Flow exchanges in multi-reservoir systems with spillbacks

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Transportation Research Part B: Methodological, Elsevier, 2019, 122, pp.327-349.

ContentslistsavailableatScienceDirect

Transportation

Research

Part

B

journalhomepage:www.elsevier.com/locate/trb

Flow

exchanges

in

multi-reservoir

systems

with

spillbacks

Guilhem Mariotte

∗

, Ludovic Leclercq

Univ. Lyon, ENTPE, IFSTTAR, LICIT, Lyon, F-69518, France

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 22 December 2017 Revised 19 February 2019 Accepted 25 February 2019 Keywords:

Macroscopic fundamental diagram Multi-reservoir systems Congestion propagation Trip lengths Network traffic Accumulation-based model Trip-based model

a

b

s

t

r

a

c

t

Large-scaletrafficflowmodelsbasedontheNetworkMacroscopicFundamentalDiagram (MFD)areusuallygroundedonthebathtubanalogyand aconservationequationfor ve-hicleaccumulation insidea givenurban area.Recent studies haveproposed adifferent approachwheretheMFDdefinesthespatialmeanspeedthatissharedbyallvehiclesin aregionwhiletheirtravelingdistanceistrackedindividually.Theformerapproachisalso referredtoas“accumulation-based” whilethelatterisusuallynamed“trip-based”.While extensivestudiesofbothmodelpropertieshavebeencarriedoutforthesinglereservoir case(auniqueregion),themulti-reservoirsettingstillrequiressomeresearcheffortin par-ticulartoclearlyunderstandhowinflowmergeatareservoirentryandoutflowdivergeat exitshouldbe managed.Thesetwocomponents playasignificantroleinthe evolution ofthewholesystem,whenflowsareexchangedbetweenmultiplereservoirs.Oneofthe crucialquestionsis toensure thatcongestionproperly propagates backwardsthrougha successionofreservoirswhenoversaturatedsituationsareobserved.

Inthispaper,weproposeathoroughanalysisofhowtohandlecongestionpropagation intheaccumulation-basedframeworkwithseveraltriplengthsorcategories,e.g.internal andexternaltrips.Thisallowstoderiveacongestionpropagationmodelforthetrip-based approachinamulti-reservoirsetting. Basedontheoreticalconsiderationsandsimulation studies,wedevelopaconsistentframeworktorestricttheinflowandadaptto oversatu-ratedtrafficconditionsinareservoir includingseveraltriplengths.Twoinflowmerging schemesareinvestigated.Thefirstoneisinspiredfromtheexistingliteratureandshares theavailablesupplybasedonthedemandflowratioattheentryboundary.Itiscalled “ex-ogeneous” incontrasttothesecond“endogenous” scheme,whichsharesthesupplywith respecttotheinternalaccumulationratioonthedifferentroutes.Atthereservoirexit,a newoutflowdivergingschemeisalsointroducedtobetterreproducetheeffectofqueuing vehiclesthatarepreventedfromexitingthe reservoirwhencongestionspillsbackfrom neighboringreservoirs.Comparedtothe conventionaloutflowmodel,ournewapproach provestoavoidunrealisticgridlockswhenthereservoirbecomesoversaturated.Both en-tryand exitflowmodelsareinvestigatedindetailsconsideringthe accumulation-based andtrip-based frameworks.Finally,the mostconsistentapproachiscomparedwithtwo otherexistingMFD modelsformultiplereservoirs.Thisdemonstratesthe importanceof properlyhandlingentryandexitflowsatboundaries.

© 2019TheAuthors.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. (http://creativecommons.org/licenses/by-nc-nd/4.0/)

∗ _{Corresponding author.}

E-mail address: guilhem.mariotte@ifsttar.fr (G. Mariotte). https://doi.org/10.1016/j.trb.2019.02.014

0191-2615/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

trip length assignedto all travelers.Some authorshave extended thisapproachto account formultiple triplengths ina reservoir, eitherto develop newapplications likemodeling search-for-parking (Geroliminis, 2009; 2015) andmacroscopic routing (withcontributions inYildirimogluetal., 2015;Ramezanietal., 2015);ortohighlight inaccuraciesin MFD-based modelsduetotheconstanttriplengthhypothesis(YildirimogluandGeroliminis,2014;Leclercqetal.,2015).Morerecently, a “trip-based” formulation ofthe singlereservoir model has gaineda new interest inthe community. Based on an idea ofArnott(2013),thisapproachhasbeenexploitedinDaganzo andLehe(2015) andthenLamotteandGeroliminis(2018),

Mariotteetal.(2017) andLeclercq etal.(2017).The principle isthat all usersin areservoir sharethe samespace-mean speed (given bythe MFD) ata giventime, andexitonce they havecompleted their individually assignedtriplength. As showninathoroughcomparisonwiththeaccumulation-basedmodelbyMariotteetal.(2017),thetrip-basedapproachgives moreaccurateresultsduringtransientphases,especiallyintermsofexperiencedtraveltime.Nevertheless,someauthorslike

HaddadandMirkin(2016)andHaddadandZheng(2017)suggestthattheinaccuraciesoftheaccumulation-basedmodelcan be takenintoaccount directlybyimplementing delayseitherinthecontrol inputsorinthe stateofthedynamicsystem. Aproperinvestigationaboutthepropertiesofthesenewmodelingapproachesisstillmissing, inparticulartoassesstheir differenceswiththetrip-basedmodel.Whilewechoosetofocusontheaccumulation-basedandtrip-basedmodelsinthis paper,integratingthesenewapproachescouldbeinterestingforfurtherresearchdirections.

However, fromthemodeler’s perspective,despiteall theserecentadvances inMFD-basedsimulation,congestion prop-agation ina multi-reservoir framework isnot fully understoodyet. Notably,the questionsof ifandhow boundary flows shouldbelimitedwhenareservoirisoversaturated,andhowtodistributeinflowsandoutflowshavenot beencompletely addressed. Indetails, asthe widemajority ofMFD-basedsimulators are developedforcontrol applications,mostauthors argue,withreason,thatthecontrollerswillnotallowthereservoirtoreachhighlycongestedstates,sothatthe aforemen-tionedconcernsmaybeeclipsedintheirtrafficflowmodels(seee.g. Kouvelasetal.,2017).However,other applicationsof MFD-basedmodelsshouldnotignorethem.Actuallysomeinterestingworksalreadyproposeincompletebutviablesolutions todealwithcongestionpropagation. Hajiahmadietal.(2013) andLentzakisetal.(2016),whosesimulatorisbasedonthe NetworkTransmissionModel(NTM) ofKnoopandHoogendoorn(2014),considerexogenous boundarycapacities between reservoirsandaglobalentrysupplyfunctionperreservoir,similartotheCellTransmissionModel(CTM)ofDaganzo(1994). Theirapproachensuresaperfectprotectionofthereservoirsfromglobalgridlock,neverthelessthisonecanhardlybe ex-tended toheterogeneous triplengthswithin reservoirs. YildirimogluandGeroliminis(2014) certainly developedthe most advancedtool inMFD-basedsimulation,asthey accountfordifferenttriplengthsandmanageflow exchangeswitha Dy-namicTrafficAssignment(DTA)procedureonmacroscopicroutes(aroutebeingasuccessionofreservoirs,sometimescalled aregionalpath).Theirframeworkserved asabasis forrecentandpromisinglarge-scalecontrol schemes(Ramezanietal., 2015;Yildirimogluetal.,2015;SirmatelandGeroliminis,2017;Yildirimogluetal.,2018).However,theyhandleeach bound-arybetweentwoadjacentreservoirsseparatelywithademandpro-ratainflowmerge,anddonotprovideanyfurther infor-mationabouttheconsistencyoftheglobaltrafficdynamicswhensomereservoirsareoversaturated.

