A Review on the Hybridization of the LM Algorithm for PV Modeling and Parameter Estimation

A Review on the Hybridization of the LM Algorithm for PV Modeling and Parameter Estimation | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Systematic Review A Review on the Hybridization of the LM Algorithm for PV Modeling and Parameter Estimation Marouan BEN EL HAJ This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8787012/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The modeling of photovoltaic (PV) systems serves as a critical component in assessing their electrical performance and enhancing their integration into broader energy frameworks. To tackle the challenges stemming from the highly nonlinear characteristics of PV model parameter identification, this paper offers an in-depth review of hybrid strategies that integrate the Levenberg-Marquardt (LM) algorithm with global metaheuristic methods. This study examines the theoretical basis of hybrid approaches and their advantages over pure metaheuristics, motivating the use of techniques like GA-LM, SA-LM, PSO-LM, and ABC-LM. A classification framework for these hybrids is presented, including evolutionary algorithms and swarm-based methods. Simulation results for the single-diode PV equivalent circuit demonstrate enhanced performance, with RMSE improvements ranging from 10% to 33% for hybrids like ABC-LM and PSO-LM compared to pure ABC and PSO, particularly under varying irradiances (600, 800, and 1000 W/m²). Hybrids also maintain low computational costs relative to high-CPU pure methods, offering better compromise scores. Finally, key challenges related to scalability, computational complexity, and irradiance-dependent accuracy are addressed, while highlighting emerging trends toward multi-objective and adaptive optimization frameworks. This review thus provides practical guidance for developing robust hybrid optimization methods tailored to PV system modeling and parameter identification. Energy Engineering Levenberg-Marquardt algorithm hybrid optimization photovoltaic modeling parameter estimation Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Introduction The escalating concerns over global warming, the depletion of fossil fuel resources, and the imperative to curb greenhouse gas emissions have, in recent decades, fueled a surge in interest toward sustainable and environmentally friendly energy technologies. Among these, photovoltaic energy stands out due to its renewable nature, abundant availability, and versatility for integration into modern energy systems 1 . However, achieving optimal utilization of PV systems heavily depends on the accuracy of their mathematical modeling and the reliable identification of their electrical parameters 2 3 4 5 . PV system modeling is typically expressed through nonlinear and nonconvex formulations 6 7 , where parameters must be estimated from experimental current-voltage data. This identification process poses a complex optimization challenge, marked by multimodal search spaces and high sensitivity to initial conditions 8 . In this setting, conventional local optimization methods are often favored for their rapid convergence and low computational overhead. Yet, their effectiveness is constrained in intricate cost landscapes, frequently resulting in suboptimal solutions. The Levenberg-Marquardt (LM) algorithm ranks among the most widely employed local techniques for solving nonlinear least-squares problems, particularly in extracting PV parameters 9 10 . While it delivers swift local convergence, the algorithm exhibits a strong reliance on initial conditions and a tendency to converge to local minima. These shortcomings have spurred growing interest in hybrid optimization frameworks that merge LM's local exploitation capabilities with the global exploration features of broader optimization methods. In recent years, numerous studies have investigated the hybridization of the LM algorithm with evolutionary, swarm intelligence, bio-inspired, stochastic, or deterministic approaches to bolster the robustness and accuracy of PV parameter estimation 11 12 13 . Despite this wealth of contributions, the existing literature remains fragmented and often lacks thorough comparative analyses to clearly delineate the advantages, limitations, and preferred application domains of each hybrid strategy. Against this backdrop, this paper aims to examine and compare hybridization strategies for the LM algorithm with global optimization methods in the context of PV model parameter identification. The hybrid approaches under review are evaluated via simulations based on the single-diode PV model, with their performance benchmarked in terms of estimation accuracy and resilience against local minima 14 . The goal is also to furnish a clear methodological framework for designing and applying these hybrid optimizers in complex engineering systems. The first section outlines the fundamental principles of the LM algorithm along with its primary limitations. The second section explores various hybridization strategies and their rationales. The third section demonstrates the outcomes through case studies on PV modeling, comparing hybrid configurations and evaluating gains in RMSE and CPU time. Finally, conclusions are drawn to synthesize the observations, discuss open challenges, and propose future research directions in LM-based hybrid optimization. 1 Levenberg-Marquardt Algorithm The LM algorithm represents a cornerstone method for nonlinear least-squares optimization, relying on a quadratic approximation of the cost function to iteratively refine parameter estimates 15 16 . LM blends gradient descent principles with the Gauss-Newton method through an adaptive damping parameter λ, which modulates the step size to ensure both stability and efficiency. The LM parameter adjustment at step k follows (1) 17 18 . $$\:{\varDelta\:x}_{k}={-({J}^{T}J+\lambda\:I)}^{-1}{J}^{T}r$$ 1 where J is the Jacobian, r represents residuals, I is the identity, and λ controls damping. This setup mimics a trust-region strategy, shifting from gradient-based steps (high λ) to Newton-like updates (low λ) near solutions. This formulation enables LM to exhibit trust-region-like behavior, transitioning from steep descent for global stability (large λ) to quadratic convergence near the optimum (small λ) 19 . As illustrated in Fig. 1 , the iterative process of LM hinges on damping adjustments that, while locally effective, highlight the algorithm's vulnerabilities in expansive search spaces. The LM method offers advantages including superior stability relative to Gauss-Newton, supporting convergence from distant initials; smooth integration of gradient and Newton steps for faster progress in well-conditioned problems; and simple implementation. These features support its use in PV-related tasks, such as fitting I-V curves from experimental solar data, tuning models for irradiance-dependent performance, and optimizing parameters in renewable energy simulations 20 21 . These strengths account for its widespread use in practical applications requiring precise parameter fitting, such as curve fitting in experimental data processing, neural network training in machine learning where it efficiently optimizes weights for complex curvatures, orbit determination in astronomy and space navigation, lens design in optics, inverse problems in geophysics for seismic data analysis, or model optimization in aeronautical engineering based on real aircraft datasets. However, LM is hampered by its dependence on initial conditions, which can steer convergence toward local minima in multimodal cost landscapes 22 . These constraints are especially evident in highly nonlinear or high-dimensional problems, where the risk of stagnation or divergence increases. Although damped or scaled variants have been developed to mitigate some issues, they do not address the fundamental lack of global exploration capabilities, underscoring the need for hybridization 23 . To address this, hybrid approaches typically combine an initial global exploration phase with local refinement via LM 24 25 , forming the conceptual backbone of the optimization strategies reviewed here (Fig. 2 ). 