Optimal traffic control at smart intersections

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ContentslistsavailableatScienceDirect

Transportation Research Part B

journalhomepage:www.elsevier.com/locate/trb

Optimal traﬃc control at smart intersections: Automated network fundamental diagram

Mahyar Amirgholy

^a

, Mehdi Nourinejad

^b

, H. Oliver Gao

^c^,^d^,^∗

aDepartment of Civil and Construction Engineering, Kennesaw State University, Atlanta, GA, 30060, United States

bDepartment of Civil Engineering, York University, Toronto, ON, M3J1P3, Canada

cSchool of Civil and Environmental Engineering, Cornell University, Ithaca, NY, 14853, United States

dSino-US Global Logistics Institute, Antai College of Economics and Management, Jiaotong University, Shanghai, China, 20 0 030

a r t i c l e i n f o

Article history:

Received 21 March 2019 Revised 15 July 2019 Accepted 10 October 2019 Available online xxx Keywords:

Communicant autonomous vehicles Cooperative traﬃc control Stochastic operational error Platoon size

Inter-platoon headway

Macroscopic fundamental diagram

a b s t r a c t

Recent advancesin artiﬁcialintelligenceand wireless communicationtechnologieshave createdgreat potentialtoreducecongestioninurbannetworks.Inthisresearch,wede- velopastochasticanalyticalmodel foroptimalcontrolofcommunicantautonomousve- hicles(CAVs)atsmartintersections.Wepresenttheautomatednetworkfundamentaldia- gram(ANFD)asamacro-levelmodelingtoolforurbannetworkswithsmartintersections.

Intheproposedcooperativecontrolstrategy,wemakeuseoftheheadwaybetweenthe CAVplatoonsineachdirectionforconsecutivepassageoftheplatoonsinthecrossingdi- rectionthroughnon-signalizedintersectionswithnodelay.Forthistohappen,thearrival anddepartureofplatoonsincrossingdirectionsneedtobesynchronized.Toimprovesys- temrobustness(synchronizationsuccessprobability), weallowamarginalgap between arrivalanddepartureoftheconsecutiveplatoonsincrossingdirectionstomakeupforop- erationalerrorinthesynchronizationprocess.Wethendevelopastochastictrafficmodel forthesmartintersections.Ourresultsshowthattheeffectsofincreasingtheplatoonsize andthemarginalgaplengthonthenetworkcapacityarenotalwayspositive.Infact,the capacitycanbemaximizedbyoptimizingthesecooperativecontrolvariables.Weanalyt- icallysolvethetrafficoptimizationproblemfortheplatoonsizeandmarginalgaplength andderiveaclosed-formsolutionforanormaldistributionoftheoperationalerror.The performanceofthenetworkwithsmartintersectionsispresented byastochasticANFD, derivedanalyticallyandverifiednumericallyusingtheresultsofasimulationmodel.The simulationresults show that optimizing the control variables increasesthe capacityby 138%whentheerrorstandarddeviationis0.1s.

1. Introduction

Autonomousvehiclesareexpectedtobeintroducedtotheconsumermarketinthenearfuture.Theartiﬁcialintelligence andwireless communication technologies embedded in these vehicles make “driving” more convenient and roads safer (ZhangandIoannou,2004;Van Aremetal., 2006; FernandesandNunes, 2011,2012;Ariaetal., 2016; Shabanpouretal., 2018). Improvementinthetraﬃccondition, however,wouldbetrivialinurban networkswithoutupgradingconventional

∗ Corresponding author.

E-mail addresses: [email protected] (M. Amirgholy), [email protected] (M. Nourinejad), [email protected] (H.O. Gao).

Pleasecitethisarticleas:M.Amirgholy,M.NourinejadandH.O.Gao,Optimaltraﬃccontrolatsmartintersections:Auto-

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traﬃc control systems (Mahmassani, 2016). In this research, we propose a cooperative traﬃc control strategy for smart intersectionstoreducecongestioninurbannetworks.

Theconcept ofself-drivingvehicleswasintroducedinthe1930s.However, onlyrecentadvancesincomputation, communication, and automation technologieshave made it feasible to realize the dream of autonomous vehicles. Currently, majorcarmanufacturers,alongwithhightechnologycompanies,aremakingprototypestobeintroducedby2025(Shiand Prevedouros,2016;Kockelmanetal.,2017).Theradar-basedautopilottechnologyofautonomousvehicles enablesreal-time monitoringoftheenvironmentandautomaticindependentactionsonroads(BoseandIoannou,2003;Nietal.,2010;Zohdy etal.,2015;Ariaetal.,2016;Kockelmanetal.,2017).Inadditiontoautopilottechnology,thecapabilityofcommunicantau- tonomous vehicles (CAVs) to exchange information withboth predecessors andinfrastructure, through vehicle-to-vehicle (V2V)andvehicle-to-infrastructure(V2I)communicationtechnologies,respectively,enablescooperativetraﬃccontrolinau- tomated highwaysandnetworks(Bekiaris-Liberisetal., 2016;Shi andPrevedouros,2016;Ghiasi etal., 2017;Lioris etal., 2017).

Cooperativetrafficcontrol cansubstantially increasethe throughputofautomatedhighways bysafelyincreasing speed anddecreasingtheheadwaybetweentheCAVsmovinginplatoons.(FernandesandNunes,2011,2012;LamandKatupitiya, 2013;Roncolietal.,2014;Ghiasietal.,2017).Improvinghighwaythroughputs,however,increasesnetworkinflowaswell, whichcan worsenthetraffic conditioninurban regions byoverloadingthe networkover thepeaks,ultimatelycausinga complete gridlock (hypercongestionphenomenon). Hence, the overall performance of the integratedsystem of highways andurbannetworkscanbe improvedbydynamicallycontrolling thespeedandsizeofCAVplatoons inhighways tokeep network inflowoptimizedovertime (Amirgholyetal.,2020). Overall,thelimitedcapacityofurban networksisthemain barrier to improving the traffic condition, even in interregional highways. In this research, we aim to improve network capacitybyenablingcooperativetrafficcontrolatsmarturbanintersections.

