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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)
*+,"-"."$,/ /%0(-
12345
Eulerian coordinates
Lagrangian coordinates
T coordinates
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Equilibrium macroscopic model (1)
The PDE expression in Eulerian coordinates
in Lagrangian coordinates
in T coordinates
flow
density
speed
spacing
kt+Q(k)x = 0
st+V(s)x = 0
rt+H(r)x = 0headway
pace
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Equilibrium macroscopic model (2)
The Hamilton-Jacobi (HJ) expression In Eulerian coordinates
In Lagrangian coordinates
In T coordinates
q=Q(k)
v=V(s)
h=H(r)
Appropriate expression ofthe FD
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Overview of first order model solutions
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A
A
A
B
A
CO
t
x
flow
density
O
C
B
A
Solutions for an unsaturated traffic signal (1)
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Solutions for an unsaturated traffic signal (2)
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The Variational Theory
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General considerations on the variations of N
y(t)
tA tB
"
A
Bx !NAB = dtN
tA
tB
" = #tN+y '(t)#nNtA
tB
"
!NAB = Q k(t)( )$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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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
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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