research article an improved mathematical model for

Research ArticleAn Improved Mathematical Model for ComputingPower Output of Solar Photovoltaic Modules

Abdul Qayoom Jakhrani1 Saleem Raza Samo1 Shakeel Ahmed Kamboh2

Jane Labadin3 and Andrew Ragai Henry Rigit4

1 Energy and Environment Engineering Department Quaid-e-Awam University of EngineeringScience and Technology (QUEST) Nawabshah Sindh 67480 Pakistan

2Department of Mathematics and Computational Science Faculty of Computer Science and Information TechnologyUniversiti Malaysia Sarawak Kota Samarahan 94300 Sarawak Malaysia

3 Faculty of Computer Science and Information Technology Universiti Malaysia Sarawak Kota Samarahan94300 Sarawak Malaysia

4 Faculty of Engineering Universiti Malaysia Sarawak Kota Samarahan 94300 Sarawak Malaysia

Correspondence should be addressed to Abdul Qayoom Jakhrani aqunimashotmailcom

Received 31 May 2013 Revised 26 December 2013 Accepted 9 January 2014 Published 17 March 2014

Academic Editor Ismail H Altas

Copyright copy 2014 Abdul Qayoom Jakhrani et alThis is an open access article distributed under theCreative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

It is difficult to determine the input parameters values for equivalent circuit models of photovoltaic modules through analyticalmethods Thus the previous researchers preferred to use numerical methods Since the numerical methods are time consumingand need long term time series data which is not available in most developing countries an improved mathematical model wasformulated by combination of analytical and numerical methods to overcome the limitations of existing methods The valuesof required model input parameters were computed analytically The expression for output current of photovoltaic module wasdetermined explicitly by Lambert W function and voltage was determined numerically by Newton-Raphson method Moreoverthe algebraic equations were derived for the shape factor which involves the ideality factor and the series resistance of a single diodephotovoltaic module power output model The formulated model results were validated with rated power output of a photovoltaicmodule provided by manufacturers using local meteorological data which gave plusmn2 error It was found that the proposed modelis more practical in terms of precise estimations of photovoltaic module power output for any required location and number ofvariables used

1 Introduction

The photovoltaic (PV) modules are generally rated understandard test conditions (STC) with the solar radiation of1000Wm2 cell temperature of 25∘C and solar spectrumof 15 by the manufacturers The parameters required forthe input of the PV modules are relying on the meteoro-logical conditions of the area The climatic conditions areunpredictable due to the random nature of their occurrenceThese uncertainties lead to either over- or underestimationof energy yield from PV modules An overestimation up to40 was reported as compared to the rated power outputof PV modules [1 2] The growing demand of photovoltaicstechnologies led to research in the various aspects of itscomponents from cell technology to the modeling size

optimization and system performance [3ndash5] Modeling ofPV modules is one of the major components responsiblefor proper functioning of PV systems Modeling providesthe ways to understand the current voltage and powerrelationships of PV modules [6ndash8] However the estimationofmodels is affected by various intrinsic and extrinsic factorswhich ultimately influence the behavior of current andvoltage Therefore perfect modeling is essential to estimatethe performance of PV modules in different environmentalconditions Hernanz et al [9] compared the performance ofsolar cells with different models and pointed out that themanufacturers did not provide the values of the resistance inseries and parallel of the manufactured cell Andrews et al[10] proposed an improved methodology for fine resolutionmodeling of PV systems using module short circuit current

Hindawi Publishing CorporationInternational Journal of PhotoenergyVolume 2014 Article ID 346704 9 pageshttpdxdoiorg1011552014346704

2 International Journal of Photoenergy

(119868sc) at 5min time scales Their work was a modified versionof the Sandia array performance model by incorporatingnew factors for the calculation of short circuit current (119868sc)to justify errors (including instrumentation alignment andspectral andmodule power tolerance errors) Chakrasali et al[11] investigated the performance of Nortonrsquos circuit model ofsolar PV module with the existing models using Matlab andreported that it is a well-suited way to predict the behavior ofPV modules operated for longer periods of time Chouder etal [12]modeled a PVmodule by a single diode lumped circuitand evaluated its main parameters by considering the powerconversion efficiency Chouder et al [13] presented a detailedcharacterization of the performance and dynamic behaviorof PV systems by using the LabVIEW platformThe LambertW function was applied for the solution of equations by Jainand Kapoor [14] Jain et al [15] Ortiz-Conde et al [16] andothers [17ndash19] Picault et al [17] presented a novel methodto forecast existing PV array production in diverse environ-mental conditions and concluded that Lambert W functionfacilitates a direct relationship between current and voltage ofmodules as it significantly reduces calculation time Chen etal [18] proposed an optimized method based on polynomialcurve fitting and Lambert W function for extraction ofparameters from the current-voltage (I-V) characteristics ofcommercial silicon solar cells The Lambert W function wasused for translation of transcendental equation into explicitanalytical solution Fathabadi [19] presented a novel methodfor characterization of silicon solar cells modules and plasticsolar cells Artificial neural network together with LambertW function was employed for determination of I-V and P-Vcurves of silicon and plastic solar cells and modules [20]

Moreover Krismadinata et al [21] used a single diodeelectrical equivalent circuit model for determination of PVcell characteristics and found that output of PV moduleswere strongly affected by the intensity of solar irradiationand ambient temperature Lu et al [22] investigated variousPV module layouts using full size as well as halved solarcells The performance of module layouts was investigated bypartially shading the PV cells using a solar cell equivalentcircuit model with SPICE software They found that theseries-parallel hybrid connection of cells within a modulehas a significant improvement on the power output of thePV module under partial shading conditions Mellit et al[23] employed a methodology for estimation power profileof a 50Wp Si-polycrystalline PV module by developing twoartificial neural networks (ANNs) for cloudy and sunny daysand found that the ANN-models performed better than theexisting models and also did not need more parametersunlike implicit models Singh [24] reviewed various modelsof PV cells and concluded that the accuracy of modelscan be improved by including series and shunt resistanceinto the model In addition the author also discoveredthat the estimation of models can further be improved byeither introducing two parallel diodes with independent setsaturation current or considering the diode quality factoras a variable parameter instead of fixed value like 1 or 2Thevenard and Pelland [25] reported that the uncertainties ofmodel predictions can be reduced by increasing the reliabilityand spatial coverage of solar radiation estimates appropriate

familiarity of losses due to dirt soiling and snow anddevelopment of better tools for PV system modeling Tian etal [26] presented a modified I-V relationship for the singlediode model The alteration in the model was made in theparallel and series connections of an array The derivation ofthe adapted I-V relationship was begunwith a single solar celland extended up to a PV module and finally an array Themodified correlation was investigated with a five-parametermodel based on the data provided by the manufacturers Theperformance of themodel was examined with a wide range ofirradiation levels and cell temperatures for prediction of I-Vand P-V curves maximum power point values short circuitcurrent and open circuit voltage Vincenzo and Infield [27]developed a detailed PV array model to deal explicitly withnonuniform irradiance and other nonuniformities across thearray and it was validated against data from an outdoortest system However the authors reduced the complexity ofthe simulations by assuming that the cell temperatures arehomogeneous for eachmodule Yordanov et al [28] presenteda new algorithm for determination of the series resistanceof crystalline-Si PV modules from individual illuminatedI-V curves The ideality factor and the reverse saturationcurrent were extracted in the typical way They found thatthe ideality factor at open circuit is increased by about 5It was established from the review that Lambert W functionis a simple technique to give the analytical explicit solutionof solar photovoltaic module characteristics as compared tothe other methods However some problems still exist for thederivation of the required model equations

Equivalent electrical circuit model is one of the keymodels under study since the last few decades It is configuredwith either single or double diode for investigation of current-voltage relationships The single diode models usually havefive four or three unknown parameters with only oneexponential term The five unknown parameters of a singlediode model are light-generated current (119868

119871) diode reverse

saturation current (119868119900) series resistance (119877

119904) shunt resistance

(119877sh) and diode ideality factor (119860) [29 30] The four-parameter model infers the shunt resistance as infinite andit is ignored [31] The three-parameter model assumes thatthe series resistance is zero and shunt resistance is infiniteand thus both of these parameters are ignored whereas thedouble diode models have six unknown parameters with twoexponential terms [32 33]

