Solar Now

by Jayne Elliott
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Imagine having a neighbor who is out to get you. What would you do if this neighbor kept reporting you to the city for violations you’re not violating?
In this story, one homeowner is dealing with this exact situation, and he’s so over it.
It all started when he had solar panels installed on his roof. Ever since then, the neighbor has decided that they’re enemies for some reason.
It’s really crazy!
Let’s read all about it.
My neighbor decided that I am his sworn enemy because I had solar panels installed on my roof.
He originally called the city to inspect the installation which I actually didn’t mind, after all I want the solar company to do a proper installation and the city inspectors would know what to look for.
All was good.
How annoying!
Since then he has called every city agency to try to get me in trouble for anything he can think of.
This has been going on for 3 years.
All his complaints end up getting dismissed. It is just out & out harassment.
The neighbor’s complaints are pretty ridiculous!
The latest is for safety violations including not having water!!!!, smoke detectors or fire extinguishers.
Of course I have water, that is just crazy and just last year I replaced all the smoke detectors and fire extinguishers.
I know this will all be dismissed when they inspect but it’s getting more and more ridiculous.
OP is just the latest target.
He has gotten himself worked up about other people in the past before he switched over to me as his target.
I truly think there is something mentally wrong with him and I don’t want to exacerbate the issue so I just deal with each item as it comes up.
All the city employees have been quite pleasant. I wonder if he has a history with them.
He’s actually not getting me into any trouble and that may actually be riling him up more.
Yikes! The neighbor sounds crazy.
Let’s see how Reddit responded.
Here’s a suggestion to call a lawyer.
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I like this idea!

Another person would sue.

If you enjoyed this story, check out this post about a woman who refused to let her neighbor have access to his spot, telling him to drive over the curb instead.
Here’s a funny suggestion!

The city officials are probably just as annoyed with the neighbor as the homeowner who wrote this story. He’s wasting their time.
I wonder why the neighbor is making up these crazy complaints. Is he just bored?
That neighbor needs a better hobby than picking on his neighbors.
Author
Jayne Elliott | Contributing Writer, Life & Drama
Jayne Elliott is a contributing writer and editor for TwistedSifter specializing in human interest stories, internet culture, and family dynamics. With over 12 years of editorial experience in digital publishing, Jayne excels at analyzing complex online communities and transforming viral social debates into thoughtful, highly engaging narratives.
Rather than simply aggregating internet drama, Jayne brings a sharp, empathetic editorial eye to everyday dilemmas. She has a unique talent for unpacking the nuances of pop culture and online conflicts, providing readers with relatable, well-researched commentary.
Based in California, Jayne spends her free time outside the newsroom exploring theme parks with her family or beach-combing along the coast.
Follow Jayne’s adventures and connect with her on Instagram, Facebook, and YouTube.
Categories: Life & Drama, Neighbors & Community
Tags: · ENTITY, neighbor, neighborhood, neighbors from hell, picture, reddit, solar panels, top
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India’s Power Capacity to Surpass 2,000 GW by 2047, Solar to Cross 1,100 GW: ENCIS Outlook  SolarQuarter
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Solar, Data Centres & Manufacturing: This Week in Energy  Energy Digital
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For a handful of days each year, Berlin turns into the tech capital of the world. This year, IFA has close to 2,000 brands showcasing the latest technology from air fryers to robot lawn mowers to sleep earbuds and, of course, robots. Aside from CES in January, it’s the best opportunity to check out what’s on the horizon from top brands like Dreame, DJI, Anker, Dyson, and plenty more.
Some of the tech on display teeters on the useless side or focuses on grand ideas that are unlikely to become mainstream. Others, like a ceiling light that can mimic natural outdoor light and stick vacuums with more suction power than ever before, are well worth bringing home.
As expected, this year’s event has a major emphasis on AI. Plenty of the AI advancements on display at IFA in 2026 are pretty great and streamline life where we need it most.
In an email, IFA CEO Leif Lindner explained, “What excites me most is seeing entire categories become more intuitive because AI is solving real problems, not simply being added for its own sake,’ he said. “That’s what’s driving the innovations I’m most excited about this year, from robotics moving closer to practical everyday applications, to wearables that help you build better habits, to connected kitchen appliances that are genuinely helpful.”
Below, we’ve rounded up the best products Mashable has spotted at IFA so far. We’ll keep this list updated as the show continues on through Sept. 8. Some of the items are already available for sale in the U.S. while a few others will be worth the wait.
The Soundcore Sleep A30 Special are Mashable’s favorite sleep earbuds, and now we have two new iterations: Sleep Earbuds 4 and Sleep Earbuds 4 Pro. Coming in at 20% slimmer than the Sleep A30 model, the new Sleep Earbuds 4 are designed to be even more comfortable for side sleepers. If you share a bed with someone who snores, the AI-generated dynamic snore masking system in the Sleep Earbuds 4 could make a major difference in your sleep quality.
The fancier Sleep Earbuds Pro 4 use the same earbud design as the standard Sleep Earbuds 4 but come with a more advanced case, among other features. Similar to the Soundcore Liberty 5 Pro earbuds, the Sleep Earbuds Pro 4 have an on-case display that lets sleepers check sleep metrics upon waking.
The screen also allows users to adjust the volume, change sleep tracks, or set an alarm from the case itself, rather than reaching for a phone.
The Anker Sleep Earbuds 4 are priced at $229.99 with a launch set for the beginning of October. The Anker Sleep Earbuds Pro 4 will sell for $349.99 come November. Preorders at Anker are currently eligible for a $30-off coupon that comes from putting down a $1 deposit.
The EcoFlow River 3 Plus is one of my favorite portable power stations for camping, so I’m thrilled we got introduced to the EcoFlow River Gen4 at IFA. Completely redesigned from the River 3, the River Gen4 features a compact design that’s significantly smaller than its predecessor. The River 260 Gen4 comes with a capacity of 256Wh, 300W sustained output, and a peak of 600W. At about 6.5 pounds, it’s highly portable for a weekend of camping.
Going bigger, the EcoFlow River 520 Gen4 has 512Wh of capacity, 500W of AC output, and a 1,000W peak. The larger of the River 520 doubles the power capacity but doesn’t double the weight, coming in at just 10 pounds. Again, that’s manageable enough to pack along on a camping trip for most people. Both models have a 10 millisecond UPS, fast-charging USB-C ports, and quick recharging.
We’re still waiting on the U.S. launch date information and pricing. EcoFlow also showcased a compact 60W solar panel at IFA that seems like it’ll be the perfect companion for these two new models.
It’s a sad time to be a DJI fan in the U.S. The brand packed the new DJI Osmo 360 II for IFA this year, but as of now, it’s not coming to the U.S. thanks to current regulations. The new model replaces last year’s Osmo 360 and comes with improvements such as the ability to shoot 8K panoramic footage. It also doubles the brightness range, offering 14.5 stops of dynamic range compared to the previous 13.5.
We’re keeping a close eye on availability in the United States. At the same time, we’re still waiting for the U.S. to see the launch of the DJI Power 1000 Mini, too.
Dreame continues to make waves with its product expansion, and at IFA 2026, the brand revealed the Dreame Leaptic Cube 8K action camera. With a U.S. launch set for November and a competitive price of $449, Dreame is coming for the likes of DJI and Insta360.
When hands-on with the Dreame Leaptic Cube, I found it remarkably lightweight, and the gesture options for taking still photos and starting and stopping recording are both easy and incredibly responsive. The modular design offers plenty of benefits, including the ability to add extra batteries.
Dreame’s also ready with accessory packages for cycling, underwater exploration, and even mounting to the dog. We’ll continue to monitor preorder opportunities before the launch in November 2026.
Topics Robot Vacuums Cameras Vacuums
Lauren Allain is a freelance journalist covering deals at Mashable. She graduated from Western Washington University with a B.A. in journalism and holds an M.B.A from Webster Leiden. You can find more of her work online from publications including Reader’s Digest, U.S. News & World Report, Seattle Refined, and more. When she’s not writing, Lauren prefers to be outside hiking, bouldering, swimming, or searching for the perfect location for all three.

