Convexity and Robustness of Dynamic Traffic Assignment and Freeway Network Control

TL;DR

This paper analyzes the convexity and robustness of optimal traffic flow controls in freeway networks using the Cell Transmission Model, demonstrating conditions where ramp metering alone achieves optimality and providing bounds on system perturbations.

Contribution

It extends previous methods to arbitrary networks, proving ramp metering sufficiency for optimality and establishing robustness bounds for traffic control systems.

Findings

Ramp metering alone suffices for optimal control in certain network types.

Bounds on traffic system perturbations are derived and validated.

Variable speed limits are unnecessary for optimality in the studied network class.

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 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.