Particle Filtering Based Remaining Useful Life Prediction for Electromagnetic Coil Insulation

sensors

Article Particle Filtering Based Remaining Useful Life Prediction for Electromagnetic Coil Insulation

Haifeng Guo 1,2,3,4,5 , Aidong Xu 1,2,3, Kai Wang 1,2,3,*, Yue Sun 1,2,3,4, Xiaojia Han 1,2,3,4, Seung Ho Hong 6 and Mengmeng Yu 6

1 Key Laboratory of Networked Control Systems, Chinese Academy of Sciences, Shenyang 110016, China; [email protected] (H.G.); [email protected] (A.X.); [email protected] (Y.S.); [email protected] (X.H.) 2 Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China 3 Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of Sciences, Shenyang 110169, China 4 University of Chinese Academy of Sciences, Beijing 100049, China 5 Liaoning Institute of Science and Technology, Benxi 117004, China 6 Department of Electronic Engineering, Hanyang University, Ansan 15588, Korea; [email protected] (S.H.H.); [email protected] (M.Y.) * Correspondence: [email protected]; Tel.: +86-24-23970266

Abstract: Electromagnetic coils are one of the key components of many systems. Their insulation failure can have severe effects on the systems in which coils are used. This paper focuses on insulation degradation monitoring and remaining useful life (RUL) prediction of electromagnetic coils. First, insulation degradation characteristics are extracted from coil high-frequency electrical parameters. Second, health indicator is deﬁned based on insulation degradation characteristics to indicate the health degree of coil insulation. Finally, an insulation degradation model is constructed, and coil insulation RUL prediction is performed by particle ﬁltering. Thermal accelerated degradation experiments are performed to validate the RUL prediction performance. The proposed method

presents opportunities for predictive maintenance of systems that incorporate coils. Keywords: insulation degradation; insulation failure; inter-turn short; resonant frequency; Citation: Guo, H.; Xu, A.; Wang, K.; Sun, Y.; Han, X.; Hong, S.H.; Yu, M. PF; prognostics Particle Filtering Based Remaining Useful Life Prediction for Electromagnetic Coil Insulation. Sensors 2021, 21, 473. https:// 1. Introduction doi.org/10.3390/s21020473 Electromagnetic coil is the energy conversion component of electromagnetic valve, motor, transformer, and other relevant equipment, and its insulation failure is prominent. Received: 27 October 2020 The study from Oak Ridge National Laboratory [1] shows that over 50% of solenoid valve Accepted: 7 January 2021 failures of nuclear power plants in U.S. were attributed to electromagnetic coil failures Published: 11 January 2021 (e.g., coil open, coil short); in the field of automobiles, about 40% of motor failures are caused by stator insulation [2]; in the field of generators, 56% of generator failures are Publisher’s Note: MDPI stays neu- related to insulation failures [3]; according to the statistics in literature [4], the failures tral with regard to jurisdictional clai- of transformer winding account for 70–80% of the total transformer failures, and among ms in published maps and institutio- which the winding insulation failure rate is the highest. Therefore, it is important to study nal affiliations. the online monitoring approaches for the insulation state of electromagnetic coils, this will benefit for the reliable operation and predictive maintenance of equipment and reducing the maintenance cost of factories. Copyright: © 2021 by the authors. Li- Prognostics and health management (PHM) technology is one of the key technologies censee MDPI, Basel, Switzerland. in smart manufacturing. PHM turned equipment management from traditional failure This article is an open access article management into degradation management. Continuous and reliable operation of equip- distributed under the terms and con- ment are realized through predictive maintenance [5,6]. The accurate acquisition of health ditions of the Creative Commons At- information of key components is the base for predicting the health status of the equipment. tribution (CC BY) license (https:// In this paper, the PHM technology based on key components is used to study the trend creativecommons.org/licenses/by/ prediction for degradation status of electromagnetic coils. 4.0/).

Sensors 2021, 21, 473. https://doi.org/10.3390/s21020473 https://www.mdpi.com/journal/sensors Sensors 2021, 21, 473 2 of 14

The failure of electromagnetic coils is usually caused by the inter-turn insulation failure. The classic methods used to characterize the quality of coil insulation are based on visual inspection, electrical, physical, and chemical measurement. In the literature [7–12], offline detection methods based on temperature, current, neutral point voltage, magnetic flux leakage, and other parameters are introduced, and these methods can only detect the failures of electromagnetic coils, but they cannot detect the degradation process of insulation performance of electromagnetic coils. Considering that insulation failure usually occurs suddenly and has catastrophic effects, insulation degradation monitoring of the electromagnetic coil is preferred to enable predictive maintenance prior to a failure detection that could cause catastrophic damage. With the degradation of stator winding, twisted pair, and electromagnetic coil, F. Perisse and N.J.Jameson et al. [13–18] found that the parameters of parasitic capacitance, impedance, and reactance would change under high frequency conditions, and these characteristic parameters can be used to characterize the insulation degradation of electromagnetic coils, however, a large amount of data is needed to determine the parameters at a specific frequency which can assess the degradation of the solenoid. The process of insulation degradation is not systematically analyzed and modeled in the paper, and it is difficult to predict the degradation of the insulation performance of electromagnetic coils. In this paper, the insulation performance degradation of electromagnetic coils is stud- ied under the condition of constant high temperature. Through analyzing the insulation degradation mechanism and the data of insulation degradation of electromagnetic coils, resonance frequency is used as a characteristic parameter to characterize the degradation state of insulation. Based on the degradation state data, the insulation performance degradation model of electromagnetic coil is established. The insulation performance degradation prediction algorithm of electromagnetic coil is proposed based on particle filtering. This paper is organized as follows. In Section2, the failure mechanism of an electromagnetic coil in a high temperature environment is analyzed, and resonant frequency is selected as a health indicator for electromagnetic coil degradation monitoring based on the equivalent circuit model and lumped capacitance network analysis. The accelerated degradation experimental platform is described to obtain characteristic data. In Section3, the model of insulation performance degradation is described with characteristic data collected through experiment. In Section4 , particle filtering based prediction algorithm is introduced for degradation of insulation performance of electromagnetic coil. In Section5, the conclusion of this paper is given.

