Tolling scheme for traffic flow networks: modelling and control of positive systems

This article addresses modelling and control of traffic flow networks for congestion mitigation. We first model the traffic flow of an urban area and derive a state-space model composed of multiple compartments. Each compartment expresses the dynamics of the traffic demand that depends on drivers' decision-making on route choices influenced by tolling. Two practical properties are found in the proposed state space model: one is the positivity of the model, and the other is the conservation law in the number of vehicles. Based on the positivity of the model, we next propose the optimal tolling design aiming at the peak demand mitigation, which is expressed by nonlinear programming. We transform the nonlinear programming into a linear one, which is numerically tractable. Finally, we show that the proposed tolling design mitigates the peak demand in designated compartments in a numerical experiment and present a demonstration of the proposed control system.