Abstract: To use photovoltaic systems, it is necessary to measure precisely the light output that affects each unit of surface in the locations to be installed, which is essential to assess your energy efficiency. For this, these cells must be calibrated and their volt-ampere characteristics (I-V) known. The error made in determining these characteristics was evaluated using a simplified mathematical model and comparing the result of both models with the experimental result of 95 Si photocells from the Combined Electronic Components of the Pinar del Río province; facilitating the necessary calculation in the use of said cells as solar radiation sensors. The experimental points were extracted using the OriginPro 2017 program and processed with the Mathcad 15.0 program and the mathematical theory of numerical methods. The quality of the adjustment is determined by comparing the statistical coefficients R2 corresponding to the complete and simplified equation, for which it has been found so far that said coefficient R2 of the simplified equation is better than that given by the equipment with which experimental values and results are determined as good as with the complete equation.

Keywords: Si photocell, I-V characteristic, mathematical model, numerical methods.


Resumen: Para utilizar sistemas fotovoltaicos es necesario medir con precisión el flujo luminoso que incide en cada unidad de superficie en los lugares a instalar, lo cual es fundamental para evaluar su eficiencia energética. Para ello, estas celdas deben estar calibradas y conocer sus características voltamperio (I-V). El error cometido en la determinación de estas características se evaluó utilizando un modelo matemático simplificado y comparando el resultado de ambos modelos con el resultado experimental de fotocélulas de 95 Si de Componentes Electrónicos Combinados de la provincia de Pinar del Río; facilitando el cálculo necesario en el uso de dichas celdas como sensores de radiación solar. Los puntos experimentales se extrajeron con el programa OriginPro 2017 y se procesaron con el programa Mathcad 15.0 y la teoría matemática de métodos numéricos. La calidad del ajuste se determina comparando los coeficientes estadísticos R2 correspondientes a la ecuación completa y simplificada, para lo cual se ha encontrado hasta ahora que dicho coeficiente R2 de la ecuación simplificada es mejor que el dado por el equipo con el que se obtienen valores experimentales y los resultados se determinan tan buenos como con la ecuación completa.

Palabras clave: fotocélula de Si, característica I-V, modelo matemático, métodos numéricos.

1. INTRODUCTION

Fossil fuels are finite resources that will inexorably run out . The current energy system is based on these sources that lead to a series of environmental and sustainability problems, among which we can mention the greenhouse effect . Fortunately, there are also renewable energies; within this group we find wind energy, photovoltaic solar energy, thermal solar energy, hydraulic energy, energy from biomass and organic waste .

Solar energy, the emblem of renewable energies, has seen a progression in recent years due to improvements in technology, associated with cost reduction and mainly thanks to the interest shown by a large number of countries ; . In photovoltaic systems it is essential to measure the efficiency of the solar cells, since this allows knowing how much power of solar radiation (W/m2) each one absorbs, for which it is also necessary to know precisely how much solar energy reaches each location per area unit in one second. Then, the amount of solar energy and its conversion efficiency into electrical energy are the two fundamental parameters to be measured and controlled in any photovoltaic system that uses solar energy . This efficiency is the ratio between the number of electrons released for electrical conduction and the number of light photons reaching the cell, called the internal quantum efficiency ; .

To determine how much solar radiation affects 1 m2 of surface, equipment with very expensive and slow technologies is used, since it cannot be put into practice in real time (online), which represents a great difficulty for its use in the country; it is for this reason that calibrated solar cells are used instead. For these cells to function correctly it is necessary to determine their I-V characteristics; and the mathematical model of a photovoltaic solar cell has a complex transcendent equation, this makes it very difficult to evaluate the cells as solar radiation sensors. However, when making certain approximations to such an equation, a simpler one is reached that facilitates the calculation of the I-V characteristics, but as a problem, it is unknown how precise it is. Therefore, the objective of the present work is to simplify a mathematical model for the determination of the volt-ampere characteristics of solar photocells, which facilitates the calculation of efficiency, which is necessary in the use of said cells as solar radiation sensors.

2. MATERIALS AND METHODS

2.1 Theoretical Foundation

The PV solar cell can be modeled by means of the equivalent circuit shown in figure 1, using the laws of circuit theory, the following mathematical model for the solar cell is obtained .

\[ I = I_L - I_D \left[ e^{\frac{q(V + R_s I)}{mKT}} - 1 \right] - \frac{V + R_s I}{R_{sh}} \hspace{1cm} (1)\] Then, for a photovoltaic module or panel, the characteristic equation is: \[ I = I_L - I_D \left[ e^{\frac{V + R_s I}{a}} - 1 \right] - \frac{V + R_s I}{R_{sh}} \hspace{1cm} (2)\] Where [a=mKTNs/q], where m is the dimensionless recombination coefficient of the cell (between 1 and 3); K is Boltzmann's constant (1.38∙10-23 J/K); q is the charge of the electron (1.6∙10-19 C); T is the temperature in Kelvin and Ns is the number of cells in the module.



