Traffic Flow Modeling on Road Networks Using Hamilton-Jacobi - - PowerPoint PPT Presentation

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]

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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]

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HJ on networks Berkeley, Oct. 09 2015 3 / 79

Motivation

Outline

1

Introduction to traffic

2

Micro to macro in traffic models

3

Variational principle applied to GSOM models

4

HJ equations on a junction

5

Conclusions and perspectives

• G. Costeseque (Inria)

HJ on networks Berkeley, Oct. 09 2015 4 / 79

Introduction to traffic

Outline

1

Introduction to traffic

2

Micro to macro in traffic models

3

Variational principle applied to GSOM models

4

HJ equations on a junction

5

Conclusions and perspectives

• G. Costeseque (Inria)

HJ on networks Berkeley, Oct. 09 2015 5 / 79

Introduction to traffic Macroscopic models

Convention for vehicle labeling

N x t Flow

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Introduction to traffic Macroscopic models

Three representations of traffic flow

Moskowitz’ surface

Flow x t N x

See also [Makigami et al, 1971], [Laval and Leclercq, 2013]

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HJ on networks Berkeley, Oct. 09 2015 7 / 79

Introduction to traffic Macroscopic models

Notations: macroscopic

N(t, x) vehicle label at (t, x) the flow Q(t, x) = lim

∆t→0

N(t + ∆t, x) − N(t, x) ∆t ,

x N(x,t ± ∆t)

the density ρ(t, x) = lim

∆x→0

N(t, x) − N(t, x + ∆x) ∆x ,

x ∆x N(x ± ∆x,t)

the stream speed (mean spatial speed) V (t, x).

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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:      ∂tρ(t, x) + ∂xQ(t, x) = 0 conservation equation Q(t, x) = ρ(t, x)V (t, x) definition of flow speed (1)

x x + ∆x

ρ(x,t)∆x

Q(x,t)∆t Q(x + ∆x,t)∆t

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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 ∂tρ(t, x) + ∂xF (ρ(t, x)) = 0 (2) with F : ρ(t, x) → Q(t, x) =: Fx (ρ(t, x))

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Introduction to traffic Focus on LWR model

Overview: conservation laws (CL) / Hamilton-Jacobi (HJ)

Eulerian Lagrangian t − x t − n CL Variable Density ρ Spacing r Equation ∂tρ + ∂xF(ρ) = 0 ∂tr + ∂nV (r) = 0 HJ Variable Label N Position X N(t, x) = +∞

x

ρ(t, ξ)dξ X(t, n) = +∞

n

r(t, η)dη Equation ∂tN + H (∂xN) = 0 ∂tX + V (∂nX) = 0 Hamiltonian H(p) = −F(−p) V(p) = −V (−p)

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Introduction to traffic Focus on LWR model

Fundamental diagram (FD)

Flow-density fundamental diagram F Empirical function with

ρmax the maximal or jam density, ρc the critical density

Flux is increasing for ρ ≤ ρc: free-flow phase Flux is decreasing for ρ ≥ ρc: congestion phase

ρmax Density, ρ ρmax Density, ρ Flow, F Flow, F ρmax Flow, F Density, ρ

[Garavello and Piccoli, 2006]

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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

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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)

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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      ∂tρ + ∂x(ρv) = 0 Conservation of vehicles, ∂t(ρI) + ∂x(ρvI) = 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) → ρI(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      ∂tρ + ∂x(ρv) = 0 Conservation of vehicles, ∂tI + v∂xI = ϕ(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) → ρI(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      ∂tρ + ∂x(ρv) = 0 Conservation of vehicles, ∂tI + v∂xI = 0 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) → ρI(ρ, I).

