Microscopic traffic flow model

Car-following models

Also known as time-continuous models, all car-following models have in common that they are defined by ordinary differential equations describing the complete dynamics of the vehicles' positions x_\alpha and velocities v_\alpha. It is assumed that the input stimuli of the drivers are restricted to their own velocity v_\alpha, the net distance (bumper-to-bumper distance) s_\alpha = x_{\alpha-1} - x_\alpha - \ell_{\alpha-1} to the leading vehicle \alpha-1 (where \ell_{\alpha-1} denotes the vehicle length), and the velocity v_{\alpha-1} of the leading vehicle. The equation of motion of each vehicle is characterized by an acceleration function that depends on those input stimuli:

:\ddot{x}\alpha(t) = \dot{v}\alpha(t) = F(v_\alpha(t), s_\alpha(t), v_{\alpha-1}(t), s_{\alpha-1}(t))

In general, the driving behavior of a single driver-vehicle unit \alpha might not merely depend on the immediate leader \alpha-1 but on the n_a vehicles in front. The equation of motion in this more generalized form reads:

:\dot{v}\alpha(t) = f(x\alpha(t), v_\alpha(t), x_{\alpha-1}(t), v_{\alpha-1}(t), \ldots, x_{\alpha-n_a}(t), v_{\alpha-n_a}(t))

Examples of car-following models

• Optimal velocity model (OVM)
• Velocity difference model (VDIFF)
• Wiedemann model (1974)
• Gipps' model (Gipps, 1981)
• Intelligent driver model (IDM, 1999)
• DNN based anticipatory driving model (DDS, 2021)
• Rakha-Pasumarthy-Adjerid model (RPA model)
• Fadhloun-Rakha model (FR model)

Cellular automaton models

Cellular automaton (CA) models use integer variables to describe the dynamical properties of the system. The road is divided into sections of a certain length \Delta x and the time is discretized to steps of \Delta t. Each road section can either be occupied by a vehicle or empty and the dynamics are given by updated rules of the form:

:v_\alpha^{t+1} = f(s_\alpha^t, v_\alpha^t, v_{\alpha-1}^t, \ldots) :x_\alpha^{t+1} = x_\alpha^t + v_\alpha^{t+1}\Delta t

(the simulation time t is measured in units of \Delta t and the vehicle positions x_\alpha in units of \Delta x).

The time scale is typically given by the reaction time of a human driver, \Delta t = 1 \text{s}. With \Delta t fixed, the length of the road sections determines the granularity of the model. At a complete standstill, the average road length occupied by one vehicle is approximately 7.5 meters. Setting \Delta x to this value leads to a model where one vehicle always occupies exactly one section of the road and a velocity of 5 corresponds to 5 \Delta x/\Delta t = 135 \text{km/h}, which is then set to be the maximum velocity a driver wants to drive at. However, in such a model, the smallest possible acceleration would be \Delta x/(\Delta t)^2 = 7.5 \text{m}/\text{s}^2 which is unrealistic. Therefore, many modern CA models use a finer spatial discretization, for example \Delta x = 1.5 \text{m}, leading to a smallest possible acceleration of 1.5 \text{m}/\text{s}^2.

Although cellular automaton models lack the accuracy of the time-continuous car-following models, they still have the ability to reproduce a wide range of traffic phenomena. Due to the simplicity of the models, they are numerically very efficient and can be used to simulate large road networks in real-time or even faster.

Examples of cellular automaton models

• Rule 184
• Biham–Middleton–Levine traffic model
• Nagel–Schreckenberg model (NaSch, 1992)

References

References

1. Gipps, P. G.. (1981). "A behavioural car-following model for computer simulation". Transportation Research Part B: Methodological.
2. (August 2000). "Congested traffic states in empirical observations and microscopic simulations". Physical Review E.
3. (July 2021). "A DNN Based Driving Scheme for Anticipatory Car Following Using Road-Speed Profile".
4. Rakha, Hesham. (April 2009). "A simplified behavioral vehicle longitudinal motion model". Transportation Letters.
5. Fadhloun, Karim. (March 2020). "A novel vehicle dynamics and human behavior car-following model: Model development and preliminary testing". International Journal of Transportation Science and Technology.