Traffic Flow Modeling on Road Networks Using Hamilton-Jacobi Equations Guillaume Costeseque Inria Sophia-Antipolis M´ editerran´ ee ITS seminar, UC Berkeley October 09, 2015 G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 1 / 79
Motivation Traffic flows on a network [Caltrans, Oct. 7, 2015] G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 2 / 79
Motivation Traffic flows on a network [Caltrans, Oct. 7, 2015] Road network ≡ graph made of edges and vertices G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 2 / 79
Motivation Breakthrough in traffic monitoring Traffic monitoring “old”: loop detectors at fixed locations (Eulerian) “new”: GPS devices moving within the traffic (Lagrangian) Data assimilation of Floating Car Data [Mobile Millenium, 2008] G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 3 / 79
Motivation Outline Introduction to traffic 1 Micro to macro in traffic models 2 Variational principle applied to GSOM models 3 HJ equations on a junction 4 Conclusions and perspectives 5 G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 4 / 79
Introduction to traffic Outline Introduction to traffic 1 Micro to macro in traffic models 2 Variational principle applied to GSOM models 3 HJ equations on a junction 4 Conclusions and perspectives 5 G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 5 / 79
Introduction to traffic Macroscopic models Convention for vehicle labeling N Flow x t G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 6 / 79
Introduction to traffic Macroscopic models Three representations of traffic flow Moskowitz’ surface N Flow x t x See also [Makigami et al, 1971], [Laval and Leclercq, 2013] G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 7 / 79
Introduction to traffic Macroscopic models Notations: macroscopic N ( t , x ) vehicle label at ( t , x ) N ( t + ∆ t , x ) − N ( t , x ) the flow Q ( t , x ) = lim , ∆ t ∆ t → 0 x N ( x , t ± ∆ t ) N ( t , x ) − N ( t , x + ∆ x ) the density ρ ( t , x ) = lim , ∆ x ∆ x → 0 x ∆ x N ( x ± ∆ x , t ) the stream speed (mean spatial speed) V ( t , x ). G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 8 / 79
Introduction to traffic Macroscopic models Macroscopic models Hydrodynamics analogy Two main categories: first and second order models Two common equations: ∂ t ρ ( t , x ) + ∂ x Q ( t , x ) = 0 conservation equation (1) Q ( t , x ) = ρ ( t , x ) V ( t , x ) definition of flow speed Q ( x + ∆ x , t )∆ t Q ( x , t )∆ t ρ ( x , t )∆ x x + ∆ x x G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 9 / 79
Introduction to traffic Focus on LWR model First order: the LWR model LWR model [Lighthill and Whitham, 1955], [Richards, 1956] Scalar one dimensional conservation law ∂ t ρ ( t , x ) + ∂ x F ( ρ ( t , x )) = 0 (2) with F : ρ ( t , x ) �→ Q ( t , x ) =: F x ( ρ ( t , x )) G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 10 / 79
Introduction to traffic Focus on LWR model Overview: conservation laws (CL) / Hamilton-Jacobi (HJ) Eulerian Lagrangian t − x t − n Variable Density ρ Spacing r CL Equation ∂ t ρ + ∂ x F ( ρ ) = 0 ∂ t r + ∂ n V ( r ) = 0 Variable Label N Position X � + ∞ � + ∞ N ( t , x ) = ρ ( t , ξ ) d ξ X ( t , n ) = r ( t , η ) d η x n HJ Equation ∂ t N + H ( ∂ x N ) = 0 ∂ t X + V ( ∂ n X ) = 0 Hamiltonian H ( p ) = − F ( − p ) V ( p ) = − V ( − p ) G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 11 / 79
Introduction to traffic Focus on LWR model Fundamental diagram (FD) Flow-density fundamental diagram F Empirical function with ρ max the maximal or jam density, ρ c the critical density Flux is increasing for ρ ≤ ρ c : free-flow phase Flux is decreasing for ρ ≥ ρ c : congestion phase Flow , F Flow , F Flow , F Density , ρ Density , ρ Density , ρ ρ max ρ max ρ max 0 0 0 [Garavello and Piccoli, 2006] G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 12 / 79
Introduction to traffic Second order models Motivation for higher order models Experimental evidences fundamental diagram: multi-valued in congested case [S. Fan, U. Illinois], NGSIM dataset G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 13 / 79