Forautomatednetworks, wecoordinateCAV platoonstosafelypassthrougheach otheratnon-signalizedintersections withnointerruption.Forthistohappen,theinter-platoonheadway(thetimegapbetweenthepassageoftherearbumper of thelast vehicle ina platoon andthe front bumper ofthe leader ofthe next platoon,froma referencepoint) ineach directionneeds tobesuﬃcient forthesafepassageoftheconsecutiveplatoonsin thecrossingdirection. Thus,theeffect of increasing the size ofthe platoons on the capacity ofthe network is not always positive, as opposed to the case in automated highways.¹ In thisresearch,we maximize networkcapacityby optimizingplatoonsize (numberof vehiclesin eachplatoon)asoneoftheprimarycooperativecontrolvariablesofthesystem.

Networkcapacitylargelydependsontheprecisionandspeedofsensors,computationprocessing,vehicle-to-vehicleand vehicle-to-infrastructurecommunicationtechnology,andtheactuationsystem.Operationalerrorincoordinatingthearrival anddepartureoftheplatoonsatanintersectioncancauseafailure(interruption)inthesynchronizationprocess.Forresyn- chronization,theapproaching platoonstopsattheintersectionupon anearly/latearrivalandwaitsforthenextupcoming spacingbetweenthesuccessiveplatoons (spacingbetweentherearbumper ofthelast vehicleina platoonandthefront bumperoftheleaderofthenextplatoon)inthecrossingdirectiontopassthroughtheintersection.Inthiscase,theinter- platoonheadwayalsoneedstobeadjustedforthesafepassageofthestoppedplatoonthroughtheintersection.Whenthe synchronizationprocessfailsrepeatedly,thecapacitysigniﬁcantly drops.Hence,we maximizethenetworkcapacitybyal- lowingamarginalgap(extratimegap)ofanoptimallengthbetweenthearrivalanddepartureoftheconsecutiveplatoons incrossingdirections.

Inthisresearch,wedevelopastochasticanalyticalmodelforoptimaltraﬃccontrolatsmartintersections.Weformulate synchronizationfailure probability asafunction ofmarginal gaplengthfora generalstatisticaldistribution oftheopera- tionalerror.Wethenderivetheintersectioncapacitybyaccountingfortheprobabilisticimpactsofsynchronizationfailure.

Our analytical resultsshow that the intersectioncapacity can be maximizedby optimizingthe size of platoons andthe length ofthe marginalgap.We analytically solvethe optimalcontrol problemfortheplatoon sizeandthe marginal gap lengthandderiveaclosed-formsolutionforageneral(bell-shaped)statisticaldistributionoftheoperationalerror.Toshow the generalityofthe analyticalderivations, we alsoreformulate theclosed-formsolution foranormaldistribution ofthe operationerror.Theperformanceofthenetworkwithsmartintersectionsisalsopresentedbytheautomatednetworkfun- damentaldiagram(ANFD).ThestochasticANFDrevealsthattheperformanceofthenetworkinthe“highlyhypercongested”

state canbe improvedby alteringthe patternof thesynchronizedoperation fromapproach-and-pass tostop-and-passin one of the directions. In the end, we verifythe analytical results using a double-ring simulation model. The simulation resultsshowthatoptimizingthecontrolvariablesincreasesthecapacityby138%whentheerrorstandarddeviationis0.1s.

Theremainderofthepaperisorganizedasfollows:Section2developsastochastictraﬃcmodelforthesmartintersec- tions.Section 3formulatestheoptimalcontrolproblem. Section4presentstheanalyticalANFD.In Section5,weevaluate theanalyticalmodelwiththeresultsofasimulationmodel.Lastly,conclusionsofthepaperaresummarizedinSection6. 2. Cooperativetraﬃccontrolinautomatednetworks

Cooperativetraﬃccontrolcan substantiallyimprovetheperformanceofurban networks.Onthelinklevel,itimproves capacitybysafelyincreasingthespeedanddecreasingtheheadwaybetweentheCAVsmovinginplatoons.Atintersections,

1In automated highways, the capacity is an increasing function of the platoon size ( Varaiya, 1993 ; Michael et al., 1998 ; Fernandes and Nunes, 2012 , 2015 ; Chen et al., 2017 ).

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Fig. 1. Cooperative traﬃc control in automated networks.

thedelaycanbe entirelyeliminated bycoordinatingarrivalanddepartureofplatoonsincrossingdirections,asillustrated inFig.1.

Intheproposedcooperativecontrolstrategy,wemakeuseofthespacingbetweensuccessiveplatoonsineachdirection for the consecutive pass of the platoons in the crossing direction. For this to happen, an approaching platoon should be synchronized to arrive at the intersection when the departing platoon in the crossing direction has already cleared the intersection. Meanwhile, the time gap to the arrival of next approaching platoon should also be equal to or larger than the minimum time required for the safe maneuveringof the platoon through the intersection. The platoon arrival timeto theintersectionandthe headwaybetweentheplatoons,however,are bothsubjecttoan operationalerror,which cancauseoccasionalinterruptionsinthesynchronized(approach-and-pass)operation oftheintersection.Inthiscase, the platoonthat arrivesattheintersectionearly (when thedeparting platoonhasnot clearedtheintersection) orlate(when thetimegaptothearrivalofnextapproaching platoonisinsuﬃcientforthesafepassoftheentirelengthoftheplatoon through the intersection) has to stop at the intersection. The synchronization failure requires some adjustments in the operation of the intersection for resynchronizing the platoons. In the adjustment operation (stop-and-pass),a late/early approaching platoonhas to stop atthe intersectionandwait forthe next upcoming spacingin the crossing directionto pass through. Forthis to happen, the inter-platoonheadway in the crossing direction alsoneeds to be adjusted for the safepassageof theentire lengthof theplatoon (thatstartsmoving from restwitha constant accelerationrate) through the intersection. The resynchronization adjustments, however, adversely affect the performance of the intersection if synchronizationfailsrepeatedly.Toreducethe failureprobability,weallow amarginalgapbetweenarrival anddeparture oftheconsecutiveplatoonsincrossingdirectionstomakeupfortheoperationalerror.Inthefollowingsection,wedevelop ananalytical modelfortheintersectioncapacityby accountingfortheprobabilistic impacts ofsynchronizationfailure on theperformance oftheintersection.Forsimplicityofformulation,weconsider thecaseofone-wayintersectionswithno turningtraﬃc;however,themodelcanbefurthergeneralizedbysynchronizingtheplatoonsalongthemultilaneroadsand accountingforthepassingtimeofvariousturningmovementsandthecorrespondingprobabilitiesintheformulations.

