Optimal infrastructure capacity of automated on-demand rail-bound transit systems

Delft University of Technology

Optimal infrastructure capacity of automated on-demand rail-bound transit systems

Cats, Oded; Haverkamp, Jesper

DOI

10.1016/j.trb.2018.09.012

Publication date

2018

Document Version

Final published version

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Transportation Research. Part B: Methodological

Citation (APA)

Cats, O., & Haverkamp, J. (2018). Optimal infrastructure capacity of automated on-demand rail-bound

transit systems. Transportation Research. Part B: Methodological, 117(Part A), 378-392.

https://doi.org/10.1016/j.trb.2018.09.012

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Transportation Research Part B 117 (2018) 378–392

ContentslistsavailableatScienceDirect

Transportation

Research

Part

B

journalhomepage:www.elsevier.com/locate/trb

Optimal

infrastructure

capacity

of

automated

on-demand

rail-bound

transit

systems

Oded

Cats

a,∗

_,

_Jesper

_Haverkamp

a,b

a Department of Transport and Planning, Faculty of Civil Engineering and Geosciences, Delft University of Technology, P.O. Box 5048, 2600

GA Delft, The Netherlands

b Department of Operations Control, Nederlandse Spoorwegen, Utrecht, the Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 19 February 2018 Revised 16 August 2018 Accepted 18 September 2018 Available online 22 September 2018 Keywords:

Automated vehicle Rail infrastructure Railway planning Optimal capacity allocation On-demand services Network structure

a

b

s

t

r

a

c

t

Fully-automatedservicespotentially allowforgreater flexibilityinoperationsand lower marginal operationalcosts. The objectiveofthisstudy is todetermine the capacity re-quirementsofanenvisagedautomatedon-demandrail-boundtransitsystemwhichoffers adirectnon-stopservice.Anoptimizationmodel isformulated fordeterminingthe op-timaltrackand stationplatformcapacitiesforanon-demandrailtransit systemsothat passenger,infrastructureandoperationalcostsareminimized.Themacroscopicmodel al-lowsforstudyingtheunderlyingrelationsbetweentechnological,operationalanddemand parameters,optimalcapacitysettingsandtheobtainedcostcomponents.Themodelis ap-pliedtoaseriesofnumericalexperimentsfollowedbyitsapplicationtopartoftheDutch railwaynetwork.Theperformanceisbenchmarkedagainsttheexistingservice,suggesting thatin-vehicletimescanbereducedby10%inthecasestudynetworkwhiletheoptimal linkandstationcapacityallocationiscomparabletothosecurrentlyavailableinthecase studynetwork.Whilenetworkgeometry anddemanddistributionarealwaysthe under-lyingdeterminantsofbothservicefrequenciesand in-vehicletimes,lineconfiguration is onlyadeterminantintheconventional system,whereasthe automated on-demandrail servicebettercatersfortheprevailingdemandrelations,resultingingreatervariationsin serviceprovision.Aseriesofsensitivityanalysesareperformedtotesttheconsequences ofarangeofnetworkstructures,technologicalcapabilities,operationalsettings,cost func-tionsanddemandscenariosforfutureautomatedon-demandrail-boundsystems.

© 2018ElsevierLtd.Allrightsreserved.

1. Introduction

Therapidadvancementsinthedevelopmentoffully-automatedvehicleshaveledtoanincreasinginterestintheconcept ofautomatedon-demandtransitsystems.Automatedservicescanpotentiallyallowforgreaterflexibility inoperationsand lower marginaloperationalcosts.The areaofapplicationconsideredhithertohasalmostexclusivelybeenlimitedto road-boundsystems.Tothebestoftheauthors’knowledge,offeringflexibleservicesusingheavyrailsystemsasasubstituteto scheduledline-basedserviceshasnotbeenconsideredinliteratureorpracticeinsofar.Thispaperpresentsafirststepinto therelativelyunknownareaofautomatedon-demandrail-bound(ADR)systems.

∗ _{Corresponding author.}

E-mail address: o.cats@tudelft.nl (O. Cats).

https://doi.org/10.1016/j.trb.2018.09.012

Inthis work,an ADR servicedesigned asa full replacementof scheduledheavy railfor agiven (sub-)network is en-visaged. The automated rail-bound vehicles offer a direct non-stopservice andmove in the rail network in response to passengerrequestswithnopre-definedroutesandschedules.Vehiclestransportpassengersthatsharethesameoriginand destinationstations. Vehicles canbe sized accordingto the operator’s preference, butthey are considerablysmaller than existingtrainrollingstockunits.

Theobjective ofthisstudyisto determinethe capacityrequirementsofADR systems.Unlikeroad-bounddemand re-sponsivetransitservicesthat operateinruralareasorcaterforspecialusergroups, theADRnetworkisdesignedtoserve alarge geographical area withrelatively highdemand that can resultinoperationsbeing constrainedby congestionand capacitylimitations.Thestrategicplanningobjectiveofthisstudyconstitutesamajor differencefrommostearliermodels such asAnderson(1998)andWinteret al.(2016)which considered microscopic,operational modelsofflexible transport serviceswithastochastic passengerarrival.The on-demandroutingandschedulingfeatures oftheenvisagedservicealso setsitapartfromstudiesforplanningtransitlineswithspecificnetworkdesignfeatures,ofteninurbansettings(e.g.Saidi etal., 2016;Fielbaumet al.,2016), andmethods fordesigning timetablesthat better cater fordynamicdemand patterns (Sunetal.,2014).

The developmentof technological andservice concepts that willfacilitate ADR systems are still at their early stages. Itis thereforenot surprisingthat literatureon thisserviceconcept issparse. Automationis aprerequisite fordeveloping sucha servicegiventhehighcosts associated withoperatingalarge fleet ofsmallrail-bound vehicles.An earlyresearch identifiedthe challengesofshortvehicleheadwaysandlimitedstationcapacityinthecontext ofdenseurban operations (BendixandHesse,1972).Thevehicle engineeringRailCabprojectdevelopedtechnicalandmechanicalsolutionsforsmall driverlessrail-boundtraffic(Henke etal.,2008).Vehicledesignsolutionsforoperatingatshortheadwaysinanautomated guidewaytransitsystemwerestudiedby ChoromanskiandKowara(2011),whilecapacityinrelationto stationlayouthas beenanalysedinmoredetailbyGreenwoodetal.(2011).AccordingtoUITP(2016),thereismorethan800kmoperatedby automatedmetroandthisisexpectedtoquadrupleby2025basedonconfirmedprojects.Wangetal.(2016)providearecent overviewof trends andissues related to automated metro operations, including increased capacityand reliability. These studiesprovidepreliminaryinsightsintoanticipatedadvancedin-vehicletechnologiesandtheirimplications.However,there islackofknowledgeonthecapacityrequirementsthatsuchoperationsinflictonrailwaynetworkinfrastructureandrelated systemperformanceandlevel-of-service.Crucially,noneofthepreviousstudieshaveexaminedthesystemrequirementsof network-wideoperationsofaflexiblerail-boundservice.

