1.1. Motivation

The global energy sector is confronted with a lot of challenges, including an increase in the demand for electricity, the necessity to reduce carbon emissions, and the need to take Sustainable Development into account ^{[1, 2]}. Consequently, many countries are turning to the use of clean energy sources, including solar PV power. According to the International Renewable Energy Agency, the share of electricity generated from Renewable Energy Sources (RES) worldwide will be 90% of the total by 2050 ^{[3, 4]}. Solar energy is the main reason behind these predictions. Despite the rapid growth in the implementation of solar PV power designs, solar PV's efficiency drops due to environmental factors like partial shading of solar panels that create various Maximum Power Points (MPP) besides the Global MPP (GMPP) ^{[5, 6]}. The problem gets aggravated as a result of rapidly varying solar irradiances. In such conditions, classic MPPT techniques, including the Perturb and Observe (P&O) technique and the INC Method, face difficulties in distinguishing MPPs from the GMPP ^{[7, 8]}.

To overcome these issues, intelligent optimization algorithms like metaheuristics have been used to boost the efficacy of the MPPT system. Swarm intelligence algorithms like Particle Swarm Optimization (PSO) ^{[9, 10]}, Whale Optimization Algorithm (WOA), and GWO ^{[11, 12]}, have been found promising as far as improving the tracking error and the execution speed with lesser complexity. The above algorithms perform well under complex environmental scenarios where traditional algorithms are not able to find optimal solutions. Although there are many strengths in algorithms like adaptive moment estimation and stochastic gradient descent that apply to this scenario, some limitations exist in achieving a balance between tracking accuracy and implementation ability under dynamic shading scenarios ^{[13, 14]}.

The study regarding the hybridization of MPPT techniques that benefit from the strengths of different optimizing methods has gained extensive attention ^{[15, 16]}. In this regard, some studies in which GA is hybridized with the GWO and INC methods have demonstrated encouraging results in improving the accuracy of global searches as well as localized optima to boost the reliability of an MPPT system. The hybrid approaches demonstrated potential to handle changes in environmental dynamics and exhibited superior tracking abilities under partially shaded conditions. However, there are still challenges in its convergence speed and stability in environmental dynamics that represent a significant R&D field ^{[17, 18]}.

1.2. Literature review

The reason for the growing demand for the adoption of photovoltaic (PV) systems is the need for more sustainable energy sources ^[19]. There remain challenges in tracing GMPP in the presence of partial shading. Variations in the incidence of solar energy can cause the presence of harmonics in the system. Moreover, connecting a PV system to a power grid increases concerns associated with Total Harmonic Distortion (THD) in the system. The need for addressing all these concerns and identifying innovative ways for efficient monitoring of GMPP and enhanced performance of a PV system has been stressed in a study conducted by Sharma and Mishra ^[20]. The study focused on various AI-based MPPT techniques and their features and limitations. It has even reasoned upon techniques for lessening THD in a PV system and recommended Adaptive Filters (AF) as a successful technique for eliminating noise and interference. The review article has brought out fresh aspects of identifying superior AI-based MPPT techniques for making optimizations in PV systems.

Due to the need for optimized MPPT techniques, Bhukya and Shanmugasundaram recommended a study on the performance analysis of a solar-based PV system through a conventional MPPT technique in ^[21]. The researchers evaluated the current-voltage curve and power-voltage curve for a solar-based PV system. The researchers applied a Perturb and Observe technique to validate MPPT in a different environmental setting. The study concluded that a theoretical comparison of MPPT techniques is required to validate the applicability of a Perturb and Observe MPPT technique in a real-time setting for a solar-based PV system. Et-Torabi et al. in ^[22] recommended a comparison between ANN and a Perturb and Observe MPPT technique in a solar-based PV system that consisted of a solar-based PV generator and a boost DC-DC converter with a resistive load. The research mainly studied the difference of MPPT methods in a solar-powered system under unchanging and changing environmental conditions. To verify the results of MATLAB Simulink modeled MPPTs, a real-world setup with LAUNCHPAD-XL 28F379D and a C2000 embedded coder was used. It has been found by the researchers that an ANN-based MPPT technique yields more power in a stable environment as well as in a varying one. The ANN-based MPPT method can easily accommodate a change in situation, and simultaneously, it is capable of producing a steady power output.

