Discrete-time day-to-day dynamic congestion pricing scheme considering multiple equilibria

In this study, we focus on the discrete-time day-to-day dynamic congestion pricing scheme which varies the toll on a day-to-day basis and aims to drive the traffic system to a given objective traffic equilibrium state. As is well known, due to the asymmetric nature of the travel cost functions, multiple equilibria exist. In this case, without external force, the traffic system cannot converge to the traffic equilibrium state as desired by traffic management through a day-to-day adjustment process if the initial traffic state does not fall into its attraction domain (Bie and Lo, 2010). Therefore, it is imperative for traffic management to propose a traffic control measure to ensure the desired traffic state can be achieved regardless of the initial traffic state. Previous studies on the day-to-day dynamic congestion pricing, either worked on continues-time day-to-day pricing scheme, or took the form of discrete-time day-to-day pricing scheme but did not guarantee the convergence to the desired objective traffic state for the cases when multiple traffic equilibria exist. Both are undesirable. This study aims to develop a discrete-time day-to-day pricing scheme so as to direct the traffic evolution to reach the desired equilibrium from any initial traffic state when multiple traffic equilibria exist. Based on the very general formulation of day-to-day traffic dynamics model, we present a general formulation of such day-to-day pricing schemes and propose a method to obtain one specific road pricing scheme. Moreover, we present rigorous proofs and numerical tests to verify the proposed pricing scheme.