Calculation of the PV Modules Angular Losses

Solar Energy Materials & Solar Cells 70 (2001) 25}38

Calculation of the PV modules angular lossesunder "eld conditions by means

of an analytical model

N. Martin*, J.M. RuizCIEMAT-DER, Avenida Complutense 22, E-28040 Madrid, Spain

UPM-Instituto de Energia Solar, E.T.S.I.T., Ciudad Universitaria s/n, E-28040 Madrid, Spain

Received 30 March 2000; received in revised form 10 July 2000

Abstract

Photovoltaic (PV) modules in real operation present angular losses in reference to theirbehaviour in standard test conditions, due to the angle of incidence of the incident radiationand the surface soil. Although these losses are not always negligible, they are commonly nottaken into account when correcting the electrical characteristics of the PV module or estimatingthe energy production of PV systems. The main reason of this approximation is the lack ofeasy-to-use mathematical expressions for the angular losses calculation. This paper analysesthese losses on PV modules and presents an analytical model based on theoretical andexperimental results. The proposed model "ts monocrystalline as well as polycrystalline andamorphous silicon PV modules, and contemplates the existence of super"cial dust. With itangular losses integrated over time periods of interest can be easily calculated. Monthly andannual losses have been calculated for 10 di!erent European sites, having diverse climates andlatitudes (ranging from 323 to 523), and considering di!erent module tilt angles. 2001Published by Elsevier Science B.V.

Keywords: PV modules; Optical losses; Angular losses; Re#ectance

1. Introduction

The optical losses of photovoltaic (PV) modules working in "eld conditions havebeen reported in several recent publications [1}7]. In most PV applications, the

*Corresponding author. Tel.:#34-91-346-6672; fax:#34-91-346-6037.E-mail address: [email protected] (N. Martin).

0927-0248/01/$ - see front matter 2001 Published by Elsevier Science B.V.PII: S 0 9 2 7 - 0 2 4 8 ( 0 0 ) 0 0 4 0 8 - 6

The over bar-on R indicates weighted by the product of the spectral response of the PV module by thespectral distribution of the solar radiation, AM15G (ASTM Standard E892-87, Annual Book of ASTMStandards 12.02, 1987). All the considered re#ectances or transmittances in this paper are weighted ones.

angles of incidence of solar radiation often di!er substantially from normal incidence,which is assumed at standard test conditions [8]. As a consequence, re#ection lossescan become signi"cant when calculating the electrical PV generation. The importanceof this e!ect strongly depends on the module orientation, as well as on local latitudeand climate characteristics. In spite of the interest of this question, there arefew theoretical studies of the optical behaviour of PV modules applied to di!erenttechnologies [2,3], and even these are merely systematic applications of Fresnelformulae that include, in some cases, the matrix thin "lm theory. Other authors[4}6] consider simpli"ed calculations of these analyses. As a consequence, a lackof easy-to-use mathematical tools for integrated re#ectance e!ects calculations isobserved.Although there is a mathematical model initially proposed by Souka and Safat [9]

and adopted by ASHRAE [10] (American Society of Heating, Refrigeration and AirConditioning) and thus known as the ASHRAE incidence modi"er [11] which isconsidered by some authors [6,7]. It is not speci"cally obtained for PV modules butcalculates transmittance as a function of the angle of incidence of solar radiation by"tting one parameter. Nevertheless, it presents problems like a discontinuity at 903and not good "tting results for high angles of incidence.This paper proposes an alternate mathematical model, which is speci"cally ob-

tained from the optical analysis of commercial PV modules of di!erent technologies.It avoids the above-mentioned problems having good "tting results in all cases. Its"tting parameter supplies direct information about the angular behaviour of a PVmodule, and is easily obtained from measurements. With it analytical expressions forthe calculation of the global e!ect of the angle of incidence have been obtained. Theresult is a useful tool with which optical losses of PV modules under "eld conditionscan be quanti"ed.

2. The PV module angular losses

The angular losses (AL) of a PV module are calculated in reference to normalincidence of radiation and clean surface, that are the conditions at which commonlythe electrical characteristics of a PV module are supplied. Being RM (0) the weightedre#ectance of the module in such reference conditions, andM (0),AM (0) the correspond-ing air-to-solar cells transmittance and absorptance (within the glass, encapsulant,etc.), respectively, the angular losses at an angle of incidence can be calculated by thefollowing formula:

AL()"1!M ()M (0)"1!1!RM ()!AM ()1!RM (0)!AM (0)1!

