A Novel Improved Puma Optimizer to Boost Photovoltaic Array Production in Partially Shaded Environments

1  Introduction

Rapid population growth and increasing industrialization exert significant pressure on the world’s limited resources, resulting in shortages in various sectors such as water, soil, and energy [1]. With global energy demand surging and concerns over energy security intensifying, transitioning to renewable energy sources has become an urgent priority for achieving sustainable development [2]. Among the various renewable energy technologies, solar energy stands out as one of the most promising alternatives. Solar power sourced from an abundant and inexhaustible resource provides a clean and sustainable approach to addressing increasing energy needs while minimizing greenhouse gas emissions [3]. Regarding multiple renewable energy sources, solar power, particularly photovoltaic technology (PV), is the most promising sustainable source due to its scalability, low operational costs, and environmental benefits. Photovoltaic systems, which directly convert sunlight into electricity, have gained significant attention as a result of these advantages [4]. However, the performance and effectiveness of PV systems can be significantly influenced by environmental factors, including ambient temperature and solar radiation. These factors vary significantly with changing climatic conditions, highlighting the need for optimized PV performance under diverse environments [5]. A critical challenge in PV technology is partial shading (PS), which results from obstructions such as trees, buildings, or clouds. Partial shading leads to significant energy losses by reducing the overall power output and causing current mismatches across modules, affecting the efficiency of the system [6]. Numerous reconfiguration strategies have been indicated to optimize PV system performance to mitigate the effects of partial shading [7,8]. Conventional static methods, such as Total Cross Tied (TCT), HoneyComb (HC), and Bridge-Linked (BL) layouts, alter electrical connections within photovoltaic arrays to reduce mismatch losses and enhance power extraction [9]. However, these static methods are limited in their ability to react in real-time, as shading conditions fluctuate dynamically throughout the day. Therefore, researchers have investigated dynamic optimization strategies that use advanced algorithms to continuously adjust PV module connections for optimal energy production [10]. Techniques such as Arnold’s Cat Map Algorithm, which applies mathematical transformations to reorganize PV module connections, and the African vultures optimization algorithm (AVOA), a metaheuristic optimization inspired by scavenging behaviour, have demonstrated promising results in improving PV efficiency [11].

Numerous researchers have explored and implemented various reconfiguration strategies, categorized as either static or dynamic, to optimize PV array performance under partial shading conditions. These techniques are designed to enhance energy generation by mitigating power losses caused by shading effects, ensuring a more efficient distribution of power across the array. Dynamic reconfiguration methods, in particular, provide real-time adaptability by altering the electrical connections between PV modules based on shading patterns, whereas static approaches focus on predefined configurations to improve overall power output. The authors in [12] developed a hybrid optimization method to improve the PV system performance under normal and dynamic irradiation circumstances. Teaching learning-based optimization (TLBO) succeeded in exploration, whereas the equilibrium optimizer (EO) has been incorporated at exploitation. The authors in [13] established a grid-connected solar PV system using a 3-phase shunt active power filter (APF) and the AVOA-based recurrent neural network for maximum power point tracking (MPPT). They also presented a four-square sudoku method for reconfiguring a 9 × 9 total cross tied (TCT) photovoltaic array under conditions of partial shadowing. An extensive comparative analysis of ten distinct controllers for PV systems, including both traditional MPPT methods and artificial intelligence (AI)-based controllers, has been conducted [14]. The research assessed many criteria, such as voltage, current, power, weather dependency, cost, complexity, reaction time, periodic tuning, stability, partial shade, and accuracy, to identify the most efficient MPPT controllers under diverse settings. A two-step module replacement approach has been presented to mitigate the effects of partial shading on PV systems [15]. The methodology highlights modifying the electrical connections of PV modules without dependence on complex algorithms or switches, providing a cost-efficient and low-complexity solution. An innovative reconfiguration method incorporated for PV modules has been presented to mitigate the detrimental impacts of shade [16]. The method utilized the simulated annealing process to modify the electrical connections among photovoltaic modules in a 9 × 9 TCT array to reduce current discrepancies between rows and enhance power production. A hybrid AVOA and recurrent neural network MPPT has been established to boost PV power generating efficiency and improve solar array power tracking [17]. The authors in [18] maximized the generation of the PV array under partial shade operation through an optimal mileage reconfiguration approach combining reinforcement learning and interior point method. Also, the additional mileage payments caused by power fluctuations have been minimized. A multi-agent negotiation method has been developed to reconfigure the PV array under partial shade operation to enhance its harvested energy [19].

