## 2 Materials and Methods
### Data Relationship Modeling of the Wind Turbine Operating State
#### 2.1.2 Sliding Window Model  
The wind turbine SCADA system data are a data sequence arranged in chronological order:  
\[D=\{(X_{1},t_{1}),\ (X_{2},t_{2}),\ \ldots,\ (X_{i},t_{i}),\ \ldots\} \tag{1}\]  
where \(X_{i}\) is the dataset recorded at time \(t_{i}\) (\(i\) = 1, 2, 3, \(\ldots\) ). The SCADA sampling frequency \(\tau\) is 1/(\(t_{i}-t_{i-1}\)), and:  
\[X_{i}=[\begin{array}{cccccc}x_{i1}&x_{i2}&\ldots&x_{ij}&\ldots&]\end{array} \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & **Time (hh:mm)** & **Wind Speed (m/s)** & **Rotor Speed (rpm)** & **Power (kW)** \\ \hline
1 & 14:20 & 5.9 & 12.82 & \(\ldots\) & 434 \\
2 & 14:20 & 6.0 & 12.72 & \(\ldots\) & 432 \\
3 & 14:20 & 6.1 & 12.72 & \(\ldots\) & 435 \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \hline \end{tabular}
\end{table}
Table 1: Raw SCADA data.  
where \(x_{ij}\) is the \(j\)th wind turbine state parameter in \(X_{i}\) (\(j=1,\,2,\,3,\,\dots\) ), such as wind speed, rotation speed, and power. During wind turbine operation, the SCADA system creates and stores a stream of data for later analysis.  
To evaluate the real-time running status of wind turbines, a sliding window model based on time is used to process the data stream, and the status online identification is determined by updating the SCADA data in the window in real time, as shown in Figure 1. The data recording frequency is defined by \(\tau\), the data length contained in the window by \(h\), the time length by \(h\tau\), the increment by \(q\), and the time increment by \(q\tau\). Thus, the SCADA data processed at time \(t_{k}\) (\(k>h\)) is a data matrix recorded from time \(t_{k-h}\) to \(t_{k}\) (i.e., sliding window data \(D_{k}\)):  
\[\begin{split} D_{k}=\left[\begin{array}{cccc}(X_{k-h},t_{k-h})& (X_{k-h+1},t_{k-h+1})&\dots&(X_{k},t_{k})\\ =\left[\begin{array}{cccc}(x_{k-h,1},t_{k-h})&(x_{k-h+1,1},t_{k-h+1})&\dots &(x_{k,1},t_{k})\\ (x_{k-h,2},t_{k-h})&(x_{k-h+1,2},t_{k-h+1})&\dots&(x_{k,2},t_{k})\\ \dots&\dots&\dots&\ddots\\ (x_{k-h,j},t_{k-h})&(x_{k-h+1,j},t_{k-h+1})&\dots&(x_{k,j},t_{k})\\ \end{array}\right]\end{split} \tag{3}\]  
where \(k\) is related to the defined window width \(h\) and increment \(q\), \(k=h+nq\); \(n\) is the number of time steps, and \(n=1,\,2,\,\dots\). When data processing finishes at time \(t_{k}\), both ends of the window move \(q\) along the positive time direction simultaneously, and the data matrix to be processed changes to \(D_{k+q}\).  
Changing the window width \(h\) adjusts the amount of data in the window, and changing the increment \(q\) adjusts the update frequency of the data in the window. The data scale corresponding to the window width \(h\) must ensure that the relationship between these data can accurately and stably describe the essential relationship, while the update frequency corresponding to the increment \(q\) must ensure that there is sufficient calculation processing time to determine the essential relationship between these data. In general, the larger the window width h is, the larger the corresponding data scale is, and the more accurate and stable the relational model obtained by data processing mining is. However, excessive window width \(h\) will lead to an increase in data processing time. Additionally, a smaller increment \(q\) yields a faster update frequency and a shorter time (period) between the two states, which means the online state identification has better real-time performance.