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Equilibrium Properties of the Cellular Automata Models for Traffic Flow in a Single Lane

187

In the near future, we would like to be able of extending this equilibrium theory to the

cellular automata models for multi-lane traffic flow and 2D traffic networks, which we hope

would be setting us in the way towards the application of these models within the context of

the urban air pollution problems.

7. Acknowledgements

We acknowledge the disinterested help and invaluable contributions we received from A.

Fierros Palacios (Director, IIE-DEA, Mexico), R. Merino (TECALCO, Mexico), and A. T.

Celada Murillo (IIE-DEA, Mexico) through their encouragements, comments, and

enlightening discussions during the preparation of this work.

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Cellular Automata for Traffic Modelling and

Simulations in a Situation of Evacuation from

Disaster Areas

Kohei Arai, Tri Harsono and Achmad Basuki

Saga University

Japan

1. Introduction

The traffic flow studies using microscopic simulations (micro traffic model) have leap with the

occurrence of the advancement of computer technology in the last one and half decade, as

shown in (Nagel, K. & Schreckenberg, M., 1992); (Nagel, K., 1996); (Bando, M., et al., 1995).

The evacuation system in the micro traffic simulation model has been studied and reported

a couple of years ago. In the early stage, some examples of micro traffic simulation models

related with emergency evacuation are investigated by (Sugiman, T. & Misumi, J., 1988);

(Stern, E. & Sinuany-Stem, Z., 1989). The modelling system of emergency evacuation in the

traffic stated by (Sheffi, Y., et al., 1982); (Hobeika, A.G. & Jamei, B., 1985); (Cova, T.J. &

Church, R.L., 1997) has chosen to estimate evacuation time from an affected area using static

analysis tools at macroscopic or microscopic levels.

Another research of emergency evacuation at the micro traffic scale was conducted by (Pidd,

M., et al., 1996). They developed a prototype of spatial decision support system that can be

used for emergency planners to evaluate contingency plans for evacuation from disaster

areas. It does not take the interactions between individual vehicles into consideration.

Two basic components of agent-based modeling, i.e. (1) a model of agents and (2) a model of

their environment were introduced by (Deadmann, P.J., 1999). An individual agent makes a

decision based on the interaction between him and the other agents together with localized

knowledge (Teodorovic, D.A., 2003).

How evacuation time can be affected under different evacuation scenarios, such as opening

an alternative exit, invoking traffic control, changing the number of vehicles leaving a

household was observed based on agent-based simulation techniques (Church, R.L. &

Sexton, R., 2002). Neighbourhood evacuation plans in an urbanized wild land interface were

described by (Cova, T.J. & Johnson, J.P., 2002) using agent-based simulation model. They

were able to assess spatial effects of a proposed second access road on household evacuation

time in a very detailed way.

The effectiveness of simultaneous and staged evacuation strategies using agent-based

simulation for three different road network structures were presented by (Chen, X. & Zhan,

F.B., 2008). They measured the effectiveness based on total time of evacuation from affected

areas.

Cellular Automata - Simplicity Behind Complexity

194

Aforementioned studies described how to evacuate all residents in affected area whereas

this study evacuates vehicles in affected road using agent-based modelling. We conduct

micro traffic agent-based modelling and simulation for assessment of evacuation time with

and without agents from the suffered area of the Sidoarjo hot mudflow situated in the East

Java Indonesia called LUSI that occurred on 29

of May, 2006. Even now, the mud volcano

remains high flow rates (Rifai, R., 2008).

One of the key elements of evacuation from the mudflow disaster is the road as main traffic

connection surrounding disaster area and dike of the hot mudflow is very close with the

road (Indahnesia.com, 2007). The vehicles density on the road is high (Mediacenter, 2007).

Our micro traffic agent-based modelling and simulation, other than with/without agent;

road traffic (vehicle density) and road networks; driving behaviour such as lane changing,

car-following; and unpredictable disturbance due to a difference between disaster speed

and vehicle speed are taken into account. Although the proposed simulation is based on the

Nagel-Schreckenberg proposed by traffic Cellular Automata (CA) as shown in (Nagel, K. &

Schreckenberg, M., 1992) and (Maerivoet, S. & De Moor, B., 2005), lane changing and car-

following parameters are specific to the proposed simulation.