The main objectiveof thisstudyis to properlyaddress congestionspillbackin multi-reservoir systems.Our contribu-tion isto identify, analyzeandeventually modify thecritical componentsof MFD-basedmodels tobuild up a consistent simulationframeworkinbothunder-andoversaturation.Inparticular,wewillensurethatflowexchangesatinterfacesare fulfillingthemainassumptionsofMFD-basedmodeling(trafficstatehomogeneity)andtheclassicalprinciplesofkinematic wave theory (flow conservation, supply limitation, etc). For the accumulation-based model, we will refer to the frame-workofYildirimogluandGeroliminis(2014)andGeroliminis(2015).Forthetrip-basedmodel,wewillpursuetheeffortof

Mariotteetal.(2017)whoprovideafirstattempttohandlespillbacksinthisformulation.Wenotablyextendandpropose newinflowandoutflowmodelingapproachesthatprovetoovercomesomeshortcomingswenoticedabouttheconventional modelsfromtheliterature.Thisstudyfocusesonmethodologicalcontributionsandaimstoprovideathoroughdescription ofhow trafficdynamics inreservoirs areinfluenced bythe modeldesign.Inparallel, theauthorsare workingon design-ingspecifictestcasesusingmicroscopicsimulationtovalidatetheproposed approaches.Theresultswillbepresentedina differentpaper.

Thispaperisorganizedasfollows.Section2presentsthecaseofasinglereservoirwithauniquetriplength.Thissection allows to set the background of ourstudy andits assumptions, andalso discusses and proposesan efficientmethod to accountforspillbacksinthetrip-basedapproach.Section3dealswithinflowandoutflowallocationbetweenmultipletrip categoriesinonereservoir,especiallywhencongestionpropagatesthroughthereservoir.Numericalexamplesarethengiven

Fig. 1. (a) Representation of the single reservoir in a multi-reservoir context, (b) typical production-MFD for the reservoir, and (c) representation of flow modeling for the two trip category, transfer trips and internal trips.

inSection4fora singlereservoircrossed bytwotripcategories.Finally,Section5illustratesthedifferencesinsimulation betweenanewapproachweproposedandtwoexistingmodelsfromtheliteratureforamulti-reservoirsystem.

2. Flowtransferinasinglereservoirwithauniquetriplength 2.1. Accumulation-basedmodeling

Inthissection,we focusontheinteractionsbetweenonereservoiranditsneighborsinthecontextofamulti-reservoir representationofacity,asrepresentedinFig.1(a).

The concept of the single reservoir model has been first presented in Daganzo (2007) and Geroliminis and Da-ganzo (2007). It corresponds to a selection of connected links and nodes of an urban network where traffic states are characterizedbyawell-definedproduction-MFDP(n) (in[veh.m/s]),orequivalently,aspeed-MFDV

(

n

)

₌P

(

n

)

_/n(in[m/s]), wheren(in [veh])is thetotalaccumulation (numberof circulatingvehicles inthe reservoir). Theproduction-MFDis de-finedthroughits jamaccumulationnj,criticalaccumulationnc atwhich themaximumproductionorcapacityisreached:

Pc=P

(

nc

)

andfree-flowspeed u,seeFig.1(b).The reservoirentryborder(alsoconceptually called“upstreamboundary”)

gathersallindividual entrynodesofthenetwork;similarlythereservoirexitborder(or“downstreamboundary”)contains alltheexitnodes.Becauseourfocusisonflowexchangesbetweendifferentreservoirsinamulti-reservoircontext,forthe singlereservoirmodelwewilldistinguishbetween“external” and“internal” origin/destinationoftrips.Atripwithan exter-nalorigin(resp.destination)crossestheentry(resp.exit)border,whereas atripwithaninternalorigin(resp.destination) originates(resp.ends)insidethereservoir.AlotofstudiesdealingwithMFD-basedaggregateddynamicsdothedistinction between“internal” trips,i.e.originatingandendinginsidethereservoir,and“transfer” trips,i.e.crossingthereservoirfrom theentryto theexitborder.However theyare oftenhandledby thesamemodelingapproach,sothat aproperdefinition ofinflowandoutflowissomehowmissing.Incontrol-orientedworks,someauthorslikeAboudolasandGeroliminis(2013),

Ampountolasetal.(2017) andKouvelasetal.(2017)split theinflowintothe receivingflow fromadjacentreservoirs (for which the controllers apply), and the “uncontrolled demand” which may be frominside or outside.In this section, we proposeaconsistentmodelingapproachesforasinglereservoirwithtransfertripsandinternaltrips.

2.1.1. Trafficdynamics

In a reservoir, traffic dynamics are essentially described by the conservation of vehicles, first mentioned in

function” (or sometimesthe“receivingcapacity”).Itsexistence andshape arediscussedinthefollowingsubsection.Ifthe demandforinternaltripsremains quitelowcomparedto theMFDcapacity,itisreasonabletoassumethatthesetripsdo not experience any delayin departure time inside the reservoir. Thisis also discussed inthe following. In thiscasethe effectiveinflowqint

in

(

t

)

equalsitsdemand

λ

int(t).Inshort,wehave:



qext in

(

t

)

=min[

λ

ext

(

t

)

; I

(

n

(

t

))

] qint in

(

t

)

=

λ

int

(

t

)

(effectiveinflows) (2)

Notethatinthisframework,theaccumulationofvehicleswaitingforenteringthereservoirincaseofan inflowrestriction is takeninto account witha point-queue model atentry. Hence the upstream demand

λ

ext_{(t) is setto a maximum—for} instance,themaximumofI(n)—everytimethatthisqueueisnotempty.

Atexit,thesystempotentialoutflowisdeterminedbythereservoirinternaldynamics,whicharedescribedeitherbythe accumulation-basedmodelorthetrip-basedmodel.Weinvestigateheretheaccumulation-basedmodelfirst.Initssimplest version,auniquetriplengthL(in[m])isconsideredforallthetravelers.ByapplyingthequeuingformulaofLittle(1961), the total potential outflow G(n) (in [veh/s]), also called “trip completion rate”, is defined as G

(

n

)

=n/L· V

(

n

)

=P

(

n

)

/L

(Daganzo,2007).G(n)isthusalsonamedtheoutflow-MFD(LamotteandGeroliminis,2018).Thisquasi-staticapproachhas severallimitationsasdetailedinMariotteetal.(2017).AccordingtoGeroliminisandDaganzo(2007)andGeroliminis(2009), thepotential outflowofeachtrip categoryiscalculatedwithits ratioofpartialaccumulationoverthetotalaccumulation, whichisalsoanapplicationofLittle’sformula:



Gext

₍

_nint_,next

₎

₌next

n G

(

n

)

= next

n P(n)

L

Gint

(

_nint_,next

)

₌nint

n G

(

n

)

= nint

n P(n)

L

(tripcompletionrate) (3)

Here again,because thetransfertrips mayexperience aflow limitationwhenentering thenext reservoiratdownstream, theeffectiveoutflowqext

out

(

t

)

istheminimumbetweenits outflowdemandGext(nint(t),next(t))andanexogenous restriction

μ

ext_{(t).Theinternaltripscanreachtheirdestinationsaccordingtotheirpotentialoutflowdefinitionwithoutanyadditional} restriction.Thuswehave:



qext

out

(

t

)

=min[

μ

ext

(

t

)

; Gext

(

nint

(

t

)

,next

(

t

))

] qint

out

(

t

)

=Gint

(

nint

(

t

)

,next

(

t

))

(effectiveoutflows) (4)

Whilebeingsatisfyingfortheoutflowmodelofinternaltrips,thecurrentdefinitionoftheoutflowdemandGext_(nint_,next₎ forthetransfertripshassomecriticalshortcomingsthatarediscussedlaterinSection2.1.3.

2.1.2. Theentrysupplyfunction

When the reservoir isoversaturated, the entrysupply functionI(n) dynamically restrictsthe incomingflow to mimic the propagationof congestioninside the reservoir.1 _{Its general shape isthus a decreasing function of thetotal}

accumu-lation n. Its existence wasfirst noticed in Geroliminis and Daganzo (2007) with simulation loadings on the network of downtown SanFransisco, USA.Then, authors like Ramezani etal. (2015) andSirmatel andGeroliminis (2017) used the-oretical piecewise linear decreasing functions to avoid creating gridlocks in their multi-reservoir modeling. These func-tions wereratherimplemented forpractical reasonsbecausetheauthorsprovided fewdetails abouthowtodesignthem.

