macroscopic ode models of traffic flow zhengyi zhou 04/01/2010

Macroscopic ODE Models of Traffic Flow

Zhengyi Zhou

04/01/2010

Introduction

Traffic Flow Models Microscopic – ODE Macroscopic – PDE

Macroscopic ODE Models?

Basics

Red light:

Green light:

Goal: find y(t) MATLAB ODE numerical solver “ode15s”

uInflow, vOutflow,

vudt

dy

Total Link Volume = y

udt

dy

vudt

dy

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 Red Light:

Green Light:

u0 > u0 - v0: linear growth

u0 = u0 - v0 :equilibrium

u0 < u0 - v0 :linear decay

0udt

dy

00 vudt

dy

RL = GL = 20; u0 =1

v0 =2 v0 =1 v0 =3

Outline

Constant Model


Case-by-Case Model



McCartney & Carey’s Model: Intro McCartney & Carey (1999) Logistic outflow

, when

v = 0, when y > J

v = outflow; y = link vol

J = jam vol; tau = trip time

)1(J

yyv

Jy

M-C Model

Red: Green:

y > J

0udt

dy )1(0 J

yyu

dt

dy

Jy

0udt

dy

J = 800 J = 900 u0 = 10, τ = 10RL = GL = 25

M-C Model: Equilibrium

Green Light equilibrium:

Or:

Green Light equilibrium exists when

System equilibrium

Equilibrium range

Exists when

eg: does not exist when J = 800, u0 = τ = 10 (J/4u0τ = 2)

exists when J = 900, u0 = τ =10 (J/4u0τ = 2.25)

0)1(0 J

yyu

dt

dy

04uJ

04uJ

2

4 02 JuJJ

y

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


Case-by-Case Model



Case-by-Case Model

Cruising speed = c

Max outflow

vmax = (1 vehicle)/ (time for it to exit) = = c / lcl /

1

0u

L

l

Case-by-Case Model: Three cases1. No waiting line

When

Call N = (no waiting line volume)

2. Maximum waiting line

When

Call J = (jam volume)

3. Some waiting line

When N < y < J

00 uc

Ly

c

Lu0

l

Ly

l

L

y0

N

J

No waiting line

Some waiting line

Max waiting line

Case-by-Case Model: u & v

y

N

0

J

Inflow Outflow

u = 0 v = vmax

v = vmaxu = min (u0, vmax)

u = u0 v = min (u0, vmax)No waiting line

Some waiting line

Max waiting line

Case-by-Case Model: Equations

Red Light: if y N

if N < y < J

if y = J

Green Light: if y N

if N < y < J

if y = J

0udt

dy

),v(udt

dymax0min

0dt

dy

),v(uudt

dymax00 min

maxmax0min v),v(udt

dy

maxvdt

dy

Case-by-Case Model: PlotRL = GL = 20; L = 600; l = 6; c = 30 J = 100; vmax = 5

u0 = 2 u0 = 4 u0 = 6

Case-by-Case Model: Analysis

No Congestion

Cyclical Congestion

Constant Congestion/ “Crawling”

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


Case-by-Case Model



Density-Dependent Model: Intro Drivers continuously & spontaneously adjust

to existing traffic on link

Inflow and outflow are both density-dependent

DD Model: u & v

Inflow: ↓ linearly as link volume ↑ Outflow: ↑ linearly as link volume ↑

y/J)(uu 10 /J)yvv max(

DD Model: Equations

Red Light: if y < J

if

Green Light: if y < J

if

y/J)(udt

dy 10

0dt

dyJy

(y/J)vy/J)(udt

dymax0 1

maxvdt

dy Jy

DD Model: Plot

RL = GL = 20; J = 50; vmax = 5

u0 = 5u0 = 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 ModelDensity-Dependent Model

DD Model: Analysis Equilibrium range of link volume

Independent of initial volume on link

y0 = 100y0 = 50y0 = 0

J = 100; u0 = 4; vmax = 5; RL=GL=20

DD Model: Analysis

Red: if y < J; if

Green: if y < J; if

y/J)(udt

dy 10 0

dt

dy

(y/J)vy/J)(udt

dymax0 1 maxv

dt

dy

Jy Jy

Non-dimensionalizationJyy /~

Ju

t

/0

Red:

Green:

where r =

dτ

yd~y~1

dτ

yd~yry ~~1

0

max

u

v

Equilibria:

