EFFECT OF STATIC ECCENTRICITY AND STATOR INTER-TURN SHORT CIRCUIT
COMPOSITE FAULT ON ROTOR VIBRATION CHARACTERISTICS OF GENERATOR
Yu-Ling He, Meng-Qiang Ke, Fa-Lin Wang, Gui-Ji Tang and Shu-Ting Wan
Department of Mechanical Engineering, North China Electric Power University, Baoding, China
E-mail: heyuling1@163.com
Received January 2014, Accepted June 2015
No. 14-CSME-13, E.I.C. Accession Number 3675
ABSTRACT
This paper investigates the radial rotor vibration characteristics under static air-gap eccentricity and stator
inter-turn short circuit composite faults. The air-gap magnetic flux density is firstly deduced to obtain the
unbalanced magnetic pull (UMP) on rotor. Then the rotor vibration characters, as well as the developing
trend between the faulty parameters and the vibration amplitudes, are analyzed. Finally, the experiments
are taken on a SDF-9 type simulating generator. It is shown that the radial deformation possibility, the 2nd,
4th, and 6th harmonic vibrations will be caused by the composite faults. Besides, the development of the
inter-turn short circuit, the increment of the static eccentricity, and the rise of the exciting current will all get
the deformation trend and the vibration amplitudes increased.
Keywords: turbo-generator; static air-gap eccentricity; stator inter-turn short circuit; composite fault; rotor
vibration.
LES EFFETS DES DÉFAILLANCES COMPOSITES D’EXCENTRICITÉ
STATIQUE ET DE COURT-CIRCUIT INTER-SPIRES DANS LE STATOR
SUR LA VIBRATION D’UN ROTOR DE GÉNÉRATEUR
RÉSUMÉ
Cet article porte sur la recherche des caractéristiques de vibration d’un rotor radial sous excentricité statique
de l’entrefer, de court-circuit inter-spires et de défaillances composites. Pour commencer, on détermine la
densité du flux de l’entrefer magnétique pour obtenir la force de l’attraction magnétique sur le rotor. Ensuite
les caractères des vibrations du rotor, de même que le développement de la tendance entre les paramètres
en défaut et les amplitudes de la vibration, sont analysés. Finalement, on fait des expériences de simulation
sur un générateur de type SDF-9. Il est démontré que les possibilités de déformation radiale, la 2ième, la
4ième et la 6ième, et des vibrations harmoniques seront causées par des défaillances composites. En outre,
le développement de court-circuit inter-spires, l’augmentation de l’excentricité statique, et la montée du
courant d’excitation feront tous augmenter la tendance de déformation et de vibration.
Mots-clés : générateur-turbo; excentricité statique de l’entrefer; court-circuit inter-spires du stator; défaillance composite; vibration du rotor.
Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 4, 2015
767
NOMENCLATURE
B
e
f
F
g
I
q
t
magnetic flux density (T)
electromotive force (V)
magnetomotive force (A), and frequency (Hz)
amplitude value of magnetomotive forces (A), and unbalanced magnetic pull (N)
radial air-gap length (m)
electricity (A)
magnetic pull per unit area (N/m2 )
time (s)
Greek symbols
α
ω
ψ
δ
Λ
µ
circumferential angle (rad)
angular frequency (rad/s)
internal power-angle of generator (rad)
relative air gap eccentricity (mm)
magnetic permeance (H)
magnetic permeability (H/m)
Subscripts
a
b
c
d
f
m
r
s
x
y
0
2
+
−
denotes phase A
denotes phase B
denotes phase C
denotes parameters related to stator interturn short circuit
denotes parameters with respect to exciting system
denotes mechanical parameter
denotes rotor
denotes static air-gap eccentricity or stator
denotes parameters in X direction
denotes parameters in Y direction
denotes constant parameter or referenced parameter
denotes 2nd harmonic component
denotes forwardly rotating direction
denotes inversely rotating direction
1. INTRODUCTION
In many cases, the air-gap of a generator is of asymmetry due to the disaccord of the rotor with the stator, and
the deformation of the stator core. This asymmetry, which is usually called static air-gap eccentricity, will
deteriorate the working condition of the bearing, deform the stator and the rotor, and damage the windings.
The definition and the preventive maintenance of the static eccentricity were early presented by Rosenberg in
1955 [1]. Thereafter, researchers carried out many significant studies and obtained a number of achievements
on this fault, such as the magnetic flux density and the force distribution [2], the branch inductance change
[3], the circulating current variation inside the parallel branch loop [4], the modeling and diagnosis method
[5, 6], the unbalanced magnetic pull (UMP) calculation [7–12], the stator and rotor vibration characteristics
[13], etc. Among these studies, most of them are about hydro-generators and motors, while few of them are
about doubly-fed induction generators or turbo-generators.
