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Scientific Reports volume 15, Article number: 44554 (2025) Cite this article
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Optimizing photovoltaic tilt angles to maximize solar radiation capture remains a critical and challenging task. This paper proposes HMWOAIGWO, a novel hybrid metaheuristic algorithm that integrates the improved grey wolf optimizer (IGWO) with the whale optimization algorithm (WOA). The proposed algorithm aims to optimize tilt angles on daily, monthly, and annual scales while addressing the limitations of individual methods, including limited population diversity, susceptibility to local optima, and slow convergence rates. The performance of HMWOAIGWO was rigorously evaluated against ten state-of-the-art algorithms using 23 benchmark suites and the CEC 2019 test functions. Results indicate that HMWOAIGWO achieved the highest accuracy on 19 out of 33 functions and ranked within the top two for convergence speed in 78% of the test functions (18/23). In addition, across five real-world optimization problems, the algorithm attained the lowest standard deviation in all cases(100%) and outperformed competitors in mean performance on 60% of the problems. Statistical validation via Wilcoxon and Friedman tests confirm that the statistical results significantly improve the optimality of the solutions obtained by HMWOAIGWO. Applied to photovoltaic systems, it yielded improvements in solar radiation capture of 4%, 1.76%, and 0.96% for daily, monthly, and annual tilt optimizations, respectively. These findings demonstrate the algorithm’s capability to effectively balance exploration and exploitation, making it a robust tool for complex, real-world photovoltaic optimization challenges.
As global awareness of environmental protection intensifies, the significance of photovoltaic (PV) power generation has grown substantially. Consequently, enhancing the efficiency of PV systems has become a subject of critical interest in practical applications^1,2,3,4,5,6. Given the variability of solar radiation depending on climate and geographical conditions, optimizing the tilt angle of PV panels to maximize energy output has become a primary focus of recent research. Numerous studies have been conducted in various regions around the world to determine the optimal tilt angle and azimuth for PV systems, such as Iberian Peninsula⁷, Saudi Arabia⁸, Algerian Big South⁹, Uganda¹⁰, India¹¹. Considering the location of the Iberian Peninsula, the optimal PV tilt angle for maximizing annual production is 34°. To optimize seasonal PV production, the optimal tilt angles are 20° in summer and 55° in winter⁷. In Saudi Arabia, the optimal annual tilt angles differ due to the geographic and latitudinal variations among the cities. In Dhahran, the optimal annual tilt angle is 27.3°; in Riyadh, it is 26.0°; in Jeddah, it is 22.7°; and in Arar and Abha, they are 32.7° and 20.1°, respectively⁸. Tests have been conducted on fixed tilt angles at different time scales in the Algerian Big South⁹. Compared to horizontally positioned solar PV modules, solar energy incident upon the surface saw a rise of 20.61% for monthly adjustments, 19.58% for seasonal, 19.24% for semi-annual, and 13.78% for annual. To fill the gap in the literature on the optimal tilt angle and orientation of solar PV systems on pitched rooftops in the equatorial region (12°S to 12°N)¹⁰, determined the most effective inclination for photovoltaic panels in this equatorial zone, using Uganda as a case study. They found that the annual optimal tilt angle ranges from 0.0° to 5.1°, while the average monthly optimal angle varies from 2.1° to 11.2°. To investigate the relationship between the optimal tilt angle and maximizing PV array capacity¹¹, selected India as the case study location. The results found that the optimal tilt angle in India varies from 63° to 0° throughout the year, with the maximum occurring in December. Since the maximum output power of solar photovoltaic systems changes with the tilt angle, each location should have an optimal tilt angle to maximize the annual energy production of solar photovoltaic power generation. Other studies have been conducted in various regions, such as Canada^12,13, United States of America¹⁴, Germany and Australia¹⁵, to investigate how adjusting the tilt angle of PV modules can optimize energy production. Numerous empirical models leveraging climate data have been devised to assess photovoltaic power generation. Researchers have also introduced various models and algorithms to calculate solar radiation on inclined surfaces under diverse geographic and climatic conditions. These transposition models usually can be classified into isotropic and anisotropic models. Isotropic models assume that the diffuse radiation is uniformly distributed and independent of azimuth and zenith angles, while the anisotropic model accounts for the circumsolar diffuse and/or the horizontal-brightening components on a sloped surface⁸. On the other hand, with the technological advancements in artificial intelligence, there is a trend in using tools such as metaheuristic algorithms to optimization the PV tilt angle.
In recent years, algorithms that employ metaheuristic optimization have become increasingly prominent due to their effectiveness in addressing complex optimization challenges across multiple domains. Employing metaheuristic algorithms for optimal tilt angle calculation or optimization can greatly reduce computational effort. Meta-heuristic optimization algorithms can be classified into several categories according to their underlying principles and search strategies: (1) evolutionary algorithms, which include genetic algorithms (GA)¹⁶, differential evolution (DE)¹⁷, evolution strategy (ES)¹⁸, genetic programming (GP)¹⁹ and so on.(2) swarm intelligence algorithms, which include particle swarm optimization (PSO)²⁰, coati optimization algorithm (COA)²¹, harris hawks optimization (HHO)²², dung beetle optimizer (DBO)²³ and so on. (3) human learning-based algorithms, including social learning optimization algorithm (SLOA)²⁴, social engineering optimizer (SEO)²⁵, cultural evolution algorithm (CEA)²⁶ and so on.(4) physics-based methods, which include equilibrium optimizer (EO)²⁷, golden ratio optimization algorithm (GROA)²⁸, Young’s Double-Slit experiment (YDSE)²⁹, atom search optimization (ASO)³⁰ and so on. Within the vast landscape of meta-heuristic optimization algorithms, an intriguing duality emerges, epitomized by the dynamic interplay between exploration (global search) and exploitation (local search). Numerous studies have been conducted to determine the optimum tilt angle and orientation of solar PV systems using artificial intelligence tools such as PSO^10,12,31, GA and simulated annealing (SA)^32,33,34, and artificial neural- network (ANN)³⁵, among others.
Among the pantheon of meta-heuristic optimization algorithms, the whale optimization algorithm (WOA) stands as a testament to nature’s ingenuity and the inspiration it provides to algorithmic design. The algorithm was proposed by Mirjalili et al. in 2016³⁶. WOA exhibits favorable characteristics such as requiring fewer number of parameters to control since it includes only two primary internal parameters for adjustment, along with straightforward implementation and notable flexibility. However, it also suffers from a slow convergence speed, low convergence accuracy, and a propensity for becoming trapped in local solutions³⁷. These issues are particularly relevant when determining the optimal tilt angle for photovoltaic (PV) systems, where efficient optimization is crucial for maximizing energy output. In light of this, in recent years, numerous scholars have endeavored to propose enhanced algorithms to further refine the performance of the WOA. Ling et al. proposed levy flight trajectory based WOA to increase diversity in solution for WOA in 2017³⁸ and Xu et al. added Levy flight and quadratic interpolation into WOA in 2018, while also introducing a non-linear dynamic strategy based on a cosine function to enhance the accuracy of solutions³⁹. In addition to the Levy flight strategy, chaotic maps are also a commonly used approach to enhance the performance of the WOA algorithm. Kaur et al. introduced chaos to control the status of WOA and enhance the performance of WOA⁴⁰. Chen et al. developed synchronous Levy flight and chaotic local search into WOA in 2018 to address its premature convergence issue⁴¹. Similar to Xu et al., Abdel-Basset et al. introduced the largest order value and mutation phase on top of incorporating levy flight and chaotic maps into WOA to enhance the algorithm’s diversity and exploration capability⁴². Besides introducing the levy flight strategy and chaotic maps, adopting an adaptive weight factor is also a strategy to enhance the WOA. Chen et al. proposed a reinforced WOA, which uses a double adaptive weight strategy to enhance the convergence speed of the WOA⁴³.
