research article comprehensive analysis and experimental validation of an improved mathematical...

Research ArticleComprehensive Analysis and Experimental Validation ofan Improved Mathematical Modeling of Photovoltaic Array

Satarupa Bal1 Anup Anurag2 Mrutyunjaya Nanda3 and Suman Sourav4

1Department of Electrical and Computer Engineering National University of Singapore 4 Engineering Drive 3 Singapore 1175832Department of Electrical Engineering and Information Technology ETH Zurich Ramistrasse 101 8092 Zurich Switzerland3Department of Electrical Engineering Asian Institute of Technology Pathumthani 12120 Thailand4School of Computing National University of Singapore 13 Computing Drive Singapore 117417

Correspondence should be addressed to Anup Anurag anuprana123gmailcom

Received 26 August 2014 Revised 30 November 2014 Accepted 1 December 2014

Academic Editor Jose Antenor Pomilio

Copyright copy 2015 Satarupa Bal et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a simple accurate and easy to model approach for the simulation of photovoltaic (PV) array and also providesa comparative analysis of the samewith two other widely usedmodels It is highly imperative that themaximumpower point (MPP)is achieved effectively and thus a simple and robustmathematicalmodel is necessary that poses lessmathematical complexity as wellas low data storage requirement in which themaximumpower point tracking (MPPT) algorithm can be realized in an effective wayFurther the resemblance of the P-V and I-V curves as obtained on the basis of experimental data should also be taken into accountfor theoretical validation In addition the study incorporates the root mean square deviation (RMSD) from the experimental datathe fill factor (FF) the efficiency of the model and the time required for simulation Two models have been used to investigate theI-V and P-V characteristics Perturb and Observe method has been adopted for MPPT The MPP tracking is realized using fieldprogrammable gate array (FPGA) to prove the effectiveness of the proposed approach All the systems are modeled and simulatedin MATLABSimulink environment

1 Introduction

Electrical energy from photovoltaic is currently regarded asthe prerequisite sustainable resource for both stand-alone aswell as grid connected applications since it is abundant andclean offers zero input fuel cost and is distributed through-out the earth [1] In practical cases photovoltaic modulesoperate over a highly intermittent nature of temperatureand irradiance but the electrical parameters provided in thedatasheet are only for the standard test conditions (STC)Moreover in power generation from PV optimal utilizationof the available solar energy is imperative due to the high costsof PV modules It is also seen that mathematical models offew individual components of PV system are represented andsimulated for better understanding of their performances [2]

This calls for a simple accurate and easy to modelapproach for the simulation of photovoltaic (PV) moduleto track the maximum power point and to predict PV

energy production under varying atmospheric conditions[3] In order to increase the accuracy the following can beincorporated but it leads to the increase in complexity of themodeling [4]

(i) temperature dependence of the diode saturation cur-rent

(ii) temperature dependence of the photo current(iii) inclusion of series resistance for more accurate shape

between the MPP and the open circuit (OC) voltage(iv) inclusion of shunt resistance in parallel with the

diode(v) variability of diode quality factor(vi) introduction of two or more parallel diodes

The accuracy of the simulation of a PV model largelydepends on the estimation of the characteristic I-V and P-V

Hindawi Publishing CorporationAdvances in Power ElectronicsVolume 2015 Article ID 654092 11 pageshttpdxdoiorg1011552015654092

2 Advances in Power Electronics

curves Furthermore factors such as efficiency field factorand simulation time affect the effectiveness of the model Asimplistic and easy to model approach is preferred so as toavoid unwanted complexity due to additional parameters

So far among the mathematical models of PV arrayproposed in the literature the simplest is the ideal singlediode model which involves only three parameters namelyshort circuit current open circuit voltage and the diodeideality factor [5] Improvement has been made with thesimplified single diode model (SSDM) being proposed in [6]which takes the effect of the series resistance (119877

119904) which is

the sum of several types of structural resistance of the deviceinto consideration [7ndash15] The influence of 119877

119904only becomes

dominant when the PV device operates in the voltage sourceregion Also it lacks the accuracy when subjected to largetemperature variations [16] Since the value of 119877

119904is very low

some authors neglect its effect [5 17ndash19] Further improve-ment has been done by the introduction of the single diodemodel (SDM) which includes the additional shunt resistance(119877119901) along with the series resistance [2] The shunt resistance

exists mainly due to the leakage current of the p-n junctionThe effect of 119877

119901is dominant when the PV device operates at

current source region of operation However since the valueof 119877119901is very high many authors [4 8 9 20 21] neglect it

in order to simplify the model Although it is much moreaccurate than the previous models it is not preferred onaccount of its computational complexity It is also reportedin [16] that the accuracy of this model deteriorates at lowirradiance levels In order to mitigate the inaccuracies offeredby the previous models the two-diode model was proposedin [22] However this leads to more model complexity andthusmore simulation time due to the involvement of a greaternumber of parameters A new mathematical PV model hasalso been proposed in [23] that includes the advantages ofprevious models combining the three main considerationsnamely simplicity ease of modeling and accuracy Howeverit doesnt take into consideration the effect of diode saturationcurrent on temperature which results in model errors at thevicinity of open-circuit voltage and consequently at otherregions

This paper proposes a new simple accurate and easyto model approach for the simulation of PV array andalso provides a comparative analysis of the same with theconventional single diode model and the improved idealsingle diode model As the PV systems are generally inte-grated with specific control algorithms in order to extractthe maximum possible power it is highly imperative that theMPP is achieved effectively and thus it is needed to designa model from which the MPPT algorithm can be realized inan effective way SomeMPPT techniques have been proposedin [1 3 4 10] However for simplicity this paper adopts thePerturb and Observe (PampO) method for MPPT

The proposed theoretical model is verified and validatedwith experimental data of commercial PV array RMSD fromthe experimental data maximum efficiency of the designthe fill factor (FF) and the simulation time has also beencalculated In addition theMPP tracking is realized in digitalenvironment using FPGA kit to prove the effectiveness of theproposed approach All the systems here are modeled and

simulated inMATLABSimulink environmentThe proposedmodeling method can be useful for users who require simplefast and accurate models in simulation of PV systems

2 Mathematical Models fora Photovoltaic Module

The major issue of real-time identification is basically theselection of a proper model It is therefore necessary to have aproper mathematical model that can represent accurately thecurrent-voltage characteristics of the PV array and which canbe solved by analytical methods in a simplified manner [24]In addition to this to maximize the power extracted from aPV array with the help of MPPT control the understandingand modeling of PV cell are also important [25]

Assuming the semiconductor diode equation and theKirchhoff laws the 119868-119881 characteristics for a PVmodule com-posed of series connected cells based on single exponentialmodel are expressed as follows [26]

119868119897= 119868119901V minus 1198680

[exp(119902 (119881 + 119868

119897119877119904)

119873119904119860119896119879

minus 1)] minus (119881 + 119868119871119877119904119873119904

119873119878119877119901

)

(1)

where 119896 is the Boltzmann constant (13806 times 10minus23 JK) and

119902 the electron charge (160217 times 10minus19 C) 119879 gives the module

temperature The parameter 119868119901V gives the photocurrent 119868

0

represents the diode saturation current and 119868119897gives the

output current 119877119904and 119877

119901give the series resistance and the

shunt resistance 119860 and 119873119904represent the diode ideality factor

and the number of cells connected in series respectivelyThe first term 119868

119901V gives the photocurrent and the secondpart is the ideal dark current that models the emitter andbase recombination All the parameters are mostly calculatedthrough sets of nonlinear equations [27]

21 Single Diode Model (SSDM) The single diode modeltakes into account both the series resistance as well as theshunt resistance unlike the ideal single diode model or thesimplified single diode model as shown in Figure 1 Thisresistance is the sum of several types of structural resistanceof the device 119877

119904depends mainly on four factors namely [2]

(i) contact resistance of the metal base with p-layer(ii) resistance of p-layer and n-layer(iii) contact resistance of the metal grid with the n-layer(iv) resistance of the grid

The shunt resistance exists mainly due to the leakagecurrent of the p-n junction It depends basically on thefabrication method of the PV cell