2. Framework for Coil Insulation Remaining Useful Life (RUL) Prediction Based on Data-Driven Methods Prognostics and health management (PHM) is a method that permits the reliability of a system to be evaluated in its actual life-cycle conditions, and life prediction and health management are the core technologies in PHM. PHM methodology is based on monitoring parameters that are sensitive to impending failure precursors. The precursor is usually a change in a measurable parameter that can be associated with impending failure. Life prediction is considered as the foundation and core content of PHM, which can be roughly divided into a physical failure model based on the method and data-driven method. Since it is difﬁcult to obtain the physical failure mechanisms, data-driven prediction methods have become mainstream techniques in recent years. The life prediction methods based on data-driven include the traditional methods based on failure data, the methods based on degradation data, and the methods based on multi-source data fusion. The data- driven method based on degradation data can obtain the characteristic parameters of equipment health state by analyzing the monitored data, and then realize the life prediction of equipment, which has been developed rapidly. It is helpful for accelerated fatigue test to obtain degradation data of the equipment quickly and effectively. The electromagnetic coil, which is generally referred to as magnet wire, is constructed of conductor (usually copper) coated by insulation material (usually some kind of polymer). The electromagnetic coil is susceptible to coupled electrical/thermal failure mechanisms. Sensors 2021, 21, x FOR PEER REVIEW 3 of 15

health state by analyzing the monitored data, and then realize the life prediction of equipment, which has been developed rapidly. It is helpful for accelerated fatigue test to obtain Sensors 2021, 21, 473 degradation data of the equipment quickly and effectively. 3 of 14 The electromagnetic coil, which is generally referred to as magnet wire, is constructed of conductor (usually copper) coated by insulation material (usually some kind of polymer). The electromagnetic coil is susceptible to coupled electrical/thermal failure Whenmechanisms. current isWhen passing current through is passing the wire, throug Jouleh the heating wire, willJoule cause heating the will increase cause of the wire in- temperature,crease of wire this temperature, will lead the this expansion will lead of the the expansion conductor, of placing the conductor, mechanical placing and thermalmechan- stressesical and on thermal the insulation stresses on material. the insulation The mechanical material. The and mechanical thermal stresses and thermal then leadstresses to insulationthen lead degradationto insulation and degrada insulationtion and failure. insulation failure. TakingTaking the the electromagnetic electromagnetic coil coil as as the the research research object, object, with with the the help help of of accelerated accelerated fatiguefatigue test, test, the the whole whole life life cycle cycle data data of of the the electromagnetic electromagnetic coil coil are are collected; collected; the the param- param- eterseters which which can can represent represent the the degradation degradation characteristic characteristic of of the the electromagnetic electromagnetic coil coil are are determineddetermined byby analyzinganalyzing the collected collected data; data; th thee life life degradation degradation model model of the of theelectromag- electro- magneticnetic coil coil is established is established bas baseded on the on theabove above characteristic characteristic parameters; parameters; then, then, the thetrend trend pre- predictiondiction and and RUL RUL prediction prediction of of the the electromagne electromagnetictic coil are realized, andand thethe health health status status of the electromagnetic coil is analyzed; eventually, the health assessment and predictive of the electromagnetic coil is analyzed; eventually, the health assessment and predictive maintenance are realized, as shown in Figure1. maintenance are realized, as shown in Figure 1.

FigureFigure 1. 1.The The prognostics prognostics and and health health management management (PHM) (PHM) implementation implementation scheme scheme of theof the electromag- electro- neticmagnetic coil. coil. 3. Electromagnetic Coils Data Acquisition Platform Construction and Data Analysis 3. Electromagnetic Coils Data Acquisition Platform Construction and Data Analysis 3.1. Data Acquisition Platform Construction 3.1. Data Acquisition Platform Construction In order to further study the degradation of electromagnetic coil insulation performance,In the order experimental to further platform study the is setdegradation up with constant of electromagnetic high temperature coil insulation stress, as shownperfor- inmance, Figure the2. Theexperimental experimental platform scheme is set is determinedup with constant as follows: high Thetemperature electromagnetic stress, as coilshown was in put Figure into the 2. The chamber experimental which is scheme set at 220 is ◦determinedC, and the electricalas follows: data The (inductance, electromag- resistance,netic coil was impedance, put into reactance) the chamber could which be collected is set at by220 sweeping °C, and the frequencies electrical fromdata 50(induct- kHz toance, 1 MHz resistance, every 2 h,impedance, which is the reactance) frequency could range be providedcollected byby thesweeping LCR meter frequencies (Keysight’s from E4980A50 kHz inductance-capacitance-resistance); to 1MHz every 2 h, which is the the frequency electrical range data areprovided collected by by the NI LCR computer; meter and(Keysight’s direct-current E4980A resistance inductance-capacitance-resista (DCR) of the coil wasnce); collected the electrical 8 times data per cycle,are collected and the by averageNI computer; of DCR and value direct-current is taken as resistance the final value(DCR) of of DCR, the coil which was is collected used to determine8 times per whethercycle, and the the coil average is in the of degradation DCR value is phase taken (there as the is final no turn-to-turn value of DCR, short which circuit is used in the to coil)determine or failure whether stage duringthe coil the is in aging the degradation test. phase (there is no turn-to-turn short circuit in the coil) or failure stage during the aging test. 3.2. Data Acquisition and Analysis With the help of thermal acceleration experiment, the life cycle data of electromagnetic coil are obtained. In this experiment, the direct current resistance (DCR) of the electromagnetic coil used is 76.713 Ω, and its single turn DCR is 0.093 Ω. In the degradation process, the DCR of the electromagnetic coil is constant, there is no inter turn or inter layer short circuit in the electromagnetic coil, but in the failure process, the DCR decreases significantly, and there is an inter turn or inter layer short circuit, as shown in Table1. During the acceleration experiment, the DCR of the electromagnetic coil is monitored in real time. As shown in Figure3, the DCR of the electromagnetic coil remains unchanged before the 99th cycle, however, the data after 100 cycles show that the DCR decreases significantly (i.e., short circuit and hot spot formation), and then the failure leads to more hot spots and a more serious short circuit. When the DCR is detected to decrease, this indicates that the whole life data of the electromagnetic coil have been obtained. Sensors 2021, 21, 473 4 of 14 Sensors 2021, 21, x FOR PEER REVIEW 4 of 15

FigureFigure 2. 2.Experimental Experimental platform platform of of the the electromagnetic electromagnetic coil: coil: Aging Aging experimental experimental platform platform is usedis used for acceleratedfor accelerated aging aging test attest constant at constant high high temperature, temperature, experimental experimental platform platform of data of data measurement measure- is usedment to is measure used to measure the electrical the electrical parameters parameters of the electromagnetic of the electromagnetic coil. coil.

