paradox occurs or not is the OD pattern of the underlying networks.
1. INTRODUCTION Local improvements in a transportation network do not necessarily lead to the improvement of the global performance of the network. This fact has been well recognized as "Braess's
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paradox"(Braess (1968)) or "Smith's paradox"(Smith (1978)). The paradoxes stimulated many researchers in the field, and a considerable number of studies have been made on the relevant topics such as the network design problem or the sensitivity analysis of the equilibrium traffic assignment. Almost all the studies are, however, based on the framework of static (equilibrium) traffic assignment; only a few attempts have so far been made to study non-stationary (dynamic) traffic flow patterns with queues. Since the properties of the dynamic flow with queues are significantly different from those of the static flow without queues, many basic problems on the paradox under non-stationary settings are yet to be investigated. The purpose of this paper is first to demonstrate that the capacity increasing paradox does also occur under non-stationary settings, in particular, under dynamic traffic assignment with endogenous, time-varying origin-destination (OD) demands. The paper also aims to capture the conditions that determine whether the paradox is likely to occur or not; we disclose that the OD pattern of the underlying networks is one of the crucial conditions. In order to achieve the purpose, we first disclose that the analytical solution of the dynamic user equilibrium (we call this DUE) traffic assignment with elastic OD demands (i.e. the assignment considering users' departure-time choice behavior) can be obtained explicitly in a particular type of network satisfying some conditions. The solutions are derived for two kinds of network: (i) networks with single origin and multiple destinations (regarded as an "Evening rush hour" on a network of a city with a single CBD; we refer to this "E-net" hereafter); and (ii) networks with single destination and multiple origins (obtained by reversing the direction of all links and origin/destinations of the E-net, we may regard it as a "Morning rush hour" on the same network above; we refer to this "M-net"). Through the analyses of the two cases, we see the significant difference in the properties of the two dynamic flow patterns for not only the case where time-varying OD demands are given but also for the case of elastic OD demand due to user's departure time choice. These basic results for the DUE assignment then enables us to demonstrate the dynamic version of the capacity increasing paradox and to discuss the significant effect of OD pattern on the occurrence of the paradox. The organization of this paper is as follows. In the second chapter, we briefly explain the basic properties of dynamic user equilibrium assignment, restricting ourselves to the minimum knowledge required for considering our problem. The third chapter explores the structure of the dynamic equilibrium assignment with exogenous OD demands for E-net and M-net. The analytical solution formulae of the equilibrium flow patterns for E-net and M-net are derived. The fourth chapter extends the analyses to the model with endogenous OD demand; not only the route choice but also the departure time choice are simultaneously considered in the model. For an appropriate set of boundary conditions, the explicit equilibrium flow patterns are derived for E-net and M-net. By using the results obtained in Chapters 3 and 4, we demonstrate a dynamic version of Braess's paradox in the fifth chapter. We first discuss the paradox for the model with exogenous OD demand; the analysis on a simple network exhibits that the paradox arises only on a
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particular condition for the network with a one-to-many OD pattern, while the corresponding paradox always arises for the reversed network with a many-to-one OD pattern. We then show that the same results also hold for the model with endogenous OD demand. Finally, the last chapter summarizes the results and remarks on some further research topics.
2. DECOMPOSITION OF DYNAMIC EQUILIBRIUM ASSIGNMENT 2.1 Networks Our model is defined on a transportation network G[N, L, W\ consisting of the set L of directed links with L elements, the set N of nodes with N elements, and the set W of origindestination (OD) nodes pairs. The origins and the destinations are the subset of N, and we denote them by R and S, respectively. In this paper, we deal with only networks with a one-to-many OD (i.e. the element of R is unique) or those with a many-to-one OD (i.e. the element of S is unique). Sequential integer numbers from 1 to N are allocated to N nodes. A link from node / toy is denoted as link (ij). We also use the notation to indicate a link by the sequential numbers from 1 to L allocated to all the links in the set L. The structure of a network is represented by a node-link incidence matrix A*, which is an N X L matrix whose (n, a) element is 1 if node n is an upstream-node of link a, -1 if node n is a downstream-node of link a, zero otherwise. The rank of this matrix is N-l since the sum of rows in each column is always zero. Hence, it is convenient in representing our model to use the reduced incidence matrix A (instead of A*), which is an (N-l) XL matrix eliminating an arbitrary row of A*. We call the node corresponding to the elimination "reference node". It is also convenient to "split" the matrix A into a pair of matrices, A _ and A + , defined as follows: A_ is a matrix that can be obtained by letting all the +1 elements of A be zero (i.e. the (n, d) element is -1 if link a arrives at node n, zero otherwise); A + is a matrix that can be obtained by letting all the -1 elements of A be zero (i.e. the (n, a) element is +1 if link a leaves node n, zero otherwise); it is needless to say that A = A_ + A + holds.
2.2. Link Model and Dynamic Equilibrium Assignment For a link model in our dynamic assignment, we employ a First-In-First-Out (FIFO) principle and the point queue concept in which a vehicle has no physical length: it is assumed that the arrival flow at link (ij) leaves the link after the free flow travel time m:j if there exists no queue on the link, otherwise it leaves the link by the maximum departure rate (capacity) ^. Concerning the assignment principle, we assume the dynamic user equilibrium (DUE)
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assignment, which is a natural extension of the static user equilibrium assignment; the DUE is defined as the state where no user can reduce his/her travel time by changing his/her route unilaterally for an arbitrary time period.
2.3. Decomposition Property of Dynamic Equilibrium Assignment Under the DUE state, the users who depart their origin at the same time, regardless of their routes, have the same arrival time at any node that is commonly passed through on the way to their destination. Furthermore, under the DUE state, the order of departure from the origin must be kept at any node through destinations. From these property, we can define the unique equilibrium arrival time at each node for each departure time from the origin. As defined in the previous section, link travel time ctj(f) depends only on the vehicles which arrived at the link before time /. Therefore, together with the above discussion on the order of arrivals at a node, it is concluded that the travel time experienced by the vehicle that departs from an origin at time s is independent of the flows of the vehicles that depart from the origin after time s. Consequently, we can consider the assignment sequentially in the order of departure from the single origin. That is, the assignment can be decomposed with respect to the departure time from the single origin provided that the OD pattern is one-to-many. Similarly, for a many-to-one OD pattern, we can easily conclude that the assignment can be decomposed with respect to the arrival time at the single destination. For the detailed discussions on this property, see Kuwahara and Akamatsu (1993) and Akamatsu and Kuwahara (1994).
3. EQUILIBRIUM FLOW PATTERNS ON SATURATED NETWORKS - FIXED DEMAND CASE In general, the DUE assignment is formulated as a non-linear complementarity problem (NCP) or a variational inequality problem (VIP), which implies that it is difficult to obtain the analytical properties of the assignment. Hence, instead of exploring the properties of the DUE assignment under general settings, we confine our analysis to "saturated networks" where we can obtain the analytical solution. The "saturated networks" are the networks satisfying the following two conditions: a) there exist inflows on all links over the network, b) there exist queues on all links over the network. The first condition a) is not very restrictive, since we can constitute the networks satisfying this condition after knowing the set of links with positive flows. Although the second condition b) may not be satisfied in many cases, we nevertheless employ this assumption because this assumption, as shown below, gives us the explicit formula for the solution of the DUE assignment, which enables us to understand the qualitative properties of interest.
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We will first show the formulation for E-net and derive the solution in 3.1; and then the formulation and the solution for M-net will be examined in 3.2.
3.1. Equilibrium on Saturated Networks with a One-to-Many Pattern (1) Formulation The DUE assignment on a network with a one-to-many OD pattern can be decomposed with respect to the origin departure-time as mentioned in chapter 2. Hence, once we know the method of solving the equilibrium pattern for one particular departure-time, we can obtain the equilibrium pattern for whole time periods by successively applying the same procedure at the order of the departure-time. In the following, we consider the problem of obtaining the equilibrium pattern for vehicles departing from origin o at time s, assuming that the solutions for vehicles departing before time s are already given. In the decomposed formulation with origin departure time s, two kinds of variables, ( y]} ,T' ), play a central roll: r,s is the earliest arrival time at node i for a vehicle departing from origin o at time s; y", is the link flow rate with respect to s, that is, y*l = dFtJ (T- ) / ds , where FtJ(f) denote the cumulative number of vehicles entered into link ij at time t. In addition, we denote the number of vehicles with destination d departing from origin o until time s (cumulative OD demand by departure-time) by Qoli (s) . In the DUE state, each user choose his/her route whose travel time is (ex post) minimum over the network.
In other words, the links with positive inflows should be on the minimum path tree.
In our saturated networks, all the links have positive inflows, and therefore the minimum path condition for users with origin departure-time s is written as c(s) + A 7 T = 0 , where c(s) is an L dimensional column vector with elements c* =c r:/ (r; v ), r(s) is an (N-l) dimensional column vector with elements r,v .
