# Parameter Estimation of Photovoltaic Models via Cuckoo Search

Authors: Jieming Ma; T. O. Ting; Ka Lok Man; Nan Zhang; Sheng-Uei Guan; Prudence W. H. Wong
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Since conventional methods are incapable of estimating the parameters of Photovoltaic (PV) models with high accuracy, bioinspired algorithms have attracted significant attention in the last decade. Cuckoo Search (CS) is invented based on the inspiration of brood parasitic behavior of some cuckoo species in combination with the Lévy flight behavior. In this paper, a CS-based parameter estimation method is proposed to extract the parameters of single-diode models for commercial PV generators. Simulation results and experimental data show that the CS algorithm is capable of obtaining all the parameters with extremely high accuracy, depicted by a low Root-Mean-Squared-Error (RMSE) value. The proposed method outperforms other algorithms applied in this study.


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## Body

## 1. Introduction

Photovoltaic (PV) cells, normally assembled into modules or arrays on mounting systems, are capable of producing electrons when photons strike its surface. Taking the advantages of many promising features like renewability, less pollution, and ease of installation, PV generators are envisaged to be an important energy source for the future.Due to the high initial cost of a PV-supplied system, predictive performance tools are widely used by engineers to optimize the system performance [1, 2]. PV manufacturers normally provide limited tabular data measured under the Standard Test Conditions (STCs), which correspond to a cell temperature of 25°C and an irradiance of 1000 W/m2 at 1.5 air mass spectral distributions. As reported in [3], PV generators always operate under environments far from the STCs. Due to this reason, the data available in the datasheet usually fail to fulfill the engineering requirements.PV model, with the ability to predictI-V characteristics of PV generators under an operating environment other than the STCs, is a predictive performance tool that allows consumers to maximize the cost effectiveness of the system before installation [2]. They are generally analytical equations based on a physical description that formulate PV generated current (I) with the most crucial technical characteristics and the environmental variables, such as the operating voltage (V), the ambient temperature (T), and the irradiance (G). Over the years, significant research efforts have been contributing to the development of the behavioral models [4–8]. Among numerous modeling approaches, the Single-Diode Model (SDM) is the most widely utilized PV model in the literature. A general SDM includes five parameters, namely, photocurrent (Ipv), saturation current (Io), diode ideality constant (n), series resistance (Rs), and shunt resistance (Rp). In order to adapt PV model behavior to different operating conditions, de Blas et al. [9] suggested to apply the procedure described in the International Standard IEC 891 that relates current and voltage of the PV characteristics at given values of T and G with the corresponding values at different operating environments. The Improved Single Diode Model (ISDM) presented by De Soto et al. [5] includes the dependence of the PV parameters on operating conditions. The normal parameters at the STCs are necessary to be determined in this model. Both SDM and ISDM are adopted in this study of parameter estimation.Analytical methods [5, 10–12] are common approaches in estimating the parameters by mathematical equations. Although having the merit of simplicity, it is hard to further reduce the errors of the estimated values. Furthermore, analytical methods utilize the I-V curve features or semiconductor parameters that are unavailable in the datasheet. This problem often reduces its feasibility. Recently, PV parameter estimation is deemed as a multidimensional optimization problem. Several computational intelligence methods, such as Genetic Algorithms (GA) [13], Chaos Particle Swarm Optimization (CPSO) [14], Firefly [15], and Pattern Search (PS) [16], were proposed in the literature. These algorithms usually extract relevant parameters by minimizing the Root Mean Square Error (RMSE) as the objective function in the optimization process. Askarzadeh and Rezazadeh [17] reported that the optimization methods produce better results than analytical methods.Cuckoo Search (CS) is a nature-inspired optimization algorithm based on the fascinating breeding behavior such as brood parasitism of certain species of cuckoos. In [18, 19], Yang and Deb reported that the CS algorithm outperforms Particle Swarm Optimization (PSO) and GA algorithms for various standard test functions. In this paper, a CS-based parameter estimation method for the SDM and ISDM is presented. Simulation and experimental results show superior accuracy and feasibility of the proposed parameter estimation method.The rest of the paper is organized as follows. Section2 explains both PV models (SDM and ISDM) used in this work. The objective function formulation is given in Section 3. This is followed by results and discussions in Section 4. The results comparison is also available here, and finally the conclusions are derived in Section 5.


