Jorge A. Laval Workshop: Mathematical Foundations of Traffic IPAM, September 2015 1.

A single-parameter model for the formation of oscillations within car-

following models

Jorge A. Laval

Workshop: Mathematical Foundations of TrafficIPAM, September 2015

1

2

Multilane instabilities – FD scatter

3

3

Introduction: NGSIM US-101Introduction: NGSIM US-101

0

200

400

600

7:50 7:53 7:56 7:59 8:02 8:05time, min

2 4grade

lane-changes

dist

ance

, m

0

200

400

600

dist

ance

, m

0 3 6 9 12 15time, min

2 4grade

(NGSIM data)

(Simulation)

4

Measurement methodMeasurement method

5

6

Timid and aggressive behaviors Timid and aggressive behaviors

• Laval and Leclercq, Phil. Trans. Royal Society A, 2010.

7

Today’s hypothesisToday’s hypothesis

• the random error in drivers acceleration processes may be responsible for most traffic instabilities:– Formation and propagation of oscillations– Oscillations growth– Hysteresis

• Laval, Toth, and Zhou (2015), A parsimonious model for the formation of oscillations in car-following models. Transportation Research Part B 70, 228-238.

8

OutlineOutline

• stochastic desired acceleration model– for a single unconstrained vehicle

• plugin to Newell’s car-following model– upgrade simulation experiment– car-following experiment

9

ScopeScope

• Car-following only, no lane changes• Single lane• Homogeneous drivers, no trucks

10

Stochastic desired accelerationsStochastic desired accelerations

• Data collected with android app

• Platoon leader accelerating at traffic lights; i.e., an unconstrained vehicle

a(v) = -0.0615 v + 1.042R² = 0.6627

0

1

2

0 5 10 15 20

Acc

eler

atio

n (m

/s/s

)

Speed (m/s)

- b

vc

11

Stochastic desired accelerationsStochastic desired accelerations

a(v) = -0.0615 v + 1.042R² = 0.6627

0

1

2

0 5 10 15 20

Acc

eler

atio

n (m

/s/s

)

Speed (m/s)

- b

vc

Normal 𝑎 (𝑣 )=(𝑣𝑐−𝑣 )β +white noise

• desired acceleration vehicle downstream does not constrain the motion

12

The SODEThe SODE

13

Solution of the SODESolution of the SODE

• Speed and position are Normally distributed:

14

Dimensionless formulationDimensionless formulation

15

d im e n s io n le s s t im e , t~

dim

ensi

onle

ss s

peed

, v~

0

5

1 0

1 5

2 0

0 1 0 2 0tim e , [ s ]t

spee

d,

[m

/s]

v

8 5 % -p ro b a b ility b o u n d s

~}(a ) (b )

An example acceleration processAn example acceleration process

datamodel

16

Coefficient of Variation / s2Coefficient of Variation / s2

• Parameter-free• most variability at the beginning and for low speeds

dimensionless time, t

v0 = 0~

v0 = 0.1

~

v0 = 0.2~

v0 = 0.5~

v0 = 1~

v0 = 3~

C[

()]

/v

t 2~

to 8

~~

~

~

v0 = 0~v0 = 0.1~

v0 = 0.2~

v0 = 0.5~

v0 = 1~

v0 = 3~

C[

()]

/x

t

2~

~

to 8

~

dimensionless time, t~

(c) (d)

to 0to 2√

17

OutlineOutline

• stochastic desired acceleration model– for a single unconstrained vehicle

• plugin to Newell’s car-following model– upgrade simulation experiment– car-following experiment

18

Plugin to Newell’s car-following modelPlugin to Newell’s car-following model

19

The upgrade simulation experimentThe upgrade simulation experiment

• Single lane, 100m-100G% upgrade

speed, v

acce

lera

tion

, a

u

gG

0 0crawl speed

- b

vc

20

21

Model captures oscillation growthModel captures oscillation growth

(a) (b)

t

x

t

x

22

Model captures hysteresisModel captures hysteresis

(a) (b)

t

x

t

x

Trajectory Explorer (trafficlab.ce.gatech.edu)

