Two-diode Model Research Articles

Parameter Estimation and Preliminary Fault Diagnosis for Photovoltaic Modules Using a Three-Diode Model

Accurate estimation of photovoltaic (PV) power generation can ensure the stability of regional voltage control, provide a smooth PV output voltage and reduce the impact on power systems with many PV units. The internal parameters of solar cells that affect their PV power output may change over a period of operation and must be re-estimated to produce a power output close to the actual value. To accurately estimate the power output for PV modules, a three-diode model is used to simulate the PV power generation. The three-diode model is more accurate but more complex than single-diode and two-diode models. Different from the traditional methods, the 9 parameters of the three-diode model are transformed into 16 parameters to further provide more refined estimates. To accurately estimate the 16 parameters in the model, an optimization tool that combines enhanced swarm intelligence (ESI) algorithms and the dynamic crowing distance (DCD) index is used based on actual historical PV power data and the associated weather information. When the 16 parameters for a three-diode model are accurately estimated, the I–V (current-voltage) curves for different solar irradiances are plotted, and the possible failures of PV modules can be predicted at an early stage. The proposed method is verified using a 200 kWp PV power generation system. Three different diode models that are optimized using different ESI algorithms are compared for different weather conditions. The results affirm the reliability of the proposed ESI algorithms and the value of creating more refined estimation models with more parameters. Preliminary fault diagnosis results based on the differences between the actual and estimated I–V curves are provided to operators for early maintenance reference.

A test of meta-heuristic algorithms for parameter extraction of next-generation solar cells with S-shaped current–voltage curves

Identifying parameters of photovoltaic (PV) models based on measured current–voltage (IV) characteristic curves is critical for simulating, evaluating, and controlling PV systems. IV characteristics of the latest-generation solar cells (SCs) often display an S-shaped deformation. In this paper, we explore the potential of meta-heuristic algorithms to address the parameter estimation problems associated with PV cells that exhibit S-shaped IV characteristics. This estimation is performed within the framework of the opposed two-diode model. We implemented a total of 14 algorithms from various classes to extract the SC parameters from synthetic IV curves, which were generated using a range of parameter values. The results were compared by using nonparametric statistical methods. These methods include the Wilcoxon signed-rank test for pairwise comparisons, and the Friedman, Friedman Aligned, and Quade tests for multiple comparisons. Comprehensive results and analyses show that the STLBO (Simplified teaching–learning based optimization algorithm) and ADELI (Adaptive differential evolution with the Lagrange interpolation argument) algorithms demonstrate highly competitive performance in terms of accuracy and reliability. This paper underscores the efficacy of advanced meta-heuristic algorithms in solving complex non-linear optimization problems in the domain of photovoltaic research, particularly concerning the unique challenges posed by S-shaped IV characteristics of new-generation solar cells.

Stochastic approach for maximum power point determination for bifacial PV modules by implementing two different cell models

PurposeThis paper presents the determination of the maximum power point of a bifacial photovoltaic (PV) system using two different cell models. The optimal power point is determined by using genetic algorithm (GA), as an optimisation tool. The purpose of this paper is to find which of the two analysed models gives better results in the determination of the maximum power point of a bifacial PV system for different solar irradiations. The quality of the results gained from both models is analysed based on the value of the objective function.Design/methodology/approachIn this research work, the maximum power point of bifacial PV modules is determined by using two different PV cell models, such as the simplified and two-diode models of PV cells. Based on the input electrical data for the analysed bifacial PV module as well as the mathematical model of the two PV cell presentations, the values for the current and the voltage at the maximum power point for a given solar irradiation and working temperature are determined by the algorithm for each solution in the population and generation.FindingsFrom the presented results and the performed analysis, it can be concluded that GA is quite appropriate for this purpose and gives adequate results for both models and for all working conditions. The two-diode model was found to be more suitable compared with the simplified model due to its complexity. Therefore, although the power difference for each of the scenarios for the two compared models does not differ significantly among the two models, it is in favour of the two-diode model. Which implicates that the for fast and simple calculation the simplified model can also do the job.Practical implicationsThis approach can be very successfully applied in the design process of a PV plant to forecast the output characteristics of the PV system if there is enough information about the weather conditions for a given location. This procedure can be very helpful in the process of selection of right PV module and inverter for a given location.Originality/valueAn optimisation technique using GA as an optimisation tool has been developed and successfully applied in the determination of the maximum power point for a bifacial PV module using to different models of solar cell. The results are compared with the analytically determined values as well as with the values given from the producer and they show good agreement.

