#### Abstract

This article presents a real photovoltaic module with modeling and simulations starting from the model of a photovoltaic (PV) cell. , , and characteristics are simulated for different solar irradiation, temperatures, series resistances, and parallel resistances. For a real photovoltaic module (ALTIUS Module AFP-235W) there are estimated series and parallel resistances for which the energetical performances of the module have optimal values for a solar radiation of 1000 W/m^{2} and a temperature of the environment of 25°C. Temperature influence over the PV module performances is analyzed by using a thermal model of the ALTIUS Module AFP-235W using the finite element method. A temperature variation on the surface of the PV module is starting from a low value 40.15°C to a high value of 52.07°C. Current and power estimation are within the errors from 1.55% to about 4.3%. Experimental data are measured for the photovoltaic ALTIUS Module AFP-235W for an entire daylight.

#### 1. Introduction

Solar energy is a renewable and no pollution energy with a significant annual riseup and this trend follows an ascending path. Locally produced renewable energy is more and more promoted as a key to the affordable and sustainable energy in remote area, in small communities, or for small consumers even into the large cities. Photovoltaic systems are realized form photovoltaic (PV) cells; one of the major advantages of photovoltaic cells is that they are highly modular and by proper scaling, they can be expected to provide adequate power for various loads [1]. These cells depend on photovoltaic effect for converting solar radiation into electricity and the generated photocurrent is proportional to the solar irradiation [2]. The output characteristics of a PV cell are nonlinear and fluctuate with solar radiation, cell temperature, series and parallel resistance, and other parameters of the mathematical model [3].

The modeling and simulation of the photovoltaic modules began long ago, but improvements of these models are analyzed and presented continually [2, 4–6]. The implementation of mathematical model of photovoltaic cell into specialized software MATLAB-Simulink is widely used. The computing models are realized by using the equations for the parameters as thermal voltage, photovoltaic current, diode saturation current, and ideality factor. The main characteristics can be obtained, such as the current-voltage (-) and power-voltage (-) characteristics [1, 4, 7–10]. The series and parallel resistances have an important influence over the PV cell parameters and these resistances are considered into the mathematical models, but in some cases [11–13] the series resistance is neglected into the model and in some cases [13–17] the parallel resistance is neglected. Thermal analysis of the photovoltaic panels is an important part of the studies in order to estimate the temperature distribution in a PV module and to determine the operating temperature of solar system accurately. Optical parameters for the reflectivity, transmissivity, and absorptivity for the relevant layers of the module are taken into account to determine the heat dissipation in the areas exposed directly to sunlight. Heat losses by convection and radiation are also included in simulations [18]. Finite element analysis is considered by many researchers [18–21]. Also empirical models are used for the photovoltaic devices [22].

Different studies consider cooling system in order to reduce the operating temperature of the PV panel and to increase the performance of it [19, 23, 24].

In this article a model of the photovoltaic module is developed and the influence of some parameters are analyzed as series and parallel resistances, temperature, and solar irradiation over the PV module output characteristics. Simulations of module characteristics are realized considering the real data of a PV module ALTIUS AFP 235 and a thermal analysis and thermal simulation is also realized. There are also some experimental data analyzed and registered for an ALTIUS 235 module which is in regular use. The authors outline the variation of main electrical characteristics of a real PV module depending on solar radiation and also the correlation of the suitable functionality of the PV module with the temperature.

#### 2. Modeling the Photovoltaic Module

##### 2.1. Ideal PV Cell

The photovoltaic (PV) cell is considered as a semiconductor diode which is exposed to the solar radiation (Figure 1). A part of this radiation is absorbed by the junction creating pairs of electron hole. These electrical charges are separated by the electric field E: the electrons migrate in the “” area and the holes migrate in the “” area. This separation is the photovoltaic effect and it generates an electric current. This situation can be modeled into a simplified model having a current source in parallel with an ideal diode, with a photovoltaic current and a diode current . Practical aspects of the PV cell can be studied considering a single-diode PV model with a series resistance and a parallel resistance (Figure 2).

