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 Examples and numerics 3 4 Traffic flows on networks Example on network 5 Markus Stachl Follow-the-leader models on networks 29.06.2016 1 / 48

Introduction Outline 1 Introduction 2 Micro to macro Examples and numerics 3 4 Traffic flows on networks Example on network 5 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 Microscopic model Density of vehicles Positions of n single vehicles Partial differential equations Equations of motion Conservation laws System of ordinary differential equations Markus Stachl Follow-the-leader models on networks 29.06.2016 3 / 48

Introduction Introduction Macroscopic model Microscopic model Density of vehicles Positions of n single vehicles Partial differential equations Equations of motion Conservation laws 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 � u t + ( f ( u )) x = 0 u ( x , 0) = u 0 ( x ) u ∈ L 1 ( R , [0 , 1]) : � � � R = 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 ℓ n := 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 ℓ n := L n and total length L car locations y i = y i ( 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 ℓ n := L n and total length L car locations y i = y i ( t ) for i = i ... n at time t distances δ i = y i +1 − y i Markus Stachl Follow-the-leader models on networks 29.06.2016 5 / 48

Introduction Discrete microscopic model Consider n cars of length ℓ n := L n and total length L car locations y i = y i ( t ) for i = i ... n at time t distances δ i = y i +1 − y i � y n = v max ˙ motion equations: y i = w ( δ i ( 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 ℓ n := L n and total length L car locations y i = y i ( t ) for i = i ... n at time t distances δ i = y i +1 − y i � y n = v max ˙ motion equations: y i = w ( δ i ( t )) ˙ for i = 1 ... n − 1 ⇓ First order Follow-the-leader-model  y n = v max ˙   y i = w ( δ i ( t )) ˙ for i = 1 ... n − 1  y i (0) = y i , 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 ℓ n := L n and total length L car locations y i = y i ( t ) for i = i ... n at time t distances δ i = y i +1 − y i � y n = v max ˙ motion equations: y i = w ( δ i ( t )) ˙ for i = 1 ... n − 1 ⇓ First order Follow-the-leader-model  y n = v max ˙   y i = w ( δ i ( t )) ˙ for i = 1 ... n − 1  y i (0) = y i , 0 for i = 1 ... n  Y n = { ( y 1 , ..., y n ) ∈ R n : δ i := y i +1 − y i ≥ ℓ n ∀ 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 ∈ C 1 ([0 , 1] , [0 , v max ]) v ′ ( u ) < 0 v (0) = v max , 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 ∈ C 1 ([0 , 1] , [0 , v max ]) v ′ ( u ) < 0 v (0) = v max , v (1) = 0 e.g. v ( u ) = v max (1 − u ) v ( u ) = v max (1 − u n ) (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 ∈ C 1 ([ ℓ n , ∞ ] , [0 , v max ]) w ′ ( δ ) > 0 for δ > ℓ n w ( ℓ n ) = 0 , lim δ →∞ w ( δ ) = v max Markus Stachl Follow-the-leader models on networks 29.06.2016 7 / 48

Micro to macro Outline 1 Introduction 2 Micro to macro Examples and numerics 3 4 Traffic flows on networks Example on network 5 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: �  u t + ( uv ( u )) x = 0 y n = v max ˙   u ( x , 0) = u 0 ( x ) y i = w ( δ i ( t )) ˙ for i = 1 ... n − 1  y i (0) = y i , 0 for i = 1 ... n  w ( δ ) = v ( ℓ n δ ) Markus Stachl Follow-the-leader models on networks 29.06.2016 9 / 48

Micro to macro Discretization operators E n I ✎ ☞ ✎ ☞ → Traffic density u discrete vehicle positions y ✍ ✌ ✍ ✌ Take any given n . Then � y n = max ( supp ( u )) E n ( u ( · )) := y = � y i +1 � � y i = max z ∈ R : u ( x ) dx = ℓ n , i = n − 1 ... 2 , 1 z Markus Stachl Follow-the-leader models on networks 29.06.2016 10 / 48

Micro to macro Discretization operator E n II Markus Stachl Follow-the-leader models on networks 29.06.2016 11 / 48

Micro to macro Anti-discretization operator C n I ✎ ☞ ✎ ☞ discrete vehicle positions y → Traffic density u n ✍ ✌ ✍ ✌ Take a given n. Then n − 1 ℓ n � C n [ y ] =: u n = χ [ y i , y i +1 ] δ i i =1 Markus Stachl Follow-the-leader models on networks 29.06.2016 12 / 48

Micro to macro Anti-discretization operator C n 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 u 0 = u (0 , · ) ∈ R ∩ BV ( R ; [0 , 1]) and y = y (0) = E n [ u 0 ]. Let y ( · ) be the solution of the discrete model with initial condition y 0 . Define u n ( t , · ) = C n [ y ( t )]. Then u n converges almost everywhere and in L 1 loc ([0 , + ∞ ) × R ) to the unique entropy solution u to the PDE problem with initial condition u (0 , · ) = u 0 . 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: u 0

Micro to macro The macro-micro limit The theorem before states that the following diagram holds: E n u 0 y 0

Micro to macro The macro-micro limit The theorem before states that the following diagram holds: E n u 0 y 0 ODE solution y

Micro to macro The macro-micro limit The theorem before states that the following diagram holds: E n u 0 y 0 ODE solution C n y ˜ u n

Micro to macro The macro-micro limit The theorem before states that the following diagram holds: E n u 0 y 0 ODE solution C n n → ∞ y u ˜ u n

Micro to macro The macro-micro limit The theorem before states that the following diagram holds: E n u 0 y 0 solve PDE ODE solution C n n → ∞ y u ˜ u n 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 C n [ y ] to ρ w.r.t. Wasserstein distance in L 1 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 C n [ y ] to ρ w.r.t. Wasserstein distance in L 1 loc using Hell’s compactness theorem [1] d 1 ( C n [ y ] , ρ ) = || F C n [ y ] − F ρ || L 1 loc ( F i cumulated densities) Markus Stachl Follow-the-leader models on networks 29.06.2016 16 / 48