1!RM ()1!RM (0), (1)

26 N. Martin, J.M. Ruiz / Solar Energy Materials & Solar Cells 70 (2001) 25}38

where RM (), AM (), M (), are the re#ectance, transmittance and absorptance at the angle. The complement to unity of the angular losses, named by the following angularfactor, f

, represents, according to the "rst equality in Eq. (1), the relative angular lighttransmission of the module. The experimental value of such a parameter can beobtained by dividing the short-circuit current (I

) at an angle by the product of the

short-circuit current at normal incidence ("0) and the cosine of the angle:

f"

I()

I(0)

1

cos 1!RM ()1!RM (0). (2)

The angular factor thus calculates the optical losses relative to the normal incidencesituation.The frequently ful"lled condition of negligible absorption within the air-to-cell path

is the obvious one leading to the last approximate equality in both (1) and (2)equations. It is, however, to be noted that the approximation should hold, even ifabsorption is not negligible, provided the absorptance ratio, AM ()/AM (0), is not muchdi!erent than the transmittance ratio (or the angular factor), a not unrealistic case.

3. Analytical model for the re6ectance of a PV module

3.1. The model expression

From the optical analysis of di!erent PV modules con"gurations, consideringcrystalline (x-Si) and amorphous silicon (a-Si) technologies, with or without antire#ec-tive coatings, and looking for a simple analytical expression, the following formula forthe re#ectance of a PV module has been obtained:

RM ()"RM (0)#[1!RM (0)]exp(!cos a/a

)!exp(!1/a

)

1!exp(!1/) , (3)

where is the irradiance angle of incidence and athe angular losses coezcient, an

empirical dimensionless parameter to "t in each case. The model has been applied tothe analysis of di!erent x- and a-Si modules. In all cases, the results are verysatisfactory. With it the expression of the angular losses becomes

AL()"1!1!exp(!cos a/a

)

1!exp(!1/) . (4)

3.2. Model-xtting performance with analytical results

The results of applying the proposed model to di!erent x- and a-Si modulecon"gurations are plotted in Fig. 1 and summarised in Table 1. The model describesvery accurately all the analysed con"gurations, as can be deduced from the "gure (seealso the high coe$cients of determination, r, in Table 1).It can be observed that there are modules that show low values of RM (0), like the ones

having the optimised triple coating or ZnS "lm, and other with quite high re#ectance

N. Martin, J.M. Ruiz / Solar Energy Materials & Solar Cells 70 (2001) 25}38 27

Fig. 1. Re#ectance calculated values (dots) and modelled ones (lines) for each indicated con"guration ofcrystalline and amorphous silicon modules.

Table 1RM (0) and a

values for each di!erent con"guration, relative asymptotic standard errors () and coe$cients

of determination (r) obtained in each regression analysis. The triple coating consists on an optimised(SiO

/Ta

O/ZnS) one. The a-Si :H thickness is 400nm, d1"92 nm and d2"62 nm

Con"guration RM (0) a

(a) r

Air/glass 0.043 0.173 2.0E-03 1.000Air/glass/Si 0.225 0.157 1.6E-02 0.998Air/glass/SiO

/Si 0.260 0.155 5.4E-03 0.999

Air/glass/triple coat./Si 0.113 0.179 1.4E-02 0.999Air/glass/ZnS/Si 0.085 0.168 3.1E-03 1.000Air/glass/a-Si:H/Ag 0.358 0.136 1.8E-02 0.998Air/glass/ITO(d1)/a-Si:H/Ag 0.267 0.138 1.4E-02 0.999Air/glass/ITO(d2)/a-Si:H/Ag 0.203 0.163 1.4E-02 0.999

values at normal incidence, like the one with no antire#ective coating or with a layerof SiO

. Nevertheless, angular losses depend on the shape of the spectral re#ectance

curve versus the angle of incidence. This e!ect is characterised by the parameter a,

which increases for good relative angular responses. Considering our example cases,although the &air/glass/triple coat/Si' con"guration presents low re#ectance at normalincidence it has a not so good angular response. The opposite e!ect is observed witha con"guration like &air/glass/SiO

/Si'.

