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Productivity and radiation use efficiency of lettuces
grown in the partial shade of photovoltaic panels
Hélène Marrou, Jacques Wéry, Lydie Dufour, Christian Dupraz
To cite this version:
Hélène Marrou, Jacques Wéry, Lydie Dufour, Christian Dupraz. Productivity and radiation use
efficiency of lettuces grown in the partial shade of photovoltaic panels. European Journal of Agronomy,
Elsevier, 2013, 44, pp.54-66. �10.1016/j.eja.2012.08.003�. �hal-01137075v2�
Europ.J.Agronomy44 (2013) 54–66
ContentslistsavailableatSciVerseScienceDirect
European
Journal
of
Agronomy
j our na l h o me p a g e :w w w . e l s e v i e r . c o m / l o c a t e / e j a
Productivity
and
radiation
use
efficiency
of
lettuces
grown
in
the
partial
shade
of
photovoltaic
panels
H.
Marrou
a,b,∗,
J.
Wery
a,
L.
Dufour
a,
C.
Dupraz
aaINRA,UMRSystem,2,PlaceViala,34060MontpellierCedex,France bSun’RSAS,7ruedeClichy,75009Paris,France
a
r
t
i
c
l
e
i
n
f
o
Articlehistory: Received18April2012
Receivedinrevisedform3August2012 Accepted5August2012
Keywords: Shade
RadiationInterceptionEfficiency RadiationConversionEfficiency Leafarea
Lettuce Agrivoltaicsystem
a
b
s
t
r
a
c
t
Combiningphotovoltaicpanels(PVPs)andcropsonthesamelandunitwererecentlyproposedasan alternativetotheconversionofcroplandintophotovoltaicplants.Thiscouldalleviatetheincreasing competitionforlandbetweenfoodandenergyproduction.Insuchagrivoltaicsystems,anupperlayerof PVPspartiallyshadescropsatgroundlevel.Theaimofthisworkwasto(i)assesstheeffectoncropyield oftwoPVPsdensities,resultingintwoshadelevelsequalto50%and70%oftheincomingradiationand (ii)identifymorphologicalandphysiologicaldeterminantsoftheplantresponsetoshade.Experiments wereconductedonfourvarietiesoflettuces(twocrispheadlettucesandtwocuttinglettuces),duringtwo seasons.Inallcases,therelativelettuceyieldatharvestwasequalorhigherthantheavailablerelative radiation.LettuceyieldwasmaintainedthroughanimprovedRadiationInterceptionEfficiency(RIE)in theshade,whileRadiationConversionEfficiency(RCE)didnotchangesignificantly.EnhancedRIEwas explainedby(i)anincreaseinthetotalleafareaperplant,despiteadecreaseinthenumberofleavesand (ii)adifferentdistributionofleafareaamongthepoolofleaves,themaximalsizeofleavesincreasing intheshade.Ourresultprovidesaframeworkfortheselectionofadaptedvarietiesaccordingtotheir morphologicaltraitsandphysiologicalresponsestoPVPshade,inordertooptimizeagrivoltaicsystems.
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
Aconsensushasemergedontheemergencytofindalternative
energiesfromrenewablesources,inordertosatisfyan
increas-ingdemandforenergy(Escobaretal.,2009).Today,biofuelsare
claimedtobeapromisingsubstitutetofossilfuel(Hoogwijketal.,
2003).However,growingbiomasscropsonarablelandraisesdeep
concernsaboutfoodsecurity(Pimenteletal.,2009;Tilmanetal.,
2009;Walker,2009).Forexample,5.9%ofarablelandsinSouth
Americawereoccupiedin2007withenergeticcrops(Rathmann
etal.,2007).InSouthernEurope,thedevelopmentofhuge
ground-mounted solar power plants makes local farmers protest, and
alarmsauthorities(Nonhebel,2005).
Toconciliatethesetwocompetingusesofland,i.e.tosupplythe
planetwithbothenergyandfood,agrivoltaicsystemswererecently
proposed(Duprazetal.,2010).Theycombinephotovoltaicpanels
(PVPs)andfoodcropsonthesamelandunitandatthesametime.
Thefirstagrivoltaicarray(AVA)inFrancewasbuiltin2010,on
asimpledesignasproposedlongtimeagobyGoaetzbergerand
∗ Correspondingauthorat:INRA,UMRSystem,2,PlaceViala,34060Montpellier Cedex,France.Tel.:+33499612684.
E-mailaddress:marrou@supagro.inra.fr(H.Marrou).
Zastrow (1982).Photovoltaic panels(PVPs) weresettledwitha
clearancethatallowsmechanicalcultivationbelow.Therefore,the
layerofcropatgroundlevelispartiallyandintermittentlyshaded
bythecoverofPVPs.
Originally,anykindofcropcanbeconsideredforcultivationin
AVAsystemsbutprioritywasgiventohorticulturalproductionsas
alleycroppingismorecompatiblewiththegeometricalconstraints
resultingfromthesupportingstructure.Thesmallsizeof
mechani-calenginesusedinvegetableproductionalsomotivatedthischoice.
AmongthemajorvegetableproductionsinSouthernEurope,
let-tucewasaprioriparticularlyadequateforthesepioneersystems.It
canbeplantedatanyseasonoftheyear,bothinopenfieldsandin
greenhousesorshadehouses(Thicoïpé,1997),andcanthereforebe
consideredasadaptedtoawiderangeofradiativeenvironments.
ThekeypointforoptimizingAVAsystemsistomanagethe
limit-ingresource,i.e.light,sothatthecropcanmakethebestprofitfrom
thereducedlightavailablebelowthesolarpanels.Optimization
canbeachievedintwomanners.Firstly,playingwiththedensity
ofPVPs,itispossibletomodulatethedegreeofshadingappliedto
thecrop.Secondly,abetterknowledgeoftheresponseofcropto
lightwouldleadtoidentify(i)lightpatternsthatcropcanmanage
with,(ii)functionaltraitstobeselected.
Shadehas beenshown toaffectcrop yieldbyslowingdown
growth(Cantagalloetal.,2004;Workuetal.,2004)andreducing
1161-0301/$–seefrontmatter © 2012 Elsevier B.V. All rights reserved.
Fig.1. Experimentaldesign.B+|FC+|B−|FC−|indicateshowvarietiesweredistributedin2011fromEasttoWest.In2010,varietyB0wasplantedinplaceofB+andFC+, whileFC+wasplantedinplaceofB− andFC−.FD,FullDensityplot;HD,HalfDensityplot;CP-E,EastControlplot;andCP-W,WestControlplots.
totalbiomassproductionatmaturity(Dapoignyetal.,2000;Kitaya
etal.,1998;Wacquantetal.,1995).Shadealsoaffectsleafareaand
leafcharacteristicssuchastheleafweighttoarearatio(Bensink,
1971).Similarchangeswereobservedonpasturewhen
intermit-tentshadingiscausedbyatreeandshrublayer(Perietal.,2007).
