# Research on Key Technology of Wind Turbine Drive Train Fault Diagnosis System Based on Digital Twin
## 3 Digital Twin Model Construction of Wind Turbine Transmission System
### Lightweighting of Wind Turbine Drive Train Models  
Due to the highly complex physical structure of wind turbine generators, accurately constructing models in virtual space would demand extremely high computer performance, consuming vast amounts of memory and GPU. This would result in computer lag and delays. Given that digital twins require low latency, it is crucial to implement lightweight processing on the models. Generally speaking, the lightweight processing of physical equipment models is divided into two steps. The first step involves the lightweight processing of the quantity of component features. Components that are invisible during the virtual module visualization process or have lower weight during operation are deleted or hidden, with only the primary parts such as the rotor hub, main drive shaft, planetary gearbox, parallel shaft gearbox, and generator retained after lightweight processing. The second step is the lightweight processing of the mesh features of the parts. Classic methods for lightweight mesh features include vertex clustering, region merging, and geometric deletion. The edge-collapse algorithm within the geometric deletion method is a mainstream mesh lightweight algorithm, as illustrated in Figure 3. However, this method can lead to model deformation and feature loss when simplifying the original model. Consequently, [PERSON] et al. [16; 17] proposed the Quadric Error Metrics (QEM) algorithm to improve the edge-collapse algorithm. It aims to minimize model deformation and feature loss as much as possible, enabling the rapid generation of high-quality simplified models. The Quadric Error Metrics (QEM) algorithm defines the error before and after lightweight processing as the sum of squared distances from a vertex to the original adjacent edges. It then calculates the quadric error matrix for each vertex to obtain the edge-collapse weights, prioritizing the collapse of edges with lower weights during the edge-collapse operation. The specific calculation process for weights is as follows: assume a plane in three-dimensional space is represented by \(Ax+By+Cz+D=0\), where \(A^{2}+B^{2}+C^{2}=1\), \(D\) are constants; given any point \(v=(m,n,o)^{T}\) in space, the squared distance from \(V\) to the plane is as follows:  
\[d^{2}=\frac{\left(Am+Bn+Co+D\right)^{2}}{\left(A^{2}+B^{2}+C^{2}\right)}= \left(Am+Bn+Co+D\right)^{2} \tag{2}\]  
And \(n=(A,B,C)\) is a unit normal vector to the plane, then the equation of the plane in space can also be represented as  
\[n^{T}\cdot v+d=0 \tag{3}\]  
The Equations (2) and (3) implies the following:  
\[d^{2}=\left(n^{T}\cdot v\right)^{2}=\left(v^{T}\cdot n\right)^{2} \tag{4}\]  
Since the quadratic error at each vertex in the grid is defined as the sum of the squares of the distances to the adjacent faces, namely:  
Figure 3: Model lightweight schematic. \[\Delta_{\rm{p}} =\sum_{n\in\alpha(v)}\left(v^{\rm{T}}\cdot n\right)^{2}\] \[=\sum_{n\in\alpha(v)}\left(v^{\rm{T}}\cdot n\right)(n^{\rm{T}}\cdot v)\] \[=\sum_{n\in\alpha(v)}v^{\rm{T}}(nn^{2})v \tag{5}\] \[=v^{\rm{T}}\left(\sum_{n\in\alpha(v)}K_{n}\right)v\]  
\(\alpha(v)\) represents the geometry of the adjacent triangles of the vertex, and T denotes any triangle plane in the set. Let \(Q_{v}\) be the quadratic error matrix for point \(v\). Since the quadratic error matrix is a 4 \(\times\) 4 matrix, denoted as \(v=\left(m,n,o,1\right)^{\rm{T}}\), \(n=\left(A,B,C,D\right)\), there is:  
\[Q_{v}=\sum_{n\in\alpha(v)}K_{n} \tag{6}\]  
\[d^{2}=\frac{\left(Am+Bn+Co+D\right)^{2}}{\left(A^{2}+B^{2}+C^{2}\right)}= \left(Am+Bn+Co+D\right)^{2} \tag{2}\]0 \[= \begin{bmatrix}A^{2}&AB&AC&AD\\ AB&B^{2}&BC&BD\\ AC&BC&C^{2}&CD\\ AD&BD&CD&D^{2}\end{bmatrix}