markets using a weekly differentiated approach Dual technology energy storage system applied to two complementaryelectricity Journal of Energy Storage

Dual technology energy storage system applied to two complementary electricity markets using a weekly differentiated approach

Helder Lopes Ferreira

^a,

*, Kate rina Sta nková

^b,c

, João Peças Lopes

^d,e

, Johannes Gerlof (Han) Slootweg

^a

, Wil L. Kling

^a,1

aDepartmentofElectricalEngineering,EindhovenUniversityofTechnology,DenDolech2,Eindhoven,TheNetherlands

bDepartmentofKnowledgeEngineering,MaastrichtUniversity,Bouillonstraat8–10,Maastricht,TheNetherlands

cDelftInstituteofAppliedMathematics,DelftUniversityofTechnology,Mekelweg4,Delft,TheNetherlands

dElectricalEngineeringDepartment,UniversityofPorto,RuaDr.RobertoFrias378,Porto,Portugal

eINESCTEC–TechnologyandScience&FEUP,InstituteforSystemsandComputerEngineeringofPorto,RuaDr.RobertoFrias465,Porto,Portugal

ARTICLE INFO

Articlehistory:

Received6June2016

Receivedinrevisedform19March2017 Accepted17April2017

Availableonlinexxx

Keywords:

Energystoragesystems Electricitymarkets

Renewableenergysourcesintegration Optimalcontrol

Optimization

ABSTRACT

Thispaperdealswithintegrationofenergystoragesystemsintoelectricitymarkets.Weexplainwhythe energystoragesystemsincreaseflexibilityofbothpowersystemsandenergymarketsandwhysuch flexibilityisdesirable,particularlywhenvariablerenewableenergysourcesarebeingusedinexisting powersystems.As opposedtotheexistingliterature,ourmodelincludesadualtechnologyenergy storagesystem,actingintwodifferentmarkets.Weintroduceamathematicalformulationforthismodel appliedtotwoDutchelectricitymarkets.Adoptingoptimalcontrolapproachwiththegoaltomaximize theyearlybenefit,weshowthatthedualenergystoragesystemcanbeprofitablealreadywhenthesame buying/sellingstrategiesareadoptedfortheworkingdaysandweekends.Weshowthattheprofitability (slightly)increaseswithdifferentbuying/sellingstrategiesfortheweekdaysandweekends.Finally,we demonstratehowtheyearlybenefitvarieswithsizeandefficiencyofthedeviceschosenandmarket prices.

©2017PublishedbyElsevierLtd.

1.Introduction

1.1.Motivation,backgroundandliteraturereview

Theworldwideenergypolicygoalsincludefurtherintegration oftherenewablegenerationtechnologiesintotheenergymarkets.

For example,the EuropeanUnion is striving toachieve 20% of energygeneratedfromrenewableenergysources(RES)by2020 andtoreachaminimumof27%ofrenewablegeneratedenergyby 2030,whilereducinggreenhousegasemissionsbyatleast40%by 2030comparedtotheirlevelin1990[1].Objectivesfor2050are evenmorechallenging,withareductionofthecarbonemissions by80–95%[2].Allaroundtheworld(e.g.inChina[3],Japan[4], NewZealand[5],UnitedStatesofAmerica[6,7]andTurkey[8])the power systems are being prepared for an increasing level of deploymentofrenewablegenerationtechnologies.

In conjunction with RES, the integration of other recent technologies,suchaselectricvehicles(EV),butalsotheunbun- dling and modification in the regulation of the power sector, influence the paradigm and structure of the power sector. As electricityhastobedealtwithwhengenerated,eitherbybeing consumedorstored,matchingthelevelsofgenerationandloadat all times is fundamental. The fact that most RES are weather- dependentwillcausethegenerationoutputtovarymorelikely with the climate conditions than with the market needs. The increasing integration of electric vehicles also increases the likelihood of high load variations during the day. The novel technologiesareexpectedtobeappliedtoanextentwhichwill certainlyamplifytheeffectofthesevariations.

The abovementioned technologicaland regulatory develop- mentscallforadjustmentofplanningandoperationofthepower systems– theyneed tobemoreflexible.Thisflexibilitycanbe achievedthroughseveraltechnologiesandtechniques(e.g.energy storagesystems(ESSs’),cross-borderinterconnectioncapacity,RES management, more flexibility from conventional generation, activedemandside managementand vehicle-to-grid)and their combinations[9].Amongthese,ESSisseenasoneofthelongterm mostfeasibleoptionstoachievethatgoal[10].

