
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39300125
72596
10.1038/s41598-024-72596-5
Article
Permanent magnet synchronous motor demagnetization fault diagnosis based on PCA-ISSA-PNN
Yu Yinquan 12
Li Yang 1259303741@qq.com

12
Zeng Dequan 12
Hu Yiming 12
Yang Jinwen 12
1 https://ror.org/05x2f1m38 grid.440711.7 0000 0004 1793 3093 Institute of Precision Machining and Intelligent Equipment Manufacturing, Key Laboratory of Conveyance and Equipment of Ministry of Education, East China Jiaotong University, Nanchang, 330013 China
2 https://ror.org/05x2f1m38 grid.440711.7 0000 0004 1793 3093 School of Mechatronics and Vehicle Engineering, East China Jiaotong University, Nanchang, 330013 China
20 9 2024
20 9 2024
2024
14 2192111 3 2024
9 9 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Aiming at the demagnetization fault problem of the permanent magnet synchronous motor (PMSM), a demagnetization fault diagnosis method based on the combination of the principal component analysis (PCA) algorithm, the improved sparrow search algorithm (ISSA), and the probabilistic neural network (PNN) algorithm is proposed. First, the principal components of phase currents are extracted using PCA. Second, ISSA is used to optimize the smoothing coefficients of the PNN algorithm, and the optimized PNN algorithm is combined with PCA to obtain the PCA-ISSA-PNN fault diagnosis model. Finally, the established fault diagnosis model was tested using the current data collected from the experiments and compared with the fault diagnosis indexes and optimization performance of the conventional PNN, PCA-PNN, PCA-GA (genetic algorithm)-PNN, PCA-DA (dragonfly algorithm)-PNN, PCA-GTO (artificial gorilla troop optimizer)-PNN, PCA-AHA-PNN, and PCA-SSA-PNN. The test results show that the fault diagnosis accuracy of PCA-ISSA-PNN reaches 95.83%, and the fault diagnosis indexes are significantly higher than those of PNN, PCA-PNN, PCA-GA-PNN, and PCA-DA-PNN; its optimization performance is also significantly better than that of PCA-GTO-PNN, PCA-AHA-PNN, and PCA-SSA-PNN, which verifies the accuracy and efficiency of the proposed method.

Keywords

Permanent magnet synchronous motor
Principal component analysis algorithm
Improved sparrow search algorithm
Probabilistic neural network
Subject terms

Electrical and electronic engineering
Mechanical engineering
the National Natural Science Foundation of China52067006 Yu Yinquan the High-end Foreign Expert Talents Project of the Ministry of Science and Technology of ChinaG2021022002L Yu Yinquan the Natural Science Foundation of Jiangxi Province20232BAB214092 Zeng Dequan The Ganpo Talent Support Program-Leading Academic and Technical Personnel in Major Disciplines of Jiangxi Province20232BCJ23091 Zeng Dequan The Key R & D Program of Jiangxi Province20224BBE51048 Zeng Dequan The 03 Special Program and 5G Project of Jiangxi Province20232ABC03A30 Zeng Dequan issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Permanent magnet synchronous motor (PMSM) is widely used in new energy vehicles, aerospace, transportation, and other fields because of its simple structure, high efficiency, high power density, and low noise1–3. However, due to the environment and temperature during operation, PMSM inevitably produces faults that affect the normal operation of the machine or even burn out, paralyzing the whole system and causing economic losses. Therefore, the study of PMSM fault diagnosis methods is of enormous significance for early detection of motor faults and saving maintenance costs.

Currently, there are numerous permanent magnet synchronous motor fault diagnosis methods, such as the fault diagnosis method based on the analytical model of the motor, the fault diagnosis method based on feature signal extraction, the motor fault diagnosis method based on artificial intelligence, and so on. The fault diagnosis method is based on the motor's analytical model4, which analyzes the fault using an accurate mathematical model. This method has the advantage of being able to diagnose a certain type of fault in the motor with high accuracy, but it is difficult to establish accurate mathematical models for different faults in different motors5. For example, S. Moon et al. proposed a method to diagnose demagnetization faults based on the structure of PMSM, addressing the challenge of accurately diagnosing the degree of demagnetization due to the change in inductance after motor demagnetization. They achieved this by calculating the inductance, taking into account the magnetic saturation effect, and combining it with the method of least squares. The experimental results demonstrate that this method can accurately track the change in inductance and accurately diagnose the degree of demagnetization of the permanent magnets6. A study by D.S.B. Fonseca et al. looked at how well the motor works when it's healthy and when it has a turn-to-turn short-circuit fault. They created a dynamic model of the motor's fault in MATLAB/Simulink and simulated and tested it with turn-to-turn short-circuit faults. The results show that the simulation results and experiment results match well, which shows that the proposed model can accurately simulate the PMSM fault state7. The fault diagnosis method relies on feature signal extraction8, utilizing signal processing to extract fault features from the voltage, current, and other running motor data. For instance, J. Gao et al. extracted the phase currents in the permanent magnet synchronous motor, converted the current signal from the time domain to the position domain with constant amplitude and frequency, and then used FFT analysis to diagnose demagnetization faults in both smooth and non-smooth conditions9. However, the process of converting the current signals proved to be complex and challenging to implement. J. Hang et al. utilized the wavelet transform to extract fault features from the cost functions found in PMSM model predictive control (MPC) systems. They then diagnosed turn-to-turn short-circuit faults by monitoring the normalized energy-dependent eigenvectors derived from the wavelet transform coefficients10. However, the selection of wavelet basis functions significantly influences the results, and determining these wavelet basis functions can be challenging. An artificial intelligence-based fault diagnosis method employs intelligent classification algorithms to pinpoint faults11. For example, C.-S. Wang et al. can achieve the detection of demagnetization faults, bearing faults, and rotor eccentricity faults in motors by analyzing the combined features of motor torque and current signals using a one-dimensional convolutional neural network12. The advantage of this method is that feature extraction can extract multi-scale features from complex conditions, but the amount of experimental data used is large, and when the amount of experimental data is small, overfitting phenomena occur, so it is not applicable to small samples. X. Song et al. used the S-transform to perform time–frequency decomposition of the back EMF signal of a permanent magnet linear synchronous motor in order to extract the feature parameters, establish the feature vectors by comparing the standard deviation values and similarity of different parameters, then use the particle swarm algorithm to optimize the regularization parameter and the width of the kernel function in the least-squares support vector machine, and finally use the optimized algorithm to perform the fault diagnosis, which can accurately identify the position of the demagnetization of the motor and the degree of demagnetization13, but the acquisition of the back EMF signals usually involves installing additional sensors, which increases the unnecessary cost.