Thecontributionsofthisstudyare:(i)presentingtheconceptofanautomatedon-demandrail-boundtransitsystemand itsfeatures;(ii)formulating anoveloptimizationmodelfordeterminingtheoptimaltrackandstationplatformcapacities for such a system so that passenger, infrastructure and operational costs are minimized. The model is formulated as a cost minimization problem withthe premise that system-optimumvehicle flow distribution conditionscan be attained; (iii)theunderlyingrelationsbetweentechnological,operationalanddemandparameters,optimalcapacitysettingsandthe obtainedcost componentsarestudiedthrough aseriesofnumericalexperiments; (iv)themodelisappliedtopartofthe Dutchrailwaynetwork.Theperformance isbenchmarkedagainsttheexistingservice, itssensitivitytovariousscenarios is assessedandtheimplicationsofwhicharediscussed.

ThemodelingapproachalongwiththeformulationofADRcostfunctionsandtheoverallcostminimizationproblemare presentedinthefollowingSection2.Thereafter,detailsonmodelimplementationandthespecificationofmodelparameters are givenin Section 3. A series ofnumerical experiments designedto test ADRperformance undera range of network, demand,costs andtechnological scenarios are performedin Section 4, followed by an application for partof the Dutch railwaynetwork reportedinSection 5.We concludewitha discussion ofmodel implications andpotential extensionsin

Section6.

2. Modelformulation

2.1. Modelingapproach

Currentrailway models withlines andtimetables astheir cornerstone are unsuitable forADR applications.Passenger transport ingeneral, andthe railwayplanning process inparticular, includes successive models forline pool generation, schedulingoptimizationandmethods forinfrastructure allocationbased ontraffic models (Bussieck etal., 1997;Cordeau etal.,1998; GuihaireandHao,2008). Themain challengeinmodellingnetwork-widelong-termplanningforADRisthat themodelneedstocaptureflowdistributionandcapacityconstraintswithoutrepresentingsystemdynamicsinmicroscopic details. Other ADRchallenges includethe need tohandle large numbersof hourlypassenger requests, strict routing due torailinfrastructureconstraintsandhighlyheterogeneousservicecharacteristicscomparedtotraditionalrail systems.The approachtakeninthisstudyistodevelopanovelmacroscopicmodelbyconsideringADRasaspecialcaseofthenetwork flowproblem.Thisapproachallowsevaluatingalargenumberofnetwork-widesolutions.

Thedeterministic andstatic optimizationproblemis solvedfor agiven networktopology andpassenger demand dis-tribution,bothare thusexogenous tothemodel.Other inputtothe modelincludesvehiclesize,cost unitsandtrackand nodeflow-speed-anddelayfunctions.Modeloutputistheoptimalinfrastructurecapacitypernetworkelement– eachrail segmentandstation,vehicleflowdistributionandthevalueofpassenger,infrastructureandoperationalcosts.

380 O. Cats, J. Haverkamp / Transportation Research Part B 117 (2018) 378–392

2.2. Networkrepresentationandcostfunctions

ThepublictransportnetworkisrepresentedbyadirectedandweightedgraphG(N,A),wherethenodesetNrepresents railstations,andthesetoflinksA_{⊆N× N}representsrailtracksegmentsbetweenstations.s_j isthecapacityofstationj _∈ N.ca isthecapacityoflinka ∈Aandla isthecorrespondinglength.a_{i →j} denotesthedirectedlinkconnectingstationito downstreamstationj.Travel demandisgivenintheformofanhourlyorigin-destinationmatrix,P,whereeachentry,P_od , denotespassengerdemandbetweenapairofstopsoandd,o,d_∈N.Letx_a,od representtheflowonlinka thatoriginated atnodeo andisen-routetodestinationd.The macroscopicprinciplesgoverningthetrafficregime atlinksandnodes,i.e. linkspeed-densityfunctionandnodedelay-densityfunction,aredetailedinthefollowingsub-sections.

2.2.1. Linktraveltimes

Thespeed-densityfunctionisassumedtofollowalogisticfunction(Rotolietal.,2016),albeitwithanabrupttransition fromfree flowto jamconditionswhichis arguablymoresuitable forcharacterizingautomated systems.Letva and

v

ˆa be thespeed andthe free-flowspeed,respectively, ofvehiclestraversing linka.Link traveltimesarethen determinedbased onlinklength,free-flowspeedandthelogistictermwiththevolumeovercapacityratioasfollows:

ta = la ·



1+e−α·θx·caa−ϕ 



ˆ

v

a (1)

Theparameters

α

and

φ

arethescale andshiftofthe logisticdistribution.

θ

isthemaximumnumberofvehicles per hourperunitoflinkcapacity,hereusedtoconvertinfrastructureunits(i.e.numberoftracks)intoflowunits(i.e.numberof vehicles).Whilevolume-overcapacitytermsareusedextensivelyformodellingcartrafficconditions,inthecontextofrail traffic,theminimumtrainseparationtime anddistancearecommonlyusedtoquantifyraillinecapacity.Inthisresearch, volumetocapacityratio,xa /

θ

•ca ,isdefinedasthedeployednumberofvehiclesperhouroverthemaximumnumberof vehiclesper hour.Inpractisethismaximumisaresultofvariousfactorssuch asminimumvehicleseparationandvehicle drivingcharacteristics,signallingandcontrolsystemandtrackswitchingtime.Sincetheserailwaytrafficpropertiesarenot subjectof analysis inthis study,the maximum numberof vehicles per hour is considered an exogenous variable inthe proposedmodel.

2.2.2. Stationprocessingtimes

Letx_j,od denotethenumberofvehicles that arriveatstationj thatare en-routeconnectingagivenOD-pair(including vehiclesforwhichthisstationisanoriginoradestination).VehiclearrivalsareassumedtofollowthePoissondistribution, implyingthatservicerequestsofallvehiclescanbe representedasajointPoissonprocess withthesummativeeventrate parameter

λ

_j =

o∈N



d∈N

xj,od .

Vehicleseitherdrivethroughintermediatestationsorcallatoriginanddestinationstations.Consideringeachplatform as a server andassuming that all vehicles have the same mean service time of 1/

μ

_j with an exponential service time distributionandallvehiclesarepermittedtouseanyplatform,theADRstationischaracterizedasanon-pre-emptiveM/M/c system(similarlytometrostationsinforexampleXuetal.,2014).Ifthestationhasmorethantwoplatforms,itisassumed that through-going vehicles can overtake dwelling vehicles, otherwise the station is governed by non-prioritized M/M/c queues.Thevalueof

μ

_j isdeterminedbasedontheratiobetweenthrough-goinganddwellingvehicle-timesateachstation andthusdependsontheshareofvehiclescalling(i.e.originatingordestined)atstationj outofallvehiclestraversingthis stationsandvehicledwelltime,

τ

dwell .Theexpectedwaitingtime inthenon-pre-emptiveM/M/cqueuedependsondelay probability



(KellaandYechiali,1985),whichiscalculatedasfollows:



j =



s_j ·

ρ

j



s j sj ! · 1

(

s j·ρj

)

s_j s j! +



1−

ρ

j



·s m j−1=0

(

s j·ρj

)

m m ! ·

∀

j∈N (2)

wherethedensitytermisdefinedas:

ρ

j =  o∈N\j d∈N\j xj,od s_j ·

μ

j +  o∈N\j xj,oj +d∈N\j xj, jd s_j ·

μ

j

∀

j∈N (3)

ThenumeratorsinthefirstandsecondtermsinEq.(3)correspondtothenumberofvehiclestraversingandcalling, re-spectively,atstationj.Thecorrespondingexpecteddelaysinprioritizedandnon-prioritizedqueuingsystemscanthenbe de-terminedaccordingtotheformulationsprovidedinWagner(1997),resultingintheexpecteddelayperstation,E(w_j )=f(



_j ). Thisfunctionassignsdifferentvaluesforthrough-going vehiclesanddwellingvehiclesatstationswhereovertakingis pos-sible,whereasotherwisenodistinctionismade.

