Application of Symbiotic Organisms Search Algorithm for Parameter Extraction of Solar Cell Models

: Extracting accurate values for relevant unknown parameters of solar cell models is vital and necessary for performance analysis of a photovoltaic (PV) system. This paper presents an effective application of a young, yet efﬁcient metaheuristic, named the symbiotic organisms search (SOS) algorithm, for the parameter extraction of solar cell models. SOS, inspired by the symbiotic interaction ways employed by organisms to improve their overall competitiveness in the ecosystem, possesses some noticeable merits such as being free from tuning algorithm-speciﬁc parameters, good equilibrium between exploration and exploitation, and being easy to implement. Three test cases including the single diode model, double diode model, and PV module model are served to validate the effectiveness of SOS. On one hand, the performance of SOS is evaluated by ﬁve state-of-the-art algorithms. On the other hand, it is also compared with some well-designed parameter extraction methods. Experimental results in terms of the ﬁnal solution quality, convergence rate, robustness, and statistics fully indicate that SOS is very effective and competitive.


Introduction
Solar energy is considered as a promising tool to fight environmental pollution and fossil energy consumption. As the main application of solar energy, solar photovoltaic (PV) has recently achieved leapfrog development. Solarpower Europe reveals that only seven countries installed over 1 GW PV in 2016. That number was changed to nine in 2017, and in 2018, the number keeps increasing and should reach 14 [1]. China, as the country with the biggest capacity of PV power, installed 24.3 GW, which was about 38% of the world's newly installed capacity PV power, in the first half of 2018 [2]. According to data from the International Energy Agency, by 2040, the fast-developing market of PV in China and India will cause solar to be the largest source of low-carbon capacity [3]. A PV system is a multi-component power unit utilized to directly convert solar energy into electricity. As the core device of a PV system, a solar cell's accurate modelling and parameter extraction are very important for the performance analysis of the PV system [4]. For solar cells, their current-voltage (I-V) characteristics are widely simulated by the most popular single diode model and double diode model [5], which have five and seven unknown parameters, respectively, that need to be extracted.
Extracting accurate value for these relevant unknown model parameters is vital and necessary, and has drawn researchers' attention in recent years [6,7]. The propounded parameter extraction Appl. Sci. 2018, 8,2155 3 of 18 some noticeable merits such as being free from tuning algorithm-specific parameters, good equilibrium between exploration and exploitation, and being easy to implement [49,50]. These merits encourage researchers to apply SOS to a host of engineering problems.
SOS has proven itself a worthy competitor and alternative in many optimization problems. Nonetheless, the promising method has not been employed to solve the problem considered here. The aim of this paper is first to present experimental results validating the performance of SOS in dealing with the parameter extraction problem of solar cell models. Three test cases consisting of the single diode model, double diode model, and PV module model are served to evaluate the effectiveness of SOS along with necessary comparisons. The experimental results comprehensively indicate that SOS behaves competitively compared with other methods.
The rest of this paper is organized as follows. The problem formulation is briefly presented in Section 2. In Section 3, the SOS is provided. Then, the results are analyzed in Section 4 and this paper is concluded in Section 5.

Single Diode Model
Single diode model is a very popular model used to simulate the I-V characteristic of a solar cell. The output current I L (A), as depicted in Figure 1, can be formulated as follows according to Kirchhoff's current law.
where I ph , I d , and I sh are the photo generated current (A), diode current (A), and shunt resistor current (A), respectively. I d and I sh are calculated by Equations (2) and (3), respectively [24,35,[51][52][53]. (2) where V L and V t represent the output voltage (V) and thermal voltage (V), respectively. I sd is the reverse saturation current (A). R s and R sh denote the series resistance (Ω) and shunt resistance (Ω), respectively. n is the diode ideal factor. k = 1.3806503 × 10 −23 J/K is the Boltzmann constant. q = 1.60217646 × 10 −19 C is the electron charge. T denotes the cell temperature in Kelvin. Substituting Equations (2)-(4) into Equation (1), the output current I L can be written as follows: It is observed from Equation (5) that if we know the values of I ph , I sd , R s , R sh , and n, then the I-V characteristic of this model can be constructed. Therefore, accurate extraction of these five unknown parameters is the core of this study.

Single Diode Model
Single diode model is a very popular model used to simulate the I-V characteristic of a solar cell.

