numercial simulation of j-v for shaded cells

7/30/2019 Numercial Simulation of J-V for Shaded Cells

1/8

PergamonSolar Energy Vol. 56, No. 6, pp. 513-520, 1996

PII: SOO3&092X( %)00006-O Copyright 0 1996 Elsevier Science LtdPrinted in Great Britain. All rights reserved0038-092X/96 $15.00+0.00

NUMERICAL SIMULATION OF CURRENT-VOLTAGECHARACTERISTICS OF PHOTOVOLTAIC SYSTEMS WITH SHADED

SOLAR CELLSVOLKER QUASCHNINGT and ROLF HANITSCHT

Electrical Machines Institute, Berlin University of Technology, Secretariat EM 4, Einsteinufer 11,D-10587 Berlin, Germany

(Communicated by JOACHIM LUTHER)Abstract-In photovoltaics the actual curve of the current-voltage characteristic of a solar generator isoften needed, for example if the maximum power point is to be determined. It is convenient to use amodel for the description of the relationship between current and voltage, which is able to describe thesolar cell in the generation region as well as the breakdowns at positive and negative voltages. As themathematical expression cannot be given in an explicit form, a set of suitable numerical algorithms tocompute the currents for a given voitage and vice versa is given. For more complex solar generators, itcan neither be assumed that the solar cells have identical electrical characteristics, nor that they are evenlyilluminated. A general model for the description of solar generators is proposed that gives all voltagesand currents as well as the voltage and current at the outputs of the solar generator itself. This new modelis also implemented in a numerical algorithm. The characteristics of partially shaded solar generatorswere calculated and discussed using the proposed models. Copyright 0 1996 Elsevier Science Ltd.

1. INTRODUCTION 2. SOLAR CELL MODELFor the description of the electrical behav ior ofa solar cell the two diodes model is commonlyused, see e.g. Overstraeten (1986).

For cells that are driven in the negativevoltage range another model is necessary, whichdescribes the breakdow n region at high negativevoltage s. Neg ative voltages for solar cells canoccur at non-uniform illuminated photovoltaicgenerators, especially during partial shadin g ofthe generator. One model w as described byRauschen bach (1980). A more precise model,based on the one diode model, was given byBishop (19 88). This model can be adapted tothe two diodes model and offers optimal condi-tions for the description of the solar cellcharacteristics.

The two diodes model is widely used for anequivalent circuit. Sh aded cells of a solar mod uleor a solar generator can be driven into thenegative voltage region. If there are no bypassdiodes for cell protection, a diode breakdow nmay happen during high negative voltages. Thisbreakdow n is not taken into account in the twodiodes model. Therefore another model isdescribed in this paper.

This model includes an extension term, whichdescribes the diode breakdow n at high negativevoltages. T he equivalent of this model is givenin Fig. 1.

Several simulation programs for photovoltaicgenerators were developed during the last twodecades. D ue to the development of more po w-erful compu ters, they became more complex andmore efficient. These programs, however, stilloffer limited possibilities to simulate the electri-cal behavior of photovoltaic generators undermism atched conditions or partial shading. Forthis purpose, a more general electrical mo delwhich allows simu lations at all conditions hasbeen developed. N umerical method s, such asdescribed by Press e t a l . (1992) are used toobtain a solution for this model.

Kirchhoffs first law yields the following rela-tionship between cell current and cell voltage:

O=f(v,U= IPh - lS1 (w(s) - 1)

-Isz(exp(s) -1) -y

-z-o(V+z~s)(l-~)-k Y J

Extension term for thenegative diode breakdown

t ISES member.This equation differs from the usual two diodesmodel by the extension term marked in the

(1 )

513


2/8

514 V. Quaschning and R. Hanitsch

Fig. 1. Equivalent circuit of the solar cell, 1(1/,) generatesthe avalanche breakdown at high negative voltages.

equation above. This extension term can gener-ate cell currents above the photocurren t. Sucha current appe ars during the avalanche break-down at high negative voltages (Miller, 1957;Bishop, 1988).

The photocurrent, the band gap and the diodesaturation currents are temp erature dependent.These pa rameters have to be changed if thetempe rature varies. The photocurren t is alsoproportional to the irradiance.