Yieldvariationshavealsobeenexplainedbyamodificationofthe
radiationuseefficiency(Dapoignyetal.,2000;Rizzallietal.,2002).
In this study,we aimat understanding biomassproduction,
lightinterceptionandlightconversionintobiomassinthespecific
shadedconditionscreatedbyPVPs.
2. Materialsandmethods
2.1. Experimentalagrivoltaicarray(AVA)
Experimentswereconductedontheprototypeofagrivoltaic
systemdescribed inDupraz etal.(2010)in Montpellier,France
(43◦6′N,3.8′E).InthisAVA,PVPswerearrangedinEast–West
orien-tated,4maboveground.Thesestripswere0.8mwideandinclined
southwardwithatiltangleof25◦.Theprototypewasdividedinto
2subsystemsthatdifferedbythedistancebetweentwosuccessive
rowsofsolarpanels:1.6minthe“FullDensity”(FD)plotand3.2m
inthe“HalfDensity”plot(HD).Insideeachsub-system,the
mea-surementareawasfarenoughfromthebordersoftheAVAsothat
theycanbeconsideredhomogenous.
Two200m2controlplotsweresetup10mapartfromtheAVA,
ontheeastern(CP-E)andwestern(CP-W)sides.Controlplotswere
farenoughfromtheprototypesoasnottobeshaded,butclose
enoughtobeonsimilarsoil(27%clay,40%siltsandmorethan1m
depth).Foreachvariablemeasuredonthefield,theaverageofdata
collectedbothinCP-EandinCP-Wwasusedasthefullsun
ref-erence.Duetothelargesizeofanyagrivoltaicsystemrequiredto
avoidunwantedbordereffects(shadeprojectionsoncontrolplots,
andsunpenetrationfromthesideundershadedplots),
design-ingreplications would haveinvolved a huge landarea (several
hectares)andaverycostlyinvestment(>1millioneuros).Thiswas
notpossibleatthisearlystageoftheprototypedevelopment.We
howeverdesignedthesystemwithtwoimportantfeaturesforthe
statisticalvalidity oftheexperiment:(1)theshadedtreatments
werelargeenoughtoinvestigatespatialheterogeneityunderthe
panels;and(2)shadedtreatmentsweresurroundedbyseveralfull
suncontrolplotsthatallowedustocontrolsoilandcrop
manage-mentheterogeneity.Besides,thehomogeneityofsoilproperties,
likelytobehighinthisdeeploamysoilhadbeencheckedinthe
sameareaformerly(Ruelle,1995,PhDthesis).Theexperimental
fieldhadbeenuniformlycroppedwithcerealsandnotillagesince
then.Soilhygrometryinthe0–30cmfirstlayerwasevaluatedby
gravimetryon28soilsamplestakenevenlyoverthetotalareaofthe
experimentalfieldbeforethefirstplanting,in2010.Thevariation
coefficientofsoilhygrometrywasequalto9.4%only.Inaddition,
theexperimentwasconductedundernon-limitingconditionsfor
waterandnitrogentolimittheeffectofsoilfactorsoncrop.A
mon-itoringofthesoilwaterstatuswasperformedweeklyinallplots
(tensiometers andneutronprobemeasurements) tocontrolthe
uniformityoftheirrigationscheme.Avarianceanalysisoflettuce
yieldatfinalharvestwasperformedonthe2fullsunplotslocated
ontheEastern(CP-E)andWestern(CP-W)sidesoftheAVA.Each
plotwasdividedintothreeblocksintheNorth-Southdirection.
Varianceanalysisshowedthattherewasneithersignificantblock
norploteffects.Thesepreliminarycontrolsentitleustoconsider
thattheeffectsofnon-monitoredenvironmentalfactors–ifany–
werenegligible.
2.2. Cropmanagement
Twolettucevarieties(B0andFC+)wereplantedonJuly22,2010.
ThevarietyFC+wasplantedagainonMarch23,2011alongwith
threeothervarieties:B+,B−andFC−.VarietiesB0,B+andB−belong
tothesubspecies“Batavia”L.sativaacephalasp.,andcorrespondto
varieties“Tourbillon”(B0),“Model”(B+),and“Emocion”(B−).
Vari-etiesFC+andFC−belongtothesubspecies“FeuilledeChêne”(L.
sativa.acephala sp.)and correspondtovarieties“Kiribati”(FC+)
and“Bassoon”(FC−).
SeedlingswereplantedundertheAVAinblocksofrowswith
EasttoWestorientation.Eachblockwascomposedofsevenrows
in2010andsixrowsin2011(Fig.1).Thedistancebetweentwo
plantingrowswas33cmandthedistancebetweentwolettuceson
arowwas27cm.
Irrigationwasprovidedbysprinklers(2010)ordriplines(2011)
andnitrogenfertilizerswereappliedimmediatelybeforeplanting
andduringthecycle.Waterandnitrogenstatusweremonitored
weeklybytensiometersandchlorophyllmeterreadings(SPAD-502,
KonicaMinoltaInc.,Japan)inordertoensurethatlettuceswerenot
submittedtowaterornitrogenstressinanytreatments.
2.3. Microclimatemonitoring
Availableradiationatcropcanopylevelwasmeasuredhourly
ineach treatmentwithpyranometers(Quantumsensors–Skye
56 44 (2013) 54–66
sensorsconnectedtodataloggers(CR1000–CampbellScientific
Inc.,USA).Sensorswereinstalledjustabovelettuceheads,bothin
theFDandintheHDplotsalongNorth–Southtransectsbetween
twopanelstrips.In2010,3pyranometers,regularlyspacedwere
usedpertransectinFDandinHD.In2011,onesensorwassetabove
eachplantingrankinatransect.PARsensorsandpyranometers
wereconsideredtobeequivalentafterwecheckeditina
comple-mentaryexperimentduringwhichseveralpyranometersandPAR
sensorswereplacedtogetheratthesamelocations,inthefullsun,
andintheshadedtreatments.Bycomparingthehourlyrecordsof
thePARsensorsandtherelatedpyranometers,wecheckedthat
PAR/Globalratiowasfairlyclosedto0.48bothintheshadeandin
thefullsun,atanytimeoftheday(datanotshown).
Hemisphericalphotographswerealsotakenaboveeachrowof
lettuces.Decadalpercentagesofavailabletotalradiationwere
cal-culatedfromthephotographswiththeGapLightAnalysersoftware
(Frazeretal.,1999)forthetimeperiodwhensensorswereinthe field.
Meandailyair temperatureand relativehumidity2mabove
groundaswellascroptemperaturewerecheckedtobeuniform
betweenthetreatments(data notshown).Airtemperatureand
airrelativehumiditywererecordedhourly(HMP35andHMP45
probes,CampbellScientificInc.,USA).In2011,croptemperatures
were measured hourly by thermocouples (Copper-Constantan
thermocouples)directlyinsertedintotheplants,closetotheplant
axis,betweenthelargerlettuceleaves.
