Lifetime Expectancy of Li-Ion Batteries used for Residential Solar Storage

Abstract

This paper analyses the degradation that is experienced by different types of Li-ion batteries when used as home solar storage systems controlled to minimize the electricity bill of the corresponding household. Simulating the annual operation of photovoltaic (PV) residential systems with batteries at different locations was undertaken to perform the study and it uses actual consumption values and real PV production profiles, as well as validated semi-empirical ageing models of the batteries. Therefore, the work provides a realistic prognosis around the lifetime expectancies for the different Li-ion chemistries.

Full text

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Energies2020,13,568;doi:10.3390/en13030568www.mdpi.com/journal/energies
Article
LifetimeExpectancyofLi‐IonBatteriesusedfor
ResidentialSolarStorage
HectorBeltran*,PabloAyusoandEmilioPérez
DepartmentofIndustrialSystemsEngineeringandDesign,JaumeIUniversity,SosBaynatAvenue,
CastellódelaPlana,12071Castelló,Spain
*Correspondence:hbeltran@uji.es;Tel.:+34‐964‐72‐81‐78(H.B.)
Received:21November2019;Accepted:21January2020;Published:24January2020
Abstract:ThispaperanalysesthedegradationthatisexperiencedbydifferenttypesofLi‐ion
batterieswhenusedashomesolarstoragesystemscontrolledtominimizetheelectricitybillofthe
correspondinghousehold.Simulatingtheannualoperationofphotovoltaic(PV)residential
systemswithbatteriesatdifferentlocationswasundertakentoperformthestudyanditusesactual
consumptionvaluesandrealPVproductionprofiles,aswellasvalidatedsemi‐empiricalageing
modelsofthebatteries.Therefore,theworkprovidesarealisticprognosisaroundthelifetime
expectanciesforthedifferentLi‐ionchemistries.
Keywords:Li‐Ionbatteries;residentialPVstorage;calendarageing;cycleageing.
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1.Introduction
PVtechnologyhasbecomethemostimportantpowergenerationsourceworldwideintermsof
addedcapacityperyearsince2016,overtakingwindpower,whichwastheleadingtechnologyupto
then[1].Inall,newPVinstallationsrepresentedapproximately100GWin2018,achievingan
accumulatedcapacitythatgoesbeyond500GW[2].Thisevolutionhasbeenpossiblethankstothe
hugepricedecreaseexperiencedbythisindustryduetotheeconomiesofscaleinthelastdecade.In
accordance,thecostforresidentialgrid‐connectedPVsystemshasworldwide‐averageddropped
downtoaround1.28€/W,beingapproximately20%higherthaninEurope,where1.13€/Wcanbe
generallyfound,and30%higherthaninAustralia,wherepricesplummetedto0.95€/W[3].Thanks
toit,theresidentialmarkethasalsoexperiencedabigdeploymentofPVinstallationsmainly
envisagedforself‐generation,whichachievesveryhighsharesoftheenergy
production‐consumptionincertainlocallowvoltagegrids(Figure1).
However,largeamountsofbehind‐the‐metersolarinstallationsthatproducestochastically
intermittentenergycanimplystabilityproblemsatthelowvoltagegridlevel[4].Hence,new
technicalandregulatorysolutionshavetobeimplementedtoavoidrunningintotheproblemof
systematicallyhavingtocurtailpartoftheirproduction.Theintroductionofenergystorage(ES)
systems,usuallybatteries,withinthehouseholds[5–7].Batteriesforenergystorageinbuildings
havebeenaroundforalongtimeinbothstand‐alone(off‐grid)andcommercialback‐uppower
systems(UPS).However,evenwiththestillrelativelyhighcostofbatteries,theircombinedusewith
solarsystemshasalsostartedtogainmomentumoverthelastfewyears.Thisisbecausesuch
systemsnotonlyallowforreducingtheintermittencyofthelocalproduction,stabilizingthelow
voltagegrid,butalsomakeitpossibletooptimizetheenergybillfortheinstallationownersand
evenprovideextendedsupplysecurity,whichcannotbetakenforgrantedeverywhere.Inthisway,
somespecializedconsultantsindicatethatoutofthe4GWofbatterieswithan8GWhcapacitytobe
deployedgloballyin2019(withcalculationstoachieve15GWwitha44GWhcapacityin2024and
between180and420GWhin2030),morethanhalfcouldbedistributed[8],i.e.,behindthemeterPV
residentialsystemswithbatteries,alsoknownashomesolarstoragesystems.Amongthem,these

Energies2020,13,5682of18
areclearlysettospreadacrossEurope.Italyseems,inthissense,tobeconsolidatingitspositionas
thesecond‐largestmarketafterGermanyandSpainiswellpositionedtodevelopitsownsizable
market,althoughthisisalwaysverydependentonthechangingpoliticalframeworkofthecountry.
FranceandtheUnitedKingdomkeepbeingpromisingmarkets,althoughtheyhavenotclearly
emergedyet[9].BeyondEurope,theUSAisaconsolidatedmarket,mainlyinstates,suchas
California,andalsoAustraliaisstartingtotakeoff.Notehow,inthelatter,thereisgreatincentiveto
storesolarenergyasthesolarfeed‐intariffhasbeenlatelyreducedtoaslittleas5¢perkWh
(¢—cent),whilethecosttopurchaseelectricityiscloserto30¢perkWh.Thishasbecomeadriving
forceandgreatincentivetostoresolarenergy,ratherthansendittothepowergridforlittlereturn.

Figure1.CommonwealthGamesAthleteʹsVillageinGlasgow(Scotland),bySolarTradeAssociation
(CCBY‐SA2.0).
Ascanbeunderstoodinsuchacontext,agreatbunchofbehindthemeterPVresidential
systemswithbatteriesarecommerciallyavailable[10]andtheyhavebeenlargelydiscussedand
analyzedintheliterature.However,mostofthepreviousacademicworksarefocusedon
optimizationsthatarerelatedtothesizingofthebatterysystem[11–13]ortomaximizingthe
economicincomeofthePVinstallations[14–17].However,fewamongthepreviousworkshave
alreadytakenthedegradationofthebatteriesduringthelifespanofthesystemintoaccount.
Degradationevolutionisveryimportantfortheeconomicanalysisofthesesystems,whicharestill
costly,asalreadyindicated,andwhoseprofitabilityisstillunderdiscussion[18].Infact,degradation
isalreadyakeyparameterintheprofitabilityanalysesofbatteriesbeingproposedinapplications,
suchasrampratecontroloflargerenewablepowergenerationplants[19,20],ancillaryservices
[21,22],energyarbitrage,andpeakshavingprovidedbylargeenergystoragesystems[23,24],and
evenforelectricvehicleintegration[25].Amongthosepublishedfortheresidentialsolarstorage
sector,somecanbehighlighted.Forinstance,theauthorsin[26]analyzetheimpactoftheinternal
batterypackcircuitdesignonthesystemoperationanditscorrespondingdegradation.Another
approachisproposedin[27]andin[28],inwhichforecastingmethodologiesareincludedto
optimizetheoperationofthesystemwiththeaimtoenhancethebatterylifetime.Finally,an
economicoptimizationofthebatteriessizingtakingintoaccounttheageingofthebatteriesis
performedin[29].However,theyuseeithersimplifiedbatterydegradationmodels[27,28]orgeneric
onesthattheyadapttodifferenttypesofbatteriesbeingvalid,evenforleadacidmodels[29].
Thisworkassumesthatthesizingandcontrolofthesystemisamaturedomainnowadaysand
itfocusesontheanalysisofdegradationthatthedifferentLi‐ionbatteriesusedinsuchan
applicationwouldexperienceunderawiderangeofoperationalconditions.Theanalysisisbasedon
well‐knownaccepteddetaileddegradationmodelsforthespecificchemistriesunderdiscussion,
whichhavealsobeenadaptedheretoactualstate‐of‐the‐artcommercialbatterypacks,whichallows
forprovidingaccurateandrealisticlifetimeexpectanciesforthoseparticularchemistriesincludedin

