Hybrid walrus optimization algorithm techniques for optimized parameter estimation in single, double, and triple diode solar cell models

Vujošević, Snežana; Ćalasan, Martin; Micev, Mihailo

doi:10.1063/5.0223492

Among all renewable energy sources, solar energy holds the greatest potential for electricity production. This transformation from solar to electrical energy is facilitated by solar cells, typically modeled using single-diode, double-diode, and triple-diode representations. In this study, we evaluate the effectiveness of the Walrus Optimization Algorithm (WOA) for estimating the parameters of these models. Furthermore, we introduce three innovative hybrid variants of WOA that incorporate chaotic sequences, adaptive modifications, and integration with the Simulated Annealing (SA) algorithm, thereby enhancing the parameter estimation process. Our research was conducted on two well-documented types of solar cells/modules, with additional tests on the performance of these algorithms on a solar panel under varying insolation and temperature conditions. The results underscore the superior efficiency, accuracy, and practicality of the hybrid algorithms, particularly the variant augmented with chaotic sequences, over traditional parameter estimation methods in solar cell technologies. This paper highlights significant advancements in algorithmic approaches, paving the way for more precise and reliable solar energy technologies.

I. INTRODUCTION

The electric power sector has undergone significant transformations over the past few decades.¹ With the increasingly complex demands of electrical distribution and consumption, there is also a heightened focus on environmental considerations—specifically, the need to reduce the ecological footprint.² Consequently, modern electrical power systems represent highly complex and dynamic entities designed to meet the continuous, reliable, and efficient supply demands of a diverse array of consumers. The pronounced technological advancements have led to the proliferation of sensitive devices, elevating the standards for the quality of electricity supply.³ Modern systems have become more sophisticated, employing smart grids and digital technologies to optimize performance.

To reduce greenhouse gas emissions and mitigate climate changes, electrical power systems are progressively incorporating renewable energy sources alongside conventional ones. Among these, solar energy is the most prevalent.⁴ It is extensively utilized for water heating and desalination, but its most critical application is in the production of electrical energy. Solar energy conversion is primarily achieved in two ways:

Solar thermal power plants utilize a system of mirrors to concentrate solar rays, heating a working fluid to generate steam that drives turbines and produces electricity.
Photovoltaic (PV) devices, or solar cells, directly convert solar radiation into electrical energy. Due to its efficiency and growing affordability, the PV technology is now the dominant and fastest-expanding form of renewable energy worldwide, accounting for about 60% of total renewable energy investments in 2022. Thus, it is clear that the PV technology plays an increasingly significant role in the global energy transformation.

While the utilization of solar energy as a renewable source presents numerous benefits, including reduced greenhouse gas emissions, less reliance on fossil fuels, and versatile applications, it also poses several challenges. The most significant difficulties include its non-stationary nature, dependence on weather conditions, energy storage challenges, and the high initial costs associated with the use of relatively expensive inverter equipment.⁵ Additionally, to enhance efficiency, maximum power point tracking is necessary, which further increases production costs.⁶

In terms of electrical energy production via photovoltaic modules, the electrical output of a solar cell (modules and panels) is best represented by its current–voltage (I–V) characteristic curve.⁷ Accurate modeling of this curve is crucial and must consider all relevant factors, such as the inherent parameters of the cell and external influences, such as radiation (G) and temperature (T).⁸ The pronounced nonlinearity of the I–V characteristic means that various models are employed to represent the equivalent circuit of a solar cell, each differing in complexity and accuracy.

Solar cells can be modeled using three types of equivalent circuits: the single diode model (SDM), the two-diode model (DDM), and the three-diode model (TDM) (see Fig. 1).⁹ The simplest SDM includes a DC source (I_PV), a diode in parallel with the source, and two resistors (R_P and R_S) in parallel and series, respectively. This model is defined by five parameters: source current (I_pv), reverse saturation current (I₀), diode ideality factor (n), parallel resistance (R_P), and series resistance (R_S), represented as a decision vector DV = [I_PV, I₀, n, R_P, R_S].

FIG. 1.

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(a) SDM, (b) DDM, and (c) TDM equivalent circuits.

The main limitation of the SDM is its inability to account for recombination current. To address this, the DDM includes an additional diode, with parameters represented by a seven-element vector DV = [I_PV, I₀₁, n₁, I₀₂, n₂, R_P, R_S], where I₀₁ and I₀₂ are the reverse saturation and recombination currents, respectively, and n₁ and n₂ are the ideality factors of each diode.¹⁰ The TDM further extends this model by adding a third diode to account for leakage and surface recombination currents, represented by a nine-element vector DV = [I_PV, I₀₁, n₁, I₀₂, n₂, I₀₃, n₃, R_P, R_S], where I₀₃ and n₃ are additional parameters.¹¹

The I–V characteristics of solar cells are highly nonlinear.^7,9 While exact analytical relationships exist for the SDM,⁷ the DDM and TDM are typically solved numerically due to the absence of straightforward analytical expressions.¹² However, recent literature has proposed an analytical relationship for the DDM, providing satisfactory results despite some mathematical approximations.¹³

Determining the parameters of solar cells can be approached in two main ways:

Classical methods, which may be either analytical¹⁴ or numerical.¹⁵
Modern metaheuristic methods.^16–42

Analytical methods, initially predominant in the analysis of solar cell operation, offer precise descriptions of the physical behaviors of the models they study. They provide exact solutions for problems that can be mathematically defined by specific equations.¹⁴ However, due to the complex nature of the equations describing solar cells’ I–V characteristics, various approximations must be introduced. These approximations, while necessary, can significantly diminish the accuracy of the solutions.

With advances in the computer technology, numerical methods have gained prominence. These methods have largely surpassed analytical methods in terms of accuracy, as they handle the complexity of solar cell models more effectively.¹⁵ Their use is, however, bounded by the computational power available, the accuracy of results based on the iteration step, and the time required for computations.

Metaheuristic methods today offer the most promising results in determining the parameters of the equivalent circuits of solar cells, including SDM, DDM, and TDM. These methods are recognized for their efficiency in optimizing highly complex and nonlinear problems. Their principal advantage lies in their ability to explore a vast array of potential solutions extensively, thereby identifying optimal or near-optimal parameter sets for the models in question. A comprehensive list of the most frequently employed metaheuristic methods in this domain is presented in Table I.^13,16–42

TABLE I.

Commonly used methods for determining parameters in solar cell equivalent circuits.

Number	References	Method	Year	First author	Journal
1	16	SDM	2022	Rawa	Mathematics
2	17	SDM	2021	Calasan	Sol. Energy
3	18	SDM	2021	Saadaoui	Energy Convers. Manag.
4	19	SDM	2021	Xiong	Energy Rep.
5	20	SDM	2021	Weng	Energy Convers. Manag.
6	21	SDM	2021	Ndi	Energy Rep.
7	22	SDM	2021	Naeijan	Energy Rep.
8	23	SDM	2020	Calasan	Energy Convers. Manag.
9	24	SDM	2020	Liang	Energy Convers. Manag.
10	25	SDM	2019	Li	Energy Convers. Manag.
11	26	SDM	2019	Chen	Energy
12	27	SDM	2019	Yu	Appl. Energy
13	28	SDM	2018	Beigi	Sol. Energy
14	29	SDM	2018	Bana	Energy Rep.
15	30	SDM	2018	Szabo	Appl. Sci.
16	31	SDM	2016	Silva	IEEE J. Photovoltaics
17	16	DDM	2022	Rawa	Mathematics
18	17	DDM	2021	Calasan	Sol. Energy
19	32	DDM	2021	Long	Energy
20	20	DDM	2021	Weng	Energy Convers. Manag.
21	19	DDM	2021	Xiong	Energy Rep.
22	18	DDM	2021	Sadaou	Energy Convers. Manag.
23	21	DDM	2021	Ndi	Energy Rep.
24	33	DDM	2020	Kumar	Optik
25	34	DDM	2020	Gude	Sol. Energy
26	35	DDM	2020	Jiao	Energy
27	36	DDM	2019	Calasan	Energies
28	28	DDM	2018	Beigi	Sol. Energy
29	13	DDM	2015	Lun	Sol. Energy
30	37	DDM	2014	Hejri	IEEE J. Photovoltaics
31	16	TDM	2022	Rawa	Mathematics
32	17	TDM	2021	Calasan	Sol. Energy
33	38	TDM	2021	Liu	Energy
34	22	TDM	2021	Naeijian	Energy Rep.
35	39	TDM	2020	Quais	Energy Convers. Manag.
36	40	TDM	2020	Omnia	IET Renew. Power Gener.
37	41	TDM	2020	Premkumar	Optik
38	33	TDM	2020	Kumar	Optik
39	42	TDM	2018	Abd Elaziz	Energy Convers. Manag

Number	References	Method	Year	First author	Journal
1	16	SDM	2022	Rawa	Mathematics
2	17	SDM	2021	Calasan	Sol. Energy
3	18	SDM	2021	Saadaoui	Energy Convers. Manag.
4	19	SDM	2021	Xiong	Energy Rep.
5	20	SDM	2021	Weng	Energy Convers. Manag.
6	21	SDM	2021	Ndi	Energy Rep.
7	22	SDM	2021	Naeijan	Energy Rep.
8	23	SDM	2020	Calasan	Energy Convers. Manag.
9	24	SDM	2020	Liang	Energy Convers. Manag.
10	25	SDM	2019	Li	Energy Convers. Manag.
11	26	SDM	2019	Chen	Energy
12	27	SDM	2019	Yu	Appl. Energy
13	28	SDM	2018	Beigi	Sol. Energy
14	29	SDM	2018	Bana	Energy Rep.
15	30	SDM	2018	Szabo	Appl. Sci.
16	31	SDM	2016	Silva	IEEE J. Photovoltaics
17	16	DDM	2022	Rawa	Mathematics
18	17	DDM	2021	Calasan	Sol. Energy
19	32	DDM	2021	Long	Energy
20	20	DDM	2021	Weng	Energy Convers. Manag.
21	19	DDM	2021	Xiong	Energy Rep.
22	18	DDM	2021	Sadaou	Energy Convers. Manag.
23	21	DDM	2021	Ndi	Energy Rep.
24	33	DDM	2020	Kumar	Optik
25	34	DDM	2020	Gude	Sol. Energy
26	35	DDM	2020	Jiao	Energy
27	36	DDM	2019	Calasan	Energies
28	28	DDM	2018	Beigi	Sol. Energy
29	13	DDM	2015	Lun	Sol. Energy
30	37	DDM	2014	Hejri	IEEE J. Photovoltaics
31	16	TDM	2022	Rawa	Mathematics
32	17	TDM	2021	Calasan	Sol. Energy
33	38	TDM	2021	Liu	Energy
34	22	TDM	2021	Naeijian	Energy Rep.
35	39	TDM	2020	Quais	Energy Convers. Manag.
36	40	TDM	2020	Omnia	IET Renew. Power Gener.
37	41	TDM	2020	Premkumar	Optik
38	33	TDM	2020	Kumar	Optik
39	42	TDM	2018	Abd Elaziz	Energy Convers. Manag

Based on the literature reviewed, the following conclusions can be drawn:

Various metaheuristic algorithms are employed in recent studies for determining the parameters of solar cell models.
The diversity of methods suggests that no single approach has yet been established as definitively superior for determining solar cell parameters. This area remains ripe for further research.
Hybrid metaheuristic algorithms are highlighted in the literature as providing superior solutions. They enhance the convergence curves of existing algorithms, achieve quicker convergence to solutions, require fewer iterations, and demonstrate better statistical performance than non-hybrid algorithms.

This paper focuses on parameter determination for all three models of solar cells using three hybrid variants of the Walrus Optimization Algorithm (WaOA). These variants will incorporate chaotic sequences, undergo adaptive modifications, and integrate with the Simulated Annealing (SA) algorithm. This study will be the first to assess the efficacy of these hybrid models in estimating solar cell parameters.

Additionally, all hybrid variants will be implemented across the three solar cell models. The Double-Diode Model (DDM) will be explored in two scenarios: using a numerical approach to describe the current–voltage relationship¹² and an analytical approach proposed by Lun et al.¹³

The main contributions of this paper are as follows:

It estimates parameters for all three models of solar cells, which is not commonly addressed in a single study.
It utilizes the Lambert W function along with iterative Lambert W approaches for parameter estimation.
It proposes three variants of a hybrid metaheuristic algorithm for parameter estimation of solar cells.
It conducts tests of the proposed algorithms on a solar panel under varying insolation and temperature conditions.

This paper is structured as follows: Sec. II describes the solar cell models. Section III introduces the new algorithm for estimating solar cell parameters. Section IV presents numerical results of the parameter estimation for literature-based solar cells and compares these results with existing approaches. Section V discusses tests of the proposed algorithm on a solar module under various conditions. The conclusion summarizes the main findings of this study and suggests avenues for future research.

