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A variational formulation for
higher order macroscopic traffic
flow models of the GSOM family
J.P. Lebacque
UPE-IFSTTAR-GRETTIA
Le Descartes 2, 2 rue de la Butte Verte
F93166 Noisy-le-Grand, France
Jean-patrick.lebacque@ifsttar.fr
M.M. Khoshyaran
E.T.C. Economics Traffic Clinic
34 Avenue des Champs-Elysées,
75008, Paris, France
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Outline of the
presentation
1. Basics
2. GSOM models
3. Lagrangian Hamilton-Jacobi and variational
interpretation of GSOM models
4. Analysis, analytic solutions
5. Numerical solution schemes
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Scope
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The GSOM model is close to the LWR model
It is nearly as simple (non trivial explicit solutions
fi)
But it accounts for driver variability (attributes)
More scope for lagrangian modeling, driver
interaction, individual properties
Admits a variational formulation
Expected benefits: numerical schemes, data
assimilation
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The LWR model
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The LWR model
in a nutshell
(notations)
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Introduced by Lighthill, Whitham (1955), Richards
(1956)
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The equations:
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Or:
 q

 0 Conservaton
i equation
t x
q  v
definit ionof v
v  Ve , x  Behavioural equation
 
 Qe , x   0
t t
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Solution of the inhomogeneous
Riemann problem
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Analytical solutions, boundary conditions,
node modeling
Density
Position
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The LWR model: supply /
demand
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the equilibrium supply e and demand  e
functions (Lebacque, 1993-1996)
Demand
Supply
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The LWR model: the
min formula
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The local supply and demand:
( x, t )  e x , t , x  
( x, t )   e x , t , x  
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The min formula
Qx, t   Min x, t , x, t 
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Usage: numerical schemes, boundary conditions
→ intersection modeling
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Inhomogeneous Riemann
problem (discontinuous FD)
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Solution with Min formula
Can be recovered by the variational formulation of LWR (Imbert Monneau
Zidani 2013)
(a,0 )
Time
q  min  a  a ,  b  b 
(b,0 )
(a )
(b )
Position
 a ,0   a1 q  if
q   a  a 
if
q   a  a 
 b,0  b1 q  if
q   b  b 
 a ,0   a
b
a  q
 a,0  a
b,0  b
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 b,0   b
if
q   b  b 
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GSOM models
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GSOM (Generic second
order modelling) models
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(Lebacque, Mammar, Haj-Salem 2005-2007)
In a nutshell
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Kinematic waves = LWR
Driver attribute dynamics
Includes many current macroscopic models
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GSOM Family: description
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Conservation of vehicles
(density)
t   x v   0
Variable fundamental diagram,
dependent on a driver attribute
(possibly a vecteur) I
Equation of evolution for I
following vehicle trajectories
(example: relaxation)
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v  , I 
I  t I  v  x I   f I 
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GSOM: basic equations
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Conservation des véhicules
 v

0
t
x
Dynamics of I along
trajectories
I Iv

   f I 
t
x
 I   t I  v x I   f I 
Variable fundamental Diagram
def
v  , I   v  , I 
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Example 0: LWR (Lighthill, Whitham,
Richards 1955, 1956)
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No driver attribute
One conservation
equation
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 v

 0 Equationde conservation
t
x
v  Ve  , x 
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Diagramme fondamental
 
 Qe , x   0
t t
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Example 1: ARZ (Aw,Rascle
2000, Zhang,2002)
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Lagrangien attribute I = difference between
actual and mean equilibrium speed
I  v  Ve   v  , I   I  Ve 
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Example 2: 1-phase Colombo model
(Colombo 2002, Lebacque Mammar Haj-Salem
2007)
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Variable FD (in the congested
domain + critical density)
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
q* 
 v0  
, I    I 
 


v0    1 
 max
Increasing
values of I
 The attribute I is the
parameter of the family
of FDs
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

def 
v  , I   



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Vma x 
Vma x  Vcrit

crit I 

q*  
 
 I   1 
   ma x 

Fundamental
Diagram (speeddensity)
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if
if
  crit I 
  crit I 
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Example 2 continued
(1-phase Colombo model)
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1-phase vs 2-phase: Flow-density FD
modéle 2-phase
modéle 1-phase
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Example 3: CremerPapageorgiou
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Based on the CremerPapageorgiou FD (Haj-Salem
2007)
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 

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

, I   V f I 1  
  max 

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l 32 I 




m
Example 4:
« multi-commodity » models
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GSOM Model
+ advection (destinations, vehicle
type)
= multi-commodity GSOM
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Example 5: multi-lane model
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Impact of multi-lane traffic
Two states: congestion (strongly
correlated lanes) et fluid (weakly
correlated lanes)
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 2 FDs separated by the phase
boundary R(v)
Relaxation towards each regime
 eulerian source terms
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Example 6: Stochastic GSOM
(Khoshyaran Lebacque 2007-2008)
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Idea:
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Conservation of véhicles
Fundamental Diagram
depends on driver
attribute I
I is submitted to
stochastic perturbations
(other vehicles, traffic
conditions, environment)
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 v

0
t
x
def
v  , I   v  , I 
 dBt 

I   I ,

dt 

 I  N , t; 
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Two fundamental properties of the
GSOM family
(homogeneous piecewise constant case)
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1. discontinuities of I propagate with the speed v of
traffic flow
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2. If the invariant I is initially piecewise constant, it
stays so for all times t > 0
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⇒ On any domain on which I
is uniform the GSOM
model simplifies to a translated LWR model (piecewise
LWR)
 