Red:

stable

Green: stablery

1

1~

1~ y0~

dτ

yd

01~1~ )y(yd

d

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

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600 1.800 2.000r

link

vo

l (d

ime

ns

ion

les

s)

RL=GL=1

RL=GL=2

predictedWeighted average of equilibriums

• Center lower; approach to green equilibrium is faster

• RL/GL ↑, center↓

DD Model: Solutions

Solve ODEs by discretization

Red:

……………………………(1)

Green:

……………….(2)

LBy )0(~ UBRLy )(~

RLeLBUB )1(1

LBGLy )(~

}1]1)1{[(1

1 )1(

rGLeUBrr

LB

UB

LB

ydτ

yd ~1~

UBy )0(~

yrydτ

yd ~~1~

DD Model: Solutions

)1(

)1(

11

1)

1

11(

rGLRL

rGLRL

er

er

eLB

)1(

)1(

11

1

11

rGLRL

rGLRLRL

e

er

er

r

UB

Outline

Constant Model


Case-by-Case Model



DD Model Application: Light Synchronization

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

Link 1 Link 2

Light 1 Light 2

Two Lights: Equations

L1 & L2 are red:

L1 is red & L2 is green:

L1 is green & L2 is red:

L1 & L2 are green:

)J

y(u

dt

dy

1

10

1 1 02 dt

dy

)J

y(u

dt

dy

1

10

1 12

2max

2

J

yv

dt

dy

1

1max

1

10

1 1J

y)-v

J

y(u

dt

dy

1

1max

2

J

yv

dt

dy

1

1max

1

10

1 1J

y)-v

J

y(u

dt

dy

2

2max

1

1max

2

J

yv

J

yv

dt

dy

Two Lights: Plot

u0 = 5J1 = J2 = 100

φ = 0

φ = π

φ = π/2

φ = 3π/2

Three Lights in Phase

All red: ; ;

All green:

)J

y(u

dt

dy

1

10

1 1 02 dt

dy03

dt

dy

1

1max

1

10

1 1J

yv)

J

y(u

dt

dy

2

2max

1

1max

2

J

yv

J

yv

dt

dy

3

3max

2

2max

3

J

yv

J

yv

dt

dy

Three Lights: Plot

Link 1 (RL/GL)Link 2 (RL/GL)Link 3 (RL/GL)

u0 = 6, vmax = 5, J1 = J2 = J3 = 100 and RL = GL = 20

Three Lights in Phase Delay Effect Smoothing Effect

Nested equilibrium ranges

Three Lights in Phase Independent of initial link volumes


Three Lights in Phase Independent of jam vol (link length) on different

links


Three Lights in Phase Non-Dimensionalization

Red:

Green:

Integrating Factor =

Later link’s y = integral of previous link’s y

Smoothing

11 ~1

~y

dτ

yd 0

~2

dτ

yd0

~3

dτ

yd

111 ~~1

~yry

dτ

yd 21

2 ~~~

yryrdτ

yd

323 ~~

~yryr

dτ

yd

212 ~~

~yryr

dτ

yd 12

2 ~~~

yryrd

yd

re

12~)~( yeye

d

d rr

dyeey rr 12

~~

Outline

Constant Model


Case-by-Case Model



DD Application 2: Traffic Junction

Traffic Junction: Equations

Light12 is green, Light34 is red:

Light12 is red, Light34 is green:

J

yv)

J

y(u

dt

dy 1max

10

1 1 J

yv

J

yαv

dt

dy 2max

1max

2

)J

y(u

dt

dy 30

3 1 J

yv

J

yα)v(

dt

dy 4max

1max

4 1

)J

y(u

dt

dy 10

1 1J

yv

J

y)v-(

dt

dy 2max

3max

2 1

J

y)-v

J

y(u

dt

dy 3max

30

3 1 J

yv

J

yβv

dt

dy 4max

3max

4

Traffic Junction: Plot1

Link 1 Link 3

Link 2 Link 4

α = β = 0.9

u0 = 6, vmax = 5, J = 100, RL = GL = 20

Traffic Junction: Plot2

Link 1 Link 3

Link 2 Link 4

α = β = 0.6

Conclusions & Further ResearchSummary 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

macroscopic ode models of traffic flow zhengyi zhou 04/01/2010

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