Research Article | Open Access
Impact of Single and Combined Faults Composed of Rotor Eccentricity and Stator Interturn Short Circuit on Electromagnetic Torque Ripples in Synchronous Generator
In order to comprehensively study and identify the electromagnetic torque (EMT) difference among the single static air-gap eccentricity (SAGE) fault, the single stator interturn short circuit fault (SISC), and the combined fault composed of these two, this article investigates the EMT ripple properties due to the mentioned three faults. Different from other studies, this paper considers not only the effect of the single fault types but also the impact of the single fault combinations on the EMT ripple characteristics. Detailed EMT expressions for each fault are firstly derived on the basis of the magnetic flux density (MFD) analysis. Then, finite element calculation and experimental study on a CS-5 prototype generator with two poles at 3000 rpm, which is specifically designed and manufactured ourselves, are carried out to validate the analysis result. It is found that the three faults will induce different ripple components in EMT. The combined faults have the most intensive impact sensitivity on the EMT ripples, while the single SAGE fault ranks the last in the impact effect.
In synchronous generators, the electromagnetic torque (EMT) ripple is a primary parameter for the energy conversion. Under the stable running conditions, the input mechanical moment should be equivalent to the electromagnetic torque (ignoring the energy loss). Since EMT is the resisting moment, its ripple properties are closely related to the output stability of the generator. On the one hand, large EMT ripples will lead to the overfrequent adjustment of the automatic control system to modify the input mechanical moment, and obviously this is harmful for the generator set . On the other hand, intensive EMT ripples will cause the torsional vibration of the shaft and may lead to the fatigue of the shaft material after a long-term performance . More seriously, it may even lead to a disaster .
By far, the EMT ripple properties in normal condition have been well studied by scholars [4, 5]. Since the magnetic flux density (MFD), which is closely related to EMT, will be changed due to different faults, the EMT ripple characteristics will be also varied in different fault cases. For instance, Hao et al. found that the occurrence of rotor interturn short circuit will increase each odd and even harmonic of EMT . In the meantime, Song et al. found that the segmented armature winding can effectively restrain the torque ripples under the stator short circuit fault . However, the EMT properties due to varied faults, especially the combined faults such as the static air-gap eccentricity (SAGE) and stator interturn short circuit (SISC) composite fault, have not been fully studied.
For SAGE, scholars have investigated both the mechanical and the electrical faulty properties. For instance, He et al. investigated the unbalanced magnetic pull (UMP) , the vibration characteristics of the rotor , and the circulating current inside the parallel branches . Besides the mechanical characteristics, it is also found that the stator phase current is an effective tool to identify the broken magnet and SAGE in permanent magnet synchronous motors , while at the same time, the shaft voltage measurements have been found as a better means for the optimal measurement modality that leads to a minimal identification error of SAGE in synchronous machines . Meanwhile, a method to quantify the rotor eccentricities in synchronous machines by exploiting the unbalance caused in the split-phase currents was proposed by Bruzzese . Also, an alternative method based on the analysis of line neutral voltage taking place between the supply and the stator neutrals in induction motor was presented by Oumaamar et al. .
For SISC, Nadarajan et al. presented a hybrid approach to set up the synchronous generator by combining the dq0 modeling with the winding function . This approach improves the accuracy and reduces the modeling complexity. In addition, Moon et al. proposed a method that uses voltage and current residual components to diagnose the exact phase and the severity of SISC in permanent-magnet synchronous machine . It is found that the occurrence of SISC will significantly impact on the zero-sequence current , the negative-sequence current , the whole stator current , the rotor current , and the electromagnetic torque .
The above studies employed different parameters to describe the fault characteristics for either the SAGE fault or the SISC fault, while few of them have paid attention to the combined faults composed of these two. However, the actual operating conditions can be far more complicated than just one single fault. A generator may have two different faults at the same time . It is found that the faulty characteristics under the composite fault are different from the composed single faults . Then, the following key point is how to distinguish the single and the combined faults by proper faulty characteristics.
In this paper, we proposed a comprehensive study on the impact of the single and the combined faults which are composed by SAGE and SISC on the electromagnetic torque ripples. The remainder of this paper is constructed as follows. The theoretical analysis model is derived in Section 2, while the FEA calculation and the experimental study are presented in Section 3. Finally, the primary conclusions from the investigation of this paper are drawn in Section 4.
