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Scientific Reports volume 16, Article number: 4670 (2026) Cite this article
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The increasing adoption of solar energy as a clean and sustainable power source has intensified research efforts toward developing more efficient photovoltaic (PV) cells. These cells exhibit nonlinear characteristics that are significantly influenced by variations in irradiance and temperature. Accurate estimation of PV model parameters plays a crucial role in maximizing performance, particularly via precise maximum power point (MPP) tracking. This paper presents a new hybrid metaheuristic algorithm that combines the global exploration capability of the Bat Algorithm (BA) with local exploitation efficiency of the Crow Search Algorithm (CR) to optimize PV parameter estimation. The proposed approach is tested using Single-Diode (SDM), Double-Diode (DDM), and Triple-Diode (TDM) models based on the RTC France dataset. The hybrid model demonstrates better convergence behavior and robustness compared to conventional approaches. Qualitatively, it effectively manages parameter uncertainty; quantitatively, it achieves RMSE values of 0.00077299 (SDM), 0.0008215 (DDM), and 0.0008068 (TDM), outperforming traditional algorithms such as Particle Swarm Optimization (PSO) and Genetic Algorithm (GA).
Energy plays a significant role in enhancing economic and social well-being, serving as the foundation for industrial and commercial production¹. It is vital for reducing poverty, enhancing human welfare, and raising living standards^2,3. Energy is the driving force to channel towards sustainable development^3,4.
Global warming and the depletion of fossil fuels reserves have driven the transformation of electric energy systems, shifting towards renewable energies^5,6,7. Over the past decade, global wind energy capacity has expanded by 6.58 times, increasing from 74 gigawatts in 2006 to 487 gigawatts in 2016. Simultaneously, PV power capacity has skyrocketed by 43.14 times, surging from 7 gigawatts in 2006 to 302 gigawatts in 2016⁸. The unique characteristics of renewable energy sources, such as environmental sustainability and abundant availability, have driven increased demand for these alternatives. Predictions indicate that clean energy sources will likely dominate traditional energy methods in the near future⁹.
Among renewable resources, solar energy stands out as one of the most promising options, widespread attention for its role in energy generation. However, the effective utilization of solar energy, like other renewable resources, depends significantly on geographical factors^10,11. Given the power plant construction is generally a one-time investment, strategic site selection critical in the development process¹². Additionally, thermal power plants, increasingly powered by diverse renewable sources, are being commissioned and decommissioned more frequently than ever before. Before a thermal power plant can supply electricity to the grid, it must undergo an initiation phase to ensure it meets the minimum required production capacity¹³.
Photovoltaic power is a key form of renewable energy, harnessed through PV systems to convert sunlight into electricity. These systems offer several advantages, contributing to their widespread adoption. Their initial experimental implementations have demonstrated exceptional efficiency, and with a lifespan of approximately 20 years, PV systems provide a flexible and reliable energy solution. They can be installed and operation in diverse environments, including remote or mountainous regions. Their adaptability extends to mobile applications, easy maintenance, and both off-grid and grid-connected functionality, setting them apart from other renewable energy sources. PV installations instantly convert solar energy into electrical power using solar cells through the photovoltaic effect^14,15. Photovoltaic systems are classified into two main categories: distributed and centralized. Distributed systems are installed near consumers to provide local energy supply, whereas centralized systems resemble traditional power plants, connecting to higher voltage grids. The majority of currently installed photovoltaic units are small-scale and connected to distribution systems. However, as PV penetration into medium and low-voltage grids increases, challenges such as overgeneration, overvoltage, and reverse power flow have emerged, potentially causing damage to consumers and the grid. Among these, overvoltage is a particularly significant issue for distribution companies managing low-voltage grids with photovoltaic generators¹⁶.
Another major challenge with PV systems is their high initial costs, especially during construction phase. Consequently, many researchers have explored methods to maximize PV energy output, such as employing maximum power point tracking (MPPT) to reduce the payback period. The MPPT algorithm must function effectively under various irradiance conditions, whether uniform or non-uniform¹⁷. A photovoltaic cell generates power energy by integrating (p and n-type) semiconductor materials. The output of a PV module, formed by connecting multiple cells, depends on factors the number of cells and environmental conditions like temperature and irradiance¹⁸. Given the rapid expansion of PV systems, comprehensive studies are required from multiple perspectives. Researchers have investigated various issues related to PV panels to increase efficiency while reducing overall costs.
Precise PV modeling is necessary for the optimization of integration into distribution networks¹⁹. One important part of this modeling is determining the optimal parameters of solar cells, which have a direct effect on power output and efficiency²⁰.
