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Scientific Reports volume 16, Article number: 7864 (2026) Cite this article
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Today, the need for photovoltaic (PV) energy is increasing due to its abundance and hazard-free nature. A PV system is described by its mathematical models. These models contain intrinsic parameters, not provided by the manufacturer. Therefore, accurately estimating these parameters is crucial for improving the reliability and efficiency of PV systems. However, the nature of PV system is highly non-linear thus finding these parameters is quite challenging. Several conventional optimization approaches have been applied but they suffer from premature convergence and stagnation. Although Quantum behaved Particle Swarm Optimization (QPSO) has been utilized for many optimization problems, its strengths and weaknesses in the context of PV parameters estimation is not explored. This paper presents an improved version of the QPSO called Modified Quantum inspired particle swarm method (MQPSO). The proposed MQPSO incorporate dual attractors, elitism strategy and local refinement thus enhancing the convergence behavior and performance. Quantitative analysis clearly shows that MQPSO outperforms conventional QPSO on all three PV models and modules. Compared with the standard QPSO, MQPSO has an average RMSE reduction of 24.74% for the SDM, 59.32% for the DDM, and 14.99% for the TDM. The results demonstrate the merit and efficiency of the proposed MQPSO as compared to other well-developed Optimization methods.
The demand of green energy is growing rapidly due to its abundance and hazard-free nature. Photovoltaic (PV) in contrast to other renewable energy sources is considered most favorable as it offers lower maintenance cost and comparatively longer life span. However, owing to the highly non-linear nature of solar cells, precisely determining the intrinsic parameters of a PV cell is crucial for accurate simulation, efficient energy capture and optimization of solar energy systems. Even small inaccuracies in parameter identification can lead to significant deviations in energy yield estimation, system control decisions, and fault diagnosis. Therefore, developing fast and reliable parameter-estimation techniques is of vital importance for both research and industrial PV applications. Consequently, parameter estimation has become a central research challenge in PV system modeling¹. Parameter identification of PV systems usually falls into three main categories: Analytical, Numerical and Soft Computing². Analytical approaches, although computationally simple, are prone to assumptions and approximations that result in accuracy losses, such as the omission of shunt resistance in Khanna et al.³ and series resistance in Ramadan et al.⁴. Numerical methods have been developed to address these limitations, with advantages in terms of convergence rate and computational efficiency. However, they too, remain sensitive to initial guesses, prone to being trapped in local optima^5,6,7. Research is still going on improving the performance of such methods as presented in⁸. Recently in another study by Saidi et.al⁹, also proposed a dimension learning-based modified grey wolf optimizer that overcomes the limitations of analytical and numerical techniques. From these it can be incurred that metaheuristic algorithms have recently gained interest because they can address complex nonlinear optimization problems. For instance, Chaotic chicken swarm optimization (CCSO) Gupta et al.¹⁰, Particle Swarm Optimization and Gravitational Search Algorithm (PSOGSA) Gupta et al.¹¹, Crayfish Optimization Algorithm (COA), Abdelminaam et al.¹², Generalized Normal Distribution Optimization (GNDO), Abdel-Basset et al.¹³, Kumari et al.¹⁴, Sharma, P. et al.¹⁵ and Mohamed.R et al.¹⁶ have all been proposed with the objective of minimizing the root mean square error (RMSE) between modeled and experimental I–V curves.
Recently, Premkumar et al.¹⁷ utilized a new physics-inspired metaheuristic algorithm, the resistance–Capacitance Optimization Algorithm (RCOA), for the first time. The algorithm is based on the concepts of resistance and capacitance and is more accurate in terms of parameter estimation than the other algorithms. Similarly, Xiao et al.¹⁸ utilized Coati-improved snow ablation optimization (CSAO) with a Weibull distribution and elite retention to achieve better accuracy and efficiency in parameter identification.
