42
L. Leclercq (2013) Traffic Flow Theory Ludovic Leclercq, University of Lyon, IFSTTAR / ENTPE July, 16th 2013 ISTTT20 - Tutorials Mijn presentatie spreekt over de verkeersstroom theorie !

Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

  • Upload
    others

  • View
    4

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

Traffic Flow Theory

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

ISTTT20 - Tutorials

Mijn presentatie spreekt over de verkeersstroom theorie !

Page 2: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

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

Page 3: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

Experimental evidences

Page 4: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 5: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 6: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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)

Page 7: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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,…)

Page 8: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

Impact of the lane agregation on the FD

Simulations are figures were kindly provided by Prof. Jorge Laval

Page 9: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

Traffic Modelling

Page 10: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 11: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 12: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

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  

Page 13: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 14: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 15: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 16: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

Overview of first order model solutions

Page 17: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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)

Page 18: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

Solutions for an unsaturated traffic signal (2)

Page 19: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

The Variational Theory

Page 20: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 21: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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)

Page 22: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 23: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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)

Page 24: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

1/κ

-w

Page 25: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

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

Page 26: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

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)

Page 27: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

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

Page 28: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 29: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

Extensions to the theory

Page 30: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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)

Page 31: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 32: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 33: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

The network fundamental diagram

(Daganzo & Geroliminis, 2008)

Page 34: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

Thank you for your attention LICIT – Ifsttar/Bron and ENTPE 25, avenue François Mitterand 69675 Bron Cedex - FRANCE [email protected]

Page 35: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

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.

Page 36: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

Exercices

Page 37: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 38: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

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]

Page 39: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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]

Page 40: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 41: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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

Page 42: Traffic Flow Theory - victorknoop.eu · three traffic representations for the LWR model • A unique model that leads to three solution methods (numerical scheme) ... 2006b. In traffic

L. Leclercq (2013)

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