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Control of mainstream traffic flow: variable speed limit and lane change
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Control of mainstream traffic flow: variable speed limit and lane change
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Control of Mainstream Traffic Flow: Variable Speed Limit and Lane Change by Faisal Hasan Alasiri A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy (ELECTRICAL ENGINEERING) May 2022 Copyright 2022 Faisal Hasan Alasiri Dedication to my beloved father, may he rest in peace ii Acknowledgements First and foremost, I would like to sincerely thank my advisor, Professor Petros Ioannou, for his encouragement, guidance, academic rigor, and patience over the past years. This research would not have been possible without his support and insightful comments. I consider myself very fortunate to be one of his students. My sincere thanks and deepest appreciation go to the committee members, Professor Maged Dessouky, Professor Ketan Savla, Professor Pierluigi Nuzzo, and Professor Paul Bogdan, for their helpful feedback and guidance. I would like to acknowledge the valuable assistance and guidance provided on several occasions by my former lab-mate, Dr. Yihang Zhang. His support is ap- preciated, and his friendship is treasured. I also would like to thank all my world- widefriendsandlab-mates, especiallyTianchenYuanandFernandoValladares, for whom I am deeply indebted for their unconditional help and encouragement. I also would like to express my gratitude to Shane Goodoff and Diane Demetras for helping me navigate my journey at USC. Finally, I would like to extend my deepest gratitude and appreciation to my family; I could not have achieved so much without their unconditional and infinite love, as well as their emotional and financial support. iii Table of Contents Dedication ii Acknowledgements iii List of Tables vii List of Figures viii Abstract xi 1 Introduction 1 1.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Existing Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Contribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . 19 2 Macroscopic Models for Freeway Traffic Flow 22 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 First-Order Macroscopic Traffic Models . . . . . . . . . . . . . . . . 24 2.2.1 The Lighthill-Whitham-Richards (LWR) Model . . . . . . . 24 2.2.2 The Cell Transmission Model (CTM) . . . . . . . . . . . . . 26 2.2.2.1 Modified Versions of the Cell Transmission Model . 29 2.2.3 Multiple-Lane Cell Transmission Model . . . . . . . . . . . . 34 2.3 Second-Order Macroscopic Traffic Models . . . . . . . . . . . . . . . 36 2.3.1 The Payne Model . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.2 The METANET Model . . . . . . . . . . . . . . . . . . . . . 38 2.3.2.1 Extended Versions of METANET . . . . . . . . . . 39 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 ComparisonofFeedbackLinearizationandModelPredictiveTech- niques for Variable Speed Limit (VSL) Control 42 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 iv 3.2.1 Effects of Forced Lane Changes at Highway Bottleneck . . . 43 3.2.2 Cell Transmission Model . . . . . . . . . . . . . . . . . . . . 46 3.3 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3.1 Desired Equilibrium Points . . . . . . . . . . . . . . . . . . . 48 3.3.2 Feedback Linearization (FL) Controller . . . . . . . . . . . . 50 3.3.3 Nonlinear Model Predictive Control (NMPC) . . . . . . . . 51 3.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.1 Scenario setup . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.2 PerformanceandRobustnessAnalysiswithMacroscopicSim- ulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4.3 PerformanceandRobustnessAnalysiswithMicroscopicSim- ulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4 The Stability Analysis of Cell Transmission Model (CTM) with Disturbance 62 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2 The Cell Transmission Model (CTM) with Disturbance . . . . . . . 63 4.3 Stability Analysis of the Open-Loop CTM with Disturbance . . . . 65 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5 Robust Variable Speed Limit (VSL) Control of the CTM with Disturbance 69 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2 Design of Robust Variable Speed Limit (VSL) Control . . . . . . . 69 5.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.3.1 Simulation Network and Fundamental Diagram . . . . . . . 81 5.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 83 5.3.2.1 Macroscopic Simulation . . . . . . . . . . . . . . . 83 5.3.2.2 Microscopic Simulation. . . . . . . . . . . . . . . . 85 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6 Evaluation of Integrated Robust Variable Speed Limit and Lane Change Control Considering VSL Zone Distance 90 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.2 Multi-Section Cell Transmission Model with Disturbance . . . . . . 90 6.3 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.3.1 Robust Variable Speed Limit Control for multi-sect CTM . . 92 6.3.2 The Effect of L 0 . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.3.3 Lane Change Control . . . . . . . . . . . . . . . . . . . . . . 97 6.3.4 Lane-Changing Recommendation Messages . . . . . . . . . . 98 6.3.5 Length of LC Controlled Segment . . . . . . . . . . . . . . . 99 6.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 101 v 6.4.1 Network Configuration . . . . . . . . . . . . . . . . . . . . . 101 6.4.2 Parameter Selections . . . . . . . . . . . . . . . . . . . . . . 102 6.4.3 Performance Measurements . . . . . . . . . . . . . . . . . . 102 6.4.4 Uncertainties and Robustness Analysis . . . . . . . . . . . . 106 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 7 Per-LaneVariableSpeedLimitandLaneChangeControlforCon- gestion Management at Bottlenecks 110 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.2 Freeway Traffic Behavior at Bottlenecks . . . . . . . . . . . . . . . 112 7.3 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.3.1 Multiple-Lane Cell Transmission Model Based VSL Control 115 7.3.2 Per-Lane Variable Speed Limit (VSL) Control Design . . . . 119 7.3.3 Lane Changing (LC) Control . . . . . . . . . . . . . . . . . 123 7.3.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 125 7.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.4.1 Simulation Network Setup . . . . . . . . . . . . . . . . . . . 127 7.4.2 Microscopic Simulation Results . . . . . . . . . . . . . . . . 128 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 8 Conclusions and Further Research 135 8.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 135 8.2 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 References 139 Appendix A The Proof of Lemma 5.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Appendix B The Proof of Theorem 7.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 154 vi List of Tables 3.1 Evaluation Results with Original Parameters . . . . . . . . . . . . . 59 3.2 Evaluation Results under Different w 0 . . . . . . . . . . . . . . . . . 60 3.3 Evaluation Results under Different ρ c,d . . . . . . . . . . . . . . . . 60 3.4 Evaluation Results under Different w b . . . . . . . . . . . . . . . . 61 6.1 Performance Evalutions d = 7000 veh/h . . . . . . . . . . . . . . . 104 6.2 Performance Evalutions d = 6000 veh/h . . . . . . . . . . . . . . . 104 6.3 Uncertainties in Measured Densities ˜ ρ i . . . . . . . . . . . . . . . . 107 6.4 Uncertainties in Measured Flows ˜ q i . . . . . . . . . . . . . . . . . . 107 6.5 Uncertainties in Model Parameter w . . . . . . . . . . . . . . . . . 108 7.1 Definition of the Model Parameters . . . . . . . . . . . . . . . . . . 117 vii List of Figures 1.1 Example of Integrating VSL and LC Signs on Highway . . . . . . . 3 1.2 The Supply and Demand Functions . . . . . . . . . . . . . . . . . . 15 2.1 Parabolic Fundamental Diagram . . . . . . . . . . . . . . . . . . . . 25 2.2 Triangular Fundamental Diagram . . . . . . . . . . . . . . . . . . . 26 2.3 Cell Representation in the CTM Framework . . . . . . . . . . . . . 27 2.4 Original Supply and Demand Functions . . . . . . . . . . . . . . . . 28 2.5 Supply and Modified Demand Function with Capacity Drop . . . . 31 2.6 Supply and Modified Demand Function with Bounded Acceleration 31 2.7 Supply and Modified Demand Function with Bounded Acceleration and Capacity Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.8 Supply and Modified Demand Function with Bounded Acceleration and Lane-Changing . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.9 Cell Representation in the METANET Framework . . . . . . . . . . 38 2.10 Rendered flow-density curves (1). . . . . . . . . . . . . . . . . . . . 40 2.11 Rendered flow-density curves (2). . . . . . . . . . . . . . . . . . . . 41 3.1 Highway Bottleneck . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2 Effects of LC and VSL on Fundamental Diagrams . . . . . . . . . . 44 3.3 Configuration of the Highway Segment . . . . . . . . . . . . . . . . 46 3.4 Desired Equilibrium Point . . . . . . . . . . . . . . . . . . . . . . . 48 3.5 Geometry of the Simulation Network . . . . . . . . . . . . . . . . . 53 3.6 Block Diagram of the System . . . . . . . . . . . . . . . . . . . . . 54 3.7 Perturbation on the Model Parameters . . . . . . . . . . . . . . . . 55 viii 3.8 ρ 7 with FL and NMPC . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.9 PerformanceSensitivityofnoControl(black),FL(blue),andNMPC (red) to Perturbations on Demand d . . . . . . . . . . . . . . . . . 56 3.10 PerformanceSensitivityofnoControl(black),FL(blue),andNMPC (red) to Perturbations on C b . . . . . . . . . . . . . . . . . . . . . . 57 3.11 PerformanceSensitivityofnoControl(black),FL(blue),andNMPC (red) to Perturbations on ρ d,c . . . . . . . . . . . . . . . . . . . . . 58 3.12 Performance Sensitivity of FL (blue) and NMPC (red) to Increasing Levels of Standard Deviation in Measurement Noise . . . . . . . . . 59 4.1 Single Road Section . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 TheSupplyandDemandFunctionsCorrespondingtotheTriangular Fundamental Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.1 Road Section with VSL Control . . . . . . . . . . . . . . . . . . . . 70 5.2 Fundamental Diagram of the VSL Zone . . . . . . . . . . . . . . . . 71 5.3 Design Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.4 Simulation Network of the I-710 Freeway . . . . . . . . . . . . . . . 82 5.5 Fundamental Diagrams of the Road Section with VISSIM Data Points 82 5.6 Macro/Micro Behavior of q1 & q2 of the Closed-loop System . . . . 84 5.7 Density of Discharging Section . . . . . . . . . . . . . . . . . . . . . 85 5.8 VSL Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.9 Discharging Flow Rate . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.10 Density of Discharging Section . . . . . . . . . . . . . . . . . . . . . 87 5.11 Growth and Discharge of the Queue . . . . . . . . . . . . . . . . . . 88 6.1 Multi-Section CTM . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.2 Multi-Section CTM with VSL Control . . . . . . . . . . . . . . . . 93 6.3 Minimal L 0 Under Different Demands . . . . . . . . . . . . . . . . . 95 6.4 Density Curves with Different L 0 . . . . . . . . . . . . . . . . . . . 96 6.5 The Effect of L 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.6 Lane Change Control . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.7 Relationship between ξ and Traffic Demands . . . . . . . . . . . . . 100 ix 6.8 Combine LC with Well-Tuned VSL . . . . . . . . . . . . . . . . . . 100 6.9 Combine LC with Not Well-Tuned VSL . . . . . . . . . . . . . . . . 101 6.10 Simulation Road Network . . . . . . . . . . . . . . . . . . . . . . . 102 6.11 Fundamental Diagram from Microscopic Simulations . . . . . . . . 103 6.12 Fundamental Diagram with Integrated Controller . . . . . . . . . . 105 7.1 Highway Bottleneck . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.2 Fundamental Diagram Per lane . . . . . . . . . . . . . . . . . . . . 114 7.3 Representation of Highway Single Lane under VSL and LC control . 116 7.4 TheSupplyandDemandFunctionsCorrespondingtotheTriangular Fundamental Diagram without/with VSL . . . . . . . . . . . . . . 118 7.5 Simulation Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.6 VSL Commands (Lanes 1, 2,3, and 4) . . . . . . . . . . . . . . . . . 129 7.7 Measured Inflows (Green), Measured Outflows (Blue), and Esti- mated Lane Changes (Black) (Lanes 1, 2,3, and 4) . . . . . . . . . . 131 7.8 Density of Discharging Section (Lanes 1, 2,3, and 4) . . . . . . . . . 132 7.9 Discharging Flow Rate for Open Loop and Closed Loop (Lanes 2,3, and 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 A.1 State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 x Abstract Thewell-knownmacroscopicCellTransmissionModel(CTM)hasbeenwidelyused to develop several Intelligent Transportation Systems (ITS) to mitigate highway traffic congestion. Variable Speed Limit (VSL) and Lane Change (LC) control techniques are the most commonly used and studied ITS applications for regulat- ing the mainstream traffic flow, notably near highway bottleneck locations. While most of the reported macroscopic simulation results showed significant improve- ments in traffic mobility, the resulting data from microscopic simulations and the deployment of such technologies in real traffic environments were somewhat con- troversial; some microscopic simulations and field tests demonstrated inconsistency under different traffic conditions and incident scenarios. This raises the question of whether the CTM needs to be modified to accurately capture the traffic dynam- ics at the bottleneck locations, especially under congested conditions, or whether the proposed mainstream traffic control designs are not robust enough to reject disturbances. In this dissertation, the CTM version, which considers the effect of both capac- ity drop and bounded acceleration, is modified to include a constant disturbance term to account for the uncertainties related to modeling and measurement errors. Motivated by the open-loop stability analysis of the modified model, a robust VSL controller is proposed, one which rejects external disturbances while guaranteeing that the traffic density converges to the desired equilibrium state as demonstrated by both macroscopic and microscopic simulations. The robust VSL control design xi is then extended to a multi-section CTM and combined with an LC control at the discharging section. The section length covered by the most upstream VSL sign is treated as a variable. Via extensive microscopic simulations, the integrated control scheme demonstrates promising improvements in the average travel time, the aver- age number of stops, and the average emission rates when being compared to the cases of no control actions. Furthermore,amultiple-laneCTM-basedVSLcontrolisintroduced,whereeach lane of a motorway is treated as a separate stream; the net lane-changing flow (lat- eralflow)ismodeledasanadditionalunknownterminthetrafficflowconservation equation. This unknown net flow is estimated in real-time, and its estimate, which is developed based on Lyapunov stability analysis, is used at each time in calculat- ing a per-lane VSL control command for each lane of the motorway. Then, a Lane Change (LC) controller is combined with the VSL to prevent creating a queue in the vicinity of the bottleneck. The stability properties of the closed-loop system are analyzed, where the integrated control scheme guarantees that the lane traffic density operates within the free-flow region of the fundamental diagram. There- fore, traffic congestion is relieved, and the per-lane outflow is maximized, except for the blocked lane, when the highway bottleneck is active, as demonstrated by microscopic simulations. xii Chapter 1 Introduction The increasing demand for capacity in urban road networks puts pressure on city plannerstofindwaystomeetthedemandandalleviatecongestion. Thetraditional approachesofexpandingroadnetworksbybuildingmoreroadsandhighwaysareno longer feasible in most areas due to high cost and societal constraints. Instead, an effective and cost-efficient solution is to manage traffic flow via suitable traffic con- trol measures and strategies. This can be done through the Intelligent Transporta- tion Systems (ITS), where advanced communications technologies are integrated into transportation infrastructure, road signs, and individual vehicles. Intelligent Transportation System (ITS) techniques, such as ramp metering, dynamic routing, driver information systems, variable speed limit (VSL), Lane Change (LC) control, andmanyothers, havebeenproposedaspromisingapproachesforimprovingtraffic flow [1, 2, 3, 4, 5, 6]. Theapplicationofvariablespeedlimit(VSL)controltoregulatethemainstream traffic flow and improve both safety and traffic mobility at congested bottlenecks hasevolvedtobecomeapromisingcontroltechniquesincethe1990s[7,8]. TheVSL controllerdynamicallyadjuststhespeedlimitupstreamaccordingtocurrenttraffic states at a downstream highway bottleneck and, thus, it relieves traffic congestion at the location of the bottleneck by improving the throughput. Besides increasing 1 traffic efficiency near bottlenecks, the variable speed limit is expected to overcome the well-known ”capacity drop” phenomenon, defined as the substantial reduction of the maximum discharging flow rate due to a formation of a queue upstream the location of an active bottleneck [9, 10]. Unfortunately, the unmanaged forced lane- changingbehaviorinthevicinityofanactivebottleneckmayresultinunsatisfactory performance of the VSL control; this behavior decreases the speed of vehicles in neighboring lanes, which makes it difficult, if not impossible, for VSL to eliminate traffic congestion. Zhang and Ioannou demonstrated that without applying a lane change (LC) control along with VSL, the VSL is limited to improve the discharging flow rate at the bottleneck location [5]. The underlying idea of the LC control is to pro- vide lane-changing recommendations to upstream vehicles to switch lanes before approaching closed lane(s) in order to avoid the formation of a queue as a result of the bottleneck, resulting in improving the road capacity. Therefore, an inte- grated approach of both VSL and LC controllers leads to fully benefits from the two strategies [11, 5, 12, 13]. With the rapid emergence in the area of communica- tion for autonomous and connected vehicle technologies, the design of a combined VSL and LC control scheme has become more convenient to implement and able to gain higher compliance rates, see Figure 1.1 for example. In turn, it increases the efficiency of existing road networks. A first step and core requirement for designing effective traffic flow control strategies, including VSL and LC controllers, is a reliable macroscopic model that accurately describes the dynamical traffic behavior, yet it is simple enough for design and analysis. The significant challenge in macroscopic traffic modeling is to reproduce the traffic behavior of a given motorway under congested conditions. Developing an accurate traffic flow model has been a topic of interest and intrigue 2 Figure 1.1: Example of Integrating VSL and LC Signs on Highway since 1955 when the seminal Lighthill-Whitham-Richards (LWR) kinematic wave traffic flow model was introduced [14, 15]. Since then, the LWR model has paved the way for the majority of the work on macroscopic traffic flow modeling. In 1994, Daganzo proposed the highly-regarded Cell Transmission Model (CTM), which is a simplified and more practical version of the LWR model [16, 17]. Due to its simplicity and its ability to capture most of the traffic flow characteristics, the CTM has been widely used to develop various traffic control techniques, including VSLcontrols,LCcontrols,oranintegratedschemeofboth. Additionally,theCTM parameters have a physical meaning, which makes them easily calibrated using real traffic data. Being a first-order model, the CTM has several desirable properties in addition to higher computational efficiency with fewer calibration efforts. The model is able toreproducethetrafficdynamicsnotonlyunderfree-flowconditionsbutalsounder congested conditions. Nevertheless, the cell transmission model in its original form suffers from a few limitations; it does not realistically reflect more complex traffic flowcharacteristics,suchasadischargingqueueandthecapacitydropphenomenon. 3 Moreover, the model does not capture the impact of forced lane-changing behavior observed at a congested bottleneck, assuming that lane-changing vehicles adjust their speed instantaneously, allowing for infinite acceleration rates (unbounded ac- celeration). Therefore, diverse modeling approaches and assumptions have been proposed in the transportation literature to incorporate the before-mentioned ob- servationsintotheCTMframework,makingitmoreconsistentwiththemicroscopic traffic flow observations [18, 19, 20, 21, 22]. Since the cell transmission model is an idealization of the actual traffic flow, it does not take into account the uncertainties that the model is highly subjected to when traffic is congested. This may result in an inaccurate calibration of the model parameters. Furthermore, the CTM assumes that the measurements of inflows and outflowsofaconsideredroadsectionareexact. Thisassumptionisnotalwaysvalid in a natural traffic environment because traffic sensors and loop detectors are sub- jecttoerrorsinmeasurement. Disturbancesandinaccurateflowratemeasurements may lead to deteriorating the performance of traffic flow controllers, including VSL andLC,whensuchcontrollersaredeployedinanactualtrafficsituation. Therefore, the issue of robustness of VSL with respect to uncertainties such as disturbances due to modeling and measurement errors is an important topic to investigate. Another related and essential subject to explore is the way of developing, im- plementing, and integrating the VSL technique with a lane change (LC) controller in order to maximize the throughput of a highway bottleneck. Since the vast ma- jority of Variable Speed Limit (VSL) control strategies developed in the literature are based on the macroscopic cell transmission model (CTM), it is assumed that traffic conditions are balanced across all lanes within a section at the bottleneck lo- cation (discharging section). This assumption does not accurately reflect the traffic behavior since every lane behaves differently depending mainly on the intensity of 4 forced lane changes observed at the bottleneck. Treating a multi-lane roadway as a single lane when VSL controllers are being designed does not adequately recognize the lateral flows (lane-changing maneuvers). In addition, this assumption tends to not fully utilize the motorway during the activation of the bottleneck. On one hand, one may conclude that the macroscopic CTM needs to be im- proved, to some extent, to account for the uncertainties due to the inevitable para- metric modeling and measurement errors in order to design reliable traffic flow controllers. On the other hand, perfect macroscopic models of traffic flow systems to be controlled are rarely if ever found when traffic controllers are being designed. As a result, the designed traffic flow controllers must be tolerant of the uncertain- ties in the model; that is, they must be robust and able to function properly under all possible operating traffic conditions and scenarios. Considering all these issues mentioned above when developing a traffic control strategy leads to a reliable and robust scheme, which minimizes the discrepancies between the mathematical re- sults (obtained from the macroscopic CTM) and the actual results (collected from real data or macroscopic simulations). Therefore, based on the first-order CTM framework, this research focuses on the design and analysis of mainstream traffic flow controllers, which not only show significant improvements in macroscopic simulations but also demonstrate similar and consistent results in microscopic simulations. More specifically, this research aims to develop robust VSL control strategies combined with LC controllers to alleviate traffic congestion and its detrimental implications at highway bottlenecks. The stability properties of the closed-loop systems are analyzed with associated proofs. Macroscopic and microscopic (using the commercial software VISSIM) are used to evaluate the performance and demonstrate the effectiveness and robustness 5 of the proposed control schemes and associated benefits under different levels of traffic demand and various traffic scenarios. 1.1 Problem Description Traffic congestion in freeway networks has turned into a frustratingly common oc- currence,mostnotablyatbottlenecklocations. Itleadstoanintensedegradationin freeway infrastructure utilization, in particular during peak hours. Due to limited economic and physical resources in most metropolitan areas, expanding existing freeway infrastructure cannot continue as the only long-term remedy to alleviate traffic congestion. An alternative, effective, and cost-efficient solution is to regu- late traffic flow through the Intelligent Transportation Systems (ITS). Implement- ing ITS techniques and integrating them into transportation infrastructure, road signs, and individual vehicles have become more convenient and reliable with the available advanced communications technologies. The most commonly used ITS applications when it comes to regulating the highway mainstream traffic flow are Variable Speed Limit (VSL) and Lane Change (LC) control strategies. In fact, a combinedapproachofthetwo controllers has shownpromising results in improving traffic mobility and maximizing the throughput at highway bottleneck locations. Over the years, efforts have been made to use the well-known macroscopic Cell Transmission Model (CTM) to design VSL and LC controllers or an integrated scheme of both in order to alleviate traffic congestion and its detrimental impli- cations at freeway bottlenecks. While most of the macroscopic simulation results demonstrate a promising improvement in traffic mobility [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], some of the microscopic simulation results and field tests disagree with the macroscopic simulations, notably under congested traffic conditions, and, 6 in some cases, microscopic simulations and field tests demonstrate inconsistency under different traffic conditions or incident scenarios [33, 34, 35, 36, 37]. Some researchers blame the inconsistencies in the highly disorganized and stochastic be- havior of traffic flow observed at congested highway bottlenecks, which makes it challenging to predict, capture and regulate at a macroscopic level [35, 38, 39, 40]. While these arguments are acceptable in the sense that they contribute to the inconsistent results, the following questions arise: 1. What are the main reasons behind the disorganized traffic behavior observed at active freeway bottlenecks? Under what circumstances this unmanaged chaotic traffic behavior starts to negatively influence the mobility of traffic flow? Is it possible to manage it via VSL and LC controllers? 2. Does the macroscopic Cell Transmission Model (CTM) realistically capture the negative impact of the disorganized traffic flow behavior, or does it need to be modified? 