## 3 Control Algorithm
### Neutral-Point Voltage Balance Control  
It is important to note that if the NPV is unbalanced in the 3L-NPC inverter, several issues will limit its employment, such as current distortion, increased torque ripple, and even the serious damage of semiconductor devices. Therefore, this paper attempts to address the unbalanced NPV issue of a 3L-NPC inverter through the investigation of the relationships between various voltage vectors and the NPV.  
Initially, the drive circuit of a 3L-NPC-inverter-fed PMSM can be equivalent to a simplified circuit model, as shown in Figure 4. Here, \(C_{1}\) and \(C_{2}\) donate the voltage-dividing capacitors, whose flowing currents are \(i_{c1}\) and \(i_{c2}\), respectively, and \(C_{1}=C_{2}\); \(i_{a}\), \(i_{b}\) and \(i_{c}\) represent the three-phase currents, and the sum of the all three is zero; \(i_{np}\) is the neutral-point current; \(S_{a}\), \(S_{b}\) and \(S_{c}\) correspond to the switch states of the simplified three-phase bridge arms, which can only be set as \(-1\), \(0\), or \(1\) when the switch is connected to N, O, and P, respectively. Intrinsically, the reason for the unbalanced NPV is because the \(i_{np}\) has failed to remain at 0A, in conjunction with the charge and discharge of the dc-bus capacitors. In addition, the neutral-point current \(i_{np}\) can be expressed as  
\[i_{np}=\sum_{x=a,b,c}{(1-|S_{x}|)i_{x}} \tag{7}\]  
On the condition of \(S_{x}=-1\) or \(S_{x}=1\), namely, \(1-|S_{x}|=0\), \(i_{np}\) with zero value has no effect on the neutral-point voltage. When \(S_{x}=\)0, namely, \(1-|S_{x}|=1\), it is obvious that \(i_{np}\) fails to remain at zero, resulting in the charge-discharge operation of \(C_{2}\). In this way, uneven partial pressure occurs between \(C_{1}\) and \(C_{2}\), indicating the problem of an unbalanced NPV.  
Furthermore, the effects of different switching vectors on the NPV are shown in Table 1. The symbols \"+\", \"-\", and \"=\" indicate the states of the NPV in the following order: increase, decrease, and invariant. \(i_{np}\) is decreased when employing medium vectors, and remains unchanged when employing large vectors and zero vectors, while the positive and negative redundant small vectors have opposite impacts on the neutral-point current. Additionally, each set of positive and negative redundant small vectors have the same magnitude and direction, indicating that their application leads to the same control effect on motor control. As such, when the \(V_{min}\) outputting from the cost function is a small vector, the balance control of the NPV in the 3L-NPC inverter can be realized through replacing the positive and negative redundant small vectors. For example, if \(V_{1}=[0-1-1]\) is the applied vector to the inverter, \(i_{np}\) will be equal to \(i_{a}\) (assuming \(i_{a}>0\)); according to Equation (7), \(C_{2}\) will be discharged, leading to the decrease in the NPV. In this time, if the sampled neutral-point voltage \(U_{0}\) is less than zero, the counterpart of \(V_{1}\), namely \(V_{2}=[100]\), should be adopted, which causes \(C_{2}\) to be charged, and the NPV therefore increases.  
Figure 4: An equivalent circuit model of the 3L-NPC inverter-fed PMSM.