# Simulation and Comparison of Mathematical Models of PV Cells with Growing Levels of Complexity

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Equivalent Circuit with Five Parameters

#### 2.1. Representative Equations

_{p}. Unlike the series resistance (R

_{s}) it does not interfere directly with the power delivered to the load. However, it penalizes the operation of the cell by providing an alternative path for a portion of the photoelectric current. It is called a leakage current because it reduces the amount of current flowing at the PN junction [33], thereby affecting the voltage to the terminals of the photovoltaic cell. The five-parameter electrical circuit is the most widely used model in the analytical study of the photovoltaic cell. This model offers a good compromise in terms of complexity and performance [34], thus being the choice of several authors in this area of research [25,26,28,35,36,37,38]. The model of the equivalent circuit with five parameters of the photovoltaic (PV) module can be observed in Figure 1.

_{D}with the diode expression and I

_{p}with V

_{d}/R

_{p}the following equation is obtained:

_{s}is the current created by photoelectric effect, I

_{is}is the reverse saturation current, q is the charge of the electron, K is the Boltzmann constant (1.38 × 10

^{−23}J/°K), T is the temperature of the junction, m is the reality parameter, R

_{s}is the parasite resistance in series and R

_{p}is the parallel parasite resistance. The value of R

_{p}is usually quite high in the manufactured photovoltaic cells. However, several authors with regard to this finding, consider useless the inclusion of this resistance [39,40,41,42,43,44]. On the other hand there are authors who consider R

_{s}negligible when the value is very low [45,46,47].

#### 2.2. Analytical Extraction of Parameters

_{is}, R

_{s}, R

_{p}, and m through the analytical solution. For this reason, appropriate numerical methods must be used.

#### 2.3. Simulation

#### 2.3.1. Assessing Equations

_{ca}with Equation (3) set under open circuit conditions:

_{n+}

_{1}the estimated value in the present iteration, x

_{n}the value obtained in the previous iteration, f(x) the function initialized with x

_{n}and the f’(x) the derivative initialized with x

_{n}.

_{ca}can be assessed by:

_{ca}becomes a null value.

_{n}is the temperature, both under Standard Test Conditions (STC) reference conditions. Additionally, in this study, the following simplification was taken into account, where G is the incident radiation in W/m

^{2}:

#### 2.3.2. Comparison between Constant R_{s} and Variable R_{p}

_{p}(R

_{s}= 0). The R

_{p}resistance was adjusted with 10 Ω, 200 Ω, and 1000 Ω, respectively. In the second scenario a fixed value of 10 mΩ was established for the R

_{s}resistor. The simulations can be observed in Figure 2.

_{p}does not interfere in the region of influence of the junction diode. In the region where the influence of the photoelectric current source predominates, the lowest value tested does not show a significant disturbance: the plot is very similar to the set of points estimated with R

_{p}= ∞.

_{p}in this range of values does not compromise the estimated maximum power. In this context of temperature and solar radiation this conclusion becomes valid.

#### 2.3.3. Characteristic Curves in Function of Temperature and Radiation

_{p}resistors, three data set scenarios are established, as can be observed in Figure 5, Figure 6 and Figure 7. Each scenario is simulated with a specific solar power, 100 W/m

^{2}, 500 W/m

^{2}, and 1000 W/m

^{2}, respectively, and having in common the same interval of test temperatures (10 °C, 25 °C, 50 °C, and 75 °C).

^{2}) the I-V curves are literally identical. Consequently, the P-V curves do not experience significant changes between 10 °C and 75 °C. The peak power is then identical, regardless of whether the cell is manufactured with a high parallel resistance (1000 Ω) or with considerably low resistance (10 Ω).

^{2}) the performance is matched to that observed in the power ceiling of 1000 W/m

^{2}. With the solar radiation reduced to a tenth (100 W/m

^{2}) of the highest power range, finally, there is some deviation in the I-V curve, characterized by R

_{p}= 10 Ω. In thermal terms, there is no correlation with R

_{p}: the difference with the versions with higher R

_{p}losses is apparently constant for the analyzed temperature scale.

