# Circulating Current Reduction in MMC-HVDC System Using Average Model
## 6 Analytical and Simulation Results
Figure 6: Results of (**a**) grid side voltage (peak line-to-line); (**b**) grid side current; (**c**) converter AC (peak line-to-line) side voltage; and (**d**) converter AC side current. Figure 7: \(Cont.\)  
The distortion in the peak values of the arm currents during the proposed direct modulation method is removed due to a reduction in the circulating current without affecting the input and output current waveforms of the converter. The grid side output current waveform of the converter in phase \(a\) is shown in Figure 7g. The addition of even order harmonics does not influence the AC side output current of the converter. Figure 8a,b shows the three-phase arm voltages for the direct modulation and proposed direct modulation methods. Due to the presence of the DC component at the top and bottom of arm voltages, there is a reduction in capacitor energy variations as DC voltage is constant (smooth) in a part of a period that reduces arm voltage oscillations in each leg of a three-phase MMC. The peak arm voltages (640 kV) in the proposed modulation method are lowered and flatter compared to peak arm voltages in the direct modulation method, which results in less oscillation. Figure 8: Results of (**a**) arm voltages-direct modulation and (**b**) arm voltages-proposed direct modulation. Figure 7: Results of (**a**) reference signal-direct modulation; (**b**) reference signal-proposed direct modulation; (**c**) upper arm current-direct modulation; (**d**) lower arm current-direct modulation; (**e**) upper arm current-proposed direct modulation; (**f**) lower arm current-proposed direct modulation; and (**g**) AC side current in phase \(a\). As per Equation (9), Figure 9a,b shows the \(i_{diff}\) currentin one leg. In Figure 8a, due to direct modulation, a circulating current is generated. In Figure 8b, due to a reduction in arm voltage oscillations by constant DC voltages, the circulating current amplitude is also reduced. A change in capacitor energy \(W_{c(t)}\) over half a period using the traditional direct modulation method and the proposed method can be calculated using Equation (51) based on simulation results and parameters available in Table 2:  
\[W_{c(t)}=\int_{-0.01}^{t}I_{leg(t)}.V_{leg(t)} \tag{51}\]  
(a) Direct modulation  
\[I_{leg(t)}=AC\ current+DC\ current\]  
Figure 5:0 (b) Proposed method  
Figure 5:1  
Figure 5:2  
where \(I_{leg(t)}\) is the leg current and \(V_{leg(t)}\) is the leg voltage. Less capacitor energy is witnessed during the proposed method compare to the direct modulation method for the same input and output power as shown in Figure 10a,b. Figure 5:3 Results of (**a**) circulating current-direct modulation and (**b**) circulating current-proposed direct modulation method. Analytical and simulation results are compared in Figure 5:4 Due to the proposed modulation, method RMS arm current is reduced compared to the direct modulation method, which is an important factor for power loss minimization in a three-phase MMC. The circulating current amplitude is also minimized due to the proposed method. The HVDC system parameters are given in Figure 5:5 \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Quantity** & **Symbol** & \multicolumn{2}{c}{**Direct Modulation**} & \multicolumn{2}{c}{**Proposed Direct Modulation**} \\ \hline  & & Amplitude & RMS Current & Amplitude & RMS Current \\ \hline Upper Arm Current & \(i_{1u}\) & 1.5 kA & 0.75 kA & 1.5 kA & 0.65 kA \\ \hline Lower Arm Current & \(i_{1l}\) & 1.5 kA & 0.75 kA & 1.5 kA & 0.65 kA \\ \hline Output Current in Phase \(a\) & \(i_{1}\) & 1.5 kA & - & 1.5 kA & - \\ \hline Differential Current & \(i_{diff}\) & 1 kA & - & 0.2 kA & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of analytical and simulation results. Figure 10: Results of (**a**) change in capacitor energy-direct modulation and (**b**) change in capacitor energy-proposed method.