Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Locomotion-based Hybrid Salp Swarm Algorithm for Parameter Estimation of Fuzzy Representation-based Photovoltaic Modules  PDF

  • Rizk M. Rizk-Allah
  • Aboul Ella Hassanien
Faculty of Engineering, Menoufia University, Shebin El-Kom, Egypt; Scientific Research Group in Egypt,Cario, Egypt; Faculty of Computers and Artificial Intelligence, Cairo University, Cairo, Egypt

Updated:2021-03-16

DOI:10.35833/MPCE.2019.000028

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Abstract

Identifying the parameters of photovoltaic (PV) modules is significant for their design and simulation. Because of the instabilities in the weather action and land surface of the earth, which cause errors in measuring, a novel fuzzy representation-based PV module is formulated and developed. In this paper, a novel locomotion-based hybrid salp swarm algorithm (LHSSA) is presented to identify the parameters of PV modules accurately and reliably. In the LHSSA, better leader salps based on particle swarm optimization (PSO) are incorporated to the traditional salp swarm algorithm (SSA) in a serialized scheme with the aim of providing more valuable information for the leader salps of the SSA. By this integration, the proposed LHSSA can escape the local optima as well as guide the seeking process to attain the promising region. The proposed LHSSA is investigated on different PV models, i.e., single-diode (SD), double-diode (DD), and PV module in crisp and fuzzy aspects. By comparing with different algorithms, the comprehensive results affirm that the LHSSA can achieve a highly competitive performance, especially on quality and reliability.

I. Introduction

BECAUSE of the rapid increase in air pollution caused by generating the power using fossil fuels, and greenhouse gas emissions, industrialized countries and governments need to rely on clean sources such as renewable energy sources. Presently, one of the most popular and promising renewable energy sources is solar energy (SE). SE is converted to electrical energy using photovoltaic (PV) systems [

1]. PV components are easily deteriorated because they operate in harsh outdoor environments, significantly affecting the efficiency of the use of SE. As a result, the process of designing accurate models of PV systems becomes an important and challenging task due to the behavior evaluation of PV cells. Different mathematical models have been introduced to describe the nonlinear performance of PV systems [2], [3]. The most popular models are the single-diode (SD) and double-diode (DD) models [4], [5]. The accuracy of any model is mainly affected by adjusting the model parameters. However, these parameters have two problems: ① they are usually unavailable and change due to faults, volatile operation conditions, and aging; ② they have a certain degree of uncertainty due to the measurement process. Hence, the creation of an accurate scheme for parameter identification is a crucial and challenging task for the effective evaluation, simulation, and control of PV systems.

Some approaches based on deterministic techniques have been proposed for PV parameter identification based on deterministic techniques. Such approaches were developed based on the Newton approach [

6], Lambert formulation [7], and so on [8]. However, some restrictions such as differentiability and convexity that lead to trapping in the local optima are present in the deterministic techniques.

The advent of metaheuristic approaches has provided promising alternative approaches for solving complex optimization tasks [

9]-[12] and overcoming the limitations of the deterministic ones as well. Several approaches such as differential evolution (DE) have been introduced for the parameter identification problem [13]. In [14], artificial bee colony (ABC) was developed for identifying solar cell parameters, and in [15], a bacterial foraging approach (BFA) was used for parameter estimation of solar cells and other systems [16]-[18]. Salp swarm algorithm (SSA) is a new optimization approach that was proposed in [19] for solving optimization tasks. This algorithm is based on the behavior of salps, which form a chain through some leaders. Furthermore, through a rapid, coordinated change strategy, this behavior can achieve better convergence. However, as a new approach, SSA has some disadvantages: ① they are only guided by leaders, which leads to unsatisfactory outcomes; ② there is no strategy for the diversity and improving the best location with each generation, which may lead to trapping in the local optima. Besides, to the best of our knowledge, no attempts in the SSA literature have been employed to solve the parameter estimation of PV models.

Identifying the parameters of PV modules is significant for their design and simulation. Because of instabilities in the weather action and land surface of the earth, which cause errors in measuring, a novel fuzzy representation-based PV module is formulated and developed. In this sense, the PV modules are proposed in two aspects: the first aspect addresses crisp PV modules, while the second one concentrates on fuzzy modeling of PV modules in which the input data of terminal (output) voltage and solar cell terminal current are the fuzzy numbers. Through each fuzzy PV module, the terminal (output) voltage as well as the solar cell terminal current is characterized with the membership function. To effectively handle the crisp and fuzzy aspects of the PV modules, this paper hybridizes the SSA with the particle swarm optimization (PSO), in a scheme called the locomotion-based hybrid salp swarm algorithm (LHSSA), to effectively achieve robust identification of the parameters of the PV modules. In this approach, the SSA operates as an explorer tool for the solution vector while PSO is integrated to modify the locations of the leader salps. The performance of the proposed LHSSA is investigated and evaluated through different PV models. Comprehensive results affirm that the LHSSA exhibits superior performance compared with different algorithms, especially on the quality and reliability.

The main contributions of this paper are as follows.

1) An LHSSA-based opposition learning scheme is introduced to refine the quality of leaders.

2) A PSO algorithm-based leading method is incorporated to enhance exploitation capability and avoid trapping in the local optima.

3) Fuzzy representation-based PV models are introduced as a novel sight, and different level schemes are conducted.

