Introduction

PSCs are an emerging, efficient technology for converting solar energy. They may surpass traditional silicon cells in efficiency and cost-effectiveness due to their distinctive crystalline structure, which enables efficient sunlight absorption and longer charge lifetimes1. PSCs can be made with energy-efficient solution-based methods, potentially reducing costs to $0.05 per watt (NREL, 2020)2. Their remarkable efficiency, exceeding 25% (26.1% as of June 2023) at commercial and laboratory scales, presents opportunities for large-scale, cost-effective production3,4,5. Lead-halide PSCs hold promise but face stability and efficiency challenges. Researchers are developing new lead-halide perovskite materials to address these issues6. These novel materials exhibit enhanced stability and efficiency, making them appealing for commercial use. Strategies like replacing lead with less toxic metals, such as tin or germanium, improve stability7. Optimizing material composition has led to power conversion efficiency (PCE) exceeding 25%, comparable to traditional solar cells8,9,10,11,12. Lead-halide PSCs excel due to their high absorption coefficients, contributing to their efficiency and superior stability in various environmental conditions, unlike lead-free PSCs. NiOx-based PSCs gain interest for their affordability and stability when compared to other materials like PEDOT: PSS, Spiro-MeOTAD, CuSCN, Cu2O, MoO3, and CZTSe13,14.

NiOx-based inverted PSCs have achieved an impressive PCE exceeding 23%, ranking among the highest reported. These devices exhibit exceptional operational stability, boasting a T90 lifetime of over 1,200 h. The high PCE of these NiOx-based inverted PSCs can be attributed to factors such as the utilization of a high-quality NiOx hole transport layer (HTL), optimized perovskite layer, and the excellent interface between the HTL and perovskite layers. NiOx serves as an ideal HTL for PSCs due to its balanced properties, including high hole mobility, air stability, and cost-effectiveness15. Lead halide PSCs exhibit remarkable versatility, offering flexible and transparent formats suitable for applications like wearable electronics, sensors, windows, and skylights. Their potential extends to large-scale deployment, contributing to future energy landscapes16. PSCs boast fast response times, ideal for scenarios with rapid light intensity changes. While concerns regarding reliability, stability, quantum behavior, and toxicity exist, the minimal lead content in lead halide PSCs limits the risk of lead exposure17. Ongoing research areas encompass alternate materials, fabrication methods, stability enhancement, degradation mechanisms, performance optimization, and durability testing. Recent advancements in PSCs leverage artificial intelligence techniques for innovative applications like tandem, flexible, and polymer-coated solar cells18.

Various characterization techniques enhance PSCs understanding and reliability. These include I-V curve analysis, photoluminescence spectroscopy (PL), X-ray diffraction (XRD), scanning electron microscopy (SEM), transmission electron microscopy, impedance spectroscopy, transient-electroluminescence (T-EL), and steady-state electro- luminescence (S-EL). Transient electroluminescence provides insights into PSCs’ light emission, aiding in IV parameter analysis, material optimization, defect identification, and degradation dynamics. Photoluminescence (PL) spectroscopy to analyze defect density in PSCs has been used extensively. However, the adoption of EL-based FWHM as a standard metric for understanding lead-based PSCs remains limited. FWHM values obtained from PL and EL in PSCs are an area of ongoing research19,20,21,22. FWHM is a metric that quantifies defect size and distribution in solar cells23,24,25,26,27. PL spectroscopy and EL spectroscopy are distinct methods to measure FWHM in PSCs, and their results may vary28,29. The integration of machine learning and artificial intelligence tools has enhanced material discovery, degradation analysis, condition monitoring, reliability forecasting, defect detection, and problem-solving in the field of PSCs, bringing them closer to commercial viability30,31,32,33.

A recent study in 2023 established correlations between EL efficiency, EL spectrum, and EL images with the IV parameters of NiOx-based PSCs using steady-state electroluminescence (S-EL) spectroscopy34. Continuing from S-EL spectroscopy, this study employs supervised machine learning, specifically multiple linear regression, to estimate IV parameters from EL data. Such non-destructive imaging techniques offer valuable insights into these cells, enhancing IV parameter prediction accuracy. In EL measurements, critical parameters like exposure time, sample-sensor distance, EL bias voltage/current, and sample thickness are kept constant to simulate real-world conditions, ensuring accurate PV performance prediction.

The elementary and traditional method of using single diode model of predicting or forecast IV curves of solar PV modules is simple. A study on using this traditional method by splitting the transmission spectrum of solar cells/films and using the irradiation spectrum of sun is performed and observed that, the forecasting model is able to predict the IV curves of solar PV modules at installation site for long time. This study proves that without the use of advanced techniques, the model seems to be both simple and achieve a reasonable accuracy to predict and at the same time to understand the behaviour of performance of solar cells/modules35. The traditional method of estimating the IV curve parameters of a solar PV modules are based on equivalent circuit method but they have large errors in the estimation. In a study to predict IV curves from environmental factors such as irradiance, temperature, humidity and atmospheric pressure using convolutional neural network (CNN) and multilayer perceptron (MLP) neural network for the solar PV modules, it is observed that the both models predict better than equivalent circuit method with CNN sowing better results compared to MLP models. Using advance machine learning techniques such as CNN or MLP does not provide us much scope to analyse the outcome of the models due to absence of mathematical equations36.

A study on EL behaviour from PL excitation is was carried out on LEDs suggest that the behaviour of light emission from solar cells under excitation or LEDs during photoexcitation of charge carriers are mainly from recombination process and it is of dynamic in nature due to presence of defect, traps, hot spots, and leakage paths in the active, confinement and contact areas of the device37. EL & PL based characterization are one of the most used method to identify defects in the PSCs and lot of research has already been carried out. Using, PL spectral emission characteristics, the predicting the performance of IV curves for LEDs provides deeper correlation between strength of the intensity and ideality factor. This method is used where direct current injection is not possible (i.e. where EL could not be carried out). Very few research works have been on the prediction of IV parameters through PL or EL techniques38. Most of the researchers have worked on transient EL and PL studies for PSCs. A study of steady state absolute PL in PSCs were conducted by article to understand the loss mechanisms, radiative/non-radiative recombination at various operating voltages and to identify the charge extraction deficits to limited depth. This type of study is called as thermodynamic approach of identifying few IV parameters of PSCs39. A study on typical nip architecture based PSCs were carried out to understand the non-radiative losses by measuring the open circuit voltage deficit using external quantum efficiency (EQE) of EL spectroscopy technique with varied thickness and concentrations of absorber (perovskite) layer. The thermodynamic approach by using the \(\:Voc=Voc\_rad\:+\:ln\left(ELQE\right)*\frac{kT}{e}\) relation is the most common method of predicting or estimating the IV curve parameters such as Voc, and Jsc, (by incorporating AM 1.5 Sun Spectrum and actual Voc) of the PSCs40.

