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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 !
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
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
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
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)
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,…)
L. Leclercq (2013)
Impact of the lane agregation on the FD
Simulations are figures were kindly provided by Prof. Jorge Laval
L. Leclercq (2013)
Traffic Modelling
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
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
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
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
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
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
L. Leclercq (2013)
Overview of first order model solutions
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)
L. Leclercq (2013)
Solutions for an unsaturated traffic signal (2)
L. Leclercq (2013)
The Variational Theory
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
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)
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
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)
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
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
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
L. Leclercq (2013)
Extensions to the theory
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)
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
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
L. Leclercq (2013)
The network fundamental diagram
(Daganzo & Geroliminis, 2008)
L. Leclercq (2013)
Thank you for your attention LICIT – Ifsttar/Bron and ENTPE 25, avenue François Mitterand 69675 Bron Cedex - FRANCE [email protected]
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.
L. Leclercq (2013)
Exercices
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
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]
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]
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
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
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