2. Theoretical Analysis Model
2.1. Effect of SISC on MMF
The magnetomotive force (MMF) and the permeance per unit area (PPUA) are the basic parameters for calculating the EMT of the generator. Their values are directly dependent on the operating state of the generator. Once the SISC fault occurs, the MMF will be changed immediately. According to , the normal MMF can be written aswhere Fs and Fr are the stator MMF and the rotor MMF, respectively, αm is the mechanical angle to indicate the circumferential position of the air gap, and ψ is the internal power angle of generator.
When SISC takes place, as shown in Figure 1, the additional circulating current in the short circuit ring will produce a pulsating magnetic field which centers on the short circuit winding axis. This new magnetic field has the same frequency as the short circuit current, namely, 50 HZ. It can be expressed aswhere Fd+ and Fd− are the forwardly and the inversely rotating MMF amplitude, respectively, and Fd+ = Fd- = Fd/2.
Since the rotating speed difference between the reversed MMF and the rotor is 2ω, an extra reversed electromotive force (EMF) at 2ω will be induced in the field windings. Then, the final excitation current in the field windings can be expressed aswhere If0 is the DC component in the field windings produced by the exciting system, while If2 is the current induced due to SISC.
Further, the air gap magnetic flux density (MFD) can be written aswhere N is the turn number of each pole and Λ0 is the permeance of the air gap.
The EMFs of the three phases can be written aswhere , q is the slot number for each phase per pole, is the turn number of each coil, is the fundamental winding factor, τ is the polar distance, f is the fundamental frequency, and L is the length of the stator core.
As indicated in (5), the induced EMF for each phase is composed of three parts. The first parts will form a forwardly rotating MMF at the fundamental frequency, since the phase angles for the three phases are 0, −120°, and +120°, respectively. On the contrary, the second parts will form a backwardly rotating MMF, while the third parts will form forwardly rotating MMF at 3ω. Consequently, the MMFs under SISC can be expressed aswhere Fs1+ and Fs3+ are the forwardly rotating MMFs of stator at ω and 3ω, respectively, Fs1− is the backwardly rotating MMF of stator at ω, and Fs3+ = Fs1−. Expanding the first item in equation (6), the MMF under SISC can be modified towhere
2.2. Impact of SAGE on PPUA
SAGE will deform the air gap and further affect PPUA (see Figures 2(a) and 2(b)). According to Figure 2(b), OB = Rs, which is the inner radius of the stator core, and O1A = Rr, which is the outer radius of the rotor core. The radial air-gap length can be written aswhere based on the cosine law it haswhich can be further modified aswhere is the average value of the radial air-gap length and δs is the relative SAGE in the radial direction. Then, the radial air-gap length can be obtained as
Since Rr >> , and the air-gap length should be smaller than Rs, (12) can be further simplified as
Finally, PPUA in normal condition and SAGE case can be expressed aswhere the second equation is expanded by power series.
2.3. Impact of SISC and SAGE on MMF
When the combined fault occurs, i.e., when SISC and SAGE take place at the same time, both MMF and PPUA will be changed. Simply multiplying equation (7) by (14), the final MFD can be obtained. However, another factor, namely, the short circuit position, will actually impact on the final MFD as well. This can be explained by Figures 1 and 3(a). When there is no SAGE, i.e., in the ideally normal condition, the short circuit position should have no impact on the MMF/MFD due to the symmetric distribution of the armature windings. However, as long as SAGE exists, the short circuit position will act on the amplitude of the short circuit current. When the short circuit turns (indicated by the central axis position of the turns) are closer to the minimum air gap (see Figure 3(a), αm = 0), the induced short circuit current Is (see Figure 1) will be larger since the PPUA has a larger value (see equation (14)).
By considering the short circuit position, equation (2) modifies towhere ρ = (1 − δs)/(1 − δs cos αsm) which is a coefficient of the MMF at the short circuit position to the MMF at the minimum air-gap point. Obviously, this coefficient depends on SISC position (see Figures 1 and 3(a)). For more clarity, the relation of ρ vs. αsm is illustrated in Figure 3(b).