II. SOLAR CELL EQUIVALENT CIRCUITS

In this section, three solar cell equivalent circuits are described.

A. SDM

The single diode model of a solar cell represents a standard approach when it comes to representing the equivalent circuit of solar cells. Mathematically, the relationship between current and voltage in this model can be expressed by the following equation:

I = I_{P V} - I_{0} (e^{\frac{V + I R_{S}}{n \times V_{t h}}} - 1) - \frac{V + I R_{S}}{R_{P}},

(1)

where I_PV represents the photogenerated current, I₀ represents the diode reverse saturation current, and n is the diode ideality factor, while V_th = K_BT/q is the thermal voltage (K_B is the Boltzmann constant, T is the temperature in kelvin, and q is the electron charge).

It is important to emphasize that an analytical solution for the mentioned relationship can be determined using the Lambert W function.²³ Therefore, the current–voltage relationship in the SDM of a solar cell can be represented as follows:

I = \frac{R_{P} (I_{P V} + I_{0}) - V}{R_{S} + R_{P}} - \frac{a V_{t h}}{R_{S}} W (\frac{I_{0} R_{S} R_{P}}{a V_{t h} (R_{S} + R_{P})} e^{\frac{R_{P} (R_{S} I_{P V} + R_{S} I_{0} + V)}{a_{1} (R_{P} + R_{S})}}),

(2)

where W represents the solution to the Lambert W function. It should be noted that in practice, when it comes to the current–voltage characteristics of solar cells, each individual voltage value corresponds to a precisely determined current value. Therefore, using Eq. , it is possible to calculate current values for each voltage value that corresponds to data obtained through measurements.

B. DDM

The relationship between current and voltage in the equivalent circuit of a solar cell with two diodes can be presented as follows:

I = I_{P V} - I_{01} (e^{\frac{V + I R_{S}}{n_{1} \cdot V_{t h}}} - 1) - I_{02} (e^{\frac{V + I R_{S}}{n_{2} \cdot V_{t h}}} - 1) - \frac{V + I R_{S}}{R_{P}},

(3)

where I_PV denotes the photogenerated current, I₀₁ and I₀₂ are the reverse saturation currents of diodes 1 and 2, respectively, and n₁ and n₂ are the ideality factors of the diodes, respectively.

Unlike the SDM, due to the pronounced nonlinearity of the observed characteristics and the great complexity of the corresponding expression, there is no exact analytical solution in this case. The complexity of the model significantly influences the choice and application of the solution methodology. Therefore, to determine the current in the DDM and TDM, the application of appropriate iterative procedures is necessary.

It should be particularly noted that Eq. (3) represents the key mathematical expression that provides the basis for various methods of calculating the current–voltage characteristics of the equivalent circuit model of a solar cell with two diodes. In this work, two characteristic methods will be presented: Explicit Double-Diode Modeling Method based on Lambert W-function (EDDMMLW)¹³ and iterative DDM Lambert W approach.¹⁷

1. Explicit double-diode modeling method based on Lambert W-function (EDDMMLW)¹³

The analytical approach presented in Ref. 13 was carried out with appropriate mathematical approximations related to the saturation current values and the ideal factor. According to this model, the current–voltage equation of the DDM of solar cells has the following form:

\begin{align} I & = \frac{R_{P} (I_{P V} + I_{01} + I_{02}) - V}{R_{P} + R_{S}} \\ - \frac{n_{1} \cdot V_{t h}}{2 R_{P}} W (\frac{R_{S} R_{P} (I_{01} + I_{02})}{n_{1} \cdot V_{t h} (R_{P} + R_{S})} e^{\frac{R_{P} (R_{S} I_{P V} + R_{S} I_{01} + R_{S} I_{02} + V)}{n_{1} \cdot V_{t h} (R_{P} + R_{S})}}) \\ - \frac{n_{2} \cdot V_{t h}}{2 R_{P}} W (\frac{R_{S} R_{P} (I_{01} + I_{02})}{n_{2} \cdot V_{t h} (R_{P} + R_{S})} e^{\frac{R_{P} (R_{S} I_{P V} + R_{S} I_{01} + R_{S} I_{02} + V)}{n_{2} \cdot V_{t h} (R_{P} + R_{S})}}) . \end{align}

(4)

2. Iterative DDM Lambert W approach

The current is obtained from the following equation:

I = \frac{R_{P}}{R_{P} + R_{S}} (I_{P V} + I_{01} + I_{02} - \frac{V}{R_{P}} - y \cdot \frac{n_{1} V_{t h}}{R_{S}} (1 + \frac{R_{S}}{R_{P}})),

(5)

in which the variable y is calculated from the following iterative equation [where (p) represents the current iteration]:

^{(p)} y = W (α + β \exp (δ \cdot^{(p - 1)} y)),

(6)

whose stopping criterion is defined as follows:

ε = |\exp (δ \cdot^{(p)} y) - \exp (δ \cdot^{(p - 1)} y)|,

(7)

where ε is a small real number (for example, ε = 10⁻¹⁰).

In the previous equation, variables α, β, and δ are as follows:

\begin{align} α & = \frac{R_{P} R_{S}}{R_{P} + R_{S}} \frac{I_{01}}{n_{1} V_{t h}} (\exp (\frac{V}{n_{1} V_{t h}})) \\ \times \exp (\frac{R_{P} R_{S}}{R_{P} + R_{S}} \frac{1}{n_{1} V_{t h}} (I_{P V} + I_{01} + I_{02} - \frac{V}{R_{P}})), \end{align}

(8)

\begin{align} β & = \frac{R_{P} R_{S}}{R_{P} + R_{S}} \frac{I_{02}}{n_{2} V_{t h}} \exp (\frac{V}{n_{2} V_{t h}}) \\ \times \exp (\frac{R_{P} R_{S}}{R_{P} + R_{S}} \frac{1}{n_{2} V_{t h}} (I_{P V} + I_{01} + I_{02} - \frac{V}{R_{P}})), \end{align}

(9)

δ = 1 - \frac{n_{1}}{n_{2}} .

(10)

C. TDM

For TDM, there is no analytical relationship between current and voltage. Therefore, their relationship is determined by an iterative approach. In this work, the iterative approach described in Ref. 17, which is analogous to the iterative approach for DDM, will be used. The current is obtained from the following equation:

I = \frac{R_{P}}{R_{P} + R_{S}} (I_{P V} + I_{01} + I_{02} + I_{03} - \frac{V}{R_{P}} - y \cdot \frac{n_{1} V_{t h}}{R_{S}} (1 + \frac{R_{S}}{R_{P}})),

(11)

in which the variable y is calculated from the following iterative equation [where (p) represents the current iteration]:

^{(p)} y = W (α + β \exp (δ \cdot^{(p)} y) + γ \exp (σ \cdot^{(p)} y)),

(12)

whose stopping criterion is defined as follows:

\begin{align} ε & = |β \exp (δ \cdot^{(p)} y) + γ \exp (σ \cdot^{(p)} y) - β \exp (δ \cdot^{(p - 1)} y) \\ - γ \exp (σ \cdot^{(p - 1)} y)|, \end{align}

(13)

where ε is a small real number (for example, ε = 10⁻¹⁰).

In the previous equation, variables α, β, and δ are as follows:

\begin{align} α & = \frac{R_{P} R_{S}}{R_{P} + R_{S}} \frac{I_{01}}{n_{1} V_{t h}} \exp (\frac{V}{n_{1} V_{t h}}) \\ \times \exp (\frac{R_{P} R_{S}}{R_{P} + R_{S}} \frac{1}{n_{1} V_{t h}} (I_{P V} + I_{01} + I_{02} + I_{03} - \frac{V}{R_{P}})), \end{align}

(14)

\begin{align} β & = \frac{R_{P} R_{S}}{R_{P} + R_{S}} \frac{I_{02}}{n_{2} V_{t h}} \exp (\frac{V}{n_{2} V_{t h}}) \\ \times \exp (\frac{R_{P} R_{S}}{R_{P} + R_{S}} \frac{1}{n_{2} V_{t h}} (I_{P V} + I_{01} + I_{02} + I_{03} - \frac{V}{R_{P}})), \end{align}

(15)

\begin{align} γ & = \frac{R_{P} R_{S}}{R_{P} + R_{S}} \frac{I_{03}}{n_{3} V_{t h}} \exp (\frac{V}{n_{3} V_{t h}}) \\ \times \exp (\frac{R_{P} R_{S}}{R_{P} + R_{S}} \frac{1}{n_{3} V_{t h}} (I_{P V} + I_{01} + I_{02} + I_{03} - \frac{V}{R_{P}})), \end{align}

(16)

δ = 1 - \frac{n_{1}}{n_{2}},

(17)

σ = 1 - \frac{n_{1}}{n_{3}} .

(18)

III. HYBRID VARIANTS OF WAOA

A. Original WaOA algorithm

The original version of the Walrus Optimization Algorithm (WaOA) is presented in Ref. 43. The basics of this algorithm are found in the natural behavior of walruses. This behavior consists of three phases:

Guiding individuals to feed.
Migration of walruses to rocky beaches.
Fight or escape from predators.

Mathematically, the population of WaOA consists of searcher members-walruses. Concretely, each walrus is a potential solution to the optimization problem. Every walrus of the population is actually a vector composed of m variables, where m stands for the dimension of the optimization problem. Therefore, the population of N walruses is described with matrix X, whose dimensions are Nxm,

X = {[\begin{matrix} \begin{matrix} X_{1} \\ ⋮ \\ X_{i} \end{matrix} \\ ⋮ \\ X_{N} \end{matrix}]}_{N x m} = {[\begin{matrix} x_{1, 1} & \dots & x_{1, j} & \dots & x_{1, m} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ x_{i, 1} & \dots & x_{i, j} & \dots & x_{i, m} \\ ⋮ & ⋱ & ⋮ & ⋱ & ⋮ \\ x_{N, 1} & \dots & x_{N, j} & \dots & x_{N, m} \end{matrix}]}_{N x m},

(19)

where X_i represents the ith walrus in the population and x_i,j is the value of the jth optimization variable suggested by the ith walrus.

The quality of each potential solution (each walrus from the population) is measured by the objective function (OF) value F_i = F(X_i),

F = {[\begin{matrix} \begin{matrix} F_{1} \\ ⋮ \\ F_{i} \end{matrix} \\ ⋮ \\ F_{N} \end{matrix}]}_{N x 1} = {[\begin{matrix} \begin{matrix} F (X_{1}) \\ ⋮ \\ F (X_{i}) \end{matrix} \\ ⋮ \\ F (X_{N}) \end{matrix}]}_{N x 1},

(20)

where F stands for the objective function.

Before the iteration procedure begins, the initial population of walruses is randomly initialized between the lower bound (LB) and upper bound (UB) of optimization variables,

X_{i} = L B + r a n d \cdot (U B - L B),

(21)

where rand is the vector of random numbers between 0 and 1, generated separately for each walrus.

The first phase of the algorithm depicts the global search of the complete search space, so-called exploration. This phase is named feeding strategy. According to the natural behavior of walruses, the strongest walrus guides other walruses in the group to find food. The strongest, or the best walrus, is the one with the lowest value of objective function. Mathematically, this phase is modeled by updating the position of each walrus X_i^P1 as follows:
$x_{i, j}^{P 1} = x_{i, j} + r a n d_{i, j} \cdot (S W_{j} - I_{i, j} \cdot x_{i, j}),$
(22)
where x_i,j^P1 is the jth dimension of the newly generated position of the ith walrus, SW denotes the best walrus of the population, and I_i,j are integer numbers that take value 1 or 2. If the improved value of objective function is obtained, then the old position is replaced by newly generated, as follows:
$X_{i} = \{\begin{cases} X_{i}^{P 1}, F_{i}^{P 1} < F_{i}, \\ X_{i}, e l s e . \end{cases}$
(23)
The second phase is used to describe the migration of the walruses to outcrops or rocky beaches, caused by the warming of the air in late summer. During this phase, the assumption is made that each walrus migrates to the another walrus randomly chosen from the population. Mathematical formulation to update the position of each walrus X_i^P2 is given as follows:
$x_{i, j}^{P 2} = \{\begin{cases} x_{i, j} + r a n d_{i, j} \cdot (x_{k, j} - I_{i, j} \cdot x_{i, j}), F_{k} < F_{i}, \\ x_{i, j} + r a n d_{i, j} \cdot (x_{i, j} - x_{k, j}), e l s e, \end{cases}$
(24)
where X_k (k≠i) is the randomly selected walrus and F_k is its’ objective function value. Similarly to the first phase, it needs to be checked whether newly generated position provides better objective function value than the old one,
$X_{i} = \{\begin{cases} X_{i}^{P 2}, F_{i}^{P 2} < F_{i}, \\ X_{i}, e l s e . \end{cases}$
(25)
In the third phase, escaping of the walruses and their fighting against predators is presented. The strategy used to escape and find the predators means that the walruses change the position, but only in the vicinity of the position in which they were previously located. The newly generated position of each walrus X_i^P3 can be calculated as follows:
$x_{i, j}^{P 3} = x_{i, j} + \frac{L B_{j}}{t} + \frac{U B_{j}}{t} - r a n d \cdot \frac{L B_{j}}{t},$
(26)
where t is the current iteration number. Furthermore, if the new position is better than the old one, it takes the following place:
$X_{i} = \{\begin{cases} X_{i}^{P 3}, F_{i}^{P 3} < F_{i}, \\ X_{i}, e l s e . \end{cases}$
(27)
After completing the third phase, one iteration of WaOA is completed. Furthermore, the described iterative procedure is repeated until the predefined maximum number of iterations is reached. Finally, the optimal solution of the optimization problem is given with the best walrus, i.e., the walrus with the best objective function value.