_{p}and under weak incident solar radiation.

## 3. Equivalent Circuit with Seven Parameters

#### 3.1. Representative Equations

_{p}. The seven-parameter equivalent electrical circuit with two diodes can be observed in Figure 8.

_{s}is the photoelectric equivalent current, I

_{is}

_{1}and I

_{is}

_{2}are the saturation currents of diode 1 and diode 2 respectively, m

_{1}and m

_{2}the ideality parameters of diode 1 and diode 2, respectively. As in the previous model, q represents the charge of the electron, K is the Boltzmann constant (1.38 × 10

^{−23}J/°K), T is the temperature of the junction, m is the reality parameter, R

_{s}is the parasite resistance in series and R

_{p}is the parallel parasite resistance.

_{1}and m

_{2}[55,60,61,62]. Other authors opt for complete identification through elaborated methodologies such as particle examination optimization [63], the estimation based on neural networks [64], on genetic algorithms [65] or through algebraic relations as a function of temperature [62].

#### 3.2. Analytical Extraction of Parameters

_{s}is excluded from the system since the linear dependence with temperature is known) the system is:

_{s}is more pronounced in the vicinity of V

_{ca}, an orderly relation to this variable is determined through the derivative of the characteristic expression of the seven-parameter model:

_{ca}and I with 0, the R

_{s}is as assessed as follows:

_{p}is as follows:

_{p}it becomes as follows:

_{p}in the vicinity of the short-circuit operating point is represented as follows:

#### 3.3. Assessing the Simulation Equations

_{ca}is given by:

_{ca}is the open circuit current. The current I is given by the following equation:

## 4. Comparison between the One-Diode Model and the Two-Diode Model

#### 4.1. Characteristic Curves in Function of the Solar Radiation and the Parallel Resistance R_{p}

_{p}completes the number of variables that characterize the equivalent circuit of two diodes. Similarly to what was done with the equivalent representation of a diode, the importance of this resistive loss in the formation of the typical curves was examined, giving natural attention to the maximum power point. The structure was simulated with five different R

_{p}values, exposed to progressively higher levels of solar radiation, between the 100 W/m

^{2}and 1000 W/m

^{2}. Figure 9 and Figure 10 show the generated curves.

_{p}is very small if the resistance is simulated with 5000 Ω—from this value the parallel branch approaches an infinite resistance. Then, since the resistances of 10 Ω and 200 Ω lead to characteristic curves identical to those found with 5000 Ω, it can be stated that R

_{p}is negligible if its value is equal to or greater than 10 Ω. The same is no longer true with R

_{p}reduced to 1 Ω. The current I starts to decrease to values close to the short-circuit voltage instead of remaining constant until the measurements of the maximum power electrical coordinates.

#### 4.2. Characteristic Curves as a Function of Temperature and Parallel Resistance R_{p}

_{p}(10 Ω and 200 Ω) is analyzed. The series resistance is equal and constant in both models. The parameters m, m

_{1}, and m

_{2}are initialized with 1.5, 1, and 2, respectively. The simulation was carried out with three scenarios of solar radiation with 100 W/m

^{2}, 500 W/m

^{2}and 1000 W/m

^{2}, respectively, and each with four levels of temperature, (10 °C, 25 °C, 50 °C, and 75 °C). The results can be witnessed in Figure 11, Figure 12 and Figure 13.

^{2}the open circuit voltage does not show any deviation between the two models. The same does not happen with 100 W/m

^{2}and 500 W/m

^{2}aggravating with the decrease of incident solar exposure. As for maximum power, the two-diode model is always at an advantage whatever the scenario. The difference is visible between 100 W/m

^{2}and 500 W/m

^{2}, with a tendency to increase in the downward direction of the sun exposure. Over the same range of solar radiation, the temperature tends to maintain the constant difference.

