nario is a hypothetical driving process aiming at testing these models under various driving regimes (such as free flow and car following); the

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3 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

4 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

5 G.F. Newell. Theory of Highway Traffic Flow, Course Notes UCB-ITS-CN A. D. May. Traffic Flow Fundamentals. Prentice-Hall, C. F. Daganzo. Fundamentals of Transportation and Traffic Operations, Pergamon-Elsevier, N. Gartner et al. Revised Monograph on Traffic Flow Theory. A Stateof-the-Art Report. TRB D. L. Gerlough and M. J. Huber. Traffic Flow Theory - A Monograph. TRB Special Report 165, D. L. Gerlough and D. G. Capelle. An Introduction to Traffic Flow Theory. HRB Special Report 79, D. R. Drew. Traffic Flow Theory and Control. McGraw Hill, Inc W. Leutzbach. Introduction to the Theory of Traffic Flow. Springer- Verlag, M. Treiber and A. Kesting. Traffic Flow Dynamics, Springer, L. Elefteriadou. An Introduction to Traffic Flow Theory, Springer, B. S. Kerner. Introduction to Modern Traffic Flow Theory and Control, Springer, 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,

6 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.

7 Contents I Traffic Flow Characteristics 14 1 Traffic Sensing Technologies Traffic Sensors Inductive-Loop Detectors Video Image Processing System Pneumatic Tubes Global Positioning System (GPS) Acoustic/Ultrasonic Sensors Aerial/Satellite imaging RFID Technology Traffic Sensor Classification Data Sources GA400 Data NGSIM Data Traffic Flow Characteristics I Mobile sensor data Point sensor data Space sensor data Time-space diagram and characteristics Relationships among characteristics Flow, speed, and density Flow and headway Density and spacing Time-mean speed and space-mean speed Occupancy and density Desired traffic flow characteristics Determining space-mean speed from point sensor data 46 6

8 2.6.2 Determining density from point sensor data Traffic flow characteristics II Generalized definition D representation of traffic flow Equilibrium Traffic Flow Single-regime models Greenshields model Other single-regime models Multi-regime models The state-of-the-art models Can we go any further? II Macroscopic 87 5 Conservation Law The continuity equation First-order dynamic model Waves Wave phenomena Mathematical representation Notation Terminology Traveling waves Traveling Wave Solutions Wave Front and Pulse General solution to wave equations Characteristics Solution to Wave Equation Method of characteristics Some properties Properties of characteristic Properties of the solution

9 7 Shock and Rarefaction Waves Gradient catastrophes Shock waves Rarefaction waves Entropy condition Summary of wave terminology LWR Model The LWR model Example: LWR with Greenshields model Shock wave solution to LWR model Riemann problem LWR model with general q-k relationship Shock path and queue tail Properties of flow-density relationship Flow-density relationship and speeds Flow-density relationship observed by a moving observer Example LWR Problems A bottleneck with varying traffic demand A moving bottleneck Numerical Solutions Discretization scheme FREFLO FREQ KRONOS Cell Transmission Model Minimum principle Mainline scenario Merge scenario Diverge scenario Simplified Theory of K-Waves Triangular flow-density relationship Forward wave propagation Backward wave propagation Local capacity Minimum principle

10 10.6 Single bottleneck Computational algorithm Further note on K-Waves Theory High-Order High-order models Relating continuum flow models Relative merits of continuum models Taxonomy of macroscopic models III Microscopic Microscopic Scope and Time Frame Notation Benchmarking Scenarios Microscopic Benchmarking Macroscopic Benchmarking Pipes and Forbes Pipes Model Applications of Pipes Model Properties of Pipes Model Forbes Model Benchmarking Microscopic Benchmarking Macroscopic Benchmarking General Motors Development of GM GM GM GM GM GM Microscopic Benchmarking Micro-Macro Bridge

11 14.4 Macroscopic Benchmarking Limitations of GM models Gipps Model Model Formulation Properties of Gipps model Benchmarking Microscopic Benchmarking Macroscopic Benchmarking More Single-Regime Newell Nonlinear Model Properties of Newell Nonlinear Model Benchmarking Newell Simplified Model IDM Model Properties of IDM Model Benchmarking Van Aerde Model Properties of Van Aerde Model Benchmarking More Intelligent Psycho-Physical Model Carsim Model Rule-Based Model Neural Network Model Summary of Car-Following IV Picoscopic Picoscopic Driver, Vehicle, and Environment Applications of Picoscopic Interactive Highway Safety Design Connected Vehicle Technology Transportation Forensics

12 Emergency Management Engine Introduction Review of Existing Engine Simple Mathematical Engine Model I: Polynomial Model Model II: Parabolic Model Model III: Bernoulli Model Validation and Comparison of the Engine Conclusion Appendix A cross-comparison of engine models Parameter Values Vehicle Overview of the DIV Model Longitudinal Movement Acceleration Performance Braking Performance Aerodynamic Drag Grade Resistance Lateral Movement Model Calibration and Validation The Field Theory Motivation Physical Basis of Traffic Flow Mechanics Phenomena Electromagnetic Phenomena Wave Phenomena Statistical Mechanics Phenomena The Field Theory Simplification of the Field Theory Motion in the Longitudinal Direction Motion in the Lateral Direction Discussion of the Field Theory Tentative definition of two vague terms

13 Connection to Existing Knowledge Base Summary Longitudinal Control Model Introduction The Longitudinal Control Model Microscopic model Macroscopic model Model Properties Boundary conditions Model flexibility Empirical Results Applications An illustrative example Macroscopic approach - graphical solution Microscopic approach - deterministic simulation Microscopic approach - random simulation Related Work Summary V The Unified Perspective The Unified Diagram Motivation A Broader Perspective Highlight of the Field Theory Relating microscopic car-following models Relating macroscopic equilibrium models The Unified Diagram Description of the Unified Diagram List of Connections in the Unified Diagram Summary Multiscale Traffic Flow Introduction The Spectrum of Scales The picoscopic scale

14 The microscopic scale The mesoscopic scale The macroscopic scale Issues of multiscale modeling The Multiscale Approach Picoscopic modeling Microscopic modeling Mesoscopic modeling Macroscopic modeling Summary