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Electric Power Systems Research 81 (2011) 751–766
Contents lists available at ScienceDirect
Electric Power Systems Research
journal homepage: www.elsevier.com/locate/epsr
Experimental validation of doubly fed induction machine electrical faults
diagnosis under time-varying conditions
Yasser Gritli a,b , Andrea Stefani b , Claudio Rossi b , Fiorenzo Filippetti b,∗ , Abderrazak Chatti a
a
b
National Institute of Applied Sciences and Technology, Tunis, Tunisia
University of Bologna, Department of Electrical Engineering, Bologna, Italy
a r t i c l e
i n f o
Article history:
Received 3 May 2010
Received in revised form 9 October 2010
Accepted 9 November 2010
Keywords:
Doubly fed induction machine
Time-varying conditions
Frequency sliding
Wavelet transforms
a b s t r a c t
This paper investigates a new diagnosis technique for incipient electrical faults in doubly fed induction
machine for wind power systems under time-varying conditions. The proposed method is based on
currents frequency sliding pre-processing, and discrete wavelet transform thereby. The mean power
calculation of wavelet signals, at different resolution levels, is introduced as a dynamic fault indicator
for quantifying the fault extents. The approach effectiveness is proved for both stator and rotor faults
under speed and fault varying conditions. Simulation and experimental results show the validity of the
developed method, leading to an effective diagnosis procedure for stator and rotor faults in doubly fed
induction machines.
© 2010 Published by Elsevier B.V.
1. Introduction
For many modern large wind farms, wind turbines equipped
with doubly fed induction machine (DFIM) are a well established
technology. Different diagnosis methods have been proposed for
wind turbines using DFIM [1–4]. Investigations on different failure
modes in variable speed induction motors done by industrials and
experts have revealed that 45% of motor failures are related to the
stator and rotor parts [5]. A detailed analysis of this type of faults
can be found in [6]. More concretely, each electrical fault that occurs
in the stator/rotor side of a DFIM (short circuits or increasing resistance) give rise to a phase dissymmetry because the impedances of
the windings are not longer equal or because of a distortion in the
airgap flux. Thus the simplest way to emulate a phase unbalance
in order to test the effectiveness of diagnosis methods is to insert
an additional resistance in series to one phase stator/rotor winding
[4] to provoke a phase unbalance.
Increasing resistance, or as commonly known in the literature
“High-Resistance Connections”, is a common problem that can
occurs in any power connections of industrial motor [4,7]. This
failure mode can be initiated by gradual abrasion, corrosion and
∗ Corresponding author. Tel.: +39 051 20 93 564; fax: +39 051 209 3588.
E-mail addresses: [email protected] (Y. Gritli),
[email protected] (A. Stefani), [email protected]
(C. Rossi), fiorenzo.fi[email protected] (F. Filippetti),
[email protected] (A. Chatti).
0378-7796/$ – see front matter © 2010 Published by Elsevier B.V.
doi:10.1016/j.epsr.2010.11.004
fretting, leading generally to a local heating which in turn leads
to insulation damage. Consequently if the evolution of this type of
faults is not detected at an incipient stage, its propagation can lead
to more serious failure modes. Several diagnostic methods, such as
motor current signature analysis (MCSA), and more recently, flux
signature analysis (FSA) and rotor modulation signature analysis
(RMSA) have been proposed to detect stator and rotor faults [8–13].
Depending on wind speed, the induction machine operates continuously in time-varying condition. In this context, the classical
application of Fourier analysis (FA) for processing the above signals fails as slip and speed vary. Thus the fault components are
spread in a bandwidth proportional to the variation. Among different solutions, high resolution frequency estimation [12] and more
recently signal demodulation (SD) technique [13] have been developed to reduce the effect of the non periodicity on the analyzed
signals. These techniques, based on FA gives high quality discrimination between healthy and faulty conditions but don’t provide
time-domain information. This shortcoming in the Fourier analysis can be overcome to some extent by analyzing a small section
of the signal at a time by means of short-time Fourier transform
(STFT). This method was widely used to detect stator and rotor failures in induction motor. As an advanced use of the FFT algorithm, it
assumes local periodicity within continuously translated time window. However the fixed size of the chosen window, the difficulties
in quantifying the faults extent and the high computational cost
required to obtain a good resolution still remain the major drawbacks of this technique [14–16]. wavelet transform (WT), on the
other hand, provides greater resolution in time for high frequency
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Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
components of a signal and greater resolution in frequency for low
frequency components. In this sense, wavelets have a window that
automatically adjusts to give the appropriate resolution developed
by its approximation and detail signals.
