Demagnetization Monitoring and Identification in PM Generators With Concentrated Windings During Transient Conditions

Direct drive permanent magnet machines used in renewables typically have high numbers of poles leading to low speeds. This poses a challenge for detection of faults during transients, while few periods of the operating waveforms can be captured. Moreover, it has been recently shown that the numbers of poles and stator coils determine the demagnetization harmonic generation mechanism and consequently the location of the fault signatures. This leads to different signatures in every permanent magnet machine. Additionally, the traditional signatures may not be clear in the spectrograms due to the low rotor speed. Under such conditions, the monitoring of specific higher harmonics can lead to reliable detection and identification of the demagnetization fault. The authors prove that the fault can be reliably distinguished from other rotor-related faulty conditions with the use of specific higher signatures in the stator current. Finally, the computed harmonics are experimentally verified in the lab, while the detection of demagnetization under transient conditions is achieved in this paper, that studies a double rotor axial flux PM generator as a case subject.


I. INTRODUCTION
R EMOTE renewable energy harvesting such as off-shore wind, tidal and wave gave space to direct-drive Permanent Magnet (PM) generators, in order to minimize the maintenance needs of gearboxes typically applied with induction generators [1]. These generators are typically built with a high number of poles leading to very low shaft speeds [2].
Different types of faults may appear in PM machines such as stator inter-turn faults [3], demagnetizations of the PMs [4] and a plethora of different types of mechanical faults such as eccentricities and misalignments [5], [6]. Among the rotor faults, the demagnetization one can be severe since it leads to the reduction of the air-gap magnetic flux density. Operation under fixed load conditions though will lead to operation under higher current, leading to additional thermal stress and further demagnetization severity until the machine collapses [7]. It is therefore crucial to detect the fault at an incipient level and plan fault tolerance and maintenance actions appropriately.
Partial demagnetization usually originates from defective manufacturing of the PMs or some damage during the assembly of the generator [8], [9]. High loading or thermal stress will build on those defects and lead to the permanent demagnetization of the PMs [10].
In the literature, one can find works dealing with the detection of demagnetization faults. Mostly applied methods are the Motor Current Signature Analysis (MCSA) [11], [12] and the magnetic flux analysis, either air-gap or stray [13], [14]. However, most papers deal with high speed PM machines where the number of poles is small and the fault related signatures are specific and few. More importantly, the demagnetization fault signatures coincide with the rotor mechanical fault ones leading to identification issues [15], [16]. Moreover, past papers focus on the analysis of the PM machines at steady state.
Frequency signature analysis is severely influenced by the winding configuration [20], and the slot number as these harmonics are influenced by the harmonics, which are generated by the slotting effect [21]. Under healthy conditions, the spatial distribution of the coil/pole slot combination significantly affects the harmonic content of voltage through the winding factor [22], in machines with and without armature core. Specifically, in machines with parallel path windings under fault conditions 0093-9994 © 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
the magnetic field along the air-gap circumference ceases to be symmetric, as a result circulating currents on the branches of a phase generate, distorting the current waveform [23] and influencing the unbalanced magnetic pull [24]. Therefore, the amplitudes of the fault signatures are much dependent on the design parameters of the machine, rending the diagnostic process unreliable. However, PM machines utilized in renewables, experience continuous transients making the steady state techniques unreliable or not easily applied. Moreover, the well-known frequencies adopted in the literature are sub-harmonics of the fundaments creating resolution issues since the low speed does not permit the capturing of enough periods for significant spectral resolution. Fault analysis under transient operating conditions can be detected using time-frequency based signal processing algorithms in which every algorithm uses different transformation functions.
Furthermore, in such cases the captured signals cease to be periodic, so other signal processing algorithms have been proposed for condition monitoring. Short time Fourier transform (STFT) [25] has a steady window length, so fast transients cannot be analyzed with high resolution and can be combined using Neuro-Fuzzy approach [26]. Therefore, for signals with rapid alterations, Continuous and Discrete Wavelet Transforms (CWT, DWT) are more appropriate as the window length is variable [27], [28]. Other time-frequency distributions methods based on the quadratic time-frequency distribution [29], Hilbert-Huang transform, which are capable of analyzing the signal from a time-frequency-energy aspect. Moreover, the demagnetization fault can also be identified using features extracted by CWT and analyzed by the box-counting method and a threshold [30]. Finally, a recent paper [31] utilizes the recursive least-squares (RLS) method to detect the demagnetization fault in a high speed motor and offers some identification with respect to the dynamic eccentricity fault.
Aiming to answer those needs, this paper which is the continuation of the early work [32], investigates the use of the stator current for condition monitoring during transient operation of PM generators. Firstly, analytical calculations will demonstrate the expected higher harmonics as a function of the machines' stator coils and numbers of poles. Then, the case of an axial flux PM generator will be examined with the use of Finite Element Analysis (FEA) and the stator current waveforms will be extracted and analysed with a high-resolution spectral decomposition. The machine will be subjected to a variety of faults such as partial demagnetization, axial and radial misalignments and eccentricities. The results will show that the monitoring of specific higher harmonics in the stator current can be an excellent method to detect and identify the partial demagnetization fault.

