Effect of Informa.on Availability on Stability of Traffic Flow: Percola.on Theory Approach

Alireza Talebpour, Hani S. Mahmassani and Samer H. Hamdar

Mo.va.on

•  Connected and automated vehicles (CAVs) will soon join the current vehicle fleet.

•  Their introduc.on will re-shape the transporta.on industry: •  Safety •  Conges.on •  Emissions and energy consump.on •  Freight •  etc.

•  Understanding CAVs’ full impact on traffic performance requires accurate modeling of their dynamics/movement.

Source:h*p://planningforreality.org/

Mo.va.on

•  Safety and efficiency are the main mo.va.ons behind bringing these technologies to market.

•  Developing string stable connected and automated systems is the key to achieve the desired safety and efficiency.

StringStablePlatoon StringUnstablePlatoon

Background

•  Many studies inves.gated string stability in connected, automated systems (mostly CACC systems):

•  The benefits of CACC over ACC systems were iden.fied (Zhang and Orosz, 2015; Lu et al., 2002).

•  The immediate leader and platoon head were found to be the key to ensuring string stability (Swaroop, 1997).

•  The possibility of developing decentralized CACC systems was shown (Naus et al., 2010).

Problem Statement

DecisionMakingHuman:lessefficientAV:moreefficient

Telecommunica@ons

VehicularMovements

TrafficPa*ernInform

a@on

Behavior

Telecommunica@onsaffectTrafficPa*ernTrafficPa*ernaffectsconnec@vity(Telecommunica@ons)

Problem Statement

•  Communica.ons dynamics and informa.on loss was not captured in these studies.

Objec.ve

•  The main objec.ve of this study is to

Develop a methodology to incorporate signal

interference and informa.on loss into analy.cal inves.ga.ons of string stability.

•  This study introduces a methodology based on Percola.on Theory.

Percola.on Theory

•  The propaga.on of informa.on within the vehicular network is similar to the

•  Transport of a fluid through porous media •  Spread of a disease among people

Source:Khosravian,H.,Joekar-Niasar,V.,&Shokri,N.(2015).Effectsofflowhistoryonoilentrapmentinporousmedia:Anexperimentalstudy.AIChEJournal,61(4),1385-1390.

Percola.on Theory

•  Con.nuum Percola.on: There exists a cri.cal density (Pc), above which there is certainly a cluster with an infinite size in the system.

•  Two types of Con.nuum Percola.on: •  Boolean model •  Random Connec.on model

•  Boolean model: Each point (x) in the space is a center of a circle with radius r (C(x,r) ).

•  Combina.on of these circles divide the space into two regions: occupied region and vacant region.

•  A pair of nodes is connected if both nodes belong to the same occupied/vacant region.

Defini.ons

•  Communica.ng Vehicles

•  Homogeneous Poisson Point Process

•  Communica.on Path

P(Xλ (L) = k) =λSL( )k

k!e−λSL

h(xi, x j ) =1 if xi − x j <min Ri,Rj{ }0 Otherwise

⎧⎨⎪

⎩⎪

Defini.ons

•  Connected k-component (CCk): A set of k communica.ng vehicles that is not a subset of another set of communica.ng vehicles.

•  Distance between two CCks is the minimum distance between pairs of the vehicles in the CCks.

Percola.on of Vehicular Ad-Hoc Networks

Connected Components: Length Es.ma.on

[ ][ ] L

k

RR

RLk

X

RX

RLX

LX

RLX

RX

kRLX

RX

L ekRL

ee

ekRL

dxxe

dxxe

dxxe

k

dxx

kNP λλλ

λλ

λ

λλ

λλ

−−−

+−

−

++

+

++

−

++

−

+=

+

=−−

−⎟⎠

⎞⎜⎝

⎛

==

∫∫

∫∫

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.!)2(

)(.

)(

)(

!

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)(

)2(

P(NL = k) = prob RI  contains a CCK | RII ∪RIII( )is empty{ }

Percola.on of Vehicular Ad-Hoc Networks

Cri.cal Density of Connected Components:

At the percola.on point, the density of connected k-components within a circle with radius R is given by,

Combining all the above equa.ons:

Where

[ ] Rk

ccc

cekRk λλ

λλ −=!)3(

)(

( ) ( )[ ] [ ] 01!

