Comparative Study of White Shark Optimization and Combined Meta-Heuristic Algorithm for Enhanced MPPT in Photovoltaic Systems

Abstract

This paper proposes a novel hybrid metaheuristic algorithm (MHA) for maximum power point tracking (MPPT), integrating particle swarm optimization (PSO), the differential evolution algorithm (DEA), and the grey wolf optimizer (GWO). The proposed method is inspired by the structural phases of the white shark optimizer (WSO), a recently introduced MHA. This study evaluates the MPPT performance of WSO and compares it with the proposed hybrid approach to provide insights into optimal MPPT selection. The key contributions include an in-depth analysis of the WSO framework, benchmarking its performance against the hybrid model. Both algorithms are implemented in an MPPT system and assessed based on tracking speed, accuracy, and adaptability. The results indicate that the WSO achieves a faster convergence due to its biologically inspired design, whereas the hybrid model, despite requiring additional coordination time, ensures comprehensive search space exploration. Notably, the proposed method excels in dynamic tracking efficiency, which is crucial for accurately following time-varying P-V curves. This study underscores the trade-off between tracking speed and efficiency, demonstrating that while WSO is advantageous for rapid tracking, the hybrid approach enhances overall MPPT performance under dynamic conditions. These findings offer valuable insights for optimizing MPPT strategies in renewable energy systems.

1. Introduction

The application of meta-heuristic algorithms (MHAs) in maximum power point tracking (MPPT) for photovoltaic (PV) systems has gained significant attention due to their capability to efficiently navigate the nonlinear and complex behavior of PV arrays, especially under partial shading conditions (PSCs). Various MHAs have been developed to tackle these challenges, including the genetic algorithm (GA) [1], horse herd optimization (HHO) [2], the chimp optimization algorithm (ChOA) [3], the horse racing algorithm (HRA) [4], and dung beetle optimization (DBO) [5]. Despite their effectiveness, conventional MHAs often struggle with maintaining a balance between exploration and exploitation, which can result in lower tracking accuracy or extended convergence times.

To overcome these shortcomings, researchers have proposed enhancements to MHAs to improve MPPT performance [6]. In general, five approaches are used for enhancement. The first approach is to combine the MHA with a deterministic method such as perturb and observe (P & O) and incremental conductance (InC). For example, Lian et al. [7] combined the P & O and PSO methods. The P & O method was used to identify the nearest local maximum. From that point onward, the PSO method was applied to locate the global maximum power point (GMPP). This approach reduced the search space for PSO, significantly decreasing the time required for convergence. In ref. [8], the grey wolf optimization (GWO) technique was combined with the conventional P & O method, where GWO was utilized in the initial phase of MPPT, followed by P & O to achieve faster convergence to the GMPP. Nugraha et al. [9] introduced a new MPPT algorithm by integrating the CS algorithm with the golden section search (GSS) to leverage the advantageous features of both methods.

The second approach is to combine various MHAs. For instance, Sathasivam et al. [10] optimized a boost converter’s duty cycle using a hybrid approach that combined the artificial bee colony (ABC) and PSO algorithms, maximizing power extraction efficiency. Liao et al. [11] introduced a hybrid MPPT method that integrates the bat algorithm (BA) and the discard strategy of the cuckoo search (CS) algorithm. Their findings demonstrated that this combination outperformed conventional BA and other advanced MPPT strategies in terms of efficiency and tracking performance. Chao et al. [12] proposed a dual-population MPPT approach that synergized GA with ant colony optimization (ACO), leveraging the advantages of both techniques. GA provided efficient global search capabilities, preventing premature convergence, while ACO enhanced local exploration and escape from local optima. Experimental results showed that this hybrid strategy outperformed the traditional approaches like P&O and standard ACO in terms of speed and efficiency. Chtita et al. [13] investigated a hybrid GWO and PSO algorithm for MPPT in PV arrays, demonstrating superior tracking accuracy, faster GMPP convergence, and improved overall efficiency compared to conventional methods.

