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Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 1
ExaminingtheInfluenceofSolarPanelInstallersonDesignInnovationandMarket
PenetrationEkaterinaSinitskaya1MechanicalEngineering,StanfordUniversityBuilding530,440EscondidoMall,Stanford,CA94305-3030,[email protected],StanfordUniversityBuilding530,440EscondidoMall,Stanford,CA94305-3030,[email protected],MassachusettsInstituteofTechnology77MassachusettsAvenue,Cambridge,MA02139,[email protected],MassachusettsInstituteofTechnology77MassachusettsAvenue,Cambridge,MA02139,[email protected],StanfordUniversityBuilding530,440EscondidoMall,Stanford,CA94305-3030,[email protected] accepted to the 2017 ASME IDETC & CIE Conference Paper DETC2017- 68338
1 Corresponding author
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 2
ABSTRACT
Thisworkusesanagent-basedmodeltoexaminehowinstallersofphotovoltaic(PV)panelsinfluencepanel
designandthesuccessofresidentialsolarenergy.Itprovidesanovelapproachtomodelingintermediary
stakeholderinfluenceonproductdesign,focusingoninstallerdecisionsinsteadofthetypicalfociofthefinal
customer(homeowners)andthedesigner/manufacturer.Installersrestricthomeownerchoicetoasubset
ofallpaneloptionsavailable,and,consequentially,determinemedium-termmarketdynamicsintermsof
quantity and design specifications of panel installations. This model investigates installer profit-
maximizationstrategiesofexploringnewpaneldesignsofferedbymanufacturers(arisk-seekingstrategy)
vs. exploiting market-tested technology (a risk-averse strategy). Manufacturer design decisions and
homeowner purchase decisions are modeled. Realistic details provided from installer and homeowner
interviews are included. For example, installers must estimate panel reliability instead of trusting
manufacturerstatistics,andhomeownersmakepurchasedecisionsbased inparton installerreputation.
Wefindthatinstallerspursuenewandmore-efficientpanelsoversticking-withmarket-testedtechnology
under a variety of panel-reliability scenarios and two different state scenarios (California and
Massachusetts).Resultsindicatethatitdoesnotmatterif installersarepredisposedtoanexplorationor
exploitationstrategy—bothtypeschoosetoexplorenewpanelsthathavehigherefficiency.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 3
NOMENCLATURE
PV Photovoltaic𝑁"#$%&' Marketsize𝑁(#)&*+ NumberofpanelsinPVsystem𝑁(&$ Numberofperiodsforestimatingreputation𝑁($,-&.'+ Numberofactiveprojectsforinstaller𝑁/#'' STCpowerratingforPVmodule𝑇1,$&.#+' Forecastinghorizonforexpectedprofit𝑽𝟎,𝜽 Bayesianpriorforthevarianceofthedistribution
fordemandfunction𝒁𝟎,𝒊𝒏𝒔𝒕 Priorvaluesfordemandfunctionestimation𝑏<,= Initialvalueforparameterforpriorfordemand
functiondistribution𝑏= ParameterofaBayesianpriorofthedemand
functiondistribution𝑐#="?)?+'$#'?,) Administrativecostsforinstaller𝑐=&+?@) CostofdesigningPVsystem𝑐?)+'#**#'?,) CostsofinstallingPVsystem𝑐"#?)'&)#).& Maintenancecostsforinstaller𝑐"#'&$?#*+ CostofmaterialforPVsystem𝑐"#$%&'?)@ Marketingcostsforinstaller𝑐(&$"?' CostofobtainingpermitsforPVsystem𝐶'BC 𝑝 𝑚𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 , 𝑞 LM
'BC Designcostsattimet+τ𝑒𝑓< Initiallevelofefficiency𝑒𝑓' EfficiencyofaPVmodule𝑓. 𝑞 LM
'BC, 𝑝 𝑚𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 Estimatedlaborcostsofmaintenance
𝑖𝑟𝑟? Internalrateofreturnofferedbydesignbyfirmi𝑖𝑟𝑟P? Internalrateofreturnofferedbydesignbyother
firms𝑛< Parameterforreputationstickiness𝑛1 Numberoffailures𝑝?)+',+/?'.Q Probabilityofswitchingtonewdesignforinstaller𝑝 𝑚𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 Probabilitydistributionforexpectedmaintenance
costsgivencurrentandexpectedportfolioofprojects
𝑝Q,+/?'.Q Probabilityofacceptingdesignforhomeownerprice PriceofaPVsystem𝑝𝑟𝑖𝑐𝑒",=R*&,STU Manufacturer’spriceofaPVmodule𝑝𝑟𝑖𝑐𝑒/#'' PriceperwattforPVmodule𝑝𝑟𝑜𝑑',? Productionforprojectiintimet𝑞'BC 𝑝𝑟𝑖𝑐𝑒 Demandforcurrentdesignattimet+τforpricep
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 4
𝑟𝑒𝑝? Reputationofafirmi𝑟𝑒𝑝P? Reputationofotherfirms𝑤' Prevailinglaborwage𝑧?)+' Demandfunctionparameters𝛼 ParameterofaBayesianpriorforprobability
distributionformaintenance𝛼< Initialvalueforpriorforprobabilitydistributionfor
maintenance𝛼<,= Initialvalueforparameterforpriorfordemand
functiondistribution𝛼<,1 Initialvalueforpriorforfailuredistribution𝛼= ParameterofaBayesianpriorfordemandfunction
distribution𝛼1 ParameterofaBayesianpriorforfailure
distribution𝛼',$&( Parameterofinstallerreputationattimet𝛽 ParameterofaBayesianpriorforprobability
distributionformaintenance𝛽< Initialvalueforpriorforprobabilitydistributionfor
maintenance𝛽<,1 Initialvalueforpriorforfailuredistribution𝛽1 ParameterofaBayesianpriorforfailure
distribution𝛽$&( Fixedintimeparameterofinstallerreputation𝜖<,] RandomvariablewithN(0,1)distribution𝜃#="?)?+'$#'?_& Parameterforadministrativecosts𝜃.,"(*&`?'a?)+'#** Parameterforcomplexityofinstallation𝜃=&+?@) Parameterfordesigncosts𝜃?,& Parametervaluesforexplorer/exploiterdecision
process𝜃"#$%&'?)@ Parameterformarketingcosts𝜃(&$"?'@&)&$#* Parameterforcostestimationofpermitting,
generalpart𝜃(&$"?'+(&.?1?. Parameterforcostestimationofpermitting,design
specificpart𝜃= Parametersofanestimateddemandfunction𝜃Q,? Parameteriofhomeowner’sdecisionfunction𝜆< Initialvalueforreliability𝜆1 Parameterforfailuredistribution𝜇 ParameterofaBayesianpriorforprobability
distributionformaintenance𝜇< Initialvalueforpriorforprobabilitydistributionfor
maintenance
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 5
𝜇<,"#?)' Initialvalueforparametersforcomplexityofmaintenancedistribution
𝝁𝟎,𝜽 Bayesianpriorforthemeanofthedistributionfordemandfunction
𝜇&1 Parameterforefficiencydistribution𝜇"#?)' Parameterofprobabilitydistributionfor
maintenance𝝁𝝐𝒎𝒂𝒊𝒏𝒕 Parametersforcomplexityofmaintenance
distribution𝜇i Parameterforreliabilitydistribution𝜎<,"#?)'k Initialvalueforparameterforcomplexityof
maintenancedistribution𝜎&1k Parameterforefficiencydistribution𝜎"#?)'k Parameterofprobabilitydistributionfor
maintenance𝜎ik Parameterforreliabilitydistribution𝑣 Priorforparameterofprobabilitydistributionfor
maintenance𝑣< Initialvalueforpriorforparameterofprobability
distributionformaintenanceΠ' Totalexpectedfutureprofitattimet𝑥.,? Complexityofthefailure𝑥1,? Timebetweenfailuresforprojecti𝚺𝝐𝒎𝒂𝒊𝒏𝒕 Parametersforcomplexityofmaintenance
distribution
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 6
1INTRODUCTION
TheUnitedStates(U.S.)residentialsolarenergymarketismorethan20yearsold
andisbeginningtomature.Inthepasttenyears,thesolarPVpenetrationratesinthe
residentialsegmenthaveincreasedfromvirtuallyzeroto0.8%ofallU.S.households[1].
Withmaturitycomesnewchallenges.TheDepartmentofEnergy(DOE)hasrecognized
theneedfornovelapproachestopushingpenetrationrateshigher,forexample,studying
solarinstallationasasocialphenomenonviatheSunShotInitiative[2].Ifthegrowthrate
declines from the current optimistic industry projections, solar installation and
equipmentbusinessesbuilt onanassumptionof constant growthwill fail, asperhaps
foreshadowedbytherecentrestructuringandacquisitionofSolarCitybyTesla[3].
Asthemechanicaldesigncommunityincreasinglyviewsproductsassystems,the
designconcernsofstakeholderswithinthesystem, inadditiontofinalconsumers,will
receiveincreasingattentioninresearch.Thispaperprovidessuchaninvestigationforthe
residentialPVmarketusinganagent-basedmodeltounderstandthedesignconcernsof
differentstakeholders.Here,weinvestigateasystemthatincludesmanufacturingdesign
andpositioning installers’material-selection strategieswith regard to the adoptionof
newtechnologies,andfinalconsumerbehavior.
Themainquestionsaskedinthisstudyare:
1. How are market outcomes shaped by different decision processes used by
stakeholders?Specifically,howdopanelinstallersdecidewhattechnologiestooffer
forsale?
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 7
Here,insteadofcreatingadetailedexpressionofend-customerneeds,wesimplify
theseneedsandfocusonadetailedexpressionofinstallerdecisionsandinteractionswith
manufacturersandhomeowners.Weexpectthatthiswillidentifynewrecommendations
forimprovingpaneladoption.
