Minimal parameter formulations of the dynamic user equilibrium using macroscopic urban models: Freeway vs city streets revisited

This paper investigates the dynamic user equilibrium (DUE) on a single origin-destination pair with two alternative routes, a freeway with a fixed capacity and the surrounding city-streets network, modeled with a network macroscopic fundamental diagram (NMFD). We find using suitable transformations that only a single network parameter is required to characterize the DUE solution, the freeway to NMFD capacity ratio. We also show that the stability and convergence properties of this system are captured by the constant demand case, which corresponds to an autonomous dynamical system that admits analytical solutions. This solution is characterized by two critical accumulation values that determine if the steady state is in free-flow or gridlock, depending on the initial accumulation. Additionally, we also propose a continuum approximation to account for the spatial evolution of congestion, by including variable trip length and variable NMFD coverage area in the model. It is found that gridlock cannot happen and that the steady-state solution is independent of surface network parameters. These parameters do affect the rate of convergence to the steady-state solution, but convergence rates appear virtually identical when time is expressed in units of the NMFD free-flow travel time.