Convexity and robustness of dynamic traffic assignment and freeway network control |
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| Institution: | 1. Department of Automatic Control, Lund University, Sweden;2. Volvo Cars Corporation, Gothenburg, Sweden;3. Sonny Astani Department of Civil and Environmental Engineering at the University of Southern California, Los Angeles, CA, United States;1. State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China;2. Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong;3. Key Laboratory of Road and Traffic Engineering, Tongji University, Shanghai 201804, China;1. School of Economics and Management, Beijing Jiaotong University, Beijing 100044, PR China;2. School of Economics and Management, Beihang University, Beijing 100191, PR China;1. School of Transportation Engineering, Hefei University of Technology, Hefei 230009, China;2. Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong;3. School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China;4. School of Economics and Management, Beihang University, Beijing 100191, China;1. Partners for Advanced Transportation Technologies (PATH), University of California, Berkeley, 3 McLaughlin Hall, Berkeley, CA, 94720-1720, USA;2. Department of Mechanical Engineering, University of California, Berkeley, 6141 Etcheverry Hall, Berkeley, CA, 94720-1740, USA |
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| Abstract: | We study the use of the System Optimum (SO) Dynamic Traffic Assignment (DTA) problem to design optimal traffic flow controls for freeway networks as modeled by the Cell Transmission Model, using variable speed limit, ramp metering, and routing. We consider two optimal control problems: the DTA problem, where turning ratios are part of the control inputs, and the Freeway Network Control (FNC), where turning ratios are instead assigned exogenous parameters. It is known that relaxation of the supply and demand constraints in the cell-based formulations of the DTA problem results in a linear program. However, solutions to the relaxed problem can be infeasible with respect to traffic dynamics. Previous work has shown that such solutions can be made feasible by proper choice of ramp metering and variable speed limit control for specific traffic networks. We extend this procedure to arbitrary networks and provide insight into the structure and robustness of the proposed optimal controllers. For a network consisting only of ordinary, merge, and diverge junctions, where the cells have linear demand functions and affine supply functions with identical slopes, and the cost is the total traffic volume, we show, using the Pontryagin maximum principle, that variable speed limits are not needed in order to achieve optimality in the FNC problem, and ramp metering is sufficient. We also prove bounds on perturbation of the controlled system trajectory in terms of perturbations in initial traffic volume and exogenous inflows. These bounds, which leverage monotonicity properties of the controlled trajectory, are shown to be in close agreement with numerical simulation results. |
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