In fact both single and double diode models require theknowledge of all unknown parameters which is usually notprovided bymanufacturers Nevertheless the current-voltageequation is a transcendental expression It has no explicitanalytical solution It is also time consuming to discover itsexact analytical solution due to the limitation of availabledata for the extraction of required parameters [34ndash36] Forthat reason the researchers gradually focused on searchingout the approximate methods for the calculation of unknownparameters The analytical methods give exact solutions bymeans of algebraic equations However due to implicit natureand nonlinearity of PV cell or module characteristics itis hard to find out the analytical solution of all unknownparameters Analytical methods have also some limitationsand could not give exact solutions when the functions

International Journal of Photoenergy 3

are not given Thus numerical methods such as Newton-Raphson method or Levenberg-Marquardt algorithm werepreferred It is because of the fact that numerical methodsgive approximate solution of the nonlinear problems withoutsearching for exact solutions However numerical methodsare time consuming and need long term time series datawhich is not available in developing countries

It was revealed from the review that a wide variety ofmodels exist for estimation of power output of PV modulesHowever these were either complicated or gave approximatesolutions To overcome the limitations of both numericaland analytical methods an improved mathematical modelusing combination of numerical and analytical methods ispresented It makes the model simple as well as compre-hensive to provide acceptable estimations for PV modulepower outputs The values of required unknown parametersof I-V curve namely light-generated current diode reversesaturation current shape parameter and the series resistanceare computed analytically The expression for output currentof PV module is determined explicitly by Lambert W func-tion and voltage output is computed numerically by Newton-Raphson method

2 Formulated Model for Computing PowerOutput of PV Modules

The power produced by a PV module depends on intrinsicelectrical characteristics (current and voltage) and extrinsicatmospheric conditions The researchers generally incorpo-rate the most important electrical characteristics and influ-ential meteorological parameters in the models for the sakeof simplicity It is almost unfeasible to obtain a model thataccounts each parameter which influences the performanceof PV modules The models generally include those parame-ters which are commonly provided by manufacturers suchas the electrical properties of modules at standard ratingconditions [37] The standard equivalent electrical circuitmodel of PV cell denoted by a single diode is expressed as[38]

119868 = 119868119871minus 119868119863minus 119868sh (1)

where 119868119871is light-generated current 119868

119863is diode current and

119868sh is shunt currentThe diode current (119868119863) is expressed by the

Shockley equation as [39 40]

119868119863= 119868119900[119890120585(119881+119868119877

119904)minus 1] (2)

The shunt current (119868sh) is defined by Petreus et al [41] as

119868sh =119881 + 119868119877

119904

119877sh (3)

Therefore the final structure of five-parameter one diodeelectrical equivalent circuit model is graphically shown inFigure 1 It is also algebraically expressed as [40 42ndash44]

119868 = 119868119871minus 119868119900[119890120585(119881+ 119868119877

119904)minus 1] minus

119881 + 119868119877119904

119877sh (4)

V

I

IoIL

Rs

Rsh

Figure 1 Equivalent electrical circuit model of a PV module

where 119868119900is the reverse saturation current and 120585 is a term

incorporated for the simplicity of (4) which is expressed as

120585 =119902

120582119896119879119888

(5)

where 119902 is electronic charge 119896 is Boltzmannrsquos constant 119879119888is

the cell temperature and 120582 is the shape factor which is givenas

120582 = 119860 times 119873CS (6)

where 119860 is the ideality factor and 119873CS is the number ofseries cells in a module The ideality factor (119860) does notdepend on the temperature as per definition of shape factor(120582) from semiconductor theory [45] The five unknownparameters namely 119868

119871 119868119900 119877119904 119877sh and 119860 can only be found

through complicated numerical methods from a nonlinearsolar cell equation It requires a close approximation of initialparameter values to attain convergence Otherwise the resultmay deviate from the real values [32 43] It is impracticalto find a method which can properly extract all requiredparameters to date [40 46] Thus for simplicity the shuntresistance (119877sh) is assumed to be infinity Hence the last termin (4) is ignored [47] Therefore the simplified form of four-parameter single diode equivalent circuit model which isused for this study is defined as [48]

119868 = 119868119871minus 119868119900[119890120585(119881+119868119877

119904)minus 1] (7)

From (7) a continuous relationship of current as functionof voltage for a given solar irradiance cell temperature andother cell parameters can be obtained

21 Determination of Unknown Parameters of FormulatedModel The expressions for unknown parameters such aslight-generated current (119868

119871) and reverse saturation current

(119868119900) were adopted from previous available models The

expressions for the shape factor (120582)which involve the idealityfactor (119860) and the series resistance (119877

119904) were algebraically

derived from existing equations The PV module used forthis study was NT-175 (E1) manufactured by Sharp EnergySolution Europe a division of Sharp Electronics (Europe)GmbH Sonninstraszlige 3 20097 Hamburg Germany

211 Light-Generated Current (119868119871) Light-generated current

(119868119871) is a function of solar radiation and module temperature


if the series resistance (119877119904) and the shape factor (120582) are

taken as constants For any operating condition 119868119871is related

to the light-generated current measured at some referenceconditions as [14 29]

119868119871= (

119878119879

119878119879119903

) [119868119871119903

+ 120583119868sc(119879119888minus 119879119888119903)] (8)

where 119878119879and 119878

119879119903are the absorbed solar radiation and 119879

119888

and 119879119888119903

are the cell temperatures at outdoor conditionsand reference conditions respectively 119868

119871119903is light-generated

current at reference conditions and 120583119868sc

is coefficient oftemperature at short circuit currentThe cell temperature (119879

119888)

can be computed from the ambient temperature and otherdata tested at nominal operating cell temperature (NOCT)conditions provided by the manufacturers

212 Reverse Saturation Current (119868119900) The reverse saturation

current (119868119900) is a function of temperature only [49] It is given

as

119868119900= 119863119879

3

119888119890120585120576119892119860 (9)

The reverse saturation current (119868119900) is a diminutive number

but its value increases by a factor of two with a temperatureincrease of 10∘C [50] It is actually computed by taking theratio of (9) at two different temperatures thereby eliminatingthe diode diffusion factor (119863) It is related to the temperatureonly Thus it is estimated at some reference conditions as thesame technique used for the determination of light current[49] Consider

119868119900= (

119879119888

119879119888119903

)

3

119890120585120582120576119892(119879119888119903minus119879119888)119860

(10)

213 Shape Factor (120582) Mostly manufacturers provide infor-mation of I-V characteristic curve at three different pointsusing reference conditions at open circuit voltage (119881oc) atshort circuit current (119868sc) and at optimum power point forboth current and voltageThe correlation for the given pointsare 119868 = 0 and 119881 = 119881oc at open circuit conditions 119868 = 119868scand 119881 = 0 at short circuit conditions and 119868 = 119868mp and119881 = 119881mp at maximum power point [48] By substituting theseexpressions in (7) it yields

119868sc119903 = 119868119871119903

minus 119868119900119903

[119890(120585119903119868sc119903119877119904)

minus 1] (11)

where

120585119903=

119902

120582119896119879119888119903

(12)

119868119871119903

minus 119868119900119903

[119890(120585119903119881oc119903)

minus 1] = 0 (13)

119868mp119903 = 119868119871119903

minus 119868119900119903

[119890120585119903(119881mp119903+119868mp119903119877119904)

minus 1] (14)

The reverse saturation current (119868119900) is a very small quantity

on the order of 10minus5 to 10minus6 A [47] It lessens the influence

of the exponential term in (11) Hence it is assumed to be

equivalent to 119868sc [32] One more generalization can be maderegarding the first term in (13) and (14) which could beignored Regardless of the system size the exponential term ismuch greater than the first termThus the equations become

119868119871119903

cong 119868sc119903 (15)

119868sc119903 minus 119868119900119903

[119890(120585119903119868sc119903119877119904)

] cong 0 (16)

119868mp119903 cong 119868119871119903

minus 119868119900119903

[119890120585119903(119881mp119903+119868mp119903119877119904)

] (17)

Solving (16) for reverse saturation current at reference condi-tions (119868

119900119903) is obtained as

119868119900119903

= 119868sc119903 [119890minus(120585119903119881oc119903)

] (18)

By substituting the value of reverse saturation current atreference conditions (119868

119900119903) from (18) into (17) it yields

119868mp119903 cong 119868sc119903 minus 119868sc119903 [119890120585119903(119881mp119903minus119881oc119903+119868mp119903119877119904)

] (19)

The Equation (17) can also be solved for 120585119903 which is given as

120585119903=

ln (1 minus 119868mp119903119868sc119903)

119881mp119903 minus 119881oc119903 + 119868mp119903119877119904 (20)