The aforementioned enhancements are primarily achieved by integrating various strategies into the existing framework of the WOA algorithm. Moreover, incorporating other algorithms during the exploration and exploitation phases of WOA to improve its performance is another effective method to enhance the efficiency of WOA during the photovoltaic optimization process. Trivedi et al. proposed a hybrid PSO-WOA algorithm which utilizes PSO for the exploitation phase but overlooks the sensitivity of PSO algorithm to initial parameters such as learning factors, and its tendency to fall into local optima. Therefore, using the PSO algorithm during the exploitation phase is not the optimal choice⁴⁴. Tang et al. proposed a hybrid WOA with artificial bee colony to improve the global performance of the WOA, however, this improvement method only compares the optimal outcomes yielded by two distinct algorithms and selects the most favorable one to serve as the starting point for the subsequent iteration of the WOA algorithm⁴⁵. Mafarja et al. proposed a hybrid WOA with simulated annealing (SA) algorithm, which utilized SA to enhance exploitation within WOA. Similarly, this approach overlooks the drawbacks of the SA algorithm, such as slow convergence speed and significant sensitivity to key parameters like initial temperature and cooling rate⁴⁶. Revathi et al. and Nishant et al. respectively proposed the brainstorm-based WOA⁴⁷ and the WOA with DE algorithm⁴⁸. However, they only provided comparisons of the proposed algorithms with others in the applied domains, without presenting standard test function validations for these two algorithms, which hinders further exploration of the performance of the proposed algorithms. Mohammed proposed a WOA-BAT algorithm which utilizes the bat algorithm for the exploration phase⁴⁹. This approach has to some extent improved the performance of the WOA algorithm. However, during testing with the 2019 CEC functions, its accuracy was not sufficient, indicating shortcomings in escaping from various local optima. Based on this, the same author proposed the grey wolf optimization (GWO)-WOA algorithm⁵⁰ to enhance the performance of the WOA algorithm.
GWO was proposed in 2014 by Mirjalili et al.⁵¹, while it possesses advantages such as simplicity, ease of implementation, and few control parameters, it also suffers from drawbacks such as premature convergence. Hence, numerous scholars have proposed a variety of improvement strategies to enhance it. E. Emary et al.⁵² proposed a binary grey wolf optimization algorithm for feature selection. The algorithm demonstrates better separability in feature selection; however, it was only compared with PSO and GA, and the limited number of algorithms used for comparison on standard test functions fails to fully showcase its superiority. Similarly, H. Joshi et al.⁵³ introduced an enhanced grey wolf optimization algorithm for global optimization and compared it with multiple algorithms; however, its accuracy is not sufficiently high, achieving the best mean value in only 10 out of 25 test functions. Moreover, D. Alsadie et al.⁵⁴ developed a modified grey wolf optimization algorithm for energy-efficient internet of things task scheduling in fog computing, which effectively improves the makespan; however, it was not validated on standard test functions, hindering a more effective demonstration of its superiority. Nadimi-Shahraki et al. introduced an improved GWO (IGWO) by using a dimension learning-based hunting (DLH) to improve both the exploration and exploitation phases of GWO in 2021⁵⁵. In studies determining the optimal tilt angle for PV systems, such improved algorithms can significantly enhance the efficiency and accuracy of the optimization process. Additionally, many scholars have fused the WOA algorithm with the GWO algorithm, introducing hybrid WOA-GWO algorithms or improved versions thereof. Similar to³⁶, Ababneh proposed a hybrid GWO and WOA to address engineering problems⁵⁶. However, as it merely combines the existing GWO with the WOA without any enhancement, it fundamentally fails to address the issue of premature convergence commonly observed in the GWO. In determining the optimal tilt angle for PV systems, this method may lead to local optima, thereby failing to achieve the best energy output. Korashy et al. introduced a hybrid WOA and GWO algorithm, which is optimized by updating the positions of whales using GWO’s leadership hierarchy⁵⁷. However, akin to^33,34, the lack of validation with standard test functions precludes a thorough assessment and exploration of the algorithm’s performance, hence its application in PV tilt angle optimization remains to be fully verified. Singh et al. introduced a hybrid WOA with mean strategy of GWO⁵⁸, the algorithm exhibits good accuracy and convergence rate when tested on multi-modal functions, but converges significantly slower when tested on unimodal functions. Therefore, while the algorithm performs well in overcoming the limitations of local optima, there is still room for improvement in terms of the speed of global search. Asghari et al. proposed a chaotic and hybrid gray wolf-whale algorithm, which integrates the e roulette wheel selection method and chaotic maps mechanism alongside the hybridization of GWO and WOA⁵⁹. The algorithm demonstrates a commendable convergence rate when tested on both unimodal and multimodal functions. However, its precision does not present notable advantages compared to the existing published WOA-GWO algorithms. Therefore, in PV tilt angle optimization, while this algorithm shows potential, it still requires further improvements to enhance its precision and practicality.
To address these weaknesses, a hybrid modified WOA-IGWO (HMWOAIGWO) is proposed in this paper to determine the optimal tilt angle of solar systems. Since the distance control parameter in the original WOA and IGWO algorithms is linear and cannot adapt to the search process of WOA and IGWO, which is nonlinear and complex, we propose a non-linearly reduced function to improve its performance. Additionally, since the (overrightarrow C) vector represents the impact of natural obstacles on approaching prey and the update of the search agent’s position frequently relies on the alpha wolf, a modified control parameter ({overrightarrow C _1}) for the alpha wolf (α) is proposed to control the area of wolves near the prey. Furthermore, due to the inadequate convergence speed and accuracy of the position update formula for the search agent after the (t + 1) iteration in the IGWO, we propose a condition and a modified formula for updating the next generation individuals’ positions. The HMWOAIGWO algorithm has the following advantages: (1) Striking an excellent balance between exploration and exploitation than the original WOA and GWO. (2) Outstanding convergence accuracy and unparalleled convergence speed. (3) Sufficient ability to maintain diversity and capacity to escape local optima. (4) Strong robustness and stability. By comparing the application results of 10 state-of-the-art algorithms on 23 benchmark functions, CEC 2019 benchmark suite and five real-world engineering problems, the algorithm proposed in this paper demonstrates the ability to accelerate convergence speed, enhance convergence accuracy, and effectively evade local optima. Based on these advantages, the HMWOAIGWO algorithm was applied for optimizing photovoltaic tilt angles to maximize radiation. The algorithm was tested for daily, monthly, and annual tilt adjustments, leading to the selection of the most effective optimization strategy for enhanced solar energy capture.
