The modeling of a solar PV system is challenging due to its nonlinear current vs. voltage characteristics. Although various optimization techniques have been applied for the parameter estimation of the solar PV system, there is still a scope to attain the best-optimized results. This paper uses a new meta-heuristic optimization algorithm and a classical technique named Ali Baba and the Forty Thieves (AFT) with Newton Rapson (NR) method to estimate solar PV system parameters. The well-known story of Ali Baba and the Forty Thieves has inspired the AFT. Besides, the inappropriate objective function used in earlier research to extract parameters from solar PV models is recognized. The experimental findings demonstrate that the suggested approach performs better when compared to state-of-the-art algorithms. Between the measured data and the computed data for AFT, the root mean square error values for the five PV models, such as single diode model (SDM), double diode model (DDM), Photowatt-PWP201, STM6-40/36, and STP6-120/36, are respectively 7.72 × 10−04 ± 6.121 × 10−16, 7.412 × 10−04 ± 9.52 × 10−06, 2.052 × 10−03 ± 3.05 × 10−17, 0.001721922 ± 2.19 × 10−17, and 0.014450817 ± 3.42 × 10−16. In terms of accuracy, the obtained results indicate that the proposed AFT algorithm is more efficient than the other optimization techniques available in the literature. The excellent correlation between the estimated parameters from characteristic curves and observed data for SDM, DDM, Photowatt-PWP201, STM6-40/36, and STP6-120/36 demonstrates that the proposed AFT is a potential option among the techniques available in the literature. The Friedman and Wilcoxon tests have been used to assess the statistical validity of the proposed algorithms.

1. Introduction

Various environmental and energy-related considerations have contributed to the increased use of alternative energy sources. In recent years, large solar PV plant installations have been used to generate electric power. PV systems are generally installed in open spaces exposed to adverse weather like gales and torrential rain [1, 2]. In response to these challenges, a much more precise information framework is formulated to pinpoint the crucial characteristics of PV systems in the solar sector. Assessments of PV electricity generation, productivity estimations, voltage regulation, MPPT, and better power monitoring of the PV arrays benefit from thoroughly examining PV input parameters retrieval. A mathematical equation is formulated first, followed by determining such variables throughout the detailed modeling of PV systems. The single diode model (SDM) is commonly applied in almost all modeling techniques in all real-world scenarios. Moreover, if such estimates are exposed to contingency system aging, these undefined variables will most likely have an unreliable and failure effect on the characteristics of the PV models. A crucial modeling task is to estimate the PV cell characteristics beforehand accurately. The PV framework is a dynamic system with a non-convex relation, the solution of which has several problems and obstacles. Researchers and scientists have recently put a lot of effort into accurately estimating unknown values [3]. The identification of accurate values of such parameters, such as photo current (IPh), saturated current (I0), shunt resistance (RSh), series resistance (RS), and diode identity factor (N), is essential. Due to solar cells’ nonlinear current-voltage (I-V) characteristics, estimating their parameters is considered a nonlinear optimization problem [4].

In the literature, various methods for estimating PV parameters are being used. And these approaches are classified into three groups. They are analytic, determinism, and meta-heuristic (MH) optimization techniques [5]. Analytical techniques calculate factors such as the open-circuit voltage, short-circuit current, MPPT, and I-V characteristics based on the supplier’s information. Some of the data sets on the I-V characteristic curve are employed in the analytic method for finding the variables that minimize the discrepancy between the expected and calculated values. The concept of “Using every exact quantitative measurement for a coherent system” underlies deterministic approaches, which recover the independent variables using only a sizable quantity of data [6]. Depending on a differential among practical but estimated data sets, an optimization problem serves as its foundation. Those procedures might have the localized optimal solution since they depend on different parameters. Since they both operate on the principle of “Using every exact value for the complete solution,” MH approaches are comparable to deterministic approaches. They are regarded as the world’s greatest optimization technique due to their many benefits, including resilience, efficiency, stability, elegance, and easy implementation [7]. There are different literary categories in which MH optimization algorithms have been classified, including swarm, evolutionary, human behavior, and physics-based optimization algorithms. Particle swarm optimization (PSO) [8], simulated annealing (SA) [9], mutative scale parallel chaos optimization (MPCOA) [10], bee pollinator flower pollination algorithm (BPFPA) [11], artificial bee swarm optimization (ABSO) [12], generalized oppositional teaching learning-based optimization (GOTLBO) [13], flower pollination algorithm (FPA) [14], genetic algorithm (GA) [15], harmony search (HS) [16], Levenberg–Marquardt algorithm with SA (LMSA) [17], artificial bee colony optimization (ABCO) [18], artificial immune system (AIS) [19], cuckoo search (CS) [20], honey badger algorithms (HBA) [21], heap-based optimizer (HBO) [22], tree seed algorithm (TSA) [23], ranking-teaching-learning-based optimization (RTLBO) [24], bacterial foraging algorithm (BFA) [25], teaching-learning-based optimization (TLBO) [26], gaining–sharing knowledge (GSK) [27], and whale optimization algorithm (WOA) [28, 29] are few examples of well-known MH algorithms that were developed to evaluate the solar PV system’s parameters.