Motivated by the above proprieties, WT was used with different
approaches for the diagnosis of anomalies in induction machine
such as: undecimated discrete wavelet transform [17], wavelet
ridge method [18] and wavelet coefficients analysis [1,19,20] for
stator and rotor fault detection. More intensive research efforts
have been focused on the use of approximation and detail signals
for extracting the contribution of fault frequency components in
case of broken bars [14,21–23], inter-turn short-circuits [14,22],
mixed eccentricity [22,24], and increasing resistance in stator phase
[4,17,25,26] or rotor phase [4,27]. Most of the reported contributions are based on wavelet analysis of the currents during start-up
or load variation for diagnosis purposes. In this context, the frequency components are spread in a wide bandwidth as slip and
speed vary considerably. The situation is more complicated under
rotor faults due to the proximity of the fault components to the fundamental one. These facts justify the common use of multi-detail
or/and approximation signals resulting from wavelet decomposition, whose levels are imposed by the sampling frequency.
This dependency on the appropriate choice of the sampling frequency and tracking multi-fault components on multi-frequency
bands complicate the diagnosis process. Moreover, the use of large
frequency bands subjects the detection procedure to erroneous
interpretations due to possible confusion with other harmonics
related to the common use of gearboxes [8] in wind turbines. In
order to quantify the fault severity, the energy content of approximation and/or detail signals resulting from wavelet decomposition
were used in [14,22]. But this attempt reduces each time–frequency
band to a single value. In such a way the time-domain information is
lost. Motivated by the above discussion, the possible improvements
can be formulated as follows:
• A low sampling frequency can be used to reduce the memory
required.
• Successive frequency sampling should be avoided in order to
reduce latency in time processing.
• More precision in removing the effects of the fundamental component and other harmonic effects around the most relevant fault
component is required.
• The contribution of the most relevant fault frequency components under time-varying conditions can be clamped in a single
frequency band.
• Monitoring the fault severity evolution dynamically over time is
mandatory for variable speed-constant frequency control strategy using DFIM.
In this paper, a simple and effective method is presented to
solve the above open points for the diagnosis of electrical faults in
DFIM under time-varying conditions. A new approach based on currents pre-processing by frequency sliding (FS) and discrete wavelet
transform (DWT) thereby is here proposed [26,27]. Once the state
of the machine has been qualitatively diagnosed, a dynamic mean
power calculation, at the resolution level of interest, is introduced as a diagnostic index for fault quantification over time. The
efficiency of the proposed approach for fault detection and quantification is proved by simulations and experiments. The results on
stator faults presented in this paper have to be considered as an
extension to those presented in [26]. Moreover, in this work, new
stator fault configurations are investigated and the method is effectively applied also for the detection of rotor faults in time-varying
conditions.
The paper is organized as follows. In Section 2 the fault phenomenon is described in time and frequency domains. Section 3
presents the proposed approaches based on wavelet transform.
Simulation and experimental results are presented and commented
in Sections 4 and 5 for stator faults and in Section 6 for rotor faults
under time-varying conditions. Once the state of the machine has
been qualitatively diagnosed under different fault configurations,
the corresponding quantitative evaluation results are presented
and discussed in Section 7. Finally, conclusions are given in
Section 8.
2. Modeling and phenomenon description
A doubly fed induction machine three-phase model has been
implemented in Matlab–Simulink. As described in [28], this model
is based on the representation of the DFIM as a rotating transformer.
The model was adapted to allow a great flexibility in managing
all the machine parameters in order to simulate any stator or
rotor asymmetry configuration during speed transients. In order
to validate the results from the simulation model, an experimental
investigation was conducted using a doubly fed induction machine.
The main characteristics of the tested motors were: rated stator
voltage: 380 V, rated rotor voltage: 186 V, rated power: 5.5 kW, 2
pair of poles, nominal stator current: 15.3 A, nominal stator current:
19.5 A, rated speed: 1400 rpm, stator phase resistance: 0.531 , and
rotor phase resistance: 0.31 .
The induction machine is coupled to a 9 kW separately excited
by a DC machine, supplied via a commercial DC/DC chopper used to
realize speed transients. Stator and rotor currents are sampled with
a 3.2 kHz sampling rate by means of a DS1103 dSpace Board. The
unbalances on stator and rotor side were obtained by additional
resistances (Radd ) connected in series to one phase winding both in
simulation and experimental tests.
The DFIM, like any other rotating electrical machine, is
subjected to both electromagnetic and mechanical forces symmetrically repartitioned. In healthy condition the three stator and rotor
phase impedances are identical, and then currents are symmetrically generated. Under these normal conditions, only fundamental
frequencies f and sf exist respectively on stator and rotor currents
(f: supply frequency, s: slip). If the stator part is damaged, the
stator symmetry of the machine is lost producing a reverse rotating
magnetic field. This dissymmetry generates magnetic forces on
the rotor, caused by the change in the magneto-motive force
from the unbalanced stator phase. More precisely, in the case of
stator asymmetry, the stator currents produce a counterrotating
magnetic field at the frequency −f. This component induces a
rotor current component at (2 − s)f. These frequency components
generate electromagnetic and mechanical interactions between
stator and rotor (Fig. 1). Consequently, a torque and speed ripples
that modulate the rotating magnetic flux are generated at frequency 2f. This modulation leads to an additional component at
frequency (2 + s)f.
