Differential Equation Models forForecasting Highway Traffic Flow
R. Eddie Wilson, University of Bristol
EPSRC Advanced Fellowship EP/E055567/1http://www.enm.bris.ac.uk/staff/rew
Differential Equation Models for Forecasting Highway Traffic Flow – p. 1/32
Outline of talk
Data part I (fairly standard)
Differential equation models
Data part II (revolutionary)
Differential Equation Models for Forecasting Highway Traffic Flow – p. 2/32
Inductance Loop Infrastructure
Loop pairs every 500m, sometimes closer
Measure lane, time, speed, vehicle length
Results usually time-averaged (1 minute)
Part of UK Highways Agency Controlled Motorwaysproject and similar systems in many other countries
Differential Equation Models for Forecasting Highway Traffic Flow – p. 3/32
Active Traffic Management
Intelligent Transport Systems on highwaysQueue Ahead warning systemsTemporary speed limitsLane managementRamp metering
Spacing of inductance loop sites is in range 30m to100m – usual operation gives 1 minute averages
Differential Equation Models for Forecasting Highway Traffic Flow – p. 4/32
Birmingham Box motorway system
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J5
J6
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TDifferential Equation Models for Forecasting Highway Traffic Flow – p. 5/32
Patterns at a Single Detector
Data from single loop site (3 lanes active):
0 5 10 15 200
20
40
60
80
100
120
5 10 15 200
20
40
60
80
100
120
140total flowper minute(3 lanes)
time (hrs) time (hrs)
lanes 1 2 3
spee
d (k
m/h
)
Very similar pictures all over the world
No simple relation between speed and flow:consequences for system versus driver optimality
Differential Equation Models for Forecasting Highway Traffic Flow – p. 6/32
Speed-Flow Relationship
0 5 10 15 20 25 30 35 40 450
20
40
60
80
100
120
140
160
flow (vehs/min)
velo
city
(km
/h)
free flow (Poissonian statistics etc.)
congestion
vehicles interactoscillations
lane 2 records only
Differential Equation Models for Forecasting Highway Traffic Flow – p. 7/32
Speed-Density and Flow-Density
0 10 20 30 40 50 60 70 800
20
40
60
80
100
120
140
160
0 10 20 30 40 50 60 70 80 900
5
10
15
20
25
30
35
40
45velocity (km/h)
density (veh/km) density (veh/km)
flow
(ve
hs/m
in)
lane 2 measurementsonly
oscillations
free
flow
congestioncongestion
freeflow
Suggests state relationship v = V (ρ), flow = ρV (ρ)v is velocity, ρ is density
Why the high flow variation in congested conditions?To explain, need to consider spatial effects also
Differential Equation Models for Forecasting Highway Traffic Flow – p. 8/32
Spatiotemporal data from the M42
distance in km against time in hrs.
Differential Equation Models for Forecasting Highway Traffic Flow – p. 9/32
Zoom into structure
distance in km against time in hrs.
Differential Equation Models for Forecasting Highway Traffic Flow – p. 10/32
Outline of talk
Data part I (fairly standard)
Differential equation models
Data part II (revolutionary)
Differential Equation Models for Forecasting Highway Traffic Flow – p. 11/32
Some facts and conclusions
Ignition of stop-and-go waves is irregularneeds full noisiness of microscopic description (butpredictions can only be probabilistic)
Wavelength is much longer than vehicle separationhow to capture the upscaling effect?
General idea here: identify families of models which arequalitatively ok and throw away models which arequalitatively inadequate
IN FUTUREFit models to microscopic dataUse emergent macroscopic dynamics for predictions
Differential Equation Models for Forecasting Highway Traffic Flow – p. 12/32
‘Whole-link’ Models ( ≃ most macro)
General idea, travel time model:
τ = f(x(t))
τ transit time for vehicles entering link at time t
x is number of vehicles on link at time t
Conservation:x(t) = qin(t) − qout(t),
qout(t + τ(t)) =qin(t)
1 + τ(t)
Some type of neutral delay differential equation
Useless for studying spatial pattern, maybe ok fornetwork equilibrium problems
Differential Equation Models for Forecasting Highway Traffic Flow – p. 13/32
PDE Modelling Approach (I)
Work with continuous density ρ(x, t), velocity v(x, t)
Conservation of vehicles
ρt + (ρv)x = 0.
Lighthill-Whitham-Richards (LWR) 1950s
v = V (ρ) (speed-density function)
Typical choice, Greenshield’s model
V (ρ) = vmax
(
1 −ρ
ρmax
)
, (1)
gives unimodal fundamental diagram Q(ρ) := ρV (ρ),characteristics give shock-waves, fans etc.