Energies2020,13,5683of18
currentcommercialproducts.Therefore,theresultsofthisworkareavaluablereferencefor
analyzingtheprofitabilityofthesestoragesolutions.
Thestructureofthispaperisasfollows.Section2isdevotedtothedefinitionofLi‐ion
chemistriestobeusedinresidentialPVstorage,totheintroductionoftheircorresponding
degradationmodels,andtodescribethemodeofoperationofthebatteriesthataredesignedto
minimizetheoperationalcostofthesystem.Subsequently,Section3summarizesthedegradation
resultsexperiencedbythebatteriesaccordingtooneyearlongsimulationsthatwereperformed
whileusingMATLAB®(Natick,MA,USA)andthediscussionontheexpectedlifetimeestimations.
Finally,someconcludingremarksareintroduced.
2.Methods
ThecombineduseofLi‐ionbatterieswithsolarPVsystemsisgainingmomentum,becausesuch
systemsnotonlyallowreducingtheintermittencyofthelocalproduction,butalsoprovideaboost
inthehandlingoflocalsolargeneration 𝑃 , i.e.,increasingtheamountofself‐productionthat
becomesself‐consumed[30].Moreover,solarstoragesystemsmakeitpossibletooptimizethe
energybillfortheinstallationownersbypeakshaving[5],[24]orbyprofitingatime‐of‐use
electricityratestructureforenergyarbitrage[31,32].Thisis,ascoupledwithsolar,batterysystems
canstoretheexcessgenerationandallowthecustomertodispatchthestoredenergyduringpeak
loadhourswhenelectricityismoreexpensive.Inthecomingfuture(alreadyarealityintheso‐called
VirtualPowerPlants,VPP)additionalrevenuescanbepassedontothebatteryuserswhenthe
systemiscentrallycontrolledbytheVPPoperatorandtheenergythatisstoredinthebatteriesis
accordinglydispatchedtoprovidegridsupportforhomeownersthatareinterestedinsharingthe
resourcewiththeutility.
Nonetheless,amongthedifferentpotentialapplicationsthataredefinedforhomestorage
systems,thesearemainlyinstallednowadaysforenergyarbitrageandpeakshaving(alsoback‐up
serviceincertainlocationssuchasCalifornia).Therefore,thisworkanalyzesthelifetimeexpectancy
thatisassociatedwithtwodifferentLi‐ionchemistrieswhenusedundersuchcontrolstrategy.Todo
so,thedifferentLi‐ionchemistriesarereviewedinthefollowingandsomearehighlighted.The
variousdegradationmodelsusedintheanalysisarethenintroduced.Finally,theoptimization
introducedcontrollingtheoperationofthehomesolarstoragesystemispresented.Theoperationof
differenthomestoragesystems,intermsofpowerandenergyratings,willbethensimulatedinan
annualcontextatdifferentlocations(radiationpatterns)andwithindifferenthouseholds(load
patterns).
2.1.Li‐ionBatteriesandtheirDegradationModels
Nowadays,thegenericlabelLi‐ionbatteriescoveruptosixdifferentbatterychemistrieswhile
usingsometypeoflithiumalloyintheirelectrodes.Thesecorrespondto:LithiumCobaltOxide
(LiCoO2),LithiumManganeseOxide(LiMn2O4),LithiumIronPhosphate(LiFePO4),LithiumNickel
CobaltAluminumOxide(LiNiCoAlO2),LithiumNickelManganeseCobaltOxide(LiNiMn‐CoO2),
andLithiumTitanate(Li4Ti5O12).ThevariousLi‐ionfamiliespresentdifferentpropertiesand
characteristicsasafunctionoftheirinternalstructureandcomposition[33],whichinvolvevarying
capabilitiesintermsofspecificpower(W/kg),specificenergy(Wh/kg),thermalstability,cyclability,
andperformance.Theyalsodifferincostandlifetimeexpectancy.
Dependingontheapplicationandtheassociatedbatteryrequirements,adifferentchemistry
willbeofchoice.Forhomestorageapplications,wherebatterieshavebeendefinedtobemainly
usedtomaximizethePVproductionexploitationwhilereducingtheeconomiccostoftheelectricity
consumptioninthehouseholdviaenergyarbitrageorpeakshaving,safety,andspacelimitations
aregenerallyassumedtobethemostdefiningorlimitingparameters.Inthiscontext,LiFePO4(LFP)
andLiNiMn‐CoO2(NMC)cellsconstitutethebatteriesbestfittingtheseconstraints,whichcanbe
confirmedbyanalyzingthedifferentmodelscommerciallyavailablenowadays,asinFigure2.


Energies2020,13,5684of18
HomeStorageSystemLi‐ionCells

(1) LGChemRESUNMC
(2) TeslaPowerwall2NMC
(3) VartaPulse6NMC
(4) SENECHomeLi10NMC
(5) AmpereEnergySquareNMC
(6) SonnenECOLFP
(7) BYDB‐BOXProLFP
(8) SimpliphiPHILFP
(9) PylonUS2000LFP
(10) PowerPlusEnergyL.Pr.LFP
Figure2.ListofmainhomestoragesystemsavailableinthemarketandtypeLi‐ioncells
implemented.
Amongthem,althoughLFPbatterieshavebeentraditionallymoreexpensivethantheNMC
ones,nowpriceshavematched.Therefore,vendorsarelookingfavorablyatthemduetotheirlackof
Cobaltasfiresafetyregulationsbecomestricter,eventhoughenergydensityisadrawbackforLFP.
Attendingtothedifferentconsiderationsintroduced,thedegradationandthelifetimeexpectancy
forthesetwochemistriesareanalyzedinthiswork.Inthissense,althoughdifferenttypesofmodels
forpredictingthelifetimeexpectanciesareavailableintheliterature[34–37],semi‐empiricalmodels
(equationsbasedonreallabmeasurements)areconsideredtobethebestapproachintermsof
complexityandreliabilitytradeoffinthiswork,asitisusuallyassumedinstudiesanalyzingthe
degradationoftheLi‐ionbatteriessubjecttorealoperationscenarios[38].Mostofthesemodelstake
differenttypesofstressfactors[34,39,40]thatdegradethecellsviathesocalledcalendarcycle
(associatedtotemperatureandstateofchargeduringstand‐byoperation)orcycleageing(also
associatedtotemperature,andtothenumber,depthofdischarge,andaveragestateofchargeforthe
cycles,aswellastheC‐rate)intoaccount[39,41,42].Amongthedifferentproposalsofsemi‐empirical
degradationmodels,twodifferentmodelsthatdisregardtheC‐ratestressfactorandfocusonthe
othershavebeenusedasreferencemodels.Thisisduetothefactthatmosthomesolarstorage
systemsusuallylimitviasoftwaretheirpowerexchangecapabilitytovaluesbelow1Cthatwould
beacertainlimitbeyondwhichC‐ratebecomesasignificantstressfactor[43,44].Thesemodels,
togetherwiththecorrespondingimprovedmodelsthatareupdatedtostate‐of‐the‐artcellsforeach
ofthechemistries,areintroducedinthefollowing.
2.2.ReferenceDegradationModelforLiFePO
4
Cells
Thesemi‐empiricaldegradationanalysisproposalusedasreferenceorbasemodelinthiswork
toestimatethelifetimeofLFPcellsisthatsuggestedbyStroeetal.[45].Thispaper,whichwas
publishedin2014,proposesamodelthatwasdevelopedforcylindricalbatterycellsasthosefrom
SAFTBatteries,modelVL41M[46].Themodelanalyzesthelossofbatterycapacityduetoboththe
useofthebattery(cycleageing)andthepassofthetimeitself(calendarageing).Thevalueofthe
capacityreduction(C
fade
)thatisassociatedtothelatterisdefinedby:
𝐶