With the a-Si con"gurations the obtained "tting results are also good. As it occurswith x-Si modules, the interface &air/glass' is a "rst approach to describe the angularin#uence of the re#ectance of an a-Si module. Nevertheless, the angular response


Fig. 2. Experimental set-up for the measurement of the angular factor f of equivalent technology solar

cells (ETC).

Table 2Regression analysis "tting results with the experimental data

a

Technology Estimated value (a) r

m-Si 0.169 1.2E-02 0.999p-Si 0.159 1.5E-02 0.999a-Si 0.163 1.8E-02 0.998

improves in general when considering the rest of the module materials. Also, goodperformance at normal incidence can imply worse relative angular responses, as itmay occur with the transparent conducting oxide of indium-tin oxide (ITO) layer.

3.3. Validation of the model with experimental results

3.3.1. Experimental methodIn order to validate the proposed model with experimental data, a testing system

has been developed. With it the angular factor f (Eq. (2)) of an equivalent technology

solar cell (ETC) of each PVmodule has been measured. As light source, a class A solarsimulator has been used and the samples have been mounted on a rotary structure onwhich any angle of incidence of the radiation can be obtained. The whole "xture hasbeen located inside a big black box where multiple re#ections between ETC and wallsare avoided (see Fig. 2). The angle of incidence and the irradiance at normal incidencehave been measured for calculating the angular e!ect. Also ETC temperatures havebeen registered for correcting the obtained currents to the same temperature and thusavoiding temperature e!ects.

3.3.2. Regression analysis with the experimental dataThe experimental values of the angular factor have been "tted to the proposed

model equation. The obtained angular losses coe$cients (a) for a representative

module of each technology are included in Table 2, together with the calculated


Fig. 3. Experimental f data (dots) of three PVmodules of di!erent technologies and "tted curves obtained

with the proposed model.


coe$cients of determination. From these values and looking at Fig. 3 it can beconcluded that the model is in very good agreement with the experimental results anddescribes correctly, for all possible angles of incidence, the angular variation of a PVmodule re#ectance. Another signi"cant result is that the module technology hasa second-order in#uence in the angular response ("rst-order determined by the&air/glass' interface), although better relative results are obtained in general withpolycrystalline (p-Si) and amorphous silicon PV modules. This fact does not meanthat with monocrystalline silicon (m-Si) technology the optical losses increase, inabsolute terms, which is mainly characterised byRM (0). In fact, most m-Si modules havetextured solar cells and antire#ective coatings and, as a consequence, low RM (0) values.But this also may imply a not so good angular response, specially regarding texturedsolar cells. This e!ect is characterised by the coe$cient a

, which increases with

angular losses.

3.4. The ewect of superxcial dust on the model parameters

As in real operating conditions the PVmodule contains a certain degree of dust, it isinteresting to check if the proposed model is also valid for these conditions. With thisaim, angular factor ( f

) measurements have been performed on PV modules havingdi!erent dust degrees. Although the dust characteristics can vary depending on itsnature and external factors and this could a!ect the optical transmittance of themodule, an approximate but e!ective way to characterise the dust thickness is bymeans of the relative transmittance at normal incidence in reference to the cleansurface condition value.But dust also modi"es the angular performance of the PV module by increasing

its angular losses. This fact is characterised by an increase of the angular lossescoe$cient, a

. Typical values of 0.17 for a m-Si module become 0.20 if a moderate dust

quantity is on its surface (I(0)

/I

(0)

"0.98) or 0.27 for a high amount of dust

(I(0)

/I

(0)

"0.92). The energetic consequences of these increments shall be

discussed afterwards in this paper.

