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Preface For years, I have been thinking about writing an introductory book on traffic flow theory. The main purpose is to help readers like me who are new to this subject and do not have much preparation in mathematics and traffic flow. To serve this purpose, I try to make the contents self-containing and assume minimal requirement on mathematics and traffic flow. This book is derived from my lecture notes for CEE520 Traffic Flow Theory and Simulation I (formerly offered as CEE590T Traffic Flow Theory on an experimental basis before it was assigned a permanent course number) at UMass Amherst. Hence, the chapters are more like lectures with focused topics, each of which fits in a class meeting. The book takes a unified perspective on traffic flow modeling and consists of five parts which are coherently connected. Each part is briefly described as follows. Part I focuses on traffic flow characteristics. It starts with Intelligent Transportation Systems (ITS) and traffic sensing technologies to illustrate how to quantify traffic flow and collect such data. This is followed by three chapters with in-dept discussion on traffic flow characteristics, based on which their relationships are developed and a few equilibrium traffic flow models are introduced. Part II is all about traffic flow modeling at the macroscopic level. The goal is to solve for temporal-spatial evolution of traffic flow characteristics given initial and boundary conditions. The first few chapters provide a jump-start on mathematical modeling, especially partial differential equations. With such a preparation, the domain knowledge of traffic flow is integrated into mathematical modeling, resulting in a first-order quasi-linear partial differential equation problem known as LWR model in traffic flow community. Solutions to the problem are introduced including a graphical technique using method of characteristics and numerical techniques involving a few discretization schemes. Part III is devoted to traffic flow modeling at the microscopic level. The emphasis is drivers car-following behavior involving operational control in the longitudinal direction. A series of car-following models are introduced with varying modeling philosophies and complexity. In order to provide an opportunity to cross-compare the relative performance of these models, a common ground is set up so that these models can demonstrate themselves. Such a process is called benchmarking and the common ground consists of two scenarios, one microscopic and the other macroscopic. The microscopic sce- 2
nario is a hypothetical driving process aiming at testing these models under various driving regimes (such as free flow and car following); the macroscopic scenario is a set of empirical data focusing on examining the macroscopic properties of these models (e.g., how their implied fundamental diagrams compare to the observed). Part IV extends traffic flow modeling to the picoscopic level. A modeling framework called driver-vehicle-environment closed-loop system is introduced to capture ultra-fine level of detail of traffic flow. Such a framework involves a driver model, a vehicle model, and the driving environment. The driver model collects and processes information from its vehicle and the driving environment and make control decision on motion in longitudinal and lateral directions. The vehicle model executes its driver s control decision and move dynamically on the road. The driver-vehicle unit constitutes one of the entities in the environment whose dynamic change affects driver control in the next step. As an example of this modeling framework, a simple engine model and further a dynamic interactive vehicle model are proposed and a field theory is formulated to model the driver. All things come together in Part V. Using the field theory as the basis, a unified perspective can be casted on traffic flow theory. Macroscopic models and microscopic models introduced thus far can be related to each other, all linked directly or indirectly to the field theory. Hence, a unified diagram is constructed to highlight such relations. In addition, a benchmarking effort is made to cross-compare the performance of some of the macroscopic models and microscopic models in the diagram. Meanwhile, a multi-scale modeling approach is presented which involves traffic flow modeling at a spectrum of four levels of detail, namely macroscopic, mesoscopic, microscopic, and picoscopic. The emphasis of multi-scale is to ensure modeling consistency, i.e., how less detailed models are derived from more detailed models and, conversely, how more detailed models are aggregated to less detailed models. The proposed approach may establish the theoretical foundation for traffic modeling and simulation at multiple scales seamlessly within a single system. This book is ideal for use by entry-level graduate students in transportation engineering as a textbook of traffic flow theory course. In addition, civil engineering juniors and seniors may find some in-dept information about traffic flow fundamentals in this book. Further more, applied math majors may find concrete examples of mathematical modeling with specific domain knowledge. Advanced readers are referred to other traffic flow theory books for in-depth coverage and the following are a few of them: 3