2 Hybridization of Levenberg-Marquardt Algorithm Hybridizing LM enables the fusion of its rapid local exploitation with global exploration strategies, overcoming limitations in multimodal search spaces and sensitivity to initial conditions 22 . This integrative approach enhances robustness, accuracy, and reproducibility of solutions, even amid noise, nonlinearity, or incomplete data. Global search algorithms like GA and PSO excel at traversing intricate objective functions and avoiding suboptimal solutions. However, they typically converge more slowly and achieve less precise results compared to local optimizers. Integrating them with LM creates a synergistic system that capitalizes on global exploration followed by local fine-tuning 26 27 . Hybridization with LM thus provides a complementary framework, leveraging the strengths of each to achieve an optimal balance between exploration and refinement. Studies demonstrate reduced initial-condition dependence and improved efficiency, with applications in PV impedance modeling, solar irradiance forecasting, and dynamic PV system emulation. Thus, hybridization establishes a balanced methodological framework, where global search precedes or alternates with LM's local refinement, surmounting the drawbacks of purely local methods and boosting performance in PV-relevant domains such as solar material characterization, environmental impact assessment for PV installations, and mathematical modeling of solar cell dynamics. Table 1 presents the primary LM hybrid strategies documented in the literature, comparing their methodologies, advantages, and performance, while highlighting significant gains in accuracy, convergence speed, and robustness relative to non-hybrid approaches. Table 1 Taxonomy and Comparative Analysis of LM Hybridization Strategies Across Optimization Paradigms Category Associated Algorithm Synergy Mechanism Scientific Gains and Advantages Initialization Optimization (Global-to-Local) Genetic Algorithm (GA) 28 GA delivers a robust initial parameter estimate; LM refines the solution for utmost precision. Achieves the lowest RMSE values for photovoltaic parameter extraction. Differential Evolution (DE) 29 Parameters are initialized by DE under physical constraints, then precisely solved by LM. 100% success rate in reaching the global solution for GPS time series (versus 22% for LM alone). Evolutionary Strategy (ES) 30 Mutation and crossover for global initialization; LM damping for refinement. Evades global minima; provides robustness to noise and initialization; scalable for high-dimensional problems. Cuckoo Search (CS) 31 CS trains the network in the first stage to select optimal weights; LM completes the minimization. Avoids stagnation and local minima; outperforms ABC-LM hybridization in convergence speed on XOR problems. Bat Algorithm (BAT) 32 Employs echolocation to identify the global optimum region prior to LM refinement. Enhances classification accuracy (99.99%) on medical datasets (cancer, diabetes) while reducing CPU time. Luus-Jaakola (LJ) 33 LJ performs direct gradient-free search for initial approximation; LM handles final refinement. Essential for convergence in MESH equation systems (distillation) where standalone LM diverges without a nearby starting point. Whale Optimization Algorithm (WOA) 34 WOA conducts global space exploration; LM performs high-precision local mining. Reduces reprojection errors in stellar sensor calibration and eliminates dependence on initial values. Quantum-Behaved PSO (QPSO) 35 QPSO ensures initial global convergence to initialize LM. Reduces computation time (under 3s) for real-time PET clinical diagnostics compared to PSO or DE alone. Firefly Algorithm (FA) 36 Foraging and echolocation for initialization; LM for local refinement. Derivative-free handling in biological systems; improved convergence speed. Bacterial Foraging Optimization (BFO) 37 Foraging and echolocation for initialization; LM for local refinement. Derivative-free handling in biological systems; improved convergence speed. Opposition-based Squirrel Search (OSS) 38 Opposition-based learning integrated with squirrel search for global exploration; LM for local refinement in sequential mode. High reliability in ill-posed or noisy optimization; superior positioning accuracy in robotic systems with reduced kinematic errors. Gravitational Search Algorithm (GSA) 39 Adaptive or distributed quantum/policy-based exploration; LM local optimization. Multi-objective handling; quantum speedups. Structural and Adaptive Hybridizations Artificial Bee Colony (ABC) 40 Adaptive switching between exploration (ABC) and exploitation (LM) based on population distances. Unifies exploration and exploitation capabilities; drastically reduces computational load for constrained engineering problems. Reinforcement Learning (RL-enhanced) 41 Adaptive or distributed quantum/policy-based exploration; LM local optimization. Privacy in distributed data; multi-objective handling. Federated PSO (FedPSO) 42 Adaptive or distributed quantum/policy-based exploration; LM local optimization. Privacy in distributed data; multi-objective handling. Multi-Objective Evolutionary Algorithm (MOEA) 43 Adaptive or distributed quantum/policy-based exploration; LM local optimization. Privacy in distributed data; multi-objective handling. Backpropagation (B) 44 Alternating: gradient or Newton steps for refinement; LM trust-region control. Superlinear convergence; error sampling without global search overhead. Quasi-Newton (QNewton) 45 Alternating: gradient or Newton steps for refinement; LM trust-region control. Superlinear convergence; error sampling without global search overhead. Gradient Descent (GD) 45 Alternating: gradient or Newton steps for refinement; LM trust-region control. Superlinear convergence; error sampling without global search overhead. Newton 46 Alternating: gradient or Newton steps for refinement; LM trust-region control. Superlinear convergence; error sampling without global search overhead. Swarm-Gradient Synergy Ant Colony Optimization (ACO) 47 Each ant-generated solution is enhanced by a local LM iteration. Performance on par with specialized algorithms for training medical neural networks (classification). Particle Swarm Optimization (PSO) 48 Combines PSO search directions and LM gradient in a weighted update rule. Capability for tracking dynamic processes and rapid convergence even with reduced particle populations. Firefly PSO (FPSO) 49 Parallel or sequential: velocity and pheromone updates for exploration; LM trust-region refinement. Multimodal landscape handling; accuracy in noisy data; feature optimization efficiency. Dynamic Control of Internal Parameters Simulated Annealing (SA) 50 SA searches for the optimal damping factor (λ) at each LM iteration. Smooth transition between gradient descent and Gauss-Newton; far superior robustness compared to classical tuning. Grey Wolf Optimizer (GWO) 51 GWO dynamically optimizes λ within the LM process. Outperforms pure deterministic methods in precision (RMSE of 9.8610 − 04 ) for solar cell identification. Fuzzy Logic (FL) 52 Fuzzy rules evaluate error variations and dynamically adjust the damping factor (λ) in LM iterations. Boosts stability and convergence for nonlinear and uncertain systems; effective in robust control for grid-connected PV under faults. The proposed categorization stems from reviewing approximately 40 distinct studies, organizing hybrids into optimization de l'initialisation (global-to-local), contrôle dynamique des paramètres internes, hybridations structurelles et adaptatives, and synergie swarm-gradient groups (Fig. 3 ). Hybrids focused on initialization optimization merge global exploration mechanisms with LM's local refinement. Notable examples include Genetic Algorithm-LM (GA-LM), applied to PV cell impedance and solar model training; Differential Evolution-LM (DE-LM), used for PV modeling and solar radiation transport inversion; Evolution Strategy-LM (ES-LM), for robust parameter optimization; Cuckoo Search-LM (CS-LM), enhancing global search in multimodal objective functions; Bat Algorithm-LM (BAT-LM), improving PV ANN performance; Luus-Jaakola-LM (LJ-LM), employed for PV differential equation fitting; Whale Optimization Algorithm-LM (WOA-LM), used for PV sensor data processing; Quantum-Behaved PSO-LM (QPSO-LM), for high-dimensional PV optimization; Firefly Algorithm-LM (FA-LM), aiding in solar anomaly detection; Bacterial Foraging Optimization-LM (BFO-LM), handling nonlinear PV models; Opposition-based Squirrel Search-LM (OSS-LM), enhancing solar tracker calibration and robotic positioning; and Gravitational Search Algorithm-LM (GSA-LM), applied to global PV optimization. These methods offer strong robustness at a moderate to high computational cost and are commonly deployed in parameter identification and solar energy applications. Structural and adaptive hybridizations draw from structural and adaptive processes to escape local minima. Artificial Bee Colony-LM (ABC-LM) is applied to PV neural training and solar function optimization; Reinforcement Learning-enhanced-LM (RL-LM) for PV energy management; Federated PSO-LM (FedPSO-LM) for distributed solar data processing; Multi-Objective Evolutionary Algorithm-LM (MOEA-LM) for sustainable PV systems; Backpropagation-LM (B-LM), applied to PV time-series neural models; Quasi-Newton-LM (QNewton-LM), targeting PV nonlinear fitting; Gradient Descent-LM (GD-LM), for PV time-series neural models; and Newton-LM, offering reliable exploration for solar optimization. These methods entail increased algorithmic complexity but ensure high global search capacity. Swarm-gradient synergy hybrids incorporate collective behaviors and gradient-based elements for progressive exploration. Ant Colony Optimization-LM (ACO-LM) supports PV array layout optimization; Particle Swarm Optimization-LM (PSO-LM), utilized for PV neural network initialization, solar resource prediction, and tracker system calibration; and Firefly PSO-LM (FPSO-LM), aiding in solar fault diagnostics. Hybrids emphasizing dynamic control of internal parameters leverage adaptive tuning for enhanced convergence. Simulated Annealing-LM (SA-LM) is combined with ANNs for PV inverse modeling; Gray Wolf Optimizer-LM (GWO-LM) enhances PV parameter extraction; and Fuzzy-LM (F-LM) provides robust control in grid-connected PV systems under unbalanced conditions. These approaches provide rapid convergence but require careful parameter tuning. 