In order to sublime the steps of the WaOA, the pseudo-code is given below.

Pseudo-code of WaOA algorithm
1: Input of the information of the optimization problem
2: Enter the population size N and maximum number of iterations t_max
3: Initialize the population randomly
4: For t = 1 to t_max
5: Update strongest walrus based on the value of objective function
6: For i = 1 to N
7: Apply phase 1—Feeding strategy
8: Calculate the positions X_i^P1
9: Update the position of each walrus
10: Apply phase 2—Migration
11: Calculate the positions X_i^P2
12: Update the position of each walrus
13: Apply phase 3—Escaping and fighting against predators
14: Calculate the positions X_i^P3
15: Update the position of each walrus
16: endfor
17: Save the best walrus so far
18: endfor
19: Global optimal solution is the walrus with the best objective function value

B. Chaotic modification of WaOA

The previously presented approach has one drawback, which is the random initialization of the population. Therefore, the first novel approach presented in this paper is to avoid random initialization, but to apply logistic chaotic maps in order to initialize the population. By this way, one kind of optimization algorithm (chaotic algorithm) is used to initialize the population, and the high quality of initial population is secured. According to the proposed approach, the population is initialized as follows:

\begin{gathered} Y_{1} = r a n d, Y_{i + 1} = 4 \cdot Y_{i} \cdot (1 - Y_{i}), i = 1, 2, \dots, N - 1, \\ X_{i} = L B + Y_{i} \cdot (U B - L B) . \end{gathered}

(28)

After the initialization is completed, the process of updating the walruses’ postions is carried out as described in Sec. III A. Such obtained chaotic version of WaOA is called chaotic-WaOA or C-WaOA.

C. Simulated annealing-WaOA (SA-WaOA)

The second approach presented in this work is to develop the hybrid SA-WaOA algorithm, in which the well-known simulated annealing (SA) algorithm is used to initialize the population.⁴⁴ This approach is very similar to the previous one, but the chaotic algorithm is substituted with the SA algorithm. Significant difference between these two algorithms is their type—chaotic is population-based algorithm, while SA is single-solution-based algorithm. Therefore, the developed SA-WaOA algorithm employs the SA algorithm to initialize the population, instead of random initialization as described in Sec. III A. The SA algorithm is described with the following pseudo-code:

Pseudo-code of SA algorithm
1: Input data: k = 0, c_k = c₀, L_k = L₀
2: Randomly initialize each staff of the population
3: For each staff of the population
4: while c_k > 0
5: For l = 0 to L_k
6: Generate solution x_j from neighborhood of current solution x_i
7: if f(p_j)<f(p_i) then p_i = p_j
8: else p_j becomes current solution with probability exp.((f(p_i)- f(p_j))/c_k)
9: end if
10: end for
11: k = k+1
12: Update values L_k and c_k
11: end while
12: end for

D. Adaptive WaOA (A-WaOA)

The third modified variant of WaOA is the adaptive modification, called A-WaOA. This type of modification is significantly different from the previous two and is based on the very important property of every metaheuristic algorithm. Namely, it is well-known that the optimal performances of any metaheuristic algorithm are achieved if global search (exploration) is applied in early iterations, and local search around the existing optimal solution (exploitation) is performed in later iterations. According to the description of the original WaOA algorithm, parameter I is used to provide balance between exploration and exploitation. However, this parameter takes random integer values that can be either 1 or 2. By this way, it is completely unpredictable how the algorithm will act in every iteration. In other words, it cannot be known if it will support local or global search. The higher values of this parameter indicate that the algorithm is focused on global search, while the lower values cause the algorithm to do the local search. Therefore, in this work, the adaptive change of this parameter is proposed. Precisely, the initial value of the parameter I is set to 2, and the algorithm then carries out the global search. Afterward, during the iterations, this parameter linearly decreases, and at the last iteration, it drops to 1, which means that the focus is to do local search. By this way, the optimal balance between global and local search is secured. Mathematically, the adaptive law for the change of the parameter I is given as follows:

I = 2 - \frac{t - 1}{t_{\max} - 1} .

(29)

IV. NUMERICAL RESULTS

This section consists of three parts. In the first part, a comparison of the current–voltage and power–voltage characteristics has been conducted, where the parameters were determined using approaches from the literature for SDM, DDM, and TDM for two well-known cells/modules. In the second part, a comparison of results obtained using the proposed hybrid variants was performed. In the third part, a comparison between the proposed methods and the best methods from the literature was conducted. Table II presents data on the parameter values of solar cells, which were obtained using numerous literary approaches for all three models of solar cells for the RTC France solar cell. Table III presents analogous results for the MSX60 solar module. Tables II and III also include data on the root mean square error (RMSE) values.

TABLE II.

The values of parameters determined using the cited literature approaches for all three models of solar cells—RTC France solar cell.

References	Model	No.	I_PV (A)	I₀₁ (µA)	n₁	R_S (Ω)	R_P (Ω)	I₀₂ (µA)	n₂	I₀₃ (µA)	n₃	RMSE
16	SDM	1	0.760 775 37	0.320 741 243	1.480 469 87	0.036 400	53.539 718 4	⋯	⋯	⋯	⋯	7.746 557 808 × 10⁻⁴
16	SDM	2	0.760 777 00	0.323 564 000	1.481 244 00	0.036 370	53.742 465 0	⋯	⋯	⋯	⋯	7.750 239 999 × 10⁻⁴
21	SDM	3	0.760 759 703 7	0.326 288 93	1.482 193 00	0.036 340 99	54.206 594 00	⋯	⋯	⋯	⋯	7.768 659 613 × 10⁻⁴
22	SDM	4	0.760 775 51	0.323 020 31	1.481 108 08	0.036 377 10	53.718 674 07	⋯	⋯	⋯	⋯	7.915 026 811 × 10⁻⁴
18	SDM	5	0.760 774 00	0.325 595 40	1.482 096 00	0.036 340 20	53.896 860 00	⋯	⋯	⋯	⋯	8.126 216 289 × 10⁻⁴
16	DDM	1	0.760 780 1	0.841 62 × 10⁻⁶	1.999 999	0.036 79	55.73	0.215 450 5 × 10⁻⁶	1.447 06	⋯	⋯	7.561 295 767 × 10⁻⁴
16	DDM	2	0.760 78	0.841 611 × 10⁻⁶	2.000 00	0.036 790 5	55.728 35	0.215 450 1 × 10⁻⁶	1.447 04	⋯	⋯	7.559 104 364 × 10⁻⁴
19	DDM	3	0.760 800 00	0.259 500 × 10⁻⁶	1.462 700 00	0.036 600 00	54.933 000 00	0.479 100 0 × 10⁻⁶	1.998 30	⋯	⋯	7.653 474 079 × 10⁻⁴
18	DDM	4	0.760 827 00	0.322 452 46 × 10⁻⁶	1.481 028 00	0.036 364 40	53.110 790 00	0.000 273 92 × 10⁻⁶	1.470 101	⋯	⋯	7.955 404 200 × 10⁻⁴
21	DDM	5	0.767 920 00	0.399 990 0 × 10⁻⁶	2.000 00	0.036 590 00	54.176 140 00	0.266 050 00 × 10⁻⁶	1.464 510	⋯	⋯	0.006 348 583 736
16	TDM	1	0.760 760 2	0.876 504 × 10⁻⁶	1.995 04	0.036 920 1	55.6798	0.204 41 × 10⁻⁶	1.442 401	0.000 1805 × 10⁻⁶	1.890 01	7.518 795 429 × 10⁻⁴
16	TDM	2	0.760 760 1	0.876 499 × 10⁻⁶	1.995 01	0.036 920 2	55.680 1	0.204 401 × 10⁻⁶	1.442 41	0.000 1801 × 10⁻⁶	1.890 01	7.520 538 313 × 10⁻⁴
41	TDM	3	0.760 790	0.320 000 × 10⁻⁶	1.866 6	0.036 70	55.441 1	0.230 000 × 10⁻⁶	1.452 10	0.740 0 × 10⁻⁶	2.394 90	0.002 471 743 406 2
41	TDM	4	0.760 763	0.280 0 × 10⁻⁶	1.468 4	0.036 50	55.382 1	0.000 670 × 10⁻⁶	1.546 80	1.000 0 × 10⁻⁶	2.322 50	7.795 840 314 4 × 10⁻⁴
41	TDM	5	0.760 770	0.235 3 × 10⁻⁶	1.454 3	0.036 68	55.444 8	0.221 300 × 10⁻⁶	2.000 00	0.457 3 × 10⁻⁶	2.000 00	8.231 360 2101 × 10⁻⁴

References	Model	No.	I_PV (A)	I₀₁ (µA)	n₁	R_S (Ω)	R_P (Ω)	I₀₂ (µA)	n₂	I₀₃ (µA)	n₃	RMSE
16	SDM	1	0.760 775 37	0.320 741 243	1.480 469 87	0.036 400	53.539 718 4	⋯	⋯	⋯	⋯	7.746 557 808 × 10⁻⁴
16	SDM	2	0.760 777 00	0.323 564 000	1.481 244 00	0.036 370	53.742 465 0	⋯	⋯	⋯	⋯	7.750 239 999 × 10⁻⁴
21	SDM	3	0.760 759 703 7	0.326 288 93	1.482 193 00	0.036 340 99	54.206 594 00	⋯	⋯	⋯	⋯	7.768 659 613 × 10⁻⁴
22	SDM	4	0.760 775 51	0.323 020 31	1.481 108 08	0.036 377 10	53.718 674 07	⋯	⋯	⋯	⋯	7.915 026 811 × 10⁻⁴
18	SDM	5	0.760 774 00	0.325 595 40	1.482 096 00	0.036 340 20	53.896 860 00	⋯	⋯	⋯	⋯	8.126 216 289 × 10⁻⁴
16	DDM	1	0.760 780 1	0.841 62 × 10⁻⁶	1.999 999	0.036 79	55.73	0.215 450 5 × 10⁻⁶	1.447 06	⋯	⋯	7.561 295 767 × 10⁻⁴
16	DDM	2	0.760 78	0.841 611 × 10⁻⁶	2.000 00	0.036 790 5	55.728 35	0.215 450 1 × 10⁻⁶	1.447 04	⋯	⋯	7.559 104 364 × 10⁻⁴
19	DDM	3	0.760 800 00	0.259 500 × 10⁻⁶	1.462 700 00	0.036 600 00	54.933 000 00	0.479 100 0 × 10⁻⁶	1.998 30	⋯	⋯	7.653 474 079 × 10⁻⁴
18	DDM	4	0.760 827 00	0.322 452 46 × 10⁻⁶	1.481 028 00	0.036 364 40	53.110 790 00	0.000 273 92 × 10⁻⁶	1.470 101	⋯	⋯	7.955 404 200 × 10⁻⁴
21	DDM	5	0.767 920 00	0.399 990 0 × 10⁻⁶	2.000 00	0.036 590 00	54.176 140 00	0.266 050 00 × 10⁻⁶	1.464 510	⋯	⋯	0.006 348 583 736
16	TDM	1	0.760 760 2	0.876 504 × 10⁻⁶	1.995 04	0.036 920 1	55.6798	0.204 41 × 10⁻⁶	1.442 401	0.000 1805 × 10⁻⁶	1.890 01	7.518 795 429 × 10⁻⁴
16	TDM	2	0.760 760 1	0.876 499 × 10⁻⁶	1.995 01	0.036 920 2	55.680 1	0.204 401 × 10⁻⁶	1.442 41	0.000 1801 × 10⁻⁶	1.890 01	7.520 538 313 × 10⁻⁴
41	TDM	3	0.760 790	0.320 000 × 10⁻⁶	1.866 6	0.036 70	55.441 1	0.230 000 × 10⁻⁶	1.452 10	0.740 0 × 10⁻⁶	2.394 90	0.002 471 743 406 2
41	TDM	4	0.760 763	0.280 0 × 10⁻⁶	1.468 4	0.036 50	55.382 1	0.000 670 × 10⁻⁶	1.546 80	1.000 0 × 10⁻⁶	2.322 50	7.795 840 314 4 × 10⁻⁴
41	TDM	5	0.760 770	0.235 3 × 10⁻⁶	1.454 3	0.036 68	55.444 8	0.221 300 × 10⁻⁶	2.000 00	0.457 3 × 10⁻⁶	2.000 00	8.231 360 2101 × 10⁻⁴

TABLE III.

