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Scientific Reports volume 16, Article number: 14016 (2026) Cite this article
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This work introduces a metaheuristic (MH) optimization method, which is inspired by the red-tailed hawks’ predatory behavior is Improved Red-tailed Hawk (IRTHA) Algorithm. The algorithm uses a dynamic adjustment method which uses the combined effect of nonlinear decay and chaotic mapping to enhance the convergence efficacy and accuracy of outcomes. This enhancement affects the search radius of the algorithm and creates diversity in the dive speed of hawks, hence adaptively balancing exploration and exploitation, enhancing diversity and convergence. IRTHA’s efficacy is examined for single, double, and triple diode models of various photovoltaic (PV) cells and modules, such as RTC France, PVM 752, STP 120/36, STM 40/36, and Photowatt-PWP201. A comparative analysis of IRTHA with other advanced MH optimization techniques indicates that IRTHA exhibits considerably lower RMSE values: 7.72986E-04 for SDM-RTC France, 7.41918E-04 for DDM-RTC France, 7.34782E-04 for TDM-RTC France, 1.59243E-04 for PVM 752, 1.44508E-02 for STP 120/36, 1.72192E-03 for STM 40/36, and 2.05285E-03 for the Photowatt-PWP201 module, respectively. The reliability of IRTHA is futher validated by statistical analyses, including non-parametric tests (Friedman and Wilcoxon rank-sum tests), convergence curve assessments, and graphical representations with boxplots, which collectively confirm its potential to deliver robust and computationally efficient optimization. From the outcomes, it is observed that the IRTHA demonstrates superior performance compared to other existing MH algorithms. The results obtained by IRTHA show exceptional performance in PV system modeling and parameter estimation in solar PV applications.
In a world increasingly preoccupied with climate change, and requiring a sustainable energy option. Solar Photovoltaic (PV) systems have become a leading renewable energy option, offering a clean and sustainable alternative to traditional fossil fuel-based power generation^1,2. The PV systems convert sunlight into electrical energy and are widely used in applications ranging from off-grid setups to large-scale solar farms^3,4. The performance and efficiency of PV systems depend on the accurate modeling and characterization of the fundamental PV cells and modules^5,6,7. As the complexity of PV systems is increasing, there is an increasing need for detailed and comprehensive models. It may lead to considerable variations in the performance as well as energy generation of a PV system, affected by environmental factors like sun intensity, temperature, and shade conditions⁸. The PV module is an important component of the PV power generation. The PV module is a crucial element in PV power generation. The design of effective models and the acquisition of precise model parameters are essential for assessing and monitoring the real performance of simulated PV modules and forecasting PV power output⁹. The accurate and reliable estimation of parameters from PV models is important to improve their efficiency as well as maximize power generation.
Recently, widely adopted photovoltaic models include equivalent circuit models for single, double, and triple diode models (SDM, DDM, and TDM), which are used to determine the I-V characteristics of the PV cell. The I–V characteristics provide an in-depth depiction of the PV cell, illustrating the correlation among its output characteristics. Nevertheless, the equivalent circuit model accurately demonstrates the internal characteristics of the PV cell. The solar energy production system has been considerably affected by several external environmental conditions, including temperature and radiation intensity. Therefore, optimizing the use of solar energy for maximum efficiency and the effective implementation of photovoltaic models is essential^10,11,12.
To address the challenge of parameter estimation in photovoltaic models, different approaches have been developed, that are primarily classified into two categories: deterministic as well as MH optimization approaches. Deterministic approaches are highly sensitive to initial solutions and assume that models demonstrate properties of convexity as well as differentiability. But the MH approach, inspired by the biological principle of survival of the fittest, can effectively mitigate these limits with more flexibility, resulting in superior accuracy and durability^13,14,15.
