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

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

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

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Governing equation: Lighthill Whitham 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 x ab Time = 0 x ab Time = t X [Bardos Leroux Nedelec, 1979; LeFloch,1988; Strub, Bayen 2006]

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

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Strong boundary conditions Big shock forward Shock forward Expansion forward and backward Expansion forward 0 Expansion backward Small shock backward Big shock backward Link 1Link 2a Link 2 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

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

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Road networks Link 1 Link 2 Link 3 Example: 1 incoming roadway, 2 outgoing roadways 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

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

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

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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]

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

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

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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]

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

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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

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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]

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Experimental Validation: Mobile Century 18 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 100 rental cars 70+ Support Staff 165 UC Berkeley Graduate Student Drivers

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Postmile time 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 LWR with EnKF Speed Contour [Work, Blandin, Tossavainen, Jacobson, Bayen, 2009]

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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

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