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

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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.

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Traffic model
Equation of motion on uphill
About
for
? 8
? 1
sensitivity
? 0
for
? 0
depends on the gradient of
Extended Optimal velocity Function
We extend the OV model and obtain the following
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?
?
?
?
?OV function on normal section
? Extended OV function on uphill section
?Extended OV function on downhill section
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Simulation method

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

Values of parameters

• LN1LD1LU1LN2L/4
• Time interval is?t1/128
• Vf,max2.0,xc4.0

• Number of cars N200
• Length of road LN?x

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Fundamental diagram( XcXdown,bXup,b)
Sensitivitya3.0gtac2.0(critical
value) Sensitivitya1.5ltac2.0(critical value)
Traffic jam induced by sag
Velocity profile(?0.17)
Traffic jam induced by sag oscillating jam at
low sensitivity
High sensitivity?3 traffic states Low sensitivity
?5 traffic states
Velocity profile ( ?0.19 )
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Relationship between headway profile and
theoretical current(XcXup,bXdown,b)
Headway profile(?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(?0.20)
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Fundamental diagram( XcXdown,b?Xup,b)
Velocity profile(?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(?0.16)
xcxup,b?xdown,b?the different
case?(case1)xcxup,bxdown,b ?the same
case?(case2)
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Relationship between headway profile and
theoretical current ( XcXdown,b?Xup,b)
Headway profile(?0.16)
In the case of XcXdown,b?Xup,b
The length of jam shorten. Headway get narrow.
Headway profile(?0.20)
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The dependence of traffic flow on the gradient
Velocity profile(?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(?0.20)
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Fundamental diagram of traffic flow with two
uphills
Headway profile(?0.20)
The traffic jam occurs just before the highest
gradient.
Headway profile(?0.20)
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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.