Many MH algorithm optimization methods have been studied for extracting unknown parameters of the solar PV model such as modified Exponential Distribution Optimization Algorithm (mEDOA), Modified electric eel foraging optimization (MEEFO), Enhanced differential evolution (EDE), Equilibrium optimizer-single candidate optimizer (EO-SCO), Grey Wolf Election-Based Optimization algorithm (GWEBO), Multi-strategy gaining-sharing knowledge-based algorithm (MSGSK), Improved Artificial Protozoa Optimizer (iAPO), Adapted human evolutionary optimization (AHEO), Hippopotamus optimizer (HOA), Multi-strategy Nutcracker Optimization Algorithm (EMNOA), Opposition-based Learning White Shark Optimizer (IWSO), Dynamic oppositional learning strategy and Sorting Teaching-Learning-Based Optimization (DSTLBO), Improved Walrus Optimizer (m WO), Frilled Lizard Optimization (FLO), Mean Differential Evolution with Newton-Raphson (MDE-NR), Differentiated Creative Search combined with Newton-Raphson (DCS-NR), Coati Improved Snow Ablation Optimization (CSAO), Chaotic Differential Variation Snake Optimization (CDVSO), Four Vector Intelligent Metaheuristic Differential Evolution (FVIM-DE), Leveraging the opposition-based Exponential Distribution Optimizer (OBEDO), Enhanced Artificial Hummingbird Algorithm (enAHA), Improved JAYA (Sjaya), Enhanced Prairie Dog Optimizer (En-PDO), Improved Simultaneous Heat Transfer Search (ISHTS), Modified version of Mountain Gazelle Optimization (MGPS), Multi-strategy-based Tree Seed Algorithm (MS-TSA), Modified RIME (MRIME), Robust Newton–Raphson method integrated improved Differential Evolution (RoNRIDE), Bio-dynamics Grasshopper Optimization Algorithm (BDGOA), Walrus optimization algorithm (WaOA), Chaos-inspired Invasive Weed Optimization (CIIWO), Adaptive Sine Cosine Particle Swarm Optimization Algorithm (ASCA-PSO), Improved Marine Predators Algorithm and Equilibrium Optimizer (IMPAEO), Enhanced Snake algorithm (ISASO), Hybrid Chaotic Particle Swarm Optimization and Slime Mould Algorithm (HCPSOSMA), Improved Crayfish Optimization Algorithm (ICOA), Modified Bare-Bone Imperialist Competitive Algorithm (MBB-ICA), Improved Kepler Optimization Algorithm (IKOA), Developed JAYA Algorithm (DIWJAYA), Hybrid Cuckoo Search-Gorilla Troop Optimization (CS-GTO), Tiki Taka Algorithm Mean Differential Evolution based on Weibull distribution (TTA-MDEW), Self-adaptive Enhanced Learning Differential Evolution (SaELDE), Weighted Velocity-Guided Grey Wolf Optimizer (WVGGWO), Artificial Hummingbird Technique (AHT), Enhanced Sine–Cosine Algorithm (ESCA), Nutcracker optimizer algorithm (NOA), Improved Snake Optimization Algorithm (ISOA), Growth Optimization (GO), Squirrel Search Algorithm (SSA), Chaos Game Optimization-Least Squares (CGO-LS), Fractional Henon Chaotic Harris Hawks Optimization (FCHHHO), Hybrid White Shark Optimize Artificial Rabbits Optimization (hWSO-ARO), Improved Moth Flame algorithm with Local escape operators (IMFOL), Northern Goshawk Optimization (NGO), and roved Archimedes Optimization Algorithm (IAOA) are given in Table 1. A detailed summary of the latest advancements in various MH algorithms for determining unknown parameters in solar PV cells/modules is presented in Table 1. The comparison has been conducted to highlight the methodological variation, modeling reliability, author names, journal title, publication year, objective functions, additional metrics, and validation approaches. In the literature, it is clearly demonstrates the applicability of several MH optimization methods in addressing the solar PV parameter estimation challenge. The no free lunch (NFL) theory claims that no optimization algorithm is globally superior for all engineering optimization scenarios¹⁶. It is essential to consider different MH techniques and frameworks for altering and enhancing solutions according to the challenge presented. However, there is potential to enhance the present frameworks rather than developing new ones. The NFL theorem motivated researchers to develop innovative MH optimization methods or enhance current ones, facilitating their application in solving real-world problems across several domains. The literature study indicates that the identification of solar PV parameters is a current research area. Moreover, recently developed MH algorithms must be evaluated for improved modelling related to error minimization, faster convergence, and enhanced statistical metrics. As a result, many MH algorithms have been studied in the literature to address the same problem and present an opportunity for the development of new methods that offer more accurate results. In this paper, a novel improved optimization approach for parameter extraction of PV models is applied, known as the Improved Red-Tailed Hawk Algorithm (IRTHA).
The IRTHA is inspired by the predatory behavior and flight patterns of red-tailed hawks. In this study, the performance of IRTHA has been systematically compared against other established MH optimization techniques such as Horned Lizard Optimization Algorithm (HLOA)¹⁷, Pelican Optimization Algorithm (POA)¹⁸, Zebra Optimization Algorithm (ZOA)¹⁹, Hybrid Particle Swarm Optimization and Grey Wolf Optimizer (PSOGWO)²⁰, Whale Optimization Algorithm (WOA)²¹, Pelican Optimization Algorithm (POA)¹⁸, Hippopotamus Optimization (HO)²², Osprey Optimization Algorithm (OOA)²³, Harris Hawks Optimization (HHO)²⁴, Grey Wolf Optimizer (GWO)²⁵, and Coati Optimization Algorithm (COA)²⁶ to evaluate its efficacy.
The outcomes demonstrate that the IRTHA algorithm competes effectively with other MH algorithms across key measures like as convergence speed, accuracy, and robustness. These key points of the algorithm’s capability as an effective tool for parameter estimation in PV cell/models and other complicated optimization challenges. Using its advanced features, IRTHA presents a crucial step toward improving optimization methods for modern energy problems. The key contributions of the paper are summarized as follows.
An enhanced MH optimization algorithm namely the Improved Red-tailed Hawk Algorithm (IRTHA), has been applied for estimating solar PV parameters by combining the features of the RTH algorithm with the Nonlinear Decay, Chaotic Map strategy, and the Newton-Raphson (NR) Method.
The IRTHA is applied for parameter estimation of solar PV models, including RTC France (SDM, DDM, and TDM), Photowatt-PWP201, PVM-752-GaAs, and STM6 40/36 PV, STP6 120/36 panels.
The outcomes of the IRTHA algorithm have been compared with 10 advanced MH optimization techniques, including HLOA, ZOA, PSOGWO, WOA, POA, HO, OOA, HHO, GWO, and COA, as well as additional parameter estimations of solar PV techniques reported in the literature.
