## 2 Conventional Predictive Control Method
Therefore, it can be seen from (21) that the capacitor voltage balance can be maintained by adjusting the rectifier switching states, \(S_{a},~{}S_{b},~{}\text{and}~{}S_{c}\), by considering the three-phase input currents and the difference of the two capacitor voltage variations. In order to control the rectifier based on the reference voltage in the proposed method, the Equation (2) can be expressed as follows using the reference currents. \[v_{x}^{*}(k)=v_{sx}(k)-R_{i}i_{x}(k)-L_{i}\frac{i_{x}^{*}(k+1)-i_{x}(k)}{T_{s} }.~{}(x=a,~{}b,~{}c) \tag{22}\]  
where \(v_{x}^{*}\) is the reference voltage for each phase. To apply the delay compensation technique, the future reference voltage at the \((k+2)\)th step can be obtained by shifting (22) by one step forward as  
\[v_{x}^{*}(k+1)=v_{sx}(k+1)-R_{i}i_{x}(k+1)-L_{i}\frac{i_{x}^{*}(k+2)-i_{x}(k+1 )}{T_{s}}~{}.~{}(x=a,~{}b,~{}c) \tag{23}\]  
Switching state selections according to positions of the reference voltage is decided as  
\[S_{x}=\left\{ \begin{array}{ll}~{}1~{}(\text{P state})&\text{if}~{}\frac{V_{ dc}}{4}\leq v_{x}^{*}\leq\frac{V_{dc}}{2}\\ ~{}0~{}(\text{O state})&\text{if}~{}-\frac{V_{dc}}{4}<v_{x}^{*}<\frac{V_{dc}}{4} \\ ~{}-1~{}(\text{N state})&\text{if}~{}-\frac{V_{dc}}{2}\leq v_{x}^{*}\leq-\frac{V_{dc} }{4}\end{array} \right.~{}~{}~{}~{}(x=a,~{}b,~{}c). \tag{24}\]  
Figure 2 shows sectors, input current polarity, switching states and maximum and minimum reference voltages of the three-phase NP rectifiers operating with unity power factor, where \(v_{max}^{*}\) and \(v_{min}^{*}\) represent the maximum and minimum values of three-phase reference voltages, respectively. As shown in Figure 2, the voltages, \(v_{max}^{*}\) and \(v_{min}^{*}\), change every \(120^{\circ}\). Furthermore, the rectifier changes the switching state every \(60^{\circ}\) on the basis of the position of the reference voltages and (24). It can be known that the rectifier selects six switching states as shown in Figure 2, PNO, PON, OPN, NPO, NOP and ONP, according to (24). In addition to the reference voltage \(v_{x}^{*}\) for the reference current tracking in (22), a new reference voltage proposed in this paper is defined with an offset voltage injection as  
\[v_{x,ref}=v_{x}^{*}+v_{offset}(x=a,~{}b,~{}c) \tag{25}\]  
where \(v_{x,ref}\) and \(v_{offset}\) are a new reference voltage proposed in this paper and an offset voltage to be injected to maintain the capacitor voltage balance, respectively. The range of the offset voltage is, because the new reference voltage \(v_{x,ref}\) with the offset voltage injection must be lower than \(\frac{V_{dc}}{2}\) and higher than \(-\frac{V_{dc}}{2}\), as  
\[-\frac{V_{dc}}{2}-v_{x}^{*}\leq v_{offset}\leq\frac{V_{dc}}{2}-v_{x}^{*}~{}.~{ }(x=a,~{}b,~{}c) \tag{26}\]  
The maximum and minimum values of the offset voltage are defined as  
\[-\frac{V_{dc}}{2}-v_{min}^{*}\leq v_{offset}\leq\frac{V_{dc}}{2}-v_{max}^{*}. \tag{27}\]In cases where the positive offset voltage is injected, the magnitude of the reference voltage increases. For instance, if an initial switching state is N, a new state O can be selected after the positive offset voltage is injected. However, a state P cannot be selected after the offset voltage is injected, because the maximum offset voltage is limited as (27). Similarly, when the injected offset voltage is lower than zero, the reference voltage reduces. For instance, if a switching state before the negative offset voltage injection is P, the negative offset voltage injection enables a new state O to be selected after the injection. In addition, a switching state O without the negative offset voltage injection can change into a new state N after the offset voltage injection. Table 2 shows the switching states that can be selected after injecting a positive or negative offset voltage, when the rectifier operates at the sector 1 in Figure 2. As shown in Figure 2, the switching state in the sector 1, before the offset voltage is injected, is (PON).