## 1 Introduction  
The field of integrating renewable energy sources into the electricity grid presents both significant challenges and opportunities. The challenges include the need to ensure the efficient integration of renewable energy while maintaining the stability and reliability of the electricity grid [1, 2]. Fluctuations in renewable energy generation can lead to rapid variations in the power injected into the grid, requiring advanced control strategies to ensure effective regulation [3]. Modular multilevel converters (MMCs) play a key role in the integration of renewable energies, offering an optimal solution for connecting these resources to the power grid [4, 5, 6]. Their modular and scalable structure enables energy to be converted efficiently while improving grid stability [7, 8]. The present study focuses on medium-voltage applications in a grid-connected system, where the MMC serves as a critical interface connecting renewable energy sources to the power grid, facilitating efficient energy transfer and integration, using an additional converter (chopper or rectifier). In this configuration, renewable energy sources are linked to the DC side, efficient energy conversion to AC for seamless and stable grid integration. This approach tackles the challenges of variability in renewable energy output while maintaining system stability and reliability. Notably,due to their excellent scalability, MMCs can also be specifically adapted and optimized to meet the demands of high-voltage applications, such as high-voltage direct current (HVDC) transmission systems [9; 10]. However, this integration comes with challenges, such as the need to achieve optimum performance in terms of power regulation, reducing unwanted circulating currents, minimizing disturbances on the DC side, and balancing the voltages of the SM capacitors. Studies highlight the importance of MMC energy control in system dynamics during grid formation and tracking operations [11]. Furthermore, the complexity of MMC topologies and the demand for efficient control strategies present additional obstacles [5; 12]. These challenges require innovative control approaches capable of addressing the complexities of MMCs while optimizing their dynamic behavior. The proposed combined control strategy, based on Virtual Synchronous Generator (VSG) control and Capacitor Voltage Balancing (CVB), aims to overcome these shortcomings by improving stability and performance [13; 14]. This innovative approach attempts to overcome the limitations and complexities identified in existing MMC control methods [15]. The VSG control focuses on power management and inertia imitation, which helps to improve the dynamic response and overall stability of the system [16; 17]. By successfully managing power sharing, VSG enhances the MMC's ability to respond dynamically to power variations, ensuring optimal operation under a range of conditions [18; 19]. The CVB method is a crucial element in maintaining uniform voltage levels and equalizing voltages between capacitors within the SM in MMC. Notable for its sensorless operation, the CVB uses estimation techniques, based on mathematical models, to derive capacitor voltages [20]. This sensorless approach improves control efficiency by reducing reliance on physical sensors, simplifying hardware complexity and potentially reducing overall system costs. In addition, the CVB method aims to strategically minimize the number of switching events in the MMC, optimizing switching schemes to balance capacitor voltages with a minimum number of events [21; 22]. This reduction in switching events not only enhances system reliability but also contributes to a reduction in switching losses, leading to an overall improvement in energy efficiency [23]. To calculate the switching losses of the converter, it is essential to obtain the energy losses at each turn-on and turn-off stage of the insulated gate bipolar transistors (IGBTs) and freewheeling diodes (FWDs). Consequently, reducing switching events has a direct impact on reducing switching losses [23].