Besides the static air-gap eccentricity, the stator inter-turn short circuit will also take place from time to
time due to the insulation deterioration caused by over-voltage pulses, mechanical vibrations, and coupled
stresses. This fault greatly affects the performing security of the generator set and meanwhile needs a high
fixing cost. Therefore, much attention has always been paid to this fault since 1952 [14]. Penman et al. [15]
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Fig. 1. Static air-gap eccentricity of turbo-generator.
used the axial leakage flux to detect the location of the inter-turn shorted coils. Bo et al. [16] used a multichannel transient detection unit to obtain the transient current signals for the protection. Nandi and Toliyat
[17] monitored certain rotor-slot-related harmonics at the terminal voltage of the generator to detect the very
fault. Bouzid et al. [18] used a feed-forward multilayer-perceptron neural network (NN) which was trained
by back propagation to detect and locate automatically an inter-turn short-circuit fault in the stator windings
of the induction machine [18]. De Angelo [19] et al. used a state observer to get the vectorial residual
which was generated from a decomposition of the current estimation error for the fault detection. Dallas et
al. [20] analyzed the effect of the interturn stator fault on the currents and the electromagnetic torque via
FEM [20]. People now have developed different kinds of protection schemes and diagnosis methods for this
fault. However, most of the studies pay attention to the electrical parameter variation such as the voltages
and currents under the fault, while few of them takes the vibration characters into account.
By far, researchers have taken a lot of investigations either on the static air-gap eccentricity fault or the
stator inter-turn short circuit fault, while these two faults have been rarely considered occurring together.
Actually, since most of the generators have a static air-gap eccentricity, there is a composite fault occurring
when the stator inter-turn short circuit takes place. Thus, studying on the corresponding characteristics
under this composite fault is of significance. The intent of this paper is to investigate the rotor vibration
characteristics of turbo-generator when the static air-gap eccentricity fault and the stator inter-turn short
circuit fault occurring at the same time, and analyze the effect of the faulty parameter development on the
vibration characteristics.
2. THEORETICAL ANALYSIS
2.1. Magnetic Flux Density Analysis
The magnetic flux density is composed of the unit permeance and the magnetomotive force (MMF) through
a multiplying operation. When under the composite fault, the static air-gap eccentricity mainly affects the
unit permeance while the stator inter-turn short circuit primarily affects the MMF. Further, the magnetic flux
density will be affected by either of these two faults.
The static air-gap eccentricity of a turbo-generator is indicated in Fig. 1. For the sake of convenience, the
X axis of the coordinate system is set along the direction that crosses the minimum air-gap, and the radial
length of the air-gap can be expressed as
g(αm ) = g(1 − δs cos αm )
(1)
where αm is the angle to indicate the circumferential position of the air-gap, g is the average value of the
radial air-gap length, and δs is the relatively static air-gap eccentricity. Ignoring the higher harmonics, the
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Fig. 2. Short circuit of the stator winding.
air-gap permeance per unit area after the expansion by power series is
Λ(αm ) =
µ0
= Λ0 (1 + δs cos αm ) = Λ0 + Λs cos αm
g(αm )
(2)
where Λ0 is the constant component, and Λs = Λ0 δs is the component caused by the static air-gap eccentricity.
