## 7 Conclusions  
In this paper, we designed three nonlinear PWM-based control strategies to eliminate MMC circulating currents, keeping the MMC stable under varying nonlinear loads. The MMC output currents and voltage harmonics were utilized to drive the first control law. Then, the control parts pertaining to the circulating currents' second-order harmonic were added to the first control law, leading to the proposed second control law. The third control law was designed by controlling the magnitudes and phase angles of all circulating current harmonics to force them to become practically zero. By assessing the switching signals under the proposed control laws, it was proved that the second-order and other harmonics caused an increment in MMC losses and decrement in the MMC lifetime, respectively. In addition, using the proposed control laws, the instantaneous equivalent SM capacitors were obtained. It was intended for each arm to have its own special capacitor in which the average and fluctuation values of these capacitors were noticeably decreased by the elimination of the harmonics of the circulating currents through the proposed control laws. The simulation results in MATLAB/SIMULINK software clearly demonstrated that the third control-law-based method provided better steady and  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Error of** & \begin{tabular}{c} **Method1** \\ **(SS)** \\ \end{tabular} & \begin{tabular}{c} **Method2** \\ **(SS)** \\ \end{tabular} & \begin{tabular}{c} **Method3** \\ **(SS)** \\ \end{tabular} & \begin{tabular}{c} **Method1** \\ **(DS)** \\ \end{tabular} & \
\begin{tabular}{c} **Method2** \\ **(DS)** \\ \end{tabular} &
\begin{tabular}{c} **Method3** \\ **(DS)** \\ \end{tabular} \\ \hline Upper capacitor voltage & \(1.5\%\) & \(1.25\%\) & \(1.09\%\) & \(2\%\) & \(1.625\%\) & \(1.34\%\) \\ Lower capacitor voltage & \(1.3\%\) & \(1.252\%\) & \(1.125\%\) & \(2.25\%\) & \(2.02\%\) & \(2.01\%\) \\ Circulating current magnitude & \([-3,2.1]\) & \([-1.35,1.15]\) & \([-0.7,0.6]\) & \([-3.9,2.4]\) & \([-1.5,1.3]\) & \([-0.86,0.7]\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: A comparison between the steady-state (SS) and dynamic (DS) performance of the three proposed control strategies.  
Figure 17: The lower arm currents of phase “a” for the (**a**) first controller, (**b**) second controller, and (**c**) third controller.  
dynamic responses for all MMC currents and voltages when compared to the other two control techniques by its ability to control all harmonic components of the circulating currents.  
M.M.: Writing the manuscript and simulation; E.P.: conceptualization and methodology; R.G.: editing and data analyzing; E.M.G.R.: editing and funding, J.P.S.C.: editing and supervision. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The authors gratefully acknowledge the contributions of [PERSON]. [PERSON] acknowledges Fundacao para a Ciencia e a Tecnologia (FCT--MCTES) for its financial support via the project UIDB/00667/2020 (UNIDEMI). [PERSON] acknowledges the support by FEDER funds through COMPETE 2020 and by Portuguese funds through FCT, under POCI-01-0145-FEDER-029803 (02/SAICT/2017) and POCI-01-0145-FEDER-006961 (UID/EEA/50014/2019).  
The authors declare no conflict of interest.