a simple accurate model for the calculation of shading power losses in photovoltaic generators

7/27/2019 A Simple Accurate Model for the Calculation of Shading Power Losses in Photovoltaic Generators

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A simple accurate model for the calculation of shading power lossesin photovoltaic generators

P. Rodrigo a,, Eduardo F. Fernandez a, F. Almonacid b, P.J. Perez-Higueras b

a Center of Advanced Studies in Energy and Environment, Jaen University, Las Lagunillas Campus, C6 Building, 23071 Jaen, Spainb Electronic Engineering and Automatic Department, Jaen University, Las Lagunillas Campus, A3 Building, 23071 Jaen, Spain

Received 13 February 2013; received in revised form 10 April 2013; accepted 19 April 2013Available online 20 May 2013

Communicated by: Associate Editor Jan Kleissl

Abstract

This paper presents a simple accurate model that allows calculating the shading power losses in photovoltaic generators in a fast, easyto implement way. Calculated losses for different generators and shading scenarios are compared to those obtained with a detailed modelbased on solving the whole currentvoltage curve of the generators. Results show a good agreement between the proposed model and thedetailed model. The proposed model is useful for the calculation of the instantaneous power losses and also gives good results in thecalculation of the energy yield, requiring less information as input than other detailed models and requiring low computational effort. 2013 Elsevier Ltd. All rights reserved.

Keywords: Photovoltaic generators; Shading power losses

1. Introduction

The presence of shading in photovoltaic (PV) generatorscauses disproportional power losses that are difficult toquantify (Drif et al., 2008, 2012; Quaschning and Hanitsch,1995; Woyte et al., 2003). In order to calculate these losses,two approaches can be used. The first one is to use a modelbased on solving the whole currentvoltage curve of thegenerator. These kinds of models are accurate, but are dif-

ficult to implement, require a great deal of information asinput and require extensive computation. The secondoption is to use a simplified model, easy to implementand fast. This option is preferable in practical applications

and can be easily integrated in common yield estimationsoftware.

Nowadays, we can find in the scientific literature modelsfor shading losses calculation based on solving the wholecurrentvoltage curve of the generator (Alonso-Garcaet al., 2006; Brecl and Topi, 2011; Kajihara and Harakawa,2005; Karatepe et al., 2007; Kawamura et al., 2003; Quasch-ning and Hanitsch, 1995, 1996; Silvestre and Chouder,2008). These models require as inputs all the details concern-

ing the basic components of the generator (solar cells andbypass diodes), as well as the details about the electricalinterconnections between them. Knowledge of the amountof shading on each solar cell is also required. Once the behav-ior of the basic components is known, the behavior of thewhole generator is calculated by applying the laws of the Cir-cuit Theory. This implies dealing with non-linear equationssystems that must be numerically solved.

The main advantage of these models is their accuracy.However, their practical application presents difficulties.First, a great deal of information is required as input in

0038-092X/$ - see front matter 2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.solener.2013.04.009

Abbreviations: PV, photovoltaic; RMSE, Root Mean Square Error; MCS,Most Concentrated Shadow; MDS, Most Distributed Shadow; SEM, si-ngle exponential model; RMSD, Root Mean Square Difference. Corresponding author. Tel.: +34 953213518; fax: +34 953212183.

E-mail address: [email protected] (P. Rodrigo).

www.elsevier.com/locate/solener

Available online at www.sciencedirect.com

Solar Energy 93 (2013) 322333


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order to be able to reproduce the IV curves of the solarcells and bypass diodes. This information is not easy toget. On the other hand, dealing with the whole IV curveof the basic components is problematic from a computa-tional point of view, taking into account that a minimumof 50 points is required to determine an IVcurve (Blaesser

and Munro, 1995) and that a PV generator is composed ofthousands of solar cells. In addition, solving the non-linearequations that govern the behavior of the generator is timeconsuming. Finally, the implementation of these modelsand the numerical algorithms is not easy. So these modelsare oriented to research works and not to their commonuse by the PV professionals in yield estimation (Klise andStein, 2009).

When it is not possible to have a set of IV curves thatcharacterize the components of the generator, the user isforced to apply a simplified model. Existing software pack-ages rely on simplified expressions that allow a fast estima-tion of the shading losses (Institute of Environmental

Sciences, 2012; Laplace systems, 2012; National RenewableEnergy Laboratory, 2012). However, an experimental cam-paign carried out by Martnez-Moreno et al. (2010) showedthat these expressions lead to appreciable errors. Theseauthors proposed a more accurate model based on an empir-ical equation that improves the previous methods. It showsgood results in the calculation of the energy yield. However,it still gives important deviations in the calculation of theinstantaneous power losses. Depending on the consideredgenerator and its particular configuration, the authorsreported Root Mean Square Errors (RMSEs) between themodeled and the experimental power losses of up to 26%.