2. Howdoesthestrategyofinstallerschangewithtimeandenvironmentalconditions?
Theagent-basedmodelexplorestheinteractionsbetweenmultiplestakeholders.
Inthemodel,installerscanchoosebetweentwostrategies:explorationorexploitation.
Theformerisarisk-seekingbehaviorofchoosingtosellanewtechnology,andthelatter
arisk-aversebehaviorofstickingwiththecurrentoffering.Bothstrategiesaredistinct
waysofmanagingsituationswithincompleteinformation.
In addition, two scenarios simulate different environmental and market
conditions:onerepresentsahighsolarirradiationlevelwithahighPVpenetrationlevel,
andonerepresentsa lowsolar irradiation levelwitha lowerPVpenetration level.We
expect to reveal insights about stakeholder strategy changes under different
environmentalconditions,definedbythelevelsofsolarirradiation,andinnewormature
markets,definedbythesolarPVpenetrationlevel.
Thispaperdetailsthestructureoftheagent-basedmodelandtherationalefor
thedecisionsmadeonwhattoincludeandexclude.Itprovidestheexplanationofmodel
calibrationtocreatereasonablebehaviorfortheU.S.PVmarket.Theworkthenmakesa
number of assumptions in order to examine some broad-level conclusions and
recommendationsforincreasingPVadoptionrates.Leversmanipulatedinclude:technical
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 8
properties(solarpanelefficiency),environmentalfactors(levelofsolarirradiation),and
economicfactors(incomelevelsofhomeowners).
The paper proceeds as follows: Section 2 reviews existing studies on solar PV
adoption,includingmodelingeffortsforthePVmarket;Section3presentsthesimulation
methodsofdecisionprocessesmodelingandengineeringmodeling;Section4presents
thesimulationresultsanddiscussion;andSections5providesconclusions.
2BACKGROUND
This literaturereviewprovidesanoverviewofthedifferentstakeholders inthe
solar PVmarket and focuses on the important role that installers play. In addition, it
introducesthemodelingtoolusedinthisstudy.
2.1StakeholdersintheresidentialPVmarketandtheimportantroleofinstallers
There are many stakeholders in the residential PV market, all of who play
importantrolesinthediffusionofthetechnology.Otherthanhomeowners,whomake
thefinaldecisionstoadopttheproduct,therearemanufacturers,regulatoryagencies,
and installers.Manufacturersproduceequipment forPVsystems.Regulatoryagencies
areresponsibleforissuingregulations,suchasbuildingrequirementsandgrid-connection
requirements.Installersconfiguresystemstosatisfycustomerneedsandinstallsystems
onhouses.
ResearchonresidentialsolarPVadoptiontraditionallyfocusesonhomeownersas
final consumersof theproduct. Forexample, KarakayaandSriwannawit [4] identified
barrierstohomeownersadoptingsolarpanels,suchasthehighpriceofPVsystems,the
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 9
complexityoftheinteractionbetweenpeopleandthePVsystem,andineffectivepolicy
measures.IslamandMeade[5]studieduserpreferenceforsolarPVandproposedthat
aneducationalcampaignforhomeownersmightbeeffectiveatincreasingadoptionrates.
Fromthemanufacturers’perspective,efforthasbeenputintoimprovingthetechnology
andthePVpanelproduction.
ForanoverviewofPVsystemstechnologyresearch,seetheNationalRenewable
EnergyLaboratory(NREL)publications[6],andforimprovementsinmanufacturingover
the lastdecade,see [7].BothmanufacturersandDOEallocatesignificantresourcesto
increasingPVpanelefficiencyandexpandingtherangeofenvironmentalconditions in
whichtheydeliveroptimalefficiency.Foranoverviewofrecentadvancesinthatarea,
see[6].Overall,manufacturerstailordistincttypesofPVpanelsthatexistonthemarket
toworkbetterindifferentdeploymentscenarios.Thispaperfocusesontheresidential
PVmarket,whereinstallersofferprimarilymonoandpolyPVpanels.Thinfilmtechnology
ismainlythedomainofutility-scaleinstallationsandthusfallsoutsidethescopeofour
work.
TheimpactofinstallersonthePVmarketisextensive.BothChenetal.[8]andour
interviews of installers, conducted at the Intersolar North America 2016 conference,
confirmed that installers make product design choices and offer homeowners only a
subsetofallpanelsavailableonthemarket[9].Therefore,installersarealsoconsumers,
in a business-to-business relationship. Installers specialize in specific designs from a
manufacturer’scatalogueandofferalimitedrangeoftheseselectionstohomeowners.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 10
Weareexploringtheroleofinstallerchoicesinaworkingpaper[9]usinglinkedjourney
maps for installers and homeowners. Traditionally a journey map is a visual
representationofacustomer’send-to-endexperiencewithabusiness;typicallythisisa
flowchartwith branch points representing customer decision points. A linked journey
maprepresentsacombinationofanumberofjourneymapsanddescribesthecollective
experience of multiple agents; this linked map can highlight major issues in their
integratedexperience.Wediscoveredthatinstallers’decisionsregardingtheirportfolio
of products guide and limit homeowners’ choices and effectively determine which
technologieswillbedeployedonthemarket.Inthissense,installersserveasgatekeepers
forthenewdevelopmentsofPVsystemmanufacturers.
Installersfacecomplexdecisions,notonlywhilechoosingfromalimitedrangeof
offeredtechnologies,butalsowhendesigningaPVsystemforhomeowners.Theirchoices
for products reflect a desire tomaximize their own profits andmay not be perfectly
alignedwiththehomeowners'preferences,themanufacturers'drivetopushtechnology
forward,orthegovernment’sgoalsforincreasingadoptionatamanageablerate.While
theinstallers'roleinsolarmarketdynamicsisapparent,thereisverylittleresearchinto
theiractualdecisionpatterns.Researchconcentratesoneithergeneraltrends,suchas
report[1]whichprovidesanoverviewofrecenttrendsinmarketpricesandvolumes,or
on modeling energy production, such as in [10, 11], which detail models of energy
production,includingrenewableenergysources,toassessimpactonutilitycosts.Another
exampleofthislineofresearchis[12],whereJankoetal.focusonestimatingeffectson
netloadsunderchangingenvironmentalfactors.Theyfoundthatthecoveragearea,the
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 11
direction of the dust storm, and the time of day affected the net load differently.
FrischknechtandWhitefootcreatedastaticmodel thatcapturesasingleperiodofPV
panelmarketsensitivitiestochangesinengineeringparameters[13].Theyfoundthatan
early-stageengineeringdesignperformancemodelcouldbeincorporatedintoadecision
framework.
2.2Investigatingsolarmarketdynamicswithanagent-basedmodel
There are many modeling tools available to investigate the penetration of
technologies, such as the system dynamicmodel or technology diffusionmodels. For
example, Islam [14] predicts the probability of time to adoption of solar panels by
householdsusingdiscretechoiceexperimentsandaBassdiffusionmodel.Thesemodels
aregoodatcapturingtheoveralltrendsofthemarket;however,theylackthecapacityto
model all individual agents’ behavior and information flows.Over the past ten years,
agent-basedmodelshaveemergedasapreferredmethodforamoretargetedapproach
to modeling technology adoption [15], [16]. This research focuses on capturing the
uncertainty of deploying complex engineering products through the multi-stage sale
process. The residential PV market is one of the few markets where a complex
engineering system is being utilized by a final consumer while needing constant
maintenance and monitoring, thus providing detailed, lifelong information to the
maintenancecontractor.Thus,itisaperfectcandidateforthemoremodern,agent-based
modelingapproach.
Wechosetheagent-basedmodelasourmaintoolbecausewewanttobeableto
predicttheeffectsofindividualinstallers’choicesonamarketwithcomplexofferingsthat
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 12
aretailoredtospecifichomeownersandexplicitlymodelthedistributionoftechnological
andsocio-economicparametersthatmayinfluenceinstallers’choices.Theagent-based
approachallowsforexplicitmodelingofpenetrationdynamics.Asiscustomaryforthis
approach,thechoicesforindividual-levelrulesultimatelycreatediffusiondynamicsfrom
thebottomup.ForexampleNasrinpouretal.[17]useanagent-basedmodeltoexplore
thediffusionofinformation.
Anagent-basedmodelissimulation-basedandexplicitlymodelsindividualagents’
actionswhilealsomodelingthenetworkofinteractionsandinterdependenciesbetween
theagents.Simulationoftheassociatedphysicalenvironmentiscustomary.Forabrief
introductiontoagent-basedmodels,see[18].Someoftheiradvantagesareanabilityto
capture multiple complex distributions and an ability to naturally model network
interactions.Thelatterisanintegralpartofenvironmentalconsiderationsasarguedin
[19].
Agent-basedmodelshavebeensuccessfullyusedforevaluatingdemand-oriented
policies,forexample,by[20–24].Zhaoetal.[20]concentratedonmulti-levelmodeling
ofsolarenergygenerationandthewaythatconsumersmakepurchasedecisionstaking
into consideration thepotential energyproduction,which is basedon the geographic
characteristicsofwheretheylive.Zhaoetal.concludedthatsensitivityofhomeowners
tochangesinincentivesisdifferentfordifferentgeographicalregions.RobinsonandRai
expandedonspatialagent-basedmodelingfortheanalysisofPVsystemsadoptionrates
[21].Theyconcludedthatagent-levelbehaviorandsocialinteractionsareimportantfor
explaining patterns of adoption. For other examples of this line of work, see [15].