The root mean square error (RMSE) value is used to assess the proposed optimization technique’s correctness. Additionally, data from recent MH algorithms are compared to the characteristics obtained from the AFT. All these researches offer an assessment of the suggested algorithm’s precision for parameter estimation of solar PV. This study concentrates on SDM, double diode model (DDM), Photowatt-PWP201, STM6-40/36, and STP6-120/36 solar PV models. Temperature and irradiance affect the solar PV model’s parameters. Therefore, a reliable parameter estimate is necessary to model solar PV. The proposed algorithm is very efficient for solving nonlinear problems. The famous story of Ali Baba and the Forty Thieves has inspired this algorithm. In the story, Ali Baba witnessed forty criminals enter a mysterious cave brimming with valuables. The performance of the AFT algorithm has been evaluated on a collection of fundamental benchmark test functions, including simple and sophisticated test functions with varied dimensions and levels of complexity. In addition, the AFT algorithm has been subjected to a thorough comparison with other well-researched algorithms, and statistical test techniques have been used to demonstrate the relevance of the findings. This paper’s main contribution is as follows:(1)A new MH optimization algorithm (AFT) is introduced to solve the solar PV parameter extraction(2)The proposed AFT performance is successfully applied for the SDM, DDM, Photowatt-PWP201, STM6-40/36, and STP6-120/36, with the significant similarity between the simulated and experimental P-V as well as I-V curves, clearly demonstrated(3)The proposed AFT demonstrates superiority by achieving the lowest RMSE objectives(4)The result of the AFT with other MH optimization techniques is compared

This paper is organized as follows: Section 2 illustrates the solar PV models such as SDM and DDM. Section 3 explains AFT comprehensively. The objective function is presented in Section 4. Section 5 lucidly presents the results and critically discusses the inferences considering five PV modules: SDM, DDM, Photowatt-PWP201, STM6-40/36, and STP6-120/36. Statistical validations (Friedman and Wilcoxon test) are presented in Section 6. And the final section summarizes the concluding remarks and the future scope of the paper.

2. Solar PV Models

This section elaborates on SDM and DDM comprehensively.

2.1. Single-Diode Model (SDM)

Several equivalent circuits have been developed to illustrate the I-V characteristics of PV modules. The SDM is the most widely utilized PV equivalent circuit model. SDM is simple and incredibly accurate [30, 31]. It is the most famous mathematical concept, but as its title indicates, it establishes the relationship among the variables using a single-diode approximation [32]. I is the output current, ISh is the shunt resistance current, Rs is the series resistance, IPh is the photo-current, ID is the diode current, and RSh is the shunt resistance [33, 34], as shown in Figure 1.

The mathematical equations for the and are given by equations (2) and (3), respectively.

The PV module output current is given by the following equation:where , Boltzman constant (K), temperature (T), Np, and Ns, respectively, are the number of cells in parallel and series, and the diode ideality factor is n. These parameters (IPh, Io, Rsh, Rs, and N) can be extracted to determine an accurate PV model [35].

2.2. Double-Diode Model (DDM)

Since the SDM excludes the effect of recombining energy loss in circuitry, the loss problem is solved by employing the more precise model DDM [30]. The circuit diagram of DDM is shown in Figure 2.

