research article parameter estimation of photovoltaic...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 362619 8 pageshttpdxdoiorg1011552013362619

Research ArticleParameter Estimation of Photovoltaic Models via Cuckoo Search

Jieming Ma12 T O Ting2 Ka Lok Man2 Nan Zhang2

Sheng-Uei Guan2 and Prudence W H Wong1

1 Department of Computer Science University of Liverpool Ashton Building Ashton Street Liverpool L69 3BX UK2Xirsquoan Jiaotong-Liverpool University 111 Renrsquoai Road Suzhou Dushu Lake HET Jiangsu Province 215123 China

Correspondence should be addressed to T O Ting totingxjtlueducn

Received 27 June 2013 Accepted 11 July 2013

Academic Editor Xin-She Yang

Copyright copy 2013 Jieming Ma 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

Since conventionalmethods are incapable of estimating the parameters of Photovoltaic (PV)models with high accuracy bioinspiredalgorithms have attracted significant attention in the last decade Cuckoo Search (CS) is invented based on the inspiration ofbrood parasitic behavior of some cuckoo species in combination with the Levy flight behavior In this paper a CS-based parameterestimation method is proposed to extract the parameters of single-diode models for commercial PV generators Simulation resultsand experimental data show that the CS algorithm is capable of obtaining all the parameters with extremely high accuracy depictedby a low Root-Mean-Squared-Error (RMSE) value The proposed method outperforms other algorithms applied in this study

1 Introduction

Photovoltaic (PV) cells normally assembled into modules orarrays on mounting systems are capable of producing elec-trons when photons strike its surface Taking the advantagesof many promising features like renewability less pollutionand ease of installation PV generators are envisaged to be animportant energy source for the future

Due to the high initial cost of a PV-supplied systempredictive performance tools are widely used by engineers tooptimize the system performance [1 2] PV manufacturersnormally provide limited tabular data measured under theStandard Test Conditions (STCs) which correspond to a celltemperature of 25∘C and an irradiance of 1000Wm2 at 15 airmass spectral distributions As reported in [3] PV generatorsalways operate under environments far from the STCs Dueto this reason the data available in the datasheet usually failto fulfill the engineering requirements

PV model with the ability to predict 119868-119881 characteristicsof PV generators under an operating environment otherthan the STCs is a predictive performance tool that allowsconsumers to maximize the cost effectiveness of the systembefore installation [2]They are generally analytical equationsbased on a physical description that formulate PV generatedcurrent (119868) with the most crucial technical characteristics

and the environmental variables such as the operatingvoltage (119881) the ambient temperature (119879) and the irradiance(119866) Over the years significant research efforts have beencontributing to the development of the behavioralmodels [4ndash8] Amongnumerousmodeling approaches the Single-DiodeModel (SDM) is the most widely utilized PV model in theliterature A general SDM includes five parameters namelyphotocurrent (119868pv) saturation current (119868

119900) diode ideality

constant (119899) series resistance (119877119904) and shunt resistance (119877

119901)

In order to adapt PV model behavior to different operatingconditions de Blas et al [9] suggested to apply the proceduredescribed in the International Standard IEC 891 that relatescurrent and voltage of the PV characteristics at given values of119879 and 119866 with the corresponding values at different operatingenvironments The Improved Single Diode Model (ISDM)presented by De Soto et al [5] includes the dependenceof the PV parameters on operating conditions The normalparameters at the STCs are necessary to be determined inthis model Both SDM and ISDM are adopted in this studyof parameter estimation

Analytical methods [5 10ndash12] are common approachesin estimating the parameters by mathematical equationsAlthough having the merit of simplicity it is hard to fur-ther reduce the errors of the estimated values Further-more analytical methods utilize the 119868-119881 curve features or

2 Journal of Applied Mathematics

semiconductor parameters that are unavailable in thedatasheetThis problem often reduces its feasibility RecentlyPV parameter estimation is deemed as a multidimensionaloptimization problem Several computational intelligencemethods such as Genetic Algorithms (GA) [13] ChaosParticle Swarm Optimization (CPSO) [14] Firefly [15] andPattern Search (PS) [16] were proposed in the literatureThese algorithms usually extract relevant parameters byminimizing the Root Mean Square Error (RMSE) as theobjective function in the optimization process Askarzadehand Rezazadeh [17] reported that the optimization methodsproduce better results than analytical methods

Cuckoo Search (CS) is a nature-inspired optimizationalgorithm based on the fascinating breeding behavior suchas brood parasitism of certain species of cuckoos In [18 19]Yang and Deb reported that the CS algorithm outperformsParticle Swarm Optimization (PSO) and GA algorithms forvarious standard test functions In this paper a CS-basedparameter estimation method for the SDM and ISDM is pre-sented Simulation and experimental results show superioraccuracy and feasibility of the proposed parameter estimationmethod

The rest of the paper is organized as follows Section 2explains both PVmodels (SDMand ISDM) used in this workThe objective function formulation is given in Section 3Thisis followed by results and discussions in Section 4The resultscomparison is also available here and finally the conclusionsare derived in Section 5

2 PV Modeling

21 Single Diode PV Model (SDM) PV cells are made ofa variety of semiconductor materials using different man-ufacturing processes The working principle of PV cells isessentially on the basis of the PV effect which refers tothe generation of a potential difference at the 119875-119873 junctionin response to visible or other radiation When a PV cellis exposed to light the semiconductor materials absorbphotons and accordingly charged carriers are generatedPotential difference and current in the external circuit lead tothe separation of carriers in the internal electric field createdby the 119875-119873 junction and collection at the electrodes Thephotogenerated charge carriers can be subsequently capturedin the form of an electric current that is electricity 119868pvEliminated the PV effect a PV cell behaves like a conventionaldiode that does not depend on any light parameters TheShockley diode equation is generally used to describe thecurrent flowing through the diode (119868

119889)

119868119889= 119868119900(119890119881119889119899119881119905 minus 1) (1)

In (1) 119868119900is the normal diode current and 119881

119889represents

the electrical potential difference between the two ends of thediode The ideality factor 119899 is assumed to be independent ofthe environment variables 119879 and 119866 119881

