Influence of illumination conditions on the design of thin-film modules

Solar Energy Materials&Solar Cells57(1999)85—95Influence of illumination conditions onthe design of thin-film modulesM.Burgelman*,A.NiemegeersVakgroep Elektronica en Informatiesystemen(ELIS),Uni v ersiteit Gent,Informatiesystemen Pietersnieuwstraat41,B-9000Gent,BelgiumReceived16January1998;accepted10July1998AbstractThe most important,and most accessible design parameter of a series integrated thin-film photovoltaic module is the cell length l.Its optimal value l is a trade offbetween area loss, and series and shunt losses.If the design is performed at standard reporting conditions(i.e. constant one sun illumination),too low a value of lresults.When the design is done for a realistic histogram of the incoming sun power P ,a larger value of l follows,and an efficiency gain of a few tenths of a percent(absolute)is obtained.A climate model,enabling to calculate the histogram of Pfor given cloud conditions is presented here.A model to calculate the module parameters was presented before(Burgelman and Niemegeers,Sol.Energy Mater. Sol.Cells,June1998).Both models are then applied together.As a result,the optimal value land the efficiency gain are calculated as a function of the location(geographical latitude)of the panel,both in clear sky and in clouded conditions. 1999Elsevier Science B.V.All rights reserved.Keywords:Thin-films modules;Climate model;Illumination condition1.IntroductionMonolithic series integration of single solar cells into a thin-film module generally introduces extra contributions to series resistance and shunt conductance;also,some area needed to isolate and to interconnect the layers is lost.An example of a common*Corresponding author.E-mail:burgelman@elis.rug.ac.be0927-0248/99/$—See front matter 1999Elsevier Science B.V.All rights reserved.PII:S0927-0248(98)00177-986M.Burgelman,A.Niemegeers/Solar Energy Materials&Solar Cells57(1999)85—95Fig.1.A common geometry used for isolation and series integration of adjacent cells in a submodule, shown for the case of CuInSe cells.The electrical series resistances and shunt conductances are indicated.geometry for series integration is shown in Fig.1.In designing the module,various loss mechanisms have to be traded offagainst each other:taking a large unit cell length l will minimize the area loss and shunt loss,whilst a small cell length will minimize the series resistance loss;also,the electrical resistivity Rᮀ8 -of the window layer(ZnO in the case of Fig.1)is to be traded offagainst its transmission[1].In practical designs,the total lost length g(Fig.1)is imposed by technological con-straints,and only l and Rᮀ8 -have to be optimized.Such an optimization almost always is done for standard reporting conditions i.e. illumination with the AM1.5G spectrum,an incident power of1000W/m ,and an operating temperature of25°C.However,real modules are illuminated with an intensity varying between0W/m and a maximum value which is less than or equal to 1000W/m ,and an operating temperature which may be subject to large variations [2,3].In this article,we will propose a module design which is optimized for the real operating conditions of the modules.This design depends on the geographic location of the module,on its orientation,and on the climate conditions of the location.To this purpose,we have extended a calculation model for module efficiencies,presented before[1],with an empirical climate model.2.Calculation of the module efficiencyAn exact calculation of the I—V curve of a module necessitates the solution of a non-linear differential equation[1,4,5].The result can be obtained at any accuracy required,but at the cost of computing time and programming effort.In the previous work,we have proposed a quick and accurate model[1]based on a Taylor expansion of the exact solution.This expansion is three dimensional,the variables being normalized(dimensionless)discrete series resistance r ,normalized distributed series resistance r and normalized shunt conductance g .The Taylor expansion is of third order in r ,r and g .The coefficients of the expansion themselves depend on the normalized open circuit voltage v "q» /nk¹,where n is the diode ideality factor. The series expansion of all coefficients as a function of1/v was given.Also,the algorithm was implemented