A Novel Metaheuristic Jellyfish Optimization Algorithm for Parameter Extraction of Solar Module

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Introduction
One of the major sources of renewable energy is solar PV systems (PVs), which play a vital role in today's world. Deep knowledge of the solar module and its parameters are governing factors to derive maximum solar energy. To accurately predict the generation of solar photovoltaic output, there is a need for accurate modeling tools. Tis forecast appears to be based on the mathematical model being designed and implemented for the PV cell/module and the information that is currently available to be determined by the corresponding modeling technique [1,2], in addition to environmental conditions. It is necessary to have a suitable mathematical model of PV solar cells/modules that is efective in all environmental conditions. Several mathematical models in the literature are available (one-diode, two-diode, and three-diode models) for transforming solar energy into electricity [3][4][5][6][7][8]. However, mainly found models in the literature are based on fve parameters [9][10][11] because of ease and acceptance. According to the review, the characteristics of photovoltaic cells can be derived from information in the manufacturer's datasheet using equations or mathematical models [1,5,[12][13][14][15] or from experimentally recorded currentvoltage (I-V) characteristics [16][17][18][19][20]. Te sensitivity analysis also plays an important role in analyzing the efect of environmental conditions on the PV module, as there are many variables that are used for the sensitivity study such as solar irradiation, temperature, dust, interconnection of cells, degradation of a solar cell, direct normal irradiation (DNI), difused horizontal irradiation (DHI), global horizontal irradiation (GBI), wind speed, and albedo. Tis sensitivity analysis is performed by comparing the model output obtained from changing one specifc value of the model. Comparison is done with the base value of the same parameter. Manufacturers provide only some electrical and thermal metrics (datasheets) that are available on STC. Tus, to develop a proper photovoltaic model, it is required to examine a modeling technique that would give the desired characteristics.
Diferent authors have used diferent approaches to extract the PV parameters, in terms of analytical, numerical, and hybrid methods [1], further [9,12] categorized numerical methods into deterministic and stochastic methods, whereas hybrid methods were termed as metaheuristic approaches. Te main problem with analytical approaches is their slowness, which makes them less accurate, especially as the number of unknown factors rises [21,22] and their function approximations give rise to lower accuracy [21,23]. In order to deal with the shortcomings of analytical methods, recent few researchers have used numerical (deterministic and stochastics) methods for explaining the mathematical models for the cell/module. Te Levenberg-Marquardt algorithm (LM) [20] and least-square methods [24] are examples of deterministic techniques. Numerical methods are computationally expensive, especially for large or complex problems, and may require signifcant computational resources. For implementation, they need to be in a state of continuity, convexity, and diferentiability. It can be much faster than analytical methods. But sometimes fnding a solution may be time-consuming and sometimes fail to converge to a solution or may converge to a solution that is not physically meaningful [13]. Stochastic techniques have some limitations, such as premature convergence, behavior deteriorates when the algorithm approaches multimodal functions. Finding solutions may be time-consuming, and it may be difcult to identify the parameter's sensitivity when deciding the optimal modifcation strategy for a particular problem [25]. So, to deal with the difculties of analytical and numerical approaches, the authors shifted toward the metaheuristic methods. Tese are distinguishable by their search space and their soft computing techniques. Tese methods are useful because they can use global search mechanisms to solve nonlinear problems without requiring complicated objective function calculations. Several authors [1,[8][9][10][11][12] have used diferent metaheuristic algorithms techniques such as particle swarm optimization (PSO), adaptive diferential evolution (DE) algorithm, sooty tern optimization algorithm, cuckoo search, grey wolf optimization (GWO), genetic algorithm (GA), artifcial bee colony (ABC), whale optimization algorithm (WOA), and many more [13,[26][27][28] for parameter extraction of PV module. Recently, hybrid approaches have been utilized to enhance one or both analytical and numerical techniques with some other optimization techniques [4,[29][30][31][32].
