Dwarf Mongoose Optimizer for Optimal Modeling of Solar PV Systems and Parameter Extraction

Abstract

This article presents a modified intelligent metaheuristic form of the Dwarf Mongoose Optimizer (MDMO) for optimal modeling and parameter extraction of solar photovoltaic (SPV) systems. The foraging manner of the dwarf mongoose animals (DMAs) motivated the DMO’s primary design. It makes use of distinct DMA societal groups, including the alpha category, scouts, and babysitters. The alpha female initiates foraging and chooses the foraging path, bedding places, and distance travelled for the group. The newly presented MDMO has an extra alpha-directed knowledge-gaining strategy to increase searching expertise, and its modifying approach has been led to some extent by the amended alpha. For two diverse SPV modules, Kyocera KC200GT and R.T.C. France SPV modules, the proposed MDMO is used as opposed to the DMO to efficiently estimate SPV characteristics. By employing the MDMO technique, the simulation results improve the electrical characteristics of SPV systems. The minimization of the root mean square error value (RMSE) has been used to compare the efficiency of the proposed algorithm and other reported methods. Based on that, the proposed MDMO outperforms the standard DMO. In terms of average efficiency, the MDMO outperforms the standard DMO approach for the KC200GT module by 91.7%, 84.63%, and 75.7% for the single-, double-, and triple-diode versions, respectively. The employed MDMO technique for the R.T.C France SPV system has success rates of 100%, 96.67%, and 66.67%, while the DMO’s success rates are 6.67%, 10%, and 0% for the single-, double-, and triple-diode models, respectively.

1. Introduction

Over the last few decades, in response to the challenges posed by global warming, and the diminishing availability of traditional fossil fuels, there has been a growing emphasis on the usage of renewable energy resources (RERs) [1,2]. According to the projections made by the International Renewable Energy Agency, it is anticipated that RERs will account for around 85% of the total energy generated in the year 2050 [3]. Among these RERs, solar PV systems are the second most utilized worldwide, following wind resources [4]. Furthermore, there is a targeted projection that solar energy will account for 20% of the overall energy production by the year 2050 [3]. In the contemporary period, numerous investments have been made in the field of renewable energy, particularly in the solar power sector. One notable example is the Benban Solar Park, located in Egypt, which boasts a significant capacity of 1.8 GW for photovoltaic power plants [5].

In this context, several challenges have emerged, necessitating a thorough analysis and precise resolution to enhance the efficacy of the solar PV systems. Hence, to boost the effectiveness of these systems prior to their installation, it is imperative to employ appropriate mathematical models that can efficiently simulate their performance under various operational circumstances.

In the literature, it is common to find three distinct circuit models for solar PV cells that are considered similar. The first model has one diode and is known as the single-diode PV model (SDPVM) [6,7,8,9,10,11,12,13,14]. It is the most utilized solar PV model and is quite simple, as it has only five parameters. The second model contains two diodes and is called the double-diode PV model (DDPVM) [6,7,8,9,10,11,13,14,15,16]. In this model, the parameters are increased by two because of the additional diode. The additional parameters made the formulation more complicated but at the same time more accurate. Finally, the third model has three diodes and is known as the triple-diode PV model (TDPVM) [4,17,18,19,20,21]. This model is the most complicated and accurate model, as it has nine parameters.

M. S. Ismail et al., in 2013, presented a study on the characterization and optimization of photovoltaic (PV) panel model parameters using a genetic algorithm [22]. While the paper provides valuable insights and contributions, it also has the potential drawbacks of a limited sample size, lack of comparative analysis, and insufficient validation. This study did not explicitly mention the number of PV panels or datasets used for the characterization and optimization process, which could raise concerns about the representativeness of the results. The absence of rigorous validation against uncertainty could raise concerns about the accuracy and reliability of the optimized model parameters. A fast method was introduced by A. Laudani et al. in 2014 to efficiently determine the five parameters of the one-diode model using the experimental I-V curve of a photovoltaic (PV) panel [23]. The method utilized reduced forms of the original five-parameter model, dividing the parameters into independent and dependent unknowns. This division helped reduce the search space dimensions, resulting in significant advantages in terms of convergence, computational costs, and execution times. However, this study derived several simplifications, and so the optimality of the obtained parameters are questionable, whereas only a single diode model was utilized. In 2015, an approach was presented by L. Lim et al. to solve a set of nonlinear equations using a limited number of points on an I-V curve [24]. The study established a connection between the diode model parameters and the transfer function of the dynamic system after applying the Laplace transform. Notably, this study focused solely on a single-diode model, limiting the scope of its analysis.

Furthermore, the genetic algorithm (GA) has been proposed to extract solar PV systems’ parameters [13,25]. In [25], the aim of the study was to enhance the design variables of the solar PV cell in the context of SDPVM. In [13], the authors improved the efficacy of the traditional GA by integrating novel mutation approaches and crossover methods. The utilization of non-uniform mutation and mixed crossover is employed in this context. The improved GA’s performance is evaluated according to a comparison with the experimental data. The newly implemented GA produces estimated curves that exhibit a lower number of errors compared to the experimental curves.

The grasshopper algorithm (GOA) has been suggested to define optimal factors for solar PV systems [6,20]. In [20], the suggested GOA was applied to extract the TDPVM parameters with consideration of the solar irradiation and cell temperature variations. Indeed, it was implemented to find the parameters of both SDPVM and DDPVM [20]. Additionally, the authors suggested the manta ray foraging optimization (MRFO) technique to handle the PV parameter extraction computational problem [26]. The MRFO method has been tested to determine the parameters of SDPVM, DDPVM, and TDPVM. Both solar irradiance and temperature changes have been considered in the problem’s formulation.

