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.