Advanced optimization techniques, such as the hybrid Bat and Crow Algorithm (BA-CR), have improved the accuracy of PV parameter estimation²¹. Through reducing modeling uncertainties and enhancing MPPT, these techniques minimize overvoltage issues, reverse power flow, and power quality disturbances in distribution networks²².
The sun is not only a vast source of energy; it is the fundamental source of nearly all the energy resources. PV panels convert sunlight directly into electricity without any mechanical devices. These panels have developed as a revolutionary technology²³. PV panels are commonly used around the world, and their demand is increasing due to their regular performance²⁴. To maximize the performance of PV panels, solar cells are strategically arranged in series and parallel groups, so-called PV arrays. The cells are made primarily of silicon. This material is commonly available in desert sands²⁵. However, despite their vast potential, photovoltaic panels face limitations in energy output due to the relatively low energy density of sunlight²⁶. This research aims to address the challenge of optimizing solar panel parameters through the implementation of a novel hybrid metaheuristic algorithm, which combines bat and crow methodologies.
Limitations of Current Approaches:
Weak Global Search Mechanisms: Many existing methods lack a robust global search capability, which can result in local optima, preventing the accurate identification of the optimal parameters for photovoltaic cell estimation.
Slow Convergence speed: Traditional optimization techniques have a tendency to have slow convergence, taking longer and requiring more computational resources to achieve reliable parameter estimation.
Sensitivity to Initial Conditions: Some methods are sensitive to initial conditions, especially in the presence of noisy or uncertain conditions.
Complicated Model Issues: The existing methods fail to estimate parameters of more complex models, like the TDM models, because such models have more variables.
Absence of Hybrid Methods: The majority of existing methods are isolated and fail to integrate the strengths of various optimization techniques, resulting in less efficient solutions.
Challenges Addressed by the Study:
Hybrid Method: The study suggests a novel hybrid method that combines the BA and CR algorithms. This method improves global search (Bat) and local search (Crow), which overcomes the limitations of existing methods.
Improved Convergence and Accuracy: The new strategy enhances convergence speed and accuracy of parameter estimation compared to traditional methods, which makes it more suitable for real-time applications.
Management of Complexity and Uncertainties: The study provides a better solution for the estimation of parameters in complex models (SDM, DDM, TDM) and uncertainties in systems.
Reducing Sensitivity to Initial Conditions: The hybrid metaheuristic approach minimizes sensitivity to initial conditions, leading to more stable and reliable results in parameter estimation.
A viable solution to this problem involves employing subclasses of metaheuristic algorithms, specifically global and local optimization techniques. By analyzing their unique characteristics and potential to improve solar panels efficiency, researchers aim to advance photovoltaic technology further.
Local optimization strategies focus on identifying a local optimum—either a maximum or minimum—within a confined search space²⁷. The objective is to determine an initial point that optimizes the function under consideration. Thus “Local optimization” or “local search” denotes to the process of finding a locally optimal solution. These algorithms iteratively refine a signal candidate solution, evaluating each modification to determine whether it improves the objective function. If an improvement is detected, the updated solution becomes the new candidate.
under certain conditions, local search algorithms may identify the global optimum:
If the local optimum is unique and aligns with the global optimum.
If the search trajectory includes the global optimum, causing the algorithm to converge at that point.
A quasi-Newton method that approximates the Hessian matrix to find local minima. Known for its efficiency in handling large-scale optimization problems. Widely applied in machine learning and statistical modeling²⁸.
A simple iterative algorithm that continuously moves toward increasing value. effective in scenarios with a clear path to the optimum but can get stuck in local optima. Applied in various applications, including robotics and artificial intelligence²⁹.
Recent studies have explored several hybrid metaheuristics structures to enhance global optimization performance. Reference³⁰suggested the Quadratically Interpolated Hybrid Pathfinder Algorithm (QHIPFA), which includes the Pathfinder Algorithm with Quadratic Interpolation (QI) to improve local exploitation and incorporates the Social Spider Algorithm (SSA) to improve global exploration. Reference³¹proposed the Dynamic Gold Rush Optimizer (DGRO), integrating the Salp Navigation Mechanism (SNM) for adaptive global–local balance with the Worker Adaptation Mechanism (WAM) to enhance localized search. Similarly;³²developed the Dynamic Golden Sine–Sand Cat Swarm Optimization (DGS-SCSO) algorithm, which uses Dynamic Pinhole Imaging (DPI) to enhance exploration diversity and the Golden Sine Algorithm (Gold-SA) to improve exploitation convergence. Building upon these developments, the BACR algorithm further refines the BA–CR hybridization framework by introducing modifications particularly designed for photovoltaic (PV) parameter estimation, thus achieving greater robustness and stability under PV-specific operating conditions.