Mathematically the current equation can be written inaccordance with Kirchhoff rsquos current law

119868119901V = 119868

119889+ 119868119901

+ 119868119897 (2)

where 119868119901V is the photocurrent generated due to the incident

light and 119868119889is the diode current (Shockley diode equation)

Advances in Power Electronics 3

Ip Id IpIl

Rp

Rs

D

Solar cell Load

Figure 1 PV cell modeled as single diode model circuit

119868119897gives the output current and there is an additional term 119868

119901

which represents the leakage current of the p-n junction Inthis model the diode is given by

119868119889

= 1198680

[exp(119902 (119881 + 119868

119897119877119904)

119873119904119860119896119879

minus 1)] (3)

where the module saturation current (1198680) is given by

1198680

(119879) =119868scr (119879ref) + 119870

119894Δ119879

exp [119902 (119881oc (119879ref) + 119870VΔ119879) 119860119896119879] (4)

where 119868scr is the PV module short circuit current (in A) andcan be found in the product datasheet 119870

119894is the short circuit

current temperature coefficient (in AK) Thus from (3) and(4) the diode current can be calculated Now 119868

119901V can becalculated by

119868119901V = 119866 times [119868scr + 119870

119894Δ119879] (5)

where 119866 is the PVmodule incident illumination (in kWm2)An additional current 119868

119901is introduced here

119868119901

=119881 + 119873

119904119868119897119877119904

119873119904119877119901

(6)

Therefore there are three basic unknown parameters 119860 119877119904

and 119877119901 Here the value of 119860 is assumed (11 lt 119860 lt 16) The

value of the resistance can be calculated from the informationavailable in the datasheet

For the calculation of the series resistance and the shuntresistance an iterative method is employed Some authorsvary the resistance independent of each other leading to pooraccuracy of results Here a concept of simultaneously varyingboth the values is shown The idea is taken from the fact thatthe maximum calculated power should be equal to the powermentioned in the datasheet From (2) (3) and (6) the currentequation can be written and the equation at MPP gives thevalue of 119877

119901to be

119877119901

=

119881mpp (119881mpp + 119873119904119868mpp119877

119904)

119873119904(119881mpp (119868

119901V minus 119868119889mpp) minus 119875max)

(7)

where

119868119889mpp = 119868

0[exp(

119902 (119881mpp + 119868mpp119877119904)

119873119904119860119896119879

)] (8)

Solar cell Load

IpId

Il

D

Figure 2 PV cell modeled as improved ideal single diode modelcircuit

The value of 1198680can be calculated as from (4) and 119868

119901V canbe calculated from (5) However some simplifications have tobe made in order to find the unknown parameters An initialguess is taken as

119877119904initial = 0

119877119901initial =

119881mpp

119868scr (119879ref) minus 119868mppminus

119881oc (119879ref) minus 119881mpp

119868mpp

(9)

Using an iterative procedure by increasing the value ofthe series resistance and simultaneously updating the shuntresistance so as to match the maximum power the outputcurrent equation can be found out The major drawback ofthis model lies on its assumption of ideality factor Also thecomputational complexity and number of iterations requiredto obtain the output current equation are considerable

22 Improved Ideal Single Diode Model The improved idealsingle diodemodel is basically based on the ideal single diodemodel as shown in Figure 2 However the modeling involvesa set of mathematical equations which produces sufficientlyaccurate results but with much reduced complexity

The series and shunt resistances are neglected for math-ematical simplicity However the method of deriving theparameters is of reduced complexity The computation ofthese equations avoids the use of a nonlinear solver [28]

The current relation can be found by applying KCL

119868119901V = 119868

119889+ 119868119897 (10)

Here the current through the diode is given by

119868119889

= 1198680

[exp (119902119881

119860119896119879) minus 1] (11)

The derivation of the saturation current 1198680begins by

119881oc (119866 119879) minus 119881oc (119866 119879ref) = minus1003816100381610038161003816120573

1003816100381610038161003816 Δ119879 (12)

where 119881oc(119866 119879) and 119881oc(119866 119879ref) represent the open circuitvoltages at a temperature 119879 and at the reference temperature119879ref 120573 gives the voltage temperature coefficient and that canbe found from the product datasheetThe open circuit voltagecan be found out by putting 119868

119897= 0 in (10) and equating the

value of 119868119889as in (11)

119881oc =119860119896119879

119902ln(

119868119901V

1198680

+ 1) (13)


Solar cell Load

Ip IdIl

D

Rs

Figure 3 PV cell modeled as improved simplified single diodemodel circuit

Now finding the values of 119881oc at both the temperatures using(13) and replacing in (12) the following is obtained

119896119860

119902[119879 ln(

119866 (119868scr + 119870119894Δ119879)

1198680

+ 1)

minus119879ref ln(119866119868scr

1198680

(119879ref)+ 1) ] = minus

10038161003816100381610038161205731003816100381610038161003816 Δ119879

(14)

Rearranging (14) the following is found

1198680

=exp (119902

10038161003816100381610038161205731003816100381610038161003816 Δ119879119860119896119879) times 119866 times (119868scr + 119870

119894Δ119879)

(119866119868scr1198680

(119879ref + 1)119879ref119879

) minus exp (1199021003816100381610038161003816120573

1003816100381610038161003816 Δ119879119860119896119879)

(15)

The 1198680(119879ref) can be calculated according to (4) Now writing

(11) at MPP

119868mpp = 119868119901V minus 1198680

[exp(

119902119881mpp

119860119896119879) minus 1] (16)

and here according to (5) at reference temperature

119868119901V = 119868scr (17)

The ideality factor A can be derived by substituting (15) and(17) into (16)

119868mpp

119868scr= exp(

119902119881mpp

119860119896119879ref) minus (

119868scr minus 119868mpp

119868scr) exp(

119902119881oc119860119896119879ref

) (18)

Thus the unknown parameters can be found out from theabove equations and from the product datasheet information

However the temperature dependence of the saturationcurrent has not been considered Also there is a slightdeviation of the 119868-119881 characteristics from the experimentalcurve due to the approximation 119877

119904= 0

23 Improved PV Modeling Approach In the proposedimproved PV modeling approach in order to avoid thecomplexity offered by the single diode model it relies onthe simplified single diode model by neglecting the shuntresistance as shown in Figure 3 Applying KCL to the abovecircuit the current relation is found to be the same as (10)

The diode current equation incorporates the additionalvoltage drop across the series resistance

119868119889

= 1198680

[exp(119902 (119881 + 119868

119897119877119904)

119860119896119879) minus 1] (19)

119868119901V is calculated according to (5)The current equation for thesimplified single diode model is given by

119868119897= 119868119901V minus 1198680

[exp(119902 (119881 + 119868

119897119877119904)

119860119896119879) minus 1] (20)

Now there are basically four unknown parameters 119868119901V 1198680 119860

and 119877119904 119868119901V can be determined from the information available

in the manufacturer s datasheet by applying (5)The value of the series resistance can be calculated from

the information available in the product datasheet Since thecurrent at maximum power point at reference temperatureis available in the product datasheet (20) is solved formaximum power point conditions

119868mpp = 119868119901V minus 1198680

[exp(

119902 (119881mpp + 119868mpp119877119904)

119860119896119879) minus 1] (21)

Thus the series resistance can be found from

119877119904

=

(119860119896119879119902) times ln ((119868119901V minus 119868mpp) 119868

0+ 1) minus 119881mpp

119868mpp (22)

In order to find the absolute value of all these parameters theideality factor and the saturation current should be calculatedeffectively In the earlier works the value of the ideality factorhas been assumed which leads to the degradation of thecurve This model has the novelty of calculating the valueof the ideality factor from the datasheet parameters so thatthe simulated curves coincide with the experimental data toa larger extent Also an expression has been provided forfinding the saturation current from the information given inthe datasheet

For the calculation of the ideality factor advantage hasbeen taken of the fact that the derivative of powerwith respectto voltage at MPP is zero