Electrical characteristics and trap signatures for Schottky barrier diodes on 4H-SiC, GaN-on-GaN, AlGaN/GaN epitaxial substrates

The forward and reverse current transport mechanisms, temperature dependence of Schottky barrier height (SBH) and ideality factor, barrier inhomogeneity analysis, and trap parameters for Schottky barrier diodes (SBDs) fabricated on 4H-SiC, GaN-on-GaN and AlGaN/GaN epitaxial substrates are reported. High SBH is identified for Ni/4H-SiC (1.31 eV) and Ti/4H-SiC (1.18 eV) SBDs with a low leakage current density of <10−8 A cm−2 at −200 V. Thermally stimulated capacitance detects the well-known Z1/2 electron trap at E C—0.65 eV in both 4H-SiC SBDs, while an additional deep-level trap at E C—1.13 eV is found only in Ni/4H-SiC SBDs. The vertical Ni/GaN SBD exhibits a promising SBH of 0.83 eV, and two electron traps at E C—0.18 eV and E C—0.56 eV are identified from deep-level transient Fourier spectroscopy. A peculiar two-diode model behavior is detected at metal/GaN/AlGaN/GaN interface of high-electron mobility transistor (HEMT); the first diode (SBH-1 of 1.15 eV) exists at the standard Metal/GaN Schottky junction, whereas the second diode (SBH-2 of 0.72 eV) forms due to the energy difference between the AlGaN conduction band and the heterojunction Fermi level. The compensational Fe-doping-related buffer traps at E C—0.5 eV and E C—0.6 eV are determined in the AlGaN/GaN HEMT, through the drain current transient spectroscopy experiments.

Modified Rime-Ice Growth Optimizer with Polynomial Differential Learning Operator for Single- and Double-Diode PV Parameter Estimation Problem

A recent optimization algorithm, the Rime Optimization Algorithm (RIME), was developed to efficiently utilize the physical phenomenon of rime-ice growth. It simulates the hard-rime and soft-rime processes, constructing the mechanisms of hard-rime puncture and soft-rime search. In this study, an enhanced version, termed Modified RIME (MRIME), is introduced, integrating a Polynomial Differential Learning Operator (PDLO). The incorporation of PDLO introduces non-linearities to the RIME algorithm, enhancing its adaptability, convergence speed, and global search capability compared to the conventional RIME approach. The proposed MRIME algorithm is designed to identify photovoltaic (PV) module characteristics by considering diverse equivalent circuits, including the One-Diode Model (ONE-DM) and Two-Diode Model TWO-DM, to determine the unspecified parameters of the PV. The MRIME approach is compared to the conventional RIME method using two commercial PV modules, namely the STM6-40/36 module and R.T.C. France cell. The simulation results are juxtaposed with those from contemporary algorithms based on published research. The outcomes related to recent algorithms are also compared with those of the MRIME algorithm in relation to various existing studies. The simulation results indicate that the MRIME algorithm demonstrates substantial improvement rates for the STM6-40/36 module and R.T.C. France cell, achieving 1.16% and 18.45% improvement for the ONE-DM, respectively. For the TWO-DM, it shows significant improvement rates for the two modules, reaching 1.14% and 50.42%, respectively. The MRIME algorithm, in comparison to previously published results, establishes substantial superiority and robustness.

Chaos inspired invasive weed optimization algorithm for parameter estimation of solar PV models

High performance solar photovoltaic models require precise knowledge of solar PV cell parameters. Numerous methods based on both deterministic and meta-heuristics have been developed for identifying solar cell parameters. However, the presented methods in the literature have a heavy computational load and limited ability to extract crucial parameters due to nonlinear dynamics of solar PV systems. In addition, because they rely on approximations to determine the objective function, the preceding state-of-the-art parameter estimation techniques do not provide accurate results. Thus, a novel chaos-inspired invasive weed optimization (CIIWO) has been developed for accurate solar PV system parameter estimation. Adding a chaotic map to IWO improves the performance of suggested method by expanding the search space globally. Moreover, to cope with the inadequacy in state-of-art objective functions, Newton Raphson approach has been combined with proposed CIIWO algorithm. The suggested approach for solar cell parametric identification has been tested on one-diode, two-diode, and three-diode models. By contrasting the outcomes with nine contemporary optimization strategies for parameter estimation, the superiority of the suggested algorithm has been demonstrated. Commercial PV cell RTC France has been used for the experimental validation. Comprehensive study of experimental data validates the efficacy and stability of the suggested algorithm.

