EURO Advanced Tutorials on Operational Research. Series editors M. Grazia Speranza, Brescia, Italy José Fernando Oliveira, Porto, Portugal

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1 EURO Advanced Tutorials on Operational Research Series editors M. Grazia Speranza, Brescia, Italy José Fernando Oliveira, Porto, Portugal

2 More information about this series at

3 Femke Kessels Traffic Flow Modelling Introduction to Traffic Flow Theory Through a Genealogy of Models 123

4 Femke Kessels Department of Transport & Planning Delft University of Technology Delft, The Netherlands Logistics, Tourism and Service Management German University of Technology Muscat, Sultanate of Oman Additional material to this book can be downloaded from ISSN X ISSN (electronic) EURO Advanced Tutorials on Operational Research ISBN ISBN (ebook) Library of Congress Control Number: Springer International Publishing AG, part of Springer Nature 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

5 To Emma, Julia and Sofie

6 Preface This book shows the history of traffic flow modelling from the perspective of modern day applications. Traffic flow models describe how vehicles travel over roads, at which speeds, what the distance is between them, how long they take to travel over a certain road section, etc. Combining the models with other information supports estimations about current and future traffic states. This allows answering questions about the presence and duration of congestion, travel times and travel time delays, emissions and safety assessment. In turn, the information can be used in a variety of applications including transportation planning and traffic control. This book shows the historical development of traffic flow theory by means of a genealogical tree of traffic flow models. The tree, included on page 15, shows the main developments in traffic flow modelling. The focus on the history of traffic flow models gives the reader insight into the basics of traffic flow modelling all through to the most advanced models that are currently under development. In addition, the book discusses numerical methods which are applied to create computer simulations based on the traffic flow models. The history of traffic flow modelling starts in the 1930s. Bruce D. Greenshields presented his findings about the relationship between vehicle speed and the distance between vehicles at the Annual Meeting of the Highway Research Board (United States). Even though he had some predecessors doing similar research, Greenshields is often considered the founder of Traffic Flow Theory. The 1940s were a relatively quiet time for traffic flow theory, but from the mid 1950s many new models were introduced. Most of these new models include dynamics of traffic flow, i.e. they describe how traffic flows change over time, due to for examples changes in inflow of vehicles or traffic lights changing colour. Different types of models were developed and applied for road design. However, research in this area mostly stalled again in the 1980s. Faster and easier to use computers brought a new era of traffic flow research from the mid-1990s, resulting in most of the models that are still applied today. Many of today s applications require efficient numerical methods for fast and accurate predictions. Applications include transportation planning, road design, safety assessment, environmental assessment, traffic management, evacuation planning and route advice. vii

7 viii Preface Previous versions of this work were published as the state-of-the-art chapter in the author s PhD dissertation (2013) and as a review article in the EURO Journal on Transportation and Logistics (2015). The book format gives more space to provide more basics, to go more in depth into the most important aspects of traffic flow modelling and simulation, and to include problem sets that will reinforce the newly gained knowledge and insights of the reader. Furthermore, the most recent developments in the field of traffic flow theory have been included. The book aims at students (MSc and PhD), researchers and practitioners who want to learn more about the background of the models they are applying. No preliminary knowledge about traffic is assumed. Some background in calculus and differential equations is required, but references will be given for those who need to refresh their knowledge. Problem sets are included at the end of each chapter, with answers to selected problems in Chap. 8. Some of the exercises require the reader to perform simulations, for which software is provided online, at the publisher s website ( Some previous experience with Octave (or Matlab) is useful for these exercises. After reading the book and exploring some of the problems, readers will understand the main concepts in traffic flow modelling and simulation in such a way that they can (1) choose an appropriate model for their research or other application and build a simple but useful simulation tool based on this model; (2) understand a newly published scientific article that builds on traffic flow theory, modelling and simulation presented in this book, review that article and apply the models/methods that are presented; (3) start developing their own models and numerical methods to create new branches of the model tree. Delft, The Netherlands June 2018 Femke Kessels

8 Acknowledgements The seeds of expanding the idea of the genealogical model tree into something to be used in education were planted during my PhD research. When the editors of this series, Grazia Speranza and José Fernando Oliveira, invited me to write a proposal, I was reminded of the encouragements from Serge Hoogendoorn and Robert Bertini and I decided to accept. After my proposal was accepted, also Christian Rauscher from Springer was always quick to respond to any of my questions. Also thanks to the colleagues in Delft who gave valuable feedback on draft versions of (parts of) this book, Serge Hoogendoorn, Hans van Lint, Simeon Calvert and Meng Wang. When working on this book, I have been writing at many different places, surrounded by even more different people. Many days I found inspiration and motivation from going to the shared office of our Personal Development Network and working in the same room as likeminded other expat partners. Thank you Steven, Eveline, Tina, Chris, Hanna, Emma and Maryam for just being there and the many coffees outside! On other days I went to the German University of Technology in Oman, to be around other academics: it was always worth the long drive and I made good progress among my now colleagues to be. Thank you Heba, Amjaad, Osman, Amaani and all interns for the hospitality! Still, all this time spent outside the house was not enough and I often worked at home, only to be disturbed in case of blood or fire, which fortunately never happened. I am forever thankful to Rosa who has been a great help and caretaker for our children during these times. The last bit of progress was made not in Muscat, but in Delft, among colleagues and friends at Delft University of Technology. Also thank you for your hospitality! Finally, a big thanks to my family, mam, pap, Mia, Giel, Manon and Niels, who have been encouraging me and giving me space to grow during calm and tumultuous times. And of course thanks and hugs to those who helped me keep a balance between all this serious stuff and love, fun, play and adventure: Emma, Julia and Sofie. ix

