Intelligent driver model

As a car-following model, the IDM describes the dynamics of the positions and velocities of single vehicles. For vehicle ${\displaystyle \alpha }$, ${\displaystyle x_{\alpha }}$ denotes its position at time ${\displaystyle t}$, and ${\displaystyle v_{\alpha }}$ its velocity. Furthermore, ${\displaystyle l_{\alpha }}$ gives the length of the vehicle. To simplify notation, we define the net distance ${\displaystyle s_{\alpha }:=x_{\alpha -1}-x_{\alpha }-l_{\alpha -1}}$, where ${\displaystyle \alpha -1}$ refers to the vehicle directly in front of vehicle ${\displaystyle \alpha }$, and the velocity difference, or approaching rate, ${\displaystyle \Delta v_{\alpha }:=v_{\alpha }-v_{\alpha -1}}$. For a simplified version of the model, the dynamics of vehicle ${\displaystyle \alpha }$ are then described by the following two ordinary differential equations:

${\displaystyle {\dot {x}}_{\alpha }={\frac {\mathrm {d} x_{\alpha }}{\mathrm {d} t}}=v_{\alpha }}$

${\displaystyle {\dot {v}}_{\alpha }={\frac {\mathrm {d} v_{\alpha }}{\mathrm {d} t}}=a\,\left(1-\left({\frac {v_{\alpha }}{v_{0}}}\right)^{\delta }-\left({\frac {s^{*}(v_{\alpha },\Delta v_{\alpha })}{s_{\alpha }}}\right)^{2}\right)}$

${\displaystyle {\text{with }}s^{*}(v_{\alpha },\Delta v_{\alpha })=s_{0}+v_{\alpha }\,T+{\frac {v_{\alpha }\,\Delta v_{\alpha }}{2\,{\sqrt {a\,b}}}}}$

${\displaystyle v_{0}}$, ${\displaystyle s_{0}}$, ${\displaystyle T}$, ${\displaystyle a}$, and ${\displaystyle b}$ are model parameters which have the following meaning:

• desired velocity ${\displaystyle v_{0}}$: the velocity the vehicle would drive at in free traffic
• minimum spacing ${\displaystyle s_{0}}$: a minimum desired net distance. A car can't move if the distance from the car in the front is not at least ${\displaystyle s_{0}}$
• desired time headway ${\displaystyle T}$: the minimum possible time to the vehicle in front
• acceleration ${\displaystyle a}$: the maximum vehicle acceleration
• comfortable braking deceleration ${\displaystyle b}$: a positive number

The exponent ${\displaystyle \delta }$ is usually set to 4.

The acceleration of vehicle ${\displaystyle \alpha }$ can be separated into a free road term and an interaction term:

${\displaystyle {\dot {v}}_{\alpha }^{\text{free}}=a\,\left(1-\left({\frac {v_{\alpha }}{v_{0}}}\right)^{\delta }\right)\qquad {\dot {v}}_{\alpha }^{\text{int}}=-a\,\left({\frac {s^{*}(v_{\alpha },\Delta v_{\alpha })}{s_{\alpha }}}\right)^{2}=-a\,\left({\frac {s_{0}+v_{\alpha }\,T}{s_{\alpha }}}+{\frac {v_{\alpha }\,\Delta v_{\alpha }}{2\,{\sqrt {a\,b}}\,s_{\alpha }}}\right)^{2}}$

• Free road behavior: On a free road, the distance to the leading vehicle ${\displaystyle s_{\alpha }}$ is large and the vehicle's acceleration is dominated by the free road term, which is approximately equal to ${\displaystyle a}$ for low velocities and vanishes as ${\displaystyle v_{\alpha }}$ approaches ${\displaystyle v_{0}}$. Therefore, a single vehicle on a free road will asymptotically approach its desired velocity ${\displaystyle v_{0}}$.
• Behavior at high approaching rates: For large velocity differences, the interaction term is governed by ${\displaystyle -a\,(v_{\alpha }\,\Delta v_{\alpha })^{2}\,/\,(2\,{\sqrt {a\,b}}\,s_{\alpha })^{2}=-(v_{\alpha }\,\Delta v_{\alpha })^{2}\,/\,(4\,b\,s_{\alpha }^{2})}$.

This leads to a driving behavior that compensates velocity differences while trying not to brake much harder than the comfortable braking deceleration ${\displaystyle b}$.

• Behavior at small net distances: For negligible velocity differences and small net distances, the interaction term is approximately equal to ${\displaystyle -a\,(s_{0}+v_{\alpha }\,T)^{2}\,/\,s_{\alpha }^{2}}$, which resembles a simple repulsive force such that small net distances are quickly enlarged towards an equilibrium net distance.

Let's assume a ring road with 50 vehicles. Then, vehicle 1 will follow vehicle 50. Initial speeds are given and since all vehicles are considered equal, vector ODEs are further simplified to:

${\displaystyle {\dot {x}}={\frac {\mathrm {d} x}{\mathrm {d} t}}=v}$

${\displaystyle {\dot {v}}={\frac {\mathrm {d} v}{\mathrm {d} t}}=a\,\left(1-\left({\frac {v}{v_{0}}}\right)^{\delta }-\left({\frac {s^{*}(v,\Delta v)}{s}}\right)^{2}\right)}$

${\displaystyle {\text{with }}s^{*}(v,\Delta v)=s_{0}+v\,T+{\frac {v\,\Delta v}{2\,{\sqrt {a\,b}}}}}$

For this example, the following values are given for the equation's parameters, in line with the original calibrated model.

Variable	Description	Value
${\displaystyle v_{0}}$	Desired velocity	30 m/s
${\displaystyle T}$	Safe time headway	1.5 s
${\displaystyle a}$	Maximum acceleration	0.73 m/s2
${\displaystyle b}$	Comfortable Deceleration	1.67 m/s2
${\displaystyle \delta }$	Acceleration exponent	4
${\displaystyle s_{0}}$	Minimum 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.

Comparison of differential equation solutions for intelligent driver model using RK1,3,5

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. Nonetheless, when introducing heterogeneous vehicles and both jam distance parameters, this observation could not suffice.

• Gipps' model
• Newell's car-following model
• Microscopic traffic flow model
• List of Runge–Kutta methods
• Traffic simulation

Treiber, Martin; Hennecke, Ansgar; Helbing, Dirk (2000), "Congested traffic states in empirical observations and microscopic simulations", Physical Review E, 62 (2): 1805–1824, arXiv:cond-mat/0002177, Bibcode:2000PhRvE..62.1805T, doi:10.1103/PhysRevE.62.1805, PMID 11088643, S2CID 1100293

• Interactive JS & HTML5 implementation of the intelligent driver model showing signalized intersections
• Interactive JS & HTML5 implementation showing stop & Go waves on a ring road. Links to other scenarios can be found there
• Interactive JS & HTML5 implementation showing the effect of different traffic rules at roundabouts
• Common values of the IDM parameters and hints for the simulation
• Textbook on traffic flow dynamics with a free downloadable chapter on the IDM