# Fuzzy and Neural Network Approaches to Wind Turbine Fault Diagnosis
## 3 Data-Driven Strategies for Fault Diagnosis
### Fault Estimators via FS and NN Tools  
This section recalls the procedure for developing the fault estimators modelled as TS FSs. In this way, the unknown dynamic relations between the selected input and output measurements of the WT plant and the faults are represented by means of FSs, which rely on a number of rules, antecedent and consequent functions. These rules are used to represent the inference system for connecting the measured signals from the WT system to its faults, in form of IF \(\Longrightarrow\) THEN relations, implemented as Fuzzy Inference System (FIS), see e.g., [17]. According to this modelling strategy, the general TS fuzzy prototype has the form of Equation (6):  
\[\hat{f}(k)=\frac{\sum_{i=1}^{n_{C}}\lambda_{i}(\mathbf{x}(k))\left(\mathbf{a}_ {i}^{\mathsf{T}}\,\mathbf{x}(k)+b_{i}\right)}{\sum_{i=1}^{n_{C}}\lambda_{i}( \mathbf{x}(k))} \tag{6}\]  
Using this approach, in general, the \(i\)-th fault signal \(\hat{f}(k)\) is reconstructed by using suitable data taken from the WT process under diagnosis. In this case, the fault function \(\hat{f}(k)\) is represented as a weighted average of affine parametric relations \(\mathbf{a}_{i}^{\mathsf{T}}\,\mathbf{x}(k)+b_{i}\) (consequents) depending on the input and output measurements collected in \(\mathbf{x}(k)\). These weights are the fuzzy membership degrees \(\lambda_{i}(\mathbf{x})\) depending on the system inputs. The parametric relations of the consequents are affine functions of the unknown variables \(\mathbf{a}_{i}\) and \(b_{i}\), which are estimated by means of an identification approach. The rule number is assumed equal to the cluster number \(n_{C}\) exploited to partition the data via a clustering algorithm with respect to regions where the parametric relations (consequents) are valid [17]. Note that the system under diagnosis corresponds to a WT plant, which is described by a dynamic model. Therefore, the vector \(\mathbf{x}(k)\) in Equation (6) contains both the current and the delayed samples of the system input and output measurements. Therefore, the consequents include discrete-time linear Auto-Regressive with eXogenous (ARX) input structures of order \(o\). This regressor vector is described in form of Equation (7):  
\[\mathbf{x}(k)=\left[\ldots,\,y_{l}(k-1),\,\ldots,\,y_{l}(k-o),\,\ldots\,u_{j} (k),\,\ldots,\,u_{j}(k-o),\,\ldots\right]^{T} \tag{7}\]  
where \(u_{l}(\cdot)\) and \(y_{j}(\cdot)\) represent the \(l\)-th and \(j\)-th components of the actual WT input and output vectors \(\mathbf{u}(k)\) and \(\mathbf{y}(k)\). These components are selected according to the results reported in Table 3.  
The consequent affine parameters of the \(i\)-th model of the Equation (6) can be organised into a vector:  
\[\mathbf{a}_{i}=\left[\alpha_{1}^{(i)},\,\ldots,\,\alpha_{o}^{(i)},\,\delta_{1}^{ (i)},\,\ldots,\,\delta_{o}^{(i)}\right]^{T} \tag{8}\]  
where the coefficients \(\alpha_{j}^{(i)}\) are usually associated to the delayed output samples, whilst \(\delta_{j}^{(i)}\) to the input ones. The approach proposed in this paper for the derivation of the generic \(i\)-th fault FIS approximator starts with the fuzzy clustering of the data \(\mathbf{u}(k)\) and \(\mathbf{y}(k)\) from the WT process. This paper exploits the well-established Gustafson-Kessel (GK) algorithm [17]. Moreover, the estimation of the FIS parameters is solved as a system identification problem from the noisy data of the WT process. Once the data are clustered, the identification strategy proposed in this work uses the methodology developed by the authors in [28]. Another point not addressed in this work concerns the selection of the optimal cluster number \(n_{C}\). This issue was investigated and developed by the authors in [28], which leads to the estimation of the membership degrees \(\lambda_{i}(\mathbf{x}(k))\) required in Equation (6) and solved as a curve fitting problem [17].