Despite the major advancements in parameter estimation algorithms, several challenges still remain. For instance, computational burden is a major issue with most metaheuristics¹⁹ and performance validation of modeled or estimated parameters against real-world experimental data²⁰. Although, QPSO introduced by Sun et al.²¹, have faster convergence they are still susceptible to getting stuck in local optima, especially under complex engineering optimization problems²². Furthermore, in our preliminary experiments and a study as presented in²³, also highlighted the two major drawbacks of standard QPSO that are: relatively low accuracy and a significant computational burden²³. In the context of nonlinear multimodal problems such as PV parameter estimation, these issues manifest better when the model complexity increases. Motivated by these challenges, this study investigates the application of basic QPSO to photovoltaic-parameter extraction problems across various models and modules. Our findings reveal that its performance is significantly compromised in more complex models such as the DDM and TDM. To mitigate this limitation, we propose a Modified QPSO (MQPSO) that is strategically designed to achieve a better balance between exploration–exploitation, stabilized convergence and improved performance. Extensive experiments have revealed that MQPSO not only reduces the computational burden but also improves the estimation accuracy, making it a more practical choice for PV modeling tasks. In this paper, the key novelties are.
Latin Hypercube Sampling (LHS) based initialization that avoids particle clustering and ensures a better coverage of the search space.
Enabling strong early exploration and precise late-stage exploitation using Adaptive contraction–expansion coefficient.
Elitism reinjection scheme, improving convergence and avoiding stagnation.
Dimension-wise dual attraction mechanism that assigns a unique random balance to each parameter that adds finer stochastic variation.
Local refinement strategy beyond global exploration to fine-tune elite solutions.
The remainder of this paper is organized as follows. Section “PV system modeling” describes the modeling of the PV system and the formulation of the parameter estimation as an optimization problem. Section “Preliminaries” outlines the proposed methodology for addressing this issue. Section “Results and discussion” presents and discusses the experimental results of this study. Section “Limitations and future work” outlines the limitations and future work. Finally, Section “Conclusion” concludes the paper with the key findings and insights.
The SDM is the simplest model for studying the behavior of PV cells. Its equivalent circuit consists of a diode (D₁) in parallel with a light-generated source (Ig), which is in series with a resistor, as shown in Fig. 1. The output current of such a model can be calculated using Eq-1²⁴
Single diode model of PV cell.
Substituting the diode current and shunt current Eq-1 can be redefined as
where ({I}_{g}) represents the light-generated current, ({I}_{d1}) is the diode current, ({I}_{sh}) is the shunt resistor current, ({I}_{s1}) and denotes the reverse saturation current of diode D_1.  (n) is the ideality factor of the diode. where k = 1.3806503 × 10^–23 J/K is the Boltzmann constant, and T represents the cell temperature in Kelvin (◦K). From Eq. (2) The five unknown parameters (({I}_{g}), ({I}_{s1}), ({R}_{s}), (Rsh), and (n)) must be accurately estimated in the SDM.
The DDM is an extended version of the SDM that considers both diffusion and recombination losses within a cell. Figure 2 shows the mathematical model of DDM. The inclusion of D₂ accounts for recombination losses in the depletion region. The current at the output terminal can be derived using Eqs. 3 and 4, respectively²⁵.
where ({I}_{s1}) and ({I}_{s2}) denote the reverse saturation currents of D₁ and D₂ respectively. Similarly,({n}_{1}) and ({n}_{2}) represent the diode ideality factors of D₁ and D₂, respectively. DDM involves seven unknown parameters that are: ( ({I}_{g}), ({I}_{s1}), ({I}_{s2}), (Rsh), ({R}_{s}) (,{n}_{1}) and ({n}_{2})) that need to be estimated accurately.
Dual diode model of PV cell.
The three-diode model (TDM) captures additional recombination losses that represent the leakage current caused by surface defects and grain boundary effects, thereby providing an even more accurate model and understanding of the behavior of a solar cell. Figure 3 shows the TDM of the PV cells. The equations governing the output are given by Eqs. 5 and 6 as given in²⁶
where ({I}_{s1}) (, {I}_{s2}) (and {I}_{s3}) denote reverse saturation currents of D₁, D₂ and D₃ respectively, ({n}_{1}) (,{n}_{2}) and ({n}_{3}) is the ideality factors of the three diodes namely D₁, D₂ and D₃ respectively. TDM contains nine unknown parameters (({I}_{g}), ({I}_{s1}), ({I}_{s2}), ({I}_{s3},) (Rsh), ({R}_{s}) (,{n}_{1}), ({n}_{2}, {n}_{3})) that should be identified accurately.
Three diode model of PV cell.
A PV MM generally consists of a series and/or parallel configuration of solar cell (SC) strings. Figure 4 illustrates the mathematical model of a PV MM. To prevent excessive current flow back to the string, several blocking diodes are used. In the case of one or more SCs failing or being blocked for any reason, a bypass diode is incorporated to avoid disturbance and preserve the output current flow. The output for a typical SDM-based MM is given by Eq-7²⁷.
where (Np) and (Ns) represent the number of SCs in parallel and series, respectively.
