## 3 SOC Balancing
### SOC Balancing Control Strategy  
SOC imbalance among ESMs presents a critical challenge to system operational stability and performance. Traditional balancing approaches often require numerous controllers and exhibit high computational complexity, limiting the practical implementation. To overcome these limitations, this paper proposes an enhanced SOC balancing strategy based on a three-level control system involving phase units, arms, and submodules. The system implements adaptive power allocation during both charge and discharge states. During discharge, ESMs with higher SOC are allowed to deliver more power, while during charging, those with lower SOC are allowed to absorb more power. Additionally, the SOC imbalances in ESMs per arm through direct regulation of submodule capacitor voltages capitalize on the proportional relationship between supercapacitor SOC and submodule capacitor voltage. By assigning distinct power references to each arm andexploiting the voltage-SOC correlation principle, the enhanced control strategy ensures SOC convergence across all ESMs:  
\[\left\{ \begin{array}{l}S_{\text{soc\_},j\bar{i}}=\frac{1}{N}\sum\limits_{i=1}^{N }S_{\text{soc\_},j\bar{i}}\\ S_{\text{soc\_},j}=\frac{1}{2}\frac{1}{\tau}S_{\text{soc\_},j\bar{j}}\\ S_{\text{soc\_},\text{ave}}=\sum\limits_{j}^{\Delta\frac{\Delta\text{c}}{S_{ \text{soc\_},j}}}\end{array} \right.. \tag{17}\]  
As shown in Equation (17), \(S_{\text{SOC\_},j\bar{i}}\) is the SOC of the \(i\)th ESM in either the upper arm or lower arm of the \(j\) phase. Additionally, \(S_{\text{SOC\_},j\bar{i}}\) and \(S_{\text{SOC\_},j}\) are defined as the average SOC values of the submodules in the arm and the phase, respectively. \(S_{\text{SOC\_},\text{ave}}\) is the average SOC value of the entire MMC-SCES. SOC imbalances among phases can be effectively addressed through controlled injection of unbalanced power. Figure 3 illustrates the proposed control for SOC balancing among phases. \(k_{\text{ph}}\) represents the proportional coefficient for phase balancing, and \(G_{\text{ph}}\) is the transfer function relating phase energy storage power to the average SOC. The control calculates the deviation between each phase's average SOC and the system mean value, then generates phase unbalanced power through proportional control. The unbalanced power is superimposed on the nominal phase powers while rigorously maintaining the zero-sum constraint across all phases. The power adjustments in each phase's supercapacitors induce corresponding variations in the submodule capacitor voltages. This results in the DC circulating currents being actively redistributed among phases. The arm SOC balancing control strategy ensures uniform SOC values between the upper and lower arm within each phase through dynamic power allocation. As illustrated in Figure 4, the proportional coefficient \(k_{\text{arm}}\) and transfer function \(\text{G}_{\text{arm}}\) regulate the SOC balancing between arms. The unbalanced power is proportional to the SOC difference between the corresponding upper and lower arms. These power adjustments follow the principle that arms with higher SOC deliver increased power during discharge while receiving reduced power during charge states. The resulting power differential between the upper and lower arms induces fundamental frequency circulating currents. Similarly, the sum of the unbalanced power of the upper and lower arm within each phase is zero. Thus, injecting unbalanced power into the arms does not affect the total energy storage power of the system. Figure 4: SOC balancing between arms. Figure 3: SOC balancing among phase units. The submodule SOC balancing control maintains uniform SOC values among ESMs within the same arm through the regulation of submodule capacitor voltage. As shown in Equation (16), when the duty cycle of switch T\({}_{3}\) in each ESM is identical, the SOC of each ESM exhibits a direct proportional relationship with its corresponding submodule capacitor voltage.