Optimal Decision-Making Strategy of an Electric Vehicle Aggregator in Short-Term Electricity Markets

20 Pages·2017·2.2 MB·English

energies
Article
Optimal Decision-Making Strategy of an Electric
Vehicle Aggregator in Short-Term Electricity Markets
HomaRashidizadeh-Kermani1,HamidRezaNajafi1,AmjadAnvari-Moghaddam2,* and
JosepM.Guerrero2
1 DepartmentofElectrical&ComputerEngineering,UniversityofBirjand,Birjand9856,Iran;
[email protected](H.R.-K.);h.r.najafi@birjand.ac.ir(H.R.N.)
2 DepartmentofEnergyTechnology,AalborgUniversity,AalborgEast9220,Denmark;[email protected]
* Correspondence:[email protected];Tel.:[phone]
(cid:1)(cid:2)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)
Received:11August2018;Accepted:10September2018;Published:12September2018
Abstract:Thispaperproposestheproblemofdecisionmakingofanelectricvehicle(EV)aggregatorina
competitivemarketinthepresenceofdifferentuncertainresources. Intheproposedmodel,abi-level
problem is formulated where, in the upper-level, the objective of the aggregator is to maximize its
expectedprofitthroughitsinteractionsand,inthelower-level,theEVownersminimizetheirpayments.
Therefore,theobjectivesoftheupperandthelower-levelarecontrary.Tosolvetheobtainednonlinear
bi-level program, Karush-Kuhn-Tucker (KKT) optimality conditions and strong duality are applied
to transform the initial problem into a linear single-level problem. Moreover, to deal with various
uncertainties,includingmarketprices,EVscharge/dischargedemandsandthepricesofferedbyrivals,
ariskmeasurementtoolisincorporatedintotheproblem. Theproposedmodelisfinallyappliedtoa
testsystemanditseffectivenessisevaluated.Simulationresultsshowthattheproposedapproachhas
thepotentialtooffersignificantbenefitstotheaggregatorandEVownersforbetterdecision-makingin
anuncertainenvironment.Duringdifferentsituations,itisobservedthatwithincreasingrisk-aversion
factor,astheaggregatortriestohedgeagainstvolatilities,itspurchasesfromday-aheadandnegative
balancingmarketsdecreasessignificantly.However,theparticipationofEVaggregatorinthepositive
balancingmarketincreasesaccordinglytomakemoreprofit.
Keywords: aggregator; competitive trading floor; electric vehicle (EV); energy management;
riskmeasurement
1. Introduction
Amassivepenetrationofelectricvehicles(EVs)inthesmartgridrequiresoptimalscheduling
of their participation in the electricity market [1]. In order to manage the interaction between
EVsandthenetwork, anEVaggregatoractsasamiddlemanandschedulesEVcharge/discharge
processes[2,3]. Inthedecision-makingframework,anEVaggregatorencountersuncertaintiesforEV
management,includingmarketpricesandEVowners’behavior,aswellasthepricesofferedbythe
rivalaggregators[4,5]. TheproblemsofdecisionmakingofanEVaggregatorinatradingfloorhave
beenpresentedinseveralresearchworks[6–11]. Forexample,in[6]singlesupplier’sdecision-making
problemissolvedbyparticleswarmoptimizationforbothblockbidandlinearbidelectricitymarket
models. Anintegratedmodelofplug-inEVsandrenewabledistributedgeneratorsinatradingfloor
isproposedin[7],inwhichtheEVownerscansellbacktheenergygeneratedfromtheirrenewables
ortheenergystoredintheirEVs. Also,anoptimalstructureforparticipationofanEVaggregator
in the day-ahead (DA) and ancillary services markets is presented in [8]. In [9], a multi-objective
day-aheadmarket-clearingmechanismwithdemandresponseoffersisdevelopedinwhichenergy
anddemand-sidereservesareclearedthroughaco-optimizationprocess. Aninteractiveprogramfor
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Energies2018,11,2413 2of20
thechargingprocessofEVs,aswellasinterruptibleloadpricing,issuggestedin[10],inwhichthe
aggregatorwillreplytothecommandsfromthenetwork. Also,in[11],authorsproposeaprobabilistic
model for a bidding strategy of an aggregator in DA and regulating trading floors. In situations
whereadecisionmakerisconfrontedwithuncertaintieswhicharemodeledasstochasticprocesses,
a risk controlling measurement is normally suggested to control the risk of profit/cost variability.
TheproposedoptimizationmodelincludesinevitabledeviationsbetweenDA-clearedbidsandactual
real-time energy delivery. Energy deviations are characterized as “uninstructed” or “instructed”,
depending on whether or not the responsibility is with the aggregator. Some research works that
addressedtheaggregatordecision-makingproblemusedriskmeasurementtoolstoavoidundesirable
outcomesduetouncertainties. In[12], astochasticprogrammingstructureforEVaggregatorsfor
participationinenergyandreservemarketsisproposedthatconsidersconditionalvalueatrisk(CVaR).
Moreover,in[13],value-at-riskisappliedtoastochasticoptimizationproblemforexchangingenergy
ofEVsintheelectricitytradingfloor. Amethodologytomaximizeaggregatorprofits,basedonmaking
decisionsinDAandbalancingmarkets,isdevelopedin[14]. Underuncertainmarketpricesandfleet
mobility,theproposedtwo-stagelinearstochasticprogramfindsoptimalplug-inEV(PEV)charging
schedulesatthevehiclelevel. Moreover,astochasticlinearprogramwiththeapplicationofaCVaR
termasarisk-aversiontoolisproposedfortheparticipationoftheaggregatorinelectricitymarkets.
