Technical note An approximation to solar cell equation for determination of solar cell parameters

\ PERGAMON

Renewable Energy 06 "0888# 124Ð130

Technical note

An approximation to solar cell equation for determination of solar cell parameters Engin Kiran\ Demir þnan I Hacettepe University\ Physics En`ineerin` Department\ Ankara\ Turkey Received 00 March 0887^ accepted 09 April 0887

Abstract An approximation was applied to a solar cell equation to eliminate reverse saturation current[ IÐU curves obtained from the approximated equation were _tted to experimentally measured data by using the iteration technique[ All instrumentation errors were taken into account in the _tting procedure[ The Filling Factor "FF# and the series resistance "Rs# appeared to be important for _tting accuracy[ The _tting procedure was repeated for di}erent illuminations and solar cell temperatures[ Þ 0888 Published by Elsevier Science Ltd[ All rights reserved[ Keywords] Solar cell

Nomenclature U I Rs Isc I9 FF k T Upmax Ipmax Uoc

solar cell output voltage solar cell output current series resistance short circuit current reverse saturation current _lling factor of solar cell Boltzmann number "0[27×09−12# temperature\ K voltage at maximum power point current at maximum power point open circuit voltage

 Corresponding author 9859Ð0370:88:, ! see front matter Þ 0888 Published by Elsevier Science Ltd[ All rights reserved PII] S 9 8 5 9 Ð 0 3 7 0 " 8 7 # 9 9 0 0 7 Ð 8

125

DUJ =s=

E[ Kiran\ D[ þ Inan : Renewable Ener`y 06 "0888# 124Ð130

voltage error for corresponding data point j absolute error

0[ Introduction The solar cell equation parameters are important criteria in solar cell applications for the following reasons] "0# IÐU characteristics are used in designs and calculations of photovoltaic appli! cations[ "1# These parameters are important in scienti_c research\ development and production[ Solar cell parameters such as short circuit current "Isc#\ open circuit voltage "Uoc#\ current and voltage at maximum power point "Ipmax\ Upmax# are usually found from the IÐU curve[ This curve is mathematically expressed by an equation which is known as the solar cell equation ð0Ł[ This equation contains four parameters namely\ series resistance of the solar cell "Rs#\ short circuit current "Isc#\ reverse saturation current "I9# and a term "L# which contains the _lling factor of the solar cell "FF#[ In general\ experimentally determination of parameters I9 and Rs is di.cult[ Other parameters are usually found from the IÐU curve of the solar cell[ To avoid di.culties in the determination of Rs and I9\ an approximation method can be applied and an approximated equation can be found which is independent of I9 and contains only Rs[ The approximated equation can then be _tted to the empirical data[ In the _tting procedure\ instrumentational errors must be taken into account[ The relative root mean square error of the empirical data and the approximated equation errors must be calculated and compared[ Fitting procedure is said to be good\ if the solar cell approximated equation errors are within the experimental error range proposed by Braunstein et al[ ð1Ł[ In this work\ experiments were made by using a solar simulator ð2Ł which was built in Hacettepe University\ Physics Engineering Department[ Data logging was achieved by a computer data acquisition system[ Solar cell parameters were determined and _tting procedure was applied under di}erent illuminations and working temperatures[

1[ The solar cell equation An equivalent solar cell circuit is shown in Fig[ 0a\ and IÐU characteristics of a cell under di}erent radiation levels in Fig[ 0b[ The relation between the load voltage U and the load current I of the cell\ U"I#\ is usually given by the solar cell equation]

E[ Kiran\ D[ þ Inan : Renewable Ener`y 06 "0888# 124Ð130

126

Fig[ 0[ "a# Equivalent circuit of a solar cell^ "b# solar cell IÐU characteristics[