Some of the applications in diferent areas where optimization techniques have played a vital role are for the reliability analysis of earth slopes using hybrid chaotic particle swarm optimization [33], Gravitational search algorithm for the coordinated design of PSS, and TCSC as damping controller [34], bacterial-foraging particle swarm optimization for Simultaneous Tuning of Static Var Compensator and Power System Stabilizer [35,36], and tuning of power system stabilizers using particle swarm optimization with the passive congregation.
According to the literature review, research towards extracting the performance characteristics of PV cells or modules is attracting many researchers' attention. Tis issue is frequently turned into an optimization problem and investigated in many ways using optimization methods. Many researchers have been making attempts over many past years to create new approaches which can overcome the disadvantages of analytical, numerical, and existing heuristic methods such as sensitivity toward the parameters, long exploration time, high computational cost, and premature convergence. A persistent attempt is made by the researchers to develop or design an efective method or algorithm which can determine PV cell/module parameters with accuracy so that accurate modeling and prediction of solar module output can be done. Te fve parameters that characterize the single-diode model are the ideality factor of the diode (α), the series resistance (R s ), the photoelectric current (I ph ), the reverse saturation current of the diode (I 0 ), and the shunt resistance (R sh ) [20][21][22][23][24]. Te series and shunt resistances are used to measure resistive losses in real solar cells. Several models have been modeled to analyze the behavior of PV modules under various operating situations. A step-by-step calculation of the equation and its dependency has been given by [14,15].
Artifcial jellyfsh [37] is a novel metaheuristic swarmbased optimization algorithm inspired by jellyfsh behaviors for fnding food in the ocean. Jellyfsh being the most effcient swimmers of all aquatic animals having umbrellashaped bells and trailing tentacles generally utilizes ocean currents at frst and thereafter motions inside jellyfsh swarms to locate food. JFO algorithm's advantage over other algorithm is, it provides a good balance between exploration and exploitation strategy and reaches optimal solutions in less time.
Te key fndings of this paper are as follows: In this paper, the JFO algorithm is used for the parameter extraction (I ph , α, I o , R s , and R sh ) of PV cells/modules as the JFO algorithm achieves the optimal solution without being trapped in local solutions in less time. Te approach is simple and does not need any experimental setup for the extraction of PV module parameters, whereas previous researchers have used experimental data for feature extraction. Te material and methods section, represents the single-diode model working and formulation of equations, and its subsections include an introduction to jellyfsh optimization, explains the step-by-step implementation of the JFO technique with the help of a fowchart, and the objective function formulation section, an objective function is formulated for feature extraction of the PV single-diode model by rearranging the equations in term of fve parameters. In the result and discussion, Section 3, the JFO technique is applied on two diferent solar modules Soltech-1STH-215 (60-cells) and PWP-201 (36-cells), to verify its performance with the manufacture data sheet, and the sensitivity analysis is done by checking its response at diferent irradiation and temperature value. Te results obtained from the JFO technique (proposed method) for the PWP-201 module are compared with the twenty-two optimization methods available in the papers [10,11,38,39]. MATLAB software 2022a is used for designing the mathematical modeling of the PV module and for the performance analysis of the proposed technique.

Single-Diode Model.
Te single-diode equivalent model of the PV solar cell is extensively used for many years for analytical purposes because it ensures the balance between precision and reliability [1]. An ideal solar cell may be modeled by a current source in parallel with a diode; in practice, no solar cell is ideal, so a shunt resistance and a series resistance component are added to the model. An ideal solar cell includes a single-diode current (I D ) paired in parallel with a light-generated current source (I Ph ), with both R s and R sh [7] as illustrated in Figure 1. R S is introduced to consider the voltage drops and internal losses due to fow of current. R sh considers the leakage current to the ground when diode is in reverse biased. Te manufacturer often only supplies a handful of values V max , I max , V oc , I sc , cell type, power, and efciency at STC. A set of lumped circuit parameters is necessary to simulate this model. Te solar module does not work on STC in real-time, instead is repeatedly exposed to diferent extents of irradiance and temperature. Other parameters of a cell are to be determined by using set of equations; therefore, a model that accurately predicts the characteristics of PV module in all operating conditions needs to be developed. Te relationship can be established by using the KCL equation for the circuit [7,12].
where I D and I sh are given by Equation (5) is obtained by using equations (2), (3), and (1), where k is Boltzmann's constant, q is the electron charge, T is the temperature in Kelvin and α is the ideality factor, I pv is the module current, I ph and I o are the module photogenerated and diode reverse saturation current, V pv is the module voltage, R s is the series resistance, R sh is the shunt resistance, N s and V t are the number of cells and junction thermal voltage [5][6][7][8][9]. For open-circuit conditions I pv � 0 and V pv � V oc , For short-circuit conditions I pv � I sc and V pv � 0. Ten, equation (5) turns to be as follows: Te value of I sc can be obtained by combining equations (7) and (8).