This limitation highlights the importance of hybrid approaches that combine local and global optimization techniques to enhance overall performance³³. Every objective function inherently possesses at least one global optimum (otherwise, optimizing its value would be meaningless). Additionally, as previously noted, it may also contain local optima, where the function’s value differs from that of the global optimum. The presence of local optimal illustrates the challenge of global optimization, as identifying a local optimum is generally easier than finding the global optimum. Global search algorithms may adopt either a single solution or a population of solutions. Regardless, they continually refine candidate solutions, generating new samples or generations for exploration. Several well-known global optimization and search algorithms including Genetic Algorithm (GA), Simulated Annealing (SA), Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO), Crow Search Algorithm (CR), and Bat Algorithm (BA).
The comparison of global and local optimization approaches reveals that:
Local search algorithms boast versatility in solving a myriad of problems.
Local optimization algorithms are effective when the region or pathway to the global optimum is well understood, or when the objective function has a single optimum.
Employing a local search algorithm for a problem requiring global search may yield poor outcomes, as the search can become trapped in local optimum. Hence:
Local search is appropriate when the global optimum is nearby or when only one optimum exists.
Global search is necessary when multiple suboptimal point exist within the search space, and distinguishing them from the global optimum is essential.
Local search algorithms, when their underlying assumptions hold, offer guarantees regarding computational complexity. This ensures a predictable level complexity when navigating towards the optimum.
In contrast, global search algorithms lack specific guarantees on computational complexity and the time required to research the global optimum. The reliability is lower because they involve extensive exploration, making the timing and certainty of reaching the best solution unpredictable. While in global optimization problems often demand the absent best solution, if finding the global optimum is too challenging, the focus shifts to identifying the best feasible solution within an acceptable complexity range.
In many cases, knowledge about the objective function’s behavior and solution space is limited. It may be unclear whether a local or global search is appropriate for a given problem. A practical approach is to define a baseline performance target and begin with a local search algorithm. If the local search result is unsatisfactory, a global search can be initialed from that point. If further improvement is achieved, it suggested that either the problem is inherently unimodal or that the initial conditions were suitable for the local search.
Local optimization algorithms are typically then global optimization due to its more predictable nature and reduced computational demands. Local search algorithms, such as Guided Local Search (GLS) and Simulated Annealing (SA), are easier to implement and converge faster a solution³⁴.
Both the BA and CR algorithms are powerful metaheuristic optimization techniques. The BA³⁵, , is inspired by bat’s echolocation strategies during food search. By dynamically adjusting frequency and amplitude, BA effectively balances exploration (global search) making it a robust optimization model³⁶. Conversely, the CR provides a simpler yet highly effective optimization approach. With only two adjustable parameters—flight length and awareness probability—CR is widely recognized its versatility across various engineering applications. The awareness probability parameter directly influences algorithm diversity, requiring fewer adjustments then other methods like GA, PSO, and HS. Inspired by crow’s nesting habits, CR operates on a population-based strategy when hidden resources (solutions) are selectively needed³⁷. While existing literature provides valuable into PV system optimization and modeling, several gaps remain. Most studies focus on individual aspects, such as parameter estimation or optimization algorithms, without addressing broader system integration and grid compatibility. Additionally, limited research explores the challenges of integrating solar energy into thermal power plants and optimizing system performance under diverse environmental conditions. Furthermore, existing optimization algorithms may struggle to solve the complex problems associated with PV systems, highlighting the need for novel approaches and hybrid techniques. Table 1 provides a comparative summary of recent studies on solar cell parameter estimation with various optimization and metaheuristic algorithms. It highlights the key features of each work, including their objectives, methodologies, advantages, limitations, and obtained results. The reviewed studies primarily focus on enhancing the accuracy and robustness of parameter estimation through hybrid frameworks, adaptive mechanisms, and enhanced search strategies. Nevertheless, despite these advancements, most approaches continue to face challenges related to computational complexity and parameter tuning.
This study contributes to the advancement of photovoltaic (PV) technology through several key innovations: (1) optimizing PV integration in thermal power plants using a hybrid optimization technique; (2) introducing an enhanced BA–CR algorithm to achieve more accurate parameter estimation; (3) improving model precision by incorporating nonlinear characteristics of solar cells; and (4) presenting effective strategies to mitigate grid-related operational challenges.