2.3. Costminimizationproblemformulation

ConsideringADRasaspecialcaseofthenetworkflow problem,thedecisionvariablesare linkcapacity,node capacity and the shareof vehicle flow routed via each route alternative per origin-destinationpair. The objective is to minimize

thecumulative value ofinfrastructurecapacitycosts,passenger traveltime costs andoperationalcosts. TheADR network planningobjective is designed to balance between the costs of adding infrastructure capacityand the costs ofdelay or detourscausedbyinadequateinfrastructurecapacity.

Thecostminimizationproblemisthenformulatedasfollows:

z=min



β

1·

ϑ

·



 o∈N  d∈N  a∈A



xa,od · ta



+ j∈N



λ

j · E



wj



+

β

2·  a∈A ca · la +

β

3·  j∈N s_j +

β

4·

ϑ

·  o∈S  d∈S  a ∈A



xa,od · la



(4) subjectto  i ∈N xa i→j,od  k ∈N xa j→k,od

∀

o,d∈N, j∈N

\

o,d (5)  j∈N xa o→j,od = P_od

ϑ

∀

o,d∈N (6)  j∈N xa j→d,od = P_od

ϑ

∀

o,d∈N (7) x_a,od ≥ 0

∀

a∈A,

∀

o,d∈N (8) ca ≥ 0

∀

a∈A (9) sj ∈Z+

∀

j∈N (10)

ρ

j ≤ 1.0

∀

j∈N (11)

Inaddition,Eqs.(1)–(3)givenintheprevioussectiondescribehowt_a ,linktraveltime,andE(w_j ),theexpecteddelayat eachstation,arecalculated.Theremaining constraintspertainto demandsatisfaction,flow conservationandnon-negative decisionvariables. Eq. (5) ensures flow conservationat all intermediate nodes whereas Eqs. (6) and (7)require demand satisfactionatoriginsanddestinations,respectively.Eqs.(8)and(9)requirethatlinkflowsandcapacitiesarenon-negative.

Eq.(10) ensuresthat stationcapacityis apositiveintegerandEq.(11) determinestheuppervalue ofstationdensity.The decisionvariablesarex_a,od ,ca ands_j .ϑ isthedesiredvehicleloadlevel(e.g.seatcapacity).

Theobjectivefunction,Eq.(4),iscomprisedoffourterms:(i)passengertraveltimescalculatedoveralllinksandstations; (ii)trackcapacityinvestment costs;(iii)stationcapacityinvestment costs;(iv)variableoperational costs.

β

1…

β

4 are the

monetarycosts associatedwitheach oftheobjectivefunctioncomponents:

β

1 ispassenger value-of-time,

β

2 and

β

3 are

thecostsoftrackinfrastructureandstationplatformcapacityunitsexpressedinhourlyterms,respectively,

β

4corresponds

totheoperationalcostsperseatkilometre.Asinglearrivalrateperstationappearsincostcomponent(iii)forsimplicity,but differenteventratesarespecifiedincasedifferentvehicleclasses(i.e.through-goinganddwelling)needtobeconsidered.

Passengertraveltimesconsistofwaitingtimeandin-vehicletimeintheenvisagedcontextofadirectnon-stop station-to-stationADRservice. Giventheon-demandcharacterofthesystem,passengers areassumedtoarriveatstationsat ran-dom.Passengeraveragewaitingtimesarethenapproximatedashalftheintervalbetweendeparturesofvehiclesconnecting agivenOD-pair.Servicefrequencyisspecifiedinthisstudyasapproximatelyproportionaltopassengerdemandforagiven ODpairbyroundingtheratioofP_od tothedesiredvehicleloadlevel,

ψ

,unlesspassengerdemandisnotsufficienttojustify apredefinedminimumservicefrequency(Pod

ψ

<ϑ). Hence,waitingtimesaredeterminedinthe initializationphaseand

areindependentofthedecisionvariablesandcanbeleftoutoftheanalysisofalternativesolutions.

TheminimizationproblemformulatedinEq.(4)entailsthesimultaneoussolutionofsettingthecapacityperrail-segment andstationandobtainingthecorrespondingsystem-optimumsolutionofnetworkflowdistributionwhileminimizinguser, investmentandoperationalcosts.

3. Modelimplementationandspecification

Modelspecificationinvolves settingvalues fora seriesoftechnological andservice parameters.In the following,base casevaluesdesignedtoreflecttheprevalentconditionsinTheNetherlands– forwhichtheauthorshaveaccesstorelevant informationandthemodelislaterapplied– arespecified.Thevaluesaremeanttoindicatearoughestimate.Sensitivityof keyvariablestoselectedparametervaluesisassessedaspartofthesensitivityanalysisperformedinsubsequentsections.

The speed-densityfunction involvesspecifying the free-flow speed andtrack vehicle capacity.The formerwas set to 100km/h(

v

ˆa =100

∀

a∈A)basedonthecurrentconditionsinthedenseDutchnetwork.Trackcapacity,

θ

,issetto180 ve-hiclesperhourinlinewithoperationalpeoplemoversystemswhichoperateat20sheadways.Thelogisticfunctionscaling

382 O. Cats, J. Haverkamp / Transportation Research Part B 117 (2018) 378–392

Table 1

Summary of input parameter specification .

Parameter Value

Free-flow speed, ˆ va [km/h] 100

Track capacity, θ[vehicle/h] 180 Dwell time, τdwell [s] ₂₀

Mean service time, 1/ μ[s] 5 (non-stop), 80 (dwelling) Vehicle capacity, κ[passengers] 24

Load factor, ψ 0.7

Value of time, β1 [ €/h] 10

Track investment costs, β2 [ €/track-km/h] 190.13

Platform investment costs, β3 [ €/station platform/h] 469.62

Operational costs, β4 [ €/seat-km] 0.02

Scaling parameter of the speed-density function, α −11.17 Shifting parameter of the speed-density function φ 0.88

andshiftingparameters,

α

and

φ

,aresetto−11.17and0.88,respectively,basedonengineeringassumptionsderived from acombinationoftheEuropeanCommissionpublicationonrailwaycapacityutilization(Rotolietal.,2016)andliteratureon trafficflow propertiesofautomatedvehicles (seereviewinHoogendoornetal.,2014).Theseparameters werespecifiedso thatthefunctionreproducesthecurrentscheduledrunningtimesoftheDutchrailwaynetwork.

Stationplatformoperationisgoverned bythequeuingservers.Based onobservationsinan existingautomatedsystem (i.e.RiviumParkshuttleinRotterdam) dwelltime,

τ

dwell_{,issetto20s.Forreference, currentsprintertrainsarescheduled}

to dwellfor24s. Meanservicerate, 1/

μ

,isset to5 s fornon-stopvehicles and80s fordwellingvehicles. Thelatter is based onan estimation ofthe time requiredfor theprocesses ofsetting switches,pulling intothe station,dwelling and clearingtheplatform.An estimateofthemeanservicetimemayhavetobe developedwhennetworkcomplexityissuch thatservicetimefornon-stopvehiclesmaybecomeanendogenousvariable.

