Modiﬁed Search Strategies Assisted Crossover Whale Optimization Algorithm with Selection Operator for Parameter Extraction of Solar Photovoltaic Models

: Extracting accurate values for involved unknown parameters of solar photovoltaic (PV) models is very important for modeling PV systems. In recent years, the use of metaheuristic algorithms for this problem tends to be more popular and vibrant due to their e ﬃ cacy in solving highly nonlinear multimodal optimization problems. The whale optimization algorithm (WOA) is a relatively new and competitive metaheuristic algorithm. In this paper, an improved variant of WOA referred to as MCSWOA, is proposed to the parameter extraction of PV models. In MCSWOA, three improved components are integrated together: (i) Two modiﬁed search strategies named WOA / rand / 1 and WOA / current-to-best / 1 inspired by di ﬀ erential evolution are designed to balance the exploration and exploitation; (ii) a crossover operator based on the above modiﬁed search strategies is introduced to meet the search-oriented requirements of di ﬀ erent dimensions; and (iii) a selection operator instead of the “generate-and-go” operator used in the original WOA is employed to prevent the population quality getting worse and thus to guarantee the consistency of evolutionary direction. The proposed MCSWOA is applied to ﬁve PV types. Both single diode and double diode models are used to model these ﬁve PV types. The good performance of MCSWOA is veriﬁed by various algorithms.


Introduction
Solar energy is an inexhaustible and carbon emission-free energy source to promote sustainable development. Solar photovoltaic (PV) is becoming the preferred choice for meeting the rapidly growing power demands globally [1,2]. It is a clean energy according to the principle of sustainability. Take China as an example, according to the latest data from the National Energy Administration, PV added 5.20GW capacity, which was more than that of wind (added 4.78GW) in the first quarter of 2019 [3]. In addition, by the end of the first quarter of 2019, the total installed PV capacity had reached 180GW, accounting for 24.3% of renewable energy, only 0.09GW below that of wind, and the gap is narrowing. Along with the increasing installed capacity of PV, its impact on the connected power system is growing, and thereby, analyzing PV systems' dynamic conversion behavior is quite important and necessary. Thereinto, accurate modeling of the PV system's basic device, i.e., the PV cell or module, is the premise and crux. The most widely used modeling tool is the single diode (SDM) and double diode (DDM) WOA/rand/1 and WOA/current-to-best/1 inspired by DE. The former uses one random weighted difference vector to perturb a randomly selected individual and thus to improve the exploration; while the latter simultaneously adopts one current-to-best weighted difference vector and one random weighted difference vector to perturb the current individual and thereby to maintain the exploitation. In addition, in the original WOA, the values of all dimensions of each offspring completely come from a vector generated by one search strategy, which cannot meet the exploration and exploitation performance requirements of different dimensions. In this case, a crossover operator based on the modified search strategies is designed. It adopts two different search strategies to generate each offspring simultaneously, which can further promote the balance between exploration and exploitation. Moreover, the original WOA preserves the generated vector regardless of its quality. This "generate-and-go" strategy may result in retrogression or oscillation in evolutionary process. To prevent this phenomenon from occurring, a selection operator instead of the "generate-and-go" operator is implemented to guarantee the consistency of evolutionary direction. The resultant improved variant of WOA, referred to as MCSWOA, is applied to five PV types modeled by both SDM and DDM.
The main contributions of this paper are the following: (1) An improved variant of WOA, i.e., MCSWOA, is presented to parameter extraction of PV models. In MCSWOA, three improved components, including two modified search strategies, a crossover operator, and a selection operator are developed and integrated well to enhance its performance. (2) MCSWOA is applied to five PV types, including RTC France cell, Photowatt-PWP201 module, STM6-40/36 module, STP6-120/36 module, and Sharp ND-R250A5 module. Both SDM and DDM are used to model these five PV types. (3) The good performance of MCSWOA in extracting accurate parameters of PV models is fully verified through comparison with other 31 algorithms in terms of the parameter accuracy, convergence speed, robustness, and statistics.
The rest of this paper is organized as follows. In Section 2, the mathematical formulation of the parameter extraction problem is described. Section 3 introduces the original WOA. Section 4 gives the proposed MCSWOA. Section 5 presents the experimental results and comparisons. The discussions are provided in Section 6. Finally, the paper is concluded in Section 7.