 , I   0
t x
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Inhomogeneous
Riemann problem
I = Il in all of sector (S)
I  I l in ( S )
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I = Ir in all of sector (T)
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vm  r m , I l   vr

I  I r in (T )
r 1
 m  
U l  U m : LWR kinematicwave
U m  U r : discontinuity of I
r
vr
l
vl
FDl
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vr , Il 
x
FDr
Generalized (translated) supply- Demand
(Lebacque Mammar Haj-Salem 2005-2007)
Example: ARZ model
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Translated Supply / Demand = supply resp demand
for the « translated » FD (with resp to I )
def

 , I   Max0 , I 
def

 , I   Max , I 
 

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Example of translated
supply / demand: the
ARZ family
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Solution of the
Riemann problem
(summary)
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Define the upstream demand,
the downstream supply (which
depend on Il ):
The intermediate state Um is
given by
The upstream demand and the
downstream supply (as
functions of initial conditions) :
Min Formula:
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def
def
l  e,l l , I l  , r  e,r m , I l 
Im  Il
vm  vr
i.e. r m , I l   vr  r r , I r 
 def
 l   l l , I l 
 def
1
1
 r   r m , I l   r r , vr , I l , I l   r r , r r , I r , I l , I l 
q0  Minl , r 
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Applications
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Analytical solutions
Boundary conditions
Intersection modeling
Numerical schemes
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Lagrangian GSOM; HJ and
variational interpretation
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Variational formulation of
GSOM models
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Motivation
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Numerical schemes (grid-free cf Mazaré et al 2011)
Data assimilation (floating vehicle / mobile data cf
Claudel Bayen 2010)
Advantages of variational principles
Difficulty
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Theory complete for LWR (Newell, Daganzo, also
Leclercq Laval for various representations )
Need of a unique value function
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Lagrangian conservation law
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Spacing r
The rate of variation of
spacing r depends on the
gradient of speed with
respect to vehicle index N
t
r
t r   N v  0
x
def
v  Ve 1 / r   V r 
The spacing is the inverse of density
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Lagrangian version of
the GSOM model
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Conservation law in lagrangian coordinates
Driver attribute equation (natural lagrangian expression)
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We introduce the position of vehicle N:
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Note that:
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Lagrangian Hamilton-Jacobi
formulation of GSOM (Lebacque
Khoshyaran 2012)
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Integrate the driver-attribute
equation
Solution:
FD Speed becomes a function of
driver, time and spacing
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Lagrangian Hamilton-Jacobi
formulation of GSOM
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Expressing the
velocity v as a
function of the
position X
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Associated optimization
problem
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Define:
Note implication: W concave with respect to r (ie
flow density FD concave with respect to density
Associated optimization problem
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Functions M and W
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Illustration of the
optimization problem:
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Initial/boundary
conditions:
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N T , T 
Blue: IC + trajectory
of first vh
Green: trajectories of t
vhs with GPS
Red: cumulative flow
on fixed detector
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C
N
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Elements of resolution
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Characteristics
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Optimal curves 
characteristics (Pontryagin)
Note: speed of
charcteristics: > 0
Boundary conditions
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Initial conditions for
characteristics
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N T , T 
Data  N , t0 
Data  N0 , t 
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IC
Vh trajectory
t
C
N
r0 t   W 1 N0 ,  t  N0 , t , t 
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Initial / boundary
condition
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Usually two solutions r0 (t )
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Data  N t , t 
N T , T 
t
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C
Example: Interaction of a shockwave
with a contact discontinuity
Eulerian view
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Initial
conditions:
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Top: eulerian
Down:
lagrangian
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Example: Interaction of a shockwave with
a contact discontinuity
Lagrangian view
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Another example:
total refraction of characteristics
and lagrangian supply
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???
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Decomposition property
(inf-morphism)
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The set of initial/boundary conditions
is union of several sets
Calculate a partial solution
(corresponding to a partial
set of IBC)
The solution is the min of partial
solutions
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C   C

The optimality pb can be
solved on characteristics only
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This is a Lax-Hopf like formula
Application: numerical schemes based on
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Piecewise constant data (including the system
yielding I )
Decomposition of solutions based on decomposition
of IBC
Use characteristics to calculate partial solutions
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Numerical scheme based on
characteristics (continued)
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If the initial condition on I is piecewise
constant 
the spacing along characteristics is
piecewise constant
Principle illustrated by the example:
interaction between shockwave and contact
discontinuity
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Alternate scheme: particle
discretization
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Particle discretization of HJ
Use charateristics
Yields a Godunov-like scheme (in
lagrangian coordinates)
BC: Upstream demand and downstream
supply conditions
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Numerical example
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Model: Colombo 1-phase, stochastic
Process for I : Ornstein-Uhlenbeck, two levels (high at the
beginning and end, low otherwise)  refraction of charateristics and
waves
Demand: Poisson, constant level
Supply: high at the beginning and end, low otherwise  induces
backwards propagation of congestion
Particles: 5 vehicles
Duration: 20 mn
Length: 3500 m
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Downstream supply
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I dynamics
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Particle trajectories
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Position of particles
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Conclusion
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Directions for future work:
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Problem of concavity of FD
Eulerian source terms
Data assimilation pbs
Efficient numerical schemes
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