Optimizing Multi-Layer Perovskite Solar Cell Dynamic Models with Hysteresis Consideration Using Artificial Rabbits Optimization

Perovskite solar cells (PSCs) exhibit hysteresis in their J-V characteristics, complicating the identification of appropriate electrical models and the determination of the maximum power point. Given the rising prominence of PSCs due to their potential for superior performance, there is a pressing need to address this challenge. Existing solutions in the literature have not fully addressed the hysteresis issue, especially in the context of dynamic modeling. To bridge this gap, this study introduces Artificial Rabbits Optimization (ARO) as an innovative method for optimizing the parameters of an enhanced PSC dynamic model. The proposed model is constructed based on experimental J-V data sets of PSCs, ensuring that it accounts for the hysteresis characteristics observed in both forward and backward scans. The study conducted a rigorous statistical analysis to validate the Modified Two-Diode Model performance with that of the Energy Balance (MTDM_E) optimized using the innovative ARO algorithm. The performance metric utilized for validation was the Root mean square error (RMSE), a widely recognized degree of the differences between values predicted by a model and the values observed. The statistical analysis encompassed 30 independent runs to ensure the robustness and reliability of the results. The summary statistics for the MTDM_E model under the ARO algorithm demonstrated a minimum RMSE of 4.84E−04, a maximum of 6.44E−04, and a mean RMSE of 5.14E−04. The median RMSE was reported as 5.07E−04, with a standard deviation of 3.17E−05, indicating a consistent and tight clustering of results around the mean, which suggests a high level of precision in the model’s performance. Validated using root mean square error (RMSE) across 30 runs, the ARO algorithm showcased superior precision in parameter determination for the MTDM_E model, with a mean RMSE of 5.14E−04, outperforming other algorithms like GWO, PSO, SCA, and SSA. This affirms ARO’s robustness in optimizing solar cell models.

Parameter identification of the photovoltaic panel’s two-diode model

This work deals with the two-diode model of a photovoltaic (PV) panel. It provides the per-unit energy and current representations in addition to identifying its unknown parameters. Mathematical equations have been proposed and built using the MATLAB/Simulink simulator to achieve this goal. Only one variable has been adjusted to get all the unknown parameters simultaneously at standard test conditions (STC). Three variables have to be adjusted under any other atmospheric conditions. Two different technologies have been used. The accuracy of the proposed mathematical model has been provided using the absolute relative error between the simulated value and the measured one. The biggest values of the absolute relative error are 0.000788%, 0.0006157%, and 0.000616% under STC, nominal cell temperature condition (NOCT), and random daily atmospheric conditions, respectively.

Current invariant as fundamental relation between saturation currents and band gaps for semiconductor solar cells

The optimization of present multijunction junction solar cells and the development of new concepts is an important task for modem photovoltaics. For its development, special attention should be paid to the potential quality of photoactive p-n junctions, as well as to their properties when operating at various temperatures. The present research has shown that for two-diode model of the p-n junction it is convenient to use current invariants Jz1 and Jz2 for calculating the diffusion and recombination saturation dark currents J01 and J02 for p-n junctions based on semiconductor materials with any band-gap energy. It has been experimentally found that Jz1 and Jz2 are constants relating the semiconductor material band gap energy and temperature with J01 and J02. It has also been shown that Jz1 and Jz2 determine the open circuit voltage and the value of voltage offset WOC. Current invariant suggested to be useful tool for evaluating the quality of a p-n junction and for modeling the characteristics of solar cells because it allows making calculations taking into account both main mechanisms of current flow and the operating temperature of the device.