2.4. Simulationofradiationavailabletoplants
A radiative model was written and implemented in R
(http://cran.r-project.org/)usingaray-tracingalgorithmona3D
scene.The model used daily global radiation measured onthe
experimentalsiteasanentry,aswellasthelatitudeofthesite.
Globalradiationisallocatedtosectorsoftheskyaccordingtoa
StandardOvercastSkydistribution.Thescenerepresentsthepanel
stripswiththesamesizeandorientationasinthefieldprototype.
However,HDandFDsub-systemsaremodeledseparatelyandthe
supportingstructureisnottakenintoaccountbythemodel.Daily
availableradiationatagivenpointofthestageiscalculatedby
inte-gratingtheinterceptionofrayslaunchedtowardthispointfrom
eachskysectors.Calculationsareiteratedwithadailytimestepfor
eachpositiononthescenewithaspatialgridof10cm×10cm.The
outputradiationisexpressedasGlobalRadiationorPAR–
depend-ingonthespecificationoftheuser–usingaPAR/Globalratioequal
to0.48.ModelalgorithmsaredescribedinAppendixA.The
radia-tionmodelwasvalidatedin2010bycomparingmodeloutputsto
GLAcomputationsandpyranometersrecords.
2.5. Lettucesgrowthmeasurements
Lettuces were sampled for destructive measurements of
biomassatthreedatesin2010(21,34,and47daysafterplanting–
DAP),andin2011(23,44,and63DAP).Thelastharvestcorresponds
tothematuritystageforcommercialselling.
Threetofive(dependingonthesamplingdate)lettucesofeach
varietywerepickedrandomlyinCP-EandCP-Wateachsampling
date.InFDandHD,threetofivelettuceswerepickedineach
plant-ingrow(discardingexternalrows)inordertogetastratifiedsample
thattakeintoaccounttheradiativespatialheterogeneityunder
theAVA.Totalaerialdrymatterandleafnumberweremeasured
foreach lettuce.The lengthandthewidthof eachleaves were
measuredforatleastonelettuceperrowandpertreatment.In
2011,leaveswerestrippedofffivebyfive,foreachsampled
let-tuce.Eachgroupofleaveswasdriedupseparatelyaftermeasuring
thelengthandwidthofoneleaf,inordertocalculatespecificleaf
areaofeachgroupoffiveleaves.Onasubsampleoflettuces,adirect
measurementofeachleafareawasobtainedbyhorizontal
photog-raphyandimagetreatment(ImageJ,MD,USA)tochecktherelation
betweenlength,widthandexact areaoflettuce leaves.As
sug-gestedbyGay(2002),alinearregressionwasfittedbetweenexact
leafareaandwidth×length.Asingleallometricrelationwasfitted
forallvarietiestogetherwitharelativeRMSEof3.2%.
2.6. InterceptedPAR
LettuceCRwasassessedperiodically. In2011,threelettuces
pervarietyandpertreatmentweremonitoredwithvertical
pho-tographsataweeklyinterval.Afterimageprocessing(ImageJ,MD,
USA)thedynamicofcoverrateevolutionwithtimewasobtained
foreach lettuce. In2010,verticalphotographs weretakenonly
once,on18thAugustbutthediameteroftheplantswasmeasured
atthreedatesinthefieldoronsampledplants.Alinearrelation
(R2=0.83)wasfoundbetweenCRestimatedfromthephotographs
ononehandandfromthediametersmeasurementsinthesame
day,ontheotherhand.Thecorrespondinglineartransformation
wasappliedtoalltheCRestimates calculatedfromthe
diame-termeasurementsin2010,togetahomogenousandconsistent
datasetacrossthetwo years.In 2011,adirect measurementof
interceptedPARwasrealizedinthefieldeverytwoweeksonpartof
themarkedlettuces.Todoso,aradiationmeasurementwastaken
aboveeachlettucehead,atsolarnoon,byuseofaportablePAR
sen-sor(JYP1000–SDEC,France).Immediatelyafterthat,nineother
measurementsweretakenatgroundlevelsdistributedregularlyon
a27cm×33cmsquarearoundthesamelettuce.InterceptedPAR
wascalculatedastheratiobetweenmeangroundlevel
measure-mentandtheabovelettucecanopymeasurement.
2.7. Calculationofradiationuse
2.7.1. Generalframework
Biomassaccumulationwasanalyzedthroughalightcentered
approach.AccordingtoMonteith(1972,1977),biomass
produc-tionistheresultoftwosuccessiveprocesses:radiationinterception
ofincidentPAR(PARinc)andtheconversionofintercepted
radia-tion(PARint)intodrymatter.Theefficienciesoftheseprocesses
arenamedRadiationInterceptionEfficiency (RIE)and Radiation
ConversionEfficiency(RCE)respectively.InthecaseofAVA,
trans-missionofsunradiation(PAR0)throughsolarpanelsresultsinto
PARincandshouldbeaddedasthefirststepoftheprocess.
Radia-tionTransmissionEfficiency(RTE)dependsonPVPsarrangement
anddensity.
Hence,accumulationofdrymatter(DM)isgivenbyEq.(1).
DM=PAR0×RTE×RIE×RCE (1)
TocompareFDandHDtreatmentstofullsun,relativevariables
arecalculatedastheratiobetweenthevaluesinshadedtreatments
andinthefullsuncontrol.ConsideringthatPAR0isidenticalinfull
sunandunderPVPsandthatRTEinfullsunequalsoneleadsto
Eq.(2),inthecaseofFD.ThesamerelationcanbewrittenforHD.
Relativevariablesarenotedwiththesuffixletterr.
rDMFD=RTEFD×rRIEFD×rRCEFD (2)
Applying logarithmictransformation, log-relative dry matter
(LDM)intheshadecanbewrittenasthesumofthecontributionsof
lighttransmission,interceptionandconversion(Eq.(3)).This
equa-tionreferstoFDandasimilarequationcanbewrittenforHD.While
LRTEisalwaysnegativeunderPVPs,LRIEandLRCEcanbepositive
ornegative,dependingonwhetherinterceptionand conversion
efficienciesincreaseordecreaseinthePVPshade,comparedtothe
fullsuncontrol.
Table1
Estimatedparametersforalogisticaladjustmentofthecoverrateasafunctionofthermaltime(Eq.(8)).
