Microscopic traffic flow model

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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 x α {\displaystyle x_{\alpha }} and velocities v α {\displaystyle v_{\alpha }} . It is assumed that the input stimuli of the drivers are restricted to their own velocity v α {\displaystyle v_{\alpha }} , the net distance (bumper-to-bumper distance) s α = x α − 1 − x α − ℓ α − 1 {\displaystyle s_{\alpha }=x_{\alpha -1}-x_{\alpha }-\ell _{\alpha -1}} to the leading vehicle α − 1 {\displaystyle \alpha -1} (where ℓ α − 1 {\displaystyle \ell _{\alpha -1}} denotes the vehicle length), and the velocity v α − 1 {\displaystyle 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:

x ¨ α ( t ) = v ˙ α ( t ) = F ( v α ( t ) , s α ( t ) , v α − 1 ( t ) , s α − 1 ( t ) ) {\displaystyle {\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 α {\displaystyle \alpha } might not merely depend on the immediate leader α − 1 {\displaystyle \alpha -1} but on the n a {\displaystyle n_{a}} vehicles in front. The equation of motion in this more generalized form reads:

v ˙ α ( t ) = f ( x α ( t ) , v α ( t ) , x α − 1 ( t ) , v α − 1 ( t ) , … , x α − n a ( t ) , v α − n a ( t ) ) {\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))}

Examples of car-following models

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 Δ x {\displaystyle \Delta x} and the time is discretized to steps of Δ t {\displaystyle \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 α t + 1 = f ( s α t , v α t , v α − 1 t , … ) {\displaystyle v_{\alpha }^{t+1}=f(s_{\alpha }^{t},v_{\alpha }^{t},v_{\alpha -1}^{t},\ldots )}
x α t + 1 = x α t + v α t + 1 Δ t {\displaystyle x_{\alpha }^{t+1}=x_{\alpha }^{t}+v_{\alpha }^{t+1}\Delta t}

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

The time scale is typically given by the reaction time of a human driver, Δ t = 1 s {\displaystyle \Delta t=1{\text{s}}} . With Δ t {\displaystyle \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 Δ x {\displaystyle \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 Δ x / Δ t = 135 km/h {\displaystyle 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 Δ x / ( Δ t ) 2 = 7.5 m / s 2 {\displaystyle \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 Δ x = 1.5 m {\displaystyle \Delta x=1.5{\text{m}}} , leading to a smallest possible acceleration of 1.5 m / s 2 {\displaystyle 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

See also

References

  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. 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. arXiv:cond-mat/0002177. Bibcode:2000PhRvE..62.1805T. doi:10.1103/physreve.62.1805. ISSN 1063-651X. PMID 11088643. S2CID 1100293.
  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). 2021 IEEE Intelligent Vehicles Symposium (IV). pp. 496–501. doi:10.1109/IV48863.2021.9575314.