Since the equation above should hold for any s, taking the derivative
with respect to 5, we have
f£) + A ' * £ » = 0 as as
V,,
(3.1)
where dc(s)/ds is an L dimensional column vector with elements dcs(l I ds , and ar(s)/ ds is an N-l dimensional column vector with elements dr* I ds . In our link model, the point queue and the FIFO principle are assumed, and therefore, the rate of change in link travel time is given by dc (t) dt
\(^,j (0 / /^ ) ~ 1 if there is a queue 0
otherwise
where /i y (/) is the standard link flow rate defined as dFtJ(t)ldt. Hence, in our saturated
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Theory
networks where all links have queues, the rate of change in the time needed to traverse link if for users with origin departure time s, dc^ I ds , can be represented as:
dr*
ds
_
ds
j ds
Jly
I
Noticing here the definitional relationship y*'. = A, (r/ ) • dr* I ds
we see that the dc^lds
reduces to a function of y*. and r/:
ds
fitj
(3.2a)
ds
or equivalently ds
= M~'y(s)-A+
\fs.
ds
(3.2b)
where M is a diagonal matrix whose oth diagonal element represents the maximum capacity of link a, y (s) is an L dimensional column vector with elements yy. Substituting (3.2) into (3.1), we obtain A-( ~\
= 0,
Vs
(3.3)
and rearranging this yields Vs.
ds
(3.4)
On the other hand, in the decomposed DUE formulation, the flow constraints that consist of the FIFO condition for each link and the flow conservation at each node over a network reduce to the following equations (for the detail, see Kuwahara and Akamatsu (1993), Akamatsu and Kuwahara(1994)): -Ay(5)-^ = 0 ds
V,.
(3.5)
where dQ(s)/ds is defined as an (N-l) dimensional vector with elements dQod(s)lds (given). Combining (3.5) with (3.4), 7 (AMA )^L^1 V
-' ds
ds
V,.
(3.6)
Thus, we see that the DUE assignment has a unique solution (dt(s)/ds) if the rank of the matrix AMA r is N-l. (2) Solution The rank of the matrix AMA^ generally depends on the choice of a reference node. For a
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307
network with a one-to-many OD, the rank of AM A [can be less than N-l when we choose an arbitrary node that is not an origin as the reference node. The rank, however, is always N-l when an origin is employed as the reference node. Furthermore, since the value of drt (s) I ds for an origin node is always 1 from the definition of rt ( s ) , it is natural to choose an origin as the reference node. Thus, by setting an origin as the reference node, we obtain the equilibrium solution, dr(s)/ ds, by the following formula:
ds
v
~'
(3-7)
ds
In addition, we can obtain the equilibrium link flow pattern, y(s), by substituting (3.7) into (3.4).
3.2. Equilibrium on Saturated Networks with a Many-to-One Pattern (1) Formulation The DUE assignment on a network with a many-to-one OD pattern can be decomposed with respect to the destination arrival-time as shown in chapter 2. In the following, we consider the problem of obtaining the equilibrium pattern for vehicles arriving at a destination at time u, assuming that the solutions for vehicles arriving before time u are already given. For the networks with a many-to-one OD pattern, by decomposing with respect to the arrival time at a single destination, the discussions almost parallels to those in the previous section, hi the decomposed formulation with destination arrival time u, two kinds of variables, (>"//» r , ), play a central roll: T" is the latest arrival time at node / for a vehicle reaching destination d at time u; yl is the link flow rate with respect to u, that is, y>l = dFl} (T" }l du. hi addition, we denote the number of vehicles with origin o arriving at destination d until time u (cumulative OD demand by arrival-time} by Q^ (u). The formulation almost parallels the discussions in 3.1. First, the minimum path conditions for saturated networks reduces to the following conditions:
du
Vw.
du
(3.8)
Then the link travel time with a point queue for saturated networks also should satisfy
du
(3.9)
du
Substituting (3 .9) into (3.8), we obtain ^M
v«.
(3.10)
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On the other hands , the link flow y should satisfy the flow constraints: Ay(w)
du
=0
(3.11)
Vw,
Combining (3.10) with (3.11), we reach
-
du
.
du
Thus, we see that the DUE assignment has a unique solution (dt(u)l du and y(u)) if the rank of AMA r isN-1. (2) Solution An arbitrary network with a many-to-one OD pattern can be obtained by reversing the direction of all links and origin/destinations of a network with a one-to-many OD pattern. Therefore, it is natural to expect that, "reversing" the result in 3.1, the rank of AMA^ become N-l when a destination is chosen as the reference node. However, it is not the case for this problem; the rank become less than N-l even if-we set the destination as the reference node; furthermore, we can prove that the rank is less than N-l for any choice of the reference node. The reason why the rank of the matrix AMA[ becomes less than N-l is that there exist particular origins (we call this "pure origins") that are not traversal nodes (i.e. the origin which has no links arriving at the origin). Letting B, be the (/'/) element of A*MAl , we easily see that 'f
(3.13)
Hence, the column vectors of AMA[ corresponding to the pure origin are always zero, and the rank of AMAT_ necessarily decreases by the number of pure origins. To see this fact more precisely, we divide the node set N into two sub-sets: the set of pure origins, N,, and the set of the other nodes, N2. Then, we divide A*, A*_, d r (u)ldu and dQ(u)/du into the two blocks corresponding to N, and N 2 , respectively:
A =
A =
A,
dt(u] du
"<&,(")" du dr2(u) du
dQ(u) du
~dQ}(u)~ du dQ2(u) du
where i th element of dQ2(u) /du is defined as - ^ {dQad (u) I du} = -^ jukd if / is an orign, dQij(u)/du if / is a destination, zero otherwise. Note that A, _, which is the first block of A_ corresponding to N,, is always 0 according to the definition of the pure origins. Rewriting (3.12) with these partitioned variables, we have
A Paradox For A Dynamic Equilibrium Assignment
du dQ2(u)
= A'MA
.r dr(u) du
du
dr2(u)
du
309
(3.14)
du
That is, du
(3.15a)
du
dr2(u) T *?2(") = -A,MA'_
du
du
(3.15b)
This means that no condition which determines the dr} / du for the pure origins is included in the equilibrium condition (3.12), while the dr2 / dufor the traversal nodes can be obtained by
du
du
(3.16)
Thus we see that the solution of the DUE assignment with a many-to-one OD pattern can not be unique and that for the problem to have a unique solution we should add appropriate conditions to resolve the indeterminacy of the dt\ I du .
4. EQUILIBRIUM FLOW PATTERNS ON SATURATED NETWORKS - ELASTIC DEMAND CASE The previous chapter analyzed the solution of the DUE assignment where only user's route choice is endogenously described given time-varying OD demands.
This chapter extends the analyses to the
case where the time-dependent OD demands are endogenously determined (we call the model "DUE assignment with Elastic demand") by incorporating the user's departure time choice. The model employed here is the simplest one that consistently unifies the two kind of dynamic equilibrium models: the dynamic equilibrium assignment presented in the previous chapter and the dynamic equilibrium model of departure time choice as is well known since Vickrey (1969) or Hendrikson and Kocur (1980). For expositional brevity, the following assumptions are made in this paper: 1) The users with the same OD pair are homogeneous, that is, their utility functions are all the same and their desired arrival time is unique; 2) The users who arrive later than the desired arrival time do not exist [This is not a restrictive assumption but one just to make the exposition as simple as possible; it is easy to extend to the case where late arrival is permitted.]. 3a) For the problems with one-to-many OD pattern (i.e. when we consider the problem on the basis of the origin departure-time), the disutility function for the users with destination d leaving origin at time s,
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VJs), is given as the linear combination of their travel time from the origin to destination d and their "schedule delay":
Vod (s) = a} {rd (5) - 5} + a2 {td - Td (5)},
(4.1)
where a,, a2 are positive parameters that satisfy al > a2, Td(s) is the destination arrival-time for the users who start from origin at time s, and td is the users' desired arrival time. 3b) For the problems with many-to-one OD pattern (i.e. when we consider the problem on the basis of the destination arrival-time), the disutility function for the users with origin o arriving at the destination at time u, Vj(u), is given as the linear combination of their travel time from origin o to the destination and their "schedule delay":
Vod («) = a} {u -
TO
(u)} + a2 {td - u],
(4.2)
where TO(U) is the origin departure-time for the users who arrive at destination at time u. 4) The networks can be regarded as "saturated networks" that is defined in the previous chapter.