The current I flows through the DDM circuit can be computed as follows:where is the first diode’s current and is the second diode’s current.where and are the ideal factor of the diode. From (6) and (7), I can be computed as

These parameters (IPh, I01, I02, RSh, Rs, N1, and N2) can be extracted from DDM.

3. Ali Baba and Forty Thieves (AFT) Optimization Technique

This work’s main objective is to apply an optimization technique that uses the story of Ali Baba and the Forty Thieves as unified modeling of human social interaction. The primary presumptions of this method are satisfied by the concepts that follow, which are drawn by this story [36]:(i)To locate Ali Baba’s home, the forty thieves work together as a team to receive directions to someone or another thief. As a result, such data may be inaccurate(ii)From an initial position, the forty thieves will travel a range till they reach Ali Baba’s house(iii)Marjaneh may fool her robber’s several ways by using cunning techniques to keep Ali Baba partially protected from their approach

It is possible to connect the actions of Marjaneh and the robbers to an objective function that needs to be optimized. This allows the development of the new MH method described as follows.

3.1. Random Initialization

The AFT process is started by initializing people’s position in such a -dimensional state space at random, as given as follows:where x = position of all thieves, d = number of variables of a given problem, and  = jth dimension of the ith thief.

Marjaneh’s wit ranking concerning all thieves could be initiated as follows:where  = astute level of Marjaneh concerning the ith thief at the jth dimension which indicates Marjaneh’s level of intelligence in comparison to the jth dimension’s ith thief.

3.2. Fitness Evaluation

A user-defined fitness value that is assessed to every thief’s location contains the choice independent variables. The related objective functions are kept in a collection and are presented as follows:where  = dimension position of thief.

The effectiveness of the response is assessed to every thief’s different destination in the AFT technique model using a predefined objective function. If the new destination’s solutions performance is higher than the old one, the position is then upgraded. On the other hand, if his present solutions grade is superior to the current one, every thief remains there.

3.3. Proposed Mathematical Model

Three primary cases discussed above may arise. At the same time, robbers look for Ali Baba. Every time, it is assumed that the thieves do thorough searches all through the surroundings. At the same time, a percentage happens due to Marjaneh’s cleverness, which compels the robbers to look in unlikely places. Finally, it is earlier looking; the following behavior can be analytically modeled:

Case 1. The robbers might use details they learned from others to find Ali Baba. Under this instance, the robbers’ actual places can be discovered as follows:

Case 2. The thieves might realize that they have really been misled and begin searching haphazardly about Ali Baba. Throughout this instance, the thieves’ actual destinations can be discovered as shown in the following equation:

Case 3. This work further analyses the investigation in locations of which some might be found by equation (11) in terms of improving the exploratory and exploitative aspects of the AFT algorithms. In this instance, the thieves’ exact location can be discovered as shown in the following equation:The iterative procedures provided in the method can be used to simplify the pseudocode of AFT algorithms as discussed in Algorithm 1.

(1) are the upper and lower bounds of dimension, is the position, and is the number of thieves.
(2) Initialization of best position and global best position.
(3) Evaluate the fitness function
(5)while () do do
(12)    Else
(14)    end if
(16)   end if
(17)  end for
(19)   Check, evaluate and update the new positions
(20)   Evaluate and update the solutions for &
(21)  end for
(23)end while

4. Objective Function

The main goal is always to lessen the overall discrepancy between the measured and theoretical findings while evaluating the PV cell characteristics. An objective function called the RMSE can be constructed in a way as to yield the optimal parameters for the PV model’s variables [37]. This paper aims to evaluate the characteristics of a PV based on the current and voltage data using MH optimization approaches. The RMSE values between the observed and anticipated actual rates form the basis of the goal function. Two optimization parameters are taken into consideration in this work for evaluating the PV characteristics given in the following equation:

The individual absolute error (IAE) of current and power is given in (17) and (18). The relative error (RE) is shown in the following equation:where Imes is the measured current, Icalc is the calculated current, Pmes is the measured power, Pcalc is the calculated current, and N is the number of samples.

The fundamental method for deriving the solar PV model’s parameters from observed measurements is to compute the RMSE using the essential functions F1. The estimated current may be determined by determining the nonlinear formulae. Because this approach requires a lot of calculation, the authors introduced a novel method during which objective function equations roughly estimate the genuine fault. As a result, a solution is used in the first way to calculate the RMSE of the erroneous values [27].

