#### Abstract

In this work, an innovative approach based on the estimation of the photovoltaic generator (GPV) parameters from the Bald Eagle Search (BES) optimization algorithm, associated with a support vector machine (SVM) classification algorithm, allowed to highlight a new tool for the classification of the signatures of shading and moisture PV defects. It recognizes signatures generated by the GPV in healthy and erroneous operation using the optimized parametric vector and classifies defects using the same optimized vector. The technique emphasizes the resilience of parameter estimate in terms of error on all parameters. The classification accuracy is 93%. The residuals between the estimated curve in healthy operation with a minimum error of the order of 10^{-4} and the one at fault are used as an indicator of faults.

#### 1. Introduction

In order to preserve the ecosystems and the environment in general, in terms of energy, the modern world has set itself the goal of emission reduction and damaging chemicals. The use of decreasing quantities of fossil fuels is becoming a major interest. Therefore, the exploitation of alternative and renewable energy sources becomes a competitive way to achieve this goal. The sun nowadays offers a great energy potential and can be used in an environmentally friendly way. At any given moment, the energy used in the world is 10^{5} times less than that received by the hemisphere exposed to the sun, whose power is over 50,000 TW. But then, with a yield of roughly 25%, solar energy received from a photovoltaic solar module is not abundant and consistent. Owing to the poor efficiency of the panels, the unpredictability of the power production of photovoltaic modules due to the vagaries of the weather, and the temperature of the cells, one of the primary issues is to maximize the use of solar energy [1]. However, solar devices will undeniably fall into failures during their operation. Hence, it is needed to detect and classify these failures in order to clearly dissociate the operation in degraded mode in spite of the weather vagaries. For this reason, there is a requirement to optimize and increase the yield of photovoltaic cells. It leads researchers to find methods to remedy this. One of the tracks involves determining the intrinsic characteristics of these cells. Several approaches for extracting parameters have been presented in the literature [2]. Each of these systems has its own set of disadvantages, whether in terms of ease of use, accuracy, or convergence and speed. Analytical, numerical, and metaheuristic approaches are the three types of methods available [3].

The analytical methods are based on their computational speed and the acceptable quality of their results. They have a low computation time. Sometimes, only one iteration is needed to obtain the result [4]. Despite the popularity of this approach, it is not always easy to implement. It consumes all the data points of the I-V feature and makes the calculation tedious [5]. Computational methods using Lambert’s W-function [6], Taylor series expansion [7], and Chebyshev polynomials [8, 9] are necessary. It should be noted that some approaches extract only a series and shunt resistance, while others estimate five parameters. Moreover, analytical methods have a weakness, and they are only suitable for standard conditions and degrade with changing conditions [10]. Compared to analytical methods, numerical methods with curve fitting techniques are by far better and provide algorithms with accurate results that evaluate all points of the PV-IV curves using the combined Newton-Raphson algorithm and Drone Squadron optimization [11].

Metaheuristic algorithms are strategies for optimization algorithms. They do not limit problem formulation and can answer a wide range of complicated queries [11]. We encounter in the literature, for the extraction of intrinsic parameters of photovoltaic cells, several metaheuristic algorithms, such as the bacterial foraging optimization algorithm (BFO) [12, 13], cuckoo search algorithm (CS) [11, 14], whale optimization algorithm (WOA) [15–17], genetic algorithm (GA) [12, 13], differential evolution algorithm (DE) [18], teaching learning-based algorithm (TLBO) [19–22], biogeography-based optimization algorithm (BBO) [23], artificial bee swarm optimization algorithm (ABSO) [24], Jaya optimization algorithm (JAYA) [25, 26], floral pollination algorithm (FPA) [27, 28], bird mating optimization algorithm (BMO) [29], the salp swarm algorithm (SSA) [30], Harris hawk optimization algorithm (HHO) [31], artificial bee colony algorithm (ABC) [19, 32, 33], particle swarm optimization (PSO) [34–36], backtracking search algorithm (BSA) (BSA) [37], cosine algorithm (SCA) [38], imperialist competitive algorithm (ICA) [39, 40], the shuffled frog leap algorithm (SFL) [41], ant-lion optimizer (ALO) algorithm [42, 43], teaching-learning artificial bee colonies (TLABC) [19], harmony search (HS) [44], firefly algorithm (FA) [45], simplified swarm optimization (SSO) [46], enhanced particle vibration system (EVPS) butterfly flame optimization (MFO) algorithm [47], water cycle algorithm (WCA) [48], cat swarm optimization (CSO) [49], multiverse optimizer (MVO) [50], and eagle strategy (ES) [51]. Despite the fact that metaheuristic algorithms are still the subject of much research due to some drawbacks, we can make a flat comparison between metaheuristic algorithms and analytical and/or numerical methods on the extraction of intrinsic panel values. This shows that metaheuristic algorithms provide satisfactory solutions on the estimation of panel parameters. CS suffers from slow convergence [11]. In the context of fault classification, several heuristic methods have also been implemented. This is the case of support vector machines [52, 53].

