# Practical Submodule Capacitor Sizing for Modular Multilevel Converter Considering Grid Faults
## 3 Transient SM Capacitor Voltage during Grid Faults
### Grid Fault SM Voltage Estimation  
Based on the dynamics of the SM voltage shown in Equation (11), the SM voltage stress of each phase-leg is governed by the arm currents (\(i_{uj}\) and \(i_{lj}\)) and its insertion indexes (\(n_{uj}\) and \(n_{lj}\)). According to Equation (12), the arm currents are composed of the circulating current and phase output current. As shown in Figure 4, the state functions of the positive current sequence can be represented as  
\[\dot{\mathbf{X}}=\mathbf{AX}+\mathbf{BU} \tag{17}\]  
Figure 4: Positive- and negative-sequence current control of the MMC. Figure 5: The circulating current control of the MMC. where  
\[\dot{\mathbf{X}}=\left[ \begin{array}{cccc}i_{d+}&i_{q+}&\dot{a}_{d+}&\dot{a}_{q+ }\end{array} \right]^{\mathsf{T}}\]  
\[\mathbf{A}=\left[ \begin{array}{cccc}-\frac{2K_{op}}{L_{a}}&\omega_{1}&K_{i}&0 \\ -\omega_{1}&-\frac{2K_{op}}{L_{a}}&0&K_{i}\\ -1&0&0&0\\ 0&-1&0&0\end{array} \right]\]  
\[\mathbf{X}=\left[ \begin{array}{cccc}i_{d+}&i_{q+}&\alpha_{d+}&\alpha_{q+} \end{array} \right]^{\mathsf{T}}\]  
\[\mathbf{B}=\left[ \begin{array}{cccc}K_{op}&0&-1&0\\ 0&K_{op}&0&-1\\ 1&0&0&0\\ 0&1&0&0\end{array} \right]\]  
\[\mathbf{U}=\left[ \begin{array}{cccc}i_{d+}^{*}&i_{q+}^{*}&v_{d+}&v_{q+}\\ \end{array} \right]^{\mathsf{T}}\]  
in which \(\alpha_{d+}=i_{d+}^{*}-i_{d}\) and \(\alpha_{q+}=i_{q+}^{*}-i_{q}\). According to Formula (17), the transfer functions of the positive-sequence current can be written as \(\mathbf{X}(s)=\mathbf{C}\mathbf{U}\) with matrix \(\mathbf{C}=(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}\). Based on this method, the transfer functions of output current control can be represented as  
\[ \begin{cases}i_{d+}(s)=H_{1}(s)i_{d+}^{*}+H_{2}(s)i_{q+}^{*}-H_{3}(s)v_{d+}-H_ {4}(s)v_{q+}\\ i_{q+}(s)=-H_{2}(s)i_{d+}^{*}+H_{1}(s)i_{q+}^{*}+H_{4}(s)v_{d+}-H_{3}(s)v_{q+}\\ i_{d-}(s)=H_{1}(s)i_{d-}^{*}-H_{2}(s)i_{q-}^{*}-H_{3}(s)v_{d-}+H_{4}(s)v_{q-}\\ i_{q-}(s)=H_{2}(s)i_{d-}^{*}+H_{1}(s)i_{q-}^{*}-H_{4}(s)v_{d-}-H_{3}(s)v_{q-} \end{cases}  \tag{18}\]  
in which \(H_{1}(s)\), \(H_{2}(s)\), \(H_{3}(s)\), \(H_{4}(s)\) are given in Formula (19), respectively:  
\[ \begin{cases}H_{1}(s)=\frac{K_{op}L_{a}^{2}s^{3}+(2K_{op}^{2}L_{a}+K_{i}L_{a}^ {2})s^{2}+(K_{op}K_{i}L_{a}^{2}+2K_{op}K_{i}L_{a})s+K_{i}^{2}L_{a}^{2}}{L_{a}^{ 2}s^{4}+4K_{op}L_{a}s^{3}+(4K_{op}^{2}+L_{a}^{2}\omega_{1}^{2}+2K_{i}L_{a}^{2} )s^{2}+4K_{op}K_{i}L_{a}s+K_{i}^{2}L_{a}^{2}}\\ H_{2}(s)=\frac{K_{op}\omega_{1}L_{a}^{2}s^{2}+K_{i}\omega_{1}L_{a}^{2}s}{L_{a}^ {2}s^{4}+4K_{op}L_{a}s^{3}+(4K_{op}^{2}+L_{a}^{2}\omega_{1}^{2}+2K_{i}L_{a}^{2 })s^{2}+4K_{op}K_{i}L_{a}s+K_{i}^{2}L_{a}^{2}}\\ H_{3}(s)=\frac{L_{a}^{2}s^{3}+2K_{op}L_{a}s^{2}+K_{i}L_{a}^{2}s}{L_{a}^{2}s^{4} +4K_{op}L_{a}s^{3}+(4K_{op}^{2}+L_{a}^{2}\omega_{1}^{2}+2K_{i}L_{a}^{2})s^{2}+ 4K_{op}K_{i}L_{a}s+K_{i}^{2}L_{a}^{2}}\\ H_{4}(s)=\frac{L_{a}^{2}\omega_{1}s^{2}}{L_{a}^{2}s^{4}+4K_{op}L_{a}s^{3}+(4K_{ op}^{2}+L_{a}^{2}\omega_{1}^{2}+2K_{i}L_{a}^{2})s^{2}+4K_{op}K_{i}L_{a}s+K_{i}^{2}L_{a} ^{2}}\end{cases}  \tag{19}\]  
Meanwhile, the dynamic response of the circulating current can be derived according to Figure 5:  
\[i_{cj}(s)=\frac{K_{cp}s^{2}+K_{r}s+4K_{cp}\omega_{1}^{2}}{L_{a}s^{3}+K_{cp} s^{2}+(K_{r}+4L_{a}\omega_{1}^{2})s+4K_{cp}}i_{cj}^{*} \tag{20}\]  
\[v_{cj}(s)=\frac{K_{cp}L_{a}s^{3}+K_{r}L_{a}s^{2}+4K_{cp}L_{a}\omega_{1}^{2}s} {L_{a}s^{3}+K_{cp}s^{2}+(K_{r}+4L_{a}\omega_{1}^{2})s+4K_{cp}}i_{cj}^{*} \tag{21}\]  
To describe the system dynamics during grid faults, Table 1 indicates the operation point of the MMC system during normal state and grid fault conditions. By applying the reverse Laplace transformation on Equation (18), the time-domain transient response of the system in the \(dq\)-frame can be obtained as \[ \begin{cases}i_{d+,\mathit{tran}}(t)=I_{d+}+\mathcal{L}^{-1}[i_{d+}(s)]\\ i_{q+,\mathit{tran}}(t)=\mathcal{L}^{-1}[i_{q+}(s)]\\ i_{d-,\mathit{tran}}(t)=\mathcal{L}^{-1}[i_{d-}(s)]\\ i_{q-,\mathit{tran}}(t)=\mathcal{L}^{-1}[i_{q-}(s)]\\ i_{cj\_tran}(t)=\mathcal{L}^{-1}[i_{cj}(s)]\\ v_{cj\_tran}(t)=\mathcal{L}^{-1}[v_{cj}(s)]\end{cases}