The values of parameters determined using the cited literature approaches for all three models of solar cells—MSX60 solar module.

References	Model	No.	I_pv (A)	I₀₁ (µA)	n₁	R_S (Ω)	R_P (Ω)	I₀₂ (µA)	n₂	I₀₃ (µA)	n₃	RMSE
16	SDM	1	3.8127	0.140 51	1.332 513	0.223 51	1105.586 9	⋯	⋯	⋯	⋯	0.012 105 788 990 909
16	SDM	2	3.812 68	0.14	1.3325	0.223 5	1155.625 8	⋯	⋯	⋯	⋯	0.012 120 811 620 561
29	SDM	3	3.8084	4.8723 × 10⁻⁴	1.000 3	0.3692	169.047 1	⋯	⋯	⋯	⋯	0.101 844 933 535 061
30	SDM	4	3.808	1.22 × 10⁻³	1.045	0.316	146.08	⋯	⋯	⋯	⋯	0.030 722 505 654 028
31	SDM	5	3.7983	6.79 × 10⁻²	1.28	0.251	582.727	⋯	⋯	⋯	⋯	0.018 106 628 879 805
16	DDM	1	3.812 527	0.123 12	1.322 88	0.226 805	805.46	07.30 × 10⁻⁵	1.988 00	⋯	⋯	0.011 955 126 049 429
16	DDM	2	3.812 526 89	0.123 119 9	1.322 86	0.226 801	807.11	7.299 × 10⁻⁵	1.988 1	⋯	⋯	0.011 896 989 581 563
16	DDM	3	3.812 527	0.123 11	1.322 90	0.226 800	800	7.299 90 × 10⁻⁵	1.988 00	⋯	⋯	0.012 028 665 165 887
13	DDM	4	3.8086	1.8952 × 10⁻⁴	0.992 47	0.376 59	166.485 4	1.8941 × 10⁻⁵	0.992 47	⋯	⋯	0.053 947 147 964 604
37	DDM	5	3.8046	4.12335 × 10⁻⁴	1	0.3392	280.202 22	3.981 11	2	⋯	⋯	0.030 522 230 174 973
16	TDM	1	3.812 52	0.123 14	1.322 74	0.226 756	831.010 00	7.299 90 × 10⁻⁵	1.988 88	1.24 × 10⁻⁴	1.930 00	0.011 683 038 647 976
16	TDM	2	3.812 53	0.123 12	1.322 71	0.226 758 6	827.510 00	7.300 0 × 10⁻⁵	1.990 60	1.2488 × 10⁻⁴	1.930 00	0.011 660 925 987 728
39	TDM	3	3.801 9	3.3525 × 10⁻¹	1.934 6	0.227 24	450.13	1.00 × 10⁻⁶	1.720 8	6.457 × 10⁻²	1.276 4	0.017 884 722 562 835
17	TDM	4	3.812 53	0.123 11	1.322 70	0.226 760	823.400 00	7.299 85 × 10⁻⁵	1.990 0	1.25 × 10⁻⁴	1.930 00	0.011 653 036 757 559
40	TDM	5	3.755 264	2.18714 × 10⁻¹	1.375 876	0.110 955 7	349.8458	2.294 004 × 10⁻⁴	1.074 414	2.210 856 × 10⁻⁴	1.094 849	0.091 223 197 368 868

References	Model	No.	I_pv (A)	I₀₁ (µA)	n₁	R_S (Ω)	R_P (Ω)	I₀₂ (µA)	n₂	I₀₃ (µA)	n₃	RMSE
16	SDM	1	3.8127	0.140 51	1.332 513	0.223 51	1105.586 9	⋯	⋯	⋯	⋯	0.012 105 788 990 909
16	SDM	2	3.812 68	0.14	1.3325	0.223 5	1155.625 8	⋯	⋯	⋯	⋯	0.012 120 811 620 561
29	SDM	3	3.8084	4.8723 × 10⁻⁴	1.000 3	0.3692	169.047 1	⋯	⋯	⋯	⋯	0.101 844 933 535 061
30	SDM	4	3.808	1.22 × 10⁻³	1.045	0.316	146.08	⋯	⋯	⋯	⋯	0.030 722 505 654 028
31	SDM	5	3.7983	6.79 × 10⁻²	1.28	0.251	582.727	⋯	⋯	⋯	⋯	0.018 106 628 879 805
16	DDM	1	3.812 527	0.123 12	1.322 88	0.226 805	805.46	07.30 × 10⁻⁵	1.988 00	⋯	⋯	0.011 955 126 049 429
16	DDM	2	3.812 526 89	0.123 119 9	1.322 86	0.226 801	807.11	7.299 × 10⁻⁵	1.988 1	⋯	⋯	0.011 896 989 581 563
16	DDM	3	3.812 527	0.123 11	1.322 90	0.226 800	800	7.299 90 × 10⁻⁵	1.988 00	⋯	⋯	0.012 028 665 165 887
13	DDM	4	3.8086	1.8952 × 10⁻⁴	0.992 47	0.376 59	166.485 4	1.8941 × 10⁻⁵	0.992 47	⋯	⋯	0.053 947 147 964 604
37	DDM	5	3.8046	4.12335 × 10⁻⁴	1	0.3392	280.202 22	3.981 11	2	⋯	⋯	0.030 522 230 174 973
16	TDM	1	3.812 52	0.123 14	1.322 74	0.226 756	831.010 00	7.299 90 × 10⁻⁵	1.988 88	1.24 × 10⁻⁴	1.930 00	0.011 683 038 647 976
16	TDM	2	3.812 53	0.123 12	1.322 71	0.226 758 6	827.510 00	7.300 0 × 10⁻⁵	1.990 60	1.2488 × 10⁻⁴	1.930 00	0.011 660 925 987 728
39	TDM	3	3.801 9	3.3525 × 10⁻¹	1.934 6	0.227 24	450.13	1.00 × 10⁻⁶	1.720 8	6.457 × 10⁻²	1.276 4	0.017 884 722 562 835
17	TDM	4	3.812 53	0.123 11	1.322 70	0.226 760	823.400 00	7.299 85 × 10⁻⁵	1.990 0	1.25 × 10⁻⁴	1.930 00	0.011 653 036 757 559
40	TDM	5	3.755 264	2.18714 × 10⁻¹	1.375 876	0.110 955 7	349.8458	2.294 004 × 10⁻⁴	1.074 414	2.210 856 × 10⁻⁴	1.094 849	0.091 223 197 368 868

From the results shown in Table II for the RTC France solar cell, we have the following:

The best parameters for the SDM are provided by the model shown in Ref. 16.
The best parameters for the DDM are given by the model shown in Ref. 16.
The best parameters for the TDM are provided by the model shown in Ref. 16.

Based on the results shown in Table III for the MSX60 solar module, we have the following:

The best parameters for the SDM are provided by the model shown in Ref. 16.
The best parameters for the DDM are given by the model shown in Ref. 16.
The best parameters for the TDM are provided by the model shown in Ref. 17.

Based on the analysis of results obtained by different methods, it is observed that the application of the TDM of the solar cell provides the most accurate parameters. DDM has a lower level of precision, while SDM results in the least accuracy compared to the other models.

The graphical results are shown in Figs. 2–7. Specifically, Fig. 2(a) shows a comparison of current–voltage characteristics for all methods from Table II, for the SDM. In Fig. 2(b), the error in current calculation is displayed. The power–voltage characteristics for all methods, as well as the error in power calculation as a function of voltage for all used methods, are given in Figs. 2(c) and 2(d), respectively. Analogous results for the DDM of the solar cell are shown in Fig. 3 and for the TDM in Fig. 4. Corresponding graphical results for the MSX60 solar module are given in Figs. 5–7.

FIG. 2.

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SDM of the RTC France solar cell. (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

FIG. 3.

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DDM of the RTC France solar cell. (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

FIG. 4.

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TDM of the RTC France solar cell. (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

FIG. 5.

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SDM of the MSX60 solar module. (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

FIG. 6.

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DDM of the MSX60 solar module. (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

FIG. 7.

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TDM of the MSX60 solar module. (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

The figures with I–V and P–V characteristics show the sum values of individual characteristic points. A few conclusions can also be drawn on the basis of all the presented results.

All methods, whether for SDM, DDM, or TDM, show good matching of measured and simulated I–V and P–V characteristics.
With the RTC France solar cell, as well as with the MSX-60 solar module, the very good matching of the curves represented by the error value is more than obvious.
All tested methods, in terms of graphical comparison, are very good.
The largest deviation of the measured and simulated curves for RTC France solar cells, for the DDM, is given by the parameters determined by method 5, and for TDM method 3. With the MSX-60 solar module, the largest deviation of the simulated and measured curves is given by the application of method 3 for SDM, method 4 for DDM, and method 6 for TDM. All previously mentioned methods are numbered in Tables II and III

In the rest of this section, the results of the parameter estimation of the previously mentioned two cells using the walrus optimization algorithm and its hybrid variants—chaotic WAOA (WaOA with chaotic sequences), adaptive WaOA (WaOA with adaptive modifications), and hybrid SA-WaOA (SA—Simulated Annealing)—are given. However, in order to test the detailed testing of the efficiency of the proposed algorithm and its hybrid variants, the results of the respective results were made by the application of well-known algorithms—BFO (Bacterial Foraging Optimization),⁴⁵ GOA (Grasshopper Optimization Algorithm),⁴⁶ and ROA (Red kite Optimization Algorithm).⁴⁷

For each algorithm, 30 repetitions were performed with a set population number of 100 and a number of iterations set at 100. The best results (parameter values) for all tested algorithms are shown in Table IV for the RTC France solar cell and in Table V for the MSX-60 solar module. In Tables IV and V, the value of RMSE is also presented. Note that, for DDM, two approaches were used—approach based on the iterative procedure described in Ref. 17 and approach based on the work of Lun¹³ (analytical approach).

TABLE IV.

The estimated values of parameters for all three models of solar cells—RTC France solar cell.

