## 2 Conventional FCS-MPC for the 3-Phase 2L-VSI Based on SVPWM
For such a system, the main control target is to fix the inverter output voltage, therefore the CF will be defined as follows:  
\[J=(V_{C_{f_{k}}}^{*}-V_{C_{f_{k}}}(k+1))^{2}+(V_{C_{f_{k}}}^{*}-V_{C_{f_{k}}}(k+1 ))^{2} \tag{15}\]  
where \(V_{C_{f}/\alpha}^{*}\) and \(V_{C_{f}/\beta}^{*}\) are the real and imaginary parts of the reference voltage, respectively. In space vector modulation, it is possible to define each available vector in the 2L-VSI in (\(\alpha\beta\)) plane, as shown in Figure 2. The modulated MPC technique calculates the prediction of the two active vectors that compose each sector at every sampling time. The cost function, defined by (15), is evaluated for each sector. Each prediction is evaluated based on (11), where the duty-cycle for the two active vectors (\(d_{i}\)) are calculated by solving the following equation:  
\[d_{i}=\frac{\delta}{I_{i}} \tag{16}\]  
where \(\delta\) is the constant of proportionality, the subscript \(i\) denotes the adjacent vectors, in this case (\(i=1;2\)) and \(i=0\) corresponds to the duty-cycle of a zero vector, which is evaluated only one time, and \(J_{i}\) is equivalent cost function that the sum of product of all possible two cost function combination from the three vectors cost functions. The sum of the duty-cycle associated with the two active vectors and zero vector equals to unity as:  
\[d_{1}+d_{2}+d_{0}=1 \tag{17}\]  
By solving the previous equations, it is possible to obtain the duty-cycle for each vector as:  
\[d_{1}=\frac{J_{2}J_{0}}{J_{1}J_{0}+J_{1}J_{2}+J_{2}J_{0}} \tag{18}\]  
\[A_{q}=exp^{AT_{s}} \tag{12}\]0  
\[A_{q}=exp^{AT_{s}} \tag{12}\]1  
Considering these expressions, the new cost function that is evaluated at every \(T_{s}\) for the sector selection, can be defined as:  
\[A_{q}=exp^{AT_{s}} \tag{12}\]2  
The two vectors that minimize the CF are selected and applied to the 2L-VSI in the next sampling interval beside the zero vector. After selecting the optimal two vectors and calculating the corresponding duty-cycle, the dwell time associated with each vector can be calculated as in (22). The complete algorithm of M\({}^{2}\)PC will be executed sequentially as given in Algorithm 1. \[ \begin{split} T_{1}&=d_{1}T_{s}\\ T_{2}&=d_{2}T_{s}\\ T_{0}&=d_{0}T_{s}\end{split}  \tag{22}\]  
The conventional MPC control strategy is carried out in the following order: (1) Measuring of MPC input variables; (2) Predicting the output voltage and for the next sample period; (3) Calculating the cost function; (4) Choosing the switching signals that minimize the cost function; (5) Applying the optimized corresponding signals. In reality, taking into consideration the computation time and applying the selected switching state after the next sample instant is a simple way to compensate for the expected delay. As a result, the control algorithm is updated in the following way: (1) Variables are measured. (2) Application of the switching state (calculated in the previous interval). (3) Estimation of the voltage values at time \(t_{k}+1\), taking into account the switched state. (4) For all conceivable switching states, prediction of load voltages for the following sampling instant (\(t_{k}+2\)). (5) For each prediction, the cost function is evaluated. (6) The switching state that minimizes the cost function is chosen. It can be assumed that \(V^{*}_{C_{f}}(k+1)=V^{*}_{C_{f}}(k)\) for the predictive voltage control example, and the cost function can be rewritten as in (15). This approximation will lead to a one-sample delay in the reference tracking of the reference currents. If compensation of the calculation time delay is considered, the reference \(V^{*}_{C_{f}}(k+2)\) is required.