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Continuum Modeling of the Deceleration Transient State In CONTINUUM MODELING OF THE DECELERATION TRANSIENT STATE IN STOCHASTIC TRAFFIC FLOW By Alma Cemerlic Dr. Henry McDonald Dr. Timothy Swafford (Chair) (Committee Member) Dr. Ramesh Pankajakshan Dr. Joe Dumas (Committee Member) (Committee Member) CONTINUUM MODELING OF THE DECELERATION TRANSIENT STATE IN STOCHASTIC TRAFFIC FLOW By Alma Cemerlic A Dissertation Submitted to the Faculty of the University of Tennessee at Chattanooga in Partial Fulfillment of the Requirements of the Degree of Doctor of Philosophy in Computational Engineering The University of Tennessee at Chattanooga Chattanooga, Tennessee August 2015 ii Copyright © 2015 By Alma Cemerlic All Rights Reserved iii ABSTRACT Due to individual driver behavior, a traffic system is subject to many stochastic factors. Deterministic partial differential equations and their extensions, traditionally used in traffic flow modeling, may not be sufficient in applications such as real-time estimation and prediction of traffic flow conditions. In previous studies, the issue of physical relevance has received little attention in efforts to introduce a stochastic component into deterministic equilibrium-based traffic models. In this work, the stochastic component is derived directly from realistic driver behavior implemented as a fully discrete fine-grained agent-based model and combined into the deterministic Aw-Rascle system of equations via ensemble averaging. The same approach can be applied to any second-order traffic model of a similar form. Solutions for the stopping and deceleration cases are obtained using the second-order Lax-Wendroff scheme and compared to the results obtained from the agent-based simulation. iv TABLE OF CONTENTS LIST OF TABLES........................................................................................................................viii LIST OF FIGURES........................................................................................................................ix CHAPTER 1. INTRODUCTION...............................................................................................................1 2. LITERATURE REVIEW....................................................................................................4 2.1 Continuum Traffic Models.......................................................................................4 2.1.1 Lighthill-Whitham-Richards Traffic Model (LWR).....................................6 2.1.2 Aw-Rascle Traffic Model (AR)....................................................................7 2.2 Car-Following Models........................................................................................... 13 2.2.1 Intelligent Driver Model (IDM)................................................................. 15 2.2.2 Nagel-Schreckenberg Particle-Hopping Model......................................... 16 2.3 Stochastic Traffic Flow........................................................................................... 17 3. METHODOLOGY............................................................................................................ 19 3.1 Aw-Rascle System of Equations - Summary......................................................... 22 3.1.1 Convergence of the Aw-Rascle to the Lighthill-Whitham-Richards Traffic Model............................................................................................. 23 3.1.2 The Influence of the Speed Relaxation Source Term on the Solution of the Aw-Rascle System of Equations...................................................... 24 3.1.3 Second-Order Lax-Wendroff Numerical Scheme...................................... 28 3.2 Fine-Grained Agent-Based Model......................................................................... 30 3.2.1 Nagel-Schreckenberg Particle-Hopping Model and the Fine-Grained Agent-Based Model............................................................ 30 3.2.2 Car-Following Intelligent Driver Model and Fine-Grained Agent-Based Model.................................................................................. 32 3.2.2.1 Effects of Discretization on the Intelligent Driver Model............. 34 v 3.3 Multi-Class Driving Behavior and Driver Population.......................................... 37 3.3.1 Fundamental Diagrams for the Homogenous and Heterogeneous Driver Populations..................................................................................... 40 3.4 Ensemble Averaged Aw-Rascle Equations............................................................ 43 3.4.1 Ensemble Averaging.................................................................................. 43 3.4.2 Ensemble Averaged Aw-Rascle Equation.................................................. 49 3.4.3 Ensemble Averaged Values in the Fine-Grained Agent-Based Model...... 51 4. RESULTS.......................................................................................................................... 54 4.1 Homogeneous Driver Population........................................................................... 56 4.1.1 Fine-Grained Agent-Based Model; Case: Stopping.................................. 57 4.1.2 Aw-Rascle Model; Case: Stopping............................................................ 60 4.2 Inhomogeneous Driver Population........................................................................ 63 4.2.1 Correlation Terms...................................................................................... 63 4.2.1.1 Functional Approximation for the Ensemble Averaged Terms Used in the Test Cases........................................................ 68 4.2.2 Case 1: Stopping....................................................................................... 71 4.2.2.1 Fine-Grained Agent-Based Solution.............................................. 71 4.2.2.2 Numerical Results of the Aw-Rascle Model and the Aw-Rascle Model with Ensemble Averaged Source Term............ 76 4.2.2.3 Evaluation...................................................................................... 78 4.2.3 Case 2: Bottleneck (Merging with a Slower Traffic Stream)..................... 82 4.2.3.1 Fine-Grained Agent-Based Solution.............................................. 83 4.2.3.2 Numerical Results and Evaluation of the Aw-Rascle Model and the Aw-Rascle Model with Ensemble Averaged Source Term............................................................................................... 85 4.3 Discussion.............................................................................................................. 89 5. CONCLUSION AND FUTURE WORK........................................................................... 92 APPENDIX A. FUNDAMENTAL DIAGRAM.......................................................................................... 95 B. EFFECTS OF DISCRETIZATION ON THE INTELLIGENT DRIVER MODEL....... 100 C. NUMERICAL VERIFICATION OF THE SECOND-ORDER LAX-WENDROFF SCHEME......................................................................................................................... 107 vi D. COMPARISONOF THE NUMERICAL SOLUTIONS OF THE HOMOGENEOUS AR MODEL AND THE LWR MODEL......................................................................... 111 REFERENCES............................................................................................................................ 122 VITA............................................................................................................................................ 127 vii LIST OF TABLES 3.1 Driver safety classification based on behind-the-wheel operating behaviors with assumed Gaussian distribution population percentages [53]................................ 38 3.2 IDM parameters used to distinguish different classes of drivers [50]................................ 38 4.1 Wave propagation speed comparison for Aw-Rascle and Aw-Rascle with ensemble averaging................................................................................................................ 78 4.2 Wave propagation speed for the stopping problem; model comparison............................ 79 viii LIST OF FIGURES 3.1 Fundamental Diagram; empirical data and curve fit [18].................................................. 20 3.2 Initial condition: ρL = 0:28;ρR = 1:0; vR = 0:1(vR = Veq (ρR) for the equilibrium case), vL = 0:0........................................................................................................ 25 3.3 Nonphysical solution as the result of the non-equilibrium initial condition and no speed relaxation source term................................................................................ 25 3.4 Adaptation from low speed to the equilibrium speed, adaptation time τ 10 and 30 seconds, simulation time: 500 time steps (115 seconds)....................................... 26 3.5 Initial condition: initial density: ρL = 0:28;ρR = 1:0; initial velocity: vR = 0:5(vR = Veq (ρR) for the equilibrium case), vL = 0:0........................................................... 27 3.6 Adaptation from high speed to the equilibrium speed, adaptation time τ 10 and 30 seconds, simulation time: 500 time steps (115 seconds)....................................... 28 3.7 Convergence of the IDM acceleration function solved using the explicit first-order Euler method. , L2-norm with respect to the solution obtained with ∆t = 1:0E − 10............................................................................................................... 36 3.8 Assumed Gaussian distribution for the synthetic driver population in heterogeneous case.......................................................................................................................
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