Sincetheelectricityindustryisevolvingintoadistributedandcompetitiveindustry, someof
the research works focused on a competitive environment to allow the EV owners to choose the
properaggregator[15–17]. In[15],theauthorsproposedanoperationaldecision-makingmodelfora
distributioncompany(DisCo)inacompetitiveenvironmentassociatedwithdistributedgeneration
(DG)unitsandinterruptibleloadoptions.ThecompetitionbetweenDisCosismodelledusingabi-level
optimizationmethod. TheDisCo’sobjectiveistominimizethecostofmarketpurchasesandDGunit
dispatchwhileconsideringtheresponsesofotherDisCos. Also,competitivebehaviorofsuppliers
inanelectricitytradingfloorismodeledin[16]usingaMarkovgameapproach. Likewise,in[17],
acooperativegamemodelisproposedtoobtaintheexchangesbetweenEVparkinglotsandutilitiesin
aspinningreservemarkettocompensatetheuncertaintiesduetothestochasticnatureofrenewables.
Atheoreticalmodelofthecompetitionbetweendemandresponse(DR)aggregatorsforsellingenergy
previouslystoredinanaggregationofstoragedevicesgivensufficientdemandfromotheraggregators
throughanincompleteinformationgameisproposedin[18].
Although the competitive environment is considered in some of the reviewed references,
thepreferencesoftheEVownersaslowerlevelplayersintheelectricitymarketarenotaddressedfrom
theviewpointofEVaggregators. However,inafewrecentstudies,thedecision-makingframework
of an EV aggregator is investigated as a stochastic bi-level problem. Also, a bi-level problem is
addressed in [19] where the aggregator wants to maximize its expected profit while considering
EVs’preferences. TheinteractionsofthePEVownersandtheparkinglotsarealsomodeledasthey
impose restrictions on the parking’s behavior. Therefore, a bi-level problem is constructed where,
intheupper-level, theobjectiveoftheaggregatoristomaximizeitsprofitthroughitsinteractions
and,inthelower-level,theparkinglotmaximizesitsownprofitlimitedtothepreferencesofPEVs.
Althoughtheuncertaintiesofresourcesareconsideredinthesamework,nomechanismisusedto
controluncertainties. In[20],amathematicalprogramwithequilibriumconstraintsispresentedto
obtainoptimalprofitofanEVaggregator,aswellastominimizethepaymentspaidbytheEVowners.
Tothisend,theformulationisproposedasabi-levelproblem,inwhichtheaggregator’supper-level
decisionsonretailpricesandoptimalbiddinginelectricitymarketsareconsidered,andthelower-level
client-sideminimizationofPEV-chargingschedulecost,includinganaffinedemandresponsetothe
retailprices,isalsotakenintoaccount. Inthatstudy,althoughtheschedulingfortheEVaggregatoris
formulatedasabi-levelproblem,thecompetitionbetweentheaggregatorsisnotconsidered. In[21],
incontrasttotheusualcontracts,theaggregatorissupposedtobidinsuchawayastoinfluencethe
market prices. Also, the impact of the aggregator’s bidding strategy on the prices is analyzed via
abi-levelprogram;however,theeffectofthepricesofferedbytherivalsisnotconsidered. In[22],
Energies2018,11,2413 3of20
abi-levelproblemofofferingpricesbyanEVaggregatorinacompetitivemarketisproposed. Inthis
work,onlygridtovehiclemodeisconsideredforthedecision-makingprocessandtheinfluenceof
EVsdischargingmodeswasnotaddressed.
In the research to date, the investigation of simultaneous charging/discharging modes in
the decision-making problem of an EV aggregator in a competitive market is very limited.
Also,theinfluenceofofferingpricesofrivalsontherisk-aversionbehavioroftheaggregator,aswellas
considerationofthevehicletogridmode,isnotaddressed. Therefore,inthispaper,forschedulingof
anEVaggregator,astochasticbi-levelmodelisproposedinacompetitivetradingfloor. Intheupper
level,theaggregatortriestomaximizeitsexpectedprofitand,inthelowerlevel,theEVownersaimat
minimizingtheirpayments. Then,byusingKarush-Kuhn-Tucker(KKT)optimalityconditionsand
dualitytheory,theequivalentsinglelevelformoftheproblemisobtained. Thesourcesofuncertainties,
suchasDAandbalancingprices,aswellasEVcharge/dischargedemands,aremodeledbasedontime
series. Moreover,thepricesofferedbyrivalsaremodeledwithaprobabilitydensityfunction(PDF).
Inaddition,CVaRisusedintheproposeddecision-makingproblem. Therefore,themaincontributions
ofthispapercouldbesummarizedasfollows:
1. To develop a stochastic decision-making model for scheduling of an EV aggregator in a
competitive trading floor, as well as to determine the optimal level of the aggregator’s
involvementinDAandbalancingmarkets.
2. Toinvestigatetheeffectsofarisk-aversionparameteronthedecision-makingoftheaggregator.
Also, the reaction of EV owners to the prices offered by the aggregators under different
risk-aversioncircumstancesisassessed.
3. To effectively solve the stochastic optimization framework with sources of uncertainties,
the proposed model is reformulated to be an expectation optimization problem with CVaR
constraintstoreducetheunfavorableeffectofundesiredscenarios.
Therestofthispaperisstructuredasfollows. Section2describestheproblemofschedulingof
anEVaggregatorinanelectricitymarket. Also,inthissection,thementionedproblemismodelled
asabi-levelstochasticprogrammingstructure. Section3presentsnumericalresultsanddiscussion.
Finally,theconclusionisbroughtinSection4.
