## 4 Evaluation of the Proposed Control Laws  
The proposed control laws have a key role in reaching stable responses of the MMC under study. In this section, the three proposed control laws are compared to each other. Using the parameters in Table 1, the upper and lower control laws given in (8), (11), and (14) are drawn in Figure 5.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline MMC Inductance (H) & \(1\times 10^{-3}\) & SM Number & 10 \\ MMC resistance (Ohm) & 1 & Fundamental Frequency (Hz) & 60 \\ MMC Arm Inductance (H) & \(2.5\times 10^{-3}\) & Switching Frequency (kHz) & 20 \\ MMC Arm Resistance (Ohm) & 0.1 & The Impedance of First Nonlinear Load (Ohm) & 81 \\ SM Voltage (V) & 40 & The Impedance of Second Nonlinear Load (Ohm) & 92 \\ dc-Link Voltage (V) & 400 & Maximum Output Voltage (V) & 200 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameters of the proposed MMC in load change conditions.  
Figure 5: The proposed upper and lower control laws (CL) for the (**a**) first controller, (**b**) second controller, and (**c**) third controller.  
All figures were obtained without the harmonic compensation. Under zero circulating currents, the proposed control laws of phase \"a\" are shown in Figure 5a. Since the magnitude of the second-order harmonic of circulating current used for drawing Figure 5b is considered small, the two figures of Figure 5a,b seem to be the same. As can be seen from this figure, the harmonic components of the output currents and voltages cause the control laws to have waveforms with greater nonlinearity. On the other hand, the harmonic components due to the circulating currents generate control laws with more distortions, as shown in Figure 5c. In addition, the existence of the harmonic components in the circulating currents can raise the magnitude of the control laws by more than 2, leading to SLPWM with over-modulation, as depicted in Figure 5c. Also, according to this figure, a negative magnitude is achieved for the third upper control laws. From Figure 5a,b, it seems that both the first and second control laws have similar waveforms. In practice, the effect of the second-order harmonic component of the circulating currents can be realized by making a comparison between the switching signals.  
Figure 6 shows a 10-level shifted PWM used for the considered MMC with 10 SMs in each arm. Based on this figure, each triangle waveform is employed for its corresponding SM. By use of the 10-level triangle waveforms given in Figure 6, the SLPWM strategy was applied to the first and second control laws of phase \"a\" in Figure 5a,b, respectively. In comparison with the switching signals of the first control laws in Figure 7a, more switching operations are executed by the second control-law-based SLPWM, as depicted in Figure 7b. More switching operations causes greater switching losses. Thus, if a second-order harmonic component exists in the MMC circulating currents, the MMC switching losses are noticeably increased.  
To further assess the switching signals, a distributed pattern of switching signals was driven by the third control-law-based SLPWM, as demonstrated by Figure 7c. In comparison with the switching signals of the first control laws in Figure 7a, inappropriate switching signals were achieved by the third control laws, as shown in Figure 7c. Furthermore, due to the over-modulation condition caused by the harmonic components of circulating currents, the third control laws cannot provide accurate switching signals in a certain operating period.  
Figure 6: The 10-level triangle waveforms used.  
Figure 7: The signals of upper and lower switches of the MMC for the (**a**) first controller, (**b**) second controller, and (**c**) third controller.