Furthermore, the accuracy and reliability of the IRTHA in PV parameter extraction is validated by using other statistical metrics, including Mean Absolute Error (MAE), Mean Square Error (MSE), Sum of Square Error (SSE), Individual Absolute Error (IAE), the Root Mean Square Error (RMSE), and the Friedman and Wilcoxon rank-sum test.
The results demonstrate that the IRTHA algorithm exhibits the minimal difference between the observed and estimated values. This illustrates the efficacy of the IRTHA algorithm for the parameter estimation problem of Solar PV.
This paper is structured into five subsections: Section “Mathematical modelling” presents the mathematical framework of solar PV systems, incorporating the Single, Double, and Triple Diode Model (SDM, DDM, and TDM), together with the corresponding objective function. Section “Improved red tailed hawk algorithm (IRTHA)” explains a detailed representation of the IRTHA algorithm, which includes a mathematical modeling, flowchart, and pseudocode of the algorithm. Section “Result and discussion” discusses the test results for parameter extraction of solar PV cells/modules, along with extensive validation to confirm the efficacy and robustness of the IRTHA algorithm from multiple perspectives. Finally, the last section of the paper outlines numerous conclusive outcomes, remarks, and observations, with the potential directions for future research.
The equivalent model of solar PV SDM, comprising a current source ((I_{ph})) which is connected in parallel with a diode ((D_{1})), a parallel resistor to account for leakage current ((I_{sh})), as well as a series resistor to model losses from the load current (I) is depicted in Fig. 1. According to Kirchhoff’s Current Law (KCL), the output current (I) of the SDM is calculated by utilizing the following Eq. 1^81,82.
Equations 2 and 3 present the mathematical formulations for (I_{d}) as well as (I_{sh}), respectively.
The output current I is shown in the given Eq. 4.
where (I_{sc}) denotes the reverse saturation current for SDM, the Kelvin temperature of the solar cell (T), the shunt resistance ((R_{sh})), the series resistance ((R_{s})), the charge of the electron ((q=1.60217646 times 10^{-19} , C)), the Boltzmann constant ((k=1.3806503 times 10^{-23} , J/K)), and the ideality factor of the diode (n) are used. Since current is not explicitly represented as a function of voltage in 4. A precise PV model can be constructed by extracting these parameters ((I_{ph}), (I_{sc}), (R_{sh}), (R_{s}), and n). The exact estimation of these factors directly influences the effectiveness of optimization as well as the maximum power point tracking of solar cells.
Equivalent circuit of SDM of solar PV.
Figure  2 illustrates the equivalent model for the photovoltaic double diode model (DDM). This model comprises a current source ((I_{ph})) in parallel with two diodes ((D_{1}) and ((D_{2})), a parallel resistor representing leakage current ((I_{sh})), and a series resistor to account for losses due to the load current (I). According to KCL, the output current of the DDM is given by the Eq. 5^9,83.
Equations 6 and 7 present the mathematical formulations for (I_{d1}) and (I_{d2}), respectively.
The output current I is illustrated in the given Eq. 8.
where (I_{sc1}) and (I_{sc2}) represent reverse saturation currents for DDM, and diode ideality factors ((n_{1})) and ((n_{2})), are used. A precise PV model can be constructed by extracting seven unknown parameters such as (I_{ph}), (I_{sc1}), (I_{sc2}), (R_{sh}), (R_{s}), (n_{1}), and (n_{2}).
Equivalent circuit of DDM of solar PV.
Figure 3 illustrates the configuration of the triple diode model (TDM), where three diodes ((I_{d1}), (I_{d2}), and (I_{d3})) as well as a photo-generated current source ((I_{ph})) are arranged in parallel with a shunt resistor ((R_{sh})). Mathematically, the TDM solar PV is expressed as follows^46,84,85.
(I_{sh}) and (I_{d}) can be calculated using Eqs. 10 and 11, respectively.
Finally, I can be determined using the following Eq. 12.
where (I_{sc1}), (I_{sc2}), and (I_{sc3}) are the reverse saturation currents for TDM. From Eq. 12, there are 9 unknown parameters in TDM, such as (I_{ph}), (I_{sc1}), (I_{sc2}), (I_{sc3}), (R_{s}), (R_{sh}), (n_{1}), (n_{2}), and (n_{3}). The accuracy of the TDM model can be evaluated by accurately determining these unknown parameters.
Equivalent circuit of TDM of solar PV.
The identification of solar PV parameters generally requires minimizing the error between measured PV current (I_{k,measured}) (reference) and estimated PV current (I) determined utilizing the selected model (SDM, DDM, and TDM). In this work, the root mean square error (RMSE) is adopted as the objective function. This work examines three categories of models defined by Eqs. 4, 8, and 12. Each model is associated with a distinct set of parameters. This objective function is expressed in Eq. 13^33,86,87.
Subject to:
Where C is the number of measured data samples.
The RMSE is a measure to accurately model the solar PV cell/module by comparing the values calculated with the experimental results. The imposed boundaries limit the algorithm from exploring infeasible spaces, hence preserving computational time.
