Convexity and Robustness of Dynamic Network Traffic Assignment for Control of Freeway Networks

We study System Optimum Dynamic Traffic Assignment (SO-DTA) for realistic traffic dynamics controlled by variable speed limits, ramp metering, and routing controls. We consider continuous-time cell-based Dynamic Network Loading models that include the Cell Transmission Model with FIFO rule at the diverge junctions as well as non-FIFO diverge rules. We consider SO-DTA formulations in which the total inflow into and the total outflow from the cells are independently constrained by concave supply and demand functions, respectively, thus preserving convexity. We design open-loop controllers that guarantee that the optimal solutions are feasible with respect to realistic traffic dynamics, and we develop this methodology for two variants of the SO-DTA problem, one of which accounts for exogenous turning ratios. Robustness of the system trajectory under the proposed controllers is evaluated by deriving bounds on the deviations induced by perturbations of the initial condition and of the external inflows.