## 2. PV Modeling

### 2.1. Single Diode PV Model (SDM)

PV cells are made of a variety of semiconductor materials using different manufacturing processes. The working principle of PV cells is essentially on the basis of the PV effect, which refers to the generation of a potential difference at theP-N junction in response to visible or other radiation. When a PV cell is exposed to light, the semiconductor materials absorb photons, and accordingly charged carriers are generated. Potential difference and current in the external circuit lead to the separation of carriers in the internal electric field created by the P-N junction and collection at the electrodes. The photogenerated charge carriers can be subsequently captured in the form of an electric current, that is, electricity Ipv. Eliminated the PV effect, a PV cell behaves like a conventional diode that does not depend on any light parameters. The Shockley diode equation is generally used to describe the current flowing through the diode (Id):
(1)Id=Io(eVd/nVt-1).In (1), Io is the normal diode current, and Vd represents the electrical potential difference between the two ends of the diode. The ideality factor n is assumed to be independent of the environment variables T and G. Vt denotes the thermal voltage of the PV, and its value can be written as a function of T:
(2)Vt=kTq,
where k and q represent the Boltzmann constant (1.380650 × 10−23 J/K) and the electron charge (1.602176 × 10−19 C), respectively.SDM assumes that the superposition principle holds; that is, the total characteristic is the sum of the dark and illuminated characteristics [3–5]. As expressed in (3) below, the terminal current I is therefore equal to Ipv subtracting the current diverting through the diode and Rs. The equivalent circuit of the SDM is shown in Figure 1. 								(3)I=Ipv-Io(e(V+IRs)/nVt-1)-V+IRsRp.Figure 1
The equivalent circuit of the SDM.PV module is a particular case of the PV cells connected in series. If the number of the connected cells is up toNs, Vt is scaled to Ns times. The model equation is then rewritten as
(4)I=Ipv-Io(e(V+IRs)/nNsVt-1)-V+IRsRp.In this sense,Ipv, Io, Rs, and Rp are the corresponding parameters of a PV module.


### 2.2. Improved Single Diode Model (ISDM)

The traditional SDM ignores the operating conditions effect on these parameters. However, some studies have shown that the parameters, such asIpv and Io, vary under different environmental conditions. These are due to changes of temperature T and irradiance G. Aiming to evaluate the PV behavior at environmental conditions other than the normal values Tn and Gn, the relations between the operating parameters and the normal parameters are studied by numerous researchers.In [4], the value of light-generated Ipv is reported to be linearly dependent on the solar irradiation under the influence of temperature:
(5)Ipv=GGn(Ipvn-KiΔT),whereIpvn is the light-generated current at the STCs. Ki, the short-circuit current coefficient, is one of the ISDM parameters. The difference between the standard test temperature Tn and T is denoted by ΔT.Based on the diode theory, Messenger and Ventre [20] presented an approximate linear expression for the diode saturation current Io, which can be expressed as
(6)Io=Ion(TTn)e[(qEg/nk)(1/Tn-1/T)],
where Eg is the material band gap. Usually, Eg is set at a reasonable level depending on the semiconductor materials (1.12 eV for crystalline silicon and 1.75 eV for amorphous silicon) in simulation and design tools [21]. De Soto et al. [5] present an estimation method for Eg in a wide temperature range:
(7)Eg=Egn(1-0.0002677ΔT),whereEgn is a normal value at the STCs (Egn=1.12 eV for silicon cells and Egn=1.6 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 of the experimental data, the values of Rs and Rp are observed varying in inverse linear modes with G:
(8)Rs=GnGRsn,Rp=GnGRpn,where the values of the resistancesRsn and Rpn are evaluated under the STCs.By using the aforementioned relations, the ISDM described in [5] is able to analytically describe the I-V characteristic of a PV generator for each generic condition of operative temperature and solar irradiance.


## 2.1. Single Diode PV Model (SDM)

PV cells are made of a variety of semiconductor materials using different manufacturing processes. The working principle of PV cells is essentially on the basis of the PV effect, which refers to the generation of a potential difference at theP-N junction in response to visible or other radiation. When a PV cell is exposed to light, the semiconductor materials absorb photons, and accordingly charged carriers are generated. Potential difference and current in the external circuit lead to the separation of carriers in the internal electric field created by the P-N junction and collection at the electrodes. The photogenerated charge carriers can be subsequently captured in the form of an electric current, that is, electricity Ipv. Eliminated the PV effect, a PV cell behaves like a conventional diode that does not depend on any light parameters. The Shockley diode equation is generally used to describe the current flowing through the diode (Id):
(1)Id=Io(eVd/nVt-1).In (1), Io is the normal diode current, and Vd represents the electrical potential difference between the two ends of the diode. The ideality factor n is assumed to be independent of the environment variables T and G. Vt denotes the thermal voltage of the PV, and its value can be written as a function of T:
(2)Vt=kTq,
where k and q represent the Boltzmann constant (1.380650 × 10−23 J/K) and the electron charge (1.602176 × 10−19 C), respectively.SDM assumes that the superposition principle holds; that is, the total characteristic is the sum of the dark and illuminated characteristics [3–5]. As expressed in (3) below, the terminal current I is therefore equal to Ipv subtracting the current diverting through the diode and Rs. The equivalent circuit of the SDM is shown in Figure 1. 								(3)I=Ipv-Io(e(V+IRs)/nVt-1)-V+IRsRp.Figure 1
The equivalent circuit of the SDM.PV module is a particular case of the PV cells connected in series. If the number of the connected cells is up toNs, Vt is scaled to Ns times. The model equation is then rewritten as
(4)I=Ipv-Io(e(V+IRs)/nNsVt-1)-V+IRsRp.In this sense,Ipv, Io, Rs, and Rp are the corresponding parameters of a PV module.