Double Diode Model
The above model performs well for almost all types of solar cells [5]. However, its performance is unsatisfactory at low irradiance for thin films based solar cells. The problem can be handled well by the double diode model [24,35]. The output current in Figure 2 is formulated as follows [52,54,55]: where I sd1 and I sd2 represent the diffusion current (A) and saturation current (A), respectively. n 1 and n 2 are the diode ideal factors. Compared with the single diode mode, this model adds two more unknown parameters (I sd2 and n 2 ) and thereby the total number of unknown parameters that need to be extracted is seven (I ph , I sd1 , I sd2 , R s , R sh , n 1 and n 2 ).
where L V and t V represent the output voltage (V) and thermal voltage (V), respectively. the I-V characteristic of this model can be constructed. Therefore, accurate extraction of these five unknown parameters is the core of this study.

Double Diode Model
The above model performs well for almost all types of solar cells [5]. However, its performance is unsatisfactory at low irradiance for thin films based solar cells. The problem can be handled well by the double diode model [24,35]. The output current in Figure 2 is formulated as follows [52,54,55]: where sd1 I and sd2 I represent the diffusion current (A) and saturation current (A), respectively.

PV Module
In general, a PV module is used to raise the output voltage. The corresponding output current is calculated as follows [19,28,56,57]: where N s and N p denote the number of solar cells in series and in parallel, respectively.

Objective Function
Accurate extracted values for the involved unknown parameters of solar cell models should make the constructed model coincide with the real model. Namely, by using the constructed model, the calculated data should match the measured data well. Therefore, the difference between the measured current and the calculated current can be used to reflect the agreement degree. In general, the root mean square error (RMSE) is highly preferred [18,[20][21][22][23][24][25].
where N is the number of measured data and x is the solution vector. For the abovementioned three models, the objective functions f (V L , I L , x) and the solution vectors x are as follows:

Symbiotic Organisms Search (SOS) Algorithm
SOS [48] is a young, yet effective metaheuristic inspired by the symbiotic interaction ways employed by organisms to improve their overall competitiveness in the ecosystem. Each organism (i.e., population individual) is represented as a D-dimensional vector X i = [x i,1 , x i,2 , . . . , x i,D ], where i = 1, 2, . . . , ps, ps is the number of organisms in the ecosystem (i.e., population size). SOS contains mutualism, commensalism, and parasitism phases.

Mutualism Phase
In this phase, two organisms establish a good interaction relationship in which they can obtain what they need, and thus their mutual survival advantage can be increased simultaneously. For each organism X i of the ecosystem, a random distinct organism X j is selected to interact with X i by the following formulations: where X i,new and X j,new are new candidate solutions for X i and X j , respectively. rand(a, b) is a random number generated uniformly in (a,b). BF 1 and BF 2 are benefit factors with the random value 1 or 2. X best represents the best organism of the ecosystem. MV = (X i + X j )/2 is the relationship characteristic.

Commensalism Phase
In this phase, two organisms build a unidirectional relationship where one organism X i benefits from the other organism X j as shown in Equation (14), whereas X j gets nothing from X i .

Parasitism Phase
In parasitism, one organism X i improves its survivability through harming the other organism X j . In SOS, this relationship is modeled as follows. An organism X i is copied and used to create an artificial parasite AP. Then, some random dimensions of AP are selected and modified by a random number generated within the corresponding bounds. The other organism X j , selected randomly from the ecosystem, serves as a host to the parasite AP. If AP is better than X j , then X j will be replaced by AP; otherwise, AP will be discarded.
The pseudo-code of SOS is presented in Algorithm 1. It can be seen that apart from the common parameter, that is, the population size used in all metaheuristic algorithms, SOS has no algorithm-specific parameters that need to be well-tuned.

Algorithm 1:
The pseudo-code of SOS 1: Initialize an ecosystem X with ps organisms randomly 2: Calculate the fitness value of each organism 3: Set the iteration number t = 1 4: While the terminating criterion is not met do 5: Select the fittest organism X best of the ecosystem 6: For i = 1 to ps do 7: /* mutualism phase */ 8: Select a random organism X j (j = i) from the ecosystem 9: Generate the i-th new organism X i,new using Equation (12) 10: Generate the j-th new organism X j,new using Equation (13)

11:
Calculate the fitness value of X i,new and X j,new 12: Replace the old organism if it is defeated by the new one 13: /* commensalism phase */ 14: Select a random organism X j (j = i) from the ecosystem 15: Generate the i-th new organism X i,new using Equation (14)

16:
Calculate the fitness value of X i,new 17: Replace the old organism if it is defeated by the new one 18: /* parasitism phase */

19:
Select a random organism X j (j = i) from the ecosystem 20: Generate an artificial parasite AP = X i 21: Select a random number of dimensions of AP

22:
Replace the selected dimensions using a random number 23: Calculate the fitness value of the modified AP

24:
Replace X j if the modified AP is better than X j 25: End for 26:

Test PV Models
In this work, SOS is applied to three cases including single diode, double diode, and PV module models. The datasets are derived from the literature [58]. The measurements are conducted on an RTC France silicon solar cell and a Photowatt-PWP201 solar module. The former operates under 1000 W/m 2 at 33 • C. The latter contains 36 polycrystalline silicon cells connected in series operating under 1000 W/m 2 at 45 • C. The boundaries of extracted parameters are presented in Table 1.