4) The efficiency of the LHSSA is investigated by comprehensive experiments and the comparison on different PV models.

The remainder of this paper is organized as follows. Section II describes the formulation of the PV models. Section III provides some basics regarding the methodology. The proposed LHSSA is developed in Section IV. The results and the comparison of the different PV models are provided in Section V. Finally, the conclusions and potential for future study are exhibited in Section VI.

II. Problem Formulation

In practice, there are different PV models that describe the characteristics of the current-voltage performance of solar cells and PV modules. The formulation of these models associated with their objective functions is described in this section.

A. PV Models

1) SD Model

The SD model uses one diode that is parallel to a current source. The structure of this model includes a current source that is parallel to the circuit diode, one series resistor to represent the losses of load current, and a shunt resistor to represent the leakage current. The current of this cell is calculated using:

IL=Iph-Id-Ish (1)
Id=IsdexpqVL+RSILnkT-1 (2)
Ish=VL+RSILRsh (3)

where IL is the terminal current of solar cell; Iph is the photo-generated current; Id is the diode current; Ish is the shunt branch current; Isd is the reverse saturation current; VL is the terminal (output) voltage; Rsh and RS are the shunt and series resistances, respectively; n is the diode ideal factor; q is the charge of an electron (1.60217646×10-19 C); k is the Boltzmann constant (1.3806503×10-23 J/K); and T is the temperature of the cell. According to (2) and (3), the terminal (output) current in (1) can be rewritten as:

IL=Iph-IsdexpqVL+RSILnkT-1-VL+RSILRsh (4)

Although the SD contains five unknown parameters (Iph,Isd,RS,Rsh,n), [

1] and [2] have shown that this optimization task has high multi-modal and noisy characteristics, thus this fact requires robust search strategies.

2) DD Model

The DD model considers two parallel diodes with a shunt resistance and current source. The terminal (output) current can be described as follows:

IL=Iph-Id1-Id2-Ish=Iph-Isd1expqVL+RSILn1kT-1-Isd2expqVL+RSILn2kT-1-VL+RSILRsh (5)

where Id1 and Id2 are the first-diode and second-diode currents, respectively; Isd1 and Isd2 are the diffusion and saturation currents, respectively; and n1 and n2 are the ideal factors of diffusion and recombination diodes, respectively. In this case, seven unknown parameters (Iph,Isd1,Isd2,RS,Rsh,n1,n2) need to be estimated to obtain the accurate performance of the solar cell.

3) PV Module Model

The PV module model is structured from several solar cells that are linked in series and/or in parallel. The output (terminal) current can be considered as follows:

ILNp=Iph-IsdexpqVL/NS+RSIL/NpnkT-1-VL/NS+RSIL/NpRsh (6)

where NS and Np are the numbers of solar cells in series and parallel, respectively. This model contains five unknown parameters (Iph,Isd,RS,Rsh,n) that need to be identified.

B. Objective Function for PV Modules: Crisp Aspect

The main aim of the PV models is to minimize the difference between the experimental and estimated data. In this regard, the error function is represented and defined by (7) and (8) for the SD and DD, respectively.

fk(VL,IL,x)=Iph-IsdexpqVL+RSILnkT-1-VL+RSILRsh-ILx=Iph,Isd,RS,Rsh,n (7)
fk(VL,IL,x)=Iph-Isd1expqVL+RSILn1kT-1-Isd2expqVL+RSILn2kT-1-VL+RSILRsh-ILx=Iph,Isd1,Isd2,RS,Rsh,n1,n2 (8)

where x is the solution set of the unknown parameters.

To objectively evaluate the performance of the proposed methodology, the objective function of the PV modules is formulated by quantifying the root mean square error (RMSE) as:

RMSE=1Mk=1MfkVL, IL, x2 (9)
Iph,minIphIph,maxRS,minRSRS,maxRsh,minRshRsh,maxIsd,minIsdIsd,maxIsd1,minIsd1Isd1,maxIsd2,minIsd2Isd2,maxnminnnmaxn1,minn1n1,maxn2,minn2n2,max (10)

where M is the number of experimental data; and the subscripts min and max represent the minimum and maximum values of related variables, respectively.

The main aim of the optimization process is to minimize the fitness function with respect to the bounds of the parameters. A smaller RMSE output implies a smaller deviation between the experimental and simulated data by the proposed algorithm.

C. Objective Function for PV Modules: Fuzzy Aspect

The electrical energy generated by PV modules involves many controlled parameters such as terminal (output) voltage and solar cell terminal current whose possible values are uncertain and ambiguous due to weather fluctuations or instabilities in the measuring process. Thus, the fuzzy model of the PV modules can be formulated as:

min Ζ˜=1Mk=1M fkV˜L,I˜L,x2s.t.  (10) (11)

where Ζ˜ is the fuzzy objective form of RMSE; V˜L is the fuzzy terminal (output) voltage; and I˜L is the fuzzy solar cell terminal current. δ˜=V˜L,I˜L is the variable of the fuzzy parameters that are represented as fuzzy number. Each fuzzy number component of δ˜ is associated by its own degree of membership function μδ˜(δ) [

20].

Definition 1   (fuzzy number): the fuzzy number δ˜ is a fuzzy subset of the real line R, which is associated with the membership function μδ˜(δ) that has the following features:

1) μδ˜(δ): R0,1 is continuous.

2) μδ˜(δ)=0,δ(-,δ1)(δ3,+).

