Pe

Ma

b

a

ARR2AA

KPFPEEH

1

etdftfcedPt

pk[aAa

m

1h

Applied Soft Computing 13 (2013) 4608–4621

Contents lists available at ScienceDirect

Applied Soft Computing

j ourna l ho me page: www.elsev ier .com/ locate /asoc

arameter estimation for five- and seven-parameter photovoltaiclectrical models using evolutionary algorithms

.U. Siddiquia,∗, M. Abidob

Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi ArabiaDepartment of Electrical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

r t i c l e i n f o

rticle history:eceived 28 June 2012eceived in revised form1 December 2012ccepted 7 July 2013vailable online 22 July 2013

a b s t r a c t

Equivalent electric circuit modeling of PV devices is widely used to predict PV electrical performance. Thefirst task in using the model to calculate the electrical characteristics of a PV device is to find the modelparameters which represent the PV device. In the present work, parameter estimation for the modelparameter using various evolutionary algorithms is presented and compared. The constraint set on theestimation process is that only the data directly available in module datasheets can be used for estimatingthe parameters. The electrical model accuracy using the estimated parameters is then compared to severalelectrical models reported in literature for various PV cell technologies.

eywords:hotovoltaicive parameter modelarameter estimationquivalent circuit modelvolutionary algorithmsybrid evolutionary algorithms

© 2013 Elsevier B.V. All rights reserved.

. Introduction

Accurate modeling tools are a key to designing efficient and cost-ffective PV systems. In total, three different models are requiredo model the electrical power output of a PV system for given irra-iance and ambient temperature. These include a thermal modelor finding the PV cell temperature, a radiation model for findinghe solar energy absorbed in the PV cells and an electrical modelor calculating the electrical characteristics of the PV system for thealculated absorbed radiation and cell temperature. Over the years,lectrical models for varied complexities and accuracies have beeneveloped for PV system. These include analytical models based onV cell physics, empirical models and a few models which combinehese two approaches.

An empirical model was developed by King et al. [1] capable ofredicting the electrical current and voltage of a PV device at fiveey points on the I–V curve. Hishikawa et al. [2] and Marion et al.3] used interpolation techniques for determining the I–V curves

t the input conditions using the I–V curves at known conditions.

different approach for modeling PV devices is to represent it byn equivalent electric circuit. Any PV device can be modeled using

∗ Corresponding author. Tel.: +966 38602096.E-mail addresses: musiddiqui@kfupm.edu.sa (M.U. Siddiqui),

abido@kfupm.edu.sa (M. Abido).

568-4946/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.asoc.2013.07.005

the equivalent circuit model by using the correct model parame-ters (IL, Io, a, Rs, Rsh). Townsend [4] presented an electric circuitmodel called the Four Parameters model for predicting the perfor-mance of PV devices in which he assumed the shunt resistance (Rsh)to be infinite which reduced the nonlinearity of the model. Duffieand Beckman [5] improved the model by Townsend by includingan additional parallel resistance in the electric circuit. Their fiveparameters model gave improved prediction accuracies for thinfilm PV cell types. De Soto et al. [6] developed a methodology forfinding these five model parameters using only manufacturer data.Valerio et al. [7] modified the five parameters model to capturechanges in operating temperature and solar irradiance more accu-rately.

In order to use the electric circuit models, the model parame-ters (IL, Io, a, Rs, Rsh) which are different for every PV device needto be first determined. A variety of techniques have been used tofind these electrical model parameters. For the single diode fiveparameters model, De Soto et al. [6] and Boyd et al. [8] used a spe-cialized non-linear equation solver to get a solution. Villalva et al.[9] explicitly defined one parameter, a, and then solved for theremaining parameters by minimizing the error in the maximumpower prediction. Townsend [4] simplified the model by assuming

the shunt resistance to be infinite which reduces the non-linearityof the system. He then solved for the remaining parameters iter-atively. Carrero et al. [10] used an iterative procedure to find allfive parameters, (IL, Io, a, Rs, Rsh). Their method only requires the

M.U. Siddiqui, M. Abido / Applied Soft C

Nomenclature

a modified diode ideality factor (V)A diode ideality factorAM air massAOI angle of incidenceE irradiance (W/m2)Eg band-gap energy of PV cell material (eV)FF fill factorfd fraction of diffuse radiation absorbed in the moduleI PV module output current (A)IL light current (A)Io diode reverse saturation current (A)k Boltzmann’s constant, 1.38066E−23 (J/K)m irradiance dependence parameter for ILn temperature dependence parameter for aNCS number of cells in series in a module’s cell-stringNp number of cell-strings in parallel in moduleNs number of cells in series in a module’s cell-stringP electrical power (W)q elementary charge, 1.60218 × 10−19 (coulomb)Rs series resistance (�)Rsh shunt resistance (�)S plane-of-array absorbed solar radiation at operating

conditions (W/m2)T temperature (◦C)V voltage (V)

Greek letters˛imp Temperature coefficient of maximum power point

current˛Isc temperature coefficient of short circuit currentˇVoc temperature coefficient of open circuit voltageˇVmp temperature coefficient of maximum power point

voltageı(Tc) thermal voltage per cell at temperature Tc

�Voc temperature coefficient of open circuit voltage� overall diode ideality factor of PV module

Subscripts0 reference cell conditionamb ambientb beam radiationc PV celldiff diffuse radiatione effective radiation; experimentalm module back surface; modeledmp maximum power pointoc open circuit pointref reference cell conditionsc short circuit pointx IV point at module voltage equal to half of open

circuit voltagexx IV point at module voltage equal to average of max.

power and open circuit voltages

Imtc

mL

amb ambient

–V data of three points i.e. the short circuit, open circuit andaximum power points. Their method uses simplified forms of

he I–V equation written at the three points and provides fast

onvergence.

