Figure 2 illustrates the initial performance comparison of optimization methods. The equivalent circuit for SDM is elucidated in Figure 8a. The current output from this model is computed using the following equation (Saleh Alluhaidan et al., 2025; Ghanim et al., 2024; AbdElminaam et al., 2024; AbdElminaam et al., 2022). The single-diode model parameters are mathematically described in Equation 1: I = I p v − I D 1 − I s h (1)

Where:

• I : Output current of the PV cell.

Where:

• I : Output current of the PV cell.

The SDM produces a current denoted as I , where I p v represents the current generated by light, I s h is the leakage current, and I D 1 is the dark saturation current. Shunt and series resistances are denoted by R p and R s , respectively. Additionally, n 1 signifies the diode ideality factor, K represents Boltzmann’s constant, q denotes the charge of an electron, and T c indicates the cell temperature. According to the provided mathematical formula, the parameters to be estimated in the SDM encompass I p v , I o 1 , n 1 , R s , and R p .

In Figure 3, the electrical diagram for the DDM is presented, utilizing two diodes to enhance output quality (Saleh Alluhaidan et al., 2025; Ghanim et al., 2024; AbdElminaam et al., 2024; AbdElminaam et al., 2022). The current output in this model is derived from the following equations: I = I p v − I D 1 − I D 2 − I s h (3)

Where:

• I : Output current of the PV cell.

Where:

• I : Output current of the PV cell.

Where I D 2 represents the dark saturation current of the second diode, and n 2 denotes the ideality factor of the second diode. The model encompasses seven parameters for estimation: I p v , I o 1 , n 1 , R s , R p , I o 2 , and n 2 .

Figure 4 demonstrates the convergence behavior of the i_MGO algorithm. The three-diode model (TDM) displayed in Figure 8c introduces an alternative approach for designing PV modules by incorporating three diodes (Saleh Alluhaidan et al., 2025; Ghanim et al., 2024; AbdElminaam et al., 2024; AbdElminaam et al., 2022). The computation of the current output in this model is carried out using Equation 5: I = I p v − I D 1 − I D 2 − I D 3 − I s h (5)

Where:

• I : Output current of the PV cell.

Where:

• I : Output current of the PV cell.

Where I D 3 represents the dark saturation current of the third diode, and n 3 denotes the ideality factor of the third diode. Estimating nine parameters is essential for the TDM: I p v , I o 1 , n 1 , R s , R p , I o 2 , n 2 , I o 3 , and n 3 .

  I p v ,   I o 1 ,   n 1 ,   R s ,   R p ,   I o 2 ,   n 2 ,   I o 3 and   n 3 .

The evaluation of TPVM’s performance involves objective functions, particularly those centered on root mean square error (RMSE). These functions quantify the difference between the computed current using estimated parameters and the actual current obtained from the dataset. The RMSE is precisely defined by Equations 7, 8: J V ,   I ,   X = I − I exp (7) R M S E = 1 N ∑ N i = 1 J V , I , X 2 (8)

In this context, I exp denotes the analysis current, N signifies the number of data readings, and X encompasses the set of decision variables.

The vector of decision variable for SDM is X = I p v ,   I o 1 ,   n 1 ,   R s   a n d   R p   .

The vector of decision variable for DDM is   X = I p v ,   I o 1 ,   n 1 ,   R s ,   R p ,   I o 2   a n d   n 2   .

The vector of decision variable for TDM is X = I p v ,   I o 1 ,   n 1 ,   R s ,   R p ,   I o 2 ,   n 2 ,   I o 3   a n d   n 3 .

Upon attaining maturity and sufficient strength, male mountain gazelles establish an isolated territory characterized by a high degree of territoriality. In a gazelle territorial conflict, the adult males fight for control of the female’s territory. In contrast, younger males attempt to take over, and this process can be modeled using Equation 9. The initialization of model parameters is described using Equation 10. TSM = male gazelle − Δ TSM × Cof r (9) Δ TSM = r i 1 × B H − r i 2 × X t × F (10)

where:

•The parameters r i 1 and r i 2 are randomly assigned values of 1 or 2, and  male  gazelle  denotes the position vector of an adult male gazelle in the global solution, and X ( t ) is the current position of gazelle.