Moving on to the future of MPPT, Gopalakrishnan et al. introduced a new idea to enhance the performance of PSO based MPPT in the PV systems, which are partially shaded ^[23]. The authors embraced the foundational principle of a meticulous study of bird flocking behavior for their invention. The research finding was that in bird flocking behavior, the individual birds initially fly in random patterns, but ultimately they turn to the central point. Hence, there was a creation of a modified PSO method that allows particles to continue their random search in the solution space for a longer time, but finally, they come to a global optimal point. The recommended technique significantly improved the MPPT locus-tracking capabilities of a flock and can therefore be applied to improve MPPT in partially shaded PV systems. Both calculated and experimental MPPT curves were given to validate this technique.

Zhou and Zhang in ^[24] developed a high-precision engineering mathematical (HPEM) model for a PV array to overcome limitations in traditional mathematical models that are unable to simulate some MPPs accurately in a shading situation. The HPEM mathematical model, developed for a single-diode model with a compensation coefficient ($\beta$), ranging between 0.5 and 1.2, is a substantial improvement over traditional methods where a fixed value of $\beta = 0.5$ is considered. The model is more efficient and more accurate for MPPT. To represent multiple peaks in a PV output curve, the scientists created a piecewise mathematical model grounded on series-parallel circuit theory for shaded areas. According to the HPEM simulation, the model gets very close to zero-error MPPT MPP and power values when temperature and irradiation vary from 25◦C to 80◦C and 200-1000 W/m², respectively, with the relative errors being 0.08% and 0.07%. It is very close to traditional MPPT in that it accurately predicts multimodal distributions with less than 1% extremum localization error and thus overcomes the common problem of local trapping. These theoretical concepts and models constitute a milestone framework for comprehending complex photovoltaic behavior under extreme lighting and the design of large-scale array configurations.

To address the multi-peak characteristics of the P-V curve under partial-shading scenarios, Hao Wu et al. in ^[13] developed a new MPPT control tactic combining a chaos-based grey wolf optimizer (CGWO) and a variable step INC (VINC) algorithm. The CGWO optimizes traditional random numbers in GWO to improve diversification capabilities in searching for better positions. The incorporation of VINC within this novel strategy increases MPPT accuracy. Comparisons between this strategy and traditional MPPT approaches like INC-GWO, Cuckoo Search (CS), PSO, and Perturb & Observe (P&O) revealed better performance. The efficiency and accuracy were improved in this strategy as it minimized average computation time for convergence by 53.30% and boosted accuracy by 1.37% in comparison to traditional strategies. The strategy optimized total solar utilization steadily and efficiently in a PV system when operated under different partial-shading scenarios.

In conclusion, an optimized MPPT control strategy has recently been recommended by Bian et al. ^[25] to overcome a deficiency that traditional MPPT control strategies have in failing to track the global MPPT when a partial shading scenario exists. The optimized strategy relies on an improved bacterial foraging algorithm (BFA), in which a leader approach is added to promote improvements in chemotaxis. In addition to this, a similar triangle assignment strategy has also been recommended to improve the elimination-dispersal process according to the optimal individual. The parameters of the optimization model, including its initial location and restart strategy, have been designed according to its characteristics in relation to generating a maximum power output in a photovoltaic array. The simulation and real-world experiments have proved that this optimized BFA-based MPPT strategy performs well in rapidly identifying and locating the global MPPT while simultaneously greatly improving the efficiency of a photovoltaic array.

Despite the promising performance of existing hybrid MPPT strategies, several limitations remain. Most GA-GWO or GWO-INC methods rely on fixed parameter settings, which limits their adaptability under rapidly changing environmental conditions. In addition, many approaches adopt a loosely coupled or sequential structure, where global and local optimization stages operate independently, resulting in insufficient interaction and suboptimal performance. Furthermore, the lack of a clear coordination mechanism between exploration and exploitation may lead to premature convergence or oscillatory behavior near the MPP. These limitations motivate the development of a more tightly integrated and adaptive hybrid optimization framework.