_

𝑇,𝑡  𝛼
_
 𝑒

 𝑡.(1)
where𝛼_and𝛽aretwoparameterswhosevaluesdependontheprecisemodelofcell
beinganalysed,Tisthetemperature(inKelvin),andtisthetime(inmonths).Moreover,the
capacityreductionthatisassociatedtotheoperation(cycling),respondsto:
𝐶

_
𝑇,𝑁𝐶  𝛼
_
 𝑒

 𝑁𝐶.(2)
where𝛼_and𝛽areagainothertwoparameterswhosevaluesdependonthemodelofcell
beinganalysed,andNCrepresentsthenumberofequivalentfullcycles,whicharecalculatedfrom
theannualevolutionofthestateofcharge(SOC)ofthebatteryregisteredwhenoperatedinahome

Energies2020,13,5685of18
storageapplication.ThisSOCevolutionisintroducedtotheRainflowCountingAlgorithm(RFC)
[47],whichiscapableofgroupingthosecycleswithanequaldepthofdischarge(DoD)andaverage
SOC.Afterwards,bymeansofthePalmgren–Minerruleandthecapacityevolutioncurvesofthe
batteriesprovidedbymanufacturers,theNCvaluecanbeobtained,assummarizedinFigure3.

Figure3.Definitionoftheequivalentnumberoffullcycles(NC)fromapartiallycycledstateof
charge(SOC)evolution.
OncetheNCaredefined,itsvalueisintroducedintoEquation(1)and(2)togetherwiththeT
valuesandthetimerangeanalysed.TheresultingCfadevaluesarecombinedtofinallyprovidethe
lifetimeestimationprognosis,inyears,bymeansofthefollowingequation:
EOL% 𝐶
𝐶
year,𝑇 𝐶𝑁𝐶,𝑇𝑦𝑒𝑎𝑟𝑠
(3)
Notehowthisequationtakesintoaccountthatbatterymanufacturersusuallydefinethe
end‐of‐life(EOL)oftheLFPbatteriesasthemomentwhenthecapacityretainedbythecellisequal
toagivenpercentageofitsinitialcapacity(C0)that,stilldependingonthecellmodel,canrangefrom
60%to80%(70%forthecellunderanalysis).Thesolution,inyears,tothisequationistheestimated
lifetimeexpectancyforthebattery.
2.3.ReferenceDegradationModelforLi(NiMnCo)O2Cells
Thesemi‐empiricaldegradationmodelused,asreference,analyzingthelifetimeofNMCtype
cellsisthatproposedbySchmalstiegetal.[48].Notehow,alsointhiscase,theworkpublishedinthe
sameyear2014focusesonthebehaviorofcylindricalcells.Moreprecisely,itisdevelopedforthe
SanyoUR18650Ecell[49].Again,asfortheLFPmodel,thedegradationisanalyzedasthelossof
batterycapacityassociatedtobothcalendarandcycling.Thefirstofthemisdescribedaccordingto:
𝐶
_
𝑉,𝑇,𝑡𝛼

_
 𝑉3.15𝑒
𝑡.(4)
where𝛼_isaparameterthatdependsonthemodelofcellunderconsideration,Visthe
averagedailyvoltage(inVolts),Tisthetemperature(inKelvin),andtisthetime(indays).
Conversely,thedegradationthatisassociatedtotheuseofthebatteryiscalculatedby:
𝐶Q, Ø𝑉,∆𝐷𝑂𝐷𝛼

_
 1.8 Ø𝑉3.667∆𝐷𝑜𝐷0.1862Q.(5)
Itcanbestatedagainthat𝛼_isaparameterthatdependsonthemodelofcellunder
consideration,Qstandsforthechargethroughput(inampere‐hour),
∅
Vistheaveragevoltagefor
eachcycle(inVolts),andΔDoDisthedepthofdischargeofthecycle(inrangeof0–1).Subsequently,
𝑁𝐶  𝐷%∙𝑁𝐶
𝑀𝑎𝑥
100%
𝐷% 𝑁𝑐𝑦𝑐𝐷𝑜𝐷
𝑁𝑚𝑎𝑥𝐷𝑜𝐷
𝐷𝑜𝐷100
𝐷𝑜𝐷1
RFC
Palmgren
Miner
𝑁𝐶  𝐷%∙𝑁𝐶
𝑀𝑎𝑥
100%
𝐷%