3.2. Data Acquisition and Analysis Table 1. Part of direct-current resistance (DCR) collected by LCR meter. With the help of thermal acceleration experiment, the life cycle data of electromagnetic coil are obtained. In this experiment, the direct current resistance (DCR) of theDCR elec- tromagneticCycle DCR1 coil used DCR2 is 76.713 DCR3 Ω, and DCR4 its single DCR5 turn DCR DCR6 is 0.093 DCR7 Ω. In DCR8 the degradation(Aver- age) process, the DCR of the electromagnetic coil is constant, there is no inter turn or inter layer short1 circuit 76.71 in the electromagnetic 76.71 76.71 coil, 76.71 but in 76.71 the failure 76.71 process, 76.71 the DCR 76.71 decreases 76.71 significantly,10 76.71and there 76.71 is an inter 76.71 turn or 76.71 inter layer 76.71 short circuit, 76.71 as 76.71shown in 76.71 Table 1. 76.71During 20 76.71 76.71 76.71 76.71 76.71 76.71 76.71 76.71 76.71 the30 acceleration 76.71 experiment, 76.71 76.71the DCR 76.72of the electromagnetic 76.71 76.70 coil 76.71is monitored 76.71 in real 76.71 time. As40 shown 76.72in Figure 76.71 3, the DCR 76.72 of the 76.72 electromagnetic 76.71 coil 76.72 remains 76.72 unchanged 76.72 before 76.72 the 99th50 cycle, 76.72 however, 76.72 the data 76.722 after 100 76.72 cycles 76.72show that 76.72 the DCR 76.72 decreases 76.72 significantly 76.72 (i.e.,60 short 76.72circuit and 76.71 hot spot 76.72 formation), 76.71 and 76.72then the 76.72failure leads 76.71 to more 76.72 hot spots 76.72 and a more70 serious 76.72 short 76.71 circuit. 76.72When the 76.72 DCR is 76.72detected 76.72to decrease, 76.71 this indicates 76.72 76.72that the 80 76.71 76.71 76.71 76.71 76.71 76.70 76.72 76.72 76.71 whole life data of the electromagnetic coil have been obtained. 90 76.71 76.71 76.71 76.72 76.71 76.72 76.72 76.71 76.71 Sensors 2021, 21, x FOR PEER REVIEW 5 of 15 100 76.62 76.62 76.62 76.63 76.63 76.63 76.64 76.63 76.63 Table106 1. Part of 74.35 direct-current 74.35 resistance 74.37 (DCR) 74.34 collected 74.35 by LCR 74.35 meter. 74.36 74.33 74.35 Cycle DCR1 DCR2 DCR3 DCR4 DCR5 DCR6 DCR7 DCR8 DCR (Average) 1 76.71 76.71 76.71 76.71 76.71 76.71 76.71 76.71 76.71 10 76.71 76.71 76.71 76.71 76.71 76.71 76.71 76.71 76.71 20 76.71 76.71 76.71 76.71 76.71 76.71 76.71 76.71 76.71 30 76.71 76.71 76.71 76.72 76.71 76.70 76.71 76.71 76.71 40 76.72 76.71 76.72 76.72 76.71 76.72 76.72 76.72 76.72 50 76.72 76.72 76.722 76.72 76.72 76.72 76.72 76.72 76.72 60 76.72 76.71 76.72 76.71 76.72 76.72 76.71 76.72 76.72 70 76.72 76.71 76.72 76.72 76.72 76.72 76.71 76.72 76.72 80 76.71 76.71 76.71 76.71 76.71 76.70 76.72 76.72 76.71 90 76.71 76.71 76.71 76.72 76.71 76.72 76.72 76.71 76.71 100 76.62 76.62 76.62 76.63 76.63 76.63 76.64 76.63 76.63 106 74.35 74.35 74.37 74.34 74.35 74.35 74.36 74.33 74.35

FigureFigure 3. 3. MeasurementMeasurement value value of of DCR DCR of of the the electromagnetic electromagnetic coil coil du duringring the the aging aging experiment. experiment.

LCRLCR meter meter is is used used to to collect collect impedance, impedance, resistance, resistance, inductance, inductance, and and reactance reactance data data in ain specific a speciﬁc frequency frequency range, range, as asshown shown in inTable Table 2.2 .Figures Figures 4–64–6 show thethe impedanceimpedance andand resistance,resistance, inductance inductance data data of of 106 106 aging aging cycles cycles,, in in the the degradation degradation phase, phase, the the data data of of each each cycle is not clearly distinguished, while in the failure phase, the three kinds of electrical data have significant changes compared with their respective data in the degradation phase. As shown in Figure 7, the reactance spectrum data of each aging period are collected. With the increase of aging time, when the electromagnetic coil is in heathy and degradation state, the reactance spectrum curves move from the right to the left regularly, the color of the curve gradually changes from black to red. After the 100th cycle, the reactance spectrum curve changes significantly (the green curve in Figure 7). Combined with DCR data, it indicates that the electromagnetic coil has been short-circuited.

Table 2. Part of the data collected by LCR meter under the frequency of 314,050 Hz.