Finally the value of the shape factor (120582) can be obtained bycomparing (12) and (20) as

120582 =

119902 (119881mp119903 minus 119881oc119903 + 119868mp119903119877119904)

119896119879119888119903ln (1 minus 119868mp119903119868sc119903)

(21)

214 Series Resistance (119877119904) The series resistance (119877

119904) is an

essential parameter when the module is not operating nearthe reference conditionsThis characterizes the internal lossesdue to current flow inside the each cell and in linkagesbetween cells It alters the shape of I-V curve near optimumpower point and open circuit voltage however its effect issmall [29 36] I-V curve without considering 119877

119904would be

somewhat dissimilar than the curves outlined including itsvalue On the basis of annual simulation the predicted poweroutput from PV systems will be 5 to 8 lower when correctseries resistance is not used [32 51] It can be determined as

119889119881

119889119868

10038161003816100381610038161003816100381610038161003816119881oc119903

minus 119877119904= 0 (22)

To obtain differential coefficient for (22) first the current (119868)can be extracted explicitly as a function of voltage (119881) byusing Lambert W function from (7) and is expressed as

119868 = (119868119871+ 119868119900) minus

119882(120585119877119904119890120585(119881+119868

119871119877119904+119868119900119877119904))

120585119877119904

(23)

By differentiating (23) with respect to 119881 and taking itsreciprocal at 119881 = 119881oc119903 and 119868

119871= 119868119871119903

and substituting into(22) it gives

119882(120585119877119904119868119900119890120585(119881oc119903+119868119871119903119877119904+119868119900119877119904)

)

[1 +119882(120585119877119904119868119900119890120585(119881oc119903+119868119871119903119877119904+119868119900119877119904))] 119877

119904

minus 119877119904= 0 (24)


A number of simplifications have beenmade in order to solve(24) analytically for 119877

119904 For example 119868

119900is usually taken in the

order of 10minus5 to 10minus6 [47] Its value for this study was taken

as the order of 10minus6 Similarly the values of 119868119871119903

and119881oc119903 weretaken from the manufacturers data The expression for 119877

119904is

obtained based on the above simplifications and by puttingthe value of 120585 in (24) as

119877119904

= 18Re(119879119888119882(minus15 times 10

71198900022(47+(17times10

5

119879119888)))

119879119888119882(minus15times 10

71198900022(47+(17times10

5119879119888)))+4400

)

(25)

where Re represents the real part because the negativeexpression inside the Lambert W function results in acomplex number However in practical problems only realvalues are to be considered

22 Determination of Optimum Power Output Parameters ofProposed Model The optimum power output parameters ofmodel were determined by deriving the equations for current(119868) and voltage (119881) by putting the values of unknown parame-ters namely 119868

119871 119868119900119877119904 and 120582 in the respective equationsThe

power (119875) is the product of current (119868) and voltage (119881) [48]therefore it can be expressed as

119875 = 119868119881 (26)

By substituting 119868 from (23) into (26) the value of 119875 iscomputed as [52]

119875 = (119868119871+ 119868119900) minus

119882[120585119877119904119890120585(119881+119868

119871119877119904+119868119900119877119904)]

120585119877119904

119881 (27)

Mathematically the optimum power occurs at the point119881maxof P-V curve where the slope of tangent line is equal to zeroas follows

119889119875

119889119881

10038161003816100381610038161003816100381610038161003816119881max

= 0 (28)

By differentiating (27) with respect to voltage (119881) and takingthe RHS equal to zero

119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)]119881

1 +119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)] 119877119904

+1

120585119877119904

times 119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)] minus 120585119877

119904(119868119871+ 119868119900) = 0

(29)

Newton-Raphson method is applied to (29) in order tofind the critical value of 119881 for 119881max The value of 119881max issubstituted into (27) in order to solve the maximum power(119875max) Consider

119875max = (119868119871+ 119868119900) minus

119882[120585119877119904119890120585(119881max+119868119871119877119904+119868119900119877119904)

]

120585119877119904

119881max (30)

0 5 10 15 20 25 30 35 40

25

2

15

1

05

0

Curr

ent (

A)

Voltage (V)

Vmax

Imax

Isc

Voc

Simulated dataManufactured data

Optimum powerpoint

Figure 2 Typical I-V characteristic curve of a PV module

Consequently the desired maximum current (119868max) can beobtained from (30) by dividing the maximum power (119875max)with maximum voltage (119881max) Consider

119868max =119875max119881max

(31)

3 Simulation of I-V and P-V CharacteristicCurves of a Selected PV Module

The familiarity of current and voltage relationship of photo-voltaic modules under real operating conditions is essentialfor the determination of their power output Normally thecells are mounted inmodules andmultiple modules are usedin arrays to get desired power output Individual modulesmay have cells connected in series and parallel combinationsto obtain the required current and voltage Similarly thearray of modules may be arranged in series and parallelconnections When the cells or modules are connected inseries the voltage is additive and when they are attached inparallel the currents are additive [52ndash56] The power outputof PV modules could be predicted from the behavior ofcurrent-voltage I-V and power-voltage P-V characteristiccurves The current-voltage and power-voltage characteristiccurves are graphically shown in Figures 2 to 9

The current-voltage I-V characteristic of a typical PVmodule is shown in Figure 2When the output voltage119881 = 0the current is the short circuit current (119868sc) and when thecurrent 119868 = 0 the output voltage is the open circuit voltage(119881oc) Mostly the current decreases slowly at a certain pointand then decreases rapidly to the open circuit conditionsThe power as a function of voltage is given in Figure 3 Themaximum power that can be obtained corresponds to therectangle of maximum area under I-V curve At the optimumpower point the power is 119875mp the current is 119868mp and thevoltage is 119881mp Ideally the cells would always operate at theoptimum power point that matches the I-V characteristic ofthe load Hence the load matching is essential for extractingthe maximum power from the solar photovoltaic modules


00

5 10

10

15 20

20

25 30

30

35 40

40

50

60

70

80

Voltage (V)


Vmax

Optimum power

Pow

er (W

)

Pmax

Figure 3 Typical P-V characteristic curve of a PV module

00

1

2

3

4

5

5 10 15 20 25 30 35 4540

Voltage (V)


Curr

ent (

A)

GT = 1000 (Wm2)

GT = 800 (Wm2)

GT = 600 (Wm2)

GT = 400 (Wm2)

GT = 200 (Wm2)

Optimum powerpoint

Figure 4 I-V characteristic curves at various solar radiation levels

Therefore the maximum power point tractors are preferredto optimize the output power from solar PV systems

I-V characteristic curves at various solar irradiation levelsand temperatures are shown in Figures 4 and 5 respectivelyThe locus of maximum power point is indicated on thecurves The short circuit current increases in proportion tothe solar radiation while the open circuit voltage increaseslogarithmically with solar radiation As long as the curvedportion of the I-V characteristic does not intersect theshort circuit current is nearly proportional to the incidentsolar radiation If the incident solar radiation is assumedto be a fixed spectral distribution the short circuit currentcan be used as a measure of incident solar radiation I-Vcharacteristics curves for the combination of irradiance andtemperatures are illustrated in Figure 6 It was observedthat the temperature linearly decreases the output voltage ascompared to current Consequently the decrease of voltagelowers the power output of PV module at constant solarirradiation level However the effect of temperature is smallon short circuit current but increases with the increase ofincident solar radiation

00

1

2

3

4

5

5 10 15 20 25 30 35 40

Voltage (V)


Curr

ent (

A)

Optimum powerpoint

Ta = 0∘C

Ta = 25∘C

Ta = 50∘C

Ta = 75∘C

Figure 5 I-V characteristic curves at various temperatures

00

1

2

3

4

5

5 10 15 20 25 30 35 40

Voltage (V)


Curr

ent (

A)

Optimum powerpoint

GT = 500Wm2

Ta = 20∘C

Ta = 60∘C

GT = 1000Wm2

Figure 6 I-V characteristic curves for various set of solar radiationand temperature

The P-V characteristics curves for various solar irradi-ation levels at constant temperature of 25∘C and at severaltemperatures with constant solar irradiance of 1000Wm2is illustrated in Figures 7 and 8 respectively Increasingtemperature leads to decreasing the open circuit voltageand slightly increasing the short circuit current Operatingof cell temperature at that region of the curve leads to asignificant power reduction at high temperatures The P-Vcharacteristics curves for the combination of irradiance andtemperatures are shown in Figure 9

