Follow-the-leader models on networks From micro to macro Markus - - PowerPoint PPT Presentation

Follow-the-leader models on networks

From micro to macro

Markus Stachl <markus.stachl@mytum.de>

TU M¨ unchen 29.06.2016

Outline

1 Introduction 2 Micro to macro 3 Examples and numerics 4 Traffic flows on networks 5 Example on network

Markus Stachl Follow-the-leader models on networks 29.06.2016 1 / 48

Introduction

Outline

1 Introduction 2 Micro to macro 3 Examples and numerics 4 Traffic flows on networks 5 Example on network

Markus Stachl Follow-the-leader models on networks 29.06.2016 2 / 48

Introduction

Introduction

Macroscopic model Density of vehicles Partial differential equations Conservation laws

Markus Stachl Follow-the-leader models on networks 29.06.2016 3 / 48

Introduction

Introduction

Macroscopic model Density of vehicles Partial differential equations Conservation laws Microscopic model Positions of n single vehicles Equations of motion System of ordinary differential equations

Markus Stachl Follow-the-leader models on networks 29.06.2016 3 / 48

Introduction

Introduction

Macroscopic model Density of vehicles Partial differential equations Conservation laws Microscopic model Positions of n single vehicles Equations of motion System of ordinary differential equations Is there an interrelation between the two representations?

Markus Stachl Follow-the-leader models on networks 29.06.2016 3 / 48

Introduction

Continuous macroscopic model

Consider traffic density u on a road average velocity v(u) of the traffic traffic flux f (u) = u · v(u)

Markus Stachl Follow-the-leader models on networks 29.06.2016 4 / 48

Introduction

Continuous macroscopic model

Consider traffic density u on a road average velocity v(u) of the traffic traffic flux f (u) = u · v(u) ⇓ The Lighthill-Whitham-Richards (LWR)-model

• ut + (f (u))x = 0

u(x, 0) = u0(x) R =

• u ∈ L1(R, [0, 1]) :
• R u(x)dx = L and supp(u) is compact
• Markus Stachl

Follow-the-leader models on networks 29.06.2016 4 / 48

Introduction

Discrete microscopic model

Consider n cars of length ℓn := L

n and total length L

Markus Stachl Follow-the-leader models on networks 29.06.2016 5 / 48

Introduction

Discrete microscopic model

Consider n cars of length ℓn := L

n and total length L

car locations yi = yi(t) for i = i...n at time t

Markus Stachl Follow-the-leader models on networks 29.06.2016 5 / 48

Introduction

Discrete microscopic model

Consider n cars of length ℓn := L

n and total length L

car locations yi = yi(t) for i = i...n at time t distances δi = yi+1 − yi

Markus Stachl Follow-the-leader models on networks 29.06.2016 5 / 48

Introduction

Discrete microscopic model

Consider n cars of length ℓn := L

n and total length L

car locations yi = yi(t) for i = i...n at time t distances δi = yi+1 − yi motion equations:

• ˙

yn = vmax ˙ yi = w(δi(t)) for i = 1...n − 1

Markus Stachl Follow-the-leader models on networks 29.06.2016 5 / 48

Introduction

Discrete microscopic model

Consider n cars of length ℓn := L

n and total length L

car locations yi = yi(t) for i = i...n at time t distances δi = yi+1 − yi motion equations:

• ˙

yn = vmax ˙ yi = w(δi(t)) for i = 1...n − 1 ⇓ First order Follow-the-leader-model      ˙ yn = vmax ˙ yi = w(δi(t)) for i = 1...n − 1 yi(0) = yi,0 for i = 1...n

Markus Stachl Follow-the-leader models on networks 29.06.2016 5 / 48

Introduction

Discrete microscopic model

Consider n cars of length ℓn := L

n and total length L

car locations yi = yi(t) for i = i...n at time t distances δi = yi+1 − yi motion equations:

• ˙

yn = vmax ˙ yi = w(δi(t)) for i = 1...n − 1 ⇓ First order Follow-the-leader-model      ˙ yn = vmax ˙ yi = w(δi(t)) for i = 1...n − 1 yi(0) = yi,0 for i = 1...n Yn = {(y1, ..., yn) ∈ Rn : δi := yi+1 − yi ≥ ℓn∀i = 1...n}

Markus Stachl Follow-the-leader models on networks 29.06.2016 5 / 48

Introduction

The macroscopic velocity v

Assume the following properties of v: v ∈ C1([0, 1], [0, vmax]) v′(u) < 0 v(0) = vmax, v(1) = 0

