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
uv
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
yJ
M-C Model: Equilibrium
Green Light equilibrium:

Or:
J  J 2  4u0 J
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 (u 0 ,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 (u 0 ,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
yJ
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:
yJ
Non-dimensionalization
~
y  y/J

t
u0 / J
Equilibria:
d~
y
y
 1 ~
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  RLGL(1 r )
r  RL
1  RLGL(1 r )
1
e 
e
1 r
1 r
UB 
1  e  RLGL(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
dy2
0
dt
dy2
y2
dy1
y1


v
 u0( 1 )
max
dt
J2
dt
J1
dy1
y
y
 u0( 1  1 )-v max 1
dt
J1
J1
dy2
y
 vmax 1
dt
J1
dy1
y
y
 u0( 1  1 )-v max 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:
;
dy2
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 y2 )  e r ~
y1
d
~
y 2  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

dy2
y
y
 αv max 1  vmax 2
dt
J
J
dy4
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
dy 2
y
 ( 1-)v max 3  vmax 2
dt
J
J
dy3
y
y
 u0( 1 3 )-v max 3
dt
J
J
y
dy4
y
 βvmax 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
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