2.1. Synchronizedoperation:approach-and-passpattern

Platoonsynchronization can entirely eliminate the queueand delay by using the spacing between the CAV platoons forthe consecutive passing of theseplatoons by each other withno interruption. When synchronizationis accurate, an approachingplatoonindirectioni∈ {X,Y}arrivesattheintersectionwhenthedepartingplatoonindirectionj∈ {X,Y},i

=j,hasalreadyclearedtheintersection,asillustratedinFig.2a.

We calculatethe platoon passing time through the intersectionin directioni,

τ

_i^S^,^as^the ^required ^time ^for^the ^entire

lengthofaplatoonofn_ivehiclestocleartheintersectionofwidthw_iwithaconstantspeedofv_i:

τ

i^S=n_il_v+

(

ⁿi−1

) ( δ

^o+

δ v

i

)

+w_i

v

i

. (1)

Here,the platoonlength is thesummation ofthe averagevehicle length,lv, andtheintra-platoon spacing,sp.The intra- platoonspacingisthebumper-to-bumperspacingbetweenthevehicles inaplatoon,expressedasalinearfunctionofthe Pleasecitethisarticleas:M.Amirgholy,M.NourinejadandH.O.Gao,Optimaltraﬃccontrolatsmartintersections:Auto-

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Fig. 2. Cooperative traﬃc control at the smart intersections.

speed witha constant buffer distance,i.e., intra-platoonjam spacing,

δ

ô ^(unitôf ^length),ândâ ^fixed incremental rate,

δ

(unitoftime):s_p(v_i)=

δ

o+

δ

^vi.Theclearancegap²isalsocalculatedasw_i/v_i.

Toimprovethe systemrobustness(synchronizationsuccessprobability),we allow amarginal gapbetweenarrival and departure of theconsecutiveplatoons in crossing directions.Bydoing so, we reduce the probability ofearly/latearrivals atthe intersection(synchronization failure probability),as we explain inSection 2.3.In a synchronizedcycle, the arrival anddeparture of theconsecutive platoons are successfullycoordinated andplatoons can approachand passthrough the intersection withno interruption. The length of the synchronized cycle(time span requiredfor the passof one platoon ineachofthedirections),T^S,isequaltothesummationoftheplatoonpassingtimesthroughtheintersectionincrossing directions,

τ

X^S and

τ

Y^S,plusthemarginalgaplength,G,asshowninFig.3:T^S=

τ

X^S+

τ

Y^S+G.

Thelanecapacity(numberofvehiclesthatpassthroughtheintersectionper laneperunitoftime)indirectioni,q^S_i,is thencalculatedastheplatoonsizeindirectioni,n_i,dividedbythelengthofthesynchronizedcycle,T^S:

q^S_i=

v

X

v

Y

(

ⁿ^X

v

Y+nY

v

X

)

^lv+

( (

ⁿ^Y⁻¹

) v

X+

(

ⁿ^X⁻¹

) v

Y

) δ

^o⁺

(

ⁿ^X⁺ⁿ^Y⁻²

) δ v

X

v

Y+wX

v

Y+wY

v

X+G

v

X

v

Y

n_i, (2)

Here,indicesiandjaresubstitutedwithXandYintheﬁrsttermofcapacityfunction(2)sincetheyareinterchangeable.We furthersimplify thecapacityfunction forthe caseofsymmetricintersections (n=n_X=n_Y,v=v_X=v_Y,w=w_X=w_Y) where theplatoonpassingtimesthroughtheintersectionbecomeequalincrossingdirections,

τ

^S=

τ

X^S=

τ

Y^S:

q^S= n

v

2

(

ⁿ^lv+

(

ⁿ−1

) ( δ

^o+

δ v )

+w

)

+G

v

^. ⁽³⁾

The intersection capacity, in a synchronizedcycle, is always a strictly increasing function of the speed and a strictly decreasingfunctionofthemarginalgaplength,astheﬁrst-order(partial)derivativesofcapacityfunction(3) withrespect tovandGalwayshavepositiveandnegativevalues,respectively:

∂

^q^S

∂ v

⁼

2n

(

ⁿ^lv+

(

ⁿ⁻¹

) δ

^o⁺^w

)

(

²

(

ⁿ^lv+

(

ⁿ⁻¹

) ( δ

^o⁺

δ v )

+w

)

⁺^G

v )

² ^>⁰^, ⁽⁴⁾

∂

^q^S

∂

^G ⁼ ⁻ⁿ

v

²

(

²

(

ⁿ^lv+

(

ⁿ⁻¹

) ( δ

^o⁺

δ v )

+w

)

⁺^G

v )

² ^<⁰^, ⁽⁵⁾

where 2n(nl_v+(n−1)

δ

o+w) > 0 and−nv² < 0 for

∀

ⁿ ≥ 1 and

∀

^v^, ^lv > 0. The behavior of capacityfunction (3) with respecttotheplatoonsizeisalsomonotonic;however,itsmonotonicitydependsonthemarginalgaplength.Asgraphically illustratedinFig.4a,theintersectioncapacitycanbeastrictlyincreasing,decreasing,orevenaconstantfunctionofnunder thefollowingconditions:

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎩

∂

^q^S

∂

ⁿ ^>⁰^, ^G^>^G^S^∗

∂

^q^S

∂

ⁿ ⁼⁰^, ^G⁼^G^∗^S

∂

^q^S

∂

ⁿ ^<⁰^, ^G^<^G^∗^S

, (6)

2Clearance gap refers to the extra time it takes a platoon to clear the intersection after the last vehicle of the platoon entered the intersection. By definition, the instantaneous throughput of the intersection is zero during the time a platoon is “clearing” the interstation, as graphically shown in the flow profiles of Fig. 3 .

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Fig. 3. Automated traffic flow profiles in crossing directions.