This paper is structured into six sections. Section 2 delves into Solar Cell Modeling and Parameter Identification, while Sect. 3 explores Optimization Algorithms. In Sect. 4, Simulation and Results are detailed. The Conclusions drawn from this study are encapsulated in Sect. 5, followed by a discussion on future directions in Sect. 6.
Solar cells are an essential component of photovoltaic systems, responsible for converting sunlight into electricity. Accurately modeling their behavior is vital to maximize performance, especially in different environmental conditions. Diode-based models, such as the Single Diode Model (SDM), Double Diode Model (DDM), and Three Diode Model (TDM), are typically to simulate the characteristics of photovoltaic cells. Each model has its strengths, depending on the level of accuracy required and the complexity that can be handled in real-world applications.
The SDM is widely utilized for its simplicity of implementation. It represents the solar cell as a current source in parallel with a diode (to model the p-n junction), and resistances to account for losses due to series (Rs) and shunt (Rsh) resistance. The model is ideal for general applications where basis approximations are sufficient, such as in initial performance evaluation or when dealing with relatively uniform irradiance and moderate temperature variations as shown in Fig. 1.
Equivalent circuit of a single-diode photovoltaic cell system.
The output current of a photovoltaic (PV) cell in the Single-Diode Model (SDM) is expressed as:
where (:{I}_{ph}:)denotes the photocurrent, (:{I}_{d}) represents the diode current, and (:{I}_{sh})corresponds the shunt current. This equation describes the current balance in the PV cell by accounting for photocurrent generation, diode losses, and shunt resistance effects, as provided in³⁰. The SDM provides a simplified yet accurate representation of PV behavior, with parameter estimation further enhanced by the hybrid BA–CR algorithm.
The shunt resistance (:{I}_{sh}:)is obtained from this formulation:
The diode current (:{I}_{d}:) can be obtained from the Shockley model as follows:
By introducing the above equations, the expression for the output current (:{I}_{L})is given in the subsequent form:
To validating let’s consider the SDM, which consists of five unknown parameters(::theta:=left[{I}_{ph}::{R}_{s}::{R}_{sh}::{I}_{SD}::nright]) that should be determined correctly for better modeling.
Practical Benefit of SDM: The SDM is computationally efficient and requires fewer parameters (five unknowns) for estimation, making it suitable for quick estimations and lower cost simulations in environments where high precision is not the primary concern.
The DDM enhances upon the SDM by incorporating and additional diode. This model better accounts for recombination currents, especially under low-light conditions, where the single diode approach might be inadequate. The DDM includes two diodes in parallel with a current source, one representing diffusion current and accounting for recombination effects in the space charge region. Figure 2 depicts the diffusion and saturation current, represented by (:{I}_{SD1}) and (:{I}_{SD2}:)respectively, and (:{n}_{1}) and (:{n}_{2}) are the ideality factor of the diodes⁴³.
Equivalent circuit of the DDM model.
Switching to the DDM is beneficial when more accurate performance is needed, particularly at lower light intensities. The additional diode and parameters improve the model’s ability to reproduce real behaviors, especially for PV cells operating under suboptimal conditions. However, this increased accuracy comes at the cost of more complex parameter estimation. The output current of a DDM is expressed as.
where (:{I}_{ph}:)denotes the photocurrent, (:{I}_{{d}_{1}})and(:{I}_{d2}) represent the current through the two diodes, and (:{I}_{sh})corresponds the shunt current. This formulation captures additional recombination losses within the photovoltaic cell, providing a more detailed representation than the SDM, as reported in³⁰. The hybrid BA–CR algorithm is used to enhance the accuracy of parameter estimation for this model.
Rearranging the equation in the form of a single-diode model (SDM), obtain:
Validating let’s consider the single diode model or SDM, which consists of five unknown parameters(::theta:=left[{I}_{ph}::{R}_{s}::{R}_{sh}::{I}_{SD1}:{I}_{SD2}::{n}_{1}:{n}_{2}right]) that should be determined correctly for better modeling.
The TDM takes the accuracy a step further by incorporating a third diode, which enables the model to capture more detailed behaviors, such as leakage current and efficiency boundaries. This model, while the most accurate, introduces significant complexity. As illustrated in Fig. 3, this model incorporates three diodes in parallel with the current source.
Equivalent circuit of the TDM model.
Practical Benefit of TDM: The TDM is ideal when the highest accuracy is needed, particularly in advanced applications like optimizing the power output under dynamic environmental conditions. This model is suitable for research or high-performance systems where small efficiency differences can have a great impact on system performance. However, the added complexity and need for accurate parameter estimation make it less practical for general use.
