parameter identification for solar cell models using harmony search-based algorithms

Available online at www.sciencedirect.com

www.elsevier.com/locate/solener

Solar Energy 86 (2012) 3241–3249

Parameter identification for solar cell models using harmonysearch-based algorithms

Alireza Askarzadeh ⇑, Alireza Rezazadeh

Faculty of Electrical and Computer Engineering, Shahid Beheshti University, G.C., Evin 1983963113, Tehran, Iran

Received 4 May 2012; received in revised form 28 August 2012; accepted 29 August 2012Available online 26 September 2012

Communicated by: Associate Editor Nicola Romeo

Abstract

Recently, accurate modeling of current vs. voltage (I–V) characteristics of solar cells has attracted the main focus of variousresearches. The main drawback in accurate modeling is the lack of information about the precise values of the models parameters,namely, photo-generated current, diode saturation current, series resistance, shunt resistance and diode ideality factor. In order to makea good agreement between experimental data and the models results, parameter identification with the help of an optimization techniqueis necessary. Because I–V curve of solar cells is extremely non-linear, an excellent optimization technique is required. In this paper, har-mony search (HS)-based parameter identification methods are proposed to identify the unknown parameters of the solar cell single anddouble diode models. Simple concept, easy implementation and high performance are the main reasons of HS popularity to solve com-plex optimization problems. For this aim, three state-of-the-art HS variants are used to determine the unknown parameters of the mod-els. The effectiveness of the HS variants is investigated with comparative study among different techniques. Simulation results manifestthe superiority of the HS-based algorithms over the other studied algorithms in modeling solar cell systems.� 2012 Elsevier Ltd. All rights reserved.

Keywords: Solar cell; Modeling; Parameter identification; Harmony search algorithm

1. Introduction

Thanks to the cost increase of fossil fuels and their prob-able depletion, air pollution, global warming phenomenonand severe environmental laws, renewable energy sourceshave gained the attention of many nations in producingelectricity. Solar energy is currently being employed world-wide to contribute in meeting the growing demand of elec-trical power. Solar cell (also called photovoltaic (PV) cell)is one of the most popular and promising renewable energysources in the world, not only because world needs moreenergy sources as the demand for energy grows, but also

0038-092X/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.solener.2012.08.018

⇑ Corresponding author. Tel.: +98 21 29904178; fax: +98 21 22431804.E-mail addresses: [email protected] (A. Askarzadeh), a-rezazade@

sbu.ac.ir (A. Rezazadeh).

because of a variety of advantages such as emission-free,no noise, little maintenance and easy installation.

For a better understanding of the characteristics, evalu-ation of the performance and therefore optimization of aPV cell system, an accurate mathematical model is animportant tool for researchers. Solar cell modeling primar-ily involves the formulation of the non-linear current vs.voltage (I–V) curve. Several models have been developedto represent the behavior of the system under differentoperating conditions (Han et al., 2004; Villalva et al.,2009; Huld et al., 2010). They vary from models with sim-ple assumptions to advanced models accompanying withmany physical variables.

In practice, there are two major equivalent circuit mod-els to portray the behavior of a solar cell system: the singleand double diode models. The solar cell models comprise agroup of parameters, namely, photo-generated current,


mailto:[email protected]




Fig. 1. The double diode model of solar cell.

3242 A. Askarzadeh, A. Rezazadeh / Solar Energy 86 (2012) 3241–3249

diode saturation current, series resistance, shunt resistanceand diode ideality factor. It is essential to determine theseparameters because solar cell performance parameters,namely, open circuit voltage, short circuit current, maxi-mum power, fill factor and conversion efficiency, that areuseful in analyzing performance losses, are derived fromthe I–V curve and the information provided by the modelparameters (Macabebe et al., 2011). Therefore, the mainproblem is the identification of the optimal values of theparameters by which the model can produce very good fitsto the experimental data.