#### 4.3. Comparative Table of Peak Power in the Set of Models

_{p}. This trend remains with the temperature varying between 10 °C and 75 °C. Decreasing the light exposure to one tenth shows a significant deviation in any of the models with the R

_{p}reduced to 10 Ω. The two-diode equivalent circuit is in any case more generous at peak power. The power deficit between the one-diode model versus the two-diode model is constant over the entire temperature range for the same level of radiation. However, it tends to worsen with the weakening of the incidence variable. The two-diode model tends to approximate the ideal one-diode model with the progressive reduction of incident radiation. With the limited incidence at 100 W/m

^{2}the equivalent circuit performance (R

_{s}= 10 mΩ and R

_{p}= 200 Ω) is comparable, as it can be observed in Figure 16.

## 5. Conclusions

^{2}of radiation and 10 °C of temperature, the deviation approaches 6.1%. By reducing the exposure to one-tenth the deviation reaches 12%. At the other end of the temperature range, the deviation reaches 10.4% at full sun exposure, and worsens up to 20.3% with exposure limited to the maximum. The most important contribution deduced form this study is that the two-diode model tends to approximate to the ideal PV cell model (one-diode model) with the progressive reduction of incident radiation. With the incidence limited to 100 W/m

^{2}the equivalent circuit performance (R

_{s}= 10 mΩ and R

_{p}= 200 Ω) is almost identical to the ideal one-diode model. This means that, for regions were the solar incident radiation is lower, the ideal one-diode model behaves similarly to the more complex seven parameter equivalent circuit, thus allowing the user to opt for this circuit in detriment to the other more complex one which allows using a less complex software tool.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Cucchiella, F.; D’Adamo, I.; Gastaldi, M. Economic Analysis of a Photovoltaic System: A Resource for Residential Households. Energies
**2017**, 10, 814. [Google Scholar] [CrossRef] - Sansaniwal, S.K.; Sharma, V.; Mathur, J. Energy and exergy analyses of various typical solar energy applications: A comprehensive review. Renew. Sustain. Energy Rev.
**2018**, 82, 1576–1601. [Google Scholar] [CrossRef] - Li, Q.; Liu, Y.; Guo, S.; Zhou, H. Solar energy storage in the rechargeable batteries. Nano Today
**2017**, 16, 46–60. [Google Scholar] [CrossRef] - Teo, J.C.; Tan, R.H.G.; Mok, V.H.; Ramachandaramurthy, V.K.; Tan, C. Impact of Partial Shading on the P-V Characteristics and the Maximum Power of a Photovoltaic String. Energies
**2018**, 11, 1860. [Google Scholar] [CrossRef] - Jacobi, N.; Haas, W.; Wiedenhofer, D.; Mayer, A. Providing an economy-wide monitoring framework for the circular economy in Austria: Status quo and challenges. Resour. Conserv. Recycl.
**2018**, 137, 156–166. [Google Scholar] [CrossRef] - Yousif, J.; Kazem, H.; Boland, J. Predictive Models for Photovoltaic Electricity Production in Hot Weather Conditions. Energies
**2017**, 10, 971. [Google Scholar] [CrossRef] - Chang, B.; Starcher, K. Evaluation of Wind and Solar Energy Investments in Texas. Renew. Energy
**2018**. [Google Scholar] [CrossRef] - Gimeno, J.Á.; Llera, E.; Scarpellini, S. Investment Determinants in Self-Consumption Facilities: Characterization and Qualitative Analysis in Spain. Energies
**2018**, 11, 2178. [Google Scholar] [CrossRef] - Beránek, V.; Olšan, T.; Libra, M.; Poulek, V.; Sedláček, J.; Dang, M.