## 2.2. Improved Single Diode Model (ISDM)

The traditional SDM ignores the operating conditions effect on these parameters. However, some studies have shown that the parameters, such asIpv and Io, vary under different environmental conditions. These are due to changes of temperature T and irradiance G. Aiming to evaluate the PV behavior at environmental conditions other than the normal values Tn and Gn, the relations between the operating parameters and the normal parameters are studied by numerous researchers.In [4], the value of light-generated Ipv is reported to be linearly dependent on the solar irradiation under the influence of temperature:
(5)Ipv=GGn(Ipvn-KiΔT),whereIpvn is the light-generated current at the STCs. Ki, the short-circuit current coefficient, is one of the ISDM parameters. The difference between the standard test temperature Tn and T is denoted by ΔT.Based on the diode theory, Messenger and Ventre [20] presented an approximate linear expression for the diode saturation current Io, which can be expressed as
(6)Io=Ion(TTn)e[(qEg/nk)(1/Tn-1/T)],
where Eg is the material band gap. Usually, Eg is set at a reasonable level depending on the semiconductor materials (1.12 eV for crystalline silicon and 1.75 eV for amorphous silicon) in simulation and design tools [21]. De Soto et al. [5] present an estimation method for Eg in a wide temperature range:
(7)Eg=Egn(1-0.0002677ΔT),whereEgn is a normal value at the STCs (Egn=1.12 eV for silicon cells and Egn=1.6 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 of the experimental data, the values of Rs and Rp are observed varying in inverse linear modes with G:
(8)Rs=GnGRsn,Rp=GnGRpn,where the values of the resistancesRsn and Rpn are evaluated under the STCs.By using the aforementioned relations, the ISDM described in [5] is able to analytically describe the I-V characteristic of a PV generator for each generic condition of operative temperature and solar irradiance.


## 3. Parameter Estimation

### 3.1. Formulation of Parameter Estimation Problem

PV parameter estimation is a process that minimizes the difference between the measured data and the calculated current by adjusting the normal PV parameters. When the number of experimental data is up toN, the objective function can be formulated by RMSE as
(9)RMSE=1N∑i=1N(fi(V,I,x))2,
where x=[Ipv,Ion,Rs,Rp] for SDM and x=[Ipvn,Ion,n,Rsn,Rpn,Ki,Eg] for ISDM. f(V,I,x) is the homogeneous form of (4) which expresses the I-V characteristics of the SDM:
(10)f(V,I,x)=Ipv-Io(e(V+IRs)/nNsVt-1)-V+IRsRp-I.For the case of ISDM,Ipv,Io,n,R, and Rp satisfy the relational expressions discussed in the previous subsection, namely, (5)–(8).


### 3.2. Cuckoo Search

The CS algorithm [18, 19], proposed by Yang and Deb, is a nature-inspired stochastic global search algorithm that follows three idealized behavior rules.(i)
A cuckoo lays an egg and dumps it randomly into other bird species’ nests.(ii)
The best nests with high quality eggs will be carried forward to the next generation.(iii)
There are a fixed number of available host nests. If a host bird discovers that the eggs are not its own, it will either throw these alien eggs away, or it may abandon the nest and build a brand new nest at a nearby location.Based on the three rules, the basic steps of CS can be summarized by the pseudocode shown in Pseudocode1. In the CS algorithm, a pattern corresponds to a nest while each individual attribute of the pattern corresponds to an egg laid by the cuckoo. On the basis of random-walk algorithms, the general system equation of the CS algorithm is given in:
(11)Xg+1;i=Xg;i+α⊗Lévy(β),
where g and i denote the generation number (g=1,2,3,…,MaxGen) and the pattern number (i=1,2,…,n), respectively. The product ⊗ means entry-wise multiplications. Here α>0 is the step size scaling factor which should be related to the scales of the problem of interest [19]. The jth attributes of the ith pattern is initiated by using (12),
(12)Xg=0;j,i=rand·(Ubi-Lbi)+Lbi,Pseudocode 1:Pseudocode of the Cuckoo Search (CS) [19].
Cuckoo Search via Lévy Flights
Initialization ofn host nests (population)
whilewithin the stopping criterion,
Choose a cuckoo egg by Lévy flights and evaluate
its fitness(Fi);
Choose an egg in other’s nest randomly and
calculate its fitness(Fj);
IfFi>Fj, replace jth egg by ith egg;
A fraction(pa) of worse nests are demolished and
replaced by new ones;
Preserve good nests (best solutions).
end whilewhereUbi and Lbi are the upper and lower bounds of the jth attributes, respectively. In each computation step, the CS algorithm checks whether the value of an attribute exceeds the allowed search range. If this happens, the value of the related attribute will be updated with the corresponding boundary value.Before the searching process, the CS algorithm detects the most successful pattern asxbest pattern. Among the existing algorithms exist for generating Lévy flights in the literature, Yang and Deb [18, 19] reported that Mantegna’s algorithm [22] works well in most of the optimization problems. Accordingly the evolution phase of the pattern initialized with the detection step of ϕ, which is given by (13) [23]:
(13)ϕ=(Γ(1+β)·sin(π·β/2)Γ(((1+β)/2)·β·2(β-1)/2))1/β,whereβ is 1.5 in the standard software implementation of the CS algorithm [30]. Γ denotes the gamma function.After initialization, the evolution phase of thexi pattern starts by defining the donor vector v, where v=xi. The required step size of the jth attributes can be calculated by the following equation:
(14)sj=0.01·(ujvj)1/β·(v-xbest),
where u=ϕ·randn[D] and v=randn[D]. The randn[D] function generates a uniform integer between [1,D] [25]. The donor pattern v is then randomly adjusted by
(15)v=v+sj·randn[D].The CS algorithm will evaluate the fitness of the random pattern. If a better solution is caught, thexbest pattern will be updated. The unfeasible patterns are revised by the crossover operator given in (16) as follows:
(16)vi={xi+rand·(xr1-xr2),randi>p0,xi,others,
where p0 is the mutation probability value (p0=0.25 in the standard software implementation [30]). The final step of a generation is to check if the revised infeasible patterns deliver a better solution.


