Static and dynamic eccentricity fault diagnosis of large salient pole synchronous generators by means of external magnetic field

Although synchronous generators are robust and long ‐ lasting equipment of power plants, consistent electricity production depends on their health conditions. Static and dynamic eccentricity faults are among the prevalent faults that may have a costly effect. Although several methods have been proposed in the literature to detect static and dynamic eccentricity faults in salient pole synchronous generators (SPSGs), they are non ‐ sensitive to a low degree of failure and require a predefined threshold to recognise the fault occurrence that may vary based on machine configuration. This article presents a detailed magnetic analysis of the SPSGs with static and dynamic eccentricity faults by focusing on the external magnetic field. The external magnetic field was measured using two search coils installed on the backside of the stator yoke. Also, advanced signal processing tools based on wavelet entropy were used to analyse the induced electromotive force ( emf ) in search coils to extract the fault index. The proposed index required no threshold to recognise the starting point of fault occurrence and was sensitive to a low degree of fault. It was also non ‐ sensitive to load variation and noise that may induce a false alarm.


| INTRODUCTION
Periodic evaluation of critical components of large synchronous generators provides a reliable condition monitoring system that prevents severe unexpected failure in power plants [1]. The complex configuration of the salient pole synchronous generator (SPSG) requires an accurate condition monitoring system to avoid an unplanned stoppage of the power plant. The eccentricity fault is one of the common faults in SPSG where air-gap length varies. The main reasons for static eccentricity (SE) and dynamic eccentricity (DE) faults in hydropower generators are their vertical installation and imported forces to the body of the generator from the movement of the rock/cement, especially for power plants located inside mountains. More than 97% of electricity production in Norway is generated by hydropower plants, which are primarily located inside the mountains. Therefore, precise fault detection is required to reduce economic loss either for the producers or consumers.
The eccentricity fault creates subharmonics in the voltage and current of the machine that feeds into the grid and vibration on the machine's frame. The ultimate consequence of severe eccentricity is that the rotor rubs the stator core and winding [2]. Therefore, early-stage detection of machine fault can avoid costly damages to the machine and economic loss. Detecting eccentricity fault is mostly based on methodologies relying on analysing stator current [3][4][5]. In that approach, the Fourier transform is applied to the phase current, and the harmonic components of the phase current are assigned as an index to detect eccentricity fault. In contrast, the sensitivity of this approach is quite low, and it requires a high degree of eccentricity to show slight changes. Besides, the proposed index is sensitive to load harmonics, which with certain harmonic loads could induce a false alarm. The split-phase current is used to detect the SE and DE faults in SPSG [6,7]. The split-phase signature analysis is based on measuring the current in parallel branches of the windings. The current passing through the parallel branches is due to the distorted air-gap magnetic field. Although this approach can distinguish severe DE and SE faults, it applies only to a synchronous machine with parallel branches. SE and DE faults produce 2f s and kf s /p components in the rotor current of SPSG, where f s and p are stator electric frequency and number of pole pairs, respectively [8]. Although the mentioned feature can detect 50% SE and 50% DE faults, the unbalanced load, short circuit and the broken damper bar faults also have the same effects on the rotor current [9,10]. Fault detection based on parameter identification was also proposed in [11,12] for induction motors. The same approach based on the machine parameter was applied to SPSG. It was shown in [13,14] that self-inductance and mutual inductance of stator and rotor winding change under eccentricity fault. However, the variation rate is insignificant under a low degree of fault. Therefore, the introduced methods are not sensitive enough to detect the fault in its early stage. It has been shown that eccentricity can produce harmonic components of the no-load line to line or line to neutral voltage [8,15,16]. In [15], the subharmonics of the noload voltage are used to predict the SE and DE faults. Although the harmonic components of a no-load voltage can detect the failure, they are sensitive to machine configuration since the amplitude of the nominated harmonics varies in different machines based on their geometrical configuration, winding layout and material properties. The type of winding connection significantly affects the harmonic content of noload voltage. The sensitivity of the diagnostic approach is high in a machine with windings connected in series rather than in parallel [16]. Although the introduced index based on subharmonics of no-load voltage depends only on the number of machine poles [15], it also needs a threshold value to predict fault occurrence.
In [17][18][19][20][21], the air-gap magnetic field was used to diagnose the SE and DE faults in a synchronous generator. For the eccentricity fault, the air-gap magnetic field is distorted and contains subharmonics. Although the air-gap magnetic field is the most reliable source for fault detection regardless of fault type, it is an invasive approach. It is not a practical approach for a generator under operation because sensors need to be installed at a standstill and cope with the environment in the air gap. Furthermore, fixing a hall-effect sensor, search coil around the stator slot [18] or core in the axial direction through the radial ducts [19] is impractical for a synchronous generator with small air-gap length, which is used in the run-off river type power plants. The magnetic noise could also affect the performance of the induced voltage in the search coils (sensor) installed inside the machine.
The effectiveness of applying the external magnetic field to induction motors has been validated and explained in [22][23][24][25][26][27] for broken rotor bar, eccentricity, short circuit and bearing fault, respectively. Various types of advanced signal processing tools are used to extract the novel features that can recognise the type and severity of fault based on the external magnetic field captured on the induction motor frame. However, applying the external magnetic field to recognise the fault in SPSG is only limited to detecting interturn short circuit fault in the field winding in [28] and [29]. The acquired electromotive force (emf) is analysed using a fast Fourier transform, and it shows that the amplitude of the harmonic component of the signal is increased in the case of fault.
This article provides a detailed magnetic analysis of SPSGs under SE and DE faults using induced emf in search coils located on the backside of the stator yoke. The effect of the fault on the external magnetic field is studied, and how selfinductance and mutual inductance of the stator and rotor link with the search coil winding is shown. To improve the diagnostic technique, a new way to treat the emf is introduced by finding the difference in the induced emf in the sensors at opposite sides of the machine. In this way, the amplitude of the emf for a machine in a healthy case becomes almost zero. The trend of the emf is investigated using statistical tools such as mean, standard deviation (STD) and the energy of the signal in healthy and under SE and DE faults from no load to full load. To quantify the occurrence or evaluation of the fault, an advanced signal processing tool based on wavelet entropy is introduced.