-Q.; Tyukhov, I. New Monitoring System for Photovoltaic Power Plants’ Management. Energies
**2018**, 11, 2495. [Google Scholar] [CrossRef] - Gelsor, N.; Gelsor, N.; Wangmo, T.; Chen, Y.-C.; Frette, Ø.; Stamnes, J.J.; Hamre, B. Solar energy on the Tibetan Plateau: Atmospheric influences. Sol. Energy
**2018**, 173, 984–992. [Google Scholar] [CrossRef] - Todde, G.; Murgia, L.; Carrelo, I.; Hogan, R.; Pazzona, A.; Ledda, L.; Narvarte, L. Embodied Energy and Environmental Impact of Large-Power Stand-Alone Photovoltaic Irrigation Systems. Energies
**2018**, 11, 2110. [Google Scholar] [CrossRef] - Huang, Y.-P.; Ye, C.-E.; Chen, X. A Modified Firefly Algorithm with Rapid Response Maximum Power Point Tracking for Photovoltaic Systems under Partial Shading Conditions. Energies
**2018**, 11, 2284. [Google Scholar] [CrossRef] - Merzifonluoglu, Y.; Uzgoren, E. Photovoltaic power plant design considering multiple uncertainties and risk. Ann. Oper. Res.
**2018**, 262, 153–184. [Google Scholar] [CrossRef] - Shah, S.W.A.; Mahmood, M.N.; Das, N. Strategic asset management framework for the improvement of large scale PV power plants in Australia. In Proceedings of the 2016 Australasian Universities Power Engineering Conference (AUPEC), Brisbane, QLD, Australia, 25–28 September 2016; pp. 1–5. [Google Scholar]
- Arefifar, S.A.; Paz, F.; Ordonez, M. Improving Solar Power PV Plants Using Multivariate Design Optimization. IEEE J. Emerg. Sel. Top. Power Electron.
**2017**, 5, 638–650. [Google Scholar] [CrossRef] - Peng, Z.; Herfatmanesh, M.R.; Liu, Y. Cooled solar PV panels for output energy efficiency optimisation. Energy Convers. Manag.
**2017**, 150, 949–955. [Google Scholar] [CrossRef] - Elibol, E.; Özmen, Ö.T.; Tutkun, N.; Köysal, O. Outdoor performance analysis of different PV panel types. Renew. Sustain. Energy Rev.
**2017**, 67, 651–661. [Google Scholar] [CrossRef] - Zhou, Z.; Macaulay, J. An Emulated PV Source Based on an Unilluminated Solar Panel and DC Power Supply. Energies
**2017**, 10, 2075. [Google Scholar] [CrossRef] - Seyedmahmoudian, M.; Horan, B.; Rahmani, R.; Maung Than Oo, A.; Stojcevski, A. Efficient Photovoltaic System Maximum Power Point Tracking Using a New Technique. Energies
**2016**, 9, 147. [Google Scholar] [CrossRef] - Wang, F.; Wu, X.; Lee, F.C.; Wang, Z.; Kong, P.; Zhuo, F. Analysis of Unified Output MPPT Control in Subpanel PV Converter System. IEEE Trans. Power Electron.
**2014**, 29, 1275–1284. [Google Scholar] [CrossRef] - Wang, Z.; Das, N.; Helwig, A.; Ahfock, T. Modeling of multi-junction solar cells for maximum power point tracking to improve the conversion efficiency. In Proceedings of the 2017 Australasian Universities Power Engineering Conference (AUPEC), Melbourne, VIC, Australia, 19–22 November 2017; pp. 1–6. [Google Scholar]
- Das, N.; Wongsodihardjo, H.; Islam, S. A Preliminary Study on Conversion Efficiency Improvement of a Multi-junction PV Cell with MPPT. In Smart Power Systems and Renewable Energy System Integration; Jayaweera, D., Ed.; Studies in Systems, Decision and Control; Springer International Publishing: Cham, Switzerland, 2016; pp. 49–73. ISBN 978-3-319-30427-4. [Google Scholar]
- Das, N.; Wongsodihardjo, H.; Islam, S. Modeling of multi-junction photovoltaic cell using MATLAB/Simulink to improve the conversion efficiency. Renew. Energy
