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. 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

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

Equation of motion on uphill

About

→ 1

for

→ ∞

sensitivity

→ 0

for

→ 0

depends on the gradient of

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

  • Length of road L=N×Δx


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)

Headway profile(ρ=0.16)

Steady state: Headways are the same.

Velocities are Optimal Velocity.

Theoretical current

( in the case of no jam at high sensitivity)

Headway profile(ρ=0.20)


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

Headway profile(ρ=0.16)

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)

Headway profile(ρ=0.16)

In the case of

Xc=Xdown,b≠Xup,b

The length of jam shorten.

Headway get narrow.

Headway profile(ρ=0.20)


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.

Headway profile(ρ=0.20)


Fundamental diagram of traffic flow with two uphills

Headway profile(ρ=0.20)

The traffic jam occurs

just before the highest gradient.

Headway profile(ρ=0.20)


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.


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