The electrical behavior of the solar cell isdescribed over the whole voltage range. This isshown in Fig. 2 for a polycrystalline solar ce ll.

3. SIMULATION OF THE SOLAR CELLIn order to model the solar cell curve, the

current I, for a given voltage V, must be com-puted individually for each value. Because theequation of the solar cell curve is not given inexplicit form, numerical meth ods are normallyused to determine the character istic curve.

4

An equation for the solar cell f( V, I) = 0 isgiven in eqn (1). For a given voltage V thecurrent I is determined by the root of eqn (l),whe re I is expe cted to be the single root. Inorder to determine the single root, it is suggeste dto use Newton-Raphsons method, describede.g. by Mathews (1987), as it offers considerableadvantages over other methods. The main ideasof this algorithm will be described shortly.Starting with the initial value I,, the followingiteration is executed :z,,,=z,-tg

2az

The iteration stops if a suitable condition ismet. In the case above the absolute differenceof two succeeding values of the iteration is used:lZi+ 1 - Ii1 < E. The derivative of the solar cellequation can be given as shown below:

1

n(V + ZR,)-V+ZRs- V,, >

The Newton-Raphson algorithm can easilybe implemented in a computer program. The

-r---r--20 - 16 - 12 - 6 -4 0 4

cel l voltage in VFig. 2. Typical characteristic curves of a 10 x 10 cm polycrystalline solar cell at different irradiances (T=

300 K). The parameters are given in Fig. 3.


3/8

Numerical simulation of current-voltage characteristics 515

ton-Raphson algorithm has the advantage of avery quick, quadratic convergence for initialvalues near the root, so that a good solutioncan be calculated within a few iteration steps.In the proximity of the diode breakdownsdifficulties occur. In this region the New ton-Raph son algorithm converges very slowly andcan even diverge for badly chosen startingpoints.

In this section a metho d is described for thecalculation of large network s. In addition, thecharacte ristic curve of each componen t will bedetermined for each point of the generatorcurve. This is done to determine the pow erlosses o f a partly or fully shaded solar cell.

To speed up the convergence, in this case, thebisection meth od can be used to determine aninitial value for the Newton -Raphso n meth od.Then, a more accura te solution can quickly becalculated by the Newton-Raphson method.The bisection me thod converge s only linearly,but it cannot fail if the roo t is in the choseninterval. The interval is bisected and the part ofthe interval that does not contain the root iseliminated until the desired accuracy is reach ed.

Kirchhoffs laws are applied to provide anequation system relating all currents and volt-ages in the network . A continuously varyingsystem of nodes and m eshes is difficult todescribe for a com puter. This is why a generalmode l for an element interconnection isdescribed as shown in Fig. 4.

The elements can be placed vertical (Ai,j) orhorizontal (Bi,j) on a regular grid. Altogethe rthere are z elements.

The following compone nts can be used aselements:

Figure 3 show s a typical current-voltagecharac teristic of an unilluminated 10 x 10 cmpolycrystalline solar cell over a wide range. Thealgorithms proposed above coincide with themea sured points, proving to be an excellentmethod to describe the solar cell.

4. SOLAR GENERATOR MODEL

solar cell (two diodes equivalent circuitincluding negative diode breakdow n descrip-tion term);diode;resistance;ideal condu ctor (in this case Rcond = 10-20 0);insulation resistance (in this case Rinsul =1020 Q)

etc.In the previous section it was shown how to It is also po ssible to use a whole assembly

calculate the charac teristics of a solar cell. Now , instead of a single compon ent, for exam ple athis model can be used for various interconnec- string of multiple solar cells. In this case thetions between solar cells, diodes, cables, and current-voltage curve and the derivative of theother components of solar generators. assembly must be known.

61 I

-6-I . . . . ! . . . . ! . . . . ! . . . . ! .!. . . 1-20 -15 -10 -5 0 5

cell voltage in VFig. 3. Comparison between measurements and simulation of the characteristic curve of an unilluminated10 x 10 cm polycrystalline solar cell (manufacturer ASE, Germany) with a high series resistance R, dueto line resistance. The following parameter set was used for the simulation: (T= 300 K; I,, = 3 x IO-A;

m,=1;I,,=6x10~6A;m,=2;R,=0,13R;R,=30R;V,,=-18V;a=2,3x10~3R~;n=1,9).