2. MaterialsandMethods
Inacompetitivemarket,thereactionofEVownerstothepricesofferedbytherivalaggregators
shouldbeappropriatelymodelled. Inthispaper,abi-levelprogramforschedulingofanEVaggregator
ispresentedinordertodeterminethesellingpricesofferedtotheEVownersinashort-termmarket.
Thebi-leveloptimizationproblemisusedtomaximizetheexpectedprofitoftheaggregatorwhile
minimizingthepaymentsoftheowners. Therefore,theplayersofthebi-levelproblemarecategorized
intotwomaingroupsbasedonthebenefittheyreceive. ThefirstcategoryincludestheEVaggregators
whotrytotakepartintheelectricitymarket. Here,itisassumedthattheaggregatorsbuyenergy
at volatile prices on the electricity market and resell itat fixed tariffs to the EV owners. The main
challengefortheaggregatorsisthelossestheyincurduetothemarketpricechanges. Toprovide
relieffromthis,theaggregatorappliesaproperofferingstrategytomaximizeitsprofitconsideringthe
uncertaintiesaswellasthebehaviorofrivalaggregators. ThesecondplayersareEVownerswhowant
tominimizetheirpayments. Figure1showsthebi-levelstructureofthementionedproblem.
2.1. ProblemDescription
As shown in Figure 1, the bi-level problem has two optimization levels. In the upper level,
the aggregator aims at maximizing its expected profit from participating in a pool-based short-term
electricitytradingfloor,includingDAandbalancingmarkets. Inthislevel,scheduledtransactionsfor
the next day are determined and then, the energy deviations are calculated and eliminated through
actionstakeninthebalancingmarket.Moreover,theaggregatorofferssuitablecharge/dischargeprices
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Energies 2018, 11, x FOR PEER REVIEW 4 of 20
totheEVownerstoencouragethemmakinginteractiveenergytrading. Also,itisreasonablethatthe
it is reasonable that the aggregator wants to purchase energy from the network with lower prices and
aggregatorwantstopurchaseenergyfromthenetworkwithlowerpricesandtoofferchargepricestothe
to offer charge prices to the EV owners with the highest possible prices to make more profit. However,
EVownerswiththehighestpossiblepricestomakemoreprofit.However,sincethemarketenvironment
since the market environment is competitive, it should consider different price scenarios that the rival
iscompetitive,itshouldconsiderdifferentpricescenariosthattherivalaggregatorsmayoffertotheEV
aggregators may offer to the EV owners since, in a competitive environment, the decision making of
ownerssince,inacompetitiveenvironment,thedecisionmakingofoneaggregatorwillaffecttheothers.
one aggregator will affect the others.
Under study aggregator
Upper Level
Scenarios of
DA prices Maximize the expected profit
Scenarios of
balancing prices
1-Determining the optimal amount
of energy traded with the grid.
Scenarios of 2- Determining the optimal amount
Charge/Discharge load Scenarios of prices of charge/discharge energy of EVs. O
of EVs offered by rivals ffered
EVs demand share EVs owners Lower Level charge/discharge price
Minimize charge cost of EVs
Maximize discharge achievements
Subject to:
Technical constraints
Charge prices Discharge prices
Input Data Forecasted Data

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balancing markets and rival aggregators’ prices, as well as the required energy of EVs. These
positive/negativebalancingmarketsandrivalaggregators’prices,aswellastherequiredenergyof
uncertainties can be modeled by using scenario generation and reduction techniques.
EVs. Theseuncertaintiescanbemodeledbyusingscenariogenerationandreductiontechniques.
The proposed framework considers three main units: a data collection and storage unit, a
Theproposedframeworkconsidersthreemainunits:adatacollectionandstorageunit,aprediction
prediction module, and an optimization unit. The data collection and storage unit governs the
module,andanoptimizationunit.Thedatacollectionandstorageunitgovernsthecollectionofinformation
collection of information related to the demand of EVs and the prediction module provides accurate
relatedtothedemandofEVsandthepredictionmoduleprovidesaccurateforecastsoffutureEVdemand.
forecasts of future EV demand. The input to the prediction unit includes information about the
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predircivtiaolns’ upnriicteiss atnhde tphree ddeicmteadndm oafr EkVets pwriitche sa, prrivoablasb’iplirtyic oefs parnedditchtieond eemrroarn. dBaosfeEdV osn wthiitsh inafoprrmobaatiboinli,t yof
preditchteio onpteirmroizra.tBioans eudniot nshtohuisldin sfoolrvme tahteio bni-,ltehveelo oppttiimmiizzaattiioonn purnoibtlsehmo utold msaoxlivmeizthe ethbei -elxepveecltoepd tpimroifzita tion
probloefm thteo amggarxeigmatizoer wthheileex mpeinctimedizpinrogf tithoe fptahyemaegngtrse ogfa tthoer owwhnileersm. Tinhiem oiuztinpugtt ohfe tphea yompteimntiszaotfiothne uonwit ners.
is the optimal bidding in the electricity market and offering of proper prices to the EVs while
Theoutputoftheoptimizationunitistheoptimalbiddingintheelectricitymarketandofferingofproper
satisfying system constraints.
pricestotheEVswhilesatisfyingsystemconstraints.