| Eccentricity fault
The eccentricity fault is divided into static, dynamic and mixed eccentricities [30]. For SE fault, the rotor symmetrical axis coincides with the rotor rotational axis, and it is displaced from the stator symmetrical axis. Although the air-gap distribution is not uniform, it is time invariant observed from the stator frame. For DE fault, the stator symmetrical axis and rotor rotational axis are identical, but the rotor symmetrical axis is displaced with respect to them. Here, the position of the minimum air gap depends on the rotor angular position. DE is time dependent, unlike SE, and the minimum air-gap length varies with time. The mixed eccentricity fault is the combination of SE and DE faults.
Severe eccentricity faults induce an unbalanced magnetic force called an unbalanced magnetic pull (UMP) that exerts mechanical stress on moving parts, such as the shaft and bearings. The prolonged operation of the machine under faulty conditions induces moving part breakage and eventually rubbing the rotor on the surface of the stator core. However, in a large synchronous machine with a damper circuit, parallel windings and saturation can significantly reduce the UMP effect. The damper circuit, regardless of the eccentricity direction, can lessen UMP, while winding layout and orientation of eccentricity concerning the winding configuration can either reduce or do not change the UMP amplitude [15,31,32].

| Finite element modelling
Precise and detailed modelling of the machine is the first step in the fault detection process. The detailed and real parameters of the machine considerably affect reliable fault detection. Figure 1(a) represents the finite element (FE) modelling of 22 MW SPSG. The detailed specification of the machine is provided in Table 1. Furthermore, the non-linearity of the stator and rotor core materials, the rotor shaft, the spatial distribution of the stator winding, the physical properties of the stator and rotor winding and damper bars and end rings are considered. To avoid additional computation complexity, the eddy effect is neglected, except for the damper winding, because a current passes through the damper winding even in the steady state whether the generator operates in a healthy or faulty situation [9]. The external circuit with rotor field windings, stator winding and damper circuits is used. A DC voltage is applied to the field winding terminals. The magnetic field equations are combined with differential equations of the external circuits. The motion equation of the machine is finally combined with magnetic and external circuit equations in the FE model. In this paper, Ansys Maxwell 2-D package is used to model the SPSG [33].