Various optimization techniques also been used for deter-ining the five model parameters. Ikegami et al. [11] used the

evenberg–Marquardt multi-variable optimization technique with

omputing 13 (2013) 4608–4621 4609

experimentally determined I–V curve to determine the modelparameters. The objective function used by Ikegami et al. was theerror in the current prediction at known voltages as calculated byEq. (1).

error =√∑n

i=1(Im(Vi) − Ie(Vi))2

n(1)

Siddiqui [12] used the simplex search algorithm to find the fivemode parameters by minimizing the fitness function defined byEq. (2). The objective function calculates two errors whose sum isminimized. First, the currents (I) at short circuit, maximum powerpoint and open circuit conditions are calculated using Eq. (4) andknown voltages from the module datasheet.

error =

√∑3i=1(Im(Vi) − Ie(Vi))

2

3+

∣∣∣ dP

dV

∣∣∣MPP

(2)

With the recent advancement in computing, the use of intel-ligent computing techniques has increased greatly. Intelligenttechniques include Fuzzy control [13–18], evolutionary algorithms[19–21] and Neural Networks [22,23] which are being appliedto engineering problems ranging from control to optimizationwith impressive success. For the estimation of PV electrical modelparameters, genetic algorithm is the most widely used evolution-ary algorithm [24–26]. The objective function used in all theseworks was the error in the current prediction at known volta-ges. Moldovan et al. [24] and Zagrouba [26] carried out similarworks in which they used genetic algorithm to minimize the errorin the current prediction at known voltages. Jervase [25] usedgenetic algorithm to find seven parameters for the two-diodeelectric circuit model. Ishaque et al. [27,28] used several evolu-tionary algorithms to find the model parameters for a two-diodeequivalent electric circuit PV model and found that penalty-baseddifferential evolution showed good accuracy and consistency ofsolution, good speed of convergence and required very low controlparameters.

For their Seven Parameters model, Siddiqui [12] used thesimplex-search algorithm to first find the five parameters. The twoadditional parameters m and n were found using a secondary opti-mization in which the original five parameters did not change. Eq.(3) was used as the objective function for the secondary optimiza-tion process.

error =√

(Pm − Pe)2∣∣∣Low Irradiance

+√

(Pm − Pe)2∣∣∣High Temeprature

(3)

In this paper, a methodology to estimate the model parametersusing only manufacturer supplied electrical performance data ispresented and the effectiveness of various evolutionary algorithms,including standard evolutionary algorithms as well as hybrid meth-ods, in estimating the model parameters is evaluated. Finally, theaccuracy of the parameter estimation methodology is checked bycomparing the results of the electrical model to other models fromliterature as well as the same model using different parametersestimation methodologies.

2. Electric circuit modeling of PV devices

Any PV device can be represented by an equivalent electric cir-cuit [5]. The equivalent circuit, shown in Fig. 1, comprises of a light

dependent current source, a p-n junction diode and two resistances.

I–V relationship in the equivalent circuit of Fig. 1 is governedby Eq. (4). The characteristic of any PV device are included in themodel by five model parameters (IL, Io, a, Rs and Rsh). The model that

4610 M.U. Siddiqui, M. Abido / Applied Soft Computing 13 (2013) 4608–4621

dF

I

PkPlpP

a

I

I

R

R

ptoPtca

E

sEe

I

a

3

mfiemp

aa

Table 1Assumed electrical model parameters.

Parameter Value

IL,ref 2.31 AIo,ref 2 × 10−10 Aaref 1 V−1

Rs,ref 1 �Rsh,ref 75,000 �

Table 2Electrical performance at reference condition for assumed parameters.

Characteristics Value

Short circuit current (Isc) 2.3099 AOpen circuit voltage (Voc) 23.1698 VMPP current (I ) 2.1737 A

Fig. 1. Equivalent circuit of a PV cell.

escribes the electrical performance of a PV device represented byig. 1 using Eq. (4) is called the five parameter model.

= IL − Io ·(

exp(

V + I · Rs

a

)− 1

)− V + I · Rs

Rsh(4)

As stated before, the five model parameters are vary with eachV device. Once the parameters at the reference condition arenown, they can be used to calculate the I–V characteristics of theV system at any other condition. Eqs. (5)–(9) are used to calcu-ate the parameters at conditions other than reference. Using thesearameters, Eq. (4) can be used to predict the performance of theV system at any temperature and irradiance.

= aref

(Tc

Tref

)(5)

L = S

Sref(IL,ref + �Isc(Tc − Tref)) (6)

o = Io,ref

(Tc

Tref

)3e((NCS.Tref/aref)((Eg,ref/Tref)−(Eg/Tc))) (7)

sh = Sref

SRsh,ref (8)

s = Rs,ref (9)

The subscript ref represents the parameters at STC, T is the tem-erature of the PV panel, S is the absorbed solar radiation, �Isc is theemperature coefficient of short circuit current, NCS is the numberf cells connected in series and Eg is the band-gap energy of theV cell material. Eq. (10) can be used to calculate Eg at the newemperature. The constant 0.0003174 is for mono-crystalline sili-on and De Soto [6] has suggested that this value can be used forll technologies with little error.

g = Eg,ref

(1 − 0.0003174

(Tc

Tref

))(10)

Siddiqui [12] modified Eqs. (5) and (6) based on the results of aensitivity analysis to improve the accuracy of the electrical model.qs. (11) and (12) are the modified translation equations for param-ters IL and a.