3 Methodology Parameter estimation for the single-diode model of the MSX60 solar photovoltaic module was conducted under controlled temperature (25°C) and different irradiances conditions (600, 800 and 1000 W/m²) using experimental I-V data. The model parameters, I ph , I 0 , R s , R sh , and a, were optimized within predefined bounds using the LM algorithm as the baseline, hybridized with various strategies: GA, SA, ABC and PSO. The single-diode model is given by Eq. 2: \(\:I=\:{I}_{ph}-{I}_{0}\left({e}^{\frac{V+I{R}_{s}}{a{V}_{t}}}-1\right)-\frac{V+I{R}_{s}}{{R}_{sh}}\) (2) where V t is the thermal voltage. Parameter fitting minimized RMSE, given by Eq. 3, between observed and predicted currents, solved implicitly via zero-finding routines. \(\:\text{R}\text{M}\text{S}\text{E}=\sqrt{\frac{1}{\text{N}}\sum\:_{\text{i}=1}^{\text{N}}{\left({\text{I}}_{\text{m}\text{e}\text{a}\text{s}\text{u}\text{r}\text{e}\text{d},\text{i}}-{\text{I}}_{\text{m}\text{o}\text{d}\text{e}\text{l}\text{e}\text{d},\text{i}}\right)}^{2}}\) (3). Hybrids employed metaheuristic phases to generate refined initials for LM, with evaluations via RMSE, runtime, and a trade-off metric (RMSE multiplied by CPU time, where lower values indicate superior balance). Performance was evaluated using multiple metrics: RMSE, CPU time, I-V/P-V curve fits, and individual absolute errors. All simulations and optimizations have been implemented in MATLAB, with initial conditions unified to ensure a realistic comparison. This comprehensive approach ensures the robustness and comparability of hybrid optimization methods applied to the estimation of PV parameters, providing a solid basis for model analysis and validation. 4 Results and Discussion To validate the efficacy of hybrid optimization methods for estimating parameters in the single-diode solar cell model, simulation results were analyzed using comparative metrics and visualizations. Each method's performance was assessed in terms of RMSE, CPU time, and a compromise score defined as the product of RMSE and CPU time, representing a balance between accuracy and computational efficiency. Simulation outcomes show ABC-LM and ABC yielding the lowest mean RMSEs of approximately 0.08 and 0.087, respectively, demonstrating superior accuracy compared to other methods. PSO-LM and GA-LM follow closely with mean RMSEs around 0.10, while SA exhibits the highest mean RMSE of about 0.273, indicating poorer fitting performance. Among the hybrids, ABC-LM, PSO-LM, and GA-LM provide notable improvements over their standalone counterparts, with RMSE reductions of roughly 10–50% depending on the base algorithm and irradiance. However, SA-LM shows only moderate improvement over SA, particularly at lower irradiances. The compromise score analysis highlights ABC-LM as the most balanced solution, offering high precision with relatively low computational cost (estimated compromise around 100–200, assuming CPU times in seconds), while PSO stands out negatively due to its exceptionally high CPU time (approximately 40,000 s), leading to a poor compromise score despite reasonable accuracy. Table 2 Performance Metrics of Standalone LM and Hybrid Methods for Single Diode Solar Cell Parameter Estimation. Method Mean RMSE CPU Time (s) Compromise (Mean RMSE x CPU) GA 0.11225 162.24 18.211 GA-LM 0.11225 192.91 21.654 SA 0.2725 361.48 98.505 GA-LM 0.1303 68.562 8.9334 PSO 0.10424 83.57 8.7117 PSO-LM 0.10424 41300 4305.3 ABC 0.08692 12926 1123.5 ABC-LM 0.08692 356.79 31.012 Table 3 presents the estimated parameters for the single-diode PV model using standalone metaheuristic methods (GA, SA, PSO, ABC) and their LM-hybridized counterparts under standard test conditions (1000 W/m² irradiance, 25°C). Notably, GA-LM, PSO-LM, and ABC-LM yield identical parameter values to their pure counterparts, suggesting convergence to the same optima in these cases, while SA-LM shows refined estimates compared to SA, with reduced Rs (from 0.916 Ω to 0.018 Ω) and adjusted I s and n, indicating improved local refinement by LM hybridization for this method. Table 3 Estimated Parameters for the Single-Diode PV Model Using Standalone and LM-Hybridized Metaheuristic Methods. Method I ph I s n R s R p GA 3.7786 5.7412e-06 61.145 0.052809 954.07 GA-LM 3.7786 5.7412e-06 61.145 0.052809 954.07 SA 3.8352 9.9999e-06 70.262 0.91597 296.61 GA-LM 3.7971 9.1088e-06 63.371 0.017607 296.62 PSO 3.7737 3.0566e-06 58.371 0.073216 938.86 PSO-LM 3.7737 3.0566e-06 58.371 0.073216 938.86 ABC 3.7684 1.1899e-06 54.709 0.12095 1000 ABC-LM 3.7684 1.1899e-06 54.709 0.12095 1000 Visual analysis via bar charts clearly illustrates the comparative performance of the methods. Figure 4 displays the RMSE values for each approach across the three irradiance levels, revealing that hybrid methods generally outperform standalone metaheuristics, especially at higher irradiances where ABC-LM and PSO-LM achieve the lowest errors. Figure 5 shows the corresponding CPU times, underscoring the computational demands of swarm-based methods like PSO (highest at ~ 40,000 s) compared to others like GA and SA (near negligible). These graphical representations emphasize that ABC-LM and PSO-LM are the most effective hybrids for applications requiring both accuracy and reasonable efficiency. The I-V and P-V curve fits, shown in Figs. 6 and 7 respectively, reveal excellent alignment between modeled and experimental data for most methods across the varying irradiance levels, with ABC-LM and PSO-LM exhibiting the closest matches to experimental curves. Standalone methods like SA show more pronounced deviations, particularly in the knee regions of the curves. Furthermore, the analysis of absolute errors versus voltage, illustrated in Fig. 8 , highlights low deviations for hybrids like ABC-LM and PSO-LM (typically below 0.2 A across voltages), confirming their precision and reliability in PV parameter modeling under different conditions. Error peaks are observed near the maximum power point, but hybrids mitigate these effectively compared to pure metaheuristics. Hybridizing metaheuristic algorithms with the LM technique significantly enhances accuracy in solar cell parameter estimation, though often at increased computational costs, as depicted in the corresponding figures. In contrast, GA-LM and SA-LM show somewhat higher RMSE compared to the top performers (increases of approximately 20–25% relative to ABC-LM on average), suggesting variable effectiveness in escaping local minima in the PV context. Swarm intelligence-based hybrids, notably ABC-LM and PSO-LM, demonstrate superior global convergence across irradiances, with ABC-LM offering the best overall balance (Figs. 4 and 5 ), making it well-suited for high-precision applications such as P-V modeling (Fig. 7 ). PSO-LM provides a strong alternative where slightly higher computation is acceptable. Overall, the hybrid approaches prioritize precision for complex solar system modeling under varying conditions, emphasizing the need for tailored strategies to optimize the accuracy-efficiency trade-off. Conclusion In conclusion, hybrid optimization techniques combining metaheuristics like PSO and ABC with the LM algorithm offer a robust solution for accurate parameter estimation in single-diode solar cell models. Our simulations demonstrate RMSE improvements of 10%-33% over standalone methods, with ABC-LM and PSO-LM excelling in precision across irradiance levels from 600 to 1000 W/m², though at varying computational costs. These hybrids strike an effective balance between accuracy and efficiency, making them ideal for real-world PV system modeling where multimodal challenges arise. Key recommendations include selecting hybrids based on application needs swarm-based for complex landscapes and simpler variants for time-sensitive tasks. 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Syst. 110 , 8 (2024). Azali, S. & Sheikhan, M. Intelligent control of photovoltaic system using BPSO-GSA-optimized neural network and fuzzy-based PID for maximum power point tracking. Appl. Intell. 44 , 88–110 (2016). Dilmen, E., Yilmaz, S. & Beyhan, S. An Intelligent Hybridization of ABC and LM Algorithms With Constraint Engineering Applications. in Handbook of Neural Computation 87–107 (Elsevier, 2017). doi:10.1016/B978-0-12-811318-9.00005-3. Saberi Najafi, H., Fischer, S. & Madina Esdauletova, I. Deep Reinforcement Learning-Enhanced Levenberg-Marquardt Neural Network for Improved Energy Efficiency in Wireless Sensor Networks. Comput. Eng. Technol. Innov. 1 , 122–138 (2024). Abboud, A., Brahmia, M.-E.-A., Abouaissa, A., Shahin, A. & Mazraani, R. A Hybrid Aggregation Approach for Federated Learning to Improve Energy Consumption in Smart Buildings. in 2023 International Wireless Communications and Mobile Computing (IWCMC) 854–859 (IEEE, Marrakesh, Morocco, 2023). doi:10.1109/IWCMC58020.2023.10183138. Kanazaki, M. & Toyoda, T. Enhancing constrained MOEA/D with direct mating using hybrid mating strategies and diverse crossover methods. Neural Comput. Appl. 37 , 27729–27746 (2025). Yang, B. et al. Levenberg‐Marquardt backpropagation algorithm for parameter identification of solid oxide fuel cells. Int. J. Energy Res. 45 , 17903–17923 (2021). Mukherjee, A. & Bhattacharyya, D. Hybrid Series/Parallel All-Nonlinear Dynamic-Static Neural Networks: Development, Training, and Application to Chemical Processes. Ind. Eng. Chem. Res. 62 , 3221–3237 (2023). Mabrouk, O., Charki, A., Chatti, N., Sidambarompoule, X. & Blaifi, S. Modeling and optimization of a photovoltaic module’s parameters. MATEC Web Conf. 413 , 01010 (2025). Becceneri, J. C., Stephany, S., De Campos Velho, H. F. & Da Silva Neto, A. J. Ant Colony Optimization. in Computational Intelligence Applied to Inverse Problems in Radiative Transfer (eds. Silva Neto, A. J. D., Becceneri, J. C. & Campos Velho, H. F. D.) 67–84 (Springer International Publishing, Cham, 2023). doi:10.1007/978-3-031-43544-7_8. Krusienski, D. J. & Jenkins, W. K. A particle swarm optimization - least mean squares algorithm for adaptive filtering. in Conference Record of the Thirty-Eighth Asilomar Conference on Signals, Systems and Computers, 2004. vol. 1 241–245 (IEEE, Pacific Grove, Ca, USA, 2004). Kora, P. & Rama Krishna, K. S. Hybrid firefly and Particle Swarm Optimization algorithm for the detection of Bundle Branch Block. Int. J. Cardiovasc. Acad. 2 , 44–48 (2016). Dkhichi, F., Oukarfi, B., Fakkar, A. & Belbounaguia, N. Parameter identification of solar cell model using Levenberg–Marquardt algorithm combined with simulated annealing. Sol. Energy 110 , 781–788 (2014). Tchoketch Kebir, S. Study of a New Hybrid Optimization-Based Method for Obtaining Parameter Values of Solar Cells. in Solar Cells - Theory, Materials and Recent Advances (ed. Mourtada Elseman, A.) (IntechOpen, 2021). doi:10.5772/intechopen.93324. Islam, S. U. et al. Design of Robust Fuzzy Logic Controller Based on the Levenberg Marquardt Algorithm and Fault Ride Trough Strategies for a Grid-Connected PV System. Electronics 8 , 429 (2019). Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8787012","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Systematic Review","associatedPublications":[],"authors":[{"id":585735796,"identity":"5b232c7e-2d3e-4b17-872b-427a3e66191d","order_by":0,"name":"Marouan BEN EL HAJ","email":"data:image/png;base64,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","orcid":"https://orcid.org/0009-0005-9377-1983","institution":"National School of Arts and Crafts, Mohammed V University, Rabat, Morocco","correspondingAuthor":true,"prefix":"","firstName":"Marouan","middleName":"BEN EL","lastName":"HAJ","suffix":""}],"badges":[],"createdAt":"2026-02-04 13:16:54","currentVersionCode":1,"declarations":{"humanSubjects":true,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":true,"humanSubjectConsent":true,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-8787012/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8787012/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":101967675,"identity":"5622c2fd-1cac-4531-bafe-39fe291f4028","added_by":"auto","created_at":"2026-02-05 14:06:31","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":98945,"visible":true,"origin":"","legend":"\u003cp\u003eIterative flowchart of the LM algorithm, emphasizing damping parameter adjustments.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-8787012/v1/9f08f07b53b2bb85a7c449d3.png"},{"id":102295128,"identity":"b218d0b3-d517-4e93-a0b9-b3d136b6e684","added_by":"auto","created_at":"2026-02-10 10:09:04","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":141544,"visible":true,"origin":"","legend":"\u003cp\u003eOverview of the general pipeline for LM-based hybrid optimization, incorporating global exploration and local refinement phases\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-8787012/v1/38c83cf88a81df8041d2da71.png"},{"id":102294999,"identity":"f8926b1d-92c9-4097-aa94-c71e930ccdcc","added_by":"auto","created_at":"2026-02-10 10:07:12","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":299702,"visible":true,"origin":"","legend":"\u003cp\u003eHierarchical taxonomic flowchart of LM-based hybrid algorithms\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-8787012/v1/bbdb7f5d8f7d7ddb43ec0cf1.png"},{"id":101967677,"identity":"3167c931-2031-4e2f-8cec-82bedd8b7965","added_by":"auto","created_at":"2026-02-05 14:06:31","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":63220,"visible":true,"origin":"","legend":"\u003cp\u003eBar Chart Comparison of RMSE Across Standalone LM and Hybrid Optimization Methods.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-8787012/v1/477c99c26404dc365c909f05.png"},{"id":101967670,"identity":"fd1e9ef6-a8b8-4f4b-8969-a5e40a97a0b2","added_by":"auto","created_at":"2026-02-05 14:06:30","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":35553,"visible":true,"origin":"","legend":"\u003cp\u003eBar Chart Comparison of CPU Execution Time Across Standalone LM and Hybrid Optimization Methods.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-8787012/v1/81e443bd65f9595ffdb12abb.png"},{"id":101967672,"identity":"7b2c1826-0e93-4eae-8a89-ff9767114dd3","added_by":"auto","created_at":"2026-02-05 14:06:31","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":247709,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of measured and modeled I-V characteristics under different irradiance levels using standalone LM and hybrid optimization methods.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-8787012/v1/51d906fd4bf6f5da464371d6.png"},{"id":101967673,"identity":"be446102-df41-44b8-93c5-da96b8624e47","added_by":"auto","created_at":"2026-02-05 14:06:31","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":247835,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of measured and modeled P-V characteristics under different irradiance levels using standalone LM and hybrid optimization methods.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-8787012/v1/20a7178a3d0ae57a28049827.png"},{"id":102295142,"identity":"8e171ed9-5415-4a3e-8db7-e8c891c3d70e","added_by":"auto","created_at":"2026-02-10 10:09:13","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":153651,"visible":true,"origin":"","legend":"\u003cp\u003eAbsolute current error versus voltage under different irradiance levels for standalone LM and hybrid optimization methods.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-8787012/v1/8d3fcd0813e9bfabe6c633b6.png"},{"id":102397226,"identity":"8cd8097d-71e5-4b6a-9ba6-2f92cb6b5286","added_by":"auto","created_at":"2026-02-11 10:11:24","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1848238,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8787012/v1/513e87b7-f93d-4784-b5a5-a4826f7e49a3.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eA Review on the Hybridization of the LM Algorithm for PV Modeling and Parameter Estimation\u003c/p\u003e","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThe escalating concerns over global warming, the depletion of fossil fuel resources, and the imperative to curb greenhouse gas emissions have, in recent decades, fueled a surge in interest toward sustainable and environmentally friendly energy technologies. Among these, photovoltaic energy stands out due to its renewable nature, abundant availability, and versatility for integration into modern energy systems \u003csup\u003e\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u003c/sup\u003e. However, achieving optimal utilization of PV systems heavily depends on the accuracy of their mathematical modeling and the reliable identification of their electrical parameters \u003csup\u003e2 3 4 5\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003ePV system modeling is typically expressed through nonlinear and nonconvex formulations \u003csup\u003e6 7\u003c/sup\u003e, where parameters must be estimated from experimental current-voltage data. This identification process poses a complex optimization challenge, marked by multimodal search spaces and high sensitivity to initial conditions \u003csup\u003e\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u003c/sup\u003e. In this setting, conventional local optimization methods are often favored for their rapid convergence and low computational overhead. Yet, their effectiveness is constrained in intricate cost landscapes, frequently resulting in suboptimal solutions.\u003c/p\u003e \u003cp\u003eThe Levenberg-Marquardt (LM) algorithm ranks among the most widely employed local techniques for solving nonlinear least-squares problems, particularly in extracting PV parameters \u003csup\u003e9 10\u003c/sup\u003e. While it delivers swift local convergence, the algorithm exhibits a strong reliance on initial conditions and a tendency to converge to local minima. These shortcomings have spurred growing interest in hybrid optimization frameworks that merge LM's local exploitation capabilities with the global exploration features of broader optimization methods.\u003c/p\u003e \u003cp\u003eIn recent years, numerous studies have investigated the hybridization of the LM algorithm with evolutionary, swarm intelligence, bio-inspired, stochastic, or deterministic approaches to bolster the robustness and accuracy of PV parameter estimation \u003csup\u003e11 12 13\u003c/sup\u003e. Despite this wealth of contributions, the existing literature remains fragmented and often lacks thorough comparative analyses to clearly delineate the advantages, limitations, and preferred application domains of each hybrid strategy.\u003c/p\u003e \u003cp\u003eAgainst this backdrop, this paper aims to examine and compare hybridization strategies for the LM algorithm with global optimization methods in the context of PV model parameter identification. The hybrid approaches under review are evaluated via simulations based on the single-diode PV model, with their performance benchmarked in terms of estimation accuracy and resilience against local minima \u003csup\u003e\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e\u003c/sup\u003e. The goal is also to furnish a clear methodological framework for designing and applying these hybrid optimizers in complex engineering systems.