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 Tordeuxa Michael Hertyb Armin Seyfriedc

• 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 backward1

1Empirics: 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

xi xi+1 xi+1 − xi ˙ xi ˙ xi+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 model2

1st-order

˙ xi = W

• xi+1 − xi, ˙

xi+1

• (1)

∗ Acceleration model3

2nd-order

¨ xi = A

• xi+1 − xi, ˙

xi, ˙ xi+1

• (2)

2E.g. models by Pipes, 1953; Gipps, 1981; Newell, 1961 and 2002. 3E.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

Spacing Speed ℓ ℓ + TV0 V0

V0 Desired speed T Time gap ℓ Minimal spacing

1/T

Pursuit situation Free driving

‘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

→ Example: OVM (1995) ¨ xi = 1

τ

• W (xi+1 − xi ) − ˙

xi

• unstable if

τ > (2W ′)−1

◮ 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-order4 distance-based following models

→ Local over-damped and global unstable incompatible, e.g. over-damped OVM τ ≤ (4W ′)−1

◮ Adding of the speed difference term necessary

FVDM, IDM, ATG models

4or delayed first order

Stop-and-go in traffic models: Phase transition

◮ Stop-and-go waves by means of global instability of the homogeneous solution

→ Example: OVM (1995) ¨ xi = 1

τ

• W (xi+1 − xi ) − ˙

xi

• unstable if

τ > (2W ′)−1

◮ 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-order4 distance-based following models

→ Local over-damped and global unstable incompatible, e.g. over-damped OVM τ ≤ (4W ′)−1

◮ 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

4or 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) ˙ xi(t) = W (∆xi(t − τ)) (3) with τ ≥ 0 the reaction time, ∆xi(t) = xi+1(t) − xi(t) the spacing and W (·) the equilibrium (or optimal) speed function

◮ Linear approximation for small τ in the delayed distance spacing

∆xi(t − τ) = ∆xi(t) − τ[ ˙ xi+1(t) − ˙ xi(t)] + o(τ) (4)

◮ First-order OV model with reaction time

˙ xi(t) = W

• ∆xi(t) − τ
• W (∆xi+1(t)) − W (∆xi(t))
• (5)

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1st-order collision-free OV model

◮ Microscopic model:

˙ xi(t) = W

• ∆xi(t) − τ
• W (∆xi+1(t)) − W (∆xi(t))
• (6)

◮ Speed (1st order) OV model with two predecessors in interaction and a reaction

time parameter

◮ Collision-free dynamics

∆xi(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, ∆xi(t) = 0 then d∆xi(t)/dt = ˙ xi+1(t) − W (−τW (∆xi+1(t))) ≥ 0 since ˙ xi+1(t) ≥ 0 while W (−τW (∆xi+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) = −αy(t) ∗ Exponential convergence to zero (over-damped) if α = (1 + τV ′)V ′ > 0

V ′ > 0

◮ Global stability analysis

Periodic boundary condition

∗ Linear expansion gives the circulant dynamics ˙ yi(t) = α∆yi(t) + β∆yi+1(t) ∗ Eigenvalues of the linear system λw = −α(1 − ejw) + τ(V ′)2ejw(1 − ejw) ℜ(λw) = V ′(1 − cos(w))(2τV ′ cos(w) − 1) < 0, ∀w ∈ (0, 2π)

τ < (2V ′)−1

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Linear stability analysis

◮ Local stability analysis

Assigned leader speed

∗ Linear expansion around equilibrium (d, V (d)) gives the dynamics ˙ y(t) = −αy(t) ∗ Exponential convergence to zero (over-damped) if α = (1 + τV ′)V ′ > 0

V ′ > 0

◮ Global stability analysis

Periodic boundary condition

∗ Linear expansion gives the circulant dynamics ˙ yi(t) = α∆yi(t) + β∆yi+1(t) ∗ Eigenvalues of the linear system λw = −α(1 − ejw) + τ(V ′)2ejw(1 − ejw) ℜ(λw) = V ′(1 − cos(w))(2τV ′ cos(w) − 1) < 0, ∀w ∈ (0, 2π)