In recent years, there has been a significant rise in interest surrounding complex real-world optimization problems that are challenging to resolve across multiple disciplines, particularly in areas related to computer science, data analytics, wireless communication, and engineering design [20–22]. These optimization challenges typically encompass elements such as constraints, decision variables, and objectives. Current research, particularly in the fields of engineering and manufacturing control, places a strong emphasis on the utilization of optimization algorithms [23]. The approaches to solving real-world engineering problems can be broadly categorized into analytical or numerical programming and heuristic or metaheuristic methods [24]. Metaheuristic algorithms (MAs) can be classified into two distinct groups: single-solution and population-based algorithms, with single-solution algorithms typically demonstrating superior efficiency. Real-world problems can often be classified as single-objective, multi-objective, or multimodal optimization problems, each presenting unique complexities [25]. Traditional mathematical techniques, such as gradient descent and conjugate gradient methods, often struggle to effectively address these complex scenarios [26]. Metaheuristic strategies are fundamentally grounded in two key principles: intensification and diversification, which play a crucial role in the quest for optimal solutions. Intensification refers to the process of concentrating efforts on the most promising areas of the search space through evolving the elite candidate agents. Simultaneously, diversification facilitates a broad and effective observation of the search space, primarily achieved through the incorporation of stochastic walks. Achieving a careful equilibrium between these two strategies is vital for ensuring effective convergence towards globally optimal solutions while circumventing the pitfalls associated with local optima [27,28]. The consistent progress of metaheuristic tactics, marked by a growing variety of approaches, highlights the vigorous involvement and persistent commitment of the scientific community to this domain. This intellectual endeavor fosters the implementation of novel approaches and enhances existing ones [29]. This ongoing cycle of invention markedly broadens the spectrum of solutions for handling complex issues [30,31], generating new chances across many application domains. The fundamental versatility of these techniques permits their adjustment to different contexts, reinforcing their importance in confronting contemporary optimization and problem-solving challenges. The Puma optimization algorithm (POA) has recently been introduced for solving optimization problems [32]. This approach is based on the behaviors demonstrated by pumas in their natural habitats, specifically their hunting strategies and territorial exploration methods. The POA adopts these behaviors to create a methodology that balances the exploration of new solutions with the exploitation of existing ones. The primary motivation for using the POA is its robust and flexible design, which allows it to efficiently solve diverse optimization problems. Unlike many other algorithms that struggle to transition between exploration and exploitation, POA incorporates a hyper-heuristic phase change mechanism. This unique feature allows the algorithm to automatically and intelligently switch between phases based on the problem’s nature. The POA outperformed several other optimizers in a majority of the 33 benchmark functions and demonstrated superior results in practical applications such as clustering, community detection, feature selection, and machine learning problems (MLP). This empirical evidence indicates that POA is a highly reliable and effective tool for solving real-world problems. The authors present the enhanced Puma optimization algorithm (IPuma), a refined version of the conventional POA. This algorithm employs the Newton-Raphson Search Rule (NRSR) to enhance the exploration process, particularly in search spaces characterized by numerous local regions. It generates random parameters, facilitating a more dynamic exploration of the search space and refining various algorithmic components. Thus, authors have become interested in the POA algorithm due to its applicability in optimizing diverse problems, especially in the field of engineering applications. Study in [33] introduced a variant of the POA algorithm through the incorporation of chaotic maps, aimed at enhancing the temporal efficiency and stability of nonlinear wheeled mobile robots. Authors in [34] utilized the POA to manage the allocation of diverse distributed generations (DGs) in a radial distribution problem and reduce the power losses. The research detailed in [35] applied the POA to optimize hyperparameters, to enhance the computational efficiency and accuracy of dynamic models related to ship designs. In the study referenced in [36], the POA was applied to derive and optimize the parameters associated with nine three-diode PV modules. The analysis conducted in [37] utilized the POA to determine the effective distribution of power generation and environmental emissions among various units, with a focus on integrating economic and environmental factors. This prompts authors to present an improved IPuma algorithm. The improved IPuma enhances the performance of the traditional POA algorithm by strengthening the exploration phase with a Newton-Raphson-based strategy and improving exploitation through the introduction of adaptive operators. The effectiveness of the proposed IPuma is evaluated using the IEEE CEC’2020 test suite CEC [38]. The IPuma is also utilized to improve the production of solar arrays that are operated under different partial shade patterns by rearranging its panels. The outcomes produced by the IPuma are compared to several established and recently developed meta-heuristics, including the Whale optimization algorithm (WOA) [39], Harris hawks optimization (HHO) [40], Grey Wolf Optimizer (GWO) [41], Particle Swarm Optimization (PSO) [42], Biogeography-Based Optimization (BBO) [43], Gravitational Search Algorithm (GSA) [44], Sine Cosine Algorithm (SCA) [45], Equilibrium optimizer (EO) [46], and original POA [32]. The findings indicate that the IPuma algorithm demonstrates significant statistical promise and surpasses the performance of competing algorithms.

This study proposes a novel optimization method that incorporates real-time monitoring, adaptive module reallocation, and intelligent decision-making. This method improves energy output and system flexibility under partial shade situations by integrating heuristic optimization algorithms with data-driven control mechanisms. The indicated technology enhances the performance of photovoltaic systems while providing a more affordable, cost-effective, and efficient approach to maximizing the generation of energy in real-world situations. This study aims to considerably develop solar energy technology, providing useful insights that can facilitate the broad utilization of photovoltaic systems and raise a sustainable energy future.

The primary contributions of this research are as follows:

•   A novel improved Puma optimization algorithm (IPuma) is introduced to increase the production of a shaded PV array via reconfiguring its architecture.

•   The Newton-Raphson search rule is proposed to boost the exploration process and adapt the exploitation parameters.

•   A 9 × 9 PV array operating under LT, LW, and SW shade patterns is analyzed, while a comparison to other architectures is conducted.

•   The recommended IPuma is evaluated using various indices, such as peak-to-mean ratio, efficiency improvement ratio, maximum efficiency improvement, and mismatch power loss.

•   IPuma’s effectiveness and dependability in generating optimal results are confirmed.

This paper is structured into multiple sections: Section 2 introduces the photovoltaic array model, while Section 3 details the Puma optimization algorithm. Section 4 presents the enhanced Puma optimization algorithm (IPuma), while Section 5 examines the results and assesses the performance of IPuma through the CEC’2020 test suite. Section 6 outlines the use of the IPuma algorithm for reconfiguring the PV array in conditions of partial shade. Section 7 provides a summary of the findings presented in the study.