## 3.1. Formulation of Parameter Estimation Problem

PV parameter estimation is a process that minimizes the difference between the measured data and the calculated current by adjusting the normal PV parameters. When the number of experimental data is up toN, the objective function can be formulated by RMSE as
(9)RMSE=1N∑i=1N(fi(V,I,x))2,
where x=[Ipv,Ion,Rs,Rp] for SDM and x=[Ipvn,Ion,n,Rsn,Rpn,Ki,Eg] for ISDM. f(V,I,x) is the homogeneous form of (4) which expresses the I-V characteristics of the SDM:
(10)f(V,I,x)=Ipv-Io(e(V+IRs)/nNsVt-1)-V+IRsRp-I.For the case of ISDM,Ipv,Io,n,R, and Rp satisfy the relational expressions discussed in the previous subsection, namely, (5)–(8).


## 3.2. Cuckoo Search

The CS algorithm [18, 19], proposed by Yang and Deb, is a nature-inspired stochastic global search algorithm that follows three idealized behavior rules.(i)
A cuckoo lays an egg and dumps it randomly into other bird species’ nests.(ii)
The best nests with high quality eggs will be carried forward to the next generation.(iii)
There are a fixed number of available host nests. If a host bird discovers that the eggs are not its own, it will either throw these alien eggs away, or it may abandon the nest and build a brand new nest at a nearby location.Based on the three rules, the basic steps of CS can be summarized by the pseudocode shown in Pseudocode1. In the CS algorithm, a pattern corresponds to a nest while each individual attribute of the pattern corresponds to an egg laid by the cuckoo. On the basis of random-walk algorithms, the general system equation of the CS algorithm is given in:
(11)Xg+1;i=Xg;i+α⊗Lévy(β),
where g and i denote the generation number (g=1,2,3,…,MaxGen) and the pattern number (i=1,2,…,n), respectively. The product ⊗ means entry-wise multiplications. Here α>0 is the step size scaling factor which should be related to the scales of the problem of interest [19]. The jth attributes of the ith pattern is initiated by using (12),
(12)Xg=0;j,i=rand·(Ubi-Lbi)+Lbi,Pseudocode 1:Pseudocode of the Cuckoo Search (CS) [19].
Cuckoo Search via Lévy Flights
Initialization ofn host nests (population)
whilewithin the stopping criterion,
Choose a cuckoo egg by Lévy flights and evaluate
its fitness(Fi);
Choose an egg in other’s nest randomly and
calculate its fitness(Fj);
IfFi>Fj, replace jth egg by ith egg;
A fraction(pa) of worse nests are demolished and
replaced by new ones;
Preserve good nests (best solutions).
end whilewhereUbi and Lbi are the upper and lower bounds of the jth attributes, respectively. In each computation step, the CS algorithm checks whether the value of an attribute exceeds the allowed search range. If this happens, the value of the related attribute will be updated with the corresponding boundary value.Before the searching process, the CS algorithm detects the most successful pattern asxbest pattern. Among the existing algorithms exist for generating Lévy flights in the literature, Yang and Deb [18, 19] reported that Mantegna’s algorithm [22] works well in most of the optimization problems. Accordingly the evolution phase of the pattern initialized with the detection step of ϕ, which is given by (13) [23]:
(13)ϕ=(Γ(1+β)·sin(π·β/2)Γ(((1+β)/2)·β·2(β-1)/2))1/β,whereβ is 1.5 in the standard software implementation of the CS algorithm [30]. Γ denotes the gamma function.After initialization, the evolution phase of thexi pattern starts by defining the donor vector v, where v=xi. The required step size of the jth attributes can be calculated by the following equation:
(14)sj=0.01·(ujvj)1/β·(v-xbest),
where u=ϕ·randn[D] and v=randn[D]. The randn[D] function generates a uniform integer between [1,D] [25]. The donor pattern v is then randomly adjusted by
(15)v=v+sj·randn[D].The CS algorithm will evaluate the fitness of the random pattern. If a better solution is caught, thexbest pattern will be updated. The unfeasible patterns are revised by the crossover operator given in (16) as follows:
(16)vi={xi+rand·(xr1-xr2),randi>p0,xi,others,
where p0 is the mutation probability value (p0=0.25 in the standard software implementation [30]). The final step of a generation is to check if the revised infeasible patterns deliver a better solution.