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 13
RobinsonandRaibroughttogethermultiplesocial,economic,andenvironmentalfactors
into a complex agent-based model to analyze penetration rates arising from
homeowners’ possible choices [21], suggesting that explicit modeling of agents’
characteristicsiscrucialforaccuratepolicyanalysis.However,noneofthisagent-based
modelingworkinsolaradoptionprovidedahigh-resolutionarticulationoftheinstallers'
roleinthesystem,andwearguethatthisabsence,infact,underestimatesthetypeand
levelofexistingbarrierstoadoption.
Thedevelopedmodelhasextensivesocio-economicelements,whichrequirethe
use of probabilistic methods and lead to unavoidable error bands on parameter
specialization. Another property of the PVmarket is the high level of unpredictability
regardingthedevelopmentofnewPVpanels. It issimplynotpossibletobeasprecise
abouthowapersonmightbehaveorhowresearchwillprogressasitistobeaboutthe
amountofelectricityproducedbyasolarpanel.Themodelaimstokeeptheminimum
level of complexity while preserving the ability to explore installer decisions. This
approachofworkingwithaprobabilisticmodelandstudyingtrendsisvaluableanduseful
forlearningaboutsocio-technicalmodels.Notethatthisisnotanoptimizationstudythat
recommends one final design outcome. Instead, our discussion is structured around
trendsandchangesoftrendsundermodelmanipulations.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 14
3MODELDESCRIPTION
3.1Generalflowofthemodel
The dynamic agent-based model simulates market dynamics for a number of
years. The simulation is run for a fixed number of steps, each step representing
approximatelyoneyearofactualtime.
This section provides an overview of important agent actions, with detailed
descriptionsgiven in the sections to follow. Figure1describes theoverall flowof the
model.
Figure1.Majorattributesofthemodelandagents’decisionprocesses
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 15
Eachagentinthemodelhasadifferentgoal.Manufacturers,representedbyan
iconofafactory,playapassiveroleinthismodel.Installers,representedbyahardhat
icon, evaluate newpanels frommanufacturers and propose systems to homeowners.
Homeowners,representedbyahouseicon,evaluatethefinancialviabilityofinvestingin
PVsystemsgiventheirspecificsetofparameters.Governmentdecisions,insteadofbeing
modeleddirectly,areembeddedinthepricingstructureofthepanels.Allprojectsare
connectedtothegridaftercompletion,whichremovestheneedforexplicitmodelingof
utilitydecisions.
Installers.Themodelfocusesonthedecisionbehaviorofinstallers.Atthestartof
eachyear(modelstep),installerschoosewhatequipmenttouseintheirinstallationbids
tohomeowners.Installerscanchoosetokeepusingtheircurrenttechnologyorexplore
newoptionsandswitchtoanotherdesignfromthesameordifferentmanufacturer.Some
installersaremoreinclinedtoexploreexistingofferingsonthemarket;andsomemay
switchtoanewtechnologyiftheexpectedgainishighenough.Afteraninstallersettles
onaspecificdesign,shecustomizesittothehomeowner’sneeds,creatingaspecificPV
systemfortheelectricitydemandlevelandhousesize.
AvailabletechnologiesforPVmodulesdifferintheirefficiencylevelsandintheir
reliability. The efficiency of the panel is its ability to convert a given amount of solar
energy into electricity per squaremeter and is immediately known to everyone. The
reliability,orprobabilityofexperiencingabreakdown,isrevealedaftertheprojectsare
deployed.Reliability,inpart,determinestheproductionofenergyforeachprojectper
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 16
year:ifthesystemgoesdown,itwillrequiremaintenanceforaperiodoftime,anditwill
notgenerateelectricityduringthistime.
Manufacturers represent exogenous technological progress and price their
current panel offerings on their expected efficiency. As time goes by, manufacturers
investinresearchanddevelopmenttoimprovetheefficiencyoftheirpanelsinagradual
manner.Astheydonotfocusonimprovingreliability,itmayeitherdecreaseorincrease
permodelstep.TheresidentialPVmarkethastheclassicformofasignalingmarketwith
hiddeninformation.Inthiscase,thehiddeninformationistherealizedperformanceof
thePVpanel.FudenbergandTirolein[25]provideequilibriumsolutionsforsuchmarkets.
A 2014 IEA report [26] highlights very high levels of reliability. These reliability levels
suggestthatasolutionwillfocusonefficiency.
HomeownersevaluatethefinancialviabilityofinvestinginPVsystemsandmake
thedecisiontoadoptornot.Afixedportionofallhomeownersrespondtomarketing
information during each year and decide to accept or reject installation offers from
installers,dependingontheofferedrateofreturnofinvestmentandthereputationof
theinstallers.
Thispaperfocusesontheresidentialsolarpanelmarket.Withinthismarket,there
aretwomodesofowningsolarpanels:hostownership(56%ofmarket)andthird-party
ownership(44%ofmarket).Hostownershipiseitherfinancedorpaidforupfrontwith
cash.Thereisnoharddataonthelatteroption;butaDOEreport[27]impliesthatcash
upfrontisthemostcommonmodeofacquiringhost-ownedsystems,whilefinancinghas
been replaced by third-party ownership schemes. Adding lease or external financing
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 17
optionswould greatly complicate the analysis, so the paper focuses only on the cash
upfrontmodeoffinancing.
Generally,manufacturershaveaccesstoactualperformanceinformationforPV
panelswhile installershave to relyon their fieldobservationof theperformance.We
modeledoneofthewaysinformationmightflowduringthereal-lifedeploymentofan
extended-life engineering project, and we did it at the level of individual projects.
Installers gather the information about each deployed PV system and use it in an
aggregatewaytogetupdatedestimatesforexpectedmaintenancecosts.Theyalsohave
estimates of homeowners’ demand.Homeowners can only observe the reputation of
installers and do not have access to raw efficiency and reliability data. Figure 2
summarizestheavailabilityoftheinformationtoeachparty.
Figure2.Informationavailabletomanufactures,installers,andhomeowners
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 18
The model was tested under two different scenarios corresponding to two
geographic regions within the U.S. General parameters represented two distinct
environments.TheCAscenariorepresentsSanJose,California,withhighlevelsofsolar
irradiation and a 5% PV market penetration; and the MA scenario represents
Massachusetts,withlowerlevelsofsolarirradiationanda0.5%PVmarketpenetration.
Levelsofirradiationweresetto6.0kWh/m2/dayforCAand4.0forMA,takenfrom[28].
Other scenarioswithvarying levelsof solar irradiationandPVmarketpenetrationare
possible,butareoutsidethescopeofthiswork.
Inthefollowingsections,weprovidedetailsofthedecisionprocessforinstallers,
manufacturers,andhomeowners.
3.2Installer’sdecisionprocess
Thebestapproachformodelinginstallerswithinanagent-basedmodelistouse
intelligentlearningagents[18],astheyallowawaytorepresenttheprocessofconstantly
weighingthebenefitsandrisksofchangingPVofferings.Intelligentlearningagentsare
agentsthatareallowedtoupdatetheirbeliefsinresponsetotheobserveddynamicsof
theenvironment.
There are different approaches tomodeling the decision process of intelligent
learning agents. Wilson and Dowlatabadi [29] provided an overview of existing
approachesinthefieldofenvironmentalresearch.Inthecaseofinstallers,areasonable
approach is tomaximize profit. Sinitskaya and Tesfatsion argue in [30] that this is an
appropriate decision procedure for learning agents in a highly volatile environment.
Bayesianmethodsareusedbecauseoftheamountofinformationthatinstallersgetfrom
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 19
serving already installed PV systems. As installers observe the actual output of the
installed systems, they canupdate theirestimatesof reliabilityof thepanel.Bayesian
methodsprovideawayofcombiningtheirinitialguessesregardingreliabilityofthePV
systemandtheobserveddata.
Research intoBayesian learning shows that it effectively captures the learning
dynamic[31,32].Gillin[33]providesdetailsofthestandardlearningtechniques.While
operating in the market, installers constantly acquire new information that they
incorporate into theirdecisionprocess.Wecapture these features inouragent-based
model by allowing installers to update their expectations after observing market
outcomesoftheirdecisions.
We allow installers to pursue different decision processes that can be either
explorativeorexploitive.The formerassumes thatagentsaremoreopentoexploring
other panels on themarket if it seems that itmight be beneficial to them. The later
strategydescribesagentsthatarelessinclinedtoexplorenewoptionsandprefertostay
with their current choices for longer. The exploration vs. exploitation question has
traditionallybeenakeypartof learning.Theclassicapproachusesmulti-armedbandit
problems,asexplained in[34].Amultitudeofmethodsforreinforcement learningare
describedin[35].Experimentsshowthatinconditionswhenthelearningenvironmentis
notextremelynoisy,itisplausibletoassumethatpeopleusedynamicoptimizationwith
Bayesianlearningtoarriveatoptimalstrategiesforexplorationvs.exploitation[36],and
thisistheapproachusedhere.Weextendouranalysistothecaseofappliedproblem
solvinginadistributed-agentsenvironment.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 20
3.2.1Installer’sdecisionsregardingPVmodulesandpricing
Inourmodel,eachinstalleroffersonlyonetypeofpanelatatimetohomeowners.
Thisassumption is theresultofour interviewswithhomeowners,whoconfirmedthat
theywereofferedone,andrarelymorethanone,optionfortheirPVsystem.Additionally,
duringourinterviewswithinstallers,theystatedthattheylimitthetypesofsolarmodules
becauseofavailabilityissues.Weinterviewedatotalof12homeowners(4inCalifornia
and8inMassachusetts)and9installers(allfromCalifornia).Paper[9]providesdetailed
questionsandproceduresforthoseinterviews.