The RTH algorithm draws inspiration from the hunting nature of red-tailed hawk’s⁸⁸. The IRTHA utilizes a dynamic adjustment strategy or method, which includes a hybrid approach (nonlinear decay as well as a chaotic map)⁸⁹. This method aims to achieve equilibrium between the exploration stage as well as exploitation stage, thereby enhancing the search process. Incorporating a hybrid methodology which integrates non-linear decay alongwith a chaotic map in the Transition Function Factor (TRF), the IRTHA can dynamically modify the hawks’ movement size of step. This adjustment improves the search radius of the algorithm and creates diversity in the hawk’s dive speed, thus influencing the convergence behavior of the IRTHA algorithm⁸⁹. This algorithm is made up of three different phases: high soaring phase, low soaring phase, and stooping and swooping phase.
The red-tailed hawk dives to significant elevations in search of optimal locations with abundant food resources. Equation 15 illustrates the mathematical formulation of this phase.
The red-tailed hawk’s location at iteration t is indicated by the symbol X(t). (X_{best1}) denotes the optimum location obtained, while (X_{mean1}) signifies the mean of all positions. The distribution function (LevyF) utilised in the calculations outlined with Eqs. 16 and 17, while TRF(t) signifies the transition factor function derived from Eq. 17.
Here, (p_1 = 0.01) signifies a constant valued, D specifies the problem’s dimension, (delta ‘_{01} = 1.5) is a constant set, while s and (r_1) are random numbers within the range of 0 to 1.
The hybrid approach, including nonlinear decay along with a chaotic map, is an effective approach for enhancing the RTH algorithm’s performance by modifying the transition function factor (TRF). Incorporating both nonlinear decay and a chaotic map mechanism into the TRF to adaptively control sparsity over time. This methodology proves particularly advantageous for datasets that display seasonal patterns or in instances where the ideal level of sparsity fluctuates over time. The formula for the modified TRF of IRTHA can be determined by the subsequent equation 19⁸⁹.
where (r_1) stands for a constant parameter that regulates the growth rate and ranges from 0 to 4, (T_{max}) denotes the maximum number of iterations, whereas (T_{iter}) indicates the current iteration count.
The hawk spirals downward towards their target while flying closest to the surface of the ground. T denotes a phase that can be depicted through the subsequent model.
where (SS_1(t)) signifies the step size as well as the parameters (y_{11}) as well as (z_{11}), which indicate direction coordinates, that can be determined through the subsequent equations.
where (R_0) denotes the initial radius value inside the interval of [0.5,3], (AG_1) denotes the spectrum of angel gain, which spans from 5 to 15, while (RG_1) signifies a random gain that can assume values between [0,1]. The variable (r_1) denotes a control gain, which may assume the values of 1 or 2. These variables facilitate the hawk’s movement surrounding the prey using spiral movements.
Pseudocode of IRTHA
Flowchart of IRTHA algorithm.
In this step, the hawk rapidly drops as well as strikes the target from the optimal position attained at the low soaring stage. This phase may be denoted by the subsequent equation 23.
The computation for every step size can be ascertained via Eqs. 24 and 25.
The parameters (beta _1) and (G_1) denote the acceleration and gravitational factors, correspondingly. They can be described as follows.
The symbol (beta _1) signifies the hawk’s acceleration, which rises with time to enhance the convergence speed, while (G_1) suggests the gravitational force, which reduces the exploitation diversity as the hawk or predator approaches its target. The pseudocode for IRTHA is presented in Algorithm 1. Also, the flowchart of the IRTHA algorithm is illustrated in Fig. 4.
This section evaluates the performance of the IRTHA algorithm through solar PV parameter estimation challenges. For analysis, the five common types of solar PV cells/modules, such as RTC France (SDM, DDM, and TDM), Photowatt-PWP201, PVM-752-GaAs, STM6 40/36, and STP6 120/36 PV panels, are used. The performance of the IRTHA algorithm is compared with different MH algorithms like HLOA¹⁷, ZOA¹⁹, PSOGWO²⁰, WOA²¹, POA¹⁸, HO²², OOA²³, HHO²⁴, GWO²⁵, and COA²⁶. Also, the parameter details of different MH optimization is given in the Table 2. The findings indicate that IRTHA exhibits superior accuracy, achieves faster convergence, and demonstrates computational efficiency. Table 3 indicate the search boundaries for each unidentified parameter of solar PV associated with the parameter extraction techniques^9,90,91. The assessment of the algorithm’s efficacy is conducted through standard metrics including RMSE, MAE, IAE, SSE, MSE, along with the analysis of the convergence curve. Additionally, a low RMSE value signifies that the parameters have been effectively determined, as RMSE seeks to minimize the variance between observed as well as predicted data. The statistical robustness of the obtained results is evaluated through evaluating the standard deviation, as well as the worst and mean error values across 30 independent runs, in addition to minimizing the RMSE. A non-parametric Wilcoxon rank-sum test is also conducted to validate the accuracy of IRTHA’s outcomes over the other compared algorithms. Furthermore, the box plots and convergence graphs are presented to visually highlight the stability and accuracy of the IRTHA algorithm.
All simulations were executed in MATLAB R2021a on a Windows 11 laptop featuring an Intel Core i5-1035G1 processor with 8GB of RAM. The algorithms employ a population size of 50, with a maximum of 1000 iterations for each of the five PV models. Each algorithm is carried out autonomously 30 times for every PV model.
