*2.1. Mathematical Model of Photovoltaic Cells*

A PV cell is a semi-conductor material that absorbs energy from sunlight and allows its electrons to jump to higher energy states. The liberated electrons subsequently undergo free movement along connected conductive wires, resulting in the generation of an electric current. This phenomenon of PV conversion is called the PV effect [35]. It harnesses the PV effect to directly transform solar energy into electrical energy. Figure 1 represents the single-diode model of a solar cell.

**Figure 1.** Single-diode model of a PV cell.

Applying Kirchhoff's current law [36], the output current is represented as follows:

$$I = I\_{\rm ph} - I\_{\rm D\_s} - I\_{\rm sh} = I\_{\rm ph} - I\_{\rm D\_s} - \frac{V + IR\_s}{R\_{\rm sh}} \tag{1}$$

where *I* represents the current through series resistor *Rs*, *Ish* is the current through shunt resistor *R*sh, *Iph* is the photo-generated current, *VD*<sup>s</sup> is the voltage across *Ds*, and *ID*<sup>s</sup> is the current through the diode *Ds*. The output voltage of the solar cell is represented by *V*.

The following Shockley equation [37] can be used to express the electric current:

$$I\_{D\_8} = I\_0 \left( e^{\frac{qV D\_8}{\eta kT}} - 1 \right) = I\_0 \left( e^{\frac{q(V + R\_S I)}{\eta kT}} - 1 \right) \tag{2}$$

where *<sup>η</sup>* is the ideality factor of the diode; *<sup>K</sup>* is the Boltzmann constant, *<sup>K</sup>* = 1.38 × <sup>10</sup>−<sup>23</sup> *<sup>J</sup>*/*K*; *<sup>T</sup>* is the ambient temperature; *<sup>q</sup>* is the electronic charge constant, *<sup>q</sup>* = 1.6 × <sup>10</sup>−<sup>19</sup> C; *Rs* is the equivalent series resistance; *V* is the output voltage of the PV array; and *I* is the output current of the PV array.

By substituting Equation (2) into Equation (1), the I-V characteristic model is derived and represented as follows:

$$I = I\_{\rm pl} - I\_0 \left( e^{\frac{q(V + R\_S I)}{\eta KT}} - 1 \right) - \frac{V + IR\_S}{R\_{\rm sh}} \tag{3}$$

A PV module commonly consists of numerous solar cells arranged in a series or in parallel to effectuate increased power, voltage, and current output levels. This unique design also allows PV modules to adapt to various system requirements and environmental conditions. The series configuration enables the module to generate higher voltages, making it suitable for applications that demand higher voltage levels. In contrast, the parallel arrangement ensures that sufficient current is provided, which is ideal for situations where higher current output is essential. The output characteristic is influenced by both its internal parameters and external factors, such as temperature and light intensity [38]. The equivalent circuit is depicted in Figure 2.

**Figure 2.** Equivalent circuit diagram of PV cells.

The current-voltage characteristics of a PV module model [39] are represented by Equation (4):

$$I = n\_p I\_{ph} - n\_p I\_{\rm sc} \left( \exp\left[\frac{q\left(V + I\frac{n\_s}{n\_p}\right)}{n\_s \eta KT}\right] - 1\right) - \frac{\frac{n\_p V}{n\_s} + IR\_s}{R\_{sh}}\tag{4}$$

where *np* is the number of lateral PV panels, *ns* is the number of vertical PV panels, *Isc* is the saturation current of the diode.

The value of *Iph* is dependent on the intensity of the light source and temperature.

$$I\_{ph} = I\_{ph\\_STC} + K\_i \left( T - T\_{ref} \right) \frac{G}{G\_{STC}} \tag{5}$$

where *Iph*\_*STC* is the short-circuit current under standard temperature and irradiance intensity; *Ki* is the temperature coefficient of current change, *Ki* = 0.003; *Tref* is the standard

temperature, *Tref* = 25 ◦C; and *G* is the current irradiance intensity, while *GSTC* is the standard irradiance intensity, *GSTC* = 1000 W/m2.

As stated in Equation (4), the I-V characteristic of PV modules undergoes significant alterations by external factors, such as solar irradiance and ambient temperature. The I-V and P-V curves presented in Figure 3a illustrate how different temperatures, while maintaining the same irradiance level, can influence the performance of the system. With increasing temperature, the I-V curve shifts toward a lower voltage, whereas a decrease in temperature causes the I-V curve to shift toward a higher voltage. Additionally, as indicated by the P-V curve, the power exhibits a negative correlation with the increase in temperature. Figure 3b shows the I-V curve and P-V curve at constant temperatures with varying irradiance. It is evident that a rise in solar irradiance leads to a corresponding increase in current and power. Observing Figure 3a,b, it is apparent that the MPP varies with changes in irradiance and module temperature. Hence, it is imperative to consider the impact of temperature and irradiance on PV solar systems.

**Figure 3.** I-V and P-V curves of PV modules. (**a**) Diverse temperatures and (**b**) different irradiances.