Theinstallersdon’twanttowaitonthemanufacturerbeforebeginningajob;ifall
theircustomersusethesamemodules,thentheycanmaintainsomeinventory,knowing
itwillbeusedforthenextcustomer.Duringeachstepinthesimulation,eachinstaller
investigatesreplacingtheircurrentofferingwitharandomlyselectedPVmodulethatis
available from manufacturers. Whether or not they decide to adopt a different PV
moduledependsontheirpropensitytoexplore(ratherthanexploit). If theydecideto
adopt, they estimate the expected profit of PV system designs for the panel under
consideration. The expected profit equals the expected revenue minus the costs of
services.Therevenuedependsontheinstallationpricethattheinstallerdecidestooffer
duringeachstep.Next,wedescribethisdecisionprocessindetail.
InstallersmaximizetheirprofitgiventhespecificPVpanelthattheyareoffering
asabasisfortheirdesign.
𝑚𝑎𝑥($?.&Π' (1)
Theexpectedprofitattimetiscalculatedovertheforecastinghorizon𝑇1,$&.#+':
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 21
Π' = 𝑞'BC 𝑝𝑟𝑖𝑐𝑒 𝑝𝑟𝑖𝑐𝑒 − 𝐶'BC 𝑝 𝑚𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 , 𝑞 LM'BC
CrLstuvwxyz
Cr<
(2)
wheredemand𝑞'BC(𝑝𝑟𝑖𝑐𝑒) = 𝑞isestimatedbasedontheofferedrateofreturn
andreputationoftheinstaller.Wewillfirstexplainthistermindetailandthenthecost
term,C.Eq.(2)isastandardprofitmaximizationformulation;forfurtherdetails,see,for
example,[37].
Simple specification in the form of linear regression provides the demand
estimation:
𝑞 = 𝒛?)+'𝜽= + 𝜖 N"#$%&' (3)
where𝑍?)+' = 𝒛?)+',' isthecollectionofobservationsattimetthat isusedin
sequential updating of Bayesian estimates of regression coefficients. Eq. (3) uses a
Bayesian linear regression specification foragent learning; [38]elaboratesondetailed
estimationforthistypeoflearning.𝒛?)+',' includesthemaindemandparametersthat
influencehomeownerchoice:theinternalrateofreturnforinstaller𝑖,itsreputation𝑟𝑒𝑝?,
the internal rate of return (irr) of other installers 𝑖𝑟𝑟P?, and the reputation of other
installers(excludinginstaller𝑖)𝑟𝑒𝑝P?.irriscalculatedinthestandardway.
𝒛?)+',' = [1, 𝑖𝑟𝑟?,', 𝑟𝑒𝑝?,', 𝑖𝑟𝑟P?,', 𝑟𝑒𝑝P?,'] (4)
𝜽= are randomly distributed regression coefficients for estimated demand.
Bayesianprioron𝜽= hasnormal-inverse-gammadistribution.
𝑝 𝜽= = 𝑝 𝜽= 𝜎k 𝑝 𝜎k (5)
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 22
= 𝑁(𝝁�, 𝑉�)×𝐼𝐺(𝑎=, 𝑏=) = 𝑁𝐼𝐺(𝝁�, 𝑉�, 𝑎=, 𝑏=)
Under these assumptions, the posterior predictive distribution is
𝑀𝑉𝑆𝑡k�∗(𝝁∗,��∗
#�∗(1 + 𝑍
~𝑉∗𝑍
~L))whereMVSstandsforMulti-variatestudentdistribution.
Themeanofthisdistributionisusedasapredictive:𝑞~= 𝑍
~𝝁∗.AppendixIdescribesthe
componentsofMVS;italsocontainsinitialparametersrepresentingexpectationofequal
marketsharesforinstallersattheprevailingratesofreturn.
There isnoassumptionof laborandequipmentconstraints.Thisassumption is
reasonablebecausethetimehorizonformaximizingtheexpectedprofitisfiveyears,and
onestepinthemodelisequivalenttooneyear.Overthesetimeintervals,installerscan
useaflexibleamountoflaborandequipment.
NowwewillexplainCfromEq.(2).Eq.(6)introducesallincorporatedcosts:
𝐶'BC 𝑝, 𝑞 LM'BC = 𝑐?)+'#**#'?,) + 𝑐=&+?@) + 𝑐(&$"?'
+𝑐"#'&$?#*+ + 𝑐#="?)?+'$#'?,) + 𝑐"#$%&'?)@ + 𝑐"#?)'&)#).&(6)
Installers have both fixed per period costs and variable costs. Variable costs
dependonthesizeandspecificationsofeachinstallation.Eqs.(7)–(13)givedetailsofcost
calculations:
𝑐?)+'#**#'?,) = 𝜃',.,"(*&`?'a?)+'#**×𝑤'×𝑞 (7)
𝑐=&+?@) = 𝜃=&+?@)×𝑤'×𝑞 (8)
𝑐(&$"?' = 𝜃(&$"?'@&)&$#*×𝑤' + 𝜃(&$"?'+(&.?1?.×𝑤'×𝑞 (9)
𝑐"#'&$?#*+ = 𝑁(#)&*+×𝑞×𝑝𝑟𝑖𝑐𝑒",=R*&,STU (10)
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 23
𝑐#="?)?+'$#'?,) = 𝜃#="?)?+'$#'?,)×𝑤' (11)
𝑐"#$%&'?)@ = 𝜃"#$%&'?)@×𝑤' (12)
𝑐"#?)'&)#).& = 𝑓.({𝑞}LM'BC, 𝑝(𝑚𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒)) (13)
Maintenancecostsin(13)dependontheexpectedprobabilityoffailureforthe
installed system 𝑝(𝑚𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒) and required labor costs to repair the systems
𝑓.({𝑞}LM'BC, 𝑝(𝑚𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒)). Eq (13) includes standard cost estimation, as well as
detailedmaintenanceestimationthatusesagent-specificdatatoupdatetheprobability
distributionfunctionformaintenancecosts.Agent-basedmodelshaveanadvantageof
beingabletouseagentspecificinformationinsimulation,forexample[39]utilizesitto
makeaforecastforpenetrationlevels.𝑝𝑟𝑖𝑐𝑒",=R*&,STUisthepriceofaPVmodulethat
isdeterminedbythemodule’smanufacturer.𝑤'istheprevailinglaborwageattimet.
𝑁(#)&*+isthenumberofpanelsthatisrequiredbythespecificdesign.
AppendixIprovidesspecificfixedcostlevelswhichcorrespondtoaveragecostsof
operating in the U.S. residential PV market for a large-scale installer. The model
verificationandvalidationsectionelaboratesfurtheronspecificchoicesforcostlevels.
Each𝜃? parameterizespartoftheoverallcostsasspecifiedinEqs.(7)–(13).Foreachtime
period𝑡 + 𝜏,where𝑡isthecurrenttimeperiodand𝜏istheforecastingoffset,totalcosts
include allmentioned parts in Eq. (6). One of the parameters that defines a variable
portionofthecostsis𝑁(#)&*+.Itisthenumberofsolarmodulesthatprovides“enough”
electricitytothehomeowner.Inprofitcalculations,“enough”means100%ofelectricity
consumption for an average homeowner, under the assumption that there is enough
physicalspaceontherooffortheinstallationandthatallotherconditionsarefavorable
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 24
forinstallingPVpanels.Whentheactualdesignisofferedtothehomeowner,roofsize
considerationswillbecomepartoftheactualoffer.
Explorationvs.exploitation.Eachinstallerhastheoptiontoswitchtoofferinga
differentPVpanel.Thedecisiontoswitchisbasedontheexpecteddifferenceinprofit
andthepropensitytoswitch.Thepropensitytoswitchisspecifictoeachinstaller,whois
classifiedaseitheranexploreroranexploiter.Inthismodelsimulationofthreeinstallers,
one installer isanexplorerand theother twoareexploiters.Thenumberof installers
reflected the number of big installers present on themarket today. Also, during our
interviewsinstallersexpressedrisk-averseattitudes;thustwooutofthreeinstallerswere
assignedrisk-aversebehavior.
Eq. (14) uses a logistic function to calculate the probability for switching
𝑝?)+',+/?'.Q:
𝑝?)+',+/?'.Q =1
1 + expP
��v��t��
P�M,v��,v
(14)
Appendix I has parameter values for 𝜃?,&. Each set of parameters
𝜃 &`(*,$&$,&`(*,?'&$ (for explorer and exploiter) specifies the propensity to switch to a
differentdesign.Π,*=iscalculatedforexpectedmaintenance,demand,andefficiencyof
thecurrentpanel;Π)&/ iscalculatedbasedontheexpectedmaintenanceforthenew
panel and is subject to the same demand estimations as the current panel. Eq. (14)
ensuresthataninstallerswitchespanelsonlyifexpectedprofitsaresignificantlyhigher,
represented as a continuous function rather than a discrete cut-off. The baseline
assumptionwasthattheprobabilityofswitchingisproportionaltothedistancebetween
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 25
the current and the expected profit. This functional form implicitly includes the
assumption that other considerations, such as market uncertainty, enter into the
installers’ decision-making procedure. Eq. (14) generally specifies randomized action
selection for an agentwho is utilizing reinforcement learning (Bayesian learning); the
coefficientsreflecttwolevelsofrisk-aversionandwereselectedtoberepresentativeof
the observed levels of risk-aversion in real situations. The expected profit, given the
portfolioofprojects,istheagent’sreward,andtheagentisassumedtousethe𝜀-greedy
algorithmforselectinghisactions.Forexamplesofsuchalgorithmssee[40].
