## 2 Analysis of Unbalanced NPV
### Basic Principles  
As shown in Table 1, the positive vectors and negative vectors have opposite effects on NPV. Therefore, according to the direction and amplitude of NPV fluctuation, NPV can be controlled by dynamically adjusting the selection method of the small-voltage vector. In the control of the three-level converter, a power frequency cycle is usually divided into six equal segments of 60-degree intervals each, while based on the correspondence between three-level vector and two-level vector, the three-level vector is corresponded to the two-level vector distribution in each 60-degree interval, and the three-level converter is equivalent to a three-phase two-level converter for control in the interval [23]. In this process, the paired small-voltage vector corresponds exactly to the zero-voltage vector of the three-phase two-level converter, as shown in Figure 2 below. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Zero Voltage** & \(i_{N}\) & **Large Voltage** & \(i_{N}\) & **Medium Voltage** & \(i_{N}\) & **Small Positive** & \(i_{N}\) & **Small Negative** & \(i_{N}\) \\
**Vector** & & **Vector** & & **Vector** & & **Voltage Vector** & & **Voltage Vector** & \\ \hline \multirow{4}{*}{OOO} & \multirow{4}{*}{0} & PNN & \multirow{4}{*}{0} & OPN & \(i_{a}\) & ONN & \(i_{a}\) & POO & \(-i_{a}\) \\  & & PPN & & PON & \(i_{b}\) & POP & \(i_{b}\) & ONO & \(-i_{b}\) \\  & & NPN & & NPO & \(i_{c}\) & NNO & \(i_{c}\) & OOP & \(-i_{c}\) \\  & & NPP & & ONP & \(i_{a}\) & OPP & \(i_{a}\) & NOO & \(-i_{a}\) \\  & & NNP & & NOP & \(i_{b}\) & NON & \(i_{b}\) & OPO & \(-i_{b}\) \\  & & PNP & & PNO & \(i_{c}\) & PPO & \(i_{c}\) & OON & \(-i_{c}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Corresponding \(i_{N}\) of different vectors. The two-level zero-voltage vector \(\mathbf{V}_{0}\) includes \(\mathbf{V}_{00}[000]\) and \(\mathbf{V}_{07}[111]\). If the control logic determines that the zero-voltage vectors are to be executed, the next step should also be selected between the two zero-voltage vectors. Generally, in order to reduce the switching frequency of each arm and the current THD, the selection of the two zero vectors is based on the principle of \"minimum number of switching changes (MNSC)\". That is, according to the switching state of each phase in the previous control cycle, when making selection between the two zero vectors in the current cycle, only the switching state of one phase changes. The principle of \"MNSC\" is implemented by the following equation:  
\[s_{a}=s_{b}=s_{c}=\big{(}s^{\prime}_{a}\big{|}\big{|}s^{\prime}_{b}\big{)} \&\&\big{(}s^{\prime}_{b}\big{|}\big{|}s^{\prime}_{c}\big{)}\&\&\big{(}s^{ \prime}_{c}\big{|}\big{|}s^{\prime}_{a}\big{)} \tag{7}\]  
where \(\parallel\) is the logical operation \"or\", \(\&\&\) is the logical operation \"add\", and \(s^{\prime}_{n}\) and \(s_{n}\) are the switching states of each phase of the previous cycle and the current cycle, respectively, in the control process. Based on the analysis, the process of NPV control using small-voltage vectors of the VIENNA circuit can be equivalent to the process of selective control for operating times of two zero vector in each 60-degree interval (corresponding to the two-level vector distribution). Therefore, an NPV control strategy based on the dynamical adjustment for selection method of the small-voltage vector is proposed. By judging whether the NPV is within the allowable range of fluctuation, it is decided whether the NPV needs to be controlled, and the appropriate vector is selected for control according to its fluctuation direction. The allowable range of fluctuation [\(\Delta v_{\text{dc-}}\), \(\Delta v_{\text{dc+}}\)] and symbolic variable \(k\) of the NPV are defined, with \(k\) meeting:  
\[\left\{ \begin{array}{ll}k=0,&\text{When}\,\Delta v_{\text{dc-}}<\Delta v_{ \text{dc}}<\Delta v_{\text{dc+}}\\ k=1,&\text{When}\,\Delta v_{\text{dc}}>\Delta v_{\text{dc+}}\\ k=-1,&\text{When}\,\Delta v_{\text{dc}}<\Delta v_{\text{dc-}}\end{array} \right.