The third approach is to adjust parameters such as the mutation rate in the GA based on the search progress. For instance, Refaat et al. [14] developed a self-tuning particle swarm optimization (ST-PSO) algorithm that dynamically adjusted its parameters, improving tracking performance. This method integrated two additional parameters—the cognitive and social components—enhancing the adaptability of the tracking process. Furthermore, an inverse tangent function was employed to constrain the search space, ensuring stability and preventing excessive perturbations. Gundogdu et al. [15] introduced an improved GWO algorithm designed to increase tracking speed. In conventional GWO, wolves approach their target by progressively reducing the search radius at a fixed rate, potentially delaying convergence. The modified approach implemented a decision mechanism that dynamically adjusted the movement speed based on the distance to the target, thereby improving efficiency. Chao et al. [16] refined the cat swarm optimization (CSO) algorithm to enhance tracking performance under variable shading and sudden irradiance fluctuations. Their method utilized a slope-based decision mechanism to dynamically modify the step size, reducing it when near the global maximum power point (GMPP) for precision tuning and increasing it when further away to accelerate convergence. Jabbar et al. [17] proposed an enhanced water cycle optimization (WCO) algorithm for MPPT under PSCs and varying load conditions. By integrating a dynamic control parameter with a chaotic map function, their technique achieved superior tracking efficiency, faster convergence, and reduced power losses compared to the conventional methods. Teshome et al. [18] proposed a modified firefly algorithm (FA) aimed at efficiently identifying the maximum power point in P-V curves. Unlike the standard FA, which directs fireflies toward all brighter counterparts, the improved version calculated a representative position, reducing computational complexity and enhancing convergence speed. Mohammed et al. [19] improved the snake optimizer (SO) algorithm by introducing a fast-response mechanism to identify the difference among uniform shading conditions (USCs) and PSCs. This approach reduced unnecessary search operations, minimizing redundant computations and expediting MPPT tracking. Mo et al. [20] refined the mayfly algorithm (MF) for MPPT applications by addressing common limitations such as slow convergence, low precision, and instability. To enhance tracking speed and accuracy while minimizing oscillations, they introduced a gravity coefficient, analogous to the inertia weight in PSO. The authors from [17] presented the initialization of the raindrops, the updated stream and river positions, raining process, and the stopping criterion of the cycle optimization algorithm for fast-tracking the GMPP under PSCs, along with a new strategy to enhance the convergence speed of the MPPT method during load variations.

The fourth approach is to enhance MHAs based on real-world biological behaviors. For instance, Millah et al. [21] presented an enhanced GWO algorithm by adding the pouncing strategy to improve MPPT accuracy and response time.

The fifth approach is to combine MHAs with artificial intelligence methods such as an artificial neural network or fuzzy logic. Note that fuzzy logic has been developed for MPPT, as presented in [22,23]. A novel approach for deploying the particle swarm optimization algorithm as an S-function was developed and applied to optimize a fuzzy logic controller (FLC) for maximum power tracking in [24]. In ref. [25], fuzzy particle swarm optimization (FPSO) was implemented to improve system performance by refining both the fuzzy membership sets and the decision-making rules. Compared to the conventional PSO, the FPSO-based MPPT algorithm was less oscillatory and tracked the GMPP faster.

In this paper, we propose a new hybrid method, which falls into the category of the second approach. The proposed method combines PSO, the differential evolution algorithm (DEA), and the GWO algorithm. These three methods are quite similar to the three phases of a recently proposed MHA, the white shark optimizer (WSO), which is a newly developed MHA that was introduced by Braik et al. in 2022 [26]. Consequently, this research aims to analyze the MPPT performance of the WSO and compare it with the proposed hybrid approach, offering insights into the selection of an optimal MPPT method. The key contributions of this work are summarized as follows:

• Investigating the structural framework of the WSO and benchmarking its performance against the proposed hybrid MHA techniques;
• Implementing both the WSO and the proposed hybrid model in an MPPT system and analyzing their effectiveness in terms of tracking speed, accuracy, and adaptability.