119905denotes the thermal

voltage of the PV and its value can be written as a function of119879

119881119905=119896119879

119902 (2)

minus

+

V

Rs

Rp

I

Id

Ipv

Figure 1 The equivalent circuit of the SDM

where 119896 and 119902 represent the Boltzmann constant (1380650times 10minus23 JK) and the electron charge (1602176 times 10minus19 C)respectively

SDM assumes that the superposition principle holdsthat is the total characteristic is the sum of the dark andilluminated characteristics [3ndash5] As expressed in (3) belowthe terminal current 119868 is therefore equal to 119868pv subtracting thecurrent diverting through the diode and 119877

119904 The equivalent

circuit of the SDM is shown in Figure 1

119868 = 119868pv minus 119868119900 (119890(119881+119868119877119904)119899119881119905 minus 1) minus

119881 + 119868119877119904

119877119901

(3)

PV module is a particular case of the PV cells connectedin series If the number of the connected cells is up to119873

119904 119881119905

is scaled to119873119904timesThemodel equation is then rewritten as


119881 + 119868119877119904

119877119901

(4)

In this sense 119868pv 119868119900 119877119904 and 119877119901are the corresponding

parameters of a PV module

22 Improved Single Diode Model (ISDM) The traditionalSDM ignores the operating conditions effect on these param-eters However some studies have shown that the parameterssuch as 119868pv and 119868

119900 vary under different environmental

conditions These are due to changes of temperature 119879

and irradiance 119866 Aiming to evaluate the PV behavior atenvironmental conditions other than the normal values 119879

119899

and 119866119899 the relations between the operating parameters and

the normal parameters are studied by numerous researchersIn [4] the value of light-generated 119868pv is reported to

be linearly dependent on the solar irradiation under theinfluence of temperature

119868pv =119866

119866119899

(119868pv119899 minus 119870119894Δ119879) (5)

where 119868pv119899 is the light-generated current at the STCs 119870119894 the

short-circuit current coefficient is one of the ISDM parame-ters The difference between the standard test temperature 119879

119899

and 119879 is denoted by Δ119879Based on the diode theory Messenger and Ventre [20]

presented an approximate linear expression for the diodesaturation current 119868

119900 which can be expressed as

119868119900= 119868119900119899(119879

119879119899

) 119890[(119902119864119892119899119896)(1119879119899minus1119879)] (6)

Journal of Applied Mathematics 3

where 119864119892is the material band gap Usually 119864

119892is set at a

reasonable level depending on the semiconductor materials(112 eV for crystalline silicon and 175 eV for amorphoussilicon) in simulation and design tools [21] De Soto et al [5]present an estimation method for 119864

119892in a wide temperature

range

119864119892= 119864119892119899(1 minus 00002677Δ119879) (7)

where 119864119892119899

is a normal value at the STCs (119864119892119899

= 112 eVfor silicon cells and 119864

119892119899= 16 eV for the triple junction

amorphous cells)In [3] Lo Brano et al study how the series and shunt

resistances are affected by the solar irradiance On the basis ofthe experimental data the values of 119877

119904and 119877

119901are observed

varying in inverse linear modes with 119866

119877119904=119866119899

119866119877119904119899

119877119901=119866119899

119866119877119901119899

(8)

where the values of the resistances 119877119904119899and 119877

119901119899are evaluated

under the STCsBy using the aforementioned relations the ISDM

described in [5] is able to analytically describe the 119868-119881characteristic of a PV generator for each generic condition ofoperative temperature and solar irradiance

3 Parameter Estimation

31 Formulation of Parameter Estimation Problem PVparameter estimation is a process that minimizes the differ-ence between the measured data and the calculated currentby adjusting the normal PV parameters When the numberof experimental data is up to119873 the objective function can beformulated by RMSE as

RMSE = radic1

119873

119873

sum

119894=1

(119891119894(119881 119868 119909))

2

(9)

where 119909 = [119868pv 119868119900119899 119877119904 119877119901] for SDM and 119909 =

[119868pv119899 119868119900119899 119899 119877119904119899 119877119901119899 119870119894 119864119892] for ISDM 119891(119881 119868 119909) is thehomogeneous form of (4) which expresses the 119868-119881characteristics of the SDM

119891 (119881 119868 119909) = 119868pv minus 119868119900 (119890(119881+119868119877119904)119899119873119904119881119905 minus 1) minus

119881 + 119868119877119904

119877119901

minus 119868 (10)

For the case of ISDM 119868pv 119868119900 119899 119877 and 119877119901satisfy the

relational expressions discussed in the previous subsectionnamely (5)ndash(8)

32 Cuckoo Search The CS algorithm [18 19] proposed byYang and Deb is a nature-inspired stochastic global searchalgorithm that follows three idealized behavior rules

(i) A cuckoo lays an egg and dumps it randomly intoother bird speciesrsquo nests

Cuckoo Search via Levy FlightsInitialization of 119899 host nests (population)

whilewithin the stopping criterionChoose a cuckoo egg by Levy flights and evaluateits fitness (119865

119894

)Choose an egg in otherrsquos nest randomly andcalculate its fitness (119865

119895

)If 119865119894

gt 119865119895

replace jth egg by ith eggA fraction (119901

119886

) of worse nests are demolished andreplaced by new onesPreserve good nests (best solutions)

endwhile

Pseudocode 1 Pseudocode of the Cuckoo Search (CS) [19]

(ii) The best nests with high quality eggs will be carriedforward to the next generation

(iii) There are a fixed number of available host nests Ifa host bird discovers that the eggs are not its ownit will either throw these alien eggs away or it mayabandon the nest and build a brand new nest at anearby location

Based on the three rules the basic steps of CS can besummarized by the pseudocode shown in Pseudocode 1 Inthe CS algorithm a pattern corresponds to a nest while eachindividual attribute of the pattern corresponds to an egg laidby the cuckoo On the basis of random-walk algorithms thegeneral system equation of the CS algorithm is given in

119883119892+1119894

= 119883119892119894+ 120572 otimes Levy (120573) (11)

where 119892 and 119894 denote the generation number (119892 =

1 2 3 MaxGen) and the pattern number (119894 = 1 2 119899)respectivelyTheproductotimesmeans entry-wisemultiplicationsHere 120572 gt 0 is the step size scaling factor which should berelated to the scales of the problem of interest [19] The 119895thattributes of the 119894th pattern is initiated by using (12)