2011 2010
Shade(FD,HD) Fullsun Shade(FD,HD) Fullsun
B0 FC+ B0 FC+
A 0.94 0.63 0.737 0.849 0.708 0.841
0.007 0.010 0.013 0.010 0.012
445.4 557.0 544.5 544.8 567.6
Fig.2.RelationbetweenthemeasuredfractionofinterceptedPARatsolarnoon andcoverrateestimatedfromphotographs,in2011.o,,♦,1standforvarieties B+,B−,FC+,FC−,respectively.Closedblacksymbols,closedgraysymbols,andopen symbolsrepresentplantintheFD,HD,andfullsunplots,respectively.
Fromfielddataandradiationsimulations,wecalculatedLRTE,
LRIE, andLRCE,foragiventime period(e.g.thetotalcropping
season),andagivenarea(e.g.thetotalmeasurementareaofHD,
FDandCPplots).
2.7.2. Radiationtransmissionefficiency
Overthecroppingseason,RTEwascalculatedaccordingtothe
followingformula:
RTE =
P
t=harvestt=planting(PARinc,t)
P
t=harvestt=planting(PAR0,t)
(4)
StandarderrorofRTEwascalculatedbypropagationofthe
stan-darderroronPARincsimulatedforeachplantingrowinFDandHD
plots.
2.7.3. RadiationInterceptionEfficiency
Overthecroppingseason,RIEiscalculatedaccordingtoEq.(5).
RIE =
P
t=harvest t=plantingPARint,tP
t=harvestt=plantingPARinc,t
(5)
Inthecaseoflettuce,thedailypercentageofincidentPAR
inter-cepted(PARint,t/PARinc,t)canbeassessedasalineartransformation
ofthecrop cover rate(CR),or CRn (De Tourdonnet,1998;Gay,
2002;Huntetal.,1984;Teietal.,1996).Wefittedourdatawitha
uniquelinearadjustment(Fig.2)withforcedto0origin,between
PARint,t/PARinc,tandCR,yieldingaRMSEof0.107(Eq.(7)).Unicity
oftheadjustmentamongstvarietiesandtreatmentswaschecked
withtestsofthemaximumof likelihoodperformedwiththeR
software(Eqs.(6)and (7)).Thevalueofthecoefficient(0.85)is
inthesamerangeasthosepreviouslyreported(seeabove)andis
consistentwithacropreflectanceof5–15%.
PARint,t PARinc,t =0.85×CRt (6) Then, RIE=
P
t=harvestt=planting(0.85×CRt×PARinc,t)
P
t=harvestt=plantingPARinc,t
(7)
CRtwasfittedasafunctionofthermaltimewithVanHolsteijn
equation(DeTourdonnet,1998;Gay,2002;Holsteijn,1980)using
measurementsofcoverratecollectedinthefieldin2010and2011
(Eq.(8)).
CRt= A
1+e(−×(TTt−)) (8)
whereTTisthethermaltimewithabasetemperatureof3.5◦C.
Thermaltimewascalculatedfromair temperature
measure-mentabovecontrolplots(CPs):indeednosignificantdifference
incumulated thermaltime wasmeasured betweenFD,HDand
thecontrolplots,whateverreferencetemperatureis used(crop
orairtemperature)(notshown).Datafrom2010and2011were
processedseparatelyasCRdynamicwasverydifferentinthetwo
years,duetoseasonaleffect.ParametersA,,and(Table1)were
adjustedwiththegnlsprocedureofR(http://cran.r-project.org/).
Testsofthemaximallikelihoodshowedthat,forboth2010and
2011,asingleadjustmentcanbeusedtopredictCRtfortreatments
FDandHDtogether,whereasadifferentsetofparametersmust
befittedfortreatmentCP(Fig.2).Concerningthefactorvariety,it
waspossibletofitasinglemodelforallvarietiesin2011,butnot
in2010.Standarderrorsofestimatedcoverrateswerecalculated
foreverydayaccordingtoSeberandWild(Pellegrinoetal.,2006;
SeberandWild,2003)(Fig.3).
UncertaintiesrelatedtomodeladjustmentsforCRtandPARint
andspatialvariabilityforPARincwerepropagatedthroughthe
cal-culation to getthe standard error of RIEand RCE. DM and CR
wereassumedtobeuncorrelatedasthecorresponding
measure-mentswererealizedondifferentplants,whileautocorrelationof
cumulatedvariables (PAR)wasaccountedfor inthecalculation
ofuncertainties.Model error(CRasa functionofTT) and
mea-suredvariabilitywereassumedtobeadditive.Uncertaintieswere
propagatedthroughnon-lineartransformationsbyderivation(Ku,
1966).
Studenttests wereperformedonmean estimatesof RIEfor
eachvariety,usingintra-treatmentvariabilityasvariance
estima-torsand(n−1)degreesoffreedom,wherenistheminimalsample
sizeforallthemeasuredorsimulatedvariablesinvolved inthe
58 44 (2013) 54–66
Fig.3. Dynamicsofthecoverrate(CR)versusthermaltimein2010(a)and2011(b)forshadedtreatments(solidline)andfullsun(dashedline).,♦,o,1,+,×represent coverratemeasuredforvarietiesB0andFC+(a),B+,B−,FC+,FC−(b)intheshade(closedsymbols)andinthefullsun(openedsymbols),respectively.Grayishstripsrepresent 95%confidenceintervaloftheadjustment.
2.7.4. RadiationConversionEfficiency
RCEwascalculatedaccordingtoEq.(9),overthecropping
sea-son.
RCE= D×DMt=harvest
P
t=harvest t=plantingPARint,t(9)
whereDisplantingdensity(D=1/(0.27×0.33)=11.2plantm−2).
Drymatteroflettucewassupposedtobenegligibleatplanting
date(four leavesstage). StandarderrorofRCEwasobtainedby
propagatingstandarderrorofDMmeasuredonlettucesamplesat
harvestandstandarderrorofPARintcalculatedasexplainedabove.
3. Results
3.1. RadiationtransmittedbelowanAVA
3.1.1. Validationoftheradiativemodel
We comparedmodeloutputstofielddatabycalculatingthe
rootmeansquareerror(RMSE)andR2 coefficientbetween
sim-ulationsandfielddata(pyranometerrecordsandhemispherical
photographs).The analysis wasrepeated for differentlevels of
spatialandtemporalintegrationfromindividuallettuceonaday
tothetotalareaofeachtreatmentplotoverthewholeseason,to
evaluatethesensitivityofthemodeltothevariabilityintimeand
inspacewithineachshadedtreatments(Table2).Thecorrelation
betweenfield dataand theradiationmodel increasedwiththe
levelofintegrationwithtimeandspace.Exceptforthecoarsest
integration level using GLA computations, R2 was above 0.70
andtherelativeRMSEwasbelow18%.Theradiationmodelwas
thereforeconsideredasafairpredictoroftheavailableradiation
bellowthePVPs,withnoriskofbiaswhencomparingHDandFD
treatments.Model outputs,averagedspatially and/orover time
wereusedforfurthercalculations.