So, it would be desirable to have a model that combinessimplicity, good results in the calculation of the energy yieldand a better characterization of the instantaneous shadinglosses. Such a model is proposed in this paper. The modelconsists on several analytical equations that can be easilyimplemented, require less information than existing detailedmodels and require low computational effort.

In this paper, the proposedmodel is comparedto a detailedmodel based on solving the whole currentvoltage curve ofthe generator. The detailed model is very accurate, so that itis used as a baseline to compare the proposed model. Thereis wide experience that the detailed model is able to representa real PV generator (Kawamura et al., 2003; Quaschning andHanitsch, 1996; Silvestre and Chouder, 2008).

Results of the present research show that the proposedmodel provides an adequate characterization of the instan-taneous power losses, as well as a good estimation of theenergy yield.

The paper is organized as follows: in the second sec-tion, some definitions related to the research are stated;in the third section, a detailed model based on solvingthe whole currentvoltage curve of the generator isbriefly summarized; in the fourth section, the proposedmodel is explained; the fifth section presents the resultsof applying the proposed model and the detailed model

to the same shading scenarios, and the differencesbetween both models are discussed; finally, in the sixthsection, the conclusions and future actions related tothe research are commented.

2. Definitions

The generic structure of a PV generator as considered inthe present research is shown in Fig. 1. The generator isdivided into blocks, where a block is composed of severalseries-connected solar cells protected by a bypass diode.The dimensions of the generator are defined by the param-eters NCELL, NBLOCK and NSTR. A block is composed ofNCELL solar cells in series, a string is composed of NBLOCKblocks in series and the generator is composed of NSTRstrings in parallel. This way, the generator containsNBLOCKNSTR diode-protected blocks. The total numberof cells is NCELLNBLOCKNSTR.

The following definitions will be used:

Geometric shading factor (sG): it is the ratio between theshaded area (ASH) and the total area (A).

sG ASH=A 1

The geometric shading factor can be defined for a solarcell (sG,CELL), for a diode protected block (sG,BLOCK), for astring of series-connected blocks (sG,STRING) or for a gener-ator (sG,GEN).

Mean global irradiance on the plane of the shaded PV

device (GSH): it can be determined from the componentsof the global irradiance (G) and the geometric shadingfactor (sG). The global irradiance G can be written asthe sum of four components: a direct component (B),a diffuse circumsolar component (DCIR), a diffuse isotro-pic component (DISO) and an albedo component (R).The geometric shading factor only affects the two direc-tional components: B and DCIR (Quaschning and Han-itsch, 1995, 1998). So, the following can be written:

GSH 1 sGB DCIR DISO R 2

In practice, DISO + R can be measured with a calibratedmodule placed on the shaded area and G, with a calibrated

module placed on the illuminated area. This way, B+ DCIRis obtained by subtracting G (DISO + R) and GSH isobtained from Eq. (2).

The mean global irradiance can be defined for a cell(GSH,CELL), a block (GSH,BLOCK), a string (GSH,STRING)or a generator (GSH,GEN).

Shading factor (s): it is defined as the ratio between theglobal irradiance loss due to shading and the total globalirradiance:

s G GSH=G 3

P. Rodrigo et al./ Solar Energy 93 (2013) 322333 323


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Again, it can be defined for a cell (sCELL), a block(sBLOCK), a string (sSTRING) or a generator (sGEN).

From (3), an alternative way of expressing the mean glo-bal irradiance is:

GSH 1 sB DCIR DISO R; 4

i.e. the factor s affects the four components of the global

irradiance.

Shading intensity factor (sI): it is the ratio between s andsG:

sI s=sG 5

It also can be expressed as the ratio between the direc-tional components of the global irradiance and the totalglobal irradiance:

sI B DCIR

G6

Shading power losses factor (LSH): it is the ratio betweenthe power loss due to shading and the theoretical powerthat would be produced in absence of shadows:

LSH P PSH=P 7

PSH being the power produced by the considered PV deviceunder shading and Pthe theoretical power produced in ab-sence of shadows. It can be defined for a cell ( LSH,CELL), ablock (LSH,BLOCK), a string (LSH,STRING) or a generator(LSH,GEN). In PV systems, the value of LSH is greater thanthe value ofs, i.e. the relation between the irradiance lossesand the power losses is not linear.

Number of shaded cells in a block (nS,ij): for a particularblock (i, j), it is the number of cells in the block with ageometric shading factor sG,CELL > 0.