L =(

G

Gref

)m

(IL,ref + �Isc(Tc − Tref)) (11)

= aref

(Tc

Tref

)n

(12)

. Estimation problem formulation

Evolutionary algorithms provide enhanced optimization perfor-ance for non-convex problems. The objective functions used for

nding the electrical model parameters are non-convex. Therefore,volutionary algorithms are expected to provide better perfor-ance than conventional optimization techniques this estimation

roblem.The fitness function used in the algorithms is given by Eq. (13)

nd is equal to the sum of absolute errors in current and volt-ge predictions at short circuit, open circuit and maximum power

mp

MPP voltage (Vmp) 18.1634 VNumber of cells in series (NCS) 36

points and the slope of the power-voltage curve at maximum powercondition.

normalized error = abs

(IMP,mod − IMP,exp

IMP,exp

)

+ abs

(VMP,mod − VMP,exp

VMP,exp

)

+ abs

(Isc,mod − Isc,exp

Isc,exp

)

+ abs

(Voc,mod − Voc,exp

IMP,exp

)

+ abs(Ioc) + abs(

dP

dV

)MP

(13)

In order to compare various evolutionary algorithms, a test casewith known parameters was assumed. The electrical characteristicsat the reference condition were calculated using the parameters.Various evolutionary algorithms were used to estimate the param-eters using the electrical characteristics at the reference conditionand their performance was compared. The assumed parameters forthe PV module and its characteristics at the reference condition aregiven in and respectively (Table 1).

4. Comparing the performance of evolutionary algorithms

Using the test case developed in section 3, the performance ofvarious standard and hybrid evolutionary algorithms was evalu-ated and compared. For comparison, several runs of each algorithmwere performed and the value of the fitness function and itsstandard deviation was compared.

4.1. Performance of standard evolutionary algorithms

Using the data available in Table 2, the electrical model param-eters were estimated using genetic algorithm [29], differentialevolution [30] and particle swarm optimization [21]. The popu-lation size was set to 500 members and the maximum allowablegenerations were set to 500 as well. The simulation parametersused for the three algorithms are listed in Table 3. It is worth men-tioning that these parameters are the best among several trials.

Five runs were conducted for each algorithm to check its accu-racy and repeatability. The results are presented in Table 4. The bestperformance was shown by differential evolution with an averagefitness function value of 0.004208 at the end of 500 iterations. It

M.U. Siddiqui, M. Abido / Applied Soft Computing 13 (2013) 4608–4621 4611

Table 3Simulation parameters for GA, DE and PSO.

Genetic algorithm Differential evolution Particle swarm optimization

Population 500 Population 500 Population 500Generations 500 Generations 500 Generations 500Pcrossover 0.9 Crossover ratio 0.5 C1 1Pmutation 0.01 Mutation factor 0.8 C2 1Ptournament 0.8 wmax 1% Elitism 0.05 wmin 0.7

Table 4Fitness function values in five runs for GA, DE and PSO.

Genetic algorithm Differential evolution Particle swarm optimization

Minimum 0.02918 Minimum 0.000452 Minimum 0.008898Maximum 0.039875 Maximum 0.007184 Maximum 0.018042Average 0.03429 Average 0.004208 Average 0.011307Std. deviation 0.004266 Std. deviation 0.002788 Std. deviation 0.005288

Table 5Simulation parameters for hybrid techniques.

TS assisted DE PSO assisted DE DE assisted TS

Population 500 Population 500 TS generations 50Generations 200 Generations 250 Local DE population 50Crossover ratio 0.5 Crossover ratio 0.5 DE generations 100Mutation factor 0.8 Mutation factor 0.8 Local search interval 0.5TS generations 200 PSO pop. size (initial sent by DE) 100 DE crossover ratio 0.5TS trial solutions 10 PSO generations 50 DE mutation factor 0.8Unchanged generations to start TS 5 C1 1

wtmwhtTr

4

ptiade

4

owSapwTST

123

Top individuals toimprove 5

C2

wmax

wmin

as also the most consistent with the lowest standard deviation ofhe three algorithms. Genetic algorithm showed the worst perfor-

ance of the three with an average fitness function value of 0.03429hile particle swarm optimization was the most inconsistent withighest standard deviation value of 0.005288. The simulation timeaken by the three algorithms with the simulation parameters ofable 3 were 1302.1 s, 1015.1 s and 3720.7 s for GA, PSO and DEespectively.

.2. Hybrid techniques

In order to further improve the accuracy of the estimationrocess, three hybrid techniques were tried. The objective waso further reduce the value of the objective function while notncreasing the simulation time significantly. In total, three hybridlgorithms were developed by hybridizing Tabu Search [20] withifferential evolution and particle swarm optimization with differ-ntial evolution.

.2.1. Tabu Search assisted differential evolutionThe problem faced during the estimation process was that the

ptimal solution remained unchanged for many generations thusasting valuable computational time. In order to avoid this, Tabu

earch was incorporated in the differential evolution algorithm. Inddition to the differential evolution parameters and Tabu Searcharameters, two additional parameters needed to be defined. Theseere the number of unchanged optimal solution iterations to start

abu Search and the number of top solutions to include in Tabuearch. The methodology of the TS DE algorithm is presented below.he process flowchart is presented in Fig. 2.

. Start conventional DE

. Mutate and crossover to generate trial solution.

. Form new population using conventional DE methodology.

110.7

4. Update optimal solution if required. Else, update counter forunchanged iterations.

a. If the optimal solution remains unchanged for a certain pre-determined number of generations, ntabu,start, call Tabu Search.Tabu Search is activated with each of the top ntabu,pop solutionsin the current DE generation as the starting point for TS.

4.2.2. Particle swarm optimization assisted differential evolutionIn the PSO assisted DE algorithm, the PSO algorithm is activated

after every 5 generations. The fraction of the best solutions in thecurrent generation is passed to the PSO algorithm as the initial pop-ulation and the PSO algorithm returns the final population at theend of the algorithm. In order to diversify the population, the worst5% solutions in every generation are replaced in random solutionsduring every iteration. The process flow chart of the PSO assistedDE algorithm is shown in Fig. 3.

4.2.3. Differential evolution assisted Tabu SearchIn the differential evolution assisted Tabu Search algorithm,

differential evolution is used to search for the optimal solutionwithin a subset of the whole search space while the Tabu Search isused to move the local search within the global space. The processflowchart of the technique is shown in Fig. 4. The main steps of thealgorithm are as follows.

1. The algorithm starts by assuming an initial solution xcurrent

within the global limits of the variables.2. The local gene limits are adjusted to be around xcurrent. A local

population is generated within the local gene limits. The current

solution xcurrent and the global optimal point are included in thepopulation since they may include important genetic features.