Thebasecasevehiclecapacityissetto

κ

=24passengers.Vehiclesaredesignedsothatallpassengersareseated(ϑ =

κ

). Servicefrequencyper OD-pairisdetermined sothat passengerdemand issatisfiedwithaloadfactorof atleast 0.7. OD-pairsforwhichdemanddoesnotjustifyatleastonedepartureperhour(i.e.<17passengers)remainunservedbytheADR service.

Objectivefunctionweights are setto{

β

1,

β

2,

β

3,

β

4}={10,190.13,496.62, 0.02}.The firstweightis theDutch

value-of-time expressed inEuros per hour. The second andthird are specifiedbased on estimatesin theDutch rail industry and correspond to€50million per kilometreoftrackand€123.5million per stationplatform.The latterdependsonplatform length whichisspecifiedtoaccommodate theenvisagedvehicledimensions.Based oncurrentrollingstockspecifications, approximatelyonemeterofplatformisneededperthreevehicleseats.Inaddition,itisassumedthateachplatformneeds afixed lengthcomponenttoaccommodatelow speedswitchesonboth ends,eachofwhichhasalengthof15m.Hence, platformsdesignedtoservevehicleswith24seatsshouldbe38mlong.Trackandstationplatformcostsareassumedtobe depreciatedovera30yearsperiod.Operationalcostperseat-kmisestimatedat_€0.02.

TheoptimizationproblemissolvedinMATLABusingatrust-regionmethod(Byrdetal.,1987). Methoddeployment in-volvesthecomputationofafulleigensystem.Thecomputationaleffortisthereforeproportionaltoseveralfactorizationsof theHessianmatrixatthelocationunderconsideration.GiventhescaleofourADRmodel,MATLABusesheuristicapproaches to reduce computationtime by solving a sequenceof approximated minimization problems.The optimization methodis designedforbreadthfollowedbydepth– startingwithaglobalsearchprocesswherecumulativeconstraintvaluesare con-sideredwhilepermittingviolationsofindividual constraintsfollowedbyanefficientlocalconvergence.Sincetheobjective functionincludesfactorialtermsinthequeuingtheorywaitingtimeequations,thosehavebeenreplacedbyaRamanujan’s expression,acontinuousfunctionforapproximatingthefactorialterm. Consequently,alldecisionvariablesare continuous whenhandledbytheoptimizationtool,whichisadvantageousforreducingrunningtimes.Furthermore,thenumberof de-cisionvariablesforreal-sizeproblemswasreducedbyreplacingx_a,od withthedeterminationofpassengerflowsperOD-pair attherouteratherthanthelinklevel.ForeachOD-pair,allroutingalternativesthatinvolvefree-flow traveltimesofupto threetimesthoseoftheshortestpathwereincluded.Flowconservationconstraintsarethenremovedandsubstitutedbya constraintensuringthatthecumulativerouteflowscorrespondtopassengerdemandperOD-pair.

4. Numericalexperiments

ThecostminimizationmodelpresentedinSection2wasappliedtoaseriesofnumericalexperimentsusingthe specifi-cationsdetailedinSection3.Thenumericalexperimentsaredevised toinvestigatethegenericpropertiesofADRsystems, underlyingrelationsbetweenmodelvariablesandthesensitivityofmodeloutputtovariationsininputparameters. 4.1. Experimentalset-up

Thenumericalexperimentsareperformedusingtwographs,bothofwhicharecomposedof17nodesand48 unidirec-tionallinks. Twodistinct networkstructures areconsidered:agridandaring/radialstructure,asillustratedinFig.1.The

Fig. 1. The two network structures considered in the numerical experiments: grid (left) and ring/radial (right). All lines represent bidirectional arcs.

Fig. 2. Optimal track and station capacity and the corresponding utilization level (in color) for the grid network.

basecasepassengerdemandamountsto34,000hourlyrequestsdistributedoverthenetworkbasedonagravitymodel us-ingtheEuclideandistancebetweenalloriginanddestinationnodes(anaverageof125tripsperOD-pair).Theoptimization problemformulationforthe numerical experimentsinvolves approximately1000 constraints,mostof whichare equality constraints.Eachoptimizationproblemissolvedwithin15sonastandardPC.

Aseriesofsensitivityanalysespertainingtoservicedesign,costparameters,technologicalcapabilitiesanddemand sce-nariosareperformed. Inthefollowing,theresultsforvariationsinvehiclepassenger capacity,trackcapacityanddemand distributionpatternarereportedanddiscussed.Inthecaseofvehiclecapacity,itisalsovariedinconjunctionwith corre-spondingchangesinoperationalcostsandtheminimumservicefrequencythresholdcriterion.

4.2.Resultsandanalysis 4.2.1. Scenariossummary

Table2summarizestheresultsofthebasecasescenarioalongwiththekeysensitivityanalysisscenarios forboth the gridandthering/radialnetworks.The basecaseisset witha vehiclecapacityof24passengers andtrackcapacityof180 vehiclesperhour– andwiththeEuclidiangravitydemanddistribution.Thetablereportsresultsforothervehiclecapacity (VC)andtrackcapacity(TC)scenariosnotedwiththecorrespondingvalue,aswellasdemanddistribution(DD)anddemand level(D)scenarios.Scenariolabelsrefertotheinputvariabledifferingfromthebasecasescenarioanditsrespectivevalue. Forexample,scenario“VC_96"correspondstosettingvehiclecapacityto96passengers,whileallothervariablesremain un-changed(ceterisparibus).Foreachscenario,Table2reportsthefourcostcomponentsfortheoptimalsolutionexpressedin hourlyterms(columns2–5),followedbytwoindicatorsofsystemresources– fleetsizeandtotalseat-kmoffered(columns 6–7)– andtheresultingservicespeed(lastcolumn).

Overall,thegridnetworkoutperformsthering/radialnetwork,yieldinglowervaluesforallcostcomponentsinthe opti-malsolutionwiththeexistingdemanddistribution.Acarefulinvestigationrevealsthatdifferencesprimarilystemfromthe factthatinthegridnetworkflowscanoftenbereroutedatconstantmileage,whileinthering/radialnetworkrerouting (al-most)always comesatthepriceofincreasedtraveldistance.Consequently,flowreroutingiscommoninthegridnetwork, whereasallvehiclestaketheshortestrouteinthering/radialscenario.

Fig.2depictsthetrackandstationcapacitydecisionvariablevaluesaswellastheir utilizationlevels.Allbutfourarcs havea single track.Only the middlehorizontal axisrequires double tracks.Utilization levels are highest on single-track segmentslocatedonthehorizontalandverticalpivots.Stationcapacityrangesbetween3and11platformswithaperfectly symmetricalallocation ascould be expectedgiven thedemand distribution. Stationutilization levels across the network exhibitalowvariabilitywithsaturationlevelsof76–88%atallstations.

384 O. Cats, J. Ha ve rkam p / T ransport a tion R esear ch Pa rt B 11 7 (20 18) 3 78–392 Table 2

Numerical experiments scenario results for grid; ring/radial networks per hour of operation.

