Explicit Numerical Model of Solar Cells to Determine Current and Voltage

Journal of Physics Conference Series PAPER OPEN ACCESS Explicit Numerical Model of Solar Cells to Determine Current and Voltage To cite this article M. Rasheed et al 2021 J. Phys. Conf. Ser. 1795 012043 View the article online for updates and enhancements. This content was downloaded from IP address 123.139.57.166 on 28/06/2021 at 1208 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the authors and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd ICMLAP 2020 Journal of Physics Conference Series 1795 2021 012043 IOP Publishing doi10.1088/1742-6596/1795/1/012043 1 Explicit Numerical Model of Solar Cells to Determine Current and Voltage M. Rasheed1, O. Y. Mohammed2, S. Shihab3, Aqeel Al-Adili4 1,3,4 University of Technology, Applied Sciences Department, Iraq 2 University of Anbar, College of Education for Pure Sciences, Iraq *Corresponding author rasheed.mohammed40yahoo.com, 10606uotechnology.edu.iq Abstract This paper deals with the extraction of the physical parameters of solar cells a single diode model from the equivalent circuit of the cell. The extraction is carried out by three different numerical methods with a comparison between them. The first and second methods are a difference of the Newton Raphson algorithm NRM and Aitkens extrapolation algorithm AEM, respectively. The roots of the nonlinear equation of this cell have been described and solved using the three methods. The proposed method is tested to solve the output voltage; current and power of this cell from the roots of this equation with the various values of load resistance RL. Keywords Aitkens extrapolation algorithm; three step method; iterations; load resistance; parameters of solar cell. 1. Introduction Numerical methods are a class of methods used to solve a wide range of mathematical problems whose origins can be mathematical models of physical conditions. These methods are unique in that they only use calculations and logic, which can be used directly on a digital computer. Numerical analysis can solve many kinds of nonlinear equations such as differential and partial equations; linear systems; Taylor series; integral equations; optimal control problems and physical problems like solar cells [1-5]. A photovoltaic cell is a specialized semiconductor diode that converts light into direct current electricity DC. Depending on the optical band gap in the light absorbing range, photovoltaic cells can also convert low-energy, infrared IR or high-energy, ultraviolet UV photons into DC electricity. These cells are made of semiconductor materials such as silicon. PV cells can be described corresponding to its manufacturing technology and the material used such as monocrystalline, polycrystalline, and amorphous solar cells; in addition; thin films solar cells [9-27]. This work aims to propose and characterize a new numerical method in order to find the real roots of the single-diode nonlinear equation of the solar cells based on three different techniques with the comparison between them. It is organized as follows section 2 characterizing the analytical model of a single-diode design of the solar cell; Section 3 establishing the root-finding Newton Raphson Method NRM; Aitkens extrapolation algorithm AEM and three step method TSM; section 4 results and discussion; section 5 conclusions of the obtained results. MATLAB program is achieved for all the acquired results. 2. Characteristics of Single-Diode Solar Cells Equation The simple equivalent electric circuit of a solar cell is shown in Figure 1. ICMLAP 2020 Journal of Physics Conference Series 1795 2021 012043 IOP Publishing doi10.1088/1742-6596/1795/1/012043 2 Figure 1. Single-diode model electrical circuit. Kirchhoffs current law is applied for single-diode model to calculate the current , the equation is given by 1 2 3 where is the photocurrent A; is reverse saturation current of the diode A; and are the delivered current and voltage, respectively. is thermic voltage 27.5≈ 26 mV at 25 oC Air-Mass 1.5; m is the recombination factor closeness to an ideal diode 1 m 2, is Boltzmann constant1.3810-23 J/K; is P-n junction temperature K; is the electron charge1.610-19 C. IphIsource 4 5 Substituting Eq. 4 in Eq. 5 we get 6 where Is reverse saturation current 10-12 A, suppose n is ideality factor1.2 in our case normally between 1 and 2. In parallel, According to Eq. 6 one can calculate of the cell numerically based on the first derivative of this equation. 