International Transactions on Electrical Energy Systems
Assuming exp(V oc /N s * (V t )) >> exp I sc * R s /N s * (V t ) and solving for I o from equation (9), we get To calculate the derivative of the current, equation (5) is diferentiated, Solving for dI/dV in equation (11) gives Applying relationship dI/dV � (−1/R sh ) [14], equation (12), can be written as follows: To obtain the maximum power point, substitute I pv � I mp and V pv � V mp , in equation (5).
At the maximum power point, dP mp /dV mp � 0.
To calculate the value of I mp , equation (15) is rearranged.
Considering equations (12) and (16), substituting V pv � V mp and I pv � I mp and rewriting the equation in terms of I mp results, we get Substituting the value of I ph from equation (7), and I o from equation (10) into equation (14), and I mp can be calculated as follows: Substituting the value of I o from equation (10), into equation (13), result is as follows:

International Transactions on Electrical Energy Systems
Also, by putting I o from equation (10), in equation (17), I mp is obtained as follows: Above all fve equations (7), (10), (18), (19), and (20), are utilized to calculate the other unknown parameters of the solar PV module. All these nonlinear equations can also be solved by using the Gauss-Seidel iteration method [14,15] to fnd the value of I ph , R sh , R s , α, and I o .

Jellyfsh Optimization [JFO]
. JFO is a new metaheuristic optimization algorithm motivated by jellyfsh behavior for seeking food in the ocean. Tis method is motivated by the exploration behavior and movement patterns of jellyfsh in the ocean [40]. In JFO, the amount of food at diferent locations varies; therefore, comparing food proportions by jellyfsh, the optimal location can be traced easily. Tis algorithm gives a better balance between exploration and exploitation approaches, and thus, reaches optimal solutions in a short duration of time [37,40,41]. To simulate jellyfsh search behavior, it is required to know about ocean currents' active and passive movements inside a jellyfsh swarm. Figure 2 depicts the JFO algorithm's fow chart. Tis algorithm uses three idealized rules [37,40,41].
(i) A "time control mechanism" allows jellyfsh to switch between the ocean current and going inside the swarm (ii) Availability of food in the ocean (iii) Te area and its dependent objective function have a signifcant impact on how much food is available Ocean current can be calculated by using the following equation [37,40]: where the relevant updated position is determined by X i (t + 1), β is a distribution coefcient (β > 0), and the mean location is µ. In a swarm motion, jellyfsh are passive (type A, i) and active (type B, j). Te motion of jellyfsh in their places is referred to as type A motion, and an updated location is given by equation (22), where c > 0 is the motion coefcient. L b represents the lower bound and U b represents the upper bound values for the defned objective functions. Type B is the motion of a jellyfsh (j) opposite to the type A motion. Equation (23) is considered to calculate the direction of motion, and equation (24), is used for updating the location.
Time control functions c(t) and c o are part of the time control mechanism. c(t) value varies between 0 and 1 and t represents the time period at a particular instant. Direction is calculated by using equation (23).
Population initialization is done by using the following: where X i is the logistic chaotic value of i th jellyfsh, X 0 ∈ (0, 1), and the value of "a" is chosen as 4.0 [37]. To use it in the script fle (MATLAB) few steps are needed to get the optimized value of the function. Equation (25) represents the time control mechanism in the JFO, and equation (26), represents the "initialization phase" [40], which is the frst step in JFO optimization. Te second step is setting the boundary condition for diferent functions. Te boundary condition is important so that jellyfsh does not move outside the boundary search area. Equation (27) is used to give the limit of jellyfsh in a search space area or boundary condition to jellyfsh [37]. A jellyfsh is located at X i,d in the d th dimension. Its upper and lower bounds are defned in search spaces as U b,d, and L b,d .