The main contributions of this paper are summarized as follows:
An advanced hybrid HMWOAIGWO algorithm, which combines WOA and IGWO with the DLH search strategy, has been developed and implemented to effectively and accurately calculate the optimization tilt angles of photovoltaic systems, with utmost precision and accuracy.
The algorithm proposed a modified control parameter for the alpha wolf (α) to control the area of wolves near the prey, aiming to enhance the algorithm’s global exploration capabilities. The algorithm also replaced the linearly reduced distance control parameter in both the original WOA and IGWO optimizers with a non-linearly decreasing function to improve performance. Furthermore, a modified formula and condition were proposed for updating the next generation’s individual positions in IGWO, focusing on strengthening its convergence speed and accuracy.
The performance of our algorithm on 23 benchmark suites and CEC 2019 test functions was evaluated and compared with 10 other algorithms, and five real-world engineering problems were used to evaluate the effectiveness of the HMWOAIGWO algorithm in solving practical problems.
The performance of the proposed HMWOAIGWO algorithm is assessed using Wilcoxon and Friedman tests, and its results are rigorously compared with those of ten well-established meta-heuristic algorithms. Additionally, daily, monthly, and annual tilt optimization cases are conducted, and the optimal tilt angles that maximize radiation are selected.
This paper is organized as follows. The introduction to the photovoltaic transposition models and the optimization objective function is presented in the “Photovoltaic Transposition Models and Formulation” section. The review of the WOA and IGWO algorithms is provided in the “Overview of WOA and IGWO Algorithms” section. The improved version of IGWO and the proposed HMWOAIGWO algorithm are described in the “Proposed HMWOAIGWO Algorithm” section. The “Experimental Verification and Results” section reports the outcomes and performance of the proposed HMWOAIGWO algorithm on 23 benchmark suites and the CEC 2019 test functions, and presents a comparison with ten state-of-the-art optimization algorithms, supplemented by the Wilcoxon and Friedman statistical tests. In addition, five real-world engineering problems are analysed in the “Real-World Optimization Application” section, and the results of tilt angle optimization for maximizing the annual, monthly, and daily energy production of solar PV power generation are analysed in the “Application of Photovoltaic Tilt Angle Optimization” section. Finally, the conclusions and future work are presented in the “Conclusions and Future Works” section.
Accurate estimation of solar radiation distribution is vital for performance assessment, designing, and economically evaluating PV panels and collector systems across diverse climatic conditions. Simultaneously, the tilt angle optimization of PV systems plays a pivotal role in ensuring the efficient capture of solar radiation, which is fundamental for their optimal performance and economic feasibility. This section will primarily focus on the modeling of the photovoltaic tilt angle.
The total radiation incident on a tilted surface of the investigated location is the sum of beam radiation (({H_B})), diffuse radiation (({H_D})) and ground reflected radiation (({H_G})), which can be expressed as follows⁸:
where H is global solar radiation on a horizontal surface, ({H_d})is diffuse radiation on a horizontal surface, (rho)is the ground reflectivity,(beta)is the tilt of the surface from horizontal. ({R_b})and ({R_d})denote the respective proportions of direct and diffuse solar radiation on an inclined plane. ({R_b})is the ratio of the cosine of the incidence angle to the cosine of the zenith angle. Its calculation formula is as follows⁸:
The angle of incidence is calculated as follows⁶⁰:
({R_d})can be expressed by using different isotropic and anisotropic models, as detailed in the following subsection.
Isotropic models posit that the intensity of diffuse sky radiation is consistent across the entire celestial hemisphere. This means that the diffuse radiation is distributed evenly in all directions, without any preferential direction. The isotropic models presented by researchers are listed as follows:
Badescu model⁶¹:
Liu and Jordan model⁶²:
Tian et al.⁶³:
Koronakis⁶⁴:
Anisotropic models consider the non-uniform distribution of diffuse sky radiation. They take into account the anisotropy (directional dependence) of the diffuse sky radiation, especially in the circumsolar region (the area around the sun), and combine it with an isotopically (evenly) distributed diffuse component from the rest of the sky dome, the anisotropy models presented by researchers are listed as follows:
Reindl et al. model⁶⁵:
Where ({H_0})is extraterrestrial radiation and (sqrt {(H_{b} /H_{g} )})is a horizontal-brightening term.
Skartveit and Olseth model⁶⁶:
Klucher model⁶⁷:
Hay model⁶⁸:
Perez3 model⁶⁹. Perez’s model delineates the sky hemisphere into three distinct regions: the circumsolar disc, the horizon band, and the isotropic background. Perez derived a straightforward correlation based on the circumsolar F1 and horizon brightness coefficients, and postulated that all circumsolar energy emanates from a single point source:
where (a=hbox{max} (0,cos theta )), (b=hbox{max} (cos {text{ }}{85^ circ },cos {theta _z})), for further information on the parameterized by the coefficients F1 and F2, please see the ref⁶⁹.
The optimization objective of this paper is to optimize the tilt angle to maximize the annual total solar radiation:
where N represents the number of days in a year, t represents the time of sunrise, while K represents the time of sunset, the optimization variable is the tilt angle (beta).
The absolute change in incident solar radiation on a flat surface, influenced by tilt angle optimization, is defined as the difference between the solar energy on the optimally tilted surface and that on a horizontal surface. The percentage change (ΔE) in incident solar energy, which is the ratio of this absolute difference to the solar energy on a horizontal surface, assesses the effects of the specified time periods on incident solar energy. In this paper, the percentage change is used to evaluate the optimization results:
To hunt prey, humpback whales adjust their movement directions to encircle potential prey by using the following equations:
where (overrightarrow {{X^ * }}) is the position vector of the best solution, (overrightarrow X (t+1))is the current position, t is the current number of iterations.(overrightarrow D)is the distance vector between whale and prey. The (overrightarrow A) and (overrightarrow C) are coefficient vectors calculated as follows:
where rand is a random vector in [0,1] and (overrightarrow a) is linearly decreased from 2 to 0 as the number of iterations increases until it reaches the maximum iteration ((Max_iter)) as follows:
Beyond the encircling mechanism, a spiral-based position update method has been devised to enable the whale to spiral around the optimal search agent in order to seize prey as follows:
where (overrightarrow {{D^{prime}}})indicates the distance from the ith whale to the optimal individual, b is a constant for defining the shape of the logarithmic spiral, l is a random number in [-1,1].
During the optimization process, a 50% probability is employed to choose between the shrinking encircling and the spiral-shaped path, as part of the modeling approach as follows:
where p and l are random numbers in [0,1], b serves as a constant to determine the shape of the logarithmic spiral.
Additionally, to enhance the global search capability of the WOA algorithm, whales are also capable of randomly choosing the positions of other whales within the feasible area for exploration. The random search formula is outlined as follows:
where (overrightarrow {{X_{rand}}})is a random whale selected from the present population.
The pseudo-code of the WOA algorithm is available in³⁶.
