Car-Following Models as Dynamical Systems and the ...

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


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


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


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


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


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


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

��

��

��

��

��

��

��

��

��

��

��

��

��

��

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.


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)


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


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


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


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


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’


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


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?


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


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


Search for new dynamics

This explanation still requires uniform flow to beunstable in some parameter regime. Is a fix possible?


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)


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)


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+


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


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


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.


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