Differentiating (20) with respect to voltage gives

119889119868119897

119889119881= minus1198680

[exp (119902V

119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879) times

119889119868119897

119889119881

+ exp (119902119868119897119877119904

119860119896119879) times

119902

119860119896119879times exp(

119902119881

119860119896119879) ]

(23)

Now it is known at MPP the derivative power with respect tovoltage is zero And hence the derivative of load current withrespect to voltage is given by

119889119868119897

119889119881

10038161003816100381610038161003816100381610038161003816MPP= minus

119868mpp

119881mpp (24)

So at MPP

minus

119868mpp

119881mpp= minus1198680

[exp (119902119881

119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879) times minus

119868mpp

119881mpp

+ exp (119902119868119897119877119904

119860119896119879) times

119902

119860119896119879times exp(

119902119881

119860119896119879)]

(25)


0 2 4 6 8 10 12 14 16 18 20 220

05

1

15Cu

rren

t (A

)

T = 348KT = 308KT = 328K

Voltage (V)

G = 1000Wm2

(a)

Curr

ent (

A)

0 2 4 6 8 10 12 14 16 18 20 220

05

1

15T = 308K

Voltage (V)

G = 1000Wm2

G = 800Wm2

G = 600Wm2

G = 400Wm2

(b)

Figure 4 (a) 119868-119881 model curves with experimental data for TBP-1237 solar array at different temperatures and (b) 119868-119881 model curves withexperimental data for TBP-1237 solar array at different irradiations

T = 308K T = 348K

0 2 4 6 8 10 12 14 16 18 20 220

15

3

45

Curr

ent (

A)

Voltage (V)

G = 1000Wm2

(a)

0 2 4 6 8 10 12 14 16 18 20 220

05

1

15

Voltage (V)

T = 308K

Curr

ent (

A) G = 1000Wm2

G = 800Wm2

G = 600Wm2

(b)

Figure 5 (a) 119868-119881 model curves with experimental data for MSX60 solar array at different temperatures and (b) 119868-119881 model curves withexperimental data for MSX60 solar array at different irradiations

Thus the value of the saturation current equals

1198680

(119879)

= 119868mpp

times (119881mpptimes [exp(119902119881

119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879) times minus

119868mpp

119881mpp

+ exp (119902119868119897119877119904

119860119896119879) times

119902

119860119896119879times exp(

119902119881

119860119896119879)])

minus1

(26)

The reverse saturation current is given by (4) Equating (4)and (26) at reference temperature

119868scr (119879ref + 119870119894Δ119879)

exp [ 119902 (119881oc (119879ref) + 119870VΔ119879) 119860119896119879]

= 119868mpp times (119881mpp times [exp (119902119881

119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879)

times minus

119868mpp

119881mpp+ exp (

119902119868119897119877119904

119860119896119879)

times119902

119860119896119879times exp(

119902119881

119860119896119879) ])

minus1

(27)

119877119904119860 can be obtained from (22) and is replaced in (27) Now

the equation has only one unknown parameter 119860 Also thesaturation current can be calculated from (26)

This modification aims at smoothing the curve betweenthe MPP and the open circuit point and also to matchthe open circuit voltages for a large range of temperaturevariations

3 Validating the Model

The PV array model is simulated in the MATLAB Simulinkenvironment in order to validate the claims Tables 1 and 2give a comparative view on the parameters on the datasheetand the parameters obtained from the proposed simulationmodel It is seen that the three main points coincide withthat of the datasheet The characteristic curves developedfrom themodel have been plotted with the experimental dataat four different temperature conditions and three differentirradiation values This has been done so as to effectivelyverify the notion that the proposed model gives acceptableresults even if the conditions are considerably far from STCas shown in Figures 4 and 5

Figure 4 gives the 119868-119881 curves for TBP-1237 at differenttemperatures and different irradiations respectively

However in order to test the validity efficiently it isrequired to test the model with other solar panels Figures


Table 1 Parameters of TBP-1237 solar module at STC (Temp 25∘CAM 15 119866 = 1000WM2)

Parameter Label Value from Value fromdatasheet model

Maximum power 119875max 20W 20WOpen circuit voltage 119881oc 2091 V 2091 VShort circuit current 119868scr 13 A 13 AVoltage at MPP 119881mpp 171 V 171 VCurrent at MPP 119868mpp 117 A 117 A

Table 2 Parameters of MSX60 solar module at STC (Temp 25∘CAM 15 119866 = 1000WM2)



5(a) and 5(b) give the 119868-119881 curves for MSX60 at differenttemperatures and different irradiations respectively

31 Effect of Variation in Temperature and Irradiation It canbe seen that the values of the parameters found can be usedfor irradiation and temperature close to the STCWith a largechange in the temperature or irradiation a smallmodificationof the algorithm can be put forward to account for the changein the model parameters

Although there is an increase of the photo current withthe increase in temperature increase owing to the slightdecrease of the band gap energy 119864

119892 the main output

characteristics such as efficiency are negatively influencedby high temperature The series resistance increases withincrease in temperature However a decrease is observedwith the increase in irradiation [28] On the other hand thejunction characteristic parameters such as ideality factor andsaturation current are highly temperature dependent Thevariation of ideality factor is almost linear with tempera-ture It increases with increase in temperature [29] This isexplained from the fact that at the increased temperaturesimperfections of basic material are more pronounced andthus create defects in the lattice structure

The short-circuit current from a solar cell depends lin-early on light intensity which results in the increase in PVoutput power as the solar radiation increases The seriesresistance is basically derived from the slope of the 119868-119881 curvebetween the MPP and the open-circuit point It is hencetreated as fitting parameter rather than fixed panel s effectiveseries resistance

The slope is strongly dependent on the panel short-circuitcurrent andmaximumpower point current and therefore alsoon the irradiation Hence it tends to increase with increase inlight intensity

It has to be pointed out that the variation of theparameters with respect to temperature can be accounted forthe calculation of 119868

119901V directly from (5) The dependence ofresistance with respect to temperature has been shown in(22) However the change in the series resistance and theideality factor of the modeling method can be accounted forThe series resistance can also be given by [30]

119877119904

= 2 (119881oc119868sc

minusAr1198682sc

minus119860119881119905

119868sc) (28)

The values of 119881oc and 119868sc can be determined at any giventemperature and irradiation according to [31] Ar gives thearea under the curve Now an iterative procedure is doneso as to find the exact values for the ideality factor andthe series resistance This can be described as Figure 6 ldquo119890rdquocan be defined as any small value close to zero The idealityfactor shows a linear behavior and thus it can be calculatedmanually by the linear nature as in [29] for the practicalworking temperatures

119860

119860STC=

119879

119879STC (29)

where 119860STC and 119879STC are the ideality factor and the tempera-ture at STC This model thus proves to be an efficient modelas compared to the previous approaches even though itneglects the shunt resistance on account of the fact that all theparameters are found from the datasheet parameters insteadof taking any assumptions into account The improved idealsingle diode model also finds the parameters but it neglectsthe series resistance which leads to increased deviation fromthe experimental values Since none of the parameters areassumed this approach provides a better model than itspredecessors which rely on intelligent assumptions so as tofit the curve

4 Comparative Results and Analysis

With the purpose of comparison between the three differentapproaches of mathematical modeling of PV array andvalidating them experimentally a small-scale module of hasbeen consideredThe accuracy of themodel is experimentallyvalidated using TBP-1237 and MSX60 The proposed PVcircuit model is implemented using a current controlledsource and simple computational blocks The inclusion ofseries resistance and ideality factor as unknown parametersaids in the smoothness of the output characteristics betweenMPP and open-circuit voltage and thus the curve coincidesclosely with the experimental data obtained not only at thethree main points but throughout the region Table 3 givesthe components used in the prototype For experimentalpurpose three arrays of bulbs with three bulbs of 200W ineach row are used as artificial sun The conventional DC-DC boost converter with a control tracker is implemented forMPP tracking

41 Analysis on Basis of 119875-119881 and 119868-119881 Curves The 119868-119881 and119875-119881 output curves for different mathematical models are