3.2.2InstallationandmaintenanceofPVpanels
Installation.Theinstalleruseshercurrently-offeredpaneltocreatespecifications
forinstallation,giventhehomeowner’sparameters.Fixedparametersforallagentsand
allsimulationsincludeenvironmentalparameters,suchaslevelofsolarirradiation,and
difficultyinacquiringpermits.Homeowner-specificparametersareroofsize,electricity
consumption,andhouseholdincome.Whenahomeownerapproachesaninstallerfora
proposal,theinstallerdesignsasystemtoprovideenoughelectricitytocoverdemand
underidealconditions,constrainedbytheroofsize.Todeterminetheprice,theinstaller
workswithintheconstraintsofthecostofinstallationandthepriceperwatt;theprice
perwatt isdeterminedduringtheprofitoptimizationprocedure.Thepriceperwatt is
multipliedbythetotalwattageproducedbythesystemtocalculatethepriceofferedto
thehomeowner.Ifthehomeowneracceptstheproposal,thentheprojectisinstalled.
Maintenance.Duringeachmodelstep,installedprojectsmightexperiencefailure
accordingtotheprobabilitydistributionspecifictoeachPVpaneldesign.Thecomplexity
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 26
ofthefailureisalsorandomlydistributed.Bothdeterminethecostofmaintenancethat
isspecifiedinEq.(13).
Reliability from installer's perspective.Unlikemanufacturers, installers donot
haveperfectinformationonpanelreliabilityinthemodel.Thisassumptionstemsfrom
(a) conversations thatwehadwith installerswho suggestedmanufacturers' reliability
statisticsweresometimes inflatedorbasedon idealconditions,and(b)reportedsolar
panelfailings,whicharehigherthanwarrantyinformationwouldsuggest.Additionally,
reliabilitycanvarywithenvironmentalconditions,suchassolarradiationlevelsandgrid
stability. The installers must determine the reliability of two panels: the one they
currentlyuse(current)andthenewonethatamanufactureroffersthem(new)ineach
model cycle. For the current panel, the installer determined its reliability using the
strategydescribedbelow.
Oncetheinstallerknowsthenumberoffailures𝑛1,theperiodfromonefailureto
another𝑥1,? forproject𝑖,andtheseverityoffailures𝑥.,?,theinstaller/agentcanupdate
hisinternalestimatefortheprobabilitydistributionforfailuresandtheircomplexity.
Under the standard distribution assumption for failure rates of the system,
reliabilityoftheinstallationhasanexponentialdistribution.
𝜆1𝑒Pis` (15)
TheinstallerdoesnotknowtheexactparameterofthedistributionofEq.(15)but
learns itbyobservingtheperformanceof installedsystemsduringthesimulation.The
priordistributionforparameter𝜆1isthegammadistribution
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 27
𝛽1�s
Γ(𝛼1)𝑥�sP]𝑒P�s`
(16)
Prior parameters𝛼<,1 and𝛽<,1 correspond to the optimistic assessment of an
actual system reliability, such as estimated for the example in [41], a study which
investigated different sources and frequencies of PV module failures. To get some
intuitiveunderstandingfortheparametervaluesinAppendixI,itispossibletothinkabout
thepriorvaluesasdescribingasituationwhen1failurein25yearsisexpected.
Forassessingthereliabilityofnewpanels,weinvestigatetheresultsofinstallers
usingoneofthreeestimationstrategies.1)Optimistic:Installersassumethatthepanels
failonlyonceevery25years,usingparametervaluesfromAppendixI;2)Sameascurrent:
Installersestimatethatthenewpanelwillhavethesamereliabilityastheircurrentpanel,
bytheirownassessment(Eq.15);or3)Average:Installersestimatethatthenewpanel
willhaveareliabilityequivalenttotheaverageinstaller-reportedreliabilityofthepanels
onthemarketnow.Section4presentstheresultsofexploringeachofthesescenarios.
Regardlessof thescenario,𝛼1and𝛽1 inEq. (16)updatewithnew information
using standard formulas for a gamma distribution. The resulting posterior predictive
distribution for the expected time before the next failure follows a Pareto Type II
distribution. Complexity of maintenance has a normal distribution, with parameters
𝜇"#?)' and𝜎"#?)'k . The prior distribution for these parameters is the normal-inverse-
gammawiththeparameters𝜇<,𝑣,𝛼,and𝛽,whichupdateusingstandardformulas.The
resultingposteriorpredictivedistributionis
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 28
𝑡k��(𝑥~𝜇�,𝛽�(𝑣� + 1)𝑣�𝛼�
) (17)
whichisanon-standardt-distributionwithscaleandlocationparameter:
𝑋 = 𝜇� +𝛽�(𝑣� + 1)𝑣�𝛼�
𝑇 (18)
where𝑇 ∼ 𝑡��,whichisastandardt-distributionwith𝛼�degreesoffreedom.
AppendixIprovidesinitialvaluesforpriorparametersofthedistribution:𝜇<,𝑣<,𝛼<,𝛽<.
Installer reputation. An installer's reputation relies on the uptime (productive
energycreation)oftheirexistingprojects,asequipmentfailuresresultindowntime.The
update procedure for estimates of reputation uses total production from all of an
installer's projects, 𝑝𝑟𝑜𝑑. Reputation follows the inverse-gamma distribution. This
distributioncorrespondstotheassumedexponentialdistributionforfailuresinEq.(15).
Thisdistributionprovidesthebestfitforthecasewheninstaller’sreputationdependson
𝑝𝑟𝑜𝑑only.The𝛽$&(scaleparameterofthedistributionisfixedatthelevel1.0.Theprior
value forparameter𝛼$&( of shape is1.0. Eq. (19) is the resultof applyingmethodof
momentstoestimatingtheparameter𝛼',$&(.Foreveryotherperiod,excepttheinitial,
Eq.(19)describesthewaytoupdatetheshapeparameter:
𝛼',$&( = 1 +1
1𝛼'P],$&(
𝑁(&$𝑁(&$ + 1
+ 𝑝𝑟𝑜𝑑'𝑁(&$ + 1
(19)
Where
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 29
𝑝𝑟𝑜𝑑' =1
𝑁($,-&.'+𝑝𝑟𝑜𝑑',?
¤¥ut¦vwzy
?r]
(20)
istheaverageuptimeproductionoverallofaninstaller'sprojects.𝑁(&$ istheadjusted
numberofperiodsforestimation,whichisequalto𝑛< + 𝑡.𝑡isthecurrentperiodofthe
simulation, and 𝑛< = 10 defines initial reputation "stickiness," meaning that realized
failuresaffectestimatedreputationwithaweightoflessthanone.
3.3Manufacturer’sdecisionprocess(passive)
Themanufacturerresearches,designs,andpricesnewpanels,representedbythe
modelaspassiveactionsfollowingrulesforpricingandexogenousspeedoftechnological
progress.
3.3.1ResearchinganddesigningnewPVpanels
Ineveryperiod,eachmanufacturerupdatestheirPVpaneldesigniftheirresearch
efforts are fruitful. The design of the new panel begins with drawing a randomly-
determined efficiency improvement, as well as expected reliability (time between
failures)andmaintenancecomplexity.Thisassumptionisasignificantsimplificationofan
actualdesignforreliability.Forexamplesanddiscussionofproblemsthatfacedesigners
whodesignforreliability,see[42].Evenasimplifiedmodelcanstillprovideinsightsinto
optimaldesignchoicesbymanufacturers,asarguedin[43].
Efficiency,ef, improvesoverallmanufacturersatan individualrandomrate,so
thatinthetimeperiod𝑡foreachmanufacturer
𝑒𝑓' = 𝑒𝑓'P]𝑒§vsB¨vs©M,� (21)
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 30
and
𝜖<,] ∼ 𝑁(0,1) (22)
Bothmanufacturerandinstallerknowtheefficiencyofthenewpanel.Expected
reliability is formed in the samewayasexpectedefficiency,but is only known to the
manufacturer(seeSection3.2.2andEq.(15)forinstaller'sequations):
𝜆' = 𝜆'P]𝑒§ªB¨ª©M,� (23)
Generally,allpaneldesignparameterscanincreaseordecreasewitheachmodel
step.Themanufacturerdecidestoofferthepaneltoinstallersonlyifitoffersabenefit
overtheirexistingoffering,intermsofefficiency.Reliabilitydoesnotaffectthisdecision.
Expected maintenance costs, known only to the manufacturer, are
𝑁(𝜇"#?)', 𝜎"#?)'k ),andupdatewitheachmodelstepinthesamefashionasefficiencyand
reliability, with an appropriate adjustment for multivariate generation. Let 𝜃"#?)' =
(𝜇"#?)', 𝜎"#?)'k )bethecombinationofparametersfordistributionofmaintenancecosts.
Eq.(24)andEq.(25)describenewvaluesfor𝜃"#?)':
𝜃',?,"#?)' = 𝜃'P],?,"#?)'𝑒©«,¬x«�z (24)
𝜖"#?)' ∼ 𝑵(𝝁©¬x«�z, 𝚺©¬x«�z) (25)
AppendixIhaslevelsforotherparametersofthedistribution:𝜇&1,𝜎&1k ,𝜇i,𝜎ik,
𝝁©¬x«�z,𝚺©¬x«�z.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 31
3.3.2Manufacturer’spricingscheme
Eq.(26)andEq.(27)describetheinitialcalculationsofprices.Theyuseestimated
priceperefficiencyunit.
𝑝𝑟𝑖𝑐𝑒/#'' = 0.65 (26)
𝑝𝑟𝑖𝑐𝑒",=R*&,STU = 𝑝𝑟𝑖𝑐𝑒/#''𝑁/#'' (27)
𝑁/#''ispeakproductioninwattsunderstandardtestconditions.Aftertheinitialperiod,
𝑝𝑟𝑖𝑐𝑒/#'' decreases along a learning rate; data from [6] determined the choice of a
specificlearningrateof8%.
