Induction generator model

This section shows the induction generator model developed without considering the magnetic saturation effect. In this case the voltage equations which characterize the induction machine written in the rotor reference frame are given by the following system:

$\frac{?}{\text{b}\text{a}\text{r}}(v{)}^{,s}=\frac{r^s}{b}ar(i{)}^{,s}+\frac{d}{(d)t}\frac{?}{\text{b}\text{a}\text{r}}({)}^{,s}+j\frac{{}^{\text{'}}}{b}ar({)}^{,s}\frac{?}{b}$

$ar(v{)}^r=\frac{r^r}{b}ar(i{)}^r+\frac{d}{(d)t}\frac{?}{\text{b}\text{a}\text{r}}({)}^r\frac{?}{b}$

$ar(v{)}^a=\frac{r^a}{b}ar(i{)}^a+\frac{d}{(d)t}\frac{?}{\text{b}\text{a}\text{r}}({)}^a$

where $r$ represents the per phase winding resistances. $\frac{?}{\text{b}\text{a}\text{r}}({)}^{\text{'}z}\frac{,}{b}ar({)}^r$ and $\frac{?}{\text{b}\text{a}\text{r}}({)}^a$ represent the fluxes linked by the stator $($defined in the rotor reference frame$),$ the rotor and the fictitious windings, respectively, expressed by:

$\frac{?}{\text{b}\text{a}\text{r}}({)}^{\text{'}s}=\frac{L^s+l^s}{b}ar(i{)}^{\text{'}s}+\frac{M^{sr}}{b}ar(i{)}^r+\frac{M^{sa}}{b}ar(i{)}^a\frac{?}{b}$

$ar({)}^r=\frac{L^r+l^r}{b}ar(i{)}^r+\frac{M^{rs}}{b}ar(i{)}^{,s}+\frac{M^{ra}}{b}ar(i{)}^a\frac{?}{b}$

$ar({)}^a=\frac{L^a+l^a}{b}ar(i{)}^a+\frac{M^{as}}{b}ar(i{)}^{,s}+\frac{M^{ar}}{b}ar(i{)}^r$

where $l^z$ and $l^r$ are, respectively, the rotor and stator leakage inductances, $l^a$ is the leakage inductance of the considered fictitious windings. $M^{sr},M^{3a}$ and $M^{ra}$ are the mutual inductances between the stator, the rotor and the fictitious windings. It will be considered that $M^{sr}=M^{rs},M^{as}=M^{sa}$ and $M^{ra}=M^{ar}.$ The $L$ quantities appearing in $(2)$ are the main winding cyclic inductances and $M$ is the mutual inductance between the different windings. It is assumed that the induction machine turn ratio at standstill is close to: $\frac{n^s}{n^r}=\sqrt[\phantom{2}]{2},$ with $L^r=\frac{L^s}{2}.$ Let us consider the main cyclic winding inductance of the fictitious windings of the tested machine $L^a=\frac{(M^{sa}{)}^2}{L^s}.$ This value of $L^a$ has been chosen in such a way that the fictitious winding can provide the starting current for the SEIG. In this case the M inductance can be expressed as:

$M^{sr}=\sqrt[\phantom{2}]{L^s{L}^r}=\frac{L^s}{\sqrt[\phantom{2}]{2}}$

$M^{sa}=\sqrt[\phantom{2}]{L^s{L}^a}$

$M^{ra}=\sqrt[\phantom{2}]{L^r{L}^a}$

For this model where the magnetic saturation effect is neglected, all the main and the mutual inductance coefficients are constant. Using equations $(1)$ and $(2),$ the stator voltage defined in the $d^r$ reference frame is given by:

$(\frac{\text{:}}{b}ar(v{)}^s=\frac{r^s+j{}^{\text{'}}(L^s+l^s)}{b}ar(\frac{?}{\text{b}\text{a}\text{r}}(i){)}^s+(L^s+l^s)\frac{\frac{d}{b}ar(i{)}^{,s}}{(d)t}+j{}^{\text{'}}{M}^{s\frac{r}{b}ar(i{)}^r}+M^{sr}\frac{\frac{d}{b}ar(i{)}^r}{(d)t}+j{}^{\text{'}}{M}^{sa}{l}^4$