PHOTOVOLTAIC (PV), energy generation has emerged as the most favored and preferred source of energy, playing a significant role in advancing the global energy scenario [1]. However, PV system encounters a significant challenge, namely, operating under non-uniform irradiance, which greatly affects power generation, P-V (power-voltage) characteristics, and overall reliability [2],[3]. Non-uniform irradiance, caused by shadows in partial shading conditions (PSC), is a major concern that leads to multiple peaks in the PV curve [4]. Consequently, the PV array experiences a significant difference in current between rows within the PV array [5]. To address these issues, array reconfiguration techniques like static and dynamic have been proposed as a solution to mitigate the impact of shading [6]. The static reconfiguration process involves physical relocation of PV modules within the array to disperse shade [7]. Different static reconfiguration techniques such as sudoku [8], arithmetic sequence pattern reconfiguration technique [9], magic square view [10], novel shade dispersion technique [11] etc. have been proposed in literature to relocate the PV modules. However, these physical relocation methods have inherent drawbacks [12], including labour requirement, lengthy cables, dependence on sub-arrays, need for arbitrary assumptions during implementation and incompatibility with asymmetrical PV arrays.

The electrical array reconfiguration (EAR) is a dynamic process that employs simultaneous switching patterns to reconfigure the interconnections of a PV array, aiming to achieve row equalization based on shade patterns [13]. As a result, EAR represents a promising solution when compared to conventional PV array reconfiguration (PVAR) approaches. Numerous research studies have utilized EAR for power enhancement in PV arrays [14]. However, achieving the desired switching patterns for shade dispersion in the EAR method is often accomplished using a trial-and-error approach, which may not be highly efficient [15]. Consequently, the processing time required to achieve the optimal switching matrix increases. Furthermore, the frequent switching involved in the reconfiguration process can have a negative impact on the lifespan of relay. To overcome the aforementioned drawback, optimization algorithms are used as an alternative solution, such as, the butterfly optimization algorithm [16], artificial ecosystem-based optimization [17], improved equilibrium optimization [18] and particle swarm optimization (PSO)[19], have been recently employed for shade dispersion in the PV array. Optimization methods based on dynamic PVAR offer several advantages including: 1) They are designed to efficiently converge towards optimal solutions, thereby reducing the time required to achieve the desired outcome. 2) Aim to minimize the difference in current among rows in the PV array. However, the existing algorithms involve complex mathematical calculations, sensitive to tuning parameters, it involves multiple iterations in the evaluation of fitness functions, which can lead to a computational burden on the system, algorithms can converge to local minima instead of the global optimum, and they are sensitivity to initial conditions.

In this paper, an accelerated linear convergence spotted hyena optimization (ALC-SHO) based dynamic PVAR technique is proposed, with the enhanced capability of shade dispersion. The proposed ALC-SHO algorithm accomplishes two objectives: it reconfigures the PV modules into their optimal positions to achieve irradiance equalization as the first objective, and simultaneously, it maximizes power extraction by tracking global maximum power point (GMPP) while minimizing the steady-state oscillations and tracking time under PSCs, as the second objective. The accelerated linear convergence factor is implemented to achieve the minimum row current difference with fewer module relocations. Consequently, this approach reduces the required number of switch activations. In addition, by employing a single tuning parameter, the complexity associated with finetuning multiple parameters is effectively circumvented. The performance of the proposed algorithm is more effective than the conventional SHO (CSHO) algorithm. The proposed algorithm is implemented on a $3 \times 3\mathrm{{PV}}$ array and compared with before-reconfiguration, and after-reconfiguration methods using PSO [19], and CSHO [20]. The proposed method is simulated using Matlab. Further, the proposed algorithm is verified using an experimental prototype under various PSCs. The comparison is based on several factors, including power enhancements, mismatch losses, percentage power loss, and irradiance equalization (IE) under different PSCs.

The main contributions of this paper are as follows:

1) The first contribution is proposed on ALC-SHO-based reconfiguration to re-arrange the PV modules to achieve IE between each rows, thereby enhancing the power.

2) The second contribution is the extraction of enhanced power using the proposed ALC-SHO-based maximum power point tracking (MPPT) algorithm, which ensures fast convergence with negligible steady-state error.