Several optimization algorithms have also been proposed to solve the problem of parameter extraction in a solar PV system. These include the supply and demand optimizer [27], the Harris hawk optimization [17], the arithmetic optimization approach [28], the improved estimation of distribution algorithm [29], the heap-based optimizer [30], artificial parameterless optimization [31], the black widow optimization algorithm [32], hunter–prey-based optimization [33], the improved generalized normal distribution algorithm [5], the biogeography-based teaching learning algorithm [34], hybrid variants of artificial gorilla troops and honey badger algorithm techniques [35], a hybrid adaptive Jaya algorithm [1], and particle swarm optimization [36].

The use of metaheuristic algorithms for parameter estimation in PV systems has been widely adopted.

Table 1. Merits and demerits of some of the optimization techniques from the literature.

Optimization Technique	Reference	Merits	Demerits
GA	[13,22]	It is capable of handling a vast number of variables. Good for exploration. Implicit parallelism.	High computational burden. Poor exploitation.
DE	[38]	Easy to implement. Fast convergence. The mutation process is employed for the first exploration. DE and its improved versions present competent and accurate outcomes.	A higher probability of premature convergence is associated with faster convergence. The computational burden is increased due to the high population size required for DE to effectively work on a given problem.
PSO	[36]	Few controlling parameters. Easy to implement. Presents competent computational efficiency outcomes compared with GA.	Susceptible to premature convergence. Provides low-quality solution. High computational efforts are required.
CSO	[9]	Effective in exploration and finding global optima. By including Lévy walk, CSO is able to effectively explore the search space.	The number of tuning parameters is four, which leads to a tedious tuning process. The inherent randomness of the Lévy walk approach may cause the method to exceed parameter boundaries, resulting in a significant decrease in the accuracy of the obtained results.
Jaya algorithm	[39]	It is flexible, as it requires only two parameters, population size and the number of generations, for its operation. Less parameter-tuning complexity.	Improper parameter tuning increases the computational burden and leads to premature convergence.
Ant Lion Optimizer	[40]	Simple and easy to implement. Fast convergence. Accelerated process in finding excellent solutions. Higher feasibility and efficiency in obtaining global optima. Few adjustable parameters.	Suffers from premature convergence. No theoretical converging characteristic. Likelihood distribution changes by generations.
Whale Optimization Algorithm	[19]	Few adjustable parameters. Ability to search the entire problem space. Its exploration abilities can be enhanced by hybridization with other algorithms.	Slow convergence. Susceptible to premature convergence with high-dimensional challenges.

tabulates the merits and demerits of some of the literature optimization techniques. However, it is important to note that no single algorithm can solve all optimization problems, as stated in Wolpert and Macready’s “No Free Lunch” theorem [37]. Over the past 30 years, researchers have made significant advancements in metaheuristic algorithms and have introduced new ones. Despite these improvements, many studies aiming to enhance the performance of these algorithms have highlighted their limitations in accurately and reliably estimating parameters for various photovoltaic models. Therefore, there is a need to develop new approaches that result in simple and efficient metaheuristic algorithms capable of solving practical optimization problems without the need for additional parameter adjustments and modifications.

J. O. Agushaka et al. [41,42] introduced the Dwarf Mongoose Optimizer (DMO), a revolutionary approach inspired by the foraging behavior of the dwarf mongoose animals (DMAs), in 2022. The foraging behavior of the DMAs motivated the DMO’s primary design. It makes use of distinct DMA societal groups including the alpha category, scouts, and babysitters. The alpha female initiates foraging and chooses the foraging path, bedding places, and distance travelled for the group. A combination of its strong global exploring capacity and robustness motivates the implementation of the DMO for handling a variety of genuine engineering optimization difficulties [43,44,45,46,47,48,49,50]. A modified intelligent metaheuristic version of the DMO (MDMO) is proposed in this article for the optimal modelling and parameter extraction of solar photovoltaic (SPV) systems. The newly introduced MDMO features an additional alpha-directed knowledge-gaining strategy to boost seeking expertise, and its modifying technique has been partially guided by the modified alpha.

The newly presented MDMO has an extra alpha-directed knowledge-gaining strategy to increase searching expertise, and its modifying approach has been led to some extent by the amended alpha. For diverse SPV modules, such as the Kyocera KC200GT and R.T.C. France SPV modules, the proposed MDMO is used as opposed to the DMO to efficiently estimate SPV characteristics. The simulation findings enhance the electrical properties of SPV systems by using the MDMO technique. In terms of efficiency and efficacy, the proposed MDMO outperforms the standard DMO. Two primary contributions of this paper can be summarized as follows:

• A novel MDMO is presented with an extra alpha-directed knowledge-gaining strategy to increase searching expertise.
• The proposed MDMO is designed to acquire the best SPV features by taking into account three distinct modelling types: SDPVM, DDPVM, and TDPVM.

Based on the simulation results, the proposed MDMO is employed in SPV technologies, including Kyocera KC200GT and R.T.C. France SPV modules. Furthermore, the proposed MDMO is statistically evaluated in comparison to previously reported optimization processes in the literature.

2. SPV Systems Representation

Numerous mathematical models are available that analyze the functioning and physical characteristics of solar PV systems. This section provides an overview of the mathematical representations of the solar PV cell that can be modeled by SDPVM, DDPVM, and TDPVM.

2.1. SDPVM

Figure 1 depicts the equivalent circuit representation of the SDPVM. This model consists of single-diode, irradiance sources modeled as current source, series resistance, and shunt resistance. The output current (I) is derived by applying the Shockley diode equation, represented as Equation (1) [51,52].

$$I=I_{PV}-I_{r1}\left[{\exp }^{\left(\frac{I{R}_S}{{\gamma }_1\hspace{0.17em}{V}_{th}}+\frac{V}{{\gamma }_1\hspace{0.17em}{V}_{th}}\right)}-1\right]-\frac{I{R}_S}{R_{sh}}+\frac{V}{R_{sh}}$$

(1)

where I_r1 represents the reverse saturation current of D₁, γ₁ characterizes the ideality coefficient of D₁, and R_sh and R_s denote the shunt and series resistances, respectively. Additionally, I_PV and I express the cell photocurrent and the output current, respectively. Furthermore, V is the terminal voltage and V_th represents the PV cell thermal voltage that can be calculated as follows [53]:

$$V_{th}=k_{B}T/q_c$$

(2)

where k_B denotes the Boltzmann’s constant, q_c represents the electron’s charge and T stands for the absolute temperature.

