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University of Southern California Dissertations and Theses
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Microscopic traffic control: theory and practice
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Microscopic traffic control: theory and practice
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MICROSCOPIC TRAFFIC CONTROL: THEORY AND PRACTICE by Milad Pooladsanj 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 AND COMPUTER ENGINEERING) December 2024 Copyright 2024 Milad Pooladsanj Dedication To Mehrnaz, for her unlimited support and encouragement. ii Acknowledgements My time at USC has been truly wonderful, and I have many people to thank for their help and support along the way. First and foremost, I want to thank my advisors, Petros Ioannou and Ketan Savla. Beyond being incredible mentors and teachers, they have always encouraged me to do my best and given me a great deal of freedom to explore in my research. I seem to run out of words as I try to thank them for all their efforts, help, and support, but I hope every student gets advisors as great as them. I also want to give a huge thanks to Mihailo Jovanovic, who has been like a third advisor to me. He has influenced the way I think about technical problems to a great extent, and has been very generous in helping me find my way when I have been unsure. I am also really grateful to Ken Alexander for being such an extraordinary math teacher. His way of teaching and tackling problems has fundamentally re-shaped my own thinking and teaching style. I would also like to extend my sincere thanks to Ruolin Li for her time and effort in serving on my defense committee. I sincerely appreciate her help in improving this thesis. Lastly, I want to express my appreciation to my family for their unwavering support, not just during my Ph.D., but throughout my entire education. And to my love, Mehrnaz, thank you for being who you are. Your immense faith in me, your patient listening ∗ , and your constant encouragement gave me the strength and reason to see this journey through to the end. ∗ I figured out the crux of the proof of Theorem 1 while she was patiently listening to my random thoughts. iii Table of Contents Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Ground Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Air Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Chapter 2: Ramp Metering to Maximize Freeway Throughput under Vehicle Safety Constraints . . 6 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Vehicle-Level Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Probabilistic Demand Model and Throughput . . . . . . . . . . . . . . . . . . . . . 14 2.2.3 Ramp Metering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Ramp Metering Policies and Performance Analysis . . . . . . . . . . . . . . . . . . . . . . 19 2.3.1 Renewal Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.2 Dynamic Release Rate Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.3 Distributed Dynamic Release Rate Policy . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.4 Dynamic Space Gap Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.5 Local and Greedy Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.6 An Outer Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.7 Discussion of the Straight Road Geometry . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.1 Greedy Policy for Low Merging Speed . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.2 Effect of Cycle Length on Queue Size . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4.3 Relaxing the V2X Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.4 Comparing the Total Travel Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 iv Chapter 3: Vehicle Following On A Ring Road Under Safety Constraints: Role of Connectivity and Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.1 Basic Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.2 Modes of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.3 Model and Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 Vehicles On a Ring Road Without Coordination . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.1 No V2V Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.2 V2V Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4 Coordination of Vehicles On A Ring Road . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5.1 High Density Traffic Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5.2 Low Density Traffic Regime: No Coordination . . . . . . . . . . . . . . . . . . . . . 58 3.5.3 Low Density Traffic Regime: With Coordination . . . . . . . . . . . . . . . . . . . 59 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Chapter 4: Throughput Maximizing Takeoff Scheduling for eVTOL Aircraft in On-Demand Urban Air Mobility Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2.1 UAM Network Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2.2 Operational Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.3 Demand and Performance Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3 Network-Wide Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3.1 VertiSync Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3.2 Size of VertiSync Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3.3 VertiSync Throughput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3.4 Fundamental Limit on Throughput . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4.1 Comparison of Theoretical Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4.2 Comparison with First-Come First-Serve Policy . . . . . . . . . . . . . . . . . . . . 84 4.4.3 Computation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Chapter 5: Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 A.1 Network Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 A.1.1 Dynamics in the Speed Tracking Mode . . . . . . . . . . . . . . . . . . . . . . . . . 97 A.1.2 Communication Cost of a Ramp Metering Policy . . . . . . . . . . . . . . . . . . . 98 A.2 Performance Analysis Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 A.3 Proofs of the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 A.3.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 A.3.2 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 v A.3.3 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 A.3.4 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 A.3.5 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 B.1 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 B.2 Proof of Theorem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 C.1 Proof of Theorem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 C.2 Proof of Theorem 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 vi List of Tables 2.1 Summary of the RM policies studied in this chapter. . . . . . . . . . . . . . . . . . . . . . 19 2.2 Performance of the policies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1 Computational results for symmetric demand. . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2 Computational results for asymmetric demand. . . . . . . . . . . . . . . . . . . . . . . . . 87 vii List of Figures 2.1 Example of a (a) straight road, (b) ring road. . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 A merging scenario. For each vehicle, the dotted arrow indicates its leading vehicle while the solid arrow indicates its virtual leading vehicle. Note that vehicle l is both the leading and the virtual leading vehicle of vehicle e. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 An illustration of the under-saturation region of some policy π (dark grey area) and the DRR policy (dark + light grey areas) from Example 2 . . . . . . . . . . . . . . . . . . . . . 16 2.4 An inner estimate of the under-saturation region of the Renewal policy (grey area) from Example 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 An illustration of the update rule for the minimum time gap g in Algorithm 1. . . . . . . . 24 2.6 An inner estimate of the under-saturation region (dark grey area) of the DRR policy from Example 4. The light grey area indicates the additional under-saturation region if we use the Renewal policy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7 On-ramp queue size profiles when all the on-ramps are long, under the FCQ policy. . . . . 35 2.8 Effect of cycle length Tcyc on the long-run expected queue size (a) for different ρ when both on-ramps are long, (b) for a fixed ρ when on-ramp 2 is short. Both plots are under the FCQ policy. In plot (b), the logarithm of the expected queue size is set to 20 whenever the freeway becomes saturated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.9 The total travel time (TTTn), where n is the number of completed trips, under the fixed demand ρ = 0.82. The vertical lines in the Safe-ALINEA, Renewal, and DRR policies represent the standard deviation. The standard deviation of the DisDRR, DSG, and greedy policies are similar to the DRR policy and are omitted for clarity. . . . . . . . . . . 39 3.1 Example of three vehicles on a closed ring road setup . . . . . . . . . . . . . . . . . . . . . 43 3.2 Logic diagram for determining the mode of operation . . . . . . . . . . . . . . . . . . . . . 44 3.3 Fundamental diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 viii 3.4 Control structure in the presence of a coordinator and/or V2V communication . . . . . . . 53 3.5 Logic diagram of a desired leader for creating the desired platoon . . . . . . . . . . . . . . 55 3.6 Simulation results for the high density traffic regime . . . . . . . . . . . . . . . . . . . . . 55 3.7 Simulation results for the low density traffic regime with no coordination . . . . . . . . . 56 3.8 Simulation results with coordination and 2-platoon symmetrical desired configuration . . 56 3.9 Steady state configuration of the vehicles with (a) no coordination, (b) coordination with 2-platoon symmetrical desired configuration (orange: leader, blue: follower) . . . . . . . . 57 4.1 A top-view sketch of a UAM network with three modes of aircraft operation: idle aircraft (red), in-service aircraft that transport passengers (black), and rebalancing aircraft that fly without passengers to high-demand areas (purple). . . . . . . . . . . . . . . . . . . . . 62 4.2 A top-view sketch of a UAM network with |V| = 4 vertiports (blue circles) and |P| = 8 O-D pairs P = {(1, 3),(1, 4),(2, 3),(2, 4),(3, 1),(4, 2),(1, 2),(2, 1)}. . . . . . . . . . . . . 65 4.3 Sector configuration for a UAM network, with sector capacity of 1 vehicle. Hence, at most 1 UAM vehicle can occupy any sector at any time. Moreover, if a UAM vehicle occupies sector A, then it moves to sector B after one time step. . . . . . . . . . . . . . . . . . . . . 67 4.4 An illustration of the under-saturation region of some policy π ′ (dark grey area) and a throughput maximizing policy π (dark + light grey areas). . . . . . . . . . . . . . . . . . . 68 4.5 Sector configuration for a UAM network. The green sector belongs to routes (1, 3),(1, 4),(2, 3),(2, 4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.6 A UAM network with 2 vertiports (blue circles) and 2 O-D pairs (1, 2) and (2, 1) sharing a single route (shown as a double-headed arrow). . . . . . . . . . . . . . . . . . . . . . . . 78 4.7 A top-view sketch of a UAM network for Los Angeles. The blue circles show the vertiports and the orange arrows show the links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.8 The sufficient (from Theorem 7), necessary (from Theorem 8), and actual (from simulations) bounds on λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.9 The rate of trip requests per τ minutes (λ(t)). . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.10 A top-view sketch of an expanded UAM network for Los Angeles with 12 vertiports and 27 O-D pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.11 The travel time under the VertiSync and FCFS policies for the demand λ(t). . . . . . . . . 83 4.12 The travel time under the VertiSync policy when the demand is increased to 1.2λ(t) (over-saturated regime), and the ground transportation travel time. . . . . . . . . . . . . . 84 ix B.1 Block diagram of a platoon of m vehicles with vehicle m as the leader . . . . . . . . . . . 121 B.2 Block diagram of a platoon with no leader . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 B.3 Block diagram of the k-platoon symmetrical configuration with vehicle n as the desired leader of a platoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 x Abstract Traffic congestion is one of the fundamental challenges in modern transportation systems, which occurs due to an imbalance between demand and infrastructural resources. To alleviate congestion, traffic controllers are often employed. In ground transportation, traffic controllers are traditionally designed using macroscopic traffic flow models, which are obtained by spatio-temporal averaging of vehicle interactions. However, these models lack the granularity to account for safety protocols under different connectivity and automation scenarios, which could lead to suboptimal performance. The first objective of this thesis is to systematically design and analyze traffic controllers for ground transportation at the microscopic level, where such nuances can be better addressed. Another approach to mitigate traffic congestion is by introducing new modes of transportation, such as Urban Air Mobility (UAM), which uses urban airspace for on-demand mobility. However, without proper traffic control, UAM systems may face the same congestion challenges as ground transportation. The secondary objective of this thesis is to design and analyze traffic control protocols for on-demand UAM systems at the microscopic level. In the second chapter of this thesis, we focus on a particular type of traffic controllers– Ramp Metering (RM)– which regulates the inflow of vehicles from on-ramps to freeway to improve traffic flow. In particular, we design RM policies at the microscopic level under vehicle following safety constraints and analyze their system-level performance. The performance of an RM policy is measured in terms of its throughput, which is characterized by the set of on-ramp arrival rates for which on-ramp queues remain bounded in expectation. The throughput of the proposed policies is characterized by studying stochastic stability of xi the induced Markov chains, and is proven to be maximized when the merging speed of all on-ramps equal the free flow speed. In the third chapter, we explore vehicle controllers, as another method to enhance traffic flow. Specifically, we design a longitudinal vehicle controller and study the dynamics of a system of homogeneous vehicles on a single-lane ring road to understand the interplay of limited space, speed, and safety. The proposed controller guarantees safety and comfort, provides smooth reference tracking, and attenuates disturbances upstream of traffic flow. Furthermore, we extend the controller to use vehicle-to-vehicle (V2V) communication or receive instructions from a central coordinator to further improve the traffic flow. Finally, in Chapter 4, we provide a microscopic-level scheduling policy for on-demand UAM aircraft, and analyze its performance. The proposed policy schedules the UAM aircraft for either servicing trip requests or rebalancing in the system under aircraft safety margins and energy requirements. Similar to Chapter 2, we measure performance in terms of the system-level throughput, which determines the demand threshold at which passenger waiting times transition from being stabilized to being increasing over time. We show that the proposed policy is able to maximize throughput for sufficiently large fleet sizes. xii Chapter 1 Introduction 1.1 Problem Description Traffic congestion arises when high demand competes for limited infrastructural resources. In response to the need to alleviate congestion and leverage advancements in vehicle and infrastructure technologies, this thesis focuses on designing microscopic-level traffic controllers for both ground and air systems. 1.1.1 Ground Transportation Recent advancements in Connected and Automated Vehicles (CAVs) offer promising tools to improve mobility. CAVs can mitigate human errors through automation and provide more information about traffic conditions by communicating with each other and the infrastructure [1, 2, 3]. However, CAVs must also adhere to strict safety rules, prompting the need to investigate how their capabilities and constraints can be integrated into the design of traffic controllers to maximize their impact on overall traffic flow. In Chapters 2 and 3, we focus specifically on using CAVs to combat freeway congestion. One of the major causes of freeway congestion is the merging of traffic from on-ramps. Ramp Metering (RM) is an effective tool for mitigating this congestion by regulating freeway inflows to balance supply and demand, thereby improving performance measures such as throughput [1, 4, 5]. Traditionally, RM policies are designed by using macroscopic traffic flow models. While these models are broadly applicable, they lack 1 the granularity needed to address safety protocols under different connectivity and automation scenarios. A more appropriate approach is the microscopic-level design. The RM design problem at the microscopic level is typically framed as determining the merging sequence at on-ramps, i.e., the order in which vehicles on the mainline and at the on-ramp cross the merging point. The goal is to choose a merging sequence that optimizes performance measures such as fuel consumption while considering vehicle constraints like speed, safety, and comfort [1, 6]. However, most existing studies focus on isolated on-ramps and overlook the broader implications for overall freeway performance. Indeed, it is possible that a greedy RM policy, where each on-ramp operates independently, limits entry from downstream on-ramps and creates long queues or congestion. Our objective in Chapter 2 is to design RM policies under vehicle following safety constraints and different communication protocols and analyze their system-level performance. Another contributor to freeway congestion is human drivers’ poor response to disturbances, which can cause congestion even in the absence of bottlenecks or lane-changes. CAVs have the potential to address this issue and prevent congestion [7]. The impact of CAVs in this case is often studied by analyzing platoons of automated vehicles on an unbounded single-lane road without passing. Most research studies assume that the platoon leader follows a predefined speed trajectory, and the dynamical analysis of following vehicles provide insights into collision avoidance, error attenuation, ride comfort, and communication protocols integration [8, 9, 10, 11, 12, 13, 14]. In these studies, increasing platoon size does not impact speed or density, as the leader’s trajectory is assumed to be uninterrupted. While the impacts of platoon size on speed or density can be examined using steady state capacity analysis [15], it is more practical to study them when vehicles are not at steady state. This will be the subject of Chapter 3. 1.1.2 Air Transportation Another potential solution to congestion is introducing a new transportation mode, such as Urban Air Mobility (UAM), which envisions using urban airspace for on-demand mobility [16]. Although the idea 2 of using flying vehicles in urban areas for transportation purposes is not new, e.g., the use of helicopters from 1940s to 1970s [17],– it has gained renewed interest due to advances in electric Vertical Takeoff and Landing (eVTOL) aircraft technology and significant industry investments [18]. While recent research efforts has focused mostly on the aircraft themselves, there has been limited attention paid to controlling the air traffic [19, 20]. The goal of air traffic control is to efficiently use the limited UAM resources, such as the airspace, takeoff and landing areas, and the aircraft, to meet the demand. The UAM traffic control problem can be viewed as a natural extension of the Air Traffic Flow Management (ATFM) problem for commercial aircraft. However, unlike commercial air traffic, where demand is highly predictable even weeks in advance, UAM systems will be designed to provide on-demand services, which poses a significant tactical challenge. Although existing studies offer valuable insights into UAM operations, two critical aspects have been largely overlooked. First is the concept of rebalancing: UAM aircraft will need to be continuously redistributed across the network as demand fluctuates between destinations. Efficient rebalancing is essential for ensuring the sustainability and effectiveness of on-demand UAM systems. While rebalancing has been extensively studied in the context of ground transportation [21], these studies predominantly use macroscopic flow-level models, which are not suitable for capturing the safety and separation requirements associated with aircraft operations. The second aspect lacking attention in UAM literature is a thorough characterization of the system-level throughput. To address these gaps, we present a centralized microscopic-level traffic control framework in Chapter 4 and analyze its performance. 1.2 Contributions Motivated by the problems described in Section 1.1, this thesis is dedicated to the design and performance analysis of microscopic-level traffic controllers. The main contributions of this thesis are summarized as follows: 3 1. Designing microscopic-level RM controllers under vehicle following safety constraints. The proposed policies are traffic-responsive and only require real-time traffic measurements obtained from vehicle-to-infrastructure (V2I) communication. However, they do not require vehicle autonomy, which makes them a suitable choice in mixed-autonomy scenarios. Our microscopic-level approach allows us to understand the impact of different safety, connectivity, and automation protocols on freeway throughput (Chapter 2). 2. Designing a longitudinal vehicle controller under vehicle following safety constraints. The proposed controller can smoothly track a given reference position and/or speed, attenuates disturbances upstream of the traffic flow, and provides comfort. The proposed controller is extended to cases where there is vehicle-to-vehicle (V2V) or vehicle-to-infrastructure (V2I) communication, in order to improve freeway capacity and/or receive instructions from a coordinator to form a desired configuration (Chapter 3). 3. Developing a centralized microscopic-level air traffic controller for on-demand UAM networks under aircraft safety margins and energy requirements. The proposed controller incorporates the aspect of rebalancing into the UAM scheduling framework (Chapter 4). 4. Formalizing the notion of throughput as a performance measure for freeway and UAM systems. Throughput measures the highest demand that an RM policy or air traffic controller can handle without creating long waiting times (Chapters 2 and 4). 1.3 Outline The rest of this thesis is organized as follows: Chapter 2 presents the results of [22] on the design and analysis of microscopic-level RM policies under vehicle following safety constraints. Chapter 3 presents the results of [23] on studying the interplay of limited road space, safety, and speed in the context of CAVs. 4 Chapter 4 presents an extension of [24] on microscopic UAM traffic control. Chapter 5 concludes the thesis and discusses future directions. 5 Chapter 2 Ramp Metering to Maximize Freeway Throughput under Vehicle Safety Constraints 2.1 Introduction As introduced in Chapter 1, Ramp Metering (RM) is an effective tool to combat freeway congestion. There is an overwhelming body of literature on designing RM policies using macroscopic traffic flow models. We review them here only briefly; interested readers are referred to [4] for a comprehensive review. RM policies can be generally classified as fixed-time or traffic-responsive [4]. Fixed-time policies such as [25] are fine-tuned offline and operate based on historical traffic data. Due to the uncertainty in the traffic demand and the absence of real-time measurements, these policies would either lead to congestion or under-utilization of the capacity of the freeway [26]. Traffic-responsive policies, on the other hand, use real-time measurements. These policies can be further sub-classified as local or coordinated depending on whether the on-ramps make use of the measurements obtained from their vicinity (local) or other regions of the freeway (coordinated) [4, 27]. A well-known example of a local policy is ALINEA [28] which has been shown, both analytically and in practice, to yield a (locally) good performance. A caveat in employing local policies is that there is no guarantee that they can improve the overall performance of the freeway 6 while providing a fair access to the freeway from different on-ramps [26]. This motivates the study of coordinated policies such as the ones considered in [26, 29, 30]. The aforementioned studies adopt macroscopic traffic flow models for design purposes, which lack the resolution to distinguish between safety protocols under different connectivity and automation scenarios. An alternative is the microscopic-level approach which can be used to study the interplay between safety, connectivity and freeway performance. In the context of RM, this approach is limited to heuristics or simulations in the literature. The objective of this chapter is to address this gap. We consider a freeway with arbitrary number of on- and off-ramps. The freeway geometry can be modeled either as a ring road to represent a freeway in which every entry point is controlled, or a straight road, in which case the upstream entry point is not controlled. Previous studies on a ring road with no on/off-ramps have suggested that this setup has some theoretical advantages over the straight road network [23, 31, 32, 33, 34]. For example, the creation and dissipation of stop-and-go waves can be captured using a ring road [31, 32]. For the sake of completeness, we consider both geometries in this chapter. However, our results are not specific to the choice of the road geometry. Vehicles in the network are assumed to have the same length, same acceleration and braking capabilities, and are equipped with Vehicle-to-Vehicle (V2V), or Vehicle-to-Infrastructure (V2I) communication systems. Each vehicle follows the standard rules for safety and speed: it accelerates and maintains the free flow speed when it is sufficiently far away from the leading vehicle, or maintains a safe gap if it gets close. We do not specify the exact transient behavior for maintaining a safe gap, nor require vehicles to adopt the same behavior during the transient. However, at the steady state free flow speed, we assume that each vehicle keeps a safe constant time headway plus an additional constant gap from its leading vehicle [9]. The entry points to the freeway are the on-ramps and, in the case of the straight road geometry, the (uncontrolled) upstream entry point. Vehicle arrivals to each on-ramp is modeled by a Bernoulli process that is independent across different on-ramps. The destination of each vehicle is one of the off-ramps, and, in the case of the straight road geometry, also the 7 downstream exit point. The destination of each vehicle is sampled independently from a routing matrix. It should be emphasized that our main results do not depend on this specific demand model. Once a vehicle enters the freeway, it follows the aforementioned safety and speed rules until it reaches its destination, at which point it exits the network. We design traffic-responsive RM policies that maximize the freeway throughput subject to vehicle following safety constraints. For a given routing matrix, the throughput of a RM policy is characterized by the set of on-ramp arrival rates for which the queue sizes at all the on-ramps remain bounded in expectation. Roughly, the throughput of a RM policy is the highest traffic demand that it can handle without creating long queues at the on-ramps. The proposed RM policies work in synchronous cycles during which an on-ramp does not release more vehicles than the number of vehicles waiting in its queue, i.e., its queue size, at the beginning of the cycle. Furthermore, all of the proposed policies operate under vehicle following safety constraints, where the on-ramps release new vehicles only if there is enough gap on the mainline at the moment of release. We provide three policies under which each on-ramp: (i) pauses release for a time-interval at the end of a cycle, or (ii) modulates the time between successive releases during a cycle, or (iii) adopts a conservative safe gap criterion for release during a cycle, all based on the traffic state. None of the policies however require information about the on-ramp arrival rates or the routing matrix, i.e., they are reactive. The throughput of these policies is characterized by studying stochastic stability of the induced Markov chains, and is proven to be maximized when the merging speed of all the on-ramps equals the free flow speed. The rest of the chapter is organized as follows: in Section 2.2, we state the problem setup, vehiclelevel rules, demand model, and a summary of the RM policies studied in this chapter. The design and performance analysis of the proposed RM policies takes place in Section 2.3. We simulate the proposed policies in Section 2.4 and conclude the chapter in Section 2.5. 