When the stator inter-turn short circuit takes place, there will be an extra circulating current Id induced, as
indicated in Fig. 2. This circulating current will form a pulsating magnetic field at f (the power frequency),
and the pulsating MMF can be written as
fd (αm ,t) = Fd cos ωt cos(αm − αm0 ) = Fd+ cos[ωt − (αm − αm0 )] + Fd− cos[ωt + (αm − αm0 )]
(3)
where αm has the same meaning as previously mentioned, αm0 is the circumferential angle to indicate the
central position of the short circuit turns, and Fd+ = Fd− = Fd /2. As indicated in the equation, there are two
items, of which the first one stands for the MMF that synchronously rotates as the rotor and will therefore
not induce electromotive force (EMF) in the rotor windings. However, the MMF indicated by the second
item reversely rotates as the rotor and will induce an extra EMF at 2ω in the rotor windings. Then the
exciting current can be written as
I f = I f 0 + I f 2 cos 2ωt,
(4)
where I f 0 is the current generated by the exciting system, while I f 2 is the current induced by the stator
inter-turn short circuit. The amplitude of the main magnetic flux density produced by the exciting currents
is
Bm (t) = (I f 0 + f f 2 cos 2ωt)N(Λ0 + Λs cos αm )
(5)
where N is the number of the exciting turns for each pole. Further, the EMFs induced in the stator windings
for each phase are


e (α ,t) = (I f 0 + I f 2 cos 2ωt)KN(Λ0 + Λs cos αm ) cos ωt

 a m


= I f 0 KN(Λ0 + Λs cos αm ) cos ωt + 0.5I f 02 KN(Λ0 + Λs cos αm ) cos ωt + 0.5I f 2 KN(Λ0 Λs cos αm ) cos 3ωt





eb (αm ,t) = (I f 0 + I f 2 cos 2ωt)KN(Λ0 + Λs cos αm ) cos(ωt − 120◦ )



= I f 0 KN(Λ0 + Λs cos αm ) cos(ωt − 120◦ ) + 0.5I f 2 KN(Λ0 + Λs cos αm ) cos(ωt + 120◦ )
+ 0.5I f 2 KN(Λ0 + Λs cos αm ) cos(3ωt − 120◦ )






ec (αm ,t) = (I f 0 + I f 2 cos 2ωt)KN(Λo + Λs cos αm ) cos(ωt + 120◦ )




= I f 0 KN(Λ0 + Λs cos αm ) cos(ωt + 120◦ ) + 0.5I f 2 KN(Λ0 + Λs cos αm ) cos(ωt − 120◦ )


 + 0.5I KN(Λ + Λ cos α ) cos(3ωt + 120◦ )
f2
0
s
m
(6)
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Fig. 3. Relation between rotor MMF and stator MMF.
where the coefficient K = 2qwc kw1 τl f . It can be seen in the formula that each EMF has three items, of
which the first ones will form a MMF synchronously rotating as the rotor at 1ω, the second ones will form
a MMF reversely rotating as the rotor at 1ω, and the third ones will form a MMF forwardly rotating as the
rotor at 3ω. Then the composite MMF of the air-gap can be written as
f (αm ,t) = (I f 0 + I f 2 cos 2ωt)N cos(ωt − αm + ψ + τ/2) + Fs1+ cos(ωt − αm )
+ Fs1− cos(ωt + αm ) + Fs3+ cos(3ωt − αm )
(7)
where FS1+ and FS3+ are the amplitudes of the MMFs which forwardly rotates at 1ω and 3ω, respectively,
FS1− is the amplitude of the MMF that reversely rotates at 1ω, and ψ is the internal power-angle of the
generator. The vector relation between the rotor MMF and the stator MMF is indicated in Fig. 3.
2.2. UMP Analysis
Based on Eqs. (2) and (7), the magnetic pull per unit area under the composite fault can be obtained through
q(αm ,t) =
B(αm ,t)2 [ f (αm ,t)Λ(αm )]2
=
2µ0
2µ0
(8)
and the UMPs in the horizontal direction and the vertical direction on the rotor are
Z 2π
FX =
0
q(αm ,t) cos αm dαm
2
2
2
= LRπ{[(I 2f 0 N 2 + I 2f 2 N 2 + Fs1+
+ Fs1−
+ Fs1+ Fs1− + Fs3+
)Λ0 Λs − 0.5I f 0 I f 2 N 2 Λ0 Λs cos 2ψ
2
+ (I f 2 NFs1− + 2I f 0 NFs1− + 0.5I f 2 NFs1+ )Λ0 Λs cos(ψ + π/2)] + [(0.5Fs1+
+ 2Fs1+ Fs3+ + 2Fs1+ Fs1−
2
+ Fs1− Fs3+ + 2i f 0 I f 2 N 2 + 0.5Fs1−
)Λ0 Λs cos 2ωt + (I f 0 NFs1− + I f 2 NFs1+ + 0.5I f 2 NFs1−
+ 0.5I f 2 NFs3+ )Λ0 Λs cos(2ωt + ψ + 0.5π) + (2I f 0 NFs3+ + I f 2 NFs1+
+ 0.5I f 2 NFs1− )Λ0 Λs cos(2ω − ψ − 0.5π) − (0.5I 2f 0 N 2 Λ0 Λs
+ 0.25I 2f 2 N 2 Λ0 Λs ) cos(ωt + 2ψ) − 0.25I 2f 2 N 2 Λ0 Λs cos(2ωt − 2ψ)]