119883119892=0119895119894

= rand sdot (119880119887119894minus 119871119887119894) + 119871119887

119894 (12)

where 119880119887119894and 119871119887

119894are the upper and lower bounds of the

119895th attributes respectively In each computation step the CSalgorithm checks whether the value of an attribute exceedsthe allowed search range If this happens the value of therelated attribute will be updated with the correspondingboundary value

Before the searching process the CS algorithmdetects themost successful pattern as 119909best pattern Among the existingalgorithms exist for generating Levy flights in the literatureYang and Deb [18 19] reported that Mantegnarsquos algorithm[22] works well in most of the optimization problemsAccordingly the evolution phase of the pattern initializedwith the detection step of 120601 which is given by (13) [23]

120601 = (Γ(1 + 120573) sdot sin (120587 sdot 1205732)

Γ (((1 + 120573) 2) sdot 120573 sdot 2(120573minus1)2))

1120573

(13)


where 120573 is 15 in the standard software implementation of theCS algorithm [30] Γ denotes the gamma function

After initialization the evolution phase of the 119909119894pattern

starts by defining the donor vector V where V = 119909119894 The

required step size of the 119895th attributes can be calculated bythe following equation

119904119895= 001 sdot (

119906119895

V119895

)

1120573

sdot (V minus 119909best) (14)

where 119906 = 120601 sdot rand119899[119863] and V = rand119899[119863] The rand119899[119863]function generates a uniform integer between [1 119863] [25]Thedonor pattern V is then randomly adjusted by

V = V + 119904119895sdot rand119899 [119863] (15)

The CS algorithm will evaluate the fitness of the randompattern If a better solution is caught the 119909best pattern will beupdated The unfeasible patterns are revised by the crossoveroperator given in (16) as follows

V119894=

119909119894+ rand sdot (119909

1199031minus 1199091199032) rand

119894gt 1199010

119909119894 others

(16)

where 1199010is the mutation probability value (119901

0= 025 in the

standard software implementation [30]) The final step of ageneration is to check if the revised infeasible patterns delivera better solution

4 Results and Discussions

With the aim of providing a thorough evaluation of the CSalgorithm in estimating the PV parameters both SDM andISDM are considered in this paper Two case studies aredesigned to estimate the CS algorithm in model parametersestimation

(i) a commercial 57mm diameter solar cell (RTCFrance [26]) operating at the standard irradiancelevel

(ii) a PV module (KC200GT Multicrystal PhotovoltaicModule) operating under varied environment condi-tions

During the parameter extraction process the objectivefunction 119891(119881 119868 119909) is minimized with respect to the param-eters range In theory the value of 119868pv119899 is slightly larger thanthat of 119868sc119899 119864119892119899 is in a loose range from 1 eV to 2 eV 119870

119894is

around the value provided by the datasheet (normally lessthan 002∘C)The 119868

119900119899is usually less than 50120583AAs stated in

[27] the ideality factor ranges between 1 and 2 PV modulesproduced by most manufacturers have 119877

119904less than 05Ω and

119877119901between 5 and 170Ω [8 28] As for PV cell the ranges of

119877119904and 119877

119901can be scaled by simply dividing119873

119904[29]

Statistical analysis is performed to evaluate the quality ofthe fitted models to the experimental data Besides RMSEother two fundamental measures namely Individual Abso-lute Error (IAE) and the Mean Absolute Error (MAE) are

Table 1 A comparison between the parameter results obtained bythe CS algorithm and that of other algorithms from the SDM

CS CPSO [14] GA [13] PS [16]119868pv 07608 07607 07619 07617119868119900

323119864ndash07 400119864ndash07 809119864ndash07 998119864ndash07119899 14812 15033 15751 16119877119904

00364 00354 00299 00313119877119901

537185 59012 423729 611026RMSE 00010 00014 00191 00149

applied to evaluate in this paper Equations (17) and (18)preset the IAE and MAE respectively

IAE =1003816100381610038161003816119868calculated minus 119868measured

1003816100381610038161003816 (17)

MAE =1

119873

119873

sum

119894=1

IAE119894 (18)

The optimization algorithms applied in this paper areprogrammed in MATLAB Similar simulation conditionsincluding population sizemaximumgeneration number andsearch ranges are set to ensure a fair evaluation (populationsize = 25 maximum generation number = 5000)

41 Case Study 1 Parameter Estimation for a PV Cell at theCertain Irradiance Level Table 1 lists the model parametersof the RTC France PV cell at 33∘C which are extractedfrom the experimental data in [26] The parameters obtainedfrom the CS algorithm are compared with three differentparameter estimation approaches CPSO [14] GA [13] andPS [16] From the RMSEs of these methods which are listedin the last row of Table 1 the CS algorithm [30] outperformsthe other three optimization methods CS obtained slightlylower RMSE recording 00010 in numerical value

During the parameter estimation process for the SDMthe values of the objective function in different optimizationalgorithms are shown in Figure 2 The function ldquogardquo inMATLAB [31] whose crossover rate 119875

119888= 08 and mutation

rate 119875119898= 02 is utilized for the convergence process test As

for PSO implementation [24] the algorithm parameters areset as learning factors 119888

1= 1198882= 2 inertia factors 119908max = 09

119908min = 04 and velocity clamping factor 119881max = 05 InFigure 2 no further improvement by GA is observed after500 iterations On the contrary the CS algorithm showedcontinuous improvement until themaximumgenerationTheCS algorithm whose convergence speed is slightly faster thanPSO shows the best accuracy result in the minimization taskafter 5000 iterations

Table 2 lists the parameters of the ISDM obtained by theCS algorithm In order to evaluate the accuracy of the CS-based estimation these parameters are substituted into theISDM Since the 119868-119881 demonstrates nonlinear characteristicsthe PV terminal current 119868 is solved by the Newton-Raphsonmethod [32] in this paper In Table 3 the calculated results119868ISDM are compared with the experimental data 119868measured toobserve the agreement between themThe notations IAESDMand IAEISDM denote the IAE for SDMand ISDM respectively


0 1000 2000 3000 4000 5000Iteration

Fitn

ess (

RMSE

)

100

10minus1

10minus2

10minus3

GA [31]PSO [24]CS [30]