3) Strictly increasing on δ[δ1,δ2].

4) μδ˜(δ)=1 for δ=δ2.

5) Strictly decreasing on δ[δ2,δ3].

Therefore, each measured value of the terminal voltage and solar cell terminal current can be represented by a certain “grade of membership” that ranges from grade 0 to 1, where 0 and 1 indicate the lowest and highest grades of membership, respectively. Thus, each fuzzy parameter is expressed by a given interval of real numbers, associated with grades of membership between 0 and 1.

Definition 2   (α-level set): the α-level set (α-cut) of the fuzzy numbers δ˜ is represented by the ordinary set Lαδ˜ that contains the values of δ when the degree of membership functions exceeds a certain level α, α[0,1].

Lαδ˜=δ|μa˜(δ)α (12)

The fuzzy representation shown in Fig. 1 has been extracted according to expert suggestions or from observing weather fluctuations or instabilities in the measuring process. Based on the α-cut concept of the fuzzy numbers, the problem with the fuzzy parameters aspect can be converted into a crisp aspect. The membership function μδ˜(δ) which defines the fuzzy parameter δ˜ is formulated as:

μδ˜(δ)=1                   δ=δ˜20δδ˜-19    0.95δ˜δ<δ˜21-20δδ˜    δ˜<δ1.05δ˜0                   δ<0.95δ˜  or  δ>1.05δ˜ (13)

Fig. 1 Fuzzy numbers for PV modules.

According to the membership function and the concept of the α-cut level, the fuzzy parameter is transformed to a crisp scheme through two end points which are induced by the α-cut level, and they represent the upper and lower bounds for the crisp parameter, as shown in Fig. 1, where δ˜L and δ˜U are the lower and upper bounds induced by α-cut for parameter δ, respectively. Thus, the optimization task of the PV modules can be converted to a non-fuzzy (crisp) form as:

min Ζ=1Mk=1MfkVLk,ILk,x2s.t.  (10)       V˜LkLVLkV˜LkU    k=1,2,...,M       I˜LkLILkI˜LkU        k=1,2,...,M (14)

where superscripts L and U represent the lower and upper bounds of related variables, respectively.

Definition 3   (α-global minimum): x* is defined as the α-global minimum to the problem (14) if and only if xΨ, δLαδ˜: Ζx*,δ*Ζx,δ, where Ψ is the feasible region of the search space and δ* is the α-level optimal parameter.

D. Overview of SSA

The SSA is a meta-heuristic optimization algorithm presented in [

19]. It mimics the natural behavior of salps living in the deep parts of oceans. Salps are a type of Salpidae and look like jelly fish with a transparent body. Salps have an intelligent behavior during their navigation and foraging as a salp chain. In this context, the SSA is a recent type of swarm algorithm developed to model the salp chain [19]. In the salp chain, the salp population is split into two categories: the leader salp which is the first salp in the chain, and the follower salps that follow the leader salp while reaching for the food source. The position set of each salp in the search space is xi={x1i,x2i,...,xdi},i=1,2,...,N, where  N is the number of salps. Thus, the leader salp updates its position as:

xj1=Fj+h1ubj-lbjh2+lbj    h30.5Fj-h1ubj-lbjh2+lbj    h3<0.5 (15)

where ubj and lbj are the upper and lower bounds in the jth dimension, respectively; Fj is the position of the target source in the jth dimension; and h1, h2, and h3 are the algorithm parameters. The first parameter h1, which is responsible for balancing the exploration and the exploitation mechanisms, is the most important parameter, and is defined as:

h1=2e-4tL2 (16)

where t and L are the current iteration and maximum iteration, respectively. h2 and h3 are random numbers between 0 and 1 whose values are uniformly generated. Furthermore, each follower salp updates its position based on Newton’s law of motion using the following equation.

xji=12xji+xji-1    2i>N (17)

where xji is the position of the ith follower salp in the jth dimension.

E. PSO

PSO is a population-based algorithm introduced in [

21]. It is inspired by the cooperative behavior of some birds, fishes, and insects. In PSO, each particle improves its position by considering its best personal and global positions. The position of each particle xi is updated as:

vit+1=wvit+c1r1pit-xit+c2r2git-xit (18)
xit+1=xit+vit+1 (19)

where xit and vit are the current position of the ith particle and its velocity at iteration t, respectively; pit is the best position of the ith particle; git is the best solution among all particles; c1 and c2 are the cognitive and social parameters, respectively; w is the inertia weight; and r1 and r2 are the random numbers between 0 and 1.

III. Proposed LHSSA Algorithm

The motivation behind developing the LHSSA is to achieve two features: ① improving the locomotion of the leader salps by memorizing the track of the leader; ② enhancing the seeking process using the hybridization-based PSO.

A. Locomotion-based Internal Memory

An internal memory is incorporated for the leader salps to keep track of their positions. During each iteration, the best agents between the salps and particles are saved and denoted by personal leaders SB,leader and the global leader is denoted by SG,leader.

B. Hybridization-based Iteration

Hybridization-based iteration is a straightforward approach for executing two algorithms iteratively through a certain sequence to enhance optimization performance [

22]. Here, SSA works as an explorer using (14), while PSO is responsible for exploiting the previous leader salps to obtain more refined leaders. Thus, the modified position and velocity equations of PSO are defined as:

vit+1=wvit+c1r1SB,leader,it-xit+c2r2SG,leadert-xit (20)
xit+1=xit+vit+1 (21)

where SB,leader,it is the personal best for the ith particle of PSO; and SG,leadert is the global best of the swarm in the iteration t of the PSO.

