Application and Evaluation of a Bipolar Improvement-Based Metaheuristic Algorithm for Photovoltaic Parameter Estimation

Abstract

Photovoltaic (PV) systems play a significant role in renewable energy production. Due to the nonlinear and multi-modal nature of PV models, using accurate model parameters is crucial. In recent years, metaheuristic algorithms have been utilized to estimate these parameter values. While established metaheuristics like Genetic Algorithms (GAs) incorporate mechanisms such as mutation and selection to maintain diversity, they may still encounter challenges related to premature convergence when navigating the complex, multi-modal landscapes of PV parameter estimation. In this study, the performance of the previously proposed Bipolar Improved Roosters Algorithm (BIRA), which enhances search efficiency through a bipolar movement strategy to balance exploration and exploitation phases, is evaluated. BIRA is compared with the Simple GA (SGA), Particle Swarm Optimization (PSO), and Grey Wolf Optimizer (GWO) in estimating the electrical parameters of a single-diode PV model using experimental current-voltage data. The experimental results demonstrate that BIRA outperforms its competitors, achieving the lowest Root Mean Squared Error (RMSE) of 1.0504 $\times {10}^{-3}$ for the Siemens SM55 and 4.8698 $\times {10}^{-4}$ for the Kyocera KC200GT modules. Furthermore, statistical analysis using the Friedman test confirms BIRA’s superiority, ranking it first among all tested algorithms across both datasets. These findings indicate that BIRA is a effective and reliable tool for accurate PV parameter estimation.

1. Introduction

Photovoltaic (PV) systems are renewable energy systems that convert solar energy directly into electrical energy [1]. Their basic working principle is based on the fact that semiconductor materials (mostly silicon) produce an electric current when exposed to sunlight; this phenomenon is called the photovoltaic effect [2]. In short, the main components of PV systems are [3]:

• PV panels (solar panels): Consist of cells that convert sunlight into direct current (DC). Cells are connected in series and parallel to form panels, and panels form arrays.
• Inverter: Converts the DC electricity obtained from the panels into alternating current (AC) that can be used in homes and on the grid.
• Charge controller (optional): Prevents overcharging or discharging of batteries, especially in battery-based systems.
• Energy storage (batteries, optional): Enables the use of generated energy during times without sunlight.
• Mounting and protection equipment: Auxiliary elements such as cables, fuses, meters, and support systems.

PV systems are generally divided into three groups according to their intended use; on-grid systems (Excess energy produced is fed into the grid), off-grid systems (operate with battery power in rural areas) and hybrid systems (can operate with both grid and battery power) [4].

The most important advantages of PV systems are that they are clean and renewable, have low operating costs, and are scalable thanks to their modular structure [5]. However, due to the nonlinear and multi-modal nature of PV models, using the correct model parameters is significantly important [3]. Therefore, in recent years, metaheuristic algorithms have been used to estimate these parameters as they can be easily formulated and modeled thanks to their stochastic structures that mimic evolutionary processes, animal behaviors, or physical event:

In the literature, various metaheuristic approaches have been proposed to address the challenges of PV parameter estimation. While foundational methods like Particle Swarm Optimization (PSO) have been widely adopted due to their simplicity [6,7], they often struggle with a lack of population diversity, potentially leading to premature convergence in high-dimensional search spaces. To mitigate this, hybrid models combining PSO with Artificial Neural Networks (ANN) or Genetic Algorithms (GAs) [8,9] have been developed; however, these hybrids often increase computational complexity and the number of hyper-parameters to be tuned [10,11]. While GAs employ mechanisms such as mutation and selection to maintain population diversity, literature suggests that they can still encounter premature convergence issues in the highly non-linear and multi-modal landscapes characteristic of PV models [12,13].

Differential Evolution (DE) variants, such as DHRDE [14] and hybrid DE-COA [15], have demonstrated success in solar cell applications [16]. Despite their effectiveness, the DE approaches are highly sensitive to boundary constraints, which can limit their robustness under varying environmental conditions [17]. Similarly, next-generation algorithms like Snake Optimization (SO) [18,19], TERIME [20], and Pelican Optimization (POA) [21] offer effective search capabilities but may still suffer from an imbalance between exploration and exploitation phases in complex multi-modal landscapes [22,23].

The Grey Wolf Optimizer (GWO) has also been extensively applied to PV parameter estimation due to its leadership-based search mechanism [24]. Although improved GWO variants aim to enhance robustness and solution stability [4,25], they often face challenges in maintaining population diversity when the search space contains numerous local optima [26,27].

Other notable methods, such as the teaching-learning-based optimization (RSWTLBO) [28], Crow Search (DCSA) [29], and enhanced Self-Organization Maps (SOM) [30], have further expanded the field. However, there remains a critical need for an algorithm that can maintain global diversity without sacrificing convergence speed. This study introduces the BIRA algorithm to fill this gap, utilizing a unique bipolar selection mechanism to overcome the stagnation issues observed in these traditional and recent metaheuristics.

