## 3 Results and Discussion
### Ultimate Load Analysis under Aerodynamic Imbalance  
Ultimate load analysis was performed under aerodynamic imbalance conditions for an individual pitch angle of each reference wind speed using the GH-BladedTM V4.1 The ultimate loads (i.e., maximum force (\(F_{x,y,pu}\), \(F_{y,pu}\), \(F_{z,pu}\)), and maximum moment (\(M_{xc}\), \(M_{yc}\), \(M_{zc}\)) were obtained through the ultimate load analysis. Using these results, resultant force and moment were calculated by Equations (3) and (4), respectively.  
\[F_{R,pu}=\sqrt{\left(F_{x,y,pu}\right)_{n1}^{2}+\left(F_{y,pu}\right)_{n2}^{2}+ \left(F_{z,pq}\right)_{n3}^{2}} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Steady Wind Speed (m/s)** & \(M_{Rc}\) (kNm)** & \(F_{Rc}\) (kN) & **Pitch Angles** \\ \hline
11 & 3677 & 1280 & 0\({}^{\circ}\) \\
14 & 4469 & 1227 & 6.3\({}^{\circ}\) \\
17 & 4516 & 1179 & 11.6\({}^{\circ}\) \\
20 & 4581 & 1154 & 15.9\({}^{\circ}\) \\
22 & 4632 & 1143 & 18.5\({}^{\circ}\) \\
24 & 4706 & 1133 & 20.8\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Ultimate loads and pitch angles for CPC.  
Figure 4: Coordinate systems for stationary hub loads.  
\[M_{R\_ipa}=\sqrt{\left(M_{x\_ipa}\right)_{n1}^{2}+\left(M_{y\_ipa}\right)_{n2}^{2}+ \left(M_{z\_ipa}\right)_{n3}^{2}} \tag{4}\]  
where, \((n1=0^{\circ},2^{\circ},4^{\circ},\cdot\cdot\cdot,30^{\circ})\), \((n2=0^{\circ},2^{\circ},4^{\circ},\cdot\cdot\cdot,30^{\circ})\), \((n3=0^{\circ},2^{\circ},4^{\circ},\cdot\cdot\cdot,30^{\circ})\)  
Figures 5 and 6 show the matrix of the pitch angles of a1 (blade) and a2 (blade 2) against a3 (blade 3) for each average wind speed for the resultant moment \(M_{R\_ipa}\) and the resultant force \(F_{R\_ipa}\), respectively. Here, each cell of the matrix represents a load distribution contour plot for the pitch angles of a1 (blade 1) and a2 (blade 2) against a3 (blade 3). Additionally, red dots indicate the minimum resultant moment and force. From these results, the ultimate load characteristics were identified for the individual pitch movement of each blade at specific wind speeds. For Figure 5 (i.e., resultant moment), referring to the vertical scale bar of each matrix, we can identify that the differences between the minimum resultant moment and the maximum resultant moment become more than four-fold as the difference of relative pitch angles between the blades increases. On the other hand, the difference in the load is smaller as the relative difference between the pitch angle gets smaller. However, we can also identify that the position of the minimum resultant moment (set of individual pitch angle) according to the increase in the wind speed does not change significantly. In detail, the pitch angles (set of individual pitch angle) of the a1 and a2 blades that generate the minimum resultant moment are not sensitive to the wind speed when the pitch angle of the a3 blade is constant. This behavior similarly appears in the resultant force in Figure 6. However, in the case of the resultant force, even though the relative difference between pitch angle of the blades is large, the difference between the minimum and the maximum resultant force is insignificant compared to the resultant moment. It is, therefore, reasonable to say that the influence of the resultant moment is dominant when aerodynamic imbalance occurs.  
Figure 5: Aerodynamic imbalance loads of \(M_{R\_ipa}\).