Differential Equation Models for Forecasting Highway Traffic Flow – p. 14/32
PDE Modelling Approach (II)
Much later: pressure, diffusion effects
vt + vvx = α {V (ρ) − v} − βρx
ρ+ µ
vxx
ρ
Pattern formation (good) via linear instability
Problems with well-posedness, backward flowing traffic
Solution is to express pressure gradient in Lagrangianframe (Aw-Rascle, Zhang, Lebacque)
It + vIx = −f(I),
where v = g(I) and I is Lagrangian marker(but then no pattern unless it is built into initial data)
Differential Equation Models for Forecasting Highway Traffic Flow – p. 15/32
Car following models
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x
xnxn+1 xn−1
vn vn−1vn+1
hn
Typical form
xn = vn,
vn = f(hn, hn, vn) and generalisations
E.g. Bando model (1995) or extensions
f = α {V (hn) − vn} + βhn, α, β > 0
V is Optimal Velocity or Speed-Headway function
Differential Equation Models for Forecasting Highway Traffic Flow – p. 16/32
Equilibrium curves
0 0.5 1 1.5 2 2.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
speed
headway
speed
density
density
flow
no observationsdue to sensing method
Differential Equation Models for Forecasting Highway Traffic Flow – p. 17/32
Empirical equilibrium curves (again)
0 10 20 30 40 50 60 70 800
20
40
60
80
100
120
140
160
0 10 20 30 40 50 60 70 80 900
5
10
15
20
25
30
35
40
45velocity (km/h)
density (veh/km) density (veh/km)
flow
(ve
hs/m
in)
lane 2 measurementsonly
oscillations
free
flow
congestioncongestion
freeflow
Validates state relationship v = V (ρ), flow = ρV (ρ)v is velocity, ρ is density
Note high flow variation in congested conditionsIs our explanation correct?(two-phase vs three-phase arguments)
Differential Equation Models for Forecasting Highway Traffic Flow – p. 18/32
Outline of talk
Data part I (fairly standard)
Differential equation models
Data part II (revolutionary)
Differential Equation Models for Forecasting Highway Traffic Flow – p. 19/32
Three Key Ideas
Intelligent Transport Systems infrastructure is routinelycollecting data which we should exploit to rationaliseand improve traffic models
Lots of dumb data (cheap) plus clever analysis may beas good as a little clever data (expensive)
I want to advertise a new data set which I am makingavailable to the community
http://www.enm.bris.ac.uk/trafficdata
Differential Equation Models for Forecasting Highway Traffic Flow – p. 20/32
NGSIM project / helicopter photography
Expensive data!!!!
Differential Equation Models for Forecasting Highway Traffic Flow – p. 21/32
Individual Vehicle Data: pair of loops
1 2 3lanes
1 2 3lanes
time,
6 s
ecs
12589
108
96
117
113
89
129
95
119
111
113
location A location B Individual vehicledata gives‘helicopter view’(speeds km/h)
Location B is 100mdownstream oflocation A: notelane change
Further spatialextent?
Differential Equation Models for Forecasting Highway Traffic Flow – p. 22/32
Ongoing trial
Piggy-backs on government trial of IDRIS hardware system— removes need for engineer’s terminals
Started January 2008
16 sites at 100m spacing near J4 on M42 including amerge (one carriageway only)
Exercise is ongoing (with some breaks)
Data is generally much cleaner than other IVD
Differential Equation Models for Forecasting Highway Traffic Flow – p. 23/32
Aerial View
http://www.enm.bris.ac.uk/trafficdata
Differential Equation Models for Forecasting Highway Traffic Flow – p. 24/32
6 sites× 20s of Individual Vehicle Data
1 2 3
5.4
5.4002
5.4004
5.4006
5.4008
5.401
5.4012
5.4014
5.4016
5.4018
5.402
x 104
85
103 117104
87
8998
107
107
10791
105
1 2 3
98
86
117101
107
88
89101
108
107
108
91111
1 2 3
104
89
119
100
107
88
10793
108
109
111
11487
117
1 2 3
104
88 119
108
101
89108
93
109
109
109
111
116
901 2 3
107
11988
113
99
10988
94
107
111
114
109
1 2 3
93
105
119
90 113
104
111
8789
101 117
109
113
time
(s)
Differential Equation Models for Forecasting Highway Traffic Flow – p. 25/32
Patterns in instrumented section (1)di
stan
ce (
km)
speed (km/h)
time (hrs)Differential Equation Models for Forecasting Highway Traffic Flow – p. 26/32
Patterns in instrumented section (2)
time (hrs)
speed (km/h)
dist
ance
(km
)
Differential Equation Models for Forecasting Highway Traffic Flow – p. 27/32
Outline of re-identification algorithm (1)
NB no rocket science
First step: partition data
Given an upstream record u find downstream records dfor which
|tu + ∆x/vu − td| < ǫ
Given a downstream record d find upstream records ufor which
|td − ∆x/vd − tu| < ǫ
Defines bipartite graph — partition it into maximalconnected components
Unique possible matches: components of size 1 × 1Can be used to learn length error statistics etc.
Differential Equation Models for Forecasting Highway Traffic Flow – p. 28/32
Outline of re-identification algorithm (2)Simple case: n × n bipartite component where n is smalli.e. n upstream records U and n downstream records DSeek ‘optimal’ bijection Π : U → D
For example, seek Π for
min∑
πijcij
cij measures length discrepancy, time discrepancy. . . (messy)(Hungarian / auction algorithms)
In practice: scores must take into account lane orderpermutations and are non-additive
All this messy: neural nets etc.
Greater than 99% accuracy in Phase 1
Differential Equation Models for Forecasting Highway Traffic Flow – p. 29/32
Does it really work?
0 5 10 15 200
5
10
15
20
25
30
35
40
45
50
TRIVIAL TRIVIAL TRIVIAL
time (hrs)
spee
d (m
/s)
OK
OK
OK
OK
ON
GO
ING
ON
GO
ING
Differential Equation Models for Forecasting Highway Traffic Flow – p. 30/32
Some Ideas for Projects (an invitation)
Camera trajectory data vs IVD — can they complementeach other (in fitting car-following models)?
Lane changing / lane-effect models calibrated in detail
Complex behavioural models (with respect to theweather, the variable speed limit, . . .)
Macroscopic theory revisited (experiment with filterwidth and test continuum assumption)
Generalisations / refinements of potential models whichuse the extra spatial information
Vehicle re-identification algorithms themselves(real-time operation?)
Differential Equation Models for Forecasting Highway Traffic Flow – p. 31/32