𝑁𝑐𝑦𝑐𝐷𝑜𝐷
𝑁𝑚𝑎𝑥𝐷𝑜𝐷
𝐷𝑜𝐷100
𝐷𝑜𝐷1

Energies2020,13,5686of18
theresultingCfadevaluescorrespondinglyassociatedtocalendarandcycledegradationarecombined
tofinallyprovidethelifetimeestimationprognosis(𝑦𝑒𝑎𝑟,inyears),bymeansof:
EOL% 𝐶
𝐶
year,𝑇,𝑉 𝐶𝑄,∅𝑉,∆𝐷𝑜𝐷𝑦𝑒𝑎𝑟𝑠(6)
ThisfinalequationtakesonceagainintoaccounttheEOLofthebatteriesasapercentageofits
C0,alsobeingestimatedatthe70%forthiscell,accordingtomanufacturer’sinformation.
2.4.DegradationModelsforState‐of‐the‐ArtCells
Asindicated,previousdegradationmodelswerepublishedseveralyearsagoandthey
correspondtoLFPandNMCcylindricalcellscommercializedinthefirsthalfofthisdecade.Li‐ion
cellshaverapidlyevolvedlatelyandcurrentcommercialmodelspresentextendedcapabilitiesthat
providemuchimprovedperformances,mainlyintermsoflifespanandwarranty.Thisisduetothe
improvementinboththecurrentinternaldesignofthecells(mostofthemevolvingtowardspouch
cellsfortheseapplications)andintheirarrangementwithinthebatterypacks.Inthisway,new
batterypackspresentbetterthermalmanagementandreducedinternalresistances,whichimply
lowerinternaloperationtemperaturesandreduceddegradationduetocurrentexchanges.
Amongthedifferentcommercialmodelsthatareavailableinthemarket,thisworkfocuseson
thesolutionsbelongingtotwoofthemainglobalenergystorageplayersnowadays:theBYDB‐BOX
ProasLFPsystemandtheLGChemRESUasNMCsystem.Forthecaseofthefirstone,BYDcomes
withoneofthebestwarrantiesonthemarket,withanestimated60%retainedcapacityafter10years
anduptoaround5100fullcycles,ifoperatedinarangeoftemperaturesbetween−10°Cand50°C.
Thiscorrespondstoanoverall23.9MWhenergythroughputcapabilityfortheB‐BOXPro7.5[50].
LGChem,ontheotherhand,implementsNMCpouchcellswithanestimatedminimum60%
retainedcapacityafter10yearsifbeingoperatedwithin−10°Cand45°Candforanoverall20MWh
energythroughputcapabilityforitsRESU6.5batterypack[51],whichimplies4000fullcycles.
Therefore,apartfromthepreviouslyintroducedreferencedegradationmodels,thiswork
considersothertwomodelsthatadaptthetrendingcurvesdefinedthroughEquations(1)–(6)tothe
warrantiesthataredefinedbyBYDandLGfortheircurrentcommercialproducts,B‐BOXand
RESU,respectively.Inthisway,theweightedeffectofthedifferentstressfactorsovertheglobal
ageingofthecellsdemonstratedbypreviousstudiesiskept,butitisupdatedtothecurrent
performancesofthecellsbeinginstalledincommercialhomestoragesystemsnowadays.Theseare
labelledasState‐of‐the‐art(SoA)modelsandtheyhavebeenvalidatedwhileusingon‐the‐field‐data
anddataprovidedbysponsorsofthestudyandprotectedbyaNon‐DisclosureAgreement(NDA).
ThecorrespondingvaluesforthecoefficientsofthedifferentmodelsareintroducedinTable1.
Table1.CalendarandCyclingAgingModels’CoefficientsforthedifferentBatteryTypesanalyzed.
BatteryModel𝜶𝒄𝒂𝒍
_
𝑳𝑭𝑷𝜶𝒄𝒚𝒄
_
𝑳𝑭𝑷𝜷𝒄𝒂𝒍𝜷𝒄𝒚𝒄𝜶𝒄𝒂𝒍
_
𝑵𝑴𝑪𝜶𝒄𝒚𝒄
_
𝑵𝑴𝑪
Ref.LFP
(SaftVL41M)3.087×10−76.87×10−50.05176 0.02715  ‐ ‐
SoALFP
(BYDB‐BOX)1.985×10−7 4.42×10−50.0510 0.02676  ‐ ‐
Ref.NMC
(SanyoUR18650E)‐ ‐ ‐ ‐ 7.54×1064.081×10−3
SoANMC
(LGChemRESU6.5)‐ ‐ ‐ ‐ 3.02×1061.632×10−3
Theresultingglobaldegradationevolutions,asrepresentedascapacityfadepercentage,
togetherwiththepartialdegradationproceduresthatareassociatedtobothcalendarandcycling,
areshowninFigure4forthetwocellswithstate‐of‐the‐artwarranties(thoseoftheRESU6.5and
thoseoftheB‐BOX7.5batterypacks).Notehowbothachievetheirwarrantypoints(retainatleast
60%ofinitialcapacity)aftera10‐yearoperationperiodsubjectto45°C.However,despitealso
agreeingwiththewarranty,theNMCpresentsanoverallenergythroughputof19.9MWhinthat

Energies2020,13,5687of18
period(whatwouldrepresent1.2fullcyclesperday),whiletheLFPcellsfromtheBYDbatterypack
goupuntil23.9MWh(whichwouldrepresent1.4fullcyclesperday).

Figure4.Degradationevolutionthroughout10yearsunderwarrantyconditionsforthetwo
commercialLi‐ionbatterypacksanalyzedinthiswork.
2.5.OperationofPVSystemswithBatteries
Aspreviouslyintroduced,althoughhomesolarstoragesystemswillbepotentiallyusedfora
widevarietyofgridservicesthatwillhelpinenhancingtheprofitabilityoftheirinstallation,these
aremainlyinstallednowadaysforenergyarbitrageandpeakshaving(alsoback‐upincertain
locations).However,andregardlessoftheoperationgoalofthesystem,thecontrolalgorithmin
chargeofsolarstoragesystem’soperationmust,ateverygivenmoment,analyzethedifference
betweentwouncontrollablevariables(𝑃andtheconsumptionoftheloads, 𝑃)anddecidewhat
todoifthereisanexcessorashortageinpowerasafunctionofthemainoperationalgoal.Inthe
firstcase,itcaneitherchargethebattery(𝑃 0)orsellenergytothegrid(𝑃 0).Inthesecond,
itcansupplementthepowerthatisproducedbythePVpanelswithenergystored(ifavailable)or
bypurchasingfromthegrid.Inanelectricitymarketwithatime‐of‐useratesstructure(on‐peakand
off‐peakhours)thecontrolalgorithmmusttakethisdecisionbyconsideringnotonlythecurrent
momentintime,butalsotheexpectedevolutionof𝑃and𝑃.Otherwise,theimplemented
strategycouldavoidpurchasingenergyduringoff‐peakhours,onlytohavetobuyitinthefuture,
duringon‐peakperiods.SuchaproblemcanbeeasilyformulatedintheframeworkofModel
PredictiveControl(MPC)[52].Thisisacontrollerdesigntechniquethatisbasedonanoptimization
strategy,inwhichthefutureoutputsforagivenhorizonN,calledthepredictionhorizon,are
predictedateachinstantwhileusingappropriatemodels.Thesepredictedoutputsdependonfuture
decisionvariables,whicharecalculatedbyoptimizingadeterminedcriterionwhilefulfillingasetof
constraints.Althoughacompletesequenceoffuturedecisionvariablesiscomputed,onlythefirst
oneiseffectivelysenttothesystem,because,atthenextsamplinginstant,newinformationwillbe
available.Thisisknownasrecedinghorizon.
Fortheproblemathand,whichfocusedontheenergyarbitrage,theeconomicbalanceofthe
energyexchangedwiththegridmainlydeterminestheoptimizationcriterion.Regardingthe
constraints,thepowerbalance,includingpowersfromthePVsystem,thebattery,thegrid,andthe
loads,mustbemet;also,thestateofcharge(SOC)ofthebatterymustbekeptwithinitslimits;and,
finally,thepowerexchangedbothwiththegridandwiththebatterymustalsostaywithincertain
limits,eitherbecauseofpowerconverterslimitationsorpeakshavingstrategies.Notethat,even
though𝑃doesnotappearintheoptimizationcost,itisstilladecisionvariableanditaffectsthe
resultsbecauseoftheconstraints.