The remainder of this paper is structured as follows: Section 2 and Section 3 present an overview of the white shark optimizer and the proposed hybrid method, respectively. Section 4 outlines the experimental setup and provides a comparative analysis of the results under both static and dynamic conditions. Finally, the conclusions are presented in Section 5.

2. White Shark Optimizer

The WSO was introduced by Braik et al., taking inspiration from the hunting behavior of great white sharks, especially their exceptional auditory perception and keen sense of smell, which they use for navigation and foraging. The WSO consists of three phases, which are movement speed towards prey, movement towards optimal prey, and movement towards the best white shark. They are delineated in the following subsections.

2.1. Movement Speed Toward Prey

The white shark identifies the position of its prey by sensing disruptions in wave patterns, as described in (1):

$$v_{s,i}^{t+1}=\mu [v_{s,i}^t+p_1(X_{gbest}-X_i^t)\times c_{s,1}+p_2(X_{pbest,i}-X_i^t)\times c_{s,2}],$$

(1)

where t denotes the current iteration and $v_{s,i}^{t+1}$ represents the new velocity vector of the i th particle (that is, the candidate solution) in (t + 1) th step. $v_{s,i}^t$ defines the new velocity vector of the i th particle in t th step. $X_{gbest}$ denotes the optimal global position vector acquired up to the t th iteration. $X_i^t$ denotes the present location vector of the i th particle at the t th iteration. $X_{pbest,i}$ signifies the finest position vector identified by the swarm thus far. $c_{s,1}$ and $c_{s,2}$ are random vectors within the range $[0,1]$.

$$p_1=p_{max}+(p_{max}-p_{min})\times e^{-{(4t/T)}^2}$$

(2)

$$p_2=p_{min}+(p_{max}-p_{min})\times e^{-{(4t/T)}^2}$$

(3)

$$\mu ={\displaystyle \frac{2}{\left|2-\tau -\sqrt{{\tau }^2-4\tau }\right|}}$$

(4)

where T stands for the current and maximum number of iterations. The values $p_{min}=0.5$ and $p_{max}=1.5$ represent the minimum and maximum velocity adjustments, ensuring the effective movement of the white shark. The parameter $\tau$ serves as an acceleration coefficient.

2.2. Movement Toward Optimal Prey

When a white shark detects movement or a scent, it instinctively moves toward its prey. Depending on the situation, the prey may either flee or continue moving, leaving a scent trail. During this pursuit, the shark exhibits random roaming behaviors akin to that of fish shoals foraging. The position update mechanism that models this hunting strategy is expressed in (5), as follows:

$$X_i^{t+1}=\left\{\begin{array}{cc}{X}_i^t·\neg \oplus w_o+UB·a_s+LB·b_s;\hfill & rand\le mv\hfill \\ X_i^t+v_{s,i}^t/f\hfill & \mathrm{else}\hfill \end{array}\right.$$

(5)

where the updated location of the i th white shark is indicated by $X_i^{t+1}$. The logical negation operator ¬, combined with the bit-wise XOR operation ⊕, modifies the search behavior. In addition, rand denotes a stochastic value generated within the interval from 0 to 1, while $mv$ signifies the movement force, which grows progressively with each iteration as the white shark moves closer to its prey, which is described in (10). The binary vectors $a_s$ and $b_s$, as described in (6) and (7), help to regulate the movement within the predefined search space boundaries formed by $UB$ and $LB$, which indicate the maximum and minimum boundaries of the exploration space, respectively. The logical vector $w_o$ is defined in (8), and f denotes the wave motion frequency of the white shark, as expressed in Equation (9).

$$a_s=\mathrm{sgn}(X_i^t-UB)>0$$

(6)

$$b_s=\mathrm{sgn}(X_i^t-LB)<0$$

(7)

$$w_o=\oplus (a_s,b_s)$$

(8)

Equations (6) and (7) ensure that the search remains within valid boundaries, preventing sharks from moving beyond the designated search space.

$$f=f_{min}+{\displaystyle \frac{f_{max}-f_{min}}{f_{max}+f_{min}}}$$

(9)

where $f_{min}$ and $f_{max}$ represent the minimum and maximum oscillation frequencies, respectively.

$$mv={\displaystyle \frac{1}{(a_0+e^{(T/2-t)/a_1})}}$$

(10)

where $a_0=6.25$ and $a_1=100$ control the transition between exploration and exploitation. The parameter $mv$ quantifies the shark’s ability to detect prey through scent and movement, increasing over iterations.