PV Module Model (MM).
This section defines the methodology for framing the problem of estimating the unknown parameters of the SDM, DDM, TDM, and MM as optimization problems. In this setting, the goal was to extract a set of parameters for which there was the minimum discrepancy between the data points estimated and the ones experimentally measured. The error functions for the SDM, DDM, TDM, and MM are defined by Eqs. (8) to (11), respectively²⁷
where (X = left{ { Ig, Is_{1} , Rs, Rsh, n_{1} } right})
where (X = left{ { Ig, Is_{1} , Is_{2} , Rs, Rsh, n_{1} , n_{2} } right})
where (X = left{ { Ig, Is_{1} , Is_{2} ,Is_{3} , Rs, Rsh, n_{1} , n_{2} , n_{3} } right})
where (X = left{ { Ig, Is_{1} , Rs, Rsh, n} right})
Then, the objective function for the optimization methods is formulated based on the modeling framework as presented in²⁷ and adapted for the present PV parameter estimation problem as follows
where Z denotes the number of experimental data points, ({Im}_{i}) represent the measured current at voltage ({V}_{i}).(X) represent the number of unknown parameters for a given PV model. The root mean square error (RMSE) was selected as the primary evaluation metric in this study due to its ability to quantify the absolute magnitude of modeling errors in physical current units. Unlike the coefficient of determination (R²), which mainly reflects overall correlation, RMSE directly measures point-wise deviations between measured and modeled I–V characteristics. This is particularly relevant for nonlinear PV parameter estimation, where high correlation does not necessarily ensure accurate fitting in critical operating regions. Therefore, R² was not considered, as RMSE provides a more physically meaningful measure of model accuracy.
QPSO was first proposed by Sun et al.²¹ as a quantum-inspired reformulation of classical Particle Swarm Optimization (PSO). QPSO does not use the velocity term but instead model’s particle movement under the assumption of a quantum delta potential well. In standard QPSO, each particle represents a candidate solution in a d-dimensional space. The swarm collectively maintains two memories: the personal best of each particle and the global best of the swarm. A unique feature of QPSO is the introduction of the mean best position, which is defined as the average of all personal bests:
The position update is then carried out using
where (u) is a uniform random number in the range [0, 1] and (beta) is a contraction–expansion coefficient.
The optimization begins by initializing a swarm of particles that represents the PV model’s parameters. Unlike standard random initialization, the LHS method is employed. The reason behind such initialization is that, it ensures a well-distributed coverage of the search space, enhancing early exploration. Mathematically, for particle (i) and dimension (d,) the proposed formulation can be written as
where ({LHS}_{i, d}) is the novel Latin Hypercube sampling in the range [0,1], ({LB}_{d}) and ({UB}_{d}) are the lower and upper bounds of each parameter in a d-dimensional space respectively.
In the next step each particle or candidate solution was assessed using a fitness evaluation function, as defined in Eq. (12). Once all particles are evaluated, the selection of the personal and global best solution is made. The selection is based on the fitness values of each particle. Afterwards the MQPSO enters into the core iterative phase by employing adaptive beta as given in Eq. (16) and computation of mean best position of the swarm as defined in Eq. (17). In simple terms, each particle compares its current fitness to the best value it has discovered so far. If the new fitness value is better its position and fitness are updated accordingly. This step facilitates subsequent guiding process and allows each candidate solution or particle to retain useful information from its own search history. Meanwhile, the algorithm also identifies the best performing particle in the entire population and assigns it as the global best (gbest). This globally performing best solution acts as a central attractor that guides all particles towards the most promising regions of the search pace. In contrast to the conventional QPSO, in the proposed MQPSO implementation, to preserve the swarm diversity and influence the quantum-behavior updating mechanism efficient, the (gbest) is not only used for attraction but also contributes to the computation of the mean best position (mbest). Thus, MQPSO dynamically balances exploration- exploitation by continuously updating these personal and global best particles.
where, (beta max) is set to 1 while (beta min) to 0.5. This stochastic quantum-based update ensures a balance between global exploration and local exploitation, allowing particles to converge toward high-quality solutions.
