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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

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Jam and Fundamental Diagram in Traffic Flow on Sag and Hill

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### Jam and Fundamental Diagram in Traffic Flow on Sag and Hill

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

Equation of motion on uphill

→ 1

for

→ ∞

sensitivity

→ 0

for

→ 0

Extended Optimal

velocity Function

We extend the OV model and obtain the following

①OV function on normal section

② Extended OV function on uphill section

③Extended OV function on downhill section

### Simulation method

• Single lane

• The periodic boundary condition

• Forth-order Runge-Kutta method

Values of parameters

• LN1=LD1=LU1=LN2=L/4

• Time interval isΔｔ＝１/１２８

• Vf,max=2.0,xｃ＝4.0

• Number of cars N=200

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

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

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

Traffic jam induced by sag

Velocity profile（ρ=0.17）

Traffic jam induced by sag

＋oscillating jam at low sensitivity

High sensitivity⇒3 traffic states

Low sensitivity ⇒5 traffic states

Velocity profile （ ρ=0.19）

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

Velocities are Optimal Velocity.

Theoretical current

（ in the case of no jam at high sensitivity）

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

Velocity profile（ρ=0.16）

3 traffic states

(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

xc=xup,b≠xdown,b：「the different case」（case1）xc=xup,b=xdown,b ：「the same case」（case2）

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

In the case of

Xc=Xdown,b≠Xup,b

The length of jam shorten.

### The dependence of traffic flow on the gradient

Velocity profile(ρ=0.20)

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

The region of saturated flow extend.

The maximum current is lower.

Fundamental diagram of traffic flow with two uphills

The traffic jam occurs

### 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