**2015**, 74, 917–924. [Google Scholar] [CrossRef] - Das, N.; Ghadeer, A.A.; Islam, S. Modelling and analysis of multi-junction solar cells to improve the conversion efficiency of photovoltaic systems. In Proceedings of the 2014 Australasian Universities Power Engineering Conference (AUPEC), Perth, WA, Australia, 28 September–1 October 2014; pp. 1–5. [Google Scholar]
- Jin, Y.; Hou, W.; Li, G.; Chen, X. A Glowworm Swarm Optimization-Based Maximum Power Point Tracking for Photovoltaic/Thermal Systems under Non-Uniform Solar Irradiation and Temperature Distribution. Energies
**2017**, 10, 541. [Google Scholar] [CrossRef] - Zhao, J.; Zhou, X.; Ma, Y.; Liu, Y. Analysis of Dynamic Characteristic for Solar Arrays in Series and Global Maximum Power Point Tracking Based on Optimal Initial Value Incremental Conductance Strategy under Partially Shaded Conditions. Energies
**2017**, 10, 120. [Google Scholar] [CrossRef] - Tobón, A.; Peláez-Restrepo, J.; Villegas-Ceballos, J.P.; Serna-Garcés, S.I.; Herrera, J.; Ibeas, A. Maximum Power Point Tracking of Photovoltaic Panels by Using Improved Pattern Search Methods. Energies
**2017**, 10, 1316. [Google Scholar] [CrossRef] - Hadji, S.; Gaubert, J.-P.; Krim, F. Real-Time Genetic Algorithms-Based MPPT: Study and Comparison (Theoretical an Experimental) with Conventional Methods. Energies
**2018**, 11, 459. [Google Scholar] [CrossRef] - Cubas, J.; Pindado, S.; de Manuel, C. Explicit Expressions for Solar Panel Equivalent Circuit Parameters Based on Analytical Formulation and the Lambert W-Function. Energies
**2014**, 7, 4098–4115. [Google Scholar] [CrossRef][Green Version] - Bahrami, M.; Gavagsaz-Ghoachani, R.; Zandi, M.; Phattanasak, M.; Maranzanaa, G.; Nahid-Mobarakeh, B.; Pierfederici, S.; Meibody-Tabar, F. Hybrid maximum power point tracking algorithm with improved dynamic performance. Renew. Energy
**2019**, 130, 982–991. [Google Scholar] [CrossRef] - Rodrigues, E.M.G.; Melício, R.; Mendes, V.M.F.; Catalão, J.P.S. Simulation of a Solar Cell Considering Single-diode Equivalent Circuit Model. Renew. Energy Power Qual. J.
**2011**, 1, 369–373. [Google Scholar] [CrossRef] - Rodrigues, E.M.G.; Godina, R.; Pouresmaeil, E.; Catalão, J.P.S. Simulation study of a photovoltaic cell with increasing levels of model complexity. In Proceedings of the 2017 IEEE International Conference on Environment and Electrical Engineering and 2017 IEEE Industrial and Commercial Power Systems Europe (EEEIC/I CPS Europe), Milan, Italy, 6–9 June 2017; pp. 1–5. [Google Scholar]
- Villalva, M.G.; Gazoli, J.R.; Filho, E.R. Comprehensive Approach to Modeling and Simulation of Photovoltaic Arrays. IEEE Trans. Power Electron.
**2009**, 24, 1198–1208. [Google Scholar] [CrossRef] - Carrero, C.; Amador, J.; Arnaltes, S. A single procedure for helping PV designers to select silicon PV modules and evaluate the loss resistances. Renew. Energy
**2007**, 32, 2579–2589. [Google Scholar] [CrossRef] - Rodriguez, C.; Amaratunga, G.A.J. Analytic Solution to the Photovoltaic Maximum Power Point Problem. IEEE Trans. Circuits Syst. Regul. Pap.
**2007**, 54, 2054–2060. [Google Scholar] [CrossRef] - Patel, H.; Agarwal, V. MATLAB-Based Modeling to Study the Effects of Partial Shading on PV Array Characteristics. IEEE Trans. Energy Convers.
**2008**, 23, 302–310. [Google Scholar] [CrossRef] - Ahmed, S.S.; Mohsin, M. Analytical Determination of the Control Parameters for a Large Photovoltaic Generator Embedded in a Grid System. IEEE Trans. Sustain. Energy
**2011**, 2, 122–130. [Google Scholar] [CrossRef] - Kim, I.; Kim, M.; Youn, M. New Maximum Power Point Tracker Using Sliding-Mode Observer for Estimation of Solar Array Current in the Grid-Connected Photovoltaic System. IEEE Trans. Ind. Electron.
