Macroscopic ODE Models of Traffic Flow with Lights
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Transcript Macroscopic ODE Models of Traffic Flow with Lights
Macroscopic ODE Models
of Traffic Flow
Zhengyi Zhou
04/01/2010
Introduction
Traffic Flow Models
Microscopic – ODE
Macroscopic – PDE
Macroscopic ODE Models?
Basics
Total Link Volume = y
Inflow,u
dy
uv
dt
Outflow,v
dy
Red light: dt u
dy
Green light: u v
dt
Goal: find y(t)
MATLAB ODE numerical solver “ode15s”
Outline
Constant Model
McCartney & Carey’s Model
Case-by-Case Model
Density-Dependent Model
Applied to a sequence of lights
Applied to a traffic junction
Constant Model
dy
u0
dt
Green Light: dy
u 0 v0
dt
Red Light:
u0 > u0 - v0:
linear growth
RL = GL = 20; u0 =1
u0 = u0 - v 0 :
equilibrium
u0 < u0 - v 0 :
linear decay
v0 =2
v0 =1
v0 =3
Outline
Constant Model
McCartney & Carey’s Model
Case-by-Case Model
Density-Dependent Model
Applied to a sequence of lights
Applied to a traffic junction
McCartney & Carey’s Model: Intro
McCartney & Carey (1999)
Logistic outflow
v
y
y
(1 )
J
v = 0,
, when y J
when y > J
v = outflow; y = link vol
J = jam vol; tau = trip time
M-C Model
Red:
dy
u0
dt
Green:
dy
y
y
u0 (1 )
dt
J
dy
u0
dt
J = 800
J = 900
u0 = 10, τ = 10
RL = GL = 25
y>J
yJ
M-C Model: Equilibrium
Green Light equilibrium:
2
J
J
4u0 J
Or: y
2
dy
y
y
u0 (1 ) 0
dt
J
Green Light equilibrium exists when J 4u0
System equilibrium
Equilibrium range
Exists when J 4u0
eg: does not exist when J = 800, u0 = τ = 10 (J/4u0τ = 2)
exists when J = 900, u0 = τ =10 (J/4u0τ = 2.25)
M-C Model: Features
Predict congestion
If congested: onset time of congestion
If not: equilibrium range of link volume
No mechanism to un-jam
Outline
Constant Model
McCartney & Carey’s Model
Case-by-Case Model
Density-Dependent Model
Applied to a sequence of lights
Applied to a traffic junction
Case-by-Case Model
l
u0
L
Cruising speed = c
Max outflow
vmax = (1 vehicle)/ (time for it to exit)
1
=
l/c
=c/l
Case-by-Case Model: Three cases
1.
No waiting line
J
When 0 y L u0
Max
waiting
line
c
Call N = Lu0 (no waiting line volume)
Some
waiting
line
c
2.
Maximum waiting line
3.
L
When y
l
Call J = L (jam volume)
l
N
No
waiting
line
Some waiting line
0
When N < y < J
y
Case-by-Case Model: u & v
Inflow
Outflow
u=0
J
u = min (u0, vmax)
Max
waiting
line
Some
waiting
line
v = vmax
v = vmax
N
No
waiting
line
u = u0
0
y
v = min (u0, vmax)
Case-by-Case Model: Equations
Red Light:
dy
u0
dt
if y N
dy
min (u0 ,vmax )
dt
dy
0
dt
if N < y < J
if y = J
Green Light: dy u0 min (u0 ,vmax ) if y N
dt
dy
min (u0 ,vmax ) vmax
dt
dy
vmax
dt
if y = J
if N < y < J
Case-by-Case Model: Plot
RL = GL = 20; L = 600; l = 6; c = 30 J = 100; vmax = 5
u0 = 2
u0 = 4
u0 = 6
Case-by-Case Model: Analysis
Constant
Congestion/
“Crawling”