3. Since the macroscopic CTM is an idealization of the actual traffic flow, is it possibletoimprovethemodeltoaccountfordisturbancesduetotheinevitable parametric modeling errors and imperfect measurements of the real flows when designing traffic flow controllers? 4. Despite the presence of disturbances, is it feasible to develop efficient and ro- bust VSL and LC controllers to alleviate congestion at active freeway bottle- necks,consideringallpossibletrafficoperatingscenarios,capacityconstraints, as well as initial conditions? 5. Does treating every lane as a separate stream when developing VSL and LC controllers within the CTM framework obtain more accurate results? Will a per-lane integrated VSL and LC control scheme result in obtaining better 7 results in traffic mobility and road utilization compared to the traditional control approaches, where all the CTM variables are aggregated across the lanes when the controllers are being designed? Answering the aforementioned questions provides a better understanding of the causal effects behind the inconsistency between macroscopic and microscopic sim- ulation results. In return, some modifications and assumptions may be needed to be incorporated within the macroscopic cell transmission model framework and the trafficcontroldesigninordertoreflectandmitigatenotablemicroscopictrafficflow observations as a result of the activation of the bottleneck. 1.2 Existing Work BasedontheCellTransmissionModel(CTM),oramodifiedversionofit,numerous studies have been conducted over the years to investigate the effects of VSL and LCcontrolonimprovingtrafficmobilityatfreewaybottlenecks. Inthissection, the important and related existing works in the transportation literature are reviewed. Over the past few years, the analysis of the stability properties and behavior of the CTM has attracted considerable attention in order to have a better under- standing of the model when devoloping traffic flow controllers. In [41], Gomes et al. performed a thorough analysis of the stability properties of the CTM equi- librium points, but the authors did not take the capacity drop phenomenon into consideration. In addition, the rate of convergence to the equilibrium points was not specified. In [42], the analysis of the structure of equilibrium sets of the CTM was presented. The authors limited their investigation to the cases for which any fixed inflow results in a unique equilibrium density. Reference [43] analyzed the 8 equilibrium points and the stability properties of traffic flow in networks with a polytree topology. It was assumed that the demand function is strictly increasing. This assumption is not valid if the demand function is modified to account for the capacitydropphenomenonandthedecreasingofthedischargingflowrateofacon- gested bottleneck [22, 44, 18, 5]. Lovisari et al. examined the stability properties of the equilibrium points of CTM in terms of the connectivity of a state-dependent dual graph, and They proposed optimal control policies that use a combination of turning preferences and speed limits to optimize convex objectives for dynamic transportation networks [45]. However, their results were established based on the monotonicityoftheCTM,whichisnotthecaseifthedemandfunctionismodified. Moreover, the stability analysis of the closed-loop system was not addressed. The following research efforts, in this paragraph, specifically address the stabil- ity issues of the closed-loop CTM with either VSL control or integrated VSL and LC control. In [22], a proportional-integral VSL controller based on two traffic flow models, the LWR model and the link queue model, was proposed. The established stability properties, however, were local as a result of linearization of the closed- loop system without indication of how big the region of attraction is and how to find it, which limits its practicality. In [5], a combined VSL with Lane Changing (LC) control was developed using a feedback linearization approach based on the CTM. It was shown analytically that the proposed scheme guarantees exponential convergence to a unique equilibrium point in the case when the demand is higher than the downstream capacity. The approach did not account for the decreasing of thedischargingflowwhenthedensityexceedsthecriticalvalue. Theauthorsin[46] rigorously analyzed the stability properties of the equilibrium points of a modified CTM, which accounts for both capacity drop and bounded acceleration, under all possible demand values for both open-loop and closed-loop system based on a feed- back linearization VSL control. Recently, Guo et al. proposed an integrated VSL 9 and LC in the structure of Model Predictive Control (MPC) to alleviate congestion near bottlenecks, where a multi-lane CTM is adopted to predict the traffic states [12]. However, the considered model did not account for the bounded acceleration, meaning that vehicles are assumed to accelerate instantly. Nevertheless, the research efforts mentioned in the previous paragraph, i.e., [5, 22, 46, 36], were conducted under the assumption that perfect knowledge of the CTM parameters and accurate measurement of the flows are available. This cannot be guaranteed in a real environment. Disturbances due to inaccurate mea- surementsandothermodelinguncertaintiesmaydeterioratetheperformanceofthe trafficcontrollers, leadingtounsatisfactoryresults, asshownin[46]. Therefore, the effect of disturbances due to inaccurate measurements as well as other modeling uncertainties have to be taken into consideration in order to design reliable and ef- fective traffic flow controllers. A few studies related to the VSL control design have recentlyaddressed thisissue and proposed robust solutions to handle uncertainties. Li et al. considered uncertain vehicle demands and developed a data-driven VSL algorithm based on a Distributionally Robust Optimization (DRO) approach [47]. The authors restricted the uncertainty source to traffic flows, and the evaluation was only based on traffic densities. Du and Razavi used likelihood estimation to detect sensor faults and restore traffic states in real-time so that they could be fed into the VSL control scheme [48]. However, it was assumed that traffic conditions are roughly balanced across all lanes within a section. This assumption does not adequately reflect the dynamical traffic behavior since each lane within the sec- tion would have different behavior depending on the surrounding traffic conditions [49, 50]. Based on the the macroscopic Cell Transmission Model (CTM), a few research papershaveaddressedthestabilizationofequilibriumpointsoftheCTMsubjected 10 to uncertainties, recently. In paper [51], sufficient conditions for global asymp- totic and global exponential stability for uncertain discreet time networks, such as piece-wise linear CTM, were provided using vector Lyapunov functions. The work, however, was focused only on the Uncongested Equilibrium Point (UEP) of the network. In [52], explicit feedback control laws that guarantee a global exponential stability of general uncertain acyclic traffic networks were constructed for the UEP. ItwasassumedthatthedisturbancedoesnotaffecttheUEPofthenetwork. More- over, the control input is the flow itself, but it is not obvious how to implement the controller using variable speed limits. Previous studies have demonstrated that by adjusting the speed limit upstream a highway bottleneck, VSL is able to improve mobility in the highway [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. Especially in light of the emergence of autonomous and connected vehicle technologies, VSL strategy has become more accessible to implement and able to gain higher compliance rates. Several VSL controllers have been proposed in the literature, which the majority can be classified based on their strategies, as mentioned hereinafter in the following four paragraphs: Feedback Control based VSL: feedback VSL controllers generate the VSL com- mands based on current and previous traffic states of a system. In 2013, Carlson et al. proposed a local feedback controller and demonstrated that the simple feedback controller could provide improvement to the Total Time Spent (TTS), similar to the MPC strategy, albeit using much lower computational effort [23]. The method was then extended to multiple bottlenecks in [24]. In [22], Jin and Jin proposed a proportional-integral (PI) VSL controller to maximize the bottleneck throughput with one VSL sign, located upstream of the bottleneck, by locally stabilizing the vehicledensityatacriticalvalue. In[25],ZhangandIoannoucombinedlanechange 11 (LC) control with a feedback linearization (FL) VSL controller, which guarantees global exponential stability. Model Predictive Control (MPC) based VSL: MPC based VSL control strate- gies compute the VSL commands by predicting the system behavior with dynamic models and solving finite-horizon optimal control problems at each time step in a receding horizon manner. Muralidharan et al. proposed an MPC VSL controller based on the link-node cell transmission model (LN-CTM) [26]. The proposed con- troller was able to recover the bottleneck from the capacity drop and obtained an optimal trajectory in the absence of capacity drop. Frejo et al. proposed a hybrid MPC controller that combines VSL with ramp metering; the proposed method re- duced the computation load of the receding horizon optimization by using genetic and exhaustive algorithms while achieving a good performance in simulation [27]. In [28], an MPC VSL strategy was proposed using a car-following model to reduce both TTS and total emissions. It was shown that a reduction of TTS alone might notreducethetotalemissions. In[29], anMPC-basedVSLcontrollerwasproposed to improve traffic safety, mobility, and the environmental impact simultaneously in a connected vehicle context. Optimal Control based VSL: the approaches of optimal control based VSL gen- erate the VSL commands by solving an optimal control problem over the entire simulation time span and apply this pre-decided VSL command sequence to the system in an open-loop manner. In contrast with the real-time optimizations in MPC-VSL, optimal control based VSL solves the optimal control problem only once in an offline manner and does not consider the actual evolution of the states in real-time, which may be adversely affected by model uncertainty, disturbances, and measurement noise. An optimal control framework for highway network flow was proposed in [30], where [23] applied the framework on a VSL controller that 12 designed to minimize the TTS. In 2016, Como et al. showed the convexity of the optimal VSL control problem with a TTS cost function based on the CTM [31]. ShockwaveTheorybasedVSL: shockwavetheory-basedVSLstrategiesmodelthe traffic flows as kinematic waves and use speed limits to control the propagation of the waves. In [32], Hegyi et al. proposed the SPECIALIST VSL controller strategy based on shockwave theory. The SPECIALIST method detects the shockwave upstreamofthebottleneckandusesVSLtomaketheshockwaveaccumulateslower anddissipatefaster, resultingindampeningtheshockwaveandimprovingmobility. Another critical aspect that has been overlooked when designing a VSL con- troller is the VSL sign locations. Most studies treated the sign locations as fixed parametersandignoredthepossibilitythattheymaysignificantlyaffecttheclosed- loop performance. Some research looked into the distance required for vehicles to complete the process of accelerations and decelerations in order to place the VSL signs [53, 54]. Lately, Mart´ ınez and Jin proposed a method to estimate the optimal VSL location as a function of the speed limit in order to prevent the capacity drop [55]. Thesestudiesfocusedon either a single-section roadway or a small distancein front of the bottleneck; it is unclear how to extend the analysis to a multi-section model, i.e., a long highway stretch subdivided into small sections. Though most of the works mentioned earlier on VSL claimed a significant im- provementintrafficmobility,therewereanumberofstudiesthatquestionedthere- portedimprovementsonTTSunderdifferentincidentscenarios, see[36,35,37,39]. Zhang et al. confirmed that the unmanaged forced lane-changing activities, which take place near active bottleneck locations, were one of the main reasons that VSL could not improve traffic mobility when being implemented in a real environment [25]. The behavior of these systematic mandatory lane changes in the vicinity of 13 a bottleneck is not captured in most traffic flow macroscopic models, including the CTM. This is because executing a lane-changing maneuver is an action of an individual vehicle that is related to multiple factors, some of which are mutuality- dependent. Thus, many microscopic lane-changing models are able to represent detailed and comprehensive lane-changing behaviors, yet they usually require a large number of parameters to do so [56, 57, 58, 59, 60, 61, 62]. Unlike microscopic lane-changing models, modeling the behavior, or the influ- ence, of lane-changing vehicles at a macroscopic level is a daunting task. Thus, researchers have proposed some creative modeling approaches in order to model the lane-changing behavior or its impact on traffic flow within the CTM frame- work. Laval and Daganzo proposed a hybrid approach to model lane changes [63]. The mesoscopic model is a blend of microscopic and macroscopic features, where lane-changing vehicles are treated as moving bottlenecks; for other related studies along this line, interested readers are referred to [64, 65]. Another approach is the one that Jin proposed in [66]. In his work, the author considered modeling the impactoflane-changingvehicles ontraffic flow attheaggregate level. Based onthe observation that, in congested traffic conditions, a lane-changing vehicle influences the traffic behavior of both its current lane (longitudinal interactions) and target lane (lateral interactions), Jin proposed a new variable called ”lane-changing inten- sity”, referred to as ε, to capture the lateral impact of lane-changing vehicles on traffic stream. This variable is determined based on the number of lane-changing vehicles N LC , the lane-changing duration t LC , the time for all vehicles to traverse the lane-changing section T, and the length d LC (which may vary [67]) and traffic density ρ of the lane-changing section as follows: ε = N LC t LC T d LC ρ . (1.1) 14 Srivastava et al. adapted the lane-changing intensity variable in order to model lane-changing based on the CTM, which accounts for bounded acceleration [19]. The effect of the mandatory lane changes at a lane-drop bottleneck was modeled by scaling down the traffic density of the demand function of the section, located immediately upstream the bottleneck, by an effective density factor α ≥ 1, where α = ε+1, as demonstrated by the orange line in Figure 1.2. This effective density factor is introduced to reflect the idea that a vehicle occupies two lanes during its lane-changing process and, therefore, contributes twice as much to density. The proposed model, however, assumed that traffic flows are moving at roughly the same speed in each lane close to a congested bottleneck. This assumption does not truly reflect the traffic behavior since every lane would have different speeds depending on the surrounding conditions [49]. Figure 1.2: The Supply and Demand Functions In recent years, efforts have been made to use the CTM, or modified versions of the CTM, to design lane change (LC) controllers and develop advisory systems to 15 manage the disruptive behavior of lane-changing vehicles at bottlenecks. LC con- trollers are often integrated with other Intelligent Transportation Systems (ITS), specifically with the application of VSL, in order to make the overall integrated control system operates smoothly, resulting in maximizing the throughput by uti- lizing the capacity of the bottleneck. In [5], a combined lane change (LC) and feedback linearization VSL control technique was developed. It was shown that the combined control scheme improved traffic mobility, safety, and environmental im- pactathighwaybottlenecks. Thework wasthen furtherextended toinclude Ramp Metering (RM) in [6]. However, the authors limited their investigation to the case in which the traffic demand is strictly higher than the road capacity. In addition, the integrated control design was developed assuming that traffic conditions are roughly balanced across all lanes within the discharging section, located directly upstream of the bottleneck location. In other words, all the CTM variables are aggregated across the lanes when the integrated control system is being designed. For more effective motorway traffic control strategies involving lane changes, a limited number of studies have considered treating each motorway lane as a sepa- rate entity. Roncoli et al. proposed an optimal control formulation for integrated highway traffic control, including VSL, ramp metering (RM), and LC control [68]. In the presence of vehicle automation and communication systems (VACS), the op- timal control was developed based on a modified and extended multi-lane version of the CTM. The application of VACS allowed the controller to improve traffic per- formance. In references [69, 70, 13], lane-changing feedback controls at bottleneck locations were proposed, assuming that some vehicles are equipped with VACS. The underlying idea is to control the lateral flow (lane changes) to avoid the cre- ation of congestion. Nonetheless, the net lateral flow was considered to be in one direction only, from right to left lanes. In [12], the authors designed an integrated VSLandLCcontrolusingtheapproachofMPC,wherea2-laneCTMwasadopted 16 for the traffic states prediction. The model, however, did not reflect the impact of theboundedaccelerationbehaviorofvehicles,assumingthatlane-changingvehicles could accelerate instantaneously without hindering the following vehicles. 1.3 Contribution Driven by the questions that have been raised in Subsection 1.1, this research is devotedtothedesign,analysis,andevaluationofmainlinetrafficflowcontrolstruc- turesthatarerobustandabletoprovideconsistentimprovementsintrafficmobility near highway bottlenecks. More precisely, this research focuses on developing vari- able speed limit (VSL) control strategies combined with advisory lane change (LC) control systems that are reliable and capable of fulfilling the desired performance requirements, despite the presence of model uncertainties and/or external distur- bances. This results in traffic flow control schemes being competent in alleviating congestion and its negative consequences at active highway bottlenecks under all possible operating traffic conditions and scenarios. The main contributions of this research are summarized as follows: • Ascertain that the macroscopic Cell Transmission Model (CTM) is subjected to a high level of uncertainty under congested traffic conditions. One of the main reasons that contribute heavily to the uncertainty in the model is the mandatory disorganized behavior of unregulated lane-changing activities in the vicinity of an active bottleneck. This disruptive behavior is inadequately represented by the first-order CTM. Without accounting for this source of uncertainty, it may result in an inaccurate calibration of the model parame- ters, which shape the fundamental diagram. Consequently, the performance of traffic flow control strategies that are developed based on the macroscopic 17 CTM may deteriorate when being evaluated in microscopic simulations or a real traffic environment. • Propose feedback linearization (FL) and model predictive control (MPC) schemes for variable speed limit (VSL) control to improve traffic flow mo- bility at highway bottlenecks. The proposed VSL techniques are combined withalanechange(LC)controller,whichisassumedtobeactivatedupstream of the bottleneck location, in order to manage the forced lane-changing ma- neuvers, and, thus, it helps both VSL control strategies to operate smoothly. The performance and robustness of the proposed two control schemes are compared with regard to perturbations on traffic demand, model parameters, and measurement noise using both macroscopic and microscopic simulations. Comparisonresultsshowthatthetwocontrollersperformsimilarly, albeitthe FL approach uses much lower computational effort. • Modify the cell transmission model to incorporate the inevitable existence of model uncertainty, such as disturbances that may be introduced to the model due to parametric modeling errors and inaccurate flow-rate measure- ment. Theinfluenceofsuchdisturbancesonthestabilityofthemodelequilib- rium points is investigated considering all possible operating traffic scenarios. Under certain traffic demand conditions, the stability analysis shows that disturbancescouldshift the equilibriumstate of vehiculardensity and itscor- responding traffic flow from the uncongested region to the congested one in the fundamental diagram, triggering a capacity drop much earlier. • Propose a robust variable speed limit (VSL) control design, which rejects the effects of the disturbances while guarantees convergence to desired sta- ble equilibrium states, which are located in the uncongested region of the fundamental diagram, and then evaluate the controller using the microscopic simulator VISSIM. The results show consistency with those generated by the 18 macroscopic modified CTM used in the design and analysis. The robust VSL is then extended to a multi-section CTM and combined with an LC control at the discharging section. The section length covered by the most upstream VSL sign is treated as a variable in the design. Via extensive microscopic simulations, the integrated control scheme demonstrates promising improve- mentsintheaveragetraveltime,theaveragenumberofstops,andtheaverage emission rates compared to the cases of no control action. • DevelopaMultiple-LaneCTM-BasedVSLControl,wherethenetlane-changing flow is modeled as an additional unknown term in the conservation equation of traffic flow. Then, propose a per-lane VSL control based on the multi-lane CTM, where the unknown net flow is estimated in real-time, and its estimate at each time is used in calculating the VSL control action for each lane. The estimatedvalueofthenetlane-changingflowisdevelopedbasedonLyapunov stability analysis. A Lane Change (LC) controller is then combined with the per-lane VSL to prevent creating a queue in the vicinity of the bottleneck. The stability analysis of the closed-loop system shows that the integrated control scheme guarantees that the traffic density of the lane operates within the free-flow region of the fundamental diagram. Consequently, it alleviates traffic congestion by improving the throughput at highway bottlenecks, as demonstrated by the microscopic simulation results. 1.4 Outline of the Dissertation The remainder of this dissertation research is organized as follows: 19 Chapter 2 provides a brief discussion pertinent to the types of traffic flow mod- els. Then, it reviews the most popular and commonly used first-order and second- order macroscopic traffic models in the transportation literature, including the Cell Transmission Model (CTM), and how it has evolved to be consistent with micro- scopic observations. The CTM framework is the foundational traffic flow model for this research. Chapter 3 proposes and compares two types of the most widely used Variable Speed Limit (VSL) techniques in the literature, namely nonlinear feedback lin- earization (FL) and model predictive control (MPC). The performance and robust- ness of the proposed controllers with respect to perturbations on traffic demand, model parameters, and measurement noise are compared using the total time spent (TTS) as the performance metric. Chapter 4 presents a modified version of the CTM that accounts for the in- evitable modeling and measurement errors. These uncertainties are treated as an unknown external disturbance term that is included in the traffic flow conservation equation. The chapter also investigates the stability properties of the open-loop model, and it shows analytically how the disturbance term influences the location of the equilibrium points on the fundamental diagram. Chapter 5 presents the design of a robust VSL (RVSL) control based on the modified CTM introduced in the previous chapter. It analyzes the stability prop- erties of the closed-loop system with associated stability proofs. Then, the chapter demonstrates the effectiveness and robustness of the proposed RVSL control using both macroscopic and microscopic simulations. 20 Chapter 6 extends the RVSL control to a multiple-section CTM, where the RVSL is combined with a Lane Change (LC) control at the discharging section. The section length covered by the most upstream VSL sign is treated as a variable in the model. Moreover, the integrated control scheme is evaluated later in the chapter using the average travel time, the number of stops, and the emission rates as performance measures. Chapter 7 proposes the design of a per-lane VSL and LC control to mitigate highway traffic congestion near lane(s) drop bottleneck locations. First, the chap- ter introduced a multiple-lane CTM framework, where the net lateral flow (lane- changingflow)ismodeledasanunknowntermintheconservationequationoftraffic flow. Then, it develops an adaptive law to be used with the proposed integrated VSL and LC control to estimate the net lateral flow in real-time. The chapter also investigates the stability properties of the closed-loop system with associated stability proofs. Finally, it evaluates the combined control scheme via microscopic simulations. Chapter8concludestheworkofthisdissertationandhighlightsthemainresults. It also discusses some potential future work opportunities that this research may inspire. 21 Chapter 2 Macroscopic Models for Freeway Traffic Flow 2.1 Introduction A reliable traffic flow model is a prerequisite for developing and evaluating traffic flow control techniques to relieve congestion and improve traffic mobility. On one hand, the model should be comprehensive and accurately predict the traffic dy- namics, especially under congested conditions. On the other hand, it needs to be efficient, flexible, and simple enough for implementation and analysis. Finding a balance between these features while modeling the traffic behavior is significantly challenging. The development of a model to represent the dynamical behavior of traffic flow varies by the desired level of detail and analysis. Traffic flow models are typically classified as microscopic, mesoscopic, or macroscopic [71, 72, 73, 74, 75]. Microscopic traffic models represent in detail every vehicular traffic dynamics and itsinteractionswithothervehiclesinthetrafficnetwork. Thesetypesofmodelsare able to simulate the movements of individual vehicles in time and space. However, they require an intensive computational effort. Therefore, they are often embedded in simulation software tools for the purpose of evaluating the effectiveness of traffic control algorithms [75, 76]. 