II. ANALYTICAL INVESTIGATION
In order to calculate the signatures induced to the stator due to the demagnetization fault, the air-gap magnetic flux density is needed and it is given by (1). where: and the permeance due to one magnet demagnetized [17]: where: Therefore, the air-gap magnetic flux density due to (1), (2) and (3) is given by: where: Furthermore, the magnetic flux can be calculated by: This flux will induce an Electromotive Force (EMF) to each stator winding, which is described by (6) Depending on the stator windings number and their spatial location with respect to each other, the following signatures are generated [18]: Moreover, the harmonics obeying to: f 3ph_null = 3γλ p f s , (γ is the number of phase coils) are expected to disappear due to the three phase winding.

III. FINITE ELEMENT ANALYSIS AT STEADY STATE
For the purposes of this work, an axial flux PM generator has been simulated with FEA under steady state and transient operating conditions. The generator has a distributed winging with 3 coils per phase while p = 6. So, it becomes clear that due to its manufacturing characteristics, this machine will generate demagnetization signatures according to f dmg2 .
The PM machine and the FEA model are shown in Fig. 1. It has been analyzed under healthy and faulty condition. The level of partial demagnetization has been 35% and affects a single magnet.
The generator is simulated to operate at steady state first, aiming to validate the formulae and identify the demagnetization harmonics in the stator current spectra, which are depicted in Fig. 2. For this machine p = 6, so the expected fault-related harmonics are given for different values of the integer κ: The signature at f s + f s 2 is indeed produced in each phase but cancels out due to the three phase winding as predicted by f 3ph_null above.

A. Time-Frequency Analysis
A class of high-resolution techniques based on an eigen decomposition of an autocorrelation matrix of the data signal have been promoted in recent literature as having better resolution and better frequency-estimation characteristics than classical frequency estimators such as Fourier transform [19]. The Min-Norm algorithm is a high-resolution technique based on harmonic decomposition of multi-component signals. A multi-component signal z can be expressed as: where s m (l) denotes the m-th component in the signal, m ∈ R, v(ψ m ) the discrete frequencies vector, [ϕ ∈ R), and η(l) the noise in the measured signal. The power spectral density (PSD) of z is given by [20]: where ψ belongs to the sample noise subspace, has minimum norm, and the first element equal to 1. The Min-Norm algorithm reduces anomalies (spurious peaks) in spectral estimates exhibit in other high-resolution techniques such as Pisarenko harmonic decomposition or the Multiple signal classification algorithms. The time-frequency distribution of the stator current is computed by sectioning the data sequence into smaller segments, taking the autocorrelation of each subsequence, and evaluating the PSD. The proposed method was implemented under MATLAB in a PC with Intel Core i5 at 1.6 GHz. The methodology is composed of four stages: analog to digital conversion, sample rate conversion, signal windowing, and spectral analysis. Once the current signal is acquired for 1 second, the complete method requires 146 msec to compute its time-frequency decomposition. The block diagram of the methods application is shown in Fig. 3 below.