2.3. Movement Toward the Best White Shark

The white shark’s strategy for maintaining an advantageous position is formulated in (11), as follows:

$${\acute{X}}_i^{t+1}=X_{gbest}+r_1{\overrightarrow{D}}_s\mathrm{sgn}(r_2-0.5)$$

(11)

where ${\acute{X}}_i^{t+1}$ represents the revised location of the i th particle, while $\mathrm{sgn}(r_2-0.5)$ determines the search direction, taking values of either 1 or −1. The parameters $r_1$ and $r_2$ are stochastic vectors within the range of $[0,1]$. The term ${\overrightarrow{D}}_s$ defines the distance between the shark and its prey, as expressed in Equation (12).

$${\overrightarrow{D}}_s=\left|rand\times (X_{gbest}-X_i^t)\right|$$

(12)

Additionally, $s_s$ models the shark’s sensory intensity for tracking nearby sharks aiming at the optimal prey, as expressed in Equation (13).

$$s_s=\left|1-e^{(-a_2\times t/T)}\right|$$

(13)

where $a_2=0.0005$ governs the balance between exploration and exploitation.

In order to simulate the behavior of a school of white sharks mathematically, the top two optimal solutions are kept, and the other white sharks’ positions are modified in accordance with these ideal locations. The behavior of the school of white sharks can be represented by Equation (14):

$$X_i^{t+1}={\displaystyle \frac{X_i^t+{\acute{X}}_i^{t+1}}{2\times rand}}$$

(14)

3. Proposed Hybrid Meta-Heuristic Algorithm for MPPT

To evaluate the effectiveness of the WSO in MPPT systems, a hybrid MHA is proposed for comparison. This hybrid approach synergizes the strengths of three established MHAs—PSO, DEA, and GWO. Figure 1 shows the overall flowchart of the proposed hybrid method.

3.1. Particle Swarm Optimization

The PSO algorithm was introduced by Kennedy et al. in 1995 [27]. It was inspired by the collective foraging behaviors of birds and fish, and is mathematically formulated as follows [27]:

$$v_i^{t+1}=w\times v_i^t+c_1{r}_1(X_{gbest}-X_i^t)+c_2{r}_2(X_{pbest,i}-X_i^t)$$

(15)

$$X_i^{t+1}=v_i^{t+1}+X_i^t$$

(16)

where w is the inertia weight, and $c_1$ and $c_2$ are acceleration coefficients. It can be seen that (15) closely resembles (1). Both equations involve two random vectors and the present optimal global location of the prey (or particle) within the search space. In PSO, the constants $c_1$ and $c_2$ are typically fixed at 1.5, while in the WSO, the equivalent parameters $p_1$ and $p_2$ are dynamic, changing with each iteration t.

3.2. Differential Evolution Algorithm

Proposed by Storn et al. in 1996 [28], the DEA is a heuristic optimization method designed for continuous optimization problems. Its mutation strategy is given by the following equation [28]:

$$X_i^{t+1}=\left\{\begin{array}{cc}{X}_i^t+F·(X_j^t-X_k^t);\hfill & rand\le CR\hfill \\ X_i^t,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(17)

where $i\ne j\ne k$, ensuring that three distinct individuals are selected. The scaling factor F is set to 0.5, while the crossover rate $CR$ is set to 0.9. These parameters control the balance between exploration and exploitation.

