## 2 Model of the PV System
### Solar PV Cell Model  
A PV module consists of several solar cells connected in series and in parallel to achieve the desired voltage and current levels. A solar panel cell is essentially a p-n semiconductor junction. When exposed to light, a direct current is generated. For simplicity, the single diode model of Figure 2 is used in this document [37]. This model offers a good compromise between simplicity and precision with the basic structure. The equivalent circuit of the general model consists of a photo current (_lph_), a diode, a parallel resistance (_Rp_) expressing a leakage current, and a series resistance (_Rs_) due to the contacts between the semiconductors and the metal parts. This equivalent circuit is depicted in Figure 2.  
In Figure 2, we apply [PERSON]'s law. The current will be obtained by the following Equation:  
\[I=I_{ph}-I_{D}-I_{p} \tag{1}\]  
where \(I_{ph}\) the current generated by light or photocurrent and \(I_{p}\) the current flowing in the parallel resistor, which can be computed as:  
\[I_{p}=\frac{V+R_{s}I}{R_{p}} \tag{2}\]  
Figure 2: Simplified equivalent circuit of solar cell. \(I_{D}\) is the current of the diode, which is proportional to the saturation current. The following equation expresses the value of this magnitude  
\[I_{D}=I_{sd}\bigg{(}exp\bigg{(}\frac{q.(V+R_{s}.I)}{n.K.T}\bigg{)}-1\bigg{)} \tag{3}\]  
being \(I_{sd}\) the reverse saturation current in amperes (A), \(q\) the electron charge (1.6 \(\times\) 10-19 C), \(K\) the Boltzmann constant (1.38 \(\times\) 10-23 J/K), \(T\) the cell temperature in Kelvin (K) and \(n\) the ideal factor. We replace the voltage-current characteristic equation of a solar cell in Equation (1). So we derive that:  
\[I=I_{ph}-I_{sd}\bigg{(}exp\bigg{(}\frac{q.(V+R_{s}.I)}{n.K.T}\bigg{)}-1\bigg{)} -\frac{V+R_{s}I}{R_{p}} \tag{4}\]  
The photocurrent depends mainly on the solar radiation and the operating temperature of the cell, which is described by the following Equation:  
\[I_{ph}=\Big{[}I_{sc}+K_{i}\Big{(}T-T_{ref}\Big{)}\Big{]}\frac{G}{G_{ref}} \tag{5}\]  
where:  
\(I_{sc}\): is the short-circuit current of the cell at 25 \({}^{\circ}\)C and 1000 W/m\({}^{2}\)  
\(K_{i}\): the temperature coefficient of the cell short-circuit current,  
\(T_{ref}\): is the reference temperature of the cell, in Kelvin (K) (=25 \({}^{\circ}\)C + 273),  
\(G\): is the solar radiation in watt/square meter (W/m\({}^{2}\)),  
\(G_{ref}\): is the reference insolation of the cell (=1000 W/m\({}^{2}\)),  
On the other hand, the cell saturation current varies with the temperature of the cell, which is described as follows:  
\[I_{sd}=I_{ts}\bigg{(}\frac{T}{T_{ref}}\bigg{)}^{3}.exp\left(\frac{q.E_{S} \Big{(}\frac{1}{T_{ref}}-\frac{1}{T}\Big{)}}{K.n}\right) \tag{6}\]  
being:  
\(I_{ts}\): the reverse saturation current of the PV cell  
\(Eg\): is the gap energy of the semiconductor used in the PV cell in electron-volt (eV)  
\(n\): is the ideal factor, which depends on the PV technology listed in Table 1.  
The reverse saturation current is given by the following Equation (7):  
\[I_{ts}=\frac{I_{sc}}{exp\Big{(}\frac{q.V_{sc}}{N_{s}.u.K.T}\Big{)}-1} \tag{7}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Technology** & **N** \\ \hline Monocrystalline Silicon (Si-mono) & 1.2 \\ Polycrystalline Silicon (Si-poly) & 1.3 \\ Hydrogenated Amorphous Silicon (a-Si:H) & 1.8 \\ Hydrogenated Amorphous Silicon tande (a-Si:H tandem) & 3.3 \\ Hydrogenated Amorphous Silicon triple (a-Si:H triple) & 5 \\ Cadmium Telluride (CdTe) & 1.5 \\ Copper Indium Selenide (CIS) & 1.5 \\ Gallium arsenide (GaAs) & 1.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ideal Factor n dependence on PV cell technology. A typical PV cell produces less than 2 W at about 0.5 V so that cells must be connected in series-parallel on a module to produce enough power [38]. A PV array, like presented in Figure 3, is a group of several PV modules electrically connected in series (\(N_{s}\) cells) and in parallel (\(N_{p}\) columns) to generate the required current and voltage.