+ [(0.5I 2f 2 N 2 + Fs1+ Fs3+ + 2Fs1− Fs3+ )Λ0 Λs cos 4ωt − 0.5I f 0 I f 2 N 2 Λ0 Λs cos(4ωt + 2ψ)
+ (I f 0 NFs3+ + 0.5I f 2 NFs1+ + I f 2 NFs1− )Λ0 Λs cos(4ωt + ψ + π/2)
+ I f 2 NFs3+ Λ0 Λs cos(4ω − ψ − π/2)]
2
+ [(0.5Fs3+
Λ0 Λs cos 6ωt + 0.5I f 2 NFs3+ Λ0 Λs cos(6ωt + ψ + π/2)
− 0.25I 2f 2 N 2 Λ0 Λs cos(6ωt + 2ψ)]}/2µ0
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(9a)
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Table 1. The UMP amplitude formulas under the composite fault and their influential factors
Z 2π
FY =
0
q(αm ,t) sin αm dαm
= LRπ{[−0.5I f 0 I f 2 N 2 Λ0 Λs sin 2ψ + (I f 0 NFs1− + 0.5I f 2 NFs1+ )Λ0 Λs sin(ψ + π/2)]
2
2
+ [(0.5Fs1+
+ Ff 1− Fs3+ − 0.5Fs1−
)Λ0 Λs sin 2ωt
− 0.5I f 2 NFs1− Λ0 Λs sin(2ωt − ψ − π/2) + (I f 0 NFs1+ + 0.5I f 2 NFs1−
+ 0.5i f 2 NFs3+ )Λ0 Λs sin(2ωt + ψ + π/2) − (0.5I 2f 0 N 2 Λ0 Λs + 0.25I 2f 2 N 2 Λ0 Λs ) sin(2ωt + 2ψ)
+ 0.25I 2f 2 N 2 Λ0 Λs sin(2ωt − 2ψ)] + [Fs1+ Fs3+ Λ0 Λs sin 4ωt − 0.5I f 0 I f 2 N 2 Λ0 Λs sin(4ωt + 2ψ)
2
+ (I f 0 NFs3+ + 0.5I f 2 NFs1+ )Λ0 Λs sin(4ωt + ψ + π/2)] + [0.5Fs3+
Λ0 Λs sin 6ωt
+ 0.5I f 2 NFs3+ Λ0 Λs sin(6ωt + ψ + π/2) − 0.25I 2f 2 N 2 Λ0 Λs sin(6ωt + 2ψ)]}/2µ0
(9b)
2.3. Analysis on Rotor Vibration Characteristics
As indicated in Eqs. (9a) and (9b), there is a DC component and three pulsating components respectively
at 2ω, 4ω and 6ω in the UMP formulas. The DC component will not cause vibrations. However, it will
make the rotor form a radial deflection in some extent. The three pulsating components will bring in radial
vibrations respectively at 2ω, 4ω and 6ω to the rotor.
To further study the rotor vibration characteristics, the upper-bound amplitude formulas and the influential
factors of each UMP component are drawn up in Table 1, where f ds is the stator inter-turn short circuit
degree.
As indicated in Table 1, the static air-gap eccentricity and the stator inter-turn short circuit will affect the
amplitudes of the 2nd, 4th, and 6th harmonic components, while the exciting current I f 0 will just mainly
affect the 2nd and 4th harmonic components.
As the static air-gap eccentricity increases, the permeance component Λs will be increased. Since the
factor Λs is contained in the expressions of the 2nd, 4th and 6th harmonic UMP components, the increment
of the static air-gap eccentricity will cause the 2nd, 4th and 6th harmonic vibrations increased.
As the stator inter-turn short circuit develops, the induced current I f 2 will be increased. Further, FS1− and
FS3+ will also be enlarged. According to Eqs. (9a) and (9b) and Table 1, the 2nd, 4th and 6th harmonic UMP
components all have the influential factors I f 2 and FS3+ . Therefore, the deterioration of the stator inter-turn
short circuit will make the 2nd, 4th and 6th harmonic vibrations increased.
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In addition, it can be seen in Eqs. (9a) and (9b) and Table 1 that the 2nd and 4th harmonic UMP components both have the factor I f 0 in their expressions. Thus, as the exciting current I f 0 increases, the 2nd and
4th harmonic vibrations will be increased. For the 6th harmonic component, the factor I f 0 does not exist in
its UMP expression and amplitude formula. However, the rise of the exciting current will increase the extra
circulating current Id (see Fig. 2). Then I f 2 (see Eqs. (3) and (4)) and FS3+ (see Eqs. (6) and (7)) will be
enlarged. Therefore, the 6th harmonic vibration will also be increased.
Besides the pulsating components, the UMPs also have DC components which contain the factors
I f 0 , Λs , I f 2 , FS1− and FS3+ . Therefore, the rise of the exciting current, the increment of the static air-gap
eccentricity, and the development of the stator inter-turn short circuit will all make the DC components
increased. Further, the radial rotor deformation will be increased.