3.3. Grey Wolf Optimizer

The GWO, introduced by Mirjalili et al. in 2014 [8], took inspiration from the structured predatory behaviors of grey wolves, particularly their hierarchical hunting techniques. The position update mechanism is given in [8], as follows:

$$A_l=2a·r_1-a,l=1,2,3$$

(18)

$$C_l=2·r_2,l=1,2,3$$

(19)

$$a=2(1-{\displaystyle \frac{t}{T}})$$

(20)

$$D_{\alpha }= \mid C_1·X_{\alpha }^t-X_i^t \mid$$

(21)

$$D_{\beta }= \mid C_2·X_{\beta }^t-X_i^t \mid$$

(22)

$$D_{\delta }= \mid C_3·X_{\delta }^t-X_i^t \mid$$

(23)

$$X_1^t=X_{\alpha }^t-A_1·D_{\alpha }$$

(24)

$$X_2^t=X_{\beta }^t-A_2·D_{\beta }$$

(25)

$$X_3^t=X_{\delta }^t-A_3·D_{\delta }$$

(26)

$$X_i^{t+1}={\displaystyle \frac{X_1^t+X_2^t+X_3^t}{3}}$$

(27)

where A regulates the step size and varies within the range $[-2,2]$, and $r_2$ is a random coefficient uniformly distributed in $[0,1]$. The parameter a decreases linearly from 2 to 0 over the course of the iterations to enhance convergence by gradually shifting the search process from exploration to exploitation. The terms $D_{\alpha }$, $D_{\beta }$, and $D_{\delta }$ represent the absolute distances between the current position of an individual wolf and the adjusted positions of the three leading wolves $\alpha$, $\beta$, and $\delta$. Based on these distances, the individual estimates three candidate positions $X_1^t$, $X_2^t$, and $X_3^t$, each influenced by one of the leaders. The final update in (27) represents the average position of the three dominant wolves, guiding the rest of the population towards promising regions.

The GWO component in the hybrid algorithm focuses on the social interaction aspect of the optimization process. It emulates the hierarchical structure of a grey wolf pack, which mirrors the social dynamics observed in white shark hunting behavior. The GWO’s role is to strengthen the global search capability by achieving a balance between exploration and exploitation, preventing the search process from becoming stuck in the local optima.

4. Experiment and Results

The experimental setup comprised a programmable PV emulator, a digital signal processor (DSP), an interleaved boost converter, and a DC power supply. The PV array was produced using an AMETEK ETS600X8C-PVF programmable PV emulator (AMETEK Programmable Power, San Diego, CA, USA), which is capable of simulating numerous PV panel models under any conditions, including PSCs. The PV panel considered was formed by 11 PV modules connected in series. To avoid hot spot problems at the PSCs, bypass diodes were connected across different modules, as shown in Figure 2. As shown in

Table 1. Irradiance of each curve (W/m²).

PV Module	Scenario						Temperature (°C)
	1	2	3	4	5	6
1–7	1000	1000	1000	1000	1000	1000	25
8	1000	940	910	800	850	750	25
9	1000	900	850	800	800	850	25
10	1000	880	820	760	750	350	25
11	1000	860	790	720	500	500	25

, there were six scenarios, each representing different irradiance levels on different PV modules. For example, in Scenario 2, modules 1 to 7 received an irradiance level of 1000 W/m² while modules 8, 9, 10, and 11 received irradiance levels of 940 W/m², 900 W/m², 880 W/m², and 860 W/m², respectively.

The TMS320F28035 DSP (Texas Instruments, Dallas, TX, USA), is a 32-bit DSP with an operating reference voltage of 3.3 V. Additionally, this DSP offers 45 general-purpose input/output (GPIO) pins, 16 sets of 12-bit analog-to-digital conversion channels (A/D), seven sets of complementary enhanced pulse width modulation (ePWM) output channels, and two sets of 12-bit serial peripheral interface (SPI) channels. Note that an uncertainty analysis to address the potential errors and uncertainties inherent in the experimental measurements is beyond the scope of the paper. Papers such as [29,30] have performed uncertainty calculations to account for a broad range of experimental factors, such as sensor precision, operator handling, and variations under ambient conditions.