Te two main components of this metaheuristic method are exploration and exploitation [26,27]. Te best location for the food is determined by the jellyfsh where the quantity of food is more, as a result, an algorithmic replacement program model is created that replicates the ocean's jellyfsh's search patterns and movement.
First, objective functions are required in terms of I ph , α, I o , R s , and R sh of PV cell, which is discussed in the next section of the paper. Equations (21)-(27) will be utilized for updating, initializing, fnding the best location, controlling the time mechanism, the direction of jellyfsh, and setting the boundary conditions.

Objective Function Formulation.
In the single-diode model section, the mathematical formula for the calculation of the parameters is derived. I ph and I o can be obtained from equations (7) and (10), whereas equations (18)- (20) are simplifed to fnd the remaining dependent variables of the PV module. To obtain R S , R sh , and α, equations (18)- (20), need to be rearranged in terms of V t as they are independent of I ph and I o and are transformed into an objective function, which can be solved by using the JFO technique. Equation (18), is rearranged in terms of V t , it is assumed that, (28), is obtained from equation (18) and is arranged such that V t is a function of R s and R sh or   International Transactions on Electrical Energy Systems Now to obtain the R s parameter, equation (20) is rearranged, and a natural log is taken on both sides to get equation (29), such that R s � function (V t , R sh ).
Now to obtain the R sh , equation (19) is solved and rearranged into equation (30), such that R sh � function (V t , R sh, α).
Equation (29), can be expressed in terms of θ, where θ is given as From equation (4), the relationship between the ideality factor (α) and V t is obtained as given by the following equation: A MATLAB code is written to solve this equation and the MATLAB function takes parameters from the data sheet (see Table 1) of the PV module and returns the fve parameters I o , V t , R s , R sh, and α. Equations (28), (29), (30), and (32) will be utilized for solving V t , R s , R sh, and α. In terms of function, it can be expressed as V t = function (R s , R sh ); R sh = function (V t , R s ), R s = function of (V t , R sh ), and α = function (V t ). Te model that has been designed is independent of change in irradiation and temperature i.e., all the extracted parameter values are calculated on STC conditions. Parameters like V t , R s , and R sh are mostly independent of irradiation and temperature, but the other two parameters I o and I ph are dependent on irradiation and temperature [42] as shown in the following equations: I ph , value can be calculated by using the following equation: Te notation for the various parameters G, T, k, and E g are shown in Table 2. Te main equations that are used for parameter extractions are given by equations (28), (29), (30), and (32). Initial values are based on estimation as per the best knowledge. A set of the lower limit and the upper limit is used to fnd the best-optimized value of R sh , R s, α, and V t . Te value of R sh , R s, and α is calculated by defning the objective function given as follows: where V t is represented by V t � function, (R sh ), R s )., R sh � function, (R s V t ). R s � function (R sh , V t )., and α � function (V t ).
Te best solution and best optimal value (fbestvl) of the objective functions are calculated by using the following equation: fbestvl � JF @fobj, fnumber, Lb, Ub, dim , Max iter, N pop .
Equation (36) is the fnal output of the JFO that is in terms of fbestvl, where fobj is the objective function given by equation (35) Tables 3 and 4.
Te mathematical equations (1)-(20) and (28)-(34) are used and arranged in terms of main parameters (I o , R sh , R s , I ph, and α) in such a way that gives parameters dependency on each other. Te objective function is formulated, and the parameters are extracted and optimized using the jellyfsh optimization algorithm. Te approach is simple and does not need any experimental setup for the extraction of PV module parameters whereas previous researchers have used experimental data for feature extraction.

Results and Discussion
Te JFO technique is applied on two diferent solar modules for feature extraction and results are validated by matching the results with manufacturing data for case study 1 and with other optimization techniques for case study 2.

Case Study 1: On SOLTECH 1-STH-215P.