In this model, there are five unknown parameters (I_PV, I_r1, R_s, R_sh, γ₁) that must be calculated from the solar PV cells’ I-V data.

2.2. DDPVM

Figure 2 illustrates the equivalent circuit representation of the DDPVM. This model is considered as an improved version of SDPVM. An extra diode is added in this version to demonstrate a space charge recombination. The output current equation of the DDPVM is mathematically calculated as in Equation (3) [33,54].

$$I=I_{PV}-I_{r1}\left[{\exp }^{\left(\frac{I{R}_S}{{\gamma }_1\hspace{0.17em}{V}_{th}}+\frac{V}{{\gamma }_1\hspace{0.17em}{V}_{th}}\right)}-1\right]-I_{r2}\left[{\exp }^{\left(\frac{I{R}_S}{{\gamma }_2\hspace{0.17em}{V}_{th}}+\frac{V}{{\gamma }_2\hspace{0.17em}{V}_{th}}\right)}-1\right]-\frac{I{R}_S}{R_{sh}}+\frac{V}{R_{sh}}$$

(3)

where I_r2 represents the reverse saturation current of D₂ and γ₂ characterizes the ideality coefficient of D₂.

There are seven unknown parameters in this model that must be calculated from the solar PV cells’ I-V data, which are I_PV, I_r1, I_r2, R_s, R_sh, γ₁, and γ₂.

2.3. TDPVM

In TDPVM, the third diode is added to improve the DDPVM, as displayed in Figure 3. The third diode is added to represent recombination in the defect area. The output current equation of the TDPVM is mathematically calculated as Equation (4) [55,56].

$$I=I_{PV}-I_{r1}\left[{\exp }^{\left(\frac{I{R}_S}{{\gamma }_1\hspace{0.17em}{V}_{th}}+\frac{V}{{\gamma }_1\hspace{0.17em}{V}_{th}}\right)}-1\right]-I_{r2}\left[{\exp }^{\left(\frac{I{R}_S}{{\gamma }_2\hspace{0.17em}{V}_{th}}+\frac{V}{{\gamma }_2\hspace{0.17em}{V}_{th}}\right)}-1\right]-I_{r3}\left[{\exp }^{\left(\frac{I{R}_S}{{\gamma }_3\hspace{0.17em}{V}_{th}}+\frac{V}{{\gamma }_3\hspace{0.17em}{V}_{th}}\right)}-1\right]-\frac{I{R}_S}{R_{sh}}+\frac{V}{R_{sh}}$$

(4)

In this model, there are nine unknown parameters (I_PV, I_r1, I_r2, I_r3, R_s, R_sh, γ₁, γ₂, γ₃) that must be calculated from the solar PV cells’ I-V data.

2.4. Representations of SPV Modules

The equations of the SDPVM, DDPVM, and TDPVM may be represented by a PV module composed of N₁ cells linked in parallel and N₂ cells connected in series. As a result, for these models, Equations (5)–(7) are formulated, respectively, as follows:

$$I=N_1\left(I_{PV}-I_{r1}\left[{\exp }^{\left((V+I{N}_2{R}_S/N_p)/({\gamma }_1\hspace{0.17em}{N}_2{V}_{th})\right)}-1\right]\right)-(V+I{N}_2{R}_S/N_1)/(N_1{N}_2{R}_{sh})$$

(5)

$$\begin{array}{cc}\hfill I=& N_1\left(I_{ph}-I_{r1}\left[{\exp }^{\left((V+I{N}_2{R}_S/N_p)/({\gamma }_1\hspace{0.17em}{N}_2{V}_{th})\right)}-1\right]\right)-I_{r2}\left[{\exp }^{\left((V+I{N}_2{R}_S/N_p)/({\gamma }_2\hspace{0.17em}{N}_2{V}_{th})\right)}-1\right]\hfill \\ & -(V+I{N}_2{R}_S/N_1)/(N_1{N}_2{R}_{sh})\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill I=& N_1\left(I_{ph}-I_{r1}\left[{\exp }^{\left((V+I{N}_2{R}_S/N_p)/({\gamma }_1\hspace{0.17em}{N}_2{V}_{th})\right)}-1\right]\right)-I_{r2}\left[{\exp }^{\left((V+I{N}_2{R}_S/N_p)/({\gamma }_2\hspace{0.17em}{N}_2{V}_{th})\right)}-1\right]\hfill \\ & -I_{r3}\left[{\exp }^{\left((V+I{N}_2{R}_S/N_p)/({\gamma }_3\hspace{0.17em}{N}_2{V}_{th})\right)}-1\right]-(V+I{N}_2{R}_S/N_1)/(N_1{N}_2{R}_{sh})\hfill \end{array}$$

(7)

PV cells/modules for SDPVM, DDPVM, and TDPVM have unidentified variables that can be estimated computationally, analytically, or using optimization approaches.

2.5. Objective Function

The statistical evaluation performed in this paper relied on the root mean square error value (RMSE) [57] and is carried out as follows:

$$RMSE=\sqrt{\frac{1}{N_c}{\displaystyle \sum _{c=1}^{N_c}{\left(I_{Calculated,c}^{}(V_{Exp,c}^{},x)-I_{Exp,c}^{}\right)}^2}}$$

(8)

where N_c represents the number of measured readings; I_Exp,c and V_Exp,c are the measured readings of each record regarding, respectively, the current and voltage; and I_Calculated,c is the calculated output current, which is a nonlinear function in terms of the experimental voltage (V_Exp,c) readings of each record and the unknown parameters (x).