The following section describes overview of the traffic models followed by car-following

models. Then the introduction of CA model, major topic of the chapter of evacuation

simulation will be explained. Agent-based approach of CA traffic model for the case of

evacuation is presented. Furthermore, the results of traffic survey are described. The traffic

survey is conducted on the specific roadway situated surrounding the hot mudflow disaster

area, Sidoarjo, Indonesia. It is found two different types of driver, i.e. usual diver and

diligent one. The next section, modified car-following model is investigated by taking the

behavior of usual driver and diligent one into account. Finally, concluding remarks and

some discussions are described.

2. Overview of the traffic models

One important portion of the micro traffic model is Car-Following (CF). Studies of car-

following models have been proposed to describe the interaction between drivers and

vehicles. CF became an important evaluation parameter for intelligent transportation system

strategies since 1990. CF theories based on the assumption that each driver reacts in some

specific way to be stimulated with the vehicles in front. A very attractive microscopic traffic

model called the Optimal Velocity Model (OVM) is proposed by (Bando, M., et al., 1995). It

was based on the idea that each vehicle has an optimal velocity, which depends on the

following distance of the preceding vehicle. Despite its simplicity and its few parameters,

the OVM can describe many properties of real traffic flows, such as the instability of traffic

flow, the evolution of traffic congestion, and the formation of stop-and-go waves.

OVM used to be calibrated using empirical data provided by (Helbing, D. & Tilch, B., 1998).

They stated that when empirical data used in OVM has weaknesses, OVM has a too high

acceleration and an unrealistic deceleration. (Helbing, D. & Tilch, B., 1998) have overcome

these problems by using a Generalized Force Model (GFM).

(Jiang, R., et al., 2001) stated that GFM cannot describe the delay time δt and the kinematics

wave speed at jam density cj properly. They so that proposed the Full Velocity Difference

Model (FVDM). FVDM has too high deceleration since empirical deceleration and

acceleration are restricted between the following regions, from -3m/s

to 4m/s

Cellular Automata for Traffic Modelling and Simulations

in a Situation of Evacuation from Disaster Areas

195

On the other hand about improving the OVM and considering the Intelligent Transportation

System (ITS) application, (Ge, X.H., et al., 2008) proposed a new model taking into account

the velocity difference Δv

and Δv

n+1

, where Δv

≡ v

n+1

− v

. They obtain a more useful model

called the Two Velocity Difference model (TVDM). TVDM has shown that unrealistic high

deceleration does not appear when they simulate the deceleration progress of two cars.

The CF models (Bando, M., et al., 1995); (Helbing, D. & Tilch, B., 1998); (Jiang, R., et al., 2001)

and (Ge, X.H., et al., 2008) are known as time-continuous models. They have in common that

they are defined by ordinary differential equations describing the complete dynamics of the

vehicle’s positions x and velocity v.

Vehicle dynamics system can also be expressed by CA model. The road is divided into

sections of a certain length Δx and the time is discretised to steps of Δt. Each road section

can either be occupied by a vehicle or empty and the dynamics are given by update rules of

the form:

(

)

111

,,,

tttt

nnnn

vfxvv

−−−

= " and

1tt t

nn n

xx v

−

+ , the simulation time t is measured in

units of Δt and the vehicle positions x

is measured in units of Δx.

In this study, we proposed the vehicle dynamics system based on Stochastic Traffic Cellular

Automaton (STCA) expressed in (Nagel, K. & Schreckenberg, M., 1992). It has the ability to

reproduce a wide range of traffic phenomena. Due to the simplicity of the models, it is

numerically very efficient and can be used to simulate large road networks or even faster. A

new characteristic of driving behaviour has been proposed, it is about diligent driver.

