## 5 Results and Discussion
### Clustering with One Parameter  
The results for sunny and cloudy days at each location are indicated in Figure 6, and the corresponding centroid values are listed in Table 4. For example, the Golden (Colorado) location has sunny and cloudy days with centroid values of 0.69 and 0.38, respectively. If the clearness index is high, then the day is more likely to be sunny and vice versa. After classifying the days as sunny or cloudy based on the centroid value at each location, the LSTM-RNN, FFNN, SVM, and persistence models are each trained for cloudy and sunny days. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Location** & **Cloudy Days Centroid** & **Sunny Days Centroid** \\ \hline Golden & 0.38 & 0.69 \\ Jena & 0.24 & 0.55 \\ Basel & 0.34 & 0.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The centroid values for sunny and cloudy days at the study locations. The results obtained using the various models are presented and evaluated according to their RMSE, NRMSE, MAE, and NMAE values in Table 5. In addition, the average NRMSEs of the four models for the three geographical locations are indicated by the bar charts in Figure 7. Thus, it can be seen that the proposed LSTM model gives the best performance for all three locations, followed by the FFNN and SVM models, while the persistence model gives the least accurate forecasting results. Figure 6: The classification of days via k-means clustering for the study locations. In detail, the LSTM provides minimum and maximum average NRMSEs 4.15 and 5.93%, respectively, over both types of days. While the average NMAE error percentage is \(\sim\)2%. With respect to the three locations, all four models give their best performances for the data from Golden (Colorado). Overall, the proposed model gives a better performance on sunny days as compared to cloudy days. In addition, the SVM and FFNN giving closely comparable performances and low deviations. Therefore, since more time is required to train the FFNN than the SVM, the computational burden at the training stage can be reduced by selecting the SVM rather than the FFNN. The measured SI and the corresponding values for cloudy and sunny days predicted by the proposed and comparison models at the three locations are plotted graphically in Figure 8. The results for sunny days are presented in the left-hand column of the figure, while those for cloudy days are presented in the right-hand column. Thus, it can be seen that the persistence model gives the worst performance for both day types, while the LSTM gives better results than all other models. Comparatively, the results for cloudy days are less accurate than those for sunny days. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Location**} & \multirow{2}{*}{**Model**} & \multicolumn{3}{c}{**RMSE (W/m\({}^{2}\))**} & \multicolumn{3}{c}{**NRMSE (\%)**} & \multicolumn{3}{c}{**MAE (W/m\({}^{2}\))**} & \multicolumn{3}{c}{**NMAE (\%)**} \\ \cline{3-13}  & & **Sunny** & **Cloudy** & **Avg. ** & **Sunny** & **Cloudy** & **Avg. ** & **Sunny** & **Cloudy** & **Avg. ** & **Sunny** & **Cloudy** & **Avg. ** \\ \hline \multirow{4}{*}{Golden} & LSTM & 44.55 & 42.27 & 43.41 & 4.11 & 4.19 & 4.15 & 21.23 & 13.97 & 17.6 & 1.95 & 1.38 & 1.67 \\  & FFNN & 52.93 & 46.08 & 49.51 & 4.88  
The results of the proposed model for the three locations are also compared with those obtained using the conventional LSTM model in Table 6 and Figure 9. The conventional LSTM-RNN model is trained, validated, and tested for the whole dataset without classifying the days into sunny and cloudy. Therefore, it can be referred to as the all-sky model. As expected, these results indicate that the hybrid LSTM provides the better performance, with a minimum error of 4.15% for the Golden (Colorado) prediction and a maximum error of 5.93% for the Basel prediction. By contrast, the conventional LSTM gives errors of 5.56 and 7.98% at Golden and Basel, respectively.