Parameter identification for solar cell models using harmony search-based algorithms
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energy sources as the demand for energy grows, but also 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,
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sbu.ac.ir (A. Rezazadeh).
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Solar Energy 86 (2012)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 more
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 (IV) curve. Several models have been developedto represent the behavior of the system under dierentoperating conditions (Han et al., 2004; Villalva et al.,2009; Huld et al., 2010). They vary from models with sim-Abstract
Recently, accurate modeling of current vs. voltage (IV) 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 identication with the help of an optimization techniqueis necessary. Because IV curve of solar cells is extremely non-linear, an excellent optimization technique is required. In this paper, har-mony search (HS)-based parameter identication 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 eectiveness of the HS variants is investigated with comparative study among dierent 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 identication; Harmony search algorithmParameter identication for sosearch-base
Faculty of Electrical and Computer Engineering, Shahid
Received 4 May 2012; received in revised foAvailable online
Communicated by: Asso0038-092X/$ - see front matter 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.solener.2012.08.018r cell models using harmonyalgorithms
heshti University, G.C., Evin 1983963113, Tehran, Iran
28 August 2012; accepted 29 August 2012September 2012
e Editor Nicola Romeo
IGHS have been developed by the authors to improvethe search capability of the original HS by maintainingthe algorithms 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 nally, conclusion is presented is Section 5.
shunted with another diode to consider the space charge
I t Iph Id1 Id2 I sh 1
olar Energy 86 (2012) 32413249diode 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, ll factor and conversion eciency, that areuseful in analyzing performance losses, are derived fromthe IV curve and the information provided by the modelparameters (Macabebe et al., 2011). Therefore, the mainproblem is the identication of the optimal values of theparameters by which the model can produce very good tsto 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 dierentiability 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 IV curvemakes the traditional optimization techniques unable toeectively solve the parameter identication 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 identication 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 musicians improvisation processcan be compared with the search process in optimization.
The simplicity and eectiveness 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-
3242 A. Askarzadeh, A. Rezazadeh / Sters of a 57 mm diameter commercial (R.T.C. France) sili-con solar cell (Easwarakhanthan et al., 1986). GGHS andrecombination current and a shunt leakage resistor to takeinto account the partial short circuit current path near thecells 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 gure, the terminal current, It, is obtained asfollows:2. Problem formulation
2.1. Solar cell models
Many models have been developed in the literature todescribe the IV 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 bebriey 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 alsoFig. 1. The double diode model of solar cell.
where Iph is the photo-generated current, Id1 denotes therst diode current, Id2 is the second diode current and Ishdenotes 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 simplied to Eq. (2).
I t Iph Isd1 exp qV t RsItn1kT
Isd2 exp qV t RsItn2kT
V t RsItRsh
where Isd1 and Isd2 are the diusion and saturation currents,respectively, Vt is the terminal voltage, Rs and Rsh are theseries and shunt resistances, q is the electronic charge, k de-
As can be seen, in the single diode model, ve parame-ters need to be identied, namely, Rs, Rsh, Iph, Isd and n.
2.2. Optimization process
When an optimization algorithm is used to solve theparameter identication problem of the solar cell models,
A. Askarzadeh, A. Rezazadeh / Solar Energy 86 (2012) 32413249 3243notes the Boltzmann constant, n1 and n2 are the diusionand recombination diode ideality factors and T (K) is thecell temperature.
The model must be able to reect 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 identied, 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 IV characteristic and t the experimentaldata. The equivalent circuit of this model is indicated inFig. 2.
The representation of this model can be formulated asfollows:
I t Iph Isd exp qV t RsItnkT
V t RsItRsh
3Fig. 2. The single diode model of solar cell.the following problems should be regarded: (1) how todene a solution, (2) how to determine the search range,and (3) how to construct the objective function. In thispaper, each solution is dened 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 denition of an objective function is the last stage.At rst, Eq. (2) and Eq. (3) are rewritten in their homoge-neous forms as follows:
f V t; I t; x I t Iph Isd1 exp qV t RsItn1kT
Isd2 exp qV t RsItn2kT
V t RsItRsh
f V t; I t; x I t Iph Isd exp qV t RsItnkT
V t RsItRsh
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 dierence between the modelresults and the experimental data. RMSE is dened by thefollowing equation.
fiV t; I t; x2vuut 6
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 (ve 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 1
Isd (lA) 0 1n 1 2
olation 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 signicant 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 nd a perfect state of harmony in music isanalogous to nding the global optimal in an optimizationprocess. In other words, musicians 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 specied 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 wors...