2  Photovoltaic Array Model

The following section determines the photovoltaic array concept and the associated circuit used for the installation of the array’s modules, as seen in Fig. 1. The most widely used similar circuit for simulating the physical behavior of PV modules is the one diode circuit, which is simpler as shown in Fig. 1. One panel’s electric output current can be calculated using Kirchhoff current law in the manner shown below [47]:

IPV=Iph−Io(exp⁡(q×(VPV+IPV×Rs)a×K×T)−1)−(VPV+IPV×RsRp)(1)

where Iph is the photogenerated current, Io is the reverse saturation current, VPV is the output voltage of the PV cell, Rs and Rp are the circuit series and parallel resistances, respectively, q is the electron charge, a is the diode ideality factor, and K is the Boltzmann constant. The incidence irradiance levels hitting the PV panel surface determine the photogenerated current. Therefore, the following formula is utilized to alter its value:

Iph=Isc_STC(GGSTC)(1+βi×(T−TSTC))(Rs+RpRp)(2)

where Isc_STC is the panel short circuit current at standard test condition (STC), G is the incident irradiance, while GSTC is the irradiance at STC, βi is the temperature coefficient for current, T and TSTC are the panel’s actual temperature and that at STC, respectively. When the panel is completely lighted, the solar irradiance is 1000 W/m2, but when it is partially shaded, it is less than 1000 W/m2, which lowers the power generated by the panel. A set of panels must be connected to produce the necessary amount of power. A well-known reconfiguration is the total cross-tied (TCT) with architecture displayed in Fig. 1, which involves connecting parallel-connected panels in each row while the rows are linked in series. Each one has an index of (ij) to indicate the panel place in the ith row and jth column.

Figure 1: (a) TCT architecture of m × n PV array, (b) single diode diagram (SDM) equivalent circuit

Since the PV panel current is dependent on the levels of incident irradiance, as previously stated, the current of each (ij) panel can be calculated as [48],

Iij=(GijGSTC)×Im_ij(3)

where Im_ij is the ij panel generated current, the ith row current, and the m × n array current can be computed as follows [48]:

Iri=∑j=1n(Iij),i=1,2,…,m(4)

IA=∑j=1n(Iij−I(i+1)j)=0,i=1,2,…,m−1(5)

The Kirchhoff voltage rule is utilized to calculate the array voltage as [48],

VA=∑i=1mVi(6)

where Vi is the ith row voltage. The power-voltage and current-voltage curves for PV panels operating in both normal and partial shadow (PS) circumstances are displayed in Fig. 2. There are several local power peaks with a single global peak power (GPP) due to operating under PS. Shade conditions have a deleterious impact on the GPPs. The reconfiguration procedure effectively reduces the detrimental impacts of the shadow by rearranging the shadowed panels on the same column without altering the array’s electrical connections.

Figure 2: (a) Power-voltage, (b) Current-voltage curves of PV module in normal and PS operations

In the event of partial shade (PS), the TCT connection produces more power than the conventional series-parallel (SP) connection. However, the multi-peak problem in the PV output curves is not resolved by the TCT connection. Researchers proposed ways to reorganize the PV array to address the multi-peak problem and optimize the output power. The SudoKu puzzle, depicted in Fig. 3, is another well-liked layout. The primary problem with SudoKu is that it cannot dynamically reconfigure the PV array. As a result, this study proposes an innovative approach to dynamically rearrange the PV operated in a PS situation to increase the generated PV output power using the improved puma optimizer (IPuma).

Figure 3: SudoKu architecture of m × n PV array

3  The Puma Optimization Algorithm (POA)

Developed in 2024, the Puma Optimization Algorithm (POA) represents a sophisticated metaheuristic approach inspired by the hunting strategies of pumas in their natural habitats. This optimization technique, which is based on population dynamics, conceptualizes each solution as a puma, where the optimal solution is similar to the dominant male, and the optimization landscape is viewed as its territory. A feature of POA is its distinctive approach of alternating between exploration and exploitation phases. This deliberate and strategic transition at every iteration enables the algorithm to adjust effectively to various stages of the optimization process. Thus, POA provides significant benefits compared to many alternative metaheuristic algorithms, positioning itself as an essential instrument for tackling diverse optimization problems across several fields. The fundamental stages of the POA algorithm are defined as follows:

3.1 Phase Change Process

In relation to the POA, the hunting behaviors exhibited by pumas are illustrated to show their capacity to traverse uncharted areas while simultaneously utilizing familiar territories. This modeling process is contingent upon the experiences gathered from prior encounters, simulating the manner in which pumas modify their hunting tactics in response to previous achievements.

3.2 Inexperience Phase

The commencement of the POA encompasses a crucial initialization phase that endures through the initial three iterations. During this phase, the algorithm simultaneously engages in exploration and exploitation activities, laying a solid groundwork for the subsequent stages. The initialization process is regulated by the phase transition mechanism, which is dependent on two fundamental functions, f1 and f2, as defined in Eqs. (7) and (8)

f1ExplorExploit=P1⋅(Seq1costExplor/costEploitSeqTime)(7)

f2Explor/Exploit=P2⋅(∑j=13SeqjcostExplor/costEploit∑j=13SeqjTime)(8)

where the constant P1 and P2 specify the order of f1 and f2, the parameter SeqTime is set to 1. The variable SeqTime is computed for exploration and exploitation processes as follows:

Seq1costExplor=|Cost initbest−Cost 1Explor|(9)

Seq2costExplor=|Cost 2Explor−Cost 1Explor|(10)

Seq3costExplor=|Cost 3Explor−Cost 2Explor|(11)

Seq1costExploil=|Cost initbest−Cost1Exploit|(12)

Seq2costExploil=|Cost2Exploit−Cost1Exploit|(13)

Seq3costExploil=|Cost3Exploit−Cost2Exploit|(14)

where Cost initbest is the cost of the initial population the best solution, Cost1,2,3costExplor/costEploit is the cost of the best solution during the first three iterations, either in the exploration or exploitation process. After that, the POA computes the score parameter that represents the effectiveness for exploration and exploitation to decide which phase to apply:

ScrExplor=P1×f1Explor+P2×f2Explor(15)

ScrExploit=P1×f1Exploit+P2×f2Exploit(16)

Thus, either score is superior, and decides the following process to begin with.