Various techniques have been used to identify the opti-mal values of the unknown parameters. In some references(Easwarakhanthan et al., 1986; Chan et al., 1986; Jian andKapoor, 2004; Saleem and Karmalkar, 2009), traditionaloptimization methods have been employed. These methodsthat require continuity, convexity and differentiability con-ditions for being applicable, involve heavy computations,are usually local in nature and converge to a local solutionrather than a global one. The non-linearity of the I–V curvemakes the traditional optimization techniques unable toeffectively solve the parameter identification problem. Asan alternative, heuristic optimization algorithms such asgenetic algorithm (GA) (AlRashidi et al., 2011), particleswarm optimization (PSO) (Wei et al., 2011) and simulatedannealing (SA) (El-Naggar et al., 2012) have beenemployed to solve the parameter identification issue.Though these algorithms produce better results than tradi-tional methods, they have their respective limits (Dai et al.,2010). As a result, there is the possibility to increase themodels accuracy using more capable algorithms.

Harmony search (HS), originally invented in (Geemet al., 2001), is a metaheuristic optimization technique, try-ing to imitate the improvisation process of musicians. Inmusic improvisation process, the ultimate goal is to reacha pleasing harmony (a perfect state). Hence, the musicianplays a harmony and attempts to achieve a better one byadjusting the pitches of his (or her) instrument. In a similarway, an optimization algorithm tries to minimize (or max-imize) an objective function by adjusting the decision vari-ables. In other words, a musician’s improvisation processcan be compared with the search process in optimization.

The simplicity and effectiveness of HS algorithm has ledto its application to optimization problems in various areas(Askarzadeh and Rezazadeh, 2012, 2011; Cheng et al.,2011; Erdal et al., 2011; Fourie et al., 2010). In comparisonwith traditional optimization methods, HS has severaladvantages. It is stochastic, imposes fewer mathematicalrequirements, produces a new solution after consideringthe existing solutions and needs no derivative information.

In this paper, HS and two state-of-the-art HS variants,namely, grouping-based global harmony search (GGHS)(Askarzadeh and Rezazadeh, 2011) and innovative globalharmony search (IGHS) (Askarzadeh and Rezazadeh,2012) algorithms, are used to identify the optimal parame-ters of a 57 mm diameter commercial (R.T.C. France) sili-con solar cell (Easwarakhanthan et al., 1986). GGHS and

IGHS have been developed by the authors to improvethe search capability of the original HS by maintainingthe algorithm’s diversity to avoid premature convergence.In order to evaluate the performance of HS-based algo-rithms, the obtained results are compared with those ofgenetic algorithm (GA), chaos particle swarm optimization(CPSO), simulated annealing (SA) and pattern search (PS).

The rest of this paper is arranged as follows: Section 2provides a description of the solar cell models; In Section3, HS and its two variants (GGHS and IGHS) areexplained; Simulation results and discussions are given inSection 4 and finally, conclusion is presented is Section 5.

2. Problem formulation

2.1. Solar cell models

Many models have been developed in the literature todescribe the I–V characteristics of solar cells, but onlytwo models are practically used. These models are simple,easy to solve and suitable for electrical engineering applica-tions. In the following subsections, these two models will bebriefly introduced.

2.1.1. Double diode model

Under illumination, an ideal solar cell is modeled as alight generated current source in parallel with a rectifyingdiode. However, in practice the current source is alsoshunted with another diode to consider the space chargerecombination current and a shunt leakage resistor to takeinto account the partial short circuit current path near thecell’s edges related to the semiconductor impurities andnon-idealities. Besides, a series resistor is connected withthe cell shunt elements due to the solar cell metal contactsand the semiconductor material bulk resistance (Wolfet al., 1977). Fig. 1 indicates the equivalent circuit of a dou-ble diode model.

From the figure, the terminal current, It, is obtained asfollows:

I t ¼ Iph � Id1 � Id2 � I sh ð1Þ

A. Askarzadeh, A. Rezazadeh / Solar Energy 86 (2012) 3241–3249 3243

where Iph is the photo-generated current, Id1 denotes thefirst diode current, Id2 is the second diode current and Ish

denotes the shunt resistor current.Using Schockley diode equation, relating the diode cur-

rent to its voltage and replacing the current of the shuntresistor, Eq. (1) is simplified to Eq. (2).

I t ¼ Iph � Isd1 expqðV t þ RsItÞ

n1kT

� �� 1

� �

� Isd2 expqðV t þ RsItÞ

n2kT

� �� 1

� �� V t þ RsIt

Rshð2Þ

where Isd1 and Isd2 are the diffusion and saturation currents,respectively, Vt is the terminal voltage, Rs and Rsh are theseries and shunt resistances, q is the electronic charge, k de-notes the Boltzmann constant, n1 and n2 are the diffusionand recombination diode ideality factors and T (K) is thecell temperature.

