Second-order continuum traffic flow model

A second-order traffic flow model is derived from microscopic equations and is compared to existing models. In order to build in different driver characteristics on the microscopic level, we exploit the idea of an additional phase-space variable, called the desired velocity originally introduced by Paveri-Fontana [Trans. Res. 9, 225 (1975)]. By taking the moments of Paveri-Fontana's Boltzmann-like ansatz, a hierarchy of evolution equations is found. This hierarchy is closed by neglecting cumulants of third and higher order in the cumulant expansion of the distribution function, thus leading to Euler-like traffic equations. As a consequence of the desired velocity, we find dynamical quantities, which are the mean desired velocity, the variance of the desired velocity, and the covariance of actual and desired velocity. Through these quantities an alternative explanation for the onset of traffic clusters can be given, i.e., a spatial variation of the variance of the desired velocity can cause the formation of a traffic jam. Furthermore, by taking into account the finite car length, Paveri-Fontana's equation is generalized to the high-density regime eventually producing corrections to the macroscopic equations. The relevance of the present dynamic quantities is demonstrated by numerical simulations.