G.F. Newell. Theory of Highway Traffic Flow, 1945-1965. Course Notes UCB-ITS-CN-95-1. 1996. A. D. May. Traffic Flow Fundamentals. Prentice-Hall, 1989. C. F. Daganzo. Fundamentals of Transportation and Traffic Operations, Pergamon-Elsevier, 1997. N. Gartner et al. Revised Monograph on Traffic Flow Theory. A Stateof-the-Art Report. TRB 2001. D. L. Gerlough and M. J. Huber. Traffic Flow Theory - A Monograph. TRB Special Report 165, 1975. D. L. Gerlough and D. G. Capelle. An Introduction to Traffic Flow Theory. HRB Special Report 79, 1964. D. R. Drew. Traffic Flow Theory and Control. McGraw Hill, Inc. 1968. W. Leutzbach. Introduction to the Theory of Traffic Flow. Springer- Verlag, 1988. M. Treiber and A. Kesting. Traffic Flow Dynamics, Springer, 2013. L. Elefteriadou. An Introduction to Traffic Flow Theory, Springer, 2014. B. S. Kerner. Introduction to Modern Traffic Flow Theory and Control, Springer, 2009. I would like to thank Professor John D. Leonard at Georgia Institute of Technology and Professor Billy M. Williams at North Carolina State University who introduced me to this field and sparkled my interest in traffic flow theory. Thanks also go to former students in my traffic flow theory classes - their insightful discussion and kind encouragement made this work possible. Finally, I should acknowledge my limitations. Though I try hard to ensure the quality and accuracy of information, I can make mistakes. Therefore, readers should use this book with discretion. Daiheng Ni Amherst, MA June, 2015 4
Traffic Flow Theory A Unified Perspective Part I Part II Part III Part IV Part V Unified Perspective Picoscopic Microscopic Mesoscopic Macroscopic ITS Traffic Sensing Technologies Driver-Vehicle- Environment Closed-Loop System Route-Choice Kinetic Cellular- Automata Conservation Law Traffic Flow Characteristics Lane-Changing Engine Gap-Acceptance TRANSIMS Prigogine -Herman Hi-order 1 st order Equilibrium Traffic Flow 5 Vehicle Car-Following Driver LWR Model The Unified Diagram Field Theory Pipes/Forbes GM Gipps Newell Non-linear Newell Simplified IDM Van Aerde Psycho-Physical CARSIM Rule-Based Neural Networks Numerical Solutions Analytical Solution Single-Regime: - Greenshields - Greenberg - Underwood - Drake (NW) - Pipes-Munjal - Drew Multi-Regime: - Edie - 2-Regime - 3-Regime Stochastic Multiscale FREQ, KRONOS, CTM Method of Characteristics, K-Waves Longitudinal Control Model (LCM) Note: gray areas are part of traffic flow theory but not covered in this book.
Contents I Traffic Flow Characteristics 14 1 Traffic Sensing Technologies 15 1.1 Traffic Sensors........................... 16 1.1.1 Inductive-Loop Detectors................. 16 1.1.2 Video Image Processing System............. 18 1.1.3 Pneumatic Tubes..................... 20 1.1.4 Global Positioning System (GPS)............ 21 1.1.5 Acoustic/Ultrasonic Sensors............... 22 1.1.6 Aerial/Satellite imaging................. 23 1.1.7 RFID Technology..................... 25 1.2 Traffic Sensor Classification................... 26 1.3 Data Sources........................... 27 1.3.1 GA400 Data........................ 27 1.3.2 NGSIM Data....................... 28 2 Traffic Flow Characteristics I 32 2.1 Mobile sensor data........................ 32 2.2 Point sensor data......................... 35 2.3 Space sensor data......................... 38 2.4 Time-space diagram and characteristics............. 39 2.5 Relationships among characteristics............... 40 2.5.1 Flow, speed, and density................. 40 2.5.2 Flow and headway.................... 40 2.5.3 Density and spacing................... 41 2.5.4 Time-mean speed and space-mean speed........ 42 2.5.5 Occupancy and density.................. 44 2.6 Desired traffic flow characteristics................ 45 2.6.1 Determining space-mean speed from point sensor data 46 6
