Stability analysis, simulation and continuum derivations of OV - PowerPoint PPT Presentation

Crowds: models and control CIRM Marseille, France — 3-7.7.19 Stability analysis, simulation and continuum derivations of OV following models Antoine Tordeux a Michael Herty b Armin Seyfried c � � aUniversity of Wuppertal (BUW) tordeux@uni-wuppertal.de vzu.uni-wuppertal.de bRWTH Aachen University herty@igpm.rwth-aachen.de rwth-aachen.de/cms/Mathematik cForschungszentrum J¨ ulich & BUW a.seyfried@fz-juelich.de asim.uni-wuppertal.de

Stop-and-go waves ◮ Traffic flow: Complex system of self-driven agents interacting locally ◮ Stop-and-go waves: Complex (self-organized) spontaneous formation (emergence) of collective waves of slow-down traffic propagating backward 1 1 Empirics: Papers by Treiterer and Meyers 1974, Kerner et al. 1997, Chowdhury et al. 2000, Helbing et al. 2000, Sugiyama et al. 2008, Stern et al. 2018

Following model x i +1 − x i x i ˙ x i +1 ˙ x i x i +1 ◮ Microscopic models for uni-directional motions of agents in 1D ◮ Non-linear speed or acceleration function depending on the distance spacing to the next vehicle ahead and the speed ∗ Speed model 2 1st-order � � x i = W ˙ x i +1 − x i , ˙ x i +1 (1) ∗ Acceleration model 3 2nd-order � � x i = A ¨ x i +1 − x i , ˙ x i , ˙ (2) x i +1 2 E.g. models by Pipes, 1953; Gipps, 1981; Newell, 1961 and 2002. 3 E.g. models by Kometani and Sasaki 1958; Greenberg, 1959; Helly, 1961, Chandler et al. 1963; Bando el al. 1995; Helbing et al. 1998; Treiber et al. 2000; Jiang et al. 2001. See Wageningen-Kessels et al. 2015 for a review.

OV models: Main parameters V 0 Desired speed V 0 T Time gap ℓ Minimal spacing Speed 1 / T Pursuit situation Free driving 0 0 ℓ ℓ + TV 0 Spacing ‘Optimal Velocity’ function + Relaxation time (inertia or reaction time) → 4 fundamental parameters 4.7.19 – CIRM Marseille, France Slide 4 / 22

Stop-and-go in traffic models: Phase transition ◮ Stop-and-go waves by means of global instability of the homogeneous solution τ > (2 W ′ ) − 1 x i = 1 → Example: OVM (1995) ¨ � W ( x i +1 − x i ) − ˙ x i � unstable if τ ◮ Models having unstable homogeneous solutions may describe phase transition and periodic stationary solutions (limit-cycle) with stop-and-go But... ◮ Collision (unbounded density) in 2nd-order 4 distance-based following models τ ≤ (4 W ′ ) − 1 → Local over-damped and global unstable incompatible, e.g. over-damped OVM ◮ Adding of the speed difference term necessary FVDM, IDM, ATG models 4 or delayed first order

Stop-and-go in traffic models: Phase transition ◮ Stop-and-go waves by means of global instability of the homogeneous solution τ > (2 W ′ ) − 1 x i = 1 → Example: OVM (1995) ¨ � W ( x i +1 − x i ) − ˙ x i � unstable if τ ◮ Models having unstable homogeneous solutions may describe phase transition and periodic stationary solutions (limit-cycle) with stop-and-go But... ◮ Collision (unbounded density) in 2nd-order 4 distance-based following models τ ≤ (4 W ′ ) − 1 → Local over-damped and global unstable incompatible, e.g. over-damped OVM ◮ Adding of the speed difference term necessary FVDM, IDM, ATG models ◮ In general (i.e. micro or macro OV models), stop-and-go waves by adding mechanisms (# parameters > 4) + fine tuning of the parameters 4 or delayed first order