3.3 Exploration Phase

During the exploration, the POA employs a systematic approach to improve and broaden the range of possibilities. The method starts with the sorting of the total population in ascending order. Then, the algorithm proceeds to update the positions of the current solutions as follows:

If rand1>0.5,Mi,G=RndDim∗(Ubound−Lbound)+LboundOtherwise,Mi,G=T1,G+G⋅(T1,G−T2,G)+G⋅(((T1,G−T2,G)−(T3,G−T4,G))+((T3,G−T4,G)−(T5,G−T6,G)))(17)

where Ubound/Lbound are the upper/lower bounds for solution positions. RndDim is a vector of dimension Dim generated randomly in the period [0, 1], T(1/2/3/4/5/6),G are solutions selected randomly from the population. The value of G is computed with a random value rand2 as follows:

G=2×rand2−1(18)

In this exploratory stage, the POA systematically improves the given solutions, utilizing the following equations to direct each subsequent iteration.

Tnew={Mi,G, ifj=jrandorrand3≤UT1,G,otherwise(19)

c=1−UNpop(20)

ifCostTnew<CostTi,U=U+c(21)

Ta,G=Tnew,ifTi,new<T1,G(22)

where Mi,G is the newly generated candidate solution, rand3 is a random value in period [0, 1], N is the population size, and jrand is the random number selected between 1 and N. The parameter U is initialized with U∈[0,1].

3.4 Exploitation Phase

At the current stage, the Puma optimization approach method utilizes unique hunting puma strategies that enhance answers: ambush and swift running. The mathematical representation of these actions is encapsulated in the form of the following formula:

Tnew={ifrand4≥0.5,Tnew=(mean(Soltotal)Npop)×T1r−(−1)β×Xi1+(α×rand5)otherwise,ifrand6≥L,Tnew=Pumamale+(2⋅rand7)⋅exp⁡(randn1)⋅T2r−Tiotherwise,Tnew=(2×rand8)×(F1×R×T(i)+F2×(1−R)×Pumamale)(2×rand9−1+randn2)−Pumamale(23)

where Tnew is the newly generated solution, Soltotal is the sum of solutions, T1r is the population solution selected randomly, β is a randomly set parameter in {0, 1}, αandL are constants, Pumamale is the best solution, rand4/5/6/7/8/9 indicate random numbers generated in period [0, 1], Npop is the size of the population. The T2r is a randomly selected solution based on the following formula:

round((Npop−1)×rand10+1)(24)

The parameters R, F1 and F2 are adjusted as follows:

R=2×rand11−1(25)

F1=randn3×exp⁡(2−Iter×(2MaxIter))(26)

F2=randn4×(randn5)2⋅cos⁡((2×rand12)⋅randn4)(27)

where rand11/12 are random numbers in [0, 1], randn3/4/5 are numbers selected randomly within the population size.

4  The Improved Puma Optimization Algorithm (IPuma)

The original version of the POA employs pure random variables to traverse the solution space, potentially leading to a less directed exploration and a preference towards premature convergence to local minima. An advanced algorithm referred to as IPuma was developed to rectify those deficiencies and boost the pursuit of global solutions. The application of the Newton-Raphson Search Rule (NRSR) within the IPuma algorithm provides benefits compared to a purely random approach. The NRSR search rule exhibits systematic behavior, marked by ergodic features that facilitate a comprehensive and varied investigation of the solution space. This reduces the likelihood of early convergence and prevents the algorithm from getting into non-promising regions. The IPuma enhances exploitation in the original POA, utilizing adaptive parameters instead of random parameters. This integration enhances navigation of the algorithm in the search space, optimizing the balance between global exploration and local exploitation. As a result, IPuma demonstrates significant progress in solution identification while minimizing the number of iterations needed to achieve the global optimum. The primary stages of the proposed IPuma are detailed as follows:

4.1 Exploration Phase

The IPuma uses the inherited Newton-Raphson Search Rule [49] to boost exploration and provide significant advantages over the purely randomly generated solutions:

If u1<0.7,Mi,G=X_TAOi,G

otherwise,Mi,G=T1,G+G⋅(T1,G−T2,G)+G⋅(((T1,G−T2,G)−(T3,G−T4,G))+((T3,G−T4,G)−(T5,G−T6,G)))(28)

The conditional parameter u1 is defined as follows:

u1=beta×3×rand+(1−beta)(29)

parameter beta is 0 or 1 with formula beta=rand<1/2, rand is a random value generated in [0, 1], and T(1/2/3/4/5/6),G are solutions selected randomly from the population.