The probabilistic neural network (PNN) algorithm is an efficient supervised pattern classification method that has the advantages of a simple network structure, fast computational speed, high computational accuracy, good classification effect, etc. It is particularly well-suited for fault diagnosis and classification problems with small samples, and it has a wide range of applications in the field of fault diagnosis14. X. Dai et al. used variational modal decomposition (VMD) on PMSM current signals to find the intrinsic modal component (IMF). They then used the energy value of the IMF as the eigenvector, the fuzzy C-mean algorithm to find the clustering centers of the fault data, and the PNN to sort the faults into two groups: turn-to-turn short-circuit faults and demagnetization faults15. In their study, J. Ding et al. used a variational particle swarm algorithm to find the best number of decompositions and penalty factors in the VMD. They then used the sample entropy of the IMF as the feature vector and combined it with PNN to correctly classify gear faults16. Nagi et al. extracted the color features of glucose leaves using fuzzy color histograms, and then used the PNN algorithm to accurately classify grapevine diseases, thereby preventing yield loss17. However, the above study's use of empirically set smoothing coefficients in the PNN significantly impacted the diagnostic accuracy.

Based on the above research, this paper proposes a PCA-ISSA-PNN-based demagnetization fault diagnosis method to improve the accuracy of PMSM fault diagnosis and reduce its cost. This method combines feature signal extraction and artificial intelligence, using the interior PMSM as the research object. The method first uses the principal component analysis algorithm to extract the principal components of the current signal, then uses ISSA to optimize the smoothing coefficients in the PNN, and combines the optimized PNN algorithm with PCA to get the fault diagnosis model of PCA-ISSA-PNN. Finally, the established fault diagnosis model was tested using the current data collected from the experiments and compared with the fault diagnosis indexes and optimization performance of the conventional PNN, PCA-PNN, PCA-GA (genetic algorithm)-PNN, PCA-DA (dragonfly algorithm)-PNN, PCA-GTO (artificial gorilla troop optimizer)-PNN, PCA-AHA-PNN, and PCA-SSA-PNN. The test results show that the fault diagnosis accuracy of PCA-ISSA-PNN reaches 95.83%, and the fault diagnosis indexes are significantly higher than those of PNN, PCA-PNN, PCA-GA-PNN, and PCA-DA-PNN; its optimization performance is also significantly better than that of PCA-GTO-PNN, PCA-AHA-PNN, and PCA-SSA-PNN, which verifies the accuracy and efficiency of the proposed method.

The method of this paper has the following advantages compared with previous research methods:The PCA-ISSA-PNN method is a fault diagnosis method that combines feature signal extraction with artificial intelligence. There is no need to establish an accurate mathematical model.

The PCA extraction of fault signals method has simple steps, which can reduce the data dimension and improve operational efficiency without the need for complex data conversion.

The method uses a small number of samples, which can improve operational efficiency. The use of current signals for diagnosis does not require the installation of additional sensors, which reduces costs.

The use of ISSA to optimize the smoothing coefficient in the PNN improves diagnostic accuracy.

PMSM mathematical model

The voltage equation of the PMSM on the d-q axis in the synchronous rotating coordinate system can be expressed18 as:1 ud=Rid+Lddiddt-ωeLqiquq=Riq+Lqdiqdt-ωe(Lqiq+ψ)

where ud,uq,id,iq are the stator voltage and stator current of d-q, respectively. Ld,Lq are the stator inductance of the d-axis and q-axis, respectively, and R is the winding resistance. ωe is the electrical angular velocity, and ψ is the magnetic chain of the permanent magnet.

The electromagnetic torque equation can be expressed18 as:2 Te=32pniqidLd-Lq+ψ

where Te is the electromagnetic torque and pn is the number of pole pairs.

When demagnetization of the permanent magnet occurs, the magnetic chain in the three-phase winding of the PMSM, which is generated by the no-load air-gap magnetism of the permanent magnet, can be expressed15 as:3 ccψA=∑k=1∞ψkpncoskpnθψB=∑k=1∞ψkpncoskpnθ-23πψC=∑k=1∞ψkpncoskpnθ+23π

where ψA, ψB, ψC are the magnetic chains of the three-phase windings, k is a positive integer, and θ is the angle between the air-gap magnetic field axis of the permanent magnet and phase A of the motor.

The magnetic chain equation of the PMSM in the d-q axis can be expressed15 as:4 ψd=ψ+Ldidψq=Lqiq

where ψd, ψq are the magnetic chains of the d-axis and q-axis, respectively.

The above model shows that when the PMSM demagnetizes, the magnetic chain changes and affects the current, allowing us to select the current signal as the fault feature vector.

Relevant theories

Principal component analysis algorithm

Principal component analysis (PCA) is a technique commonly used to analyze large datasets and can be used to reduce high-dimensional data to the best feasible set of features19–21. PCA can significantly reduce the complexity of the neural network, thus improving the network recognition rate and prediction accuracy. Therefore, this paper uses PCA to extract data features to enhance fault diagnosis accuracy.

The specific steps of PCA are as follows:

Select test samples and training samples to build the initial feature matrix X as follows:X=X11⋯X1m⋯⋯⋯Xn1⋯Xnm

where n is the number of samples and m is the initial feature dimension.