3. Newton Raphson Method The following algorithm suggestion for solving Eq. 5 by using NRM see Figure 3 INPUT initial approximate solution x01, tolerance , maximum number of iterations N. OUTPUT approximate solution Step 1 Set Step 2 while Step 3 Calculate ́ for Step 4 if | | ; then OUTPUT and stop. Step 5 Set ; and go to Step 2. Step 6 OUTPUT 4. Aitkens Extrapolation Algorithm AEM Given ,  , , , IL R SH RSID I V - A ISH ICMLAP 2020 Journal of Physics Conference Series 1795 2021 012043 IOP Publishing doi10.1088/1742-6596/1795/1/012043 3 Step 1 For to 2 Step 2 Calculate ̅ ̅ ̅ ̅ ̅ ̅ ̅ for Step 3 If or , then go to Step 6 Step 4 Set ̅ ̅ Step 5 , , go back to Step 2. Step 6 OUTPUT and stop iteration. 5. Three Step Method TSM Six-order convergences with three steps are investigated. Let is a nonlinear equation, suppose x0 as an initial value, so the iteration results xn1 can be calculated using the following scheme ́ , ́ ́ ́ 7 Eq. 7 has a six-order convergence called a three-step method TSM; the proposed method. 6. Results and Discussion Consider the Eq. 6 is modeled in the form single-diode solar cell; has obtained the following approximate solutions and three numerical methods are applied first, Newton-Raphson methods NRM with initial value x_0. second, the Aitkens extrapolation algorithm AEM with the initial values from NRM and ; third the proposed method three-step method TSM with the initial value from AEM. In Table 1 the Aitkens extrapolation algorithm AEM and three-step method TSM of the solution results voltage ; current and power of the solar cell are given and listed in the last columns of this table when the load resistance . Table 1. The , , and  values using AEM and TSM Iterations R -AEM -AEM -AEM -TSM -TSM -TSM 1 1 0.947037857 0.947037857 0.896880703 0.922220699 0.922220699 0.850491018 2 1 0.930012729 0.930012729 0.864923676 0.921919557 0.921919557 0.84993567 3 1 0.923271149 0.923271149 0.852429615 0.922412266 0.922412266 0.850844388 4 1 0.922434357 0.922434357 0.850885144 0.922423132 0.922423132 0.850864435 5 1 0.922423136 0.922423136 0.850864443 0.922423135 0.922423135 0.850864439 6 1 0.922423135 0.922423135 0.850864439 0.922423135 0.922423135 0.850864439 7 1 0.922423135 0.922423135 0.850864439 Figure 2 presents the obtained solutions of the study result. ICMLAP 2020 Journal of Physics Conference Series 1795 2021 012043 IOP Publishing doi10.1088/1742-6596/1795/1/012043 4 Figure 2. Obtained solutions of the study result at the load resistance . In Table 2 the Aitkens extrapolation algorithm AEM and three-step method TSM of the solution results voltage ; current , and power of the solar cell are given and listed in the last columns of this table when the load resistance . Table 2. The , , and  values using AEM and TSM Iterations R -AEM -AEM -AEM -TSM -TSM -TSM 1 2 0.945750417 0.472875208 0.447221925 0.917699118 0.458849559 0.421085836 2 2 0.927013023 0.463506512 0.429676573 0.916335587 0.458167793 0.419835454 3 2 0.918476227 0.459238113 0.421799289 0.91700519 0.458502595 0.420449259 4 2 0.917067904 0.458533952 0.42050677 0.917035365 0.458517683 0.420476931 5 2 0.917035399 0.4585177 0.420476962 0.917035382 0.458517691 0.420476946 6 2 0.917035382 0.458517691 0.420476946 0.917035382 0.458517691 0.420476946 7 2 0.917035382 0.458517691 0.420476946 Figure 3 presents the obtained solutions of the study result. ICMLAP 2020 Journal of Physics Conference Series 1795 2021 012043 IOP Publishing doi10.1088/1742-6596/1795/1/012043 5 Figure 3. Obtained solutions of the study result at the load resistance . In Table 3 the Aitkens extrapolation algorithm AEM and three-step method TSM of the solution results voltage ; current , and power of the solar cell are given and listed in the last columns of this table when the load resistance . Table 3. The , , and  values using AEM and TSM Iterations R -AEM -AEM -AEM -TSM -TSM -TSM 1 3 0.944437431 0.472218715 0.44598103 0.912686741 0.45634337 0.416498543 2 3 0.92381119 0.461905595 0.426713557 0.909522637 0.454761319 0.413615714 3 3 0.912938978 0.456469489 0.416728789 0.910316762 0.455158381 0.414338303 4 3 0.910504334 0.455252167 0.414509071 0.910403208 0.455201604 0.414417 5 3 0.910403537 0.455201768 0.4144173 0.910403374 0.455201687 0.414417152 6 3 0.910403374 0.455201687 0.414417152 0.910403374 0.455201687 0.414417152 7 3 0.910403374 0.455201687 0.414417152 Figure 4 presents the obtained solutions of the study result. ICMLAP 2020 Journal of Physics Conference Series 1795 2021 012043 IOP Publishing doi10.1088/1742-6596/1795/1/012043 6 Figure 4. Obtained solutions of the study result at the load resistance . In Table 4 the Aitkens extrapolation algorithm AEM and three-step method TSM of the solution results voltage ; current , and power of the solar cell are given and listed in the last columns of this table when the load resistance . Table 4. The , , and  values using AEM and TSM Iterations R -AEM -AEM -AEM -TSM -TSM -TSM 1 4 0.943098312 0.235774578 0.222358607 0.907097754 0.226774439 0.205706584 2 4 0.92038679 0.230096697 0.211777961 0.9010338 0.22525845 0.202965477 3 4 0.90644763 0.226611907 0.205411826 0.901608561 0.22540214 0.203224499 4 4 0.90208766 0.225521915 0.203440537 0.901753616 0.225438404 0.203289896 5 4 0.901742565 0.225435641 0.203284913 0.90174069 0.225435173 0.203284068 6 4 0.901740602 0.225435151 0.203284028 0.901740602 0.22543515 0.203284028 7 4 0.901740602 0.22543515 0.203284028 Figure 5 presents the obtained solutions of the study result. ICMLAP 2020 Journal of Physics Conference Series 1795 2021 012043 IOP Publishing doi10.1088/1742-6596/1795/1/012043 7 Figure 5. Obtained solutions of the study result at the load resistance . In Table 5 the Aitkens extrapolation algorithm AEM and three-step method TSM of the solution results voltage ; current , and power of the solar cell are given and listed in the last columns of this table when the load resistance . Table 5. The , , and  values using AEM and TSM Iterations R -AEM -AEM -AEM -TSM -TSM -TSM 1 5 0.941732458 0.188346492 0.177372004 0.900883039 0.180176608 0.16231805 2 5 0.916716819 0.183343364 0.168073945 0.889778347 0.177955669 0.158341101 3 5 0.898705719 0.179741144 0.161534394 0.8884201 0.17768402 0.157858055 4 5 0.890512633 0.178102527 0.15860255 0.889061297 0.177812259 0.158085998 5 5 0.889126783 0.177825357 0.158109287 0.889092694 0.177818539 0.158097164 6 5 0.889092735 0.177818547 0.158097178 0.889092715 0.177818543 0.158097171 7 5 0.889092715 0.177818543 0.158097171 0.889092715 0.177818543 0.158097171 8 5 0.889092715 0.177818543 0.158097171 Figure 6 presents the obtained solutions of the study result. ICMLAP 2020 Journal of Physics Conference Series 1795 2021 012043 IOP Publishing doi10.1088/1742-6596/1795/1/012043 8 Figure 6. Obtained solutions of the study result at the load resistance . The obtained solution plot in the no. of iteration--plane and the initial-output values proves that the proposed method TSM has seven iterations indicated a fast behaviour. Parallel to this feature, it is noticed that the proposed method TSM has a behaviour of the solution in the initial values -AEM has the smallest error tolerance compared with NRM and AEM with initial value . Results of tables 1 to 5 are showing that the suggested method TSM have low error after relatively view iterations are computed and this in turn is demonstrating their efficiency 7. Conclusion This paper, give three numerical solutions for single-diode for PV cells mathematical model. The basic advantages of the proposed method TSM are simplicity and high accurate approximate solution which was achieved using a few numbers of iterations. The obtained numerical results were compared with two other methods NRM and AEM. 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