In article, for predicting FF of PSCs, voltage dependent PL is carried out to understand the recombination and resistive losses without use of optical simulations. Upon prediction of Voc, Jsc, and FF through the thermodynamic approach, the efficiency of solar cell is calculated (estimated)41. A recent study in 2022 on silicon heterojunction solar cells by the researchers of Fraunhofer Institute for Solar Energy Systems (ISE), Germany conducted contactless measurements and prediction of IV curve parameters by adopting CNN based empirical regression model from using PL data, reflectance spectra, and EL images. This approach of predicting IV curve parameters of a solar cell without actually performing IV characterization is showing a need for contactless measurements for solar cells to have lesser measurement times, measurement errors and cost savings without mismatch errors and damages to the solar cells due to mechanical stress42.

In the past year, a systematic review in the article shows that the there is a lack of available research works in ML-based prediction of IV parameters and there is a lot of scope of improvement in the parameters estimation performance. The research also indicates that the Neural Network (NN) is the most used followed by random vector functional link (RVFL) and support vector machine (SVM) based ML methods. Within the category of individual ML techniques, ANN emerged as the most widely used approach, followed by back propagation neural network (BPNN), genetic neural network (GNN), and long short-term memory (LSTM). With all these advanced ML approaches, unlike the said above, the simplest MLLR techniques are most useful due to its mathematical equation based prediction and behavioural analysis43. The study referenced in the article focused on silicon-based PV modules for predicting IV curve parameters using supervised ML algorithm like CNN from EL imaging and spectroscopic data. This study has proven that EL is a powerful tool for characterizing solar cell/modules, offering high-speed, high-resolution insights into device performance. By analyzing EL image and spectral data, it is possible to predict solar cells/module’s electrical properties, including IV curve parameters, with remarkable accuracy. The similar principles and techniques can be directly applied to PSCs. Perovskite materials exhibit unique degradation mechanisms and exhibit distinct EL signatures. By carefully selecting and extracting appropriate EL image/spectral data, it is possible to develop accurate predictive models for perovskite solar cells. This paper focuses more on EL based prediction of IV curve parameters44.

EL-based prediction offers several advantages over traditional characterization methods. It is a high-speed, non-destructive technique that enables real-time monitoring of module performance. By providing detailed spatial information, EL images allow for precise localization of degradation. Additionally, EL-based prediction can provide insights into the underlying causes of performance loss by quantifying specific EL image features related to degradation mechanisms. Moreover, this technique can detect early signs of degradation, enabling timely maintenance or replacement of modules.

The primary goal here is to estimate PSCs’ IV parameters using EL experiments. Therefore, FWHM and CCT extracted from EL spectra are crucial for accurately developing the predictive MLLR model. This study conducted three predictive MLLR models to estimate IV parameters from EL spectra: (a) FWHM-based, (b) CCT-based, and (c) Hybrid (FWHM-CCT-based) models. These models were compared, evaluated, and validated to determine the most effective one.

In the context of PSCs, FWHM of EL emission spectrum can provide valuable insights into the properties of the perovskite material and its performance under EL conditions. A narrower FWHM from EL suggests a more homogeneous distribution of carrier energies within the perovskite layer. This can lead to reduced spectral broadening and a smaller angle-dependent spectral shift45. In contrast, a wider FWHM suggests a broader distribution of carrier energies, which can result in more pronounced spectral changes with viewing angle due to light trapping and waveguide effects within the device structure. This can be correlated with improved carrier transport and reduced non-radiative recombination, leading to higher efficiencies in both solar cell and LED modes. Hence, FWHM can be considered as a proxy for carrier dynamics46.

With respect to the impact of FWHM on IV curve parameters, a narrower FWHM might indicate a higher proportion of carriers that contribute to the photocurrent, leading to a higher Jsc, a more homogeneous carrier distribution could result in a higher Voc due to reduced recombination losses, and a narrower FWHM might suggest improved charge carrier collection efficiency, resulting in a higher FF47. Therefore, a linear regression model can be employed to establish a quantitative relationship between FWHM and IV curve parameters. By training the model on a dataset of FWHM values and corresponding IV curve parameters, we can predict the IV curve parameters of a new PSC based on its measured FWHM of EL. By correlating FWHM with IV curve parameters using a machine learning-based linear regression model, it is possible to gain valuable insights into the relationship between the spectral properties of PSCs and their electrical performance48. This approach can be used to optimize the fabrication and characterization of PSCs, ultimately leading to improved device efficiency and stability.

The CCT of emitted light from PSCs is directly linked to the bandgap of the perovskite material. This means that devices with different bandgaps would emit light with different CCTs. In a study, PSCs with bandgaps ranging from 1.60 eV to 1.97 eV were investigated, and each bandgap exhibited a distinct CCT. The study revealed an inverse relationship between CCT and PCE. The CCT of emitted light from PSCs are dependent of the bandgap of the device such as 1.6 eV, 1.72 eV, 1.82 eV, 1.97 eV, etc. It is observed that CCT would decrease in case of high efficient PSCs and increase in the case of low efficiency PSCs49. In the case of the 1.60 eV devices, increasing the CCT (from 1700 K to 6500 K) led to a decrease in PCE. This suggests that for these devices, optimizing the bandgap for higher efficiency might involve targeting a lower CCT. Conversely, for the 1.97 eV devices, increasing the CCT improved the PCE. This indicates that for these devices, a higher CCT might be beneficial for achieving higher efficiency. The findings suggest that by carefully tailoring the bandgap of perovskite materials, it might be possible to design PSCs that are optimized for efficient low-energy harvesting under specific conditions. For example, devices with lower bandgaps and higher CCTs could be suitable for applications where harvesting light in the blue and green regions of the spectrum is desired. On the other hand, devices with higher bandgaps and lower CCTs might be more efficient for harvesting light in the red and near-infrared regions. By understanding the relationship between CCT, bandgap, and PCE, researchers can develop strategies for designing PSCs with tailored optical properties for specific energy harvesting needs.

When considering steady-state EL spectroscopy, the changes in CCT or Voc will be minimal as the devices under consideration of this study are of same hole transport layer (HTL)50. A study on perovskite-based LEDs indicated that CCT tends to increase when the injected voltage is increased. The same would also appear in case of PSCs operating under LED mode51. A recent study in 2022 on PSC behaviour with regard to different climatic temperature conditions were carried out. The CCT of PSCs varies with operating temperature and it is found that at low temperatures, the CCT values are relatively higher compared to high temperature in the range of 0–75 °C52. This model using both FWHM and CCT parameters were not carried out so far. This paper has slightly explored (in supplementary information) the possibility of combining these parameters for possible prediction of IV curve parameters using simple linear-regression based machine learning method.