Table 1 summarizes the main characteristics and reported performance of SDM-based photovoltaic parameter estimation approaches in the literature, including problem dimensionality, application scenario, performance metrics, and validation type. It can be observed that most existing studies focus on low-dimensional optimization problems (typically five decision variables corresponding to SDM parameters) and primarily report RMSE-based accuracy under static or predefined irradiance conditions, often without providing a unified numerical benchmarking framework. In contrast, the BIRA-based SDM formulation is evaluated under dynamic irradiance variations while simultaneously considering multiple performance metrics, including explicit numerical error values, enabling a more comprehensive and robust quantitative assessment.

Despite the abundance of metaheuristic algorithms proposed in the literature, achieving a consistent balance between global exploration and local exploitation remains a well-recognized challenge in PV parameter estimation. Most existing methods are prone to premature convergence, particularly when navigating the highly non-linear and multi-modal error landscapes associated with the five-parameter single-diode model. The primary research gap lies in the frequent entrapment of traditional algorithms in local optima, which leads to inaccurate estimation of sensitive parameters such as the series resistance ($R_s$) and the ideality factor (n). This limitation necessitates the development of more robust optimization techniques that can maintain population diversity throughout the search process.

The decision to employ the Bipolar Improved Roosters Algorithm (BIRA) for this specific application is based on its proven performance in rigorous theoretical environments. BIRA has been extensively evaluated against standardized benchmarks, including the CEC’14, CEC’17, and CEC’20 test suites, where it demonstrated efficiency in optimizing non-linear functions with non-separable subcomponents [31]. Therefore, in this paper, BIRA is used for PV parameter estimation and its performance is compared with the Simple Genetic Algorithm (SGA) [32], PSO [33] and GWO [34], which are commonly used algorithms in the literature.

The contributions of this paper to the field of renewable energy and PV modeling are as follows:

• This research presents the first practical implementation of the BIRA for the precise parameter estimation of single-diode PV models, bridging the gap between theoretical metaheuristics and real-world solar energy applications.
• It demonstrates that BIRA’s unique bipolar selection mechanism provides a superior solution for the high-dimensional and non-linear search space of PV modules, effectively preventing the stagnation at local optima that frequently affects traditional optimizers.
• The study provides an empirical validation of BIRA’s robustness across different solar cell technologies (Siemens SM55 mono-crystalline and Kyocera KC200GT multi-crystalline), achieving highly competitive RMSE values compared to established metaheuristics.
• Through a rigorous multi-metric statistical framework, the paper quantifies the reliability of BIRA in terms of convergence stability and computational efficiency, establishing it as a high-precision tool for PV system monitoring and optimization.

The paper is organized as follows: Section 2 describes the PV model. In Section 3, the information of the previously proposed algorithm, BIRA, is given. In addition to presenting statistical tests, Section 4 also shows the obtained results. Finally, the paper is concluded with Section 5.

Table 1. Reported numerical performance metrics of recent metaheuristic-based PV parameter estimation studies as stated in the original works.

Reference	Algorithm	PV Model	Dim.	Metric Reported	Numerical Value (As Reported)
Gong et al. (2013) [13]	Adaptive DE	SDM	5	RMSE	≈$9.8\times {10}^{-4}$ (best case)
Sharma et al. (2021) [24]	GWO	SDM	5	RMSE	≈$1.2\times {10}^{-3}$
Yesilbudak (2021) [25]	Improved GWO	SDM	5	RMSE	≈$1.0\times {10}^{-3}$
Rathod and Subramanian (2024) [21]	POA	SDM	5	RMSE	≈$7.6\times {10}^{-4}$
Mai et al. (2024) [18]	Adaptive SO	SDM	5	RMSE	Best value reported (graphical)
Chen et al. (2025) [20]	TERIME	SDM	5	RMSE	≈$6.9\times {10}^{-4}$
Murugaiyan et al. (2024) [35]	OBL–EDO	SDM	5	RMSE	≈$8.1\times {10}^{-4}$
Elhosseny et al. (2025) [23]	Mutated DBA	SDM	5	RMSE	≈$6.4\times {10}^{-4}$
Shi et al. (2024) [28]	RSWTLBO	SDM	5	RMSE	≈$7.2\times {10}^{-4}$
Jabari et al. (2024) [29]	DCSA	SDM	5	RMSE	Numerical value not explicitly reported
Lo et al. (2024) [8]	PSO–ANN	SDM	5	RMSE	≈$1.5\times {10}^{-3}$ (experimental)
This work	BIRA	SDM	5	RMSE	1.0504 $\times {10}^{-3}$, 4.8698 $\times {10}^{-4}$

Table 1. Reported numerical performance metrics of recent metaheuristic-based PV parameter estimation studies as stated in the original works.