3. The differential evolution algorithm is called to determine thelocal optimal point.

4612 M.U. Siddiqui, M. Abido / Applied Soft Computing 13 (2013) 4608–4621

Fig. 2. Process flowchart of TS assisted DE algorithm.

Fig. 3. Process flowchart of PSO assisted DE algorithm.

M.U. Siddiqui, M. Abido / Applied Soft C

4

56

4

pfif

ntnaawta

5

oascfi

Fig. 4. Process flowchart of DE assisted TS algorithm.

. If local optimal point is better than previous global optimal point,the global optimal point is set equal to the local optimal point.

. The local optimal point is set as xcurrent.

. The process is repeated until a stopping criterion is met.

.2.4. Performance of hybrid techniquesAll three hybrid techniques were used to estimate the model

arameters using the data in Table 2. The process was repeatedve times for each of the algorithms. The simulation parameters

or the three hybrid techniques are listed in Table 5.The simulation results are presented in Table 6. All three tech-

iques provided lower average fitness values than the standardechniques and with higher consistencies. Between the three tech-iques, DE assisted TS technique provided the lowest fitness valuesnd the highest consistency. The simulation time taken by TSssisted DE and PSO assisted DE using the parameters of Table 5ere 4079.9 s and 3135.1 s, respectively, which are comparable to

he simulation time for conventional DE. On the other hand DEssisted TS algorithm took 240.0 s for completion.

. Validation of parameter estimation methodology

In order to validate the proposed parameter estimation method-logy and to compare its accuracy with the accuracy of models

vailable in literature, a total of six PV modules, three crystallineilicon modules and three thin film modules, were selected. The I–Vurves available in the datasheets of the modules were digitized andve key I–V points were extracted from each curve. These points are

omputing 13 (2013) 4608–4621 4613

short circuit (SC), open circuit (OC), maximum power point (MPP),point with voltage equal to half the voltage at maximum powerpoint (X) and point with voltage equal to the average of maximumpower point voltage and open circuit voltage (XX).

5.1. Selected PV modules

The PV modules selected for model validation are listed below.The manufacturer supplied electrical performance characteristicsof the modules at STC and the I–V points extracted from the I–Vcurves on the module datasheets are listed in Appendix A. Theselected modules and their cell technologies are listed below.

1. Astro Power AP110 (mc-Si)2. Shell Solar S36 (pc-Si)3. Kyocera KC-40T (pc-Si)4. Shell Solar ST36 (CIS)5. Solarex MST43LV (2-a-Si)6. Unisolar PVL-124 (3-a-Si)

5.2. Parameters estimation for the selected modules

Before the electrical performance of the modules can be simu-lated, the model parameters for the modules were determined. Thefive model parameters for the selected PV modules were estimatedusing the DE assisted TS algorithm using the PV module electri-cal characteristics from Table A1 and are listed in Table 7. Thetwo additional parameters for the seven parameters model werethen estimated in a secondary optimization routine in the first fiveparameters remained fixed. The estimated values of the additionalparameters are listed in Table 8.

5.3. Electrical models selected for comparison

Five electrical performance models were selected for compari-son with the five parameter model using the proposed parametersestimation methodology.

The selected models are:

1. Four parameter electric circuit model [4]2. Sandia Labs model [1]3. Villalva et al. electric circuit model [9]4. Five parameter model by Siddiqui [12]5. Seven parameter model by Siddiqui [12]

It is important to note here that two of the models selected forcomparison, four parameter electric circuit model and Villalva et al.electric circuit model, are variations of the five parameter modeland differ from the proposed modeling methodology in either theassumptions involved or the parameter estimation process.

5.3.1. Four parameter PV modelFour parameter model, developed by Townsend [4] assumes

that the shunt resistance is high enough to be considered infiniteand neglected. This assumption greatly simplifies the parameterestimation process by reducing the model nonlinearity. The I–Vcharacteristics in the simplified electrical circuit of the four param-eter model is described by Eq. (14).

I = IL − I0

[exp

(q(V + IRS)

�kTC

)− 1

](14)

The parameters IL, Io and Rs are the same as in the five parameter

model. The parameter � is related to the parameter a in the 5 param-eter model by the relation a = �kTc/q. The parameter estimationprocess can be found in [31] and the estimated model parametersare listed in Table B1.

4614 M.U. Siddiqui, M. Abido / Applied Soft Computing 13 (2013) 4608–4621

Table 6Fitness function values in five runs for hybrid techniques.

TS assisted DE PSO assisted DE DE assisted TS

Minimum 0.001764 Minimum 0.0016165 Minimum 2.78E−5Maximum 0.002768 Maximum 0.0020454 Maximum 7.37E−5Average 0.002382 Average 0.0017524 Average 4.37E−5Std. deviation 0.000375 Std. deviation 0.0001748 Std. deviation 2.16E−5

Table 7Parameters for the five parameter model at STC.

Parameter AP-110 S-36 KC-40T ST-36 MST-43LV PVL-124

IL,ref [A] 7.5118 2.3113 2.6502 2.7136 3.5047 5.4479Io,ref [A] 1.64E−6 2.067E−10 5.5685E−9 9.9728E−7 5.7667E−16 6.185E−24aref [1/V] 1.355 0.925 1.0861 1.5325 0.62953 0.7654Rs,ref [�] 0.0702 1.0652 0.5239 1.4725 1.6128 2.2618Rsh,ref [�] 43.7946 99999.99 7185.0838 117.6139 26.0083 33.1596

Fig. 5. I–V and P–V curves for AP-110 mono-crystalline silicon module.

Fig. 6. I–V and P–V curves for S-36 m

ulti-crystalline silicon module.

M.U. Siddiqui, M. Abido / Applied Soft Computing 13 (2013) 4608–4621 4615

Fig. 7. I–V and P–V curves for KC-40T mono-crystalline silicon module.

