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Modeling and stability analysis of three- and six-phase asymmetrical grid-connected induction generator

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Abstract

This paper presents the development of linearized model of asymmetrical six-phase grid-connected induction generator (GCIG) for small-signal stability. The developed linearized model has been used to obtain the eigenvalue to evaluate generator stability with respect to parameter variation of the machine in comparison with its three-phase counterpart. During the analysis, suitability of six-phase induction generator for higher output power has been accessed. Instability tendency during low electrical output power can be avoided through closed loop operation of the machine. Hence, the indirect field-oriented control scheme for asymmetrical six-phase GCIG has been proposed for stable operation with higher reliability. The key analytical results have been demonstrated with its experimental validation.

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Authors and Affiliations

  1. Department of Electrical Engineering, Rajkiya Engineering College Ambedkar Nagar, Akbarpur, India

    Arif Iqbal

  2. Department of Electrical Engineering, Indian Institute of Technology Roorkee, Roorkee, India

    Girish Kumar Singh

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  1. Arif Iqbal
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  2. Girish Kumar Singh
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Appendix

Appendix

During mathematical modeling of induction machine, neutral points of both three-phase stator winding were assumed separate for the prevention of fault propagation. Equations of induction machine are used to develop the equivalent circuit as shown in Fig. 

Fig. 15
figure 15

Equivalent circuit of asymmetrical six-phase induction machine

Full size image

15.

$$ {\varvec{v}}_{{{\text{qdk}}}} = \left[ {\begin{array}{*{20}c} {v_{{{\text{qk}}}} } & {v_{{{\text{dk}}}} } \\ \end{array} } \right]^{T} $$
$$ {\varvec{v}}_{{{\text{qdr}}}} = \left[ {\begin{array}{*{20}c} {v_{{{\text{qr}}}} } & {v_{{{\text{dr}}}} } \\ \end{array} } \right]^{T} $$
$$ \phi_{{{\text{qdk}}}} = \left[ {\begin{array}{*{20}c} {\phi_{{{\text{qk}}}} } & {\phi_{{{\text{dk}}}} } \\ \end{array} } \right]^{T} $$
$$ \phi_{{{\text{dqk}}}} = \left[ {\begin{array}{*{20}c} {\phi_{{{\text{dk}}}} } & { - \phi_{{{\text{qk}}}} } \\ \end{array} } \right]^{T} $$
$$ \phi_{{{\text{qdr}}}} = \left[ {\begin{array}{*{20}c} {\phi_{{{\text{qr}}}} } & {\phi_{{{\text{dr}}}} } \\ \end{array} } \right]^{T} $$
$$ \phi_{{{\text{dqr}}}} = \left[ {\begin{array}{*{20}c} {\phi_{{{\text{dr}}}} } & { - \phi_{{{\text{qr}}}} } \\ \end{array} } \right]^{T} $$
$$ {\varvec{i}}_{{{\text{qdk}}}} = \left[ {\begin{array}{*{20}c} {i_{{{\text{qk}}}} } & {i_{{{\text{dk}}}} } \\ \end{array} } \right]^{T} $$
$$ {\varvec{i}}_{{{\text{qdr}}}} = \left[ {\begin{array}{*{20}c} {i_{{{\text{qr}}}} } & {i_{{{\text{dr}}}} } \\ \end{array} } \right]^{T} $$
$$ {\varvec{v}} = \left[ {\left( {{\varvec{v}}_{{{\text{qdk}}}} } \right)_{k = 1, 2} {\varvec{v}}_{{{\text{qdr}}}} } \right]^{T} $$
$$ {\varvec{i}} = \left[ {\left( {{\varvec{i}}_{{{\text{qdk}}}} } \right)_{k = 1, 2} {\varvec{i}}_{{{\text{qdr}}}} } \right]^{T} $$

where \({\text{k = }}\left\{ {\begin{array}{*{20}c} {\text{1, for winding set abc}} \\ {\text{2, for winding set xyz}} \\ \end{array} } \right.\)