3. Proposed MDMO for Optimal Modeling of SPV Systems and Parameter Extraction

The dwarf mongoose optimizer (DMO) is created by studying the foraging behavior of dwarf mongoose animals (DMAs). In the presented meta-heuristic technique (DMO), the population of DMAs is initially generated as follows [56,58]:

$$X_{{}_m}(0)=X_{{}_{\min }}.(1-R)+R.X_{{}_{\max }}, m=1:N_{DMA}$$

(9)

where m is an integer as a counter, which is equal to 1, 2, 3, 4, 5,………, N_DMA; N_DMA is the entire population of the dwarf mongoose animals. In Equation (9), the symbol “.” represents the dot product, which is a fundamental way of combining two vectors, indicating the product of each element in the vector and the respective one in the other vector with same dimension. X_m specifies the position of each DMA (m), and X_min and X_max signify the lowest and maximum bounds. R is a randomized vector of dimension (D) in relation to the entire number of control variables and N_DMA signifies the total size of DMAs group.

Next, in the initialization of the DMAs’ positions, the fitness rating (FS_m) of every solution (X_m) is evaluated. Following that, the alpha female is chosen depending on the probable value (α_m) of every group’s fitness as follows:

$${\alpha }_m=\frac{F{S}_m}{{\displaystyle \sum _{m=1}^{N_{DMA}}F{S}_m}}$$

(10)

where α_m is the probability value of each animal in the group.

In the alpha organization, the number of DMAs is proportional to the population size minus the number of babysitters (Bs). The symbol (peep) observes the alpha’s vocalization, which keeps the DMAs on track. Each DMA rests within the first sleeping area which has been assigned to them. To generate the next position towards the projected food position, the DMO employs the equation described in Equation (11).

$$X_{{}_m}(T+1)=R\times peep+X_{{}_m}(T), m=1:N_{DMA}-Bs$$

(11)

where Bs is the number of babysitters in the entire group and T represents the present iteration.

The young dwarf mongooses are moved from one sitting mound to a different one rather than having a home built for them. In addition to searching for food, the alpha group scouts for an alternative mound to attend after the babysitting exchange requirement is satisfied. In order to replicate this, the average value of the sitting mound is estimated for each iteration, which may be represented in the following manner:

$$S{M}_{{}_m}=\frac{F{S}_{{}_m}(T+1)-F{S}_{{}_m}(T)}{\max \left(\left|F{S}_{{}_m}(T)-F{S}_{{}_m}(T+1)\right|\right)}$$

(12)

where FS_m(T) is the fitness score of the current solution (X_m) at the current iteration (T) and FS_m(T + 1) is the fitness score of the updated solution (X_m) at the consequent iteration (T + 1).

The mean value (ψ) of the observed sitting mound is given as follows:

$${\psi }_{{}_m}=\frac{{\displaystyle \sum _{m=1}^{N_{DMA}}S{M}_m}}{N_{DMA}}$$

(13)

Based on the overall success of the DMAs, the progression that follows is represented as a success or failure assessment while creating a new mound [59]. The following equation can be used to model the scout mongoose:

$$X_{{}_m}(T+1)=\left\{\begin{array}{c}{X}_{{}_m}(T)+CF\times R\times \left(M-X_{{}_m}(T)\right)\\ X_{{}_m}(T)-CF\times R\times \left(M-X_{{}_m}(T)\right)\end{array}\right.\begin{array}{c}if {\psi }_{{}_{m+1}}>{\psi }_{{}_m}\\ if {\psi }_{{}_{m+1}}<{\psi }_{{}_m}\end{array} m=1:N_{DMA}$$

(14)

where CF reduces gradually as its iterations progress, which is demonstrated in Equation (15), and M appears to be a vector which impacts the DMAs’ eventual sleeping area migration and is calculated in Equation (16). The CF factor represents the value of the parameter which controls the DMA organization’s collective–volitive motion.

$$CF={\left(1-\frac{T}{T_{\max }}\right)}^{\left(\raisebox{1ex}{$2\times T$}\!\left/ \!\raisebox{-1ex}{$T_{\max }$}\right.\right)}$$

(15)

$$M={\displaystyle \sum _{m=1}^{N_{DMA}}\frac{X_m\times S{M}_m}{X_m}}$$

(16)

where T_max represents the total number of iterations.

In order to increase the searching expertise, the alpha-directed knowledge-acquiring technique is paired with the formula described in Equation (11) to construct a likely food location with the goal of increasing the seeking skills:

$$X_{{}_m}(T+1)=\left\{\begin{array}{c}{X}_{Alpha}(T)+R.\left(X_{{}_m}(T)-X_{{}_{Rd}}(T)\right)\\ X_{{}_m}(T)+R\times peep\end{array}\right. \begin{array}{c}if r_1<PSF\\ Else\end{array} m=1:N_{DMA}-Bs$$

(17)

where X_m(T + 1) indicates the updated solution (X_m) at the consequent iteration (T + 1). X_Alpha(T) corresponds to the alpha location having the smallest objective worth; R is a randomized vector of dimension (D); X_m(T) indicates the current solution (X_m) at the current iteration (T); X_Rd refers to the position of a randomly selected DMA; r₁ represents a randomly generated number within range [0, 1]; and PSF indicates the probability of the selection factor. Figure 4 depicts the critical steps of the projected MDMO.

4. Results

This part delves into two PV systems, Kyocera KC200GT and R.T.C France, that implement the suggested MDMO method rather than the usual DMO technique. The first assessment is centered upon R.T.C France, which operates at a sun radiation and a temperature of 1000 w/m² and 33 °C, respectively. The second case studied involves the Kyocera KC200GT module, which consists of 54 multi-crystalline cells linked in series and that operates at 25 °C [60,61]. Additionally, in comparison to current optimizing methodologies recently disclosed in the literature, the suggested MDMO technique has been evaluated and applied to parameter extraction problems across multiple PVSDM, PVDDM, and PVTDM arrangements. A full Appendix A (

Table A1. Selected parameters of the proposed MDMO.

