## 1 Introduction  
To reduce fossil fuel consumption, lower carbon dioxide emissions, and mitigate the greenhouse effect, the world is accelerating the transformation of energy systems. This transition also prompts reforms in power system architecture, with the installed capacity of renewable energy sources steadily increasing globally. Building large-scale smart grids that integrate clean energy sources, such as wind and solar power, with extensive energy storage systems is a common goal for countries worldwide [1, 2, 3]. As a result, the use of dc power in the grid has become more prevalent, with examples including high-voltage direct current (HVDC) systems, AC-DC hybrid distribution networks, and large-scale energy storage batteries [4, 5, 6]. Consequently, the integration of DC-AC converters into the power grid has increased. Among these, a voltage source converter (VSC) is relatively straightforward to configure and helps reduce system costs [7]. Compared to a current source converter (CSC), VSC can achieve bidirectional current flow, simplifying the reversal of power flow direction [8, 9]. Additionally, VSCs do not rely on an ac power network as a line commutated converter (LCC) does, allowing them to supply power to passive loads and providing black-start capability (i.e., it can be started by itself without a signal from the grid) [10]. Pulse-width modulation (PWM) converters, known for their excellent controllability, flexible operation, and simple structure, are dominant in applications such as reactive power compensation, renewable energy grid integration, and AC-DC hybrid distribution networks [11, 12]. However, inherent characteristics of two-level PWMthree-phase converters lead to drawbacks, including switching losses and harmonic distortion [13]. Specifically, the switching devices in PWM converters endure repeated on-off cycles under high voltage conditions, resulting in energy loss and voltage surges within the devices [14]. Consequently, high-capacity snubber circuits are required to absorb surge energy and reduce device stress. Additionally, two-level PWM three-phase converters necessitate large-capacity filters at the ac terminal to suppress harmonics, which increases the size and cost of the equipment. Moreover, these filters may induce resonance, potentially compromising the stability of the power grid [15; 16; 17]. The reinjection multilevel voltage source converter (RMVSC) effectively reduces output harmonics and enhances power quality without the need for filters by cascading an additional reinjection circuit at the dc side of a traditional 12-pulse converter. This modification shapes the dc voltage into a linear multilevel staircase waveform [18]. Additionally, this approach allows the power electronic switches to operate at lower switching frequencies [19]. Furthermore, the reinjection circuit ensures zero voltage switching (ZVS) conditions for the power electronic switches in the main bridge, significantly reducing their losses, ensuring safe device operation, and improving the converter efficiency and lifespan. This also reduces the switching voltage rating and equipment maintenance costs [20]. Consequently, RMVSC is well suited for widespread application in smart grids. In the development of smart grids, integrating new energy sources such as photovoltaics and wind power--characterized by their randomness and intermittency--presents challenges to the safe and stable operation of the power system, power quality, and real-time power balance when scaled up [21]. As distributed energy sources with higher penetration rates are increasingly incorporated into the grid, voltage limit violations at the grid connection points, caused by voltage drops at the output side of new energy power plants, impede the advancement of these sources [22; 23; 24]. Moreover, with the rise in AC-DC hybrid distribution networks and the extensive integration of generalized dc loads, such as electric vehicles and data centers, the demand for low-voltage reactive power compensation is growing annually [25; 26; 27; 28].