In real-world situations, it is assumed that the EV charging stations are equipped with smart
Inreal-worldsituations, itisassumedthattheEVchargingstationsareequippedwithsmart
energy management controllers (SEMC) and the owners are able to respond to the charge/discharge
energymanagementcontrollers(SEMC)andtheownersareabletorespondtothecharge/discharge
prices. To this end, the SEMC of each charging station can choose a proper aggregator by monitoring
pricerse.aTl-otimtheis pernicdes, tahned ScEanM sCwiotcfhe atoc hthceh marogsitn cgomstpaetitoitnivcea anggcrheogoastoera inp srohopretr-taergmgr secghaetdourlibnyg.m Tohnisi tios ring
real-tfiemaseibpler ibcyes daenvdelocapninsgw ai tfcahst tocotmhemmunoiscattcioonm mpeetditiaiv weiatgh gbriedgiraetcotrioinnasl hdoartta- tterramnsfsecrh beedtuwleineng .thTeh isis
feasiabglegrbegyadtoervs ealnodp itnheg EaVf acshtarcgoimngm stuantiiocnast.i oItn shmoeudldi abew nitohtedb itdhiarte cthtieo onwalnderast awitlrl annostf ebre binevtowlveeedn the
aggreeagcaht odrasya inn dthteh peroEcVescs;h raartgheinr,g thsitsa aticot niss d.oIntes hboy uthlde SbEeMnCo tseydstethma atnthd ethoewrenfoerres itw isi lnlonto dtifbfiecuinltv oorl ved
eachbduarydeinnstohmeep irno pcerascst;icrae tfhoer rt,hteh oiswancetriss [2d3o]n. ebytheSEMCsystemandthereforeitisnotdifficultor
burd ensomeinpracticefortheowners[11].
Energies2018,11,2413 5of20
Energies 2018, 11, x FOR PEER REVIEW 5 of 20
Data collection/ storage and
prediction unit
EVs charge/
Rivals’ prices Market prices
discharge demand
Generate scenarios Generate scenarios Generate scenario
based on PDF based on relation (1) based on time series
Scenario Reduction
Combine scenarios
Optimization unit
of First Level:
ermine charge/discharge percentageEvs supplied by the aggregators TSEunobe STSmrjegeuoT123accyb ---xomtoO j iCTb etndmiofachnhdef ltieeaitt a zmhetL rrnoeeireg ncein:m etzcevgeh/e io edr e,n cgnRlti e:hhesysicxaets trhr pkrapgaea iaecrdncyg/ottidmeesnni:d gssee tc npnrwhaertasioirnt rgfhogtiys ftet hoo ,op…effwr iaEmncgVeeagr.rsrsk,e:egta,tor: Offering charge/discharge prices offered by under study aggregator
et
d 1- To supply the required energy of EVs,
o
T 2-To satisfy technical constraints of EVs.
1- To Combine the first and second level
2- To linearize the problem
To maximize the expected profit of the
aggregator:
Subject to:
The two level constraints,
The obtained mathematical constraints

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anpdamymaxeinmtsi zeanthde dmisacxhimarigzee acthheie vdeimscehnatrsg.eH earceh,itehveemsceenntasr. ioHserreela, tetdhet ostcheencahriaorsg er/edlaistecdh artgoe pthreic es
offcehraerdgeb/yditshceharrigvea lpsraicrees goeffneerreadt ebdy twheit rhivaalnso arrme gaelnPeDraFt.eTdh weieths taim noartmioanl ePrDroFr. sThoef tehsteimriavtaiolnp reircreosrsa re
of the rival prices are modeled with intervals as the standard deviation. In addition to
modeledwithintervalsasthestandarddeviation. Inadditiontocharge/dischargepricesofferedby
charge/discharge prices offered by the rivals, there exist other uncertain resources in the problem,
therivals, thereexistotheruncertainresourcesintheproblem, suchasDAandpositive/negative
such as DA and positive/negative imbalance prices and the EVs’ charge/discharge requests. Here, an
imbalancepricesandtheEVs’charge/dischargerequests. Here,anautoregressiveintegratedmoving
autoregressive integrated moving average (ARIMA) model [24] is used to generate the scenarios
average (ARIMA) model [23] is used to generate the scenarios associated with DA and balancing
associated with DA and balancing prices. In this study, it is supposed that EVs demand is correlated
prices. Inthisstudy,itissupposedthatEVsdemandiscorrelatedtotheDApricessuchthateachEV
to the DA prices such that each EV demand scenario is generated based on a DA price scenario as
demandscenarioisgeneratedbasedonaDApricescenarioasfollows[23]:
follows [24]:
EEtD,ωctDh,ωc=h E=ˆtDcEhˆtD+cEhˆt+DchΠEˆt[D(cChtΠD,ωA[(−CCˆtDt,DωAA)−/CˆCtDˆtDA]A )/CˆtDA] (1) (1)
whwehreerCeˆ tDCAˆtDisAt hise tehxep eexcpteecdteDdA DApr picreicien inp epreiroioddt ta annddt thhee rreellaattiioonnsshhiipp bbeetwtweeeenn ththe eDDAA prpicreic aenadn tdhet he
EV demand in each scenario, ωωis represented by ΠΠ, which is considered to be 0.2 based on [24].
EV demand in each scenario, is represented by , which is considered to be 0.2 based on [24].
Also, the discharge load of EVs is supposed as a percentage of EVs demand. In total, each set of
Energies2018,11,2413 6of20
scenariosconsistsofDAandpositive/negativebalancingpricesandEVscharge/dischargerequests
asfollows:
(cid:110) (cid:111)
Scenarioω = CDA, Cpos,B, Cneg,B,EDch,EDdch (2)
t,ω t,ω t,ω t,ω t,ω
2.2. Bi-LevelModel
Theproposeddecision-makingproblemoftheEVaggregatorisformulatedastwoupperand
lowerlevels.
