2D Traffic Flow Modeling via Kinetic Models

Transcription

1 Modeling via Kinetic Models Benjamin Seibold (Temple University) September 22 nd, 2017 Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 1 / 18

2 Overview Different Descriptions of Physical Systems Microscopic description (N-particle system) ẋ j = v j, v j = F j, with force F j = F j (x 1,..., x N, ẋ 1,..., ẋ N ). Kinetic (=mesoscopic) description t f + v x f + F v f = C[f ] Boltzmann equation (if collision term C[f ] = 0: Vlasov equation) f = f (x, t, v) phase space density = probability to find particle at position x, time t, flying with velocity v. Macroscopic description ρ t + (ρu) = 0 u t + (u )u =... Key question conservation of mass (continuity equation) balance of momentum Can two particles at the same position have different velocities? Yes: Kinetic description; No: Macroscopic description Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 2 / 18

3 Overview Different Descriptions of Physical Systems Microscopic description ẋ j = v j, v j = F j. Kinetic description t f + v x f + F v f = C[f ] Macroscopic description ρ t + (ρu) = 0, u t + (u )u =... Key question (slightly relaxed) Can two particles at the same position have different velocities? Or: Can two particles very close to each other have vastly different velocities? Yes: Kinetic description required; No: Macroscopic does the job. Example F j = 0: free flight of particles without interaction ( photons); cannot be described macroscopically; need kinetic description. Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 3 / 18

4 Overview 1D Traffic Flow 1D Traffic Flow Can two vehicles be at the same position, while having different velocities? Strictly speaking: No. Yet, in lane-aggregated models: Yes (same x-position, but different lanes). Traditional role of kinetic descriptions in traffic modeling Not useful as stand-alone models. But very useful to derive (new and existing) macroscopic models via a systematic procedure. Yields insight into the phenomenological connection between car-following dynamics and large-scale emergent phenomena. [Klar, Guenther, Wegener, Materne, SIAM J. Appl. Math., 2004] [Borsche, Kimathi, Klar, Computers Math. Appl., 2012] [Visconti, PhD thesis, 2016] Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 4 / 18

5 Fundamental Questions 2D Traffic Flow Premise Describe large-scale flow of vehicles in large metropolitan area via a time-dependent PDE in 2D. Advantages Conceptual simplification over flow on network. PDE natural for up- and down-scaling (lower computational cost). PDE framework allows for more direct application of control theory. Key questions How many field quantities are (at minimum) needed to capture the true flow behavior of vehicles on the road network? Can one get away with a scalar (conservation) law for ρ(x, y, t)? Answer: Hard to conceive as this would not distinguish East-bound from West-bound traffic... Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 5 / 18

6 Four-Velocity Model Four-Velocity Model (for Manhattan-type Cities) Four 2D densities: (i) ρ E (x, y): vehicles heading East (towards x ) (ii) ρ W (x, y): vehicles heading West (towards x ) (iii) ρ N (x, y): vehicles heading North (towards y ) (iv) ρ S (x, y): vehicles heading South (towards y ). Defined as ρ E (x, y) #vehicles in C h(x, y) heading East h 2 where C h (x, y) = [x h 2, x + h 2 ] [y h 2, y + h 2 ] and h reasonable length scale. Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 6 / 18

7 Simplest Model Simplest Model Vehicles move with a fixed speed s > 0 in their direction of heading. Moreover, vehicles never change their heading, and do not impede each other. On domain Ω = [x L, x R ] [y L, y R ], this situation is described via the system of PDE t ρ E + s x ρ E = 0 t ρ W s x ρ W = 0 t ρ N + s y ρ N = 0 t ρ S s y ρ S = 0 with b.c. ρ E = ρ in E for x = x L with b.c. ρ W = ρ in W for x = x R with b.c. ρ N = ρ in N for y = y L with b.c. ρ S = ρ in S for y = y R For low densities (free-flow traffic), this linear transport model is actually not too bad (see later). Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 7 / 18

8 Changes of Direction Changes of Direction Real vehicles may take turns at intersections. 2D continuum description: changes in heading occur at any point (x, y). t ρ E + s x ρ E = α E N ρ E α E S ρ E + α N E ρ N + α S E ρ S t ρ W s x ρ W = α W N ρ W α W S ρ W + α N W ρ N + α S W ρ S t ρ N + s y ρ N = α N E ρ N α N W ρ N + α E N ρ E + α W N ρ W t ρ S s y ρ S = α S E ρ S α S W ρ S + α E S ρ E + α W S ρ W Turning rates α old new : rate of vehicles heading in direction old switching to direction new. Model parameters, same idea as split or turning ratios at off-ramps in 1D network models. Generally α(x, t), or even α(x, t, ρ j ) (fewer left-turns possible if opposing flow at high density). Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 8 / 18

9 Kinetic Description Kinetic Description Can introduce additional directions (diagonal roads), all the way to a continuum of directions. Yields kinetic model: direction of travel = polar angle θ w.r.t. East. Phase space density of vehicles φ(t, x, y, θ). t φ + s cos(θ) x φ + s sin(θ) y φ = 2π 0 2π 0 σ(t, x, y, θ, θ)φ(t, x, y, θ ) dθ σ(t, x, y, θ, θ ) dθ φ(t, x, y, θ) Here σ(t, x, y, θ, θ) is collision kernel = probability that a vehicle heading in the direction θ changes its heading to the new direction θ. Multi-density PDE model is discretization (in θ) of kinetic model via finite set of directions. Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 9 / 18

