11850 FACTA UNIVERSITATIS Series: Electronics and Energetics Vol. 36, No 4, December 2023, pp. 567 - 576 https://doi.org/10.2298/FUEE2304567O © 2023 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper ALGORITHM TO EXTRACT MODEL PARAMETERS OF PARTIALLY SHADED PHOTOVOLTAIC MODULES Adelmo Ortiz-Conde, Francisco J. García-Sánchez Solid State Electronics Laboratory (LEES), Simón Bolívar University (USB), Caracas 1080A, Venezuela) Abstract. Uneven irradiation, due to partial shading, can produce hot spots in photovoltaic modules. A classical solution to avoid hot spot consists in using bypass diodes in antiparallel to series-connected cell groups. This solution brings a new problem: the presence of multiple local maximum power points. We present a simple algorithm for fast extraction of the model parameters of partially shaded photovoltaic panels with bypass diodes. An example of the application of the proposed algorithm is illustrated using the data from a real monocrystalline silicon technology photovoltaic module measured under uniform illumination and partial shading conditions. The possibility of using the algorithm as a practical approximate solution is also discussed. The simulations, using only four parameters, represent reasonably well the measured data. Key words: Maximum Power Point (MPP), Partial shading, Photovoltaic module, Solar cell. 1. INTRODUCTION Photovoltaic (PV) arrays frequently consist of strings of series-connected PV cells that are connected in parallel to achieve given power requirements [1-3]. PV panels made up of such arrays can be seriously affected by partial shading conditions which occur when clouds, buildings, trees, dirt or any other opaque body partially obstruct the radiation falling on the panel. In a series connection configuration, partial shading conditions can cause shaded cells to become reverse biased and start to generate heat instead of producing energy. In order to avoid this situation, bypass diodes are customarily connected in antiparallel to series-connected cell groups. This use of such bypass diodes causes the emergence of multiple local maximum power points that complicate deciding where the global maximum is. Several PV array reconfiguration techniques have been proposed to mitigate partial shading effects. An outstanding review article on the topic was authored by Belhachat and Larbes in 2021 [4]. Received May 21, 2023; revised July 20, 2023; accepted August 26, 2023 Corresponding author: Adelmo Ortiz-Conde Solid State Electronics Laboratory (LEES), Simón Bolívar University (USB), Caracas 1080A, Venezuela E-mail: ortizc@usb.ve 568 A. ORTIZ-CONDE, F J. GARCÍA-SÁNCHEZ Parameter extraction of uniformly illuminated solar cells (SCs) and photovoltaic (PV) cell arrays’ models is an important subject of continuous interest to device designers and PV systems engineers [5-11]. At the same time, partial shading (PS) of solar cell arrays is a frequent occurrence in actual outdoors operation. Because of its grave consequences on overall PV system performance deterioration, this topic has been and continues to be the subject of serious concern and intense study [12-15]. It is important to understand the effect of partial shading in order to improve the performance of the solar panels. In 2020, Kermadi et al. presented a general analytic approach to study partial shading effects based on modeling each component of the panel and then obtaining the total current-voltage characteristic for each region of operation [16]. In the last three years, many articles have been dedicated to partial shading