## 4. Results and Discussions

With the aim of providing a thorough evaluation of the CS algorithm in estimating the PV parameters, both SDM and ISDM are considered in this paper. Two case studies are designed to estimate the CS algorithm in model parameters estimation:(i)
a commercial 57 mm diameter solar cell (R.T.C. France [26]) operating at the standard irradiance level;(ii)
a PV module (KC200GT Multicrystal Photovoltaic Module) operating under varied environment conditions.During the parameter extraction process, the objective functionf(V,I,x) is minimized with respect to the parameters range. In theory, the value of Ipvn is slightly larger than that of Iscn. Egn is in a loose range from 1 eV to 2 eV. Ki is around the value provided by the datasheet (normally less than 0.02%/°C). The Ion is usually less than 50 μA. As stated in [27], the ideality factor ranges between 1 and 2. PV modules produced by most manufacturers have Rs less than 0.5 Ω and Rp between 5 and 170 Ω [8, 28]. As for PV cell, the ranges of Rs and Rp can be scaled by simply dividing Ns [29].Statistical analysis is performed to evaluate the quality of the fitted models to the experimental data. Besides RMSE, other two fundamental measures, namely, Individual Absolute Error (IAE) and the Mean Absolute Error (MAE), are applied to evaluate in this paper. Equations (17) and (18) preset the IAE and MAE, respectively:
(17)IAE=|Icalculated-Imeasured|,(18)MAE=1N∑i=1NIAEi.The optimization algorithms applied in this paper are programmed in MATLAB. Similar simulation conditions, including population size, maximum generation number, and search ranges, are set to ensure a fair evaluation (population size = 25; maximum generation number = 5000).


