Effect of a Thermal Nonlinear Absorption Coefficient on the Dynamics of a Photovoltaic Panel

In this article, we propose a theoretical study of the effect of a thermal nonlinear absorption coefficient on the dynamics of a photovoltaic (PV) panel. This coefficient is expressed as a quadratic function of temperature of a PV module known as active zone of a PV panel. Based on the energy balance at each layer of a PV panel, the mathematical expressions of the temperature on each layer are derived. A comparative study of the thermal behavior of a PV panel for a constant and nonlinear coefficient is made, thanks to the numerical investigations carried out in the MATLAB environment. We reveal that, this nonlinear absorption coefficient induces the cooling of a PV panel as a whole, and optimize the electrical efficiency of a PV panel around 2.5%.


Introduction
Energy production today, is a huge challenge. Indeed, as the energy needs of industrialized countries increase exponentially over time, they need more energy to develop their countries. Today, most of the world's energy production is provided by fossil fuels and nuclear known as non renewable source of energy. The consumption of these non renewable sources of energy, impacts on the environment by spoiling the fauna, flora and the climatic changes. To face this issue of pollution, the development of new cleaning non-polluting sources of energy must be explored. Fortunately, thanks to the renewable energies, our expectations can be met (Rahul et al., 2014). This type of energy has revolutionized the energy sector worldwide. It can be subdivided into several branches such as solar energy, wind, geothermal energy just to name a few (Nottom et al., 1996). The above-mentioned energies come mainly from solar energy (Sanjeev et al., 2016) which is a second form of energy used by human being after the muscular energy. In addition, the energy coming from the sun is very important and presents a solar potential of about 1340W/m 2 . This potential can be directly transformed into electricity thanks to the PV cells and into thermal energy through the solar collector depending of needing. However, in the view of solving the need of electrical energy in isolated area, people used more than more PV panels based on photovoltaic conversion. Since the pioneering work of Becquerel (1939) showing that photovoltaic effect results from the interaction between a photon (quantum of light) and the matter (electron) which produces current and electric voltage, this concept has been intensively studied. Latter Einstein (1917) explained the photovoltaic effect using the basic principle of quantum mechanics. Depending on the current or voltage sought, solar cells can be grouped in series, parallel, and compound to form a PV module (Emmanuel, 2006). Despite the conversion of the solar radiation into electricity thank to a PV cell, its electrical efficiency is still very low due to the factors that affect negatively its dynamics. It is the reason why since the past decades many researches are continued to be done aiming to optimize its electrical performances and maintain it in operating temperature. This can be explained by many research works. Indeed, Siyu et al. (2016) studied the local effects of inhomogeneity of a silicon solar cell. They showed that, the inhomogeneity causes the decrement of the open circuit voltage of a PV cell while the short circuit current increases. Osueke (2015) optimized the parameters of the solar cells through an experiment based on the angle of tilt, the angle of azimut. He showed that the angle of tilt increases with the electrical efficiency. Following the same goal, Mohamed et al. (2008) conducted studies on an exponential model of a series resistance. Their work revealed that this profile of series resistance optimizes the electrical efficiency of a solar cell. Still in the line to improve the performances of a solar cell, Shaharim et al. (2011) established that the accumulation of dust on the surface of a solar cell causes the drop of its electrical parameters of about 50%. On the order hand, Garg et al. (1973) demonstrated that when a PV module is inclined under an angle of 45 degrees, its transmission coefficient and its performance both decrease. In 2010, Nottom et al. showed that when a PV module is shaded, its electrical performances decrease. Moreover, Mohammad et al. (2016) and Xu et al. (2014) established that, when a solar cell is exposed to the sun, about 9-12% of the solar radiation is converted into electricity. The rest is absorbed by the solar cell, leading to the growth of its operating temperature and therefore to the reduction of its electrical efficiency about 0.45% per degree Celsius (Davud et al., 2013;Gao et al., 2010;Kalogirou et al., 2006;Latifa et al., 2014;Skoplaki et al., 2009). Furthermore, as the nature is essentially nonlinear, Dongo et al. (2014) worked on the effects of a thermal nonlinear series resistance, based on an electrical model of a solar cell. They showed that the nonlinear character of the series resistance optimizes of about 3.6% the power of a PV cell. These previous mentioned works traduce the great desire to deeply master the dynamics of the PV cell in order to propose new approaches that assure their stabilities and improve their yield.
Highly motivated by the nonlinearity like Dongo et al. (2014) and the amount of energy dissipated in the PV module, we plan to study theoretically the effect of a nonlinear absorption coefficient of a PV module. This coefficient is expressed as a quadratic function of the temperature across it, in replacement of the constant absorption coefficient commonly used in the scientific literature. To achieve our goal, the structure of this paper is organized as follows. In section 2, we present the different parts of a PV panel which lead to the mathematical formulations thanks to the energy balance. In section 3, simulations of the mathematical expressions are made exploiting the MATLAB environment. Comparison and discussion of the results obtained are carried out in the cases of constant and nonlinear absorption coefficients of a solar cell, ended by discussions and concluding remarks.