Figure 3 illustrates the interleaved DC-DC boost converter, which consisted of two pairs of inductors, diodes, and power switches. Compared to the conventional boost converters, the interleaved design improved efficiency by minimizing fluctuations in the input current and output voltage, component stress, and electromagnetic interference while enhancing transient response. The Chroma 62012P-600-8 DC power supply (Chroma ATE Inc., Taoyuan City, Taiwan), shown in Figure 4, regulated $V_{dc}$, which was set to 450 V in the experiment. The load, modeled as $R_{dc}$, had a 600 W rating.

A total of six P-V curves were produced in the PV emulator to assess the performance of the algorithms under both static and dynamic conditions, with power values transitioning from an initial 453.38 W to a final 312.82 W. Figure 5 depicts the curve transitions, while Figure 2 lists the corresponding irradiance levels for each PV array configuration.

The proposed method is compared to the WSO and three other MHAs, which are BA, sand cat swarm optimization (SCSO), and the whale optimization algorithm (WOA).

Table 2. Method parameters.

Algorithm		Parameters
Hybrid	PSO	w = 0.3; $c_1$ = 1.5; $c_2$ = 1.5
	DEA	F = 0.5; $CR$ = 0.9
	GWO	a = 2
BA		$f_{max}$ = 2; $f_{min}$ = 0; A = 0.9; r = 0.1
SCSO		$r_G$ = 2
WOA		a = 2; b = 1;
WSO		$p_{max}$ = 1.5; $p_{min}$ = 0.5; $a_0$ = 6.25; $a_1$ = 100; $a_2$ = 0.0005

presents the parameter values utilized across all methods.

4.1. Static Case Results

Figure 6, Figure 7 and Figure 8 present the measured waveforms for scenarios 1, 5, and 6, respectively. In scenario 1, the WSO reaches the GMPP in just 3.93 s, whereas the proposed hybrid method requires 6.06 s. The BA, SCSO, and WOA methods track the GMPP in 6.02 s, 6.01 s, and 5.93 s, respectively. In scenario 5, the WSO achieves the GMPP in 4.53 s, while the proposed hybrid method takes 6.01 s. The BA, SCSO, and WOA methods exhibit tracking times of 5.99 s, 5.99 s, and 5.93 s, respectively. In scenario 6, the WSO attains the GMPP in 4.25 s, whereas the proposed hybrid method requires 6.01 s. The BA, SCSO, and WOA methods complete the tracking in 6 s, 6.02 s, and 5.9 s, respectively.

The statistical performance measures, such as the average, the standard deviation (STD), and confidence interval (CI), of the accuracies and tracking times of several MPPT algorithms for various scenarios are summarized in

Table 3. Comparison of tracking accuracy and time.

Algorithm		Scenario			Average
		No. 1	No. 5	No. 6
BA [31]	Accuracy STD CI	98.00% 0.016 (97.986, 98.014)	98.78% 0.068 (98.720, 98.840)	98.58% 0.558 (98.091, 99.069)	98.45%
SCSO [32]	Accuracy STD CI	98.97% 0.118 (98.867, 99.073)	98.70% 0.086 (98.625, 98.775)	98.53% 0.171 (98.380, 98.680)	98.73%
Proposed	Accuracy STD CI	98.87% 0.136 (98.703, 99.041)	99.17% 0.042 (99.116, 99.220)	98.92% 0.109 (98.824, 99.016)	98.99%
WOA [33]	Accuracy STD CI	98.48% 0.086 (98.405, 98.555)	98.92% 0.008 (98.913, 98.927)	98.36% 0.260 (98.132, 98.588)	98.59%
WSO [34]	Accuracy STD CI	98.31% 0.139 (98.188, 98.432)	99.07% 0.267 (98.836, 99.304)	98.47% 0.479 (98.050, 98.890)	98.62%
BA [31]	Time (s) STD CI	4.73 1.463 (3.448, 6.012)	5.98 0.632 (5.426, 6.534)	5.27 0.588 (4.755, 5.785)	5.33
SCSO [32]	Time (s) STD CI	5.99 0.007 (5.984, 5.996)	6.00 0.015 (5.987, 6.013)	6.03 0.025 (6.008, 6.052)	6.01
Proposed	Time (s) STD CI	5.66 0.233 (5.456, 5.864)	5.90 0.471 (5.487, 6.313)	5.63 0.186 (5.467, 5.793)	5.73
WOA [33]	Time (s) STD CI	5.17 0.196 (4.998, 5.342)	5.89 0.082 (5.818, 5.962)	5.01 0.599 (4.485, 5.535)	5.36
WSO [34]	Time (s) STD CI	4.40 0.498 (3.963, 4.837)	4.14 0.326 (3.854, 4.426)	4.53 0.321 (4.249, 4.811)	4.36