Introduction to traffic Second order models Motivation for higher order models Experimental evidences fundamental diagram: multi-valued in congested case phenomena not accounted for: bounded acceleration, capacity drop... Need for models able to integrate measurements of different traffic quantities (acceleration, fuel consumption, noise) G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 14 / 79
Introduction to traffic Second order models GSOM family [Lebacque, Mammar, Haj-Salem 2007] Generic Second Order Models (GSOM) family ∂ t ρ + ∂ x ( ρ v ) = 0 Conservation of vehicles , ∂ t ( ρ I ) + ∂ x ( ρ vI ) = ρϕ ( I ) Dynamics of the driver attribute I , v = I ( ρ, I ) Fundamental diagram , (3) Specific driver attribute I the driver aggressiveness, the driver origin/destination or path, the vehicle class, ... Flow-density fundamental diagram F : ( ρ, I ) �→ ρ I ( ρ, I ) . G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 15 / 79
Introduction to traffic Second order models GSOM family [Lebacque, Mammar, Haj-Salem 2007] Generic Second Order Models (GSOM) family ∂ t ρ + ∂ x ( ρ v ) = 0 Conservation of vehicles , (3) ∂ t I + v ∂ x I = ϕ ( I ) Dynamics of the driver attribute I , v = I ( ρ, I ) Fundamental diagram , Specific driver attribute I the driver aggressiveness, the driver origin/destination or path, the vehicle class, ... Flow-density fundamental diagram F : ( ρ, I ) �→ ρ I ( ρ, I ) . G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 15 / 79
Introduction to traffic Second order models GSOM family [Lebacque, Mammar, Haj-Salem 2007] Generic Second Order Models (GSOM) family ∂ t ρ + ∂ x ( ρ v ) = 0 Conservation of vehicles , (3) ∂ t I + v ∂ x I = 0 Dynamics of the driver attribute I , v = I ( ρ, I ) Fundamental diagram , Specific driver attribute I the driver aggressiveness, the driver origin/destination or path, the vehicle class, ... Flow-density fundamental diagram F : ( ρ, I ) �→ ρ I ( ρ, I ) . G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 15 / 79
Micro to macro in traffic models Outline Introduction to traffic 1 Micro to macro in traffic models 2 Variational principle applied to GSOM models 3 HJ equations on a junction 4 Conclusions and perspectives 5 G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 16 / 79
Micro to macro in traffic models Homogenization Setting Space ( i − 1) ( i ) t �→ x i ( t ) trajectory of vehicle i i = discrete position index ( i ∈ Z ) Spacing n = continuous (Lagrangian) variable Speed x Headway n = i ε and t = ε s ε > 0 a scale factor t Time G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 17 / 79
Micro to macro in traffic models Homogenization Setting Space ( i − 1) ( i ) t �→ x i ( t ) trajectory of vehicle i i = discrete position index ( i ∈ Z ) Spacing n = continuous (Lagrangian) variable Speed x Headway n = i ε and t = ε s ε > 0 a scale factor t Time Proposition (Rescaled positions) Define x i ( s ) = 1 � t � ε X ε ( ε s , i ε ) ⇐ ⇒ X ε ( t , n ) = ε x ⌊ n ε ⌋ ε G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 17 / 79
Micro to macro in traffic models Homogenization General result Let consider the simplest microscopic model; x i ( t ) = F ( x i − 1 ( t ) − x i ( t )) ˙ (4) the LWR macroscopic model (HJ equation in Lagrangian): ∂ t X 0 = F ( − ∂ n X 0 ) (5) G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 18 / 79
Micro to macro in traffic models Homogenization General result Let consider the simplest microscopic model; x i ( t ) = F ( x i − 1 ( t ) − x i ( t )) ˙ (4) the LWR macroscopic model (HJ equation in Lagrangian): ∂ t X 0 = F ( − ∂ n X 0 ) (5) Theorem ((Monneau) Convergence to the viscosity solution) � t � with ( x i ) i ∈ Z solution of (4) and X 0 the unique If X ε ( t , n ) := ε x ⌊ n ε ⌋ ε solution of HJ (5), then under suitable assumptions, | X ε − X 0 | L ∞ ( K ) − ε → 0 0 , → ∀K compact set . G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 18 / 79
Micro to macro in traffic models Multi-anticipative traffic Toy model Vehicles consider m ≥ 1 leaders First order multi-anticipative model m � x i ( t + τ ) = max ˙ 0 , V max − f ( x i − j ( t ) − x i ( t )) (6) j =1 f speed-spacing function non-negative non-increasing G. Costeseque (Inria) HJ on networks Berkeley, Oct. 09 2015 19 / 79
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