A study on PSC’s EL characterization revealed that relatively larger average Vi (injected EL voltage) and smaller FWHM of the passivated sample are strong evidence of substantially suppressed nonradiative recombination and improved homogeneity. This also indicates that the larger Vi the PSC takes, more the presence of defective region which tends to forward bias the surrounding regions and draws more current to emit more light in such regions. This was observed in the EL images.

This study exclusively examines forward IV characteristics, pioneering the estimation of IV parameters from EL spectroscopy with a set 25% tolerance level. The study also delves into modeling with discontinuous datasets, challenging due to perovskite’s regenerating properties, making the correlation of IV parameters with CCT and FWHM in EL spectroscopy intricate. The primary focus lies on the proposed FWHM-based predictive MLLR model.

This approach involves training the proposed model on a dataset that includes known IV and EL parameters (EL-FWHM & Peak Wavelength of Emission). The model learns to associate a few hundred EL-FWHM & Emission Wavelength values with distinct IV parameters. The dataset was divided into training and test sets, with the former used for model training and the latter for evaluation. This study demonstrates the feasibility of a supervised ML-based optoelectronic characterization model in predicting PSC IV characteristics, establishing relationships between EL and IV parameters, and validating the results with test datasets. A novel predictive ML model is proposed and evaluated, offering potential utility for solar cell research and development.

Materials and methods

Methodology

A Spectroradiometer-based (Make: Ocean optics) PSCs using a 400 µm optical fiber cable. Ocean optics capture EL emissions via the 400 µm fiber cable. Concurrently, the EL imaging process is conducted by a camera situated 150 mm away from the PSC surface and positioned at a 45º angle.

Raw materials

Nickel acetate tetrahydrate for the hole transport layer was obtained from Sigma Aldrich. The precursors for Methylammonium lead iodide, like Lead iodide (PbI2 with purity ~ 99.9985%) and Methylammonium iodide (MAI with purity > 99.99%), were procured from Greatcell solar materials and Alfa aesar respectively. PC60BM from Nano-C was used for the electron transport layer. All solvents used in the fabrication, like chlorobenzene (CB), N,N- dimethylformamide (DMF), and dimethyl sulfoxide (DMSO) were procured from Sigma Aldrich. Perovskite precursors lead iodide (PbI2) and methylammonium iodide (MAI) were purchased from Alfa Aesar and Greatcell Solar Materials.

Fabrication

Nickel acetate tetrahydrate was used as the precursor to make a solution of 0.5 M in ethanol. 30µl volume of ethanolamine was added to it to make a stable dispersion. The solution was stirred for 2 h and finally spin-coated at 1500 rpm. The NiOx films were annealed at 450 °C for 30 min and cooled naturally. Devices have a planar p-i-n architecture like FTO/ NiOx/ CH3NH3PbI3/ PCBM/ Ag. Patterned FTO-coated glass was used as substrates. They were cleaned using soap solution, de-ionized water, acetone, and isopropyl alcohol by ultrasonication. The FTOs (Fluorine-doped Tin Oxide) were dried and exposed to UV-Ozone for 30 min to improve the surface’s wettability. 1.3 M perovskite solution was prepared using precursors PbI2 and MAI in mixed solvent. The perovskite solution was pin coated in two sequential steps at 1500 and 5000 rpm, followed by annealing at 60 °C and 100 °C. After that, the electron transport layer was deposited by spin coating Hot solution of PCBM. Ag was deposited by the thermal evaporation process for the back contacts.

Characterizations

I-V measurement and capacitance behavior were investigated using Keithley 4200 SCS. AM 1.5 G illumination condition was simulated employing an AAA class Newport solar simulator. The PL measurements were done using a HORIBA spectrometer. EL experiments were carried out using a spectrometer and fiber optic cable under controlled temperature and humidity conditions. Figure S1 provides the experimental setup used in this study.

ML-based optoelectric modelling of PSCs

This study uses a log-log-based linear relationship to model the relationship between IV & EL parameters. The log-log-based linear relationship is a mathematical function often used to model the relationship between two variables on a logarithmic scale. This type of relationship is often found in material science, where the logarithm of its composition can describe the properties of a material. Given the wide dynamic ranges of both IV parameters and EL-FWHM measurements, a log-log transformation enables a more effective linear representation of their relationship. This log-log-based color relationship accurately models the correlation between IV parameters and EL-FWHM, suggesting its potential as a valuable tool for researchers studying PSCs. Continuous datasets are used to correlate EL findings and I-V parameters properly. EL spectroscopy provides the values of CCT, wavelength (λ), andFWHM. I-V curves provide the values of open-circuit voltage (Voc), short-circuit current (Jsc), fill factor (FF), and photovoltaic efficiency (PV-Eff). The datasets are divided into two types: training data and test data. Artificial intelligence/supervised machine learning-based log-transformed multiple linear regression is performed on the training dataset using input EL parameters and output I-V parameters to derive a predictive model. This predictive model can predict the I-V parameters from EL imaging for a PSC. Figure S2 explains the basic experimental flowchart of this study and optoelectric characterization modeling for lead halide PSCs.

Limitations of advanced ML techniques

While advanced ML techniques have shown promise in other domains, their application to IV parameter prediction remains limited. Simple MLLR models, on the other hand, offer several advantages, including interpretability, computational efficiency, robustness, and simplicity. By leveraging these advantages, MLLR models can provide a valuable tool for predicting IV parameters in clinical settings. Many advanced ML techniques, especially deep neural networks, are often referred to as “black box” models because they can be difficult to interpret. This lack of interpretability can hinder our understanding of the underlying relationships between predictors and outcomes, making it challenging to trust the model’s predictions. Advanced ML techniques often require large datasets to train effectively. In the context of IV parameter prediction, where data collection can be challenging, this can limit their applicability. Complex ML models can be difficult to implement and maintain, requiring specialized expertise. This can increase the cost and time associated with developing and deploying these models.

Advantages of simple MLLR techniques

MLLR models provide transparent relationships between IV parameters and their predictors. The coefficients in these models can be directly interpreted, offering a clear understanding of how each predictor contributes to the outcome. This interpretability is crucial for prediction applications where understanding the underlying mechanisms of PSCs are essential for clinical decision-making. MLLR models are computationally efficient compared to many ML algorithms, especially deep neural networks. This efficiency is particularly advantageous in real-time applications where quick predictions are required. MLLR models are generally less prone to overfitting than complex ML models, especially when the data is limited. Overfitting can lead to poor generalization performance, which is undesirable in unstable PSCs performance where the model needs to perform well on new, unseen data. The simplicity of MLLR models makes them easier to implement and understand, reducing the risk of errors and facilitating collaboration between domain experts and data scientists.