10 Density Propagation in 1D Density Propagation in 1D Long road with many traffic lights. Want large-scale averaged law of traffic flow: ρ t + q x = 0. Low density: Negligible queuing at lights. Vehicles advance at effective average speed s f (depending on signal phase properties) independent of ρ. Hence q = s f ρ. Macroscopic free flow. q q c ρ c ρ s ρ m ρ Medium density: Traffic lights limit flux. Green light lets 1 vehicle pass per time τ. Average (long time scale) throughput independent of ρ. Hence q = q c β/τ, where 0 β 1 is green portion. High density: Spillback effects (from an intersection to the one upstream) limit throughput. In practice, rather than modeling from first principles, these large-scale fundamental diagrams should be determined from data. Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 10 / 18

11 Density Propagation in 1D Density Propagation in 1D: Solution Behavior ρ c = 0.4, ρ s = 0.8, ρ m = 1, s f = 1 Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 11 / 18

12 Density Propagation in 1D 2D Model with Nonlinear 1D Fluxes Yields nonlinear model with diagonal fluxes: t ρ E + x Q(ρ E ) = α E N ρ E α E S ρ E + α N E ρ N + α S E ρ S t ρ W x Q(ρ W ) = α W N ρ W α W S ρ W + α N W ρ N + α S W ρ S t ρ N + y Q(ρ N ) = α N E ρ N α N W ρ N + α E N ρ E + α W N ρ W t ρ S y Q(ρ S ) = α S E ρ S α S W ρ S + α E S ρ E + α W S ρ W In general, flux function Q depends on position (x, y), time t, and on the direction of travel: Q E, Q W, Q N, and Q S. q c Q(ρ) ρ c ρ s ρ m ρ Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 12 / 18

13 Modeling Spillback Modeling Spillback Highly congested traffic: throughput reduction on city network due to spillback; one flow direction impedes another flow direction due to vehicles blocking the intersection. Coupling between fluxes at high densities. ψ 1 0 Example ρ s ρ m ρ Only ρ E and ρ N present: Q E (ρ E, ρ N ) = Q(ρ E )ψ(ρ N ) Q N (ρ E, ρ N ) = Q(ρ N )ψ(ρ E ) with Q(ρ) is 1D flux function. Exceeding spillback density (ρ > ρ s ), flow impedes other directions. Friction function ( ( ) ) ψ(ρ) = min max 1 ρ ρs ρ m ρ s, 0, 1 All four directions present Q E = Q(ρ E ) min{ψ(ρ N ), ψ(ρ S )} Q W = Q(ρ W ) min{ψ(ρ N ), ψ(ρ S )} Q N = Q(ρ N ) min{ψ(ρ E ), ψ(ρ W )} Q S = Q(ρ S ) min{ψ(ρ E ), ψ(ρ W )} Only perpendicular fluxes are reduced. Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 13 / 18

14 Modeling Impedance of Turning Rates Modeling Impedance of Turning Rates Right turn: Friction from new direction of travel. Replace α E S by α E S = α E S ψ(ρ S ). Left turn: α E N receives friction from ρ N (new direction), from ρ S (squeezing through cars blocking the intersection), and from ρ W (if no specific left turn phase): α E N = α E N min{ψ(ρ N ), ψ(ρ S ), ψ(ρ W )}. [Could also use different friction functions for different effects.] Full model t ρ E + x Q E (ρ) = α E N (ρ)ρ E α E S (ρ)ρ E + α N E (ρ)ρ N + α S E (ρ)ρ S t ρ W x Q W (ρ) = α W N (ρ)ρ W α W S (ρ)ρ W + α N W (ρ)ρ N + α S W (ρ)ρ S t ρ N + x Q N (ρ) = α N E (ρ)ρ N α N W (ρ)ρ N + α E N (ρ)ρ E + α W N (ρ)ρ W t ρ S x Q S (ρ) = α S E (ρ)ρ S α S W (ρ)ρ S + α E S (ρ)ρ E + α W S (ρ)ρ W. Here ρ = (ρ E, ρ W, ρ N, ρ S ) T. Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 14 / 18

15 Model Results No Interaction Between Directions Model Results: No Interaction Between Directions Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 15 / 18

16 Model Results Interaction Between Directions Model Results: Interaction Between Directions Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 16 / 18

17 Model Results Blocking of Roads Model Results: Blocking of Roads Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 17 / 18

18 Model Calibration for Kinetic 2d Traffic Models Model Calibration for Kinetic 2d Traffic Models Conclusion: These types of models look intriguing, worth investigating further. Lots of questions, most fundamentally: Coupled models well-posed? A key open research step is how to calibrate (data-fit) and also validate these models. At first glance, this looks extremely challenging, given the high-dimensional structure of the scattering kernel σ(t, x, y, θ, θ). However, (1) data are abundant to fit 5D functions; and (2) The 2D PDE model structure is actually really nice: We have removed the nasty network from the equations Hence, rather than equipping roads and intersections with sensors, we can directly use vehicle trajectory data (GPS) in (longitude, latitude); no mapping onto the road network needed. Benjamin Seibold (Temple University) 2D Traffic Modeling via Kinetic Models 09/22/2017, ERC Scale-FreeBack 18 / 18