3) An experimental study is performed on the proposed method and compared with the conventional methods.

4) Comparative analysis is performed to show the effectiveness of the proposed system in terms of mismatch power loss, and percentage of power loss.

The rest of the paper is structured as follows: Section II presents the introduction to the novel ALC-SHO algorithm and is followed by implementation steps for PVAR and MPPT in Section III. In Section IV, the simulation result and discussion for the proposed dynamic PVAR method under various shading conditions are summarized. Section V presents the experimental results. Finally, Section VI summarises and concludes the key findings.

The spotted hyena optimization (SHO)[21], draws inspiration from the social behavior and hunting strategies of spotted hyenas. Spotted hyenas engage in four key operations: search, encirclement, hunting, and attacking their prey. This section presents the mathematical models for each of these processes.

1) Encircling: Assuming that the target prey represents the current best solution, the remaining search agents will adjust their positions toward this best solution. The mathematical model for the encircling process can be described as follows:

where $B$ and $E$ denote the coefficients, ${X}_{\mathrm{P}}$ is the position of prey, X indicates the position of hyenas, ${D}_{\mathrm{h}}$ represents the distance between the prey and the spotted hyena, and $t$ denotes the current iteration. The coefficient vectors $B$ and $E$ are given in (3) and (4) respectively.

where ${r}_{1}$, and ${r}_{2}$ are random variables in the range $\left\lbrack {0,1}\right\rbrack$, itr is denoted as the iteration number, ${it}{r}_{\max }$ is the maximum iteration required, and $a$ is a convergence factor that falls linearly from 5 to 0.

This paper proposes an accelerated linear convergence factor $\left({a}_{\mathrm{{alc}}}\right)$, which plays a significant role in finding an optimum point. It is essential to coordinate with exploration and exploitation to determine the optimum solution.

where, ${}^{\iota }n$’ is the accelerated convergence parameter ranging between $\left\lbrack {0,4}\right\rbrack$. The key role of'$n$’ is to navigate and converge towards an optimal solution with less number of iterations. When'$n$’ is set to lower values, exploration takes place. The algorithm aims to discover new possibilities by stepping into unknown areas of the solution space. When'$n$’ is set to higher values (closer to 4), the algorithm exploits and tries to refine its search around promising solutions to find the best possible solution. The impact of'$n$’ parameter on algorithm behavior is shown in Fig. 1 and demonstrates the transition from exploration to exploitation as'$n$’ increases. It plays a crucial role in optimizing the convergence speed of the algorithm. The larger the value of $a$, the stronger the global search capability. Thus, hyenas avoid settling into a local optimum. A smaller value of $a$ indicates better exploitation, which speeds up the convergence. However, in the existing SHO method, the convergence factor $a$ decreases linearly from 5 to 0 as the number of iterations increases, as given in (5). Further, the linearly decreasing approach of the convergence factor a has an good global search ability, but the convergence rate is slow. Hence, in this paper, an accelerated linear convergence factor ${a}_{\text{alc }}$ is proposed, which decreases linearly from 1 to 0, as given in (6). Therefore, the accelerated linearly decreasing quantity is better than the linearly decreasing quantity, thereby showing better global search ability with a rapid convergence rate and reduced iteration count as depicted in Fig. 1(b). The outcomes of the modified convergence factor ${a}_{\mathrm{{alc}}}$ is better than those of the existing CSHO algorithm.

2) Hunting: The spotted hyenas hunt in packs and have the ability to locate the prey. The hunting mechanism is determined using (7) and (8). The other search agents form a cluster $\left({C}_{\mathrm{h}}\right)$ towards the best search agent and save the best solutions in each iteration to update their positions.

3) Attacking: It is feasible to assault the prey mathematically by decreasing the ${a}_{\text{alc }}$ value. The variation in $E$ is also responsible for changing the value of ${a}_{\mathrm{{alc}}}$.

4) Search for prey: According to the cluster $\left({C}_{\mathrm{h}}\right)$, the spotted hyenas search for prey. They move farther apart to hunt and assault the prey. The coefficients $B$ and $E$ are used to explore and avoid local optima.