Parameter	Value	Parameter	Value
Maximum number of iterations (Tmax)	1000	Number of dwarf solutions per iteration (NDMA)	200
Number of babysitters (Bs)	3	Number of Alpha group	197
Number of Scouts	197	Probability of the selection factor (PSF)	0.5
Alpha’s vocalization (peep)	2

) detailing the selected parameters of the proposed technique is provided.

4.1. R.T.C France SPV System

4.1.1. Application for PVSDM

For the SPV system of R.T.C. France, the PVSDM arrangement is used in conjunction with the prescribed MDMO and standard DMO methodologies to decrease the RMSE function. The characteristics resulting from the recommended MDMO and DMO are manifested in

Table 2. Parameters derived by MDMO and DMO for R.T.C France SPV system (PVSDM).

Parameters	DMO	MDMO
IPV (A)	0.7605558260	0.7607755103
Rsh (Ω)	0.0358427018	0.0363769511
RS (Ω)	58.9141185038	53.7195239178
Ir1 (A)	0.0000003677	0.0000003230
γ1	1.4943327807	1.4811871929
RMSE	0.0010212370	0.0009860219

.

According to Table 2, the recommended MDMO yields a significantly smaller RMSE rating of 9.8602 × 10⁻⁴ than the typical DMO technique, which yields an RMSE level of 0.00102. Hence, the proposed MDMO method beats the existing DMO method by 3.45%. Additionally, Figure 5 shows the generated RMSE objectives throughout the individual executions employing the recommended MDMO and regular DMO techniques. This figure shows the Root Mean Square Error (RMSE) values obtained through the application of two different optimization techniques: MDMO (proposed) and DMO (regular). The study focuses on the SPV (solar photovoltaic) system of R.T.C France. It illustrates the RMSE objectives obtained during individual executions using both MDMO and DMO techniques. The RMSE is a measure of the difference between the predicted values and the actual values, reflecting the accuracy of the optimization results. It highlights that the recommended MDMO technique demonstrates a substantial advantage over the regular DMO technique. It indicates a significant improvement in performance. Specifically, the average acquired RMSE score for MDMO is reported as 9.86025 × 10⁻⁴ (scientific notation for 0.000986025), while DMO achieves an RMSE score of 0.001126. This suggests that MDMO outperforms DMO by achieving a 12.4% improvement in terms of RMSE. In summary, it emphasizes the superior performance of the recommended MDMO technique over the regular DMO technique in terms of RMSE. MDMO achieves a lower average RMSE score, indicating better accuracy and optimization results for the SPV system under investigation.

Based on the acquired RMSE described in Figure 6 with consideration of a +5% increase over the optimum RMSE score (9.86025 × 10⁻⁴), the success rate of the proposed MDMO is 100%, whereas the standard DMO records a success rate of 6.67%.

Moreover, Figure 6 exhibits the mean convergence trends for the indicated MDMO and classic DMO. It can be manifested that the proposed MDMO approach achieves progress in lessening the RMSE value in contrast to the DMO. As demonstrated, great improvement in finding a smaller RMSE is demonstrated via the application of the proposed MDMO compared to the DMO through the various iterations. It shows improvements of 25.71%, 19.84%, 14.47%, and 12.4% after one-quarter, half, three-quarters, and all of the iterations, respectively.

Table 3. Comparisons of MDMO and DMO for R.T.C France SPV system (PVSDM).

Algorithms	RMSE	Difference to MDMO	Improvement Percentage
Proposed MDMO	9.8602 × 10−4	0	-
DMO	1.021237 × 10−3	3.52 × 10−5	3.572%
Improved DE [66]	9.89 × 10−4	2.98 × 10−6	0.302%
Chaotic PSO [66]	13.8607 × 10−4	0.0004	40.572%
CSA [66]	9.91184 × 10−4	5.16 × 10−6	0.524%
Comprehensive Learning PSO [38]	9.9633 × 10−4	1.03 × 10−5	1.046%
IGA [13]	9.8618 × 10−4	1.6 × 10−7	0.016%
TLBO [66]	9.8733 × 10−4	1.31 × 10−6	0.133%
HSBA [63]	9.95146 × 10−4	9.13 × 10−6	0.926%
ABC [65]	10 × 10−4	1.4 × 10−5	1.418%
BBO with mutation [64]	9.8634 × 10−4	3.2 × 10−7	0.032%
Jaya optimizer [39]	9.8946 × 10−4	3.44 × 10−6	0.349%
GWO [62]	75.011 × 10−4	0.006515	660.745%

contrasts the suggestion for MDMO to numerous PVSDM optimization approaches which have been presented. It compares the RMSE rating of the current DMO technique to those of known optimization approaches such as comprehensive learning PSO [38], GWO [62], HSBA [63], BBO with mutation [64], ABC [65], the Jaya optimizer [39], TLBO [66], and IGA [13].

Based on the provided comparisons in Table 3, which evaluates the performance of various optimization algorithms for the R.T.C France SPV (solar photovoltaic) system, we can make the following observations.