3.4Homeowner’sdecisionprocess
Ineverymodelstep,installerspresenthomeownerswithaPVsystemproposal,
whichhomeownersacceptornot.Thepromisedinternalrateofreturnandtheinstaller's
reputationguidethisdecision,whichisalso,inpart,determinedbytheincomelevelof
thehomeowner.Higherincomelevelsrequirelowerlevelsofexpectedreturn,withthe
assumptionthatatacertainlevelofincome,peoplechoosePVsystemsforreasonsother
thanfinancial,suchasenvironmentalconcernsorpropensitytobeanearlyadopter.This
nuanceisincludedbasedontheresultsofourinterviewswithcurrentPVsystemowners.
During those interviews, homeowners who belonged to the highest income bracket
expressedtheirdesiretocontributetothegreenmovementandwerelessstimulatedby
thepromisedreturnsontheirinvestment,whilestillrequiringpositivereturns.Another
assumption is the importance of the installer’s reputation,which includes howmuch
hassleisinvolvedforthehomeownerinmaintainingtheequipment.Theinternalrateof
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 32
return incorporates the installer’s reputation, which allows us to include the
homeowners’concernsexpressedduringinterviews.
Theprobabilityofahomeowneracceptinganygivenproposalisalogisticfunction
withthefollowingspecification:
𝑝Q,+/?'.Q =1
1 + expP?$$×�°,±P
]�°,�
(]B�°,M�°,�
²]<<<)
³ ��´µ°,M
�°,¶
(28)
Figure3providesintuitionregardingtheresponseofEq.(28)tochangesinincome
level,from$10Kto$100K,andinternalratesofreturn.Notethatthethresholdvalueof
therequiredrateofreturndependsonthehomeowner'sincome.AppendixIhasother
parametersforthelogisticfunction.Figure4illustratestheresponseofthedistribution
tochanges inotherparameters.Homeownerswillmakeadecision toadoptPV if the
promisedrateofreturn ishighenough,withthecaveatthathomeownerswithhigher
levelsofincomearehappywithlowerratesofreturntoconsideradoption.Parameters
inEq.(28)controlthegeneralslopeofthefunctionandthelocationoftheswitchpoint.
Eq. (28) is based on a number of heuristics that homeowners expressed during their
interviews,aswellasonstudies,suchas[44],thatusedreturnoninvestment.Thespecific
form is one of the possible ways of aggregating both those heuristics and modeling
uncertaintywithrespecttootherparametersofagent’sdecision-making.Thechoicethat
a homeowner makes is probabilistic, and there exists no specific threshold for his
decisions.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 33
Figure3.ProbabilityofhomeowneracceptingPVproposal,givenrateofreturn(irr)andlevelofincome
Figure4.ProbabilityofhomeowneracceptingPVproposal,givenrateofreturn(irr)anddifferentlevelsofparameters
Ahomeowner,whodoesnotknowthereliabilityofthepanel,usesthereputation
ofaninstaller𝑟𝑒𝑝? toadjusttheexpectedrateofreturnfortheoffereddesign𝑖𝑟𝑟&.The
resultingrateofreturnis𝑖𝑟𝑟& ⋅ 𝑟𝑒𝑝?.BasedonassumptionsofSection3.2.2andEq.(19),
Eq.(29)usesthemeanoftheestimateddistributiontocalculatereputation:
𝑟𝑒𝑝? =1
𝛼',$&( − 1𝛽$&( (29)
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 34
The internal rate of return can be negative in themodel. The figures show a
simplified visual presentation of a range of [0, 1], but this does not represent the
constrainton the internal rateof return.Anegative internal rate couldhappenwhen
savingsontheelectricitybillarenotoffsettoahighenoughdegreebythepurchaseprice
ofthePVsystem,orwhennet-meteringpriceswillresultinlowrealizedsavings.
We further explain our reasons for choosing specific parameters in the next
section.
3.5Modelverificationandvalidation
ThispaperispartofanongoingprojecttoinvestigatetheresidentialPVmarket,
andourprimaryfocusatthisstageistotestasmall-scalemodelthatdemonstratesthe
possibilitiesoftheagent-basedapproach.Wehaveperformedavarietyofcalibrationand
validationactivitiestothisend.
3.5.1Calibration
We used data frommultiple sources for calibrating parameters of themodel.
StatementsfromSolarCityCorporationservedasastartingpointforcalibratinginstallers’
profitfunctionparameters.Weusedthecoststructurefrom[45]forcross-checkinginitial
estimates.WedidnotusethecurrentPVsystemprices,reportedin[45],toinitializethe
model;insteadweallowedtheinstallertoofferprofitmaximizingprices.Weensuredthat
NREL-reportedpriceswereclosetothosecomingfromtheprofitmaximizingsolution.
We used actual data regarding the total number of installations and each
installer’smarketshare,givenin[1],todefinethemarketsizeforoursimulationandto
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 35
assignequalportionsofthemarkettoeachinstaller.Weassumedmoderateriskaversion
tocalibratethelevelofincentivesforexploringorexploiting.
WeusedtheIEAreport[26]asafoundationforsettingparametersgoverningthe
reliabilityofthepanels.Weusedinformationfrom[6]asasourceofforecastsforfuture
pricesfornewPVpanels.Wealsousedthissourcetodiscovermanufacturers’research
processesfordevelopingnewpanels. Itshouldbenotedthattheresearchprocessfor
developingnewpanelsishighlyunpredictableforatimehorizonof5years.Whileresults
of the simulation provide insights into market dynamics, the precision of the results
significantlydecreasesovertime.
Homeowner preferences reflect current market returns on investment with
adjustmentforperceivedriskofinvestinginPVpanels.Homeowners’physicalandsocio-
economic parameters replicate distributions inferred from RECS [46]. For all model
scenarios, residentialelectricitypricesare fixedat$0.15perkilowatthour.Each solar
purchasereceivestheactualcurrentfederalinvestmenttaxcreditof30%ofthepurchase
price.Theinflationratewas2%.
The number of agents in our model (7 manufacturers, 3 installers, and 1000
homeowners)reflectstheactualnumberofmajormanufacturersandinstallerscurrently
inthemarket.Wechosethisset-upbasedonananalysisoftheNEMdatabase[47]anda
DOEreport[48],whichhighlightseightmajorproducersofPVpanels,withfourofthem
havingmuchsmallershares,andthreeofthembeingmajorinstallers(Vivint,SolarCity
Corporation, Sunrun) that operate in theU.S. residential solarmarket.Wemodel the
leadersinthemarketandassumethatsmallerinstallerswillfollowtheirdecisionpatterns
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 36
and pricing. We picked the conservative assumption with respect to their decision
patternsandleavetheexplorationofalternativestofuturework.
3.5.2Verification
Verifying the software code involved multiple steps. The first step was
documentingthecodebase; thesecondstepwasconductingmultiplecodereviewsby
otherteammembers.Debuggingwasamajorpartofourverificationefforts,aswellasa
step-by-stepverificationofeachmethod.Anotherpartwasfunctionaltestingofthemain
simulation blocks, such as manufacturers’ decisions, installers’ decisions, and
homeowners’ decisions. Independent implementation of the engineering model in
PythonservedasaverificationfortheC++counterpart.
3.5.3Qualitativeandempiricalvalidation
Webasedourqualitativevalidationonarangeofassumptionsaboutthereliability
andefficiencyofPVpanels.Ourmodelperformed ina reasonableway for the tested
range.Islam[14]testedtherangeofhypothesisontheadoptiontimeprobabilitiesfor
households.While our model represents a different methodological approach and is
focusedoninstallers,someoftheresultscouldbevalidatedagainstthosepresentedby
Islam.Islam’sresultssupportthehypothesisthathouseholdsthatarelesssensitivetothe
financialbenefitsofinstallingPVsystemswillhaveahigherprobabilityofadopting.The
sameresultholdsinourmodel,wherehouseholdswithhigherincomesadoptPVsystems
more.Anotherhypothesisthat[14]confirmedisthehigheradoptionratesforhouseholds
thathavehigherpreferencesforenergycostsavings. Inourmodel,higherenergycost
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 37
savingsarealsoassociatedwiththehigherprobabilityofadoptingaPVsystem.Sections
4and5belowpresentour resultsbasedon the rangeofassumptions thatmatch the
historicaldata.
Industryspecialistsconfirmedthat industryparticipantsgavelessweighttothe
reliabilityofPVpanels than to theexpected financialbenefits fromdeploying specific
typesofPVpanels.
4RESULTSANDDISCUSSION
This section describes the results of running two scenarios (CA, MA) under a
number of different conditions. For each scenario, there were 7 manufacturers, 3
installers,and1000homeowners.Themodelverificationandvalidationsectionabove
providesthereasonsforourchoiceofspecificnumbers.Each15-yearruntook5minutes
tocompleteincompiledC++.Theresultspresentedhererepresenttheaverageof100
runsusingdifferentseeds.2
Therearethreemainindicatorsthatgiveasenseofmarketconditionsoverthe
years:
1. Hitpercent.ThepercentofhomeownerswhoadoptaPVsystemeachyear.Notethat
Hitpercentisconstrainedtoamaximumof10%,whichisthemaximumnumberof
homeownersapproachedbyinstallerseachyear.HitpercentindicatesPVpenetration
foragivenyear.
2 The modelisavailableatthepublicgithubrepositoryhttps://github.com/wilfeli/ABMIRISLab.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 38
2. Accumulated percentage of installations, total number of installations. The
percentage(ornumber)ofallthehomeownersthathaveeverinstalledPVsystems
ontheirroof,whichindicatestheaccumulatedPVpenetrationlevel.
3. Priceperwatt.Theaveragepurchasepriceperwattforsystemsinstalledinagiven
year.