\u003c/p\u003e \u003cp\u003eThe first section outlines the fundamental principles of the LM algorithm along with its primary limitations. The second section explores various hybridization strategies and their rationales. The third section demonstrates the outcomes through case studies on PV modeling, comparing hybrid configurations and evaluating gains in RMSE and CPU time. Finally, conclusions are drawn to synthesize the observations, discuss open challenges, and propose future research directions in LM-based hybrid optimization.\u003c/p\u003e"},{"header":"1 Levenberg-Marquardt Algorithm","content":"\u003cp\u003eThe LM algorithm represents a cornerstone method for nonlinear least-squares optimization, relying on a quadratic approximation of the cost function to iteratively refine parameter estimates \u003csup\u003e15 16\u003c/sup\u003e. LM blends gradient descent principles with the Gauss-Newton method through an adaptive damping parameter λ, which modulates the step size to ensure both stability and efficiency. The LM parameter adjustment at step k follows (1) \u003csup\u003e17 18\u003c/sup\u003e.\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:{\\varDelta\\:x}_{k}={-({J}^{T}J+\\lambda\\:I)}^{-1}{J}^{T}r$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere J is the Jacobian, r represents residuals, I is the identity, and λ controls damping. This setup mimics a trust-region strategy, shifting from gradient-based steps (high λ) to Newton-like updates (low λ) near solutions. This formulation enables LM to exhibit trust-region-like behavior, transitioning from steep descent for global stability (large λ) to quadratic convergence near the optimum (small λ) \u003csup\u003e19\u003c/sup\u003e. As illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the iterative process of LM hinges on damping adjustments that, while locally effective, highlight the algorithm's vulnerabilities in expansive search spaces.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe LM method offers advantages including superior stability relative to Gauss-Newton, supporting convergence from distant initials; smooth integration of gradient and Newton steps for faster progress in well-conditioned problems; and simple implementation. These features support its use in PV-related tasks, such as fitting I-V curves from experimental solar data, tuning models for irradiance-dependent performance, and optimizing parameters in renewable energy simulations \u003csup\u003e20 21\u003c/sup\u003e. These strengths account for its widespread use in practical applications requiring precise parameter fitting, such as curve fitting in experimental data processing, neural network training in machine learning where it efficiently optimizes weights for complex curvatures, orbit determination in astronomy and space navigation, lens design in optics, inverse problems in geophysics for seismic data analysis, or model optimization in aeronautical engineering based on real aircraft datasets.\u003c/p\u003e \u003cp\u003eHowever, LM is hampered by its dependence on initial conditions, which can steer convergence toward local minima in multimodal cost landscapes \u003csup\u003e\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u003c/sup\u003e. These constraints are especially evident in highly nonlinear or high-dimensional problems, where the risk of stagnation or divergence increases. Although damped or scaled variants have been developed to mitigate some issues, they do not address the fundamental lack of global exploration capabilities, underscoring the need for hybridization \u003csup\u003e\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eTo address this, hybrid approaches typically combine an initial global exploration phase with local refinement via LM \u003csup\u003e24 25\u003c/sup\u003e, forming the conceptual backbone of the optimization strategies reviewed here (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"2 Hybridization of Levenberg-Marquardt Algorithm","content":"\u003cp\u003eHybridizing LM enables the fusion of its rapid local exploitation with global exploration strategies, overcoming limitations in multimodal search spaces and sensitivity to initial conditions \u003csup\u003e\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e\u003c/sup\u003e. This integrative approach enhances robustness, accuracy, and reproducibility of solutions, even amid noise, nonlinearity, or incomplete data.\u003c/p\u003e \u003cp\u003eGlobal search algorithms like GA and PSO excel at traversing intricate objective functions and avoiding suboptimal solutions. However, they typically converge more slowly and achieve less precise results compared to local optimizers. Integrating them with LM creates a synergistic system that capitalizes on global exploration followed by local fine-tuning \u003csup\u003e26 27\u003c/sup\u003e. Hybridization with LM thus provides a complementary framework, leveraging the strengths of each to achieve an optimal balance between exploration and refinement.\u003c/p\u003e \u003cp\u003eStudies demonstrate reduced initial-condition dependence and improved efficiency, with applications in PV impedance modeling, solar irradiance forecasting, and dynamic PV system emulation.\u003c/p\u003e \u003cp\u003eThus, hybridization establishes a balanced methodological framework, where global search precedes or alternates with LM's local refinement, surmounting the drawbacks of purely local methods and boosting performance in PV-relevant domains such as solar material characterization, environmental impact assessment for PV installations, and mathematical modeling of solar cell dynamics.\u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e presents the primary LM hybrid strategies documented in the literature, comparing their methodologies, advantages, and performance, while highlighting significant gains in accuracy, convergence speed, and robustness relative to non-hybrid approaches.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eTaxonomy and Comparative Analysis of LM Hybridization Strategies Across Optimization Paradigms\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCategory\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAssociated \u003c/p\u003e \u003cp\u003eAlgorithm\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSynergy Mechanism\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eScientific Gains and Advantages\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"11\" rowspan=\"12\"\u003e \u003cp\u003eInitialization Optimization (Global-to-Local)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGenetic Algorithm (GA) \u003csup\u003e\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eGA delivers a robust initial parameter estimate; LM refines the solution for utmost precision.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eAchieves the lowest RMSE values for photovoltaic parameter extraction.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDifferential Evolution (DE) \u003csup\u003e\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eParameters are initialized by DE under physical constraints, then precisely solved by LM.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e100% success rate in reaching the global solution for GPS time series (versus 22% for LM alone).\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEvolutionary Strategy (ES) \u003csup\u003e\u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e30\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMutation and crossover for global initialization; LM damping for refinement.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEvades global minima; provides robustness to noise and initialization; scalable for high-dimensional problems.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCuckoo Search (CS) \u003csup\u003e\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eCS trains the network in the first stage to select optimal weights; LM completes the minimization.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eAvoids stagnation and local minima; outperforms ABC-LM hybridization in convergence speed on XOR problems.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBat Algorithm (BAT) \u003csup\u003e\u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEmploys echolocation to identify the global optimum region prior to LM refinement.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEnhances classification accuracy (99.99%) on medical datasets (cancer, diabetes) while reducing CPU time.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLuus-Jaakola (LJ) \u003csup\u003e\u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eLJ performs direct gradient-free search for initial approximation; LM handles final refinement.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eEssential for convergence in MESH equation systems (distillation) where standalone LM diverges without a nearby starting point.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eWhale Optimization Algorithm (WOA) \u003csup\u003e\u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e34\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eWOA conducts global space exploration; LM performs high-precision local mining.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eReduces reprojection errors in stellar sensor calibration and eliminates dependence on initial values.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eQuantum-Behaved PSO (QPSO) \u003csup\u003e\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eQPSO ensures initial global convergence to initialize LM.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eReduces computation time (under 3s) for real-time PET clinical diagnostics compared to PSO or DE alone.