Phenomenological Study of Dynamical Model of Traffic Flow

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Phenomenological Study of Dynamical Model of Traffic Flow

Masako Bando, Katsuya Hasebe, Ken Nakanishi, Akihiro Nakayama, Akihiro Shibata, Yūki Sugiyama

To cite this version:

Masako Bando, Katsuya Hasebe, Ken Nakanishi, Akihiro Nakayama, Akihiro Shibata, et al.. Phe-

nomenological Study of Dynamical Model of Traffic Flow. Journal de Physique I, EDP Sciences, 1995,

5 (11), pp.1389-1399. �10.1051/jp1:1995206�. �jpa-00247145�

Classification

Physics

Abstracts

02.60Cb 05.70Fh 05.70Jk

Phenomenological Study of Dynamical Model of llkaflic Flow

Masako

Bandoi~), Katsuya Hasebei~),

Ken

Nakanishii~),

Akihiro

Nakayamai~),

Akihiro

Shibatai~)

and Yùki

Sugiyamai~)

(~)Physics Division,

Aichi

University, Miyoshi,

Aichi 470-02,

Japan (~) Department

of

Physics, Kyoto University, I(yoto

606-01,

Japan

(~) ^{Gifu Keizai} University,

Ohgaki,

Gifu 503,

Japan

(~)

Department

of

Physics, Nagoya University, Nagoya

464-01,

Japan

(~) ^{Division of} Mathematical

Science, City College

of Mie,

Tsu,

Mie

514-01, Japan (Received

26 June 1995, received in final form 12

July 1995, accepted

21

July 1995)

Abstract. We show that the Car

Following

Model with

Optimal Velocity (Optimal Velocity

Model),

whicl~ _was

proposed

in _our previous paper, is a successful traliic model in

reproducing

the characteristic features of observed traffic flow data. In _our model the transition from free flow to

congested

flow _occurs

spontaneously by

the collective motion of

vehides,

wl~ich

obey

to tl~e _same

dynamical equation.

Tl~e observed

specific discrepancy

of traffic flow

vs- car

density grapl~

is well understood in _terms of tl~e

phase

transition in _our model.

1. Introduction

Trailic flow has been studied from _many different points ^{of view} and there have been

essentially

two different

types

^of

approaches, macroscopic

and

microscopic

ones

idynamical models).

We

are here

mainly

concerned with the latter

approach,

in which the

global

^features of trailic flow

are to be

explained

from the collective motion of individual vehicles.

Among

those

approaches

one of the most traditional _treatment is the

study

of the Car

Following

Model

[1-3].

^These

models _are based _on the reliable

assumption

that each driver controls his vehicle

only according

to his

preceding

vehicle's motion. Most models have taken the

assumption

that the acceleration

is

proportional

to the relative

velocity

of two successive

vehides,

and therefore have used

essentially

the first-order differential

equation

of motion of _an individual vehide.

However,

these models have

diiliculty

in

explaining

the

specific property

^{of trailic}

flow,

which has two

distinct

behaviors,

free and

congested

flows.

Conventionally

trallic flow is characterized

by

three basic

quantities: velocity i~),

flow

iQ)

and

density (k).

The main mterests of trailic engineers are rather

practical problems

such _as how many vehides _are allowed to _run in the

freeway

traflic fanes with

specified capacities-

Measurements _are made to search the

relationship

between the flow

(or velocity)

and the

density.

The

dependence

of the

velocity

_on the

density,

which is

usually

called the k-~ _curve,

generally

shows that

velocity

_{is a}

decreasmg

^function of

density-

The

typical

feature of trailic

@

Les Editions de

Physique

1995

flow

Q velocity

_v

density

k

density

k

Fig.

1. Illustration of tl~e

flow-density

and

velocity-density relationsl~ip

of traffic flow.

flow has been

expressed using

the

Q-k graph,

which shows the relation between the flow and the

density

_[4].

From accumulated data of measurement on

freeways

the

following

facts _are

commonly

_rec-

ognized.