• G. Costeseque (Inria)

HJ on networks Berkeley, Oct. 09 2015 15 / 79

Micro to macro in traffic models

Outline

1

Introduction to traffic

2

Micro to macro in traffic models

3

Variational principle applied to GSOM models

4

HJ equations on a junction

5

Conclusions and perspectives

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HJ on networks Berkeley, Oct. 09 2015 16 / 79

Micro to macro in traffic models Homogenization

Setting

Speed Time

(i − 1)

Space x Spacing Headway t

(i)

t → xi(t) trajectory of vehicle i i = discrete position index (i ∈ Z) n = continuous (Lagrangian) variable n = iε and t = εs ε > 0 a scale factor

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Micro to macro in traffic models Homogenization

Setting

Speed Time

(i − 1)

Space x Spacing Headway t

(i)

t → xi(t) trajectory of vehicle i i = discrete position index (i ∈ Z) n = continuous (Lagrangian) variable n = iε and t = εs ε > 0 a scale factor Proposition (Rescaled positions) Define xi(s) = 1 εX ε(εs, iε) ⇐ ⇒ X ε(t, n) = εx⌊ n

ε ⌋

t ε

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Micro to macro in traffic models Homogenization

General result

Let consider the simplest microscopic model; ˙ xi(t) = F (xi−1(t) − xi(t)) (4) the LWR macroscopic model (HJ equation in Lagrangian): ∂tX 0 = F(−∂nX 0) (5)

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Micro to macro in traffic models Homogenization

General result

Let consider the simplest microscopic model; ˙ xi(t) = F (xi−1(t) − xi(t)) (4) the LWR macroscopic model (HJ equation in Lagrangian): ∂tX 0 = F(−∂nX 0) (5) Theorem ((Monneau) Convergence to the viscosity solution) If X ε(t, n) := εx⌊ n

ε⌋

t ε

• with (xi)i∈Z solution of (4) and X 0 the unique

solution of HJ (5), then under suitable assumptions, |X ε − X 0|L∞(K) − →

ε→0 0,

∀K compact set.

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Micro to macro in traffic models Multi-anticipative traffic

Toy model

Vehicles consider m ≥ 1 leaders First order multi-anticipative model ˙ xi(t + τ) = max  0, Vmax −

m

• j=1

f (xi−j(t) − xi(t))   (6) f speed-spacing function

non-negative non-increasing

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Micro to macro in traffic models Multi-anticipative traffic

Homogenization

Proposition ((Monneau) Convergence) Assume m ≥ 1 fixed. If τ is small enough, if X ε(t, n) := εx⌊ n

ε⌋

t ε

• with (xi)i∈Z solution of (6)

and if X 0(n, t) solves ∂tX 0 = F

• −∂nX 0, m
• (7)

with F(r, m) = max  0, Vmax −

m

• j=1

f (jr)   , then under suitable assumptions, |X ε − X 0|L∞(K) − →

ε→0 0,

∀K compact set.

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Micro to macro in traffic models Multi-anticipative traffic

Multi-anticipatory macroscopic model

χj the fraction of j-anticipatory vehicles traffic flow ≡ mixture of traffic of j-anticipatory vehicles χ = (χj)j=1,...,m , with 0 ≤ χj ≤ 1 and

m

• j=1

χj = 1 GSOM model with driver attribute I = χ            ∂tρ + ∂x (ρv) = 0, ∂t(ρχ) + ∂x(ρχv) = 0, v :=

m

• j=1

χjF(1/ρ, j) = W (1/ρ, χ). (8)

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Micro to macro in traffic models Numerical schemes

Numerical resolution

Godunov scheme in Eulerian t − x with (∆t, ∆xk) steps ⇒ CFL condition Variational formulation and dynamic programming techniques [2] Particle methods in the Lagrangian framework t − n    xt+1

n

= xt

n + ∆tW

xt

n−1 − xt n

∆n , χt

n

• χt+1

n

= χt

n

(9)

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Micro to macro in traffic models Numerical schemes

Numerical example

5 10 15 20 25 30 35 40 5 10 15 20 25

Spacing r (m)

Speed−spacing diagrams

Speed v (m/s)

1 2 3 4 5

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Micro to macro in traffic models Numerical schemes