The model must be able to reflect the solar cell behavioras well as the actual performance. Hence, the main draw-back in accurate modeling is the extraction of the optimalvalues of the unknown parameters. For such a model thereare seven unknown parameters to be identified, namely, Rs,Rsh, Iph, Isd1, Isd2, n1 and n2. This aim can be done by thehelp of a capable optimization technique and a set of exper-imental data obtained from the real system.

2.1.2. Single diode model

Single diode model is the most common mathematicalrepresentation of the solar cell behavior. This model is builtby combining together both diode currents, under theintroduction of a non-physical diode ideality factor, n. Inrecent years, this model has been widely used to describethe solar cell I–V characteristic and fit the experimentaldata. The equivalent circuit of this model is indicated inFig. 2.

The representation of this model can be formulated asfollows:

I t ¼ Iph � Isd expqðV t þ RsItÞ

nkT

� �� 1

� �� V t þ RsIt

Rshð3Þ

Fig. 2. The single diode model of solar cell.

As can be seen, in the single diode model, five parame-ters need to be identified, namely, Rs, Rsh, Iph, Isd and n.

2.2. Optimization process

When an optimization algorithm is used to solve theparameter identification problem of the solar cell models,the following problems should be regarded: (1) how todefine a solution, (2) how to determine the search range,and (3) how to construct the objective function. In thispaper, each solution is defined by a vector, x, wherex = [Rs Rsh Iph Isd1 Isd2 n1 n2], when we consider the doublediode model and x = [Rs Rsh Iph Isd n] if the single diodemodel is considered. The upper and lower bounds of theparameters, provided by the literature survey, are shownin Table 1.

The definition of an objective function is the last stage.At first, Eq. (2) and Eq. (3) are rewritten in their homoge-neous forms as follows:

f ðV t; I t; xÞ ¼ I t � Iph þ Isd1 expqðV t þ RsItÞ

n1kT

� �� 1

� �

þ Isd2 expqðV t þ RsItÞ

n2kT

� �� 1

� �

þ V t þ RsIt

Rshð4Þ

f ðV t; I t; xÞ ¼ I t � Iph þ Isd expqðV t þ RsItÞ

nkT

� �� 1

� �

þ V t þ RsIt

Rshð5Þ

The value of f is calculated for each pair of the experi-mental data. We use the root mean square error (RMSE)as a criterion to quantify the difference between the modelresults and the experimental data. RMSE is defined by thefollowing equation.

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XN

i¼1

ðfiðV t; I t; xÞÞ2vuut ð6Þ

where N is the number of the experimental data.During optimization process, the objective function is to

be minimized with respect to the parameters range. Themodel parameters (five or seven based on the selectedmodel) are successively adjusted by the optimization algo-rithm, until a termination criterion is met. It is clear thatthe smaller the objective function value, the better the solu-

Table 1Upper and lower range of the solar cell parameters.

Parameter Lower Upper

Rs (X) 0 0.5Rsh (X) 0 100Iph (A) 0 1Isd (lA) 0 1n 1 2


tion is. It must be noticed that when the experimental dataare used, there is no information about the precise values ofthe parameters; therefore, any reduction in the objectivefunction value is significant since it results in improvementin the knowledge about the real values of the parameters.

3. Harmony search (HS)-based algorithms

In this section, HS algorithm and two of its state-of-the-art variants, namely, grouping-based global harmonysearch (GGHS) and innovative global harmony search(IGHS), are explained.

3.1. HS algorithm

In order to achieve a pleasing harmony, musicians try tooptimally adjust the pitches of their instruments. In musicimprovisation process, each musician sounds any pitchwithin the possible range. The pitches make together a har-mony. The quality of the improvised harmony is evaluatedusing aesthetic standard and the musicians try to sound abetter state of the harmony by punishing the instrumentspitch. This process continues until a perfect state of har-mony is obtained.

The attempt to find a perfect state of harmony in music isanalogous to finding the global optimal in an optimizationprocess. In other words, musician’s improvisation processcan be compared with the search process in optimization.The pitch of each musical instrument determines the aes-thetic quality, just as the value of decision variables willdetermine the quality of an objective function.

In analogous with the musician improvisation process,the optimization process seeks to discover a global opti-mum which the evaluation of a solution is carried out inputting values of decision variables into the objective func-tion. Based on this similarity, HS algorithm was devised.