Fig. 8. I–V and P–V curves for MST43-LV amorphous silicon module.

Fig. 9. I–V and P–V curves for ST36 CIS module.

4616 M.U. Siddiqui, M. Abido / Applied Soft Computing 13 (2013) 4608–4621

Fig. 10. I–V and P–V curves for PVL-1

Fig. 11. Maximum power RMS error comparison with conventional parameter esti-mation technique (Module 1 = AP-110, Module 2 = S-36, Module 3 = KC-40T, Module4 = MST-43LV, Module 5 = ST-35, and Module 6 = PVL-124).

Table 8Parameters m and n for the seven parameters model.

Module m n

AP-110 1.2271 1.2418S-36 1.0749 1.1918KC-40T 0.8754 1.073ST-36 1.0756 1.0987

5

dpmfacf

I

I

V

MST-43LV 1.0564 0.01238PVL-124 1.0511 0.1469

.3.2. Sandia Labs PV modelKing et al. [1] while working at the Sandia National Labs

eveloped an empirical model to accurately predict the electricalerformance of PV devices. The model includes its own thermalodel and takes into account the solar spectral and optical effects

or photovoltaic modules. The electrical characteristics at any oper-ting conditions are predicted in terms of five I–V points on the I–Vurve using Eqs. (15)–(21). The model parameters have been takenrom Sandia Labs database and are listed in Table B3.

sc = Isc0.f1(AMa){

Ebf2(AOI) + fdEdiff}

{1 + ˛isc(Tc − T0)} (15)

E0

mp = Imp0{C0Ee + C1E2e }{1 + ˛imp(Tc − T0)} (16)

oc = Voc0 + Nsı(Tc)ln(Ee) + ˇvoc(Ee)(Tc − T0) (17)

24 amorphous silicon module.

Vmp = Vmp0 + C2Nsı(Tc)ln(Ee) + C3Ns{ı(Tc)ln(Ee)}2

+ ˇvmp(Ee)(Tc − T0) (18)

Pmp = ImpVmp (19)

Ix = Ix0{C4Ee + C5E2e }{1 + ˛isc(Tc − T0)} (20)

Ixx = Ixx0{C6Ee + C7E2e }{1 + ˛imp(Tc − T0)} (21)

5.3.3. Villalva et al. electric circuit modelVillalva et al. [9] developed a parameter estimation method-

ology for the five parameters model based on the equivalentelectric circuit of Fig. 1. In order to simplify the estimation pro-cess, one of the five parameters, aref, was explicitly specified. Theremaining parameters were then found out by minimizing the errorin maximum power prediction. The methodology to find the modelparameters can be seen in [9]. The estimated model parameters arelisted in Table B2.

5.3.4. Five parameters model by SiddiquiLike Villalva et al., Siddiqui [12] developed a methodology for

finding the model parameters for the five parameters model. Theused a simplex search optimization technique to minimize theobjective function represented by Eq. (2) which is sum of the slopeof the power–voltage curve and the errors in current predictions atshort circuit, open circuit and maximum power points.

5.3.5. Seven parameters model by SiddiquiSiddiqui [12], based on the results of a sensitivity analysis,

improved the equations to vary two of the five parameters forchanging input conditions based on the results of a sensitivity anal-ysis. The sensitivity analysis showed that the five parameters modelwas much more sensitive to variations in IL and a than the otherthree parameters. The modified translations equations are given byEqs. (11) and (12). The two additional parameters were found usingthe objective function given by Eq. (3).

5.4. Method of comparison

The accuracies of the various models were compared using twometrics, the root mean square error and the mean bias error, givenby Eqs. (22) and (23), respectively. In these equations, y is the

M.U. Siddiqui, M. Abido / Applied Soft Computing 13 (2013) 4608–4621 4617

Fig. 12. Current and voltage RMS error comparison with conventional parameter estimation technique.

Fig. 13. Crystalline silicon modules current and voltage errors.

4618 M.U. Siddiqui, M. Abido / Applied Soft Computing 13 (2013) 4608–4621

Fig. 14. Thin film modules current and voltage errors.

F odule

mv

R

M

ig. 15. Maximum power prediction errors (Module 1 = AP-110, Module 2 = S-36, M

odeled value, x is the measured value and n is the number ofalues.

MSE =(1/ntotal)

√∑ntotali=1 (yi − xi)

2

(1/ntotal)∑ntotal

i=1 xi

(22)

BE = (1/ntotal)∑ntotal

i=1 (yi − xi)

(1/ntotal)∑ntotal

i=1 xi

(23)

3 = KC-40T, Module 4 = MST-43LV, Module 5 = ST-35 and Module 6 = PVL-124).

6. Results and discussion

Using the estimated parameters presented in Table 7, thecurrent-voltage curve and power-voltage curves of the six selectedmodules were calculated at reference condition and are presentedin Figs. 5–10 along with measured I–V points. It can be seen fromFigs. 5–7 and 9 that for all the crystalline silicon modules and the

CIS module, the I–V curve was accurately captured. For the amor-phous silicon cell modules, some error in the initial slope of thecurve can be seen. This can be attributed to a lack of data used forthe estimation. But since PV systems usually operate around the

M.U. Siddiqui, M. Abido / Applied Soft Computing 13 (2013) 4608–4621 4619

Table A1Electrical characteristics of the selected modules at STC.

Astro Power AP-110 Shell Solar S-36 Kyocera KC-40T Shell Solar ST-36 Solarex MST43-LV Unisolar PVL-124

Short circuit current (Isc) 7.5 2.3 2.65 2.68 3.3 5.1Open circuit voltage (Voc) 20.7 21.4 21.7 22.9 22.7 42MPP current (Imp) 6.6 2.182 2.48 2.28 2.6 4.13MPP voltage (Vmp) 16.7 16.5 17.4 15.8 16.5 30Number of cells in series (NCS) 36 36 36 42 16 20Isc temperature coefficient (�Isc) 0.0034 0.001 0.00106 0.00032 0.00066 0.001Voc temperature coefficient (�Voc) −0.08 −0.076 −0.0821 −0.1 −0.1 −0.0038

Table A2Extracted I–V points for crystalline silicon modules.