*Correspondingauthor.

E-mailaddress:h.m.lopes.ferreira@gmail.com(H.L.Ferreira).

1Deceased.

http://dx.doi.org/10.1016/j.est.2017.04.005 2352-152X/©2017PublishedbyElsevierLtd.

ContentslistsavailableatScienceDirect

Journal of Energy Storage

j o u r n a lh o m e p ag e :w w w . e l s e vi e r . c o m / l o c a t e / e s t

ESSscanprovideuptotwicetheirrating(sumofchargeand discharge capacities) to balance the electricity grid. This is accomplishedbyswitchingbetweenthetwomodesofcharging anddischarging,ineitherdirection(fromchargingtodischarging orfromdischargingtocharging).Therefore,ESSshelptobalance the electricity system when there is a generation surplus or a deficit.ESSscanprovidevariousservices,mostimportantofwhich belongtooneofthetwomajorcategories:

powermarketarbitrage ancillaryservicesandbalancing

Power market arbitrage is an energy service provided via charginganenergystoragedevicewhentheelectricitypricesare lowanddischarging itwhentheprices arehigh[11].Theprice variationsarecausedbydaily,weeklyorseasonalcycles.Lately, alsovariationsinrenewablepowergeneration,e.g.windandsolar energy,areaffectingtheenergymarketstoadegreedependingon their level of market penetration and the flexibility of the underlying conventional generation fleet. The most adequate marketsexercisingarbitrageareday-aheadandintra-daymarkets [12].

Inunbundledmarkets,thesystemoperatorsarenotallowedto own energygeneration assets. Therefore, they need toprocure severalancillaryservices.Examplesoftheseancillaryservicesare balancingsupportandcongestionmanagement.

OtherservicescanbesuppliedbyESSs[6,7,11],dependingon thecharacteristics of thespecific energystoragetechnologies.

The problem of energy storage integration into existing electricitymarketswasstudiedin[13–15].Theliteratureimplies thatinmost markets,withcurrentprice differences,arbitrage provision is not sufficient to make energy storage profitable.

Hybridenergystoragesystemsusingtwoenergystoragedevices arepresentintheliterature.However,theseareassociatedwith electric vehicle power system or variable renewable energy generationsiteintegrationintothegrid[16].Nonetheless,tothe best of our knowledge, no models including two electricity markets and two ESS technologies operating in parallel have beendevelopedsofar.

Thispaperfocusesonacombinationofenergymarketarbitrage and provision of balancing support by the same dual energy storagesystem.Themodelthatweintroduceinthispaperdiffers fromthemodelsanalysedintheliteratureintwomajoraspects.

Firstly,we considerasystem combiningarbitrageand ancillary services.Withthiscombinationweexpecthigheryearlybenefits thanusingarbitrageonly.Secondly,theenergystoragesystemthat weproposeusestwoenergystoragetechnologiessimultaneously.

Thedualtechnologysystemwaschosenin ordertoprofitfrom characteristicsofboth devicesandmarketpricevariations.This paperextendsourresearchpresentedin[17].

InordertoseehowprofitabletheESScouldbe,inthispaperwe seekoptimalstrategyintermsofpricethresholdsforbuyingand sellingelectricityattheDutchday-aheadandbalancingelectricity markets.Mathematically,weformulatetheproblemasanoptimal control problemwiththe goal tomaximize theyearly benefit.

Firstly, we consider the situation when buying and selling thresholds may vary between working days and weekends.

Secondly, we consider a situation when theworking days and weekendthresholdsarethesame.Weusepatternsearchtofind theoptimalstrategyandmotivatethechoiceofthismethod.

Theremainderofthispaperiscomposedasfollows.Section2 introduces electricity markets in The Netherlands. Section 3 explains the background of the model we put forward. The problem dealt within this paper is defined mathematically in Section4.Implementationofthemodelandasolutionmethodare describedinSection5.Section6presentsanddiscussestheresults

of the case studies. Section 7 finalizes the paper with the conclusionsanddirectionsforfutureresearch.

1.2.Notation

Tables1and2describethemainsymbolsusedinthispaper.