The proposed evacuation modelling and simulations above (Sugiman, T. & Misumi, J.,

1988); (Stern, E. & Sinuany-Stem, Z., 1989); (Sheffi, Y., et al., 1982); (Hobeika, A.G. & Jamei,

B., 1985); (Cova, T.J. & Church, R.L., 1997); (Church, R.L. & Sexton, R., 2002); (Cova, T.J. &

Johnson, J.P., 2002) and (Chen, X. & Zhan, F.B., 2008) describe how to evacuate all residents

from affected areas, the proposed modelling and simulation here is for vehicle evacuation

from affected road based on an agent-based modelling. Namely, we conduct micro traffic

agent-based modelling and simulation for assessment of evacuation time from the suffered

area. In the micro traffic agent-based modelling and simulation, road traffic (probability of

vehicle density), driving behaviour such as probability of lane changing, car-following are

taken into account. The specific parameter in the proposed modelling and simulation is car-

following under a consideration of agents. Diligent driver is also added to the proposed car-

following parameter. The number of diligent drivers will be probabilistically determined.

Although the proposed simulation is based on the Nagel-Schreckenberg traffic cellular

automata, lane changing and new car-following parameters are specific to the proposed

modelling and simulation.

3. Overview of the car-following models

There are two major methods of car-following, continuous and discrete models. Some of

continuous models are based on the optimal velocity model (OVM), generalized force model

(GFM), full velocity difference model (FVDM), and two velocity difference model (TVDM).

Meanwhile, cellular automaton model is used for the discrete model.

3.1 Continuous models

In the micro traffic model, the drivers are stimulated their own velocity v

, the distance

between the car and the car ahead x

, and the velocity of the vehicle in front v

n-1

. The

equation of vehicle motion is characterized by the acceleration function which depends on

the input stimuli,

Cellular Automata - Simplicity Behind Complexity

196

(

)

() () (), (), ()

n n nnn

xt vt Fvtxtv t

−

 

(1)

3.1.1 Optimal Velocity Model (OVM)

The OVM is a dynamic model of traffic congestion based on a vehicle motion equation. In

this model, the optimal velocity function of the headway of the preceding vehicle is

introduced. Congestion may occur due to induce by a small perturbation without any

specific origin such as a traffic accident or a traffic signal. The OVM can regard to this

congestion phenomenon as the instability and the phase transition of a dynamical system

(Bando, M., et al., 1995).

The vehicle motion equation is expressed with the assumption that each vehicle driver

responds to a stimulus from other vehicles in some specific fashion. The response is

followed by acceleration. The sensitivity of acceleration to stimulus is a function of the

vehicle position, his time derivatives (or the distance), and so on. This function is

determined by the drivers’ characteristics, whether or not the vehicle drivers obey

postulated traffic regulations at all times in order to avoid traffic accidents.

There are two major types of drivers. The first type is based on the idea that each vehicle

must maintain the legal safe distance of the preceding vehicle, which depends on the

relative velocity of these two successive vehicles. The second type is that each vehicle has

the optimal velocity, which depends on the following distance of the preceding vehicle. In

the OVM, the traffic dynamics equation is proposed based on the latter assumption results

in a realistic traffic flow model. In this proposed model, the stimulus is assumed to be a

function of a following distance and the sensitivity is a constant. The OVM ignores the

vehicle length and assumes that all the drivers have common sensitivities. Then each vehicle

has the optimal velocity V and that each vehicle driver responds to a stimulus from the

vehicle ahead. Driver has to control their speed for maintaining the legal safe velocity in

accordance with the motion of the preceding vehicle.

The dynamical equation of the system is obtained (based on the acceleration equation) as,

()

() ()

()

dx t dx t

aV x t

dt dt

⎡

⎤

=Δ−

⎢

⎥

⎣

⎦

(2)

where

() () ()

nn n

xt x t xt

=−

(3)

n denotes vehicle number (n = 1, 2, …, N). N is the total number of vehicles, a is a constant

representing the driver’s sensitivity (which has been assumed to be independent of n), and

is the location of the nth vehicle. The OVM assumes that the optimal velocity V(Δx

) of n

vehicle depends on the distance between the n

vehicle and the (n-1)

vehicle (preceding

vehicle) (a distance-dependent optimal velocity). When the headway becomes short the

velocity must be reduced and become small enough to avoid crash. On the other hand,

when the headway becomes longer the driver accelerates under the speed limit, the

maximum velocity. Thus, V is represented as to have the following properties,

a monotonically increasing function, and

ii.

()VxΔ has an upper bound.

(

)

max

VVx

≡

Δ→∞.