8 The following standard notations are used throughout the chapter. Let N, N0, and R respectively denote the set of positive integers, non-negative integers and real numbers. For m ∈ N, [m] denotes the set {1, . . . , m}. 2.2 Problem Formulation Consider a simple model of a freeway, where the mainline is abstracted either as a straight road or a ring road of length P, as illustrated in Figure 2.1. We use the ring road geometry to formulate the problem and state the main results; the discussion of the straight road geometry can be found in Section 2.3.7. The freeway has m on- and off- ramps; they are placed alternately, and are numbered in an increasing order along the direction of travel, such that, for all i ∈ [m], off-ramp i comes after on-ramp i ∗ . The section of the mainline between the i-th on- and off- ramps is referred to as link i. Vehicles arrive at the on-ramps from outside the freeway and join the on-ramp queues. We assume a point queue model for vehicles waiting at the on-ramps, with the queue on an on-ramp co-located with its ramp meter. The on-ramp vehicles are released into the mainline by the ramp meters installed at each on-ramp. Upon release, each vehicle follows the standard speed and safety rules until it reaches its destination off-ramp at which point it leaves the freeway without creating any obstruction for the upstream vehicles. Vehicles are equipped with V2V and V2I communication systems. They use these systems to communicate their state, e.g., speed, to nearby vehicles and the on-ramp control units. The exact information communicated by a vehicle will be specified in later sections. The objective is to design RM policies that perform well under vehicle following safety constraints. The performance of a RM policy is evaluated in terms of its throughput defined as follows: let λi be the arrival rate to on-ramp i, i ∈ [m], and λ := [λi ] be the vector of arrival rates. Let R = [Rij ] be the routing ∗By this numbering scheme, we do not mean to imply that the location of an on-ramp must be close to the next off-ramp as in practice, the opposite is usually the case; see Figure 2.1. In fact, our setup is quite flexible and can also deal with cases where the number of on- and off-ramps are not the same. For simplicity of notation, we have used the same number of on- and off-ramps in this chapter. 9 matrix, where Rij specifies the fraction of arrivals to on-ramp i that want to exit from off-ramp j. For a given routing matrix R, the under-saturation region of a RM policy is defined as the set of vector of arrival rates λ for which the queue sizes at all the on-ramps remain bounded in expectation † . The boundary of the under-saturation region is called the throughput. We are interested in finding RM policies that “maximize" the throughput for any given R. We will formalize this in Section 2.2.2. The remainder of this section is organized as follows: in Section 2.2.1, we discuss the vehicle-level rules. We specify the demand model and formalize the notion of throughput in Section 2.2.2. We summarize the RM policies considered in this chapter in Section 2.2.3. (a) (b) Figure 2.1: Example of a (a) straight road, (b) ring road. †Note that λ and R are macroscopic quantities. In order to specify vehicle arrivals and their destination at the microscopic level, a more detailed (probabilistic) demand model is required; see Section 2.2.2. The expected queue size is defined with respect to the probabilistic demand model. 10 2.2.1 Vehicle-Level Objectives We consider vehicles of length L that have the same acceleration and braking capabilities, and are equipped with V2V and V2I communication systems. We use the term ego vehicle to refer to a specific vehicle under consideration, and denote it by e. Consider a vehicle following scenario and let ve (resp. vl ) be the speed of the ego vehicle (resp. its leading vehicle), and Se be the safety distance between the two vehicles required to avoid collision. We assume that Se satisfies Se = hve + S0 + v 2 e − v 2 l 2|amin| , (2.1) which is calculated based on an emergency stopping scenario with details given in [9]. Here, h > 0 is a safe time headway constant, S0 > 0 is an additional constant gap, and amin < 0 is the minimum possible deceleration of the leading vehicle. For simplicity, we assume a third-order vehicle dynamics throughout the chapter. We consider two general modes of operation for each vehicle: the speed tracking mode and the safety mode. The main objective in the speed tracking mode is to adjust the speed to the free flow speed Vf when the ego vehicle is far from any leading vehicle; see Appendix A.1.1. The main objective in the safety mode is to avoid collision when the ego vehicle gets close to a leading vehicle. We define the acceleration lane of an on-ramp as the section of the network starting immediately downstream of the ramp meter and ending on the mainline, such that if the ego vehicle is in the speed tracking mode throughout the entire section, it achieves the speed Vf at the end of it. With this definition, the acceleration lane may or may not overlap with the mainline, depending on the distance from the ramp meter to the point where the merging occurs on the mainline, i.e., the merging point; see Figure 2.2. We assume that all the vehicles coming from an on-ramp merge at a fixed merging point. We define the merging speed of an on-ramp as the speed of the ego vehicle at the merging point, if the ego vehicle is in the 11 Figure 2.2: A merging scenario. For each vehicle, the dotted arrow indicates its leading vehicle while the solid arrow indicates its virtual leading vehicle. Note that vehicle l is both the leading and the virtual leading vehicle of vehicle e. speed tracking mode between release and reaching the merging point. Note that the merging speed of an on-ramp is at most Vf . Consider a merging scenario as shown in Figure 2.2. An ego vehicle entering the merging area is assigned a virtual leading vehicle, which is the vehicle that is predicted to be in front of the ego vehicle once the ego vehicle has crossed the merging point. The virtual leading vehicle is determined (and continuously updated) as follows: the ego vehicle predicts (and continuously updates) its speed and the time it will cross the merging point, and communicates this information to all other vehicles in the merging area using V2V communication. Other vehicles communicate the same information to the ego vehicle. We assume that the ego vehicle uses the following speed prediction rule: if in the speed tracking mode, the ego vehicle assumes that it remains in this mode in the future; if in the safety mode, it assumes a constant speed trajectory. At each time t before crossing the merging point, let tm|t be the time the ego vehicle predicts to cross the merging point based on the information available at time t. According to the prediction received from other vehicles at time t, the virtual leading vehicle is the vehicle that is to be in front of the ego vehicle at time tm|t. We assume that vehicles obey the following rules: (VC1) the ego vehicle maintains the constant speed Vf at time t if: 12 (a) it is at a safe gap with respect to its leading vehicle, i.e., ye(t) ≥ Se(t), where ye is the gap between the two vehicles. Note that if the leading vehicle is also at the constant speed Vf , then ye ≥ Se = hVf +S0. This gap is equivalent to a time headway of at least τ := h+ (S0+L)/Vf between the front bumpers of the two vehicles. This rule is believed to be widely adopted by human drivers as well as standard adaptive cruise control systems [9]. (b) it predicts to be at a safe gap with respect to its virtual leading vehicle in merging areas, i.e., yˆe(tm|t) ≥ Sˆ e(tm|t) := hvˆe(tm|t) +S0 + vˆ 2 e (tm|t)−vˆ 2 ˆl (tm|t) 2|amin| , where yˆe(tm|t) is the predicted gap between the two vehicles at the moment of merging based on the information available at time t ≤ tm. Similarly, vˆe and vˆˆl are the predicted speeds of the ego vehicle and its virtual leading vehicle, respectively. (VC2) the ego vehicle is initialized to be in the speed tracking mode upon release. It changes mode at time t if ye(t) < Se(t) or yˆe(tm|t) < Sˆ e(tm|t). Also, if two vehicles are released from the same on-ramp at least τ seconds apart, and neither change mode because of other vehicles, then the following vehicle remains in the speed tracking mode. (VC3) there exists Tfree such that for any initial condition, vehicles reach the free flow state after at most Tfree time if no other vehicle is released from the on-ramps. The free flow state refers to a state where: (i) if a vehicle is in the safety mode, then it moves at the constant speed Vf and maintains this speed in the future, and (ii) if a vehicle is in the speed tracking mode, then it remains in this mode in the future. Remark 1. Note that we intentionally do not specify the total number of submodes within the safety mode, the exact control logic within each submode, or the exact logic for switching back to the speed tracking mode. Such details will be introduced only if and when needed for performance analysis of RM policies 13 in the chapter. Also, note that the control logic in the safety mode during the transient is allowed to be different for different vehicles. 2.2.2 Probabilistic Demand Model and Throughput It will be convenient for performance analysis later on to adopt a discrete time setting. Let the duration of each time step be τ , representing the minimum safe time headway between the front bumpers of two consecutive vehicles that are moving at the constant speed Vf ; see (VC1) in Section 2.2.1. Let vehicles arrive to on-ramp i ∈ [m] according to an i.i.d. Bernoulli process with parameter λi ∈ [0, 1] independent of the other on-ramps. That is, in any given time step, the probability that a vehicle arrives at the i-th on-ramp is λi independent of everything else. Note that λi specifies the arrival rate to on-ramp i in terms of the number of vehicles per τ seconds. Let λ := [λi ] be the vector of arrival rates. The destination off-ramp for individual arriving vehicles is i.i.d. and is given by the routing matrix R = [Rij ], where 0 ≤ Rij ≤ 1 is the probability that an arrival to on-ramp i wants to exit from off-ramp j. Note that Rij specifies the (long-run) fraction of arrivals at on-ramp i that want to exit from off-ramp j. Naturally, for every on-ramp i we have P j Rij = 1. Finally, we let R˜ = [R˜ ij ] be the cumulative routing matrix, where R˜ ij is the fraction of arrivals at on-ramp i that need to cross link j in order to reach their destination. Note that λiR˜ ij is the rate of arrivals at on-ramp i that need to use link j in order to reach their destination, i.e., the load induced on link j by on-ramp i. Let ρj := P i λiR˜ ij be the total load induced on link j by all the on-ramps, and let ρ := maxj∈[m] ρj be the maximum load. Remark 2. The current demand model, i.e., Bernoulli arrivals and Bernoulli routing, is chosen to simplify the technical details in the proofs. We believe that our results are far more general and hold for more practical demand models used in the literature, e.g., see [35] for an example of arrival models. 14 Example 1. Let the routing matrix for a 3-ramp freeway, i.e., m = 3 as in Figure 2.1b, be given by R = R11 R12 R13 R21 R22 R23 R31 R32 R33 . Then, the corresponding cumulative routing matrix is R˜ = 1 1 − R11 1 − (R11 + R12) 1 − (R22 + R23) 1 1 − R22 1 − R33 1 − (R31 + R33) 1 . We now formalize the notion of “throughput" which is the key performance metric in this chapter. For i ∈ [m], let Qi(t) be the vector of destination off-ramps of the vehicles waiting at on-ramp i, arranged in the order of their arrival, at t. We use |Qi(t)| to denote the queue size at on-ramp i at time t. Let |Q(t)| = [|Qi(t)|] be the vector of queue sizes at all the on-ramps at time t. For a given routing matrix R, the under-saturation region of a RM policy π is defined as follows: Uπ,R = {λ : lim sup t→∞ E [|Qi(t)|] < ∞ ∀i ∈ [m] under policy π}. This is the set of λ’s for which the queue sizes at all the on-ramps remain bounded in expectation. The boundary of this set is called the throughput of the policy π. We are interested in finding a RM policy π such that for every R, Uπ′ ,R ⊆ Uπ,R for all policies π ′ , including those that have information about λ and R. In other words, for every R, if the freeway remains under-saturated using some policy π ′ , then it also remains under-saturated using the policy π. In that case, we say that policy π maximizes the freeway 15 Figure 2.3: An illustration of the under-saturation region of some policy π (dark grey area) and the DRR policy (dark + light grey areas) from Example 2 throughput. One of our main results is to introduce policies that maximize the saturation limit for all practical purposes but do not require any information about λ or R, i.e., they are reactive. Remark 3. A rigorous definition of the throughput should also include its dependence on the initial condition of the vehicles on the freeway and the initial queue sizes. We have removed this dependence for simplicity as the performance results given for our proposed policies do not depend on the initial condition. Example 2. Consider a 3-ramp freeway with R = 0.2 0.7 0.1 0 0.8 0.2 0.5 0 0.5 , λ3 = 0.5 [veh/time step], and suppose that the merging speed at all the on-ramps is Vf . Let us consider one of the policies introduced in Section 2.3, called the Dynamic Release Rate (DRR) policy. According to Theorem 2, the under-saturation region of this policy is given by UDRR,R = {(λ1, λ2) : λ1 < 0.75, 0.8λ1 + λ2 < 1}. 16 This set is illustrated in Figure 2.3. Moreover, according to Theorem 4, for any policy π we have Uπ,R ⊆ UDRR,R, except maybe at the boundary of UDRR,R; see Figure 2.3. Also, note that the boundary of UDRR,R has zero volume, which implies that the vector (λ1, λ2) lies either inside or outside UDRR,R in practice. More generally, Theorem 2 and Theorem 4 state that the previous conclusions hold for any other choice of the routing matrix. Therefore, the DRR policy maximizes the throughput for all practical purposes. 2.2.3 Ramp Metering To conveniently track vehicle locations in discrete time, we introduce the notion of slot. A slot is associated with a particular point on the mainline or acceleration lanes at a particular time. We first define the mainline slots. Let nc be the maximum number of distinct points that can be placed on the mainline, such that the space gap between adjacent points is hVf + S0 + L. This gap is governed by the safety distance Se as explained in Section 2.2.1. In particular, nc is the maximum number of vehicles that can safely travel at the constant free flow speed Vf on the mainline. Consider a configuration of these nc points at t = 0. Each point represents a slot on the mainline that moves at the free flow speed; without loss of generality, we let the length of the mainline P be such that each slot replaces the next slot at the end of each time step τ . We next define the acceleration lane slots. Suppose that the ego vehicle is released from the i-th onramp at t = 0 such that it remains in the speed tracking mode in the future. Consider the ego vehicle’s location at the end of each time step τ until it exits the acceleration lane. Each of these location points represents a slot for the i-th acceleration lane at t = 0. For example, if the ego vehicle exits the acceleration after 2.5τ seconds upon release, there are three slots corresponding to its location at times τ , 2τ , and 3τ . Let ni be the number of i-th acceleration lane slots, and na = P i ni . Note that by definition, the last acceleration lane slot of every on-ramp is on the mainline. Without loss of generality, we consider a 17 configuration of slots at t = 0, such that all the last acceleration lane slots coincide with a mainline slot. The details to justify the no loss in generality are given as follows: for a given configuration of mainline slots at t = 0, there exists ti ∈ [0, τ ) such that the last acceleration lane slot of on-ramp i coincides with a mainline slot at time ti . Thereafter, the last acceleration lane slot coincides with a mainline slot at the end of every time step τ , i.e., at times kτ +ti for all k ∈ N0. The times kτ +ti , k ∈ N0, are the release times of on-ramp i in the proposed RM policies. Therefore, the assumption that all the last acceleration lane slots initially coincide with a mainline slot (which corresponds to ti = 0 for all i ∈ [m]) only means a shifted sequence of release times, which justifies the no loss in generality. Consider an initial condition of the vehicles on the freeway, where the vehicles are in the free flow state such that the location of each vehicle coincides with a slot for all times in the future. For this initial condition and under the proposed RM policies, the following sequence of events occurs during each time step: (i) the mainline slots rotate one position in the clockwise direction and replace the next slot; similarly, the acceleration lane slots of each on-ramp replace the next slot with the last slot replacing the first slot; the numbering of the slots is reset with the new first mainline slot after on-ramp 1 numbered 1, and the rest of the mainline slots numbered in an increasing order; the acceleration lane slots are numbered similarly; (ii) vehicles that reach their destination off-ramp exit the freeway without obstructing the upstream vehicles; (iii) if permitted by the RM policy, a new vehicle is released; for the given initial condition and under the proposed RM policies, the location of the newly released vehicle coincides with a slot for all times in the future. The information available to the proposed policies will be a combination of |Q| and the state (or part of the state) of all the vehicles X := (xe)e∈[n] , where n is the number of vehicles on the mainline and acceleration lanes, and xe = (pe, ve, ae, Ie) is the state of the ego vehicle, where pe is its location, ve is the speed, ae is the acceleration, and Ie is a binary variable which is equal to one if the ego vehicle is in the safety mode, and zero otherwise. These information are communicated to the on-ramps using V2I 18 communication systems. Table 2.1 provides a summary of the communication cost of the RM policies considered in this chapter. The notion of communication cost is explained in Appendix A.1.2. In Table 2.1, nm is the total number of slots in all the merging areas, Tper is a design update period, and Qi , i = 1, 2, is the contribution of the queue size to the communication cost. Table 2.1: Summary of the RM policies studied in this chapter. Ramp Metering Policy Worst-case Communication Cost Renewal (nc + na)m + Q1 Dynamic Release Rate (DRR) m(nc + na)/Tper + nm + Q2 Distributed Dynamic Release Rate (DisDRR) (nc + na)/Tper + nm + Q2 Dynamic Space Gap (DSG) (nc + na)m + Q2 Greedy nm 2.3 Ramp Metering Policies and Performance Analysis In this section, we provide traffic-responsive RM policies subject to vehicle following safety constraints and analyze their performance. This is done in Sections 2.3.1-2.3.6. In Section 2.3.7, we discuss the extension of the results to the straight road geometry. For easier navigation, we briefly review the proposed policies here. The policies in Sections 2.3.1, 2.3.2, and 2.3.4 are coordinated, the one in Section 2.3.3 is a distributed version of the coordinated policy in Section 2.3.2, and the one in Section 2.3.5 is a local policy. In all of our policies, the on-ramps work in synchronous cycles during which an on-ramp does not release more vehicles than its queue size at the beginning of the cycle. The synchronization of cycles is done in real time in Section 2.3.1, whereas it can be done once offline in Sections 2.3.2-2.3.5. Furthermore, all of our policies operate under vehicle following safety constraints, where the on-ramps release new vehicles only if there is enough gap on the mainline at the moment of release (cf. (VC1)). The policies differ in using the vehicle information to either pause release for a time interval at the end of a cycle (Section 2.3.1), to modulate the time between successive releases during a cycle (Sections 2.3.2-2.3.3), or to adopt a conservative dynamic safe gap criterion for 19 release during a cycle (Section 2.3.4). The actions of all the three policies in Sections 2.3.2-2.3.4 at the free flow state is equivalent to that of the local policy in Section 2.3.5. All the policies are reactive, meaning that they do not require any information about the vector of arrival rates or the routing matrix. An inner estimate to the under-saturation region is provided for each policy, and is then compared to an outer estimate in Section 2.3.6. 2.3.1 Renewal Policy The first policy is inspired by the queuing theory literature in the context of communication networks, e.g., see [36, 37]. Once an on-ramp releases all the vehicles waiting at the beginning of a cycle, it pauses release until all other on-ramps have done so, and these vehicles exit the freeway, i.e., until the mainline and acceleration lanes are empty – hence we refer to it as the Renewal policy. Definition 1. (Renewal ramp metering policy) No vehicle is released until all the initial vehicles exit the freeway, i.e., until the mainline and acceleration lanes are empty, say at time t1. Thereafter, the policy works in cycles of variable length. At the beginning of the k-th cycle at time tk, each on-ramp allocates itself a “quota" equal to the queue size at that on-ramp at tk. At time t during the cycle, an on-ramp releases the ego vehicle if: (M1) t = kτ for some k ∈ N0. (M2) the on-ramp has not reached its quota. (M3) ye(t) ≥ Se(t), i.e., there is a safe gap in front of the ego vehicle (cf. (VC2)). (M4) it predicts that the ego vehicle will be at a safe gap with respect to its virtual leading and following vehicles between merging and exiting the acceleration lane. Once an on-ramp reaches its quota, it does not release a vehicle during the rest of the cycle. The next cycle begins when the mainline and acceleration lanes are empty. 20 Remark 4. A simpler form of this policy, called the Quota policy, is analyzed in [36]. Direct adaptation of the Quota policy to the current transportation setting requires additional analysis because of the vehicle dynamics. We need an additional notation for future results. Consider a situation where on-ramp i releases the ego vehicle under (M1)-(M4), its virtual leading and following vehicles are at the constant speed Vf , and their location coincides with a mainline slot. We let τi be the minimum time headway between the front bumpers of the virtual leading and following vehicles such that they maintain a safe gap with respect to the ego vehicle after merging. Note that τi is an exact multiple of τ and τi ≥ 2τ , where the equality holds if and only if the merging speed of on-ramp i is Vf . Theorem 1. For any initial condition, the Renewal policy keeps the freeway under-saturated if ( τi τ − 1)ρi − ( τi τ − 2)λi < 1 for all i ∈ [m]. Proof. See Appendix A.3.1. V2I communication requirements: the Renewal policy uses information about |Q| and X. Its worst-case communication cost is calculated as follows: at each time step during a cycle, any vehicle that is on the mainline or an acceleration lane must communicate its state to all on-ramps. After a finite time, the number of these vehicles is no more than nc + na. Furthermore, at the beginning of every cycle, all the vehicles in an on-ramp queue must communicate their presence in the queue to that on-ramp. The contribution of the queue size to the communication cost is Q1 := lim sup K→∞ 1 K X tk≤Kτ X i∈[m] |Qi(tk)|, where tk is the beginning of the k-th cycle in the Renewal policy. Hence, the communication cost C is upper-bounded by (nc + na)m + Q1 . 