. For better visualization, a box plot for the different MPPT algorithms is drawn in Figure 9. The proposed method yields a relatively small box, indicating that the middle $50\%$ of the data, represented by the interquartile range, is tightly clustered. This suggests low variability within the central portion of the dataset, meaning that most values near the median are relatively close to each other, reflecting uniformity in tracking accuracy and time. The proposed hybrid method consistently delivers the most accurate results, followed by SCSO, WSO, WOA, and BA, with average tracking accuracy values of $98.78\%$, $98.73\%$, $98.62\%$, $98.59\%$, and $98.45\%$, respectively. Consequently, the proposed hybrid method achieves the highest tracking accuracy. However, compared to the other algorithms, the WSO demonstrates a significantly faster tracking time.

To demonstrate that the proposed hybrid method can be applied to more challenging scenarios, a P-V curve with seven LMMPs is generated in the PV emulator, with the GMPP located at (154 V, 304 W), as shown in Figure 10. Figure 10 contains seven local peaks in the P-V curve. As demonstrated in [35], it is more challenging for a maximum power point tracker to track such a P-V curve because it may easily get trapped in a local maximum power point.

The number of particles for the BA, SCSO, proposed hybrid method, WOA, and WSO remains at five, and their initialization settings remain unchanged.

Figure 11 presents the measured waveforms of the BA, SCSO, proposed hybrid method, WOA, and WSO. The proposed hybrid method consistently achieves the highest accuracy, followed by the SCSO, WSO, WOA, and BA, with tracking accuracy values of 99.64%, 99.37%, 99.20%, 99.14%, and 95.17%, respectively. These results indicate that the proposed hybrid method outperforms the others even in this challenging scenario.

Figure 12 analyzes the convergence behavior of the algorithms. The convergence graph (fitness value vs. iteration number) illustrates how quickly each algorithm reaches an optimal solution.

To further compare the proposed hybrid method with some other MPPT methods, we evaluate it against an AI-based method proposed in [36], which combines ANN with P&O. The tracking accuracy of this AI method under Scenario 1, 5, 6, and the challenging scenario is 98.7905%, 93.783%, 95.8859%, and 95.119%, respectively. The average accuracy in static cases is 95.8946%, which is lower than that of the other MHAs. The overall static tracking performance of AI and the five MHAs is compared in Figure 13, where the proposed hybrid method achieves the highest tracking performance among all the algorithms.

The average performance of several MPPT algorithms for the different scenarios is summarized in Table 3. The proposed hybrid method consistently delivers the most accurate results, followed by the SCSO, WSO, WOA, and BA, with tracking accuracy values of $98.99\%$, $98.73\%$, $98.62\%$, $98.59\%$, and $98.45\%$, respectively. Consequently, the proposed hybrid method achieves the highest tracking accuracy. However, compared to the other algorithms, the WSO demonstrates a significantly faster tracking time.