Markus Stachl Follow-the-leader models on networks 29.06.2016 6 / 48

Introduction

The macroscopic velocity v

Assume the following properties of v: v ∈ C1([0, 1], [0, vmax]) v′(u) < 0 v(0) = vmax, v(1) = 0 e.g. v(u) = vmax(1 − u) v(u) = vmax(1 − un) (Greenshield)

Markus Stachl Follow-the-leader models on networks 29.06.2016 6 / 48

Introduction

The microscopic velocity w

Assume the following properties of w: w ∈ C1([ℓn, ∞], [0, vmax]) w′(δ) > 0 for δ > ℓn w(ℓn) = 0, limδ→∞ w(δ) = vmax

Markus Stachl Follow-the-leader models on networks 29.06.2016 7 / 48

Micro to macro

Outline

1 Introduction 2 Micro to macro 3 Examples and numerics 4 Traffic flows on networks 5 Example on network

Markus Stachl Follow-the-leader models on networks 29.06.2016 8 / 48

Micro to macro

Connecting micro and macro

Microscopic and macroscopic models are related through particle paths:

• ut + (uv(u))x = 0

u(x, 0) = u0(x)      ˙ yn = vmax ˙ yi = w(δi(t)) for i = 1...n − 1 yi(0) = yi,0 for i = 1...n w(δ) = v( ℓn

δ )

Markus Stachl Follow-the-leader models on networks 29.06.2016 9 / 48

Micro to macro

Discretization operators En I

✎ ✍ ☞ ✌

Traffic density u →

✎ ✍ ☞ ✌

discrete vehicle positions y Take any given n. Then En(u(·)) := y =

• yn = max(supp(u))

yi = max

• z ∈ R :

yi+1

z

u(x)dx = ℓn

• , i = n − 1...2, 1

Markus Stachl Follow-the-leader models on networks 29.06.2016 10 / 48

Micro to macro

Discretization operator En II

Markus Stachl Follow-the-leader models on networks 29.06.2016 11 / 48

Micro to macro

Anti-discretization operator Cn I

✎ ✍ ☞ ✌

discrete vehicle positions y →

✎ ✍ ☞ ✌

Traffic density un Take a given n. Then Cn[y] =: un =

n−1

• i=1

ℓn δi χ[yi,yi+1]

Markus Stachl Follow-the-leader models on networks 29.06.2016 12 / 48

Micro to macro

Anti-discretization operator Cn II

Markus Stachl Follow-the-leader models on networks 29.06.2016 13 / 48

Micro to macro

Connecting micro and macro (ctd.)

As the number of cars (particles) tends towards infinity: Discrete ODE solution n→∞ − − − → Continuous PDE solution Theorem Let the velocities v and w be defined as before. Choose u0 = u(0, ·) ∈ R ∩ BV (R; [0, 1]) and y = y(0) = En[u0]. Let y(·) be the solution of the discrete model with initial condition y0. Define un(t, ·) = Cn[y(t)]. Then un converges almost everywhere and in L1

loc([0, +∞) × R) to

the unique entropy solution u to the PDE problem with initial condition u(0, ·) = u0.

Markus Stachl Follow-the-leader models on networks 29.06.2016 14 / 48

Micro to macro

The macro-micro limit

The theorem before states that the following diagram holds: u0

Micro to macro

The macro-micro limit

The theorem before states that the following diagram holds: u0 y0 En

Micro to macro

The macro-micro limit

The theorem before states that the following diagram holds: u0 y0 En y ODE solution

Micro to macro

The macro-micro limit

The theorem before states that the following diagram holds: u0 y0 En y ODE solution ˜ un Cn

Micro to macro

The macro-micro limit

The theorem before states that the following diagram holds: u0 y0 En y ODE solution ˜ un Cn u n → ∞

Micro to macro

The macro-micro limit

The theorem before states that the following diagram holds: u0 y0 En y ODE solution ˜ un Cn u n → ∞ solve PDE

Markus Stachl Follow-the-leader models on networks 29.06.2016 15 / 48

Micro to macro

Proof of main theorem

Elements of the proof: (see [6])

Markus Stachl Follow-the-leader models on networks 29.06.2016 16 / 48

Micro to macro

Proof of main theorem

Elements of the proof: (see [6]) let y be a solution to the discrete FTL model

Markus Stachl Follow-the-leader models on networks 29.06.2016 16 / 48

Micro to macro

Proof of main theorem

Elements of the proof: (see [6]) let y be a solution to the discrete FTL model show convergence of Cn[y] to ρ w.r.t. Wasserstein distance in L1

loc using Hell’s compactness theorem [1]

