## 2 Converter Structure and Operation Principle
### Power Flow Analysis  
The power flow directions are schematically illustrated in Figure 1. Neglecting losses, a portion of the DC power is transferred to the AC link, while the remainder is exchanged with the supercapacitor energy storage system. The power relationship can be expressed as:  
\[P_{\rm dc}=P_{\rm ac}+P_{\rm sc}, \tag{5}\]  
where \(P_{\rm sc}\) represents the total energy storage power of all ESMs in the MMC-SCES, denoted as:  
\[P_{\rm sc}=\sum_{j}^{\rm a,b,c}\sum_{r}^{\rm p,n}\sum_{i=1}^{N}P_{\rm sc,\gamma j }. \tag{6}\]  
The MMC-SCES operates as a three-port power conversion system, connected to the AC grid, the DC system, and the supercapacitor energy storage system. It can simultaneously perform energy storage and AC-DC conversion functions. Neglecting the effect of the inductor, the voltages for the upper and lower arms of each phase in the MMC-SCES are given by:  
\[\left\{ \begin{array}{ll}u_{\rm pj}=\frac{U_{\rm dc}}{2}-u_{\rm vj}=\frac{U_ {\rm dc}}{2}-U_{\rm vj}\cos\omega t\\ u_{\rm nj}=\frac{U_{\rm dc}}{2}+u_{\rm vj}=\frac{U_{\rm dc}}{2}+U_{\rm vj}\cos \omega t\end{array} \right.. \tag{7}\]  
In the MMC-SCES, the second- and higher-order harmonic components of the circulating current do not affect the active power transmission. Consequently, the analysis considers only the DC component and fundamental frequency component of the circulating current, which can be mathematically expressed as:  
\[i_{\rm cj}=I_{\rm dcj}+I_{\rm 1j}\cos(\omega t+\theta_{j}), \tag{8}\]  
where \(I_{\rm dcj}\) is the DC component of the circulating current, and \(I_{\rm 1j}\) and \(\theta_{j}\) are the amplitude and phase of the fundamental component of the circulating current, respectively. Assuming symmetrical arm impedances, the AC current divides equally between the upper and lower arms within each phase unit. The arm currents can be defined as:  
\[\left\{ \begin{array}{l}i_{\mathrm{p}j}=i_{\mathrm{c}j}+\frac{i_{\mathrm{v}j}}{2} =I_{\mathrm{d}cj}+I_{\mathrm{I}j}\cos(\omega t+\theta_{j})+\frac{I_{\mathrm{v}j }}{2}\cos(\omega t+\varphi)\\ u_{\mathrm{n}j}=i_{\mathrm{c}j}-\frac{i_{\mathrm{v}j}}{2}=I_{\mathrm{d}cj}+I_{ \mathrm{I}j}\cos(\omega t+\theta_{j})-\frac{I_{\mathrm{v}j}}{2}\cos(\omega t+ \varphi)\end{array} \right., \tag{9}\]  
where \(\varphi\) is the power factor angle. By multiplying Equations (7) and (9), the average power of the upper arm, lower arm, and entire phase can be obtained, as expressed by:  
\[\left\{ \begin{array}{l}P_{\mathrm{p}j}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}u_{ \mathrm{p}j}t_{\mathrm{p}j}\mathrm{d}t\\ =\frac{U_{\mathrm{d}c}I_{\mathrm{d}cj}}{2}-\frac{U_{\mathrm{v}j}I_{ \mathrm{I}j}\cos\theta_{j}}{2}-\frac{U_{\mathrm{v}j}I_{\mathrm{v}j}\cos\varphi }{4}\\ P_{\mathrm{n}j}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}u_{\mathrm{n}j}u_{ \mathrm{n}j}\mathrm{d}t\\ =\frac{U_{\mathrm{d}c}I_{\mathrm{d}cj}}{2}+\frac{U_{\mathrm{v}j}I_{ \mathrm{I}j}\cos\theta_{j}}{2}-\frac{U_{\mathrm{v}j}I_{\mathrm{v}j}\cos\varphi }{4}\\ P_{j}=P_{\mathrm{p}j}+P_{\mathrm{n}j}=U_{\mathrm{d}c}I_{\mathrm{d}cj}-\frac{U_{ \mathrm{v}j}I_{\mathrm{v}j}\cos\varphi}{2}\end{array} \right.. \tag{10}\]  
Under steady-state conditions, capacitor voltage stability requires the submodule capacitor to absorb equal average power from both the MMC and supercapacitor energy storage system. This power balance ensures that the average power in each arm equals the total supercapacitor power for that arm, and the average power per phase matches the total supercapacitor power within the respective phase:  
\[\left\{ \begin{array}{l}P_{\mathrm{r}j}=P_{\mathrm{sc},rj}=\sum\limits_{i=1}^{ N}P_{\mathrm{sc},rji}\\ P_{j}=P_{\mathrm{sc},j}=\sum\limits_{r}^{\mathrm{p,n}}\sum\limits_{i=1}^{N}P_{ \mathrm{sc},rji}\end{array} \right.. \tag{11}\]  
In a three-phase balanced operation, the power of each phase is affected by the DC component of the circulating current. Supercapacitor energy storage system power imbalances within phases induce corresponding imbalances in this DC circulating current component.