Mathematically this step can be expressed as
One of the major issues with population-based algorithms is premature convergence and stagnation. To address these issues the proposed MQPSO incorporates a novel mechanism called the elitism reinjection. This mechanism makes the present study different from other conventional approaches. Elitism reinjection works such that at predefined intervals, a portion of the best performing particles known as “Elite Set (E)” and an equal portion of the worst-performing particles called the “Weak Set (W)” were identified. This identification was made on the basis of their respective fitness error values. The worst particles are then pushed towards the Elite particles ensuring that the underperforming particles are guided toward promising regions of the search space without completely discarding their current positions. This step efficiently accelerates convergence by rescuing the underperforming particles while retaining the swarm diversity because each worst particle is paired with a different elite solution. During elitism updates, the worst-performing particles were adjusted according to a weighted combination of their previous position ((Xw))​ and an elite particle ((Xe)). This process has been formulated and can be expressed as
where (Xw) represents the worst particle, (Xe) denotes the elite particle, ({X}_{w}^{New}) represents the new position of the worst particle, (alpha) is the weighting factor, set as 0.5 to strike a balance between retaining the exploratory characteristics of the original particle and incorporating the high-quality guidance of elites.
In standard QPSO, the particles position update process is mainly governed by a combination of each particle’s personal and global best. However, this approach depends on these two attractors, which is susceptible to reducing the diversity and may lead to premature convergence. In contrast, the proposed MQPSO introduces two additional guiding attractors called the intermediate point (I{P}_{d})  and the mean best ({mbest}_{d}) . The proposed formulation (I{P}_{d})  is given in Eq. (20)²¹ and defined as, a dimension-wise weighted combination of the particle’s personal best and the global best. This combination is controlled by a random factor ({F}_{d}). Thus, the particle’s position is updated so that each dimension of each particle samples a different balance between the personal and global best. Equation (17)²⁸ is the proposed formulation that provides a stable and collective reference that promotes efficient swarm search behavior. Our formulated MQPSO differs from standard QPSO in that it balances randomized exploration through the (I{P}_{d}) with stable convergence from the mean best ({mbest}_{d}).
where ({F}_{d}) is a random number [0,1].
The candidate position for each particle in dimension d is then updated using a novel modified version of the basic QPSO as
where ({s}_{d} in [-1,+1]) introduces a random direction of movement, preventing premature stagnation, and (beta left(tright)) governs the balance between exploration and exploitation as provided in Eq. (16).
Once all the particles moved to their new positions, the algorithm re-evaluated their fitness. Each particle then compares its current performance with its previously recorded best performance. If the new position offers a lower error, the personal best is updated to reflect this. Similarly, the global best is updated whenever any particle outperforms the current best solution known to the entire swarm, as given in Eq. (22–23)²¹. Thus, both individual experience and collective knowledge are continuously refreshed as the search progresses. This mechanism ensures that useful information is never lost; it remains stored in memory and serves as a reference point for guiding further updates.
The optimization process was iteratively continued until a predefined limit was reached. To further enhance the accuracy, a local refinement step was applied using a Nelder–Mead search, starting from the global best position. This additional step fine-tunes the parameters within the neighborhood of the current best solution, potentially reducing the error further. The working mechanism of the proposed MQPSO is depicted in Fig. 5.
Flowchart of the proposed MQPSO.
This section presents the comparative evaluations carried out to validate the performance of the proposed MQPSO algorithm. All simulations were performed using MATLAB R2021b, Intel Core i5 (7th generation), 2.71 GHz and 8 GB of RAM. The comparative experiments highlight several strengths of the proposed MQPSO relative to conventional QPSO and other metaheuristic optimizers. The experiments were conducted on three commercial photovoltaic (PV) cell and modules: the RTC France cell, the Photowatt-PWP201 module, and the STM6-40/36 module. The I–V characteristics of the RTC France silicon solar cell and the Photowatt-PWP201 module were obtained from⁵, while the data for the STM6-40/36 module were taken from²⁹. To ensure statistical reliability, each algorithm was executed independently 30 times for every cell and module. To make a fair comparison, a two-stage non-parametric statistical analysis was conducted to ensure a robust comparison among the algorithms. First, the Friedman test was applied to examine whether statistically significant differences exist among all algorithms across multiple independent runs and test cases. Second, a Wilcoxon signed-rank statistical test was also conducted at an alpha level of α = 0.001 using MQPSO as the baseline algorithm. The Wilcoxon signed-rank test provides detailed pairwise insights between MQPSO and each competitor. Accordingly, the convergence analysis and RMSE statistics provide complementary perspectives on optimization dynamics and final solution accuracy. Moreover, to further show the superiority of the proposed method, the Boxplot analysis was also carried out on all competing algorithms. The performance of the proposed MQPSO was benchmarked against four other metaheuristic algorithms. The parameter settings and user-defined configurations of all algorithms are summarized in Table 1, whereas the lower and upper bounds of the parameters for the SDM, DDM, TDM, and PV module model (MM) are provided in Table 2.