A simplified three-diode model for photovoltaic module: cell modeling and performance analysis

Purpose Estimation of solar cell parameters, mathematical modeling and the actual performance analysis of photovoltaic (PV) cells at various ecological conditions are very important in the design and analysis of maximum power point trackers and power converters. This study aims to propose the analysis and modeling of a simplified three-diode model based on the manufacturer’s performance data. Design/methodology/approach A novel technique is presented to evaluate the PV cell constraints and simplify the existing equation using analytical and iterative methods. To examine the current equation, this study focuses on three crucial operational points: open circuit, short circuit and maximum operating points. The number of parameters needed to estimate these built-in models is decreased from nine to five by an effective iteration method, considerably reducing computational requirements. Findings The proposed model, in contrast to the previous complex nine-parameter three-diode model, simplifies the modeling and analysis process by requiring only five parameters. To ensure the reliability and accuracy of this proposed model, its results were carefully compared with datasheet values under standard test conditions (STC). This model was implemented using MATLAB/Simulink and validated using a polycrystalline solar cell under STC conditions. Originality/value The proposed three-diode model clearly outperforms the earlier existing two-diode model in terms of accuracy and performance, especially in lower irradiance settings, according to the results and comparison analysis.

Matlab-Simulink of photovoltaic system based on a two-diode model simulator with shaded solar cells

This paper proposes a photovoltaic system modeling and characterization software based on Matlab-Simulink in order to estimate the parameters and the characteristic and electrical behavior of a cell/module with respect to changes in environmental parameter regarding irradiance, temperature, and surface conditions (partial shading). The first objective is to find the parameters of the nonlinear I–V equation by adjusting the curve regarding three particular points: open circuit, maximum power, and short circuit. Given these three points, which are provided by all commercial module datasheets, the method finds the best I–V equation for the two-diode photovoltaic (PV) model, with the best series and parallel resistances estimates. The accurateness of the simulator is verified by applying the model to different manufacturer’s modules. Thanks to an interactive graphic interface, this software offers a great flexibility to PV professional and to researchers.

An Advanced Bio-Inspired Mantis Search Algorithm for Characterization of PV Panel and Global Optimization of Its Model Parameters.

Correct modelling and estimation of solar cell characteristics are crucial for effective performance simulations of PV panels, necessitating the development of creative approaches to improve solar energy conversion. When handling this complex problem, traditional optimisation algorithms have significant disadvantages, including a predisposition to get trapped in certain local optima. This paper develops the Mantis Search Algorithm (MSA), which draws inspiration from the unique foraging behaviours and sexual cannibalism of praying mantises. The suggested MSA includes three stages of optimisation: prey pursuit, prey assault, and sexual cannibalism. It is created for the R.TC France PV cell and the Ultra 85-P PV panel related to Shell PowerMax for calculating PV parameters and examining six case studies utilising the one-diode model (1DM), two-diode model (1DM), and three-diode model (3DM). Its performance is assessed in contrast to recently developed optimisers of the neural network optimisation algorithm (NNA), dwarf mongoose optimisation (DMO), and zebra optimisation algorithm (ZOA). In light of the adopted MSA approach, simulation findings improve the electrical characteristics of solar power systems. The developed MSA methodology improves the 1DM, 2DM, and 3DM by 12.4%, 44.05%, and 48.88%, 28.96%, 43.19%, and 55.81%, 37.71%, 32.71%, and 60.13% relative to the DMO, NNA, and ZOA approaches, respectively. For the Ultra 85-P PV panel, the designed MSA technique achieves improvements for the 1DM, 2DM, and 3DM of 62.05%, 67.14%, and 84.25%, 49.05%, 53.57%, and 74.95%, 37.03%, 37.4%, and 59.57% compared to the DMO, NNA, and ZOA techniques, respectively.

Parameter identification of solar cells using improved Archimedes Optimization Algorithm

The parameters of solar cells for five PV models are identified using an Improved Archimedes Optimization Algorithm (IAOA) in this paper. Two modifications are made to the original Archimedes Optimization Algorithm (AOA). To control the unequal exploration and exploitation phases, the initial adjustment is to incorporate an augmented density decreasing factor. A random average calculation between the current object position and the best object position is implemented for the second modification to solve the local optima issue. The proposed IAOA is then used to tackle the problem of identifying PV model parameters from experimental I-V data. Different PV models, such as the one-diode model (ODM), the two-diode model (TDM), and the PV module model (PMM), have been distinguished using the suggested IAOA. The proposed IAOA outperforms other present algorithms and even outperforms the original AOA based on the revealed results. As closely as feasible to the experimental I-V data of real PV solar cells and module models, the proposed IAOA can choose the best parameter values for PV models.