| Magnetic field analysis
The magnetic field in electric machines contains all information about its detailed specification, which could be used to monitor the machine condition. The eccentricity fault causes asymmetries in the air-gap magnetic field distribution. The airgap magnetic field is caused by the stator and rotor magnetomotive force, the stator and rotor slot permeance, and rotor pole saliency permeance. The eccentricity fault feeds additional subharmonics to the air-gap magnetic field. The magnetic field fluctuation depends on the type and severity of SE or DE fault. The distorted magnetic field distribution considerably affects machine parameters such as inductance, the magnetic field in the core and subsequently the external magnetic field.
The magnetic field of a machine under eccentricity fault trivially influences self-inductance and mutual inductance of the stator and the rotor winding. The amplitudes of selfinductance and mutual inductance between the stator and rotor of SPSG vary in the range of mH. Consequently, any perturbation due to a small degree of eccentricity fault up to 20% does not remarkably change their amplitudes. Therefore, all quantities such as the stator phase voltage and current that are correlated with the self-inductance and mutual inductance of SPSG are unreliable signals for SE or DE fault detection with a low degree of severity. It is, however, possible to use phase voltage or current if the fault severity is high [5].
There is always an external magnetic field outside the electric machine, whether in the radial or axial direction of the machine. The location of the sensor significantly affects the captured external magnetic field [34,35]. There are two options to locate the sensor in the vicinity of the machine core, as shown in Figure 1(b). In position A, axial flux is measured by the sensor. In position B, the sensor may trap both radial and axial magnetic fields with the radial field as the dominant field in the captured signal. Regarding the sensor location, since the SE and DE faults have a radial nature with significant effects in the radial direction, the external field sensors are located on the backside of the stator yoke, as shown in Figure 1(c). Therefore, they are vulnerable to capture more radial signals, whether in a healthy or faulty situation.
The amplitude of the magnetic field is reduced when moving away from the air gap. Eventually, its amplitude  becomes negligible in order of μT at the outer radius of the backside of the stator yoke. Figure 2 depicts the airgap magnetic field and the magnetic field at the middle and outer side of the stator yoke. Although the amplitude of the external magnetic field is remarkably smaller than the air-gap magnetic field, its shape and periodicity are similar. Moreover, the external magnetic field is the mirror of the air-gap magnetic field that includes harmonic components of both the stator and rotor magnetic fields.
The induced voltage in the installed sensor on the vicinity of the backside of the stator yoke is proportional to the sensor cross-section, the number of turns and external magnetic field. The number of turns and the cross-section of the copper wire in the sensor are 3000 turns and 0.12 mm 2 . The dimension of sensor is 80 mm � 80 mm � 10 mm. The resistivity and inductance of the sensor in its terminal are 912 Ω and 714 mH. Figure 3 depicts the induced emf in the sensor caused by an external magnetic field. For SE fault, the emf shape does not change, whereas the amplitude of the signal based on its location changes slightly, in a way that the amplitude of induced emf increases for a sensor located on the side of the machine that the air-gap length is reduced and vice versa. Both the amplitude and emf shape in sensor dramatically change for DE fault.
The fluctuation of the induced emf in the sensors is due to varying mutual inductance between the stator and rotor windings with the coils of the sensors. Unlike stator and rotor self-inductance and mutual inductance, the eccentricity fault considerably affects self-inductance and mutual inductance of the sensor coils located on the backside of the stator yoke. Figure 4 depicts the mutual inductance between the rotor winding and the sensor in a healthy, 20% SE and 20% DE faults. Observably, there are no changes regarding the shape of mutual inductance for SE fault compared with the healthy case, while its amplitude is decreased. The reason is that the reluctance of the path for the linkage flux is increased and it reduces the mutual inductance between the rotor and sensor. For DE fault, as shown in Figure 4, the amplitude and shape of the mutual inductance change considerably since the DE fault varies in location, and time simultaneously changes the magnitude and shape of the mutual inductance. In addition to the oscillation, the mean value of the signal is also changed under DE fault.
The above argument is also valid by considering mutual inductance between the stator phase windings with a sensor coil. Figure 5 demonstrates the mutual inductance of stator phases A, B and C winding with sensor coil in the healthy, 20% SE and 20% DE faults, respectively. A comparison between the mutual inductance in the healthy case for all three phases reveals that the mutual inductance depends on the location and distribution of the winding with respect to the sensor coil. As seen, the amplitude of the mutual inductance between the phase winding and the sensor coil under SE fault decreases considerably more than the mutual inductance between the rotor and sensor coils because the path of the linkage flux is shorter in this case. Also, the air gap could not change and reduce the flux. The oscillation of the mutual inductance between the phase winding and sensor coil is significant under DE fault, and a comparison between the envelope of the signals in Figure 5 reveals that DE fault makes additional subharmonics.