4.1. Equilibrium on Saturated Networks with a One-to-Many Pattern (1) Formulation In this section we consider the networks with a one-to-many OD pattern where all nodes except the origin are destination, i.e., there are no nodes that are neither origin nor destination. [This is simply for the convenience of expositional brevity. The appropriate division of the node set easily extends our analyses to the general case where there are some nodes that are neither origin nor destination. See Appendix.] The elastic demand DUE employed in this chapter is defined as the state where no one can improve his/her utility by changing either his/her route or their departure-time unilaterally. To formulate this, consider users who choose time s as departure time. Since the users choose their optimal route (conditional on the optimal departure time) in the DUE state, the equilibrium conditions for the route choice should be represented by the following differential equations as shown in Chapter 3:
ds
(4.3)
ds
where the origin node is selected as a reference node as discussed in 3.1 . Then, the condition that no user can improve his utility by changing his/her departure-time in the DUE state can be represented by
\ f s , Md.
ds
(4.4)
Substituting the definition of disutility function (4.1) into this, we obtain the equilibrium rate of change in the destination arrival-time as follows:
ds
=
_ a, -flu
Vj>
Md
(4.5)
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[We are assuming that networks can be regarded as "saturated networks" and all OD pairs have positive OD flows during the period of time. In general we should consider the analysis period to include the time where some OD pairs have no generation of OD flows. By introducing appropriate classification, however, the general case can be reduced to the combination of our basic case (the case where all OD pairs have positive OD flows during the period for our analysis) and the case presented in Appendix.]. Thus, the elastic DUE conditions are represented as the following system of differential equations:
ds
=E
dQ(s)_ ds "
a
\ a, -a
(4.6a)
(4.6b)
-' ds
where E is an (N-l) dimensional column vector whose elements are all equal to 1. It is worthwhile to compare the equilibrium conditions with those for the fixed demand case. In the fixed demand DUE model, eq.(4.3) with a given constant vector dQ(syds determines dr(s)/ds . On the contrast, in the elastic demand DUE, dr(s}/ds is first determined from the departure-time equilibrium condition, and then eq.(4.3) with fixed dr(s)lds determines dQ(s)/ds. (2) Solution By setting appropriate boundary conditions, we can obtain the solution (t(s),QCs)) for the differential equation (4.6). For the boundary conditions, we first set the initial time Ss of the time period (measured with respect to the origin departure-time) during which eq.(4.6) holds (i.e. the networks can be regarded as "saturated networks" and all OD pairs have positive OD flows). Then we give the value of cumulative OD flows for the time ss and for the final time of the period: Vd
(4.7a)
Vd
(4.7b)
where s(td) is an origin departure-time of the final users who arrive at destination d at time td (note that we do not have to give the value of s(td} explicitly). Integrating the second equation of (4.6) from time ss to s with the initial condition (4.7a), we have
a, - a2 where Q is an (N-l) dimensional vector with elements g .
*,),
(4-8)
We then solve (4.6) with respect tot. Integrating the first equation of (4.6) from time Ss to time s(td } reduces to t-t(lI) = -^— M/J-EfJ
W.
(4.9)
where t, t(.sv ), and s^ are (N-l) dimensional vectors with elements td, rd (ss), and s(t^, respectively.
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312
The length of the time period that appears in the right hand side of (4.9), s(td )-ss , can be obtained by substituting (4.7b) into (4.8): -'(Q-Q) ~ .
(4.10)
Hence, from (4. 1 0) and (4.9), we can determine the initial equilibrium arrival time corresponding to ss : T(s 5 ) = t-(AMA[)-'(Q-Q)
(4.11)
Thus, the equilibrium pattern (-t(s), Q(s)) with the boundary condition (4.7) is given by
Q(.?)=Q+AMA[E
a
'
a, -a.
Vs
(4.12)
Vs .
(4.13)
(s-ss)
and the corresponding equilibrium disutility Td (s) is calculated by p = (t - Es, ) a, + ( AMAT_ )"' (JQ - Q
-a2)
4.2. Equilibrium on Saturated Networks with a Many-to-One Pattern (1) Formulation In the following we consider the networks with a many-to-one OD pattern where all nodes except the destination are origins, i.e., there is no node that is neither origin nor destination. For the general case where there are some nodes that are neither origin nor destination, see Appendix. We divide the node set N into two sub sets: the set of origins N,, and the set of the single destination, N2.
Then, we divide A*, A*_, dr(u)/du
and dQ(u)/du into the two blocks
corresponding to N, and N2 , respectively:
*.(«)
du
du
1
dQ(u) _ '*?.(")' <& du
(4.14)
where A, is an (N—1)XL matrix, A 2 is an L dimensional column vector, dQ,(u)/du is an N-l dimensional column vector with elements dQtxl(u)ldu, and ^d = ^^tj. ijeLj
The elastic demand DUE employed here is defined as the state where no one can improve his/her utility by changing either his/her route or their departure/arrival-time unilaterally. Since the users choose their optimal route (conditional on having chosen his/her optimal departure/arrival-time) in the DUE state, the equilibrium conditions for the route choice should be represented by the following differential equations as shown in Chapter 3:
A Paradox For A Dynamic Equilibrium Assignment
v
du
313
' du
Rewriting this with the variables introduced in (4.14), we have
v
du
"' du
'
_ (AIMA[_)
(4.16)
The condition that all the users can not improve their utility by changing his/her arrival-time (or departuretime) in the DUE state can be represented as dV
°d\u> = Q
VM, Vo.
(4.17)
Substituting the definition of disutility function (4.2) into this, we obtain the equilibrium rate of change in the destination arrival-time as follows: -
*!>) = £LI£L du a}
VM, Vo
(4.18)
Thus, the elastic DUE conditions are represented as the following system of differential equations: du
a,
(4.19) du
It is worthwhile to compare the equilibrium conditions with those for the fixed demand case. In the fixed demand DUE model, we tried to determine dr(u) I du from the eq.(4.15) with a given constant vector dQ(u)/du.
Then we encountered the indeterminacy of dr(u) I du due to the decrease in the rank
of matrix A*MA*-. On the contrast, in the elastic demand DUE, the indeterminacy problem is resolved since dt(u)l du is first determined from the departure-time equilibrium condition, and then eq.(4.16) with fixed dt(u)l du determines dQ(u)/du. (2) Solution As in the case of one-to-many OD pattern, we can obtain the solution (i(s), Q(s)) for the differential equation (4.19) by giving appropriate boundary conditions. For the boundary conditions, we first set the initial time us of the time period (measured with respect to the destination arrival-time) during which eq.(4.19) holds (i.e. the networks can be regarded as "saturated networks" and all OD pairs have positive OD flows). Then, on a parallel with the discussion in 4.1, it is natural to give the value of cumulative OD flows from us and for the final time td : Qod («,) = Qod = S'ven iven
Qod (*u } = Qod = g
Vo Vo
-
(4.20a)
»
(4.20b)
The conditions (4.20) in conjunction with (4.19) can be solved with respect to Q(w). However, these
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314
Theory
conditions are not enough to determine the value of T . Hence, instead of (4.20a), we give the time needed to travel from origin o to the destination at the initial time us as a new boundary condition: us - TO (us ) = rod = given Vo . (4.20c) Integrating the second equation of (4. 19) from time u to td with the initial condition (4.20c), we have -«)
VM
(4.21)
We next solve (4.19) with respect tot . Integrating the first equation of (4.19) from time us to time u with the initial condition (4.20c), we obtain ~ - ^ ) a,
VM.
(4.22)
and the corresponding equilibrium disutility rd (s) is calculated by p = fl2-(/rf-MJ)E +
fl1r
(4.23)
5. PARADOXES Having derived the formulae for the solution of the dynamic traffic equilibrium assignment so far, now we can discuss the capacity increasing paradox. The paradox presented here is a situation such that improving the capacity of a certain link on a network worsen the total travel cost over the network; this is a dynamic version of Braess's paradox which is well known in the static assignment. Using the results obtained in Chapters 3 and 4, we derive the necessary conditions for the occurrence of the paradox for E-net and M-net, which are shown to be significantly different.
5.1. A Paradox for a Network with a One-to-Many OD Pattern We consider the paradox for the network shown in Fig. 5.1, where node 1 is a unique origin; nodes 2 and 3 are destinations; the maximum departure rate of link a (a = 1,2,3) is given by //„.
Fig.5.1. Example Network with Single Origin and Two Destinations
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315
For the brevity of notation, we employ the superscript" • " as the derivative operation with respect to origin departure-time s in this section, (e.g. t,(s) = dr^lds, Q^s) = dQod(s)lds}. (1) Fixed Demand Case For the network in Fig. 5.1, the origin (i.e. node 1) should be the reference node; the incidence matrix A*, the reduced incidence matrix A, and the corresponding A_ are given as follows:
1
1 0
A = -1
0
0
1
A=
-1 -1
-1
0
1
0
-1
-1
A =
- 1 0 0 0 -1 -1
(5-D
Hence,
*>
(5.2) 0
The equilibrium pattern for the vehicles with the departure time 5 from a single origin can be calculated using the results of Chapter 3.
From (3.6), we first obtain the rate of change in
equilibrium arrival time: fiuW.
