## 3 Impact of Yaw Operation on Wind Turbine Blades
### Prediction of the Blade Loads during Yaw Operation
Then the mathematical models of \(T_{1}\) and \(T_{2}\) are established using multiple regression analysis. As opposed to the linear regression analysis, nonlinear regression analysis is more informative. Therefore, a quadratic regression analysis technique is employed here to construct the models. According to the principle of statistical analysis, a complete quadratic regression equation including constant term, linear terms, cross-product terms, and square terms can be expressed as  
\[\left\{ \begin{array}{l}T_{1}=c_{0}+\sum\limits_{i=1}^{4}c_{i}x_{i}+\sum \limits_{i=1}^{3}\sum\limits_{j=i+1}^{4}c_{i}x_{i}x_{j}+\sum\limits_{i=1}^{4}c _{i}x_{i}^{2}\\ T_{2}=d_{0}+\sum\limits_{i=1}^{4}d_{i}x_{i}+\sum\limits_{i=1}^{3}\sum\limits_{j =i+1}^{4}d_{i}x_{i}x_{j}+\sum\limits_{i=1}^{4}d_{i}x_{i}^{2}\end{array} \right. \tag{1}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Level** & \(\gamma\)/\({}^{o}\) & \(\
u\)/(m/s) & \(\omega\)/(s/min) & \(\beta\)/\({}^{o}\) \\ \hline
1 & 0 & 3 & 3 & 0 \\
2 & 4 & 7 & 7 & 3 \\
3 & 8 & 11 & 11 & 6 \\
4 & 12 & 13 & 15 & 9 \\
5 & 16 & 15 & 17 & 12 \\
6 & 20 & 19 & 19 & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Factor levels for the orthogonal experiments. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Experiment** & **Level of** & **Level of** & **Level of** & **Level of** & **Tj/(kN-m)** & \(T_{1}\)/(kN-m) \\ \hline
1 & 1 & 1 & 1 & 1 & 14.13 & 92.404 \\
2 & 1 & 2 & 2 & 2 & 116.052 & 515.703 \\
3 & 1 & 3 & 3 & 3 & 295.077 & 1116.128 \\
4 & 2 & 1 & 2 & 3 & \(-\)26.301 & \(-\)11.035 \\
5 & 2 & 2 & 3 & 1 & 145.982 & 1036.561 \\
6 & 2 & 3 & 1 & 2 & 58.571 & 273.8 \\
7 & 3 & 1 & 3 & 2 & \(-\)54.02 & 26.203 \\
8 & 3 & 2 & 1 & 3 & 33.573 & 139.406 \\
9 & 3 & 3 & 2 & 1 & 78.193 & 598.994 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The first set of orthogonal experiments. where \(c_{ij},d_{ij}\) are the coefficients to be determined. Then, together with the standard error, the ratio of regression coefficient \(t\) can be readily obtained using the least-squares method. The probability of sample observation (or extreme result when the original hypothesis is true) \(P\) is calculated by substituting the sample data into the test statistic formula when H0 (also called invalid hypothesis) is assumed to be true. The smaller the value of P, the more likely that the test statistic calculated from the sample point will fall into the rejection domain. Generally, if the value of \(P\) is less than 0.05, it is considered that it falls into the rejection domain, and the small probability event occurs. Then, the original assumption can be rejected. The quadratic regression parameter estimates for \(T_{1}\) and \(T_{2}\) are listed in Tables 4 and 5, respectively. Substituting the values of these parameters into (1) gives  
\[ \begin{array}{l}T_{1}=-109.989+3.1879x_{1}-18.7357x_{2}+31.5959x_{3}+35.5136x _{4}-0.4873x_{1}x_{2}-0.1132x_{1}x_{3}\\ +0.2654x_{1}x_{4}+3.5584x_{2}x_{3}+1.4949x_{2}x_{4}.3281x_{3}x_{4}-0.66x_{1}^{2 }+0.4057x_{2}^{2}-2.1032x_{3}^{2}-1.7072x_{3}^{2}\\ T_{2}=-539.501+5.6244x_{1}+6.407x_{2}+95.956x_{3}+65.6491x_{4}-1.2143x_{1}x_{2}- 0.7268x_{1}x_{3}\\ +1.2093x_{1}x_{4}+9.5762x_{2}x_{3}-2.1539x_{2}x_{4}-1.7124x_{3}x_{4}-0.1948x_{1 }^{2}+0.8518x_{2}^{2}-2.8406x_{3}^{2}+0.8189x_{4}^{2}\end{array}  \tag{2}\]