The new rotor harmonic component (2 + s)f interacts with the
arising torque and speed ripples at frequency 2f and give rise
to new stator current harmonics at the frequencies ±3f. Which
in turn induces reaction on rotor parts and generate a new frequency component at (4 − s)f on the rotor side. This chain of
interactions leads to the appearance of new harmonic components
((fksa )s = ±kf)k=1,3,5,. . . and ((fkra )s = (2k ± s)f)k=1,2,3,. . . in the stator and
rotor currents respectively.
Practically, in squirrel cage induction machine whose rotor currents are not accessible, the focus has been always on the tracking of
the 3rd and the 5th harmonic components ((fksa )s = ±kf)k=3,5 using
DWT [14,24]. But these components are naturally damped by the
effect of machine-load inertia on high order fault harmonics (Fig. 1).
In Section 3, a simple new method for an efficient tracking of the
first and the most relevant stator fault component ((fksa )s = −kf)k=1
is proposed using DWT.
Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
753
Fig. 1. Time–frequency propagation of a stator fault: the sign (−) in time domain is corresponding to the inverse current sequence component.
The effects of the stator unbalance on the electromagnetic
torque due to the first fault frequency components can be summarized as follows. In healthy condition the torque Te can be expressed
as:
Te = −pMsr
sin (isa ira + isb irb + isc irc )
+ sin −
+ sin +
2
3
2
3
(isa irc + isb ira + isc irb )
(isa irb + isb irc + isc ira )
(1)
where isabc and irabc are respectively stator and rotor phase currents.
A more detailed development of the torque expression leads to:
Te =
9
pMsr Is Ir sin(˛ − ˇ − )
4
(2)
where ˛ and ˇ are the initial phases for stator and rotor currents
respectively and is the angular displacement between stator and
rotor references.
In faulty stator operating condition, stator and rotor phase currents can be expressed as:
isa = Is cos(ωs t + ˛) + Isl cos(ωsl t + ˛sl )
ira = Ir cos(ωr t + ˛) + Irl cos(ωrl t + ˛rl )
(3)
For the shake of clarity and because of the damping effect due
to the motor inertia, only the first and the most relevant frequency components (ωsl = −ωs ) and (ωrl = (2 − s)ωs ) respectively on
stator and rotor currents were considered in these expressions.
The products Isl Irl , Isl Isl and Irl Irl can be neglected in comparison with the terms containing the fundamentals. Hence (1)
becomes:
here can be observed on simulation results depicted in Fig. 2. For
the sake of clarity, only simulation results are presented, although
experimental validation of this phenomenon was reported
in [25].
In healthy condition, the speed and the torque show a constant
value during the four seconds of simulation in steady state condition. However, under stator unbalance (Radd = Rs ) where Rs is the
value of the stator phase resistance) the torque and speed ripples
are evidenced at 2fs = 100 Hz.
Similar considerations can be made under a rotor unbalance. In this case, a rotor current inverse-sequence component
−sf arises, generating a single harmonic component on the
stator side at the frequency (1 − 2s)f, which in turn leads to
electromagnetic and mechanical interactions at frequency 2sf.
Following this interaction process, a chain of harmonic components ((fkra )r = ±ksf)k=1,3,5,. . . and ((fksa )r = (1 ± 2ks)f)k=1,2,3,. . . occurs
in rotor and stator currents respectively. A detailed description of the propagation of these chains of fault components
these fault frequencies chain in time–frequency domain can be
found in [27]. It is worth noting that the most relevant fault
components in rotor and stator currents are −sf and (1 − 2s)f
respectively, due to the damping effect of the machine load and
inertia on higher order fault harmonics. Thus with reference to
stator currents, only the fault component (1 − 2s)f was investigated in Section 6, using the proposed method based on FS and
DWT. Although the proposed technique can be efficiently applied,
considering also the inverse-sequence component −sf in rotor currents.