Kouvelasetal.(2017)mentionedthattheentrysupplyfunctionisnotalwaysrequired:itcanbeignoredforareservoirthat remains undersaturatedorwhich isprotectedby aperimetercontrol system.Inthe generalcase, i.e.withoutsucha pro-tection,areservoircangotogridlockwhenloadedwithaninflowdemandhigherthanitsMFDflowcapacityPc/L,as

high-lightedbyLavaletal.(2018).BecausetheoutflowfollowstheevolutionofG(n)bydefinition,itcannotsustainanequilibrium flowhigherthanPc/L.Then,aneffectivewaytoprotectthereservoirmaybetoadoptashapeofI(n)similartothesupply

functionoftheCTM:I

(

n

)

=Pc/Lifn≤ ncandI

(

n

)

=G

(

n

)

otherwise.ThiswasnotablythechoiceofHajiahmadietal.(2013),

Knoop andHoogendoorn (2014) andLentzakis etal. (2016). Nevertheless,this approach is clearlytoo restrictive inMFD

1 Used along the whole paper, this notion of propagation actually means the evolution of traffic states towards oversaturation, i.e. the increase of accu-

simulationasitpreventsanyinflowsurgefromentering thereservoir. Ifthisweretrue,theinflowwouldbealways auto-regulatedandtherewouldbenoneedforperimetercontrolatall.Actually,Pc/Lcorrespondstothereservoircapacityina

stationarystate.When thiscapacityisdriven by internalbottlenecksin themiddleofthe reservoir,it wouldnotbe sur-prisingto observehigherinflowvaluesduringanetwork loadingbeforeall thesebottlenecksareactivated andmake the congestionspillingbacktotheperimeter.Thechoiceofthecriticalaccumulationnc todistinguishunder-andoversaturated

conditionsinI(n) is alsoa matterof debate.Unlikethe above-mentionedstudies, Geroliminis andDaganzo(2007) found withsimulationresultsthatthecriticalaccumulationofI(n)maybehigherthannc.

Basedon theseapproachesfrom theliterature, we furtherinvestigatethe shape ofthe entrysupplyfunction here. In themostgeneralcase, theinflowwillingtoentera givenreservoirmaybe firstlimitedbyaphysicalconstraint, whichis thesumofthe capacities ofall thelinksthat define thereservoir entryborder.Thisidea ofa fixed capacityassignedto areservoirborderwasalsomentionedinKnoopandHoogendoorn (2014)andRamezanietal.(2015).Letusconsiderthe simplecaseofasingleentryborderfirst,inthissituationthemaximumofI(n)preciselycorrespondstothisphysical con-straint,whichapplies forlow accumulations(undersaturation).Then,abovesome criticalaccumulation,thelimitingfactor becomesthe reservoirinner state whichundergoes saturation or oversaturation: the higherthe accumulation,the lesser thetotalinflow.Howtheinflowlimitationshouldbeclosetothereservoirsaturationisstillunclear,butitisobviousthat thislimitationshouldexactlyfollows theoutflow-MFD G(n) atsome pointto ensurea consistentflow equilibriuminthe system.Itisworthnoticingthat thisnotionofequilibriumalsodescribesthetransientstateofthesystem,becauseofthe underlyingquasi-statichypothesisin theaccumulation-basedmodel.Moreover, theinflowlimitationshould beconsistent withtheoutflowwheninternaltripsareconsideredaswell.Underourassumptionthat alow demandforinternaltripsis unrestricted,theeffectivelimitationthatappliestothetransfertrips isthereforeI

(

n

(

t

))

₋

λ

int

₍

_t

₎

_{.Thisapproachshouldbe} reasonablewhenthedemand

λ

int_{(t) remainslow andevolvesslowlywithtime.Butthisisprobablynotthecaseforhigh} andfast-varyingpeaksforinternaltrips,asthiswouldmeanhighandinstantaneouschangesintheavailableinflowsupply fortransfertrips,justasifthereservoirinternaldynamicswouldinstantaneouslyadapttoademandsurgeofinternaltrips. Then,abetterwaytohandlethesetripsistorestricttheirinflowsimilarlytotransfertrips,byusingeitherthesameentry supplyfunctionoradifferentfunctionforinternaltrips.Amergingschemecould bealsousedtoallocatetheright limita-tiontoeach tripcategory, butthisfallsintotheframeworkofmultipletriplengthsinthesinglereservoirmodel,whichis investigatedlaterinSection3.

Toillustrate the applicationof the entry supply function, we simulate severalnetwork loadings in a single reservoir withthe characteristics: nj = 1000 veh, nc = 400 veh, Pc = 3000 veh.m/s, L= 2.5km and u = 15 m/s. The reservoir

includesinternaltripswithalowconstantdemandof

λ

int_{(t)=0.3veh/s,andtransfertripswithdemandlevelsrangingfrom}

λ

ext_{(t) =0.8to 1.3veh/s.The reservoirtotal flowcapacityequals P}

c/L=1.2veh/s, whichmeans thatatsome pointsthe

totalinflowdemandwillexceedthiscapacity.Fig.2presentsthesimulationresultsinthetotalaccumulation-totalinflow plane (n(t), q_in(t)) forthree different function models ofI(n). Model 1 corresponds to the modeladopted by Knoop and Hoogendoorn(2014)withahigherlimitinundersaturation.Inthisexamplethislimit isarbitrarysetto1.7veh/s,butthis value should comefromthe aggregationofall entering link capacities ina realsituation. Model2 issimilar to model1 exceptthatitscriticalaccumulationncsishigherthannc.Finally,model3doesnotmatchtheoutflow-MFDG(n)atalland

allows higherinflows to enter the reservoir. Fig. 2(a)-(c)consist of a demandsurge while Fig.2(d)-(f) presentthe same simulationswithapeakdemandprofile:thehighdemandleveliskeptforamuchshorterperiodinthiscase.Thissimple exampleallowsustoexhibitobviousdrawbacksofsomemodels.Whilepotentiallyallowinghighinflowsinundersaturation, model1is not satisfyingbecausethe accumulationcannot exceednc whichistoo restrictive comparedto whatwe may

observe inreality, see Fig. 2(a) and(d). Actually, we knowfrom empirical data that a network can reach oversaturated states while beingloaded withboth transfer andinternal trips (Geroliminis and Daganzo, 2008). As shownin Fig. 2(f), model3 caneffectively reproducereliable network loadingandunloading withthe demand peak profiles.However, it is unable to protect the reservoir fromgridlock ifa high level ofdemand above the MFD capacity ismaintained during a longerperiod,see alsoFig.2(c). In thiscase,apart fromsettinga maximuminflowlimit, theentrysupply functiondoes notseemofgreatutility.Model2iscertainlythebestcompromisebetweenmodel1andmodel3,andprovestoreproduce reliabletraffic state evolutionsinall demandsettings asshowninFig.2(b)and(e).Thisillustrates that atsome pointin oversaturationI(n)mustbeconfoundedwithG(n) tostabilizethesystemandavoidafastunrealisticattractiontogridlock inmanysituations.Interestingly,theuseofashapeofI(n)similartomodel2impliesthatthereisatheoreticalmaximum accumulation ncs that cannot be exceeded when loading a network, which is precisely the critical accumulation of I(n).

Furtherempiricalstudiesorcomparisonswithmicrosimulationareneededtovalidatethisproperty,aswellastheshapeof

I(n)ensuringthetransitionbetweenunder-andoversaturatedstates.Suchvalidationsarepartoffurtherresearchandout ofthescopeofourtheoreticalconsiderationsdevelopedinthispaper.