Figure 2 Convergence process of different optimization algorithmsduring the parameter estimation process of the SDM

Table 2 Parameters of the ISDM obtained by the CS algorithm

119868pv119899 119868119900119899

119899 119877119904119899

119877119901119899

119870119894

119864119892119899

07361 184119864ndash07 15009 00355 578394 00031 10020

Although the RMSE of the ISDM is less than that of CPSOGA and PS it is similar to the RMSE of the conventionalSDM under a certain environmental condition

42 Case Study 2 Parameter Estimation for a PV Moduleunder Different Environment Conditions In this section thevalidity of the CS algorithm is evaluated using KC200GT PVmodule operating under different environment conditionsThe estimated parameters both in the SDM and ISDMare shown in Table 4 As illustrated in Section 1 the mainapplication of the parameter extraction is to predict the119868-119881 characteristics for design purpose It is worth pointingout that the SDM parameters can only be extracted by theexperimental data measured under a certain test conditionSignificant errors may occur as the experimental data aremeasured under varying operating conditions In the com-mercial simulation tool like PSIM [21] the PV parameters ofthe SDM are firstly estimated at the STCs then the equations(given in the appendix) are applied to calculate the electricalcharacteristics of different operating conditions The ISDM-based parameter estimation however can be performed bythe data measured under any conditions

Figure 3 shows the 119868-119881curves generated using the param-eters obtained by the CS algorithmThe simulated results arecompared with the experimental data which are collectedat five different irradiance levels (1000Wm2 800Wm2600Wm2 400Wm2 and 200Wm2) and three differenttemperature levels (25∘C 50∘C and 75∘C) It can be seenthat the 119868-119881 curves of the ISDM fit the whole range of theexperimental dataset On the other hand the errors of SDMseem larger at lower irradiance and higher temperature levelsWith the experimental data the RMSE of the current 119868 in

Table 3 A comparison between the errors of ISDM and SDM Theparameters are extracted by the CS algorithm

No 119881measured 119868measured 119868ISDM IAEISDM IAESDM

1 minus02057 07640 07639 00001 000012 minus01291 07620 07626 00006 000073 minus00588 07605 07614 00009 000094 00057 07605 07602 00003 000035 00646 07600 07592 00008 000096 01185 07590 07583 00007 000107 01678 07570 07574 00004 000018 02132 07570 07565 00005 000099 02545 07555 07555 00000 0000410 02924 07540 07540 00000 0000311 03269 07505 07517 00012 0000912 03585 07465 07476 00011 0000913 03873 07385 07402 00017 0001614 04137 07280 07273 00007 0000615 04373 07065 07066 00001 0000516 04590 06755 06748 00007 0000217 04784 06320 06304 00016 0001118 04960 05730 05717 00013 0000919 05119 04990 04994 00004 0000520 05265 04130 04137 00007 0000521 05398 03165 03176 00011 0000722 05521 02120 02127 00007 0000123 05633 01035 01033 00002 0000824 05736 minus00100 minus00089 00011 0000825 05833 minus01230 minus01244 00014 0001426 05900 minus02100 minus02095 00005 00009MAE 00007 00007RMSE 00010 00010

Table 4 Parameters of the KC200GT PV module obtained by theCS algorithm

(a) SDM parameters (extracted by the CS algorithm)

119868pv 119868o 119899 119877s 119877119901

81729 423Endash10 10090 02665 1404875

(b) ISDM parameters (extracted by the CS algorithm)

119868pv119899 119868119900119899

119899 119877119904119899

119877119901119899

119870119894

119864119892119899

81847 512Endash10 10170 02574 1179224 00028 12474

SDM is calculated as 02837 while the RMSE of 119868 in ISDMis only 00776

Figure 4 shows the absolute current errors of differentperformance predicting methods under different operatingconditionsThe curves denoted by the label ldquoanalytical SDMrdquoare obtained from the analytical SDM model [4] Ignoringthe effect of incidence angle and air mass the curves labeledby ldquoanalytical ISDMrdquo denote the 119868-119881 curves from De Sotorsquos


0 5 10 15 20 25 300123456789

Experimental dataDSMISDM

V (V)

I(A

)

(a)

0 5 10 15 20 25 300123456789


V (V)

I(A

)

(b)

Figure 3 The simulated 119868-119881 characteristic curves of the KC200GT PV module (a) under different irradiance levels (b) under differenttemperature levels

0 5 10 15 20 25 30

Analytical ISDMCS estimation-based SDMCS estimation-based ISDM

V (V)

Analytical SDM[4]

200400

600800

10000

02040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)

G (Wm 2)

(a)

0 5 10 15 20 25 30

025

5075

0

05

1

15

2

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)T ( ∘C)

Analytical SDM[4]

(b)

Figure 4 A comparison of the individual absolute errors among different PV modeling methods (a) under different irradiance levels (b)under different temperature levels

analytical ISDM model [5] It is evident the ISDM with theparameters extracted by the CS algorithm is more accuratethan the analytical model As for the SDM the CS algorithmis capable of extracting a set of PV parameters with a goodfit for the experimental data at the STCs However the SDMwith the equations in the appendix does not exhibit a goodprediction performance under other operating conditions

To further validate the accuracy of the CS algorithmthe extracted parameters are compared to the ones obtainedusing GA in Figure 5 In general the CS algorithm gives thebetter performance than GA for all cases The MaximumPower Point (MPP) usually locating around 74 of theopen circuit voltage is an important technical data in PVmodeling However a negative point of the GA-based ISDMis that the errors in the high voltage range are relatively high

The maximum absolute error of the GA-based ISDM is up toabout 08 A while the absolute error of the CS is kept below02 A

5 Conclusion

In this work the Cuckoo Search (CS) algorithm is appliedto estimate the parameters of two PV models namely SingleDiode Model (SDM) and its improved version (ISDM) Thefeasibility of the proposed method has been validated byestimating the parameters of two commercial PV generatorsThe simulation and experimental results showed that the CSalgorithm is capable of not only extracting all the parametersof the SDM under a certain condition but also successfully


0 5 10 15 20 25 30

200400

600800

10000

02040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)

CS estimation-based ISDMGA estimation-based ISDM

V (V)G (Wm 2)

(a)