Similar to the WOA algorithm, the original GWO algorithm consists of three main stages: encircling, hunting, searching and attacking the prey. The mathematical expression for encircling the prey is as follows:
where (overrightarrow X (t))represents the position vector of the grey wolf, (overrightarrow {{X_p}} (t))denotes the position vector of the prey, (overrightarrow A)and (overrightarrow C)are coefficient vectors calculated as Eqs. (18) and (19). The elements of the elements of the vector (overrightarrow a) is calculated as in Eq. (20).
When the grey wolf hunts its prey, it assumes that α, β, and δ possess superior knowledge regarding the prey’s location. Consequently, the remaining wolves are compelled to trail behind these three leading solutions. The following equations delineate the hunting behavior:
where (overrightarrow {{A_1}}),(overrightarrow {{A_2}}),(overrightarrow {{A_3}}) are calculated by equations (18) and (overrightarrow {{C_1}}),(overrightarrow {{C_2}}),(overrightarrow {{C_3}})are calculated by equations (19). (overrightarrow {{X_alpha }}),(overrightarrow {{X_beta }}),(overrightarrow {{X_delta }})are the first best solutions. The searching of the prey location relies on the dispersion of search agents, occurring when (left| {overrightarrow A } right|>1). Attacking the prey can be accomplished through the convergence of search agents when (left| {overrightarrow A } right|<1).
Different from the original GWO, IGWO incorporates a dimension learning-based hunting (DLH) search strategy. In the DLH search strategy, a radius (overrightarrow {{R_i}} (t)) is determined through the Euclidean distance computation between the current position of (overrightarrow {{X_i}} (t))and the candidate position (overrightarrow {{X_{i – GWO}}} (t+1)) by using the following equation:
Then, based on the radius (overrightarrow {{R_i}} (t)), the neighbours of (overrightarrow {{X_i}} (t))are constructed by the following equation:
where (overrightarrow {{D_i}}) is Euclidean distance between (overrightarrow {{X_i}} (t)) and (overrightarrow {{X_j}} (t)).Next, the new position of the wolf (overrightarrow {{X_{i – DLH,d}}} (t)) in each dimension is determined, where the individual wolf learns from its various neighbors and a randomly chosen wolf from the population:
where (overrightarrow {{X_{n,d}}} (t))is the d-th dimension of a random neighbor selected from (overrightarrow {{N_i}} (t))and (overrightarrow {{X_{r,d}}} (t))is a random wolf from population. After calculated (overrightarrow {{X_{i – DLH,d}}} (t)), the new position of (overrightarrow {{X_i}} (t+1))is calculated by evaluating the fitness values between two candidates (overrightarrow {{X_{i – GWO}}} (t+1)) and (overrightarrow {{X_{i – DLH}}} (t+1)):
Finally, if the fitness value of the chosen candidate is lower than that of (overrightarrow {{X_i}} (t)), (overrightarrow {{X_i}} (t)) is updated with the selected candidate; otherwise, (overrightarrow {{X_i}} (t)) remains unchanged within the population.
The pseudo-code of the IGWO algorithm can be found in the literature⁵⁵.
The WOA has been widely applied since its inception due to its straightforward principles and strong operability. However, despite its proficiency in locating optimal solutions, the ability of WOA to refine these solutions in each iteration is not sufficiently robust. To address this, there is a need to integrate WOA with an algorithm that excels in exploitation capabilities to enhance its performance and yield better solutions. On the other hand, IGWO, as an enhancement of the original GWO, introduces a DLH strategy to bolster both the exploration and exploitation phases of GWO, addressing the issues of premature convergence and lack of population diversity.
Thus, the motivation behind merging WOA with IGWO lies in harnessing IGWO’s advanced exploitation techniques to complement WOA’s weaknesses in the exploitation phase, while preserving WOA’s strengths in global search. Essentially, integrating WOA with IGWO is driven by the desire to leverage the strengths of both algorithms, address their respective limitations, and capitalize on the advancements made in the field of optimization. This hybrid approach holds the promise of achieving superior performance in terms of convergence speed, solution quality, and robustness across a wide range of optimization problems. By combining these algorithms, a balance between global and local search strategies can be better achieved, effectively avoiding local optima and offering a novel perspective and methodology for tackling complex optimization challenges.
In the original WOA and IGWO algorithms, the coefficient (overrightarrow A) undergoes continuous changes with variations in the distance control parameter(overrightarrow a)throughout the iteration process affects the exploration and exploitation capabilities of both algorithms. However, this linear reduction strategy may not effectively respond to the complex and variable landscape of optimization problems. Initially, a larger distance control parameter(overrightarrow a)enhances exploration, preventing premature convergence. However, as the iteration progresses, reducing (overrightarrow a)gradually focuses the population search, improving exploitation, and accelerating convergence speed. Nonetheless, in real optimization problems, a linearly decreasing(overrightarrow a) strategy might not adequately adapt to the dynamic search environment. Therefore, the necessity to replace the linearly reduced distance control parameter(overrightarrow a)with a non-linearly decreasing function arises from the need to enhance the adaptability of the optimization algorithms to realistic search scenarios. The formula for calculating (overrightarrow a)is as follows:
where (Max_iter) is the maximum number of iterations.
The (overrightarrow C) vector represents the impact of natural obstacles on approaching prey. Consequently, these obstacles manifest along the hunting paths of wolves, effectively impeding them from swiftly and conveniently closing in on their prey. The value of (overrightarrow C) is adjusted to randomly increase ((overrightarrow C >1)) or decrease ((overrightarrow C <1)) the difficulty of the wolf pack approaching the prey. However, there is currently scarce literature on the study of the control parameter (overrightarrow C). Hence, in this paper, we propose a modified control parameter ({overrightarrow C _1}) for the alpha wolf (α) to control the area of wolves near the prey. In the initial phases of the algorithm’s search process, compared to (overrightarrow C <1), (overrightarrow C >1) has a stronger global exploration capability⁷⁰, Yet, as the algorithm progresses into its later iterations, encountering local optima becomes a notable challenge, often trapping the algorithm. At this juncture, the stochastic nature of (overrightarrow C)becomes crucial for navigating away from these local optima, particularly during the critical iterations aimed at securing the global optimum solution. Consequently, to adapt to these changing requirements throughout the algorithm’s execution, we maintain the original computation formula for ({overrightarrow C _1})in alpha wolf (α) later stages. However, for the early stages, we introduce a modification ({overrightarrow C _1}) for the alpha wolf (α), aiming at enhancing the algorithm’s global exploration capabilities. The revised formula for ({overrightarrow C _1}) is detailed below:
In this section, we introduce a new condition and search agent’s position updating method for the hunting and decision-making stage of IGWO. Within the IGWO algorithm, the parameter (overrightarrow A) plays a pivotal role. It dictates the mechanism by which each search agent updates its position, relying either on the best solution identified up to that point ((left| {overrightarrow A } right|<1)) or on a search agent chosen at random ((left| {overrightarrow A } right|>1)). This implies that exploiting (attacking) the prey occurs through the convergence of search agents, examined when ((left| {overrightarrow A } right|<1)), while exploring (searching) the prey’s location is achieved through the divergence of search agents, occurring when ((left| {overrightarrow A } right|>1)).Therefore, this study utilizes the random variable to determine whether the hunting strategy in the IGWO will rely solely on the alpha and beta wolves or will also involve the delta wolves, based on whether (overrightarrow A)falls within the range of -1 to 1.By considering Eqs. (18) and (35), it becomes apparent that in the later stages of iteration, the majority of (overrightarrow A) values are less than 1. During this phase, fewer guiding wolves result in stronger exploitation capabilities. Consequently, we exclusively utilize the alpha and beta wolves to update the positions of the succeeding generation individuals. Conversely, in the earlier stages of iteration, the majority of (overrightarrow A) values are greater than or equal to 1. During this period, having more guiding wolves enhances exploration capabilities. Therefore, we employ the alpha, beta, and delta wolves to update the positions of the next generation individuals. Hence, if the absolute value of (overrightarrow {{A_1}} ,overrightarrow {{A_2}} ,overrightarrow {{A_3}}) is less than 1 (overrightarrow {(left| {{A_1}} right|} <1& & overrightarrow {left| {{A_2}} right|} <1& & overrightarrow {left| {{A_3}} right|} <1)), the formula for updating the positions of the next generation individuals is as follows:
Conversely, the formula for updating the positions of the next generation individuals is:
The dimension learning-based hunting (DLH) search strategy is the same as the original IGWO, which is presented in 3.2.2.