A young male solution X r a is generated at random from the interval r a with a total number of gazelles N (i.e., r a = N 3 … N ). The average number of randomly selected search agents, denoted as M p r , is equal to N 3 while r 1 and r 2 are two generated random numbers within the range from 0 to 1.

• F can be computed according to Equation 12.

N 1 is a random number drawn from the normal distribution of the problem variable’s dimension D . While exp ,  t  , and  Max_t  are the exponential function, the current iteration, and the maximum number of iterations, respectively.

•A randomly chosen coefficient vector C o f r is employed and modified with each iteration to enhance the search capability and calculated according to Equation 13.

The value of a based on the current and maximum number of iterations (i.e., a = − 1 + t × − 1  Max_t  ). r 3 and r 4 ∈ ( 0,1 ) with cos represent the cosine function. N 2 , N 3 , and N 4 are random numbers within the normal range of the problem’s dimensions D .

The life cycle of mountain gazelles isn’t complete without maternity herds, which are responsible for giving birth to healthy male gazelles. Adult male gazelles can also be involved in mating and the birth of young males attempting to mate with females, modeled as follows: M H = B H + C o f 2 , r + Δ M H × C o f 3 , r (14) Δ M H = r i 3 × male gazelle − r i 4 × X rand (15) where the influence factor of young males is B H ’s vector according to Equation 11. The coefficient vectors C o f 2 , r and C o f 2 , r are computed independently and are picked at random according to Equation 13.  male  gazelle  and X rand  are the best solution so far and the randomly selected solution from the current population, respectively. r i 3 and r i 4 are two random numbers ∈ ( 0,1 ) . The loss functions applied during optimization are expressed in Equations 15, 17.

As male gazelles reach adulthood, they create territories and compete for control over female gazelles, resulting in aggressive conflicts that involve younger male gazelles. This behavior can be modeled as in Equation 16. B M H = X t − D + Δ BMH × C o f r (16) Δ BMH = r i 5 ×  male  gazelle  − r i 6 × B H (17) D = | X t | + | male  gazelle | × 2 × r 6 − 1 (18) where X ( t ) represents the current position of the gazelle vector at iteration t and D is a vector computed based on the best solution as in Equation 18. r i 5 and r i 6 are randomly selected numbers that can only be 1 or 2, while r 6 ∈ ( 0,1 ) . Other parameters are computed as defined previously.

Mountain gazelles continuously seek new food sources and will travel great distances to migrate and seek food. Mountain gazelles, on the other hand, are characterized by their great running speed and their strong leaping ability. Their migration behavior can be modeled mathematically according to Equation 19. M S F = u b − l b × r 7 + l b (19) where u b and l b are the upper and lower boundaries of the problem, respectively, and r 7 is a selected random number between 0 and 1.

Every gazelle undergoes the same reproductive process every time: T S M , M H , B M H , and M S F . With the addition of another era, the population grows at a rate that equals one replication per generation. After each age, the gazelles are also sorted from lowest to highest. In each given population, the best gazelles—those with low costs, excellent quality, and potential solutions—are the ones who survive. Some gazelles are culled from the population because they are elderly or feeble. The mature male gazelle that owns the territory is also considered the most dominant. The overall structure of the MGO algorithm is presented in Algorithm 1 with an associated flowchart that describes the basic steps as shown in Figure 5.

Algorithm 1

The algorithmic steps of MGO algorithm.

1 Initialize the population size n , and maximum number of iterations Max_t .

The Gradient Search Operator (GSO), first presented in the Gradient-based Optimizer (GBO), is intended to instill stochastic dynamics within the optimization procedure, enhancing the search space’s exploration while preventing local minima entrapment. The GBO algorithm is further refined with the inclusion of the Directional Movement (DM) component, which directs the local search trajectory, thus accelerating the GBO’s convergence.

To evolve the current vector’s position, the succeeding formulas are employed (refer to Algorithm 2). The irradiance and temperature dependencies are given in Equations 20–22: S 1 , t i = S t i − G S O + rand × k 2 × S b − S t i (20) S 2 , t i = S b − G S O + rand × k 2 × S t , r 1 − S t , r 2 (21) S 3 , t i = S t i − k 1 × S 2 , t i − S 1 , t i (22)

The subsequent iteration’s solution S t + 1 i is determined through the coordinates of S 1 , t i , S 2 , t i , and S 3 , t i , combined with the current position S t i , and Equation 23 defines the fitness evaluation in the optimizer algorithm: S t + 1 i = r a × r b × S 1 , t i + 1 − r b × S 2 , t i + 1 − r a × S 3 , t i (23)

Algorithm 2

Gradient Search Operator (GSO).