1.3. Research Gap and Contribution

To address the limitations of existing hybrid MPPT methods, this paper proposes a GA-GWO-INC cooperative optimization framework with tightly coupled global-local interaction and dynamic parameter adaptation. Compared with existing GA-GWO, GWO-INC, and other hybrid strategies, the main contributions of this work are summarized as follows:

(1) Dynamic parameter adaptation mechanism: A time-varying crossover probability and mutation factor are introduced, enabling the algorithm to maintain strong exploration ability in early stages while ensuring convergence stability in later stages.

(2) Embedded GA-GWO interaction mechanism: Unlike conventional cascade structures, GA operators (crossover and mutation) are directly embedded into each GWO iteration, forming a deeply coupled population evolution process, which effectively prevents premature convergence.

(3) Two-stage coordinated optimization framework: A structured global-local optimization strategy is developed, where the GA-GWO hybrid performs global exploration, and the INC method is adaptively activated for monotonic local refinement near the MPP, improving tracking accuracy.

(4) Mechanism-level convergence synergy: The proposed method integrates the contractive convergence of GWO, the diversity enhancement of GA, and the monotonic adjustment property of INC into a unified optimization framework.

These features distinguish the proposed method from existing hybrid MPPT approaches and provide a more robust solution under dynamic partial shading conditions.

3.1. Application of GA in MPPT Optima

In MPPT for PV systems, GA employs three fundamental genetic operations: selection-replication, crossover, and mutation to optimize the population fitness. This approach facilitates the swift identification of the MPP while effectively mitigating the challenges associated with local optima.

3.2. Application of the Grey Wolf Optimizer (GWO) in MPPT

In MPPT for a PV system using GWO, a leadership hierarchy is created among four kinds of GWs: alpha ($\alpha$), beta ($\beta$), delta ($\delta$), and omega ($\omega$). The alpha has generally better fitness and is known as the leader of the group. The level of fitness for a beta is second to that of the alpha and is considered to aid in some decisions made in conjunction with the alpha. The fitness level for a delta is third and is known to support alpha and beta. The rest of its group is known as omega and follows alpha, beta, and delta when they are out for a hunt.

3.3. The Application of INC in MPPT

In the MPPT of photovoltaic systems, the INC method determines the operating point more precisely by calculating the incremental relationship between power and voltage in real time. This power-voltage incremental relationship is derived using Eq. (12).

In Eq. (12), $P_{new}$ is the power value after the position update, $P_{old}$ is the power value before the update, $U_{new}$ is the voltage after the update, and $U_{old}$ is the voltage before the update.

Next, by substituting Eq. (12) into Eq. (13), we obtain the updated duty cycle.

In Eq. (13), $D(t + 1)$ is the duty cycle after the update, $D(t)$ is the duty cycle before the update, $k$ is the step size for adjusting the duty cycle, $\text{sign}\left(\frac{\Delta P}{\Delta U}\right)$ denotes the sign function.

3.4. Design of the GA-GWO-INC Hybrid Algorithm for MPPT

In this work, the proposed GA-GWO-INC algorithm is not a simple sequential hybridization, but a deeply integrated cooperative optimization framework. Specifically, GA-based crossover and mutation operators are embedded within each GWO iteration, enabling continuous population diversification during the global search process. This design allows the algorithm to dynamically balance exploration and exploitation, rather than relying on fixed-stage switching. Specifically, after the fitness evaluation and leader ($\alpha, \beta, \delta$) updating in each GWO iteration, GA-based crossover and mutation are applied to the candidate positions to generate a diversified population for the next GWO update. The dynamic mutation factor (see Eq. (4)) helps adjust the search strategy, making it more exploratory at the beginning of the cycles, while Eq. (3) gradually weakens the crossover operation in later stages to enhance convergence stability. By substituting Eqs. (4) and (3) into Eqs. (7) and (9), we get

By substituting Eqs. (14) and (15) into Eq. (10), the new vector values of $X_1^*$, $X_2^*$, and $X_3^*$ (as displayed in Eq. (16)) are obtained. From these, an update strategy (Eq. (17)) is derived that effectively integrates the advantages of multiple solutions. Eq. (17) is expressed as follows:

In Eqs. (16) and (17), $r$ is a random number, and $X_c$ is the updated position vector.