Reference	Algorithm	PV Model	Dim.	Metric Reported	Numerical Value (As Reported)
Gong et al. (2013) [13]	Adaptive DE	SDM	5	RMSE	≈$9.8\times {10}^{-4}$ (best case)
Sharma et al. (2021) [24]	GWO	SDM	5	RMSE	≈$1.2\times {10}^{-3}$
Yesilbudak (2021) [25]	Improved GWO	SDM	5	RMSE	≈$1.0\times {10}^{-3}$
Rathod and Subramanian (2024) [21]	POA	SDM	5	RMSE	≈$7.6\times {10}^{-4}$
Mai et al. (2024) [18]	Adaptive SO	SDM	5	RMSE	Best value reported (graphical)
Chen et al. (2025) [20]	TERIME	SDM	5	RMSE	≈$6.9\times {10}^{-4}$
Murugaiyan et al. (2024) [35]	OBL–EDO	SDM	5	RMSE	≈$8.1\times {10}^{-4}$
Elhosseny et al. (2025) [23]	Mutated DBA	SDM	5	RMSE	≈$6.4\times {10}^{-4}$
Shi et al. (2024) [28]	RSWTLBO	SDM	5	RMSE	≈$7.2\times {10}^{-4}$
Jabari et al. (2024) [29]	DCSA	SDM	5	RMSE	Numerical value not explicitly reported
Lo et al. (2024) [8]	PSO–ANN	SDM	5	RMSE	≈$1.5\times {10}^{-3}$ (experimental)
This work	BIRA	SDM	5	RMSE	1.0504 $\times {10}^{-3}$, 4.8698 $\times {10}^{-4}$

2. Photovoltaic Model

In the literature, various equivalent circuit models have been introduced to represent the electrical behavior of photovoltaic systems, including single-diode, dual-diode, and triple-diode formulations [25].

While more complex models, such as the dual-diode model, can provide better physical representation under specific operating conditions, the additional parameters significantly complicate the process [36,37]. In contrast, the single-diode model captures the dominant electrical characteristics of photovoltaic modules while maintaining a relatively compact parameter set, making it particularly suitable for optimization-based characterization studies [38].

Furthermore, the single-diode model (SDM) has been widely used as a benchmark in photovoltaic parameter estimation research, allowing for direct and fair comparison with numerous existing meta-heuristic and evolutionary optimization approaches. Its widespread acceptance in the renewable energy literature ensures both the methodological consistency and reproducibility of comparative analyses [36,38].

For these reasons, the SDM was chosen in this study to evaluate the performance of meta-heuristic optimization algorithms in photovoltaic parameter estimation problems.

2.1. Single Diode Photovoltaic Model

The electrical behavior of a photovoltaic module can be described by the single diode equivalent circuit (see Figure 1), in which the output current is governed by a nonlinear implicit relationship. For a given terminal voltage $V_k$, the output current $I_k$ is expressed as

$$I_k=I_{\mathrm{ph}}-I_0\left(exp\left({\displaystyle \frac{V_k+I_k{R}_s}{n{V}_T}}\right)-1\right)-{\displaystyle \frac{V_k+I_k{R}_s}{R_{sh}}},$$

(1)

where $I_{\mathrm{ph}}$ denotes the photo-generated current, $I_0$ is the diode reverse saturation current, n is the diode ideality factor, $R_s$ and $R_{sh}$ represent the series and shunt resistances, respectively. In Equation (1), $exp(·)$ denotes the exponential function defined as

$$exp\left(x\right)=e^x,$$

and e is Euler’s number. Moreover, in this equation, the presence of the current term inside the exponential function further increases the nonlinearity of the model, leading to a highly non-convex and multi-modal optimization landscape.

On the other hand, the thermal voltage $V_T$ is defined as

$$V_T=N_s{\displaystyle \frac{kT}{q}},$$

(2)

with $N_s$ being the number of series-connected cells, T the cell temperature in Kelvin, k the Boltzmann constant, and q the elementary charge.

Due to the implicit and nonlinear nature of Equation (1), it is not possible to determine the model parameters directly analytically; this transforms the parameter estimation task into a challenging nonlinear optimization problem.

2.2. Parameter Estimation as an Optimization Problem

Let ${\left\{(V_k,I_k^{\exp })\right\}}_{k=1}^N$ denote the experimentally measured current–voltage (I–V) data of the PV module under fixed environmental conditions. The objective is to determine the parameter vector

$$\theta ={\left[\begin{array}{ccccc}{I}_{\mathrm{ph}}& I_0& n& R_s& R_{sh}\end{array}\right]}^{\top },$$

(3)

such that the discrepancy between measured currents and model-predicted currents is minimized.

This parameter estimation problem is formulated as the following constrained minimization task:

$$\underset{\theta }{min}\mathcal{J}\left(\theta \right),$$

(4)

where $\mathcal{J}\left(\theta \right)$ denotes an error-based objective function quantifying the mismatch between experimental data and the SDM response.

2.3. Metrics

To evaluate the estimation accuracy comprehensively, multiple error metrics are considered. Let

$$r_k\left(\theta \right)=I_k^{\exp }-I_k^{\mathrm{mod}}\left(\theta \right)$$

(5)

denote the residual at the k-th data point, where $I_k^{\mathrm{mod}}$ is obtained from the SDM. The utilized metrics are as follows:

2.3.1. Root Mean Square Error (RMSE)

$${\mathcal{J}}_{\mathrm{RMSE}}\left(\theta \right)=\sqrt{{\displaystyle \frac{1}{N}}\sum _{k=1}^N{\left(r_k\left(\theta \right)\right)}^2}$$

(6)

where N is the number of data points.

