Transcript traws1 13172

A single-parameter model for the
formation of oscillations within carfollowing models
Jorge A. Laval
Workshop: Mathematical Foundations of Traffic
IPAM, September 2015
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Multilane instabilities – FD scatter
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Introduction: NGSIM US-101
(NGSIM data)
lane-changes
distance, m
600
400
200
0
7:50
7:53
7:56
time, min
7:59
8:05 2 4
grade
(Simulation)
600
distance, m
8:02
400
200
0
0
3
3
6
time, min
9
12
15 3 2 4
grade
Measurement method
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Timid and aggressive behaviors
• Laval and Leclercq, Phil. Trans. Royal Society A, 2010.
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Today’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, 228238.
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Outline
• 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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Scope
• Car-following only, no lane changes
• Single lane
• Homogeneous drivers, no trucks
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Stochastic desired accelerations
Acceleration (m/s/s)
2
• Data collected with android
app
• Platoon leader accelerating
at traffic lights; i.e., an
unconstrained vehicle
a(v) = -0.0615 v + 1.042
R² = 0.6627
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vc
-b
0
0
10
5
10
15
Speed (m/s)
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Stochastic desired accelerations
• desired acceleration  vehicle downstream does not
constrain the motion
Acceleration (m/s/s)
2
a(v) = -0.0615 v + 1.042
R² = 0.6627
𝑎 𝑣 = 𝑣𝑐 − 𝑣 β+white noise
Normal
1
vc
-b
0
0
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5
10
15
Speed (m/s)
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The SODE
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Solution of the SODE
• Speed and position are Normally distributed:
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Dimensionless formulation
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(a) model
}~
85%-probability bounds
speed, v [m/s]
dimensionless speed, ~v
An example acceleration process
(b) data
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15
10
5
0
0
~
dimensionless time, t
15
10
time, t [s]
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Coefficient of Variation / 2
0.
1
v~0
(c)
=0
to √2
v~0 = 0.2
~v = 0.5
0
~ =1
v0
~
v0 = 3
~
dimensionless time, t
to
v~0 =
0 .1
~ = 0.2
v0
(d)
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=
~
to
~2
C[x( t~ )]/ 
~2
C[v~( t~ )]/ 
v0~
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~
~
v0 = 0
v~0 = 0.5
~
v0 = 1
to 0
~
v0 = 3
dimensionless time, t~
• Parameter-free
• most variability at the beginning and for low speeds
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Outline
• 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 model
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The upgrade simulation experiment
acceleration, a
• Single lane, 100m-100G% upgrade
0
vc
u speed, v
0
crawl
speed
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-b
gG
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Model captures oscillation growth
(a)
x
x
t
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(b)
t
Model captures hysteresis
(b)
(a)
x
x
t
t
Trajectory Explorer (trafficlab.ce.gatech.edu)
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Model captures “concavity”
• Tian et al, Trans. Res. B (2015)
• Jian et al, PloS one (2014)
model,
0.12
speed SD
2.5
2.0
1.5
1.0
0.5
0
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5
10
15
20
25
veh
The upgrade simulation experiment cont’d:
analysis of oscillations
distance, m
600
400
200
0
0
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3
6
time, min
9
12
15
2 4
grade
Fourier spectrum analysis
period = 3.3 min
amplitude = 21.5 km/hr
x
t
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Oscillations period and amplitude
• Large variance
• PDF not symmetric
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avg. speed [km/hr]
Average speed at the botlleneck
(a)
~
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Oscillations period and amplitude
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Outline
• 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 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 experiment
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Example data
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Trailing vehicle speed peak
(a)
v1
x [m]
v6
5t
v6
(b)
v [km/hr]
v1
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-w
Trailing vehicle speed peak: oblique trajectories
v1
v6
-w*
v6
v1
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Trailing vehicle speed peak: oblique trajectories
v1
v6
v6
v1
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Car-following experiment #1


th
leader, i =1 (data)
95 percentile (model)
median
veh i = 6 (data)
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5th percentile
Car-following experiment #2

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
Car-following exp. #2–Social force model

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
Car-following experiment #3
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
Car-following experiment
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
Q&A
THANK YOU !
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90%-probability interval


h(v1,)
95th percentile
median
h(v1,)
5th percentile
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Car-following experiment


h(v1,)
95th percentile
median
h(v1,)
5th percentile
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Trajectory Explorer
• www.trafficlab.ce.gatech.edu
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Models 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)
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• Heterogeneous behavior in congestion (Laval and Leclercq, 2010, Chen et
al, 2012a,b)
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Source: NGSIM (2006)
Parameter-free representation
• 𝑡 ′ = 𝑡 𝛽,
• 𝑥 ′ = 𝑥/(𝐾𝑄/𝛽)
• 𝑛′ = 𝑛/(𝑄/𝛽)
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-b
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vc