*2.1. 6PIM Model in α* − *β Subspace*

As already mentioned, only the *α* − *β* subspace components contribute to the electrical energy conversion. Using the VSD strategy, the normal six-dimensional electrical components of the 6PIM are mapped into the *α* − *β*, *z*<sup>1</sup> − *z*2, and *o*<sup>1</sup> − *o*<sup>2</sup> subspaces by an appropriate matrix named *T*6, which is presented in Appendix A. The *α* − *β* voltage equations in the stationary reference frame are as follows:

$$\begin{cases} \begin{array}{c} \overline{V\_{s}} = R\_{S}\overline{I\_{s}} + \rho \overline{\mathbf{V}\_{s}} \\ 0 = R\_{r}\overline{I\_{r}} + \rho \overline{\mathbf{V}\_{r}} - j\omega\_{r}\overline{\mathbf{V}\_{r}} \end{array} \tag{1} $$

where, *Vs* = *vs<sup>α</sup>* + *jvsβ*, *Is* = *is<sup>α</sup>* + *jisβ*, *Ir* = *ir<sup>α</sup>* + *jirβ*, Ψ*<sup>s</sup>* = *ψs<sup>α</sup>* + *jψsβ*, Ψ*<sup>r</sup>* = *ψr<sup>α</sup>* + *jψrβ*, *RS* is the stator resistance, *Rr* is the rotor resistance, *ω<sup>r</sup>* is the angular speed, and *ρ* is the derivative operator. The stator flux linkage (Ψ*s*) and rotor flux linkage (Ψ*r*) can be expressed as:

$$
\begin{bmatrix}
\overline{\Psi}\_s \\
\overline{\Psi}\_r
\end{bmatrix} = \begin{bmatrix}
L\_s & M \\
M & L\_r
\end{bmatrix} \begin{bmatrix}
\overline{I}\_s \\
\overline{I}\_r
\end{bmatrix} \tag{2}
$$

where, *Ls*, *Lr*, and *M* are the stator, rotor and magnetizing inductances, respectively.