Energies2020,13,5688of18
Takingalloftheseconsiderationsintoaccount,andfollowinganapproachsimilartothatin
[53],theoptimizationproblemtobesolvedintheMPCframeworkcanbeformulatedas:
min 𝐽𝑇𝑐
𝑡𝑘𝑃
𝑡𝑘


(7)
subjectfor𝑘0…𝑁 to:
𝑃
𝑡𝑘𝑃
𝑡𝑘𝑃
𝑡𝑘𝑃
𝑡𝑘
𝐸𝑡𝑘1𝐸
𝑡𝑘𝑇𝑃
𝑡𝑘
𝐸, 𝐸
𝑡𝑘𝐸
,
𝑃, 𝑃
𝑡𝑘𝑃
,
𝑃, 𝑃
𝑡𝑘𝑃
,
where:
 Tisthesamplingperiod.
 𝑃
𝑡𝑘and𝑃
𝑡𝑘arethepredictionsfor𝑃𝑡𝑘and𝑃𝑡𝑘,respectively.
 𝑃𝑡𝑘isthepowerexchangedwiththegridatinstant𝑡𝑘,with𝑃𝑡𝑘0when
energyispurchased.
 𝑃𝑡𝑘isthepowerexchangedbythebatteryatinstant𝑡𝑘,with𝑃𝑡𝑘0when
discharging.
 𝐸𝑡𝑘istheenergyavailableinthebatteryatinstant𝑡𝑘.
 𝑃,and𝑃,arethelowerandupperboundsforthepowerexchangedwiththeESS.
Theseconstraintsareduetothelimitationsonthepowerconvertersand,therefore:𝑃, 
𝑃,.
 𝑃,and𝑃,arethelowerandupperboundsforthepowerexchangedwiththegrid
(𝑃, 𝑃
,).Thesecanalsorepresentlimitationsonaconverterorcanbeusedin
ordertoimplementpeakshaving.
 𝐸,and𝐸,arethelimitsinbetweenwhichthebatterySOCmustbekept.
 𝑐 
𝑐ℎ 𝑓𝑜𝑟 𝑃 0
𝑐 𝑓𝑜𝑟 𝑃 0aretheelectricityprices.𝑐changesitsvalueswiththehour
ofthedayℎdependingontherateperiod(on‐peakoroff‐peak),while𝑐isconsidered
constantand𝑐ℎ𝑐
foranygivenℎ.
Inthiswork,thedescribedoptimizationisperformedhourly,updatingtheproblemwiththe
newmeasurementsof𝐸𝑡ateachstep,andalso 𝑃
𝑡𝑘whennewinformationisavailable.
Althoughmostoftheequationsintheaboveproblemsarelinear,suchaprobleminitscurrent
formulationisstilldifficulttosolvebecauseofthepiecewisenatureof𝑐.Thiskindoffunctions
canbedealtwithbyintroducingbinaryvariables,whichleadtoamixedintegerlinearprogram
(MILP),whichiscomputationallyprohibitivefortheproblemsize.Therefore,adifferent
formulationisproposed.Theideaistoreplace𝑃bytwonewvariables,𝑃and𝑃,forthe
caseswhenitispositiveornegative,whichyieldsthefollowingproblem:
min
𝐽
𝑇𝑐
𝑡𝑘𝑃𝑡𝑘𝑐𝑡𝑘𝑃𝑡𝑘


(8)

subjectfor𝑘0…𝑁 to:
𝑃
𝑡𝑘𝑃
𝑡𝑘𝑃
𝑡𝑘𝑃
𝑡𝑘𝑃
𝑡𝑘
𝐸𝑡𝑘1𝐸
𝑡𝑘𝑇𝑃
𝑡𝑘

Energies2020,13,5689of18
𝐸, 𝐸
𝑡𝑘𝐸
,
0𝑃
𝑡𝑘𝑃
,
0𝑃
𝑡𝑘𝑃
,
𝑃, 𝑃
𝑡𝑘𝑃
,
𝑃𝑡𝑘𝑃
𝑡𝑘0,
wherethelastquadraticconstraintistheonlyonenotlinear,beingintroducedtoavoidthe
suboptimalsolutioninwhichenergyissimultaneouslyboughtandsoldfromthegrid.
Then,letusconsideranewoptimizationproblembydroppingthislastconstraintthatinvolves
𝑃and𝑃.Becauseofthestructureofthisnewproblem,itsoptimalsolutionissuchthateither
𝑃
 0or𝑃
 0.Indeed,if𝑃
 0and𝑃
 0, as c c 0,therewouldexista
newfeasiblesolution𝑃′
 𝑃

 𝑃

andP′
 0 withalowervalueofthecostfunction,
rendering𝑃
and𝑃
asbeingsuboptimal.Furthermore,thenewproblem,withboththecost
functionandconstraintsdefinedaslinearfunctions,isalinearprogram(LP),whichisaconvex
optimizationproblem.Convexityimpliesthattheoptimizationalgorithmwillfind𝑃′
 and
𝑃′
 insteadofthesuboptimalsolutions.Therefore,thisnewlinearprogram(LP)canreplacethe
originaloptimizationproblem,whichachievesthesameoptimalsolutionsandiseasilysolvedwith
standardoptimizationtools.
Finally,itisimportanttoremindthat,fortheapplicationoftheMPCstrategy,prediction
models𝑃
and𝑃
areneeded.
Regarding𝑃
,itispredictedinthisworkfromthePVsystemratedpowerandefficiencyin
combinationwithirradianceforecastedvaluesdownloadedfromthemodelsavailableonlineatthe
EuropeanCentreforMedium‐RangeWeatherForecasts(ECMWF)[54].ECMWFproducesan
ensembleofpredictionsthatindicatethelikelihoodofarangeoffutureweatherscenarios.Forecasts
aremadewhileusingNWPmethods,whichhavegoodperformancesforday‐aheadforecastsand
beyond.Theforecastspredictthenext10dayswithone‐hourtimesteps,andupdatedforecastsare
availableevery12hours,at12a.m.and12p.m.Thespatialresolutionoftheforecastsis0.1°forboth
latitudeandlongitude.
Ontheotherhand,giventheSouthernEuropeanlocationofthehouseholdsunderanalysis,
𝑃
isobtainedasoneofthereferenceloadprofilesdefinedbytheSpanishMinistryofIndustryfor
theresidentialsector[55].EnergyretailersintheIberianMarketusetheseprofilestobillthose
consumersnotupdatedyetwithconsumptiontelemetry,andarethereforesupposedtorepresent
theaveragetrends.Amongthevariousprofilesthataredefinedin[55],thisworkusesthat
designatedforuserswithtime‐of‐useelectricityrates,whicharethosethatarepronetobeeligibleby
consumerswithhomesolarstoragesystems.ThebluecontinuouslineinFigure4representsthe
annuallyaverageddailydistributionofthatloadprofile.
3.ResultsandDiscussion
TheanalysisofthelifetimeexpectancythatisofferedbythedifferenttypesofLi‐ionbatteries
hasbeenperformedbymeansofsimulationswhileusingMATLAB®.Accordingtotheintroduced
controlalgorithm,theoperationofthehomestoragesystemproducessuccessivechangesinthe
batterySOCthatbasicallydependonthepowerbalancetobegrantedhourafterhour.Therefore,the
continuousPVproductionandthehouseholdloadvaluesstronglycharacterizetheSOCevolution.
Inthissense,withtheaimtotrytogeneralizetheresults,theoperationofthehomestorage
systemsatthreedifferentlocationsinSouthernEuropeduringuptothreeyearswasexplored(2016,
2017,2018).Thus,uptonineannualirradiancepatterns,presentingallofthemaround1500peak
sunhours(PSH)peryearonthehorizontalplane,wereavailableandevaluated.Thethreeplaces
usedforanalysiswere:alocationintheeasternpartofSpain(ontheMediterraneancoastatsea
level,labelledasA),alocationinthecentralplateau(800mabovesealevel,labelledasB),anda

