jorge a. laval workshop: mathematical foundations of traffic ipam, september 2015 1
TRANSCRIPT
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A single-parameter model for the formation of oscillations within car-
following models
Jorge A. Laval
Workshop: Mathematical Foundations of TrafficIPAM, September 2015
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Multilane instabilities – FD scatter
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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)
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Measurement methodMeasurement method
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Timid and aggressive behaviors Timid and aggressive behaviors
• Laval and Leclercq, Phil. Trans. Royal Society A, 2010.
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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.
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OutlineOutline
• stochastic desired acceleration model– for a single unconstrained vehicle
• plugin to Newell’s car-following model– upgrade simulation experiment– car-following experiment
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ScopeScope
• Car-following only, no lane changes• Single lane• Homogeneous drivers, no trucks
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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
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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
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The SODEThe SODE
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Solution of the SODESolution of the SODE
• Speed and position are Normally distributed:
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Dimensionless formulationDimensionless formulation
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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
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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√
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OutlineOutline
• stochastic desired acceleration model– for a single unconstrained vehicle
• plugin to Newell’s car-following model– upgrade simulation experiment– car-following experiment
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Plugin to Newell’s car-following modelPlugin to Newell’s car-following model
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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
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Model captures oscillation growthModel captures oscillation growth
(a) (b)
t
x
t
x
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Model captures hysteresisModel captures hysteresis
(a) (b)
t
x
t
x
Trajectory Explorer (trafficlab.ce.gatech.edu)
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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
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The upgrade simulation experiment cont’d:analysis of oscillationsThe upgrade simulation experiment cont’d:analysis of oscillations
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200
400
600
dist
ance
, m
0 3 6 9 12 15time, min
2 4grade
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Fourier spectrum analysisFourier spectrum analysis
t
xperiod = 3.3 minamplitude = 21.5 km/hr
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Oscillations period and amplitudeOscillations period and amplitude
• Large variance• PDF not symmetric
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Average speed at the botlleneckAverage speed at the botlleneck
avg.
spe
ed [
km/h
r]
~
(a)
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Oscillations period and amplitudeOscillations period and amplitude
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OutlineOutline
• stochastic desired acceleration model– for a single unconstrained vehicle
• plugin to Newell’s car-following model– upgrade simulation experiment– car-following experiment
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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)
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Car-following experimentCar-following experiment
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Example dataExample data
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Trailing vehicle speed peakTrailing vehicle speed peak
-wv1
v6
5t
x [m
]
v1
v6
v [k
m/h
r]
(a)
(b)
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Trailing vehicle speed peak: oblique trajectoriesTrailing vehicle speed peak: oblique trajectories
-w*
v1
v6
v1
v6
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Trailing vehicle speed peak: oblique trajectoriesTrailing vehicle speed peak: oblique trajectories
v1
v6
v1
v6
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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)
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Car-following experiment #2Car-following experiment #2
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Car-following exp. #2–Social force modelCar-following exp. #2–Social force model
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Car-following experiment #3Car-following experiment #3
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Car-following experimentCar-following experiment
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Q & AQ & A
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THANK YOU !
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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 )
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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 )
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Trajectory ExplorerTrajectory Explorer
• www.trafficlab.ce.gatech.edu
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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)
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Parameter-free representationParameter-free representation
• )• )
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- bv c