3. Fault frequency tracking: the proposed approach
9
9
Te = pMsr Is Ir sin(˛−ˇ − )− pMsr Ir Isl sin(2ωs t + + ˇ + ˛sl ) (4)
4
4
3.1. Wavelet decomposition
The first term of this expression is related to the torque value
in healthy condition (2). The second term has a periodic variation of twice the stator frequency supply (2ωs ). This pulsating
torque previously described in Fig. 1 and analytically developed
The principal feature of wavelet transform is its high multiresolution analysis (HMRA) capability. Wavelet analysis is a signal
decomposition using a combination of approximation Caj,p and
detail Cdj,p coefficients, via a scaling function ϕJ,p at level J and a
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Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
a
b
1405
Speed (rpm)
Torque (Nm)
40
35
42
40
30
38
1400
1405.5
1395
1405
36
25
34
1.98
0
1.99
1
2
2.01
2
1404.5
1.98
2.02
3
1390
4
0
1.99
1
Time (s)
2.01
2.02
3
4
Time (s)
c
d
40
1405
Speed (rpm)
Torque (Nm)
2
2
35
42
40
38
30
1400
1404
1395
1403.5
36
25
34
1.98
0
1.99
1
2
2
2.01
1403
1.98
2.02
3
1390
4
Time (s)
0
1.99
1
2
2
2.01
2.02
3
4
Time (s)
Fig. 2. Instantaneous values of (a) torque and (b) speed in healthy condition (Radd = 0). The corresponding (c) torque and (d) speed under stator unbalance (Radd = Rs ) for
steady-state conditions. Simulation results.
mother wavelet function
i(n) =
CaJ,p · ϕJ,p (n) +
p
j,p
at level j.
J
j=1
Cdj,p ·
j,p (n)
(5)
p
More detailed analytical bases of the wavelet technique can
be found in [29]. Generally, the multiresolution analysis (MRA) is
represented by a hierarchical successive and complementary filterbank operations in which the original signal i(n) is decomposed
into approximation and detail signals [25]. The decomposition is
repeated until the signal is analyzed at a pre-defined J level [30]. The
level of decomposition J is related to the sampling frequency (fsam )
of the signal being analyzed. Commonly, a high level decomposition J leading to a HMRA covering all the range of frequencies along
which the fault components varies during transient conditions is
chosen. The level of decomposition is given by [31]:
J>
can be written as in (7) considering a stator dissymmetry.
√
√
2Is cos(ωs t + s ) + 2I−f cos(ω−f t + −f )
√
+ 2I3f cos(ω3f t + 3f )
ia (t) =
log(fsam /f )
+1
log(2)
(6)
Hence, these bands can’t be changed unless a new acquisition
with different sampling frequency is made, which complicate any
fault detection based on DWT, particularly in time-varying conditions.
With regard to the type of mother wavelet, a 10th order of
Daubechies family db10 was chosen, although other families (Symlet and Coiflet) also allow a clear detection of the phenomenon
(stator or rotor unbalance). Actually, a high order of mother wavelet
is recommended for minimizing the overlapping effect in expense
of a higher computation time [23,31]. But thanks to the robustness
of the proposed method, the use of low order mother wavelet is
able to provide satisfactory results.
3.2. The frequency sliding methodology
Neglecting other effects like slotting and saturation it can be
assumed that the major components of one stator phase current
(7)
where Is is the rms value of the fundamental component, I−f and I3f
are those relatively to the first and second levels of fault harmonic
components, ω−f = −ωs , ω3f = 3ωs , ϕs , ϕ−f and ϕ3f are the corresponding frequencies and angular displacements respectively. The
stator current space vector referred to the stator reference frame is
computed by applying the well-known instantaneous symmetrical
components transformation (8):
is =
=
√
2
[isa (t) + isb (t)ej2/3 + isc (t)e−j2/3 ]
3
3[Is ejωs t + I−f ej(ω−f t−−f ) + I3f ej(ω3f t−3f ) ]
(8)
A simple processing of (8), allows shifting the fault component
−f to a chosen frequency band. More in detail, a frequency sliding
is applied at each time slice to the stator current space vector as in
(9).
(Isli )(Isli )S (t) = Re[is (t)e−j2fsli t ]
(9)
where fsli is the value in Hertz of the desired shift in frequency. In
this way the fault component can be moved to one of the intervals [0:2−(J+1) .fsam ] or [2−(J+1) .fsam :2−J .fsam ] Hz, corresponding to the
approximation and details signals frequency bands respectively
[29,30].
Then the real part of the shifted signal is analyzed by means
of DWT, leading to an effective isolation of the component in the
chosen frequency band.