Fig. 4. The capacity function behavior with respect to the platoon size.

where

∂

^q^S

∂

ⁿ ⁼

(

G

v

−2

v δ

⁻²

δ

^o⁺²^w

) v

(

²

(

ⁿ^lv+

(

ⁿ⁻¹

) ( δ

^o⁺

δ v )

+w

)

⁺^G

v )

²^, ⁽⁷⁾

andG^∗_S isthesolutionoftheﬁrst-orderconditionequation,

∂

^q^S^/

∂

ⁿ=0:

G^∗_S=2

( v δ

+

δ

o−w

)

v

^. ⁽⁸⁾

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Note that the cyclelength is an increasing function of theplatoon size since the inter-platoon headway required for the safepass of theconsecutive platoons through each other atthe intersection alsoincreases withthe platoon size.In a synchronizedcycle, theeffectof increasingthe platoonsize onthe capacityispositive whenG>G^∗_S inwhichcasethe cyclelengthgrowsslowerthan theplatoonsize(

∂

^q^S^/

∂

ⁿ>0),negativewhenG<G^∗_S inwhichcasethecyclelengthgrows fasterthantheplatoon size(

∂

^q^S^/

∂

ⁿ<0), andneutralwhenG=G^∗_S inwhich casethecyclelengthandplatoonsize grow proportionally (

∂

^q^S^/

∂

ⁿ=0).Althoughtheeffectofincreasing theplatoonsizeontheintersectioncapacitycanbepositive, negative,orneutralinasynchronizedcycle,thelimitvalue isﬁxedandindependentoftheplatoonsizeandthemarginal gaplengthinallthecasesastheplatoonsizegoestoinﬁnity:

˜ q^S= lim

n→∞q^S=

v

2

(

^lv+

v δ

⁺

δ

^o

)

^. ⁽⁹⁾

2.2. Adjustmentoperation:stop-and-passpattern

The operationalerrorofthe control systemincoordinatingthe platoons causesa stochasticityintheir platoon arrival times atthe intersection.This error, iflarge enough, can causean interruption inthe synchronized (approach-and-pass) operationoftheintersection(synchronizationfailure)whenaplatoonarrivesattheintersectionatthepointintimethat:

(i) thedeparting platoon inthecrossing directionhasnot clearedtheintersection yet(earlyarrival),or(ii)the time gap to thearrival ofthe nextapproaching platoon inthecrossing directionisinsufficient forthesafepassage oftheplatoon throughtheintersection(latearrival).Notethattheconceptoftheearliness/latenessisbydefinitionrelativeinthecrossing directions;anearlyarrivalinonedirectioncanbealsoseenasalatedepartureinthecrossingdirectionandviceversa.So, we choosea primary direction,X,asthe reference,anddefine earliness/lateness inthesecondary direction,Y. Forresyn- chronization, the approaching platoon indirection Y stopsat the intersectionupon an early/late arrival, asillustrated in Fig.2b.ThestoppedplatoonthenpassesthroughthenextupcomingspacingbetweenthesuccessiveplatoonsindirectionX withthemaximumallowableaccelerationrate,³a_v.Forthistohappen,theinter-platoonheadwayindirectionXalsoneeds tobeadjustedforthesafepassageoftheentirelengthofaplatoonofn_YvehiclesindirectionYthroughtheintersectionof widthw_Y,asdescribedbythefollowingequationofmotionwithconstantaccelerationfromkinematics:

nYl_v+

(

ⁿ^Y⁻¹

) δ

^o⁺

δ v

^¯p

τ

Y^A

+wY =1 2a_v

τ

Y^A

2

. (10)

NotethatthelengthofthepassingplatoonindirectionYcontinuouslyincreasesovertimeastheintra-platoonspacing increaseswiththeriseinspeedduringtheaccelerationmaneuverthroughtheintersection.Here,theintra-platoonspacing is a linear function of the instantaneous speed, and the speed also increases linearly over time. Therefore, the platoon passing time through the intersectionin secondary directionY,

τ

Y^A, can be equivalentlyderived for the average passing speedoftheplatoon,

v

¯p(

τ

Y^A)=a_v

τ

Y^A/2,bysolvingmotionEq.(10):

τ

Y^A=a_v

δ (

ⁿY−1

)

+c₁

(

ⁿY

)

2a_v , (11)

with

c1

(

ⁿ^Y

)

⁼

(

^av

δ (

ⁿ^Y⁻¹

) )

²⁺⁸^av

(

ⁿ^Y^lv+

δ

^o

(

ⁿ^Y⁻¹

)

⁺^w^Y

)

^, ⁽¹²⁾

where thetermunderthe square rootis positivefor

∀

ⁿY ≥1. Theplatoon passing time through theintersectionin pri- marydirectionX,however,remains unchangedfromthesynchronizedcycle,

τ

X^A=

τ

X^S,andcan bederived using(1). Inan adjustmentcycle,werecalculatethepassingtimethroughtheintersectioninsecondarydirectionYforthecasewhenapla- toonhasstoppedattheintersectionuponan early/latearrival.Thelength oftheadjustmentcycle(timespanrequiredfor thepassageofoneplatoonineach ofthedirections), T^A,isequaltothesummationoftheplatoonpassingtimesthrough the intersection in crossing directions,

τ

X^S and

τ

Y^A, plus the marginal gap length, G,as shownin Fig. 3:T^A=

τ

X^S+

τ

YÂ+G. The adjustedlanecapacityineach directioni ∈ {X,Y}, qÂ_i,isthen derivedasthesize oftheplatoon(passesthroughthe intersectionpercycle)indirectioni,n_i,dividedbythelengthoftheadjustmentcycle,TÂ:

q^A_i = 2

v

Xa_v

v

Xc1

(

ⁿ^Y

)

⁺^c²

(

ⁿ^X^,ⁿ^Y

)

⁺²

v

Xa_vGni, (13) where

c2

(

ⁿ^X^,ⁿ^Y

)

⁼^av

(

²ⁿ^X^lv+2

(

ⁿ^X⁻¹

) δ

^o⁺

(

²ⁿ^X⁺ⁿ^Y⁻³

) δ v

X+2wX

)

^. ⁽¹⁴⁾

Inasymmetricintersection,capacityfunction(13)issimpliﬁedas:

q^A= 2n

v

a_v

v

C₁

(

ⁿ

)

⁺^C2

(

ⁿ

)

⁺²

v

a_vG, (15)

3Maximum acceleration rate that is tolerable and safe for the passengers (see remarks of Li et al. (2014) and Le Vine et al. (2015) ).