21 Figure 2.4: An inner estimate of the under-saturation region of the Renewal policy (grey area) from Example 3. Remark 5. Under the constant time headway safety rule in Section 2.2.1, the flow capacity of the mainline is 1 vehicle per τ seconds. Theorem 1 provides an inner estimate of the under-saturation region in terms of the induced loads ρi , arrival rates λi , and the mainline capacity. In particular, if for every i ∈ [m], (τi/τ − 1)ρi − (τi/τ − 2)λi is less than the capacity, then the Renewal policy keeps the freeway undersaturated. Remark 6. The interplay between safety during merging and throughput is captured using the parameter τi , i ∈ [m]. In particular, as the merging speed of an on-ramp decreases, the required safety distance Se in (2.1) increases. This puts a limit on the rate at which the on-ramp can release new vehicles under the vehicle following safety constraint, which in turn decreases the throughput. Remark 7. Under the Renewal policy, vehicles that arrive during a cycle cannot enter the freeway until the next cycle begins. This increases the chance that these vehicles enter the freeway in platoons, rather than individually, once the next cycle begins. The platoon formation increases the release rate of other on-ramps as compared to the individual formation, when the merging speeds are less than Vf . As a consequence, the inner estimate of the under-saturation region of the Renewal policy is larger than the ones of other policies described in the following sections. 22 Example 3. Consider a 3-ramp freeway with τ1 = τ3 = 2τ , τ2 = 3τ , i.e., the merging speed of on-ramps 1 and 3 are Vf and is lower for on-ramp 2. Suppose that R and λ3 are the same as Example 2. Then, the inner estimate of the under-saturation region given by Theorem 1 is illustrated in Figure 2.4. 2.3.2 Dynamic Release Rate Policy This policy imposes dynamic minimum time gap criterion, in addition to (M1), between release of successive vehicles from the same on-ramp. Changing the time gap between release of successive vehicles by an on-ramp is akin to changing its release rate, and hence the name of the policy. Definition 2. (Dynamic Release Rate (DRR) ramp metering policy) The policy works in cycles of fixed length Tcycτ , where Tcyc ∈ N. At the beginning of the k-th cycle at tk = (k − 1)Tcycτ , each on-ramp allocates itself a “quota" equal to the queue size at that on-ramp at tk. At time t ∈ [tk, tk+1] during the k-th cycle, on-ramp i releases the ego vehicle if (M1)-(M4), and the following condition is satisfied: (M5) at least g(t) time has passed since the release of the last vehicle from on-ramp i, where g(t) is a piecewise constant minimum time gap, updated periodically at t = Tper, 2Tper, . . . as described in Algorithm 1. Once an on-ramp reaches its quota, it does not release a vehicle during the rest of the cycle. In Algorithm 1, Xf (t) := Xf1 (t) + Xf2 (t), where Xf1 and Xf2 are defined as follows: let Xf1 (t) := P e∈[n] (|ve(t) − Vf | + |ae(t)|)Ie(t), where n is the number of vehicles on the mainline and acceleration lanes. Furthermore, for t ≥ Tper, let Xf2 (t) := P(δe(t) + ˆδe(t)), where the sum is over all the vehicles that either: (i) violated the safety distance Se at some time in [t − Tper, t], or (ii) predicted at some time in 23 Figure 2.5: An illustration of the update rule for the minimum time gap g in Algorithm 1. [t − Tper, t] that they would violate the safety distance once they reach a merging point. The terms δe and ˆδe are, respectively, the maximum error and predicted error in the relative spacing, and are given by δe(t) = max t ′∈[t−Tper,t] |ye(t ′ ) − Se(t ′ )|1{ye(t ′)<Se(t ′)} , ˆδe(t) = max t ′∈[t−Tper,t] |yˆe(tm|t ′ ) − Sˆ e(tm|t ′ )|1{yˆe(tm|t ′) 0, γ1 > 0, γ2 > 0, θ ◦ > 0, β > 1 initial condition: g(0) = 0, θ(0) = θ ◦ , Xf (0) = Xf1 (0) for t = Tper, 2Tper, · · · do if Xf (t) ≤ max{Xf (t − Tper) − γ1, 0} then θ(t) ← θ(t − Tper) g(t) ← max{g(t − Tper) − γ2, 0} else θ(t) ← βθ(t − Tper) g(t) ← g(t − Tper) + θ(t) end if end for Remark 8. Recall the safety distance Se in (2.1). Since the speed of the vehicles are bounded, Se is also bounded. Hence, δe (and similarly ˆδe) in Xf2 is bounded. We use this in the following theorem. Theorem 2. For any initial condition, Tcyc ∈ N, and design constants in Algorithm 1, the DRR policy keeps the freeway under-saturated if ( τi τ − 1)ρi < 1 for all i ∈ [m]. Proof. See Appendix A.3.2. V2I communication requirements: This policy uses information about |Q| (for Tcyc > 1) and X. Its worst-case communication cost is calculated as follows: after a finite time, X is communicated to all on-ramps only at the end of each update period Tper. After such finite time, the number of vehicles that constitute X is no more than nc + na. Furthermore, at each time step during a cycle, the vehicles in the merging area of an on-ramp communicate their state to that on-ramp for safe gap evaluation (cf. (M3)-(M4)). The number of vehicles in all the merging areas is at most nm. Finally, if Tcyc > 1, then all 25 Figure 2.6: An inner estimate of the under-saturation region (dark grey area) of the DRR policy from Example 4. The light grey area indicates the additional under-saturation region if we use the Renewal policy. the vehicles in an on-ramp queue must communicate their presence in the queue to that on-ramp at the beginning of every cycle. The contribution of the queue size to the communication cost is Q2 := lim sup K→∞ 1 K X k≤K/Tcyc+1 X i∈[m] |Qi((k − 1)Tcycτ )| 1{Tcyc>1} . Hence, C is upper bounded by m(nc + na)/Tper + nm + Q2 . Remark 9. Similar to Theorem 1, Theorem 2 provides an inner estimate of the under-saturation region of the DRR policy in terms of the induced loads ρi and the mainline capacity. The region specified by this estimate is the same for all Tcyc ∈ N and is contained in the one given for the Renewal policy in Theorem 1. However, this does not necessarily mean that the throughput of the DRR policy is the same for different cycle lengths, or the Renewal policy gives a better throughput as the inner estimates may not be exact. A simulation comparison of the throughput of the DRR policy for different cycle lengths is given in Section 2.4.2. Example 4. Let the freeway parameters be as in Example 3. Then, the inner estimate of the under-saturation region given by Theorem 2 is illustrated in Figure 2.6, and is compared with the one in Figure 2.4. Also, 26 comparing with Example 2 where the merging speed of on-ramp 2 was also Vf , one can see that the region specified in this example is smaller. 2.3.3 Distributed Dynamic Release Rate Policy This policy imposes dynamic minimum time gap criterion, just like its coordinated counterpart. However, each on-ramp only receives information from the vehicles in its vicinity and the downstream on-ramps. Definition 3. (Distributed Dynamic Release Rate (DisDRR) ramp metering policy) The policy works in cycles of fixed length Tcycτ , where Tcyc ∈ N. At the beginning of the k-th cycle at tk = (k − 1)Tcycτ , each on-ramp allocates itself a “quota" equal to the queue size at that on-ramp at tk. At time t ∈ [tk, tk+1] during the k-th cycle, on-ramp i releases the ego vehicle only if (M1)-(M4), and the following condition is satisfied: (M5) at least gi(t) time has passed since the release of the last vehicle from on-ramp i, where gi(t) is a piecewise constant minimum time gap, updated periodically at t = Tper, 2Tper, . . . according to Algorithm 2. Once an on-ramp reaches its quota, it does not release a vehicle during the rest of the cycle. For i ∈ [m], let Xi f be the part of Xf associated with all the vehicles located between the i-th and (i + 1)-th on-ramps. Thus, Xf = P i∈[m] Xi f . We assume that Xi f is available to on-ramp i. Furthermore, if gi+1(t) > Tmax for some design constant Tmax, then all the on-ramps j downstream of on-ramp i communicate X j f to on-ramp i. In that case, P j>i X j f is available to on-ramp i, where the notation “j > i" means that on-ramp j is downstream of on-ramp i. Note that for the ring road geometry, all the on-ramps downstream of on-ramp i is equivalent to all the on-ramps. Hence, P j>i X j f = Xf for the ring road geometry. Informally, when gi+1 ≤ Tmax, gi depends only on the traffic condition in the vicinity of on-ramp i, whereas when gi+1 > Tmax, it also depends on the downstream traffic condition. Naturally, 27 higher values of Tmax make the policy more “decentralized", but it may take longer to reach the free flow state. Algorithm 2 Update rule for the minimum time gap between release of vehicles under the DisDRR policy Input: design constants: Tper > 0, Tmax > 0, γ1 > 0, γ2 > 0, θ ◦ i > 0, β > 1 initial condition: (gi(0), gi(Tper)) = (0, 0), (θi(0), θi(Tper)) = (θ ◦ i , θ◦ i ), Xi f (0) = Xi f1 (0), i ∈ [m] for t = 2Tper, 3Tper, · · · do the following for each on-ramp i ∈ [m] if Xi f (t) ≤ max{Xi f (t − Tper) − γ1, 0} then if gi+1(t − Tper) ≤ Tmax or P j>i X j f (t) ≤ max{ P j>i X j f (t − Tper) − γ1, 0} then θi(t) ← θi(t − Tper) gi(t) ← max{gi(t − Tper) − γ2, 0} else θi(t) ← βθi(t − Tper) gi(t) ← gi(t − Tper) + θi(t) end if else if Xi f (t − Tper) ≤ max{Xi f (t − 2Tper) − γ1, 0} then θi(t) ← βθi(t − Tper) else θi(t) ← θi(t − Tper) end if gi(t) ← gi(t − Tper) + θi(t) end if end for 28 Proposition 1. For any initial condition, Tcyc ∈ N, and design constants in Algorithm 2, the DisDRR policy keeps the network under-saturated if ( τi τ − 1)ρi < 1 for all i ∈ [m]. Proof. See Appendix A.3.3. V2I communication requirements: This policy uses information about |Q| (for Tcyc > 1) and X. Its worst-case communication cost is calculated similar to the DRR policy, except that at the end of each update period Tper, X is not communicated to all on-ramps. Instead, for all i ∈ [m], only the part X associated with the vehicles between the i-th and i + 1-th on-ramps is communicated to on-ramp i. The contribution of the queue size to the communication cost is the same as the DRR policy because of the same cycle mechanism. Thus, the worst-case communication cost is reduced to (nc + na)/Tper + nm + Q2 . 2.3.4 Dynamic Space Gap Policy In this policy, on-ramps require an additional space gap, in addition to the safe gaps in (M3)-(M4), before releasing a vehicle. This additional space gap is updated periodically based on the state of all vehicles. Recall that the DRR policy enforces an additional time gap between release of successive vehicles, which is updated based on the current state of the vehicles as well as their state in the past. The dynamic space gap policy only requires the current state of the vehicles. However, it requires additional assumptions on the vehicle-level rules: consider n vehicles over a time interval during which at least one vehicle is in the speed tracking mode, and no vehicle leaves the freeway. Then, during this time interval: (VC4) each vehicle changes mode at most once. (VC5) if no vehicle changes mode, then Xg := Xg1 + Xg2 converges to zero globally exponentially, where Xg1 and Xg2 are defined as follows: let Xg1 (t) := P e∈[n] (|ve(t) − Vf | + |ae(t)|)Je(t), where Je is 29 a binary variable which is equal to zero if the ego vehicle has been in the speed tracking mode at all times since being released from an on-ramp, and one otherwise. Moreover, Xg2 := X e∈[n] |ye(t) − Se(t)|1{ye(t)<Se(t)} + max s∈[t ′ ,tf ] |yˆe(s|t) − Sˆ e(s|t)|1{yˆe(s|t) 1) and X. Except for an initial finite time, X is communicated to every on-ramp at each time step. Moreover, the contribution of the queue size to the communication cost is the same as the DRR policy because of the same cycle mechanism. Thus, the communication cost is upper bounded by (nc + na)m + Q2 . Remark 10. The choice of the additional gaps fi , i = 1, 2, 3, in the DSG policy is not limited to the expressions found in the proof of Theorem 3. An alternative expression is simulated in Section 2.4.4 2.3.5 Local and Greedy Policies The performance analysis of the three policies in Sections 2.3.2-2.3.4 can be divided into two phases. The first phase concerns the transient from the initial condition to the free flow state where each vehicle occupies some slot, and the second phase is from this free flow state onward. Since the performance index of throughput is an asymptotic notion, it is natural to examine the policies specifically in the second phase. Indeed, in the second phase, the actions of all the three policies can be shown to be equivalent to the following policy: Definition 5. (Fixed-Cycle Quota (FCQ) ramp metering policy) The policy works in cycles of fixed length Tcycτ , where Tcyc ∈ N. At the beginning of the k-th cycle at tk = (k − 1)Tcycτ , each on-ramp allocates itself a “quota" equal to the queue size at that on-ramp. During a cycle, the i-th on-ramp releases the ego vehicle if (M1)-(M4) are satisfied. Once an on-ramp reaches its quota, it does not release a vehicle during the rest of the cycle. V2I communication requirements: This policy uses information about |Q| (for Tcyc > 1) and part of X. At each time step during a cycle, the vehicles in the merging area of every on-ramp communicate their state to that on-ramp. Moreover, the contribution of the queue size to the communication cost is the same as the DRR policy because of the same cycle mechanism. Hence, C is upper bounded by nm + Q2 . 31 Note that the FCQ policy is local, except for synchronizing the beginning of the cycles which can be done once offline. In the special case of Tcyc = 1, the FCQ becomes the simple greedy (and local) policy under which the on-ramps do not need to synchronize cycles with other on-ramps or keep track of their quota. Therefore, the communication cost of the greedy policy is upper bounded by nm. One can see from the proof of the DRR policy that, starting from the aforementioned second phase, the freeway is undersaturated using the FCQ policy if (τi/τ − 1)ρi < 1 for all i ∈ [m]. It is natural to wonder if such a result holds if we start from any initial condition, or under any other greedy policy (not just the slot-based). In Section 2.4.3, we provide intuition as to when the former statement might be true. 2.3.6 An Outer Estimate We now provide an outer estimate to the under-saturation region of any policy, against which we benchmark the inner estimates derived in the previous sections for the proposed policies. This outer estimate can be thought of as the network analogue of the flow capacity of a single on-ramp. To formalize this, let Dπ,p(kτ ) be the cumulative number of vehicles that has crossed point p on the mainline up to time kτ , k ∈ N0, under the RM policy π. Then, the crossing rate at point p is defined as Dπ,p(kτ )/k and the “long-run" crossing rate is lim supk→∞ Dπ,p(kτ )/k. Note that Dπ,p((k + 1)τ ) − Dπ,p(kτ ) represents the traffic flow at point p in terms of the number of vehicles per τ seconds. Macroscopic traffic models suggest that the flow is no more than the mainline capacity, which is 1 vehicle per τ seconds. Hence, Dπ,p((k + 1)τ ) − Dπ,p(kτ ) ≤ 1 for all k ∈ N0 and any RM policy π. This implies that lim sup k→∞ Dπ,p(kτ )/k ≤ 1, (2.2) for any RM policy π. We take (2.2) as an assumption to the next Theorem. 32 Theorem 4. If a policy π keeps the freeway under-saturated and satisfies (2.2) for at least one point on each link, then the demand must satisfy ρ ≤ 1. Proof. See Appendix A.3.5. Remark 11. In all of the policies studied in previous sections, vehicles move at the constant speed Vf at some point pi on the i-th link, i ∈ [m], after a finite time. This and the constant time headway rule imply that the number of vehicles that can cross pi at each time step is no more than one. Therefore, lim supk→∞ Dπ,pi (kτ )/k ≤ 1 for all i ∈ [m], and the long-run crossing rate condition in (2.2) holds. Remark 12. If the merging speed at all the on-ramps is Vf , then the minimum time headways τi , i ∈ [m], are all equal to 2τ . In this case, one can check that the inner estimate of the under-saturation region given in Theorem 1, Theorem 2, and Proposition 1 become ρ < 1. Comparing with Theorem 4, this implies that the Renewal, DRR, and DisDRR policies give the maximum possible throughput for all practical purposes. 2.3.7 Discussion of the Straight Road Geometry In this section, we discuss how the previous results can be extended to the straight road geometry with m − 1 on- and off-ramps. The on- and off-ramps are placed alternatively, and they are numbered in an increasing order along the direction of travel. The on-ramps are numbered from 2 to m, and the off-ramps are numbered from 1 to m − 1. The upstream entry point and the downstream exit point are numbered 1 and m, respectively; see Figure 2.1a. The vehicle-level rules discussed in Section 2.2.1 remain unchanged, except that, at the free flow state, we require the vehicles from the upstream entry point to enter at the constant speed Vf . One can also relax (VC3) as follows: (VC3) there exists Tfree such that for any initial condition, if no vehicle is released from on-ramps 2, . . . , j for some j ∈ {2, . . . , m}, and all the vehicles downstream of link j are at the free flow state, then all the vehicles reach the free flow state after at most Tfree time. 33 Note that condition (VC3) imposes macroscopic-level constraints on the inflow from the upstream entry point before reaching the free flow state. This is justified if, e.g., the on-ramps upstream of the entry point are also controlled. We also need to specify vehicle arrivals and their destination at the microscopic level after reaching the free flow state. To this end, we may adopt an i.i.d Bernoulli process with parameter λ1 and an i.i.d Bernoulli routing process that are independent of the on-ramp processes discussed in Section 2.2.2. The routing matrix, the cumulative routing matrix, and the induced loads are defined accordingly. However, this choice of demand model may be generalized to include practical cases as noted in Remark 2. Since the vehicle arrivals from the upstream entry point is not controlled, e.g., by a ramp meter, the mainline may never become empty even if all the on-ramps pause release. Hence, we need to modify the description of the Renewal policy. In particular, the first cycle of the Renewal policy begins when the vehicles reach the free flow state. Thereafter, the k-th cycle, k > 1, begins when all quotas of the (k−1)-th cycle reach their destination. The description and performance of the DRR policy remains the same. In the description of the DisDRR policy, X2 f is modified to include all vehicles between the upstream entry point and on-ramp 2. The performance of this policy also remains the same. 2.4 Simulations The following setup is common to all the simulations in this section. We consider a ring road with P = 1860 [m] and m = 3 on- and off-ramps. Let h = 1.5 [s], S0 = 4 [m], L = 4.5 [m], and Vf = 15 [m/s]. For these parameters, we obtain nc = 60. The on-ramps are located symmetrically at 0, P/3, and 2P/3; the off-ramps are also located symmetrically 155 [m] upstream of each on-ramp. The initial queue size of all 34 the on-ramps is set to zero. Vehicles arrive at the on-ramps according to i.i.d Bernoulli processes with the same rate λ; their destinations are determined by R = 0.2 0.7 0.1 0 0.8 0.2 0.5 0 0.5 . Note that, on average, most of the traffic wants to exit from off-ramp 2. Thus, one should expect that on-ramp 3 finds more safe gaps on the mainline than the other two on-ramps. As a result, on-ramp 3’s queue size is expected to be less than the other two on-ramps. 2.4.1 Greedy Policy for Low Merging Speed Figure 2.7: On-ramp queue size profiles when all the on-ramps are long, under the FCQ policy. The mainline and acceleration lanes are assumed to be initially empty in this section and Section 2.4.2. The on-ramps use the greedy policy, i.e., the FCQ policy with Tcyc = 1, from Section 2.3.5. When the merging speed of all the on-ramps is Vf , then for the given R, the throughput is given by λ = 5/9 [veh/time step]. Note that since the arrival rates are the same, the throughput is a single point. The 35 (a) (b) Figure 2.8: Effect of cycle length Tcyc on the long-run expected queue size (a) for different ρ when both on-ramps are long, (b) for a fixed ρ when on-ramp 2 is short. Both plots are under the FCQ policy. In plot (b), the logarithm of the expected queue size is set to 20 whenever the freeway becomes saturated. queue size profiles for λ = 0.5 [veh/time step], which corresponds to ρ = 0.9 (heavy demand), is shown in Figure 2.7. As expected, |Q3(·)| is generally less than the other two on-ramps. We next consider the case when the merging speed of on-ramps 1 and 3 are Vf , i.e., τ1 = τ3 = 2τ , and is approximately Vf /3 = 5 [m/s] for on-ramp 2 which corresponds to τ2 = 3τ . In the heavy demand regime, i.e., ρ = 0.9, |Q2(·)| increases steadily which suggests that the freeway becomes saturated even though ρ < 1. The throughput in this case is estimated from simulations to be λ = 0.44 [veh/time step]. The estimates of the throughput of the Renewal and FCQ policies, given by Theorems 1 and 2 are λ = 0.5 [veh/time step] and λ = 0.33 [veh/time step], respectively. Combining this with the simulation results, we see that the Renewal policy performs better than the greedy policy in terms of its throughput. 2.4.2 Effect of Cycle Length on Queue Size The long-run expected queue size, i.e., Q := lim supk→∞ E [ P i |Qi(kτ )|], are compared under the FCQ policy for different Tcyc. The expected queue size is computed using the batch means approach, with 36 warm-up period of 105 , i.e., the first 105 observations are not used, and batch size of 105 . In each case, the simulations are run until the margin of error of the 95% confidence intervals are 1%. Figure 2.8a shows Q vs. ρ for different Tcyc when the merging speed of all the on-ramps is Vf . The plot suggests that for all ρ, Q increases monotonically with Tcyc. However, this does not hold true when the merging speeds are low. For example, Figure 2.8b shows Q in log scale vs. Tcyc for λ = 0.455, which corresponds to ρ ≈ 0.82, when the merging speed of on-ramps 1 and 3 is Vf , and is Vf /3 for on-ramp 2. The plot shows that the freeway may become saturated depending on the choice of Tcyc, and that the dependence of Q on Tcyc is not monotonic. 2.4.3 Relaxing the V2X Requirements In this scenario, we evaluate the performance of the greedy policy for ρ = 0.9 (heavy demand) when the merging speed of all the on-ramps is Vf , and the mainline is initially not empty. The safety mode only consists of the following submode: in a non-merging scenario, the ego vehicle follows the leading vehicle by keeping a safe time headway as in, e.g., [23]. In a merging scenario, it applies the aforementioned control law with respect to both its leading and virtual leading vehicles. The most restrictive acceleration is used by the ego vehicle. For this control law, platoons of vehicles are stable and string stable [23]. The initial number of vehicles on the mainline, their location, and their speed are chosen at random such that the safety distance is not violated. The initial acceleration of all the vehicles is set to zero. We conducted 10 rounds of simulation with different random seed for each round. It is observed in all scenarios that vehicles reach the free flow state after a finite time using the greedy policy. We conjecture that this observation holds for any initial condition if platoons of vehicles are stable and string stable. The intuition behind this statement is as follows: using the greedy policy, vehicles are released only if there is enough space gap on the mainline, and successive releases are at least τ seconds apart. In between releases, vehicles on the mainline try to reach the free flow state because of platoon stability. Once a 37 vehicle is released, it may create disturbance, e.g., in the acceleration of the upstream vehicles. However, string stability implies that such disturbance is attenuated. Therefore, it is natural to expect that vehicles will reach the free flow state after a certain finite time. Thereafter, by Theorem 2, the greedy policy keeps the freeway under-saturated if (τi/τ − 1)ρi < 1 for all i ∈ [m]. 2.4.4 Comparing the Total Travel Time We evaluate the total travel time under the Renewal, DRR, DisDRR, DSG, and greedy policies and compare them with the ALINEA ramp metering policy under vehicle safety constraints. The ALINEA policy was introduced in [28] and, for a given on-ramp, can be expressed as follows: r(k) = r(k − 1) + Kr(ˆo − o(k)). Here, r(·)is the on-ramp outflow, Kr is a positive design constant, o(·)is the occupancy of the mainline downstream of the on-ramp, and oˆ is the desired occupancy. For the ALINEA policy, we use time steps of size 60 [s], Kr = 70 [veh/h] [28], and oˆ = 13 % corresponding to the capacity flow. We also add the following safety filter on top of ALINEA: the ego vehicle is released only if it predicts to be at a safe gap with respect to its virtual leading and following vehicles at the moment of merging. We use the name Safe-ALINEA to refer to his policy. For the DRR and DisDRR policies, we use the following parameters: Tcyc = 12, Tper = 2τ , γ1 = 50, γ2 = 10, θ ◦ i = 0.1, β = 1.01, and Tmax = 100 [s]. Informally, γ1 is set to a high value so that the release times gi(·) increase if the traffic condition, i.e., the value of Xf , does not improve significantly in between the update periods; γ2 and θ ◦ i are set to low values so that the jump sizes in gi(·) are small; Tmax is set to a high value so that on-ramps update their release time in a completely decentralized way. For the DSG policy, we set Tcyc = 12 and use the additional gap κ(Xf1 + P e∈[n] |ye − (hve + S0)|Ie) on top of (M3)-(M4), where κ = 0.01 is chosen so that the additional gap is relatively small (≈ 8 [m] for the initial condition described next). 