Navier-Stokes-like equations applicable to adaptive cruise control traffic flows

Abstract

Under the scenario in which, within a traffic flow, each vehicle is controlled by adaptive cruise control (ACC), and the macroscopic one-vehicle probability distribution function fits the Paveri-Fontana hypothesis, a set of reduced Paveri-Fontana equations considering the ACC effect is derived. With the set, by maximizing the specially defined informational entropy deviating from a certain reference homogeneous steady state, the Navier-Stokes-like equations considering ACC are introduced. For a homogeneous steady traffic flow in a single circular lane, when the steady velocity or density is perturbed along the lane, numerical simulations indicate that ACC-controlled vehicles require less time for re-equilibration than manually driven vehicles. The re-equilibrated steady densities for ACC and manually driven traffic flows are all close to the original values; the same is true for the re-equilibrated steady velocity for manually driven traffic flows. For ACC traffic flows, the re-equilibrated steady velocity may be higher or lower than the original value, depending upon a parameter ω (introduced to solve the distribution function of the reference steady state), and the headway time (introduced in ACC models). Also, the simulations indicate that only an appropriate parameter set can ensure the performance of ACC; otherwise, ACC may result in low traffic running efficiency, although traffic flow stability becomes better.