Calculate the correlation coefficient matrix;

Calculate the eigenvalues λi(i=1,⋯,m) of correlation coefficient matrix and arrange λi in ascending order. Calculate the accumulative contribution ∑i=1hλi/∑i=1mλi, and determine the number of principal component dimensions h.

Calculate the principal components.

Probabilistic neural network algorithm

Probabilistic neural network (PNN) is a feed-forward neural network model based on the Bayesian minimum risk criterion and the Parzen window function probability density estimation method proposed by Specht in 199014, which has been widely used in different scientific fields22. The PNN method is swift and more accurate compared to other neural network methods23.

A PNN network usually consists of four layers: the input layer, the pattern layer, the summation layer, and the output layer24. The structure of PNN network is shown in Fig. 1, assuming that the input test sample is x=(x1,x2,⋯,xk), the training sample of pattern layer is xij=(x1j,x2j,⋯,xij), and the output is Typei.Fig. 1 The structure of PNN.

Layer 1 of the PNN is the input layer, which receives the feature vectors of the test samples that need to be classified for fault classification and passes them to the network. Layer 2 is the pattern layer, which is used to compute the similarity between the input feature vectors and the patterns of each category in the training set. Its input–output relationship can be determined by Eq. (5):5 δijx=12πk2σkexp-x-xijTx-xij2σ2

where xij is the feature vector of the jth sample of the ith class in the training sample, x is the feature vector of the prediction sample, k is the sample dimension, and σ is the smoothing coefficient.

Layer 3 is the summation layer, which serves to compute the probability accumulation of a particular class to obtain the probability of belonging to a particular class of faults. The summation layer is calculated by Eq. (6), which sums up the mode layer units in each class.6 fiNix=1Ni∑j=1iδijx

where fiNix is the probability density function of x belonging to the ith class, and Ni is the number of training samples in the ith class.

Layer 4 is the output layer, which serves to output the fault category of the motor.

Combining Eq. (5) with Eq. (6), the formula for the probability density is obtained as:7 fiNix=12πk2σk·1Ni·∑j=1iexp-x-xijTx-xij2σ2

Usually, the smoothing coefficient σ in PNN is set based on experience. If it is not set properly, it will affect the accuracy of recognition. Therefore, in order to improve recognition accuracy, this paper adopts the ISSA algorithm to optimize the smoothing coefficient σ in PNN.

Improved sparrow search algorithm

The sparrow search algorithm (SSA) is an intelligent optimization algorithm proposed in 2020 to mimic the predatory and anti-predatory behavior of sparrows25. During sparrow foraging, sparrows are categorized into discoverers and followers26. Discoverers are responsible for finding and guiding the way to food areas; they have high energy and provide foraging information to the entire population. Followers, on the other hand, are lower in energy and rely on the guidance of the finders for food27,28.

During each iteration, the position update formula of the discoverer can be described29 as:8 Xi,jt+1=Xi,jt·exp-iα∙itermax,ifR2<STXi,jt+Q·L,ifR2≥ST

where t is the current iteration number, itermax is the maximum iteration number, and Xi,jt+1 is the position information of the ith sparrow in the jth dimensional space .α is a uniform random number of 0,1, R2 is an alarm value of 0,1, and ST is a safety value of 0.5,1. Q is a random number and obeys a normal distribution, and L is a 1×d dimensional matrix with all 1 elements. When R2<ST, it means that there is no predator around the population, and the discoverer can search widely for food. When R2≥ST, it means that a sparrow has discovered the predator and alarmed the police, then all sparrows have to fly quickly to a safe place to forage.

The follower's position update formula can be described as:9 Xi,jt+1=Q·expXworstt-Xi,jti2,i>n/2Xpt+1+Xi,jt-Xpt+1A+·L,i≤n/2

where Xpt+1 is the optimal position that the discoverer is currently positioned at, and Xworstt is the worst position that the discoverer is currently positioned at. A is a 1×d dimensional matrix with elements equal only to 1 or -1 and A+=AT(AAT)-126 . When i>n/2, it means that the ith follower cannot grab food and has to go to other regions to feed. When i≤n/2, it means that the follower is feeding around the optimal individual Xp.

When there is a danger, the sparrow moves towards a safe area. The position update equation can be described as:10 Xi,jt+1=Xbestt+β·Xi,jt-Xbestt,fi>fgXi,jt+K·Xi,jt-Xworsttfi-fw+ε,fi=fg

where Xbestt is the global optimal position, and Xworstt is the worst position that the discoverer is currently positioned at. β is a parameter controlling the step size and has mean 0 and variance 1, obeying a normal distribution. K is -1,1 the number of random numbers determining the direction of sparrow movement. fi is the current individual fitness value, fg is the current global optimal fitness value, fw is the current global worst fitness value, and ε is the smallest constant that avoids a denominator of zero26. When fi>fg, sparrows are at the edge of the population, they are more vulnerable to predation. When fi=fg, sparrows in the middle of the population realize the crisis and need to move closer to their peers to reduce the likelihood of predation.

SSA outperforms traditional optimization algorithms in terms of convergence speed, stability, and robustness30. However, the initial population of SSA is randomly initialized with great uncertainty, which has the disadvantages of low population diversity and making it easy to fall into local optimal solutions31. Therefore, the main improvement of SSA should focus on refining the population initialization32. The random generation of sparrows during the initialization stage makes the initial solution susceptible to aggregation, resulting in low coverage in the solution space and a low individual difference. However, the chaotic mapping initialization of the population can effectively address this issue. The circle mapping of chaos has the characteristics of randomness, uniformity, and order, which are more stable, and the coverage of the chaotic values is high. Therefore, in this paper, circle mapping is chosen to optimize the initial population to obtain the improved sparrow search algorithm (ISSA). The formula for generating the initial population by circle mapping is shown in Eq. (11):11 xn+1=mod(xn+b-a2πsin2πxn,1)

where n is the dimension of the solution, a and b are the control parameters, xn is the chaotic solution, and mod denotes the residual.

