12
Jam and Fundamental Diagram in Traffic Flow on Sag and Hill K.Komada S.Masukura T.Naga tani Shizuoka Univ. Japan
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.