C. Experience-based Opposition Learning Scheme

The opposition learning scheme based on the behavior of follower salps is developed to improve swarm diversity with the aim of increasing exploration ability. To be specific, for each follower salp, the opposition solution is defined as in (22).

xi,j,t'=randLBj+UBj-xi,j,t    xi,j,t'Bj,UBjrandLBj,UBj                  otherwise (22)

where xi,j,t' is the jth element of the ith opposition solution set on the tth iteration; and LBj and UBj are the dynamic bounds of the jth variable defined as:

LBj=minxi,j,tUBj=maxxi,j,t (23)

LBj and UBj are updated every 50 generations and if the obtained solution does not lie within the bounds, a random solution is generated. The flow chart of the proposed scheme is given in Fig. 2.

Fig. 2 Flow chart of proposed LHSSA.

IV. Experimental Results and Analysis

The performance of LHSSA is investigated to identify the parameters of different PV models such as the SD, DD, and PV module. Thus, the benchmark data set for the solar cells and solar module is utilized [

23]. To guarantee accurate comparison, the bounds for the parameters are given in Table I [23]. In this context, the superior performance of the proposed LHSSA is validated by comparing with other well-established algorithms [24] such as PSO, fire fly algorithm (FFA), grey wolf algorithm (GWO), dragonfly algorithm (DA), and standard SSA. For fair comparisons, the same maximum number of iterations in each run with 20 independent runs for every problem is adopted in the comparative algorithms. Further, the parameter configurations of the compared algorithms are listed in Table II and are suggested as in [24].

Table I Parameters Range for SD, DD, and PV Module
ModelBoundIph (A)Isd,Isd1,Isd2 (μA)RS (Ω)Rsh (Ω)n,n1,n2
SD Lower 0 0 0.0 0 1
Upper 1 1 0.5 100 2
DD Lower 0 0 0.0 0 1
Upper 1 1 0.5 100 2
PV module Lower 0 0 0.0 0 1
Upper 2 50 2.0 2000 50
Table II Parameter Configurations for Comparative Algorithms
ParameterAlgorithm
Population size PS=20, acceleration coefficients c1=c2=2, inertia weight w: 0.2-0.9 PSO
PS=20, initial attractiveness β0=1, randomization parameter α=0.2, absorption coefficient γ=1 FFA
PS=20, a[0,2],A=2ar2-a,C=2r1,r1,r2U(0,1), U represents a uniform distribution GWO
PS=20, w: 0.2-0.9, separation s=0.1, alignment  a=0.1, cohesion c=0.7, food f=1, enemy  e=1 DA
PS=20, h1[0,2], h2,h3[0,1] SSA
PS=20, acceleration coefficients: c1=c2=2, inertia weight w: 0.2-0.9, h1[0,2], h2,h3[0,1] LHSSA

A. Results of SD Model

In this section, the comparison results of the SD model including the extracted parameters and RMSE are shown in Table III, where the best RMSE value among all comparative algorithms is highlighted in boldface.

Table III Comparisons with Various Algorithms for SD Model
AlgorithmRs (Ω)Rsh (Ω)Iph (A)Isd (μA)nRMSE (mA)
Proposed LHSSA 0.03640 53.7185 0.7607 0.32300 1.48170 0.98602
PSO 2.22040×1016 1.1489 0.8368 2.22000×108 1.00000 19.58100
FFA 0.01540 47.0033 0.5697 3.09000 1.89920 141.63000
GWO 0.02970 32.6211 0.7652 1.10700 1.61860 222.86000
DA 2.22040×1016 1.1489 0.8368 2.22000×108 1.00000 222.86000
SSA 4.73800×104 5.6237 0.7443 11.02000 1.98690 39.90600
IJAYA 0.03640 53.7595 0.7608 0.32280 1.48110 0.98603
JAYA 0.03640 54.9298 0.7608 0.32810 1.48280 0.98946
GOTLBO 0.03630 53.3664 0.7608 0.32970 1.48330 0.98856
LETLBO 0.03630 53.7429 0.7608 0.32597 1.48210 0.98738
LBSA 0.03640 54.1083 0.7609 0.32583 1.48200 0.99125
CLPSO 0.03610 54.1965 0.7608 0.34302 1.48730 0.99633
BLPSO 0.03590 60.2845 0.7607 0.36620 1.49390 1.02720
DE/BBO 0.03640 55.2627 0.7605 0.32477 1.48170 0.99922
CMM-DE/BBO 0.03640 53.8753 0.7608 0.32384 1.48140 0.98605
IADE 0.03621 54.7643 0.7607 0.33613 1.48520 0.98900
IGHS 0.03610 53.2845 0.7608 0.34350 1.48740 0.99306
ABSO 0.03659 52.2903 0.7608 0.30623 1.47878 0.99124

The results of all the comparative algorithms and the results taken from [

23] are listed in Table III, where the meaning of the abbreviations can be found in [23]. Regarding Table III, it can be observed that the proposed LHSSA gives the lowest RMSE value (0.98602 mA) compared with the other comparative algorithms. Additionally, to further emphasize the quality of the obtained results, the best extracted parameters of the LHSSA are employed to replot the I-V and P-V characteristic curves as depicted in Fig. 3. The depicted curves show that the calculated data acquired by the LHSSA highly coincides with the measured one based on the voltage range. Additionally, the individual absolute error (IAE) is introduced as a quality index to determine the absolute difference between the experiment data Itm and the simulated one Ite as in Table IV. The obtained values of the IAE are less than 1.62×10-3 and the total sum is 2.58×10-5, which affirms the accuracy of the parameters estimated by the LHSSA.