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whereC₁(n)andC₂(n)are thesymmetricversionsofc₁(n_Y) andc₂(n_X,n_Y)inwhichn_i,v_i,andw_iarereplacedbyn,v,and w,respectively.Notethattheadjustmentoperationcausesadeclineinthecapacityoftheintersection(q^A ≤q^S)sincethe cyclelengthincreasesintheadjustmentcyclessince

τ

^A ≥

τ

^S^.⁴

Similartoasynchronizedcycle,theintersectioncapacityinanadjustmentcycleisalsoastrictlyincreasinganddecreas- ingfunctionofthespeedandthemarginalgaplength,respectively:

∂

^q^A

∂ v

⁼

4a²_vn

(

ⁿ^lv+

δ

^o

(

ⁿ⁻¹

)

⁺^w

)

( v

C1

(

ⁿ

)

+C2

(

ⁿ

)

+2

v

a_vG

)

² ^>⁰^, ⁽¹⁶⁾

∂

^q^A

∂

^G ⁼ ⁻⁴^a

2vn

v

²

( v

C1

(

ⁿ

)

⁺^C²

(

ⁿ

)

⁺²

v

a_vG

)

² ^<⁰^, ⁽¹⁷⁾

since4a²_vn(ⁿ^lv+

δ

^o(ⁿ−1)+w)>0and−4a²_vn

v

²<0for

∀

ⁿ≥1and

∀

^v^,^l^v^,^w>0.Thebehaviorofthecapacityfunctionwith respecttotheplatoonsize,however,alsodependsontheparametersandvariablesotherthantheplatoonsize.Asdepicted inFig.4b,capacitymodel(15)canbeastrictlyincreasing,decreasing,orevenanon-monotonicfunctionoftheplatoonsize, n,underthefollowingconditions:

⎧ ⎪

⎨

⎪ ⎩

∂

^q^A

∂

ⁿ ^>⁰^, ^G^>^G^∗^A

(

ⁿ

)

∂

^q^A

∂

ⁿ ^<⁰^, ^G^<^G^∗^A

(

ⁿ

)

, (18)

where

∂

^q^A

∂

ⁿ ⁼

2a_v

v

² C1

(

ⁿ

)

⁻ⁿ^a^v

(

⁴⁽^δ^o⁺^l^v⁾⁺^a²^v^δ²⁽ⁿ⁻¹⁾

)

C1(n)

−

(

²

δ

^o⁺³

δ v

−2

(

^G

v

+w

) )

( v

C₁

(

ⁿ

)

⁺^C2

(

ⁿ

)

⁺²

v

a_vG

)

² ^, ⁽¹⁹⁾

andG^∗_A(ⁿ)îs^the^solutionôf^the^first-order^conditionêquation,

∂

^q^A^/

∂

ⁿ=0:

G^∗_A

(

ⁿ

)

⁼^a^v

δ

²

(

ⁿ⁻¹

)

⁻⁴

(

ⁿ^lv+

δ

^o

(

ⁿ⁻²

)

⁺²^w

)

2C1

(

ⁿ

)

⁺

δ

^o

v

⁺

3

δ

2 −2w. (20)

Theoptimalplatoonsizeinthenon-monotoniccasecanbederivedas:

n^∗=G^∗−_A ¹

(

^G

)

^, ⁽²¹⁾

whereG^∗−_A ¹(·)^denotes^the^inverse^function^of^G^∗_A(·)^.

Inanadjustmentcycle,increasingtheplatoonsizecausestheinstantaneousspeedoftheplatoontofurtherincreasedur- ingtheacceleratingmaneuverthroughtheintersectionastheplatoongetslonger.Therefore,theoveralleffectofincreasing theplatoonsize ontheintersectioncapacity,inan adjustmentcycle, ispositive whenG>G^∗_A(ⁿ)^,ⁱⁿ^which^case^the^cycle lengthgrowsslowerthantheplatoonsize(

∂

^q^A^/

∂

ⁿ>0),andnegativewhenG<G^∗_A(ⁿ)^,ⁱⁿ^which^case^the^cycle^length^grows fasterthantheplatoonsize(

∂

^q^A^/

∂

ⁿ<0).Inthenon-monotoniccase,however,theintersectioncapacitygetsmaximizedat stationarypointn^∗wherethecyclelengthvariesproportionallywiththeplatoonsizesothatthemarginaleffectofchanging theplatoonsizeontheintersectioncapacitybecomeszero(

∂

^q^A^/

∂

ⁿ=0).Althoughtheintersectioncapacitycanbeastrictly increasing,decreasing,orevenanon-monotonicfunctionoftheplatoonsizeinanadjustmentcycle,thelimitvalueisﬁxed andindependentoftheplatoonsizeandthemarginalgaplengthinallthesecasesastheplatoonsizegoestoinﬁnity:

˜ q^A= lim

n→∞q^A=

v

l_v+2

v δ

⁺

δ

^o^. ⁽²²⁾

2.3.Synchronizedoperationwithaprobabilityoffailure

Interruptioninthesynchronizedoperationoftheintersection,ifitoccursrepeatedly,canhavenegativeimpactsonthe overallperformanceofthesystem.Atthesmartintersections,therobustness(successprobability)ofthecontrolsystem(in coordinatingtheapproachoftheplatoonsinthecrossingdirections)canbesigniﬁcantlyimprovedbyallowinga marginal gapbetweenarrivalanddepartureoftheconsecutiveplatoonsincrossingdirectionstoreducethefailureprobabilityatthe costofincreasingthecyclelength.Inthiscase,anapproachingplatoonindirectionYcansimultaneouslypassthroughthe intersectionwithnointerruptioninasynchronizedcycleonlyifitarrivesattheintersectionwithinthemarginalgap,after thedeparting platoon indirection Xhas clearedthe intersection.However, an early/late platoon(one that arrivesat the intersectionbefore/afterthemarginalgap)hastostopattheintersectionforresynchronizationinanadjustmentcycle.

4Theoretically, τÂ^≥τ^S ^{for 1 ≤}^{n ≤}ⁿUwhere n U= (2 v ²+ a v( δo+ δ^v⁻^w^))/(âv( δo+ δ^v⁺^lv)) is the solution of equation τÂ⁼τ^S^{for n}^∈^R>0. In practice, n U≥20 for a reasonable range of values for the model variables and parameters, which is outranged by the platoon size safety/stability criteria (See the remarks of Biswas et al. (2006) , Robinson et al. (2010) , and Amoozadeh et al. (2015) ), even in absence of an upper limit for the instantaneous speed of the platoons during the accelerating maneuver through the intersection.

