Fakultät für Maschinenbau und Sicherheitstechnik

Car-following models

Car-following models are microscopic traffic flow modelling approaches developed since the 1950s. The models are continuous in time and space. They are first or second-order differential equation systems describing the speed or the acceleration of a vehicle according to the distance spacing and speed of the predecessor. Car-following models are notably used for the longitudinal motion planning of automated vehicles (adaptive cruise control systems).

ATG model

The time gap is a fundamental variable for the modelling of car-following situations. Classical regulation policies recommend keeping a constant time gap during a pursuit (ranging between 0.8 and 2.2 s, see the ISO 15622 Standard for ACC systems). In the adaptive time gap (ATG) model, the time gap is relaxed to a safety time gap parameter.  The ATG model and its extensions can describe stable and collision-free dynamics.

CFOV model

The optimal velocity (OV) model is a famous car-following model introduced in 1995 by Bando et al. Despite its simplicity, the model describes stop-and-go dynamics for parameter fine-tuning. Unfortunately, unrealistic collisions and backward motion are not to exclude. The collision-free optimal velocity (CFOV) model is a minimal first-order version of the optimal velocity model. The CFOV model describes stop-and-go dynamics for large reaction time as the OV model does, but without the drawback of collisions or backward movement (see the simulations #1, #2, #3). Derivation of the microscopic model results in a parabolic convection-diffusion partial differential macroscopic equation.


M. Bando, K. Hasebe, A. Nakayama, A. Shibata, Y. Sugiyama. Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E, 51(2), 1035 (1995).

M. Treiber, A. Hennecke, D. Helbing. Congested traffic states in empirical observations and microscopic simulations. Phys. Rev. E, 62(2), 1805 (2000).

A. Tordeux, S. Lassarre, M. Roussignol. An Adaptive Time Gap Car-Following Model. Transp. Res. B-Meth. 44, 1115 (2010).

R. Monneau, M. Roussignol, A. Tordeux. Invariance and homogenization of an adaptive time gap car-following model. NoDEA Nonlinear Differential Equations Appl. 21, 491 (2014).

P. Khound, P. Will, A. Tordeux, F. Gronwald. Extending the adaptive time gap car-following model to enhance local and string stability for adaptive cruise control systems. J. Intell. Transp. Syst. (2021).

A. Tordeux, A. Seyfried. Collision-free nonuniform dynamics within continuous optimal velocity models. Phys. Rev. E 90, 042812 (2014).

A. Tordeux, G. Costeseque, M. Herty, A. Seyfried. From traffic and pedestrian follow-the-leader models with reaction time to first order convection-diffusion flow models. SIAM J. Appl. Math. 78, 63 (2018).

NetLogo online simulation module of car-following models


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