Fig. 3 Comparison between experimental and simulated data obtained by LHSSA for SD model. (a) I-V characteristics. (b) P-V characteristics.

Table IV IAE of LHSSA for Each Measurement on SD Model
ItemMeasured voltage (V)Measured current (A)Calculated current (A)IAE
1 -0.2057 0.7640 0.7640 -0.0000872
2 -0.1291 0.7620 0.7626 -0.0006600
3 -0.0588 0.7605 0.7613 -0.0008500
4 0.0057 0.7605 0.7601 0.0003500
5 0.0646 0.7600 0.7590 0.0009500
6 0.1185 0.7590 0.7580 0.0009600
7 0.1678 0.7570 0.7571 -0.0000911
8 0.2132 0.7570 0.7561 0.0008600
9 0.2545 0.7555 0.7551 0.0004100
10 0.2924 0.7540 0.7536 0.0003400
11 0.3269 0.7505 0.7513 -0.0008900
12 0.3585 0.7465 0.7473 -0.0008500
13 0.3873 0.7385 0.7401 -0.0016200
14 0.4137 0.7280 0.7273 0.0006200
15 0.4373 0.7065 0.7069 -0.0004700
16 0.4590 0.6755 0.6752 0.0002200
17 0.4784 0.6320 0.6307 0.0012400
18 0.4960 0.5730 0.5719 0.0010700
19 0.5119 0.4990 0.4996 -0.0006100
20 0.5265 0.4130 0.4136 -0.0006500
21 0.5398 0.3165 0.3175 -0.0010100
22 0.5521 0.2120 0.2121 -0.0001500
23 0.5633 0.1035 0.1022 0.0012500
24 0.5736 -0.0100 -0.0087 -0.0012800
25 0.5833 -0.1230 -0.1255 0.0025100
26 0.5900 -0.2100 -0.2084 -0.0015200

B. Results of DD Model

The parameters of the DD model associated with the RMSE of the different methods are recorded in Table V. The results of the compared algorithms are also presented in Table V. It is obvious that the proposed LHSSA outperforms the comparative algorithms because it provides the best RMSE value (0.98249 mA). The characteristic curves for I-V and P-V of the measured data and the one estimated by the LHSSA are depicted in Fig. 4, whereas the IAE values are given in Table VI. Figure 4 shows that the data estimated by the LHSSA are in good congruence with the measured data. From Table VI, the sum of errors is 3.84×10-6 and all the IAE values are smaller than 1.239×10-3, indicating the high accuracy of the identified parameters.

Table V Comparisons with Various Algorithms for DD Model
AlgorithmRs (Ω)Rsh (Ω)Iph (A)Isd1 (μA)Isd2 (μA)n1n2RMSE (mA)
Proposed LHSSA 0.036740 55.4824 0.76080 0.7473 0.2259 2.0000 1.4515 0.98249
PSO 2.220400×1016 1.1487 0.83680 2.2204×1010 2.2204×1010 1.0000 1.0000 222.86000
FFA 2.220400×1016 1.1904 0.86410 2.2204×1010 2.2204×1010 1.8384 1.3378 226.05000
GWO 0.048295 22.6562 0.76170 8.3152×103 2.4189×1010 1.1862 1.5192 6.61280
DA 2.220400×1016 1.1489 0.83680 2.2204×1010 2.2204×1010 1.0000 1.1676 222.86000
SSA 5.613300×104 1.9814 0.83910 2.2204×1010 3.9483 1.9963 1.8427 82.60300
IJAYA 0.037600 77.8519 0.76010 0.0050 0.7509 1.2186 1.6247 0.98293
JAYA 0.036400 52.6575 0.76070 0.0061 0.3151 1.8436 1.4788 0.98934
GOTLBO 0.036500 53.4058 0.76080 0.1389 0.2621 1.7254 1.4658 0.98742
LETLBO 0.036500 54.3021 0.76080 0.1739 0.2266 1.6585 1.4578 0.98565
LBSA 0.036500 56.0524 0.76070 0.2487 0.2744 1.8817 1.4682 0.98751
CLPSO 0.036700 57.9422 0.76070 0.2584 0.3862 1.4625 1.9435 0.99894
BLPSO 0.036600 61.1345 0.76080 0.2719 0.4351 1.4674 1.9662 1.06280
DE/BBO 0.038500 58.4018 0.76060 0.0012 0.3722 1.8791 1.4956 1.02550
CMM-DE/BBO 0.036000 57.9882 0.76070 0.3537 0.0256 1.4907 1.8835 1.00880
IGHS 0.036900 53.8368 0.76080 0.9731 0.1679 1.9213 1.4281 0.98635
ABSO 0.036600 54.6219 0.76077 0.2671 0.3819 1.4651 1.9815 0.98344

Fig. 4 Comparison between experimental and simulated data obtained by LHSSA for DD model. (a) I-V characteristics. (b) P-V characteristics.