2.1.3. Modelingexitratefortransfertrips

Wenow discussthe description ofvehicles exiting thereservoir andpropose a newoutflowmodel fortransfer trips. Toourbestknowledge,all thestudies fromtheliteratureconsider thatthe systemtotaldemandforoutflow(ortotal ef-fectiveoutflowiftheydonotapplysupplylimitations)alwaysequalsG(n).Inouropinionhowever,webelievethat thisis onlytruetomodelinternalcongestionfortheoutflowofinternaltrips Gint_(nint_,next_{),butthatadistinctionbetweenwhat} we callthe exitdemandfunction Oext_(nint_{, n}ext_{) andG}ext_(nint_,next_{) should bemade fortransfertrips. Notethat O}ext_(nint_,

next_{) isnot theeffectiveoutflowq}ext

Fig. 2. Network loading of the single reservoir with transfer and internal trips for different shapes of the entry supply function. (a) Demand surge for transfer trips with entry supply model 1, (b) model 2 and (c) model 3. (d) Demand peak for transfer trips with entry supply model 1, (e) model 2 and (f) model 3. The legend indicates the demand value for the gap or the peak.

Eq.(6).Withthe traditionalapproach,the reservoircan easily convergetoheavily congested situations withoutany pos-sibility torecover, even ifthe next reservoirs have plenty ofcapacity left. Thisphenomenon was notably highlighted by

Mariotteetal.(2017)withasupplyreduction scenarioatexit.ThisissuehappensonlyifOext_(nint_,next_{)describesthe} de-mand foroutflow andnot the effectiveoutflow. In classical kinematic wave theory, the observed flow at the headof a queueisalways equaltotheavailablecapacity.Thisismathematicallyexpressedby ademandvalue equaltothemaximal capacityfor saturatedand oversaturatedregimes. We should observethe samebehavior when a reservoir isdischarging without externaldownstream constraints.IfOext_(nint_{, n}ext_{) always equals G}ext_(nint_,next_)forn>n

c, theoutflowdemand is

verylow forahighaccumulationnascongestionspillsback froma downstreamreservoir(represented bytheexogenous limitation

μ

ext_{).Theconsequenceofthislowoutflowisthatthequeuecannotemptywhencongestiondisappearfromthe} downstreamreservoir(i.e.when

μ

ext_{isremoved).Thisevenleadstoanincreaseofaccumulationandthusaloweroutflow} andso on until the systemstabilizes once inflow equals outflow.At this point,the reservoir cannot retrieve a free-flow situation,eventhough thedownstream constraintdoesnotexistanymore. Thiscan beavoided ifwe adoptthefollowing definitionofoutflowdemandfortransfertrips:

Oext

₍

_nint_,next

₎

₌



_next

n P(n)

L =Gext

(

nint,next

)

ifn<nc next

n Pc

L otherwise

(exitdemandfunction) (5)

Hencethenewmodelofoutflowforthesinglereservoirwithtransferandinternaltripsbecomes:



qext

out

(

t

)

=min[

μ

ext

(

t

)

; Oext

(

nint

(

t

)

,next

(

t

))

] qint

out

(

t

)

=Gint

(

nint

(

t

)

,next

(

t

))

(effectiveoutflows) (6)

TheideabehindtheexpressioninEq.(5)isthatthepotentialoutflowoftransfertripsisnowconstantandindependent fromthereservoirstateinoversaturation(n>nc).Thismimicsthebehaviorofusersthatarephysicallyqueuingattheexit

definedbyitsMFD,themaximumrateatwhichuserscanexitthenetworkisgivenbythereservoircapacityormaximum throughput Pc/L. Without havingaccess to the groundtruth, we cannot claim that the latter expression is more orless

realisticthanthetraditionalapproachOext

₍

_nint_,next

₎

_=Gext

₍

_nint_,next

₎

_{.Nevertheless,ourpropositionisspecificallydesigned} tohandleflowexchangesduringcongestionperiodsinmulti-reservoirsystems,accordingtowhatisalreadywell-knownin link-scaletrafficflowtheory.Becausetemporarilysupplylimitationsareverylikelytooccuratthereservoirperimeterwhen congestionpropagates, assumingOext

₍

_nint_,next

₎

_=Gext

₍

_nint_,next

₎

_{wouldoftenlead togridlock,making such a modelquite} uselessin practiceto studyoversaturatedsituations overa large-timehorizon. The traditionalapproach isactually based onobservations fromeither simulation studies(Geroliminis andDaganzo, 2007) orempirical evidences (Geroliminis and Daganzo,2008),butwhattheseauthorsobservedistheeffectiveoutflow.Whereasourmodelingframeworkisbasedonthe demandforoutflow,aquantitywhichisnotmeasurableinsimulationnorinrealfield.Thisconcept provestobeefficient toavoidtheextremesituationmentionedaboveandillustratedfurtherinSection2.3whenconsideringtransfertripsonly.

2.2.Trip-basedmodeling

Wenow focuson themodeling ofspillbacksinthe trip-basedapproach. Thetheoretical backgroundofthe trip-based modelhasbeenfirstintroducedby Arnott(2013).Letusconsidera singlereservoir witha uniquetriplength L.In afirst step,we onlyconsidertransfertripsinthetrip-basedmodel,sothat n=next_{,becausetheformulationofthisapproachis} much morecomplex than theaccumulation-based one. The generalcasewith multipletrips for both accumulation-and trip-basedmodels willbe presentedinSection 3.Ateach time t,all the vehiclesare travelingatthesamespeed V(n(t)). Auserexitingthereservoiratthasexperienced atraveltimeofT(t).Thisuserthusenteredthereservoiratt− T

(

t

)

,and his/hertripdistancewasL.Thetrip-basedmodelconsidersthattheaccumulationandthereforethemeanspeedmaychange duringtheuser’strip,whichismathematicallyexpressedas:

L=

t



t−T(t)

V

(

n

(

s

))

ds (7)

Byusingbasicrelationshipsbasedonenteringandexitingcountcurves,itcanbeshownthatthederivativeofEq.(7)leads to(seee.g.Arnott,2013;Mariotteetal.,2017):

qout

(

t

)

=qin

(

t− T

(

t

))

·

V

(

n

(

t

))

V



n

(

t− T

(

t

))



(8)

Using Eq. (8) to solve the conservation Eq. (1) either for internal or transfer trips leads to a differential equation with endogenousdelay.Despitebeingmathematicallyintractable,thisformulationoftheoutflowcanallowthedevelopmentof efficientnumericalresolutionschemes(continuousapproximationonvehicleindexesorevent-basedresolutionmethod),as showninMariotteetal.(2017). Asimplerformulationintegrating staticdelayshasbeenproposed forthe accumulation-based model by Haddadand Zheng (2017). However, a comparison study is currently missingto assess the differences betweenthese both approaches, so that in this paper we decided to focus on the more refined trip-based formulation. Theseresolutionmethodsofthe latterworkinfree-flow only,whereqin(t) isthe input,equaltothe inflowdemand

λ

(t), andwhereqout(t) isthe consequenceofthe systemevolution.In congestionhowever,the role ofinflowand outfloware switched,asqout(t) becomesthegivenboundarycondition, equaltotheoutflowsupply

μ

(t),andqin(t) hasnowtoadapt tothe systemevolutionduetothe restrictionatexit.It canbe shownthat Eq.(8)canbe reversed toexpressqin(t) asa functionofqout(t):

qin

(

t

)

=qout

(

t+T∗

(

t

))

·

V

(

n

(

t

))

V



n

(

t+T∗

₍

_t

₎₎



(9)

whereT∗(t)istheexactpredictivetraveltime, i.e.thetime duringwhichtheuserenteringattwilltravel,seealsoFig.3. Byconstructionwehave:T

(

t

)

₌T∗

(

t_{− T}

(

t

))

.ButEq.(9)meansthattocalculatetheeffectofanexitflowlimitationonthe entry,oneneedstoknowthefutureofthesystem,whichisproblematic.First,itisnotpossibletodeducetheinflowwhen downstreamsupplyrestrictionshouldapply,andsecond,ifthiswaspossiblewehavenoclueonhowtomaketheswitch.