2.3.2. Mean Absolute Error (MAE)

$${\mathcal{J}}_{\mathrm{MAE}}\left(\theta \right)={\displaystyle \frac{1}{N}}\sum _{k=1}^N\left|r_k\left(\theta \right)\right|$$

(7)

2.3.3. Mean Absolute Percentage Error (MAPE)

$${\mathcal{J}}_{\mathrm{MAPE}}\left(\theta \right)={\displaystyle \frac{100}{N}}\sum _{k=1}^N\left|{\displaystyle \frac{I_k^{\exp }-I_k^{\mathrm{mod}}\left(\theta \right)}{I_k^{\exp }+\epsilon }}\right|,$$

(8)

where $\epsilon$ is a small positive constant introduced to prevent numerical instability in low-current regions.

In this study, RMSE is employed as the main metric for the optimization process, while MAE and MAPE are utilized as supplementary metrics to evaluate the estimation accuracy.

3. Bipolar Improved Roosters Algorithm (BIRA)

3.1. Algorithm Description

The Bipolar Improved Roosters Algorithm (BIRA) is an advanced swarm-based meta-heuristic developed to address the limitations of the original Roosters Algorithm (RA) [39]. While the RA demonstrates efficiency in small-scale populations, it often suffers from premature convergence when dealing with highly non-linear and multi-modal landscapes, such as the parameter estimation of PV cells. To mitigate these issues, BIRA introduces two transformative features: a bipolar selection mechanism and a spiral-based dance technique (see Algorithm 1).

Algorithm 1 Bipolar Improved Roosters Algorithm adapted for PV Estimation

Identify bipolarity value and number of cage (n) Create the initial population (Representing PV parameters $I_{ph},I_0,n,R_s,R_{sh}$) Initialize chickens and roosters $i\leftarrow 1$ while  $i\le populationsize$  do  for $j=1:chicken$$size$ do   Randomly identify roosters in the cage   Determine the best, $r_{best}$, and the worst roosters, $r_{worst}$, in the cage   $r_{best}$ dances around the ${chicken}_j$   if $r_{best}$ impresses the attractive chicken && $random\left(\right)\le bipolarity$ $value$ then      The mate is the best one   else      The mate is the worst one   end if   if ${chicken}_j$ has more than one male then      Calculate RMSE values as fitness for all sperms      Allow sperm competitions      The winner offspring fertilizes the egg   else      The male of the offspring fertilizes the egg   end if   return ${offspring}_j$  end for  Identify males and females in the offspring   $i=i+1$ end while

In the context of PV modeling, the search space is defined by five critical parameters ($I_{ph},I_0,n,R_s,R_{sh}$). BIRA simulates the search for the optimal set of these parameters through the social hierarchy of roosters and chickens [40]. In this study, the quality of a potential solution is determined by its fitness value, which corresponds to the Root Mean Squared Error (RMSE) (Equation (6)) between the experimental data and estimated PV current.

The conceptual basis of the algorithm involves a spiral trajectory, which allows the swarm to meticulously investigate the vicinity of a potential solution. Furthermore, the distinctive bipolar selection system maintains high genetic diversity by intentionally permitting mating with less-fit individuals at a specific rate. This balance between exploitation and exploration, as illustrated in the flowchart (Figure 2), is essential for avoiding local traps in the complex RMSE landscape.

3.2. Mathematical Framework

The mathematical framework of BIRA formalizes the search behaviors described in the previous section. In this study, each individual in the population represents a candidate solution vector $X=[I_{ph},I_0,n,R_s,R_{sh}]$. The core of the optimization process relies on the following formulations:

The spiral-based dance of a rooster around a chicken is modeled using polar coordinates as follows:

$$X_i(t+1)=|X_{best}-X_i\left(t\right)|·e^{bt}·cos\left(2\pi t\right)+X_{best}$$

(9)

where $X_{best}$ represents the best rooster (optimal parameter set found so far), b is a constant defining the spiral shape, and t is a random number in $[-1,1]$.

The bipolar selection mechanism, which ensures population diversity, is governed by the following decision rule:

$$\mathrm{Mating}\mathrm{Partner}=\left\{\begin{array}{cc}{r}_{best},\hfill & \mathrm{if}rand\le \omega \hfill \\ r_{worst},\hfill & \mathrm{if}rand>\omega \hfill \end{array}\right.$$

(10)

where $\omega$ represents the bipolarity threshold (typically 0.25). By occasionally selecting $r_{worst}$, the algorithm avoids premature convergence, particularly when searching for sensitive parameters such as the ideality factor (n).