2.6.2 Determining density from point sensor data...... 46 3 Traffic flow characteristics II 51 3.1 Generalized definition....................... 51 3.2 3D representation of traffic flow................. 56 4 Equilibrium Traffic Flow 66 4.1 Single-regime models....................... 66 4.1.1 Greenshields model.................... 69 4.1.2 Other single-regime models............... 73 4.2 Multi-regime models....................... 73 4.3 The state-of-the-art models................... 74 4.4 Can we go any further?...................... 81 II Macroscopic 87 5 Conservation Law 88 5.1 The continuity equation..................... 89 5.2 First-order dynamic model.................... 96 6 Waves 101 6.1 Wave phenomena......................... 101 6.2 Mathematical representation................... 102 6.2.1 Notation.......................... 102 6.2.2 Terminology........................ 103 6.3 Traveling waves.......................... 104 6.4 Traveling Wave Solutions..................... 105 6.5 Wave Front and Pulse...................... 106 6.6 General solution to wave equations............... 106 6.7 Characteristics.......................... 109 6.8 Solution to Wave Equation.................... 111 6.9 Method of characteristics..................... 113 6.10 Some properties.......................... 115 6.10.1 Properties of characteristic................ 115 6.10.2 Properties of the solution................ 117 7
7 Shock and Rarefaction Waves 121 7.1 Gradient catastrophes...................... 121 7.2 Shock waves............................ 122 7.3 Rarefaction waves......................... 127 7.4 Entropy condition......................... 132 7.5 Summary of wave terminology.................. 135 8 LWR Model 139 8.1 The LWR model......................... 139 8.2 Example: LWR with Greenshields model............ 141 8.3 Shock wave solution to LWR model............... 143 8.4 Riemann problem......................... 144 8.5 LWR model with general q-k relationship............ 145 8.6 Shock path and queue tail.................... 147 8.7 Properties of flow-density relationship.............. 148 8.7.1 Flow-density relationship and speeds.......... 148 8.7.2 Flow-density relationship observed by a moving observer149 8.8 Example LWR Problems..................... 150 8.8.1 A bottleneck with varying traffic demand........ 150 8.8.2 A moving bottleneck................... 151 9 Numerical Solutions 160 9.1 Discretization scheme....................... 161 9.2 FREFLO............................. 163 9.3 FREQ............................... 165 9.4 KRONOS............................. 165 9.5 Cell Transmission Model..................... 168 9.5.1 Minimum principle.................... 168 9.5.2 Mainline scenario..................... 169 9.5.3 Merge scenario...................... 170 9.5.4 Diverge scenario...................... 174 10 Simplified Theory of K-Waves 179 10.1 Triangular flow-density relationship............... 180 10.2 Forward wave propagation.................... 180 10.3 Backward wave propagation................... 182 10.4 Local capacity........................... 182 10.5 Minimum principle........................ 183 8
10.6 Single bottleneck......................... 183 10.7 Computational algorithm..................... 186 10.8 Further note on K-Waves Theory................ 188 11 High-Order 195 11.1 High-order models........................ 195 11.2 Relating continuum flow models................. 198 11.3 Relative merits of continuum models.............. 199 11.4 Taxonomy of macroscopic models................ 200 III Microscopic 204 12 Microscopic 205 12.1 Scope and Time Frame................ 205 12.2 Notation.............................. 208 12.3 Benchmarking Scenarios..................... 209 12.3.1 Microscopic Benchmarking................ 209 12.3.2 Macroscopic Benchmarking............... 212 13 Pipes and Forbes 215 13.1 Pipes Model............................ 215 13.1.1 Applications of Pipes Model............... 216 13.1.2 Properties of Pipes Model................ 218 13.2 Forbes Model........................... 219 13.3 Benchmarking........................... 221 13.3.1 Microscopic Benchmarking................ 221 13.3.2 Macroscopic Benchmarking............... 224 14 General Motors 228 14.1 Development of GM................... 228 14.1.1 GM1............................ 229 14.1.2 GM2............................ 230 14.1.3 GM3............................ 230 14.1.4 GM4............................ 231 14.1.5 GM5............................ 231 14.2 Microscopic Benchmarking.................... 232 14.3 Micro-Macro Bridge....................... 235 9