Model	Algorithm	Method	I_pv (a)	I₀₁ (a)	n₁	R_S (Ω)	R_P (Ω)	I₀₂ (a)	n₂	I₀₃ (µA)	n₃	RMSE
SDM	BFO	⋯	0.760 795 54	0.310 623 47	1.477 249 53	0.036 546 77	52.804 896 46	⋯	⋯	⋯	⋯	7.730 195 3501 · 10⁻⁴
SDM	GOA	⋯	0.761 109 35	0.309 898 04	1.477 100 14	0.036 461 69	48.825 954 40	⋯	⋯	⋯	⋯	8.069 368 879 9 · 10⁻⁴
SDM	ROA	⋯	0.760 700 00	0.302 599 99	1.474 600 00	0.036 800 00	54.491 100 00	⋯	⋯	⋯	⋯	7.887 712 627 3 · 10⁻⁴
SDM	AWaOA	1	0.760 787 84	0.310 620 30	1.477 247 03	0.036 547 86	52.888 623 27	⋯	⋯	⋯	⋯	7.730 063 830 8 · 10⁻⁴
SDM	CWaOA	2	0.760 787 98	0.310 682 30	1.477 267 05	0.036 546 96	52.889 267 70	⋯	⋯	⋯	⋯	7.730 062 693 7 · 10⁻⁴
SDM	SAWaOA	3	0.760 787 95	0.310 684 51	1.477 267 76	0.036 546 94	52.889 826 40	⋯	⋯	⋯	⋯	7.730 062 690 1 · 10⁻⁴
SDM	WaOA	4	0.760 787 96	0.310 684 04	1.477 267 61	0.036 546 95	52.889 696 19	⋯	⋯	⋯	⋯	7.730 062 690 2 · 10⁻⁴
DDMa	BFO	⋯	0.760 832 27	0.128 549 65	1.407 401 82	0.037 376 72	54.000 000 00	0.800 000 00	1.839 408 20	⋯	⋯	7.498 286 041 3 · 10⁻⁴
DDMa	GOA	⋯	0.760 780 40	0.098 531 12	1.402 962 51	0.037 251 76	52.575 514 38	0.315 202 86	1.607 081 98	⋯	⋯	7.718 353 491 0 · 10⁻⁴
DDMa	ROA	⋯	0.760 700 00	0.126 700 00	1.407 900 00	0.037 300 00	53.352 400 00	0.615 399 99	1.783 000 00	⋯	⋯	7.650 683 805 1 · 10⁻⁴
DDMa	AWaOA	1	0.760 860 38	0.116 236 55	1.400 286 27	0.037 434 36	54.000 000 00	0.800 000 00	1.820 315 30	⋯	⋯	7.483 440 787 1 · 10⁻⁴
DDMa	CWaOA	2	0.760 858 38	0.124 205 34	1.405 018 38	0.037 388 88	53.999 945 94	0.799 959 40	1.832 092 75	⋯	⋯	7.483 709 998 3 · 10⁻⁴
DDMa	SAWaOA	3	0.760 871 55	0.092 972 94	1.385 085 34	0.037 544 50	54.000 000 00	0.798 724 94	1.784 359 26	⋯	⋯	7.481 111 488 8 · 10⁻⁴
DDMa	WaOA	4	0.760 861 96	0.114 729 58	1.399 456 09	0.037 436 79	54.000 000 00	0.799 998 06	1.817 280 17	⋯	⋯	7.482 161 974 4 · 10⁻⁴
DDMb	BFO	⋯	0.760 500 46	0.421 285 96	1.992 027 76	0.019 683 80	54.172 233 99	0.301 079 85	1.460 248 47	⋯	⋯	7.835 723 845 6 · 10⁻⁴
DDMb	GOA	⋯	0.760 868 52	0.254 121 56	1.453 076 18	0.020 632 98	49.564 869 37	0.396 824 35	1.878 700 74	⋯	⋯	8.200 835 772 1 · 10⁻⁴
DDMb	ROA	⋯	0.760 400 00	0.139 800 00	1.452 700 00	0.020 800 00	56.190 500 00	0.515 300 00	1.886 300 00	⋯	⋯	9.153 351 6739 · 10⁻⁴
DDMb	AWaOA	1	0.760 841 18	0.080 569 08	1.458 633 15	0.036 205 14	52.196 121 33	0.229 292 22	1.496 992 89	⋯	⋯	7.775 282 305 7 · 10⁻⁴
DDMb	CWaOA	2	0.760 546 96	0.602 923 18	1.980 422 54	0.019 717 07	52.627 193 80	0.092 071 02	1.456 505 81	⋯	⋯	7.575 840 849 3 · 10⁻⁴
DDMb	SAWaOA	3	0.760 563 92	0.195 696 07	1.442 268 84	0.020 889 13	51.607 313 10	0.393 288 88	1.869 520 15	⋯	⋯	7.646 977 930 8 · 10⁻⁴
DDMb	WaOA	4	0.760 550 84	0.360 956 24	1.991 998 01	0.019 632 25	52.595 728 64	0.342 000 88	1.457 468 11	⋯	⋯	7.576 014 864 2 · 10⁻⁴
TDM	BFO	⋯	0.760 881 12	0.819 703 51	1.981 798 83	0.036 727 39	55.850 927 47	0.210 925 05	1.445 813 50	0.028 724 73	1.898 681 62	7.619 616 916 0 · 10⁻⁴
TDM	GOA	⋯	0.761 616 48	0.821 140 12	1.760 561 56	0.035 887 82	52.897 997 68	0.029 217 16	1.322 534 24	0.339 728 61	1.654 388 06	1.234 593 159 2 · 10⁻³
TDM	ROA	⋯	0.761 000 00	0.826 700 00	1.985 300 00	0.036 300 00	59.021 600 00	0.213 400 00	1.448 100 00	0.254 400 00	1.998 000 00	8.797 168 933 2 · 10⁻⁴
TDM	AWaOA	1	0.761 149 61	1.481 668 63	1.822 460 80	0.037 826 45	50.952 673 14	0.041 527 12	1.327 292 47	0.007 423 91	1.833 401 50	0.102 798 482 3 · 10⁻³
TDM	CWaOA	2	0.760 766 46	1.440 285 52	1.996 675 31	0.038 027 53	59.517 273 13	0.081 112 85	1.368 470 84	0.663 862 73	1.949 161 96	7.344 414 875 1 · 10⁻⁴
TDM	SAWaOA	3	0.760 772 48	1.814 777 38	1.993 304 59	0.037 651 00	58.760 298 75	0.111 537 86	1.393 209 98	0.005 026 51	1.996 954 57	7.366 785 087 3 · 10⁻⁴
TDM	WaOA	4	0.760 820 95	1.476 449 79	1.931 136 65	0.038 386 05	59.332 041 31	0.058 862 95	1.344 844 96	0.631 887 44	1.988 294 69	7.368 722 511 8 · 10⁻⁴

Model	Algorithm	Method	I_pv (a)	I₀₁ (a)	n₁	R_S (Ω)	R_P (Ω)	I₀₂ (a)	n₂	I₀₃ (µA)	n₃	RMSE
SDM	BFO	⋯	0.760 795 54	0.310 623 47	1.477 249 53	0.036 546 77	52.804 896 46	⋯	⋯	⋯	⋯	7.730 195 3501 · 10⁻⁴
SDM	GOA	⋯	0.761 109 35	0.309 898 04	1.477 100 14	0.036 461 69	48.825 954 40	⋯	⋯	⋯	⋯	8.069 368 879 9 · 10⁻⁴
SDM	ROA	⋯	0.760 700 00	0.302 599 99	1.474 600 00	0.036 800 00	54.491 100 00	⋯	⋯	⋯	⋯	7.887 712 627 3 · 10⁻⁴
SDM	AWaOA	1	0.760 787 84	0.310 620 30	1.477 247 03	0.036 547 86	52.888 623 27	⋯	⋯	⋯	⋯	7.730 063 830 8 · 10⁻⁴
SDM	CWaOA	2	0.760 787 98	0.310 682 30	1.477 267 05	0.036 546 96	52.889 267 70	⋯	⋯	⋯	⋯	7.730 062 693 7 · 10⁻⁴
SDM	SAWaOA	3	0.760 787 95	0.310 684 51	1.477 267 76	0.036 546 94	52.889 826 40	⋯	⋯	⋯	⋯	7.730 062 690 1 · 10⁻⁴
SDM	WaOA	4	0.760 787 96	0.310 684 04	1.477 267 61	0.036 546 95	52.889 696 19	⋯	⋯	⋯	⋯	7.730 062 690 2 · 10⁻⁴
DDMa	BFO	⋯	0.760 832 27	0.128 549 65	1.407 401 82	0.037 376 72	54.000 000 00	0.800 000 00	1.839 408 20	⋯	⋯	7.498 286 041 3 · 10⁻⁴
DDMa	GOA	⋯	0.760 780 40	0.098 531 12	1.402 962 51	0.037 251 76	52.575 514 38	0.315 202 86	1.607 081 98	⋯	⋯	7.718 353 491 0 · 10⁻⁴
DDMa	ROA	⋯	0.760 700 00	0.126 700 00	1.407 900 00	0.037 300 00	53.352 400 00	0.615 399 99	1.783 000 00	⋯	⋯	7.650 683 805 1 · 10⁻⁴
DDMa	AWaOA	1	0.760 860 38	0.116 236 55	1.400 286 27	0.037 434 36	54.000 000 00	0.800 000 00	1.820 315 30	⋯	⋯	7.483 440 787 1 · 10⁻⁴
DDMa	CWaOA	2	0.760 858 38	0.124 205 34	1.405 018 38	0.037 388 88	53.999 945 94	0.799 959 40	1.832 092 75	⋯	⋯	7.483 709 998 3 · 10⁻⁴
DDMa	SAWaOA	3	0.760 871 55	0.092 972 94	1.385 085 34	0.037 544 50	54.000 000 00	0.798 724 94	1.784 359 26	⋯	⋯	7.481 111 488 8 · 10⁻⁴
DDMa	WaOA	4	0.760 861 96	0.114 729 58	1.399 456 09	0.037 436 79	54.000 000 00	0.799 998 06	1.817 280 17	⋯	⋯	7.482 161 974 4 · 10⁻⁴
DDMb	BFO	⋯	0.760 500 46	0.421 285 96	1.992 027 76	0.019 683 80	54.172 233 99	0.301 079 85	1.460 248 47	⋯	⋯	7.835 723 845 6 · 10⁻⁴
DDMb	GOA	⋯	0.760 868 52	0.254 121 56	1.453 076 18	0.020 632 98	49.564 869 37	0.396 824 35	1.878 700 74	⋯	⋯	8.200 835 772 1 · 10⁻⁴
DDMb	ROA	⋯	0.760 400 00	0.139 800 00	1.452 700 00	0.020 800 00	56.190 500 00	0.515 300 00	1.886 300 00	⋯	⋯	9.153 351 6739 · 10⁻⁴
DDMb	AWaOA	1	0.760 841 18	0.080 569 08	1.458 633 15	0.036 205 14	52.196 121 33	0.229 292 22	1.496 992 89	⋯	⋯	7.775 282 305 7 · 10⁻⁴
DDMb	CWaOA	2	0.760 546 96	0.602 923 18	1.980 422 54	0.019 717 07	52.627 193 80	0.092 071 02	1.456 505 81	⋯	⋯	7.575 840 849 3 · 10⁻⁴
DDMb	SAWaOA	3	0.760 563 92	0.195 696 07	1.442 268 84	0.020 889 13	51.607 313 10	0.393 288 88	1.869 520 15	⋯	⋯	7.646 977 930 8 · 10⁻⁴
DDMb	WaOA	4	0.760 550 84	0.360 956 24	1.991 998 01	0.019 632 25	52.595 728 64	0.342 000 88	1.457 468 11	⋯	⋯	7.576 014 864 2 · 10⁻⁴
TDM	BFO	⋯	0.760 881 12	0.819 703 51	1.981 798 83	0.036 727 39	55.850 927 47	0.210 925 05	1.445 813 50	0.028 724 73	1.898 681 62	7.619 616 916 0 · 10⁻⁴
TDM	GOA	⋯	0.761 616 48	0.821 140 12	1.760 561 56	0.035 887 82	52.897 997 68	0.029 217 16	1.322 534 24	0.339 728 61	1.654 388 06	1.234 593 159 2 · 10⁻³
TDM	ROA	⋯	0.761 000 00	0.826 700 00	1.985 300 00	0.036 300 00	59.021 600 00	0.213 400 00	1.448 100 00	0.254 400 00	1.998 000 00	8.797 168 933 2 · 10⁻⁴
TDM	AWaOA	1	0.761 149 61	1.481 668 63	1.822 460 80	0.037 826 45	50.952 673 14	0.041 527 12	1.327 292 47	0.007 423 91	1.833 401 50	0.102 798 482 3 · 10⁻³
TDM	CWaOA	2	0.760 766 46	1.440 285 52	1.996 675 31	0.038 027 53	59.517 273 13	0.081 112 85	1.368 470 84	0.663 862 73	1.949 161 96	7.344 414 875 1 · 10⁻⁴
TDM	SAWaOA	3	0.760 772 48	1.814 777 38	1.993 304 59	0.037 651 00	58.760 298 75	0.111 537 86	1.393 209 98	0.005 026 51	1.996 954 57	7.366 785 087 3 · 10⁻⁴
TDM	WaOA	4	0.760 820 95	1.476 449 79	1.931 136 65	0.038 386 05	59.332 041 31	0.058 862 95	1.344 844 96	0.631 887 44	1.988 294 69	7.368 722 511 8 · 10⁻⁴

^a

Iterative approach.¹⁷

^b

Analytical approach.¹³

TABLE V.

The estimated values of parameters for all three models of solar cells—MSX-60 solar module.