5. Results and Discussions

The aim of the paper is to evaluate the SDM of a PV cell’s five unmeasured variables (Iph, Io, RSh, Rs, and N) and the DDM of a PV cell’s seven unmeasured variables (Iph, Io1, Io2, RSh, Rs, N1, and N2). The AFT algorithm parameters for determining the parameters of the PV model are as follows: maximum iteration = 700; number of thieves (N) = 700; random number (rand, r1, r2, and r4) [0, 1] and r3 ≥ 0.5. The configurations for both the lower bound (LB) and upper bound (UB) for SDM, DDM, Photowatt-PWP201, STM6-40/36, and STP6-120/36 cell/module are displayed in Table 1 [38].

5.1. Case Study 1: R. T. C France Solar Cell (Single-Diode Model)

Table 2 displays the comparison of actual and experimental values for SDM. Table 3 presents the estimated parameters with the RMSE errors from AFT minimum, average, and standard deviation[39]. Figure 3 shows the convergence curves for the formulated objective functions. Figures 4 and 5 display, respectively, the I-V and P-V properties of a PV cell under real-world and speculative conditions. It is clear from Table 2 that the absolute individual error (IAE) values are less than 3.399 × 10−03 and that the relative error (RE) values range from 6.91 × 10−03 to 2.65897 × 10−01, proving the SDM’s high efficiency as identified by AFT. Furthermore, Table 3 demonstrates that using the proposed AFT algorithm for the SDM provides the minimum RMSE value of 7.72 × 10−04 as compared with the recent MH optimization methods in the literature. Also, it presents the optimal value of the unknown parameters (Iph, Io, RSh, Rs, and N) calculated using AFT and other MH optimization methods in the literature as shown in Table 3.

5.2. Case Study II: Double-Diode Model R. T. C France Solar Cell

The values of RE and IAE for both current and power are given in Table 4. Seven parameters (Iph, Io1, Io2, RSh, Rs, N1, and N2) must be evaluated as in the case of DDM, and the estimated parameters along with the minimal level, mean, and standard deviation are shown in Table 5. The converging curve for DDM is shown in Figure 6. The I-V and P-V curves are shown in Figures 7 and 8. It is clear from Table 4 that IAE values are less than 0.0169 and that the relative error (RE) values range from 8.04971 × 10−05 to 0.055186158, proving the DDM’s high efficiency as identified by the AFT method. Also, Table 5 demonstrates that using the proposed AFT algorithm for the DDM provides the minimum RMSE value of 0.00074 as compared with the recent MH optimization methods in the literature.

5.3. Case Study III: STM6-40/36 Monocrystalline PV Module

The Schutten Solar STM6-40/36 PV module is obtained using the AFT technique. It has 36 series-connected polycrystalline cells, each measuring 156 mm in diameter. Twenty data points have been collected at T = 51°C for the data set. Table 6 displays the comparison of actual and experimental values for SDM. It is clear from Table 4 that IAE values are less than 0.02179 and that the relative error (RE) values range from −0.00044to 0.001823, proving the DDM’s high efficiency as identified by the AFT method. Table 7 shows the results of the parameters acquired by MH optimization techniques in the literature for the STM6-40/36 PV module design. Figures 911 depict the convergence, I-V, and P-V curves to support the overall precision of the derived values. In the operating voltage of both the I-V and P-V curves, it is clear that the simulated data from the AFT agree well with the observed data. Table 7 also demonstrates that using the proposed AFT algorithm for the STM6-40/36 PV modules provides the minimum RMSE value of 0.001722 as compared with the recent MH optimization methods in the literature. Also, it presents the optimal value of the unknown parameters (Iph, Io, RSh, Rs, and N) calculated using AFT and other MH optimization techniques in the literature, as shown in Table 7.