This section extends the concepts of WOA and IGWO by integrating their hybrid form to enhance the algorithm’s efficiency in achieving superior solutions. The hybridization of modified WOA and IGWO aims to enhance performance during the exploitation phase primarily conducted by IGWO. After initializing the populations (overrightarrow {{X_i}} (t)), the distance control parameter (overrightarrow a) and alpha wolf coefficient vectors (overrightarrow {{C_1}}) undergo refinement through adjustments to the control parameter strategy and accelerates the convergence of the algorithm. In the IGWO, when (left| {overrightarrow A } right|)is less than 1, the wolf pack converges to attack prey trapped in local optima, while when (left| {overrightarrow A } right|) is greater than 1, the grey wolves diverge from the prey in search of a more suitable global optimum. This suggests that attacking prey (when (left| {overrightarrow A } right|<1)) involves the convergence of search agents, while exploring prey locations (when (left| {overrightarrow A } right|>1)) is achieved through the divergence of search agents. As the value of (overrightarrow {left| A right|}) dictates whether the algorithm is in the attacking or searching phase, in the latter stages of iteration, when (left| {overrightarrow A } right|) below 1 and the algorithm shifts into the attacking mode, the presence of fewer guiding wolves leads to enhanced exploitation capabilities. On the other hand, when (left| {overrightarrow A } right|)exceeds 1, moving the algorithm into the searching mode, the inclusion of more guiding wolves boosts its exploration potential. Thus, deviating from the original WOA algorithm, which assigns a 50% probability to opt between the shrinking encircling mechanism and the spiral model for updating the whales’ positions during optimization, we introduce a novel condition and an alternative approach for updating the next generation wolves’ positions. This approach supersedes the spiral model in the original WOA algorithm to amplify its performance. Specifically, when the coefficient vectors of the top three wolves (overrightarrow {(left| {{A_1}} right|} <1& & overrightarrow {left| {{A_2}} right|} <1& & overrightarrow {left| {{A_3}} right|} <1)) meet the criteria, we leverage the positions of the alpha and beta wolves for the positional updates of the next generation of individuals. Conversely, if this condition is unmet, we continue to rely on the positions of the three lead wolves for updating the next generation’s positions, thereby strengthening the algorithm’s capability for global search. Finally, DLH strategy is used to boost both exploration and exploitation, signifying an improvement in balancing local and global search strategies and facilitating the escape from local optima.
The specific implementation approach for the HMWOAIGWO algorithm is described as follows. Firstly, HMWOAIGWO begins by initializing the population size for the whales and wolves search agents. After that, if a search agent strays beyond the search space, the population will adjust it and calculate the function’s fitness. Next, the value of the distance control parameter (overrightarrow a) and the coefficient vectors (overrightarrow {{C_1}})is adjusted through Eq. (34) and Eq. (35), respectively. Subsequently, the values of the remaining parameters (left| {overrightarrow A } right|), p and l are updated accordingly. Following entering the WOA exploration stage, if the randomly generated number p is less than 0.5, another conditional statement(left| {overrightarrow A } right|<1) is activated. If this condition be met, the position and fitness are computed using Eq. (17). Consequently, if the new position surpasses the old one in fitness, the old position gets updated. Otherwise, the position and fitness are calculated through Eq. (24), the current position and the previous position are compared, and the superior value of the two is utilized for updating. On the another hand, if the value of p is more or equal than 0.5, A new condition is introduced to improve the algorithm’s search process, which is to determine whether .(overrightarrow {left| {{A_1}} right|} <1& & overrightarrow {left| {{A_2}} right|} <1& & overrightarrow {left| {{A_3}} right|} <1). If this condition is true, we use the alpha and beta wolves for the positional updates via Eq. (36), and if not, the three lead wolves via Eq. (37) is used to updated the positions. After all the previous position updates are finalized, the DLH strategy is employed to expand the global search domain through multi-neighbor learning. The synergy between the movement strategy and the DLH strategy enhances both global and local search capabilities of the algorithm.
The pseudo code of the HMWOAIGWO algorithm is presented in Table 1 and the flowchart of the algorithm is shown in Fig. 1.
Thirty-three standard benchmark functions were utilized to assess the HMWOAIGWO’s ability to address diverse objective functions, including a 23 benchmark suite found in the literature³⁶, CEC 2019 as reported in the literature⁴⁹. To evaluate the effectiveness of HMWOAIGWO in finding the optimal solution, its performance is compared against ten established algorithms, including: PSO, HHO, DBO, COA, GWO, WOA, WOAGWO, IGWO, WOABAT, YDSE. Optimization results are presented using five indicators: best, mean, standard deviation (std), execution time (ET) and rank. Table 2 outlines the parameters for alternative comparative algorithms. The benchmark functions were assessed across various dimensions through 30 independent runs. Each run consisted of a maximum of 500 iterations and a population size of 50. All experiments were conducted on a CPU, specifically an Intel Core (TM) i3-8100 running at 3.6 GHz with 8.00 GB of RAM. Programming was carried out using MATLAB R2018a.
Flow chart of the HMWOAIGWO algorithm.