22 Mesoscopicmodelsincorporatethefeaturesofbothmicroscopicandmacroscopic into one model. Vehicle flow is described in aggregate terms at the mesoscopic level, butdriverbehavioralrulesareestablishedforindividualvehicles. Mesoscopic modelsofferanintermediatelevelofdetail; however,theuseofthesetypesoftraffic models for designing traffic control strategies is limited. Onthecontrary, macroscopictrafficmodelsarethebasisforthedesignoftraffic controllers. Macroscopic models represent the traffic dynamics at an aggregate level, in analogy with the flow of fluids or gases, instead of modeling the behavior of individual vehicles in the network. The traffic behavior at the macroscopic level is described using traffic flow, density, and speed. Researchers have developed a wide variety of macroscopic traffic models in order to capture most of the traffic phenomena observed in real environments [14, 15, 16, 17, 77, 78, 79, 20, 80]. Based on the order of system dynamics, macroscopic traffic flow models can be classified into first-order and second-order models [74, 75]. First-order models capture the dynamics of only one aggregate variable, namely, the traffic density, whereas second-order traffic models, besides considering the dynamics of the traf- fic density, explicitly introduce a dynamic equation for the speed. Hereinafter, the mostpopularfirst-orderandsecond-ordermodelswillbediscussed,consideringalso the extended versions related to this work. We will be focusing on the general un- derstanding of the theory and the formulation of the models. However, any further details regarding the boundary conditions and stability analysis of the models will be discussed when used in the following chapters. 23 2.2 First-Order Macroscopic Traffic Models First-order traffic models are known for their simplicity and computational effi- ciency. Therefore, they are commonly used to develop several traffic control strate- gies. First-order models can be categorized as either continuous or discrete. Con- tinuous models represent the dynamical behavior of traffic with differential equa- tions since space and time are considered as continuous variables, whereas discrete models, in which space and time are discretized, the system dynamics is given by difference equations. A significant first-order macroscopic model for both theoret- ical analysis and practical applications is the Lighthill–Whitham–Richards (LWR) model. LWR is a continuous model. A discretized version of the LWR model is the so-called Cell Transmission Model (CTM), which is widely accepted in the communities of mathematicians and traffic engineers. 2.2.1 The Lighthill-Whitham-Richards (LWR) Model The LWR is one of the earliest continuous first-order macroscopic traffic models, developed by Lighthill and Whitham [14] and by Richards [15] in the 50s. Since then,theLWRmodelhaspavedthewayforthemajorityoftheworkonmacroscopic traffic modeling. Referring to a generic location x and time t, the LWR model is described by the following three equations: ∂ρ (x,t) ∂t + ∂q(x,t) ∂x = 0 (2.1) q = ρv (2.2) q = Q(ρ ) (2.3) 24 where ρ , q, and v denote the vehicle density, flow-rate, and speed, respectively. Equation (2.1) is a first-order partial differential equation that describes the con- servationlawofvehicleflows. Equation(2.2),knownasthehydrodynamicequation, gives the relationship between flow-rate, vehicle density, and speed. The flow-rate q, bydefinition, istheproductofvehicledensity ρ andflowspeed v. Equation(2.3) refers to the flow-density relation under steady-state traffic conditions; that is, the equilibrium flow-rate q is a function of the density ρ . This relationship is usually referred to as the fundamental diagram. Greenshield formulated the first fundamental diagram based on real traffic flow measurements [81, 82]. In this formulation, speed and density are linearly related, leading to a parabolic flow-density relationship, as shown in Figure 2.1. Figure 2.1: Parabolic Fundamental Diagram Greenshield’s fundamental diagram shows that when the density is zero, the flow-rate is zero since there are no vehicles on the roadway. As the density in- creases, the flow also increases until the density is equal to some critical density ρ c . When the traffic density exceeds the value of the critical density, i.e. ρ > ρ c , the flow decreases as the density keeps increasing until it reaches the maximum value, 25 referred to as the jam density ρ j , where the flow-rate must be zero (vehicles are lining up end to end) because the speed is zero. Another widely accepted fundamental diagram other than the parabolic one is the triangular fundamental diagram, see Figure 2.2. This bi-linear fundamental diagram assumes that under uncongested traffic conditions, the speed of the flow is maximumfree-flowspeed, v f ,whereas,undercongestedconditions,thespeedofthe flow is the speed of the congested shockwave w. It is important to point out that solving the LWR model using shockwaves is tedious and difficult. An alternative waytosolvethemodelisusingdiscretizationschemes, asexplainedinthefollowing subsection. Figure 2.2: Triangular Fundamental Diagram 2.2.2 The Cell Transmission Model (CTM) TheCTM,developedbyDaganzo[16,17], isaconvergentnumericalapproximation totheLighthill-Whitham-Richards(LWR)model[14,15],whichlaysthefoundation for most of the proposed macroscopic traffic models. By adopting the triangular 26 flow-density curves (triangular fundamental diagrams) [83, 84], CTM updates the densityofagivenroadsectionbasedontheconservationoftheinflowsandoutflows, determined by the supply (or receiving) and demand (or sending) functions. In the CTM framework, a considered road segment is divided into I homoge- neous cells, consecutively numbered in the direction of the flow starting with the upstream end of the road, from i = 1,2,...,I, as illustrated in Figure 2.3. Figure 2.3: Cell Representation in the CTM Framework Each cell is characterized by the following variables: • The density of cell i, which is the number of vehicles per unit length, denoted by ρ i . • The inflow, which is the traffic volume entering cell i from upstream, denoted by q i . • The outflow, which is the traffic volume leaving cell i, denoted by q i+1 . • The length of cell i, denoted by L i . AgraphicalrepresentationofthesevariablesispresentedinFigure2.3. Thedensity in cell i is then updated according to the following ordinary differential equation: dρ i (t) dt = 1 L i [q i (t)− q i+1 (t)] (2.4) 27 where q i (t) = min{D i− 1 ,S i } D i = min n v f i ρ i ,C i o S i = min C i ,w i (ρ j i − ρ i ) (2.5) D i is the demand function; S i is the supply function; the constants v f i , C i , w i , and ρ j i denote the free-flow speed, the maximum capacity, the back propagation speed, and the jam density of cell i, respectively, as shown in Figure 2.4. From Figure 2.4: Original Supply and Demand Functions equation(2.5),theflow q i isdictatedbythedesiredflowfromtheupstreamcell i− 1 attempting to enter cell i and the maximum capacity of the cell C i . If the density ρ i ≤ ρ c i , where ρ c i denotes the critical density, then cell i is able to receive the desired upstream demand, which is min = n v f i− 1 ρ i− 1 ,C i o . If ρ i >ρ c i , however, the available capacity of the cell decreases at a rate w i as ρ i increases until the cell is fully congested ρ i =ρ j i , where q i = 0. 28 Though the original form of the CTM is able to reproduce the traffic dynamics not only under free-flow conditions but also under congested conditions, it does not allow for capturing more complex traffic flow phenomena, such as the capacity drop, bounded acceleration, and forced lane-changing effects. 2.2.2.1 Modified Versions of the Cell Transmission Model Freewaybottleneckisalocalizedsectionofthemotorwaywheretrafficencountersa reductioninthespeedduetoconstructionzones, accidents, changesinroadgeome- tries, and many other factors. These bottlenecks are the most vulnerable locations in a traffic network since if the arriving demand is higher than the capacity of the bottleneck, congestion occurs, which consequently leads to the so-called capacity drop phenomenon, in which the maximum discharging flow-rate decreases due to a formation of a queue [9, 10]. This phenomenon significantly damages highways efficiencyandmakesthedynamicsofthetrafficflowatthebottleneckhighlyunsta- ble. Understanding the causal factors of the capacity drop is critical for developing macroscopic models and designing effective control strategies. In[10],thecapacitydropwasassociatedwiththepresenceoftheupstreamqueue formation, which consequently led to a reduction in the discharging flow as drivers accelerating away from the queue. Based on detailed observations, Cassidy et al. reported that the capacity drop was triggered by a queue that formed upstream of the active bottleneck and changing-lane vehicles that maneuver around slow traffic [85]. Other following studies confirmed that forced lane changes at an active bottleneck reduced the flow rate, resulting in a capacity drop at the bottleneck [66, 67, 63]. Evidence in past studies strongly suggests that the combination of bounded acceleration due to the queue formation [86, 87] and forced lane changes 29 [63] are the main reasons for the drop in the discharge flow rate at a congested lane-drop bottleneck. Diverse modeling approaches have been proposed in the research community to incorporatethecapacitydropphenomenonintotheCTMframework. Oneapproach is to reduce the outflow of a cell to a fixed value that is lower than the value of the maximum capacity [5, 22, 26, 88]. This modification was incorporated with the CTM by modifying the demand part of the discharging flow-rate in [5]. Let the maximum capacity of the downstream cill i + 1 be C i+1 = C d , where C d ≤ C i . According to [5], the demand function D i is modified as follows: D i = min{v f i ρ i ,(1− ϵ i (ρ ))C d }, if C d <C i min{v f i ρ i ,C i }, otherwise , ϵ i (ρ ) = 0 if 0≤ ρ i ≤ C d v f i ϵ 0 i otherwise (2.6) where the parameter ϵ 0 i referred to as the capacity drop factor (which may depend on microscopic result) to model the drop of the capacity, as shown in Figure 2.5. If C d < C i and ρ i > C d v f i , congestion happens at the bottleneck as a result of a queue formation and forced lane changes, which leads to a reduction of the speed of the discharging flow, leading to a capacity drop. If C d =C i , then the original demand function is applied. The original form of the CTM and the aforementioned modified version do not realistically capture the the behavior of a discharging queue at a congested bottle- neck. Vehiclesinthesetwomodelswereassumedtoadjusttheirspeedimmediately, allowing for infinite acceleration rates (unbounded acceleration). The authors in 30 Figure 2.5: Supply and Modified Demand Function with Capacity Drop [18] proposed a modified demand function, which accounts for bounded accelera- tion, to be used with the CTM. The modified function updates the slope of the demand function under congested conditions to make it linearly decreasing with a rate of ˜ w i , where ˜ w i < w i , of as the density increases, as presented in Figure 2.6. Therefore, to take into consideration the bounded acceleration effects when Figure 2.6: Supply and Modified Demand Function with Bounded Acceleration modeling the capacity drop, equation (2.6) is updated as follows [46]: 31 D i = min{v f i ρ i , ˜ w i (˜ ρ j i − ρ i ),(1− ϵ i (ρ ))C d }, if C d <C i min{v f i ρ i , ˜ w i (˜ ρ j i − ρ i )}, otherwise , (2.7) where ˜ ρ j is the projected demand jam density due to the bounded acceleration effect, ϵ i (ρ ) follows the same definition in (2.6). Figure 2.7 show this modification on the demand function, which is consider later to investigate the stability analysis of the CTM with respect to uncertainties. Figure2.7: SupplyandModifiedDemandFunctionwithBoundedAccelerationand Capacity Drop Another approach proposed in [19] incorporated the forced lane-changing ma- neuvers and bounded acceleration effects together to model the capacity drop at a lane-drop bottleneck. As shown in Figure 2.8, the disruptive impact of lane changes on traffic flow was captured by scaling down the demand function using a new parameter α i , referred to as the effective density factor ( α i ≥ 1), to reflect the intensity of lane-changing, where α i = ε i +1 (see equation (1.1)). This factor was introduced based on the observation that a changing-lane vehicle influences the traffic conditions on both its lane and target lane [66, 67]. Therefore, to take 32 Figure2.8: SupplyandModifiedDemandFunctionwithBoundedAccelerationand Lane-Changing into consideration the effects of both bounded acceleration and forced lane changes when modeling the capacity drop, the demand function is updated as follows [19]: D i = min ( v f i ρ i , ˜ w i ˜ ρ j i α i − ρ i !) , if α i > 1 min{v f i ρ i , ˜ w i (˜ ρ j i − ρ i )}, otherwise . (2.8) Note that if α i = 1, then the resulting fundamental diagram is the same as the one in Figure 2.6. There are several different CTM modifications that have been proposed in the research community; however, the approaches reviewed in this section are ones related to this research. For other CTM-based approaches, interested readers are referred to [20, 21, 22]. 33 2.2.3 Multiple-Lane Cell Transmission Model Those first-order macroscopic models based on the CTM mentioned earlier com- monly treat traffic flow across multiple lanes in a motorway as one aggregated stream in a single lane. In other words, traffic conditions are assumed to be bal- anced across all lanes, although a motorway may consist of multiple lanes, and each lane may experience different behaviors depending on the surrounding condi- tions, notably near bottleneck locations [49, 50]. Treating a multi-lane roadway as a single lane, especially when traffic flow controllers are being designed, does not adequately recognize the lateral flows (lane-changing maneuvers). In addition, this assumption tends not to utilize the full motorway capacity when the bottleneck is active. Therefore, efforts have been made to extend the CTM framework to rec- ognize lane-changing behavior between lanes by treating every lane as a separate stream. In [63], Laval and Daganzo presented a multi-lane hybrid version of the CTM that recognizes the effect of lane-changing vehicles in freeway traffic streams. They modeled the freeway traffic flow as a set of interacting streams connected by the lane-changing flows. The authors included a term to represent the one-directional net lane-changing rate from lane ℓ to laneℓ ′ , refereed to asϕ ℓ ′ ℓ , whereℓ ′ ̸=ℓ, in the conservation equation of the lane. In addition to the supply and demand functions, they introduced three functions, assuming that lane-changing flows move laterally to the downstream section, i.e., lanes changes do not take place between adjacent lanes within the same section. These three functions are defined as follows: (i) the demand for lane-changing from lane ℓ to lane ℓ ′ , denoted by L ℓℓ ′, (ii) the desired set of through flows on lane ℓ, denoted by T ℓ , and (iii) the available capacity on lane ℓ, denoted by M ℓ . These functions are computed as follows: 34 L ℓℓ ′ =π ℓℓ ′D ℓ , ∀ℓ ′ ̸=ℓ T ℓ = 1− X ℓ ′ ̸=ℓ π ℓℓ ′ D ℓ , ∀ℓ M ℓ =S ℓ , ∀ℓ (2.9) whereπ ℓℓ ′ isanewparameterthatrepresentsthefractionofchoice-makersperunit time wishing to change from lane ℓ to lane ℓ ′ ; this parameter is assumed to be proportional to the positive difference of the speed between lanes. D ℓ and S ℓ are the same supply and demand functions introduced earlier in equation (2.5), but they are considered here for the lane instead of the entire section, i.e., replacing the subindex i by ℓ. Then, using the so-called the Incremental-Transfer (IT) principle in [89], the actual lane-changing and through flows are calculated based on a first- come-first-served basis. That is, if the total demand for the inflows is less than or equal to the available capacity of the lane, i.e., T ℓ + P ℓ̸=ℓ ′ L ℓ ′ ℓ ≤ M ℓ , then all demands are met and able to enter the lane. Otherwise, the available space of the lane is proportionally assigned to inflows according to their demands. This can be translated mathematically as follow: γ ℓ = min ( 1, M ℓ T ℓ + P ℓ̸=ℓ ′ L ℓ ′ ℓ ) (2.10) whereγ ℓ representsthefractionofthedemandthatcanadvance. Hence, theactual lane-changing flow ϕ ℓ ′ ℓ , and through flow, referred to as q ℓ , that can both advance are calculated as follows: ϕ ℓ ′ ℓ =γ ℓ L ℓ ′ ℓ , ∀ℓ ′ ̸=ℓ q ℓ =γ ℓ T ℓ (2.11) A similar model formulation has been proposed in reference [50]. Neverthe- less, the authors assumed that the lateral traffic flow moving from lane j to lane 35 occurs within the same section, and they introduced the concept of the ”attractive- ness rate” to reflect the ”aggressiveness” in lane changes, especially near off/on- ramps. In addition, the model also considers the off-ramp and on-ramp flows in the conservation equation of the lane, and it gives a higher priority to lateral flows. Other multi-lane macroscopic/mesoscopic traffic flow models can be found in [17, 20, 78, 90, 89, 91, 92, 93] 2.3 Second-Order Macroscopic Traffic Models The main difference between first-order and second-order models is that the first- order models are restricted to the equilibrium flow-density relationship, which is knownasthefundamentaldiagram. Thefundamentaldiagramrepresentsthetraffic in equilibrium states. In other words, all vehicles in the first-order models are assumed to move and adjust their behavior according to the equilibrium state. That is, the transition behavior of the speed from one state to the other is too coarse. In contrast, second-order traffic flow models describe the non-equilibrium states by introducing a second dynamic equation, which describes the dynamics of the mean speed of vehicles. Similar to the first-order traffic flow models, second- order models can also be classified as either continuous or discrete. In the 70s, the first continuous second-order traffic flow model was introduced, known as the Payne model [77]. 2.3.1 The Payne Model In an attempt to capture the transient behavior of the speed, The PW model transformed the LWR model to a second-order model by adding the acceleration equation (2.12). 36 ∂v(x,t) ∂t =− v(x,t) ∂v(x,t) ∂x | {z } convection + 1 τ h V e (ρ (x,t))− v(x,t) | {z } relaxation − µ ∂ρ (x,t) ∂x ρ (x,t) | {z } anticipation i (2.12) where τ > 0 is a constant referred to as speed adaptation time, and µ is a model parameter. The following three terms are the main ingredients of equation (2.12): 1. The Convection term: this term describes the influence of the downstream speed on the acceleration/deceleration of the traveling vehicles gradually, im- plying that a higher downstream speed leads to an increase in the speed of the traffic flow upstream and vice versa. 2. The Relaxation term: this term is proportional to the difference between v(x,t) and the desired equilibrium speed V e (ρ (x,t)) provided by the funda- mental diagram. It reflects the idea that traveling vehicles tend to adjust their speed to the steady-state speed. 3. The Anticipation term: this term describes the effect of downstream density on the acceleration/deceleration of the traveling vehicles by assuming that drivers look ahead and adjust their actual speed accordingly. If the down- stream density is lower than ρ (x,t), then drivers accelerate and vice versa. This term is also inversely proportional to ρ (x,t) for safety concerns. To sum up, equations (2.1)-(2.3) and (2.12) gives the complete Payne’s second- order traffic flow model. The main difference between this one and the LWR model is that the speed of the traffic flow is considered to be dynamic rather than static. 37 2.3.2 The METANET Model METANET, an acronym for Mod` ele d’ ´Ecoulement de Trafic surAutorouteNET- works,isadiscreteversioninspaceandtimeoftheaforementionedPayne’ssecond- order traffic flow model [94, 79, 95]. The METANET is a quite popular model for designing and simulating traffic control strategies in the transportation literature. The model considers new terms in the conservation equation to model the impact of on-ramp and off-ramp flows on the mainstream dynamical behavior. Each cell in the METANET Framework is characterized by the density of cell i, denoted by ρ i , themeantrafficspeedinincell i, denotedbyv i , themainstreaminflow, denotedby q i− 1 , the mainstream outflow, denoted by q i , the on-ramp flow, denoted by r i , the off-ramp flow, denoted by s i , and the length of cell i, denoted by L i . A graphical representation of these variables is illustrated in Figure 2.9 Figure 2.9: Cell Representation in the METANET Framework Let T be the discrete-time step; then the previously defined traffic variables are calculated for each cell i at each time step k by the following equations: v i (k+1) = v i (k) + T L i h v i (k)v i− 1 (k)− v i (k) 2 i + T τ h V e (ρ i (k))− v i (k) i 38 − µT τL i ρ i+1 (k)− ρ i (k) ρ i (k)+κ − δT L i λ i r i (k)v i (k) ρ i (k)+κ (2.13) ρ i (k+1) = ρ i (k)+ T L i λ i h q i− 1 (k)− q i (k)+r i (k)− s i (k) i (2.14) q i (k) = ρ i (k)v i (k)λ i (2.15) V e (ρ i (k)) = v f exp h − 1 α m ρ i (k) ρ c α mi (2.16) where α m > 0 is a parameter of the parabolic fundamental diagram, and λ i is the number of lanes per cell i. τ , δ , and µ are model parameters. The second-order model described above is extended in the literature to incorporate the impact of applying Variable Speed Limits (VSL) control strategy on traffic flow behavior. 2.3.2.1 Extended Versions of METANET OneapproachtoincludetheimpactofVSListoadjusttheequilibriumspeedcurve V e (ρ i (k)) by scaling the parameters of the curve, such as the free-flow speed v f and the critical density ρ c , by the ratio of imposing the VSL command to the original speedlimit[96,23]. Thatis,ThethreeparametersofV e (ρ i (k))includedinequation (2.16), i.e., v f , ρ c and α m , are rendered the by the following functions: v f [b i (k)] = v f ∗ b i (k) ρ c [b i (k)] = ρ c ∗ {1+A m [1− b i (k)]} (2.17) α m [b i (k)] = α m ∗ [E m − (E m − 1)b i (k)] where the parameters v f ∗ , ρ c ∗ , and α m ∗ denote the non-VSL values, and the other parameters A m and E m are constants to be estimated from available real data. An illustration of the rendered fundamental diagram using (2.17) is shown in Figure 2.10. 39 Figure 2.10: Rendered flow-density curves (1) Another extension of the METANET model to describe the effect of VSL on the traffic flow behavior is to extend equation (2.16) to be the minimum between the original value of V e (ρ i (k)) given by (2.16) and the desired speed limit caused by implementing the VSL control action [97]; accordingly, equation (2.16) is modified as follows: V e (ρ i (k)) = min v f exp h − 1 α m ρ i (k) ρ c α mi ,(1+σ )V control (k) (2.18) where where V control is speed limit imposed on cell i, and (1 + σ ) is the non- compliance factor. This factor reflects the behavior of drivers who usually do not fully comply with the speed limit displayed by the VSL signs. A representation of the rendered fundamental diagram using equation (2.18) is depicted in Figure 2.11. 40 The earlier mentioned second-ordered models, with some of their extensions, discussed here in this section are some of the most popular ones in the transporta- tion literature. For more detail about second-order models, interested readers are referred to [75, 77, 79, 80, 98, 99, 94, 95]. Figure 2.11: Rendered flow-density curves (2) 2.4 Conclusion Thischapterreviewsthemostpopularandwidelyusedfirst-orderandsecond-order macroscopic traffic flow models, where the general understanding of the theory and theformulationofthemodelsarebrieflypresented. Thechapterfocusesonthefirst- order Cell Transmission Model (CTM) and reviews its extended versions related to this dissertation. 41 Chapter 3 Comparison of Feedback Linearization and Model Predictive Techniques for Variable Speed Limit (VSL) Control 3.1 Introduction Different types of variable speed limit (VSL) controllers have been proposed over the years. Although most of them claim a significant improvement in traffic mobil- ity, there is a number of studies that question the reported improvements on travel time spent (TTS) under different incident scenarios [35, 36, 37, 39]. In [5], it was demonstrated that the inconsistency between analysis and microscopic simulations is mainly caused by the forced lane-changing vehicles in the vicinity of an active bottleneck, which, consequently, led to a capacity drop. This unmanaged forced lane-changing behavior decreases the speed of vehicles in neighboring lanes, mak- ing it difficult, if not impossible, for VSL to eliminate highway congestion. The authors in [5] also confirmed that by providing lane change recommendations to vehicles upstream of the bottleneck, it is possible to make most of the lane changes happen away from the active bottleneck. Thus, the capacity drop is dramatically reduced. In this chapter, we propose feedback linearization (FL) and model predic- tive control (MPC) schemes for VSL-actuated highway traffic, where a lane change 42 (LC) controller is assumed to be active upstream of the bottleneck. Both con- trollers are designed with a CTM-based model representing the ideal system. Via simulation studies, we compare the performance and robustness of the proposed controllers with respect to perturbations on traffic demand, model parameters, and measurement noise. 3.2 System Modeling 3.2.1 EffectsofForcedLaneChangesatHighwayBottleneck A bottleneck is a localized section of a highway with the lowest flow capacity. The highway bottleneck can be introduced by a lane drop, a merge point, an incident that causes lane(s) blockage, or other road conditions. Due to the activation of the bottleneck, a queue of vehicles forms as traffic demand increases. Since the discharging flow rate of the bottleneck determines the throughput of the entire highway segment, the modeling of the bottleneck traffic flow is crucial to designing an efficient traffic control strategy. Figure 3.1 shows a highway segment with 3 lanes, where a bottleneck is in- troduced by an incident that blocks the middle lane. The bottleneck’s length is denoted by L b , which is assumed to be small enough that the effect of the density within L b is negligible and does not affect the bottleneck flow. The discharging flow rate q b at the bottleneck is determined by the vehicle density ρ d of the imme- diate upstream section of the bottleneck, referred to as the discharging section. If the speed limit of the discharging section is set to be equal to the free-flow speed, v f = 65 mi/h, then the relationship between the density of the vehicles ρ d and the discharging flow q b is shown by the dashed black curve in Figure 3.2 (a). As can be noticed from the figure, when ρ d is less than some critical density ρ d,c , the 43 Figure 3.1: Highway Bottleneck value of q b increases with ρ d . However, when ρ d exceeds the critical density ρ d,c , a queue forms at the discharging section as a result of forced lane-changing behavior performed by vehicles in the queue. This behavior results in a capacity drop, which makes it difficult for VSL controllers to increase the value of the discharging flow at the bottleneck. VSL regulates the average density ρ d in the discharging section and cannot do anything related to the lane changes at the bottleneck. (a) w/ and w/o LC Control (b) w/ and w/o VSL Control Figure 3.2: Effects of LC and VSL on Fundamental Diagrams Allowing upstream drivers to change their lanes before approaching the bottle- neck location by providing appropriate lane-changing (LC) recommendations, as demonstrated in Figure 3.1, leads to avoiding the capacity drop [5]. As shown by 44 the solid red curve in Figure 3.2 (a), the fundamental diagram becomes continuous at the critical density when applying LC control, which makes it possible for the VSL to stabilize ρ d at ρ d,c , where the maximum possible q b is achieved. Looking more closely at the fundamental diagram with LC control, it can be noticed that it is close to its triangular approximation, represented by the dashed red curve, when the value ofρ d is small. Then, the speed of the flow decreases as ρ d approachesρ d,c . The authors in [5] concluded that the speed reduction is due to modeling error, de- lay in following the speed limits, and drivers’ behavior when passing by an incident zone. With respect to traffic mobility, the deviation of the speed would not harm the benefit of VSL when designing it based on the triangular fundamental diagram as long as ρ d is stabilized at ρ d,c . If the speed limit of the upstream segment of the bottleneck is assumed to be the free-flow speed v f = 65 mi/h, drivers need to accelerate and then decelerate whenapproachingthebottleneck,resultinginshockwavesthatpropagateupstream. Therefore, decreasing the upstream speed limit to v d , such that 0<v d <v f , shifts the critical density in the fundamental diagram to a higher value and decreases the slope of the under-critical part to be close to a straight line [5, 96]. The solid blue line in Figure 3.2 (b) depicts the fundamental diagram under a speed limit of 40 mi/h, which shows that the bottleneck’s capacity does not decrease although the upstream speed limit has decreased, and the critical density has increased. Here the fundamental diagram is very close to its triangular approximation, and the deviation of the speed limit is minimal at ρ d =ρ d,c . By forcing the VSL command toconvergetov d attheequilibriumstate,theshockwaveupstreamofthebottleneck wouldbeattenuated. Underthespeedlimitv d ,andtheassumptionofthetriangular fundamental diagram, the model of highway bottleneck is described as follows: q b = v d ρ d , ρ d ≤ ρ d,c w b (ρ j,d − ρ d ), ρ d >ρ d,c (3.1) 45 where ρ j,d is the jam density (ρ j,d = (v d ρ d,c )/w b +ρ d,c ), and w b is the backward propagating wave speed. 