Single Diode Model (SDM)
The equivalent circuit of SDM is presented in Figure 1.
Remote Sens. 2019, 11, x FOR PEER REVIEW 3 of 24 also reveal that the use of both improved search strategies and DE can enhance the performance of WOA significantly. Having noticed this, in this paper, we propose two modified search strategies named WOA/rand/1 and WOA/current-to-best/1 inspired by DE. The former uses one random weighted difference vector to perturb a randomly selected individual and thus to improve the exploration; while the latter simultaneously adopts one current-to-best weighted difference vector and one random weighted difference vector to perturb the current individual and thereby to maintain the exploitation. In addition, in the original WOA, the values of all dimensions of each offspring completely come from a vector generated by one search strategy, which cannot meet the exploration and exploitation performance requirements of different dimensions. In this case, a crossover operator based on the modified search strategies is designed. It adopts two different search strategies to generate each offspring simultaneously, which can further promote the balance between exploration and exploitation. Moreover, the original WOA preserves the generated vector regardless of its quality. This "generate-and-go" strategy may result in retrogression or oscillation in evolutionary process. To prevent this phenomenon from occurring, a selection operator instead of the "generate-and-go" operator is implemented to guarantee the consistency of evolutionary direction. The resultant improved variant of WOA, referred to as MCSWOA, is applied to five PV types modeled by both SDM and DDM.
The main contributions of this paper are the following: (1) An improved variant of WOA, i.e., MCSWOA, is presented to parameter extraction of PV models. In MCSWOA, three improved components, including two modified search strategies, a crossover operator, and a selection operator are developed and integrated well to enhance its performance.
(3) The good performance of MCSWOA in extracting accurate parameters of PV models is fully verified through comparison with other 31 algorithms in terms of the parameter accuracy, convergence speed, robustness, and statistics.
The rest of this paper is organized as follows. In Section 2, the mathematical formulation of the parameter extraction problem is described. Section 3 introduces the original WOA. Section 4 gives the proposed MCSWOA. Section 5 presents the experimental results and comparisons. The discussions are provided in Section 6. Finally, the paper is concluded in Section 7.

Single Diode Model (SDM)
The equivalent circuit of SDM is presented in Figure 1. The output current L I can be achieved according to the Kirchhoff's current law: The output current I L can be achieved according to the Kirchhoff's current law: where I ph , I sh and I d are the photogenerated current, shunt resistor current, and diode current, respectively. I d and I sh are calculated as follows [4,6]: where I sd is the saturation current, V L is the output voltage, R s and R sh are the series and shunt resistances, respectively, n is the diode ideal factor, k is the Boltzmann constant (1.3806503 × 10 −23 J/K), q is the electron charge (1.60217646 × 10 −19 C), and T is the cell temperature (K). The output current I L can be obtained by substituting Equations (2) and (4) into (1): From Equation (5), it can be seen that the SDM has 5 unknown parameters (i.e., I ph , I sd , R s , R sh , and n) that need to be extracted.

Double Diode Model (DDM)
When considering the effect of the recombination current loss in the depletion region, we can get the equivalent circuit of DDM, as shown in Figure 2. It performs well in some applications [4].
where sd I is the saturation current, L V is the output voltage, s R and sh R are the series and shunt resistances, respectively, n is the diode ideal factor, k is the Boltzmann constant (1.3806503×10 -23 J/K), q is the electron charge (1.60217646×10 -19 C), and T is the cell temperature (K).
The output current L I can be obtained by substituting Equations.
Error! Reference source not found. and Error! Reference source not found. into Error! Reference source not found.: From Equation Error! Reference source not found., it can be seen that the SDM has 5 unknown parameters (i.e., ph sd s sh , , , , and I I R R n ) that need to be extracted.