The main equations to describe the - characteristics of an ideal PV cell are [1, 4, 7]

##### 2.2. Modeling the PV Module

Many researches [1, 5, 9, 10] show that, for the analysis of the PV cells, there are some necessary supplementary parameters to be used into (3):whereIn order to assure the parameters required on the consumers, a photovoltaic system has to be made by a sufficient number of PV cells, connected in series or in parallel, usually called modules or arrays. Figure 3 presents the equivalent circuit model of a photovoltaic system.

For the structure module in Figure 3, (4) will be [5, 8]Based on the same characteristic, the following items are to be considered as well: the open circuit voltage/temperature coefficient (), the short circuit current/temperature coefficient (), and the maximum experimental peak output power () [5]. These aspects are related to the nominal condition or to the Standard Test Conditions (STC). In these conditions the photovoltaic device has series resistances with a strong impact when the system operates in the voltage source area and a parallel resistance with a strong influence when the system operates in the current source area. In some cases [11–13] the resistance is neglected into the model and in some cases [13–17] the resistance is neglected.

For some points on the characteristic in Figure 4 (the short circuit (), open circuit (), and Maximum Power Point ()), the relations it can be written as follows[9]:The current generated by the photovoltaic panel depends directly on the solar irradiation and is influenced by the temperature according to the next relation [26, 27]:whereThe nominal open voltage can be influenced by the temperature according to the relation [14]:The saturation current of the diode depends also on the temperature [5, 28, 29]:where is the band gap energy of the semiconductor ( = 1.12 eV for the polycrystalline Si at 25°C). The value of the saturated nominal current is given by [5]Some researches [5, 30, 31] suggest an improving of the saturation current on the diode established by (10). The current can be assumed to be approximately equal to , which is a very common assumption in PV modeling [5]. The assumption gives a good approximation because the series resistance is usually very small and the parallel resistance is large [5, 10]. In these conditions the saturation current on diode becomesThe performances of a PV cell depend also on the fill factor (FF), which is the ratio between the Maximum Power of a cell and the power of an ideal PV cell in the same operating conditions.

For a variable resistive load connected at the cell output the power will be at maximum when the resistance will have an optimal value equal to the ratio between the voltage and the current , Figure 4. Thus, the theoretical Maximum Power generated by the PV cell will be equal to the product between the and .

In these conditions, the fill factor (FF) is defined byThe efficiency is calculated as the ratio between the power in MPP (for a specific temperature) and the power of the solar irradiation:

#### 3. The Effect of the Series and Parallel Resistances on the PV Cell Performances

The series and parallel resistances ( and ) have an important influence on the fill factor FF decreasing. Its value will decrease with increasing and decreasing. It is noteworthy that the series resistance depends mainly on the internal resistances of the semiconductor devices, on the contact resistances, and on the electrical wires resistances and it has to be as low as possible. In the same time, the parallel resistance depends on the metallic bridges between the edges of the junction, on the material defects (where losses’ currents can appear, which can short-circuit the junction). Usually, its value is high, from thousands to tenth of thousands of ohms.

The relation between and can be estimated starting from the equality , resulting in the relation for the resistance [5, 10]:This equation is solved by successive iterations until it results in the best solution for the PV device. The solution has to concede with the maximum of the points and , on the - characteristic (Figure 4). In these conditions, the following with be the results [10]:The initial value for is considered as zero, while the value for is given by [7]

#### 4. Numerical Modeling and Simulations of a Real Photovoltaic Module

Using the parameters of the ALTIUS AFP 235 Module (Table 1) and the adjusted parameters at nominal operating conditions (Table 2) simulations were realized in MATLAB-Simulink.

The results of the simulations are presented as -, -, and - characteristics in Figures 5–16 for different solar irradiation, temperatures, series resistances, and parallel resistances.

Figures 5, 6, and 7 present the -, -, and - characteristics for a constant temperature (25°C) and for different solar radiations (from 200 W/m^{2} to 1000 W/m^{2}). From Figure 5, with the increase of the solar radiation the current will increase as a result and from Figure 6 the voltage will increase, which will increase the generated power. Figures 8, 9, and 10 present the -, -, and - characteristics for a constant radiation (1000 W/m^{2}) and for different temperatures of the environment (from −20°C to 60°C). It is to observed that, with the increasing of the temperature, the current will have a low increase while the voltage will decrease significantly. In this case the power delivered by the PV module will decrease significantly.