effects [12-19]. For example, in [3,17,18], different array configurations have been evaluated under partial shading conditions. In particular, dynamic array reconfiguration seems to be a good solution to disperse the partial shading over the entire array and hence mitigate the partial shading effects [17,18]. On the other hand, bifacial photovoltaic modules have also been studied under partial shading conditions [19]. In this article we present a simple algorithm to extract the model parameters of partially shaded photovoltaic modules with bypass diodes to reduce the possible occurrence of hot spots in shaded cells. The proposed procedure is based on an effective array model that mimics the well-known five parameter Single Diode (SD) solar cell model. The effective model’s parameters are described through analytical equations expressed in terms of the PV module’s short circuit current Isc, open circuit voltage Voc, and the voltage and current coordinates (Vmppx, Imppx) of whatever multiple Maximum Power Points (MPPs) happen to appear on the measured I-V characteristics. This article is organized as follows. Section 2 presents the model equation for a uniformly illuminated array connected in series-parallel. Section 3 illustrates measurements of a partially shaded module having two maxima power points. Section 4 presents a simple model for partial shading conditions. Finally, section 5 contains the conclusions. 2. MODEL EQUATION FOR A UNIFORMLY ILLUMINATED ARRAY The I-V characteristics of any illuminated solar cell may be most simply represented by the SD model and its corresponding lumped-element equivalent circuit [5-11]. This circuit, shown in Fig. 1, consists of a single exponential-type non-ideal diode defined by its two parameters, the reverse saturation current I0, and the ideality factor n, a photo-generated constant current source of intensity Iph dependent on the illumination level, and possibly including two types of parasitic linear losses, a junction shunting parallel conductance of constant magnitude Gp and a resistance in series with the terminals of constant magnitude Rs. Gp I RS V + - n, Io Iph Fig. 1 Single diode lumped-element equivalent circuit model with parasitic parallel and series resistive losses valid for solar cells and photovoltaic arrays Algorithm to Extract Model Parameters of Partially Shaded Photovoltaic Modules 569 The mathematical description of this lumped element equivalent circuit model is given by the following implicit equation: 𝐼 = −𝐼0 [𝑒𝑥𝑝 ( 𝑉+𝐼𝑅𝑠 𝑛𝑉𝑡ℎ )] − (𝑉 + 𝐼𝑅𝑠)𝐺𝑝 + 𝐼𝑝ℎ (1) where I is the terminal current, V is the terminal voltage, and Vth = kBT/q is the thermal voltage. By using the Lambert W function, the previous implicit equation was first solved, for the particular case of Iph=0, by Banwell and Jayakumar in 2000 [20]; and also, by Ortiz- Conde [21]; and the general case by Jain and Kapoor in 2004 [22]. Approximate solutions of multi-exponential solar cell models, using the Lambert W function, were presented in 2012 [23]. Consider now a PV array composed of a certain number Np of parallel connected solar cell strings, where each of the strings contains a number Ns of series connected solar cells, as illustrated in Fig. 2. Assuming that all the cells are identical and uniformly illuminated, the current at the array’s terminals may be described by a lumped-element equivalent circuit model for a single solar cell, like the one shown in Fig. 1. Likewise, its corresponding equation is similar to (1) for a single solar cell, but instead of the five solar cell parameters (Iph, n, Io, Rs