## 2 Model of the Voltage Source Inverter  
The system under investigation is illustrated in Figure 1, while diagrams depicting the loads are provided in Figure 2. The configuration relates to a three-phase, two-level voltage source inverter (VSI) that incorporates an _LC_ filter that supplies the isolated load.  
To assess the efficacy of the control schemes within the scope of real-time results, a non-linear load was considered. This configuration comprises a three-phase rectifier connected in parallel with a capacitance (\(C_{NL}\)) and a resistance (\(R_{NL}\)), as shown in Figure 2. Furthermore, the linear load, depicted in Figure 2, is made up of a three-phase load (\(RL\)).  
Then, considering the operator (\(\mathbf{a}=e^{\jmath 2\pi/3}\)), the currents and voltages shown in Figure 1 can be articulated within the \(\alpha\beta\) reference frame in vectorial format as follows:  
\[\mathbf{i_{f}}=\frac{2}{3}\Big{(}i_{fa}+\mathbf{a}i_{fb}+\mathbf{a}^{2}i_{fc} \Big{)} \tag{1}\]  
\[\mathbf{v_{c}}=\frac{2}{3}\Big{(}v_{ca}+\mathbf{a}v_{cb}+\mathbf{a}^{2}v_{cc} \Big{)} \tag{2}\]  
\[\mathbf{i_{o}}=\frac{2}{3}\Big{(}i_{oa}+\mathbf{a}i_{ob}+\mathbf{a}^{2}i_{oc} \Big{)} \tag{3}\]  
\[\mathbf{v_{i}}=\frac{2}{3}\Big{(}p_{aN}+\mathbf{a}v_{bN}+\mathbf{a}^{2}v_{cN} \Big{)} \tag{4}\]  
where \(\mathbf{v_{i}}\) denotes the voltage vector produced by the switching states of the converter. Applying [PERSON]'s law, the system's dynamics can be described as follows.  
\[\frac{Ld\mathbf{i_{f}}}{dt}=\mathbf{v_{i}}-\mathbf{v_{c}} \tag{5}\]  
\[\frac{Cd\mathbf{v_{c}}}{dt}=\mathbf{i_{f}}-\mathbf{i_{o}} \tag{6}\]  
Figure 2: Diagrams of the implemented loads for the real-time tests.  
Table 1 presents the eight possible switching states of the converter. Furthermore, states \(S_{a1}\), \(S_{b1}\), and \(S_{c1}\) can be expressed as \(\alpha\)-\(\beta\) components in vector notation.  
\[\mathbf{S}=\frac{2}{3}(S_{a1}+\mathbf{a}S_{b1}+\mathbf{a}^{2}S_{c1}) \tag{7}\]  
The voltage vector (\(\mathbf{v_{i}}\)), as indicated in Figure 3, is defined by Equation (8) using Equations (4) and (7).  
\[\mathbf{v_{i}}=\mathbf{S}\times V_{dc} \tag{8}\]  
Equations (5) and (6) are currently expressed in continuous time and need to be converted into discrete time for application in FCS-MPC. Using the discrete derivative defined over the interval of \(T_{s}\), the following formulations can be established:  
\[\mathbf{i_{f}}(k+1)=\mathbf{i_{f}}(k)+\frac{T_{s}}{L_{f}}(\mathbf{v_{i}}(k)- \mathbf{v_{c}}(k)) \tag{9}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Vector** & \(S_{a1}\) & \(S_{b1}\) & \(S_{c1}\) & \(S_{a2}\) & \(S_{b2}\) & \(S_{c2}\) \\ \hline \(v_{0}\) & 0 & 0 & 0 & 1 & 1 & 1 \\ \hline \(v_{1}\) & 1 & 0 & 0 & 0 & 1 & 1 \\ \hline \(v_{2}\) & 1 & 1 & 0 & 0 & 0 & 1 \\ \hline \(v_{3}\) & 0 & 1 & 0 & 1 & 0 & 1 \\ \hline \(v_{4}\) & 0 & 1 & 1 & 1 & 0 & 0 \\ \hline \(v_{5}\) & 0 & 0 & 1 & 1 & 1 & 0 \\ \hline \(v_{6}\) & 1 & 0 & 1 & 0 & 1 & 0 \\ \hline \(v_{7}\) & 1 & 1 & 1 & 0 & 0 & 0 \\ \hline \end{tabular}
\end{table}
Table 1: Available switching states of the two-level, three-phase VSI.  
Figure 3: Possible voltage vectors of the two-level, three-phase VSI (\(\mathbf{v_{i}}\)).  
\[\mathbf{v_{c}}(k+1)=\mathbf{v_{c}}(k)+\frac{T_{s}}{C_{f}}(\mathbf{i_{f}}(k)- \mathbf{i_{o}}(k)) \tag{10}\]  
It can be considered feasible to assume that \(\mathbf{i_{o}}(k)\) is approximately equivalent to \(\mathbf{i_{o}}(k-1)\) due to the minimal variations in behavior, as indicated by [57]. Using Equation (10), the load current value can be estimated, thereby obviating the need for direct measurements. The expression below delineates the value of \(\mathbf{i_{o}}(k)\) employed within this model.  
\[\mathbf{i_{o}}(k)\approx\mathbf{i_{o}}(k-1)=\mathbf{i_{f}}(k-1)-\frac{C_{f}}{ T_{S}}(\mathbf{v_{c}}(k)-\mathbf{v_{c}}(k-1)) \tag{11}\]