When the

density

is very

low,

each driver _can control his vehide almost

freely,

which

can be called "free-flow". On the other

hand,

when the

density

is _very

high,

each driver is

strongly

forced to decrease his vehide's

velocity

and _a

"congested

^flow" _appears. A

typical

k-~

curve and

Q-k graph

are illustrated in

Figure 1, neglecting

the difference of

quantitative

details

among several data. A

possible discontinmty

_is observed between the

high

and low

density regions

and there _seems to exist _a critical

density

which

separates

^the two

regions [5-7].

We

can

identify

these

high

and low

density regions

as the

congested

and the free flow.

Figure

2 is _an

example

of the accumulated observed data taken

by

the

Japan Highway

Public

Corporation (JHPC)

_[4]. You would find such

types

^of

graphs

in most of textbooks of trailic theories. In

Figure 2,

_we ^find a

possible discontinuity

at the _occupancy P _cf

25%

,

which _may be called _a critical

density

_{as we} mentioned before.

In trie traditional studies of trailic

flow,

the behavior of _two distinct

region

are

separately explained

with

independent

formula _or models. No

early

works of Car

Following

Model could

reproduce

the

discontinuity

between free and

congested

flows in unified _way.

In _our

previous

paper

[8, 9],

we

proposed

_{a new} model

(hereafter

_we call

"Optimal Velocity

Model"

).

^{The essential} difference from traditional Car

Followmg

Models _is the introduction of

an

optimal velocity

^of a

vehicle,

which value is

changed accordîng

to the

headway

distance. In

our model the trailic

congestion

_occurs

spontaneously

and this

phenomenon

_can be understood

as a sort of

phase

transition from free flow state to

congested

flow state.

Actually,

the

dynamical equation

of _our model has two different kind of solutions. One _is the

homogeneous

flow solution and the other _is the

congested

^flow

solution,

which _consists of the two distinct

regions; congested

regions

ihigh density)

and smooth moving regions, or free _regions

ilow density).

The

specific property

^{of observed trallic}

flow, (Fig.

l and

Fig. 2),

will be

explained

in this _context.

The purpose of this paper is to

apply

_ouf model to

explain

observed data and to show how well _ouf model

reproduces

the

Q-k graph

of traflic

flow, especially

trie behavior _at the cntical

density.

In Section 2 _we

explore

_some charactenstics of observed trallic flow data. After

doing

a brief _review of

Optimal Velocity

Model in Section

3,

_we shall

apply

_our model to realistic trallic

phenomena

_using the data of Chuo

Motorway

in Section 4.

Summary

and discussions will be given m Section 5.

la) 16)

~y

-

/~

_

~

© ~

Î

_£ _~

fi

~

>

.fi

CÎ '

(

_',

~ >

O GZ

'. ,j=.

20 60 80 100

occupancy P

(9b)

_occupancy P

(9b)

Fig.

2- Example of

flow-occupancy (a)

and

velocity-occupancy (b)

relations obtained from the observed data taken

by Japan Highway

Public

Corporation.

