Microscopic Models

Following Ward ([61]) we identify two main types of microscopic models: car follow-ing models and cellular automata.

The setup of a car-following model is as follows (see figure 1.7): to describe the movement of individual vehicles in time, we consider a single lane of vehicles following each other labelled 1, 2, etc. in the upstream direction. We define the position and velocity of each vehicles to be x_n and ˙x_n := v_n > 0 respectively. We can then define the front-to-front spacing between vehicles to be h_n:= x_n−1− x_n, and the speed difference between a vehicle and its leader as ∆v_n= v_n−1− v_n. (from [63])

Figure 1.7: car-following model setup.

Bando et al. in [3] defined an optimal velocity function V (h) based on the obser-vation from the fundamental diagram that there exists a relationship between a vehicles speed and the spacing to the preceding vehicle. This function is increasing in spacing

to represent that one chooses a higher speed for larger spacings. They proposed in their paper the following S-shaped function:

V(h_n) = tanh(h_n− 2) + tanh(2) (1.6)

Their model, named the Optimal Velocity Model (OVM) represents the acceleration function of the n^th vehicle as follows (with dot denoting differentiation with respect to time):

˙

v_n= α(V (h) − vn) (1.7)

The parameter α := _τ¹ is to be calibrated to observations, and can be interpreted as

”sensitivity”, or the inverse of the speed adaptation time τ. This latter variable models how quickly a vehicle will adapt its present velocity to its desired velocity.

Although this model has a very simple form it is able to replicate qualitatively stop-and-go waves (as defined in section (1.1.1)). However, this model has some draw-backs: Treiber and Kesting in [58] performed a simulation of OVM and found that they were obliged to use a speed adaptation time that was much too small compared with experimental values by a factor of about 10. Increasing it only slightly would result in negative spacings (which correspond to accidents), and decreasing it slightly would cause the system to not be able to exhibit stop-and-go waves. Thus the simulation out-come is not robust to the fine tuning of the parameter. Another result of this unrealistic parameter value is that the periods of the resulting stop and go waves are much too low.

This behaviour is mainly due to the fact that drivers in this model are short sighted:

they only react to space headways and not the difference in speed with the preceding vehicle, and thus need to react unreasonably quickly to avoid accidents.

Nonetheless, the simulation outcomes are qualitatively correct: the model repro-duced stop an go waves in simulations as is discussed in [58], and a bifurcation analysis of the model reveals some very interesting dynamics [20] which can provide an expla-nation for stable stop and go traffic. This suggests that the assumptions behind the model, namely the existence of a relationship between velocity and spacing and the relaxation of a vehicle to its ”optimal velocity” is a fundamentally correct one.

1.1. Traffic flow modelling 41 To correct for the short sighted quality of drivers in the OVM, one can include the speed difference ∆v_nbetween a vehicle’s speed and the speed of the preceding vehicle.

This results in the following acceleration function ([58]):

˙

v_n= α(V (hn) − v_n) + β ∆vn (1.8)

One can further refine the model by letting the additional parameter β depend on the headway rather than be constant. An example of such a model is the Intelligent Driver Model[57]. Its acceleration function is given by:

˙

In this model, a is the maximum acceleration, b is the comfortable deceleration, vmax

is the desired maximum velocity, h_stop is the desired stopping distance, T_gap is the desired time gap. As well as considering the velocity difference in a more sophisticated way than the FVDM, this model has an intelligent breaking strategy which models how drivers desire to break with a certain comfortable deceleration, but will decelerate more if they are in a so-called ”critical situation” (for example if breaking with comfortable deceleration would not be enough to avoid a collision). The parameters of this model therefore have a clear meaning and the results fit empirical data as expected. However, it does have some drawbacks: a platoon of identical drivers will disperse more than in observed traffic. Furthemore, if the time gap between two vehicles is too small (due to a vehicle overtaking for example) the deceleration of the following vehicle will be too strong and unrealistic (as most drivers in real traffic would not consider a vehicle merging into their lane just in front of them as a critical situation). Extensions to this model have been proposed to address these issues, and are used in adaptive cruise controls (ACC) in vehicles. However, IDM is on the whole a model that reproduces observed spatio-temporal dynamics well (a discussion can be found in [58]).

All these different models can be considered as a special case of the following general formulation [58]:

˙

x_n= v_n v˙_n= f (h_n, v_n, ∆vn) (1.10)

The acceleration function should model the fact that drivers accelerate if their front-to-front spacing or relative velocity increases, but that they decelerate if their velocity increases. This is reflected in the following general criteria for sensible microscopic models ( f_x denoting the partial derivative of f with respect to the variable x):

f_v(h, v, ∆v) < 0 (1.11)

f_h(h, v, ∆v) ≥ 0 (1.12)

f_∆v(h, v, ∆v) ≥ 0 (1.13)

An important property of these models is that they accept a uniform flow in which the velocities and headways are time independent:

h_n(t) = h_e ∆v_n(t) = 0 v_n(t) = v_e (1.14)

Other extensions of car-following models not considered here include delay terms and anticipation terms (the headway of several preceding vehicles is considered).

The second main branch of microscopic models, cellular-automata models con-sider a discretised space and velocity (and sometimes time). They are concon-sidered mi-croscopic as they are representing the movement of individual vehicles, and are gener-ally computationgener-ally more efficient than other microscopic models. A popular example is the stochastic model due to Nagel & Schreckenberg ([45]).

On the whole, these micro models are used to describe interactions between ve-hicles on a small scale, and can therefore be used to model small networks or ACC systems. However, it can be seen from (1.10) that one needs n equations to describe a system of n vehicles: modelling a large network can quickly become computationally heavy. In these scenarios, using macroscopic models which consist of a few equations can be preferable.

1.1. Traffic flow modelling 43