As illustrated in Fig. 3, the DWT analysis divides the frequency
band of the original signal into logarithmically spaced frequency
bands. From this bandwidth segmentation, it is possible to realize
that the approximation aJ and detail dJ have the same and the smallest frequency band width equal to (fsam /2J+1 ). On the other hand,
Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
aJ
dJ
dJ-1
755
d1
[0 : fsam/2J+1] [fsam/2J+1 : fsam/2J] [fsam/2J : fsam/2J-1]
[fsam/4 : fsam/2]
Fig. 3. The DWT filtering process.
wavelets are far from behaving as ideal filters. The presence of a
transition bandwidth whose width is non negligible, cause partial
overlapping between frequency bands [23,31]. This causes some
distortion if a certain frequency component of the signal is close
to the limit of a band. As shown in Fig. 3, the detail dJ is subjected
to the overlapping effect, imposed by the approximation aJ (on the
lower limit) and detail dJ−1 (on the upper limit). However, approximation aJ is subjected only to the overlapping effect of the detail dJ .
So, approximation aJ is half a time less subjected to the overlapping
effect phenomenon than detail dJ . Subsequently, approximation aJ
was chosen for tracking the fault component −f. The value of the
frequency sliding needed to shift the −f frequency in the center of
the frequency interval [0:2−(J+1) .fsam ] Hz of the approximation aJ is
calculated as:
fsli = −f − 2−J .fsam
(10)
With a sampling frequency fsam = 3.2 kHz, the application of (6)
leads to an eight level decomposition (J = 8). The frequency bands
associated with each wavelet signal are the ones shown in Table 1.
Finally, the fault component −f will be tracked in the frequency
band [0:6.25] Hz corresponding to the approximation signal a8 .
According to (10) the value of fsli is −53.125 Hz.
4. Stator fault analysis under speed-varying conditions
The first fault diagnosis condition in this work is a fixed stator
unbalance under speed variations. The induction motor has been
initially simulated in healthy conditions (Radd = 0) in order to perform a comparison with faulty cases during two speed transients.
The two first simulations were done respectively under speed variations from 1410 rpm to 1495 rpm (Fig. 4a) and from 1486 rpm to
1410 rpm (Fig. 4b), corresponding to slip ranges from s = 0.0593 to
0.0033 and from s = 0.0093 to 0.0593. The wavelet decomposition
results obtained under healthy machine condition during acceleration and deceleration are respectively used as a reference in
comparison with the faulty cases.
4.1. Fault-frequency tracking in rotor currents
Stator Faults were firstly detected through the analysis of rotor
currents without frequency sliding methodology, but only exploiting the capability of DWT to isolate the contribution of the fault
harmonic components [26]. As reported in Table 1, the contributions of the fault frequencies (2 − s)f and (2 + s)f in rotor currents
can be observed respectively on details d5 and d4 . Hence, the MRA
was done for only five decomposition levels. For the sake of clarity,
details d3 , d2 and d1 are omitted from figures. Observing the 4th
and 5th level details, resulting from wavelet decomposition of a
rotor phase current, we find that large slip range variations during
acceleration or deceleration for a healthy machine has no effect on
the details of interest: d4 and d5 (Fig. 4e–h).
However, in faulty condition (Radd = Rs ) these detail signals show
significant variations with respect to the speed range evolutions
(Fig. 5e–h), and greater magnitudes than those registered in healthy
condition.
Experimental results under healthy and faulty conditions
(Radd = Rs ) for the above considered speed transients are reported
in Figs. 6 and 7. These results corroborate simulations although the
magnitude evolution in some cases is even bigger than in simulation. The 4th and 5th detail levels issued from the experimental
results, show the sensitivity and the effectiveness of these particular details to reproduce the evolution of the stator fault frequencies
(2 − s)f and (2 + s)f under stator unbalance (Radd = Rs ).
4.2. Fault-frequency tracking in stator currents
For this second advanced approach, a HMRA was adopted (J = 8)
to isolate the contribution of the fault frequency −f issued from the
stator current space vector. Under speed-varying conditions, only
the magnitude of this component change under stator fault conditions. The harmonic component −f is fixed at −50 Hz. As explained
in the previous section, for an optimal use of the wavelet analysis,
the contribution of the negative sequence −f will be observed on
approximation a8 ([0:6.25] Hz) after the shift in frequency.
In Fig. 8e and f, the 8th approximations for the healthy simulated machine are depicted. It is possible to notice that no effect
on these signals has been registered during the above considered
acceleration or deceleration transients. However, in faulty condition (Radd = Rs ) the same signals show significant variation in
magnitude (Fig. 9e and f). The corresponding experimental results
under healthy and faulty conditions (Radd = Rs ) are reported in
Figs. 10 and 11. The results obtained experimentally corroborate
simulations although the magnitude evolutions in some cases are
even bigger than in simulation. The 8th approximation a8 issued
from the experimental results, prove the effectiveness of the proposed approach based on frequency sliding to detect dynamically
over time the contribution of the negative sequence −f under stator
fault unbalance (Radd = Rs ).
5. Stator fault analysis under fault-varying conditions
Table 1
Frequency band of each level.