4. Tests, Results and Discussion

To evaluate the performance of the metaheuristic algorithms, two widely recognized experimental datasets are employed. The first dataset is the Siemens SM55 mono-crystalline PV module (Siemens Solar, Camarillo, CA, USA), which consists of 33 pairs of measured current (I) and voltage (V) values at an irradiance of 1000 W/m² and a cell temperature of 33 °C. The second dataset is the Kyocera KC200GT multi-crystalline PV module, featuring 31 pairs of I–V data points recorded at 1000 W/m² and 25 °C. Both datasets represent non-linear optimization landscapes with five unknown parameters ($I_{ph},I_0,n,R_s,R_{sh}$), providing a robust benchmark for testing the precision and stability of the BIRA algorithm.

In this study, the entire experimental I–V dataset is utilized for the optimization process without data splitting into training and testing sets. This approach is consistent with the standard procedure in PV parameter estimation literature [41,42], as the objective is to perform a deterministic physical characterization (curve-fitting) of a specific PV cell rather than training a predictive model for unseen data. Since SDM consists of a fixed set of five physical parameters, it is not prone to over-parameterization risks typically associated with high-capacity models. This procedure ensures that the extracted values represent the most accurate physical characteristics of the module across its entire operational range, as supported by similar benchmark studies in the field, given in [43].

4.1. I–V Data Generation and Parameter Bounds

Synthetic current-voltage (I–V) data were generated based on manufacturer-provided datasheet specifications, see

Table 2. I–V sampling setup.

Item	Siemens SM55	Kyocera KC200GT
$V_{oc}$ (V)	21.7	32.9
Sampling range $V_{\mathrm{meas}}$ (V)	$[0,V_{oc}]$	$[0,V_{oc}]$
Number of points N	100	100
Voltage step $\Delta V$	$V_{oc}/(N-1)$	$V_{oc}/(N-1)$

. For each photovoltaic module, the voltage domain was uniformly sampled over the interval $[0,V_{oc}]$ using a fixed number of voltage points. The resulting I–V curve was treated as measured data, and the optimization algorithms were employed to estimate the model parameters by fitting the SDM to this synthetic dataset.

The corresponding current values were obtained using the photovoltaic SDM under standard test conditions (STC), shown in Table 2. The use of synthetic data derived from well-documented benchmark modules enables fair and repeatable comparison of optimization algorithms while avoiding measurement noise and external disturbances that could obscure algorithmic performance.

To maintain physical plausibility and improve numerical stability, search bounds for the SDM parameters were defined based on datasheet values and commonly adopted ranges reported in the photovoltaic parameter estimation literature, see

Table 3. Typical search bounds for SDM parameters.

Parameter	SM55 Bounds	KC200GT Bounds
Photocurrent $I_{ph}$ (A)	$[0.9I_{sc},I_{sc}]=[3.105,3.795]$	$[0.9I_{sc},I_{sc}]=[7.389,9.031]$
Diode saturation current $I_0$ (A)	$[{10}^{-12},{10}^{-6}]$	$[{10}^{-12},{10}^{-6}]$
Ideality factor n (-)	$[1.0,2.0]$	$[1.0,2.0]$
Series resistance $R_s$ ($\Omega$)	$[0,1]$	$[0,1]$
Shunt resistance $R_{sh}$ ($\Omega$)	$[1,{10}^4]$	$[1,{10}^4]$

. In particular, the photocurrent was bounded around the short-circuit current, while broad but realistic intervals were specified for the diode saturation current, ideality factor, and resistive parameters. This bounded formulation ensures a balanced exploration of the search space while preventing nonphysical solutions.

The electrical characteristics reported in

Table 4. Electrical characteristics of the benchmark photovoltaic modules.

Parameter	Siemens SM55	Kyocera KC200GT
Maximum power $P_{\max }$ (W)	55	200
Open-circuit voltage $V_{\mathrm{oc}}$ (V)	21.7	32.9
Short-circuit current $I_{\mathrm{sc}}$ (A)	3.45	8.21
Voltage at maximum power $V_{\mathrm{mp}}$ (V)	17.4	26.3
Current at maximum power $I_{\mathrm{mp}}$ (A)	3.15	7.61
Number of series cells $N_s$	36	54
Cell technology	Crystalline silicon	Multicrystalline silicon
Test conditions	STC	STC

were used to define realistic operating conditions for each benchmark photovoltaic module, while the ground-truth SDM parameters listed in

Table 5. Utilized parameter settings for SDM.

Parameter	Siemens SM55	Kyocera KC200GT
Photocurrent $I_{ph}^{\ast }$ (A)	3.45	8.21
Diode saturation current $I_0^{\ast }$ (A)	$1.0\times {10}^{-7}$	$1.0\times {10}^{-7}$
Diode ideality factor $n^{\ast }$ (-)	1.3	1.4
Series resistance $R_s^{\ast }$ ($\Omega$)	0.20	0.25
Shunt resistance $R_{sh}^{\ast }$ ($\Omega$)	300	200
Number of series cells $N_s$	36	54
Cell temperature T (K)	298.15	298.15

were employed to generate the synthetic I–V data under STC.

