Diagnostics Relevant Modeling of Squirrel-Cage Induction Motor: Electrical Faults

Abstract

In this paper, simplified SCIM models are formulated based on stationary, rotor, and synchronous reference frames. All these models are compared and analyzed in terms of their diagnostic relevance to major electrical faults (stator inter-turn short-circuit and broken rotor bars). Ability to develop distinct residual signatures is a key for any model-based fault diagnosis method. The performance of various models in terms of their ability to generate a distinct residual and best-suited model is recommended based on discriminatory ability index proposed in this manuscript. Extended Kalman filter is the most commonly used estimator for nonlinear systems. The SCIM, being a nonlinear system, extended Kalman filter is considered for state estimation. As an extension, parameter sensitivity analysis is carried out for the best-suited model. Efforts are made to convey which parameters have a significant effect in case there is plant-model mismatch. Analytical computations are carried out for a 3 kW SCIM motor using MATLAB software. The results show that the most effective squirrel-cage induction motor model for model-based fault diagnostics.

Appendix

All the parameters of the SCIM model are given here. Also, $\theta_{l} = \left( {\theta_{e} - n_{p} \theta_{r} } \right)$. Here, $\theta_{e}$ and $\theta_{r}$ are transformation angular position and rotor angle, respectively.

$$\begin{aligned} {\mathbf{K}}_{s} & = \frac{2}{3}\left[ {\begin{array}{*{20}c} {\cos \theta _{e} } & {\cos \left( {\theta _{e} - 2\pi /3} \right)} & {\cos \left( {\theta _{e} + 2\pi /3} \right)} \\ {\sin \theta _{e} } & {\sin \left( {\theta _{e} - 2\pi /3} \right)} & {\sin \left( {\theta _{e} + 2\pi /3} \right)} \\ {1/2} & {1/2} & {1/2} \\ \end{array} } \right], \\ {\mathbf{K}}_{r} & = \frac{2}{3}\left[ {\begin{array}{*{20}c} {\cos \theta _{l} } & {\cos \left( {\theta _{l} - 2\pi /3} \right)} & {\cos \left( {\theta _{l} + 2\pi /3} \right)} \\ {\sin \theta _{l} } & {\sin \left( {\theta _{l} - 2\pi /3} \right)} & {\sin \left( {\theta _{l} + 2\pi /3} \right)} \\ {1/2} & {1/2} & {1/2} \\ \end{array} } \right] \\ \end{aligned}$$

$$\begin{aligned} {\mathbf{L}}_{{abc}}^{s} & = \left[ {\begin{array}{*{20}c} {\left( {L_{{ls}} + L_{{ms}} } \right)} & { - \frac{1}{2}L_{{ms}} } & { - \frac{1}{2}L_{{ms}} } \\ { - \frac{1}{2}L_{{ms}} } & {\left( {L_{{ls}} + L_{{ms}} } \right)} & { - \frac{1}{2}L_{{ms}} } \\ { - \frac{1}{2}L_{{ms}} } & { - \frac{1}{2}L_{{ms}} } & {\left( {L_{{ls}} + L_{{ms}} } \right)} \\ \end{array} } \right], \\ {\mathbf{L}}_{{abc}}^{r} & = \left[ {\begin{array}{*{20}c} {\left( {L_{{lr}} + L_{{mr}} } \right)} & { - \frac{1}{2}L_{{mr}} } & { - \frac{1}{2}L_{{mr}} } \\ { - \frac{1}{2}L_{{mr}} } & {\left( {L_{{lr}} + L_{{mr}} } \right)} & { - \frac{1}{2}L_{{mr}} } \\ { - \frac{1}{2}L_{{mr}} } & { - \frac{1}{2}L_{{mr}} } & {\left( {L_{{lr}} + L_{{mr}} } \right)} \\ \end{array} } \right] \\ \end{aligned}$$