Parameters are extracted for the PV module SOLTECH-1STH-215P taken from MATLAB. Specifcations for the 213.15-Watt PV module taken for the parameter extraction are shown in Table 1. To obtain an accurate mathematical model of the solar module it is necessary to extract the unknown parameters. Te parameter's value set for the JFO technique [40] is shown in Table 3. Initial values are based on estimation as per the best knowledge. Te jellyfsh main loop was operated for a maximum iteration value of 1000. To calculate the time control mechanism c (t) equation (25) is used, movement in ocean current was calculated by using equation (21), and a passive and active motion was calculated using equations (22) and (24).
Te mean and RMS values of R sh , R s, and α obtained by using JFO are shown in Table 2. Te specifcation used for the PV module design is shown in Table 5. R sh , R s, and α (mean value) are used in the Simulink model and I ph and I o values are obtained from simulation. Te model is created by using the mathematical formula used in the single-diode model section.
Te simulation model of the PV module is shown with manufacturer data and extracted data (see Figure 3(a)). Figure 3(b) shows the convergence plot for the extracted parameter. It can be seen, the convergence rate of the proposed algorithm is fast. I-V and P-V characteristics are obtained for the manufacturer and extracted data. Te mean      a cell), the I-V characteristic is shown in Figure 4(a)), for the manufacturer and extracted data. It is observed that I sc and V oc values remain constant, while there is a slight change in the values of I max and V max due to the change in the value of the extracted parameters. Te P-V characteristic of manufacturer data and the extracted (mean and RMS) data are shown in Figure 4(b). Te graph shows the comparison with the manufacturer data in terms of the P-V curves. Te designed model values I max , V max, and P max are shown in Table 6, and it is observed that mean extracted data are found much closer to the manufacturer data and give a better result, with very less error. As JFO works on the mean values so mean error indicates the high accuracy of the proposed method. In this study, it is extremely helpful to judge a model's performance as the error was close to zero.
RMSE is a standard way to measure the error of a model in predicting quantitative data.
For a large-scale solar photovoltaic (SPV) based system, an ofine characterization study viz. sensitivity analysis needs to be performed at the design stage. A sensitivity study is made to fnd the behavior of a system due to variation in infuential parameters viz insolation and temperature. On open-circuit voltage and short-circuit current, the impact of temperature and irradiation variation is shown in Figure 5. Te response is shown at STC, at 20°C with 1000 watt/m 2 irradiation, and 25°C with 800 watt/m 2 irradiation. As temperature and irradiation are the primary input to the panel. Te efect of irradiation and temperature on the V oc and I sc values of the PV panel can be seen in the magnifed view.
Parameters of the module are given in MATLAB as per the manufacturer and the extracted data using the JFO algorithm are shown in Table 7. It is seen that for manufacturer data V max is 29.02 volts and P max is 213.1 watt at STC, whereas in extracted data V max comes out to be 29.37 volts with P max of 214.1 watt for the mean value of data. As per the results obtained, it was concluded that the module parameters extracted using the JFO algorithm are found to be near the manufacturer data, and the total execution time taken by the JFO technique was 3.9552 sec to track the parameters. Te mean error (manufacture and designed model) is 0.001, which is close to zero which shows that parameters are efciently extracted by using the proposed method.
When the temperature (T) is changed while keeping irradiance (G) constant, the PV module's I-V and P-V characteristics for extracted mean values are shown in Figure 6). It is observed that there is negligible change in I sc value but there is a signifcant change in V oc with the change in temperature at constant irradiation (see Figure 6(a)). As per the PV curves (see Figure 6(b)) it is observed that P max and V max are much afected due to changes in temperature with constant irradiation and less impact on I max . Te efect of irradiation and temperature on photocurrent, diode saturation current, maximum power, max.voltage, and max.current values are shown in Table 8.
With a change in solar irradiation value at a constant temperature, there is a negligible change in V oc but a signifcant change in I sc (see Figure 7(a)), whereas a change in irradiation with constant temperature signifcant impact on I max and P max values has and less impact on V max can be seen in Figure 7(b)). From Table 8 it is seen that the diode saturation current (I o ) is not afected by the change in irradiation and temperature. It is also observed that I ph changes slightly with a change in temperature while the change in irradiation afects I ph very much.