$$ z = \left| {\begin{array}{*{20}l} {r_{1} + \frac{p}{{\omega_{b} }}\left( {x_{l1} + x_{lm} + x_{m} } \right) } \hfill & {\frac{{\omega_{e} }}{{\omega_{b} }}\left( {x_{l1} + x_{lm} + x_{m} } \right)} \hfill & {\frac{p}{{\omega_{b} }}\left( {x_{lm} + x_{m} } \right)} \hfill & { \frac{{\omega_{e} }}{{\omega_{b} }}\left( {x_{lm} + x_{m} } \right)} \hfill & {\frac{p}{{\omega_{b} }}x_{m} } \hfill & {\frac{{\omega_{e} }}{{\omega_{b} }}x_{m} } \hfill \\ { - \frac{{\omega_{e} }}{{\omega_{b} }}\left( {x_{l1} + x_{lm} + x_{m} } \right)} \hfill & {r_{1} + \frac{p}{{\omega_{b} }}\left( {x_{l1} + x_{lm} + x_{m} } \right)} \hfill & { - \frac{{\omega_{e} }}{{\omega_{b} }}\left( {x_{lm} + x_{m} } \right)} \hfill & {\frac{p}{{\omega_{b} }}\left( {x_{lm} + x_{m} } \right)} \hfill & { - \frac{{\omega_{e} }}{{\omega_{b} }}x_{m} } \hfill & {\frac{p}{{\omega_{b} }}x_{m} } \hfill \\ {\frac{p}{{\omega_{b} }}\left( {x_{lm} + x_{m} } \right)} \hfill & {\frac{{\omega_{e} }}{{\omega_{b} }}\left( {x_{lm} + x_{m} } \right)} \hfill & {r_{2} + \frac{p}{{\omega_{b} }}\left( {x_{l2} + x_{lm} + x_{m} } \right)} \hfill & {\frac{{\omega_{e} }}{{\omega_{b} }}\left( {x_{lm} + x_{m} } \right) } \hfill & {\frac{p}{{\omega_{b} }}\left( {x_{lm} + x_{m} } \right)} \hfill & {\frac{{\omega_{e} }}{{\omega_{b} }}x_{m} } \hfill \\ { - \frac{{\omega_{e} }}{{\omega_{b} }}\left( {x_{lm} + x_{m} } \right)} \hfill & {\frac{p}{{\omega_{b} }}\left( {x_{lm} + x_{m} } \right)} \hfill & { - \frac{{\omega_{e} }}{{\omega_{b} }}\left( {x_{lm} + x_{m} } \right)} \hfill & {r_{2} + \frac{p}{{\omega_{b} }}\left( {x_{l2} + x_{lm} + x_{m} } \right)} \hfill & { - \frac{{\omega_{e} }}{{\omega_{b} }}x_{m} } \hfill & { \frac{p}{{\omega_{b} }}x_{m} } \hfill \\ { \frac{p}{{\omega_{b} }}x_{m} } \hfill & { \frac{{s\omega_{e} }}{{\omega_{b} }}x_{m} } \hfill & {\frac{p}{{\omega_{b} }}x_{m} } \hfill & { \frac{{s\omega_{e} }}{{\omega_{b} }}x_{m} } \hfill & { r_{r} + \frac{p}{{\omega_{b} }}\left( {x_{lr} + x_{m} } \right)} \hfill & { \frac{{s\omega_{e} }}{{\omega_{b} }}\left( {x_{lr} + x_{m} } \right)} \hfill \\ { - \frac{{s\omega_{e} }}{{\omega_{b} }}x_{m} } \hfill & { \frac{p}{{\omega_{b} }}x_{m} } \hfill & { - \frac{{s\omega_{e} }}{{\omega_{b} }}x_{m} } \hfill & {\frac{p}{{\omega_{b} }}x_{m} } \hfill & { - \frac{{s\omega_{e} }}{{\omega_{b} }}\left( {x_{lr} + x_{m} } \right) } \hfill & {r_{r} + \frac{p}{{\omega_{b} }}\left( {x_{lr} + x_{m} } \right)} \hfill \\ \end{array} } \right| $$
$$ X = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\left( {x_{l1} + x_{lm} + x_{m} } \right)} & 0 & {\left( {x_{lm} + x_{m} } \right)} \\ 0 & {\left( {x_{l1} + x_{lm} + x_{m} } \right)} & 0 \\ {\left( {x_{lm} + x_{m} } \right)} & 0 & {\left( {x_{l2} + x_{lm} + x_{m} } \right)} \\ \end{array} \begin{array}{*{20}c} 0 & {x_{m} } & 0 \\ {\left( {x_{lm} + x_{m} } \right)} & 0 & {x_{m} } \\ 0 & {x_{m} } & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & & { \left( {x_{lm} + x_{m} } \right) 0} \\ {x_{m} } & 0 & {x_{m} } \\ 0 & {x_{m} } & 0 \\ \end{array} \begin{array}{*{20}c} {\left( {x_{l2} + x_{lm} + x_{m} } \right)} & 0 & {x_{m} } \\ 0 & {\left( {x_{lr} + x_{m} } \right)} & 0 \\ {x_{m} } & 0 & {\left( {x_{lr} + x_{m} } \right)} \\ \end{array} } \\ \end{array} } \right] $$
$$ u = \left[ {\begin{array}{*{20}c} {\left( {\Delta {\varvec{v}}_{{{\text{qdk}}}} } \right)_{k = 1, 2} } & {\Delta {\varvec{v}}_{{{\text{qdr}}}} } & {\Delta \tau_{l} } \\ \end{array} } \right]^{T} $$