The performance of ALC-SHO-based PVAR is evaluated for three different sizes of PV array. In the initial case, a $3 \times 3$ PV array is subjected to two different shading patterns. The objective is to assess the ability of ALC-SHO algorithm to disperse shade effectively in a PV array and analyze its effectiveness in maximizing power output. In the second case, two asymmetric $3 \times 4$ PVAR is evaluated, further demonstrating the adaptability of the algorithm in addressing the complex PVAR. The proposed ALC-SHO algorithm-based PVAR is applied for a larger size $\left({6 \times 6}\right)\mathrm{{PV}}$ array lastly. This case aims to verify the performance of the algorithm on a larger-size PV array. The specifications of the PV module is as follows: Open circuit voltage $\left({V}_{\mathrm{{oc}}}\right)$ is 44.4 V, short circuit current $\left({I}_{\mathrm{{sc}}}\right)$ is 8.6 A, the voltage at $\operatorname{MPP}\left({V}_{\mathrm{{mpp}}}\right)$ is ${36.6}\mathrm{\;V}$, current at $\operatorname{MPP}\left({I}_{\mathrm{{mpp}}}\right)$ is ${8.2}\mathrm{\;A}$ and the power at $\operatorname{MPP}\left({P}_{\mathrm{{mpp}}}\right)$ is 300.12W. Four different shading patterns are considered to verify the effectiveness of the proposed ALC-SHO algorithm.

1) PSC-1: Column shade pattern.

2) PSC-2: Triangular shade pattern.

3) PSC-3: Corner shade pattern.

4) PSC-4: L-shape shade pattern.

The proposed ALC-SHO ${}_{\mathrm{R}/\mathrm{M}}$ algorithm-based PV system is depicted in Fig. 12. The PV array characteristics are emulated using the solar array simulator. Additionally, the Opal-RT 4500 is employed to generate PWM signals for the DC-DC converter. The voltage and current sensors used in the setup are the LV-20p and LA-25p, respectively. A mixed signal oscilloscope (MSO46) records system responses. The PV system specification is as follows: ${V}_{\mathrm{{oc}}}= {80}\mathrm{\;V},{V}_{\mathrm{{mp}}}= {65}\mathrm{\;V}$, ${I}_{\mathrm{{mp}}}= {2.43}\mathrm{\;A},{I}_{\mathrm{{sc}}}= 2\mathrm{\;A},{P}_{\mathrm{{mp}}}= {150}\mathrm{\;W}$. The followings are the specifications of DC-DC boost converter, ${C}_{\text{in }}= {100\mu }\mathrm{F},{C}_{\text{out }}$ $={250\mu }\mathrm{F}, L = 2\mathrm{{mH}},{f}_{\mathrm{s}}= {10}\mathrm{{kHz}}$. Two shading conditions are considered, to experimentally validate the performance of the proposed algorithm. Furthermore, the proposed algorithm is compared with the ${\mathrm{{TCT}}}_{\mathrm{C}}/{\mathrm{{PO}}}_{\mathrm{M}},{\mathrm{{PSO}}}_{\mathrm{R}/\mathrm{M}}$ and ${\mathrm{{CSHO}}}_{\mathrm{R}/\mathrm{M}}$ algorithms. The total number of particles used for reconfiguring the PV modules by the algorithms is 12. Increasing the number of particles in the algorithm typically leads to a larger exploration of the search space. However, it’s important to note that this expanded exploration leads to increased computational time. With more particles, the algorithm requires additional iterations to converge towards GMPP. Moreover, ALC-SHO is able to balance between exploration and exploitation with a fewer number of particles. Indeed, in the pursuit of the global optimum, PSO algorithm often employs a larger number of particles. Thus, while more exploration is beneficial, the longer computational time can be considered a drawback. Furthermore, with fewer particles, there is a higher chance of being trapped in a local optimum solution.

