Continuum Traffic Equations from Microscopic Car-Following Models

We present a method to derive macroscopic traffic models through a direct coarse graining procedure from microscopic car-following models. The method is similar to that developed by Helbing and coworkers to derive the gas-kinetic theory (GKT), but our resulting macroscopic equations are local and contain no empirical parameters other than those inherited from the microscopic models. Fluctuation effects are not included. When we work in the so-called linear regime, the derived momentum equation for the optimal velocity model consists of the relaxation term, the anticipation term, and the diffusion term, as in the previous local continuum model of Kerner and Konhäuser (KK). However, the origin of the last two terms are different and their density dependent coefficients are fixed from the microscopic behavior of individual vehicles. Properties of the resulting macroscopic models are compared with those of the optimal velocity model through numerical simulations, and reasonable agreement is found although there are deviations in the quantitative level. The phase diagram in the presence of an on-ramp also shows qualitatively similar features as in the case of the GKT and KK models.