2.ElectricitymarketsinTheNetherlands

InTheNetherlandsmostoftheelectricityisstilltradedinthe bilateral market, where the generation companies sell the electricity directly to large consumers, traders and supply companies.Theremainingelectricitygeneratedistraded inone ofthetwospotmarkets:theday-aheadandintra-daymarkets.For balancingpurposesalsoadedicatedmarketexists,managedbythe Dutchtransmissionsystemoperator(TSO)TenneT.Theday-ahead andintra-daymarketshavedistinctdimensions.For2011,about 40TWhofelectricityweretradedintheday-aheadmarketandless than 1% of that value, 278GWh, were traded in the intra-day market [19]. TheNetherlands hasbeen identified as“themost promising[electricitymarket]formassstorage”[18].

2.1.Day-aheadmarket

TheDutchday-aheadmarketisactiveeverydaypriortotheday ofoperationandclosesatnoon.Thismarkethasanhourlytime unit.Unlessstateddifferently,inthispaperweusepricedatafrom 2014.Forthis year,wecalculatedthemeanpriceofenergy per MWhfortheDutchday-aheadmarket:41.18s/MWh.Fig.1depicts the averageprices for 2014 and both day-aheadand balancing market.Itispossibletoobservetheweekendvariationintheday- aheadmarketinthelasttwodays,wherepricestendtobelower thanduringtheweekdays.

2.2.Balancingmarket

Balancing marketsare volatile, and are used tobalance the unattended mismatch between generation and load. In The Netherlands,thebalancingmarket,alsocalledimbalancemarket, workswithatimeunitof15min.Thisunitisalsocalledprogram timeunit(PTU).ThismarketismanagedbyTenneT,thenational transmissionsystem operator (TSO). TheTSO triestoavoidthe mismatchmentioned asmuch as possiblebysharingbalancing responsibilities with balancing responsible parties (BRPs). Each BRP aggregates a partof the consumers and generators in the network.TheBRPssubmittheirdailyzero-sumconsumptionand generation plans ex-ante. Each of these plans include their expectednetenergy exchange withtheotherBRPstothe TSO.

Afterwards,inrealtime,theTSOverifiesifthereisanyimbalance inthesystem.

TherearetwotypesofBRPs,thosespecificallyaskedtoprovide balancing capacity by active contributions (Balancing Service Providers–BSPs)andthoseeitherusingtheimbalancesettlement system for theirown imbalance or being activewithout being selected[20].Bybiddingontheimbalancemarket,eachBRPgives theTSOtheright(butnottheobligation)tobuybalancingenergy.

Loadforecastingisnotexactandenergygenerationforecasting withincreasingintegration ofvariablerenewable-basedgenera- tionis hardertoachieve.Thus, thebalancingmarketisusedto solvetheseunexpectedvariations,bytradingflexibility.Tradition- ally,thiswasachievedbyincreasingordecreasinggeneration[21].

Recently, whenever available, also demand side response and energystoragemaybeused[21],aslongasthetechnologiesused cancopewiththeresponsetimerequiredbythesystemoperator.

The Dutch imbalance market has4 possible modes: down- wards,upwards,upwards/downwardsandnocontribution,which

wewilldenoteby1,1,2and0,respectively(seeTable3).These modesarecalculatedbytheTSOintherealtime.Inmode1there isanexcessofpowerinthesystem.Thisexcessofpowerisalso called“long”andrequiresdownwardregulation.Inmode1thereis alackofpowerinthesystem.Thislackof powerisalsocalled

“short” and requires upward regulation. In mode 2 there are periodsofbothexcessandlackofpowerinthesystemwithinthe timestepof15min,whileinmode0thereisnoimbalance.

Based ondatafrom2014,wecalculatedtheaveragepriceof energy per MWh for the Dutch balancing market: for upward regulation (mode 1) it is 38.31s/MWh and for downward regulation(mode1)itis11.12s/MWh.Thepricesinthismarket varyduringtheweekasshowninFig.1.Fig.2showsthefrequency of2014pricesforbothday-aheadandbalancingmarkets(upward anddownward).Thetwofiguressuggestthatthepricesvarymore inthebalancingmarketthanintheday-aheadmarket.

Astheyearlybenefitofthedualenergystoragesystemdepends onthepricefluctuationsinbothmarkets,wehavecalculatedthe historicalpricevolatilityoftheprices,whichisameasureofprice fluctuationsobservedoveragiventimeperiod(e.g.hourly,daily, weeklyoryearly) [21,22],seealsoAppendixA for detailsofits

calculation. Fig. 3 shows the 2014 price volatility towards the previoustimeslotof thesameday(onehour inthedayahead marketandonePTUinthebalancingmarket)forthethreetypesof pricesofthetwomarkets.Clearly,thebalancingmarketismore volatilethantheday-aheadmarket

3.ModelBackground

Inthissectionweintroducethebackgroundofourmodel.The two energy storage technologiesconsidered are a highenergy (bulk)andahighpowertechnology,tradingintheday-aheadand balancingmarkets,respectively.