| SIGNAL PROCESSING
Fault detection based on unprocessed signal is a difficult task since the variation of the signal does not give meaningful information regarding the machine condition. Several signal processing tools are used to extract useful patterns inside the signals for fault detection that can be divided into three categories: � Time domain [36,37] � Frequency domain [2][3][4]13,14] � Time-frequency domain [22,24,34,38] In this section, statistical tools such as mean value, STD, energy, frequency domain and time-frequency domain

| Preprocessing-definition of differential electromotive force
Defining the threshold for processed data from the signal processing part is the most challenging part of the fault detection process. To overcome this challenge, two sensors are installed on two sides of the machine in a radial direction exactly opposite of each other. The acquired emf from each sensor in a healthy case must be the same. Therefore, the induced emf in both sensors is subtracted and the resultant differential electromotive force (demf ) is almost zero for a healthy case and non-zero under SE and DE faults. The construction tolerance of the large SPSG in the hydropower plant is tight, and it is almost impossible to have an imbalance due to the machining of the stator or rotor core. However, in large SPSG with segmented stator core, the stator ovality is detectable by installing four sensors perpendicular to each other. Therefore, the assumption of considering demf equal to zero in large healthy SPSG is valid.

| STD and mean value of demf
The dispersion or variation of the data set is measured by the statistical term STD. According to its definition, the low value of STD indicates its tendency to the mean value of the data set, and the high value of STD suggests that the value is scattered over a wide range [39]. This definition could be used to analyse demf in a healthy or under SE and DE faults. The amplitude of demf in a healthy case is expected to be zero, which is not the case in reality due to tolerance in the manufacturing process. Therefore, the mean value and consequently the STD of demf may not be zero. Figure 6 demonstrates the variation of STD and the mean value of demf under SE and DE faults for noload and full-load cases. As seen, either mean value or STD is almost equal to zero in a healthy case, and its amplitude increases by increasing fault severity degree. The variation rate under DE fault is higher than that in SE fault since the fluctuation rate under DE fault is higher than that in SE fault. Therefore, the amplitude of STD under 20% DE tends to 1, which shows high degree of data dispersion, but it is 0 in healthy case. Besides, under SE fault, only the amplitude of the demf is altered, while under DE fault both the amplitude and some extra harmonics are also involved in demf waveform According to Figure 6, it is possible to identify the severity of the fault based on STD and mean value of demf acquired by external field sensors. Besides, the value of the feature for the  healthy case is almost zero and while increasing the fault severity, the amplitude of the STD and mean value significantly increase. In addition, the magnitude of the features in no-load case exceeds that of full-load case because the magnetic field created by parallel windings in the stator tries to balance the distorted air-gap magnetic field. The amplitude of the features in DE exceeds that of SE either in STD or mean value indicator because the waveform of the signal for DE is under considerable fluctuation. In contrast, the amplitude of the signal under SE fault depends on the location of the sensor which is increased or decreased. Alternatively, the emf of DE contains more harmonics. Indeed, the difference between the value of STD and mean value indicators for healthy and faulty conditions is significant, which demonstrates the effectiveness of the method.

| Energy of demf
The energy of a signal represents the strength of the signal since it gives the covered area under the curve of the power at any time interval [40]. Therefore, when the signal goes under any variation, it varies the energy as well. The energy of demf is derived as follows: From Figure 3, the amplitudes of demf under SE and DE faults compared with the healthy case have increased, which induce energy level increment. According to Table 2, the energy of demf in the no-load case is increased from 1.5 in healthy situation to 81.5 and 66.6 under 5% SE and 5% DE faults, respectively. By increasing the severity of the fault, the amount of signal energy is also increased. Although under fullload condition, the amplitude of demf is increased, the loading decreases the sensitivity of the signal energy as the fault progresses. However, it has a high degree of sensitivity to the occurrence of the fault since under a low degree of a fault, the amplitude is increased significantly compared with the healthy case. Although the amplitude of the signal energy shows no significant increment by increasing the fault severity, the sudden increase in the energy value by the fault occurrence is a fingerprint to demonstrate fault occurrence.