Number of totally shaded cells in a block(nT,ij): for a par-ticular block (i, j), it is the number of cells in the blockwith a geometric shading factor sG,CELL = 1. It is obvi-ous that, for a block, the following is true:

0 6 nT;ij 6 nS;ij 6 NCELL 8

Number of shaded blocks in a string(nS,j): for a particularstring j, it is the number of blocks in the string with ashading power losses factor LSH,BLOCK > 0.

Different kinds of shadows will be considered in order todevelop the proposed model for shading power losses cal-culation. For each block in the generator, four shadowsare defined: the real shadow, the Most Concentrated Sha-dow (MCS), the Most Distributed Shadow (MDS) andthe approximated real shadow. Fig. 2 shows the values of

the cell shading factors for the four defined shadows in ageneric block.

The four shadows for a block are defined in the follow-ing way:

Real shadow: the real shadow has a geometric shadingfactor of 1 for the first nT,ij cells. Hence, the shading fac-tor for these cells is sij,k= sI sG,ij,k= sI 1 = sI. The cellsin the positions from nT,ij + 1 to nS,ij have shading fac-tors s1, s2, . . . , sh. The remaining cells are unshaded.

MCS shadow: the Most Concentrated Shadow of a blockis obtained by concentrating the shaded area in the mini-mum possible number of solar cells. This way, the MCS

Fig. 1. Structure of a generic PV generator.

324 P. Rodrigo et al./ Solar Energy 93 (2013) 322333


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shadow has a shading factor of 1 for the first Int(sBLOCK,ij-NCELL) cells, where Int(sBLOCK,ijNCELL) designates theinteger part of the product sBLOCK,ijNCELL. Afterwards,there is one partially shaded cell with a shading factor sr,ij.The remaining cells are unshaded.

The value of sr,ij can be obtained taking into accountthat the block shading factor can be expressed as the meanvalue of the individual cells shading factors:

sBLOCK;ij 1

NCELL

XNCELLk1

sij;k IntsBLOCK;ijNCELL 1 sr;ij

NCELL

! sr;ij sBLOCK;ijNCELL IntsBLOCK;ijNCELL

9

MDS shadow: the Most Distributed Shadow of a blockis obtained by distributing the shaded area between thenS,ij cells in a way that every cell is equally shaded. Thisway, the MDS shadow has a constant shading factor sd,ijfor the first nS,ij cells and the rest of the cells remainunshaded. The value of sd,ij can be obtained by:

sBLOCK;ij 1

NCELL

XNCELLk1

sij;k nS;ijsd;ij

NCELL! sd;ij

sBLOCK;ijNCELL

nS;ij10

Approximated real shadow: in order to simplify the final

equations of the model, an approximated real shadow isdefined in the following way: the first nT,ijcells have a shad-ing factor sIand the cells in the positions from nT,ij+ 1 tonS,ijhave a constant shading factor sa,ij. The restof the cellsremain unshaded. The value ofsa,ijcan be obtained by:

sBLOCK;ij 1

NCELL

XNCELLk1

sij;k nT;ij sI nS;ij nT;ij sa;ij

NCELL

! sa;ij sBLOCK;ijNCELL nT;ij sI

nS;ij nT;ij11

On the other hand, for a given string j in the generator,three kinds of shadows are defined: the real shadow, the

Most Concentrated Shadow (MCS) and the Most

Distributed Shadow (MDS). In this case, each block in thestring is replaced by an equivalent PV device whose shadingfactor is equal to the shading power losses factor of theblock. Hence, if we name sBLOCK EQ,ij the shading factor ofthe equivalent PV device for the block (i, j), this parameteris defined as:

sBLOCK EQ;ij LSH;BLOCK;ij 12

and the equivalent shading factor of the string sSTRING EQ,jis defined as:

sSTRING EQ;j 1

NBLOCK

XNBLOCKi1

sBLOCK EQ;ij 13

Fig. 3 shows the values of the block equivalent shadingfactors for the three defined shadows in a generic string.

The values of sr,j and sd,j for the case of a string can beobtained in a similar way than for the case of a block:

sr;j sSTRING EQ;jNBLOCK IntsSTRING EQ;jNBLOCK 14

sd;j sSTRING EQ;jNBLOCK

nS;j15

3. Model based on solving the whole IV curve of the

generator

Several authors have proposed models based on solving thewhole currentvoltage curve of the generator for the calcula-tion of shading power losses from similar points of view(Alonso-Garca et al., 2006; Brecl and Topi, 2011; Kajiharaand Harakawa, 2005; Karatepe et al., 2007; Kawamuraet al., 2003; Quaschning and Hanitsch, 1995, 1996; Silvestreand Chouder, 2008). In the present research, a detailed modelinspired in some of these works has been implemented. As itwas commented in Section 1, there is experimental evidencethat this model is very accurate and provides a good character-ization of real PV generators (Kawamura et al., 2003; Quasch-ning and Hanitsch, 1996; Silvestre and Chouder, 2008).