4. Corrected expression of Isc with angular losses

The expression of the short-circuit current of a PV module considering the angularlosses of each radiation component (direct, di!use and albedo) is

I"IM

GM B cos [1!F ()]#D1#cos

2[1!F

()]

#A 1!cos2

[1!F()], (5)

where IM

is the short-circuit current at standard test conditions, GM the standard1000W/m irradiance, B the direct irradiance, D the di!use irradiance on the


Table 3Results of "tting equations (6b) and (6c) to the &exact+ F

and F

values for typical a

values of silicon PV

modules

a

c1 c (c

) r

0.16 4/(3) !0.074 !1.3E-02 0.9990.17 4/(3) !0.069 !1.3E-02 0.9990.18 4/(3) !0.064 !1.4E-02 0.999

horizontal plane, A the ground-re#ected irradiance on a horizontal plane facing theground, the module's tilt angle, the angle of incidence of solar radiation, F

the

angular losses factor of the solar radiation direct component, Fthe angular losses

factor of the solar radiation di!use component and Fthe angular losses factor of the

solar radiation ground-re#ected (albedo) component.The angular losses factors F

, F

and F

are obtained by the formulae

F()"exp(!cos a/a )!exp(!1/a )

1!exp(!1/)

, (6a)

F()exp!

1

acsin #

!sin 1!cos

#csin #

!sin 1!cos

(6b)

F()exp!

1

acsin #

!!sin 1#cos

#csin #

!!sin 1#cos

(6c)While the factor F

is directly calculated from the model expression, both F

and

Fare calculated by solving two integrals that consider the contribution of each solid

angle unit incident on the PV module (assuming an isotropic distribution of di!useand albedo radiation). Expressions (6b) and (6c) are two approximate analyticalsolutions of these integrals, with c

"4/(3) and c

as a "tting parameter (see Table 3).

The coe$cient of determination of such approximation is always greater than 0.999for the typical a

values, as it is shown in Table 3. Fig. 4 represents F

and F

versus

the tilt angle of the PV module for two representative avalues.

In good approximation it can be demonstrated [12] that relative power variationsare proportional to short-circuit current ones, being the proportionality factor slightlybigger than unity, commonly in the range [1, 1.1]. This fact permits to calculate therelative power (or energy) losses in PV generation due to current losses and, in mostcases, to consider that they are practically similar.


Fig. 4. Angular losses factors of the di!use and albedo radiation components for typical avalues of a m-Si

module (0.17 for clean surface and 0.2 for a medium dust quantity).

5. Angular losses calculation at di4erent European sites

With the aim of applying the proposed model to some illustrative cases of interest,the angular losses of a south-oriented standard monocrystalline PV module shall becalculated at di!erent sites and considering several tilt angles. For that purpose, thetypical meteorological year of ten di!erent European sites [13}16] having diverseclimates and latitudes has been considered. Table 4 summarises the geographical andclimatic characteristics of each location.

5.1. Monthly average losses

PV modules angular losses are mainly determined by the angle of incidence ofdirect radiation, and thus, the module tilt angle, local latitude and solar positionare the most in#uencing parameters. Considering Europe, in southern sitesPV modules su!er the highest monthly average losses in June}July for verticalposition, while in locations with higher latitudes the maximum angular losses areobtained in December at horizontal position. On the other hand, minimum valuesare obtained in all considered cases in December}January (at 803 tilt angle innorthern sites and 703 in the southern ones). Table 5 summarises the monthlyaverage values of the angular losses calculated on each considered site andFig. 5 represents the yearly evolution of the monthly losses for some selectedcases.


Table 4Geographic and climatic characteristics of the considered sites for the calculation of the PV angular losses

Site name Latitude (deg) Longitude (deg) Altitude (m) Clime KoK ppen [17]

Betdagan 32.0N 34.82E 30 BSw (semiarid)Seville 37.4N 6.0W 10 BSh (warm steppe)Murcia 37.79N 0.80W 3 BSh (warm steppe)Athens 37.97N 22.72E 7 Csa (mediterranean)Madrid 40.45N 3.71W 664 Csa (mediterranean)Logron o 42.5N 2.5W 384 Csb (mediterranean, moderate summer)Nice 43.65 7.2E 4 Csa (mediterranean)Paris}Trappes 48.78N 2.0E 123 Cfb (mild winters fresh summers)Ucle 50.8N 4.35E 100 Cfb (marine west coast)De Bilt 52.1N 5.18E 3 Cfb (marine west coast)

Table 5Monthly average angular losses calculated at each di!erent considered site. The table shows the minimumand maximum values and the correspondent tilt angle.