Figures 11, 12, and 13 present the -, -, and - characteristics for a constant value of the parallel resistance Ω, for different values for the series resistance ( Ω, Ω, and Ω), and for standard testing conditions STC (1000 W/m^{2} and 25°C). It is to observed that, for a high value of the resistances , the results will be a decrease in the current and the voltage and, accordingly, a decrease in the generated power. This influence is better to observe near the Maximum Power Point, where both the Maximum Power and the fill factor FF have low values. In this situation it is necessary to have a series resistance as low as possible, very close to zero value.

Figures 14, 15, and 16 present the -, -, and - characteristics for a constant series resistance Ω, for different values for the parallel resistance ( Ω, Ω, and Ω), and for standard testing conditions STC (1000 W/m^{2} and 25°C). It is to observed that, for a low value of the parallel resistance, the results will be decreasing the value for the fill factor and power. For a better operation of the PV cell, it is necessary to have a parallel resistance as high as possible.

From all the above characteristics we can say that the energetical performances of the ALTIUS Module AFP-235W have optimal values when the series resistance is and the parallel resistance is Ω, for a solar radiation of 1000 W/m^{2} and a temperature of the environment of 25°C.

#### 5. Temperature Influence over the PV Module Performances

The increase in the environment temperature and in the thermal radiation has a negative influence over the energetical performances of the PV system. To study these aspects, a thermal model of the ALTIUS Module AFP-235W using the finite element method was realized. The properties of the materials used on the module are presented in Table 3.

The photovoltaic panel is a capsuled system realized from many successive layers. Thus, it is necessary to have a high thermal conductivity in order to assure good cooling of the PV cells. Considering this construction in layers, the thermal transfer is realized between the layers by conduction, convection, and radiation. Thermal analysis by the finite element method supposes an establishment of the thermal equilibrium for every volume zone .The left term of the equation is the heating power from the current flow, . It is in balance with the heat stored by temporal change of temperature , the power removed from the element by thermal conduction , and the thermal power dissipated to the surrounding area by the surface convection, . For , , , and , the following equations can be written:

Thus,The material density, specific heat, and thermal conductivity do not have an important temperature variation; thus they can be regarded as constants. On the other hand, the electrical resistivity has a significant temperature variation and can be estimated through a parabolic variation or a linear one. The experimental tests concluded that the difference between these two types of variation is not so important. For the electrical resistivity a linear variation with the temperature has been considered [32]:with the notation:Figure 17 shows the result of the thermal simulation for the ALTIUS Module AFP-235W for a nominal load of 235 W considered with a uniform distribution over the surface of the photovoltaic panel.

A temperature variation on the surface of the PV module is observed starting from a low temperature of 40.15°C to a high value of 52.07°C on the top of the surface of the PV module. This is explained because of variable thermal convection coefficient. This coefficient includes the circulation of the air flow from the lower side of the panel to the top. It was considered only natural to cool of the PV module and for the temperature of the environment to be 25°C.

#### 6. Experimental Data for the PV Module

For the experimental analysis the photovoltaic panel ALTIUS Module AFP-235W was considered, oriented to the South and with an inclination of 45°. The experimental tests were realized on May 26, 2016 between 7.00 a.m. and 8 p.m. The solar irradiation was measured using a Solar Survey100/200R instrument, which is in compliance with the IEC-62446 international standard on PV systems. It measures the solar radiation to a maximum of 1500 W/m^{2}, with a resolution of 1 W/m^{2}. The temperature was measured with Extech 42545 IR thermometer (range: 50⋯1000°C, resolution: 0.1°C). A digital multimeter Fluke 115 was used to measure the voltage and current (resolution of 1 mV and 1 mA). Figure 18 shows the evolution of the temperature of the environment, the temperature on the surface of the photovoltaic module, and the solar irradiation variation along the day. As the solar irradiation increases and is absorbed at the PV module surface, its temperature is increasing due to the photons absorption. At the maximum temperature of the environment of 28.3°C and a solar irradiation of 672 W/m^{2}, the temperature of the module surface reaches the value of 41.1°C.