An Innovative MPPT Method Based on the Estimated Model of the Photovoltaic Cell considering the Environmental Conditions

In this paper, a novel adaptive photovoltaic maximum power point tracking (MPPT) circuit is proposed, called AT2MPPT. The AT2MPPT circuit is based on a type two fuzzy logic set and utilizes a detailed two-diode model of a solar cell, as well as an innovative hybrid algorithm for designing the MPPT circuit. The effects of environmental conditions such as azimuth, wind speed, wind direction, and irradiation, as well as the uncertainty of radiation intensity, are taken into account in the solar cell model. It is demonstrated that changes in environmental conditions can cause up to a 20% change in the PV parameters. To optimize the AT2MPPT parameters and fuzzy membership functions (MFs), a hybrid lightning search algorithm and whale optimization algorithm (hLSA-WOA) are proposed. The accuracy of the hLSA-WOA is evaluated by comparing its results with those of other optimization algorithms on 10 standard benchmarks. The simulation results indicate that the hLSA-WOA has high accuracy and is nontrapping in local optima. Finally, the proposed AT2MPPT circuit is used to track the MPPT of a PV array in partial shade, and its performance is compared with that of P&amp;O and fuzzy MPPT circuits. The results confirm the superior performance of the AT2MPPT circuit, which produces 1.1% more power than the other two circuits. Additionally, the AT2MPPT circuit has lower power ripple and tracking time compared to perturb, observation, and fuzzy MPPT circuits.

Parameter Estimation Techniques for Photovoltaic System Modeling

In improving PV system performance, the parameters associated with electrical photovoltaic equivalent models play a pivotal role. However, due to the increased mathematical complexities and non-linear traits of PV cells, the precise prediction of these parameters is a challenging task. To estimate the parameters associated with PV models, a reliable, robust, and accurate optimization technique is needed. This paper introduces a new algorithm, Rat Swarm Optimizer (RSO), for obtaining the optimum PV cell and module parameters. The proposed method maintains an adequate balance between the exploration and exploitation phases to overcome premature particle issues. The results obtained using RSO are compared with those of other algorithms, i.e., Particle Swarm Optimization (PSO), Ant Lion Optimizer (ALO), Salp Swarm Algorithm (SSA), Harris Hawks Optimization (HHO), and Grasshopper Optimization (GOA), in this work. The modified one-diode model (MODM) and modified two-diode model (MTDM) are used to analyze the parameters of the mono-crystalline PV cell using the suggested RSO. The obtained findings imply that the parameters estimated by the suggested RSO are more accurate than those calculated by the other algorithms taken into consideration in the paper. The statistical results are compared, and it is clear that RSO is a very accurate, fast, and dependable approach for the parameter estimation of PV cells.

An efficient method for predicting PV modules performance based on the two-diode model and adaptable to the single-diode model

This paper introduces a simplified and accurate modeling approach to model photovoltaic modules. The current method uses a hybrid technique combining numerical and analytical approaches to predict the current-voltage characteristics of PV generators by calculating the parameters of the double-diode model. First, to reduce the complexity of the current equation and minimize the calculation time, some simplifications reducing the research space from seven to only five unknowns are taken into consideration. Second, three of these five parameters are extracted using simple analytical equations derived from the output current's equation based on the available values of the typical points on the manufacturer's datasheet. Then, the remaining two parameters are extracted using a combination of numerical approach and slope adjustment technique at the short-circuit point. The technique is adapted to the single-diode model in order to study the effect of the second diode on the simulating accuracy. The effectiveness is tested over six PV modules matching different PV technologies, and the results are compared with a great number of modeling methods based on different approaches, and the adjustment technique has shown the highest accuracy levels (RMSE values less than 0.045A) at the least computational times (compilation time less than 0.084 s).