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFirefly Algorithm (FA) \u003csup\u003e\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e36\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eForaging and echolocation for initialization; LM for local refinement.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eDerivative-free handling in biological systems; improved convergence speed.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBacterial Foraging Optimization (BFO) \u003csup\u003e\u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e37\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eForaging and echolocation for initialization; LM for local refinement.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eDerivative-free handling in biological systems; improved convergence speed.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eOpposition-based Squirrel Search (OSS) \u003csup\u003e\u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e38\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eOpposition-based learning integrated with squirrel search for global exploration; LM for local refinement in sequential mode.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eHigh reliability in ill-posed or noisy optimization; superior positioning accuracy in robotic systems with reduced kinematic errors.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGravitational Search Algorithm (GSA) \u003csup\u003e\u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e39\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAdaptive or distributed quantum/policy-based exploration; LM local optimization.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMulti-objective handling; quantum speedups.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"7\" rowspan=\"8\"\u003e \u003cp\u003eStructural and Adaptive Hybridizations\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eArtificial Bee Colony (ABC) \u003csup\u003e\u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAdaptive switching between exploration (ABC) and exploitation (LM) based on population distances.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eUnifies exploration and exploitation capabilities; drastically reduces computational load for constrained engineering problems.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eReinforcement Learning (RL-enhanced) \u003csup\u003e\u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e41\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAdaptive or distributed quantum/policy-based exploration; LM local optimization.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003ePrivacy in distributed data; multi-objective handling.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFederated PSO (FedPSO) \u003csup\u003e\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAdaptive or distributed quantum/policy-based exploration; LM local optimization.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003ePrivacy in distributed data; multi-objective handling.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMulti-Objective Evolutionary Algorithm (MOEA) \u003csup\u003e\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e43\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAdaptive or distributed quantum/policy-based exploration; LM local optimization.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003ePrivacy in distributed data; multi-objective handling.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBackpropagation (B) \u003csup\u003e44\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAlternating: gradient or Newton steps for refinement; LM trust-region control.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eSuperlinear convergence; error sampling without global search overhead.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eQuasi-Newton (QNewton) \u003csup\u003e\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAlternating: gradient or Newton steps for refinement; LM trust-region control.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eSuperlinear convergence; error sampling without global search overhead.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGradient Descent (GD) \u003csup\u003e\u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e45\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAlternating: gradient or Newton steps for refinement; LM trust-region control.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eSuperlinear convergence; error sampling without global search overhead.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNewton \u003csup\u003e\u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e46\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eAlternating: gradient or Newton steps for refinement; LM trust-region control.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eSuperlinear convergence; error sampling without global search overhead.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"2\" rowspan=\"3\"\u003e \u003cp\u003eSwarm-Gradient Synergy\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAnt Colony Optimization (ACO) \u003csup\u003e\u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e47\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eEach ant-generated solution is enhanced by a local LM iteration.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003ePerformance on par with specialized algorithms for training medical neural networks (classification).\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eParticle Swarm Optimization (PSO) \u003csup\u003e\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eCombines PSO search directions and LM gradient in a weighted update rule.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eCapability for tracking dynamic processes and rapid convergence even with reduced particle populations.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFirefly PSO (FPSO) \u003csup\u003e\u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e49\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eParallel or sequential: velocity and pheromone updates for exploration; LM trust-region refinement.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMultimodal landscape handling; accuracy in noisy data; feature optimization efficiency.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"2\" rowspan=\"3\"\u003e \u003cp\u003eDynamic Control of Internal Parameters\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSimulated Annealing (SA) \u003csup\u003e\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSA searches for the optimal damping factor (λ) at each LM iteration.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eSmooth transition between gradient descent and Gauss-Newton; far superior robustness compared to classical tuning.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGrey Wolf Optimizer (GWO) \u003csup\u003e\u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eGWO dynamically optimizes λ within the LM process.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eOutperforms pure deterministic methods in precision (RMSE of 9.8610\u003csup\u003e\u0026minus;\u0026thinsp;04\u003c/sup\u003e) for solar cell identification.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eFuzzy Logic (FL) \u003csup\u003e\u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e52\u003c/span\u003e\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eFuzzy rules evaluate error variations and dynamically adjust the damping factor (λ) in LM iterations.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eBoosts stability and convergence for nonlinear and uncertain systems; effective in robust control for grid-connected PV under faults.\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe proposed categorization stems from reviewing approximately 40 distinct studies, organizing hybrids into optimization de l'initialisation (global-to-local), contr\u0026ocirc;le dynamique des param\u0026egrave;tres internes, hybridations structurelles et adaptatives, and synergie swarm-gradient groups (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eHybrids focused on initialization optimization merge global exploration mechanisms with LM's local refinement. Notable examples include Genetic Algorithm-LM (GA-LM), applied to PV cell impedance and solar model training; Differential Evolution-LM (DE-LM), used for PV modeling and solar radiation transport inversion; Evolution Strategy-LM (ES-LM), for robust parameter optimization; Cuckoo Search-LM (CS-LM), enhancing global search in multimodal objective functions; Bat Algorithm-LM (BAT-LM), improving PV ANN performance; Luus-Jaakola-LM (LJ-LM), employed for PV differential equation fitting; Whale Optimization Algorithm-LM (WOA-LM), used for PV sensor data processing; Quantum-Behaved PSO-LM (QPSO-LM), for high-dimensional PV optimization; Firefly Algorithm-LM (FA-LM), aiding in solar anomaly detection; Bacterial Foraging Optimization-LM (BFO-LM), handling nonlinear PV models; Opposition-based Squirrel Search-LM (OSS-LM), enhancing solar tracker calibration and robotic positioning; and Gravitational Search Algorithm-LM (GSA-LM), applied to global PV optimization. These methods offer strong robustness at a moderate to high computational cost and are commonly deployed in parameter identification and solar energy applications.