In Figures 19, 20, and 21 the simulated and experimental , , and, respectively, characteristics for the module PFV ALTIUS 235W for a medium temperature of 25°C and a solar irradiation of 800 W/m^{2} are plotted. The value of the nominal short circuit current is of A for the experimental data and of A for the simulation, with an error of 1.55%. The nominal open circuit voltage is of V for the experimental characteristic and V for the simulated one, with an error of −2,6%. In the Maximum Power Point the experimental values for the voltage and current are V, A, and a power of W, while the simulated resulted values are V, A, and a power of W. The error for the power estimation is about 4.3%. In these characteristics there are some differences between the simulated and the experimental data. For example on the point (), there is a difference of 0.13 A, on the point (), there is a difference of 2.42 V, and on the MPP point, there is a difference of 2.24 V and 0.982 A (7.98 W). These differences are due to the variation of the real temperature of the environment, of the solar irradiation, and consequently to the surface panel heating. Also the dust deposited on the panel surface and the wind has an important influence over the experimental data and thus on the difference between the simulated data and the experimental ones.

#### 7. Conclusions

In this article a model of the photovoltaic module is developed and the influences of some parameters are analyzed as series and parallel resistances, temperature, and solar irradiation over the PV module output characteristics (, and characteristics). For a real photovoltaic module (ALTIUS Module AFP-235W) there are estimated series and parallel resistances for which the energetical performances of the module have optimal values for a solar radiation of 1000 W/m^{2} and a temperature of the environment of 25°C. Temperature influence over the PV module performances is analyzed by using a thermal model of the ALTIUS Module AFP-235W using the finite element method. A temperature variation on the surface of the PV module is estimated as a difference of about 12°C between the bottom and the top surface of the PV module. Experimental data are measured for the photovoltaic ALTIUS Module AFP-235W for an entire daylight. Differences between the simulated and recorded data are due to the variation of the real temperature of the environment, of the solar irradiation, and consequently to the surface panel heating. The dust deposited on the panel surface and the wind has an important influence over the experimental data and thus on the difference between the simulated data and the experimental ones.

#### Nomenclature

: | The surface of the cell |

: | The ideality factor of the diode, |

: | Specific heat |

: | The incident irradiation |

: | The nominal irradiation (usually 1000 W/m^{2}) |

: | The reverse saturation current of the diode |

: | The nominal saturation current |

: | The diode current |

: | The current at the MPP point |

: | The photovoltaic current of the PV cell |

: | The photovoltaic current at nominal irradiation and temperature |

: | The nominal short circuit current |

: | Current density |

: | The open circuit voltage temperature coefficient of the module |

: | The short circuit current temperature coefficient of the module |

: | The Boltzmann constant |

: | Total transfer coefficient |

: | Perimeter length of the external surface |

MPP: | Maximum Power Point |

: | The number of the cells connected in parallel |

: | The number of the cells connected in series |

: | The thermal power dissipated to the environment area by the surface convection |

: | The heating power from the current flow |

: | The maximum experimental power |

: | The maximum calculated peak output power |

: | The power removed from the element by thermal conduction |

: | The heat stored by temporal change of temperature |

: | The electron charge |

: | The equivalent parallel resistance of the module |

: | The equivalent series resistance of the module |

: | Surface convection |

: | The ambient temperature |

: | The nominal temperature (usually 25°C) |

: | The open circuit voltage |

: | The nominal open circuit voltage of the module |

: | The thermal voltage of the module |

: | The voltage at the MPP |

: | Coefficient of electrical resistivity variation with temperature |

: | Material density |

: | Thermal conductivity |

: | Electrical resistivity |

: | Electrical resistivity at the temperature |

: | Temperature |

: | The initial temperature of the photovoltaic module |

: | The temperature of the environment |

: | As notation. |

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.