Microscopic traffic flow model

Microscopic traffic flow models are a class of scientific models of vehicular traffic dynamics.

In contrast, to macroscopic models, microscopic traffic flow models simulate single vehicle-driver units, so the dynamic variables of the models represent microscopic properties like the position and velocity of single vehicles.

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 [math]\displaystyle{ x_\alpha }[/math] and velocities [math]\displaystyle{ v_\alpha }[/math]. It is assumed that the input stimuli of the drivers are restricted to their own velocity [math]\displaystyle{ v_\alpha }[/math], the net distance (bumper-to-bumper distance) [math]\displaystyle{ s_\alpha = x_{\alpha-1} - x_\alpha - \ell_{\alpha-1} }[/math] to the leading vehicle [math]\displaystyle{ \alpha-1 }[/math] (where [math]\displaystyle{ \ell_{\alpha-1} }[/math] denotes the vehicle length), and the velocity [math]\displaystyle{ v_{\alpha-1} }[/math] of the leading vehicle. The equation of motion of each vehicle is characterized by an acceleration function that depends on those input stimuli:

[math]\displaystyle{ \ddot{x}_\alpha(t) = \dot{v}_\alpha(t) = F(v_\alpha(t), s_\alpha(t), v_{\alpha-1}(t), s_{\alpha-1}(t)) }[/math]

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

[math]\displaystyle{ \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)) }[/math]

Examples of car-following models

Optimal velocity model (OVM)
Velocity difference model (VDIFF)
Wiedemann model (1974)
Gipps' model (Gipps, 1981)^[1]
Intelligent driver model (IDM, 1999)^[2]
DNN based anticipatory driving model (DDS, 2021)^[3]

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 [math]\displaystyle{ \Delta x }[/math] and the time is discretized to steps of [math]\displaystyle{ \Delta t }[/math]. Each road section can either be occupied by a vehicle or empty and the dynamics are given by updated rules of the form:

[math]\displaystyle{ v_\alpha^{t+1} = f(s_\alpha^t, v_\alpha^t, v_{\alpha-1}^t, \ldots) }[/math]

[math]\displaystyle{ x_\alpha^{t+1} = x_\alpha^t + v_\alpha^{t+1}\Delta t }[/math]

(the simulation time [math]\displaystyle{ t }[/math] is measured in units of [math]\displaystyle{ \Delta t }[/math] and the vehicle positions [math]\displaystyle{ x_\alpha }[/math] in units of [math]\displaystyle{ \Delta x }[/math]).

The time scale is typically given by the reaction time of a human driver, [math]\displaystyle{ \Delta t = 1 \text{s} }[/math]. With [math]\displaystyle{ \Delta t }[/math] 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 [math]\displaystyle{ \Delta x }[/math] 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 [math]\displaystyle{ 5 \Delta x/\Delta t = 135 \text{km/h} }[/math], 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 [math]\displaystyle{ \Delta x/(\Delta t)^2 = 7.5 \text{m}/\text{s}^2 }[/math] which is unrealistic. Therefore, many modern CA models use a finer spatial discretization, for example [math]\displaystyle{ \Delta x = 1.5 \text{m} }[/math], leading to a smallest possible acceleration of [math]\displaystyle{ 1.5 \text{m}/\text{s}^2 }[/math].

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

References

↑ Gipps, P. G. (1981). "A behavioural car-following model for computer simulation". Transportation Research Part B: Methodological 15 (2): 105–111. doi:10.1016/0191-2615(81)90037-0. ISSN 0191-2615. https://dx.doi.org/10.1016/0191-2615%2881%2990037-0. Retrieved 2022-02-17.
↑ Treiber, null; Hennecke, null; Helbing, null (August 2000). "Congested traffic states in empirical observations and microscopic simulations". Physical Review E 62 (2 Pt A): 1805–1824. doi:10.1103/physreve.62.1805. ISSN 1063-651X. PMID 11088643. Bibcode: 2000PhRvE..62.1805T.
↑ Isha, Most. Kaniz Fatema; Shawon, Md. Nazirul Hasan; Shamim, Md.; Shakib, Md. Nazmus; Hashem, M.M.A.; Kamal, M.A.S. (July 2021). "A DNN Based Driving Scheme for Anticipatory Car Following Using Road-Speed Profile". 2021 IEEE Intelligent Vehicles Symposium (IV). pp. 496–501. doi:10.1109/IV48863.2021.9575314.

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[1] Gipps, P. G. (1981). "A behavioural car-following model for computer simulation". Transportation Research Part B: Methodological 15 (2): 105–111. doi:10.1016/0191-2615(81)90037-0. ISSN 0191-2615. https://dx.doi.org/10.1016/0191-2615%2881%2990037-0. Retrieved 2022-02-17.

[2] Treiber, null; Hennecke, null; Helbing, null (August 2000). "Congested traffic states in empirical observations and microscopic simulations". Physical Review E 62 (2 Pt A): 1805–1824. doi:10.1103/physreve.62.1805. ISSN 1063-651X. PMID 11088643. Bibcode: 2000PhRvE..62.1805T.

[3] Isha, Most. Kaniz Fatema; Shawon, Md. Nazirul Hasan; Shamim, Md.; Shakib, Md. Nazmus; Hashem, M.M.A.; Kamal, M.A.S. (July 2021). "A DNN Based Driving Scheme for Anticipatory Car Following Using Road-Speed Profile". 2021 IEEE Intelligent Vehicles Symposium (IV). pp. 496–501. doi:10.1109/IV48863.2021.9575314.

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