Start

Find all parameters

at STC

Plot curve for another temperature and

irradiation using same values

Redraw curveusing new values

No

StopYes

Find areaunder I-V curve

Use (28) and (29)to find Rs

If Rsnew minus Rsold gt e

Figure 6 Flowchart for determining the fianl values of the parameters at any temperature and irradiation

(a) (b)

Figure 7 (a) Bulbs used as artificial sun for PV panel and (b) prototype of the conventional boost converter used

Table 3 Components used in the study

Parameter Label ValueequipmentInductance 119871 17mHOutput capacitor 119862out 300 120583FLoad 119866119863 IR2213A to D converter ADC THS1030Current sensor CS LEM LA-55P

compared with the experimental results for determining theaccuracy of themodels and their closeness to the actual curveFor the single diode model the graph obtained coincides

with the experimental result approximately at 119868sc and 119881oc butthe MPP deviates from the one obtained from experimentalresult For the improved ideal single diode model thesimulation result at MPP points exhibit less deviation withimprovement in 119881oc point as compared to the former It takesadvantage of the simplicity of ideal models and enhances theaccuracy by deriving a mathematical representation capableof extracting accurate estimates of the model parametersdirectly related to manufacturer datasheets The inclusionof the calculation of ideality factor affects the curvatureof the 119868-119881 curve and thus expresses the ideality of thediode The mathematical derivation of the same improvesthe model fitting However owing to the assumption of noseries or shunt resistance the model shows some deviation


M

2

Power

Voltage

Current

(a)

Maximum power points

0 2 4 6 8 10 12 14 16 18 20 2202

Pow

er (W

)

10864

1214161820

Voltage (V)

G = 900Wm2

G = 800Wm2

G = 600Wm2

(b)

R2

R1

AB

CBBBBB

ABC

G = 600Wm2

G = 800Wm2

G = 900Wm2

(c)

Figure 8 (a) Experimental waveforms of power voltage and current for TBP-1237module (119884-Axis-Vdiv10 VdivAdiv1 AdivWdiv10Wdiv) (b) simulation PV waveforms and (c) experimental power points fromMPP tracking using P and O method

from the actual curve between the MPP and the open circuitpoint Extracting the advantage of the above two models animproved simplified single diode model has been proposedhere where the result closely matches not only at the threepoints the open circuit point the MPP and the short circuitpoint but also throughout the curve

42 Analysis on Basis of Simulation Time The simulationtime taken by the models has been shown in Table 4 Moresimulation time indicates a higher complexity of the modelAs expected the single diode model takes the highest timeon account of the computational complexity offered by it Itis then followed by the proposed model and the lowest timeis taken by the improved ideal single diode model Howeverthe time difference offered by the latter ones is very less andhence no significant time is elapsed during simulation

43 Analysis of Basis of Fill Factor Another factor whichdetermines the quality of a model is the fill factor This factorgives the ratio of the power produced at the maximum powerpoint to the maximum theoretical power that should bepossible to extract from the module For good cells the valueshould generally be more than 07 It is known that the higher

the fill factor the better the model Table 4 shows the fillfactors for various models Much difference is not observedbetween the fill factors However it is seen that the proposedmodel gives the highest fill factor It is followed by the singlediode model and the least fill factor is seen in the improvedideal single diode model It is due to its deviation in the curvefrom the actual experimental data Although the curve at theMPP deviates towards the lower side in case of single diodemodel it shows a higher fill factor on account of its greaterdeviation towards the higher side at the other regions

44 Analysis on Basis ofMaximumEfficiency Thesubsequentparameter used for the analysis is the maximum efficiencywhich is the ratio between the maximum power and theincident power At an irradiation of 1000Wm2 and a temper-ature of 35∘C the maximum efficiency offered by the modelsin is illustrated in Table 4

45 Analysis on Basis of RMSD The accuracy of a particularmodel can be known from the RMSD it shows Higher theRMSD value less accurate the model Table 4 shows theRMSD comparison between the various PV models Theleast RMSD is shown by the proposed model since it takes


0005001

0015002

0025003

0035004

Abso

lute

erro

r (A

)

Proposed modelImproved ideal single diode modelSingle diode model

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

(a)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(b)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(c)

Figure 9 Comparison of absolute errors among three different approaches for TBP-1237 module at (a) STC (Temp 25∘C AM 15 119866 =

1000Wm2) (b) Temp 35∘C AM 15 119866 = Wm2 and (c) Temp 35∘C AM 15 119866 = 1000Wm2

into account all the parameters The improved ideal singlediode model also shows a low RMSD value even if someof the parameters are ignored since it is not based on anyassumption in calculation The single diode model shows abetter RMSD than the improved ideal single diode modelHowever it is not true in all cases In some cases of varyingtemperature and irradiation the improved ideal single diodemodel proves to be a better model than the single diodemodel The absolute errors for these models are comparedin Figure 9(a) for STC It is seen that the proposed modelprovides better results when compared with the results foundby different approaches for the same module

In order to verify the notion that the proposed modelgives better accuracy even far from STC comparisons havebeen made based on the absolute values of error as shown inFigures 9(b) and 9(c)

46 Experimental Validation for MPP Tracking The pro-posed method being an improvement over the conventionalmethod and exhibiting closeness to the real cell at MPP isvalidated by comparingwith the experimental result Figure 7depicts the prototype used with PV panel as the input tothe converter and battery of 45W as its output The currentand voltage sensed by the sensors are used to generate dutyratio in the FPGA environment With battery as the outputof the converter the input is regulated by the PampO control

algorithm where irrespective of the alteration in irradiationMPP point for each 119875-119881 curve is tracked efficiently Figure 8shows the experimentalwaveformof theMPPpoints for threedifferent irradiation levels which perfectly matches with theone found from the modeling

5 Conclusions

This paper analyses the development of an effective approachto modeling of PV modules used for simulation purposesThe proposed modeling fits the mathematical characteristicmodel equations to the experimental curve In addition toit it avoids complexities and achieves better accuracy by theinclusion of the series resistance The dependency of diodesaturation current on the temperature as well as on the MPPhelps in making the equation effective and allows the userto adjust the output curve at the open-circuit voltage shortcircuit current point and the MPP at different temperaturesand irradiations Also the inclusion of the calculation of theideality factor from the information available in the datasheetimproves the accuracy of the curve and fits the curve with theexperimental one The results are experimentally validatedto prove the effectiveness of the proposed method ThePerturb and Observe MPPT algorithm implemented alongwith the boost converter at different irradiation substantiatethe closeness of the MPP points of the proposed model to


Table 4 Parameters used for comparative analysis

Model MPP points RMSD FF Max eff (in ) Simulation time

Single diode Model 17092V1172A 00561 07256 157366 214 s

Improved ideal single diode model 17087V1169A 00587 07244 156881 086 s

Proposed model 171 V117 A 00531 07263 157862 089 s

the real time observed points Along with it the comparisonof the proposed model with the conventional along with theimproved single diode model in terms of the fundamentalquantities such as fill factor RMSD and maximum efficiencygives a better insight into the advantage of the proposedmodel to the previous ones

The proposed mathematical model is easy to implementin various simulation platforms for PV power system studiesand it avoids the underlying complexities involved in PVparameter identificationThis paper is intended to be a usefultool serving the cause of both beginners as well as seasonedusers

Conflict of Interests

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

References

[1] M A G de Brito L P Sampaio L G Junior and C A CanesinldquoEvaluation of MPPT techniques for photovoltaic applicationsrdquoin Proceedings of the IEEE International Symposium on Indus-trial Electronics (ISIE rsquo11) pp 1039ndash1044 Gdansk Poland June2011

[2] M G Villalva J R Gazoli and E R Filho ldquoComprehensiveapproach to modeling and simulation of photovoltaic arraysrdquoIEEE Transactions on Power Electronics vol 24 no 5 pp 1198ndash1208 2009

[3] M Sokolov T C Green P D Mitcheson and D ShmilovitzldquoDynamic analysis of photovoltaic system with MPP locusemulationrdquo in Proceedings of the IEEE 26th Convention ofElectrical and Electronics Engineers in Israel (IEEEI rsquo10) pp 212ndash215 Eliat Israel November 2010