T3(S) = —^— 013 (j) /" 2 +y" 3
(5.3)
Substituting these into (3.3), we have the following equilibrium link flow pattern:
(5.4) To discuss the "capacity increasing paradox", we employ the total travel time for the users departing from an origin from time 0 to T as an indicator for measuring the efficiency of the network flow pattern:
s
X f y*
=Z f Q
(5.5)
We then refer to the situation "paradox" if increasing the capacity of a certain link, //0, causes the increase of TC (i.e. dTC/d/na > 0 implies "paradox"). Let us examine whether the paradox arises or not for the network in Fig. 4.1. (5.3) into (5.5), we obtain TC:
Substituting
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(5.6) From (5.6), we easily see that the increase of //, or ju2 always decreases TC (note that both //, and ju2 appear in only the denominator of TC), that is, the paradox does not arise for links 1 and 2. Increasing ju3, however, causes the paradox. The reason is that since
if the condition: '° ' 2 ^
" > -*0
'3 ^
"
S
*
(5.8)
is always positive , this means the occurrence of the paradox. The (5.8) is the condition that the paradox occurs for a certain time period 0 ^ T. From this, we can also derive the condition under which the paradox occurs for an arbitrary time period: QM' *>&&)!
to-
(5-9)
The meaning of this inequality is simple. Since the increase of /J3 always results in the increase of y3 (see (5.4)), suppose 1 unit of increase in flow on link 3 (=^3). This means that the number of users with destination 3 who pass through link 1 increases by 1 unit. The increase in flow on link 1 then causes Qn (s)l //, of increases in total travel time for the users with destination 2 ("User-2"). On the other hand, total travel time for the users with destination 3 ("User-3") decreases by Q}3 (s)/ ju2 , since the flow on link 2 decreases 1 unit. Therefore, the 1 unit of increase in flow on link 3 causes the increase of total travel time by Qn (5V P\ ~ Q\i (s}/ -"2 • Thus, we see that (5.9) means the condition that the "net benefit" for User-2 and User-3 (User-3's benefit minus User-2's loss) due to the increase of ju3 becomes positive. (2) Elastic Demand Case
The equilibrium pattern for the network in Fig. 5.1 can be calculated from the results of Chapter 4. From (4. 1 2), we first obtain the equilibrium arrival times and OD flows:
(5.10) at-a2
a,-0 2
where Qad = Qod - Q . Then (4. 1 3) gives the equilibrium disutility for each origin:
We define the sum of disutility experienced by all users over a network, TC, as an indicator for measuring the efficiency of the network usage: .
(5-13)
d
The TC for the network in Fig.5. 1 is given by TC = a,lt2 -SS]QU + fe -SM+h -a^-^-+ -^-j
(5.14)
To check the occurrence of the paradox, we calculate dTC/dju^:
*rc
0,2
0,3
-v 2
(5.15)
Note that the capacity of link 1 should be greater than that of link 3 (i.e. //, > /J3) in order for (5.11) to satisfy the (physically evident) condition Q}2 (s) - Qt 2 (s v ) > 0. Hence dTC/dju3 > 0 holds only if Qn /(//, -// 3 ) > jQ13 /(// 2 +^3) •
(5.16a)
We see from (5.16a) that the paradox arise (with the capacity increase of link 3) independent of the value of/^3 if the following condition hold:
It is noteworthy that the condition (5.1 6b) is identical in form to the condition for the fixed demand case.
5.2. A Paradox for a Network with a Many-to-One OD Pattern We consider the paradox for the network in Fig. 5. 2, where node 1 is a unique destination; nodes 2 and 3 are origins; the maximum departure rate of link a (a = 1 ,2,3) is given by jua. For the brevity of notation, we employ the superscript " • " as the derivative operation with respect to destination arrival time u in this section, (e.g. T^^dr^u}/ du ,Qod(u) = dQad(u)/du)
*•—'
Fig. 5.2. Example Network with Two Origins and Single Destination
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Theory
(1) Fixed Demand Case For the network in Fig. 5.2, node 3 is the pure origin; we divide the incidence matrix A*, the corresponding A*_ and the OD flow vector as follows: A,=[0
A, =
1
l ] (node3)
A,_=[0
-1
-1
0~\(node\),
1
0
-\\(node2)
-1
0
0
-1
0
(5.17)
e2,(«)
0 0 - 1
Hence, 0
~
A MA - =
0
(5.18)
//3 The equilibrium pattern for the vehicles with the arrival time u at a single destination can be calculated from the results in Chapter 3.
From (3.16), we first obtain the rate of change in
equilibrium arrival time for nodes 1 and 2: T2(u) = -
(5.19)
Substituting these into (3.10) yields the link flow rates (with respect to u): (5.20) Note that this flow pattern is significantly different from that for the reversed network (see (5.4)). hi order to determine the rate of change in equilibrium arrival time for node 3 (= the pure origin), adding an appropriate condition is required. Here we assume for node 3 that the OD flow rate measured at the origin,
031 (") = ^ 1 + ^ 2 - 021 («)
and qod :
Substituting this into the definitional relationship between qod(u)
du
dT0(u)
du
^
we obtain the rate of change in equilibrium arrival time at node 3:
~
f
\
q3](u)
(5.22)
Defining the total travel time for the users arriving at an destination from time 0 to T as an indicator for measuring the efficiency of the network flow pattern: TC
*
(5.23)
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319
let us examine whether the paradox arises or not in the network in Fig. 5.2. (5.21) and (5.22) into (5.23), we obtain the TC for this network:
Substituting (5.19),
(5.24) du
where Q^(u}= \ qod(u)du.
We see from this equation that the increase in /^, or ju2 will
always decrease TO, the paradox does not arise for links 1 and 2. However, the increase in the capacity of link 3 always results in the occurrence of the paradox. This fact can be easily examined as follows. Calculating the derivative of TC with respect to ju3, we have
Note that f2 (u) should be positive in the DUE state. The reason is that if f 2 («) is not positive the users with the destination arrival time if> u must depart from their origin before the users with arrival time u, and this contradict the assumption that the state is in the DUE. Therefore, from the (5.25) and the fact that f 2 («)>0 for any u, the inequality dTC I dju3 > 0 always holds; we see that the paradox for link 3 takes place without any additional conditions. (2) Elastic Demand Case
The equilibrium pattern for the network in Fig. 5.2 can be calculated from the results of Chapter 4. For the network in Fig.5 .2, the matrices A , M A \_ and A2 MA 2_ defined in 42 are
oi
T
(5.26)
Hence, from (4.21) and (4.22), we obtain the equilibrium arrival times and OD flows: / -, a^-a2 a2 L a, /, J , \ a,-a 2 a2 L a, /„L J ^(«) = -a-L" a+ — \usa -- Lr2l(u,)\, T3(u) =a ^-2a-u + -±\u s -- r 31 (wJ a \ \ ( 2 I \ \ [ 2 }
,
(5.27)
,
We also get the equilibrium disutility from (4.23): P2=a2(t-us)+a}r2}(us),
p3 = a2(t -us)+a^(us).
(5.29)
Let us define the sum of disutility experienced by all users over a network, TC, as an indicator for measuring the efficiency of the network usage:
320
Transportation and Traffic Theory TC^p0Qod
(530)
Substituting (5.29) into the definition (5.30), we get the TC for the network in Fig.5.2:
(531)
To check the occurrence of the paradox, we calculate dTC/d/u^: ^ = a}(a,-a2](t-us}{r3,(us)-r2l(us)}. a//3
(5.32)
Note that the relationship r*(u,)>r2l(«,)
(5-33)
7 2 («,)>^(«J
(5-34)
or equivalently, should holds as long as the network in Fig.5.2 is a saturated network. The reason can be proved by contradiction: consider two users with origin 2 and 3, denoted as U2 and U3, who arrive at the destination at the same time us; suppose that the (5.34) does not hold, then it implies that U2 should leave his origin earlier than U3 does; this clearly contradict the assumption of the saturated network. Thus, from (5.32) and (5.33), we see that dTC/dju3 > 0 always holds; in other words, the occurrence of the paradox is inevitable when the capacity of link 3 is increased. It is worth noting that we eventually obtained the same result as in the fixed demand case.
6. Concluding Remarks This paper discussed a capacity increasing paradox under a dynamic equilibrium assignment with elastic OD demands: the paradox is a situation such that improving the capacity of a certain link on a network worsen the total travel cost over the network. Our analysis in a simple network disclosed that the paradox arises only on a particular condition for a network with a one-to-many OD pattern, while the corresponding paradox always arises for the reversed network with a manyto-one OD pattern. This is the asymmetrical result that can not be seen in the classical static assignment framework; it is particular to the dynamic assignment with queue. Furthermore, we show that this property holds not only for the assignment with fixed OD demands but also for the assignment with elastic OD demands.
A Paradox For A Dynamic Equilibrium Assignment
3 21
In this paper, particular simple networks were employed to demonstrate the paradox. Note, however, that the examples presented here are not the exceptional ones that can hardly be observed in practical situations but the ones that can be seen universally if we regard the example networks as a macroscopic representation of real road networks. Therefore, we think that the examples, despite their simplicity, describe one of the essential points that should be considered in deciding practical traffic management operations such as ramp metering or addition of lanes in freeways. We recognize that there are still several relevant topics to be studied. First, we should extend our analysis to the paradox in a more complex network by exploiting the analytical formula of the DUE solution derived in this paper; it may be possible to obtain systematic methods for general networks that detect (without computing the equilibrium patterns) the links where the paradox takes place; the exploration of this possibility would be an interesting future topic. Secondly, we should analyze more realistic case where the assumption of "saturated networks" are relaxed; the exploration would be achieved by employing not only the analytical approach just as shown in this paper but also the numerical approach based on the recent convergent algorithms for the DUE assignment (see Akamatsu (1998)). Finally, we should explore the case where physical queues are explicitly incorporated into the analysis. Though the incorporation of physical queues may cause very complex phenomena as shown in Daganzo(1998), comprehensive studies on this topic would be indispensable for a clear understanding of the properties of dynamic network flows.