14.4 Macroscopic Benchmarking.................... 240 14.5 Limitations of GM models.................... 241 15 Gipps Model 246 15.1 Model Formulation........................ 246 15.2 Properties of Gipps model.................... 250 15.3 Benchmarking........................... 251 15.3.1 Microscopic Benchmarking................ 253 15.3.2 Macroscopic Benchmarking............... 253 16 More Single-Regime 257 16.1 Newell Nonlinear Model..................... 257 16.1.1 Properties of Newell Nonlinear Model.......... 258 16.1.2 Benchmarking....................... 258 16.2 Newell Simplified Model..................... 261 16.3 IDM Model............................ 263 16.3.1 Properties of IDM Model................. 263 16.3.2 Benchmarking....................... 264 16.4 Van Aerde Model......................... 266 16.4.1 Properties of Van Aerde Model............. 267 16.4.2 Benchmarking....................... 267 17 More Intelligent 272 17.1 Psycho-Physical Model...................... 272 17.2 Carsim Model........................... 274 17.3 Rule-Based Model......................... 276 17.4 Neural Network Model...................... 277 17.5 Summary of Car-Following............... 279 IV Picoscopic 286 18 Picoscopic 287 18.1 Driver, Vehicle, and Environment................ 288 18.2 Applications of Picoscopic............... 292 18.2.1 Interactive Highway Safety Design............ 292 18.2.2 Connected Vehicle Technology.............. 292 18.2.3 Transportation Forensics................. 293 10
18.2.4 Emergency Management................. 293 19 Engine 296 19.1 Introduction............................ 296 19.2 Review of Existing Engine................ 298 19.3 Simple Mathematical Engine.............. 299 19.3.1 Model I: Polynomial Model............... 299 19.3.2 Model II: Parabolic Model................ 300 19.3.3 Model III: Bernoulli Model................ 301 19.4 Validation and Comparison of the Engine....... 304 19.5 Conclusion............................. 309 19.6 Appendix............................. 309 19.6.1 A cross-comparison of engine models.......... 309 19.6.2 Parameter Values..................... 309 20 Vehicle 313 20.1 Overview of the DIV Model................... 313 20.2 Longitudinal Movement................ 314 20.2.1 Acceleration Performance........... 314 20.2.2 Braking Performance............. 315 20.2.3 Aerodynamic Drag............... 316 20.2.4 Grade Resistance............... 316 20.3 Lateral Movement................... 316 20.4 Model Calibration and Validation................ 318 21 The Field Theory 321 21.1 Motivation............................. 321 21.2 Physical Basis of Traffic Flow.................. 322 21.2.1 Mechanics Phenomena.................. 323 21.2.2 Electromagnetic Phenomena............... 326 21.2.3 Wave Phenomena..................... 327 21.2.4 Statistical Mechanics Phenomena............ 328 21.3 The Field Theory......................... 329 21.4 Simplification of the Field Theory................ 334 21.4.1 Motion in the Longitudinal Direction.......... 335 21.4.2 Motion in the Lateral Direction............. 338 21.5 Discussion of the Field Theory.................. 339 21.5.1 Tentative definition of two vague terms......... 339 11
21.5.2 Connection to Existing Knowledge Base........ 341 21.6 Summary............................. 344 22 Longitudinal Control Model 346 22.1 Introduction............................ 346 22.2 The Longitudinal Control Model................. 348 22.2.1 Microscopic model.................... 348 22.2.2 Macroscopic model.................... 350 22.3 Model Properties......................... 351 22.3.1 Boundary conditions................... 351 22.3.2 Model flexibility...................... 352 22.4 Empirical Results......................... 355 22.5 Applications............................ 365 22.5.1 An illustrative example.................. 365 22.5.2 Macroscopic approach - graphical solution....... 367 22.5.3 Microscopic approach - deterministic simulation.... 368 22.5.4 Microscopic approach - random simulation....... 370 22.6 Related Work........................... 372 22.7 Summary............................. 373 V The Unified Perspective 377 23 The Unified Diagram 378 23.1 Motivation............................. 378 23.2 A Broader Perspective...................... 380 23.2.1 Highlight of the Field Theory.............. 381 23.2.2 Relating microscopic car-following models....... 382 23.2.3 Relating macroscopic equilibrium models........ 390 23.3 The Unified Diagram....................... 394 23.3.1 Description of the Unified Diagram........... 394 23.3.2 List of Connections in the Unified Diagram....... 394 23.4 Summary............................. 397 24 Multiscale Traffic Flow 400 24.1 Introduction............................ 400 24.2 The Spectrum of Scales................ 402 24.2.1 The picoscopic scale................... 403 12
24.2.2 The microscopic scale................... 404 24.2.3 The mesoscopic scale................... 405 24.2.4 The macroscopic scale.................. 406 24.2.5 Issues of multiscale modeling............... 406 24.3 The Multiscale Approach..................... 407 24.3.1 Picoscopic modeling................... 407 24.3.2 Microscopic modeling................... 410 24.3.3 Mesoscopic modeling................... 412 24.3.4 Macroscopic modeling.................. 414 24.4 Summary............................. 415 13