2.2.1. Upper-LevelFormulation
TheEVaggregatortakespartinDAandbalancingmarketsinordertoobtainmaximumexpected
profitduringtheschedulinghorizon. Theaggregatorprofitconsistsoftherevenueachievedfrom
sellingenergyforthechargeprocessandfromloadreductioninthenegativebalancingmarket,minus
thepaymentsduetothepurchasesfromDAandpositivemarkets,aswellasbuyingenergyfromEVs
dischargingprocess. Therefore,theupperlevelproblemisformulatedasfollows:
Maximize ∑ π(ω)∑ [(Ech Cch −EdchCdch)+(−EDACDA−Epos,BCpos,B+Eneg,BCneg,B)]
ω∈Ω t∈T t,ω s0,t t,ω s0,t t,ω t,ω t,ω t,ω t,ω t,ω
(cid:20) (cid:21) (3)
+β ζ− 1 ∑ π(ω)ι(ω)
1−α
ω∈Ω
whereαandι(ω)representtheconfidencelevelandauxiliaryvariableforriskcalculations,respectively.
Equation(3)representstheobjectivefunctionfromtheaggregator’sviewpoint. Theleftexpression
of the first line of this equation denotes the charge revenue minus the discharge payments of the
aggregatorunderstudy. TherightexpressionexpressesthepaymentsinDAandpositivebalancing
marketsand,also,thenegativebalancingrevenue. Thesecondlinedemonstratestherisk-aversion
term. Thetradeoffbetweentheobjectivefunctionandrisk-aversiontermisprovidedbyparameter
β as the risk parameter [24]. This parameter models the tradeoff between the expected profit and
risk, andthereforedependsonthepreferencesoftheaggregator. Ariskaverseaggregatorprefers
minimizingrisk;thus,itchoosesalargevalueofβtoincreasetheweightoftheriskmeasurementtool
intheobjectivefunction. Incontrast,whentheaggregatordisplayslessriskaversion,itchooseshigh
valuesofβand,consequently,itobtainshigherprofitvalues. Therelatedconstraintsofthislevelare
providedasfollows:
Ech −Edch = EDA+Epos,B−Eneg,B (4)
t,ω t,ω t,ω t,ω t,ω
(cid:104) (cid:105)
− ∑ Ech Cch −EdchCdch−EDACDA−Epos,BCpos,B+Eneg,BCneg,B +ζ−ι(ω) ≤0 (5)
t,ω s0,t t,ω s0,t t,ω t,ω t,ω t,ω t,ω t,ω
t∈T
ι(ω) ≥0 (6)
Ech = EDch ∑ ρch(ξ)Xch (7)
t,ω t,ω S0,t,ξ
ξ∈Ξ
Edch = EDdch ∑ ρdch(ξ)Xdch (8)
t,ω t,ω S0,t,ξ
ξ∈Ξ
EDA = EDA (9)
t,ω t,ω(cid:48)
Epos/neg,B ≤ P (10)
t,ω
Constraint(4)explainsthebalanceforenergyateachscenarioandateachschedulinghour[14].
Constraints (5) and (6) are associated with the CVaR term [14]. Moreover, EVs’ charge/discharge
demand that is supplied by the aggregator is obtained from constraints (7) and (8), respectively.
Non-anticipativity is provided in (9) and imposes that identical DA bids have to be made in all
Energies2018,11,2413 7of20
scenarioswithequalDAprices[20]. Theenergytradedinthepositiveandnegativebalancingmarket
islimitedbasedon(10).
2.2.2. Lower-LevelFormulation
ThelowerlevelproblemexplainsthepreferencesofEVownersandisformulatedasinEquation(11).
Basedonthisequation,theEVownersseekthecheapestchargeandthehighestdischargepricesoffered
bythemostcompetitiveaggregatortominimizetheirpayments.
(cid:95)Dch
Minimize{E [Cch Xch + ∑ Cch Xch ]
t s0,t s0,t,ξ s,t,ξ s,t,ξ
s ∈ N
S
s (cid:54)=0
+ ∑ ∑ (cid:95)EDchWs,s(cid:48)ZS,S(cid:48) + ∑ ∑ (cid:95)EDdchWs,s(cid:48)ZS,S(cid:48) )
t ch ch,t,ξ t dch dch,t,ξ
s∈NS s(cid:48) ∈ N s∈NS s(cid:48) ∈ N (11)
S S
s(cid:48) (cid:54)= s s(cid:48) (cid:54)= s
(cid:95)Ddch
−(E [CdchXdch + ∑ CdchXdch]}
t s0,t s0,t,ξ s,t,ξ s,t,ξ
s ∈ N
S
s (cid:54)=0
wheresands(cid:48) refertothetransferofEVownersbetweentheaggregators,andindexs=0denotes
the aggregator under study. Equation (11) investigates the objective function of the lower level
problem,whichexpressesthecostofpurchasedenergyfromboththeaggregatorunderstudyandrival
aggregators. ThefirstlineoftheequationrepresentstheEVchargecostspaidtotheaggregatorunder
studyandrivalaggregators,respectively. Thesecondlinedenotesthecostsduetotheshiftofcharge
anddischargeofEVsbetweentheaggregators,respectively. ThelastrepresentstherevenueofEVs