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Fig. 5. General probability distribution of the operational error.

Toaccountforthestochasticityassociatedwiththesynchronizationprocess,weconsiderastatisticaldistribution,with a general probability density function (PDF), f, for the random operational errorin the platoon arrival time (withmean zeroandstandarddeviation

σ

ε)andthemarginalgaplength(withmeanG¯andstandarddeviation

σ

G):

ε

∼ f(^t;0,

σ

_ε²)^and G∼ f(^t;G¯,

σ

G²)^.^Theoperational errorgenerallyhasasymmetric⁵ (bell-shaped)statisticaldistributionandthemean error in the coordinatedarrival time ofthe platoons at the intersectionis zero.Hence, the synchronizationfailure probability canbe equallyminimizedfortheearly andlate arrivalsbycoordinatingtheplatoons toarrivetotheintersectionrightat the midpointofthe marginal gap,withmeanlength ofG¯,asillustrated inFig. 3a,b.In thiscase, themaximumsuccess probabilityofthesynchronizedprocess,Pr(−G/2≤

ε

≤G/2),canbeformulatedintermsofG¯,asshowninFig.5:

Ps

_¯

G

= G¯/2

−G¯/2

z

( ω )

^d

ω

=Z

_¯

G 2

−Z

−G¯ 2

. (23)

Here,Zdenotesthecumulativedistributionofz∼f(^t;0,

σ

ε²+

σ

G²/4)^.^Theprobabilityofthesynchronizationfailure(followed bytheadjustmentoperation)isthecomplementaryprobabilityofsuccess,1−Pr(−G/2≤

ε

≤G/2):

P_A

_¯

G

=1−Ps

_¯

G

=1+Z

−G¯ 2

−Z

_¯

G 2

. (24)

Theexpectedlengthofthecyclesiscalculatedasthesummationofthelengthsofthesynchronizedandadjustmentcycles, weightedby thesuccess andfailureprobabilities: E[T]=T^SPs(^G^¯)+T^AP_A(^G^¯)^.^The ^expected^capacityⁱⁿ^directionⁱ ∈ {X,Y}

isthen formulated asthesize oftheplatoon (passesthroughthe intersectionper cycle)indirectioni,n_i,divided bythe expectedlengthofthecycles,E[T]:

qi= ni

τ

X^SPs

_¯

G

+

τ

X^APA

_¯

G

+

τ

Y^S+G¯ . (25)

Here,

τ

i^S and

τ

X^A can be plugged in from (1) and (11), respectively. In a symmetric intersection, capacity model (23) is simpliﬁedas:

q= 2n

v

a_v

(

^C²

(

ⁿ

)

⁻

v

C1

(

ⁿ

)

⁻²^av

(

ⁿ⁻¹

) δ v )

Ps

_¯

G

+

v

C1

(

ⁿ

)

⁺^C²

(

ⁿ

)

⁺²

v

a_vG¯ . (26) Asisthecaseinbothsynchronizedandadjustmentcycles,theexpectedcapacityisalsoastictlyincreasingfunctionof speed,

∂

^q^/

∂

^v >0for

∀

ⁿ≥1and

∀

^v^,^l^v^,^w>0.Thebehaviorofthecapacityfunctionwithrespecttotheplatoonsizealso followsa patternsimilar tothat oftheadjustmentcycle,while thelimitvalue canbe recalculatedforthestochastic case as:

˜ q= lim

n→∞q=

v

2

v δ

⁺

( δ

^o⁺^lv

)

1+PS

_¯

G

. (27)

Theintersectioncapacity,however,isnotamonotonicfunctionofthemarginalgaplengthanymorewhenthesynchroniza- tionprocessissubjecttoprobabilisticfailure.Inthenextsection,wemaximizetheexpectedcapacityoftheintersectionby optimizingthetwoprimarycontrolvariablesofthesystem:platoonsizeandmarginalgaplength.

5A probability distribution is symmetric if and only if ∃x o|^f(x o−)= f(x o+ ), ∀∈ R .

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3. Optimalplatooncontrolproblem

Besidesthe physicalandtechnological characteristics ofthevehicles, infrastructure,andthe control system, theinter- sectioncapacitylargelydependsonadjustmentsofthecontrolsystemsettings.Toachievethehighestperformanceofthe system,wemaximize theexpectedcapacityoftheintersectionbysolvingthefollowingoptimizationproblemforthepla- toonsize,n,andthemarginalgapmeanlength,G¯:

maxn,G¯

q= n

τ

^S

1+Ps

_¯

G

+

τ

^A^P^A

_¯

G

+G¯, (28)

where P_A(^G^¯)=1−P_S(^G^¯)^, ^and

τ

^S ^and

τ

^A ^can ^be ^plugged ⁱⁿ ^from ^relations ⁽¹⁾ ^and ⁽¹¹^), respectively, by replacing n_i, v_i, andw_iwith n,v, andwfora symmetric intersection.The expectedcapacityis directlyproportional to theplatoon size, andinversely proportional to the expected cyclelength, which is an increasing function of the platoon size. Therefore, theoverall effectof increasing theplatoon size onthe expectedcapacityis positivefor smallerplatoons when thecycle lengthstillgrowsslowerthantheplatoonsize(

∂

^q^/

∂

ⁿ>0),andbecomesnegativeforlargerplatoonswhenthecyclelength growsfasterthantheplatoonsize(

∂

^q^/

∂

ⁿ<0).So,theintersectioncapacityismaximizedatstationarypointn^∗wherethe cyclelength variesproportionallywiththeplatoonsize,wherethemarginal effectofachangeintheplatoonsize onthe intersectioncapacitybecomeszero.Therefore,thesizeoftheplatoonscanbeoptimizedbysolvingtheﬁrst-ordercondition equation(

∂

^q^/

∂

ⁿ=0)fornas:

n^∗= a_v

δ

²−4

( δ

^o+l_v

)

²

1−PS

_¯

G

2

v

²

( δ

^o−w

)

