Car-Following Models as Dynamical Systems and the Mechanisms for Macroscopic 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
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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
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
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