Double Diode Model (DDM)
When considering the effect of the recombination current loss in the depletion region, we can get the equivalent circuit of DDM, as shown in Figure 2. It performs well in some applications [4]. The output current L I is calculated as follows: where sd1 I and sd2 I are diode currents, 1 n and 2 n are diode ideal factors. The DDM has 7 unknown parameters (i.e., ph sd1 sd2 s sh 1 2 , , , , , and I I I R R n n ) that need to be extracted. The output current I L is calculated as follows:

PV Module Model
where I sd1 and I sd2 are diode currents, n 1 and n 2 are diode ideal factors. The DDM has 7 unknown parameters (i.e., I ph , I sd1 , I sd2 , R s , R sh , n 1 and n 2 ) that need to be extracted.

PV Module Model
For a PV module with N s × N p solar cells in series and/or in parallel, its output current I L can be formulated as follows: For the SDM based PV module: Remote Sens. 2019, 11, 2795

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For the DDM based PV module:

Objective Function
One way to extract the unknown parameters of PV models is to construct an objective function to reflect the difference between the measured data and the calculated data. Commonly, the root mean square error (RMSE) between the measured current I L,measured and the calculated current I L,calculated as shown in Equation (9) is recommended [6,8,9,30,31].
where N is the number of measured data, x is the vector of unknown parameters.

Whale Optimization Algorithm
WOA [25] is an effective metaheuristic inspired by the special spiral bubble-net hunting behavior of humpback whales. In WOA, the position of each whale (i.e., population individual) is represented as . . , ps, t = 1, 2, . . . , t max , ps is the population size, t max is the maximum number of iterations, and D is the dimension of one individual. WOA contains the following three parts: (1) Encircling prey WOA defines the position of a current best humpback whale as the target prey, and other whales encircle the prey using the following formulation: where x t g is the best position found so far. A and C are coefficient parameters and calculated for each individual using the following method: where a linearly decreases from 2 to 0 with the increasing of iterations. r is a random real number in (0,1).
(2) Bubble-net attacking method WOA employs both shrinking encircling and spiraling to spin around the prey with the same probability as follows: where b is a constant for defining the shape of the logarithmic spiral, l and p are random real numbers in (0,1).
(3) Searching for prey Before finding the prey, humpback whales swim around and select a random whale to search for prey. This behavior is formulated as follows and continues if |A| ≥ 1. where r ∈ {1, 2, . . . , ps is different from i.

Modified Search Strategies
It is well-known that balancing exploration and exploitation is very important for a metaheuristic algorithm. For the original WOA, it emphasizes the exploitation excessively and thus easily suffers from premature convergence [28]. In order to solve this issue, one active method is to modify its search strategy.
Differential evolution (DE) [32] has proved its efficiency in solving different real-world problems. The efficiency of DE comes largely from its versatile mutation strategies. The following are 2 popular mutation strategies widely used in the literature: where r1, r2 and r3 are random distinct integers selected from 1, 2, · · · , ps and are also different from i, the parameter F is the scaling factor. The former, i.e., DE/rand/1 strategy, usually presents good exploration while the latter, i.e., DE/current-to-best/1 strategy exhibits good exploitation. Inspired by the mutation strategies of DE, in this paper, two modified search strategies are proposed to generate new donor individuals as follows: The above-modified search strategies are employed to replace Equations (15) and (13), respectively.

Modified Search Strategies Assisted Crossover Operator
In the original WOA, the random parameter p is generated for each individual, indicating that all dimensions would perform the same search strategy. For example, on the premise of |A| ≥ 1, if p < 0.5, then the current individual would perform Equation (15). According to Equation (15), WOA updates the current individual around a random individual x t r , which is beneficial for the exploration but harmful to the exploitation. In fact, different dimensions of an individual have different performance requirements for exploration and exploitation. For one dimension, it is wise to perform the exploration-oriented search strategy if the population diversity associated with this dimension is high; otherwise, it is wise to perform the exploitation-oriented search strategy. In order to meet the performance requirements of different dimensions, a crossover operator based on the abovementioned modified search strategies is proposed and shown in Figure 3. In the crossover operator, for each dimension of each individual, the random parameter p is regenerated, and thereby the target dimension of the donor individual has the same chance of deriving from 2 search strategies, which is able to promote the balance between the exploration and exploitation. This crossover operator can be formulated as follows: Remote Sens. 2019, 11, 2795