PMSM demagnetization fault diagnostic model

In this paper, we use the error value between the real result and the predicted result of PCA-PNN as the fitness function of ISSA, and the optimized parameters of ISSA as the smoothing coefficients of PCA-PNN. This allows us to combine the three algorithms of PCA, ISSA, and PNN, thereby improving the performance of fault diagnosis. The flowchart of the PCA-ISSA-PNN demagnetization fault diagnosis model is shown in Fig. 2. Firstly, the initial parameters of ISSA are set: the initial number of sparrow individuals is set to 10, the number of iterations is set to 30, and the proportion of the initial discoverer is set to 20% of the number of sparrow individuals. Then it inputs the normalized training samples and test samples, uses PCA-PNN for pattern recognition, calculates the classification error, and obtains the optimized parameter of ISSA through continuous iteration. Finally, the optimized parameter is brought into the smoothing coefficient to obtain the PCA-ISSA-PNN model for testing and validation.Fig. 2 Flowchart of PCA-ISSA-PNN diagnosis model.

Experimental platform

In order to verify the accuracy of the method in this paper and to obtain the PMSM normal and faulty phase current data, the experiment is carried out using the permanent magnet synchronous motor fault experimental platform, as shown in Fig. 3. The platform mainly includes an experimental motor, a load motor, a dynamometer, a power cabinet, a frequency converter, and a control computer. The experimental motors are normal motors with 50% unipolar demagnetization and 50% overall demagnetization, for a total of three motors. The structures are interior type. The related parameters of PMSM are shown in Table 1, and the related parameters of the load motor are shown in Table 2. Power cabinet input is three-phase 380-500 V voltage, frequency is 50 Hz, output is three-phase 380-500 V voltage, the frequency is 0-320 Hz, input and output current are 46A, rated power is 22 kW. The frequency converter model is SP332201C, the rated voltage is 3300 V, the rated current is 200A, the frequency test range is 0.1–1500 Hz, and the power is 1.5 kW. The function of the control computer is to control the platform to energize and de-energize, set the load torque, and motor speed.Fig. 3 Permanent magnet synchronous motor fault experimental platform.

Table 1 Related parameters of PMSM.

Model number	TZQ160-5-72	
Rated power	5 kW	
Rated voltage	72 V	
Rated speed	3000 rpm	
Peak speed	6800 rpm	
Rated torque	19.9 N∙m	
Number of pole pairs	4	

Table 2 Related parameters of load motor.

Model number	DSM-38C015K-E0F3A24S	
Rated power	15 kW	
Rated voltage	380 V	
Rated speed	1500 rpm	
Peak speed	8000 rpm	
Rated current	29.2 A	
Reference frequency	50 Hz	
Number of pole pairs	4	

The data acquisition process involves the following steps: First, open the switch on the power cabinet and wait for its voltage to stabilize to 380 V. Next, the control computer activates the platform power supply; it then initiates the experimental and load motors, adjusting their rotational speeds to 500–3000 rpm and the load to a range of 1–10 N∙m. Finally, the WP4000 dynamometer, with a sampling frequency of 125 kHz, collects the data. It has a voltage measurement range of 100 μV–15 kV, a current measurement range of 100 μA–15 kA, and a measurement accuracy of 0.1%. We collect data once for every 100 rpm increase in rotational speed and every 1 N∙m increase in load torque.

Experimental results and analysis

Fault features extraction

The experiment selected 260 sets of current data under different working conditions: normal motor, 50% unipolar demagnetization, 50% overall demagnetization, motor speed range of 500–3000 rpm, load range of 1–10 nm, a total of 780 sets of data, each with an acquisition time of 0.15 s and 18,750 sample points. The current-load waterfall diagram of the PMSM for a speed of 2000 rpm and a load of 1–10 nm is shown in Fig. 4. The current waveforms of the three motors at a load of 9 Nm and a speed of 2000 rpm are shown in Fig. 5. From the figure, it can be seen that when the demagnetization fault occurs in the motor, the current will increase; the overall demagnetization 50% motor current amplitude is the largest; the unipolar demagnetization motor is the second largest; and the normal motor amplitude is the smallest.Fig. 4 PMSM current-load waterfall diagram (2000 rpm).

Fig. 5 Current waveforms (9N · m: 2000 rpm).

To extract features using PCA, first create the initial matrix and calculate the correlation coefficient matrix. Second, calculate the eigenvalues corresponding to the correlation coefficient matrix and arrange them from largest to smallest, as shown in Fig. 6. Next, calculate the cumulative contribution and the contribution of individual principal components, as shown in Fig. 7. Figures 6 and 7 reveal that the subsequent principal components, after the 25th, have low eigenvalues and contribution degrees. Therefore, we can select the first 25 principal components to represent the majority of the data, with a corresponding cumulative contribution degree of 94.4%. Figure 8 shows the calculation of the last retained principal components.Fig. 6 Eigenvalue ordering diagram.

Fig.7 Contribution and accumulative contribution chart.

Fig. 8 Results chart of PCA (9N · m: 2000 rpm).

Experimental results analysis

In the experiment, 220 groups of normal motor, unipolar demagnetized 50% motor, and overall demagnetized 50% motor samples are selected as training samples, and the remaining 40 groups are used as test samples, with a total of 660 groups of training samples and a total of 120 groups of test samples, and the label designations are set to be 1, 2, and 3 in order. The pattern recognition is performed by using the PNN, PCA-PNN, PCA-GA-PNN, PCA-DA-PNN, PCA-GTO-PNN, PCA-AHA-PNN, PCA-SSA-PNN, and PCA-ISSA-PNN, respectively.

From the experimental results, it can be seen that the accuracy of the PNN test set is only 70.83%, and the test set has 18 recognition errors at class 1, 13 recognition errors at class 2, and 4 recognition errors at class 3, as shown in Fig. 9. The accuracy of the PCA-PNN test set was 88.33%, and the test set showed 6 recognition errors at class 1, 5 recognition errors at class 2, and 3 recognition errors at class 3, as shown in Fig. 10. The test set's accuracy is improved by 17.5% over PNN.Fig. 9 PNN verification results.