$$ x = \left[ {\begin{array}{*{20}c} {\left( {\Delta i_{{{\text{qdk}}}} } \right)_{k = 1, 2} } & {\Delta i_{{{\text{qdr}}}} } & {{\raise0.7ex\hbox{${\Delta \omega_{r} }$} \!\mathord{\left/ {\vphantom {{\Delta \omega_{r} } {\omega_{b} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\omega_{b} }$}}} \\ \end{array} } \right]^{T} $$
$$ W_{1p} = \left| {\begin{array}{*{20}l} {\left( {x_{l1} + x_{{{\text{lm}}}} + x_{m} } \right)} \hfill & 0 \hfill & {\left( {x_{{{\text{lm}}}} + x_{m} } \right)} \hfill & { 0} \hfill \\ 0 \hfill & {\left( {x_{l1} + x_{{{\text{lm}}}} + x_{m} } \right)} \hfill & 0 \hfill & {\left( {x_{{{\text{lm}}}} + x_{m} } \right)} \hfill \\ {\left( {x_{{{\text{lm}}}} + x_{m} } \right)} \hfill & 0 \hfill & {\left( {x_{l2} + x_{{{\text{lm}}}} + x_{m} } \right)} \hfill & 0 \hfill \\ 0 \hfill & {\left( {x_{{{\text{lm}}}} + x_{m} } \right)} \hfill & 0 \hfill & {\left( {x_{l2} + x_{{{\text{lm}}}} + x_{m} } \right)} \hfill \\ \end{array} } \right| $$
$$ W_{2p} = \left[ {\begin{array}{*{20}c} {x_{lr} + x_{m} } & 0 \\ 0 & {x_{lr} + x_{m} } \\ \end{array} } \right] $$
$$ X_{1p} = \left[ {\begin{array}{*{20}c} {x_{m} 0} \\ {0 x_{m} } \\ {x_{m} 0} \\ {0 x_{m} } \\ \end{array} } \right] $$
$$ X_{2p} = \left[ {\begin{array}{*{20}c} {x_{m} 0 x_{m} 0 } \\ {0 x_{m} 0 x_{m} } \\ \end{array} } \right] $$
$$ Y_{1p} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right] $$
$$ Y_{2p} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right] $$
$$ S_{p} = \left[ { - 2P\omega_{b}^{2} /J} \right] $$
$$ Q_{1p} = \left[ {0 0 0 0} \right] $$
$$ Q_{2p} = \left[ {0 0} \right] $$
$$ W_{1k} = \left| {\begin{array}{*{20}l} {r_{1} } \hfill & { \left( {x_{l1} + x_{lm} + x_{m} } \right)} \hfill & 0 \hfill & {\left( {x_{{{\text{lm}}}} + x_{m} } \right)} \hfill \\ { - \left( {x_{l1} + x_{{{\text{lm}}}} + x_{m} } \right)} \hfill & { r_{1} } \hfill & { - \left( {x_{{{\text{lm}}}} + x_{m} } \right)} \hfill & 0 \hfill \\ 0 \hfill & {\left( {x_{lm} + x_{m} } \right)} \hfill & {r_{2} } \hfill & {\left( {x_{l2} + x_{{{\text{lm}}}} + x_{m} } \right)} \hfill \\ { - \left( {x_{{{\text{lm}}}} + x_{m} } \right)} \hfill & 0 \hfill & { - \left( {x_{l2} + x_{{{\text{lm}}}} + x_{m} } \right)} \hfill & {r_{2} } \hfill \\ \end{array} } \right| $$
$$ X_{1k} = \left[ {\begin{array}{*{20}c} {0 x_{m} } \\ { - x_{m} 0 } \\ {0 x_{m} } \\ { - x_{m} 0 } \\ \end{array} } \right] $$
$$ X_{2k} = \left[ {\begin{array}{*{20}c} {0 {\text{sx}}_{m} 0 {\text{sx}}_{m} } \\ { - {\text{sx}}_{m} 0 - {\text{sx}}_{m} 0 } \\ \end{array} } \right] $$
$$ Y_{1k} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right] $$
$$ Y_{2k} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right] $$
$$ Q_{1k} = \left[ {x_{m} i_{{{\text{dr}}}} - x_{m} i_{{{\text{qr}}}} x_{m} i_{{{\text{dr}}}} - x_{m} i_{{{\text{qr}}}} } \right] $$
$$ Q_{2k} = \left[ { - x_{m} \left( {i_{d1} + i_{d2} } \right) x_{m} \left( {i_{q1} + i_{q2} } \right) } \right] $$
$$ S_{k} = \left[ 0 \right] $$

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Iqbal, A., Singh, G.K. Modeling and stability analysis of three- and six-phase asymmetrical grid-connected induction generator. Electr Eng 103, 1169–1181 (2021). https://doi.org/10.1007/s00202-020-01145-w

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