A Calculation Model to Estimate the Electrical Performance of a Photovoltaic Panel

An accurate estimate of the temperature of photovoltaic panels is essential to assess their electrical performance. Therefore, in this work, one of the main objectives is to illustrate a simple and very accurate method for predicting the temperature of the PV panels, on which the electrical power supplied and the electrical efficiency depend.

Being able to predict the producibility of a photovoltaic field is becoming increasingly important for their interconnection within smart-grids [1]. In these networks, on which investments are being made for the future, it is necessary to estimate the electrical loads, the electrical generations and the electrical storage systems installed in a decentralized manner on the territory [2-4]. Consequently, the prediction of the behavior of small-scale systems also becomes important [5], especially as regards plants that use renewable sources with high variability of the primary source.

In this work, a calculation model will be developed and validated with a photovoltaic panel installed at the DIMEG of the University of Calabria. Bevilacqua et al. [6] have validated their calculation method on the same photovoltaic system. They developed a finite difference model by defining many nodes within the panel. In the present work, with a similar methodology, we want to investigate the possibility of obtaining good results with a simpler finite difference model, defining only 5 nodes, as also proposed by Notton et al. [7].

In the literature there are several models to estimate the electrical producibility of a photovoltaic panel.

The technique applied in this study consist in a non-stationary model, therefore able to also consider capacitive effects and reliably simulate the operating conditions relating, for example, to the passage of a cloud or to abrupt local climatic variations [8, 9].

Several scientific papers investigate the link between electrical efficiency and climatic variables such as irradiance and temperature. Chaibi et al. [10] analyzes the current/voltage (I-V) characteristics for Si-crystalline PV modules under non-standard condition by using single-diode and double-diode models. Rodziewicz et al. [11] analyzes the influence of changes in the solar radiation spectrum distribution on the properties of various photovoltaic modules, with particular emphasis on the scattered component.

A complex fluid dynamics calculation model was developed by Jaszczur et al. [12]. However, the calculation requires a computational effort and is not easy to use.

The aim of this paper will be to use a simpler procedure for estimating the quantities of interest, but with still accurate results.

The thermal model developed in the present study has two functions: the first is to estimate the temperature of the photovoltaic cells, which is used to predict the electrical energy production which varies with operating conditions; secondly, it provides the temperatures in the different layers along the thickness of the photovoltaic panel, which helps to better understand the function of each of them in heat dissipation. After having analyzed numerous other scientific articles on this topic [13-17], the correlation used to estimate the electrical efficiency is the Evans equation [18]. In this equation the most important term is the PV temperature. Therefore, it is necessary to determine it precisely.

The idea is to carry out the calculations, in a transient manner, as a function of solar irradiation, temperature, wind speed, relative humidity [19] in order to be used for predictive purposes.

In the thermal model, the photovoltaic panel is divided into five separate layers, for each of them a point is considered and the energy conservation equations are written in the form of thermal balance equations. Starting from top to bottom, the first point in the glass layer is taken on the surface and has a temperature $T_{f g}$; the second point in the middle of the upper EVA layer has a temperature $T_{e v, s u p}$, the third point in the center of the panel and the PV cell layer has a temperature $T_{p v}$, the fourth in the center of the lower EVA layer has a temperature $T_{e v, s u p}$ and, finally, the fifth point is located on the lower surface of the tedlar layer and has a temperature $T_{\text {ted }}$. Figure 1 shows the layers with their thicknesses and the point temperatures for each layer.

1.png

Figure 1. Subdivision into layers of the PV panel and nodes

For each node the relative thermal balance equation can be written taking into account the conductive resistances between each of them, the convective and radiative resistances at the boundary, the accumulated thermal power and the absorbed thermal power.