3.2.2 Cell Transmission Model As shown in Figure 3.3, the upstream highway segment of the bottleneck is divided intoN+1sections. TheVSLsignsareplacedatthebeginningofsection0through section N− 1. ρ i , v i , q i , and L i are the vehicle density, VSL command, and inflow, and the length of section i, respectively, where i = 0,1,...,N − 1. In section N, the speed limit is constant, denoted by v d . The discharging flow rate is represented Figure 3.3: Configuration of the Highway Segment by q b . By the conservation law, the dynamics of densities ρ i are described by the differential equations: ˙ ρ i = 1 L i (q i − q i+1 ), for i = 0,1,...,N− 1 ˙ ρ N = 1 L N (q N − q b ). (3.2) Under the assumption of the triangular fundamental diagram, the flow rate q i can be calculated as follows: 46 q 0 = min{d,C 0 ,w 0 (ρ j,0 − ρ 0 )} q i = min{v i− 1 ρ i− 1 ,C i ,w i (ρ j,i − ρ i )}, i = 1,...,N (3.3) wheredisthedemandflowofthishighwaysegment. Here,weassume disconstant. ρ j,i is the jam density of section i, at which q i would be 0. w i is the backward propagating wave speed in section i, C i is the capacity, i.e. the maximum possible flow-rate in section i, given by the following equation C i = v i w i ρ j,i (v i +w i ) . Section N works as the discharging section in Figure 3.3. By applying the lane change(LC)controller, capacitydropatthebottleneckcanberemoved. According to (3.1), we have q b = v d ρ N , ρ N ≤ ρ d,c w b (ρ j,d − ρ N ), ρ N >ρ d,c (3.4) where ρ j,d = (v d ρ d,c )/w b +ρ d,c , and the capacity of the bottleneck is C b =v d ρ d,c . In order to track the number of vehicles lined up at the entrance of section 0, we introduce a new state Q, that is ˙ Q =d− q 0 , (3.5) with Q = 0 at t = 0. Therefore, if the number of vehicles upstream of section 0 is greater than the number at time 0, Q > 0, otherwise Q ≤ 0. We should note here that the introduction of Q is only for the purpose of evaluating the TTS. Both the FL and MPC controllers are implemented based on system (3.2)-(3.4). The performance metric travel time spent (TTS) is defined as follows: TTS = Z T 0 Q(t)+ N X i=0 ρ i (t)L i dt (3.6) 47 3.3 Control Design 3.3.1 Desired Equilibrium Points As discussed in subsection 3.2.1, the equilibrium speed limit is selected to be v d , such that 0 < v d < v f , in order to reduce the speed deviation. We consider the case when the demand is higher than the downstream capacity d > C b , which may introduce congestion at the bottleneck. Figure 3.4, illustrates the triangular fundamental diagrams of both the bottleneck section (represented by the green curve) and the upstream sections, the non-bottleneck sections, (represented by the blue curve). The desired equilibrium points should be the ones at which maximum possible flow-rate C b is achieved, the upstream traffic flow is homogenized, and the average travel time is minimized. Considering both the mobility of the network Figure 3.4: Desired Equilibrium Point and homogeneity of the traffic flow, we set the equilibrium points to be ρ e i =ρ d,c , v e i =v d , for i = 1,...,N. (3.7) 48 ρ e 0 =ρ j,0 − C b /w 0 ,v e 0 =C b w 0 /(ρ j,0 − C b ). (3.8) As shown in Figure 3.4, at the desired equilibrium points (3.7) - (3.8), the inflow and outflow of each section are the same and equal the bottleneck capacity C b . Aroundtheequilibriumpoint(3.7)-(3.8),system(3.2)-(3.4)canbeexpressed as follows: ˙ ρ 0 = (w 0 (ρ j,0 − ρ 0 )− v 0 ρ 0 )/L 0 ˙ ρ i = (v i− 1 ρ i− 1 − v i ρ i , for )/L i , i = 1,...,N− 1 ˙ ρ N = (v N− 1 ρ N− 1 − v d ρ N )/L N , ρ N ≤ ρ d,c (v N− 1 ρ N− 1 − w b (ρ j,b − ρ N ))/L N , ρ N >ρ d,c (3.9) We define the error system as: e i =ρ i − ρ e i for i = 0,2,,...,N and u i =v i − v e i for i = 0,1,...,N− 1; substituting into (3.9), we have ˙ e 0 = (− w 0 e 0 − v e 0 e 0 − u 0 ρ 0 )/L 0 ˙ e i = (v e i− 1 e i− 1 +u i− 1 ρ i− 1 − v e i e i − u i ρ i )/L i for i = 0,1,...,N− 1 ˙ e N = (v e N− 1 e N− 1 +u N− 1 ρ N− 1 − v d e N )/L N , e N ≤ 0 (v e N− 1 e N− 1 +u N− 1 ρ N− 1 +w b e N )/L N , e N > 0 (3.10) The transformation of (3.9) to (3.10) shifts the nonzero equilibrium states of (3.9) to the zero equilibrium point of (3.10). Letting e = [e 0 ,e 1 ,...,e N ] T , and u = [u 0 ,u 1 ,...,u N− 1 ] T , we can implicitly express system (3.10) as: ˙ e =f(e,u) (3.11) 49 3.3.2 Feedback Linearization (FL) Controller According to [25], we design the FL-VSL controller based on system (3.10) as follows: u i = (− v e i e i − λ i L i e i+1 )/ρ i , for i = 0,...,N− 2 u N− 1 = − λ N− 1 L N e N − v e N− 1 e N− 1 +v d e N ρ N− 1 ,e N ≤ 0 − λ N− 1 L N e N − v e N− 1 e N− 1 − w b e N ρ N− 1 ,e N > 0 (3.12) where λ i > 0 for i = 0,...,N− 1 are design parameters. With the FL controller (3.12), the closed-loop system becomes: ˙ e 0 =− w 0 L 0 e 0 +λ 0 e 1 ˙ e i =− λ i− 1 L i− 1 L i e i +λ i e i+1 ,for i = 1...,N− 2 ˙ e N− 1 = − λ N− 2 L N− 1 L N− 1 e N− 1 +(λ N− 1 − v d L N− 1 )e N , e N ≤ 0 − λ N− 2 L N− 1 L N− 1 e N− 1 +(λ N− 1 + w b L N− 1 )e N , e N > 0 ˙ e N =− λ N− 1 L N− 1 L N e N (3.13) Theorem1in[25]showsthatwiththeFLcontroller(3.12), thezeroequilibrium point of the closed-loop system (3.13) is guaranteed to be globally exponentially stable. Takingintoconsiderationthedriver’sacceptanceandsafetyintherealworld,we keep the VSL commands constant within the time interval (kT c ,(k+1)T c ], where k = 0,1,2,... and T c is the control step size. The following constraints are also applied to the VSL command. Let u i (k) denote u i computed by equation (3.12) at t =kT c . Then, we have, ¯v i (k) = [v e i +u i (k)] 5 (3.14) 50 ˜ v i (k) = max{¯v i (k),v i (k− 1)− C v ,v i− 1 (k)− C v } (3.15) v i (k) = v max , if ˜ v i (k)>v max v min , if ˜ v i (k)<v min ˜ v i (k), otherwise (3.16) for i = 0,2,...,N− 1,k = 0,1,2,.... In(3.14),thesymbol[· ] 5 istheoperatorwhichroundsarealnumbertoitsclosest multipleof5. Equation(3.15)describesthesaturationontheamountofdecreasing the VSL commands between successive control steps and highway sections, C v is the maximum decrease allowed. In (3.16), v max and v min are the upper and lower bounds of VSL commands, respectively. 3.3.3 Nonlinear Model Predictive Control (NMPC) Here, we formulate the problem of finding the VSL commands u(· ) that try to maintain system (3.11) at the desired equilibrium points as the following finite- horizon constrained optimal control problem (OCP): minimize u(· ) Z kTc+Tp kTc e(τ ) T ˜ Qe(τ )+u(τ ) T ˜ Ru(τ )dτ subject to e(kT c ) = ˆ e(kT c ) ˙ e =f(e,u), ∀τ ∈ [t,t+T p ] v min − v e ≤ u(τ )≤ v max − v e , (3.17) where t is the current control sampling instant in time, ˆ e(t) is the measurement on error states taken at that instant, ˜ Q and ˜ R are weighting matrices on error and control input, respectively, whereas T p is the prediction horizon. The optimization problem is solved at the beginning of each control step kT c , with ˆ e(kT c ) as the initial condition. Constraint (3.16) has already been included in the constraints 51 of the optimization problem. (3.14) and (3.15) are also applied to the MPC VSL commands before applied to the system. Duetothecontinuous-timedynamics, theOCP(3.17)isaninfinitedimensional optimization problem. We resort to approximating it as a finite dimensional non- linear program (NLP) via the direct multiple shooting method [100]. Details on direct methods from numerical optimal control literature can be found in [101]. 3.4 Numerical Simulation In this section, macroscopic and microscopic simulations are used to evaluate the performance and robustness of the FL and NMPC. 3.4.1 Scenario setup The proposed controllers have evaluated on a southbound segment on the I-710 freeway (between I-105 junction and the Long Beach Port), California, United States of America. The segment of the highway is divided into 8 sections, as shown in Figure 3.5, where the VSL signs are placed at the beginning of each section. In order to create a bottleneck, an incident that blocks the middle lane is introduced at the end of section 7. The LC controller is applied at the beginning of section 7 alongwiththeVSL.Thesimulationnetworkiscalibratedwithrealworlddatafrom the PeMs system (California Department of transportation (2015)). Without an incident, thecapacityofthehighwaysegmentis6800veh/hr. Theidealcapacityof thebottleneckaftertheincidentoccursisabout4500veh/hr. Inoursimulation,the incidenthappens5minutesafterthesimulationstarts, anditlastsfor30 min. The nominal demand is 6000 veh/hr. The desired equilibrium points of this network is calibrated to be: 52 Figure 3.5: Geometry of the Simulation Network ρ e 0 = 278 veh/mi ρ e 1 =ρ e 2 =··· =ρ e 7 =ρ d,c = 110 veh/mi v e 0 = 15.8 mi/h v e 1 =v e 2 =··· =v e 7 = 40 mi/h For the FL controller, we choose λ i = 50 for i = 0,1,...,6. The NMPC controllerisimplementedusingthedirectmultipleshootingmethodviatheCasADi toolbox [102] in MATLAB 8.5.0 (R2015a), on a 64-bit Windows PC with 3.4-GHz Intel Core i7 processor and 8-GB RAM, where IPOPT [103] is used for solving the NLPs. In our simulation, we choose the prediction horizon T p = 10 min, which is muchgreaterthanthecontroltimestepT c =30sec. Weightmatricesarechosenas ˜ Q =Iand ˜ R = 0.1I,withIdenotingtheidentitymatrixofappropriatedimensions. The computation time of NMPC is around 0.35 sec, whereas it is negligible for FL. The NMPC scheme is still computationally tractable, as its computation time of 0.35 sec per step is negligible with respect to the control time step of 30 sec. 53 3.4.2 PerformanceandRobustnessAnalysiswithMacroscopic Simulations To compare the performance and robustness of the FL and MPC VSL controllers, weevaluatethefollowingcriteriaforthetwocontrollers: 1)Totaltimespent(TTS) as defined in (3.6), and sensitivity of TTS with respect to 2) perturbation on traffic demand, 3) perturbation on model parameters and 4) measurement noise. In the simulation,theFLandMPCcontrollersaresynthesizedwiththeidealmodel(3.11), but the control commands are applied on a perturbed model. The structure of the simulation system is shown in Figure 3.6. For the traffic demand, we add Figure 3.6: Block Diagram of the System up to ± 20% perturbation on the nominal demand 6000 veh/hr. For the model parameters,asshowninFigure3.7,werespectivelyaddupto± 20%perturbationon the nominal value of ρ d,c and C b , which directly alter the shape of the fundamental diagram of the bottleneck section. For the measurement noise, we use Gaussian white noise with different levels of standard deviation up to σ = 0.1ρ cb to match the scale of the density measurements. 54 Figure 3.7: Perturbation on the Model Parameters Figure 3.8 shows the behavior of the vehicle density in the discharging section under FL and MPC controller. Both controllers are able to maintain the density around the desired value ρ e 7 = 110 veh/hr after the incident occurs at t = 5 min. The oscillation is introduced by the roundup-to-5 constraint. However, the MPC controller introduces higher frequency chattering and a sharp decrease at the beginning of the incident. Figure 3.8: ρ 7 with FL and NMPC 55 A series of simulation experiments are conducted with different levels of per- turbation and measurement noise. Figure 3.9 shows how TTS varies with varying demand levels. The figure shows that both controllers are able to function properly undervariouslevelsofdemand; theTTSapproximatelyincreasesanddecreaseslin- earlywiththedemand. This demonstrates that both MPC and FL VSL controllers are robust with respect to the variation of demand, which is due to the selection of the desired equilibrium points (3.7) - (3.8), since the speed limit in section 0 is decreased accordingly to prevent excessive traffic demand from entering the entire control segment. Therefore, the discharging flow is not affected. Under different levels of perturbation, the TTS of the FL is always slightly lower than that of the MPC, which shows that the MPC fails to beat the FL in TTS even though the MPC control commands are generated by solving an optimization problem in a receding horizon fashion. Figure 3.9: Performance Sensitivity of no Control (black), FL (blue), and NMPC (red) to Perturbations on Demand d 56 In Figures 3.10 and 3.11, the change in TTS is plotted with respect to different values of perturbation on C b and ρ d,c , respectively. Both controllers achieve signif- icant improvements over the no control case and are able to operate properly even under situations with a high level of uncertainty in perturbed model parameters. With perturbation on C b , the TTS under FL and MPC are increased by 45% and 43% in the worst case, respectively. Considering the fact that in this case, the bottleneck capacity is decreased by 20% as a baseline, the TTS does not increase too much due to the modeling error and is still much lower than that in the no control case. The worst-case for the perturbation on ρ d,c is 27% worse than the non-perturbed value for FL, and 16% for NMPC. Figure 3.10: Performance Sensitivity of no Control (black), FL (blue), and NMPC (red) to Perturbations on C b The sensitivity of TTS performance in the case of varying levels of standard deviation in measurement noise is given in Figure 3.12, which shows that the TTS under both controllers increases with the standard deviation of measurement noise. 57 Figure 3.11: Performance Sensitivity of no Control (black), FL (blue), and NMPC (red) to Perturbations on ρ d,c However, the system does not diverge as the no control case. The performance of FL is always better than that of NMPC in this case. 3.4.3 PerformanceandRobustnessAnalysiswithMicroscopic Simulations The microscopic simulator VISSIM is used to evaluate and compare the perfor- mance of the two controllers with respect to TTS. Table 3.1 shows the microscopic simulation results with calibrated model parameter set: w 0 = 14 mi/h, w b = 40 mi/h, ρ c,d = 110 veh/mi. Both controller show significant improvement compared to the no control case. The performance of the MPC controller is slightly better than the FL controller 58 Figure 3.12: Performance Sensitivity of FL (blue) and NMPC (red) to Increasing Levels of Standard Deviation in Measurement Noise by 2%. Table 3.2 - Table 3.4 demonstrate the simulation results of NMPC and FL controller under different values of model parameters. From the results, we can see that the FL controller is robust with respect to the perturbations on w 0 , w b , and ρ c,d . As to the MPC, the mobility performance is significantly adverse by the perturbationsonw 0 andρ c,d , whichbothchangethevalueoftheequilibriumpoint. However, the MPC is robust with respect to the perturbations on w b , which, in fact, does not change the value of equilibrium point and could be compensated by the control input. Table 3.1: Evaluation Results with Original Parameters TTS (hr) No Contro mean± std 1270± 42 Improvement - FL mean± std 1036± 36 Improvement 18% NMPC mean± std 1018± 41 Improvement 20% 59 Table 3.2: Evaluation Results under Different w 0 TTS (hr) FL w 0 =9 mean± std 1036± 36 Improvement 18% w 0 =14 mean± std 1036± 36 Improvement 18% w 0 =6 mean± std 1036± 36 Improvement 18% NMPC w 0 =9 mean± std 1096± 55 Improvement 14% w 0 =14 mean± std 1018± 41 Improvement 20% w 0 =6 mean± std 1226± 61 Improvement 3% Table 3.3: Evaluation Results under Different ρ c,d TTS (hr) FL ρ c,d =100 mean± std 1024± 44 Improvement 19% ρ c,d =110 mean± std 1036± 36 Improvement 18% ρ c,d =120 mean± std 1031± 43 Improvement 19% NMPC ρ c,d =100 mean± std 1236± 41 Improvement 3% ρ c,d =110 mean± std 1018± 41 Improvement 20% ρ c,d =120 mean± std 1242± 35 Improvement 2% 60 Table 3.4: Evaluation Results under Different w b TTS (hr) FL w b =20 mean± std 1025± 36 Improvement 19% w b =40 mean± std 1036± 36 Improvement 18% w b =60 mean± std 1042± 34 Improvement 18% NMPC w b =20 mean± std 1098± 58 Improvement 14% w b =40 mean± std 1018± 41 Improvement 20% w b =60 mean± std 1092± 53 Improvement 14% 3.5 Conclusion Withthetotaltimespent(TTS)asaperformancecriterion, weevaluatedandcom- paredtheperformanceandrobustnessofnon-linearfeedbacklinearization(FL)and model predictive VSL controllers, where an LC controller is assumed to be active just upstream of the bottleneck, with respect to perturbation on traffic demand, model parameters, and measurement noise. Simulation results show that both controllers work properly under different levels of perturbation and measurement noise. Although the synthesis of the model predictive control (MPC) follows an optimization-basedroutineand,inreturn,shouldprovidethe‘optimal’performance tosomeextent, itconsumesmuchlargercomputationalresourcesanddoesnotpro- vide better performance than the feedback linearization. On the other hand, the proposed FL controller guarantees exponential stability of the equilibrium points. Therefore, by tuning the feedback gains, the FL controller should be able to force the system to converge as fast as possible, only limited by the saturation of the control inputs while improving the throughput. 61 Chapter 4 The Stability Analysis of Cell Transmission Model (CTM) with Disturbance 4.1 Introduction Even though the CTM captures most of the traffic flow characteristics, it is an ide- alization of a far more complex dynamical system. When traffic is congested, the model is subjected to a high level of uncertainty that may result in an inaccurate calibration of the model parameters, which shape the fundamental diagram. The CTM assumes that the measurements of the inflows and outflows of a considered road section are exact, which is not always true in practice since traffic sensors and loop detectors are subject to errors in measurement. Disturbances or mea- surement noise may drive the equilibrium density from the uncongested region in the fundamental diagram to the congested region. Moreover, faulty measurements may lead to an oscillatory behavior of the closed-loop system, as shown in [46]. This chapter investigates the effect of disturbances in the CTM that may be due to measurement flow and parameter errors. The disturbances are assumed to be approximatelyconstantoverlargeintervalsoftime,andtheycouldshiftthedensity and flow equilibrium states to the congested region under certain traffic demand conditions. 62 4.2 The Cell Transmission Model (CTM) with Disturbance The CTM needs to be modified to account for the uncertainties related to the fundamentaldiagramaswellastovariousparametersandmeasurements. Consider a single road section of unit length under the assumption that vehicle density ρ is uniform along the section. The road section is expected to meet a demand of flow d. Let ˜ q 1 and ˜ q 2 represent the true value of the inflow and outflow, respectively. Then, according to the conservation law of traffic flow, we have ˙ ρ = ˜ q 1 − ˜ q 2 . In practice, however, we can only measure the corrupted values of ˜ q 1 and ˜ q 2 due to the inevitable parametric modeling errors and imperfect measurements of the true flows. Let q 1 and q 2 denote the measured inflow and outflow, respectively, and let µ account for the uncertainties in the model related to modeling and measurement errors. Then, the evolution of the road section density ρ with respect to time, shown in Figure 4.1, is given by the following differential equation: Figure 4.1: Single Road Section ˙ ρ =q 1 − q 2 +µ, 0≤ ρ (0)≤ ρ j (4.1) where 63 q 1 = min{d,C,w(ρ j − ρ )}, q 2 = min{v f ρ, ˜ w(˜ ρ j − ρ ),(1− ϵ (ρ ))C d }, if C d <C min{v f ρ, ˜ w(˜ ρ j − ρ ),C d }, otherwise , v f ρ c =w(ρ j − ρ c ) = ˜ w(˜ ρ j − ρ c ) =C, 0<ρ c <ρ j ,0< ˜ w <w,v f > 0, ϵ (ρ ) = 0 if 0≤ ρ ≤ C d v f ϵ 0 otherwise , (4.2) and the parameters in equation (4.2) are defined as follows: • C: the capacity of the considered road section, measured in veh/hr. • C d : the downstream capacity, measured in veh/hr. • v f : the free flow speed of the road section, measured in mile/hr. • w: the back propagation speed, measured in mile/hr. • ˜ w: theratethattheoutflow q 2 decreaseswiththeroaddensityρ ,whenρ ≥ ρ c , measured in mile/hr. • ρ c : the critical density of the road section, at which v f ρ c = w(ρ j − ρ c ) = ˜ w(˜ ρ j − ρ c ) =C, measured in veh/mile. • ρ j : the jam density; the highest possible density, at which the inflow q 1 = 0, measured in veh/mile. • ˜ ρ j : the jam density associated with outflow q 2 , measured in veh/mile. • ϵ 0 : the capacity drop factor, ϵ 0 ∈ (0,1), which is dimensionless. 64 Figure 4.2 shows the aforementioned parameters on the fundamental diagram, defined by the supply and demand functions, based on the CTM that accounts for bounded acceleration effects and the capacity drop phenomenon, which is in- troduced when the demand is high and the bottleneck capacity is lower than the main section capacity, i.e. C d < C. In model (4.1)-(4.2), the parameter µ is an unknowndisturbancethataccountsforalluncertaintiesinthemodel, andq 1 ,q 2 are the measured flows. We assume that µ is bounded by a constant µ m and satisfies |µ |≤ µ m ≪ C d . In other words, compared to the bottleneck capacity, the magni- tudeofthedisturbanceµ issmall, whichalsoguaranteesthat0≤ ρ (t)≤ ρ j ,∀t≥ 0. We also assume that µ is approximately constant for large intervals of time. In the following section, the equilibrium points of the open-loop system de- scribed by (4.1)-(4.2) are identified and their stability properties are analyzed. 4.3 Stability Analysis of the Open-Loop CTM with Disturbance In order to analyze the stability properties of the dynamical model presented in (4.1)-(4.2), all possible operating scenarios are investigated. These scenarios are defined by the capacity of the road section C, the capacity of the downstream section C d , the capacity drop factor ϵ 0 (which may depend on microscopic results [44, 104, 5]), and the level of both demand d and disturbance µ . The following theorem presents the results of the analysis. Theorem 4.3.1. Consider the open loop system (4.1)-(4.2), where the disturbance term µ is assumed to be constant but otherwise unknown. We have the following results: 65 (a) Supply & Demand if C d <C (b) Supply & Demand if C d ≥ C Figure 4.2: The Supply and Demand Functions Corresponding to the Triangular Fundamental Diagram 1. If (d+µ )< (1− ϵ 0 )C d and C >C d , then ρ (t) converges exponentially fast to d+µ v f ,∀ρ (0)∈ [0,ρ j ]. 2. If (d+µ ) = (1− ϵ 0 )C d and C >C d , then • ρ (t) converges exponentially fast to d+µ v f = (1− ϵ 0 )C d v f ,∀ρ (0)∈ [0, C d v f ]. • ρ (t) =ρ (0),∀ρ (0)∈ ( C d v f ,ρ j − d+µ w ]. • ρ (t) converges exponentially fast to ρ j − d+µ w = ρ j − (1− ϵ 0 )C d w , ∀ρ (0) ∈ (ρ j − d+µ w ,ρ j ]. 66 3. If (1− ϵ 0 )C d < (d+µ )≤ C d and C >C d , then • ρ (t) converges exponentially fast to d+µ v f ,∀ρ (0)∈ [0, C d v f ]. • ρ (t) converges exponentially fast to ρ j − (1− ϵ 0 )C d w ,∀ρ (0)∈ ( C d v f ,ρ j ]. 4. If (d+µ ) > C d and C > C d , then ρ (t) converges exponentially fast to ρ j − (1− ϵ 0 )C d w ,∀ρ (0)∈ [0,ρ j ]. 5. If (d+µ ) < C and C ≤ C d , then ρ (t) converges exponentially fast to d+µ v f , ∀ρ (0)∈ [0,ρ j ]. 6. If (d + µ ) ≥ C and C ≤ C d , then ρ (t) converges exponentially fast to C v f , ∀ρ (0)∈ [0,ρ j ]. Proof. Under the assumption that (d + µ ) ≥ 0, the proof follows from that of Theorem 1 in [46] for all six cases by replacing d with (d+µ ). Theorem4.3.1showsthattheequilibriumdensitystatesoftheopen-loopsystem aredirectlyaffectedbytheexternaldisturbance. Infact,thedisturbancetermhasa directeffectontheactualdemandandcouldtriggerthecapacitydropmuchearlier. Depending on the initial condition of the density, the system would reach a state of equilibrium when (q 1 +µ ) = q 2 . While some cases have more than one isolated equilibrium point associated with high density, in case 2, when (d+µ ) = (1− ϵ 0 )C d and C >C d , thereisaninfinite number of equilibrium points. The aim is to design acontrollersuchthatthetrafficflowoftheroadsectionoperateswithinthefreeflow region in the fundamental diagram despite the presence of the disturbance term µ . The control input is variable speed limit commands to vehicles upstream in order to protect the section under consideration and maximize the throughput under 67 different demands, initial density conditions, and constant disturbances. Such a controller is introduced in the next chapter. 4.4 Conclusion Inthischapter,theCellTransmissionModel(CTM)withcapacitydropandbounded acceleration is modified to include a constant disturbance term, which accounts for the uncertainties in the model due to the inevitable parametric modeling and mea- surement errors. The stability properties of the equilibrium points of the modified CTM are analyzed under all possible operating scenarios. The analysis shows that the disturbance influences the location of the equilibrium points on the fundamen- tal diagram. This results in shifting the equilibrium states from the uncongested regiontothecongestedone,whichleadstotheactivationofthecapacitydropmuch sooner than expected in some operating scenarios. 68 Chapter 5 Robust Variable Speed Limit (VSL) Control of the CTM with Disturbance 5.1 Introduction After modifying the CTM, in the previous chapter, to account for a constant dis- turbancethatmayintroduceduetomodelingandmeasurementerrors, thepurpose of this chapter is to design a variable speed limit (VSL) controller that rejects the constant disturbance µ , guarantees convergence to the desired density located in the free-flow region in the fundamental diagram, and improves the throughput of an active bottleneck. 