Numerical example: (χj)j

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Micro to macro in traffic models Numerical schemes

Numerical example: Lagrangian trajectories

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Micro to macro in traffic models Numerical schemes

Numerical example: Eulerian trajectories

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Variational principle applied to GSOM models

Outline

1

Introduction to traffic

2

Micro to macro in traffic models

3

Variational principle applied to GSOM models

4

HJ equations on a junction

5

Conclusions and perspectives

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HJ on networks Berkeley, Oct. 09 2015 27 / 79

Variational principle applied to GSOM models LWR model

LWR in Eulerian (t, x)

Cumulative vehicles count (CVC) or Moskowitz surface N(t, x) Q = ∂tN and ρ = −∂xN If density ρ satisfies the scalar (LWR) conservation law ∂tρ + ∂xF(ρ) = 0 Then N satisfies the first order Hamilton-Jacobi equation ∂tN − F(−∂xN) = 0 (10)

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Variational principle applied to GSOM models LWR model

LWR in Eulerian (t, x)

Legendre-Fenchel transform with F concave (relative capacity) M(q) = sup

ρ

[F(ρ) − ρq]

M(q) u w

Density ρ

q q

Flow F

w u

q Transform M −wρmax

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Variational principle applied to GSOM models LWR model

LWR in Eulerian (t, x)

(continued)

Lax-Hopf formula (representation formula) [Daganzo, 2006]

N(T, xT) = min

u(.),(t0,x0)

T

t0

M(u(τ))dτ + N(t0, x0),

• ˙

X = u u ∈ U X(t0) = x0, X(T) = xT (t0, x0) ∈ J (11)

Time Space J (T, xT)

˙ X(τ)

(t0, x0)

Viability theory [Claudel and Bayen, 2010]

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Variational principle applied to GSOM models LWR model

LWR in Eulerian (t, x)

(Historical note)

Dynamic programming [Daganzo, 2006] for triangular FD (u and w free and congested speeds)

Flow, F w u ρmax Density, ρ u x w t Time Space (t, x)

Minimum principle [Newell, 1993] N(t, x) = min

• N
• t − x − xu

u , xu

• ,

N

• t − x − xw

w , xw

• + ρmax(xw − x)
• ,

(12)

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Variational principle applied to GSOM models LWR model

LWR in Lagrangian (n, t)

Consider X(t, n) the location of vehicle n at time t ≥ 0 v = ∂tX and r = −∂nX If the spacing r := 1/ρ satisfies the LWR model (Lagrangian coord.) ∂tr + ∂nV(r) = 0 Then X satisfies the first order Hamilton-Jacobi equation ∂tX − V(−∂nX) = 0. (13)

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Variational principle applied to GSOM models LWR model

LWR in Lagrangian (n, t)

(continued)

Dynamic programming for triangular FD

1/ρcrit Speed, V u −wρmax Spacing, r 1/ρmax

−wρmax n t (t, n) Time Label

Minimum principle ⇒ car following model [Newell, 2002] X(t, n) = min

• X(t0, n) + u(t − t0),

X(t0, n + wρmax(t − t0)) + w(t − t0)

• .

(14)

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Variational principle applied to GSOM models GSOM family

GSOM in Lagrangian (n, t)

From [Lebacque and Khoshyaran, 2013], GSOM in Lagrangian      ∂tr + ∂Nv = 0 Conservation of vehicles, ∂tI = 0 Dynamics of I, v = W(N, r, t) := V(r, I(N, t)) Fundamental diagram. (15) Position X(N, t) := t

−∞

v(N, τ)dτ satisfies the HJ equation ∂tX − W(N, −∂NX, t) = 0, (16) And I(N, t) solves the ODE

• ∂tI(N, t) = 0,

I(N, 0) = i0(N), for any N.

