car-following models as dynamical systems and the
TRANSCRIPT
Car-Following Models as DynamicalSystems and the Mechanisms forMacroscopic Pattern Formation
R. Eddie Wilson, University of Bristol
EPSRC Advanced Research Fellowship EP/E055567/1
http://www.enm.bris.ac.uk/staff/rew
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.1/25
Macroscopic Traffic Data
stop
-and
-go
waves
110
0
average sp
eed (km
/h)
M25 anticlockwise carriageway 1/4/2000
06:40 time 11:00
spac
e (1
7km
)
veh
icle
tra
ject
ori
es
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.2/25
Some facts and conclusions (I)Propagation of stop-and-go is (fairly) regular
so can be captured by macroscopic deterministicmodels?
v
x
Downstream interface does not spread (Kerner 90s) —problem for LWR and I believe ARZ / Lebacqueframework
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.3/25
Some facts and conclusions (II)
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: 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
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.4/25
Active Traffic Management system
Aim, reduce: accidents, (variance of) journey timesQueue Ahead warning systemsTemporary speed limitsLane management
Spacing of inductance loop pairs is in range 30m to100m
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.5/25
Individual Vehicle Data from ATM system
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
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.6/25
Zoom-view and future scope
1 2 3lanes
1 2 3lanes
tim
e, 6
sec
s
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
Propose toreconstruct vehicletrajectories over55×100m×1 week
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.7/25
Jam formation in simulations
0 50 100 150 200 250 300 3500
20
40
60
80
100
120
140
160
180
200
dimensionless time
dim
ensi
on
less
sp
ace
simulation of Optimal Velocity model
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.8/25
Car-following models
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x
xn
xn+1 xn−1
vn vn−1vn+1
hn
Typical form
xn = vn,
vn = f(hn, hn, vn) and generalisations
E.g. Bando model (1995)
f = α {V (hn) − vn} , α > 0
V is Optimal Velocity or Speed-Headway functionCar-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.9/25
Linear stability framework
General car-following model
vn = f(hn, hn, vn),
Equilbrium condition, there exists V (h) so that
f(h∗, 0, V (h∗)) = 0 for all h∗ > 0.
Linearisation yields
˙vn = (Dhf)hn + (Dhf) ˙hn + (Dvf)vn,
with sensible sign constraints
Dhf, Dhf ≥ 0 and Dvf ≤ 0.
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.10/25
Linear stability, part 2
Re-arrangement hn = vn−1 − vn gives
¨hn = (Dhf)(hn−1 − hn) + (Dhf)( ˙hn−1 −
˙hn) + (Dvf) ˙hn.
Then try exponential ansatz hn = real(
ceinθeλt)
θ is perturbation’s discrete wavenumberreal(λ) is growth rate
to obtain quadratic
λ2 +{
(Dhf)(1 − e−iθ) − (Dvf)
}
λ + (Dhf)((1 − e−iθ) = 0.
Then derive results for λ(θ) in quite general terms(proofs omitted)
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.11/25
Technical detailsShort wavelength analysis, θ = π
λ2 +{
2(Dhf) − (Dvf)
}
λ + 2Dhf = 0
All coeffs positive, therefore stable roots
Long wavelength analysis, θ > 0 small, λ = λ1θ + λ2θ2
gives λ1 = i(Dhf)/(Dvf) and
λ2 =(Dhf)
(Dvf)3
{
1
2(Dvf)2 − (Dhf) − (D
hf)(Dvf)
}
Can show neutral stability λ = iω for general θ isequivalent to λ2 = 0.Therefore: need only analyse λ2
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.12/25
Onset from infinite wavelength
0 0.1 0.2 0.3 0.4 0.5 0.6−4
−3
−2
−1
0
1
2
3x 10
−3
onset of in
stability w
ith change in
parameters
infinite
growth
discretewavenumberwavelength
rate
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.13/25
Onset at medium densities
0 0.5 1 1.5 2 2.5 3 3.5 4−0.15
−0.1
−0.05
0
0.05
0.1
chan
ge
in p
aram
eter
s
long wavelengthgrowth parameter
nondimensional headway
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.14/25
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
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.15/25
Other types of linear (in)stability
Notional experiment in semi-infinite column of vehicleswhere second vehicle is instantaneously perturbed out ofequilibrium
Linearised dynamics of nth vehicle
¨hn+[
(Dhf) − (Dvf)
] ˙hn+(Dhf)hn = (Dhf)hn−1+(Dhf) ˙hn−1
Solve resonant oscillators inductively, large t
hn(t) ∼tn−1
(n − 1)!
