L. Leclercq (2013)

Traffic Flow Theory

Ludovic Leclercq, University of Lyon, IFSTTAR / ENTPE July, 16th 2013

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Mijn presentatie spreekt over de verkeersstroom theorie !

L. Leclercq (2013)

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

L. Leclercq (2013)

Experimental evidences

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Traffic flow on a motorways (M6 in England) 6:00 6:30 7:00 7:30 8:00 8:30 9:00 9:30 10:00

t

x

Vit

ess

e [

km

/h]

E!

S"

M7135

M7125

M7107E! : entrée

S" : sortie

(a)

0

50

100

6:00 6:30 7:00 7:30 8:00 8:30 9:00 9:30 10:00

t

x

Vit

ess

e [

km

/h]

S"

E!

S"

E!

S"

S"

E!

M7135

M7125

M7107

(b)

0

20

40

60

80

100

120

Spe

ed [k

m/h

]

Data were kindly provided by the Highway Agency

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Traffic representation 5

t

x NGSIM Data – I80/lane 4 – USA

Vehicle dynamics position x speed v

density k flow q

Microscopic vision

Macroscopic vision

30 km/h

Interactions spacing s

Headway 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 (%)

Déb

it (v

eh/h

)

Occupancy [%] – proxy for density

Flow

[veh

/h]

The fundamental diagram (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 (%)

Déb

it (v

eh/h

)

Occupancy [%] – proxy for density

Flow

[veh

/h]

FD for monitoring FD for simulation q

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 = ∂t N

k = −∂xN

Space (X) Time (T)

Veh

num

ber (

N) Eulerian coordinates

N(t,x)

T coordinates T(t,n)

Lagrangian coordinates X(t,n)

Moskowitz’s surface

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From micro to macro: Edie’s definitions

I 1 I 2

1 2 3 4 5 6 7 8 9 10

200 m

nodes with traffic signals

(c)

I/Oï1

I/Oï7

I/Oï13

I/Oï19

I/Oï2

I/Oï8

I/Oï14

I/Oï20

I/Oï3

I/Oï9

I/Oï15

I/Oï21

I/Oï4

I/Oï10

I/Oï16

I/Oï22

I/Oï5

I/Oï11

I/Oï17

I/Oï23

I/Oï6

I/Oï12

I/Oï18

I/Oï24

1

7

13

19

25

31

2

8

14

20

26

32

3

9

15

21

27

33

4

10

16

22

28

34

5

11

17

23

29

35

6

12

18

24

30

36

200 m

(d)

(a)

Virtual loop detector

0

li

t t+6t

link i

tk(x)

tk(0)

tkïng(l

iïx)(li)

veh fx

x+6xS’w

(b)

x

t

- -o= 6 6 = 6 6q

d

x t k x t

' and

'i

kk

i

kk

Δx

Δt

L. Leclercq (2013)

The three representation of traffic flow

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

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

T(n,x) time vehicle n crosses location x

Eulerian T coordinates Lagrangian

(Laval and Leclercq, 2013, part B)

q Micro  Macro   Meso  

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Classical classification of traffic models

•  Macroscopic models –  Continuous representation –  Mainly deterministic –  Global behavior (may be distinguished per class) –  Equilibrium (1st order) / + transition states (2nd order)

•  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 tree from (van Wageningen-Kessels, PhD, 2013)

Equilibirum  model  (LWR)  

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 = 0 headway

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 of the FD

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Overview of first order model solutions

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17 k

q

�kc

A

B

C

O

(a1)

(a2)

A

A

A

B

A

CO

t

xrouge vert

OA

B

C

s

v

� sc

(b1)

(b2)

A

A A

CB

A

0 0

t

n

n0

x

rouge vertx

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

Γ A

B x

ΔNAB = dtNtA

tB

∫ = ∂t N + y '(t)∂n NtA

tB

∫

ΔNAB = 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

Legendre’s transformation:

r(y '(t),k) ≤ R(y '(t)) = supk

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

y’(t) R(y’(t))

This makes costs independent from traffic states 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) ⇔ ∂t k =Q −∂x k( )HJ Equation:

t

y’(t)

tO(Γ) tP

Γ O(Γ)

P x Ψ

NP = minΓ∈DP

NO Γ( ) + Δ Γ( )( )Δ Γ( ) = r(y '(t),k)dt

tO Γ( )

tP∫General expression for the solutions:

NP = minΓ∈DP

NO Γ( ) + Δ ' Γ( )( )Δ ' Γ( ) = R(y '(t))dt

tO Γ( )

tP∫

Key VT result using the Legendre’s transformation

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

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VT in Eulerian – The Highway Problem Li

nk

x

t

N(x,t)

u

-w

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

+  0  

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

+  κ(xd-­‐x)  

N (x,t) = min N xu ,t −(x − xu )

u⎛⎝⎜

⎞⎠⎟

free− flow

, N xd ,t −(xd − x)

w⎛⎝⎜

⎞⎠⎟+κ (xd − x)

congestion

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

Newell’s model (1993) !!!