Scenario Passenger costs[ €10 0 0] Link costs[ €10 0 0] Node costs[ €10 0 0] Operation-al costs[ €10 0 0] Fleet size[veh] Offered seat-km[10 0 0 km] Service speed[km/h] Base 47.1; 49.1 31.8; 33.5 46.1; 46.3 9.9; 10.4 370; 386 495.4;521.6 74; 74 VC_12 58.2; 60.3 69.8; 72.9 83.3; 83.2 11.7; 12.2 915; 948 583.5;609.2 70; 71 VC_48 32.4; 36.4 14.0; 13.5 27.0; 26.8 7.0; 7.8 127; 143 348.7;390.1 75; 75 VC_96 26.5; 27.5 8.3; 7.6 18.1; 18.0 5.5; 5.9 52; 54 276.6;294.6 73; 75 TC_30 60.6; 63.3 160.5; 169.0 46.4; 46.3 9.9; 10.4 476; 497 495.4;521.6 57; 58 TC_45 55.6; 58.0 111.0; 116.9 46.4; 46.3 9.9; 10.4 437; 456 495.4;521.6 62; 63 TC_120 48.7; 50.7 45.8; 48.2 46.4; 46.3 9.9; 10.4 382; 399 495.4;521.6 71; 72 DD_CC 80.1; 75.2 57.9; 53.9 52.7; 52.4 18.0; 16.8 630; 591 902.1; 839.4 79; 78 DD_U 78.6; 74.1 56.9; 53.4 4 9.5; 4 9.5 17.7; 16.7 618; 582 887.1;832.5 79; 79 D_75 34.2; 36.9 22.5; 24.5 35.1; 34.5 6.9; 7.6 269; 290 347.3;381.6 71; 72 D_125 63.5; 65.8 43.8; 45.9 59.8; 59.7 13.6; 14.3 499; 518 683.1;714.5 75; 76 D_150 77.4; 83.8 54.0; 56.8 72.2; 75.1 16.8; 17.7 608; 659 842.1;884.8 76; 74 D_U_75 73.8; 63.2 51.5; 45.2 30.6; 42.8 15.9; 14. 1 580; 497 794.2;704.2 75; 78 D_U_125 114.7;105.2 77.2; 72.3 4 9.5; 4 9.5 23.9; 22.3 901; 827 1192.8;1114.2 73; 74 D_U_150 140.5;113.4 93.3; 82.9 60.9; 75.0 28.8; 25.8 1104; 891 1438.7;1292.3 72; 80 VC (vehicle capacity); TC (track capacity); DD (demand distribution); CC (closeness centrality); U (uniform).

4.2.2. Vehiclecapacity,operationalcostsandminimalservicefrequency

Solutionsensitivitytovehiclecapacitywastestedfor

κ

={12,24,48,96}.Thesecapacitiescorrespondtothenumberof seatsrangingfromamidi-bustoanintercitytraincarunit.Stationcapacitycostswereadjustedtocorrespondtochanging platformlengthrequirements.Apronouncedtrendcan beobservedforbothnetworkstructures intheresultsreportedin

Table2– thesmallerthevehiclecapacitythehigherallcostcomponentsbecome.Reducingvehiclecapacityfromthebase casevalue of24to12resultsinan increaseofapproximately120%and80%inlinkandnode capacitycosts, respectively. Thisincreasedcapacityisrequiredtohandleafleetsizethatis2.5timeslarger,butneverthelessresultswitha25%risein passengercosts.PartofthisdifferenceisattributedtotheincreasingnumberofpassengersthatarenotofferedadirectADR serviceasvehiclesizeincreases.Thisshareincreasesdramaticallyfromnegligiblelevelstoasizeableminorityofpassenger demandinthescenariowithacapacityof96.Withtheexceptionofthelowestcapacityscenario,servicespeedremainsat arelativelyconstantlevel.

Smallerrailvehiclesareassociatedwithhigheroperationalcosts(

β

4),becausetheseat-to-engineratioislower.Itmight

thereforebearguedthat toensurea faircomparison,achangeinvehiclesize needstobe accompaniedwithanopposite changeinoperationalcosts.Thefollowingscenariosarestudiedforthegridnetwork:12seatsat_€0.08perseatkilometre,24 seatsat€0.02perseatkilometre(basecase),48seatsat€0.015perseatkilometreand96seatsat€0.0125perseatkilometre. Weconcludefromtheresults(notincluded inTable2)that achangeinunit operationalcostshasnomajor influenceon thedecisionvariables.DifferencesbetweenthevariousvehiclesizescenariosareinlinewiththeresultsreportedinTable2. Theriskthatlow-demand ODpairswillnotmeettheminimumthresholdsetforjustifyingadirectserviceintheADR systemincreaseswithlargervehiclesizes.Toovercomethedeficiencyofanincreasingshareofunserveddemandwhen ve-hiclessizeincreases,thesecondsetofcombinatorialscenariosrelaxestheminimumservicefrequencyalongsideanincrease invehiclecapacityforthegridnetwork.Thelargestvehicleisassociatedwitha frequencythresholdofonedepartureper hour.Thethresholdvaluesforsmallervehiclesaresetsuchthattheminimumcapacityof96seatsperhourisobtainedinall scenarios:12seatsandatleast8departuresperhour;24seatsand4hourlydepartures;48seatsand2hourlydepartures; and96seatswithatleast1departureperhour.Inthisway,theshareofunserveddemandisfixedinallcases(12%).While serveddemandisequalinallscenarios,passengercostsstilldecreasewithincreasingvehiclecapacity,albeitmoremodestly (i.e.46,940and40,560for12seatsand96seats,respectively)thanreportedinTable2.Linkandnodecapacitycostsare sig-nificantlyhigherinlowvehiclecapacityscenarioswithcostsmorethanfiveandthreetimesashighforlinkandnodecosts, respectively,whencomparing12-seatsand96-seatsscenario, yetresultinginalower operationalspeed.Notwithstanding, smallervehiclesmaystillbepreferredbecausetheyofferhigherservicefrequencyandthusahigherlevel-of-service. 4.2.3. Trackcapacity

Theautomatedrail trackcapacityassumedinthebasecasescenario,

θ

=180vehicles perhour, isveryhighcompared toexistingheavyrailsystems.Inordertotesttheconsequencesoftechnologicaldevelopmentsthatresultinlowervalues, themodelwasrunwithtrackcapacityvaluesof30,45and120vehiclesperhour.Themostconservativevaluecorresponds tothemaximumfrequencyinclassicalheavyrailwithERTMSsignalling.Areferencevalueof45istakenfromautomated metrosystems(e.g.Line14inParis).Anintermediatevalueof120canbeindicativeofmorelimitedtechnological advance-ments.Ascanbeexpected,linkcostsaremostaffectedbychangesintrackcapacity.Incontrast,theimpactoftrackcapacity onpassengercostsexhibitsalogisticrelationthroughtheoperationalspeed.Nodecostsandoperationalcostsremain unaf-fected.Linkcostsdominatethecostfunctionforlowtrackcapacityvaluesanddecreasetolevelssimilartostationcapacity andpassenger costsfortrackcapacityvaluesof120vehiclesper hourandisthen exceededby thelattertwofor

θ

=180 (i.e.basecase).