The proposed MDMO algorithm achieved an RMSE of 9.8602 × 10⁻⁴. This serves as the baseline for comparison. The DMO algorithm obtained an RMSE of 1.021237 × 10⁻³. It has a difference of 3.52 × 10⁻⁵ (0.0352%) compared to the proposed MDMO. This indicates that DMO performed slightly worse than MDMO, with a 3.572% deterioration in performance. The improved DE algorithm achieved an RMSE of 9.89 × 10⁻⁴. It has a difference of 2.98 × 10⁻⁶ (0.00298%) compared to MDMO, representing a 0.302% improvement over MDMO. The chaotic PSO algorithm obtained an RMSE of 13.8607 × 10⁻⁴. It has a significant difference of 0.0004 (0.04%) compared to MDMO, indicating a 40.572% deterioration in performance. The CSA algorithm achieved an RMSE of 9.91184 × 10⁻⁴. It has a difference of 5.16 × 10⁻⁶ (0.00516%) compared to MDMO, representing a 0.524% improvement over MDMO. The comprehensive learning PSO algorithm obtained an RMSE of 9.9633 × 10⁻⁴. It has a difference of 1.03 × 10⁻⁵ (0.0103%) compared to MDMO, indicating a 1.046% improvement over MDMO. The IGA algorithm achieved an RMSE of 9.8618 × 10⁻⁴. It has a difference of 1 × 10⁻⁷ (0.00016%) compared to MDMO, indicating a minor improvement of 0.016% over MDMO. The TLBO algorithm obtained an RMSE of 9.8733 × 10⁻⁴. It has a difference of 1.31 × 10⁻⁶ (0.00131%) compared to MDMO, representing a 0.133% improvement over MDMO. The HSBA algorithm achieved an RMSE of 9.95146 × 10⁻⁴. It has a difference of 9.13 × 10⁻⁶ (0.00913%) compared to MDMO, indicating a 0.926% improvement over MDMO. The ABC algorithm obtained an RMSE of 10 × 10⁻⁴. It has a difference of 1.4 × 10⁻⁵ (0.014%) compared to MDMO, representing a 1.418% improvement over MDMO. The BBO with mutation algorithm achieved an RMSE of 9.8634 × 10⁻⁴. It has a difference of 3.2 × 10⁻⁷ (0.00032%) compared to MDMO, indicating a minor improvement of 0.032% over MDMO. The JAYA optimizer algorithm obtained an RMSE of 9.8946 × 10⁻⁴. It has a difference of 3.44 × 10⁻⁶ (0.00344%) compared to MDMO, representing a 0.349% improvement over MDMO. The GWO algorithm achieved an RMSE of 75.011 × 10⁻⁴. It has a significant difference of 0.006515 (0.6515%) compared to MDMO, indicating a substantial deterioration in performance with a 660.745% increase in the RMSE compared to MDMO. These results suggest that the proposed MDMO algorithm performed relatively well compared to most of the other optimization algorithms in terms of the RMSE metric. However, it is important to consider additional factors and performance metrics, as well as conduct further analysis, to make a comprehensive assessment of the algorithms’ suitability for the R.T.C France SPV system.

4.1.2. Application for PVDDM

The proposed MDMO and classic DMO techniques are used for the PVDDM, and

Table 4. Parameters derived by MDMO and DMO for R.T.C France SPV system (PVDDM).

Parameter	DMO	MDMO
IPV (A)	0.761086003	0.760777046
Rsh (Ω)	0.036452844	0.03658083
RS (Ω)	56.0407128	54.7047585
Ir1 (A)	3.81141 × 10−7	4.27843 × 10−7
γ1	1.83357911	1.991913976
Ir2 (A)	2.38858 × 10−7	2.63353 × 10−7
γ2	1.458364626	1.463888853
RMSE	0.001028696	0.000983217

shows the corresponding estimated parameter values. Table 4 shows that the suggested MDMO has a smaller RMSE score of 0.000983217 than the traditional DMO with an RMSE level of 0.001028696. Hence, the proposed MDMO outperforms the traditional DMO by 4.42%.

Figure 7 displays the RMSE objectives obtained from individual executions utilizing both the recommended MDMO and regular DMO techniques. The results clearly demonstrate the significant advantage of the recommended MDMO, representing a noteworthy improvement over the conventional DMO approach. Regarding the average acquired RMSE, MDMO achieves an RMSE score of 9.98962 × 10⁻⁴ (scientific notation for 0.000998962), while DMO obtains a score of 0.001397, indicating a 29.2% improvement. Based on the acquired RMSE described in Figure 7 with a consideration of a +5% increase over the optimum RMSE score (9.8602 × 10⁻⁴), the success rate of the proposed MDMO is 96.67%, whereas the standard DMO records a success rate of 10%.

In addition to that, Figure 8 displays the mean convergence trends for the indicated MDMO and DMO techniques. It can be seen that the proposed MDMO achieves progress in lessening the RMSE target in contrast to the DMO. As demonstrated, great improvement in finding smaller a RMSE is demonstrated via the application of the proposed MDMO compared to the DMO throughout the iterations. It shows improvements of 39.58%, 35.55%, 33.3%, and 29.2% after one-quarter, half, three-quarters, and all of the iterations, respectively.

Table 5. Comparisons of MDMO and DMO for R.T.C France SPV system (PVDDM).

Algorithms	RMSE
Proposed MDMO	0.000983217
Standard DMO	0.001028696
Generalized oppositional TLBO [67]	4.43212 × 103
Cat swarm algorithm [68]	1.22 × 103
Sine cosine approach [69]	9.86863 × 104
Flower pollination algorithm [70]	1.934336 × 103
Teaching–learning-based ABC [71]	1.50482 × 103
TLBO [72]	1.52057 × 103
ABC [73]	1.28482 × 103
Comprehensive learning PSO [74]	1.3991 × 103

contrasts the anticipated MDMO with numerous PVDDM optimization strategies that have been reported. It compares the proposed MDMO approach’s RMSE rating to that of existing optimization strategies. As demonstrated, the suggested MDMO outperforms other alternatives in terms of identifying the lowest RMSE.

4.1.3. Application for PVTDM

For the PVTDM, the proposed MDMO and DMO techniques are used, and

Table 6. Parameters derived by MDMO and DMO for R.T.C France SPV system (PVTDM).

Parameter	DMO	MDMO
IPV (A)	0.760598554	0.760761654
Rsh (Ω)	0.035617495	0.036502802
RS (Ω)	71.78477566	54.55034588
Ir1 (A)	4.56677 × 107	4.57851 × 108
γ1	1.862177339	1.981650008
Ir2 (A)	4.77356 × 107	1.92461 × 107
γ2	1.685843012	1.957369883
Ir3 (A)	1.05595 × 107	2.85309 × 107
γ3	1.410321073	1.470762385
RMSE	0.001233216	0.00098448

shows the corresponding estimated parameter values. It illustrates that the suggested MDMO has a smaller RMSE score of 0.00098448 than the DMO, which has an RMSE level of 0.0012332. Thus, the proposed MDMO outperforms the traditional DMO method by 20.17%.

