Traffic flow on networks: conservation laws models

Daniel WORK, UC Berkeley

Benedetto PICCOLI, IAC-CNR

Outline

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

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

ab

a b

Density Evolution

Traffic

vehicle density

Flu

x (v

eh /

min

)

Fundamental Diagram

[Greenshield, 1935; Lighthill-Whitham, 1955; Richards, 1956]

Governing equation: Lighthill Whitham Richards PDE

• Model features– Shocks develop in finite time, even

from smooth initial dataResult:– 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

a b

Time = 0

x

a b

Time = t

X

[Bardos Leroux Nedelec, 1979; LeFloch,1988; Strub, Bayen 2006]

• Weak boundary conditions can be defined considering the solution to the Riemann Problem between the boundary data and trace

Weak boundary conditions

Big shockforward

Shockforward

Expansionforward and backward

Expansion forward

a

0

Expansionbackward

Small shockbackward

Big shock backward

b

Strong boundary conditions

Big shockforward

Shockforward

Expansionforward and backward

Expansion forward

0

Expansionbackward

Small shockbackward

Big shock backward

Link 1 Link 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

Outline

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

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

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:

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

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]

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

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

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]

Outline

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

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

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]

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

SanFransisco

Bay

165 UC Berkeley Graduate Student Drivers

100 rental cars 70+ Support Staff

165 UC BerkeleyGraduate Student Drivers

Po

stm

ile

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 EnKFSpeed Contour

[Work, Blandin, Tossavainen, Jacobson, Bayen, 2009]

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