in a user program,available with the authors.M.Burgelman,A.Niemegeers/Solar Energy Materials&Solar Cells57(1999)85—9587 3.Climate modelThe purpose of the climate model in this work is to establish the histogram of incident power on a solar panel during a given period,e.g.a year.We start from a model presented in the literature[6];it is based on standard formulae of the spherical astronomy to describe the position of the sun relative to the panel,and on some simple assumptions about scattering and reflection of sunlight.The model is summarized here.The subscript p refers to the panel,and s to the sun.The panel orientation is given by its zenith distance z (z "0for a horizontal panel),its azimuth A (A "0means north;A is measured in the direction N P E P S P W),and the geographical latitude ( "0at the equator, "# /2at the north pole).The sun’s declination (here expressed in degrees)is given by a series development[7]:"0.33281!22.984cos t!0.34990cos2 t!0.13980cos3 t,#3.7872sin t#0.03025sin2 t#0.07187sin3 t,(1) where t is the day of the year,and "2 /365.When is the solar time( "0at solar noon),the sun’s position(z ,A )is calculated from and from:cos z "sin sin #cos cos cos ,sin z cos A "!cos sin #sin cos cos ,sin z sin A "cos sin .(2) This formulation is more elegant than that given,e.g.in Refs.[6—8],since it easily allows to determine A uniquely by means of the C or Fortran routine atan2(e.g. Ref.[9]).The angle of incidence of the sunlight,i,is then calculated ascos i"cos z cos z #sin z sin z cos(A !A )(3) Illumination not normal to the panel introduces a correction factor c: c" cos i for cos i*0,(4)0for cos i)0.The direct solar radiation P is given byP "cP f(t)w exp(!km).(5) Here m is the airmass(m"sec z ),and f(t)is a correction factor taking in account the yearly variation of the distance sun—earth[7]:f(t)"1!0.0335cos t!6.505;10\ cos2 t!3.4;10\ cos3 t#1.333;10\ sin t#7.20;10\ sin2 t#1.3608;10\ sin3 t.(6)88M.Burgelman,A.Niemegeers/Solar Energy Materials&Solar Cells57(1999)85—95The parameters P and k are two climatological parameters;estimations are given in the literature for various conditions[7].For an urban to industrial region in a cool-temperate climate zone(the author’s site),appropriate values are P "1050W/m and k"0.3[6].The cloud factor w in Eq.(5)is another climatologi-cal factor,which will be further treated below.The diffuse sky radiation is given byP " (1#cos Z )sP f(t)cos z [1!w exp(!km)],(7) where s is the sky parameter(0)s)1)[6].Finally,the reflected radiation P is given byP " (1!cos z )rP f(t)cos z [s#(1!s)w exp(!km)],(8) where r is the albedo or the reflection coefficient of the area surrounding the solar panel.Typical values of s and r for various conditions are given in Ref.[7].We take s"0.4and r"0.3.The total incoming radiation P is thenP "P #P #P .(9) The climatological model now proceeds as follows:the parameters P ,k,s and r are chosen from literature data for typical climate conditions.Then,the cloud parameter w is assumed to have a simple variation over the year:w"w (t)"w #w cos t(10) or a third-order Fourrier expansion as in Eq.(1)and(6)(then seven coefficients w through w are needed).For each month,the total incoming energyE " P (t)d tis calculated;the coefficients w then are chosen such that the calculated E valuesfit the data for a typical year,as published by the local meteorological institutes.In the calculation,the solar time is varied between sunrise and sunset in sufficiently small time increments( )10min).The solar time of sunrise(! )and sunset( )follows from Eq.(2)by putting z " /2:cos "!tan tan .(11) During calculation,a histogram of the incoming power P is built up:P is discretized in equidistant intervals P,and the energy content of radiation with power in the range P $ P/2is recorded.With the model presented in Ref.[1],the output energy of the panel is computed for each incoming power P ,weighed with the energy content,and summed.Typically we use P"0.02sun"50W/m .To obtain a realistic histogram of P ,the climate model has to be refined.Afirst refinement is to allow a seasonal statistical variation of the cloud factor:the year is divided in short reference periods(one or a few days),and in each reference period statistical guess is made for the cloud factor,subject to the following restrictions:the expectation value of w is w (t)as given by Eq.