As indicated in Figure 3(b), the developing regularity of ρ changes similarly to the cosine function. When asm = 0, ρ will obtain its max value. This means when the short circuit takes place in the winding whose central line coincides with the minimum air-gap point, SISC will have the maximum impact on MFD/MMF, i.e., MFD/MMF will be decreased most. Moreover, the more severe the SAGE degree is, the larger the amplitude variation of ρ will be. In general, the effect of the closer short circuit position to the minimum air-gap point on MFD is similar to the more severe SISC degree; this conclusion has been also obtained in . Although a generator may have the SAGE degree to different extent, such law still exists even when the SAGE degree is 2% or smaller. Consequently, the final MMF in the air gap for the combined fault is modified to
Since the impact of the closer short circuit position to the minimum air-gap point can be similarly treated as a severe SISC degree, for the sake of concision, in this paper, we mainly focus on the influence of the fault types and fault degrees on the EMT ripples, while the variation difference due to varied SISC position will not be fully expanded.
2.4. EMTs in Different Running Conditions
In this paper, we employ the principle of virtual power to derivate the EMT expression. The air-gap magnetic energy can be calculated via where L is the axial length of the stator core and R (αm, t) is the average radius of the air gap which can be written as
According to the MMF and PPUA derived previously (see (7), (14), and (16)), the EMTs in normal condition, SAGE fault, SISC fault, and the CF cases can be written aswherewhere e and ν represent the degrees of SAGE and SISC, respectively. The greater their values are, the more serious the fault will be. As indicated in (20), there is only a DC component in normal condition. However, the occurrence of SAGE will bring in extra 2nd harmonic to EMT, while SISC will primarily induce the 2nd and 4th harmonics. For the combined faults, an extra 6th harmonic will be newly produced in addition to the 2nd and 4th ones induced by the two single faults. Significantly, this ripple component difference can be employed to identify and diagnose the fault types. To be more distinct, the upper amplitude formulas, the influential factors, and the developing tendencies are listed in Table 1.
3. FEA and Experiment Validation
3.1. FEA and Experiment Setup
In this paper, the CS-5 prototype generator, which was designed and manufactured ourselves, is used for the 2D finite element calculation and the experimental test. The main parameters are shown in Table 2.
The generator rotor is kept stable by the bearing blocks, while the stator can be moved along both the radial and the axial directions. The movements are controlled by four dial indicators and confirmed by the high-precision filler gauges (see Figures 5(a) and 5(b)). Two indicators are for the radial movement while the other two are for the axial movement. On the generator, there is a plate with several short circuit taps in phases A and B (see Figures 5(c) and 5(d)). By setting varied SAGE values and connecting different short circuit taps, different running conditions including single and combined faults can be simulated.
The established finite element model is illustrated in Figures 5(e) and 5(f). During FEA, the rotor core as well as the field windings is shifted along X axis (to the right) to simulate SAGE (see Figure 5(e)). The SISC fault is simulated by both the external coupling circuit (the switch S-A1 in Figure 5(e)) and the physical model (the failure windings are divided into two parts: one is to simulate the healthy windings while the other is to simulate the short circuit part (see Figure 5(g)). By adjusting the turn numbers of LA1 (short circuit part) and LPhaseA1 (the healthy part) indicated in Figure 5(g), different SISC fault degrees can be simulated.
During the FEA calculation, the parameters are set to the same as the experiments. Four groups of experiments and FEA calculations as follows are carried out, respectively.(1)Normal condition: there is neither SAGE nor SISC. EMT values are collected as reference.(2)SAGE cases: 10%, 20%, and 30% SAGE conditions are set, respectively.(3)SISC cases: 3%, 6%, and 9% SISC conditions are taken, respectively.(4)CF cases: (1) combined faults of stable SAGE and varied SISC degrees, including 10% SAGE and 3%, 6%, and 9% SISC, 20% SAGE and 3%, 6%, and 9% SISC, and 30% SAGE and 3%, 6%, and 9% SISC and (2) combined faults of stable SISC and varied SAGE degrees, including 3% SISC and 10%, 20%, and 30% SAGE, 6% SISC and 10%, 20%, and 30% SAGE, and 9% SISC and 10%, 20%, and 30% SAGE are taken, respectively.