B. FEA Analysis and Examined Cases of Rotor Faults
In this section the test-machine is modelled using 3-D FEA software, named Simcenter MagNet by Siemens digital industries. Axial-flux machines, unlike radial-flux machines, can only be modelled reliably using 3-D modeler, since the scalar and vector magnetic potentials alter along all directions. Another advantage of the 3-D simulations is that the stator-end winding fields are accounted in the field solution. Hence, this method of modelling is considered the most accurate, with the disadvantages that is time-consuming and occasionally divergent due to the high complexity of the volume mesh grid. By solving the model using the transient with motion analysis the phase quantities in the time domain are obtained. A Fast-Fourier Transform (FFT) algorithm is then used to calculate the frequency spectra for the stationary signals.
The faults are implemented as follows. The partial demagnetization fault is inserted in the machine by reducing the slope of the PM's magnetic characteristic. The magnetic characteristic of rare-earth magnets is approximately linear on the second quadrant at 20 o [C]. By reducing the slope of the line, common point of the load line and characteristic reduces, consequently a demagnetization fault which is equal with the percentage of the slope reduction can be implemented. The axis misalignment faults are implement by displacing the rotor disc parallel to the stator disc. For static axis misalignment, the rotor axis of rotation should be the same with the axis of the displacement. For dynamic axis misalignment, the rotor axis of rotation should be the same with the stator's axis after the displacement. In this fault, a portion of the magnetic flux tends to leaves the surface of the stator coils. With the same concept the angular misalignment faults can be implemented with the only difference that a rotor disc is twisted relatively to the stator disc. In this case the air-gap becomes non-uniform. Fig. 4 illustrates the angular and axis misalignment for each eccentricity case. Table I illustrates the major specifications for the studied axial-flux generator.

C. Analysis Results of Rotor Faults
The simulation scheme allows a detailed analysis of the machine behavior under several fault conditions. A Time-frequency (t-f) analysis based on the Min-Norm technique is applied to the stator current signals for tracking the presence of fault-related Under healthy condition where there is no abnormality in the PM generator, the largest magnitude component of the analyzed current is the fundamental mode f s , whose instantaneous frequency changes as time evolves with positive slope of 75 Hz/sec as can be seen in Fig. 5. The spectrogram also exhibits the presence of a spectral line evolution from 100 to 250 Hz corresponding to the 5th harmonic of f s . This result coincides with the steady state analysis in Fig. 2(a).
The behavior of the PM generator under partial demagnetization of a single magnet is presented in Fig. 6., the t-f decomposition shows the appearance of a high-power signature below the fundamental component f s , which seem to be the f s -f s /2 fault-related harmonic, but the spectral signature is not accurately localized in the t-f plane because it is very close to the f s . However, above f s a demagnetization harmonic located at f s +3f s /2 can be easily identified. Although this fault-related harmonic has lower energy concentration than the 0.5f s mode, it does not suffer from interference problem of the fundamental component. As expected, f s +f s /2 is not present in the signal due to the winding configuration. Also note that the t-f decomposition generates a weaker harmonic component between the 5f s Fig. 5. T-f decomposition of the stator current at transient state: healthy condition. Fig. 6. T-f decomposition of the stator current at transient state: 20% Partial Demagnetization. and 2.5f s trajectories that appears to be the 3.5 f s fault-related and is the result of f s +5f s /2. Figs. 7 and 8. depict the stator current's t-f decompositions under two simulated dynamic misalignment fault conditions. Fig. 7 depicts a plot of the t-f decomposition for 2 mm of dynamic axis misalignment, whereas Fig. 8. shows the t-f distribution for 40% of dynamic angular misalignment. In both cases, the spectral contents of the currents are very similar except that the axis misalignment case (Fig. 7) presents spectral shapes crossing the 5th harmonic, but they are almost indistinguishable in the time-frequency distribution.
Finally, the transient analysis technique was implemented to the simulated PM generator under two static misalignment faults. The results of the analysis of axial and angular misalignments are presented in Figs. 9 and 10, respectively. The fundamental component and the 5th harmonic are present and with large amplitudes in both time-frequency decompositions. If the spectrogram of the axial misalignment condition (Fig. 9) is compared with the healthy case (Fig. 5), it is observed an increase in spectral energy above the 5th component. Fig. 7. T-f decomposition of the stator current at transient state: 2 mm dynamic axis misalignment. Fig. 8. T-f decomposition of the stator current at transient state: 40% dynamic angular misalignment. Fig. 9. T-f decomposition of the stator current at transient state: 1 mm static axis misalignment. On the other hand, in the case of angular misalignment condition, the PM generator generates different signatures in the stator current spectrogram, components with abrupt changes in frequency are observed in the spectral band between 50 and 300 Hz (see Fig. 10), indicating the presence of very small spikes in the stator current signal.
Among the spectrograms, results of several fault conditions, the transient analysis of the PM generator under partial demagnetization, has been shown to exhibit a particular signature pattern in the time-frequency domain. The transient analysis of the stator current permitted distinguish unequivocally the demagnetization condition not only from the healthy state, but also from other fault conditions.

D. Stator Inter-Turn Faults and Analysis
The generator model has been modified to represent stator inter-turn faults of two different severities (2 and 3 shorted turns) in phase A. The contact resistance is 0.1 Ω. The stator current has been extracted from both models and their spectrograms are shown in Fig. 11 below. It is clear that, the 5th harmonic increases in amplitude, however there is absolutely no presence of any of the demagnetization signatures (no signature at 0.5f s, 2.5f s , or 3.5f s ).