(10);always0)w)1;and theFig.2.Module efficiency and optimal cell length l of the integrated series connection,as a function of a constant illumination intensity P .The full lines are for a cell temperature ¹ "25°C,the dashed lines for ¹"70°C.standard deviation is a given parameter.We used the beta distribution to obtain this (see Appendix A).A second refinement was to impose a deterministic daily variation of w around its day average:this models the normal occurrence of morning and/or evening haze,or noon clouding at the panel’s location (see Appendix B).The third refinement is to allow a daily statistical variation of w ,in a reference period typical for the climate (e.g.a fraction of an hour to some hours).The three refinements can be applied separately or together.4.Results and discussionThe geographical and climatological location of the panel influences its electrical output mainly through variation of the illumination intensity P and the solar cell temperature ¹ .At each value of P and ¹ the trade-offbetween area loss,shunt loss and series loss gives rise to a different optimal value l of the cell length l .This is shown in Fig.2for the cell and module parameters relevant for CIGS modules we used in previous work (Table 2of Ref.[1]).The optimization was done for a fixed value of R ᮀ8 -"12 ,and not for an optimized value of R ᮀ8 -.The underlying idea is that it is feasible that a manufacturer adapts the scribing geometry (thus the cell length l)to the planned location of the panel,but that it is not probable that he will M.Burgelman,A.Niemegeers /Solar Energy Materials &Solar Cells 57(1999)85—9589Fig.3.Histogram of the illumination intensity P ,for three geographical locations of the panel: "0°, "30°,and "60°.The full lines are for the clear sky condition,the dashed lines for the clouding conditions of Table 1.adapt the ZnO deposition process.It is clear from Fig.2that a lower level illumina-tion necessitates a higher cell length (this holds if the losses due to distributed series resistance are dominating shunt losses).It is also clear that,while the operating temperature of the cells strongly influences the panels output,it has only a minor influence on the optimal value l .The histogram of incoming sunpower depends on the geographical position of the panel.In Fig.3,we calculated the histogram as a function of the latitude for two climate conditions:clear sky (w "1),and a statistical cloud model suited to describe the author’s site [10].The parameters of this climate model are listed in Table 1.It is no surprise to see that the average illumination intensity,weighed over the incoming energy,is substantially higher at equatorial sites.The shape of the histogram of P however is sensitive to the parameter values of the cloud model.The parameters of Table 1yield a good agreement with a measured histogram [11].We will use the histograms calculated for Belgian cloud conditions to obtain a rough idea of the influence of geographical latitude on the module design.For better calculations,a measured P histogram,or carefully chosen cloud parameters should be used.Fig.4shows the calculated module efficiency versus cell length l for various illumination conditions:constant illumination with P "1sun,with P "0.338sun (the average illumination intensity in Belgium),with P "0.25,and with a histogram of Pcalculated for a panel situated in Belgium.It is clear that a module design 90M.Burgelman,A.Niemegeers /Solar Energy Materials &Solar Cells 57(1999)85—95Table 1Input data for the climatological model.The parameters are relevant to the author’s site (Belgium)Latitude51°Panel orientationZenith distancez 0(horizontal)AzimuthA0Extinction law P 1050W/m k0.3Sky parameter s0.4Albedo r0.3Clouding:Seasonal average w 0.345w!0.136Seasonal statisticsNew statistical value each 2days0.8 Daily statisticsNew statistical value each 30min 0.8 Daily deterministic Sunrisew "0.5w Noonw "w #0.5(1!w )Sunset w "0.5wFig.4.Module efficiency as a function of the cell length l of the integrated series connection.The calculation is for a constant illumination P "1,0.338and 0.25sun,and for a histogram of P calculated for the geographical location of Belgium and with the clouding conditions of Table 1.M.Burgelman,A.Niemegeers /Solar Energy Materials &Solar Cells 57(1999)85—9591Fig.5.Module