The IRTHA algorithm is tested using a single, double, and triple diode model (SDM, DDM, and TDM) of the RTC France PV cell under standard test conditions of 33 °C, 1000 W/(hbox {m}^2). Tables 4, 5, 6, 7, 8, 9, 10, 11, and 12 present the detailed analysis of the RTC France solar cell (SDM, DDM, and TDM). Tables 4, 7, and 10 display the measured as well as estimated data points of current for SDM, DDM, and TDM, respectively. Also, the Table 4, 7, and 10 present in-depth statistical measures like MAE, MSE, RMSE, SSE, MBE, and IAE values. A comparative analysis of the efficacy of eleven optimization algorithms is shown in Table 5, 8, and 11 respectively. Also, Table 5, 8, and 11 present a detailed overview of the best, worst, mean, min, standard deviation, and optimal values achieved by each MH algorithm across 1000 iteration and 30 run for SDM, DDM, and TDM, respectively. The results of these experiments indicate that the IRTHA algorithm shows outstanding results regarding both the mean objective function value and the best objective function value when compared to other algorithms. The optimal RMSE solutions for SDM, DDM, and TDM achieved with the IRTHA algorithm are 7.72986E-04, 7.41918E-04, and 7.34782E-04. The graphical characteristics demonstrated in Figs. 5a , 6a and 7a highlights the effectiveness of the IRTHA algorithm. The measured and estimated values on the I-V curves of SDM, DDM, and TDM align precisely, indicating that the model accurately reflects the performance of the RTC France PV cell. Also, Figs. 5b , 6b, and 7b present the effectiveness of the IRTHA algorithm. The measured and estimated values on the P-V curves of SDM, DDM, and TDM are in exact match, demonstrating that the model correctly represents the effectiveness of the RTC France PV cell. The data presented in these figures clearly indicate a strong connection between the experimental polarization curves and those derived from the identified model. Figures 5c , 6c, and 7c provide the behavior of convergence for 1000 iterations, in which IRTHA has a more rapid and stable convergence curve than other MH algorithms.
Figures 5d , 6d, and 7d shows the comparative boxplot analysis achieved by IRTHA in comparison with other algorithms across the RTC France Solar PV (SDM, DDM, and TDM). Furthermore, Figures 5e , 6e , and 7e illustrate a radar chart which indicates the ranking of the 11 MH optimization methods for the RTC France Solar PV (SDM, DDM, and TDM). It has been noted that IRTHA exhibits the smallest shaded area, clearly illustrating its enhanced performance relative to the other algorithms. The shaded areas of POA and HO are positioned in 2nd and 3rd place, indicating that POA and HLOA are in close competition with the IRTHA algorithm. On the basis of the Wilcoxon rank test in Table 6, 9, and 12, IRTHA obtained the first rank for SDM, DDM, and TDM, hence highlighting IRTHA’s superiority in accuracy and convergence performance. This demonstrates that the effectiveness of the IRTHA algorithm, implemented as an optimization technique for parameter estimation from solar PV, significantly outperforms that of other algorithms.
The RTC France SDM (a) I-V characteristic (b) P-V characteristic (c) convergence curve characteristic (d) Boxplot characteristic (e) Radarchart characteristic.
The RTC France TDM (a) I-V characteristic (b) P-V characteristic (c) convergence curve characteristic (d) Boxplot characteristic (e) Radarchart characteristic.
The IRTHA algorithm demonstrated exceptional performance in optimizing the PVM-752-GaAs thin film Solar PV, providing accurate parameter estimations, quick convergence, and dependable results. The precision and effectiveness of the IRTHA are evaluated by estimating the unknown model parameters of PVM-752-GaAs solar PV. Table  13 presents the measured outcomes for 44 voltage-current data points along with corresponding measured and estimated power. Additionally, the statistical metrics, such as MAE, MSE, RMSE, and MBE, are also presented in Table  13. A comparative analysis of the efficacy of eleven optimization algorithms, including IRTHA, HLOA, ZOA, PSOGWO, WOA, POA, HO, and OOA, is shown in Table 14. From the numerical simulation outcomes demonstrated in Table 14 it is observed that the proposed IRTHA achieves the lowest RMSE value of 1.59243E-04 in comparison with other MH algorithms. The I-V and P-V characteristic curves presented in Fig. 8a and b demonstrate a high correlation between the experimental and estimated data, thus validating the IRTHA performance in simulating the PVM752 module. Additionally, Fig.  8c illustrates the convergence curves of all algorithms used for comparison across the PVM-752-GaAs Solar PV, providing insight into the performance of algorithms. The figure shows that the convergence curves of the IRTHA algorithm demonstrate superior performance compared to other comparative algorithms. This simple and effective convergence illustrates IRTHA’s capability to rapidly and effectively optimize fitness values, a significant benefit in optimization. The Fig.  8d shows the comparative boxplot analysis achieved by IRTHA in comparison with other algorithms across the PVM-752-GaAs Solar PV.