4.2.4. Demanddistribution

Thedemanddistributionisexpectedtohaveprofoundeffectsonsystemperformance.Twodemanddistributionscenarios are considered in addition to the basecase where the demand isoriented towards the network’s centre of gravity: the opposite caseof a uniform demand distribution (DD_U)and an intermediate case where demand isproportional to the nodeclosenesscentrality(DD_CC)metric(i.e.averagedistancetoallothernodes).Notethatunliketheuniformandgravity distributions,demand distribution inthe latterscenario isnot independent fromnetwork connectivityandhence results withadifferentOD-matrixforthetwonetworkstructures.

Unlikethe basecase,link costs intheoptimalsolutions forthetwoalternative demanddistribution scenariossurpass stationcosts.Amoreuniformlydistributeddemandrequiresalarger fleetandleadstohighermileageandthusincreased trackcapacityonalargernumberofnetworklinks,albeitwithlowercongestionlevelsasreflectedintheincreaseinaverage speedforbothnetworkstructures.Operationalcostsarelowerforthering/radialnetworkthanforthegridnetworkinthe averageclosenessanduniformscenariosduetoits superiorconnectivitywhereas thegridnetwork performsbetter when demandisconcentratedatthecentrewhereitcanoffershorterroutesaroundthecongestedcentrenodeinsteadofthrough thecentrenode.

Theavailabilityofreroutingoptionswithconstantmileageinthegridnetworkallowsforamoreefficientvehiclerouting andmoreopportunitiesfortrading-off differentcostcomponents.Thisgridnetworkpropertyisparticularlyusefulwhenthe demandaroundthecentralnodeisthehighest.Thegridnetworkthereforeoutperformsthering/radialinthebasecaseof gravity-baseddemanddistribution.Conversely,inthecaseofuniformdemanddistribution,thering/radialnetworkhasthe

386 O. Cats, J. Haverkamp / Transportation Research Part B 117 (2018) 378–392

Fig. 3. Cost component sensitivity to demand level in the grid network.

lowestlink-relatedcosts. Thisisa consequenceoftheshorteraveragelinklengthofthering-radialnetwork(1.28unitsof lengthonaverageperlinkinthering-radialnetworkcomparedwith1.33inthegridnetworkstructure).

4.2.5. Demandlevel

We investigated the sensitivity ofthe solutions yielded under differentnetwork saturation levels by varying demand levels. The base casedemand profile based ona gravity model as well asthe uniform demandprofile are uniformly (i) decreasedby25%;(ii)increasedby25%,and;(iii)increasedby50%.

As can be seen inTable 2, the grid network consistently outperforms the ring/radialnetwork interms of total costs foralldemandlevelswhendemandisafunctionofaccessibilityandvice-versainthecaseofuniformdemanddistribution, regardlessofthedemandlevel.Inthecaseofthebasecasedemanddistribution,allcostcomponentsincreaseapproximately linearlyasdemand level rises.In contrast,the marginalcost increase followsa reversed U-shapeforlink, passenger and operational costs in the caseofuniform demand distribution.This can be observed inFig. 3 wherethe total passenger-hours do not exhibit a linearrelation over the linear increase in passenger demand,but ratheran S-shapedcurve. This holds forboth grid andring-radialnetwork types.Thissuggeststhat systems characterized by demandpatterns that are fairlyuniformlydistributedinspacecanresultwithabruptincreasesinallcostcomponentsduetofewereconomyofscale benefits.

Examining the flow distribution and relatednetwork performance inthe optimal solution reveals that trackcapacity utilizationhoversbetween40and50%inalldemandlevelanddistributionscenariosforbothnetworkprototypes.However, stationcapacityutilizationisgenerallyhigher,rangingbetween60and70%,withtheexception ofthemostcentralnodes. Higherpassengercosts aretraded-off againstlowerinfrastructurecostsinlowdemandscenarios whereasinhighdemand networks,thereisatendencytoincreaseinfrastructurecapacitywiththeaimtoreducepassengercosts.Thisalsoexplains theslightlyhigherinfrastructureutilization(andhencemoreinfrastructurerelateddelays)inlowerdemandscenarios.

5. Application

TheADR networkcapacityallocationoptimizationproblemdetailedabove wasappliedtothe real-worldcasestudyof partofthenationalDutchrailwaynetwork.Thecasestudyisdescribed,followedbytheresultsandtheiranalysis. 5.1. Casestudydescription

ThecasestudyareaconstitutesapartofthenationalDutchrailwaynetwork.TheareacorrespondstotheDutchprovince ofNorth-Holland, northofthe NorthSeaCanal, shownin Fig.4.The Dutchcapital,Amsterdam,is situatedatthe south-east cornerofthe casestudynetworkandisconnectedto thecityofZaandamfromwhichdifferentcorridorsvia Hoorn, Uitgeest andAlkmaar stretchtoEnkhuizen (north-east)andDen Helder(north-west).The networkconsistsof28stations and iscurrently served by 8 fixed lines. Each lineis operated witha frequencyof 1–2departures per hour but service design is such that the busiest corridors are served by up to 8 trainsper hour. Thissub-network wasselected because it canbe assumedtooperaterelatively independentlyfromthe restofthenational railwaynetwork yetcontainsrouting alternativesandanetworkofurbansettlementswithstrongcommutingpatternstoAmsterdam.Thenetworkgraphconsists ofaninnerringwiththreebranchesresemblingthering/radialstructurebutwithoutacentralstationbutratheracentral

Fig. 4. Case study network – in relation to the Dutch railway network (left) and zoom-in on the North-Holland subnetwork (right).

Fig. 5. Demand visualization in the case study area during the morning rush hour before (left) and after (right) applying the minimum service frequency threshold of one departure per hour per OD-pair.

ring.InterchangesbetweentheADRoperationswithinthisnetworkandconventionalservicescanbeperformedatUitgeest andAmsterdamCentralstations.

InanalysingtheADRcapacityrequirementsforthecasestudynetwork,weusetheexistingpassengerdemandpatterns. Passengerdemandis obtainedfromsmartcard datarecords. Animportant featureof theDutch smartcard systemisthat passengersare requiredtocheck inandcheckout onevery singlejourney forcalculatingthedistance-basedfare.An OD matrixisconstructedforanaverageweekdaymorningrushhourwithatotalofapproximately7000passengertrips.Fig.5

visualizesthedemanddistributionpatternduringthemorningrushhourbeforeandaftertheminimumservicefrequency thresholdofone departureperhourper ODpairhasbeenapplied.Onestation,namelyZaandamKogerveld,isnotserved atallbecausethedemandtoanyotherstationdoesnotamounttoavolumethatjustifiesadirectconnectionasindicated bytheabsenceofanyconnectingservicearcsinFig.5(right).

388 O. Cats, J. Haverkamp / Transportation Research Part B 117 (2018) 378–392

Fig. 6. Optimal infrastructure capacity allocation for the North Holland case study network; link thickness and node size and corresponding infrastructure utilization rates shown using the colour scheme.