Table VI IAE of LHSSA for Each Measurement on DD Model
ItemMeasured voltage (V)Measured current (A)Calculated current (A)IAE
1 -0.2057 0.7640 0.7639 0.0000165
2 -0.1291 0.7620 0.7626 -0.0006200
3 -0.0588 0.7605 0.7613 -0.0008400
4 0.0057 0.7605 0.7602 0.0003300
5 0.0646 0.7600 0.7591 0.0008900
6 0.1185 0.7590 0.7581 0.0008800
7 0.1678 0.7570 0.7572 -0.0001900
8 0.2132 0.7570 0.7562 0.0007600
9 0.2545 0.7555 0.7552 0.0003200
10 0.2924 0.7540 0.7537 0.0002800
11 0.3269 0.7505 0.7514 -0.0009100
12 0.3585 0.7465 0.7473 -0.0008000
13 0.3873 0.7385 0.7400 -0.0015100
14 0.4137 0.7280 0.7272 0.0007500
15 0.4373 0.7065 0.7069 -0.0003500
16 0.4590 0.6755 0.6752 0.0002900
17 0.4784 0.6320 0.6308 0.0012400
18 0.4960 0.5730 0.5720 0.0010100
19 0.5119 0.4990 0.4998 -0.0007100
20 0.5265 0.4130 0.4137 -0.0007300
21 0.5398 0.3165 0.3175 -0.0010500
22 0.5521 0.2120 0.2121 -0.0001200
23 0.5633 0.1035 0.1022 0.0013400
24 0.5736 -0.0100 -0.0088 -0.0012100
25 0.5833 -0.1230 -0.1255 0.0025400
26 0.5900 -0.2100 -0.2084 -0.0016300

C. Results of PV Module Model

In this model, five parameters are estimated and the RMSE values are obtained and reported in Table VII.

Table VII Results Among Comparative Techniques on PV Module
AlgorithmRs (Ω)Rsh (Ω)Iph (A)Isd (μA)nRMSE (mA)
Proposed LHSSA 0.0334 27.2773 1.0305 3.4822 1.3517 2.4250
PSO 0 2000.0000 1.1741 3046.3000 2.8761 77.5850
FFA 0 586.7000 1.4650 226590.0000 9.1139 220.4100
GWO 0.0005 1064.5000 1.0467 680.5200 2.3241 23.5320
DA 0 2000.0000 1.0484 765.8700 2.3616 23.8810
SSA 0 1931.5000 1.3032 54173.0000 5.3692 144.9300
IJAYA 1.2016 977.3700 1.0305 3.4703 48.6298 2.4251
JAYA 1.2014 1022.5000 1.0302 3.4931 48.6531 2.4278
GOTLBO 1.1995 969.9300 1.0307 3.5124 48.6766 2.4266
LETLBO 1.2015 974.6100 1.0306 3.4705 48.6301 2.4251
LBSA 1.2010 987.7800 1.0305 3.4901 48.6513 2.4252
CLPSO 1.1978 1017.0000 1.0304 3.6131 48.7847 2.4281
BLPSO 1.2002 992.7900 1.0305 3.5176 48.6815 2.4252
DE/BBO 1.1969 1015.1000 1.0303 3.6172 48.7894 2.4283
CMM-DE 1.2013 981.9800 1.0305 3.4823 48.6428 2.4251
PS 1.2053 714.2800 1.0313 3.1756 48.2889 11.8000
SA 1.1989 833.3300 1.0331 3.6642 48.8211 2.7000

The results of the proposed LHSSA are compared with different algorithms, some of which are taken from literature for comparison [

23]. The overall IAE values are given in Table VIII.

Table VIII IAE of LHSSA for Each Measurement on PV Module
ItemMeasured voltage (V)Measured current (A)Calculated current (A)IAE
1 0.1248 1.0315 1.02912 0.0023800
2 1.8093 1.0300 1.02738 0.0026200
3 3.3511 1.0260 1.02574 0.0002600
4 4.7622 1.0220 1.02411 -0.0021100
5 6.0538 1.0180 1.02229 -0.0042900
6 7.2364 1.0155 1.01993 -0.0044300
7 8.3189 1.0140 1.01636 -0.0023600
8 9.3097 1.0100 1.01049 -0.0005000
9 10.2163 1.0035 1.00063 0.0028700
10 11.0449 0.9880 0.98455 0.0034500
11 11.8018 0.9630 0.95952 0.0034800
12 12.4929 0.9255 0.92284 0.0026600
13 13.1231 0.8725 0.87260 -0.0001000
14 13.6983 0.8075 0.80728 0.0002200
15 14.2221 0.7265 0.72834 -0.0018400
16 14.6995 0.6345 0.63714 -0.0026400
17 15.1346 0.5345 0.53622 -0.0017100
18 15.5311 0.4275 0.42951 -0.0020100
19 15.8929 0.3185 0.31877 -0.0002700
20 16.2229 0.2085 0.20739 0.0011100
21 16.5241 0.1010 0.09616 0.0048400
22 16.7987 -0.0080 -0.00833 0.0003300
23 17.0499 -0.1110 -0.11095 -0.0000538
24 17.2793 -0.2090 -0.20926 0.0002600
25 17.4885 -0.3030 -0.30088 -0.0021200

It is clear that the proposed LHSSA outperforms the other algorithms as it gives the best RMSE value (2.42507×10-3) among all the compared algorithms. The comparison shows that the LHSSA performs well. Due to space limitations, the I-V and P-V characteristics are not depicted. The obtained IAE values are all smaller than 4.837×10-3 and the total sum of error is 4.3172×10-5. The parameters with high accuracy are achieved again by the LHSSA.