The power output of photovoltaic module by formulatedmodel gave aplusmn2 error when comparedwith the rated powerof PV module provided by manufacturers on average basisHowever at higher solar radiation and temperature valuesthemodel simulated results were somehow deviated from therated power of PV module Since the shunt resistance (119877sh)was assumed to be infinity in the proposed model It wasfound from the analysis that the increase of temperature anddecrease of incident solar radiation levels lead to lower poweroutput and vice versa The power output from PV modules


0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power

GT = 1000 (Wm2)

GT = 800 (Wm2)GT = 600 (Wm2)GT = 400 (Wm2)GT = 200 (Wm2)

180

160

140

120

100

80

60

40

20

0

Pow

er (W

)

Figure 7 P-V characteristic curve at constant temperature of 25∘C

0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power180

160

140

120

100

80

60

40

20

0

Pow

er (W

) Ta = 0∘C

Ta = 25∘C

Ta = 50∘C

Ta = 75∘C

Figure 8 P-V characteristic curve at constant solar radiation of1000Wm2

0 5 10 15 20 25 30 35 40

Voltage (V)


Optimum power160

140

120

100

80

60

40

20

0

Pow

er (W

)

Ta = 60∘C

Ta = 20∘C

GT = 1000Wm2

GT = 500Wm2

Figure 9 P-V characteristic curve at various set of solar radiationand temperature

approaches zero if the amount of solar radiation tends todecrease and the temperature goes up

4 Conclusions

The proposed mathematical model is formulated by inte-gration of both analytical and numerical methods Therequired parameters of current-voltage (I-V) curve such asthe light-generated current diode reverse saturation currentshape parameter and the series resistance are computedanalytically The expression for output current from PVmodule is determined explicitly by Lambert W function andvoltage output is computed numerically by Newton-Raphsonmethod The main contribution of this study is algebraicderivation of equations for the shape factor (120582)which involvethe ideality factor (119860) and the series resistance (119877

119904) of single

diode model of PV module power output These equationswill help to find out the predicted power output of PVmodules in precise and convenient manner

The current-voltage (I-V) and the power-voltage (P-V)characteristic curves obtained from the proposed modelwere matching with the curves drawn from the PV moduleat standard test conditions The variation of incident solarradiation and temperature were found to be themain cause ofmodifications in the amount of PV module power output Alinear relationship between the power output of PV moduleand the amount of incident solar radiation were observed ifother factors were kept constant

The estimated results of the proposedmodel are validatedby PVmodule rated power output provided bymanufacturerwhich gave a plusmn2 error The model is found to be morepractical in terms of the number of variables used andpredicted satisfactory performance of PV modules

Nomenclature

119878119879 absorbed solar radiation Wm2

119896 Boltzmannrsquos constant 1381 times 10minus23 JK

119879119888 Cell temperature at actual conditions K

119868 Current output of cell A119868119863 Diode current or dark current A

119863 Diode diffusion factor -119868119900 Diode reverse saturation current A

119860 Ideality factor 1 for ideal diodes andbetween 1 and 2 for real diodes

119882 Lambert W function -119868119871 Light-generated current A

120576119892 Material band gap energy eV 112 eV for

silicon and 135 eV for gallium arsenide119868mp Maximum current of PV module A119875mp Maximum power of PV module W119881mp Maximum voltage of PV module V119873cs Number of cells in series -119881oc Open circuit voltage of PV module V120585 Parameter 119902119896119879

119888120582

119877119904 Series resistance Ω

120582 Shape factor of I-V curve -


119868sc Short circuit current of PV module A119877sh Shunt resistanceΩ120583119868sc Temperature coefficient of short circuitcurrent -

120583119881oc Temperature coefficient of voltage VK

119881 Voltage output of cell V119866119879 Solar radiation Wm2

119902 Electron charge 1602 times 10minus19C

119879119886 Ambient temperature ∘C

The Subscript

119903 In any notation the corresponding value ofparameter at reference conditions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] W Durisch D Tille A Worz and W Plapp ldquoCharacterisationof photovoltaic generatorsrdquo Applied Energy vol 65 no 1ndash4 pp273ndash284 2000

[2] A Q Jakhrani A K Othman A R H Rigit S R Samo andS R Kamboh ldquoA novel analytical model for optimal sizing ofstandalone photovoltaic systemsrdquoEnergy vol 46 no 1 pp 675ndash682 2012

[3] A N Celik ldquoA simplified model based on clearness index forestimating yearly performance of hybrid pv energy systemsrdquoProgress in Photovoltaics Research and Applications vol 10 no8 pp 545ndash554 2002

[4] A H Fanney and B P Dougherty ldquoBuilding integrated photo-voltaic test facilityrdquo Journal of Solar Energy Engineering vol 123no 3 pp 194ndash199 2001

[5] M A Green ldquoShort communication priceefficiency correla-tions for 2004 photovoltaic modulesrdquo Progress in PhotovoltaicsResearch and Applications vol 13 pp 85ndash87 2005

[6] A Q Jakhrani A K Othman A R H Rigit and S R SamoldquoModel for estimation of global solar radiation in SarawakMalaysiardquo World Applied Sciences Journal vol 14 pp 83ndash902011

[7] A Q Jakhrani A K Othman A R H Rigit and S R SamoldquoAssessment of solar and wind energy resources at five typicallocations in Sarawakrdquo Journal of Energy amp Environment vol 3pp 8ndash13 2012

[8] R Ramaprabha andB LMathur ldquoDevelopment of an improvedmodel of SPV cell for partially shaded solar photovoltaic arraysrdquoEuropean Journal of Scientific Research vol 47 no 1 pp 122ndash1342010

[9] J A R Hernanz J J C ampayoMartin I Z Belver J L LesakaE Z Guerrero and E P Perez ldquoModelling of photovoltaicmodulerdquo in Proceedings of the International Conference onRenewable Energies and Power Quality (ICREPQ rsquo10) GranadaSpain March 2010

[10] R B Andrews A Pollard and J M Pearce ldquoImproved para-metric empirical determination of module short circuit currentfor modelling and optimization of solar photovoltaic systemsrdquoSolar Energy vol 86 pp 2240ndash2254 2012

[11] R L Chakrasali V R Sheelavant andHNNagaraja ldquoNetworkapproach tomodeling and simulation of solar photovoltaic cellrdquoRenewable and Sustainable Energy Reviews vol 21 pp 84ndash882013

[12] A Chouder S Silvestre N Sadaoui and L Rahmani ldquoMod-eling and simulation of a grid connected PV system basedon the evaluation of main PV module parametersrdquo SimulationModelling Practice andTheory vol 20 no 1 pp 46ndash58 2012

[13] A Chouder S Silvestre B Taghezouit and E KaratepeldquoMonitoring modeling and simulation of PV systems usingLabVIEWrdquo Solar Energy vol 91 pp 337ndash349 2013

[14] A Jain and A Kapoor ldquoA new approach to study organic solarcell using Lambert W-functionrdquo Solar Energy Materials andSolar Cells vol 86 no 2 pp 197ndash205 2005

[15] A Jain S Sharma and A Kapoor ldquoSolar cell array parametersusing Lambert W-functionrdquo Solar Energy Materials and SolarCells vol 90 no 1 pp 25ndash31 2006

[16] A Ortiz-Conde F J Garcıa Sanchez and JMuci ldquoNewmethodto extract the model parameters of solar cells from the explicitanalytic solutions of their illuminated I-V characteristicsrdquo SolarEnergy Materials and Solar Cells vol 90 no 3 pp 352ndash3612006

[17] D Picault B Raison S Bacha J de la Casa and J AguileraldquoForecasting photovoltaic array power production subject tomismatch lossesrdquo Solar Energy vol 84 no 7 pp 1301ndash1309 2010

[18] Y Chen X Wang D Li R Hong and H Shen ldquoParametersextraction from commercial solar cells I-V characteristics andshunt analysisrdquo Applied Energy vol 88 no 6 pp 2239ndash22442011

[19] H Fathabadi ldquoNovel neural-analytical method for determiningsiliconplastic solar cells and modules characteristicsrdquo EnergyConversion and Management vol 76 pp 253ndash259 2013


[21] Krismadinata N A Rahim H W Ping and J Selvaraj ldquoPho-tovoltaic module modeling using simulinkmatlabrdquo ProcediaEnvironmental Sciences vol 17 pp 537ndash546 2013

[22] F Lu S Guo T M Walsh and A G Aberle ldquoImproved PVmodule performance under partial shading conditionsrdquo EnergyProcedia vol 33 pp 248ndash255 2013