3.1.2. PredictedradiationundertheAVA
Therelativetransmitted(availableatplantlevel)radiation(RTE,
Eq.(4))duringthecroppingseasonaveraged53%inFDforboth
2010and2011(Fig.4).Itvariedfrom48%to56%dependingonthe
plantingrow,forthetwoyears.InHD,itvariedfrom68%insummer
2010to73%inspring2011.Variabilityofavailablelightbetween
plantingrowswashigherinthistreatment:itrangedbetween63%
and72%in2010andbetween71%and74%in2011.Besides,the
hourlypatternofradiationvariedfromonerowtoanotherand
differedbetweendaysforthesamerow(datanotshown).Inthe
fol-lowingsteps,wethereforeusethespatialaverageofPARinc,which
integratethisvariability,inordertocharacterizetheFDandHD
Table2
Qualityofthepredictionforradiationavailabilityatcroplevelatdifferenttimeandspaceintegratedscalesbytheray-tracingradiationmodel,comparedtotwofielddata sources:hemisphericalphotographsprocessedwithGLAsoftwareandpyranometerrecords.
Spaceintegration Timeintegration GLA Pyranometers
R2 CV(RMSE)% R2 RRMSE%
1sensorlocation 1day 0.80 22.0
Totalplotarea 1day 0.92 11.2
1sensorlocation 10days 0.56 27.3 0.80 17.8
Totalplotarea 10days 0.89 11.4 0.91 9.5
1sensorlocation 76days 0.93 6.6
Fig.4. CumulatedincidentPARinthefullsun(measurements)andforeachplanting rowposition(Rk2–Rk6in2010andRk2–Rk5in2011)intheFDandHDtreatments (simulations),duringthetwocroppingseasons(summer2010andspring2011). IncidentPARisexpressedinpercentofthecumulatedPARinthefullsun,attheend ofthecroppingseason.
3.2. Lettuceyieldandlightuse
3.2.1. YieldoflettucesintheshadeofanAVA
Theaverageyieldinthefullsunapproximatelyreached25gof
drymatterperplantinthetwoseasons.Freshweightinthefull
Fig.5.RelativeDrymatteroflettucesplottedversusrelativeavailablelight (PAR-inc/PAR0)atharvestingdate(47DAPin2010,and63DAPin2011).Filledsymbols correspondtoFDtreatment,whileopensymbolsrepresentHDtreatments.
sunwas561gperplantinsummer2010and312gin2011,for
allvarieties.AccordingtoThicoïpé(1997)minimalfreshweight
forlettucecommercializationinFranceis280gperplantinwinter
and400ginsummer.Thisisafurtherconfirmationthatplantswere
grownclosetotheirpotentialinthefullsunandthatlightwasthe
mainlimitingfactorintheFDandHDplots,astargetedwithour
monitoringandmanagementofwaterandnitrogenstatusofthe
crop(seeSection2.2).
In2010,yieldwasreducedsignificantlyto58%ofcontrol(all
varietiestogether),whenplantsweresubmittedtoheavyshading
(FD).InHD,yieldswereat81%ofthecontrolyieldforthesame
year.In2011,yieldreductionswerelower:theyequaled79%offull
suninFDand99%inHD.Thisindicatesthat,atleastinthespring
plantingconditionsof2011,yieldwashardlyaffectedwhenthe
shadewasmoderate(i.e.intheHDdesign).In2011,forvarietiesFC+
andB−,yieldinHDwereevenabovethatofthefullsun.Moreover
biomassreductionwaslessthantheavailablelightreduction,for
everysamplingdatewiththeexceptionofB0in2010(Fig.5).This
showsthecapacityoflettucetoproducebiomassmoreefficiently
whenthelightresourceisreduced.
ThespatialheterogeneityoflightatcroplevelinFDandHD
(seeSection3.1.2)wasnottranslatedinlettuceyield.Differencesin
yieldbetweenplantingrowswithinthesametreatmentwere
sig-nificantbutnegligiblecomparedtodifferencesbetweentreatments
(testofmaximumlikelihood).
Thecomparisonbetweenvarietieshighlighteddifferencesinthe
tolerancetoshade:in2011varietiesFC+andB−appearedtobeless
affectedbyshadethanthetwoothers.In2010,varietyFC+gave
60 44 (2013) 54–66
Fig.6. Contributionoftransmission,interceptionandconversionoflightintobiomassinFDandHDtreatments.Horizontalbarsrepresentlog-relativeefficienciesLRTE, LRIE,andLRCE.Errorbarsfeaturestandarddeviationofrespectiveefficiencies.‘++’indicatesvarietiesforwhichdrymatterinHDexceededdrymatterinthefullsun.
treatment,varietyFC+havingaloweryieldintheshadeinsummer
2010butahigheryieldintheshadein2011.
3.2.2. RIEandRCEinthePVPsshade
Therelativelightavailability(RTE)wassimilarin2010and2011
(Fig.6). In2010,RIEwasnotsignificantlydifferentinthePVPs
shade(neitherFDnorHD),comparedtothefullsun,accordingto
aStudent’stest.In2011,itwassignificantlyincreasedintheshade
(p-value=0.0036inFDand0.0023inHD,Student’stest),andwas
higherinFD(0.460)thaninHD(0.454).RIEisproportionaltoCR
(Eq.(6)),whichwassignificantlyhigherintheshade(0.94)than
inthefullsun(0.63)atfinalharvest(Fig.2).Wecouldnotshow
significantdifferencesinRCEundershadeforanyofthevarieties.
However,forvarietyFC+,whichisthemostproductive,RCEtended
toincreaseintheshadebothin2010and2011.
3.3. YieldcomponentsoflettuceinthePVPsshade
3.3.1. Numberofleaves
In 2010,thenumberofleaves(longerthan1cm)was
signif-icantlyreducedin theshade for eachsamplingdate(Fig.7). In
2011,thenumberofleaveswasalsoreducedsignificantlyinFDfor
eachvarietyandeverysamplingdate.InHD,thenumberofleaves
wasunchangedformostvarieties,butitdecreasedsignificantlyfor
varietyFC−(notshown).
3.3.2. Leafareaandthickness
In2011,thetotalleafareaincreasedsignificantlyinFD,
com-paredtoCP,foreachvariety,from43DAPto63DAPunlikethe
numberofleaves(Fig.8).InHD,leafareatendedtoincreasetoo,
butchangewassignificantonlyforvarietiesFC+andB−,at63DAP.
In2010,at47DAP,totalleafareawassmallerintheshade,butthe
differencewasnotsignificant.
Foreachvariety,in2011,themaximumofthelength×width
productaveragedoveragroupoffivecontinuousleaves(called
hereacrown)inaplant,wassignificantlyhigherintheshadethan
infullsun(Fig.9).Thereforethehigherplantleafareaintheshade
observedinspringplantingwastheconsequenceofanincreased
sizeof“matureleaves”oflettuces(Fig.8).Wecall“matureleaves”
thegroupofleaveswhoseareaoverpassed75%ofthesizeofthe
biggestleafoftheplant.Matureleavesarerepresentativeofthe
growthpotentialoftheplant(Bensink,1971)andtheydetermine
thecircumferenceofthelettucehead.