3. EXPERIMENT RESEARCH
3.1. Equipment and Method
The experiments are taken on a SDF-9 type non-salient pole fault simulating generator, as indicated in
Fig. 4a. The generator is of double-layer pitch-shortening windings, and the corresponding parameters are
as follows:
• rated capacity: 7.5 kVA
• rated exciting current: 1.5 A
• number of pole pairs: p = 1
• radial air-gap length: 0.8 mm
• pitch-shortening value: 0.966
• factor of pitch to polar distance: 0.83
• number of turns for each phase: 100
• number of exciting winding turns for each pole: 480
On the generator there is a lug plate with 3, 8 and 15% inter-turn short circuit taps in phase A and 2,
6 and 12% short circuit taps in phase B, respectively, as indicated in Fig. 4b. The stator of the generator
can be horizontally moved along the radial direction while the rotor is fixed to the floor pedestal. There
are two adjusting screws respectively in the front and at the back of the generator. The stator movement
can be performed by the four adjusting screws and controlled by the two dial indicates so that the static
air-gap eccentricity can be simulated, as shown in Figs. 4c and d. Different composite fault conditions can
be simulated by setting different static air-gap eccentricities and connecting different short circuit taps.
During the experiment, the generator is connected to the power grid. The general testing method is
shown in Fig. 5a. A velocity sensor of CD-21S type (made by Beijing Vibration Instrument Co., Ltd,
and the sensitivity is 30 mV/mm/s) is set to the bearing pedestal to test the rotor vibration, as indicated in
Fig. 5b. Meanwhile, a U60116C type data collector made by Beijing BOPU Co., Ltd is used. The sampling
frequency is set at 10 kHz.
The detailed experiment schemes are as follows:
1. Normally performing: there is neither static air-gap eccentricity nor stator inter-turn short circuit. The
exciting current I f 0 = 0.8 A. The rotor vibration data is collected as reference samples.
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Fig. 4. SDF-9 type fault simulating generator set: (a) general outlook of the generator, (b) method to set stator interturn short circuit, and (c) and (d) method to set static air-gap eccentricity.
2. Composite fault performing with certain static air-gap eccentricity and different short circuit degrees:
0.1 mm static air-gap eccentricity is firstly set, and then the taps A1 –A11 , A1 –A12 and A1 –A13 are
respectively connected to set 3, 8 and 15% stator inter-turn short circuit. The exciting current I f 0 =
0.8 A.
3. Composite fault performing with certain short circuit degree and different static air-gap eccentricity
conditions: the taps A1 –A11 are firstly connected to set 3% stator inter-turn shot circuit, and then 0.1,
0.3 and 0.5 mm static air-gap eccentricities are respectively set. The exciting current I f 0 = 0.8 A.
4. Composite fault performing with different exciting currents: 0.1 mm static air-gap eccentricity and
3% stator inter-turn short circuit are firstly set, and then the exciting currents are set to 0.8, 0.9 and
1.0 A each time, respectively.
3.2. Results and Discussion
3.2.1. Rotor vibration in normal condition
According to Wan et al. [13], the rotor should theoretically have no radial vibrations in normal condition.
However, the experiment shows that the rotor has radial vibration components of each harmonic. This is
mainly caused by the asymmetry inside the generator and the random factors surrounding the generator.
To remove the influence of the mismatch between the theoretical analysis and the experimental result
on the further study, the amplitudes of each vibration component in normal condition are treated as the
null-shifts of the generator system. Then the tested vibration amplitudes under the composite faults are
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Fig. 5. Rotor vibration test method: (a) the general testing method, and (b) method to set the velocity sensor.
Fig. 6. Rotor vibration spectrum under the composite faults with 0.1 mm static eccentricity and different short circuit
degrees: (a) normal condition, (b) 3% short circuit, (c) 8% short circuit, and (d) 15% short circuit.
subtracted by the null-shifts, so that the true vibration values caused by different composite fault conditions
can be obtained.
3.2.2. Rotor vibration under composite faults with different short circuit degrees
The radial rotor vibration spectrum under the composite faults with 0.1 mm static air-gap eccentricity and
different stator inter-turn short circuit degrees is shown in Fig. 6, and the corresponding amplitudes of each
component are shown in Table 2.
It can be seen in Fig. 6 and Table 2 that the 2nd, 4th, and 6th harmonic components have a more obvious
change than other components. This indicates that the composite fault will mainly cause the rotor to vibrate
at 2 f , 4 f and 6 f . Moreover, as the stator inter-turn short circuit develops, the 2nd, 4th, and 6th harmonic
components have a regular increment while other components do not.
