## 2 Analysis of Unbalanced NPV  
### Causes of Unbalanced NPV  
The topology of the VIENNA circuit is shown in Figure 1.  
The dashed box in the right half of Figure 1 shows the DC portion of the VIENNA circuit, in which the NPV problem will be analyzed. Taking the neutral point \(N\) on the DC  
Figure 1: Topology of VIENNA circuit.  
side as the voltage reference point, the current flowing into the point \(N\) is defined as \(i_{N}\); the currents flowing through \(C_{1}\) and \(C_{2}\) are \(i_{C1}\) and \(i_{C2}\) respectively; the capacitance values of the capacitors \(C_{1}\) and \(C_{2}\) are equal, i.e., \(C_{1}=C_{2}=C\); the DC bus voltage is defined as \(v_{\text{dc}}\), and the capacitor voltages are defined as \(v_{\text{dc1}}\) and \(v_{\text{dc2}}\) respectively. It can be seen from the theory of charging process of the current to the capacitor that:  
\[\left\{\begin{array}{l}v_{\text{dc1}}=\frac{1}{C}\int_{0}^{t}i_{\text{c1}} dt+v_{\text{dc1}\_0}\\ v_{\text{dc2}}=\frac{1}{C}\int_{0}^{t}i_{\text{c2}}dt+v_{\text{dc2}\_0}\end{array}\right. \tag{1}\]  
where \(v_{\text{dc1}\_0}\) and \(v_{\text{dc2}\_0}\) are the zero-time voltages of \(C_{1}\) and \(C_{2}\) respectively.  
Based on Kirchhoff's Current Law (KCL), it can be derived that:  
\[i_{N}=i_{\text{C2}}-i_{\text{C1}} \tag{2}\]  
According to Equations (1) and (2), the NPV deviation \(\Delta v_{\text{dc}}\) can be obtained as:  
\[\Delta v_{\text{dc}}=v_{\text{dc1}}-v_{\text{dc2}}=\frac{1}{C}\int_{0}^{T_{ \text{s}}}{(i_{\text{C1}}-i_{\text{C2}})dt}=-\frac{1}{C}\int_{0}^{T_{\text{s} }}{i_{N}dt} \tag{3}\]  
According to Equation (3), it is clear that the root cause of unbalanced NPV is the non-zero integral value of \(i_{N}\) during a control cycle T\({}_{s}\), which in turn leads to unequal charging and discharging voltages of capacitors \(C_{1}\) and \(C_{2}\). Therefore, in case of NPV deviation, the NPV can be balanced again by adjusting the flow direction or size of \(i_{N}\) to act on the NPV.  
As shown in Figure 1, the three-phase AC current \(i_{n}\) (\(n=a,\,b,\,c\)) and the three-phase voltage \(e_{n}\) are defined, with the output direction of the three-phase voltage \(e_{n}\) as the positive direction of \(i_{n}\). The three-phase switching function \(s_{n}\) is also defined, \(s_{n}\) satisfying:  
\[\left\{\begin{array}{l}s_{n}=1;\quad\text{the $n^{\text{th}}$--phase arm switch $s_{n}$ is turned on.}\\ s_{n}=0;\quad\text{the $n^{\text{th}}$--phase arm switch $s_{n}$ is turned off.}\end{array}\right. \tag{4}\]  
As can be seen in Figure 1, only when a certain phase or multi-phase switch of the three-phase switches \(S_{n}\) is turned on, i.e., the corresponding switch function \(s_{n}=1\), there is current flowing through the branch corresponding to the neutral point current \(i_{N}\) and \(i_{N}\) will not be zero. If all the three-phase switches are turned off, i.e., the corresponding three-phase switch functions all satisfy \(s_{n}=0\), there is no current flowing through the branch corresponding to the neutral point current \(i_{N}\) and the value of \(i_{N}\) will also be zero. Therefore, the value of the neutral point current \(i_{N}\) on the DC side of the VIENNA circuit depends on the three-phase switch \(s_{n}\) and the three-phase current \(i_{n}\), i.e., it satisfies:  
\[i_{N}=s_{u}i_{a}+s_{b}i_{b}+s_{c}i_{c} \tag{5}\]  
Therefore, it can be seen from Equation (5) that when the NPV is unbalanced, different voltage vectors, i.e., different combinations of three-phase switch states, can be selected to adjust the NPV to balance again.