## 3 Data Derivation System of Electric Field Modeling
### Multidimensional Array Modeling of Insulator  
The primary design of the MMC valve currently adopts the supported valve tower structure, in which the interlayer insulators and ground insulators are important components of the valve tower. The electric field modeling parameters matrix of the MMC valve insulators are defined as follows:  
\[\mathbf{Y}_{\mathrm{insulator}L}=\left[ \begin{array}{cccc} \mathbf{Y}_{\mathrm{insulator}L11}&\mathbf{Y}_{\mathrm{insulator}L12}&\cdots&\mathbf{Y}_{\mathrm{insulator}L11\mathrm{NL}}\\ \mathbf{Y}_{\mathrm{insulator}L21}&\mathbf{Y}_{\mathrm{insulator}L22}&\cdots& \mathbf{Y}_{\mathrm{insulator}L2\mathrm{NL}}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{Y}_{\mathrm{insulator}L11}&\mathbf{Y}_{\mathrm{insulator}L12}&\cdots& \mathbf{Y}_{\mathrm{insulator}L11\mathrm{NL}}\\ \mathbf{Y}_{\mathrm{insulator}G1}&\mathbf{Y}_{\mathrm{insulator}G2}&\cdots& \mathbf{Y}_{\mathrm{insulator}GNG}\end{array} \right] \tag{17}\] \[\mathbf{E}(\mathrm{insulator})_{ij}\] \[\mathbf{G}(\mathrm{insulator})_{ij}\] \[\mathbf{S}\mathbf{P}(\mathrm{insulator})_{ij}\] \[\mathbf{E}(\mathrm{insulator}L/\mathrm{G})=MC_{\mathrm{ insulator}}=MC_{\mathrm{insulator}}=\varepsilon_{\mathrm{insulator}}\]In Equation (17), \(\mathbf{Y}_{\text{insulator,L}}\) is the modeling parameters of interlayer insulators, \(\mathbf{Y}_{\text{insulatorG}}\) is the modeling parameters of ground insulators, NIG is the number of ground insulators, NIL is the number of interlayer insulators in one layer. The electrical parameter matrix of insulators are solely characterized by the dielectric constant of the insulating material. The geometric parameters matrix and positional parameters matrix are defined as follows:  
\[\left\{ \begin{array}{l}\mathbf{G}(\text{insulator}\text{L})_{ij}=EC(\text{ insulator}\text{L})=\left[\begin{array}{c}Form\\ R_{1}\\ R_{2}\\ NP\\ hL\end{array}\right]\\ hL=f_{V}(N_{L}\cdot\mathbf{u}_{\text{smd}})\\ NP=f_{N}(hL,F)\\ \mathbf{SP}(\text{insulator}\text{L})_{ij}=\left[\begin{array}{c}\chi_{ \text{SC11}}+xm_{ij}\\ Y_{\text{SC11}}+ym_{ij}\\ Z_{\text{SC11}}+zm_{ij}\end{array}\right]\\ \end{array} \right. \tag{18}\]  
In Equation (18), \(EC\)(insulatorL) is a computational matrix for the geometric dimensions of the interlayer insulators, \(Form\) is the shape structure, \(R_{1}\) is the shed radius, \(R_{2}\) is the core radius, \(hL\) is the height of the insulator, and \(NP\) is the number of shed layers, \(X_{\text{SC11}}\), \(Y_{\text{SC11}}\) and \(Z_{\text{SC11}}\) are the displacement coordinates of interlayer insulator in the MMC valve, \(xm_{ij}\), \(ym_{ij}\) and \(zm_{ij}\) are the displacement coordinates of insulator L \(ij\) relative to the insulator L\({}_{11}\). The \(hL\) is correlated with the voltage difference between layers, which is equal to the sum of the voltages of the sub-modules in each layer. The geometric parameters matrix and positional parameters matrix of ground insulators are defined as follows:  
\[\left\{ \begin{array}{l}\mathbf{G}(\text{insulatorG})_{i}=GEC(\text{ insulatorG})=\left[\begin{array}{c}Form\\ R_{1G}\\ R_{2G}\\ NPG\\ hG\end{array}\right]\\ NPG=f_{N}(hG,Form)\\ hG=f_{V}[SINL(VTG),LIWL(VTG)]\\ \mathbf{SP}(\text{insulatorG})_{i}=\left[\begin{array}{c}\chi_{\text{SC1}}+ xgm_{i}\\ Y_{\text{SC1}}+ygm_{i}\end{array}\right]\\ \end{array} \right. \tag{19}\]  
In Equation (19), \(EC\)(insulatorG) is a computational matrix, \(Form\) is the shape structure, \(hG\) is the height of the insulator, and \(NPG\) is the number of shed layers on the insulator, \(X_{\text{SC1}}\) and \(Y_{\text{SC1}}\) are the displacement coordinates of ground insulator in the MMC valve, \(xgm_{i}\) and \(ygm_{i}\) are the displacement coordinates of insulatorG\({}_{i}\) relative to the insulator G1. The height of the ground insulator corresponds to the air clearance between the MMC valve and the ground, which is determined by the withstand overvoltage of MMC-HVDC system. The materials used in the MMC valve, including insulators, supports, sub-module enclosures, and shielding covers, are all linear materials.