Considering the equivalent circuit, the output current of this model can be expressed as:
Here, (:{I}_{ph}:) is the photocurrent, (:{I}_{{d}_{1}}), (:{I}_{d2}), and (:{I}_{d3}) represent the currents through the three diodes, and (:{I}_{sh}) represent the shunt current. This formulation captures complex recombination and leakage effects, providing the most detailed representation among photovoltaic models, as reported in³⁰. The hybrid BA–CR algorithm is applied to improve the accuracy of parameter estimation for TDM. Using a similar method as before, Eq. (7) can be written as:
Adding a third diode makes the model more complex by integrating two more parameters, which increases the total parameters to nine⁴³. The amended parameters are illustrated as follows:
This innovation more than doubles the size of this model, so it can only be used after all nine parameters are determined for accurate modelling and performance.
The selection between SDM, DDM, and TDM is determined by the required accuracy and the level complexity of the system. The SDM is simple and efficient, which makes it useful for general applications where higher accuracy is not needed. The DDM offers improved accuracy, particularly under low-light conditions where the SDM is not work well. The TDM is ideal for critical applications that require high accuracy. Table 2 presents a comparative summary of these cell models, highlighting their accuracy, typical applications, and main advantages.
The SDM, DDM, and TDM are used in various PV panels depending on the required accuracy and computational complexity. The selection of an appropriate model depends on parameters like irradiance levels, shading conditions, and system optimization. A comparison of three diode-based models (SDM, DDM, and TDM) used in PV, including their applications and advantages, is provided in Table 3.
When solving the problem of solar cell parameter identification with an optimization algorithm, many important factors should be considered:
Problem Formulation: Defining the objective of the optimization, like maximizing efficiency or minimizing power losses.
Range Search Determination: Establish valid ranges for the parameters to generate meaningful values.
Objective Function Construction: Develop a mathematical function that measures how the model predictions match with experimental data. This function guides the optimization algorithm in finding optimum parameter values.
The objective function used in this study is the Root Mean Square Error (RMSE), which quantifies the deviation between the measured and predicted current values in the I–V characteristics. It is expressed as:
where (:N) is the total number of data point (:{{rm:I}}_{meas,k}:)represents the measured current, and (:{{rm:I}}_{est,k}:)is the current estimated by the PV model (SDM, DDM, or TDM). This function acts as the optimization criterion for the hybrid BA–CR algorithm, guiding it to minimize estimation errors and identify the optimal model parameters.
Accurate modeling and parameter identification are crucial for the optimization of PV panel performance. Both the two-diode and single-diode models offer effective solutions, with parameter identification tending to involve the use of optimization algorithms and experimental data fitting techniques.
In this study, BA and CR are used as optimization algorithms. Therefore, the following sections provide a definition and explanation of these algorithms.
The Bat Algorithm (BA) is a metaheuristic algorithm inspired by the echolocation behavior of bats, being the only mammals capable of flight, rely heavily on their acute hearing to navigate, hunt, and locate roosts. They use sound emission and feedback to perceive their surroundings, effectively utilizing this system for prey hunting. When approaching prey, bats increase their emitted frequency while simultaneously decreasing the wavelength. The BA, proposed by Yang, utilizes this behavior to create an innovative optimization algorithm. Three simple assumptions are made to adapt bat behavior for complex problem-solving:
Each bat automatically obtains the distance from prey through sound emission and reception.
Bats possess a velocity vector, position, and frequency within a range denoted by min ​ and max​.
Each bat’s emitted sound level varies between initial and final values denoted by 0 and min​, respectively.
The algorithm models two types of search strategies—global and local—for each bat, wherein initially, a bat conducts a global search followed by a probabilistic local search around optimal solutions. The frequency update mechanism for sound emission during global search is governed by Eq. (11), where(:{f}_{i}) represents the bat’s frequency, (:{f}_{min}) and (:{f}_{max}) denote the minimum and maximum emission frequencies, respectively, and β is a random number in the range [0,1]³¹. The algorithm’s effectiveness is validated through benchmarking against various optimization algorithms, demonstrating its superior accuracy in extracting optimal solutions compared to methods like PSO and GA.
The CR is a metaheuristic inspired by crow’s behavior, utilizing two parameters including flight length and awareness probability. It involves group crows that update their position based on past experiences to search for optimal solution⁴⁴. During this process, two scenarios can arise:
In the first scenario: the leading crow , not realizing that there is another crow behind it, gets to where crow is hiding. To move crow , the following equation is applied⁴⁵.