and Gp), it contains five effective parameters (Iphef, nef, Ioef, Rsef and Gpef) that describe this array as an effective solar cell. These effective parameters are defined in terms of the single cell parameters and the total number of cells connected in series Ns and cell strings in parallel Np, as follows: 𝐼𝑝ℎ𝑒𝑓 = 𝑁𝑝𝐼𝑝ℎ , 𝑛𝑒𝑓 = 𝑁𝑠𝑛 , 𝐼𝑜𝑒𝑓 = 𝑁𝑝 𝐼𝑜 𝑅𝑠𝑒𝑓 = 𝑁𝑠 𝑁𝑝 𝑅𝑠 , 𝐺𝑝𝑒𝑓 = 𝑁𝑝 𝑁𝑠 𝐺𝑝 (2) We wish to point out here that the above mentioned effective model and parameter definitions can be readily demonstrated; and that effective SD model equations of this type are routinely used without overtly referring to their parameters as “effective” [24-26]. Here, although we will be talking about array models, for the sake of conciseness, from now onward we will drop the “ef” subscript when writing the effective parameters’ names, keeping in mind that their true meaning is defined in (2). 3. EXAMPLE OF A PARTIALLY SHADED MODULE Experimental data of a mono-crystalline silicon technology PV module with bypass diodes, measured at constant ambient temperature and under various natural terrestrial sunlight illumination conditions, was obtained from the US National Renewable Energy Laboratory (NREL). Figure 3 presents six I-V characteristics of this module measured at six instants during the time span from 7:00 to 10:00am on 24 April 2013, in the city of Golden, Colorado, U.S.A. [27]. More than 180 non-uniformly spaced data points were recorded at the module’s external terminals within the range from (V=0, I=Isc) to (V =Voc, I=0). 570 A. ORTIZ-CONDE, F J. GARCÍA-SÁNCHEZ Fig. 2 A generic PV array with Np parallel connected cell strings, where each string has Ns series connected cells The five power (P=IV) vs voltage curves, presented in the lower pane of Fig. 3, which were calculated from their respective I-V characteristics data measured every half hour from 7:00am to 9:00am, exhibit two distinct maxima, whereas the one P-V curve, corresponding to the later 10:00am measurement, exhibits only one maximum. The presence of two power maxima is a clear indication of partial shading. It seems that during the time of the first five measurements (7:00am to 9:00am) part of the PV module was shaded, receiving much less illumination than the other part, which was fully illuminated. The change from two power maxima to the single maximum observed for the 10:00am measurement indicates that by that time the partial shading had ceased and the whole module had become fully illuminated. The hourly progression of the measured I-V characteristics, presented in the upper pane of Fig. 3, further confirms that this particular PV module contains two distinguishable series connected parts, and that they most likely are equal halves, each one consisting of an array of solar cells shunted by a bypass diode [28-30]. Fig. 3 NREL data of mono-crystalline silicon technology PV module, measured on April 24, 2013 at various times in Golden, Colorado, USA [24] Algorithm to Extract Model Parameters of Partially Shaded Photovoltaic Modules 571 To analyze the I-V and P-V characteristics shown in Fig. 3, we will first examine the curves that correspond to the data measured at 10:00am, which is the time when the whole PV module was uniformly illuminated and both bypass diodes were in the off state. For convenience we will assume that the module has negligible parasitic losses, i.e. Rs =0 and Gp=0. In such case we may use a simplified approximate effective solar cell SD model to describe the whole module using just 3 effective parameters [31-32]: Iph, n, and I0. Thus, (1) reduces to: 