## 2 Conventional Predictive Control Method  
Figure 1 shows a three-phase three-level NP rectifier. The ac input side of the circuit structure in Figure 1 gives an equation as  
\[v_{\text{sx}}=R_{i}i_{x}+L_{i}\frac{di_{x}}{dt}+v_{x}(x=a,\ b,\ c) \tag{1}\]  
where \(v_{\text{sx}}\), \(i_{x}\), \(R_{i}\), \(L_{i}\) and \(v_{x}\) represent the source voltage, the input current, the input resistance, the input inductance, and the phase voltage of the rectifier, respectively. Using a constant sampling period \(T_{s}\) and [PERSON]'s formula, the Equation (1) is described in the discrete time domain as  
\[v_{\text{sx}}(k)=R_{i}i_{x}(k)+L_{i}\frac{i_{x}(k+1)-i_{x}(k)}{T_{s}}+v_{x}(k).\ (x=a,\ b,\ c) \tag{2}\]  
Figure 1: Three-phase three-level NP rectifier. By rearranging (2), one can obtain  
\[i_{x}(k+1)=\left(1-\frac{R_{i}T_{s}}{L_{i}}\right)i_{x}(k)+\frac{T_{s}}{L_{i}}(v_ {sx}(k)-v_{x}(k)).(x=a,\ b,\ c) \tag{3}\]  
where \(v_{x}\) which is the rectifier pole voltage with respect to the node \(O\) in Figure 1, can be determined based on the switching states as presented in Table 1.  
The predictive control technique evaluates all switching state combinations to select an optimal switching state. Therefore, computational time included in the evaluation algorithm is compensated in the delay compensation techniques, in which the future current at the (\(k+2\))th step is used for the predictive control method [24; 25]. The future current at the (\(k+2\))th step is acquired through shifting the model of (3) for one step forward as  
\[i_{x}(k+2)=\left(1-\frac{R_{i}T_{s}}{L_{i}}\right)i_{x}(k+1)+\frac{T_{s}}{L_{i }}(v_{sx}(k+1)-v_{x}(k+1)).\ (x=a,\ b,\ c) \tag{4}\]  
A switching function according to switching operation of each phase in the rectifier can be defined as  
\[S_{x}=\left\{ \begin{array}{l}S_{x}=1\ \ \ \ \ (S_{x1},S_{x2}:\ ON)\\ S_{x}=0\ \ \ \ (S_{x2},S_{x3}:\ ON)\ \ \.\ (x=a,\ b,\ c)\\ S_{x}=-1\ \ \ (S_{x3},S_{x4}:\ ON)\end{array} \right.\ \ (x=a,\ b,\ c) \tag{5}\]  
where \(S_{ax},\ S_{bx},\) and \(S_{cx}\ (x=1,\ 2,\ 3,\ 4)\) refer to switching states of the four switches located on the \(a,\ b\) and \(c\) phase. The rectifier switching states and the input currents can be used to describe the currents flowing through the upper and lower capacitors as shown in (6) and (7). \[i_{c1}=\frac{S_{a}(S_{a}+1)}{2}i_{a}+\frac{S_{b}(S_{b}+1)}{2}i_{b}+\frac{S_{c} (S_{c}+1)}{2}i_{c}. \tag{6}\]  
\[i_{c2}=-\frac{S_{a}(S_{a}-1)}{2}i_{a}-\frac{S_{b}(S_{b}-1)}{2}i_{b}-\frac{S_{c }(S_{c}-1)}{2}i_{c}. \tag{7}\]  
where \(i_{c1}\), and \(i_{c2}\) represent the currents flowing through the upper and lower capacitors, respectively. In addition, the upper capacitor voltage and the lower capacitor voltage can be expressed as  
\[\frac{dv_{ci}}{dt}=\frac{1}{C}i_{c1}\ (i=1,\ 2) \tag{8}\]  
where \(C\) is the capacitance of the upper capacitor and lower capacitor. \(v_{c1}\) and \(v_{c2}\) represent the voltage of the upper and lower capacitors, respectively. With a sampling period \(T_{s}\), the variation of the capacitor voltage is expressed in the discrete time domain as follows. \[\frac{dv_{ci}}{dt}\approx\frac{v_{ci}(k+1)-v_{ci}(k)}{T_{s}}.\ (i=1,\ 2) \tag{9}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Switching** & & **Switching Status (\(\mathbf{r}=a\),\(b\),\(c\))** & & **Pole Voltage** \\
**State** & \(S_{\mathbf{x1}}\) & \(S_{\mathbf{x2}}\) & \(S_{\mathbf{x3}}\) & \(S_{\mathbf{x4}}\) & \(v_{\mathbf{x}}\) \\ \hline P & ON & ON & OFF & OFF & \(V_{dc}/2\) \\ O & OFF & ON & ON & OFF & \(0\) \\ N & OFF & OFF & ON & ON & \(-V_{dc}/2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: States and pole voltage of NP rectifier depending on switch status. Using (17), (20) can be written by the difference of the two capacitor voltage variations \(v_{diff}\) and a function related with the input currents and the switching functions of the rectifier \(M\), as  
\[\frac{v_{diff}}{C}M\leq 0 \tag{21}\]  
where \(M=S_{a}^{2}i_{a}+S_{b}^{2}i_{b}+S_{c}^{2}i_{c}\).