Markus Stachl Follow-the-leader models on networks 29.06.2016 16 / 48

Micro to macro

Proof of main theorem

Elements of the proof: (see [6]) let y be a solution to the discrete FTL model show convergence of Cn[y] to ρ w.r.t. Wasserstein distance in L1

loc using Hell’s compactness theorem [1]

d1(Cn[y], ρ) = ||FCn[y] − Fρ||L1

loc (Fi cumulated densities) Markus Stachl Follow-the-leader models on networks 29.06.2016 16 / 48

Micro to macro

Proof of main theorem

Elements of the proof: (see [6]) let y be a solution to the discrete FTL model show convergence of Cn[y] to ρ w.r.t. Wasserstein distance in L1

loc using Hell’s compactness theorem [1]

d1(Cn[y], ρ) = ||FCn[y] − Fρ||L1

loc (Fi cumulated densities)

show that the limit ρ ∈ L∞ is bounded from above by R := maxi=1...n−1

• ℓ

y0,i+1−y0,i

• Markus Stachl

Follow-the-leader models on networks 29.06.2016 16 / 48

Micro to macro

Proof of main theorem

Elements of the proof: (see [6]) let y be a solution to the discrete FTL model show convergence of Cn[y] to ρ w.r.t. Wasserstein distance in L1

loc using Hell’s compactness theorem [1]

d1(Cn[y], ρ) = ||FCn[y] − Fρ||L1

loc (Fi cumulated densities)

show that the limit ρ ∈ L∞ is bounded from above by R := maxi=1...n−1

• ℓ

y0,i+1−y0,i

• show that ρ is a weak solution to the PDE problem

Markus Stachl Follow-the-leader models on networks 29.06.2016 16 / 48

Examples and numerics

Outline

1 Introduction 2 Micro to macro 3 Examples and numerics 4 Traffic flows on networks 5 Example on network

Markus Stachl Follow-the-leader models on networks 29.06.2016 17 / 48

Examples and numerics

Example I: slow-moving traffic

Consider the LWR-model

• ut + (f (u))x = 0

u(0) = u0 where f (u) = vmax · u · (1 − u) and u0 =

• 0.8

for x > 0 0.4 for x < 0

Markus Stachl Follow-the-leader models on networks 29.06.2016 18 / 48

Examples and numerics

Analytical solution

Analytically with u− = 0.4 and u+ = 0.8 we have that f (u) is strictly concave f ′(u) is strictly decreasing u+ > u−

Markus Stachl Follow-the-leader models on networks 29.06.2016 19 / 48

Examples and numerics

Analytical solution

Analytically with u− = 0.4 and u+ = 0.8 we have that f (u) is strictly concave f ′(u) is strictly decreasing u+ > u− ⇒ the solution consists of a single shock u(t, x) =

• u+

for x > λt u− for x < λt

Markus Stachl Follow-the-leader models on networks 29.06.2016 19 / 48

Examples and numerics

Shock speed λ

The shock speed can be derived via the Rankine-Hugoniot equation: λ = f (u+) − f (u−) u+ − u− =

Markus Stachl Follow-the-leader models on networks 29.06.2016 20 / 48

Examples and numerics

Shock speed λ

The shock speed can be derived via the Rankine-Hugoniot equation: λ = f (u+) − f (u−) u+ − u− = vmax · 0.8(1 − 0.8) − vmax · 0.4(1 − 0.4) 0.8 − 0.4

Markus Stachl Follow-the-leader models on networks 29.06.2016 20 / 48

Examples and numerics

Shock speed λ

The shock speed can be derived via the Rankine-Hugoniot equation: λ = f (u+) − f (u−) u+ − u− = vmax · 0.8(1 − 0.8) − vmax · 0.4(1 − 0.4) 0.8 − 0.4 = −1 5vmax

Markus Stachl Follow-the-leader models on networks 29.06.2016 20 / 48

Examples and numerics

Shock speed λ

The shock speed can be derived via the Rankine-Hugoniot equation: λ = f (u+) − f (u−) u+ − u− = vmax · 0.8(1 − 0.8) − vmax · 0.4(1 − 0.4) 0.8 − 0.4 = −1 5vmax [shock speed] = [jump in the flux]

[jump in the state]

