## 2 FSympscc of a Six-Phase Induction Drive Using VVV  
Figure 1 shows the FSympscc diagram of a VSI-driven 6 phase IM using a set of VVV. An outer loop, indicated as the PI controller in the diagram, is responsible for torque/speed regulation. The reference currents in the \(d-q\) plane are translated into the \(\alpha-\beta\) subspace using the Park transformation given by the matrix \(D\) as follows  
\[D=\left( \begin{array}{cc}\cos\theta_{s}&\sin\theta_{s}\\ -\sin\theta_{s}&\cos\theta_{s}\end{array} \right), \tag{1}\]  
where the flux position \(\theta_{a}\) is obtained as \(\theta_{a}=\int\omega_{c}dt\). Flux and electrical torque are calculated using reference currents \(i_{sd}^{*}\) and \(i_{sq}^{*}\). These quantities are set by the outer speed loop using typically a PI controller, although different control methods, such as MBPC, can be used [23]. The electrical frequency can be found by considering the mechanical rotor speed and the slip frequency from \(\omega_{e}=\omega_{r}+\omega_{sl}\), where field orientation is assumed and \(\omega_{sl}=\frac{R_{L}}{L_{r}}\frac{i_{sq}^{*}}{S_{sl}}\). The measured phase currents \(i_{s}=(i_{sq},i_{sb},i_{sc},i_{sd},i_{se},i_{sf})\) are converted to \(\alpha-\beta\) using the Clarke transformation matrix \(T\) based on power-invariant decoupling and defined as  
\[T=\frac{1}{\sqrt{3}}\left( \begin{array}{cccc}1&-1/2&-1/2&\sqrt{3}/2&-\sqrt{ 3}/2&0\\ 0&\sqrt{3}/2&-\sqrt{3}/2&1/2&1/2&-1\end{array} \right). \tag{2}\]  
At each discrete time \(k\), the reference signal for the tracking of the stator current \(i^{*}(k)\) uses an amplitude \(I_{s}^{*}\) calculated as \(I_{s}^{*}=\sqrt{i_{sd}^{*2}+i_{sq}^{*2}}\). As a result, the references for stator  
Figure 1: Diagram of FSympscc of a 6-phase IM using a single VVV set. currents are \(i_{sa}^{*}(t)=I_{s}^{*}\sin\omega_{e}t\), \(i_{SB}^{*}(t)=I_{s}^{*}\cos\omega_{e}t\), \(i_{xx}^{*}(t)=0\), \(i_{sy}^{*}(t)=0\). In this way, the FSMPSCC receives as input the \(\alpha-\beta\) reference currents \(i_{\alpha-\beta}^{*}\), the actual \(\alpha-\beta\) currents \(i_{a-\beta}\) and the sensed mechanical speed \(\omega_{m}\). With this information, the FSMPSCC obtains the optimal VVV to be applied during the next sampling period \(VVV^{opt}(k+1)\). This VVV is achieved by the appropriate selection of the VSI states \(S\) for their respective application times \(T_{1}\), \(T_{2}\) over the duration of the sampling period \(k+1\). As a result, the stator currents of the IM, \(i_{s}\), evolve so that the \(\alpha-\beta\) components move to their reference values \(i_{\alpha-\beta}^{*}(k+2)\). The selection of the optimal VVV is made by exhaustive optimization over all possible control moves (elements of the VVV set). The procedure is repeated at each sampling period following the receding-horizon rule typical of predictive controllers. In the present case, the multi-phase drive incorporates an asymmetrical six-phase IM. The machine has distributed windings where two independent and isolated neutral points are created. Two three-phase two-level VSIs are connected to a single DC link. The switching state of every VSI leg is associated with the can be defined using a binary variable \(S_{ij}\), being \(S_{ij}=0\) if the lower switch is ON and the upper switch is OFF, and \(S_{ij}=1\) if the opposite situation occurs. In Figure 2, voltages are represented in \(\alpha-\beta\) and \(x-y\) sub-spaces. To obtain those voltages aligned with the different phases. A decimal representation is used to identify the stator voltage in the \(\alpha-\beta\) plane. Such a number is derived from the binary number corresponding to the switching state \(S\), where \(S=(S_{a1},S_{b1},S_{c1},S_{a2},S_{b2},S_{c2})\), where the six phases have been numbered as \(a_{1}\), \(a_{2}\), \(b_{1}\), \(b_{2}\), \(c_{1}\), and \(c_{2}\). The predictive model needed by the FSMPSCC is a discrete-time model for the evolution of \(i_{s}\) derived from the IM equations in the phase variables and from the VSI structure.