Results

IV and EL parameters of seven PSCs, which exhibit efficiencies of 14% ± 1%, were measured continuously for 8 days. Parameters extracted from the IV curve include Voc, Jsc, FF, and PV-Eff, and EL parameters include Intensity, emission wavelength, and FWHM. The dataset of three devices (D1, D2, and D3) is considered for training the MLLR model. The remaining four devices (P1, P2, P3, and P4) were individually and combined as test datasets for validating the MLLR model. Figure 1a–c show the IV curves and EL spectra of D1, D2, and D3 for the continuous measurements from Day 1 to Day 8. Figure 1d shows the day-wise variation of EL images for D1, D2, and D3. The FWHM, peak emission wavelength and the measurement day of the EL spectrum are significant factors affecting PSC’s Voc, Jsc, FF, and PV-Eff parameters. The EL intensity is the radiant power emitted by a PSC per unit solid angle when acting as a light-emitting diode (LED). The emission wavelength is the light output color close to the magenta color (or reddish pink). The graph shows that the EL intensity peaks at a wavelength of around 775 nm in the near-infrared (NIR) region (end of the visible region). It can be inferred that the graph also shows that the EL intensity increases over the 8 days. The increase in EL intensity could be due to several factors, such as the degradation of the PSC materials (decrease in performance of the PSC as a solar cell), localization of charge carriers, increase in hot spots, or fabrication issues during the preparation process.

Figure 1d, e shows the day-wise EL images captured during the EL spectroscopy measurements for Day 1 to Day 8 for Devices D1, D2, and D3, along with respective light-emitting and non-emitting areas. The images show the degradation of PSCs over a period of 8 days. The first image shows the cells on day 1, and the last image shows them on day 8. We can observe that cells have become significantly less efficient over time (in light-emitting areas) and increased non-emitting areas over the course of 8 days for the devices. The cells have also changed from a bright pink to a duller pink. It is observed that the dark/opaque areas of the cells increase over time, while the non-dark areas decrease. The dark/opaque areas are due to the degradation of the perovskite material. While perovskite materials are susceptible to degradation from light, heat, and real-world stresses, the images from this 8-day experiment suggest remarkable stability in the PSCs. This is encouraging, as degradation remains a major hurdle for widespread PSC deployment. With time, the propagation of inactive regions increases, starting from the edges. It is well-known that any edge is more prone to degradation because of defects (presence of dangling bonds) and humidity ingress. These signs of degradation could depend on the fabrication process and the purity of materials used in the PSC.

Fig. 1
figure 1

IV Curves and EL spectrum of D1 (a), D2 (b), and D3 (c) PSCs along with (d) Captured EL Images and (e) Light-emitting and non-emitting areas from Day 1 to Day 8.

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With the efficiency of D1, D2, and D3 being 13.29%, 13.94%, and 14.19%, respectively, it can be observed that the more the device sustains to act as an LED, the lesser its solar power-conversion efficiency. However, the overall trend is that the cells are relatively stable and less likely to degrade over time. This is a promising sign for the future of PSCs.

Tables S1, S2, S3, S4, S5, and S6 provide the measured IV and EL parameters for devices D1, D2, and D3, respectively, for 10 mA and 20 mA current injection in order. The correlation between EL Images, IV Curves, EL Spectra, and CCT reveals important insights about the aging process of PSCs. As PSCs age, the light-emitting area within them decreases, and this reduction is directly related to the IV and EL parameters. Specifically, the IV curve parameter FF demonstrates a strong positive correlation with the light-emitting areas observed in EL Images. Similarly, EL Spectra analysis shows that the FWHM correlates comparatively more strongly with light-emitting areas. These findings indicate that light-emitting areas indicate the functional portion of the PSC, and to some extent, they provide insights into PV-Eff by revealing defects or the proportion of low-efficiency areas. However, the average EL Intensity alone does not offer significant insights because it considers the entire light-emitting area, which is non-uniform. A more informative approach would be to study the distribution of EL Intensity across the area, although this aspect is not covered in this study. It is worth noting that FWHM is constant, depends only on the inherent properties of the PSC, and does not depend on aging, indicating that the relationship between the two can be formed easily and can help understand the reliability and stability of the PSC. Conversely, EL and IV parameters exhibit an inverse relationship with CCT values. CCT values are negatively correlated with PV-Eff when compared to other IV parameters. In other words, as PV-Eff declines, CCT values tend to increase. In summary, CCT and FWHM values can serve as valuable indicators of PSCs’ reliability and performance, offering insights into these solar cells’ aging process and efficiency.

The training dataset (from D1, D2, and D3) is processed for a simple ML regression to obtain R2 values as close to 1. Figure 2 shows four separate equations each for Voc, Jsc, FF, and PV-Eff predictions, along with their R-square values.

Equation (S1) provides the matrix representation for the obtained predictive MLLR Model equations for respective IV parameters. The equations of this MLLR model relate the Voc, Jsc, FF, and PV-Eff of the PSC from the FWHM and wavelength (λ) of its EL spectrum. The coefficients and constants in the equations represent the strength of the relationship between the input and output variables. The equations can be used to predict the IV curve parameters of a PSC from its EL spectrum. The equations can also be used to understand the factors that affect the IV curve parameters of the PSC. The constant and coefficient of the ‘t’ term have the same sign for Jsc and FF, whereas the opposite for Voc and PV-Eff equations. It could be said that the similarity in signs of the constant term and coefficient of ‘t’ indicates that the model could predict relatively better for Jsc and FF compared to Voc and PV-Eff values.

Fig. 2
figure 2

Log-transformed multiple linear regression equations of the supervised machine learning-based predictive model for IV parameters from FWHM values (forward characteristics). (a) Voc, (b) Jsc, (c) FF, and (d) PV-Eff.

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Out of the seven PSCs, three PSCs were used as training datasets and four other additional devices were used as test datasets for validating the obtained predictive MLLR model. P1, P2, P3, and P4 are the four PSCs for which IV and EL measurements were taken continuously from Day 1 to Day 8. Tables S7, S8, S9, S10, S11, S12, S13, and S14 show the EL and IV measurement data of P1, P2, P3, and P4 devices for 10 mA and 20 mA current injections, which are used as test datasets in this study for estimating the prediction accuracy of the proposed model. Figure 3a–d shows the IV curves and corresponding EL spectra for devices P1 to P4. Figure S3 provides the details of devices’ non-emitting (Dark region) areas and light-emitting areas during the course of 8 days, along with the day-wise EL images. Similar to the training dataset from D1, D2, and D3, the pattern of degradation is the same for test dataset devices P1, P2, P3, and P4.

Fig. 3
figure 3

IV curves and EL Spectra of test dataset PSCs: P1 (a), P2 (b), P3 (c), and (d) P4 from Day 1 to Day 8.