For adjusting the pitch of their instruments, musiciansuse one of the following three rules: (1) playing a pitchfrom the memory, (2) playing a pitch close to a pitch fromthe memory and (3) playing a pitch from the possible rangeat random. In HS algorithm, each solution, named har-mony, is specified by a vector, x, including d elements equalto the problem dimension. At the beginning of the algo-rithm, a population of harmony vectors are randomly gen-erated in the possible range and stored in the harmonymemory (HM). A new harmony is then improvised. Eachdecision variable is adjusted by one of the three rules: (1)selecting a value from the HM, (2) selecting a value closeto one value from the HM and (3) selecting a value fromthe possible range at random. The worst harmony of theHM is removed and replaced by the new one if the qualityof the improvised harmony is better than that of the worstharmony.

The key parameters which have a profound effect on theHS performance are harmony memory considering rate(HMCR), pitch adjusting rate (PAR) and bandwidth ofgeneration (bw). These parameters can be potentially useful

in adjusting convergence rate of the algorithm to optimalsolution. A harmony is selected from the HM with theprobability of 1-HMCR. It is introduced to get away fromlocal optima when all parts of the global solution do notexist in the HM. The value of PAR will determine the prob-ability of generating a value near-by one value chosen fromthe HM. This parameter is used to improve the quality ofthe HM harmonies and bw provides a balance betweenlocal and global search. HMCR value varies between 0and 1. PAR is linearly decreased and the value of bw isexponentially decreased with the number of iterations asfollows (Mahdavi et al., 2007; Geem and Sim, 2010):

PARðtÞ ¼ PARmax �PARmax � PARmin

tmax

� t ð7Þ

bwðtÞ ¼ bwmax exp Lnðbwmin=bwmaxÞ �t

tmax

� �ð8Þ

where PARmax and PARmin are the maximum and mini-mum pitch adjusting rates, respectively, t denotes the iter-ation index, tmax is the maximum number of iterationsand bwmax, bwmin are the maximum and minimum band-widths, respectively.

The steps of the HS algorithm used in this paper, are asfollows:

Step 1. At First, harmony memory size (HMS), HMCR,PARmax, PARmin, tmax, bwmax and bwmin are valued.Step 2. The HM is initialized with HMS randomly har-monies in the search range using Eq. (9).

xiðjÞ ¼ lðjÞ þ a� ðuðjÞ � lðjÞÞ ð9Þwhere i = 1,2, . . . ,HMS is harmony’s index, j = 1,2, . . . , d

denotes the decision variable’s index, a is a random numberuniformly distributed from the interval [0,1] and l(j), u(j)are, respectively, the lower and upper bounds of jth deci-sion variable. Hence, the HM is a HMS � d matrix.

Step 3. The objective function value for each harmony iscalculated.

Step 4: A new harmony, xnew, is improvised as follows:

for j = 1:dif r1 P HMCR

xnewðjÞ ¼ lðjÞ þ r2 � ðuðjÞ � lðjÞÞ;else

n = the index of a harmony from HM

xnewðjÞ ¼ HMðn; jÞ;if r3 < PAR

xnewðjÞ ¼ xnewðjÞ þ ðr4 � r5Þ � bw� juðjÞ � lðjÞj;end

end

end

where r1, r2, r3, r4 and r5 are random numbers from theinterval [0, 1].

olar Energy 86 (2012) 3241–3249 3245

Step 5: If the new improvised harmony is in the search
space, its objective function value is computed. If it isbetter than the worst harmony of the HM, the worstharmony is removed and the new one is replaced, other-wise, it is abandoned.Step 6: Step 4 and step 5 are repeated until a predefinednumber of iterations, tmax, is met.
A. Askarzadeh, A. Rezazadeh / S

3.2. GGHS and IGHS algorithms

With the aim of better use from the high-quality HMharmonies, GGHS and IGHS have been developed bythe authors. GGHS attempts to provide an efficient wayfor using the best harmonies of the HM by consideringthe fact that some of the worse harmonies may have usefulinformation and by using them the optimization algorithmcan more easily reach the optimal solution. In GGHS, HMharmonies are divided equally into three groups accordingto their quality. When a new harmony needs to be impro-vised, two probabilistic approaches are employed. The firstapproach is employed to specify the interesting group andthe second one is used to choose the interesting harmony, e,from the selected group. Tournament selection and roulettewheel are used as the first and second probabilisticapproaches, respectively. In the tournament selectionapproach, weak groups have a smaller chance of beingselected. In roulette wheel approach, as the quality of aharmony increases, the probability of its selectionincreases, too. Hence, HM harmonies with better qualityhave a more chance of being selected for improvisationprocess. A more detailed description of the algorithm canbe found in the literature (Askarzadeh and Rezazadeh,2011). The steps of the GGHS are the same as those ofHS except that the step 4 is replaced as follows:

Table 2A comparison between the results obtained by the HS-based algorithmsand the other ones for the double diode model parameter identification.



select interesting group;select interesting harmony;xnewðjÞ ¼ eðjÞ;if r3 < PAR


end

end
Item HS GGHS IGHS SA PS
Rs (X) 0.03545 0.03562 0.03690 0.0345 0.0320Rsh (X) 46.82696 62.7899 56.8368 43.1034 81.3008Iph (A) 0.76176 0.76056 0.76079 0.7623 0.7602Isd1 (lA) 0.12545 0.37014 0.97310 0.4767 0.9889Isd2 (lA) 0.25470 0.13504 0.16791 0.0100 0.0001n1 1.49439 1.49638 1.92126 1.5172 1.6000n2 1.49989 1.92998 1.42814 2.0000 1.1920RMSE 0.00126 0.00107 9.8635e�4 0.01664 0.01518

IGHS algorithm proposes that a predefined number ofHM harmonies with the best quality are selected as the eliteharmonies. Then, in order to improvise a new harmony, aprobabilistic approach (roulette wheel) is used to select oneof them as the interesting elite harmony, e, for improvisa-tion process. By taking this way into account the probabil-

ity of generating a harmony with better quality increases,since the new harmony is improvised using the informationof the best harmonies. A more detailed description of thedevised algorithm can be found in the literature (Askar-zadeh and Rezazadeh, 2012). The steps of the IGHS aresame as those of HS except that the step 4 is replaced asfollows:



select elite harmonies;select interesting elite harmony;xnewðjÞ ¼ eðjÞ;if r3 < PAR


end

end

4. Results and discussion

In order to study the efficiency of the HS-based algo-rithms in parameters identification of the solar cell models,a 57 mm diameter commercial (R.T.C. France) silicon solarcell is considered. The experimental data has been adoptedfrom the system under 1 sun (1000 W/m2) at 33 �C (Easwa-rakhanthan et al., 1986). The parameter setting for HS-based algorithms is HMS = 30, HMCR = 0.95, PAR-

max = 0.7, PARmin = 0.1, bwmax = 1, bwmin = 0.0001,tmax = 5000 and the number of the elite harmonies usedin IGHS is set to 5. It is worthwhile to mention that theparameter setting is based on the trial and no attempthas made to optimize it.

The algorithms are coded and executed in the Matlabenvironment to identify the cell parameters using the singleand double diode models. Table 2 summarizes the resultsof HS, GGHS and IGHS for double diode model in com-parison with the results found by simulated annealing (SA)(El-Naggar et al., 2012) and pattern search (PS) (AlHajriet al., 2012). In this table, the optimal parameters have

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.2057

-0.0588

0.0646

0.1678

0.2545

0.3269

0.3873

0.4373

0.4784

0.5119

0.5398

0.5633

0.5833

Vt (V)

It (

A)

It measured (A)It calculated (A)

Fig. 3. Comparison between the I–V characteristics obtained by theexperimental data and the double diode model identified by IGHSalgorithm.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-0.2057

-0.0588

0.0646

0.1678

0.2545

0.3269

0.3873

0.4373

0.4784

0.5119

0.5398

0.5633

0.5833

Vt (V)

Pt

(W)

Pt measured (W)Pt calculated (W)

Fig. 4. Comparison between the P–V characteristics obtained by theexperimental data and the double diode model identified by IGHSalgorithm.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Iteration

Obj

ecti

ve f

unct

ion

Fig. 5. Convergence process of IGHS algorithm during the identificationprocess.

Table 3Mean absolute error (MAE) based on the extracted parameters (doublediode model).