The Production Operator (PO) in the MGO is inspired by the movement of gazelles in their natural habitat, where the producer (the worst-performing individual) is updated to explore new regions in the solution space. The primary role of the PO is to facilitate the exploration process during optimization, allowing the algorithm to search broader areas in the initial stages and later refine the search around promising solutions. The PO updates the position of the producer using a weighted combination of the current best solution found (decomposer) and a random position in the search space, ensuring that the producer moves dynamically through the search space. The update equation for the producer’s position is, Equations 24–27 present iterative update rules used by i_MGO: x new = 1 − α ⋅ x best + α ⋅ x rand (24) where x new is the updated position of the producer, x best is the position of the best individual (decomposer), and x rand is a randomly generated position. The parameter α is the weight factor that controls the balance between exploration and exploitation, defined as: α = 1 − t T ⋅ r 1 (25) where t is the current iteration, T is the total number of iterations, and r 1 is a random number in the range [ 0,1 ] . Initially, α is large, promoting exploration, but as the iterations increase, α decreases, leading to more focused exploitation around the best solution.

As the optimization progresses, the Production Operator shifts the balance from exploration (searching broader regions of the solution space) to exploitation (refining the best solutions). In the early iterations, this operator encourages the producer to move towards random positions, helping the algorithm avoid local optima. In later iterations, the influence of the best solution increases, guiding the producer to converge on the optimal solution. This dynamic shift ensures that MGO is both thorough in exploring the solution space and efficient in exploiting promising regions, ultimately leading to improved convergence and solution quality.

Introduced within the Gradient-based Optimizer framework, the Local Escaping Operator (LEO) functions as a nuanced local search strategy to augment GBO’s efficacy in real-world problem spaces. The operator’s role is pivotal in enhancing the solution position updates and facilitating rapid convergence while circumventing local optima.

LEO generates potential high-performing solutions S LEO by synergistically harnessing optimal positions S b , the solutions S 1 , t i and S 2 , t i , alongside two randomly derived solutions S r 1 and S r 2 , in addition to a newly formulated solution S k . This interplay is quantitatively captured in the following formulation: S t + 1 i = S t i + k 1 ⋅ m 1 ⋅ S b − m 2 ⋅ S k + k 2 k 1 ⋅ m 3 ⋅ S t i − S 1 , t i + m 2 2 ⋅ S r 1 − S r 2 , if rand < 0.5 S b + k 1 ⋅ m 1 ⋅ S b − m 2 ⋅ S k + k 2 k 1 ⋅ m 3 ⋅ S 2 , t i − S 1 , t i + m 2 2 ⋅ S r 1 − S r 2 , otherwise (26)

In this expression, k 1 is uniformly distributed in the interval [-1, 1], k 2 follows a normal distribution, and m 1 , m 2 , and m 3 are stochastic variables described as, Equations 28–30 describe the convergence checks and stopping conditions: m 1 = F 1 ⋅ 2 ⋅ rand + 1 − F 1 (27) m 2 = F 1 ⋅ rand + 1 − F 1 (28) m 3 = F 1 ⋅ rand + 1 − F 1 (29)

Here, F 1 is a binary switch, activating with a probability less than 0.5, while rand ( ) yields a random number between 0 and 1. The variable k 1 maintains a dynamic equilibrium between exploitation and exploration, modulated by the sinusoidal rhythm of ϕ , computed as, finally, Equations 31–34 summarize the derived expressions for validating model performance: k 1 = 2 ⋅ rand − ϕ (30) ϕ = h ⋅ sin 3 π 2 + sin h ⋅ 3 π 2 (31) h = h min + h max − h min ⋅ 1 − t t max 3 (32)

The interval [ h min , h max ] is defined between 0.2 and 1.2. The new solution S k is derived from either a random selection or the current population based on the outcome of a binary event with a probability of 0.5, formalized as: S k = S rand , if  f 2 < 0.5 S t , p , otherwise (33)

The calculation results for the Single-Diode model (SDM) in the paper demonstrate that the Improved Mountain Gazelle Optimizer (i_MGO) significantly outperforms other optimization algorithms in terms of accuracy and efficiency. The i_MGO achieved the lowest Root Mean Square Error (RMSE) values, indicating superior precision in parameter estimation compared to competitors such as Harris Hawks Optimization (HHO), Lightning Attachment Procedure Optimization Algorithm (LAPO), Sine Cosine Algorithm (SCA), and Grey Wolf Optimizer (GWO). The i_MGO also showed remarkable consistency and reliability across multiple calculation runs, with a rapid convergence rate that enhances computational efficiency. These attributes confirm the i_MGO’s effectiveness in optimizing parameters for the Single-Diode PV model, making it a valuable tool for researchers and engineers working in solar energy technologies.