In this paper, we use the Bald Eagles Search algorithm in PV modules, to estimate intrinsic parameters with the objective of detecting faults during operation [54] and a SVM for classification. This work’s key contribution is the fault diagnosis from the best estimated parameter vector in a PV panel using the BES algorithm and SVM. The continuation of the paper is laid out as follows: Section 2 presents the modeling of the single-diode PV field for PV cells and panel; the principles and applications of BES for detection are presented in Section 3; the principles and applications of SVM for classification are shown in Section 4; the results and discussions are presented in Section 5, and the conclusion is presented in Section 6.

#### 2. Modeling of the PV Field

##### 2.1. Single-Diode Configuration

The single-diode model of a photovoltaic cell shown in Figure 1 is the simplest model for implementing a photovoltaic panel, module, or field. where

: Saturation current of the diode,

: Charge of the electron (),

: Ideality factor of the diode,

: Boltzmann constant (),

: Temperature in Kelvin.

Equation (1) is referred to as the five-parameter implicit equation: , , , , and have already been the subject of several parameter estimation studies [10, 55].

##### 2.2. Modeling of a Photovoltaic Field

The PV module’s model from [56–58] is depicted in Figure 2.

A photovoltaic panel consists of a set of photovoltaic cells. In a photovoltaic panel, all the cells are uniform: this implies the uniformity of currents and voltages for each cell of the panel. Equation (2) essentially gives us the mathematical model of the panel. The panel is either a serial, a parallel, or a mixed configuration of cell. For the highlighting of our panel, in order to model our faults in the Matlab/Simulink environment, we use the model in Figure 3, scalable according to the desired operation.

#### 3. Detection of Defects by the BES Optimization Algorithm

##### 3.1. Principle

We have in nature species that are on top of the food chain like the bald eagles. Predators are for the most part aggressive always at the top of the life chain, which gives bald eagles a majestic status for hunting (fresh meals) and collecting food (carrion). However, these eagles love the easy pittance, but better still alive or not. The favourite fish is salmon. From the air or from a perch, they spot prey and look for the best area and position to engage in the hunt, by crazy dives with a spiral-like movement.

However, whether the prey is dead or alive, white-tailed eagles have the same hunting technique. They choose a direction and a privileged zone to search for the prey, while spotting the trajectories of the other birds. The description of hunting in the middle of the salmon season is illustrated in Figure 4. Eagles are fierce hunters.

The steps of the BES algorithm were inspired by the behaviour of bald eagles during hunting. It is broken into three sections: selecting a search space, searching inside that space, and scanning. The three primary steps of the BES quest are depicted in Figure 5.

##### 3.2. Selecting a Stage

Our large bird optimizes its hunt by favouring food-abundant regions within the set hunting limits where the probability of getting prey is high. During the selection stage, the hunt has already begun. This is what our equation (3) shows us. where is the control element for the direction change, which has a range of 1.5 to 2, is a random number with a range of 0 to 1, is the previous search area, is the average search area, translates the best search area, and is the new search area. The selection stage allows bald eagles to choose a location based on the information gathered in the preceding stage. Eagles randomly choose another search area, different from the one they just searched, but still close. The function indicates the newly chosen search area by them based on the best position found during the previous search. The eagles randomly search all the locations along the previous search area. shows that these eagles have utilised all the knowledge from the preceding points during this period.