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Variational principle applied to GSOM models GSOM family

GSOM in Lagrangian (n, t)

(continued)

Legendre-Fenchel transform of W according to r M(N, c, t) = sup

r∈R

{W(N, r, t) − cr}

M(N, p, t) pq W(N, q, t) W(N, r, t) q r p p

u

c Transform M

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Variational principle applied to GSOM models GSOM family

GSOM in Lagrangian (n, t)

(continued)

Lax-Hopf formula X(NT , T) = min

u(.),(N0,t0)

T

t0

M(N, u, t)dt + c(N0, t0),

• ˙

N = u u ∈ U N(t0) = N0, N(T) = NT (N0, t0) ∈ K (17)

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Variational principle applied to GSOM models GSOM family

GSOM in Lagrangian (n, t)

(continued)

Optimal trajectories = characteristics ˙ N = ∂rW(N, r, t), ˙ r = −∂NW(N, r, t), (18) System of ODEs to solve Difficulty: not straight lines in the general case

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Variational principle applied to GSOM models Methodology

General ideas

First key element: Lax-Hopf formula Computations only for the characteristics X(NT , T) = min

(N0,r0,t0)

T

t0

M(N, ∂r W(N, r, t), t)dt + c(N0, r0, t0),

• ˙

N(t) = ∂rW(N, r, t) ˙ r(t) = −∂NW(N, r, t) N(t0) = N0, r(t0) = r0, N(T) = NT (N0, r0, t0) ∈ K (19) K := Dom(c) is the set of initial/boundary values

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Variational principle applied to GSOM models Methodology

General ideas

(continued)

Second key element: inf-morphism prop. [Aubin et al, 2011] Consider a union of sets (initial and boundary conditions) K =

• l

Kl, then the global minimum is X(NT , T) = min

l

Xl(NT , T), (20) with Xl partial solution to sub-problem Kl.

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Variational principle applied to GSOM models Numerical example

Fundamental Diagram and Driver Attribute

20 40 60 80 100 120 140 160 180 200 500 1000 1500 2000 2500 3000 3500

Density ρ (veh/km) Flow F (veh/h)

Fundamental diagram F(ρ,I)

5 10 15 20 25 30 35 40 45 50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Label N

Initial conditions I(N,t0)

Driver attribute I0

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Variational principle applied to GSOM models Numerical example

Initial and Boundaries Conditions

5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50

Label N Initial conditions r(N,t0) Spacing r0 (m)

5 10 15 20 25 30 35 40 45 50 −1400 −1200 −1000 −800 −600 −400 −200

Label N Position X (m) Initial positions X(N,t0)

20 40 60 80 100 120 12 14 16 18 20 22 24 26 28 30

Time t (s) Spacing r(N0,t) Spacing r0 (m.s−1)

20 40 60 80 100 120 500 1000 1500 2000 2500

Time t (s) Position X0 (m) Position X(N0,t)

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Variational principle applied to GSOM models Numerical example

Numerical result (Initial cond. + first traj.)

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Variational principle applied to GSOM models Numerical example

Numerical result (Initial cond. + first traj.)

20 40 60 80 100 120 −1500 −1000 −500 500 1000 1500 2000 2500

Location (m) Time (s)

Vehicles trajectories

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Variational principle applied to GSOM models Numerical example

Numerical result (Initial cond.+ 3 traj.)

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Variational principle applied to GSOM models Numerical example

Numerical result (Initial cond. + 3 traj.)

20 40 60 80 100 120 −1500 −1000 −500 500 1000 1500 2000 2500

Location (m) Time (s)

Vehicles trajectories

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Variational principle applied to GSOM models Numerical example

Numerical result (Initial cond. + 3 traj. + Eulerian data)

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HJ equations on a junction

Outline

1

Introduction to traffic

2

Micro to macro in traffic models

3

Variational principle applied to GSOM models

4

HJ equations on a junction

5

Conclusions and perspectives

• G. Costeseque (Inria)

HJ on networks Berkeley, Oct. 09 2015 47 / 79

HJ equations on a junction

Motivation

Classical approaches: Macroscopic modeling on (homogeneous) sections Coupling conditions at (pointwise) junction For instance, consider          ∂tρ + ∂xQ(ρ) = 0, scalar conservation law, ρ(., t = 0) = ρ0(.), initial conditions, ψ(ρ(x = 0−, t)