| Spectrum analysis
The fast Fourier transform is the most prevalent signal processing tool used in fault detection of electric machines. Its low computational complexity makes it a straightforward tool, especially for real-time assessment. Figure 7   A comparison between the healthy and 5% DE fault in Figure 7 (b) for frequency components of 25, 75 and 125 Hz shows that their magnitudes are increased to 68 52 and 51.8 dB. There is the same pattern of frequency component increment by increasing the fault severity. A comparison between the magnitudes of sidebands of spectrum density for SE and DE faults exhibits that the variation of DE components is significantly higher than SE components. Although spectrum density of emf under SE and DE faults shows that the sideband components are increased due to the fault, the variation of each frequency component does not follow the same pattern by increasing the fault severity. For instance, some of the amplitudes of the faulty sidebands are the same as a healthy case or even less, which is misleading for faulty data interpretation. In addition, the machine specification and operating environment of SPSG considerably affect the amplitude of frequency components because white Gaussian noise can mask or change the sideband magnitudes. Moreover, the machine specification determines the amplitude of sidebands, indicating that a threshold level is required to determine fault occurrence, which is difficult to propose and it needs expert knowledge.

| Time-frequency analysis
The wavelet transform (WT) is a useful signal processing tool used in various fields like power systems [41] and electrical machines [38]. The time localisation of different frequency components of a signal is used in WT. The WTs, unlike traditional frequency-domain signal processing tools, do not use a fixedwidth window. The wavelet analysing function adjusts its time widths according to the frequency component of a given signal, in which lower frequencies are in the broader window and higher frequencies in the narrower one. Alternatively, signals with oscillations and localised impulses could be treated using WT in a way that high-frequency and low-frequency components are decomposed in the short and long-time intervals, respectively. The signal is decomposed to its components by filtering the signal with high-pass (HPF) and low-pass filters (LPF). The output of the HPF is called details, while the output of the LPF is called approximations. The bandwidth of the two filters must be the same. After each step of decomposition, the sampling frequency of the signal is halved. The output of the LPF is decomposed recursively to produce the next sub-band of the wavelet. Equation 2 demonstrates the summation of all components of the demf signal into multiresolution decomposition as details and approximations: where j is the number of decomposition level, D(n) and A (n) are details and approximations of wavelet.
In this paper, Daubechies-8 (n = 8) is used as a mother wavelet. A higher order wavelet, similar to D-8, has a higher resolution that could improve the quality of fault detection in electrical machines. Figure 8 shows the procedure of signal decomposition using discrete WT, where S is the input signal, and LPF and HPF are low-pass and high-pass filters. Preliminarily, a given signal to WT is divided into two halves, which are the inputs of the LPF and HPF. The output of the firstlevel LPF is then separated into half the frequency bandwidth. This procedure is continued until the given signal is decomposed into the predefined value of that level. The sampling frequency in this paper is 10 kHz, and based on Nyquist's theorem, the highest frequency that the signal could contain would be 5 kHz. Consequently, the frequency bandwidth of the first sub-band of WT must be between 5 and 2.5 kHz. Figure 9 shows the applied discrete wavelet model to the demf in H, 20% SE and 20% DE faults. By comparing the demf in all three cases, it shows that the amplitudes of demf under faulty situations are 100 and 200 times more significant than the healthy case for SE and DE faults, respectively. A comparison between the detailed signal of H and SE shows that the frequency contents of the detailed signal from D1 to D6 must be the same, and the only difference must be their amplitude. However, the comparison between H-and DEdetailed signals of demf (Figure 9(a) and (c)) shows that due to the nature of DE fault that rotation of the rotor and consequently magnetic field is a function of space and time, the shape and amplitude differ. Unlike some signals like air-gap magnetic field, torque, current or voltage, where only one of the wavelet sunbands shows remarkable deviation from the healthy case, the wavelet-detailed level of demf predominantly changes under SE and DE faults compared with H situation. load as in the no-load case. However, the loading condition under SE fault reduces wavelet sub-band amplitude. Generally, SE and DE faults predominantly affect demf either in no-load or full-load cases. Moreover, results prove that demf has adequate information about irregularities due to the fault in SPSG. However, an index must be introduced to quantify fault severity.