The various behaviors of HMWOAIGWO during the iterative process in solving the 23 benchmarks suite test are presented in Fig. 2. This analysis section employs four key metrics to assess HMWOAIGWO’s performance: search history, the trajectory of the first search agent in the 1st dimension, the mean fitness of the population, and the convergence curve. The search history metric describes the distribution of the position of search agents in the search space during the algorithm iterations, showcasing HMWOAIGWO’s adeptness in globally and locally scanning the search space and swiftly converging towards the optimal solution after identifying the primary optimal region. Trajectory diagrams of the first search agent indicate substantial changes in population positions during initial iterations, driven by exploration capabilities to identify the main optimal area. Subsequent iterations witness smaller changes in population positions, converging towards potentially superior solutions based on exploitation capabilities. Mean fitness diagrams depict the population’s trend towards the optimal solution across algorithm repetitions. The convergence curve metric illustrates the improvement of candidate solutions and HMWOAIGWO’s progress toward the solution across iterations. The 23 benchmarks suite, as one of the most popular test sets, comprises seven unimodal benchmark functions (F1-F7), six multimodal different dimensions benchmark functions (F8-F13), and ten multimodal fixed-dimension multimodal benchmark functions (F14-F23). The unimodal benchmarks demonstrate the algorithm’s overall exploitation ability, while the multimodal benchmarks assess its exploration capability. The optimization results, obtained by utilizing HMWOAIGWO to optimize the 23 benchmarks suite and comparing them with those of ten other state-of-the-art algorithms, are displayed in Table 3; Fig. 3. The data in Table 3 clearly indicates that MHWOAIGWO consistently achieves highly competitive solutions in comparison to other latest outstanding algorithms. Meanwhile, the optimal values of each function in the table are highlighted in bold, and the optimizers are ranked according to their mean value. If multiple algorithms have the same mean value, they are sorted based on their standard deviation (STD). For seven unimodal functions (F1-F7), the proposed HMWOAIGWO algorithm ranks first in four functions (F1-F4) and ranks second in three functions (F5-F7) among all eleven algorithms. Therefore, the superior exploitation ability of the HMWOAIGWO among other compared algorithms is confirmed. For six multimodal different dimensions benchmark functions, HMWOAIGWO hit four best results (F8-F11) out of six functions and ranks second (F12-F13) among all algorithm.
The various behaviors of HMWOAIGWO in the iterative process.
For ten multimodal fixed-dimension multimodal benchmark functions, HMWOAIGWO sorted first in six (F15-F17, F21-F23) out of ten test functions. The competitive results showcase the excellent exploitation ability of HMWOAIGWO. Among all 23 benchmarks suite, HMWOAIGWO hit the best solution of fourteen out of twenty three functions (14/23), while the COA algorithm hit (10/23) and the WOAGWO hit (5/23). The exceptional outcomes illustrate the high ability of HMWOAIGWO in exploitation and exploration, which has contributed to the perceived effectiveness of the proposed approach compared to other algorithms. However, the calculation time of HMWOAIGWO algorithm is the longest. This is because the DLH strategy is utilized and the objective function values are calculated more times compared to the original algorithm in each iteration update. Compared with the O ((Max_iter cdot N cdot Dim)) complexity of the GWO and WOA algorithms, the HMWOAIGWO algorithm has an O ((N^2 cdot Dim)) cost due to neighborhood search and distance calculation. The HMWOAIGWO algorithm’s computational complexity stems from several core operations. The initial setup entails position generation and fitness evaluation for all agents, incurring O ((N cdot Dim)) complexity, where N represents the population size and Dim denotes the problem dimensionality. With the introduction of the DLH strategy, the calculation of the radius based on Euclidean distance and the computation of the distance matrix between positions incur O ((N^2 cdot Dim)) complexity. Updating the candidate positions and recalculating their fitness values has a complexity of O ((N cdot Dim)). Therefore, the overall computational complexity of HMWOAIGWO per iteration is O ((N^2 cdot Dim)). Over Max_iter iterations, the total complexity of the algorithm is O ((Max_iter cdot {N^2} cdot Dim)).
Comparison chart of convergence characteristics of HMWOAIGWO algorithm.
In this section, CEC 2019 benchmark functions are used to evaluated the performance of the proposed algorithm compare it with other ten latest outstanding algorithms. The results of algorithms are presented in Table 4. HMWOAIGWO clearly achieves optimal values in 5 (F1, F3, F7, F8, F10) of the 10 functions (5/10), while the IGWO hit (3/10) and HHO hit (2/10), thus outperforming all the algorithms in Table 4. The box plot comparison of results of HMWOAIGWO with other competitor algorithms in CEC-2019 is presented in Fig. 4, which confirms the stability of the algorithm while finding the optimum results.
The convergence curves of HMWOAIGWO and other approaches in handing all 23 benchmark functions are presented in Fig. 3, and the convergence curves are plotted using the mean of the best solution from 30 independent runs in each iteration. From Fig. 3, it shown that HMWOAIGWO converges to superior results over time. With more iterations, HMWOAIGWO can approximate increasingly accurate solutions in the vicinity of the optimum. More importantly, compared to other latest outstanding algorithms, HMWOAIGWO exhibits faster convergence trends during its iteration process. It is evident from Fig. 3, HMWOAIGWO demonstrates the fastest convergence speed among ten functions out of a total of 23 benchmark functions (9/23), encompassing F1-F4, F8-F11, and F15, outperforming other algorithms by a significant margin. This superiority is particularly evident in functions F1-F4. In functions F1 and F3, HMWOAIGWO achieves an optimal value of 0 within just 100 iterations, while other algorithms require at least 400 iterations to reach the same value. Similarly, in functions F2 and F4, HMWOAIGWO converges to 0 within 200 iterations, whereas other algorithms need 500 iterations to converge to values of 4.17E-212 and 3.00E-205, respectively. Neither in convergence speed nor in convergence accuracy can other algorithms match the proposed approach. Additionally, HMWOAIGWO ranks second in both convergence speed and accuracy among all algorithms in functions F5-F7, F12-F13, and F15. Especially noteworthy is its ranking in functions F21-F23, where it ranks second in convergence speed and first in convergence accuracy. The remarkable performance of HMWOAIGWO in both convergence speed and accuracy highlights its capability to significantly enhance the fitness values of the population, ensuring the exploitation of improved results. Especially, it achieves a better balance between exploration and exploitation throughout the iteration process compared to other state-of-the-art algorithms.
Box plot comparison of results of HMWOAIGWO with other competitor algorithms in optimization of the CEC-2019.
To further compare the difference between the proposed HMWOAIGWO and other ten algorithms, both Wilcoxon rank sum non-parametric statistical test and Friedman test are utilized to assess the algorithms. The optimal values obtained from 30 independent runs by all the algorithms are used for conducting the Wilcoxon rank sum test and Friedman test. The p-value is used to define and assess the significance of the statistical results of the algorithm in Wilcoxon rank sum test. If the p-value is smaller than 0.05, there is a significance between the two algorithms. The results of overall performance of the HMWOAIGWO in both 23 benchmark function and CEC-2019 function used Wilcoxon rank sum test is shown in Table 5 and 7, respectively. In those table, bold indicates results without significant differences and #AI implies that the difference between the two algorithms approaches zero indefinitely. The number of algorithms exhibiting significant differences compared to HMWOAIGWO in each function is presented at the far right of the table, while the count of functions displaying significant differences within each algorithm, in comparison to the 10 algorithms, is listed at the bottom. Notably, it is observed that every algorithm, when compared to HMWOAIGWO, demonstrates significant differences in 16–22 functions. The mean rank results calculated used the Friedman test of the 23 benchmark function and CEC-2019 function is listed in Tables 6 and 8, respectively. The Friedman rankings of HMWOAIGWO among the 11 algorithms are presented to the right in the table. Remarkably, HMWOAIGWO secures the top position in 10 out of 23 benchmark tests (10/23) and claims first place in 5 functions across the CEC 2019 tests (5/10). According to the comprehensive rankings from both the Wilcoxon rank sum test and the Friedman test, HMWOAIGWO emerges as one of the top-performing algorithms among its competitors.