Fig. 10 PCA-PNN verification results.

The smoothing coefficient σ optimization improved the accuracy of the PCA-GA-PNN, PCA-DA-PNN, PCA-GTO-PNN, PCA-AHA-PNN, PCA-SSA-PNN, and PCA-ISSA-PNN test sets. We ran the algorithms several times each, and obtained the average iteration curves of the six algorithms, as shown in Fig. 11. Figure 11 demonstrates that both GA and DA iterations achieve local optimality, resulting in corresponding values σ of 0.7874 and 0.6456. GTO, AHA, SSA, and ISSA all reach the optimal solution, which is 0.089.Fig. 11 Iteration curves of six optimization algorithms.

The accuracy of the test set corresponding to PCA-GA-PNN is 92.5%, and the test set has 3 recognition errors in class 1, 4 recognition errors in class 2, and 2 recognition errors in class 3, as shown in Fig. 12. The accuracy of the test set corresponding to PCA-DA-PNN is 91.67%, and the test set has 4 recognition errors in class 1, 4 recognition errors in class 2, and 2 recognition errors in class 3, as shown in Fig. 13. The corresponding test set of PCA-(GTO/AHA/SSA/ISSA)-PNN has an accuracy of 95.83%, and it has 3 recognition errors in class 1, 1 recognition error in class 2, and 1 recognition error in class 3, as shown in Fig. 14.Fig. 12 PCA-GA-PNN verification results.

Fig. 13 PCA-DA-PNN verification results.

Fig. 14 PCA-(GTO/AHA/SSA/ISSA)-PNN verification results.

The parameters related to the PNN network in the eight algorithms are shown in Table 3. From Table 3, it can be seen that all the PNN networks have a 4-layer structure with 660 nodes in layer 2, 3 nodes in layer 3 and 1 node in layer 4. However, the original PNN network has 18,750 nodes in layer 1. Due to dimensionality reduction using PCA, all other 7 algorithms have 25 nodes in layer 1. In this paper, only the smoothing coefficients σ are optimized, so the values of the corresponding smoothing coefficients σ are different after using the optimized algorithms, and the accuracy of the different algorithms will be different.Table 3 Parameters associated with PNN networks in eight algorithms.

Algorithm	Layers	Layer 1
nodes	Layer 2
nodes	Layer 3
nodes	Layer 4
nodes	Smoothing
coefficient σ	
PNN	4	18,750	660	3	1	1.92	
PCA-PNN	25	1.92	
PCA-DA-PNN	25	0.6456	
PCA-GA-PNN	25	0.7874	
PCA-(GTO/AHA/SSA/ISSA)-PNN	25	0.089	

The performance comparison of the six optimization algorithm models is shown in Table 4. From Table 4, it can be seen that GA and DA fall into the local optimum during optimization with average fitness of 0.075 and 0.0833, respectively, and GTO, AHA, SSA, and ISSA all reach the optimal solution with fitness of 0.4167. However, the average optimization time of ISSA is only 7.8 s, and the convergence takes place at the 3rd time on average, which is significantly lower than that of GTO, AHA, and SSA, as shown in Fig. 15, proving the high efficiency of ISSA.Table 4 Performance comparison of six optimization algorithms.

Algorithm	Average optimization time	Average optimal fitness	Average number of iterations to optimization	
PCA-GA-PNN	Not optimal	0.075	Not optimal	
PCA-DA-PNN	Not optimal	0.0833	Not optimal	
PCA-AHA-PNN	23.79s	0.4167	13th	
PCA-GTO-PNN	42.68s	0.4167	12th	
PCA-SSA-PNN	16.15s	0.4167	6th	
PCA-ISSA-PNN	7.8s	0.4167	3rd	

Fig. 15 Performance comparison of four optimization algorithms.

A comparison of the eight algorithms is shown in Table 5. From Table 5, it can be seen that PNN has the lowest recognition accuracy, precision per class, recall per class, and F1 score. When the PCA algorithm was added, the recognition accuracy reached over 88%, the precision rate for each class reached over 82%, the recall rate for each class reached over 85%, and the F1 scores reached over 83%, which demonstrated the high efficiency of PCA feature extraction. Among them, the PCA-ISSA-PNN accuracy reached 95.83%, the precision of each class reached more than 94.87%, the recall of each class reached more than 92.5%, and the F1 scores reached more than 93.67%, which were significantly higher than the other models. It proves that the PCA-ISSA-PNN model has good fault diagnosis indexes and can identify demagnetization faults in motors with high precision and efficiency.Table 5 Comparison results of eight algorithms.

Algorithm	Accuracy (%)	Precision (%)	Recall (%)	F1 score (%)	
		1	2	3	1	2	3	1	2	3	
PNN	70.83	75.86	87.1	60	55	67.5	90	63.77	76.06	72	
PCA-PNN	88.33	82.93	87.5	94.8	85	87.5	92.5	83.95	87.5	93.64	
PCA-DA-PNN	91.67	87.8	94.74	92.68	90	90	95	88.89	92.32	93.83	
PCA-GA-PNN	92.5	88.1	94.74	95	92.5	90	95	90.54	92.32	95	
PCA-(GTO/AHA/SSA/ISSA)-PNN	95.83	94.87	97.5	95.12	92.5	97.5	97.5	93.67	97.5	96.30	

The experimental dataset used in this paper is only a segment of the real dataset selected for experiments and dimensionality reduction, so the sample size of the experimental dataset is smaller compared to the real dataset, which may have the following effects on the model.The sample dimension of PCA after dimensionality reduction is smaller compared with the real dataset, which may lose some information, thus affecting the accuracy and recall of the model.

The number of samples and the length of samples selected for the experiment may not fully reflect all the information in the real dataset, so the model trained on the experimental dataset may not generalize well to the real dataset.

Real datasets are large and complex, and using real datasets for training and evaluation may require more computational resources and time. Experimental datasets, on the other hand, can reduce these costs to some extent.