$$\begin{aligned} {\mathbf{K}}_{\omega } & = \left[ {\begin{array}{*{20}c} {010} \\ { - 100} \\ {000} \\ \end{array} } \right],{\mathbf{v}}_{{qd0}}^{s} = {\mathbf{K}}_{s} {\mathbf{v}}_{{abc}}^{s} ,{\mathbf{R}}_{{qd0}}^{s} = r_{s} {\mathbf{I}}_{{3 \times 3}} ,{\mathbf{R}}_{{qd0}}^{r} = r_{r} {\mathbf{I}}_{{3 \times 3}} \\ {\mathbf{L}}_{{abc}}^{{sr}} & = L_{{sr}} \left[ {\begin{array}{*{20}c} {\cos \left( {n_{p} \theta _{r} } \right)\cos \left( {n_{p} \theta _{r} + 2\pi /3} \right)\cos \left( {n_{p} \theta _{r} - 2\pi /3} \right)} \\ {\cos \left( {n_{p} \theta _{r} - 2\pi /3} \right)\cos \left( {n_{p} \theta _{r} } \right)\cos \left( {n_{p} \theta _{r} + 2\pi /3} \right)} \\ {\cos \left( {n_{p} \theta _{r} + 2\pi /3} \right)\cos \left( {n_{p} \theta _{r} - 2\pi /3} \right)\cos \left( {n_{p} \theta _{r} } \right)} \\ \end{array} } \right], \\ {\mathbf{L}}_{{abc}}^{{rs}} & = \left( {{\mathbf{L}}_{{abc}}^{{sr}} } \right)^{{\text{T}}} \\ {\dot{\mathbf{L}}}_{{abc}}^{{sr}} & = - \omega _{r} L_{{sr}} \left[ {\begin{array}{*{20}c} {\sin \left( {n_{p} \theta _{r} } \right)\sin \left( {n_{p} \theta _{r} + 2\pi /3} \right)\sin \left( {n_{p} \theta _{r} - 2\pi /3} \right)} \\ {\sin \left( {n_{p} \theta _{r} - 2\pi /3} \right)\sin \left( {n_{p} \theta _{r} } \right)\sin \left( {n_{p} \theta _{r} + 2\pi /3} \right)} \\ {\sin \left( {n_{p} \theta _{r} + 2\pi /3} \right)\sin \left( {n_{p} \theta _{r} - 2\pi /3} \right)\sin \left( {n_{p} \theta _{r} } \right)} \\ \end{array} } \right], \\ {\dot{\mathbf{L}}}_{{abc}}^{{rs}} & = \left( {{\dot{\mathbf{L}}}_{{abc}}^{{sr}} } \right)^{{\text{T}}} \\ \end{aligned}$$

$$\begin{aligned} T_{{em}} & = - n_{p} L_{{ms}} \left\{ \begin{gathered} \left( \begin{gathered} i_{{as}} \left( {i_{{ar}} - \frac{1}{2}i_{{br}} - \frac{1}{2}i_{{cr}} } \right) + i_{{bs}} \left( {i_{{br}} - \frac{1}{2}i_{{ar}} - \frac{1}{2}i_{{cr}} } \right) \hfill \\ + \left. {i_{{cs}} \left( {i_{{cr}} - \frac{1}{2}i_{{br}} - \frac{1}{2}i_{{ar}} } \right)} \right)\sin \left( {n_{p} \theta _{r} } \right) \hfill \\ \end{gathered} \right. \hfill \\ + \frac{{\sqrt 3 }}{2}\left[ {i_{{as}} \left( {i_{{br}} - i_{{cr}} } \right) + i_{{bs}} \left( {i_{{cr}} - i_{{ar}} } \right) + i_{{cs}} \left( {i_{{ar}} - i_{{br}} } \right)} \right]\cos \left( {n_{p} \theta _{r} } \right) \hfill \\ \end{gathered} \right\}\left( {{\text{or}}} \right) \\ T_{{em}} & = n_{p} \left( {{\mathbf{i}}_{{abc}}^{s} } \right)^{T} \frac{\partial }{{\partial \theta _{r} }}\left[ {{\mathbf{L}}_{{abc}}^{{sr}} } \right]\left( {{\mathbf{i}}_{{abc}}^{r} } \right). \\ \end{aligned}$$