38 Figure 2.9: The total travel time (TTTn), where n is the number of completed trips, under the fixed demand ρ = 0.82. The vertical lines in the Safe-ALINEA, Renewal, and DRR policies represent the standard deviation. The standard deviation of the DisDRR, DSG, and greedy policies are similar to the DRR policy and are omitted for clarity. We let the merging speed of on-ramps 1 and 3 be Vf , and Vf /3 for on-ramp 2. We let λ = 0.455, and consider a congested initial condition where the initial number of vehicles on the mainline is 100 > nc; each vehicle is at the constant speed of 6.7 [m/s], and is at the distance h × 6.7 + S0 ≈ 14.1 [m] from its leading vehicle. We evaluate the Total Travel Time (TTTn), which is the average travel time of the first n completed trips. In order to show consistency, for each policy we conducted 10 rounds of simulations with different random seed for each round. In each round, we used the same seed across different policies. We use the average and standard deviation of the 10 simulation rounds for illustration. For the aforementioned setup, the freeway becomes saturated under the Safe-ALINEA and greedy policies, but remains under-saturated under the other policies. Figure 2.9 and Table 2.2 show the simulation results. From the results, we can see that the DRR policy provides significant improvement in the TTT as compared to the Safe-ALINEA (approximately 83%) and Renewal (approximately 80%) policies. The DisDRR, DSG, and greedy policies show similar improvements. However, as shown in Section 2.4.2, the 39 freeway becomes saturated under the greedy policy for long enough simulation time (not shown in Figure 2.9). This implies that the current improvement in the TTT will disappear as the number of completed trips grows. On the other hand, the choice of Tcyc = 12 in the DRR, DisDRR, and DSG policies will show steady improvement. Table 2.2 also provides the time average queue size, i.e., Q(Kτ ) = 1 mK P i,j |Qi(jτ )|, where |Qi | is the average queue size over the 10 simulation rounds, and Kτ is the simulation time. It can be seen that the Safe-ALINEA policy has the largest queue size as compared to the other policies. This is mainly because of the safety filter added on top of the ALINEA policy. Without the safety filter, ALINEA releases vehicles more frequently, which results in shorter on-ramp queues on average. However, the safety distance is violated more often and more congestion is created on the mainline, which results in higher travel time compared to the DRR policy. In conclusion, with proper choice of design parameters, our proposed policies create and maintain the free flow state on the mainline, which improves the travel time. Table 2.2: Performance of the policies. Ramp Metering Policy Renewal DRR DisDRR DSG Greedy Safe-ALINEA max n TTTn [min] 10.2 2 2 1.8 1.7 11.7 Average queue size 101 6 10 7 12 127 2.5 Conclusion We provided a RM framework at the microscopic level subject to vehicle following safety constraints. This allows to explicitly take into account the V2X communication scenarios into the RM design, and study the impact on the freeway performance. We specifically provided RM policies for a few such scenarios, and analyzed performance in terms of the throughput of the freeway. s 40 Chapter 3 Vehicle Following On A Ring Road Under Safety Constraints: Role of Connectivity and Coordination 3.1 Introduction In Section 2.2, we assumed that vehicles follow certain “rules" without specifying the vehicle controller that accomplishes this task. In particular, we assumed that vehicles maintain their steady state speed if they are at a safe distance from their leading vehicles. In practice, vehicles may deviate from the free flow state, e.g., due to external disturbance. Moreover, the disturbance on one vehicle may propagate upstream of traffic and, in extreme cases, create congestion. In fact, [31] has used human drivers on a closed ring road setup to show that this type of congestion can occur even in the absence of lane changes or bottlenecks such as on-ramps. Hence, it is important to design vehicle controllers that can mitigate this issue and study the trade-offs between limited road space, safety, and speed in the context of CAVs. Similar to the experiment in [31], we use a closed ring road setup in this chapter to study the aforementioned trade-offs. While studying vehicles on a fixed-length section of a straight road with boundary conditions is also a possibility, it is analytically easier to study the dynamics of vehicles on a ring road [33, 38, 39]. Furthermore, the ring road setup can capture the formation of stop-and-go waves in the absence 41 of bottlenecks/lane changes [31]. However, our results in this chapter are consistent with previous results on a straight road. Inspired in part by [31] and the theoretical advantages of a ring road geometry, there has been recent dynamical analysis on this setup for mixed-autonomy [32, 33, 38, 39, 40, 41]. The foci of the analytical aspects of these works, however, is on the formation and dissipation of traffic jams using autonomous vehicles for a high density scenario, without explicit consideration of safety. In this chapter, we consider homogeneous automated vehicles on a closed single lane ring road. We design state feedback control laws and consider the following three scenarios: in the first scenario, we assume that vehicles do not communicate and there is no coordination. Each vehicle either follows a constant speed trajectory, or safely follows the vehicle ahead by keeping a safe time headway. Transitioning between modes of operation is determined by a combination of relative spacing and speed signals and is handled by the vehicle’s supervisory controller similar to [42]. In the second scenario, we assume that vehicles are also able to communicate their braking capabilities and their instantaneous acceleration with their following vehicles. In the final scenario, we assume that a coordinator communicates the desired platoon formations and inter-platoon spacings to certain vehicles. We show that the designed controllers guarantee robust speed tracking and/or vehicle following, and attenuation of errors in the desired relative spacing, speed, and acceleration upstream a platoon while satisfying desired acceleration bounds for all three scenarios. The rest of this chapter is outlined as follows. In section 3.2, we state the problem formulation and control objectives. In Section 3.3, we first consider the case where vehicles do not communicate and there is no coordination on the ring road. We next extend the analysis for the case where V2V communication is possible. In Section 3.4, the role of coordination on the ring road is evaluated and suitable transition logics are proposed. Section 3.5 provides simulation results for these scenarios. We conclude the chapter in section 3.6. 42 Figure 3.1: Example of three vehicles on a closed ring road setup 3.2 Problem Formulation 3.2.1 Basic Notations Consider n homogeneous vehicles of length L, on a closed ring road of length P > nL. We assign coordinates over the distance interval [0, P] to the ring road in the clock-wise direction. The vehicles are indexed such that vehicle i is the i-th closest vehicle to point 0 at time t = 0. Let xi(t) ∈ [0, ∞) be the distance traveled by the i-th vehicle with respect to a fixed reference point on the road, e.g., point 0, and vi(t), ai(t) denote the speed and acceleration at time t ≥ 0, respectively. Let yi(t) = xi+1(t)−xi(t)−L be the relative spacing of the i-th vehicle with respect to the vehicle i+ 1 ahead at time t ≥ 0 (xn+1 ≡ P +x1 due to the periodicity of the ring road). For simplicity of notations, we formulate the vehicle model and controller design for an ego vehicle with subscript e and use the subscript l for its leading vehicle ∗ . Note that by definition, Pn i=1 yi = P − nL. This constraint is the main contrast to a straight line with no space limitation. An illustration of this setup for three vehicles is depicted in Figure 3.1. ∗The reader should not confuse the leading vehicle with the leader of a platoon. 43 Figure 3.2: Logic diagram for determining the mode of operation 3.2.2 Modes of Operation Each vehicle operates in one of the following two modes of operation: cruise control or vehicle following † , see Figure 3.2. If there is no coordination, the ego vehicle operates in the cruise mode if no valid vehicle is ahead that is within its sensing range. The validity of the leading vehicle is determined by comparing the relative spacing to a design threshold value. The ego vehicle is in the vehicle following mode, i.e., it follows the leading vehicle by keeping a safety distance, as long as the leading vehicle’s speed is within the allowable speed limit. The speed limit is taken to be equal to the free flow speed Vf when there is no coordination. The platoon formation state in Figure 3.2 is activated in order to achieve a desired platoon formation when a coordinator is present. This state will be discussed in detail in Section 3.4. †The cruise control mode in this chapter is more general than the speed tracking mode in Chapter 2 as it allows the ego vehicle to track different speed references. Furthermore, the vehicle following mode can be considered as a submode of the safety mode in Chapter 2 which is activated when the vehicle is away from the merging areas. 44 3.2.3 Model and Control Objectives We use the following input-output linearized vehicle model which is derived from the Newton’s second law of motion [43]: a˙ e = ue. (3.1) Here, ue is the control input which is to be designed to meet the following control objectives: 1. Safety: no rear-end collision under a worst-case stopping scenario as explained in [9]. 2. Smooth longitudinal maneuver: Smooth position and/or speed tracking in the two modes of operation as well as a smooth transition between these modes. 3. String error attenuation: attenuation of the amplitude of errors, e.g., in the position, upstream a platoon. 4. Passenger comfort: amin ≤ ae ≤ amax, except in an emergency braking scenario, and small jerk a˙ e [8]. In the following sections, we design and analyze control laws that can meet the objectives with and without V2V communication and in the presence of a coordinator. 3.3 Vehicles On a Ring Road Without Coordination 3.3.1 No V2V Communication In this section, we assume that vehicles do not communicate with each other and obtain the necessary information by using their own sensing capabilities. When the mode of operation is determined as explained in Section 3.2.2, the sensing data are passed through appropriate filters [42] in order to generate continuous-time signals passed to the longitudinal controller ue designed as follows: 45 1. Cruise: ue = Kaae + Cv(vr − ve) + Z t 0 [Cs(vr − ve)]dτ, (3.2) v˙r = sat[p(Vs − vr)], vr(0) = ve(0), (3.3) sat[x] = amax if x ≥ amax x if amin < x < amax amin if x ≤ amin . (3.4) 2. Vehicle following: ue = Kaae + Cp(t)δe + Cv(vr − ve) + Z t 0 [Cq(τ )δe + Cs(vr − ve)]dτ, (3.5) vr = vl + (vr(0) − vl)e −κt , (3.6) δe = ye − (hve + S0), (3.7) where Ka < 0, Cv, Cs, p, κ > 0 are design constants, and Vs = Vf is the speed limit. Moreover, the threshold distance ∆d for switching from the cruise control mode to the vehicle following mode is chosen as follows: ∆d = hve + S0 + r(ve − vl) if ve ≥ vl hve + S0 otherwise , (3.8) where r > 0 is a design constant. If the relative spacing of the ego vehicle with respect to the leading vehicle is greater than ∆d at t = 0, the ego vehicle starts to operate in the cruise control mode and the controller (3.2) is used. The reference speed vr in this case is generated by passing the desired speed limit Vs through the nonlinear acceleration limiter filter (3.3) with the saturation function described in (3.4). The acceleration limiter prevents the acceleration outside the comfortable range when there is a large initial 46 speed error Vs − ve(0) [8]. If the relative spacing becomes less than ∆d at some time t0 ≥ 0, the ego vehicle switches to the vehicle following mode and the controller (3.5) is used. The design parameters Cp(t), Cq(t) are smoothly increased from zero to some positive design constants Cp, Cq > 0, i.e., Cp(t) = Cp(1 − e −κ(t−t0) ), Cq(t) = Cq(1 − e −κ(t−t0) ), t ≥ t0. Moreover, the reference speed vr is smoothly changed from the initial value to the speed of the leading vehicle vl (see (3.6)), and the reference relative spacing is set to hve + S0 (see (3.7)), where S0 > 0 is a constant standstill separation distance and h is a safe time headway constant. This is a well-known safe vehicle following strategy where the following vehicles try to keep a safe constant time headway from the leading vehicle [9]. Accordingly, the switching distance ∆d is chosen to be equal to the safety distance hve + S0 plus an additional non-negative term r(ve − vl) if the ego vehicle is travelling at least as fast as the leading vehicle (see (3.8)). The ego vehicle keeps operating in the vehicle following mode as long as the leading vehicle’s speed is within its allowable speed limit Vs. Remark 13. The objective is to design the control parameters such that (3.2) - (3.8) ensures stability, string error attenuation, and, lastly, satisfies comfort. We should emphasise that the designed longitudinal controller is only responsible for smoothly adjusting the spacing and/or speed. Other operations such as emergency braking are assessed by a higher-level supervisory controller and operated by different control laws which are not addressed in this paper. However, this problem is resolved in other papers, see for example [8],[42]. We define nc = P hVf +S0+L as the critical number of vehicles on the ring road (nc can be non-integer). We also define configuration as the vector of relative spacings on the ring road. Theorem 5. There exist design parameters such that the following hold: (i) The controller (3.2) - (3.8) guarantees smooth vehicle following and/or speed tracking in all modes of operation and string error attenuation, i.e., the attenuation of the amplitude of errors with respect to the desired relative spacing, speed, and acceleration upstream a platoon. 47 (ii) If n < nc, there is an infinite number of vehicle configurations on the ring road; however the equilibrium speed is Vf in each of these configurations. (iii) If n ≥ nc, there is a unique vehicle configuration where all vehicles are symmetrically distributed around the ring road and their speed converges to an equilibrium speed of 1 h ( P n − S0 − L) ≤ Vf . Proof. A sketch of the proof of part (i) with essential equations is provided here. Parts (ii) and (iii) can be derived from part (i). Details are included in Appendix B.1. (i) (sketch) We consider the following three cases. First, if the ego vehicle is in the cruise control mode, we show that its speed converges to Vf if the poles of K(s) = Cvs + Cs s 3 − Kas 2 + Cvs + Cs (3.9) lie in the open left-half of the complex plane. Furthermore, assuming zero initial condition, we show that if K(s) satisfies |K(jω)| ≤ 1, ∀ω ≥ 0 and k(t) ≥ 0, ∀t ≥ 0, then, |ae(t)| ≤ amax, ∀t ≥ 0. (3.10) We next consider two possible cases where the ego vehicle switches to the vehicle following mode and joins a platoon of vehicles. In the first case, the platoon has a leader that operates in the cruise control mode, whereas in the second case the platoon has no leader. In the first case, the relative spacing and speed of the ego vehicle converges to the desired equilibrium points if the poles of G(s) = Cvs 2 + (Cp + Cs)s + Cq F(s) , (3.11) 48 lie in the open left-half of the complex plane, where F(s) = s 4 − Kas 3 + (hCp + Cv)s 2 + (Cp + hCq + Cs)s + Cq. In the second case, an additional condition is required which is satisfied if we have |G(jω)| ≤ 1, ∀ω ≥ 0. Moreover, the string error attenuation is guaranteed in both cases if |G(jω)| ≤ 1, ∀ω ≥ 0, and g(t) ≥ 0, ∀t ≥ 0. We also show that if G(s) satisfies the aforementioned constraints, then |ae(t)| ≤ |al(t)| + ˜e0(t), (3.12) where e˜0(t) is an exponentially vanishing term. Remark 14. Simulations (and practical considerations) suggest that a vehicle in the vehicle following mode does not switch to cruise control under the proposed logic-based controller (see Figure 3.2). This implies that there is no switch in the overall mode of the system after a finite time. Furthermore, within each mode, we show in the proof of Theorem 5 that the saturation constraints on the acceleration are not binding, and hence thereafter the system can be treated as an LTI system. In this sense, linear system tools are sufficient for asymptotic analysis. Remark 15. Equations (3.10) and (3.12) imply that a vehicle satisfies the comfort requirements except, maybe, for an exponentially vanishing time. Remark 16. According to Theorem 5, for a given number of vehicles n < nc, vehicles can form platoons of (possibly) different sizes with different inter-platoon spacing at steady state, which depends on the initial condition. We discuss in Section 3.4 the role of coordination in achieving a unique desired configuration in order to improve efficiency in utilizing the limited space. 49 Figure 3.3: Fundamental diagram Remark 17. Macroscopic traffic flow interpretation of Theorem 5: Let v ∗ be the equilibrium speed of vehicles, φ = n P be the space-mean density, φc = nc P be the critical density, and q ∗ = φv∗ be the equilibrium space-mean flow. It follows from Theorem 5 that v ∗ = min{Vf , 1 h ( P n − S0 − L)}. Therefore, q ∗ = Vfφ if φ < φc 1 h (1 − φ(S0 + L)) if φ ≥ φc . In other words, when the density is less than the critical density, the flow increases linearly with increasing density. However, when the density exceeds the critical density, the flow decreases linearly with increasing density. This gives rise to the well-known triangular fundamental diagram (see Figure 3.3). The maximum value of q ∗ , i.e., the capacity C of the ring road, is then found to be C = Vf hVf +S0+L . Remark 18. According to the proof of Theorem 5, in the cruise control mode the poles of K(s) in (3.9) must lie in the open left half of the s-plane. This condition is satisfied if KaCv + Cs < 0. (3.13) Moreover, for position/speed tracking and string error attenuation in the vehicle following mode, the design parameters must be chosen such that poles of G(s) in (3.11) have negative real parts and |G(jω)| ≤ 50 1, ∀ω ≥ 0. The former can be guaranteed by using pole placement. Additionally, |G(jω)| ≤ 1, ∀ω ≥ 0 is satisfied if K2 a − 2(hCp + Cv) ≥ 0, (hCp + Cv) 2 + 2Cq + 2Ka(Cp + hCq + Cs) − C 2 v ≥ 0. (3.14) We provide a set of parameters in Section 3.5 (see (3.18)) that satisfies (3.13), (3.14), and the stability criterion for G(s). 3.3.2 V2V Communication We now assume that vehicles are able to communicate their braking capabilities as well as their instantaneous acceleration and deceleration to their following vehicles. This feature allows for accurate reference tracking when vehicles are outside the sensing range and also smaller safe time headway constant between the vehicles [44]. With V2V communication, the longitudinal control law in the vehicle following mode (3.5) is modified as follows: ue = Kaae + Cp(t)δe + Cv(vr − ve) + Ca(t)(al − ae) + Z t 0 [Cq(τ )δe + Cs(vr − ve) + Cb(τ )(al − ae)]dτ, (3.15) where Ca(t), Cb(t) ≥ 0 are additional control parameters which behave similar to Cp(t), Cq(t). Note that the only difference between (3.5) and (3.15) is the additional acceleration terms Ca(t)(al − ae) and Cb(t)(al − ae) in (3.15). Since by choosing Ca(t) = Cb(t) = 0, ∀t ≥ 0, (3.15) becomes identical to the control law in (3.5), all of the results for stability, string error attenuation, and comfort holds when V2V communication is possible. As mentioned earlier, V2V communication reduces the minimum safe time headway constant h. Thus, the critical number of vehicle nc for which vehicles can operate at the free flow speed increases. As a 51 result, the critical density φc = 1 hVf +S0+L and the capacity C = Vf hVf +So+L in Remark 17 are increased. Therefore, V2V communication expands the free-flow region of the Fundamental diagram . Furthermore, by using V2V communication, vehicles can be organized in platoons and decide among themselves certain configurations. Moreover, it allows for accurate tracking of the position, speed, and acceleration of vehicles ahead even when they are outside the sensing range. This feature expands the number of possible configurations that can be achieved in the presence of a coordinator. We discuss this in the next section. 3.4 Coordination of Vehicles On A Ring Road In the previous section, we assumed that vehicles travel without coordination, i.e., their action to achieve the speed limit or follow a vehicle in front was determined by their own sensors and/or V2V communication. According to Theorem 5, if n < nc, there is an infinite number of configurations in which the system of vehicles can occupy the limited space but, in all of them, they travel with the free flow speed Vf . It may so happen that some configurations on the road are more desirable than others from the point of view of a coordinator. For example, vehicles may be organized in platoons in order to decrease air drag and thus fuel consumption, or use the bandwidth of the coordinator-to-vehicle communication system more effectively by only communicating to the leaders of platoons [42]. Furthermore, when entry/exit to the road is possible, those configurations which minimize the waiting time of the on-ramp vehicles are more desirable. In this section, we show that the proposed controller is able to steer the vehicles to a specified configuration when coordination is possible. We assume that vehicles use their own sensors and/or V2V communication in combination with commands from a coordinator, see Figure 3.4. The coordinator chooses a configuration from the following set of desired configurations and communicates it along with the perimeter of the road P, and the number of vehicles n to certain vehicles: 52 Figure 3.4: Control structure in the presence of a coordinator and/or V2V communication 1. 1-platoon asymmetrical: a single platoon of n vehicles, 2. Symmetrical: all vehicles sharing the limited space equally, 3. k-platoon symmetrical: k platoons of vehicles, 1 < k ≤ n 2 , sharing the limited space equally. These configurations are chosen in order to illustrate the main idea. The following discussion can be easily applied to other desired configurations as well. Upon receiving commands from the coordinator, the ego vehicle calculates the appropriate reference speed and spacing according to the desired configuration and passes it to the longitudinal controllers in (3.2), (3.5) with the reference speed and spacing modified as follows: Vs = Vf if desired platoon formed behind αVf otherwise , (3.16) δe = ye − yd, yd = (hd + (h − hd)e −κt)ve + S0 if requires spacing adjustment hve + S0 otherwise . (3.17) 53 Upon receiving initiation commands from the coordinator at t = 0, the ego vehicle uses the platoon formation flow chart in Figure 3.5 in order to calculate the reference speed and spacing. According to Figure 3.5, when the platoon for which the ego vehicle is its desired leader has not yet formed, the ego vehicle changes its speed limit Vs from Vf to αVf for some α ∈ (0, 1) (see (3.16)). According to the switching logic explained in Section 3.2.2, it starts to decelerate, when it is safe, so that the vehicles behind catch up. When the desired platoon has formed, the ego vehicle is notified by the coordinator and/or V2V communication and the speed limit is reset to Vf (see (3.16)). Moreover, if the ego vehicle needs to adjust its relative spacing depending on the desired configuration, the controller smoothly tracks vl and changes the reference relative spacing from the initial value to hdve + S0 (see (3.17)). The reference time headway constant hd is calculated by the ego vehicle such that hdVf +S0 is equal to the reference relative spacing, e.g., when the desired configuration is symmetrical, the reference relative spacing is P n − L, and hd = 1 Vf ( P n − L − S0). Since the reference time headway constant hd is different than h, the design parameters Cp(t), Cq(t) in (3.2) are also smoothly changed, if necessary, such that the controller maintains good tracking performance. Remark 19. Note that “coordination" in this chapter is a one-off message to the vehicles to specify the desired configuration. In this sense, the coordination is "centralized". However, upon receiving the message, the vehicles achieve the desired configuration in a “decentralized" manner. Note that in Section 3.3, the vehicles used homogeneous time headway constants h, and constant desired speed Vf in the cruise control mode. However, with the coordinator in the loop, the desired time headways are (possibly) heterogeneous and time-varying, and the reference speed in the cruise control mode is, in general, piece-wise constant. Therefore, additional analysis is required in order to establish stability. 54 Figure 3.5: Logic diagram of a desired leader for creating the desired platoon Figure 3.6: Simulation results for the high density traffic regime Theorem 6. There exist design parameters such that the longitudinal controller (3.2)-(3.8) with the reference speed/spacing specified in (3.16), (3.17), guarantees that the system of vehicles converges to the configuration specified by the coordinator. Proof. See Appendix B.2. Remark 20. The equilibrium distance between adjacent platoons in the k-platoon symmetrical configuration, 1 < k ≤ n 2 , is d = n k ( P n − L − hVf − S0) + hVf + S0. Since P > n(L + hVf + S0), i.e., n < nc, d decreases when k is increased from 2 to n 2 . In other words, as the size of the platoon increases in the 55 Figure 3.7: Simulation results for the low density traffic regime with no coordination Figure 3.8: Simulation results with coordination and 2-platoon symmetrical desired configuration k-platoon symmetrical configuration, the desired inter-platoon relative spacing increases. Since the sensing range of vehicles are limited, V2V communication allows for accurate tracking of the position of the vehicle ahead even when it is outside of the sensing range. In other words, V2V communication expands the number of achievable desired configurations when there is coordination. This can be especially useful when vehicles are scattered on the road and it is desirable to organize vehicles in tight platoons. 56 (a) (b) Figure 3.9: Steady state configuration of the vehicles with (a) no coordination, (b) coordination with 2-platoon symmetrical desired configuration (orange: leader, blue: follower) 3.5 Simulations In this section, we illustrate the performance of the designed control laws by simulating a few scenarios. In all scenarios, the control parameters are chosen as follows, Ka = −9, Cp = 2, Cv = 6, Cq = 0.01, Cs = 0.03, h = 1.5 [s], S0 = 4 [m], p = 10, amin = −0.2g, amax = 0.1g, r = 1, κ = 0.5. (3.18) For this choice of design constants, the stability, string error attenuation, and comfort conditions are satisfied. Other parameters are chosen as follows: P = 320 [m], L = 4.5 [m], Vf = 29 [m s ]. Therefore, the critical number of vehicles is nc = 6.04. 