Most of the reviewed papers rely on the standard Single

Exponential Model (SEM) in order to characterize the

Fig. 2. Shading factors of the individual cells in a generic block for four cases of shadows: real shadow, MCS shadow, MDS shadow and approximatedreal shadow.



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behavior of a solar cell. This model is composed of a cur-rent source, a diode, a series resistance and a shunt resis-tance. As, in shading scenarios, the operation of the cellwith high reverse voltage is possible, the SEM model mustbe extended to take into account the avalanche effect. Sev-eral authors have proposed models for solar cells working

in reverse voltage (Hartman et al., 1980; Lopez Pineda,1986; Spirito and Abergamo, 1982). For the purposes ofthe present research, we have followed the formulation ofBishop (1988). It is based on including a voltage sourcein series with the shunt resistance. Fig. 4 shows the electri-cal circuit proposed for modeling the solar cell.

The M term is defined as the exponential functionM a1 Vj=VBR

n, Vj being the pn junction voltage.

This way, the equation of the solar cell IV curve can be

expressed as:

I IPH I0fexpV IRS=mVT 1g V

IRS=RPf1 a1 V IRS=VBRn

g 16

equation that depends on eight parameters: IPH (the photo-current), I0 (the saturation current), m (the diode idealityfactor), RS (the series resistance), RP (the shunt resistance),VBR (the avalanche breakdown voltage), a (the factor ofthe Bishops term) and n (the exponent of the Bishopsterm). VT is the semiconductor thermal voltage, VT= kT/q, being k the Boltzmann constant, T the absolute temper-ature of the junction and q the electron charge.

In general, these parameters depend on the operatingconditions of the cell: G (global irradiance on the planeof the cell) and TC (cell temperature). In the present study,

VBR is assumed to grow linearly with the cell temperature(Alonso-Garca and Ruiz, 2006; McKay, 1954), parametersa, n and RPare kept constant and the rest of the parametersare allowed to vary with G and TC. There are differentmethods for determining the variable parameters of themodel IPH, I0, m and RS (Chan et al., 1986; Charleset al., 1981; Chegaar et al., 2006; Easwarakhantan et al.,1986; Jain and Kapoor, 2004; Ortiz-Conde et al., 2006;Phang et al., 1984; Ranuarez et al., 2000). In this case, anumerical method has been chosen based on building a sys-tem of four non-linear equations: three equations are

derived by applying the solar cell equation, Eq. (16), tothe short-circuit point, the open-circuit point and the max-imum power point; the fourth equation is obtained byapplying the condition that the derivative of the power withrespect to the voltage must be zero at the maximum powerpoint voltage (Kennerud, 1969). The four non-linear equa-tions are then solved by means of a trust-region optimiza-tion method (Powell, 1970).

Once the solar cell parameters have been determined,the generator model is based on building a system of equa-tions that govern the behavior of the generator. Theseequations involve: (1) the equations of the partially shadedsolar cells, (2) the equations of the bypass diodes and (3)the equations that relate the system currents and voltagesobtained from the Kirchhoffs laws.

The equations for the solar cells are:

1 sij;kIPH I0 expVij;k IijCRS

mVT

1

!

Vij;k IijCRS

RP1 a 1

Vij;k IijCRS

VBR !

n

& ' IijC 017

sij,k being the shading factor of the cell in the block (i, j)position k.

The equations for the bypass diodes are:

I0D exp Vij

mDVTD

1

! IijD 0 18

the bypass diode parameters being I0D (saturation current),mD (diode ideality factor) and VTD (thermal voltage).

Fig. 3. Equivalent shading factors of the individual blocks in a generic string for three cases of shadows: real shadow, MCS shadow and MDS shadow.

Fig. 4. Solar cell model including the avalanche effect.



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These equations in combination with the equationsderived from the Kirchhoffs laws are solved by means ofthe multivariable NewtonRaphson method.

Finally, we have obtained two IV curves: the curve ofthe unshaded solar cell and the curve of the partiallyshaded generator. From the first curve, we calculate the

maximum power point power of the unshaded cell (PCELL)and, from the second curve, the maximum power pointpower of the shaded generator (PSH,GEN). The shadingpower losses factor of the generator is then obtained by:

LSH;GEN 1 PSH;GEN

NCELLNBLOCKNSTRPCELL19

4. Proposed model

The explanation of the proposed model is organized inthree subsections:

Calculation of the shading power losses factors of theisolated blocks, LSH,BLOCK,ij (considering that eachblock is directly connected to an inverter, i.e. it is iso-lated from the rest of the blocks).