4/8


Fig. 4. General model for a connection of various elements in a photovoltaic assembly. (A: vertical ele-ments, B: horizontal elements, M: meshes, N: nodes).

Each element is described by the mathemati- The vector u is the vector of the unkno wncal relationship between current and voltage. sub-voltages and the unkno wn total current:These eq uations can be given in the followingform: uT=(Y 41,iY., Y4A yJ, f31,1,*.., &iBy-lJ-l,Z)

S(vAi,j,ai.j)= 0 and f(Vei,j,1ei.j)= 0 The to tal voltage is given , for exam ple, by thefollowing equation:In this case the current for a given voltage hasto be determined by a num erical method. Formany elements (e.g. resistance) there is an ri(u) = E V,,,i - I/= 0i=l (4)explicit equation for the current:

ZAi,j=g(JL,j) and Z~j.j=g(l/si,j)In the circuit there are altogether m meshesMi,j, according to Kirchhoffs second law. Thesum of the voltages in each m esh equals zero:

5. SIMULATION OF THE GENERATORThe total voltage I/ is to be given for the

determ ination of all characteristic curves. Therelating total current I, all sub-voltages andsub-currents, can be calculated for this totalvoltage. T his means the total voltage V can beinterpreted as a know n quantity and is keptconstant for all calculations. For each sub-voltage the relating sub-current can be deter-mined by the relations indicated above, so thatthe calculation of the sub-voltages is sufficient.Now , there are z + 1 unkn own s. These are zsub-voltages of each element plus the totalcurrent I.

r2...m+lW= bi,j- YBi ,j - I/ai,j+l+ hi-l,j=O (5 )for i = 1. . .y; j = 1 . . . x - 1 with Vi,j = 0 if i = 0or i = y. For the total current Z results theequation:

rm+Z(u)= 5 Z,i,j-I=0j= l (6)Altogether there are n nodes iVj,j n the circuit.According to Kirchhoffs first law, the sum of

the currents equals zero.rm+3...m+n+Z(U)=z.4i,j=zBi,j-z,4i+I,j-zBi,j-l=o (7)


5/8


for i= 1 . . . y - 1; j = 1 . . .x with Zsi,j = 0 if j = 0or j = x.

For the z + 1 unkn own s, represented by thevector u, z+ l independent equationsri (u) . . . Y, +,, +2(u) are given. These equationsform the non-linear equation system r(u).

r(u) =

rm+n+2(u)Yzl Y4i.l v

v,l,l - f3l.l - L*VAy,x-1 T/Ay,x T/By-1,x-l

i lA1.j - zj=l

ZAy-1,x-ZAy,x_ZBy-1,x-l

(8 )

Wh en using the solution vector u for a giventotal voltage V, all results of the equations ofthis equation system sh ould be zero (ri(u) =r2(u) = . . .rm+n+ 2(u) = 0). The root of the equa-tion system, represented by the vector u, whichsatisfies the equation r(u) = 0, is to be found.

This equation system shall also be solved byapplying numerical methods. The Newton-Raph son iteration requirement of a single equa-tion with one unknown can be widened on theequation system. This results in:

ui + 1 = ui - ( J(u,))- Y(ui) (9 )Beginning with the starting vector II,, this

iteration should be performed until the stoppingcondition 11i + 1 - Ui(I


6/8


1u.z 0.655 0.6a,s 0.4z module curve with one

partially shaded cell0

0 5 10 15 20module voltage in V

Fig. 5. Measurements and simulations for a solar module SM50 with 36 monocrystalline silicon solarcells with no bypass diodes. At the second curve 75% of one solar cell was shaded. The measurementswere done under real sky conditions (E = 407 W/m, T= 298 K). The following parameter set was usedfor the simulation: (I,,, = 1,27A; I,,=2,4x 10-A; m,=l; Is,=3,6x 10e6A; m,=2; R,=14mR;

R,=225R,I/,,=-41,5V;a=0,22x10-3~-;n=3).