[23] A Mellit S Saglam and S A Kalogirou ldquoArtificial neuralnetwork-based model for estimating the produced power ofa photovoltaic modulerdquo Renewable Energy vol 60 pp 71ndash782013

[24] G K Singh ldquoSolar power generation by PV (photovoltaic)technology a reviewrdquo Energy vol 3 pp 1ndash13 2013

[25] D Thevenard and S Pelland ldquoEstimating the uncertainty inlong-term photovoltaic yield predictionsrdquo Solar Energy vol 91pp 432ndash445 2013

[26] H Tian F Mancilla-David K Ellis E Muljadi and P JenkinsldquoA cell-to-module-to-array detailed model for photovoltaicpanelsrdquo Solar Energy vol 86 pp 2695ndash2706 2012

[27] M C D Vincenzo and D Infield ldquoDetailed PV array model fornon-uniform irradiance and its validation against experimentaldatardquo Solar Energy vol 97 pp 314ndash331 2013

[28] G H Yordanov O-M Midtgard and T O Saetre ldquoSeriesresistance determination and further characterization of c-SiPV modulesrdquo Renewable Energy vol 46 pp 72ndash80 2012


[29] W De Soto S A Klein andW A Beckman ldquoImprovement andvalidation of amodel for photovoltaic array performancerdquo SolarEnergy vol 80 no 1 pp 78ndash88 2006

[30] E Karatepe M Boztepe and M Colak ldquoNeural network basedsolar cell modelrdquo Energy Conversion and Management vol 47no 9-10 pp 1159ndash1178 2006

[31] Y-C Kuo T-J Liang and J-F Chen ldquoNovel maximum-power-point-tracking controller for photovoltaic energy conversionsystemrdquo IEEE Transactions on Industrial Electronics vol 48 no3 pp 594ndash601 2001

[32] R Chenni M Makhlouf T Kerbache and A Bouzid ldquoAdetailed modeling method for photovoltaic cellsrdquo Energy vol32 no 9 pp 1724ndash1730 2007

[33] A Wagner Peak-Power and Internal Series Resistance Mea-surement under Natural Ambient Conditions EuroSun Copen-hagen Denmark 2000

[34] M Chegaar G Azzouzi and P Mialhe ldquoSimple parameterextraction method for illuminated solar cellsrdquo Solid-State Elec-tronics vol 50 no 7-8 pp 1234ndash1237 2006

[35] K Bouzidi M Chegaar and A Bouhemadou ldquoSolar cellsparameters evaluation considering the series and shunt resis-tancerdquo Solar Energy Materials and Solar Cells vol 91 no 18 pp1647ndash1651 2007

[36] J C Ranuarez A Ortiz-Conde and F J Garcıa Sanchez ldquoA newmethod to extract diode parameters under the presence of para-sitic series and shunt resistancerdquoMicroelectronics Reliability vol40 no 2 pp 355ndash358 2000

[37] M A De Blas J L Torres E Prieto and A Garcıa ldquoSelecting asuitable model for characterizing photovoltaic devicesrdquo Renew-able Energy vol 25 no 3 pp 371ndash380 2002

[38] D Petreus C Frcas and I Ciocan ldquoModelling and simulationof photovoltaic cellsrdquo Acta Technica Napocensis Electronica-Telecomunicatii vol 49 pp 42ndash47 2008

[39] V D Dio D La Cascia R Miceli and C Rando ldquoA mathe-matical model to determine the electrical energy production inphotovoltaic fields under mismatch effectrdquo in Proceedings of theInternational Conference on Clean Electrical Power (ICCEP rsquo09)pp 46ndash51 Capri Italy June 2009

[40] W Kim andW Choi ldquoA novel parameter extractionmethod forthe one-diode solar cell modelrdquo Solar Energy vol 84 no 6 pp1008ndash1019 2010

[41] D Petreus I Ciocan and C Farcas An Improvement onEmpirical Modelling of Photovoltaic Cells ISSE Madrid Spain2008

[42] M S Benghanem and N A Saleh ldquoModeling of photovoltaicmodule and experimental determination of serial resistancerdquoJournal of Taibah University for Science vol 1 pp 94ndash105 2009

[43] A N Celik ldquoEffect of different load profiles on the loss-of-loadprobability of stand-alone photovoltaic systemsrdquo RenewableEnergy vol 32 no 12 pp 2096ndash2115 2007

[44] Q Kou S A Klein and W A Beckman ldquoA method forestimating the long-term performance of direct-coupled PVpumping systemsrdquo Solar Energy vol 64 no 1ndash3 pp 33ndash40 1998

[45] J A Duffie and W A Beckman Solar Engineering of ThermalProcesses John Wiley amp Sons 3rd edition 2006

[46] A H Fanney B P Dougherty and M W Davis ldquoEvaluat-ing building integrated photovoltaic performance modelsrdquo inProceedings of the 29th IEEE Photovoltaic Specialists Conference(PVSC rsquo02) New Orleans La USA May 2002

[47] R L Boylestad L Nashelsky and L Li Electronic Devices andCircuit Theory Prentice-Hall 10th edition 2009

[48] E Saloux A Teyssedou and M Sorin ldquoExplicit model ofphotovoltaic panels to determine voltages and currents at themaximum power pointrdquo Solar Energy vol 85 no 5 pp 713ndash722 2011

[49] J Crispim M Carreira and C Rui ldquoValidation of photovoltaicelectrical models against manufacturers data and experimentalresultsrdquo in Proceedings of the International Conference on PowerEngineering Energy and Electrical Drives (POWERENG rsquo07) pp556ndash561 Setubal Portugal April 2007

[50] F M Gonzalez-Longatt Model of Photovoltaic Module inMatlab II CIBELEC Puerto La Cruz Venezuela 2006

[51] D Sera and R Teodorescu ldquoRobust series resistance estimationfor diagnostics of photovoltaic modulesrdquo in Proceedings of the35thAnnual Conference of the IEEE Industrial Electronics Society(IECON rsquo09) pp 800ndash805 Porto Portugal November 2009

[52] A Q Jakhrani A K Othman A R H Rigit R Baini SR Samo and L P Ling ldquoInvestigation of solar photovoltaicmodule power output by various modelsrdquo NED UniversityJournal of Research pp 25ndash34 2012

[53] A Q Jakhrani A K Othman A R H Rigit and S RSamo ldquoComparison of solar photovoltaic module temperaturemodelsrdquoWorld Applied Sciences Journal vol 14 pp 1ndash8 2011

[54] A Q Jakhrani A K Othman A R H Rigit and S R SamoldquoDetermination and comparison of different photovoltaicmod-ule temperaturemodels for Kuching Sarawakrdquo inProceedings ofthe IEEE 1st Conference on Clean Energy and Technology (CETrsquo11) pp 231ndash236 Kuala Lumpur Malaysia June 2011

[55] A Q Jakhrani S R Samo A R H Rigit and S A KambohldquoSelection of models for calculation of incident solar radiationon tilted surfacesrdquoWorld Applied Sciences Journal vol 22 no 9pp 1334ndash1341 2013

[56] A Q Jakhrani A K Othman A R H Rigit S R Samo andS A Kamboh ldquoSensitivity analysis of a standalone photovoltaicsystem model parametersrdquo Journal of Applied Sciences vol 13no 2 pp 220ndash231 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


(119868sc) at 5min time scales Their work was a modified versionof the Sandia array performance model by incorporatingnew factors for the calculation of short circuit current (119868sc)to justify errors (including instrumentation alignment andspectral andmodule power tolerance errors) Chakrasali et al[11] investigated the performance of Nortonrsquos circuit model ofsolar PV module with the existing models using Matlab andreported that it is a well-suited way to predict the behavior ofPV modules operated for longer periods of time Chouder etal [12]modeled a PVmodule by a single diode lumped circuitand evaluated its main parameters by considering the powerconversion efficiency Chouder et al [13] presented a detailedcharacterization of the performance and dynamic behaviorof PV systems by using the LabVIEW platformThe LambertW function was applied for the solution of equations by Jainand Kapoor [14] Jain et al [15] Ortiz-Conde et al [16] andothers [17ndash19] Picault et al [17] presented a novel methodto forecast existing PV array production in diverse environ-mental conditions and concluded that Lambert W functionfacilitates a direct relationship between current and voltage ofmodules as it significantly reduces calculation time Chen etal [18] proposed an optimized method based on polynomialcurve fitting and Lambert W function for extraction ofparameters from the current-voltage (I-V) characteristics ofcommercial silicon solar cells The Lambert W function wasused for translation of transcendental equation into explicitanalytical solution Fathabadi [19] presented a novel methodfor characterization of silicon solar cells modules and plasticsolar cells Artificial neural network together with LambertW function was employed for determination of I-V and P-Vcurves of silicon and plastic solar cells and modules [20]

