A New Traffic Kinetic Model Considering Potential Influence

Shoufeng LuChangsha University of Science

and Technology

Outline

Introduction The heterogeneous Delitala-Tosin

model The weight function considering

potential influence An example and conclusion

Introduction Traffic flow are composed of many

driver-vehicle units. Their physical state includes, in addition

to position and speed variables, driver behaviour (activity variable).

The mathematical approach of active particles is called mathematical kinetic theory for active particles. [Bellomo, N. 2008. Modelling complex living systems: A kinetic theory and stochastic game approach , Springer publisher ,Boston .]

The difference, with respect to the classical theory, is that interactions follow stochastic rules technically related to the strategy developed by individuals.

The strategy of drivers dominates the interaction rules, and consequently vehicles may accelerate or decelerate according such a strategy.

There are two important mathematical tools in this approach: interaction rate and table of games.

About the vehicle interaction, binary interactions are used in the table of games. For example, Delitala and Tosin proposed a weighted binary interactions function. Its main feature is that interactions are distributed over the visibility zone of the drivers.

Almost all the related papers using this approach are about highway traffic flow. Little is studied about the city traffic flow.

There are more affecting factors in city traffic flow than in highway traffic flow, for example, intersection signal control, pedestrian crossing, bus station, etc. In real traffic flow observation, the vehicles in the multi-lane city traffic flow adjust its speed not only base on the single leader vehicle, but also other factors, e.g. the density ahead, queue length, and the color of an intersection signal.

The paper aims to extend the weight function using potential influence to model city traffic flow.

The heterogeneous Delitala-Tosin model

In Delitala-Tosin model, all quantities employed to describe the system are in dimensionless form.

Where is the rate of interactions among vehicles weighted over visibility zone and increases with density.

The weight function considering potential

influence The function weights the interactions of the candidate vehicle with each field vehicle located in front of it in the visibility zone. Different choices of can be made, according to different possible criteria to evaluate the effectiveness of the interactions. Delitala-Tosin proposed the piecewise constant function ,

and the piecewise decreasing function with y increasing.

When Wiedemann and Reiter proposed the microscopic traffic simulation model, they mentioned actual influence and potential influence, which is based on human perception.

Actual influence is from a front vehicle. The potential influence is from those situations when a driver is not yet influenced by a front vehicle.

Following this term, we use potential influence to represent the affecting factors in city traffic, e.g. signal control, pedestrian crossing, bus station.

We propose the following weight function. We divide the interaction zone into actual influence zone and potential influence zone, which is discontinuous between them.

Potential influence zone is a position of affecting factors in city traffic. Velocity class of potential influence zone can be set according to influence level. With the vehicles moving, its actual influence zone moves towards potential influence point. When a vehicle moves in the potential influence zone, a vehicle interacts with both actual influence zone and potential influence zone.

An example

In this paper, the hyperbolic conservation laws with source terms of Delitala-Tosin model is solved by splitting schemes (see Toro[6] for further details).

The attraction of splitting schemes is in the freedom available in choosing the numerical operators. In general, one may choose the best scheme for each type of problems. By splitting schemes, the solution of the Delitala-Tosin model can be found by solving the following pair of Initial Value Problems (IVP).

The initial condition of IVP (1) is the actual initial condition for the original IVP(3) and the initial condition for IVP (4) is the solution r(x,t) of IVP (3). Godunov method is used to solve PDE (3). Matlab solver ode45 is used to solve ODE (4).

We intend to simulate the formation of a queue due to red light of signal control. As initial condition, we set traffic light always red at the boundary x=1 of the spatial domain. At the inflow boundary x=0, we imagine a group of incoming vehicles entering the domain with a certain positive velocity, that we choose as the maximum possible according to our velocity grid.

Figure 1. Evolution of a queue at different times. Solid line refers to the density profile of the proposed model. Dash line refers to the density profile of D-T model.

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

position

dens

ity

t=200t=300t=400

t=500

t=600

Figure 2 Evolution of velocity distribution in position 42 at different times. Solid line refers to the velocity distribution profile of the proposed model. Dash line refers to the velocity distribution profile of Delitala-Tosin model.

1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

velocity class

density

t=100

t=200

For this example, we can expect that a queue will form and propagate backwards as illustrated in Figure 1.

For Delitala-Tosin model, the vehicles do not interact with the red light at x=1 before they reach there.

So the velocity can obtain the maximum velocity class, as illustrated in Figure 2. But for the proposed model, the vehicles in the potential influence zone interact with front vehicles and red light. So the velocity is lower. This is the reason why backward propagation speed of the proposed model is slower than the Delitala-Tosin model.

REFERENCES Bellomo, N. 2008. Modelling complex living systems: A kinetic theory and

stochastic game approach , Springer publisher ,Boston . Bellomo, N., Dogbe, C., 2011. “On the modelling of traffic and crowds: A

survey of models, speculations, and perspectives”, SIAM Review, 53(3), 409-463.

Tosin,A., 2008. Discrete kinetic and stochastic game theory for vehicular traffic: modeling and mathematical problems, Ph.D Thesis, Politecnico Di Torino.

Delitala,M., Tosin,A., 2007. “Mathematical modelling of vehicular traffic: A discrete kinetic theory approach”, Mathematical Models and Methods in Applied Sciences, 17, 901-932.

Wiedemann,R., Reiter, U., 1992. Microscopic traffic simulation: the simulation system MISSION, background and actual state, CEC Project ICARUS (V1052), Final Report, vol.2, Appendix A. Brussels: CEC.

Toro, Eleuterio F. 2009. Riemann solvers and numerical methods for fluid dynamics, A practical introduction, third edition, Springer-Verlag Berlin Heidelberg.