U  −IRs¦

6

7

Isc−I 0 ¦0 ln L I9

"0#

where\ L

q IpmaxUpmax \FF  FF = kT I=U

In eqn "0#\ Rs is the series resistance of cell\ I9 is the reverse saturation current and Isc is the short circuit current[ FF is the _lling factor of the cell[ The voltage U in eqn "0# contains four parameters] Rs\ Isc\ I9 and L[ Equation "0# is acceptable for a high shunt resistant solar cells but di.cult to use due to its four parameters[ IÐU characteristics of a solar cell are suitable to determine Isc and L experimentally\ but no use for determining I9[ Rs can be determined by a method of Swanson proposed by Beghi ð3Ł and Negro ð4Ł\ however\ in this method at least two IÐU characteristics in di}erent illuminations are necessary[ As Rs is dependent on illumination\ only an approximate Rs value can be found from this method[ An approximate equation can be derived from eqn "0# which is independent of the reverse saturation current[ At open circuit condition\ I  9 and U  Uoc and from eqn "0# we get I9 

Isc exp"LUoc#−0

For regular cells\ exp"LUoc# × × 0\ therefore I9 becomes\ I9 

Isc # exp"LUoc

Substituting eqn "1# into eqn "0# yields\

"1#

127

E[ Kiran\ D[ þ Inan : Renewable Ener`y 06 "0888# 124Ð130

U  −IRs¦

6$ %

0 ln L

7

Isc−I exp"LUoc#¦0 Isc

"2#

Substituting exp"LUoc#×exp"−LUoc#  0 into eqn "2# yields\ U  Uoc−IRs¦

6

7

Isc−I 0 ¦exp"−LUoc# ln L Isc

"3#

Equation "3# does not contain I9[ For short circuit condition\ "I  Isc# in eqn "3#\ we get U ³ 9 and in order to impose U  9\ a coe.cient "B# will be added to eqn "3#\ U  Uoc−IRs¦

6

Isc−I 0 ¦B exp"−LUoc# ln L Isc

7

"4#

Setting I  Isc and U  9\ 9  Uoc−IscRs¦

0 ln"B exp"−LUoc##\ B  exp"LIscRs# L

Substituting B into eqn "4#\ one gets an approximated equation which is independent of the reverse saturation current "I9#] U  Uoc−IRs¦

6

$

Isc−I 0 ln ¦exp L"IscRs−Uoc# L Isc

%7

"5#

Equation "5# contains three parameters] Uoc\ Isc\ L[ From a solar cell "UÐI# curve\ all parameters contained in eqn "5# can be found from experimental data[ Rs can be found from approximated equation by using the iteration method[

2[ Fitting procedure In order to check the consistency of the curve obtained from eqn "5# with measured values\ a set of experiments was carried out for a solar cell ð5Ł at di}erent temperatures and illuminations[ For each temperature and for each illumination\ measured values were plotted "Fig[ 1#[ Afterwards\ by substituting experimental values of Isc\ Uoc and FF into the approximated equation\ a suitable curve for each data set was _tted[ In the _tting procedure\ the series resistance of the solar cell "Rs# is an important parameter for _tting accuracy and the _tting procedure was accomplished by the iteration of Rs values "Fig[ 1#[ Another important point in the _tting procedure is the error limits[ The root mean square error of the empirical data was calculated from the following equation] 1 0:1

$ 0 1% 0 1 s DUexp j j

sexp 

s Uexp j j

E[ Kiran\ D[ þ Inan : Renewable Ener`y 06 "0888# 124Ð130

128

Fig[ 1[ Experimental data points of IÐU "tick marks# and approximated equation _tting curves "straight lines# of solar cell at di}erent working conditions[ "a# IÐU for constant temperature "29>C# at di}erent illuminations[ "b# IÐU for constant illumination "099 mW cm−1# at di}erent temperatures[ Experimental values were obtained by using a solar simulator built at Hacettepe University[

139

E[ Kiran\ D[ þ Inan : Renewable Ener`y 06 "0888# 124Ð130

app Fig[ 2[ Instrumentational errors "DUexp j # and approximation equation errors "DUj # of one of the solar cells for corresponding data points j at 099 mW cm−1 and 29>C[