Themodelisbuiltfromthepointofviewoftheownerofthe energy storagesystem, withthe goalof maximizingthe yearly benefit.Theday-aheadmarket isused toperformenergy price arbitrageand thebalancingmarketis usedtoprovideancillary servicesupport.

Fig.4depictstherelationshipbetweentheconsideredmarkets andthetwotypesofenergystoragedevices.Wehavebuilttwo partialmodels,eachofthemdescribingthebehaviourofoneof thesedevices(seeSection5fordetails).

Table1

Mainsymbolsusedinthispaper,theirmeaning,andunits(ifapplicable).

Symbol Description Unit

nX numberofelementsofanysetX

J setofelectricitymarketsandofenergystoragedevices,J={1,2}

j electricitymarket/storagedevice,elementofsetJ

D setofdaysunderanalysis

d day,elementofsetD

Tj setoftimestepsformarketjwithinaday^a

tj elementofTj;timestepformarketjwithinaday

M^j setofpossiblemodesformarketj,M¹={0},M²={1,0,1,2}

m^j,d,tj modeofmarketjondaydandtimesteptj,elementofsetM^j k(d) daytype;indicationofworkingdays/weekends,k(d)2{1,2}

h^j;kð_B ^dÞ relativebuyingthresholdformarketj

anddaytypek(d),h^j;kð_B ^dÞ2½0;1

h^j;kð_S ^dÞ relativesellingthresholdformarketj

anddaytypek(d),h^j;kð_S ^dÞ2½0;1

u vectorofbuyingandsellingpricethresholds(tobeoptimized)

Z(u) yearlybenefit

whensetofthresholdsuisadopted s

u^* vectorofoptimalbuyingandsellingpricethresholds

maximizingZ(u) q^j;d;tj;m

j;d;tj S

energyquantitysoldinmarketj,

modem^j,d,tj,daydandtimesteptj MWh

q^j;d;tj;mj;

d;tj B

energyquantityboughtinmarketj,

modem^j,d,tj,daydandtimesteptj MWh

p^j;d;tj;mj;

d_;tj S

sellingenergypriceinmarketj,

modem^j,d,tj,daydandtimesteptj s/MWh p^j;d;tj;m

j;d;tj B

buyingenergypriceinmarketj,

dayd,timesteptjandmodem^j,d,tj s/MWh

p^j;_B^{d minimumbuyingpriceformarketjonday}d _s/MWh

p^j;_S^{d minimumsellingpriceformarketjonday}d s/MWh

x^{j;d;tj;mj;d;tj} stateofchargeofdevicejondayd,timesteptjandmodem^j,d,tj MWh q^j;d;tj;mj;

d;tj D

quantityofenergydischargedbydevicej,

ondayd,timesteptjandmodem^j,d,tj MWh

q^j;d;tj;mj;

d_;tj C

quantityofenergychargedbydevicej,

ondayd,timesteptjandmodem^j,d,tj MWh

h^j_D dischargingefficiencyofdevicej,h^j_D2½0;1

h^j_C chargingefficiencyofdevicej,h^j_C2½0;1

q^j_C; max maximumamountofenergythat

devicejcanchargeinonetimestep MWh

q^j_D; max maximumamountofenergythat

devicejcandischargeinonetimestep MWh

x^j

min

minimumstateofchargeofdevicej MWh

x^j max maximumstateofchargeofdevicej MWh

aTypicalvalues:24timeunitsforanhourlymarket,96timeunitsformarketwithquartersofhourasabasictimeunit.

Forthesakeofsimplicity,inourmodelweassumebothperfect electricitypriceforecastandapricetakerapproach,basedontwo assumptions:

Thestoragesizeisnotbigenoughtomodifymarketprices[23].

There is a perfect forecast window, more or less extended accordingtothestudy[23].

These two assumptions are very standard when analysing potential profitabilityof energystoragesystems in amodelling framework.

