Transcript Validating a Hamilton-Jacobi Approximation to Hybrid

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
a
(LWR) PDE:
Density Evolution
b
x
a
–
is the density of
vehicles on the road
–
is the flux, given
by:
b
Traffic
– Example (Greenshield) flux
function:
[Greenshield, 1935; Lighthill-Whitham, 1955; Richards, 1956]
Flux (veh / min)
Fundamental Diagram
vehicle density
Governing equation: Lighthill Whitham Richards PDE
• Model features
– Shocks develop in finite time, even
from smooth initial data
Result:
– Weak (distributional) solutions:
Time = 0

x
a
b
Time = t

X
– 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
[Bardos Leroux Nedelec, 1979; LeFloch,1988; Strub, Bayen 2006]
a
x
b
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
Big shock
backward
Small shock
backward
a
0
b
Expansion
backward
Expansion
forward and
backward
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
Link 1
a
Link 2
Expansion
backward
0
Expansion
forward and
backward
Outline
•
•
•
Conservation laws model of traffic
Extension to networks
Mobile Millennium implementation
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 1
Example: 1 incoming roadway, 2 outgoing roadways
Conservation of vehicles: solution 1
Initial density distribution:
Link 1
One Solution: All traffic goes to Link 2
Link 1
Conservation of vehicles, solution 2
Initial density distribution:
Link 1
Conservation not sufficient for uniqueness
Another Solution: All traffic goes to Link 3
Link 1
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
•
Assume a traffic
distribution matrix:
Link 1
One Solution: All traffic goes to Link 3
Link 1
Applying Rule (A), solution 2
•
Assume a traffic
distribution matrix:
Link 1
Another Solution: No traffic crosses the junction
Link 1
•Derivatives vanish on each link,
so PDE is satisfied.
•Similarly, with no flow,
rule (A) is satisfied
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 highway traffic
Visualizer
– Real-Time data assimilation performed
using nonlinear Ensemble Kalman
Filtering algorithm
[Work, Blandin, Tossavainen, Piccoli, Bayen, 2009]
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
165 UC Berkeley
Graduate Student Drivers
100 rental cars
70+ Support
Staff
Revealing the previously unobservable (daily)
Postmile
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]
time
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