## 3 Performance Analysis and Simulations
### System Efficiency
#### 3.2.2 System Conversion Efficiency  
Conversion efficiency is a measure used to analyze and evaluate the performance of a CPV system. Conversion efficiency is defined as the ratio of energy output from the CPV system to input energy from the sun. The efficiency of a CPV system depends on the conversion efficiency of solar cells used in the system, the optical efficiency of the CPV units, and the spectrum and intensity of the incident sunlight.  
The total solar radiation on a horizontal surface, denoted by \(G\), is the sum of horizontal beam (direct) solar radiation and diffuse solar radiation [1]:  
\[G=G_{\mathrm{B}}+G_{\mathrm{D}}, \tag{10}\]  
where \(G_{\mathrm{B}}\) is beam radiation on a horizontal surface (direct normal irradiation), \(G_{D}\) is diffuse radiation on a horizontal surface, and \(G\) is global horizontal irradiation.  
The conversion efficiency of a conventional CPV system, denoted by \(\eta_{\mathrm{C}}\), is given by [28]:  
\[\eta_{\mathrm{C}}=\frac{\eta_{\mathrm{opt\_CPV}}\eta_{\mathrm{cell\_CPV}}G_{ \mathrm{B}}}{G}, \tag{11}\]  
where \(\eta_{\mathrm{opt\_CPV}}\) and \(\eta_{\mathrm{cell\_CPV}}\) are the optical efficiency of the CPV based on direct solar radiation and the conversion efficiency of the MJ solar cell, respectively.  
The proposed CPV system captures both direct and diffuse solar radiation, so its conversion efficiency depends on not only direct solar radiation, but also diffuse solar radiation. The conversion efficiency of the designed CPV system, denoted by \(\eta_{D}\), can be calculated by [28]:  
\[\eta_{\mathrm{D}}=\frac{\eta_{\mathrm{opt\_CPV}}\eta_{\mathrm{cell\_CPV}}G_{ \mathrm{B}}+\eta_{\mathrm{opt\_PV}}\eta_{\mathrm{cell\_PV}}G_{\mathrm{D}}}{G}, \tag{12}\]  
where \(\eta_{\mathrm{opt\_PV}}\) and \(\eta_{\mathrm{cell\_PV}}\) are the optical efficiency of the Si low-cost solar cell based on diffuse solar radiation and the conversion efficiency of the Si solar cell, respectively.  
An improvement factor of system efficiency \(f\) was used to analyze and evaluate the system efficiency of the proposed CPV system with the Si low-cost solar cell compared with that of the conventional CPV system without the low-cost solar cell.  
The improvement factor \(f\) is determined as follows:  
\[f=\frac{\eta_{\mathrm{D}}-\eta_{\mathrm{C}}}{\eta_{\mathrm{C}}}. \tag{13}\]  
Figure 12: (**a**) comparison of incident solar power received at the solar cells of CPV units; and (**b**) comparison of the optical power ratio between CPV units.  
By substituting (11) and (12) into (13), the improvement factor \(f\) is re-written as follows:  
\[f=\frac{\eta_{\text{opt\_PV}\eta_{\text{cell\_PV}}}}{\eta_{\text{opt\_CPV}\eta_{ \text{cell\_CPV}}}\times\frac{G_{\text{D}}}{G_{\text{B}}}}. \tag{14}\]  
Let \(\tau\) and \(\rho\) be:  
\[\tau=\frac{\eta_{\text{opt\_CPV}\eta_{\text{cell\_CPV}}}}{\eta_{\text{opt\_PV} \eta_{\text{cell\_PV}}}}, \tag{15}\]  
\[\rho=\frac{G_{\text{D}}}{G}, \tag{16}\]  
where \(\tau\) is the ratio of the conversion efficiency of the MJ solar cell to that of the Si solar cell in the CPV system, and \(\rho\) is the diffuse-to-global ratio, or the ratio of diffuse solar radiation to the global solar radiation.  
The improvement factor \(f\) is then determined as follows:  
\[f=\frac{\rho}{\tau\left(1-\rho\right)}. \tag{17}\]  
Figure 13 shows the improvement factor \(f\) values for seven \(\rho\) values (0.1 \(\div\) 0.7) with \(\tau=1\), 2, and 3, respectively. For \(\tau=2\), an improvement \(f\) of 33.3% is expected for a medium DNI region with \(\rho\cong 0.4\), whereas an improvement \(f\) of 12.5% is expected for a high DNI region with \(\rho\cong 0.2\).