The experimental results of the SDM, DDM, and TDM-based RTC France solar cells are presented in Tables 3–5, which present the best-optimized parameters of all the tested algorithms along with their respective RMSE values. Table 3 lists the results of the SDM, which contains five unknown parameters and their respective RMSE values. It is clear that the performance of the MQPSO is better than that of all other algorithms. Table 4, which reports the DDM based seven unknown parameters, further suggests that MQPSO has superior performance compared to the competing algorithms. Table 5 lists the nine parameters corresponding to the TDM. It can be observed that MQPSO has the best performance, followed by QPSO and DEDIWPSO. In addition, to further show the performance of the MQPSO, the error curves of individual absolute error (IAE) for the current and power based on the SDM, DDM, and TDM are presented in Fig. 6a and b, respectively. From them, it can be observed that the maximum IAE of current and power for SDM are 1.585E-3 and 0.794E-3 for DDM these are 1.350E-3 and 0.787E-3 while TDM has 1.350E-3 and 0.787E-3, respectively. Additionally, the run time and functions evaluations (FEs) curves between QPSO and MQPSO, for SDM. The DDM and TDM are shown in Fig. 7a and b, respectively. It can be observed from Fig. 7b that the FEs of MQPSO are slightly high but it is attributed to the local refinement phase that resulted in enhanced accuracy as compared to the standard QPSO. Moreover, the I-V and P-V curves for the SDM, DDM, and TDM based on the optimized parameters are presented in Fig. 8a, b, and c, respectively. These curves reveal that the simulated and measured data are highly consistent.
(a) Individual absolute error (current) for SDM, DDM and TDM (b) Individual absoulte error(Power) for SDM, DDM and TDM.
(a) Run time comparison between QPSO and MQPSO for SDM, DDM and TDM (b) Function evaluations (FEs) between QPSO and MQPSO for SDM, DDM and TDM.
(a) Single diode model (SDM) (b) Dual diode model (DDM) (c) Three diode model (TDM).
The results of parameter estimation of STM6-40/36 and Photowatt-PWP201 are given in Tables 6 and 7, respectively. Both tables affirm the fact that the performance of MQPSO is improved compared to all others. Along with this, error curves of the current and power of the STM6-40/36 and PWP201 MMs are also illustrated in Fig. 9a and b, respectively. Max. IAE values of current and power for STM6-40/36 were calculated to be 6.077E-3 and 9.043E-2, respectively, and for PWP201 MM, the IAE values of current and power were calculated to be 3.752E-3 and 3.517E-2, respectively. Figure 10a and b are showing the comparison of run time and Function Evaluations (FEs) of QPSO and MQPSO for STM6-40/36 and PWP201 MMs, respectively. Along with this, I-V and P–V curves of parameters calculated using the best results of STM6-40/36 and Photowatt-PWP201 MMs using MQPSO are also illustrated in Fig. 11a and b, respectively.
(a) Individual absolute error for STM6-40/36 MM (b) Individual absolute error for PWP201 MM.
(a) run time comparison between QPSO and MQPSO for STM6-40/36 and PWP201 MM (b) FEs comparison between QPSO and MQPSO for STM6-40/36 and PWP201 MM.
(a) STM6-40/36 module model (MM) (b) Photowatt-PWP201 module model (MM).
This section describes the statistical results to further evaluate the performance of the MQPSO. The comparison was made on the basis of 30 independent runs, observing the maximum (Max), minimum (Min), average (Mean) values, and the standard deviation (Std) of the RMSE. In addition, the Friedman’s and the Wilcoxon signed-rank test results keeping the proposed MQPSO as baseline are shown in Tables 8, 9, 10, 11. Although all comparisons with MQPSO were statistically significant, some p-values were identical, which is attributed to the discrete and rank-based nature of the Wilcoxon signed-rank statistics. From the statistical results listed in Tables 8–11, we can draw the following conclusions:
For the SDM, the proposed MQPSO achieved the lowest and most consistent RMSE as evident from its Mean = 7.7301 × 10^–4 and almost negligible value of SD ≈ 1.43 × 10^–14. These results highlight its high accuracy and stable convergence behavior across all runs. The Friedman test yielded a p-value of 3.75 × 10⁻²⁰, confirming statistically significant differences among all algorithms. Furthermore, the Wilcoxon signed-rank test results also showed that MQPSO significantly outperformed all competitors.