Therearetwoadditionalindicatorsthatareusefultofollow,whilenotingthatthey
arepartiallydeterminedbymodelparameters,asindicatedinSection3.1:
4. Efficiency.Efficiencyisthepercentageofenergyfromthesunthatapanelconverts
intoelectricity.TheefficiencyofmodernpanelsintheU.S.currentlyhoversaround
20%.
5. Reliability. In the results, reliability is reported as the number of years with one
failure, from the manufacturer's point of view (as opposed to the installers' and
homeowners'estimates).Forexample,areliabilityof"20"meansthatthepanel is
predictedtofailoncein20years.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 39
Figure5.OverallmarketbehaviorsinbothCAandMAscenarios
AnoverviewofthemarketbehaviorisshowninFigure5.Thehitpercent,taken
togetherforbothCAandMA,fluctuatesyearoveryearanddoesnotshowadefinitive
trend,althoughitseems,ingeneral,todecreaseforCA;howeverforMA,itseemstopeak
and then decline. The total number of installations trends slightly higher in the CA
scenario compared to the MA scenario. For both scenarios, the penetration level
increasestoaround14%(140ofthe1000homeowners)bytheendofthesimulation,as
shownbythetotalnumberofinstallationsattime15.Overall,roofsizespresentaphysical
limitation to thenumberofpossibleeffective installations,but this shouldbeat least
partially offset by increases in panel efficiency, which decreases the effective size of
installations.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 40
Figure6.CAscenario:threeapproachestoestimatingpanelreliabilityandtheefficiencychoicesbyinstallers
Figure6showsthatefficiencyincreasesovertime.Itpresentsthethreedifferent
scenarios that installers can use to gauge panel reliability: market average, same-as-
currentpaneloffering,andoptimistic.Whilethesethreestrategiesdoaffecttheactual
reliabilityofthepanelsofferedbymanufacturers,thestrategieshavenoeffectonthe
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 41
pushforefficiency.Manufacturershaveastrongtendencytoswitchtodesignswitha
higherlevelofefficiency,evidentfromtheupwardtrendinefficiencypresentedinFigure
6.Installerswhoareexplorers(er)andexploiters(el)allpursuepanelswithhigherlevels
ofefficiency,aseffectsoflowerreliabilityareverylimited.Installerreputationhaslittle
effect on homeowner decisions. The growing market provides a good incentive for
adoptingnewtechnologyanddecreasesitsrisks.
Thus, when the benefits of improving efficiency are well-known to all market
participants, and the knowledge (or reality) of system-failure is low, it pays for
manufacturerstoinvestinefficiencyimprovements.Forinstallers,itisbettertopursue
anexplorationstrategywhenbenefitsarehighandgeneralrisklevelsarelow—evenfor
installersassignedtopreferanexploitationstrategy.Thisispromisingmodelbehavior,as
itmatchesbothintuitionandactualindustryperformance.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 42
Figure7.CAscenario:thechangesofhitpercentandpriceperwatt(toppanel),andaccumulatedpercentageofinstallations(lowerpanel)overtime
Figure 7 presents results for price-per-watt dynamics and penetration level by
income category and by level of electricity consumption in the CA scenario. The hit
percent decreases gradually, while price per watt remains steady. Homeowners with
more income and electricity consumptionmake up themajority of those that opt to
install.
Figure 8 shows the simulated results for theMA scenario.Over time, partially
controlled by parameterization, price perwatt remains stable in the CA scenario but
decreasesintheMAscenario.PricesstabilizeathigherlevelsintheCAscenariocompared
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 43
totheMAcase.HighCAenergypricesallowhomeownerstoaccepthigherpricesandstill
receive a reasonable return on their investment. In theMA scenario, price per watt
stabilizesatlowerlevelsthaninCA,becauselesssolarenergymeanshomeownersrequire
ahigherrateofreturn.
Figure8.MAscenario:thechangesofhitpercentandpriceperwatt(toppanel),andaccumulatedpercentageofinstallations(lowerpanel)overtime
Themodelpricesarehigher than those reported in theDOEanalysis [6],$3.0-
4.0𝑊=. inourmodelvs.$1.8𝑊=. foryear2020in2020prices.TheDOEreportprojects
that those pricesmight be achieved if the strong push from the government for the
developmentofphotovoltaicscontinues.Ourmodeldoesnotincludenewmeasuresand
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 44
initiativesthatthegovernmentmightimplementtoachievethatgoal;andthusourmodel
givesaconservativepriceestimate.
A number of assumptions would shift the modeled forecasted price, such as
assumptions about exogenous price dynamics. But the qualitative conclusions would
remainthesame.
Agent-basedmodelingallowsustolookatthedynamicsofthepenetrationlevel
byincomegroupandelectricitybill.AsresultsinFigure7andFigure8areaveragedacross
simulationrunswithdifferentseeds,itisinstructionaltoinvestigatethegeneraldynamics
ofincreasingpenetrationssharesforhigherincomeandelectricitybillgroups.Eachbarin
thebottompanelofFigure8,forexample,haslowerincome(andlowerelectricitybill)
groupsatthebottomandhighincome(andhighelectricitybill)atthetop.Itisnosurprise
that the relative penetration level is much higher for high income groups and those
familiesthathavehighelectricitybillsforbothCAandMAscenarios.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 45
5CONCLUSION
Figure9.Factorsthatdeterminemarketoutcomesinthepresenceofdifferentdecisionprocessesbyinstallersandhomeowners
Thematrix in Figure 9 summarizes, in qualitative form, the sensitivity analysis
resultsofthemodel.Inthematrix,allfactorsthatinfluencethecurrentpenetrationlevel
("immediate characteristics") or expected future sales belong to a few categories:
physical characteristics (including design characteristics), market characteristics, and
preferences characteristics. All factors were evaluated in terms of changes of total
penetrationlevelswithrespecttothebaselinescenarioforCalifornia.Sensitivityinthe
tablebelowprovidedthedataforthesummaryFigure9.
Table1.Sensitivityvaluesforthemodelcharacteristics.Sensitivityisdefinedaschangeofpenetrationratewithrespecttothebaselinescenariowhenthemodel
characteristicisvariedaroundthebaselinescenariovalueandotherparametersarefixedattheirbaselinevalues.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 46
Characteristic Sensitivity Impact
Maintenance cost 0.002 Low
Market size 0.027 Low
Reliability 0.062 Moderate
Propensity of firms to switch 0.065 Moderate
Roof size 0.079 Moderate
Efficiency 0.213 High
Hard costs 0.233 High
Solar irradiation 0.598 High
Propensity of homeowners to choose 0.978 High
Soft costs 1.419 High
We explicitly model the design parameters of the solar PV systems as their
efficiency and reliability. We find that between efficiency and reliability, efficiency
dynamicsshapethemarketoutcomesthemost,whilereliabilityparametersonlyguide
someofthedecision-makingandhavealess-profoundeffect.Maintenancecosts,tiedto
thereliabilityofthesystem,donotrepresentamajordecisionfactorforhomeownersor
installers,duetotherelativerarityofmaintenanceeventsandtheirlowcost.Installers
trackallof theirPVsystem installations throughtheir lifetimetodirectlyestimatethe
maintenance costs and other performancemeasures. The Installers usemaintenance
information on existing projects to update their predictions of expected cost of
maintenance,asshowninFigure1.Wemodelthistodemonstratehowacomplicated
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 47
technologyproductthathasanextendedlifespancanbeintegratedintoasystemmodel
andprovideadditionalinformationfordecision-makingindesign.
Wealsofindthatroofsizeintroducesrestrictionsonthepossiblesystemsizeand,
assuch,thepotentialmarketsize.Interviewswithinstallerssupportthisfinding.Installers
stated that the physical properties of roofs contribute directly to the estimated
installationprice.Futureimprovementsinpanelefficiencywillpartiallymitigatetheroof-
sizeeffect.Also,exogenouschangesinthelevelofsolarirradiation,asexhibitedintheCA
vs.MAscenarios,altertherealizedmarketpriceperwatt.Inbothscenarios,thetendency
tofollowexplorationstrategydominates.
Asformarketcharacteristics,thepotentialmarketsizedeterminesthefinancial
viabilityofanystrategyandthusisimportanttothedecision-makingagents.However,
costs(hardandsoft)aremoreimportantfactorssincetheydirectlydecidetheexpenses
anddeterminetheprofitabilityofthesystemsandatwhatpricepoint.Costs,representing
theeconomicsofcurrentlyavailablepanels,areparticularly important intheanalyzed
scenarios.Thisconclusionissupported,forexample,bytheNRELreportbyFuetal.[45],
whoanalyzed installationdynamicsandPVsystems.Theefficiency levelsofPVpanels
directlyinfluencetheattainablehardcostslevels.Thisfindingexplainswhyinstallerstrack
efficiencylevelssoclosely.
OtherfactorsmightinfluencethechoicesthatinstallersmakeonthePVmarket.
For example, shocks to government policy might shift demand curves and result in
temporaryout-of-stockevents.Testingthosescenariosispossiblewiththecurrentmodel
butisoutsidethescopeofthework.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 48
Apossiblelimitationofthequalitativeconclusionsisthatwefixedtheelectricity
priceinsteadofitfluctuatingovertime.Ifthepriceofelectricityincreased,theadoption
ratemayhaveincreased.However,allowingtoomanyparameterstovarymakesitmore
difficulttounderstandthereasonsfortheoutput;thus,thispotentialvariablewasheld
fixed.