The conductive resistances of each layer are indicated with $R_{i}=\frac{s_{i}}{k_{i} A_{i}}$ with reference to the whole thickness s. The resistance of half the thickness of the layer is indicated with $\frac{1}{2} R_{i}$, accordingly. The equations for the five nodes are:

$\frac{T_{a}-T_{f g}}{R_{\text {conv }}}+\frac{T_{s k y}-T_{f g}}{R_{r a d, f g-s k y}}+\frac{T_{g r}-T_{f g}}{R_{r a d, f g-g r}}+\frac{T_{e v, s u p}-T_{f g}}{R_{f g}+\frac{1}{2} R_{e v}}+\dot{Q}_{f g}=\rho_{f g} c_{f g} V_{f g} \frac{d T_{f g}}{d t}$                       (1)

$\frac{T_{f g}-T_{e v, \text { sup }}}{R_{f g}+\frac{1}{2} R_{e v}}+\frac{T_{p v}-T_{e v, s u p}}{\frac{1}{2} R_{e v}+\frac{1}{2} R_{p v}}=\rho_{e v} c_{e v} V_{e v} \frac{d T_{e v, \text { sup }}}{d t}$                                (2)

$\frac{T_{e v, s u p}-T_{p v}}{\frac{1}{2} R_{e v}+\frac{1}{2} R_{p v}}+\frac{T_{e v, i n f}-T_{p v}}{\frac{1}{2} R_{p v}+\frac{1}{2} R_{e v}}+\dot{Q}_{p v}=\rho_{p v} c_{p v} V_{p v} \frac{d T_{p v}}{d t}$                          (3)

$\frac{T_{p v}-T_{e v, i n f}}{\frac{1}{2} R_{p v}+\frac{1}{2} R_{e v}}+\frac{T_{t e d}-T_{e v, i n f}}{\frac{1}{2} R_{e v}+R_{t e d}}=\rho_{e v} c_{e v} V_{e v} \frac{d T_{e v, i n f}}{d t}$                              (4)

$\frac{T_{a}-T_{t e d}}{R_{\text {conv }}}+\frac{T_{s k y}-T_{\text {ted }}}{R_{\text {rad,ted-sky }}}+\frac{T_{g r}-T_{t e d}}{R_{\text {rad,ted }-g r}}+\frac{T_{e v, i n f}-T_{t e d}}{\frac{1}{2} R_{e v}+R_{t e d}}=\rho_{t e d} c_{t e d} V_{t e d} \frac{d T_{t e d}}{d t}$                                      (5)

The total solar radiation absorbed in the PV cells is calculated by means of radiation models and, in the literature, there are several available [20, 21].

The formula proposed by Aly et al. [22] which has been validated with experimental results is:

$\dot{Q}_{p v}=\alpha_{P V} \cdot \tau(i) \cdot G \cdot A \cdot\left(1-\eta_{p v}\right)$                      (6)

in which $\alpha_{P V}$ is the effective absorption coefficient of the photovoltaic cells, which in our case assumes a value equal to 0.93 [23]; G is the total radiation incident on the inclined plane of the PV panel, τ(i) is the transmission coefficient of the front glass for the direct component which depends on the angle of incidence i. The latter is obtained from the following formula [24].

$\tau(i)= e^{-\left(\frac{K \cdot s_{f g}}{\cos i_{r}}\right)}\left[1-\frac{1}{2}\left(\frac{\sin ^{2}\left(i_{r}-i\right)}{\sin ^{2}\left(i_{r}+i\right)}+\frac{\tan ^{2}\left(i_{r}-i\right)}{\tan ^{2}\left(i_{r}+i\right)}\right)\right]$                            (7)

in which K is the extinction constant of the glass, which assumes a value of 4 m^-1 in the case of tempered glass used in PV panels [20], $i_{r}$ is the refraction angle of the direct radiation incident through the glass and can be derived from Snell's law [20].

The last term of Eq. (6) represents the inefficiency of converting solar energy into electricity as $\eta_{p v}$ is the yield of the PV cells. The latter depends on the temperature of the cells $T_{P V}$ and by irradiance G:

$\eta_{p v}=\eta_{p v, r e f} \cdot\left[1-\beta_{r e f}\left(T_{p v}-T_{r e f}\right)+\gamma \log _{10} \frac{G}{1000}\right]$                                     (8)

with $\beta_{r e f}=0.006^{\circ} \mathrm{C}^{-1}$ and $\gamma=0.085$ for the PV panel (Tonui and Tripanagnostopoulos [25]) and $T_{\text {ref }}=25^{\circ} \mathrm{C}$.