Measurement Vt (V) It measured(A)

It calculated(A)

Relativeerror

1 �0.2057 0.764 0.76391 0.0001182 �0.1291 0.762 0.762563 �0.000743 �0.0588 0.7605 0.761327 �0.001094 0.0057 0.7605 0.760191 0.0004075 0.0646 0.76 0.759149 0.0011196 0.1185 0.759 0.758184 0.0010757 0.1678 0.757 0.757266 �0.000358 0.2132 0.757 0.756331 0.0008849 0.2545 0.7555 0.755261 0.00031610 0.2924 0.754 0.753789 0.00027911 0.3269 0.7505 0.751429 �0.0012412 0.3585 0.7465 0.747282 �0.0010513 0.3873 0.7385 0.739926 �0.0019314 0.4137 0.728 0.727156 0.00115915 0.4373 0.7065 0.706708 �0.0002916 0.459 0.6755 0.675113 0.00057217 0.4784 0.632 0.630808 0.00188618 0.496 0.573 0.572108 0.00155719 0.5119 0.499 0.49958 �0.0011620 0.5265 0.413 0.41359 �0.0014321 0.5398 0.3165 0.317282 �0.0024722 0.5521 0.212 0.212113 �0.0005323 0.5633 0.1035 0.102689 0.00783924 0.5736 �0.01 �0.00929 0.070625 0.5833 �0.123 �0.1244 �0.0113426 0.59 �0.21 �0.20915 0.004039MAE 0.0044


been tabulated along with the RMSE values. As can beseen, HS-based algorithms outperform the other ones sincethey produce smaller RMSE value than SA and PS.Though, among the HS algorithms, the best performancebelongs to IGHS with RMSE = 9.8635e�4, the RMSE val-ues found by HS and GGHS are slightly worse than that ofIGHS.

In order to confirm the accuracy of the identified param-eters, the optimal parameters found by the HS algorithmsare returned to the double diode model and the I–V char-acteristic is reconstructed. This is simply performed byapplying Newton method when It is unknown while Vt isknown. The obtained results are plotted along with theexperimental data to observe the agreement between them.Since the characteristics are very close to each other, thecharacteristic found by IGHS algorithm is only illustrated.Figs. 3 and 4 indicate that the I–V and P–V characteristics

are in accordance with the experimental data. In Fig. 3,correlation coefficient and root mean square percentagedeviation between the calculated values and the measuredvalues are 1 and 0.076, respectively. These values are 1and 0.033 for Fig. 4. Based on the correlation coefficientsand root mean square percentage deviations, it is observedthat there is good agreement between the experimental dataand the model results. The convergence rate of the IGHS

Table 4A comparison between the results obtained by the HS-based algorithms and the other ones for the single diode model parameter identification.

Item HS GGHS IGHS SA PS GA CPSO

Rs (X) 0.03663 0.03631 0.03613 0.0345 0.0313 0.0299 0.0354Rsh (X) 53.5946 53.0647 53.2845 43.1034 64.1026 42.3729 59.012Iph (A) 0.76070 0.76092 0.76077 0.7620 0.7617 0.7619 0.7607Isd (lA) 0.30495 0.32620 0.34351 0.4798 0.9980 0.8087 0.4000n 1.47538 1.48217 1.48740 1.5172 1.6000 1.5751 1.5033RMSE 9.9510e-4 9.9097e-4 9.9306e-4 0.01900 0.01494 0.01908 0.00139

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.2057

-0.0588

0.0646

0.1678

0.2545

0.3269

0.3873

0.4373

0.4784

0.5119

0.5398

0.5633

0.5833

Vt (V)

It (

A)

It measured (A)It calculated (A)

Fig. 6. Comparison between the I–V characteristics obtained by theexperimental data and the single diode model identified by GGHSalgorithm.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-0.2057

-0.0588

0.0646

0.1678

0.2545

0.3269

0.3873

0.4373

0.4784

0.5119

0.5398

0.5633

0.5833

Vt (V)

Pt

(W)

Pt measured (W)

Pt calculated (W)

Fig. 7. Comparison between the P–V characteristics obtained by theexperimental data and the single diode model identified by GGHSalgorithm.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Iteration

Obj

ecti

ve f

unct

ion

Fig. 8. Convergence process of GGHS algorithm during the identificationprocess.


algorithm during the identification process is also plottedin Fig. 5, which represents the best objective function valueat each iteration.