Cyclical
Congestion
No Congestion
u0 = 2
u0 = 4
u0 = 6
Case-by-Case Model: Features
All features of M-C Model
3 congestion levels
Specific time periods of congestion
No permanent congestion
Disadvantage: discrete cases
Critical link vol (N or J) for behavioral changes
Outline
Constant Model
McCartney & Carey’s Model
Case-by-Case Model
Density-Dependent Model
Applied to a sequence of lights
Applied to a traffic junction
Density-Dependent Model: Intro
Drivers continuously & spontaneously adjust
to existing traffic on link
Inflow and outflow are both densitydependent
DD Model: u & v
Inflow: ↓ linearly as link volume ↑
Outflow: ↑ linearly as link volume ↑
u u0( 1 y/J)
v (vmax /J)y
DD Model: Equations
Red Light:
dy
u0( 1 y/J)
dt
dy
0 if y
dt
Green Light:
if y < J
J
dy
u0( 1 y/J) vmax(y/J)
dt
dy
vmax
dt
if
yJ
if y < J
DD Model: Plot
RL = GL = 20; J = 50; vmax = 5
u0 = 5
u0 = 20
DD Model vs. Case-by-Case Model
DD Model
Superior: model driver’s behaviors better
Constant adjustment less likely to jam
Fewer cars get through
Same parameter values:
J = 100; u0 = 4
vmax = 5
RL = GL = 20
Case-by-Case Model
Density-Dependent Model
DD Model: Analysis
Equilibrium range of link volume
Independent of initial volume on link
J = 100; u0 = 4; vmax = 5; RL=GL=20
y0 = 100
y0 = 50
y0 = 0
DD Model: Analysis
dy
dy
0 if y J
u0( 1 y/J) if y < J;
dt
dt
dy
dy
Green: u0( 1 y/J) vmax(y/J) if y < J; dt vmax if
dt
Red:
yJ
Non-dimensionalization
~
y y/J
t
u0 / J
Equilibria:
d~y
1 ~
y
dτ ~
Red:
Green:
dy
~
1 ~
y
r
y
dτ
where r =
vmax
u0
d~y
Red: dτ 0 ~y 1
d
~
(
1
y ) 1 0 stable
d~
y
Green:
1
~
y
1 r
stable
DD Model: Rate of approach
Switch to approach 2 stable equilibria
stable equilibrium range
Approach at the same rate?
If yes, center of equilibrium range = weighted
average of 2 equilibrium points
Numerical simulations:
DD Model: Rate of Approach
1.000
0.900
link vol (dimensionless)
0.800
0.700
0.600
0.500
0.400
RL=GL=1
• Center lower; approach to
0.300
RL=GL=2
green equilibrium is faster
0.200
predicted
Weighted average of
equilibriums
• RL/GL ↑, center↓
0.100
0.000
0.200 0.400 0.600 0.800 1.000 r 1.200 1.400 1.600 1.800 2.000
DD Model: Solutions
Solve ODEs by discretization
Red:
UB
d~
y
1 ~
y
dτ
~
y (0) LB
LB
~
y ( RL) UB
UB 1 (1 LB)e RL
……………………………(1)
d~
y
1 ~
y r~
y
dτ
Green:
~
y (0) UB
~
y (GL) LB
1
LB
{[(1 r )UB 1]e GL (1 r ) 1}
1 r
……………….(2)
DD Model: Solutions
LB
(1 e
RL
1
1
GL (1 r )
)e
1 r
1 r
1 e RLGL (1 r )
r RL
1 RLGL (1 r )
1
e
e
1 r
1 r
UB
1 e RLGL (1 r )
Outline
Constant Model
McCartney & Carey’s Model
Case-by-Case Model
Density-Dependent Model
Applied to a sequence of lights
Applied to a traffic junction
DD Model Application: Light Synchronization
Light 1
Link 1
Light 2
Link 2
Outflow of Link 1 = Inflow of Link 2