Selection Operator
In the original WOA, the target individual is directly replaced by the newly generated vector regardless of its quality. This "generate-and-go" operator is not very effective because the newly generated vector may be worse than the target individual. In order to guarantee the consistency of evolutionary direction, a selection operator is employed to determine whether the target individual or the donor individual survives to the next iteration. This selection operator is formulated as follows: Hence, the prerequisite of using the donor individual to replace the target individual is that the donor individual achieves an equal or better fitness value; otherwise, the donor individual is

Selection Operator
In the original WOA, the target individual is directly replaced by the newly generated vector regardless of its quality. This "generate-and-go" operator is not very effective because the newly generated vector may be worse than the target individual. In order to guarantee the consistency of evolutionary direction, a selection operator is employed to determine whether the target individual or the donor individual survives to the next iteration. This selection operator is formulated as follows: Hence, the prerequisite of using the donor individual to replace the target individual is that the donor individual achieves an equal or better fitness value; otherwise, the donor individual is abandoned, and the target individual is retained and passed on to the next iteration. Consequently, the population either gains quality improvement or maintains the current quality level, but never gets worse. Update a, A, and l 8: for d = 1 to D do 9: Update p 10: if p < 0.5 then 11:

Test Cases
In this work, the proposed MCSWOA was applied to five PV types, including RTC France cell, Photowatt-PWP201 module, STM6-40/36 module, STP6-120/36 module, and Sharp ND-R250A5 module. Both the SDM and DDM were adopted to model them, and thus we could get 10 test cases. The detailed information about these 10 test cases is tabulated in Table 1. The search ranges of involved parameters are presented in Table 2. They are kept the same as those used in [6,9,10].

Experimental Settings
In this work, the maximum number of fitness evaluations (Max_FEs) setting as 50,000 [15,17,24,33] was employed as the stopping condition. All involved algorithms used the same population size with the value ps = 50 [14,24]. With regard to other parameters associated with the compared algorithms, the same values in their original literature were used for a fair comparison. In addition, 50 independent runs for each algorithm on each test case were performed in MATLAB 2017a.

Comparison of MCSWOA with WOA
In this subsection, the proposed MCSWOA was compared with the original WOA to demonstrate its effectiveness. The experimental results tabulated in Table 3 contain the minimum (Min), maximum (Max), mean, and standard deviation (Std Dev) values of the RMSE values over 50 independent runs. The best results on each case are highlighted in boldface. It can be seen that MCSWOA was significantly better than WOA in all terms of RMSE values in all cases, indicating that the proposed modified components could improve the performance of WOA considerably.
The extracted values corresponding to the minimum RMSE given by MCSWOA for the involved unknown parameters are presented in Table 4. By using these extracted parameters, the output current could be easily calculated and given in Tables 5-9, respectively. Two error metrics, i.e., individual absolute error (IAE) and the sum of individual absolute error (SIAE) were used to evaluate the fitting results between the calculated current and the measured current. Tables 5-9 only provide the detailed calculated current of MCSWOA due to the space limitation, while for WOA only the SIAE values were listed. It is obvious that MCSWOA achieved smaller SIAE values than WOA on all cases. Namely, the calculated current obtained by MCSWOA fitted the measured current better than that of WOA, meaning that the parameters extracted by MCSWOA were more accurate. In addition, it can be observed that the DDM obtained slightly smaller SIAE values on the RTC France solar cell and Photowatt-PWP201 module, while the SDM yielded somewhat better results on the STM6-40/36, STP6-120/36 and Sharp ND-R250A5 modules. However, the differences were very small, which could be confirmed by some representative reconstructed I-V and P-V characteristic curves illustrated in Figure 4. Figure 4 also shows that the calculated data given by MCSWOA with both SDM and DDM were highly in agreement with the measured data throughout the entire voltage range.              Table 9. Calculated results of MCSWOA for the Sharp ND-R250A5 module.   It can be seen from Section 4 that the proposed MCSWOA has three improved components, i.e., modified search strategies, crossover operator, and selection operator. In this subsection, the influence of these three components on MCSWOA was assessed. Six variants of MCSWOA were considered here:  Table 10. The Wilcoxon's rank sum test was employed to compare the significance between MCSWOA and other algorithms. It is clear that MCSWOA performed significantly better than all of the other algorithms on all cases. Comparing WOAwM, WOAwC, and WOAwS with the original WOA, they won on 7, 10 and 5 cases while lost on 3, 0, and 5 cases, respectively. Additionally, comparison with WOAwM, WOAwC, and WOAwS, MCSWOAwoM beat them on all cases; MCSWOAwoC was better on 9, 4, and 9 cases, respectively; and MCSWOAwoS outperformed WOAwM and WOAwS on all cases, while just lost on cases 9 and 10 when compared with WOAwC. The comparison result indicated that the crossover operator contributed the most to MCSWOA, followed by the selection operator and modified search strategies. Besides, the absence of any improved component would deteriorate the performance of MCSWOA.