4.2. Dynamic Case Results

The efficiency of dynamic tracking is defined as follows:

$${\eta }_d={\displaystyle \frac{{\sum }_i{V}_{pv,i}·I_{pv,i}·\Delta T_i}{{\sum }_i{P}_{pvm,i}·\Delta T_i}}\times 100\%$$

(28)

where $\Delta T_i$ represents the sampling interval, and $V_{pv,i}$ and $I_{pv,i}$ denotes the PV voltage and current, respectively. The term $P_{pvm,i}$ represents the threshold for the GMPP. Note that it is assumed that there are a total of i different sections containing their individual GMPPs.

To detect PV power change, the following criterion is applied:

$${\displaystyle \frac{|P_{pv}-P_{pv,last}|}{P_{pv,last}}}\ge \Delta P_{pv}$$

(29)

where $P_{pv,last}$ is the previously identified GMPP, and the change threshold $\Delta P_{pv}$ is set to 6%. When a new GMPP is detected, the system resets and initiates a fresh tracking process.

4.2.1. Dynamic Case 1

The measured waveforms for the BA, SCSO, proposed hybrid method, WOA, and WSO are shown in Figure 14. The proposed hybrid method exhibits the highest dynamic tracking efficiency at $93.52\%$, followed by the WSO at $93.02\%$, BA at $91.49\%$, SCSO at $90.33\%$, and WOA at $90.25\%$. The superior dynamic efficiency of the hybrid method results from its extended duration in precisely identifying and following the peak power point. In contrast, the WSO method, although faster in tracking, tends to converge prematurely to a local optimum. This explains why the hybrid method excels in tracking efficiency, despite taking more time, whereas the WSO method sacrifices some accuracy for speed.

4.2.2. Dynamic Case 2

Compared to the first dynamic case, dynamic case 2 consists of a very sharp power change. The starting power level is 453.38 W, the halfway power level is 312.82 W, and the ending power level is back to 453.38 W. Figure 15 are the measured waveforms for the BA, SCSO, proposed method, WOA, and WSO, respectively. As shown in Figure 15, the proposed hybrid method performs with the highest dynamic tracking accuracy, followed by the WSO, BA, WOA, and SCSO, which are 94.15%, 93.89%, 91.01%, 88.89%, and 88.15%, respectively. The overall dynamic tracking performance of five MHAs is compared in Figure 16. The proposed hybrid method achieves the highest tracking performance among all the algorithms.

4.3. Discussions

The proposed approach integrates PSO, the DEA, and the GWO to form a hybrid MHA, selected for their structural resemblance to the three operational phases of WSO. The comparative analysis yields the following key observations:

• WSO exhibits a faster convergence than the hybrid model due to its inherently connected phases, inspired by natural biological processes. In contrast, the hybrid approach requires additional coordination time, as it combines distinct algorithms with differing optimization mechanisms.
• The hybrid method effectively explores the search space through PSO, the DEA, and the GWO, ensuring comprehensive optimization. While this enhances tracking efficiency, findings indicate that efficiency is more critical than speed in dynamic conditions. Notably, the proposed method achieves the highest dynamic tracking efficiency, which measures the ability of an MPPT approach to adapt to time-varying P-V curves. A higher dynamic tracking efficiency reflects superior MPPT performance.
  These insights underscore the trade-off between speed and efficiency, suggesting that while WSO provides rapid convergence, the hybrid approach offers enhanced adaptability and precision in dynamic environments.

5. Conclusions

This paper presents a novel hybrid MHA for maximum power point tracking (MPPT) by integrating PSO, the DEA, and the GWO. These algorithms were selected due to their resemblance to the three operational phases of the recently introduced WSO. The study evaluates the MPPT performance of the WSO and benchmarks it against the proposed hybrid approach to determine optimal tracking strategies. The proposed hybrid approach excels in dynamic tracking efficiency, which is crucial for handling time-varying P-V curves. Comparative analyses further highlight its superiority over the other algorithms, including BA, SCSO, and WOA, in terms of dynamic tracking accuracy. In the future, we will extend the study to real-world implementations in large-scale photovoltaic (PV) systems with diverse environmental disturbances, providing a more comprehensive evaluation of the proposed approach.