Calculation of the shading power losses factors of theisolated strings, LSH,STRING,j (considering that eachstring is directly connected to an inverter, i.e. it is iso-lated from the rest of the strings).

Calculation of the final shading power losses factor ofthe generator, LSH,GEN.

4.1. Shading power losses factors of the isolated blocks

Two cases are considered for a partially shaded block;they are named Most Concentrated Shadow (MCS) andMost Distributed Shadow (MDS). The shading powerlosses factor of the block is calculated for each one of thesecases, obtaining LSH(MCS)ij and LSH(MDS)ij. It will beshown that the real shading power losses factor of theblock, LSH,BLOCK,ij, will have a value between LSH(MDS)ijand LSH(MCS)ij:

LSHMDSij 6 LSH;BLOCK;ij 6 LSHMCSij

Therefore, once the MCS and MDS cases are calculated,a procedure is proposed for calculating LSH,BLOCK,ij fromthese two values.

The shading power losses factor of the block for theMCS case is modeled as:

LSHMCSij minfsBLOCK;ijNCELL; 1g 20

This approximate expression considers that the shadingpower losses factor grows linearly with the shading factorand gets a value of 1 when sBLOCK,ij= 1/NCELL, i.e. thepower generated by the block cancels when one cell is fullyshaded.

The shading power losses factor of the block for the

MDS case is modeled as:

LSHMDSij sBLOCK;ijNCELL=nS;ij 21

This approximate expression considers that the shadingpower losses factor grows linearly with the shading factorand gets a value of 1 when sBLOCK,ij = nS,ij/NCELL, i.e. thepower generated by the block cancels when the nS,ij cellsare fully shaded.

Fig. 5 shows an example of the LSH(MDS) and LSH(MCS)functions for a block containing 4 cells with 2 shaded cells.The double arrows indicate the possible values of the blockshading power losses factor for two given shading factorssBLOCK1 and sBLOCK2.

Once the LSH(MCS)ij and the LSH(MDS)ij shadingpower losses factors have been determined, the procedurefor calculating the shading power losses factor of the block,LSH,BLOCK,ij, is as follows:

(1) Calculation of the mean value of the shading factorfor the Most Distributed Shadow case (lij):

lij 1

nS;ij

XnS;ijk1

sij;k nS;ijsd;ij

nS;ij sd;ij

sBLOCK;ijNCELL

nS; ij22

(2) Calculation of the standard deviation of the MostConcentrated Shadow shading factors with respectto lij (rMCS,ij):

rMCS;ij

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi1

nS;ij

XnS;ijk1

sij;klij2

vuut

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiIntsBLOCK;ijNCELL 1 lij

2 sr;ij lij2 nS;ij IntsBLOCK;ijNCELL 1 0 lij

2

nS;ij

s

23

(3) Calculation of the standard deviation of the approx-imated real shadow shading factors with respect to lij(rij):

rij

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

nS;ij

XnS;ijk1

sij;k lij2

vuut

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinT;ij sI lij

2 nS;ij nT;ij sa;ij lij2

nS;ij

s24

(4) Calculation of the shading power losses factor of theblock (LSH,BLOCK,ij):

LSH;BLOCK;ij LSHMDSij

rij

rMCS;ij LSHMCSij

LSHMDSij 25

As can be seen, the standard deviation with respect to lijis an indicator of how far is the considered shadow withrespect to the Most Distributed Shadow case. A standarddeviation of zero corresponds to the MDS case, in whichwe have a minimum value of the block losses. A standarddeviation ofrMCS,ij corresponds to the MCS case, in whichwe have a maximum value of the block losses. The stan-dard deviation of the approximated real shadow will have

a value between 0 and rMCS,ij and the model supposes that



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the shading power losses factor of the block grows linearlywith this value of the standard deviation.

4.2. Shading power losses factors of the isolated strings

For a given string j in the generator, each block in the

string is replaced by an equivalent PV device whose shad-ing factor is equal to the shading power losses factor ofthe block as calculated in Section 4.1. Again, two casesare considered for the isolated string: the Most Concen-trated Shadow (MCS) and the Most Distributed Shadow(MDS).