### 4.1. Case Study 1: Parameter Estimation for a PV Cell at the Certain Irradiance Level

Table1 lists the model parameters of the R.T.C France PV cell at 33°C, which are extracted from the experimental data in [26]. The parameters obtained from the CS algorithm are compared with three different parameter estimation approaches: CPSO [14], GA [13], and PS [16]. From the RMSEs of these methods, which are listed in the last row of Table 1, the CS algorithm [30] outperforms the other three optimization methods. CS obtained slightly lower RMSE, recording 0.0010 in numerical value.Table 1
A comparison between the parameter results obtained by the CS algorithm and that of other algorithms from the SDM.
CS
CPSO [14]
GA [13]
PS [16]
I
pv
0.7608
0.7607
0.7619
0.7617
I
o
3.23
E
–
07
4.00
E
–
07
8.09
E
–
07
9.98
E
–
07
n
1.4812
1.5033
1.5751
1.6
R
s
0.0364
0.0354
0.0299
0.0313
R
p
53.7185
59.012
42.3729
61.1026
RMSE
0.0010
0.0014
0.0191
0.0149During the parameter estimation process for the SDM, the values of the objective function in different optimization algorithms are shown in Figure2. The function “ga” in MATLAB [31], whose crossover rate Pc = 0.8 and mutation rate Pm=0.2, is utilized for the convergence process test. As for PSO implementation [24], the algorithm parameters are set as learning factors c1=c2=2, inertia factors wmax=0.9, wmin=0.4, and velocity clamping factor Vmax=0.5. In Figure 2, no further improvement by GA is observed after 500 iterations. On the contrary, the CS algorithm showed continuous improvement until the maximum generation. The CS algorithm, whose convergence speed is slightly faster than PSO, shows the best accuracy result in the minimization task after 5000 iterations.Figure 2
Convergence process of different optimization algorithms during the parameter estimation process of the SDM.Table2 lists the parameters of the ISDM obtained by the CS algorithm. In order to evaluate the accuracy of the CS-based estimation, these parameters are substituted into the ISDM. Since the I-V demonstrates nonlinear characteristics, the PV terminal current I is solved by the Newton-Raphson method [32] in this paper. In Table 3, the calculated results IISDM are compared with the experimental data Imeasured to observe the agreement between them. The notations IAESDM and IAEISDM denote the IAE for SDM and ISDM, respectively. Although the RMSE of the ISDM is less than that of CPSO, GA, and PS, it is similar to the RMSE of the conventional SDM under a certain environmental condition.Table 2
Parameters of the ISDM obtained by the CS algorithm.
I
pv
n
I
o
n
n
R
s
n
R
p
n
K
i
E
g
n
0.7361
1.84
E
–
07
1.5009
0.0355
57.8394
0.0031
1.0020Table 3
A comparison between the errors of ISDM and SDM. The parameters are extracted by the CS algorithm.
No
V
measured
I
measured
I
ISDM
IA
E
ISDM
IA
E
SDM
1
-
0.2057
0.7640
0.7639
0.0001
0.0001
2
-
0.1291
0.7620
0.7626
0.0006
0.0007
3
-
0.0588
0.7605
0.7614
0.0009
0.0009
4
0.0057
0.7605
0.7602
0.0003
0.0003
5
0.0646
0.7600
0.7592
0.0008
0.0009
6
0.1185
0.7590
0.7583
0.0007
0.0010
7
0.1678
0.7570
0.7574
0.0004
0.0001
8
0.2132
0.7570
0.7565
0.0005
0.0009
9
0.2545
0.7555
0.7555
0.0000
0.0004
10
0.2924
0.7540
0.7540
0.0000
0.0003
11
0.3269
0.7505
0.7517
0.0012
0.0009
12
0.3585
0.7465
0.7476
0.0011
0.0009
13
0.3873
0.7385
0.7402
0.0017
0.0016
14
0.4137
0.7280
0.7273
0.0007
0.0006
15
0.4373
0.7065
0.7066
0.0001
0.0005
16
0.4590
0.6755
0.6748
0.0007
0.0002
17
0.4784
0.6320
0.6304
0.0016
0.0011
18
0.4960
0.5730
0.5717
0.0013
0.0009
19
0.5119
0.4990
0.4994
0.0004
0.0005
20
0.5265
0.4130
0.4137
0.0007
0.0005
21
0.5398
0.3165
0.3176
0.0011
0.0007
22
0.5521
0.2120
0.2127
0.0007
0.0001
23
0.5633
0.1035
0.1033
0.0002
0.0008
24
0.5736
-
0.0100
-
0.0089
0.0011
0.0008
25
0.5833
-
0.1230
-
0.1244
0.0014
0.0014
26
0.5900
-
0.2100
-
0.2095
0.0005
0.0009
MAE
0.0007
0.0007
RMSE
0.0010
0.0010


### 4.2. Case Study 2: Parameter Estimation for a PV Module under Different Environment Conditions

In this section, the validity of the CS algorithm is evaluated using KC200GT PV module operating under different environment conditions. The estimated parameters, both in the SDM and ISDM, are shown in Table4. As illustrated in Section 1, the main application of the parameter extraction is to predict the I-V characteristics for design purpose. It is worth pointing out that the SDM parameters can only be extracted by the experimental data measured under a certain test condition. Significant errors may occur as the experimental data are measured under varying operating conditions. In the commercial simulation tool like PSIM [21], the PV parameters of the SDM are firstly estimated at the STCs, then the equations (given in the appendix) are applied to calculate the electrical characteristics of different operating conditions. The ISDM-based parameter estimation, however, can be performed by the data measured under any conditions.Table 4
Parameters of the KC200GT PV module obtained by the CS algorithm.
(a)
SDM parameters (extracted by the CS algorithm)
I
pv
I
o
n
R
s
R
p
8.1729
4.23E–10
1.0090
0.2665
140.4875
(b)
ISDM parameters (extracted by the CS algorithm)
I
pv
n
I
o
n
n
R
s
n
R
p
n
K
i
E
g
n
8.1847
5.12E–10
1.0170
0.2574
117.9224
0.0028
1.2474Figure3 shows the I-Vcurves generated using the parameters obtained by the CS algorithm. The simulated results are compared with the experimental data, which are collected at five different irradiance levels (1000 W/m2, 800 W/m2, 600 W/m2, 400 W/m2, and 200 W/m2) and three different temperature levels (25°C, 50°C, and 75°C). It can be seen that the I-V curves of the ISDM fit the whole range of the experimental dataset. On the other hand, the errors of SDM seem larger at lower irradiance and higher temperature levels. With the experimental data, the RMSE of the current I in SDM is calculated as 0.2837, while the RMSE of I in ISDM is only 0.0776.The simulatedI-V characteristic curves of the KC200GT PV module: (a) under different irradiance levels; (b) under different temperature levels.
(a)
(b)Figure4 shows the absolute current errors of different performance predicting methods under different operating conditions. The curves denoted by the label “analytical SDM” are obtained from the analytical SDM model [4]. Ignoring the effect of incidence angle and air mass, the curves labeled by “analytical ISDM” denote the I-V curves from De Soto’s analytical ISDM model [5]. It is evident the ISDM with the parameters extracted by the CS algorithm is more accurate than the analytical model. As for the SDM, the CS algorithm is capable of extracting a set of PV parameters with a good fit for the experimental data at the STCs. However, the SDM with the equations in the appendix does not exhibit a good prediction performance under other operating conditions.A comparison of the individual absolute errors among different PV modeling methods: (a) under different irradiance levels; (b) under different temperature levels.
(a)
(b)To further validate the accuracy of the CS algorithm, the extracted parameters are compared to the ones obtained using GA in Figure5. In general, the CS algorithm gives the better performance than GA for all cases. The Maximum Power Point (MPP), usually locating around 74% of the open circuit voltage, is an important technical data in PV modeling. However, a negative point of the GA-based ISDM is that the errors in the high voltage range are relatively high. The maximum absolute error of the GA-based ISDM is up to about 0.8 A, while the absolute error of the CS is kept below 0.2 A.A comparison of the individual absolute errors between CS- and GA- based ISDM: (a) under different irradiance levels; (b) under different temperature levels.
(a)
(b)