To strengthen the theoretical rigor of the proposed hybrid GA-GWO-INC algorithm, an explanation of its convergence characteristics is provided as follows. In the GWO component, the convergence factor $\alpha$ decreases linearly from its initial value to zero, causing the coefficient vector $A = 2\alpha r - \alpha$ to satisfy $|A| < 1$ in later iterations. This ensures that the position update of each wolf becomes a contractive mapping toward the leader positions, forming the theoretical basis for global convergence.

In addition, the genetic crossover mechanism integrated into the position update employs a bounded and gradually decreasing crossover probability. This design provides stronger exploration during the early iterations while reducing the perturbation effect in the later stage, ensuring that the genetic operator remains stable and does not disrupt the contractive behavior of the GWO search. Furthermore, the INC refinement step monotonically adjusts the duty cycle based on the sign of $\Delta P/\Delta U$. Convergence occurs when $\Delta P/\Delta U = 0$, corresponding exactly to the MPP condition. Therefore, the INC stage operates as a monotonic local refinement mechanism capable of correcting small deviations that may arise during global search.

By combining the contractive global search of GWO, the bounded and decreasing perturbation of the genetic operator, and the monotonic refinement of the INC adjustment, the overall iterative update satisfies $\|X(t + 1) - X(t)\| \to 0$. This indicates that the hybrid method converges toward a stable region around the global MPP. The simulation results in Tables 6 and 7 further confirm this conclusion by demonstrating fast convergence, strong tracking accuracy, and reliable performance under both static and dynamic partial shading conditions.

From a computational perspective, the proposed GA-GWO-INC algorithm maintains relatively low complexity because it employs a small population size and a limited number of iterations in the global search stage. The computational complexity of the GWO-based search is approximately $O(NT)$, where $N$ represents the population size and $T$ denotes the number of iterations. In this study, $N = 10$ and $T = 5$, resulting in a very small computational burden. After global exploration, the INC method performs lightweight local refinement with minimal arithmetic operations. Therefore, the overall computational load remains suitable for real-time implementation in embedded photovoltaic controllers.

In addition, the proposed algorithm does not require large-scale matrix operations, deep neural network inference, or high-dimensional optimization calculations. Most computations involve basic arithmetic operations and iterative position updates, which can be efficiently executed on commonly used DSP, MCU, or FPGA-based photovoltaic control platforms. Due to the small iteration count and low memory occupancy, the proposed method demonstrates favorable real-time implementation potential for practical photovoltaic energy conversion systems under dynamic partial shading conditions.

4.1. Power Tracking Test Under Static Shading

The comparison between INC, GWO, GWO-INC, and GA-GWO-INC approaches under partial shading (PS) is done through simulation. The corresponding graphs for the output power waveform under static shading are given in Figs. 5-7.

Fig. 5. Output power waveforms for scenario I. (a) INC. (b) GWO. (c) GWO-INC. (d) GA-GWO-INC.

Fig. 6. Output power waveforms for scenario II. (a) INC. (b) GWO. (c) GWO-INC. (d) GA-GWO-INC.

Fig. 7. Output power waveforms for scenario III. (a) INC. (b) GWO. (c) GWO-INC. (d) GA-GWO-INC.

The convergence time is defined as the time instant when the tracked power first reaches and remains at or above 95% ($\ge 95\%$) of the final steady-state maximum power. The corresponding values for time and accuracy are displayed below in Table 5.

As shown in Table 5, in tracing accuracy, GA-GWO-INC has always demonstrated a superior accuracy above 99.989% for Scenarios II and III. In comparison to the INC, GWO, and GWO-INC algorithms, there exist possibilities of stability problems and even occurrences of entrapment in a local optimum. As for convergence time, GA-GWO-INC has not only demonstrated superior accuracy but has also optimized tracing time. In tracing times for Scenarios I and II in comparison to other methods, GA-GWO-INC has reduced tracing times between 9.4% and 34%, thus significantly improving system response times.

4.2. Power Tracking under Dynamic Shading

The simulation is done for validating power tracking in a scenario that has a time-varying lighting condition. The process operates in Scenario I for $t=0$ to 0.5 seconds. Later, for $t=0.5$ to 1 second, it operates in Scenario II. Furthermore, for $t=1$ to 1.5 seconds, it operates in Scenario III. The output power of all four algorithms is displayed in Fig. 8.