| Wavelet entropy
Combining WT with entropy can provide a novel tool to analyse the transient behaviour of the faulty signals that have a non-stationary trend. The application of wavelet entropy in different fields like physiology [42], power systems [41] and condition monitoring of induction motors [43] shows that it could provide useful information. Therefore, wavelet entropy is unprecedentedly used to extract SPSG information under SE and DE faults. The wavelet entropy of the signal represents the degree of disorder in the wavelet sub-bands. The entropy is measured between 0 and 1, with 0 showing the perfect order, and 1 shows a high degree of disorder. However, the entropy value is not necessarily limited to an upper limit of 1, and it could be more significant than that, which means a higher degree of disorder.
Shannon entropy provides a practical index for evaluating and analysing the probability distribution [44]. Shannon entropy that measures the uncertainty and disorder of wavelet sub-bands is defined as follows: where P i is a relatively normalised value of each wavelet sub-band energy (E j ) to the total energy of the signal (E). Figure 11 displays the rate of entropy changes under SE, and DE fault varies from healthy up to 20% eccentricity in no-load and full-load cases. From Figure 11, the entropy of wavelet subbands shows a high degree of sensitivity to occurrence and progression of fault. According to the entropy definition, the entropy value in the healthy case must be almost equal to zero, which reveals the less or none degree of disorder in wavelet subbands. By increasing the degree of SE or DE fault, the rate of signal disorder is increased inducing high entropy value. The entropy value for detailed signals D7 and D8 demonstrates minimal degree of changes compared with other detailed subbands. The magnitudes of entropy for D7 and D8 in no-load and full-load healthy cases are 0.34 and 0.22, which under 20% SE and 20% DE faults increase to 27.6 and 3.1, respectively. By comparing the entropy of different wavelet sub-bands, it is found that D2, D3 and D4 have a higher degree of sensitivity to fault progress, whether under SE or DE fault. In a full-load case, the rate of change for D5 is increased compared with D3, which is due to circulating third harmonic in a machine winding.
Although a specific value of the threshold for a fault indicator has been proposed for fault occurrence [38], the introduced index in this article requires no specific threshold. The method proposed in this paper requires no specific threshold to indicate the fault appearance since increasing the entropy value from zero (healthy case) to any value indicates fault. A high degree of index sensitivity to failure induces discrimination of fault in a low degree of severity even less than 10% eccentricity. For instance, by having 10% SE, the magnitude of wavelet entropy is increased from 0 to 85.9, which shows that the index can detect low severe fault with high precision.

| Load effects on proposed index
Figures 12 and 13 depict the variation of wavelet entropy under load variation from no load to full load under different degrees of SE and DE faults, respectively. Comparisons of the amplitude of entropy in no load and full load show that load reduces the entropy amplitude even by increasing the fault severity level. For no load, the most contributing magnetic field is produced by the rotor, and even a small degree of eccentricity causes a high degree of distortion in demf, whereas in loading condition, both the stator and rotor magnetic fields synergistically influence the air-gap magnetic field and consequently, the external magnetic field. Therefore, the machine, especially with a parallel winding layout, tries to balance the magnetic field in a faulty case. Hence, the degree of disorder in wavelet entropy of sub-bands under loading conditions must be reduced. In addition, the ratio of tangential to the radial magnetic field under loading conditions is increased, while the radial magnetic field is the dominant field captured by sensors in the no-load case. However, wavelet entropy of sub-band D5 is robust to load variation, while the variations of the other sub-bands (D1, D2, D3, D4 and D6) are in an acceptable range.

| Noise effects on proposed index
The term 'signal' in the field of fault diagnosis means only the desirable data that are measured [40]. However, the signal is F I G U R E 1 1 The entropy of wavelet sub-bands (D1 to D6) for different degrees of SE and DE faults in no-load and full-load cases. DE, dynamic eccentricity; SE, static eccentricity EHYA ET AL. vulnerable to various noises during the process of data acquisition, storage and conversion. Different noises could specifically affect data, which can demonstrate its effect in signal processing. White Gaussian noise is one of the prevalent noises in the industry and power plant that could mask or even hide fault features for a high signal-to-noise ratio (SNR). The  Table 3. The various rates of SNR from 60 dB as the lowest level up to 20 dB as the highest level are applied to demf signal in the healthy and under SE and DE faults. As shown in Table 3, by increasing SNR from 60 to 20 dB, the amplitude for the healthy case in D1 to D6 increases with the highest value in D6. However, the minimum amplitude of various wavelet entropy sub-bands is much higher than the maximum value of the healthy case under 20 db SNR. Although the proposed feature is robust to noise, the wavelet entropy of sub-band D6 among other sub-bands is vulnerable to serious variation in a noisy environment.