## 4.1. Case Study 1: Parameter Estimation for a PV Cell at the Certain Irradiance Level

Table1 lists the model parameters of the R.T.C France PV cell at 33°C, which are extracted from the experimental data in [26]. The parameters obtained from the CS algorithm are compared with three different parameter estimation approaches: CPSO [14], GA [13], and PS [16]. From the RMSEs of these methods, which are listed in the last row of Table 1, the CS algorithm [30] outperforms the other three optimization methods. CS obtained slightly lower RMSE, recording 0.0010 in numerical value.Table 1
A comparison between the parameter results obtained by the CS algorithm and that of other algorithms from the SDM.
CS
CPSO [14]
GA [13]
PS [16]
I
pv
0.7608
0.7607
0.7619
0.7617
I
o
3.23
E
–
07
4.00
E
–
07
8.09
E
–
07
9.98
E
–
07
n
1.4812
1.5033
1.5751
1.6
R
s
0.0364
0.0354
0.0299
0.0313
R
p
53.7185
59.012
42.3729
61.1026
RMSE
0.0010
0.0014
0.0191
0.0149During the parameter estimation process for the SDM, the values of the objective function in different optimization algorithms are shown in Figure2. The function “ga” in MATLAB [31], whose crossover rate Pc = 0.8 and mutation rate Pm=0.2, is utilized for the convergence process test. As for PSO implementation [24], the algorithm parameters are set as learning factors c1=c2=2, inertia factors wmax=0.9, wmin=0.4, and velocity clamping factor Vmax=0.5. In Figure 2, no further improvement by GA is observed after 500 iterations. On the contrary, the CS algorithm showed continuous improvement until the maximum generation. The CS algorithm, whose convergence speed is slightly faster than PSO, shows the best accuracy result in the minimization task after 5000 iterations.Figure 2
Convergence process of different optimization algorithms during the parameter estimation process of the SDM.Table2 lists the parameters of the ISDM obtained by the CS algorithm. In order to evaluate the accuracy of the CS-based estimation, these parameters are substituted into the ISDM. Since the I-V demonstrates nonlinear characteristics, the PV terminal current I is solved by the Newton-Raphson method [32] in this paper. In Table 3, the calculated results IISDM are compared with the experimental data Imeasured to observe the agreement between them. The notations IAESDM and IAEISDM denote the IAE for SDM and ISDM, respectively. Although the RMSE of the ISDM is less than that of CPSO, GA, and PS, it is similar to the RMSE of the conventional SDM under a certain environmental condition.Table 2
Parameters of the ISDM obtained by the CS algorithm.
I
pv
n
I
o
n
n
R
s
n
R
p
n
K
i
E
g
n
0.7361
1.84
E
–
07
1.5009
0.0355
57.8394
0.0031
1.0020Table 3
A comparison between the errors of ISDM and SDM. The parameters are extracted by the CS algorithm.
No
V
measured
I
measured
I
ISDM
IA
E
ISDM
IA
E
SDM
1
-
0.2057
0.7640
0.7639
0.0001
0.0001
2
-
0.1291
0.7620
0.7626
0.0006
0.0007
3
-
0.0588
0.7605
0.7614
0.0009
0.0009
4
0.0057
0.7605
0.7602
0.0003
0.0003
5
0.0646
0.7600
0.7592
0.0008
0.0009
6
0.1185
0.7590
0.7583
0.0007
0.0010
7
0.1678
0.7570
0.7574
0.0004
0.0001
8
0.2132
0.7570
0.7565
0.0005
0.0009
9
0.2545
0.7555
0.7555
0.0000
0.0004
10
0.2924
0.7540
0.7540
0.0000
0.0003
11
0.3269
0.7505
0.7517
0.0012
0.0009
12
0.3585
0.7465
0.7476
0.0011
0.0009
13
0.3873
0.7385
0.7402
0.0017
0.0016
14
0.4137
0.7280
0.7273
0.0007
0.0006
15
0.4373
0.7065
0.7066
0.0001
0.0005
16
0.4590
0.6755
0.6748
0.0007
0.0002
17
0.4784
0.6320
0.6304
0.0016
0.0011
18
0.4960
0.5730
0.5717
0.0013
0.0009
19
0.5119
0.4990
0.4994
0.0004
0.0005
20
0.5265
0.4130
0.4137
0.0007
0.0005
21
0.5398
0.3165
0.3176
0.0011
0.0007
22
0.5521
0.2120
0.2127
0.0007
0.0001
23
0.5633
0.1035
0.1033
0.0002
0.0008
24
0.5736
-
0.0100
-
0.0089
0.0011
0.0008
25
0.5833
-
0.1230
-
0.1244
0.0014
0.0014
26
0.5900
-
0.2100
-
0.2095
0.0005
0.0009
MAE
0.0007
0.0007
RMSE
0.0010
0.0010