During the experiment, the electromagnetic torque can be usually measured in two means in general. One is the direct means which is primarily based on the torquemeter, and the other is the indirect means primarily based on the equivalent energy conversion calculation. In this paper, we employ the indirect means. The key thought is that the energy of the electromagnetic torque should be equivalent to the output power when the generator is under a steady running state, no matter healthy or faulty. The electromagnetic torque energy conversion model is illustrated in Figure 5(f), where the power of the rotor is FtRΔθ = Tω, and it should be equivalent to the power of the stator (UAIA + UBIB + UCIC) cos φ. Then, the EMT is calculated viawhere UA, UB, and UC are the three-phase voltages, IA, IB, and IC are the three-phase currents, n is the rotating speed of the rotor, and cosφ is the power factor.
3.2. Results and Discussion for SAGE Faults
The EMTs before and after SAGE by FEA are illustrated in Figure 6, while the results by experiment are indicated in Figure 7. It is shown that the occurrence of SAGE will increase the general EMT. The severer the SAGE degree is, the more the EMT increment will be.. For the spectrum components, both the FEA result and the experimental data show that the 2nd harmonic is the prominent one whose amplitude will be greatly intensified by SAGE. Although the amplitude values of the FEA and the experimental results are not exactly equal, their developing tendencies are generally consistent with each other. Also, both the FEA results and the experiment data follow the previously theoretical analysis conclusion (see equation (20)).
Actually, in addition to the 2nd harmonic, there are still 4th and 6th harmonics showing the increasing tendency. The reason for this is the presence of higher-order MMF harmonics. In Figure 2 and equation (20), only the 1st harmonic MMF is taken into account, while other higher-order harmonics are ignored. If we consider the higher-order harmonics such as the 3rd and 5th, the square of the MMF (see equation (17)) will lead to the appearance of the even EMT harmonics including 2nd, 4th, 6th, etc. Since the fundamental-frequency MMF component has the largest amplitude, the 2nd EMT harmonic which is mainly the square of the 1st MMF will be the most prominent one.
It is also suggested from Figures 6 and 7 that the occurrence of SAGE will generally increase the DC value of EMT (the curves are shifting upward to the larger value direction). This result well follows equation (14). According to (14), for the SAGE case, there is an extra factor 0.5δs2 which will of course increase the DC value (see equation (17)) as the occurrence and increment of SAGE.
Theoretically, there should be only even harmonics in both normal and SAGE cases since the MMF has only odd harmonics (the square operation in equation (17) will produce only even harmonics). However, the experimental result indicated in Figure 7 appears to be odd harmonic. This is primarily caused by two factors. On the one hand, the exciting current is transferred by the rectifier from the AC current and actually includes residual harmonics. In other words, the exciting current is not an ideal DC current. On the other hand, the actual rotor surface is not a strict circle. The slots and the uneven roundness inform a dynamic eccentricity which will produce odd harmonics in EMT (there will be an extra ωt factor in the permeance expression in equations (14) and (17)).
The FEA and the experimental results are compared in Figure 8. It is shown that although the test EMT amplitude increment cannot exactly match the FEA dada, the general developing tendencies are still in accordance. To confirm the data differences, we have redone the experiments and carried out the FEA calculation three times, but the data gap shows the same result. The probable reason may be related to both the manufacturing level and the calculation error.
3.3. Results and Discussion for SISC Faults
The time-domain waves as well as the spectra by FEA and experiment are illustrated in Figures 9 and 10, respectively. It is shown that the EMT ripples, especially the 2nd and the 4th harmonics, will be increased as SISC takes place. The more severe the SISC is, the larger the harmonic amplitudes will be.
As indicated in Figure 10, there are still odd harmonics in the experiment result. The very reason is just the same as previously mentioned in Section 3.2. In general, the FEA results are in accordance with the experiment data. Also, both the FEA and the experimental results are consistent with the theoretical analysis conclusion drawn from equation (20) and Table 1.
The result comparison between the FEA result and the test data is illustrated in Figure 11. Similar to SAGE case, the EMT amplitude increment by FEA and experiment still has a data gap, while the overall developing tendencies of the two data types still match with each other.