Conclusion

This paper proposes a PMSM demagnetization fault diagnosis method based on PCA-ISSA-PNN, using the interior PMSM as the research object. The established fault diagnosis model is tested and validated using the current data collected from the experiment. The validation results show that:The accuracy of the PMSM fault diagnosis model based on PCA-ISSA-PNN reaches 95.83%, which is higher than the fault diagnosis accuracies of 70.83% for the PNN network, 88.33% for the PCA-PNN network model, 92.5% for the PCA-GA-PNN network model, and 91.67% for the PCA-DA-PNN network model. The accuracy of the PCA-ISSA-PNN class has higher accuracy, recall, and F1 score than them, and its fault diagnosis index is clearly optimal.

The PCA-ISSA-PNN-based PMSM fault diagnosis model has a significantly lower average running time and average number of convergences than PCA-GTO-PNN, PCA-AHA-PNN, and PCA-SSA-PNN. PCA-ISSA-PNN has better fault diagnosis performance.

The accuracy and efficiency of the proposed PCA-ISSA-PNN-based fault diagnosis method are demonstrated. The method solves the problem of unsatisfactory diagnostic effects due to the artificial setting of smoothing coefficients in the PNN network, improves the accuracy of PMSM demagnetization fault diagnosis, and has good prospects for popularization and application.

Future work

In this paper, we only improve the initial population of SSA, and in the future, we will improve the optimization strategy of SSA to further optimize it. Due to the limitations of experimental conditions, the method proposed in this paper is only applied to three kinds of fault classification at present, and in the future, we will try to diagnose the multiple kinds of faults in PMSM by using this method. Additionally, readers can apply this method to other fields and experiment with various feature extraction methods, which could potentially lead to improved classification results.

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant No. 52067006), the Natural Science Foundation of Jiangxi Province (Grant No. 20232BAB214092; Grant No. 20224BAB214045), the Ganpo Talent Support Program-Leading Academic and Technical Personnel in Major Disciplines of Jiangxi Province (Grant No. 20232BCJ23091), and the National Natural Science Foundation of China (Grant No. 52462053), and the Key R & D Program of Jiangxi Province (Grant No. 20224BBE51048), and the 03 Special Program and 5G Project of Jiangxi Province (Grant No. 20232ABC03A30) and the High-end Foreign Expert Talents Project of the Ministry of Science and Technology of China (Grant No. G2021022002L).

Author contributions

Author contributions: Conceptualization, Y.Y. and Y.L.; review and supervision, D.Z., Y.Y., Y.H. and J.Y. ; methodology, Y.Y. and Y.L.; writing, Y.Y. and Y.L.; software, Y.Y. and Y.L.; Writing— original draft, Y.Y. and Y.L.; Writing—review and editing, Y.Y. and Y.L. All authors have read and agreed to the published version of the manuscript.

Data availability

The datasets used and analysed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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References