Thusinpractice,thismodelneedstobecoupledwithanothermodelforreproducingcongestionpropagation.Afirst at-tempthasbeenmadebyMariotteetal.(2017).Theyassumeafree-flowevolutionofthesystemandthenapplytheoutflow reductionandtheminimumprincipleofNewell(1993)ontheinflow.Thismethodwithoff-linecalculationsissufficientfor theanalysisofasinglereservoir,butnot suitableinamulti-reservoircontextwheretrafficstatesinthereservoirs depend fromeach otherall thetime.Inourstudy,weproposeasimplewaytoperformin-line computations,i.e.stepbystep,of inflowlimitationsinthetrip-basedmodel.Itconsistsinswitchingtotheaccumulation-basedframeworkincongestion,by usingthesameentrysupplyfunctionI(n) whichrestrainsinflowforhighvaluesofaccumulation n.We showinthe next sectionwithsimplesimulationscenariosthatsuchamethodworkswellinpractice.

2.3.Numericalimplementation

Inthefollowingofthispaper,wewillusetheevent-basedschemepresentedinMariotteetal.(2017)tosolvenumerically thetrip-basedmodel.Incongestion,thereservoir exitflow oftransfertripsislimitedto

μ

ext_{(t)ateachtime byretaining}

Fig. 3. Cumulative count curves of the single reservoir with the total accumulation n ( t ), the experienced travel time T ( t ) and the exact predictive travel time T ∗_{( t ).}

the vehiclesinside thezone untilthe next exittime,even ifthey havealreadycompleted their triplength. Atentry,the inflowlimitationisensuredwiththedefinitionofaminimumorsupplytimeforenteringthereservoir:

tNin

entrysupply=tN

in₋₁

entry +

1

I

(

n

)

(entrysupplytime) (10)

wheretNin

entrysupplyisthesupplytimefortheN

in_{thvehicletoenterthereservoir,t}Nin₋₁

entry istheentering timeoftheprevious vehicle,andI(n)istheentrysupplyfunctionoftheaccumulation-basedmodelintroducedinSection2.1.

The application of this method is illustrated with two test cases withtransfer trips only. We have therefore: n

(

t

)

=

next

₍

_t

₎

_,q

in

(

t

)

=qextin

(

t

)

, qout

(

t

)

=qextout

(

t

)

,

λ

(

t

)

=

λ

ext

(

t

)

and

μ

(

t

)

=

μ

ext

(

t

)

. Theentrysupplyfunctionhasa shapesimilar to model1intheseexamples.The firsttest caseisaboutademand peaktemporarilyexceeding theexitsupply,andthe second one concerns a supply reduction atexit below the demandlevel at entry. Thesenumerical examples consider a singlereservoirwiththesamecharacteristicsasinSection2.1.

Fig.4(a1)showsthedemand

λ

(t) andsupply

μ

(t)profiles forthedemandpeakcase.Thesimulationscenariohasbeen designedtolet thecongestionreachtheentrybeforethedemanddecreases. Thereservoirstate evolutionispresentedin

Fig.4(b1)and(c1)withtheinflow/outflowandaccumulation.Thebluecurvescorrespondtotheaccumulation-basedmodel, thegreen onesto thetrip-basedmodel.Notethata queueatthereservoirentryistakenintoaccountwhen vehiclesare waitingtoenteriftheinflowislimitedinbothmodels,thoughnotpresentedhere.Allgraphsshowsimilarresultsforboth modelingapproaches.Thiswasactuallyexpected,since themodeling ofspillbacksishandledinthesamemannerinboth models.Thisalsoprovesthattheswitchtotheaccumulation-basedmodelworkswellinthetrip-basedframeworkwithfew modificationsintheevent-basedresolutionscheme.

Fig.4(a2)showsthedemand

λ

(t)andsupply

μ

(t)profilesforthesupplyreductioncase.Similarly,thesimulationscenario hasbeendesignedtoletthe congestionreach theentrybeforetheexitsupplyincreasesagain.InFig.4(b2)and(c2),the red andyellowcurvescorresponds to theevolution ofinflow/outflow andaccumulationwhen Oext _{always equals G}ext _as itistraditionallyassumedintheliterature.Intheaccumulation-basedmodel(inred),weobservethatthesystemreaches an equilibriumpoint once inflow equals outflowshortlyafter 4000 s. Then,the reservoir state doesnot evolve anymore becauseafterthispointthe outflowcorresponds totheexitdemandOext_,andthusq

out(t) isnotimpactedbyan increase of

μ

(t),seeEq.(6).Inthetrip-basedapproach(in yellow),theuserstravelatalowmeanspeedafter4000stoadaptthe exitsupplyreduction.Butwhen thislimitationdisappears,the vehicleexitrateisstillthesame becausethemeanspeed remainslow,andconsequentlythesystemcannotrecoverfromcongestioninthisframeworktoo.Wecanfixthisproblem ifwe keep the outflowdemand Oext _{maximumduring severecongestion periods.This canbe modeled in thetrip-based} frameworkonlyifweforcethetravelerstocompletetheir triplengthatapace thatmatchestheexittime definedbythe outflowwhenn_{≥ n}c.Theoretically,itimpliesthattherelateduserswillhaveaspeed differentfromV(n) duringcongested

situations.ThisformulationhappenstobeequivalenttotheoutflowdemanddefinitionofEq.(5).Thisisillustratedbythe blueandgreencurvesinFig.4(b2)and(c2),whichalsoshowsimilarresultsforbothmodels.

3. Flowtransferinasinglereservoirwithmultipletriplengths 3.1. Accumulation-basedframework

This section aims at extending the previous considerations on inflow and outflowmodeling when a single reservoir includesmultipletripcategories.Thisisthelaststeptowardsourfinalpurpose:proposingarobustmodelingframeworkfor congestionpropagationinamulti-reservoirenvironment.Like YildirimogluandGeroliminis(2014),we considerthatusers areassignedtoasetofgiven“macroscopicroutes” orsometimescalled“regionalpaths”,definedassuccessionsofreservoirs, asillustratedinFig.5(a).Followingtheapproachoftheseauthors,itisassumedthat thesystemstatecanbedescribedat

Fig. 4. (a1) Demand peak at the reservoir entry, demand λ( t ) and supply μ( t ) profiles where the gray area indicates when demand exceeds supply, (b1) inflow q in ( t ) and outflow q out ( t ) and (c1) accumulation n ( t ) for the accumulation and trip-based models. (a2) Supply reduction at the reservoir exit,

demand and supply profiles where excess of demand compared to supply is indicated by the gray area, (b2) inflow and outflow and (c2) accumulation for the accumulation- and trip-based models, where model “2” corresponds to the situation when O ext always equals G ext . (For interpretation of the references

to color in this figure legend, the reader is referred to the web version of this article.)

thelevel of themacroscopic routes, later simply referred to as“routes”. The readercan refer to Batista etal.(2019) for moredetails aboutthe concept of macroscopicroutes andthe methods to estimate their lengths. As a reservoir can be crossedbydifferentrouteswithdifferenttriplengths, thethoroughunderstandingofflowdynamicsinonereservoirwith heterogeneoustriplengthsisessentialtobuildapropermulti-reservoirsimulationtool.

The extension of thesingle reservoir modelwith one trip length to severaltrip lengths hasbeen first established in

Geroliminis (2009, 2015). This theoretical framework has then been used in various studies withmore complex multi-reservoirsettings (e.g.YildirimogluandGeroliminis, 2014;Ramezanietal., 2015;Yildirimoglu etal., 2018). Notethat the

Fig. 5. (a) Examples of three routes ( i, j, k ) for a macroscopic OD ( R o , R d ) in a multi-reservoir system, (b) representation of the reservoir R crossed by the

routes in the accumulation-based framework.

presentstudyalsoapplies forareservoir withdifferentaccumulationcategories(usersare distinguishedby theirrouteor destination)butwithauniquetriplength(usersareassumedtotravelthesamedistance).

3.1.1. Trafficdynamics

Letusconsider asinglereservoir withN triplength categoriesLi,orN routeswithlengthLi,aspresentedinFig.5(b).