velocity

v llow Q

(a)

~~~

density k density k

Fig.

3.

a)

Drake's k-u _curve;

b) Q-k diagram corresponding

to Drake's k-u _curve.

2. Behavior of the

Q-k Graph

at Critical

Density

In

early

works

explaining

the

shape

of

Q-k

_curve, several functions have been

proposed

^for

the

density-dependence

of the

velocity ~ik), by

Greenshields

[loi, Greenberg [iii,

Edie

[5],

Underwood

[12],

^Drew

[13],

^{Drake et ai.}

[14].

^{Some of them} are introduced

semi-empirically

and the others _are derived from calculations of _some

underlying

models.

Among

these

works,

the Drake _curve, which is obtained from fluid

dynamics,

has been

thought

to

reproduce

the

global

features of observed data

fairly

well

[14].

^{We show the} illustration of Drake's _curves in

Figure

3. He derived the

Q-k

_curve

corresponding

to a certain

~(k)

with the

assumption

Q

= k~. Trie _curve does not

reproduce

_a

striking wedge

at the critical

density

_m the observed

data

(see Fig. 2).

No _{one can} obtain such

wedge shape

_so

long

as he _uses any continuous function for the k-~ _curve.

Among

the above works Edie's model

only

showed the

wedge shape

_[5].

However,

he introduced the critical

point by

hand.

Our model _can

explain

the features of trailic

phenomena

for both

congested

and free flow _m

a unified way, 1-e-, to understand them from the _same

microscopic dynamical

law. We know _a lot of

physical phenomena

that the basic

equation

has _a different kind of

solutions,

which have

qmte

^different

macroscopic aspects.

This _can be

explained

in terms of

phase

transition.

3.

Optimal Velocity

Model

Let _us make _a brief _review of _our

dynamical

model [8]. We

investigate

_our model in the most

simple

situation where N vehides _{move on a}

single

fane circuit with _a circumference L. Here

we assume that road conditions _are uniform

along

the circuit and drivers _are identical.

We _use the

following notations; position

_x,

velocity

fl and

headway

AT _as the basic vari- ables for convenience. The

dynamical equation

for each vehide _is the

following

second order

differential

equation:

in

= a

(ViAxn) in), 11)

where _xn is the

position

of n-th vehide and

Axn

denotes the

headway

of the n-th

vehide, Axn

= xn+i xn and _{a is} the

sensitivity.

Each driver controls the acceleration of his vehide in such _a way that its

velocity

_is maintained at an

optimal

value

V(Az) according

to the

headway

AZ. We call this

"Optimal Velocity" (~).

In _contrast to earlier

works,

_we do not introduce the discontinuous function

V(Ax)

in order to

reproduce

trie observed

discontinuity

in k-~ and

Q-k

_curves.

The

Optimal Velocity

function

ViAx)

should have _some

properties

for

safety driving _[8].

We

adopt

_a

tanhix)-type

_{curve as a} candidate. For the convenience to fit the function with observed

data,

_we rewrite

ViAx)

_as

V(Ax)

= Vo

[tanh m(A~ bf)

^tanh

m(bc bf)] 12)

We have four parameters

Vo,m,

bf and bc m this

function,

which should be determined from observed data of the individual vehide behavior. The maximum

velocity

for _a

large enough headway

is

given by

Vmax ₌

Vo[1- tanhmibc _bf)]

The

headway,

AZ

=

bf, corresponds

to the inflection

point

of the

Optimal Velocity function,

where

Vibf)

= Vmax

j.

The

Optimal Velocity

becomes _zero at AZ

=

bc,

whicl~ is

regarded

_{as an} effective _car

lengtl~

and is _a httle

larger

tl~an tl~e average

lengtl~

of

vel~ides,

_{l~, smce} eacl~ driver

stops

^{bis vehicle before}

crasl~ing:

b~ = l~ + à.

13)

We bave _a

l~omogeneous

flow solution of

equal spacing

b

=

LIN

and _constant

velocity V(b)

for

equation (1),

~n(t)

=

b(n ₁₎

₊

Vib)t in

=

1,.

,

N). 14)

This solution _is unstable _m the _case of

2V'lb)

_{> a,} whicl~

corresponds

to tl~e _area between two dashed fines _as shown in

Figure

5 and traflic

congestion

_occurs

spontaneously

_[8].

Figures

4a and 4b show

typical

results for N

= 100 and L

=

200,

which _are the

snapshots

at t ₌ 100 and 1000 of the velocities of each

vehicle, respectively (~).

The

homogeneous

