ISTTT Tutorial Leclercq 20130716

8/12/2019 ISTTT Tutorial Leclercq 20130716

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Traffic Flow Theory

Ludovic Leclercq, University of Lyon, IFSTTAR / ENTPEJuly, 16th 2013

ISTTT20 - Tutorials

Mijn presentatie spreekt over de verkeersstroom theorie !


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Outline

Experimental evidences Traffic behavior on freeways The fundamental diagram

Traffic modeling The three representation of traffic flow The three kinds of traffic models Equilibrium (first order) model

Overview of first order model solutions The variational theory

General basis Connections between the three traffic representations

Some extensions to the theory


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Experimental evidences


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Traffic flow on a motorways (M6 in England)

Vitesse[km

/h]

S"

E!

S"

E!

S"

S"

E!

M7135

M7125

M7107

20

40

60

80

100

120

Speed[km

/h]

Data were kindly provided by the Highway Agency


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Traffic representation

5

t

xNGSIM Data I80/lane 4 USA

Vehicle dynamicspositionx

speed v

density kflow q

Microscopic vision

Macroscopic vision

30 km/h

Interactions

spacingsHeadway h

s

h

Shockwaves


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Flow / occupancy plot on a motorway (M6)

6

0 10 20 30 40 50 60 700

1000

2000

3000

4000

5000

6000

7000

8000

TO (%)

Dbit(veh

/h)

Occupancy [%] proxy for density

Flow

[veh/h]

The fundamentaldiagram (FD)


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Different definitions of the FD

0 10 20 30 40 50 60 700

1000

2000

3000

4000

5000

6000

7000

8000

TO (%)

Dbit(

veh/h)

Occupancy [%] proxy for density

Flow

[veh/h]

FD for monitoring FD for simulationq

k

Aggregation / impacts of local behavior (lane-changing, traffic composition,!)


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Impact of the lane agregation on the FD

Simulations are figures were kindly provided by Prof. Jorge Laval


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Traffic Modelling


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From discrete to continuous representations

q =! tN

k =!"xN

Space (X)Time (T)

Vehnum

ber(N) Eulerian coordinates

N(t,x)

T coordinatesT(t,n)

Lagrangian coordinatesX(t,n)

Moskowitzs surface


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The three representation of traffic flow

N(t,x) # of vehicles that have crossed locationxby time t

X(t,n) position of vehicle nat time t

T(n,x) time vehicle ncrosses locationx

Eulerian T coordinates Lagrangian

(Laval and Leclercq, 2013, part B)

q !"#$%!'#$% !()%


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Classical classification of traffic models

Macroscopic models Continuous representation Mainly deterministic Global behavior (may be distinguished per class) Equilibrium (1storder) / + transition states (2ndorder)

Microscopic models Discrete representation Mainly stochastic Local interactions (car-following)

Mesoscopic models Discrete or semi-discrete representation Intermediate level for traffic representation

(vehicle clusters or link servers)

For further details see the model treefrom (van Wageningen-Kessels, PhD, 2013)

*+,"-"."$,/ /%0(-

12345

Eulerian coordinates

Lagrangian coordinates

T coordinates


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Equilibrium macroscopic model (1)

The PDE expression in Eulerian coordinates

in Lagrangian coordinates

in T coordinates

flow

density

speed

spacing

kt+Q(k)x = 0

st+V(s)x = 0

rt+H(r)x = 0headway

pace


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Equilibrium macroscopic model (2)

The Hamilton-Jacobi (HJ) expression In Eulerian coordinates

In Lagrangian coordinates

In T coordinates

q=Q(k)

v=V(s)

h=H(r)

Appropriate expression ofthe FD


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Overview of first order model solutions


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17

A

A

A

B

A

CO

t

x

flow

density

O

C

B

A

Solutions for an unsaturated traffic signal (1)


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Solutions for an unsaturated traffic signal (2)


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The Variational Theory


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General considerations on the variations of N

y(t)

tA tB

"

A

Bx !NAB = dtN

tA

tB

" = #tN+y '(t)#nNtA

tB

"