The thermal power absorbed by the glass is equal to:

$\dot{Q}_{f g}=\alpha_{f q} \cdot G \cdot A$                             (9)

in which $\alpha_{f g}=0.05$ is the absorption coefficient of the front glass.

To solve the problem, the technique of finite differences FD with respect to time is used. The calculations will then be carried out with the aid of a simple spreadsheet (Microsoft Excel). For each instant in time, the incoming data are the total radiation incident on the surface of the photovoltaic panel, the ambient temperature and the wind speed. First of all, the equations are rewritten by dividing both sides of each equation by the surface area of the corresponding layer and discretizing the time derivative referring it to finite time intervals:

$h_{\text {conv }}\left(T_{a}-T_{f g}\right)+h_{r a d, f g-s k y}\left(T_{s k y}-T_{f g}\right)+\frac{T_{e v, \text { sup }}-T_{f g}}{\frac{s_{f g}}{k_{f g}}+\frac{1}{2} \frac{s_{e v}}{k_{e v}}}+h_{r a d, f g-g r}\left(T_{g r}-T_{f g}\right)+\alpha_{f g} G=\rho_{f g} c_{f g} s_{f g} \frac{T_{f g}-T_{f g}^{p}}{\Lambda t}$                        (10)

$\frac{T_{f g}-T_{e v, \sup }}{\frac{s_{f g}}{k_{f g}}+\frac{1}{2} \frac{s_{e v}}{k_{e v}}}+\frac{T_{p v}-T_{e v, s u p}}{\frac{1}{2} \frac{s_{e v}}{k_{e v}}+\frac{1}{2} \frac{s_{p v}}{k_{p v}}}=\rho_{e v} c_{e v} s_{e v} \frac{T_{e v, s u p}-T_{e v, s u p}^{p}}{\Delta t}$                   (11)

$\frac{T_{e v, s u p}-T_{p v}}{\frac{1}{2} \frac{s_{e v}}{k_{e v}}+\frac{1}{2} \frac{s_{p v}}{k_{p v}}}+\frac{T_{e v, i n f}-T_{p v}}{\frac{1}{2} \frac{s_{p v}}{k_{p v}}+\frac{1}{2} \frac{s_{e v}}{k_{e v}}}+\alpha_{P V} \cdot \tau(i) \cdot G \cdot\left(1-\eta_{P V}\right)=\rho_{p v} c_{p v} s_{p v} \frac{T_{p v}-T_{p v}{ }^{p}}{\Delta t}$                       (12)

$\frac{T_{p v}-T_{e v, i n f}}{\frac{1}{2} \frac{s_{p v}}{k_{p v}}+\frac{1}{2} \frac{s_{e v}}{k_{e v}}}+\frac{T_{t e d}-T_{e v, i n f}}{\frac{1}{2} \frac{s_{e v}}{k_{e v}}+\frac{s_{t e d}}{k_{t e d}}}=\rho_{e v} c_{e v} s_{e v} \frac{T_{e v, i n f}-T_{e v, i n f}^{p}}{\Delta t}$                   (13)

$h_{\text {conv }}\left(T_{a}-T_{t e d}\right)+h_{\text {rad,ted-sky }}\left(T_{s k y}-T_{\text {ted }}\right)+\frac{T_{e v, i n f}-T_{t e d}}{\frac{1}{2} \frac{s_{e v}}{k_{e v}}+\frac{s_{t e d}}{k_{t e d}}}+h_{r a d, t e d-g r}\left(T_{g r}-T_{t e d}\right)=\rho_{t e d} c_{t e d} s_{t e d} \frac{T_{t e d}-T_{t e d}^{p}}{\Delta t}$                      (14)

In the equations, all temperatures and variables refer to the present time t, while temperatures with apex p refer to the previous time $t-\Delta t$.