4.Modelformulation

Our goalis tofindtheoptimal chargeanddischargerelative priceboundaries,perdevicetypeandday,sothattheyearlybenefit obtainedismaximized.Mathematicallyspeaking,theproblemof finding the optimal strategy for the energy storage system operation,composedofthefinitesetJofstoragedevices,canbe formulated as an optimal control problem. For the sake of simplicity, we assume that each device hasa uniquetype and thatthistypeuniquelyidentifiesthetypeofmarketitisusedfor.² ThispaperconsidersJ={1,2},wherej=1identifiesbothDutchday- aheadelectricitymarketandbulkenergystoragedevice,whilej=2 identifiesbothDutchbalancingelectricitymarketandhighpower energystoragedevice.Forclarificationonthemeaningofthemain variablesusedinourmodel,pleaseseeTables1and2.

Mathematicalformulationoftheoptimalcontrolproblemdealt withinthispaperreadsasfollows:

u¼arg max

u2U ZðuÞ; ð1Þ

where the yearly benefit Z(u) for the vector of relative price thresholdsu¼ðh¹_B^;1;h¹_S^;1;:::;h²_B^;2;h²_S^;2Þisdefinedas

X

j2J

X d2D

X

t_j2Tj

X

m^j;d;tj2M^j

ðq^j;_S^d;tj;mj;d;tj

p^j;d;tj;mj;

d;tj

S q^j;d;tj;mj;

d;tj

B

p^j;d;tj;mj;

d;tj

B Þ; ð2Þ

withrespecttoEqs.(3)–(26).HereU=([0,1])^2nK

nJandh^j_B^;kðdÞand h^j_S^;kðdÞ belongtotheset[0,1]andarerelativebuyingandselling Fig.1.Averageweeklypricesforbothday-aheadandbalancingmarkets2014.

Table3

BalancingmodesinTheNetherlands,basedon[20].

Balancingmode(m^2,d,t2) 1 0 1 2

Condition Long No Short Bothlong

(Downward) Imbalance (Upward) andshort Table2

Main symbols used in this paper, their meaning, and units (if applicable), continuationofTable1.

Symbol Description Unit

l^{2;d;t2;m2;d;t2 energyreceivedbydevice2,}transferredfrom

device1,ondayd,timestept2andmodem^2,d,t2 MWh g^{1;d;t1;m1;d;t1 energyreservedbydevice1tobe}transferred

todevice2ondayd,timestept1andmodem^1,d,t1 MWh f^{1;d;t1;m1;d;t1 energyreservedondevice1andnotusedto}

supplydevice2ondayd,timestept1andmodem^1,d,t1 MWh s^j;_tj^d_t

j1

historicalpricevolatilityinmarketjfordayd andtimedifferencetjt_j1

v^j;tj^dt^;tjj1

pricereturn,

ratiobetweenpricesattimesteptjand attimesteptj1,formarketjanddayd v^j;d meanpricereturninmarketj,dayd c^j minimumpaybackperiod

forthetechnologyjunderanalysis years

r^j powerratingofdevicej kW

m^j costperunitofpowerofdevicej s/kW

e^{j energyratingofdevicej kW}

j^j costperunitofenergyofdevicej s/kWh

o^j yearlyfixedoperation s

andmaintenancecostsfordevicej

k^j variableoperationandmaintenance s/kWh

costsfordevicej

i internalrateofreturn %

Y totalnumberofyears years

y elementofY years

*PleasechecknotezinTable1.

2However,thisassumptiondoesnotchangethemainideasbehindthemodel andcanbeeasilyrelaxed.

thresholds,respectively. The real pricethresholds

p

^j_B^;d and

p

^j_S^;d, which are the maximal buying and minimal selling prices, respectively,arecalculatedasfollows:

p

^j;_B^d¼ P

tj2Tj

P

m^j;d;tj2Mjp^j;d;tj;mj;

d_;tj B

n^m_T^j;d;tj

j

ð1h^j_B^;kðdÞÞ; ð3Þ

p

^j_S^;d¼ P

tj2Tj

P

m^j;d;tj2Mjp^j;d;tj;m

j;d;tj

S

n^m_T^j;d;tj

j

ð1þh^j_S^;kðdÞÞ; ð4Þ

where kð

d

Þ¼ 1; if modð

d

_;7Þ2f1;2;3;4;5g; 2; otherwise:

Market1isalwaysinthesamemode0,i.e.,M¹=0,whileM²={1, 0, 1,2}(seeTable3foroverviewofthesemodes).Inmodem^2,d,t2=2 onlyone action (buyingor selling)is allowed for device2. We assumethatinsuchasituationdevice2sells,becausesellingis moreadvantageousfortheenergystorageowner,asshownin[17].