57 3.5.1 High Density Traffic Regime Let n = 8 > nc, i.e., a high-density traffic regime, with two platoons of sizes 3 and 5 initially at rest. The first platoon consists of vehicles 1 − 3 with vehicle 3 as the leader, and the second platoon consists of vehicles 4 − 8 with vehicle 8 as the leader. The distance between the first and second platoons is initially 100 meters, i.e., y3(0) = 100 [m], and all the following vehicles are assumed to be at the desired spacing at t = 0. According to Theorem 5, the system of vehicles converges to a unique configuration with the equilibrium relative spacing of P n − L = 35.5 [m], and speed of 1 h ( P n − S0 − L) = 21 [m s ]. The speed, acceleration, and relative spacing profiles for sample vehicles are shown in Figures 3.6. As can be seen from the acceleration and speed profiles, the leader of the first platoon, i.e., vehicle 3, operates in the cruise control mode until t ≈ 15 [s]. At this point it switches to the vehicle following mode and the two platoons become connected. Furthermore, the leader of the second platoon switches to the vehicle following mode at t ≈ 26 [s], and a platoon with no leader ‡ is formed. It is clear that the speed and acceleration profiles are smooth and within the comfort range in both modes of operation as well as during the transition between the two modes. 3.5.2 Low Density Traffic Regime: No Coordination In this scenario, let n = 4 < nc, i.e., a low-density traffic regime, and all the vehicles are initially at rest. We assume that vehicles 1 − 3 are initially in platoon formation with the following vehicles at the desired spacing, and vehicle 4 is 100 meters ahead. The speed, acceleration, and relative spacing profiles of the vehicles are shown in Figure 3.7. It can be seen from the relative spacing profiles that vehicles 3 and 4 operate in the cruise control mode at all times, thus at steady state, we have one platoon of three vehicles with vehicle 3 as the leader, and a single vehicle, i.e., vehicle 4, operating in the cruise control mode, see Figure 3.9a. Moreover, it can be easily verified that all the vehicles reach the free flow speed Vf while ‡The term “platoon with no leader" refers to the situation where each vehicle follows its leading vehicle. 58 satisfying the the acceleration bounds during the transient. Also, it can be seen from the acceleration profiles that the errors in acceleration are not magnified upstream a platoon, i.e., string error attenuation. 3.5.3 Low Density Traffic Regime: With Coordination We again consider n = 4. We assume that the vehicles are travelling at steady state speed of Vf and initial configuration of the previous scenario. Let the coordinator’s desired configuration be 2-platoon symmetrical with vehicles 2 and 4 as the desired leaders. The simulation results for this scenario are shown in Figure 3.8. The coordinator communicates the desired configuration to vehicles 2 and 4 at t = 10 [s]. Since the platoon consisting of vehicles 3 and 4 has not formed at t = 10 [s], vehicle 4 sets its speed limit to Vs = 0.8Vf , and starts to decelerate until vehicle 3 catches up. At the same time vehicle 2 smoothly increases its time headway constant to the desired value hd ≈ 3.43 [s], and starts decelerating in order to adjust its relative spacing. At t ≈ 20 [s], vehicle 3 switches to the vehicle following mode, and vehicle 4 resets its speed limit while smoothly increasing its desired relative spacing to hdv1 + S0. Due to large initial positive relative spacing and speed error at t ≈ 20 [s], this introduces acceleration outside the comfortable bounds for vehicle 4. It can be seen from the relative spacing profiles at t ≈ 35 [s], that the desired configuration is achieved asymptotically. The qualitative steady state configuration of this scenario is shown in Figure 3.9b. 3.6 Conclusion In this chapter, we designed longitudinal controllers for homogeneous vehicles travelling on a single-lane ring road under safety and comfort constraints to evaluate the impact of limited space on the speed of flow. We showed that if the number of vehicles is less than a certain critical number nc, which depends on the perimeter of the ring road, free flow speed limit, and safety spacing, the vehicles can organize themselves 59 around the ring road in an infinite number of different configurations. When the number of vehicles increases to be greater than or equal to nc, all vehicles converge to a unique equilibrium configuration where the equilibrium speed decreases as the number of vehicles increases. When we add V2V communications, the controller is modified for faster action during vehicle following and safety can be guaranteed under lower intervehicle spacing. As a result, the critical number of vehicles nc that can operate at the maximum allowable speed increases. We also showed that if a coordinator dictates the configuration of the vehicles around the ring road, the proposed controllers can force the vehicles to converge to the desired coordination while maintaining safety and passenger comfort. Simulations are used to demonstrate the performance of the controllers. 60 Chapter 4 Throughput Maximizing Takeoff Scheduling for eVTOL Aircraft in On-Demand Urban Air Mobility Systems 4.1 Introduction In this chapter, we transition from ground transportation to air transportation as an alternative way of mitigating traffic congestion. Specifically, we consider Urban Air Mobility (UAM) and systematically design a traffic management protocol, which efficiently uses limited UAM resources. The starting point of this chapter is the well-known Air Traffic Flow Management (ATFM) problem for commercial airplanes, which was formalized in [45]. The key idea behind ATFM is to proactively manage congestion by anticipating traffic demand and manage the usage of various airspace and airport resources. To this end, an integer program formulation, called the Traffic Flow Management Problem (TFMP), is solved to assign desired flight trajectories to each airplane subject to operational constraints [46, 47]. The UAM traffic management problem can be considered a natural extension of ATFM and its integer program formulation. However, unlike commercial air traffic, where the demand is highly predictable even weeks in advance, the UAM systems will be designed to provide on-demand services. This poses a significant tactical challenge. 61 Figure 4.1: A top-view sketch of a UAM network with three modes of aircraft operation: idle aircraft (red), in-service aircraft that transport passengers (black), and rebalancing aircraft that fly without passengers to high-demand areas (purple). The UAM traffic management methods can be generally classified as either centralized or decentralized [48]. In decentralized methods, each UAM vehicle can select its preferred route while being responsible for its separation margins with other vehicles using onboard technology. These methods can be based on cooperative multi-agent negotiation [49], mixed-integer linear programming [50], decentralized model predictive control [51], or Markov decision processes [52]. While it is reasonable to expect that the decentralized approach is feasible for low-volume UAM operation, it can lead to a significant loss in efficiency, and even gridlock when the UAM systems mature and demand grows [53]. Therefore, the UAM traffic management system should be centralized, especially in high-demand scenarios. This means that there should be a central authority to assign flight trajectories to each UAM vehicle or to resolve all conflicts. Centralized methods are typically modeled as an optimization problem. In [54], the authors consider a two-phase approach, where in the first phase, an integer program is solved to determine a conflict-free trajectory for a given flight. The solution of the optimization problem is then passed to a velocity profile smoothing model in the second phase. The work in [55] considers a combination of pre-computed optimal paths and integer program to determine conflict-free trajectories for all 62 flight requests. Recent works such as [56] extend the existing TFMP formulation to accommodate the ondemand nature of UAM and incorporate fairness. Solution methods other than optimization formulations include heuristic methods such as first-come first-served scheduling [57] and simulations [58]. The previous works provide valuable insights into the operation of UAM systems. However, most, if not all, of them do not explicitly address two critical aspects. First is the concept of rebalancing: the UAM aircraft will need to be constantly redistributed in the network when the demand for some destinations is higher than others; see Figure 4.1. Efficient rebalancing ensures the effectiveness and sustainability of on-demand UAM systems. The concept of rebalancing has been explored extensively in the context of ondemand ground transportation [21]. However, these studies predominantly use flow-level formulations which do not capture the safety and separation considerations associated with aircraft operations. The second aspect which has not been addressed in the UAM literature is a thorough characterization of the system-level throughput. Similar to the discussion in Chapter 2, throughput of a given traffic management policy determines the highest demand that the policy can handle. In the context of UAM, throughput is tightly related to the notion of passenger waiting time. In particular, the throughput determines the demand threshold at which the expected passenger waiting times transition from being stabilized to being increasing over time. Therefore, it is desirable to design a policy that achieves the maximum possible throughput. To address these gaps, we present a centralized framework to determine conflict-free takeoff scheduling for UAM aircraft. In particular, we propose a policy, called VertiSync, which synchronously schedules the aircraft for either servicing trip requests or rebalancing in the network. The proposed policy modifies and extends the TFMP by incorporating elements of Chapter 2 on on-ramp metering in ground transportation. Recall that in Chapter 2, we used queuing theory to design algorithms that maximize freeway throughput without prior knowledge of demand. Although the methods for characterizing throughput in this chapter are conceptually similar to those in Chapter 2, the underlying systems are significantly 63 different. Specifically, in Chapter 2, we modeled the entire freeway as a single static "server." In contrast, in this chapter, we deal with multiple dynamic servers– aircraft moving around in the system—- adding complexity to the problem. The rest of this chapter is organized as follows: in Section 4.2, we describe the problem formulation. We provide our traffic management policy and characterize its throughput in Section 4.3. We provide the Los Angeles case study in Section 4.4, and conclude the chapter in Section 4.5. 4.2 Problem Formulation We use the following standard notations throughout this chapter. We let N be the set of positive integers, and N0 be the set of non-negative integers. For n ∈ N, we use [n] to denote the set {1, 2, · · · , n}. For a set A, we let |A| denote its cardinality. 4.2.1 UAM Network Structure We describe the UAM network by a directed graph G. A node in the graph G represents either a vertiport, i.e., take-off/landing area, or an intermediate point where two or more routes cross paths. A link in the graph G represents a section of a route (or multiple routes). We let V be the set of vertiports. We let Nv be the total number of vertipads, i.e., takeoff/landing pads, at vertiport v ∈ V . An Origin-Destination (O-D) pair p is an ordered pair p = (op, dp) where op, dp ∈ V and there is at least one route from op to dp. We let P be the set of O-D pairs; see Figure 4.2. To simplify the network representation, we assume that each vertiport has exactly one outgoing link exclusively used for takeoffs from that vertiport, and a separate incoming link exclusively used for landings. We also assume that there is at most one route between any two vertiports and that the UAM routes do not conflict with any existing airspace. Remark 21. Given an O-D pair p = (op, dp), the opposite pair q = (dp, op) may or may not be an O-D pair. However, to enable rebalancing, it is natural to assume that there exists a sequence of routes connecting 64 Figure 4.2: A top-view sketch of a UAM network with |V| = 4 vertiports (blue circles) and |P| = 8 O-D pairs P = {(1, 3),(1, 4),(2, 3),(2, 4),(3, 1),(4, 2),(1, 2),(2, 1)}. dp to op, with the first route originating from dp and the last route ending in op. We assume that this holds for any two vertiports. In graph theory language, this means that the graph G is strongly connected. In the next section, we will discuss the operational constraints of UAM aircraft. 4.2.2 Operational Constraints In this section, we describe the constraints and assumptions related to UAM vehicle flight operations. We assume that all UAM vehicles have the same characteristics so that these constraints are the same for all of them. Let A be the set of UAM vehicles in the system. Each vehicle’s flight operation consists of the following three phases: • takeoff: During this phase, the UAM vehicle is positioned on a departure vertipad and passengers (if any) are boarded before the vehicle is ready for takeoff. To position the vehicle on the departure vertipad, it is either transferred from a parking space or directly lands from a different vertiport. Let τ denote the takeoff separation, which is the minimum time required between successive takeoffs from the same vertipad. In other words, the takeoff operations are completed in a τ -minute time window 65 for every flight, which implies that the takeoff rate from each vertipad is at most one vehicle per τ minutes. • airborne: To ensure safe operation, all UAM vehicles must maintain minimum horizontal and vertical safety margins from each other while airborne. To this end and according to the FAA standards [53, 48], UAM corridors are divided into three-dimensional sectors, with each sector having a certain capacity. The flight trajectories must then comply with the sector capacity constraints. Without loss of generality, we let the capacity of each sector be 1 vehicle; we restrict capacity to 1 to avoid the need for tactical deconfliction within a sector. We also discretize time into time steps of length τc such that in each time-step, a vehicle moves to the next sector; see Figure 4.3. We assume that τ ≥ τc, i.e., the takeoff separation is more restrictive than the separation imposed by airborne safety margins, and kτ := τ /τc is integer-valued. These assumptions are based on the current technological limitations [58, 59]; future work can extend our results if the technology or regulations change over time. • landing: Once a UAM vehicle lands, passengers (if any) are disembarked, new passengers (if any) are embarked, and the vehicle undergoes servicing if needed. Thereafter, the vehicle is either transferred to a parking space, e.g., for re-charging, or, if it has boarded new passengers or needs to be rebalanced, takes off to another vertiport. Similar to takeoff operations, we assume that the landing operations are completed within a τ -minute time window for every flight. That is, once a vehicle lands, the next takeoff or landing can occur after τ minutes. We assume that the parking capacity at each vertiport is at least |A| so that an arriving UAM vehicle always clears the vertipad after landing. To take into account battery limitations of UAM vehicles, we also assume that the number of charging stations at each vertiport is at least |A|. 66 Figure 4.3: Sector configuration for a UAM network, with sector capacity of 1 vehicle. Hence, at most 1 UAM vehicle can occupy any sector at any time. Moreover, if a UAM vehicle occupies sector A, then it moves to sector B after one time step. Without loss of generality, we assume that different links of the graph G are at a safe horizontal and vertical distance from each other, except in the vicinity of the nodes where they intersect. In addition to the above assumptions, we consider an ideal case where there is no external disturbance such as adverse weather conditions. As a result, if a UAM vehicle’s flight trajectory satisfies the safety margins and the separation requirements, then the vehicle follows it without deviating from the trajectory. Since we only focus on conflict-free takeoff scheduling policies in this paper, we do not specify the low-level vehicle controller that follows a given trajectory. 4.2.3 Demand and Performance Metric In an on-demand UAM network, the demand is likely not known in advance. We use exogenous stochastic processes to model the unpredictable nature of the demand. It will be convenient for performance analysis later on to adopt a discrete time setting. Let the duration of each time step be τc, which is the time needed for a UAM vehicle to move to an adjacent sector as described in Section 4.2.2. The number of trip requests for an O-D pair p ∈ P is assumed to follow an i.i.d Bernoulli process with parameter λp independent of other O-D pairs. That is, at any given time step, the probability that a new trip is requested for the O-D 67 Figure 4.4: An illustration of the under-saturation region of some policy π ′ (dark grey area) and a throughput maximizing policy π (dark + light grey areas). pair p is λp independent of everything else. Note that λp specifies the rate of new requests for the O-D pair p in terms of the number of requests per τc minutes. Let λ := (λp) be the vector of arrival rates. For each O-D pair, the trip requests are queued up in an unlimited capacity queue until they are serviced at which point they leave the queue. The queues in this case do not represent any “physical" queue; rather, they serve as lists of trip requests in the order they arrive. Therefore, the unlimited capacity assumption is not binding. In order for a request to be serviced, a UAM vehicle serving that request must take off from the verriport ∗ . A scheduling policy is a rule that schedules the UAM vehicles in the system for either servicing trip requests or rebalancing, i.e., taking off without passengers to service trip requests at other vertiports. The objective of the paper is to design a policy that can handle the maximum possible demand under the operational constraints discussed in Section 4.2.2. The key performance metric to evaluate a policy is the notion of throughput which we will now formalize. For p ∈ P, let Qp(t) be the number of outstanding ∗By using this notion of service, queue length at time t is a function of takeoff at time t. Alternatively, one could use UAM vehicles landing as the notion of service by appropriately shifting this function in time. 68 trip requests for O-D pair p at time t. Let Q(t) = (Qp(t)) be the vector of outstanding trip requests for all the O-D pairs at time t. We define the under-saturation region of a policy π as Uπ = {λ : lim sup t→∞ E [Qp(t)] < ∞ ∀p ∈ P under policy π}. This is the set of λ’s for which the network remains under-saturated, meaning that the expected number of outstanding trip requests remain bounded for all the O-D pairs. The boundary of this set is called the throughput of policy π. We are interested in finding a policy π such that Uπ′ ⊆ Uπ for all policies π ′ , including those that have information about the demand λ. In other words, if the network remains undersaturated using some policy π ′ , then it also remains under-saturated using policy π. In that case, we say that policy π maximizes the throughput for the UAM network. In the next section, we introduce one such policy. Remark 22. A rigorous definition of throughput should also include its dependence on the initial condition of the UAM vehicles and the initial queue sizes. We have removed this dependence for simplicity since the throughput of our policy does not depend on the initial condition. Example 5. Consider the UAM network in Figure 4.2, and suppose that a policy π is able to maximize the throughput; that is, for any other policy π ′ , we have Uπ′ ⊆ Uπ. Suppose further that λp is fixed for every O-D pair p except (1, 3) and (2, 4). An illustration of Uπ and Uπ′ is shown in Figure 4.4. From the figure, one can see that if (λ(1,3), λ(2,4)) ∈ Uπ′, then (λ(1,3), λ(2,4)) ∈ Uπ, i.e., if the expected number of outstanding trip requests remain bounded under the policy π ′ , then it also remains bounded under policy π. 69 4.3 Network-Wide Scheduling 4.3.1 VertiSync Policy We now introduce our scheduling policy, called VertiSync, which is inspired by the queuing theory literature [37] and the classical TFMP [46]. Informally, VertiSync works in cycles during which only the trips that were requested before the start of a cycle are serviced during that cycle. To this end, at the start of a cycle, the central planner determines conflict-free takeoff schedules to service all outstanding trip requests. Once the takeoff schedules are determined, they are communicated to the UAM vehicles and vertiport operators responsible for handling takeoff and landing operations at each vertiport. When all outstanding trip requests are serviced, i.e., some UAM vehicle serving those requests has taken off from the vertiport, the cycle ends and the next cycle starts. It is assumed that the central planner knows the state of each UAM vehicle as well as the number of outstanding trip requests for each O-D pair. As discussed in Section 4.2.2, we divide the UAM corridors into sectors of capacity 1, as shown in Figure 4.5, and we also discretize time into time steps of length τc such that in each time-step, a UAM vehicle moves to an adjacent sector. Recall that these sectors are constructed such that adjacent sectors satisfy the airborne safety margins. We also extend sectors to include the origin and destination vertiports. In particular, the first sector for O-D pair p ∈ P is located at the origin vertiport op, meaning that the UAM vehicle is located on a vertipad at op, and the last sector is located at the destination vertiport dp, meaning that the UAM vehicle has landed on a vertipad at dp. Without loss of generality, we assume that if a link is common to two or more routes, then the sectors associated with those routes coincide with each other on that link, e.g., the green sector in Figure 4.5. We assign a unique identifier to each sector, with overlapping sectors belonging to different routes having different identifiers. For example, the green sector in Figure 4.5 has four different identifiers each belonging to routes (1, 3),(1, 4),(2, 3), and (2, 4). We do 70 Figure 4.5: Sector configuration for a UAM network. The green sector belongs to routes (1, 3),(1, 4),(2, 3),(2, 4). this intentionally for more clarity in the mathematical formulation of the problem later on. Let Sp be the set of sectors associated with O-D pair p. Let tk be the start time of the k-th cycle, n ∈ N0, p ∈ P, and i ∈ Sp. A key decision variable in the VertiSync formulation is w a,p i,n , which represents the number of times that vehicle a ∈ A has visited sector i of route p in the time interval (tk, tk +nτc]. For brevity, tk is dropped from w a,p i,n , as we conduct scheduling one cycle at a time. By definition, w a,p i,n is non-decreasing with respect to n. Moreover, if w a,p i,n −w a,p i,n−1 = 1 for some n ≥ 1, then it means that vehicle a must have occupied sector i at some time in the interval (tk + (n − 1)τc, tk + nτc]. Note that this occupation time is not necessarily a multiple of τc. We also use the variable w a,p o,n to represent the number of times that vehicle a with route p has taken off from vertiport op in the interval (tk, tk + nτc]. Similarly, w a,p d,n denotes the number of times that vehicle a with route p has landed on vertiport dp in the interval (tk, tk + nτc]. Finally, we use the variable w a v,n to denote the number of times that vehicle a has visited vertiport v in the interval (tk, tk + nτc]. For sector i, we use the notation i + 1 to specify the next sector along a UAM vehicle’s route. Moreover, given two O-D pairs p, q ∈ P and sectors i ∈ Sp and j ∈ Sq, we use the notation i = j to specify that sector i coincides with sector j. We are now in a position to formally introduce VertiSync. 71 Definition 6. (VertiSync Policy) The policy works in cycles of variable length, with the first cycle starting at time t1 = 0. At the beginning of the k-th cycle at time tk, each vertiport communicates to the central planner the number of trip requests for each O-D pair that originates from that vertiport, i.e., the vector of trip requests Q(tk) = (Qp(tk)) is communicated to the central planner. During the k-th cycle, only these requests will be serviced. The k-th cycle ends once all these requests have been serviced, i.e., right after the last takeoff. The central planner solves the following optimization problem to determine the takeoff schedules during the k-th cycle. The objective of the optimization is to minimize the total flight time of all UAM vehicles † . That is, we minimize: X a∈A X p∈P (w a,p o,Mk − w a,p o,0 )Tp, (4.1) † In fact, we are minimizing the total rebalancing component of all UAM vehicles. 