!NAB = Q k(t)( )$y '(t)k(t)r(y '(t),k)

! "### $###tA

tB

"

t

k

q

k0

q0 y(t)

r(y(t),k0)-w

u

Same costs

Legendres transformation:

r(y '(t),k) !R(y '(t))=supk

r(y '(t),k)( )

y(t)R(y(t))

This makes costs independent from trafficstates but no longer from the paths

Equality is observed on the optimal wave paths


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Variational theory (VT) in Eulerian

General basis

q = Q(k) ! "tk = Q # "x k( )HJ Equation:

t

y(t)

tO(") tP

"

O(")

Px #

NP =min!"DP

NO !( ) +

# !( )( )

# !( ) = r(y '(t),k)dttO !( )

tP

$General expression for the solutions:

NP =min!"DP

NO !( ) + # ' !( )( )

# ' !( ) = R(y '(t))dttO !( )

tP

$

Key VT result usingthe Legendres transformation

VT is really useful with PWL FD(and especially triangular one)


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VT in Eulerian The Highway Problem

Link

x

t

N(x,t)

u

-w

N(xu,t-(x-xu)/u)

6 7

N(xd,t-(xd-x)/w)

6 !1"#$"5

N(x,t) = min N xu,t!

(x!xu)

u

"

#$%

&'

free!flow! "### $###

,N xd,t!

(xd!x)w

"

#$%

&'+((x

d!x)

congestion

! "##### $#####

)

*

++++

,

-

.

.

.

.

Newells model (1993) !!!

Triangular FD


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23

NNux(t)

Ny(t)(y-x)/w

(y-x).!

Nx(t)

t

x yxNx(t)

q

-wu

!

k

(x-x)/u

Classical formulation of the Newells N-curve model

Well-known as the three dectectors problem

Ndx(t)


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VT in Lagangian - the IVP problem

t

n

i-1

i

#Vehicle i-1 trajectory

B"0

"w!

XB =min X

O(!0)+ "(!0),XO(!w#) + "(!w#)( )

t0

XB = X(t,i) =min X t0 ,i( )+ u.(t! t0 )free!flow

! "### $###

,X t!1 / (w"),i!1( )!w.(1/ w"congestion

! "##### $#####

)#

$%%

&

'((

Newells model again !!! (2002)

speed

spacing

u

w!

1/!

-w


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Classical formulation of Newells car-following model

t

x

Leader veh i-1

veh i u!

w 1 / (w!),"1 /!( )

!w Effective trajectory

(Newell, 2002)

The simplest car-following ruleAccount for driver reaction time


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VT in T coordinates

headway

pace

n

x

xu

xd

T(x,n)

Passing time

T(xu,n)

T(xd,n-!(xd-x))

6 1"8"&59&

6 1"#$"59'

T(x,n) =max T xu ,n( )+(x!xu)

ufree!flow

! "### $###

,T xd,n !"(xd!x)( )!(xd! x)

wcongestion

! "##### $#####

#

$

%%

&

'

((

Mesoscopic model(Mahut, 2000; Leclercq & Becarie, 2012)


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The mesoscopic LWR model

!

!

"#$%

"

#

$$

"!

%&&'"

!&&'!'

!((

)*+&

)*+&'

!

"!&&-' "

!&&-./'


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Variational theory - summary

Variational theory exhibits the connections between thethree traffic representations for the LWR model

A unique model that leads to three solution methods(numerical scheme) corresponding to the three different

vision on traffic flow(macroscopic / mesoscopic / microscopic)

Some previous models appears to be particular cases forthe LWR model and a triangular fundamental diagram in

different systems of coordinates


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Extensions to the theory


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Diverge: Newells FIFO model30

Nx(t)N

t

Dy(t) Dy1(t)

Dy2(t)

1

Ny1(t)

m1

80%

20%x y

1

2

1

Ny2(t)q2

q'2

FIFO => Travel times should be equal whatever the destination is

(Newell, 1993)


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Merge: Daganzos model31

"1

2

"2

"

y1

2

Free-flow

congestion

(Daganzo, Transportation Research part B, 1995)