• upstream

, ρ(x = 0+, t)

• downstream

) = 0, coupling condition. (21) See Garavello, Piccoli [3], Lebacque, Khoshyaran [6] and Bressan et al. [1]

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HJ equations on a junction HJ junction model

Star-shaped junction

JN J1 J2 branch Jα x x x x

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HJ equations on a junction HJ junction model

Junction model

Proposition (Junction model [IMZ, ’13]) That leads to the following junction model (see [5])      ∂tuα + Hα (∂xuα) = 0, x > 0, α = 1, . . . , N uα = uβ =: u, x = 0, ∂tu + H

• ∂xu1, . . . , ∂xuN

= 0, x = 0 (22) with initial condition uα(0, x) = uα

0 (x) and

H

• ∂xu1, . . . , ∂xuN

= max

α=1,...,N

• H−

α (∂xuα)

• from optimal control

.

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HJ equations on a junction HJ junction model

Basic assumptions

For all α = 1, . . . , N, (A0) The initial condition uα

0 is Lipschitz continuous.

(A1) The Hamiltonians Hα are C 1(R) and convex such that:

p H−

α (p)

H+

α (p)

pα

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HJ equations on a junction Numerical scheme

Presentation of the scheme

Proposition (Numerical Scheme) Let us consider the discrete space and time derivatives: pα,n

i

:= Uα,n

i+1 − Uα,n i

∆x and (DtU)α,n

i

:= Uα,n+1

i

− Uα,n

i

∆t Then we have the following numerical scheme:        (DtU)α,n

i

+ max{H+

α (pα,n i−1), H− α (pα,n i

)} = 0, i ≥ 1 Un

0 := Uα,n

, i = 0, α = 1, ..., N (DtU)n

0 +

max

α=1,...,N H− α (pα,n

) = 0, i = 0 (23) With the initial condition Uα,0

i

:= uα

0 (i∆x).

∆x and ∆t = space and time steps satisfying a CFL condition

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HJ equations on a junction Mathematical results

Stronger CFL condition

−m0

pα p Hα(p) pα

As for any α = 1, . . . , N, we have (gradient estimates) pα ≤ pα,n

i

≤ pα for all i, n ≥ 0 Then the CFL condition becomes: ∆x ∆t ≥ sup

α=1,...,N pα∈[pα,pα]

|H′

α(pα)|

(24)

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HJ equations on a junction Mathematical results

Existence and uniqueness

Theorem (Existence and uniqueness [IMZ, ’13]) Under (A0)-(A1), there exists a unique viscosity solution u of (22) on the junction, satisfying for some constant CT > 0 |u(t, y) − u0(y)| ≤ CT for all (t, y) ∈ JT. Moreover the function u is Lipschitz continuous with respect to (t, y).

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HJ equations on a junction Mathematical results

Convergence

Theorem (Convergence from discrete to continuous [CML, ’13]) Assume that (A0)-(A1) and the CFL condition (24) are satisfied. Then the numerical solution converges uniformly to u the unique viscosity solution

• f the junction model (22) when ε := (∆t, ∆x) → 0

lim sup

ε→0

sup

(n∆t,i∆x)∈K

|uα(n∆t, i∆x) − Uα,n

i

| = 0

Proof

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HJ equations on a junction Traffic interpretation

Setting

J1 JNI JNI+1 JNI+NO x < 0 x = 0 x > 0 Jβ γβ Jλ γλ

NI incoming and NO outgoing roads

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HJ equations on a junction Traffic interpretation

Links with “classical” approach

Definition (Discrete car density) The discrete vehicle density ρα,n

i

with n ≥ 0 and i ∈ Z is given by: ρα,n

i

:=      γαpα,n

|i|−1

for α = 1, ..., NI , i ≤ −1 −γαpα,n

i

for α = NI + 1, ..., NI + NO, i ≥ 0 (25)

