## 4 Results and Discussion
### Kalman Filtered Forecasts of Solar Radiation
#### 4.3.2 KF_Kt Forecasts  
The comparison of the WRF model and KF_Kt forecasts of solar radiation under clear sky conditions is plotted in Figure 7. It presents hourly WRF raw forecasts (black line) and observed solar radiation at the target site (red line) as well as the KF_Kt predictions (blue line) for a period of 14 days. It is observed from Figure 7a that the WRF model demonstrates the capability of predicting the day-ahead solar radiation at the Maizuru site. Nevertheless, the KF_Kt can improve the raw prediction of the WRF model significantly, with reductions of MBE by 90.1% and RMSE by 43.4%. As expected, from Figure 7b, we also can see the positive impact of KF_Kt on the WRF raw solar prediction for the Qinghai site. Figure 7: The solar radiation (W/m\({}^{2}\)) series over 14 days under clear sky condition for Maizuru (**a**) and Qinghai sites (**b**). The solid black, blue and dotted red lines stand for WRF raw forecasts, KF_Kt forecasts and the corresponding observations, respectively. \begin{table}
\begin{tabular}{l l c c c c} \hline \hline
**Site** & **Case** & **MBE (W/m\({}^{2}\))** & **MAE (W/m\({}^{2}\))** & **RMSE (W/m\({}^{2}\))** & **nRMSE** \\ \hline \multirow{2}{*}{Maizuru} & Raw & 70.2 & 117.3 & 213.8 & 0.21 \\  & KF\_TS & 13.9 & 87.9 & 164.8 & 0.17 \\ \hline \multirow{2}{*}{Qinghai} & Raw & 27.1 & 97.6 & 183.4 & 0.08 \\  & KF\_TS & \(-\)10.0 & 79.4 & 158.1 & 0.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance statistics of hourly WRF raw and KF_TS forecasts of solar radiation, respectively. Based on the above discussions, we can primarily conclude that KF_TS and KF_Kt both can make the performance of WRF raw forecasts better. To help to illustrate the priority between those two, the comparison of MBE and RMSE under three sky conditions for both ways of conducting the Kalman filter, are plotted in Figures 8 and 9. Figure 8a depicts the MBE of the KF_TS and KF_Kt forecast concerning the WRF raw prediction for three sky conditions at the Maizuru site. It can be easily found that the WRF model overestimates the solar radiation under clear and cloudy sky conditions while it underestimates the solar radiation of an overcast day. The most massive MBE value of raw forecasts is found for clear sky conditions (124.8 W/m\({}^{2}\)), followed by the cloudy day. Both the KF_TS and KF_Kt can mainly alleviate the systematic error tendency in the raw forecasts of the solar radiation. In the case of clear sky conditions, the reduction of MBE from the KF_TS correction around 54.5%, while that of the KF_Kt is 95.7%, which indicates that KF_Kt performs significantly better than KF_TS under clear sky conditions in removing the systematical errors. Nearly the same positive effects for both KF_TS and KF_Kt can be found for cloudy sky conditions. Note that the Kalman filter without considering the clearness index hardly corrects the systematical errors of solar radiation forecasts under overcast sky conditions. Figure 8b displays the results of RMSE for the Maizuru site. Similar to MBE, the largest value of RMSE (247.6 W/m\({}^{2}\)) and meanwhile the smallest value (130.0 W/m\({}^{2}\)) are found for clear and overcast sky conditions, respectively. As expected, for clear sky conditions, the KF_Kt reduces RMSE by 42.2%, which is more significant compared to that of KF_TS forecasts (23.4%). The improvements in RMSE of both KF_TS and KF_Kt are around 12.0% for the cloudy days. KF_Kt can improve the RMSE under overcast sky conditions by 3.2%; however, a decrease of 7.1% can be found for KF_TS. This means that neither Kalman filter nor clearness index-based Kalman filter can improve the day-ahead solar radiation largely under overcast conditions, due maybe to the complexity of WRF outputs. The characteristics of MBE and RMSE for the Qinghai site are summarized in Figure 9. The significant advantage of KF_Kt against KF_TS for reducing the systematical errors can be found in Figure 9a. The corrections of MBE by KF_Kt are all around 90% for three sky conditions.