Modeling traffic dynamics in periphery-downtown urban networks combining Vickrey's theory with Macroscopic Fundamental Diagram: user equilibrium, system optimum, and cordon pricing

This paper constructs a dynamic model for a periphery-downtown urban network combining Vickrey's theory with Macroscopic Fundamental Diagram (MFD) to capture interactions between the peripheral traffic and the downtown, and to examine how travelers on different corridors compete for the downtown road resources in the context of dynamic user equilibrium. In no-toll equilibrium if existing, when there exists no congestion delay in the downtown area, the departures for each corridor are regulated only by its bottleneck's capacity on the periphery of the downtown area, as occurred in Vickrey's bottleneck model, whereas when the downtown accumulation goes beyond the critical value, the departures for each corridor are constrained not only by its bottleneck's capacity but also by the downtown congestion level, regulated by all corridors’ departures in turn. We show that under specific assumptions, at the system optimum, the entrances at the boundary of the downtown area should be running at their full capacities before the onset of the downtown congestion, and the downtown network should be operating upon the onset of the downtown congestion at the critical accumulation with maximum production and highest traveling speed. Two optimal time-varying cordon pricing schemes (entrance-independent and entrance-dependent respectively) are developed to minimize the total social cost, which includes the queuing time costs concerning all corridors, the moving time cost in the interior of the downtown area, and schedule delay cost. What's more, the two time-varying cordon tolls to support the system optimum are analyzed from practical views. In the end, analytical results are illustrated and verified with numerical experiments.