# Intelligent Fault Diagnosis Techniques Applied to an Offshore Wind Turbine System
## 4 Performance and Robustness Analysis
### Simulation Results
In order to perform the fault detection task, the diagnostic residuals \(r_{i}(k)=\hat{f}_{i}(k)\) were compared according to the threshold logic of Equation (4). The parameter \(\delta\) has to be selected in order to optimize the fault diagnosis performance: for example, in terms of missed faults and false alarm rates [22]. Table 6 summarizes the values of this parameter for each fault estimator \(i\). In the following, the simulation results are reported, particularly for Fault Cases 1, 4, 8, and 9. The estimated faults \(\hat{f}_{i}\) depicted in Figure 5 demonstrate that the fault detection task was achieved, as they exceeded the threshold levels only when the corresponding fault was active, as reported in Table 4.  
Figure 5 depicts the reconstructed fault functions \(\hat{f}_{i}(k)\) generated by the fuzzy estimators in faulty conditions (black continuous line) with respect to the fault-free residuals (gray line). The fixed thresholds of Equation (4) are depicted by dotted lines. It is worth noting that in fault-free conditions, the estimated fault functions \(\hat{f}_{i}(k)\) are not zero due to the model-reality mismatch and the measurement error. The results also highlight the robustness and reliability characteristics of the developed fault diagnosis technique, which relies on the proposed fuzzy tool. For the fuzzy systems, nine NARX NN models were designed according to the scheme in Figure 3. The NN structure selected in this study consisted of 3 layers, with 3 neurons in the input layer, 8 in the hidden one, and 1 neuron in the output layer. Furthermore, in this case, a trial and error procedure was used to determine the optimal number of delays \(d_{u}\) and \(d_{y}\), as well as the number of neurons, that led \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Residual \(r_{i}(k)\)** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline \(\delta\) & 3.8 & 4.3 & 4.2 & 4.5 & 3.7 & 4.4 & 4.3 & 3.5 & 3.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Optimal value of the parameter \(\delta\). Figure 5: The estimated faults \(\hat{f}_{i}\) for Cases 1, 4, 8, and 9. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Fault Estimator \(i\)** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline
**RMSE** & 0.016 & 0.023 & 0.021 & 0.020 & 0.019 & 0.021 & 0.017 & 0.021 & 0.019 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Fault estimator performance in terms of RMSE. to the minimization of the fault estimation error. In particular, \(d_{u}=d_{y}=4\) delays were selected in the relation of Equation (9). According to Table 3 and Figure 3, the NN models have three inputs. The prediction capabilities of the neural fault estimators are summarized in Table 7, which reports the values of the RMSEs obtained by comparing the estimated faults with the simulated ones. Furthermore, in this case, the fault detection task was achieved by comparing the residuals \(r_{i}=\hat{f}_{i}(k)\) from the neural fault estimators with the optimized thresholds of Equation (4). The values of the parameter \(\delta\) are reported in Table 8.  
As an example, with reference to Fault Cases 1, 2, 3, and 4, Figure 6 depicts the residuals \(\hat{f}_{i}(k)\) generated in faulty conditions by the NN estimators (continuous line) compared with the fixed thresholds (dashed line). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Fault Estimator \(i\)** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline
**RMSE** & 0.009 & 0.009 & 0.009 & 0.012 & 0.011 & 0.011 & 0.009 & 0.009 & 0.014 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: NN performance in terms of RMSE. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Residual \(r_{i}(k)\)** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline \(\delta\) & 4.2 & 4.9 & 4.7 & 5.1 & 4.2 & 4.6 & 4.8 & 4.1 & 4.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: \(\delta\) values for the threshold logic.