τ < (2V ′)−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

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Rewriting the microscopic model

◮ Same methodology as in [Aw et al. (2002)] considering the density at vehicle i

and time t, ρi(t), as the inverse of the spacing ρi(t) = 1 ∆xi(t) (8)

◮ The microscopic model becomes

˙ xi = W 1 ρi − τ

• W
• 1

ρi+1

• − W

1 ρi

• = ˜

V (ρi+1, ρi) (9)

◮ Then

∂t 1 ρi = ∂t∆xi(t) = ˜ V (ρi+2, ρi+1) − ˜ V (ρi+1, ρi) (10)

→ Semi–discretized version of PDE in the space of vehicle indices

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Derivation in Lagrangian coordinates

◮ y ∈ R such that yi = i∆y. We will construct ρ(t, y) at the limit N → ∞, ℓ → 0,

1 ρi(t) = 1 ∆y yi + ∆y

2

yi − ∆y

2

1 ρ(t, z) dz (11)

◮ Rescaling of time t → t∆y and reaction time τ → τ∆y, with V (x) = W ( 1

x )

∂t 1 ρi(t) − 1 ∆y

• V
• ρi+1

1 − ρi+1τ V (ρi+2)−V (ρi+1)

∆y

• − V
• ρi

1 − τρi

V (ρi+1)−V (ρi ) ∆y

• = 0

(12)

◮ We obtain in the limit ∆y → 0 the macroscopic equation in Lagrangian

coordinates ∂t 1 ρ − ∂yV

• ρ

1 − τρ∂yV (ρ)

• = 0

(13)

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Derivation in Eulerian coordinates

◮ Using the coordinate transformation (t, y) → (t, x) where y =

x

−∞ ρ(t, x)dx,

the macroscopic model reads in Eulerian coordinates ∂tρ + ∂x

• ρV
• ρ

1 − τ∂xV (ρ)

• = 0

(14)

∗ Extension of the LWR model with OV-function V : ρ → V

• ρ/(1 − τ∂xV (ρ))
• ∗ Classical LWR for constant densities ρ(x, t) or for τ = 0

◮ Taylor expansion for small τ yields in a convection/diffusion model

∂tρ + ∂x(ρV (ρ))

• convection
• LWR model

= −τ∂x

• (ρV ′(ρ))2∂xρ
• diffusion

(15)

4.7.19 – CIRM Marseille, France Slide 14 / 22

Fundamental diagram

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0

ρ V

• ρ/(1 − τ∂xV (ρ))
• τ∂xV =

0.3

• 0.3

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8

ρ ρV

• ρ/(1 − τ∂xV (ρ))
• Abbildung: Fundamental diagram for constant inhomogeneity τ∂xV (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

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Numerical schemes for the macroscopic model

◮ Explicit Euler scheme for the density

ρi(t + δt) = ρi(t) + δt δx

• fi−1(t) − fi(t)
• (16)

◮ Truncated Godunov scheme for the flow

fi = G(ρi, ρi+1)

• convection

+ τ δx ρiV ′(ρi)

• G(ρi+1, ρi+2) − G(ρi, ρi+1)
• diffusion

(17) with G(x, y) = min{∆(x), Σ(y)} (18) the Godunov scheme, ∆(·) and Σ(·) are the traffic demand and supply functions extracted from the fundamental diagram

4.7.19 – CIRM Marseille, France Slide 17 / 22

General linear stability analysis

◮ Linearisation of the perturbed system around (ρE , V (ρE ))

εi(t + δt) = α εi(t) + β εi+1(t) + γ εi+2(t) + ξ εi−1(t) with α, β, γ, ξ the partial derivative of the numerical scheme at equilibrium