1. Liu S Wang Q Wang G Li B Ding D Zhang G Xu D Virtual-Axis injection based online parameter identification of PMSM considering cross coupling and saturation effects IEEE Trans. Power Electron. 2023 38 5 5791 5802 10.1109/tpel.2023.3236031
Liu, S. et al. Virtual-Axis injection based online parameter identification of PMSM considering cross coupling and saturation effects. IEEE Trans. Power Electron. 38(5), 5791–5802. 10.1109/tpel.2023.3236031 (2023).
2. Yu Y Gao H Chen Q Liu P Niu S Demagnetization Fault detection and location in PMSM based on correlation coefficient of branch current signals Energies 2022 10.3390/en15082952
Yu, Y., Gao, H., Chen, Q., Liu, P. & Niu, S. Demagnetization Fault detection and location in PMSM based on correlation coefficient of branch current signals. Energies10.3390/en15082952 (2022).
3. Zeng C Huang S Lei J Wan Z Yang Y Online rotor fault diagnosis of permanent magnet synchronous motors based on stator tooth flux IEEE Trans. Ind. Appl. 2021 57 3 2366 2377 10.1109/tia.2021.3058541
Zeng, C., Huang, S., Lei, J., Wan, Z. & Yang, Y. Online rotor fault diagnosis of permanent magnet synchronous motors based on stator tooth flux. IEEE Trans. Ind. Appl. 57(3), 2366–2377. 10.1109/tia.2021.3058541 (2021).
4. Chowdhury, M.H. (2016) Modeling of faults in permanent magnet synchronous machines, In 2016 IEEE Transportation Electrification Conference and Expo, Asia-Pacific (ITEC Asia-Pacific), IEEE, pp. 246-250.
5. Zhan H Zhu ZQ Odavic M Li Y A novel zero-sequence model-based sensorless method for open-winding PMSM With common DC bus IEEE Trans. Ind. Electron. 2016 63 11 6777 6789 10.1109/tie.2016.2585465
Zhan, H., Zhu, Z. Q., Odavic, M. & Li, Y. A novel zero-sequence model-based sensorless method for open-winding PMSM With common DC bus. IEEE Trans. Ind. Electron. 63(11), 6777–6789. 10.1109/tie.2016.2585465 (2016).
6. Moon S Lee J Jeong H Kim SW Demagnetization fault diagnosis of a PMSM based on structure analysis of motor inductance IEEE Trans. Ind. Electron. 2016 63 6 3795 3803 10.1109/tie.2016.2530046
Moon, S., Lee, J., Jeong, H. & Kim, S. W. Demagnetization fault diagnosis of a PMSM based on structure analysis of motor inductance. IEEE Trans. Ind. Electron. 63(6), 3795–3803. 10.1109/tie.2016.2530046 (2016).
7. Fonseca DSB Santos CMC Cardoso AJM Stator faults modeling and diagnostics of line-start permanent magnet synchronous motors IEEE Trans. Ind. Appl. 2020 56 3 2590 2599 10.1109/tia.2020.2979674
Fonseca, D. S. B., Santos, C. M. C. & Cardoso, A. J. M. Stator faults modeling and diagnostics of line-start permanent magnet synchronous motors. IEEE Trans. Ind. Appl. 56(3), 2590–2599. 10.1109/tia.2020.2979674 (2020).
8. Chen Y Liang S Li W Liang H Wang C Faults and diagnosis methods of permanent magnet synchronous motors: A review Appl. Sci. 2019 10.3390/app9102116 34113463
Chen, Y., Liang, S., Li, W., Liang, H. & Wang, C. Faults and diagnosis methods of permanent magnet synchronous motors: A review. Appl. Sci.10.3390/app9102116 (2019).34113463
9. Gao, J., Gui, W., Yang, C., Tao, H., Peng, T. Demagnetization Fault Detection based on Position-Frequency Analysis for PMSMs in Nonstationary Conditions, In 2021 CAA Symposium on Fault Detection, Supervision, and Safety for Technical Processes (SAFEPROCESS), 2021, pp. 1–5.
10. Hang J Zhang J Xia M Ding S Hua W Interturn fault diagnosis for model-predictive-controlled-pmsm based on cost function and wavelet transform IEEE Trans. Power Electron. 2020 35 6 6405 6418 10.1109/tpel.2019.2953269
Hang, J., Zhang, J., Xia, M., Ding, S. & Hua, W. Interturn fault diagnosis for model-predictive-controlled-pmsm based on cost function and wavelet transform. IEEE Trans. Power Electron. 35(6), 6405–6418. 10.1109/tpel.2019.2953269 (2020).
11. Orlowska-Kowalska T Wolkiewicz M Pietrzak P Skowron M Ewert P Tarchala G Krzysztofiak M Kowalski CT Fault diagnosis and fault-tolerant control of pmsm drives-state of the art and future challenges IEEE Access 2022 10 59979 60024 10.1109/access.2022.3180153
Orlowska-Kowalska, T. et al. Fault diagnosis and fault-tolerant control of pmsm drives-state of the art and future challenges. IEEE Access 10, 59979–60024. 10.1109/access.2022.3180153 (2022).
12. Wang C-S Kao IH Perng J-W Fault diagnosis and fault frequency determination of permanent magnet synchronous motor based on deep learning Sensors 2021 10.3390/s21113608 35009852
Wang, C.-S., Kao, I. H. & Perng, J.-W. Fault diagnosis and fault frequency determination of permanent magnet synchronous motor based on deep learning. Sensors10.3390/s21113608 (2021).35009852
13. Song X Zhao J Song J Dong F Xu L Zhao J Local demagnetization fault recognition of permanent magnet synchronous linear motor based on S-transform and PSO–LSSVM IEEE Trans. Power Electron. 2020 35 8 7816 7825 10.1109/tpel.2020.2967053
Song, X. et al. Local demagnetization fault recognition of permanent magnet synchronous linear motor based on S-transform and PSO–LSSVM. IEEE Trans. Power Electron. 35(8), 7816–7825. 10.1109/tpel.2020.2967053 (2020).
14. Cui, J., Zhang, N., Cui, X., Wang, J., Yu, M., Liu, D., Jiang, L. Fault Diagnosis Method of Aircraft Anti-skid Brake System Based on GWO-PNN, In 2021 33rd Chinese Control and Decision Conference (CCDC), 2021, pp. 5402–5406.
15. Dai, X., Zhang, Y., Qiao, L., Sun, D., Fault diagnosis of permanent magnet synchronous motor based on improved probabilistic neural network, In 2021 40th Chinese Control Conference (CCC), IEEE, 2021, pp. 2767-2772.
16. Ding J Xiao D Li X Gear fault diagnosis based on genetic mutation particle swarm optimization VMD and probabilistic neural network algorithm IEEE Access 2020 8 18456 18474 10.1109/access.2020.2968382
Ding, J., Xiao, D. & Li, X. Gear fault diagnosis based on genetic mutation particle swarm optimization VMD and probabilistic neural network algorithm. IEEE Access 8, 18456–18474. 10.1109/access.2020.2968382 (2020).
17. Nagi, R., Tripathy, S.S., Disease identification in grapevine leaf images using fuzzy-PNN, In 2022 2nd International Conference on Artificial Intelligence and Signal Processing (AISP), 2022, pp. 1–4.
18. Xu B Zhang L Ji W Improved non-singular fast terminal sliding mode control with disturbance observer for PMSM Drives IEEE Trans. Transp. Electr. 2021 7 4 2753 2762 10.1109/tte.2021.3083925