As is known, the radial vibration is the response of the rotor system to the exciting UMP, and the increment
of the UMP will obviously intensify the rotor vibration. However, this does not mean that, the UMP is the
only reason for what the vibration increases. In other words, the UMP is a sufficient condition but not a
necessary condition to cause the rotor vibration. Since the vibration is a mechanical parameter which is
hard to be calculated in theory, it is not sensible to directly compare the theoretical vibration with the tested
one. The middle course to solve the problem for a verification is to compare the calculated UMP developing
curve with the tested vibration tendency cure, so that the correlation between these two can be indicated.
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Table 2. The rotor vibration amplitudes of each component.
Performing condition
Vibration speed (mm/s)
50 Hz 100 Hz 150 Hz 200 Hz 250 Hz
Normal
5.0
0.9
0.4
1.15
1.1
Composite fault with 3% short circuit
5.2
4.2
0.6
0.6
0.5
Composite fault with 8% short circuit
5.2
4.8
0.6
0.9
0.6
Composite fault with 15% short circuit
5.5
5.6
0.7
1.5
0.6
300 Hz
0.3
0.4
0.5
0.7
Fig. 7. Comparison between theoretical UMP and rotor vibration increment percentage under the composite faults
with different short circuit degrees: (a) 2nd harmonic comparison, (b) 4th harmonic comparison, and (c) 6th harmonic
comparison.
Relating to the SDF-9 type fault simulating generator, the amplitudes of the UMP components are calculated and compared with the tested rotor vibration, as shown in Fig. 7. The comparison shows that the
two curves generally have the same increasing trend except at the 4th harmonic. The mismatch appearing
at the 4th harmonic is probably because: (1) the generator has somewhat initial asymmetry in the winding
distribution, and (2) the 3% stator inter-turn short circuit offsets this asymmetry to a larger extent.
3.2.3. Rotor vibration under composite faults with different eccentricity conditions
The rotor vibration spectrums under the composite faults with different static air-gap eccentricity conditions
are shown in Fig. 8, and the corresponding amplitudes of each frequency are shown in Table 3.
As indicated in Fig. 8 and Table 3, the components except the 2nd, 4th, and 6th harmonics do not have an
obvious change in the vibration amplitudes. This is in accordance with the results in 3.2.2. Moreover, it still
shows that as the static air-gap eccentricity increases, the amplitudes of the 2nd, 4th, and 6th components
will also be increased, especially the 2nd component. Its vibration amplitude increments from 0.1 mm static
eccentricity to 0.3 and 0.5 mm static eccentricities are respectively 2.9 and 5.4 mm/s. The amplifications are
69 and 128.6%, respectively.
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Fig. 8. Rotor vibration spectrum under composite faults with 3% short circuit and different static eccentricities:
(a) normal condition, (b) 0.1 mm static eccentricity, (c) 0.3 mm static eccentricity, and (d) 0.5 mm static eccentricity.
Fig. 9. Comparison between theoretical harmonic UMP and tested rotor vibration increment percentage with different
eccentricities: (a) 2nd harmonic comparison, (b) 4th harmonic comparison, and (c) 6th harmonic comparison.
The calculated 2nd, 4th, and 6th UMP amplitudes in the X direction for the SDF-9 type fault simulating
generator are compared with the tested vibration amplitudes, as indicated in Fig. 9, where the 4th harmonic
vibration is decreased when the static air-gap eccentricity is increased from 0 to 0.1 mm. This might be
mainly caused by the inner asymmetry of the generator, and the small eccentricity partly compensates this
asymmetry. However, as the static eccentricity increment goes on, the vibration amplitudes are increased to
match the change of the calculating UMP. Generally, except the singular point in the 4th harmonic component, the calculated and the tested curves have the same increasing trend.
In addition, the UMP values in Fig. 8 are larger than those in Fig. 6. Relating to the tested results, the
vibration amplitudes in Table 3 are also larger than those in Table 2. This can indicate that the calculated
data and the tested data are reasonable.
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Table 3. Rotor vibration amplitudes under composite faults with 3% short circuit and different static eccentricities.
Performing condition
Vibration speed (mm/s)
50 Hz 100 Hz 150 Hz 200 Hz 250 Hz 300 Hz
Normal
5.0
0.9
0.4
1.15
1.1
0.3
Composite fault with 0.1 mm eccentricity
5.2
4.2
0.6
0.6
0.5
0.4
Composite fault with 0.3 mm eccentricity
5.4
7.1
0.8
1.1
0.5
0.6
Composite fault with 0.5 mm eccentricity
5.4
9.6
0.8
1.8
0.7
0.9
Fig. 10. Rotor vibration spectrum with different exciting currents: (a) normal condition with 0.8 A, (b) composite fault
with 0.8 A, (c) composite fault with 0.9 A, and (d) composite fault with 1.0 A.