The most accurate parameter values and the corresponding Root Mean Square Error (RMSE) values are presented in Table 4. The calculation results following the execution of each optimizer 30 times on the SDM-based RTC France are detailed in the table.

The results indicate that i_MGO is the most effective algorithm. Its Best RMSE performance is on par with specific algorithms and exceeds others in every performance metric. On the contrary, i_MGO exhibits parity with fifty percent of the algorithms as determined by the worst RMSE results. Furthermore, the standard deviation is documented as a supplementary metric for assessing performance. In all cases, i_MGO consistently outperforms alternative algorithms or maintains parity, never demonstrating subpar performance. In addition, examining the fill factor and Iphoto parameters highlights discrepancies between the results generated by i_MGO and those generated by alternative algorithms.

As illustrated in Figure 6a, the convergence curve is utilized in our analysis investigations to assess the rate at which the i_MGO algorithm achieved stability and reduced Root Mean Square Error (RMSE). In contrast to its claim of being the quickest algorithm to converge, i_MGO demonstrated a similar and satisfactory rate compared to the other algorithms, according to our findings. Nonetheless, the i_MGO was identified as having the minimum RMSE value. Significantly, the optimal performance threshold was successfully achieved after 120 iterations, demonstrating that i_MGO effectively achieves increased precision in a relatively short computational period.

Furthermore, the P–V and I–V curves derived from the optimal parameters obtained via the i_MGO algorithm are depicted in Figures 6b,d, respectively. The visual depictions illustrate the correspondence between the projected and observed measurements. The parameters deduced by i_MGO enable the achievement of current and power levels highly consistent with the empirical data, as evidenced by the figures.

Figure 6c presents the voltage-current characteristics for the double-diode model. The superior performance of the proposed i_MGO algorithm over other optimization techniques, such as GWO, HHO, and PSO, can be attributed to several scientific and mathematical factors:

1. Balanced Exploration and Exploitation: The i_MGO algorithm integrates a dynamic Gradient Search Operator (GSO) and a Modified Local Escaping Operator (LEO). The GSO ensures effective exploitation by refining solutions in the later stages of optimization, while the LEO prevents premature convergence by allowing the search to escape local optima. Mathematically, this balance is modeled using the following transition:

In summary, the proposed algorithm’s success lies in its ability to dynamically balance search mechanisms, adaptively tune parameters, and robustly navigate complex solution landscapes, ensuring accurate and reliable parameter identification for the Single-Diode Model.

The analysis evaluation of the Double-Diode model using the Improved Mountain Gazelle Optimizer (i_MGO) revealed significant improvements in parameter estimation accuracy and efficiency. The i_MGO consistently achieved lower Root Mean Square Error (RMSE) values compared to other advanced optimization algorithms such as Harris Hawks Optimization, Lightning Attachment Procedure Optimization, Sine Cosine Algorithm, and Grey Wolf Optimizer. Notably, the i_MGO demonstrated robust performance with enhanced convergence speeds, ensuring quick attainment of optimal solutions. The algorithm’s ability to reliably and precisely estimate the complex parameters of the Double-Diode model underscores its potential as a powerful tool in PV model optimization, facilitating more accurate simulations and designs in solar energy systems.

Following thirty iterations of each algorithm execution on the DDM-based RTC France, the RSME values for the best and worst outcomes are recalculated and are displayed in 5. The data presented in the table indicates that i_MGO achieves the highest root mean square error (RMSE) among the evaluated algorithms. Notably, the most egregious root mean square error (RMSE) value recorded in Table 4 indicates that i_MGO outperforms 90% of the alternative algorithms. The convergence curve of the implemented DDM-based algorithms is depicted in 7-a. i_MGO outperforms the alternative algorithms in terms of speed and attaining the smallest Root Mean Square Error (RMSE). Although not the most expeditious, the convergence rate of i_MGO is deemed satisfactory, as it reaches saturation after an estimated 130 iterations. Its consistent ability to attain the smallest RMSE values is significantly offsetting this convergence behavior.