J1 JNI JNI+1 JNI+NO x < 0 x > 0

−2 −1 2 1 −2 −2 −1 −1 1 1 2 2

Jβ Jλ ρλ,n

1

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HJ equations on a junction Traffic interpretation

Traffic interpretation

Proposition (Scheme for vehicles densities) The scheme deduced from (23) for the discrete densities is given by: ∆x ∆t {ρα,n+1

i

− ρα,n

i

} =      F α(ρα,n

i−1, ρα,n i

) − F α(ρα,n

i

, ρα,n

i+1)

for i = 0, −1 F α

0 (ρ·,n 0 ) − F α(ρα,n i

, ρα,n

i+1)

for i = 0 F α(ρα,n

i−1, ρα,n i

) − F α

0 (ρ·,n 0 )

for i = −1 With    F α(ρα,n

i−1, ρα,n i

) := min

• Qα

D(ρα,n i−1), Qα S (ρα,n i

)

• F α

0 (ρ·,n 0 ) := γα min

• min

β≤NI

1 γβ Qβ

D(ρβ,n 0 ), min λ>NI

1 γλ Qλ

S(ρλ,n 0 )

• incoming
• utgoing

ρλ,n ρβ,n

−1

ρβ,n

−2

ρλ,n

1

x x = 0

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HJ equations on a junction Traffic interpretation

Supply and demand functions

Remark It recovers the seminal Godunov scheme with passing flow = minimum between upstream demand QD and downstream supply QS.

Density ρ ρcrit ρmax Supply QS Qmax Density ρ ρcrit ρmax Flow Q Qmax Density ρ ρcrit Demand QD Qmax

From [Lebacque ’93, ’96] and [Daganzo ’95]

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HJ equations on a junction Numerical simulation

Example of a Diverge

An off-ramp:

J1 ρ1 J2 ρ2 ρ3 J3

with                γe = 1, γl = 0.75, γr = 0.25

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HJ equations on a junction Numerical simulation

Fundamental Diagrams

50 100 150 200 250 300 350 500 1000 1500 2000 2500 3000 3500 4000 (ρc,fmax) (ρc,fmax)

Density (veh/km) Flow (veh/h)

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HJ equations on a junction Numerical simulation

Initial conditions (t=0s)

−200 −150 −100 −50 10 20 30 40 50 60 70

Road n° 1 (t= 0s)

Position (m) Density (veh/km) 50 100 150 200 10 20 30 40 50 60 70

Road n° 2 (t= 0s)

Position (m) Density (veh/km) 50 100 150 200 10 20 30 40 50 60 70

Road n° 3 (t= 0s)

Position (m) Density (veh/km)

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HJ equations on a junction Numerical simulation

Numerical solution: densities

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HJ equations on a junction Numerical simulation

Numerical solution: Hamilton-Jacobi

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HJ equations on a junction Numerical simulation

Trajectories

1 2 3 4 5 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13

Trajectories on road n° 1

Position (m) Time (s) −200 −150 −100 −50 5 10 15

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 12

Trajectories on road n° 2

Position (m) Time (s) 50 100 150 200 5 10 15

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 10 11 12

Trajectories on road n° 3

Position (m) Time (s) 50 100 150 200 5 10 15

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HJ equations on a junction Recent developments

New junction model

Proposition (Junction model [IM, ’14]) From [4], we have      ∂tuα + Hα (∂xuα) = 0, x > 0, α = 1, . . . , N uα = uβ =: u, x = 0, ∂tu + H

• ∂xu1, . . . , ∂xuN

= 0, x = 0 (26) with initial condition uα(0, x) = uα

0 (x) and

H

• ∂xu1, . . . , ∂xuN

= max flux limiter

• L

, max

α=1,...,N

• H−

α (∂xuα)

• minimum between

demand and supply

• .
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HJ equations on a junction Recent developments

Weaker assumptions on the Hamiltonians

For all α = 1, . . . , N, (A0) The initial condition uα

0 is Lipschitz continuous.