In this section, five typical engineering optimization problems are used to test HMWOAIGWO, including Three-bar truss design problem⁷¹, Gear train design problem⁷², Gas Transmission Compressor Design problem⁷³, Cantilever beam design problem⁷⁴, Piston Lever problem⁷⁵. The detailed description for these five typical engineering problems is presented in the Appendix. Solving real-world optimization problems is challenging due to numerous constraints and the complexity of the objective function. There are several strategies to deal with the constraint problems. In this paper, the death penalty function is used to handle the constraint problems. This function assigns a large fitness value to solutions violating constraints, discarding the infeasible ones. The results provided by HMWOAIGWO are compared with the original WOA, GWO, WOAGWO, PSO and HHO algorithms. The population size and the maximum iterations of all algorithms are set to 50 and 500, respectively. All the results are attained over 20 independent runs. The performance of all algorithms in terms of mean and STD of the results is presented in Table 9, note that the best values are bolded. The overall results of Wilcoxon test used optimal values are presented in Table 10, the values with no significant difference are highlighted in bold. Based on Table 9, it can be seen that the HMWOAIGWO demonstrates excellent performance on the majority of test functions (RT2, RT3 and RT5), whether in absolute values or standard deviation values. While the PSO performed best in two cases (RT1, RT4). Note that the results of Wilcoxon’s test confirm significant differences in the behaviors of HMWOAIGWO compared to WOA, GWO, WOAGWO, PSO, and HHO across the majority of the comparisons. The statistical results of HMWOAIGWO confirm a significant enrichment in the optimality of solutions. Based on the above, the proposed HMWOAIGWO demonstrates satisfactory performance in addressing real-world engineering problems and can be viewed as one of the most competitive algorithms when compared with others (Tables 6 and 7).
This paper combines the proposed HMWOAIGWO algorithm to optimize the tilt angle of photovoltaic systems for maximizing solar radiation, along with a comparative analysis against other algorithms. In this paper, meteorological data are collected at Changsha (latitude: 28.07° N, longitude: 112.59° E), which is located in the central-southern part of China, using PVsyst software. Subsequently, detailed elaborations will be provided for daily, monthly, and annual tilt optimization. Due to the complexity of photovoltaic tilt angle modeling, the proposed HMWOAIGWO algorithm will be demonstrated to outperform the WOA, GWO, and WOAGWO algorithms in terms of its performance in this section. The maximum iteration count is limited to 500, while the population size is defined as 30. The daily and monthly solar radiation for global horizontal radiation, horizontal diffuse radiation, and horizontal beam radiation is presented in Figs. 5 and 6. The simulation results of the proposed algorithm optimization are analyzed (Table 8).
Daily solar radiation for global horizontal, diffuse, and beam radiation.
Monthly solar radiation for global horizontal, diffuse, and beam radiation.
It is evident that optimizing the tilt angle based on different time periods can significantly increase the amount of captured solar radiation, consequently, in this part, the daily tilt angle for solar radiation is focused on being optimized. Given the multitude of optimization variables involved in the daily tilt angle optimization, the effects of this process are more pronounced than those of the monthly and annual tilt angle optimizations. This distinction more vividly highlights the superiority of the algorithms proposed in this paper. The optimized tilt angles throughout the year under various algorithms are presented in Fig. 7. Notably, during the summer months, and especially in June, the optimal tilt angle is nearly zero. Conversely, in winter, the tilt angle reaches its annual maximum. The corresponding daily solar radiation values are depicted in Fig. 8. Interestingly, these values peak in the summer season and diminish during the winter, which is the opposite trend to that of the optimized tilt angles. The monthly solar radiation, calculated based on the daily average tilt angle, is displayed in Fig. 9. As indicated, July experiences the peak radiation levels, followed by August and September, while the minimum radiation occurs in January, followed by December, February, and March, aligning with the patterns observed in daily radiation data. The annual radiation data, derived from the daily mean tilt angle, is illustrated in Fig. 10. According to the figure, the algorithm proposed herein, which optimizes the daily tilt angle, achieves the maximum annual radiation of 4105.1965 kWh/m², marking a 4.0% enhancement over the unoptimized condition. Comparatively, the optimizations performed by the WOAGWO, GWO, and WOA algorithms result in more modest increases of 3.24%, 3.96%, and 3.79%, respectively, when contrasted with the unoptimized radiation levels (Tables 9 and 10).
The optimal daily tilt angle of different algorithms.
Daily radiation calculated using the daily mean tilt angle.
Monthly radiation calculated using the daily mean tilt angle.
Annual radiation calculated using the daily mean tilt angle.
In this section, 15 individual runs will be conducted, the best fitness value, mean, and standard deviation (STD) of annual total solar radiation from 15 individual runs using various algorithms are shown in Table 11, the results indicate that HMWOAIGWO yields the best outcomes in terms of best, average, as well as standard deviation values. Figure 11 presents the convergence evolution of the HMWOAIGWO algorithm in comparison to WOA, GWO, and WOAGWO over iteration index. From Fig. 11, it can be seen that the HMWOAIGWO algorithm demonstrates rapid convergence. In each iteration of the algorithm, the photovoltaic tilt adjustment system follows the best search agent to update the calculated optimal tilt angle, thereby obtaining maximum solar irradiance. The mean optimum tilt angle for 15 independent runs various different algorithm is shown in Fig. 12. From Fig. 12, it can be seen that the optimal tilt angle for receiving maximum solar radiation varies by month, the optimum tilt angle decreases to its minimum value in summer, even below the latitude of the location, while it increases during the winter months. The Annual total solar radiation calculated using the mean optimum tilt angle is presented in Fig. 13. From Fig. 13, it can be seen that since this is a maximization optimization problem, the annual solar irradiance obtained through optimization with the HMWOAIGWO algorithm is also maximized, which is 4014.4713 MJ/m², the percentage change in incident solar energy after annual tilt angle optimization compared to the radiation when the tilt angle is not optimized (i.e., the tilt angle is 0°), calculated by Eq. (15), is 1.70%. The monthly solar radiation calculated using the HMWOAIGWO algorithm is presented in Fig. 14, the results indicate that the solar radiation hitting a flat surface gradually increases as it is tilted from a 0° horizontal position to a certain angle of inclination. Beyond this point, further increases in the tilt angle of the flat surface led to a decrease in received solar radiation. Meanwhile, the amount of solar radiation obtained is highest during summer, with a subsequent decrease in other seasons accordingly.
Convergence curves of different algorithms.
Monthly mean optimum tilt angle for 15 independent runs.
Annual radiation calculated using the monthly mean optimum tilt angle.
Monthly radiation calculated using the HMWOAIGWO algorithm.
When optimizing the annual tilt angle, since there is only one variable, the differences between algorithms are not significant, the average annual radiation obtained by the HMWOAIGWO, WOA, GWO, and WOAGWO algorithms across 15 independent runs has arrived at the same result, which are 3985.7011 MJ/m², and the optimal annual tilt angle is 11.3272°. The monthly radiation values under annual tilt angle optimization are depicted in Fig. 15, similar to the optimization of the monthly tilt angle, the maximum monthly radiation is obtained in July and August, while the minimum monthly radiation is obtained in January. The percentage change in incident solar energy after annual tilt angle optimization compared to the radiation when the tilt angle is not optimized (i.e., the tilt angle is 0°), calculated by Eq. (15), is 0.97%.