Priority to link 1

Priority to link 2

Shared priority

This model has been proved consistent with experimentalobservations a multitude of times


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Other extensions

:%,;0(0'##(-($''";F($)(#


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The network fundamental diagram

(Daganzo & Geroliminis, 2008)


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Thank you for your attention

LICIT Ifsttar/Bron and ENTPE

25, avenue Franois Mitterand

69675 Bron Cedex - FRANCE

[email protected]


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References

Daganzo, C.F., 2006b. In traffic flow, cellular automata = kinematic waves. Transportation Research B, 40(5), 396-403. Daganzo, C.F., 2005. A variational formulation of kinematic waves: basic theory and complex boundary conditions.

Transportation Research B, 39(2), 187-196.

Daganzo, C.F., 2005b. A variational formulation of kinematic waves: Solution methods. Transportation Research B, 39(10),934-950.

Daganzo, C.F., 1995. The cell transmission model, part II: network traffic. Transportation Research B, 29(2), 79-93. Daganzo, C.F., 1994. The cell transmission model: A dynamic representation of highway traffic consistent with the

hydrodynamic theory. Transportation Research B, 28(4), 269-287.

Daganzo, C.F., Menendez, M., 2005. A variational formulation of kinematic waves: bottlenecks properties and examples. In:Mahmassani H.S. (Ed.), 16

th

ISTTT, Elsevier, London, 345-364. Edie, L.C., 1963. Discussion of traffic stream measurements and definitions. In: J. Almond (Ed.), 2ndISTTT, OECD, Paris,

139-154.

Laval, J.A., Leclercq, L. The Hamilton-Jacobi partial differential equation and the three representations of traffic flow,Transportation Research part B, 2013, accepted for publication.

Leclercq, L., Laval, J.A., Chevallier, E., 2007. The Lagrangian coordinates and what it means for first order traffic flowmodels. In: Allsop, R.E., Bell, M.G.H., Heydecker, B.G. (Eds), 17thISTTT, Elsevier, London, 735-753.

Gerlough, D.L. et Huber M.J. Traffic flow theory a monograph. Special report n165, Transportation Research Board,Washington. 1975.

Lighthill, M.J et Whitham, J.B. On kinematic waves II. A theory of traffic flow in long crowded roads. Proceedings of the RoyalSociety, 1955, VolA229, p. 317-345.

Newell, G.F., 2002. A simplified car-following theory: a low-order model. Transportation Research B, 36(3), 195-205. Newell, G.F., 1993. A simplified theory of kinematic waves in highway traffic, I general theory; II queuing at freeway; III multi-

destination. Transportation Research B, 27(4), 281-313.

Richards, P.I., 1956. Shockwaves on the highway. Operations Research, 4, 42-51.


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Exercices


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Problem statement

Let consider a freeway with two lanes and the following FD:u=30 m/s ; w=4.28 m/s ; !=0.28 veh/m. Two points a et bare respectively located atx=0 m andx=3600 m.

The flow at ais constant and equal to 3000 veh/h. At timet=120 s, the capacity at is reduced from 1800 veh/h during10 minutes.

Draw the fundamental diagram Determine the N-curve atx=3600,x=1800,x=600 and

x=0 m

Provide an estimate for the maximal length of thecongestion


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The fundamental diagram

0 50 100 150 200 250 3000

500

1000

1500

2000

2500

3000

3500

4000

density [veh/km]

flow[

veh/h]


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N-curve atx=3600 m

0 500 1000 1500 20000

200

400

600

800

1000

1200

1400

1600

1800

Time [s]

N[

veh]


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N-curve atx=1800 m

0 500 1000 1500 20000

200

400

600

800

1000

1200

1400

1600

1800

Time [s]

N[

veh]

Nx


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N-curve atx=600 m

0 500 1000 1500 20000

200

400

600

800

1000

1200

1400

1600

1800

Time [s]

N[

veh]

Nx


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N-curve atx=0 m

0 500 1000 1500 20000

200

400

600

800

1000

1200

1400

1600

1800

Time [s]

N[

veh]

Nx

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ISTTT Tutorial Leclercq 20130716