Thespecificleafarea(SLA=leafarea/leafdryweight)ofmature
leaveswassignificantlyhigherintheshadeforallvarieties.AsSLA
hasbeenshowntobecloselyrelatedtoleafthickness(Wilsonetal.,
1999),wecanmakethehypothesisthatleafthicknesshasbeen
reducedbytheshadeconditionscreatedbythePVPs.Onthe
con-trary,thelength/widthratioofmatureleaveswasnotmodifiedin
theshade.
4. Discussion
4.1. ContributionofRCEandRIEtothetoleranceoflettucesto
PVPsshade
Wefoundthatlettucescanmaintainrelativelyhighyieldsunder
PV,intheHDshadetreatmentand,forsomevarieties,intheFD
treatmentinspringgrowingconditions(Table3).Byuseof
Mon-teithequationadaptedtoAVA(Eq.(1)),weshowedthattolerance
toPVPsshademainlyreliedontheabilityoflettucestoimprove
theircapacitytointerceptlight(Fig.6).Thisresultisconsistent
withSinclairet al. (1999)and Gimenezet al.(2002).
Intercep-tionwasassumedtobeaphysicalprocess,directlyproportional
tothepercentageofgroundcover.Therelationbetweencoverrate
Fig.7.Numberofleaveslongerthan1cmperlettuce,forallvarietiestogetherat 21,34and47DAPin2010,and23,44and63DAPin2011.Verticalerrorbarsfeature standarddeviations.FDandHDbarsaremarkedwith‘**’whenthenumberofleaves issignificantlydifferentfromtheCPatthesamedate,accordingtoLSDtestwitha risk˛=5%.
grownintheshade.Consequently,improvedRIEcanbedirectly
attributedtoabetterabilityoftheplantfora higherandmore
rapidsoilcoveringunderthePVPsshade.Thiswasachievedwith
morphologicalchangescontributingto(i)anincreaseinthetotal
plantleaf area and (ii) an optimized leaf area arrangement to
harvest light more efficiently. The various plant strategies to
achievethisaresummarizedinFig.10andanalyzedbelowonthe
basisofourobservations.
4.1.1. Strategy1:increasingtotalplantleafarea
Thiscanbeobtained withan increasedmean sizeof leaves
and/ornumberofleaves.
Weshowedthattheindividualareaofmatureleavesincreased
whereasthenumberofleavesdecreasedsignificantlyintheshade
(FDandHD).Inthefullsun,CRcanbepredictedbyallometryas
alogistical functionof thenumber ofleaves (Gay, 2002). From
Fig.8.Totalleafarea,forallvarietiestogetherat21,34and47DAPin2010,and 23,44and63DAPin2011.Totalleafareaiscalculatedthroughdestructive mea-surementasthesumofthelength×widthproductofeveryleaflongerthan1cm. Verticalerrorbarsfeaturestandarddeviations.FDandHDbarsaremarkedwith‘**’ whenthetotalfoliarareaissignificantlydifferentfromtheCPatthesamedate, accordingtoLSDtestwitharisk˛=5%.
fielddata,twodifferentmodelswerefittedforplantsinthesun,
andplantsintheshade.Theyweretestedtobedifferentwiththe
likelihoodratiotest(p-value=4.7×10−4).Thisconfirmedthatthe
numberofleavescannotexplaintheincreaseofCRintheshade.
However,reducingleafnumbercouldhaveallowedtheproduction
oflargerleavesforthesameamountofcarbonassimilated.Bensink
(1971)showedthatintheshade,leafemissionratedecreasedand
leafgrowthduration increased,therebyresulting inlongerand
widerleaves.Thissuggeststhat,intheshade,carbonallocationis
preferentiallydirectedtofurtherleafgrowthinsteadofleaf
emis-sion(Bensink,1971).
4.1.2. Strategy2:betterarrangementofthelightharvestingarea
4.1.2.1. Increasing headdiameter. Groundprojectionof alettuce
headcanbeapproximatedasadisktheradiusofwhichisequal
Table3
Drymatter(g/plant),foreachvarietytestedinthetwoexperiments(2010,2011).Thestandarddeviation(inbrackets)isexpressedasapercentageofmeanyieldscalculated onthecorrespondingsample(i.e.threetofivelettuces).
2010(Summer) 2011(Spring)
B0 FC+ B− B+ FC− FC+
FD 13.9a(17%) 16.0a(12%) 17.8a(33%) 18.4a(21%) 18.9(23%) 21.6(25%)
HD 19.4a(26%) 21.7a(21%) 26.1a(17%) 23.9(32%) 21.9(13%) 23.7(24%)
CP 26.2(21%) 24.9(16%) 23.7(33%) 26.2(23%) 24.9(18%) 22.0(27%)
62 44 (2013) 54–66
Fig.9.Distributionofleafarea(meanofwidth×heightproduct)fromthebottomtothetopoftheplantforthefourlettucevarieties(B−,B+,FC−andFC+)testedin2011. Eachpointofthecurverepresentsthemeansizeofaleafwithinacrownoffivecontiguousleaves.Crownsarenumberedfromtheolderleaves(crown1)totheyoungest. Verticalbarrepresent95%confidenceintervalcalculatedforthecorrespondingcrownfromasampleofthreelettuces.Theblackhorizontalmarksthecrownscorresponding tomatureleaves.
tothemeanlengthofthematureleaves(Fig.9). Wefoundthat
these leaves are both wider and longer in the shade than in
thefull sun. Changesin leaf angle(i.e. the angle betweenthe
soiland theleaf midrib) inlettuce headcouldalsobe
respon-siblefor differences in CR. Leaf anglereduction in response to
shadingwasreportedfor grassesintercroppedinorchards(Peri
etal.,2007),asadirectconsequenceofleavesgettingbothlonger
and thinner, and thus less rigid under shade. Lettuces with a
leafangleclosetohorizontalorientationwouldbeabletocover
alargestproportion ofground and thereforetointercept more
light.Similarly,Wurret al.(1992)reportedlettuce headstobe
lesstightwhenplantaresubmittedtohighlevelsofsolar
radi-ation in thesecond part of thecycle (after hearting).In 2011,
weobservedthatplantaxesweresignificantlytallerandthinner
intheshade(7.4cmand2.0gofdrymatterperaxis)compared
to full sun (6.6cm and 2.5g per axis). Changes in axis shape
couldalsohavecontributedtoleafcrownbeinglooserandmore
horizontal.