In this formulation, (:{r}_{i}) is a uniform random variable in the [0, 1] interval, ((:f{l}^{i,:iter})) shows the distance flown by crow i at the iter iteration and (:left({m}^{j,:iter}right)) denotes the position which the best crow has achieved.
In the second scenario: when a given crow starts to follow the crow , the crow either moves to a different location in the search space or loses the crow . here, the new position of crow is defined by Eq. (12) which has a variable called random number and describes the awareness probability and flight length. In addition, Eq. (7) explains how to update the fight length ((:fl)) of crow , as a time approach, without violating the maximum and minimum limits which are provided⁴⁵. In the same way, Eq. (13) alters the awareness probability (AP) of crow j between the iterations, with limits determined by the maximum and minimum values⁴⁵.
Where (:{r}_{j})​ is a random variable uniformly bounded between 0 and 1 for crow , and (:A{P}^{j,:iter}:)is the participant awareness probability of crow j at the iteration.
In this equation, (:{fl}^{i,:iter+1}) represents the flight length of crow at iteration (:iter+1). (:f{l}_{max}) is the maximum flight length, and (:f{l}_{min}) is the minimum flight length. (:iter) denotes the current iteration, while (:ite{r}_{max}:)refers to the maximum number of iterations.
where, (:{AP}^{j,:iter+1}) indicates the awareness probability of crow at iteration (:iter+1), while (:{AP}_{max}) and (:{AP}_{min}:)are the maximum and minimum awareness probability values respectively.
As previously mentioned, this study utilizes a combined crow and bat algorithm to estimate the parameters of solar panels. In the previous sections, the CR and BA were introduced. In this section, the combination of these two algorithms is explained:
In the search space, to connect the considered variable to its corresponding point, equivalent to a crow, with the origin of coordinates, thereby obtaining the path direction and thus eliminating problem constraints. The distance between the center of the problem space and the extreme state function is the reliability index in that direction. Therefore, a reliability index can be calculated for each crow, corresponding to the sensitivity along the path stated by that crow. In fact, to calculate the reliability index of each direction, crow parameters increase in the desired direction until the extreme state function becomes zero. At this moment, the obtained parameters create the design point, and the distance between the design point and the center of the problem space is the reliability index.
In the proposed algorithm, each group member behaves like a crow, meaning they observe other group members to see where they hide their food. In this regard, like bats, each system member uses their own solution during the search. Each member has an awareness parameter like CR. The awareness parameter allows a participant to detect if another participant is tracking it, influencing its subsequent action. When a participant realizes it is being tracked, it takes evasive measures by misleading the tracker, changing directions, and seeking shelter elsewhere. The system members use optimization equations to find the optimal solution while monitoring other members’ positions. In order to improve the process of searching, local search methods are used to enable every member to optimize its solution by considering neighbors.
The hybridization of the BA and the CR uses complementary strengths of both methods, which provides a good balance between searching and focusing on the best solution. BA excels at global search, helping it efficiently search large solution spaces. Meanwhile, CR enhances solutions by local search to focus on favorable regions. This suggested approach uses the strengths of the two algorithms, so it becomes more reliable and effective.
The motivation for integrating the BA with the (CR) is to leverage BA’s frequency-adjusted global search (Eq. 11) for extensive exploration and CR’s position-update mechanism (Eqs. 11–14) for targeted local exploitation. This integration produces a modified hybrid BA–CR algorithm that balances exploration and exploitation while accelerating convergence. Within this framework, BA’s exploration capability is strengthened global search, enabling efficient traversal of large solution spaces (as discussed in Eq. 11). In contrast, CR’s exploitation mechanism utilizes social interactions via position updates and awareness probability (Eqs. 11–14 for flight length and awareness adjustments), efficiently refining solutions in promising fields. The BA–CR hybrid strategically uses BA-driven global updates in early iterations for wide exploration and CR-driven local refinements in later iterations, achieving a well-balanced optimization process with enhanced convergence speed. To verify this, comparative analyses were made with conventional methods, showing higher enhancements in convergence speed and parameter estimation accuracy. The algorithm was experimented with various models (SDM, DDM, and TDM), and the improved results closely followed the experimental data. The computational complexity of the hybrid BA–CR algorithm is (:{rm:O}left({rm:N}times:itertimes:dright)) where (:text{{rm:N}}) is the population size, (:iter) is the total number of iterations, and (:d) represents the number of problem dimensions. Though this complexity is comparable to that of the individual BA and CR algorithms the hybrid approach enhances overall efficiency by effectively balancing exploration and exploitation, thereby reducing the number of iterations required and improving performance in PV parameter estimation.
