Convex Formulations 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 as special cases the Cell Transmission Model (CTM) with FIFO rule at the diverge junctions as well as non-FIFO diverge rules. While a straightforward consideration of traffic dynamics and control variables in the SO-DTA is known to lead to a non-convex program, and hence is computationally expensive for real-time applications, we consider SO-DTA formulations in which the total inflow into and the total outflow from the cells are independently constrained to be upper bounded by concave supply and demand functions, respectively, thus preserving convexity. We then design open-loop controllers that guarantee that the optimal solutions under the relaxed constraints are feasible with respect to realistic traffic dynamics. We develop this methodology for three variations of the SO-DTA problem that impose constraints on turning ratios to varying degrees.