D. Statistical Measures and Convergence Behavior

In this subsection, statistical measures are calculated for the proposed and comparative algorithms over 20 independent runs and are recorded in Table IX. These measures include the mean RMSE that quantifies the average accuracy and also confirms the stability of the algorithm runs, and St.dev that represents the standard deviation of the RMSE values that defines the reliability of the parameter estimation. For each model, the overall best RMSE values among the comparative algorithms are highlighted in boldface. Table IX demonstrates that the proposed LHSSA performs much better than all the other comparative algorithms for all the models in terms of reliability and accuracy. In this regard, the convergence curves for the comparative algorithms are depicted in Fig. 5 and the box-plot representations are used to show the distribution of the results obtained by those algorithms over 20 independent runs, as shown in Fig. 6. It is noted that the LHSSA has a faster convergence rate than the other algorithms in all models.

Table IX Statistical Measures of Different Techniques for Three Models
ModelAlgorithmRMSE (mA)
MinMeanMaxSt.dev
SD LHSSA 0.98602 0.98602 0.98602 6.269700×10-10
PSO 19.58100 212.69000 222.86000 45.454000
FFA 141.64000 225.21000 241.84000 20.333000
GWO 222.86000 222.86000 222.87000 4.254900×10-3
DA 222.86000 222.86000 222.86000 1.042400×10-13
SSA 39.90600 121.35000 222.86000 44.665000
IJAYA 0.98603 0.99204 1.06220 1.403300×10-2
JAYA 0.98946 1.16170 1.47830 0.187960
GOTLBO 0.98856 1.04500 1.20670 0.502180
LETLBO 0.98738 1.03330 1.15930 0.469460×10-2
LBSA 0.99125 1.14660 1.48620 0.134820
CLPSO 0.99633 1.05810 1.31960 7.485400×10-2
BLPSO 1.02720 1.31390 1.79280 0.211660
DE/BBO 0.99922 1.29480 2.22580 0.250740
CMM-DE/BBO 0.98605 1.04860 1.34750 8.167900×10-2
DD LHSSA 0.98249 0.98337 0.98602 1.494500×10-3
PSO 222.86000 230.57000 299.95000 24.378000
FFA 226.06000 244.84000 266.29000 11.980500
GWO 6.61290 159.45000 222.87000 102.160000
DA 222.86000 222.86000 222.86000 1.072200×10-4
SSA 82.60300 147.24000 165.79000 24.236000
IJAYA 0.98293 1.02690 1.40550 0.098325
JAYA 0.98934 1.17670 1.47930 0.193560
GOTLBO 0.98742 1.14750 1.39470 0.113300
LETLBO 0.98565 1.08690 1.48700 0.153600
LBSA 0.98751 1.25450 1.73430 0.222360
CLPSO 0.99894 1.14580 1.54940 0.143670
BLPSO 1.06280 1.48210 1.74110 0.177890
DE/BBO 1.02550 1.55710 2.40420 0.362970
CMM-DE/BBO 1.00880 1.54870 2.05890 0.294130
PV module LHSSA 2.42500 2.42500 2.42500 3.525700×10-9
PSO 77.58500 202.35000 274.25000 87.493000
FFA 220.41000 242.59000 261.92000 15.209000
GWO 23.53200 99.18200 274.29000 120.820000
DA 23.88100 213.11000 274.25000 96.186000
SSA 144.93000 186.23000 246.43000 37.828000
IJAYA 2.42510 2.42890 2.43930 3.775500×10-3
JAYA 2.42780 2.45370 2.59590 3.456300×10-2
GOTLBO 2.42660 2.47540 2.56380 2.938800×10-2
LETLBO 2.42510 2.44070 2.58210 2.949000×10-2
LBSA 2.42520 2.46740 2.53440 2.910900×10-2
CLPSO 2.42810 2.45490 2.54330 2.581000×10-2
BLPSO 2.42520 2.43790 2.48830 1.372400×10-2
DE/BBO 2.42830 2.46160 2.52560 2.925100×10-2
CMM-DE/BBO 2.42510 2.42520 2.42680 3.554800×10-4

Fig. 5 Convergence curves for two models. (a) SD model. (b) DD model.

Fig. 6 Boxplot representations of RMSE for SD model and DD model. (a) SD model. (b) DD model.

E. Study of Fuzzy Representation

The imprecise descriptions of the solar cell models are often caused by weather fluctuations or instabilities in the measuring process. Thus, new insight from the operating point of view is presented by incorporating this impreciseness using the fuzzy concept in the solar cell models. The fuzzy number representation is illustrated in Section II. Additionally, the fuzzified value is transformed into a crisp value based on the α-cut level, using upper and lower bound values.

A searching process is carried out to identify the optimal values for the parameters in terms of the α level. This strategy is investigated on the SD, DD and PV module models at different levels, but the results are reported for α=0.8 only due to space limitation. The estimated parameters are recorded in Tables X-XII. The convergence curves and box plot are demonstrated for the DD model only due to space limitation in Figs. 7 and 8, respectively. The statistical measures for these models are also listed in Table XIII.