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Discussion

The trained MLLR model is validated using a test dataset obtained from EL spectroscopy. Four devices, P1, P2, P3, and P4, are considered individually and combined as separate test datasets for validation. Firstly, based on the test dataset, the validation of the proposed predictive MLLR model is accessed by comparing IV parameters predicted and actual values when corresponding EL parameters are provided as input. Figure 4 provides predicted vs. actual Values of IV parameters from Day 1 to Day 8 obtained using the proposed FWHM-based predictive MLLR model using the test dataset (combined).

Fig. 4
figure 4

Predicted vs. Actual Values of IV parameters (a) Voc, (b) Jsc, (c) FF and (d) PV-Eff from Day 1 to Day 8 obtained using the proposed FWHM-based predictive MLLR model using continuous test dataset.

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Table S15 shows the prediction accuracy (%) of individual devices (test devices) using the proposed MLLR Model for various IV Parameters. Figure S4 shows the prediction vs. measured values of IV parameters from Day 1 to Day 8 for individual devices P1, P2, P3, and P4 using the FWHM-based predictive MLLR model.

Secondly, the model’s accuracy is computed to estimate its effectiveness in predicting the IV parameters. Figure S5.1 shows the average accuracy of the proposed FWHM-based predictive MLLR Model (when all test datasets are combined) at 25% tolerance. Equation (1) is used to obtain the accuracy of the model. The accuracy of the model for Voc is the highest. This could be because Voc is a simple factor unaffected by as many other factors as Jsc and FF. The accuracy of the model for Jsc is also high. However, the model’s accuracy for Jsc & FF is slightly lower than that for Voc. This is because Jsc & FF are more complex factors that are affected by several other factors, such as temperature, relative humidity, device fabrication, defect densities, and stray light exposure. The model’s accuracy for PV-Eff is lower than Voc, Jsc, and FF models. This is because PV-Eff is a function of Voc, Jsc, and FF, and the model’s accuracy. Overall, the accuracy of the models for Voc and Jsc is high. However, the model’s accuracy for FF is lower, and the model’s accuracy for PV-Eff is the lowest.

At 25% tolerance levels for individual devices, the model’s prediction accuracy is high for Voc, Jsc, and FF and average for PV-Eff. Similarly, when the dataset is combined, the overall prediction accuracy is 100%, 96%, 96%, and 73% for Voc, Jsc, FF, and PV-Eff, respectively. From Table 1, the accuracy is higher for devices P1 and P4 compared to devices P2 and P3. Due to higher non-linearities inside the data, P2 and P3 have poor performance in predicting compared to P1 and P4.

$$\:Accuracy\:\left(\%\right)=\frac{\text{N}\text{u}\text{m}\text{b}\text{e}\text{r}\:\text{o}\text{f}\:\text{t}\text{r}\text{u}\text{e}\:\text{v}\text{a}\text{l}\text{u}\text{e}\text{s}\:}{Number\:of\:total\:values}*\:100$$
(1)

The prediction accuracy results of the proposed MLLR model from predicting PSCs IV parameters from EL experiments are very promising at 25% tolerance. The average prediction accuracy (indicated as “Average” in Table 1) for all devices is 91.5%, which is quite good for this complex prediction task. The combined prediction accuracy (indicated as “Combined” in Table 1) for all devices is also 91.5%, which indicates that the model is able to generalize well to new data and cross-validate the results of the proposed model.

Table 1 Prediction accuracies of IV parameters at 25% tolerance level when individual devices are predicted separately (individual test datasets each for P1, P2, P3, and P4).
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The average prediction accuracy is calculated by averaging the individual prediction accuracies for all devices. The combined prediction accuracy is calculated by combining all devices’ datasets and then predicting the IV parameters. The prediction accuracy for individual devices (P1, P2, P3, and P4) is even higher, with all devices having an accuracy of at least 90%. This suggests that the model is able to learn the complex relationship between IV parameters and can predict the IV parameters for most devices accurately. Overall, the prediction accuracy of the proposed ML model for predicting solar IV parameters from EL experiments of PSCs is quite good. An analysis of how the accuracy changes with different tolerance levels (e.g., 20%, 15%, 10%, and 5%) are provided in Table S19 indicating the scope of improvement in prediction accuracy of the proposed model. However, it is important to note that the prediction accuracy can vary depending on the specific device and the conditions under which it is tested. The model could speed up the development of new PSC materials and devices. However, more research is needed to improve the prediction accuracy for different types of PSCs.

It is important to note that the last two columns of Table 1 show the predicted PV-Eff and calculated PV-Eff values from the proposed MLLR Eq. (1) and conventional Eq. (3), respectively. When considered individually, the predicted efficiency values are almost the same, with a maximum of 10% deviation. When considered together/combined, the prediction accuracies are the same for the average prediction accuracy and combined prediction accuracy. Both indicate that the proposed model estimates the efficiency values accurately irrespective of using the proposed or conventional equation for efficiency. This is another type of cross-validation to the proposed MLLR model.

$$\:Ca{l}_{-}PV-Eff=\frac{Voc\:*\:Jsc\:*\:FF\:}{\left(G*A\right)}$$
(2)

Where G = input solar irradiance = 100 mW/cm2, A = area of the PSC = 0.1cm2, Voc = open- circuit voltage, Jsc = short-circuit current density, and FF represented in percentage (%).

After validation and optimization of the proposed machine learning model, it needs to be evaluated for its performance. There are different evaluation metrics for several types of problems to evaluate the model objectively. For regression models, the commonly used metrics for regression problems are mean absolute error (MAE), root mean squared error (RMSE), mean absolute percentage error (MAPE), and R-squared53,54,55,56,57,58. Figure S5.2 shows the model’s MAE, RMSE, and MAPE for respective IV parameters for the combined test dataset. The MAE, RMSE, and MAPE are all measures of the error between the predicted and actual values. The MAE is the average of the absolute errors, the RMSE is the square root of the average of the squared errors, and the MAPE is the average of the absolute errors expressed as a percentage of the actual values. Equations (S2), (S3), and (S4) represent the equations for MAE, RMSE, and MAPE, respectively, that are used to evaluate the proposed MLLR model.

The accuracy of the model closely matches the results obtained from MAPE values. Therefore, MAPE is the best tool to see the effectiveness of the model compared to MAE and RMSE indices. MAPE of 0.04 V, 0.11 mA/cm2, 0.12, and 0.18% for Voc, Jsc, FF, and PV-Eff, respectively, indicate that this model predicts the corresponding IV parameters for 95% of the time within 4%, 11%, 12%, and 18% of the actual values. Overall, the MAE, RMSE, and MAPE for the MLLR model are all relatively low, indicating that the model is performing well.