Table 4. Rotor vibration amplitudes under the composite faults with different exciting currents.
Performing condition
Vibration speed (mm/s)
50 Hz 100 Hz 150 Hz 200 Hz 250 Hz 300 Hz
Normal condition with 0.8A exciting current
5.0
0.9
0.4
1.15
1.1
0.3
Composite fault with 0.8A exciting current
5.2
4.2
0.6
0.6
0.5
0.4
Composite fault with 0.9A exciting current
5.4
5.9
0.5
1.0
0.6
0.55
Composite fault with 1.0A exciting current
5.3
7.5
0.6
1.6
0.5
0.8
3.2.4. Rotor vibration under composite faults with different exciting currents
The rotor vibration spectrum under the composite faults with different exciting currents is shown in Fig. 10,
and the corresponding amplitudes of each frequency are shown in Table 4, while the comparison between
the theoretical UMP and the tested vibration is indicated in Fig. 11.
The data in Figs. 10 and 11 and in Table 4 shows that as the exciting current increases, the 2nd, 4th and
6th harmonic components of the rotor vibration will be increased while other components change little. The
experiment result complies with the previous theoretical analysis.
4. CONCLUSIONS
This paper investigates the rotor vibration characteristics under the static air-gap eccentricity and stator interturn short circuit composite faults, and meanwhile analyzes the effect of the faulty parameter development
on the rotor vibration characteristics. The corresponding experimental study is also provided. The main
results drawn from the theoretical and experimental research can be given as follows:
1. Theoretically, the rotor should have no radial vibration in normal condition. However, vibration of
each harmonic exists due to the asymmetry inside the generator and the random factors surrounding
the generator.
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Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 4, 2015
Fig. 11. Comparison between theoretical 2nd, 4th and 6th harmonic UMP and tested rotor vibration increment percentage with different exciting currents: (a) 2nd harmonic comparison, (b) 4th harmonic comparison, and (c) 6th harmonic
comparison.
2. When the static air-gap eccentricity and stator inter-turn short circuit composite fault happens, there
will be 2nd, 4th and 6th harmonic vibrations caused. Meanwhile, the rotor will also have a radial
deformation possibility.
3. Under the composite fault, the development of the stator inter-turn short circuit, the increment of the
static eccentricity, and the rise of the exciting current will all cause the deformation trend and the 2nd,
4th and 6th harmonic amplitudes of the rotor vibration increased.
The investigation proposed in this paper is probable to be used as a basis for the corresponding monitoring
applications, for example, the rotor vibration characters of the composite fault (vibrates at 2 f , 4 f , and 6 f )
are different from that of the single static air-gap eccentricity fault (vibrates only at 2 f [13]) and the single
stator inter-turn short circuit fault (vibrates only at f [21]). So the proposed work can be beneficial to the
failure criterion and the diagnosis of the static air-gap eccentricity and stator short circuit composite fault of
turbo-generator.
ACKNOWLEDGEMENTS
This work is supported by the National Natural Science Foundation of China (No. 51307058 and No.
51177046), the Natural Science Foundation of Hebei Province, China (E2014502052, E2015502013), and
the Chinese Fundamental Research Funds for the Central Universities (2015ZD27).
REFERENCES
1.
Rosenberg, L.T., “Eccentricity, Vibration, and Shaft Currents in Turbine Generators”, Transactions of the American Institute of Electrical Engineers, Vol. 74, pp. 38–41, 1955.
Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 4, 2015
779
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
780
DeBortoli, M.J., Salon, S.J., Burow, D.W. and Slavik, C.J., “Effects of rotor eccentricity and parallel windings
on induction machine behavior: a study using finite element analysis”, IEEE Transactions on Magnetics, Vol. 29,
No. 2, pp. 1676–1682, 1993.
Jiahui, Z. and Arui, Q., “Inductance calculation of large salient-pole synchronous generator with air-gap eccentricity”, in Proceedings 2006 12th Biennial IEEE Conference on Electromagnetic Field Computation, pp. 264,
2006.
Shuting, W. and Yuling, H., “Analysis on stator circulating current characteristics of turbo-generator under eccentric faults”, IEEE 6th International Conference on Power Electronics and Motion Control, pp. 2062–2067,
2009.