Efficiency-loss analysis of monolithic perovskite/silicon tandem solar cells by identifying the patterns of a dual two-diode model’s current-voltage curves

In this work, we developed a simple and direct circuit model with a dual two-diode model that can be solved by a SPICE numerical simulation to comprehensively describe the monolithic perovskite/crystalline silicon (PVS/c-Si) tandem solar cells. We are able to reveal the effects of different efficiency-loss mechanisms based on the illuminated current density-voltage (J-V), semi-log dark J-V, and local ideality factor (m-V) curves. The effects of the individual efficiency-loss mechanism on the tandem cell’s efficiency are discussed, including the exp(V/V T) and exp(V/2V T) recombination, the whole cell’s and subcell’s shunts, and the Ohmic-contact or Schottky-contact of the intermediate junction. We can also fit a practical J-V curve and find a specific group of parameters by the trial-and-error method. Although the fitted parameters are not a unique solution, they are valuable clues for identifying the efficiency loss with the aid of the cell’s structure and experimental processes. This method can also serve as an open platform for analyzing other tandem solar cells by substituting the corresponding circuit models. In summary, we developed a simple and effective methodology to diagnose the efficiency-loss source of a monolithic PVS/c-Si tandem cell, which is helpful to researchers who wish to adopt the proper approaches to improve their solar cells.

Using GPUs to simulate photovoltaic power plants: Special cases of performance loss

The installed capacity of renewable energy is soaring, and Photovoltaic Power is a sizeable part of it. Photovoltaic panels are being installed on both buildings and in large Photovoltaic Power Plants (PVPPs), reaching up to 2 GW of installed capacity. We need to perform simulations of the state of PVPPs to ensure the stability of the power system and their profitability. Traditional simulations that take into consideration each panel can accurately predict the generated power but are too computationally taxing. To lower the computing cost of these simulations, a GPU-based solver that processes the dynamic model of a PVPP connected to a power system has been developed. It simulates the output of the whole photovoltaic power plant down to the photovoltaic panel level. The characteristics of the power system to be simulated were analyzed in order to parallelize the system. The two-diode model is used for the PV panels, and this model was linearized, resulting in a large sparse matrix with millions of columns that must be solved. Our proposal can simulate a model of the PVPP with both accuracy and low computational cost with no PV panel aggregation required. The system was solved using an adaptive step and helper conductances to aid convergence of the matrix. Parallel column factorization and column grouping is key to achieving a high performant simulation, which makes it feasible to simulate events such as cloud occlusion efficiently. We also studied how the wiring of PV panels affects the global power plant output when panels are shaded and/or broken. It was found that the aggregated model returns a higher generated output, which is key to predicting its output and its profitability. PVPP simulations on the GPU solver are up to 14.3 times faster than on the CPU.

Accelerated aging of unencapsulated flexible GaInP/GaAs/InGaAs solar cells by means of damp heat and thermal cycling tests

The extended damp heat and thermal cycling tests were performed on unencapsulated flexible thin-film GaInP/GaAs/InGaAs solar cells to assess the long-term stability. The solar cells were subjected to 85 °C/85% damp heat test for more than 1000 h and 420 cycles of thermal cycling test between −60 °C and 75 °C, respectively. The performance attenuations of flexible solar cells were less than 2% in both cases, which were due to the slow decline of the open-circuit voltage with aging time. The slight decrease in open voltage was attributed to the increase in reverse saturation current due to the enhanced recombination, which was in good agreement with the calculation based on the two-diode model. The good performance of the unencapsulated flexible GaInP/GaAs/InGaAs solar cells in severe environment indicated the stable and reliable device fabrication art in the experiment.

Development of a Photovoltaic Module Two-Diode Model for Studying Transients in the Solar Power Plant DC Network

The aim of the study is to develop and verify the analysis model of a photovoltaic module (PVM) based on a two-diode mathematical model in the MATLAB/Simulink software environment. The trends of and forecasts for the further development of photovoltaic power plants are considered. The PVM mathematical model is described, and a procedure for obtaining the parameters that are not included in the PVM datasheet but necessary for implementing the analysis model is described. The developed analysis model is verified based on the datasheet parameters of the PVM simulated and with reference to the values of PVM currents, voltages and power at the maximum power point (MPP) that were obtained on the test installation in the course of its operation. A classification of faults in the PVM arrays is given. The obtained results will subsequently be used in the development of a photovoltaic power plant analysis model for calculating transients in PVM arrays during faults with taking into account the MPP tracking device operation for correctly selecting the protection devices in the PVM circuits and developing algorithms for detecting and isolating these faults. Based on the obtained study results, a conclusion has been drawn about the need to take into account the influence of the insolation level, temperature, and PVM degradation in calculating transients in a photovoltaic array.