duetotheirdischargingprocesstotheaggregatorunderstudyandrivalaggregators. Theconstraints
areasfollows:
Xch = X0 + ∑ ZS(cid:48),S − ∑ ZS,S(cid:48) (12)
s,t,ξ sch,t,ξ ch,t,ξ ch,t,ξ
s(cid:48) ∈ N s(cid:48) ∈ N
S S
s(cid:48) (cid:54)= s s(cid:48) (cid:54)= s
Xdch = X0 + ∑ ZS(cid:48),S − ∑ ZS,S(cid:48) (13)
s,t,ξ sdch,t,ξ dch,t,ξ dch,t,ξ
s(cid:48) ∈ N s(cid:48) ∈ N
S S
s(cid:48) (cid:54)= s s(cid:48) (cid:54)= s
(cid:95)EDch = ∑ π(ω)EDch (14)
t t,ω
ω∈Ω
(cid:95)EDdch = ∑ π(ω)EDdch (15)
t t,ω
ω∈Ω
∑ Xch/dch =1 (16)
s,t,ξ
s∈NS
Xch/dch ≥0 (17)
s,t,ξ
Zs,s(cid:48) ≥0,∀s,s(cid:48) ∈ S,s (cid:54)= s(cid:48) (18)
ch/dch,t,ξ
TheshareofeachaggregatortosupplyEVsforthechargeanddischargeprocessisrepresentedin
constraints(12)and(13),respectively. Constraints(14)and(15)explainthetotalexpectedchargeand
dischargedemandofEVsateachhour. Also,constraint(16)explainsthatalloftheaggregatorssupply
thecharge/dischargeprocessofEVsateachhour. Moreover,constraints(17)and(18)explainthe
limitationforthevariablesthatshowthepercentageofdemandtobesupplied[25]. Finally,constraints
Energies2018,11,2413 8of20
(19)-(24)describethetechnicalconstraintstokeepstateofcharge(SOC)ofEVswithinitslimitation
andtheamountofenergythattheycanobtainorinjectfrom/tothenetwork[17]. Inthefollowing
equations,µch/dch,µch/dch,γch/dch,γch/dch,µs andµs representtheauxiliaryvariablesthatareused
t,ω t,ω t,ω t,ω t,ω t,ω
forKKToptimalityconditions.
0≤ Ech ≤ Pch : µch ,µch (19)
t,ω t,ω t,ω
0≤ Edch ≤ Pdch : µdch,µdch (20)
t,ω t,ω t,ω
1
SoC −SoC −ηch×Ech + ×Edch =0: λs (21)
t,ω t−1,ω t,ω ηdch t,ω t,ω
SoC×ECap ≤ SoC ≤ SoC×ECap : µs ,µs (22)
t,ω t,ω t,ω
0≤ ηch×Ech ≤ (SoC×ECap)−SoC : γch ,γch (23)
t,ω t−1 t,ω t,ω
1
0≤ ×Edch ≤ SoC : γdch, γdch (24)
ηdch t,ω t−1,ω t,ω t,ω
TheupperlevelproblemgivestheamountofenergypurchasedfromDAandbalancingmarkets
andthecharge/dischargepricesignalsofferedtotheEVowners. Incontrast,thelowerlevelproblem
givesthepercentageofEVcharge/dischargeloadthatwillbesuppliedbytheaggregators. Itshould
bementionedthattheproblemdiscussedaboveisnonlinearduetotheinclusionoftermsEch Cch and
t,ω s0,t
EdchCdch inrelations(3)and(5).
t,ω s0,t
Theequivalentsingle-levelformoftheproblemisobtainedwiththeapplicationofKKToptimality
conditionsofthelowerlevelproblem.Then,thebilinearproductsofEch Cch andEdchCdcharereplaced
t,ω s0,t t,ω s0,t
withequivalentsingle-levelexpressionsusingthestrongdualitytheorem[25]. Finally,theequivalent
formofthebi-levelproblemexplainedin(3)–(24)isobtainedasasingle-levelmixed-integerlinear
programming(MILP)problem,whichcanbeefficientlysolvedbycommerciallyavailablesoftware.
This equivalent problem includes the objective function of the upper level as represented in (25),
theconstraintsofbothlevels,andtheequivalentexpressionofthelowerlevelobjectivefunctionthatis
representedasfollows:
 
Revch −Revdch (cid:34) (cid:35)
Maximize ∑ π(ω)∑ −tE,ωDACDAt,ω +β ζ− 1 ∑ π(ω)ι(ω) (25)
 t,ω t,ω  1−α
ω∈Ω t∈T −Epos,BCpos,B+Pneg,BCneg,B ω∈Ω
t,ω t,ω t,ω t,ω
The bi-level programming problem is transformed into its equivalent single-level nonlinear
optimization problem using the KKT optimality conditions of each lower-level problem.
KKT conditions apply here since the lower-level problems are convex. In Equation (25), Revch
t,ω
andRevdch representthechargerevenuesanddischargepaymentsfortheaggregatorunderstudy,
t,ω
respectively. The other terms are as explained earlier. Equation (25) is obtained subject to the
constraints(4)–(10),(12)–(24)andtheconstraintsthatobtainedfromKKTanddualitytheory,asgiven
inAppendixA.
3. NumericalResultsandDiscussion
Inordertoevaluatetheapplicabilityandeffectivenessofthepresentedbi-levelmodel,atestcase
studywithrealisticmarketpricesisused.