**2006**, 53, 1027–1035. [Google Scholar] [CrossRef] - Xiao, W.; Dunford, W.G.; Capel, A. A novel modeling method for photovoltaic cells. In Proceedings of the 2004 IEEE 35th Annual Power Electronics Specialists Conference (IEEE Cat. No. 04CH37551), Aachen, Germany, 20–25 June 2004; Volume 3, pp. 1950–1956. [Google Scholar]
- Yusof, Y.; Sayuti, S.H.; Latif, M.A.; Wanik, M.Z.C. Modeling and simulation of maximum power point tracker for photovoltaic system. In Proceedings of the PECon 2004 National Power and Energy Conference, Kuala Lumpur, Malaysia, 29–30 November 2004; pp. 88–93. [Google Scholar]
- Khouzam, K.; Ly, C.; Koh, C.K.; Ng, P.Y. Simulation and real-time modelling of space photovoltaic systems. In Proceedings of the 1994 IEEE 1st World Conference on Photovoltaic Energy Conversion—WCPEC (A Joint Conference of PVSC, PVSEC and PSEC), Waikoloa, HI, USA, 5–9 December 1994; Volume 2, pp. 2038–2041. [Google Scholar]
- Glass, M.C. Improved solar array power point model with SPICE realization. In Proceedings of the 31st Intersociety Energy Conversion Engineering Conference, Washington, DC, USA, 11–16 August 1996; Volume 1, pp. 286–291. [Google Scholar]
- Altas, I.H.; Sharaf, A.M. A Photovoltaic Array Simulation Model for Matlab-Simulink GUI Environment. In Proceedings of the 2007 International Conference on Clean Electrical Power, Capri, Itlay, 21–23 May 2007; pp. 341–345. [Google Scholar]
- Matagne, E.; Chenni, R.; Bachtiri, R.E. A photovoltaic cell model based on nominal data only. In Proceedings of the 2007 International Conference on Power Engineering, Energy and Electrical Drives, Setubal, Portugal, 12–14 April 2007; pp. 562–565. [Google Scholar]
- Tan, Y.T.; Kirschen, D.S.; Jenkins, N. A model of PV generation suitable for stability analysis. IEEE Trans. Energy Convers.
**2004**, 19, 748–755. [Google Scholar] [CrossRef] - Kajihara, A.; Harakawa, A.T. Model of photovoltaic cell circuits under partial shading. In Proceedings of the 2005 IEEE International Conference on Industrial Technology, Hong Kong, China, 14–17 December 2005; pp. 866–870. [Google Scholar]
- Benavides, N.D.; Chapman, P.L. Modeling the effect of voltage ripple on the power output of photovoltaic modules. IEEE Trans. Ind. Electron.
**2008**, 55, 2638–2643. [Google Scholar] [CrossRef] - Sera, D. Real-Time Modelling, Diagnostics and Optimised MPPT for Residential PV Systems; Institut for Energiteknik, Aalborg Universitet: Aalborg, Denmark, 2009; ISBN 978-87-89179-76-6. [Google Scholar]
- Bashirov, A. Transcendental Functions. In Mathematical Analysis Fundamentals; Bashirov, A., Ed.; Elsevier: Boston, MA, USA, 2014; Chapter 11; pp. 253–305. ISBN 978-0-12-801001-3. [Google Scholar]
- Danandeh, M.A.; Mousavi, G.S.M. Comparative and comprehensive review of maximum power point tracking methods for PV cells. Renew. Sustain. Energy Rev.