Acquisition Frequency Impedance Resistance Inductance Reactance Cycle Time (Hz) (Ω) (Ω) (H) (Ω) 1 2020-4-28 08:02:45 3.14 × 105 4.98 × 102 4.97 × 102 3.4 × 10−5 2.85 × 101 10 2020-4-29 01:37:15 3.14 × 105 4.65 × 102 4.63 × 102 3.5 × 10−5 −4.92 × 101 20 2020-4-29 19:53:08 3.14 × 105 5.39 × 102 5.28 × 102 5.7 × 10−5 −1.11 × 102 30 2020-4-30 20:01:41 3.14 × 105 5.51 × 102 5.38 × 102 6.1 × 10−5 −1.67 × 102 40 2020-5-1 16:13:13 3.14 × 105 4.45 × 102 4.41 × 102 3.2 × 10−5 −6.36 × 101 50 2020-5-4 15:41:46 3.14 × 105 4.56 × 102 4.52 × 102 3.0 × 10−5 −5.91 × 101 60 2020-5-5 21:28:30 3.14 × 105 5.68 × 102 5.53 × 102 6.5 × 10−5 −1.28 × 102 70 2020-5-7 17:37:42 3.14 × 105 4.57 × 102 4.52 × 102 3.6 × 10−5 −7.03 × 101 80 2020-5-9 20:35:33 3.14 × 105 4.71 × 102 4.56 × 102 5.9 × 10−5 −1.15 × 102 90 2020-5-11 00:46:51 3.14 × 105 5.45 × 102 5.13 × 102 6.4 × 10−5 −1.84 × 102 100 2020-5-12 00:28:57 3.14 × 105 5.26 × 102 2.62 × 102 2.3 × 10−5 −4.56 × 102 106 2020-5-12 13:33:31 3.14 × 105 3.07 × 102 6.06 × 101 1.5 × 10−5 −3.01 × 102 Sensors 2021, 21, 473 5 of 14

cycle is not clearly distinguished, while in the failure phase, the three kinds of electrical data have signiﬁcant changes compared with their respective data in the degradation phase. As shown in Figure7, the reactance spectrum data of each aging period are collected. With the increase of aging time, when the electromagnetic coil is in heathy and degradation state, the reactance spectrum curves move from the right to the left regularly, the color of the curve gradually changes from black to red. After the 100th cycle, the reactance spectrum curve changes signiﬁcantly (the green curve in Figure7). Combined with DCR data, it indicates that the electromagnetic coil has been short-circuited.

Table 2. Part of the data collected by LCR meter under the frequency of 314,050 Hz.

Acquisition Frequency Impedance Resistance Inductance Reactance Cycle Time (Hz) (Ω) (Ω) (H) (Ω) 2020-4-28 1 3.14 × 105 4.98 × 102 4.97 × 102 3.4 × 10−5 2.85 × 101 08:02:45 2020-4-29 10 3.14 × 105 4.65 × 102 4.63 × 102 3.5 × 10−5 −4.92 × 101 01:37:15 2020-4-29 20 3.14 × 105 5.39 × 102 5.28 × 102 5.7 × 10−5 −1.11 × 102 19:53:08 2020-4-30 30 3.14 × 105 5.51 × 102 5.38 × 102 6.1 × 10−5 −1.67 × 102 20:01:41 2020-5-1 40 3.14 × 105 4.45 × 102 4.41 × 102 3.2 × 10−5 −6.36 × 101 16:13:13 2020-5-4 50 3.14 × 105 4.56 × 102 4.52 × 102 3.0 × 10−5 −5.91 × 101 15:41:46 2020-5-5 60 3.14 × 105 5.68 × 102 5.53 × 102 6.5 × 10−5 −1.28 × 102 21:28:30 2020-5-7 70 3.14 × 105 4.57 × 102 4.52 × 102 3.6 × 10−5 −7.03 × 101 17:37:42 2020-5-9 80 3.14 × 105 4.71 × 102 4.56 × 102 5.9 × 10−5 −1.15 × 102 20:35:33 2020-5-11 90 3.14 × 105 5.45 × 102 5.13 × 102 6.4 × 10−5 −1.84 × 102 00:46:51 2020-5-12 100 3.14 × 105 5.26 × 102 2.62 × 102 2.3 × 10−5 −4.56 × 102 00:28:57 2020-5-12 Sensors 2021106, 21, x FOR PEER REVIEW 3.14 × 105 3.07 × 102 6.06 × 101 1.5 × 10−5 −3.01 × 610 of2 15 13:33:31

FigureFigure 4. 4. TheThe measurement measurement value value of of impedance impedance spectra spectra of of the the electromagnetic electromagnetic coil in the whole life.

Figure 5. The measurement value of resistance spectra of the electromagnetic coil in the whole life.

Figure 6. The measurement value of inductance spectra of the electromagnetic coil in the whole life.

SensorsSensors 2021 2021, ,21 21, ,x x FOR FOR PEER PEER REVIEW REVIEW 66 of of 15 15

Sensors 2021, 21, 473 6 of 14 FigureFigure 4. 4. The The measurement measurement value value of of impedance impedance spectra spectra of of the the electromagnetic electromagnetic coil coil in in the the whole whole life. life.

FigureFigure 5. 5. The The measurement measurement value value of of resistance resistance spectra spectra of of the the electromagnetic electromagnetic co coilcoilil in in the the whole whole life. life.life.

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Figure 6. The measurement value of inductance spectra of the electromagnetic coil in the whole life. FigureFigure 6. 6. TheThe measurement measurement value value of of induct inductanceance spectra spectra of of the the electromag electromagneticnetic coil coil in in the the whole whole life. life.

FigureFigure 7. 7. TheThe measurement measurement value value of of Reactance Reactance spectra spectra of of the the electromagnetic electromagnetic coil coil with with aging aging time. time.

