# Research on Key Technology of Wind Turbine Drive Train Fault Diagnosis System Based on Digital Twin
## 4 Fault Diagnosis Method Based on Digital Twin
### Particle Swarm Optimization Algorithm Optimizing Least Squares Support Vector Machine (PSO-LSSVM)  
Support vector machine (SVM) [22] is a supervised learning algorithm widely used for classification and regression tasks. It is based on the principle of structural risk minimization in statistical learning, aiming to find an optimal decision boundary called the maximum margin hyperplane to maximize the margin distance between different classes. SVM deploys kernel techniques to handle nonlinear problems by mapping the data into a higher-dimensional space, making data that are nonlinearly separable in the original space linearly separable in the new space. SVM, as a classical machine learning method, has been successful in many fields, but it also has some drawbacks. First, SVM has high computational complexity when dealing with large-scale datasets, especially for nonlinear kernel functions, which require a lot of computational resources and time. Second, SVM is more sensitive to parameter selection, including the selection of kernel function and the setting of regularization parameters, which need to be repeatedly tested and adjusted. In addition, when the number of positive and negative samples in the training dataset varies greatly, the classification performance of SVM may be affected, which is prone to the classification bias problem caused by sample imbalance. In order to address some of the shortcomings of SVM, the least squares support vector machine (LSSVM) [23] proposes a series of improvements. First, LSSVM transforms the original problem into a convex quadratic optimization problem with constraints by introducing a regularization term, which reduces the computational complexity and is more efficient, especially when dealing with large-scale datasets. Secondly, LSSVM can automatically select the optimal parameters using methods such as cross-validation, which reduces the subjectivity of parameter adjustment and improves the generalization ability and stability of the model. In addition, LSSVM introduces more kinds of kernel functions, such as polynomial kernel function and radial basis function (RBF) kernel function, which makes the model more suitable for dealing with nonlinear problems and improves the classification performance. In summary, LSSVM improves computational efficiency, parameter selection, and nonlinear problem handling compared to traditional SVM, which makes the model more suitable for real-world scenarios and shows better performance in some specific application scenarios. The core idea of LSSVM is that, for a set of training samples, \(H=\left\{(x_{i},y_{i})|i=1,2,\ldots,n\right\}\), where \(x_{i}\in R_{n}\) is the input data, and \(y_{i}\in(-1,1)\) is the output data. When projecting sample data into a high-dimensional feature space, consider the nonlinear mapping \(f(x_{i})\) to construct the optimal decision function \(y_{i}=(\omega^{T}f(x_{i})+\varepsilon)\), where \(\omega\) and \(\varepsilon\) are model parameters solved optimally as follows:  
\[\left\{ \begin{array}{l}f_{\min}(\omega,\varepsilon_{i})=\frac{1}{2}\|\omega \|^{2}+\frac{\varepsilon}{2}\sum\limits_{i=1}^{n}\varepsilon_{i}^{2}\\ \text{s.t.}y_{i}=\left(\omega^{T}f(x_{i})+\varepsilon\right)+\varepsilon_{i} \end{array} \right. \tag{27}\]  
In Equation (27), \(\omega\) and \(\varepsilon\), respectively, represent the weight vector and bias; \(e_{i}\) represents the error variable; \(C\) represents the regularization parameter; \(f(x)\) is the mapping function corresponding to the input data \(x_{i}\). The solution for \(\omega\) and \(\varepsilon\) can be expressed as follows:\[L(\omega,\varepsilon,e_{i},\lambda)=\frac{1}{2}\omega^{2}+\frac{C}{2}\sum_{i=1}^{n }e_{i}^{2}-\sum_{i=1}^{n}\lambda_{i}\Big{(}\Big{(}\omega^{\rm T}f(x_{i})+ \varepsilon\Big{)}+e_{i}-y_{i}\Big{)} \tag{28}\]  
Taking partial derivatives with respect to \(\omega,\varepsilon,e_{i}\), and \(\lambda\) (where \(\lambda_{i}\in R\) is a Lagrange multiplier), it can be expressed as follows:  
\[