5.1. Progressive stator unbalance condition
Approximations «aj »
Frequency
bands (Hz)
Details «dj »
Frequency
bands (Hz)
a8
a7
a6
a5
a4
[0–6.25]
[0–12.5]
[0–25]
[0–50]
[0–100]
d8
d7
d6
d5
d4
[6.25–12.5]
[12.5–25]
[25–50]
[50–100]
[100–200]
The first sub-part of this section is focused on the analysis of low
range stator unbalance degradation obtained in simulation through
a ramp of stator resistance increment from Radd = 0 to Radd = 1.1Rs
during 8.5 s at constant speed. An experimental set up was designed
to emulate this situation. The fault was achieved by inserting an
external variable resistance in series to one stator phase.
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Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
b
1500
1450
1400
2
3
4
5
1400
c
50
d
50
0
ira (A)
1
1450
ira (A)
0
1500
Speed (rpm)
Speed (rpm)
a
0
1
2
3
4
-50
5
e
5
f
5
0
d5
0
d5
-50
0
1
2
3
4
-5
5
g
5
h
5
0
d4
0
d4
-5
0
-5
0
1
2
3
4
-5
5
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
Time (s)
Time (s)
Fig. 4. Instantaneous values of speed (a and b) and rotor current (c and d) in healthy condition during acceleration and deceleration transients. DWT results of a rotor phase
current: (e and f) detail d5 and (g and h) detail d4 . Simulation results.
b
Speed (rpm)
1500
1450
1400
1
2
3
4
1450
1400
5
c
50
d
50
0
ira (A)
0
1500
ira (A)
Speed (rpm)
a
0
-50
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
-50
2
3
4
5
e
5
f
5
0
d5
1
d5
0
0
-5
-5
2
3
4
5
g
5
h
5
0
d4
1
d4
0
0
-5
-5
0
1
2
3
Time (s)
4
5
Time (s)
Fig. 5. Instantaneous values of speed (a and b) and rotor current (c and d) under stator fault condition (Radd = Rs ) during acceleration and deceleration transients. DWT results
of a rotor phase current: (e and f) detail d5 and (g and h) detail d4 . Simulation results.
Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
a
757
b
ira (A)
c
1450
1400
0
1
2
3
4
5
Speed (rpm)
1500
1450
1400
d
50
0
-50
d5
0
1
2
3
4
5
f
5
0
-5
d4
0
1
2
3
4
5
h
5
d4
5
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
-5
0
g
2
0
d5
5
1
-50
0
e
0
50
ira (A)
Speed (rpm)
1500
0
-5
-5
0
1
2
3
4
5
Time (s)
Time (s)
Fig. 6. Instantaneous values of speed (a and b) and rotor current (c and d) under healthy condition during acceleration and deceleration transients. DWT results of a rotor
phase current: (e and f) detail d5 and (g and h) detail d4 . Experimental results.
b
1500
Speed (rpm)
Speed (rpm)
a
1450
1400
ira (A)
c
1
2
3
4
1450
1400
5
50
d
50
0
ira (A)
0
1500
0
-50
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
-50
2
3
4
5
e
5
f
5
0
d5
1
d5
0
0
-5
-5
2
3
4
5
g
5
h
5
0
d4
1
d4
0
0
-5
-5
0
1
2
3
Time (s)
4
5
Time (s)
Fig. 7. Instantaneous values of speed (a and b) and rotor current (c and d) under stator fault condition (Radd = Rs ) during acceleration and deceleration transients. DWT results
of a rotor phase current: (e and f) detail d5 and (g and h) detail d4 . Experimental results.
758
Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
b
1500
Speed (rpm)
Speed (rpm)
a
1450
1400
2
3
4
1400
5
c
30
d
30
0
isa (A)
1
1450
isa (A)
0
1500
0
2.5
0
-2.5
0
1
2
3
4
-30
5
f
0
1
2
3
4
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
2.5
a8
e
a8
-30
0
0
-2.5
5
Time (s)
Time (s)
Fig. 8. Instantaneous values of speed (a and b) and stator current (c and d) in healthy conditions during acceleration and deceleration transients. DWT results of the shifted
stator current space vector Isli : (e and f) approximation a8 . Simulation results.
5.1.1. Fault-frequency tracking in rotor currents
Firstly rotor currents were examined for stator fault detection. For the sake of brevity only the contribution of the (2 − s)f
fault harmonic is considered in this section. Instantaneous current waveform retrieved form simulations and its corresponding
5th detail signal resulting from MRA are represented in Fig. 12a
for the machine running at the nominal speed. During the first
time period (t = 0–2.5 s) under the healthy operating machine, the
5th detail signal, representative of the stator fault frequency com-
b
a
1500
Speed (rpm)
Speed (rpm)
ponent (2 − s)f do not have any variation. Following the ramp
evolution of the stator resistance, detail d5 shows particular magnitude escalation proportional to the progressive stator unbalance
evolution.