Markus Stachl Follow-the-leader models on networks 29.06.2016 20 / 48

Examples and numerics

Travelling shock wave

Markus Stachl Follow-the-leader models on networks 29.06.2016 21 / 48

Examples and numerics

Demo

Markus Stachl Follow-the-leader models on networks 29.06.2016 22 / 48

Examples and numerics

Micro to macro convergence

Markus Stachl Follow-the-leader models on networks 29.06.2016 23 / 48

Examples and numerics

Numerics

Markus Stachl Follow-the-leader models on networks 29.06.2016 24 / 48

Examples and numerics

Example II: traffic lights red → green

Consider the LWR-model

• ut + (f (u))x = 0

u(0) = u0 where f (u) = vmax · u · (1 − u) and u0 =

• for x > 0

0.8 for x < 0

Markus Stachl Follow-the-leader models on networks 29.06.2016 25 / 48

Examples and numerics

Analytical solution

Analytically with u− = 0.8 and u+ = 0 we have that f (u) is strictly concave f ′(u) is strictly decreasing u+ < u−

Markus Stachl Follow-the-leader models on networks 29.06.2016 26 / 48

Examples and numerics

Analytical solution

Analytically with u− = 0.8 and u+ = 0 we have that f (u) is strictly concave f ′(u) is strictly decreasing u+ < u− ⇒ the solution consists of a rarefaction wave u(t, x) =      u+ for x > f ′(u+)t u− for x < f ′(u−)t ω if x

t = f ′(ω) for some ω ∈ [u+, u−]

Markus Stachl Follow-the-leader models on networks 29.06.2016 26 / 48

Examples and numerics Markus Stachl Follow-the-leader models on networks 29.06.2016 27 / 48

Examples and numerics

Demo

Markus Stachl Follow-the-leader models on networks 29.06.2016 28 / 48

Examples and numerics

Micro-macro convergence

Markus Stachl Follow-the-leader models on networks 29.06.2016 29 / 48

Examples and numerics

Numerics

Markus Stachl Follow-the-leader models on networks 29.06.2016 30 / 48

Traffic flows on networks

Outline

1 Introduction 2 Micro to macro 3 Examples and numerics 4 Traffic flows on networks 5 Example on network

Markus Stachl Follow-the-leader models on networks 29.06.2016 31 / 48

Traffic flows on networks

Network setup I

Directed graph G of NR edges (roads) and NJ nodes (junctions) LR length of road R for R ∈ {1...nr}

Markus Stachl Follow-the-leader models on networks 29.06.2016 32 / 48

Traffic flows on networks

Network setup II

IN(j) incoming roads into node j OUT(j) outgoing roads from node j

Markus Stachl Follow-the-leader models on networks 29.06.2016 33 / 48

Traffic flows on networks

Network setup II

IN(j) incoming roads into node j OUT(j) outgoing roads from node j j ∈ origins O if IN(j) = ∅ j ∈ destinations D if OUT(j) = ∅

Markus Stachl Follow-the-leader models on networks 29.06.2016 33 / 48

Traffic flows on networks

Network setup II

IN(j) incoming roads into node j OUT(j) outgoing roads from node j j ∈ origins O if IN(j) = ∅ j ∈ destinations D if OUT(j) = ∅ at time t: R(i, t) denotes road of car i at time t yi(t) distance of car i from origin of road R(i, t)

Markus Stachl Follow-the-leader models on networks 29.06.2016 33 / 48

Traffic flows on networks

Particle flow on networks

General assumptions: preassigned paths p for all cars i = 1...n

Markus Stachl Follow-the-leader models on networks 29.06.2016 34 / 48

Traffic flows on networks

Particle flow on networks

General assumptions: preassigned paths p for all cars i = 1...n car is leader if there is no car ahead at their path ⇒ multiple leaders are possible

Markus Stachl Follow-the-leader models on networks 29.06.2016 34 / 48

Traffic flows on networks

Distances on networks

Problem: the vehicle ahead of car i is not necessarily on the same street

Markus Stachl Follow-the-leader models on networks 29.06.2016 35 / 48

Traffic flows on networks

Distances on networks

Problem: the vehicle ahead of car i is not necessarily on the same street → define NEXT(i) as the car ahead of car i ⇒ ˆ δi(t) =

• yNEXT(i)(t) − yi(t)

if R(i, t) = R(NEXT(i), t) (LR(i,t) − yi(t)) +

s Ls + yNEXT(i)(t)

if R(i, t) = R(NEXT(i), t)

Markus Stachl Follow-the-leader models on networks 29.06.2016 35 / 48

Traffic flows on networks

Distances on networks

Problem: the vehicle ahead of car i is not necessarily on the same street → define NEXT(i) as the car ahead of car i ⇒ ˆ δi(t) =

• yNEXT(i)(t) − yi(t)

if R(i, t) = R(NEXT(i), t) (LR(i,t) − yi(t)) +

s Ls + yNEXT(i)(t)

if R(i, t) = R(NEXT(i), t)