Allaccumulationsniineachrouteishouldsatisfythefollowingsystem(Geroliminis,2015):

∀

i∈

{

1,...,N

}

, dni

dt =qin,i

(

t

)

− qout,i

(

t

)

(11)

whereqin,i(t)andqout,i(t)arerespectivelytheeffectiveinflowandoutflowforroutei.Bydefinitionwehave:n=Ni=1ni(the

totalaccumulation),qin=Ni=1qin,i(thetotaleffectiveinflow),andqout=Ni=1qout,i(thetotaleffectiveoutflow).Depending

on the route originand destination (outside orinside the reservoir), its inflow and outflowtreatment may be different. Therefore,toincludeanypossiblecase,wedefinefoursetsofroutes:

• _Pext

in contains all the routes that originate outside the reservoir,

• _Pext

outcontains all the routes that end outside the reservoir,

• _Pint

in contains all the routes that originate inside the reservoir,

• _Pint

outcontains all the routes that end inside the reservoir.

Anyrouteiisincludedintwooftheabove-mentionedsets.Forinstance,transfertripsgather alltheroutesinPext in and

Pext

out,whileinternaltripsgather theroutesinPinint andPoutint (whichisonlyonerouteinthiscase,becausethesequenceof reservoirsdefiningthisrouteisreducedtoonereservoir).

Conceptually,thereservoirissplitinto“sub-reservoirs” governedbytheaccumulationni.Thesesub-reservoirsare

cou-pledtogether bythe meanspeed V(n)orthetotalproductionP(n).It isassumedthat inslow-varyingconditions,thetrip completionrateGiofeachrouteisatisfiesLittle’squeuingformula(Geroliminis,2015):

Gi

(

n1,...,nN

)

=Gi

(

ni,n

)

= ni Li V

(

n

)

= ni n P

(

n

)

Li

(tripcompletionrate) (12)

Based onthisformulation,a dynamicaverage triplength L(t) can bedefinedby applyingLittle’s formulaatthereservoir scale:G

(

n

)

=n/L· V

(

n

)

.Itcomes(Geroliminis,2009):

L

(

t

)

=_Nn

(

t

)

i=1niL(it)

=

N

i=1Gi

(

ni

(

t

)

,n

(

t

))

Li

Theaimofthissectionisthedevelopmentofacongestionmodulethatisrobustandconsistentwiththeunderlyingsteady stateassumption ofthisframework, transcribedinabove Eqs.(12)and(13). Thehypothesis ofa quasi-staticevolutionof thesystemdoesnotaccommodate easilywiththefast-varying transientphenomenathat mayappearduringacongested event.OneofthemajorchangescomparedtotheframeworkinSection2isthedifferenceintriplengths.Theconversionof productionquantitiessuchasthecapacityPcintoflowunitsisnolongertrivial,whichiscriticaltodefinetheentrysupply

functionisthiscase.

3.1.2. Entrysupply

Asdiscussed inSection 2,the routesi∈Pint

in originatinginside thereservoir shouldnot havetheir inflow restrictedif theirdemand

λ

i(t)islowenoughandevolvesslowly:

∀

i∈Pint

in, qin,i

(

t

)

=

λ

i

(

t

)

(14)

Wealreadyexplainedthisassumptionintheprevioussection,butthisneedstobefurtherinvestigatedinupcomingstudies withsimulationcomparisons.Inparticular,thequalitativeattributesof“low” and“slowlyevolving” mustbequantified,but thisisoutofthescopeofthepresentwork.

Alltheotherroutesi∈Pext

in mayexperienceaninflowlimitationastheycrosstheentryborderofthereservoir.However, unlikeSection2,amergingschememustbeusedtoallocatetheavailableentrycapacitytotheroutescomingfromoutside. Thedefinitionofaproperinflowmergehasbeenpoorly investigatedintheliterature.Previous studiesusuallyconsider a globalsupplyfunctionI(n) atthe reservoirentrythat appliesforall the routesenteringthe reservoir,andthen splitthis supplyusingademandpro-ratarule(Geroliminis,2009;KnoopandHoogendoorn,2014;YildirimogluandGeroliminis,2014; Ramezanietal.,2015).Butnothingissaid(i)ontheshapeofI(n)andhowitrelatestotheproduction-MFDP(n)providing that thereare differenttriplengths insidethe reservoir,and(ii) ifanothermerging rulewouldbe satisfyingaswell and why.

Inthissubsection,we look fortheproper formulationofthe individual supplylimitationIi

(

n1,. . .,nN

)

appliedto the

entryofeachroutei.Wefirstconsiderthatalltheroutescrossthereservoirentryborderforthebeginningofourdiscussion. WehaveseenwithsomegeneraltestcasesinSection 2.1.2thattheconsistencybetweeninflowandthereservoirinternal dynamicsiscriticalforthesystemstabilityinoversaturation.Intheaccumulation-basedmodel,theinternal dynamicsare expressedthroughthesteadystateapproximationwithLittle’sformula.Thiscanbewritteneither(i)attheroutelevelwith

Eq.(12),or(ii)atthereservoirlevelthrough thedynamicaveragetriplength L(t) withEq.(13).Assuminga givencritical accumulationncs>ncasinthesingletriplengthcase,for(i)eachindividuallimitationshouldbeIi

(

ni,n

)

=ni/n· P

(

n

)

/Lifor

n>ncs, andfor(ii)the globallimitationshould beNi=1Ii

(

ni,n

)

=P

(

n

)

/L forn>ncs.Note that thefirst definitionimplies

thesecond one,but that the reversecase isnot true. Here, one can seethat it is convenientto define the globalentry supplyfunction inproductionunits toencompass both approaches.Letusthen define theentryproduction supplyPs(n)

withacriticalaccumulationncs,andwhichshapecanbedesignedlikemodel2accordingtoourdiscussioninSection2.1.2.

Wenotablyhave:Ps

(

n

)

=P

(

n

)

forn>ncs.ButwhenappliedtotheroutesinP_inext only(thateffectivelycrossthereservoir

entryborder), the entryproduction supply hasto be modifiedaccordingly to account forthe internal origins: Pext

s

(

n

)

=

Ps

(

n

)

−_i∈Pint in

Li

λ

i.Theaccumulationratioismodifiedintoni/nextwherenext=i∈Pext

in ni.Thesecond approachmustalso

be rewrittenfor theroutes inPext in only:



i∈Pext

in Ii

(

ni,n

)

=P

ext

s

(

n

)

/Lext for n>ncs, whereLext=next



_

i∈Pext in

ni

Li. Itfollows thatthetwoabove-mentionedapproachesfortheexpressionoftheconsistencybetweeninflowlimitationandtheinternal dynamicsleadtotwopossibledefinitionsoftheindividualinflowsupply:

∀

i∈Pext in , Ii

(

ni,n

)

= ni next Pext s

(

n

)

Li

(endogenousentrysupply) (15)

∀

i∈Pext

in , Ii

(

ni,n

)

=

α

i

Pext

s

(

n

)

Lext (exogenous entrysupply) (16)

wherethecoefficients

{

α

_i

}

_i∈Pext

in onlyverify



i∈Pext

in

α

i=1.Thesearecalledthemergingcoefficientsandmaybeindependent

ofthe reservoir state, so that the approach in Eq.(16) is called“exogenous”. As inthe studies mentioned earlier,these coefficientscancorrespondto ademandpro-rata rule:

∀

i_∈Pext

in ,

α

i=

λ

i/j∈Pext

in

λ

j, whichisacommonschemein traffic

flowtheory(JinandZhang,2003).Inthefollowingofthepaper,theexogenousformulationwillbealwaysassociatedwith ademandpro-ratarule.Onthecontrary,theapproach inEq.(15)isreferredtoas“endogenous” becauseitsmergingrule dependsontheratioofaccumulations,thatis,theinnerstateofthereservoir.WhenrewrittenasLiIi

(

ni,n

)

=ni/n· Psext

(

n

)

,

thisapproachconceptuallycorrespondstoa mergeinproduction,withtheavailableproductionforroutei beingLiIi(ni,n)

andthetotalavailableproductionPext

s

(

n

)

.