00 5 10 15 20 25 30

002040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)25

5075

T ( ∘C)

(b)

Figure 5 A comparison of the individual absolute errors between CS- and GA- based ISDM (a) under different irradiance levels (b) underdifferent temperature levels

estimating all the parameters of ISDMunder different operat-ing conditions In statistical analysis CS algorithm recordedthe lowest RMSE value compared to other algorithms such asGA PSO and PS

Appendix

PV Physical Model Adopted in PSIM

By using the parameters extracted at the STCs the 119868-119881 char-acteristics of a PV generator under nonstandard operatingconditions can be calculated via the following equations

119868 = 119868pv minus 119868119889 minus 119868119877

119868pv = 119868sc119899 sdot119866

119866119899

minus 119870119894sdot (119879 minus 119879

119899)

119868119889= 119868119900sdot (119890119902119881119889119899119896119879 minus 1)

119868119900= 119868119888119899sdot (

119866

119866119899

)

3

sdot 119890(119902119864119892119899119896)sdot(1119879minus1119879119899)

119868119877=119881119889

119877119901

119881119889=

119881

119873119904

+ 119868 sdot 119877119904

(A1)

Acknowledgments

The authors are grateful to Professor Xin-She Yang for thesharing of Cuckoo Search source code online Without hisgenerosity this work would not be possible This researchis supported by the National Natural Science Foundation ofChina under Grant 61070085

References

[1] B Amrouche A Guessoum and M Belhamel ldquoA simplebehavioural model for solar module electric characteristicsbased on the first order system step response for MPPT studyand comparisonrdquo Applied Energy vol 91 no 1 pp 395ndash4042012

[2] A Orioli and A di Gangi ldquoA procedure to calculate the five-parameter model of crystalline silicon photovoltaic modules onthe basis of the tabular performance datardquo Applied Energy vol102 pp 1160ndash1177 2013

[3] V Lo Brano A Orioli G Ciulla and A di Gangi ldquoAn improvedfive-parameter model for photovoltaic modulesrdquo Solar EnergyMaterials and Solar Cells vol 94 no 8 pp 1358ndash1370 2010

[4] 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

[5] 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

[6] K Ishaque Z Salam and H Taheri ldquoSimple fast and accuratetwo-diodemodel for photovoltaicmodulesrdquo Solar EnergyMate-rials and Solar Cells vol 95 no 2 pp 586ndash594 2011

[7] V Lo Brano A Orioli and G Ciulla ldquoOn the experimentalvalidation of an improved five-parameter model for siliconphotovoltaic modulesrdquo Solar Energy Materials and Solar Cellsvol 105 pp 27ndash39 2012

[8] 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

[9] 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

[10] J P Charles G Bordure A Khoury and P Mialhe ldquoConsis-tency of the double exponential model with physical mecha-nisms of conduction for a solar cell under illuminationrdquo Journalof Physics D vol 18 no 11 pp 2261ndash2268 1985

[11] D S H Chan and J C H Phang ldquoAnalytical methods forthe extraction of solar-cell single- and double-diode model


parameters from I-V characteristicsrdquo IEEE Transactions onElectron Devices vol 34 no 2 pp 286ndash293 1987

[12] J CH PhangD SH Chan and J R Phillips ldquoAccurate analyt-ical method for the extraction of solar cell model parametersrdquoElectronics Letters vol 20 no 10 pp 406ndash408 1984

[13] A J Joseph B Hadj and A L Ali ldquoSolar cell parameterextraction using genetic algorithmsrdquoMeasurement Science andTechnology vol 12 no 11 pp 1922ndash1925 2001

[14] W Huang C Jiang L Xue and D Song ldquoExtracting solar cellmodel parameters based on chaos particle swarm algorithmrdquo inProceedings of the International Conference on Electric Informa-tion and Control Engineering (ICEICE rsquo11) pp 398ndash402 April2011

[15] I Fister I Fister Jr X S Yang and J Brest ldquoA comprehensivereview of firefly algorithmsrdquo Swarm and Evolutionary Compu-tation 2013

[16] M F AlHajri K M El-Naggar M R AlRashidi and A KAl-Othman ldquoOptimal extraction of solar cell parameters usingpattern searchrdquo Renewable Energy vol 44 pp 238ndash245 2012

[17] A Askarzadeh and A Rezazadeh ldquoParameter identificationfor solar cell models using harmony search-based algorithmsrdquoSolar Energy vol 86 pp 3241ndash3249 2012

[18] X Yang and S Deb ldquoEngineering optimisation by cuckoosearchrdquo International Journal of Mathematical Modelling andNumerical Optimisation vol 1 no 4 pp 330ndash343 2010

[19] X Yang and S Deb ldquoCuckoo search via Levy flightsrdquo inProceedings of the World Congress on Nature and BiologicallyInspired Computing (NABIC rsquo09) pp 210ndash214 December 2009

[20] R AMessenger and J Ventre Photovoltaic Systems EngineeringCRC Press New York NY USA 2nd edition 2004

[21] PSIM User Manual Powersim Woburn Mass USA 2001[22] R N Mantegna ldquoFast accurate algorithm for numerical sim-

ulation of Levy stable stochastic processesrdquo Physical Review Evol 49 no 5 pp 4677ndash4683 1994

[23] A H Gandomi X Yang and A H Alavi ldquoCuckoo searchalgorithm a metaheuristic approach to solve structural opti-mization problemsrdquoEngineeringwith Computers vol 29 pp 17ndash35 2013

[24] B Birge Particle Swarm Optimization Toolbox httpwwwmathworkscommatlabcentralfileexchange7506-particle-swarm-optimization-toolbox

[25] P Civicioglu and E Besdok ldquoA conceptual comparison of theCuckoo-search particle swarmoptimization differential evolu-tion and artificial bee colony algorithmsrdquo Artificial IntelligenceReview vol 39 pp 315ndash346 2013

[26] T Easwarakhanthan J Bottin I Bouhouch and C BoutritldquoNonlinear minimization algorithm for determining the solarcell parameters with microcomputersrdquo International Journal ofSolar Energy vol 4 pp 1ndash12 1986

[27] M Bashahu and P Nkundabakura ldquoReview and tests of meth-ods for the determination of the solar cell junction idealityfactorsrdquo Solar Energy vol 81 no 7 pp 856ndash863 2007