Fig. 8. Power output waveforms under dynamic shading. (a) INC. (b) GWO. (c) GWO-INC. (d) GA-GWO-INC.

The settling time is defined as the time required for the tracked power to reach and remain at or above 95% ($\ge 95\%$) of its final steady-state peak value. The table below shows a summary of experimental track time and accuracy for all algorithms.

As Table 6 reveals, the GA-GWO-INC hybrid algorithm achieves the highest accuracy under all test conditions. Here, particularly in Scenario III, it achieves 99.998%, which is much higher compared to GWO (99.854%) and GWO-INC (99.985%). Contrarily, the INC algorithm indicates extremely high accuracy oscillations and gets trapped in a local optimum in Condition III, and indicates extremely low precision.

As far as time tracing is concerned, GA-GWO-INC provides superior accuracy and involves considerable time savings. In Scenario II, for instance, tracing time is reduced by about 45% compared to GWO-INC, demonstrating excellent responsiveness. Meanwhile, the proposed method maintains tracking accuracy above 99.989% under all tested conditions, indicating a favorable balance between steady-state precision and transient response performance.

4.3. Discussion

A PV system has huge potential and has immense promise to meet the growing energy demand and address some environmental issues to a large extent. However, there remain some concerns about effectively implementing it in a situation where partial shading is involved. The presence of more than one point in the power-voltage curve in a scenario where partial shading is involved makes it a very challenging task for the traditional MPPT technique to track the Global MPP. In this exploration, a fresh GA-GWO-INC MPPT tactic has been recommended to address this challenging scenario. The recommended technique is a combination of GA, GWO, and INC methods.

The recommended method is a combination of GA crossover and mutation methods and optimized methods in GWO, and has additional exploitation capabilities in the vicinity through INC. The adaptive cross probability and step size in INC are added to this optimized method to ensure that this optimized method has better search capability and exploitation in the vicinity. The recommended optimized method has been developed in a simulation scheme for a PV system for different scenarios to analyze its better performance in comparison to traditional methods like INC, GWO, and GWO-INC.

The results mentioned earlier have clearly proved that for GA-GWO-INC, there is a huge improvement regarding other techniques in MPPT for MPP. In all scenarios that were considered for testing, GA-GWO-INC has maintained its accuracy to 99.998%, wherein there is a decrease of 45% in convergence time. In all scenarios that contain dynamic shades, GA-GWO-INC has proved its better capability and ability to resist and track the global MPP without entrapment in a local optimum. The above results indicate that GA-GWO-INC can assist in boosting the efficiency of a PV system even in a complex scenario. From a practical application perspective, the proposed method not only improves tracking accuracy, but also reduces transient power loss during rapid irradiance variation. In practical photovoltaic systems, environmental conditions such as moving clouds and partial shading frequently cause operating-point deviations. Faster convergence enables the PV array to return more quickly to the global maximum power point, thereby improving dynamic energy capture capability. In addition, the proposed method maintains tracking accuracy above 99.989% under all tested shading conditions while reducing convergence time by up to 45%, demonstrating favorable real-time performance for photovoltaic MPPT applications.

Although it has some promising results, some limitations are associated with this work. Firstly, it can easily be concluded that this algorithm is still heavily dependent on variables like probability and step size. But in the near future, this issue can easily be resolved in more in-depth works related to either optimizing this existing algorithm or using adaptive techniques. The scalability of this hybrid technique for large-scale scenarios in relation to PV and complex shading patterns may provide some valuable information. In a nutshell, it can easily be concluded that even though this recommended technique has brought iconic development in accuracy and efficiency in solar path-tracking techniques, its computational complexity remains moderate and suitable for real-time implementation. Moreover, because the proposed algorithm only requires a small population size, limited iteration cycles, and simple arithmetic operations during the local refinement stage, it can be implemented on practical embedded photovoltaic controllers with relatively low processor and memory requirements. This characteristic further improves its engineering applicability for real-time MPPT control under dynamic partial shading environments. But its development in relation to stability and efficiency in partial-shading patterns for a photovoltaic system has supreme significance in solar technology advancement.