5.2 DesignofRobustVariableSpeedLimit(VSL) Control The underlying idea of the VSL control is to reduce the incoming flow by informing the upstream vehicles to follow a speed limit so that the density and flow rate of the road section converge to the desired possible values, which correspond to the maximum possible throughput at the bottleneck. The control problem would have been trivial if one could directly control the inflow via traffic light control. Since 69 such an approach is not feasible in most traffic situations, controlling the inflow via the upstream speed is the only feasible choice. The nonlinear relationship between the inflow and upstream speed makes the design and analysis of VSL control more challenging. As shown in Figure 5.1, the VSL action v is applied to the upstream road section called the ”VSL zone”. All vehicles in the VSL zone are asked to follow the speed limit v and then follow the free flow speed v f within the road section under consideration. Figure 5.1: Road Section with VSL Control If the VSL command v is less than the free flow speed v f , then the fundamental diagram of the VSL zone is distorted, as shown in Figure 5.2, assuming that the VSL zone has similar characteristics as the road section [105, 106, 22]. From the geometry of the fundamental diagram, it follows that the parameters ρ j , w, and ˜ w remain unchanged, while the maximum possible flow rate the VSL zone could send to the considered road section is given by the term vwρ j v+w . The use of VSL control will affect the inflow q 1 , which in addition to upstream demand d will also depend on how much flow is allowed by the VSL. Therefore, the model (4.1)-(4.2) with VSL control inputs is given by: ˙ ρ =q 1 − q 2 +µ, 0≤ ρ (0)≤ ρ j , (5.1) where 70 Figure 5.2: Fundamental Diagram of the VSL Zone q 1 = min{d, vwρ j v+w ,C,w(ρ j − ρ )}, q 2 = min{v f ρ, ˜ w(˜ ρ j − ρ ),(1− ϵ (ρ ))C d }, v f ρ c =w(ρ j − ρ c ) = ˜ w(˜ ρ j − ρ c ) =C, 0<ρ c <ρ j ,0< ˜ w <w,v f > 0, ϵ (ρ ) = 0 if 0≤ ρ ≤ C d v f ϵ 0 otherwise , (5.2) In equation 5.2, the term that could be influenced by the VSL is vwρ j v+w since it is the only term that depends on the control action, namely the upstream speed of flow v. Let q 1v = vwρ j v+w , and, without loss of generality, assume d < C. Then, system (4.1)-(4.2) can be rewritten as follows: ˙ ρ =q 1 − q 2 +µ, 0≤ ρ (0)≤ ρ j , (5.3) where 71 q 1 = min{d,q 1v ,w(ρ j − ρ )}, q 2 = min{v f ρ, ˜ w(˜ ρ j − ρ ),(1− ϵ (ρ ))C d }, q 1v = median{0,¯q 1v ,C}, (5.4) and the parameters in equation (5.4) follow the same definition as in (5.2). The median function is used to guarantee that when mapping the controlled flow rate q 1v into the VSL command, i.e., v = wq 1v wρ j − q 1v , the speed does not become less than zero or exceed the free flow speed limit. ¯ q 1v is the unconstrained control variable to be designed. The following constants 0 < ρ L ≤ ρ ⋆ < ρ U < C d v f , shown in Figure 5.3, are defined to help design the controller, where ρ ⋆ is the desired value to which we want the density of the road section to converge. ρ L and ρ U denote the lower and upper bounds, respectively, of ρ ⋆ . Figure 5.3: Design Constants 72 Selecting the value of ρ ⋆ is critical for designing the controller. ρ ⋆ needs to be chosensothatthebottleneckcongestionisreducedorpreventedandthedischarging flow rate is improved as compared to the case without applying a control action. Intuitively, ρ ⋆ should be in the uncongested region of the fundamental diagram to take advantage of the free-flow speed in that region. A trivial choice is to make ρ ⋆ = C d v f . This choice corresponds to the highest discharging flow in this case. However, small disturbances or faulty measurements may push the density towards the capacity-drop region, resulting in an oscillatory behavior of the closed-loop system, as shown in [46]. The values of ρ ⋆ and ρ U should be chosen so that we do not lose excessive potential capacity. Therefore, ρ ⋆ and ρ U can be arbitrarily close to the value C d v f as long as the inequality 0 < ρ L ≤ ρ ⋆ < ρ U < C d v f holds, as will be shown later in Theorem 5.2.1. ρ L andρ U are introduced to prevent unwanted rapid switching when applying the controller. When the road section is congested due to the activation of the bottleneck, we decrease the inflow q 1 in order to bring ρ to the uncongested region in the fundamental diagram by letting ¯q 1v =q s (5.5) where q s is a small constant flow such that q s < min{v f ρ L ,(1− ϵ 0 )C d , ˜ w(˜ ρ j − ρ j )}, which guarantees that ∀ρ ≥ ρ L , q s < q 2 , implying that ˙ ρ < 0. A trivial selection is q s = 0. Theoretically, this selection is valid. However, the implication of such a choice leads to sacrificing the potential road capacity. Therefore, the constant flow q s should be selected in a way that we do not significantly lose some of the potential capacity. Thus, q s can be arbitrarily close to the value min{v f ρ L ,(1− ϵ 0 )C d , ˜ w(˜ ρ j − ρ j )} as long as the inequality q s < min{v f ρ L ,(1− ϵ 0 )C d , ˜ w(˜ ρ j − ρ j )} is satisfied. Since the outflow is greater than the inflow in this case, the density 73 of the road section is decreasing. Thus, there exists a finite time instant t 0 > 0 at which ρ (t 0 ) =ρ L , and we set ¯q 1v =q 2 − λ 1 (ρ − ρ ⋆ )− λ 2 Z t t 0 (ρ − ρ ⋆ )dτ +c (5.6) whereλ 1 > 0,λ 2 > 0andcaredesignconstantstobeselectedinordertoguarantee thatρ asymptoticallyconvergestoρ ⋆ . Equation(5.6)isaproportional–integral(PI) controller, whichrejectsthedisturbanceµ andstabilizesthedensityatρ =ρ ⋆ . The integral action enables the PI controller to eliminate the offset with respect to ρ ⋆ introducedbythedisturbanceµ . Iftheintegralpartisremovedfromequation(5.6), the steady-state error with respect to ρ ⋆ may result in capacity drop, as shown in [46]. We can increase the value of λ 1 to suppress the steady-state error in this case. However, the controlled flow rate q 1v is constrained by the saturation of the VSL as well as the road capacity. Therefore, the proportional plus integral control is essential for stability. It rejects the effect of the disturbances and drives the road density to the desired point. Ifρ decreasestotheuncongestedregionρ ≤ C d v f , wedonotwantthedisturbance to push back the density to the capacity drop region. Therefore, if ρ increases and reaches ρ U , we switch back to (5.5). In order to avoid undesirable stability phe- nomena and oscillations as a result of frequent switching, we introduce a hysteresis to make the switching continuous. Therefore, the unconstrained flow ¯ q 1v with hys- teresis characteristics is described as follows: 74 ¯q 1v =k(t) k(0) = k 1 (0) if ρ (0)>ρ L k 2 (0) otherwise k(t) = k 1 (t) if k(t − ) =k 2 and ρ (t) =ρ U k 2 (t) if k(t − ) =k 1 and ρ (t) =ρ L k(t − ) otherwise ,∀t> 0 (5.7) where k 1 (t) =q s , k 2 (t) =q 2 − λ 1 (ρ − ρ ⋆ )− λ 2 Z t t 0 (ρ − ρ ⋆ )dτ − c ,t≥ t 0 , and the constants in the controller are defined as follows: q s < min{v f ρ L ,(1− ϵ 0 )C d , ˜ w(˜ ρ j − ρ j )}, λ 2 > 0, λ 1 > max 2 p λ 2 ,v f + λ 2 v f > 0, c = λ 1 (ρ (t 0 )− ρ ⋆ )− µ m λ 2 . In equation (5.7), k 1 (t) = q s , where q s is the constant flow rate that satisfies q s < min{v f ρ L , (1− ϵ 0 )C d , ˜ w(˜ ρ j − ρ j )} in order to decrease the road density and keep it operating within the free flow region. Since the goal is to improve the throughput, the value of q s should be arbitrarily close to the value min{v f ρ L , (1− ϵ 0 )C d , ˜ w(˜ ρ j − ρ j )}aslongastheinequalityholdsinordertotakeadvantageofsome of the road capacity potential. k 2 (t) is a PI controller that rejects the disturbances µ and forces the road density to converge to the desired density ρ ∗ . Depending on thevalueoftheroaddensityatthecurrenttimeρ (t)andonwhetherattheprevious 75 time k 1 (t − ) or k 1 (t − ) is activated, the controller k(t) switches according to the ’if conditions’ in equation (5.7). Note that while k 2 is active, ρ (t) < ρ L . To prevent the disturbances from pushing the road density towards the capacity drop region, the controller switches from k 2 to k 1 at ρ (t) = ρ L in order to reduce the value of the road density and maintain it within the uncongested region of the fundamental diagram. Once ρ (t) = ρ U , the controller switches to the PI control k 2 to drive the densityoftheroadsectiontothepredetermineddensityρ ∗ . Inthek 2 (t)equation,if k(0) =k 2 (0)att = 0, thent 0 = 0. Ifk(t)switchesfromk 1 (t)tok 2 (t)att̸= 0, then t 0 = t is the switching time instant. The proportional gain λ 1 , the integral gain λ 2 , and c are design constants used to guarantee that ρ asymptotically converges to ρ ⋆ , as we will show later. After obtaining the value of the unconstrained control flow ¯ q 1v , we map the constraint controlled flow rate q 1v = median{0,¯q 1v ,C} into the VSL control action via v = wq 1v wρ j − q 1v . (5.8) When k 2 (t) is activated in (5.7), the value of the section density is strictly less than ρ U , i.e. ρ < ρ U < C d v f . Therefore, q 2 = v f ρ and q 1 = median{0,d,¯q 1v }, due to d<C. Let Φ( t) = Z t t 0 (ρ (τ )− ρ ⋆ )dτ − λ 1 (ρ (t 0 )− ρ ⋆ )− µ m λ 2 ,t≥ t 0 . (5.9) Then, we have ˙ Φ( t) = ρ (t)− ρ ⋆ . Since Φ( t) = Φ( t 0 )+ R t t 0 ˙ Φ( τ )dτ, t ≥ t 0 , we have that Φ( t 0 ) =− λ 1 (ρ (t 0 )− ρ ⋆ )− µ m λ 2 . Therefore, with k(t) = k 2 (t), system (5.3)-(5.4) can be written as: 76 ˙ ρ (t) = median{0,d,¯q 1v }− v f ρ +µ ˙ Φ( t) =ρ (t)− ρ ⋆ , ∀t≥ t 0 ρ (t 0 )≤ ρ L ,Φ( t 0 ) =− λ 1 (ρ (t 0 )− ρ ⋆ )− µ m λ 2 (5.10) where ¯q 1v =k 2 (t) =v f ρ − λ 1 (ρ − ρ ⋆ )− λ 2 Φ , and λ 2 > 0,λ 1 > max{2 √ λ 2 ,v f + λ 2 v f } > 0. Let us first assume that capacity drop is ignored, i.e. (5.10) holds for all ρ (t) ∈ℜ. The following lemma describes the stability properties of (5.10). Lemma 5.2.1 will then be subsequently used to analyze the closed-loop system (5.3)-(5.4) with the controller (5.7), taking into account the capacity drop effect. Lemma 5.2.1. Consider system (5.10), if d + µ ≥ v f ρ ⋆ , we have the following results: 1. System (5.10) has a unique equilibrium point [ρ (t) e ,Φ( t) e ] T = h ρ ⋆ , µ λ 2 i T . 2. ∀(ρ (t 0 ),Φ( t 0 ))∈ℜ 2 , [ρ (t),Φ( t)] T asymptotically converges to h ρ ⋆ , µ λ 2 i T . 3. ∀(ρ (t 0 ),Φ( t 0 ))∈S ={(ρ, Φ) |v f ρ ⋆ − µ − d≤ (λ 1 − v f )(ρ − ρ ⋆ )+λ 2 (Φ − µ λ 2 )≤ v f ρ ⋆ − µ }∩{(ρ, Φ) |− v f ρ ⋆ − µ v f < (ρ − ρ ⋆ ) < − v f ρ ⋆ − µ − d v f }, then (ρ (t)− ρ ⋆ ) ∈ S,∀t≥ t 0 . Proof. The proof of Lemma 5.2.1 is in Appendix A. Based on the value of the inflow q 1 , Lemma 5.2.1 shows that the trajectories of system(5.10)asymptoticallyconvergetotheuniqueequilibriumpoint[ρ (t) e ,Φ( t) e ] T = 77 h ρ ⋆ , µ λ 2 i T . It follows from the proof that if system (5.10) holds for ∀ρ ∈ ℜ, then the unique equilibrium point [ρ (t) e ,Φ( t) e ] T = h ρ ⋆ , µ λ 2 i T is globally asymptotically stable. However, the proof also indicates that if the initial conditions (ρ (t 0 ),Φ( t 0 )) are in the region S 3 ={(ρ, Φ) |(λ 1 − v f )(ρ − ρ ⋆ )+λ 2 (Φ − µ λ 2 )}, it is possible that the trajectory of ρ (t) approaches− v f ρ ∗ − µ − d λ 2 , which is already in the capacity drop region, where (5.10) does not hold. Fortunately, the initial conditions in system (5.10) are set to be in a certain region. Therefore, we only need to show that for some specific ρ (t 0 ), system (5.10) holds for all t ≥ t 0 , and ρ (t) converges to the predetermined density ρ ⋆ . The results of Lemma 5.2.1 are used as a stepping stone to help analyzing the stability properties of the closed-loop system (5.3)-(5.4) with (5.7) in the following theorems. Theorem 5.2.1. Consider the system in (5.3)-(5.4) with the controller (5.7), if (d+µ )≥ v f ρ ⋆ , then ρ (t) asymptotically converges to ρ ⋆ ,∀ρ (0)∈ [0,ρ j ]. Proof. ∀ρ (0) < ρ L , k(0) = k 2 (0). If ∀t ≥ 0, ρ (0) < ρ U , then (5.10) holds for all t≥ 0,thusρ (t)convergestoρ ⋆ asymptoticallyaccordingtoLemma5.2.1. If∃t> 0, such that ρ (t) = ρ U , then k(t) switches to k 1 (t) and for all ρ (t) > ρ L , ˙ ρ (t) < 0, thus ∃t 0 > 0, such that ρ (t 0 ) = ρ L and k(t 0 ) = k 2 (t 0 ). Similarly, ∀ρ (0) > ρ L , k(0) = k 1 (0), then ∃t 0 > 0, such that ρ (t 0 ) = ρ L and k(t 0 ) = k 2 (t 0 ). Therefore, we only need to consider the case when ρ (t 0 ) = ρ L and k(t 0 ) = k 2 (t 0 ). To shift the equilibrium point of the system to the origin, define x 1 (t) = ρ (t)− ρ ⋆ and x 2 (t) = Φ( t)− µ λ 2 . As long as k(t) = k 2 (t), system (5.3)-(5.4) can be written as follows: ˙ x 1 = median{0,d,¯q 1v }− (v f ρ ⋆ +v f x 1 )+µ ˙ x 2 =x 1 , ∀t≥ t 0 x 1 (t 0 ) =ρ L − ρ ⋆ ,x 2 (t 0 ) =− λ 1 x 1 (t 0 )− µ m +µ λ 2 78 Since x 2 (t 0 ) =− λ 1 x 1 (t 0 )− µ m+µ λ 2 , we have that λ 1 x 1 (t 0 )+λ 2 x 2 (t 0 ) =µ m − µ> 0, and because of µ m ≪ v f ρ ⋆ , we also have λ 1 x 1 (t 0 )+λ 2 x 2 (t 0 ) < v f ρ ⋆ − µ . Furthermore, x 1 (t 0 ) = ρ L − ρ ⋆ > − v f ρ ⋆ − µ m v f > − v f ρ ⋆ − µ v f . Similarly, x 1 (t 0 ) < − v f ρ ⋆ − µ − d v f . As a result, x(t 0 )∈S. According to Lemma 5.2.1, as long as x 1 (t)<ρ U − ρ ⋆ we have ˙ x =Ax x 1 (t 0 ) =ρ L − ρ ⋆ ,x 2 (t 0 ) =− λ 1 x 1 (t 0 ) λ 2 where A = − λ 1 − λ 2 1 0 . Since λ 1 > 2 √ λ 2 , A has two real negative roots, i.e. p 1 = − λ 1 + p λ 2 1 − 4λ 2 2 , p 1 = − λ 1 − p λ 2 1 − 4λ 2 2 where 0>p 1 >p 2 . We can calculate that e At =L − 1 [(sI− A) − 1 ] = a 11 (t) a 12 (t) a 21 (t) a 22 (t) where a 11 (t) = 1 2 (e p 1 t +e p 2 t )− λ 1 p 1 − p 2 (e p 1 t − 2e p 2 t ) and a 12 (t) =− λ 2 p 1 − p 2 (e p 1 t − e p 2 t ). Therefore, x(t) =x 1 (t 0 )a 11 (t− t 0 )+x 2 (t 0 )a 12 (t− t 0 ) = 1 2 x 1 (t 0 )(e p 1 (t− t 0 ) +e p 2 (t− t 0 ) )− λ 1 x 1 (t 0 )+λ 2 x 2 (t 0 ) p 1 − p 2 (e p 1 (t− t 0 ) − e p 2 (t− t 0 ) ) 79 Sincep 1 >p 2 ,x 1 (t 0 )≤ 0andλ 1 x 1 (t 0 )+λ 2 x 2 (t 0 )> 0,wehavethatx(t)< 0, ∀t≥ t 0 . Consequently, ˙ x 1 =− v f x 1 − v f ρ ⋆ +µ +d and ˙ x 2 =x 1 , for all t>t 0 . According to Lemma 5.2.1, x 1 (t) converges to 0 asymptotically, which implies that ρ (t) asymp- totically converges to ρ ⋆ ,∀ρ (0)∈ [0,ρ j ]. Theorem 5.2.1 shows that if (d+µ )≥ v f ρ ⋆ , then controller (5.7) drives ρ (t) to ρ ⋆ asymptotically. In the case when (d+µ )<v f ρ ⋆ , the dynamics and the stability properties of the closed-loop system are given by the following theorem. Theorem 5.2.2. Consider the system in (5.3)-(5.4) with the controller (5.7), if (d+µ )<v f ρ ⋆ , then ρ (t) asymptotically converges to (d+µ ) v f ,∀ρ (0)∈ [0,ρ j ]. Proof. Since d+µ < v f ρ ⋆ , then ∃η > 0, such that d+µ ≤ v f ρ ⋆ − η . Similar to the analysis in Theorem 5.2.1, we only need to consider the case where ρ (t 0 )≤ ρ L and k(t 0 ) =k 2 (t 0 ). Let x 1 (t) =ρ (t)− ρ ⋆ and x 2 (t) = Φ( t)− µ λ 2 in order to shift the equilibrium point of the system to (0,0). Then, according to (5.3)-(5.4), we have ˙ x 1 ≤ d− q 2 +µ =d+µ − (v f ρ ⋆ +v f x 1 ) Thus,∀x 1 (t 0 )≤ ρ L − ρ ⋆ , x 1 (t)≤ d+µ v f − ρ ⋆ ,∀t≥ t 0 . Therefore, x 2 (t) =x 2 (t 0 )+ Z t t 0 x 1 (t)dτ − µ λ 2 ≤ x 2 (t 0 )− µ λ 2 − η (t− t 0 ),t≥ t 0 which decreases to negative infinity as t increases. As a result, q 1 saturates at the valueofd,andthus ˙ x 1 =d+µ − (v f ρ ⋆ +v f x 1 ). Therefore,x 1 convergesto (d+µ ) v f − ρ ⋆ , which implies that ρ (t) asymptotically converges to (d+µ ) v f ,∀ρ (0)∈ [0,ρ j ]. 80 In summary, with the proposed controller (5.7), if the sum of the upstream demand d and the disturbance µ is greater than or equal to the predetermined equilibrium flow, the density in the section will converge to the equilibrium point ρ ⋆ . If the sum of the upstream demand d and the disturbance µ is less than the predetermined equilibrium flow, the density converges to d+µ v f , at which the steady stateflow d+µ isthemaximumpossiblevalue. Notethattheselectionofρ L affects the distance from the switching point to the desired equilibrium point. According to the proof of Theorem 5.2.1, we can select ρ L =ρ ⋆ , which still guarantees conver- gence. In addition, since ρ (t) always converges to ρ ⋆ from the left side, ρ ⋆ can be arbitrarilychosentobecloseto C d v f ,aslongastheinequality0 <ρ L ≤ ρ ⋆ <ρ U < C d v f holds. 5.3 Numerical Simulations In this section, the commercial microscopic simulator VISSIM is used to evaluate the performance of the proposed robust VSL control. The results are compared with those generated by the macroscopic CTM used in the design and analysis. 5.3.1 Simulation Network and Fundamental Diagram A 10 mile (16 km)-long southbound segment of the I-710 freeway in Long Beach, California, UnitedStates, isbuiltusingthemicroscopicsimulatorVISSIM,without considering the on-ramps and off-ramps. The considered section of the freeway to be controlled has 3 lanes and is divided into two segments of length 0.71 mile and 0.36 mile upstream of a bottleneck location, as shown in Figure 5.4. Under different levels of traffic demand and by taking into account the high volume of heavy trucks (since the freeway is close to the port of Long Beach), the data of the density and flow rate of the 0.71 mile-long segment are collected in order to 81 Figure 5.4: Simulation Network of the I-710 Freeway obtain the fundamental diagram for the case of no bottleneck. Then, a bottleneck is created by introducing an incident that blocks the middle lane, and the data of the density and flow rate are gathered to demonstrate the fundamental diagram for this case. As shown in Figure 5.5, the blue line describes the fundamental diagram of the 0.71 mile-long segment without incident. Figure 5.5: Fundamental Diagrams of the Road Section with VISSIM Data Points 82 The maximum capacity of the road segment is 8460 veh/hr, which corresponds to a critical density ρ c = 140 veh/mile. The free flow speed is v f = 60 mile/hr. If the bottleneck is activated, then the resulting fundamental diagram is shown by the red line in the same figure, where the maximum capacity in this case is 5200 veh/hr. 5.3.2 Simulation Results 5.3.2.1 Macroscopic Simulation The resulting data from the microscopic simulator are used to validate the macro- scopic model. The macroscopic simulations are performed on a single-section road network with the following parameters: C = 8460 veh/hr, v f = 60 mile/hr, ρ j =460 veh/mile, ˜ ρ j = 980 veh/mile, w = 21 mile/hr, ˜ w = 10 mile/hr, C d = 5200 veh/hr, and the demand d = 6000 veh/hr. The controller is applied to the single-sectionsystemwiththefollowingdesignconstants: λ 1 =200,λ 2 =900,µ m = 350 veh/hr, q s = 3200 veh/hr, ρ ∗ = ρ L = 75 veh/mile, and ρ U = 84 veh/mile. It is assumed that the incident takes a place 25 minutes after the simulation starts and remains for 30 minutes. Figures 5.6, 5.7 and 5.8 show the behavior of the flow rates, densityandtheVSLcommands, respectively, withboththemacroscopicand microscopic simulations shown together for the sake of comparison. The solid red lines illustrate the macroscopic results. As seen from the Figure 5.7, when the in- cident happens at 1500 sec, the density is 100 veh/mile, which is greater than ρ L . It results in reducing the inflow q 1 , represented by the red line in Figure 5.6 (a), since q 1v = ¯q 1v = q s = 3200 veh/hr. Thus, the VSL command is v = 10 mile/hr. The outflow q 2 , represented by the red line Figure 5.6 (b), decreases as the value of ρ keeps decreasing until ρ ≤ ρ L . 83 (a) Inflow q1 (b) Outflow q2 Figure 5.6: Macro/Micro Behavior of q1 & q2 of the Closed-loop System At that moment, the PI controller takes over and eventually forces the outflow to converge to 4500 veh/hr, which corresponds to the predetermined density value ρ ∗ = 75 veh/mile, where the VSL commands are v = 15 mile/hr. During the incident time (from 1500 to 3300 sec), there is a difference of 300 veh/hr between the values of q 1 and q 2 since q 1 − q 2 +µ = 0. After the incident is removed, the VSL displays the value of 60 mile/hr, and both inflow and outflow converge to the constant demand d = 6000 veh/hr. 84 Figure 5.7: Density of Discharging Section Figure 5.8: VSL Commands 5.3.2.2 Microscopic Simulation The proposed controller is evaluated using the commercial software VISSIM by applying it to the same simulated segments of the I-710 freeway in Long Beach. The freeway is injected with demand d = 6000 veh/hr, with the consideration that the traffic flow in that area has a high volume of trucks since it is near Long Beach Port. The middle lane is blocked with an accident in order to create a bottleneck. The accident takes a place 25 minutes after the simulation starts and lasts for 85 30 minutes (1500 to 3300 sec). The VSL commands are given to vehicles at the beginning of each section, as illustrated in Figure 5.4. ThebluecurvesinFigures5.6, 5.7and5.8showthemicroscopicbehaviorofthe inflow, the outflow, the density of the discharging section, and the VSL commands. When the accident happens at 1500 sec, the VSL control is activated by gradually reducing the speed in steps of 5 mile/hr to ensure safety for the drivers, until it reaches 10 mile/hr. Then it increases to 15 mile/hr where it stays the same until the incident is cleared, as represented by the blue curve in Figure 5.8. The outflow q 2 reduces dramatically from around 6500 veh/hr to approximately 3600 veh/hr. Then, it slightly increases and fluctuates around 4500 veh/hr. However, once the incident is removed, the flow jumps immediately to around 8400 veh/hr andthengraduallydecreaseswithtime, whenitfluctuatesaround6000 veh/hr. As illustrated by the blue curve in Figure 5.7, the VSL stabilizes the density around ρ ∗ = 75 veh/mile and stays less than ρ U = 84 veh/mile. The resulting VISSIM curves of the inflow, the outflow, the density of the discharging section, and the VSL commands are similar to the ones carried out by the macroscopic simulations. The microscopic VISSIM simulator is also used to compare the open loop to the closed-loop system for the outflow, density, and the queue length. Figure 5.9 represents the discharging flow of the bottleneck with and without control. In the case of no control, illustrated by the green curve, the flow rate at the bottleneck drops to around 3380 veh/hr due to the lane closure and capacity drop. When the incident is cleared, the flow rate increases directly to 7500 veh/hr. Then, it concentrates around 6000 veh/hr. When the VSL controller is applied, illustrated by the blue curve, the flow rate decreases to around 4500 veh/hr, which is higher by 33% compared to that in the no control case during the time of the incident. It then increases to approximately 6000 veh/hr after the incident is over. 86 Figure 5.9: Discharging Flow Rate AsseeninFigure5.10,thedensityofthedischargingsection(greencurve)starts increasing due to the incident from around 100 to 350 veh/mile since no control action is applied. Then the density stays around 350 veh/mile until the incident is Figure 5.10: Density of Discharging Section removed. When the VSL control is applied, the density (blue curve) is stabilized around the preset density value ρ ∗ = 75 veh/mile, which is much lower than the scenario without control. 87 The definition that the authors used in [107] to track the number of vehicles lined up at the entrance of the controlled sections is used here to measure the queue length. Let Q represent the number of vehicles in the queue. Using the flow conservationequation,wehave ˙ Q =d− q 1 . NotethatQonlytracksthelengthofthe upstream queue. Therefore, the stability of the closed-loop system is not affected. Figure 5.11 shows the time evolution of the queue length for both open-loop and closed-loop systems, represented by the green and blue curves, respectively. In the case of closed-loop system, the queue length increases less rapidly, reaches a lower maximum and decreases fast when the incident is removed compared to the no control case. Figure 5.11: Growth and Discharge of the Queue 5.4 Conclusion BasedonamodifiedversionoftheCellTransmissionModel(CTM),whichconsiders a constant disturbance term that accounts for the uncertainties in the model, a robust variable speed limit (VSL) control is proposed to reject the disturbance while improving the throughput. The closed-loop stability analysis shows that the 88 proposed VSL controller guarantees that the density of the considered road section asymptotically converges to the predetermined desired density, located in the free flow region in the fundamental diagram, despite the presence of the disturbance. Using both macroscopic and microscopic (VISSIM) simulations by applying it to a 1.07 mile-long southbound segment of the I-710 freeway in Long Beach, California, United States, the simulation results show that the controller stabilizes the density of the road section at the desired density. In addition, the discharging flow rate is improved by 33% by using the robust VSL controller when compared to the case of no control action. 89 Chapter 6 Evaluation of Integrated Robust Variable Speed Limit and Lane Change Control Considering VSL Zone Distance 6.1 Introduction In this chapter, the Robust VSL control proposed in the previous chapter is ex- tended to multi-section. The Robust VSL is combined with LC control to alleviate highway traffic congestion caused by lane drop and reject possible uncertainties. The length covered by the most upstream VSL sign is treated as a variable (L 0 ) and its impact is analyzed on the closed-loop convergence. The controller is eval- uated under different levels of measurements and model uncertainties in order to examine the robustness of the proposed controller. 6.2 Multi-Section Cell Transmission Model with Disturbance In Multi-Section CTM framework, a considered road segment is divided into N homogeneous cells/sections, consecutively numbered in the flow direction starting with the upstream end of the road, from i = 1,2,...,N, as illustrated in Figure 6.1. 90 Figure 6.1: Multi-Section CTM ρ i and q i are the vehicle density and inflow of section i, respectively. As demon- strated in the previous chapter, the measurement and model uncertainties could be represented as an additional termµ in the conservation law of traffic flow. Without loss of generality, it is assumed that the geometry of all sections is identical, and each section is of a unit length. Let C denote the maximum capacity of each sec- tion, whereas C d denote the capacity at the outlet. The road segment is expected to meet a demand of flow d, which appears at the entrance of section 1. Therefore, the density ρ i is updated based on the following equation of conservation of traffic flow: ˙ ρ i =q i − q i+1 +µ, 0≤ ρ i (0)≤ ρ j , fori = 1,...,N (6.1) where q 1 = min d,C,w(ρ j − ρ 1 ) q i = min v f ρ i− 1 , ˜ w(˜ ρ j − ρ i− 1 ),C,w(ρ j − ρ i ) , fori = 2,...,N q N+1 = min v f ρ N , ˜ w(˜ ρ j − ρ N ),(1− ϵ (ρ N ))C d , ifC d <C min v f ρ N , ˜ w(˜ ρ j − ρ N ),C d , otherwise (6.2) 91 ϵ (ρ N ) = 0 if 0≤ ρ N ≤ C d v f ϵ 0 otherwise , and the the parameters in equations (6.2) and (6.1) follow the same definitions as in chapters 4 and 5. Since it is assumed that all the sections, from 1 to N, have the same maximum capacity C, the capacity drop can only occur at the outlet of section N, when C d <C, which affects the value of q N+1 , as expressed in (6.2). 6.3 Control Design ThissectionaimstodevelopacombinedVSLandLCcontroldesignbasedonmulti- sect CTM. The section length covered by the most upstream VSL sign is treated as a variable in the design and its impact on the performance of the closed-loop system is investigated. 6.3.1 Robust Variable Speed Limit Control for multi-sect CTM Thecontrolobjectiveistorejectuncertaintiesandmakethedensitiesofallsections converge to a desired value, denoted as ρ ∗ . In perfect traffic conditions (without any uncertainty), a trivial choice is to make ρ ∗ = C d /v f , which corresponds to the highest discharging flow-rate in this situation. However, a small disturbance may move the density towards the capacity-drop region, which introduces unwanted os- cillatorybehavioroftheclosed-loopsystemandnegativelyimpactstheconvergence rate. On one hand, the value of ρ ∗ needs to be comprised for the sake of robust- ness, i.e., ρ ∗ < C d /v f . On the other hand, it should be chosen so that not to lose excessive potential road capacity. 