𝐼 = 𝐼𝑝ℎ − 𝐼0 [𝑒𝑥𝑝 ( 𝑉 𝑛𝑉𝑡ℎ ) − 1] (3) Next, we will find the values of these three effective model parameters Iph, n, and I0 by extracting them from the known coordinates of the three most prominent points in a PV module’s I-V characteristics: the short circuit current Isc, the open circuit voltage Voc and the MPP (Vmpp, Impp), as measured in this case at 10:00am under uniform illumination. Evaluation of (3) at the short circuit condition (V=0, I=Isc) tells us that in this ideal case with Rs0 and Gp0. the effective photo-generated current is equal to the measured short circuit current: Iph = Isc. Please be reminded that this is the “effective” photo-current generated by the whole module, and that according to (2), it is equal to the photo-current generated by a single cell multiplied by the module’s total number of cell strings connected in parallel Np. The derivative of the current through the PV module’s terminals with respect to the voltage across them gives the slope at any point on the I-V characteristics, and is obtained by differentiating (3): 𝑑𝐼 𝑑𝑉 = −𝐼0 𝑛𝑉𝑡ℎ 𝑒𝑥𝑝 ( 𝑉 𝑛𝑉𝑡ℎ ) (4) To be able to achieve maximum output power any photovoltaic cell, array or module must be presented with an optimal load that allows it to operate at its MPP. In this sense, the MPP may be defined as the point (Vmpp, Impp) on the I-V characteristics where the curve’s slope and the ratio of its two coordinates Impp/Vmpp become equal in magnitude but opposite in sign [33], i.e.: − 𝐼𝑚𝑝𝑝 𝑉𝑚𝑝𝑝 = 𝑑𝐼 𝑑𝑉 | 𝑀𝑃𝑃 = −𝐼0 𝑛𝑉𝑡ℎ 𝑒𝑥𝑝 ( 𝑉𝑚𝑝𝑝 𝑛𝑉𝑡ℎ ) (5) Evaluating (3) at the MPP, ignoring the -1 term, recalling that Iph = Isc, substituting it into (5) and solving for the effective ideality factor n yields: 𝑛 ≈ (𝐼𝑠𝑐−𝐼𝑚𝑝𝑝)𝑉𝑚𝑝𝑝 𝑉𝑡ℎ𝐼𝑚𝑝𝑝 (6) Finally, the effective I0 is calculated using (3) evaluated at the open circuit condition (V=Voc, I=0), ignoring the -1 term and recalling that Iph  Isc: 𝐼0 ≈ 𝐼𝑠𝑐𝑒𝑥𝑝 (− 𝑉𝑜𝑐 𝑛𝑉𝑡ℎ ) (7) Figure 4 presents the module’s data under uniform illumination, measured at 10:00 am, together with the playback of the approximate SD model without series or parallel resistive losses, calculated with the three parameter values: Iph=6.18A, n=81.3 and Io=190 A, as extracted from the coordinate values of Isc=6.18A, Voc=21.90V, Impp=5.50A and Vmpp=16.98V, using equations Iph=Isc, (6) and (7). The thermal voltage was taken to be 0.0259V. This simple approximate SD model with just 3 parameters appears to be good enough to simulate, within a reasonably small error, a close description of the module’s I- V characteristics under uniform illumination, as can be seen in Fig. 4. 572 A. ORTIZ-CONDE, F J. GARCÍA-SÁNCHEZ Fig. 4 Module’s measurements at 10:00 am under uniform illumination and the SD model simulation using only 3 extracted parameters: Iph=6.18A, n=81.3 and Io=190 A Fig. 5 Proposed lumped-element equivalent circuit model of the studied PV module, where its two parts (arrays) are represented by two ideal SD “effective cells,” each one shunted by a bypass diode Algorithm to Extract Model Parameters of Partially Shaded Photovoltaic Modules 573 4. MODELING UNDER PARTIAL SHADING The I-V characteristics with two power maxima (two MPPs), shown in Fig. 3 suggest the use of the model presented in Fig. 5, in which two arrays are represented by two “effective cells,” with 4 effective parameters n, Io, Iph1 and Iph2, and two bypass diodes with parameters nb and Ib. In order to extract the 4 effective parameters, n, Io, Iph1 and Iph2, we will only use Isc, Voc, and the coordinates of the MPPs: Vmpp1, Impp1, Vmpp2, and Impp2. It is assumed in this analysis that Iph1 < Iph2 and that there are two regions of operation: a) I < Iph1 where both bypass diodes are reversed biased (turned off); and b) Iph1< I < Iph2 where bypass diode across array 1 is forward biased (turned on) while the other is reversed biased (turned off). In region of operation b) where Iph1< I < Iph2 , neglecting the forward-biased voltage drop of the bypass diode, yields: 𝐼𝑝ℎ2 = 𝐼𝑠𝑐 (8) Let us now analyze the first MPP at Impp1, which is located in region of operation a) where I < Iph1. Within this region there will be a nearly constant voltage drop across array 2, because Iph2> Impp1. This constant voltage (Vct) may be approximated neglecting the bypass diode and using equation 3 with the values I =Impp1, V=Vct and Iph= Iph2. That is: 𝐼0𝑒𝑥𝑝 ( 𝑉𝑐𝑡1 𝑛𝑉𝑡ℎ ) ≈ 𝐼𝑝ℎ2 − 𝐼𝑚𝑝𝑝1 (9) Considering the open circuit voltage (I =0) and the constant voltage Vct across array 2, we can substitute I =0, V=(Voc-Vct) and Iph= Iph1 into (3) to obtain: 0 ≈ 𝐼𝑝ℎ1 − 𝐼0𝑒𝑥𝑝 ( 𝑉𝑜𝑐−𝑉𝑐𝑡 𝑛𝑉𝑡ℎ ) (10) Now, combining (9) and (10), and solving for I0, yields: 𝐼0 = √𝐼𝑝ℎ1(𝐼𝑝ℎ2 − 𝐼𝑚𝑝𝑝1)𝑒𝑥𝑝 (− 𝑉𝑜𝑐 𝑛𝑉𝑡ℎ ) (11) At I=Impp1 and array 2 with an approximately constant voltage Vct across it, the following relationship is obtained: 𝑑𝐼 𝑑𝑉 | 𝑉=𝑉𝑚𝑝𝑝1 = − 𝐼𝑚𝑝𝑝1 𝑉𝑚𝑝𝑝1 ≈ − 𝐼0 𝑛𝑉𝑡ℎ 𝑒𝑥𝑝 ( 𝑉𝑚𝑝𝑝1−𝑉𝑐𝑡 𝑛𝑉𝑡ℎ ) (12) Using (3) with I =Impp1 , V= Vmpp1-Vct and Iph= Iph1: 𝐼𝑚𝑝𝑝1 = 𝐼𝑝ℎ1 − 𝐼0𝑒𝑥𝑝 ( 𝑉𝑚𝑝𝑝1−𝑉𝑐𝑡 𝑛𝑉𝑡ℎ ) (13) Combining (12) and (13), and solving for Iph1: 𝐼𝑝ℎ1 = 𝐼𝑚𝑝𝑝1 (1 + 𝑛 𝑉𝑡ℎ 𝑉𝑚𝑝𝑝1 ) (14) At I=Impp2, the bypass diode across array 1 is forward biased (turned on) so that array 1 is effectively eliminated. Then, using equation (6) with Impp =Impp2 and Vmpp =Vmpp2: 𝑛 ≈ (𝐼𝑠𝑐−𝐼𝑚𝑝𝑝2)𝑉𝑚𝑝𝑝2 𝑉𝑡ℎ 𝐼𝑚𝑝𝑝2 (15) 574 A. ORTIZ-CONDE, F J. GARCÍA-SÁNCHEZ Figure 6 shows the data measured at 9:00 am (when there was partial shading) and the AIM-SPICE model playback simulations [34], using the lumped-element equivalent circuit model shown in Fig. 5 with the four “effective cell” parameters: Iph2, Io, Iph1 and n, as extracted by equations (8), (11), (14) and (15), respectively. The power and the absolute error in power are also illustrated in the figure. Fig. 6 Measurements at 9:00 am with two maxima power points and AIM-SPICE simulations using the 4 extracted parameters: Iph1=1.22 A, Iph1=4.35 A, n=35.5 and Io=16.8 A 5. CONCLUSION We have presented a simple algorithm to extract the model parameters of uniformly illuminated and partially shaded photovoltaic module. The algorithm has been tested with measured data from where the model parameters where extracted. The data from the photovoltaic module used was obtained from the USA National Renewable Energy Laboratory (NREL). These NREL measurements present two maxima power points, which Algorithm to Extract Model Parameters of Partially Shaded Photovoltaic Modules 575 allows us to assume that the panel is using bypass diodes. Our simple model is based on two effective cells with two bypass diodes. Comparison between the original data and the playback of the model, as simulated using the extracted parameters, indicate that the proposed algorithm, although approximate, provides a fast and fairly accurate procedure. The simulations represent reasonably well the measured data, considering that this model uses only four parameters, which were calculated from the known values of Isc, Voc, and the coordinates of the two MPPs. We consider that the present approach could be extended to cases with 3 or more maxima power points. Acknowledgement: The authors would like to thank Dr. Bill Marion and coworkers and the National Renewable Energy Laboratory (NREL), for