In addition, Fig. 8a presents a radar chart illustrating the ranking of the 11 MH optimization techniques for the PVM 752 GaAs thin-film cell. The results reveal that IRTHA exhibits the smallest shaded area, hence illustrating its greater efficacy relative to the other MH algorithms. The shaded portions of POA and HHO are positioned in (hbox {2}^{nd}) and (hbox {3}^{rd}) place, indicating that POA and HHO are in close competition with IRTHA. Finally, rank analysis is carried out using the Wilcoxon signed-rank test, which is illustrated in Table 15. The IRTHA ranked the highest among all the algorithms, followed by POA, HHO, and HO. Conversely, algorithms such as OOA, COA, and GWO exhibited inferior rankings, hence highlighting IRTHA’s superiority in accuracy and convergence performance. The comparison has been conducted by considering the evaluation of statistical parameters such as standard deviation, worst, mean, and minimum RMSE values, and other indicators that highlight convergence, and solution quality metrics. Interestingly, IRTHA exhibits significant stability and consistency as marked by its low standard deviation and low mean, min RMSE.
The RTC France TDM (a) I-V characteristic (b) P-V characteristic (c) convergence curve characteristic (d) Boxplot characteristic (e) Radarchart characteristic.
The PVM-752-GaAs (a) I-V characteristic (b) P-V characteristic (c) convergence curve characteristic (d) Boxplot characteristic (e) Radarchart characteristic.
The STP-120/36 PV module was utilized to evaluate the efficacy of the proposed IRTHA algorithm in comparison to eleven sophisticated optimization techniques. Table 16 presents the measured outcomes for 24 voltage-current data points along with corresponding measured and estimated power. Furthermore, the statistical metrics, such as MAE, MSE, RMSE and MBE, are also presented in Table 16. Completing the simulation of the algorithms in Matlab, the statistical metrics of the final RMSE values derived from the MH optimization methods are shown in Table 17. Table 17 provides a detailed overview of the best, worst, mean, min, standard deviation, and optimal values achieved by each MH algorithm across 1000 runs. The results of these experiments indicate that the IRTHA algorithm shows outstanding results regarding both the mean objective function value and the best objective function value when compared to other algorithms. The best RMSE result obtained utilizing the IRTHA algorithm is 1.44508E-02. The graphical characteristics show in Fig. 9a , and b demonstrate the efficacy of the IRTHA algorithm. The measured as well as estimated values on the I-V and P-V curves match exactly, which demonstrates that the model accurately represents the performance of the STP-120/36 PV module. The figures demonstrate a clear correlation between the experimental polarization curves and the model-derived curves. Figure 9c shows a direct comparison of the algorithms’ effectiveness in improving their search methodologies to attain the minimal RMSE value. The IRTHA algorithm illustrates its superiority in Fig. 9c by attaining optimal fitness values. The optimization process of the IRTHA algorithm demonstrates constant performance, whereas other MH algorithms display either delayed convergence or unstable fluctuations, highlighting challenges with balancing the exploration as well as exploitation stages. Figure  9e shows the comparative boxplot analysis conducted by IRTHA compared to other MH optimization techniques for the PVM-752-GaAs Solar PV module. Additionally, Fig. 9e depicts a radar chart which displays the ranking of the 11 MH optimization algorithms for the STP6 120/36 PV module. The findings indicate that IRTHA displays the smallest shaded area, showcasing its exceptional performance compared to the other MH techniques. The shaded areas of POA and HO are in (hbox {2}^{nd}) and (hbox {3}^{rd}) positions, indicating that POA and HO are in close competition with the IRTHA algorithm. On the basis of Wilcoxon rank test in Table  18, IRTHA obtained the (hbox {I}^{st}) rank followed by POA and HO while OOA and COA exhibited inferior rankings, hence highlighting IRTHA’s superiority in accuracy and convergence performance.This highlights the efficacy of the IRTHA algorithm, implemented as an optimization technique for parameter identification from Solar PV, significantly outperforming other algorithms.
The STP6 120/36 PV module (a) I-V characteristic (b) P-V characteristic (c) convergence curve characteristic (d) Boxplot characteristic (e) Radarchart characteristic.