For natural convection the Nusselt number is calculated as if the plate were horizontal, with the following expression for the top surface [26]:

$N u_{\text {conv, free }}=0.13 \cdot R a^{\frac{1}{3}}$                   (15)

And the following for the lower surface [27]:

$N u_{\text {conv, free }}=0.27 \cdot R a^{\frac{1}{4}}$                     (16)

The Rayleigh number is calculated using the average length of the two panel dimensions.

For forced convection, in turbulent outflow, the Nusselt number is calculated with the following relation due to Sparrow [28]:

$N u_{\text {conv,forced }}=0.86 \cdot R e_{L, c}^{\frac{1}{2}} \cdot P r^{\frac{1}{3}}$                      (17)

The Reynolds number is calculated at the characteristic length $L_{c}$ which is four times the plate area divided by the perimeter. It is not a length that considers the wind direction.

When the flow is laminar, however, it is preferred to use the classical expression coming from the boundary layer theory [29]:

$N u_{\text {conv,forced }}=0.664 \cdot R e_{Lc}^{\frac{1}{2}} \cdot \operatorname{Pr}^{\frac{1}{3}}$                  (18)

The overall value of the convection is calculated taking into account the combined natural and forced flow and is estimated using the Churchill expression [30]:

$h_{\text {conv }}=\sqrt[3]{{h_{\text {conv,free }}}^{3}+{h_{\text {conv,forced }}}^{3}}$              (19)

Many authors in the scientific literature, such as Notton et al. [7] and Barroso et al. [31], use the formulas of the coefficients of heat transmission by radiation, both for the upper and lower surfaces, in linearized form. In our model, to be more precise, we will use the exact non-linearized formulas and apply them separately for both surfaces:

$h_{r a d, i-s k y / g r}=\frac{\sigma \cdot\left(T_{i}^{2}+T_{s k y / g r}^{2}\right)\left(T_{i}+T_{s k y / g r}\right)}{\frac{1-\varepsilon_{i}}{\varepsilon_{i}}+\frac{1}{F_{i-s k y / g r}}}$                        (20)

in which $T_{i}$ is the surface temperature of the outer layer for which we want to calculate the heat exchange coefficient, $T_{s k y / g r}$ represents the temperature of the celestial vault or the temperature of the ground. The temperature of the celestial vault can be calculated with the following formula [32], widely used in scientific literature:

$T_{s k y}=0.0552 \cdot T_{a}^{1.5}$                 (21)

where $T_{a}$ is the air temperature.

The ground temperature can be set equal to the ambient temperature:

$T_{g r}=T_{a}$                       (22)

The view factors are:

$F_{f g-s k y}=F_{t e d-g r}=\frac{1+\cos \beta}{2}$           (23)

$F_{f g-g r}=F_{t e d-s k y}=\frac{1-\cos \beta}{2}$                  (24)

$\varepsilon_{f g}$ is the emissivity of the glass for which various values have been proposed in the scientific literature including 0.86 proposed by Hegazy [33] and 0.85 by Notton et al. [7] taken from experimental tests verified and disclosed by the manufacturer Photowatt. We will use the latter value. For the lower surface of the tedlar $\varepsilon_{\text {ted }}$ will be taken equal to 0.85.

The aim of the work was the thermal modeling of a photovoltaic panel to analyze its performance, for example for predictive purposes. The physical and optical-radiative modeling of the photovoltaic panel made it possible to evaluate the spatial and temporal temperature profile within the PV. In particular, the convective and radiative exchanges were estimated for both the upper and lower surfaces with the relative view factors, evaluating the most accurate and reliable equations in the scientific literature.

All the input data made it possible to obtain, through thermal balance equations, a system of equations in which the unknowns are the temperatures, variable over time, of the points of the internal layers of the PV panel. For the resolution, the finite difference method was used.

The modeling results were validated with experimental data obtained from an experimental set-up located on the roof of a building of the Department of Mechanical, Energy and Management Engineering (DIMEG) of the University of Calabria.

For clear days, a square error of about 1.5°C is observed with regard to the temperatures reached, an RMSE of about 3 W for the electrical power supplied and an RMSE of about 0.25% for the electrical efficiency. The results are very satisfying. The improvements on which the authors are studying concern the behavior in cloudy day conditions and in the case in which the panel is cooled by spray.