Case Study 2: On PWP-201 Solar Module.
To further validate the efectiveness of JFO, the results were computed on the PWP-201 module (36 cells) taken from [38]. Other researchers [10,39,43,11,38] have also done the parameter extraction of the same module using diferent optimization techniques and the results are compared. In [10,11,38,39] two modules i.e., at the RTC Francies and PWP-201 are taken for analysis purposes. Authors [10,11,38,39,43] have not considered the JFO technique for parameter extraction. In [39] the U b value for R sh is set to 2000, with an ideality factor range of 0-50, and the comparison is done on PWP-201. As per the available specifcation of the PWP-201 module (see Table 9, JFO technique is used to identify the fve unknown parameters, and their values are compared (see Tables 10 and 11) with the existing twenty-two techniques [11,38,39,43] for a diferent set of parameters value set 1 and 2.  Te parameters of JFO are taken the same as in [10,11,38,39] so that the results obtained using JFO can be compared and validated with the results of existing twentytwo optimization techniques [10,11,38,39] the values of the parameter (see Table 4) are set for the JFO algorithm. In [43] the extracted parameters of the ideality factor of the panel are not taken into consideration. In this proposed research work, no experimental setup is used for the extraction of parameters, only by using the mathematical equations the unknown parameters are extracted and the results obtained are comparable to the other existing techniques.
Te total execution time taken by the JFO technique was 3.9166 sec for set-1 and 5.073 sec for set-2 value. Te estimated parameters obtained from the JFO technique are almost comparable with other techniques [10,11,38,39], with some minor deviations (see Tables 10 and 11). Comparison is done in terms of the I ph , I o , R s , R sh, and α values of the other algorithms. Te convergence plot for the JFO algorithm is given for set 1 and set 2 in Figures 8(a) and 8(b) for the PWP-201 model. It can be seen, the convergence rate of the proposed algorithm is fast. To prove the efectiveness of the JFO algorithm its module I-V, and P-V curves on the STC (see Figure 9) and at variable irradiation (see Figures 10 and 11) and at variable temperature (see Figures 12 and 13) are evaluated by comparing them with the GA and PSObased I-V and P-V curves [38].
It is seen that the curves obtained by the mathematical model made from extracted parameters using JFO are   Figure 5: Efect on V oc and I sc value due to change in irradiation and temperature.    showing variation at diferent temperature and irradiation and matches with the response of GA and PSO (see         constant (1000 w/m 2 ). From the I-V and P-V curves (see  it is observed that the JFO is giving a similar response to GA and PSO in all environmental conditions.

Conclusions
Te efciency of a PV module depends exclusively on the précised value of its parameters. As the data provided by the manufacturer (V max , I max, V oc , I sc ) is not sufcient for simulation so it is important to obtain the extracted parameters correctly to evaluate the performance of the PV module. In this paper, fve important parameters (I ph , I o , R sh , R s, and α) are extracted using the JFO technique for two diferent PV solar modules (Soltech-1STH-215P and PWP-201). In case study 1, the results are verifed by comparing it with the manufacturer data available in MATLAB and found comparable with negligible deviation. Te results of case study 2 are validated by comparing the results of JFO with other existing twenty-two techniques and also with the I-V and P-V curves are compared for JFO, PSO, and GA techniques. Te analysis shows that the JFO approach achieve good optimization, which led to similar performance as recorded in prior research. Te impact of change in temperature and irradiance are also observed on the I-V and P-V characteristics of PV solar modules for the extracted parameters and compared with other techniques' PV module characteristics at diferent irradiation and temperature. It is observed that a change in temperature afects the V oc , while the change in irradiation afects I sc and P max values. From the various results and characteristics, it is concluded that using the JFO technique, the parameters of the PV solar module can be extracted accurately in less time and can be used for simulation purposes. Since no experimental measurement is to be put up, it is a very efcient technique to predict solar cell parameters for all commercial modules. Te model obtained from the extracted parameters can be utilized for fnding the maximum power point for any PV module.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon reasonable request.