6. MODEL EVALUATIONFigure 5 shows a typical current-voltage

characteristic of a photovoltaic modu le with36 mono crystalline silicon solar cells withoutany bypass diodes at two different illuminationsituations. One curve presents the current-volt-age characteristic of a uniform illuminated solarmod ule under real sky conditions. The othercurve shows the current-voltage characteristicof the same mod ule at the same irradiance andtemperature, but 75% of one solar cell of themod ule are shaded. Th e simulated curve of thephotovoltaic modu le coincides just as well asthe simulated curve of the single solar cell inSection 3 with the measured points. Therefore,the described m ethod for the photovoltaic gen-erator is perfectly suited to obtain simu lationsfor the current-voltage characteristics.

The influence of mod ule shading on themod ule performance can also be seen in Fig. 5.The performance loss is 70% although only 2%of the module area is shaded. Part of the perfor-mance difference has to be dissipated by theshaded cell. To avoid cell dam aging the modulemanu facturers integrate byp ass diodes over cellstrings of 16 up to 24 cells. Figure 6 show s thatthe modu le p erformance loss can only bereduced insignificantly by bypass diodes overlarge cell strings. In the case of the 75% shadedcell the performance loss decreases from 70 to55% by the use of bypass diodes over 18 cells.

Only separate bypass diode over every cellcan minimize the performance losses. Therefore,Green e t a l . (1981, 1984) and Shepard andSugim ura (198 4) proposed to integrate bypassdiodes into so lar cells. Figure 7 show s a simula-tion of the same modu le used in Fig. 6 withbypass diodes over different n umb ers of solarcells. Again 75% of one solar cell are shaded.The performance losses can be reduced to 5%if a byp ass diode is used over every cell.

7. CONCLUSIONSA model for the description of the total

characteristic curve for a solar cell, based o nthe two diodes model and the model of Bishop(1988) is described. N umerical methods arenecessary to determine the current-voltagecurve. Some well-known numerical algorithmsand their implementation are described.Furthermore, a general mo del for the determina-tion of a solar generator curve as well as allsub-voltages and sub-currents of the solar gener-ator are given. The suitable algorithms for thesolution of the equation system are introducedfor this model.

By analysing the sub-curves, statementsregarding possible power losses at mism atchedor shaded solar cells can be made. This isnecessary to minimize the power losses resultingfrom shading, and to avoid the damaging of


7/8


% call is shaded

5 10 15 20module voltage in V

Fig. 6. Measurements and simulations for a solar module SM50 with 36 monocrystalline silicon solarcells with two bypass diodes. One cell was shaded differently. The measurements were done under realsky conditions (5 = 574 W/m, T= 300 K). The following parameter set was used for the simulation: (I,,, =1,79A; Is,=3,3~lO-~A; m,=l; !,,=7,8~lO-~A; m,=2; Rs=l4mR; R,=lSOR; Vsr=-30V;

a = 8 X lo-4R-1; n = 1,9).

Figcell

bypass diode over 9 cebypass diode over 6 cebypass diode over 4

0 5 10 15 20module voltage in V

7. Simulations for the same module as used in Fig. 6. The bypass diodes were connected over differentstring sizes. 75% of one cell was shaded. As parameters E = 1000 W/m, T= 300 K, I,, = 3, 11 A were

used. The other parameters were the same as used for Fig. 6.

solar cells by hot spots during high power losses.The performance losses can drastically bereduced by using a bypass diode over every cell.This should b e an important aspect to pick upthe shading problem again and to think aboutnew shadow -tolerant mod ules for terrestrialapplications.