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Tracer World Invites Global Investment For 200 MW Mumbwa Solar PV + 180 MWh BESS Project In Zambia  SolarQuarter
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Scientific Reports volume 16, Article number: 4467 (2026) Cite this article
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Accurate identification of photovoltaic (PV) cell and module parameters is essential for reliable electrical modeling, performance assessment, and long-term energy yield prediction. This task is commonly formulated as an optimization problem, where the root mean square error (RMSE) between measured and estimated current-voltage characteristics is minimized. While numerous metaheuristic algorithms have been proposed to solve this problem, most existing studies focus primarily on algorithmic modifications, with limited attention given to enhancing the problem formulation itself. In this work, a recently introduced metaheuristic, the Starfish Optimization Algorithm (SFOA), is employed for PV parameter extraction and systematically evaluated against four contemporary optimization algorithms. In addition, a novel secant-based reformulation of the objective function is proposed to improve the accuracy of the parameter estimation process beyond the conventional RMSE-based approach. The proposed framework is validated on multiple PV models, including the single-diode (SDM), double-diode (DDM), and three-diode (TDM) models for PV cells, as well as the single-diode model of a PV module (PVM). Two widely used benchmark datasets, RTC France and Photowatt-PWP201, are used for experimental verification. The results demonstrate that integrating the secant-based objective function significantly enhances estimation accuracy and robustness across all considered models. In particular, the SFOA-Secant configuration achieves the lowest RMSE values of (7.6579 times 10^{-4}) for SDM, (7.4192 times 10^{-4}) for DDM, (7.3218 times 10^{-4}) for TDM, and (2.0489 times 10^{-3}) for PVM, outperforming all competing methods. These findings confirm that reformulating the objective function using the secant method constitutes an effective and complementary strategy for improving PV parameter extraction accuracy.
Solar energy is one of the most important energy sources used in the field of renewable energy^1,2. In recent years, solar energy has been used in several fields, such as military and medical applications^3,4. It is increasingly applied in residential and industrial sectors as well, providing a sustainable alternative to fossil fuels^5,6. Solar panels are now widely installed on rooftops, powering homes, schools, and businesses with clean electricity⁷. Moreover, the development of grid-connected photovoltaic (PV) systems has significantly enhanced the efficiency and reliability of solar energy utilization⁸, enabling excess energy to be integrated into the grid and guaranteeing a consistent power supply, even during periods of limited solar generation.
The efficiency of PV cell technology has steadily improved through advancements in cell design, material selection, and fabrication techniques^9,10,11. To accurately analyze and predict the performance of PV cells under various operating conditions, researchers commonly employ mathematical modeling¹². In the literature, three primary models are widely used for simulation: the single-diode model (SDM), the double-diode model (DDM), and the three-diode model (TDM)¹³. However, these models contain different unknown parameters that must be identified before they can be effectively used for simulation and practical applications¹⁴.
In the literature, the identification of PV parameters is commonly formulated as an optimization problem, where the root mean square error (RMSE) is typically employed as the objective function to minimize the discrepancy between the experimental data and the estimated data obtained from the extracted parameters.
In¹⁵, the authors proposed the mother tree optimization with climate change (MTO-CL) algorithm to enhance parameter estimation in the three-diode PV model. The method achieved greater accuracy and robustness than seven other algorithms by adding elimination and distortion phases based on climate dynamics. This resulted in remarkably low RMSE and power errors across different modules. But this increase in accuracy comes with a longer computational time, which makes it less advantageous for real-time applications. In¹⁶, a modified electric eel foraging optimization (MEEFO) algorithm incorporating fractional-order calculus, fitness distance balance, and quasi-opposition-based learning to improve PV parameter estimation. The method demonstrated superior exploration and exploitation capabilities, avoiding premature convergence and achieving the lowest RMSE values across single, double, and triple diode models under varying meteorological conditions. However, the added strategies increase algorithmic complexity, which may limit its adaptability for real-time applications. In¹⁷, the research introduced the Levy flight and mutation-enhanced artificial rabbit optimization (LMARO) algorithm for PV parameter extraction. By combining swarm-elite learning, Levy flight, and mutation strategies, LMARO improved global exploration, population diversity, and convergence speed, achieving competitive RMSE values across single-diode, double-diode, and PV module models. While the method demonstrated strong accuracy and stability compared to several advanced algorithms, its adaptability to broader optimization problems still requires further improvement.
In¹⁸, the authors presented the bio-dynamics grasshopper optimization algorithm (BDGOA) aimed at improving photovoltaic parameter identification. The method enhanced convergence speed, exploration, and robustness relative to GOA, effectively estimating parameters of various commercial PV modules across different conditions; however, temperature-dependent variations in specific parameters persisted. In¹⁹, the paper suggested the enhanced prairie dog optimizer (En-PDO), which integrates random learning and logarithmic spiral search to improve PV parameter identification. Compared with the original PDO and eighteen recent algorithms, En-PDO consistently achieved lower RMSE values across single-, double-, and triple-diode models as well as PV module models under diverse conditions. While the method proved robust and accurate, future improvements are needed to extend its adaptability for dynamic environments and real-time applications. In²⁰, the work proposed a hybrid multi-population gorilla troops optimizer and beluga whale optimization (HGTO-BWO) to improve PV parameter extraction. By combining multi-population strategies with exploration-exploitation mechanisms such as Levy flight and synchronized motion, the method achieved the lowest RMSE values across double- and triple-diode models for various PV cells and modules under different operating conditions. Despite its high accuracy and robustness, the approach is computationally demanding and complex to implement, limiting its practicality without further simplification.
In²¹, a hybrid kepler optimization algorithm (HKOA) was introduced by enhancing KOA with ranking-based update and exploitation-improvement mechanisms. These additions strengthen exploration to avoid local optima and improve exploitation for faster convergence. HKOA outperformed several recent algorithms on the RTC France cell and multiple PV modules, showing strong accuracy and stability. However, its reliance on extra control parameters and relatively high computational cost remain notable limitations. In²², a hybrid flower grey differential (HFGD) algorithm combining flower pollination algorithm (FPA), grey wolf optimizer (GWO), and differential evolution (DE) algorithm; was proposed to improve PV parameter estimation. With added Newton-Raphson refinement, HFGD achieved the lowest RMSE and strong robustness across several PV models. Its main drawback is the increased algorithmic complexity arising from multi-hybrid integration. In²³, a kangaroo escape optimization (KEO) algorithm was proposed by modeling kangaroos’ escape behavior through chaotic energy adaptation, zigzag exploration, long-jump motions, and decoy-based exploitation. KEO achieved high accuracy and robustness across SDM, DDM, and TDM on the RTC France and Photowatt-PWP201 datasets, outperforming several recent optimizers. Its improved exploration-exploitation balance makes it effective for complex nonlinear PV models, though its multi-stage update strategy adds complexity that may affect computational efficiency in real-time scenarios.
In²⁴, the authors introduced reconfigured single- and double-diode models (Reconfig-SDM and Reconfig-DDM) by adding a small series resistance to the diode branches to better capture PV nonlinearities. Using the squirrel search algorithm for parameter extraction, the proposed models achieved notably lower RMSE than classical SDM/DDM on RTC France and CS6P-220P modules. However, the improved accuracy comes at the cost of increased model complexity, requiring more parameters and computational effort during estimation. In²⁵, an enhanced differential evolution (EDE) algorithm was introduced using stage-specific mutation and crossover strategies to improve exploration and exploitation during PV parameter estimation. EDE achieved the lowest RMSE across SDM, DDM, and TDM for multiple cells and modules, showing strong accuracy and convergence reliability. Its main limitation is the increased algorithmic complexity from adaptive control parameters, which may hinder real-time deployment. In²⁶, the improved sinh cosh optimizer (ISCHO) enhances SCHO with trigonometric operators to strengthen exploitation and avoid local optima. It achieves low RMSE and reliable performance across multiple PV models and modules, outperforming several recent methods. However, the added operators increase computational complexity, limiting its application in real-time settings.
In recent years, researchers have employed various metaheuristic optimization algorithms to tackle the challenge of estimating PV parameters. These algorithms include: a hybrid optimization technique combining the analytical Newton-Raphson-based optimization (Ana-NRBO) algorithm with an analytical initialization method²⁷, wild horse optimizer²⁸, shuffled puma optimizer²⁹, frilled lizard optimization³⁰, snake optimization with sine-cosine algorithm³¹, PID-based search algorithm (PSA)³², generalized normal distribution optimization based on neighborhood search strategies (NSGNDO)³³, pelican optimization algorithm³⁴, drone squadron optimization³⁵, lungs performance-based optimization (LPO) algorithm³⁶, gold rush optimizer³⁷, archimedes optimization algorithm³⁸, improved simultaneous heat transfer search³⁹, butterfly optimization algorithm with chaos learning strategy⁴⁰, northern goshawk optimization algorithm⁴¹, tree seed algorithm⁴², nutcracker optimization algorithm⁴³, landscape-aware particle swarm optimization (LaPSO)⁴⁴, adaptive slime mould algorithm (ASMA)⁴⁵, weighted mean of vectors (INFO)⁴⁶, chaotic-gradient-based optimizer⁴⁷, modified elephant herding optimization⁴⁸, improved bonobo optimizer⁴⁹, gradient-based optimizer⁵⁰, and coyote optimization algorithm⁵¹.
Although numerous metaheuristic algorithms have been proposed for PV parameter extraction, most existing studies concentrate primarily on algorithmic improvements while relying on the same conventional problem formulation based on the RMSE objective function. This narrow focus overlooks the fact that the accuracy and convergence behavior of any optimizer are strongly influenced not only by the search mechanism but also by the mathematical structure of the objective function itself. As a result, even recently introduced high-performance optimizer may still exhibit slow convergence, stagnation, or sensitivity to initial conditions when the underlying formulation remains unchanged.
A limited number of works have attempted to modify the problem formulation directly^13,52. For example, in¹³ proposed the Flood Algorithm (FLA) along with a Newton–Raphson based objective function. While their modified formulation enhanced accuracy, it introduced substantial computational overhead due to derivative evaluations, making it unsuitable for real-time or embedded PV applications. This highlights two key limitations in the current literature: derivative-based objective reformulations increase computational cost and sensitivity to numerical instabilities, and metaheuristic algorithms enhancements alone cannot fully address the inherent nonlinearity of PV models when the traditional RMSE formulation remains unchanged.
Motivated by these gaps, this work introduces a secant-based objective function modification as a derivative-free reformulation of the PV parameter extraction problem. Unlike the traditional formulation of the objective function, the secant mechanism approximates the estimated current used in the RMSE formula without requiring explicit derivatives, thereby maintaining numerical stability. This modification enabling faster convergence toward high-accuracy solutions.
In parallel, and in accordance with the No Free Lunch (NFL) theorem, which states that no single optimizer performs best across all problem classes⁵³. A recently developed metaheuristic, the starfish optimization algorithm (SFOA), is employed. SFOA offers a well-balanced exploration-exploitation structure inspired by starfish sensory-driven movement and regeneration behavior⁵⁴. Its multi-arm search mechanism enables effective coverage of high-dimensional spaces, while its regeneration phase helps escape local minima; an essential capability for the highly multimodal PV parameter extraction problem. Compared to traditional nature-inspired metaheuristic algorithms, SFOA does not rely on delicate control parameters, making it robust and easier to implement.
This paper investigates the secant method to enhance problem formulation, aiming to achieve a small RMSE value compared to the traditional formulation of the PV parameters extraction objective function. The key contributions of this paper can be summarized as follows:
Application of the starfish optimization algorithm: this recently introduced metaheuristic is employed for accurate extraction of parameters in PV cell and module models.
Comparative performance analysis: the results obtained using SFOA are rigorously compared with those from four other state-of-the-art optimization algorithms, educational competition optimizer (ECO)⁵⁵, hippopotamus optimization algorithm (HO)⁵⁶, osprey optimization algorithm (OOA)⁵⁷, and zebra optimization algorithm (ZOA)⁵⁸.
Objective function enhancement: the objective function is reformulated using the secant method, and its performance is evaluated against the conventional objective functions commonly reported in the literature.
The combination of SFOA with the secant-based formulation achieves superior RMSE performance across SDM, DDM, TDM, and PV module models, confirming the effectiveness of both the new problem formulation and the optimizer.
This paper is organized as follows. The section on PV Models and Problem Formulation outlines the photovoltaic cell and module models and formulates the parameter extraction problem, including the secant-based objective function. The Starfish Optimization Algorithm section presents a detailed description of the proposed optimization method. The Experimental Setup section describes the benchmark datasets, error metrics, and comparative optimization algorithms. The Results and Discussion section presents and analyzes the obtained results. Finally, the Conclusion section summarizes the main findings and highlights potential directions for future research.
Accurate design of PV cells and modules requires both a precise mathematical model and an efficient metaheuristic algorithm to estimate the model’s unknown parameters. In the literature, the single-diode, double-diode, and three-diode models are widely used for PV cells, while the single-diode model (SDM) is typically employed for PV modules. These models are described in this section.
The electrical equivalent circuit of the SDM is shown in Fig. 1a. It consists of a constant current source, a diode, a shunt (parallel) resistance, and a series resistance. The diode models the p-n junction of the solar cell, representing the diffusion current due to carrier transport across the junction. By applying Kirchhoff’s current law, the output current of the SDM can be expressed as¹³:
In this equation, (I_{pv}) is the output current of the photovoltaic (PV) cell, and (I_{ph}) is the photocurrent generated by the incident light. The term (I_{sd}) denotes the reverse saturation current of the diode. The constants k and q represent the Boltzmann constant and the elementary charge, with values 1.38064852 (times 10^{-23}) J/K and 1.6021764 (times 10^{-19}) C, respectively.
The parameter n is the diode ideality factor, and T is the temperature of the PV cell in Kelvin. (V_{pv}) denotes the output voltage of the PV cell. The terms (R_s) and (R_{sh}) represent the series resistance and the shunt resistance of the PV cell, respectively.
PV modules typically consist of (N_s) solar cells connected in series to meet the desired power output, as shown in Fig. 1b. The output current of the PV module model (PVM) is given by¹³:
After excluding the known parameters, both the SDM and PVM have five unknown parameters that require estimation, defined as (X = left[ I_{ph}, I_{sd}, R_{sh}, R_{s}, n right]).
Equivalent circuit of (a) SDM, (b) PVM, (c) DDM, and (d) TDM.
The Double-Diode Model (DDM), shown in Fig. 1c, contains the same elements as the SDM but includes a second diode connected in parallel with the first. The second diode models the recombination current in the depletion (space-charge) region, while the first diode represents the diffusion current as in the SDM. This model provides better accuracy under low-irradiance conditions or for high-quality cells where recombination effects in the depletion region are significant. The output current of the DDM is¹⁸:
In this model, (I_{sd1}) and (I_{sd2}) denote the saturation currents due to diffusion and recombination, respectively, while (n_1) and (n_2) represent the corresponding ideality factors of the diodes.
By excluding the known parameters, the DDM is characterized by seven unknown parameters that need to be estimated: (X = left[ I_{ph}, I_{sd1}, I_{sd2}, R_{sh}, R_{s}, n_1, n_2 right]).
The Three-Diode Model (TDM), shown in Fig. 1d, extends the DDM by adding a third diode in parallel. This additional diode captures further nonlinear effects, such as junction breakdown, defect-related leakage, or recombination at grain boundaries in polycrystalline cells. The TDM is particularly useful for modeling cells under low-irradiance conditions. The output current of the TDM is given by⁴⁶:
In this model, (I_{sd3}) represents the additional saturation current associated with a third recombination mechanism, often included to capture more complex carrier dynamics and improve the accuracy of the PV cell modeling under various operating conditions.
After excluding the known parameters, the triple-diode model (TDM) comprises nine unknown parameters that must be estimated: (X = left[ I_{ph}, I_{sd1}, I_{sd2}, I_{sd3}, R_{sh}, R_{s}, n_1, n_2, n_3 right]).
Parameter identification of a PV cell/module is treated as an optimization problem, aiming to determine the set of parameters that most closely matches the experimental I–V curve of the PV cell/module. Modeling a PV cell/module involves the precise determination of the parameters in Eqs. (1), (2), (3), and (4). For that, an objective function is defined as follows:
where X is the vector of parameters to be extracted for each model, (N_{m}) is the number of experimental data points, and M is the model type. The function (f_M) for each model is defined as follows:
In this approach, (V_{pv}) and (I_{pv}) are replaced by the measured data to compute the error between the measured current (I_{m}) and the estimated current (I_{e}) based on the candidate parameter set X.
For all PV models considered in this study (SDM, PVM, DDM, and TDM), the current–voltage relationship can be expressed in the implicit nonlinear form (f_M(V_{pv}, I_{pv}, X) = 0), Due to the coexistence of linear and exponential current terms, this equation does not admit a closed-form solution for (I_{pv}). The conventional objective function minimizes the algebraic residual of equation (5). This approach does not enforce physical consistency and results in a rugged fitness landscape due to the strong nonlinearity of (f_M).
In the literature, this approach is the most common in papers on parameter extraction of PV cells/modules. However, in this paper, a new approach is proposed based on finding the estimated current (I_{e}) using the secant method, then calculating the RMSE.
To overcome the limitation of conventional approach, the estimated current (I_{e}) is obtained by explicitly solving (f_M(V_{pv}, I_{pv}, X) = 0). for each voltage point using the secant method. The objective function is then defined as:
which ensures that the PV model equation is satisfied for every candidate solution.
The secant method is adopted due to its derivative-free nature and superlinear convergence, making it numerically robust when embedded within population-based metaheuristic optimization.
The secant method is an iterative, derivative-free algorithm for finding roots of nonlinear equations g(x). Lets consider
It starts with two initial approximations, (x_0) and (x_1), and at each step computes a new approximation (x_{n+1}) as the x-intercept of the secant line passing through the points
The iteration formula is:
In contrast to Newton’s method, which requires the exact derivative (g'(x)), the secant method approximates it by a finite difference:
This avoids derivative evaluations while still achieving superlinear convergence under suitable smoothness conditions.
For each model, (f_{M}) is replaced by the difference between the measured current (I_{m}) and the estimated current (I_{e}) as in (10), where (I_{e}) is the root of (f_{M}), found using the secant method. The whole process of this approach is shown in Fig. 2.
The process of objective function formulation based on Secant-method.
The Starfish Optimization Algorithm (SFOA) was proposed in⁵⁴. The inspiration behind the SFOA comes from the unique biological and behavioral characteristics of starfish. Specifically, SFOA draws from starfish’s abilities to explore their surroundings using five arms (each equipped with light-sensitive eyes), their distinctive preying method of everting their stomachs to digest food externally, and their remarkable capacity for regeneration.
During the initialization phase of SFOA, starfish positions are randomly generated within the bounds of the design variables and represented in matrix form:
where, X denote the matrix representing the positions of the starfish, with dimensions (N times D), where N is the population size and D is the number of design variables. In the initialization phase, the position of each starfish is computed according to the following equation:
here, (X_{ij}) denotes the position of the ith starfish in the jth dimension, r is a uniformly distributed random number in the range (0, 1), and (U_j) and (L_j) represent the upper and lower bounds of the design variable in the jth dimension, respectively. Once the initial position matrix is generated, the fitness values of all starfish can be computed using the objective function and stored in a vector:
here, F is a matrix of size (N times 1) used to store and update the fitness values. After initialization, SFOA enters its main loop, beginning the exploration and exploitation phases.
The exploration phase of the SFOA algorithm simulates the starfish’s search capability, inspired by its five arms, each ending with an eye to aid in environmental sensing. In the exploration phase of SFOA, a novel search strategy is introduced, which combines a five-dimensional search pattern for cases where (D>5), and a unidimensional search pattern when (Dle 5), depending on the nature of the optimization problem. The dimensional threshold is inspired by the biological structure of starfish, which possess five arms (or eyes), serving as a natural basis for this design.
If the dimension of the optimization problem exceeds 5 ((D>5)), the search space becomes significantly large, requiring the starfish to utilize all five arms to explore its surroundings effectively. Moreover, each arm relies on knowledge of the best position found by the search agents to guide its movement. Based on this, a mathematical model is developed to represent this phase as follows:
here, (Y_{i,p}^{T}) and (X_{i,p}^{T}) represent the updated and current positions of the i-th starfish in the p-th dimension, respectively. (X_{best,p}^{T}) denotes the p-th dimension of the current best position. The vector p consists of five randomly selected dimensions from the total D dimensions, and r is a random number in the range (0, 1). The parameters (a_1) and (theta) are computed as follows:
here, T denotes the current iteration, while (T_{max}) is the maximum number of iterations. The sine and cosine components reflect the equal probability of a starfish arm twisting left or right in its attempt to approach food.
For optimization problems where (D>5), the five-dimensional search pattern from (15) is applied to update only five dimensions of each position. This approach enhances search capability and improves efficiency compared to a full vector-based search pattern.
If an updated position falls outside the boundaries of the design variables, the starfish’s arms tend to remain at their previous positions rather than adopting the invalid update. This behavior can be mathematically expressed as follows:
where p denotes the updated dimension, (L_{b,p}) and (U_{b,p}) represent the bounds of design variables, respectively.
If the dimension of the optimization problem is less than or equal to 5 ((Dle 5)), the exploration phase adopts a unidimensional search pattern to update positions. In this case, only one arm of the starfish moves to search for a food source, guided by the positional information of other starfish. The updated position in this scenario is defined as follows:
here, (X_{k_1,p}^{T}) and (X_{k_2,p}^{T}) represent the p-th dimensional positions of two randomly selected starfish. (A_1) and (A_2) are random numbers within the range ((-1, 1)), and p is a randomly selected dimension from the D total dimensions. (E_t) denotes the energy of the starfish, calculated as follows:
Similar to the previous update rule, if the newly obtained position of a starfish falls outside the boundary, the starfish will remain at its previous position rather than adopting the updated one.
In SFOA, the exploitation phase focuses on searching for global solutions through two distinct updating strategies: preying and regeneration.
The parallel two-directional search strategy is used to model the preying phase of starfish; in this strategy, the SOFA needs to utilize the information from other starfish and the current best position of the population. First, five distances are computed between the best-known position and those of other starfish. Then, two of these distances are randomly selected to guide the position update of each starfish using a parallel two-directional search strategy. These distances are calculated as follows:
here, (d_m) represents the five distances between the global best starfish and five other selected starfish, while (m_{p}) denotes five randomly chosen starfish. Based on this, the position update rule during the starfish’s preying behavior is defined as follows:
where, (r_1) and (r_2) are random numbers in the range (0, 1), while (d_{m_{1}}) and (d_{m_{2}}) are randomly selected values from the set (d_m).
In addition, starfish are vulnerable to predators during predation due to their slow movement. When threatened, a starfish may escape by shedding an arm, a defensive mechanism used to evade capture.
Consequently, the regeneration phase in SFOA is applied exclusively to the last starfish in the population ((i=N)). Since regeneration in nature takes several months, this phase is modeled with a very slow movement speed. Accordingly, the position update rule for the regeneration phase is defined as follows:
If the position obtained from (22) or (23) exceeds the boundaries of the design variables, it is adjusted as follows:
The detailed procedure of SFOA is illustrated in Fig. 3, which presents the algorithm’s flowchart.
The SFOA begins with the initialization phase, where algorithm parameters and problem-specific information are provided. The population is then randomly generated within the design variable boundaries using (12), followed by the evaluation of fitness values. After initialization, SFOA proceeds into the main optimization loop.
In the main loop, the decision to proceed with either the exploration or exploitation phase is based on comparing a random number in the range (0, 1) with the algorithmic parameter (G_p), which is set to 0.5 in SFOA based on the original paper⁵⁴. When the maximum iteration criterion is met, the main loop terminates, and the final global solution is returned.
At the initialization phase, the computational complexity of SFOA is (O(N times D)). During the exploration phase, a hybrid search pattern is employed depending on the dimensionality: for (D>5), the complexity is (O( 1/2 times T_{max} times N times 5)), and for (D le 5), it is (O( 1/2 times T_{max} times N times 1)). In the exploitation phase, the complexity is given by (O( 1/2 times T_{max} times N times D)).
For an optimization problem where (D>5), the total computational complexity of SFOA is calculated as follows:
For the case where (Dle 5), the total computational complexity of SFOA is calculated as follows:
To determine the computational complexity of the proposed method, the time complexity of SFOA is combined with that of the Secant-based objective function, denoted as (O(text {Secant-OF})). Accordingly, the overall time complexity of the proposed method can be expressed as:
The computational complexity of the Secant-based objective function is approximated by
where (N_m) represents the size of the dataset used for each model, and (L_s) denotes the number of iterations required by the Secant method. In the other hand, the space complexity of the proposed method is identical to that of SFOA and is approximated as (O left( N times D right)).
The flowchart of SFOA.
The benchmark data used for the PV cell are from the RTC France monocrystalline silicon cell, which has a diameter of 57 mm and consists of a single cell. Its experimental I–V curve was obtained under an incident irradiance of 1000 W/m² at an operating temperature of 33 °C, and is characterized by 26 pairs of current–voltage data points. The PV module data are from the Photowatt-PWP201 polycrystalline module, which is composed of 36 cells connected in series. Its experimental I–V curve was measured under an irradiance of 1000 W/m² at an operating temperature of 45 °C, and is characterized by 25 pairs of current–voltage data points. The experimental data are shown in Table 1.
Table 2 lists the parameter ranges for each PV cell/module model, which are identical to those reported in previous studies^13,18.
To evaluate the accuracy of the estimated current values in comparison to the measured data, the following error metrics are used:
Absolute error (AE) represents the total magnitude of the deviation between the measured current values (I_{m,i}) and the estimated current values (I_{e,i}), summed over all data points.
Mean absolute error (MAE) measures the average magnitude of the absolute differences between measured and estimated currents, providing a general indication of prediction accuracy.
Maximum absolute error (MaxAE) is the maximum absolute difference between the measured and estimated currents over all data points. It represents the worst-case error.
Median bias error (MBE) represents the average signed difference between measured and estimated currents. A positive MBE indicates underestimation by the model, while a negative value indicates overestimation.
In this study, to maintain conciseness, only brief summaries of each algorithm are presented in Table 3, instead of providing detailed explanations.
For consistency in benchmarking, all algorithms were configured with an identical population size of 50 and 1000 iterations. All simulations were conducted in MATLAB R2021a on a personal computer equipped with an Intel Core i5-3230M CPU at 2.60 GHz and 8.00 GB of RAM. Each algorithm was executed 30 times in independent runs. For the SFOA, the parameter Gp was set to 0.5, while for the ECO, the learning habit boundary was set to (H = 0.5). The remaining algorithms were used in their original form without any control parameter adjustments, as they do not incorporate tunable control parameters. For the initial guess of secant method, (x_0) is set to 0 and (x_1) is set to (I_{ph}).
Figure 4 illustrates the convergence curves of the RMSE, without secant modification, for the SFOA, ECO, HO, OOA, and ZOA algorithms applied to the SDM, DDM, and TDM models, based on RTC France and PVM for the Photowatt-PWP201 module. Across all models, the SFOA algorithm demonstrates the fastest and most stable convergence, consistently reaching the lowest fitness values well before the maximum iteration limit, except in the case of PVM. For SDM, DDM, TDM, and PVM, SFOA achieves near-optimal solutions within the first 600–650 iterations, 800–850 iterations, 800–900 iterations, and 900–1000 iterations, respectively. Followed closely by ECO, ZOA, and HO, which converge more slowly and to slightly higher final fitness values.
OOA perform significantly worse across all cases, with OOA maintaining an almost constant and high fitness value throughout the iterations, indicating poor search capability and an inability to improve over time. The PVM case in Fig. 4d highlights these performance differences even more clearly; SFOA reaches the lowest fitness value with a sharp initial drop, while ECO, HO, and ZOA exhibit slower and less stable convergence patterns, and OOA once again shows negligible improvement.
Convergence curves for (a) SDM, (b) DDM, (c) TDM, and (d) PVM for different algorithms without secant modification.
Figure 5 shows the convergence curves of the different algorithms for the various models in the case of modification with the secant method. The inclusion of the secant modification significantly accelerates convergence for all methods, with SFOA showing the most remarkable improvement. In this case, SFOA reaches the lowest fitness value, lower than (10^{-3}), well before 750 iterations and maintains this advantage throughout the run. ECO and HO also benefit from the modification, converging faster and to lower final fitness values than in the non-secant case.
In contrast, ZOA experiences modest gains, converging more slowly and stabilizing at higher fitness levels, while OOA remains almost unchanged, stuck at a constant, poor fitness value. The PVM case in Fig. 5d highlights the advantage of the secant modification for SFOA, which quickly reaches near-optimal solutions, whereas other algorithms converge more gradually and to worse solutions.
Convergence curves for (a) SDM, (b) DDM, (c) TDM, and (d) PVM for different algorithms with secant modification.
To highlight the differences in terms of convergence and to provide a clearer view of the SFOA, Fig. 6 illustrates its convergence curves for the various models, in both cases with and without secant modification. Across all four models, the secant-enhanced SFOA (green curve) consistently converges faster and achieves lower fitness values than the standard version (blue curve).
Convergence curves for SFOA with and without secant modification for (a) SDM, (b) DDM, (c) TDM, and (d) PVM.
Using five optimization algorithms and their corresponding secant-based hybrid variants, Table 4 summarizes the ideal parameters found for the SDM, DDM, and TDM based on the RTC France reference cell and the Photowatt-PWP201 PVM. The photocurrent (I_{ph}), saturation current(s) ((I_{sd}), (I_{sd1}), (I_{sd2}), (I_{sd3})), shunt resistance (R_{sh}), series resistance (R_{s}), and ideality factor(s) (n, (n_1), (n_2), (n_3)), as well as the RMSE as a measure of fitting accuracy, are among the presented parameters for each model. The RMSE values of the secant-hybrid approaches are lower than those of their standalone counterparts, especially for the SDM and DDM. The most reliable accuracy across various PV models is provided by SFOA-Secant and ECO-Secant among the tested configurations, indicating the resilience of hybridization in parameter extraction.
The best results are obtained with the SFOA-Secant configuration for the TDM and DDM, yielding RMSE values of (7.3218times 10^{-4}) and (7.4162times 10^{-4}), respectively. These are closely followed by ECO-Secant for the DDM ((7.5727times 10^{-4})) and SFOA-Secant for the SDM ((7.6579times 10^{-4})). Such minimal error values indicate highly accurate parameter estimation and an excellent match between the modeled and experimental I–V characteristics. For the PVM, SFOA-Secant achieves the lowest RMSE of (2.0489times 10^{-3}).
In contrast, the poorest performance is observed for the OOA and OOA-Secant configurations in the PVM case, with RMSE values of (3.4675times 10^{-1}) and (1.9808times 10^{-1}), respectively, followed by OOA in the DDM ((2.4222times 10^{-2})) and OOA-Secant in the same model ((2.0863times 10^{-2})). These large deviations suggest that these settings fail to converge effectively to the optimal parameters, resulting in poor model fitting and significant mismatches with the measured data.
Table 5 shows that SFOA-Secant achieves the lowest RMSE across all PV models (SDM, DDM, TDM, and PVM), outperforming recent optimization methods reported in the literature. Several competing algorithms provide results only for simpler models, indicating limited scalability when handling higher-order PV models. Moreover, although some methods yield competitive RMSE values for SDM and DDM, their accuracy degrades for more complex models, particularly for PV modules. These results indicate that most existing approaches are constrained by the traditional RMSE-based objective formulation. In contrast, the proposed secant-based modification significantly enhances convergence accuracy and numerical stability, demonstrating that improving the problem formulation is essential for achieving consistently high-precision PV parameter extraction.
Table 6 presents the performance metrics of the five optimization algorithms and their Secant-based hybrid variants in estimating the optimal parameters of the SDM, DDM, and TDM for RTC France reference cell, as well as the PVM for Photowatt-PWP201. The evaluation metrics include AE, MAE, MaxAE, MBE, and the average computation time ((avg t(s))) as performance indicators. Across all models, the Secant-based variants generally outperform their standalone counterparts in terms of AE, MAE, and MaxAE, reflecting improved fitting precision. However, this accuracy gain often comes at the cost of increased computation time, particularly for the HO-Secant approach, which exhibits the longest runtimes in all cases.
The SFOA-Secant and ECO-Secant configurations consistently yield the best results. For instance, SFOA-Secant achieves some of the lowest values recorded for this model in the SDM case, with an AE of 0.017501793, an MAE of 0.000673146, and a MaxAE of 0.001528391. Similarly, SFOA-Secant produces an AE of 0.017008394 and an MAE of 0.000654169 for the DDM and an AE of 0.016697426 and an MAE of 0.000642209 for the TDM, both of which demonstrate extremely accurate parameter estimation. ECO-Secant outperforms all other methods in the PVM case, yielding the lowest AE of 0.04216968 and an MAE of 0.001686787.
The OOA, on the other hand, exhibits the worst results, especially in the PVM case, where AE reaches 7.61536214, MAE rises to 0.304614486, and MaxAE peaks at 0.481289424. Other models, like the DDM, also show this pattern, with OOA producing an AE of 0.602298626 and an MAE of 0.023165332. With AE values of 0.367535968 for the TDM and 4.133594452 for the PVM, OOA-Secant also performs poorly in some cases, showing that hybridization may not always lead to better performance when the base algorithm itself shows poor convergence characteristics.
From the average computation time reported in Table 6, it is evident that all Secant-based variants require a longer execution time compared to their conventional counterparts. This increase is attributed to the additional iterative root-finding process introduced by the Secant method during objective-function evaluation. The effect is consistent across all PV models and optimization algorithms. These results experimentally confirm the computational time approximation formulated in (28), demonstrating that the improved accuracy achieved by the Secant-based formulation is obtained at the cost of increased computational effort.
The statistical analysis of RMSE values for the SDM, DDM, TDM, and PVM based on 30 independent runs using five optimization algorithms and their Secant-hybrid variants is shown in Table 7. Insight into the accuracy and consistency of the methods is provided by reporting the minimum, maximum, mean, median, and standard deviation for each case.
The findings show that the Secant-hybrid versions, SFOA-Secant in particular, achieve consistently low RMSE values with little variation across all PV models. For example, SFOA-Secant exhibits near-perfect repeatability in the SDM, maintaining an exceptionally stable RMSE of roughly (7.6579times 10^{-4}) with a standard deviation of just (1.3793times 10^{-9}). Both SFOA-Secant and ECO-Secant demonstrate their robustness in parameter extraction by achieving low mean RMSE values and small standard deviations in the DDM and TDM.
In contrast, specific baseline algorithms exhibit substantial variability and higher error values. Notably, OOA shows fluctuations, as observed in the SDM (mean RMSE of (1.2286times 10^{-1}), maximum of (2.5212times 10^{-1})) and PVM (mean RMSE of (4.4056times 10^{-1})). In the PVM case, the standard deviation for OOA reaches (1.7721times 10^{-2}), indicating a strong sensitivity to initialization or being stuck in local minima. By comparison, the best-performing hybrid variants consistently combine accuracy with stability, making them more suitable for reliable PV model parameter estimation.
Figure 7 presents box plots of the fitness for the different optimization algorithms, with and without the secant modification, applied to SDM, DDM, TDM, and PVM. Across all subfigures, the Secant-based variants, especially SFOA-Secant exhibit extremely compact distributions with minimal interquartile range (IQR) and negligible outliers, indicating both high accuracy and exceptional stability. In particular, SFOA-Secant maintains median RMSE values close to zero with no visible spread, confirming its repeatable performance across independent runs.
In contrast, specific baseline algorithms, such as OOA and HO, show significant variance and higher median RMSE values, as clearly visible in the elongated box and whiskers, along with numerous outliers. This effect is most pronounced in SDM of Fig. 7a and TDM in Fig. 7c, where OOA’s performance fluctuates considerably between runs. For PVM in Fig. 7d, OOA and ECO display substantial dispersion, whereas SFOA-Secant, ECO-Secant, and other hybrid variants retain consistently low error values.
Box plot for (a) SDM, (b) DDM, (c) TDM, and (d) PVM for different algorithms with and without secant modification.
Therefore, the figure graphically supports the previous statistical conclusions: the Secant-hybrid methods, particularly SFOA-Secant, are consistently the most accurate and dependable, yielding strongly clustered, low-error results for all PV models.
The p-values from the Wilcoxon rank-sum test, which compares the RMSE distributions of SFOA-Secant to each of the other tested algorithms for SDM, DDM, TDM, and PVM, are shown in Table 8. At the (5%) significance level, SFOA-Secant significantly outperforms the compared method, as indicated by the p-values, which are incredibly small in all cases and range from (10^{-10}) to (10^{-11}), accompanied by a “+” symbol. The robustness and superiority of the SFOA-Secant approach in obtaining lower RMSE values with high confidence are highlighted by this consistent statistical dominance across all models and algorithms.
Table 9 presents the results of the Friedman ranking test, showing the mean ranks and the corresponding rank orders for each algorithm across all four PV models. In every case, SFOA-Secant obtains the best possible mean rank (close to 1.0) and is ranked first, confirming its consistent top performance. The second-best performers vary depending on the model, SFOA in SDM, DDM, and TDM, and ECO-Secant in PVM; although their mean ranks are notably higher than SFOA-Secant’s. Conversely, OOA and OOA-Secant occupy the bottom ranks in nearly all cases, reflecting their poor optimization stability and accuracy observed in earlier tables.
With results that are statistically significant when compared to all competitors, SFOA-Secant is not only the most accurate configuration but also the most consistently dominant, according to the statistical tests, which generally support the conclusions from the error analysis.
Figures 8 and 9 present the I–V and P–V characteristic curves, respectively, for RTC France based on SDM, DDM, TDM, and Photowatt-PWP201 based on PVM, using different algorithms with and without the secant modification. In both sets of plots, the measured experimental curves serve as the reference, while the estimated curves from each optimization approach are plotted for comparison.
Across the first three models, Figs. 8a–c and 9a–c, the secant-based variants, especially SFOA-Secant achieve almost perfect overlap with the measured curves, demonstrating precise parameter estimation and accurate reproduction of both the current and power profiles. The differences between the measured and estimated curves are practically imperceptible for these cases, indicating negligible modeling error.
In contrast, some baseline algorithms, particularly OOA for PVM, exhibit noticeable deviations. The secant modification effectively reduces these discrepancies, with SFOA-Secant and ECO-Secant producing near-perfect fits even in the more challenging PVM case.
The I–V characteristic curves of RTC France based on (a) SDM, (b) DDM, and (c) TDM and Photowatt-PWP201 based on (d) PVM using different algorithms.
The P–V characteristic curves of RTC France based on (a) SDM, (b) DDM, and (c) TDM and Photowatt-PWP201 based on (d) PVM using different algorithms.
These figures support the previous statistical and error-metric findings; the Secant-based hybrids, particularly SFOA-Secant, exhibit the lowest numerical errors and the highest fidelity in accurately representing the actual electrical behavior of the PV models under all operating conditions.
The extraction of unknown parameters in photovoltaic (PV) models is a fundamental task, as it enables a rigorous, physics-based interpretation of PV cell and module behavior. Such an understanding directly supports practical applications, including accurate performance prediction, real-time monitoring, fault diagnosis, and the development of effective strategies for system optimization and efficiency enhancement. In this paper, the parameters of several PV models were extracted using a recently developed metaheuristic, namely the Starfish Optimization Algorithm (SFOA). Its performance was systematically evaluated and compared with four recent state-of-the-art algorithms: the Educational Competition Optimizer (ECO), Hippopotamus Optimization Algorithm (HO), Osprey Optimization Algorithm (OOA), and Zebra Optimization Algorithm (ZOA). Two objective function formulations were investigated: the conventional RMSE-based formulation and a novel secant-based reformulation. The primary contribution of this work lies in the development and successful integration of the secant-based objective function into the PV parameter extraction problem.
The results demonstrate that the secant-hybrid variants consistently achieve lower RMSE values with negligible variance across all considered PV models. An ablation analysis, conducted by replacing the proposed secant-based objective with the classical RMSE formulation while keeping all optimization settings unchanged, confirms that the observed performance gains originate from the proposed reformulation rather than from algorithmic parameter tuning. Furthermore, statistical validation using the Wilcoxon rank-sum test verifies that the improvements are statistically significant and not attributable to random effects.
Among all tested configurations, the SFOA-Secant approach exhibits the fastest and most stable convergence behavior, reliably attaining the lowest fitness values well before reaching the maximum number of iterations. The best overall results were obtained using SFOA-Secant for the three-diode model (TDM) and double-diode model (DDM), achieving RMSE values of (7.3218 times 10^{-4}) and (7.4162 times 10^{-4}), respectively. These results are closely followed by ECO-Secant for the DDM ((7.5727 times 10^{-4})) and SFOA-Secant for the single-diode model (SDM) ((7.6579 times 10^{-4})). Comprehensive statistical analyses further confirm that SFOA-Secant consistently outperforms all competing methods with high precision and robustness across different PV models.
In contrast, the OOA and OOA-Secant configurations exhibit inferior performance in most scenarios, particularly for the PV module (PVM) case. Although the secant-based hybridization improves accuracy in some instances, it also introduces additional computational overhead. This increase in computation time is mainly attributed to the iterative root-finding process embedded within the objective function evaluation. Moreover, the I–V and P–V characteristic curves obtained using OOA, with and without secant modification, show noticeable deviations from the experimental data, indicating poor convergence behavior.
The main limitation of the proposed secant-based objective function lies in its increased computational complexity, as evidenced by the higher average runtimes observed across different algorithms. While this overhead is acceptable for offline parameter identification, it may restrict the direct applicability of the method in strict real-time or embedded environments without further optimization.
Future research can address these limitations through several promising directions.
The proposed secant-based objective function can be evaluated on additional benchmark datasets and under varying environmental conditions to further assess its generalization capability.
Computational efficiency may be improved by developing adaptive or simplified variants of the secant method that preserve accuracy while reducing runtime.
Enhancing poorly performing algorithms, such as OOA, particularly for PV module parameter extraction, represents another important research avenue.
The real-time implementation of SFOA-based parameter extraction in operational PV systems, possibly through hardware acceleration or reduced-order modeling, constitutes a valuable direction for future work.
All data generated or analyzed during this study are included in this published article.
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The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TU-DSPP-2024-128).
This research was funded by Taif University, Saudi Arabia, Project No. (TU-DSPP-2024-128).
Department of Electrical Engineering, University of Science and Technology Houari Boumediene, P.O. Box 32, El-Alia, Algiers, 16111, Algeria
Yacine Bouali
Department of Electrical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif, 21944, Saudi Arabia
Basem Alamri
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Search author on:PubMed Google Scholar
Yacine Bouali: conceptualization, methodology, software, validation, formal analysis, investigation, data curation, original draft preparation, visualization. Basem Alamri: validation, resources, review and editing, supervision, project administration, funding acquisition. Both authors have read and approved the final version of the manuscript.
Correspondence to Yacine Bouali or Basem Alamri.
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Bouali, Y., Alamri, B. Parameter extraction of photovoltaic cell/module models using starfish optimization algorithm with a secant-based objective function modification. Sci Rep 16, 4467 (2026). https://doi.org/10.1038/s41598-025-34584-1
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Interest in rooftop solar has surged in recent months, as Europeans scramble to protect themselves from the spiralling costs of fossil fuels.
According to an EU-wide analysis published in the Nature Energy journal, rooftop solar photovoltaics (solar PV) could supply around 40 per cent of Europe’s electricity by 2050. However, only around 10 per cent of Europe’s building rooftops are currently equipped with PV.
Governments across the continent are trying to bolster uptake by offering generous subsidies or financial perks. For example, in Ireland homeowners can receive grants of up to €1,800 through the country’s sustainable energy scheme – while public grants in Hungary can cover up to two-thirds of solar panel costs for homeowners providing they meet specific requirements.
Many other countries such as Germany and the Netherlands have implemented benefits such as 0 per cent VAT on solar panel sales and installation to make the switch more affordable. However, heavy upfront installation costs remain a huge barrier – particularly among low-income households.
But, could ‘solar bonds’ be the magic solution?
The cost of installing solar panels can vary significantly, as there are many factors that can affect the pricing.
According to solar firm LOGI, a single family home in Europe typically needs between 6 and 15 kWp of solar power, depending on household size, heat pump use and electric car charging, which costs anything from €7,000 to €30,000. In the UK, a typical 4.5 kWp system is about £7,600 (€8,831).
While solar panels can save homeowners hundreds of euros on their energy bills every year, it can take time before you will see a return on your investment. Again, this can vary depending on multiple factors – including how much you have electrified your home, whether you work from home, and what type of tariff you are on.
Many experts, including the UK’s Energy Savings Trust, say it could take a typical home at least 10 years to recoup the costs of installing panels. It also requires hefty capital for installation, which in itself can cost up to £10,000 (€11,621). For many homeowners, this means taking out a loan subject to interest.
This is why Dr Donal Brown, a senior researcher in energy policy and political economy at the Environmental Change Institute (ECI) is calling for government-backed finance to make rooftop solar more affordable.
Published by the Common Wealth thinktank, the report, titled A Right to the Sun, proposes a universal Solar Bond scheme under which all households with a suitable roof would be eligible to access solar panels without credit checks or other eligibility requirements that are usually associated with conventional loans.
This cost would be repaid over a 25-year period through household energy bills, with the finance attached to the property rather than the homeowner. This means if you move out, the loan and finance stays with the property, so the panels and their remaining repayments would be passed onto the next occupier.
The report found that such a scheme could save households at least £83 (€96) a year (or almost double with a battery) even when the cost of finance repayments are taken into account.
“Delivering this programme would also provide an opportunity to create jobs and improve livelihoods in communities across the UK,” the report says.
“A healthy market for rooftop solar installations already exists, with the Solar Trade Association estimating more than 42,000 new jobs could be created by an expanded rollout by 2030.”


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A research group in China has investigated how PV installations reshape soil erosion patterns and hydrodynamic processes on hillslopes. Using simulated rainfall experiments, the researchers examined the effects of PV panel installation height, tilt angle, and array configuration on runoff and erosion dynamics.
“PV panels disrupt the natural rainfall-surface interaction through two concurrent mechanisms,” the researchers said. “They intercept rainfall over a substantial portion of the slope, reducing splash detachment and surface sealing, while simultaneously concentrating intercepted water along panel lower edges as high-energy drip flow. This redistribution creates a distinctive spatial pattern in which covered zones receive no direct rainfall while drip lines receive concentrated flow with substantially enhanced erosive energy.”
The researchers conducted rainfall simulation experiments using a 2.0 m × 1.0 m × 0.4 m adjustable soil flume. They filled the flume with clay-loam agricultural soil collected from the southern Loess Plateau in China and set it at a fixed 10-degree slope. Four custom-made PV modules, each measuring 57 cm × 48 cm, were installed above the soil in either a linear (1 × 4) or block (2 × 2) configuration.
The team tested combinations of three installation heights – 0.4 m, 0.6 m, and 0.8 m – and three tilt angles – 30 degrees, 35 degrees, and 40 degrees. The researchers subjected each configuration to simulated rainfall at an intensity of 80 mm/h for 60 minutes. They also tested equivalent bare-slope controls and repeated each scenario twice.
During each trial, the researchers collected runoff and sediment samples and measured flow velocity and depth across covered and uncovered sections of the slope. After each experiment, they recorded the initiation time, length, width, and depth of erosion channels that formed beneath the panel drip lines. They also conducted eight mitigation trials using either 20-cm-wide turf mats or gravel strips.
“Slope-scale sediment export fell by up to 56.6%, with the strongest mean reduction at 0.4 m installation height (46.3%) and 35° tilt (36.3%),” the researchers said.
Transverse rills, or small erosion channels running across the slope, developed beneath the drip lines in every PV configuration, while none formed on the bare slopes.
“Erosive energy accumulated longitudinally, with flow velocity rising by 148% from the uppermost to the lowermost covered zone under laminar (Re <500) and subcritical (Fr <1.0) conditions, so the downslope-most panel row constitutes the critical location for foundation scour,” the researchers said. “Rill location and spacing (48–57 cm) were dictated by panel layout, with rill width and depth increasing with installation height to 3.44 cm and 1.84 cm at 0.8 m, evidencing an externally imposed erosion geometry absent on natural slopes.”
The researchers said the findings show that PV arrays can reduce overall sediment loss while concentrating erosion at specific locations beneath panel drip lines, particularly around the lowest panel rows.
The results were presented in “Photovoltaic panel arrays reshape soil erosion patterns and hydrodynamic processes on hillslopes,” published in the Journal of Hydrology. Researchers from China’s Northwest A&F University collaborated on the study with scientists from the Chinese Academy of Sciences (CAS).

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Five articles into this series, the shape of the US solar manufacturing supply chain is by now familiar: strong at the module stage, progressing yet dependent on imports at the cell level, with very limited capacity at the wafer level creating a choke point for the growing polysilicon capacity behind it. This final article shifts from charting the industry’s current position to projecting where PV Tech Research anticipates it will arrive by 2030 – and what this trajectory means for market participants. As with the other articles in this series, the data and insights in this final part come from PV Tech Research’s new ‘US Domestic Solar Manufacturing Tracker’ report. 
By 2030, PV Tech Research expects the US to have built out most of its currently announced capacity, and for the gap between module and cell capacity documented across this series to have substantially closed. That is a meaningful shift from today’s market, where cell capacity accounts for 33% of module capacity (including thin film), depending on the technology, as detailed in Day 4. If the pipeline converts as expected, the module-cell mismatch that currently forces import dependency at the cell stage should no longer be the binding constraint it is today. 
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The situation remains less definitive further up the supply chain. US polysilicon capacity is projected to reach approximately 50GW by 2030, yet only about half of that volume, roughly 25GW, is anticipated to serve PV applications rather than semiconductor-grade production, reinforcing the shared-capacity dynamic identified in Part 1 of this series. Wafer capacity is expected to expand to around 15GW, a substantial increase from the current 5GW, though it will remain the smallest segment in the chain by a considerable margin. According to PV Tech Research, the most significant structural weakness in the US value chain is expected to emerge by 2030: the constraint identified in Parts 2 and 3 – domestically manufactured polysilicon ingots that frequently lack domestic wafering options and must be processed overseas – will not be fully addressed within the forecast timeframe.
The resulting supply chain by 2030 will still be downstream-concentrated: the bulk of manufacturing investment and capacity growth continues to be concentrated in cells and modules, while polysilicon and, especially, wafer capacity remain the stages most likely to require imports to complete the chain. It is also worth noting that the core value chain, polysilicon through to modules, is not the entire ecosystem. Equipment, materials and the broader supporting infrastructure that this series has not covered in detail remain comparatively underbuilt relative to the core manufacturing stages, and represent a second, less visible gap alongside the wafer shortfall. For example, you can read more about the shortfall in US-made solar glass in our in-depth report on PV Tech last week (subscription required). 
No forecast for this industry can be separated from the policy environment that has driven it, and that environment has been defined as much by risk as by support. The winners of the past few years’ policy volatility have been domestic manufacturers, who have benefited from trade protection and manufacturing credits even as the rules around them have shifted. The cost of that volatility, however, has fallen largely on developers and end buyers, who have repeatedly found themselves on the losing side of a market shaped more by trade policy than by demand. 
The clearest recent example came in August 2026, when Minimum Import Pricing under Section 232 was announced – the price floor mechanism discussed on Day 2 that will extend across the entire solar value chain. MIPs are an effective short-term correction, giving domestic manufacturers room to compete against import pricing that would otherwise undercut the build-out this series has documented. But a price floor is a short-term tool, not a long-term industrial policy, and it does nothing on its own to support the deployment side of the market. The survival of the 45X manufacturing credit is encouraging for the same reason highlighted throughout this series: it is durable and directly supports the manufacturing base. What remains missing is an equivalent downstream policy to govern and incentivise renewable energy deployment, a gap made more conspicuous by the fact that solar is now the cheapest and fastest-to-deploy source of new electricity generation in terms of levelised cost of energy and, paired with batteries, is increasingly dispatchable rather than purely intermittent. 
Across five articles, the data points to a consistent conclusion: the United States has built, and is continuing to build, a substantial domestic solar manufacturing base, concentrated most successfully at the module and, increasingly, the cell stage, with polysilicon and wafer capacity still lagging behind the ambitions implied by the announcement tracker. Manufacturers positioned in the four hub regions, particularly at the wafer stage where competition is thinnest and the gap is widest, stand to benefit most as the market converges toward the 2030 forecast.  
What could derail that outcome is less a manufacturing question than a policy one: a Minimum Import Pricing regime and a 45X credit that support production but not deployment leave the demand side of the market exposed, and it is ultimately demand, not announced capacity, that determines whether the 2030 forecast in this report is realised or not. The foundation has been laid. Whether it supports a thriving domestic industry or becomes stranded capacity will depend on sustaining robust end-market demand for modules in the coming years, particularly as existing tax credits phase out and the industry transitions from policy-driven growth to market-driven momentum. 
Learn more about the topics discussed in this article series at our PV CellTech USA conference in San Francisco on 13-14 October. For details and booking, click here. 