The parameter optimization of the STM6 40/36 PV module was carried out utilizing the IRTHA algorithm as well as was compared with ten other MH optimization algorithms, which include HLOA, ZOA, PSOGWO, WOA, POA, HO, OOA, HHO, GWO, and COA. The preciseness of the IRTHA optimized model has been demonstrated by comparing the measured and estimated values of current ((I_{m}) and (I_{e})) and power ((P_{m}) and (P_{e})) across various current densities (IE), as presented in Table 19. Furthermore, the statistical metrics, such as MAE, MSE, RMSE, and MBE, are also presented in Table 19. A comparative analysis of the efficacy of eleven optimization algorithms is shown in Table 20. Also, Table 20 provides a detailed overview of the best, worst, mean, min, standard deviation, and optimal values achieved by each MH algorithm across 1000 iterations and 30 runs. The results of these experiments indicate that the IRTHA algorithm shows outstanding results regarding both the mean objective function value and the best objective function value when compared to other algorithms. The optimal RMSE solution achieved with the IRTHA algorithm is 1.72192E-03. The I–V and P-V characteristic graphs of the STM6 40/36 PV module are illustrated in Fig. 10a, and b, demonstrating a significant similarity between the measured as well as simulated current-voltage values obtained through the IRTHA algorithm. This indicates that the IRTHA algorithm provides a significant level of accuracy in modelling the STM6 40/36 PV module. Figure 10cillustrates the convergence plots of the objective function (RMSE) achieved by the IRTHA algorithm in comparison with the other algorithms. The diagram illustrates that the IRTHA algorithm exhibits rapid and consistent convergence closer to the optimal solution when compared to the other algorithms. Figure 10dpresents the boxplot graphs for the STM6 40/36 PV module. It is evident that HO, POA, and HLOA exhibit a close relationship regarding the distribution range and fluctuations. It is evident that the data derived from the IRTHA algorithm exhibits narrower distribution ranges and upper/lower bands compared to the other MH algorithms. This indicates that the IRTHA algorithm can attain the lowest RMSE while maintaining the highest stability. Additionally, Fig. 10ashows a radar chart demonstrating the position of 11 MH optimization techniques for the STM6 40/36 PV module. The IRTHA presents the minimal shaded area, clearly highlighting its superior performance compared to other MH techniques. The shaded areas of HLOA and POA are positioned in (hbox {2}^{nd}) and (hbox {3}^{rd}) positions, indicating that HLOA and POA are in close competition with IRTHA. Finally, rank analysis is carried out utilizing the Wilcoxon signed-rank test, which is shown in Table 21.IRTHA achieved the top ranking among all algorithms, followed by HLOA, POA, HO, and PSOGWO. On the other hand, algorithms like HHO, COA, and WOA demonstrated lower rankings, hence demonstrating IRTHA’s superiority in accuracy as well as convergence performance. The comparison has been carried out by evaluating statistical parameters, including standard deviation, worst, mean, and minimum RMSE values, along with other indicators that demonstrate convergence and solution quality indicators. Interestingly, IRTHA exhibits significant stability and consistency as marked by its low standard deviation, mean, and min RMSE.
The STM6 40/36 PV module (a) I-V characteristic (b) P-V characteristic (c) convergence curve characteristic (d) Boxplot characteristic (e) Radarchart characteristic.
The IRTHA algorithm demonstrated exceptional performance in optimizing the Photowatt-PWP201 PV module, providing accurate parameter estimations, quick convergence, and dependable results. The precision and effectiveness of the IRTHA are evaluated by estimating the unknown model parameters of Photowatt-PWP201 PV module. Table  22 presents the measured outcomes for 25 voltage-current data points along with corresponding measured and estimated power. Additionally, the statistical metrics, such as MAE, MSE, RMSE, and MBE, are also presented in Table  22. Furthermore, Table 23 displays the optimal parameter values along with the RMSE. The experimental findings were recorded after the 30-time run of every optimizer. The findings reveal that the IRTHA optimization method surpasses other MH algorithms, as shown by its optimal RMSE performance presented in Table 23. Additionally, Fig. 11a and b shows the P-V and I-V characteristic curves, which are obtained from the optimal parameters determined by the IRTHA algorithm. The graphical representations present the relationship between estimated and actual measurements. The data indicates that the parameters obtained from the IRTHA algorithm achieve current and power levels that closely match the observed outcomes. Figure 11d illustrates a direct comparison of the algorithms’ effectiveness in optimizing their search methodologies to attain the minimal RMSE value. The IRTHA algorithm demonstrates its superiority in Fig. 11d by achieving optimum fitness values. Figure 11d presents a direct comparison of the algorithms’ efficacy in optimizing their search techniques to achieve the smallest RMSE value. The IRTHA algorithm reveals its superiority in Fig. 11dby attaining optimum fitness values. The optimization process for IRTHA algorithm exhibits consistent performance, while other MH algorithms reveal either delayed convergence or unstable variations, indicating challenges in balancing exploration as well as exploitation. The Fig. 11d presents the comparative boxplot analysis performed by IRTHA compared to other MH techniques for the Photowatt-PWP201 PV module.
Additionally, Fig. 11a illustrates a radar chart that indicates the position of the 11 MH optimization algorithms for the Photowatt-PWP201 PV module. The findings indicate that IRTHA demonstrates the least shaded area, effectively highlighting its superior performance compared to the other algorithms. The shaded areas of POA and HLOA are positioned in the (hbox {2}^{nd}) and (hbox {3}^{rd}) positions, indicating that POA and HLOA are in close competition with the proposed algorithm. On the basis of Wilcoxon rank test in Table  24, IRTHA obtained (hbox {I}^{st}) rank followed by POA and HLOA while GWO and PSOGWO exhibited inferior rankings, hence highlighting IRTHA’s superiority in accuracy and convergence performance. This illustrates that the efficacy of the IRTHA algorithm, implemented as an optimization technique for parameter identification from Solar PV, significantly outperforms that of other algorithms.
The Photowatt-PWP201 PV module (a) I-V characteristic (b) P-V characteristic (c) convergence curve characteristic (d) Boxplot characteristic (e) Radarchart characteristic.