[28] S J Jun and L Kay-Soon ldquoPhotovoltaic model identificationusing particle swarm optimization with inverse barrier con-straintrdquo IEEE Transactions on Power Electronics vol 27 pp3975ndash3983 2012

[29] K Ishaque Z Salam S Mekhilef and A Shamsudin ldquoParam-eter extraction of solar photovoltaic modules using penalty-based differential evolutionrdquo Applied Energy vol 99 pp 297ndash308 2012

[30] X S Yang Cuckoo Search Algorithm (Source Code) httpwwwmathworkscommatlabcentralfileexchange29809-cuckoo-search-cs-algorithm

[31] Optimization Toolbox The MathWorks Inc httpwwwmathworkscomproductsoptimizationindexhtml

[32] R L Burden and J D Faires Numerical Analysis CengageLearning Singapore 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


semiconductor parameters that are unavailable in thedatasheetThis problem often reduces its feasibility RecentlyPV parameter estimation is deemed as a multidimensionaloptimization problem Several computational intelligencemethods such as Genetic Algorithms (GA) [13] ChaosParticle Swarm Optimization (CPSO) [14] Firefly [15] andPattern Search (PS) [16] were proposed in the literatureThese algorithms usually extract relevant parameters byminimizing the Root Mean Square Error (RMSE) as theobjective function in the optimization process Askarzadehand Rezazadeh [17] reported that the optimization methodsproduce better results than analytical methods

Cuckoo Search (CS) is a nature-inspired optimizationalgorithm based on the fascinating breeding behavior suchas brood parasitism of certain species of cuckoos In [18 19]Yang and Deb reported that the CS algorithm outperformsParticle Swarm Optimization (PSO) and GA algorithms forvarious standard test functions In this paper a CS-basedparameter estimation method for the SDM and ISDM is pre-sented Simulation and experimental results show superioraccuracy and feasibility of the proposed parameter estimationmethod

The rest of the paper is organized as follows Section 2explains both PVmodels (SDMand ISDM) used in this workThe objective function formulation is given in Section 3Thisis followed by results and discussions in Section 4The resultscomparison is also available here and finally the conclusionsare derived in Section 5

2 PV Modeling

21 Single Diode PV Model (SDM) PV cells are made ofa variety of semiconductor materials using different man-ufacturing processes The working principle of PV cells isessentially on the basis of the PV effect which refers tothe generation of a potential difference at the 119875-119873 junctionin response to visible or other radiation When a PV cellis exposed to light the semiconductor materials absorbphotons and accordingly charged carriers are generatedPotential difference and current in the external circuit lead tothe separation of carriers in the internal electric field createdby the 119875-119873 junction and collection at the electrodes Thephotogenerated charge carriers can be subsequently capturedin the form of an electric current that is electricity 119868pvEliminated the PV effect a PV cell behaves like a conventionaldiode that does not depend on any light parameters TheShockley diode equation is generally used to describe thecurrent flowing through the diode (119868

119889)

119868119889= 119868119900(119890119881119889119899119881119905 minus 1) (1)

In (1) 119868119900is the normal diode current and 119881

119889represents

the electrical potential difference between the two ends of thediode The ideality factor 119899 is assumed to be independent ofthe environment variables 119879 and 119866 119881

119905denotes the thermal

voltage of the PV and its value can be written as a function of119879

119881119905=119896119879

119902 (2)

minus

+

V

Rs

Rp

I

Id

Ipv

Figure 1 The equivalent circuit of the SDM

where 119896 and 119902 represent the Boltzmann constant (1380650times 10minus23 JK) and the electron charge (1602176 times 10minus19 C)respectively

SDM assumes that the superposition principle holdsthat is the total characteristic is the sum of the dark andilluminated characteristics [3ndash5] As expressed in (3) belowthe terminal current 119868 is therefore equal to 119868pv subtracting thecurrent diverting through the diode and 119877

119904 The equivalent

circuit of the SDM is shown in Figure 1


119881 + 119868119877119904

119877119901

(3)

PV module is a particular case of the PV cells connectedin series If the number of the connected cells is up to119873

119904 119881119905

is scaled to119873119904timesThemodel equation is then rewritten as


119881 + 119868119877119904

119877119901

(4)

In this sense 119868pv 119868119900 119877119904 and 119877119901are the corresponding

parameters of a PV module

22 Improved Single Diode Model (ISDM) The traditionalSDM ignores the operating conditions effect on these param-eters However some studies have shown that the parameterssuch as 119868pv and 119868

119900 vary under different environmental

conditions These are due to changes of temperature 119879

and irradiance 119866 Aiming to evaluate the PV behavior atenvironmental conditions other than the normal values 119879

119899

and 119866119899 the relations between the operating parameters and

the normal parameters are studied by numerous researchersIn [4] the value of light-generated 119868pv is reported to

be linearly dependent on the solar irradiation under theinfluence of temperature

119868pv =119866

119866119899

(119868pv119899 minus 119870119894Δ119879) (5)

where 119868pv119899 is the light-generated current at the STCs 119870119894 the

short-circuit current coefficient is one of the ISDM parame-ters The difference between the standard test temperature 119879

119899

and 119879 is denoted by Δ119879Based on the diode theory Messenger and Ventre [20]

presented an approximate linear expression for the diodesaturation current 119868

119900 which can be expressed as

119868119900= 119868119900119899(119879

119879119899

) 119890[(119902119864119892119899119896)(1119879119899minus1119879)] (6)



119892is set at a



range

119864119892= 119864119892119899(1 minus 00002677Δ119879) (7)

where 119864119892119899






119904and 119877

119901are observed


119877119904=119866119899

119866119877119904119899

119877119901=119866119899

119866119877119901119899

(8)







RMSE = radic1

119873

119873

sum

119894=1

(119891119894(119881 119868 119909))

2

(9)




119881 + 119868119877119904

119877119901

minus 119868 (10)







119894


119895

)If 119865119894

gt 119865119895


119886


endwhile





119883119892+1119894

= 119883119892119894+ 120572 otimes Levy (120573) (11)



119883119892=0119895119894

= rand sdot (119880119887119894minus 119871119887119894) + 119871119887

119894 (12)

where 119880119887119894and 119871119887




120601 = (Γ(1 + 120573) sdot sin (120587 sdot 1205732)

