# Rotor Faults Diagnosis in PMSMs Based on Branch Current Analysis and Machine Learning

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## Abstract

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## 1. Introduction

_{s}is the power frequency, k is an integer constant and p is the number of pole pairs. From Formula (1), the fault characteristics of different fault types share the same frequency, so it is difficult to effectively distinguish these three types of faults by using these harmonic components. However, it is noteworthy that when fixing these three rotor faults, different methods are to be taken and different components are to be replaced, and any misdiagnosis or misjudgment of faults can waste maintenance time and even lead to equipment damage. Hence, it is of great importance to carry out a study on the diagnosis and identification of these three faults for rapid and accurate diagnosis and repair of motor faults.

## 2. Theoretical Analysis on PMSM Rotor Faults

#### 2.1. Analysis of Induced Electromotive Force

_{s}, as shown in Formula (2).

_{s}is a function correlated to the location of the rotor angle θ

_{r}and can be expressed as:

_{n}is the function amplitude of the stator winding distribution with a reference angle θ

_{s}; B

_{n}is the density amplitude of the flux covering the surface of a single slot winding. At a constant rotational speed, θ

_{r}= ω

_{r}t, in which ω

_{r}is the mechanical angular velocity, and the induced electromotive force E(t) can be expressed as:

#### 2.2. Influence of Eccentricity Fault on Induced Electromotive Force

_{h}is the length of the radial air gap when the motor is healthy and e is the eccentric distance.

_{h}. Since ε and the cosine function are both smaller than one, contents below the third item in Formula (6) can be ignored. The air gap permeability can also be expressed as:

_{s}= pw

_{r}, w

_{s}is the electromechanical angular velocity and p denotes the number of pole pairs in a permanent magnetic motor. According to Ampere’s Law, B

_{s}, the air gap flux density of the motor stator, can be defined as:

_{0}is the vacuum magnetic permeability; j

_{s}is the current density of the inner surface of the stator; j

_{0}is the peak value of the current density. Formulas (7) and (9) are substituted in Formula (8) to get Formula (10).

_{s}, the air gap flux density of the motor’s stator, can be expressed as:

_{s}when the motor suffers dynamic eccentricity, where n is a positive integer. On the main magnetic circuit of the motor, the air gap flux relates to the stator surface flux. The existence of stator reluctance can lead to a magnetic potential drop, so B

_{n}can be estimated as:

_{n}corresponding to B

_{s}. It is usually in the range of [0, 1]. When Formula (12) is substituted in Formula (11), the stator surface flux density B

_{n}of the motor with DE fault can be expressed as:

_{s}harmonic components in B

_{n}. Due to the change in B

_{n}, such harmonic components will also appear in the induced electromotive force of the motor, which can be evidenced by Formula (4). Therefore, we consider the higher power part neglected in Formula (6). When the motor suffers DE, it is for sure that various electrical signals will appear at the specific frequency of the fault harmonics. The frequency of these faulty harmonics can be expressed as:

_{s}is the power frequency of the motor. E

_{ec}(t), the induced electromotive force when eccentricity fault occurs, can be expressed as:

#### 2.3. Influence of Demagnetization Fault on Induce Electromotive Force

_{de_slot}, the counter electromotive force generated by the fault motor rotor in a stator slot, can be expressed as [14]:

_{slot}is the amplitude of a counter electromotive force generated by a healthy motor in a stator slot; K

_{de}indicates the demagnetization degree of a single permanent magnet; p is the number of pole pairs of the motor and f

_{e}is the fundamental frequency of the motor.

_{N}

_{1}is the coefficient of motor winding; G is the winding coefficient, which can be expressed in Equation (18). q is the number of slots per pole per phase, and q = Q/2mp; a is the number of the branch of the motor. Taking Equation (16) in Equation (18) into Equation (17) for calculation, the branch current E

_{de_b}is expressed as:

#### 2.4. Influence of Rotor Fault on Current

_{fa}(counter electromotive force of the fault motor) and i

_{fa}(current of the fault motor) is expressed as:

## 3. Signal Analysis and Characteristic Extraction

#### 3.1. Numerical Calculation

#### 3.2. Signal Preprocessing

#### 3.3. Signal Characteristic Extraction

_{x}is the standard deviation of x(n). Eleven characteristic factors were extracted from a group of preprocessed residual value signals of currents from three types of faults to construct fault characteristic vectors. A comparison of the characteristic factors of a set of three types of faults is shown in Figure 7.

