## 4 Results and Discussion
### FCS-MPC: Variable-Switching-Frequency Operation with Switching-Frequency Minimization  
In contrast to software-based simulations, real-time analysis confers notable advantages in terms of speed and precision. Dedicated and specialized hardware, such as the implementation of field-programmable gate arrays (FPGAs) in real-time simulation and hardware-in-the-loop (HIL) platforms, facilitates exceedingly high processing velocities, thereby permitting simulations to be conducted in microsecond or even nanosecond intervals. This capability is essential to ensure accuracy in capturing rapid transient events and dynamic behaviors within power electronics, phenomena that slower simulations might overlook or represent inaccurately [60]. The aforementioned becomes apparent when examining the waveforms generated via the OPAL-RT platform, as depicted by the transitions of the weighting factor (\(\lambda_{min}\)) from 0 to 0.5 and from 0.5 to 1.0, which are illustrated in Figures 10a-d and 11a-d below. Figure 10: Real-time results of FCS-MPC variable-frequency operation with minimization of switching frequency. (**a**) Linear load: capacitor voltages with weighting factors (\(\lambda_{min}\)) ranging from 0 to 0.5; (**b**) linear load: capacitor voltages with weighting factors (\(\lambda_{min}\)) ranging from 0.5 to 1.0; (**c**) non-linear load: capacitor voltages with weighting factors (\(\lambda_{min}\)) ranging from 0 to 0.5; (**d**) non-linear load: capacitor voltages with weighting factors (\(\lambda_{min}\)) ranging from 0.5 to 1.0. In Figure 10a, the waveforms obtained for the capacitor voltage of the \(LC\) filter are shown. For this case, the used load corresponds to a linear \(RL\) load. The weighting factor (\(\lambda_{min}\)) ranges from 0 to 0.5. It can be seen that once the \(F_{min}\) factor begins to affect the cost function (\(g_{min}\)) of Equation (15), distortion appears in the waveforms. For the same case of using a linear load, as the weighting factor (\(\lambda_{min}\)) increases from 0.5 to 1.0, the distortion in the waveforms for the capacitor voltages of the \(LC\) filter in the system also increases. This can be seen in Figure 10b. An increased impact of the factor with respect to the minimization of switching (\(F_{min}\)) contributes to a less refined operation--in this case, of the predictive controller for capacitor voltages. As illustrated in Figure 10a,b, there is an observable degradation in the quality of the waveforms, demonstrating the influence of the augmented weighting factor (\(\lambda_{min}\)) within the cost function of the defined predictive control scheme. Unlike software-based simulations, the use of real-time results enables researchers to investigate the effects of varying the weighting factor (\(\lambda_{min}\)) on the control output of the three-phase, two-level VSI with an isolated load. A slightly different situation occurred when applying the same control algorithm to the three-phase 2L-VSI but connected to a non-linear load through the \(LC\) filter. The degradation of the quality of the waveforms increased with an increasing weighting factor (\(\lambda_{min}\)), but this change was more prominent than in the linear load case. This can be seen in Figure 10c,d, where the weighting factor (\(\lambda_{min}\)) ranges from 0 to 0.5 and from 0.5 to 1.0, respectively. In Figure 10c, as a non-linear load is connected to the system, it can be seen that the predictive controller functions properly, with the capacitor voltages of the \(LC\) filter as its control objective. However, once the weighting factor (\(\lambda_{min}\)) has a value of 0.5, the additional term (\(F_{min}\)) begins to influence the cost function of Equation (15), and excessive distortion appears in the voltage waveforms of the capacitor (from \(t=0.2\) s). Figure 11: Real-time results of FCS-MPC variable-frequency operation with minimization of switching frequency.