## 4 Case Analysis  
### Case Analysis of Wind-Power Probability Density Prediction  
The historical measured data of a wind farm in China in July 2017 is selected as the sample for short-term wind-power probability density prediction. The rated capacity of the wind farm is 150 MW, the data sampling interval is 10 min, and a total of 4464 pieces of data are used, as shown in Figure 4.  
Figure 3: Solution flow of closing angle control model based on NSGA-II.  
The dataset is divided. The data of the previous 30 days comprise the training set, and the data of the 31 st day comprise the test set, with an interval of 10 min at each time point. An LSTM network with three hidden layers is built through the Keras deep learning framework. The number of neurons in the hidden layer is 32. The optimization algorithm uses adaptive time estimation (Adam). The maximum number of iterations for the LSTM network under each quantile is set to be 100, the initial learning rate is 0.005, the penalty parameter is 1, the kernel function of the quantile regression parameter is Gaussian function, the quantile is \(\tau\in(0,1)\), and the interval is 0.1. The training process of the LSTM network is shown in Figure 5 (taking \(\tau=0.5\) as an example).  
Figure 4: Historical data of wind power.  
Figure 5: Training convergence process of LSTM network.  
According to Figure 5, after 60 training rounds are completed, the root mean square error (RMSE Blue data in the figure) and loss values of the mean square error (MSE Orange data in the figure) tend to be stable. According to the above modeling conditions, run the established model to obtain the wind-power output prediction results of the test set data under the 90% confidence interval, as shown in Figure 6.  
As shown in Figure 6, when the wind power fluctuates greatly, the model can also show good fitting ability. The evaluation index is used to evaluate the model prediction results. The interval coverage rate of the model prediction is 91.67%, and the average interval width is 22.64. Most of the actual values can fall within the confidence interval, which can meet the prediction requirements.  
The probability density function curves at different time points in the prediction period are obtained by kernel density estimation. The probability density function curves at 6 time points of 4, 12, 18, 24, 27, and 34; these are randomly selected, as shown in Figure 7.  
As shown in Figure 7, from the predicted probability density function, the real values fall on both sides near the middle of the density function, which can better explain the probability density of wind-power output at the predicted time point in the future.  
To further cope with the uncertainty of wind-power output, the wind-power output interval at the prediction time is reasonably divided. As shown in Figure 8, taking the probability density function of the 25 th prediction point as an example, the average value in the interval is taken as the output value of wind power, and the area of each interval in the total area of the selected area is taken as the probability of the output scenario, forming multiple scenarios of wind-power output at that time. See Table 2 for specific values.  
Figure 6: Forecast results of wind power interval at 36 time points in the future.  
Figure 7: The probability density function of wind power at different prediction time points.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Wind Power (MW)** & **Probability** \\ \hline
96 & 0.1 \\
99 & 0.2 \\
103 & 0.52 \\
107 & 0.08 \\
116 & 0.1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Wind-power output value and corresponding probability.