Experimental results, depicted in Fig. 12b, are in complete agreement with simulation (Fig. 12a) results. The effectiveness of the 5th
detail signal in reproducing the contribution of the stator fault frequency component (2 − s)f in rotor currents under a low range of
stator unbalance degradation is evident.
1450
1400
2
3
4
1400
5
c
30
d
30
0
isa (A)
1
1450
isa (A)
0
1500
0
a8
e
0
1
2
3
4
-30
5
2.5
f
2.5
0
a8
-30
0
-2.5
0
1
2
3
Time (s)
4
5
-2.5
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
Time (s)
Fig. 9. Instantaneous values of speed (a and b) and stator current (c and d) under stator fault conditions (Radd = Rs ) during acceleration and deceleration transients. DWT
results of the shifted stator current space vector Isli : (e and f) approximation a8 . Simulation results.
Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
b
1500
Speed (rpm)
Speed (rpm)
a
1450
1400
2
3
4
1450
1400
5
c
30
d
30
0
isa (A)
1
1500
isa (A)
0
0
a8
0
1
2
3
4
-30
5
2.5
f
2.5
0
a8
-30
e
759
0
-2.5
0
1
2
3
4
-2.5
5
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
Time (s)
Time (s)
Fig. 10. Instantaneous values of speed (a and b) and stator current (c and d) in healthy conditions during acceleration and deceleration transients. DWT results of the shifted
stator current space vector Isli : (e and f) approximation a8 . Experimental results.
5.1.2. Fault-frequency tracking in stator currents
Stator currents were processed under the same stator fault evolution. After a suitable frequency sliding the 8th approximation
signals resulting from the MRA of Isli is depicted in Fig. 13a and b, for
simulation and experimental tests respectively. Following the ramp
of the stator resistance, approximation a8 shows particular magnitude escalation proportional to the progressive stator unbalance
evolution. Experimental results (Fig. 13b) are in complete agreement with the simulation results (Fig. 13a).
The evolution of the 8th approximation a8 retrieved from
the experimental tests prove the effectiveness of the proposed
approach based on frequency sliding to detect the contribution of
5.2. Intermittent stator unbalance escalation
The second sub-part of this section is focused on the analysis
of intermittent stator unbalance degradation. To simulate this fault
configuration, an intermittent increment of additional resistances
connected in series to one stator phase was performed. The experimental set up was designed to reproduce this situation by means of
an external variable resistance in series to one stator phase whose
value was intermittently increased during short time intervals.
b
1500
Speed (rpm)
Speed (rpm)
a
the negative sequence −f under low range of progressive stator
unbalance.
1450
1400
2
3
4
1400
5
c
30
d
30
0
isa (A)
1
1450
isa (A)
0
1500
0
1
2
3
4
-30
5
e
2.5
f
2.5
0
a8
0
a8
-30
0
-2.5
0
1
2
3
Time (s)
4
5
-2.5
0
1
2
3
4
5
0
1
2
3
4
5
0
1
2
3
4
5
Time (s)
Fig. 11. Instantaneous values of speed (a and b) and stator current (c and d) under stator fault conditions (Radd = Rs ) during acceleration and deceleration transients. DWT
results of the shifted stator current space vector Isli : (e and f) approximation a8 . Experimental results.
760
Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
40
20
0
-20
-40
b
40
20
0
-20
-40
ira (A)
ira (A)
a
0
1
2
3
4
5
6
7
8
9
10
0
-5
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
5
d5
d5
5
0
1
2
3
4
5
6
7
8
9
0
-5
10
Time (s)
Time (s)
Fig. 12. Instantaneous values of rotor current and its 5th decomposition detail signal d5 under progressive stator unbalance condition: (a) simulation and (b) experimental
results.
5.2.1. Fault-frequency tracking in rotor currents
Also for this fault configuration rotor currents were examined
at first. Instantaneous current waveform and its corresponding 5th
detail signal resulting from MRA are represented in Fig. 14a for the
simulated machine running at the nominal speed.
During healthy time sequences, the 5th detail signal, representative of the stator fault frequency component (2 − s)f does
not have any variation. With the inception of the fault, detail d5
shows particular abrupt magnitude escalations proportional to the
four intermittent stator phase resistance increases. The magnitude
increase at these four points, meets the criterion for detection.
Experimental results (Fig. 14b) are in total agreement with the
simulation results (Fig. 14a) although the magnitude evolutions
in some cases are even bigger than in simulation. In this configuration too, the details d5 reproduce clearly the contribution
of the stator fault frequency component (2 − s)f dynamically over
time.