The value of X_TAO is generated as follows:

X_TAOi,G={Xu+G×(T1,G−u1×Ti,G)+G×δ(u1×Tavg−u2×Ti,G)If u1>0.5X1,G+G×(u1×X1,G−u2×Xi,G)+G×δ(u1×Tavg−u2×Xi,G)Otherwise(30)

where

Xu=a1×(a1×X1+(1−a2)×X2)+(1−a2)×X3(31)

X3=Xi,G−δ×(X2−X1)(32)

where operator δ is adjusted with optimization progress with variables (Iter, MaxIter) as current/total iterations,

δ=(1−Iter⋅(2MaxIter))12(33)

The a1 and a2 are random values in [0, 1], X1 and X2 are generated as follows:

X1=Ti,G−NSR+rho(34)

X2=T1,G−NSR+rho(35)

where the step size rho is generated as follows:

rho=rand×(T1,G−Ti,G)+rand×(T1,G−T2,G)(36)

T(1/2),G are solutions specified randomly from the population. Ti,G is the current solution. The NSR is computed as follows:

NSR=randn×(Xw−Xb)×Δx2×(Xw+Xb−2×xn)(37)

where randn is a randomly distributed value with variance and mean values (0, 1), the solutions Xw/Xb are the worst and best solutions, Δx is the distance between the best solution Xb and the current solution xn as below,

Δx=randdim×(Xb−xn)(38)

The Role of the Newton-Raphson Search Rule in Enhancing the IPuma Algorithm:

The Newton-Raphson Search Rule (NRSR) is introduced in the IPuma algorithm to improve its exploration capabilities. The core idea is to guide the search agents toward promising regions more effectively. (i) Generating new positions: The NRSR is utilized to calculate new candidate solutions, denoted as X1 and X2, as provided in Eqs. (34) and (35). These new positions are not random; they are influenced by the best solution found, the worst solution, and the current agent’s position (x). This creates a direct search path. (ii) Integrating a dynamic step size: The NRSR is part of the calculation for a dynamic step size rho. This rho is a vector that guides the search by considering the difference between the best solution and the current solution, as well as the difference between two randomly selected solutions. (iii) Improving the position update: The enhanced IPuma algorithm applies the NRSR to update the solutions’ positions. Further, the update incorporates a dynamic parameter delta that decreases over iterations. This causes the search to be broader in the beginning with a high delta value and more refined and localized toward the end when the delta value becomes low, which is a common and effective strategy in optimization algorithms. In essence, the NRSR provides a more sophisticated and calculated way for the algorithm to explore the search space, moving beyond simple random movements to a more informed and adaptive strategy. Algorithm 1 shows the Pseudo-code of the suggested IPuma exploration phase.

4.2 Exploitation Phase

Regarding the exploitation factors and conditions, the original POA uses a constant parameter that needs fine-tuning when applying to different optimization problems; on the other hand, the IPuma employs adaptive parameters. (i) The exploitation condition operator L: In the original POA, L is constant, meaning the balance between different exploitation sub-strategies remains fixed throughout the optimization process. In the IPuma, the condition operator L is updated with optimization progress as depicted in Eq. (40). This means the algorithm starts with a higher chance of choosing the more aggressive attack strategy and shifts toward a more conservative, population-based approach as it converges. (ii) The γ is updated adaptively based on the variable’s current iteration and total iterations as in Eq. (41). This causes the step size to increase over time, raising larger jumps in the exploitation phase as the algorithm progresses, which can help escape local optima.

The exploitation process performed to refine explored solutions is updated as follows:

Tnew={ifrand4≥0.5,Tnew=(mean(Soltotal)Npop)×T1r−(−1)β×Ti1+(α×rand5)otherwise,ifrand6≥L,Tnew=Pumamale+γ×exp⁡(randn1)×T2r−Tiotherwise,Tnew=γ×(F1×R×T(i)+F2×(1−R)×Pumamale)(2×rand9−1+randn2)−Pumamale(39)

where Tnew is the newly generated solution, Soltotal is the sum of solutions, The (T1r,T1r) are population solutions selected randomly from the best third of the population after sorting, instead of random selection over the population in the original POA algorithm, β is a randomly set parameter in {0, 1}, α is set to 2, Pumamale is the best solution, rand5/9 is the random numbers generated in period [0, 1], Npop is the size of the population, and Ti is the current solution. The condition operator L is updated with optimization progress variables current iteration Iter, and total iterations MaxIter,

L=(1−(2×Iter)/MaxIter))2(40)

The γ is updated adaptively based on variables current iteration Iter, and total iterations MaxIter,

γ=rand+2×IterMaxIter(41)

The suggested exploration phase of the IPuma is clarified in the Pseudo-code given in Algorithm 2. The full Pseudo-code of IPuma is given in Algorithm 3.

The flowchart of the proposed IPuma optimization methodology is illustrated in Fig. 4.

Figure 4: Proposed IPuma optimization methodology

4.3 Parameter Settings

In the experiments conducted, each algorithm is executed 30 times, comprising 1000 iterations, with 30,000 function evaluations (FEs) to guarantee that the comparison is both equitable and dependable. The configurations of the comparative optimizers are detailed in Table 1. The authors adhered to standard settings for each algorithm, which is considered a best practice and mitigates the potential for parameterization bias, as noted by [50]. This study utilized a varied array of performance metrics to evaluate the effectiveness of each algorithm.

4.4 Time Complexity

The execution time of the IPuma algorithm is largely contingent upon the period needed to update the positions of the solutions. This can be articulated in the following manner:

O(IPuma)=(O(position update))=O(NT)(42)

where N is the size of the population, and T is the number of iterations. The big-O analysis for both POA and IPuma reveals a comparable complexity, indicating that IPuma is on par with the original Puma regarding big-O performance.

5  Experimental Series 1: CEC’2020 Test Suite Analysis

The effectiveness of IPuma is illustrated through the CEC’20 benchmark.

5.1 Statistical Results

The effectiveness of the IPuma algorithm is assessed through a statistical approach that involves determining the average and standard deviation of the leading solutions from each execution. Accordingly, Table 2 provides a detailed comparison of IPuma’s performance alongside alternate algorithms on test functions of dimension 10, with the optimal results emphasized in bold. Reviewing the findings in Table 2 shows that the IPuma algorithm reliably outperforms other algorithms in most test functions. In addition, as shown in Table 2, the IPuma algorithm attains the highest rank in the Friedman test.