3.4. Results and Discussion for Combined Faults
The detailed EMT results under combined faults are illustrated in Figures 12–15. To be more comprehensive, two groups of combined faults are simulated by both FEA and experiment. One is to keep the SAGE degree as stable while changing the SISC degrees, and the other is to keep the SISC degree as stable while changing the SAGE degrees. More details about the fault simulation can be sound in.
As indicated in Figure 12, in the combined fault cases, the increment of SAGE will increase the 2nd, 4th, and 6th EMT harmonics. Moreover, comparing Figures 12(a)–12(c) with Figure 9(b), it shows that the harmonic amplitude increment under the combined fault is larger than that under the single SISC fault. On the contrary, the comparison between Figures 12(d)–12(f) and 6(b) suggests that the EMT harmonic increment in the combined fault cases is smaller than that of the single SAGE fault. The comparison result in the experimental data shows the similar result (see Figures 13, 10, and 7). The experimental results indicated in Figure 13 generally follow the developing tendencies obtained by FEA. The primary difference between the experimental and the FEA results is that there are odd harmonics, especially the 1st harmonic, existing in the spectra. The very reason, which has been mentioned previously, is that there is somewhat dynamic eccentricity due to the slots and the uneven roundness of the rotor, and the exciting current transferred by the rectifier is not standard DC current.
Comparing Figure 13 with Figure 12, it is shown that the ripple amplitudes by the experiment are smaller than those obtained by FEA. The reason for this mainly lies in the constraint difference between the FEA calculation and the experiment. During FEA, it is assumed under an ideal condition. For instance, the energy conversion is treated as no extra loss, while actually part of the energy will be consumed due to different kinds of losses. This means, the sum of the actual EMT energy and the loss energy equals to the input mechanical torque energy, while in FEA, the mechanical torque energy is generally treated as equivalent to the EMT energy. Therefore, the actual EMT as well as the ripples should be smaller than the FEA results.
More details about the difference between the FEA result and the experimental data are illustrated in Figures 14 and 15. Although there is still data gap between the FEA and the experimental results, the developing tendencies match well. Comparing Figure 14 with Figure 15, it is suggested that the gradients of the curves in Figure 15 (for the SISC increased cases) are larger than those in Figure 14. This means in the combined fault cases, SISC has a more intensive impact on the EMT ripple amplitude increment.
3.5. Comparison between Single and Combined Faults
As indicated in Figure 16, the gradients of the SISC curves are larger than those of the SAGE cases (the polylines for the SISC faults are sharper). This means the impact sensitivity of the single SISC fault is more intensive than that of the SAGE fault. Although the fault degrees for these two single faults are different, the sensitivity index is calculated with the same formula which can be written aswhere ΔT is the variation of EMT and ΔFd is the variation of the fault degree.
On the contrary, the comparison between the combined faults and the single SAGE faults indicated in Figure 17, where the curves marked with “SAGE 0%” denotes the single SISC fault, suggests that the combined faults generally have a more intensive sensitivity than the single SISC faults. Similarly, the comparison result illustrated in Figure 18 shows that the impact sensitivity of the combined faults is more intensive than that of the single SAGE faults (the single SAGE faults are the curves marked as “SISC 0%”).
Based on the previous analysis, the final sensitivity rank order among the single and the combined faults should be: (1) the combined faults, (2) the single SISC fault, and (3) the single SAGE fault.
In this paper, the EMT ripple characteristics in a synchronous generator caused by the single SAGE fault, the single SISC fault, and the combined fault composed of SAGE and SISC are analyzed and compared. The novelty of this paper primarily lies in the following three issues. (1) We investigate the EMT ripple properties not only under the single SAGE fault and the single SISC fault but also in the combined fault cases which are rarely taken into account by other studies. (2) We specifically design and manufacture a novel fault simulating generator which has been rarely reported and is able to simulate both the single and the combined faults composed of SAGE and SISC at the same time. Since it is hard to sample the EMT data from the actual generator set, the experimental data by the newly designed generator can be used as a reference for further related research studies. (3) We found out the impact sensitivity rank among the single and the combined faults on the EMT ripples. This is of significance and is potentially beneficial for the further studies related to the single and the combined faults mentioned in this paper.
