## 2 Conventional Predictive Control Method
After the positive offset voltage is injected to the state (PON) in the sector 1, the two states, (POO) and (PON), can be possible because of the boundary conditions of the offset voltage in (27) and the locations of the reference voltages in Figure 2. Likewise, the negative offset voltage injection in the sector 1 can result in the two states, (OON) and (PNN), from (27) and Figure 2. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(v_{offset}\) & \(S_{a}\) & \(i_{a}\) & \(S_{b}\) & \(i_{b}\) & \(S_{c}\) & \(i_{e}\) \\ \hline \multirow{2}{*}{\(+\)} & P & & O & & & O & \\  & P & & O & & & N & \\ \multirow{2}{*}{\(-\)} & O & & N & & N & & \\  & P & & N & & N & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: States after the offset voltage is injected, the sign of the injected offset voltage, and the sign of input currents in a case of sector 1. Figure 2: Sectors, input current polarity, switching states, maximum and minimum reference voltages of the rectifier with unity power factor. In cases where the positive offset voltage is injected into the reference voltage \(v_{x}^{*}\) in the sector 1, new switching states can be (POO) and (PON) as shown in Table 2. The function M in (21) for the switching states (PPO) and (PON), by considering the polarity of the input currents in the sector 1, are given by (28) and (29), respectively. \[S_{a}^{2}i_{a}+S_{b}^{2}i_{b}+S_{c}^{2}i_{c}=i_{a}+i_{b}>0 \tag{28}\]  
\[S_{a}^{2}i_{a}+S_{b}^{2}i_{b}+S_{c}^{2}i_{c}=i_{a}+i_{c}>0 \tag{29}\]  
Therefore, when \(v_{diff}<0\), meaning \(v_{c1}<v_{c2}\), injecting the offset voltage higher than zero satisfies (21). Figure 3a,b show the capacitor current flow when the switching states are (POO) and (PON). In Figure 3a, the current flows only to the upper capacitor. Thus, the associated capacitor is charged. As shown in Figure 3b with \(i_{a}=i_{b}+i_{c}\), the upper capacitor charges. Thus, the new switching states after injecting the positive offset voltage in the sector 1, when the voltage difference \(v_{diff}\) is lower than zero, can reduce the voltage difference by increasing the upper capacitor voltage, which can maintain the capacitor voltage balance. On the other hand, injecting the negative voltage \(v_{offset}\) into the reference voltage \(v_{x}^{*}\) in the sector 1 can generate new switching states (ONN) and (PNN) as shown in Table 2. The function M for the switching states (ONN) and (PNN) by considering the polarity of the input currents in the sector 1 are (30) and (31), respectively. \[S_{a}^{2}i_{a}+S_{b}^{2}i_{b}+S_{c}^{2}i_{c}=i_{b}+i_{c}<0 \tag{30}\]  
\[S_{a}^{2}i_{a}+S_{b}^{2}i_{b}+S_{c}^{2}i_{c}=i_{a}+i_{b}+i_{c}=0 \tag{31}\]  
Therefore, if the negative offset voltage is injected when the voltage difference \(v_{diff}>0\), meaning \(v_{c1}>v_{c2}\), the new switching states generated after the offset voltage injection satisfies (21). Figure 4a,b show the capacitor current path with the switching states (ONN) and (PNN), respectively. As shown in Figure 4a, the lower capacitor is charged by the current flowing and the voltage \(v_{c2}\) is increased. As a result, the voltage difference \(v_{diff}\), which was greater than zero, can be reduced, leading to the capacitor voltage balancing. Figure 4b shows the switching state (PNN), and the capacitor voltage does not change because no current flows through the neutral point, which gives no harmful effects on the capacitor voltage balancing. Figure 3: Capacitor current flow of sector 1 with switching status: (**a**) (POO); (**b**) (PON). Based on the same approach used in sector 1, relationships between the capacitor voltage difference and the sign of the offset voltage can be established in the other sectors from 2 to 12 in Figure 2. The sign of the offset voltage to retain the balance of the capacitor voltage can be defined, with the opposite sign of the capacitor voltage difference.