Triangular FD

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23

N Nu

x’(t)

Ny(t) (y-x’)/w

(y-x

’).κ

Nx’(t)

t

x y x’ Nx(t) q

-w u

κ

k

(x’-x)/u

Classical formulation of the Newell’s 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

Ψ Vehicle i-1 trajectory

B Γ0

Γwκ

XB = min XO(Γ0 )+ Δ(Γ0 ),XO(Γwκ )

+ Δ(Γwκ )( )

t0

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

,X t −1/ (wκ ),i −1( )−w.(1 /wκ

congestion )

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Newell’s model again !!! (2002)

speed

spacing

u

wκ

1/κ

-w

L. Leclercq (2013)

Classical formulation of Newell’s car-following model

t

x Leader – veh i-1

veh i u w 1 / (wκ ),−1 /κ( )

w

Effective trajectory

(Newell, 2002)

The simplest car-following rule Account for driver reaction time

L. Leclercq (2013)

VT in T coordinates

headway

pace

n

x

xu

xd

T(x,n)

Passing time

T(xu,n)

T(xd,n-κ(xd-x))

+  (x-­‐xu)/u  

+  (xd-­‐x)/w  

T (x,n) = max T xu ,n( ) + (x − xu )ufree− flow

,T xd ,n −κ (xd − x)( )− (xd − x)w

congestion

⎛

⎝⎜⎜

⎞

⎠⎟⎟

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

L. Leclercq (2013)

The mesoscopic LWR model

(a)

0

L

Link

t

x

W ïW

t0s(i)t

L(iïngL)

L/w

Veh i

Veh iïngL

tL(i’) t

L(i’+1)

(b)

0

L

Link

t

x

Wi

ïWi

t0s(i)t

L(iïng

mL)

L/wm

Veh i

Veh iïngmL

tL(i’) t

L(i’+1)

set A

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Variational theory - summary

•  Variational theory exhibits the connections between the three 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 for the 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: Newell’s FIFO model 30

Nx(t) N

t

Dy(t) Dy1(t)

Dy2(t) µ1

Ny1(t)

µm1

80%

20% x y

1

2

µ1

Ny2(t) q2

q'2

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

(Newell, 1993)

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Merge: Daganzo’s model 31

λ1

2

λ2

α

µ

y 1

2

µ

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 experimental observations a multitude of times

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Other extensions

Bounded  accelera<on  

Mul<class  

Fixed  and  moving  

boAlenecks  Mul<lanes  

Lane-­‐changing  

and  relaxa<on  

Complex  intersec<ons  

Hybrida<on  

Need to be coupled with a route choice model to simulate a network

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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 François Mitterand 69675 Bron Cedex - FRANCE leclercq@entpe.fr

L. Leclercq (2013)

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.), 16th ISTTT, Elsevier, London, 345-364. •  Edie, L.C., 1963. Discussion of traffic stream measurements and definitions. In: J. Almond (Ed.), 2nd ISTTT, 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 flow

models. In: Allsop, R.E., Bell, M.G.H., Heydecker, B.G. (Eds), 17th ISTTT, Elsevier, London, 735-753. •  Gerlough, D.L. et Huber M.J. Traffic flow theory – a monograph. Special report n°165, 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 Royal

Society, 1955, Vol A229, 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 ; κ=0.28 veh/m. Two points a et b are respectively located at x=0 m and x=3600 m.

The flow at a is constant and equal to 3000 veh/h. At time t=120 s, the capacity at is reduced from 1800 veh/h during 10 minutes.

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

x=0 m •  Provide an estimate for the maximal length of the

congestion

L. Leclercq (2013)

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 at x=3600 m

0 500 1000 1500 20000

200

400

600

800

1000

1200

1400

1600

1800

Time [s]

N [v

eh]

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N-curve at x=1800 m

0 500 1000 1500 20000

200

400

600

800

1000

1200

1400

1600

1800

Time [s]

N [v

eh]

Nx

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N-curve at x=600 m

0 500 1000 1500 20000

200

400

600

800

1000

1200

1400

1600

1800

Time [s]

N [v

eh]

Nx

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N-curve at x=0 m

0 500 1000 1500 20000

200

400

600

800

1000

1200

1400

1600

1800

Time [s]

N [v

eh]

Nx