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Nature Communications volume 17, Article number: 4785 (2026) Cite this article
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Photovoltaic-thermoelectric (PV-TE) hybrid systems offer a platform for enhancing the energy conversion efficiency of photovoltaic devices. However, they still suffer from energy losses and limited efficiency improvements owing to underutilized parasitic thermal energy and electrical parameters mismatches between PV and TE components. Here, we presented a comprehensive theoretical analysis and simulation based on a PV-TE Thermo-Electrical Coupling Model, predicting that the maximum efficiency of the system could reach 60.34% with state-of-the-art PV and commercial TE technologies. Following this model, we fabricated hybrid systems with organic and perovskite solar cells coupled with thermoelectric cells, achieving record-high efficiencies of 34.85% and 42.03% at 298 K, and 43.16% and 50.28% at 313 K, respectively, under AM 1.5 G illumination, with optimal thermal utilization and current matching between series-connected PV and TE modules. This work highlights the potential of PV-TE hybrid systems and could offer guidance for designing higher-efficiency systems, driving future advancements in photovoltaics.
Sunlight, as a clean and sustainable energy, has long been harnessed through photovoltaic technology on a global scale to generate electricity. Recently, significant advancements have sparked renewed excitement in the field. Notable progress has been made recently in areas such as organic solar cells (OSCs)^1,2 and perovskite solar cells (PSCs)^3,4, as well as in thermophotovoltaics (TPV)^5,6 and concentrating photovoltaics (CPV)^7,8. TPV and CPV technologies have garnered considerable interest due to their relatively high efficiencies^5,9. However, they require ultrahigh-temperature heat sources¹⁰ and condensers¹¹, which complicates their structural design.
As emerging photovoltaic technologies, solution-processed solar cells such as OSCs and PSCs are catching more interest due to their many unique advantages, such as simple fabrication process, low cost, lightweight, and potential for roll-to-roll large-area production^4,12,13. To date, the best power conversion efficiency achieved for single-junction OSCs and PSCs and silicon-based solar cells has risen to 21%^2,14,15, 27%^4,16,17, and 28%¹⁸, respectively. However, further efficiency improvement remains challenging, as these cells can only utilize a limited portion of sunlight in the visible and near-infrared regions, and also the parasitic heat, mainly generated from the sub-bandgap photons and thermalization loss, has been wasted^19,20. For any photovoltaic technology, following the law of conservation of energy, to achieve the maximum electricity output, the sunlight must be used in the maximum manner and the parasitic heat generated in the process should be minimized or used at the maximum^21,22.
Meanwhile, as it is well-known, the heat generated during the operation of solar cells, primarily from infrared sunlight^21,23 and the parasitic heat^19,20, not only reduces the efficiency (Supplementary Fig. 1) but also impacts their operational lifespan. This issue is particularly critical for emerging OSCs and PSCs technologies. Therefore, converting the parasitic heat from solar cells into electricity, while maintaining a low operating temperature, could not only enhance the utilization of solar energy and boost power conversion efficiency but also extend the lifespan of the solar cells^22,24.
Thermoelectric (TE) technology, which can directly convert thermal energy into electricity via Seebeck effect^25,26, has been long used for parasitic heat recovery and utilization in many scenarios^27,28,29. Thus, it would be a perfect fit to combine PV and TE cells together^21,22. Indeed, since the concept of PV-TE hybrid system was first proposed in the 1970s³⁰, many pioneering theoretical simulations and experimental studies have been carried out^22,31,32,33. But surprisingly, a comprehensive investigation of the literatures finds that, up to date, the best reported simulated efficiency is only 33.8% for PV-TE hybrid systems^9,34. Regarding the experimental studies, unfortunately, the highest reported efficiency of PV-TE hybrid systems is merely approximately 23% under AM 1.5 G solar illumination^32,35. These surprisingly low efficiencies in literatures for PV-TE hybrid systems in terms of both theoretical and experimental studies indicate that there must be some fundamental issues, and thus some serious and comprehensive analysis is warranted. It should be noted that the prevailing efficiency measurement for PV-TE hybrid systems is typically defined as the ratio of electrical energy output to incident solar energy input, which is consistent with the calculation of the power conversion efficiency (PCE) for standalone PV cells^21,22,35. Contributions from ambient thermal energy or additional cooling sources are typically not considered. This method enables a direct comparison of the performance improvement of PV-TE hybrid systems relative to standalone PV cells.
Above the obvious requirement that the PV and TE cells themselves should be state-of-the-art in the PV-TE hybrid systems, it is crucial that the parasitic heat generated from the sunlight must be efficiently converted into electricity through the TE module to maximize overall system efficiency^36,37,38. Moreover, the output power (efficiency) of any PV-TE hybrid system must adhere to basic physics principles for multi-cell systems to achieve the maximum or energy lossless coupling output^39,40. Clearly, the possible maximum output power of the PV-TE hybrid system is the sum of the two given individual PV and TE cells/modules according to the energy conservation law, and this requires the two modules of PV and TE in the hybrid system to have electrically matchable characteristic parameters (note the PV and TE cells are very different in their electrical parameters)^41,42,43. Therefore, to achieve the best-performed PV-TE hybrid system, it is essential to optimize the overall configuration of the hybrid system to simultaneously maximize parasitic heat utilization and achieve energy lossless coupling output between the PV and TE submodules. Considering the overwhelming complexity, systematic modeling is first required. Thus, we constructed a PV-TE Thermo-Electrical Coupling Model (PT-TECM) to analyze the conditions of maximizing heat utilization and matching electrical parameters, and then simulated the efficiency of PV-TE hybrid systems through the COMSOL Multiphysics® platform.
Following the proposed model above, we predicted that the maximum efficiency of the optimized PV-TE hybrid system could reach 60.34% by using the best single-junction PSCs^4,16,17 and commercialized TE materials^26,44,45. Guided by these simulations, our fabricated optimized solution-processed solar cells (OSCs and PSCs) and thermoelectric hybrid systems connected in series achieved record efficiency of 34.85% / 43.16% and 42.03% / 50.28% at environmental temperatures (T_atm) of 298 K/313 K (with the active area of 0.24/0.28 cm²) under AM 1.5 G solar illumination, respectively. To demonstrate scalability, we further fabricated a larger-area 1.0 cm² OSC-TE hybrid system using the same strategy. This larger system also exhibited a high efficiency of 33.48% at 298 K, with an efficiency similar to that of the small-area OSC-TE hybrid system. Furthermore, we fabricated a large-area, flexible and wearable OSC-TE hybrid system capable of directly powering a sensor for real-time pulse monitoring, while the individual OSC module could not achieve this due to its relatively low output power. We believe the results of both theoretical modeling and experimental results demonstrated in this work would provide valuable design guidelines for high-performance PV-TE hybrid systems, particularly for solution-processed PV cells, and significantly expand their application potential in various fields.
As mentioned above, to address the overwhelming complexity and guide the fabrication and optimization of such high-performed hybrid systems, the PT-TECM was established (Supplementary Notes 1 and 2) to achieve both the conditions for optimal parasitic heat utilization and electrical parameter matching and further simulate the efficiency (η) of the PV-TE hybrid systems. The simulation involved the development of heat transfer and equivalent circuit models using the COMSOL Multiphysics® platform (Supplementary Note 2). For efficient harvesting of parasitic thermal energy, TE cells are configured as a stacked module located beneath the PV module in the PV-TE hybrid system (Supplementary Note 1.1). In terms of electrical parameter matching, the maximum output power of the hybrid system can be achieved either by connecting two different types of batteries in parallel or in series (Supplementary Note 1.2). However, as detailed in Supplementary Note 1.3, due to the changing temperature and light intensity of the actual application environment, the voltage and current of both PV and TE cells (modules) vary correspondingly. This would lead to severe charging/discharging between the PV and TE cells if connected in parallel, even leading to overheating or battery damage. But for the system connected in series, the PV and TE cells (modules) can still operate relatively stable. Therefore, the connection of PV and TE modules in series would be the best choice and was thus used in this study. For a series-connected PV-TE hybrid system, current matching must be achieved to maximize output power. This means that the PV and TE modules must produce the same output current at their respective maximum output power points to ensure the highest efficiency of the entire system.
Guided by the theoretical analysis above, a heat transfer model of stacked TE modules was used to optimize thermal energy utilization (Supplementary Note 2.1). Current matching between the PV and TE modules was achieved through the appropriate series and parallel configuration of their respective sub-modules, and then the efficiency of the PV-TE hybrid system was calculated using an equivalent circuit model (Supplementary Note 2.2). Importantly, without maximizing parasitic heat utilization and electrical matching, the series-connected OSC or PSC and TE hybrid systems showed a very limited efficiency improvement of ~2–3% (Supplementary Fig. 2), similar to previous reports^21,24. Using the PT-TECM, it is possible to simulate and optimize the output power and efficiency of any combination of PV and TE cells with different types, and the simulated results subsequently were used to guide us to carry out relevant experiments. Figure 1a illustrates the structure of the PV-TE hybrid system used in both simulations and experiments, where the PV (OSC or PSC) module (top) is connected to the TE module (bottom) in series with a thermal conductive layer between them, and the effective area of the PV and TE modules keeps always the same. Figure 1b shows the detailed structure of the OSC, PSC, and TE cell units used in this study. The TE cells are made of bismuth telluride (Bi₂Te₃)-based thermoelectric materials, due to their high thermoelectric performance and stability at room temperature^46,47. The equivalent electrical circuits for the PV-TE hybrid systems are shown in Fig. 1c. AFM images of the OSC and PSC films reveal smooth surface topography. GIWAXS analysis of the OSC indicates well-defined molecular packing and orientation, while XRD patterns of the PSC confirm the formation of a well-crystallized perovskite phase (Supplementary Figs. 3, 4). UV-vis absorption spectra demonstrate the spectral complementarity between the PV and TE components for efficient solar energy harvesting (Supplementary Fig. 5). These characterizations demonstrate the reproducibility of device fabrication and provide a solid physical foundation for the integration of the PV-TE hybrid system.
a Schematic illustration of the PV-TE hybrid system. PV and TE modules are connected in series, and all TE cells are connected in series. b Architecture of OSC, PSC, and TE cells. For the OSC, the active layer is PM6:L8-BO, with ZnO/NMA and MoOx serving as the electron transport layer and hole transport layer, respectively. In the PSC, the active layer is the perovskite material, while SnO₂ and Spiro-OMeTAD/MoOx function as the electron transport layer and hole transport layer, respectively. The TE cell unit consists of multiple p/n Bi₂Te₃ legs connected in series, with a copper (Cu) layer serving as electrodes. c Equivalent electric circuit of the OSC-TE or PSC-TE hybrid systems. The OSC/PSC is modeled as a single-diode equivalent circuit, which consists of a photocurrent source in parallel with a diode and a shunt resistance (R_sh), together with a series resistance (R_s). The TE cell is modeled as a voltage source (V_TE) with a series resistance (R_TE). I_p is the photogenerated current, I_sh is the current through the shunt resistance, I_d is the current through the diode, and I is the output current.
To optimize the utilization of parasitic thermal energy generated by solar cells, a heat transfer model of the stacked PV-TE hybrid structure was established (Supplementary Note 2.1). Using this model, the thermoelectric temperature difference (ΔT) and voltage (V_TE) can be obtained by solving Equations S3-S8 (Supplementary Note 2.1). The established general heat transfer framework was then applied to a specific OSC-TE hybrid configuration, consisting of a single OSC (with a reported efficiency of 18% in our previous work⁴⁸) and multiple commercial TE cells (see Supplementary Note 2.1 and Supplementary Fig. 6 for details). Both the OSC and each individual TE cell possess an identical active area of 0.04 cm².
To find the best conditions for heat utilization, the ΔT with different numbers (k) of TE layers in the system was simulated using the model, where k ∈ [1, 10] (Supplementary Fig. 7). The hot-side temperature of the PV-TE hybrid system is determined by the heat generated from the solar cell under AM 1.5 G solar illumination, while the cold-side temperature could be controlled by various cooling methods. To achieve a large temperature difference and evaluate the maximum efficiency of the hybrid system, the cold-side temperature of the system was maintained at 0 °C (273 K). In the likely practical applications scenarios, without external energy input, the condition (0 °C) might be achieved when the PV-TE hybrid system is deployed in marine environments with seawater cooling (from the equator to the poles, seawater temperature decreases gradually from ~ 25 °C to ~ 0 °C or even below 0 °C), as well as in polar regions or space, as detailed in the final applications section.
Detailed simulation results for ΔT and V_TE of the 0.04 cm² OSC-TE hybrid systems with different numbers of TE layers are shown in Fig. 2a (the blue shaded area) and Supplementary Fig. 8. In this case, since the overall performance of the series-connected hybrid system is mainly determined by ΔT and V_TE from the TE module²², the relationship between the V_TE and the number of TE layers (k) were selected to evaluate the efficiency of hybrid system. The shaded region in Fig. 2a represents the performance fluctuation range of the system due to changes in the temperature of the surrounding air, which exchanges heat convectively with the system (Supplementary Note 3). As the k increases from 1 to 10, the ΔT and V_TE gradually increase and reach approximately constant when k exceeds 6. Then, the efficiency of the OSC-TE hybrid system was obtained by inserting the V_TE into the equivalent circuit model and solving Equations S9-S13 (Supplementary Note 2.2). As a result, the OSC-TE hybrid system reaches the maximum efficiency improvement (Δη = η_OSC-TE – η_OSC) at k = 6 (the red shaded area in Fig. 2a), compared to the individual OSC.
a Experimental and simulated TE voltage (V_TE) and performance enhancement (Δη) of 0.04 cm² OSC-TE hybrid systems with varying numbers of TE layers (k ∈ [1, 10]), compared to the individual OSC. The cold-side temperature was maintained at 273 K. The shaded regions represent the fluctuation range of V_TE and Δη due to the variation in the temperature of the air that conducting convective heat transfer with the systems. b The current ratio between OSC and TE modules, and Δη of OSC-TE hybrid systems relative to the standalone OSC with varying areas of 0.04 × n cm², where n ∈ [1, 9]. n represents the number of 0.04 cm² OSC units connected in parallel, and m represents the number of 0.04 cm² TE units connected in series in each layer. n is equal to m due to the same areas of the PV and TE units, as well as the final module, to optimize heat transfer. The PV or TE current is measured at maximum power points when modules operate independently. The shaded band indicates efficiency fluctuation of the systems, due to the variation in the air temperature conducting convective heat transfer with the systems. In both (a, b) the center of the error bar is defined as the average value, and error bar is defined as the standard deviation, which is calculated from the statistical results of at least five individual devices. c, d Experimental champion J-V curves of OSC-TE (c) and PSC-TE (d) hybrid systems (n = m = 6, k = 6) at 298 K, corresponding to a system area of 0.24 cm². T_atm is the environment temperature and η is the power conversion efficiency. e J-V and P-V characteristics of the TE module in the optimized OSC-TE and PSC-TE hybrid systems at 298 K. f Power contribution of the PV and TE components in the optimized hybrid systems at 298 K.
Guided by the simulation results above, we fabricated a series of PV-TE hybrid systems using the OSCs with a reported efficiency of 18% in our previous work⁴⁸ and commercial TE cells with measured total Seebeck coefficient of 9.72 mV/K (Supplementary Fig. 6). The bottom temperature of the PV-TE hybrid system was maintained at 273 K. The experiment results of ΔT, V_TE, and Δη of the system with different values of k are shown in Fig. 2a and Supplementary Fig. 9a.
Note the overall performance enhancement (Δη) of the series-connected hybrid system is mainly determined by ΔT and V_TE from the TE module. At k = 6, the ΔT and V_TE reach their maximum value (approximately 28 K and 0.28 V, respectively), indicating that the TE module achieves optimal thermal energy utilization. At this condition, the 0.04 cm² OSC-TE hybrid system achieved its highest efficiency of 23.99% (Supplementary Fig. 9b), corresponding to the Δη of 6.17%. Thus, we can conclude that the PV-TE hybrid system could maximize thermal energy utilization at k = 6. The consistency between the experimental and simulation results demonstrates the accuracy of our model and the reliability of the experimental data, highlighting the potential of our PT-TECM for performance optimization.
Besides maximizing thermal energy utilization, current matching between the PV and TE modules is also a must for achieving maximum efficiency. Note the current at the maximum output power point of the TE module (k = 6) was approximately 7.0 mA, significantly larger than the current of the 0.04 cm² OSC (1.0 mA). Therefore, to achieve the current matching, the PV module must be constructed with n solar cells connected in parallel to increase its current (Supplementary Fig. 10). Meanwhile, the areas of the PV and TE modules must be kept the same. Thus, each layer of the TE submodule was built with m TE cells connected in series, and then each layer of the TE submodule is again connected in series to form the final TE module, maintaining a constant current in the TE module. The effective area of the PV and TE modules must be equal to ensure maximum utilization of parasitic heat, therefore n = m. The corresponding heat transfer model and equivalent circuit model for current matching are shown in Supplementary Notes 2.1 and 2.2. We can obtain the J-V curve and efficiency of the PV-TE hybrid system with different areas (corresponding to n = m ∈ [1, 9], k = 6) by solving the Equations S9-S13 in Supplementary Note 2.2. The simulated results of Δη of the entire hybrid systems with different electrical matching are presented in the shaded area of Fig. 2b. Again, the shaded region reflected the performance fluctuation of the system due to changes in the temperature of the surrounding air, which exchanges heat convectively with the system (Supplementary Note 4). Note the Δη first increases and then decreases gradually. The PV and TE modules achieved electrical matching (the same current) and thus the highest simulated system efficiency when n = m = 6, k = 6.
Guided by these simulation results, corresponding experiments were conducted using the OSCs and TE cells mentioned above. To achieve the best and also reliable experimental results, we prepared various OSC parallel modules with effective areas ranging from 0.04 to 0.36 cm², corresponding to n ∈ [1, 9], to have a complete and systematic experimental evaluation. As indicated by the above results and discussion on heat utilization, the number of TE layers was fixed at 6 (k = 6) to maximize heat utilization. Infrared thermal imaging of the PV-TE hybrid systems illustrates the thermal energy harvesting and the cooling effect provided by the TE modules, highlighting the thermal synergy between the PV and TE components (Supplementary Fig. 11). The experimental current ratio between OSC and TE modules and Δη of the OSC-TE hybrid system with different areas were shown in Fig. 2b. As the system area increases (n = m ∈ [1, 9], k = 6), the measured current ratio of the OSC and TE modules increased from 0.13 to 1.49, with Δη initially increasing and then decreasing, peaking when the current ratio reached 1.0 (Fig. 2b and Supplementary Fig. 12a). At the peak point, the 0.24 cm² OSC-TE hybrid system (n = m = 6, k = 6) achieved the maximum heat utilization and current matching simultaneously, resulting in the highest Δη of approximately 16.0% with energy lossless coupling output (Supplementary Fig. 12b), which is consistent with the simulated results (Fig. 2b and Supplementary Note 4). It is noted that the efficiency calculation method of the PV-TE hybrid system (Equation S13, Supplementary Note 2.2) follows the widely used approach in this field^21,22,35, where the efficiency is the ratio of electrical energy output to solar energy input. Thus, Δη represents the ratio of the electricity generated by the TE module to the input solar energy. Figure 2c shows that the OSC-TE hybrid system (n = m = 6, k = 6) achieved the maximum efficiency of 34.85% at 298 K under AM 1.5 G solar illumination (Supplementary Table 1), with statistical distribution confirming reproducibility (Supplementary Fig. 13a). Furthermore, the efficiency of the 0.24 cm² OSC-TE hybrid system with different TE layers (keep n = m = 6, k ∈ [5, 7]) was also evaluated (Supplementary Fig. 13b), and the experimental results further confirm that k = 6 is indeed the optimal value for maximizing thermal energy utilization.
Following the design guideline for optimal OSC-TE hybrid systems, we further fabricated PSC-TE hybrid systems using high-efficiency PSCs (approximately 26% efficiency) reported in our previous work⁴⁹. As shown in Fig. 2d, the 0.24 cm² PSC-TE hybrid system (n = m = 6, k = 6) achieved the maximum efficiency of 42.03% at 298 K (Supplementary Table 2), with statistical distribution confirming reproducibility (Supplementary Fig. 14). Under optimal coupling conditions, Figure 2e, f show that the total output power of the integrated PV-TE hybrid system is nearly equal to the sum of the maximum output powers of the PV and TE modules when operating independently. This demonstrates that the system achieved energy lossless coupling between the PV and TE components.
Note that all the above simulated and experimental results of the system are based on basic PV and TE cells with an area of 0.04 cm². To investigate the scalability of the system, we fabricated a larger-area 1.0 cm² OSC-TE hybrid system using a single OSC and a six-layer TE module, with both the OSC and TE units having the same area of 1.0 cm². As shown in Supplementary Fig. 15 and Table 3, the 1.0 cm² OSC-TE hybrid system also exhibited similar efficiency improvement from 17.13% to 33.48%. The similar efficiency, independent of the device area, would be important for advancing larger-area PV-TE hybrid systems with high efficiency, as the large-area fabrication would be one of the critical factors for practical applications. So far, both the PV (OSC and PSC) and TE devices have achieved large-area module fabrication. A 204 cm² OSC module with a certified power conversion efficiency of 14.5% was achieved in 2024⁵⁰. The PSC module, with a size of 2 m², achieved a certified record efficiency of 20.05% in 2025⁵¹. Bismuth telluride-based thermoelectric generators have also been widely produced and applied at large scale⁵². These results indicate that the PV-TE hybrid system could have excellent potential for scalability and would be capable of large-scale fabrication and application as technology progresses.
Stability would be also one of the critical factors for practical applications. The PV (OSC and PSC) and TE units have demonstrated excellent stability under continuous illumination and temperature fluctuations. For the photostability, to date, the OSC could retain about 93% of their initial efficiency after 2000 hours of continuous maximum power point tracking (MPPT) under 1 sun condition⁵³. The PSC could also retain about 82% of their initial efficiency after 2500 h of continuous MPPT under 1 sun condition⁵⁴. For the thermal stability, to date, the OSC could retain 94% of their initial efficiency after 1032 hours of 85 °C/85% relative humidity damp heat and 200 thermal cycles (−40 °C to 85 °C) tests⁵⁵. The PSC could also keep 92% of the initial efficiency when ageing at 85 °C for 1800 h and 94% after 200 thermal cycles between −40 °C and + 85 °C⁵⁶. Furthermore, bismuth telluride-based thermoelectric generators have successfully achieved commercial production and application due to their exceptional stability^52,57.
To assess the stability of the PV-TE hybrid system, we compared the stability of the integrated system with that of each standalone unit under the same testing conditions. The photostability of standalone PV (OSC and PSC) and PV-TE (OSC-TE and PSC-TE) hybrid systems was evaluated under the ISOS-L-1 protocol. The experimental results indicated that both the OSC-TE and PSC-TE hybrid systems exhibited enhanced photostability compared to the standalone OSC and PSC due to the ability of TE module to harvest heat and provide cooling (Supplementary Fig. 16). Thermal stability was assessed under the ISOS-T-2 protocol by subjecting standalone PV (OSC and PSC) and PV-TE (OSC-TE and PSC-TE) hybrid systems to thermal cycles between 25 °C and 65 °C. The OSC and PSC integrated within the hybrid system exhibited similar thermal stability compared to the standalone OSC and PSC after 100 thermal cycles (Supplementary Fig. 17a, b). The TE module exhibited negligible change in output voltage and current after 100 thermal cycles (Supplementary Fig. 17c). In summary, the enhanced photostability and similar thermal stability of PV-TE hybrid system compared with the standalone PV units indicate that the integrated system could achieve or even exceed the reported stability of the PV units, demonstrating the system’s potential for reliable, long-term practical applications.
To investigate the key factors influencing the efficiency of PV-TE hybrid system, a parametric study was conducted using the PT-TECM. The simulated results indicated that the efficiency of the PV-TE hybrid system increases with higher TE material figure of merit (zT) and higher PV cell efficiency (Fig. 3a and Supplementary Note 5). Moreover, a larger ΔT across the TE module, achieved by a lower cold-side temperature or a higher environmental temperature, can further improve the system efficiency (Fig. 3b and Supplementary Note 6). Additional parameters that might influence system performance, such as the convective heat transfer coefficient and thermal interface material properties, are presented in Supplementary Fig. 18.
a Effect of the TE-material figure of merit (zT) and efficiency of PV cells on the efficiency of the 0.04 cm² OSC-TE hybrid system (n = m = 1, k = 6) at 298 K under AM 1.5 G solar illumination, with a cold-side temperature of 273 K. b Effect of cold-side and environmental temperatures on the efficiency of the 0.04 cm² OSC-TE hybrid system (n = m = 1, k = 6) under AM 1.5 G solar illumination. The PT-TECM analysis is based on steady-state thermal equilibrium and does not incorporate transient variations in irradiance, wind speed, or ambient temperature. c Simulated J-V curves of optimized PV-TE hybrid systems at 298 K and 313 K under AM 1.5 G solar illumination, with a cold-side temperature of 273 K, based on the state-of-the-art single-junction OSCs^2,14,15 and PSCs^4,16,17 reported in literatures and the best commercial Bi₂Te₃-based TE materials^26,44,45. d Experimental J-V curves of 0.28 cm² PV-TE hybrid systems (n = m = 7, k = 6) at 313 K under AM 1.5 G solar illumination, with a cold-side temperature of 273 K. Note that these peak efficiencies (e.g., > 50%) are obtained under controlled steady-state conditions, including AM1.5 G solar illumination and a regulated cold-side temperature. For performance deviations under outdoor conditions, see Supplementary Figs. 27 and 28. e Comparison of efficiency in this work with representative OSC-TE^24,31, PSC-TE^{21,22,32,35,58}, Inorganic PV-TE^33,40,59,60 hybrid systems under AM 1.5 G solar illumination reported in literatures.
Therefore, to explore the efficiency limit of the PV-TE hybrid system, we further carried out the simulation using the state-of-the-art PV cells (OSC with ~ 21% efficiency^2,14,15 and PSC with ~27% efficiency^4,16,17) and the best commercial Bi₂Te₃-based TE materials (zT ≈ 1.0 at room temperature^26,44,45) (Fig. 3c and Supplementary Note 7). At 298 K (room temperature), the simulated optimal OSC-TE and PSC-TE hybrid systems (n = m = 6, k = 6) could achieve maximum efficiency of 40.67% and 47.06% under AM 1.5 G solar illumination, respectively. At 313 K (the environment temperature in a typical hot summer), the optimal OSC-TE and PSC-TE hybrid systems (n = m = 8, k = 6) could reach maximum efficiency of 53.88% and 60.34%, respectively. The reason that n and m are larger at 313 K compared to 298 K is that the higher environment temperature increases the ΔT and current of the TE module, thereby requiring more PV units connected in parallel for current matching.
To verify the simulated results at 313 K, based on the existing OSC (18% efficiency) ⁴⁸ and PSC (26% efficiency)⁴⁹ in our laboratory, we further measured the efficiency of the OSC-TE and PSC-TE hybrid systems at 313 K. As shown in Fig. 3d, the OSC-TE and PSC-TE hybrid systems (n = m = 7, k = 6) achieved the maximum efficiency of 43.16% and 50.28% at 313 K under AM 1.5 G solar illumination, respectively (Supplementary Tables 4, 5), with statistical distribution confirming reproducibility (Supplementary Figs. 19, 20). Under optimal coupling conditions, the total output power of the integrated PV-TE hybrid system is also nearly equal to the sum of the maximum output powers of the PV and TE modules when operating independently at 313 K (Supplementary Fig. 21). Compared to the simulation results (Fig. 3c), the lower efficiencies observed in experiments (Fig. 3d) can be attributed to non-ideal practical factors, primarily a lower zT of the employed TE module (zT ≈ 0.9, compared to zT ≈ 1.0 in simulation) and the neglect of interfacial thermal resistance in the simulation model. Figure 3e compares the simulated and experimental efficiency of our PV-TE hybrid systems in this work with that of previously reported representative OSC-TE, PSC-TE, and inorganic PV-TE hybrid systems in literatures. Note our optimal OSC-TE and PSC-TE hybrid systems have achieved record-high efficiency of 43.16% and 50.28% at 313 Kunder AM 1.5 G solar illumination, respectively, significantly exceeding the efficiency of previously reported OSC-TE^24,31, PSC-TE^{21,22,32,35,58}, and inorganic PV-TE^33,40,59,60 hybrid systems in literatures (Supplementary Table 6). Our PV-TE hybrid systems also show great advantages and competitiveness, compared with reported tandem PV devices, PV-TE hybrid systems with radiative cooling, and concentrated PV-TE hybrid systems (Supplementary Table 7). Furthermore, as indicated by the simulation in Fig. 3c, it is expected that with continued advancements in PV and TE technologies, the maximum achievable efficiency of PV-TE hybrid systems should be even better.
To explore the application potential of PV-TE hybrid systems, we developed a flexible, large-area OSC-TE hybrid system, and integrated it into wearable clothing (Fig. 4a, Supplementary Figs. 22–24). The integrated system achieved a higher V_oc, enabling direct operation of a pulse sensor for physiological signal monitoring, whereas the individual OSC could not power the sensor (Fig. 4b, Supplementary Fig. 25). Furthermore, the flexible OSC-TE hybrid system maintained stable performance after 1000 bending cycles, demonstrating its potential as a wearable power source (Supplementary Fig. 26).
a Photograph of the flexible and wearable OSC-TE hybrid system. b Demonstration of the OSC-TE hybrid system driving a sensor for real-time pulse monitoring under AM 1.5 G solar illumination, while the individual OSC cannot achieve this. c, Potential applications in ocean environments. From the equator to the poles, seawater temperature decreases gradually from ~ 25 °C to ~ 0 °C or even below 0 °C, which ensures a large and stable temperature difference across TE modules to produce more electricity. d Potential applications in space. The sunny side temperature of the solar panel can reach 120 to 150 °C, while the dark side temperature can drop to −100 to −200 °C. This could create a significant ΔT across the TE module, which can enhance the output power and efficiency of PV-TE hybrid systems in space.
Beyond wearable applications in daily life, our PV-TE hybrid systems might hold potential for deployment in some extreme environments. As shown in Fig. 3b, a lower cold-side temperature can increase the ΔT of the TE-module and thereby improve the efficiency of the PV-TE hybrid system. This prompts us to think about some possible practical scenarios. For instance, seawater serves as a massive constant-temperature source, and Antarctic and Arctic seawater is much cooler in particular. Under solar irradiation, a large ΔT across the TE module can be achieved when the PV-TE hybrid system floats at the seawater surface, providing a possible ideal application scenario for PV-TE hybrid systems (Fig. 4c). To estimate the power generation potential of the PV-TE hybrid system in marine environments, outdoor performance testing was conducted under sufficient sunlight and wind-free conditions, on one day in June (22 June 2025) and three days in September (7 – 9 September 2025) in Tianjin, China (39.1076° N, 117.1734° E). (Supplementary Fig. 27). The OSC-TE hybrid system achieved maximum average efficiencies of 35.03% in June and 29.16% in September, while the PSC-TE hybrid system reached 40.71% in June and 35.83% in September (Supplementary Fig. 28). The higher efficiency of the hybrid systems in June compared to September was attributed to higher environment temperature in June, which is consistent with our laboratory results (Fig. 3b). The outdoor maximum efficiency of the PV-TE hybrid system experienced an efficiency loss of approximately 9% in June and 14% in September relative to laboratory measurements under ideal steady-state conditions (Fig. 3d).
Additionally, space provides abundant solar and thermal energy, with sunlight unobstructed by the atmosphere, enabling solar cells to operate at high efficiency. When integrated into a satellite, the PV-TE hybrid systems could experience a significant ΔT between the sunny and dark sides, enabling the TE module to generate substantial extra amounts of electricity (Fig. 4d). Thus, the greater potential demonstrated in this work indicates that PV-TE hybrid systems might be much more competitive than previously thought for many possible applications and even to support human activities in ocean and space environments in the future (Supplementary Note 8).
It is important to note that these scenarios remain conceptual, and real-world deployment will require corrosion-resistant encapsulation, mechanical durability, and long-term validation. Further specific hurdles that must be addressed include the efficiency and stability limitations of OSCs and PSCs, large-area fabrication and integration of PV and TE modules, and performance fluctuations of the PV and TE units in dynamic environments. Cost consideration also remains a potential issue.
Possible strategies to overcome these issues include material design and interfacial engineering^56,61,62, optimizing the fabrication process for both the PV and TE modules^63,64,65, and dynamic matching circuit design for the integrated systems⁶⁶ (Supplementary Note 9). The PV-TE hybrid system currently exhibits a significantly higher LCOE than standalone PV and TE modules, but reducing the cost of TE modules and extending annual equivalent operating hours of the hybrid system could make the system a more cost-effective option in the future (Supplementary Note 10). It is also important to note that the disadvantage of its unfavorable LCOE might be tuned down for some special environments (e.g., in aerospace, offshore, or payload-limited environments), where higher power density is prioritized over cost per watt.
In this work, through theoretical analysis and simulation based on a PV-TE Thermo-Electrical Coupling Model, we have found the requirements and conditions for such a PV-TE hybrid system to have maximum efficiency with energy lossless coupling output. Guided by the model above, we predicted that, using the best available PV modules and commercialized TE materials, the maximum efficiency of PV-TE hybrid systems could reach 60.34%. Following these theoretical guidelines, we have fabricated such a hybrid system and achieved the highest efficiency with energy lossless coupling output using solution-processed solar cells (OSCs and PSCs). The optimized OSC-TE hybrid systems achieved high efficiency of 34.85% and 43.16% at 298 K and 313 K under AM 1.5 G solar illumination, respectively. Likewise, the optimized PSC-TE hybrid systems exhibited remarkable efficiency of 42.03% and 50.28% at 298 K and 313 K under AM 1.5 G solar illumination, respectively.
Although the results presented in this work demonstrate great potential for the practical application of PV-TE hybrid systems, further research remains necessary to advance toward real-world applications. These efforts include the use of more advanced solar cells and TE modules to enhance the performance and stability of PV-TE hybrid systems, further studies on large-area fabrication and integration of PV and TE modules, and exploration of system performance under varied application scenarios. These aspects are currently being pursued in our group.
Donor polymer PM6 was purchased from Solarmer Materials, Inc. Acceptor molecule L8-BO was provided by Jiaxing Hyper Optoelectronic Technology Co., Ltd. Indium tin oxide (ITO) was purchased from Liaoning Advanced Election Technology. Zinc oxide (ZnO) nanoparticles were synthesized the methods reported in literature⁶⁷. The organic molecule named 2-(3-(dimethylamino)propyl)-1,3-dioxo-2,3-dihydro-1H-benzo[de]-isoquinoline-6,7-dicarboxylic acid (NMA) were synthesized using the methods reported in literature⁴⁸.
Tin(IV) oxide, 15% in H₂O colloidal dispersion, was provided by Alfa Aesar. PbI₂ (99.99%) was purchased from TCI. Cesium iodide (CsI, 99.999%) was purchased from Advanced Election Technology Co., Ltd. 2,2’,7,7’-Tetrakis[N,N-di(4-methoxyphenyl)amino]-9,9’-spiro-bifluorene (Spiro-OMeTAD) was provided by Woerjiming (Beijing) Technical Development Institute. Lithium bis (trifluoromethylsulfonyl)-imide (99.9%) was purchased from Sigma-Aldrich, 4-tertbutylpyridine (96.0%) was purchased from TCI. Poly[bis(4-phenyl)(2,4,6-trimethylphenyl)amine] (PTAA) were all purchased from Xi’an Polymer Light Technology in China. FAI, MACl, and ThPyI were synthesized using the methods reported in literatures^68,69.
The micro thermoelectric cells (model 1MD02-024-03, N = 24) were purchased from RMT Co., Ltd. Each thermoelectric cell, with dimensions of 2 mm × 2 mm × 0.9 mm, consists of 24 pairs of p-n junctions made from bismuth telluride (Bi₂Te₃) material with copper electrodes. The dimensions of a single Bi₂Te₃ thermoelectric leg were 0.2 mm × 0.2 mm × 0.3 mm, and the internal resistance and total Seebeck coefficient of each thermoelectric cell were approximately 3.32–3.5 Ω and 10.0 mV/K, respectively. The Bi₂Te₃ semiconductor grains used for fabricating large-area flexible thermoelectric devices were sourced from Changsha KunYong New Materials Co., Ltd, with dimensions of 1.0 mm × 1.0 mm × 2.0 mm.
The OSCs were fabricated following our previous work with an inverted structure of ITO/ZnO/NMA/PM6:L8-BO/MoO_x/Ag⁴⁸. Firstly, the indium tin oxide (ITO)-coated glass substrates were cleaned by ultrasonic treatment in detergent, deionized water, acetone, and isopropyl alcohol under ultrasonication for 15 min each and subsequently dried by a nitrogen blow. Subsequently, a 15 nm thick layer of ZnO was deposited by spin-coating a ZnO precursor solution (prepared by dissolving 100 mg zinc acetate dihydrate and 28 µL ethanolamine in 4 mL of 2-methoxyethano) on the top of the ITO glass substrates at 3000 rpm for 40 s. After being baked at 200 °C in the air for 1.0 h, the ZnO-coated substrates were transferred into a nitrogen-filled glove box. For the hybrid ETL of ZnO/NMA, the NMA film was deposited upon ZnO film from its solution with a concentration of 0.4 mg/mL in methyl alcohol with ammonium hydroxide of 2% volume (to ensure good solubility) and then baked on a hot plate at 120 °C for 10 min in ambient atmosphere. For the active layer of PM6:L8-BO (weight ratio of 1:1.2), the PM6:L8-BO blend film was generated by spin-coating the blend solution at a spin-coating rate of 3,000 rpm upon the corresponding ETLs, and then was thermally annealed at 110 °C for 5 min. Then, MoO_x (~2 nm) and Ag (~150 nm) were successively evaporated onto the active layer through a shadow mask. The effective area for the devices was 0.04–0.36 cm² (step: 0.04 cm²), determined using an optical profilometer.
The PSCs were fabricated according to our previous work⁴⁹. The ITO substrate was washed sequentially with distilled water, acetone, and isopropanol. Before use, the ITO was cleaned with ultraviolet ozone for 20 min. The SnO₂ electron transport layer (2.5 wt%, diluted by water) was coated on the ITO substrate and annealed in air at 170 °C for 30 minutes. After cooling to room temperature, the substrate was treated with ultraviolet ozone for 10 minutes before spin-coating the perovskite solution. Usually, 1.5 M PbI₂ with 2 mol % CsI was dissolved in DMF/DMSO (v/v, 94/6), and then stirred at 70 °C for 4 hours. The PbI₂ solution was then deposited by spin coating at 1500 rpm for 30 seconds, dried at 70 °C for 1 minute, and then cooled to room temperature. A solution of 2D spacer (2 mg/mL) in isopropanol (IPA) was spin-coated on the PbI₂ film at a spin rate of 2000 rpm for 30 seconds, and annealed at 70 °C for 30 seconds. After cooling to room temperature, a solution of FAI/MACl (90:9 mg mL⁻¹) in IPA was spin-coated on top of the PbI₂ layer at a rotation speed of 2000 rpm for 35 seconds, followed by thermal annealing at 160 °C for 12 min in the air (relative humidity about 65%). After perovskite formation, the samples were transferred to a nitrogen-filled glove box for further processing. For the passivation layer, the ThPyI solution was dissolved in IPA with 4 mg/mL and spin-coated onto the perovskite surface at a spin rate of 5,000 rpm, without any further processing. Then, spin-coated the Spiro-OMeTAD solution (80 mg of Spiro-OMeTAD, 30 µL of 4-tert-butylpyridine and 35 µL of lithium bis(trifluoromethylsulfonyl)-imide (Li-TFSI, 260 mg/mL in acetonitrile) in 1 mL of chlorobenzene) on the perovskite layer with 6000 rpm for 30 seconds. For the thermal stability of devices, 20 mg PTAA was dissolved in 1 mL toluene, then 10 μL Li-TFSI (260 mg/ml in acetonitrile) and 10 μL 4-tert-butylpyridine were added. The PTAA solution was spin-coated on the surface of the perovskite layer at 2000 rpm for 30 s. Finally, a 12 nm MoO₃ layer and 100 nm Ag layer were deposited by thermal evaporation under a pressure of 1.0 × 10⁻⁴ Pa. The effective area for the devices was 0.04–0.36 cm² (step: 0.04 cm²), determined using an optical profilometer.
In the PV-TE hybrid systems, the photovoltaic (PV) solar cells (OSCs and PSCs) parallel module was connected in series with the thermoelectric (TE) module. All TE cell units were connected in series in the TE module. The TE electrodes were soldered with copper wires, while the electrodes of the solar cells were connected to copper wires using conductive silver paste. Subsequently, different electrodes were interconnected via copper wires.
Initially, for optimal thermal energy utilization, a 0.04 cm² PV-TE hybrid system with a single PV and multiple TE cells was constructed. The PV solar cell and each TE cell have an equal area of 0.04 cm². In the PV-TE hybrid systems, the TE cells were attached to the bottom of the PV cell using a phase change thermal interface material (PCM). When multiple TE cells were employed, they were added one by one in the stack using the PCM (Supplementary Fig. 7). The PCM (PTM7950 series from Honeywell Co., Ltd.), with a thickness of 0.2 mm and a thermal conductivity of ~8.5 W/mK, can effectively fill interfacial gaps and reduces thermal contact resistance. Polished brass plates (5 mm × 5 mm) were welded to both sides of each TE cell (2 mm × 2 mm × 0.9 mm) to homogenize the temperature distribution and limit adverse thermal radiative losses.
To achieve current matching, parallel modules of OSCs and PSCs with areas ranging from 0.04 to 0.36 cm² were fabricated (Supplementary Fig. 10). Then multiple TE units were employed in the xy-plane to match the effective area of the PV module. For each layer, the total area of the TE cells was equal to the effective area of the PV module. In the z-plane, the TE cells were stacked layer by layer using the PCM.
The fabrication of a flexible large-area OSC module was performed following a previously reported procedure⁷⁰. The flexible large-area OSC module was fabricated by applying an inverted architecture of PET/ITO/ZnO/NMA/active layer/MoOx/Ag. ITO-coated PET substrate was ultrasonically treated in detergent, deionized water, acetone, and isopropyl alcohol in sequence for 15 min, followed by blowing dry using argon gas. Afterward, the ZnO layer was blade-coated on the pre-cleaned ITO-coated glass at 50 °C in air with a coating velocity of 10 mm/s and a blade-substrate gap of 200 μm, followed by annealing at 120 °C for 15 mins in atmospheric air. After that, a thin film of NMA was blade-coated on ZnO in the air with a coating velocity of 10 mm/s and a blade-substrate gap of 150 μm. PM6: BTP-BO-4CI (1:1.2, D: 9 mg/mL) in chlorobenzene with 0.3 vol% 1,8-diiodooctane (DIO) was blade-coated at 60 °C with a coating velocity of 20 mm/s and a blade-substrate gap of 400 μm in air. Finally, MoOx (~ 6 nm) and Ag (~150 nm) were successively evaporated onto the active layer through a shadow mask (2 × 10⁻⁴Pa). The schematics of the large-area module consisting of four sub-cells connected in series using the methods reported in literature⁷¹. The photograph of the flexible large-area OSC module with a 14.4 cm² effective area is shown in Supplementary Fig. 22.
The dimensions of the p-type and n-type thermoelectric legs based on bismuth telluride materials are 1.0 mm × 1.0 mm × 2.0 mm. An array comprising multiple p-n junction pairs, thermally in parallel and electrically in series, was fabricated with dimensions of 55 mm × 55 mm. Copper wires were used as electrodes to connect different legs. The assembled thermoelectric array was then cast using PDMS, which was prepared by mixing PDMS precursor with curing agent at a mass ratio of 10:1. The embedded array was subsequently cured in an oven at 80 °C for 4 h. Finally, a 1.5 mm-thick PDMS composite layer, containing BN and Al₂O₃ to enhance thermal conductivity, was cast and cured on both sides of the thermoelectric array (Supplementary Fig. 23).
The PDMS/BN/Al₂O₃/DM fabrics were prepared using a 3D extrusion process and a woven process (Supplementary Fig. 24a, b). Initially, the PDMS precursor and curing agent were mixed at a mass ratio of 10:1. Subsequently, BN (500 nm), Al₂O₃ (50 nm), and DM (n-Docosane microcapsule) were incorporated into the PDMS mixture in a container. Then, the mixture was placed in a vacuum mixer for 5 minutes at room temperature to achieve a homogeneous phase. The resulting PDMS/BN/Al₂O₃/DM mixture was transferred to a syringe, extruded, and subsequently cured at 80 °C for 4 h. The fabricated composite fibers had a diameter of 0.6 mm. Finally, the fibers were woven into a fabric, which was integrated into clothing for cooling and heat dissipation applications in flexible PV-TE hybrid devices.
The current density-voltage (J-V) characteristics of PV-TE hybrid systems were obtained using a Keithley 2400 source measure unit, the scan speed and dwell times of the J-V curves were fixed at 0.02 V s⁻¹ and 20 ms, respectively. The photocurrent was measured under AM 1.5 G illumination using an SS-X50 solar simulator, calibrated with a standard Si solar cell (Enli Technology CO., Ltd., Taiwan, and calibrated report can be traced to NREL).
The thermoelectric signals were obtained using a Keithley 2400 source measure unit. Temperature variation and distribution in this work were recorded through a thermocouple, and the temperature can be read out in real-time.
The photostability testing was conducted following the ISOS-L-1 protocol. The standalone PV (OSC and PSC) and the PV-TE hybrid system (OSC-TE and PSC-TE) were encapsulated and tested under continuous 1-sun LED illumination at the open-circuit state in ambient air (20–30 °C, 40–60% RH). The LED spectrum is shown in Supplementary Fig. 16c. J-V characteristics were recorded using a solar simulator after each 48 h light-soaking period.
The thermal cycling stability testing was conducted following the ISOS-T-2 protocol. The standalone PV (OSC and PSC) and the PV-TE hybrid system (OSC-TE and PSC-TE) were encapsulated, then placed in an oven and thermally cycled between 25 °C and 65 °C under dark, open-circuit conditions. J-V characteristics were recorded using a solar simulator after every 20 thermal cycling periods.
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
All data are available in the main text or the Supplementary Materials. Additional information can be obtained from corresponding authors upon request. Source data are provided with this paper.
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This work is supported by the National Key R&D Program of China (2022YFB4200400) (Y.C.) and the National Natural Science Foundation of China (Grant No. 52025033, 52273248, 52303238, and 52473215) (Y.C., R.M., and D.Z.).
These authors contributed equally: Zhanzhao Yin, Ding Zhang.
State Key Laboratory of Elemento-Organic Chemistry, Frontiers Science Center for New Organic Matter, Nankai University, Tianjin, China
Zhanzhao Yin, Longyu Li, Yuping Gao, Yongsheng Liu, Xiangjian Wan & Yongsheng Chen
Key Laboratory of Functional Polymer Materials, Institute of Polymer Chemistry, College of Chemistry, Nankai University, Tianjin, China
Zhanzhao Yin, Longyu Li, Yuping Gao, Yongsheng Liu, Xiangjian Wan & Yongsheng Chen
School of Materials Science and Engineering, Nankai University, Tianjin, China
Ding Zhang & Rujun Ma
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Z.Y. and D.Z. contributed equally to this work. Y.C. and R.M. conceived and designed the project. Z.Y., D.Z., L.L., and Y.G. performed the device fabrication. Z.Y. and D.Z. carried out the performance measurements and simulation calculations. Y.C., R.M., Z.Y., D.Z., Y.L., and X.W. analyzed all experimental and simulated data. Z.Y. and D.Z. prepared the manuscript under the supervision of Y.C. and R.M. All the authors contributed to the revision and comments to the manuscript.
Correspondence to Rujun Ma or Yongsheng Chen.
The authors declare no competing interests.
Nature Communications thanks Lin Jiang, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.
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Yin, Z., Zhang, D., Li, L. et al. Solution-processed photovoltaic and thermoelectric hybrid systems with efficiency exceeding 50%. Nat Commun 17, 4785 (2026). https://doi.org/10.1038/s41467-026-71389-w
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Received: 06 April 2025
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DOI: https://doi.org/10.1038/s41467-026-71389-w
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Surrounded by energy crises, Bihar eyes Rs 1.38 lakh crore solar investment  indianexpress.com
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News of two utility-scale solar PV developments in Texas, where construction began on a 347MW project in Wharton County and finances secured for a 310MW/250MWh pair of projects.
US independent power producer (IPP) OCI Energy and Israeli IPP Areva Power have broken ground on the 347MW SunRoper solar PV project in Texas.
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The project in Wharton County, Texas, is expected to begin operations in December 2027. The companies said that the site is backed by a long-term power purchase agreement (PPA) with an unnamed Fortune 100 company and will add “significant new generating capacity” to the Houston metropolitan area, one of the largest demand areas in the US.
The SunRoper project is being financed by ING and will be built by construction firm WHC.
“SunRoper demonstrates how strategic partnerships can help meet Texas’ growing demand for electricity through investments in critical energy infrastructure,” said Sabah Bayatli, president of OCI Energy, which is owned by the US subsidiary of Korean chemicals manufacturing firm OCI Holdings.
OCI, Areva and ING closed construction financing for the SunRoper project in February 2026, which included funds from US bank BHI and the Israeli Bank of Hapoalim.
The two IPPs have had a relationship in the Texas solar market since 2021, when Areva acquired the 270MW SUNRAY project in Ulvade County from OCI Energy. In May this year, Areva acquired a 50% stake in OCI Energy’s La Salle solar project in Texas. The 670MW project is expected to start commercial operations in 2028, when it will be the largest single-site solar project in OCI’s portfolio.
Sunraycer, the Maryland-based IPP, has closed financing on two Texas solar-plus-storage projects with a combined 310MW/250MWh of solar PV and energy storage capacity.
The closing funds were delivered by Monarch Private Capital, an investment firm which focuses on projects tied to federal tax credits. Monarch provided tax equity financing to both projects, though did not disclose the respective values.
The projects in question are the 127MW/100MWh Midpoint solar project in Hill County, Texas, and the 183MW/150MWh Gaia project in Navarro County. Both projects are backed in part by Environmental Attribute Purchase Agreements (EAPA) with tech giant Meta, signed in May 2025. They began full-scale operations in the first half of this year.
“The completion of Midpoint and Gaia represents another important milestone in Monarch’s mission to finance energy infrastructure that delivers both attractive investor returns and meaningful long-term impact,” said Bryan Didier, partner and managing director of energy, at Monarch Private Capital. “Our partnership with Sunraycer demonstrates what’s possible when sophisticated developers marry with disciplined tax equity investors to bring complex utility-scale long-term energy solutions online.”
“Monarch’s tax equity expertise and engagement have been instrumental in bringing these projects to full-scale operation, and we’re proud of what our teams have accomplished together,” added David Lillefloren, chief executive officer of Sunraycer.
Back in April 2025, the company secured US$475 million in project financing for the Gaia and Midpoint projects.
In addition to these, it is also developing the 400MW Lupinus project in Texas, which is backed by a power purchase agreement (PPA) with Google.

