## 2 Analysis of Unbalanced NPV
### Basic Principles
\tag{8}\]  
where \(k\) is the variable that determines whether the NPV is outside the fluctuation range. The acquisition method of symbolic variable \(k\) is shown in Figure 3.  
Figure 2: Equivalent process from three-level small-voltage vector to two-level zero-voltage vector. As can be seen from the definitions in Table 2, \(k=0\) means that the NPV does not exceed the specified minimum fluctuation range, and in this case, \(k_{1}=1\) and \(k_{2}=0\) respectively, i.e., there is no need to add the NPV control function. Additionally, \(k=-1\) or \(k=1\) means that the NPV exceeds the minimum allowable fluctuation range, and in this case, \(k_{1}=0\), i.e., it is necessary to add the NPV control function. When \(k_{1}=0\), \(k_{2}\) is defined as 0 and 1, respectively, and \(k_{2}\) is used to determine whether \(\mathbf{V}_{00}[000]\) or \(\mathbf{V}_{07}[111]\) is finally selected. If \(k_{2}=1\), the NPV is deviated and \(\Delta v_{\text{dc}}>\Delta v_{\text{dc+}}\), \(\mathbf{V}_{07}[111]\) is selected; in contrast, if \(k_{2}=0\), the NPV is deviated and \(\Delta v_{\text{dc}}<\Delta v_{\text{dc}}\), \(\mathbf{V}_{00}[000]\) is selected. Therefore, the NPV of the VIENNA circuit can be controlled by adjusting or controlling its own six pairs of small-voltage vectors, i.e., the two zero-voltage vectors corresponding to its two-level vector distribution. The NPV of the VIENNA circuit can be controlled in each 60-degree equal interval by the following equation:  
\[s_{a}=s_{b}=s_{c}=[\,(s^{\prime}_{a}||s^{\prime}_{b})\&\&(s^{\prime}_{b}||s^{ \prime}_{c})\&\&(s^{\prime}_{c}||s^{\prime}_{a})]\&\&k_{1}||k_{2} \tag{9}\]  
where, \([(s^{\prime}_{a}|s^{\prime}_{b})\&\&(s^{\prime}_{b}|s^{\prime}_{c})\&\&(s^{ \prime}_{c}|s^{\prime}_{a})]\) is Equation (7); \(k_{1}\) is the variable used to determine whether it is necessary to add the NPV control function; and \(k_{2}\) is the variable that determines whether \(\mathbf{V}_{00}[000]\) or \(\mathbf{V}_{07}[111]\) is finally selected when it is necessary to add the NPV control function. Figure 4 shows the control flow of the VIENNA rectifier. The part in the dotted line box in Figure 4 is the realization process of the NPV control strategy proposed in this paper. It can be seen clearly in Figure 4 that the balance control of NPV is realized through real-time optimizing the selection method of positive and negative small-voltage vectors \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Category**} & **NPV Control** & \multirow{2}{*}{\(\mathbf{k}\)} & \multirow{2}{*}{\(\mathbf{k_{1}}\)} & \multirow{2}{*}{\(\mathbf{k_{2}}\)} \\  & **Requirements** & & & \\ \hline \(\Delta v_{\text{dc}}\)\(<\Delta v_{\text{dc}}<\)\(-\Delta v_{\text{dc+}}\) & Not required & 0 & 1 & 0 \\ \hline \(\Delta v_{\text{dc}}>\Delta v_{\text{dc+}}\) & Required & 1 & 0 & 1 \\ \hline \(\Delta v_{\text{dc}}<\Delta v_{\text{dc}}\) & Required & \(-1\) & 0 & 0 \\ \hline \end{tabular}
\end{table} 
Table 2: Judgment rule of NPV control requirements. Figure 3: Acquisition method of the symbolic variable \(k\). dynamically. With the principle of \"MNSC\" adopted together, the proposed NPV control strategy can control the NPV with the minimum switching times, and the implementation process is simple and convenient. Theoretically, it can effectively reduce the equivalent switching frequency, minimizing the influence on the AC current quality. Figure 4: Control flow of VIENNA rectifier. Figure 2: VIENNA circuit is equivalent to a three-phase two-level converter to control in each 60-degree interval.