The results of this study demonstrate the potential of using EL spectra to predict IV parameters of PSCs. Future work could expand the dataset to include more PSCs with varied materials and device architectures. This would allow for further validation of the model and improvement of its accuracy. Additionally, investigating the physical mechanisms underlying the relationship between EL spectra and IV parameters could provide a deeper understanding of how PSCs work and could lead to the development of new, more efficient devices. Finally, developing this model using EL spectra to predict IV parameters of PSCs, such as open-circuit voltage, short-circuit current, fill factor, and efficiency, would be a valuable tool for PSC optimization. This work is the first to carry out such a study with a few hundred datasets, which is large in material science40. This could lead to new insights into PSCs’ underlying physics and help improve their performance.

Comparison of MLLR models

The study has successfully completed the electro-optical modeling of PSCs based on EL spectroscopy to estimate the device (PSC) IV parameters (Voc, Jsc, FF, and PV-Eff) using continuous datasets of similar efficiency devices. studies and FWHM-CCT-based hybrid model prediction of IV parameters were conducted. Figure S6 shows the Log-transformed Multiple Linear Regression Equations and plots of the supervised machine learning-based predictive CCT model for estimating IV parameters from CCT values (for forward characteristics). It was found that all of them are CCT-based MLLR Models with an R-square over 0.99. Here, the pattern could be observed from the coefficients of log (CCT/T), t (day of measurement), and the constant terms, which are positive, negative, and negative, respectively. This means that this equation is very good in estimating the degradation characteristics of PSC compared to IV parameters, as the “t” term refers to the same efficiencies. Figure S8 shows the CCT range of values for each day.

Using the test dataset devices P1 to P4, the CCT-based MLLR Model is used to predict the IV parameters. This model predicts Voc and FF values better than PV-Efficiency values and cannot predict Jsc values. Figure S9 shows predicted vs. measured values of IV parameters (a) Voc, (b) Jsc, (c) FF, and (d) PV-Eff from Day 1 to Day 8 obtained using the proposed CCT-based predictive MLLR model using the test dataset. This indicates that the CCT value could be used to understand the device FF and, therefore, the degradation performance on par with the FWHM-based MLLR model. Figure S10 shows the accuracy of the proposed CCT-based predictive MLLR Model (when all test datasets are combined) at 25% tolerance. It can be observed that the accuracy of estimating the IV parameters Voc and FF are almost the same as that of the proposed FWHM-based model and fairly comparable in estimating the PV-efficiency values. In general, Voc and Jsc depend on the device’s inherent properties, whereas FF and PV-Eff are performance properties of the device. Therefore, the FF estimation or FF values are expected to be proportional in nature. If the number of datasets were large enough, then the estimation of PV-Eff and FF would be almost similar for both types of models (FWHM or CCT-based). The lower accuracy of FF and PV-Eff values estimation from the CCT-based model could be due to fewer datasets than the proposed FWHM-based model.

From the above two models, it is clear that Voc estimation is slightly better by the CCT-based MLLR model and Jsc and FF estimation by the FWHM-based MLLR model. PV-Eff estimation is almost the same for both models, but the FWHM-based model outperforms the CCT-based model by 12.6%. Considering the same and the analysis being already explained using the FWHM- based model, a hybrid model is also studied but with the use of estimating the CCT-based model for Voc, PV-Eff estimation, and Jsc and FF estimation through the FWHM-based model. Figure S11 shows the Log-transformed Multiple Linear Regression Equations of the supervised machine learning-based predictive hybrid model for IV parameters from CCT & FWHM values (for forward characteristics). The equations are obtained from FWHM-based and CCT-based models for analysis and are shown in equations S2, S3, S4, and S5. When estimated for their IV parameters from this model, the accuracy of predicting Voc, Jsc, FF, and PV-Eff were 100%, 96.4%, 96.4%, and 64%, respectively. Figure S12 shows the accuracy of the proposed predictive Hybrid MLLR Model (when all test datasets are combined) at 25% tolerance. After understanding the three types of MLLR models using forward IV characteristics datasets, the reverse characteristics datasets were also analyzed using ML regression and corresponding equations were obtained for estimating the IV parameters from the same EL and CCT parameters. Table S15 shows the prediction accuracy of three types of ML regression models for forward and reverse IV characteristics of PSCs in predicting the IV parameters from EL parameters (FWHM and CCT).

Discontinuous datasets analysis

To further explore the validation of this model, discontinuous datasets are considered. A few (4 Nos.) additional fresh devices were prepared on the same day, but their IV and EL measurement sets were taken with a 7-day gap. Further, individual devices were purposefully stored in the glove box or clean room in between continuous measurements to have discontinuous datasets for each of the PSCs. Table S16 shows the EL images corresponding to the measurement day for each PSC. Figures S13 to S28 show the results of various measurements performed on these PSC devices nos. 1, 2, 3, and 4 over 22 days with a 20 mA current injection. Each of the device’s EL images and IV curves for each measurement day is analyzed, and the correlation between PV-Eff and FWHM values is also shown. As defect density increases, the PV-Eff reduces, which is the same for all devices. It can be observed that the relation of FWHM and PV-Eff with time are inversely proportional to each other. Further to FWHM-based analysis, chromaticity coordinates of all the devices at 20 mA current injection in EL with time are analyzed, correlating PV-Eff and CCT values. The PSCs’ regeneration capability (self-healing property) was clearly observed; therefore, the prediction of IV parameters was difficult. It is observed that by applying the FWHM-based and CCT-based MLLR models on the discontinuous datasets for predicting their IV parameters from EL parameters FWHM & wavelength and CCT, respectively, the prediction accuracies are good for both the models, but the FWHM-based model shows better accuracy compared to CCT based model at 25% tolerance level.

EL Intensity, EL emission wavelength, and EL efficiency studies need to be explored further in their relation to material properties of the absorber layer and transport layers with several types of substrates. FWHM from EL spectra is a useful parameter related to the defect density of the PSC, like the PL spectroscopy method. CCT and FWHM values are inclusive parameters that include the variation of IV parameters and could be used to better estimate the PV-Eff of PSC. Future works will concentrate more on similar studies with many datasets by evaluating highly efficient PSCs with different substrates (ITO, flexible, etc.), absorber layers, and environmental conditions.

When such discontinuous datasets are used for estimating the IV parameters using the equations obtained in Eq. (1), a correction factor must be added. Table S17 shows the values of the CFs (Correction Factors) for the respective models of regression equations for estimating IV parameters such as Voc, Jsc, FF, and PV-Eff values. The CFs are used to adjust the measured values of the IV parameters to account for the effects of inherent properties or environmental factors. There is no clear relation between the FWHM-based and CCT-based models. The CFs for the two models are different for all IV parameters. This suggests that the two models are not directly comparable.