Outline Minimal collision-free OV model Macroscopic derivation Numerical schemes Simulation results 4.7.19 – CIRM Marseille, France Slide 6 / 22

Outline Minimal collision-free OV model Macroscopic derivation Numerical schemes Simulation results 4.7.19 – CIRM Marseille, France Slide 7 / 22

1st-order collision-free OV model ◮ The microscopic following model comes from the generalized Newell’s model (1961) x i ( t ) = W (∆ x i ( t − τ )) ˙ (3) with τ ≥ 0 the reaction time, ∆ x i ( t ) = x i +1 ( t ) − x i ( t ) the spacing and W ( · ) the equilibrium (or optimal) speed function ◮ Linear approximation for small τ in the delayed distance spacing ∆ x i ( t − τ ) = ∆ x i ( t ) − τ [ ˙ x i +1 ( t ) − ˙ x i ( t )] + o ( τ ) (4) ◮ First-order OV model with reaction time � �� � x i ( t ) = W ˙ ∆ x i ( t ) − τ W (∆ x i +1 ( t )) − W (∆ x i ( t )) (5) 4.7.19 – CIRM Marseille, France Slide 8 / 22

1st-order collision-free OV model ◮ Microscopic model: � �� � x i ( t ) = W ˙ ∆ x i ( t ) − τ W (∆ x i +1 ( t )) − W (∆ x i ( t )) (6) ◮ Speed (1st order) OV model with two predecessors in interaction and a reaction time parameter ◮ Collision-free dynamics ∆ x i ( t ) ≥ ℓ ∀ i , t as soon as W ( · ) ≥ 0 positive and W ( s ) = 0 for all s ≤ ℓ (7) Formal proof (by continuity) Suppose ∃ i , t , ∆ x i ( t ) = 0 then d ∆ x i ( t ) / dt = ˙ x i +1 ( t ) − W ( − τ W (∆ x i +1 ( t ))) ≥ 0 since ˙ x i +1 ( t ) ≥ 0 while W ( − τ W (∆ x i +1 ( t ))) = 0 by construction 4.7.19 – CIRM Marseille, France Slide 9 / 22

Linear stability analysis ◮ Local stability analysis Assigned leader speed ∗ Linear expansion around equilibrium ( d , V ( d )) gives the dynamics y ( t ) = − α y ( t ) ˙ V ′ > 0 ∗ Exponential convergence to zero (over-damped) if α = (1 + τ V ′ ) V ′ > 0 ◮ Global stability analysis Periodic boundary condition ∗ Linear expansion gives the circulant dynamics y i ( t ) = α ∆ y i ( t ) + β ∆ y i +1 ( t ) ˙ λ w = − α (1 − e jw ) + τ ( V ′ ) 2 e jw (1 − e jw ) ∗ Eigenvalues of the linear system ℜ ( λ w ) = V ′ (1 − cos( w ))(2 τ V ′ cos( w ) − 1) < 0, ∀ w ∈ (0 , 2 π ) τ < (2 V ′ ) − 1 4.7.19 – CIRM Marseille, France Slide 10 / 22

Linear stability analysis ◮ Local stability analysis Assigned leader speed ∗ Linear expansion around equilibrium ( d , V ( d )) gives the dynamics y ( t ) = − α y ( t ) ˙ V ′ > 0 ∗ Exponential convergence to zero (over-damped) if α = (1 + τ V ′ ) V ′ > 0 ◮ Global stability analysis Periodic boundary condition ∗ Linear expansion gives the circulant dynamics y i ( t ) = α ∆ y i ( t ) + β ∆ y i +1 ( t ) ˙ λ w = − α (1 − e jw ) + τ ( V ′ ) 2 e jw (1 − e jw ) ∗ Eigenvalues of the linear system ℜ ( λ w ) = V ′ (1 − cos( w ))(2 τ V ′ cos( w ) − 1) < 0, ∀ w ∈ (0 , 2 π ) τ < (2 V ′ ) − 1 ◮ Same global linear stability condition as original Newell or OVM models but systematically locally over-damped as soon as V ′ > 0 4.7.19 – CIRM Marseille, France Slide 10 / 22