The key conclusions drawn from the theoretical derivation, the FEA calculation, and the experiments are as follows.
The detailed EMT expressions for both the single and the combined faults are derived based on the analysis of the magnetic flux density variation. It is shown that the fundamental-frequency magnetomotive force will induce only the DC component in EMT in normal condition. However, the occurrence of SAGE will introduce the 2nd harmonic, while SISC fault will bring in both 2nd and 4th harmonics to EMT. Interestingly, the EMT characteristics of the combined faults are not the simple superposition of the two single faults. When the combined fault takes place, the new components include not only the 2nd and the 4th harmonics but also the 6th harmonic which is newly produced in addition to those induced by the two single faults. The more severe the fault degree is, the more the corresponding ripple amplitudes will be increased. Therefore, the harmonic component construction of the EMT can be employed as a tool for the identification and diagnosis among these three types of faults.
The finite element calculation and the experimental studies are carried out on the CS-5 prototype generator which is of two poles and 3000 rpm. This prototype generator is specifically designed and manufactured ourselves and is able to simulate both the single and the combined faults composed of SAGE and SISC. The FEA result as well as the experimental data generally follows the theoretical analysis conclusion. It is shown that the combined fault has the highest rank in the impact sensitivity on the EMT ripples, while following it is the single SISC fault. The single SAGE fault has the lightest impact on EMT among these three types of faults. This result is highly potential to be further applied in the future studies related to the mentioned faults in this paper.
The further work of this paper will lie in the EMT ripple restrain and the compensation method under the single and the combined faults composed of SAGE and SISC.
|SAGE:||Static air-gap eccentricity|
|SISC:||Stator interturn short circuit|
|FEA:||Finite element analysis|
|MFD:||Magnetic flux density|
|PPUA:||Permeance per unit area|
|Fr, Fr1+:||Rotor MMF in normal condition and SISC case, respectively|
|Fs, Fs1+:||Stator MMF in normal condition and SISC case, respectively|
|ω:||Electrical angular frequency|
|αm:||Mechanical circumferential angle|
|Is:||Short circuit current|
|Ir:||Current in the remaining route of short circuit turns|
|Ia1, Ia2:||Branch current of phase A|
|Ψ:||Internal power angle of generator|
|Fd:||Pulsating MMF amplitude generated by SISC|
|Fd+ , Fd−:||Forwardly and inversely rotating MMF amplitude|
|αsm:||Angle of central position of short circuit turns|
|If0:||Exciting current produced by exciting system|
|If2:||Induced current in exciting windings by SISC|
|N:||Number of exciting turns per pole|
|q:||Number of slots per pole per phase|
|:||Number of turns in serial for each phase|
|kw1:||Fundamental winding factor|
|f:||Fundamental electrical frequency|
|L:||Axial length of stator core|
|Fr1−, Fs1−:||Inversely rotating rotor and stator MMFs at ω, respectively|
|Fr3+, Fs3+:||Inversely rotating rotor and stator MMFs at 3ω, respectively|
|W:||Magnetic energy in air gap|
|R0:||Average radius of air gap|
|Rs:||Inner radius of stator core|
|Rr:||Outer radius of stator core|
|:||Average length of radial air gap|
|μ0:||Permeability of air|
|δs:||Relatively static air-gap eccentricity|
|ρ:||MMF proportion depending on SISC position|
|p:||Number of pole pairs|
|T, TNormal, TSAGE, TSISC, TCF:||EMT and EMT in normal, SAGE, SISC, and CF case, respectively|
|φ:||Power factor angle|
|P:||Output power of generator|
|Ua, Ub, Uc:||Three-phase voltages|
|Ia, Ib, Ic:||Three-phase currents.|
The data used to support the findings of this study are available from the corresponding author upon request (wangxiaolong126.com).
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
This work was supported in part by the National Natural Science Foundation of China (51777074), the Hebei Provincial Natural Science Foundation (E2019502047), the Chinese Fundamental Research Funds for the Central Universities (2018YQ03), and the 3rd Top Youth Talent Support Program of Hebei Province (2018)-27).
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