Markus Stachl Follow-the-leader models on networks 29.06.2016 35 / 48

Traffic flows on networks

Overlapping cars at junctions I

Markus Stachl Follow-the-leader models on networks 29.06.2016 36 / 48

Traffic flows on networks

Overlapping cars at junctions I

Observations: # overlapping cars at junction i is bounded by |IN(i)| area where overlapping can happen:

• LR − ℓn, LR

→ ∅ for n → ∞

Markus Stachl Follow-the-leader models on networks 29.06.2016 36 / 48

Traffic flows on networks

Handling the overlapping

Allowing ”mild” overlap by adjusting w(δ): w∗(δ) =

• w(δ)

if δ > ℓn if δ ≤ ℓn

Markus Stachl Follow-the-leader models on networks 29.06.2016 37 / 48

Traffic flows on networks

Handling the overlapping

Allowing ”mild” overlap by adjusting w(δ): w∗(δ) =

• w(δ)

if δ > ℓn if δ ≤ ℓn Extended Follow-the-leader-model      ˙ yi = w∗(ˆ δi(t)) if NEXT(i) = ∅ ˙ yi = vmax if NEXT(i) = ∅ yi(0) = yi,0 for i = 1...n

Markus Stachl Follow-the-leader models on networks 29.06.2016 37 / 48

Traffic flows on networks

Macroscopic model on networks

We can define the macroscopic model on networks as a system:

• µp

t + (µpv∗(ωp))x = 0

µp(xp, 0) = µp

0(xp)

where µp is the density of cars from population p along path p ωp = P

q=1 µq(xp, t) is the density of all cars along path p

Markus Stachl Follow-the-leader models on networks 29.06.2016 38 / 48

Traffic flows on networks

Micro to macro convergence on networks

Theorem Let the velocities v∗ and w∗ be defined as before. Fix p ∈ {1...P} and choose µp

0 = µp(0, ·) ∈ R ∩ BV (R; [0, 1]) and

yp = yp(0) = En[µp

0]. Let yp(·) be the solution of the discrete

model with initial condition yp

0 (assuming the other solution yq(·)

are given). Define µp

n(t, ·) = Cn[yp(t)].

Then µp(t, x) := limn→∞ µp

n(t, x) is a weak solution to the PDE

problem with initial condition µp

0.

Markus Stachl Follow-the-leader models on networks 29.06.2016 39 / 48

Example on network

Outline

1 Introduction 2 Micro to macro 3 Examples and numerics 4 Traffic flows on networks 5 Example on network

Markus Stachl Follow-the-leader models on networks 29.06.2016 40 / 48

Example on network

Example: Merge

Initial densities: u1(x, 0) = 0.5, u2(x, 0) = 0.3, u3(x, 0) = 0

Markus Stachl Follow-the-leader models on networks 29.06.2016 41 / 48

Example on network

Merge: numerics

Figure: Merge, result of the simulation at final time. Total macroscopic density redefined on roads (red line) and density of microscopic vehicles (blue circles) (cf. Cristiani et al., 2015, [4])

Markus Stachl Follow-the-leader models on networks 29.06.2016 42 / 48

References

References

• A. Bressan.

Hyperbolic conservation laws: An illustrated tutorial. Rendiconti del Seminario Matematico della Universit` a di Padova, pages 217–235. Gabriella Bretti, Maya Briani, and Emiliano Cristiani. An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments. Discrete and Continuous Dynamical Systems - Series S, 7(3):379–394, 2014. Rinaldo Colombo and E. Rossi. On the micro-macro limit in traffic flow. Rendiconti del Seminario Matematico della Universit` a di Padova, 131:217–235, 2014.

• S. Sahu E. Cristiani.

On the micro-ta-macro limit for first-order traffic flow models on networks. Network and Heterogeneous Media 2016 (in press)., 11. Mauro Garavello and Benedetto Piccoli. Traffic flow on networks: Conservation laws model, volume v. 1 of AIMS series on applied mathematics. American Institute of Mathematical Sciences, Springfield, MO, 2006.

Markus Stachl Follow-the-leader models on networks 29.06.2016 43 / 48

References

References

• M. di Francesco, M.D. Rosini.

Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit. Archive for Rational Mechanics and Analysis September 2015, 217:pp 831–871.

Markus Stachl Follow-the-leader models on networks 29.06.2016 44 / 48

Questions?

Backup

References

Macroscopic model on networks II

Markus Stachl Follow-the-leader models on networks 29.06.2016 47 / 48

References

Density functions

Markus Stachl Follow-the-leader models on networks 29.06.2016 48 / 48