Traffic flow on networks: conservation laws models

1. Traffic flow on networks: conservation laws models Daniel WORK, UC Berkeley Benedetto PICCOLI, IAC-CNR

2. Outline • Conservation laws models of traffic • Extension to networks • Mobile Millennium implementation

3. Flux (veh / min)‏ vehicle density Governing equation: LighthillWhitham Richards PDE • Governing equation • First order hyperbolic conservation law – LighthillWhitham Richards (LWR)PDE: • is the density of vehicles on the road • is the flux, given by: • Example (Greenshield) flux function: Density Evolution  b a x a b Traffic Fundamental Diagram [Greenshield, 1935; Lighthill-Whitham, 1955; Richards, 1956]

4. Governing equation: LighthillWhitham Richards PDE • Model features • Shocks develop in finite time, even from smooth initial data Result: • Weak (distributional) solutions: • Implementation of the boundary conditions in a strong sense (i.e., trace of the solution takes the value of the boundary data) can lead to an ill posed problem Time = 0 Time = t   x x a a b b X • [Bardos Leroux Nedelec, 1979; LeFloch,1988;Strub, Bayen 2006]

5. Weak boundary conditions • Weak boundary conditions can be defined considering the solution to the Riemann Problem between the boundary data and trace Shock forward Big shock forward Expansion forward a Big shock backward Small shock backward 0 b Expansion forward and backward Expansion backward

6. Strong boundary conditions • On a network, a neighboring link gives the “boundary data” • For mass conservation across neighboring links, strong boundary conditions must hold for all links • Strong boundary conditions define admissible fluxes between links Link 2 Strong Boundary Conditions Shock forward Big shock forward Expansion forward Big shock backward Small shock backward 0 Link 1 a Link 2 Expansion forward and backward Expansion backward

7. Outline • Conservation laws model of traffic • Extension to networks • Mobile Millennium implementation

8. Road networks • Road networks can be modeled as a directed graph • Each road is a link • Each intersection is a junction • Problem: how to define solution to the Riemann Problem at the junctions Link 3 Link 1 Link 2 Example: 1 incoming roadway, 2 outgoing roadways

9. Conservation of vehicles: solution 1 Initial density distribution: Link 3 Link 1 Link 2 One Solution: All traffic goes to Link 2 Link 3 Link 1 Link 2

10. Conservation of vehicles, solution 2 Initial density distribution: Link 3 Link 1 Link 2 Conservation not sufficient for uniqueness Another Solution: All traffic goes to Link 3 Link 3 Link 1 Link 2

11. Rule (A) traffic distribution matrix • (A) There are prescribed preference of drivers, i.e. traffic from incoming roads distribute on outgoing roads according to fixed (probabilistic) coefficients • Rule (A) implies conservation of cars: [Outgoing links flux] = A * [Incoming links flux]

12. Applying Rule (A), solution 1 • Assume a traffic distribution matrix: Link 3 Link 1 Link 2 One Solution: All traffic goes to Link 3 Link 3 Link 1 Link 2

13. Applying Rule (A), solution 2 • Assume a traffic distribution matrix: Link 3 Link 1 Link 2 Another Solution: No traffic crosses the junction Link 3 • Derivatives vanish on each link, • so PDE is satisfied. • Similarly, with no flow, • rule (A) is satisfied Link 1 Link 2

14. Rule (B) Maximize Flow • Rule (B) drivers behave as to maximize flow • Combining rules (A) and (B) yields the following linear program: Max: St: • Bounds: , are given by maximal values of admissible fluxes for strong boundary conditions • [Coclite, Garavello, and Piccoli, 2005; Garavello and Piccoli, 2006]

15. Outline • Conservation laws model of traffic • Extension to networks • Mobile Millennium implementation

16. Mobile Millennium traffic estimation • Mobile Millennium is a field operational test • Participating users download Mobile Millennium Traffic Pilot (available at traffic.berkeley.edu) on a GPS and java enabled phone • Deployment of thousands of cars in Northern California, Launched Nov. 2008 • Phones receive live information on map application

17. Network traffic estimation in Mobile Millennium • Network modelled as a directed graph (automatically generated from Navteq map database) • We cover all the major highways in Northern California • 4164 links • 3639 junctions • Networked LWR PDE is discretized using generalized Godunov scheme • Nonlinear discrete dynamical system for density is transformed into a velocity evolution equation • phones measure velocity • Real-Time data assimilation performed using nonlinear Ensemble Kalman Filtering algorithm Real Time highway traffic Visualizer • [Work,Blandin, Tossavainen, Piccoli,Bayen, 2009]

18. Experimental Validation: Mobile Century • Prototype System • Run Feb. 8, 2008 • Multi-lane highway with heavy morning and evening congestion • Ground truth: Loop detectors, HD film crew on bridges. • Rich data set for future traffic modelling and estimation research San Fransisco Bay 165 UC Berkeley Graduate Student Drivers 70+ Support Staff 165 UC Berkeley Graduate Student Drivers 100 rental cars 18

19. Revealing the previously unobservable (daily) • 5 car pile up accident (not Mobile Century vehicles) • Captured in real time • Delay broadcasted to the system in less than one minute Loop Detectors Speed Contour Postmile LWR with EnKF Speed Contour time • [Work,Blandin, Tossavainen, Jacobson,Bayen, 2009]

20. Summary • Lighthill Whitham Richards PDE – conservation of vehicles • Riemann Solver at junctions: • Traffic distribution matrix • Maximize flux • Mobile Millennium – Traffic estimation using GPS cell phones: http://traffic.berkeley.edu