92 Figure 6.2: Multi-Section CTM with VSL Control As shown in Figure 6.2, the use of VSL affects the inflow q i since the upstream speed (v i− 1 ≤ v f ) is adjusted so that the traffic density ρ i converges to the desired valueρ ∗ . The following equations describe the dynamics of the flows when the VSL controller is activated: q 1 = min d, v 0 wρ j v 0 +w , v 1 wρ j v 1 +w ,w(ρ j − ρ 1 ) q i = min v i− 1 ρ i− 1 , v i− 1 wρ j v i− 1 +w , v i wρ j v i +w ,w(ρ j − ρ i ) fori = 2,...,N− 1 q N = min v N− 1 ρ N− 1 , v N− 1 wρ j v N− 1 +w ,C,w(ρ j − ρ N ) q N+1 = min v f ρ N ,(1− ϵ (ρ N ))C d , ˜ w(˜ ρ j − ρ N ) (6.3) The term v i wρ j v i +w in (6.3) is the maximum possible flow governed by the VSL control action v i . Therefore, the speed limit commands for each section can be computed as following: v 0 = wq 1v wρ j − q 1v v i− 1 = q iv ρ i fori = 2,...,N v N =v f (6.4) 93 where q iv is the desired inflow of section i. Assume that the disturbance term µ in (6.1) is bounded by a constant µ m and satisfies |µ | ≤ µ m ≪ C d . In order to reject µ and guarantee the convergence of the closed-loop system, the following proportional-integral (PI) controller equation, introduced earlier in 5.2, is adopted in here in the design of q iv : q iv =q i+1 − λ 1 (ρ i − ρ ∗ )− λ 2 Z t t 0 (ρ i − ρ ∗ )dτ − λ 1 (ρ i (t 0 )− ρ ∗ )− µ m λ 2 (6.5) whereq i+1 andρ i aremeasurementssubjecttouncertainties, λ 1 > 0andλ 2 > 0are the proportional and integral gains, respectively. These parameters are initialized with the empirical values from [22] and tuned based on simulation results. t 0 denotes the time when the controller is activated. To ensure safety and feasibility in real world, we also incorporate the following constraints on the speed limit computations: • v i is rounded to be a multiple of 10 km/h • The bounds of v i : 20 km/h≤ v i ≤ 100 km/h. • v i can be increased or decreased by at most 10 km/h in each control cycle. 6.3.2 The Effect of L 0 Determining the length of the most upstream section L 0 , as shown in Figure 6.5, playsanessentialroleinwhethertheVSLcontrollerisabletostabilizethedensities across all sections around the predetermined density or not. For a specific highway segment, the value of L 0 required under different traffic demands could be found by simulations. 94 To determine the value of L 0 in our case, the traffic flow on the southbound segment of the I-710 freeway in Long Beach, California, United States, is simulated using the commercial microscopic traffic software VISSIM. The considered segment of the freeway has 3 lanes and subdivided into 6 sections of length 1.6 km, i.e., L i = 1.6 km (1 mi) for i = 1,...,6. The road capacity of each section C = 7200 veh/h. The bottleneck capacity C d = 2 3 C = 4800 veh/h when one lane is closed out of three. Figure 6.3 presents the smallest L 0 in integer miles that guarantees the convergence under different demand levels within [ C d ,C]. When d < C d , no control action is needed. When d > C, the actual vehicle input is restricted by C, which produces the same results as d = C. The simulation results indicate that a higher demand requires a longer L 0 . Figure 6.3: Minimal L 0 Under Different Demands To show the impact of L 0 on convergence, the traffic density behavior with respect to time of sections 1 and 5 under two choices of L 0 with the same level of demand d = 7000 veh/h are plotted in Figure 6.4. The dash-dotted curves 95 represent the unstable behavior of the densities caused by a too-small L 0 choice, while the solid curves depict the stable density behavior produced by setting L 0 properly. Figure 6.4: Density Curves with Different L 0 To understand the effect of L 0 , we need to consider the moment when the lane drop happens, and the VSL controller, consequently, is activated. In Figure 6.5, the black vehicles represent the ones that have entered the considered segment of the freeway before the lane drop takes place, so they are not affected by the VSL command v 0 , while the yellow vehicles are the ones entering the network after the lane drop happens, so they are influenced by v 0 as the controller has already been activated. When d > C d , the value of v 0 is usually less than those downstream v i , i = 1,2,...,6 because it is being utilized to reduce the incoming flow down to v f ρ ∗ . The downstreamspeedlimitsareresponsibleformaintainingasteadyflow,whichmeans they can be equal to v f as long as its successive section is not congested. Although 96 the black vehicles quickly approach the bottleneck and create the congestion, the yellow vehicles are held by v 0 , which produces a low-density space that allows the bottleneck congestion to be removed and the shockwaves to be smoothed out. The solid density curve of section 1 in Figure 6.4 confirms the existence of the low- density space. As the demand increases, congestion and shockwaves become more severe. This requires increasing L 0 so that these yellow vehicles are held longer by v 0 . Figure 6.5: The Effect of L 0 6.3.3 Lane Change Control In order to improve the bottleneck throughput and relieve congestion, a Lane Change (LC) control is implemented in the discharging section N. As shown in Figure 6.6, vehicles moving in the closed lane within a distance (denoted as d LC ) from the incident location will be advised to make lane changes to the neighboring lanes. The idea of Lane Change (LC) Control is to provide LC recommendations to upstream vehicles, while in motion, before arriving at a queue formed as a result of the bottleneck. The LC control framework involves notifying vehicles of the up- coming bottleneck and suggesting appropriate lanes to which they could change. 97 Figure 6.6: Lane Change Control Another critical component in designing LC control is determining at what dis- tance from the bottleneck such recommendations are given. The details of the LC controller are presented below: 6.3.4 Lane-Changing Recommendation Messages Consider a general multi-lane highway with I lanes, with lane 1 (lane I) being the most right (left) lane. Let R i denote the lane-changing recommendation for lane i, where i = 1,2,...,I, using the following rules: 1. If lane i, 1≤ i≤ I, is open, then R i = ”Keep Straight”. 2. If lane i, i = 1 (i =I), is closed, then R i = ”Change to Left (Right) Lane”. 3. If lane i, 1 < i < I, is closed, and both lane i− 1 and lane i+1 are open, then R i = ”Change to Either Lane”. 4. Iflanei, 1<i<I, isclosed, andlanei− 1(lanei+1)isclosed, butlanei+1 (lane i− 1) is open, then R i = ”Change to Left (Right) Lane”. 5. If lane i, 1 < i < I, is closed, and both lane i− 1 and lane i+1 are closed, then, check R i− 1 and R i+1 : • If R i− 1 =R i+1 , then R i =R i− 1 =R i+1 . • If R i− 1 ̸=R i+1 , then R i = ”Change to Either Lane”. 98 The five rules mentioned above cover all possible operation scenarios. They are well-defined, mutually disjoint, and self-consistent. The lane-changing recom- mendations depend on the location of the bottleneck. Therefore, the length of the lane-changing control segment plays an essential role in the LC control design so that vehicles could safely switch lanes. 6.3.5 Length of LC Controlled Segment The length of the LC controlled segment, d LC , needs to be long enough to provide sufficient space and time for upstream vehicles to change lanes safely. On one hand, if more lanes are blocked at the bottleneck, a longer LC control distance is needed. On the other hand, a more extended LC controlled segment may lead to road surface underutilization. Therefore, using extensive microscopic simulation studies, the LC controlled segment length d LC is decided by the following empirical equation: d LC =ξn (6.6) ξ is a design parameter that depends on the values of both the bottleneck capacity and upstream demand. n is the number of closed lanes at the bottleneck. The model (6.6) is empirical and more spacial than temporal despite the dependence of ξ on demand, which may be time-varying. For more details about the the model, interested readers are referred to [11]. For a specific highway segment, the minimum value of ξ that is required under different traffic demands could be found by simulations. In our case, We set d LC = 800 meters based on the simulation results shown in Figure 6.7. 99 Figure 6.7: Relationship between ξ and Traffic Demands ToshowtheeffectofLCcontrol,wecomparethedensitycurvesofsections1and 5 using only VSL control versus the combination of VSL and LC control. As can be seen in Figure 6.8, when the VSL controller is well-tuned, adding LC control reduces the time to convergence. In Figure 6.9, when the VSL controller is not well-tuned or there exists a large disturbance, the convergence cannot be achieved by using only VSL control, whereas adding LC control may drive the closed-loop system to the desired value. Therefore, incorporating LC control is beneficial for the closed-loop performance, especially when there are uncertainties in the model. Figure 6.8: Combine LC with Well-Tuned VSL 100 Figure 6.9: Combine LC with Not Well-Tuned VSL 6.4 Numerical Simulations The commercial microscopic simulator PTV VISSIM 10 is used to evaluate the performance of the proposed controller in this section. 6.4.1 Network Configuration We use the commercial software PTV VISSIM 10 to carry out the microscopic simulations. The road network in Figure 6.10 is built along a 14.4-km (9 mi) segment of the I-710 freeway from I-105 to the Long Beach Port in California, United States. As previously mentioned, the length of section 0 depends on the demand d with a maximum value of 4.8 km (3 mi) when d ≥ 6500 veh/h. The remaining 6 sections have a unified length of 1.6 km (1 mi). The network has a fixed lane number of 3, and no on-ramps or off-ramps are considered. An incident triggers middle lane closure and creates a bottleneck at the end of the last section. 101 Figure 6.10: Simulation Road Network 6.4.2 Parameter Selections According to the fundamental diagram shown in Figure 6.11, the road capacity C = 7200veh/handthebottleneckcapacityC d = 4800veh/h. Thedensityleading to maximum bottleneck throughput is C d /v f = 48 veh/km. To ensure the stability with uncertainties, we select ρ ∗ = 45 veh/km. When the capacity drop happens, the dropped capacity (1− ϵ 0 )C d is around 4300 veh/h. Thus, the level of capacity drop can be determined as ϵ 0 = 0.1. The remaining model parameters are v f = 100 km/h, w = 30 km/h, ˜ w = 15 km/h, ρ j = 312 veh/km, ˜ ρ j = 552 veh/km. Each simulation run lasts for 90 minutes. The incident happens at the 10th minute and is removed at the 80th minute. We take the measurements and update the control commands every 30 seconds during the simulation, which is consistent with the data collection rate from the Highway Performance Measurement System (PeMS). 6.4.3 Performance Measurements In this section, we evaluate the performance of the proposed controller in a one- lane-drop scenario under different levels of demands. The following methods have been adopted for the evaluations [5]: 102 Figure 6.11: Fundamental Diagram from Microscopic Simulations • Average Travel Time (ATT): the average time spent for each vehicle to travel through the whole network. ATT = 1 N v Nv X i=1 (t i,out − t i,in ) (6.7) where N v is the number of vehicles passing through the network, t i,in and t i,out are the time vehicle i enters and exits the network. • Average number of stops: the average number of stops performed by each vehicle when traveling in the network. ¯s = 1 N v Nv X i=1 s i (6.8) where s i is the number of stops performed by vehicle i. 103 • Average emission rates of CO 2 : the calculation of emission rates is based on the MOVES model provided by the Environment Protection Agency (EPA). ¯ R = Nv X i=1 E i / Nv X i=1 d i (6.9) whereE i is the emission produced by vehicle i andd i is the travelled distance of vehicle i Table 6.1: Performance Evalutions d = 7000 veh/h Evaluations No Control VSL Only VSL+LC ATT (min) 22.2 18.7 18.6 ¯s 43.5 2.0 1.3 CO 2 (g/veh/km) 399.4 319.3 313.8 Table 6.2: Performance Evalutions d = 6000 veh/h Evaluations No Control VSL Only VSL+LC ATT (min) 19.0 18.6 18.4 ¯s 31.7 2.2 1.2 CO 2 (g/veh/km) 366.0 317.2 311.6 For each entry in Table 6.1 and 6.2, we take the average results of 5 microscopic simulations to reduce the randomness and increase the reliability. The proposed controller (VSL+LC) reduces the ATT by 16.2%, average number of stops by 97% and emission rates of CO 2 by 21.4% compared with the openloop system under the demand of 7000 veh/h. When the demand drops down to 6000 veh/h, the proposed controller can still improve the ATT by 3.2%, average number of stops by 96.2% and emission rates of CO 2 by 14.9%. The evaluation results demonstrate the effectiveness of the proposed controller in solving the congestion caused by lane 104 drop. Inaddition,thehigherdemandoftheincomingtraffic,themoreimprovement we obtain. The single VSL controller cannot keep up with the integrated controller in terms of the average number of stops, because LC controller prevents the queue formulation ahead of the bottleneck. Figure 6.12 is a fundamental diagram depicted from the closed-loop simulation data under multiple demands with the incident. The proposed controller is only activated when d > C d and deactivated otherwise. We can observe a clear con- vergence for all high demand scenarios where the controller being applied. The open-loop convergence can also be achieved when d < C d . The data points of d = 4000 veh/h is more scattered because the actual vehicle input generated in VISSIM is randomly distributed around 4000. A high input may drive the system to the capacity drop region for a short time. Figure 6.12: Fundamental Diagram with Integrated Controller 105 6.4.4 Uncertainties and Robustness Analysis There are two types of uncertainties we may encounter, uncertainties in measure- ments and uncertainties in model parameters. The corrupted measurements can be represented as following: ˜ ρ i =ρ i (1+µ ρ ) (6.10) ˜ q i =q i (1+µ q ) (6.11) where ρ i , q i are the true measurements and µ ρ , µ q are the levels of uncertainties. On the other hand, many model parameters are determined by the fundamental diagram or empirical values, which is likely to introduce uncertainties. w is the mostsensitiveamongtheseparametersbecauseitisinvolvedinthecomputationsof controlcommands. Althoughρ j isalsoinvolved,ρ j canbecomputedasC/v f +C/w, whereC andv f areconsideredasdeterministicvalues. Othernoisyparameterssuch as ˜ w and ˜ ρ j arenotinvolvedinthecomputations. Therefore, weareonlyinterested in the effect of w. With the existence of uncertainties, our primary concern is whether the closed- loop system can still be stabilized around the desired equilibrium. In order to evaluate the robustness of the proposed controller, we collect density and flow data for all sections and compute the relative root mean square error (RRMSE) with respect to the desired equilibrium as following: RRMSE(ρ i ) = 1 ρ ∗ v u u t 1 T e − T c +1 Te X T=Tc (ρ i (T)− ρ ∗ ) 2 (6.12) RRMSE(q i ) = 1 v f ρ ∗ v u u t 1 T e − T c +1 Te X T=Tc (q i (T)− v f ρ ∗ ) 2 (6.13) 106 whereT c isthetimestepofconvergencebeginningandT e isthetimestepofincident ending. In general, weconsider the convergence is acceptable when RRMSE(ρ i )≤ 20% and RRMSE(q i ) ≤ 20% for i = 1,...,N. Table 6.3, 6.4 and 6.5 show the evaluation results when there exist uncertainties in measured densities, measured flows and model parameter w respectively. Table 6.3: Uncertainties in Measured Densities ˜ ρ i µ ρ RRMSE (ρ 1 ) RRMSE (ρ 5 ) RRMSE (q 1 ) RRMSE (q 5 ) 0 6.7% 13.4% 7.8% 12.3% -0.2 6.2% 14.4% 8.5% 13.4% -0.1 6.3% 14.7% 8.7% 13.6% 0.1 6.7% 12.2% 7.2% 11.7% 0.2 268.3% 23.0% 12.3% 22.3% Table 6.4: Uncertainties in Measured Flows ˜ q i µ q RRMSE (ρ 1 ) RRMSE (ρ 5 ) RRMSE (q 1 ) RRMSE (q 5 ) 0 6.7% 13.4% 7.8% 12.3% -0.2 344.1% 22.9% 23.6% 21.4% -0.1 6.8% 14.1% 9.1% 12.7% 0.1 6.0% 14.2% 9.0% 12.9% 0.2 6.8% 13.9% 8.2% 12.6% In Table 6.3 and 6.4, the convergence deteriorates as we increase µ ρ from 0.1 to 0.2 or decrease µ q from -0.1 to -0.2. In both cases, the actual speed limit commands are lower than the ones computed from true measurements. It implies thatslowingdownthetrafficexcessivelyproducesworseperformancethanspeeding up the traffic for the proposed controller. The intuitive explanation is that we create more shockwaves as we slow down the traffic unnecessarily, which prevents 107 Table 6.5: Uncertainties in Model Parameter w w RRMSE (ρ 1 ) RRMSE (ρ 5 ) RRMSE (q 1 ) RRMSE (q 5 ) 30 6.7% 13.4% 7.8% 12.3% 24 6.3% 12.7% 8.6% 11.3% 27 6.6% 15.6% 8.3% 14.1% 33 6.6% 13.1% 8.2% 11.8% 36 6.9% 14.7% 8.5% 13.7% the convergence. The evaluation results indicate the proposed controller is able to tolerate 10% of uncertainties in measurements and 20% of variations in the value of w, so the robustness of the closed-loop system is satisfactory. 6.5 Conclusion In this chapter, a robust integrated VSL-and-LC controller to alleviate highway bottleneck congestion caused by lane drop is proposed. A modified multi-section CTM is used to describe the traffic behavior and capture more complex traffic flow phenomena, such as the capacity drop and bounded acceleration. Since the length of the most upstream section L 0 , located just upstream the first section of the con- sideredhighwaystretch,istreatedasaVSLcontrolparameterandoptimizedbased on extensive simulation results, the results show that higher demand levels require more extended L 0 since the incoming traffic flow needs to be held longer within L 0 in order to give the downstream sections more time and space to clear congestion and shockwaves. The LC controller is implemented at the beginning of the dis- charging section to prevent queue formulation due to forced lane changes, resulting in improving the throughput. The integrated controller presented in this chapter 108 shows significant improvements in traffic mobility, safety, and environmental sus- tainability, as our microscopic simulations demonstrated. In addition, it is able to tolerate10%ofuncertainties in measurements and 20% of variations in the valueof back propagation speed. The closed-loop system is prone to instability as we either overestimate the traffic density or underestimate the traffic flow because, in both cases, we produce shockwaves by reducing the VSL control action excessively. 109 Chapter 7 Per-Lane Variable Speed Limit and Lane Change Control for Congestion Management at Bottlenecks 7.1 Introduction For more effective VSL control strategies involving lane changes, treating every lane as an independent entity is a key procedure in order to increase highway ca- pacity utilization at the bottleneck location. Undoubtedly, modeling the behavior of lane-changing vehicles at a macroscopic scale is extremely complex. This is be- cause multiple factors, which some of them are mutuality-dependent, are needed to be considered. To name a few, for example, human driving behavior is entirely different from one to another. In fact, it varies depending on the vehicle type and road geometry, which may add a level of uncertainty and intricacy. In addition, surroundingtrafficandenvironmentalconditionsareothersourcesofdifficultythat highly contribute to lane-changing characteristics. The motivation behind execut- ing a lane-changing maneuver, categorized generally as either a mandatory or a discretionary lane change, is another factor that results in a variety of different lane-changing characteristics. This is often translated macroscopically within the 110 CTM framework to the demand for lateral flow. It is usually assumed and repre- sented as a fixed fraction of the total demand wanting to leave the section (or cell). Nevertheless, determining, calibrating, and tuning this fraction is unclear given the aforementioned variety and complexity factors of lane changes. Accordingly, the performance of traffic flow controllers involving lane changes may deteriorate when being evaluated in microscopic simulations or a real traffic environment. In this chapter, therefore, we estimate in real-time the net lateral flows (lane changes) in each lane based on the available online measurements of the traffic density and longitudinal flows, which are well-defined in the CTM. By treating everylaneasaseparatestream,thenetlane-changingflow,definedasthedifference between the lateral inflows and outflows within the same segment/cell, is modeled as an additional term in the traffic conservation equation. The estimated value of this term is developed based on Lyapunov stability analysis. Then, a per-lane VSL control strategy is developed by taking into account the estimated value of the net lane changes while alleviating traffic congestion at an active highway bottleneck. Since the benefits of VSL may deteriorate by the unmanaged forced lane-changing behavior in the vicinity of the bottleneck [5], a lane change (LC) controller is combined with the proposed VSL. The paper analyzes the stability properties of the closed-loop system, where the integrated control scheme guarantees that the lane traffic density will be operating within the free-flow region of the fundamental diagram. Thus, the bottleneck traffic congestion is relieved, and the throughput per lane is maximized, except for the blocked lane, as demonstrated by microscopic simulations. 111 7.2 Freeway Traffic Behavior at Bottlenecks Ahighwaybottleneckisalocalizedsectionofthemotorwaywheretrafficencounters areductionindischargingflowrate. Itisthemostvulnerablelocationinanytraffic network since, if the arriving demand is higher than the bottleneck’s capacity, congestion occurs, which leads to the capacity drop phenomenon [9, 10]. One of the most common bottlenecks on highways is the one caused by lane(s) drop. This kind of bottleneck could be observed, for instance, at accident locations and work zones, where one or more lanes may temporarily become unavailable. When the bottleneck is triggered, there exists a discharging section located directly upstream of the lane-drop location, where the majority of lane-changing maneuvers take place to avoid the blocked lane(s), as illustrated in Figure 7.1. Furthermore, there is an acceleration section downstream of the bottleneck, where vehicles start to accelerate from the congested speed to the free-flow speed. Let X b be the length of the lane-drop bottleneck. If X b is very small compared to the discharging section length, denoted by L, then the effect of the density within the space of the lane-drop is negligible. During the activation of the bottleneck, traffic flow behavior is different from one lane to another in terms of speed, lane-changing intensity, available capacity, and other surrounding traffic conditions. Lanes that are neighboring the blocked lane(s) are the most impacted by the interaction be- tween vehicles due to forced lane changes. To demonstrate the impact of unmanaged forced lane changes on the traffic flow near an active bottleneck, the commercial microscopic traffic software PTV VISSIMisused, whereahypothetical5.3-mile(8.5-kilometer)highwaystretchwith four lanes is constructed. VISSIM is calibrated with typical freeway road geometry anddrivingbehavior,anditgivesrealisticanddetailedresultsoftrafficbehavioron a microscopic level. The considered highway is subdivided into sections of length 112 Figure 7.1: Highway Bottleneck L = 0.31 miles (500 meters). The maximum capacity of each lane within the section is around 2900 veh/h/lane, which corresponds to a 48 veh/mi/lane critical density, defined as the density at which the maximum possible flow is achieved. A bottleneckisformedbyanincident,whichblockslane1(themostrightlane)atthe end of section 16, which is considered to be the discharging section. The network is injected with different levels of traffic demand, and the relationship between the flow rate and the vehicle density (known as the fundamental diagram) of the 0.31 mile-long discharging section is plotted per lane, as shown in Figure 7.2. Considering lane 1 is unavailable due to the incident, it is clear that the flow rate of lanes 1 and 2 are the most affected ones compared to the others, since vehicles are forced to move to lane 2 from lane 1 when approaching the bottleneck location. When the demand is roughly less than 1000 veh/h/lane, we have light traffic conditions. Thus, finding enough space in the target lane for the lane- changing vehicle is relatively easy; accordingly, drivers could easily switch lanes while driving at the free-flow speed. Once the demand starts increasing to be higherthan1000veh/h/lane,theflowrateoflane1approximatelysaturatesaround that value, which corresponds to a very low speed, while the vehicle density keeps increasing until the lane is nearly fully congested. This reflects the behavior of drivers slowing their speed down to a point where some have to completely stop, waiting for the right moment to execute a lane-changing maneuver safely. 113 0 50 100 150 200 250 300 density (vehicle/mile (veh/mi)) 0 500 1000 1500 2000 2500 3000 flow-rate (vehicle/hour (veh/h)) Fundamental Diagram (lane 4) Fundamental Diagram (lane 3) Fundamental Diagram (lane 2) Fundamental Diagram (lane 1) Figure 7.2: Fundamental Diagram Per lane Eventually, the interaction between vehicles in the other lanes increases as the intensity of unmanaged mandatory lane-changing activities increases, which nega- tivelyinfluencesthespeedsincesomevehiclesarenolongertravelingatthefree-flow speed. Drivers are responsible for adjusting their speed in order to change lanes safely or to allow gaps for merging vehicles. This results in the data points being densely scattered until the traffic density per lane exceeds some high density value, roughly around 165 veh/mi, 90 veh/mi, 70 veh/mi, and 60 veh/mi, for lanes 1, 2, 3,and4,respectively. Notethatallthesevaluesarehigherthanthecriticaldensity. Indeed, traffic congestion per lane begins to form and propagates upstream from the bottleneck location. The speed of vehicles passing the bottleneck is reduced due to forced lane-changing, queuing, and bounded acceleration. Thus, the follow- ing vehicles have to decelerate since the reduction in speed cascades upstream. In short, observing all resulting fundamental diagrams in Figure 7.2, it becomes clear 114 that every lane experiences a different traffic flow behavior at the bottleneck loca- tion. Therefore, every lane in this paper is treated as an independent entity when designing the combined VSL and LC control in order to increase highway capacity utilization at the bottleneck location. 7.3 System Modeling This section aims to develop a per-lane VSL control strategy based on a multi- lane CTM and combine it with LC control to alleviate congestion and its negative consequences at active highway bottlenecks. The VSL controller is expected to worktogetherwithLCcontrolinordertomaketheper-lanevehicledensityoperate within the uncongested region of the fundamental diagram and, thus, improve the capacity of the discharging section. 7.3.1 Multiple-Lane Cell Transmission Model Based VSL Control Consider a generic multi-lane highway stretch with the index i = 1,2,3,...,I rep- resentsthenumberoflanes, wherelane1(laneI)beingthemostright(left)lanein the direction of the traffic flow. Assume that, without loss of generality, the high- way is partitioned into small identical sections of length L, as depicted in Figure 7.3, and all the lanes of the highway have similar characteristics. Every lane within the discharging section is characterized by the following variables: • ρ i : the traffic density, defined as the number of vehicles per unit length. • q in i : the longitudinal inflow, defined as the traffic volume entering lane i from upstream. • q out i : the longitudinal outflow, defined as the traffic volume leaving lane i to the acceleration section. 