Acknowledgements The authors gratefully acknowledge stimulating discussions with Nozomu Takamatsu on the topic of this paper. Thanks are also due to Benjamin Heydecker and three anonymous referees for their helpful comments and suggestions.
REFERENCES Akamatsu T. (1996). The Theory of Dynamic Traffic Network Flows, Infrastructure Planning Review 13, 23-48. Akamatsu T. (1998). An Efficient Algorithm for Dynamic User Equilibrium Assignment for a Oneto-Many OD Pattern, submitted to Transportation Science. Akamatsu T. and M. Kuwahara (1994). Dynamic User Equilibrium Assignment on Over-saturated Road Networks for a One-to-Many / Many-to One OD Pattern, JSCE Journal of Infrastructure Planning and Management IV-23, 21-30. Arnott R., De Palma A., and R. Lindsey (1993). Properties of Dynamic Traffic Equilibrium Involving Bottlenecks, Including a Paradox and Metering, Transportation Science 27, 148-160. Bernstein D., T.L. Friesz, R.L. Tobin, and B.W. Wie (1993). A Variational Control Formulation
322
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of the Simultaneous Route and Departure Choice Equilibrium Assignment, Proc. of the 12th International Symposium on Transportation and Traffic flow Theory, 107126. Braess D. (1968). U her ein Paradox in der Verkehsplanung, Unternehmensforshung 12,258-268. Daganzo C.F. (1997). Fundamentals of Transportation and Traffic Operations, Elsevier Science, Oxford. Daganzo C.F. (1998). Queue Spillovers in Transportation Networks with a Route Choice, Transportation Science 32,1-11. Hendrikson C. and G. Kocur (1981). Schedule Delay and Departure Time Decisions in a Deterministic Model, Transportation Science 15, 62-11. Heydecker B. G and J.D. Addison (1996). An Exact Expression of Dynamic Traffic Equilibrium, In J.-B. Lesort (Ed.) Proc. of the 13th International Symposium on Transportation and Traffic Theory, 359-383. Kuwahara M. (1990). Equilibrium Queueing Patterns at a Two-Tandem Bottleneck during the Morning Peak, Transportation Science 24,217-229. Kuwahara M. and T. Akamatsu (1993). Dynamic Equilibrium Assignment with Queues for a One-to-Many OD Pattern. In C. Daganzo (Ed.) Proc. of the 12th International Symposium on Transportation and Traffic Theory, 185-204. Kuwahara M., and GF. Newell (1987). Queue Evolution on Freeways Leading to a Single Core City during the Morning Peak. Proc.of the 10th International Symposium on Transportation and Traffic Theory, 21-40. Murchland J.D. (1970). Braess's Paradox of Traffic Flow. Transportation Research 4, pp.391-394. Smith M.J. (1978). In a Road Network, Increasing Delay Locally Can Reduce Delay Globally. Transportation Research 12B, 419-422. Smith M.J. (1993). A New Dynamic Traffic Model and the Existence and Calculation of Dynamic User Equilibria on Congested Capacity-constrained Road Networks. Transportation Research 27B, 49-63. Vickrey, W.S. (1969). Congestion Theory and Transportation Investment. American Economic Review 59. Yang H. and M.GH. Bell (1998). A Capacity Paradox in Network Design and how to Avoid it. Transportation Research 32A, 539-545.
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Appendix In Chapter 4 it is assumed that all the OD pairs have always positive flows during the period of analysis. In this appendix, we briefly demonstrate how the formulation can be extended to the case where some OD pairs have no OD flows. The formulations for the one-tomany OD problem and the many-to-one are presented in turn. (1) One-to-Many OD pattern We first divide the node set N (where the origin is excluded as a reference node) into two sub sets: the set of destinations with positive OD flows, N,, and the set of the other destinations, N2. Then, we divide A, A_, dn(s)lds and dQ(s)/ds into the two blocks corresponding to N, and N 2 , respectively:
~dTi(sY A=
A =
ctr(s v /}
A,.
ds
as/
dT2(S)
~dQi(s)~ as dQ2(s) ds
dO(s] ' ds
ds
r
/ xn
^i V / ds
o
(A-l)
For the destinations with positive OD flows, the arrival times are governed by the departuretime equilibrium condition (4.5):
ds
(A-2)
=E
For the other destination nodes, the arrival times should be determined from the route choice equilibrium condition (3.6). Rewriting the condition (3.6) with the variables defined above,
ds 0
(A-3)
dr2(s)
ds
or equivalently,
ds 0 =
ds
ds
ds
ds
(A-4a)
(A-4b)
Thus, the elastic DUE conditions are represented as the following system of differential equations: = 7
«5
dt7(s) ft , . . r V i / ' . , , A r V , #1 = -(A —, 2 MA 2 _) (A 2 MA,_jE as a, -a-,
Ea, — a2
J
\
^
^
~
/
\
'
i
l
(A-5) r
ds
r
r
= fA,MA, _)- (A,MA2 _ \A2MA2_)"' (A2MA, _
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324
(2) Many-to-One OD pattern We first divide the node set N into two sub sets: the set of origins with positive OD flows, N15 and the set of the other nodes (including the destination), N2.
Then, we divide A, A_, dr(u)/du
and dQ(u)/du into the two blocks corresponding to N] and N 2 , respectively: ~drt(u}~ dQ(u) du dr2(u) ' du du
dr(u) du
A
A-| '.A = A, " A ,
fdQiMl du dQ2(u) du
(A-6)
where dQi(u)/du is a column vector with element dQ0j(u)/du, dQ2/du is a column vector whose element is nd = ^ jukd if it corresponding to the destination, otherwise zero. k
For the origins with positive OD flows, the departure times are governed by the departure-time equilibrium condition (4.18): s-hf
If/ I
du
:-!—2sy
/-f
(A-?)
a}
For the other destination nodes, the arrival/departure times should be determined from the route choice equilibrium condition (3.12). Rewriting the condition (3.12) with the variables defined above,
fdQiMl
[
du dQ2(u) du
"rfE,(«)~
A, 1 r T r ' A;JM[A[. AL.
du dr2(u) du
(A-8)
or equivalently,
du
(A-9a)
du
du
(A-9b)
v
Thus, the elastic DUE conditions are represented as the following system of differential equations: du
a,
du
a,
du
(A-10)
du du
CHAPTER 5 TRAFFIC ASSIGNMENT
Making the simple complicated is commonplace; making the complicated simple, awesomely simple, that's creativity. Everything should be made as simple as possible, but not simpler. (Albert Einstein) To get nowhere, follow the crowd.
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A Cell Based Dynamic Traffic Assignment Formulation
327
A Dynamic Traffic Assignment Formulation that Encapsulates the Cell-Transmission Model Hong K. Lo Department of Civil Engineering, Hong Kong University of Science and Technology Clear Water Bay, Hong Kong. E-mail: [email protected]
ABSTRACT This study develops an analytical dynamic traffic assignment (DTA) formulation based on a dynamic extension of Wardrop's Principle, referred to as dynamic user optimal (DUO) (Ran and Boyce, 1996). We develop a gap function for the corresponding nonlinear complementarity prolem (NCP) and prove that minimizing the gap function produces a solution that fulfills the ideal DUO conditions. Existing analytical DTA formulations mostly use macroscopic link travel time functions to model traffic. In general it is difficult for such functions to capture traffic interactions across multiple links such as queue spill-back and dynamic traffic phenomena such as shock-wave. Instead, traffic in this formulation is modeled after the CellTransmission Model (CTM) (Daganzo, 1994, 1995a). CTM provides a convergent approximation to the Lighthill and Whitham (1955) and Richards (1956) (LWR) model and covers the full range of the fundamental diagram. This study transforms CTM in its entirely to a set of mixed-integer constraints. The significance of this is that it opens up CTM to a wide range of dynamic traffic optimization problems, such as the DUO formulation developed herein, dynamic signal control, and possibly other applications.
1.