4. Determination of Insulation Degradation Characteristic Parameters and HI Con- struction The resonant frequency is the frequency at which reactance (imaginary part of impedance) is zero, that is, the inductive reactance is equal to the capacitive reactance. With the accelerated fatigue test, the resonance frequency has the decreasing gradually trend, as shown in Figure 7. According to the references [19,20], when the imaginary part is equal to zero, then the resonant frequency can be expressed as:

𝑓 = 𝐿𝐶⁄ −𝑅 𝐿 (1)

where: 𝑓 is the resonant frequency. According to the reference [19–22], with the degradation of electromagnetic coil, the R and inductance remain constant, and the 𝐶 changes with the thickness of insulation material, and then the resonance frequency also changes. Therefore, the resonance frequency, as an inherent index of electromagnetic coil, can be used to evaluate the health status of electromagnetic coil. According to formula (1), the resonance frequency will change with the degradation of the electromagnetic coil. The resonant frequency of each cycle can be used as an index to evaluate the health status of the electromagnetic coil. By fitting the reactance data collected from the experiment under multiple frequencies, the resonant frequency could be obtained under the current state. In the process of degradation, the resonance frequency generally shows a downward trend, cycle 1 to cycle 99 is the degradation phase, cycle 100 to cycle 106 is the failure phase, as shown in Figure 8. The test result demonstrates that the resonance frequency could be used as an effective characteristic parameter to evaluate the healthy status of the electromagnetic coil.

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4. Determination of Insulation Degradation Characteristic Parameters and HI Construction The resonant frequency is the frequency at which reactance (imaginary part of impedance) is zero, that is, the inductive reactance is equal to the capacitive reactance. With the accelerated fatigue test, the resonance frequency has the decreasing gradually trend, as shown in Figure7. According to the references [19,20], when the imaginary part is equal to zero, then the resonant frequency can be expressed as: q 2 2 fr = L/Cg − R /L (1) where: fr is the resonant frequency. According to the reference [19–22], with the degradation of electromagnetic coil, the R and inductance remain constant, and the Cg changes with the thickness of insulation material, and then the resonance frequency also changes. Therefore, the resonance frequency, as an inherent index of electromagnetic coil, can be used to evaluate the health status of electromagnetic coil. According to formula (1), the resonance frequency will change with the degradation of the electromagnetic coil. The resonant frequency of each cycle can be used as an index to evaluate the health status of the electromagnetic coil. By ﬁtting the reactance data collected from the experiment under multiple frequencies, the resonant frequency could be obtained under the current state. In the process of degradation, the resonance frequency generally shows a downward trend, cycle 1 to cycle 99 is the degradation phase, cycle 100 to cycle 106 is the failure phase, as shown in Figure8. The test result demonstrates that the resonance Sensors 2021, 21, x FOR PEER REVIEW 8 of 15 frequency could be used as an effective characteristic parameter to evaluate the healthy status of the electromagnetic coil.

Figure 8.Figure Degradation 8. Degradation trend of resonant trend of resonantfrequency frequency of the electromagnetic of the electromagnetic coil. coil.

The health indicatorThe health (HI), indicator which is (homogenizedHI), which is based homogenized on resonant based frequency, on resonant is in- frequency, is troduced to indicateintroduced the todegree indicate of insulation the degree degradation of insulation of degradation the electromagnetic of the electromagnetic coil, as coil, as shown in formulashown (1). in Therefore, formula (1). HITherefore, is equal toHI 0.45,is equalwhich to is 0.45, the failure which isthreshold the failure of the threshold of the electromagneticelectromagnetic coil. coil. 𝐻𝐼(𝑘) =(𝑓(𝑘) − 𝑓 )/(𝑓 −𝑓 ) (2) HI(k) =( f (k) −ffailure)/( fhealth − ffailure) (2) where: where: 𝑓 is resonant frequency of the health coil; f is resonant frequency of the health coil; 𝑓 is resonanthealth frequency of the failure coil; and f is resonant frequency of the failure coil; and 𝑓(𝑘) is the resonantfailure frequency of cycle 𝑘. f (k) is the resonant frequency of cycle k. 5. Modeling of the Degradation of the Insulation Performance of the Electromagnetic Coils and RUL Prediction Based on Particle Filtering 5.1. Wavelet Threshold Denoising Method In the accelerated degradation experiment, high-frequency data are collected. The accuracy of real data is undermined because of the interference of environmental conditions in the measurement process and the sensitivity of high frequency data. In order to obtain the high frequency data accurately, using filtering technology to filter out the noise interference is necessary. Wavelet denoising has a wide range of adaptability, and this will improve the accuracy of the data. After the use of the wavelet de-noising method, the degradation curve based on health index is obtained. It shows that the trend of the degradation curve is more obvious, which is more beneficial for the enhancement of the accuracy of degradation trend prediction and RUL prediction, as shown in Figure 9.

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5. Modeling of the Degradation of the Insulation Performance of the Electromagnetic Coils and RUL Prediction Based on Particle Filtering 5.1. Wavelet Threshold Denoising Method In the accelerated degradation experiment, high-frequency data are collected. The accuracy of real data is undermined because of the interference of environmental conditions in the measurement process and the sensitivity of high frequency data. In order to obtain the high frequency data accurately, using ﬁltering technology to ﬁlter out the noise interference is necessary. Wavelet denoising has a wide range of adaptability, and this will improve the accuracy of the data. After the use of the wavelet de-noising method, the degradation Sensors 2021, 21, x FOR PEER REVIEWcurve based on health index is obtained. It shows that the trend of the degradation curve9 of is 15

more obvious, which is more beneﬁcial for the enhancement of the accuracy of degradation trend prediction and RUL prediction, as shown in Figure9.

FigureFigure 9. 9.Degradation Degradation trend trend of of health health index index of of the the electromagnetic electromagnetic coil coil after after wavelet wavelet denoising. denoising.