Energies2020,13,56810of18
locationinthenorthernpartofthecountry(420mabovesealevel,labelledasC),respectively.With
thesamegoal,threetypesofresidentialloadpatterns,thosewhoseannuallyaveragedhourly
distributionsareintroducedinFigure5,wereanalyzed.Notehowoneofthemispractically
coincidentwiththereferenceloaddistributionmodelthatisprovidedin[55],thatofhousehold3,
whiletheothertwoloadpatternsrepresentuserswithsignificantconsumptionduringthe
mornings,household1,orduringtheevenings,household2.Alsonotethat,althoughbeing
differentlydistributedthroughouttheday,theactualstochasticconsumptionofeachofthe
householdsdayafterdayaddsuptoanoverallannualconsumptionofaround5000kWh/year.
Moreover,anactualtime‐off‐usetariffforresidentialconsumerspresentingdailyoff‐peak
(from10pmtillnoon)andon‐peak(fromnoontill10pm)hours,regardlessofthedifferentseasons
oftheyearwasimplemented.ThiscomprehendsthecostsorpricesthataresummarizedinTable2
forthepurchaseofelectricityfromthegridorfortheinjectionofpowerintoit.Notehowthe
acquisitioncostisalwayshigherthanthegenerationone.
Table2.Costs,inc€/kWh,oftheelectricityfortheresidentialelectricitytariffconsidered.
PeriodOn‐peakOff‐peak
Purchaseprice(c€/kWh)2211
Saleprice(c€/kWh)55

Figure5.Loaddistributionprofileforthethreehouseholdsanalyzed,andreferenceloadcurveused
asmodel.
Afterwards,simulationshavebeenrunforeachofthecombinationsexplainedabove.Inall,if
thethreelocations,thedifferentyearsavailable,andthevariousloadprofilesareanalyzedwitheach
ofthealternativehomestoragetechnologicalsolutions,i.e.,considering2kWor4kWratedPV
installationsateachhousehold,andcombiningeachofthesePVsystemswiththevariousbattery
typesanalyzed(LFP,NMC…),which,inturn,areintroducedwiththreedifferentenergycapacities
(3.3kWh,6.5kWh,and10kWh)andaroundtripefficiencyof95%,thesesumup648combinations.
NotethatthevaluesofratedpowerforthePVplantssimulatedandthethreedifferentenergy
capacityratingsaccountedforthebatteriestryingtoreflectthemostpopularPVcasesforresidential
installationsinEuropeandtheusualbatterystoragecommercialmodelsbeinginstallednowadays.
3.1.AnnualEvolutionoftheGlobalDegradation
Foreachofthecases,anannualhourlysimulationofthehomesolarstoragesystemoperation
hasbeenperformed,providingtheSOCevolutionofthecorrespondingbatterypackasmainresult.
AsexplainedinSection2.1,thedifferentSOCevolutionsthatweregeneratedforthevariousbattery

Energies2020,13,56811of18
capacitiesconsideredatagivenhouseholdinoneofthelocationsunderstudywere,inturn,
introducedintotheRFCalgorithm.Thisalgorithmreturnedthehistogramofpartial
charge‐dischargecyclesthatwereannuallyregisteredbythedifferentbatteriesandtheaverageSOC
levelforeachcycle.Figure6representssuchhistogramsgenerated,inthatcase,forhousehold1in
locationAwhena4kWrated‐PVisassumedandforeachofthethreebatterycapacitiesconsidered
(3.3kWh,6.5kWh,and10kWh).Itisimportanttonoticeinthosedepictedcyclingpatternshowthe
maximumdepthofthecharge‐dischargecyclesis95%,limitedinthiswaybythecontrolsystemin
agreementwithmanufacturer’sindicationtoavoidexcessivedegradationofthebatterythatis
associatedwithverydeepcycles.Apartfromthismaincycle,nosignificantpartialcyclesintermsof
DoDareexperiencedbythehomestoragesystems.Mostofthepartialcyclesregisteredcorrespond
to25%andbelowDoDmaneuvers.Addingthemall,thesedonotevenachieveinnumberthe95%
DoDcycles.Therefore,homestorageapplicationscanbeconcludedtobeverymonotonousand
sortableintermsofDoDpatterns.Additionally,notehow,asthebatterycapacitygetslarger,the
numberoffullcycles(thoseof95%DoD)getssmalleranditisredistributedalongalltherangeof
partialDoDs.Thisdirectlyimpactsthecycledegradationofthedifferentbatteries.Fromthese
histograms,theamountofQ(chargethroughput)thatisexperiencedbythebatterycanbealso
estimated.Inthesameway,the
∅
VandVvaluesrequiredforthecalculationsinequations(4)and(5)
canbederivedfromtheresultingaverageSOCprovidedbytheRFCalgorithmandfromtheSOC
evolutionitself,respectively.Thisisdonebymeansofanexperimentalequationobtainedforthe
batterypacksmeasuredinourlabthatcorrelatesthevoltageofthebatterypackwiththe
correspondingSOC.Therefore,accordingtoEquations(2)and(5),suchamethodologyisusefulin
thedegradationanalysisassociatedwiththecycling.

Figure6.Histogramforthenumberofcyclesforeachdepthofdischarge(DoD)forthreedifferent
batterycapacities.
Conversely,notbeingtheonlystressfactor,temperatureisconsideredtobethemainonethatis
associatedtocalendarageingthough.Forthecasesthatweresimulatedinthiswork,thebatteries
havebeensupposedtobestressedtocelloperationtemperaturesalwayswithinthewarrantylimits
andrangingfrom35°Cto45°Calongtheyear.Thesearedefinedforthecellsasthetemperature
thatwouldbeachievedwithinthebatterypackduringregularoperationwheninstalledindoorsat
locationswithairtemperaturesrangingfrom20°Cto32°C.Therefore,thecalendarageingofthe
batterypacksalongtheyearcanbeequallyestimatedwhileusingEquations(1)and(4).
Calendarandcycleageingundertheseoperationconditionsarebothrepresentedforthefirst
yearofoperationinFigure7forthecommercialbatterypacksfromBYDandLGChem,respectively.