efficiency gain and optimal cell length l of the integrated series connection,as a function of the geographical latitude ,calculated with the clouding conditions of Table 1.The efficiency gain is defined in the text.performed at standard reporting conditions (1sun)will definitely result in too low a value of l "4mm.The efficiency gain in choosing a cell length l "6.2mm,appropriate to Belgian illumination conditions,is about 0.4%absolute: (l "4mm)"12.95%and (l "6.2mm)"13.33%.It is also clear that a design carried out with a constant averaged or effective illumination intensity is not appro-priate.In Fig.5,the optimal cell length l and the efficiency gain are calculated as a function of the geographical latitude ,for clear sky and for Belgian clouding conditions.Here, is defined as " (l )! (l "4mm).It is seen that an efficiency gain of a few tenths of a percent (absolute)can be obtained by adapting the design of the module to the geographical and climate conditions.This efficiency gain increases as the site is further away from the equator.5.ConclusionsThe most important,and most accessible design parameter of a series integrated thin film photovoltaic module is the cell length l :often it can be adapted simply and accurately by changing the computer driven settings of the laserscriber.If the design is performed at standard reporting conditions (i.e.constant one sun illumination),too92M.Burgelman,A.Niemegeers /Solar Energy Materials &Solar Cells 57(1999)85—95low a value of l results.When the design is done for a realistic histogram of the incoming sunpower P ,a larger value of l follows,and an efficiency gain of a few tenths of a percent (absolute)is obtained.This gain might seem low at first sight,but it is obtained at virtually no cost:one just has to adapt the settings of the laser scriber to the destination of the panel,and this constitutes an organizational problem rather than a technical problem.A climate model,enabling to calculate the histogram of P at given cloud conditions is presented here.A model to calculate the module parameters was presented before [1].Both models are implemented together in one Windows-based user program,which is available with the authors.AcknowledgementsWe acknowledge the support of the Fund for Scientific Research,FWO Vlaanderen (M.B.,research associate)and of the IWT-Institute,Vlaanderen (A.N.,research assistant).Appendix A.Statistical variations of the cloud factor wA statistical distribution which fulfills the conditions outlined in Section 3is the beta-distribution.The distribution function is the incomplete beta function I U (a ,b ):I U (a ,b )" U t ?\ (1!t )@\ d t t ?\ (1!t )@\ d t ,0)w )1,a ,b '1.(A.1)The limitation a ,b '1ensures that the probability is peaked around the mean value,and not around the two boundaries w "0and w "1.The mean value and the variance of this distribution are1w 2"w "a a #b and "ab (a #b )(a #b #1).(A.2)The conditions a ,b '1impose a maximal value of the variance:"min w (1!w )1#w ,w (1!w ) 2!w (A.3)The average value w of the clouding factor is set by the seasonal variation (Eq.(10)),and the variance is chosen as a parameter,as a fraction of .The parameters a and b of the beta-distribution then follow from Eq.(A.2):a "w (w!w ! ),b "1!w (w !w ! ).(A.4)M.Burgelman,A.Niemegeers /Solar Energy Materials &Solar Cells 57(1999)85—9593The incomplete beta function I V (a ,b )was evaluated according to the Numerical Recipes [12].To make a statistical guess of the cloud factor,the inverse of the distribution function is needed.This was implemented from I V (a ,b )with a simple Newton —Raphson algorithm.This statistical guess of w was used as a new average w ,for calculation of deterministic (see below)or statistical variations of w during the day.Appendix B.Deterministic variation of the cloud factor w during the dayThe climate model incorporates the possibility to set a deterministic variation of the cloud factor w ( )during the day.One can choose a cloud factor w in the morning (solar time "! ),w at solar noon ( "0)and w in the evening ( " ).The day average of the cloud factor w "1w ( )2is imposed by the seasonal variation (deter-ministic (Eq.(10)),with optional statistical variations (Appendix A)).We used a simple w ( )-law:m "w #w !2w 2(w !w )!1,w ( )"w #(w !w) ! 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