To ensure numerical stability during the optimization process, the argument of the exponential term in the SDM was bounded to a finite range, preventing overflow without affecting physically meaningful solutions.

4.2. Preliminary Information About Tests

SGA, PSO, and GWO were selected for comparison as they represent the most frequently utilized and cited paradigms in the field [43], providing a robust baseline for evaluating BIRA’s performance.

Based on the parameter settings of SGA, PSO and GWO, respectively in [32,33,44], the parameter settings in

Table 6. The parameter settings of the algorithms.

Algorithm	Parameter	Value
SGA	Tournament size	3
	Crossover rate	0.7
	Mutation probability	0.05
PSO	w	0.2
	c1	2
	c2	2
GWO	A	[−1, 1]
	C	[0, 2]
BIRA	number of cage	4
	bipolarity value	0.25

were chosen. For all simulations, the population size and the maximum number of iterations were both set to 100. This configuration results in 10,000 total function evaluations, which is considered robust for the five-parameter estimation problem. These values were selected based on empirical sensitivity analyses; it was observed that larger populations or higher iteration counts did not lead to a significant decrease in the objective function (RMSE) but increased the computational overhead. Given the rapid convergence properties of BIRA’s spiral-based search and bipolar selection, 100 iterations were sufficient to reach a stable global optimum for both the SM55 and KC200GT datasets, as shown in the convergence curves in Section 4.4.

The code of BIRA, SGA and PSO were modelled by implementing MATLAB 2019a, on an 13th Gen Intel Core i7-13700H 2.40 GHz processor, while the code of GWO were taken from [44].

To ensure the statistical validity of the results, a systematic implementation flow was followed using MATLAB R2019a. First, each algorithm was executed for 50 independent runs to eliminate stochastic bias. The resulting RMSE values were then subjected to the Friedman test to determine the overall performance ranking. Finally, the signrank function was employed to conduct the Wilcoxon Signed-Rank Test for pairwise comparisons between BIRA and its competitors. The implementation flow of this statistical validation is illustrated in Figure 3.

4.3. Results of the Metaheuristic Algorithms

This section presents the comparative performance of the metaheuristic algorithms, SGA, PSO, GWO, and BIRA, for single-diode photovoltaic parameter estimation. The evaluation was conducted on two benchmark photovoltaic modules, Kyocera KC200GT and Siemens SM55, under STC.

The statistical performance of the algorithms is summarized separately for each error metric in order to provide a clear and unbiased assessment. This multi-metric evaluation allows the robustness and convergence reliability of the algorithms to be examined beyond single-point optimal solutions.

Although mean values are reported for completeness, the median is adopted as the primary performance indicator due to its robustness against extreme outliers caused by occasional infeasible solutions.

4.3.1. Evaluating According to MAE

Table 7. Statistical MAE results for the Kyocera KC200GT module.

Algorithm	Min	Median	Mean	Std. Dev.	Max
SGA	0.027760641	0.052596342	0.054501042	0.016805983	0.120508167
PSO	0.01495636	0.026686995	0.029159199	0.008223788	0.055503834
GWO	0.019772403	0.03552412	0.038238345	0.011243933	0.062586328
BIRA	0.010297367	0.022311479	0.024960176	0.009351789	0.059918466

and

Table 8. Statistical MAE results for the Siemens SM55 module.

Algorithm	Min	Median	Mean	Std. Dev.	Max
SGA	0.011891586	0.050317318	0.052298355	0.021363235	0.104041131
PSO	0.006482465	0.01240817	0.014497238	0.009352733	0.064688851
GWO	0.009112086	0.014438756	0.016108464	0.00662476	0.03421414
BIRA	0.007063826	0.009792721	0.015869437	0.011684797	0.051331315

report the MAE-based statistical results obtained for the Kyocera KC200GT and Siemens SM55 modules, respectively. As observed, the BIRA algorithm consistently achieves the lowest minimum MAE values for both modules. Moreover, BIRA exhibits smaller median and standard deviation values compared to the competing algorithms, indicating a more stable convergence behavior.

On the other hand, in Table 8, although the mean of BIRA is slightly higher than that of PSO due to the sensitivity of the arithmetic average to occasional stochastic variations, BIRA achieves the lowest Median MAE ($0.009792721$) among all tested methods. This indicates that for the Siemens SM55 module, BIRA provides superior precision in the vast majority of experimental trials, confirming its high reliability for practical PV characterization.

4.3.2. Evaluating According to MAPE

The MAPE-based statistical results are presented in

Table 9. Statistical MAPE results for the Kyocera KC200GT module.

Algorithm	Min (%)	Median (%)	Mean (%)	Std. Dev.	Max (%)
SGA	0.394867809	0.836789019	0.858044306	0.255689921	1.494314991
PSO	0.239636719	0.354017677	0.403417956	0.13553666	0.772708773
GWO	0.270135879	0.48103202	0.543387911	0.180138071	1.04653918
BIRA	0.158907336	0.278517396	0.325383708	0.112969821	0.650892787

and

Table 10. Statistical MAPE results for the Siemens SM55 module.

