## 3 Data Derivation System of Electric Field Modeling
### Multidimensional Array Modeling of Submodule  
The submodule is the fundamental building block of the MMC valve and constitutes its core component of the MMC valve. Each converter valve is typically composed of multiple submodules. The parameter matrix of the submodules is defined as follows:  
\[\left\{ \begin{array}{l}\mathbf{Y}_{\text{sm}}=\left[\begin{array}{cccc} \mathbf{Y}_{\text{sm11}}&\mathbf{Y}_{\text{sm12}}&\cdots&\mathbf{Y}_{\text{ sm1 NSM}}\\ \mathbf{Y}_{\text{sm21}}&\mathbf{Y}_{\text{sm22}}&\cdots&\mathbf{Y}_{\text{ sm2 NSM}}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{Y}_{\text{smCLT1}}&\mathbf{Y}_{\text{smCLT2}}&\cdots&\mathbf{Y}_{\text{ smCLTNSM}}\end{array}\right]\\ \mathbf{Y}_{\text{sm}ij}=\left[\begin{array}{c}\mathbf{E}(\text{sm})_{ij}\\ \mathbf{G}(\text{sm})_{ij}\\ \mathbf{SP}(\text{sm})_{ij}\end{array}\right]\end{array} \right. \tag{3}\]  
In Equation (3), \(\mathbf{Y}_{\text{sm}ij}\) is the modeling parameters matrix of the submodule \(j\) in layer \(i\) of the MMC valve, MSN denotes the number of submodules in each layer of the MMC valve, and CLT denotes the number of layers in the MMC valve. As shown in Equation (3), the electric field modeling parameters of each submodule consist of three distinct sets of parameters, the electrical parameter matrix \(\mathbf{E}\)(sm), the spatial geometric parameter matrix \(\mathbf{G}\)(sm), and the spatial location matrix \(\mathbf{SP}\)(sm). Due to the metallic conductor nature of the sub-module's outer casing, the sub-module acquires a certain electric potential during the operation of the MMC valve, with the entire outer casing forming an equipotential body. The electrical parameter matrix \(\mathbf{E}\)(sm) of the submodule is defined as follows:  
\[\left\{ \begin{array}{l}\mathbf{E}(\text{sm})_{ij}=\left[\begin{array}{c} GV_{\text{sm}ij}\\ M\text{C}_{\text{sm}}\end{array}\right]\\ GV_{\text{sm}ij}=GV_{\text{sm11}}+NS\cdot\mathbf{u}_{\text{smd}}=GV_{\text{T}}+ NS\cdot\mathbf{u}_{\text{smd}}\end{array} \right. \tag{4}\]In Equation (4), \(GV_{\text{sm}ij}\) is electric potential of the submodule _ij_ in the valve, NS denotes the voltage coefficient, \(u_{\text{smd}}\) is the rated voltage of submodule, \(GV_{\text{T}}\) is the electric potential of the first submodule in the valve, and \(MC_{\text{sm}}\) is the material property of the submodule shell. The submodule is composed of power devices, capacitors, and other components. Once the type of power device and the parameters of the capacitor are specified, the geometric dimensions of the submodule can be determined. The geometric parameter matrix of the submodule is defined as follows:  
\[\mathbf{G}(\text{sm})_{ij}=EC(\text{sm})=f(\text{Csm},U_{\text{CmN}})=\left[ \begin{array}{c}Form\\ X_{\text{S0}}\\ Y_{\text{S0}}\\ Z_{\text{S0}}\end{array} \right] \tag{5}\]  
In Equation (5), \(EC(\text{sm})\) is a computational matrix for the geometric dimensions of the submodule, \(C_{\text{sm}}\) is rated capacitance of the submodule, \(U_{\text{CmN}}\) is the rated voltage of \(C_{\text{sm}}\), \(Form\) is the shape structure of the submodule. The dimensions \(X_{0}\), \(Y_{0}\) and \(Z_{0}\) denote the preliminary geometric sizes of the submodule. The shape structure of the submodule capacitor is obtained from the capacitor manufacturer and typically has a limited range of options. The preliminary geometric sizes of the submodule are generally provided by the structural design engineers. The calculation of geometric parameters adopts a database format, where the geometric parameters of the submodule are determined based on its rated voltage and capacitance. The calculation methods for the rated voltage of the submodule and the rated voltage of the capacitor are defined as follows:  
\[\left\{ \begin{array}{l}I_{\text{dc}}=\frac{P_{\text{dc}}}{2U_{\text{dc}}}\\ \upnu_{\text{PSD}}=\mathbf{DB}(\text{powerSD},\,I_{\text{dc}})\\ \upnu_{\text{smd}}<\frac{1}{2}\upnu_{\text{PSD}}\\ U_{\text{CmN}}=\text{K}_{\text{CU}}\upnu_{\text{smd}}\end{array} \right.