Outline Minimal collision-free OV model Macroscopic derivation Numerical schemes Simulation results 4.7.19 – CIRM Marseille, France Slide 11 / 22

Rewriting the microscopic model ◮ Same methodology as in [Aw et al. (2002)] considering the density at vehicle i and time t , ρ i ( t ), as the inverse of the spacing 1 ρ i ( t ) = (8) ∆ x i ( t ) ◮ The microscopic model becomes � 1 � 1 1 ��� � � � = ˜ x i = W ˙ − τ W − W V ( ρ i +1 , ρ i ) (9) ρ i ρ i +1 ρ i ◮ Then 1 = ∂ t ∆ x i ( t ) = ˜ V ( ρ i +2 , ρ i +1 ) − ˜ ∂ t V ( ρ i +1 , ρ i ) (10) ρ i → Semi–discretized version of PDE in the space of vehicle indices 4.7.19 – CIRM Marseille, France Slide 12 / 22

Derivation in Lagrangian coordinates ◮ y ∈ R such that y i = i ∆ y . We will construct ρ ( t , y ) at the limit N → ∞ , ℓ → 0, � y i + ∆ y 1 1 1 2 ρ i ( t ) = ρ ( t , z ) dz (11) ∆ y y i − ∆ y 2 ◮ Rescaling of time t → t ∆ y and reaction time τ → τ ∆ y , with V ( x ) = W ( 1 x ) 1 1 � ρ i +1 ρ i �� � � � ∂ t ρ i ( t ) − V − V = 0 1 − ρ i +1 τ V ( ρ i +2 ) − V ( ρ i +1 ) V ( ρ i +1 ) − V ( ρ i ) ∆ y 1 − τρ i ∆ y ∆ y (12) ◮ We obtain in the limit ∆ y → 0 the macroscopic equation in Lagrangian coordinates 1 � ρ � ∂ t ρ − ∂ y V = 0 (13) 1 − τρ∂ y V ( ρ ) 4.7.19 – CIRM Marseille, France Slide 13 / 22

Derivation in Eulerian coordinates � x ◮ Using the coordinate transformation ( t , y ) → ( t , x ) where y = −∞ ρ ( t , x ) dx , the macroscopic model reads in Eulerian coordinates � � �� ρ ∂ t ρ + ∂ x ρ V = 0 (14) 1 − τ∂ x V ( ρ ) � � ∗ Extension of the LWR model with OV-function V : ρ �→ V ρ/ (1 − τ∂ x V ( ρ )) ∗ Classical LWR for constant densities ρ ( x , t ) or for τ = 0 ◮ Taylor expansion for small τ yields in a convection/diffusion model � � ( ρ V ′ ( ρ )) 2 ∂ x ρ ∂ t ρ + ∂ x ( ρ V ( ρ )) = − τ∂ x (15) � �� � � �� � convection diffusion � �� � LWR model 4.7.19 – CIRM Marseille, France Slide 14 / 22

Fundamental diagram 2.0 τ∂ x V = 0.8 � � ρ/ (1 − τ∂ x V ( ρ )) ρ/ (1 − τ∂ x V ( ρ )) 0.3 1.5 0 0.6 -0.3 1.0 0.4 0.5 0.2 � � ρV V 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ρ ρ Abbildung: Fundamental diagram for constant inhomogeneity τ∂ x V ( ρ ) = ± 0 . 3. V : ρ �→ max { min { 2 , 1 /ρ − 1 }} . Bounded FD with scattered performances in congested states [Colombo (2003), Goatin (2006), Colombo et al. (2010)] 4.7.19 – CIRM Marseille, France Slide 15 / 22

Outline Minimal collision-free OV model Macroscopic derivation Numerical schemes Simulation results 4.7.19 – CIRM Marseille, France Slide 16 / 22