Itisimportanttonoticethatthemaindifferencebetweentheexogenousandtheendogenousformulationsisthe dimen-sion(timeorspace)thatisusedtodeterminethesharebetweenthedifferentroutes. Theendogenousmodelisgrounded onLittle’sformulaandthesteadystate approximationinsidethereservoir, whichisthebasicassumptionforall theMFD theory.Insteadystate,thesharebetweenroutesinsidethereservoirandattheperimeterareequivalentandthusthe ra-tiosofaccumulationsni/ncanbe usedtodeterminethefractionsofentryflowattheperimeter.The exogenousmodelis

oftheendogenoussupplyfunctionsimpliesamergeinentryproductions,whiletheoneoftheexogenoussupplyfunctions amergeininflows:

{

Liqin,i

}

i∈Pext in =Merge



{

Li

λ

i

}

i∈Pext in,

{

ni/n ext

_}

i∈Pext in,P ext s

(

n

)



(endogenousentry supply) (17)

{

qin,i

}

i∈Pext in =Merge



{

λ

i

}

i∈Pext in,

{

α

i

}

i∈Pinext, Pext s

(

n

)

Lext

(exogenousentrysupply) (18)

Forthe function Merge(·),we use themerge algorithm presentedin Leclercq andBecarie (2012) whichconsistsof an extension ofthe fairmergeofDaganzo (1995).Foranysetof Mmergingdemands {



i}1≤ i≤ M withrespective merge co-efficients {

α

_i}_{1≤ i≤ M} towards a unique entrywith capacityC, theresulting effective inflows {Q_i}_{1≤ i≤ M} are calculated as:

∀

i∈

{

1,...,M

}

, Qi=

⎧

⎪

⎨

⎪

⎩



i if



i≤

α

iC

α

i  Qj>αjC

α

j



C−  Qj≤αjC Qj



otherwise (fairmerge) (19)

ThismergealgorithmensuresthatthetotalavailablecapacityCisalwaysusedwhenonlycertaininflowsarelimitedwhile othersarenot.Itsprincipleisthatalldemands



ibelowtheirrespectivelimitation

α

iCareserved,andthentheremaining

capacityisshared amongtheremaining inflowsaccordingto theirmerge coefficients.Ifsome oftheseremaining inflows exceedtheirrespectivedemandaftersharing,thenthesedemandsareservedandtheremainingcapacityisadjusted accord-inglyandsharedamongtheremaininginflows.Thisprocessisrepeateduntilalltheinflowsareservedand/orthecapacity

Cisfullyused.Notethatthisalgorithmisappliedsimilarlyforbothproductionsandflows.

Thissinglereservoirframeworkalsoincludesapoint-queuemodelforeachroutetostorequeuingvehiclesatentrywhen thecorrespondingdemandisnotsatisfied.Onceaqueuehasformedforaspecificroutei,itsdemand

λ

iissettomaximum,

equaltothereservoirphysicalinflowcapacity(sumofallentrylinkcapacities),providedthatthequeueisnotempty.

3.1.4. Exitdemand,divergingmodelandeffectiveoutflows

The questionofoutflowdivergingconcerns allthe routeswithinthe reservoir, sinceit istheresultofits internal dy-namics.However, thedifferencebetweenthetwo setsPext

out andPoutint isthat eachroutei endingoutsidethereservoir may undergoanexogenousoutflowlimitation

μ

i(t)whencrossingtheexitboundary(representinginflowrestrictiontothenext

reservoir),whileingeneral,theoutflowofeachrouteendinginsidethereservoirisnotexogenouslylimited:

∀

i∈Pint

out,

μ

i

(

t

)

=+∞ (20)

Nevertheless,notethatfinitesupplyvalues

μ

ifori∈Poutint couldbeeventuallyusedtosimulatesaturatedparkinginsidethe reservoir.

WhenconsideringtheroutesinPext

out,theexitdemandistherateatwhichuserswanttocrossthereservoirexitboundary toreachaneighboringreservoir.Theexitdemandisnotmeasurableinpractice,butitisanessentialconcepttodefinethe reservoiroutflowinoversaturation.Whilerarelymentionedinotherworks,itisimplicitlyinvolvedassoonasthepotential outflowofareservoirmaybelimitedbytheentrycapacityofaneighboringreservoir,asinthestudies mentionedearlier (e.g.,KnoopandHoogendoorn,2014;YildirimogluandGeroliminis, 2014;Ramezani etal., 2015;SirmatelandGeroliminis, 2017).Inthesestudies,theoutflowdemandisconsideredequaltothetripcompletion rateGi(ni,n).Wethusrefertothis

approachasthe“decreasingexitdemand” (inoversaturation)fortheoutflowmodel.Inthisframeworktheeffectiveoutflows canbecalculatedas:



∀

i∈Pext

out, qout,i

(

t

)

=min[

μ

i

(

t

)

; Gi

(

ni

(

t

)

,n

(

t

))

]

∀

i∈Pint

out, qout,i

(

t

)

=Gi

(

ni

(

t

)

,n

(

t

))

(foradecreasingexitdemand) (21)

However,wepinpointedinSection2.1.3adrawbackofthisformulationfortheroutescrossingtheexitboundary.During theoffsetofacongestionpeak,weshowedwithsimplesimulationscenarios thatoversaturationinthereservoirresultsin

verylowdemandforoutflow,andthuspreventsitfromaproperrecoveryafterthepeak.AsinEq.(5),weproposethatthe exitdemandshouldbemaximuminoversaturationfortheroutesinPext

out:

∀

i∈Pext out, Oi

(

ni,n

)

=



_n i n P(n) Li =Gi

(

ni,n

)

ifn<nc ni n Pc Li otherwise

(maximumexitdemand) (22)

Thisnewoutflowmodelisthereforenamed the“maximum exitdemand”.Butunlike thedecreasing exitdemandmodel, herethecalculationoftheeffectiveoutflowsisnot straightforward.Inoversaturation,ournewformulationintroduces too manydegreesoffreedominthedescriptionofeachrouteoutflow.Actually,becauseuserstravelingondifferentroutesinthe samereservoirsharethesamemeanspeed,theirrespectiveratesatwhichtheycanexitthereservoirarenotindependent. We must have in steady state:

∀

i∈

{

1,...,N

}

,qout,i=Gi

(

ni,n

)

=ni/Li· V

(

n

)

, thus the inter-dependencyrelationships are

asfollows:

∀

i_,j_∈

{

1_,...,N

}

_,q_{out,i/qout, j=}n_{i/nj· Lj/Li}.Whensome routesi_∈Pext

outarelimitedby

μ

i,theserelationshipsare

verifiedin thedecreasing exitdemandmodel thanks toEq.(21),whereeach exitis driveneither by itslimitationor by thetrip completion rate.But inthemaximum exitdemandmodel, someroutes mayhavean outflowdisconnectedfrom themeanspeedevolutionbecauseofthemaximumweset.Whendrivenby theexitlimitations,wecould haveq_out,i=

μ

_i

andqout, j=

μ

jforsomei,j∈Poutext,buttheratiooftwoexogenouslimitations

μ

i/

μ

jhasafewchancestoequalni/nj·Lj/Liin

practice.Hence,weexplicitlyaddtheinter-dependencyrelationshipstocalculatetheeffectiveoutflowswiththemaximum exitdemandmodel:

qout,k

(

t

)

=min[

μ

k

(

t

)

; Ok

(

nk,n

)

] (mostconstrainedoutflow) (23)

where:k=argmin

1≤i≤N

μ

i Oi

(

ni,n

)

∀

i∈

{

1,...,N

}

,i= k, qout,i

(

t

)

= _nni

(

t

)

k

(

t

)

Lk Li

qout,k

(

t

)

(foramaximum exitdemand) (24)