In order to confirm the quality of the fit to experimentaldata, the statistical analysis based on mean absolute error(MAE) is made. Table 3 shows the relative error, e = (Imea-

sured � Icalculated)/Imeasured, for each measurement along withthe MAE, MAE ¼ 1

N

PNi¼1jeij. The calculated value of the

MAE denotes the high accuracy of the identificationprocess.

The results of the HS-based algorithms and those foundby simulated annealing (SA) (El-Naggar et al., 2012), pat-tern search (PS) (AlHajri et al., 2012), genetic algorithm(GA) (AlRashidi et al., 2011) and chaos particle swarmoptimization algorithm (CPSO) (Wei et al., 2011) inparameter identification of the solar cell single diode modelare listed in Table 4. It is clear that the HS algorithms out-perform the other studied algorithms. In this case, GGHSproduces better results than HS and IGHS. Just as before,the optimal parameters are put into the single diode modeland Newton method is used to reconstruct the I–V charac-teristic. Figs. 6 and 7 indicate I–V and P–V characteristicsfound by GGHS algorithm. In Fig. 6, correlation coeffi-cient and root mean square percentage deviation betweenthe calculated values and the measured values are 1 and0.078, respectively. These values are 1 and 0.033 forFig. 7. Based on the correlation coefficients and root meansquare percentage deviations, it is observed that there isgood agreement between the experimental data and themodel results. The value of the objective function duringthe identification process of the GGHS algorithm is alsoindicated in Fig. 8. Table 5 represents the relative errorand the mean absolute error (MAE) for the case of the sin-gle diode model. As can be seen, the accuracy of the param-eters identification process is high.

Table 5Mean absolute error (MAE) based on the extracted parameters (singlediode model).

Measurement Vt (V) It measured(A)

It calculated(A)

Relativeerror

1 �0.2057 0.764 0.764274 �0.000362 �0.1291 0.762 0.762832 �0.001093 �0.0588 0.7605 0.761508 �0.001334 0.0057 0.7605 0.760293 0.0002725 0.0646 0.76 0.759181 0.0010786 0.1185 0.759 0.758155 0.0011137 0.1678 0.757 0.757193 �0.000258 0.2132 0.757 0.756232 0.0010159 0.2545 0.7555 0.755167 0.00044110 0.2924 0.754 0.753734 0.00035311 0.3269 0.7505 0.751446 �0.0012612 0.3585 0.7465 0.747394 �0.001213 0.3873 0.7385 0.740129 �0.0022114 0.4137 0.728 0.727416 0.00080215 0.4373 0.7065 0.706963 �0.0006616 0.459 0.6755 0.675302 0.00029317 0.4784 0.632 0.630898 0.00174418 0.496 0.573 0.57211 0.00155319 0.5119 0.499 0.499537 �0.0010820 0.5265 0.413 0.413554 �0.0013421 0.5398 0.3165 0.317286 �0.0024822 0.5521 0.212 0.212161 �0.0007623 0.5633 0.1035 0.102756 0.00718824 0.5736 �0.01 �0.00925 0.07525 0.5833 �0.123 �0.12444 �0.0117126 0.59 �0.21 �0.2093 0.003333MAE 0.0046


When we compare the accuracy of the double and singlediode models in fitting the experimental data, it can bedrawn that the double diode model is slightly more accu-rate than the single diode model to describe the solar cellbehavior, because the smallest RMSE value in doublediode model is 9.8635e�4 found by IGHS, while this valueis 9.9097e�4 in single diode model found by GGHS. How-ever, the error of 0.47% implies the suitability and precisionof the single diode model.

5. Conclusion

This paper proposes harmony search (HS) algorithm forparameter identification of the solar cell mathematicalmodels, namely, single and double diode models. The mainaim is to acquire an accurate I–V characteristic of a realsystem, a 57 mm diameter commercial (R.T.C. France) sil-icon solar cell, by identifying the unknown parameters.Three HS variants are used to extract the optimal parame-ters of both the single and double diode models. It isrevealed that the accuracy of the double diode model isslightly more than that of the single diode model. Resultsobtained using HS variants are quite promising and supe-rior, especially when are compared to the other methods.As a result, HA-based algorithms are capable approachesand can be efficiently applied to parameter identificationof the solar cell models.

Acknowledgements

The authors would like to thank from the research vice-presidency of Shahid Beheshti University for financial sup-port under Grant Number 600/1787.

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parameter identification for solar cell models using harmony search-based algorithms

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