Four types of comparative analysis are carried out using discontinuous datasets. Firstly, the dataset is merged into a single dataset, and a combined analysis of EL injection currents at 10 mA and 20 mA (i.e., 50% and 100% of rated current density) was carried out. In this scenario, the prediction accuracies are comparatively low for FWHM and CCT-based MLLR models. Secondly, the dataset containing 20 mA EL injection current (rated short circuit current density) values are alone considered for analysis and found that the prediction accuracy is better than the first type, and the FWHM-based MLLR model showed better rate of prediction than the CCT-based MLLR model. Thirdly, the actual day of measurement is considered as the start of measurement rather than considering the day of preparation as Day 1. In this scenario with modified days of measurement, both models showed better accuracy compared to the past two types of comparison. This indicates that the PSC behavior is not affected much if they are stored immediately after the day of preparation but shows regenerative (self-healing) behaviors in the case of storage between the measurement periods. Lastly, the longest working PSC (D1) is alone considered for estimating IV parameters, i.e., single device estimation. This scenario highlighted that the single device prediction accuracy is higher than the other three types for FWHM and CCT-based MLLR models. This is expected as the dataset reduces the prediction accuracy increases. Voc and Jsc estimation is 100% in the case of the FWHM-based MLLR model and far better than the CCT-based model. If observed closely, in single-device prediction of IV parameters, the prediction of FF and PV-Efficiency is slightly better for the CCT-based MLLR model than the FWHM-based model in the case of single-device prediction. The FWHM-based MLLR model is the best-fitted model to estimate all IV parameters from EL spectroscopy.

Comparison of Voc estimation and Voc loss analysis using thermodynamic model and proposed machine learning approach

The present-day PCE of approximately 25% is undoubtedly impressive for PSCs. However, these devices have yet to realize their full potential in terms of open-circuit voltage (Voc). Despite the theoretical Voc limit of 1.32 V for lead halide perovskites, which typically exhibit a bandgap of 1.6 eV, the actual Voc values achieved under standard AM 1.5 G illumination conditions remain suboptimal. This discrepancy between the theoretical maximum and practical performance highlights a critical area for further research and development in the field of perovskite photovoltaics. One of the purpose of thermodynamic approach is to exceed the Shockley–Queisser limit for PCE and EQEEL (closer to 1) of a solar cell by optimizing and reducing the non-radiative voltage losses due to defects, traps, geometrical loss (like reflection, waveguiding, etc.). For maximizing the Voc, better EL efficiency is required59. Research that simultaneously investigates the performance of devices and their corresponding spectroscopy data is limited, particularly in the context of PSCs and the significance of EL60. The analysis of EQE through EL spectroscopy reveals fascinating insights into the fundamental characteristics of light-emitting devices. It is proven that EQE of EL of the solar cells under bias voltage with current injection in dark equal to short circuit current density of the PSC can help estimate the recombination in the devices61. PSCs limited by radiative recombination can experience a substantial boost in Voc, Jsc, and overall efficiency through optimized internal optics and the strategic utilization of internal luminescence. While internally emitted photons can escape the cell directly, effective confinement using front and back reflectors enables a significant portion to be reabsorbed, a process known as photon recycling. This phenomenon elevates the equilibrium minority carrier concentration, consequently increasing Voc62.

Nonradiative losses (NRL) to the Voc is one of the factors limiting the PSC’s PCE. This can be determined from the radiative thermodynamic limit of the photovoltage. The real Voc is the difference between the radiative limit and the total nonradiative losses (\(\:{{V}_{oc},}_{nrad}\)), defined as the sum of the intrinsic (\(\:{V}_{oc}^{intrinsic}\) ) and electrode-induced (\(\:{V}_{oc}^{electrode}\) ) losses63. For high Voc in PSCs, most researchers have linked with Shockley–Queisser limit for a specific bandgap that was determined in different ways from paper to paper. This can be easily replaced by radiative limit of Voc, which can be calculated using the EQE of the PSC. The radiative limit to the Voc can be calculated from \(\:{V}_{oc}=\:{{V}_{oc},}_{rad}-\:{{V}_{oc},}_{nrad}\:\) and \(\:{{V}_{oc},}_{rad}\:=\:{n}_{id}^{rad}*\left(\frac{kT}{q}\right)\:*\:ln\left(\right({EQE}_{EL}*\frac{{J}_{sc}}{{J}_{0},rad})\:+\:1)\)64,65, where EQE=1 to determine the Voc, rad.

Enhancing non-radiative recombination within the cell leads to a rise in saturation current and a reduction in the observed Voc. Consequently, we can identify and measure the impact of non-radiative recombination losses using \(\:{{V}_{oc},}_{nrad}\:=\:{n}_{id}^{rad}*\left(\frac{kT}{q}\right)\:*\:ln\left(\right({EQE}_{EL})\)66, where Voc rad is the radiative Voc; k is Boltzmann’s constant; T is the temperature in Kelvin (T = 293 K in this case); q is the elementary charge; Jsc under 1 Sun illumination with AM1.5 spectrum; Jo, rad is the radiative saturation current density, and most importantly, ideality factor most commonly used is replaced with ectypal factor (an alternate yet proven) for PSCs i.e. \(\:{n}_{id}^{rad}=2.6\:\)for considering a multi-diode model of PSC (\(\:{n}_{id}^{rad}\) = 1 for single diode model for ideal solar cells and mostly applicable to silicon crystalline types). Most common ideality factor used by many researchers so far for PSCs is either 1 or 2 depending on the type of recombination and diode model used as per classical theory67,68,69. Many research works on PSCs in calculating ideality factor shows that the voltage drop across each layer of PSCs is significant by including the ion migration effects within the absorber layer70. Non-ideal voltage blocking effects in ETL or HTL could be one the reasons for ideality factors going beyond 271. This is also supported by performing IMVS, OCVD, impedance and IMPS studies showing that PSCs have this ectypal factor can range from 2.6 to 5.2 for same set of manufactured batches72,73. The measurement of ideality factor or ectypal factor for PSCs is necessary to understand the type of recombination and their effects on efficiency. More research is required into the underlying recombination processes in working devices. This study has assumed the ideality factor to be 2.6 considering the literature data for PSCs74. Applying the many-diode model allows for accurately determining \(\:{{V}_{oc},}_{rad}\) from these spectral data and justifies that neglecting the defect response is a valid method to straightforwardly determine \(\:{{V}_{oc},}_{rad}\). The mathematical expressions of related parameters like Jsc, J0,rad, and Black body spectral photon flux (φbb) are provided in75,76.