## 4.2. Case Study 2: Parameter Estimation for a PV Module under Different Environment Conditions

In this section, the validity of the CS algorithm is evaluated using KC200GT PV module operating under different environment conditions. The estimated parameters, both in the SDM and ISDM, are shown in Table4. As illustrated in Section 1, the main application of the parameter extraction is to predict the I-V characteristics for design purpose. It is worth pointing out that the SDM parameters can only be extracted by the experimental data measured under a certain test condition. Significant errors may occur as the experimental data are measured under varying operating conditions. In the commercial simulation tool like PSIM [21], the PV parameters of the SDM are firstly estimated at the STCs, then the equations (given in the appendix) are applied to calculate the electrical characteristics of different operating conditions. The ISDM-based parameter estimation, however, can be performed by the data measured under any conditions.Table 4
Parameters of the KC200GT PV module obtained by the CS algorithm.
(a)
SDM parameters (extracted by the CS algorithm)
I
pv
I
o
n
R
s
R
p
8.1729
4.23E–10
1.0090
0.2665
140.4875
(b)
ISDM parameters (extracted by the CS algorithm)
I
pv
n
I
o
n
n
R
s
n
R
p
n
K
i
E
g
n
8.1847
5.12E–10
1.0170
0.2574
117.9224
0.0028
1.2474Figure3 shows the I-Vcurves generated using the parameters obtained by the CS algorithm. The simulated results are compared with the experimental data, which are collected at five different irradiance levels (1000 W/m2, 800 W/m2, 600 W/m2, 400 W/m2, and 200 W/m2) and three different temperature levels (25°C, 50°C, and 75°C). It can be seen that the I-V curves of the ISDM fit the whole range of the experimental dataset. On the other hand, the errors of SDM seem larger at lower irradiance and higher temperature levels. With the experimental data, the RMSE of the current I in SDM is calculated as 0.2837, while the RMSE of I in ISDM is only 0.0776.The simulatedI-V characteristic curves of the KC200GT PV module: (a) under different irradiance levels; (b) under different temperature levels.
(a)
(b)Figure4 shows the absolute current errors of different performance predicting methods under different operating conditions. The curves denoted by the label “analytical SDM” are obtained from the analytical SDM model [4]. Ignoring the effect of incidence angle and air mass, the curves labeled by “analytical ISDM” denote the I-V curves from De Soto’s analytical ISDM model [5]. It is evident the ISDM with the parameters extracted by the CS algorithm is more accurate than the analytical model. As for the SDM, the CS algorithm is capable of extracting a set of PV parameters with a good fit for the experimental data at the STCs. However, the SDM with the equations in the appendix does not exhibit a good prediction performance under other operating conditions.A comparison of the individual absolute errors among different PV modeling methods: (a) under different irradiance levels; (b) under different temperature levels.
(a)
(b)To further validate the accuracy of the CS algorithm, the extracted parameters are compared to the ones obtained using GA in Figure5. In general, the CS algorithm gives the better performance than GA for all cases. The Maximum Power Point (MPP), usually locating around 74% of the open circuit voltage, is an important technical data in PV modeling. However, a negative point of the GA-based ISDM is that the errors in the high voltage range are relatively high. The maximum absolute error of the GA-based ISDM is up to about 0.8 A, while the absolute error of the CS is kept below 0.2 A.A comparison of the individual absolute errors between CS- and GA- based ISDM: (a) under different irradiance levels; (b) under different temperature levels.
(a)
(b)


## 5. Conclusion

In this work, the Cuckoo Search (CS) algorithm is applied to estimate the parameters of two PV models, namely, Single Diode Model (SDM) and its improved version (ISDM). The feasibility of the proposed method has been validated by estimating the parameters of two commercial PV generators. The simulation and experimental results showed that the CS algorithm is capable of not only extracting all the parameters of the SDM under a certain condition but also successfully estimating all the parameters of ISDM under different operating conditions. In statistical analysis, CS algorithm recorded the lowest RMSE value compared to other algorithms such as GA, PSO and PS.



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*Source: 362619-2013-08-07.xml*