Algorithm 1 clearly shows how BA and CR are optimized at each iteration of the loop. The following pseudocode shows this process step by step.
Algorithm 1: Hybrid Bat and Crow Search Algorithm (BA-CR)
Figure 4 shows a hybrid optimization algorithm that uses several steps to find the best solution.
Hybrid Optimization Algorithm Flowchart.
Figure 4 shows the steps of a nature-inspired optimization algorithm that starts with an initial population and evaluates them. Then, iterates through the teacher and learner phases (possibly modified) and improves the solutions until it reaches a stopping criterion and provides the best solution.
The study presents the outcomes derived from applying the proposed method to estimate the parameters of solar cells. Notably, MATLAB version 2021 was utilized for simulation, conducted on a system equipped with a 5-core processor and 8 GB of RAM.
To evaluate the performance of the proposed BA-CR algorithm (Bat and Crow Algorithm), the characteristics of the solar cell systems, specifically the SDM, DDM, and TDM are initially defined. The data used is based on the photovoltaic cells presented in Tables 4 and 5, which are based on a commercial silicon solar cell manufactured by R.T.C. in France.
In this research compare the results with other algorithms form⁴³, to ensure consistency in the search space and parameter ranges. Table 6 presents the range of every variable of the PV system model.
In the following, the results of parameter estimation for all three SDM, DDM, and TDM models are presented. Figures 5 and 6 illustrate the I–V and P–V curves of the photovoltaic models. A strong agreement is observed between the estimated and measured values for both I–V and P–V curves. The measured data were obtained from experimental readings of the RTC France silicon solar cell under standard test conditions (1000 W/m²irradiance and 25 °C temperature). This agreement confirms the effectiveness of the hybrid BA–CR algorithm in accurately modeling PV behavior across the SDM, DDM, and TDM configurations.
Current-Voltage Graph.
Power-Voltage Graph.
Subsequently, the study investigates the SDM mode. It’s important to mention that in order to estimate the parameters and calculate the RMSE for each mode, the algorithm was executed 30 times, with the reported results being the average of the RMSE values and parameter estimations. For the SDM mode, five parameters ((:{I}_{ph},:{R}_{s},:{R}_{sh},:{I}_{sd},:n)) are ultimately regarded as the parameters of interest. The findings of this mode are outlined below, with Fig. 7 illustrating the convergence curve of the proposed algorithm.
Convergence curve of the proposed algorithm in determining the solar cell parameters for the SDM mode.
Figure 8 shows the RMSE convergence curves for the SDM across ten algorithms, like BA–CR, PSO, GA, SA, CR, BA, DE, ABC, GWO, and WOA highlighting the superior convergence speed of the BA–CR algorithm. To maintain consistency across models, the figure depicts the convergence behavior of all ten algorithms (as summarized in Tables 7,8 and 9) for the SDM. The results clearly demonstrate the superiority of the hybrid BA–CR algorithm, which achieves the lowest RMSE value of 0.00077299.
RMSE Convergence Curves for SDM across 10 Algorithms.
Figures 9 and 10 apply this analysis to the DDM and TDM, respectively. In both figures, RMSE values are plotted against the number of iterations (1 to 1000), with final RMSEs of 0.0008215 for the DDM and 0.0008068 for the TDM.
RMSE Convergence Curves for DDM across 10 Algorithms.
These results confirm the superior performance of the BA–CR algorithm. Overall, this analysis emphasizes the efficiency and robustness of the new hybrid method across all photovoltaic models.
Figure 10 shows RMSE convergence curves for the TDM across ten algorithms (BA–CR, PSO, GA, SA, CR, BA, DE, ABC, GWO, WOA), illustrating the overall superior performance of the BA–CR algorithm.
RMSE Convergence Curves for TDM across 10 Algorithms.
The results of assessed variables of the SDM model for the RTC France photovoltaic cell utilizing the proposed method are presented in Table 6.
Furthermore, an assessment of the outcomes derived from the DDM model is included. This model encompasses seven key indicators (:{I}_{ph},:{R}_{s},:{R}_{sh},:{I}_{SD1},:{I}_{SD2},:{n}_{1},:{n}_{2}), with the convergence plot depicted in Fig. 8, Fig. 11.
Convergence Plot of the Proposed Algorithm in Determining the Parameters of Solar Cells for the DDM Model.