**2018**, 82, 2743–2767. [Google Scholar] [CrossRef] - Batarseh, M.G.; Za’ter, M.E. Hybrid maximum power point tracking techniques: A comparative survey, suggested classification and uninvestigated combinations. Sol. Energy
**2018**, 169, 535–555. [Google Scholar] [CrossRef] - Uoya, M.; Koizumi, H. A Calculation Method of Photovoltaic Array’s Operating Point for MPPT Evaluation Based on One-Dimensional Newton–Raphson Method. IEEE Trans. Ind. Appl.
**2015**, 51, 567–575. [Google Scholar] [CrossRef] - Gow, J.A.; Manning, C.D. Development of a photovoltaic array model for use in power-electronics simulation studies. IEE Proc.—Electr. Power Appl.
**1999**, 146, 193–200. [Google Scholar] [CrossRef] - Walker, G.R. Evaluating MPPT converter topologies using a Matlab PV model. Aust. J. Electr. Electron. Eng.
**2001**, 21, 49–55. [Google Scholar] - Ishaque, K.; Salam, Z.; Taheri, H. Simple, fast and accurate two-diode model for photovoltaic modules. Sol. Energy Mater. Sol. Cells
**2011**, 95, 586–594. [Google Scholar] [CrossRef] - Nishioka, K.; Sakitani, N.; Uraoka, Y.; Fuyuki, T. Analysis of multicrystalline silicon solar cells by modified 3-diode equivalent circuit model taking leakage current through periphery into consideration. Sol. Energy Mater. Sol. Cells
**2007**, 91, 1222–1227. [Google Scholar] [CrossRef] - Enrique, J.M.; Durán, E.; Sidrach-de-Cardona, M.; Andújar, J.M. Theoretical assessment of the maximum power point tracking efficiency of photovoltaic facilities with different converter topologies. Sol. Energy
**2007**, 81, 31–38. [Google Scholar] [CrossRef] - Chowdhury, S.; Chowdhury, S.P.; Taylor, G.A.; Song, Y.H. Mathematical modelling and performance evaluation of a stand-alone polycrystalline PV plant with MPPT facility. In Proceedings of the 2008 IEEE Power and Energy Society General Meeting—Conversion and Delivery of Electrical Energy in the 21st Century, Pittsburgh, PA, USA, 20–24 July 2008; pp. 1–7. [Google Scholar]
- Salam, Z.; Ishaque, K.; Taheri, H. An improved two-diode photovoltaic (PV) model for PV system. In Proceedings of the 2010 Joint International Conference on Power Electronics, Drives and Energy Systems 2010 Power, New Delhi, India, 20–23 December 2010; pp. 1–5. [Google Scholar]
- Gow, J.A.; Manning, C.D. Development of a model for photovoltaic arrays suitable for use in simulation studies of solar energy conversion systems. In Proceedings of the 1996 Sixth International Conference on Power Electronics and Variable Speed Drives, Nottingham, UK, 23–25 September 1996; pp. 69–74. [Google Scholar]
- Hyvarinen, J.; Karila, J. New analysis method for crystalline silicon cells. In Proceedings of the 3rd World Conference on Photovoltaic Energy Conversion, Osaka, Japan, 1–18 May 2003; Volume 2, pp. 1521–1524. [Google Scholar]
- Bowden, S.; Rohatgi, A. Rapid and Accurate Determination of Series Resistance and Fill Factor Losses in Industrial Silicon Solar Cells. In Proceedings of the 17th European Photovoltaic Solar Energy Conference, Munich, Germany, 22–26 October 2001. [Google Scholar]
- Sandrolini, L.; Artioli, M.; Reggiani, U. Numerical method for the extraction of photovoltaic module double-diode model parameters through cluster analysis. Appl. Energy
**2010**, 87, 442–451. [Google Scholar] [CrossRef] - Dolan, J.A.; Lee, R.; Yeh, Y.; Yeh, C.; Nguyen, D.Y.; Ben-Menahem, S.; Ishihara, A.K. Neural network estimation of photovoltaic I–V curves under partially shaded conditions. In Proceedings of the 2011 International Joint Conference on Neural Networks, San Jose, CA, USA, 31 July–5 August 2011; pp. 1358–1365. [Google Scholar]
- Ishaque, K.; Salam, Z.; Taheri, H.; Shamsudin, A. A critical evaluation of EA computational methods for Photovoltaic cell parameter extraction based on two diode model. Sol. Energy
**2011**, 85, 1768–1779. [Google Scholar] [CrossRef]