5.2. Life Model Based on Health Index 5.2. Life Model Based on Health Index The main idea of degradation modeling is to describe the distribution of degradation trajectoryThe main with idea the ofdata degradation of characteristic modeling parameters. is to describe In this the paper, distribution degradation of degradation prediction trajectoryresearch of with long-life the data equipment of characteristic is carried parameters. out by studying In this paper,the change degradation of its characteristic prediction researchparameters. of long-life Through equipment the characteristic is carried data, out bywhich studying can represent the change the oftrend its characteristic of the equip- parameters.ment life, the Through degradation the characteristic model based data, on which the characteristic can represent parameters the trend of is the established equipment to life,predict the degradationthe health status model basedof the onequipment. the characteristic Several parameters common degradation is established models to predict are theshown health in formula status of (3)–(5): the equipment. Several common degradation models are shown in formula (3)–(5): HI𝐻𝐼(k()𝑘=) =𝑎∗𝑒𝑥𝑝a ∗ exp(b(∗𝑏∗𝑘k) ) (3)(3) HI𝐻𝐼(k()𝑘=) =𝑎∗𝑒𝑥𝑝a ∗ exp(b(∗𝑏∗𝑘k) +) c+𝑐∗𝑒𝑥𝑝∗ exp(d (∗𝑑∗𝑘k) ) (4)(4) HI𝐻𝐼(k()𝑘=) =𝑎∗𝑘a ∗ k3 + +𝑏∗𝑘b ∗ k2 + c+𝑐∗𝑘+𝑑∗ k + d (5)(5) where:where: a𝑎, 、b, 𝑏c,、d𝑐is、 parameter𝑑 is parameter of model; of model; k𝑘is is aging aging time time (measurement (measurement cycle); cycle); and and HI𝐻𝐼((k𝑘))is is the the health health indicator indicator of of k k measurement measurement cycle. cycle. PolynomialPolynomial and and exponential exponential model model (formula (formula (3)–(5)) (3)–(5)) are are used used to fitto thefit the life life cycle cycle data data of theof the electromagnetic electromagnetic coil, coil, and and the results the results are shown are shown in Figure in Figure 10. The 10. results The results show thatshow there that wasthere a decreasingwas a decreasing trend of trend the health of the indicator health indicator as the coil as aged the duringcoil aged the during degradation the degrada- phase. Thetion curve-fitting phase. The curve-fitting model, which model, can accurately which can describe accurately the performance describe the degradation performance trend, deg- isradation identified trend, as the is identified degradation as modelthe degradatio of the electromagneticn model of the coil.electromagnetic It is not expected coil. It to is the not twoexpected curve-fitting to the two models curve-fitting (the formula models (3) and (the (4)) formula that can’t (3) and track (4)) the that degradation can’t track data the very degradation data very well, but the polynomial model (formula (5)) is better than the first two models, as shown in Figure 10. Therefore, the polynomial model is used as the life degradation model of the insulation performance of the electromagnetic coil, and the coefficients of the polynomial degradation model are obtained by means of the fitting tool of MATLAB.

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well, but the polynomial model (formula (5)) is better than the first two models, as shown in Figure 10. Therefore, the polynomial model is used as the life degradation model of the Sensors 2021, 21, x FOR PEER REVIEW 10 of 15 insulation performance of the electromagnetic coil, and the coefficients of the polynomial degradation model are obtained by means of the fitting tool of MATLAB.

Figure 10. Curve-fittingCurve-ﬁtting model for the relationship between health indicator and aging time.

5.3. Particle Filtering 5.3. Particle Filtering Particle filter (PF) algorithm is a non-linear, non-Gaussian system filtering method basedParticle on the filterMonte (PF) Carlo algorithm idea, UKF, is a non-linear,and other Kalman non-Gaussian filters that system adopt filtering the statistical method linearizationbased on the method, Monte Carlo which idea, is based UKF, on and the other linear Kalman and Gaussian filters that assumption adopt the of statistical the system.linearization When the method, system whichis in nonlinear is based onand the non- linearGaussian and Gaussian state, the assumption prediction ofeffect the of system. UKF isWhen not obvious. the system The isresearch in nonlinear object andof this non-Gaussian paper is an electromagnetic state, the prediction coil with effect a long of UKF life, andis not the obvious. industrial The environment research object is complex, of this and paper its islife an distribution electromagnetic is non-linear coil with and a non- long Gaussian,life, and the so industrialthe PF with environment wider applicability is complex, is selected. and its Therefore, life distribution the PF is algorithm non-linear is andone ofnon-Gaussian, the effective somethods the PF to with realize wider the applicability trend and RUL is selected. prediction Therefore, based on the model PF algorithm [23]. The is twoone important of the effective steps methods of PF are: to (1) realize Random the trendparticles and are RUL extracted prediction from based the empirical on model con- [23]. The two important steps of PF are: (1) Random particles are extracted from the empirical ditional distribution; (2) the weight of each particle is calculated according to the observa- conditional distribution; (2) the weight of each particle is calculated according to the tion probability distribution, importance distribution, and Bayesian formula. As the num- observation probability distribution, importance distribution, and Bayesian formula. As ber of samples increases, the method approaches to the real posterior probability density the number of samples increases, the method approaches to the real posterior probability function of state variables, and then the prediction of the degradation trend and RUL of density function of state variables, and then the prediction of the degradation trend and the electromagnetic coil is realized. The model based on particle filter can be established RUL of the electromagnetic coil is realized. The model based on particle filter can be by formula (6): established by formula (6): 𝑥x = f𝑓((x𝑥− ) +) +𝑢u k k 1 k (6) y = h(x ) + v (6) 𝑦k =ℎ(𝑥k ) +𝑣k where: xk is the state vector; 𝑥 is the state vector; yk is the observation vector; 𝑦 is the observation vector; f (xk−1) is state transition function; 𝑓(𝑥) is state transition function; h(xk) is observation function; ℎ(𝑥) is observation function; k is the time index; 𝑘 is the time index; u is process noise sequence, the gaussian distribution with mean value of 0 and 𝑢 k is process noise sequence, the gaussian distribution with mean value of 0 and var- variance of σ2 is satisfied; and iance of σ 2 is satisfied; and 𝑣 is observation noise sequence, the gaussian distribution with mean value of 0 and variance of σ 2 is satisfied. Each particle of PF algorithm is a set of modified parameters, and the particles are carried out at the same time, therefore, the PF algorithm is a multi-point parallel algo-

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vk is observation noise sequence, the gaussian distribution with mean value of 0 and Sensors 2021, 21, x FOR PEER REVIEW 2 11 of 15 variance of σ is satisfied. Each particle of PF algorithm is a set of modified parameters, and the particles are carried out at the same time, therefore, the PF algorithm is a multi-point parallel algorithm, whichrithm, greatly which improvesgreatly improves the prediction the prediction accuracy. accuracy. The following The following Figure 11 Figure is the 11 prediction is the pre- flowdiction chart flow of particlechart of filteringparticle algorithm.filtering algorithm. In this paper, In this the paper, degradation the degradation model is usedmodel as is theused observation as the observation equation, andequation, the health and indicatorthe health of indicator the electromagnetic of the electromagnetic coil is used as coil the is observation.used as the observation.