INTRODUCTION
Existing Dynamic Traffic Assignment (DTA) models basically follow two approaches: simulation and mathematical formulation. The first approach emphasizes microscopic traffic flow characteristics. Strict adherence to traffic assignment principles, such as Wardrop's, is secondary. Earlier generations of this approach used intersection-turning ratios to split traffic without route specification. Recent models specify route choices based on the k-shortest paths criteria, which was then extended to the concept of "bounded rationality" for dynamic route switching (Chang and Mahmassani, 1988). This approach shares the following properties. Firstly, they are essentially descriptive, not prescriptive tools. They simulate the probable results of a certain traffic management strategy but do not prescribe what the strategy ought to be. Secondly, they lack well-defined solution properties. One cannot prove whether the solution
328
Transportation and Traffic
Theory
has achieved the required optimality. In each computer simulation, the model produces a realization out of a large space of probable realizations. Therefore, one must be careful in generalizing or transferring results. DTA models can also be developed through an analytical approach (examples: Ran and Boyce, 1996; Ran et al. 1996; Lo, et al., 1996a; Jayakrishnan et al., 1995; Janson, 1991; Friesz et al., 1993; Smith, 1993; Wie et al., 1990). This approach often has well-defined properties, in terms of optimality conditions and adherence to a dynamic version of Wardrop's principle (1952). Depending on how the objective functions are defined, these models may be used for prescriptive or descriptive purposes. The main difficulty with the analytical approach is adding realistic traffic dynamics to already complicated formulations. For this reason, most DTA formulations use macroscopic link travel time functions (e.g., variations of the BPR function) to approximate traffic dynamics. This lack of accurate traffic dynamics is a shortcoming. Daganzo (1995b) pointed to the potential problem of link travel time functions under dynamic loads. Our studies (Lo et al., 1996b, 1996c) confirmed that traffic dynamics is too important to be replaced or simplified by a macroscopic link travel time function. This study develops a DTA formulation to overcome this shortcoming. The formulation has well-defined solution properties and follows a dynamic extension of Wardrop's Principle, referred to as dynamic user optimal (DUO) (Ran and Boyce, 1996). It models traffic dynamics by encapsulating the network version of the Cell-Transmission Model (CTM) (Daganzo, 1994, 1995a). CTM provides a convergent numerical approximation to the Lighthill and Whitham (1955) and Richards (1956) (LWR) model and covers the full range of the fundamental diagram. This property makes it a suitable platform for modeling dynamic traffic. However, incorporating CTM as part of the constraints of a mathematical formulation is not straightforward, as we will see in Section 3. Generally, traffic assignment formulations use four approaches: (i) mathematical programming (MP) (example, Sheffi, 1985); (ii) variational inequality (VI) (examples, Dafermos, 1980; Nagurney, 1993; extensions to dynamic traffic: Ran and Boyce, 1996; Friesz et al. 1993); (iii) nonlinear complementarity problem (NCP) (Aashtiani, 1979); and (iv) fixed-point problem (FPP) (Asmuth, 1978). Lin and Lo (1998) discussed a potential problem of extending the MP approach for dynamic traffic assignment. Two summaries of these approaches are annotated by Patriksson (1994) and Florian and Hearn (1995). They showed the linkages and equivalence conditions among these approaches. One common requirement for all formulations is the demand constraint (i.e., the sum of path flows per origin-destination (OD) pair equals the given OD demand or follows the demand function). For this reason, the path flow variables are included in most formulations as part of the constraint set1. In problems where the path costs are additive of the link costs, one can use the link-path incidence matrix to express the path cost 1 One can avoid the use of the path flow variables totally by defining the link flow variables by origin or by destination and maintaining multiple copies of the node and link flow conservation constraints to form a "linkbased" multi-commodity flow problem. See LeBlanc et al. (1975) for example.
A Cell Based Dynamic Traffic Assignment Formulation
329
as the sum of link costs on the path. Hence, one can avoid the path flow variables in the objective function. Subsequently, by using a column generation method in the solution, as in most of the widely used algorithms, one can solve the problem without the need to store the path flows (see Patriksson, 1994, and Florian and Hearn, 1995, for a summary of these algorithms). This is a marked advantage, especially for large networks in which the number of paths is much more than the number of links. The outputs of these algorithms are mostly expressed in link flows, although path flows would also be available if the paths generated in each column generation procedure are stored. Recently, there is a resurgence in the interest of reformulating or solving the traffic assignment directly with path flows (examples: Jayakrishnan et al., 1994, Cascetta et al., 1997; Bell et al., 1997; Gabriel and Bernstein, 1997; Lo and Chen, 1998). This redirection of effort can be summarized by four motivations: (i) for some types of problems, obtaining the solution in path flows (but without exhaustive path enumeration) is faster (Chen and Jayakrishnan, 1998); (ii) where route preference2 is an important attribute or where turning restrictions are common hence restricting the route choices substantially (e.g, downtown), formulating and solving the problem with path flows is more convenient; (iii)where the path costs are nonadditive (i.e., the path cost is not the direct sum of link costs), it is difficult, if not impossible, to formulate and solve the problem with link flows alone (Gabriel and Bernstein, 1997; Lo and Chen, 1998). (iv)in assignment models that explicitly consider queuing, as in this study, one must track the paths of the spill-back queues, which may extend over multiple links. Moreover, path flow provides important information to model traffic at diverges and merges. For these reasons, we develop this DTA formulation based on the path-flow variables. The possible paths between each OD pair are either predefined or generated through a column generation procedure in the solution algorithm. In practice, the predefined paths could be based on travelers' preferences or interview results. The formulation will then equilibrate traffic flows according to the DUO principle. That is, all the used paths between the same OD pair will have equal travel time; while the unused ones will have equal or higher travel times. We will delineate this formulation and discuss its optimality conditions in Section 2.
2.
DYNAMIC TRAFFIC ASSIGNMENT FORMULATION
We consider a general transportation network with multiple origin-destination (OD) flows. The traffic network is represented by a set of cells and directed links. Road segments are represented by a series of cells that have physical length, while links are there merely to delineate connectivity between cells. As such, links have no physical length and cannot hold traffic. Traffic begins at an origin cell (denoted as r) and terminates at a destination cell 2
Such as route guidance in which users have specific route preferences , such as choosing a safe, scenic, or highway-based route whose distance or travel time may not be necessarily the shortest
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Transportation and Traffic Theory
(denoted as s). We consider the fixed time period [0, T] that is long enough to allow all vehicles to complete their trips. Generally, two dynamic user optimal (DUO) conditions have been defined in the literature (Ran and Boyce, 1996): Ideal and Instantaneous DUO. This paper focuses on the conditions of Ideal DUO, restated as: for each origin-destination pair at each instant of time, the actual travel times experienced by travelers departing at the same time are equal and minimal. Possible scenarios where this could happen include (i) in commuting traffic where the patterns were reproduced every day and hence commuters could modify their route choices until they could improve no further after many days of experiences; (ii) in the future when the techniques of traffic detection and prediction will become accurate. The Ideal DUO condition implies that a path p between OD pair r-s will not be used at time t if its actual travel time is longer than the shortest travel time between r-s. Conversely, any used path p at time t must have its travel time equal to the shortest travel time between r-s. Mathematically, they are:
VrstRS
(2) "
f > 0, u > 0
(3) (4)
The notations are the following: RS set of OD pairs for the whole network rs an OD pair, rs e RS Prs set of paths connecting OD pair rs p a path between an OD pair, p e P™ fp(t] path flow on p between OD pair rs departing at t f
set of {/p"(f)j with dimension n 1 = ^ P r e x n 3 where «3 is the discretized time rs
dimension Tl"(t) path cost (or disutility) on path p between OD pair rs for flows departing at t n
set of {77™ (f )} with dimension n, = ]T P"| x n3 rs
nrs(i) shortest travel cost (or disutility) between OD pair rs departing at t u
set of {^"(f)} with dimension n2 = RS xr^
qrs(t] demand between OD pair rs . It is a constant for the fixed demand case and could be a function of u and t for the elastic demand case, qrs:R? —» Rl+ q the set of {
A Cell Based Dynamic Traffic Assignment Formulation
331
Conditions (1) and (3) are the complementary slackness conditions for ideal DUO. Aashtiani (1979) observed that the user optimal conditions for static traffic assignment are equivalent to a nonlinear complementarity problem (NCP). Here we generalize this approach to the case of Ideal DUO, stated as: x-F(x) = 0 F(x)>0 x>0 by setting x = (f ,u)e R" where n = n, +n 2 and letting
F(x) =
(5)
efl".
(6)
Actually, (5) and (6) are just rewrites of (l)-(4) in NCP format. Recent advancements in the analysis of NCP and mathematical programs (MP) offer new insights into reconsidering this formulation. Furthermore, recasting this NCP formulation into a MP would make available a large number of solution techniques already developed for MP. This reformulation from NCP to MP primarily draws upon the use of a gap or merit function. Definition: Let Q be the set of solutions to the NCP formulation (5)-(6). A function G: R" —> R[ is a gap function for the NCP formulation if ii. G > 0
In essence, the gap function provides a measure of convergence of the NCP formulation at any point x . By minimizing G over x , a point in Q is obtained. That is: minG(x) s.t.xeV
where ¥ = (x>0,F(x) >0}.
(7)
Facchinei and Scares (1995) suggested three desirable properties to have when transforming a NCP formulation with a gap function, including: 1. G is smooth (or differentiable); 2. every stationary point of (7) is a global solution of the NCP formulation; 3. the level sets of L(a) = |xe /?":xe ^,F{\)< aj are bounded, where F(x) is an element of F(x) and a a finite constant These properties are important from a computational point of view. If they are satisfied, the MP can be solved efficiently by applying a number of optimization algorithms to (7). Recently, a new gap function has been proposed to solve NCP. This hinges on the key observation that this simple, two-variable, convex function (Fischer, 1992):
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332
(I)(a,b} = ja2+b2 -(a + b),
(8)
has this property:
= 0.
