In traffic flow modeling, the intelligent driver model (IDM) is a time-continuous car-following model for the simulation of freeway and urban traffic. It was developed by Treiber, Hennecke and Helbing in 2000 to improve upon results provided with other "intelligent" driver models such as Gipps' Model, which loose realistic properties in the deterministic limit.
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As a car-following model, the IDM describes the dynamics of the positions and velocities of single vehicles. For vehicle , denotes its position at time , and its velocity. Furthermore, gives the length of the vehicle. To simplify notation, we define the net distance , where refers to the vehicle directly in front of vehicle , and the velocity difference, or approaching rate, . For a simplified version of the model, the dynamics of vehicle are then described by the following two ordinary differential equations:
, , , , and are model parameters which have the following meaning:
The exponent is usually set to 4.
The acceleration of vehicle can be separated into a free road term and an interaction term:
This leads to a driving behavior that compensates velocity differences while trying not to brake much harder than the comfortable braking deceleration .
Let's assume a ring road with 50 vehicles. Then, vehicle 50 will follow vehicle 1. Initial speeds are given and since all vehicles are considered equal, vector ODEs are further simplified to:
For this example, the following values are given for the equation's parameters.
Description | Value |
---|---|
Desired velocity | 30 m/s |
Safe time headway | 1.5 s |
Maximum acceleration | 1.00 m/s2 |
Desired deceleraton | 3.00 m/s2 |
Acceleration exponent | 4 |
Jam distance | 2 m |
Vehicle length | 5 m |
The two ordinary differential equations are solved using Runge-Kutta methods of orders 1, 3, and 5 with the same time step, to show the effects of computational accuracy in the results.
This comparison shows that the IDM does not show extremely irrealistic properties such as negative velocities or vehicles sharing the same space even for from a low order method such as with the Euler's method (RK1). However, traffic wave propagation is not as accurately represented as in the higher order methods, RK3 and RK 5. These last two methods show no significant differences, which lead to conclude that a solution for IDM reaches acceptable results from RK3 upwards and no additional computational requirements would be needed. None-the-less, when introducing heterogeneous vehicles and both jam distance parameters, this observation could not suffice.
Treiber, Martin; Hennecke, Ansgar; Helbing, Dirk (2000), "Congested traffic states in empirical observations and microscopic simulations", Physical Review E 62 (2): 1805–1824, doi:10.1103/PhysRevE.62.1805