[4] GWalker ldquoEvaluatingMPPT converter topologies using amat-lab PV modelrdquo Journal of Electrical amp Electronics EngineeringAustralia vol 21 no 1 pp 49ndash55 2001

[5] D S H Chan and J C H Phang ldquoAnalytical methods forthe extraction of solar-cell single- and double-diode modelparameters from I-V characteristicsrdquo IEEE Transactions onElectron Devices vol 34 no 2 pp 286ndash293 1987

[6] J A Gow and C D Manning ldquoDevelopment of a photovoltaicarray model for use in power-electronics simulation studiesrdquoIEE Proceedings Electric Power Applications vol 146 no 2 pp193ndash200 1999

[7] G E Ahmad H M S Hussein and H H El-Ghetany ldquoThe-oretical analysis and experimental verification of PV modulesrdquoRenewable Energy vol 28 no 8 pp 1159ndash1168 2003

[8] M Veerachary ldquoPSIM circuit-oriented simulator model forthe nonlinear photovoltaic sourcesrdquo IEEE Transactions onAerospace and Electronic Systems vol 42 no 2 pp 735ndash7402006

[9] A N Celik and N Acikgoz ldquoModelling and experimentalverification of the operating current of mono-crystalline pho-tovoltaic modules using four- and five-parameter modelsrdquoApplied Energy vol 84 no 1 pp 1ndash15 2007

[10] 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

[11] M T Elhagry A A T Elkousy M Saleh T Elshatter and E MAbou-Elzahab ldquoFuzzy modeling of photovoltaic panel equiva-lent circuitrdquo in Proceedings of the 40th Midwest Symposium onCircuits and Systems vol 1 pp 60ndash63 August 1997

[12] S Liu and R A Dougal ldquoDynamicmultiphysicsmodel for solararrayrdquo IEEETransactions on Energy Conversion vol 17 no 2 pp285ndash294 2002

[13] D Sera R Teodorescu and P Rodriguez ldquoPV panel modelbased on datasheet valuesrdquo in Proceedings of the IEEE Interna-tional Symposium on Industrial Electronics (ISIE rsquo07) pp 2392ndash2396 Vigo Spain June 2007

[14] M A Vitorino L V Hartmann A M N Lima and M B RCorrea ldquoUsing the model of the solar cell for determining themaximum power point of photovoltaic systemsrdquo in Proceedingsof the European Conference on Power Electronics and Applica-tions (EPE rsquo07) pp 1ndash10 Aalborg Denmark September 2007

[15] H Patel and V Agarwal ldquoMATLAB-based modeling to studythe effects of partial shading on PV array characteristicsrdquo IEEETransactions on Energy Conversion vol 23 no 1 pp 302ndash3102008

[16] R Noroozian M Abedi G Gharehpetian and S HosseiniModelling and simulation of microturbine generation systemfor on-grid and offgrid operation modes

[17] S Bal A Anurag and B C Babu ldquoComparative analysisof mathematical modeling of Photo-Voltaic (PV) arrayrdquo inProceedings of the Annual IEEE India Conference (INDICONrsquo12) pp 269ndash274 Kochi India December 2012

[18] Y T Tan D S Kirschen and N Jenkins ldquoA model of PVgeneration suitable for stability analysisrdquo IEEE Transactions onEnergy Conversion vol 19 no 4 pp 748ndash755 2004

[19] A Kajihara and T Harakawa ldquoModel of photovoltaic cellcircuits under partial shadingrdquo in Proceedings of the IEEEInternational Conference on Industrial Technology (ICIT rsquo05) pp866ndash870 December 2005

[20] M Glass ldquoImproved solar array power point model withspice realizationrdquo in Proceedings of the 31st Intersociety EnergyConversion Engineering Conference (IECEC rsquo96) vol 1 pp 286ndash291 August 1996


[21] I H Altas and A M Sharaf ldquoA photovoltaic array simulationmodel formatlab-simulinkGUI environmentrdquo inProceedings ofthe International Conference on Clean Electrical Power (ICCEPrsquo07) pp 341ndash345 Capri Italy May 2007

[22] Z Salam K Ishaque and H Taheri ldquoAn improved two-diodephotovoltaic (PV) model for PV systemrdquo in Proceedings of theJoint International Conference on Power Electronics Drives andEnergy Systems amp Power India (PEDES rsquo10) pp 1ndash5 IEEE NewDelhi India December 2010

[23] YMahmoudW Xiao andH H Zeineldin ldquoA simple approachto modeling and simulation of photovoltaic modulesrdquo IEEETransactions on Sustainable Energy vol 3 no 1 pp 185ndash1862012

[24] D Bonkoungou Z Koalaga and D Njomo ldquoModelling andsimulation of photovoltaic module considering single-diodeequivalent circuit model in MATLABrdquo International Journal ofEmerging Technology andAdvanced Engineering vol 3 no 3 pp493ndash502 2013

[25] M Hatti A Meharrar and M Tioursi ldquoNovel approach ofmaximumpower point tracking for photovoltaicmodule neuralnetwork basedrdquo in Proceedings of the International Symposiumon Environment Friendly Energies in Electrical Applications pp1ndash6 Ghardaıa Algeria November 2010

[26] K Nishioka N Sakitani Y Uraoka and T Fuyuki ldquoAnalysis ofmulticrystalline silicon solar cells by modified 3-diode equiv-alent circuit model taking leakage current through peripheryinto considerationrdquo Solar Energy Materials and Solar Cells vol91 no 13 pp 1222ndash1227 2007


[28] S Chowdhury S P Chowdhury G Taylor and Y H SongldquoMathematical modelling and performance evaluation of astand-alone polycrystalline pv plant with mppt facilityrdquo inProceedings of the 2008 IEEE Power and Energy Society GeneralMeetingmdashConversion andDelivery of Electrical Energy in the 21stCentury pp 1ndash7 July 2008

[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] M Sabry and A E Ghitas ldquoInfluence of temperature onmethods for determining silicon solar cell series resistancerdquoJournal of Solar Energy Engineering vol 129 no 3 pp 331ndash3352007

[31] G H Yordanov O-M Midtgard and T O Saetre ldquoEquivalentcell temperature calculation for PV modules with variableideality factorsrdquo in Proceedings of the 38th IEEE PhotovoltaicSpecialists Conference (PVSC rsquo12) pp 505ndash508 Austin TexUSA June 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



curves Furthermore factors such as efficiency field factorand simulation time affect the effectiveness of the model Asimplistic and easy to model approach is preferred so as toavoid unwanted complexity due to additional parameters

So far among the mathematical models of PV arrayproposed in the literature the simplest is the ideal singlediode model which involves only three parameters namelyshort circuit current open circuit voltage and the diodeideality factor [5] Improvement has been made with thesimplified single diode model (SSDM) being proposed in [6]which takes the effect of the series resistance (119877

119904) which is

the sum of several types of structural resistance of the deviceinto consideration [7ndash15] The influence of 119877

119904only becomes

dominant when the PV device operates in the voltage sourceregion Also it lacks the accuracy when subjected to largetemperature variations [16] Since the value of 119877

119904is very low

some authors neglect its effect [5 17ndash19] Further improve-ment has been done by the introduction of the single diodemodel (SDM) which includes the additional shunt resistance(119877119901) along with the series resistance [2] The shunt resistance

exists mainly due to the leakage current of the p-n junctionThe effect of 119877

119901is dominant when the PV device operates at

current source region of operation However since the valueof 119877119901is very high many authors [4 8 9 20 21] neglect it

in order to simplify the model Although it is much moreaccurate than the previous models it is not preferred onaccount of its computational complexity It is also reportedin [16] that the accuracy of this model deteriorates at lowirradiance levels In order to mitigate the inaccuracies offeredby the previous models the two-diode model was proposedin [22] However this leads to more model complexity andthusmore simulation time due to the involvement of a greaternumber of parameters A new mathematical PV model hasalso been proposed in [23] that includes the advantages ofprevious models combining the three main considerationsnamely simplicity ease of modeling and accuracy Howeverit doesnt take into consideration the effect of diode saturationcurrent on temperature which results in model errors at thevicinity of open-circuit voltage and consequently at otherregions

