# Intelligent Fault Diagnosis Techniques Applied to an Offshore Wind Turbine System
## 3 Fault Diagnosis Techniques: Fuzzy Systems and Neural Networks  
In order to solve the fault diagnosis problem, this work assumes that the wind turbine system is affected by equivalent additive faults on the input and output measurements, as well as measurement errors, as described by the relations in Equation (2):  
\[\left\{ \begin{array}{lcl}\mathbf{u}(k)&=&\mathbf{u}^{*}(k)+\tilde{\mathbf{u} }(k)+\mathbf{f}_{u}(k)\\ \mathbf{y}(k)&=&\mathbf{y}^{*}(k)+\tilde{\mathbf{y}}(k)+\mathbf{f}_{y}(k) \end{array} \right. \tag{2}\]  
where \(\mathbf{u}^{*}(k)\) and \(\mathbf{y}^{*}(k)\) represent the actual process variables; \(\mathbf{u}(k)\) and \(\mathbf{y}(k)\) are the measurements acquired by the sensors; and \(\tilde{\mathbf{u}}(k)\) and \(\tilde{\mathbf{y}}(k)\) describe the measurement errors. Note that, according to the relations in Equation (2), it is assumed that the fault signals \(\mathbf{f}_{u}(k)\) and \(\mathbf{f}_{y}(k)\) have equivalent additive effects. These functions are different from zero only in the presence of faults. In general, the vector \(\mathbf{u}(k)\) has \(r\) components, i.e., the number of process inputs, while \(\mathbf{y}(k)\) has \(m\) elements, i.e., the number of process outputs. This work suggests exploiting fuzzy system and NN structures in order to provide an online estimation \(\hat{\mathbf{f}}(k)\) of the fault signals \(\mathbf{f}_{u}(k)\) and \(\mathbf{f}_{y}(k)\). Hence, as shown in Figure 2, the diagnostic residuals \(\mathbf{r}(k)\) are equal to the estimated fault signals, \(\hat{\mathbf{f}}(k)\), as in Equation (3):  
\[\mathbf{r}(k)=\hat{\mathbf{f}}(k) \tag{3}\]  
The variable \(\hat{\mathbf{f}}(k)\) is the fault vector, i.e., \(\hat{\mathbf{f}}(k)=\left\{\hat{f}_{1}(k),\,\ldots,\,\hat{f}_{r+m}(k)\right\}\). Therefore, the general fault estimate \(\hat{f}_{i}(k)\) is equal to the \(i^{\text{\emph{text}}}\) component of the fault vectors \(\mathbf{f}_{u}(k)\) or \(\mathbf{f}_{y}(k)\) in Equation (2), with \(i=1,\,\ldots,\,r+m\). This residual generation scheme is represented in Figure 2.  
Figure 2 shows that, in general, the residual generators are fed by the input and output measurements \(\mathbf{u}(k)\) and \(\mathbf{y}(k)\). The occurrence of the \(i^{\textit{text}}\) fault can be simply detected using the threshold logic of Equation (4) applied to the \(i^{\textit{text}}\) residual \(r_{i}(k)\)[11]:  
\[ \begin{cases}r_{i}-\delta\sigma_{r_{i}}\leq r_{i}\leq r_{i}+\delta\sigma_{i}& \text{fault--free case}\\ r_{i}<r_{i}-\delta\sigma_{r_{i}}\text{ or }r_{i}>\bar{r}_{i}+\delta\sigma_{r_{i}}& \text{faulty case}\end{cases}  \tag{4}\]  
with \(r_{i}(k)\) representing the \(i^{\textit{text}}\) component of the vector \(\mathbf{r}(k)\). Its mean \(\bar{r}_{i}\) and variance \(\sigma_{r_{i}}^{2}\) values are computed in a fault-free condition from \(N\) samples according to the relations in Equation (5):  
\[\left\{ \begin{array}{rcl}\bar{r}_{i}&=&\frac{1}{N}\sum_{k=1}^{N}r_{i}(k)\\ \sigma_{r_{i}}^{2}&=&\frac{1}{N}\sum_{k=1}^{N}\left(r_{i}(k)-\bar{r}_{i} \right)^{2}\end{array} \right. \tag{5}\]  
Note that the parameter \(\delta\) represents a variable that has to be properly tuned in order to separate the fault-free from the faulty conditions effectively, as shown in Section 4. Once the fault detection phase is complete, the fault isolation task is directly obtained by means of the bank of estimators depicted in Figure 3.  
Figure 3: The estimator scheme for the reconstruction of the equivalent input or output faults \(f_{i}(t)\). Figure 2: The residual generation scheme. According to the scheme depicted in Figure 3, the number of estimators in the bank is equal to the number of faults that have to be diagnosed, i.e., \(r+m\). In general, the \(i^{text}\) estimator is driven by the input and output signals \(\mathbf{u}(k)\) and \(\mathbf{y}(k)\). However, its inputs \(u_{j}(k)\) and output \(y_{l}(k)\) are selected in order to be selectively sensitive to the particular fault \(f_{i}(t)\).