Optimal perimeter control synthesis for two urban regions with aggregate boundary queue dynamics

Perimeter control policies for urban regions with Macroscopic Fundamental Diagram (MFD) modeling have been presented in previous works. The control policies might meter the number of transferring vehicles from one region to another, resulting in queueing vehicles at regional boundaries. Concentrated vehicles at boundaries might affect the existence of well-defined MFDs. Most previous works neglect the effect of the boundary concentrated vehicles on the traffic flow dynamics, and do not explicitly consider their effect on the perimeter control policy. This paper introduces a new MFD-based model for two-region networks with aggregate boundary queue dynamics. The dynamic flow characteristics for the two urban regions are modeled by the MFD functions, while aggregate boundary queue dynamics for both regions are modeled by input-output balance differential equations. Maximum lengths are imposed on the aggregate boundary queues, that aim at maintaining the existence of well-defined MFDs and their dynamics. Based on the developed model, the optimal control policy to maximize the total network throughput is found. Analytical solutions for the optimal perimeter control problem, with constrained perimeter control inputs and constrained lengths of aggregate boundary queues, are derived. The optimal synthesis for principal cases are found and verified by numerical tests. The numerical results demonstrate the effect of aggregate boundary queues on the optimal perimeter control policy.