72 where Mk ∈ N is such that Mkτc is a conservative upper-bound on the duration of the k-th cycle, and Tp is the flight time for route p. The following constraints must be satisfied: X a∈A (w a,p o,Mk − w a,p o,0 ) ≥ Qp(tk), ∀p ∈ P, (4.2a) w a,p i,n−1 − w a,p i,n ≤ 0, ∀n ∈ [Mk], p ∈ P, i ∈ Sp, a ∈ A, (4.2b) w a v,n−1 − w a v,n ≤ 0, ∀n ∈ [Mk], v ∈ V, a ∈ A, (4.2c) w a,p i+1,n = w a,p i,n−1 , ∀n ∈ [Mk], p ∈ P, i ∈ Sp : i ̸= d, a ∈ A, (4.2d) w a v,n = w a v,n−1 + X p∈P:dp=v (w a,p d,n − w a,p d,n−1 ), ∀n ∈ [Mk], v ∈ V, a ∈ A, (4.2e) X p∈P:op=v w a,p o,n − w a v,n−kτ ≤ 0, ∀n ∈ {kτ , · · · , Mk}, v ∈ V, a ∈ A, (4.2f) X a∈A (w a,p i,n − w a,p i,n−1 ) + (w a,q j,n − w a,q j,n−1 ) ≤ 1, ∀n ∈ [Mk], p, q ∈ P, i ∈ Sp, j ∈ Sq : i = j, i, j ̸= o, d, (4.2g) X a∈A X p∈P:op=v (w a,p o,n − w a,p o,n−kτ ) ≤ Nv, ∀n ∈ {kτ , . . . , Mk}, v ∈ V, (4.2h) X a∈A X p∈P:op=v (w a,p o,n − w a,p o,n−1 ) + X q∈P:dq=v (w a,q d,n−1 − w a,q d,n−uτ ) ≤ Nv, ∀n ∈ {kτ , . . . , Mk}, v ∈ V, (4.2i) w a,p i,n , wa v,n ∈ N0, ∀n ∈ [Mk], v ∈ V, p ∈ P, i ∈ Sp, a ∈ A. Constraint (4.2a) ensures that all outstanding trip requests are serviced by the end of the cycle. Constraint (4.2b) forces the decision variables w a,p i,n to be non-decreasing in time. Constraint (4.2d) guarantees that if vehicle a occupies sector i at some time t ∈ (tk + (n − 1)τc, tk + nτc], then it will occupy sector i + 1 at time t + τc. Constraint (4.2e) updates the number of visits to vertiport v when vehicle a lands on v, and implicitly forces the decision variables w a v,n to be non-decreasing in time. Constraint (4.2f) ensures 73 that the number of takeoffs from vertiport v is no more than the number of visits to vertiport v. Constraint (4.2g) ensures the airborne safety margins by allowing at most one UAM vehicle occupying any overlapping sector at any time. Similarly, constraints (4.2h) and (4.2i) ensure that the takeoff and landing separations are satisfied at every vertiport, respectively. We also need additional constraints to take into account UAM vehicles battery limitations. Let Ep be the amount of battery consumption while flying route p, which is calculated as the sum of the battery consumption required for takeoff, cruise, and landing. Let Ea n be vehicle a’s state of charge at time n, and let u a v,n denote the number of times vehicle a has been re-charged at vertiport v in the time interval (tk + (n − 1)τc, tk + nτc]. In addition to the constraints in (4.2), we require: E a n = E a n−kc − X p∈P (w a,p o,n − w a,p o,n−1 )Ep + X v∈V (u a v,n−kc − u a v,n−kc−1 )Einc, ∀n ∈ {kc + 1, . . . , Mk}, a ∈ A, (4.3a) u a v,n−1 − u a v,n ≤ 0, ∀n ∈ [Mk], v ∈ V, a ∈ A, (4.3b) Emin ≤ E a n ≤ 100, ∀n ∈ [Mk], a ∈ A, (4.3c) 0 ≤ Einc ≤ 100, (4.3d) u a v,n − u a v,n ≤ w a v,n − w a v,n−1 , ∀n ∈ [Mk], v ∈ V, a ∈ A, (4.3e) X p∈P (w a,p o,n − w a,p o,n−kc ) ≤ 1 − (u a v,n−kc − u a v,n−kc−1 ), ∀n ∈ {kc + 1, . . . , Mk}, v ∈ V, a ∈ A, (4.3f) u a v,n ∈ N0, ∀n ∈ [Mk], v ∈ V, a ∈ A, where Einc is battery’s charging increment per time step while being recharged, Emin is the minimum allowed battery charge, and kc is an upper-bound on the number of time steps it takes to re-charge a UAM vehicle. Constraint (4.3a) is the balance equation for vehicle a’s state of charge, and constraint (4.3b) forces the decision variable u a v,n to be non-decreasing in time. Constraint (4.3c) limits the minimum and 74 maximum state of charge for vehicle a, and constraint (4.3d) forces the charging increment to be within the allowable range. Finally, constraint (4.3e) ensures that vehicle a can be recharged at a vertiport only if it has visited that vertiport, while constraint (4.3f) ensures that a takeoff can occur only after kc time steps. The initial values of w a,p i,0 , w a v,0 , and u a v,0 at the start of a cycle are determined by the location and state of charge of vehicle a at the end of the previous cycle. In particular, if vehicle a has occupied sector i of O-D pair p at the end of cycle k − 1, then w a v,0 = 0 for all v ∈ V , w a,p j,0 = 1 for sector j = i and any other sector j ∈ Sp that precedes sector i along the UAM vehicle’s route, and w a,q j,0 = 0 for all q ̸= p and j ∈ Sq. Remark 23. Note that VertiSync only requires real-time information about the number of trip requests, but does not require any information about the arrival rate. This makes VertiSync a suitable option for an actual UAM network where the arrival rate is unknown or could vary over time. 4.3.2 Size of VertiSync Formulation In this section, we characterize the size of the optimization problem (4.1)-(4.3), and we describe a preprocessing technique to reduce its size. Recall that |V| denotes the number of vertiports, |P| denotes the number of O-D pairs, Sp denotes the number of sectors associated with O-D pair p, and |A| denotes the number of UAM vehicles. Moreover, Mk is an integer that determines an upper-bound on the length of the k-th cycle. The total number of variables w a,p i,n is Mk|A| P p∈P |Sp|, and the total number of variables w a v,n and u a v,n are 2Mk|A||V|. The number of constraints is upper-bounded by: |P| + Mk (2|A| + 1)X p∈P Sp + Mk (2|V| + 3|V||A| + 3|A|). Note that P p∈P |Sp| ≥ |P| and P p∈P |Sp| ≥ |V|since the origin and destination vertiports are always counted as sectors along each vehicle’s route. Therefore, |Sp| ≥ 2 for all p ∈ P, which implies that the 75 number of constraints is dominated by the term Mk (2|A| + 1)P p∈P |Sp|. In order to get a sense of the size of the formulation, let us consider an example. Example 6. Consider the UAM network shown in Figure 4.5. We have |V| = 4, |P| = 8, P p∈P |Sp| = 40. Let |A| = 5 and Mk = 100. Then, the number of variables is 22,000 and the number of constraints is at most 52,308. We can reduce the size of the optimization problem by concatenating some of the constraints. In particular, suppose that the route corresponding to O-D pair p does not conflict with any other routes, except at the origin or destination vertiports. Then, we may remove the variables w a,p i,n , i ̸= o, d, and their corresponding constraints and concatenate (4.2d) into the following constraint: w a,p o,n−|Sp| = w a,p d,n, ∀n ∈ {|Sp|, · · · , Mk}, p ∈ P, a ∈ A. Using the above pre-processing technique, the number of variables in Example 6 is reduced to 14,000 and the number constraints to 34,708. 4.3.3 VertiSync Throughput We next characterize the throughput of VertiSync. To this end, we introduce the notion of service vector. A service vector is a |P|-dimensional vector r = (r p ) that specifies which O-D pairs can the UAM vehicles take off from and at what rate, so that the airborne safety margins and separation requirements are not violated. In particular, if r p ̸= 0, then it means that UAM vehicles can safely takeoff at the rate r p for O-D pair p. If r p = 0, then the takeoff rate for O-D pair p is zero. Recall from the operational constraints in Section 4.2.2 that the takeoff rate from each vertipad is at most 1 per τ minutes, the takeoff rate from each vertiport is at most 1 per τc, and kτ = τ /τc is integervalued. Therefore, if vertiport v has Nv vertipads, the takeoff rate from vertiport v can be 0, τc/τ , 2τc/τ , 76 . . . , max{Nvτc/τ, 1} per time step. For example, a vector ri with r p i = τc/τ and r q i = 0 for all q ̸= p is a valid service vector since UAM vehicles can safely take off from O-D pair p at the rate of τc/τ vehicle per time step. We let R be the set of all such non-zero service vectors. Example 7. Consider the network in Figure 4.5, which has 8 O-D pairs (1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (4, 2), (1, 2), and (2, 1) that we number from 1 to 8, respectively. Let the takeoff separation be τ = 5 minutes, and τc = 0.5 minutes and suppose that each vertiport has only one vertipad. Therefore, the takeoff rate from each vertiport is at most τc/τ = 0.1 aircraft per time step. Moreover, note that O-D pairs 1 and 4 share a common link along their routes. However, if an aircraft for O-D pair 1 takes off at t = 0, then an aircraft for O-D pair 4 can take off at t = τc without violating the airborne safety margins, as it will not occupy the same sector with the first aircraft. Hence, r1 = (0.1, 0, 0, 0.1, 0, 0, 0, 0) is a service vector. Similarly, r2 = (0, 0.1, 0.1, 0, 0, 0, 0, 0) and r3 = (0, 0.1, 0, 0, 0, 0, 0, 0) are two other service vectors. However, (0.1, 0.1, 0, 0, 0, 0, 0, 0) is not a service vector since aircraft cannot simultaneously take off from vertiport 1 at the rate of 0.1 aircraft per time step. By using the service vectors ri ∈ R, a feasible solution to the optimization problem (4.1)-(4.3) can be constructed as follows: (i) activate at most one service vector ri ∈ R at any time, (ii) while ri is active, schedule available UAM vehicles to take off at the rate r p i for any O-D pair p ∈ P, (iii) switch to another service vector in R for servicing outstanding requests and/or rebalancing, provided that the safety margins with respect to airborne UAM vehicles from previous service vector are not violated, and (iv) repeat (i)-(iii) until all outstanding requests for the k-th cycle are serviced. The next theorem provides an inner-estimate for the throughput of VertiSync when the number of UAM vehicles |A| is sufficiently large and the following “reversibility" assumption holds: Assumption 1. (Reversibility) For every service vector ri ∈ R, there exists a service vector rj ∈ R such that for all p ∈ P with r p i > 0, r q j = r p i , where q = (dp, op) is the opposite O-D pair to the pair p. In other words, if all the O-D pairs in r p i are “reversed", then the resulting vector is also a service vector. 77 Figure 4.6: A UAM network with 2 vertiports (blue circles) and 2 O-D pairs (1, 2) and (2, 1) sharing a single route (shown as a double-headed arrow). We define a service vector ri ∈ R as “symmetric" if, for all p ∈ P, r q i = r p i , where q = (dp, op) is the opposite O-D pair to p. In other words, using the service vector ri , UAM vehicles can continuously take off at the same rate for both the O-D pair p and its opposite pair q, without violating the safety margins and separation requirements. We extend this definition to include service vectors that are not symmetric themselves but can be transformed into a symmetric service vector. Specifically, given ri ∈ R, if there exists a symmetric service vector rj ∈ R such that r p j = r p i for all p ∈ P with r p i > 0, then we also consider ri to be symmetric. Note that if a service vector is symmetric, it automatically satisfies the reversibility requirement in Assumption 1 but the reverse is not true as seen in the following example. Example 8. Consider the simple network in Figure 4.6, where there are only two O-D pairs (1, 2) and (2, 1) that share a single route. Suppose that each vertiport has only one vertipad, the takeoff separation is τ = 5 minutes, and τc = 0.5 minutes. Since there is only a single route, when an aircraft is traveling in one direction, then no aircraft can travel in the opposite direction. Therefore, there are only two service vectors r1 = (0.1, 0) and r2 = (0, 0.1). As can be seen, the network is reversible but not symmetric. Theorem 7. If the UAM network satisfies the reversibility Assumption 1, and the number of aircraft satisfies |A| ≥ max i∈[R] P p∈P r p i minp∈P: r p i >0 r p i , then the VertiSync policy can keep the network under-saturated for demands belonging to the set D◦ 1 = {λ : λ < X |R| i=1 rixi 1 + ci , xi ≥ 0, i ∈ [|R|], X |R| i=1 xi ≤ 1}, 78 where ci = Ii+ (1 + Ii) max p∈P max{ Tp τc + kc − |A| P q r q i − kτ , Tp τc Ii} P p r p i |A| , Ii = 1 ri non-symmetric 0 ri symmetric , and the vector inequality λ < P|R| i=1 rixi/(1 + ci) is considered component-wise. Proof. See Appendix C.1. A special case of Theorem 7 is when the UAM network is symmetric, i.e., all service vectors are symmetric, and the number of aircraft is sufficiently large such that the inner maximum in ci is equal to zero. Corollary 1. If the UAM network is symmetric, i.e., all service vectors are symmetric, and the number of aircraft satisfies |A| ≥ max i∈[R] X p∈P r p i max{ 1 minp∈P: r p i >0 r p i , Tp τc + kc − kτ }, then the VertiSync policy can keep the network under-saturated for demands belonging to the set D◦ 2 = {λ : λ < X |R| i=1 rixi , for xi ≥ 0, i ∈ [|R|], X |R| i=1 xi ≤ 1}, where the vector inequality λ < P|R| i=1 rixi is considered component-wise. Proof. If the network is symmetric and the number of aircraft satisfies the given lower bound, then ci = 0 for all i ∈ [|R|]. The result then follows from Theorem 7. Example 9. (Example 8 cont’d) Consider again the network in Figure 4.6, where we number the two O-D pairs (1, 2) and (2, 1) as 1 and 2, respectively. Let T1 = T2 = 8 minutes, |A| = 32, and kc = 10. Since both 79 service vectors r1 = (0.1, 0) and r2 = (0, 0.1) are reversible and non-symmetric, we have I1 = I2 = 1 and: c1 = c2 = 1 + 2 max p∈P max{ 8 0.5 + 10 − 32 0.1 − 10, 8 0.5 } 0.1 32 = 1.1. Therefore, the set D◦ 1 from Theorem 7 is: D◦ 1 = {(λ1, λ2) : λ1 < x1 21 , λ2 < x2 21 , x1, x2 ≥ 0, x1 + x2 ≤ 1}. 4.3.4 Fundamental Limit on Throughput In this section, we provide an outer-estimate for the throughput of any conflict-free takeoff scheduling policy. A conflict-free policy is a policy that guarantees before takeoff that each UAM vehicle’s entire route will be clear and a vertipad will be available for landing. Since the UAM vehicles have limited battery, it is desirable to use conflict-free policies for traffic management purposes [60]. Any conflict-free policy uses the service vectors in R, either explicitly or implicitly, to schedule the UAM vehicles. Although it is possible for a conflict-free policy to activate multiple service vectors at any time, we may restrict ourselves to policies that activate at most one service vector from R at any time. To justify this, we note that by activating at most one service vector at any time and rapidly switching between service vectors in R, it is possible to achieve an exact or arbitrarily close approximation of any conflict-free schedule while ensuring the safety margins and separation requirements. The next result provides a fundamental limit on the throughput of any conflict-free policy. Theorem 8. If a conflict-free policy π keeps the network under-saturated, then the demand must belong to the set D = {λ : λ ≤ X |R| i=1 rixi , for xi ≥ 0, i ∈ [|R|], X |R| i=1 xi ≤ 1}, 80 Figure 4.7: A top-view sketch of a UAM network for Los Angeles. The blue circles show the vertiports and the orange arrows show the links. where the vector inequality λ ≤ P|R| i=1 rixi is considered component-wise. Proof. See Appendix C.2. Example 10. (Examples 8 and 9 cont’d) For the parameters given in the previous examples, the set D from Theorem 8 is: D = {(λ1, λ2) : λ1 < x1 10 , λ2 < x2 10 , x1, x2 ≥ 0, x1 + x2 ≤ 1}. As expected, D◦ 1 ⊂ D. 4.4 Simulation Results In this section, we demonstrate the performance of the VertiSync policy and compare it with a heuristic scheduling policy from the literature. As a case study, we select the city of Los Angeles, which is anticipated to be an early adopter market due to severe road congestion, existing infrastructure, and mild weather [59]. All the simulations were performed using Python as the programming language, with optimization 81 Figure 4.8: The sufficient (from Theorem 7), necessary (from Theorem 8), and actual (from simulations) bounds on λ. Figure 4.9: The rate of trip requests per τ minutes (λ(t)). performed by Gurobi optimizer on a PC with Intel(R) Core(TM) i7-8700 processors, 3.2 GHz, 12 GB RAM with 64-bit Windows OS. 4.4.1 Comparison of Theoretical Bounds In this section, we evaluate the under-saturation bounds given by Theorems 7 and 8, which we will refer to as “sufficient" and “necessary" bounds, respectively. We compare these bounds against the “actual" under-saturation bound calculated through simulations. We consider the Los Angeles network shown in Figure 4.7, which consists of four vertiports located in Redondo Beach (vertiport 1), Long Beach (vertiport 2), and the Downtown Los Angeles area (vertiports 3 and 4). The choice of vertiport locations is adopted from [59]. Each vertiport is assumed to have 1 vertipad. This network has eight O-D pairs (1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (4, 2), (1, 2), and (2, 1), 82 Figure 4.10: A top-view sketch of an expanded UAM network for Los Angeles with 12 vertiports and 27 O-D pairs. Figure 4.11: The travel time under the VertiSync and FCFS policies for the demand λ(t). which we number from 1 to 8, respectively. We let the takeoff and landing separations τ be 5 [min], and the sector separation τc be 0.5 [min]. We assume that a UAM vehicle gets fully re-charged during the τ period assigned to takeoff/landing operations. The flight times for O-D pairs (1, 2) and (2, 1) are assumed to be 5 [min], and for the rest of the O-D pairs are 8 [min]. We let the trip requests for O-D pairs 1 − 4 follow a Poisson process with rate λ. The demand for other O-D pairs is set to zero. Given the number of vertiports and vertipads, this network has 40 service vectors. However, the only service vectors that play a role in computing the sufficient and necessary bounds are r1 = (0.1, 0, 0, 0.1, 0, 0, 0, 0) 83 Figure 4.12: The travel time under the VertiSync policy when the demand is increased to 1.2λ(t) (over-saturated regime), and the ground transportation travel time. and r2 = (0, 0.1, 0.1, 0, 0, 0, 0, 0). Using these service vectors, the sufficient and necessary bounds are calculated and shown in Figure 4.8 for different number of UAM vehicles. Note that the necessary bound, which represents the threshold beyond which no policy can keep the network under-saturated, may be conservative. Thus, we have also plotted the actual under-saturation bound in Figure 4.8. The actual undersaturation bound is calculated through simulations and represents the actual threshold beyond which no policy can keep the network under-saturated. As can be seen, the sufficient bound is less conservative than the necessary bound, as the necessary condition in Theorem 8 does not take into account the number of UAM vehicles. 4.4.2 Comparison with First-Come First-Serve Policy We next evaluate travel time under our policy and the First-Come First-Serve (FCFS) policy [57]. The FCFS policy is a heuristic policy which schedules the trip requests in the order of their arrival at the earliest time that does not violate the safety margins and separation requirements. We again consider the Los Angeles network from the previous section, with each vertiport having 10 vertipads. We let the number of UAM vehicles be |A| = 32, and assume that all of them are initially located at vertiport 1. We let the takeoff, landing, and sector separations be the same as the previous section. We also assume that a UAM vehicle gets re-charged during the τ period assigned to takeoff/landing operations. 84 We simulate this network during the morning period from 6:00-AM to 11:00-AM, during which the majority of demand originates from vertiports 1 and 2 and ends in vertiports 3 and 4. We let the trip requests for O-D pairs 1 − 4 follow a Poisson process with a piece-wise constant rate λ(t). The demand for other O-D pairs is set to zero during the morning period. With a slight abuse of notation, we scale λ(t) to represent the number of trip requests per τ minutes. From Theorem 8, given λ(t) = λ, the necessary condition for the network to remain under-saturated is that λ ≤ τ /4τc = 2.5 trip requests per τ minutes, i.e., ρ := 4λτc/τ ≤ 1. Figure 4.9 shows λ(t), where we have considered a heavy demand between 7:00-AM to 9:30-AM to model the morning rush hour, i.e., ρ(t) = 4λ(t)τc/τ ∈ [0.9, 1) between 7:00-AM to 9:30-AM. For the above demand and a random simulation seed, 518 trips are requested during the morning period from which the FCFS policy services 411 before 11:00 AM while the VertiSync policy is able to service all of them. Figure 4.11 shows the passenger travel time, which is computed by averaging the travel time of all trips requested within each 10-minute time interval. The travel time of a trip is computed from the moment that trip is requested until it is completed, i.e., reached its destination. As can be seen, VertiSync is able to keep the network under-saturated, while the FCFS policy fails to do so, due to its greedy use of the vertipads and UAM airspace which is inefficient. We next evaluate the demand threshold at which travel time under VertiSync becomes comparable to ground transportation. Figure 4.12 shows the travel time under VertiSync when the demand is increased to 1.2λ(t). By Theorem 8, the network is in the over-saturated regime from 6:30-AM to 10:00-AM since 1.2λ(t) > 2.5. However, as shown in Figure 4.12, the travel time is still less than the ground travel time during the morning period. The ground travel times are collected using the Google Maps service from 6:00-AM to 11:00-AM on Thursday, May 19, 2023 from Long Beach to Downtown Los Angeles (The travel times from Redondo Beach to Downtown Los Angeles were similar). 85 4.4.3 Computation Results In this section, we present the computational experience with the optimization problem (4.1)-(4.3). We consider an expansion of the Los Angeles network from previous section to the network shown in Figure 4.10. The network consists of 12 vertiports and 27 O-D pairs, with vertiport locations adopted from [59]. In this network, O-D pairs (3, 7) and (7, 3) are assumed to share a single route similar to Example 8. Moreover, note that even though (11, 4) is an O-D pair, its opposite direction is not. Therefore, for rebalancing purposes, a UAM vehicle needs to make a stop at the intermediate vertiport 10. We let the number of vertipads, takeoff, landing, and airborne separations be the same as the previous section. We also let the number of UAM vehicles be 64, and assume that they are evenly spread across the vertiports at the start of the cycle. We only consider a single cycle with a duration upper-bounded by 75 minutes, i.e., Mkτc = 75 minutes in (4.1). Similar to the previous section, we assume that a UAM vehicle gets re-charged during the τ period assigned to takeoff/landing operations. With the input data described above, the model has on the order of 2.6 million constraints and 1.4 million decision variables after applying the pre-processing technique in Section 4.3.2. Theses numbers are about 5 times larger than the number of constraints and decision variables used in TFMP for national size instances in the United States [47]. Given the size of the problem, we have taken advantage of the capability of Gurobi to stop after finding a good solution with optimality gap of 1%. However, as shall be seen, Gurobi is able to find optimal solutions within a reasonable time in all cases. Table 4.1: Computational results for symmetric demand. # Trip Requests CPU Time (sec) Cycle Len. (min) O.F. Gap (%) 27 185.0 9 238 0.00 81 185.8 16 678 0.00 135 203.6 61 1,130 0.00 270 426.6 75 2,316 0.00 324 688.6 75 2,798 0.00 378 Infeasible - - - 86 Table 4.2: Computational results for asymmetric demand. # Trip Requests CPU Time (sec) Cycle Len. (min) O.F. Gap (%) 27 179.8 75 270 0.00 83 261.0 43 1,146 0.00 125 347.7 75 2,078 0.34 173 1,415.1 75 2,794 0.00 215 Infeasible - - - We consider two cases for how the demand is spread across the network; symmetric and asymmetric. In the symmetric case, the number of trip requests are spread evenly across all O-D pairs. In the asymmetric case, 80% of the demand originates from vertiports 1, 2, 5, 6, and 11 and ends in vertiports 3, 4, and 7. The computational results for both cases are reported in Tables 4.1 and 4.2. The second column in each table shows the actual cycle length, and the third column shows the objective function value. It is clear from the results that Gurobi can compute the optimal solution within a reasonable time in all the cases with average CPU time of 337.9 seconds for symmetric demand, and 300 seconds for asymmetric demand. We observe that the computational time for the asymmetric demand is generally longer than the symmetric case. This can be explained by noting that if several UAM vehicles need to fly the same O-D pair during the same time period, then there are several orderings in which they can do so without changing the value of the objective function. A second observation is that the feasible region for the asymmetric demand is much smaller than the symmetric demand. This is due to the increasing level of congestion in vertiports with high demand, which prevents flights to occur simultaneously. A third observation is that the computational time tends to degrade when we are closer to the infeasiblity border. Indeed, by accepting a larger optimality gap, say 3%, the algorithm is able to compute a good quality solution much faster. For example, in the symmetric demand case with 324 trip request, the computational time is reduced to 430.1 seconds. 87 4.5 Conclusion In this chapter, we provided a conflict-free takeoff scheduling policy for on-demand UAM networks and analyzed its throughput. We conducted a case study for the city of Los Angeles and showed that our policy significantly improves travel time compared to a first-come first-serve policy. We also showed that our policy is computationally viable even for large instances of the problem. 88 Chapter 5 Conclusions and Future Directions This thesis was about the design and analysis of traffic control protocols at the microscopic level for ground and air transportation. Motivated by the need to alleviate traffic congestion and leverage advancements in vehicle and infrastructure technologies, one part of the thesis (Chapter 2) provided a suite of Ramp Metering (RM) controllers, which were based on ideas from queuing theory literature. The proposed policies operated under vehicle following safety constraints and took into account vehicle-to-infrastructure (V2I) communication scenarios. We analyzed the system-level throughput of the proposed policies by studying stochastic stability of the induced Markov chains, and showed that throughput is maximized when the merging speed of all on-ramps equals the free flow speed. The second part of the thesis (Chapter 3) was on designing longitudinal controllers, and studying the dynamics of a system of homogeneous vehicles on a single-lane ring road, to understand the interplay of limited space, speed, and safety. We showed that vehicles can occupy the limited space in many different configurations, if the number of vehicles is less than a certain threshold, while they converged to a unique symmetric configuration, if the number of vehicles exceeds that threshold. We also considered vehicleto-vehicle (V2V) communication and coordination and showed that they can improve freeway capacity and/or force vehicles to converge to a desired configuration. 89 In the final part of the thesis (Chapter 4), we transitioned to air transportation as an alternative resort for current traffic congestion. We provided a centralized scheduling policy– VertiSync– which scheduled the aircraft for either servicing trip requests or rebalancing in the system subject to aircraft safety margins and energy requirements. We characterized the system-level throughput of VertiSync, and showed that it can maximize throughput for sufficiently large fleet sizes. There are several avenues for generalizing the setup and methodologies initiated in this thesis. One interesting direction from Chapter 2 is to consider freeway systems where some on-ramps are not controlled. The proposed RM policies generally cannot deal with such scenarios, as they would passively look for safe gaps on the freeway instead of coordinating with the mainstream vehicles. Therefore, vehicles from the controlled on-ramps may have to wait for a long time before being released. 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Prentice hall Upper Saddle River, NJ, 2002. 