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A decade ago, Karnataka had 29 times Bihar’s renewable capacity. Today it has 39 times. That widening gap is the first half of this story. The second half is that Bihar has just committed ₹1.38 lakh crore to close it with a run to 24 GW, even as Karnataka plots a run to 66 GW of its own.
Karnataka spent the last decade turning an early lead into outright dominance. Bihar spent it almost entirely on the sidelines, and is only now, in the last two years, showing the first real signs of a state that intends to catch up.
2026 renewable capacity
27,442 MW
709.30 MW
Ten years ago, Karnataka was already the country’s runaway renewable-energy leader, with 5,673 MW installed (Till March 2016) against Bihar’s 194 MW — a near-30x gap before either state’s real growth phase had even begun.
Karnataka didn’t rest on that lead; it compounded it. More than 70% of its current renewable base, 7,131 MW, was added after April 2016 alone, anchored by the commissioning of the 2,000 MW-plus Pavagada Solar Park, one of the largest solar parks built anywhere in the world at the time. 
By 2020, Karnataka ranked first nationally with close to 10 GW of renewable capacity. Today that figure stands at 27,442 MW of renewables within a total installed base of over 39 GW. An almost two-thirds share for clean hydro and renewable power.
Bihar’s decade, by contrast, was one of near-total absence from India’s renewable buildout. From a base of 194 MW, the state added just 515 MW of renewable capacity over ten years to reach 709 MW today, a number Karnataka now adds within a matter of months.
Framed as growth rates, Bihar’s 266% increase can look respectable next to Karnataka’s 384%; framed as capacity actually built, Karnataka added roughly 30 times Bihar’s entire current renewable base in that same window. For most of the decade, Bihar simply wasn’t in the race.Karnataka added roughly 30 times Bihar’s entire current renewable base in the time it took Bihar to build what it has today.
What changes the story is the last two years, in which Bihar has moved with a speed that its previous decade gives almost no hint of. The state has set a target of 24 GW of renewable-energy capacity and 6 GWh of energy storage by 2030, backed by a ₹1.38 lakh crore power investment pipeline,  an ambition that, if realised, would mean building out roughly 34 times its entire current renewable base within this decade alone.
The intent shows up in early execution, not just targets. On rooftop solar, Bihar has committed ₹1,512 crore to a first-phase rollout covering 2.5 lakh Kutir Jyoti beneficiary households, part of a wider plan to bring 25 lakh households under the PM Surya Ghar Muft Bijli Yojana by November 2027.
On utility-scale and storage, the tendering pipeline has gone from a standing start to real activity: a 75 MW solar project with SJVN in 2026, and a 125 MW/500 MWh battery storage award at ₹4.44 lakh/MW/month.
The state’s storage ambitions trace back further than they might appear.  BSPGCL first sought developers for a 185 MW solar project paired with a 254 MWh BESS around 2024, followed by a standalone BESS procurement in March 2025. Taken together, these are the first signs of a state moving from isolated projects toward an actual renewable-energy strategy: utility-scale solar, rooftop and storage advancing together rather than in isolation.
Karnataka isn’t standing still while Bihar finds its footing. The state is now preparing a ten-year roadmap to take total installed power capacity from over 39 GW to 66 GW by 2030, with clean energy expansion central to that growth rather than incidental to it. That builds on the Karnataka Renewable Energy Policy 2022–2027, which targets 10 GW of additional renewable capacity over its five-year window and at least 1 GW of dedicated rooftop solar capacity by 2027. A deliberate push to give distributed generation a larger role alongside the state’s large-scale solar parks and hydro base.
The 2026-27 state budget reinforced that direction, with Chief Minister Siddaramaiah extending a ₹48,000 crore electricity subsidy to 36 lakh farmers alongside continued emphasis on decentralised energy systems and rural power sustainability. It’s a signal that Karnataka’s next phase isn’t only about adding gigawatts at Pavagada scale — it’s about spreading renewable and clean power more evenly across the state’s rural and agricultural base, even as the larger 66 GW roadmap takes shape.
Put side by side, these aren’t two versions of the same story — they’re two different clocks running on India’s renewable-energy transition. Karnataka’s has been ticking for a decade and is now accelerating into its next phase, from an already-massive base, with policy, rural distribution and a fresh 66 GW roadmap all reinforcing each other. Bihar’s clock effectively didn’t start until the last two years, and it now has to compress what Karnataka did over ten years into the back half of this decade — 24 GW of renewables and 6 GWh of storage from a base of well under a gigawatt. Taken along with Bengal, Assam  and Odisha plsns in East India, the East has finally woken up, many would say. Opening a whole new, and much needed demand centre for India’s solar manufacturers as well.
We are India’s leading B2B media house, reporting full-time on solar energy, wind, battery storage, solar inverters, and electric vehicle (EV)
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Despite the Wexford Joint Planning Commission’s 8-0 recommendation to deny a zoning amendment that would allow utility-scale solar projects in agricultural/forest production districts, Ranger Power continues pursuing the proposed 375-megawatt Shipyard Solar project and appears to be preparing for a potential review by the Michigan Public Service Commission under Public Act 233.