Table 25 presents the computational time complexity (in seconds) of all MH algorithms utilized for RTC France (SDM, DDM, and TDM), Photowatt-PWP201, STP6 120/36, PVM-752-GaAs, and STM6 40/36 PV panels. All simulations have been executed in MATLAB R2021a on a Windows 11 laptop containing an Intel Core i5-1035G1 CPU and 8GB of RAM. The algorithms have a population size of 50, with a maximum of 1000 iterations, and have been executed for 1 run for all the algorithms used for analysis. From the analysis, it is observed that the IRTHA continuously takes a higher execution time than other MH algorithms, such as ZOA, HO, HHO, WOA, POA, PSOGWO, OOA, COA, GWO, and HLOA. The IRTHA algorithm takes approximately 16.7799s, 18.5538s, 21.3759s, 18.6296s, 27.6161s, 18.0251s, and 16.8908s for the RTC France (SDM, DDM, and TDM), Photowatt-PWP201, PVM-752-GaAs, STP6 120/36, and STM6 40/36 PV panels, respectively. Despite the higher computational time complexity, IRTHA achieves the minimal RMSE values among all PV cells/modules. Although other MH algorithms exhibit lower time complexity, these MH algorithms fail to achieve optimal results. The RMSE achieved by IRTHA is 7.72986E-04 for SDM, 7.42740E-04 for DDM, 7.42631E-04for TDM, 2.05285E-03 for Photowatt-PW201, 1.59243E-04 for PVM-752-GaAs, 1.44508E-02 for STP6-120/36, and 1.72192E-03 for STM6 40/36, all of which are consistently lower than the results obtained using the other MH algorithms. The outcomes illustrate a distinct balance between computational complexity and estimation accuracy, with the IRTHA highlighting reliability and precision in solar PV parameter estimation.
This paper presents an improved MH algorithm, known as the Improved Red-tailed Hawk Algorithm (IRTHA), which has been proposed for the estimation of solar PV parameters by integrating the properties of RTHA with the Nonlinear Decay Chaotic Map strategy and the Newton-Raphson Method. This article focuses on the modeling of solar PV, presenting simulation results that closely align with those observed in the experimental. The objective function of this work is the RMSE, representing the variation between calculated as well as measured voltages. Three distinct varieties of PEMFCs, specifically including the RTC France (SDM, DDM, and TDM), Photowatt-PWP201, STP6 120/36, PVM-752- GaAs, and STM6 40/36 PV panels, were utilized to illustrate the robustness of the IRTHA. The outcomes show that IRTHA demonstrated a superior RMSE value in comparison to various techniques, including HLOA, ZOA, PSOGWO, WOA, POA, HO, OOA, HHO, GWO, and COA, as well as additional parameter estimations of solar PV techniques reported in the literature, in addition to a more accurate model. A comparative analysis with other advanced MH optimization techniques illustrates that IRTHA demonstrates significantly lower RMSE values: 7.72986E-04 for SDM-RTC France, 7.41918E-04 for DDM-RTC France, 7.34782E-04 for TDM-RTC France, 1.59243E-04 for PVM 752, 1.44508E-02 for STP 120/36, 1.72192E-03 for STM 40/36, and 2.05285E-03 for the Photowatt-PWP201 module, respectively. Additionally, determine the reliability and effectiveness of the IRTHA in extracting PV parameters by employing various statistical metrics, including Mean Absolute Error (MAE), Mean Square Error (MSE), Sum of Square Error (SSE), Individual Absolute Error (IAE), Root Mean Square Error (RMSE), and the Friedman and Wilcoxon rank-sum tests. In addition, the convergence to the optimal values occurs rapidly as compared to the other MH algorithms. The statistical evaluation demonstrates the superior reliability of the calculated outcomes. The solutions demonstrate an optimal alignment between the calculated and measured I–V curves using the proposed IRTHA. Therefore, the findings validate that the IRTHA algorithm is promising and acts as an effective tool for extracting PV cell parameters, as it demonstrates superior performance in addressing the nonlinear equations of the analyzed challenge.
Future research should focus on enhancing the IRTHA and other MH algorithms to achieve additional advantages in parameter estimation for fuel cell challenges, renewable energy, power systems, the scalability of the methods to large-scale PV arrays, partial shading conditions, and other real-world challenges with real-time embedded implementation.
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
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Open access funding provided by Vellore Institute of Technology. Vellore Institute of Technology, Vellore, Tamil Nadu, India.
School of Electrical Engineering, Vellore Institute of Technology, Vellore, Tamil Nadu, India
Pankaj Sharma, Asmita Ajay Rathod, Shubhi Shukla, Saravanakumar Raju & Balaji Subramanian
Department of Electrical Engineering, National Institute of Technology, Andhra Pradesh, Tadepalligudem, Andhra Pradesh, India
Pankaj Sharma & Asmita Ajay Rathod
Department of Mathematics, School of Advance Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, India
Arun Choudhary
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Pankaj Sharma: Conceptualization, Methodology, Data curation, Writing-Original draft Preparation, Resources, Reviewing and Editing, Validation, Result and Discussion. Asmita Ajay Rathod, Shubhi Shukla, : Methodology, Writing-original draft preparation, Data curation, Software, Validation, Result and Discussion, Real-world Application. Arun Choudhary, Saravanakumar Raju, Balaji Subramanian: Writing – review & editing, Visualization, Validation, Supervision, Resources, Project administration, Investigation, Formal analysis.
Correspondence to Balaji Subramanian.
The authors declare no competing interests.
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Sharma, P., Ajay Rathod, A., Shukla, S. et al. Optimized parameter estimation of solar PV models using an improved red-tailed hawk algorithm. Sci Rep 16, 14016 (2026). https://doi.org/10.1038/s41598-026-42400-7
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Received: 04 December 2025
Accepted: 25 February 2026
Published: 18 March 2026
Version of record: 01 May 2026
DOI: https://doi.org/10.1038/s41598-026-42400-7
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