flow with _a small initial disturbance

develops

into a

congested

flow with time evolution.

Figure

4b

indicates the

spontaneous generation

of

congestion.

The

profile

of the

congested

flow _is illustrated in the AZ k

plane (phase space).

After tl~e

generation

of

congestion

is

finisl~ed,

the _motion of vehides

organizes

the

specific

dosed _curve in the

phase

_space. We called this

"Hysteresis Loop"

^and the

loop

_can be understood _{as some} kind of _a hmit

cycle. Here,

_we observe that each vehide _moves from the free region

iindicated

(~) It _was called

"legal velocity"

_{m our previous papers.}

(~)

Numerical solution of

Figure

4 is obtained with parameter values, _Vo

" 1, m = 1, bi ₌ 2, bc

= 0,

and the initial condition: _~i

" o-1, _{~n =}

bi(n

₁₎

(for

_n >

1),

in ₌

V(bi)

(a) snapshot t

= 100 <b) snapshot t ₌ 1000

2 2

.M 1.5 _.M 15

~ ~

à é

1

~ l ^~ l

o.5 o.5

~

10 20 30 40 50 60 70 80 90100

~

10 20 30 40 50 60 70 80 90100

carnumbei carnumber

Fig.

4- The

snapshots

at t ₌ 100

(a)

and t ₌ 1000

(b).

A

congested

flow

(kink-like)

solution appears as time

develops.

stable unstable stable

i

1"'

:1

;"

.'

M /

~ ./

é

_/

(

_+C

> _'

f

_/

./

_."

_.i

"1

headway

Ax

Fig.

5. The motion of vehides _m the A~ 1

plane

_on the "hmit

cycle".

The dotted _curve denotes

V(A~).

The stable and unstable _regions of

homogeneous

flow solution

are also shown-

by point

A%

iAxF, ~F))

to the

congested region idenoted by

the

point

Be

(Axc, uc))

on this

"limit

cycle", iindicated by

the _{arrows m}

Fig. 5)

_[9] and ~ice _~ersa. The

shape

of the limit

cycle depends

_on the

sensitivity

_a; the smaller the

sensitivity

_is, the wider the limit

cycle

_grows.

This will be

apparent

ⁱⁿ

Figure

7-

4.

Phenomenological Study

Now _{we move} to the _main

topic

^{in this} paper. We compare the

predictions

of _our model with

freeway

trailic flow data.

,=~

." ..,

- .ô

~ _:~',

Î

~' j

'w T

~ 60

û ~;~ "'.

° .~~ ~

fl ~

É JZ~

_~~

~'

0

0 10 20 30 40 50

clearance S (m)

Fig.

6.

Velocity-clearance

data from _a

car-following

expenment on the Chuo

Motorway,

which _was taken

by

Koshi et ai.

[6,15]

Solid _curve

is the determined

Optimal Velocity

function V-

Our first task is to determine the values of

parameters

to fix trie form of _an

Optimal Velocity

function

12).

^This can be clone with the reference of the observed data of _an mdividual vehide's behavior. As the most

appropriate

data for _{our purpose, we use} the data of

car-following experiment

_on the Chuo

Motorway

taken

by

"Koshi

group" [6,15]. They

obtained data

points

on the

velocity-dearance plane (Fig. 6),

where the dearance S is defined _as the

headway

subtracted

by

the vehides

length,

l~,

isee Eq. 13));

S=Ax-l~. 15)

There _{are very} few data

points

around 55

km/h velocity

in

Figure

6. In the view of _ouf

dynamical model,

drivers feel

difliculty

to _run vehides

maintaining

with such

velocity,

which is

just corresponding

to the unstable

velocity

_region of the

homogeneous

flow solution.

So,

_{we can}

mfer that the inflection point ^is

nearby

such

velocity.

Let _us fix the parameters of the

Optimal Velocity

function from

Figure

6. The mflection point ^is taken _as

(S,

_v

=

(20

_{m, 55} km

/h).

The maximal

velocity

is Vmax = ils

km/h (dashed line)

and the minimal dearance is

Smin

= à

=

2.0 _m

iindicated by

dotted

line).

From the

above,

_we

_get

the concrete form of the

function,

V =

16.8[tanh 0.0860(Ax (20

+

l~))

+

0.913] (m/s). (6)

The

resulting

function

(6)

is also

displayed by

a sohd _{curve m}

Figure

6. For

simulations,

we

should take the

length

^{of the}

vehide,

l~ = 5 _m ~vhich _was used in the

car-following experiment

on Chuo

Motorway.

Next _we determine the value of

sensitivity

_a. For this _{purpose we} make simulations for

vanous

values;

_a

=

1.6,

2.0 and 2.8 usmg the

Optimal Velocity

function determined above.

The results _are shown in

Figure

7. We choose the

sensitivity

a = 2.0 _as the most

appropriate

value which

reproduces

the realistic