NOMENCLATUREE exactness for stopping iteration1 factor in the modified Newton iterationa correction factor (a < 1 Q-)

A,.j vertical element (i, j running position index)

B,.j horizontal element (i , j running position index)f functioni indexI current of the solar cell or the solar generator

I,,.j current at element A,.jfsi,j current at element B,,jI,, photocurrentIs, saturation current of the first diodeI,, saturation current of the second diode

I(V,)cur generated for the diode breakdown at highnegative voltages

j indexJ(u) Jacobian matrixm number of meshes; m = y(x - 1)m, deahty factor of the first diodem2 deality factor of the second diode

M,,j Kirchhoff mesh (i. j running position index)


8/8

V. Quaschning and R. Hanitsch20

nIVn1.)r(u)

ri(UlR?RS7u:UIV

vA*.jv6i . ,V6r

v,v,xYz

avalanche breakdown exponent (n = I . IO)numbcrofnodcs;n=x(y-I)Kirchhog node (i. j running position index)system of nonlinear equationspartial function of r(u)parallel or shunt resistanceseries resistancetemperaturevector of unknownsstarting vector (or the iterationelement of the vector Yvoltage of the solar all or the solar generatorvoltage at element A,,voltage at element B,.,breakdown voltage(V,,r -ISV... -40V)voltage over the parallel mistana R,temperature voltagenumber of horizontal elements of a rownumber 0r vertical elements 0r a columnnumber of all elements (vertical and horizontal);z=Zxy-n-y+I=m+n+I

BIBLIOGRAPHYAbete. A., Barbisio, E, Cane, F. and Dcmartini. P. (1990)

Analysis of photovoltaic modules with protection diodesin p&cnaeof mismatching. 2/a IEEE PV SpecialistsCon/. DD. 1005-1010. Kissimimce. U.S.A.

Engeln&titilgcs, G. (1990) Formelsammlung zur Numer-ischen Mathematik mit C-Programmen. BI-Wissen-schaltsverlag. Mannheim.

Quaschning V. and Hanitsch. R. (1995) Shade calculationsin photovoltaic systems. ISES Solur Work/ Congress.Hararc, Zimbabwe.

Quaschning, V. and Hanitsch. R. (1995) Quick determina-tion of irradiana reduction caused by shading atPV-locations. /Jr/t European Phororolroic So/or EnergyCon&. Nia France.Rauschenbach. H. S. (1971) Electrical output of shadowed

solar arrays. IEEE Trms. on Electrnn De&es. 1%483-490.

Stocr. J. (1993) Numerische Mar/tern&k 1. Springer, Berlin.Survanto Hasvim. E.. Wenham. S. R. and Green. M. A.

(1986) Shadow tolerana of modulesncorporating intcg-ral bypass diode solar alls Solar Cells 19, 109-122.

Swaleh, M. S. and Green. M. A. (1982) Effect of shuntresistance and bypass diodes on the shadow tolerance olsolar all modules. Solar Cells 5, 183 198.Wagdy. R. A. ( 1989) Partial shadowing protection withoutdiodes. Solar Cells 26, 227-239.

REFERENCESBishop. J. W. (1988) Computer simulation of the etTectsof

electrical mismatches in photovoltaic all interconncc-tion circuits. So/or Cells 2S. 73-(19.

Green. M. A., Gauja. F. and Yachamanankul, W. W. (I981 )Silicon solar fells with intcgratcd bypass diodes. SolarCells 3, 233-244.

Green, M. A.. Hasyim, H. S.. Wenham. S. R. and Willison.M. R. (1984) Silicon solar cells with integrated bypassdiodes. 17th IEEE Phorow~lruic Specialisrs Con/,pp. 513-517. Kissimimcc. U.S.A.

Mathews. J. H. (1987) Numerical Merhods jar CornpurerScience. Engineering and Marhomorics. Prentice Hall, NJ.

Miller, S. L. ( 1957) Ionization rates for holes and electronsin silicon. Phys. Rev. ltJS(4). l246- 1249.

Overstraeten, R. J. and Menens. R. P. (1986) Physics. Terh-nology und Use o/ Phorouofraics.Adam Hilger. Bristol.

Press, W. H.. Teukolsky. S. A.. Vctterling. W. T. and Flan-nery. B. P. (1992) Numerical Recipes in C. CambridgeUniv. Press.

Rauschenbach. H. S. ( 1980) So/or Ce// Array Design Hund-/rook. Van Nostrand Reinhold, New York.

Shepard, N. F. and Sugimura. R. S. (1984) The integrationof bypass diodes with terrestrial photovoltaic modulesand arrays. 17rh /LEE Phnrocnlraic Specialisrs Con/I,pp. 676-681, Kissimimcc. U.S.A.

numercial simulation of j-v for shaded cells

Documents