Determining degradation model Degredation data

Data processing

Determining Model coefficients

Stsrting point

Particle Filter k=k+1 K time particle prediction

New observations no

yes Weight and Normalization

Resampling

Degradation prediction

FigureFigure 11. 11.Prediction Prediction ﬂow flow chart chart based based on on particle particle ﬁltering. filtering.

5.4.5.4. Prediction Prediction of of Insulation Insulation Degradation Degradation Based Based on on Particle Particle Filtering Filtering Algorithm Algorithm InIn the the process process of of degradation degradation of of insulation insulation performance performance of of the the electromagnetic electromagnetic coil, coil, thethe data data collection collection process process contains contains white white noise noise with with zero zero mean mean and and unknown unknown variance. variance. TheThe state state transitiontransition equationequation describes the the functional functional relationship relationship between between the the state state of ofthe thesystem system at the at the previous previous moment moment and and the thestate state at the at thecurrent current moment. moment. Therefore, Therefore, the thedeg- degradationradation model model of ofthe the insulati insulationon performance performance of of electromag electromagneticnetic coils (Formula(Formula 5)5) is is neededneeded to to deal deal with with the the following following forms: forms: 2 HI𝐻𝐼((k𝑘))==𝐻𝐼HI((k𝑘−1− 1))++((33∗𝑎+𝑏∗ a + b))∗∗𝑘k2 ++(3(∗3∗𝑎+2∗𝑏+𝑐a + 2 ∗ b + c) ∗)k∗𝑘++ (a +(𝑎+𝑏+𝑐b + c) ) (7)(7)

where:where： HI𝐻𝐼((k𝑘))is isk time𝑘 time health health indicator; indicator; HI𝐻𝐼((k𝑘−1− 1))is isk −𝑘−11 time time health health indicator; indicator; k𝑘is is degenerate degenerate cycle cycle (a (a cycle cycle is is 2 h);2 h); and and a,a,b ,b, and andc care are the the parameters parameters of of the the degradation degradation model. model. WithWith (7), (7), the the following following formula formula can can be be obtained obtained by by integrating integrating the the upper upper constant constant term:term: 2 HI(k) = HI(k − 1) + α ∗ k +2β ∗ k + γ (8) 𝐻𝐼(𝑘) =𝐻𝐼(𝑘−1) +𝛼∗𝑘 +𝛽∗𝑘+𝛾 (8) where: ： whereα = 3 ∗ a + b; 𝛼=3∗𝑎+𝑏; β= 3 ∗ a + 2 ∗ b + c; and 𝛽= 3∗𝑎+2∗𝑏+𝑐; and γ = a + b + c. 𝛾=𝑎+𝑏+𝑐. It can be seen from the above formula that the health indicator at 𝑘-time is composed of the above two items, which are related to both the health index at 𝑘–1 time and the degeneration cycle. Therefore, the degradation model of the electromagnetic coil is taken as the observation, and the parameters of this model are taken as the state variables, as shown in formula (9):

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Sensors 2021, 21, x FOR PEER REVIEW 12 of 15 It can be seen from the above formula that the health indicator at k-time is composed of the above two items, which are related to both the health index at k–1 time and the degeneration cycle. Therefore, the degradation model of the electromagnetic coil is taken as the observation, and⎧ the parameters𝛼(𝑘 of) =𝛼 this(𝑘−1 model) +𝑢 are taken as the state variables, as shown in formula (9): ⎪ 𝛽(𝑘) =𝛽(𝑘−1) +𝑢 (9) ⎨ 𝛾(𝑘) =𝛾(𝑘−1) +𝑢  α(k) = α(k − 1) + u  ⎪𝐻𝐼(𝑘) =𝐻𝐼(𝑘−1) +𝛼∗𝑘 +𝛽∗𝑘+𝛾+𝑣α  ⎩ β(k) = β(k − 1) + u β (9)  γ(k) = γ(k − 1) + uγ  2 The prediction startingHI(k )point= HI is(k set− 1to) +beα Start∗ k += cycleβ ∗ k + 50.γ As+ v shownk in Figure 12, the prediction basically reflects the degradation trend of the insulation of the electromagnetic coil, Thewhich prediction has a certain starting guiding point role is set for to bethe Start insulation = cycle degradation 50. As shown of in the Figure electromagnetic 12, the pre- dictioncoil. basically reﬂects the degradation trend of the insulation of the electromagnetic coil, which has a certain guiding role for the insulation degradation of the electromagnetic coil.

FigureFigure 12.12.Prediction Prediction ofof particleparticle ﬁlteringfiltering algorithm algorithm at at the the beginning beginning of of cycle cycle 50 50 of of the the electromagnetic electromagnetic coil. coil.

5.5. RUL Prediction of the Electromagnetic Coil Based on Particle Filter 5.5. RUL Prediction of the Electromagnetic Coil Based on Particle Filter The above experiments and analysis show that the HI can describe the performance degradationThe above of experimentsthe electromagnetic and analysis coil, showand the that 45% the decreaseHI can describe of the HI the is performance taken as the degradationfailure threshold of the of electromagnetic the electromagnetic coil, coil. and The the 45%RUL decrease prediction of of the electromagneticHI is taken as thecoil failureis realized threshold by the of particle the electromagnetic filter mentioned coil. ab Theove, RUL that prediction is, the life of is electromagnetic terminated when coil the is realizedHI is equal by theto failure particle threshold, ﬁlter mentioned and the above,number that of cycles is, the from life is the terminated predicted when starting the HIpointis equalto theto failure failure threshold threshold, is regarded and the number as the RUL. of cycles In order from to the evaluate predicted the startingaccuracy point of RUL, to thethe failureabsolute threshold error formula is regarded of RUL as theis defined RUL. In as order follows: to evaluate the accuracy of RUL, the absolute error formula of RUL is deﬁned as follows:

𝑹𝑼𝑳𝒆𝒓𝒓𝒐𝒓 =|𝑹𝑼𝑳𝒕𝒓𝒖𝒆 −𝑹𝑼𝑳𝒑𝒓𝒆𝒅𝒊𝒄𝒕𝒊𝒐𝒏 | (10) RULerror = RULtrue − RULprediction (10)

InIn thethe processprocess of of RUL RUL prediction, prediction, the the prediction prediction starting starting point point is is set set to to be be Start Start = = cycle cycle 50,50, 60,60, andand 70,70, asas shownshown inin Figures Figures 13 13–15.–15. AccordingAccording toto thethe aboveabove threethree RUL RUL prediction prediction results,results, thethe predictionprediction errorerror ofof RULRUL isis quantitativelyquantitatively givengiven byby equationequation 11,11, asas shownshown inin TableTable3 .3. It It can can be be shown shown from from the the above above results results that that RUL RUL prediction prediction of of the the electromagnetic electromagnetic coilcoil can can be be realized realized by by PF PF algorithm, algorithm, meanwhile, meanwhile, with with the the increase increase of the of historicalthe historical data, data, the accuracythe accuracy of RUL of RUL prediction prediction is further is further improved. improved. However, However, if there if there is no is largeno large amount amount of historicalof historical data, data, it is it difﬁcult is difficult to predictto predict the the remaining remaining life, life, and and it lacks it lacks accuracy, accuracy, as shown as shown in in Figure 13. Only when a lot of historical data is provided, the RUL prediction based on data-driven can be more accurate. However, the RUL prediction based on the physics-of-

Sensors 2021, 21, 473 12 of 14 Sensors 2021, 21, x FOR PEER REVIEW 13 of 15 Sensors Sensors 2021 2021, ,21 21, ,x x FOR FOR PEER PEER REVIEW REVIEW 1313 of of 15 15

Figure 13. Only when a lot of historical data is provided, the RUL prediction based on data- failure model can be realized without a lot of data. At present, our research group is stud- failuredrivenfailure model canmodel be can morecan be be accurate. realized realized without However,without a a lot thelot of of RUL data. data. prediction At At present, present, based our our on research research the physics-of-failure group group is is stud- stud- ying the physics-of-failure model of the electromagnetic coil based on creep degradation, yingmodelying the the can physics-of-failure physics-of-failure be realized without model model a lot of of ofthe the data. electr electr Atomagneticomagnetic present, our coil coil research based based on on group creep creep is degradation, studyingdegradation, the and the subsequent establishment of the physics-of-failure model can solve the problem andphysics-of-failureand thethe subsequentsubsequent model establishmentestablishment of the electromagnetic ofof thethe phphysics-of-failureysics-of-failure coil based on modelmodel creep cancan degradation, solvesolve thethe problemproblem and the due to lack of data. duesubsequentdue to to lack lack of establishmentof data. data. of the physics-of-failure model can solve the problem due to lack of data.

Figure 13. RUL (remaining useful life) prediction based on starting point 50 cycle. FigureFigure 13. 13. RUL RUL (remaining (remaining useful useful life) life)life) prediction predictionprediction based basedbased on onon starting startingstarting point point 50 50 cycle. cycle.

Figure 14. RUL prediction based on starting point 60 cycle. FigureFigureFigure 14. 14.14. RUL RUL prediction prediction based basedbased on onon starting startingstarting point pointpoint 60 6060 cycle. cycle.cycle.

Figure 15. RUL prediction based on starting point 70 cycle. FigureFigure 15. 15. RUL RUL prediction prediction based basedbased on onon starting startingstarting point pointpoint 70 7070 cycle. cycle.cycle.

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Table 3. Prediction error analysis.

Prediction Result Absolute Error Staring Point Real Life (Cycle) (Cycle) (Cycle) 50 96 72 24 60 96 83 13 70 96 93 3

6. Conclusions The electromagnetic coils are the widely used components in many applications and systems. Their insulation is failure-prone, which can lead to disastrous consequences. Therefore, predicting RUL for electromagnetic coil insulation can effectively avoid the unexpected shut-down and thus improve the availability of the related equipment that incorporate coils. Data-driven PHM framework is proposed to perform insulation degradation monitoring and RUL prediction for coil insulation. Coil electrical parameters, which include DCR and impedance under different frequencies, were collected by thermal accelerated fatigue test. Coil resonant frequency is identified as the insulation degradation characteristic parameter by analysis of the insulation failure mechanism and collected experimental data. The insulation health indicator is thus defined based on resonant frequency to indicate the health status of the electromagnetic coil. Further, the life model of the electromagnetic coil is constructed based on the polynomial fitting for resonant frequency data. The RUL prediction of electromagnetic coils are realized by PF. In order to increase the accuracy and reduce the complexity of particle filter, the polynomial model with one parameter reduced is used as the observation equation of the PF algorithm. The prediction accuracy of RUL is gradually improved with the increase of historical data. The prediction of electromagnetic coil health status provides support for predictive maintenance of the related equipment that incorporates coils. In the future research, the accelerated fatigue test of the electromagnetic coil will be carried out to collect the degradation data under different fatigue temperature conditions. The degradation model of the electromagnetic coil is constructed under multi-temperature, which lays the foundation for the prediction of RUL on variable temperature environments on site. Meanwhile, the physics-of-failure model based on creep degradation is established, which solves the defect that RUL prediction relies seriously on historical data. Combined with the self-learning function of neural networks and the ability to find the optimal solution at high speed, the degradation model based on neural network is established to realize the RUL prediction.

Author Contributions: H.G., A.X., and K.W. conceived the approach and wrote the paper; S.H.H. and M.Y. designed experimental platform and data analysis; Y.S. and X.H. performed the experiments; all authors read and approved the final manuscript. All authors have read and agreed to the published version of the manuscript. Funding: The research is supported by National Natural Science Foundation of China (62073313). Institutional Review Board Statement: Not applicable for studies not involving humans or animals. Informed Consent Statement: Not applicable for studies not involving humans. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available due to the need for a short confidentiality period. Conflicts of Interest: The authors declare no conflict of interest. Sensors 2021, 21, 473 14 of 14

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