Duetotheefficiencylossesincharginganddischarging,onehasto buymoreelectricitythanitcanbephysicallychargedintoadevice and, similarly, one has to discharge more electricity than the amountofenergysold:

q^j;d;tj;mj;

d;tj

B ¼q^j;d;tj;mj;

d;tj

C

h

^j_C ^{; q}

j;d;tj;m^j;d;tj

S ¼q^j;d;tj;mj;

d;tj

D

h

^j_D; ð5Þ

where

h

^j_D2½0;1 and

h

^j_C2½0;1 refer to the efficiencies of dischargingandcharging,respectively,andareknownapriori.

Fig.3.Volatilitytowardsprevioustimeslot(hourorPTU,dependingonthemarket)forthe2014day-aheadandbalancingmarketprices.

Fig.2.Pricehistogramfortheday-aheadmarketpricesandbalancingmarketupwardanddownwardpricesfor2014.Eachbinhasarangesizeof2s.

Nodevicecansimultaneouslychargeanddischargeelectricity and the amount of electricity charged and discharged cannot exceeditsprespecifiedboundaries,i.e.,

q^j;d;tj;mj;

d;tj

C

q^j;d;tj;mj;

d;tj

D ¼0; q^j;d;tj;mj;

d;tj

C 2½0;q^j_C; max ;

q^j;d;tj;mj;

d;tj

D 2½0;q^j_D; max :

ð6Þ Aseachmarket/devicejcanonlybeinonemodem^j,d,tjonday

d

and time step tj, we set quantities of electricity charged and dischargedandtheirbuyingandsellingpricesforallotherbutthe currentmode,sameday

d

andtimesteptj,tozero:

q^j;_C^d;tj;b¼0 8b2M^jfm^j;d;tjg; if q^j;d;tj;mj;

d;tj

C >0;

Modem^j,d,tjandelectricitypricesforeachmarketj,day

d

andtime steptjareexogenousandassumedtobeknownapriori.

Ourmodeltakesadvantageofanylowerpricesintheday-ahead marketwhencomparedwiththebalancingmarket,bytransferring energyfromdevice1todevice2.Asdevice2hasatimestepof 15min,twocompletecyclesofcharginganddischargingmaybe performedinonehour.

Inordertoavoidanyinconsistency,sincedevice1anddevice2 are used in our model with different time steps, the energy transferablefromonedevicetoanotherisreservedapriori.This reservationis performedeveryhour, which is the time stepof device1andlargerofthetwotimesteps.Eq.(11)determinesthis energyreservedindevice1transferabletodevice2.Asdevice1is much bigger than device2, device 1 can providea temporary additionaloutputtochargedevice2whenneeded:

g

^{1;d;t1;m1;d;t1}

¼

2q²_C; max

h¹_Dh²_C ^{;ifx1;d;t11;m}

1;d;t11

þq^1;_C^d;t1;m1;d;t1 2q²_C; max

h¹_Dh²_C ^{þx1 min ;} 0; otherwise:

8>

>>

>>

><

>>

>>

>>

:

ð11Þ Theenergytransferwillonlyhappenwhendevice2isnotbeing usedinmarket2inthecurrenttimestept2,asstatedin(12).Also,

device2canonlyreceiveenergyifitispartiallyorfullydischarged.

Ifso,device2willreceivetheenergyfromdevice1(

l

^{2;d;t2;m2;d;t2}).

Thisamountwillbetheloweroftwovalues:maximumquantity q²_C; max charged by device 2 or the energy

x^{1;d;t1;m1;d;t1}x¹

min

h¹_Dh²_C thatcanbetransferredtodevice2from device 1. This transfer can occur in every 15min. For each t22{4t13, ...,4t1},

l

^{2;d;t2;m2;d;t2}

¼

min ðq²_C; max ;x² max x^{2;d;t21;m2;d;t21}Þ;

if g^{1;d;t1;m1;d;t1}6¼0;

q^2;_C^d;t2;m2;d;t2¼0 and q^2;_D^d;t2;m2;d;t2 ¼0; 0; otherwise:

8>

>>

>>

><

>>

>>

>>

:

ð12Þ The amount of energy reserved in device 1 not transferred to device2isdefinedasForeacht1=1,2, ...