##### 3.3. Search Area

Equation (4) indicates mathematically the best point of diving. At this stage, the eagle being naturally confined in its zone of research, of preys, delimited in spiral with a considerable acceleration, has strong chance to succeed its hunting. where is a parameter with a range of 5 to 10 that determines the wedge between the search point and the center point and is a parameter with a range of 0.5 to 2 that determines the number of search cycles. Figure 6 depicts how bald eagles spiral through the search space they have chosen to find the optimal spot to dive and hunt their prey:

The polar graph attribute is used in this approach to mathematically depict this motion. The BES algorithm can identify new spaces and improve variety if we take the difference between the current point and the next point and multiply it with the polar point on the -axis and add the difference between the current point and the center point with the polar point on the -axis. All the search points move to the center point, which leads to the use of the average solution. Additionally, we use a special equation for the shape of the spiral ((5a)-(5d)), where all points in the polar plot take a value between -1 and 1.

##### 3.4. Dive Stage

During the dive stage, eagles move to the best spot in the search area to find their prey. All points ideally move in the same direction. Mathematically, equation (8) illustrates this fact.

##### 3.5. Application of BES for Fault Detection

For a single-diode model, the objective function is expressed as follows: where

: Is the objective function to be minimized,

: Is the number of points that have been measured,

: Is the measured current.

The parameters are determined by minimizing the objective function , by finding the intrinsic parameter vector [11, 26, 59–61].

To accomplish this, we must minimize the difference between the measured and estimated currents as much as possible. This difference is the error function. We use the algorithm of Figure 7 to determine the vectors of intrinsic parameters.

#### 4. Supervised Classification by Support Vector Machine

##### 4.1. Principle

The method of support vector machines (SVM) is part of the data-based methods. It is derived from statistical learning theory. It is used in this article for defect classification.

The most common problems encountered in practice are those with several classes. The extension of SVM to problems with more than two classes is therefore necessary. The SVM algorithm is implemented in pseudocode of Figure 8.

##### 4.2. Application of SVM for Classification

In practice, we often have to deal with multiclass problems. The extension of SVM to problems with more than two classes is therefore necessary. [62] has put together a good review of the literature applications and theories related to multiclass support vector machines. Consider a set of training data separable into classes

In this case, it is no longer a question of finding a single separator between two classes, but it will be necessary to classify the examples into several classes, which amounts to constructing linear hyperplanes of equations given as follows:

Generalize SVMs in the multiclass case amounts to solve the following quadratic programming problem:

The decision function is then

Thus, a new individual will have as class, the class such that

#### 5. Simulation Test Results and Validation of Fault Detection and Classification Algorithms

##### 5.1. Simulink Model of the PV Panel

Table 1 shows the technical specifications of the PV panels studied in this paper.

##### 5.2. Simulation

For the experimental study, the schematic in Figure 9 realized in the Matlab/Simulink platform was used to collect the data of the current-voltage characteristic. For this purpose, four simulation steps were performed: (i)1st step: normal operation(ii)2nd step: partial shading(iii)3rd step: series resistance deflection(iv)4th step: parallel resistance deflection

The curves in Figure 8 show the four operating states of our system.

##### 5.3. Results and Discussions

In this work, the BES algorithm has allowed to estimate the parameters of all the stages and to highlight the defect by the difference that exists between the estimated curve and the measured one. This is because this algorithm has better stability and error. We have done the validation of our vectors, comparing them to the vectors of the algorithms of the Harris hawk optimizer (HHO), equilibrium optimizer (EO), and slime mould algorithm (SMA) as shown in Table 2.

After validating our parameter estimation algorithm, we have obtained after several simulations the vectors of parameters at different operation conditions of the PV panel. These different vectors serve as a database for our classification. For our detection, the curves related to these vectors serve as signatures that will be compared to those measured from the current and voltage. Some vectors are listed in Table 3.