The shading power losses factor of the string for theMCS case is modeled as:

LSHMCSj minsSTRING EQ;j

0:9; 1

n o26

i.e. the shading power losses factor grows linearly with the

equivalent shading factor of the string and gets a value of 1when sSTRING EQ,j= 0.9. In order to justify this approxima-tion Fig. 6 is presented. The graphs compare the functionLSH(MCS) = f(sSTRING EQ) obtained with the detailedmodel to the function obtained with the proposed model.It can be seen that the more the number of blocks in thestring, the better the proposed model approximates thisfunction.

The shading power losses factor of the string for theMDS case is modeled as:

LSHMDSj min1:1 sSTRINGEQ;j

0:1 nS;j=NBLOCK;1:1

nS;j

NBLOCK;1

& '27

Fig. 7 represents this function for a string containing 10blocks and four values of nS,j (nS,j= 3, 6, 8, 10). It can beseen that the proposed model approximates the functionfor any value of sSTRING EQ,j and nS,j.

Finally, the procedure for calculating the shading powerlosses factor of the string, LSH,STRING,j, consists of the fol-

lowing equations:

lj sd;j sSTRING EQ;jNBLOCK

nS;j28

rMCS;j

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIntsSTRINGEQ;jNBLOCK 1 lj

2 sr;j lj2 nS;j IntsSTRINGEQ;jNBLOCK 1

0 lj

2

nS;j

s

29

rj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

nS;j

XnS;ji1

sBLOCK EQ;ij lj2

vuut 30LSH;STRING;j LSHMDSj

rj

rMCS;j LSHMCSj

LSHMDSj 31

4.3. Shading power losses factor of the generator

Once the shading power losses factors of the isolatedstrings LSH,STRING,j have been calculated, a first approachto the calculation of the generator shading power lossesfactor would be:

LSH;GEN 1

NSTR

XNSTRj1

LSH;STRING;j 32

However, this is an optimistic estimation. Both theblocks and the strings operate in points different than themaximum power point due to mismatch effects. These mis-match losses were not considered in the previous calcula-tions. So, in order to take into account these mismatchlosses, the shading power losses factor calculated withEq. (32) must be corrected. We propose the followingexpression for obtaining the final value of the shadingpower losses factor of the generator:

LSH;GEN 1 e1 LSH;GEN

h i LSH;GEN 33

where a value of the mismatch parameter e = 0.3 has been

established.This parameter was calibrated through simulations.

While results presented in the next section have beenobtained with e = 0.3, a number of simulations were previ-ously done in order to get this value. An algorithm wasimplemented that varied the value of e and calculated ateach iteration the whole set of generators and shading con-ditions. The algorithm computed the root mean squareddifference in the shading power losses factor between theproposed model and the detailed model for each value ofe. Finally, the value ofe = 0.3 was observed to be the valuethat minimized this difference.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Shading factor of the block

Shadingpowerlossesfacto

roftheblock

Lsh(MCS)

Lsh(MDS)

sblock2sblock1

Lsh,block2

Lsh,block1

Fig. 5. Example of a block containing 4 cells (NCELL = 4) with 2 shadedcells (nS= 2). The LSH(MDS) function is a lower limit ofLSH,BLOCK, whilethe LSH(MCS) function is an upper limit of LSH,BLOCK. When the blockhas a shading factor sBLOCK1, the double arrow LSH,BLOCK1 indicates thepossible values of the block shading power losses factor. When the blockhas a shading factor sBLOCK2, the double arrow LSH,BLOCK2 indicates thepossible values of the block shading power losses factor.



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5. Results

Fig. 8 shows the generators and configurations for two

analyzed cases: A and B.The case A is based on the module A, a module com-

posed of 32 solar cells in series with 2 bypass diodes, onediode each 16 cells. A generator is defined composed ofsix modules. The seriesparallel connection of the modulesin the generator is varied in order to obtain four differentconfigurations: A1, A2, A3 and A4.

The case B is based on the module B, a module com-posed of 30 solar cells in series with 3 bypass diodes, onediode each 10 cells. A generator is defined composed ofeight modules. The seriesparallel connection of the mod-ules in the generator is varied in order to obtain four differ-

ent configurations: B1, B2, B3 and B4.

Shadows on a PV generator can be caused by a lot ofdifferent external obstacles: trees, buildings, clouds, adja-cent arrays, etc. The possible shading cases are infinite.So, in the present analysis, we have selected a set of repre-sentative shading conditions in order to make the problemaffordable. This way, different rectangular shadows havebeen reproduced for the generators. The shape of theseshadows is as shown in Fig. 9. This is the shape of the shad-ows projected on a two-axis solar tracker by an adjacenttracker. Also, these shadows could be caused by a row offixed modules on an adjacent fixed array, or by a wallplaced near the analyzed generator. Each shading profileis defined by two parameters: the horizontal geometricshading factor (sG,HOR = XSH/XGEN) and the vertical geo-metric shading factor (SG,VER = YSH/YGEN). This way, 20different shading profiles have been simulated for each gen-erator by means of the two implemented models. The val-ues ofsG,HOR and sG,VER for the 20 reproduced shadows areindicated in Table 1.

