*3.1. Mathematical Modeling of Solar Power*

The output power of the PV module is obtained by solar irradiance and with respect to PV module area. The output of the PV model is determined by [23]

$$P\_{solar} = \eta\_{\mathcal{S}} i\_{\mathcal{V}} A\_{\prime} \tag{1}$$

where, *η<sup>g</sup>* = generation efficiency, *i<sup>r</sup>* = solar irradiation (W/m<sup>2</sup> ) and *A* = area (m<sup>2</sup> ), and the PV efficiency is determined by Equation (2).

$$
\eta\_{c\varepsilon} = \eta\_{ref}\eta\_{c\varepsilon} \left[ 1 - \beta \left( T\_{cell} - T\_{cellref} \right) \right] \tag{2}
$$

where, *ηce* = power conditioning efficiency, *β* = Temperature co-efficient *C* ((0.004–0.006)/*C*), *ηre f* = reference module efficiency, *Tcellre f* = reference cell temperature, and the temperature (*TC*) is determined by Equation (3).

$$T\_c = T\_d + \left[\frac{\text{NOCT} - \text{20}}{800}\right] \text{G}\_{l\prime} \tag{3}$$

where, *T<sup>a</sup>* = temperature in *C*, *NOCT* = nominal operating cell temperature in *C*, *G<sup>t</sup>* = solar irradiation in tilted module (W/m<sup>2</sup> ).

Total radiation in the solar cell considering normal and partial solar radiation is obtained by

$$T\_I = I\_D R\_D + (I\_b + I\_d) R\_r. \tag{4}$$

System Modeling

The PV or solar cell operation is similar to the operation of PN junction diode, which converts light energy into electricity through the photovoltaic effect [24]. The PV module is grouped based on the series and parallel connection of multiple PV cell [24,25]. The single PV cell is configured into a single diode representation as in Figure 3. In this model, solar irradiance is represented by a current source, and the other circuit parameters are diode current *I<sup>d</sup>* , output current *I*, series resistance *R<sup>s</sup>* , parallel resistance *Rp*, and output voltage *V*. The output current is calculated by

$$I = N\_P \left[ I\_{ph} - I\_{rs} \left( \frac{\exp q(V + IR\_s)}{AKTN\_s} - 1 \right) \right],\tag{5}$$

$$I\_{\rm RS} = I\_{rr} - \left[\frac{1}{T\_{\rm K}} - \frac{1}{T}\right],\tag{6}$$

where, *N<sup>P</sup>* and *N<sup>S</sup>* = number of cell connected in parallel and series, the *K* = Boltzmen constant, *A* = diode ideality factor, *IR<sup>S</sup>* = reverse saturation current of cell at *T*, *T<sup>r</sup>* = referred cell temperature, and *Irr* = reverse saturation current at *T<sup>r</sup>*

$$I\_{ph} = \left[ I\_{scr} + K\_i (T - T\_r) \frac{S}{100} \right] \tag{7}$$

where, *Iscr* = short circuit current at a reference temperature of the cell, *K<sup>i</sup>* = co-efficient of the short circuit temperature, *S* = solar irradiation in (W/m<sup>2</sup> ). In this model, the shunt resistance in parallel to the ideal shunt diode and the *I*-*V* characteristics are determined by the equation as follows:

$$I = I\_{\rm ph} - I\_{\rm D} \tag{8}$$

$$I = I\_{ph} - I\_o \left[ \exp\frac{q(V + IR\_s)}{AKT} - 1 \right] - \frac{V + R\_s I}{R\_{sh}} \,\text{,}\tag{9}$$

where, *Iph* = Irradiance current (A), *I<sup>D</sup>* = diode current (A), *I<sup>o</sup>* = Inverse saturation current (A), *R<sup>S</sup>* = series resistance (Ω), *Rsh* = shunt resistance (Ω), *I* = cell current (A), *V* = cell voltage, and the output current of the PV cell using single diode model is expressed as

$$I = I\_{PV} - I\_{D1} - \frac{V + IR\_s}{R\_{sh}}.\tag{10}$$

**Figure 3.** A single diodemodel of a photovoltaic (PV) cell.

The open circuit voltage and maximum power of the PV module is obtained by the simplified PV system modeling proposed by [26]. The voltage and power with the values of series resistance (*RS*) is calculated by fill factor [27–29].

$$FF = FF\_o \left[ 1 - \left. \frac{\frac{Rs}{V\_{OC}}}{\frac{I\_{SC}}{I\_{SC}}} \right|\_{SC} \right] \tag{11}$$
 
$$FF\_o = \frac{V\_{OC} - \ln(V\_{OC} + 0.72)}{1 + V\_{VOC}} \tag{12}$$

$$P\_{\text{max}} = FF \times V\_{\text{OC}} \times I\_{\text{OC}}$$

$$P\_{\text{max}} = \frac{V\_{\text{OC}} - \ln(V\_{\text{OC}} + 0.72)}{1 + V\_{\text{OC}}} \times \left(1 - \frac{I\_{\text{SC}} \times R\_{\text{S}}}{V\_{\text{OC}}}\right) \times \frac{V\_{\text{OCO}}}{1 + \delta \ln \frac{C\_{\text{O}}}{C}} \times \left(\frac{T\_{\text{O}}}{T}\right)^{\delta} \times I\_{\text{SO}} \left(\frac{G}{G\_{\text{O}}}\right)^{a} \tag{12}$$

where, *FF* = fill factor of the ideal PV module without resistive effects and *VOC* = normalized value of the open circuit voltage to thermal voltage.

The power conversion in the PV system is obtained through the PV modules. The performance capability of the PV depends on the temperature and its characteristic curve (power & *V*, *I* curve) at standard test condition, which is shown in Figure 4. A single PV cell of any rating will not be able to generate the required power levels. Hence, several PV cells are interconnected through a series and parallel combinations that scale up to generate the required PV power. The voltage and current are obtained by scaling up of PV modules, which is expressed as

$$\begin{aligned} I\_A &= \frac{N\_p}{I\_M} \\ V\_A &= N\_\mathcal{S} \times V\_M \\ P\_A &= FF \times V\_A \times I\_A \end{aligned} \tag{13}$$

where, *I<sup>A</sup>* and *V<sup>A</sup>* = PV array voltage and current, *I<sup>M</sup>* and *V<sup>M</sup>* = PV module voltage and current, and *P<sup>A</sup>* and *P<sup>M</sup>* = PV array power and module power.

**Figure 4.** Solar cell characteristics curve (voltage vs. current and power).