115 • l i± 1,i : the lateral inflows, defined as the traffic volume moving from lane i± 1 to lane i while remaining within the discharging section. • l i,i± 1 : the lateral outflows, defined as the traffic volume moving from lane i to lane i± 1 while remaining within the discharging section. A graphical representation of these variables is illustrated in Figure 7.3. Let l n i := (l i− 1,i + l i+1,i )− (l i,i− 1 + l i,i+1 ) denote the net lane-changing flow, defined as the difference between the total lateral inflows and outflows. Then, the traffic density ρ i of lane i is updated based on the following traffic flow conservation equation: Figure 7.3: Representation of Highway Single Lane under VSL and LC control ˙ ρ i = 1 L q in i − q out i +l n i , 0≤ ρ i (0)≤ ρ j , i = 1,2,3...,I (7.1) where 116 q in i = min d u i , v vsl i wρ j v vsl i +w ,w(ρ j − ρ i ),C q out i = min v f ρ i , ˜ w(˜ ρ j − ρ i ),C d i C d i = 0, if lane i is blocked C, otherwise v f ρ c =w(ρ j − ρ c ) = ˜ w(˜ ρ j − ρ c ) =C, 0<ρ c <ρ j ,0< ˜ w <w,0<v f <v vsl i (7.2) In equation (7.2), d u i is the per-lane upstream demand, which every lane within the discharging section is expected to meet. v vsl i is the per-lane VSL control ac- tion, which will be designed later in subsection 7.3.2, and the other parameters in equations (7.1) and (7.2) are defined in table 7.1. Figures 7.4a and 7.4b show the aforementioned parameters on the fundamental diagram, constructed by the sup- ply (or receiving) and demand (or sending) functions. The triangular form of the fundamental diagram is adopted here when computing the longitudinal flows, q in i andq out i , because a lane change control is assumed to be operating at the beginning ofthedischargingsectionwhilethebottleneckisactive. Thus, thetriangularshape without the capacity drop is recovered, as demonstrated in [5, 107]. Table 7.1: Definition of the Model Parameters Parameter Definition Unit C the maximum capacity of the lane per VSL/discharging section. vehicle/hour (veh/h) C d i the downstream maximum capacity of the lane per acceleration section. vehicle/hour (veh/h) v f the per-lane free-flow speed. mile/hour (mi/h) ρ c the critical density of the lane per section, defined as the density at which the maximum possible flow with the flow-free speed is achieved. vehicle/mile (veh/mi) ρ j the jam density of the lane per section, defined as the highest possible density at which the lane inflow q in i = 0. vehicle/mile (veh/mi) ˜ ρ j the per-lane projected jam density associated with the lane outflow q out i . vehicle/mile (veh/mi) w the per-lane back propagation speed. mile/hour (mi/h) ˜ w the rate that the outflow of the lane q out i decreases with density ρ i , when ρ i ≥ ρ c [19]. mile/hour (mi/h) As equation (7.2) reads, the use of the per-lane VSL controller has an effect upon the value of the inflow q in i . In addition to the upstream demand d u i , the 117 (a) Triangular Fundamental Diagram (b) Triangular Fundamental Diagram under VSL Figure 7.4: The Supply and Demand Functions Corresponding to the Triangular Fundamental Diagram without/with VSL inflow also depends on how much flow is allowed to be sent by the upstream lane inside the VSL section. If the VSL command is less than the free-flow speed, i.e., v vsl i < v f , then the resulting fundamental diagram of the lane in VSL section is distorted [105, 106, 22, 46]. Figure7.4ashowsthegeometryofthefundamentaldiagramundertheinfluence of VSL. It follows that the parameters ρ j , w, and ˜ w remain unchanged, while the 118 maximum possible flow rate that the lane in the VSL section could send down- stream is given by the term v vsl i wρ j v vsl i +w , where the VSL control command per lane v vsl i is designed in the following subsection. 7.3.2 Per-LaneVariableSpeedLimit(VSL)ControlDesign The VSL control objective is to force the traffic density of the lane withing the discharging section ρ i to converge to a desired density value, referred to as ρ ⋆ i . This desired density should be chosen to correspond to the highest possible flow rate value and free-flow speed. In other words, the maximum possible utilization of the highway infrastructure is needed to be obtained during the activation of the bottleneck. Intuitively, the desired traffic density for the open lanes should be at the critical density, where the maximum possible flow under the flow-free speed is achieved. On the other hand, the desired density for the blocked lane(s) is required to be selected at a small value to prevent the creation of traffic congestion. Since every lane is treated as a separate stream, the desired traffic density ρ ⋆ i of lane i is defined as follows: ρ ⋆ i = ρ L i , if lane i is blocked ρ c , otherwise (7.3) ρ L i is a low-density value that satisfies 0 < ρ L i ≪ ρ c . It could be defined as the density at which the speed of the flow in the blocked lane(s) decreases beyond the value of ρ L i due to a queue formation. Figure 7.4a shows the location of the desired density values on the fundamental diagram. Choosing the desired density for the blocked lane(s) to be strictly greater than zero, i.e., ρ ⋆ i = ρ L i > 0, can be accomplishedbyintegratingtheVSLwithalanechangecontroltopreventtheVSL control from acting like a traffic light control, which is not feasible in most highway traffic situations. 119 Unlike the lateral flows, the longitudinal inflows and outflows are well-defined macroscopically in the CTM, and they are accessibly available to be measured microscopically via traffic sensors and loop detectors. Therefore, the design of the VSL control action is combined with an adaptive law that generates and adjusts a parameter called ˆ l n i , which estimates the value of the net lane-changing flow l n i in real-time. If lane i is blocked, then according to equation (7.2), the value of the longitudinal outflow is equal to zero, i.e., q out i = 0, due to C d i = 0. As a result, the value of the lateral outflow is known to be greater than lateral inflow since vehicles are advised to leave the lane, and, therefore, the sign of the net lane-changing flow is known to be negative for the blocked lane(s) and positive for the open lane(s) in the traffic flow conservation law (7.1). The following equations summarize the VSL control inputs, considering the estimated value of the net lane-changing flow ˆ l n i . Then, we develop an adaptive law that generates and adjusts the parameter ˆ l n i online using Lyapunov stability analysis. The per-lane VSL controller is computed as follows: v vsl i = wq vsl i wρ j − q vsl i q vsl i = median{0,¯q vsl i ,v f ρ ⋆ i } ¯q vsl i = ˆ l n i − λ i (ρ i − ρ ⋆ i ), if lane i is blocked, q out i − ˆ l n i − λ i (ρ i − ρ ⋆ i ), otherwise. (7.4) The median function in (7.4) is used to guarantee that when mapping the con- trolled flow rate q vsl i into the VSL command v vsl i , the speed of the lane does not become less than zero or exceed the free-flow speed limit. λ i > 0 is a design con- stant to be selected. ¯q vsl i is the unconstrained feedback controlled flow, taking into account the estimated value of the net lane-changing flow ˆ l n i . To search for an adaptive law to generate ˆ l n i on-line, we assume that the structure of the adaptive law is given by 120 ˙ ˆ l n i =f i (ρ i ,¯q vsl i ) (7.5) where f i is some function to be selected, and propose V i (ρ i , ˆ l n i ) = (ρ i − ρ ⋆ i ) 2 2 + ( ˆ l n i − l n i ) 2 2γ i (7.6) for some γ i > 0 as a potential Lyapunov-like function for the closed-loop system (7.1)-(7.4). It follows from (7.4) that q vsl i can also be rewritten as q vsl i = v vsl i wρ j v vsl i +w , and since in equation (7.2), the term v vsl i wρ j v vsl i +w is the only one in q in i that depends on v vsl i , let us first assume that q in i = min n d u i , v vsl i wρ j v vsl i +w ,w(ρ j − ρ i ),C o = v vsl i wρ j v vsl i +w and 0 < ¯q vsl i ≤ v f ρ ⋆ i , which both will be relaxed later for the general equations. Thus, we have q in i = q vsl i = ¯q vsl i and the time derivative of (7.6) along the trajectories of the system (7.1)-(7.4) is given by ˙ V i (ρ i , ˆ l n i ) = − λ i (ρ i − ρ ⋆ i ) 2 L + ( ˆ l n i − l n i )(ρ i − ρ ⋆ i ) L + ( ˆ l n i − l n i ) γ i f i , if lane i is blocked, − λ i (ρ i − ρ ⋆ i ) 2 L − ( ˆ l n i − l n i )(ρ i − ρ ⋆ i ) L + ( ˆ l n i − l n i ) γ i f i , otherwise. (7.7) Choosing f i = − γ i L (ρ i − ρ ⋆ i ) for the blocked lane(s), and f i = γ i L (ρ i − ρ ⋆ i ) for the open lane(s), i.e., ˙ ˆ l n i (t) = − γ i L (ρ i (t)− ρ ⋆ i ), ˆ l n i (0) =λ i (ρ i (0)− ρ ⋆ i )+v f ρ ⋆ i , if lane i is blocked, γ i L (ρ i (t)− ρ ⋆ i ), ˆ l n i (0) =λ i (ρ i (0)− ρ ⋆ i ), otherwise, (7.8) we have ˙ V i (ρ i , ˆ l n i ) = − λ i L (ρ i − ρ ⋆ i ) 2 ≤ 0 (7.9) 121 The function V i (ρ i , ˆ l n i ) given by equation (7.6) is positive definite, whose time derivative ˙ V i (ρ i , ˆ l n i ) in equation (7.9) is negative semi-definite, then choose c i > 0 such that the following set Ω i := {(ρ i , ˆ l n i ) ∈ ℜ 2 |V i (ρ i , ˆ l n i ) ≤ c i } is a compact positively invariant set. Then, the set E i :={(ρ i , ˆ l n i )∈ Ω i |ρ i = ρ ⋆ i } is an invariant set,andthelargestinvariantsubsetM i ofE i underthesystemdynamicsisE i itself. Therefore, M i =E i , and from LaSalle’s invariance principle [108], we conclude that every trajectory starting in Ω i approaches M i , as t → ∞, which, in turn, implies that ρ i (t)→ρ ⋆ i as t→∞,∀ρ i (0)∈ [0,ρ j ]. WehaveestablishedthattheproposedVSLcontrol(7.4)togetherwiththedevel- oped adaptive law (7.8) guarantee that the lane traffic density ρ i converges asymp- totically with time to the desired state ρ ⋆ i . The initial conditions of ˆ l n i in equation (7.8) are set to be in such a way to guarantee that all the trajectories of the closed- loopsystemconvergetothedesiredstate,aswewillshowlaterintheorem7.3.1that when (d u i ± l n i )≥ v f ρ ⋆ i , then the proposed VSL control action with the adaptive law forces the traffic density of the lane to converge to the desired point ρ ⋆ i , asymptot- ically with time, not only when q in i = q vsl i = ¯q vsl i but also for the general equations where q in i = min n d u i , v vsl i wρ j v vsl i +w ,w(ρ j − ρ i (t)) o and q vsl i = median{0,¯q vsl i ,v f ρ ⋆ i }. Fur- thermore, theorem 7.3.2 will show that when (d u i ± l n i ) < v f ρ ⋆ i , then the derived controller can still guarantee that ρ i will operate within the free-flow region of the fundamental diagram, where the throughput reaches its maximum possible value. As mentioned earlier, combining the VSL with an LC control utilizes the ca- pacity of the motorway while the bottleneck is active due to the selection of the desireddensity(7.3). TheVSLcontrolinformsvehiclesintheVSLsectiontofollow the per-lane speed limit v vsl i and then follow the free-flow speed v f within the dis- charging section, where vehicles are also advised to change lanes in order to avoid creating traffic congestion, as shown in Figure 7.3. By doing so, the integrated 122 controlschemecanforcethe traffic conditions to be lightin the closed lane(s) while maximizingthethroughputintheotherlanes. Underthesecontrolledtrafficcondi- tions, findingenoughspacein the target lane for the lane-changing vehicle becomes relatively easier, and, therefore, the discharging section acts as the acceleration section. In the following subsection, the LC control to be combined with the VSL control is presented. 7.3.3 Lane Changing (LC) Control The purpose of the LC controller is to provide LC recommendations to upstream vehicles while in motion before arriving at a queue formed due to the bottleneck. Integrating the LC control with VSL helps the VSL control to operate smoothly. In addition, it increases highway utilization during the activation of the bottleneck. The LC control framework involves notifying vehicles of the upcoming bottleneck and suggesting appropriate lanes to which they could change. Depending on the road geometry and the location of the blocked lane(s), the lane-changing recom- mendations inform drivers to ”Change to Left (Right) Lane,” ”Change to Either Lane,” or ”Keep Straight” if lane is open. These recommendations are provided to drivers per lane at the beginning of the discharging section, as shown in Figure 7.3. For more details about the LC mechanism, please see 6.3.3. To this end, an per-lane integrated VSL and LC control strategy is presented to prevent traffic congestion from happening at a motorway lane drop bottleneck. The lanes in the road are considered independent entities. Assuming that the LC control is active at the discharging section, the VSL control is designed based on the CTM, taking into account the estimated value of the net lane changes (net lateral flow). This value is assumed to be estimated online and developed using Lyapunov-based stability analysis. Combining all equations mentioned earlier in this section, the closed-loop system is formulated as follows: 123 ˙ ρ i (t) = 1 L q in i − l n i , if lane i is blocked 1 L q in i − q out i +l n i , otherwise ˙ ˆ l n i (t) = − γ i L (ρ i (t)− ρ ⋆ i ), if lane i is blocked γ i L (ρ i (t)− ρ ⋆ i ), otherwise (7.10) where 0≤ ρ i (0)≤ ρ j ˆ l n i (0) = λ i (ρ i (0)− ρ ⋆ i )+v f ρ ⋆ i , if lane i is blocked λ i (ρ i (0)− ρ ⋆ i ), otherwise q in i = min d u i , v vsl i wρ j v vsl i +w ,w(ρ j − ρ i ),C q out i = min v f ρ i , ˜ w(˜ ρ j − ρ i ),C v vsl i = wq vsl i wρ j − q vsl i q vsl i = v vsl i wρ j v vsl i +w = median{0,¯q vsl i ,v f ρ ⋆ i } ¯q vsl i = ˆ l n i − λ i (ρ i − ρ ⋆ i ), if lane i is blocked q out i − ˆ l n i − λ i (ρ i − ρ ⋆ i ), otherwise (7.11) and ρ ⋆ i = ρ L i , if lane i is blocked ρ c , otherwise v f ρ c =w(ρ j − ρ c ) = ˜ w(˜ ρ j − ρ c ) =C, (7.12) 0<ρ c <ρ j ,0<ρ L i ≪ ρ c ,0< ˜ w <w,0≤ v vsl i ≤ v f ,0<v f <λ i ,0<γ i . The following subsection analyzes the stability properties of the closed-loop traffic flow control system given by equations (7.10)-(7.12). 124 7.3.4 Stability Analysis Thebehavioroftheclosed-loopsystemdescribedbytheabove-mentionedequations (7.10), (7.11), and (7.12) is summarized in the following theorems. Theorem 7.3.1. Considering the closed-loop system given by equations (7.10)- (7.12), we have the following results: • If lane i is blocked and (d u i − l n i )≥ v f ρ ⋆ i , then the system trajectories converge asymptotically with time to a unique equilibrium point [ρ e i , ˆ l e i ] T = [ρ ⋆ i ,l n i ] T = [ρ L i ,v f ρ ⋆ i ] T ,∀(ρ i (0),l n i (0))∈ℜ 2 . • If lane i is open and (d u i +l n i )≥ v f ρ ⋆ i , then the system trajectories converge asymptotically with time to a unique equilibrium point [ρ e i , ˆ l e i ] T = [ρ ⋆ i ,l n i ] T = [ρ c ,l n i ] T ,∀(ρ i (0),l n i (0)) ∈ ℜ 2 . Proof. See Appendix B. The proof of the theorem 7.3.1 shows in an analytically rigorous manner that by considering the same Lyapunov function candidate introduced earlier in the equation (7.6), we conclude that ˙ V i (ρ i , ˆ l n i ) < 0, ∀(ρ i (t), ˆ l n i (t)) ̸= (ρ ⋆ i ,l n i ) and the equilibrium state of the considered lane, i.e., [ρ e i , ˆ l e i ] T = [ρ ⋆ i ,l n i ] T , is globally asymp- toticallystable,implyingthatρ i (t)→ρ ⋆ i ast→∞,∀ρ i (0)∈ [0,ρ j ]. Inotherwords, aslongas(d u i ± l n i )≥ v f ρ ⋆ i ,theorem7.3.1demonstratesthattheproposedcontroller drives the traffic density of the lane to the desired value asymptotically with time. In the case when (d u i − l n i ) < v f ρ ⋆ i for the blocked lane(s) and (d u i +l n i ) < v f ρ ⋆ i for the open lane(s), the dynamics and the stability properties of the closed-loop system are given by the following theorem. Theorem 7.3.2. Considering the closed-loop system given by equations (7.10)- (7.12), we have the following results: 125 • If lane i is blocked and (d u i − l n i )<v f ρ ⋆ i , then the traffic density of the closed- loop system converges to an equilibrium manifold which is the interval [0,ρ ⋆ i − η i ], where η i > 0. This equilibrium manifold corresponds to the maximum possible flow rate and free-flow speed. • If lane i is open and (d u i +l n i ) < v f ρ ⋆ i , then the traffic density of the closed- loop system converges to the point (d u i +l n i ) v f , which corresponds to the maximum possible flow rate and free-flow speed. Proof. • If lane i is blocked and (d u i − l n i ) < v f ρ ⋆ i , then ∃η i > 0, such that (d u i − l n i )≤ v f (ρ ⋆ i − η i ). ∀ρ i (0)∈ (ρ ⋆ i ,ρ j ], according to the analysis in theorem 1,∃t 1 > 0 such that ρ i (t 1 ) = ρ ⋆ i = ρ L i . Then, either ρ i (t 1 ) = ρ ⋆ i or the initial condition 0≤ ρ i (0)≤ ρ ⋆ i , we have w(ρ j − ρ i (t))≥ C >v f ρ ⋆ i ≥ min{d u i ,q vsl i }≥ v f (ρ ⋆ i − η i )≥ d u i − l n i ≥ 0 Therefore,theinflow q in i = min{d u i ,q vsl i } =d u i sinceq vsl i = median{0,¯q vsl i ,v f ρ ⋆ i } saturates at the value v f ρ ⋆ i as the time t increases; thus ˙ ρ i (t) = 1 L (d u i − l n i )≤ 1 L v f (ρ ⋆ i − η i ) ,∀t≥ 0, implying that 0≤ ρ i (t)≤ ρ ⋆ i − η i <ρ ⋆ i ,∀ρ (0)∈ [0,ρ j ] • If lane i is open, then similar to the previous analysis we will end up with ˙ ρ i (t) = 1 L (q in i − q out i +l n i ) = 1 L d u i − v f ρ i (t)+l n i . Therefore, ρ i (t) converges to the point (d u i +l n i ) v f ,∀ρ (0)∈ [0,ρ j ]. Theorem 7.3.2 shows that there exists a finite time where the traffic density of lane i will be operating within the free-flow speed region of the fundamental dia- gram. By doing so, traffic congestion is avoided since forced lane-changing vehicles are switching lanes while traveling at the free-flow speed v f . In addition, there is always enough capacity in the target lane because of the selection of the desired traffic density per lane. 126 7.4 Numerical Simulations In this section, the commercial microscopic simulator PTV VISSIM is used to evaluate the performance of the proposed VSL and LC control. 7.4.1 Simulation Network Setup Thesame4-lane,5.3-mile-long(8.5-kilometer)highwaystretchintroducedinsection 7.2 is applied here to evaluate the proposed controller. As mentioned earlier, the highwayissubdividedinto17smallsectionsoflengthL =0.31miles(500meters). A bottleneck is created by an incident that blocks lane 1 (the most right lane) at the end of section 16. The considered sections of the road to be controlled are 15 and 16, which are recognized as the VSL section and the discharging section, respectively; see Figure 7.5. Figure 7.5: Simulation Network During the time of the incident, vehicles per lane are asked to follow the speed limit v vsl i , where i = 1,2,3,4, generated by the VSL control in section 15. Then, along with the LC recommendations, they follow the free-flow speed v f once they access section 16. When entering the discharging section, the LC control advises the vehicles in lane 1 to change to the left lane. The Cooperative Lane Change 127 parameterinVISSIMisactivatedduringthesimulations. Activatingthisparameter permits vehicles to assist each other in making lane-changing maneuvers. This feature allows vehicles in the adjacent lanes to make non-necessary lane changes to create space for merging vehicles. The highway is loaded with traffic demand equal to 10000 veh/h, i.e., d u i = 2500 veh/h/lane. The traffic flow is a combination of light-duty vehicles (passenger cars) and heavy trucks. In the simulation, all four lanes are open at the beginning of the simulation. 25 minutes after the simulation starts, the incident takes place and remains for 30 minutes (1500 to 3300 seconds). Once the incident starts, the integrated VSL and LC control is activated, and it deactivates when the incident is cleared. Observing the per-lane fundamental diagram presented earlier in Figure 7.2, the VSL controller per lane is applied with the following parameters: C = 2900 veh/h, ρ c = 48 veh/mi, ρ j =200 veh/mi, ˜ ρ j = 250 veh/mi, v f = 60 mi/h, w = 19 mi/h, ˜ w = 14.3 mi/h, λ i = 120, and γ i = 50, for i = 1,2,3,4. The desired density points are chosen to be ρ ⋆ 1 = 18 veh/mi for lane 1 (the blocked lane) and ρ ⋆ 2 = ρ ⋆ 3 =ρ ⋆ 4 =ρ c = 48veh/mi for the other lanes (open lanes) to utilize the roadway as muchaspossibleatthebottlenecklocationduringtheincidenttime. Forthesafety and compliance of drivers, the speed limits increase or decrease by 10 mi/h/lane. 7.4.2 Microscopic Simulation Results To verify that the proposed controller generates consistent results, 15 sets of Monte Carlo simulations are considered. The final performance measurements are the averages of the Monte Carlo simulation results. Figure 7.6 shows the VSL control actions given per lane. It can be seen that when the incident happens at 1500 sec, the control is activated, where the speed 128 per lane in the VSL section gradually starts reducing in steps of 10 mi/h to ensure safety for the drivers. The VSL command of lane 1 stays around 10 mi/h most of the time during the incident period, while the VSL command of lane 2 fluctuates around 20 and 30 mi/h. The VSL action related to lane 3, i.e., v vsl 3 , reduces the speed to 30 mi/h when the incident takes place, then it increases to 40 mi/h and stays around that value for most of the 30-minute period. Since lane 4 is the farthest lane from the closed lane (lane 1), it has the highest speed limits in the VSL section. Its speed primarily alternates between 40 and 50 mi/h during the time of the incident. 0 1000 2000 3000 4000 5000 6000 time (second (sec)) 0 10 20 30 40 50 60 70 speed (mile/hour (mi/h)) VSL for lane 1 (a) Lane 1 0 1000 2000 3000 4000 5000 6000 time (second (sec)) 0 10 20 30 40 50 60 70 speed (mile/hour (mi/h)) VSL for lane 2 (b) Lane 2 0 1000 2000 3000 4000 5000 6000 time (second (sec)) 0 10 20 30 40 50 60 70 speed (mile/hour (mi/h)) VSL for lane 3 (c) Lane 3 0 1000 2000 3000 4000 5000 6000 time (second (sec)) 0 10 20 30 40 50 60 70 speed (mile/hour (mi/h)) VSL for lane 4 (d) Lane 4 Figure 7.6: VSL Commands (Lanes 1, 2,3, and 4) 129 The green, blue, and black curves in Figures 7.7 represent the behavior of the measured inflow, measured outflow, and the estimated net lane-changing flow per lane, respectively. During the 30-minute incident time, we can observe the impact ofthe VSLcontrolonlowering the value of the inflows related to lanes 1,2, and 3in order to let vehicles in the discharging section move away from lane 1 while driving at the free-flow speed. Therefore, traffic congestion is avoided, and, in turn, the discharging flow rates of lanes 2 and 3 are maximized to be around 2900 veh/h, while the outflow of lane 4 exhibits a slight increase to almost 2600 veh/h. All these values of the measured outflows correspond to the highest possible flow rates while lane 1 is blocked. As expected, the estimated value of the net lane changes for lane 1 converges to the value 1080 veh/h since the VSL control action forces the inflow to stay around that value. Thus, drivers have enough time and space to comply with the LC recommendations and leave the lane before they approach the bottleneck, preventing them from forming a queue. The estimated net lane-changing values of lanes 2 and 3 eventually converge to 530 veh/h and 450 veh/h, respectively. In contrast, the value of the estimated net lateral flow related to lane 4 diverges. This is because, during the incident time, the traffic density of lane 4 in the discharging section remains for the most time approximately around 44 veh/mi, which is less than the desired density, i.e., (44<48)veh/mi,seeFigure7.8d. Asaresult,theinflowoflane4saturatesaround 2500 veh/h. AsseeninFigure7.8, when the proposed controlscheme is activefrom the1500 sec to the 3300 sec, it is able to eventually stabilize the densities of lanes 1, 2, and 3 around the desired values, i.e., ρ 1 = ρ ⋆ 1 = 18 veh/mi and ρ 2 = ρ 3 = ρ c = 48 veh/mi . Though the density of lane 4 stays around 44 veh/mi, which is less than the critical density, all the densities per lane in the discharging section correspond 130 0 1000 2000 3000 4000 5000 6000 time (second (sec)) -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 flow (vehicle/hour (veh/h)) inflow of lane 1 outflow of lane 1 estimated LC lane 1 (a) Lane 1 0 1000 2000 3000 4000 5000 6000 time (second (sec)) -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 flow (vehicle/hour (veh/h)) inflow of lane 2 outflow of lane 2 estimated LC lane 2 (b) Lane 2 0 1000 2000 3000 4000 5000 6000 time (second (sec)) -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 flow (vehicle/hour (veh/h)) inflow of lane 3 outflow of lane 3 estimated LC lane 3 (c) Lane 3 0 1000 2000 3000 4000 5000 6000 time (second (sec)) -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 flow (vehicle/hour (veh/h)) inflow of lane 4 outflow of lane 4 estimated LC lane 4 (d) Lane 4 Figure 7.7: Measured Inflows (Green), Measured Outflows (Blue), and Estimated Lane Changes (Black) (Lanes 1, 2,3, and 4) tothemaximumpossibledischargingflowratesandthefree-flowspeed, despitethe lane-changing maneuvers. ThemicroscopicVISSIMsimulatorisalsousedtocomparetheresultingoutflows from the case when the proposed controller is not activated (open-loop system) to ones when it is running (closed-loop system) during the 30-minute incident period. Figure7.9representsthedischargingflowratesforlanes2,3,and4atthebottleneck with and without control. 131 0 1000 2000 3000 4000 5000 6000 time (second (sec)) 0 20 40 60 80 100 density (vehicle/mile (veh/mi)) density of lane 1 (a) Lane 1 0 1000 2000 3000 4000 5000 6000 time (second (sec)) 0 20 40 60 80 100 density (vehicle/mile (veh/mi)) density of lane 2 (b) Lane 2 0 1000 2000 3000 4000 5000 6000 time (second (sec)) 0 20 40 60 80 100 density (vehicle/mile (veh/mi)) density of lane 3 (c) Lane 3 0 1000 2000 3000 4000 5000 6000 time (second (sec)) 0 20 40 60 80 100 density (vehicle/mile (veh/mi)) density of lane 4 (d) Lane 4 Figure 7.8: Density of Discharging Section (Lanes 1, 2,3, and 4) Inthecaseofnocontrol, illustratedbytheredcurves, theoutflowoflane2dra- matically drops from 2500 veh/h to around 1370 veh/h. Then, it keeps highly fluc- tuating around 1500 veh/h during the lane closure time. Similar behavior appears on the discharging flow rate of lane 3, where it remains decreasing and increasing around2000veh/h. Lane4 is slightly impacted by the closed lane. Its outflow only decreases by roughly 4%. When the incident is cleared, it can be observed that the flow rates of lanes 2 and 3 increase directly to the maximum capacity of 2900 veh/h/lane. Then, they concentrate around 2500 veh/h/lane. When the proposed controller is applied, illustrated by the blue curves, the flow rate of lanes 2, 3, and 4 improve by approximately 93%, 45%, and 8%, respectively, compared to that in 132 0 1000 2000 3000 4000 5000 6000 time (second (sec)) 0 500 1000 1500 2000 2500 3000 3500 4000 flow (vehicle/hour (veh/h)) outflow of lane 2 (open loop) outflow of lane 2 (closed loop) (a) Lane 2 0 1000 2000 3000 4000 5000 6000 time (second (sec)) 0 500 1000 1500 2000 2500 3000 3500 4000 flow (vehicle/hour (veh/h)) outflow of lane 3 (open loop) outflow of lane 3 (closed loop) (b) Lane 3 0 1000 2000 3000 4000 5000 6000 time (second (sec)) 0 500 1000 1500 2000 2500 3000 3500 4000 flow (vehicle/hour (veh/h)) outflow of lane 4 (open loop) outflow of lane 4 (closed loop) (c) Lane 4 Figure7.9: DischargingFlowRateforOpenLoopandClosedLoop(Lanes2,3, and 4) the no-control case during the time of the incident. After the incident is over, the flow per lane stays around the value 2500 veh/h/lane. 7.5 Conclusion This chapter presents a per-lane VSL control strategy and then combines it with a LaneChange(LC)controlinoneintegratedschemetomitigatetrafficcongestionat lane(s) drop highway bottlenecks. Based on a multi-lane Cell Transmission Model 133 (CTM), the proposed integrated control design treats each lane in the highway as a separatestreamintendingtoutilizethemaximumcapacityofthefreewaywhenthe bottleneck is active. The control design involves an adaptive law, developed based onLyapunovstabilityanalysis, toestimatethenetlane-changingflow(lateralflow) of the lane online each time when computing the VSL control action. The stability analysis of the closed-loop system has shown that the combined control design guaranteesthatthetrafficdensityofthelaneoperateswithinthefree-flowregionof the fundamental diagram, which corresponds to the maximum possible discharging flow rate at the free-flow speed as demonstrated using microscopic Monte Carlo simulations. 