Jam and Fundamental Diagram in Traffic Flow on

Sag and Hill

K.Komada S.Masukura T.Nagatani

Shizuoka Univ. 　 Japan

Purpose of Study

• Proposal of traffic model including the gravitational force　　－ We extend the optimal velocity model to study the jamming transition induced by the gravitational force.

• Fundamental diagrams for the traffic flow on sag and hill　　－ We study the flow, traffic states ,and jamming transitions induced by sag and hill.

• Jam induced by sag　　－ We clarify the relationship between densities before and after the jam from the theoretical current curves.

Traffic model

dt

tdxxBmgxF

mdt

txd iiii )()(sin)()(2

2

)(sin)(

)()(

2

2

ii

ii xBmg

dt

tdxxF

dt

txdm

Equation of motion on uphill

θ

mg

mgsin θ

mgcos θ

dt

tdxxVa

dt

txd ii

i )()(

)(2

2

sensitivity

)(sin)(

)( iii

xBmgxFxV

Δ

ma

Extended Optimal velocity Function

ccif xxx

vtanhtanh

2max,

　　　 depends on the gradient of max,,upgv )(sin ixBmg

)( ixB About

ix

ix

)( ixB

)( ixB

for → ∞

→ 0for

→ 1

→ 0

We extend the OV model and obtain the following

)tanh()tanh(2 ,,

max,,bupbupi

upg xxxv

ccif

i xxxv

xV tanhtanh2max,

)tanh()tanh(2 ,,

max,,bupbupi

upg xxxv

ccif

i xxxv

xV tanhtanh2max,

)tanh()tanh(2max,

ccif

i xxxv

xV

bdownbdownidowng xxx

v,,

max,, tanhtanh2

①OV function on normal section

② Extended OV function on uphill section

③Extended OV function on downhill sectionO

ptim

al V

eloc

ity

Headway

Vf,max

xcxdown,b

Vg,down,max

Opt

imal

Vel

ocit

y

Headway

Vf,max

xc(=xup,b)

Vg,up,max

①②③①

Simulation method• Single lane • The periodic boundary condition• Forth-order Runge-Kutta method

Values of parameters• Number of cars N=20

0• Length of road L=N×Δ

x

• LN1=LD1=LU1=LN2=L/4

• Time interval isΔ ｔ＝１ / １２８• Vf,max=2.0,x ｃ＝ 4.0

LN1 LD1 LU1LN2

0.5

0.4

0.3

0.2

0.1

0.0

Cur

rent

0.60.50.40.30.20.10.0

Density

Sag(a=1.5) Sag(a=3.0) Theory

Vf,max=2.0

Vg,down,max=0.5

Vg,up,max=0.5

Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ

High sensitivity⇒3 traffic statesLow sensitivity ⇒5 traffic states

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Vel

ocit

y

10008006004002000

Position

N1 N2D1 U1

a=1.5 a=3.0

Velocity profile （ ρ=0.17 ）

Velocity profile （ ρ=0.19 ）

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Vel

ocit

y

10008006004002000

Position

N1 N2D1 U1

a=1.5 a=3.0

Traffic jam induced by sag＋ oscillating jam at low sensitivity

Sensitivity:a=3.0>ac=2.0(critical value)

Sensitivity:a=1.5<ac=2.0(critical value)

Fundamental diagram （ Xc=Xdown,b=Xup,b ）

Traffic jam induced by sag

Relationship between headway profile and

theoretical current （ X ｃ =Xup,b=Xdown,b ）

00 / xxVQth ΔΔ

Headway profile(ρ=0.16)

Headway profile(ρ=0.20)

Theoretical current

（ in the case of no jam at high sensitivity ）

Steady state ： Headways are the same.　　　　 　 Velocities are Optimal Velocity.