The boxplot analysis depicted in Fig. 5 is derived from 30 independent runs and serves to evaluate the distributions of the data. A compact shape in the boxplot signifies a high level of conformity in the algorithm results. The boxplots corresponding to the IPuma are predominantly narrower than those of other algorithms across most test methods, indicating lower value ranges. Accordingly, the IPuma shows a marked improvement in performance over its competitors across most test methods. The analysis of convergence is illustrated in Fig. 6, which presents the curves corresponding to the various comparative methods. Overall, the IPuma stands out with a rapid convergence towards optimal values. This swift convergence is critical for optimization tasks that necessitate efficiency in time, particularly in real-time optimization, highlighting the effectiveness of the IPuma algorithm in addressing such challenges.

Figure 5: Boxplot visualizations for the proposed IPuma technique, in comparison to alternative methods, were generated using the CEC’2020 test suite at a dimensionality of Dim = 10

Figure 6: Convergence visualizations for the proposed IPuma technique, in comparison to alternative methods, were generated using the CEC’2020 test suite at a dimensionality of Dim = 10

5.2 Wilcoxon Test Results

Wilcoxon is a non-parametric statistical metric to assess the significance of data obtained from algorithms. This test illustrates that the behaviour of the algorithm is consistent rather than erratic. Despite the inherent stochastic characteristics of metaheuristic algorithms, their performance is anticipated to remain reliable. For a thorough understanding of Wilcoxon’s test, readers are encouraged to consult [51]. This investigation employs the Wilcoxon rank-sum test, utilizing the average values of the optimal outcomes from 30 independent executions, to illustrate the differences between the proposed IPuma and other optimization techniques. Table 3 presents a comprehensive analysis of the results obtained from IPuma and alternative methodologies. The p-value indicates the significance level of the test, utilized to assess the rejection of the hypothesis of similarity, with a significance threshold set at 5%.

6  Real-Time Application

This section describes a practical engineering application that uses the proposed IPuma to dynamically reconfigure and boost the power output of a PV array running in PS circumstances. Double pole multi-throw (DPMT) switches are utilized to link the PV panels, and a switching matrix is utilized to control the states of these switches. DPMT switches are required for the dynamic reconfiguration of PV arrays because each PV module has two electrical terminals (positive and negative poles) that need to be reconnected simultaneously when the array topology is changed. Although single-pole switches can only reroute a single line, they run the risk of leaving modules floating or causing dangerous open circuits. The cost, losses, and durability of DPMT switches should be carefully weighed against the energy advantages from mismatch loss reduction, even if these switches are necessary for safe and flexible reconfiguration. The recommended IPuma Pseudo code provided in Algorithm 3 receives the generated power after the temperature, solar irradiance, array current, and voltage are recorded. Under PS situations, the recommended method is in charge of obtaining the ideal switching matrix that maximizes the power produced by the PV array. The indicated PV reconfiguration procedure conducted using the recommended IPuma is displayed in Fig. 7. This work presents a simplified switching matrix that employs double-pole multirow switches to reduce complexity and the number of switches. It is important to highlight that double-pole switches can only be connected to adjacent switch contacts.

Figure 7: Proposed PV reconfiguration procedure conducted using the recommended IPuma

The fitness function being examined is maximizing the PV array output power, and it can be expressed as follows:

MaximizePPV=VA×IA=∑i=1m∑j=1n(Vmi×Iij)(43)

The PV panel can only be swapped out for another one in the same column, which is the problem’s limitation. This can be stated as,

{xik∈{1,2,…,m},i=1,2,…,m,k=1,2,…,n⋃k=1nxik={1,2,…,m},i=1,2,…,m(44)

where xik is the ij switch state. This work uses a 9 × 9 PV array to investigate three PS scenarios: short wide (SW), long wide (LW), and lower triangle (LT).

In TCT and Sudoku layouts, the shaded and unshaded modules are distributed more uniformly across strings, so the mismatch loss is enhanced. This improves the homogeneity of current between parallel branches; however, these systems are static and fail in adjusting the array once the wiring is fixed, and therefore lose effectiveness when the shading is dynamic or irregular. On the other hand, the indicated IPuma actively searches for near-optimal interconnections of PV modules under any given shading condition. Unlike TCT/Sudoku, it was adapted dynamically to shifting shading instead of depending on a set arrangement, as well as avoiding local optima and finding configurations that more efficiently minimize mismatch using a population-based search with a modified exploration and exploitation balance. In addition, IPuma can deal with complex or asymmetrical shading patterns when static patterns are unable to remain consistent. Because IPuma is adaptable enough to change in real-time, it decreases mismatch power losses far more than TCT and Sudoku, particularly in situations with non-uniform, dynamic partial shade.

In this experiment, Kotak 80 W panels with the electrical specifications given in [52] are used. When fully illuminated and functioning normally, the array’s total power is 14.648545 kW. The results are compared using the indicated IPuma, TCT [9], SudoKu [53], and additional methods such as BBO, SCA, GWO [54], HHO, and the original Puma. Each algorithm is executed 30 times, 200 iterations, and a population size of 20 to guarantee a fair comparison to the employed approaches.

