## 3 Mathematical Formulation of the CPV
### Mathematical Model of a PV Cell  
A solar PV cell can be transformed into an equivalent electrical circuit to predict its current and voltage output. Ideally, a PV cell's equivalent circuit is represented by a diode connected in parallel with a photocurrent, \(\mathrm{I_{ph}}\), (i.e., a light generated current source). However, in reality, additional parameters must be considered, including the resistivity of the material, the ohmic losses and the shunt resistance's effect on the cell. These parameters can be modelled as the shunt resistance, \(\mathrm{R_{sh}}\) and series resistance, \(\mathrm{R_{s}}\). The equivalent circuit of the PV cell can be represented by a single-diode model (see Figure 3). Several researchers enhanced the single-diode model by including an additional diode into the circuit [56; 57; 58; 59]. The new representation is denoted the two-diode model. This extra diode corresponds to the recombination effects of the charge carriers [60; 61]. Despite providing a more precise representation of the PV cell's outputs, the two-diode model requires much longer computational time compared with the previous single-diode model [60].  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Angle of Incidence (°)** & **Opto-Electronic Gain, \(\mathrm{C_{opto-e}}\)** \\ \hline
0 & 4.17143 \\ \(\pm\)5 & 4.00000 \\ \(\pm\)10 & 3.89205 \\ \(\pm\)15 & 3.60465 \\ \(\pm\)20 & 3.37313 \\ \(\pm\)25 & 3.14642 \\ \(\pm\)30 & 2.78105 \\ \(\pm\)35 & 1.84722 \\ \(\pm\)40 & 1.21561 \\ \(\pm\)45 & 0.95142 \\ \(\pm\)50 & 0.69643 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Opto-electronic gain of the RADTIRC at different angle of incidences.  
Figure 3: A single-diode equivalent circuit. Adapted from [61].  
The expression for the single-diode model's output current is given by the following Equation (2) [61]:  
\[\rm I=I_{ph}-I_{d}-I_{sh} \tag{2}\]  
where \(\rm I_{d}\) is the diode current. Equation (1) can then be transcribed as [23]:  
\[\rm I=I_{ph}-I_{s}\left[\rm exp\left(\frac{q(V+IR_{s})}{n_{d}kT}-1\right) \right]-\frac{V+IR_{s}}{R_{sh}} \tag{3}\]  
where \(\rm I_{s}\) is the saturation current for the diode, while \(\rm n_{d}\) corresponds to the diode ideality factor, \(\rm q\) is the electron charge (\(1.602\times 10^{-19}\) C), is the Boltzman constant (\(1.38\times 10^{-23}\) J/K), T is the temperature of the p-n junction (in Kelvin), and V is the voltage across the PV cell.  
In this model, it is required to compute \(\rm I_{ph}\), \(\rm I_{rs}\) and \(\rm I_{s}\). These values will be determined from the following equations. \(\rm I_{ph}\) is independent of V (or \(\rm R_{s}\)), linearly dependent on the solar irradiance and is also affected by the temperature. This relationship is presented in the following Equation (4) [38].  
\[\rm I_{ph}=(I_{sc}+\alpha(T-T_{ref}))*G_{con} \tag{4}\]  
where \(\rm T_{ref}\) is \(25\ ^{\circ}\)C and \(\alpha\) is the current temperature coefficient.  
[PERSON] et al. [38] indicated that the reverse saturation current, \(\rm I_{rs}\) and the saturation current, \(\rm I_{s}\) are determined using Equations (5) and (6), where the value of the energy bandgap, \(\rm E_{g}\) is provided by the manufacturer. The key parameters for the solar PV cell are tabulated in Table 2. These parameters were provided by the manufacturer of the solar PV cell, Narec (National Renewable Energy Centre).  
\[\rm I_{rs}=I_{sc}/\left[\rm exp\left(\frac{qV_{oc}}{n_{d}kT}\right)-1\right] \tag{5}\]  
\[\rm I_{s}=I_{rs}\left[\frac{T}{T_{ref}}\right]^{3}\rm exp\left[\left(\frac{qE_ {g}}{n_{d}k}\right)\left(\frac{1}{T}-\frac{1}{T_{ref}}\right)\right] \tag{6}\]