## 2 Single-Phase NPC Converter Structure and Conventional MPC Approach
### Common-Mode Voltage Term for Capacitor Voltage Balancing  
For NP voltage balancing, the four switching states with redundancy in Table 1 need to be selected. As a result, the proposed method includes a common-mode reference voltage, which allows the reference voltage to select redundant switching states for NP voltage balancing. In the proposed MPC method, the reference voltage is defined by adding the difference-mode voltage term and the common-mode term as in:  
\[v_{ref}^{n}(k+1) = v_{diff}^{n}(k+1)+v_{comm}(k+1) \tag{24}\] \[v_{ref}^{b}(k+1) = v_{diff}^{b}(k+1)+v_{comm}(k+1). \tag{25}\]  
The difference-mode voltage terms in the reference voltages deal with sinusoidal source current generation, whereas the common-mode voltage term employs the NP voltage balancing. Figure 3 shows that the reference voltages added by the positive common-mode voltage can change the positions of the difference-mode reference voltages shown in Figure 2. For example, Figure 2(a) illustrates that the difference-mode reference voltages, \(v_{diff}^{a}(k+1)\) and \(v_{diff}^{b}(k+1)\), were placed just like Figure 1(a). By adding the positive common-mode voltage, the reference voltages will be increased, as presented in Figure 2(a). Changing the position of the reference voltages can make the operating state become \((1,\ 0)\), as known from Equation (23), which corresponds to the redundant switching state for NP voltage balance control. Note that the switching state in Figure 2(a) can remain in \((1,\ -1)\) after adding the common-mode voltage term \(v_{comm}(k+1)\) if the magnitude of the positive common-mode voltage is small. Likewise, Figure 2(c),d shows that the reference voltages with a positive common-mode voltage can result in the operating state \((0,\ 1)\), which is also a redundant switching state for NP voltage balancing. Similarly, it is seen from Figure 4 that adding a negative common-mode voltage can result in NP voltage balancing states. Because it is confirmed that the addition of the common-mode voltages to the difference-mode voltages can balance NP voltages, it is necessary to determine the polarity and the magnitude of the common-mode voltage. To ensure stable NP voltage balancing, the Lyapunov theory is employed as  
\[\frac{dV}{dt}\leq 0,\ V\geq 0 \tag{26}\]  
where \(V=v_{c(gap)}^{2}\). The difference between the upper and lower capacitor voltages should satisfy the following for NP voltage balance:  
\[2v_{c(gap)}\frac{dv_{c(gap)}}{dt}\leq 0. \tag{27}\]  
Equation (27) can be expressed in the discrete-time domain as:  
\[v_{c(gap)}(k)\frac{\Delta v_{c(gap)}(k)}{T_{s}}\leq 0. \tag{28}\]  
Using Equation (8), Equation (28) is written as  
\[v_{c(gap)}(k)\frac{i_{s}(k)}{C}S_{poly}\leq 0 \tag{29}\]  
where \(S_{poly}=S_{a}^{2}-S_{b}^{2}\). Thus, one can obtain the voltage polarity between the upper and lower capacitors, the source current, and the switching states, which should be satisfied for stable NP voltage balancing operation, as in  
\[sgn\Big{(}v_{c(gap)}(k)\,i_{s}(k)S_{poly}\Big{)}\leq 0 \tag{30}\]  
Figure 4: Possible locations of reference voltages (\(t_{ref}^{a}\) and \(v_{ref}^{b}\) ) with a negative common-mode voltage (\(v_{comm}>0\) ) corresponding to Figure 2, leading to switching states: (**a**) (\(0\), \(-1\)); (**b**) (\(0\), \(-1\)); (**c**) (\(-1\), \(0\)); (**d**) (\(-1\); \(0\)). Figure 3: Possible locations of reference voltages in phase \(a\) and \(b\) (\(t_{ref}^{a}\) and \(v_{ref}^{b}\) ) with a positive common-mode voltage (\(v_{comm}>0\) ) corresponding to Figure 2, leading to switching states: (**a**) (\(1\), \(0\)); (**b**) (\(1\), \(0\)); (**c**) (\(0\), \(1\)); (**d**) (\(0\), \(1\)). where \(sgn(. )\) is a function to output the polarity of an input variable. From Figures 3 and 4, it can be seen that the polarity of the difference-mode reference voltage, the polarity of the common-mode voltage, and the switching states are related.