Staff Writer/Reporter
Despite the Wexford Joint Planning Commission’s 8-0 recommendation to deny a zoning amendment that would allow utility-scale solar projects in agricultural/forest production districts, Ranger Power continues pursuing the proposed 375-megawatt Shipyard Solar project and appears to be preparing for a potential review by the Michigan Public Service Commission under Public Act 233.
• Despite the Wexford Joint Planning Commission’s 8-0 recommendation to deny a zoning amendment that would allow utility-scale solar projects in agricultural/forest production districts, Ranger Power continues pursuing the proposed 375-megawatt Shipyard Solar project and appears to be preparing for a potential review by the Michigan Public Service Commission under Public Act 233.
BUCKLEY — Roughly a month after a proposed zoning ordinance amendment was denied, the company looking to create a large-scale solar farm is still looking to move forward with the project.
On July 27, the Wexford Joint Planning Commission voted 8-0 to recommend denying a proposed zoning ordinance amendment that would have allowed utility uses, including large-scale solar facilities, in agricultural/forest production districts.
Commission members said the proposed amendment was too broad because it would permit a wider range of utility uses beyond solar energy projects.
More than 20 people spoke during a public hearing before the vote, with most opposing the amendment. Many said they support renewable energy but questioned placing a utility-scale solar development on productive farmland. Others urged local officials to preserve agricultural land while encouraging renewable energy in other locations.
Following the commission’s decision, the recommendation was sent to participating township boards, which have 60 days to decide whether to approve or reject the amendment.
So far, Wexford Joint Planning Commission Planning and Zoning Administrator Robert Hall said three townships have confirmed their decisions, with all three upholding the joint planning commission’s recommendation to deny the zoning ordinance amendment. The townships are Hanover, South Branch and Selma.
“I have had a couple townships call asking about the process. I expect a majority will be in by the end of the month upholding the planning commission’s recommendation,” Hall said.
Hall said Ranger Power has also reached out to Wexford Township as part of its preparation to go before the Michigan Public Service Commission. He said the project apparently has increased from 300 megawatts to 375 megawatts based on a letter sent to the township. He said the same letter was sent to Wexford County.
In the Aug. 14 letter Shipyard Solar LLC sent to Wexford County Board Chairman Gary Taylor regarding its proposed solar and energy storage project in Wexford Township, the company said it plans to develop, construct, own and operate an up to 375-megawatt solar energy facility and an up to 375-megawatt energy storage facility in Wexford Township and Grant Township in Grand Traverse County.
The letter also referenced a recent Michigan Court of Appeals opinion upholding the Michigan Public Service Commission’s authority to permit utility-scale renewable energy projects under Public Act 233 of 2023. The company asked Wexford County to confirm that it has not adopted a compatible renewable energy ordinance, or CREO, that would affect the project.
CREOs complies with statewide standards such as setbacks, decibel levels and height requirements, but a renewable energy ordinance is not considered compatible if it is more restrictive than statewide standards.
Shipyard Solar offered to meet with county officials to discuss the project and said that if it does not receive a response within 30 days, or if the county confirms it does not have a CREO, the company intends to file an application directly with the Michigan Public Service Commission seeking a siting certificate for the project.
Wexford County Administrator Joe Porterfield said he is talking with the county’s legal counsel to discuss and produce a written response to the letter.
On Tuesday, the Grant Township Board in Grand Traverse County held a special meeting to discuss the Aug. 14 notice.
The Grant Township board was scheduled to discuss hiring special legal counsel to assist the township in responding to the notice from Shipyard Solar LLC and Ranger Power and to address subsequent matters related to the project.
According to company materials, the project was originally going to generate up to 300 megawatts of electricity but that has since been increased to 375 megawatts based on the Aug. 14 letter. The project also will occupy about 1,800 fenced acres under long-term lease agreements with participating landowners. Ranger Power said it began acquiring land in 2023 and is targeting permitting in 2026, construction beginning in late 2028 and commercial operation by the second quarter of 2030.
The company has said it selected the area because of existing transmission infrastructure, willing landowners and proximity to electricity demand. Ranger Power also says the project would avoid wetlands, streams and sensitive wildlife habitat whenever possible while generating tens of millions of dollars in personal property tax revenue over the life of the project, creating hundreds of construction jobs, providing additional income for participating landowners and allowing farmland to regenerate through permanent vegetation planted beneath the solar arrays.
Despite the planning commission’s recommendation, the future of the project remains uncertain because of Michigan’s renewable energy siting law.
Attorney Catherine Kaufman, who addressed the commission during the July 27 meeting, explained that under Michigan’s Public Act 233, developers of renewable energy projects of 50 megawatts or larger may seek approval directly from the Michigan Public Service Commission if a local government does not have a compatible renewable energy ordinance.
Because the proposed Shipyard Solar project is expected to produce approximately 375 megawatts, it could qualify for that process.
Kaufman also said a local moratorium would not prevent developers from applying directly to the MPSC under state law.
She added that farmland preservation concerns can be considered during the MPSC review process and said local governments may receive up to $150,000 to hire legal counsel if they intervene in a case before the commission.
Hall previously said following the July 27 meeting that it is his understanding the developer would not necessarily have to wait for township boards to complete their review before applying to the state.
• Despite the Wexford Joint Planning Commission’s 8-0 recommendation to deny a zoning amendment that would allow utility-scale solar projects in agricultural/forest production districts, Ranger Power continues pursuing the proposed 375-megawatt Shipyard Solar project and appears to be preparing for a potential review by the Michigan Public Service Commission under Public Act 233.
rcharmoli@cadillacnews.com
Staff Writer/Reporter
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Scientific Reports volume 16, Article number: 16671 (2026) Cite this article
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Maximum Power Point Tracking (MPPT) algorithms, which are employed to extract the maximum power from photovoltaic (PV) systems, exhibit different performance characteristics under uniform irradiance and partial shading conditions (PSC) arising from nonuniform solar irradiance distribution on PV panels. Under PSC, the performance of conventional MPPT algorithms becomes inadequate, leading to increased interest in optimization-based approaches. In this study, the Grey Wolf Optimization (GWO) algorithm, commonly used in MPPT applications, was modified, and an Improved Grey Wolf Optimization (IGWO) algorithm was proposed. A PV system model consisting of four series-connected PV panels and a boost converter was developed in the MATLAB/Simulink environment to evaluate the performance of the proposed algorithm. The effectiveness of the algorithm was tested under nine distinct and complex PSC scenarios. The results obtained under these nine PSC cases were analyzed through comparisons of the proposed IGWO algorithm with GWO, the Cuckoo Search Algorithm (CSA), and the Flower Pollination Algorithm (FPA). The results demonstrate that the IGWO algorithm achieves the highest mean maximum power and exhibits superior MPPT performance compared to the other algorithms, with a mean tracking efficiency of 98.34%.
Global warming, environmental pollution, the depletion of carbon-based fuel reserves, and the steadily increasing energy demand have significantly increased interest in renewable energy sources. Among renewable energy sources, solar energy stands out due to its sustainability and wide applicability. Photovoltaic (PV) panels generate different power levels depending on variable atmospheric conditions such as irradiance and temperature. The relatively low efficiency of PV panels and the requirement to operate at maximum power under varying operating conditions necessitate the use of effective control and tracking methods. Accordingly, Maximum Power Point Tracking (MPPT) algorithms have been developed to maximize the power extracted from PV systems^1,2.
In PV systems, PV panels are connected in series and parallel configurations to achieve the desired power and voltage levels. When all panels in a PV system are subjected to identical temperature and irradiance conditions, the system is considered to operate under uniform irradiance, whereas exposure of panels to different irradiance levels is referred to as partial shading conditions (PSC). Under uniform irradiance conditions, conventional MPPT algorithms such as Perturb and Observe (P&O), Incremental Conductance (InC), and the 0.8 V_oc method provide effective and reliable performance³. However, MPPT algorithms are expected to rapidly reach the maximum power point during transient conditions while minimizing power oscillations under steady-state operation. To satisfy these requirements, advanced MPPT approaches have been proposed in the literature, including Fuzzy Logic^4,5, Artificial Neural Networks (ANNs)^6,7, modified P&O^8,9, and modified InC^10,11.
Achieving maximum power in PV systems operating under PSC constitutes a significant challenge. The primary reason for this difficulty is that, under PSC, multiple maximum power points can occur at different voltage levels depending on the number of PV panels. Among these points, only one corresponds to the highest power, known as the Global Maximum Power Point (GMPP), and operating the PV system at this point is critical for maximizing power generation. The remaining maximum power points are defined as Local Maximum Power Points (LMPPs), and operation at these points results in undesirable power losses. To achieve maximum power under PSC, modified conventional MPPT algorithms have been proposed in the literature^12,13. In addition, scanning-based MPPT algorithms, similar in principle to conventional MPPT approaches, have also been reported in recent studies. MPPT algorithms based on voltage scanning and voltage transient behavior stand out due to their high tracking speed and efficiency^14,15. While scanning-based algorithms offer the advantage of being independent of PV system parameters, high-speed and high-efficiency methods such as voltage segmentation and the modified 0.8 V_oc are known to operate in a parameter-dependent manner with respect to PV panel characteristics^16,17.
Interest in optimization-based MPPT algorithms has increased in recent years, primarily because these algorithms offer a significant advantage over conventional methods by largely eliminating power oscillations under steady-state conditions. Moreover, these approaches are capable of achieving high performance under both uniform irradiance and partial shading conditions (PSC), thereby reducing the need for separate PSC detection mechanisms. It has been demonstrated in the literature that Particle Swarm Optimization (PSO)-based MPPT algorithms provide superior performance under both uniform and PSC conditions compared to conventional hill-climbing methods, while significantly reducing steady-state power fluctuations¹⁸. The Cuckoo Search Algorithm (CSA) is another optimization algorithm widely used in MPPT applications, its effectiveness has been validated in numerous studies. CSA can be applied independently or in combination with the Golden Section Search (GSS) algorithm to achieve faster maximum power point tracking^19,20. Unlike swarm-based algorithms, the Flower Pollination Algorithm (FPA) is inspired by the natural pollination mechanism of flowers and has been extensively employed in the literature to extract maximum power from PV systems^21,22.
In recent years, a wide range of optimization-based approaches has been proposed for MPPT applications, including the honey badger algorithm²³, seagull optimization algorithm²⁴, coot optimization algorithm²⁵, dung beetle optimization algorithm²⁶, zebra optimization algorithm²⁷, search and rescue optimization algorithm²⁸, horse herd optimization algorithm²⁹, salp swarm optimization algorithm³⁰, roach infestation optimization algorithm³¹, falcon optimization algorithm³², arithmetic optimization algorithm³³, and musical chairs algorithm³⁴. The primary objective of these algorithms is to reach the maximum power point with high efficiency and short tracking time under complex PSC scenarios. However, in practical applications, low computational burden is of critical importance for microcontroller-based systems. Reducing the computational complexity of MPPT algorithms enables cost reduction through the use of simpler hardware.
The Grey Wolf Optimization (GWO) algorithm is one of the early optimization-based methods developed for MPPT applications and has become an important benchmark approach frequently used to compare the performance of newly developed algorithms in the literature. GWO models the leadership hierarchy and hunting behavior of grey wolves in nature³⁵. Although the GWO algorithm has demonstrated successful results in identifying the maximum power point in PV systems, further studies have been required to improve its tracking speed. In this context, a modified GWO algorithm in which the convergence factor a was adjusted achieved an approximately 45% improvement in tracking time³⁶. Similarly, enhanced GWO algorithms, in which the convergence factor a was restructured using trigonometric expressions, were comparatively evaluated in combination with the Salp Swarm Optimization algorithm^37,38. The results indicate that the improved GWO algorithms provide significant advantages, particularly for PV systems operating under PSC.
In another study, a newly designed two-stage MPPT algorithm combined the GWO and P&O algorithms, resulting in a substantial improvement in tracking time³⁹. Furthermore, MPPT approaches that hybridize GWO and the Whale Optimization Algorithm (WOA) with the P&O method were proposed, and their performance was compared through simulations conducted under uniform irradiance conditions and using real field data⁴⁰. A high-efficiency hybrid MPPT algorithm, developed by integrating GWO with the Equilibrium Optimizer algorithm, was experimentally compared with Particle Swarm Optimization (PSO), WOA, and FPA methods⁴¹. Similarly, high-efficiency hybrid MPPT algorithms based on CSA-GWO and PSO-GWO combinations were developed, and their effectiveness was validated under different PSC scenarios^42,43. Finally, in a recently developed GWO-based MPPT algorithm, tracking efficiency was further improved, and the experimentally obtained results were presented graphically⁴⁴.
In this study, a novel MPPT algorithm based on Improved Grey Wolf Optimization (IGWO) is proposed by redefining the operating boundaries of the conventional GWO algorithm. The proposed IGWO algorithm is designed to operate over a total of 20 iterations. In the GWO algorithm, if the targeted maximum power is reached within the first 10 iterations, the process is terminated; otherwise, when the desired power variation is not achieved, the duty cycle corresponding to the highest power obtained at the end of the 10th iteration is used as the initial reference, and the algorithm is restarted. With this approach, the search space of the GWO algorithm is constrained, aiming to reach the maximum power point more rapidly within a narrower operating range.
A PV cell can be represented by an equivalent circuit model consisting of a diode, two resistors, and a current source, as shown in Fig. 1^45,46. By connecting PV cells in series and parallel, PV panels are formed, and multiple panels combined together constitute PV arrays. In this study, a PV system model is developed in the MATLAB/Simulink environment by connecting four SunPower SPR-X19-240 PV panels in series.
Single PV cell modelling.
Each PV panel produces approximately 240 W of power at the maximum power point under nominal test conditions (1000 W/m², 25 °C), resulting in a total nominal power of 960 W for the four-panel system. A boost converter is connected to the output of the PV system to supply the load and enable the implementation of MPPT algorithms. To operate the optimization-based MPPT algorithms, the current and voltage values of the PV system are measured. The parameters of the PV panels and the boost converter are presented in Table 1.
In the PV system simulation, the sampling time is set to 1 µs, while the switching frequency of the boost converter is 40 kHz. The MATLAB/Simulink blocks of the PV system simulation are shown in Fig. 2.
MATLAB blocks of the PV system.
When all PV panels in a PV system are subjected to the same irradiance level, the operating condition is referred to as uniform irradiance. The effectiveness of conventional and modified MPPT algorithms under uniform irradiance conditions has been well established. When PV panels are exposed to irradiance levels of varying magnitudes, partial shading conditions (PSC) occur⁴⁷. The P–V curves obtained under uniform irradiance and PSC conditions are shown in Fig. 3.
P–V curves of the PV system under uniform irradiance and PSC.
As shown in Fig. 3, multiple maximum power points occur under PSC. Among these points, only one corresponds to the GMPP, while the others are LMPPs. Conventional MPPT algorithms may converge to LMPPs, which leads to unsuccessful operation under PSC. Consequently, the need to develop new and advanced MPPT algorithms has emerged. Another important aspect to consider is that the number of potential maximum power points formed in different voltage regions increases with the number of series-connected PV panels.
The GWO algorithm, proposed by Mirjalili et al.⁴⁸, mathematically models the hierarchical social structure and hunting strategies of grey wolves. In nature, grey wolves are apex predators that live and hunt in packs. To emulate this leadership hierarchy, four types of wolves are defined in the GWO algorithm: alpha (α), beta (β), delta (δ), and omega (ω). In a grey wolf pack, the alpha leads the group by directing activities such as hunting and migration. If the alpha becomes ineffective, leadership is assumed by the beta. The delta supports both the alpha and beta, while the omega represents the remaining members of the pack. The GWO algorithm is structured according to this hierarchical order. Accordingly, the alpha (α) symbolizes the optimal solution, reflecting the wolves’ leadership hierarchy. The beta (β) and delta (δ) represent the second and third best solutions, respectively, while the omega (ω) denotes all other candidate solutions. During hunting, grey wolves exhibit encircling behavior around their prey. The iterative process begins at the onset of hunting; therefore, the α, β, and δ wolves guide the remaining wolves (search agents) to surround the prey. This encircling behavior is mathematically expressed in Eq. (1).
Here (vec{X}) represents the position of the search agents, (vec{X}_{p}) denotes the prey position, and (vec{A}) is the coefficient vector at the (left( {t + 1} right){text{th}}) iteration. The coefficient (vec{D}) is defined in Eq. (2).
Here, the parameter vectors (vec{A}) and (vec{C}) are obtained using the randomly generated vectors (vec{r}_{1}) and (overrightarrow { r}_{2}) whose elements are randomly selected within the interval [0, 1], as shown in Eqs. (3) and (4).
Here, the components of (vec{a}) decrease linearly from 2 to 0 over the iterations.
During the hunting process, grey wolves update their positions by considering the location of the prey. The α, β, and δ wolves guide the ω wolves toward potential prey locations, thereby coordinating their movements to maximize hunting efficiency. The following equations describe the position-update mechanism through which the pack collectively tracks the prey and maximizes hunting success.
The position vector is modified by (vec{a}) at each iteration to guide the omega-type wolves either toward or away from the prey. The vector (vec{a}) decreases from 2 to 0 after each iteration, as expressed in Eq. (8).
Here, t represents the current iteration number, while N denotes the total number of iterations.
In the IGWO method, upon reaching a certain iteration, the initial conditions or boundaries of the search space are redefined, and the GWO algorithm is restarted to improve hunting performance. Initially, the GWO algorithm is run for 10 iterations. If, at the end of the 10th iteration, the change in power is very small ((Delta P_{pv} < 5)), it is considered that the maximum power has been approached, and the algorithm is terminated. Otherwise, the position corresponding to the closest approach to the prey is determined, and the GWO algorithm is restarted. In the second step, the position obtained in the first step ((D_{new})) is used to define the initial conditions, as shown in Eq. (9). By narrowing the search area in this second step, the success rate of locating the maximum power point is increased.
Here, (D_{1} , D_{2} , D_{3} ,) and (D_{4}) represent the initital duty cycles determined during the first execution of the GWO algorithm and are set to 0.05, 0.3, 0.5, and 0.7, respectively. The flowchart of the IGWO algorithm is shown in Fig. 4.
Flowchart of IGWO algorithm.
In GWO algorithm, if convergence toward the maximum power cannot be achieved within a specified number of iterations under steady-state conditions according to the expression given in Eq. (10) the algorithm is terminated, and the best position attained up to that point is accepted as the optimization result. In this study, after the 10th iteration, the search space was narrowed to achieve faster convergence and higher efficiency. The 10th iteration threshold was determined through extensive trial-and-error analyses. Restarting the algorithm before the 10th iteration led to unfavorable results in certain scenarios. This is because, during the initial startup phase, the assigned duty cycles operate temporarily under transient conditions, which may mislead the algorithm. To mitigate the influence of transient states during initial operation and under dynamic atmospheric conditions, the number of iterations was selected such that each duty cycle (D1–D2–D3–D4) was applied at least twice.
After the 10th iteration, redefining the initial duty cycles using a simple computational technique such that they oscillate around the newly assigned starting value facilitates convergence toward the maximum power point and enhances the algorithm’s efficiency without increasing the number of iterations. In the proposed method, the values added to and subtracted from the new initial value are obtained using 20% of the sum of the first assigned maximum and minimum duty cycles, 20% of the sum of the second and last duty cycles, and the first duty cycle. In the second stage, the newly generated initial duty cycles were associated with the initial state, and the reuse of identical values was prevented. Consequently, after the 10th iteration, the power values oscillate around the newly assigned initial point, and the subsequent position is determined based on the power levels obtained from the preceding and succeeding duty cycles.
If, after the second stage and up to the 10th iteration, the convergence criterion toward the maximum power specified in Eq. (10) is not satisfied, the algorithm is terminated, and the highest power value obtained in the second stage is accepted as the optimization result. Even in cases where convergence to the maximum power is not fully achieved, the IGWO algorithm continues the search process around the maximum power region, as it utilizes the duty cycle transferred from the first stage in the second stage. Consequently, the probability of accurately identifying the maximum power point is increased.
MPPT algorithms are expected to reach the maximum power rapidly and operate at high efficiency under PSC in PV systems. Therefore, the proposed algorithm is tested under nine different PSC scenarios, as illustrated in Fig. 5. The irradiation levels applied to each PV panel for generating the P–V curves illustrated in Fig. 5 are presented in Table 2. All partial shading condition (PSC) scenarios were established at a constant temperature of 25 °C.
Nine different PSCs used to test proposed algorithm.
A detailed examination of Fig. 5 reveals PSCs in which maximum power points of different magnitudes occur across four distinct voltage regions. Moreover, as observed in PSC2b and PSC3, there are challenging scenarios in which multiple power points within the same PSC scenario produce very similar power levels in different voltage regions. Similarly, the maximum power points in PSC1a and PSC3a occur at different voltage regions but have approximately the same power values. In PSC2, PSC2a, and PSC2b, two of the maximum power points are close to each other, while the third is lower, with all occurring within the same voltage region. The PSCs include P–V curves with two, three, or four distinct maximum power values. To evaluate the performance of the proposed algorithm, challenging PSC scenarios were created, and the proposed IGWO method was compared with the GWO, FPA, and CSA algorithms. All methods were tested under identical conditions, with the GWO and IGWO algorithms completing the same number of iterations. Each iteration was applied at equal time intervals. Simulation time and sampling time were taken as equal for all methods. Optimization stopping criteria were limited by Eq. 10 and the maximum number of iterations for GWO and IGWO, while they were limited by Eq. 10 for FPA and CSA. Important parameters of the optimization algorithms are given in Table 3.
As shown in Fig. 6, under PSC1, the PV system achieves a maximum power of 625.24 W using the FPA method. The IGWO algorithm produces a power of 623.01 W, which is very close to the FPA result. Although the GWO algorithm is faster, it achieves only 584.1 W, demonstrating lower performance. While the FPA algorithm generates slightly higher power than IGWO, its tracking speed of 0.245 s is considerably slower.
Power, current, voltage, and duty cycles obtained using GWO, FPA, CSA, and IGWO algorithms for PV system under PSC1 conditions.
As observed in Fig. 7, the IGWO algorithm achieves the highest power of 306 W while reaching the maximum power in approximately the same time as the GWO algorithm. Although the GWO algorithm has a high tracking speed, it delivers the lowest output power of 220.1 W from the system. The FPA algorithm closely approaches the maximum power but demonstrates a very low tracking speed of approximately 0.5 s. The CSA algorithm shows poor performance in both tracking speed and maximum power output. When examining the maximum power obtained under PSC1 and PSC1a conditions, both IGWO and FPA achieve the highest efficiency at the GMPP voltage region; however, the tracking time of FPA is significantly longer than that of IGWO.
Power, current, voltage, and duty cycles obtained using GWO, FPA, CSA, and IGWO algorithms for PV system under PSC1a conditions.
In Fig. 8, the algorithms were executed for the GMPP occurring in another voltage region. Except for the GWO algorithm, all other methods produced approximately the same power. Specifically, FPA, CSA, and IGWO achieved powers of 465.1 W, 465.1 W, and 464 W, respectively, while GWO delivered 461.8 W. The IGWO algorithm reached the maximum power faster than CSA and FPA, with a tracking speed of 0.11 s.
Power, current, voltage, and duty cycles obtained using GWO, FPA, CSA, and IGWO algorithms for PV system under PSC2 conditions.
As shown in Fig. 9, the performance of the algorithms under PSC2 is similar, except for GWO. The GWO algorithm delivers a significantly lower power of 392 W from the PV system. Both FPA and CSA achieve a power of 528.74 W with a tracking time of 0.23 s, showing similar performance. The IGWO algorithm reaches 528.26 W in only 0.11 s, achieving nearly the same efficiency but with a much higher tracking speed.
Power, current, voltage, and duty cycles obtained using GWO, FPA, CSA, and IGWO algorithms for PV system under PSC2a conditions.
As shown in Fig. 10, under PSC2b, the highest powers are obtained using FPA and IGWO, with values of 314.7 W and 314.5 W, respectively. In terms of tracking speed, IGWO reaches the maximum power in 0.115 s, whereas FPA requires 0.35 s, clearly demonstrating the speed advantage of IGWO. For the voltage regions corresponding to the maximum power obtained under PSC2, PSC2a, and PSC2b conditions, FPA produces slightly higher power than IGWO; however, both algorithms achieve approximately the same maximum power. On the other hand, IGWO stands out due to its superior tracking speed.
Power, current, voltage, and duty cycles obtained using GWO, FPA, CSA, and IGWO algorithms for PV system under PSC2b conditions.
Figure 11 shows the power, voltage, current, and duty cycles obtained from the PV system operating under PSC3. Compared to other voltage regions, the FPA algorithm delivers a lower power of 436.8 W, indicating reduced performance. The IGWO algorithm achieves 461.8 W, operating at slightly lower efficiency than FPA and CSA in this scenario. In terms of tracking speed, both GWO and IGWO reach the maximum power in 0.11 s, performing significantly better than the other two algorithms.
Power, current, voltage, and duty cycles obtained using GWO, FPA, CSA, and IGWO algorithms for PV system under PSC3 conditions.
Figure 12 presents the power, voltage, current, and duty cycles obtained from the PV system operating under PSC3a conditions using the different algorithms. While GWO, FPA, and CSA achieve approximately the same power from the PV system, IGWO delivers a lower power of 301.5 W. In this voltage region, the GWO algorithm demonstrates superior performance in terms of both efficiency and tracking speed.
Power, current, voltage, and duty cycles obtained using GWO, FPA, CSA, and IGWO algorithms for PV system under PSC3a conditions.
As observed in Fig. 13, all algorithms achieve the same power from the PV system operating under PSC4. Both GWO and IGWO algorithms demonstrate successful performance with a tracking speed of 0.1 s.
Power, current, voltage, and duty cycles obtained using GWO, FPA, CSA, and IGWO algorithms for PV system under PSC4 conditions.
As shown in Fig. 14, the lowest maximum power generated in the lowest voltage region is obtained using the GWO algorithm. FPA, CSA, and IGWO produce approximately the same power, while the IGWO algorithm stands out due to its superior tracking speed.
Power, current, voltage, and duty cycles obtained using GWO, FPA, CSA, and IGWO algorithms for PV system under PSC4a conditions.
The powers, efficiencies, and tracking speeds obtained under all atmospheric conditions are presented in Table 4. The efficiency of the PV system is calculated as seen in Eq. (11).
where, (P_{pv}) represents the power obtained from the PV system and is calculated as (V_{pv} times I_{pv}). (P_{GMPP}) represents the maximum power that can be obtained in the PSC scenario and is determined using P–V curves as shown in Fig. 5. Tracking speeds of the MPPT algorithms is calculated as in shown Eq. (12). Each iteration occurred at 5 ms intervals. The sampling time of the duty period is 25 µs and is compatible with the switching frequency of the boost converter.
Examining Table 4, the average power obtained across all scenarios is highest with the IGWO algorithm, reaching 367.45 W. The FPA algorithm achieves 365.26 W, coming very close to the IGWO result. This trend is also reflected in the average efficiencies, with FPA and IGWO achieving 97.88% and 98.34%, respectively, which are very similar. A notable difference is observed in tracking speeds: the average tracking time of IGWO is 0.11 s, whereas FPA requires 0.33 s. Thus, in addition to the small advantage in efficiency, IGWO reaches the maximum power approximately three times faster than FPA. The challenging selection of atmospheric conditions and the wide duty cycle range over which power is obtained further demonstrate the robustness and effectiveness of the proposed algorithm.
To evaluate the performance of the proposed algorithm under rapidly varying atmospheric conditions, four different scenarios were designed, as illustrated in Fig. 15. In each scenario, five distinct PSCs states were applied to the PV system at 0.2 s intervals using different combinations. The steady-state power values and the corresponding efficiencies obtained for each PSC state are presented in the figure. Accordingly, it was observed that high-efficiency operation was achieved in nearly all cases. In conclusion, the superior performance of the proposed MPPT algorithm under dynamic atmospheric conditions has been demonstrated.
Power obtained from a PV system operating under four different dynamic atmospheric conditions using the IGWO method.
In Scenario 1, considering the steady-state algorithm efficiencies of five different PSC conditions, a very good average efficiency of 99.6% was obtained. In Scenario 2, the average efficiency of 5 different PSCs applied sequentially was obtained as 98.64%. However, only PSC4 showed low efficiency. Currently, the efficiency of all algorithms under PSC4 is very low. In other cases, the efficiency is quite high. In Scenario 3, the average efficiency was very high at 99.89%. In Scenario 4, the average efficiency was very high at 98.12%. However, the efficiency is again reduced by the operation under PSC4. Very high efficiency was achieved under other atmospheric conditions.
Figure 15 shows that the proposed algorithm is as successful as other methods when the simulations are repeated. When the system is first energized, the voltage rise times can change in transient situations due to the system’s behavior. The duty cycles recorded in the first study can sometimes give misleading results. As seen in Fig. 15, higher power was obtained in PSC3a in Scenario 1 and Scenario 2. Overall, the superiority of the method is clearly seen in the average power obtained from nine different scenarios.
The high speed and efficiency of the IGWO method have been demonstrated through simulation studies. On the other hand, a significant advantage of the IGWO algorithm over other modified GWO and hybrid GWO algorithms is its simple structure. In the proposed approach, if steady-state operation cannot be achieved within the first 10 iterations, the initial conditions are adjusted using a simple procedure and the GWO is restarted. In³⁶, the convergence speed was improved by deriving the convergence factor through an alternative formulation; however, the use of complex mathematical operations increased the overall complexity of the optimization method. In³⁷, a nonlinear convergence factor calculation method was proposed, and the newly introduced approach involving trigonometric expressions further increased the computational burden. In^{38,39,40,41,42,43}, new hybrid optimization algorithms were developed by combining two different optimization techniques. The computational complexity of such hybrid algorithms is inevitably high. A review of the literature indicates that the IGWO method stands out due to its high efficiency, rapid convergence, and low computational complexity.
Tracking the global maximum power point rapidly and with high efficiency remains a major challenge for MPPT algorithms in PV systems operating under partial shading conditions (PSC). Therefore, the development of new MPPT algorithms that simultaneously improve tracking speed and ensure high steady-state efficiency is of great importance. In this study, a PV system consisting of four series-connected PV panels and a boost converter was modeled in the MATLAB/Simulink environment, and nine different PSC scenarios were constructed to evaluate the performance of MPPT algorithms. The widely used GWO algorithm was improved by redefining its operating boundaries, resulting in a novel Improved Grey Wolf Optimization (IGWO)-based MPPT algorithm that offers both high tracking speed and high efficiency. The performance of the proposed algorithm was comparatively analyzed against well-established algorithms in the literature, namely the Flower Pollination Algorithm (FPA) and the Cuckoo Search Algorithm (CSA). The IGWO algorithm was obtained through a simple yet effective modification of the conventional GWO approach. By restricting the search space and restarting the algorithm after a predefined number of iterations, the hunting (search) capability was enhanced, enabling faster convergence toward the maximum power point. To ensure a fair comparison, both the conventional GWO and the proposed IGWO algorithms were executed using the same number of iterations. The results demonstrate that the IGWO algorithm provides a significantly higher average tracking efficiency compared to the conventional GWO. Across the nine PSC scenarios, the IGWO algorithm achieved an average power of 367.45 W, while the FPA algorithm yielded a closely comparable average power of 365.26 W. In terms of average tracking efficiency, IGWO and FPA achieved 98.34% and 97.88%, respectively. However, a notable distinction was observed in tracking speed: the IGWO algorithm reached the maximum power point with an average tracking time of 0.11 s, whereas the FPA algorithm required 0.33 s. These findings indicate that, in addition to delivering high efficiency, the IGWO algorithm offers approximately three times faster tracking performance.
In conclusion, this study introduces a new IGWO-based MPPT algorithm capable of rapidly and efficiently extracting maximum power from PV systems operating under partial shading conditions. Simulation results obtained under challenging PSC scenarios clearly confirm the effectiveness and reliability of the proposed method. Future work will focus on supporting the IGWO algorithm with hybrid structures and further improving its average tracking efficiency, aiming to develop next-generation MPPT algorithms that simultaneously provide very high speed and ultra-high efficiency.
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
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The authors received no funding for this work.
Department of Mechatronics Engineering, Firat University, 23200, Elazig, Turkey
Resat Celikel & Omur Aydogmus
Department of Electrical and Electronics Engineering, Batman University, 72100, Batman, Turkey
Musa Yilmaz
Center for Environmental Research and Technology, Bourns College of Engineering, University of California at Riverside, Riverside, CA, 92521, USA
Musa Yilmaz
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R.C. Writing—review & editing, original draft, software, methodology, investigation, formal analysis. O.A. Writing—review & editing, formal analysis, investigation, visualization. M.Y. Review, original draft, validation, visualization.
Correspondence to Musa Yilmaz.
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Celikel, R., Aydogmus, O. & Yilmaz, M. An improved grey wolf optimization-based MPPT algorithm for photovoltaic systems under partial shading conditions. Sci Rep 16, 16671 (2026). https://doi.org/10.1038/s41598-026-45860-z
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DOI: https://doi.org/10.1038/s41598-026-45860-z
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A research group in China has fabricated a perovskite solar cell using a new molecular design approach aimed at reducing defects in the perovskite film and increasing device efficiency and stability.
“Rapid film crystallization can generate subtle defects that become concentrated at surfaces, grain boundaries, and buried interfaces,” the research’s lead author, Cong Chen, told pv magazine. “Although molecular additives can effectively mitigate these imperfections, their design requires a careful balance: strong interactions favor defect passivation, but excessive molecular packing or the formation of insulating interfacial layers can hinder charge transport.”
“Additive engineering offers a practical strategy to address these challenges by regulating crystallization while simultaneously passivating electronic defects,” he went on to say. “However, effective additives must balance strong defect binding with unhindered charge transport while ideally providing additional protection against environmental stress.”
The proposed molecular design approach is described as a steric-gated dual-site chelation (SGDC) strategy. It consists of a rigid three-dimensional adamantane scaffold bearing two carboxymethyl coordination arms that enable strong chelation of undercoordinated lead ions (Pb²⁺), while sterically regulating local molecular packing.
The scientists used an adamantane-based molecular additive known as ADA-DA, which was designed to passivate defects in perovskite films. Its two carboxymethyl groups can simultaneously coordinate with undercoordinated Pb²⁺, while its rigid three-dimensional adamantane core controls molecular packing and prevents excessive interfacial crowding.
“This combination enables steric-gated dual-site chelation, providing strong defect passivation without hindering charge transport,” Chen said. “Experimental measurements confirmed strong interactions of ADA-DA with both lead and iodide ions.”
The perovskite film resulting from the molecular strategy exhibited larger grains, fewer grain boundaries, and a more compact and smoother morphology, according to the research team. ADA-DA was distributed throughout the perovskite but preferentially accumulated near surfaces and grain boundaries, where defect passivation is most needed. Electrical measurements further confirmed a substantial reduction in both electron and hole trap densities.
The proposed perovskite solar cell was based on an inverted positive-intrinsic-negative (p-i-n) architecture consisting of an indium tin oxide (ITO) substrate, a hole transport layer (HTL) made of nickel oxide (NiOₓ) and a self-assembled monolayer, the perovskite film, a buckminsterfullerene (C60) electron transport layer (ETL), a bathocuproine (BCP) buffer layer, and a silver (Ag) metal contact. The perovskite layer incorporating ADA-DA was deposited using a vacuum-flash crystallization process.
Under standard illumination conditions, the device achieved a power conversion efficiency of 27.39%, compared with 26.24% for a reference cell built without the proposed molecular strategy. The result was verified by an undisclosed independent third-party certification body, the researchers said.
In a further step, the academics scaled up the ADA-DA-modified device from small-area cells to bifacial mini-modules using either transparent ITO or indium zinc oxide (IZO) electrodes on both sides. Under glass/ITO-side illumination, a 13-cell module with an area of 163.93 cm² reached a peak power conversion efficiency of 22.21%, while under IZO-side illumination, a 12-cell module with an area of 151.32 cm² achieved an efficiency of up to 22.33%.
“The bifacial architecture combines high efficiency with transparency and illumination from either side, making it attractive for building-integrated photovoltaics (BIPV),” Chen stated. “More importantly, the 151.32 cm² ADA-DA module maintained nearly constant power output for over 5,000 h under continuous white light-emitting diode (LED) illumination. Its output decreased only from 1,028.5 to 1,024.1 mW, corresponding to 99.6% retention, demonstrating exceptional long-term operational stability.”
Both the cells and modules were described in “Steric-gated dual-site chelation enables 27.3% efficient perovskite solar cells and ultra-stable 22% bifacial modules”, published in Joule. The research team included scientists from Hebei University of Technology, Jiaxing Nanhu University, and Tianjin University.
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Accumulation of dust on solar panels lowers performance and limits energy production, particularly in dry locations. Dust accumulation on photovoltaic panels diminishes performance and reduces energy output, especially in arid regions. This study uses four identical modules in Roorkee, India, from October to December to examine the impact of cleaning frequency on photovoltaic (PV) performance. The reference panel is cleaned daily, while the remaining panels are cleaned weekly, biweekly, and monthly. Alongside short-circuit current measurements, environmental parameters including global horizontal irradiance, ambient temperature, wind speed, and relative humidity are continuously recorded. In this study, soiling loss (%) is examined as the primary performance indicator under various cleaning intervals to observe dust accumulation progression and its impact on the performance of the solar photovoltaic module. Experimental data are utilized to develop an empirical regression model that describes the trend of dust accumulation. The daily average soiling loss ranges between 0.17 and 0.21%. Furthermore, machine learning models, including Decision Tree, K-Nearest Neighbour, support vector regression, artificial neural network, and a stacking ensemble method, are developed for accurate prediction of soiling loss from environmental variables and cleaning frequency. The stacking model consistently achieves the best performance across all months, with root mean square error as low as 0.03–0.045, mean absolute error below 0.03, and R² = 0.999 compared to other models. Moreover, statistical analyses such as Bland–Altman plots and the Wilcoxon signed-rank test are employed to validate the significance and agreement of the predicted outcomes. The study highlights the benefits of data-driven solutions for predictive operation and maintenance of solar photovoltaic systems and provides valuable insights into the impact of cleaning frequency on reducing soiling losses.
The solar energy extensively uses for heating purpose, desalination of water, cooling process and production of electrical energy, serving a diverse array of applications from home to industrial as well in agricultural sectors^1,2. About half of the world’s electricity will come from wind and solar power alone by 2050³. Until then, around two-thirds of India’s electricity will come from solar and wind. The price of solar energy has dropped by around 85% since 2010⁴. The primary factors for India’s solar PV to develop exponentially are the sharp decline in solar energy prices and Indian government policies subsidies schemes to use solar PV technology to produce power. India has abundant solar insolation due its location inside the tropical belt, enabling it to fulfil its daily electrical requirements. It gets an average solar insolation of 4 to 7 kWh/m² and around 2300 to 3200 h of sunlight annually⁵. Numerous components affect the efficiency of solar photovoltaic power generating systems as shown in Fig. 1 such as type of material, spacing of solar cell, module area, tilt angle and orientation, environmental condition, surface dust of solar photovoltaic (PV) panels. The most often occurring element influencing the solar photovoltaic panel performance is surface dust^6,7,8. The accumulation of dust on the surface of solar panels can result in changes in the electrical charecteristics of the panel array. These changes can cause the panels to have a reverse bias, which in turn can result in a loss of power generated by the panels⁹. The synthesized bio-derived TiO₂ nanoparticles using plant extract and demonstrated improved photovoltaic performance through enhanced light absorption and charge transport, highlighting the importance of material-level enhancements for solar cell efficiency¹⁰.The soiling rates vary between 0.05% and 0.55% per day in India^11,12, while in Dhaka, Bangladesh, it is 0.78%¹³.
Factors contributing dust accumulation impacts on PV modules.
Large-scale solar farms in remote locations are particularly affected by the soiling issue. This is due to the fact that regular cleaning and inspection may be challenging and costly, given the expenses of labor and long-distance travel^14,15. In order to determine the amount of power that is lost by dirty photovoltaic modules, it is desirable to have automated soiling detection. Inadequate maintenance of the cleanliness of solar photovoltaic panel surfaces will lead to significant economic losses. Consequently, utilising precise and effective techniques to identify dust buildup on the surfaces of solar photovoltaic panels is crucial. This facilitates prompt cleaning of the panels, thereby ensuring their safe and efficient functioning. Dust and ambient temperature energy losses were quantified using artificial neural network (ANN) and extreme learning machine (ELM) algorithms¹⁶. Both the ELM and ANN models predict 91.42 and 90.69% accurately. Multivariate Linear Regression (MLR) and ANN models were used to estimate dust-related energy and economic losses in solar panels¹⁷. ANN and MLR models estimated dust-related cost and energy losses at 89.97% and 86.78%, respectively. In another study Adaptive Neuro-Fuzzy Inference System (ANFIS) was used to predict the dust-exposed solar module performance¹⁸. The ANFIS model achieves root mean square error (RMSE) of 0.18719 and coefficient of determination (R²) of 0.99803 for monocrystalline silicon PV modules. In comparison, polycrystalline PV modules have an RMSE of 0.87098 and a R² of 0.99714. The study in¹⁹ predicted dust-induced PV panel performance deterioration in Qatar using ANN and MLR models. The ANN model has a R² of 0.537 and mean square error (MSE) of 0.0038, while the MLR model has a R² of 0.167 and an MSE of 0.0082. The ANN model outperformed the MLR model. Study in²⁰ estimated dust losses using artificial neural networks. Modern technology allows ANN to estimate losses with normalized root mean square errors (NRMSE) of 6.79 and correlation coefficients (R) of 0.91. Another of²¹ utilised AMM, MLR, Interactive Multivariate Linear Regression Model (MLRWI) and Response Surface Methodology (RSM), to predict the loss caused by dust on solar PV module surface. The artificial neural network produced better predictive results than the other machine learning models. The results for R² and RMSE are 0.813 and 0.026, respectively. The separate studied carried out focusing on machine learning (ML) methods implemented and their performance as shown in Table 1.
The studies summarised in Fig. 2 show that the reported soiling losses vary widely with location, exposure duration and local climatic conditions —observed rates in the literature span roughly 0.1% to 1.1% perday, with the highest values typically found in arid, dusty environments (e.g. Bahrain, Qatar) and lower values in regions with occasional rainfall or wind cleaning. Differences in measurement period, panel tilt, dust type, and cleaning practice (and the diversity of experimental protocols) make direct comparison difficult. Overall, the literature indicates a clear need for (a) longer-term, standardized measurements across diverse climates, (b) studies that relate soiling loss to measured environmental drivers (global horizontal irradiance (GHI), wind speed (WS), relative humidity (RH), dust deposition rate (DDR)), and (c) predictive models ( ML approaches) validated against controlled experiments^{36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55}. These gaps motivate the present work, which combines experimental measurements with ML models to produce robust soiling-loss predictions.
Real-world natural dust build up on PV panels under four cleaning frequencies (daily, weekly, biweekly, monthly) is studied instead of controlled dust deposition research, offering practical insights.
This work established two empirical models: one for PV short circuit current (I_sc) prediction and one for soiling loss (SL), encompassing both electrical performance and soiling effects.
Novel stacking model for SL prediction and comparison with other ML models (ANN, SVM, KNN, DT) provide strong empirical and data-driven comparisons.
Bland–Altman analysis and the Wilcoxon signed-rank test for model validation provide a unique level of statistical validity, assuring model dependability.
Soiling loss and study duration across locations.
The methodology of the present work is shown in Fig. 3 with an experimental setup consisting of four PV panels subjected to different cleaning frequencies (P1-daily, P2-weekly, P3-biweekly, and P4-monthly). Data including GHI, RH, WS, ambient temperature (AT), and the short-circuit current (Isc) of each panel were recorded using a data logger. Based on the collected data, two empirical models were developed: An Isc model as a function of environmental parameters such as global horizontal irradiance (GHI), ambient temperature (AT), wind speed (WS), relative humidity (RH) are considered in present study, and a soiling loss (SL) model as a function of RH, WS, AT, and cleaning frequency (CF). To improve predictive capability, machine learning models (ANN, SVM, KNN, DT, and stacking ensemble) were implemented for SL prediction. The performance of empirical and ML models was compared using evaluation metrics RMSE, mean absolute percentage error (MAPE), MAE, MSE, and R². The statistical validation using Bland–Altman analysis and the Wilcoxon signed-rank test was carried out to assess the significance and agreement of the models.
Methodology of the work.
The experimental work was conducted in Roorkee, India (29.86°N, 77.89°E), located in the Indo-Gangetic plain and characterised by a subtropical climate with distinct winter, summer, and monsoon seasons. The study period, October to December, represents the dry-winter season with frequent dust-laden winds, moderate humidity, and occasional foggy conditions, making it suitable for investigating soiling effects. Rainfall-induced natural cleaning was avoided to maintain experimental consistency and isolate the effects of environmental variables and manual cleaning intervals.
The experimental setup is shown in Fig. 4, consisted four identical crystalline silicon PV modules, each rated at the same electrical capacity (:{P}_{max}) 20 W, installed outdoors with a fixed tilt angle of 30⁰ and south-facing orientation to maximize solar exposure. All the PV modules are mounted on a common frame to ensure that they experienced identical environmental conditions such as GHI, AT, RH, and WS.
The present study isconducted during the dry season because the primary objectiveis to evaluate the impact of manual cleaning at fixed and predefined intervals (daily, weekly, biweekly, monthly) under controlled accumulation conditions. During the monsoon season, frequent rainfall events act as natural cleaning mechanisms. Such stochastic and uncontrolled cleaning would interfere with the predefined manual cleaning schedules and compromise the controlled comparison between different cleaning frequencies.
Experimental setup: (1–4) PV modules with different cleaning frequencies, (5) weather station, (6) data logger, and (7) PV analyser.
The data acquisition architecture shown in Fig. 4 as follows:
The ATMEGA2560-based data logger is shown measuring:
Global horizontal irradiance (GHI).
Ambient temperature (AT).
Relative humidity (RH).
PV module current and voltage.
Timestamp via real-time clock (RTC).
The Weather Station is separately indicated as the source of:
Wind Speed (WS).
The experiment is conducted under natural outdoor exposure, allowing dust to accumulate under real environmental conditions. Although dust composition may vary geographically, the measured electrical performance and environmental parameters inherently capture the net effect of dust deposition and adhesion. This approach ensures that the dataset reflects realistic soiling behaviour and provides a reproducible foundation for predictive modelling.
To evaluate the impact of cleaning frequency on soiling losses, four panels are exposed to several cleaning regimens. Panel P1 served as the clean reference and underwent daily cleaning, whilst P2, P3, and P4 were cleaned weekly, biweekly, and every four weeks, respectively. Cleaning occurred at 06:00 AM utilizing distilled water and a gentle, lint-free cloth to avert scratches or more surface contaminants. Electrical measurements encompassed the short-circuit current of each module, recorded at consistent intervals as the principal performance metric for dust build-up. Meteorological parameters were continually recorded, including global horizontal irradiance, ambient temperature, relative humidity, and wind speed. The data were obtained using calibrated sensors incorporated with a data logging system, ensuring synchronous environmental and electrical recordings. The experimental configuration included several sensors to enable precise data collection and real-time observation of solar PV system metrics presented in Table 2.
Electrical and environmental parameters (GHI, AT, RH, WS, (:{I}_{SC}), and voltage) were recorded at fixed and uniform intervals using the ATMEGA2560-based data acquisition system. Measurements were sampled at regular intervals (hourly), ensuring temporal consistency throughout the experimental period. For modeling analysis, the recorded data were aggregated into daily average values to represent the cumulative effect of dust deposition under each cleaning frequency. The Fig. 5 illustrates the cleaning schedule followed during the experimental period from October to December, clearly highlighting the systematic maintenance intervals adopted for each panel.
Cleaning schedule timeline (Oct-Dec 2024).
To account for the influence of external climatic variables, meteorological parameters were continuously recorded during the experimental study. The Fig. 6 illustrates the daily average variation of global horizontal irradiance and wind speed, while Fig. 7 presents the daily average variation of relative humidity and ambient temperature from October to December 2024. Statistical description of the collected data is shown in Table 3. These measurements highlight the dynamic environmental conditions under which the PV panels operated, providing essential context for analysing the soiling effect and validating the empirical and machine learning models.
Daily average global horizontal irradiance and wind speed during the experimental period (October–December 2024).
Figure 6 shows that the wind speeds stayed in the moderate range (~ 1–3 m/s) during the experiment. Under such conditions, particle transport and deposition mechanisms are likely to dominate over aerodynamic removal, explaining the observed positive correlation between wind speeds and soiling loss.
Daily average relative humidity and ambient temperature during the experimental study (October–December 2024).
The abrupt increase in relative humidity as shown in Fig. 7 accompanied by a drop in ambient temperature corresponds to short-duration high-humidity events commonly observed during the winter season in the Indo-Gangetic Plain. These events are typically associated with fog formation, condensation, or transient cloud cover rather than measurable rainfall.
The monthly data analysis from October to December 2024 shows how the performance of PV is affected by both changes in the environment and how often it is cleaned. The daily-cleaned panel (P1) consistently exhibited higher short-circuit current values, while progressive reductions were observed in P2, P3, and most significantly in P4, reflecting the cumulative effect of soiling at longer cleaning intervals. The variability of irradiance, indicated by higher standard deviation values, further underscores the dynamic operating environment of the panels. These findings confirm that both environmental conditions and soiling accumulation substantially impact PV output, thereby establishing the need for predictive models. Accordingly, the subsequent section develops empirical models to express.
(:{I}_{SC:})as a function of GHI, AT, RH, and WS, and to quantify soiling loss as a function of RH, WS, AT, and CF, thus providing a foundational framework for later comparison with machine learning models.
The experimental dataset was utilised to derive empirical regression models for short-circuit current and soiling loss in percentage, using global horizontal irradiance, ambient temperature, wind speed, relative humidity, and cleaning frequency as predictors. Multiple linear regression is adopted to establish these relationships.
The regression Eq. (1) obtained for (:{I}_{SC:}) is:
Table 4 presents the estimated regression coefficients for the empirical model of short-circuit current⁵⁶. The results clearly highlight GHI as the most dominant predictor (Estimate = 0.001265, p < 0.001), consistent with the physical dependence of Isc on solar irradiance. Cleaning frequency also shows a highly significant negative effect (p < 0.001), indicating the reduction in current with increasing days between cleaning. Ambient temperature has a small but significant positive effect, while relative humidity shows a minor negative influence. In contrast, wind speed was statistically insignificant (p = 0.238), confirming its limited role in determining (:{I}_{SC:}).
To further ensure model robustness, adjusted R² and residual diagnostics were evaluated before and after removing GHI. The change in adjusted R² was negligible (< 0.001), confirming that irradiance does not contribute to predictive power in the SL formulation. The removal therefore improves model parsimony without compromising explanatory strength, consistent with regression theory principles. The regression Eq. (2) for soiling loss (SL) expressed as,
Table 5 presents the estimated regression coefficients for the empirical model of of soiling loss show that cleaning frequency is the most significant predictor (Estimate = 0.18487, p < 0.001), highlighting the strong impact of longer cleaning intervals on increased soiling losses. Relative humidity also exhibits a significant positive influence (p = 0.007), which may be attributed to dust adhesion and cementing effects under humid conditions. Ambient temperature has a significant negative effect (p < 0.001), suggesting that higher temperatures may reduce relative deposition or increase self-cleaning effects. Wind speed shows a near-significant positive influence (p = 0.048), reflecting its dual role in either removing or redistributing dust. In contrast, GHI is statistically insignificant (p = 0.989), indicating that irradiance itself does not directly drive soiling loss but instead affects PV output through (:{I}_{SC}).
To further ensure model robustness, adjusted R² and residual diagnostics were evaluated before and after removing GHI. The change in adjusted R² was negligible (< 0.001), confirming that irradiance does not contribute to predictive power in the SL formulation. The removal therefore improves model parsimony without compromising explanatory strength, consistent with regression theory principles.
The performance of the four PV modules was evaluated in terms of short-circuit current (Isc), soiling ratio (SR) and soiling loss (SL%). The SR and SL can be computed using I_SC as mention below Eqs. (3) and (4) as,
Where I_{sc soiled} is the short-circuit current of the soiled panel and I_{sc clean} is that of the clean reference panel (P1). The soiling loss percentage (SL%) was calculated as,
To further improve predictive accuracy beyond the empirical formulations, machine learning algorithms were employed using the experimental dataset described in Sect. 3. The empirical SL model achieved a high determination coefficient (R² = 0.978) with low RMSE and MAE; however, the mean absolute percentage error (MAPE = 28%) remained relatively high, reflecting systematic nonlinearities and residual bias. To address these limitations, an ML-based predictive framework was developed.
The experimental dataset consist of both environmental and operational parameters: global horizontal irradiance (GHI), ambient temperature (AT), wind speed (WS), relative humidity (RH), reference short-circuit current of the clean panel ((:{I}_{SC:left(Cleanright)})) and cleaning frequency (CF). Categorical variables such as cleaning interval were encoded as hot encoding method, while all continuous features were standardized to zero mean and unit variance. To assess the relative contribution of selected input variables, a sensitivity analysis using the CAM approach was carried out under the Sect. 2.4. The Fig. 8 presents the scatter matrix of the selected features (CF, GHI, AT, WS, RH, and SL), excluding the month variable. In contrast to the off-diagonal scatter plots, which represent the pairwise correlations between the parameters, the diagonal histograms illustrate the distribution of each parameter.
Pairwise matrix of experimental features and relationship with soiling losses.
Global horizontal irradiance, ambient temperature, relative humidity, and wind speed have substantial temporal autocorrelation, making subsequent measurements not statistically independent. Randomly mixing time-dependent samples destroys temporal structure and leaks temporal data, enabling the model to indirectly learn patterns from subsequent observations. This may result in inappropriately optimistic assessment outcomes and exaggerated performance measures (e.g., R²). To evade this, the dataset was divided into 80% training and 20% testing observations using a time-ordered split. This method retains temporal causality and provides realistic model generalization in practice.
Time-ordered data splitting for temporally correlated environmental data.
Figure 9 compares random and time-ordered (blocked) data splitting for temporally auto-correlated environmental variables (GHI, RH, AT, WS). Random splitting leaks temporal data and inflates performance measures by include samples from comparable time periods in practice and testing. Time-ordered splitting conserves chronology and enables realistic generalization.
In this work, the cosine amplitude method (CAM) was used to assess the correlation between the input and output data. The mathematical formulation of CAM given in Eq. (5).
There is a connection between the cosine function and the dot product, as shown by Eq. (3). Whereas the inner product of two vectors where Xi input vector while Y is the output vector equal to zero when they are at right angles to one another, the product of two vectors that are collinear is equal to one. A greater directional similarity (and hence sensitivity) to the output is seen by features that have higher CAM scores (closer to 1) than those with lower scores as shown in Fig. 10.
Cosine amplitude scores (CAM) for feature importance.
CAM is the cosine similarity between the features (GHI, AT, WS, RH and CF) and target vectors soiling loss (%) is scale-invariant. The Fig. 9 indicate that the CF is the strongest driver of soiling loss in comparison with meteorological variables which shows moderate CAM score.
To evaluate potential multicollinearity among environmental predictors, the Variance Inflation Factor (VIF) was computed for GHI, RH, and AT. The obtained VIF values were 1.023 (GHI), 1.5987 (RH), and 1.5809 (AT), all of which are substantially below the threshold value of 5. These results confirm the absence of significant multicollinearity and demonstrate that the regression coefficients are stable and not adversely affected by linear dependency among predictors.
To further validate feature sensitivity, permutation importance analysis⁵⁷ was performed, as shown in Fig. 11 Cleaning frequency was identified as the dominant predictor of soiling loss, consistent with physical dust accumulation mechanisms. Environmental variables such as ambient temperature, relative humidity, irradiance, and wind speed exhibited smaller but meaningful contributions. The inset plot provides a detailed view of environmental feature importance. These findings confirm the robustness and physical consistency of the predictive model.
Permutation-based feature importance analysis.
After pre-processing data, soiling loss are estimated using various method of ML such as DT, KNN, SVM, ANN and Stacking. The stacking ensemble combined ANN, SVM, and DT as base learners, with a gradient boosting regressor (GBR) as the meta-learner. GBR effectively refined the base predictions by capturing residual nonlinear patterns, resulting in improved accuracy and reduced bias. MATLAB R2024 a is used to run simulations on a Dell laptop, featuring a Core i9-11900 H processor and 32 GB of RAM.
ANN is intended to replicate the neuronal organisation of the human brain by employing layers that are interconnected in order to capture complicated interactions, as seen in (Fig. 12)^58,59. Backpropagation is used to train the model in this study in order to minimize the errors. In given Eq. (6) w_n are representing weights corresponding to each inputs x_n and b is the bias, while final predicted output represented by Y.
ANN model.
Figure 13 shows how support vector machine (SVM) uses kernel functions to divide data in high-dimensional regions and capture complicated connections for classification and regression⁶⁰. It estimates SL in this work and expressed in Eq. (7) as,
where, Z is the input vector, W is the weight and B is the bias term.
Support vector machine (regressor).
RT, as seen in Fig. 14, are decision trees used to forecast continuous variables SL by segmenting data according to defined criteria and computing the mean target value for each subgroup. They are interpretable, resilient to outliers, and adept at managing non-linear connections successfully⁶¹. The proposed approach employs regression trees by segmenting the feature space and predicting the target variables SL inside each segment as mentioned in Eq. (8).
Decision tree (regressor).
In a regression tree, N is the total number of nodes (leaves), each region Z_n represents a partition of the feature space, C_n is the mean target value within that region, and I (X∈Zm) is an indicator function that equals 1 if X belongs to Z_n otherwise 0.
The K-nearest neighbours (KNN) algorithm is a simple, non-parametric method that predicts outputs based on the average of the k closest data points in the feature space as shown in Fig. 15. In the context of this work, KNN estimates soiling loss by finding similar conditions of GHI, AT, WS, CF and RH from experimental data. It is intuitive and effective for capturing local patterns without requiring an explicit training phase.
K-nearest neighbours.
In KNN regression equation where K is the number of nearest neighbours, N_K(X) is the set of those neighbours, and Y_i​ are their target values, making the prediction the average of the K closest points.
The stacking model is an ensemble learning approach that combines multiple base learners to improve predictive performance shown in Fig. 16. In this work, ANN, SVM, and DT were used as base learners to capture diverse data patterns, and their outputs were blended by a Gradient Boosting Regressor (GBR) as the meta-learner. This framework leverages the strengths of each individual model while compensating for their weaknesses. As a result, the stacking model achieved higher accuracy and robustness compared to single-model approaches.
For the L base learner the stacking prediction given in Eq. (10) as,
where m_L (x) are the base learner outputs (ANN, SVM, DT in this work) and g(⋅) is the meta-learner (GBR) that combines them to produce the final output.
Stacking ensemble model.
In this study, conventional random k-fold cross-validation was not employed because the dataset represents a physically time-ordered environmental process. Environmental variables such as irradiance, temperature, humidity, and wind speed exhibit strong temporal autocorrelation and causal continuity. Randomized cross-validation would mix past and future observations, introducing information leakage and leading to overly optimistic performance estimates, particularly for stacking models where the meta-learner learns second-order correlations. To ensure physically realistic and leakage-free validation, a time-ordered training–testing strategy was adopted. As illustrated in the Fig. 17, the dataset is divided chronologically into a training window (earlier observations) and a testing window (later unseen observations). The base learners (ANN, SVM, and DT) were trained exclusively on the training window, and their predictions were used to train the meta-learner (Gradient Boosting Regressor). The trained stacking model was then evaluated only on the testing window, which contained future unseen samples.
Time-aware training of stacking ensemble without cross-validation leakage.
The hyperparameters of all machine learning models, including ANN, SVM, DT, KNN, and the GBR used in the stacking ensemble, were selected using a systematic tuning procedure based on grid search combined with validation on the training dataset⁶². The hyperparameter combinations presented in Table 6 which is used to all machine learning model in order to minimized prediction error while avoiding overfitting. For each model, a range of candidate hyperparameters was evaluated, and the optimal configuration was selected based on minimum RMSE and stable generalization performance. The same training dataset and evaluation criteria were applied consistently across all models to ensure fair comparison.
It is important to evaluate the precision of the prediction model. A variety of measures have been used to evaluate the precision of predicting PV output power production¹², which include:
(a) Mean Absolute Error (MAE): Computes the average of absolute differences between actual and predicted values, giving equal weight to all errors as expressed in Eq. (11)
(b) Mean Square Error (MSE): Measures the average of squared differences between actual and predicted values, penalizing larger errors more, its mathematical expression mention in Eq. (12)
(c) Root Mean Square Error (RMSE): Square root of MSE, expressing as Eq. (13) prediction error in the same units as the target variable.
(d) Coefficient of Determination (R²): Indicates how much variance in the actual data is explained by the model, with values closer to 1 showing better fit.The expression shown below in Eq. (14)
(e) Mean Absolute Percentage Error (MAPE): Represents as shown in Eq. (15) the average absolute error as a percentage of actual values, useful for relative accuracy.
This section discusses the performance and findings of the suggested models. The testing findings under actual environmental condition from the Roorkee area, India, are also given according to month and cleaning frequency. The empirical model produced from the experimental data is constructed and compared with machine learning models.
The performance of the four PV modules was evaluated in terms of short-circuit current (Isc), soiling ratio (SR), soiling loss (SL%), and current–voltage (I–V) and power–voltage (P–V) characteristics. The daily-cleaned panel (P1) was taken as the clean reference, while P2, P3, and P4 represent panels cleaned at weekly, biweekly, and monthly intervals, respectively.
Short-circuit current ((:{I}_{SC:})) was adopted as the primary soiling indicator because dust accumulation predominantly reduces optical transmission, directly affecting photocurrent generation. Since Isc is approximately proportional to irradiance, it provides a linear and direct measure of optical attenuation. In contrast, power output (P_max) incorporates nonlinear temperature and fill factor effects, which may obscure pure dust-related losses.
In addition to (:{I}_{SC:})based metrics, I–V and P–V curves were generated using a calibrated PV analyzer for each panel at different cleaning intervals. These curves provide detailed insight into the effect of dust accumulation not only on the short-circuit current but also on the maximum power point (.
(:{P}_{MPP})), open-circuit voltage ((:{V}_{oc})), and fill factor (FF). Figures 18 and 19 shows daily average.
(:{I}_{SC:}) and daily soiling ratio (SR) trend over the time period of the experiment while Fig. 20 illustrate about the monthly soiling loss in percentage.
Daily average variation of short-circuit current for PV panels with different cleaning frequencies.
Daily variation of soiling ratio for PV panels cleaned at different intervals.
Monthly average soiling loss (%) for PV panels with different cleaning intervals: P2 (clean weekly), P3 (clean biweekly), and P4 (clean monthly).
The Figs. 18, 19 and 20 collectively illustrate the impact of cleaning frequency on PV performance. The daily-cleaned panel (P1) maintained the highest (:{I}_{SC:}), while P2–P4 showed progressive reductions with longer cleaning intervals. The soiling ratio (SR) exhibited a stepwise decline within each cleaning cycle, steepest for the monthly-cleaned panel (P4). Monthly average soiling losses confirmed this trend, increasing from 1 to 1.5% (P2) to 2.5% (P3) and5% (P4). These results clearly demonstrate that extended cleaning intervals accelerate dust-induced performance degradation.
PV analyser measurement of I-V and P-V characteristics at GHI 415 w/m² (a) P1:-clean daily (b) P2:- clean weekly (c) P3:- clean biweekly (d) P4:- clean monthly.
Figure 21 shows the I–V and P–V characteristics of the four PV panels under different cleaning frequencies. The clean reference panel (P1, cleaned daily) achieved the highest maximum power point ((:{P}_{MPP}) = 6.00 W) and current at MPP ((:{I}_{mpp}) = 0.370 A). Panels with reduced cleaning frequency demonstrated progressive reductions in both (:{I}_{mpp}) and (:{P}_{MPP}): P2 (weekly) produced 5.80 W, P3 (biweekly) dropped to 5.59 W, and P4 (monthly) showed the lowest performance at 5.31 W. The open-circuit voltage ( (:{V}_{oc})) remained relatively stable across all panels, indicating that dust accumulation primarily impacts the short-circuit current and the maximum power output.
Validation performance of empirical (:{I}_{SC:}) models for four PV panels under different cleaning frequencies during October–December are shown in Fig. 22 as, (a) RMSE, (b) MAE, and (c) MAPE. Panels P1–P3 (daily, weekly, and biweekly cleaning) maintained low errors (RMSE ≤ 0.009 A, MAE ≤ 0.007 A, MAPE = 1–1.5%), whereas P4 (monthly cleaning) exhibited significantly higher deviations (RMSE up to 0.020 A, MAE = 0.017 A, MAPE = 3.7% in December), highlighting the negative impact of extended cleaning intervals on model accuracy.
Monthly cleaning frequency wise performance evaluation of empirical model (a) RMSE (b) MAE (c) MAPE.
Figure 23 shows that the four PV panels have a R² value of 0.99 or above, proving that the empirical modelling framework is reliable for describing the changes in I_(SC) under various cleaning conditions. The gradual decline in R² with reduced cleaning frequency highlights the sensitivity of empirical models to dust accumulation patterns.
Experimental vs. Empirical model R-Squared plot (Oct-Dec 2024) of (a) P1:-clean daily (b) P2:- clean weekly (c) P3:- clean biweekly (d) P4:- clean monthly.
The three PV panels are compared from October to–December to analyse the measured vs. projected soiling loss (SL, %) as shown in Fig. 24. The estimated regression line explains most of the variation (monthly R² =0.97–0.98) with minor absolute errors (RMSE = 0.35, MAE = 0.27). The moderate relative error (MAPE = 26–31%) suggests systemic bias or nonlinear effects that the basic empirical fit cannot capture. The following part uses machine-learning to lessen this relative inaccuracy.
Experimental vs. empirical SL model R-squared plot for month (a) October (b) November (c) December.
The empirical SL model’s performance is shown in Fig. 25. PV Panel (a) displays the regression plot between actual and expected soiling loss values. The data points closely correspond with the fitted regression line, indicating a high coefficient of determination (R2 = 0.978). The model’s low RMSE (0.364) and MAE (0.278) validate its trend capture. However, the mean absolute percentage error (MAPE) remains greater (28%), showing relative variances, especially for lower SL values. Panel (b) shows the residual distribution, where errors are centred around zero but include outliers. This residual spread shows systematic deviations not completely represented by the empirical formulation, motivating the upcoming section to use sophisticated machine-learning algorithms to minimize relative error while maintaining high R².
Overall empirical SL model plot of (a) R-squared (b) residual.
Although the empirical SL model achieved a high coefficient of determination (R² ≈ 0.978), the MAPE value (~ 28%) appears relatively high. This is primarily due to the sensitivity of MAPE to small denominator values. Since several SL observations fall within low ranges (below 2%), even small absolute deviations lead to inflated percentage errors. Furthermore, the linear regression framework may not fully capture nonlinear dust accumulation patterns, contributing to structural bias at low SL levels. This constraint led to the introduction of machine learning algorithms to make percentage-based predictions more accurate. .
Experimental data was used to develop the machine learning model. The model inputs are AT, GHI, RH, WS, and CF, while the target variable is solar PV module SL. The prediction data size was 5 × 336 and divided 80:20 for training and testing, as shown in Table 7. SL is predicted using stacking, ANN, SVM, DT, and KNN models.
Statistical metrics are needed to assess machine learning models’ prediction performance for reliability and robustness. This research evaluated solar panel soiling loss models using MAE, RMSE, MAPE, and R2. These measures show the models’ capacity to reduce prediction errors, capture data variability, and generalize across environmental conditions.
The stacking ensemble model’s tight alignment of projected and observed responses in training and testing datasets showed high predictive performance in Fig. 26. The Fig. 27 shows residual plots with random residuals around zero, confirming the model’s dependability and lack of systematic bias. Performance metrics as shown in Table 8, which demonstrated the model’s resilience, with R² values of 0.9995 (training) and 0.9997 (testing) and low error values (RMSE: 0.0566 and 0.0456; MAE: 0.0404 and 0.0333). The stacking model generalizes effectively across datasets and outperforms individual models, making it a very accurate soiling loss prediction framework.
R² plot of (a) Training (b) Testing data set.
Residual plot of (a) Training (b) Testing data set.
To examine whether cleaning frequency (CF) dominates the learning process, a feature ablation study was conducted by evaluating models trained using (i) the full feature set, (ii) CF alone, and (iii) environmental variables alone.
The full model consistently achieved the lowest prediction error. Although CF-only models exhibit strong correlation with soiling loss due to their causal relationship, they produce significantly higher absolute errors compared to the full model. Conversely, models trained exclusively on environmental variables perform poorly. Table 9 shows that environmental characteristics give important extra information and that cleaning frequency does not hide environmental learning.
The scatter plots reveal that the ANN model accurately predicted soiling loss as shown in Fig. 28. The stacking model had somewhat less departures from the ideal prediction line than the ANN, especially at higher response levels. The Fig. 29 residual plots reflect this tendency, with residuals spreading more broadly and displaying patterns at extreme values, indicating small bias in specific ranges. Performance measurements is shown in Table 10, which supports this result, with R² values of 0.9923 (training) and 0.9806 (testing) and greater error levels (RMSE: 0.2138 and 0.2822; MAE: 0.1277 and 0.1358). The ANN model is highly predictive, but its error distribution and somewhat lower accuracy than the stacking model suggest it cannot completely capture nonlinear data variability.
R² plot of (a) Training (b) Testing data set.
Residual plot of (a) Training (b) Testing data set.
Compared to the ANN and stacking models, the DT model predicted well but had lesser accuracy. The scatter plot (Fig. 30) demonstrates that although projected responses track the actual values, deviations from the ideal prediction line are greater at higher response levels. The Fig. 31 shows residual plots with larger dispersion and predictable patterns, showing overfitting in specific areas. This is supported by performance measurements (Table 11), including R² values of 0.9767 (training) and 0.9777 (testing), and higher error levels (RMSE: 0.3766 and 0.4020; MAE: 0.1989 and 0.2090). The DT model captures the input-soiling loss connection, but its restricted generalization and higher residual spread make it less suitable than sophisticated ensemble approaches.
R² plot of (a) Training (b) Testing data set.
Residual plot of (a) Training (b) Testing data set.
The scatter plots are shown in Fig. 32 to show that the Support Vector Machine (SVM) model predicted values that matched observed responses. Significant departures from the ideal prediction line, especially at higher response levels, imply limits in catching extreme instances. The Fig. 33 residual plots show hetero-scedasticity in predictions, with errors spreading further at higher response levels. Performance measures (Table 12) indicate lower R² values (0.9527) and greater error values (RMSE: 0.5365 and 0.4418; MAE: 0.3124 and 0.2917) compared to ANN and stacking. SVM has superior generalization and testing performance than DT, but its lower accuracy and higher residual spread restrict it compared to the stacking ensemble.
R² plot of (a) Training (b) Testing data set.
Residual plot of (a) Training (b) Testing data set.
As seen in the scatter plots (Fig. 34), the k-Nearest Neighbor (kNN) model had mixed predictive performance, with projected values following the genuine responses but deviating at higher response levels. The Fig. 35 residual plots show higher error dispersion and systematic bias in extreme ranges, indicating model resilience is lowered. Performance measures (Table 13) demonstrate high generalization on test set but poor fit during training, with R² values of 0.7662 (training) and 0.9701 (testing). Our error measurements were greater, with RMSE values of 1.1926 (training) and 0.4657 (testing) and MAE values of 0.6948 and 0.3904. Despite good testing accuracy, the kNN model’s large training error and residual spread overfit local patterns and impair dependability compared to stacking.
R² plot of (a) Training (b) Testing data set.
Residual plot of (a) Training (b) Testing data set.
The slightly higher testing R² compared to training R² for the KNN model is attributed to the local interpolation nature of KNN and the distribution of samples in feature space, rather than data leakage. Since testing samples fall within well-represented regions of the training feature space, stable prediction performance is achieved.
To assess potential overfitting and validate the generalization capability of the proposed stacking ensemble model, learning curve analysis is performed in accordance with statistical learning theory. The training and validation errors were evaluated as a function of increasing training data size. The learning curves demonstrate that although the training error decreases with increasing sample size, the validation error converges to a stable and closely aligned value without divergence. The narrow gap between training and validation errors confirms that the stacking model does not suffer from overfitting and generalizes well to unseen data.
The ensemble structure successfully balances the bias-variance trade-offs, so the validation error does not increase as the model capacity increases. These results provide theoretical and empirical evidence that the high R² values achieved by the stacking model are due to robust learning rather than memorization of the experimental dataset.
Learning curve for training and testing.
Also, the fact that the validation error doesn’t go up when the model capacity goes up shows that the ensemble structure does a good job of balancing bias and variation. Learning curves showing in Fig. 36 convergence of training and testing RMSE for the stacking ensemble, indicating strong generalization and absence of overfitting.
To assess the robustness of the stacking model against potential measurement noise, controlled Gaussian perturbations (± 3%) were introduced to environmental input variables. The model was retrained using the same train–test partition to ensure consistency. Figure 37 illustrates the residual distribution comparison between the original inputs and perturbed inputs.
Residual distribution: original vs. noisy inputs (± 3%).
Model accuracy diminishes with longer cleaning intervals, with weekly cleaning reaching R² = 0.995 and low RMSE (between 0.05 and 0.1), whereas monthly cleaning drops R² to 0.964 and raises RMSE over 0.4, as shown in Fig. 38. In all intervals, the stacking model had the lowest error (e.g., MAPE < 10%, RMSE ≈ 0.05) and greatest R² (> 0.99), demonstrating its durability over individual models.
Model performance comparison (MAPE vs. RMSE, bubble ∝ R²) across cleaning frequency intervals (a) clean weekly (b) clean biweekly (c) clean monthly.
Figure 39 demonstrates that stacking had the lowest MSE at all cleaning intervals: 0.003 (weekly), 0.001 (biweekly), and 0.001 (monthly). Empirical and KNN models had the largest errors, 0.022–0.294 and 0.144–0.158, respectively, especially during longer cleaning intervals. These findings confirm that stacking provides the most accurate and consistent forecasts regardless of cleaning frequency.
Model MSE comparison across cleaning-frequency intervals (a) clean weekly (b) clean biweekly (c) clean monthly.
Figure 40 shows MAE fluctuation by cleaning interval. The stacking model had the lowest MAE values (0.030 (weekly), 0.022 (biweekly), and 0.018 (monthly), whereas empirical and KNN models had the largest errors (0.468 and 0.337, respectively). This proves stacking’s prediction error-reducing ability under protracted soiling.
Model MAE Heatmap across cleaning intervals P2:- clean weekly P3:-clean biweekly P4:- clean monthly.
These findings show that stacking is the best accurate method for soiling loss estimate across cleaning frequencies and is resilient to increasing soiling buildup.
Table 14 indicates that stacking consistently outperformed other models with RMSE ranging from 0.03 to 0.045, MAE ≤ 0.03, and R² = 0.999 throughout all months. Empirical and KNN models had the largest errors (e.g., MAPE up to 35.38% and MAE > 0.4), especially in December, proving the stacking ensemble’s better resilience and dependability.
Shewhart control charts of RMSE for October–December 2024 forecasting models are shown in Fig. 41. Control charts, or Shewhart charts, provide performance data over time to determine control limits. The upper and lower control limits (UCL and LCL) set the permitted range of variation. Values over these limits indicate instability or unexpected swings. Central line (CL) shows process mean. The stacking model showed the most consistent projected accuracy throughout all months, with an average RMSE of 0.04 and tight control limits (UCL = 0.06, LCL = 0.01). ANN showed RMSE variation between 0.13 and 0.30 (mean 0.20), DT between 0.27 and 0.33 (mean 0.29), and KNN peaked at 0.49 (mean 0.38), suggesting greater variability. The stacking ensemble predicts well because of its low errors (Figs. 42 and 43) and process stability.
Shewhart control charts of RMSE for different predictive models (Oct–Dec 2024).
The Shewhart control chart of RMSE (Fig. 41 shows that prediction errors remain well within the statistical control limits across all months, confirming stable model performance and absence of instability due to non-stationarity. The empirical regression model also maintained consistent performance, with R² values between 0.97 and 0.98 across different months, further supporting the temporal robustness of the predictive framework. The environmental variables recorded during the experimental period exhibited natural variability while remaining within the same physical operating regime, enabling the model to learn stable relationships between environmental drivers and soiling loss. Since the models rely on physically meaningful predictors such as cleaning frequency, humidity, and irradiance, the learned relationships remain consistent over time. These results confirm that the predictive models demonstrate stable performance across different months without evidence of significant parameter drift or non-stationary.
Month-wise comparison of MAPE (%) across predictive models.
Month-wise comparison of MAE across predictive models.
The Shewhart control chart analysis shows that the stacking model predicts soiling loss with the lowest prediction errors and retains stability within restricted limits, making it the most dependable strategy⁶³.
To ensure that the superior performance of the stacking model was not merely a consequence of increased model flexibility, several safeguards were employed:
Independent testing evaluation (80:20 split) demonstrated that training and testing R² values were nearly identical (0.9995 vs. 0.9997), indicating strong generalization without overfitting.
Residual analysis showed random dispersion without systematic patterns.
Month-wise and cleaning-frequency-wise evaluations confirmed consistent performance across operational conditions.
Control chart stability analysis demonstrated low variance and stable error distribution across months.
These results collectively indicate that the improved performance of the stacking model arises from its ability to capture nonlinear environmental interactions rather than merely from increased complexity.
To evaluate the effectiveness of the proposed machine learning framework, its performance was compared with the semi-empirical regression model developed in Sect. 3.2 under identical validation conditions. The empirical model represents a physics-based baseline using environmental predictors such as irradiance, temperature, humidity, wind speed, and cleaning frequency. As shown in Table 12, the stacking ensemble achieved significantly lower prediction error (RMSE = 0.03–0.045, MAE ≤ 0.03) compared to the empirical model (RMSE = 0.348–0.386, MAE = 0.266–0.294). This represents an approximately 85–90% reduction in prediction error. These results demonstrate the superior predictive capability of the proposed machine learning framework in capturing nonlinear soiling dynamics compared to conventional semi-empirical models.
After performance assessment, statistical analysis verified machine learning model dependability and robustness. Although error measurements like RMSE, MAE, MAPE, and R² assess accuracy, they do not adequately resolve bias between estimated and actual soiling loss levels. A Bland–Altman (BA) plot was utilized to visually evaluate agreement, highlight recurring deviations, and indicate prevalent prediction error boundaries. A non-parametric Wilcoxon signed-rank test was employed to see whether predicted and actual values differed significantly. These graphical and inferential methods analyse model performance comprehensively.
To evaluate the statistical independence of the experimental observations, the autocorrelation function (ACF) of the soiling loss time series was analyzed, as shown in Fig. 44 Since environmental and PV performance data are collected sequentially, temporal autocorrelation may reduce the effective sample size and affect model validity⁶⁴.
The ACF results show that autocorrelation values decrease rapidly and remain within the 95% confidence bounds for most lags. Only short-term correlations are observed at very small lags, while longer lags exhibit negligible autocorrelation. This indicates weak temporal dependence and confirms that the observations are sufficiently independent for predictive modelling.
Furthermore, the natural variability in environmental parameters, including irradiance, temperature, humidity, and wind speed, along with different cleaning frequencies, ensured diverse operating conditions across samples. This variability further supports the effective independence of observations and validates the robustness of the machine learning models.
Autocorrelation function of daily-averaged soiling loss residuals.
The Bland–Altman (BA) study assessed the agreement between anticipated and actual soiling loss values. The BA plot shows bias and limitations of agreement, indicating systematic and random model prediction deviations, unlike traditional error measures. A lower bias value and narrower ranges of agreement imply that model predictions match data. This approach helps validate if machine learning models can reproduce experimental observations across operational circumstances.
The arrangement of dots around zero illustrates the degree of concordance between predictions and actual values, with tighter clustering near the red bias line signifying enhanced consistency. The dispersion within the limits of agreement (LoA) indicates the model’s variability, while outliers situated far beyond the LoA denote instances of inaccurate predictions, as depicted in Figs. 45 and 46, respectively.
.
Bland–Altman plot for (a) Empirical model (b) Stacking ML model (c) ANN (d) DT (e) SVM (f) KNN.
Model biases with limits of agreement (vertical lines).
The Fig. 47 shows the Wilcoxon signed-rank test comparing real and forecasted soiling loss values to assess the models’ predictive ability. A statistically insignificant result (p > 0.05) suggests that model predictions match experimental results, indicating model resilience. However, a significant finding (p < 0.05) indicates consistent disparities between projected and actual values. All models’ actual and expected soiling loss values were compared using the Wilcoxon signed-rank test. Despite having the lowest median difference (0.0016), the stacking model has a substantial p-value (p = 0.0083), demonstrating its capacity to capture tiny deviations with high consistency. The empirical (p = 0.593), decision tree (p = 0.276), and KNN (p = 0.428) models had non-significant p-values, indicating no statistically significant difference between their predictions and actual values.
Wilcoxon signed-rank test plot for (a) Empirical model (b) Stacking ML model (c) ANN (d) DT (e) SVM (f) KNN.
Even with strong numerical performance, ANN (p = 2.71e-42) and SVM (p = 7.4e-24) showed substantial discrepancies, suggesting systematic prediction errors. Stacking is the most reliable technique since it has minimum bias and statistically significant consistency, whereas empirical and tree-based approaches are equivalent but less robust. Although the Wilcoxon signed-rank test yielded a statistically significant p-value (p < 0.01) for the stacking model, indicating that the median difference between predicted and actual values is not exactly zero, the magnitude of this deviation was extremely small (≈ 0.001–0.002). Given the relatively large sample size, even minor deviations can become statistically significant. However, absolute error metrics (RMSE and MAE) remained very low, suggesting that the detected bias is negligible in practical terms. Therefore, the stacking model demonstrates high predictive accuracy with minimal practical bias rather than perfect agreement.
Bland–Altman analysis and Wilcoxon signed-rank test p-values vary because they employ different statistical methods. The Bland–Altman approach estimates the p-value using a paired t-test to see whether the mean difference (bias) between actual and predicted values is substantially different from zero. The Wilcoxon signed-rank test, on the other hand, tests if the median of the paired differences deviates considerably from zero without assuming normality. BA focuses on systematic bias in the mean, whereas Wilcoxon confirms median differences, therefore p-values may vary. Two methods give a more complete statistical assessment of model performance.
Although the dataset originates from a single geographical site and season, meaningful domain shifts exist within the data due to temporal variation in environmental conditions and operational variation in cleaning frequency. Figure 48 illustrates covariate distribution shifts of global horizontal irradiance across months, confirming changes in the input feature space.
Figures 49 and 50 further demonstrate that the stacking model maintains stable RMSE across temporally distinct months and across different cleaning frequencies. The consistency of predictive performance under these distributional and operational shifts indicates robust within-domain generalization rather than simple interpolation of identical conditions. While the present study does not claim cross-climate transferability, the proposed framework demonstrates strong robustness within the studied domain.
Covariate distribution shift of GHI across months.
Temporal domain shift evaluation of stacking model.
Operational domain shift evaluation of stacking model.
To statistically validate the observed performance dominance of the stacking ensemble, paired Diebold–Mariano tests were conducted on squared prediction error sequences. The results indicate as shown in Table 15 that the stacking model significantly outperforms ANN, DT, SVM, KNN, and empirical models, with DM statistics ranging from − 3.97 to − 12.08 and corresponding p-values well below 0.05.
The negative DM statistics confirm that the stacking model consistently yields lower prediction errors than competing models. These findings demonstrate that the superior performance of the stacking ensemble is statistically significant and not attributable to random variation.
This study investigated natural soiling on solar panels subjected to different cleaning protocols and developed empirical and machine learning models for predicting soiling loss. We employ experimental analysis and data-driven methods to test, evaluate, and predict how well PV systems will work when they are dirty in the real world. Ensemble learning outperforms empirical approaches in accuracy and robustness. The experimental study on four PV panels with different cleaning frequencies (daily, weekly, biweekly, monthly) confirmed that natural soiling significantly impacts PV performance, with higher losses under longer intervals.
This study contributes to the field by establishing a validated experimental–empirical–machine learning framework for real-world soiling prediction under controlled cleaning intervals.
The proposed stacking model significantly outperforms the semi-empirical baseline, demonstrating improved predictive accuracy and robustness under identical validation conditions.
Unlike purely simulation-based studies, the proposed approach is grounded in field measurements and incorporates temporal causality-aware validation, making it both scientifically rigorous and practically deployable.
The findings provide a reproducible methodology for future PV degradation studies and open avenues for intelligent, data-driven operation and maintenance optimization in solar energy systems.
Two empirical models were created one for (:{I}_{SC:})prediction (dominated by GHI) and another for Soiling Loss (SL) (mainly impacted by RH and cleaning frequency). The SL model has R² = 0.978, but a high MAPE = 28%, suggesting insignificant nonlinear effects.
Machine learning showed considerable increases, with the stacking ensemble obtaining the highest accuracy (R² = 0.9997, RMSE = 0.0456, MAE = 0.0333) and surpassing individual models (ANN, DT, SVM, KNN), among other models.
Model accuracy falls according to decreased cleaning frequency, but stacking remains strong (MSE ≤ 0.003, MAE < 0.03) even with monthly cleaning.
Statistical validation using Bland–Altman and Wilcoxon signed-rank tests confirmed stacking’s superiority, with minimal bias and narrowest limits of agreement, although minor statistically detectable differences were observed in some cases.
Overall, the integrated experimental–empirical–ML framework demonstrates that ensemble-based data-driven models can reliably predict soiling loss, enabling optimized maintenance scheduling and predictive O&M strategies for PV systems.
While prediction error differences may appear numerically small, their operational significance becomes substantial when translated into cumulative energy losses over extended periods. As shown in Table 3, soiling losses exceeding 5% were observed under extended cleaning intervals. Accurate prediction of soiling progression enables optimized maintenance scheduling, improving energy yield and reducing operational costs. The proposed model is developed based on short-term dry-season data, during which module surface properties are assumed constant. Long-term surface degradation, coating wear, and micro-roughness evolution may influence dust adhesion behaviour and soiling accumulation rates. Such effects represent gradual structural drift and would require multi-season or multi-year datasets for comprehensive modelling. Therefore, the current framework is primarily applicable to short- and medium-term predictive maintenance planning.
The present model was developed using data collected during dry environmental conditions to ensure controlled soiling accumulation. Extreme events such as rainfall-induced natural cleaning or dust storms introduce regime shifts that require representative training data for accurate prediction. Future work will incorporate multi-season datasets including rainfall and extreme environmental conditions to enhance model robustness and generalization capability.
Data is available based on request.
AdaBoost
Autoencoder
Artificial neural network
Ambient temperature
Backpropagation neural network
Cleaning frequency
Convolutional neural network
Cell temperature
Diffuse horizontal irradiance
Atmospheric pressure
Particulate matter (PM10 / PM2.5)
Relative humidity
Direct normal irradiance
Wind speed
Global horizontal irradiance
K-nearest neighbor
Linear regression
Long short-term memory
Mean absolute error
Mean absolute percentage error
Machine learning
Multilayer perceptron
Mean square error
Root mean square error
Recurrent neural network
Seasonal auto regressive integrated moving average with exogenous variables
Sunshine hour
Soiling loss
Soiling ratio
Support vector machine
Support vector regression
Random forest
RGB images of solar panels
Decision tree
Extreme learning machine
Gated recurrent unit
Extreme gradient boosting
Coefficient of determination
Current at maximum power point
Short-circuit current
Short-circuit current of clean reference panel
Short-circuit current of soiled panel
Power output of clean panel
Maximum power output
Power output of soiled panel
Temperature of dusty panel
PV module temperature
Voltage at maximum power point
Open-circuit voltage
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This Research was conducted with the financial support provided by UPES, Dehradun, India. The authors express gratitude to the Research & Development Department at UPES, Dehradun, Uttarakhand, India for their support under Grant Number UPES/R&D-SoAE/25062025/27.
Open access funding provided by Manipal University Jaipur.
Electrical Cluster, School of Advanced Engineering, UPES, Dehradun, 248007, India
Ashutosh Shukla & Rupendra Kumar Pachauri
Miyan Research Institute, International University of Business, Agriculture and Technology, Dhaka, 1230, Bangladesh
Rupendra Kumar Pachauri
UCRD & CSE-APEX, Chandigarh University, Mohali, Punjab, India
Ranjan Walia
Department of Electrical Engineering, Manipal University Jaipur, Jaipur, India
Vinay Gupta
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Ashutosh Shukla (AS): Conceptualization, Methodology, Writing – original draft, Software, Visualization. Rupendra Kumar Pachauri (RKP): Methodology, Data curation, Writing – review and editing, Supervision. Ranjan Walia (RW): Investigation, Writing – review and editing, Supervision. Vinay Gupta (VG): Investigation, software, Visualization, data analysis, Writing – review and editing.
Correspondence to Vinay Gupta.
The authors declare no competing interests.
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Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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Shukla, A., Pachauri, R.K., Walia, R. et al. Machine learning-based prediction of soiling losses in photovoltaic modules under different cleaning frequencies: an experimental investigation. Sci Rep 16, 17416 (2026). https://doi.org/10.1038/s41598-026-45485-2
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Received: 17 December 2025
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Published: 14 April 2026
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DOI: https://doi.org/10.1038/s41598-026-45485-2
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Independent research commissioned by the Canadian Renewable Energy Association. That would not be biased… or should it be assumed to be biased, as they promote solar panels.
“Recycling solar panels involves more than simply collecting discarded equipment. Transportation can be a major factor. The recycling process is complex and often energy intensive, as it must separate and extract valuable materials like silicon, silver, copper, and aluminium frames from each panel.”
“On average, recycling a solar panel can set you back between $20 to $50 per panel.” – just a couple of google unbiased info quotes.
AB doing all things to keep people hooked on oil / gas.
I wish I could post photos of the wind turbine blade and solar panel graveyards in rural Texas. It’s a horrible mess.
They need to look at the recycling of all the electric vehicle batteries as well.
People can go green all they want but need to know the truth that those items need to be disposed of or recycled at the end of their life and that does not come cheap. If those items go to the landfills then they are not very green or protect the environment for which they were created to protect!
These create a lot of waste and the disposal costs should fall on the users, not tax payers. Good plan!
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Albertans purchasing new solar panels next month will face a new hefty fee for recycling them.
Environment Minister Grant Hunter announced Thursday a ban on landfill disposal for the panels and the $14 recycling fee per panel beginning Oct. 1.
Hunter said the move will protect municipalities and taxpayers from future costs. He estimated that by 2045, most of the solar panels currently installed in the province will reach the end of their lives and could produce nearly 73,000 tons of material.
“We’re creating the opportunity for new made-in-Alberta recycling programs and industries that recover valuable materials instead of burying them,” he told reporters in Brooks.
The government estimates a typical household of 20 panels.
But industry associations and advocates have said the $14 price tag is far too steep, punitive and being imposed without clear accounting to back it up.
They note it’s nearly five times the $2.75 cost of recycling a large television set in the province.
Heather MacKenzie, executive director of renewables advocate Solar Alberta, said for a typical panel, a $14 fee is like adding a 10 per cent tax.
In an interview, she said it’s yet another hit to the solar sector from a provincial government that appears intent on undermining it.
“It appears that the government of Alberta is artificially increasing the cost of solar, and that will enable natural gas to compete more easily in our marketplace,” she said.
She added that the impact will be significant.
“It’ll hurt all the small solar installation companies. It’ll hurt the electricians, and we will see job losses as a result.”
When word of the fee trickled down to stakeholders from the Alberta Recycling Management Authority earlier this summer, MacKenzie said they pushed back, and asked how the $14 figure was reached.
“They are not willing to show how they came to that number, which means that nobody can really critique the assumptions that are embedded in it,” said MacKenzie.
Independent research commissioned by the Canadian Renewable Energy Association put the cost of recycling a solar module at about $5.
Hunter, when asked how the province arrived at its cost, said the actual cost of shipping old panels to the United States for processing is $40 per panel.
He said it will also take time for recyclers to develop “economies of scale,” and if they can do it for cheaper, then that price will be passed on to consumers.
The announcement comes after a string of rule changes affecting renewable energy development from Premier Danielle Smith’s United Conservatives.
In 2023, the UCP put a short-term moratorium on renewable energy project development before introducing strict new rules on where they can go. Subsequent reclamation fees for decommissioned projects were later criticized as too high.
Business Renewables Centre Canada — which works to help businesses and institutions reduce their emissions by connecting buyers and sellers of renewable power — said it supports a recycling fee but Alberta’s is higher than those of any other jurisdiction in the world.
“We’re disappointed to see the government saddle the industry with another disproportionate burden,” said director Jorden Dye in a Thursday news release.
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Officially certified by Fraunhofer ISE in Germany and listed in NLR’s Best Research-Cell Efficiencies Chart
National Research Council of Science & Technology