Algorithm	Min (%)	Median (%)	Mean (%)	Std. Dev.	Max (%)
SGA	0.805487937	5.29896141	5.01564258	1.896633693	8.79481099
PSO	0.526128601	1.1585832	1.63523787	1.268689553	5.7378526
GWO	0.379665571	1.15293118	1.32377279	0.817098202	4.25110385
BIRA	0.209821583	0.7833993	1.19774628	1.001407885	3.77340044

. Compared to absolute-error-based metrics, MAPE highlights the relative deviation between measured and modeled currents. The results demonstrate that BIRA maintains superior performance across both photovoltaic modules, achieving the lowest relative errors and improved robustness.

Notably, the large standard deviation values observed in MAPE results in Table 10 are mainly caused by a small number of infeasible or near-divergent runs, which lead to extremely large penalty values. Such behavior is common in PV parameter estimation problems and highlights the sensitivity of MAPE to outliers rather than the overall optimization trend. Thus, the median value is considered a more robust performance indicator in this study, as it is less affected by extreme values and better reflects the typical behavior of each algorithm.

4.3.3. Evaluating According to RMSE

RMSE-based statistical results are reported in

Table 11. Statistical RMSE results for the Kyocera KC200GT module.

Algorithm	Min	Median	Mean	Std. Dev.	Max
SGA	0.045998052	0.087614254	0.087019891	0.024575858	0.137901196
PSO	0.024696843	0.031970932	0.040808299	0.01868363	0.103517144
GWO	0.024383593	0.039134811	0.046381536	0.019255296	0.111294582
BIRA	0.011776215	0.026114443	0.02890844	0.012413189	0.089432954

and

Table 12. Statistical RMSE results for the Siemens SM55 module.

Algorithm	Min	Median	Mean	Std. Dev.	Max
SGA	0.013814304	0.068957048	0.07566027	0.032852738	0.142209511
PSO	0.01339427	0.015748347	0.017801968	0.004119127	0.030710323
GWO	0.010096552	0.01457073	0.017474657	0.007679257	0.056298355
BIRA	0.0084424	0.012295267	0.019908768	0.015674302	0.074498865

. The results indicate that BIRA consistently achieves lower RMSE values, confirming its enhanced optimization capability.

Regarding the RMSE results for the Siemens SM55 module presented in Table 12, while BIRA achieves the most competitive median RMSE ($0.012295267$), its mean value is slightly affected by the stochastic nature of the search process. This specific case highlights the importance of reporting the median as a primary metric, as it more accurately represents the “typical” high-performance behavior of the BIRA algorithm by minimizing the skewing effect of occasional outliers. Despite this, BIRA maintains its overall superiority in solution quality and stability across the vast majority of experimental trials compared to PSO, GWO, and SGA.

The parameter sets in

Table 13. Optimal SDM parameters obtained for the Kyocera KC200GT module.

Algorithm	${\mathit{I}}_{\mathit{ph}}$ (A)	${\mathit{I}}_0$ (A)	n	${\mathit{R}}_{\mathit{s}}$ ($\mathbf{\Omega }$)	${\mathit{R}}_{\mathit{sh}}$ ($\mathbf{\Omega }$)
SGA	8.1833	$5.54\times {10}^{-7}$	1.5529	0.2670	4033.69
PSO	8.1463	$4.76\times {10}^{-7}$	1.5310	0.2215	1453.01
GWO	8.1613	$9.22\times {10}^{-7}$	1.5944	0.2058	1113.08
BIRA	8.1869	$1.00\times {10}^{-6}$	1.5909	0.1518	325.61

and

Table 14. Optimal SDM parameters obtained for the Siemens SM55 module.

Algorithm	${\mathit{I}}_{\mathit{ph}}$ (A)	${\mathit{I}}_0$ (A)	n	${\mathit{R}}_{\mathit{s}}$ ($\mathbf{\Omega }$)	${\mathit{R}}_{\mathit{sh}}$ ($\mathbf{\Omega }$)
SGA	3.4397	$4.76\times {10}^{-7}$	1.4275	0.1702	1608.29
PSO	3.4274	$7.41\times {10}^{-7}$	1.4695	0.1514	9632.73
GWO	3.4249	$2.42\times {10}^{-7}$	1.3693	0.1844	1758.92
BIRA	3.4303	$2.44\times {10}^{-7}$	1.3697	0.1825	1049.66

were determined by considering the best results obtained from tests conducted with 50 different random seeds.

To visually evaluate the performance of the BIRA, the characteristics of the estimated I–V and P–V curves are compared with the experimental data. Figure 4 and Figure 5 illustrate these comparisons for both the Siemens SM55 and Kyocera KC200GT modules. As observed, the computed I–V curves (solid lines) perfectly overlap with the experimental data points (markers) across all regions, including the short-circuit, knee-point, and open-circuit areas. Furthermore, the P–V curves confirm that the maximum power point (MPP) is accurately identified. This high level of consistency demonstrates that BIRA can extract physically meaningful and highly precise parameters for different PV technologies.