Model	Algorithm	Method	I_PV (A)	I₀₁ (µA)	n₁	R_S (Ω)	R_P (Ω)	I₀₂ (µA)	n₂	I₀₃ (µA)	n₃	RMSE
SDM	BFO	⋯	3.810 826 08	0.117 399 43	1.319 079 65	0.228 704 62	890.000 000 00	⋯	⋯	⋯	⋯	0.011 651 129 2
SDM	GOA	⋯	3.810 755 33	0.118 923 83	1.320 053 92	0.228 189 52	890.000 000 00	⋯	⋯	⋯	⋯	0.011 650 632 3
SDM	ROA	⋯	3.809 900 00	0.142 800 00	1.334 200 00	0.223 500 00	895.166 900 00	⋯	⋯	⋯	⋯	0.011 917 822 5
SDM	AWaOA	1	3.810 973 98	0.120 573 51	1.321 096 77	0.227 865 15	894.286 718 32	⋯	⋯	⋯	⋯	0.011 651 382 8
SDM	CWaOA	2	3.810 982 86	0.119 845 76	1.320 639 85	0.228 042 71	890.000 000 00	⋯	⋯	⋯	⋯	0.011 650 149 4
SDM	SAWaOA	3	3.810 973 38	0.119 741 42	1.320 574 11	0.228 070 11	890.000 000 00	⋯	⋯	⋯	⋯	0.011 650 151 5
SDM	WaOA	4	3.810 985 29	0.119 632 77	1.320 505 84	0.228 112 78	890.000 000 00	⋯	⋯	⋯	⋯	0.011 650 163 9
DDMa	BFO	⋯	3.809 996 71	0.100 827 50	1.308 677 21	0.229 711 91	895.160 760 73	0.196 535 55	1.818 720 02	⋯	⋯	0.011 655 271 7
DDMa	GOA	⋯	3.810 590 93	0.061 857 92	1.277 470 82	0.235 509 79	890.000 000 00	0.324 971 66	1.669 094 48	⋯	⋯	0.011 627 026 1
DDMa	ROA	⋯	3.818 400 00	0.160 700 00	1.344 000 00	0.216 200 00	900.000 000 00	0.488 800 00	1.979 000 00	⋯	⋯	0.012 421 769 0
DDMa	AWaOA	1	3.811 439 89	0.112 033 89	1.316 015 89	0.228 458 16	891.979 261 26	0.286 939 47	1.998 168 73	⋯	⋯	0.011 636 643 5
DDMa	CWaOA	2	3.811 450 54	0.055 918 64	1.273 337 73	0.232 232 62	899.731 349 21	0.388 007 76	1.638 250 54	⋯	⋯	0.011 584 271 3
DDMa	SAWaOA	3	3.811 415 38	0.053 046 55	1.270 865 91	0.232 160 27	892.875 466 29	0.341 282 41	1.608 530 64	⋯	⋯	0.011 585 313 3
DDMa	WaOA	4	3.811 353 40	0.062 580 67	1.278 515 69	0.232 598 59	896.469 952 91	0.605 128 61	1.742 003 75	⋯	⋯	0.011 576 405 8
DDMb	BFO	⋯	3.812 156 17	0.230 295 22	1.336 931 43	0.110 903 02	900.000 000 00	0.180 000 00	1.925 608 39	⋯	⋯	0.011 878 345 0
DDMb	GOA	⋯	3.806 905 22	0.092 371 37	1.322 601 30	0.115 946 24	890.000 167 61	0.244 397 41	1.750 414 68	⋯	⋯	0.012 041 792 5
DDMb	ROA	⋯	3.812 700 00	0.010 400 00	1.490 100 00	0.153 000 00	892.479 300 00	0.201 200 00	1.300 000 00	⋯	⋯	0.012 261 595 0
DDMb	AWaOA	1	3.810 348 34	0.136 936 75	1.316 499 39	0.115 315 16	899.368 972 07	0.185 311 30	2.000 000 00	⋯	⋯	0.011 653 480 2
DDMb	CWaOA	2	3.810 572 02	0.082 526 69	1.318 221 46	0.115 031 84	890.000 000 00	0.246 568 35	2.000 000 00	⋯	⋯	0.011 648 380 3
DDMb	SAWaOA	3	3.810 779 31	0.052 628 22	1.316 927 27	0.115 503 42	890.190 697 87	0.270 654 27	1.950 686 68	⋯	⋯	0.011 651 059 6
DDMb	WaOA	4	3.810 686 78	0.103 816 71	1.319 140 86	0.114 843 12	891.534 107 08	0.229 014 68	1.999 516 22	⋯	⋯	0.011 649 267 8
TDM	BFO	⋯	3.813 001 35	0.076 738 78	1.291 429 64	0.229 828 63	899.086 189 45	0.310 206 73	2.000 000 00	0.910 047 56	1.870 556 77	0.011 656 228 1
TDM	GOA	⋯	3.823 718 21	0.039 983 78	1.287 246 20	0.196 643 60	883.511 875 76	0.478 726 06	1.605 040 61	0.227 583 42	1.478 433 93	0.014 552 471 2
TDM	ROA	⋯	3.813 200 00	0.101 800 00	1.310 400 00	0.224 700 00	894.259 300 00	0.547 300 00	1.984 800 00	0.182 700 00	1.794 100 00	0.011 786 009 1
TDM	AWaOA	1	3.811 870 95	0.068 260 86	1.282 756 73	0.233 188 09	885.298 505 14	0.222 425 74	1.827 898 77	0.723 091 97	1.875 872 67	0.011 579 678 1
TDM	CWaOA	2	3.812 768 11	0.017 686 01	1.198 985 26	0.244 855 55	891.808 473 35	0.157 888 06	1.783 032 88	2.571 728 73	1.871 111 00	0.011 455 665 5
TDM	SAWaOA	3	3.812 571 62	0.020 257 87	1.207 489 05	0.243 346 24	888.226 145 94	0.212 841 35	1.764 718 94	1.980 896 17	1.843 096 64	0.011 465 719 1
TDM	WaOA	4	3.811 943 18	0.053 313 39	1.266 481 85	0.235 305 31	900.048 266 76	0.493 471 36	1.864 959 44	1.437 398 49	1.962 768 29	0.011 533 368 0

Model	Algorithm	Method	I_PV (A)	I₀₁ (µA)	n₁	R_S (Ω)	R_P (Ω)	I₀₂ (µA)	n₂	I₀₃ (µA)	n₃	RMSE
SDM	BFO	⋯	3.810 826 08	0.117 399 43	1.319 079 65	0.228 704 62	890.000 000 00	⋯	⋯	⋯	⋯	0.011 651 129 2
SDM	GOA	⋯	3.810 755 33	0.118 923 83	1.320 053 92	0.228 189 52	890.000 000 00	⋯	⋯	⋯	⋯	0.011 650 632 3
SDM	ROA	⋯	3.809 900 00	0.142 800 00	1.334 200 00	0.223 500 00	895.166 900 00	⋯	⋯	⋯	⋯	0.011 917 822 5
SDM	AWaOA	1	3.810 973 98	0.120 573 51	1.321 096 77	0.227 865 15	894.286 718 32	⋯	⋯	⋯	⋯	0.011 651 382 8
SDM	CWaOA	2	3.810 982 86	0.119 845 76	1.320 639 85	0.228 042 71	890.000 000 00	⋯	⋯	⋯	⋯	0.011 650 149 4
SDM	SAWaOA	3	3.810 973 38	0.119 741 42	1.320 574 11	0.228 070 11	890.000 000 00	⋯	⋯	⋯	⋯	0.011 650 151 5
SDM	WaOA	4	3.810 985 29	0.119 632 77	1.320 505 84	0.228 112 78	890.000 000 00	⋯	⋯	⋯	⋯	0.011 650 163 9
DDMa	BFO	⋯	3.809 996 71	0.100 827 50	1.308 677 21	0.229 711 91	895.160 760 73	0.196 535 55	1.818 720 02	⋯	⋯	0.011 655 271 7
DDMa	GOA	⋯	3.810 590 93	0.061 857 92	1.277 470 82	0.235 509 79	890.000 000 00	0.324 971 66	1.669 094 48	⋯	⋯	0.011 627 026 1
DDMa	ROA	⋯	3.818 400 00	0.160 700 00	1.344 000 00	0.216 200 00	900.000 000 00	0.488 800 00	1.979 000 00	⋯	⋯	0.012 421 769 0
DDMa	AWaOA	1	3.811 439 89	0.112 033 89	1.316 015 89	0.228 458 16	891.979 261 26	0.286 939 47	1.998 168 73	⋯	⋯	0.011 636 643 5
DDMa	CWaOA	2	3.811 450 54	0.055 918 64	1.273 337 73	0.232 232 62	899.731 349 21	0.388 007 76	1.638 250 54	⋯	⋯	0.011 584 271 3
DDMa	SAWaOA	3	3.811 415 38	0.053 046 55	1.270 865 91	0.232 160 27	892.875 466 29	0.341 282 41	1.608 530 64	⋯	⋯	0.011 585 313 3
DDMa	WaOA	4	3.811 353 40	0.062 580 67	1.278 515 69	0.232 598 59	896.469 952 91	0.605 128 61	1.742 003 75	⋯	⋯	0.011 576 405 8
DDMb	BFO	⋯	3.812 156 17	0.230 295 22	1.336 931 43	0.110 903 02	900.000 000 00	0.180 000 00	1.925 608 39	⋯	⋯	0.011 878 345 0
DDMb	GOA	⋯	3.806 905 22	0.092 371 37	1.322 601 30	0.115 946 24	890.000 167 61	0.244 397 41	1.750 414 68	⋯	⋯	0.012 041 792 5
DDMb	ROA	⋯	3.812 700 00	0.010 400 00	1.490 100 00	0.153 000 00	892.479 300 00	0.201 200 00	1.300 000 00	⋯	⋯	0.012 261 595 0
DDMb	AWaOA	1	3.810 348 34	0.136 936 75	1.316 499 39	0.115 315 16	899.368 972 07	0.185 311 30	2.000 000 00	⋯	⋯	0.011 653 480 2
DDMb	CWaOA	2	3.810 572 02	0.082 526 69	1.318 221 46	0.115 031 84	890.000 000 00	0.246 568 35	2.000 000 00	⋯	⋯	0.011 648 380 3
DDMb	SAWaOA	3	3.810 779 31	0.052 628 22	1.316 927 27	0.115 503 42	890.190 697 87	0.270 654 27	1.950 686 68	⋯	⋯	0.011 651 059 6
DDMb	WaOA	4	3.810 686 78	0.103 816 71	1.319 140 86	0.114 843 12	891.534 107 08	0.229 014 68	1.999 516 22	⋯	⋯	0.011 649 267 8
TDM	BFO	⋯	3.813 001 35	0.076 738 78	1.291 429 64	0.229 828 63	899.086 189 45	0.310 206 73	2.000 000 00	0.910 047 56	1.870 556 77	0.011 656 228 1
TDM	GOA	⋯	3.823 718 21	0.039 983 78	1.287 246 20	0.196 643 60	883.511 875 76	0.478 726 06	1.605 040 61	0.227 583 42	1.478 433 93	0.014 552 471 2
TDM	ROA	⋯	3.813 200 00	0.101 800 00	1.310 400 00	0.224 700 00	894.259 300 00	0.547 300 00	1.984 800 00	0.182 700 00	1.794 100 00	0.011 786 009 1
TDM	AWaOA	1	3.811 870 95	0.068 260 86	1.282 756 73	0.233 188 09	885.298 505 14	0.222 425 74	1.827 898 77	0.723 091 97	1.875 872 67	0.011 579 678 1
TDM	CWaOA	2	3.812 768 11	0.017 686 01	1.198 985 26	0.244 855 55	891.808 473 35	0.157 888 06	1.783 032 88	2.571 728 73	1.871 111 00	0.011 455 665 5
TDM	SAWaOA	3	3.812 571 62	0.020 257 87	1.207 489 05	0.243 346 24	888.226 145 94	0.212 841 35	1.764 718 94	1.980 896 17	1.843 096 64	0.011 465 719 1
TDM	WaOA	4	3.811 943 18	0.053 313 39	1.266 481 85	0.235 305 31	900.048 266 76	0.493 471 36	1.864 959 44	1.437 398 49	1.962 768 29	0.011 533 368 0

^a

Iterative approach.¹⁷

^b

Analytical approach.¹³

Based on the presented results, following conclusions can be derived:

Generally, classic WaOA produces better results than BFO, GOA, and ROA.
Results indicate that hybridization of each algorithm results in performance improvement.
Analysis shows that the most efficient hybridization is CWaOA, while AWaOA is the least efficient hybrid.
If you compare the iterative and analytical approaches in solar cell modeling, the iterative approach has greater accuracy.

A graphical comparison of the current–voltage and power–voltage characteristics of the observed solar cell and solar module, as well as the corresponding error values for each model of solar cell and module, is given in Figs. 8–15. Figures 8–11 show the results for the RTC France solar cell, and Figs. 12–15 show the results for the MSX 60 solar module. In Figs. 8–15 shown, method 1 corresponds to the parameters determined using AWaOA, method 2 corresponds to those determined using CWaOA, method 3 corresponds to those determined using SAWaOA, and method 4 corresponds to those determined using WaOA. The graphical results fully correspond to the results shown in Tables IV and V with parameter and RMSE values. This is confirmed by the graphs that show errors of the algorithms in determining current and power values. A common characteristic of all methods is that errors reach their maximum values at very low and at high current values. Likewise, on certain zoomed-in parts, it is clearly indicated that the DDM iterative approach enables better results compared to the analytical DDM approach.

FIG. 8.

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SDM of the RTC France solar cell. (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

FIG. 9.