AP-110 S36 KC-40T

Point V I V I V I

1000 W/m2 and 25 ◦C 1000 W/m2 and 25 ◦C 1000 W/m2 and 25 ◦CSC 0.00 7.50 0.00 2.30 0.00 2.65X 8.35 7.30 8.25 2.29 8.70 2.61MPP 16.70 6.60 16.50 2.18 17.40 2.48XX 18.70 4.60 18.95 1.53 19.55 1.82OC 20.70 0.00 21.40 0.00 21.70 0.00

1000 W/m2 and 60 ◦C 1000 W/m2 and 60 ◦C 1000 W/m2 and 50 ◦CSC 0.00 7.70 0.00 2.33 0.00 2.74X 6.46 7.43 7.26 2.33 7.69 2.70MPP 12.92 6.70 14.52 2.09 15.38 2.55XX 15.13 3.88 16.64 1.44 17.49 1.78OC 17.34 0.00 18.75 0.00 19.6 0.00

800 W/m2 and 45 ◦C 400 W/m2 and 25 ◦C 400 W/m2 and 25 ◦CSC 0.00 6.00 0.00 0.92 0.00 1.22X 7.33 5.88 8.25 0.89 8.32 1.21MPP 14.65 4.98 16.54 0.84 16.64 1.15XX 16.41 3.54 18.41 0.64 18.82 0.82OC 18.18 0.00 20.27 0.00 20.99 0.00

Table A3Extracted I–V points for thin film modules.

PVL-124 ST36 MST43LV

Point V I V I V I

1000 W/m2 and 25 ◦C 1000 W/m2 and 25 ◦C 1000 W/m2 and 25 ◦CSC 0.00 5.10 0.00 2.68 0.00 3.30X 15.00 4.98 7.90 2.65 8.25 3.10MPP 30.00 4.13 15.80 2.28 16.50 2.60XX 36.00 2.62 19.35 1.49 19.60 1.75OC 42.00 0.00 22.90 0.00 22.70 0.00

600 W/m2 and 25 ◦C 400 W/m2 and 25 ◦C 250 W/m2 and 25 ◦CSC 0.00 3.07 0.00 1.07 0.00 0.88X 14.7 3.03 7.77 1.06 8.15 0.77MPP 29.4 2.62 15.53 0.89 16.30 0.67XX 35.24 1.71 18.25 0.61 18.46 0.52OC 41.09 0.00 20.96 0.00 20.61 0.00

200 W/m2 and 25 ◦C 1000 W/m2 and 60 ◦CSC 0.00 1.06 0.00 2.72X 14.74 1.04 6.36 2.68MPP 29.47 0.91 12.71 2.18XX 34.04 0.66 16.10 1.35OC 38.60 0.00 19.50 0.00

Table B1Parameters for the four parameter Model at STC.

Parameter AP-110 S-36 KC-40T ST-36 MST-43LV PVL-124

IL,ref 7.5 2.3 2.65 2.68 3.3 5.1Io,ref 3.0176E−6 1.3873E−7 7.1599E−8 1.9691E−4 1.1114E−4 9.9250E−5� ref 54.7192 50.1118 48.4726 93.6517 85.8243 150.72

Rs,ref 0.1545 0.4934 0.3548 1.1070 1.0697 1.35

4620 M.U. Siddiqui, M. Abido / Applied Soft Computing 13 (2013) 4608–4621

Table B2Parameters for Villalva et al. model at STC.

Parameter AP-110 S-36 KC-40T ST-36 MST-43LV PVL-124

IL,ref 7.516 2.3015 2.6511 2.711 3.3714 5.1838Io,ref 2.50335E−7 4.289E−8 3.8505E−8 2.181E−7 2.0643E−18 2.5232aref 1.2018 1.2018 1.2018 1.4021 0.5341 0.6677Rs,ref 0.0707 0.5 0.3368 0.5 0.5 0.5Rsh,ref 33.079 769.4 807.9666 43.1282 23.092 30.43

Table B3Parameters for the Sandia Labs PV model.