6.1 Scenario (1): Operation in SW Pattern

Fig. 8a indicates that the array is distributed with solar irradiances of 0.9, 0.6, 0.4, and 0.2 kW/m2 in this shading pattern. The TCT is reflected in this setup, and the row currents can be computed as,

Irow−7=Irow−8=Irow−9=3×0.6×Im+3×0.4×Im+3×0.2×Im=3.6×Im

Irow−6=5×0.6×Im+4×0.9×Im=6.6×Im

Figure 8: Best structure of PV system (a) TCT, (b) SudoKu, (c) BBO, (d) SCA, (e) GWO, (f) HHO, (g) Puma, and (h) The proposed IPuma during SW shade pattern

Irow−1=Irow−2=Irow−3=Irow−4=Irow−5=9×0.9×Im=8.1×Im(45)

On the other hand, the original Puma generated the following row currents:

Irow−1=Irow−3=Irow−4=Irow−7=Irow−9=5×0.9×Im+2×0.6×Im+0.4×Im+0.2×Im=6.3×Im

Irow−2=Irow−5=Irow−6=Irow−8=6×0.9×Im+0.6×Im+0.4×Im+0.2×Im=6.6×Im(46)

The proposed IPuma generates the following currents:

Irow−1=Irow−5=6×0.9×Im+2×0.4×Im+0.2×Im=6.4×Im

Irow−2=Irow−4=Irow−7=6×0.9×Im+0.6×Im+2×0.2×Im=6.4×Im

Irow−9=4×0.9×Im+4×0.6×Im+0.4×Im=6.4×Im

Irow−3=Irow−8=5×0.9×Im+2×0.6×Im+2×0.4×Im=6.5×Im

Irow−6=5×0.9×Im+3×0.6×Im+0.2×Im=6.5×Im(47)

The optimal results for PV arrays running at the SW pattern with the proposed IPuma and other methods are listed in Table 4. Although the proposed IPuma received 10.441 kW of global power from the array, the original Puma extracted 10.379 kW. However, in second place behind the recommended IPuma, the BBO’s GP output was 10.413 kW. All PV array configurations’ current-voltage and power-voltage characteristics during the SW pattern are shown in Fig. 9. Out of all the curves, the ones that were produced using the recommended method are the best. The proposed approach fared better in that PS pattern than the others.

Figure 9: PV array (a) Current-voltage, (b) Power-voltage characteristics during operation in SW pattern

6.2 Scenario (2): Operation in the LW Pattern

The PV array surface in the second PS under consideration is exposed to solar irradiances of 0.9, 0.6, 0.5, 0.4, and 0.2 kW/m2, with the distribution displayed in Fig. 10a. This PS is long and wide (LW). Following the implementation of the recommended IPuma and others, Fig. 10b–h shows the fetched optimal reconfigurations of SudoKu, BBO, SCA, GWO, HHO, and the original Puma, respectively.

Figure 10: Best structure of PV system (a) TCT, (b) SudoKu, (c) BBO, (d) SCA, (e) GWO, (f) HHO, (g) Puma, and (h) The proposed IPuma during LW shade pattern

During the LW shade pattern, the proposed IPuma produces the following currents:

Irow−1=Irow−2=Irow−3=4×0.9×Im+2×0.5×Im+2×0.4×Im+0.2×Im=5.6×Im

Irow−4=Irow−5=Irow−6=4×0.9×Im+5×0.4×Im=5.6×Im

Irow−7=Irow−8=Irow−9=4×0.9×Im+2×0.6×Im+0.4×Im+2×0.2×Im=5.6×Im(48)

Table 5 lists the optimal GP acquired using the proposed IPuma layout and other methods during the LW pattern. The original Puma achieved a GP of 9.041 kW, while the proposed method retrieved the largest GP from the array under this shade pattern, with a value of 9.063 kW. With the lowest GP of 9.022 kW, the HHO is the worst of the metaheuristic techniques that were considered. In such a shadow pattern, the recommended IPuma performed better than any other configuration.

Fig. 11 displays the current-voltage and power-voltage characteristics for all PV array configurations under the LW pattern. The curves that were created utilizing the proposed procedure are the best of all of them. In that PS pattern, the recommended method performed better than the others.

Figure 11: PV array (a) Current-voltage, (b) Power-voltage characteristics during operation in LW pattern

6.3 Scenario (3): Operation in LT Pattern

The third shade pattern under consideration is the lower triangle (LT), whose solar radiation distribution is shown in Fig. 12a and includes four sun radiations of 0.9, 0.7, 0.5, and 0.3 kW/m2. The proposed IPuma and others are implemented, while the optimal fetched configurations are illustrated in Fig. 12.

Figure 12: Best structure of PV system (a) TCT, (b) SudoKu, (c) BBO, (d) SCA, (e) GWO, (f) HHO, (g) Puma, and (h) The proposed IPuma during LT shade pattern

The proposed IPuma generates the following currents under the LT shade pattern:

Irow−1=Irow−2=Irow−9=5×0.9×Im+0.5×Im+3×0.3×Im=5.9×Im

Irow−3=Irow−7=Irow−8=4×0.9×Im+2×0.7×Im+3×0.3×Im=5.9×Im

Irow−4=Irow−5=Irow−6=3×0.9×Im+2×0.7×Im+3×0.5×Im+0.2×Im=5.9×Im(49)

With a GP of 9.567 kW, the arrangement derived from the proposed IPuma, Fig. 12g, is the largest of the arrangements that were considered. The original Puma configuration in this case generates 9.522 kW of GP. The SCA-based arrangement generated the least amount of GP with a value of 9.451 kW, making it the worst of the metaheuristic techniques that were examined. The best fetched GP via the proposed IPuma arrangement and other algorithms in the LT pattern are listed in Table 6.

Fig. 13 displays the power-voltage and current-voltage characteristics for each PV array configuration in the LT pattern. In terms of quality, curves made using the proposed method are superior. The proposed approach outperformed the others in that PS pattern.