the valuable comprehensive data of photovoltaic modules provided in the public domain. REFERENCES [1] N. Agrawal, B. Bora, A. Kapoor, "Experimental investigations of fault tolerance due to shading in photovoltaic modules with different interconnected solar cell networks", Solar Energy, vol. 211, pp. 1239– 1254, Nov. 2020. [2] O. Kunz, R. J. Evans, M. K. Juhl, T. Trupke, "Understanding partial shading effects in shingled PV modules", Solar Energy, vol. 202, pp. 420–428, May 2020. [3] D. Prince Winston, S. Kumaravel, B. Praveen Kumar, S. Devakirubakaran, "Performance improvement of solar PV array topologies during various partial shading conditions", Solar Energy, vol. 196, pp. 228–242, Jan. 2020. [4] F. Belhachat, C. Larbes, "PV array reconfiguration techniques for maximum power optimization under partial shading conditions: A review", Solar Energy, vol. 230, pp. 558–582, Dec. 2021. [5] A. Ortiz-Conde, F. J. García Sánchez, J. Muci, "New method to extract the model parameters of solar cells from the explicit analytic solutions of their illuminated I-V characteristics", Solar Energy Materials and Solar Cells, vol. 90, pp. 352–361, Feb. 2006. [6] C.W. Hansen, A. Luketa-Hanlin, J.S. Stein, "Sensitivity of Single Diode Models for Photovoltaic Modules to Method Used for Parameter Estimation", In Proceedings of the 28th European PV Solar Energy Conf., Paris, France, 2013, pp. 3258–3264. [7] A.M. Humada, M. Hojabri, S. Mekhilef, H.M. Hamada, "Solar cell parameters extraction based on single and double-diode models: A review", Renewable and Sustainable Energy Reviews, vol. 56, pp. 494–509, April 2016. [8] E. Cardelli, A. Laudani, F. R. Fulginei, "Fast and simple numerical computation of Maximum Power Point in PV systems", In Proceedings of the Int. Conf. Electrical, Computer, Communications and Mechatronics Engineering, Maldives, 2022. [9] F. Montalvo-Galicia, M. T. Sanz-Pascual, P. Rosales-Quintero, M. Moreno-Moreno, "Solar Cell Parameter Extraction Method from Illumination and Dark I-V Characteristics", Nanomaterials, vol. 12, Article number 1955, June 2022. [10] V.-T. Rangel-Kuoppa, "Obtention of solar cell parameters, through convergence of iterative cycles. Part 1: Theoretical analysis and cycles proposal", Heliyon, vol. 8, Article number e10551, Sept. 2022. [11] A. Ortiz-Conde, F. J. García Sánchez, J. Muci, A. Sucre-González, "A review of diode and solar cell equivalent circuit model lumped parameter extraction procedures", Facta Universitatis,Series: Electronics and Energetics, vol. 27, pp. 57–102, Jan. 2014. [12] E. D. Chepp, A. Krenzinger, "A methodology for prediction and assessment of shading on PV systems", Solar Energy, vol. 216, pp. 537–550, March 2021. [13] Y. Zhang, et al., "Performance estimation of photovoltaic module under partial shading based on explicit analytical model", Solar Energy, vol. 224, pp. 327–340, Aug. 2021. [14] Z. Zhang et al., "A data-driven photovoltaic string current mismatch fault diagnosis method based on I-V curve", Microelectronics Reliability, vol. 138, Article number 114705, Nov. 2022. [15] O. Ragb, H. Bakr, "A new technique for estimation of photovoltaic system and tracking power peaks of PV array under partial shading", Solar Energy. vol. 268, Article number 126680, April 2023. 