From the DDM analysis, MQPSO again outperformed the other approaches by achieving the best values for all four indices, reflecting its reliability even for more complex and multimodal optimization problems. QPSO under this model exhibited very large errors and larger standard deviations, suggesting a reduction in their stability. The Wilcoxon test further validated that MQPSO is significantly better than IJAYA, DEDIWPSO, CLPSO, PSO, and QPSO.
For the complex TDM too, MQPSO consistently maintained its advantage with the lowest Min and Mean RMSEs of 7.29E-04 and 8.69E-04 respectively, underlining its robustness even in the most complex PV parameter estimation tasks. Although the QPSO and DEDIWPSO obtained the minimum RMSEs of 7.47E-4 and 7.99E-4 respectively, their other performance metrices aren’t satisfactory. The Friedman test again indicated strong statistical significance (p = 7.14 × 10⁻²²). Furthermore, the Pairwise Wilcoxon comparisons also confirmed that MQPSO significantly outperforms all other algorithms, including QPSO (p = 2.11 × 10⁻³), reinforcing its robustness across increasingly complex PV models.
The analysis of STM6-40/36 MM clearly indicates that MQPSO obtained the best values for all the performance indices. The minimum RMSE was 1.73 × 10⁻³ with a relatively small standard deviation (9.19 × 10⁻⁴), indicating both accuracy and consistency in the PV parameters estimating task. QPSO provided competitive results with a minor error range, demonstrating robustness, although it can be observed that it is slightly less accurate than MQPSO. The Friedman test confirmed statistically significant differences among the competing algorithms (p = 7.17 × 10⁻²⁰). Additionally, Wilcoxon signed-rank tests also revealed that MQPSO significantly outperforms all competing algorithms, including QPSO, IJAYA, and DEDIWPSO.
For Photowatt-PWP201 MM, MQPSO again demonstrated superior accuracy with a Min RMSE of 1.8727 × 10⁻³ and SD of 2.84E-04. This stability highlights the robustness of the algorithm across multiple runs. Although DEDIWPSO obtained lower RMSE, it can clearly be observed that they exhibit high maximum and extremely large standard deviation indicating unstable and inconsistent convergence behavior under this case. QPSO performed the worst in this case, as indicated by its mean RMSE of 1.341E-02. The Friedman test confirmed statistically significant performance differences among the algorithms (p = 4.26 × 10⁻²³). MQPSO, under this complex problem too, outperforming all competitors. Pairwise Wilcoxon signed-rank tests further verified that MQPSO significantly outperforms IJAYA, CLPSO, PSO, and QPSO.
The convergence characteristics of all the tested algorithms are shown in Fig. 12. From this, it can be observed that the proposed MQPSO maintains steady progress and reaches lower final solutions across all benchmark models although not faster enough in some cases (e.g. PWP201 MM). This nature is truly attributed to the design of the proposed method, which emphasizes broader exploration in the early phase of iteration with gradual exploitation in later stages to avoid premature convergence, as manifested by few competing algorithms. The differences in convergence behavior across various test cases is in accordance with the No Free Lunch (NFL) theorem which implies no single method can perform optimally on all optimization problems. Moreover, the convergence curves illustrate the RMSE reduction during the global search phase of the MQPSO and therefore reflect the algorithm’s convergence behavior and search dynamics. The major contribution of the MQPSO lies in its robust convergence stability and accurate final solutions, rather than rapid early convergence. However, it is also utmost important to clarify that a more comprehensive assessment can only be made by considering convergence curves in conjunction with the statistical performance indicators (Wilcoxon rank sum, Friedman’s test) as presented in Tables 8–11. From these Tables it can clearly be observed that the proposed method outperforms all other competing approaches on all performance indices. The RMSE values reported in these Tables correspond to the final solutions, which include the effect of the local refinement step. As this refinement is applied only after the completion of the main optimization loop, its impact is reflected in the final RMSE statistics but not in the convergence trajectories Fig. 13 depicts the ranking of algorithms based on Fridman’s test analysis. Lastly, the Boxplot analysis as depicted in Fig. 14, again reinforces the superiority of the proposed method as it achieves lower mean error and reduced variability, indicating more stable and accurate final solutions across all runs.