1) The performance of the ALC-SHO algorithm is investigated under PSC-1: The P-V and I-V characteristics of ${\mathrm{{TCT}}}_{\mathrm{C}}/{\mathrm{{PO}}}_{\mathrm{M}},{\mathrm{{PSO}}}_{\mathrm{R}/\mathrm{M}},{\mathrm{{CSHO}}}_{\mathrm{R}/\mathrm{M}}$ and proposed methods for symmetric $3 \times 3$ under the column shading pattern is shown in Fig. 13(a). In this shading pattern, the ${\mathrm{{TCT}}}_{\mathrm{C}}/{\mathrm{{PO}}}_{\mathrm{M}}$ method demonstrates a notable decrease in output power, tracking 127.29 W. Additionally, steady-state oscillations are also found around the MPP as shown in Fig. 13(b). The ${\mathrm{{PSO}}}_{\mathrm{R}/\mathrm{M}}$ method failed to effectively disperse the shade across the PV array. The power output achieved by the ${\mathrm{{TCT}}}_{\mathrm{C}}/{\mathrm{{PO}}}_{\mathrm{M}}$ and ${\mathrm{{PSO}}}_{\mathrm{R}/\mathrm{M}}$ methods are the same. This is due to the performance of PSO which depends on its parameters, such as the inertia weight, cognitive, and social components. Proper tuning of these parameters can help the algorithm to explore the search space effectively while preventing premature convergence to local power point, however, tuning of these parameters is complex and time consuming. Furthermore, the ${\mathrm{{PSO}}}_{\mathrm{R}/\mathrm{M}}$ algorithm requires a longer duration of 0.98s to track the GMPP. The CSHO algorithm is enhances the power to 142.85 W and tracks GMPP in 0.55 s. The proposed algorithm improves the power of 142.86 W and is able to track within 0.3s with negligible steady-state oscillations. Therefore, the ALC-SHO exhibits a lower number of peaks and a significant power enhancement of 10.89% compared to the existing configurations.

2) The performance of the ALC-SHO algorithm is investigated under PSC-2: Here, $3 \times 3$ symmetric shading conditionis considered. The ${\mathrm{{TCT}}}_{\mathrm{C}}/{\mathrm{{PO}}}_{\mathrm{M}}$ method achieves 103.15 W. Moreover, significant steady-state oscillations are observed around the MPP, as shown in Fig. 14(a). The ${\mathrm{{PSO}}}_{\mathrm{R}/\mathrm{M}}$ method enhances the power output by ${116.86}\mathrm{\;W}$. It is worth noting that the ${\mathrm{{PSO}}}_{\mathrm{R}/\mathrm{M}}$ algorithm requires a longer duration of ${1.8}\mathrm{\;s}$, to track GMPP. Inefficient initialization or parameter settings can prolong the convergence process, as the PSO may struggle to effectively explore the search space and adjust particle trajectories towards the GMPP. Further, due to repeated exploration of the particle also leads to longer convergence period. The CSHO improves the power of ${136.5}\mathrm{\;W}$ and tracks GMPP in 0.66 s. The ALC-SHO not only enhances power output by ${136.52}\mathrm{\;W}$, but it has ability to track within a ${0.35}\mathrm{\;s}$, minimizing the steady-state oscillations as shown in Fig. 14(d).

The paper presents a novel ALC-SHO algorithm for the optimal reconfiguration of a partially shaded PV array. The primary objective of the proposed method is to enhance the power from the PV array under PSC by irradiance equalization principle. The secondary objective is to extract the maximum power with negligible steady-state error. The accelerated linear convergence factor is employed to achieve irradiance equalization while minimizing the need to relocate modules, and it is accomplished within a reduced iteration count. The effectiveness of the proposed method is evaluated through simulation and experimental results. Various shading patterns are examined and compared against ${\mathrm{{TCT}}}_{\mathrm{C}}/{\mathrm{{PO}}}_{\mathrm{M}},{\mathrm{{PSO}}}_{\mathrm{R}/\mathrm{M}}$ and ${\mathrm{{CSHO}}}_{\mathrm{R}/\mathrm{M}}$. The average output power enhancement of proposed ALC-SHO ${}_{\mathrm{R}/\mathrm{M}}$ method is ${16.27}\%$ and ${11.54}\%$ compared to ${\mathrm{{TCT}}}_{\mathrm{R}}/{\mathrm{{PO}}}_{\mathrm{M}}$ and ${\mathrm{{PSO}}}_{\mathrm{R}/\mathrm{M}}$ respectively. A performance analysis is carried out to provide additional insights into the efficacy of the proposed method.