## 2 Conventional FCS-MPC for the 3-Phase 2L-VSI Based on SVPWM  
Many applications have been proposed for MPC, including electrical drivers, UPS, and grid-connected inverters [6]. FCS-MPC is based on the premise of employing a discrete model of the power converter with its associated filter to anticipate the future behavior of all potential control inputs, and then applying the best one that minimizes a predetermined cost function (CF) at each sampling time (\(T_{s}\)). The main concept is to harness the microcontroller's raw processing capability to incorporate all control loops into a single algorithm that takes into account the converter's model and the accompanying filter. The general block diagram of FCS-MPC to control a three-phase voltage source inverter with an \(LC\) filter in the output is shown in Figure 1. The FCS-MPC procedure is conducted sequentially in its most basic version, and at the start of each sampling time, it applies a new switching pattern that is determined from the previous step. It then receives new and updated measurements in order to select a new switching pattern. The three-phase 2L-VSI power circuit is considered, with the two switches in each leg operating in a complementary mode to avoid short-circuit across the input source. The switching states are represented by \(S_{a}\), \(S_{b}\), and \(S_{c}\) switching signals, which are specified as follows:  
\[S_{a}= \begin{cases}1\text{ if }S_{1}\text{ on and }S_{4}\text{ off}\\ 0\text{ if }S_{1}\text{ off and }S_{4}\text{ on}\end{cases}  \tag{1}\]  
\[S_{b}= \begin{cases}1\text{ if }S_{2}\text{ on and }S_{5}\text{ off}\\ 0\text{ if }S_{2}\text{ off and }S_{5}\text{ on}\end{cases}  \tag{2}\]  
\[S_{c}= \begin{cases}1\text{ if }S_{3}\text{ on and }S_{6}\text{ off}\\ 0\text{ if }S_{3}\text{ off and }S_{6}\text{ on}\end{cases}  \tag{3}\]  
The voltage across filter inductance (\(L_{fa}\), \(L_{fb}\), \(L_{fc}\)) can be written in a vectorial form as:  
\[L_{f}\frac{\text{d}I_{L_{f}}}{\text{d}t}=V_{i}-V_{C_{f}} \tag{4}\]  
where \(L_{f}\) is the filter inductance, \(I_{L_{f}}\) is the filter current, and \(V_{C_{f}}\) is the filter voltage, and \(V_{i}\) denotes the system's inverter voltage, which has eight different voltage vectors (i.e., six active vectors and two zero vectors), as illustrated in Figure 2. It is clear that the space  
Figure 1: Schematic of the 2L-VSI for UPS applications based on M\({}^{2}\)PC. vector diagram has six sectors. Mathematically, the equation that defines the dynamic behavior of the output voltage is as follows:  
\[C_{f}\frac{\mathrm{d}V_{C_{f}}}{\mathrm{d}t}=I_{L_{f}}-I_{o} \tag{5}\]  
where \(I_{o}\) is the load current, which can be measured or estimated, and \(C_{f}\) is the filter capacitance (\(C_{fa}\), \(C_{fb}\), \(C_{fc}\)). In the state space model, these equations can be expressed as:  
\[\frac{\mathrm{d}X}{\mathrm{d}t}=AX+B_{1}V_{i}+B_{2}I_{o} \tag{6}\]  
where  
\[X= \begin{bmatrix}I_{L_{f}}\\ V_{C_{f}}\end{bmatrix}  \tag{7}\]  
\[A= \begin{bmatrix}-R/L_{f}&-1/L_{f}\\ 1/C_{f}&0\end{bmatrix}  \tag{8}\]  
\[B_{1}= \begin{bmatrix}1/L_{f}\\ 0\end{bmatrix}  \tag{9}\]  
\[B_{2}= \begin{bmatrix}0\\ -1/C_{f}\end{bmatrix}  \tag{10}\]  
A discrete model is obtained from (6) and it can be expressed as follows:  
\[x\big{(}k+1\big{)}=A_{q}x\big{(}k\big{)}+B_{q}V_{i}\big{(}k\big{)}+B_{dq}I_{o} \big{(}k\big{)} \tag{11}\]  
where  
\[A_{q}=exp^{AT_{s}} \tag{12}\]  
\[B_{q}=\int_{0}^{T_{s}}exp^{A\tau}B_{1}d\tau \tag{13}\]  
\[B_{dq}=\int_{0}^{T_{s}}exp^{A\tau}B_{2}d\tau. \tag{14}\]  
This model is used to predict the filter current in \(L_{f}\) and voltage across \(C_{f}\) for any input voltage and different loading conditions. The evaluation of cost function influences  
Figure 2: Space vector diagram and the voltage vectors generated by the inverter (\(V_{i}\)). the choice of the optimal output voltage vector to be applied in the next sampling interval.