Γ (((1 + 120573) 2) sdot 120573 sdot 2(120573minus1)2))

1120573

(13)






119904119895= 001 sdot (

119906119895

V119895

)

1120573



V = V + 119904119895sdot rand119899 [119863] (15)


V119894=

119909119894+ rand sdot (119909

1199031minus 1199091199032) rand

119894gt 1199010

119909119894 others

(16)


0= 025 in the







119894is






119877119904and 119877


119904[29]



CS CPSO [14] GA [13] PS [16]119868pv 07608 07607 07619 07617119868119900


00364 00354 00299 00313119877119901

537185 59012 423729 611026RMSE 00010 00014 00191 00149



1003816100381610038161003816 (17)

MAE =1

119873

119873

sum

119894=1

IAE119894 (18)











0 1000 2000 3000 4000 5000Iteration

Fitn

ess (

RMSE

)

100

10minus1

10minus2

10minus3

GA [31]PSO [24]CS [30]



119868pv119899 119868119900119899

119899 119877119904119899

119877119901119899

119870119894

119864119892119899

07361 184119864ndash07 15009 00355 578394 00031 10020









119868pv 119868o 119899 119877s 119877119901

81729 423Endash10 10090 02665 1404875


119868pv119899 119868119900119899

119899 119877119904119899

119877119901119899

119870119894

119864119892119899

81847 512Endash10 10170 02574 1179224 00028 12474




0 5 10 15 20 25 300123456789


V (V)

I(A

)

(a)

0 5 10 15 20 25 300123456789


V (V)

I(A

)

(b)


0 5 10 15 20 25 30


V (V)

Analytical SDM[4]

200400

600800

10000

02040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)

G (Wm 2)

(a)

0 5 10 15 20 25 30

025

5075

0

05

1

15

2

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)T ( ∘C)

Analytical SDM[4]

(b)





5 Conclusion



0 5 10 15 20 25 30

200400

600800

10000

02040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)G (Wm 2)

(a)

00 5 10 15 20 25 30

002040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)25

5075

T ( ∘C)

(b)



Appendix



119868 = 119868pv minus 119868119889 minus 119868119877

119868pv = 119868sc119899 sdot119866

119866119899

minus 119870119894sdot (119879 minus 119879

119899)

119868119889= 119868119900sdot (119890119902119881119889119899119896119879 minus 1)

119868119900= 119868119888119899sdot (

119866

119866119899

)

3

sdot 119890(119902119864119892119899119896)sdot(1119879minus1119879119899)

119868119877=119881119889

119877119901

119881119889=

119881

119873119904

+ 119868 sdot 119877119904

(A1)

Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119892is set at a



range

119864119892= 119864119892119899(1 minus 00002677Δ119879) (7)

where 119864119892119899






119904and 119877

119901are observed


119877119904=119866119899

119866119877119904119899

119877119901=119866119899

119866119877119901119899

(8)







RMSE = radic1

119873

119873

sum

119894=1

(119891119894(119881 119868 119909))

2

(9)




119881 + 119868119877119904

119877119901

minus 119868 (10)







119894


119895

)If 119865119894

gt 119865119895


119886


endwhile





119883119892+1119894

= 119883119892119894+ 120572 otimes Levy (120573) (11)



119883119892=0119895119894

= rand sdot (119880119887119894minus 119871119887119894) + 119871119887

119894 (12)

where 119880119887119894and 119871119887




120601 = (Γ(1 + 120573) sdot sin (120587 sdot 1205732)

Γ (((1 + 120573) 2) sdot 120573 sdot 2(120573minus1)2))

1120573

(13)






119904119895= 001 sdot (

119906119895

V119895

)

1120573



V = V + 119904119895sdot rand119899 [119863] (15)


V119894=

119909119894+ rand sdot (119909

1199031minus 1199091199032) rand

119894gt 1199010

119909119894 others

(16)


0= 025 in the







119894is






119877119904and 119877


119904[29]



CS CPSO [14] GA [13] PS [16]119868pv 07608 07607 07619 07617119868119900


00364 00354 00299 00313119877119901

537185 59012 423729 611026RMSE 00010 00014 00191 00149



1003816100381610038161003816 (17)

MAE =1

119873

119873

sum

119894=1

IAE119894 (18)











0 1000 2000 3000 4000 5000Iteration

Fitn

ess (

RMSE

)

100

10minus1

10minus2

10minus3

GA [31]PSO [24]CS [30]



119868pv119899 119868119900119899

119899 119877119904119899

119877119901119899

119870119894

119864119892119899

07361 184119864ndash07 15009 00355 578394 00031 10020









119868pv 119868o 119899 119877s 119877119901

81729 423Endash10 10090 02665 1404875


119868pv119899 119868119900119899

119899 119877119904119899

119877119901119899

119870119894

119864119892119899

81847 512Endash10 10170 02574 1179224 00028 12474




0 5 10 15 20 25 300123456789


V (V)

I(A

)

(a)

0 5 10 15 20 25 300123456789


V (V)

I(A

)

(b)


0 5 10 15 20 25 30


V (V)

Analytical SDM[4]

200400

600800

10000

02040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)

G (Wm 2)

(a)

0 5 10 15 20 25 30

025

5075

0

05

1

15

2

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)T ( ∘C)

Analytical SDM[4]

(b)





5 Conclusion



0 5 10 15 20 25 30

200400

600800

10000

02040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)G (Wm 2)

(a)

00 5 10 15 20 25 30

002040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)25

5075

T ( ∘C)

(b)



Appendix



119868 = 119868pv minus 119868119889 minus 119868119877

119868pv = 119868sc119899 sdot119866

119866119899

minus 119870119894sdot (119879 minus 119879

119899)

119868119889= 119868119900sdot (119890119902119881119889119899119896119879 minus 1)

119868119900= 119868119888119899sdot (

119866

119866119899

)

3

sdot 119890(119902119864119892119899119896)sdot(1119879minus1119879119899)

119868119877=119881119889

119877119901

119881119889=

119881

119873119904

+ 119868 sdot 119877119904

(A1)

Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119904119895= 001 sdot (

119906119895

V119895

)