Experimental Settings
In this work, the maximum number of fitness evaluations (Max_FEs), which is set to 50,000 [29], serves as the terminating criterion. In addition, to verify the effectiveness of SOS, five state-of-the-art algorithms including across neighborhood search (ANS) [59], biogeography-based learning particle swarm optimization (BLPSO) [60], competitive swarm optimizer (CSO) [61], chaotic teaching-learning algorithm (CTLA) [62], and levy flight trajectory-based whale optimization algorithm (LWOA) [63] are used for performance comparison. These five methods keep the original algorithm parameters, except the population size ps, setting the same unified value 50 for fair comparison. For each case, each method runs 50 times independently.
The best extracted values for the five unknown parameters of single diode model are given in Table 3. We observe that these listed methods almost extract close values for the unknown parameters. Utilizing the extracted parameters in Table 3, we reconstruct the characteristic curves as illustrated in Figure 3. We see that both the output current and power calculated by SOS match the measured values well throughout the whole range of voltage. In addition, we also tabulate the output current data calculated by ANS, BLPSO, CSO, CTLA, LWOA, and SOS in Table 4. An error index the sum of individual absolute error (SIAE) given in Equation (15) is used to evaluate the fitting error. It is obvious that the SIAE value of SOS is the smallest, followed by that of ANS, LWOA, BLPSO, CTLA, and CSO, meaning that SOS achieves more accurate values for the relevant parameters of single diode model.      Besides, the convergence curves are presented in Figure 4. It is obvious that SOS is slightly slower than LWOA in the opening phase, however, the latter stagnates soon and then suffers from premature convergence, indicating that it has been caught in a local optimum. For the other four methods, SOS consistently converges faster than them throughout the whole evolutionary process.

Results Comparison on the Double Diode Model
The experimental results of the second case are summarized in Table 5. Similar to the comparison results on the single diode model, SOS performs better than ANS, BLPSO, CSO, CTLA, and LWOA in various RMSE indicators on the double diode model. SOS is surpassed by GOTLBO, CARO, and IJAYA, but it outperforms GGHS, PS, and SA. It is worth noting that the standard deviation value of SOS is the smallest among all compared methods, which indicates that SOS is highly robust. The extracted parameters are tabulated in Table 6. The reconstructed characteristic curves provided in Figure 5 clearly demonstrate that the calculated current and power achieved by SOS match up well with the measured values. The curve fitting results presented in Table 7 manifest once again that SOS can yield the smallest SIAE value (0.0182), followed by ANS, LWOA, BLPSO, CTLA, and CSO, which demonstrates the high accuracy of the parameters extracted by SOS for the double diode model. The convergence graph illustrated in Figure 6 reveals that SOS exhibits noticeably faster convergence rate than BLPSO, CSO, CTLA, and LWOA, but not ANS, which is slightly faster than SOS during the intermediate stage.
However, ANS is surpassed by SOS in other stages.

Results Comparison on the PV Module Model
The RMSE values of the third case listed in Table 8 indicate that SOS, together with IJAYA, can provide the smallest RMSE value (2.4251 × 10 −3 ) among all methods. Based on the optimal extracted parameters in Table 9, the corresponding characteristic curves are rebuilt and illustrated in Figure 7. It is clear that the output current and power calculated by SOS are highly in coincidence with the measured values. The SIAE results presented in Table 10 repeatedly manifest that SOS can achieve the most accurate values for the unknown parameters, followed by ANS, BLPSO, LWOA, CTLA, and CSO. The curves presented in Figure 8 state clearly that SOS is consistently faster than its competitors from beginning to end.

Statistical Analysis
The significance difference between two methods can be measured by the statistical analysis. Wilcoxon's rank sum test is a reliable and robust statistical analysis tool and is widely used in metaheuristic methods. In this paper, the Wilcoxon's rank sum test at a 0.05 confidence level is used to identify the significance difference between SOS and other compared methods on the same case. The test results are tabulated in Table 11. The symbol " †" denotes that SOS is statistically better than its competitor. The results demonstrate that SOS significantly outperforms every method on every case (p < 0.05), indicating the better performance of SOS from another perspective.