134 Chapter 8 Conclusions and Further Research This chapter presents the conclusion of this dissertation research by summarizing the main results related to the research objectives and questions. It then proposes related opportunities for future research topics. 8.1 Summary and Conclusions In this dissertation research, the well-known and commonly used macroscopic first- order and second-order traffic flow models of highway traffic flow with different fundamental diagrams are reviewed. Based on the macroscopic first-order Cell Transmission Model (CTM) framework, the design, analysis, and evaluation of sev- eral integrated mainstream traffic flow control strategies, i.e., Variable Speed Limit (VSL)andLaneChanges(LC)controls,areproposedthroughoutthisreport,where their performances are evaluated using macroscopic and microscopic simulations. ThemainreasonbehindcombininganLCcontrolwithVSListopreventtheperfor- mance of VSL control from being deteriorated when being evaluated in microscopic simulations or real traffic environments. This is because the macroscopic CTM does not adequately capture the unmanaged lane-changing behavior at the vicinity of active highway bottlenecks. 135 The most widely used mainstream traffic control strategies, which are nonlin- ear feedback linearization (FL) and model predictive control (MPC), for VSL are designed and combined with an LC control at the discharging section. The per- formance and robustness of both integrated control schemes are compared with respect to perturbations on traffic demand, model parameters, and measurement noise. The simulation results demonstrate that both controllers are able to im- prove the total time spent (TTS). Though the MPC follows an optimization-based routine, the FL-based VSL guarantees exponential stability with negligible compu- tational effort and similar robustness. The CTM with capacity drop and bounded acceleration is then modified to include a constant disturbance term, which accounts for the uncertainties in the model due to the inevitable parametric modeling and measurement errors. The stability properties of the equilibrium points of the modified CTM are analyzed under all possible operating scenarios. The analysis shows that the disturbance directly influences the location of the equilibrium points on the fundamental dia- gram, resulting in triggering the capacity drop phenomena much earlier. To reject thedisturbance,therefore,whileimprovingthethroughput,arobustVSLcontroller is introduced, and the stability properties of the closed-loop system are analyzed. Ourfindingsshowthattheproposedcontrollerguaranteesthatthetrafficdensityof the considered road section converges asymptotically with time to a predetermined desired density located in the free-flow region of the fundamental diagram. Both macroscopicandmicroscopicsimulationresultsshowthattherobustcontrolscheme stabilizes the density of the road section at the desired density, and it improves the discharging flow rate by 33%, compared to the case of no control action. Furthermore, the traffic flow behavior per lane near an active highway bottle- neckisinvestigatedusingextensivemicroscopicsimulations. Thesimulationresults 136 demonstrate that every lane experiences a different traffic flow behavior, resulting in obtaining a different fundamental diagram per lane. A per-lane combined VSL and Lane Change (LC) control strategy is developed to prevent traffic congestion from being created at lane(s) drop highway bottlenecks. Based on a multi-lane Cell TransmissionModel(CTM),theproposedcontrolschemetreatseachlaneasasepa- ratestreamaimingtoutilizethemaximumcapacityofthemotorwayinfrastructure near an active bottleneck. While the LC control suggests LC recommendations to upstream vehicles, the VSL control forces traffic conditions to be very light in the blocked lane(s). In addition, it ensures that the other lanes have enough capacity for the lane-changing vehicles, and, accordingly, vehicles can change lanes while driving at the free-flow speed. This is achieved by introducing an adaptive law, based on Lyapunov stability analysis, to estimate the net lane-changing flow (lat- eralflow)ofthelaneonlinewhendeterminingtheVSLcontrolaction. Thestability analysis of the closed-loop system shows that the proposed controller guarantees that the traffic density of the lane operates within the free-flow region of the fun- damental diagram, which corresponds to the maximum possible discharging flow rate at the free-flow speed. The simulation results demonstrate the effectiveness of the introduced controller in improving the per-lane discharging flow rate, except for the blocked lane, when they are being compared to the open-loop cases. 8.2 Further Research Additional modifications may need to be included in the CTM framework to cre- ate a more realistic model that captures the traffic characteristics, especially un- der congested traffic conditions. Incorporates modeling uncertainties in terms of unmeasured unknown disturbances, parametric uncertainties in the various param- eters of the model, density and flow measurement noise, and uncertainties in the 137 assumed fundamental diagram are still interesting research subjects to explore. With a more realistic model, understanding the analysis of its open-loop properties may result in obtaining consistent results of what is observed in practice (qualita- tively) to the ones generated from microscopic simulations (quantitatively) when mainstream traffic flow controllers are being designed. The issue of robustness in designing traffic flow control strategies, such as VSL control, developed based on the macroscopic CTM, is yet an open subject. Ex- tending the present work to be integrated with other traffic control strategies, such as on-ramp controllers, and analyzing the stability properties as well as address- ing the robustness issue could be interesting future research topics. 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When 0≤ ¯q 1v ≤ d, i.e. x∈ S 1 ={x|v f ρ ⋆ − µ − d≤ (λ 1 − v f )x 1 +λ 2 x 2 ≤ v f ρ ⋆ − µ }, we have q 1 = ¯q 1v , and the dynamics of system (5.10) can be rewritten as: ˙ x 1 =− λ 1 x 1 − λ 2 x 2 ˙ x 2 =x 1 (A.1) Case 2. When ¯q 1v < 0, i.e. x∈ S 2 ={x|(λ 1 − v f )x 1 +λ 2 x 2 > v f ρ ⋆ − µ }, we have q 1 = 0, and the dynamics of system (5.10) can be rewritten as: ˙ x 1 =− v f x 1 − v f ρ ⋆ +µ ˙ x 2 =x 1 (A.2) Case 3. When ¯q 1v > d, i.e. x∈ S 3 ={x|(λ 1 − v f )x 1 +λ 2 x 2 < v f ρ ⋆ − µ − d}, we have q 1 =d, and the dynamics of system (5.10) can be rewritten as: ˙ x 1 =− v f x 1 − v f ρ ⋆ +µ +d ˙ x 2 =x 1 (A.3) Therefore, the state space is divided as shown in Figure A.1. It is clear that system (A.1) has a unique equilibrium point x e = [0,0] T ∈ S 1 , whereas system (A.2) and (A.3) have no equilibrium points. Consider the following Lyapunov function V(x) =x T Px, (A.4) where P = 2 λ 1 λ 1 λ 2 1 +2λ 2 149 1 x 2 x Saturation Boundaries 1 S 2 S 3 S (0,0) 1 f f v x v 1 f f vd x v S Figure A.1: State Space V(x) is positive definite since the matrix P is symmetric and positive definite. For all x∈S 1 , we have ˙ x =Ax where A = − λ 1 − λ 2 1 0 The derivative of the above Lyapunov function (A.4) is ˙ V(x) =x T (A T P +PA)x =− x T Qx, where Q = 2λ 1 0 0 2λ 1 λ 2 . Thus, ˙ V(x)< 0 for all x∈S 1 \{0}. 150 Now, we are going to show that if∃t 0 ≥ 0, such that x(t 0 )∈ S 3 , then∃t 1 > t 0 , such that x(t 1 )∈ S 1 . Let α = v f ρ ⋆ − µ − d. If∃t 0 ≥ 0, such that x(t 0 )∈ S 3 , then according to (A.3), we have that dx 2 dx 1 = x 1 − v f x 1 − α =− 1 v f + α v 2 f x 1 +v f α dx 2 =− 1 v f dx 1 + α v 2 f x 1 +v f α dx 1 taking the integral of both sides, we get x 2 (t)− x 2 (t 0 ) =− 1 v f (x 1 (t)− x 1 (t 0 ))+ α v 2 f ln(v 2 f x 1 (t)+v f α )− ln(v 2 f x 1 (t 0 )+v f α ) x 2 (t) =− 1 v f x 1 (t)+ α v 2 f ln(v 2 f x 1 (t)+v f α )+x 2 (t 0 )+ 1 v f x 1 (t 0 )− α v 2 f ln(v 2 f x 1 (t 0 )+v f α ) (A.5) From (A.3), we know that x 1 (t) approaches− α v f when x∈S 3 . According to (A.5), x 2 approaches infinity as x 1 (t) approaches − α v f . Thus, at some finite time instant t 1 , x 2 (t 1 ) is large enough and (λ 1 − v f )x 1 +λ 2 x 2 =v f ρ ⋆ − µ − d, i.e. x∈S 1 . Then, we show that if ∃t 0 ≥ 0, at which x(t 0 ) lies on the boundary between S 1 and S 3 , and x(t) moves towards S 3 , then ∃t 1 > t 0 , at which x(t 1 ) lies on the boundarybetweenS 1 andS 3 , andx(t)movestowardsS 1 . Furthermore, V(x(t 0 ))> V(x(t 1 )). The normal vector of the boundary line which points to S 3 is V n = [v f − λ 1 ,− λ 2 ] T . If∃t 0 ≥ 0, at which x(t 0 ) lies on the boundary between S 1 and S 3 , and x(t) moves towards S 3 , we have the following: x(t 0 ) = [x 1 (t 0 ),x 2 (t 0 )] T and ˙ x(t 0 ) = [− v f x 1 (t 0 )− α,x 1 (t 0 )] T Since ˙ x(t 0 ) points to S 3 , V T n ˙ x(t 0 )> 0, that is [v f (λ 1 − v f )− λ 2 ]x 1 (t 0 )− (v f − λ 1 )α> 0 x 1 (t 0 )> (v f − λ 1 )α v f (λ 1 − v f )− λ 2 >− α v f due to λ 1 > v f + λ 2 v f . According to (A.3), if x(t 0 ) ∈ S 3 and x 1 (t 0 ) > − α v f , then x 1 (t) > − α v f and ˙ x 1 (t) < 0,∀t > t 0 , as long as x(t) stays in S 3 . Together with (A.5), we know that∃t 1 >t 0 , at which the trajectory of x(t) crosses the boundary 151 line and moves towards S 1 , x 1 (t 1 )>− α v f and x 1 (t 1 )<x 1 (t 0 ). For all points on the boundary line between S 1 and S 3 , the Lyapunov function is evaluated as V(x) = 2x 2 1 +2λ 1 x 1 x 2 +(λ 2 1 +2λ 2 )x 2 2 , whose partial derivative with respect to x 1 along the boundary line is ∂V ∂x 1 = 4x 1 +2λ 1 x 2 +2λ 1 x 1 ∂x 2 ∂x 1 +2(λ 2 1 +2λ 2 ) ∂x 2 ∂x 1 (A.6) Since x(t 0 ) = [x 1 (t 0 ),− λ 1 − v f λ 2 x 1 (t 0 )+ α λ 2 ] T on the boundary line, we have that ∂x 2 ∂x 1 =− λ 1 − v f λ 2 . Thus, (A.6) can be rewritten as ∂V ∂x 1 =ax 1 +b where a = 4− 4λ 1 λ 1 − v f λ 2 +2(λ 2 1 +2λ 2 )( λ 1 − v f λ 2 ) 2 and b = 2[λ 1 − (λ 2 1 +2λ 2 )] α λ 2 . Note that a = 2[1, λ 1 − v f λ 2 ] T P[1, λ 1 − v f λ 2 ]> 0 due to P been positive definite, and a(− α λ 2 )+b =− [2λ 2 1 (λ 1 − v f )+2λ 1 λ 2 ]α> 0 due to α < 0 and λ 1 > v f + λ 2 v f . Thus, ∀x 1 >− α λ 2 , ∂V ∂x 1 = ax 1 +b > 0. Therefore, − α λ 2 <x 1 (t 1 )<x 1 (t 0 ), indicating that V(x(t 0 ))>V(x(t 1 )). Similarly, we can show that if ∃t 0 ≥ 0, such that x(t 0 ) ∈ S 2 , then ∃t 1 > t 0 , such that x(t 1 ) ∈ S 1 . Moreover, if ∃t 0 ≥ 0, at which x(t 0 ) lies on the boundary line between S 1 and S 2 and x(t) moves towards S 2 , then ∃t 1 > t 0 , at which x(t 0 ) lies on the boundary between S 1 and S 2 , and x(t) moves towards S 1 . Furthermore, V(x(t 0 ))>V(x(t 1 )). Summarizing the behavior of the Lyapunov function V(x), we conclude that ∀x(t 0 ) ∈ ℜ 2 , x(t) converges to x e = [0,0] T asymptotically, which implies that ∀(ρ (t 0 ),Φ( t 0 ))∈ℜ 2 , [ρ (t),Φ( t)] T asymptotically converges to h ρ ⋆ , µ λ 2 i T . 152 It follows from (5.10) that − v f x 1 − (v f ρ ⋆ − µ )≤ ˙ x 1 ≤− v f x 1 − (v f ρ ⋆ − µ − d), thus for all− v f ρ ⋆ − µ v f < x 1 (t 0 ) <− v f ρ ⋆ − µ − d v f ,− v f ρ ⋆ − µ v f < x 1 (t) <− v f ρ ⋆ − µ − d v f ,∀t≥ t 0 . We have also shown that if x(t)∈ S 1 ∩{x|− v f ρ ⋆ − µ v f < x 1 <− v f ρ ⋆ − µ − d v f }, x(t) will not leave S 1 . Therefore, ∀x(t 0 ) ∈ S = {x|v f ρ ⋆ − µ − d ≤ (λ 1 − v f )x 1 +λ 2 x 2 ≤ v f ρ ⋆ − µ }∩{x|− v f ρ ⋆ − µ v f < x 1 < − v f ρ ⋆ − µ − d v f }, x 1 (t) ∈ S,∀t ≥ t 0 , implying that ∀(ρ (t 0 ),Φ( t 0 ))∈S,(ρ (t)− ρ ⋆ )∈S,∀t≥ t 0 . 153 Appendix B The Proof of Theorem 7.3.1 Proof. • If lane i is the blocked lane and (d u i − l n i )≥ v f ρ ⋆ i , then we have that q out i = 0 and ρ ⋆ i = ρ L i . Therefore, the closed-loop system described by equa- tions (7.10)-(7.12) in section 7.3 can be rewritten as ˙ ρ i (t) = 1 L q in i − l n i , 0≤ ρ i (0)≤ ρ j ˙ ˆ l n i (t) =− γ i L (ρ i (t)− ρ ⋆ i ), ˆ l n i (0) =λ i (ρ i (0)− ρ ⋆ i )+v f ρ ⋆ i (B.1) q in i = min d u i ,q vsl i ,w(ρ j − ρ i (t)),C v vsl i = wq vsl i wρ j − q vsl i q vsl i = v vsl i wρ j v vsl i +w = median{0,¯q vsl i ,v f ρ ⋆ i } ¯q vsl i = ˆ l n i (t)− λ i (ρ i (t)− ρ ⋆ i ) ˆ l n i (t) = ˆ l n i (0)− γ i L Z t 0 (ρ i (τ )− ρ ⋆ i ) dτ, ∀t≥ 0 (B.2) Recall that the control objective is to drive the lane traffic density to the desired value, i.e., ρ i (t) = ρ ⋆ i , which corresponds to the maximum possible flow rate and speed, i.e., q in i =l n i =v f ρ ⋆ i . ∀ρ i (0)∈ [0,ρ ⋆ i ], then as long as (d u i − l n i )≥ v f ρ ⋆ i we have the following w(ρ j − ρ i (t))≥ w(ρ j − ρ ⋆ i )>w(ρ j − ρ c ) =C≥ d u i − l n i ≥ d u i ≥ v f ρ ⋆ i ≥ q vsl i ≥ 0 Thus, q in i = min d u i ,q vsl i ,w(ρ j − ρ i (t)),C = q vsl i = median{0,¯q vsl i ,v f ρ ⋆ i }, and the the dynamics of the closed-loop system (B.1)-(B.2) can be rewritten as 154 ˙ ρ i (t) = 1 L median{0,¯q vsl i ,v f ρ ⋆ i }− l n i , 0≤ ρ i (0)≤ ρ ⋆ i ˙ ˆ l n i (t) =− γ i L (ρ i (t)− ρ ⋆ i ), ˆ l n i (0) =λ i (ρ i (0)− ρ ⋆ i )+v f ρ ⋆ i (B.3) where ¯q vsl i = ˆ l n i (t)− λ i (ρ i (t)− ρ ⋆ i ) ˆ l n i (t) = ˆ l n i (0)− γ i L Z t 0 (ρ i (τ )− ρ ⋆ i ) dτ, ∀t≥ 0 (B.4) In this case, ¯q vsl i ≤ v f ρ ⋆ i . Therefore, if 0≤ ¯q vsl i , then the dynamics of system (B.3)-(B.4) become: ˙ ρ i (t) = 1 L − λ i (ρ i (t)− ρ ⋆ i )− l n i + ˆ l n i (t) ˙ ˆ l n i (t) =− γ i L (ρ i (t)− ρ ⋆ i ) (B.5) The equilibrium state of (B.5) is ρ e i = ρ ⋆ i , ˆ l e i = l n i = v f ρ ⋆ i . To analyse the stability of the equilibrium state, consider the same Lyapunov function can- didate introduced earlier in the equation (7.6) whose time derivative in this case along the trajectories of system (B.5) is ˙ V i (ρ i , ˆ l n i ) = − λ i L (ρ i − ρ ⋆ i ) 2 ≤ 0 (B.6) V i (ρ i , ˆ l n i )> 0, i.e., positive definite, whereas ˙ V i (ρ i , ˆ l n i )≤ 0, i.e., negative semi- definite,thensimilartotheanalysisinsubsection7.3.2,onecanconcludethat theequilibriumstateisasymptoticallystablebyapplyingLaSalle’sinvariance principle [108]. If ¯q vsl i < 0, then the dynamics of system (B.3)-(B.4) become: ˙ ρ i (t) =− l n i L ˙ ˆ l n i (t) =− γ i L (ρ i (t)− ρ ⋆ i ) (B.7) The time derivative of the Lyapunov function in (7.6) along the trajectories of system (B.7) is ˙ V i (ρ i , ˆ l n i ) =− (ρ i − ρ ⋆ i ) L ˆ l n i < − λ i L (ρ i − ρ ⋆ i ) 2 < 0 (B.8) 155 dueto ˆ l n i (t)<λ i (ρ i − ρ ⋆ i )and0≤ ρ i <ρ ⋆ i .Thus, ˙ V i (ρ i , ˆ l n i )< 0,∀(ρ i (t), ˆ l n i (t))̸= (ρ ⋆ i ,v f ρ ⋆ i ). ∀ρ i (0)∈ (ρ ⋆ i ,ρ j ], then as long as (d u i − l n i )≥ v f ρ ⋆ i we have the following w(ρ j − ρ ⋆ i )>w(ρ j − ρ c ) =C≥ d u i ≥ d u i − l n i ≥ v f ρ ⋆ i ≥ q vsl i ≥ w ρ j − v f ρ ⋆ i w > 0 Therefore, the inflow q in i = min d u i ,q vsl i ,w(ρ j − ρ i (t)),C = min{q vsl i ,w(ρ j - ρ i (t))}, and the the dynamics of the closed-loop system (B.1)-(B.2) can be rewritten as ˙ ρ i (t) = 1 L min q vsl i ,w(ρ j − ρ i (t)) − l n i , ρ ⋆ i <ρ i (0)≤ ρ j ˙ ˆ l n i (t) =− γ i L (ρ i (t)− ρ ⋆ i ), ˆ l n i (0) =λ i (ρ i (0)− ρ ⋆ i )+v f ρ ⋆ i (B.9) where q vsl i = median 0,¯q vsl i ,v f ρ ⋆ i ¯q vsl i = ˆ l n i (t)− λ i (ρ i (t)− ρ ⋆ i ) ˆ l n i (t) = ˆ l n i (0)− γ i L Z t 0 (ρ i (τ )− ρ ⋆ i ) dτ, ∀t≥ 0 (B.10) If min q vsl i ,w(ρ j − ρ i (t)) = q vsl i = median 0,¯q vsl i ,v f ρ ⋆ i , then in this case we have that ¯q vsl i > 0. If ¯q vsl i ≤ v f ρ ⋆ i , then the dynamics of system (B.9)- (B.10) become: ˙ ρ i (t) = 1 L − λ i (ρ i (t)− ρ ⋆ i )− l n i + ˆ l n i (t) ˙ ˆ l n i (t) =− γ i L (ρ i (t)− ρ ⋆ i ) (B.11) The time derivative of the Lyapunov function in (7.6) along the trajectories of system (B.11) is ˙ V i (ρ i , ˆ l n i ) = − λ i L (ρ i − ρ ⋆ i ) 2 ≤ 0 (B.12) duetoρ ⋆ i ≤ ρ i ≤ ρ j . Similar tothe analysis ofsystem (B.5), we concludethat the equilibrium state ρ e i =ρ ⋆ i , ˆ l e i =l n i =v f ρ ⋆ i is asymptotically stable. If ¯q vsl i >v f ρ ⋆ i , then the dynamics of system (B.9)-(B.10) become: 156 ˙ ρ i (t) = 1 L v f ρ ⋆ i − l n i ˙ ˆ l n i (t) =− γ i L (ρ i (t)− ρ ⋆ i ) (B.13) In this case, the time derivative of the Lyapunov function in (7.6) along the trajectories of system (B.13) is ˙ V i (ρ i , ˆ l n i ) = (ρ i − ρ ⋆ i ) L v f ρ ⋆ i − ˆ l n i < − λ i L (ρ i − ρ ⋆ i ) 2 < 0 (B.14) due to ˆ l n i (t)<− λ i (ρ i − ρ ⋆ i )+v f ρ ⋆ i and ρ ⋆ i <ρ i ≤ ρ j ; therefore, ˙ V i (ρ i , ˆ l n i )< 0, ∀(ρ i (t), ˆ l n i (t))̸= (ρ ⋆ i ,v f ρ ⋆ i ). If min q vsl i ,w(ρ j − ρ i (t)) = w(ρ j − ρ i (t)), it implies that ¯q vsl i ≥ v f ρ ⋆ i ≥ q vsl i >w(ρ j − ρ i (t)). Inthiscase,thedynamicsofsystem(B.9)-(B.10)become: ˙ ρ i (t) = 1 L w ρ j − ρ i (t) − l n i ˙ ˆ l n i (t) =− γ i L (ρ i (t)− ρ ⋆ i ) (B.15) andthetimederivativeoftheLyapunovfunctionin(7.6)alongthetrajectories of (B.15) is ˙ V i (ρ i , ˆ l n i ) = (ρ i − ρ ⋆ i ) L w ρ j − ρ i − ˆ l n i < − λ i L (ρ i − ρ ⋆ i ) 2 < 0 (B.16) due to ˆ l n i (t)>w(ρ j − ρ i (t))+λ i (ρ i − ρ ⋆ i ) and ρ ⋆ i <ρ i ≤ ρ j . Thus, ˙ V i (ρ i , ˆ l n i )< 0,∀(ρ i (t), ˆ l n i (t))̸= (ρ ⋆ i ,v f ρ ⋆ i ). Summarizing the behavior of the Lyapunov function given by equation (7.6), weconcludethat ˙ V i (ρ i , ˆ l n i )< 0,∀(ρ i (t), ˆ l n i (t))̸= (ρ ⋆ i ,v f ρ ⋆ i )andtheequilibrium state [ρ e i , ˆ l e i ] T = [ρ ⋆ i ,l n i ] T = [ρ ⋆ i ,v f ρ ⋆ i ] T = [ρ L i ,v f ρ ⋆ i ] T is asymptotically stable, but since V i (ρ i , ˆ l n i ) is also radially unbounded, we infer that the equilibrium state is globally asymptotically stable, which, in turn, implies that ρ i (t)→ρ ⋆ i as t→∞,∀ρ i (0)∈ [0,ρ j ]. • If lane i is open, then we have that q out i ̸= 0 and ρ ⋆ i = ρ c . Accordingly, the closed-loop traffic flow control system described by (7.10)-(7.12) can be rewritten as 157 ˙ ρ i (t) = 1 L q in i − q out i +l n i , 0≤ ρ i (0)≤ ρ j ˙ ˆ l n i (t) = γ i L (ρ i (t)− ρ ⋆ i ), ˆ l n i (0) =λ i (ρ i (0)− ρ ⋆ i ) (B.17) q in i = min d u i ,q vsl i ,w(ρ j − ρ i (t)),C q out i = min v f ρ i (t), ˜ w(˜ ρ j − ρ i (t)),C v vsl i = wq vsl i wρ j − q vsl i q vsl i = v vsl i wρ j v vsl i +w = median{0,¯q vsl i ,v f ρ ⋆ i } ¯q vsl i =q out i − ˆ l n i (t)− λ i (ρ i (t)− ρ ⋆ i ) ˆ l n i (t) = ˆ l n i (0)+ γ i L Z t 0 (ρ i (τ )− ρ ⋆ i ) dτ, ∀t≥ 0 (B.18) ∀ρ i (0)∈ [0,ρ ⋆ i ], then as long as (d u i +l n i )≥ v f ρ ⋆ i we have the following w(ρ j − ρ i (t))≥ d u i +l n i ≥ w(ρ j − ρ c ) =v f ρ c =v f ρ ⋆ i =C≥ q vsl i ≥ 0 and ˜ w(˜ ρ j − ρ i (t))≥ w(ρ j − ρ c ) =v f ρ c =v f ρ ⋆ i =C≥ v f ρ i (t)≥ 0 As a result, q in i = q vsl i = median{0,¯q vsl i ,v f ρ ⋆ i }, and q out i = v f ρ i (t), and the closed-loop system represented in (B.17) and (B.18) can be rewritten as ˙ ρ i (t) = 1 L median 0,¯q vsl i ,v f ρ ⋆ i − v f ρ i (t)+l n i , 0≤ ρ i (0)≤ ρ ⋆ i ˙ ˆ l n i (t) = γ i L (ρ i (t)− ρ ⋆ i ), ˆ l n i (0) =λ i (ρ i (0)− ρ ⋆ i ) (B.19) where ¯q vsl i =v f ρ i (t)− ˆ l n i (t)− λ i (ρ i (t)− ρ ⋆ i ) ˆ l n i (t) = ˆ l n i (0)+ γ i L Z t 0 (ρ i (τ )− ρ ⋆ i ) dτ, ∀t≥ 0 (B.20) 158 In this case, we have ¯q vsl i ≤ v f ρ ⋆ i . Therefore, if 0≤ ¯q vsl i , then the dynamics of system (B.19) become: ˙ ρ i (t) = 1 L − λ i (ρ i (t)− ρ ⋆ i )+l n i − ˆ l n i (t) ˙ ˆ l n i (t) = γ i L (ρ i (t)− ρ ⋆ i ) (B.21) The equilibrium state of (B.21) is ρ e i = ρ ⋆ i , ˆ l e i = l n i . To analyze its stability properties,weusethesameLyapunovfunctionV i (ρ i , ˆ l n i )describedbyequation (7.6). The time derivative of V i (ρ i , ˆ l n i ) along the trajectories of system (B.21) is ˙ V i (ρ i , ˆ l n i ) = − λ i L (ρ i − ρ ⋆ i ) 2 ≤ 0 (B.22) Similar to the previous analysis of system (B.5), one can conclude that the equilibrium state (ρ ⋆ i ,l n i ) is asymptotically stable. If ¯q vsl i < 0, then the dynamics of system (B.19) become: ˙ ρ i (t) = 1 L − v f ρ i (t)+l n i ˙ ˆ l n i (t) = γ i L (ρ i (t)− ρ ⋆ i ) (B.23) The time derivative of the Lyapunov function in (7.6) along the trajectories of the system (B.23) is ˙ V i (ρ i , ˆ l n i ) = (ρ i − ρ ⋆ i ) L − v f ρ i + ˆ l n i < − λ i L (ρ i − ρ ⋆ i ) 2 < 0 (B.24) due to ˆ l n i > v f ρ i − λ i (ρ i − ρ ⋆ i ) and 0 ≤ ρ i < ρ ⋆ i . Hence, ˙ V i (ρ i , ˆ l n i ) < 0, ∀(ρ i (t), ˆ l n i (t))̸= (ρ ⋆ i ,l n i ). ∀ρ i (0)∈ (ρ ⋆ i ,ρ j ], then as long as (d u i +l n i )≥ v f ρ ⋆ i we have the following d u i +l n i ≥ w(ρ j − ρ c ) =v f ρ c =v f ρ ⋆ i =C≥ w ρ j − q vsl i w >w(ρ j − ρ j ) = 0 and v f ρ i (t)≥ v f ρ ⋆ i =v f ρ c =C≥ ˜ w(˜ ρ j − ρ i (t))≥ ˜ w(˜ ρ j − ρ j )> 0 159 Thus, q in i = min q vsl i ,w(ρ j − ρ i (t)) , and q out i = ˜ w(˜ ρ j − ρ i (t)). In this case, the closed-loop system represented in (B.17) and (B.18) can be rewritten as ˙ ρ i (t) = 1 L min{q vsl i ,w(ρ j − ρ i (t))}− ˜ w(˜ ρ j − ρ i (t))+l n i , ρ ⋆ i <ρ i (0)≤ ρ j ˙ ˆ l n i (t) = γ i L (ρ i (t)− ρ ⋆ i ), ˆ l n i (0) =λ i (ρ i (0)− ρ ⋆ i ) (B.25) where q vsl i = median 0,¯q vsl i ,v f ρ ⋆ i ¯q vsl i = ˜ w(˜ ρ j − ρ i (t))− ˆ l n i (t)− λ i (ρ i (t)− ρ ⋆ i ) ˆ l n i (t) = ˆ l n i (0)+ γ i L Z t 0 (ρ i (τ )− ρ ⋆ i ) dτ, ∀t≥ 0 (B.26) If min q vsl i ,w(ρ j − ρ i (t)) = q vsl i = median 0,¯q vsl i ,v f ρ ⋆ i , then in this case we have that 0 < ¯q vsl i ≤ v f ρ ⋆ i , and the dynamics of system (B.25)-(B.26) become: ˙ ρ i (t) = 1 L − λ i (ρ i (t)− ρ ⋆ i )+l n i − ˆ l n i (t) ˙ ˆ l n i (t) = γ i L (ρ i (t)− ρ ⋆ i ) (B.27) The time derivative of the Lyapunov function given by equation (7.6) along the trajectories of the system (B.27) is ˙ V i (ρ i , ˆ l n i ) = − λ i L (ρ i − ρ ⋆ i ) 2 ≤ 0 (B.28) Similar to the previous analysis of system (B.5), one can infer that the equi- librium state [ρ e i , ˆ l e i ] T = [ρ ⋆ i ,l n i ] T is asymptotically stable. Ifmin q vsl i ,w(ρ j − ρ i (t)) =w(ρ j − ρ i (t)),itimpliesthat ¯q vsl i ≥ v f ρ ⋆ i ≥ q vsl i > w(ρ j − ρ i (t)). In this case, the dynamics of system (B.25)-(B.26) become: ˙ ρ i (t) = 1 L w(ρ j − ρ i (t))− ˜ w(˜ ρ j − ρ i (t))+l n i ˙ ˆ l n i (t) = γ i L (ρ i (t)− ρ ⋆ i ) (B.29) The time derivative of the Lyapunov function giving by (7.6) along the tra- jectories of the system (B.29) is 160 ˙ V i (ρ i , ˆ l n i ) = (ρ i − ρ ⋆ i ) L w(ρ j − ρ i )− ˜ w(˜ ρ j − ρ i )+ ˆ l n i < − λ i L (ρ i − ρ ⋆ i ) 2 < 0 (B.30) due to ˆ l n i < − λ i (ρ i − ρ ⋆ i )− w(ρ j − ρ i )+ ˜ w(˜ ρ j − ρ i ) and ρ ⋆ i < ρ i ≤ ρ j ; thus, ˙ V i (ρ i , ˆ l n i )< 0,∀(ρ i (t), ˆ l n i (t))̸= (ρ ⋆ i ,l n i ). Itfollowsfromequations(B.22),(B.24),(B.28),and(B.30)that ˙ V i (ρ i , ˆ l n i )< 0, ∀(ρ i (t), ˆ l n i (t))̸= (ρ ⋆ i ,l n i )andtheequilibriumstate[ρ e i , ˆ l e i ] T = [ρ ⋆ i ,l n i ] T =[ρ c ,l n i ] T is globally asymptotically stable since the Lyapunov function described by equation (7.6) is also a radially unbounded function. 161
Abstract (if available)
Abstract
The well-known macroscopic Cell Transmission Model (CTM) has been widely used to develop several Intelligent Transportation Systems (ITS) to mitigate highway traffic congestion. Variable Speed Limit (VSL) and Lane Change (LC) control techniques are the most commonly used and studied ITS applications for regulating the mainstream traffic flow, notably near highway bottleneck locations. While most of the reported macroscopic simulation results showed significant improvements in traffic mobility, the resulting data from microscopic simulations and the deployment of such technologies in real traffic environments were somewhat controversial; some microscopic simulations and field tests demonstrated inconsistency under different traffic conditions and incident scenarios. This raises the question of whether the CTM needs to be modified to accurately capture the traffic dynamics at the bottleneck locations, especially under congested conditions, or whether the proposed mainstream traffic control designs are not robust enough to reject disturbances. ❧ In this dissertation, the CTM version, which considers the effect of both capacity drop and bounded acceleration, is modified to include a constant disturbance term to account for the uncertainties related to modeling and measurement errors. Motivated by the open-loop stability analysis of the modified model, a robust VSL controller is proposed, one which rejects external disturbances while guaranteeing that the traffic density converges to the desired equilibrium state as demonstrated by both macroscopic and microscopic simulations. The robust VSL control design is then extended to a multi-section CTM and combined with an LC control at the discharging section. The section length covered by the most upstream VSL sign is treated as a variable. Via extensive microscopic simulations, the integrated control scheme demonstrates promising improvements in the average travel time, the average number of stops, and the average emission rates when being compared to the cases of no control actions. ❧ Furthermore, a multiple-lane CTM-based VSL control is introduced, where each lane of a motorway is treated as a separate stream; the net lane-changing flow (lateral flow) is modeled as an additional unknown term in the traffic flow conservation equation. This unknown net flow is estimated in real-time, and its estimate, which is developed based on Lyapunov stability analysis, is used at each time in calculating a per-lane VSL control command for each lane of the motorway. Then, a Lane Change (LC) controller is combined with the VSL to prevent creating a queue in the vicinity of the bottleneck. The stability properties of the closed-loop system are analyzed, where the integrated control scheme guarantees that the lane traffic density operates within the free-flow region of the fundamental diagram. Therefore, traffic congestion is relieved, and the per-lane outflow is maximized, except for the blocked lane, when the highway bottleneck is active, as demonstrated by microscopic simulations.
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Asset Metadata
Creator
Alasiri, Faisal Hasan
(author)
Core Title
Control of mainstream traffic flow: variable speed limit and lane change
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Degree Conferral Date
2022-05
Publication Date
01/02/2022
Defense Date
12/09/2021
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
integrated mainstream traffic flow control,lane change control,multi-lane cell transmission model,OAI-PMH Harvest,stability,variable speed limit
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Ioannou, Petros (
committee chair
), Dessouky, Maged (
committee member
), Nuzzo, Pierluigi (
committee member
), Savla, Ketan (
committee member
)
Creator Email
alasiri@usc.edu,eng.faisal13@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-oUC110455245
Unique identifier
UC110455245
Legacy Identifier
etd-AlasiriFai-10325
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Alasiri, Faisal Hasan
Type
texts
Source
20220112-usctheses-batch-907
(batch),
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository e-mail address given.
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Repository Email
cisadmin@lib.usc.edu
Tags
integrated mainstream traffic flow control
lane change control
multi-lane cell transmission model
stability
variable speed limit