# Practical Submodule Capacitor Sizing for Modular Multilevel Converter Considering Grid Faults
## 2 Steady-State Analysis of SM Capacitor Voltage  
A three-phase MMC topology is shown in Figure 1. Each arm of the MMC contains \(N\) identical half-bridge SMs and one series-connected arm inductor \(L_{a}\). The SM is formed by two IGBTs and a parallel SM capacitor \(C_{sm}\). The SM voltage \(v_{sm}\) is defined as the voltage across each SM capacitor, while the DC-bus voltage is considered as a constant denoted by \(V_{dc}\).The AC output of the MMC is connected to the grid through a three-phase \(\Delta/\text{Y}\) transformer. Thus, there will not be any zero-sequence currents on the converter side during grid fault conditions.  
To simplify the analysis, it is assumed that the voltage across each SM is well-balanced. Thus, an equivalent circuit can be derived for each phase-leg of the MMC, as shown in Figure 2. All the SMs in each arm are combined into a single equivalent capacitor being \(C_{eq}=C_{sm}/N\). The equivalent capacitors hold the total voltage stress of the upper arm \(v_{uj}\) and lower arm \(v_{lj}\), where \(j\) indicates the phase (\(j=a\), \(b\), \(c\)). The switching devices of each SM are replaced by a pair of equivalent controlled-sources. The grid phase voltage and output current of each phase are represented by \(v_{j}\) and \(i_{j}\). The circulating current is denoted as \(i_{ij}\). The insertion index of the upper and lower arms is represented by \(n_{uj}\) and \(n_{lj}\), while the arm current is defined as \(i_{uj}\) and \(i_{lj}\). Due to the symmetrical characteristic of the MMC system, only the expression of upper arm is given.  
In steady-state, the grid voltage and current of each phase can be expressed as  
\[v_{j}=\hat{V}_{j}\cos\left(\omega_{1}t\right) \tag{1}\]  
\[i_{j}=\hat{I}_{j}\cos\left(\omega_{1}t-\phi_{j}\right) \tag{2}\]  
where \(\omega_{1}\) is the fundamental line frequency, \(\phi_{j}\) is the power factor angle, and \(\hat{V}_{j}\) and \(\hat{I}_{j}\) are the peak value of grid voltage and current.  
The total energy variation of the upper arm is analyzed in [17] with  
\[e_{uj}=\frac{C_{sm}V_{dc}^{2}}{2N}+\frac{V_{dc}\hat{I}_{j}}{16\omega_{1}}[4\sin( \omega_{1}t-\phi_{j})-m\sin(2\omega_{1}t-\phi_{j})-2m^{2}\sin(\omega_{1}t)\cos (\phi_{j})] \tag{3}\]  
where \(m\) is the modulation index, given by \(m=2\hat{V}_{j}/V_{dc}\).  
Meanwhile, the energy storage in the upper arm can be represented as  
\[e_{uj}=\frac{1}{2}C_{eq}v_{uj}^{2}=\frac{1}{2}NC_{sm}v_{sm}^{2} \tag{4}\]  
Combining Equations (3) and (4), the voltage stress of an SM capacitor in the upper arm can be given as  
\[v_{smj}=\sqrt{\frac{V_{dc}^{2}}{N^{2}}+\frac{V_{dc}\hat{I}_{j}}{8 NC_{sm}\omega _{1}}}\mathbf{F} \tag{5}\]  
Figure 1: Schematic of a three-phase modular multilevel converter.  
Figure 2: Equivalent circuit of each phase-leg of the MMC.  
(5)  
where  
\[\mathbf{F}=4\sin(\omega_{1}t-\phi_{j})-m\sin(2\omega_{1}t-\phi_{j})-2m^{2}\sin( \omega_{1}t)\cos(\phi_{j}) \tag{6}\]  
To study the steady-state SM voltage ripple, the waveform of \(\mathbf{F}\) function under different operating point is shown in Figure 3. Hence, the operating voltage boundary of SM capacitors in steady-state can be obtained as  
\[V_{ssmax}=\sqrt{\frac{V_{dc}^{2}}{N^{2}}+\frac{V_{dc}\hat{I}_{j}}{8 NC_{sm}\omega _{1}}\mathbf{F}_{max}} \tag{7}\]  
\[V_{ssmin}=\sqrt{\frac{V_{dc}^{2}}{N^{2}}+\frac{V_{dc}\hat{I}_{j}}{8 NC_{sm}\omega _{1}}\mathbf{F}_{min}} \tag{8}\]  
where \(\mathbf{F}_{max}\) and \(\mathbf{F}_{min}\) are the maximum and minimum value of the \(\mathbf{F}\) function.