96 Appendix A A.1 Network Specifications A.1.1 Dynamics in the Speed Tracking Mode Suppose that the ego vehicle is in the speed tracking mode for all t ≥ 0, and ve(0) = v0 ∈ [0, Vf ], ae(0) = a0 ∈ [amin, amax], where ae is the acceleration and amax is the maximum possible acceleration. Then, for the third-order vehicle dynamics, for all t ≥ 0 we have ve(t) = v0 + Z t 0 ae(ξ)dξ ae(t) = Jmaxt + a0 if 0 ≤ t < t1 amax if t1 ≤ t < t1 + t2 amax − Jmaxt if t1 + t2 ≤ t < t1 + t2 + t3 0 if t ≥ t1 + t2 + t3 , where Jmax is the maximum possible jerk, t1 is the time at which the ego vehicle reaches the maximum acceleration, t2 is the additional time required to reach a desired speed before it decelerates, and t3 is the time required to reach the zero acceleration in order to avoid exceeding the speed limit Vf . Hence, ve(t1 + t2 + t3) = Vf , and ae(t1 + t2 + t3) = 0. The dynamics for the v0 > Vf case is similar. 97 A.1.2 Communication Cost of a Ramp Metering Policy The calculation of the communication cost for a policy is inspired by the robotics literature [61, Remark 3.27]. Let cij (k) be the communication cost of vehicle i to on-ramp j at time kτ . We let cij (k) = 1 if vehicle i communicates with on-ramp j, and cij (k) = 0 otherwise, where we have normalized the cost for simplicity independent of the type of information communicated. Then, the total communication cost at time kτ is C(k) = P i∈[n],j∈[m] cij (k), where n is the total number of vehicles in the network. The communication cost of the policy is obtained as follows: C = lim sup K→∞ 1 K K X−1 k=0 C(k). A.2 Performance Analysis Tool Recall the initial condition from Section 2.2.3, where the vehicles are in the free flow state such that the location of each vehicle coincides with a slot for all times in the future. For this initial condition and under the proposed RM policies, we can cast the freeway dynamics as a discrete-time Markov chain. With a slight abuse of notation, we let t = 0, 1, . . . with time steps of duration τ whenever we are talking about the underlying Markov chain. Let M(t) be the vector of the destinations of the occupants of the nc + na slots: Mℓ(t) = j if the destination of the vehicle occupying slot ℓ at time t is off-ramp j, and Mℓ(t) = 0 if slot ℓ is empty at time t. Consider the following discrete-time Markov chain with the state Y (t) := (Q(t), M(t)). The transition probabilities of this chain are determined by the RM policy being analyzed, but will not be specified explicitly for brevity. For all the RM policies considered in Chapter 2, the state Y (t) = (0, 0) is reachable from all other states, and P (Y (t + 1) = (0, 0) | Y (t) = (0, 0)) > 0. Hence, the chain Y is 98 irreducible and aperiodic. In the the proofs of Theorem 1 and 2, we construct new Markov chains (that are also irreducible and aperiodic) by “thinning" the chain Y . The following is an adaptation of a well-known result, e.g., see [62, Theorem 14.0.1], for our setting: Theorem 9. (Foster-Lyapunov drift criterion) Let {Z(t)}∞ t=1 be an irreducible and aperiodic discrete time Markov chain evolving on a countable state space Z. Suppose that there exist V : Z → [0, ∞), f : Z → [1, ∞), a finite constant b, and a finite set B ∈ Z such that, for all z ∈ Z, E [V (Z(t + 1)) − V (Z(t))| Z(t) = z] ≤ −f(z) + b1B(z), (A.1) where 1B(z) is the indicator function of the set B. Then, limt→∞ E [f(Z(t))] exists and is finite. Remark 24. If the conditions of Theorem 9 hold true, then V is referred to as a Lyapunov function. Additionally, if f(Z(t)) = ∥Q(t)∥∞, where ∥Q(t)∥∞ denotes the sup norm of the vector of queue sizes |Q(t)|, then limt→∞ E [∥Q(t)∥∞] < ∞, and hence lim supt→∞ E [|Qi(t)|] < ∞ for all i ∈ [m]. We end this section by introducing a notation which will be used in the proofs of Theorem 1 and 2 to construct Lyapunov functions and/or prove certain results about them. Given on-ramp i ∈ [m], we let Ni(t) be the degree of on-ramp i at time t, which represents the number of vehicles in the network at time t that need to cross the merging point of on-ramp i in order to reach their destination. A.3 Proofs of the Main Results A.3.1 Proof of Theorem 1 For the sake of readability, we present proofs of intermediate claims at the end. Since vehicles reach the free flow state after at most Tfree time (see (VC3)), the mainline and acceleration lanes become empty after a finite time. Without loss of generality, let the start of the first cycle t1 be at time 0. Thereafter, we can 99 adopt the Markov chain setting from Appendix A.2 with {Y (tk)}k≥1 as the Markov chain, where tk is the beginning of the k-th cycle. Consider the function V : Y → [0, ∞) V (Y (tk)) = T 2 cyc(k), where Y is the range of values of Y , and Tcyc(k) = tk+1 − tk is the length of the k-th cycle. We let V (Y (t)) ≡ V (t) for brevity. It is shown at the end of the proof that Tcyc(k + 1) ≤ A˜(Tcyc(k)) + 2nc + na, (A.2) where A˜(Tcyc(k)) depends on the number of arrivals in the interval [tk+1, tk+1] and satisfies the following: lim t→∞ E A˜(Tcyc(k)) Tcyc(k) !2 Tcyc(k) = t = max i∈[m] ( τi τ − 1)ρi − ( τi τ − 2)λi 2 . (A.3) By the assumption of the Theorem, (τi/τ − 1)ρi − (τi/τ − 2)λi < 1 for all i ∈ [m]. Combining this with (A.2) and (A.3), it follows that lim sup t→∞ E " Tcyc(k + 1) Tcyc(k) 2 Tcyc(k) = t # ≤ lim sup t→∞ E A˜(Tcyc(k)) Tcyc(k) !2 Tcyc(k) = t = max i∈[m] ( τi τ − 1)ρi − ( τi τ − 2)λi 2 < 1. Therefore, there exists δ ∈ (0, 1) and T > 0 such that for all t > T we have E " Tcyc(k + 1) Tcyc(k) 2 Tcyc(k) = t # < 1 − δ, 100 which in turn implies that E T 2 cyc(k + 1) − T 2 cyc(k) Tcyc(k) > T < −δT2 cyc(k). Since |Qi(tk)| ≤ Tcyc(k) ≤ T 2 cyc(k) for all i ∈ [m], it follows that E T 2 cyc(k + 1) − T 2 cyc(k) Tcyc(k) > T < −δ∥Q(tk)∥∞, where ∥Q∥∞ = maxi |Qi | as defined in Appendix A.2. If Tcyc(k) ≤ T, then A˜(Tcyc(k)) is shown in (A.6) to satisfy A˜(Tcyc(k)) ≤ mT maxis∈[m] (τi/τ − 1). Combining this with (A.2) gives Tcyc(k + 1) ≤ mT maxi∈[m] (τi/τ − 1) + 2nc + na. Therefore, E T 2 cyc(k + 1) Tcyc(k) ≤ T ≤ mT maxi∈[m] (τi/τ − 1) + 2nc + na 2 + T 2 cyc(k) − δ∥Q(tk)∥∞, where we have used δ∥Q(tk)∥∞ ≤ ∥Q(tk)∥∞ ≤ T 2 cyc(k). Combining all the previous steps gives E [V (tk+1) − V (tk)| Y (tk)] ≤ −δ∥Q(tk)∥∞ + mT max i∈[m] ( τi τ − 1) + 2nc + na 2 1B, where B = {Y (tk) : V (tk) ≤ T 2} (a finite set). The result then follows from Theorem 9. Proof of (A.2) Consider the (k + 1)-th cycle. Let si ∈ [tk+1, tk+2) be the time at which the queue at on-ramp i ∈ [m] becomes empty of the (k + 1)-th cycle quota. We claim that si − tk+1 ≤ ( τi τ − 1)Ni(tk+1) − ( τi τ − 2)|Qi(tk+1)|, 10 where Ni is defined in Appendix A.2. Suppose not, i.e., si − tk+1 − ( τi τ − 1)Ni(tk+1) − ( τi τ − 1)|Qi(tk+1)| > |Qi(tk+1)|. (A.4) We will use (A.4) to reach a contradiction. If at least one upstream vehicle crosses the merging point in τi/τ − 1 time steps, then the last acceleration lane slot of on-ramp i that is not on the mainline must be empty because of the safe gap release criteria (M4). Since the number of vehicles from other on-ramps that need to cross the merging point of on-ramp i is Ni(tk+1) − |Qi(tk+1)|, the aforementioned acceleration lane slot can be empty for at most (τi/τ − 1)(Ni(tk+1) − |Qi(tk+1)|) time steps. Therefore, it must be occupied for at least si − tk+1 − (τi/τ − 1)(Ni(tk+1) − |Qi(tk+1)|) time steps, which by (A.4) is greater than |Qi(tk+1)|. This, however, contradicts the feature of the Renewal policy under which the number of vehicles released by an on-ramp during a cycle does not exceed the queue size at the beginning of the cycle. Since Tcyc(k + 1) ≤ maxi∈[m] si − tk+1 + 2nc + na, it follows that Tcyc(k + 1) ≤ max i∈[m] n ( τi τ − 1)Ni(tk+1) − ( τi τ − 2)|Qi(tk+1)| o + 2nc + na. (A.5) Let Ai(s) be the number of arrivals to on-ramp i at time s, and Aℓ,i(s) be the number of arrivals to all the on-ramps at time s that need to cross link i. We let A¯ i(tk+1 − tk) ∗ denote the cumulative number of arrivals to on-ramp i during the interval [tk + 1, tk+1], i.e., A¯ i(tk+1 − tk) = t Xk+1 s=tk+1 Ai(s). ∗ In this context, the beginning and end of each interval is specified whenever needed. Thus, the notation A¯i(tk+1 − tk) should not create any confusion. 102 We define A¯ ℓ,i(tk+1−tk) similarly. Note that |Qi(tk+1)| is precisely the cumulative number of arrivals to on-ramp i in [tk + 1, tk+1], i.e., |Qi(tk+1)| = A¯ i(tk+1 − tk). Similarly, Ni(tk+1) = A¯ ℓ,i(tk+1 − tk). This and (A.5) imply (A.2) with A˜(Tcyc(k)) = max i∈[m] n ( τi τ − 1)A¯ ℓ,i(tk+1 − tk) − ( τi τ − 2)A¯ i(tk+1 − tk) o . Note that because the number of arrivals to all the on-ramps is at most m at each time step, we have A˜(Tcyc(k)) ≤ mTcyc(k) max i∈[m] (τi/τ − 1). (A.6) Proof of (A.3) Consider the sequences {Ai(s)}∞ s=tk+1 and {Aℓ,i(s)}∞ s=tk+1. Each sequence is i.i.d and E [Ai(s)] = λi , E [Aℓ,i(s)] = ρi . By the strong law of large numbers, with probability one, lim Tcyc(k)→∞ (τi/τ − 1)A¯ ℓ,i(tk+1 − tk) − (τi/τ − 2)A¯ i(tk+1 − tk) Tcyc(k) = ( τi τ − 1)ρi − ( τi τ − 2)λi . Therefore, with probability one, lim Tcyc(k)→∞ A˜(Tcyc(k)) Tcyc(k) = lim Tcyc(k)→∞ max i∈[m] 1 Tcyc(k) ( τi τ − 1)A¯ ℓ,i(tk+1 − tk) − ( τi τ − 2)A¯ i(tk+1 − tk) = max i∈[m] lim Tcyc(k)→∞ 1 Tcyc(k) ( τi τ − 1)A¯ ℓ,i(tk+1 − tk) − ( τi τ − 2)A¯ i(tk+1 − tk) = max i∈[m] ( τi τ − 1)ρi − ( τi τ − 2)λi . Moreover, since the real function x 2 is continuous, we obtain, with probability one, lim Tcyc(k)→∞ A˜(Tcyc(k)) Tcyc(k) !2 = lim Tcyc(k)→∞ A˜(Tcyc(k)) Tcyc(k) !2 = max i∈[m] ( τi τ − 1)ρi − ( τi τ − 2)λi 2 . 103 Finally, if the sequence A˜(Tcyc(k)) Tcyc(k) 2 ∞ Tcyc(k)=1 is upper bounded by an integrable function, then the dominated convergence theorem implies lim Tcyc(k)→∞ E A˜(Tcyc(k)) Tcyc(k) !2 = E lim Tcyc(k)→∞ A˜(Tcyc(k)) Tcyc(k) !2 = max i∈[m] ( τi τ − 1)ρi − ( τi τ − 2)λi 2 , which in turn gives (A.3). The upper bound follows from the following fact: the number of arrivals to all the on-ramps is at most m at each time step. Thus, A˜(Tcyc(k)) Tcyc(k) = 1 Tcyc(k) max i∈[m] ( τi τ − 1)A¯ ℓ,i(tk+1 − tk) − ( τi τ − 2)A¯ i(tk+1 − tk) ≤ 1 Tcyc(k) max i∈[m] ( τi τ − 1)A¯ ℓ,i(tk+1 − tk) ≤ max i∈[m] ( τi τ − 1)m, as desired. A.3.2 Proof of Theorem 2 We first show that for any initial condition, there exists k0 ∈ N such that Xf (kTper) = 0 for all k ≥ k0. That is, Xf1 (kTper) = 0 for all k ≥ k0 and Xf2 (t) = 0 for all t ≥ k0Tper. The equality Xf2 (t) = 0 for all t ≥ k0Tper implies that each vehicle remains at a safe gap with respect to its leading vehicle, and predicts to be at a safe gap with respect to its virtual leading vehicle in merging areas. This and (VC2) imply that if a vehicle is in the speed tracking at time k0Tper, it remains in this mode in the future. Furthermore, if a vehicle is in the safety mode at time k0Tper, Xf1 (k0Tper) = 0 implies that it moves at the constant speed Vf . Together with (VC1) and Xf2 (t) = 0 for all t ≥ k0Tper, it follows that the vehicle maintains its speed in the future. All this would imply that the vehicles reach and remain in the free flow state at time k0Tper. Hence, after a finite time after k0Tper, g(·) = 0, the location of all the vehicles coincide with a slot, and we can use the Markov chain setting from Appendix A.2. 104 Recall from Remark 8 that ve and δe + ˆδe are bounded, say by V and δ, respectively. Let Pa be the total length of all the acceleration lanes, and amax be the maximum possible acceleration. Since successive releases from an on-ramp are at least τ seconds apart, the number of vehicles that communicate δe + ˆδe in [t − Tper, t] is at most mTper/τ + (P + Pa)/L. One can then easily show that Xf (t) is upper bounded by X¯ := (mTper τ + P + Pa L )δ + P + Pa L (max{V − Vf , Vf } + max{amax, |amin|}). To show Xf (·) = 0 after a finite time, we use a proof by contradiction. Suppose that Xf (kTper) ̸= 0 for infinitely many k ∈ N. Since Xf (kTper) ≤ X¯ for all k ∈ N, there exists an infinite sequence {kn}n≥1 such that Xf (knTper) > Xf ((kn − 1)Tper) − γ1 for all n ≥ 1. This implies that θ(knTper) = βθ((kn − 1)Tper) for all n ≥ 1. Since θ(·) is non-decreasing and β > 1, it follows that limt→∞ θ(t) = ∞, which in turn implies that lim supt→∞ g(t) = ∞. Let tf be such that g(tf ) > mTfree(1 + γ2/Tper). Note that for all t ∈ [tf , tf + mTfree], g(t) > mTfree. Thus, each on-ramp releases at most one vehicle during the interval [tf , tf +mTfree]. Hence, there exists a time interval of length at least Tfree in [tf , tf +mTfree] during which no on-ramp releases a vehicle. Condition (VC3) then implies that the vehicles will reach the free flow state after such Tfree time. This and (M4) imply that any vehicle that is released thereafter will remain in the speed tracking mode. Thus, Xf (kTper) = 0 for all k ≥ k0 for some k0 ∈ N; a contradiction to the assumption that Xf (kTper) ̸= 0 for infinitely many k. Without loss of generality, we let g(0) = 0, and we assume that the vehicles are initially in the free flow state, and their location coincide with a slot as in Appendix A.2. For the sake of readability, we present proofs of intermediate claims at the end. We adopt the Markov chain setting from Appendix A.2 with {Y (t△)}t≥0 as the Markov chain, where △ is an exact multiple of the cycle length Tcyc so that Y 105 satisfies the Markov property, and is to be determined in the rest of the proof. Consider the function V : Y → [0, ∞) given by V (t△) ≡ V (Y (t△)) := N 2 (t△), (A.7) where Y is the range of values of Y in Appendix A.2, and N(·) = maxi∈[m] Ni(·), where Ni is the degree of on-ramp i as defined in Appendix A.2. Note that (A.7) implies V ((t+ 1)△)−V (t△) = N2 ((t+ 1)△)− N2 (t△). We claim that if V (t△) is large enough, specifically if V (t△) > L := m△2 + nc + na 2 , then there exists i ∈ [m] such that Ni decreases by at least one in every τi/τ − 1 time steps in the interval [t△,(t + 1)△], without considering the new arrivals. Note that V (t△) > L implies N(t△) > m△2 + nc + na, which in turn implies that there exists at least one on-ramp, say q ∈ [m], such that |Qq(t△)| > △2 . If not, then N(t△) ≤ P i |Qi(t△)|+nc+na ≤ m△2+nc+na, which is a contradiction. Let △ > △1 := maxi∈[m]{τi/τ − 1}. If, in addition, △ ≥ 1 + p 3(⌈nq/Tcyc⌉ + 1)Tcyc, then we show at the end of the proof that for all w ∈ [t△,(t + 1)△ − τq/τ + 1] we have Nq(w + τq τ − 1) ≤ Nq(w) − 1 + w+ Xτq/τ−1 s=w+1 Aℓ,q(s), (A.8) where Aℓ,q(s) is the number of arrivals to all the on-ramps at time s that need to cross link q. By summing up (A.8) over disjoint sub-intervals of length τq/τ − 1 we obtain Nq((t + 1)△) ≤ Nq(t△) − △ τq/τ − 1 + 1 + (t X +1)△ s=t△+1 Aℓ,q(s) ≤ N(t△) − △ τq/τ − 1 + 1 + (t X +1)△ s=t△+1 Aℓ,q(s). (A.9) 10 We show at the end of the proof that a similar form of (A.9) holds for all the on-ramps. Letting △2 := maxi∈[m]{1 + p 3(⌈ni/Tcyc⌉ + 1)Tcyc}, we then show that for all △ ≥ max{△1, △2}, N((t + 1)△) ≤ N(t△) − δ△ + C + A˜(△), (A.10) where δ ∈ (0, 1), C := (Tcyc + nc + 1)m + 2na, and A˜(△) depends on the number of arrivals in [t△ + 1,(t+ 1)△] (thus, it is independent of N(t△)) and is shown to have the following properties † : there exist ε, △3 > 0 such that ε < δ and for all △ ≥ △3 we have E h A˜(△) i < ε△, E h A˜2 (△) i < 2ε(△) 2 . (A.11) The inequality (A.10) implies that N 2 ((t + 1)△) ≤ (N(t△) − δ△ + C) 2 + A˜2 (△) + 2 (N(t△) − δ△ + C) A˜(△). (A.12) By choosing △ ≥ △3 and taking conditional expectation from both sides of (A.12), we obtain E N 2 ((t + 1)△) − N 2 (t△) | V (t△) > L ≤ −2N(t△)((δ − ε)△ − C) + (δ△ − C) 2 + 2ε△(△ − δ△ + C). (A.13) †The expression for A˜(△) in this proof is different than A˜(Tcyc(k)) in the proof of Theorem 1. However, they play similar roles in the proofs which justifies the same notation. 107 By definition, the degree of each on-ramp is at least as large as its queue length. Hence, ∥Q(t△)∥∞ = maxi |Qi(t△)| ≤ N(t△). Plugging into (A.13) gives for all △ ≥ max{△1, △2, △3, C δ−ε }, E N 2 ((t + 1)△) − N 2 (t△) | V (t△) > L ≤ −2∥Q(t△)∥∞ ((δ − ε)△ − C) + (δ△ − C) 2 + 2ε△(△ − δ△ + C) = −∥Q(t△)∥∞ − 2∥Q(t△)∥∞ (δ − ε)△ − C − 1 2 + (δ△ − C) 2 + 2ε△(△ − δ△ + C). Moreover, since ∥Q(t△)∥∞ ≥ |Qq(t△)| ≥ △2 , for all △ ≥ max{△1, △2, △3, △4 := C+1/2 δ−ε } we have E N 2 ((t + 1)△) − N 2 (t△)| V (t△) > L ≤ −∥Q(t△)∥∞ − 2(δ − ε)△3 + δ 2 + 2ε(1 − δ) + 2C + 1 △2 − 2(δ − ε)C△ + C 2 . In addition to the previously stated lower bound △ ≥ max{△1, △2, △3, △4}, if we also choose △ such that −2(δ − ε)△3 + δ 2 + 2ε(1 − δ) + 2C + 1 △2 − 2(δ − ε)C△ + C 2 < 0, (A.14) then E [V ((t + 1)△) − V (t△)| V (t△) > L] ≤ −∥Q(t△)∥∞. Such a △ always exists because the −2(δ− ε)△3 term in (A.14) dominates for sufficiently large △. Finally, if V (t△) = N2 (t△) ≤ L, then ∥Q(t△)∥∞ ≤ √ L because ∥Q(t△)∥∞ ≤ N(t△). Also, V ((t + 1)△) ≤ (m△ + N(t△))2 ≤ (m△ + √ L) 2 because the number of arrivals to all the on-ramps at each time step is at most m. Combining this with the previously considered case of V (t△) > L, we get E [V ((t + 1)△) − V (t△)| Y (t△)] ≤ −∥Q(t△)∥∞ + (m△ + √ L) 2 + (1 + √ L) 1B, where B = {Y (t△) : V (t△) ≤ L} (a finite set). The result follows from Theorem 9. 1 Proof of (A.8) Let w ∈ [t△,(t+ 1)△ −τq/τ + 1]. If at least one upstream vehicle crosses on-ramp q in [w, w +τq/τ −1), then (A.8) obviously follows. If not, then the last acceleration lane slot of on-ramp q that is not on the mainline satisfies (M1)-(M5) at some time in [w, w + τq/τ − 1). So, it must be occupied if on-ramp q had non-zero quotas at the time this slot was the first acceleration lane slot. This would again give (A.8). We now show that on-ramp q indeed had non-zero quotas at the time the aforementioned slot was the first acceleration lane slot, which is at most nq time steps before w. Let t ′ be the start of the most recent cycle at least nq time steps before t△. Considering at most one arrival per time step gives |Qq(t ′ )| ≥ |Qq(t△)| − (t△ − t ′ ) ≥ △2 − (nq + Tcyc), which is at least △ + 2(⌈nq/Tcyc⌉ + 1)Tcyc if △ ≥ 1 + r 3(⌈ nq Tcyc ⌉ + 1)Tcyc. Now, let t ′′ ≥ t ′ be the start of the most recent cycle at least nq time steps before w. Since at most one vehicle is released from on-ramp q per time step, it follows that |Qq(t ′′)| ≥ |Qq(t ′ )|−(t ′′ −t ′ ) > nq +Tcyc, where the last inequality follows from t ′′ −t ′ ≤ △+ (⌈nq/Tcyc⌉+ 1)Tcyc. This gives us the non-zero quota property of on-ramp q. Proof of (A.10) Consider on-ramp i ∈ [m]. If during every τi/τ − 1 time steps in the interval [t△,(t + 1)△] at least one vehicle crosses on-ramp i, then for all w ∈ [t△,(t + 1)△ − τi/τ + 1] we have Ni(w + τi/τ − 1) ≤ Ni(w) − 1 + w+ Xτi/τ−1 s=w+1 Aℓ,i(s). 109 Hence, Ni((t + 1)△) ≤ Ni(t△) − △ τi/τ − 1 + 1 + (t X +1)△ s=t△+1 Aℓ,i(s) ≤ N(t△) − △ τi/τ − 1 + 1 + (t X +1)△ s=t△+1 Aℓ,i(s). If not, let si ∈ [t△,(t + 1)△ − τi/τ + 1] be the last time at which a vehicle does not cross onramp i in [si , si + τi/τ − 1), i.e., Ni(si + τi/τ − 1) = Ni(si) + Psi+τi/τ−1 s=si+1 Aℓ,i(s). Hence, for all w ∈ [si + 1,(t+ 1)△ −τi/τ + 1], Ni(w +τi/τ −1) ≤ Ni(w)−1 +Pw+τi/τ−1 s=w+1 Aℓ,i(s). This further gives Ni((t + 1)△) ≤ Ni(si + 1) − (t + 1)△ − si − 1 τi/τ − 1 + 1 + (t X +1)△ s=si+2 Aℓ,i(s). (A.15) Furthermore, we claim that the queue size at on-ramp i at time si + 1 cannot exceed Tcyc + ni , i.e., |Qi(si + 1)| ≤ Tcyc + ni . Note that since no upstream vehicles crosses on-ramp i in [si , si + τi/τ − 1), it must be that: (i) all the mainline slots upstream of the merging point that are at most τi/τ − 1 time steps away are empty at time si , (ii) the last acceleration lane slot of on-ramp i that is not on the mainline at time si is empty. Moreover, (i) implies that the aforementioned last acceleration lane slot satisfies (M1)-(M5) at time si . Therefore, it must be that on-ramp i had zero quotas at the time this slot was the first acceleration lane slot, which is at most ni time steps before si . Since the number of on-ramp arrivals is at most one per time step, it follows that |Qi(si + 1)| ≤ Tcyc + ni . Hence, Ni(si + 1) ≤ Ni−1(si + 1) + |Qi(si + 1)| + nc + ni ≤ Ni−1(si + 1) + Tcyc + nc + 2ni . Combining this with (A.15) gives Ni((t + 1)△) ≤ Ni−1(si + 1) − (t + 1)△ − si − 1 τi/τ − 1 + 1 + (t X +1)△ s=si+2 Aℓ,i(s) + Tcyc + nc + 2ni . (A.16) 110 Similarly, if during every τi−1/τ − 1 time steps in the interval [t△, si + 1] at least one vehicle crosses on-ramp i − 1, then Ni−1(si + 1) ≤ N(t△) − si + 1 − t△ τi−1/τ − 1 + 1 + sXi+1 s=t△+1 Aℓ,i−1(s). Otherwise, there exists si−1 ∈ [t△, si − τi−1/τ + 2] such that Ni−1(si + 1) ≤ Ni−2(si−1 + 1) − si − si−1 τi−1/τ − 1 + 1 + sXi+1 s=si−1+2 Aℓ,i−1(s) + Tcyc + nc + 2ni−1, This process can be repeated until we find an on-ramp, indexed by i−mi for some mi ∈ {0}∪[m−1], such that during every τi−mi /τ −1 time steps in the interval [t△, si−mi+1 + 1] at least one vehicle crosses it. Indeed, one such on-ramp is always q; see the argument around (A.8). Therefore, Ni−mi (si−mi+1 + 1) ≤ N(t△) − si−mi+1 + 1 − t△ τi−mi /τ − 1 + 1 + si−mXi+1+1 s=t△+1 Aℓ,i−mi (s). By following all the inequalities involved in this process, starting from Ni((t + 1)△) on the left-hand side and ending in N(t△) on the right-hand side, we obtain Ni((t + 1)△) ≤ N(t△) +Xmi p=0 A¯ ℓ,i−p(si−p+1 − si−p) − si−p+1 − si−p τi−p/τ − 1 + (Tcyc + nc + 1)m + 2na, where si+1 = (t + 1)△ − 1, si−mi = t△ − 1, and A¯ ℓ,i(si−p+1 − si−p) is the cumulative number of arrivals in the interval [si−p + 2, si−p+1 + 1] that need to cross link i. By the assumption of the theorem, (τi/τ − 1)ρi < 1 for all i ∈ [m]. So, there exists δ ∈ (0, 1) such that for all i ∈ [m], ρi + δ < 1 τi/τ − 1 . 111 Thus, for all i ∈ [m] we have Xmi p=0 A¯ ℓ,i−p(si−p+1 − si−p) − si−p+1 − si−p τi−p/τ − 1 < −δ△ + Xmi p=0 A¯ ℓ,i−p(si−p+1 − si−p) − ρi−p(si−p+1 − si−p) . Hence, (A.10) follows with A˜(△) = max i∈[m] Xmi p=0 A¯ ℓ,i−p(si−p+1 − si−p) − ρi−p(si−p+1 − si−p) . (A.17) Proof of (A.11) Consider the sequence {Aℓ,is (s)}∞ s=t△+1, where the indices is ∈ [m] are allowed to depend on time s. For a given s ∈ [t△ + 1, ∞), the term Aℓ,is (s) is independent of the other terms in the sequence, ρis = E [Aℓ,is (s)] is bounded, and σ 2 is := E h A2 ℓ,is (s) − ρ 2 is i is (uniformly) bounded for all is ∈ [m]. As a result, lim△→∞ P(t+1)△ s=t△+1(s − t△) −2σ 2 is is also bounded. From Kolmogorov’s strong law of large numbers [63, Theorem 10.12], we have, with probability 1, lim △→∞ 1 △ (t X +1)△ s=t△+1 Aℓ,is (s) − (t X +1)△ s=t△+1 ρis = 0. By following similar steps to the proof of (A.3) in Theorem 1, it follows for n = 1, 2 that lim △→∞ E " A˜(△) △ !n# = 0, which in turn gives (A.11). 112 A.3.3 Proof of Proposition 1 We show that Xf (kTper) = 0 after a finite time. This would then imply that gi(·) = 0 for all i ∈ [m] after a finite time. Thereafter, the rest of the proof follows along the lines of the proof of Theorem 2. Suppose not; then, there exists q ∈ [m] and an infinite sequence {kn}n≥1 such that X q f (knTper) ̸= 0 for all n ≥ 1. We prove that lim supk→∞ gq(kTper) = ∞ by considering the following two cases: (i) if X q f (kTper) ≤ max{X q f ((k − 1)Tper) − γ1, 0} (A.18) holds for finitely many k’s, then Algorithm 2 implies lim supk→∞ gq(kTper) = ∞. (ii) if (A.18) holds for an infinite sequence, let {k ′ n}n≥1 be the sequence of all k’s for which (A.18) holds. Then, there exists an infinite subsequence {k ′ nℓ }ℓ≥1 of {k ′ n}n≥1 for which (A.18) does not hold at k = k ′ nℓ + 1 for all ℓ ≥ 1. If not, then there exists M ∈ N such that (A.18) holds for all k ≥ kM. Since X q f ((kM − 1)Tper) is bounded, this implies that X q f (kTper) = 0 for all k sufficiently greater than kM – a contradiction to X q f being non-zero for the infinite sequence {kn}n≥1. With respect to the subsequence {k ′ nℓ }ℓ≥1, Algorithm 2 implies that θq((k ′ nℓ + 1)Tper) = βθq(k ′ nℓ Tper) for all ℓ ≥ 1. Since θq(·) is non-decreasing and β > 1, this implies limℓ→∞ gq((k ′ nℓ + 1)Tper) ≥ limℓ→∞ θq((k ′ nℓ + 1)Tper) = ∞. That is, lim supk→∞ gq(kTper) = ∞. Recall from the description of the DisDRR policy that P j>q−1 X j f (·) = Xf (·) for the ring road geometry. Also, recall from the proof of Theorem 2 that Xf (·) is bounded by X¯. Thus, if P j>q−1 X j f (kTper) ≤ max{ P j>q−1 X j f ((k − 1)Tper) − γ1, 0} for ⌈X/γ ¯ 1⌉ consecutive periods, then Xf (·) = 0 at the end of it. In addition, if Xf (·) = 0 for a large enough interval of time (say of length at least T¯), then by a similar argument to the proof of Theorem 2, one can show that the vehicles will reach the free flow state, which 113 would imply that Xf (·) = 0 thereafter – a contradiction to Xf being non-zero for the infinite sequence {kn}n≥1. Hence, in an interval of length at least ⌈X/γ ¯ 1⌉Tper + T¯, we must have X j>q−1 X j f (kTper) > max{ X j>q−1 X j f ((k − 1)Tper) − γ1, 0} (A.19) for at least one k. From lim supk→∞ gq(kTper) = ∞ and the fact that gq(·) decreases by at most γ2 in every update period, it follows that there exists time intervals of arbitrarily large length during which gq(·) > Tmax. Combining this with the argument around (A.19), it follows that (A.19) holds for infinitely many k’s while gq(·) > Tmax. Algorithm 2 implies that for any such k, θq−1(kTper) = βθq−1((k−1)Tper). Therefore, limk→∞ θq−1(kTper) = ∞, which then implies that lim supk→∞ gq−1(kTper) = ∞. By repeating the above arguments for other on-ramps, we can conclude that limk→∞ θi(kTper) = ∞ for all i ∈ [m]. We now show that there is a time interval during which all the gi ’s are simultaneously large enough so that no on-ramp releases a vehicle for at least Tfree time. Let k0 be such that gq(k0Tper) and θi(k0Tper) for i ∈ [m] are all greater than max{Tmax, mTfree(1 + γ2/Tper)} + mγ2(⌈X/γ ¯ 1⌉ + ⌈T /T ¯ per⌉). In the ⌈X/γ ¯ 1⌉ + ⌈T /T ¯ per⌉ periods after the time k0Tper, gq(·) decreases by at most γ2(⌈X/γ ¯ 1⌉ + ⌈T /T ¯ per⌉). During this time, gq(·) > Tmax and (A.19) holds at least once, which, by Algorithm 2, would increase gq−1(·) by at least βθq−1(k0Tper) > θq−1(k0Tper). Similarly, in the ⌈X/γ ¯ 1⌉+⌈T /T ¯ per⌉ periods after gq−1(·) is increased, gq−2(·) increases at least once, and so on. Therefore, after at most m(⌈X/γ ¯ 1⌉ + ⌈T /T ¯ per⌉) periods after the time k0Tper, we have gi(kfTper) ≥ max{Tmax, mTfree(1 + γ2/Tper)} for all i ∈ [m], where kf = k0 + m(⌈X/γ ¯ 1⌉ + ⌈T /T ¯ per⌉). Note that for all t in the interval [kfTper, kfTper + mTfree], we have gi(t) ≥ mTfree, i ∈ [m]. Thus, each on-ramp releases at most one vehicle during this interval, which implies that there exists a subinterval of length at least Tfree during which no on-ramp releases a vehicle. Condition (VC3) implies that the vehicle will reach the free flow state after such Tfree time. This 114 and (M4) again imply that Xf (·) = 0 after a finite time – a contradiction to Xf being non-zero for the infinite sequence {kn}n≥1. A.3.4 Proof of Theorem 3 We show that if fi(·), i = 1, 2, 3, is large enough, then: (i) no vehicle initially present on the freeway changes mode because of a vehicle that is released thereafter; (ii) no vehicle released after the time t = 0 ever changes mode to the safety mode. (i) ensures that all the initial vehicles reach the free flow state in Tfree time. Combining with (ii), it follows that all the vehicles reach and