Moreover Krismadinata et al [21] used a single diodeelectrical equivalent circuit model for determination of PVcell characteristics and found that output of PV moduleswere strongly affected by the intensity of solar irradiationand ambient temperature Lu et al [22] investigated variousPV module layouts using full size as well as halved solarcells The performance of module layouts was investigated bypartially shading the PV cells using a solar cell equivalentcircuit model with SPICE software They found that theseries-parallel hybrid connection of cells within a modulehas a significant improvement on the power output of thePV module under partial shading conditions Mellit et al[23] employed a methodology for estimation power profileof a 50Wp Si-polycrystalline PV module by developing twoartificial neural networks (ANNs) for cloudy and sunny daysand found that the ANN-models performed better than theexisting models and also did not need more parametersunlike implicit models Singh [24] reviewed various modelsof PV cells and concluded that the accuracy of modelscan be improved by including series and shunt resistanceinto the model In addition the author also discoveredthat the estimation of models can further be improved byeither introducing two parallel diodes with independent setsaturation current or considering the diode quality factoras a variable parameter instead of fixed value like 1 or 2Thevenard and Pelland [25] reported that the uncertainties ofmodel predictions can be reduced by increasing the reliabilityand spatial coverage of solar radiation estimates appropriate

familiarity of losses due to dirt soiling and snow anddevelopment of better tools for PV system modeling Tian etal [26] presented a modified I-V relationship for the singlediode model The alteration in the model was made in theparallel and series connections of an array The derivation ofthe adapted I-V relationship was begunwith a single solar celland extended up to a PV module and finally an array Themodified correlation was investigated with a five-parametermodel based on the data provided by the manufacturers Theperformance of themodel was examined with a wide range ofirradiation levels and cell temperatures for prediction of I-Vand P-V curves maximum power point values short circuitcurrent and open circuit voltage Vincenzo and Infield [27]developed a detailed PV array model to deal explicitly withnonuniform irradiance and other nonuniformities across thearray and it was validated against data from an outdoortest system However the authors reduced the complexity ofthe simulations by assuming that the cell temperatures arehomogeneous for eachmodule Yordanov et al [28] presenteda new algorithm for determination of the series resistanceof crystalline-Si PV modules from individual illuminatedI-V curves The ideality factor and the reverse saturationcurrent were extracted in the typical way They found thatthe ideality factor at open circuit is increased by about 5It was established from the review that Lambert W functionis a simple technique to give the analytical explicit solutionof solar photovoltaic module characteristics as compared tothe other methods However some problems still exist for thederivation of the required model equations

Equivalent electrical circuit model is one of the keymodels under study since the last few decades It is configuredwith either single or double diode for investigation of current-voltage relationships The single diode models usually havefive four or three unknown parameters with only oneexponential term The five unknown parameters of a singlediode model are light-generated current (119868

119871) diode reverse

saturation current (119868119900) series resistance (119877

119904) shunt resistance

(119877sh) and diode ideality factor (119860) [29 30] The four-parameter model infers the shunt resistance as infinite andit is ignored [31] The three-parameter model assumes thatthe series resistance is zero and shunt resistance is infiniteand thus both of these parameters are ignored whereas thedouble diode models have six unknown parameters with twoexponential terms [32 33]

In fact both single and double diode models require theknowledge of all unknown parameters which is usually notprovided bymanufacturers Nevertheless the current-voltageequation is a transcendental expression It has no explicitanalytical solution It is also time consuming to discover itsexact analytical solution due to the limitation of availabledata for the extraction of required parameters [34ndash36] Forthat reason the researchers gradually focused on searchingout the approximate methods for the calculation of unknownparameters The analytical methods give exact solutions bymeans of algebraic equations However due to implicit natureand nonlinearity of PV cell or module characteristics itis hard to find out the analytical solution of all unknownparameters Analytical methods have also some limitationsand could not give exact solutions when the functions






119868 = 119868119871minus 119868119863minus 119868sh (1)





119868119863= 119868119900[119890120585(119881+119868119877

119904)minus 1] (2)


119868sh =119881 + 119868119877

119904

119877sh (3)


119868 = 119868119871minus 119868119900[119890120585(119881+ 119868119877


119881 + 119868119877119904

119877sh (4)

V

I

IoIL

Rs

Rsh




120585 =119902

120582119896119879119888

(5)



120582 = 119860 times 119873CS (6)




119868 = 119868119871minus 119868119900[119890120585(119881+119868119877

119904)minus 1] (7)














119868119871= (

119878119879

119878119879119903

) [119868119871119903

+ 120583119868sc(119879119888minus 119879119888119903)] (8)

where 119878119879and 119878


119888

and 119879119888119903





119888)




as

119868119900= 119863119879

3

119888119890120585120576119892119860 (9)



119868119900= (

119879119888

119879119888119903

)

3

119890120585120582120576119892(119879119888119903minus119879119888)119860

(10)


119868sc119903 = 119868119871119903

minus 119868119900119903

[119890(120585119903119868sc119903119877119904)

minus 1] (11)

where

120585119903=

119902

120582119896119879119888119903

(12)

119868119871119903

minus 119868119900119903

[119890(120585119903119881oc119903)

minus 1] = 0 (13)

119868mp119903 = 119868119871119903

minus 119868119900119903

[119890120585119903(119881mp119903+119868mp119903119877119904)

minus 1] (14)





119868119871119903

cong 119868sc119903 (15)

119868sc119903 minus 119868119900119903

[119890(120585119903119868sc119903119877119904)

] cong 0 (16)

119868mp119903 cong 119868119871119903

minus 119868119900119903

[119890120585119903(119881mp119903+119868mp119903119877119904)

] (17)



119868119900119903

= 119868sc119903 [119890minus(120585119903119881oc119903)

] (18)




] (19)


120585119903=

ln (1 minus 119868mp119903119868sc119903)

119881mp119903 minus 119881oc119903 + 119868mp119903119877119904 (20)


120582 =

119902 (119881mp119903 minus 119881oc119903 + 119868mp119903119877119904)

119896119879119888119903ln (1 minus 119868mp119903119868sc119903)

(21)


119904) is an


119904would be


119889119881

119889119868

10038161003816100381610038161003816100381610038161003816119881oc119903

minus 119877119904= 0 (22)


119868 = (119868119871+ 119868119900) minus

119882(120585119877119904119890120585(119881+119868

119871119877119904+119868119900119877119904))

120585119877119904

(23)


119871= 119868119871119903


119882(120585119877119904119868119900119890120585(119881oc119903+119868119871119903119877119904+119868119900119877119904)

)

[1 +119882(120585119877119904119868119900119890120585(119881oc119903+119868119871119903119877119904+119868119900119877119904))] 119877

119904

minus 119877119904= 0 (24)








119904is


119877119904

= 18Re(119879119888119882(minus15 times 10

71198900022(47+(17times10

5

119879119888)))

119879119888119882(minus15times 10

71198900022(47+(17times10

5119879119888)))+4400

)

(25)





119875 = 119868119881 (26)


119875 = (119868119871+ 119868119900) minus

119882[120585119877119904119890120585(119881+119868

119871119877119904+119868119900119877119904)]

120585119877119904

119881 (27)


119889119875

119889119881

10038161003816100381610038161003816100381610038161003816119881max

= 0 (28)


119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)]119881

1 +119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)] 119877119904

+1

120585119877119904

times 119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)] minus 120585119877

119904(119868119871+ 119868119900) = 0

(29)


119875max = (119868119871+ 119868119900) minus

119882[120585119877119904119890120585(119881max+119868119871119877119904+119868119900119877119904)

]

120585119877119904

119881max (30)

0 5 10 15 20 25 30 35 40

25

2

15

1

05

0

Curr

ent (

A)

Voltage (V)

Vmax

Imax

Isc

Voc


Optimum powerpoint



119868max =119875max119881max

(31)





00

5 10

10

15 20

20

25 30

30

35 40

40

50

60

70

80

Voltage (V)


Vmax

Optimum power

Pow

er (W

)

Pmax


00

1

2

3

4

5

5 10 15 20 25 30 35 4540

Voltage (V)


Curr

ent (

A)

GT = 1000 (Wm2)

GT = 800 (Wm2)

GT = 600 (Wm2)

GT = 400 (Wm2)

