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 fine-tuning the parameters. 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.
Vehicular flows tend to describe stop-and-go waves for congested density levels (see the experiments by Sugiyama et al. or more recently by Stern et al.). Stop-and-go waves negatively impact road user safety and comfort and the environment. Indeed stop-and-go waves generate more energy consumption and pollutant emission than homogeneous streaming (see ref1).
Stop-and-go wave as phase transition
Stop-and-go waves are typical self-organized collective phenomena emerging from driver interactions. They are observed in pedestrian and bicycle flow as well. In traffic engineering, stop-and-go dynamics are explained by low reactivity of the drivers using non-linear car-following models describing instability (or even metastability) and phase transition (see, e.g., ref2, ref3, ref4). The phase transition generally induces capacity drop and hysteresis phenomena.
Noise-induced stop-and-go waves
Pioneer works with cellular automata models by Nagel, Schreckenberg, and Schadschneider have shown that stop-and-go dynamics can result from noise effects. In continuous-time, it turns out that colored noises such as Brownian noises given by the Ornstein-Uhlenbeck process allow describing stop-and-go phenomena as well. However, technically, the approach is different from a phase transition. No non-linearity, instability, and bifurcation are necessary. Stochastic effects initiate stop-and-go waves in the second order (ref5, ref6, ref7).
Stability properties of car-following models are expected features of adaptive cruise control systems in traffic engineering. Indeed, several recent experiments have shown that current ACC systems describe unstable behaviors (see ref1 and ref2 and references therein).
In the local stability analysis (also platoon stability), some vehicles follow a leader with assigned speed. The local stability is called over-damped when the convergence is oscillation-free. Such a feature allows ensuring (at least partially) the absence of oscillations in the dynamics.
The global stability analysis (also string stability) tackles vehicles on an infinite line or with periodic boundary conditions. Global stability conditions contain inductive and convective perturbations that may locally vanish. They are, in general, more restrictive than local conditions.
Local stability analysis generally relies on Laplace transform, while circulant properties or exponential ansatz allow tackling the global stability for finite periodic and infinite systems, respectively (see, e.g., ref3, ref4, ref5). The stability conditions are determined by solving the characteristic equation in the complex plane. Stable solutions only exist in the left half-plane with overdamped solutions living on the real semi-axis. The characteristic equations are polynomials for ODE systems while they are exponential-polynomials for delayed systems. Hopf bifurcation analysis can allow determining the stability conditions in this last case (see ref3). However, only few delayed models in the literature have explicit stability conditions.
- P. Khound, P. Will, A. Tordeux and F. Gronwald, "Unified framework for over-damped string stable adaptive cruise control systems" , Transportation Research Part C: Emerging Technologies, vol. 148, pp. 104039, 2023. Elsevier.
- R. Subaih, M. Maree, A. Tordeux and M. Chraibi, "Questioning the anisotropy of pedestrian dynamics: An empirical analysis with artificial neural networks" , Applied Sciences, vol. 12, no. 15, pp. 7563, 2022. MDPI.
- J. Cordes, M. Chraibi, A. Tordeux and A. Schadschneider, "Time-to-collision models for single-file pedestrian motion" , Collective Dynamics, vol. 6, pp. 1-10, 2022.
- P. Khound, P. Will, A. Tordeux and F. Gronwald, "The importance of the over-damped string stability criterion for a platooning control system" in 2022 26th International Conference on System Theory, Control and Computing (ICSTCC), 2022. pp. 1-6.
- J. Wang, M. Boltes, A. Seyfried, A. Tordeux, J. Zhang and W. Weng, "Experimental study on age and gender differences in microscopic movement characteristics of students" , Chinese Physics B, vol. 30, no. 9, pp. 098902, 2021. IOP Publishing.
- P. Khound, P. Will, A. Tordeux and F. Gronwald, "Extending the adaptive time gap car-following model to enhance local and string stability for adaptive cruise control systems" , Journal of Intelligent Transportation Systems, vol. 27, pp. 36-56, 2021. Taylor & Francis.
- M. Friesen, H. Gottschalk, B. Rüdiger and A. Tordeux, "Spontaneous wave formation in stochastic self-driven particle systems" , SIAM Journal on Applied Mathematics, vol. 81, no. 3, pp. 853-870, 2021. SIAM.
- A. Schadschneider and A. Tordeux, "Noise-induced stop-and-go dynamics in pedestrian single-file motion" , Collective Dynamics, vol. 5, pp. 356-363, 2020.
- A. Tordeux, A. Schadschneider and S. Lassarre, "Stop-and-go waves induced by correlated noise in pedestrian models without inertia" , Journal of traffic and transportation engineering (English edition), vol. 7, no. 1, pp. 52--60, 2020. Elsevier.
- A. Tordeux, J. Lebacque and S. Lassarre, "Robustness analysis of car-following models for full speed range ACC systems" in Traffic and Granular Flow 2019, Springer, 2020, pp. 571-581.
- J. Cordes, A. Schadschneider and A. Tordeux, "The trouble with 2nd order models or how to generate stop-and-go traffic in a 1st order model" in Traffic and Granular Flow 2019, Springer, 2020, pp. 45--51.
- J. Wang, M. Boltes, A. Seyfried, A. Tordeux, J. Zhang, V. Ziemer and W. Weng, "Influence of gender on the fundamental diagram and gait characteristics" in International Conference on Traffic and Granular Flow, 2019. pp. 225-234.
- A. Tordeux, A. Schadschneider and S. Lassarre, "Noise-induced stop-and-go dynamics" in International Conference on Traffic and Granular Flow, 2019. pp. 337-345.
- A. Tordeux, G. Costeseque, M. Herty and A. Seyfried, "From traffic and pedestrian follow-the-leader models with reaction time to first order convection-diffusion flow models" , SIAM Journal on Applied Mathematics, vol. 78, no. 1, pp. 63-79, 2018. SIAM.
- A. Tordeux, M. Chraibi, A. Schadschneider and A. Seyfried, "Influence of the number of predecessors in interaction within acceleration-based flow models" , Journal of Physics A: Mathematical and Theoretical, vol. 50, no. 34, pp. 345102, 2017. IOP Publishing.