This paper proposes a new simple accurate and easyto model approach for the simulation of PV array andalso provides a comparative analysis of the same with theconventional single diode model and the improved idealsingle diode model As the PV systems are generally inte-grated with specific control algorithms in order to extractthe maximum possible power it is highly imperative that theMPP is achieved effectively and thus it is needed to designa model from which the MPPT algorithm can be realized inan effective way SomeMPPT techniques have been proposedin [1 3 4 10] However for simplicity this paper adopts thePerturb and Observe (PampO) method for MPPT

The proposed theoretical model is verified and validatedwith experimental data of commercial PV array RMSD fromthe experimental data maximum efficiency of the designthe fill factor (FF) and the simulation time has also beencalculated In addition theMPP tracking is realized in digitalenvironment using FPGA kit to prove the effectiveness of theproposed approach All the systems here are modeled and

simulated inMATLABSimulink environmentThe proposedmodeling method can be useful for users who require simplefast and accurate models in simulation of PV systems

2 Mathematical Models fora Photovoltaic Module

The major issue of real-time identification is basically theselection of a proper model It is therefore necessary to have aproper mathematical model that can represent accurately thecurrent-voltage characteristics of the PV array and which canbe solved by analytical methods in a simplified manner [24]In addition to this to maximize the power extracted from aPV array with the help of MPPT control the understandingand modeling of PV cell are also important [25]

Assuming the semiconductor diode equation and theKirchhoff laws the 119868-119881 characteristics for a PVmodule com-posed of series connected cells based on single exponentialmodel are expressed as follows [26]

119868119897= 119868119901V minus 1198680

[exp(119902 (119881 + 119868

119897119877119904)

119873119904119860119896119879

minus 1)] minus (119881 + 119868119871119877119904119873119904

119873119878119877119901

)

(1)

where 119896 is the Boltzmann constant (13806 times 10minus23 JK) and

119902 the electron charge (160217 times 10minus19 C) 119879 gives the module

temperature The parameter 119868119901V gives the photocurrent 119868

0

represents the diode saturation current and 119868119897gives the

output current 119877119904and 119877

119901give the series resistance and the

shunt resistance 119860 and 119873119904represent the diode ideality factor

and the number of cells connected in series respectivelyThe first term 119868

119901V gives the photocurrent and the secondpart is the ideal dark current that models the emitter andbase recombination All the parameters are mostly calculatedthrough sets of nonlinear equations [27]

21 Single Diode Model (SSDM) The single diode modeltakes into account both the series resistance as well as theshunt resistance unlike the ideal single diode model or thesimplified single diode model as shown in Figure 1 Thisresistance is the sum of several types of structural resistanceof the device 119877

119904depends mainly on four factors namely [2]

(i) contact resistance of the metal base with p-layer(ii) resistance of p-layer and n-layer(iii) contact resistance of the metal grid with the n-layer(iv) resistance of the grid

The shunt resistance exists mainly due to the leakagecurrent of the p-n junction It depends basically on thefabrication method of the PV cell

Mathematically the current equation can be written inaccordance with Kirchhoff rsquos current law

119868119901V = 119868

119889+ 119868119901

+ 119868119897 (2)

where 119868119901V is the photocurrent generated due to the incident

light and 119868119889is the diode current (Shockley diode equation)


Ip Id IpIl

Rp

Rs

D

Solar cell Load



119901


119868119889

= 1198680

[exp(119902 (119881 + 119868

119897119877119904)

119873119904119860119896119879

minus 1)] (3)


1198680

(119879) =119868scr (119879ref) + 119870

119894Δ119879

exp [119902 (119881oc (119879ref) + 119870VΔ119879) 119860119896119879] (4)





119868119901V = 119866 times [119868scr + 119870

119894Δ119879] (5)



119868119901

=119881 + 119873

119904119868119897119877119904

119873119904119877119901

(6)





119901to be

119877119901

=

119881mpp (119881mpp + 119873119904119868mpp119877

119904)

119873119904(119881mpp (119868


(7)

where

119868119889mpp = 119868

0[exp(

119902 (119881mpp + 119868mpp119877119904)

119873119904119860119896119879

)] (8)

Solar cell Load

IpId

Il

D




119877119904initial = 0

119877119901initial =

119881mpp


119881oc (119879ref) minus 119881mpp

119868mpp

(9)





119868119901V = 119868

119889+ 119868119897 (10)


119868119889

= 1198680

[exp (119902119881

119860119896119879) minus 1] (11)



1003816100381610038161003816 Δ119879 (12)



value of 119868119889as in (11)

119881oc =119860119896119879

119902ln(

119868119901V

1198680

+ 1) (13)


Solar cell Load

Ip IdIl

D

Rs



119896119860

119902[119879 ln(

119866 (119868scr + 119870119894Δ119879)

1198680

+ 1)

minus119879ref ln(119866119868scr

1198680

(119879ref)+ 1) ] = minus

10038161003816100381610038161205731003816100381610038161003816 Δ119879

(14)


1198680

=exp (119902

10038161003816100381610038161205731003816100381610038161003816 Δ119879119860119896119879) times 119866 times (119868scr + 119870

119894Δ119879)

(119866119868scr1198680

(119879ref + 1)119879ref119879

) minus exp (1199021003816100381610038161003816120573

1003816100381610038161003816 Δ119879119860119896119879)

(15)


(11) at MPP

119868mpp = 119868119901V minus 1198680

[exp(

119902119881mpp

119860119896119879) minus 1] (16)


119868119901V = 119868scr (17)


119868mpp

119868scr= exp(

119902119881mpp

119860119896119879ref) minus (


119868scr) exp(

119902119881oc119860119896119879ref

) (18)



119904= 0



119868119889

= 1198680

[exp(119902 (119881 + 119868

119897119877119904)

119860119896119879) minus 1] (19)


119868119897= 119868119901V minus 1198680

[exp(119902 (119881 + 119868

119897119877119904)

119860119896119879) minus 1] (20)





119868mpp = 119868119901V minus 1198680

[exp(

119902 (119881mpp + 119868mpp119877119904)

119860119896119879) minus 1] (21)


119877119904

=


0+ 1) minus 119881mpp

119868mpp (22)




119889119868119897

119889119881= minus1198680

[exp (119902V

119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879) times

119889119868119897

119889119881

+ exp (119902119868119897119877119904

119860119896119879) times

119902

119860119896119879times exp(

119902119881

119860119896119879) ]

(23)


119889119868119897

119889119881

10038161003816100381610038161003816100381610038161003816MPP= minus

119868mpp

119881mpp (24)

So at MPP

minus

119868mpp

119881mpp= minus1198680

[exp (119902119881

119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879) times minus

119868mpp

119881mpp

+ exp (119902119868119897119877119904

119860119896119879) times

119902

119860119896119879times exp(

119902119881

119860119896119879)]

(25)


0 2 4 6 8 10 12 14 16 18 20 220

05

1

15Cu

rren

t (A

)

T = 348KT = 308KT = 328K

Voltage (V)

G = 1000Wm2

(a)

Curr

ent (

A)

0 2 4 6 8 10 12 14 16 18 20 220

05

1

15T = 308K

Voltage (V)

G = 1000Wm2

G = 800Wm2

G = 600Wm2

G = 400Wm2

(b)


T = 308K T = 348K

0 2 4 6 8 10 12 14 16 18 20 220

15

3

45

Curr

ent (

A)

Voltage (V)

G = 1000Wm2

(a)

0 2 4 6 8 10 12 14 16 18 20 220

05

1

15

Voltage (V)

T = 308K

Curr

ent (

A) G = 1000Wm2

G = 800Wm2

G = 600Wm2

(b)



1198680

(119879)

= 119868mpp


119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879) times minus

119868mpp

119881mpp

+ exp (119902119868119897119877119904

119860119896119879) times

119902

119860119896119879times exp(

119902119881

119860119896119879)])

minus1

(26)


119868scr (119879ref + 119870119894Δ119879)

exp [ 119902 (119881oc (119879ref) + 119870VΔ119879) 119860119896119879]