4.1.2.2. Reducingselfshading. Differentadjustmentswerefittedto
CRasafunctionofthermaltime,intheshadeandinfullsun.In2010,
theinflexionpointofthecurve(representedbytheparameterin
Eq.(8)andTable1)isreachedearlierintheshadeforvarietyFC+,
whichseemstobethemoreshadetolerant.AccordingtoHolsteijn
(1980),itdenotesareductionofself-shadingandmutualshading
withneighboringlettuces.Self-shadingreferstoleavesofthesame
plantshadingeachotherand increaseswhen thelettuce leaves
twistandturn.ThisresultisconsistentwiththefindingofBensink
(1971), thatshading encouragestheelongationof leaf’scentral
vein at the expense of lamina expansion, creating spoonshape
leaves.
4.2. Perspectiveforoptimizingphotovoltaicsystem
4.2.1. AdequatelevelofshadingbyPVPs
OurresultsshowthatunderanAVAwithhalfthenumberof
PVPs,lettuceyieldwashardlyaffected.UnderastandardAVAwith
thefulldensityofPVPs,lettuceyieldwasstronglyreduced,down
to48%ofthepotentialyieldinthefullsunformostofthevarieties.
Iflettuceproductionisexpectedtobenotsignificantlyreduced,
thentheAVAshouldbedesignedtoallowatleast70%ofPAR0at
thecroplevel(HDdesign).Asmartsuggestiontomaintainahigh
levelofelectricityproductionwouldbetomountthePVPsona
mobilestructurethatwouldallowachangeinthetiltingangleof
thepanels.Thiswouldhelptomonitorlighttransmissionunder
thepanelstomaintainitabovetherequiredthreshold.Between
cropping periods,or when thecrops need less radiation, PVPs
couldbesetbacktotheiroptimumpositionformaximalelectricity
production.Itcouldbepossibletocalculatetheyearproductionof
electricityandcropproductsresultingfromsuchamanagement,
andtooptimizethesettingstakingintoaccountthecashvalueof
bothelectricityandcropproducts.
Decreasingthedensityofpanels(astestedinourexperimental
array)isanotheroption,butitislessflexible:thiswillleadtoreduce
theelectricproductivityofthePVPlayerforthewholeyear,
includ-ingwhencropsarenotpresent.Comparingthetwooptions(mobile
panelsversusreduceddensityofpanels)wouldalsorequiretotake
intoaccounttheinvestmentsintothearray,whicharehigherfora
FDarray(includingthecostofmobilepanels)thanforaHDarray.
Finally,theproductivityofagrivoltaicsystemshouldbetested
onlongerperiods,inordertoassessifthisproductionsystemis
economically and environmentallysustainable. Our experiment
includes other vegetable species (beans, cucumber) as well as
Fig.10.SummaryofdifferentleversforincreasingRIEbyplantadaptationtoPVPsshade.Leversingrayboxeshavebeentestedinourexperimentsandidentifiedasactually playingaroleintheresponsetoshade.Leverswithadoublethinborderhavebeenshowednottoparticipateinshadetolerance.
maybehavedifferentlyintheshadeofsolarPVs.Their
combina-tioninproperrotationwillalsobeneededtoensureasustainable
managementofweeds,pestanddiseaseandsoilquality.
There-fore,acomprehensiveassessmentoftheseagrivoltaicsystemswill
requireamulticriteriaanalysisofcroprotations.
4.2.2. Towardaselectionofspecificplanttraitsforagrivoltaic
systems
Thedifferentvarietiestestedinthetwoseasons(summerin
2010andspringin2011)respondeddifferentlytothesamelevel
ofshadecreatedbyPVPs.In2011,varietiesFC+andB−appeared
to be much more tolerant to shade. This result suggests that
functionaltraitsforshadetolerancecouldbeidentified(Fig.10)in
ordertoselectadaptedgenotypesforagrivoltaicsystems.Current
literatureonshadetolerancemainlyfocusesonRCEandpointsout
netphotosyntheticrate,leafmassarea,leafchlorophyllcontent,or
leaflife-spanasindicatorsofshadetolerance(Halliketal.,2009;
Niinemets, 2010; Poorter et al., 2009; Poorter, 2001; Seidlova
etal., 2009). However,weshowed thatRCE, which is likely to
beaffectedbythesetraits,is notthemainleverofadaptionto
shade for lettuces.Our results suggestthat adaptative traits of
leafdevelopmentandexpansioncouldleadtoanincreasedRIEin
PVPsshadewhichcouldcompensateforreducedlightavailability.
AssummarizedinFig.10,totalplantleafarea,coverrateandthe
harvestinglightabilitythroughadaptiveleafmorphologyarethe
mainlevers for theplanttohandle lightreduction by PVPs.In
HD,varietiesFC+andB−exhibitedsignificantmodificationofleaf
areadistribution(Fig.9)andkeptfinalyieldssimilartoyieldsin
thefullsun.Atthesametime,thesetwovarietieshadthelowest
individualleafareainthefullsun(Fig.9),indicatingthestrong
adaptivenatureofleafareatolightavailabilityinthesevarieties.
Thissuggestthattheselectionofvarietiesadaptedtoagrivoltaic
systemsshouldnotbebasedonleafgrowthpotentialinthefull
sun,butontheirplasticityinthespecificshadeconditionscreated
byPVPspanels(i.e.intermittentlightduringtheday)whichare
likelytobedifferenttopartialbutcontinuousshade.
5. Conclusion
Toourknowledgethisworkisthefirstattempttoanalyzeplant
productioninthespecificconditionsofshadecreatedbythePVPsof
anagrivoltaicsystem.Itshowsthatsomeplantslikelettucehave
theabilitytoadapttotheseconditionsandcompensatepartially
ortotallythereductionoflightavailabilitybyahigherlight
har-vestingcapability.Wewereabletorelatethisadaptivebehavior
tomorphologicalchangesin leafdevelopmentandmorphology.
Similarworkisconductedwithotherspecies(cucumber,French
beansanddurumwheat)(Marrou,PhDthesis)inordertoidentify
croprotationsadaptedtotheseconditions.Ourresultssuggestthat
theseagrivoltaïcsystemscanbeoptimizedbothbyplantbreeding
andbyspecificarrangementsofPVPspanelsinordertofindthe
bestcompromisebetweenfoodproductionandelectricity
produc-tiononthesamepieceofland.Inthissense,thisworkopensthe
scopeforanintegrateddesignofagrivoltaicsystemstooptimize
foodandenergyproductioninagiveneconomiccontext.
Acknowledgements
TheauthorsgratefullyacknowledgetheSun’Rsocietyfor
fund-ingthisresearchproject.Specialacknowledgmentsarededicatedto
thetechnicalstaff(J.F.Bourdoncle,A.Sellier,P.Parra),andtrainees
(A.Legendre,E.deMondenard,A.Calvet,andA.Franc¸ois)fortheir
essentialcontributiontodatacollectioninthefield.ThankstoG.