The primary reasons for this performance are as follows:

• The increased complexity of the DDM, involving additional parameters and non-linear interdependencies, is effectively addressed by the adaptive Modified Transition Factor (mTF). This ensures a gradual transition from global exploration to localized refinement, enabling precise parameter estimation even in high-dimensional solution spaces.

The analysis of the Triple-Diode model using the Improved Mountain Gazelle Optimizer (i_MGO) yielded remarkable outcomes. The i_MGO demonstrated a superior capability in achieving the lowest Root Mean Square Error (RMSE) values across all tested optimization algorithms. This indicates a significant advancement in the precision of parameter estimations for complex PV models. Additionally, the i_MGO displayed a rapid convergence rate, which is crucial in reducing computational time and resource usage. Notably, the results also underscore the robustness and consistency of the i-MGO, making it an indispensable tool for researchers and practitioners seeking to enhance the efficiency and effectiveness of Triple-Diode PV systems. The consistency of the results across various calculation setups reaffirms the optimizer’s reliability and its potential to serve as a benchmark tool in the field of solar energy optimization. Table 6 shows the comparative performance metrics of the i_MGO algorithm against conventional optimization methods across different PV models.

This section utilizes the i-MGO algorithm to determine the optimal TDM parameters, as specified by RTC France, thereby enabling a thorough assessment of its performance. The results produced by different algorithms in this particular context are displayed in Table 5. It is evident that i_MGO outperforms the others and is the most effective algorithm. Furthermore, the table presents the root mean square error (RMSE) values, which contrast the performance of i_MGO with that of its competitors, significantly differentiating i_MGO from all other algorithms assessed. In addition, the convergence contours of each algorithm are illustrated in Figure 7a, highlighting the superior performance of i_MGO. An examination of this graph indicates that i_MGO attains its minimum root mean square error (RMSE) after around 120 iterations. As shown in 8b and 8d, although i_MGO does not converge the quickest, it is the most precise algorithm when compared to the others in estimating the unknown parameters of the TDM-based RTC France solar cell. The presented data indicate that the I-V and P-V curves estimated by i_MGO and the corresponding measured data exhibit high congruence.

Figure 7c highlights the effect of irradiance variation on PV output using the proposed model. The Triple-Diode Model (TDM) presents an even greater optimization challenge due to its higher dimensionality and increased number of parameters compared to the SDM and DDM. The proposed i_MGO algorithm demonstrates remarkable performance in addressing these challenges, achieving an RMSE of 0.00092975, significantly outperforming competing algorithms.

Figure 8 illustrates a comparative analysis of the execution time (run time) across different optimization algorithms employed for photovoltaic (PV) parameter estimation in the study. The figure provides valuable insight into the computational efficiency of each algorithm when applied to the three PV models: Single-Diode Model (SDM), Double-Diode Model (DDM), and Triple-Diode Model (TDM) on the RTC France cell.

The I_MGO consistently demonstrated a favorable balance between accuracy and speed, achieving shorter convergence times while maintaining high precision, as evident from its RMSE performance. Compared to traditional algorithms like the Sine Cosine Algorithm (SCA), Grey Wolf Optimizer (GWO), and Electric Eel Foraging Optimization (EEFO), the i_MGO reached optimal solutions more rapidly and stabilized sooner during iteration cycles.

In the case of the Single-Diode Model (SDM), the I_MGO displayed superior efficiency by achieving faster convergence compared to its peers. It demonstrated a significant reduction in run time while maintaining minimal RMSE, showcasing its effectiveness in solving simpler PV model structures. Traditional algorithms like SCA and GWO took longer to converge, indicating that their exploration phases may delay optimization in less complex models. The faster stabilization of i_MGO highlights the advantage of its hybrid mechanisms (GSO, PO, LEO) in quickly reaching near-optimal solutions as shown in Figure 8a.