+C3

_¯

G

+

2C4

_¯

G

a_v

δ

²⁻²

( δ

^o⁺^lv

)

²

1−PS

_¯

G

2

v

²

( δ

^o⁺^lv

)

⁺^av

δ

²^C³

_¯

G

^, ⁽²⁹⁾

where C3

_¯

G

=

a_v

( δ

^o⁺

δ v

−w

)

1+PS

_¯

G

−G¯

v

( δ

^o⁻^w

)

1+PS

_¯

G

−G¯

v

+2

δ v

, (30)

and C4

_¯

G

=a_v

2

( δ

^o⁺^lv

)

²⁻^av

δ

²

(

^lv+w

)

2

δ

^o

1+PS

_¯

G

+

δ v

3+PS

_¯

G

−2

_¯

G

v

+w

1+PS

_¯

G

2

× 2

1−PS

_¯

G

2

v

²

( δ

o−w

)

⁺^C³

_¯

G

. (31)

Amarginal gapbetweenarrival anddeparture oftheplatoons in crossingdirections cansignificantly enhance therobustness(successprobability)ofthesynchronizationprocessatthecostofincreasingthecyclelength.Hence,theexpected capacityoftheintersectionismaximizedbyallowingamarginalgapofanoptimallengthbetweenarrivalanddepartureof theconsecutiveplatoonstomakeupfortheassociatedoperationalerror.Inthiscase,theimprovementresultingfromthe enhancementofrobustnessmaximally outweighsthe declineinthethroughputof theintersection.Notethatthe success probability,P_S(^G^¯)^,îs ^generallyân încreasing ^concave ^function ôf^G^¯≥0,regardless of type ofthe operational errordistri- bution.Therefore,

τ

=

τ

^S(¹+P_s(^G^¯))+

τ

Â(¹−P_s(^G^¯)) ^becomesâ ^decreasing^convex ^functionôf ^G^¯ ^since

τ

^A ≥

τ

^S^. ^Following

fromabove, E[T]=

τ

+G¯ becomes a non-monotonic convexfunction ofG¯≥0, asillustrated in Fig. 6. The marginal gap (mean)lengthisthenoptimizedbysolvingthefollowingequationresultingfromtheﬁrst-orderconditionfortheminimum expectedcyclelength,

∂

^E[^T^]/

∂

^G^¯=0:

∂

^Ps

_¯

G

∂

^G^¯

τ

^A⁻

τ

^S

−1=0, (32)

where

∂

^P^s

_¯

G

∂

^G^¯ ⁼

∂

^Z ^G2^¯

−Z −^G₂^¯

∂

^G^¯ ⁼

1 2

z

_¯

G 2

+z

−G¯ 2

. (33)

Sincetheoperationalerrorgenerallyhasasymmetricprobabilitydistribution,z(^G^¯/2)=z(−G¯/2)^,^relation⁽³³⁾^can^be^further simpliﬁedas:

∂

^P^s

_¯

G

∂

^G^¯ ⁼^z

_¯

G 2

. (34)

Bysolving theequation resulting fromplugging

∂

^PS(^G^¯)/

∂

^G^¯ ^from^relation⁽³⁴⁾ înto^first-order ^condition⁽³²^), ^theôptimal

marginalgaplength,G¯^∗,isgenerallyderived foranygivenparametric/nonparametricprobability distributionoftheopera- tionalerroras:

G¯^∗=2z⁻¹ 1

τ

^A⁻

τ

^S

, (35)

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Fig. 6. Variations of the components of the expected cycle length with the marginal gap length ( G ¯).

wherez⁻¹( •)denotestheinversefunctionofz∼f(^t;0,

σ

_ε²+

σ

G²/4)^.^Note^that ^aclosed-formsolutioncanbe simplygen- eratedforanyinvertiblePDF oftheoperationalerrorbysubstitutingz⁻¹(•)inEq.(35).Todemonstratethegenerality of theanalytical derivations,we alsoreformulatethe closed-formsolutionfora normaldistributionoftheoperationalerror, z∼N(^t;0,

σ

_ε²+

σ

G²/4)^,^whichîsâ^well-fittedstatisticaldistributionfortheoperationalerror(Zitoetal.,1995;Dulmanetal., 2003;YanandBitmead,2005):

G¯^∗=

⁸

σ

ε²+

σ

G²/4

ln

⎛

⎝ τ

^A⁻

τ

^S

2

π

σ

ε²+

σ

G²/4

⎞

⎠

. (36)

Optimizingtheoperationofthesmartintersectionscansigniﬁcantlyimprovetheperformanceofthesystematthenetwork level,asweexplaininSection4.

4. Automatednetworkfundamentaldiagram

Inthissection,wepresenttheautomatednetwork fundamentaldiagram(ANFD) asan analyticaltoolformodelingthe dynamicsoftrafficinnetworkswithsmartintersections.Inconventional networks,themacroscopicfundamentaldiagram (MFD)approximates the interrelationshipbetweentraffic variables inlarge urban regions. Observedtraffic data fromthe city ofYokohama (GeroliminisandDaganzo, 2008) andthe resultsofthe traffic simulationofthe downtownnetwork of SanFrancisco(Geroliminis andDaganzo,2007)show thatwhencongestionhasauniformdistributionacrossthenetwork, flow (veh/s.lane) increaseswithvehiculardensity(veh/m.lane) fromzeroto itsmaximum value intheuncongested state ofthenetwork.Theflow, however,sharplydecreaseswithafurtherriseindensityinthehypercongestedstateofthenet- workuntilcompletegridlockoccurs(Daganzo,2007;DaganzoandGeroliminis,2008). Inautomated networks,cooperative trafficcontrolmakes itpossibletokeepthevehiculardensityhomogenousacrossthenetwork.Thesynchronizationfailure probability alsoremains identicalforthe intersectionscontrolled by thesame technology.The proposed analyticalmodel forthe intersectioncapacitycan be then extendedto presentthe performance ofautomated networkson amacroscopic level.