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Perovskite/CIGS tandem solar cell developed by the KIER research team
Credit: KOREA INSTITUTE OF ENERGY RESEARCH
 The Photovoltaic Research Department of the Korea Institute of Energy Research (KIER) achieved a certified world-record efficiency of 26.7% for perovskite/CIGS tandem solar cells, opening a new chapter in next-generation thin-film photovoltaics.
 This achievement was officially certified by the Fraunhofer Institute for Solar Energy Systems (ISE) in Germany and listed in the Best Research-Cell Efficiencies Chart published by the US National Laboratory of the Rockies (NLR, formerly NREL).
 The previous world-record efficiency of 26.3%, set a year earlier by a joint research team from Seoul National University and the Korea Institute of Science and Technology (KIST), was surpassed by another Korean research team at KIER, demonstrating the country's leading role in next-generation thin-film solar cell technology. 
 Silicon solar cells, currently the most widely used solar cell technology, have already reached technological maturity, leaving limited room for further efficiency improvements due to fundamental physical limitations. Against this backdrop, tandem solar cells are emerging as a promising next-generation solution for high-efficiency photovoltaics. This technology uses multiple solar cells with different characteristics stacked in layers to capture a broader range of sunlight wavelengths.
 The perovskite/CIGS tandem solar cells developed by the KIER research team have a perovskite cell at the top and a CIGS cell at the bottom. This unique configuration enables the two cells to absorb different wavelengths of sunlight simultaneously. Since both perovskite and CIGS are well suited for thin-film processing, the technology combines high efficiency, light weight, and flexibility.
 Assembling the two cells may, however, degrade the perovskite light-absorbing layer. In addition, some cell layers may absorb unwanted light, reducing the overall efficiency. To address these challenges, the KIER research team conducted a comprehensive analysis of the root causes of such efficiency losses. As a result, they developed an advanced interfacial layer material and processing technology to mitigate potential damage to the perovskite layer. The team also successfully minimized undesired light absorption and potential photocurrent loss by optimizing the structure of the top transparent electrode and charge transport layer.
 This approach resulted in a laboratory-measured efficiency of 27% and a Fraunhofer ISE-certified efficiency of 26.7%.
 The developed technology is expected to increase electricity generation per unit area and thereby expand the potential applications of photovoltaic power generation. Furthermore, owing to their lightweight and flexible thin-film design, they are promising not only for buildings and automobiles but also as power sources for small satellites and space-based data centers for future space applications, where weight and space constraints are critical.
 Inyoung Jeong, a senior researcher at KIER who led the research, said, "This achievement is significant in that both cell efficiency and stability can be enhanced by minimizing potential interfacial and optical losses during the integration of perovskite and CIGS. The resulting efficiency was also officially certified by a world-renowned institute and recognized as a world-record performance, underscoring Korea's technological competitiveness." 
 Going forward, the research team will focus on ensuring that large-area modules achieve the same efficiency as the small-area devices developed in this study. They will collaborate with industry partners interested in mass production and commercialization and pursue technology transfer. In the long term, they will expand the technology to next-generation space solar cells capable of reliable operation in space due to their light weight and high efficiency.
Disclaimer: AAAS and EurekAlert! are not responsible for the accuracy of news releases posted to EurekAlert! by contributing institutions or for the use of any information through the EurekAlert system.
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Sea trials are now underway for a combined wind-and-solar energy system aimed at cutting auxiliary engine emissions aboard a converted container vessel. The test setup features a novel horizontally oriented wind turbine, developed to overcome the difficulties of adding wind-assisted propulsion to ships where cargo takes priority for deck space.
The equipment was fitted in late August in Aarhus, Denmark, on the Samskip-operated containership Helgafell. Constructed in 2005, this vessel serves as a dependable asset for the company, running regular routes to Iceland regardless of weather. The ship has a deadweight tonnage of 11,143, measures 137.5 meters (451 feet) long, and can carry up to 909 TEU.
This retrofit unites two renewable power sources in a modular design intended for easy installation on existing vessels. SideWind’s horizontal turbines are placed inside converted freight containers, while Solbian‘s photovoltaic panels are mounted on the container tops. This configuration seeks to exploit open deck areas while keeping structural changes to a minimum. For SideWind, these sea trials signify a shift from land-based development and testing to real-world maritime validation.
SideWind CEO Oskar Svavarsson noted that the trials would yield crucial operational data on performance and control, supporting progress toward a commercially scalable retrofit solution. The Solbian solar modules were specially engineered to match the dimensions of the SideWind container roofs and withstand harsh ocean conditions. Throughout the testing period, the solar array will function in tandem with the turbines, forming a hybrid system that draws on both wind and sunlight.
Testing started right after installation, as the ship traversed the North Atlantic and reached Iceland on August 24. The evaluation will span four planned voyages over the coming three months. Guomundur Oskarsson, Samskip’s Director of Logistics, expressed expectations that the trials would offer insights into how modular renewable technologies can support greener daily shipping operations without compromising the safety, reliability, or service quality that customers and crews rely on.
This effort represents the second of two demonstrations under the WHISPER initiative, which stands for Wind Energy Harvesting for Ship Propulsion Assistance and Power. That project began in January 2023 and is scheduled to conclude on December 31, 2026. It involves 13 partners from six European nations and has secured backing from the European Union’s Horizon Europe program. The first demonstration applied the same hybrid wind-solar concept to the dry bulk carrier Maria Tipoc, where solar panels sit on hatch covers alongside OceanWings’ rigid, tilting sails.
WHISPER was conceived to examine the hurdles and prospects of equipping older ships with such hybrid systems. Its objectives include achieving roughly 30 percent fuel savings on a converted bulk carrier and up to 15 percent on a converted containership.
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Can I charge my EV for free from solar?  drive.com.au
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Stephen Barnfield, a Regulatory and Insurance Partner at Forbes Solicitors, looks at whether a maturing risk profile is required for solar panels in schools.