3.1. TestCaseStudy
Inthisstudy,toevaluatetheproposeddecision-makingmodel,fouraggregatorsareconsidered,
inwhichoneofthemisconsideredtobetheaggregatorunderstudy(S )andtheothersareconsidered
0
toberivals(S –S ). TheDAandbalancingelectricitymarketpricesforthemodelpresentedaboveare
1 3
Energies2018,11,2413 9of20
Energies 2018, 11, x FOR PEER REVIEW 9 of 20
eexxttrraacctteedd ffrroomm tthhee NNoorrddppooooll mmaarrkkeett [[2276]]. .IInn tthhiiss ssttuuddyy,, ffoorr eeaacchh ssttoocchhaassttiicc ppaarraammeetteerr,, 220000 sscceennaarriiooss
aarree ggeenneerraatteedd uussiinngg AARRIIMMAA mmooddeellss.. TThhee ggeenneerraatteedd sscceennaarriiooss ooff eeaacchh ppaarraammeetteerr aarree rreedduucceedd ttoo sseevveenn
sscceennaarriiooss bbyy uussiinngg tthhee mmeetthhoodd eexxppllaaiinneedd iinn [[2222]].. TThhee sscceennaarriiooss rreellaatteedd ttoo eeaacchh ppaarraammeetteerr aarree sshhoowwnn
iinn FFiigguurree 33.. MMoorreeoovveerr,, iinn tthhiiss ssttuuddyy,, 110000 EEVVss wwiitthh tthhee ssaammee cchhaarraacctteerriissttiiccss wwiitthh tthhee bbaatttteerryy ccaappaacciittyy
ooff 1166kkWWhh aarree ccoonnssiiddeerreedd.. TThhee iinniittiiaall SSOOCC ooff EEVVss aatt eeaacchh sscceennaarriioo iiss rraannddoommllyy ggeenneerraatteedd wwiitthhiinn iittss
tteecchhnniiccaall lliimmiittaattiioonn. .TThhee ffoorreeccaasstteedd cchhaarrggee//ddisicshcahragreg epprirciecse soofffefereredd bbyy ththrreeee rriivvaal laaggggrreeggaattoorrss aarree
oobbttaaiinneedd ffrroomm [2[278]]a nadndth etihreairs soacsisaotceidatsecde nsacreionsarairoesg aenree rgateenderwaittehda wthirtehe -as etghmreeen-tsnegomrmeanlt PnDoFrm[2a8l] .
α
PTDhFe v[2a9l]u. eTohfe avdaolupete odf is0 .9a5dobpasteedd iosn 0.[9154 ]baanseddt ohne [s1im4]u alnadti othnet ismimeusltaetpioins stiemtteo s1tehp. iTs hseet stcoh 1e dh.u Tlihneg
shcohreidzounlinisg 2h4ohri.zFoinn aisl l2y4,t hh.e Fbini-alellvye, ltshteo cbhi-alestvieclp srtoogchraamstimc ipnrgogprraomblmemingis pforormbluemlat iesd foarsmanuleaqteudiv aasl eannt
eMquILivPaplernotg MramILPa npdrosgorlvaemd awnidth soClPvLedE Xwiintht hCePGLAEXM iSns tohfetw GaAreM[2S9 s]oofntwaacroem [3p0u] toenr wa ictohm4pGuBtsero wfRitAh M4
GanBds oCfi 5RACPMU a.nd Ci5 CPU.
e
c
pri 30
e
alancWh) 25
bM
ve im(€/ 20
siti
Po 0 2 4 6 8 10 12 14 16 18 20 22 24
Time (hour)

(a) (b)
ce 50 0.4
ce pri 40 MWh) 0.3
mbalanMWh) 2300 emand ( 0.2
ative i(€/ 10 EVs d 0.1
g
Ne 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24
Time (hour) Time (hour)
(c) (d)
Figure3.Scenariosof(a)day-aheadprices;(b)positivebalancingprices;(c)negativebalancingprices;
Figure 3. Scenarios of (a) day-ahead prices; (b) positive balancing prices; (c) negative balancing prices;
(d)EVdemand.
(d) EV demand.
3.2. Simulation
3.2. Simulation
Regardingrisk-aversionintheformofCVaRweights,Figure4ashowstheexpectedprofitversus
CVaRRefgoarrddiifnfegr reinskt-laevveerlssioofnβ in. Itthies foobrmse rovfe CdVtahRat wweiitghhitns,c Freigausirneg 4aβ sthhoewags gthreeg eaxtpoerc’steedx ppreocftietd veprrsoufist
CdVecarRea fsoers .dWiffheerenntth leevagelgsr eogf aβt.o Irt bies coobmseersvveedr ythraistk waivther isnec,rtheaeslionwg eβs tth(1e- αa)g-gqrueagnattioler’osf epxrpoeficttesdce pnarorifoits
daercerieganseosr.e Wd.hAenls toh,ei tagcagnrebgeatsoere bnecthoamteisn vleorwy rriisskk -aavveerrssei,o tnhef alocwtoerss,t (C1V-αa)R-qiusannetgilae toivf ep,rwofhiti scchenmaeraionss
atrhea itgtnhoerperdo. fiAtlisno,s iotm caeno bfet hseeesnc etnhaart iions lioswn ergisakt-iavvee,rassioenxp falacitnoerds, iCnV[1a4R] .isH noewgaetvivere,, wwihthicahn mineacrnesa tshinagt
trhisek p-raovfeirt siino nsofmacet oorf, tthhee ssecenneagraiotisv ies snceegnaatriivoes, waso euxlpdlabienoedm iintt e[1d4a].n Hdoawseavreers,u wlti,tCh VaanR inmcroevaessintgo wrisakrd-
apvoesristiiovne vfaaclutoers,. tMheosree onveegra,ttihvee esxcpeneactreiods pwroofiutldv ebresu osmthiteteadv earnadg eaos fat hreessutaltn, dCaVrdaRd emvioavtieosn todwurairndg
pthoesitwivheo vlealduaeys. fMorordeioffveerre,n tthve aelxupeescotefdβ pirsoifliltu vsetrrasutesd thine aFvigeurargee4 obf. thAes ssthaonwdanr,dw dietvhiainticorne adsuinrgingβ ,
tthhee wsthaonldea drdayd feovri adtiifofneroenftt hvealpureosfi otsf dβ eicsr ielalussetsr.aItnedf aicnt ,Fwighuerne 4thbe. Aagsg srheogwatno,r wtriitehs itnochreeadsginega gβ,a itnhset
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apvreorfistes, hfaorwfreovmer,t thhoes epreoxfpitesc itne dsc.eInnaoritohse brewcoomrdes ,mwoirteh diniscpreerasseidn,g wβh,itchhe lpearodfis ttso wexitpherloiewncpinrogb parboifliittsy
faarre ferloimmi nthaotesde. eTxhpeercetfeodre. ,Itnh eotshtaenr dwarodrddse, vwiaittiho ninocfrtehaesienxgp eβc, ttehdep prorofiftitosf wthietha glogwre gpartoobradbuirliitnyg atrhee
eslcimheidnuatliendg. hTohreirzeofnorwe,o tuhled sdteacnrdeaasrde. deviation of the expected profit of the aggregator during the
scheduling horizon would decrease.