Figure 9 depicts the RMSE objectives obtained from individual executions using both the recommended MDMO and regular DMO techniques. The results clearly illustrate the substantial advantage provided by the proposed MDMO, representing a significant improvement over the conventional DMO technique. In terms of the average acquired RMSE, MDMO achieves an RMSE score of 0.001000974, while DMO attains a score of 0.0017076, indicating a remarkable 41.36% improvement.

Based on the acquired RMSE described in Figure 9 with consideration of +5% increase over the optimum RMSE score (9.8602 × 10⁴), the success rate of the proposed MDMO is 66.67%, whereas the standard DMO completely fails to achieve a success rate.

Moreover, Figure 10 signifies the mean convergence trends for the indicated MDMO and classic DMO techniques. As demonstrated, great improvement in finding smaller RMSE is demonstrated via the application of the proposed MDMO compared to the DMO through the various iterations. It shows improvements of 40.74%, 42.83%, 43.33%, and 41.36% after one-quarter, half, three-quarters, and all of the iterations, respectively.

Figure 11 additionally demonstrates the actual and anticipated P-V and I-V characteristics of the PVTDM. The suggested MDMO approach yielded data that largely corresponded with the original data from experiments, indicating that the recommended MDMO technique successfully retrieves the required SPV parameters.

4.2. KC200GT SPV System

4.2.1. Application for PVSDM

In the beginning, the PVSDM arrangement is used in conjunction with the recommended MDMO and conventional DMO methodologies to reduce the RMSE value aims for the KC200GT SPV system.

Table 7. Parameters derived by MDMO and DMO for KC200GT SPV system (PVSDM).

Applied Technique	Standard DMO	Proposed MDMO
IPV (A)	8.195463895	8.216699311
Rsh (Ω)	0.004628453	0.004824347
RS (Ω)	34.8331475	6.283367658
Ir1 (A)	5.82106 × 108	2.6348 × 108
γ1	1.264179155	1.21319155
RMSE	0.021238996	0.000647145

summarizes the derived parameter values.

As shown, the MDMO yields a drastically smaller RMSE rating of 0.0006471 than the typical DMO technique, which yields an RMSE level of 0.02123899.

In Table 7, there is a large difference in R_S (Ω) and I_r1 (A) between DMO and MDMO. To analyze this using the I-V curve of KC200GT, Figure 12 displays the I-V curve based on DMO and MDMO for the KC200GT SPV system (PVSDM). As shown in the three zoomed-in subfigures, the proposed MDMO derives better coincidence between the experimental recordings and the estimated curve. Also, the large differences in R_S and I_r1 between DMO and MDMO show significant deviations between the desired and calculated curves, especially in the first half of the recordings.

Figure 13 also shows the generated RMSE objectives throughout the individual executions employing each of the MDMO and regular DMO techniques. As shown, the MDMO derives a significant improvement over the typical DMO technique. In terms of average acquired RMSE, the MDMO obtains an RMSE score of 0.002218941, whereas the DMO achieves an RMSE score of 0.02673, a 91.7% improvement. Regarding the worst attained RMSE, the MDMO obtains an RMSE score of 0.003956, whereas the DMO achieves an RMSE score of 0.03238, an 87.78% improvement. In terms of the standard deviation of the achieved RMSE, the MDMO obtains an RMSE deviation of 0.00082, whereas the DMO achieves an RMSE deviation of 0.00258 with a 68.27% improvement.

Figure 14 also depicts the mean convergence trends for the MDMO and conventional DMO methodologies mentioned. As proven, the use of the suggested MDMO compared to the DMO through various iterations results in a significant improvement in discovering a reduced RMSE. The rate of the RMSE’s objective minimization is increased with the increase in iterations based on the proposed MDMO. It improves by 68.2%, 77.5%, 86.32%, and 91.7% after one-quarter, half, three-quarters, and all of the iterations, respectively.

Table 8. Comparisons of MDMO and DMO for KC200GT SPV system (PVSDM).

Applied Technique	RMSE
Proposed MDMO	0.000647145
Standard DMO	0.021238996
HTS [60]	0.01799763
EVO [60]	0.023069893
GO [60]	0.008515347
FPA [60]	0.011225773

contrasts the presented MDMO with numerous PVSD arrangement optimization strategies that have been reported. It compares the given MDMO approach’s RMSE score to those of known optimization approaches such as HTS [60], EVO [60], FPA [60], and GO [60]. As demonstrated, the suggested MDMO outperforms other alternatives in terms of identifying the lowest RMSE.

4.2.2. Application for PVDDM

The proposed MDMO and DMO techniques are used for the PVDDM, and

Table 9. Parameters derived by MDMO and DMO for KC200GT SPV system (PVDDM).

Parameter	DMO	MDMO
IPV (A)	8.190007547	8.217022713
Rsh (Ω)	0.004316377	0.004832051
RS (Ω)	50.79314675	6.310735656
Ir1 (A)	5.39995 × 107	7.18428 × 108
γ1	1.79153122	1.703337282
Ir2 (A)	1.25929 × 107	2.37725 × 108
γ2	1.320680216	1.207443552
RMSE	0.025515893	0.000647617

shows the corresponding estimated parameter values. Table 9 illustrates that the proposed MDMO has a smaller RMSE score of 0.0006476 than the traditional DMO, which has an RMSE level of 0.02551.

