# Intelligent Fault Diagnosis Techniques Applied to an Offshore Wind Turbine System
## 4 Performance and Robustness Analysis
### Simulation Results  
With reference to the wind turbine benchmark in Section 2, all simulations were driven by the same wind sequence \(v_{w}(t)\). It represents a real measurement of wind speed, from 5-20 m/s, with a few spikes at 25 m/s. Moreover, the rated power of the wind turbine is \(P_{r}=4.8\) MW, and the nominal generator speed is \(\omega_{nom}=162.5\) rad/s [7]. The simulations lasted for 4400 s with single fault occurrences. The measurements were acquired with a sampling frequency of 100 Hz, so \(N=440,\!000\) samples were generated for each run. Table 4 summarizes the wind turbine fault modes, as described in Section 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fault Case** & **Fault Type** & **Fault Shape** & **Occurrence (s)** \\ \hline
1 & actuator & step & 2000–2100 \\
2 & actuator & step & 2300–2400 \\
3 & actuator & step & 2600–2700 \\
4 & actuator & step & 1500–1600 \\
5 & actuator & step & 1000–1100 \\
6 & sensor & step & 2900–3000 \\
7 & sensor & trapezoidal & 3500–3600 \\
8 & sensor & step & 3800–3900 \\
9 & sensor & step & 4100–4300 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fault modes of the wind turbine simulator. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Fault Case \(f_{i}\)** & **Most Sensitive Inputs \(u_{j}\)** & **Most Sensitive Outputs \(y_{l}\)** \\ \hline
1 & \(\beta_{1,m1}\), \(\beta_{1,m2}\) & \(\omega_{g,m2}\) \\
2 & \(\beta_{1,m2}\), \(\beta_{2,m2}\) & \(\omega_{g,m2}\) \\
3 & \(\beta_{1,m2}\), \(\beta_{3,m1}\) & \(\omega_{g,m2}\) \\
4 & \(\beta_{1,m2}\) & \(\omega_{g,m2}\), \(\omega_{r,m1}\) \\
5 & \(\beta_{1,m2}\) & \(\omega_{g,m2}\), \(\omega_{r,m2}\) \\
6 & \(\beta_{1,m2}\), \(\beta_{2,m1}\) & \(\omega_{g,m2}\) \\
7 & \(\beta_{1,m2}\), \(\beta_{3,m2}\) & \(\omega_{g,m2}\) \\
8 & \(\beta_{1,m2}\), \(\tau_{g,m}\) & \(\omega_{g,m2}\) \\
9 & \(\beta_{1,m2}\) & \(\omega_{g,m1}\), \(\omega_{g,m2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The most sensitive measurements with respect to the considered fault scenario. Note that Fault Case 7 reported in Table 4 is modeled with a trapezoidal function, which is directly added to the corresponding output measurement according to the model in Equation (2). On the other hand, Fault Case 9 is generated as a step change of the parameters of the transfer function describing the drivetrain model. However, the effect of this fault on the output measurements is different from a step function. More details regarding the wind turbine fault scenario can be found in [4; 7]. As an example, in order to show different fault effects on process measurements, Figure 4 compares the results of the fault sensitivity test in terms of fault-free and faulty signals. In particular, Faults 1, 2, 3, and 8 are considered. When the FMID tool was applied to the data of the wind turbine simulator, \(n_{\mathrm{C}}=4\) clusters and \(o=3\) delays to input and output regressors of the TS fuzzy models were determined. This tool also provided the membership function points, which were fitted through Gaussian membership functions [14]. The optimal values of \(n_{\mathrm{C}}\) and \(o\) were determined in order to minimize the fuzzy model estimation errors. After data clustering, the regressands \(\alpha_{j}^{(i)}\) and \(\delta_{j}^{(i)}\) in Equation (8) were identified. The TS models in Equation (6) were thus implemented, and nine fault estimators were organized with the bank structure of Figure 3. Note that, according to Table 3, each fuzzy fault estimator in Equation (6) has three inputs. Therefore, each TS fuzzy model has a number of parameters equal to \((3+1)\times n=12\). The capabilities of the TS fuzzy estimators were assessed in terms of Root-Mean-Squared Error (RMSE), which is computed as the difference between the predicted \(\hat{f}_{i}(k)\) and the actual fault \(f_{i}(k)\), with \(i=1,\,\ldots,\,9\). Table 5 summarizes the achieved performance of the nine TS fuzzy fault estimators. Figure 4: The fault-free (gray line) signals with respect to the faulty ones (black line).