9 Contents 1 Introduction to Traffic Flow Modelling Traffic Flow Modelling Cycle Observationsand Phenomena ObservingTraffic Phenomenain Traffic Traffic Flow Models Agent-BasedModels and TheirVariables Continuum Models and Edie s Definitions Classifications of Models Traffic Flow, Fluid Flow and Other ComplexSystems Approachand Scope ofthis Book The GenealogicalModelTree Numerical Methods for Computer Simulation Other Aspects of Traffic Flow Modelling ProblemSet Further Reading The Fundamental Diagram High Densities, Low Speeds and Vice Versa Shapes of the Fundamental Diagram Fundamental Diagrams in Macroscopic Models Fundamental Diagrams in Microscopic Models Properties andrequirements Requirements Properties: Capacity, Free Flow and Congestion AdditionalRequirements Scatter in the Fundamental Diagram ProblemSet Further Reading xi

10 xii Contents 3 Microscopic Models Safe-Distance Models Safe-Distance Modelswith Delay High Speeds Versus Safety Stimulus-Response Models More Recent Stimulus-Response Models: OVM and IDM Simulation Results with a Stimulus Response Model Generic Model and Stability Action Point Models Cellular-AutomataModels Extensions Heterogeneity Multi-Anticipation Time Delay Lateral Movements Numerical Methods for Car-Following Models Advanced Numerical Methods Numerical Methods and Delay ProblemSet Further Reading Macroscopic Models Kinematic Wave Models GraphicalDerivation Methodof Characteristics SimulationResults with the Kinematic Wave Model Critique and Adaptationsof the Kinematic Wave Model Multi-Class Kinematic Wave Models Multi-Dimensional Fundamental Diagram Fastlane Models with Three Regimes Porous Flow Models Requirements of Multi-Class Models Higher-OrderModels Critique on HigherOrder Models AnisotropicHigherOrder Models Generic HigherOrderModel MovingCoordinates The LagrangianCoordinateSystem GraphicalDerivation Generic HigherOrderModel in LagrangianCoordinates Bounded Acceleration, Hysteresis and Capacity Drop Bounded Acceleration Hysteresis ProblemSet Further Reading... 81

11 Contents xiii 5 Numerical Methods for Continuum Models Finite Difference Methods and Time Stepping Explicit Time Stepping First Order Finite Difference Methods Stability of Numerical Methods Minimum Supply Demand Method for Kinematic Wave Models Methods for Higher Order Models Lagrangian Simulation Methods LagrangianMethodfor the LWR Model Simplified Lagrangian Simulation and Car-Following Models Characteristics and Numerical Methods Lagrangian Methods for Higher Order Models Discretisation of Bounded Acceleration VariationalTheoryandLink TransmissionModels ProblemSet Further Reading Mesoscopic Models Headway DistributionModels andcluster Models Gas-Kinetic Models Generic Gas-Kinetic Model Continuum Gas-Kinetic Models HybridModels Lagrangian Methods for Mesoscopic Models Interface Modelling Moving Interfaces ProblemSet Further Reading Conclusion: Convergence Versus Branching Out Modelling Scale: Microscopic vs. Macroscopic Macroscopic Modeling and the Continuum Assumption MicroscopicModels andparameters Model Choice Considerations PredictiveAccuracy Numerical Methods Current Trendsand Outlook GeneralizedModels Extensionsand Adaptationsof ExistingModels Outlook ProblemSet Further Reading Answers to Selected Problems Bibliography

12 Symbols Variables x Position (m) t Time (s) n Vehicle number (-) v Speed (m/s) a Acceleration (m/s 2 ) ρ Density (veh/m) q Flow (veh/s) s Spacing (m/veh) Model Parameters v max v crit ρ crit ρ jam s jam Maximum speed (m/s) Critical speed (m/s) Critical density (veh/m) Jam density (veh/m) Jam spacing (veh/m) a min Maximum deceleration (m/s 2 ) a max Maximum acceleration (m/s 2 ) T Minimum time headway (s) τ Time delay or reaction time (s) xv

13 xvi Symbols Numerical Parameters and Indices Δt Time step size (s) Δx Grid cell size (m) Δn Vehicle group size (# of veh s) ν CFL number (-) k Time step index n Vehicle number j Grid cell index i Vehicle group index