# Intelligent Fault Diagnosis Techniques Applied to an Offshore Wind Turbine System
## 3 Fault Diagnosis Techniques: Fuzzy Systems and Neural Networks
### Fault Sensitivity Analysis  
The design of the fault diagnosis schemes proposed in this paper and represented in Figure 3 is enhanced by the tool presented here. It consists of a fault sensitivity analysis that is performed on the measurements acquired from the wind turbine simulator. The procedure aims to define the most sensitive measurements \(u_{j}(k)\) and \(y_{l}(k)\) with respect to the general fault \(f_{i}(k)\) considered in Section 2.  
According to the assumption of Equation (2), the considered fault signals \(f_{i}(k)\) have been injected into the wind turbine simulator, and only single faults may occur. Then, the Relative-Mean-Squared Errors (RMSEs) between the fault-free and faulty signals acquired from the simulator are computed. In this way, the most sensitive signals \(u_{j}(k)\) and \(y_{l}(k)\) are selected for each fault \(i\). The achieved results are summarized in Table 2.  
In particular, the fault sensitivity analysis follows the selection algorithm, which relies on the normalized sensitivity function \(N_{x}\) of Equation (10),  
\[N_{x}=\frac{S_{x}}{S_{x}^{*}} \tag{10}\]  
with:  
\[S_{x}=\frac{\left\|x_{f}(k)-x_{n}(k)\right\|_{2}}{\left\|x_{n}(k)\right\|_{2}} \tag{11}\]  
and:  
\[S_{x}^{*}=\max\frac{\left\|x_{f}(k)-x_{n}(k)\right\|_{2}}{\left\|x_{n}(k) \right\|_{2}} \tag{12}\]  
In fact, \(N_{x}\) represents the effect of the considered fault case with respect to the measured signal \(x(k)\), with \(k=1\), \(2\),..., \(N\). The subscripts \"\(f\)\" and \"\(n\)\" indicate the faulty and the fault-free cases, respectively. Therefore, the measurement that is most affected by the considered fault is the value of \(N_{x}\), which, in this case, is equal to one. Otherwise, smaller values of \(N_{x}\) indicate that \(x(k)\) is not affected by that fault.  
The complete results of the fault sensitivity analysis are summarized in Table 3.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Fault \(f_{i}\)** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline
**Measurements \(\mathbf{u}_{j}\), \(\mathbf{y}_{l}\)** & \(\beta_{1,m1}\) & \(\beta_{2,m2}\) & \(\beta_{3,m1}\) & \(\omega_{r,m1}\) & \(\omega_{r,m1}\) & \(\beta_{2,m1}\) & \(\beta_{3,m2}\) & \(\tau_{g,m}\) & \(\omega_{g,m1}\) \\
**RMSE** & 11.29 & 0.98 & 2.48 & 1.44 & 1.45 & 0.80 & 0.73 & 0.84 & 0.77 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The most sensitive measurements \(u_{j}(k)\) and \(y_{l}(k)\) and their RMSE values with respect to the fault \(f_{i}(k)\).  
This method represents a key feature of the proposed approach to fault diagnosis. In fact, the fault estimators of the bank of Figure 3 are designed by exploiting a reduced number of input signals \(u_{j}(k)\) and \(y_{l}(k)\). It also leads to a noteworthy simplification of the complexity and the computational cost of the identification and training phases of the fuzzy and NN models, respectively.  
Note finally that the fault sensitivity analysis was performed by considering one fault at a time. The case of multiple faults was not considered here, as the wind turbine benchmark simulates the occurrence of single faults only, as described in [4, 7]. However, the case of multiple faults occurring at the same time could be considered, even if a different fault sensitivity analysis has to be executed.