(A1) The Hamiltonians Hα are continuous and quasi-convex i.e. there exists points pα

0 such that

     Hα is non-increasing on (−∞, pα

0 ],

Hα is non-decreasing on [pα

0 , +∞).

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HJ equations on a junction Recent developments

Homogenization on a network

Proposition (Homogenization on a periodic network [IM’14]) Assume (A0)-(A1). Consider a periodic network. If uε ∈ Rd satisfies HJ equation on network, then uε converges uniformly towards u0 when ε → 0, with u0 ∈ Rd solution of ∂tu0 + H

• Du0

= 0, t > 0, x ∈ Rd (27) See [Imbert, Monneau ’14] [4]

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HJ equations on a junction Recent developments

Numerical homogenization on a network

Numerical scheme adapted to the cell problem (d = 2)

Traffic Traffic eH γH eV γH γV γV i = 0 i = N 2 i = −N 2

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HJ equations on a junction Recent developments

First example

Proposition (Effective Hamiltonian for fixed coefficients [IM’14]) If (γH, γV ) are fixed, then the (Hamiltonian) effective Hamiltonian H is given by H (∂xuH, ∂xuV ) = max

• L,

max

i={H,V } H (∂xui)

• ,

(traffic flow) effective flow Q is given by F(ρH, ρV ) = min

• −L, F(ρH)

γH , F(ρV ) γV

• .
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HJ equations on a junction Recent developments

First example

Numerics: assume F(ρ) = 4ρ(1 − ρ) and L = −1.5,

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HJ equations on a junction Recent developments

Second example

Two consecutive traffic signals on a 1D road

flow l L L x1 x2 xE E xS S

Homogenization theory by [Galise, Imbert, Monneau, ’14]

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HJ equations on a junction Recent developments

Second example

Effective flux limiter −L (numerics only)

5 10 15 20 25 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Offset (s) Flux limiter

l=0 m l=5 m l=10 m l=20 m

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Conclusions and perspectives

Outline

1

Introduction to traffic

2

Micro to macro in traffic models

3

Variational principle applied to GSOM models

4

HJ equations on a junction

5

Conclusions and perspectives

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Conclusions and perspectives

Personal conclusions

Advantages Drawbacks Micro-macro link Limited time delay Homogeneous Explicit solutions Concavity of the FD link Data assimilation Exactness (2nd order) Multilane Uniqueness of the solution Fixed proportions Junction Homogenization result Multilane

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Conclusions and perspectives

Perspectives

Some open questions: Micro-macro: higher time delay? Confront the results with real data (micro datasets) Explicit Lax-Hopf formula for time/space dependent Hamiltonians?

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References

Some references I

• A. Bressan, S. Canic, M. Garavello, M. Herty, and B. Piccoli, Flows
• n networks: recent results and perspectives, EMS Surveys in Mathematical

Sciences, (2014).

• G. Costeseque and J.-P. Lebacque, A variational formulation for higher order

macroscopic traffic flow models: numerical investigation, Transp. Res. Part B: Methodological, (2014).

• M. Garavello and B. Piccoli, Traffic flow on networks, American institute of

mathematical sciences Springfield, MO, USA, 2006.

• C. Imbert and R. Monneau, Level-set convex Hamilton–Jacobi equations on

networks, (2014).

• C. Imbert, R. Monneau, and H. Zidani, A Hamilton–Jacobi approach to

junction problems and application to traffic flows, ESAIM: Control, Optimisation and Calculus of Variations, 19 (2013), pp. 129–166.

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References

Some references II

J.-P. Lebacque and M. M. Khoshyaran, First-order macroscopic traffic flow models: Intersection modeling, network modeling, in Transportation and Traffic

• Theory. Flow, Dynamics and Human Interaction. 16th International Symposium on

Transportation and Traffic Theory, 2005.

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References

Thanks for your attention Any question? guillaume.costeseque@inria.fr

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