## 2 Single-Phase NPC Converter Structure and Conventional MPC Approach  
A single-phase three-level NPC converter, operating in rectifier mode, is shown in Figure 1. In the ac input side of the converter, a voltage source \(v_{s}\) provides the source current \(i_{s}\) through an ac side filter including input inductor \(L_{s}\) and resistor \(R_{s}\). The output dc side is composed of two capacitors \(C\) and an output resistor load \(R_{\mathrm{L}}\), where \(v_{c1}\) and \(v_{c2}\) indicate the dc capacitor voltages, and \(V_{dc}\) represents the output dc side voltage. The input ac side is connected to the output dc side through two legs with eight switching devices and antiparallel diodes. The Insulated Gate Bipolar Transistors (IGBTs) and corresponding switching states in phase \(a\) and \(b\) are represented by \(S_{aj}\) and \(S_{bj}\) (\(j=1\), \(2\), \(3\), \(4\)), respectively. The switching states of each phase can be described as  
\[S_{x}=\left\{ \begin{array}{rl}1\;if\;S_{x1},S_{x2}:\;ON\\ 0\;if\;S_{x2},S_{x3}:\;ON\\ -1\;if\;S_{x3},S_{x4}:\;ON\end{array} \right.\quad x=a,\;b). \tag{1}\]  
In this, the upper and lower switches operate complementarily. The combination of switching states in Equation (1) generates nine different operating states in single-phase NPC operation. The possible switching state, the resulting switching status, the converter input voltage \(v_{ab}\), and corresponding capacitor voltages are shown in Table 1 for these nine states. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{\#} & \multicolumn{2}{c}{**Operating Status**} & \multicolumn{3}{c}{**Switching State**} & \multicolumn{3}{c}{**Input**} & \multirow{2}{*}{**Capacitor Voltage**} \\ \cline{2-2} \cline{5-10}  & \(S_{a}\) & \(S_{b}\) & \(S_{a1}\) & \(S_{a2}\) & \(S_{b1}\) & \(S_{b2}\) & \(v_{ab}\) & \(v_{c1}\) & \(v_{c2}\) \\ \hline
1 & 0 & 0 & OFF & ON & OFF & ON & 0 & - & - \\
2 & 1 & 1 & ON & ON & ON & ON & 0 & - & - \\
3 & \(-1\) & \(-1\) & OFF & OFF & OFF & OFF & 0 & - & - \\
4 & 1 & \(-1\) & ON & ON & OFF & OFF & \(V_{dc}\) & \(\uparrow\) & \(\uparrow\) \\
5 & 1 & 0 & ON & ON & OFF & ON & \(V_{dc}/2\) & \(\uparrow\) & \(\downarrow\) \\
6 & 0 & \(-1\) & OFF & ON & OFF & OFF & \(V_{dc}/2\) & \(\downarrow\) & \(\uparrow\) \\
7 & 0 & 1 & OFF & ON & ON & ON & \(-V_{dc}/2\) & \(\uparrow\) & \(\downarrow\) \\
8 & \(-1\) & 0 & OFF & OFF & OFF & ON & \(-V_{dc}/2\) & \(\downarrow\) & \(\uparrow\) \\
9 & \(-1\) & 1 & OFF & OFF & ON & ON & \(-V_{dc}\) & \(\downarrow\) & \(\downarrow\) \\ \hline \hline \end{tabular} \# items number.  
\end{table} 
Table 1: Units for magnetic properties. Figure 1: Structure of single-phase NPC converter. The nine operating states can generate the five-level input voltage \(v_{ab}\), which results in redundant operating states. As presented in Table 1, the two redundant states, (1, 0) and (0, -1), for (\(S_{a}\), \(S_{b}\) ) produce the same converter input voltage level but a different effect on the capacitor voltages. Likewise, the redundant states (0, 1) and (-1, 0) synthesize the same converter input voltage and opposite capacitor voltage behaviors. Hence, these redundant states will be used to control the capacitor voltages, \(v_{c1}\) and \(v_{c2}\). The currents \(i_{c1}\) and \(i_{c2}\), in Figure 1, are described by the operating status and the source current \(i_{s}\) in Equations (2) and (3), respectively. Therefore, these currents are acquired without other measurements [28].