remain in the free flow state after the time Tfree. This would imply that Xg(·) = 0 after a finite time, which sets the additional gaps fi to zero. The rest of the proof follows along the lines of the proof of Theorem 2. We provide a proof for (ii); the proof for (i) is similar. Suppose that (ii) does not hold, and lett ∈ [0, Tfree) be the first time that an ego vehicle released at time t0 ≥ 0 changes mode to the safety mode. Without loss of generality, let t0 = 0. To avoid tedious algebra and without loss of generality, we assume that the ego vehicle has merged into the mainline at some time t1 at the speed Vf and moved at this speed up to time t. Also, it has changed mode at time t because of a leading vehicle l that had been in the safety mode and located on the mainline at time t0. Thus, it satisfies |vl(η) − Vf | ≤ Xg(η) for all η ∈ [0, t]. Pointers for the proof of the general case is presented at the end. We have ye(t) = |ye(t1) + Z t t1 vl(η) − ve(η)dη| ≥ ye(t1) − Z t t1 |vl(η) − Vf |dη ≥ f2(X(0)) − Z t1 0 |vl(η) − vl(0)|dη − Z t t1 |vl(η) − Vf |dη ≥ f2(X(0)) − |vl(0) − Vf |t1 − Z t 0 |vl(η) − Vf |dη. (A.20) We show that, for sufficiently large f2(·), f2(X(0)) − |vl(0) − Vf |t1 − Z t 0 |vl(η) − Vf |dη ≥ Se(t), 115 which combined with (A.20) implies that the ego vehicle does not satisfy the (VC2) criterion for changing mode to the safety mode – a contradiction. Let 0 ≤ ξ1 ≤ . . . ≤ ξℓ ≤ t be the “jump" time instants. That is, when a vehicle that was initially present on the freeway either: (I) leaves the freeway, or (II) changes mode. In between these jump events, e.g., η ∈ [ξj , ξj+1], j ∈ [ℓ − 1], (VC5) implies that Xg(η) ≤ ce−r(η−ξj )Xg(ξj ) for some c, r > 0. Moreover, for all j ∈ [ℓ], in jump event (I) we have Xg(ξj ) ≤ Xg(ξ − j ), and in jump event (II), if vehicle i changes mode, then Xg(ξj ) ≤ Xg(ξ − j ) + |vi(ξ − j ) − Vf | + |ai(ξ − j )|, where the equality holds if vehicle i changes mode outside the acceleration lane of the on-ramp it has merged from. To avoid tedious algebra and without loss of generality, we assume that this is the case for all the mode changes in [0, t]. We now bound |vl(η) − Vf | in terms of Xg(0) as follows: for all j ∈ [ℓ − 1] and η ∈ [ξj , ξj+1], we have |vl(η) − Vf | ≤ Xg(η) ≤ ce−r(η−ξj )Xg(ξj ) ≤ · · · ≤ c j+1e −rηXg(0). Hence, Z t 0 |vl(η) − Vf |dη + V 2 f − v 2 l (t) 2|amin| ≤ 1 r (1 − e −rt) + Vf + V 2|amin| e −rt c ℓ+1Xg(0) ≤ 1 r + Vf + V 2|amin| c 3n(0)Xg(0), where n(0) is the initial number of vehicles on the freeway, and in the second inequality, we have used (VC4) to bound ℓ + 1 by 3n(0). Thus, by choosing f2(X(0)) = 1 r + Vf + V 2|amin| c 3n(0) + t1 Xg(0), (A.20) follows. Note that t1 is the constant time between release and exiting the acceleration lane. When the simplifying assumptions do not hold, additional terms are needed in (A.20) to ensure that the ego vehicle does not change mode between release and merging. Moreover, additional terms must be added to f2 to account for the cases where the leading vehicle is initially not in the safety mode or on the 116 mainline. Similarly, additional terms in f2 are needed to account for mode changes inside the acceleration lanes in jump event (II). All of this would result in a different expression for f2 that still satisfies f2(·) = 0 if Xg(·) = 0. A.3.5 Proof of Theorem 4 Let t = 0, 1, . . . with time steps of size τ . Without loss of generality, let the point with the long-run crossing rate no more than one be the merging point of on-ramp i for all i ∈ [m]. We have Ni(t) = Ni(0)+Pt s=1 Aℓ,i(s)−Di(t), where we have dropped the dependence on the RM policy for brevity. Note that this equation holds for any point on the i-th link, which justifies the previous no loss in generality. Since the arrival processes are i.i.d. across the on-ramps, the strong law of large numbers implies that for all i ∈ [m], with probability one, lim inf t→∞ Pt s=1 Aℓ,i(s) t = ρi , and hence lim inf t→∞ Ni(t) t = ρi − lim sup t→∞ Di(t) t . If ρi > 1 for some i ∈ [m], then, with probability one, lim inft→∞ Ni(t)/t is bounded away from zero, and hence lim inft→∞ Ni(t) = ∞. Thus, lim inft→∞ |Qj (t)| = ∞ for some on-ramp j ∈ [m]. Combining this with Fatou’s lemma imply that the average queue size grows unbounded at on-ramp j. This contradicts the freeway being under-saturated. 117 Appendix B B.1 Proof of Theorem 5 In the following, we say that a function of time x : [t0, ∞) → R n converges to x ∗ (or simply x(t) → x ∗ ) exponentially fast as t → ∞ if there exist positive constants c, λ such that ∥x(t) − x ∗ ∥ ≤ ce−λ(t−t0) , ∀t ≥ t0. We say that x(t) is exponentially vanishing if x ∗ = 0. Finally, consider the dynamical system x˙ = f(x, t), x(t0) = x0, where f is a well-behaved function and let x ∗ be its (unique) equilibrium. We say that x ∗ is (globally) exponentially stable if there exist positive constants c, λ such that ∥x(t) − x ∗ ∥ ≤ c∥x0 − x ∗ ∥e −λ(t−t0) . 118 (i) Let we = R t 0 [Cq(τ )δe+Cs(vr−ve)]dτ , where Cq(t) = 0 when the ego vehicle is in the cruise control mode. We consider the following three cases: first, let the ego vehicle be in the cruise control mode. The closed loop dynamics of the ego vehicle can be written as follows: v˙e = ae, a˙ e = Kaae + Cv(vr − ve) + we, w˙ e = Cs(vr − ve), v˙r = sat[p(Vf − vr)]. (B.1) Thus, (ve, ae, we, vr) = (Vf , 0, 0, Vf ) is the unique equilibrium of this mode. Without loss of generality, suppose that p(Vf − vr(0)) > amax, i.e., the saturation function is initially active. It follows that vr(t) = amaxt if 0 ≤ t ≤ T − amax p e −p(t−T) + Vf if T ≤ t , (B.2) where T = Vf amax − 1 p . Hence, vr → Vf exponentially fast as t → ∞. As a result, the state of the saturation function converges to its linear region and (B.1) becomes a LTI system after a finite time. By shifting the equilibrium point of (B.1) to zero and taking Laplace transform of the first three equations assuming zero initial condition, it follows that Ve(s) = K(s)Vr(s), where Ve(s), Vr(s) are Laplace transforms of ve, vr, respectively, and K(s) is given in (3.9). For stability of the equilibrium, we require that the poles of K(s) lie in the open left half of the complex plane. For analyzing the performance in achieving comfort, note that Ae(s) = sK(s)Vr(s), where Ae(s) is the Laplace transform of ae. Therefore, if we choose the design constants such that |K(jω)| ≤ 1, ∀ω ≥ 0 and 119 k(t) ≥ 0, ∀t ≥ 0, then ||k(t)||1 ≤ 1. Thus, assuming zero initial condition, it follows from (B.2) that for all t ≥ 0, |ae(t)| ≤ ||k(t)||1 sup t≥0 |v˙r(t)| ≤ amax. Similarly, if the saturation function is initially active in the other direction, i.e., p(Vf −vr(0)) < amin, then ae(t) ≥ amin. Next, let the ego vehicle switch to the vehicle following mode at time t = 0. The closed loop dynamics of the ego vehicle can be written as follows: y˙e = vl − ve, v˙e = ae, a˙ e = Kaae + Cp(t)δe + Cv(vl + (vr(0) − vl)e −κt − ve) + we, w˙ e = Cq(t)δe + Cs(vl + (vr(0) − vl)e −κt − ve). (B.3) Assuming constant vl and neglecting the exponentially vanishing terms, (ye, ve, ae, we) = (hve + S0, vl , 0, 0) is the unique equilibrium of (B.3). Let z T e = (ye ve ae we). Note that vr(0)e −κt → 0 exponentially fast as t → ∞. Thus, it has no effect on the stability and is ignored in the analysis that follows. By shifting the equilibrium to zero, (B.3) can be written in the following compact form: z˙e = (A1 + D1(t))ze + (B1 + D2(t))vl , (B.4) where A1, B1 are constant matrices and D1(t), D2(t) → 0 as t → ∞ (we have omitted these matrices for the sake of brevity). Thus, the equilibrium of (B.4) is exponentially stable if the equilibrium 120 Figure B.1: Block diagram of a platoon of m vehicles with vehicle m as the leader Figure B.2: Block diagram of a platoon with no leader of the LTI system z˙e = A1ze + B1vl is exponentially stable [64]. By taking Laplace transform of the corresponding LTI system, we arrive at the following equation: Ve(s) = G(s)Vl(s) + E0(s), (B.5) where E0(s) is due to non-zero initial condition of the ego vehicle and G(s) is given in (3.11). We consider two possible cases. In the first case, the ego vehicle has joined a platoon of k vehicles, k ∈ [n], with the leader in the cruise control mode, see Figure B.1. In this case, the stability of the equilibrium is guaranteed if poles of G(s) lie in the open left half of the s-plane. In the second case, the ego vehicle has joined a platoon with no leader, see Figure B.2. It is well-known [64] that a sufficient condition for exponential stability of the equilibrium of the system in Figure B.2 is that poles of G(s) lie in the open left half of the s-plane and |G(jω)| ≤ 1, ∀ω ≥ 0. For analyzing the performance of the controller in achieving passenger comfort, we assume that Cp(t), Cq(t) are constant and vr = vl in the vehicle following mode in order to make use of the properties of LTI systems. It follows from (B.5) that Ae(s) = G(s)Al(s) + E˜ 0(s), (B.6) 121 where G(s) is specified in (3.11) and E˜ 0(s) = sE0(s) + G(s)vl(0) − ve(0) is due to non-zero initial condition. Since |G(jω)| ≤ 1, ∀ω ≥ 0 in order to guarantee stability, if we choose the design constants such that g(t) ≥ 0, ∀t ≥ 0, we derive (3.12) in which e˜0(t) is the inverse Laplace transform of E˜ 0(s) and is exponentially vanishing. Thus, the following vehicles accelerate/decelerate at most as high as the vehicle ahead except, maybe, for an exponentially vanishing term. Finally, from (B.5) the necessary and sufficient condition for string error attenuation of the speed in the L2 sense is that |G(jω)| ≤ 1, ∀ω ≥ 0 [9]. Note that this condition is already satisfied in order to ensure stability of the equilibrium. Moreover, a sufficient condition for string error attenuation in the L∞ sense is that |G(jω)| ≤ 1, ∀ω ≥ 0, and g(t) ≥ 0, ∀t ≥ 0 [9]. This condition is also satisfied in order to provide comfort. Due to the homogeneity of the vehicles, string error attenuation extends to the position and acceleration errors . (ii) Let n < nc, or equivalently P > n(hVf + S0 + L). We claim that at least one vehicle must be operating in the cruise control mode at steady state. Suppose not; then from the equilibrium analysis of (B.3) we have for every i ∈ [n] that yi = hvi + S0, vi = vi+1. Therefore, yi = hvi + S0 = hvi+1 + S0 = yi+1. Since Pn i=1 yi = P − nL, we obtain for every i ∈ [n] that yi = P/n − L, and vi = 1 h ( P n − S0 − L) > Vf , which cannot occur because, according to the designed logic, this violates the speed limit Vf . It follows from the equilibrium analysis of (B.1), (B.3) and the stability of the equilibrium from the previous part that for every i ∈ [n], if vehicle i operates in the cruise control mode at steady state, its speed converges to Vf . Moreover, it is required from the switching logic that δi ≥ 0. Hence, the relative spacing of vehicle i converges to hVf + Si , where Si ≥ S0 depends on the initial condition. 122 On the other hand, if vehicle i operates in the vehicle following mode, its speed converges to Vf , and its relative spacing converges to hVf + S0. (iii) Let n ≥ nc, or equivalently P ≤ n(hVf + S0 + L), then all the vehicles must be operating in the vehicle following mode at steady state. Otherwise, using a similar argument as before we arrive at the contradiction P > n(hVf + S0 + L). Hence, for every i ∈ [n], the relative spacing of vehicle i converges to P/n − L, and its speed converges to 1 h (P/n − S0 − L). B.2 Proof of Theorem 6 Since the constant term S0 has no effect on the stability, we neglect it in the analysis whenever needed. We consider the following three cases: first, consider the 1-platoon asymmetrical desired configuration and let the ego vehicle be the desired leader. At t = 0, the ego vehicle sets its speed limit to αVf , α ∈ (0, 1), and starts tracking αVf . From (B.1), it follows that the equilibrium of the closed loop dynamics of the ego vehicle is ve = vr = αVf , ae = 0, and we = 0. Since at the equilibrium we must have vi = ve = αVf , i ∈ [n], and α < 1, it follows that no other vehicle can be operating in the cruise control mode at steady state. Therefore, all of the vehicles switch to the vehicle following mode after a finite time and form a platoon of n vehicles with the ego vehicle as its leader. Using αVf instead of Vf in the analysis from (B.2) - (B.5), exponential stability of the equilibrium of the platoon immediately follows. Similarly, the equilibrium state of the platoon is exponentially stable when the ego vehicle resets its speed limit to Vf . Note that the total number of switching in the reference speed and spacing is finite, thus the switching does not affect the stability. Consider the symmetrical desired configuration. Note that by construction, all the vehicles eventually switch to the vehicle following mode in order to adjust their relative spacing by using the desired time headway constant and do not switch again at future times (see Figure 3.5). Thus, the switching does not 123 affect the steady state behavior of vehicles. Without loss of generality, let the ego vehicle be in the vehicle following mode when it smoothly increases its time headway constant from h to h1 at time t = 0, where h1 is such that h1Vf + S0 = P/n − L. The closed loop dynamics of the ego vehicle can be written as follows: y˙e = vl − ve, v˙e = ae, a˙ e = Kaae + Cp(t)(ye − h(t)ve) + Cv(vl − ve) + we, w˙ e = Cq(t)(ye − h(t)ve) + Cs(vl − ve), (B.7) where h(t) = h1 + (h − h1)e −κt. The design parameters Cp(t), Cq(t) are also smoothly changed to the desired values, e.g., Cp(t) = C˜ p + (Cp − C˜ p)e −κt, where C˜ p > 0 is a design constant to be chosen. Note that if the ego vehicle was operating in the cruise control mode at t = 0, its closed loop dynamics would have been the same as (B.7) except that Cp(0) = Cq(0) = 0, and vr = vl + (vr(0) − vl)e −κt. Let z T e = (ye ve ae we). We can write the closed loop dynamics (B.7) in the following compact form: z˙e = (A2 + D3(t))ze + B2vl , (B.8) where A2 and B2 are constant matrices, and D3(t) → 0 exponentially fast as t → ∞. The expressions for these matrices are omitted for the sake of brevity. The equilibrium of (B.8) is exponentially stable if the equilibrium of the LTI system z˙e = A2ze + B2vl , (B.9) is exponentially stable [64]. In order to find the stability condition, we use another representation of (B.9) by taking Laplace transform of both sides and deriving the following equation: Ve(s) = H1(s)Vl(s) + E0(s), (B.10) 124 Figure B.3: Block diagram of the k-platoon symmetrical configuration with vehicle n as the desired leader of a platoon where E0(s) is due to non-zero initial condition of the ego vehicle, and H1(s) = C˜ ps + C˜ q F1(s) , F1(s) = s 4 − Kas 3 + (h1C˜ p + Cv)s 2 + (C˜ p + h1C˜ q + Cs)s + C˜ q. Since the desired configuration is symmetrical, i.e., hd = h1 for all the vehicles, the closed loop dynamics of each vehicle on the ring road becomes the same as (B.7) after the final switching time, with (possibly) different initial values of the design parameters and reference speed. Hence, (B.10) holds for all the vehicles in the corresponding LTI system and we have a similar block diagram as in Figure B.2. Using a similar argument to the proof of Theorem 5, a sufficient condition for stability is that poles of H1(s) lie in the left half of the s-plane and |H1(jω)| ≤ 1, ∀ω ≥ 0. It can be verified that if the design constants Ka, Cp, Cq satisfy KaCp+Cq < 0, and C˜ p, C˜ q are chosen such that h1C˜ p = hCp, h1C˜ q = hCq, the stability conditions are guaranteed. Finally, consider the k-platoon symmetrical desired configuration, 1 < k ≤ n/2, and let the ego vehicle be a desired leader. From the result for the 1-platoon asymmetrical configuration, it follows that after a finite time, all of the desired followers of the ego vehicle switch to the vehicle following mode. Also, similar to the symmetrical configuration, all the vehicles eventually operate in the vehicle following mode and do not switch at future times. Assume that the ego vehicle sets its reference spacing to hkve + S0 at t = 0, 125 where hk is the desired time headway constant at the free flow speed calculated by the ego vehicle. The closed loop dynamics of the ego vehicle can be written as (B.7) with a different hk and (possibly) different design constants and reference speed trajectory. Following a similar argument as in the previous scenario, the following equation can be found for the corresponding LTI system: Ve(s) = Hk(s)Vl(s) + E0(s), (B.11) where E0(s) is due to non-zero initial condition and Hk(s) has the same form of H1(s) in (B.10) only with different parameters. Note that the Laplace domain relationship between the speeds of the following vehicles of a platoon is different than (B.11) and was derived in (B.5). Figure B.3 shows the block diagram of this case after the final switching time. By using a similar argument as in the proof of Theorem 5, a sufficient condition for stability is that the poles of Hk(s)G(s) lie in the open left half of the s-plane, and |Hk(jω)G(jω)| ≤ 1, ∀ω ≥ 0. Since the transfer function G(s) is stable and satisfies |G(jω)| ≤ 1, ∀ω ≥ 0, a sufficient condition for the stability is that Hk(s) is stable and |Hk(jω)| ≤ 1, ∀ω ≥ 0. These conditions are automatically guaranteed if KaCp + Cq < 0, and hkC˜ p = hCp, hkC˜ q = hCq. 126 Appendix C C.1 Proof of Theorem 7 Consider the (k + 1)-th cycle. We first construct a feasible solution to the optimization problem (4.1)-(4.3) by using the service vectors in R. Consider the Linear Program (LP) Minimize X |R| i=1 Ki Subject to X |R| i=1 riKi ≥ Q(tk+1), Ki ≥ 0, i ∈ [|R|], (C.1) where the inequality P|R| i=1 riKi ≥ Q(tk+1)is considered component-wise. Let K∗ i , i ∈ [|R|], be a solution to (C.1). A feasible solution to the optimization problem (4.1)-(4.3) can be constructed as follows: 1. Choose a service vector ri ∈ R with K∗ i > 0, and, before activating ri , distribute the aircraft in the system so that for any p ∈ P with r p i > 0, there is |A|p := r p P i p∈P r p i |A| aircraft at vertiport op. Note that |A|p ≥ 1 by the assumption on the minimum number of aircraft. The initial distribution of aircraft takes at most |A|T /τc time steps, where T = maxp∈P Tp. 127 2. Once the initial distribution is completed and the airspace is empty, we would like to activate ri for a total duration of K∗ i + kτ time steps, during which, upon availability, aircraft with route p take off at the rate r p i from vertiport op. To ensure that aircraft is available at vertiport op, additional time for rebalancing may be required. Let Ci denote an upper bound on the additional rebalancing time. If ri is symmetric, then Ci = max p∈P: r p i >0 max{ Tp τc + kc − |A|p r p i − kτ , 0} r p i (K∗ i + kτ ) |A|p = Ci(K∗ i ) + C ′ i , (C.2) where Ci(K∗ i ) is the part that depends on K∗ i and C ′ i is the part that does not depend on K∗ i . The right hand side of (C.2) is the time it takes for an aircraft from the opposite direction q = (dp, op) to reach vertiport op and gets recharged before it is ready for takeoff, times r p i (K∗ i + kτ )/|A|p, which is the total number of iterations needed for ri to be used for K∗ i + kτ time steps. On the other hand, if ri is non-symmetric, then Ci = K∗ i + 2 max p∈P: r p i >0 max{ Tp τc + kc − |A|p r p i − kτ , Tp τc } r p i (K∗ i + kτ ) |A|p = Ci(K∗ i ) + C ′ i . (C.3) The right hand side of (C.3) is calculated similar to (C.2). Note that if ri is symmetric, aircraft can be rebalanced while ri is active. However, if ri is not symmetric, it must be deactivated first before all aircraft can be rebalanced to vertiport op. 3. Once step 2 is completed and the airspace is empty, repeat steps 1 and 2 for another vector in R. The amount of time it takes for the airspace to become empty at the end of step 2 is at most T /τc time steps. Once each service vector ri ∈ R with K∗ i > 0 have been activated, P|R| i=1 r p i K∗ i requests will 128 be serviced for each O-D pair p ∈ P. From the constraint of the LP (C.1), P|R| i=1 riK∗ i ≥ Q(tk+1), i.e., all the requests for the (k + 1)-th cycle will be serviced and the cycle ends. By combining the time each of the above steps takes, it follows that Tcyc(k + 1) ≤ X |R| i=1 (1 + ci)K∗ i + X |R| i=1 C ′ i + |R| τc (|A|T + T), (C.4) where ci = Ci(K∗ i )/K∗ i , which does not depend on K∗ i , and Tcyc(k + 1) = (tk+2 − tk+1)/τc. Without loss of generality, we assume that the ordering by which ri ’s are chosen at each cycle are fixed and, when a cycle ends, the next cycle starts once the airspace becomes empty and the start time of is a multiple of τc. Finally, we assume that the initial distribution of aircraft before each ri is activated takes |A|T /τc time steps. With these assumptions, we can cast the network as a discrete-time Markov chain with the state {Q(tk)}k≥1. Since the state Q(tk) = 0 is reachable from all other states, and P (Q(tk+1) = 0 | Q(tk) = 0) > 0, the chain is irreducible and aperiodic. Consider the function f : Z |P| + → [0, ∞) V (Q(tk)) = T 2 cyc(k), where Z |P| + is the set of |P|-tuples of non-negative integers. Note that Tcyc(k) is a non-negative integer from our earlier assumption that the cycle start times are a multiple of τc. We let V (Q(tk)) ≡ V (tk) for brevity. We next show that lim sup n→∞ E " Tcyc(k + 1) Tcyc(k) 2 Tcyc(k) = n # < 1. (C.5) 129 To show (C.5), let Tcyc(k) = n, and let Ap(tk, tk+1) be the cumulative number of trip requests for the O-D pair p ∈ P during the time interval [tk, tk+1). Note that Qp(tk+1) = Ap(tk, tk+1), which implies from the strong law of large numbers that, with probability one, limn→∞ Qp(tk+1) n = λp. By the assumption of the theorem, λ ∈ D◦ 1 . Hence, with probability one, there exists N′ > 0 such that for all n > N′ we have Q(tk+1)/n ∈ D◦ 1 . Since D◦ 1 is an open set, for a given n > N′ , there exists nonnegative x1, x2, . . . , x|R| with P|R| i=1 xi < 1 such that Q(tk+1)/n < P|R| i=1 rixi/(1 + ci), or equivalently, Q(tk+1) < P|R| i=1 rinxi/(1+ci). Thus, if we let Ki := nxi/(1+ci), i ∈ [|R|], then, Q(tk+1) < P|R| i=1 riKi , which implies that Ki ’s are a feasible solution to the LP (C.1). Moreover, P|R| i=1(1 + ci)Ki < n. Therefore, from (C.4) and with probability one, it follows for all n > N′ that Tcyc(k + 1) ≤ X |R| i=1 (1 + ci)K∗ i + X |R| i=1 C ′ i + |R| τc (|A|T + T) < n + X |R| i=1 C ′ i + |R| τc (|A|T + T), which in turn implies, with probability one, that lim sup n→∞ Tcyc(k + 1) n 2 < 1. (C.6) Finally, since the number of trip requests for each O-D pair is at most 1 per τc minutes, the sequence {Tcyc(k+ 1)/n}∞ n=1 is upper bounded by an integrable function. Hence, from (C.6) and the Fatou’s Lemma (C.5) follows. 130 We will now use (C.5) to show that the network is under-saturated. Note that (C.5) implies that there exists δ ∈ (0, 1) and N such that for all n > N we have E " Tcyc(k + 1) Tcyc(k) 2 Tcyc(k) = n # < 1 − δ, which in turn implies that E T 2 cyc(k + 1) − T 2 cyc(k) Tcyc(k) > N < −δT2 cyc(k). Furthermore, Qp(tk) ≤ Tcyc(k) ≤ T 2 cyc(k)for all p ∈ P, where the first inequality follows from the fact that tk+1 − tk ≥ Qp(tk)τc for any O-D pair p ∈ P. Therefore, E T 2 cyc(k + 1) − T 2 cyc(k) Tcyc(k) > N < −δ∥Q(tk)∥∞, where ∥Q∥∞ = maxp Qp. Finally, if Tcyc(k) ≤ N, then Tcyc(k+1) ≤ 2N|P|(T /τc+kτ ) =: b. Therefore, E T 2 cyc(k + 1) Tcyc(k) ≤ N ≤ b 2 + T 2 cyc(k) − δ∥Q(tk)∥∞, where we have used δ∥Q(tk)∥∞ ≤ ∥Q(tk)∥∞ ≤ T 2 cyc(k). Combining all the previous steps gives E [f(tk+1) − f(tk)| Q(tk)] ≤ −δ∥Q(tk)∥∞ + b 21B, where B = {Q(tk) : f(tk) ≤ N2} (a finite set). From this and the well-known Foster-Lyapunov drift criterion [62, Theorem 14.0.1], it follows that lim supt→∞ E [Qp(t)] < ∞ for all p ∈ P, i.e., the network is under-saturated. 131 C.2 Proof of Theorem 8 We prove by contradiction. Suppose that some conflict-free policy π keeps the network under-saturated but λ /∈ D. Then, for any non-negative x1, x2, . . . , x|R| with P|R| i=1 xi ≤ 1, there exists some O-D pair p ∈ P such that λp > P|R| i=1 r p i xi . Without loss of generality, we may assume that whenever the service vector ri becomes active, it remains active for a time interval that is a multiple of τ . Given k ∈ N0, let tk := kτ , and let xi(tk) be the proportion of time that the service vector ri has been active under the policy π up to time tk. Then, xi := lim supk→∞ xi(tk) ≥ 0 for all i ∈ [|R|] and P|R| i=1 xi ≤ 1. Therefore, there exists p ∈ P such that λp > P|R| i=1 r p i xi . Note that when the service vector ri is active, the trip requests for O-D pair p are serviced at the rate of at most r p i /τ . Hence, the number of trip requests for O-D pair p that have been serviced by ri up to time tk is at most r p i xi(tk)tk/τ . Let Ap(tk) ≡ Ap(0, tk) be the cumulative number of flight requests for the O-D pair p up to time tk. We have Qp(tk) ≥ Qp(0) + Ap(tk) − X |R| i=1 r p i xi(tk)tk τ , which implies Qp(tk) tk ≥ Qp(0) tk + Ap(tk) tk − 1 τ X |R| i=1 r p i xi(tk). By letting k → ∞, it follows from the strong law of large numbers that, with probability one, lim inf k→∞ Qp(tk) k ≥ λp − X |R| i=1 r p i xi . Since λp > P|R| i=1 r p i xi , then, with probability one, lim infk→∞ Qp(tk)/k is bounded away from zero. Hence, lim inf k→∞ Qp(tk) = ∞. 132 Combining this with Fatou’s Lemma imply that the expected number of flight requests for the O-D pair p grows unbounded. This contradicts the network being under-saturated. 133
Abstract (if available)
Abstract
Traffic congestion is one of the fundamental challenges in modern transportation systems, which occurs due to an imbalance between demand and infrastructural resources. To alleviate congestion, traffic controllers are often employed. In ground transportation, traffic controllers are traditionally designed using macroscopic traffic flow models, which are obtained by spatio-temporal averaging of vehicle interactions. However, these models lack the granularity to account for safety protocols under different connectivity and automation scenarios, which could lead to suboptimal performance. The first objective of this thesis is to systematically design and analyze traffic controllers for ground transportation at the microscopic level, where such nuances can be better addressed. Another approach to mitigate traffic congestion is by introducing new modes of transportation, such as Urban Air Mobility (UAM), which uses urban airspace for on-demand mobility. However, without proper traffic control, UAM systems may face the same congestion challenges as ground transportation. The secondary objective of this thesis is to design and analyze traffic control protocols for on-demand UAM systems at the microscopic level.
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Microscopic traffic control: theory and practice
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