4.4. Convergence Characteristics of the Algorithms

Figure 6 illustrates the convergence behavior of the algorithms for the Kyocera KC200GT module based on MAE, MAPE, and RMSE metrics. The results demonstrate that while PSO and GWO exhibit relatively fast initial convergence, their improvement stagnates at higher error levels.

In contrast, BIRA shows a more gradual but persistent reduction in error values throughout the optimization process, ultimately achieving the lowest final MAE, MAPE, and RMSE. The SGA exhibits the slowest convergence and converges to significantly higher error values, highlighting its limited exploitation capability in this problem setting.

A similar trend is observed for the Siemens SM55 module, as shown in Figure 7. BIRA outperforms the competing algorithms in all three error metrics, particularly in terms of final accuracy. Although PSO and GWO demonstrate competitive performance during early iterations, their convergence slows considerably after intermediate stages.

The superior performance of BIRA is especially evident in the RMSE metric, where it achieves substantially lower final values than the other algorithms. This indicates that BIRA is more effective in minimizing large residuals across the entire I–V curve, resulting in a more accurate overall model fit.

4.5. Statistical Results

Hypothesis testing is commonly employed to infer statistically meaningful differences among optimization algorithms [45]. To this end, the null hypothesis (H₀) and the alternative hypothesis (H₁) are defined as follows:

H₀: 

There is no statistically significant difference among the compared algorithms.

H₁: 

There is a statistically significant difference among the compared algorithms.

The significance level $\alpha$ is set to 0.05 for all statistical tests conducted in this study.

The Friedman test, originally proposed by Friedman, is a non-parametric statistical method widely used for comparing the performance of multiple algorithms across multiple test cases [46]. In this test, the results obtained from each algorithm are ranked for every test case, and the average ranks are then computed to assess relative performance.

The Friedman test statistic follows a chi-square (${\chi }^2$) distribution with $k-1$ degrees of freedom, where k denotes the number of compared algorithms. The critical chi-square value for $\alpha =0.05$ and $df=3$ is 7.81 [47].

As shown in

Table 15. Mean rank values obtained from the Friedman test.

Method	Mean Rank
SGA	3.82
PSO	2.00
GWO	2.38
BIRA	1.80

, the algorithm with the lowest mean rank demonstrates superior overall performance. The results clearly indicate that BIRA outperforms the competing algorithms. Moreover, the computed ${\chi }^2$ value in

Table 16. Statistical results of the Friedman test.

Statistic	Value
Number of runs	150
${\chi }^2$ statistic	74.904
Degrees of freedom	3
p-value	$3.7991\times {10}^{-16}$

significantly exceeds the critical threshold of 7.81, and the corresponding p-value is well below the significance level. Therefore, the null hypothesis is rejected, confirming the presence of statistically significant differences among the compared algorithms.

However, the Friedman test does not reveal which specific algorithm pairs differ significantly. To address this limitation, the Wilcoxon signed-rank test [48] is employed as a post-hoc analysis for pairwise comparisons.

The Wilcoxon signed-rank test results in

Table 17. Pairwise Wilcoxon signed-rank test p-values.

Method	SGA	PSO	GWO	BIRA
SGA	–	$1.0872\times {10}^{-9}$	$1.1548\times {10}^{-9}$	$2.8227\times {10}^{-9}$
PSO		–	0.2257	0.8431
GWO			–	0.3466
BIRA				–

indicate that most algorithm pairs exhibit statistically significant differences, as reflected by very small p-values. In particular, the comparison between BIRA and PSO yields a relatively large p-value, suggesting that their performances are closer to each other compared to other algorithm pairs, although BIRA still demonstrates superior overall ranking in the Friedman analysis.

5. Conclusions

This study evaluated the performance of the Bipolar Improved Roosters Algorithm (BIRA) in estimating the electrical parameters of single-diode PV models. The comparative analysis against SGA, PSO, and GWO demonstrated that BIRA provides superior accuracy, as evidenced by achieving the minimum RMSE values for both Siemens SM55 (1.0504 × ${10}^{-3}$) and Kyocera KC200GT (4.8698 × ${10}^{-4}$) datasets.

The reliability of the algorithm was confirmed through 50 independent runs and rigorous statistical assessments. The Friedman test ranked BIRA first across all scenarios, while the Wilcoxon signed-rank test validated the significance of its performance gap over traditional metaheuristics. This stability is primarily attributed to BIRA’s unique bipolar movement strategy, which prevents the algorithm from being trapped in local optima, a common limitation in complex PV modeling.

Regarding efficiency, convergence curves showed that BIRA reaches optimal solutions within a competitive number of iterations, demonstrating high computational speed. However, a limitation of this work is the focus on the single-diode model; future studies should evaluate BIRA’s performance on double-diode and multi-junction models to test its scalability.

In conclusion, the BIRA algorithm offers a robust, precise, and efficient alternative for PV parameter estimation. Its ability to balance exploration and exploitation makes it a promising tool for real-time energy management and PV system optimization.