Optimization Criteria for Modelling Intersections of Vehicular - PowerPoint PPT Presentation

Optimization Criteria for Modelling Intersections of Vehicular Traffic Flow Salissou Moutari 1 Michael Herty 2 and Michel Rascle 1 1 Laboratoire J. A. Dieudonné, University of Nice-Sophia Antipolis 2 Fachbereich Mathematik, TU Kaiserslautern

Outline � Introduction » Outline � The “Aw-Rascle” Traffic model Introduction “Aw-Rascle” Model � Junctions Modelling Junctions � Numerical Examples ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 2

Outline � Introduction » Outline � The “Aw-Rascle” Traffic model Introduction “Aw-Rascle” Model � Junctions Modelling Junctions � Numerical Examples ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 2

Outline � Introduction » Outline � The “Aw-Rascle” Traffic model Introduction “Aw-Rascle” Model � Junctions Modelling Junctions � Numerical Examples ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 2

Outline � Introduction » Outline � The “Aw-Rascle” Traffic model Introduction “Aw-Rascle” Model � Junctions Modelling Junctions � Numerical Examples ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 2

» Outline Introduction » Goal “Aw-Rascle” Model Introduction Junctions ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 3

Goal � Boundary Conditions and Riemann Problem for the » Outline Introduction » Goal “Aw-Rascle” Model through a junction: “Aw-Rascle” Model Junctions ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 4

Goal � Boundary Conditions and Riemann Problem for the » Outline Introduction » Goal “Aw-Rascle” Model through a junction: “Aw-Rascle” Model Junctions � Preserve the mass flux and the pseudo momentum ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 4

Goal � Boundary Conditions and Riemann Problem for the » Outline Introduction » Goal “Aw-Rascle” Model through a junction: “Aw-Rascle” Model Junctions � Preserve the mass flux and the pseudo momentum � Maximize the total flux at the junction ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 4

Goal � Boundary Conditions and Riemann Problem for the » Outline Introduction » Goal “Aw-Rascle” Model through a junction: “Aw-Rascle” Model Junctions � Preserve the mass flux and the pseudo momentum � Maximize the total flux at the junction � Holden & Risebro (1995), Coclitte, Garavello & Piccoli (2005), Garavello & Piccoli (2005), . . . ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 4

» Outline Introduction “Aw-Rascle” Model » A-R Model » A-R model on a network The “Aw-Rascle” macroscopic model » Riemann Problem » Optimization Junctions ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 5

The “Aw-Rascle” Model The “Aw-Rascle” (AR) macroscopic model of traffic flow: » Outline Introduction “Aw-Rascle” Model » A-R Model » A-R model on a network » Riemann Problem » Optimization Junctions ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 6

The “Aw-Rascle” Model The “Aw-Rascle” (AR) macroscopic model of traffic flow: » Outline Introduction “Aw-Rascle” Model � » A-R Model ∂ t ρ + ∂ x ( ρ v ) = 0, » A-R model on a network (1) » Riemann Problem ∂ t ( ρ w ) + ∂ x ( ρ vw ) = 0, » Optimization Junctions ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 6

The “Aw-Rascle” Model The “Aw-Rascle” (AR) macroscopic model of traffic flow: » Outline Introduction “Aw-Rascle” Model � » A-R Model ∂ t ρ + ∂ x ( ρ v ) = 0, » A-R model on a network (1) » Riemann Problem ∂ t ( ρ w ) + ∂ x ( ρ vw ) = 0, » Optimization Junctions where, � ρ : a dimensionless local density (the fraction of space occupied by cars), � v : the macroscopic velocity of cars � and w : a Lagrangian marker. E.g. w = v + p ( ρ ) , where ρ �− → p ( ρ ) is a known function such that ρ p ′′ ( ρ ) + 2 p ′ ( ρ ) > 0. (2) ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 6

The “Aw-Rascle” model � The system (1) is strictly hyperbolic (except for ρ = 0 ). » Outline Introduction “Aw-Rascle” Model » A-R Model » A-R model on a network » Riemann Problem » Optimization Junctions ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 7

The “Aw-Rascle” model � The system (1) is strictly hyperbolic (except for ρ = 0 ). » Outline Introduction � The eigenvalues of the 2 × 2 matrix “Aw-Rascle” Model » A-R Model » A-R model on a network λ 1 = v − ρ p ′ ( ρ ) » Riemann Problem λ 2 = v and » Optimization Junctions ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 7

The “Aw-Rascle” model � The system (1) is strictly hyperbolic (except for ρ = 0 ). » Outline Introduction � The eigenvalues of the 2 × 2 matrix “Aw-Rascle” Model » A-R Model » A-R model on a network λ 1 = v − ρ p ′ ( ρ ) » Riemann Problem λ 2 = v and » Optimization Junctions � λ 1 is genuinely nonlinear: 1-shock or 1-rarefaction whose curves coincide here (Temple system). ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 7

The “Aw-Rascle” model � The system (1) is strictly hyperbolic (except for ρ = 0 ). » Outline Introduction � The eigenvalues of the 2 × 2 matrix “Aw-Rascle” Model » A-R Model » A-R model on a network λ 1 = v − ρ p ′ ( ρ ) » Riemann Problem λ 2 = v and » Optimization Junctions � λ 1 is genuinely nonlinear: 1-shock or 1-rarefaction whose curves coincide here (Temple system). � λ 2 is linearly degenerate: 2-contact discontinuity. ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 7

The “Aw-Rascle” model on a network Basic notations: » Outline Introduction “Aw-Rascle” Model » A-R Model » A-R model on a network » Riemann Problem » Optimization Junctions ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 8

The “Aw-Rascle” model on a network Basic notations: » Outline Introduction Road Network: Finite directed graph G = ( I , N ) “Aw-Rascle” Model » A-R Model » A-R model on a network » Riemann Problem with |I| = I and |N | = N . » Optimization Junctions � Each arc i = 1 . . . I corresponds to a road. � Each vertex n = 1 . . . N corresponds to junction. ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 8

The “Aw-Rascle” model on a network Basic notations: » Outline Introduction Road Network: Finite directed graph G = ( I , N ) “Aw-Rascle” Model » A-R Model » A-R model on a network » Riemann Problem with |I| = I and |N | = N . » Optimization Junctions � Each arc i = 1 . . . I corresponds to a road. � Each vertex n = 1 . . . N corresponds to junction. For a fixed junction n , � δ − n : the set of incoming k roads to n , � δ + n : the set of outgoing roads j to n . ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 8

The “Aw-Rascle” model on a network Basic notations: » Outline Introduction Road Network: Finite directed graph G = ( I , N ) “Aw-Rascle” Model » A-R Model » A-R model on a network » Riemann Problem with |I| = I and |N | = N . » Optimization Junctions � Each arc i = 1 . . . I corresponds to a road. � Each vertex n = 1 . . . N corresponds to junction. For a fixed junction n , � δ − n : the set of incoming k roads to n , � δ + n : the set of outgoing roads j to n . Each road i is modelled by I i = [ a i , b i ] ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 8

The “Aw-Rascle” model on a network � We required the A-R system (1) to hold on each arc i ∈ I of » Outline the network. Introduction “Aw-Rascle” Model » A-R Model » A-R model on a network » Riemann Problem » Optimization Junctions ← → HYP 2006 – Lyon, July 17-21, 2006 – Salissou Moutari — Optimization in Traffic Modelling — Page 9