Thisformulationisdesignedtoensurethat noneofthepartialoutflowsq_out,i(t)exceeds itscorrespondingexogenous lim-itation

μ

i(t). To this end, it is based on the definition of the most constrained outflow. In congestion, the system will

adapttothelimitation

μ

k(t)forroutek,sothatatequilibriumwehaveGk

(

nk,n

)

=nk

(

t

)

/LkV

(

n

(

t

))

=

μ

k

(

t

)

(assumingthat

μ

k(t) isconstant after a giventime). Knowing that outflowk is chosen asthe mostconstrainedone, i.e.with the

high-estdifference betweendemandOk(nk,n) andsupply

μ

k (demand beinghigherthansupply, seeEq.(23)), wehave then:

∀

i=k,Oi/Ok=ni/nk.Lk/Li=qout,i/qout,k≤

μ

i/

μ

k,andthusqout,i≤

μ

ibecauseqout,k=

μ

k.Itresultsthatalleffectiveoutflows

iinEq.(24)willbeautomaticallylowerthantheir respectivelimitations

μ

_i(t).However,whileprotectingthedownstream reservoirsfroman excessofflow andthus fromapossiblegridlock,theconsequenceofthisapproach isthatthe flowin manyroutesmaybe actually lower than their respectivelimitations. Thisis alsoobserved whenusing Eq.(21) withthe decreasingexitdemandmodel.AllthisisillustratedintheupcomingSection4.

Interestingly,wecanshowthattheinstantaneousapplicationoftherelationshipsinEq.(24)isequivalenttothe follow-ingidentityatthereservoirscale:

N



i=1

Liqout,i

(

t

)

=L

(

t

)

qout

(

t

)

(totalexitproduction) (25)

ThelatterimpliesthattheexpressionofthedynamicaveragetriplengthL(t)isalwaysvalidtodefinethetotalexit produc-tion.

3.2.Implementationinthetrip-basedmodel

Theinflowandoutflowmodels wehavedesignedfortheaccumulation-basedmodelare straightforwardtoimplement inthe trip-basedmodel by usingthe “switching” method we presentedinSections 2.2and2.3.This isexactly what we proposeto manage inflows consideringthe entrysupplyfunctionsdescribedby Eqs. (15)and(16).Outflow management canalsobedonebyusingasimilarmethodandtransformingdemandflowsintodemandexittimesthatarematchedwith thesupplytimesatexit.However,wewanttopromotehereadifferentmethod,whichismoreinaccordancewiththe trip-basedframeworkandspirit.Basically,theideaistoensureaFirst-In-First-Out(FIFO)disciplineinvehiclearrivaltimes.This doesnotmeanthattrafficconditionsareFIFOinthereservoir,becauseofthedifferenttriplengths.Butthismeansthatas thespeedisuniformateachstepofthesimulation,theorderofvehiclesattheexitboundariesdefinedbyremainingtravel distancesshouldbepreserved.Inpractice,wemaintainaglobalwaitinglistandawaitinglistofvehiclesperroute.Whena newvehicleentersthereservoir,thearrivaltimesofthetravelingvehiclesarestillunknown,butbecausethevehiclesareall travelingatthesamespeed,theycanbesimplyorderedbytheirremainingtraveldistance.Keepingthevehicleglobalorder correspondstothenaturaldynamicsofthetrip-basedmodel,asinthisframeworktheoutflowispreciselydefinedbythe rateatwhichvehiclescompletetheirtriplength.Thisglobalordercanbeseenastheequivalentoftheaccumulationratios

ni/nthat wefindintheaccumulation-basedformulationthrough Little’sformula. Inslow-varyingconditions,both models

lead tosimilar results asthe underlyingsteady state conditionsof Little’s formulaholds (see Mariotteet al., 2017). This meansthatthesequenceofroutesintheglobalwaitinglistisinaccordancewiththeaccumulationratios.Infast-varying

Fig. 6. Representation of the reservoir R crossed by the routes ( i, j, k ) in the trip-based framework.

conditions however,we expect different resultsbecause the accumulationratios ni/ncan be instantaneously modified at

boththeentryandexitintheaccumulation-basedmodel,whereasittakestimetogetaneworderofroutes(represented bytravelingvehicles)inthewaitinglistofthetrip-basedmodel.

Toillustratethis,letusconsiderasimpleexampleofareservoircrossedbytworoutesdenoted1and2.Supposethatwe haven1/n=0.75andn2/n=0.25intheaccumulation-basedmodelwhenthesystemevolvesslowly.Then,wewould ob-servethefollowingsequenceofroutesforthetravelingvehiclesinthetrip-basedmodel:[11121112_{. . .}].Whensomecapacity restrictionsappearatoneexit,themodificationoftheaccumulationratioscanevolveaccordinglyintheaccumulation-based model,supposedlytowardsn1/n=n2/n=0.5inourexample.Thusthelimitationthatweapplyatentryisconsistentwith whatishappeningatexit,i.e.aflowofthekindn_i/n_·V(n).Butwhileweapplythesamelimitationatentryinthetrip-based model, thesituation atexit is still givenby the previous order [11121112. . .], andwe have towait for thesevehicles to exitbefore thenewselectionatentryprovides thenewexpectedorder[12121212. . .].Thisdelaybetweenentryandexit isoneofthefundamentaldifferencesbetweenboth modelingapproaches.Thisisthereasonwe prefertheoutflowmodel preservingtheexitvehicleorderinthetrip-basedframework,whichisexplicitlydesignedtoaccountforsuchdelays.

3.2.1. Effectiveentrytimes

Atthereservoirentry,intheevent-basedschemeofthetrip-basedmodel,the

{

Nin

i

}

1≤i≤N representthenumbersofthe nextvehiclestoentereachroutei.Theircorrespondingentrydemandtimesarecalculatedas:tNiin

entrydem,i=t Nin

i −1

entry,i+1/

λ

i

(

t

)

,

wheretNini −1

entry,i isthe effective entrytime of theprevious vehiclein route i. As regards theinflow merging scheme,recall

thatwe designedtwoentryflowfunctionsfortheaccumulation-basedmodel.Thefirstone,calledtheendogenoussupply formulation,provides asupply value perentryandcan be implementedin thetrip-basedmodel asfollows.Eachroute i

mayrestrain its inflowby asupplytime tNiin

entrysupply,i fortheNiinthvehiclewilling toenter. Thissupplytime iscalculated

withtheentrysupplyIi(ni,n)presentedinEq.(15):

∀

i∈Pext in , t Nin i entrysupply,i=t Nin i−1 entry,i+ 1 Ii

(

ni,n

)

(entrysupplytime) (26)

wheretNini −1

entry,i istheentrytime oftheprevious vehicleinroutei,seealsoFig.6.Incaseonlysome oftheroutesare

con-gested,we canalsoapplythefairmergeasdescribedinSection3.1.3 toensurethatthetotalenteringproductioncapacity isused.Inpractice,thisisachievedintheevent-basedresolutionschemebymodifyingtheentrysupplytimesofthe con-gestedroutes.ThismodificationisdonedirectlyintheentrysupplyfunctionIi(ni,n)usedinEq.(26):

∀

i∈Pext in , Ii

(

ni,n

)

=

⎧

⎪

⎨

⎪

⎩

λ

i if

λ

i≤ Ii

(

ni,n

)

ni  qin, j>Ij(nj,n)nj 1 Li



Psext

(

n

)

−  qin, j≤Ij(nj,n) Ljqin, j



otherwise (27) wherePext

s

(

n

)

isthemodifiedentryproductionsupplyintroducedinSection3.1.2.Then,thenextreservoirentrytimeisthe

minimumofthepossibleentrytimesofalltheroutes: tNkin

entry,k=1min≤i≤N



max



tNini entrydem,i; t Nin i entrysupply,i





(foranendogenousentrysupply) (28)

Thesecond entrysupplyformulation,referred toasexogenous,isbasedona globalsupplyvalue atentrysplitamong therouteswithexogenous mergingcoefficients.Asintheaccumulation-basedmodel,themergingcoefficientsaredefined