Monthly angular losses

Minimum Maximum

Site name Value (%) Month Tilt angle(deg)

Value (%) Month Tilt angle(deg)

Betdagan (32.0N) 1.8 Dec 70 14.7 Jul 90Sevilla (37.4N) 1.4 Dec 70 13.5 Jul 90Murcia (37.8N) 1.4 Dec 70 13.3 Jun 90Athens (38.0N) 1.5 Jan 70 14.8 Jun 90Madrid (40.5N) 1.7 Jan 70 12.6 Jun 90Logron o (42.5N) 1.6 Jan 70 11.0 Jun 90Nice (43.7N) 1.3 Dec 70 11.1 Jul 90Paris-Trappes (48.8N) 1.5 Dec 80 10.5 Dec 0Ucle (50.8N) 2.0 Dec 80 9.4 Dec 0De Bilt (52.1N) 1.9 Dec 80 10.8 Dec 0

5.2. Annual losses: The latitude and tilt angle inyuence

The angular losses for each considered place versus the module tilt angle arerepresented in Fig. 6. Highest angular losses correspond in all cases to verticalposition (facades in buildings) and di!erent minimum values are obtained for eachlocation, depending mainly on the latitude: the lowest losses in a yearly basis areachieved with tilt angles some degrees under the local latitude. This observed behav-iour is more clearly observed in Fig. 7, which represents the annual angular losses


Fig. 5. Monthly angular losses of a standardm-Si module at di!erent sites, plotted versus the number of themonth. Two extreme tilt angles are considered (03 and 903).

Fig. 6. Annual angular losses of a standard m-Si module at di!erent sites, plotted versus tilt angle.

versus the di!erence latitude-tilt angle. All the curves "t quite accurately quadraticfunctions (coe$cients of determination'0.98) and moreover they can be groupedinto two more general equations, one for the Mediterranean sites (Betdagan, Seville,Murcia, Athens, Madrid, Logron o, Nice) and the other one for central Europe ones(Paris-Trappes, Ucle, De Bilt):


Fig. 7. Annual angular losses of a standard m-Si module at di!erent sites, plotted versus latitude-tilt angle.

Mediterranean Europe:

AL

(%)"11.310(!)!11.910(!)#2.87.Coefficient of determination"0.97; standard error"0.19;

(7a)

Central Europe:

AL

(%)"8.410(!)!11.010(!)#3.49.Coefficient of determination"0.96; standard error"0.13.

(7b)

An approximate value for minimum annual angular losses at medium latitude sites is3% and is obtained with a tilt angle some degrees under the local latitude. Also animportant consequence of the above-obtained results deals with the integration of PVin buildings, where the angular losses can become especially signi"cant if PV moduleshave to be installed at orientations and tilt angles quite di!erent from the ideal ones.For medium latitude locations, especial attention should be paid to re#ection losseswhen dealing with facades integration. A good alternative is to install the PV modulesinto well-oriented roofs, or awnings.

5.3. The dust inyuence in the average angular losses

The existence of dust increases substantially the angular losses by a factor that canvary typically between 1.3 and 1.5, depending on the dust thickness (from moderate tovery signi"cant). Fig. 8 shows for the particular case of Madrid the dust e!ect on theannual angular losses, considering a typical moderate dust degree (I

(0)

/I(0)

"

0.98) and a very high one (I(0)

/I

(0)

"0.92).


Fig. 8. Annual angular losses of a standard m-Si module at Madrid, plotted versus tilt angle, consideringthree di!erent surface dust degrees.

6. Conclusions

With the proposed analytical model angular losses in PV modules working in realconditions can be easily calculated. The model depends on a "tting parameter calledthe angular losses coe$cient that characterises the relative angular response of the PVmodule. The model has been applied to the calculation of monthly and yearly losses in10 di!erent European sites, considering their typical meteorological years. Whilemodule technology does not in#uence meaningfully the angular losses, dust does:besides reducing the light transmittance at normal incidence, it also increases therelative angular losses. These losses present an annual evolution, which is verydependent on the latitude and the tilt angle, with an average value that is function ofthe di!erence latitude-tilt angle. A minimum annual value of about 3% is found for allthe considered sites.

Acknowledgements

The authors express their gratitude to Luis Zarzalejo and EstefanmHa Caaman o forproviding the typical meteorological years radiation data.

References

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Calculation of the PV Modules Angular Losses