Progressive solar power
The education sector is embracing solar technology in a move to manage both costs and carbon emissions. In recent weeks, the Department for Education (DfE) announced that a new wave of schools and colleges across England are set to save £220 million on energy bills over the lifetime of solar panels. The aim is to free up money that would have otherwise be spent on energy, so that it can be reinvested in children’s education.
According to the DfE’s July 2026 update, 245 schools and colleges already have government-funded solar panels, and 100 more will now join the Great British Solar Partnership. A further 150 schools and colleges will also be embracing Photovoltaic (PV) technology through private sector support, with a pilot programme seeing installation and maintenance of panels at no upfront cost.
It’s clear there’s progressive adoption of PV technology throughout education, but as panels become an increasingly familiar sight on school roofs, is there a danger they are being treated as a ‘fit and forget’ technology? Recent reports suggest this is a real risk, and one that could impact insurance cover for buildings, contents and business / operational interruption, as well as employer, governors’ and trustees’, and public liability.
Fire safety risks
In June this year, the BBC reported that Suffolk County Council had taken the decision to switch off solar panels at about 80 schools across the county. This came after a fire at Sidegate School in Ipswich, which the local fire service confirmed as being caused by solar panels on the roof. Thankfully, there were no reports of anyone being injured by the fire.
It’s also believed that fires at two other Suffolk schools over the last year were linked to PV panels. And in 2025, Northumberland County Council switched off solar panels on 141 of its buildings following a fire at a school, as well as a community centre. Solar panel safety issues aren’t isolated to public sector owned and managed buildings, with PV panel fires also reported across domestic dwellings and commercial properties, including a rooftop fire at a supermarket’s regional distribution centre in Peterborough in 2024.
Solar panel fire risks seem to be more commonly associated with the age and maintenance of PV technology, which should prompt wider consideration about ongoing management. Local authorities, academy trusts, schools and colleges, and insurers must treat solar panels as critical infrastructure, supported by risk management and fire safety frameworks that account for the changing risk profile of PV systems over time.
Moving away from ‘fit and forget’
Solar technology is often perceived as a relatively straightforward system, with no moving parts and low upkeep. These solutions are often positioned as long-term investments that require little maintenance to generate affordable, clean energy. Internet searches even reveal benefits such as natural rainwater cleaning panels by washing away dust and dirt, or a panel installation only needing a quick check or wash a couple of times per year.
Once installed, solar panels can quietly generate electricity for years with little visible intervention, creating a risk that they are viewed as a ‘fit and forget’ technology. Strategies and programmes must avoid this approach.
PV systems are made up of a range of electrical components, including panels, wiring, inverters and connectors, and in some cases, battery storage systems. Each part represents a potential point of failure and like many products, are subject to the ‘bathtub curve’ of reliability. Failure rates can be typically higher during installation and commissioning, before falling (the downward part of the curve) to a relatively low level during normal operation, and then rising again (upward curve) as systems age and components begin to deteriorate.
Based on reported school fires, attention seems to be increasingly focusing on ageing solar installations. The PV panels at Sidegate Primary School were installed in 2012 and Suffolk County Council’s decision to switch off panels at about 80 schools applied to systems installed between 2011 and 2016. Components can become more susceptible to wear, deterioration and potential failure as they mature.
However, risks aren’t only age-related. Solar panel problems can also be caused by faults and damage and may not always provide visible warning signs until an issue becomes more significant.

