Examining the Influence of Solar Panel Installers on...

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



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.



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



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



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



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?



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



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



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.



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



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



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].



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.



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



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



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



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



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



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.



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,$&.#+':



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)



= 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)



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



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



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



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



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



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



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)



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.



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



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.



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)



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



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



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



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.



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.



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.



Figure6.CAscenario:threeapproachestoestimatingpanelreliabilityandtheefficiencychoicesbyinstallers

Figure6showsthatefficiencyincreasesovertime.Itpresentsthethreedifferent

scenarios that installers can use to gauge panel reliability: market average, same-as-

currentpaneloffering,andoptimistic.Whilethesethreestrategiesdoaffecttheactual

reliabilityofthepanelsofferedbymanufacturers,thestrategieshavenoeffectonthe



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.



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



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



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.



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.



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



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.



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



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



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



expressedinthismaterialarethoseoftheauthorsanddonotnecessarilyreflecttheviews

oftheNationalScienceFoundation.



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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



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



𝜃Q,k 0.02 Parameter2ofhomeowner’sdecisionfunction

𝜃Q,½ 0.5 Parameter3ofhomeowner’sdecisionfunction



Tablecaptions:

Table1.Sensitivityvaluesforthemodelcharacteristics.Sensitivityisdefinedaschangeofpenetrationratewithrespecttothebaselinescenariowhenthemodelcharacteristicisvariedaroundthebaselinescenariovalueandotherparametersare fixedat theirbaselinevalues.



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

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