Comparison with Advanced WOA Variants
In this subsection, some advanced WOA variants were employed to verify the proposed MCSWOA. These advanced WOA variants included CWOA [34], IWOA [28], Lion_Whale [35], LWOA [36], MWOA [37], OBWOA [27], PSO_WOA [38], RWOA [39], SAWOA [40], WOA−CM [41], and WOABHC [42]. The experimental results are summarized in Table 11. It can be seen that MCSWOA was consistently significantly better than all of the other 11 algorithms on all cases, according to the statistical result of Wilcoxon's rank sum test. In addition, the standard deviation values of RMSE achieved by MCSWOA were also the smallest, meaning that the proposed algorithm was the most robust one among these 12 advanced WOA variants. Furthermore, the Friedman test result presented in Figure 5 manifests that MCSWOA yielded the first ranking, followed by IWOA, WOA−CM, Lion_Whale, MWOA, WOABHC, RWOA, LWOA, SAWOA, PSO_WOA, OBWOA, and CWOA. Some representative convergence curves given in Figure 6 indicate that MCSWOA had the fastest convergence speed overall, while other algorithms converged relatively slowly and had the possibility of being plunged into local optima. IWOA was slightly faster than MCSWOA at the initial stage on Case 2, but it was overtaken and surpassed quickly by MCSWOA. 1.1190E−02±8.4623E−06 † denotes MCSWOA is significantly better than the compared algorithm according to the Wilcoxon's rank sum test at 5% significance difference.   1.1190E−02±8.4623E−06 † denotes MCSWOA is significantly better than the compared algorithm according to the Wilcoxon's rank sum test at 5% significance difference.

Comparison with Advanced Non−WOA Variants
The performance of MCSWOA was further verified by some advanced non−WOA variants. Thirteen algorithms consisting of BLPSO [43], CLPSO [44], CSO [45], DBBO [46], DE/BBO [47], GOTLBO [14], IJAYA [17], LETLBO [48], MABC [49], ODE [50], SATLBO [15], SLPSO [51], and TLABC [24] were employed for comparison in this subsection. The result of Wilcoxon's rank sum test tabulated in Table 12 shows that MCSWOA performed very competitively and outperformed all of the other 13 algorithms on 9 cases except Case 4, on which MCSWOA was surpassed by ODE and DBBO, and tied by TLABC. Considering the standard deviation values, the comparison result was similar to that of the mean values of RMSE, which validated the good robustness of MCSWOA. Similarly, the Friedman test result given in Figure 7 shows that MCSWOA won the first ranking again, followed by TLABC, IJAYA, SATLBO, LETLBO, GOTLBO, ODE, DE/BBO, DBBO, CLPSO, MABC, BLPSO, SLPSO, and CSO. In addition, the convergence curves in Figure 8 reveal again that MCSWOA obtained a competitively fast convergence speed throughout the whole evolutionary process although it was temporarily surpassed by ODE at the early stage.