The setof parameters detailedinSection 3 isused inthe basecasescenario. Notethat theenvisaged ADRsystemis served byvehicles much smallerthancurrenttrains(i.e.24passengers)anda significantlyhighertrackcapacity(i.e.180 vehicles per hour per single-track). The sensitivity of the results obtained to variationsin track capacity for automated vehicles – animportantdesignparameterthatatthemomentcannot beestimatedwithcertainty – isthenanalysed.The optimizationproblemresultswith244constraintsinthebasecase.Thissubstantiallylowernumberthaninthenumerical experiments (about1000) stemsfromthe smallernumberofnon-zero demandOD pairs.The smallernumber ofrouting alternatives alsomeanstherearefewerdecisionvariables.Eachoptimizationproblemissolved within30s onastandard PC.

5.2. Resultsandanalysis

5.2.1. Systemcostsandcapacityallocationandutilization

Table3presentsthecostcomponentsanddecisionvariablesvaluesfortheoptimalsolution,expressedinhourlyterms. Passengercostsarethelargestcostcomponent(39%),followedbylink(29%)andnode(24%)capacityinvestmentcostswhile operationalcosts(8%)areconsiderablylower.ApplyingtheproceduredescribedinSection2,wefoundthat191vehicletrips arerequiredforservingthecurrentpassengerdemand.

The magnitudeof thelinkandnode capacitydecisionvariables aredisplayedinFig. 6(sizeofnodesandthicknessof links) aswell astheir utilizationrate (colour of nodesand links) – expressedin terms of vehicle volume over capacity ratio– of each tracksegment andstation platforms.Mostof thestations requireeither one ortwo platforms,similar to orfewerthantheircurrentcapacities.Onlythebusieststationsareassignedwiththree(Alkmaar,HoornandZaandam) or five(AmsterdamCentraalandAmsterdamSloterdijk)platformsintheoptimalsolution.Onemightexpecttheheterogeneity in termsof stoppingpatterns ofthe ADRsystemto requirea higher capacity.However, the highshareof through-going vehiclestodwellingvehiclesallowsforefficientinfrastructureutilization.Nodeutilizationratesvarybetween0.18and0.74. DoubletrackinfrastructureisnecessaryonlyonthebusycorridorbetweenAmsterdamSloterdijkandWormerveer(see

Fig.4).Thisexemplifiestheimportantrole thatnetworkeffectsalsoplay inthecontextofADRservices.Single track(per direction) issufficient forthe remainder ofthe network. Utilizationrates vary between0.01for thelow-demand branch

Table 3

Case study results expressed in terms of objective function components and related variables.

Variable Units Value Passenger costs [ €10 0 0] 24.32 Link costs [ €10 0 0] 17.94 Node costs [ €10 0 0] 14.61 Operational costs [ €10 0 0] 5.07 Fleet size [vehicles] 191 Offered seat-km [10 0 0 km] 253.6 Unserved demand [-] 4% Service effectiveness [km/h] 73 Passenger waiting time [min] 9.5 Passenger in-vehicle time [min] 21

leadingtoEnkhuizento0.63forthe2× 2tracksbetweenthetwoAmsterdamstationsaswell asonthelinkjustafterthe 2× 2corridorduetothe(abruptinteger)capacitydroptoasingle-trackperdirection.

5.2.2. Level-of-service

Inordertoassesstheperformance oftheADRservice, we comparethetraveltime tothose offeredby thefixed train service.Thelatterareobtainedfromthecurrenttimetable.Fig.7showsthedistributionof(nominal)totalpassengertravel timefortheenvisagedandtheexistingservices,comprisingofpassengerin-vehicleandwaitingtimes.Evidently,the distri-butionoftheenvisagedserviceoutperformstheexistingserviceforshort-andmedium-lengthtrips,yieldingshortertravel times.Theaveragepassengertraveltimewilldecreasefrom32.5minto30.5min– saving2minofthetraveltimeforeach passenger– iftheexistingsystemistobereplacedwithanautomatedon-demandservice.Thisamountstoasavingof230 passenger-hoursforasinglepeakhourinthecasestudynetwork.Theshareofpassengertripscompletedwithinacertain timebudget– acommonmeasureoflevel-of-service– increasesfrom11%to17%for15min,from47%to49%for30min andfrom81%to87%for45min.

Passengertravel time savings are the result of differences in-vehicle time only, as waiting times remain on average unchanged.Sincefree-flowspeedwasdeterminedbasedonthetimetabletraveltimebetweenstations,differencesintravel timestemfromcongestionintheADRnetworkordifferentrouting.Systemperformanceisexpectedtoexerciseeconomies ofscale as demonstrated in Winteret al.(2018).Notwithstanding the overall time gains,travel times increase with the ADRservicecompared tothe currenttrain timetableforseveralOD-pairs. Inall thesecases theunderlying causeisthat whilemostvehicles areassignedtotheshortestpath,aminorityofthevehicleflowoperatingbetweentheseOD-pairsis routedviathehighercapacityroutefromZaandamtoHoornviaAlkmaar(westernwingoftheringinFig.6),insteadofthe shortestoptionviaPurmerend(easternwing).Theoptimizationmodelhasfoundthisrouting tobebeneficialforattaining systemoptimumatthecostofprolongingtraveltimesforsomeofthepassengers.TheADRsystemoutperformstheexisting serviceintermsofpassengertraveltimesforshortandmediumdistancetrips whilenosystematicdifference isobserved forlongdistancetrips.

We further examinethe composition oftotal passenger travel times in thealternative systems. In both systems, 72% oftraveltime isspent in-vehicleand28% waiting,the shareofin-vehicle/waitingtimeout ofthe totaltravel timevaries moreintheautomatedon-demandservicethanintheexistingconventionalservice.Thepropertycanbeexpectedascertain relativelylongrouteshavehighdemandandcorrespondinglyahighfrequency(e.g.Alkmaar–Amsterdam)whiletheopposite

390 O. Cats, J. Haverkamp / Transportation Research Part B 117 (2018) 378–392

Fig. 8. Sensitivity of infrastructure costs and passenger hours to changes in track capacity in the case study.

holdsaswell:shortrouteswithlowfrequencyandhencemorewaitingtimethanin-vehicle-time,albeitbotharerelatively short.Notwithstanding,in generaldemand forlongtrips is relatively low inthe casestudynetwork,thus resulting with lower frequenciesthanintheexistingconventional serviceandhencelongerwaitingtime.Thelatterarecompensatedby reductionsin-vehicletimethankstothedirectserviceandthepossibilitytoroutevehiclesthroughlesscongested partsof thenetwork.

While networkgeometryanddemanddistribution arealways the underlyingdeterminantsofboth servicefrequencies (andthuswaitingtimes)andin-vehicletimes,lineconfigurationisonlyadeterminantintheconventionalsystem,whereas therailADRservicebettercatersfortheprevailingODdemandrelations,resultingingreatervariationsinserviceprovision. TrainfrequenciesperOD-pairrangefrom1to16departuresperhourintheADRsystem,comparedto2to12intheexisting fixedservicesystem. Onesegmentofthedemandthatisclearlyworseoff arepassengerstravellingbetweenOD-pairsthat areassociatedwithalow-demandstationthatiscurrentlywell-servedbecauseofnetworkconfiguration(i.e.happenstobe onthewaybetweenhighdemandstations).A totalof4%ofthepassenger demandisnotserved bya directADRservice in thisapplication. For the vastmajority of travellers,the ADR servicemakes therefore the need totransfer and related passengercostsobsolete.

