Simulation-based stability analysis of car-following models under

heterogeneous traffic

Hao WangSchool of Transportation

Southeast University

Aug 13, 2013Shanghai, China

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Sixth International Conference on Nonlinear Mechanics

The stability of car-following models has been studied for more than 60 years. Chandler et al. (1958) , Herman et al. (1959), Bando et al. (1995), Holland (1998), …

Introduction Background

There are two kinds of definitions of stability, i.e., the local stability and the asymptotic stability.

Local stability Asymptotic stability

Copied from Herman et al. (1959)

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Sixth International Conference on Nonlinear Mechanics

The asymptotic stabilities of car-following models were mostly discussed for the homogeneous traffic flow, based on Lyapunov stability theory.

Introduction Background

For the GM models, the generalized asymptotic stabilities can be formulated as (Holland, 1998):

1

2

Sensitive parameter Time lag

1( ) ( )n n na t v t v t Generalized GM model

where ω can be a constant or spacing (velocity) related parameter.

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Sixth International Conference on Nonlinear Mechanics

Introduction Background

However, the traffic is heterogeneous in the real world.

1 2 3 4 5 6 7 … n

1 1 2 2 3 3 4 4 5 5 6 6 7 7 n n

What is the asymptotic stability condition of heterogeneous flow like?

If not, what it should be?

1

1 n

iin

1

1 n

iin

1

2 ?

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Sixth International Conference on Nonlinear Mechanics

Introduction Objectives

This paper studies the asymptotic stability of traffic flow consisting of two kinds of vehicles with a special structure as “ABABAB…”, based on Gazis Model.

A B

a a b b A B A B A B …

a a b b a a b b a a b b

11

( ) ( )n n nn n

a t v t v tx t x t

Gazis model

1n nx t x t

with

Instead of theoretical analysis, a simulation-based method is used to analyze traffic stabilities in this study.

1

2c a b 1

2c a b

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Methodology Framework

Initial steady state

The steady state velocity-spacing relationship is obtained by integrating Gazis model as follows,

ln ( )j

xv

x

Determination of stability criterion

The stability criterion is obtained by observing the propagation of small perturbation in dense traffic flow, which is achieved by following steps:

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Methodology Framework

Step 1: 50 vehicles were simulated to move in steady state at beginning with the velocity of 12.2 m/s.

Step 2: The first vehicle in the platoon decelerated at -1 m/s2 from the 6th second to the 8th second, maintained a constant speed of 10.2 m/s for the next 4 seconds, then, accelerated at 1 m/s2 from the 12th second to the 14th second, keeping moving at a constant speed 12.2 m/s till the 200th second.

Step 3: Let and denote the maximum and minimum velocity of vehicle in the simulation respectively. can then describe the fluctuation range of vehicle n. if , then the traffic is unstable, else if , then the traffic is stable.

maxnV

minnV

max minn n nf V V

2 0n n kf f

2 0n n kf f

A B A B A B A B ……fn fn+2k

fn fn+2k

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Sixth International Conference on Nonlinear Mechanics

Methodology Framework

Step 4: For a particular combination of parameters and , let’s keep and constants, then change in the interval of [0,3s] with the step of 0.1s.

If for , we have

While for ,

Then the parameters are right on the stability criterion of the model.

,a b ,a b ,a b a b

*, , , 0.1a b a b 8 20 0 ( 8, 6)f f n k

*, , ,a b a b 8 20 0 ( 8, 6)f f n k

*, , ,a b a b

Numerical simulation

Let change in the interval of [7,17] with step of 0.5; change in the interval of [0,3s] with step of 0.1s, using the method mentioned above to obtain the stability criterion.

,a b ,a b

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Results Overview

In Fig.1(a), and , so that traffic flow can be approximately regarded as homogeneous.

Fig.1 Points on stability criterion of Gazis model (a) homogeneous traffic flow. (b) heterogeneous traffic flow.

0.2a b 0.2a b

In Fig.1(b ), are randomly selected within the given range so as to describe various possible situations of heterogeneous traffic flow.

, , ,a b a b

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Results Effect from dispersion

a bd a bd r

1 ( )( ) 0

1 ( )( ) 0a b a b

a b a b

whend

when

Define:

where and describe the discrete manner of the heterogeneous flow composed of two types of vehicles (A & B).

d d

Fig.2 Influence of characteristic parameters on stability criterion under heterogeneous traffic

0.6 0.7d

1dr 0.3 0.4d

1dr

0.5 0.6d

1dr 0.2 0.3d

1dr

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Conclusion

The stability criterion of homogeneous traffic flow does not hold for the heterogeneous flow. The asymptotic stability conditions of heterogeneous traffic flow can not be stated by stability conditions of homogeneous flow in terms of average value of model parameters.

The stability of the heterogeneous flow depends not only on the averages of the parameters of individual vehicles, but also on the dispersion of the parameters of individuals.

Large dispersion of sensitive parameter loosen the stability criterion of the heterogeneous flow, which is more strict under the large dispersion of lag time.

Findings

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Conclusion Future work

Find the theoretical stability criterion for the “ABAB” pattern heterogeneous traffic flow.

Expand the studies to some other car-following models and other patterns as well.

Questions?

Thanks