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).
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.
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.
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