_{7}(pulse factor), F

_{8}(margin factor), and F

_{9}(skewness), so these three factors can serve as good characteristics for fault classification. While the similarity between F

_{4}(peak value) and F

_{5}(waveform factor) is relatively high, it will interfere with the classification results and affect the accuracy of the classification results, so they should be removed. Therefore, this paper selected nine feature factors, such as F

_{7}(pulse factor), F

_{8}(margin factor), and F

_{9}(skewness) to establish fault characteristic vectors.

## 4. Fault Classification and Diagnosis Algorithm

#### 4.1. Diagnosis Principle Based on SVM

_{1}and H

_{2}are the parallel lines of the two types of sample points closest to the optimal classification line H, respectively. The distance between H

_{1}and H

_{2}is the classification interval. The sample points A

_{1}, A

_{2}and B

_{1}on the two lines are the points closest to H in the two types of samples, which are called support vectors.

_{1}, y

_{1}), …, (x

_{i}, y

_{i}),…, (x

_{m}, y

_{m})} (where i = 1, …, m) as the sample set and y

_{i}= {1, −1} as the classification label. If x

_{i}belongs to the first type, y

_{i}= 1; if xi belongs to the second type, y

_{i}= −1. The optimal classification hyperplane can not only realize accurate classification but also maximize the classification interval and thus converts to the problem of solving a second order convex programming.

_{i}is the slack variable; m is the total number of samples; φ(x

_{i}) is the characteristic mapping; C is the penalty factor, which is used to balance the model complexity and loss error. The kernel function of SVM is the inner product function which aims to segment the samples in high-order vectors. Whether the kernel function is suitable or not hinges on the classification effect of SVM. There are three common kernel functions: radial basis function, linear kernel function, and polynomial kernel function. Radial basis function is used as the kernel function of SVM in this paper, which is expressed as follows:

#### 4.2. SVM Parameter Optimization Based on GA

_{b}of PMSM operation under different operating conditions after the occurrence of different types of rotor faults.

_{b}are preprocessed and the time-domain features are extracted to construct the fault feature vectors for fault classification. These certain numbers of fault feature vector samples constitute the sample set, and they are divided into training set and test set.

_{c}during PMSM operation is acquired and the branch current i

_{c}is subjected to Fourier transform to obtain the frequency spectrum. The frequency spectrum is determined whether there is a significant increase in the fault harmonic component of (1 ± n/p)f

_{s}.

_{c}spectrum and then preprocess the branch current i

_{c}time-domain waveform and extract time domain features. The feature vectors are constructed and brought into the GA-SVM model for rotor fault identification.

## 5. Results and Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**MBCS waveforms of motors with different faults (

**a**) in the time domain and (

**b**) in the frequency domain.

**Figure 5.**Residual and normalized MBCS of different faults. (

**a**) Demagnetization fault, (

**b**) eccentricity fault and (

**c**) hybrid fault.

**Figure 10.**Optimizing the training process of C and γ by GA; (

**a**) iterative process; (

**b**) training set classification results.

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

Rated output power | 28.3 kW | Number of poles(2p) | 8 |

Rated speed | 1500 r/min | Number of slots(Q) | 48 |

Rated voltage | 345 V | Number of phase(m) | 3 |

Length | 200 mm | Parallel branches(a) | 2 |

Stator outer diameter | 230 mm | Magnet thickness | 4.5 mm |

Stator inner diameter | 149 mm | Embrace | 0.87 |

Rotor outer diameter | 147 mm | Magnet type | XG196/96 |

Rotor inner diameter | 60 mm |

Fault Type | Rated Speed | Torque of Load | Fault Settings |
---|---|---|---|

Eccentric fault | 1500 r/min | 180.16 N, 144.128 N, 108.096 N, 72.064 N, 36.032 N | Static eccentric 0.3 mm |

Dynamic eccentric 0.4 mm | |||

Static eccentric 0.2 mm and dynamic eccentric 0.2 mm (hybrid eccentric) | |||

Demagnetization fault | 1500 r/min | 180.16 N, 144.128 N, 108.096 N, 72.064 N, 36.032 N | No. 5 pole demagnetization 20% |

