## 4 Case Studies of Real Wind Power Data
### Experimental Results
#### 4.2.3 Distributional Forecast  
The continuous rank probability score (CRPS) is used to evaluate the performance of the proposed distributional forecasts. The CRPS is defined as:  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **30 min** & **40 min** & **50 min** & **60 min** \\ \hline AR & 6.991 & 8.871 & 11.883 & 11.996 \\ MC & 6.592 & 8.426 & 10.654 & 11.091 \\ SVM-MC & 6.591 & 8.425 & 10.654 & 11.091 \\ NEAT & 7.255 & 8.379 & 10.366 & 10.274 \\ SNEAT & 7.427 & 8.612 & 10.471 & 10.385 \\ LSTM & 7.025 & 9.255 & 11.558 & 10.915 \\ SSEN & 7.092 & 8.182 & 9.727 & 9.849 \\ ANN & 7.109 & 8.087 & 9.384 & 9.627 \\ DSN & **5.420** & **5.595** & **7.023** & **6.197** \\ \hline \hline \end{tabular}
\end{table}
Table 4: MAE of different models at different forecasting horizons over ramp events of the year 2010.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **30 min** & **40 min** & **50 min** & **60 min** \\ \hline AR & 4.837 & 6.516 & 8.160 & 9.624 \\ MC & 4.733 & 6.233 & 7.551 & 8.727 \\ SVM-MC & 4.733 & 6.233 & 7.550 & 8.727 \\ NEAT & 4.804 & 5.851 & 6.939 & 7.640 \\ SNEAT & 5.064 & 6.322 & 7.681 & 8.277 \\ LSTM & 4.664 & 6.517 & 7.681 & 8.257 \\ SSEN & 4.852 & 5.970 & 7.095 & 7.862 \\ ANN & 4.755 & 5.846 & 6.580 & 7.491 \\ DSN & **3.755** & **4.220** & **4.746** & **5.069** \\ \hline \hline \end{tabular}
\end{table}
Table 3: MAE of different models at different forecasting horizons over the whole year 2010.  
\[\text{CRPS}=\frac{1}{N_{t}}\sum_{t}\int_{0}^{p_{eg}^{max}}(\hat{F}_{t}(x)-U(x-P_{ gg}(t)))dx \tag{20}\]  
where \(\hat{F}_{t}(x)\) is the cumulative density function (cdf) obtained by using the distributional forecast. In addition, \(U(.)\) is a unit step function that equals to 1 if \(x>P_{gg}(t)\) and 0 otherwise. Generally, the lower the CRPS, the more accurate the distributional forecast is. In Tables 5 and 6, we compare the forecast of different NN-based models under different horizons using the whole year data and ramp events in the year 2010, respectively. The results of the non-NN models can be found in [37]. We observe that our model performs much better than other benchmarks for longer prediction horizons (normally longer than 30 min) where the wind ramps are large, while the performance is similar for the 10 min forecast. This indicates the superior performance of the proposed method on handling the uncertainty of the wind.  
For the 30 min ahead forecast, compared with the NN-based benchmarks (NEAT, SNEAT, LSTM, ANN, and SSEN), the proposed approach improves the CRPS at least 24.9% for the whole year data and at least 21.7% for the ramp events, respectively.  
For the 40 min ahead forecast, compared with the NN-based benchmarks (NEAT, SNEAT, LSTM, ANN, and SSEN), the proposed approach improves the CRPS at least 26% for the whole year data and at least 30% for the ramp events, respectively.  
For the 50 min ahead forecast, compared with the NN-based benchmarks (NEAT, SNEAT, LSTM, ANN, and SSEN), the proposed approach improves the CRPS at least 33.3% for the whole year data and at least 38.8% for the ramp events, respectively.  
For the 60 min ahead forecast, compared with the NN-based benchmarks (NEAT, SNEAT, LSTM, ANN, and SSEN), the proposed approach improves the CRPS at least 35.8% for the whole year data and at least 35.2% for the ramp events, respectively.