# 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)
\begin{bmatrix}0&I\\ I^{T}&T+C^{-1}E\end{bmatrix}\begin{bmatrix}\varepsilon\\ \lambda\end{bmatrix} = \begin{bmatrix}0\\ Y\end{bmatrix}  \tag{29}\]  
In the Equation (29), \(I=[1,1,\ldots,1]T\); \(Y=[y_{1},y_{1},\ldots,y_{1}]T\); \(\lambda=[\lambda_{1},\lambda_{2},\ldots,\lambda_{n}]\); \(T=f(x_{i})\cdot f(x_{j})\); \(Ei\times j\) is the identity matrix; \(K(x_{i},x_{j})=(f(x_{i}),f(x_{j}))\) is the kernel function. At this point, given a training sample set \(H=\big{\{}(x_{i},y_{j})\big{|}i=1,2,\ldots,n\big{\}}\), the equation can be solved to obtain \(\lambda\) and \(\varepsilon\), resulting in a decision function, which can be expressed as:  
\[f(x)=\text{sgn}\Bigg{[}\sum_{i=1}^{n}\lambda_{i}K(x_{i},x_{j})+\varepsilon \Bigg{]} \tag{30}\]  
Particle swarm optimization (PSO) [24] is a swarm intelligence algorithm characterized by its simplicity and fast convergence speed. Its core idea is first to define a certain number of free particles and then use an automatic iterative process to calculate the optimal solution of the objective function. Each particle represents a possible solution, and intelligent problem-solving is achieved through the simple behavior of particles and the interaction of group information. In each iteration process, in order to find parameter values that meet the given requirements, it is necessary to track the changes in two \"extremes\" and continuously adjust the positions of particles. When the value of a particle reaches the optimal solution of its iteration process, it is called the individual extreme value (\(pBest\)), and the individual extreme value is shared with random particles in the entire population. Another extreme value is the optimal solution obtained by the entire population, called the global extreme value (\(gBest\)). The optimal solution found by the particle itself means that only some of the population has found the optimal solution. After finding \(pBest\) and \(gBest\) through iteration, each random particle's position and velocity are updated according to Equation (31). \[V=wV+c_{1}rand()(pBest-p)+c_{2}rand()(gBest-p) \tag{31}\]  
\[p=p+V \tag{32}\]  
In this context, \(rand()\) is a random function value between (0,1); \(c_{1}\) and \(c_{2}\) are known as learning factors; \(w\) is referred to as the weighting factor; \(V\) is known as the velocity of the particle's movement; \(p\) is referred to as the particle's current position. As the iteration process continues, the value of \(w\) changes according to the trend of a monotonically decreasing function, causing \(w_{\text{max}}\) to transition further to \(w_{\text{min}}\). \[w=w_{\text{max}}-iter\times(w_{\text{max}}-w_{\text{min}})/iter_{\text{max}} \tag{33}\]  
Here, \"\(iter\)\" refers to the current number of iterations in PSO, and \"\(iter_{\text{max}}\)\" refers to the maximum number of iterations in PSO. This paper uses the training accuracy during the training process as the objective function for particle swarm optimization least squares support vector machine (PSO-LSSVM). The final optimization problem can be transformed into finding the optimal parameters \((\sigma^{2},C)\). Through iterative calculation, the maximum value of the objective function is finally obtained. The specific implementation process of PSO-LSSVM is shown in Figure 10.