## 1 Introduction  
Among the various kinds of multilevel converters [1; 2; 3], modular multilevel converters (MMCs) have attracted increasing attention for various industrial applications such as Static Synchronous Compensators (STATCOM) [4], energy storage systems [5], power electronic-based transformers [6], rolling mill factories [7], high-voltage direct current (HVDC) [8; 9; 10], solar photovoltaics [11], grid-connected conditions [12], and motor drives [13]; simultaneously, many new modulation-based control strategies have been recently developed in order to present more effective controllers for these kinds of multilevel converters [14]. Several publications have focused on proposing modulation techniques with the aim of reducing capacitor voltage ripples and switching losses, as well as allowing a smaller cell capacitor size for MMCs [15; 16]. In [17], a discontinuous modulation-based control strategy was designed for MMCs based on adding a zero sequence to the basic modulation signals. This strategy caused the MMC arms to be clamped with the upper or lower terminals of the dc-link bus, which can help minimize the switching losses and noticeably reduce the capacitor voltage ripples, especially for low modulation indices [17]. Another reduced-switching-frequency-based modulation technique was proposed for MMCs in [18] that can also suppress the circulating currents. On the other hand, in order to reach a smaller cell capacitor size without an increment in the total device rating and decrement in the dc current control capability, a phase-shifted square wave modulation (PS-SWM) strategy was proposed for MMCs in [19]. Moreover, for achieving reduced switching losses with good harmonic performance, a novel modulation method which consists of a combination of fundamental frequency modulation and multi-carrier-based sinusoidal pulse width modulation schemes was presented in [20]. A growing trend towards space vector (SV)-based modulations [21] for the control of MMCs has arisen in recent years. In [22], a dual space vector pulse width modulation strategy was designed for an MMC. Eliminating the arm voltage balancing control, independent controllers for the three-phase upper and lower arms employing redundant switching vectors for capacitor voltage balancing, and reducing the current sensors used by 50% were the achievements of this strategy [22]. On the other hand, space vector pulse width modulations were applied to MMCs for fault-tolerant challenges [23] and electric ship propulsion systems [24] as well. Optimization issues were involved with control of the space vector modulation-based MMC in [25]. In this referenced work, the maximum level numbers were achieved along with optimized performance of decreasing common-mode voltage, balancing of the capacitor voltage, and elimination of the circulating currents [25]. Optimal pulse width modulation (PWM) has been a popular discussion topic, leading to proposals of new modulation-based control techniques for MMCs in recent years [26]. In [27], synchronous optimal pulse width (SOP) was enhanced for MMCs in order to obtain improved quality of converter output currents, low device switching frequency, and desired values of capacitor voltages. Optimized phase disposition (PD) modulation for MMCs was proposed in [28] with the help of a single PD modulator for the entire upper and lower arms as important parts for the control of MMCs [29]. This single modulator was implemented through a state machine decoder which had the duty of distributing the balanced switching pulses to all submodules [28]. Another optimal pulse width modulation strategy for reducing the common-mode voltage of a medium-voltage modular multilevel converter that fed open-end stator winding induction motor drives was designed in [30]. The achievements of the modulation-based control technique designed in [30] are low switching frequency, minimized harmonic currents of the machine stator, elimination of common-mode voltages of the machine stator windings, and the desired capacitor voltages.