V. EXPERIMENTAL TESTING
To validate the proposed signal processing technique a test bed was created to capture the signal of the stator current under various faulty conditions on a transient state. The proposed test bed is illustrated in Fig. 12. It consists of a 3-phase voltage inverter, which drives a 3-phase induction motor. The induction motor is the prime mover to the tested axial-flux permanent-magnet generator, which supplies a variable, 3-phase resistive load. To acquire the signals of the stator current, a data-acquisitionsystem employing current probes has been adopted.
To verify the feasibility of the analysis, three different conditions are tested: healthy state, one magnet demagnetized and one with dynamic angular misalignment. The demagnetization  fault was created by removing a magnet from one of the rotor discs. Since it is a double sided disc generator, every pole pitch is occupied by two magnets, hence a 50% partial demagnetization has been created. The dynamic angular misalignment is implemented by adding washers on two out of four bolts, which tie a rotor disc with the axis of the machine. In order for the rotor discs, to be angularly rotatable, the opposite inner diameter of a disc was filled to create enough clearance. The Fig. 13. T-f decomposition of the stator current at non-stationary conditions: healthy condition. experimental tests realize an ascending speed curve starting from standstill for all the studied conditions. Spectrograms of the stator currents are presented in the same frequency bandwidth for comparison purposes, the short-time analysis have been performed by the Min-Norm algorithm with a window length of 128 samples, a leap of 2 samples between successive windows, and a low-frequency band analysis. In addition, it should be specified that the original sampling frequency (f n = 25kHz) is converted to f n = f n /20 in this study to reduce the computational burden.
In Fig. 13, the t-f decomposition of the stator current signal for the healthy case is shown. It can be seen from the result that the stator current consists of the fundamental component f s , which is embedded in a high level of noise. It is known that the instantaneous frequency of f s is directly proportional to the instantaneous speed of the PM generator, which can reflect the motion behavior of the machine. In this case, there are linear speed changes appearing as linear FM segments in the t-f distribution of the stator current.
The time-frequency decomposition of the stator current under faulty condition is shown in Fig. 14. The fundamental component reflects a time-varying behavior of the PM machine, similar to the experimental result in Fig. 13. However, the decomposition reveals supplementary information. The energy around f s and the DC component seems to have an increment. But more important is the fact that the three fault signatures predicted earlier, 0.5 fs, 2.5fs and 3.5 fs emerge from the noise floor of the t-f decomposition and displays the trajectories corresponding to the demagnetization fault at f dmg2 (for k = 1, 3, and 5). It's worth mentioning that at low speeds spectral density around f s and DC could overlap with the fault signature 0.5 f s , leading to a false diagnostic. On the other hand, high harmonics present a larger spectral distance between other components.
Finally, Fig. 15 illustrates the time-varying spectrum for the rotor misaligned case. The FM law for f s is clear from the sharp resolution of the decomposition. However, the stator current does not contain any other oscillatory component. Experimental results show that indeed the time-frequency analysis allows Fig. 14. T-f decomposition of the stator current at non-stationary conditions: demagnetized condition. Fig. 15. T-f decomposition of the stator current at non-stationary conditions: dynamic angular misalignment condition. detection and discrimination between the demagnetization condition and other PM generator conditions.
Although subspace methods improve the frequency estimation performance, computational complexity is one important factor for practical applications. In this study, high-resolution results were obtained applying the short-time minimum algorithm to stator current signals. The time-frequency plots shown in Figs. 13-15 have been computed in 11.48, 11.69, and 11.87 seconds, respectively. Considering that the analyzed signal lengths are 19 seconds, it can be concluded that, the computational time is not a problem for analyzing non-stationary signals. Time-frequency decompositions were computed in a PC with an Intel Core i7 processor.

VI. CONCLUSION
This paper demonstrates the advantageous use of higher stator current harmonics for the detection of partial demagnetizations faults in low speed direct drive PM generators. While those generators are utilized in renewable energy harvesting, they experience continuous transients therefore conventional steady state diagnostic approaches cannot be applied. It is shown in the paper that the Min-Norm decomposition is capable not only of detecting the partial demagnetization fault but reliably distinguishing it from other common rotor-related faulty conditions such as static and dynamic misalignment and eccentric conditions. A synopsis of the new elements to the diagnostics community, offered by this research paper are summarized below.