Table 9 shows a significant difference between DMO and MDMO in R_S (Ω) and I_r1 (A). Figure 15 shows the I-V curves based on DMO and MDMO for the KC200GT SPV system (PVDDM), which may be used to analyze the use of I-V curves for KC200GT. The suggested MDMO yields improved the coincidence between the calculated curve and the experimental records, as demonstrated by the three zooming subfigures. Significant variations between the desired and calculated curves are also evident from the considerable differences in Rs and I_r1 between DMO and MDMO, particularly in the first half of the recordings.

Figure 16 also shows the generated RMSE objectives throughout the individual executions employing each of the recommended MDMO and regular DMO techniques. As demonstrated, the substantial advantage is indicated due to the recommended MDMO. It represents a significant improvement over the typical DMO technique. In terms of average acquired RMSE, the MDMO obtains an RMSE score of 0.005014, whereas the DMO achieves an RMSE score of 0.03263, an 84.63% improvement. Regarding the worst attained RMSE, the MDMO obtains an RMSE score of 0.011526, whereas the DMO achieves an RMSE score of 0.037687, an 69.42% improvement. In terms of the standard deviation of the achieved RMSE, the MDMO obtains an RMSE deviation of 0.002383, whereas the DMO achieves an RMSE deviation of 0.002821, a 15.53% improvement.

Furthermore, Figure 17 establishes the mean convergence trends for the indicated MDMO and classic DMO techniques. It can be demonstrated that the proposed MDMO approach achieves progress in lessening the RMSE target in contrast to the DMO. As demonstrated, great improvement in finding a smaller RMSE is demonstrated via the application of the proposed MDMO compared to the DMO through the various iterations. It shows improvements of 56.8%, 69.18%, 78.1%, and 84.62% after one-quarter, half, three-quarters, and all of the iterations, respectively.

Table 10. Comparisons of MDMO and DMO for KC200GT SPV system (PVDDM).

Applied Technique	RMSE
Proposed MDMO	0.000647617
Standard DMO	0.025515893
HTS [60]	0.020515491
EVO [60]	0.02717656
GO [60]	0.009049475
FPA [60]	0.014006267

contrasts the presented MDMO with numerous PVDD arrangement optimization strategies that have been reported. It compares the given MDMO approach’s RMSE score to those of known optimization approaches such as HTS [60], EVO [60], FPA [60], and GO [60]. As demonstrated, the suggested MDMO outperforms other alternatives in terms of identifying the lowest RMSE.

4.2.3. Application for PVTDM

For the PVTDM, the proposed MDMO and classic DMO techniques are used, and

Table 11. Parameters derived by MDMO and DMO for KC200GT SPV system (PVTDM).

Parameter	DMO	MDMO
IPV (A)	8.212043659	8.215581534
Rsh (Ω)	0.00433185	0.004873579
RS (Ω)	65.08583028	6.522906157
Ir1 (A)	2.69786 × 107	2.14545 × 107
γ1	1.769933684	1.959407949
Ir2 (A)	1.49714 × 107	1.75659 × 108
γ2	1.332828581	1.190073125
Ir3 (A)	4.67121 × 108	5.03381 × 107
γ3	1.670933023	1.864164725
RMSE	0.029416653	0.000839604

shows the corresponding estimated parameter values. It illustrates that the suggested MDMO has a smaller RMSE score of 0.0008396 than the DMO approach, which has an RMSE level of 0.029416.

In Table 11, there is a large difference in R_S (Ω) and I_r1 (A) between DMO and MDMO. To analyze this using the I-V curve of KC200GT, Figure 18 displays the I-V curve based on DMO and MDMO for the KC200GT SPV system (PVTDM). As shown in the three zoomed-in subfigures, the proposed MDMO derives a better coincidence between the experimental recordings and the estimated curve. Also, the large difference in R_S and I_r1 between DMO and MDMO shows significant deviations between the desired and calculated curves, especially in the first half of the recordings.

Figure 19 also shows the generated RMSE objectives throughout the individual executions employing each of the recommended MDMO and regular DMO techniques. As demonstrated, the substantial advantage is stated because of the proposed MDMO. It represents a significant improvement over the typical DMO technique. In terms of average acquired RMSE, the MDMO obtains an RMSE score of 0.008754, whereas the DMO achieves an RMSE score of 0.036029, a 75.7% improvement.

Additionally, Figure 20 illustrates the mean convergence trends for the indicated MDMO and classic DMO techniques. As demonstrated, great improvement in finding smaller RMSE is demonstrated via the application of the proposed MDMO compared to the DMO through the various iterations. It shows improvements of 54.16%, 64.22%, 70.37%, and 75.63% after one-quarter, half, three-quarters, and all of the iterations, respectively.

Figure 21 additionally demonstrates the actual and anticipated P-V and I-V characteristics of the PVTDM. The suggested MDMO approach yielded data that largely corresponded with the original data from experiments, indicating that the recommended MDMO technique successfully retrieves the required SPV parameters.

5. Conclusions

In this research, a modified intelligent metaheuristic version of the Dwarf Mongoose Optimizer (MDMO) is presented for the optimal modelling and parameter extraction of solar photovoltaic (SPV) systems. The currently provided MDMO features an additional alpha-directed knowledge-gaining strategy to improve searching skills, and its modifying approach has been influenced to some extent by the modified alpha. The proposed MDMO is utilized to effectively estimate SPV parameters for two separate SPV modules, Kyocera KC200GT and R.T.C. France. In this context, different model arrangements are handled, namely PVSDM, PVDDM, and PVTDM. The simulated studies boost the electrical properties of SPV systems using the MDMO. In terms of effectiveness and robustness, the proposed MDMO beats the standard DMO, with significant improvement in the success rate and convergence characteristics. In addition, the proposed MDMO strategy beats numerous existing optimization strategies that were reported considering the SPV system of R.T.C. France. Also, for the SPV module of KC200GT, in comparison to four recently described methods, FPA, GO, EVO and, HTS, the suggested MDMO outperforms them in determining the lowest RMSE. Furthermore, the suggested MDMO approach can successfully extract the SPV parameters, with the retrieved data typically matching the experimental data.