Mathematical formulation 2.1. Description of a PV panel
A PV panel consists of several layers like glass, active zone known as PV module, Teldar and EVA as shown on figure 1. However, in practice the depth and heat capacities of Teldar and EVA are very small justifying why these layers are neglected in various studies. Therefore, the simplified PV panel is made of two layers as drawn on figure 2. On this figure, the first layer is the glass that physically protects the solar cell against the projectiles capable to destroy the PV panel. Known as the electronic part, the second layer is the PV module consisting of an association of identical solar cells in which the photogeneration, the diffusion and the recombination process of minority carriers take place.
Within our investigations, the sky is considered as a black body at constant temperature. It is also assumed that losses at the front and back surface of the PV panel are identical, and the effect of the dust at the front surface of the PV panel is negligible. We also consider that the temperature of each layer varies slowly with time. Then, the time derivative in the energy balance is neglected. We also admit that our searches are conducted in the steady state, and that the solar radiation is considered uniform on the PV panel. Based on the above assumptions, the temperature, thermodynamics constants as well as the optical properties are also uniform in each layer of the PV panel (Emmanuel, 2006).

Energy balance of glass cover
Glass is the first layer of a photovoltaic panel. Owing to , the energy balance through this layer is given by: in which the quantity c T defines the temperature of the PV module while g T and g  designate the temperature and the absorption coefficient of the glass cover, respectively. Based on Eq.
(1), one derives the temperature of the glass defined as ( ) In the previous relations, the parameter gc U stands for the coefficient of thermal loss between the glass cover and the active zone whereas ga U designates the coefficient of thermal loss between the glass cover and the ambient milieu.
In (2), the parameter ( ) G t stands for the solar radiation intensity during the whole day. In scientific reviews, there are several models of solar radiation intensity existing such as Adnot (Ionel et al., 2015), Karsten ( Kasten et al., 1989), Haurwitz (Ionel et al., 2015) and Empirical model (Paulescu et al., 2003). It has clearly appeared that the model of Adnot is the best since it describes the dynamics of the solar radiation nearest the reality. Then, this model is considered during our investigations and is expressed as follows (Ionel et al., 2015 where T represents the time.

Energy balance in the PV module
In the literature, nonlinearity generates new harmonics in a given dynamical system, offers a good advantage and is used to produce chaotic behavior, masking of the information. Beyond these applications, nonlinearity offers very often advantages in providing energy in a system (Dongo et al., 2014;Pelap et al., 2016). Motivated by this positive effect, we replace the constant absorption coefficient of the PV module by a nonlinear absorption coefficient expressed as a quadratic function of the temperature, since the variation of this coefficient due to the joule effect is usually neglected: wherein  is the quadratic coefficient consequently chosen. The quantities 0 c  , c T and  are, respectively, the normalized thermal absorption coefficient, the temperature of the PV module and the coefficient of thermal linear expansion of the material (Dongo et al., 2014;Fouad et al., 2012).
In the active zone of the PV panel known as PV module, many processes take place like photo generation, diffusion and recombination of minority carriers. The rise of the temperature of the PV panel is mainly linked to this zone. According to , the energy balance at this layer is given by Resolution of Eq. (7) yields two mathematical solutions for the temperature for the PV module and the accepted physical solution is defined below.
Fortunately, the MATLAB environment offers the possibility to forward the study and examine the contribution of the nonlinear absorption coefficient on the dynamics of the PV module

Effect of a nonlinear absorption coefficient on the glass cover
In this subsection, we conduct numerical simulations of the mathematical expression (2) figure 3), we note that the profile of the temperature of the glass cover follows that of the solar irradiation. It could be also noted that the maximum temperature is around 345 K  (i.e., 70 C  ) at the midday. However, for various values of the nonlinear absorption coefficient ( ), we remark that the shape of the obtained curves (in color on figure 3) does not change. It also appears that the value of the maximum temperature decreases as the parameter  grows. According to these results, the nonlinear character of the absorption coefficient plays the role of a cooling system for the glass cover. The presence of the nonlinear absorption coefficient of the PV module reduces the temperature of the glass cover. , we observe that the maximum temperature of the PV module is closed to 70 degrees, but remains slightly higher than the temperature of the glass cover obtained in the same conditions (Swapnil et al., 2009). Indeed, this difference of temperature is linked to the smallness of the heat capacity of the glass compared to that of the active zone known as PV module. Furthermore, several processes like recombination effect take place in a PV module. In fact, this recombination effect contributes also to the rise of the temperature of that layer thanks to the joule effect (Ibrahima et al., 2016). Figure 4: Variation of the temperature of a PV module for constant and nonlinear absorption coefficient obtained with same system parameters as in figure 3. The presence of the nonlinear absorption coefficient of the PV module reduces both the recombination effect in that zone and also its temperature. However, by taking into account the nonlinear behavior of the absorption coefficient of this layer, we note that the temperature of the PV module retains the shape of the solar radiation. Moreover, its maximum temperature decreases and reaches the minimum for  (Table 1). Indeed, this coefficient  impacts positively on the thermal behavior of the PV module by reducing both the recombination effect taking place as well as its operating temperature. In addition, the curves of figure 5 display that the electrical efficiency of the PV module grows as  diminishes (Table1). These results are in good accordance with those established by Pelap et al. (2016) in which they proved the useful of the nonlinearity in the solar cell domain. Thus, the nonlinear absorption coefficient reduces both the temperature and the recombination effect of a PV module and therefore increases it electrical efficiency.

Conclusion
In this paper, we have analyzed theoretically the effects of a thermal nonlinear absorption coefficient on the dynamics of a PV panel. This coefficient is expressed as a quadratic function of the temperature of the PV module known as the electronics active part of a PV panel. . The above results offer to the manufacturers of PV panel a new trick to improve the electrical efficiency of the PV modules and reduce both their operating temperatures and recombination effects. For future works, we plan to study other nonlinear aspects of a PV panel.