For the more complex Double-Diode Model (DDM), I_MGO continued to outperform others in terms of runtime efficiency. Although the problem complexity increased, I_MGO maintained a reasonable execution time while still achieving low RMSE values. This shows that the enhancements in I_MGO offer not just improved accuracy but also scalability in handling mid-level complexity models. In contrast, algorithms like HO and SSOA showed noticeable delays, possibly due to prolonged search phases required to explore the expanded parameter space as shown in Figure 8b.

With the Triple-Diode Model (TDM) as shown in Figure 8c being the most complex among the three, the run time naturally increased for all algorithms. However, I_MGO still achieved a faster and more stable convergence compared to most alternatives. Its runtime remained consistently lower, especially when compared to exploration-heavy algorithms like COA and AVOA, which required more time to escape local minima. Despite the complexity, I_MGO’s hybrid strategy allowed it to effectively navigate a larger solution space without significant increases in computational burden.

Across all three PV models, I_MGO consistently demonstrated the best trade-off between accuracy and execution time. Its ability to converge faster without compromising solution quality makes it highly effective for real-time PV system modeling. The inclusion of Gradient Search Operator, Production Operator, and Local Escaping Operator significantly enhanced its convergence speed and robustness. The runtime advantage of I_MGO was especially evident in higher-complexity models, proving its versatility and adaptability.

In summary, Figure 8 validates I_MGO as a computationally efficient and accurate optimizer, suitable for diverse PV model complexities—from simple SDM to highly nonlinear TDM.

Figure 9a compares model accuracy under standard conditions, while Figure 9c shows the behavior under partial shading. The reasons for this improved performance include:

The TDM involves complex parameter interactions that can lead to highly non-linear error surfaces. The adaptive behavior of the Modified Transition Factor (mTF) ensures effective navigation of this landscape by emphasizing exploration in the early stages and localized refinement in the later stages. The stochastic nature of the Modified Local Escaping Operator (LEO) is particularly crucial in the TDM, where local minima are more prevalent. This operator introduces diversity into the search process, preventing premature convergence and ensuring better global search capability.The gazelle-inspired optimization framework adapts well to the TDM’s multi-modal landscape by leveraging nature-inspired decision-making processes. This adaptability gives i_MGO a significant advantage over algorithms like PSO and GWO, which often struggle in such high-dimensional spaces.

So that that the i_MGO not only provides accurate parameter estimation but also ensures faster convergence, making it suitable for real-time PV modeling applications where computational time is a critical factor.

This section presents a detailed comparative analysis assessing the robustness and statistical performance of the Improved Mountain Gazelle Optimizer (i_MGO) against various established optimization algorithms across different PV models. The analysis focused on multiple performance metrics, including Root Mean Square Error (RMSE), convergence speed, and consistency across various runs. Statistical evaluations demonstrated that the i_MGO consistently achieved superior performance metrics, showing lower RMSE values and faster convergence rates compared to competitors such as the Grey Wolf Optimizer (GWO) and the Harris Hawks Optimization (HHO). Furthermore, the i_MGO displayed exceptional robustness in handling different parameter estimation challenges within Single-Diode, Double-Diode, and Triple-Diode models, affirming its reliability and efficiency. This comparative study not only highlights the strengths of the i_MGO but also provides critical insights into its potential for broader application in optimizing complex systems in the renewable energy sector.

In Section 7-a, 7-b, 7-c a comparison is made among the three PV models (TPVM). The efficacy of various approaches utilizing the proposed i_MGO algorithm is evaluated within the framework of TPVM. The effectiveness of each algorithm is assessed in terms of its convergence curve, minimum Root Mean Square Error (RMSE) value, and duration. The standard deviation (SD) is employed concurrently to evaluate the system’s dependability.

In the areas of accuracy and reliability, the i_MGO outcomes are superior. According to our analysis of the results, the i_MGO has the highest accuracy for the SDM, followed by the HHO, SSOA, HO, LAPO, EEFO, GWO, AVOA, MGO, COA, and SCA in that order.

The convergence curve shows satisfactory iterations to reach the least RMSE. Convergence time is very comparable relative to the fastest algorithms. On the other hand, the best accuracy is achieved.

For SDM, DDM, and TDM, the optimal RMSE values for the suggested i_MGO algorithm are 0.00081373, 0.00073908 and 0.00092975, respectively. In addition to displaying the P-V and I-V curves, Figures 6b,d, 7b,d, 9b,d show the absolute error value between the simulated and analysis results for power and current. These figures states efficiency and outperformance of the suggested algorithm i_MGO.