311,, =−⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛×+−= −− µµ µ

µλ ek

ekRfk

c

µ = − log 1− Ac (R)( ) Ac (R) =1− e−λcR

Percola.on of Vehicular Ad-Hoc Networks

•  Results indicate that percola.on first occurs at k=3 and Ac(R)=0.785.

Communica.on Probability

•  For Ac(R) ≥ 0.785: all the vehicles receive the informa.on.

•  For Ac(R) < 0.785: the probability that there is only one point in any circle with radius R is:

P(N2R =1) = 2Rλe−2Rλ

Stability Analysis Approach

In line of the defini.on of string stability, the following criteria guarantees the string instability of a heterogeneous traffic flow (Ward, 2009):

where

02

22

<⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

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≠Δ

n nm

ms

ns

nv

nv

nv fffff

Traffic Simula.on Framework

No Automa.on Not Connected

No Automa.on Connected

Self-Driving Not Connected

•  The car-following model of Talebpour, Hamdar, and Mahmassani (2011) is used.

•  Accelera.on Behavior: Probabilis.c

•  Percep.on of Surrounding Traffic Condi.on:

Subjec.ve

•  Reac.on Time: High

•  Safe Spacing: High

•  High-Risk maneuvers: Possible

•  Probabilis.c •  Recognizes two different driving regimes: •  Congested •  Uncongested

•  Consider crashes endogenously

Traffic Simula.on Framework

No Automa.on Not Connected

No Automa.on Connected

Self-Driving Not Connected

•  The Intelligent Driver Model (Treiber, Hennecke, and Helbing, 2000) is used.

•  Accelera.on Behavior: Determinis.c

•  Percep.on of Surrounding Traffic Condi.on: Accurate

•  Reac.on Time: Low

•  Safe Spacing: Low

•  High-Risk maneuvers: Very Unlikely

Traffic Simula.on Framework

No Automa.on Not Connected

No Automa.on Connected

Self-Driving Not Connected

•  Speed should be low enough so that the vehicle can react to any event outside of the sensor range ( ) (Reece and Shafer, 1993 and Arem, Driel, Visser, 2006).

vmax

Stability Analysis Results

Full Connec.vity R = 50m R = 70m

R = 90m R = 110m R = 130m

a =1.4m / s2 b = −2.0m / s2

Stability Analysis Results

Full Connec.vity R = 50m R = 70m

R = 90m R = 110m R = 130m

b = −4.0m / s2a = 3.0m / s2

Conclusion

•  This paper presents a Percola.on Theory based approach to incorporate par.al communica.on and informa.on loss into analy.cal inves.ga.on of string stability.

•  The analy.cal studies reveal that as communica.on range increases, the system becomes more stable.

•  At communica.on ranges above 130m, the system performs very similar to the system with full connec.vity assump.on.

Ques@onsandComments

Model Parameters

Parameters Typical Value

Sensitivity Exponents of the Generalized Utility γ = 0.2

Asymmetry Factor for Negative Utilities wm = 2.0

Velocity Uncertainty Variation Coefficient α = 0.08

Weighing Factor for Accidents wc =10000.0

Maximum Anticipation Time Horizon τmax = 4.0s

Logit Uncertainty Parameter (Intra-Driver Variability) β = 5.0

Maximum Acceleration amax = 4m / s2

Minimum Acceleration amin = −8m / s2

Parameters Typical Value

Free Acceleration Exponent δ = 4.0

Desired Time Gap T = 4.5s

Jam Distance s0 = 2.0m

Maximum Acceleration a =1.4m / s2

Desired Deceleration b = −2.0m / s2

•  Automated vehicle parameters are

•  Regular and connected vehicles parameters are shown in the tables.

0.1=ak , 58.0=vk , and 1.0=dk

Regular Vehicles Connected Vehicles

δ = 4.0T = 4.5ss0 = 2.0ma =1.4m / s2b = −2.0m / s2