## 3 Simulations  
### Optimization of SOE Design  
Figure 2 shows our SOE designs with different angles when placed on the TJ solar cells. At this point, we fixed the SOE height at 15 mm and maintained a constant distance between the Fresnel lens and the TJ solar cell. Using these SOEs, we calculated the irradiance on the TJ solar cell and on the 12 silicon solar cells (positions #1 to #12 in Table 1). We also analyzed the irradiance by dividing the results into an effective wavelength range (0.3-1.299 \(\upmu\)m) and a lossy wavelength range (1.3-2.5 \(\upmu\)m). The irradiance on the TJ solar cell in the effective wavelength range was minimized for a perpendicular SOE (326.8 W/m\({}^{2}\)), and maximized when \(\theta=30^{\circ}\) (606.2 W/m\({}^{2}\)); when the angle is greater than \(30^{\circ}\), the light is reflected and the intensity of the light is reduced. Although the sum of power in the effective and lossy wavelength ranges was higher in the \(\theta=40^{\circ}\) case than the \(\theta=30^{\circ}\) case, the E\({}_{\rm TJ,loss}\)/E\({}_{\rm TJ,eff}\) ratio indicates that the loss ratio of the \(\theta=40^{\circ}\) (6.9%) case was higher than that of the \(\theta=30^{\circ}\) (5.7%) case, so \(\theta=30^{\circ}\) seems more optimal. In addition, higher SOE angles in the silicon solar cell will decrease the irradiance, due to shading from the SOE.  
Next, we analyzed the irradiance with a fixed \(30^{\circ}\) SOE angle as we varied the SOE height from 0 mm to 100 mm, as shown in Figure 3 and Table 2. Since the conversion efficiency of the TJ solar cell is better than the low-cost silicon solar cells, the hybrid CPV module configuration should be adjusted for better light concentration. When the height was zero (i.e., no SOE), we calculated powers of 590.8 W/m\({}^{2}\) and 131.2 W/m\({}^{2}\) on the TJ solar cell and the silicon solar cells, respectively. Although larger heights led to higher power values on the TJ solar cell, the irradiance within the lossy wavelength range increased faster than the irradiance within the effective wavelength range, as shown by the increasing E\({}_{\rm TJ,loss}\)/E\({}_{\rm TJ,eff}\) ratio. Moreover, since taller SOEs increase the production cost and maximize shading, moderately tall SOEs are more desirable. While the irradiance on the silicon solar cells was 127.4 W/m\({}^{2}\) for h = 15 mm, the irradiances on the silicon solar cells were 124.8 W/m\({}^{2}\) and 85.4 W/m\({}^{2}\) for h = 20 and 100 mm, respectively, due to the shading on the SOEs. Therefore, we focused our subsequent simulations and experimental measurements on the case of h = 15 mm and \(\theta=30^{\circ}\).  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**SOE Angle** & **E\({}_{\rm TJ,eff}\) (W/m\({}^{2}\))** & **E\({}_{\rm TJ,loss}\) (W/m\({}^{2}\))** & **E\({}_{\rm TJ,loss}\)/E\({}_{\rm TJ,eff}\) ratio (\%)** & **E\({}_{\rm Si,eff}\) (W/m\({}^{2}\))** & **E\({}_{\rm Si,loss}\) (W/m\({}^{2}\))** \\ \hline \(0^{\circ}\) (Perpendicular) & 326.8 & 7.2 & 2.2 & 135.8 & 7.3 \\ \(10^{\circ}\) & 495.6 & 15.1 & 3.1 & 130.6 & 7.1 \\ \(20^{\circ}\) & 588.6 & 24.1 & 4.1 & 129.8 & 13.5 \\ \(30^{\circ}\) & 606.2 & 34.5 & 5.7 & 127.4 & 9.9 \\ \(40^{\circ}\) & 604.7 & 41.7 & 6.9 & 125.8 & 10.2 \\ \(50^{\circ}\