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GRETTIA 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 [email protected] M.M. Khoshyaran E.T.C. Economics Traffic Clinic 34 Avenue des Champs-Elysées, 75008, Paris, France www.econtrac.com ISTTT 20, July 17-19, 2013 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 ISTTT 20, July 17-19, 2013 GRETTIA www.econtrac.com GRETTIA Scope www.econtrac.com 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 ISTTT 20, July 17-19, 2013 GRETTIA The LWR model ISTTT 20, July 17-19, 2013 www.econtrac.com The LWR model in a nutshell (notations) GRETTIA www.econtrac.com Introduced by Lighthill, Whitham (1955), Richards (1956) The equations: Or: q 0 Conservaton i equation t x q v definit ionof v v Ve , x Behavioural equation Qe , x 0 t t ISTTT 20, July 17-19, 2013 Solution of the inhomogeneous Riemann problem GRETTIA www.econtrac.com Analytical solutions, boundary conditions, node modeling Density Position ISTTT 20, July 17-19, 2013 The LWR model: supply / demand the equilibrium supply e and demand e functions (Lebacque, 1993-1996) Demand Supply ISTTT 20, July 17-19, 2013 GRETTIA www.econtrac.com The LWR model: the min formula GRETTIA www.econtrac.com The local supply and demand: ( x, t ) e x , t , x ( x, t ) e x , t , x The min formula Qx, t Min x, t , x, t Usage: numerical schemes, boundary conditions → intersection modeling ISTTT 20, July 17-19, 2013 Inhomogeneous Riemann problem (discontinuous FD) GRETTIA www.econtrac.com 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 ISTTT 20, July 17-19, 2013 b,0 b if q b b GRETTIA GSOM models ISTTT 20, July 17-19, 2013 www.econtrac.com GSOM (Generic second order modelling) models www.econtrac.com (Lebacque, Mammar, Haj-Salem 2005-2007) In a nutshell GRETTIA Kinematic waves = LWR Driver attribute dynamics Includes many current macroscopic models ISTTT 20, July 17-19, 2013 GRETTIA GSOM Family: description 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) www.econtrac.com v , I I t I v x I f I ISTTT 20, July 17-19, 2013 GRETTIA GSOM: basic equations www.econtrac.com 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 ISTTT 20, July 17-19, 2013 Example 0: LWR (Lighthill, Whitham, Richards 1955, 1956) No driver attribute One conservation equation www.econtrac.com v 0 Equationde conservation t x v Ve , x GRETTIA Diagramme fondamental Qe , x 0 t t ISTTT 20, July 17-19, 2013 Example 1: ARZ (Aw,Rascle 2000, Zhang,2002) GRETTIA www.econtrac.com Lagrangien attribute I = difference between actual and mean equilibrium speed I v Ve v , I I Ve ISTTT 20, July 17-19, 2013 Example 2: 1-phase Colombo model (Colombo 2002, Lebacque Mammar Haj-Salem 2007) Variable FD (in the congested domain + critical density) GRETTIA www.econtrac.com q* v0 , I I v0 1 max Increasing values of I The attribute I is the parameter of the family of FDs ISTTT 20, July 17-19, 2013 def v , I Vma x Vma x Vcrit crit I q* I 1 ma x Fundamental Diagram (speeddensity) ISTTT 20, July 17-19, 2013 if if crit I crit I GRETTIA www.econtrac.com Example 2 continued (1-phase Colombo model) 1-phase vs 2-phase: Flow-density FD modéle 2-phase modéle 1-phase ISTTT 20, July 17-19, 2013 GRETTIA www.econtrac.com Example 3: CremerPapageorgiou Based on the CremerPapageorgiou FD (Haj-Salem 2007) GRETTIA www.econtrac.com , I V f I 1 max ISTTT 20, July 17-19, 2013 l 32 I m Example 4: « multi-commodity » models GRETTIA www.econtrac.com GSOM Model + advection (destinations, vehicle type) = multi-commodity GSOM ISTTT 20, July 17-19, 2013 Example 5: multi-lane model GRETTIA www.econtrac.com Impact of multi-lane traffic Two states: congestion (strongly correlated lanes) et fluid (weakly correlated lanes) 2 FDs separated by the phase boundary R(v) Relaxation towards each regime eulerian source terms ISTTT 20, July 17-19, 2013 GRETTIA Example 6: Stochastic GSOM (Khoshyaran Lebacque 2007-2008) Idea: Conservation of véhicles Fundamental Diagram depends on driver attribute I I is submitted to stochastic perturbations (other vehicles, traffic conditions, environment) www.econtrac.com v 0 t x def v , I v , I dBt I I , dt I N , t; ISTTT 20, July 17-19, 2013 Two fundamental properties of the GSOM family (homogeneous piecewise constant case) GRETTIA www.econtrac.com 1. discontinuities of I propagate with the speed v of traffic flow 2. If the invariant I is initially piecewise constant, it stays so for all times t > 0 ⇒ On any domain on which I is uniform the GSOM model simplifies to a translated LWR model (piecewise LWR) , I 0 t x ISTTT 20, July 17-19, 2013 Inhomogeneous Riemann problem I = Il in all of sector (S) I I l in ( S ) I = Ir in all of sector (T) GRETTIA www.econtrac.com 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 ISTTT 20, July 17-19, 2013 vr , Il x FDr Generalized (translated) supply- Demand (Lebacque Mammar Haj-Salem 2005-2007) Example: ARZ model GRETTIA www.econtrac.com Translated Supply / Demand = supply resp demand for the « translated » FD (with resp to I ) def , I Max0 , I def , I Max , I ISTTT 20, July 17-19, 2013 Example of translated supply / demand: the ARZ family ISTTT 20, July 17-19, 2013 GRETTIA www.econtrac.com Solution of the Riemann problem (summary) 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: GRETTIA www.econtrac.com 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 Minl , r ISTTT 20, July 17-19, 2013 GRETTIA Applications Analytical solutions Boundary conditions Intersection modeling Numerical schemes ISTTT 20, July 17-19, 2013 www.econtrac.com Lagrangian GSOM; HJ and variational interpretation ISTTT 20, July 17-19, 2013 GRETTIA www.econtrac.com Variational formulation of GSOM models www.econtrac.com Motivation GRETTIA Numerical schemes (grid-free cf Mazaré et al 2011) Data assimilation (floating vehicle / mobile data cf Claudel Bayen 2010) Advantages of variational principles Difficulty Theory complete for LWR (Newell, Daganzo, also Leclercq Laval for various representations ) Need of a unique value function ISTTT 20, July 17-19, 2013 GRETTIA Lagrangian conservation law www.econtrac.com 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 ISTTT 20, July 17-19, 2013 Lagrangian version of the GSOM model GRETTIA www.econtrac.com Conservation law in lagrangian coordinates Driver attribute equation (natural lagrangian expression) We introduce the position of vehicle N: Note that: ISTTT 20, July 17-19, 2013 Lagrangian Hamilton-Jacobi formulation of GSOM (Lebacque Khoshyaran 2012) Integrate the driver-attribute equation Solution: FD Speed becomes a function of driver, time and spacing ISTTT 20, July 17-19, 2013 GRETTIA www.econtrac.com Lagrangian Hamilton-Jacobi formulation of GSOM Expressing the velocity v as a function of the position X ISTTT 20, July 17-19, 2013 GRETTIA www.econtrac.com Associated optimization problem Define: Note implication: W concave with respect to r (ie flow density FD concave with respect to density Associated optimization problem ISTTT 20, July 17-19, 2013 GRETTIA www.econtrac.com GRETTIA Functions M and W ISTTT 20, July 17-19, 2013 www.econtrac.com Illustration of the optimization problem: Initial/boundary conditions: GRETTIA www.econtrac.com N T , T Blue: IC + trajectory of first vh Green: trajectories of t vhs with GPS Red: cumulative flow on fixed detector ISTTT 20, July 17-19, 2013 C N GRETTIA Elements of resolution ISTTT 20, July 17-19, 2013 www.econtrac.com GRETTIA Characteristics Optimal curves characteristics (Pontryagin) Note: speed of charcteristics: > 0 Boundary conditions ISTTT 20, July 17-19, 2013 www.econtrac.com Initial conditions for characteristics GRETTIA www.econtrac.com N T , T Data N , t0 Data N0 , t IC Vh trajectory t C N r0 t W 1 N0 , t N0 , t , t ISTTT 20, July 17-19, 2013 Initial / boundary condition Usually two solutions r0 (t ) GRETTIA www.econtrac.com Data N t , t N T , T t ISTTT 20, July 17-19, 2013 C Example: Interaction of a shockwave with a contact discontinuity Eulerian view ISTTT 20, July 17-19, 2013 GRETTIA www.econtrac.com GRETTIA www.econtrac.com Initial conditions: Top: eulerian Down: lagrangian ISTTT 20, July 17-19, 2013 Example: Interaction of a shockwave with a contact discontinuity Lagrangian view ISTTT 20, July 17-19, 2013 GRETTIA www.econtrac.com Another example: total refraction of characteristics and lagrangian supply ??? ISTTT 20, July 17-19, 2013 GRETTIA www.econtrac.com Decomposition property (inf-morphism) 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 ISTTT 20, July 17-19, 2013 GRETTIA www.econtrac.com C C The optimality pb can be solved on characteristics only GRETTIA www.econtrac.com This is a Lax-Hopf like formula Application: numerical schemes based on Piecewise constant data (including the system yielding I ) Decomposition of solutions based on decomposition of IBC Use characteristics to calculate partial solutions ISTTT 20, July 17-19, 2013 Numerical scheme based on characteristics (continued) GRETTIA www.econtrac.com 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 ISTTT 20, July 17-19, 2013 Alternate scheme: particle discretization GRETTIA www.econtrac.com Particle discretization of HJ Use charateristics Yields a Godunov-like scheme (in lagrangian coordinates) BC: Upstream demand and downstream supply conditions ISTTT 20, July 17-19, 2013 GRETTIA Numerical example www.econtrac.com 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 ISTTT 20, July 17-19, 2013 GRETTIA Downstream supply ISTTT 20, July 17-19, 2013 www.econtrac.com I dynamics GRETTIA www.econtrac.com ISTTT 20, July 17-19, 2013 Particle trajectories GRETTIA www.econtrac.com ISTTT 20, July 17-19, 2013 Position of particles ISTTT 20, July 17-19, 2013 GRETTIA www.econtrac.com GRETTIA Conclusion Directions for future work: Problem of concavity of FD Eulerian source terms Data assimilation pbs Efficient numerical schemes ISTTT 20, July 17-19, 2013 www.econtrac.com GRETTIA www.econtrac.com ISTTT 20, July 17-19, 2013