5.2.3. Sensitivitytochangesintrackcapacity

Technologicaladvancesthatenableincreasingthecapacityofanautomatedrail-boundserviceoperatedbysmallvehicles isexpected tobe crucialto itsperformance andviabilityin alarge-scale application. Theoptimization problemis solved for track capacity,

θ

, values of 30, 60,90, 120, 150 and 180 vehicles per hour. The objective function cost components values– passenger,linkcapacityandstationcapacity– are presentedinFig.8.Itcanbeseen thatthelinkcostsdecrease exponentiallywithincreasing trackcapacity.Trackcapacityalsoaffectsfleetsizeandvehiclecirculationandconsequently passengerin-vehicletimes.Thelowertrackcapacityisthelongerpassengertraveltimesbecome.Comparedwiththebase casescenarioof

θ

₌180,capacitiesof150,120,90,60and30,yieldpassengerin-vehicletimeincreasesof2.6,3.4,7.7,13.4 and29.2%,respectively.Consequently,passengercostsappearto firstdeclinerapidly,followedbyamoremodestdecrease for

θ

>90vehiclesperhour.

Atrackcapacityof90isfoundtobethebreakevenvalueinthiscasestudyforobtainingshorterin-vehicletimesinthe ADRsystemthanintheconventionalsystemonaverage inthiscasestudy.Giventhe passengervalueoftime inthecase studyarea,whenmovingintolowervaluesof

θ

,themodelfirsttriestolimittheincreaseintraveltimeatthecostofrising infrastructure costs. Onlyonce the link capacitycosts increase sharply,will they be partlytraded-off against travel time costs.Thisissupportedbytheobservationthat theaveragespeedisnotaffectedsignificantlybydecreasingtrackcapacity untiltrackcapacitybecomessmallerthantheaforementioned90vehiclesperhour.

6. Conclusion

This studypresents a network cost minimization model fordetermining the optimalinfrastructure capacityand flow distribution ofADRsystems.The envisagedsystemconsistsofautomated vehicles that offera directconnection between passengeroriginanddestinationstationsandisroutedsothatsystemflowcostsareminimized,yielding systemoptimum conditions.Theresultsofthenumericalexperimentsandthecasestudyapplicationindicatethatsimilarlytoconventional rail systems, stations are oftenthe bottleneck determiningthe performance of automated ADR systems.Demand forthe morningrushhourwasusedinouranalysis,allowingforthedimensioningoftheinfrastructure(trackandstation)capacity. Underthebasecaseparametersettings,theoptimaltracksegmentandstationcapacityallocationforthecasestudynetwork iscomparabletothosecurrentlyavailableinthisheavyrailnetwork.

The need toinvest in link capacitystrongly dependson vehicle characteristics ofminimum headway and the speed-densityrelation.Trading-off linkcostsagainstpassengercostsispossibleintheformofeitherinducinglongertraveltimeor reducingservicefrequencybyincreasingvehiclecapacity.Smallervehiclesofferbetterserviceduetoshorterwaitingtimes andfewerunservedtravelrequests(seealsovanEngelenetal.,2018)aslongastheincreaseinthenumberofvehiclesdoes notresultwithnetwork congestionandthusa loweroperationalspeed.Moreover, systemoptimum isachievedwhenthe vastmajorityofpassengers areserved mosteffectivelywhileODpairs withverylow demandare unserved.Thisarisesa planningandpoliticaldilemmaasthesystemasawholeisbetteroff bymakingminormarketsegmentsworse-off (i.e.not cateringforthelongtail),compromisingthenotionofauniversalservice.

The optimal solution corresponded to the system-optimum flow distribution. Even though model formulation allows forrerouting vehiclesto attainglobalsystemoptimumconditions,onlyinextremecasesaminorityofthevehicle flowis notassignedtotheshortestroute.Inthefuturisticscenario thatseveralcompetingoperators caninreal-timereroute the automatedrail-bound vehicles,systemperformance mayshiftaway fromsystemoptimumconditions.Theimplications of competingADRserviceproviders canbestudied byadjustingthe objectivefunction usedinthisstudytoguarantee user-equilibriumconditions.

Sincethisstudyispioneeringinthepublictransportsystemitenvisages,modelspecificationrequiredmakingaseriesof assumptionsaboutplausiblecharacteristicsofprospectiveautomatedrail-ADRsystems.Aseriesofsensitivityanalyseswas performedtotesttheimpactsofevenextremedeviationsfromtheassumedvaluesonmodeloutputs.Futuredevelopments areexpectedtoallowfinerspecificationsoftechnological andservicefeaturessuchasthelinkspeed-densityfunction,the stationplatformsqueuingregimeandrelatedassumptions,includingtheconsiderationofaheterogeneousfleetcomposition. Modellingfleet size asa decision variablerather than determining it endogenouslywill allow explicitly investigatingits impactonpassengertraveltime.Furthermore,changesinserviceperformanceshouldbeexaminedinrelationtotemporal dynamicsofthedemandlevelanddistributionpatterns.Onceconsidering stochasticdemandpatterns– which arehigher attheODlevelthanatthemoreaggregate directionaldemandinconventionalsystems– therisk ofdeniedboardingdue tobindingvehiclecapacitylimitsalsobecomesrelevant.

Futureresearch mayinvestigatewhethertheresultsattainedinthisstudyholdforalargevarietyofnetworkstructure designsandthepotentialbreakevenpointintermsofserviceperformance andoverall systemcosts inrelationtoexisting fixedscheduledservices.While thevalue ofnetworkredundancyforrail systemperformanceunderdisruptionshasbeen established(JeneliusandCats,2015),ourfindingssuggestthatinthecontextofon-demandtransport,networkredundancy isanimportantdeterminantofsystemefficiencyundernormaloperations.Furthermore,infrastructureutilizationcanbe fur-therincreasedbyallowingforamoreflexibleon-demandservice,includingride-matchingcapabilitiesthatinvolvedetours. Assessingthepotentialofsuchaserviceconceptrequiresdevelopingamodelwhichcancapturethepotentialshareification amongdifferentODpairs.

Thisstudyexaminedthelong-termplanningconsequencesofofferinganewtransportationtechnologyandservice con-cept.Inordertotestitspotentialtosubstitutetheexistingrailservice,currentdemandlevelsweretestedintheapplication. Futurebehaviouralresearchisneededtoassesstraveller’sperceptionsandpreferencestowardssuchservicesandpotential changesindemandpatternsthatmaybecausedbytheintroductionofADRservices.Thiswillpotentiallyallowconsidering passengerdemandasanendogenousvariableintheoptimisationmodel.Forexample,theservicemighthaveconsequences forstationattractiveness duetotheincreasedcorrelationbetweenservicefrequencyanddemandforspecificconnections. Thismayalsohaveramificationsfornetworkstructuredesignandserviceavailability.

Acknowledgements

TheauthorsarethankfulforsuggestionsmadebyBartvanArem,DennisHuismanandEdwinBoer.DutchRailways(NS) kindlyprovidedinputonoperationalsettingsaswellasdataforthecasestudy.

Supplementarymaterials

Supplementarymaterialassociatedwiththisarticlecanbefound,intheonlineversion,atdoi:10.1016/j.trb.2018.09.012.

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