## 3 Finite Control Set MPC in 5 PIM Drives
### Proposed VSTLPC Considering VVV as Control Actions  
The incorporation of VVV as control actions in the FCS-MPC has permitted the improvement of current quality in multiphase drives, thanks to the simultaneous regulation of the available subspaces [29]. These vectors are created through a combination of two or more voltage vectors in such a way that zero or reduced average voltage amplitudes are produced in the \(xy\) plane. Consequently, the regulation of secondary currents is directly carried out with the VVV application, without including a specific control loop in the algorithm. This characteristic results in the simplification of the prediction model and the predictive control algorithm. In this work, diverse sets of 10 virtual VVVs are used in the optimization problem of the VSTLPC algorithm instead of the 32 original VVs of the 5p2L-VSI. For the case under study, three sets of VVVs will be independently considered as control actions, with the objective of comparing the pros and cons of each one (Table 2 summarizes the sets of VVVs that will be considered in this work):  
Medium-large VV (ML-VVV): A large vector is combined with a medium vector with dwell ratios equal to \(\lambda_{1}\) = 0.382 and \(\lambda_{2}\) = 0.618, respectively. As seen in Figure 3, they are aligned in the \(\alpha\beta\) plane, but they are in the opposite direction in the \(xy\) plane (vectors 12 and 30, for example), and their combination leads to a VVV with zero \(xy\) voltage on average. This is the common strategy in most research, where a total of 10 VVVs can be constructed considering all possible combinations. As a disadvantage, the instantaneous value of the \(xy\) voltage components is not zero, thus producing the circulation of \(xy\) currents in the machine. Furthermore, the use of the DC link is reduced compared to the original voltage vectors, since the size of the VVV is 0.5528\(V_{dc}\). The latest has an important influence on the operating range of the drive, mainly in high-speed/torque operation. Duo of large VVs (LL-VVV): Two adjacent large voltages are combined with dwell times ratios equal to \(\lambda_{1}\) = \(\lambda_{2}\) = 0.5. In this case, the combination will not provide zero \(xy\) voltage amplitudes (see vectors 12 and 28, for example), but its magnitude is limited by applying the indicated dwell ratios. Therefore, secondary currents will not effectively be regulated to zero, but their instantaneous values will be lower than those with conventional VVVs. Additionally, the resulting VVV has a size of 0.6155\(V_{dc}\) in the primary plane, which extends the use of the DC link. As in the previous case, 10 different VVVs can be constructed. Trio of large VVs (LLL-VVV): Three adjacent large vectors are combined with \(\lambda_{1}\) = 0.382, \(\lambda_{2}\) = 0.236, and \(\lambda_{3}\) = 0.382 as dwell ratios. In this case, zero-average \(xy\) voltages are obtained with low instantaneous \(xy\) injection, thus resulting in the expectation of improved current quality in the machine compared to the application of the medium-large \begin{table}
\begin{tabular}{c c c} \hline
**VVV Group Name** & **Voltage Vectors Used** & **Dwell Ratios** \\ \hline ML-VVV & One medium + one large & \(\lambda_{1}\) = 0.382 \\  & & \(\lambda_{2}\) = 0.618 \\ LL-VVV & Two adjacent large & \(\lambda_{1}\) = 0.5 \\  & & \(\lambda_{2}\) = 0.5 \\  & & \(\lambda_{1}\) = 0.382 \\ LLL-VVV & Three adjacent large & \(\lambda_{2}\) = 0.236 \\  & & \(\lambda_{3}\) = 0.382 \\ \hline \end{tabular}
\end{table} 
Table 2: Set of VVVs considered. combination. However, the magnitude of the new VVV is \(0.5528V_{dc}\), and the use of the DC link is poorer compared to the set of VVVs formed by only two adjacent voltage vectors. Following the same algorithm described above, the VSTLPC selects the optimal VVV to be applied and its application time based on the measured currents and the established references. Only a set of VVVs and a zero vector are considered as possible control actions.