Discussions
In this work, we present modified search strategies, crossover operator, and selection operator to enhance the performance of MCSWOA. In the modified search strategies, WOA/rand/1 strategy focuses on the exploration, while WOA/current−to−best/1 strategy emphasizes the exploitation. They can cooperate well to achieve a good ratio between exploration and exploitation. In the crossover operator, each dimension of each donor individual has the same chance of deriving from two search strategies, which can further promote the balance between exploration and exploitation. In the selection operator, only comparative or better individuals can survive to the next iteration, which makes the population either gain quality improvement or maintain the current quality level, but never get worse. Experiments have been conducted on five PV types modeled by both SDM and DDM. From the experimental results and comparisons, we can summarize that: (1) MCSWOA obtains better results on most of the cases except Case 4, which can be explained by the no free lunch theorem [52]. According to the theorem, there is no "one size fits all" method that always wins all cases.
(2) The convergence curves show that MCSWOA converges the fastest overall throughout the whole evolutionary process, which indicates that it achieves an excellent balance between exploration and exploitation.
(3) The crossover operator contributes the most to MCSWOA, followed by the selection operator and modified search strategies. Nevertheless, each component is indispensable, and missing anyone will deteriorate the performance MCSWOA significantly.
(4) Comparing the results of SDM and DDM, it concludes that not every equivalent circuit model is suitable for every PV type. Notwithstanding, the differences are very small. In addition, the DDM is harder to optimize under the same stopping condition (i.e., the same value of Max_FEs) because it has seven unknown parameters whereas the SDM has only five.

Discussions
In this work, we present modified search strategies, crossover operator, and selection operator to enhance the performance of MCSWOA. In the modified search strategies, WOA/rand/1 strategy focuses on the exploration, while WOA/current−to−best/1 strategy emphasizes the exploitation. They can cooperate well to achieve a good ratio between exploration and exploitation. In the crossover operator, each dimension of each donor individual has the same chance of deriving from two search strategies, which can further promote the balance between exploration and exploitation. In the selection operator, only comparative or better individuals can survive to the next iteration, which makes the population either gain quality improvement or maintain the current quality level, but never get worse. Experiments have been conducted on five PV types modeled by both SDM and DDM. From the experimental results and comparisons, we can summarize that: (1) MCSWOA obtains better results on most of the cases except Case 4, which can be explained by the no free lunch theorem [52]. According to the theorem, there is no "one size fits all" method that always wins all cases. (2) The convergence curves show that MCSWOA converges the fastest overall throughout the whole evolutionary process, which indicates that it achieves an excellent balance between exploration and exploitation. (3) The crossover operator contributes the most to MCSWOA, followed by the selection operator and modified search strategies. Nevertheless, each component is indispensable, and missing anyone will deteriorate the performance MCSWOA significantly. (4) Comparing the results of SDM and DDM, it concludes that not every equivalent circuit model is suitable for every PV type. Notwithstanding, the differences are very small. In addition, the DDM is harder to optimize under the same stopping condition (i.e., the same value of Max_FEs) because it has seven unknown parameters whereas the SDM has only five.

Conclusions
An improved WOA variant referred to as MCSWOA by integrating modified search strategies, crossover operator, and selection operators is proposed to extract accurate values for involved unknown parameters of PV models. Five PV types modeled by both SDM and DDM are employed to validate the performance of MCSWOA. The experimental results compared with various algorithms (original WOA, 6 MCSWOA variants, 11 WOA advanced variants, and 13 non−WOA advanced variants) demonstrate that MCSWOA is better or highly competitive in terms of the solution quality, convergence performance, and statistical analysis, indicating that it can achieve more accurate and reliable parameters of PV models. Therefore, MCSWOA is a promising candidate for parameter extraction of PV models.
In this work, the proposed MCSWOA is verified at one given operating condition for a PV type, and its performance still has room to improve. In future work, on the one hand, adaptive learning and local search strategies will be used to further enhance its performance and, on the other hand, other PV types operating at different irradiances and temperatures will be employed to verify the enhanced performance.