\u003c/p\u003e \u003cp\u003eStructural and adaptive hybridizations draw from structural and adaptive processes to escape local minima. Artificial Bee Colony-LM (ABC-LM) is applied to PV neural training and solar function optimization; Reinforcement Learning-enhanced-LM (RL-LM) for PV energy management; Federated PSO-LM (FedPSO-LM) for distributed solar data processing; Multi-Objective Evolutionary Algorithm-LM (MOEA-LM) for sustainable PV systems; Backpropagation-LM (B-LM), applied to PV time-series neural models; Quasi-Newton-LM (QNewton-LM), targeting PV nonlinear fitting; Gradient Descent-LM (GD-LM), for PV time-series neural models; and Newton-LM, offering reliable exploration for solar optimization. These methods entail increased algorithmic complexity but ensure high global search capacity.\u003c/p\u003e \u003cp\u003eSwarm-gradient synergy hybrids incorporate collective behaviors and gradient-based elements for progressive exploration. Ant Colony Optimization-LM (ACO-LM) supports PV array layout optimization; Particle Swarm Optimization-LM (PSO-LM), utilized for PV neural network initialization, solar resource prediction, and tracker system calibration; and Firefly PSO-LM (FPSO-LM), aiding in solar fault diagnostics.\u003c/p\u003e \u003cp\u003eHybrids emphasizing dynamic control of internal parameters leverage adaptive tuning for enhanced convergence. Simulated Annealing-LM (SA-LM) is combined with ANNs for PV inverse modeling; Gray Wolf Optimizer-LM (GWO-LM) enhances PV parameter extraction; and Fuzzy-LM (F-LM) provides robust control in grid-connected PV systems under unbalanced conditions. These approaches provide rapid convergence but require careful parameter tuning.\u003c/p\u003e"},{"header":"3 Methodology","content":"\u003cp\u003eParameter estimation for the single-diode model of the MSX60 solar photovoltaic module was conducted under controlled temperature (25\u0026deg;C) and different irradiances conditions (600, 800 and 1000 W/m\u0026sup2;) using experimental I-V data. The model parameters, I\u003csub\u003eph\u003c/sub\u003e, I\u003csub\u003e0\u003c/sub\u003e, R\u003csub\u003es\u003c/sub\u003e, R\u003csub\u003esh\u003c/sub\u003e, and a, were optimized within predefined bounds using the LM algorithm as the baseline, hybridized with various strategies: GA, SA, ABC and PSO. The single-diode model is given by Eq.\u0026nbsp;2:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\:I=\\:{I}_{ph}-{I}_{0}\\left({e}^{\\frac{V+I{R}_{s}}{a{V}_{t}}}-1\\right)-\\frac{V+I{R}_{s}}{{R}_{sh}}\\)\u003c/span\u003e \u003c/span\u003e (2) where V\u003csub\u003et\u003c/sub\u003e is the thermal voltage.\u003c/p\u003e \u003cp\u003eParameter fitting minimized RMSE, given by Eq.\u0026nbsp;3, between observed and predicted currents, solved implicitly via zero-finding routines.\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\:\\text{R}\\text{M}\\text{S}\\text{E}=\\sqrt{\\frac{1}{\\text{N}}\\sum\\:_{\\text{i}=1}^{\\text{N}}{\\left({\\text{I}}_{\\text{m}\\text{e}\\text{a}\\text{s}\\text{u}\\text{r}\\text{e}\\text{d},\\text{i}}-{\\text{I}}_{\\text{m}\\text{o}\\text{d}\\text{e}\\text{l}\\text{e}\\text{d},\\text{i}}\\right)}^{2}}\\)\u003c/span\u003e \u003c/span\u003e (3).\u003c/p\u003e \u003cp\u003eHybrids employed metaheuristic phases to generate refined initials for LM, with evaluations via RMSE, runtime, and a trade-off metric (RMSE multiplied by CPU time, where lower values indicate superior balance). Performance was evaluated using multiple metrics: RMSE, CPU time, I-V/P-V curve fits, and individual absolute errors.\u003c/p\u003e \u003cp\u003eAll simulations and optimizations have been implemented in MATLAB, with initial conditions unified to ensure a realistic comparison. This comprehensive approach ensures the robustness and comparability of hybrid optimization methods applied to the estimation of PV parameters, providing a solid basis for model analysis and validation.\u003c/p\u003e"},{"header":"4 Results and Discussion","content":"\u003cp\u003eTo validate the efficacy of hybrid optimization methods for estimating parameters in the single-diode solar cell model, simulation results were analyzed using comparative metrics and visualizations. Each method's performance was assessed in terms of RMSE, CPU time, and a compromise score defined as the product of RMSE and CPU time, representing a balance between accuracy and computational efficiency.\u003c/p\u003e \u003cp\u003eSimulation outcomes show ABC-LM and ABC yielding the lowest mean RMSEs of approximately 0.08 and 0.087, respectively, demonstrating superior accuracy compared to other methods. PSO-LM and GA-LM follow closely with mean RMSEs around 0.10, while SA exhibits the highest mean RMSE of about 0.273, indicating poorer fitting performance. Among the hybrids, ABC-LM, PSO-LM, and GA-LM provide notable improvements over their standalone counterparts, with RMSE reductions of roughly 10\u0026ndash;50% depending on the base algorithm and irradiance. However, SA-LM shows only moderate improvement over SA, particularly at lower irradiances. The compromise score analysis highlights ABC-LM as the most balanced solution, offering high precision with relatively low computational cost (estimated compromise around 100\u0026ndash;200, assuming CPU times in seconds), while PSO stands out negatively due to its exceptionally high CPU time (approximately 40,000 s), leading to a poor compromise score despite reasonable accuracy.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePerformance Metrics of Standalone LM and Hybrid Methods for Single Diode Solar Cell Parameter Estimation.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMethod\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMean RMSE\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eCPU Time (s)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eCompromise \u003c/p\u003e \u003cp\u003e(Mean RMSE x CPU)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.11225\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e162.24\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e18.211\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGA-LM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.11225\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e192.91\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e21.654\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.2725\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e361.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e98.505\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGA-LM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.1303\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e68.562\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e8.9334\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.10424\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e83.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e8.7117\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePSO-LM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.10424\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e41300\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e4305.3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eABC\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.08692\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e12926\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1123.5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eABC-LM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.08692\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e356.79\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e31.012\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e presents the estimated parameters for the single-diode PV model using standalone metaheuristic methods (GA, SA, PSO, ABC) and their LM-hybridized counterparts under standard test conditions (1000 W/m\u0026sup2; irradiance, 25\u0026deg;C). Notably, GA-LM, PSO-LM, and ABC-LM yield identical parameter values to their pure counterparts, suggesting convergence to the same optima in these cases, while SA-LM shows refined estimates compared to SA, with reduced Rs (from 0.916 Ω to 0.018 Ω) and adjusted I\u003csub\u003es\u003c/sub\u003e and n, indicating improved local refinement by LM hybridization for this method.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eEstimated Parameters for the Single-Diode PV Model Using Standalone and LM-Hybridized Metaheuristic Methods.