GT = 200 (Wm2)

Optimum powerpoint




00

1

2

3

4

5

5 10 15 20 25 30 35 40

Voltage (V)


Curr

ent (

A)

Optimum powerpoint

Ta = 0∘C

Ta = 25∘C

Ta = 50∘C

Ta = 75∘C


00

1

2

3

4

5

5 10 15 20 25 30 35 40

Voltage (V)


Curr

ent (

A)

Optimum powerpoint

GT = 500Wm2

Ta = 20∘C

Ta = 60∘C

GT = 1000Wm2





0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power

GT = 1000 (Wm2)


180

160

140

120

100

80

60

40

20

0

Pow

er (W

)


0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power180

160

140

120

100

80

60

40

20

0

Pow

er (W

) Ta = 0∘C

Ta = 25∘C

Ta = 50∘C

Ta = 75∘C


0 5 10 15 20 25 30 35 40

Voltage (V)


Optimum power160

140

120

100

80

60

40

20

0

Pow

er (W

)

Ta = 60∘C

Ta = 20∘C

GT = 1000Wm2

GT = 500Wm2



4 Conclusions


119904) of single




Nomenclature










119888120582









The Subscript




References



































































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of






119868 = 119868119871minus 119868119863minus 119868sh (1)





119868119863= 119868119900[119890120585(119881+119868119877

119904)minus 1] (2)


119868sh =119881 + 119868119877

119904

119877sh (3)


119868 = 119868119871minus 119868119900[119890120585(119881+ 119868119877


119881 + 119868119877119904

119877sh (4)

V

I

IoIL

Rs

Rsh




120585 =119902

120582119896119879119888

(5)



120582 = 119860 times 119873CS (6)




119868 = 119868119871minus 119868119900[119890120585(119881+119868119877

119904)minus 1] (7)














119868119871= (

119878119879

119878119879119903

) [119868119871119903

+ 120583119868sc(119879119888minus 119879119888119903)] (8)

where 119878119879and 119878


119888

and 119879119888119903





119888)




as

119868119900= 119863119879

3

119888119890120585120576119892119860 (9)



119868119900= (

119879119888

119879119888119903

)

3

119890120585120582120576119892(119879119888119903minus119879119888)119860

(10)


119868sc119903 = 119868119871119903

minus 119868119900119903

[119890(120585119903119868sc119903119877119904)

minus 1] (11)

where

120585119903=

119902

120582119896119879119888119903

(12)

119868119871119903

minus 119868119900119903

[119890(120585119903119881oc119903)

minus 1] = 0 (13)

119868mp119903 = 119868119871119903

minus 119868119900119903

[119890120585119903(119881mp119903+119868mp119903119877119904)

minus 1] (14)





119868119871119903

cong 119868sc119903 (15)

119868sc119903 minus 119868119900119903

[119890(120585119903119868sc119903119877119904)

] cong 0 (16)

119868mp119903 cong 119868119871119903

minus 119868119900119903

[119890120585119903(119881mp119903+119868mp119903119877119904)

] (17)



119868119900119903

= 119868sc119903 [119890minus(120585119903119881oc119903)

] (18)




] (19)


120585119903=

ln (1 minus 119868mp119903119868sc119903)

119881mp119903 minus 119881oc119903 + 119868mp119903119877119904 (20)


120582 =

119902 (119881mp119903 minus 119881oc119903 + 119868mp119903119877119904)

119896119879119888119903ln (1 minus 119868mp119903119868sc119903)

(21)


119904) is an


119904would be


119889119881

119889119868

10038161003816100381610038161003816100381610038161003816119881oc119903

minus 119877119904= 0 (22)


119868 = (119868119871+ 119868119900) minus

119882(120585119877119904119890120585(119881+119868

119871119877119904+119868119900119877119904))

120585119877119904

(23)


119871= 119868119871119903


119882(120585119877119904119868119900119890120585(119881oc119903+119868119871119903119877119904+119868119900119877119904)

)

[1 +119882(120585119877119904119868119900119890120585(119881oc119903+119868119871119903119877119904+119868119900119877119904))] 119877

119904

minus 119877119904= 0 (24)








119904is


119877119904

= 18Re(119879119888119882(minus15 times 10

71198900022(47+(17times10

5

119879119888)))

119879119888119882(minus15times 10

71198900022(47+(17times10

5119879119888)))+4400

)

(25)





119875 = 119868119881 (26)


119875 = (119868119871+ 119868119900) minus

119882[120585119877119904119890120585(119881+119868

119871119877119904+119868119900119877119904)]

120585119877119904

119881 (27)


119889119875

119889119881

10038161003816100381610038161003816100381610038161003816119881max

= 0 (28)


119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)]119881

1 +119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)] 119877119904

+1

120585119877119904

times 119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)] minus 120585119877

119904(119868119871+ 119868119900) = 0

(29)


119875max = (119868119871+ 119868119900) minus

119882[120585119877119904119890120585(119881max+119868119871119877119904+119868119900119877119904)

]

120585119877119904

119881max (30)

0 5 10 15 20 25 30 35 40

25

2

15

1

05

0

Curr

ent (

A)

Voltage (V)

Vmax

Imax

Isc

Voc


Optimum powerpoint



119868max =119875max119881max

(31)





00

5 10

10

15 20

20

25 30

30

35 40

40

50

60

70

80

Voltage (V)


Vmax

Optimum power

Pow

er (W

)

Pmax


00

1

2

3

4

5

5 10 15 20 25 30 35 4540

Voltage (V)


Curr

ent (

A)

GT = 1000 (Wm2)

GT = 800 (Wm2)

GT = 600 (Wm2)

GT = 400 (Wm2)

GT = 200 (Wm2)

Optimum powerpoint




00

1

2

3

4

5

5 10 15 20 25 30 35 40

Voltage (V)


Curr

ent (

A)

Optimum powerpoint

Ta = 0∘C

Ta = 25∘C

Ta = 50∘C

Ta = 75∘C


00

1

2

3

4

5

5 10 15 20 25 30 35 40

Voltage (V)


Curr

ent (

A)

Optimum powerpoint

GT = 500Wm2

Ta = 20∘C

Ta = 60∘C

GT = 1000Wm2





0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power

GT = 1000 (Wm2)


180

160

140

120

100

80

60

40

20

0

Pow

er (W

)


0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power180

160

140

120

100

80

60

40

20

0

Pow

er (W

) Ta = 0∘C

Ta = 25∘C

Ta = 50∘C

Ta = 75∘C


0 5 10 15 20 25 30 35 40

Voltage (V)


Optimum power160

140

120

100

80

60

40

20

0

Pow

er (W

)

Ta = 60∘C

Ta = 20∘C

GT = 1000Wm2

GT = 500Wm2



4 Conclusions


119904) of single




Nomenclature










119888120582









The Subscript




References



































































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of





119868119871= (

119878119879

119878119879119903

) [119868119871119903

+ 120583119868sc(119879119888minus 119879119888119903)] (8)

where 119878119879and 119878


119888

and 119879119888119903





119888)




as

119868119900= 119863119879

3

119888119890120585120576119892119860 (9)



119868119900= (

119879119888

119879119888119903

)

3

119890120585120582120576119892(119879119888119903minus119879119888)119860

(10)


119868sc119903 = 119868119871119903

minus 119868119900119903

[119890(120585119903119868sc119903119877119904)

minus 1] (11)

where

120585119903=

119902

120582119896119879119888119903

(12)

119868119871119903

minus 119868119900119903

[119890(120585119903119881oc119903)

minus 1] = 0 (13)

119868mp119903 = 119868119871119903

minus 119868119900119903

[119890120585119903(119881mp119903+119868mp119903119877119904)

minus 1] (14)





119868119871119903

cong 119868sc119903 (15)

119868sc119903 minus 119868119900119903

[119890(120585119903119868sc119903119877119904)

] cong 0 (16)

119868mp119903 cong 119868119871119903

minus 119868119900119903

[119890120585119903(119881mp119903+119868mp119903119877119904)

] (17)



119868119900119903

= 119868sc119903 [119890minus(120585119903119881oc119903)

] (18)




] (19)


120585119903=

ln (1 minus 119868mp119903119868sc119903)

119881mp119903 minus 119881oc119903 + 119868mp119903119877119904 (20)


120582 =

119902 (119881mp119903 minus 119881oc119903 + 119868mp119903119877119904)

119896119879119888119903ln (1 minus 119868mp119903119868sc119903)

(21)