EEnneerrggiieess 22001188,, 1111,, 2x4 F1O3R PEER REVIEW 1100 ooff2 200
Expected profit (€) 777567 β=β0=.001.1 β=1 β=2 β=β5=9; β=1β0=6 Expected profit (€) 777567 β=10β=9β=6 β=3 β=2ββ==00.0.11
74 74
-0.4 -0.25 -0.1 0.05 0.2 0.35 0.5 0.65 14.6 14.7 14.8 14.9 15
CVaR (€) Profit standard deviation (€)
(a) (b)
Figure4.Expectedprofitversus(a)CVaR;(b)averageofstandarddeviationoftheprofitfordifferent
Figure 4. Expected profit versus (a) CVaR; (b) average of standard deviation of the profit for different
valuesofβ.
values of β.
Table1showstheexpectedDA,positive/negativeenergyexchangedbetweentheaggregatorand
Table 1 shows the expected DA, positive/negative energy exchanged between the aggregator
(cid:95)DA (cid:95)Pos  (cid:95)Neg 
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tthoe 1a0g, gtrheeg aatgogr’rsepgaartotirc’isp aptaiornticinipDatAioann dinn DegAa taivnedb anleagnactinivgem baarlkanetcsinvgar imesafrrkoemts 3v.7a8r6ieasn fdro2m.15 38.7M8W6 ahntdo
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aisg bgerecgauatsoer wbehceonm these magogrreergiasktoarv beercsoe,miteps umrcohraes reisskle asvsebrlsoec,k ist pofuerncheragsyesf rleosms bthloecmkso oref evnoelartgilye fmroamrk tehtse.
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tFhigeusrhea p5ae,co,f iEt Vcadne bmea snedenin thFaigt uthree 3trde.nAdl soof ,DwAh eanndth neeEgVatsivaere bnaolatnscaitnisgfi eednewrgityh pthuercphuarscehsa fsoeldloewnse rtghye
fsrhoampet hoef DEVA dmemaraknetd, tihne Faiggugrree g3adt.o Arwlsoil,l wprhoecnu rtehed eEvVias taioren sniont tshaetispfoiesdit iwveitmh tahrek eptu(Frcihguasreed5 ben).eIrtgiys
cfroonmce tivheed DfAro mmaFrikgeutr,e th3de athgagtredguartionrg wthiell npirgohctuarned deeavrilaytiionntsh ienm thoer npinosgi,titvhee EmVarokwetn (eFrisgturyret o5bc)h. aIrtg ies
tchoenicreEivVesdt forobmer Feiagduyref o3rdd tahialty duusrei.nCg othnes enqiughent atlny,dt ehaeralyg ginr etghaet morotrrnieinsgto, thbue yEVm oowreneenres rtgryy tfoo rchthaorgsee
hthoeuirr sEtVoss utop pbely rEeaVdsyf rfoomr dtahielyD uAsem. Carokneste,qwuheincthlyh,a tshleo awgegrrepgriacteosr tthraiens tthoe bpuoys imtivoereb eanlaenrcgiyn gfomr tahrokseet.
hours to supply EVs from the DA market, which has lower prices than the positive balancing market.
Table1.Expectedenergyexchanged(MWh)fordifferentvaluesofβ.
Table 1. Expected energy exchanged (MWh) for different values of β.
β EˆDA EˆPos EˆNeg
0.01β 3Eˆ.7D8A6 Eˆ0P.o5s2 9 EˆN2e.g1 53
0.1 3.785 0.529 2.152
0.01 3.786 0.529 2.153
1 3.747 0.530 2.118
0.1 3.785 0.529 2.152
2 3.732 0.535 2.108
1 3.747 0.530 2.118
3 3.466 0.615 2.007
2 3.732 0.535 2.108
5 3.22 0.716 1.897
6 3 33.4.2606 0.601.752 1 2.0017.8 82
105 33..12929 0.701.762 2 1.8917.8 80
6 3.20 0.721 1.882
10 3.199 0.722 1.880
Table2.Expectedenergyexchanged(MWh)fordifferentvaluesofβadoptedfrom[14].
Table 2. Expected energy excβhanged E(ˆMDAWh) foErˆ PdoisfferenEtˆ Nvaeglues of β adopted from [14].
0.01β 2Eˆ.6D5A5 Eˆ1P.o2s76 EˆN0e.g653
0.08 2.575 1.33 9 0.6 36
0.103.01 21..675753 1.227.063 1 0.6503.5 25
0.301.08 21..567054 1.323.196 6 0.6306.4 92
0.47 1.310 2.361 0.393
0.13 1.773 2.031 0.525
3.89 1.181 2.457 0.360
0.31 1.604 2.166 0.492
0.47 1.310 2.361 0.393
3.89 1.181 2.457 0.360