Managing the risks
To effectively manage the risks and potential liabilities of PV systems, organisations should implement a structured risk management framework based on hazard identification, risk likelihood and impact evaluation, and monitoring and inspections. The framework should cover the entire lifecycle of the installation, from design, installation and commissioning, through to operation, maintenance and component replacement. By aligning inspections and performance reviews with the expected lifespan of key assets, organisations can identify any emerging issues early to ensure systems remain safe and operating as intended.
System inspections though, should not be limited solely to component lifespans. Risk management frameworks need to also account for external factors that could affect the condition and performance of PV systems over time. These may include weather exposure, storms and high winds, as well as accidental damage. For example, school and college settings may mean that solar panels are subjected to repeated impacts from footballs or other sports equipment. There is also a possibility of overhanging trees, falling branches and vegetation growth causing damage.
Wildlife should also be considered as part of risk assessments. Birds’ nesting activity could cause a buildup of twigs, leaves and features beneath panels and close to components, potentially restricting ventilation, concealing defects and increasing fire risks when overheating occurs. Risk management should be questioning: how frequently are systems inspected, and are procedures in place to assess installations following events such as storms or impacts from falling objects?
Detailed, formalised records should support risk management frameworks, with documentation covering asset registers, risk assessments, maintenance schedules, inspections and performance monitoring, incident reporting, and end-of-life planning. This information will provide an audit trail, demonstrating accountability and helping organisations more clearly determine fact, responsibilities and possible liabilities if a fire occurs. The management of formalised records also helps to keep PV systems front of mind, helping avoid the ‘fit and forget’ approach.
Consideration needs also to be given to how the inspections are to be carried out and who is going to conduct them. Will it be in-house or managed by contractors? In either case, the activity itself needs to be managed safely, as it will inevitably involve work at height and proximity to electrical systems.
As the education sector increasingly embraces PV technology, it’s important that systems are not treated as one-off capital investments. Mindsets must shift towards managing solar panels as critical infrastructure that requires ongoing oversight.
About Forbes Solicitors:
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The firm has 64 partners and an overall headcount of nearly 400, advising on a wide range of commercial and personal matters. Offering services to public sector organisations, commercial entities and consumers, Forbes specialises in providing legal expertise in practice areas including litigation, commercial law, intellectual property, corporate legal services, employment law, business immigration, HR consultancy, insurance, property litigation, commercial property, insolvency and debt recovery, and individual services.
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Pro Football Hall of Famer Emmitt Smith is facing a lawsuit alleging he and his commercial real estate company misused a $2.5 million investment meant for a Texas solar farm, according to Bisnow. Kituwah LLC, a tribally owned Native American economic development agency, filed the suit in Delaware on Monday against Smith, his firm 4 13 Solutions Inc., business partner David Mosley, and Darrel Wilson of Wilson Holdings of North America.
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Kituwah says Smith and Mosley persuaded the agency to form a joint venture with 4 13 Solutions. The agency also agreed to loan the company $2.5 million toward an interest in Project Exodus, a planned Texas solar farm.
The pair allegedly told Kituwah that developer Genesis Consolidated Industries had already started securing land for the project. They said the developer needed more capital. They also claimed other investors were “clamoring” to finance the deal.
Kituwah investigated the claims and detailed its findings in the complaint. The agency concluded that 4 13 Solutions used the representations as part of a broader scheme. Smith and Mosley allegedly intended to cheat Kituwah out of its $2.5 million rather than make a legitimate investment that later underperformed.
The lawsuit claims 4 13 Solutions also promised to secure a Department of Energy loan for permanent financing. The company allegedly said the solar farm would operate and generate millions in income by the end of 2024.
Instead, Kituwah alleges that Smith and Mosley redirected the loan to Wilson Holdings of North America. The pair had partnered with the entity on other ventures.
By the end of 2024, Kituwah says it found no evidence that 4 13 Solutions had secured rights to the project. The loan also remained unpaid when it matured on Feb. 1, 2024.
Kituwah’s complaint says the promissory note remains unpaid years after maturity. The agency has not recovered any of its $2.5 million loan. That outcome highlights how long the dispute has continued before reaching litigation.
Kituwah now seeks recovery through the courts. The agency says it made repeated attempts to resolve the matter directly with Smith and Mosley, but those efforts failed.
The suit adds Smith to a growing list of high-profile figures facing CRE-related litigation. Similar cases include the JCPenney lease dispute, which turned a real estate deal into a multimillion-dollar legal fight.
The case also comes amid greater scrutiny of solar assets changing hands across the industrial and energy sectors. Investors continue to reassess renewable project valuations.
Celebrity-backed real estate and energy ventures face greater legal scrutiny across the industry. More athletes and entertainers now invest in development deals. They often rely on partners and operators to manage those projects. However, the athletes themselves may not independently vet those partners’ track records.
Smith is the NFL’s all-time leading rusher and a three-time Super Bowl champion with the Dallas Cowboys. He founded 4 13 Solutions in 2020 as part of a broader push into real estate after retirement.
If proven, the allegations could complicate Smith’s relationships with future investors. They also highlight risks for tribal and institutional lenders. Those lenders may face added exposure when they back athlete-affiliated development ventures without independently verifying land and financing claims.
4 13 Solutions did not respond to a request for comment. The case will now move through Delaware’s court system. Kituwah is seeking to recover its unpaid loan.
The outcome could influence how tribal economic development agencies structure future joint ventures with celebrity-backed real estate firms.
Watch for potential claims against Genesis Consolidated Industries or Wilson Holdings of North America. Their roles could receive more scrutiny as the litigation moves forward. Smith’s other real estate ventures could also face greater attention from prospective investors.
Discovery could provide more details about how the $2.5 million was spent. For now, the two sides continue to dispute those details.

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“You’re going to see lower panel output right when the demand is the highest.”
Photo Credit: YouTube
Very hot weather can cut into a home’s solar production, leaving systems less efficient during the same summer stretches when electricity use tends to surge.
That dynamic was highlighted in a YouTube video from Two Steps from Off-Grid (@terrahillfarm), where a homesteader in Central Texas said his off-grid panels reached about 140 degrees Fahrenheit and saw output fall by 12%-20%.
In the video’s caption, he said the drop could amount to “almost 2,500 watts” versus what the panels might produce under cooler conditions. He reported air temperatures of 100-105 degrees in the area, with the panel surfaces running far hotter than the surrounding air.
He further said that the timing is especially problematic because “You’re going to see lower panel output right when the demand is the highest.” For homes that rely on solar, including off-grid setups, that mismatch can put more pressure on power needs during air-conditioning season and increase reliance on batteries or other backup sources.
Want to go solar but not sure who to trust? EnergySage has your back with free and transparent quotes from fully vetted providers in your area.
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Commenters echoed the concern with their own real-world experiences. “Learned this for the first time on record hot days with my extra dark panels,” one wrote. “They derated by half.”
Going solar is one of the best ways to save money on home energy costs, but system design matters if you want significant savings during extreme weather. Homeowners who are considering panels can explore EnergySage to get free solar installation estimates and compare quotes.
The poster said high temperatures can also affect equipment beyond the panels themselves. Batteries and inverters kept in hot garages or sheds may wear out more quickly, and excessive heat can cause those components to perform worse or shut down to protect themselves.
Because batteries and inverters are among the priciest parts of a residential solar setup, any heat-driven loss of lifespan could translate into replacement costs sooner than expected, in addition to the energy lost when overheated panels produce less power.
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To help protect that equipment, he said he added a solar-hybrid mini-split to the room, along with his battery bank and inverters, and has kept the area under 80 degrees even during peak heat. He said, “That one change has made a real difference” by reducing wear and thermal stress.
Among the steps he recommended were maintaining airflow around panels, checking for weeds or tree growth that might cast shade, and keeping battery and inverter spaces well-ventilated. He also said extra system capacity can provide a cushion against summertime efficiency losses during periods of heavy demand.
Adding battery storage to a solar setup is one of the best ways to protect your home during outages, save money on energy, and go off grid. It can also help households store daytime solar generation for use later, which is especially useful when hot weather reduces panel efficiency. You can explore EnergySage for information about home battery storage options, including competitive installation estimates.
EnergySage’s free tools can also make the shopping process easier. With EnergySage’s help, most people can save up to $10,000 on solar purchases and installations. EnergySage’s solar map shows the average cost of a home solar panel system on a state-by-state basis, along with details on solar panel incentives in each state. Together, these resources can help you get the best price for rooftop solar panels and access available incentives.
💡Go deep on the latest news and trends shaping the residential solar landscape
As the OP put it, “don’t overlook summer heat. It’s not just an inconvenience, it’s a real performance factor.”
These stories add more context to the summer solar picture. They also touch on the incentives and financing options that can shape what people install and how much they end up paying over time.
• In Pennsylvania, a homeowner logged 144 kWh in one day, output fellow solar owners praised.
• Homeowners doing the math found solar panel leases can trim upfront costs without guaranteeing savings.
• In California, rooftop solar incentives plunged, and sales slumped in a once-booming market.
The broader point is that solar performance depends on more than one factor. Panel output in extreme heat, battery backup, incentives, and financing can all affect how well a system delivers when power demand climbs.
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India’s residential rooftop solar initiative is seen as key to easing the burden on the country’s strained grid. © AP
Despite progress, consumer barriers and differing utility incentives limit uptake
BENGALURU — It took three months of research before 25-year-old finance professional Sagar Agarwal took the plunge into rooftop solar, joining an initiative widely recognized as the largest of its kind in the world. But despite being a resident of Lucknow, which in April became the Indian city with the most residential solar installations, he was left frustrated by the friction he encountered — a situation mirrored across the nation, whose lopsided rooftop solar rollout threatens to impinge on energy goals.

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GCL Technology’s revenue remained broadly stable, but weak polysilicon and wafer prices continued to weigh on its profitability
GCL SI’s revenue fell sharply amid oversupply, with overseas markets and large orders providing some support
Both companies are expanding beyond conventional PV, with GCL Technology pursuing perovskite and GCL SI developing BC, tandem and space-solar
The prolonged downturn in the solar manufacturing market continued to weigh on the H1 2026 financial results of GCL Group’s solar companies. Both—the group’s polysilicon producer GCL Technology as well as its solar PV cells wafer-cell-module-systems production arm GCL System Integration (GCL SI)—reported losses, but are focusing future growth on newer technologies.
During H1 2026, GCL Technology reported revenue of RMB 5.77 billion, broadly unchanged from RMB 5.73 billion a year earlier, representing 0.7% year-on-year (YoY) increase.
However, the company remained under pressure from weak pricing in the polysilicon and wafer markets. Its gross loss nevertheless narrowed to RMB 434 million from RMB 700 million, with 38.1% annual decline, while the loss attributable to owners widened to RMB 2.08 billion from RMB 1.77 billion.
GCL Technology said H1 2026 half was ‘the most concentrated period of pressure’ for the PV industry in recent years. Average polysilicon and wafer prices fell below the cash costs of most producers, while high polysilicon inventories also affected selling prices and sales volumes for its granular polysilicon.
During the reporting period, the company said its average external selling price for granular polysilicon was RMB 31.97/kg, while its average production cash cost was RMB 25.23/kg.
Its solar materials business generated RMB 5.73 billion in revenue, accounting for almost all of the group’s total revenue to which polysilicon sales contributed RMB 3.88 billion. Wafer sales followed at RMB 696 million and industrial silicon at RMB 344 million.
As of June 30, 2026, GCL Technology had 480,000 metric tons (MT) of granular polysilicon production capacity.
The company is also looking beyond its traditional solar materials business. GCL Technology expanded into cathode material business through its non-wholly owned subsidiary Leshan Xinneng New Materials Technology. The latter, in which GCL holds an indirect equity interest of 30%, is manufactures and sells lithium iron phosphate cathode materials (see China Solar PV News Snippets).  
Perovskite is another technology stream GCL is expanding into, with a focus on perovskite-silicon tandem modules. In H1 2026, the world’s ‘first’ GW-scale perovskite-silicon tandem module production line, Kunshan GCL, entered stable operations, while the company advanced commercial orders and demonstration projects.
GCL Technology reported net loss for FY 2025, but managed to turn a positive adjusted EBITDA (see GCL Technology Revenue Falls, But Loss Narrows In 2025).
GCL SI, meanwhile, operates further downstream in the solar supply chain and is focused on cells, modules, storage products and PV solutions. It also faced a difficult H1 2026 as its operating revenue fell 39.7% YoY to RMB 4.64 billion, while net loss widened to RMB 430.2 million with a 31.6% jump. 
The management linked the pressure to weak end-market demand, industry overcapacity and lower prices across the PV supply chain. China’s solar PV module manufacturing capacity exceeded 1.1 TW while global module demand is expected to reach around 536 GW in 2026. This means lower capacity utilization, it pointed out.  
Solar modules remained the company’s main business. Module revenue was RMB 4.19 billion in the first half, while system integration services contributed RMB 149.5 million and cell sales RMB 243.9 million.
Overseas revenue stood at RMB1.11 billion, with markets outside China accounting for a smaller but growing part of the business. Domestic market accounted for 76.10% of the total revenue with an annual decline of 50.08%, while overseas expanded its share by 78.36% YoY to 23.90%.
GCL SI said it secured more than 1.5 GW of large orders in markets including South Asia and Europe. It is also strengthening its presence in the Middle East, Africa and East Asia.
GCL SI had 30 GW of high-efficiency N-type module capacity at the end of June. Some production lines can also manufacture XBC products, allowing the company to adjust capacity as demand shifts between technologies.
The company said its module business maintained relatively high capacity utilization during the industry downturn. GCL SI said its GPC back-contact technology reached an average cell conversion efficiency of more than 28.36%. Its GPC 3.0 all-black modules have entered mass production, while the company plans to add 8 GW of GPC production capacity from H2 2026 through H1 2027.
Perovskite is another technology stream GCL SI is expanding into. During the first half, the company continued development of GTC perovskite-silicon tandem cells with Soochow University and other research partners. Its small-area two-terminal GTC1.0 cell reached an internally tested efficiency of 33.02%, while a three-terminal GTC2.0 sample using a BC bottom cell reached 29.05%.
Space solar PV is another area of interest with the company working on materials and cell technologies designed to withstand radiation, thermal cycling and vibration. It reported an efficiency of 21.5% for an AM0 thin-film crystalline-silicon cell designed for space applications.
This gives GCL SI a technology roadmap that extends beyond conventional TOPCon modules, with BC mass production and perovskite tandem development emerging as two key areas of focus.
At the TaiyangNews Next-Gen PV Technologies Conference 2026, GCL SI presented its GPC 3.0 back-contact platform (see GCL SI Details BC And Tandem Technologies).
TaiyangNews 2024

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Japan’s Amada raising local solar generation portfolio to 6.4 MW  Renewables Now
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MANILA, Philippines — ACEN Corp. is infusing more than P9 billion into two subsidiaries to support the development of a solar park and a battery energy storage system (BESS) in Zambales.
The Ayala-backed renewable energy producer said Friday it granted a loan deal with its wholly owned unit, Sanmar Solar Inc., involving a fund of up to P6 billion.
READ: ACEN open to energize Pax Silica
The group said the funding would help Sanmar Solar build a 500-megawatt BESS project, complementing its existing solar power plant.
This allows ACEN to form an integrated renewable energy and storage system in San Marcelino town.
The Department of Energy earlier issued an order mandating developers to equip their solar and wind farms with energy storage systems.
This technology allows producers to store excess power and release it when the demand peaks.
Another subsidiary getting a capital boost is Giga Ace 8 Inc., with ACEN extending a loan of as much as P3.24 billion.
The company said the amount would finance the construction of the 300-megawatt Palauig 2 solar project. It would also support the rollout of transmission line and substation facilities.
The P16-billion large-scale solar park is the group’s second-biggest solar facility in the country.
READ: ACEN H1 net income spikes to P3.9B
Giga Ace 8 is ACEN’s special purpose vehicle for renewable energy projects in the Philippines. Aside from the local market, ACEN is present in India, Australia, Vietnam, Laos and Indonesia.
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ACEN had a strong first-semester finish, with its earnings soaring by 411 percent to P3.9 billion from P763 million a year ago. Its revenues from the sale of electricity also significantly jumped to P22.51 billion from P15.3 billion in the same period last year. INQ
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Summer 2026 brought contrasting solar resource conditions across Asia as El Niño developed and the Western Pacific Subtropical High shifted farther north than normal, according to analysis using the Solcast API. The displaced high-pressure system brought more cloud and rainfall to Northeast China while supporting sunnier conditions across central parts of the country. Farther south, the transition towards El Niño, reinforced by a positive Indian Ocean Dipole, reduced cloud across eastern Indonesia and Papua New Guinea, although smoke limited irradiance gains in western Indonesia.
The northward shift of the Western Pacific Subtropical High created opposing irradiance conditions across China. This large, semi-permanent high-pressure system over the western Pacific helps determine the position of East Asia’s summer rain belt and the paths taken by tropical weather systems. In 2026, its northward position carried moist air and persistent rainfall into Northeast China and parts of Siberia, reducing irradiance by as much as 20%. The same shift contributed to flooding in Shenyang in early to mid-July and slowed the accumulation of GHI through the season.
Farther southwest, the high-pressure ridge suppressed cloud over Sichuan, Chongqing and Guizhou, lifting irradiance 20% to 25% above average. The Sichuan Basin was particularly notable. Its enclosed terrain typically supports frequent cloud and fog, giving the basin some of China’s lowest summer GHI based on the 2007-2025 average. Against this low climatological average, relatively small absolute variations produce large percentage anomalies, as is the case with the 25% anomaly this year.
Along southeastern China and across most of Mainland Southeast Asia, irradiance was around 5% to 10% below normal. Easterly winds on the southern side of the displaced high steered tropical systems westward, rather than allowing them to turn north. Tropical Storm Maysak (June 29-July 6) and Typhoon Narra (22-24 August) contributed cloud and rainfall along this corridor.
Farther south, the developing El Niño began to exert a clearer influence across Indonesia and neighbouring island regions, reducing the rising air that normally supports tropical cloud and rainfall. This effect was reinforced by a positive Indian Ocean Dipole, a difference in ocean temperatures across the Indian Ocean that commonly reduces rainfall near Indonesia during its positive phase. Eastern Indonesia and Papua New Guinea recorded irradiance 20% to 25% above average.
Western Indonesia experienced a smaller increase despite Indonesia recording its driest July since 1991. Fire hotspots in Sumatra and Kalimantan produced smoke during August, scattering and absorbing incoming sunlight and offsetting some of the irradiance increase expected from reduced cloud.
ECMWF SEAS5 forecasts indicate that above-normal irradiance will persist across Indonesia and the surrounding island region from September to November. The dry pattern associated with El Niño and the positive Indian Ocean Dipole extends beyond the usual end of the dry season in October, with no clear recovery associated with monsoon onset.
The forecast also indicates an early monsoon withdrawal over Mainland Southeast Asia. A strong positive anomaly over Northeast China and Korea in September is expected to weaken into a broader, less concentrated area of above-normal irradiance during October and November.
Solcast produces these figures by tracking clouds and aerosols at 1-2km resolution globally, using satellite data and proprietary AI/ML algorithms. This data is used to drive irradiance models, enabling Solcast to calculate irradiance at high resolution, with typical bias of less than 2%, and also cloud-tracking forecasts. This data is used by more than 350 companies managing over 350 GW of solar assets globally.
The views and opinions expressed in this article are the author’s own, and do not necessarily reflect those held by pv magazine.
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Floating Photovoltaic Power Generation  Idaho National Laboratory (.gov)
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“The Central Government is placing the highest priority on solar and hydro power.”
Photo Credit: iStock
Tripura is turning to rooftop solar as it faces two mounting pressures at once: rising electricity demand and a dwindling supply of natural gas that has long powered much of the state’s grid.
State officials said that shift could be especially meaningful for lower-income households, since it may help keep energy bills from becoming even more difficult to manage.
During a special council meeting in Agartala on Aug. 17, 2026, Tripura Power Minister Ratan Lal Nath said the state needs broader rooftop solar adoption to meet future electricity needs without leaving vulnerable residents behind, according to Northeast Today. The session at the Agartala Municipal Corporation Conference Hall centered on rooftop solar installations under the “ULA Model.”
According to Nath, Tripura’s electricity consumer base has grown to around 10.68 lakh (about 1.07 million), up from roughly 7 lakh before the current government took office. He added that more than four lakh consumers use no more than 50 units a month, which he said suggests many belong to economically weaker sections.
Want to go solar but not sure who to trust? EnergySage has your back with free and transparent quotes from fully vetted providers in your area.
To get started, just answer a few questions about your home — no phone number required. Within a day or two, EnergySage will email you the best options for your needs, and their expert advisers can help you compare quotes and pick a winner.
“When the demand for something increases, its price goes up,” Nath said.
Nath said the challenge is not only rising demand, but also mounting pressure on the state’s conventional power system. He said a limited gas supply has reduced the Palatana plant’s output to about 500 MW, even though it was designed for around 700 MW, while Ramchandranagar is generating only 50 to 60 MW instead of roughly 100 MW.
As electricity demand rises, solar is one of the best ways to save money on home energy. Homeowners who want to explore rooftop systems can use EnergySage to get free solar installation estimates and compare quotes.
Reliable electricity is needed across the state, Nath said, including for industry, hospitals, schools, colleges, and waste-recycling facilities.
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He described solar, wind, and hydropower as greener options than gas- and coal-based generation, which he characterized as more polluting.
Tripura is also working to advance rooftop solar through coordination between the municipal corporation and power authorities. Nath said Agartala Mayor Dipak Majumder, Agartala Municipal Corporation Commissioner Saju Vaheed A, Deputy Mayor Manika Das Datta, and other officials took part in the discussion.
For households elsewhere, the economics of solar can be easier to understand with free comparison tools. With EnergySage’s help, the average person can save up to $10,000 on solar purchases and installations. EnergySage’s solar map shows the average cost of a home solar panel system on a state-by-state basis, along with solar panel incentives for each state. Together, these resources can help readers get the best price for rooftop solar panels and access available incentives.
Additionally, adding battery storage to a solar setup is one of the best ways to protect your home during outages. It can also save money on energy and make going off-grid more realistic, and readers can explore EnergySage for information about home battery storage options, including competitive installation estimates.
💡Go deep on the latest news and trends shaping the residential solar landscape
Nath said, “Gas reserves are depleting; the supply is dwindling,” and “The Central Government is placing the highest priority on solar and hydro power.”
Tripura’s rooftop solar push reflects a broader shift as households and governments look for cleaner, cheaper power while conventional fuel supplies come under strain. Across India and in nearby Pakistan, wider access to solar is already changing monthly bills, grid planning, and the pace of renewable investment.
• Across India, rooftop solar installations surged 125% as PM Surya Ghar accelerated household adoption.
• India saves billions by prioritizing solar, showing why states are leaning beyond gas.
• In Gujarat, a vast solar-wind project underscores how quickly India’s renewable buildout is scaling.
• In Pakistan, rooftop solar is shielding consumers from shocks and easing import costs.
• In Colorado, plug-in solar panels are widening home-energy access for residents shut out.
Rooftop solar is becoming a bigger piece of a much wider energy transition. These projects also show how much policy design matters when the goal is getting cleaner, lower-cost power to households that need relief most.
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Action News Now reporter Bella Barbosa reports from Butte County on new details about a large solar project that is expected to be proposed in unincorporated Butte County between the Skyway and Neal Road.
A Southern California energy company is looking to build a 300-megawatt solar facility in Butte County between the Skyway and Neal Road. The company, Cenergy Power, is planning to submit a proposal for the solar farm on 1,700 acres in unincorporated Butte County.
Currently in Chico
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Bangladesh has launched an incentive programme for rooftop solar systems paired with battery storage. The programme aims to ease the country’s worst power shortages in years and reduce its dependence on imported fuels. Consumers installing eligible systems by February 28, 2027, will receive BDT 10.50 ($0.086) per kWh for surplus electricity fed into the national grid, for three years. The government has capped the generation cost of rooftop solar power, including battery storage, at BDT 8 ($0.065) per unit. The incentive payment includes a 20% profit margin and an additional premium, while consumers installing systems below the benchmark cost can retain the savings. Bangladesh currently has about 1,559 MW of renewable capacity, mostly solar, and aims to reach 20% renewable generation, roughly 5,500 MW, by 2030. 
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India proposes mandatory battery storage for new renewable projects from July 2027  Reuters
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