Optimal synchronization for smoothest flow
Light 1: red if sin(t) > 0; green if sin(t) < 0
Light 2: red if sin(t+φ) > 0; green if sin(t+φ) <0
φ : phase difference, 0 ≤ φ < 2π
Two Lights: Equations
L1 & L2 are red: dy
1
dt
u0( 1
L1 is red & L2 is green:
L1 is green & L2 is red:
L1 & L2 are green:
y1
)
J1
dy 2
0
dt
dy2
y2
dy1
y1
v
u0( 1 )
max
dt
J2
dt
J1
dy1
y
y
u0( 1 1 )-vmax 1
dt
J1
J1
dy2
y
vmax 1
dt
J1
dy1
y
y
u0( 1 1 )-vmax 1
dt
J1
J1
dy2
y
y
vmax 1 vmax 2
dt
J1
J2
φ=0
Two Lights: Plot
φ=π
φ = π/2
φ = 3π/2
u0 = 5
J1 = J2 = 100
Three Lights in Phase
All
dy1
y1
u
(
1
red: dt 0 J1 )
All green:
;
dy 2
0
dt
;
dy1
y
y
u0( 1 1 ) vmax 1
dt
J1
J1
dy2
y
y
vmax 1 vmax 2
dt
J1
J2
dy3
y
y
vmax 2 vmax 3
dt
J2
J3
dy3
0
dt
Three Lights: Plot
u0 = 6, vmax = 5, J1 = J2 = J3 = 100 and
RL = GL = 20
Link 1 (RL/GL)
Link 2 (RL/GL)
Link 3 (RL/GL)
Three Lights in Phase
Delay Effect
Smoothing Effect
Nested equilibrium ranges
Three Lights in Phase
Independent of initial link volumes
Link 1 (RL/GL)
Link 2 (RL/GL)
Link 3 (RL/GL)
Three Lights in Phase
Independent of jam vol (link length) on different
links
Link 1 (RL/GL)
Link 2 (RL/GL)
Link 3 (RL/GL)
Three Lights in Phase
Non-Dimensionalization
d~
y1
1 ~
y1
dτ
Red:
Green:
d~
y1
1 ~
y1 r~
y1
dτ
d~
y2
r~
y1 r~
y2
dτ
d~
y3
0
dτ
d~
y2
0
dτ
d~
y2
r~
y1 r~
y2
dτ
d~
y3
r~
y2 r~
y3
dτ
d~
y2 ~
ry2 r~
y1
d
Integrating Factor = e r
d r ~
( e y 2 ) e r ~
y1
d
~
y2 e r e r ~
y1d
Later link’s y = integral of previous link’s y
Smoothing
Outline
Constant Model
McCartney & Carey’s Model
Case-by-Case Model
Density-Dependent Model
Applied to a sequence of lights
Applied to a traffic junction
DD Application 2: Traffic Junction
Traffic Junction: Equations
Light12 is green, Light34 is red:
dy1
y
y
u0( 1 1 ) vmax 1
dt
J
J
dy3
y
u0( 1 3 )
dt
J
dy 2
y
y
αv max 1 vmax 2
dt
J
J
dy 4
y
y
( 1 α)v max 1 vmax 4
dt
J
J
Light12 is red, Light34 is green:
dy1
y
u0( 1 1 )
dt
J
y
dy2
y
( 1-)vmax 3 vmax 2
dt
J
J
dy 3
y
y
u0( 1 3 )-v max 3
dt
J
J
y
dy 4
y
βv max 3 vmax 4
dt
J
J
Traffic Junction: Plot1
α = β = 0.9
Link 1
Link 3
Link 2
Link 4
u0 = 6, vmax
= 5, J = 100,
RL = GL = 20
Traffic Junction: Plot2
α = β = 0.6
Link 1
Link 3
Link 2
Link 4
Conclusions & Further Research
Summary
Case-by-Case Model
Density-Dependent Model
Applied to a sequence of lights and a junction
Further Research
Different RL/GL in DD equilibrium range analysis
Traffic junction with fewer simplifying assumptions
Compare with macroscopic PDE models
Delay differential equations
References & Acknowledgements
McCartney, M. and Carey, M. “Modeling Traffic Flow:
Solving and Interpreting Differential Equations”,
Teaching Mathematics and Its Applications 18, no. 3
(1999): 118-119.
MATLAB
Professor Gallegos, Buckmire, Cowieson &
Lawrence
Math Department
Friends