The results of estimating the parameters of the DDM model for the RTC France photovoltaic cell using the proposed method are outlined in Table 8.
This section assesses the performance of the TDM model and uses the proposed algorithm for this purpose. This model consists of nine important Factors [(:{I}_{ph},:{R}_{s},:{R}_{sh}:{I}_{sd1},{I}_{sd2},:{I}_{sd3},:{n}_{1},:{n}_{2},{n}_{3}]), In Fig. 12, a convergence plot is provided displaying the progress of the algorithm.
Convergence Plot of the Proposed Algorithm in Determining the Parameters of Solar Cells for the TDM Model.
The Assessment parameters value for the TDM model, applied to the RTC France solar cell employing the proposed method, are given in Table 7.
The results obtained from each simulation model are assessed, building upon the findings of Najijian et al. (2021)⁴³, who employed 10 different algorithms. The parameter estimation outcomes and the RMSE values are presented for each model. This evaluation is presented in Tables 10, 11 and 12.
Upon scrutinizing the comparative results provided in the aforementioned tables, it’s evident that the proposed algorithm, based on the hybrid approach of bat and crow (BA-CR), achieves a superior RMSE value compared to other algorithms across all three SDM, DDM, and TDM models.
The results from Tables 8 and 9, and 12 clearly show the superior performance of the BA-CR algorithm across all three models. In the SDM model, BA-CR achieved RMSE of 0.00077299, better than other algorithms such as PSO and GA, which achieved higher RMSE values, indicating lower accuracy. For the DDM and TDM models, the BA-CR algorithm also performed exceptionally well, with RMSE values of 0.0008215 and 0.0008068, respectively, confirming its precision in parameter estimation. A close look at the comparison outcomes shows that the hybrid proposed algorithm is the best of all other techniques consistently. It not only obtains the lowest RMSE, but also converges faster than the other optimization algorithms.
The increasing global adoption of solar energy has driven significant research into the design of efficient photovoltaic cells. These systems, being highly nonlinear, exhibit varying power and voltage outputs, particularly with changes in environmental factors such as irradiance and temperature. Identifying the MPP of photovoltaic generators is significant for optimal performance, which depends on accurately estimating the model parameters.
This paper introduces a novel hybrid optimization method that combines the crow and bat algorithms to enhance the parameter optimization of photovoltaic cells. The new algorithm is fast, global in search, robust, and capable of providing accurate predictions for SDM, DDM, and TDM models. Experimental results validate the effectiveness of this approach as estimated parameters closely match experimental data.
Among the models, the TDM consistently provides the most accurate representation of the TRANCE TRC cells, achieving the lowest RMSE values compared to other models. The hybrid BA-CR approach also outperforms other algorithms, particularly in addressing the increased uncertainty in parameters, providing more accurate results for the TDM model. The BA–CR algorithm demonstrates robust performance in applications such as grid integration and MPPT under varying conditions, thanks to its balanced exploration–exploitation strategy, as reflected in the low RMSE values across the SDM, DDM, and TDM models. These findings highlight the improved performance and reliability of the proposed method in optimizing photovoltaic cell parameters.
Future work will focus on extending the hybrid BA–CR algorithm to other renewable energy systems, such as wind turbines and integrated hybrid grids, as well as enhancing its scalability for large-scale photovoltaic installations. A current limitation is the computational complexity of the hybrid approach, which could pose challenging for real-time or large-scale PV applications, underscoring the need for further optimization to enable practical deployment.
The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.
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Thanks in advance.
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Department of Electrical & Electronics Engineering, Karabuk University, Karabuk, Turkey
Abdulsalam Ashour Mohameed Almabrouk & Selçuk Alparslan Avci
Department of Electrical and Electronics Engineering, Istanbul Topkapi University, Istanbul, Turkey
Javad Rahebi
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A.A.: Conceptualization, Methodology, Writing—Original Draft, Software, Writing—Reviewing and Editing; S.A. : Supervision, Investigation, Software; J.R. : Data Curation, Investigation, Visualization. All authors have read and agreed to the published version of the manuscript.
Correspondence to Javad Rahebi.
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Ashour Mohameed Almabrouk, A., Avci, S.A. & Rahebi, J. Parameter Estimation in photovoltaic systems using a hybrid Bat and crow metaheuristic algorithm. Sci Rep 16, 4670 (2026). https://doi.org/10.1038/s41598-025-34906-3
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Received: 29 May 2025
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Published: 06 January 2026
Version of record: 03 February 2026
DOI: https://doi.org/10.1038/s41598-025-34906-3
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