No. 3 and 5 pole demagnetization 20% | |||

No. 4 and 5 pole demagnetization 20% | |||

Hybrid fault | 1500 r/min | 180.16 N, 144.128 N, 108.096 N, 72.064 N, 36.032 N | No. 5 pole demagnetization 20% and static eccentric 0.3 mm |

No. 3 and 5 pole demagnetization 20% and Dynamic eccentric 0.4 mm | |||

No. 4 pole demagnetization 20%, static eccentric 0.2 mm and dynamic eccentric 0.2 mm |

Fault Type | High Peak | Low Peaks | Number of Wave Peaks |
---|---|---|---|

Demagnetization fault | 2 | 0 | 2 |

Eccentric fault | 1 | 2 | 3 |

Hybrid fault | 1 | 1 | 2 |

Characteristic Factor | Statistical Calculation Formula | Characteristic Factor | Statistical Calculation Formula | Meaning |
---|---|---|---|---|

Absolute-mean Deviation: F_{1} | $\frac{1}{N}{\displaystyle \sum _{n=1}^{N}\left|x\left(n\right)\right|}$ | Standard Deviation: F_{2} | $\sqrt{\frac{1}{N-1}{\displaystyle \sum _{n=1}^{N}{\left[x\left(n\right)-\overline{x}\right]}^{2}}}$ | Represents the time-domain fluctuation energy of the signal |

Root-mean-square Value: F_{3} | $\sqrt{\frac{1}{N}{\displaystyle \sum _{n=1}^{N}{x}^{2}(n)}}$ | Peak Value: F_{4} | $\mathrm{max}\left|x\left(n\right)\right|$ | |

Shape Factor: F_{5} | ${F}_{3}/{F}_{1}$ | Crest Factor: F_{6} | ${F}_{4}/{F}_{3}$ | Represents the time-domain distribution characteristics of the signal |

Impulse Factor: F_{7} | ${F}_{4}/{F}_{1}$ | Margin Factor: F_{8} | ${F}_{4}\xb7{\left(\frac{1}{N}{\displaystyle \sum _{n=1}^{N}\left|\sqrt{x\left(n\right)}\right|}\right)}^{-2}$ | |

Skewness: F_{9} | $\frac{{\displaystyle \sum _{n=1}^{N}{\left[x\left(n\right)-\overline{x}\right]}^{3}}}{(N-1){\sigma}_{x}^{3}}$ | Kurtosis: F_{10} | $\frac{{\displaystyle \sum _{n=1}^{N}{\left[x\left(n\right)-\overline{x}\right]}^{4}}}{(N-1){\sigma}_{x}^{4}}$ | |

Energy: F_{11} | $\sum _{n=1}^{N}{x}^{2}(n)$ |

Algorithm Model | KNN | LDA | SVM | BP | GA-SVM |
---|---|---|---|---|---|

Accuracy | 85.1% | 77% | 74.4% | 77.6% | 92.2% |

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## Share and Cite

**MDPI and ACS Style**

Yu, Y.; Gao, H.; Zhou, S.; Pan, Y.; Zhang, K.; Liu, P.; Yang, H.; Zhao, Z.; Madyira, D.M.
Rotor Faults Diagnosis in PMSMs Based on Branch Current Analysis and Machine Learning. *Actuators* **2023**, *12*, 145.
https://doi.org/10.3390/act12040145

**AMA Style**

Yu Y, Gao H, Zhou S, Pan Y, Zhang K, Liu P, Yang H, Zhao Z, Madyira DM.
Rotor Faults Diagnosis in PMSMs Based on Branch Current Analysis and Machine Learning. *Actuators*. 2023; 12(4):145.
https://doi.org/10.3390/act12040145

**Chicago/Turabian Style**

Yu, Yinquan, Haixi Gao, Shaowei Zhou, Yue Pan, Kunpeng Zhang, Peng Liu, Hui Yang, Zhao Zhao, and Daniel Makundwaneyi Madyira.
2023. "Rotor Faults Diagnosis in PMSMs Based on Branch Current Analysis and Machine Learning" *Actuators* 12, no. 4: 145.
https://doi.org/10.3390/act12040145