119904) is an


119904would be


119889119881

119889119868

10038161003816100381610038161003816100381610038161003816119881oc119903

minus 119877119904= 0 (22)


119868 = (119868119871+ 119868119900) minus

119882(120585119877119904119890120585(119881+119868

119871119877119904+119868119900119877119904))

120585119877119904

(23)


119871= 119868119871119903


119882(120585119877119904119868119900119890120585(119881oc119903+119868119871119903119877119904+119868119900119877119904)

)

[1 +119882(120585119877119904119868119900119890120585(119881oc119903+119868119871119903119877119904+119868119900119877119904))] 119877

119904

minus 119877119904= 0 (24)








119904is


119877119904

= 18Re(119879119888119882(minus15 times 10

71198900022(47+(17times10

5

119879119888)))

119879119888119882(minus15times 10

71198900022(47+(17times10

5119879119888)))+4400

)

(25)





119875 = 119868119881 (26)


119875 = (119868119871+ 119868119900) minus

119882[120585119877119904119890120585(119881+119868

119871119877119904+119868119900119877119904)]

120585119877119904

119881 (27)


119889119875

119889119881

10038161003816100381610038161003816100381610038161003816119881max

= 0 (28)


119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)]119881

1 +119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)] 119877119904

+1

120585119877119904

times 119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)] minus 120585119877

119904(119868119871+ 119868119900) = 0

(29)


119875max = (119868119871+ 119868119900) minus

119882[120585119877119904119890120585(119881max+119868119871119877119904+119868119900119877119904)

]

120585119877119904

119881max (30)

0 5 10 15 20 25 30 35 40

25

2

15

1

05

0

Curr

ent (

A)

Voltage (V)

Vmax

Imax

Isc

Voc


Optimum powerpoint



119868max =119875max119881max

(31)





00

5 10

10

15 20

20

25 30

30

35 40

40

50

60

70

80

Voltage (V)


Vmax

Optimum power

Pow

er (W

)

Pmax


00

1

2

3

4

5

5 10 15 20 25 30 35 4540

Voltage (V)


Curr

ent (

A)

GT = 1000 (Wm2)

GT = 800 (Wm2)

GT = 600 (Wm2)

GT = 400 (Wm2)

GT = 200 (Wm2)

Optimum powerpoint




00

1

2

3

4

5

5 10 15 20 25 30 35 40

Voltage (V)


Curr

ent (

A)

Optimum powerpoint

Ta = 0∘C

Ta = 25∘C

Ta = 50∘C

Ta = 75∘C


00

1

2

3

4

5

5 10 15 20 25 30 35 40

Voltage (V)


Curr

ent (

A)

Optimum powerpoint

GT = 500Wm2

Ta = 20∘C

Ta = 60∘C

GT = 1000Wm2





0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power

GT = 1000 (Wm2)


180

160

140

120

100

80

60

40

20

0

Pow

er (W

)


0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power180

160

140

120

100

80

60

40

20

0

Pow

er (W

) Ta = 0∘C

Ta = 25∘C

Ta = 50∘C

Ta = 75∘C


0 5 10 15 20 25 30 35 40

Voltage (V)


Optimum power160

140

120

100

80

60

40

20

0

Pow

er (W

)

Ta = 60∘C

Ta = 20∘C

GT = 1000Wm2

GT = 500Wm2



4 Conclusions


119904) of single




Nomenclature










119888120582









The Subscript




References



































































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of








119904is


119877119904

= 18Re(119879119888119882(minus15 times 10

71198900022(47+(17times10

5

119879119888)))

119879119888119882(minus15times 10

71198900022(47+(17times10

5119879119888)))+4400

)

(25)





119875 = 119868119881 (26)


119875 = (119868119871+ 119868119900) minus

119882[120585119877119904119890120585(119881+119868

119871119877119904+119868119900119877119904)]

120585119877119904

119881 (27)


119889119875

119889119881

10038161003816100381610038161003816100381610038161003816119881max

= 0 (28)


119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)]119881

1 +119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)] 119877119904

+1

120585119877119904

times 119882[120585119877119904119868119900119890120585(119881+119868

119871119877119904+119868119900119877119904)] minus 120585119877

119904(119868119871+ 119868119900) = 0

(29)


119875max = (119868119871+ 119868119900) minus

119882[120585119877119904119890120585(119881max+119868119871119877119904+119868119900119877119904)

]

120585119877119904

119881max (30)

0 5 10 15 20 25 30 35 40

25

2

15

1

05

0

Curr

ent (

A)

Voltage (V)

Vmax

Imax

Isc

Voc


Optimum powerpoint



119868max =119875max119881max

(31)





00

5 10

10

15 20

20

25 30

30

35 40

40

50

60

70

80

Voltage (V)


Vmax

Optimum power

Pow

er (W

)

Pmax


00

1

2

3

4

5

5 10 15 20 25 30 35 4540

Voltage (V)


Curr

ent (

A)

GT = 1000 (Wm2)

GT = 800 (Wm2)

GT = 600 (Wm2)

GT = 400 (Wm2)

GT = 200 (Wm2)

Optimum powerpoint




00

1

2

3

4

5

5 10 15 20 25 30 35 40

Voltage (V)


Curr

ent (

A)

Optimum powerpoint

Ta = 0∘C

Ta = 25∘C

Ta = 50∘C

Ta = 75∘C


00

1

2

3

4

5

5 10 15 20 25 30 35 40

Voltage (V)


Curr

ent (

A)

Optimum powerpoint

GT = 500Wm2

Ta = 20∘C

Ta = 60∘C

GT = 1000Wm2





0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power

GT = 1000 (Wm2)


180

160

140

120

100

80

60

40

20

0

Pow

er (W

)


0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power180

160

140

120

100

80

60

40

20

0

Pow

er (W

) Ta = 0∘C

Ta = 25∘C

Ta = 50∘C

Ta = 75∘C


0 5 10 15 20 25 30 35 40

Voltage (V)


Optimum power160

140

120

100

80

60

40

20

0

Pow

er (W

)

Ta = 60∘C

Ta = 20∘C

GT = 1000Wm2

GT = 500Wm2



4 Conclusions


119904) of single




Nomenclature










119888120582









The Subscript




References



































































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


00

5 10

10

15 20

20

25 30

30

35 40

40

50

60

70

80

Voltage (V)


Vmax

Optimum power

Pow

er (W

)

Pmax


00

1

2

3

4

5

5 10 15 20 25 30 35 4540

Voltage (V)


Curr

ent (

A)

GT = 1000 (Wm2)

GT = 800 (Wm2)

GT = 600 (Wm2)

GT = 400 (Wm2)

GT = 200 (Wm2)

Optimum powerpoint




00

1

2

3

4

5

5 10 15 20 25 30 35 40

Voltage (V)


Curr

ent (

A)

Optimum powerpoint

Ta = 0∘C

Ta = 25∘C

Ta = 50∘C

Ta = 75∘C


00

1

2

3

4

5

5 10 15 20 25 30 35 40

Voltage (V)


Curr

ent (

A)

Optimum powerpoint

GT = 500Wm2

Ta = 20∘C

Ta = 60∘C

GT = 1000Wm2





0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power

GT = 1000 (Wm2)


180

160

140

120

100

80

60

40

20

0

Pow

er (W

)


0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power180

160

140

120

100

80

60

40

20

0

Pow

er (W

) Ta = 0∘C

Ta = 25∘C

Ta = 50∘C

Ta = 75∘C


0 5 10 15 20 25 30 35 40

Voltage (V)


Optimum power160

140

120

100

80

60

40

20

0

Pow

er (W

)

Ta = 60∘C

Ta = 20∘C

GT = 1000Wm2

GT = 500Wm2



4 Conclusions


119904) of single




Nomenclature










119888120582









The Subscript




References



































































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power

GT = 1000 (Wm2)


180

160

140

120

100

80

60

40

20

0

Pow

er (W

)


0 5 10 15 20 25 30 35 4540

Voltage (V)


Optimum power180

160

140

120

100

80

60

40

20

0

Pow

er (W

) Ta = 0∘C

Ta = 25∘C

Ta = 50∘C

Ta = 75∘C


0 5 10 15 20 25 30 35 40

Voltage (V)


Optimum power160

140

120

100

80

60

40

20

0

Pow

er (W

)

Ta = 60∘C

Ta = 20∘C

GT = 1000Wm2

GT = 500Wm2



4 Conclusions


119904) of single




Nomenclature










119888120582









The Subscript




References



































































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of







The Subscript




References



































































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of







































Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

research article an improved mathematical model for

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