Energies2020,13,56812of18

(a)

(b)
Figure7.Cycling,calendar,andtotalcapacityreductioninoneyearfora3.3kWhLiFePO4(LFP)(a)
andLiNiMn‐CoO2(NMC)(b)batteryoperatedatlocationAwithinhousehold1.
InagreementwiththewarrantyevolutioncurvesthatarerepresentedinFigure4,notehowthe
maindifferenceamongthebatterydegradationmodelsisdefinedbytheirresponseintermsof
calendarageing.Itcanbeclearlyobservedhow,forthesameoperationconditionsand,accordingly,
thecorrespondingsimilarcycleageing,theglobaldegradationoftheLFPbatteryismuchmore
importantthanthatofitsNMCcounterpartduringthefirstyearofoperationduetotheaccelerated
calendarageing.However,thisbehaviorbecomesmorebalancedastimegoesby,becauseofthe
relationwithtimeofbothcalendarmodels(t0.75forNMCVs.t0.5forLFP).Thiscanbeclearlydeduced
fromFigure3anditisalsoreflectedinthelifetimeestimationvaluesthatareprovidedforthe
differentchemistriesinthedifferentcasesanalyzedandsummarizedinthefollowingsubsection.
3.2.ComparativeDiscussionamongTechnologies.
Outofthe648combinationsanalyzed,justsomeofthemareintroducedfordiscussion.The
selectedcasescoverthemaindifferentoperatingscenariosandallowforextractingthemain
conclusionsfromthisstudy.Tostartwith,Table3introducesthelifetimeexpectancyresultsthat
wereobtainedinagivenhousehold(specificloadcurve)andagivenlocation(irradianceprofile)for
thefourbatterymodels:thoseintroducedinSection2.1and2.2asreferenceforLFPandNMCcells,
respectively;and,thoseintroducedinSection2.3asstate‐of‐the‐art(SoA)models.Itisimportantto
noticethatthelifetimeexpectancyhasbeencalculatedassumingtheEOLisachievedwhenthe
batteryretains70%(asdefinedforthereferencecellsintheircatalogues)or60%(asdefinedby
manufacturersfortheSoAbatterypacks).Itcanbeobservedataglancehowallowingthecellsto
operateuntila40%capacityfadeoffersacertainlyincreasedlifetimeexpectancy.Therefore,
althoughthewarrantiesfrommanufacturersdefinedifferentEOLhorizonsforthevariouscells,for
thesakeofcomparison,bothhorizonshavebeenanalyzedforallofthem.


Energies2020,13,56813of18
Table3.Lifetimeexpectancyresultsforthedifferentbatteriesathousehold1inlocationA.
Modelof
Battery
PV
Power
(kW)
Battery
Capacity
(kWh)
Calendar
Ageing
EOL70%
Cycling
Ageing
EOL70%
Lifetime
Expect.
(years)
Calendar
Ageing
EOL60%
Cycling
Ageing
EOL60%
Lifetime
Expect.
(years)
LFPRef23.322.077.943.5629.4310.576.30
LFPRef26.522.827.193.7630.419.596.63
LFPRef21024.405.614.3832.577.447.59
LFPRef43.322.097.923.5729.4710.536.32
LFPRef46.522.397.623.6529.8910.126.45
LFPRef41023.406.604.0131.228.787.02
LFPSoA23.319.1210.8810.1625.5314.4717,80
LFPSoA26,520.0010.0111.0626.6913.3219.48
LFPSoA21022.027.9813.3829.3810.6223.53
LFPSoA43.319.1510.8510.2025.5614.4517.85
LFPSoA46.519.5210.4810.5326.0413.9618.55
LFPSoA41020.739.2711.8127.6712.3320.88
NMCRef23.315.0115.002.0521.6518.373.34
NMCRef26.517.5512.452.8724.8615.154.57
NMCRef21020.269.744.1928.4011.616.47
NMCRef43.315.6014.402.0722.3717.643.33
NMCRef46.517.4712.542.5724.6315.384.18
NMCRef41019.7610.253.0827.7312.284.65
NMCSoA23.317.7812.228.4925.2014.8013.48
NMCSoA26.519.9810.0211.5827.9312.0718.18
NMCSoA21022.437.5716.0230.989.0224.58
NMCSoA43.318.2711.738.4125.8014.2013.31
NMCSoA46.519.8010.2010.3727.7112.2916.19
NMCSoA41021.858.1511.5730.239.7717.75
ItcanbeconcludedfromresultsinTable3thattheSoAbatterymodelssignificantlyoutperform
theirreferencecounterparts.ThelifetimesfortheSoALFPbatteryareexpectedtobearoundthree
timesthoseofthereferencemodel,whileafourfoldrelationisregisteredbetweentheNMCmodels.
Theimprovedpropertiesofthenewbatteriesintermsofexpectedlifetimeactuallyenablethemasa
formalsolutionforresidentialstorage,whichwasunderdiscussionuntilrecentlyduetothetight
marginbetweentheirexpectedlifetimesandtheirpaybackperiods(around10years,withhighly
dependenceonthemarket).AccordingtotheresultsinTable3,newbatterydevelopmentswould
grantlifetimeexpectanciesbeyond10yearsformostofthecasessimulated,evenifonlya30%
capacityfadeisallowedtothebatterypriortosubstitution.
Whencomparingthechemistries,itcanbeobservedhowthecycleageingimpactsmoreNMC
cellsthanLFPones;thismakestheLFPbatteriesoutfitNMCmodelswhensmallbatterypacksare
implementedinthehomestoragesolution.Thisisbecausebatterieswillexperiencedeep
charge‐dischargecycleseverydayduetotheirlimitedcapacity.However,lifetimeexpectancies
convergeasbatterycapacitygetslarger.
Beyondthat,notehowincreasingthePVratedpowerfrom2kWto4kWdonotsignificantly
varytheexpectedlifetimewhensmallsizebatteriesareimplemented(thoseof3.3kWh).Itcan
introducea15%variationinthelifetimeifthebatterypresents10kWhthough.Thisisbecause3.3
kWhbatteriesgetregularlysaturated(profitalltheirDoD)everydayduringthenormaloperation
withanyPVinstallationbeyond2kW.Therefore,thesecannotdegradefasterduetocycling.Onthe
contrary,10kWhbatteriesdonotexperiencefulldischargedailycycleswiththe2kWsolarplant
(reducingtheircycleageing),whiletheyaremoreextensivelyprofited,intermsofcycling,withthe4
kWplant.Finally,notethat,asexpected,thelargerthebatterysizeisthelongerthelifetime
expectancyresults.ThisisduetothesamereasonjustexposedandalsoexplainedwithFigure6,as
thebatterypresentsmorecapacity,cyclesareshorter,anditsufferslessfromcycleageing.
InordertoanalyzetheimpactofthelocalPVproduction(irradiancecharacteristicsatthe
location),simulationsbothfordifferentyearsatthesamelocation(Table4)andforthethree
differentlocations(Table5)havebeencarriedoutThefirsttableintroducesthelifetimeexpectancy
thatwasobtainedfortheLFPbatterypackinstalled,inthatcase,athousehold3.Thisisanalyzed