119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879)

times minus

119868mpp

119881mpp+ exp (

119902119868119897119877119904

119860119896119879)

times119902

119860119896119879times exp(

119902119881

119860119896119879) ])

minus1

(27)
























119877119904

= 2 (119881oc119868sc

minusAr1198682sc

minus119860119881119905

119868sc) (28)


119860

119860STC=

119879

119879STC (29)






Start

Find all parameters

at STC




No

StopYes





(a) (b)







M

2

Power

Voltage

Current

(a)


0 2 4 6 8 10 12 14 16 18 20 2202

Pow

er (W

)

10864

1214161820

Voltage (V)

G = 900Wm2

G = 800Wm2

G = 600Wm2

(b)

R2

R1

AB

CBBBBB

ABC

G = 600Wm2

G = 800Wm2

G = 900Wm2

(c)









0005001

0015002

0025003

0035004

Abso

lute

erro

r (A

)


0 2 4 6 8 10 12 14 16 18 20Voltage (V)

(a)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(b)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(c)







5 Conclusions












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Ip Id IpIl

Rp

Rs

D

Solar cell Load



119901


119868119889

= 1198680

[exp(119902 (119881 + 119868

119897119877119904)

119873119904119860119896119879

minus 1)] (3)


1198680

(119879) =119868scr (119879ref) + 119870

119894Δ119879

exp [119902 (119881oc (119879ref) + 119870VΔ119879) 119860119896119879] (4)





119868119901V = 119866 times [119868scr + 119870

119894Δ119879] (5)



119868119901

=119881 + 119873

119904119868119897119877119904

119873119904119877119901

(6)





119901to be

119877119901

=

119881mpp (119881mpp + 119873119904119868mpp119877

119904)

119873119904(119881mpp (119868


(7)

where

119868119889mpp = 119868

0[exp(

119902 (119881mpp + 119868mpp119877119904)

119873119904119860119896119879

)] (8)

Solar cell Load

IpId

Il

D




119877119904initial = 0

119877119901initial =

119881mpp


119881oc (119879ref) minus 119881mpp

119868mpp

(9)





119868119901V = 119868

119889+ 119868119897 (10)


119868119889

= 1198680

[exp (119902119881

119860119896119879) minus 1] (11)



1003816100381610038161003816 Δ119879 (12)



value of 119868119889as in (11)

119881oc =119860119896119879

119902ln(

119868119901V

1198680

+ 1) (13)


Solar cell Load

Ip IdIl

D

Rs



119896119860

119902[119879 ln(

119866 (119868scr + 119870119894Δ119879)

1198680

+ 1)

minus119879ref ln(119866119868scr

1198680

(119879ref)+ 1) ] = minus

10038161003816100381610038161205731003816100381610038161003816 Δ119879

(14)


1198680

=exp (119902

10038161003816100381610038161205731003816100381610038161003816 Δ119879119860119896119879) times 119866 times (119868scr + 119870

119894Δ119879)

(119866119868scr1198680

(119879ref + 1)119879ref119879

) minus exp (1199021003816100381610038161003816120573

1003816100381610038161003816 Δ119879119860119896119879)

(15)


(11) at MPP

119868mpp = 119868119901V minus 1198680

[exp(

119902119881mpp

119860119896119879) minus 1] (16)


119868119901V = 119868scr (17)


119868mpp

119868scr= exp(

119902119881mpp

119860119896119879ref) minus (


119868scr) exp(

119902119881oc119860119896119879ref

) (18)



119904= 0



119868119889

= 1198680

[exp(119902 (119881 + 119868

119897119877119904)

119860119896119879) minus 1] (19)


119868119897= 119868119901V minus 1198680

[exp(119902 (119881 + 119868

119897119877119904)

119860119896119879) minus 1] (20)





119868mpp = 119868119901V minus 1198680

[exp(

119902 (119881mpp + 119868mpp119877119904)

119860119896119879) minus 1] (21)


119877119904

=


0+ 1) minus 119881mpp

119868mpp (22)




119889119868119897

119889119881= minus1198680

[exp (119902V

119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879) times

119889119868119897

119889119881

+ exp (119902119868119897119877119904

119860119896119879) times

119902

119860119896119879times exp(

119902119881

119860119896119879) ]

(23)


119889119868119897

119889119881

10038161003816100381610038161003816100381610038161003816MPP= minus

119868mpp

119881mpp (24)

So at MPP

minus

119868mpp

119881mpp= minus1198680

[exp (119902119881

119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879) times minus

119868mpp

119881mpp

+ exp (119902119868119897119877119904

119860119896119879) times

119902

119860119896119879times exp(

119902119881

119860119896119879)]

(25)


0 2 4 6 8 10 12 14 16 18 20 220

05

1

15Cu

rren

t (A

)

T = 348KT = 308KT = 328K

Voltage (V)

G = 1000Wm2

(a)

Curr

ent (

A)

0 2 4 6 8 10 12 14 16 18 20 220

05

1

15T = 308K

Voltage (V)

G = 1000Wm2

G = 800Wm2

G = 600Wm2

G = 400Wm2

(b)


T = 308K T = 348K

0 2 4 6 8 10 12 14 16 18 20 220

15

3

45

Curr

ent (

A)

Voltage (V)

G = 1000Wm2

(a)

0 2 4 6 8 10 12 14 16 18 20 220

05

1

15

Voltage (V)

T = 308K

Curr

ent (

A) G = 1000Wm2

G = 800Wm2

G = 600Wm2

(b)



1198680

(119879)

= 119868mpp


119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879) times minus

119868mpp

119881mpp

+ exp (119902119868119897119877119904

119860119896119879) times

119902

119860119896119879times exp(

119902119881

119860119896119879)])

minus1

(26)


119868scr (119879ref + 119870119894Δ119879)

exp [ 119902 (119881oc (119879ref) + 119870VΔ119879) 119860119896119879]


119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879)

times minus

119868mpp

119881mpp+ exp (

119902119868119897119877119904

119860119896119879)

times119902

119860119896119879times exp(

119902119881

119860119896119879) ])

minus1

(27)
























119877119904

= 2 (119881oc119868sc

minusAr1198682sc

minus119860119881119905

119868sc) (28)


119860

119860STC=

119879

119879STC (29)






Start

Find all parameters

at STC




No

StopYes





(a) (b)







M

2

Power

Voltage

Current

(a)


0 2 4 6 8 10 12 14 16 18 20 2202

Pow

er (W

)

10864

1214161820

Voltage (V)

G = 900Wm2

G = 800Wm2

G = 600Wm2

(b)

R2

R1

AB

CBBBBB

ABC

G = 600Wm2

G = 800Wm2

G = 900Wm2

(c)









0005001

0015002

0025003

0035004

Abso

lute

erro

r (A

)


0 2 4 6 8 10 12 14 16 18 20Voltage (V)

(a)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(b)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(c)







5 Conclusions












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Solar cell Load

Ip IdIl

D

Rs



119896119860

119902[119879 ln(

119866 (119868scr + 119870119894Δ119879)

1198680

+ 1)

minus119879ref ln(119866119868scr

1198680

(119879ref)+ 1) ] = minus

10038161003816100381610038161205731003816100381610038161003816 Δ119879

(14)


1198680

=exp (119902

10038161003816100381610038161205731003816100381610038161003816 Δ119879119860119896119879) times 119866 times (119868scr + 119870

119894Δ119879)

(119866119868scr1198680

(119879ref + 1)119879ref119879

) minus exp (1199021003816100381610038161003816120573

1003816100381610038161003816 Δ119879119860119896119879)

(15)


(11) at MPP

119868mpp = 119868119901V minus 1198680

[exp(

119902119881mpp

119860119896119879) minus 1] (16)


119868119901V = 119868scr (17)


119868mpp

119868scr= exp(

119902119881mpp

119860119896119879ref) minus (


119868scr) exp(

119902119881oc119860119896119879ref

) (18)



119904= 0



119868119889

= 1198680

[exp(119902 (119881 + 119868

119897119877119904)

119860119896119879) minus 1] (19)