(9)
From ( 8), a gap function is proposed based on squaring 0(a,&) (Facchinei and Soares, 1995): (p(a,b) = 2(a,b). The gap function (0,0) = (0,0) b. cp(a,b) > 0 , for all (a,b) e R2 c. (p(a,b) = Qa a>0,b>0,ab = 0 This approach can be applied to reformulate the DTA problem in the form of NCP. Specifically, we define the gap function for the traffic equilibrium problem as:
(10) where xi,Fi are the corresponding elements in (5) and (6), and R1 is a gap function for these conditions. Proof: (i) This gap function satisfies the condition: G(x) = 0<=>xeQ, where x = (f ,u)e R" , f denotes the set of {/p"(0l >and u denotes the set of {#"(*)} . Necessity: Given the Ideal DUO conditions: f"(t\il"(t)-JCa(t)] = 0, frps(t) > Qfrj^(t)-Jf"(t) a = f"(t),b = ri"(t)-n;"(t). <
> 0 , let
The "only-if condition of property c. implies that
p(/ P "(0' 7p(0~ ™(0) 0 f° ^ fs&RS and p&Prs. Given the demand conditions: r
;r
=
b = ^fpS(t)-qrs(t). p
r
= 0,
7f"(f)>0,
The
"only-if
P
2lifpS(t)-qrs(t)>0, let a = n"(t) p condition of property c. implies
and that
= Q for all rseRS. Adding these two conditions for all J
A Cell Based Dynamic Traffic Assignment Formulation
^^
/
X
^
(
333
^
}
instances yield: G(\) = Y T. « /> « I P ) since each term is zero. Therefore, given the Ideal DUO conditions, the gap function is zero. Sufficiency: Here we prove that a zero gap function implies the Ideal DUO condition. According to properties b., (p(fpS(t),rir*(t)-n"(t)) Given
that
G(x) = 0,
9\ nrs(t\^,fp(t}~^rs(t]
each
> 0 and J n"(t),^f?(t)-q"(t) v p term
> 0.
) and
must be zero. Using the "if condition of property c., one obtain
the Ideal DUO and demand conditions. (ii) The gap function satisfies this second condition: G(x)>0. From property b., both
( rs(t\2.f ^ rs(t)-qrs(t}\ } are nonnegative. Therefore, the (p\n I P ) sum of all terms, G(x), is nonnegative. (p(frs(t),r]'"<(t)-nrs(t)\and
This completes the proof. Thus, the Ideal DUO condition can be achieved by minimizing the gap function (10).
3.
TRAFFIC FLOW MODEL
The above section describes an Ideal DUO formulation based on path flow and actual path travel time. Many dynamic link travel time functions could be used for this purpose. To capture detailed traffic dynamics, this formulation encapsulates the Cell-Transmission Model (CTM) (Daganzo, 1994, 1995a) as the underlying traffic flow model. Developed as a simulation tool, CTM uses nonlinear mathematical operations and logic statements. While their inclusion as part of a simulation code does not pose any problem, casting them as constraints in a mathematical program (MP) is problematic. In this paper, we develop a transformation to convert CTM to a set of mixed-integer constraints. The significance of this is that the transformed CTM (and hence, the underlying LWR model) can be used in a general optimization context, such as dynamic signal control, dynamic traffic assignment, or other optimization problems that include a traffic model. By encapsulating CTM in this ideal DUO formulation, we improve the accuracy of traffic dynamics modeling significantly. This conversion, however, increases the complexity of the formulation, as discussed in the following.
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Transportation and Traffic
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Cell Transmission Model (CTM): Basic Principles The Lighthill and Whitham (1955) and Richards(1956) (LWR) model can be stated by the following two conditions: ^ + — = 0 and ox at
= Y(k,x,t)
(11)
where yis the traffic flow; k is the density; jc and t, respectively, are the space and time variables, and Y is a function relating y and k. The first partial differential equation states the traffic flow conservation. The relation Y defines the Fundamental Diagram. Given a set of well-posed initial conditions, one can determine y and k at any (x,t) by solving (11). This model is sometimes referred to as the hydrodynamic or kinematic wave model of traffic flow. Lighthill and Whitham (1955) and Newell (1991) developed two different solution approaches to this model. Daganzo (1994, 1995a) simplified the solution scheme by adopting the following relationship between traffic flow, y, and density, k: y = min{vk,Q,w(kjam-k)}
(12)
where kjam,Q,V,Wdenote, respectively, jam density, inflow capacity (or maximum allowable inflow), free-flow speed, and the speed of the backward shock wave (or the backward propagation speed of disturbances in congested traffic), then the LWR equations for a single highway link are approximated by a set of difference equations. Essentially, (12) approximates the fundamental diagram by a piece-wise linear model as shown in Figure 1.
w Density Figure 1
The fundamental diagram used in CTM
By discretizing the road into homogenous sections (or cells) and time into intervals such that the cell length is equal to the distance traveled by free-flowing traffic in one time interval, then the LWR results are approximated by this set of recursive equations (Daganzo, 1994, 1995a): (13) AT'-"
(14)
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335
where the subscript y refers to cell j, andj+1 (j-1) represents the cell downstream (upstream) of j. The variables « y (f),V;(f),A^(f) denote the number of vehicles, the actual inflow, and the maximum number of vehicles (or holding capacity) that can be held in cell j at time /, respectively. The variables Qj(t),V,W follow the earlier definitions. It is important to differentiate between G/(0 an^ ^y(0 • ^ former is the inflow capacity while the latter is the actual inflow. Because (13)-(14) provide a numerical approximation to the LWR equations, all the traffic phenomena demonstrated in the LWR model are replicated in CTM. The key is to determine y}(t] from the minimization (14). Once this is accomplished, n}(t) can be determined recursively from the linear equation (13). However, including (14) as a constraint would make the resultant MP difficult to solve. To overcome this problem, we transform the minimization in (14) to a set of mixed-integer constraints, as discussed in the next section. Equations (13)-(14) provide the basic principle of modeling traffic flow on a series of straight cells. To apply this principle to a general network with multiple OD pairs, three extensions are required: (a) modeling merge and diverge junctions; (b) differentiating the OD specific traffic; (c) maintaining the first-in-first-out (FIFO) property. In the next section, we discuss how CTM is extended for network traffic and the corresponding transformations.
Cell Transmission Model: Network Traffic The extension of CTM for network traffic was addressed in Daganzo (1995a). To facilitate cross-reference, this paper adopts similar notations as in there. Moreover, to make the description succinct, this paper only covers the basic concepts needed for exposition. One may refer to that paper for detailed discussions. The focus here is to cast CTM as a constraint set in this ideal DUO formulation. As mentioned in Section 2, road segments are represented by a series of cells that have physical length, while links, without physical length, are there merely to delineate connectivity between cells. In general, five types of connectivity are needed to represent a network, including (Figure 2): 1. 2. 3. 4. 5.
Ordinary connection: where two cells are joined by a straight link Origin connection: where exogenous traffic enters the cell Destination connection: where traffic terminates at the cell Merge connection: where two cells merge into one Diverge connection: where one cell diverges into two
In Figure 2, Bk and Ek refer to the beginning and ending cell of each cell-group, while Ck in the merge and diverge connections refers to the complementary (third) cell. For the origin and
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Theory
destination connections, Bk and Ek are replaced by r and s, respectively. Links are denoted by k or ck dependent on whether the link has one end at a complementary cell Ck. More complicated network connections can be simplified through a combination of one or more of these five connections. In a general network with many OD pairs, the general principle of (13)-(14) still applies to the aggregate traffic flows between cells. However, to maintain the intended paths of traffic and the first-in-first-out (FIFO) property, traffic in each cell is disaggregated by path (p) and by arrival time at the cell ( T , modeled as a discrete time index). The path variable is used to direct traffic at merges and diverges. By tracking T , the FIFO property is maintained by ensuring that earlier arrivals (with a larger T) will leave sooner. K
Ordinary
Bk
Ek
Origin
r
Ek
Destination
Bk
s
Merge
Diverge Figure 2
Five Types of Network Connections Captured in CTM
Ordinary Connection For Bk and Ek on path p, denote nBk
pr(t],
"at p r(0' respectively, as the traffic in cells
Ordinary
Bk
K
Ek
Bk and Ek at time t on path p and entered the cells in the time interval immediately after the clock tick (t - T). Essentially, T represents the waiting time of that packet of traffic in the cell. Define yk
T
Equation (15) states that the new traffic enters Ek at T = l. Equation (16) states that the remaining traffic in Bk at time t+1 (determined by the difference between the traffic at t minus those leaving) increases their waiting time from T to T +1. If the minimum wait aBk (t] in cell Bk is known, then the flow on link k is given by3: 3
Since all the variables are time-dependent, to simplify notations, we drop "(t) " from all the variables hereafter
except in instances of ambiguity.