velocity

and

headway

of

congested

_or free _regions. Then

using

^this

value,

_we will make simulations of various car-densities

(NIL)

for the _purpose of

obtaining

the

density-flow relationship (Q-k curve).

120

ioo

«

(

g

)

40 a

=28

20

~0

eadway

"., homo~eneou

jow

"., ca geste pw

... i i

simlat>on

o

"._

"..,1 A

é

~

~

H

/Î

( '.

~

i i i

i i

i i i

i i

]

Fig.

8-

Q-k

_{curves on} tl~e

flow-density plane.

Tl~e

flow-density

relations of tl~e

l~omogeneous

flow

solution and

congested

flow solution _are

depicted by

solid and dotted _curves

respectively.

A and B denote the free

region

and the

congested

_region points respectively- C corresponds to tl~e inflection point of

Optimal

Velocity function

V(A~) (see

also

Fig. 5).

By substituting

the value of

A~c

and

A~F

obtained from the simulation

result,

_we find

Q

" 318 3.36 k

(ii)

On the other

hand,

the

Q-k

relation of the

homogeneous

flow _is

analytically

derived

by using

the solution

(4)-

The flow

Q

is

easily

written _as

Q

"

~

"

kV(1/k)

= 5.04

(tanh(

^~~

2.15)

+

0.913)

k

(12)

These two curves,

equations Ill

and

(12),

_are drawn _m

Figure

8- The free

region point iA)

and the

congested region

_point

iB) correspond

to the end points ^{of the limit}

cycle

in

Figure

5- We note that the

homogeneous

flow solution is unstable in the

region

^between dashed lines

m

Figure

8 from the

analysis

of linearized

theory

_[8]- The

corresponding region

_is shown _as

region

III in

Figure

8. The

congested

flow solution _is

expected

to be realized in this

region,

therefore the simulation data will be

plotted

_on the dotted line

iii).

Now _we finished the discussion about

Q-k

relation and _we

perform

the simulations for varions

NIL

values. We _use the

Optimal Velocity

function

16)

determined from Chuo

Motorway data,

and take the initial condition of simulations _as the

homogeneous

flow with _a

tiny

disturbance.

The result of simulations is

plotted

also in

Figure

8. Each diamond mark

corresponds

to _one

simulation,

which shows _a

good agreement

with _our

prediction.

We _can

separate

the

density

mto five regions divided

by

dashed hnes _{as m}

Figure

8. In regions I and

V, only

the

homogeneous

solution is stable

ithe

solid _{curve is}

realized).

In

region III, only

the

congested

flow solution _is stable

(the

dotted line _is

realized).

On the other

hand,

in

regions

II and IV both solutions _are stable. In

Figure

8 the solid _{curve is} realized because we

started the simulation from the

homogeneous

flow. When _we start from another distributions of vehides such that

congestion already exists,

^the

congested

flow solutions _are stable _as

long

as

they obey

to the

Hysteresis loop profile.

We have

already

confirmed this result in the previous paper

[9].

^{In this} case the dotted hne _is realized _as well in

regions

II and IV.

250

200

~

$ l$0

~Î

~~

O

~ '

~

0

P (%)

Fig.

9-

Q-k diagram

_on the

flow-occupancy plane.

The result of _our simulations _is shown

by

the diamond marks

togetl~er

with the observed data of

Figure

2a.

Now _we cl~eck _our simulation data

against

_some ^observed data. In

principle,

_we sl~ould compare the simulation results with real

Q-k

data measured in the _same

freeway

that _we determined the

Optimal Velocity

function.

Unfortunately

_we have _no data of

Q-k diagram given by

the observation in Chuo

Motorway,

which should be

compared

with the result of the simulation

using

the

Optimal Velocity

function

(6).

On the

otl~erl~and,

_we bave tl~e

Q-k

data of JHPC

(Fig. 2),

but _we do not bave the data hke

Figure 6,

which _can determine tl~e

Optimal Velocity

function. Tl~us _we

apply

tl~e

Optimal Velocity

function for Chuo

Motorway

to tl~e

study

of JHPC data

(Fig. 2)

^with some modification. This _can be done

by only rescaling

trie overall factor

V,

which makes _no

change

in trie

dynamics

of _ouf model. When _we fit the

Optimal Velocity

function

VIA~)

to the JHPC data

(Fig. 2b)

in the _same way as tl~e _case of Cl~uo

Motorway (Fig. 6),