f

^{1;d;t1þ1;m1;d;t1þ1}¼

g

^{1;d;t1;m1;d;t1} P4

i¼1

l

^{2;d;ðt11Þ4þi;m2;d;ðt11Þ4þi}

h

¹D

h

²_C ^{: ð13Þ}

Herealsothelossesofdischargingdevice1andchargingdevice2 aretakenintoconsideration.Thecurrentstateofchargeofdevices 1and2dependsontheirstateofchargeintheprevioustimestep:

x^{1;d;t1;m1;d;t1} ¼x^{1;d;t11;m1;d;t11}þq¹_C^{;d;t1;m1;d;t1}q¹_D^{;d;t1;m1;d;t1}

g

^{1;d;t1;m1;d;t1}þ

f

^{1;d;t1;m1;d;t1}; ð14Þ

x^{2;d;t2;m2;d;t2} ¼x^{2;d;t21;m2;d;t21}þq²_C^{;d;t2;m2;d;t2}

q²_D^{;d;t2;m2;d;t2}þ

l

^{2;d;t2;m2;d;t2}: ð15Þ When it is not possible to charge or discharge quantities q¹_C; max orq¹_D; max ,thedevicewillchargeor discharge as much as possible, given by x¹ max x^{1;d;t11;m1;d;t11} and x^{1;d;t11;m1;d;t11}x¹

min , respectively.

As most energy storage devices cannot be fully discharged, x^j

min representstheminimumusefulstateofcharge, thelowestchargelevelthestoragedevicecanbedischargedto.

Quantityq^1;m_D ^1;d;t1;d;t1isdefinedasfollows:

q^1;_D^d;t1;m1;d;t1

¼

min ðq¹_D; max ;x^{1;d;t11;m1;d;t11}x¹

min Þ;

and p^1;_S^d;t1;m1;d;t1p^1;_S^d_;

0; otherwise: 8>

<

>:

ð16Þ Asdevice1startsdischarged,Eq. (16)isonlyvalidfort12 or when

d

2.Likewise,q^1;m_C ^1;d;t1;d;t1isdefinedasfollows:

q^1;_C^d;t1;m1;d;t1

¼

min ðq¹_C; max ;x¹ max x^1;d;t11;m^1;d;t11Þ;

if p¹_B^{;d;t1;m1;d;t1}p¹_B^;d; 0; otherwise:

8>

<

>:

ð17Þ If device 2 cannot charge the maximum quantity of energy charged q²_C; max , as it would overrun the maximum amount of charge x² max , it will charge quantity x² max x^{2;d;t21;m2;d;t21}. Similarly, when discharging, if Fig.4. Illustrationoftherelationshipbetweentheconsideredenergymarketsand

energystoragedevices.Thearrowsindicatethepossibleenergytransferdirections.

Thenumbers1and2identifythetwopartialmodels.

device 2 cannot discharge the maximum quantity of energy dischargedq²_D; max ,asitwouldoverrunthefunctional minimumamountof chargex²

min , itwill chargethe quantityx^{2;d;t21;m2;d;t21}x²

min . q^2;_D^d;t2;m2;d;t2

¼

min ðq²_D; max ;x^{2;d;t21;m2;d;t21}x² min Þ;

if m^2;d;t22f1;2g;

p²_S^{;d;t2;m2;d;t2}p²_S^;d_; 0; otherwise:

8>

>>

<

>>

>:

ð18Þ

q^2;_C^d;t2;m2;d;t2

¼

min ðq²_C; max ;x² max x^{2;d;t21;m2;d;t21}Þ;

if m^2;d;t2¼1; and p²_B^{;d;t2;m2;d;t2}p²_B^;d_; 0; otherwise:

8>

>>

<

>>

>:

ð19Þ Asdevice2startsdischarged,Eq.(18)isonlyvalidonlyfort22or when

d

2.

Themaximumdischargecapacityq¹_D;maxofdevice1isloweror equaltoitsmaximumstateofchargex¹ max minusthe transferableenergytodevice2^2q

2

C; max

h¹_Dh²_C :

q¹_D; max x¹ max

2

q²_C; max

h

¹_D

h

²_C ^{: ð20Þ}

Otherconstraints:

q^j_C;maxx^j max ; ð21Þ

q²_D; max x² max ; ð22Þ

q^j

D; min ¼q^j

C; min ¼0; ð23Þ

0x^j

min x^j max : ð24Þ

Initialconditions:

x^j;1;0;mj;1;0¼0; ð25Þ

f

^1;1;1;m1;1;1¼0: ð26Þ

In words,theproblem (1),subjectto(2)–(26),is tofindthe sellingand buyingthresholdprices perdayin aweek,for each device type/market, so that the revenue of the entire storage systemdeviceismaximized.