A Cell Based Dynamic Traffic Assignment Formulation
337
(17) if T<
where a flt + denotes the smallest integer equal to or greater than aBk. Equation (17) states that if the waiting time T of a certain traffic packet is longer than the larger integer part of the minimum wait, the whole packet advances; none of the packet leaves Bk if its waiting time is less than aBk; and lastly, for the packet with waiting time T = \aBk + , the fraction of (T - aBk) leaves. Note that the exit flow yk
If T>aBk+ or, equivalently, T — aBk>l, the middle term is nBkpr. If T =
or,
equivalently, 0 < T — a B A < l , the middle term is (^ — aBk)nBkpr. Lastly, if T < equivalently, T-aBk < 0 , the middle term is zero. This completes the verification.
or,
To incorporate (17) as part of the constraint set, the mid-value function (18) is converted to a set of mixed-integer constraints. This is achieved by the technique discussed in Appendix C. The resultant constraints are: L • v, + e < (T - aBk }nBkpr - nBk^ < U • (l - v,)
(19)
L-v2 + £<-(T-aBk)nSk^
(20)
L.v3+e
(21)
L-(v, + v3) < ykipiT -nBkpr < U -(v, + v 3 )
(22)
L.(2-v 1 -v 3 )
(23)
L-(v1+v2)p>r-(T-aBt)nB,ipT
(24)
L-(2-v 1 -v 2 )
(25) (26) (27)
where L,U are very large negative and positive constants, e is a very small positive constant, v,, v 2 , v3 are three binary variables. Refer to Appendix C for details. Without loss of generality, the disaggregate variables can be expressed in aggregate forms by summing through the path and arrival time indices, as the following:
Transportation and Traffic Theory
338
r
p
r
p
Daganzo (1995a) showed that the aggregate flow on link k, yk, observes:
yk=^Ly r k,P,r P
(29)
yk=min{SBk,REk},
(30)
SBk=min{QBk,nBk},and
(31)
REk = min{<2£* ,SEk[NEk -nEk]} .
(32)
where
The aggregate flow is the minimum between the effective capacity of the "sending" cell Bk, SBk, and that of the "receiving" cell Ek, REk. These two capacities are subject to the conditions (31) and (32), where QBk,QEk are the absolute flow capacities of the sending and receiving cells; NEk,nEkthe holding capacity and vehicle occupancy of cell Ek, respectively; therefore NEk~nEk the available space in the receiving cell, and SEk=W/V. In essence, these conditions follow from the basic principle of (14). The minimization equations (30)-(32) can each be converted to a set of linear constraints through the transformation in Appendix A. For brevity, we only show the equivalent constraints for (30): 0
Origin
A Cell Based Dynamic Traffic Assignment Formulation
339
network through cell r according to the path flow/^) 4 . Other than this change, the entire set of constraints and the associated equivalent mixed-integer constraints for the ordinary connection applies here. Vehicle occupancy in cell r is described by:
=/,"W
(36)
Destination Connection The destination connection is also similar Destination Bk s to the ordinary connection. The only difference is that a destination cell is modeled with an infinite storage capacity to act as a big parking lot. Traffic arrives at a destination cell is considered to have left the network. The only change to the set of constraints for the ordinary connection is that the holding capacity Ns is taken as infinity, where the subscript s refers to the destination cell. So instead of setting Rs =min{Qs,8s[Ns — «J}, we useRs = Qs as A^—»°°. Again all the conditions can be transformed to mixed-integer constraints through the techniques in Appendices A and B. Merge Connection Much of the principle of developing the merge connection follows from the ordinary connection. We have one set of conditions for the disaggregate traffic representation and one set for the aggregate case. Vehicle occupancies in these three cells are represented by:
Merge
6k (37)
(38) (39)
= «c*.,.t M - y«*,P,T (0
The disaggregate exit flows from cells Bk and Ck, respectively, are: >Wr=mid{« a j p T , (r-aCk)nCkpT,
OJ
(41)
Note that each of the sending cells has its own minimum wait, aBk,aa . The aggregate flows are:
4
The notation of the path flow f" follows from its earlier definition. The use of three indices (r, s, p) is
redundant because path p alone automatically defines the starting and ending cells (r and s) as well as all the cells along p. This is only for presentation clarity and can be simplified in coding.
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Theory
*,p,r
r
p
r
(42)
p
The effective capacities of the sending cells Bk and Ck and that of the receiving cell Ek are defined, respectively, by: SBk=min{QBk,nBk}, (44) SCk=min{QCk,nCk}, (45) R =m i n G , $ t f
-«
•
Up to this point, all the conditions follow from the ordinary connection. By the techniques in Appendices A and C, each of these operations can be converted to a set of mixed-integer constraints. In a merge connection, in addition, one must ensure that traffic exit from the two sending cells can be accommodated in the receiving cell. The basic idea is that if the receiving cell has sufficient capacity, all the traffic from Bk and Ck enters, hi the case of insufficient Ek capacity, traffic from Bk and Ck is apportioned according to the amount of traffic from Bk and Ck and the two priority parameters Pc and Pck associated with the exit links. Daganzo (1995a) provided a detailed treatment of this aspect and is not repeated here. The mathematical conditions are written as: if REk>SBk+SCk, if Rn
tfaen
.=S
[yck - ^ck
{ y = mid{5Bk REk } fljt , REk Ek - SCkck , P k k • Ek then \ k } \ U* = mid{5ct , REk - SBk , Pck • REk }
Pk + Pck=l
(47)
(48)
The parameters Pc and Pck (which sum to one) provide flexibility in modeling prioritized merge junctions or the action of a traffic signal. Letting: Mk = mid{5fl, , REk - Sck , Pk • REk } Mck = mid{Sct , Ra - SBk , Pck -REk}, condition (47) can be rearranged as: SBk Mk
ifREk>SBk+SCk XREk
(49) (50)
A Cell Based Dynamic Traffic Assignment Formulation
if REk -
S
Bk + SC
ifR
341
(52)
Conditions (49) and (50) can each be transformed to a set of mixed-integer constraints by the transformation in Appendix C. And (51) and (52) can be transformed by the approach in Appendix B. In summary, all the conditions in the merge connection can be transformed to mixed-integer constraints. Diverge Connection Similar to the other connections, two sets of Diverge constraints are required to fully define the diverge connection: one for disaggregate traffic and one for aggregate traffic representations. The flow conservation constraints at the disaggregate level is similar to the ordinary connection, stated as: (53)
(54) (55) The disaggregate exit flow from Bk is:
(56) (57) To maintain FIFO, a single minimum wait aBkis defined for Bk regardless of which downstream cell the traffic is heading. (Although the right-hand-sides of (56) and (57) look identical, numerically they are different because cells Ek and Ck lie on different paths.) For exposition purposes, we also define the sum of outflows from Bk: yBk,p,r = yk,p,r + yck,P,r
(58)
The aggregate traffic in each cell and link flows are then defined in the following by summing the disaggregate variables through the path and arrival time indices:
(59) (60) The aggregate link flows must observe the capacity limits. That is,
(61)
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342
Theory
Note that the flows yk, yck, and the sum yk + yck are a function of the minimum wait aBk. A lower aBk permits higher flows yk, yck, and yk +yck. In order not to violate any of the three constraints in (61), one needs to find the value of aBk such that it begins to make the constraints binding. This can be achieved by these constraints: ykpr=mid[nBkpT, (T-a'Bk)nBkpr, y'
= midln
, (T — a" }n
0}
(62)
OJ
(63)
,
(65)
(66) (67) The value of a'Bk that produces the limiting case of y^. = REk is determined by (62), the first equation in (65), and the first equation in (66). Similarly, a'Bk, a'Bk, corresponding to the limiting cases of y'ck = Rck and y'Bk = SBk, are determined from (62)-(66). Finally aBk takes the maximum among a'Bk, a'Bk, a'B'k in order not to violate any of the constraints in (61). In the above six equations, (62)-(64) can be transformed to mixed-integer constraints through the technique in Appendix C. The maximization (67) can be transformed to mixed-integer constraints by the technique in Appendix A by re-stating it as: aBk = — min(-a^, —a'Bk, -a^")- Therefore, all the conditions and operations related to the diverge connection can be cast into mixed-integer constraints. Summarizing, the network CTM involves three types of operations that are difficult to be incorporated directly as constraints, namely, "if-then" conditions, the mid-value function, and minimization. We showed in this section that each of these three operations can be transformed to mixed-integer constraints, which can then be part of the constraint set of a MP.
Determination of Path Travel Times This ideal DUO formulation uses the actual path travel time J?p(0
to
equilibrate traffic. The
r
network CTM tracks the path index in each cell. Let (Q p(t)be the cumulative vehicle count of traffic on path p in the origin cell r and cosp(t) the corresponding cumulative count in the destination cell s. According to Figure 3, one can write:
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343
Graphically, the travel time on path p is the horizontal distance between the two cumulative curves. If time is discretized and hence traffic arrives in packets, generally the cumulative counts in cells r and s will not be equal at the discretized time ticks. We estimate the path travel time in this way. If the cumulative count in r at time t is bounded by the cumulative counts in s between t' and f'+l, then the path travel time is set to be: t' - t. The maximum error in this estimation is one time interval. Mathematically, it can be stated as: if (Osp(t'}<0)rp(t)
then rfsp(t) = t'-t