The curves in blue in Figure 10(a) are obtained from the best vector provided by the BES optimization algorithm from the smallest possible error determined on the order of 10^{-3} [55] and 10^{-2} with our panel. The red line represents the voltage measurement (at the PV system) and then the current measurement.

**(a)**

**(b)**

The difference between the predicted curve and the one measured during operation is used to detect the failure. Figure 10(a) depicts the disparity between the healthy operation’s estimated curve in blue and the erroneous operation’s measured curve in red. The figure where we note the superposition of the curves reflects the healthy operation. As for the other curves, in case of the 50% shading defect, the concavity disappears completely and appears a point of inflection at 9 V and a decrease in the open circuit voltage. For series and parallel resistance (or shunt) faults, the open circuit voltage shows a considerable decrease. The same features are shown in Figure 10(b), but this time with the estimated and measured curves for each operating mode. The comparison between the estimated or simulated curve and the measured curve rightly allows for fault detection. By observing the gap existing between the two curves, the captured residue brings a clear information for the detection of the fault.

The set of characteristic vectors of the PV module operation given by the BES optimization algorithm was used as the training database for the SVM. Table 4 presents the statistical study of these data. This static analysis appropriately depicts the collected data in the workspace. It easily validates our data, as the distribution based on the selected 100-vector bundles remains uniform in the different quartiles.

For a good classification, a pairwise learning is necessary. Figure 11 shows the different learning scenarios of our classes. We observe the distribution of one class in relation to another, by a point by point data distribution.

The data distribution for the variables parallel resistance and series resistance is shown in Figure 12. The boundaries between these data show sufficiently that these defects can be classified.

From the samples in Figure 11, we proceed to the classification as shown in Figure 13. The large yellow area represents the one that classifies the shunt resistance defect or parallel. It classifies for values between 900 and 2000 ohms. The red areas represent the limits in which we find the characteristic points of the series resistance fault. The two zones are due to the dispersion of these points in the classification space. We have them easily in the intervals from 0 to 0.3 ohm and from 1.650 to 2 ohms for the abscissa axis. On the ordinate axis, the spaces are 0 to 180 ohms and then 0 to 1100 ohms. The blue we observe delineates the classification surface of the partial shading defect. It extends from 0.3 to 1,650 ohms on the abscissa and from 0 to 1200 ohms on the ordinate. However, the surface of healthy operation is materialized by the green color, confined between 0 and 0.850 ohm depending on the series resistance and 150 to 1750 ohms depending on the shunt resistance. We have points that are outside their automatic classification area. This presence of fake points gives us a classification accuracy of 93%. This accuracy, close to 100, allows us to value our result.

Table 5 presents a comparative study with other methods in the literature.

#### 6. Conclusion

In this paper, we highlight two algorithms, one for parameter estimation and detection optimization (BES) and the other for supervised fault classification (SVM). The diagnosis is robust because of its high classification accuracy of 93%, for a vector sample of about 750. The detection was done by the comparison method between the measured value and the estimated value. The estimation error of our parameters is of the order of 10-^{2}. With these two algorithms, we could classify four operating states of the PV panel, the healthy, the shaded, with fault of series, and parallel resistors. The simplicity of the method allows us to go beyond three faults on a PV, and even on other systems. We will be able to hybridize the parameter estimation algorithms to further optimize the estimation error and improve the fault detection and then also optimize the SVM or also hybridize to improve the fault classification rate of a PV panel.

#### Data Availability

The simulation data used to support the conclusions of this study is included in the supplementary information file.

#### Conflicts of Interest

The authors declare no conflict of interest regarding the publication of this article.

#### Supplementary Materials

The data from the Matlab/Simulink 2020a program is used to highlight our findings. We estimate each panel operation’s parameters. Our database for elaborating the classification is the vector of intrinsic parameters for each operation. They are numerical values. The parameter vector’s order is used to divide the data into columns. We used 15% for testing, 15% for validation, and 70% of the data for learning in this project.* (Supplementary Materials)*