Fig. 7 Convergence curves for DD model at α=0.8.

Fig. 8 Box plot for RMSE over 20 runs for DD α=0.8.

Table XIII Statistical Measures Among Different Techniques for Three Models Under Fuzzy Environment (α=0.8)
ModelAlgorithmRMSE (mA)
MinMeanMaxSD
SD PSO 220.77000 231.40000 300.55000 24.33200
FFA 189.52000 227.43000 242.79000 146.71000
GWO 217.92000 218.65000 219.02000 0.36142
DA 228.64000 273.31000 295.14000 23.96100
SSA 116.52000 161.67000 208.19000 23.10600
LHSSA 0.10029 0.25622 0.34823 0.07818
DD PSO 29.05000 219.53000 302.33000 74.49600
FFA 222.72000 238.44000 251.02000 9.48900
GWO 8.65650 118.91000 219.85000 105.79000
DA 251.53000 11504.00000 111420.00000 35109.00000
SSA 65.54200 137.93000 198.67000 45.38300
LHSSA 0.11987 0.27267 391.31000 0.07176
PV module PSO 91.95800 213.75000 276.93000 720.69000
FFA 212.37000 246.50000 264.59000 14.30600
GWO 19.14200 47.80500 266.10000 76.81400
DA 194.71000 354.88000 441.70000 96.89600
SSA 136.25000 196.20000 254.81000 51.95600
LHSSA 0.49438 0.69770 0.90773 0.13615
Table X Estimated Parameters in Fuzzy Environment (SD Model) when α=0.8
AlgorithmRs (Ω)Rsh (Ω)Iph (A)Isd (μA)nRMSE (mA)
PSO 2.2204×1016 1.1058 0.841825 2.2204×1010 2.0000 220.77000
FFA 0.0181 75.4861 0.949916 8.5959 1.9143 189.52000
GWO 4.3206×1016 1.1224 0.832974 1.7551×109 1.9233 217.92000
DA 2.6996×103 1.0181 0.899220 4.8877×1010 1.0000 228.64000
SSA 0.0269 1.6612 0.863027 4.3412 1.8747 116.52000
LHSSA 2.2204×1016 36.5013 0.766241 14.1930 1.9995 0.10029
Table XI Estimated Parameters in Fuzzy Environment (DD Model) when α=0.8
AlgorithmRs (Ω)Rsh (Ω)Iph (A)Isd1 (μA)Isd2 (μA)n1n2RMSE (mA)
PSO 2.2204×1016 100.0000 0.7426 13.238 2.220×1010 2.00 2.00 29.05
FFA 2.2204×1016 1.1034 0.8331 2.220×1010 2.220×1010 1.50 1.11 222.72
GWO 1.4317×102 73.9081 0.7673 12.968 8.489×1010 2.00 1.43 8.66
DA 2.3572×1016 2.7496 0.6802 3.754×1010 2.818×1010 1.06 1.04 251.53
SSA 1.6169×102 4.6353 0.8661 4.672×103 12.827 1.54 1.97 65.54
LHSSA 2.2561×102 65.3882 0.7539 2.024 2.220×1010 1.71 1.00 0.12
Table XII Estimated Parameters in Fuzzy Environment (PV Module) when α=0.8
AlgorithmRs (Ω)Rsh (Ω)Iph (A)Isd(μA)nRMSE (mA)
PSO 2.2204×1016 2000.0000 1.2169 6938.000 3.3448 91.95800
FFA 2.2204×1016 548.3519 1.1426 88576.000 7.3094 212.37000
GWO 2.2204×1016 905.2461 1.0513 795.350 2.3831 19.14200
DA 2.2204×1016 1659.3036 1.0943 76921.000 6.7461 194.71000
SSA 1.3802×104 1703.3602 1.3011 38435.000 4.8604 136.25000
LHSSA 0.0220 1999.9620 1.0324 65.662 1.7777 0.49438

F. Effects of α-level Schemes on RMSE

In order to clear the effects of the α-level schemes on RMSE, three cases are considered (α=1,α=0.8,α=0.4) as in Fig. 9, where α=1 represents the crisp case. Based on the obtained results, the RMSE is affected by the vagueness aspect induced by the α-level cut. Finally, we hope that this paper will inspire researchers in studying the uncertainty aspect of different solar cell performances, which is caused by various factors such as shading, weather changes and so on.

Fig. 9 Effects of α-level schemes on RMSE.

V. Conclusion

In this paper, a novel LHSSA is presented to accurately estimate the parameters of PV models. In the LHSSA, the standard SSA is conducted to search globally and explore the different areas in the search space. Afterwards, PSO is employed to guide the SSA leaders with the aim of eliciting the promising area. Additionally, a learning scheme based on the follower behavior is introduced with the aim of improving the population diversity. In this regard, the SSA emphasizes on the diversification while PSO focuses on the intensification.

The proposed LHSSA is investigated on different PV models, i.e., SD, DD, and PV module models. Comprehensive results affirm that LHSSA is able to obtain highly competitive performance in comparison with other algorithms, especially in terms of quality and reliability. In future work, we will study the effect of shading on the performance of the PV modules. Additionally, some other modification will be developed, dealing with more complex renewable energy problems and studying the use of the rough set theory for dealing with different PV models. Finally, we hope that this paper will inspire researchers in studying the uncertainty aspect of solar cell performances.

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