## 1 Introduction  
The growing global environmental awareness and depletion of fossil fuel reserves have heightened interest in renewable energy sources [1, 2]. However, renewable power generation faces several challenges [3, 4, 5, 6, 7]. Firstly, the power generated by renewable energy sources demonstrates random, intermittent, and fluctuating outputs that pose substantial challenges to grid stability. Secondly, heavy reliance on power electronic equipment leads to lower system inertia and insufficient grid voltage and frequency regulation support. Lastly, the uneven distribution of renewable energy stations and load centers results in long transmission distances and substantial losses. To address these challenges, integrating energy storage into the power system is often necessary. The supercapacitor energy storage system (SCES) provides several advantages over batteries, such as rapid charging, high power density, and environmental sustainability. These characteristics render it a practical, efficient, and environmentally friendly energy storage solution that can smooth power fluctuations, enhance power quality, and facilitate renewable energy integration [8]. In medium- and high-voltage transmission systems, the modular multilevel converter (MMC) is characterized by high modularity, low harmonic distortion, and negligible losses [9]. The integration of supercapacitors into each MMCsubmodule forms the MMC-SCES system. This system demonstrates robust fault tolerance, flexible control, and notable advantages for renewable energy grid integration and high-voltage direct-current (HVDC) transmission [10; 11]. To leverage the modularity of the MMC, supercapacitors can be integrated with each submodule through three distinct approaches: direct parallel connection [12; 13], parallel connection via a bidirectional DC/DC converter [14; 15; 16], and parallel connection through an isolated DC/DC converter [17; 18]. The first topology offers simplicity and relatively low power losses. However, this approach directly exposes the supercapacitor to second-harmonic power, necessitating large submodule capacitors for mitigation [19]. To preserve the supercapacitor's lifespan, a bidirectional DC/DC converter serves as an intermediate stage [20]. The DC/DC converter decouples the supercapacitor from the submodule capacitor, enhancing system flexibility while enabling active charge/discharge management. A notable limitation is the reduction in overall energy conversion efficiency introduced by the DC/DC converter. The isolated topology provides essential insulation between submodules in high-voltage applications while ensuring electrical isolation for safety. However, it requires additional switching devices, increasing the system complexity and cost [21]. Considering both functionality and practicality, this paper adopts the second topology. The MMC-SCES consists of multiple energy storage modules (ESMs) connected in series. Inherent parameter variations among supercapacitors lead to state-of-charge (SOC) discrepancies across ESMs. The ESM with the minimum SOC determines the system's maximum utilizable energy storage capacity. Consequently, SOC balancing among ESMs becomes essential for optimal system performance. One effective approach to balance SOC within an arm is to sort the SOCs of all submodules [14]. However, the computational complexity escalates substantially with the increasing number of submodules. While the zero-sequence voltage injection method achieves SOC balancing among different phases, its implementation demands sophisticated mathematical computations and, consequently, more advanced control hardware [20; 22]. A simpler closed-loop method can balance SOC among submodules within an arm and phases but does not address SOC imbalances between the upper and lower arms [23]. To overcome these limitations, a three-level proportional controller employing circulating current injection has been proposed for SOC balancing under carrier phase-shifted pulse width modulation (CPS-PWM).