The superscript exp stands for experimental and the subscript j stands for the data points which may be j  9\ 0\ 1\ [ [ [ [ N[ DUexp is the voltage error for the corresponding j current Ij that was calculated from instrumentational errors[ Approximated equation errors were calculated using the same procedure] 1 0:1

$ 0 1% 0 1 s DUapp j j

sapp 

s Uapp j j

Superscript app stands for approximation[ There are three parameters in the approxi! mated equation along with their errors[ In Fig[ 2\ the criteria of the _tting curve in terms of the approximated and experimental errors are shown[ The _tting curve\ evaluated by the approximated equation\ is said to be good if the approximated exp equation error "DUapp j # is inside the experimental error "DUj # for the corresponding current Ij\ "sapp ³ sexp#\ "Fig[ 2#[ Experiments were run with a computer data acquisition system that has a sensitivity of 1[3 mV at 209 Vdc and each IÐU set of data was logged in a time of about 299 ms[ All computational calculations were made with 375DX1!55 MHz computer[

130

E[ Kiran\ D[ þ Inan : Renewable Ener`y 06 "0888# 124Ð130

Table 0 Solar cell parameters from manufacturer and achieved by approximation equation with their error ranges

099 mW m−1\ 29>C

Isc "mA#

Uoc "V#

Ipm "mA#

Upm "V#

Rs "V#

FF

h ")#

Manufacturer|s data Approximation eqn Error=s=")#

16[5 17[1 1[0

9[46 9[43 4[1

14[3 15[1 2[0

9[34 9[33 1[1

* 0[4 *

9[62 9[63 0[2

00[32 00[41 9[7

3[ Conclusions The approximated equation parameters and the manufacturer|s data of a solar cell ð5Ł are given in Table 0[ The calculated open circuit voltage\ using the approximated equation\ is identical to the manufacturer|s data with an accuracy of 83[6) "9[43:9[46  83[7)#[ Rs and the _lling factor "FF# are critical parameters in the designs for solar cell applications and the approximated equation gives us good results with a high accuracy for these parameters[ It must be noted that all experiments and the approximated equation are acceptable with very high shunt resistance about 095 V owing to the assumption in eqn "1#[ If Isc\ Uoc and FF are known from the manufacturer|s catalogue\ then by substituting these values into eqn "5# the IÐU curve of a solar cell can be obtained[ Also\ the Filling Factor "FF# and the Rs values can be obtained from the approximated equation[

References ð0Ł Pankove IJ[ Optical processes in semiconductors[ Englewood Cli}s\ New Jersey] PrenticeÐHall\ Inc[\ 0858[ p[ 291Ð26[ ð1Ł Braunstein A\ Bany J\ Appelbaum J[ Determination of solar cell equation parameters from empirical data[ Energy Conversion[ Vol 06[ Pergamon Press\ 0866[ p[ 0Ð5[ ð2Ł Kiran E\ þnan I D[ Temperatured controlled solar simulator for measuring IÐV characteristics of solar cells[ Turkish Journal of Engineering and Environmental Sciences 0866^ 10[ ð3Ł Beghi G[ Performance of Solar Energy Converters] Thermal Collectors and Photovoltaic Cells[ ECSC\ EEC\ EAEC\ Brussels and Luxembourg\ 0872[ p[ 036Ð051[ ð4Ł Negro E[ PC simulation and sizing of PV systems[ Workshop on Material Science and Physics of Non! Conventional Energy Sources[ SMR[ 693Ð04^ 0882[ Ý bersicht:Technische Daten\ Cell type ð5Ł Telefunken System Technik\ Deutsche Aerospace[ Solarzellen U TZZM1449[ ð6Ł ERDA:NASA Terrestrial Photovoltaic Measurement Procedures[ Report no[ 0911!66:05\ NASA TM 62691\ June 0866[