Alsoimportantarethemodel'sassumptionsaboutthepreferencesof installers
(explorationvs.exploitation)andhomeowners (ratesof returnon investment).Future
workwillexploremoresophisticatedmodels,includingexpandinghomeowners’decision
choices to include design parameters of the PV system. We will use survey data,
specifically choice-based-conjointmodelsbuilt from surveys, to articulatehomeowner
preferences.Wealsoplantoincludeinfuturemodelsrepresentationsofgovernmentand
utilitydecisions.Additionally,thispaperdoesnotexploredifferentpurchasemodels,such
asleasing.Researcherscouldadaptthecurrentmodeltoaddressthis.Wedonotdiscuss
home renters in this study, as a typical home rental lease agreement forbids the
modification of or addition to the house structure, nor do renters carry the proper
insurance to permit installation. Renters could be modeled as an influencer on a
homeowner'sdecision.
Asthemodelcyclestomaturity,thedynamicsofthebalancebetweenexploration
andexploitationchange,and the reliability isweightedmoreheavilybyhomeowners.
Another factor that continues to shape these dynamics is low-price, low-quality
competition from manufacturers that specialize in such systems. The reason why
efficiencyisthedominantfactorindecision-makingisbecauseofourmodelassumptions
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 49
andbecauseoftheexistingcostandinformationstructureofthemarket.Theresearch
intoresidentialsolarPVownersbyRaietal.[49]supportssomeoftheseconclusions.Rai
et al. found that financial considerations are important for solar adopters. But other
results,suchashowspecificinstallersmaketheirdecisionsandwhattheiroverallroleis,
areopenquestionsforfurtherinvestigationbyresearchers.
Ourassumptionsandgeneralizationsdoimposelimitationsontheimplicationsof
the work. These limitations include not explicitly modeling manufacturers' research
priorities.Explicitmodelingofthesemightchangetheunanimousdominanceofefficiency
as the major deciding factor for installers. The restrictions on choice also limit our
conclusions.Forexample,installerscanonlyconsideronemanufacturereachcycle,and
homeownerscanonlyconsideroneinstaller,whooffersonlyonepaneltype,eachcycle.
Thesechoicerestrictionsprovideclarityontheagent-basedmodelconclusions,without
toomany"movingparts,"butitisdifficulttoperformbasicvalidationoftheresultsand
explorehigh-leveltrends,aspresentedhere.
Another potential issue lies with imposing specific functional forms on the
reliabilityandcomplexitydistributions.Whileourassumptionsareconservative,itcould
bearguedthatweshouldinvestigateotherpossibleapproaches.Wehavenotbeenable
tofindcomparableresearchpaperstoprovidethecomparativeanalysis.Therearealso
limitations on the installer when choosing potential PV panels for new proposals or
considering only profit as the goal. But it is unclear if more explicitmodeling of this
procedurewouldaltertheresults,whilethelevelofmodelcomplexitywouldsignificantly
increase.Ourmodeldoesnotincludegovernmentandutilitypermittingprocesses,but
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 50
theywillbeaddedinfuturework.Properlyaddressingsomeproblems,suchaschanges
inthePVindustrystandardsatthestateandcountylevel,requireschangingthefocusof
themodelandisthereforeoutsideofthescopeofthework.
Tosummarize,theanalysisinvestigatedthedynamicsofPVsystempenetrationin
theU.S. residentialsolarenergymarketusinganagent-basedmodel. Inparticular,we
focusedonanintermediaryagent,installers;thiswasarticulatedinthemodelasguiding
designdevelopmentsofPVpanels.Themoretraditionalapproach is todirectlymodel
homeowners as guiding these developments through their preferences/choices. The
modelarticulatedtheinstallers'decisionprocessasoneofexplorationvs.exploitation,
while maximizing profits. Whether the exploration or the exploitation technology
adoptionstrategydominatesdependsonthespecificsofthemarket,suchaseffectof
reputation,whichwaspossiblyunder-representedinthemodel.
As represented, the installersexplorenewpaneloffersmore than theyexploit
existingpanels,andthisdrivestechnologicaldevelopment(panelefficiency).Thisresult
haspotentialimplicationsforpolicy-makersatthestateandnationallevel;ifpoliciescan
alleviate risks from new panel technologies, perhaps by financially compensating
homeowners for systemdowntime, efficiency shouldbehighly-sought over reliability,
whichwouldbeaboontotheprogressofthesolarindustry.
ACKNOWLEDGMENTS
ThisresearchisbaseduponworksupportedbytheNationalScienceFoundation
underGrantNo.1363254.Anyopinions,findings,andconclusionsorrecommendations
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 51
expressedinthismaterialarethoseoftheauthorsanddonotnecessarilyreflecttheviews
oftheNationalScienceFoundation.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 52
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APPENDIXI
Parametervaluesforthemodel
Parameter Value Description
Initialparametervaluesfortheinstaller'sdemandestimationprocedure.𝝁𝟎,𝜽 (-0.002375
5.9375,0.002375-2.375,
-0.002375)
Bayesianpriorforthemeanofthedistributionfordemandfunction
𝑽𝟎,𝜽 0.5𝑰º Bayesianpriorforthevarianceofthedistributionfordemandfunction
𝛼<,= 1 Initialvalueforparameterforpriorfordemandfunctiondistribution
𝑏<,= 1 Initialvalueforparameterforpriorfordemandfunctiondistribution
𝒁𝟎,𝒊𝒏𝒔𝒕 1.0, 0.1, 1.0, 0.1, 1.0 Priorvaluesfordemandfunctionestimation
𝑁"#$%&' 50000 Marketsize
Parametervaluesfortheinstaller'sdecisionprocedure:costfunction.𝜃.,"(*&`?'a?)+'#** 100 Parameterforcomplexityofinstallation
𝜃=&+?@) 350 Parameterfordesigncosts
𝜃(&$"?'@&)&$#* 200 Parameterforcostestimationofpermitting,generalpart
𝜃(&$"?'+(&.?1?. 50 Parameterforcostestimationofpermitting,designspecificpart
𝜃#="?)?+'$#'?,) 2000000 Parameterforadministrativecosts
𝜃"#$%&'?)@ 2500000 Parameterformarketingcosts
Parametervaluesfortheinstaller'sdecisionprocedure:propensitiestoswitch.𝜃<,&`(*,$&$ 1 Parametervaluesforexplorer/exploiter
decisionprocess𝜃],&`(*,$&$ 0.25 Parametervaluesforexplorer/exploiter
decisionprocess𝜃<,&`(*,?'&$ 1.5 Parametervaluesforexplorer/exploiter
decisionprocess
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 56
𝜃],&`(*,?'&$ 0.5 Parametervaluesforexplorer/exploiterdecisionprocess
Parametervaluesfortheinstaller'sdecisionprocedure:priorsforreliabilitydistribution.𝛼<,1 1 Initialvalueforpriorforfailure
distribution𝛽<,1 25 Initialvalueforpriorforfailure
distributionParametervaluesfortheinstaller'sdecisionprocedure:priorsforcomplexitydistribution.
𝜇< 50 Initialvalueforpriorforprobabilitydistributionformaintenance
𝜐< 1 Initialvalueforpriorforprobabilitydistributionformaintenance
𝛼< 1 Initialvalueforpriorforprobabilitydistributionformaintenance
𝛽< 50 Initialvalueforpriorforprobabilitydistributionformaintenance
ParametervaluesforthedesignofPVpanelsformanufacturers.𝜇&1 0.0025 Parameterforefficiencydistribution
𝜎&1k 0.0025 Parameterforefficiencydistribution
𝑒𝑓< 0.16 Initiallevelofefficiency
𝜇i 0.0 Parameterforreliabilitydistribution
𝜎ik 0.65 Parameterforreliabilitydistribution
𝜆< 0.2 Initialvalueforreliability
𝝁𝝐,𝒎𝒂𝒊𝒏𝒕 0.0, 0.0 Parametersforcomplexityofmaintenancedistribution
𝚺𝝐,𝒎𝒂𝒊𝒏𝒕 0.01 0.010.01 0.02 Parametersforcomplexityof
maintenancedistribution
𝜇<,"#?)' 160 Initialvalueforparametersforcomplexityofmaintenancedistribution
𝜎<,"#?)'k 400 Initialvalueforparametersforcomplexityofmaintenancedistribution
Parametervaluesforthehomeowner'sdecisionprocedure.𝜃Q,< 1.5 Parameter0ofhomeowner’sdecision
function𝜃Q,] 2 Parameter1ofhomeowner’sdecision
function
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 57
𝜃Q,k 0.02 Parameter2ofhomeowner’sdecisionfunction
𝜃Q,½ 0.5 Parameter3ofhomeowner’sdecisionfunction
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 58
Tablecaptions:
Table1.Sensitivityvaluesforthemodelcharacteristics.Sensitivityisdefinedaschangeofpenetrationratewithrespecttothebaselinescenariowhenthemodelcharacteristicisvariedaroundthebaselinescenariovalueandotherparametersare fixedat theirbaselinevalues.
Journal of Mechanical Design
Ekaterina Sinitskaya MD-18-1303 59
Figurecaptions:
Figure1.Majorattributesofthemodelandagents’decisionprocesses
Figure2.Informationavailabletomanufactures,installers,andhomeowners
Figure3.ProbabilityofhomeowneracceptingPVproposal,givenrateofreturn(irr)andlevelofincome
Figure4.ProbabilityofhomeowneracceptingPVproposal,givenrateofreturn(irr)anddifferentlevelsofparameters
Figure5.OverallmarketbehaviorsinbothCAandMAscenarios
Figure6.CAscenario:threeapproachestoestimatingpanelreliabilityandtheefficiencychoicesbyinstallers
Figure7.CAscenario:thechangesofhitpercentandpriceperwatt(toppanel),andaccumulatedpercentageofinstallations(lowerpanel)overtime
Figure8.MAscenario:thechangesofhitpercentandpriceperwatt(toppanel),andaccumulatedpercentageofinstallations(lowerpanel)overtime
Figure9.Factorsthatdeterminemarketoutcomesinthepresenceofdifferentdecisionprocessesbyinstallersandhomeowners