\(I_L \approx I_{sc} \) , where \(I_L\) is the maximum intensity and \(I_{sc}\) is the short circuit intensity. \(I_D = I_L \cdot e^{\frac{V_{oc}}{a}} \) , where Voc is the open circuit voltage: \[ V_{oc} = \frac{mKT}{q} \ln \left( \frac{I_L}{I_D} \right) = mV_T \ln \left( \frac{I_L}{I_D} \right) \hspace{1cm}(3)\] \[ V_T = \frac{KT}{q} \hspace{1cm}(4)\]The series resistance is: \[ R_s = \frac{a \cdot \ln \left( 1 - \frac{I_{mp}}{I_L} \right) - V_{mp} + V_{oc}}{I_{mp}} \hspace{1cm}(5)\] Where \(I_{mp}\) and \(V_{mp}\) are the Intensity and Voltage at the point of maximum power given by the expressions: \[ I_{mp} = I_L \left( 1 - c^{-d} \right) \hspace{1cm}(6)\] \[ V_{mp} \approx V_{oc} \left( 1 - \frac{\ln c}{c} \right) \hspace{1cm}(7)\] Where: \[ c = 1 + \ln \left( \frac{I_L}{I_D} \right) \hspace{1cm} and \hspace{1cm} d = \frac{c}{c+1}\hspace{1cm}(8)\] Equation (2) is a transcendent equation, which makes the process cumbersome to determine the intensity I for the different voltage values V necessary for the evaluation of photovoltaic cells as sensors of solar radiation. To facilitate these calculations, a simpler equation can be arrived at by making an approximation, since the value of Rsh, the parallel resistance, which is found in the denominator of the term corresponding to the intensity of the current Ish, is much greater than its numerator and can be considered negligible. The characteristic equation for a PV module or panel is: \[ I = I_L - I_D \left[ e^{\frac{V + R_s I}{a}} - 1 \right] \hspace{1cm}(9)\] This equation defines I as an implicit function of V, but also defines V as an explicit function of I, obtaining the following expression. \[ V(I) = a \ln \left( \frac{I_D - I + I_L}{I_D} \right) - I R_s \hspace{1cm}(10)\]

2.2 Materials, Equipment and Methods to Follow

95 images of 95 graphs of the I-V characteristics corresponding to 95 photocells are taken as a sample. These data were supplied by the UEB for PV energy, CCE Combined for Electronic Components, located on Álvaro Barba Airport Highway Km 2 ½, Colon Avenue, Passage A, No.16 between Álvaro Barba and Passage B, Pinar del Río province, Cuba. In the conformation of the solar panels, technology from the firm EPSON or Seiko Epson Corporation is used, which is a Japanese company and one of the world's largest manufacturers of electronic components such as SCARA robots for industrial use. While the Solar Simulator HSM2 is used to test the main electro-optical characteristics of solar cells and map the I-V curve. The 95 images are digitized with the OriginPro 2017 program to extract the experimental points corresponding to the characteristic curves of intensity I versus voltage V. The program is data analysis and graph generation software. It is a leader in the scientific-technical sector; belongs to OroginLab Corporation and is used to import, graph, explore, analyze and interpret data as needed.



Once these are obtained, they are introduced in the MathCad 15.0 program, which is a software designed mainly for the verification, validation, documentation and use of engineering calculations that allows exploring problems, formulating ideas, analyzing data, modeling and checking scenarios, determine the best solution; as figure 3 shows.

3. PRESENTATION AND DISCUSSION OF THE RESULTS

The 95 images were digitized where the corresponding data to be processed in the MathCad program were obtained, which consist of its main electro-optical parameters. Taking into account their efficiency values, they are classified by colors according to the order of quality, very good (High quality), fair (Medium quality) and poor (Low quality), as shown in the table in figure 4.


Note. a Fill Factor, area of the cell that is "filled" with solar radiation and converts it into electricity. b Efficiency. c Electro-optical parameters with their mean values and absolute errors of measurements from the root mean square deviation.

Below are scatter diagrams for the main electro-optical parameters of the 95 photovoltaic solar cells studied with their mean values and their respective maximum and minimum errors.

3.1 Adjustments of the Experimental Results to the Exact and Approximate Models

The table in figure 16 shows 15 examples of the adjustments with the main electro-optical parameters of the 95 cells studied for the exact model, for the approximate model and for reported experimental values. The table in figure 16 shows 15 examples of the adjustments with the main electro-optical parameters of the 95 cells studied for the exact model, for the approximate model and for reported experimental values. The quality of the fits is determined by comparing the corresponding statistical coefficients of determination R2 (error), where the perfect fit corresponds to R2=1.



Figure 17 shows examples that are considered good, regular and bad, taking into account the quality of their adjustments according to R2 ; both for the exact model (complete equation) and for the approximate model (simplified equation) and for experimental values reported consecutively.

3.2 Analysis and Comparison

The quality of solar cells for use or not in solar panels is determined from the electro-optical parameters given by the manufacturer, and also the number of cells that are classified as good, fair and poor in quality, according to the quality of the corresponding adjustments from the R2 of each one with the main electro-optical parameters.





It can be concluded whether a batch is good, fair or bad depending on the percentage of cells classified as good, fair and bad. It can also be seen that the coefficient R2 of the simplified equation is better than the one given by the equipment with which the experimental values are determined and in the order of the R2 for the complex equation. That is to say, the results are as good as with the complete equation; in addition, the simplified equation is easier to work with, since it is possible to clear one variable as a function of another.

4. CONCLUSIONS

Experimental data were extracted from their corresponding I-V curve images of the 95 photocells.

The results of the mathematical models were compared, both with the transcendental equation and with the simplified equation, with the experimental values; the simplified model has turned out to be a good one.

The approximate model was used without making large errors.

The quality of the solar cells for their use or not in the solar panels was determined from the electro-optical parameters given by the manufacturer.

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