5.Implementation

In thissectionwedescribetheimplementationofthemodel andoptimalpricethresholds,givenby(1)–(26).Thetechnicaland economicaldataonthedevicesaretakenfrom[11].Allcasestudies definedbyequationswereimplementedusingMatlab¹.

The problem (1)–(26) could not be solved by standard optimization techniques, such as gradient-based optimization

methods, due to many local minima and even regions in the domainof theyearly benefit functionwhich correspondtothe samebenefitvalue.Afterexperimenting withheuristic solution methods, such as particle swarm optimization (PSO),we have adoptedapatternsearch (PS)algorithm includedin theMatlab optimizationtoolbox forsolvingthe problem.The mainadvan- tagesofPSareitsspeedandthefactthatitdoesnotusegradient approximationtomaximizetheprofitfunction.Therefore, PSis oftenusedformaximizingcomplicatedfunctionswhicharenon- smoothorevendiscontinuousand/orhavemanylocalminima.The method was first proposed in the literature by [25] and is extensivelydescribedin[26].

Algorithmforsolvingtheproblem(1)–(26)isdepictedinFig.5.

Forcomparisonpurposeswealsocalculatedtheresultswhenusing a single weekly set of price thresholds per device, using the algorithmdepictedinFig.6.

Weassumethatbothdevicescanbefullydischarged(xj,min=0) and start discharged (x^j;1;0;mj;1;0¼0). No self-discharge was consideredasthedevicesareworkingalmostcontinuously.Ramp rateswerealsonotconsidered.

6.Results

6.1.Findingoptimalpricethresholds

Wehavecalculated thesolutiontotheproblem(1)–(26)for year2014using thepattern searchintroducedin Section3.For 2014,theoptimalu^*containspricethresholdsdepictedinFig.7.

Thegenericbulkenergystoragepowerratingisvariedbetween 50 and 150MWfora dischargeduration of 5–10h. Theenergy ratingvariesbetween500and1500MWh.Thepowerratingisthe maximum amount of energy that the device can charge or dischargeinonehour.Unlessmentionedotherwise,theroundtrip efficiencyis80%=0.80,andthecharginganddischargingefficien- ciesareequalto ffiffiffiffiffiffiffi

0:8

p ( 0.8944=89.44%).Theelectricity market datausedtoevaluatetheprofitabilityofthisdeviceisDutchday- aheadmarketdataprovidedbythespotmarketAPX-Endex.

Forthehigh-powerdevice,thepowerratingisvariedbetween 20and60MW.Thenominaldischargedurationisof15minutes.As energy rating of the device is calculated as its power rating multipliedbyitstimeofdischargeofthatsamedevice,theenergy ratingwillvaryfrom5to15MWh.Therefore,thisenergyrating interval [5,15] MWhwas selected asthe basepowerratingof device2.TheminimumbidsizefortheDutchbalancingmarketis5 MW [20] per PTU of 15 minutes. The default charging and dischargingefficienciesofthisdeviceareboth95%(0.95),which results in a round trip efficiency of around 90% (0.90). The balancingmarket dataispubliclyavailable fromtheDutchTSO TenneT[28].Thedimensionsandefficiencylevelsofbothdevices arechosenaccordingtotechnologiesdescribedin[11].Fig.7shows thesolutiontotheproblem(1)–(26)whentwosetsofthresholds areapplied.Thesizeofthebulkandthehighpowerdevicesinthis exampleare50MW,500MWhand20MW,5MWh,respectively.

There, different results can be observed for weekdays and weekends.Fortheday-aheadmarketand bulkdevice,weekend thresholdsareusuallylower.Forthebalancingmarketandhigh power device, the difference is bigger. In this case, the yearly benefitisthehighestwhenthehighpowerdeviceisusedtobuy lessintheweekendsandwhenitsellsmoreduringtheworking daysthanintheweekends.

However,itwasrealizedthattheimprovementtowardsasingle setof thresholdswaslimited(0.21.1%)and thecomputational timewas atleastdoubled.Thisis possiblyduetotheweekend pricesbeingusuallylowerthantheonesoftheworkingdaysand thattheirvolatilityislowerthanvolatilityofthepricesforworking

Fig.6.Representationofthealgorithmsolvingtheproblem(1)–(26)withthesameweekendandworkingdayspricethresholds.

Fig.5. Representationofthealgorithmsolvingtheproblem(1)–(26).