# Fuzzy and Neural Network Approaches to Wind Turbine Fault Diagnosis
## 4 Simulation Results, Experimental Validation, Comparisons and Discussion  
With reference to the WT benchmark of Section 2, the simulations are driven by different wind sequences generated in a random way. They represent real measurements of wind speed sequences from typical WT operating conditions, with ranges varying from 5 m/s to 20 m/s. This scenario was modified by the authors with respect to the benchmark earlier proposed in [20]. The simulations consist of 4400 s, with single fault occurrences and a number of samples \(N=\) 440,000 for a sampling frequency of 100 Hz. Almost all fault signals are modelled as step functions lasting for 100 s with different commencing times. Further details can be found in [20,29]. The first part of this section reports the results achieved by means of the fuzzy prototypes used as fault reconstructors according to Section 3.1. In particular, the fuzzy \(c\)-means and the GK clustering algorithms were exploited. A number of clusters \(n_{C}=4\) and a number of delays \(o=4\) were estimated. The membership functions of the TS FS and the parameters of the consequents \(\alpha_{j}^{(i)}\) and \(\delta_{j}^{(i)}\) were derived for each cluster by following the procedure developed by the same authors in [30]. The TS FSs of Equation (6) were thus determined and 9 fault reconstructors were organised according to the scheme of Figure 2.  
The performances of the 9 TS FSs when used as fault estimators were evaluated again according to the RMSE index, computed as the difference between the reconstructed \(\hat{f}(k)\) and the actual \(f(k)\) signals for each of the fuzzy estimators. These values were reported in Table 4.  
Indeed, the RMSE values reported in Table 4 represent an average of the results obtained from a campaign of 1000 simulations, as the benchmark exploited in this work changes the parameters of the WT model at each run. Moreover, the model-reality mismatch, the measurement errors, uncertainty and disturbance effects are described as Gaussian processes with suitable distributions, as remarked in Section 2. Therefore, Table 4 reports also the values of the standard deviation (std. dev.) of the estimation errors achieved by the FS fault estimators. Note that these reconstructed signals \(\hat{f}(k)\) can be directly used as diagnostic residuals in order to detect and isolate the faults affecting the WT. Moreover, each TS FS of Equation (6) is fed by 3 inputs (according to Table 3), with a number of delayed inputs and outputs \(o=3\) and \(n_{C}=4\) clusters. As an example, Figure 5 shows the results regarding the fault cases 1, 2, 3, and 4 of the WT plant recalled in Section 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Fault Case** & **1** & **2** & **3** & **4** & **5** \\ \hline RMSE\% & 1.61\% & 2.22\% & 1.95\% & 1.87\% & 1.92\% \\ Sdt. Dev. & \(\pm\)0.02\% & \(\pm\)0.03\% & \(\pm\)0.01\% & \(\pm\)0.01\% & \(\pm\)0.01\% \\ \hline
**Fault Case** & **6** & **7** & **8** & **9** & \\ \hline RMSE\% & 2.15\% & 1.76\% & 2.13\% & 1.98\% & \\ Sdt. Dev. & \(\pm\)0.02\% & \(\pm\)0.01\% & \(\pm\)0.02\% & \(\pm\)0.01\% & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: FS fault estimator capabilities. In particular, Figure 5 reports the estimated faults \(\hat{f}(k)=r_{i}(k)\) provided by the FSs in faulty conditions (black continuous line). They are compared with respect to the actual faults (grey dashed line). The fault detection task can be easily accomplished as the residuals are significantly different from zero in faulty conditions. Note also that the reconstructed fault functions \(\hat{f}(k)=r_{i}(k)\) are different from zero also in fault-free conditions due to the measurement errors and the model-reality mismatch. This aspect serves to highlight the accuracy of the reconstructed signals provided by the estimated fuzzy models. As for the FSs, 9 NARX NNs summarised in Section 3.1 were trained to provide the reconstruction of the 9 faults affecting the WT plant.