Parameter AP-110 S-36 KC-40T ST-36 MST-43LV PVL-124

NCS 36 36 36 42 16 20NP 1 1 1 1 4 1Isc0 7.5000 2.3000 2.6500 2.6800 3.5300 5.1000Voc0 20.7000 21.4000 21.7000 22.9000 22.2700 42.0000Imp0 6.6000 2.1800 2.4800 2.2800 2.5700 4.1300Vmp0 16.7000 16.5000 17.4000 15.8000 16.6700 30.0000˛isc 3.3000E−04 4.50E−04 4.00E−04 −1.3000E−05 6.600E−04 1.00E−03˛imp −4.2000E−04 −1.40E−04 −1.40E−04 −4.5000E−04 9.500E−04 1.00E−03C0 0.9970 0.9890 1.0060 0.9720 1.0227 1.0960C1 0.0030 0.0110 −0.0060 0.0280 −0.0227 −0.0960ˇVoc −8.4000E−02 −7.60E−02 −8.21E−02 −9.0600E−02 −1.050E−01 −1.60E−01m�voc 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000ˇvmp −8.4000E−02 −7.60E−02 −8.40E−02 −7.4400E−02 −8.700E−02 −9.30E−02m�vmp 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000n 1.4040 1.3700 1.3700 1.7520 3.5480 3.7700C2 −0.1775 −0.1170 −0.1170 0.5088 −0.2203 −1.1416C3 −11.0652 −11.0820 −11.0820 −2.9540 −4.0595 −2.8912a0 9.1800E−01 9.22E−01 9.22E−01 9.2100E−01 8.937E−01 1.05E+00a1 6.8713E−02 7.09E−02 7.09E−02 7.1815E−02 1.416E−01 8.21E−04a2 −1.0438E−02 −1.43E−02 −1.43E−02 −1.4619E−02 -5.539E−02 −2.59E−02a3 7.2504E−04 1.17E−03 1.17E−03 1.2500E−03 5.613E−03 3.17E−03a4 2.0182E−05 −3.37E−05 −3.37E−05 −3.7406E−05 −1.770E−04 −1.10E−04b0 1.0000E+00 1.00E+00 1.00E+00 1.0000E+00 1.000E+00 1.00E+00b1 −2.4380E−03 −2.47E−03 −2.44E−03 −2.4380E−03 −2.438E−03 -5.02E−03b2 3.1030E−04 3.15E−04 3.10E−04 3.1030E−04 3.103E−04 5.84E−04b3 −1.2460E−05 −1.26E−05 −1.25E−05 −1.2460E−05 −1.246E−05 −2.30E−05b4 2.1120E−07 2.14E−07 2.11E−07 2.1120E−07 2.112E−07 3.83E−07b5 −1.3590E−09 −1.37E−09 −1.36E−09 −1.3590E−09 −1.359E−09 −2.31E−09�T 3.0000 3.0000 3.0000 3.0000 3.0000 1.0000Fd 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000a −3.5600 −3.5370 −3.5600 −3.4700 −3.4700 −2.8100b −0.0750 −0.0721 −0.0750 −0.0594 −0.0594 −0.0455C4 0.9840 0.9866 0.9866 0.9829 1.0124 1.0440C5 0.0160 0.0134 0.0134 0.0171 −0.0124 −0.0440Ix0 7.3000 2.2600 2.6100 2.5900 3.2100 4.7200Ixx0 4.6000 1.5300 1.7400 1.5800 1.7700 2.9000C6 1.1230 1.1183 1.1183 1.0450 1.1185 1.1300

183

mm

cpfdsa

mtTmaett

C7 −0.1230 −0.1183 −0.1

aximum power point (MPP), this error will not greatly affect theodel performance to predict the I–V points around MPP.The electrical performance of the six selected PV modules was

alculated using the five parameter model with the proposedarameter estimation methodology and the five selected modelsrom literature at the absorbed radiation and cell temperature con-itions listed in Tables A2 and A3. Using the results, root meanquare errors and mean bias errors were calculated for the currentsnd voltages at the five key points and for maximum power.

Fig. 11 shows the root mean square error in the prediction ofaximum power using the five and seven parameters model with

he new parameters calculated in the current work presented inables 7 and 8 and the parameters calculated using the estimationethod by Siddqui [12]. The two methodologies use the same data

nd differ only in the optimization algorithm applied for the param-ter estimation. It can be seen that the five parameters model usinghe proposed methodology provides improvement in the predic-ion of maximum power. The most significant improvement was

−0.0450 −0.1185 −0.1300

seen for the amorphous silicon module PVL-124 whose RMS errorfor the five parameters model decreased from 5.5% to 2.5%.

The RMS errors in the current and voltage prediction for thecrystalline silicon cell modules and the thin film cell modules arepresented in Fig. 12. For the crystalline silicon cell modules, thecurrent at point XX showed the highest errors of around 3.5%. Forthe thin film cell type modules, all models provided comparableperformance with two exceptions. First, the five parameters modelby Siddiqui gave the lowest error in current prediction at point Xwhile the proposed five parameters model gave the significantlybetter results for current prediction at MPP.

Figs. 13–15 show a comparison of the errors in current, volt-age and maximum power prediction between the proposed fiveand seven parameters models and three models from literature.

In Fig. 13, all models show comparable performance except twoinstances. First, the Sandia Labs model shows significantly lowervalues of RMS error in current prediction at point XX and second,the proposed seven parameters model shows better accuracy for

Soft C

taFtsSTpwubes

7

mmtdfit

1

2

3

A

PC

Am

A

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

M.U. Siddiqui, M. Abido / Applied

he MPP current. Overall, the Sandia Labs model provided the mostccurate results. It also showed the least bias of all the models. Fromig. 14 which shows the current and voltage prediction errors forhin film cell modules, it can be seen that the five parameters modelhowed the most accurate results for current prediction while theandia Labs model was the most accurate in voltage prediction.he worst performing models for thin film modules were the fourarameters model and the Villalva et al. model. Finally, from Fig. 15hich shows the maximum power prediction error for all six mod-les, it can be seen that the seven parameters model shows theest performance. All other models considered show predictionrrors of around 2% except for the Villalva et al. model which showsignificantly higher error for thin film cell modules.

. Conclusions

In the current work, evolutionary algorithms were used to esti-ate the model parameters for the five and seven parametersodel. The models using the estimated parameters were then used

o predict the electrical performance of six PV modules. The pre-iction accuracies were compared with the prediction accuracies ofve other electrical models from literature. From the current work,he following conclusions were drawn.

. The differential evolution assisted Tabu Search optimizationalgorithm developed in the current work provided the lowestvalues of objective function at the end of the algorithm andtherefore its estimated parameters are the most accurate of theevolutionary algorithms.

. A comparison of the accuracy of the electrical model usingthe parameters estimated by the proposed methodology andthe electrical model using parameters estimated by a conven-tional optimization algorithm showed that prediction accuracyimproved for all modules when the proposed methodology wasused.

. The electrical model using the parameters estimated by theproposed methodology also showed better results than severalmodels from literature.

cknowledgements

Part of the work was supported by King Fahd University ofetroleum and Minerals through the Center for Clean Water andlean Energy at KFUPM (DSR project # R6-DMN-08) and MIT.

ppendix A. Electrical performance data for selected PVodules

See Tables A1–A3.

ppendix B. Model parameters for models from literature

See Tables B1–B3.

eferences

[1] D.L. King, W.E. Boyson, J.A. Kratochvil, Photovoltaic Array Performance Model,Sandia National Laboratories, Albuquerque, NM, 2004.