Table 2 shows in detail the results of five example shad-ows applied to the generators A1, A2, A3 and A4. Thetable indicates the shading power losses factor obtainedby the detailed model and by the proposed model for thefive selected shadows. Also, the relative difference between

models in percentage is shown in brackets:relative difference% LSH;proposed model

LSH;detailed model 100 34

For these generators and example shadows, the relativedifferences vary between 9.87% and 15.91%. These largevalues can surprise the reader. However, it must be takeninto account that these values correspond to specific exam-ples. The advantages of the proposed model are better shownif we examine the averaged values of the whole set of simula-tions. For this purpose, the Root Mean Square Difference(RMSD) between the proposed model and the detailed

model has been calculated for each generator by means of:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Equivalent shading factor of the string

Shadingpowerlossesfactorofthestring

fortheMCScase

Nblock=10

Detailed model

Proposed model

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Shadingpowerlosses

factorofthestring

fortheMCS

case

Nblock=20

Detailed model

Proposed model

Fig. 6. MCS case: shading power losses factor of a string containing 10 blocks (left) and 20 blocks (right) as a function of the equivalent shading factor.The function obtained with the detailed model is compared to the function obtained with the proposed model.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Shad

ingpowerlossesfactorofthestring

fortheMDScase

Nblock=10

ns=3

ns=6

ns=8

ns=10

Detailedmodel

Proposed model

Fig. 7. MDS case: shading power losses factor of a string containing 10blocks as a function of the equivalent shading factor for four values of nS(nS= 3, 6, 8, 10). The function obtained with the detailed model iscompared to the function obtained with the proposed model.



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RMSD%

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XNi1

relative difference%2

vuut 35Nbeing the number of analyzed shadows (N= 20). Results

are shown in Table 3.

The global RMSD is 8.0%, what is a good value.Another existing simplified model (that developed byMartnez-Moreno et al. (2010)) provides Root MeanSquare Errors between modeled and experimental powerlosses between 12% and 26% depending on the analyzed

generator. This means that the proposed model seems to

Fig. 8. Generators A1, A2, A3, A4 and B1, B2, B3, B4 under study.



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reduce the error in the shading power losses calculation.However, it must be taken into account that the calculatedRMSDs correspond to the specific shading profiles definedin Table 1. If other profiles were simulated, the globalRMSD could be different. The presented results only indi-cate that, for the considered generators and shading pro-files, the model is accurate. In order to generalize theresults, many other generators and shadings should be sim-ulated. In this paper, we have limited the number of shad-ing scenarios and therefore, the global RMSD of 8.0%cannot be considered definitive.

The proposed model also seems to be adequate for the cal-culation of the energy yield. Next, we analyze an example ofannual energy calculation. Obviously, it is only a particularexample. The number of possible cases is infinite and thereforethe results of this example cannot be considered definitive.

A PV plant composed of 25 two-axis solar trackers hasbeen simulated. The 25 trackers are equally spaced on agrid of L L m (Fig. 10) and the shadows projected onthe central tracker by the 24 adjacent trackers have beenreproduced. The reproduced shadows are fully realistic,i.e. the exact geometric shape has been computed at each

timepoint from the plant configuration and the solar posi-tion. Each tracker supports a PV generator composed of 3strings of 12 modules type A.

Fig. 11 shows the evolution of the central tracker powerproduction for the earliest hours of a typical day in March(from sunrise at 6:05 AM to 8:10 AM) considering the typ-

ical meteorological data of Jaen, South of Spain, 37.766N3.790W and a distance between trackers L = 10 m. Threecurves are represented: the curve of the detailed model,the curve of the proposed model and the curve of the sim-plified model developed by Martnez-Moreno et al. (2010).It can be seen that the proposed model approximates betterthe curve of the detailed model, so that it better character-izes the shading power losses for this particular example.

Finally, Table 4 shows the annual shading energy lossesof the central tracker for two cases of distance betweentrackers: L = 10 m and L = 11 m. It can be seen that theproposed model provides a very good approximation ofthe energy yield in these cases. Experimental annual energy

losses in real tracking plants are in the range 56% ( Garcaet al., 2008). In the simulations, we have obtained lossesslightly higher. This can be due to the fact that, in realplants, the trackers located at the boundaries are lessaffected by shade and contribute to reduce the plant energylosses.