[
λ(Dhf) + (Dhf)
2λ + (Dhf) − (Dvf)
]n−1
eλt
where λ is stable ‘platoon’ eigenvalue
Use moving absolute space frame t = nh∗/(c + v∗) andStirling’s formula to define growth ‘wedge’
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.16/25
Problems (?) with linear instability
Setting a reduced speed limit to induce mid-rangedensity and increase flow does not induce flowbreakdown
Stop-and-go waves almost always ignite at merges orother large amplitude ‘externalities’
These problems may explain the continuing adherance
to one-phase PDE models, be they first order like LWR or
second order like ARZ/Lebacque
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.17/25
Introduction to bifurcation theory
Loss of stability of uniform flow is via a Hopf bifurcation,of which there are two types:
stable unstable
unstable jam
subcritical
stable
stable jam
unstable
parameter
no
rm
supercritical
supercritical: stable periodic solutions are bornsubcritical: unstable periodic solutions are born,branch bends back — so what is dynamics?
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.18/25
Introduction to bifurcation theory
Loss of stability of uniform flow is via a Hopf bifurcation,of which there are two types:
stable
stable jam
unstable
parameter
no
rm
supercritical
stable unstable
unstable jam
subcritical
stable jam
Subcritical bifurcation with cyclic fold gives jump to largeampitude traffic jam solution plus region of bistability
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.18/25
Computational results
Application of numerical parameter continuation tools toanalyse stop-and-go waves on the ring road
0 1 2 3 40
0.1
0.2
0.3
0.4
0.5
h∗
α Hopf (k = 1)
fold (k = 1)
Hopf (k > 1)
fold (k > 1)
stopping
collision
two
traffi
cja
ms
h∗
vamp k = 1
k = 2
k = 3
k = 4
REW, Krauskopf and Orosz, also group of Gasser
Large perturbations (lane changes at merges?) causejump to jammed state
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.19/25
Search for new dynamics
This explanation still requires uniform flow to beunstable in some parameter regime. Is a fix possible?
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.20/25
Search for new dynamics
This explanation still requires uniform flow to beunstable in some parameter regime. Is a fix possible?
‘Design’ bifurcation diagram:
always stable uniform flow
stable jam
unstable jam
headway
no
rm
Ongoing work vn = α(hn)F (V (hn) − vn)
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.20/25
Alternative: travelling wave analysisComputationally wasteful (and perhaps inappropriate)to analyse wave structures via bifurcations of periodicorbits of large systems of ODEs/DDEs
Instead: travelling wave analysis. Two methods:Weakly nonlinear continuum limit (Kim, Lee, Lee):
ρt + (ρv)x = 0, see TGF ’01
vt + vvx = α{
V (ρ) − v}
+ α
[
V ′(ρ)ρx
2ρ+
vxx
6ρ2
]
,
Single advance/delay equation, derived from
hn−1(t) = hn(t + τ), vn−1(t) = vn(t + τ)
substitution in car-following model (ongoing workwith Tony Humphries, McGill)
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.21/25
Travelling wave phase diagramSee TGF ’01
PSfrag
CR
RC
RL
CL
LC
LR
L → C
R → C
R → C
R → C
L → C
R → C
L → C
L → C
L → C
R → C
R → C
R → C
L → C
R → C
L → C
L → C
L → C
R → C
R → C
R → CL → C
R → CL → C
ρ−
ρ+
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.22/25
Recent discrete computation (stable)
0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
3.5
4
h−
h +
Solutions on (h−,h
+) plane, τ
d=0 α=2.2
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.23/25
Recent discrete computation (unstable)
0 0.5 1 1.5 2 2.5 3 3.5 40
0.5
1
1.5
2
2.5
3
3.5
4
h−
h +
Solutions on (h−,h
+) plane, τ
d=0 α=1
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.24/25
Broad conclusions
For the car-following community:Still some work to do in understanding fully patternmechanisms at the nonlinear level and on the infiniteline. Fitting models to new sources of microsopicdata.
For the PDE community:Vanilla versions of LWR/ARZ/Lebacque do notqualitatively replicate data or what car-followingmodels do generically (even at the linear level). Thisneeds a fix — NB global existence results willbecome ugly / difficult.
Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic Pattern Formation – p.25/25