◮ N cells with periodic boundary conditions (circulant system) — Eigenvalues of

the linear system λw = α + βιw + γι2

w + ξι−1 w

with ιw = eiw

◮ System linearly stable if

|λw| < 1 for all w = 2πk/N, k = 1, . . . , N − 1

Here |λw |2 = α2 + β2 + γ2 + ξ2 − 2αγ − 2βξ + 2f (cw ), f (x) = 4γξx3 + 2(αγ + βξ)x2 + (αβ + αξ + βγ − 3γξ)x and cw = cos(w)

4.7.19 – CIRM Marseille, France Slide 18 / 22

Stability analysis for the Euler/Godunov scheme

◮ Affine speed function V (ρ) = 1 T (1/ρ − ℓ), with T the time gap and ℓ the vehicle size

F(ρi, ρi+1, ρi+2, ρi−1) = ρi + δt ℓ δx T

• ρi+1 − ρi +

τ δx T ρi+1 − ρi+2 ρi − ρi − ρi+1 ρi−1

• α = 1 − A(1 + B), β = A(1 + 2B), γ = −AB and ξ = 0 with A =

δt ℓ δx T and B = τ T δx ρE ◮ Linear stability condition

Unstable

for all δt > 0 Shortest wavelength

Unstable

δt <

Tδx ℓ+ ℓτ TδxρE

Shortest wavelength

Stable

δt <

Tδx ℓ+ ℓτ TδxρE

Stable

δt < Tδx

ℓ

−

2τ ℓρE

Unstable

δt <

Tδx ℓ+ ℓτ TδxρE

Wavelength from N/2 to N/6

− 1

2 TδxρE

−TδxρE

1 2 TδxρE

τ

◮ Same conditions as the microscopic model for δt → 0 and δx = 1/ρE = mean spacing 4.7.19 – CIRM Marseille, France Slide 19 / 22

Outline

Minimal collision-free OV model Macroscopic derivation Numerical schemes Simulation results

4.7.19 – CIRM Marseille, France Slide 20 / 22

Trajectories

20 40 60 80 100 20 40 60 80 100

Jam initial configuration Space Time

Trajectories

20 40 60 80 100 20 40 60 80 100

Perturbed initial configuration Space Time

Fundamental diagram

Bounds: V −(ρ) = V

• ρ

1−τρ(V0−V (ρ))

• and

V +(ρ) = V

• ρ

1+τρV (ρ)

• 0.4

0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0

Microscopic model Speed

0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0

Macroscopic model

V (·) 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Density Flow

0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Density

V (·)

Time

100 200 300 400 500

Trajectories

20 40 60 80 100 20 40 60 80 100

Random initial configuration Space Time

Summary

◮ Minimal collision-free OV following model with two predecessors in interaction

˙ xi(t) = W

• ∆xi(t) − τ
• W (∆xi+1(t)) − W (∆xi(t))
• ∗ Relaxation operates in space with the neighbors instead of operating in time

∗ Physical (collision-free) traffic performance for any density level, positive OV function and value of the reaction time parameter

4.7.19 – CIRM Marseille, France Slide 22 / 22

Summary

◮ Minimal collision-free OV following model with two predecessors in interaction

˙ xi(t) = W

• ∆xi(t) − τ
• W (∆xi+1(t)) − W (∆xi(t))
• ∗ Relaxation operates in space with the neighbors instead of operating in time

∗ Physical (collision-free) traffic performance for any density level, positive OV function and value of the reaction time parameter

◮ Macroscopic derivation: Degenerate parabolic convection/diffusion PDE with

non constant diffusivity ∂tρ + ∂x(ρV (ρ)) = −τ∂x

• (ρV ′(ρ))2∂xρ
• ◮ Diffusion may be positive or negative according to the sign of ∂xρ

∗ Negative in deceleration phases / Positive in acceleration phases (slow-to-start) ∗ Mechanism responsible for the formation of stop-and-go waves?

4.7.19 – CIRM Marseille, France Slide 22 / 22

Many thanks for your attention!

Division for traffic safety and reliability University of Wuppertal vzu.uni-wuppertal.de

c Mike Baldwin / Cornered