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Iterative DDM of the RTC France solar cell. (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

FIG. 10.

DDM of the RTC France solar cell proposed by Lun et al.13 (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

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DDM of the RTC France solar cell proposed by Lun et al.¹³ (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

FIG. 11.

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TDM of the RTC France solar cell. (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

FIG. 12.

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SDM of the MSX60 solar module. (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

FIG. 13.

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Iterative DDM of the MSX60 solar module. (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

FIG. 14.

DDM of the MSX60 solar module proposed by Lun et al.13 (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

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DDM of the MSX60 solar module proposed by Lun et al.¹³ (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

FIG. 15.

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TDM of the MSX60 solar module. (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics.

A statistical comparison of the results, obtained for WaOa and its hybrid variants, in terms of mean, best, median, STD, and worst, is given in Tables VI and VII, for the RTC France solar cell and MSX60 solar module, respectively. Based on the presented results, the following conclusion can be derived:

The best results achieved by all algorithms in SDM are approximately the same, and the difference is extremely small.
In DDM and TDM, the best results differ and it can be seen that hybridization improves the accuracy of the solution.
The same results apply to the observed solar cell and solar module.

TABLE VI.

Statistical comparison of the results in terms of mean, best, median, STD, and worst is for the RTC France solar cell.

Model	Algorithm	Best	Worst	Mean	Median	STD
SDM	WaOA	7.730 062 690 2 · 10⁻⁴	7.731 252 114 4 · 10⁻⁴	7.730 203 925 8 · 10⁻⁴	7.730 063 619 8 · 10⁻⁴	3.053 812 351 5 · 10⁻⁸
SDM	CWaOA	7.730 062 693 7 · 10⁻⁴	7.779 280 070 4 · 10⁻⁴	7.733 257 565 2 · 10⁻⁴	7.730 070 479 9 · 10⁻⁴	1.137 498 673 7 · 10⁻⁶
SDM	AWaOA	7.730 063 830 8 · 10⁻⁴	7.906 620 335 0 · 10⁻⁴	7.741 673 024 9 · 10⁻⁴	7.730 558 305 0 · 10⁻⁴	3.269 881 726 2 · 10⁻⁶
SDM	SAWaOA	7.730 062 690 1 · 10⁻⁴	7.734 754 216 9 · 10⁻⁴	7.730 228 453 5 · 10⁻⁴	7.730 062 906 1 · 10⁻⁴	8.551 791 466 2 · 10⁻⁸
DDMa	WaOA	7.482 161 974 4 · 10⁻⁴	7.724 567 274 1 · 10⁻⁴	7.548 627 895 0 · 10⁻⁴	7.528 359 618 7 · 10⁻⁴	6.848 711 186 2 · 10⁻⁶
DDMa	CWaOA	7.483 709 998 3 · 10⁻⁴	7.847 456 188 8 · 10⁻⁴	7.573 346 718 3 · 10⁻⁴	7.535 165 974 3 · 10⁻⁴	9.832 241 888 1 · 10⁻⁶
DDMa	AWaOA	7.483 440 787 1 · 10⁻⁴	9.162 289 680 1 · 10⁻⁴	7.619 105 043 2 · 10⁻⁴	7.553 635 429 9 · 10⁻⁴	9.910 543 713 8 · 10⁻⁶
DDMa	SAWaOA	7.481 111 488 8 · 10⁻⁴	7.750 058 279 1 · 10⁻⁴	7.594 598 445 2 · 10⁻⁴	7.589 661 167 6 · 10⁻⁴	7.729 925 192 6 · 10⁻⁶
DDMb	WaOA	7.576 014 864 2 · 10⁻⁴	1.100 994 586 1 · 10⁻³	8.159 637 579 4 · 10⁻⁴	7.964 366 332 8 · 10⁻⁴	6.741 282 366 9 · 10⁻⁵
DDMb	CWaOA	7.575 840 849 3 · 10⁻⁴	1.134 162 057 2 · 10⁻³	8.157 350 718 8 · 10⁻⁴	7.784 559 450 8 · 10⁻⁴	8.973 622 901 1 · 10⁻⁵
DDMb	AWaOA	7.775 282 305 7 · 10⁻⁴	2.153 326 309 7 · 10⁻³	1.175 112 441 0 · 10⁻³	1.087 369 819 4 · 10⁻³	3.066 424 952 1 · 10⁻⁴
DDMb	SAWaOA	7.646 977 930 8 · 10⁻⁴	1.002 122 106 3 · 10⁻³	8.506 956 028 6 · 10⁻⁴	8.619 990 760 3 · 10⁻⁴	6.923 984 278 5 · 10⁻⁵
TDM	WaOA	7.368 722 511 8 · 10⁻⁴	3.317 515 344 8 · 10⁻³	1.222 554 732 2 · 10⁻³	7.816 522 148 5 · 10⁻⁴	7.992 967 083 2 · 10⁻⁴
TDM	CWaOA	7.344 414 875 1 · 10⁻⁴	3.317 511 373 4 · 10⁻³	1.103 556 873 0 · 10⁻³	7.820 647 565 1 · 10⁻⁴	7.286 074 551 8 · 10⁻⁴
TDM	AWaOA	1.027 984 823 4 · 10⁻³	5.195 656 906 1 · 10⁻³	2.965 400 319 0 · 10⁻³	3.425 108 345 6 · 10⁻³	1.044 615 534 4 · 10⁻³
TDM	SAWaOA	7.366 785 087 3 · 10⁻⁴	3.317 511 368 1 · 10⁻³	1.215 850 688 1 · 10⁻³	7.685 444 061 7 · 10⁻⁴	8.714 512 623 3 · 10⁻⁴

Model	Algorithm	Best	Worst	Mean	Median	STD
SDM	WaOA	7.730 062 690 2 · 10⁻⁴	7.731 252 114 4 · 10⁻⁴	7.730 203 925 8 · 10⁻⁴	7.730 063 619 8 · 10⁻⁴	3.053 812 351 5 · 10⁻⁸
SDM	CWaOA	7.730 062 693 7 · 10⁻⁴	7.779 280 070 4 · 10⁻⁴	7.733 257 565 2 · 10⁻⁴	7.730 070 479 9 · 10⁻⁴	1.137 498 673 7 · 10⁻⁶
SDM	AWaOA	7.730 063 830 8 · 10⁻⁴	7.906 620 335 0 · 10⁻⁴	7.741 673 024 9 · 10⁻⁴	7.730 558 305 0 · 10⁻⁴	3.269 881 726 2 · 10⁻⁶
SDM	SAWaOA	7.730 062 690 1 · 10⁻⁴	7.734 754 216 9 · 10⁻⁴	7.730 228 453 5 · 10⁻⁴	7.730 062 906 1 · 10⁻⁴	8.551 791 466 2 · 10⁻⁸
DDMa	WaOA	7.482 161 974 4 · 10⁻⁴	7.724 567 274 1 · 10⁻⁴	7.548 627 895 0 · 10⁻⁴	7.528 359 618 7 · 10⁻⁴	6.848 711 186 2 · 10⁻⁶
DDMa	CWaOA	7.483 709 998 3 · 10⁻⁴	7.847 456 188 8 · 10⁻⁴	7.573 346 718 3 · 10⁻⁴	7.535 165 974 3 · 10⁻⁴	9.832 241 888 1 · 10⁻⁶
DDMa	AWaOA	7.483 440 787 1 · 10⁻⁴	9.162 289 680 1 · 10⁻⁴	7.619 105 043 2 · 10⁻⁴	7.553 635 429 9 · 10⁻⁴	9.910 543 713 8 · 10⁻⁶
DDMa	SAWaOA	7.481 111 488 8 · 10⁻⁴	7.750 058 279 1 · 10⁻⁴	7.594 598 445 2 · 10⁻⁴	7.589 661 167 6 · 10⁻⁴	7.729 925 192 6 · 10⁻⁶
DDMb	WaOA	7.576 014 864 2 · 10⁻⁴	1.100 994 586 1 · 10⁻³	8.159 637 579 4 · 10⁻⁴	7.964 366 332 8 · 10⁻⁴	6.741 282 366 9 · 10⁻⁵
DDMb	CWaOA	7.575 840 849 3 · 10⁻⁴	1.134 162 057 2 · 10⁻³	8.157 350 718 8 · 10⁻⁴	7.784 559 450 8 · 10⁻⁴	8.973 622 901 1 · 10⁻⁵
DDMb	AWaOA	7.775 282 305 7 · 10⁻⁴	2.153 326 309 7 · 10⁻³	1.175 112 441 0 · 10⁻³	1.087 369 819 4 · 10⁻³	3.066 424 952 1 · 10⁻⁴
DDMb	SAWaOA	7.646 977 930 8 · 10⁻⁴	1.002 122 106 3 · 10⁻³	8.506 956 028 6 · 10⁻⁴	8.619 990 760 3 · 10⁻⁴	6.923 984 278 5 · 10⁻⁵
TDM	WaOA	7.368 722 511 8 · 10⁻⁴	3.317 515 344 8 · 10⁻³	1.222 554 732 2 · 10⁻³	7.816 522 148 5 · 10⁻⁴	7.992 967 083 2 · 10⁻⁴
TDM	CWaOA	7.344 414 875 1 · 10⁻⁴	3.317 511 373 4 · 10⁻³	1.103 556 873 0 · 10⁻³	7.820 647 565 1 · 10⁻⁴	7.286 074 551 8 · 10⁻⁴
TDM	AWaOA	1.027 984 823 4 · 10⁻³	5.195 656 906 1 · 10⁻³	2.965 400 319 0 · 10⁻³	3.425 108 345 6 · 10⁻³	1.044 615 534 4 · 10⁻³
TDM	SAWaOA	7.366 785 087 3 · 10⁻⁴	3.317 511 368 1 · 10⁻³	1.215 850 688 1 · 10⁻³	7.685 444 061 7 · 10⁻⁴	8.714 512 623 3 · 10⁻⁴

^a

Iterative DDM—Calasan.¹⁷

^b

Analytical DDM approach—Lun et al.¹³

TABLE VII.

Statistical comparison of the results in terms of mean, best, median, STD, and worst is for the MSX-60 solar module.

Model	Algorithm	Best	Worst	Mean	Median	STD
SDM	WaOA	0.011 650 163 9	0.011 877 258 7	0.011 692 333 2	0.011 661 490 4	6.556 651 182 5 · 10⁻⁵
SDM	CWaOA	0.011 650 149 4	0.012 027 684 1	0.011 767 859 6	0.011 678 542 4	1.410 392 147 8 · 10⁻⁴
SDM	AWaOA	0.011 651 382 8	0.012 233 109 0	0.011 825 698 1	0.011 787 354 0	1.566 434 034 1 · 10⁻⁴
SDM	SAWaOA	0.011 650 151 5	0.012 026 551 2	0.011 706 750 0	0.011 662 094 3	8.491 062 130 0 · 10⁻⁵
DDMa	WaOA	0.011 576 405 8	0.012 139 729 6	0.011 657 676 9	0.011 630 454 3	1.040 269 125 1 · 10⁻⁴
DDMa	CWaOA	0.011 584 271 3	0.012 961 646 0	0.011 759 640 1	0.011 652 914 1	2.819 459 671 6 · 10⁻⁴
DDMa	AWaOA	0.011 636 643 5	0.013 184 362 1	0.012 092 941 9	0.012 004 187 3	4.482 871 606 8 · 10⁻⁴
DDMa	SAWaOA	0.011 585 313 3	0.012 692 722 9	0.011 727 554 8	0.011 654 805 5	2.149 159 203 3 · 10⁻⁴
DDMb	WaOA	0.011 649 267 8	0.013 141 816 1	0.011 853 524 5	0.011 702 460 3	3.599 933 302 1 · 10⁻⁴
DDMb	CWaOA	0.011 648 380 3	0.014 012 680 1	0.012 108 413 0	0.011 932 345 4	5.257 616 334 7 · 10⁻⁴
DDMb	AWaOA	0.011 653 480 2	0.013 593 750 8	0.012 130 153 7	0.011 990 795 2	4.941 893 544 8 · 10⁻⁴
DDMb	SAWaOA	0.011 651 059 6	0.013 230 030 0	0.012 166 524 9	0.012 142 209 6	4.540 619 883 1 · 10⁻⁴
TDM	WaOA	0.011 533 368 0	0.015 038 193 0	0.012 137 966 0	0.011 825 659 0	8.204 308 100 5 · 10⁻⁴
TDM	CWaOA	0.011 455 665 5	0.014 224 109 9	0.011 919 068 7	0.011 694 504 5	5.913 037 278 5 · 10⁻⁴
TDM	AWaOA	0.011 579 678 1	0.015 287 164 2	0.013 006 582 3	0.012 927 760 3	9.326 587 293 7 · 10⁻⁴
TDM	SAWaOA	0.011 465 719 1	0.012 835 244 3	0.011 737 735 5	0.011 646 806 4	2.801 014 814 8 · 10⁻⁴