**Figure 2.**I-V curves as a function of the parallel resistance R

_{p}and with R

_{s}: (

**a**) 0 Ω and (

**b**) 10 mΩ; (STC).

**Figure 4.**P-V curves as a function of the parallel resistance R

_{p}and with R

_{s}: (

**a**) 0 Ω and (

**b**) 10 mΩ; (STC).

**Figure 5.**Characteristic curves in function of temperature and parallel resistance R

_{p}, with a constant resistance R

_{s}(G = 1000 W/m

^{2}).

**Figure 6.**Characteristic curves in function of temperature and parallel resistance R

_{p}, with a constant resistance R

_{s}(G = 500 W/m

^{2}).

**Figure 7.**Characteristic curves in function of temperature and parallel resistance R

_{p}, with a constant resistance R

_{s}(G = 100 W/m

^{2}).

**Figure 9.**I-V curves of equivalent 1 and 2 diode circuits in function of the solar radiation and of the parallel resistance R

_{p}, with series resistance R

_{s}constant (T = 25 °C).

**Figure 10.**P-V curves of equivalent 1 and 2 diode circuits in function of the solar radiation and of the parallel resistance R

_{p}, with series resistance R

_{s}constant (T = 25 °C).

**Figure 11.**Characteristic curves of equivalent circuits of one and two diodes in function of temperature and parallel resistance R

_{p}, with a constant resistance R

_{s}(G = 1000 W/m

^{2}).

**Figure 12.**Characteristic curves of equivalent circuits of one and two diodes in function of temperature and parallel resistance R

_{p}, with a constant resistance R

_{s}(G = 500 W/m

^{2}).

**Figure 13.**Characteristic curves of equivalent circuits of one and two diodes in function of temperature and parallel resistance R

_{p}, with a constant resistance R

_{s}(G = 100 W/m

^{2}).

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**MDPI and ACS Style**

Manuel Godinho Rodrigues, E.; Godina, R.; Marzband, M.; Pouresmaeil, E. Simulation and Comparison of Mathematical Models of PV Cells with Growing Levels of Complexity. *Energies* **2018**, *11*, 2902.
https://doi.org/10.3390/en11112902

**AMA Style**

Manuel Godinho Rodrigues E, Godina R, Marzband M, Pouresmaeil E. Simulation and Comparison of Mathematical Models of PV Cells with Growing Levels of Complexity. *Energies*. 2018; 11(11):2902.
https://doi.org/10.3390/en11112902

**Chicago/Turabian Style**

Manuel Godinho Rodrigues, Eduardo, Radu Godina, Mousa Marzband, and Edris Pouresmaeil. 2018. "Simulation and Comparison of Mathematical Models of PV Cells with Growing Levels of Complexity" *Energies* 11, no. 11: 2902.
https://doi.org/10.3390/en11112902