119868119897= 119868119901V minus 1198680

[exp(119902 (119881 + 119868

119897119877119904)

119860119896119879) minus 1] (20)





119868mpp = 119868119901V minus 1198680

[exp(

119902 (119881mpp + 119868mpp119877119904)

119860119896119879) minus 1] (21)


119877119904

=


0+ 1) minus 119881mpp

119868mpp (22)




119889119868119897

119889119881= minus1198680

[exp (119902V

119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879) times

119889119868119897

119889119881

+ exp (119902119868119897119877119904

119860119896119879) times

119902

119860119896119879times exp(

119902119881

119860119896119879) ]

(23)


119889119868119897

119889119881

10038161003816100381610038161003816100381610038161003816MPP= minus

119868mpp

119881mpp (24)

So at MPP

minus

119868mpp

119881mpp= minus1198680

[exp (119902119881

119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879) times minus

119868mpp

119881mpp

+ exp (119902119868119897119877119904

119860119896119879) times

119902

119860119896119879times exp(

119902119881

119860119896119879)]

(25)


0 2 4 6 8 10 12 14 16 18 20 220

05

1

15Cu

rren

t (A

)

T = 348KT = 308KT = 328K

Voltage (V)

G = 1000Wm2

(a)

Curr

ent (

A)

0 2 4 6 8 10 12 14 16 18 20 220

05

1

15T = 308K

Voltage (V)

G = 1000Wm2

G = 800Wm2

G = 600Wm2

G = 400Wm2

(b)


T = 308K T = 348K

0 2 4 6 8 10 12 14 16 18 20 220

15

3

45

Curr

ent (

A)

Voltage (V)

G = 1000Wm2

(a)

0 2 4 6 8 10 12 14 16 18 20 220

05

1

15

Voltage (V)

T = 308K

Curr

ent (

A) G = 1000Wm2

G = 800Wm2

G = 600Wm2

(b)



1198680

(119879)

= 119868mpp


119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879) times minus

119868mpp

119881mpp

+ exp (119902119868119897119877119904

119860119896119879) times

119902

119860119896119879times exp(

119902119881

119860119896119879)])

minus1

(26)


119868scr (119879ref + 119870119894Δ119879)

exp [ 119902 (119881oc (119879ref) + 119870VΔ119879) 119860119896119879]


119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879)

times minus

119868mpp

119881mpp+ exp (

119902119868119897119877119904

119860119896119879)

times119902

119860119896119879times exp(

119902119881

119860119896119879) ])

minus1

(27)
























119877119904

= 2 (119881oc119868sc

minusAr1198682sc

minus119860119881119905

119868sc) (28)


119860

119860STC=

119879

119879STC (29)






Start

Find all parameters

at STC




No

StopYes





(a) (b)







M

2

Power

Voltage

Current

(a)


0 2 4 6 8 10 12 14 16 18 20 2202

Pow

er (W

)

10864

1214161820

Voltage (V)

G = 900Wm2

G = 800Wm2

G = 600Wm2

(b)

R2

R1

AB

CBBBBB

ABC

G = 600Wm2

G = 800Wm2

G = 900Wm2

(c)









0005001

0015002

0025003

0035004

Abso

lute

erro

r (A

)


0 2 4 6 8 10 12 14 16 18 20Voltage (V)

(a)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(b)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(c)







5 Conclusions












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 10 12 14 16 18 20 220

05

1

15Cu

rren

t (A

)

T = 348KT = 308KT = 328K

Voltage (V)

G = 1000Wm2

(a)

Curr

ent (

A)

0 2 4 6 8 10 12 14 16 18 20 220

05

1

15T = 308K

Voltage (V)

G = 1000Wm2

G = 800Wm2

G = 600Wm2

G = 400Wm2

(b)


T = 308K T = 348K

0 2 4 6 8 10 12 14 16 18 20 220

15

3

45

Curr

ent (

A)

Voltage (V)

G = 1000Wm2

(a)

0 2 4 6 8 10 12 14 16 18 20 220

05

1

15

Voltage (V)

T = 308K

Curr

ent (

A) G = 1000Wm2

G = 800Wm2

G = 600Wm2

(b)



1198680

(119879)

= 119868mpp


119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879) times minus

119868mpp

119881mpp

+ exp (119902119868119897119877119904

119860119896119879) times

119902

119860119896119879times exp(

119902119881

119860119896119879)])

minus1

(26)


119868scr (119879ref + 119870119894Δ119879)

exp [ 119902 (119881oc (119879ref) + 119870VΔ119879) 119860119896119879]


119860119896119879) times

119902119877119904

119860119896119879times exp(

119902119868119897119877119904

119860119896119879)

times minus

119868mpp

119881mpp+ exp (

119902119868119897119877119904

119860119896119879)

times119902

119860119896119879times exp(

119902119881

119860119896119879) ])

minus1

(27)
























119877119904

= 2 (119881oc119868sc

minusAr1198682sc

minus119860119881119905

119868sc) (28)


119860

119860STC=

119879

119879STC (29)






Start

Find all parameters

at STC




No

StopYes





(a) (b)







M

2

Power

Voltage

Current

(a)


0 2 4 6 8 10 12 14 16 18 20 2202

Pow

er (W

)

10864

1214161820

Voltage (V)

G = 900Wm2

G = 800Wm2

G = 600Wm2

(b)

R2

R1

AB

CBBBBB

ABC

G = 600Wm2

G = 800Wm2

G = 900Wm2

(c)









0005001

0015002

0025003

0035004

Abso

lute

erro

r (A

)


0 2 4 6 8 10 12 14 16 18 20Voltage (V)

(a)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(b)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(c)







5 Conclusions












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























119877119904

= 2 (119881oc119868sc

minusAr1198682sc

minus119860119881119905

119868sc) (28)


119860

119860STC=

119879

119879STC (29)






Start

Find all parameters

at STC




No

StopYes





(a) (b)







M

2

Power

Voltage

Current

(a)


0 2 4 6 8 10 12 14 16 18 20 2202

Pow

er (W

)

10864

1214161820

Voltage (V)

G = 900Wm2

G = 800Wm2

G = 600Wm2

(b)

R2

R1

AB

CBBBBB

ABC

G = 600Wm2

G = 800Wm2

G = 900Wm2

(c)









0005001

0015002

0025003

0035004

Abso

lute

erro

r (A

)


0 2 4 6 8 10 12 14 16 18 20Voltage (V)

(a)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(b)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(c)







5 Conclusions












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Start

Find all parameters

at STC




No

StopYes





(a) (b)







M

2

Power

Voltage

Current

(a)


0 2 4 6 8 10 12 14 16 18 20 2202

Pow

er (W

)

10864

1214161820

Voltage (V)

G = 900Wm2

G = 800Wm2

G = 600Wm2

(b)

R2

R1

AB

CBBBBB

ABC

G = 600Wm2

G = 800Wm2

G = 900Wm2

(c)









0005001

0015002

0025003

0035004

Abso

lute

erro

r (A

)


0 2 4 6 8 10 12 14 16 18 20Voltage (V)

(a)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(b)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(c)







5 Conclusions












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










M

2

Power

Voltage

Current

(a)


0 2 4 6 8 10 12 14 16 18 20 2202

Pow

er (W

)

10864

1214161820

Voltage (V)

G = 900Wm2

G = 800Wm2

G = 600Wm2

(b)

R2

R1

AB

CBBBBB

ABC

G = 600Wm2

G = 800Wm2

G = 900Wm2

(c)









0005001

0015002

0025003

0035004

Abso

lute

erro

r (A

)


0 2 4 6 8 10 12 14 16 18 20Voltage (V)

(a)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(b)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(c)







5 Conclusions












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0005001

0015002

0025003

0035004

Abso

lute

erro

r (A

)


0 2 4 6 8 10 12 14 16 18 20Voltage (V)

(a)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(b)

0005001

0015002

0025003

0035004

0 2 4 6 8 10 12 14 16 18 20Voltage (V)

Abso

lute

erro

r (A

)


(c)







5 Conclusions












References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article comprehensive analysis and experimental validation of an improved mathematical...

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