Monthly radiation calculated using the annual mean optimum tilt angle.
The HMWOAIGWO algorithm incorporates several key innovations that jointly enhance optimization performance. It adopts a sophisticated hybrid multi-strategy framework that seamlessly integrates WOA’s exploration mechanisms, GWO’s hierarchical hunting strategies, and neighborhood search capabilities via an adaptive probability switching mechanism (p < 0.5 for WOA, p ≥ 0.5 for GWO). This design ensures a dynamic balance between exploration and exploitation while maintaining population diversity. The algorithm features improved parameter control equations combining non-linear decay strategies with oscillating coefficients, enabling more precise parameter tuning than traditional linear methods. Its adaptive GWO position update mechanism adjusts guidance strategies based on parameter values—utilizing only Alpha and Beta wolves when convergence conditions ((A1>-1 && A1 < 1) && (A2>-1 && A2 < 1) && (A3>-1 && A3 < 1)) are met, or engaging all three leaders otherwise. A dynamic neighborhood radius mechanism determines optimal neighborhood sizes from Euclidean distances between current and candidate positions, enhancing local search efficiency. Additionally, the algorithm incorporates an individual memory mechanism akin to PSO’s personal best concept, preserving valuable historical information and preventing solution degradation. A multi-level selection strategy ensures optimal candidate retention through dual-layer comparisons between GWO-DLH solutions and each agent’s historical best, while enhanced boundary-handling procedures maintain solution feasibility at every stage. A comprehensive history tracking system records position trajectories, fitness evolution, and convergence patterns, offering rich analytical data for performance evaluation and parameter tuning. These integrated innovations form a robust optimization framework that substantially improves convergence reliability and solution quality compared with conventional single-strategy approaches. In this work, the algorithm is applied to optimize photovoltaic module tilt angles over different time scales—annual, monthly, and daily—to maximize solar radiation capture. Through precise optimization, the most effective installation angles are determined. WOA, GWO, and the WOAGWO hybrid are used as benchmarks to evaluate the proposed algorithm’s performance in photovoltaic tilt angle optimization. Experimental results show that HMWOAIGWO consistently achieves superior optimal values, average values, and standard deviations. Moreover, simulations of maximum annual solar irradiance confirm its competitive edge over other algorithms. Based on these findings, the following conclusions can be drawn:
Through experiments conducted with popular benchmark suites, HMWOAIGWO has demonstrated superior performance compared to all other latest outstanding optimization algorithms discussed in this paper (such as PSO, HHO, DBO, COA, GWO, WOA, WOAGWO, IGWO, WOABAT, YDSE). HMWOAIGWO can also demonstrate exceptional accuracy and unmatched superior convergence speed.
In the context of daily tilt angle calculations, employing the HMWOAIGWO algorithm for optimization has resulted in a 4% increase in annual radiation compared to the scenario without optimization, with the annual radiation value reaching 4105.1965 kWh/m². When calculating monthly tilt angles, the annual radiation saw a 1.7% enhancement, and for annual tilt angles, the increase was 0.96%. It is evident that the shorter the time scale considered, the more pronounced the benefits of tilt angle optimization become. Consequently, the calculations for daily tilt angles have yielded the highest annual radiation figures.
Among all 23 benchmark tests, HMWOAIGWO exhibited the top-ranked or second-ranked convergence speed in 78% of the test functions (18/23). Furthermore, out of the total 33 test functions across the 23 benchmarks and CEC 2019 suite, it showcased the highest precision in 19 functions (19/33), highlighting its ability to maintain diversity, escape local optima, and strike a commendable balance between exploration and exploitation capabilities.
Non-parametric statistical tests (Wilcoxon rank-sum test and the Friedman test) confirmed a significant enhancement in solution optimality, and a study involving five real-world engineering problems was conducted.
These findings highlight HMWOAIGWO as a promising method for photovoltaic tilt angle optimization calculations. The proposed PV tilt angle optimization method is designed in a model-independent framework, making it easily adaptable to different climatic regions. Although the current study focuses on Changsha as a representative site with a subtropical monsoon climate, the optimization framework does not rely on location-specific parameters. Once local solar radiation, temperature, and geographical data are available, the same algorithmic process can be directly applied to other cities or countries without modification. The approach has strong potential for generalization in regions with distinct solar and climatic characteristics, such as arid northern areas or coastal humid zones. Future research will extend this analysis to multiple representative locations to further validate the method’s universality and regional adaptability.
Despite HMWOAIGWO’s demonstrated potential and broad applicability, its robustness and computational characteristics merit further discussion. While its capability to handle nonlinear and complex search spaces ensures high optimization accuracy, the algorithm’s structural characteristics introduce certain computational and scalability considerations, as discussed below. The algorithm consistently delivers competitive solutions on challenging tasks. Although the algorithm demonstrates promising results, certain limitations remain. The computational cost—stemming from the complexity of its iterative structure and additional expense of the DLH strategy—may restrict its applicability in time-sensitive scenarios. Furthermore, current assessments have focused solely on single-objective problems, whereas many real-world cases involve multiple, often conflicting, objectives within a unified model. Extending HMWOAIGWO to a multi-objective framework, by incorporating Pareto-dominance concepts, decomposition methods, or indicator-based selection, could substantially enhance its practical relevance for complex real-world decision-making.
Hence, the future scope of this research will focus on further reducing computational complexity by designing optimization schemes to shorten both algorithmic complexity and execution time, as well as developing a multi-objective HMWOAIGWO variant to elucidate trade-offs among conflicting criteria. In addition, integrating the proposed algorithm with deep learning techniques for power prediction, and extending its application to various domains such as wind power and microgrids, will enhance system scalability, robustness against disturbances, and decision-making accuracy.
Data sets generated during the current study are available from the first author on reasonable request.
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This work was supported by the Key Laboratory of Renewable Energy Electric-Technology of Hunan Province under Grant no. 2024ZNDL003 and the Provincial Natural Science Foundation of Hunan Province under Grant no. 2023JJ30048.
Changsha University of Science and Technology, Changsha, 410114, China
Lingqi He, Bin Zhao, Biao Zhu & Li Wang
Northeast Electric Power University, Jilin, 132012, China
Bin Zhao & Biao Zhu
School of Electrical and Information Engineering, Hunan University, Changsha, 410082, China
Fei Rong
Guizhou University of Finance and Economics, Guiyang, 550000, China
Ming Xu
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Lingqi He: methodology, software, writing-original draft. Bin Zhao: methodology design, review & editing. Fei Rong: review & editing. Biao Zhu: formal analysis. Ming Xu: resources. Li Wang: conceptualization.
Correspondence to Bin Zhao.
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He, L., Zhao, B., Rong, F. et al. An innovative metaheuristic algorithm for photovoltaic tilt angle optimization. Sci Rep 15, 44554 (2025). https://doi.org/10.1038/s41598-025-28391-x
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Received: 03 July 2025
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Published: 07 December 2025
Version of record: 24 December 2025
DOI: https://doi.org/10.1038/s41598-025-28391-x
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