Figure 13: PV array (a) Current-voltage, (b) Power-voltage characteristics during operation in LT pattern

In addition, four metrics are calculated to confirm the feasibility of the proposed method compared to the methods that are being considered: mismatch power loss (Pml), efficiency improvement ratio (EIR), maximum efficiency improvement (MEI), and peak-to-mean ratio (PMR). The computation of these terms is as follows [55]:

{Pml=PGPs−PGP%EIR=(Pave−Pave,orgPave,org)×100%MEI=(PGP−PGP,orgPGP,org)×100%PMR=(PGPPave)×100(50)

where PGPs and PGP are the array global power without PS and after reconfiguration, respectively, Pave and Pave,org are the average power of the original and configured arrays, respectively, and PGP and PGP,org are the global powers after and before reconfiguration, respectively. The PMR and EIR performance measurements are helpful in assessing how well the IPuma minimizes mismatch losses in the context of PV array reconfiguration. A lower PMR indicates that reconfiguration has effectively balanced the contributions of various modules and reduced mismatch. The PMR compares the peak power output to the mean power across the array to reflect the uniformity of power distribution among the PV strings. The EIR directly shows the increase in overall array efficiency by calculating the relative improvement in total power output following reconfiguration as compared to a reference interconnection. It can assess the net energy benefit realized as well as the equity of power sharing among modules by combining these measurements, providing a more comprehensive view of the reconfiguration algorithm’s effectiveness. Table 7 presents the values of the considered metrics under all examined scenarios. With values of 4.2072, 5.5856, and 5.0818 kW for SW, LW, and LT patterns, respectively, the proposed IPuma was able to obtain the lowest mismatch power losses among the explored approaches; in terms of maximum efficiency improvement, it achieved the best improvements with 26.86%, 21.89%, and 29.07% for the examined patterns. Using the proposed IPuma and additional techniques, the bar charts of the considered metrics for each of the examined scenarios are shown in Fig. 14. The charts demonstrate how good the parameters obtained using the proposed method are.

Figure 14: Bar chart of (a) mismatch power loss, (b) efficiency improvement ratio, (c) maximum efficiency improvement, and (d) peak-to-mean ratio for all examined scenarios

Through reconfiguration, the collected results showed that the proposed IPuma can be recommended as an efficient method for enhancing PV array generation for all examined PS conditions.

There are still a number of issues with the proposed IPuma, including the fact that it depends on simplified PV models, so noise, MPPT transients, and model mismatch can be included in the fitness assessment. In addition, regular topology changes can erode advantages due to realistic arrangement constraints, including switching hardware, finite endurance, on-state losses, thermal stress, and non-zero reconfiguration time. In addition, the analysis is performed on specific arrays and shade patterns; benefits can be less at sites with moderate or very temporary mismatches, and the expense of the reconfiguration hardware cannot always be warranted.

7  Conclusions

This study presents the Enhanced Puma Optimization Algorithm (IPuma), a novel dynamic reconfiguration tool for TCT-connected photovoltaic arrays to optimize energy production under partial shading conditions. The proposed approach uses the Newton-Raphson search rule (NRSR) to increase exploration, particularly in search spaces with more local regions. It also increases exploitation using adaptive parameters instead of the original Puma’s random parameters. Extensive testing on the CEC’20 benchmark problems has confirmed the usefulness of the proposed IPuma. It demonstrates superior performance to contemporary and established metaheuristic algorithms in terms of efficiently traversing the search space and reaching convergence towards areas that are close to ideal. In addition, the proposed IPuma is utilized to reorganize a 9 × 9 array that functions in LT, LW, and SW shade patterns. The proposed approach is compared to standard Sudoku and TCT configurations with other programmed techniques such as WOA, HHO, GWO, PSO, BBO, GSA, SCA, EO, and the original Puma. In addition, the success of the proposed method is evaluated by calculating the metrics of mismatch power loss, maximum efficiency improvement, efficiency improvement ratio, and peak-to-mean ratio.

The primary findings of the study are as follows:

•   For SW, LW, and LT, the proposed IPuma increases the generated power by 36.72%, 28.03%, and 40.97%, respectively, surpassing the TCT configuration.

•   Among the investigated methods, the proposed IPuma can achieve the lowest mismatch power losses, with values of 4.2072, 5.5856, and 5.0818 kW for SW, LW, and LT patterns, respectively.

•   Among the examined algorithms, the proposed IPuma produces the best maximum efficiency improvement, with 26.86%, 21.89%, and 29.07% for the patterns under investigation.

The outcomes demonstrate the superiority and proficiency of the proposed method in terms of convergence rates and stability, as well as its suitability for dynamically rearranging the PV system and optimizing its energy harvesting. Future work will concentrate on making the proposed approach more scalable for bigger PV plants and more complex shading scenarios. Also, field-scale validation and hardware-in-the-loop testing on bigger systems will be crucial in the next work.

Acknowledgement: This research project was funded by the Deanship of Scientific Research and Libraries, Princess Nourah bint Abdulrahman University, through the Program of Research Project Funding After Publication, grant No. (RPFAP-82-1445).

Funding Statement: This research project was funded by the Deanship of Scientific Research and Libraries, Princess Nourah bint Abdulrahman University, through the Program of Research Project Funding After Publication, grant No. (RPFAP-82-1445).

Author Contributions: The authors confirm contribution to the paper as follows: study conception and design: Ahmed Fathy, Mohamed A. Mahdy, Essam H. Houssein; data collection: Nagwan Abdel Samee; analysis and interpretation of results: Ahmed Fathy, Mohamed A. Mahdy, Essam H. Houssein; draft manuscript preparation: Maali Alabdulhafith. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: This data used in the study are not available due to institutional policies and constraints on confidentiality.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

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