In networkswith smartintersections, flow increases fromzeroto the network capacitywith a rise indensity inthe uncongestedstate withnodeclineinthe(freeflow)speed,v.Furtherincrease ofthevehiculardensity,however,requires reducing the networkspeed to enableafurther decreaseof (i) therequiredinter-platoon spacingand(ii)the safeintra- platoonspacinginorderto accommodatea largernumberofplatoons inthenetwork.Inthiscase, thenetworkflow de- creases withthe decline ofspeed inthe hypercongested state of the network asthe system movestowards a complete gridlock. The ANFD expresses the relationship betweenthe network flow, Q, and the vehicular density, k, in automated networksasshownbelow:

Q

(

^k

)

⁼

_k

v

, 0≤k≤km

k

v (

k

)

^, ^k^m^<^k^≤^k^j^, ⁽³⁷⁾

wheretheoptimaldensity,km,(foragivenplatoonsizeandmarginalgaplength)iscalculatedbypluggingthenetworklane capacityfromEqs.(3),(15),and(26)intoQminthemacroscopicﬂowequation,km=Qm/v,forapproach-and-pass(absolute Pleasecitethisarticleas:M.Amirgholy,M.NourinejadandH.O.Gao,Optimaltraﬃccontrolatsmartintersections:Auto-

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robustsynchronization),stop-and-pass(absolutefragilesynchronization),andstochastic(synchronizationwithaprobability offailure)operationscenariosasshownbelow:

km=

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎩

n

2

(

ⁿ^lv+

(

ⁿ⁻¹

) ( δ

^o⁺

δ v )

+w

)

⁺^G^¯

v

^approach⁻^and⁻^pass

2na_v

v

C1

(

ⁿ

)

⁺^C²

(

ⁿ

)

⁺²

v

a_vG¯ stop−and−pass 2na_v

(

^C²

(

ⁿ

)

⁻

v

C1

(

ⁿ

)

⁻²^av

(

ⁿ⁻¹

) δ v )

Ps

_¯

G

+

v

C1

(

ⁿ

)

⁺^C²

(

ⁿ

)

⁺²

v

a_vG¯ stochastic

. (38)

Thenetworkspeedinthehypercongestedstate(km<k≤k_j)isthenderivedfordifferentoperationscenariosbyreversing theoptimaldensityfunction,

v

₍k)=k⁻_m¹(^k)^:

v (

k

)

⁼

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

n−2k

(

ⁿ^lv+

(

ⁿ⁻¹

) δ

^o⁺^w

)

k

2

(

ⁿ−1

) δ

+G¯

^approach−and−pass

2na_v−2ka_v

(

ⁿ^lv+

(

ⁿ⁻¹

) δ

^o⁺^w

)

k

3a_v

(

ⁿ⁻¹

) δ

⁺²^avG¯+C1

(

ⁿ

)

^stop−and−pass 2na_v−2ka_v

(

ⁿ^lv+

(

ⁿ⁻¹

) δ

o+w

)

1+Ps

_¯

G

k

a_v

(

ⁿ⁻¹

) δ

3+Ps

_¯

G

+2a_vG¯+C1

(

ⁿ

)

1−Ps

_¯

G

^stochastic

. (39)

Thejamdensityindifferentoperationscenariosisalsoderivedbysolvingthezerospeedequation,v(k)=0:

kj=

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎩

n

2

(

ⁿ^lv+

(

ⁿ⁻¹

) δ

o+w

)

^approach⁻^and⁻^pass n

nl_v+

(

ⁿ⁻¹

) δ

^o⁺^w ^stop⁻^and⁻^pass

n

(

ⁿ^lv+

(

ⁿ⁻¹

) δ

^o⁺^w

)

1+Ps

_¯

G

^stochastic

. (40)

As illustrated inFig. 7a,the ANFD is boundedfromabove and belowby its extremecases, whilethe failure probability, P_A(^G^¯)=Ps(^G^¯)−1,variesbetween0 and1.Thecapacities derived forthe networkunderthe absoluterobustness(P_S(^G^¯)= 1 for

∀

^G^¯≥0) andabsolute fragility (P_S(^G^¯)=0 for

∀

^G^¯≥0) conditions, respectively, determine the upperbound, Q_m^S, and thelowerbound,Q_m^A,oftheoperationalcapacityofthenetwork,Q_m^A<Qm<Q_m^S,whenthesynchronizationprocessatthe intersectionsissubjecttoaprobabilisticfailure(0<P_S(^G^¯)<1).Foragivendistributionoftheoperationalerror,thenetwork capacitycanbe maximizedbyoptimizingthesizeoftheplatoons,n,andthelengthofthemarginalgaps,G¯,asexplained inSection 3.By pluggingn^∗ andG¯^∗ fromEqs. (29)and(35) into(38), themaximumachievable capacityofthe network understochasticity, Q_m^∗, isderived forthe adjustedoptimaldensity, k^∗_m, asshowninthe optimal-hybridANFD ofFig.7b.

In automated networks, the jam density also variesbetweenan upperbound anda lower bound when the operational errorhasastatisticaldistribution.Inthiscase,thejamdensityderivedundertheabsolutefragilitycondition(P_S(^G^¯)=1for

∀

^G^¯≥0), k^A_j,determines theupperboundandthejamdensityderived undertheabsoluterobustnesscondition(P_S(^G^¯)=0 for

∀

^G^¯≥0),k^S_j,determinesthelowerboundoftheoperationaljamdensityinautomatednetworks,k^S_j<k_j<k^A_j.

Remark. The performance of the network in the highlyhypercongested state (k > k_I) can be signiﬁcantly improved by alteringthe synchronized operation patternof the intersections from approach-and-pass to stop-and-passin one of the directions,asshowninFig. 7b.Theacceleratingmaneuveroftheplatoons throughtheintersections inthestop-and-pass operation scenario allows further increase of the density in the highly hypercongested state by reducing the minimum requiredinter-platoonspacingincomparisontotheapproach-and-passoperationscenario.

ThecriticaldensityatinterchangepointI,k_I,isderived bysolvingtheequation resultingfromequating theupperand thelowerboundsoftheANFD’sdeclininglegs:

kI= n

(

^C¹

(

ⁿ

)

⁻^av

(

ⁿ⁻¹

) δ )

2

(

ⁿ^lv+

(

ⁿ⁻¹

) δ

^o⁺^w

)

a_v

(

ⁿ⁻¹

) δ

⁺^G^¯

+C1

(

ⁿ

)

. (41)

Thecriticalspeed,v_I=v(k_I),andﬂow,Q_I=v_Ik