576 A. ORTIZ-CONDE, F J. GARCÍA-SÁNCHEZ [16] M. Kermadi, V. J. Chin, S. Mekhilef, Z. Salam, "A fast and accurate generalized analytical approach for PV arrays modeling under partial shading conditions", Solar Energy, vol. 208, pp. 753–765, Sept. 2020. [17] S. Sugumar, D. Prince Winston, M. Pravin, "A novel on-time partial shading detection technique for electrical reconfiguration in solar PV system", Solar Energy, vol. 225, pp. 1009–1025, Sept. 2021. [18] A. Srinivasan et al., "L-Shape Propagated Array Configuration With Dynamic Reconfiguration Algorithm for Enhancing Energy Conversion Rate of Partial Shaded Photovoltaic Systems", IEEE Access, 9, vol. 9, pp. 97661–97674, 9474476, 2021. [19] T. Hariharasudhan, D. Prince Winston, M. Palpandian, M. Pravin, "A comparative analysis of polycrystalline and bifacial photovoltaic module under various partial shading condition", Energy Conversion and Management, vol. 270, Oct. 2022. [20] T.C. Banwell, A. Jayakumar, "Exact analytical solution for current flow through diode with series resistance", Electronics Letters, vol. 36, pp. 291–292, Feb. 2000. [21] A. Ortiz-Conde, F.J. García Sánchez, J. Muci, "Exact analytical solutions of the forward non-ideal diode equation with series and shunt parasitic resistances", Solid-State Electronics, vol. 44, pp. 1861-1864, Oct. 2000. [22] A. Jain, A. Kapoor, "Exact analytical solutions of the parameters of real solar cells using Lambert W- function", Solar Energy Materials and Solar Cells, vol. 81, pp. 269–277, Feb. 2004. [23] A. Ortiz-Conde, D. Lugo-Muñoz, F. J. García Sánchez, "An explicit multi-exponential model as an alternative to traditional solar cell models with series and shunt resistances", IEEE J of Photovoltaics, vol. 2, pp. 261–268, March 2012. [24] F. J. Toledo, J. M. Blanes, V. Galiano, "Two-Step Linear Least-Squares Method for Photovoltaic Single-Diode Model Parameters Extraction", IEEE Trans on Industrial Electronics, vol. 65, pp. 6301–6308, Aug. 2018. [25] A. Mohapatra, B. Nayak, K. B. Mohanty, "Analytical approach to locate multiple power peaks of photovoltaic array under partial shading condition and hybrid array configuration schemes to reduce mismatch losses", Energy Sources, A, Recovery, Utilization, Environ. Effects, Article in Press, pp. 1–22, 2021. [26] A. Ul-Haq, S. Fahad, S. Gul, R. Bo, "Intelligent Control Schemes for Maximum Power Extraction from Photovoltaic Arrays under Faults", Energies, vol. 16, Article number 974, Jan. 2023. [27] B. Marion et al., "New data set for validating PV module performance models", In Proceedings of the IEEE 40th Photovoltaic Specialist Conference, Denver, Colorado, USA, pp. 1362–1366, 2014. [28] R. G. Vieira, F. M. U. de Araújo, M. Dhimish, M. I. S. Guerra, "A comprehensive review on bypass diode application on photovoltaic modules", Energies, vol. 13, Article number 2472, May 2020. [29] H. Ren, P. Han, "Necessity Analysis of Bypass Diode for AC Module under Partial Shading Condition", Energies, vol. 14, Article number 4778, Aug. 2021. [30] M. Etarhouni, B. Chong, L. Zhang, "A novel square algorithm for maximising the output power from a partially shaded photovoltaic array system", Optik - Int J Light and Electron Optics, vol. 257, Article number 168870, May 2022. [31] E. Saloux, A. Teyssedou, M. Sorin, "Explicit model of photovoltaic panels to determine voltages and currents at the maximum power point", Solar Energy, vol. 85, pp. 713–722, May 2011. [32] E. Batzelis, "Non-Iterative Methods for the Extraction of the Single-Diode Model Parameters of Photovoltaic Modules: A Review and Comparative Assessment", Energies, vol. 12, Article number 358, Jan. 2019. [33] O. Trejo, A. Ortiz-Conde, "A Simple Algorithm for High-Accuracy Maximum Power Point Calculation of Photovoltaic Systems", IEEE Journal of Photovoltaics, vol. 10, pp. 1839-1845, Nov. 2020. [34] AIM-SPICE http://www.aimspice.com http://www.aimspice.com/