## 4 Discussion  
In previous work, we have shown that super-multi-junction solar cells could solve low annual performance of concentrator photovoltaic systems affected by mismatching loss due to solar spectrum variation. Spectrum influence equally affects non-concentrating solar cells. However, the impact of the spectrum variation for non-concentrating applications needs to consider complex phenomena of direct, scattered, and reflected spectrum combined with angular effect. It was not appropriate to expand the model to non-concentrating applications. We then tried to develop annual modeling performance of multi-junction solar cells, considering the spectrum (climate pattern, atmospheric parameters, sun-angle, airmass). The spectrum-enhanced performance model of multi-junction solar cells successfully explained the strange annual performance. Then, we combined this model with previous work on optimization of the bandgap energy using the Monte Carlo method. The previous works on optimization and sensitivity to spectrum change relied on the distribution of atmospheric parameters, especially those of the worst case. This method \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Bandgap Energy (eV) from Top to Bottom Junction**} \\ \hline
**2J** & \(1.72\pm 0.03\) & \(1.12\pm 0.02\) & & & & & \\ \hline
**3J** & \(1.89\pm 0.05\) & \(1.33\pm 0.07\) & \(0.89\pm 0.08\) & & & & \\ \hline
**4J** & \(1.99\pm 0.07\) & \(1.47\pm 0.07\) & \(1.07\pm 0.09\) & \(0.73\pm 0.11\) & & & & \\ \hline
**5J** & \(2.11\pm 0.09\) & \(1.63\pm 0.07\) & \(1.27\pm 0.09\) & \(0.97\pm 0.08\) & \(0.72\pm 0.10\) & & & \\ \hline
**6J** & \(2.08\pm 0.15\) & \(1.68\pm 0.11\) & \(1.34\pm 0.11\) & \(1.07\pm 0.11\) & \(0.84\pm 0.11\) & \(0.66\pm 0.11\) & & \\ \hline
**7J** & \(2.17\pm 0.16\) & \(1.80\pm 0.11\) & \(1.48\pm 0.10\) & \(1.21\pm 0.12\) & \(0.99\pm 0.11\) & \(0.77\pm 0.12\) & \(0.62\pm 0.12\) & & \\ \hline
**8J** & \(2.19\pm 0.16\) & \(1.84\pm 0.09\) & \(1.53\pm 0.11\) & \(1.28\pm 0.10\) & \(1.05\pm 0.10\) & \(0.86\pm 0.09\) & \(0.67\pm 0.09\) & \(0.55\pm 0.09\) & \\ \hline
**9J** & \(2.25\pm 0.19\) & \(1.88\pm 0.13\) & \(1.61\pm 0.12\) & \(1.37\pm 0.11\) & \(1.13\pm 0.10\) & \(0.95\pm 0.10\) & \(0.70\pm 0.10\) & \(0.62\pm 0.08\) & \(0.52\pm 0.08\) & \\ \hline
**10** & \(2.21\pm 0.21\) & \(1.89\pm 0.14\) & \(1.63\pm 0.14\) & \(1.40\pm 0.12\) & \(1.19\pm 0.14\) & \(1.00\pm 0.11\) & \(0.82\pm 0.12\) & \(0.66\pm 0.10\) & \(0.55\pm 0.09\) & \(0.46\pm 0.09\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: List of the set of the bandgap of the super-multi-junction solar cell. Figure 13: Distribution of the bandgap energy of the optimized (to the spectrum and other climate conditions given by random numbers according to Figure 2) multi-junction solar cells under the modeled fluctuation in the climate in Miyazaki, Japan (N 31.83\({}^{\circ}\), E 131.42\({}^{\circ}\)). This is an example of 10 junctions. Please note that the histogram of the calculated optimized bandgap energy in each junction is normalized so that the integral of the range becomes unity. Also, note that the overlap of each peak does not mean that the higher bandgap junction has lower bandgap energy than that of the lower peak. It is constrained that the bandgap structure was equivalently modeled by allowing the bandgap energy of the (\(i+1\))th junction to be equal or greater than that of the (\(j\))th junction, but not allowing the bandgap energy of the (\(i+1\))th junction to be less than that of the (\(j\))th junction. The \(y\)-axis is normalized so that the integration of the distribution becomes unity: (**a**) Normal multi-junction solar cell; (**b**) Super-multi-junction solar cell. was too simple to describe the real fluctuation of the spectrum. For example, aerosol density and water precipitation had a distinct seasonal change that correlates with sun height and climate trends.