1120573



V = V + 119904119895sdot rand119899 [119863] (15)


V119894=

119909119894+ rand sdot (119909

1199031minus 1199091199032) rand

119894gt 1199010

119909119894 others

(16)


0= 025 in the







119894is






119877119904and 119877


119904[29]



CS CPSO [14] GA [13] PS [16]119868pv 07608 07607 07619 07617119868119900


00364 00354 00299 00313119877119901

537185 59012 423729 611026RMSE 00010 00014 00191 00149



1003816100381610038161003816 (17)

MAE =1

119873

119873

sum

119894=1

IAE119894 (18)











0 1000 2000 3000 4000 5000Iteration

Fitn

ess (

RMSE

)

100

10minus1

10minus2

10minus3

GA [31]PSO [24]CS [30]



119868pv119899 119868119900119899

119899 119877119904119899

119877119901119899

119870119894

119864119892119899

07361 184119864ndash07 15009 00355 578394 00031 10020









119868pv 119868o 119899 119877s 119877119901

81729 423Endash10 10090 02665 1404875


119868pv119899 119868119900119899

119899 119877119904119899

119877119901119899

119870119894

119864119892119899

81847 512Endash10 10170 02574 1179224 00028 12474




0 5 10 15 20 25 300123456789


V (V)

I(A

)

(a)

0 5 10 15 20 25 300123456789


V (V)

I(A

)

(b)


0 5 10 15 20 25 30


V (V)

Analytical SDM[4]

200400

600800

10000

02040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)

G (Wm 2)

(a)

0 5 10 15 20 25 30

025

5075

0

05

1

15

2

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)T ( ∘C)

Analytical SDM[4]

(b)





5 Conclusion



0 5 10 15 20 25 30

200400

600800

10000

02040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)G (Wm 2)

(a)

00 5 10 15 20 25 30

002040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)25

5075

T ( ∘C)

(b)



Appendix



119868 = 119868pv minus 119868119889 minus 119868119877

119868pv = 119868sc119899 sdot119866

119866119899

minus 119870119894sdot (119879 minus 119879

119899)

119868119889= 119868119900sdot (119890119902119881119889119899119896119879 minus 1)

119868119900= 119868119888119899sdot (

119866

119866119899

)

3

sdot 119890(119902119864119892119899119896)sdot(1119879minus1119879119899)

119868119877=119881119889

119877119901

119881119889=

119881

119873119904

+ 119868 sdot 119877119904

(A1)

Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1000 2000 3000 4000 5000Iteration

Fitn

ess (

RMSE

)

100

10minus1

10minus2

10minus3

GA [31]PSO [24]CS [30]



119868pv119899 119868119900119899

119899 119877119904119899

119877119901119899

119870119894

119864119892119899

07361 184119864ndash07 15009 00355 578394 00031 10020









119868pv 119868o 119899 119877s 119877119901

81729 423Endash10 10090 02665 1404875


119868pv119899 119868119900119899

119899 119877119904119899

119877119901119899

119870119894

119864119892119899

81847 512Endash10 10170 02574 1179224 00028 12474




0 5 10 15 20 25 300123456789


V (V)

I(A

)

(a)

0 5 10 15 20 25 300123456789


V (V)

I(A

)

(b)


0 5 10 15 20 25 30


V (V)

Analytical SDM[4]

200400

600800

10000

02040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)

G (Wm 2)

(a)

0 5 10 15 20 25 30

025

5075

0

05

1

15

2

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)T ( ∘C)

Analytical SDM[4]

(b)





5 Conclusion



0 5 10 15 20 25 30

200400

600800

10000

02040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)G (Wm 2)

(a)

00 5 10 15 20 25 30

002040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)25

5075

T ( ∘C)

(b)



Appendix



119868 = 119868pv minus 119868119889 minus 119868119877

119868pv = 119868sc119899 sdot119866

119866119899

minus 119870119894sdot (119879 minus 119879

119899)

119868119889= 119868119900sdot (119890119902119881119889119899119896119879 minus 1)

119868119900= 119868119888119899sdot (

119866

119866119899

)

3

sdot 119890(119902119864119892119899119896)sdot(1119879minus1119879119899)

119868119877=119881119889

119877119901

119881119889=

119881

119873119904

+ 119868 sdot 119877119904

(A1)

Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 300123456789


V (V)

I(A

)

(a)

0 5 10 15 20 25 300123456789


V (V)

I(A

)

(b)


0 5 10 15 20 25 30


V (V)

Analytical SDM[4]

200400

600800

10000

02040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)

G (Wm 2)

(a)

0 5 10 15 20 25 30

025

5075

0

05

1

15

2

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)T ( ∘C)

Analytical SDM[4]

(b)





5 Conclusion



0 5 10 15 20 25 30

200400

600800

10000

02040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)G (Wm 2)

(a)

00 5 10 15 20 25 30

002040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)25

5075

T ( ∘C)

(b)



Appendix



119868 = 119868pv minus 119868119889 minus 119868119877

119868pv = 119868sc119899 sdot119866

119866119899

minus 119870119894sdot (119879 minus 119879

119899)

119868119889= 119868119900sdot (119890119902119881119889119899119896119879 minus 1)

119868119900= 119868119888119899sdot (

119866

119866119899

)

3

sdot 119890(119902119864119892119899119896)sdot(1119879minus1119879119899)

119868119877=119881119889

119877119901

119881119889=

119881

119873119904

+ 119868 sdot 119877119904

(A1)

Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30

200400

600800

10000

02040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)G (Wm 2)

(a)

00 5 10 15 20 25 30

002040608

1

Indi

vidu

al ab

solu

te er

ror (

IAE)


V (V)25

5075

T ( ∘C)

(b)



Appendix



119868 = 119868pv minus 119868119889 minus 119868119877

119868pv = 119868sc119899 sdot119866

119866119899

minus 119870119894sdot (119879 minus 119879

119899)

119868119889= 119868119900sdot (119890119902119881119889119899119896119879 minus 1)

119868119900= 119868119888119899sdot (

119866

119866119899

)

3

sdot 119890(119902119864119892119899119896)sdot(1119879minus1119879119899)

119868119877=119881119889

119877119901

119881119889=

119881

119873119904

+ 119868 sdot 119877119904

(A1)

Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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