Equilibrium Velocity Distribution Function for Traffic Flow

1)Shanghai University China2)Wuzhou University China

Wei SHI

1 PROBLEMS

• Mesoscopic models for traffic flows are usually difficult to be employed because of the appearance of integro-differential terms in these models . Meng and Qian[2008] introduced the lattice Boltzmann equation to solve the problem. However, the model is not easy to propose special equilibrium distribution function for special traffic phenomena.

• In this paper, we propose a new kind of equilibrium distribution function for different road condition considering the specific velocity-density relation and the basic relations of the Boltzmann method.

2. MODEL AND EQUILIBRIUM VELOCITY DISTRIBUTION

• Helbing[2001,TRB] summarized some works and pointed out the equilibrium velocity distribution is in line with Gaussian distribution for single type vehicles. Thereby the equilibrium velocity distribution function with power function type can be suggested as below,

• : the density at the kth sites• :the undetermined coefficient• : the velocity of vehicles under the local

equilibrium state.

21 ( )eqi n i i eqf v v

nieqv

• Generally speaking, a series of velocity-density relation for specific road condition can be constructed through field measurement. The velocity-density relation proposed by Drake et al. is used temporarily for its superior goodness of fit and convenient parameters setting. The relation is given by the upper function.

2

0

1exp

2eq fv v

• the free velocity;• the critical density;• the local vehicle density

fv

0

Applying the following relations in the lattice Boltzmann method

max

0

veqi n

i

f

max

0

veq

i i n eqi

v f v

• For the convenience of computation, two integer velocity values are assumed, i.e., 0 and 1 respectively, that is to say each vehicle stay in the site or move forward to its nearest site.

0 1 eqv 1 1 1 eqv

3 SPECIAL EQUILIBRIUM VELOCITY DISTRIBUTION CURVES

Figure 1. Two velocity distribution curves as 0 0.15

• Further reading:

• Wei SHI, Improvement and Application of Lattice Boltzmann Models and Car Following Models for Traffic Flow, PhD Dissertation. Shanghai University, 2011.

• Thanks you!