## 1 Introduction
Other modulation strategies have also been considered to improve MMC performance in various operating conditions. In [31], a new function-based modulation control technique for MMCs was designed; it is noticeably less complex when compared to the existing control methods and is proper for the conditions of load and parameter variations. In [32], a new modulation technique based on generalized low-cost computational decoupled sampled average PWM was designed for MMCs in which two averaged nearest voltage levels from the upper and lower arms in each sample periodwere utilized to drive the reference output voltages [32]. This modulation-based control strategy was completed via a simplified voltage balancing method [32]. In addition, two novel modulation strategies were proposed for HVDC and wind farm integration applications in [33] and [34], respectively. A new modulation method, along with a capacitor voltage balancing method, was presented in [35] that was able to provide fixed switching frequency at 50 Hz. Using reference [35], located at the nearest voltage level control, the switching angles were calculated, and then every four pulse patterns were grouped together in order to assess the average of the dc and fundamental ac components in the output voltages of individual submodules [36; 37]. In this paper, three control laws are proposed in order to suppress the circulating currents and provide stable performance for MMCs under nonlinear load variation, since a study regarding linear load was already made in [38]. To design the three proposed control laws, the harmonic components of MMC output voltages, MMC output currents, and circulating currents are utilized. Three novelties are presented that cannot be found in previous works. These novelties are the following: (1) For the first time, an accurate mathematical equation is achieved for an MMC arm capacitor. This equation can be employed for approaching more stable operating conditions for MMCs. (2) In the proposed control strategy, the harmonic components of all MMC state variables are considered to reach full stable operation for the MMC under various operating conditions. (3) A comprehensive assessment related to how the second-order component and other harmonic components of circulating currents can impact the switching signals and equivalent submodule (SM) capacitors is executed. On the other hand, the following contributions are accomplished as well: (1) The proposed control strategies are designed in an a-b-c reference frame, leading to simplicity of performance and highly reducing the calculation burden of the controller design process. (2) Also, it is demonstrated that the appropriate control of circulating current harmonics can significantly reduce the total MMC losses. The paper is organized as follows. The MMC under study is described in Section 2. The proposed control strategies are discussed in Section 3. Evaluation of the proposed control laws is accomplished in Section 4. Section 5 is dedicated to obtaining the equivalent SM capacitors of the MMC's arms. Section 6 presents the simulation results in the MATLAB/SIMULINK environment. Finally, in Section 7, the conclusions are drawn.