| APPLICATION, LIMITATION AND CONSIDERATION IN FIELD TEST
The application of this method for large SPSGs in hydropower plants is possible. It is possible to attach the sensor, as shown in Figure 1, to the backside of SPSG since hydropower generator has no steel housing like a turbo generator or induction machine. However, some circumstances may induce some differences between finite element model (FEM) and real-test results as discussed below: � Lack of material data sheet in FE modelling may change the amplitude of emf and consequently, it induces variance in the measured data with simulation. � The manufacturing tolerance is disregarded, which may also affect the simulation results. � In large hydropower plants, the housing of SPSG is the wall in the generator pit that is usually made of concrete. The vertical and horizontal beams of frames are used to take up to the torsional force acting on the stator body. The distance between the horizontal and vertical beams exceeds 30 cm.
Notably, it could affect the signal since the material is iron. However, putting the sensor in the middle of the beams aids to avoid the effect of the frame. Therefore, for sensor attachment close to the frame, the modelling of the frame in FEM is mandatory. � It is almost impossible to locate the sensors exactly in front of each other in a real-field test and there is a possibility of a few centimetre errors in the sensor installation. Therefore, one of the sensors in FE modelling is moved for 5 and 10 cm concerning the other sensor in the opposite direction. Results show that location error does not significantly affect the amplitude of demf as shown in Figure 14. The wavelet entropy is also applied to the signals and their amplitudes do not change substantially, and it increases 0.1 and 0.14 for 5 and 10 cm sensor relocation, respectively. � The middle of SPSG yoke is the optimal position for the sensor installation in order to avoid the magnetic field effect on sensor due to end winding and high voltage bus bars.
� Since the working environment of SPSG in hydropower plants is vulnerable to magnetic noise, a shielded co-axial cable is required to reject the noise effect on the sensor.

| CONCLUSION
A new method is introduced to detect the SE and DE faults in SPSG. Different faults in induction machines induce significant variation in current, torque and speed waveform (characteristics). The mentioned signals in SPSG are robust to a low degree of fault; therefore, the only reliable source of fault detection is based on either the magnetic field in the air gap or external magnetic field. While the former is an invasive tool that is inappropriate for generators under operation, the latter is non-invasive, low cost, easy to design and fabricate, which makes the proposed method in this article noteworthy. The proposed methods are based on the external magnetic field analysis. The mentioned external magnetic field is captured by installing two search coils on the backside of the stator yoke precisely opposite to each other. By having two signals from two sides of the machine, the net demf in a healthy case is almost zero, and by the occurrence of the fault, based on its type, the amplitude and shape of demf differ. Various approaches and signal processing tools are used to detect the appearance and evolution of the fault. The proposed method relies on the time-frequency analysis of demf, and discrete WT is used to identify the hidden pattern under SE or DE fault. Also, the following conclusions represent the summary of achievements in this paper.
1. The detailed and accurate modelling of SPSG in the FE model shows how the external magnetic field responds to SE and DE faults. The fault occurrence induces some subharmonics in the air-gap magnetic field in emf. 2. The investigation of self-inductance and mutual inductance between the rotor and stator with search coils shows that due to the low rate of inductance variation under faulty cases, the amplitudes of the stator and rotor self-inductance and mutual inductance show no significant changes, while the mutual inductance between search coil with rotor and stator windings show a high degree of variation. 3. The demf provides a sensitive signal with respect to fault since the amplitude of demf in the healthy case must be almost zero. 4. The variation of mean value and STD of demf proves that the fault alters the signal behaviour. 5. The occurrence of SE or DE fault significantly increases the energy of the signal that could be used as an early-stage fault indicator. 6. The studies of discrete wavelet sub-bands reveal that SE and DE faults greatly affect them. To quantify the fault severity, wavelet entropy is used. It shows that this approach can find SE and DE faults even in its early stage in SPSG. Furthermore, there is no need to specify the threshold value to detect the occurrence of failure, which is an additional advantage of this method.