Sundaram, V.M. and Toliyat, H.A., “Diagnosis and isolation of air-gap eccentricities in closed-loop controlled
doubly-fed induction generators”, in Proceedings 2011 IEEE International Conference on Electric Machines
and Drives, pp. 1064–1069, 2011.
M’hamed, D. and Marques Cardoso, A.J., “Airgap-eccentricity fault diagnosis in three-phase induction motors,
by the complex apparent power signature analysis”, IEEE Transactions on Industrial Electronics, Vol. 55, No. 3,
pp. 1404–1410, 2008.
Zarko, D., Ban, D., Vazdar, I. and Jaric, V., “Calculation of unbalanced magnetic pull in a salient-pole synchronous generator using finite-element method and measured shaft orbit”, IEEE Transactions on Industrial
Electronics, Vol. 59, No. 6, pp. 2536–2549, 2012.
Wang, L., Cheung, Richard W., Zhiyun, Ma, et al., “Finite-element analysis of unbalanced magnetic pull in a
large hydro-generator under practical operations”, IEEE Transactions on Magnetics, Vol. 44, No. 6, pp. 1558–
1561, 2008.
Keller, S., Tu Xuan, M., Simond, J.-J. and Schwery, A., “Large low-speed hydro-generators-unbalanced magnetic pulls and additional damper losses in eccentricity conditions”, IET Electric Power Applications, Vol. 1,
No. 5, pp. 657–664, 2007.
Zarko, D., Ban, D., Vazdar, I. and Jarica, V., “Calculation of unbalanced magnetic pull in a salient-pole synchronous generator using finite-element method and measured shaft orbit”, IEEE Transactions on Industrial
Electronics, Vol. 59, No. 6, pp. 2536–2549, 2012.
Perers, R., Lundin, U. and Leijon, M., “Saturation effects on unbalanced magnetic pull in a hydroelectric generator with an eccentric rotor”, IEEE Transactions on Magnetics, vol. 43, no. 10, pp. 3884–3890, 2007.
Pennacchi, P. and Frosini, L., “Dynamical behaviour of a three-phase generator due to unbalanced magnetic
pull”, IEE Proceedings on Electric Power Applications, Vol. 152, No. 6, pp. 1389–1400, 2005.
Shuting, Wan and Yuling, H., “Investigation on stator and rotor vibration characteristics of turbo-generator under
air gap eccentricity fault”, Transactions of the Canadian Society for Mechanical Engineering, Vol. 35, No. 2,
pp. 161–176, 2011.
Cameron, A.W.W., “Diagnoses of A-C generator insulation condition by nondestructive tests [includes discussion]”, Power apparatus and systems, Part III. Transactions of the American Institute of Electrical Engineers,
Vol. 71, No. 1, pp. 263–274, January 1952.
Penman, J., Sedding, H.G., Lloyd, B.A. and Fink, W.T., “Detection and location of interturn short circuits in
the stator windings of operating motors”, IEEE Transactions on Energy Conversion, Vol. 9, No. 4, pp. 652–658,
December 1994.
Bo, Z.Q., Chen, Z., Redfern, M.A., et al., “A nondifferential protection scheme for generator stator windings”,
IEEE Power Engineering Review, Vol. 21, No. 1, pp. 49–51, January 2001.
Nandi, S. and Toliyat, H.A., “Novel frequency-domain-based technique to detect stator interturn faults in induction machines using stator-induced voltages after switch-off”, IEEE Transactions on Industry Applications,
Vol. 38, No. 1, pp. 101–109, January/February 2002.
Bouzid, M., Champenois, G., Bellaaj, N.M., et al., “An effective neural approach for the automatic location
of stator interturn faults in induction motor”, IEEE Transactions on Industrial Electronics, Vol. 55, No. 12,
pp. 4277–4289, 2008.
De Angelo, C.H., Bossio, G.R., Giaccone, S.J., et al., “Online model-based stator-fault detection and identification in induction motors”, IEEE Transactions on Industrial Electronics, Vol. 56, No. 11, pp. 4671–4680,
2009.
Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 4, 2015
20.
21.
Dallas, S.E., Safacas, A.N. and Kappatou, J.C., “Interturn stator faults analysis of a 200-MVA hydrogenerator
during transient operation using FEM”, IEEE Transactions on Energy Conversion, Vol. 26, No. 4, pp. 1151–
1160, 2011.
Shu-ting, W., He-ming, L., Zhao-feng, X. and Yong-gang, L., “Analysis on generator vibration characteristic on
stator winding inter-turn short circuit fault”, Proceedings of the CSEE, Vol. 24, No. 4, pp. 157–161, 2004.
Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 4, 2015
781