Model	Algorithm	Best	Worst	Mean	Median	STD
SDM	WaOA	0.011 650 163 9	0.011 877 258 7	0.011 692 333 2	0.011 661 490 4	6.556 651 182 5 · 10⁻⁵
SDM	CWaOA	0.011 650 149 4	0.012 027 684 1	0.011 767 859 6	0.011 678 542 4	1.410 392 147 8 · 10⁻⁴
SDM	AWaOA	0.011 651 382 8	0.012 233 109 0	0.011 825 698 1	0.011 787 354 0	1.566 434 034 1 · 10⁻⁴
SDM	SAWaOA	0.011 650 151 5	0.012 026 551 2	0.011 706 750 0	0.011 662 094 3	8.491 062 130 0 · 10⁻⁵
DDMa	WaOA	0.011 576 405 8	0.012 139 729 6	0.011 657 676 9	0.011 630 454 3	1.040 269 125 1 · 10⁻⁴
DDMa	CWaOA	0.011 584 271 3	0.012 961 646 0	0.011 759 640 1	0.011 652 914 1	2.819 459 671 6 · 10⁻⁴
DDMa	AWaOA	0.011 636 643 5	0.013 184 362 1	0.012 092 941 9	0.012 004 187 3	4.482 871 606 8 · 10⁻⁴
DDMa	SAWaOA	0.011 585 313 3	0.012 692 722 9	0.011 727 554 8	0.011 654 805 5	2.149 159 203 3 · 10⁻⁴
DDMb	WaOA	0.011 649 267 8	0.013 141 816 1	0.011 853 524 5	0.011 702 460 3	3.599 933 302 1 · 10⁻⁴
DDMb	CWaOA	0.011 648 380 3	0.014 012 680 1	0.012 108 413 0	0.011 932 345 4	5.257 616 334 7 · 10⁻⁴
DDMb	AWaOA	0.011 653 480 2	0.013 593 750 8	0.012 130 153 7	0.011 990 795 2	4.941 893 544 8 · 10⁻⁴
DDMb	SAWaOA	0.011 651 059 6	0.013 230 030 0	0.012 166 524 9	0.012 142 209 6	4.540 619 883 1 · 10⁻⁴
TDM	WaOA	0.011 533 368 0	0.015 038 193 0	0.012 137 966 0	0.011 825 659 0	8.204 308 100 5 · 10⁻⁴
TDM	CWaOA	0.011 455 665 5	0.014 224 109 9	0.011 919 068 7	0.011 694 504 5	5.913 037 278 5 · 10⁻⁴
TDM	AWaOA	0.011 579 678 1	0.015 287 164 2	0.013 006 582 3	0.012 927 760 3	9.326 587 293 7 · 10⁻⁴
TDM	SAWaOA	0.011 465 719 1	0.012 835 244 3	0.011 737 735 5	0.011 646 806 4	2.801 014 814 8 · 10⁻⁴

^a

Iterative DDM—Calasan.¹⁷

^b

Analytical DDM approach—Lun et al.¹³

3D graphs (objective function–number of iterations–number of starts) for the SDM, DDM, and TDM of solar cells, as well as the best curves for all algorithms, are shown in the Appendix. The best convergence curves for the SDM, obtained for all observed algorithms and hybrid variants of the WaOA algorithm, are presented in Fig. 16. Based on Fig. 16, it is quite clear that the chaotic WaOA variant of the algorithm provides the best results not only from the point of view of accuracy but also from the point of view of convergence speed (see the zoomed parts in Fig. 16).

FIG. 16.

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The best convergence curves of all the observed algorithms for the RTC France solar cell and SDM.

V. EXPERIMENTAL RESULTS

Based on the research presented in Sec. IV, it has been clearly demonstrated that the best accuracy is achieved by using the chaotic variant of the WaOA algorithm. For this reason, this algorithm was utilized to estimate the parameters of the KC200GT solar panel and to verify the application of these parameters in simulating the operation of this panel under various values of insolation and temperature.

Therefore, the application of the mentioned algorithm was conducted for the estimation of panel parameters at an irradiance of 1000 W/m² and a temperature of 25 °C. Subsequently, a procedure was used for mapping parameters for different values of temperature and insolation. Table VIII presents the results of the parameter estimation for the KC200GT solar panel for the single-diode, double-diode, and triple-diode models, at an irradiance of 1000 W/m² and a temperature of 25 °C. Figure 17 displays the IV and PV characteristics for all three models of the solar panel. It also shows the difference between measured and estimated values of current and power. As can be seen, the biggest differences between the measured and estimated values are achieved for voltage values that are close to the no-load voltage. The analysis shows that the application of the triple-diode model (TDM) of solar cells results in the most precise parameters. Additionally, from the obtained results, it can be observed that the iterative method provides a better match with the measured values compared to the analytical method used (Lun et al.¹³).

TABLE VIII.

The parameter estimation results for the KC200GT solar panel using SDM, DDM, and TDM.

	SDM	DDM—iterative	DDM—analytical	TDM
I_PV(A)	8.211 923 050 6	8.211 906 775 8	8.209 341 381 5	8.209 148 563 1
I₀₁ (μA)	0.057 524 160 6	0.004 816 330 8	0.002 684 614 9	0.253 604 672 0
n₁	1.263 380 690 2	1.200 482 433 8	1.262 147 168 1	1.433 817 542 7
R_S(Ω)	0.237 124 082 6	0.237 692 842 3	0.119 146 272 3	0.247 110 827 3
R_P(Ω)	373.467 438 183	373.842 508 315	373.630 640 990	436.916 824 264
I₀₂ (μA)	⋯	0.056 612 587 3	0.192 527 728 3	0.001 017 009 3
n₂	⋯	1.279 573 814 5	2.000 000 000 0	1.069 862 586 8
I₀₃ (μA)	⋯	⋯	⋯	1.00000 · 10⁻⁸
n₃	⋯	⋯	⋯	1.000 000 000
RMSE	0.005 103 2	0.005 079 5	0.005 141 7	0.004 583 3

	SDM	DDM—iterative	DDM—analytical	TDM
I_PV(A)	8.211 923 050 6	8.211 906 775 8	8.209 341 381 5	8.209 148 563 1
I₀₁ (μA)	0.057 524 160 6	0.004 816 330 8	0.002 684 614 9	0.253 604 672 0
n₁	1.263 380 690 2	1.200 482 433 8	1.262 147 168 1	1.433 817 542 7
R_S(Ω)	0.237 124 082 6	0.237 692 842 3	0.119 146 272 3	0.247 110 827 3
R_P(Ω)	373.467 438 183	373.842 508 315	373.630 640 990	436.916 824 264
I₀₂ (μA)	⋯	0.056 612 587 3	0.192 527 728 3	0.001 017 009 3
n₂	⋯	1.279 573 814 5	2.000 000 000 0	1.069 862 586 8
I₀₃ (μA)	⋯	⋯	⋯	1.00000 · 10⁻⁸
n₃	⋯	⋯	⋯	1.000 000 000
RMSE	0.005 103 2	0.005 079 5	0.005 141 7	0.004 583 3

FIG. 17.

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SDM, analytical DDM, iterative DDM, and TDM of the KC200GT solar module. (a) I–V characteristics, (b) P–V characteristics, (c) current difference–voltage characteristics, and (d) power difference–voltage characteristics at 1000 W/m² and 25 °C.

In Fig. 18(a), the IV and PV characteristics of the KC200GT solar panel, obtained by rescaling the estimated parameters (calculated using the TDM) for different levels of insolation, are displayed. Similarly, Fig. 18(b) represents the corresponding characteristics of the KC200GT panel, which are the result of mapping for different temperatures. The procedure for solar cell parameter rescaling for different temperature and insolation levels is described in Ref. 48. In all cases (see Fig. 18), a very high match of the obtained characteristics with the measured values is observed, indicating a small error value. Therefore, it is more than clear that the proposed algorithm for parameter estimation enables obtaining results with high efficiency and accuracy.

FIG. 18.

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I–V and P–V characteristics of the KC200GT solar module for different (a) insolation and (b) temperatures.

VI. CONCLUSION

This work addresses the precise determination of parameters for three models of solar cells (SDM, DDM, and TDM), employing both the Lambert W equation and iterative Lambert W approaches. A review of the literature indicates that the most effective solutions in this field are offered by hybrid metaheuristic algorithms. However, typically, only one recommended method or hybrid variant is presented and proposed in existing studies. Diverging from this conventional approach, our study evaluated three hybrid variants of the same metaheuristic algorithm, WaOA (CWaOA, AWaOA, and SAWaOA), to identify the optimal method. Testing was conducted on a well-documented solar cell (RTC France) and a module (MSX60). The analysis revealed that the chaotic variant of WaOA (CWaOA) yielded the best results, specifically in the accurate parameter estimation for the KC200GT solar panel under an irradiance of 1000 W/m² and a temperature of 25 °C. The close alignment of our results with the measured values substantiates the efficacy of the CWaOA hybrid variant. To further validate this approach, we rescaled the obtained parameters for the TDM, which proved to be not only the most precise but also the most demanding in terms of usage, to different irradiance and temperature settings. This adjustment achieved an exceptionally high degree of correlation with the measured data, further confirming the superiority and efficiency of the proposed CWaOA hybrid metaheuristic algorithm.

Future research could explore the application of these algorithms for estimating parameters of various solar cell models across different levels of insolation and temperature.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Snežana Vujošević: Formal analysis (equal); Investigation (equal); Software (equal); Writing – original draft (equal). Martin Ćalasan: Resources (equal); Software (equal); Supervision (equal); Writing – review & editing (equal). Mihailo Micev: Software (equal); Supervision (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

APPENDIX: CONVERGENCE CURVES

In the Appendix, the convergence curves of a different algorithm are presented (Figs. 19–21).

FIG. 19.

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SDM RTC France solar cell 3D convergence curves (objective function–iteration number–number of run) and the best convergence curves for (a) WaOA, (b) cWaOA, (c) AWaOA, (d) SAWaOA, (e) BFO, and (f) ROA.

FIG. 20.

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RTC France solar cell 3D convergence curves (objective function–iteration number–number of run) and the best convergence curves determined via cWaOA for (a) iterative DDM, (b) analytical DDM, and (c) TDM.

FIG. 21.

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SDM MSX60 solar module 3D convergence curves (objective function–iteration number–number of run) and the best convergence curves for (a) WaOA, (b) cWaOA, (c) AWaOA, (d) SAWaOA, (e) BFO, and (f) ROA.

FIG. 22.

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MSX60 solar module 3D convergence curves (objective function–iteration number–number of run) and the best convergence curves determined via cWaOA for (a) iterative DDM, (b) analytical DDM, and (c) TDM.

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© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC) license (https://creativecommons.org/licenses/by-nc/4.0/).

Hybrid walrus optimization algorithm techniques for optimized parameter estimation in single, double, and triple diode solar cell models

I. INTRODUCTION

II. SOLAR CELL EQUIVALENT CIRCUITS

A. SDM

B. DDM

1. Explicit double-diode modeling method based on Lambert W-function (EDDMMLW)¹³

2. Iterative DDM Lambert W approach

C. TDM

III. HYBRID VARIANTS OF WAOA

A. Original WaOA algorithm

B. Chaotic modification of WaOA

C. Simulated annealing-WaOA (SA-WaOA)

D. Adaptive WaOA (A-WaOA)

IV. NUMERICAL RESULTS

V. EXPERIMENTAL RESULTS

VI. CONCLUSION

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX: CONVERGENCE CURVES

REFERENCES

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Hybrid walrus optimization algorithm techniques for optimized parameter estimation in single, double, and triple diode solar cell models Open Access

I. INTRODUCTION

II. SOLAR CELL EQUIVALENT CIRCUITS

A. SDM

B. DDM

1. Explicit double-diode modeling method based on Lambert W-function (EDDMMLW)13

2. Iterative DDM Lambert W approach

C. TDM

III. HYBRID VARIANTS OF WAOA

A. Original WaOA algorithm

B. Chaotic modification of WaOA

C. Simulated annealing-WaOA (SA-WaOA)

D. Adaptive WaOA (A-WaOA)

IV. NUMERICAL RESULTS

V. EXPERIMENTAL RESULTS

VI. CONCLUSION

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX: CONVERGENCE CURVES

REFERENCES

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Submit your article

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Hybrid walrus optimization algorithm techniques for optimized parameter estimation in single, double, and triple diode solar cell models

1. Explicit double-diode modeling method based on Lambert W-function (EDDMMLW)¹³