Convergence curves of all tested algorithms.
Friedman’s test based rank of all tested algorithms.
Boxplot analysis of all tested algorithms.
Although the proposed MQPSO framework demonstrates high accuracy and robust convergence in PV parameter estimation problem, several limitations still exist. The computational cost, though reduced compared to standard QPSO, may still be significant for real-time or embedded applications. The method also relies on high-quality I–V measurements; noise, partial shading, or sensor inaccuracies can affect performance. In addition, while the algorithm presents superior performance compared to all the competing algorithms within this work, its generalization capability has been tested within the context of photovoltaic applications only. It is suggested that future work could further evaluate its generalization capability using standardized CEC benchmark test functions.
In this study, we proposed an improved Modified Quantum-behaved Particle Swarm Optimization (MQPSO) as an advanced variant of the classical QPSO framework for efficient photovoltaic parameter estimation. The algorithm’s quantum–mechanical update rules, elitism strategy, and adaptive contraction–expansion coefficient collectively contributed to superior exploration–exploitation balance and consistently lower RMSE values across 30 independent runs. The proposed MQPSO methodology shows enhancement in convergence characteristics, reduced computational burden, and higher accuracy compared to standard QPSO. MQPSO combines quantum-behaved dynamics with Latin Hypercube Sampling initialization, adaptive control parameters, dual attractor influences, and an elitism reinjection mechanism for balanced exploration–exploitation trade-offs thus avoid premature convergence and suboptimal exploration characteristic of complex optimization problems. Experimental results clearly shows that the proposed MQPSO consistently yields lower error values and exhibit a more stable and faster convergence behavior than conventional QPSO and several other contemporary state-of-the-art metaheuristics. These findings reveal the robustness of MQPSO in tackling nonlinear, multimodal, and high-dimensional optimization challenges. These challenges are indeed crucial for practical engineering applications, such as photovoltaic domain, power system optimization, and machine-learning model training. It is suggested that MQPSO may constitute a meaningful advancement within swarm intelligence by integrating theoretical rigors from quantum mechanics with practical adaptability of swarm-based learning. However, some of the aspects remain unaddressed, such as sensitivity to measurement noise, dependence on algorithmic parameter settings, and applicability to real-time or large-scale photovoltaic systems. In this regard, future research shall be directed towards adaptive hybrid metaheuristic integration, noise-robust strategies, and more comprehensive benchmarking against CEC benchmark test functions, outlining concrete pathways for further enhancements in photovoltaic parameter estimation performance.
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The authors extend their appreciation to Umm Al-Qura University, Saudi Arabia for funding this research work through the grant number: 25UQU4310138GSSR01.
This research work was funded by Umm Al-Qura University, Saudi Arabia under grant number: 25UQU4310138GSSR01.
Department of Electrical Engineering, Sarhad University of Science and IT, Peshawar, Pakistan
Zia Ur Rehman, Obaid Ur Rehman & M Abid Saeed
Department of Computer and Network Engineering, College of Computers, Umm Al-Qura University, Makkah, Saudi Arabia
Amr Munshi
Schools of Sciences, Engineering and Environment, University of Salford, Manchester, UK
Sadaqat Ur Rehman
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Zia Ur Rehman (First Author): Conceptualization, Software, Methodology, Data Curation, Writing—Original Draft. Obaid Ur Rehman (Second Author): Software, Supervision, Project Administration and Validation. Amr Munshi (Third Author): Addressing reviewers comments, Re-executing experiment and validating result, Financial support. M Abid Saeed (Fourth Author): Writing—Review & Editing, Formal Analysis, Visualization. Sadaqat Ur Rehman (Fifth Author): Software, Resources, Funding Acquisition, Final. Approval of Manuscript.
Correspondence to Amr Munshi.
The authors declare no competing interests.
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Rehman, Z.U., Rehman, O.U., Munshi, A. et al. Parameters optimization of photovoltaic systems using modified quantum inspired particle swarm method. Sci Rep 16, 7864 (2026). https://doi.org/10.1038/s41598-026-38620-6
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Received: 13 October 2025
Accepted: 30 January 2026
Published: 09 February 2026
Version of record: 02 March 2026
DOI: https://doi.org/10.1038/s41598-026-38620-6
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