Talbot(INRA,Montpellier,France)forhishelpinsimulating
radi-ation,M.Tchamitchian(INRA,Avignon,France)forhisadvicein
ecophysiology,N.Hilgert(INRA,Montpellier,France)forherhelp
ongnlsadjustments,andP.Ruelle(IRSTEA,Montpellier,France)for
hissupportallalongthetwoexperimentalcampaigns.
AppendixA. Radiationmodelmathematicalformalisms
A.1. PartA:Buildingaskymap,atdailytimestep
Thefirstpartofthemodelusesmeasuredmeandailyglobal
radiation(Gr)tocreateaskymapforeverydayoftheperiodduring
whichthesimulationwillberun.Theskyisdividedintosectors
withananglestep˛fixedbytheuser(weranthesimulationswith
ananglestep˛=5◦).Eachskysectorisdefinedbytheazimuth(az)
64 44 (2013) 54–66
Themodel distributesdiffuse,directandtotal Gramongsky
sectors.TheglobalradiationGraz,el,dcomingfromasector(az,el),
forthedayd,iscalculatedfromEq.(A.1)
Graz,el,d=Grd×(PDIFd×DIFel+(1−PDIFd)×DIRaz,el,d) (A.1)
where
[1]PDIFdistheratioofdiffuseradiationRGdif,dabovetheglogal
radiationGrdforthedayd.PDIFdisgivenbyEq.(A.2)(
Varlet-Grancher,1975).
PDIFd=1.3−1.2× Grd G0d 0.01≤PDIFd≤1 (A.2)withG0dtheextraterrestrialradiationforthedayd.
G0discalculatedaccordingtoFAO56(Allenetal.,1998)
for-malism(Eq.(A.3)).
G0d=
24×60
×Gsc×dr[ω×sin( )×sin(ı)+cos( )×cos(ı)
×sin(ω)] (A.3)
withGsc:solarconstant.Gsc=0.0820MJm−2min−1.
drinverserelativedistanceEarthSun[m]dr=1+0.0033
×cos
2365d
(A.4)
ω:sunsethourangle[rad].ω=arcos(−tan( )×tan(ı)) (A.5)
w:latitude[rad].
ı:solardecimation [rad].ı=0.409×sin
2365×d−1.39
(A.6)
[2]DIFelistheproportionofdiffuseradiationincomingfromeach
skysector (az,el). Assky wasdiscretizedfollowing a
Stan-dartOvercastSky(SOC)distribution,asdescribedbyMoonand
Spencer,1942,DIFonlydependsontheskysectorelevation.
DIFel=67×n×
(sin(el+(˛/2))2−(sin(el−(˛/2))2
2 +2×(sin(el+(˛/2)) 3 −(sin(el−(˛/2))3 3
(A.7)withnthenumberofskysectors.
[3]DIRel,az,distheproportionofdiffuseradiationincomingfrom
eachskysectors(az,el),forthedayd.Thisfractionofthedaily
radiationisincomingfromthesundirectionintwosteps:(1)
sunpositioniscalculatedanddirectradiationisallocatedto
skysectorsclosetothesunpositionwithatimestepinferioror
equalto1hand(2)thedirectradiationisintegratedoverthe
daylengthforeachskysector.
Timestepmustbecoherentwithanglestep:smallanglestep
requiresshorttimesteps.Weranthecalculationswithatime
stepof20min. DIRel,az,d=
X
t=sunset t=sunrise sin(s,t)P
t=sunset t=sunrisesin(s,t) ×P
Wel,az,t el,azWel,az,t (A.8) withs,dthesundeclinationattimet,calculatedaccordingAllen
etal.(1998),aswellassunriseandsunsettimes
Wel,az,t aweightingtermthatdistributes thedirectenergy
amongtheskysectorsthataretheclosesttothesunposition.
Wel,az,tisequaltotheintersectionoftheskysector(el,az)witha
diskofsolidangleequalto2/n(n:numberofsectors),centered
onthesun.
Wel,az,t=2× acos ˝el,az,t r ×r2−(q
(r2−˝2 el,az,t)×˝el,az,t) if ˝el,az,t≤r Wel,az,t=0 if ˝el,az,t>r (A.9) with ˝el,az,t=acos cos(||az −azs,t||)×cos(||el−els,t||) 2and||az−azs and||az−azs,t||in[−;] (A.10)
r=acos
1−1n
.
azs,t is thesunazimuth attime t, calculatedaccording to
Allenetal.(1998);els,tisthesunelevationattimet,calculated
accordingtoAllenetal.(1998).
A.2. PartB:sortinginterceptedandnoninterceptedbeams
Thesimulatedsceneis3Disorientatedina3Dorthogonal
ref-erential.xaxisisparalleltoNorth–Southdirection,pointingSouth
(azimuth=0),yaxisispointing East, and yaxisis pointing the
Zenith.
Forkvaryingfrom1tothenumberofPVPsstripsNp,PVPstrip
kisdefinedasafiniteplaneofEq.(B.1).
Pk:(x,y,z)/
sin( )×(x−Dk)+cos( )×(z−h)=0 h≤z≤h+l×sin( ) y0≤y≤y0+m (B.1)where isthetiltangleofthePVPstripsfromthehorizontallevel
[rad];Dk isthepositionofthesouthernedgeofPVPstripk[m]
Dk=x0+(k−1)×e,wherex0isthepositionofthefirstPVPstrip
(southernedge) ontheNorth Southaxis, and e is thedistance
between2 PVPstrips.histheheightofthePVPstrips(bottom
edge);y0isthepositionoftheEasternendofthePVPstripsalong
theEast–Westaxis.
Raysincomingfromskysector(az,el)andhittingtheground
atthelocation(xM,yM,0)areassimilatedtostraightlinescoming
fromthecenteroftheskysector.TheyaredefinedbyEq.(B.2).
Daz,el,M:(x,y,z)/
∃
∈R/
x−xM=cos(az) tan(el) × y−yM=−sin(az) tan(el)× z= (B.2)TransmittedradiationTrdforthedaydattheposition(xM,yM,
0)forthedaydiscalculatedasthesumoftheraysthatarenot
interceptedbyanyofthePVPstrips.
TrM=
X
Daz,el,M∈TM
whereTMisthepoolofrayshittinggroundatthelocation(xM,yM,
0),andthatarenotinterceptedbyanyPVPstrips.
TM:(az,el)/
or tan( )×cos(az)tan(el) =−1 and
∀
k in[[1;Np]],
[sin( )×(Dk−xM)+cos( )×h]=/0 or[sin( )×(Dk−xM)+cos( )×h]=0 and yM+tan(az)
×h
sin( ) <y0
or
[sin( )×(Dk−xM)+cos( )×h]=0 and yM+tan(az)
×h
sin( ) >y0+m
tan( )×cos(az)
tan(el) =/ −1 and
∀
k in[[1;Np]]
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