0.5

0.4

0.3

0.2

0.1

0.0

Cur

rent

0.60.50.40.30.20.10.0

Density

c

b

a

Current(Vg,down,max =0.5) Current(Vf,max =2.0) Current(Vg,up,max =0.5)d

e

Maximal value of the current of the Up Hill

12

10

8

6

4

2

0

Hea

dway

10008006004002000

Position

N1 N2D1 U1

lJL

b b

c

d e

Sag

12

10

8

6

4

2

0

Hea

dway

120010008006004002000

Position

N1 N2D1 U1

lJL

b b

c

a

e

Sag

Velocity profile （ ρ=0.16 ）

Headway profile （ ρ=0.16 ）xc=xup,b≠xdown,b ：「 the different case 」（ ca

se1 ） xc=xup,b=xdown,b ：「 the same case 」（ case2 ）

(3) of case2 is not consistent with that of case1 but (1) and (2) case 2 agree with those of case1. (1)Free traffic

(2)Traffic with saturated current

(3) Congested traffic

3 traffic states

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Vel

ocity

12001000800600400200

Position

N1 D1 U1 N2

lJL

case1 case2

12

10

8

6

4

2

0

Hea

ddw

ay

12001000800600400200

Position

N1 D1 U1 N2

lJL

case1 case2

B Ba

C

e E

A

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Cur

rent

0.60.50.40.30.20.10.0

Density

a=3.0 sag(xc=xup,b≠ xdown,b)

sag(xc=xup,b=xdown,b) Theory

xc=xup,b=4.0

xdown,b=2.0

xc=xup,b=xdown,b=4.0

xc=4.0

Fundamental diagram （ Xc=Xdown,b≠Xup,b ）

Relationship between headway profile and theoretical current （ Xc=Xdown,b≠Xup,b ）

Headway profile(ρ=0.16)

Headway profile(ρ=0.20) The length of jam shorten.Headway get narrow.

In the case of Xc=Xdown,b≠Xup,b

0.5

0.4

0.3

0.2

0.1

0.0

Cur

rent

0.60.50.40.30.20.10.0

Density

A

Current(Vg,down,max=0.5,

xdown,b=2.0)

Current(Vf,max=2.0)

Current(Vg,up,max=0.5,

xup,b=4.0)

B

C DE

Maximal value of the current of the Up Hill

12

10

8

6

4

2

0

Hea

ddw

ay

12001000800600400200

Position

N1 D1 U1 N2

lJL

case1 case2

B Ba

C

e E

A

12

10

8

6

4

2

0

Hea

dway

1000800600400200

Position

N1 D1 U1 N2

lJL

case1 case2

B B

C

D

eE

The dependence of traffic flow on the gradient

Velocity profile(ρ=0.20)

Headway profile(ρ=0.20)

　 As the gradient is high, the maximum velocity become lower and higher on up- and down-hills respectively. 　

0.8

0.6

0.4

0.2

0.0

Cur

rent

0.60.50.40.30.20.10.0

Density

a=3.0Sag(Vg,up,max=Vg,down,max=0.5)

Sag(Vg,up,max=Vg,down,max=1.0)

Sag(Vg,up,max=Vg,down,max=1.5) Theory

Vf,max=2.0

Vg,up,max=0.5

Vg,up,max=1.0

Vg,up,max=1.5

Vg,down,max=0.5

Vg,down,max=1.0

Vg,down,max=1.53.0

2.5

2.0

1.5

1.0

0.5

0.0

Vel

ocit

y

10008006004002000

Position

N1 N2D1 U1

Vf,max-Vg,up,max =0.5 Vf,max-Vg,up,max =1.0 Vf,max-Vg,up,max =1.5

35

30

25

20

15

10

5

0

Hea

dway

10008006004002000

Position

N1 N2D1 U1

Vf,max-Vg,up,max =0.5 Vf,max-Vg,up,max =1.0 Vf,max-Vg,up,max =1.5

The region of saturated flow extend.The maximum current is lower.

Fundamental diagram of traffic flow with two uphills

Headway profile(ρ=0.20)

Headway profile(ρ=0.20)

The traffic jam occurs just before the highest gradient.

LN1 LU2LN3

LU1LN2

LN1

0.4

0.3

0.2

0.1

0.0

Cur

rent

0.60.50.40.30.20.10.0

Density

a=3.0 Current Theory

Vmax=2.0

Vmax=1.5

Vmax=1.0

16

14

12

10

8

6

4

2

0

Hea

dway

1400120010008006004002000

Position

N1 N2 N3U1 U2

lJL

14

12

10

8

6

4

2

0

Hea

dway

10008006004002000

Position

N1 N2 N3U1 U2

lJL

Summary

●We have extended the optimal velocity model to take into account the gravitational force as an external force.

● We have clarified the traffic behavior for traffic flow on a highway with gradients

●We have showed where, when, and how the traffic jams occur on highway with gradients.

● We have studied the relationship between densities before and after the jam from the theoretical analysis.