Dynamic System Optimum Analysis of Multi-Region Macroscopic Fundamental Diagram Systems With State-Dependent Time-Varying Delays

This paper investigates the dynamic system optimum (DSO) problem with simultaneous route and departure time assignments for a general traffic network partitioned into multiple regions. Regional traffic congestion is modeled with a well-defined macroscopic fundamental diagram (MFD) mapping the trip completion rate to the vehicular accumulation. To overcome the limitation of inconsistent flow propagation between region boundaries and the corresponding travel time, the state-dependent regional travel time function is explicitly incorporated in the flow propagation of the conventional MFD dynamics. From a systems perspective, the traffic dynamics within a region can be regarded as a dynamic system with an endogenous time-varying delay depending on the system state. Equilibrium condition for the DSO problem is analytically derived through the lens of Pontryagin minimum principle and is compared against the static SO counterpart. The structure of path specific marginal cost is analyzed regarding the path travel cost and early-late penalty function. In contrast to existing analytical methods, the proposed method is applicable for general MFD systems without linearization of the MFD dynamics. Neither approximation of the equilibrium solution nor constant regional delay assumption is required. Numerical examples are conducted to illustrate the characteristics of DSO traffic equilibrium and the corresponding marginal cost together with other dynamic external costs.