23

Model captures “concavity”Model captures “concavity”

• Tian et al, Trans. Res. B (2015)• Jian et al, PloS one (2014)

0 5 10 15 20 25veh

0.5

1.0

1.5

2.0

2.5

speed SDm odel, 0.12

24

The upgrade simulation experiment cont’d:analysis of oscillationsThe upgrade simulation experiment cont’d:analysis of oscillations

0

200

400

600

dist

ance

, m

0 3 6 9 12 15time, min

2 4grade

25

Fourier spectrum analysisFourier spectrum analysis

t

xperiod = 3.3 minamplitude = 21.5 km/hr

26

Oscillations period and amplitudeOscillations period and amplitude

• Large variance• PDF not symmetric

27

Average speed at the botlleneckAverage speed at the botlleneck

avg.

spe

ed [

km/h

r]

~

(a)

28

Oscillations period and amplitudeOscillations period and amplitude

29

OutlineOutline

• stochastic desired acceleration model– for a single unconstrained vehicle

• plugin to Newell’s car-following model– upgrade simulation experiment– car-following experiment

30

Car-following experimentCar-following experiment

• 6-vehicle platoon, unobstructed leader• 5Hz GPS devices and Android app in each vehicle• two-lane urban streets around Georgia Tech campus

• Objective: – compare 6th trajectory with model prediction– given: leader trajectory and grade G=G(x)

31

Car-following experimentCar-following experiment

32

Example dataExample data

33

Trailing vehicle speed peakTrailing vehicle speed peak

-wv1

v6

5t

x [m

]

v1

v6

v [k

m/h

r]

(a)

(b)

34

Trailing vehicle speed peak: oblique trajectoriesTrailing vehicle speed peak: oblique trajectories

-w*

v1

v6

v1

v6

35

Trailing vehicle speed peak: oblique trajectoriesTrailing vehicle speed peak: oblique trajectories

v1

v6

v1

v6

36

Car-following experiment #1Car-following experiment #1

9 5 p e rc e n tileth

5 p e rc e n ti leth

le a d e r, = 1 (d a ta )i

v e h = 6i

m e d ia n

(d a ta )

(m o d e l)

37

Car-following experiment #2Car-following experiment #2

38

Car-following exp. #2–Social force modelCar-following exp. #2–Social force model

39

Car-following experiment #3Car-following experiment #3

40

Car-following experimentCar-following experiment

41

Q & AQ & A

41

THANK YOU !

42

90%-probability interval90%-probability interval

9 5 p e rc e n tileth

5 p e rc e n ti leth

m e d ia n

h v( ,1 )

h v( ,1 )

43

Car-following experimentCar-following experiment

9 5 p e rc e n tileth

5 p e rc e n ti leth

m e d ia n

h v( ,1 )

h v( ,1 )

44

Trajectory ExplorerTrajectory Explorer

• www.trafficlab.ce.gatech.edu

45

45

Source: NGSIM (2006)

Models that predict OscillationsModels that predict Oscillations

• Unstable car-following models• 2nd-order models• Delayed ODE type: oscillation period predicted ~ a few seconds

(Kometani and Sasaki, 1958, Newell, 1961)• ODE type, a few minutes (Wilson, 2008)

• Fully Stochastic Models• Random perturbations not connected with driver behavior (NaSch, 1992,

Barlovic et al., 1998, 2002, Del Castillo, 2001 and Kim and Zhang, 2008)

• Behavioral models• Human error (Yeo and Skabardonis, 2009)• Heterogeneous behavior in congestion (Laval and Leclercq, 2010, Chen

et al, 2012a,b)

46

Parameter-free representationParameter-free representation

• )• )

47

- bv c