Xu, B., Zhang, L. & Ji, W. Improved non-singular fast terminal sliding mode control with disturbance observer for PMSM Drives. IEEE Trans. Transp. Electr. 7(4), 2753–2762. 10.1109/tte.2021.3083925 (2021).
19. Udas PB Karim ME Roy KS SPIDER: A shallow PCA based network intrusion detection system with enhanced recurrent neural networks J. King Saud Univ. Comput. Inf. Sci. 2022 34 10 10246 10272 10.1016/j.jksuci.2022.10.019
Udas, P. B., Karim, M. E. & Roy, K. S. SPIDER: A shallow PCA based network intrusion detection system with enhanced recurrent neural networks. J. King Saud Univ. Comput. Inf. Sci. 34(10), 10246–10272. 10.1016/j.jksuci.2022.10.019 (2022).
20. Rodríguez-Jorge D Romero-Sánchez F Rodríguez Salgado D González AG Transmission and actuation systems in cable-driven, walking-assistance exosuits based on postural and dynamic synergies Alex. Eng. J. 2023 77 383 393 10.1016/j.aej.2023.07.004
Rodríguez-Jorge, D., Romero-Sánchez, F., Rodríguez Salgado, D. & González, A. G. Transmission and actuation systems in cable-driven, walking-assistance exosuits based on postural and dynamic synergies. Alex. Eng. J. 77, 383–393. 10.1016/j.aej.2023.07.004 (2023).
21. Romero JM Pardo YS Parra M Castillo ADJ Maury H Corredor L Sánchez I Rueda B Gonzalez-Quiroga A Improving the rotary kiln-electric furnace process for ferronickel production: Data analytics-based assessment of dust insufflation into the rotary kiln flame Alex. Eng. J. 2022 61 4 3215 3228 10.1016/j.aej.2021.08.036
Romero, J. M. et al. Improving the rotary kiln-electric furnace process for ferronickel production: Data analytics-based assessment of dust insufflation into the rotary kiln flame. Alex. Eng. J. 61(4), 3215–3228. 10.1016/j.aej.2021.08.036 (2022).
22. Khan A Hayat S Zhong Y Arif A Zada L Fang M Computational and topological properties of neural networks by means of graph-theoretic parameters Alex. Eng. J. 2023 66 957 977 10.1016/j.aej.2022.11.001
Khan, A. et al. Computational and topological properties of neural networks by means of graph-theoretic parameters. Alex. Eng. J. 66, 957–977. 10.1016/j.aej.2022.11.001 (2023).
23. Alweshah M Rababa L Ryalat MH Al Momani A Ababneh MF African buffalo algorithm: Training the probabilistic neural network to solve classification problems J. King Saud Univ. Comput. Inf. Sci. 2022 34 5 1808 1818 10.1016/j.jksuci.2020.07.004
Alweshah, M., Rababa, L., Ryalat, M. H., Al Momani, A. & Ababneh, M. F. African buffalo algorithm: Training the probabilistic neural network to solve classification problems. J. King Saud Univ. Comput. Inf. Sci. 34(5), 1808–1818. 10.1016/j.jksuci.2020.07.004 (2022).
24. Al-Nabhani Y Jalab HA Wahid A Noor RM Robust watermarking algorithm for digital images using discrete wavelet and probabilistic neural network J. King Saud Univ. Comput. Inf. Sci. 2015 27 4 393 401 10.1016/j.jksuci.2015.02.002
Al-Nabhani, Y., Jalab, H. A., Wahid, A. & Noor, R. M. Robust watermarking algorithm for digital images using discrete wavelet and probabilistic neural network. J. King Saud Univ. Comput. Inf. Sci. 27(4), 393–401. 10.1016/j.jksuci.2015.02.002 (2015).
25. Kathiroli P Selvadurai K Energy efficient cluster head selection using improved sparrow search algorithm in wireless sensor networks J. King Saud Univ. Comput. Inf. Sci. 2022 34 10 8564 8575 10.1016/j.jksuci.2021.08.031
Kathiroli, P. & Selvadurai, K. Energy efficient cluster head selection using improved sparrow search algorithm in wireless sensor networks. J. King Saud Univ. Comput. Inf. Sci. 34(10), 8564–8575. 10.1016/j.jksuci.2021.08.031 (2022).
26. Wang X Liu J Hou T Pan C The SSA-BP-based potential threat prediction for aerial target considering commander emotion Def. Technol. 2022 18 11 2097 2106 10.1016/j.dt.2021.05.017
Wang, X., Liu, J., Hou, T. & Pan, C. The SSA-BP-based potential threat prediction for aerial target considering commander emotion. Def. Technol. 18(11), 2097–2106. 10.1016/j.dt.2021.05.017 (2022).
27. Lv J Yang Y Niu L Sun X Wang L Lin W Rong H Zou L Prediction of hand grip strength based on surface electromyographic signals J. King Saud Univ. Comput. Inf. Sci. 2023 35 5 101548 10.1016/j.jksuci.2023.04.001
Lv, J. et al. Prediction of hand grip strength based on surface electromyographic signals. J. King Saud Univ. Comput. Inf. Sci. 35(5), 101548. 10.1016/j.jksuci.2023.04.001 (2023).
28. Wang Y Zhang Z Wang Y You L Wei G Modeling and structural optimization design of switched reluctance motor based on fusing attention mechanism and CNN-BiLSTM Alex. Eng. J. 2023 80 229 240 10.1016/j.aej.2023.08.039
Wang, Y., Zhang, Z., Wang, Y., You, L. & Wei, G. Modeling and structural optimization design of switched reluctance motor based on fusing attention mechanism and CNN-BiLSTM. Alex. Eng. J. 80, 229–240. 10.1016/j.aej.2023.08.039 (2023).
29. Xue J Shen B A novel swarm intelligence optimization approach: sparrow search algorithm Syst. Sci. Control Eng. 2020 8 1 22 34 10.1080/21642583.2019.1708830
Xue, J. & Shen, B. A novel swarm intelligence optimization approach: sparrow search algorithm. Syst. Sci. Control Eng. 8(1), 22–34. 10.1080/21642583.2019.1708830 (2020).
30. Shi PM Gao SM Tan H Xu XF Li RX MGGSED-SSA: An improved sparse deconvolution method for rolling element bearing diagnosis Appl. Acoust. 2024 220 109960 10.1016/j.apacoust.2024.109960
Shi, P. M., Gao, S. M., Tan, H., Xu, X. F. & Li, R. X. MGGSED-SSA: An improved sparse deconvolution method for rolling element bearing diagnosis. Appl. Acoust. 220, 109960 (2024).
31. Wei X Rao CJ Xiao XP Chen L Goh M Risk assessment of cardiovascular disease based on SOLSSA-CatBoost model Expert Syst. Appl. 2023 219 119648 10.1016/j.eswa.2023.119648
Wei, X., Rao, C. J., Xiao, X. P., Chen, L. & Goh, M. Risk assessment of cardiovascular disease based on SOLSSA-CatBoost model. Expert Syst. Appl. 219, 119648. 10.1016/j.eswa.2023.119648 (2023).
32. Hu YT Zhao E Li J Li JJ Zhao XQ Ma B Dong MR Fault diagnosis of rolling bearing based on SEMD and ISSA-KELMC Meas Sci. Technol. 2024 35 5 35 10.1088/1361-6501/ad2c4e
Hu, Y. T. et al. Fault diagnosis of rolling bearing based on SEMD and ISSA-KELMC. Meas Sci. Technol. 35(5), 35 (2024).