For PSCs, if all non-radiative recombination losses are reduced, the Voc could be increased about 150mV high. Despite good optoelectronic properties, increasing the crystallinity and reducing the non-radiative recombination with help of voltage-dependent EL of PSC could help understand the effect of interlayers, contact selectivity, and defects in these devices. This relationship manifests in a voltage-dependent behavior that significantly impacts photon emission. Having higher Voc or reaching near to Shockley-queisser limits by reducing defects and non-radiative recombination for PSC can lead to better efficiency77. At voltages below the EL intensity peak, electrons struggle to acquire sufficient energy for photon release, relying on environmental energy absorption. Conversely, when voltages surpass this peak, an extreme quasi-Fermi level separation occurs, resulting in electrons possessing excess energy beyond what’s emitted during recombination78. Despite the potential for radiative recombination above the peak EL energy, energy loss during down-conversion is unavoidable. These observations underscore the critical importance of precise voltage regulation in minimizing unnecessary energy dissipation, paving the way for more efficient optoelectronic devices79.

Table S20 to S23 (in Supplementary Information) contains mainly the Voc measured from 1 sun irradiance (using IV characterization tool) i.e. the actual Voc, predicted Voc by well-know and existing thermodynamics model, and predicted Voc of the proposed FWHM-based MLLR model, along with other parameters such as Voc_rad, Voc_nrad, EQE_EL, EG_EL, Jo_rad, for corresponding PSC devices. The comprehensive Voc loss and prediction evaluation of three predictive models – FWHM-based, CCT-based, and thermodynamic – across four PSC devices (P1-P4) revealed valuable insights into their performance and applicability. All models demonstrated commendable predictive capabilities, as evidenced by their low mean absolute error (MAE) and root mean square error (RMSE) values. Figures S29, S30, S31, S32, and S33 provides the information on comparative statistical metrics, correlation analysis, error analysis, residual analysis, and time-series analysis for test PSC devices (used for predicting from the proposed models) P1 to P4. However, the FWHM and CCT-based models consistently outperformed the thermodynamic model in terms of accuracy and consistency. These models exhibited lower average MAE (0.0346 and 0.0340, respectively) and RMSE (0.0394 and 0.0391) compared to the thermodynamic model (MAE: 0.0598, RMSE: 0.0670). Moreover, the FWHM and CCT-based models demonstrated remarkable adaptability across different devices, suggesting their robustness for general use in various PSC configurations. While the thermodynamic model exhibited lower overall performance, it may still provide valuable insights into specific physical aspects of PSCs, potentially making it useful for particular applications. In conclusion, the FWHM and CCT-based models emerged as the preferred choices for precise Voc predictions in PSCs due to their superior performance and adaptability .

Conclusions

This study demonstrated that supervised machine learning can predict IV parameters in PSCs. The log-log-based linear relationship was used to model the relationship between IV parameters and EL-FWHM, and this model could predict the IV parameters with > 90% accuracy. Considering the regression model used in this study, it can be observed that MAPE is the best-evaluating index for determining the effectiveness of ML-based regression models compared to MAE or RMSE. The model was validated, and the evaluation metric MAPE showed that the model predicts the IV parameters for 95% of the time within 4%, 11%, 12%, and 18% of the actual values for Voc, Jsc, FF, and PV-Eff, respectively. However, it is important to note that ML regression models’ performance might vary depending on the dataset, features used, and model complexity. This study has introduced a new method and a valuable tool using EL spectra and images to identify defects, damaged, and low-efficient PSCs at a solar power plant. It is low-cost, non-destructive, fast, accurate, reliable, and can be used to analyze images of PSCs taken in the field. The method can also be used to make other inferences about the health of the PSCs.

These findings have significant implications for the solar energy industry, potentially revolutionizing quality control processes and enabling more efficient maintenance strategies for large-scale solar installations.

To conclude, the supervised machine learning-based multiple linear regression models based on FWHM and CCT are good for analyzing the PSCs characteristics and estimating IV parameters. The proposed FWHM-based MLLR model can be used to best fit the model for all IV parameters, whereas the CCT-based model could also be applied to all but more for estimating Voc and FF parameters. Further work is ongoing regarding reducing the error of models and tolerance levels up to 10% or less from the 25% level. Looking towards alternate and reasonably complex methods of regressions or machine learning methods for improving the models for better prediction of IV parameters and quantification of degradation mechanisms in PSC is needed.

The pursuit of these advanced methods could lead to breakthroughs in our understanding of PSC degradation, ultimately contributing to the development of more durable and efficient solar cells.

Further analysis and evaluation of the models’ robustness would be essential to assess their predictive capabilities comprehensively. This study provides a valuable tool for researchers and manufacturers developing PSCs. It can be used to predict the IV parameters of these cells, which can help to improve their efficiency and performance. The study uses a relatively small dataset with a few hundred measurements. This is a limitation of the study, but it is still a significant achievement to be able to train an ML model on such a small dataset.

This pioneering study, one of the first to apply machine learning techniques in perovskite solar cell characterization, has far-reaching implications for the PSC community and future solar energy industry. By achieving over 90% accuracy in predicting IV parameters, this approach could revolutionize quality control processes and enable more efficient maintenance strategies for large-scale perovskite solar installations.

The pursuit of advanced regression and machine learning methods, as outlined in this study, could lead to significant breakthroughs in our understanding of PSC degradation mechanisms. This enhanced understanding is crucial for the development of more durable and efficient PSCs, addressing key challenges in their widespread adoption.

The success of this MLLR model, particularly its high accuracy with a limited dataset, demonstrates its potential scalability and real-world applicability. Future research could explore integrating this methodology with real-time monitoring systems, potentially enabling predictive maintenance and optimizing the operational efficiency of perovskite solar power plants.

The non-destructive nature of the EL technique used in this study opens up new possibilities for in-situ analysis of PSCs. This could dramatically accelerate the iterative process of PSC design and optimization, significantly reducing development cycles and associated costs in PSC research and manufacturing.

In the broader context of renewable energy adoption, the implications of this research are profound. As the world transitions towards cleaner energy sources, tools that enhance the efficiency and reliability of PSC technology play a crucial role. This study contributes to that goal by offering a method to quickly identify underperforming cells, potentially increasing the overall output and lifespan of PSCs.

Looking ahead, the integration of this ML-based approach with other emerging technologies, such as Internet of Things (IoT) devices and big data analytics, could create powerful synergies in the field of PSC. These combinations might lead to smart, self-diagnosing PSCs capable of optimizing their performance in real-time, further enhancing the viability and attractiveness of this technology as a primary power source.

In conclusion, while this study marks a significant step forward in PSC characterization and performance prediction using machine learning, it also opens up numerous avenues for future research and practical applications. As we continue to refine and expand upon these methods, the potential for improving perovskite solar cell efficiency, reducing costs, and accelerating the global adoption of this promising solar technology becomes increasingly tangible.