[2] Y. Hishikawa, Y. Imura, T. Oshiro, Irradiance-dependence and translation of theI–V characteristics of crystalline silicon solar cells, in: 2000 28th IEEE Photo-voltaic Specialists Conference (PVSC), 2000, pp. 1464–1467.

[

[

omputing 13 (2013) 4608–4621 4621

[3] B. Marion, S. Rummel, A. Anderberg, Current–voltage curve translation bybilinear interpolation, Progress in Photovoltaics: Research and Applications 12(2004) 593–607.

[4] T.U. Townsend, A Method for Predicting the Long-Term Performance ofDirectly-Coupled Photovoltaic Systems, University of Wisconsin, Madison,1989.

[5] J.A. Duffie, W.A. Beckman, Solar Engineering of Thermal Processes, 2nd ed., JohnWiley & Sons, Inc., New York, 1991.

[6] W. DeSoto, S.A. Klein, W.A. Beckman, Improvement and validation of a modelfor photovoltaic array performance, Solar Energy 80 (2006) 78–88.

[7] L.B. Valerio, A. Orioli, G. Ciulla, A. Di Gangi, An improved five-parameter modelfor photovoltaic modules, Solar Energy Materials and Solar Cells 94 (2010)1358–1370.

[8] M.T. Boyd, S.a. Klein, D.T. Reindl, B.P. Dougherty, Evaluation and validation ofequivalent circuit photovoltaic solar cell performance models, Journal of SolarEnergy Engineering 133 (2011) 021005.

[9] M.G. Villalva, J.R. Gazoli, E.R. Filho, Comprehensive approach to modeling andsimulation of photovoltaic arrays, IEEE Transactions on Power Electronics 24(2009) 1198–1208.

10] C. Carrero, D. Ramírez, J. Rodríguez, C.a. Platero, Accurate and fast convergencemethod for parameter estimation of PV generators based on three main pointsof the I–V curve, Renewable Energy 36 (2011) 2972–2977.

11] T. Ikegami, T. Maezono, F. Nakanishi, Y. Yamagata, K. Ebihara, Estimation ofequivalent circuit parameters of PV module and its application to optimal oper-ation of PV system, Solar Energy Materials and Solar Cells 67 (2001) 389–395.

12] M.U. Siddiqui, Multiphysics Modeling of Photovoltaic Panels and Arrays withAuxiliary Thermal Collectors, King Fahd University of Petroleum & Minerals,2011 (M.S. Thesis).

13] C.W. Chen, A fuzzy AHP-based fault diagnosis for semiconductor lithographyprocess, International Journal of Innovative Computing, Information and Con-trol 7 (2011) 805–816.

14] C.W. Chen, Fuzzy control of interconnected structural systems using the fuzzyLyapunov method, Journal of Vibration and Control 17 (2011) 1693–1702.

15] M.L. Lin, Application of fuzzy models for the monitoring of ecologically sen-sitive ecosystems in a dynamic semi-arid landscape from satellite imagery,Engineering Computations 27 (2010) 5–19.

16] C.W. Chen, Fuzzy control for an oceanic structure: a case study in time-delayTLP system, Journal of Vibration and Control 16 (2010) 147–160.

17] C.W. Chen, Application of fuzzy-model-based control to nonlinear structuralsystems with time delay: an LMI method, Journal of Vibration and Control 16(2010) 1651–1672.

18] F.H. Hsiao, T-S fuzzy controllers for nonlinear interconnected systems withmultiple time delays, IEEE Transactions on Circuits and Systems I: RegularPapers 52 (2005) 1883–1893.

19] M.A. Abido, Parameter optimization of multimachine power system stabilizerusing genetic local search, International Journal of Electrical Power and EnergySystems 23 (2001) 785–794.

20] M.A. Abido, Optimal power flow using Tabu Search algorithm, Electric PowerComponents and Systems 30 (2002) 469–483.

21] M.A. Abido, Optimal design of power system stabilizers using particle swarmoptimization, IEEE Transactions on Energy Conversion 17 (2002) 406–413.

22] C.W. Chen, GA-based adaptive neural network controllers for nonlinear sys-tems, International Journal of Innovative Computing 6 (2010) 1793–1803.

23] C.W. Chen, Stabilization of adaptive neural network controllers for nonlinearstructural systems using a singular perturbation approach, Journal of Vibrationand Control 17 (2011) 1241–1252.

24] N. Moldovan, R. Picos, E. Garcia-Moreno, Parameter extraction of a solar cellcompact model using genetic algorithms, in: Spanish Conference on ElectronDevices, 2009 (CDE 2009), IEEE, 2009, pp. 379–382.

25] J. Jervase, H. Bourdoucen, Solar cell parameter extraction usinggenetic algorithms, Measurement Science and Technology 12 (2001)1922–1925.

26] M. Zagrouba, A. Sellami, M. Bouaïcha, M. Ksouri, Identification of PV solar cellsand modules parameters using the genetic algorithms: application to maxi-mum power extraction, Solar Energy 84 (2010) 860–866.

27] K. Ishaque, Z. Salam, H. Taheri, A. Shamsudin, A critical evaluation of EA com-putational methods for photovoltaic cell parameter extraction based on twodiode model, Solar Energy 85 (2011) 1768–1779.

28] K. Ishaque, Z. Salam, An improved modeling method to determine the modelparameters of photovoltaic (PV) modules using differential evolution (DE),Solar Energy 85 (2011) 2349–2359.

29] F. Herrera, M. Lozano, J.L. Verdegay, Tackling real-coded genetic algorithms:operators and tools for behavioural analysis, Artificial Intelligence Review 12(1998) 265–319.

30] A.A.A. El Ela, M.A. Abido, S.R. Spea, Optimal power flow using differ-ential evolution algorithm, Electric Power Systems Research 80 (2010)878–885.

31] R. Chenni, M. Makhlouf, T. Kerbache, A. Bouzid, A detailed modeling methodfor photovoltaic cells, Energy 32 (2007) 1724–1730.