# On-Line Open-Phase Fault Detection Method for Switched Reluctance Motors with Bus Current Measurement

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## Abstract

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## 1. Introduction

- In [19], the EKF uses the four stator current measurements. On the contrary, the fault diagnosis (FD) system proposed in this article only needs the bus current knowledge.
- The signature matrix in the conference paper is different. In that article, the Flag is activated when the residual is negative. On the contrary, the FD system proposed in this article requires to define a positive threshold for the fault detection.
- In the conference paper only a one-phase fault is studied. Conversely, The FD system proposed can detect when two phases are in fault.
- This contribution presents a validation by using signals obtained from a more realistic motor model which is based on the Maxwell’s equations that are solved with FEM.
- This paper presents a comparison between our proposed FD method and a widely used FFT method.
- The FD system proposed in this article includes an optimization stage for identifying the faulty phases when two faulty phases are present.

- It is an on-line method.
- The proposed method uses only information from the measured bus current, the angular position, number of stator phases and number of rotor poles.
- The FD system does not need to store data to obtain a diagnostic.
- The proposed method is robust because stator currents are reconstructed using only measured angular position. This means the fault will not affect the AR calculation.

## 2. Preliminaries

#### 2.1. Switched Reluctance Motor Features

- A.1
- Mutual inductances are negligible: stator phases are electrically and magnetically independent.
- A.2
- Winding inductance is defined as$$\begin{array}{cc}\hfill {L}_{\mathit{j}}\left(\theta \right)& ={l}_{0}-{l}_{1}cos\left({N}_{r}\phantom{\rule{0.166667em}{0ex}}\theta \left(t\right)-(\mathit{j}-1)\frac{2\pi}{m}\right)\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$
- A.3
- Stator flux linkage ${\lambda}_{\mathit{j}}$ is defined as$$\begin{array}{c}\hfill {\lambda}_{j}={L}_{\mathit{j}}\left(\theta \right){i}_{j}\left(t\right)\phantom{\rule{0.166667em}{0ex}},\end{array}$$

#### 2.2. Observability Analysis

#### 2.3. Fault-Tolerant Controller

## 3. Main Results

#### 3.1. Extended Kalman Filter

Algorithm 1: EKF estimator for the phase currents |

- For $\alpha =1$, one obtains the usual EKF, and for $\alpha >1$, the EKF exponential data weighting. For more details check [37].
- The symbolic Jacobian matrix is obtained only once as follows:$${\mathbf{A}}_{k}={\left[{\displaystyle \frac{\partial \varphi}{\partial \mathbf{x}}}\right]}_{\mathbf{x}={\widehat{\mathbf{x}}}_{k}}.$$
- ${\mathbf{A}}_{k}$ is computed numerically in each time step.
- Increasing $\mathbf{Q}$ would indicate the presence of heavy system noise or greater uncertainty of the parameters. Increasing the elements of $\mathbf{Q}$ will also increase the EKF gain, resulting in faster filter dynamics. Increasing the values of the elements of $\mathbf{W}$ will mean that the measurements are affected by noise and are therefore unreliable.

#### 3.2. Algorithm Complexity

#### 3.3. Fault Detection and Isolation Scheme

Algorithm 2: FD method |

## 4. Numerical Model of the SRM

#### Validation of the 2D FE Model

## 5. Validation of the FD System

#### 5.1. FFT Method

#### 5.2. FD System

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AR | Analytical relations |

DC | Direct Current |

EKF | Extended Kalman Filter |

FD | Fault diagnosis |

FE | Finite Element |

FFT | Fast Fourier Transform |

PBC | Passivity Based Control |

PC | Personal Computer |

RPM | Revolutions Per Minute |

SRM | Switched Reluctance Motor |

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Operation | Complexity Order | Operations |
---|---|---|

${\mathbf{K}}_{k}={\mathbf{P}}_{k}^{-}{\mathbf{C}}^{T}{(\mathbf{C}{\mathbf{P}}_{k}^{-}{\mathbf{C}}^{T}+\mathbf{W})}^{-1}$ | $4{n}^{2}m/4{m}^{2}n$ | 6 |

${\widehat{\mathbf{x}}}_{k}={\widehat{\mathbf{x}}}_{k}^{-}+{\mathbf{K}}_{k}({\mathbf{y}}_{k}-\mathbf{C}{\widehat{\mathbf{x}}}_{k}^{-})$ | $2mn$ | 12 |

${\mathbf{P}}_{k}=(\mathbf{I}-{\mathbf{K}}_{k}\mathbf{C}){\mathbf{P}}_{k}^{-}$ | ∼$2{n}^{3}/2{m}^{2}n$ | ∼36 |

${\mathbf{A}}_{k}={\left[{\displaystyle \frac{\partial \varphi}{\partial \mathbf{x}}}\right]}_{\mathbf{x}={\widehat{\mathbf{x}}}_{k}}$ | - | ∼137 |

${\widehat{\mathbf{x}}}_{k+1}^{-}=\mathbf{\u0152}({\widehat{\mathbf{x}}}_{k},{\mathbf{u}}_{k})$ | $2{n}^{2}$ | 72 |

${\mathbf{P}}_{k+1}^{-}={\alpha}^{2}{\mathbf{A}}_{k}{\mathbf{P}}_{k}{\mathbf{A}}_{k}^{T}+\mathbf{Q}$ | $4{n}^{3}$ | 864 |

AR | No Fault | Fault |
---|---|---|

r | ≈0 | >0 |

Parameter | Value |
---|---|

Power | 550 [$\mathrm{W}$] |

Speed range | 0–1500 [RPM] |

Voltage | $300\phantom{\rule{3.33333pt}{0ex}}\left[\mathrm{V}\right]$ |

Inertia moment | $0.00149257\phantom{\rule{3.33333pt}{0ex}}\left[\mathrm{k}\mathrm{g}{\mathrm{m}}^{2}\right]$ |

Phase resistance | $4.2048\phantom{\rule{3.33333pt}{0ex}}\left[\mathsf{\Omega}\right]$ |

Leakage inductance | $0.5996\phantom{\rule{3.33333pt}{0ex}}\left[\mathrm{m}\mathrm{H}\right]$ |

Number of turns per phase | 284 |

Characteristics | FD System | FFT Method |
---|---|---|

on-line | yes | no |

number of current sensors | one | one |

bus current locations | one | three |

detection time | $5{T}_{ph}$ | ${T}_{ph}$ |

robustness | high | high |

cost | low | low |

fault diagnosis | easy | easy |

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**MDPI and ACS Style**

de la Guerra, A.; Jimenez-Mondragon, V.M.; Torres, L.; Escarela-Perez, R.; Olivares-Galvan, J.C. On-Line Open-Phase Fault Detection Method for Switched Reluctance Motors with Bus Current Measurement. *Actuators* **2020**, *9*, 117.
https://doi.org/10.3390/act9040117

**AMA Style**

de la Guerra A, Jimenez-Mondragon VM, Torres L, Escarela-Perez R, Olivares-Galvan JC. On-Line Open-Phase Fault Detection Method for Switched Reluctance Motors with Bus Current Measurement. *Actuators*. 2020; 9(4):117.
https://doi.org/10.3390/act9040117

**Chicago/Turabian Style**

de la Guerra, Alejandra, Victor M. Jimenez-Mondragon, Lizeth Torres, Rafael Escarela-Perez, and Juan C. Olivares-Galvan. 2020. "On-Line Open-Phase Fault Detection Method for Switched Reluctance Motors with Bus Current Measurement" *Actuators* 9, no. 4: 117.
https://doi.org/10.3390/act9040117