# Fuzzy and Neural Network Approaches to Wind Turbine Fault Diagnosis
## 2 Wind Turbine Benchmark
### Fault Sensitivity Analysis  
The paper proposes to exploit this tool, which was suggested earlier by the authors for a different application, see e.g., [27], as it simplifies the design of the bank of fault estimators. Moreover, it enhances the identification of the dynamic FS and NN prototypes recalled in Section 3.1. This analysis must be preliminarily performed on the WT simulator. In particular, as already remarked, it is used to select the input and output measurements \(u_{j}(k)\) and \(y_{l}(k)\) of the process that feed the dynamic FS or NN of the bank of Figure 2.  
In practice, the faults considered in the WT benchmark have been injected into the simulator, assuming that only a single fault may occur. Then, the Relative Mean Square Errors (RMSE) index computed by considering the fault-free and faulty measured signals is computed, so that, for each fault, the most sensitive signal \(u_{j}(k)\) and \(y_{l}(k)\) is determined. \begin{table}
\begin{tabular}{c c c c} \hline \hline Fault \# & 1 & 2 & 3 \\ \hline Typology & Fixed value & Scaling error & Fixed value \\ Sensor \# & Blade 1 & Blade 2 & Blade 1 \\ \hline Fault \# & 4 & 5 & 6 \\ \hline Typology & Fixed value & Scaling error & Dynamics \\ Sensor \# & Pitch 1 & Generator 2 & Actuator 2 \\ \hline Fault \# & 7 & 8 & 9 \\ \hline Typology & Dynamics & Fixed value & Dynamics \\ Sensor \# & Pitch 3 & Converter & Drive-train \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Turbine benchmark model fault scenarios. In particular, the fault sensitivity analysis relies on a selection algorithm using the normalised sensitivity function \(N_{x}\) in the form of Equation (1):  
\[N_{x}=\frac{S_{x}}{S_{x}^{*}} \tag{1}\]  
with:  
\[S_{x}=\frac{\left\|x_{f}(k)-x_{n}(k)\right\|_{2}}{\left\|x_{n}(k)\right\|_{2}} \tag{2}\]  
and:  
\[S_{x}^{*}=\max\frac{\left\|x_{f}(k)-x_{n}(k)\right\|_{2}}{\left\|x_{n}(k) \right\|_{2}} \tag{3}\]  
The factor \(N_{x}\) represents the effect of the considered fault case on the generic measured signal \(x(k)\), with \(k=1\), \(2\),..., \(N\) its sample number. The subscripts '\(f\)' and '\(n\)' indicate the faulty and the fault-free case, respectively. Therefore, the signals mainly affected by the considered fault generate a value of \(N_{x}\) equal to 1. On the other hand, values of \(N_{x}\) closer to zero indicate that \(x(k)\) is not affected by the fault. Those signals corresponding to significantly higher values of \(N_{x}\) are thus selected as the most sensitive measurements to the fault cases, and will be used to feed the residual generator modules of the bank reported in Figure 2.  
As already remarked, the WT benchmark is able to generate different typical fault cases affecting the sensors, the actuators and the process components. This scenario with the 9 faults is illustrated by means of Table 3, which reports the input and output measurements acquired from the WT process signals and mainly affected by these faults. Table 3 thus summarises the results of this fault sensitivity analysis for the case of the WT simulator. For FDI purpose, the WT benchmark of Figure 1 can be described as a nonlinear continuous-time dynamic model represented by the function \(\mathbf{f}_{\text{art}}\) of Equation (4) with state vector \(\mathbf{x}_{\text{art}}\) and fed by the driving input vector \(\mathbf{u}\):  
\[\left\{ \begin{array}{lcl}\dot{\mathbf{x}}_{\text{art}}(t)&=&\mathbf{f}_{ \text{art}}(\mathbf{x}_{\text{art}},\mathbf{u}(t))\\ \mathbf{y}(t)&=&\mathbf{x}_{\text{art}}(t)\end{array} \right. \tag{4}\]  
Equation (4) highlights that the WT simulator allows to measure the state vector signals, i.e., the rotor speed, the generator speed and the generated power of the WT process:  
\[\mathbf{x}_{\text{art}}(t)=\mathbf{y}(t)=\left[\omega_{g,m1},\,\omega_{g,m2}, \,\omega_{r,m1},\,\omega_{r,m2},\,P_{g,m}\right]\]  
The driving input vector is represented by the following signals:  
\[\mathbf{u}(t)=\left[\beta_{1,m1},\,\beta_{1,m2},\,\beta_{2,m1},\,\beta_{2,m2}, \,\beta_{3,m1},\,\beta_{3,m2},\,\tau_{g,m}\right]\]