The examples analyzed in this section show that the pro-posed model seems to provide a good characterization ofthe shading power losses as well as an accurate estimationof the energy yield. However, it will be necessary to vali-date the model for many other scenarios in order to ensurethe ability of the model to adapt to every case.

Fig. 9. Type of shadows reproduced for the generators.

Table 1Dimensions of the 20 reproduced shadows for the generators A1, A2, A3, A4, B1, B2, B3 and B4.

Shadow number sG,HOR sG,VER Shadow number sG,HOR sG,VER Shadow number sG,HOR sG,VER Shadow number sG,HOR sG,VER

1 1.000 0.094 6 0.771 0.250 11 0.313 0.688 16 0.125 1.0002 1.000 0.250 7 0.771 0.438 12 0.135 0.094 17 0.271 1.0003 1.000 0.438 8 0.490 0.094 13 0.135 0.906 18 0.333 1.0004 0.875 0.094 9 0.490 0.750 14 0.031 0.375 19 0.458 0.8755 0.771 0.094 10 0.313 0.063 15 0.031 1.000 20 0.625 0.875

Table 2Results of applying the two implemented models to the generators A1, A2, A3 and A4 for five example shadows.

Shadownumber

sG,HOR sG,VER sG LSH detailed model LSH proposed model (relative difference, %)

Gen A1 Gen A2 Gen A3 Gen A4 Gen A1 Gen A2 Gen A3 Gen A4

1 1.000 0.094 0.094 0.5477 0.5477 0.5477 0.5477 0.7068 (15.91) 0.7068 0.7068 0.7068(15.91) (15.91) (15.91)

7 0.771 0.438 0.337 0.8236 0.6950 0.7383 0.7977 0.8266 0.7436 0.7708 0.7964(0.30) (4.87) (3.25) (0.13)

12 0.135 0.094 0.013 0.1320 0.1002 0.1954 0.1647 0.2023 0.1130 0.1795 0.2023(7.03) (1.28) (1.58) (3.76)

16 0.125 1.000 0.125 0.1854 0.1432 0.2788 0.3269 0.2283 0.1782 0.2283 0.2283(4.29) (3.50) (5.06) (9.87)

20 0.625 0.875 0.547 0.7407 0.5764 0.5760 0.6334 0.7920 0.6403 0.6403 0.6798

(5.13) (6.39) (6.44) (4.64)



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6. Conclusions and future actions

A simple model that allows calculating the shading powerlosses in photovoltaic generators has been developed. It

require as inputs the geometric shading factor, the numberof shaded cells and the number of totally shaded cells foreach block in the generator and it also takes into accountthe seriesparallel interconnection of the blocks.

The model does not require solving the full IVcurve ofthe generator so that it is fast and easy to implement. Likeother existing simplified models, it requires a reducedamount of information in order to be applied, so that itis useful when it is not possible to get a set of IV curvesfor the components of the generator. It has shown goodperformance in the calculation of both the energy yieldand the instantaneous shading power losses for some ana-lyzed examples. So, it is a good candidate to be integratedin existing photovoltaic software packages and to be usedby the photovoltaic professionals.

However, as the number of possible cases of generatorsand shading scenarios is infinite, the model needs to beapplied to more cases in order to be completely validated.Also, the presented results are based on comparing the pro-

posed model to a detailed model based on solving thewhole IV curve of the generators, so that, as a futureaction, an experimental campaign will be required in orderto prove the validity of the model when applied to realshading scenarios.

Table 3Root Mean Square Difference in the shadingpower losses factor between the proposed modeland the detailed model.

Generator RMSD (%)

A1 8.5A2 7.2

A3 7.3A4 7.5B1 11.0B2 7.0B3 7.2B4 8.4

Global 8.0

Fig. 10. Configuration of a PV plant composed of 25 two-axis solartrackers.

6:00 6:10 6:20 6:30 6:40 6:50 7:00 7:10 7:20 7:30 7:40 7:50 8:00 8:100

200

400

600

800

1000

1200

Time (h:min)

Centraltrackerpowerproduction(W)

Detailed model

Martnez-Moreno model

Proposed model

Fig. 11. Central tracker power production as calculated by three models for a typical day in March in Jaen from sunrise to 8:10 AM.

Table 4Annual shading energy losses of the central tracker located at Jaen for twovalues of the distance between trackers, L.

Distancebetweentrackers, L (m)

GroundCover Ratio,GCR

Annualenergylosses, %

Annual energy losses, %(relative difference, %)

Detailed

model

Proposed model

10 0.177 7.17 7.13 (0.04)11 0.147 6.01 5.94 (0.07)



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a simple accurate model for the calculation of shading power losses in photovoltaic generators

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