tl~e maximum

velocity

of

V(Ax)

of JHPC data is about

80%

smaller tl~an tl~e Cl~uo

Motorway

_one. So _we rescale the function _as

V'

= 0.8 _x V

(13)

The time occupancy

P,

which _appears in

Figure

2 is related to the

density;

P

=

lck.

Here _we

take _l~

= 7 _{m as an} average

length

^of vehides.

We

perform

trie simulations usmg tl~is function witl~ tl~e _same

procedure

_as tl~e previous

case. We also took tl~e initial condition _as tl~e

l~omogeneous

flow with _a

tiny

disturbance.

We present trie result of _ouf simulations _m

Figure

9

by

tl~e diamond marks

togetl~er

with tl~e observed data of

Figure

2a. In the

region

where tl~e

congested

flow solution is

stable,

the series of diamond marks _{are very} dense and _may look like _a thick solid fine. Ouf result agrees

quite

well with the observation and the

specific discontinuity

between free and

congested

flow is well

reproduced.

5.

Summary

and Discussion

We have demonstrated that

Optimal Velocity

Model

reproduces

the characteristic features of the traflic flow

phenomena quite

well.

Especially

_our model

reproduce

the

specific discontinuity

near the critical

car-density

between free and

congested

^flows in the observed

Q-k graph.

This

discontinuity

is understood _as the behaviors of two different kind of solutions of _our

dynamical model,

which _means the two different

phases

of _our

dynamical system.

^Thus _we stress the traflic flow

phenomena

with

congestion

_can be well described in the

concept

of

phase

transition.

Recently

several models have been

proposed

to understand the

phenomenon

_near the critical

density.

Navin and Hall

proposed

to describe the

system

on 3-dimensional _space

ik-u-Q)

with a smooth function based _on

catastrophe theory iii.

This allows _one to

reproduce

the

discontinuity

_on

2-parameter plane by projecting

the smooth function to this

plane.

The spontaneous

generation

of

congestion,

whicl~ _we bave demonstrated

by

_our

Optimal Velocity Model, suggests wl~y

tl~e

catastrophe tl~eory

describes tl~e traflic flow

quite

well. We know

some otl~er kinds of models wl~icl~ _can

generate

^{tl~e trailic}

congestion spontaneously.

Cellular Automaton models _are

proposed by Nagel

and

Scl~reckenberg _[16], Nagatani iii]

and

Kikucl~i,

Tadaki and Yukawa

[18].

^{In tl~ese models tl~e lane} is divided into lattice sites and eacl~ _car sl~ares _a lattice site and _moves from _a site to site in discrete time

steps according

to some

rule. Kikuchi et ai. also _propose another model based _on

coupled

_map lattice model

[18].

These models have also been successful in

generating congestion spontaneously.

It would be

interesting

to

investigate

the _common feature _to their treatments and _ours, whicl~ lead _{to a} successful

description

of tl~e traflic flow. Tl~e

l~ydrodynamical approach

based _on well known

Navier-Stokes

equation

^is also

proposed by

Kerner and Konl~àuser

[19].

Under _some

special condition, tl~ey

lead to tl~e

simple equation,

^which is very similar to _our model. The

sensitivity

m our model

corresponds

to the inverse of relaxation time.

We should stress that _our model is rather

powerful

in

dealing

with realistic trailic flow

phenomena,

because it is easy ^to indude modifications to

reproduce

realistic trailic flow. For

example,

_our model _can indude tl~e effects of different bel~avior of each driver

by introducing

tl~e

driver-dependence

of tl~e

sensitivity

_an and

Optimal Velocity

function

VniAx).

Acknowledgments

Tl~e autl~ors _{express our}

appreciation

for _many

stimulating

discussions witl~ T.

Nagatani,

M.

Kikucl~i,

S.

Yukawa,

S.

Sasa,

M. Tanaka. We also would like to tl~ank H. Ozaki and M. Kawal~ara for

providing

_us witl~ valuable observed data and F.

Hall,

Y.

Nogami

and otl~er members of Mcmaster

University

for encouragmg ^comments. One of tl~e autl~or

iY.S.)

would

specially

tl~ank to K.

Nagel

and M.

Scl~reckenberg

for frmtful discussions. We also thanks to A. Bordner for careful

reading

of the

manuscript.

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