Equilibrium Manifolds in 2D Fluid Traffic Models

Abstract

The main goal of this paper is to analytically find a steady-state in a large-scale urban traffic network with known and constant demand and supply on its boundaries. Traffic dynamics are given by a continuous two-dimensional macroscopic model, where state corresponds to the vehicle density evolving in a 2D plane. Thereby, the flux magnitude is given by the space-dependent fundamental diagram and the flux direction depends on the underlying network topology. In order to find a steady-state, we use the coordinate transformation such that the 2D equation can be rewritten as a parametrized set of 1D equations. This technique allows us to obtain the curves along which the traffic flow evolves, which are essentially the integral curves of the flux field constructed from the network geometry. The results are validated by comparing the obtained steady-state with the one estimated by using a microsimulator.