Energies2020,13,56814of18
withannualirradiancedatafromthreesuccessiveyearsthatwereregisteredinthelocationlabelled
asA.NoteinTable4howusingagivenyearofannualirradiancedatadonotsignificantlyimpact
theprognosisintermsofexpectedlifetime.Thedeviationsareminimal.Ontheotherhand,itcanbe
observedinTable5howthedifferentirradiancepatternsregisteredatthevariouslocations,
althoughallofthemwithsimilarPSH,certainlyinfluencethelifetimeexpectancyofthebatteries.In
thiscase,variabilitycanrangefrom5%to20%,dependingonthelocationand,also,stronglyonthe
sizeofthebatteryunderanalysis.However,inanyofthecases,thebatterypackskeeppresenting
lifetimesbeyondthe10‐yearwarrantytimes.
Table4.LifetimeexpectancyresultsfortheLFPstate‐of‐the‐artbatterypackinstalledathousehold3
analyzedwithirradiancedatafromthreesuccessiveyearsinlocationA.
Modelof
Battery
PVPower
(kW)
Battery
Capacity
(kWh)
LifetimeExp.
Irradiance’16
(Years)
LifetimeExp.
Irradiance’17
(Years)
LifetimeExp.
Irradiance’18
(Years)
LFPSoA23.318.9618.9918.64
LFPSoA26.521.9121.8021.49
LFPSoA21025.9025.8225.52
LFPSoA43.318.1518.2118.07
LFPSoA46.519.2619.1419.01
LFPSoA41022.4822.5722.42
Table5.LifetimeexpectancyresultsfortheNMCstate‐of‐the‐artbatterypackinstalledathousehold
2andanalyzedatthethreedifferentlocations.
Modelof
Battery
PV
Power
(kW)
Battery
Capacity
(kWh)
LifetimeExp.
LocationA
(years)
LifetimeExp.
LocationB
(Years)
LifetimeExp.
LocationC
(Years)
NMCSoA23.313.5013.1212.93
NMCSoA26.518.5816.5915.71
NMCSoA21025.4321.4519.81
NMCSoA43.313.1512.4712.33
NMCSoA46.516.0914.9814.35
NMCSoA41018.2316.6715.68
Finally,thelifetimeanalysisoftheseLi‐ionbatteriesistheinfluenceoftheloaddistributionor
itsvariabilityistheotherfactortotakeintoconsideration.Theresultsinthissenseareintroducedin
Table6.Thistablesummarizeslifetimeexpectanciesthatwereobtainedforboththestate‐of‐the‐art
LFPandNMCbatterypackswheninstalled,forthiscase,atlocationBandforeachofthethree
householdsanalyzedwiththeircorrespondingdifferentlyhourly‐distributedloads,accordingto
Figure4.NotealsoagainthattheloadprofilesthatarerepresentedinFigure4arerealand,
therefore,alsopresentdifferentvariabilitiesamongthedaysthroughouttheyear.ObserveinTable5
howthedifferentloadprofiles(batteryoperationwithinthedifferenthouseholds)implylifetime
deviationsthatarebelow10%amongthecases,andforeachofthebatteries.Therefore,theeffectof
theloadcanbeconcludedtobelowerthanthatoftheradiationonthelifetimeexpectancyanalysis.
Table6.LifetimeexpectancyresultsfortheLFPandNMCstate‐of‐the‐artbatterypacksinstalledat
locationBinanyofthehouseholdsanalyzed.
Modelof
Battery
PV
Power
(kW)
Battery
Capacity
(kWh)
LifetimeExp.
Household1
(years)
LifetimeExp.
Household2
(years)
LifetimeExp.
Household3
(years)
LFPSoA23.317.6417.6817.95
LFPSoA26.518.6718.7920.13
LFPSoA21021.9922.4523.96
LFPSoA43.317.6217.6117.66
LFPSoA46.518.2818.2918.65
LFPSoA41020.7221.1622.27
NMCSoA23.313.1113.1213.66

Energies2020,13,56815of18
NMCSoA26.516.4416.5918.60
NMCSoA21020.9221.4524.28
NMCSoA43.312.5512.4712.77
NMCSoA46.514.9814.9815.62
NMCSoA41016.5516.6717.38
4.Conclusions
ThispaperintroducesthelifetimeexpectancyforfourdifferenttypesofLi‐ionbatteries
implementedashomesolarstoragesystems.Thechemistriesanalyzedarethosethatarebeing
proposednowadaysforthiskindofapplicationbymostofthemanufacturers,i.e.,LFPandNMC
typebatteries.Foreachofthem,twotypesofcellshavebeenconsidered:cellsthatwere
commercializedinthefirsthalfofthisdecadeandstate‐of‐the‐artcellsbeingusedincommercial
currentsolutions.Thefirstpresentsdegradationmodelsintheliteraturethatareusedasreference
foreachofthechemistries,whilethelatterpresentsextendedandimprovedoperationwarranties
thataretakenintoaccounttoupdateandcalibratenewdegradationmodels.Eachofthefourbattery
typeshasbeenanalyzedinvariousscenariosofloadpattern(threecasesofdistributionofthe
consumptionalongtheday),atthreedifferentlocationswiththeircorrespondingvariationsinthe
PVproductionprofile.ThisiscombinedwithtwosizesofPVinstallation(2kWand4kW)andwith
uptothreelevelsofenergystoragecapacity(3.3kWh,6.5kWh,and10kWh).Simulatingtheannual
operationofthePVresidentialsystemswithbatteriesateachofthescenariosintroducedandtaking
intoaccountbatteryoperationtemperaturesrangingfrom35°Cto45°C,dependingonthetimeof
theyear,isundertakentoperformthestudy.Theintroductioninthesemi‐empiricaldegradation
modelsofthebatteryoperationparameters(SOCevolutions,exchangedcurrent,voltagestand‐by
levels,andothers)obtainedatthesimulations,deliverslifetimeexpectanciesforthedifferent
batteriesunderthevariousscenarios.Ingeneral,itcanbeobservedhowthecycleageingimpacts
moreNMCcellsthanLFPones,whichmakesLFPbatteriesoutfitNMCmodelswhensmallbattery
packsareimplementedinthehomestoragesolution.However,asbatterycapacitygetslarger,
lifetimeexpectanciesconverge.Additionally,notethatthedegradationexperiencedforthedifferent
loadprofilesdoesnotvarysignificantly,whichimpliesthatthewarrantycanbegranted,regardless
ofthetypeofconsumer.Additionally,similarly,thelifetimeexpectancythatisobtainedatthe
differentlocationsisalsoquiteregular,whichalsoimpliesthatthewarrantycanbegrantedatany
location.Infact,noneofthecasesanalyzedprovidedlifetimesbelow10yearsbeingthebatterypack
temperatureofsimulationsalwaysbelowthatofthewarrantyforstate‐of‐the‐artmodels.Neither
chargethroughputwentbeyondthatofferedinthemanufacturers’datasheets.Therefore,itcanbe
finallyconcludedthatcurrentcommercialbatterymodelspresentlifetimeexpectanciesunderreal
operationconditionsbeyondthebreak‐evenpointthatisestimated,dependingonthecountryand
itscorrespondingelectricitytariffs’structures,between8and12years.
AuthorContributions:Conceptualization,HectorBeltran;methodology,HectorBeltran,PabloAyusoand
EmilioPerez;software,PabloAyusoandEmilioPerez;validation,PabloAyusoandHectorBeltran;formal
analysis,HectorBeltranandPabloAyuso;investigation,HectorBeltran,PabloAyusoandEmilioPerez;
resources,PabloAyusoandEmilioPerez;datacuration,PabloAyuso;writing—originaldraftpreparation,
HectorBeltranandPabloAyuso;writing—reviewandediting,HectorBeltranandEmilioPerez;visualization,
HectorBeltran,PabloAyusoandEmilioPerez;supervision,HectorBeltranandEmilioPerez;project
administration,HectorBeltran;fundingacquisition,HectorBeltran”.Allauthorshavereadandagreedtothe
publishedversionofthemanuscript.
Funding:ThisresearchwasfundedbyUniversitatJaumeI(Spain),projectUJI‐B2017‐26,andbythe
GeneralitatValenciana,projectGV‐2019‐087.
Acknowledgements:
ConflictsofInterest:Theauthorsdeclarenoconflictofinterest.


Energies2020,13,56816of18
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