## 4 Simulation
### SOC Balancing Control with Constant Proportional Coefficients  
The SOC of all ESMs was initialized between 50% and 60%. During system operation, the supercapacitor energy storage system maintained a constant 100 MW discharge power, with the SOC balancing control activated at \(t=2\) s. Figure 7 presents a comprehensive performance comparison between the proposed SOC balancing strategy and the traditional method [24], with both approaches utilizing constant proportional coefficients. Without balancing control, ESM SOC differences remained constant, potentially leading some ESMs to reach threshold limits and exit operation. This would progressively degrade the system's energy capacity utilization. When balancing control was implemented, both strategies successfully achieved SOC convergence. The maximum SOC difference of the enhanced balancing strategy was reduced from the initial 5.17% to 0.0106%, compared to a reduction of 0.0193% in the traditional strategy. The difference between the two strategies is that the enhanced SOC balancing strategy maintained the balancing of the ESMs regardless of the startup time. Moreover, the proposed strategy required only six centralized energy storage controllers, simplifying system control complexity by a factor of \(1/N\). This offers promising potential for large-scale systems containing hundreds of ESMs, where both precision and simplicity are critical. Subsequently, the impact of the enhanced SOC balancing control strategy on the average switching frequency of the system was verified. The switching frequency of each IGBT was quantified by the number of transitions between the off and on states within \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Rated capacity/MVA & 400 & AC frequency/Hz & 50 \\ AC voltage/kV & 220 & DC voltage/kV & 400 \\ Arm inductance/mH & 76 & ESM capacitance/mF & 6.7 \\ Supercapacitance/F & 8 & Carrier frequency/kHz & 5 \\ Number of ESMs & 200 & Energy storage & 10 \\ per arm & & inductance/mH & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 7: SOC for the two balancing strategies. (**a**) SOC of the enhanced balancing strategy. (**b**) SOC of the traditional balancing strategy. one fundamental frequency cycle, multiplied by 50. In this analysis, the average switching frequency of the MMC-SCES was determined using the ESMs in the upper arm of phase A [43]. Mathematically, this can be expressed as:  
\[f_{\text{ave}}=\frac{\sum\limits_{i=1}^{2N}N_{\text{on},i}}{2N}\times 50 \tag{27}\]  
where \(N_{\text{on},i}\) is the number of times the \(i\)th IGBT turned on during a fundamental frequency cycle. The proposed SOC balancing strategy demonstrated a switching frequency of 2474 Hz, closely matching the 2448 Hz frequency of traditional approaches. These high switching frequencies arose from minor voltage variations among ESMs' capacitors, necessitating continuous real-time adjustments of IGBT triggering pulses in the half-bridge modules. To enhance IGBT efficiency and minimize switching losses, the RSF voltage balancing algorithm can be implemented. It exclusively sorts capacitor voltages for extra ESMs requiring insertion or bypass. As illustrated in Figure 8, this method effectively maintained SOC balancing while accommodating various voltage unbalance thresholds. The corresponding IGBT switching frequencies in half-bridge modules are listed in Table 2. The system began with a 5.17% maximum SOC imbalance across ESMs. As the voltage unbalance threshold increased, the maximum SOC difference grew progressively from 0.23% to 0.45%. It also verified the positive correlation between SOC and submodule capacitor voltage. Notably, MMC's average IGBT switching frequency showed a substantial reduction. Even at \(\sigma_{\text{m}}=6\%\), the maximum SOC difference remained below 0.50%, satisfying the supercapacitor requirements. Additionally, the average switching frequency reduced to just 135 Hz, which is only 5.46% of the traditional control.