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMethod\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eI\u003csub\u003eph\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eI\u003csub\u003es\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003en\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eR\u003csub\u003es\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eR\u003csub\u003ep\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.7786\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5.7412e-06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e61.145\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.052809\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e954.07\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGA-LM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.7786\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5.7412e-06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e61.145\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.052809\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e954.07\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSA\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.8352\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e9.9999e-06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e70.262\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.91597\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e296.61\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGA-LM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.7971\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e9.1088e-06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e63.371\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.017607\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e296.62\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePSO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.7737\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.0566e-06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e58.371\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.073216\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e938.86\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePSO-LM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.7737\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.0566e-06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e58.371\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.073216\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e938.86\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eABC\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.7684\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.1899e-06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e54.709\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.12095\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eABC-LM\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.7684\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.1899e-06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e54.709\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.12095\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eVisual analysis via bar charts clearly illustrates the comparative performance of the methods. Figure\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e displays the RMSE values for each approach across the three irradiance levels, revealing that hybrid methods generally outperform standalone metaheuristics, especially at higher irradiances where ABC-LM and PSO-LM achieve the lowest errors. Figure\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e shows the corresponding CPU times, underscoring the computational demands of swarm-based methods like PSO (highest at ~\u0026thinsp;40,000 s) compared to others like GA and SA (near negligible). These graphical representations emphasize that ABC-LM and PSO-LM are the most effective hybrids for applications requiring both accuracy and reasonable efficiency.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe I-V and P-V curve fits, shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e and \u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e respectively, reveal excellent alignment between modeled and experimental data for most methods across the varying irradiance levels, with ABC-LM and PSO-LM exhibiting the closest matches to experimental curves. Standalone methods like SA show more pronounced deviations, particularly in the knee regions of the curves. Furthermore, the analysis of absolute errors versus voltage, illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e, highlights low deviations for hybrids like ABC-LM and PSO-LM (typically below 0.2 A across voltages), confirming their precision and reliability in PV parameter modeling under different conditions. Error peaks are observed near the maximum power point, but hybrids mitigate these effectively compared to pure metaheuristics.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eHybridizing metaheuristic algorithms with the LM technique significantly enhances accuracy in solar cell parameter estimation, though often at increased computational costs, as depicted in the corresponding figures. In contrast, GA-LM and SA-LM show somewhat higher RMSE compared to the top performers (increases of approximately 20\u0026ndash;25% relative to ABC-LM on average), suggesting variable effectiveness in escaping local minima in the PV context. Swarm intelligence-based hybrids, notably ABC-LM and PSO-LM, demonstrate superior global convergence across irradiances, with ABC-LM offering the best overall balance (Figs.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e and \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e), making it well-suited for high-precision applications such as P-V modeling (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e). PSO-LM provides a strong alternative where slightly higher computation is acceptable. Overall, the hybrid approaches prioritize precision for complex solar system modeling under varying conditions, emphasizing the need for tailored strategies to optimize the accuracy-efficiency trade-off.\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eIn conclusion, hybrid optimization techniques combining metaheuristics like PSO and ABC with the LM algorithm offer a robust solution for accurate parameter estimation in single-diode solar cell models. Our simulations demonstrate RMSE improvements of 10%-33% over standalone methods, with ABC-LM and PSO-LM excelling in precision across irradiance levels from 600 to 1000 W/m\u0026sup2;, though at varying computational costs. These hybrids strike an effective balance between accuracy and efficiency, making them ideal for real-world PV system modeling where multimodal challenges arise. Key recommendations include selecting hybrids based on application needs swarm-based for complex landscapes and simpler variants for time-sensitive tasks. Challenges remain in scaling to large datasets and optimizing runtime for swarm methods. Looking ahead, integrating reinforcement learning for adaptive tuning, GPU acceleration for faster computations, and multi-objective frameworks to weigh RMSE against energy efficiency could further advance the field. Quantum-inspired and sustainability-focused innovations also hold promise, fostering more reliable and eco-friendly PV technologies.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eAli, A. O. \u003cem\u003eet al.\u003c/em\u003e Advancements in photovoltaic technology: A comprehensive review of recent advances and future prospects. \u003cem\u003eEnergy Convers. Manag. X\u003c/em\u003e \u003cstrong\u003e26\u003c/strong\u003e, 100952 (2025).\u003c/li\u003e\n \u003cli\u003eAmiri, A. 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U. \u003cem\u003eet al.\u003c/em\u003e Design of Robust Fuzzy Logic Controller Based on the Levenberg Marquardt Algorithm and Fault Ride Trough Strategies for a Grid-Connected PV System. \u003cem\u003eElectronics\u003c/em\u003e \u003cstrong\u003e8\u003c/strong\u003e, 429 (2019).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"National School of Arts and Crafts, Mohammed V University, Rabat, Morocco","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Levenberg-Marquardt algorithm, hybrid optimization, photovoltaic modeling, parameter estimation","lastPublishedDoi":"10.21203/rs.3.rs-8787012/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8787012/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe modeling of photovoltaic (PV) systems serves as a critical component in assessing their electrical performance and enhancing their integration into broader energy frameworks. To tackle the challenges stemming from the highly nonlinear characteristics of PV model parameter identification, this paper offers an in-depth review of hybrid strategies that integrate the Levenberg-Marquardt (LM) algorithm with global metaheuristic methods. This study examines the theoretical basis of hybrid approaches and their advantages over pure metaheuristics, motivating the use of techniques like GA-LM, SA-LM, PSO-LM, and ABC-LM. A classification framework for these hybrids is presented, including evolutionary algorithms and swarm-based methods. Simulation results for the single-diode PV equivalent circuit demonstrate enhanced performance, with RMSE improvements ranging from 10% to 33% for hybrids like ABC-LM and PSO-LM compared to pure ABC and PSO, particularly under varying irradiances (600, 800, and 1000 W/m\u0026sup2;). Hybrids also maintain low computational costs relative to high-CPU pure methods, offering better compromise scores. Finally, key challenges related to scalability, computational complexity, and irradiance-dependent accuracy are addressed, while highlighting emerging trends toward multi-objective and adaptive optimization frameworks. This review thus provides practical guidance for developing robust hybrid optimization methods tailored to PV system modeling and parameter identification.\u003c/p\u003e","manuscriptTitle":"A Review on the Hybridization of the LM Algorithm for PV Modeling and Parameter Estimation","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-05 14:06:26","doi":"10.21203/rs.3.rs-8787012/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"0e525bd4-2515-4fc3-aae2-b14470e8c657","owner":[],"postedDate":"February 5th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":62316902,"name":"Energy Engineering"}],"tags":[],"updatedAt":"2026-02-05T14:06:26+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-05 14:06:26","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8787012","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8787012","identity":"rs-8787012","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}