5.4. Case Study IV: STP6-120/36 Module

The polycrystalline STP6-120/36 is less than 1000 W/m2 at 55°C and has 36 series-connected cells. Table 8 shows that the IAE sum is smaller than 4.64 × 10−02, demonstrating good agreement between the observed and extract values. Table 8 also provides the Ie, Im, V, Pe,Pm, IAE (current), IAE (power), and RE. The IAE explains the discrepancy between both the observed data and the retrieved value. The findings of various strategies from the literature are also compared in Table 9, with recent MH optimization techniques for the STP6-120/36 PV module design. The comparison proved that AFT is more successful than other MH methods. The RMSE for the AFT application used to extract the PV model’s parameters is 0.014451. The analytical findings and convergence curve are displayed in Table 9 and Figure 12. The I-V and P-V curves of the simulated data discovered by AFT are remarkably compatible with the real data, as shown in Figures 13 and 14. In other terms, when the IAE and RMSE values are low, the obtained values are better (Iph, Io, RSh, Rs, and N).

5.5. Case Study V: Photowatt-PWP201 PV Module

The Photowatt-PWP201 PV module, which is made up of 36 silicon cells that are connected and operate under operating conditions of 1000 W/m2 of solar irradiation and a cell temperature of 45°C, has been used to validate the AFT technique further and demonstrate its effectiveness in estimating the optimal parameters of various models. The outcomes have contrasted with those documented in the literature using other methods. Table 10 is a listing of the outcomes. The findings of additional MH optimization techniques from the literature for the Photowatt-PWP201 PV module design are also compared in Table 11. The comparison proved that AFT is more successful than alternative methods. Better, the RMSE based on the AFT application to extract the PV model’s parameters is equivalent to 2.052 × 10−03. The improved performance observed by AFT for the PV module is obvious from Table 10, where the IAE ratings are lower than 4.418 × 10−03, and the RE values are between 6.44 × 10−03 and 1.3167 × 10−01. The analytical findings are shown in Tables 10 and 11, and the convergence curve is shown in Figure 15. The I-V and P-V curves of the simulated data discovered by AFT are remarkably compatible with the real data, as shown in Figures 16 and 17. Also, it presents the optimal value of the unknown parameters (Iph, Io, RSh, Rs, and N) calculated using AFT techniques and other MH optimization techniques in the literature as shown in Table 11.

6. Statistical Validations

In this section, the proposed optimization algorithm is statistically tested using a series of separate runs (30) for each case/algorithm using the Friedman and Wilcoxon tests [43, 44]. Tables 12 and 13 show the average rankings for the proposed algorithms using all five solar PV cell/module problems (SDM, DDM, Photowatt-PWP201, STM6-40/36, and STP6-120/36) based on the Friedman test. The AFT algorithm is shown as the base algorithm. The results verify that the AFT algorithm is superior to other MH algorithms in solving all five solar PV cell/module problems. The results of a statistical analysis utilising Wilcoxon’s test comparing AFT and other MH algorithms are summarised in Tables 14 and 15. In general, the performance of AFT is better than other algorithms in all five solar PV cells/modules. In the AFT algorithm versus MH algorithms, the AFT algorithm gets higher R+ values than R- in all five solar PV cell/module problems. The results show that the AFT algorithm outperforms existing MH methods in solving all five solar PV cell/module problems.

7. Conclusions and Future Directions

In this paper, a well-known optimization technique, known as the AFT optimization algorithm, is used to obtain the optimum solution for solar PV cells and module parameters. Numerous scenarios have been used for SDM, DDM, and PV panel modules to show the performance of the AFT optimization algorithm. The observed and calculated data’s I-V characteristics, as well as P-V characteristics, demonstrate the suggested method’s high degree of accuracy. The results of simulation tests and comparisons to other MH optimization techniques illustrate the method’s accuracy and validity in extracting the characteristics of a PV cell and module. It offers the benefit of quickly convergent and consistent results for each test. The approach is presented using real-world information from several solar PV manufacturers (SDM, DDM, Photowatt-PWP201, STM6-40/36, and STP6-120/36). Its accuracy is demonstrated by comparing its RMSE with a variety of MH optimization techniques. As a result, the close similarity between the generated I-V and P-V curves and the measured features has validated the AFT’s accuracy. Furthermore, its application to parameter assessment and to solve the additional power system optimization issues can be used. The statistical validity of the suggested algorithms has been examined using the Friedman and Wilcoxon tests. The suggested AFT algorithm is therefore better than the existing MH optimization algorithms. In the future, hybrid optimization techniques will be used to find out the optimal value of unknown parameters and reduce the RMSE value of solar PV.

Data Availability

The figures and tables used to support the findings of this study are included in the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.