6. Rotor fault analysis under speed-varying conditions
As explained in Section 2 the presence of rotor faults gives rise
to a chain of frequencies on stator phase currents beside the fundamental. Since the most relevant is the (1 − 2s)f we will focus our
attention on its evolution. In time-varying conditions, the rotor
fault component (1 − 2s)f in stator current, whose amplitude must
be monitored, assumes different values depending on the load conditions and the rotor degradation degree. These facts complicate
considerably the diagnosis process.
In this section, the proposed approach based on FS as a preprocessing for time–frequency analysis using DWT, is tested for
rotor fault frequency tracking under a large range of speed during a
deceleration leading from zero to the nominal slip. Fig. 16 shows the
above considered transient retrieved from simulations and experimental tests. Fig. 16 shows the corresponding Instantaneous Fault
Frequency Evolution (IFFE) of the fault component (1 − 2s)f which
was computed in time domain as [27]:
5.2.2. Fault-frequency tracking in stator currents
For the same stator fault evolution the stator current space vector was processed as previously explained in Section 3. After the
shift in frequency the approximation a8 , issued from the wavelet
decomposition of Isli and representative of the fault component −f
was investigated. Simulation and experimental results are depicted
in Fig. 15a and b respectively. The 8th approximation signal does not
show any significant magnitude escalations except those relatively
to the abrupt stator fault inception.
Experimental results are in total agreement with simulations,
although the magnitude evolutions in some cases are even bigger. In this case too, the 8th approximation a8 issued from the
experimental tests, confirms the effectiveness of the proposed
approach.
a
fsli = fup − 2−(J+1) .fsam
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
0
1
2
3
4
5
6
7
8
9
10
5
6
7
8
9
10
5
a8
5
a8
0
-20
-20
0
-5
(12)
20
isa (A)
0
(11)
In this case, the frequency sliding was applied to the interval
[flow :fup ] Hz in which IFFE of the (1 − 2s)f evolves. After several
tests, the values of flow = 39.667 Hz and fup = 46.25 Hz were adopted
corresponding to 1345 and 1443.8 rpm around the nominal speed
(1400 rpm) of the motor. Although, these values can be easily
adjusted for a speed range needed for a maximum power tracking
of the wind turbine. In order to shift this frequency interval in the
approximation a8 ([0:2−(J+1) .fsam ] Hz), fsli = 40 Hz was calculated as:
b
20
isa (A)
fksa = (1 − 2ks(t)f )k=1
0
-5
0
1
2
3
4
5
6
Time (s)
7
8
9
10
Time (s)
Fig. 13. Instantaneous values of stator current and its 8th decomposition approximation signal a8 under progressive stator unbalance: (a) simulation and (b) experimental
results.
Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
40
20
0
-20
-40
b
ira (A)
ira (A)
a
0
1
2
3
4
5
6
7
8
9
10
d5
d5
10
0
-10
40
20
0
-20
-40
0
10
0
1
2
3
4
5
6
7
8
9
1
2
3
4
1
2
3
4
5
6
7
8
9
10
5
6
7
8
9
10
0
-10
0
10
761
Time (s)
Time (s)
Fig. 14. DWT of a rotor phase current under intermittent stator unbalance conditions: (a) simulation and (b) experimental results.
a
b
20
isa (A)
isa (A)
20
0
-20
-20
1
2
3
4
5
6
7
8
9
10
a8
0
a8
0
0
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
10
Time (s)
Time (s)
Fig. 15. DWT of the shifted stator current space vector under intermittent stator unbalance conditions: (a) simulation and (b) experimental results.
The application of the frequency sliding to one stator phase current leads to a similar consideration as for (9):
(Isli )r (t) = Re[isa (t)e−j2fsli t ]
(13)
Observing the IFFEs depicted in Fig. 17, before and after the
frequency sliding, it can be seen that the evolution of the rotor
fault component (1 − 2s)f is clamped in the frequency interval [06.25] Hz, which correspond to the 8th wavelet decomposition level
as mentioned in Table 1. So the frequency band of interest for tracking the contribution of the (1 − 2s)f fault frequency is again the
approximation a8 . For the sake of clarity only a8 retrieved from the
DWT decomposition of Isli are reported in the following analysis.
b
1500
Speed (rpm)
Speed (rpm)
a
Simulation and experimental analysis results are depicted in
Fig. 18. The 8th approximation a8 resulting from MRA of Isli doesn’t
show any kind of variation in healthy conditions (Fig. 18a). However, in faulty conditions (Radd = Rr ) where Rr is the rotor phase
resistance) approximation a8 shows significant variation in magnitude (Fig. 18b). It’s worth noting that during the first time period
(t = 0–1.8 s), under a constant speed of 1495 rpm, the 8th approximation, representative of the (1 − 2s)f frequency component, does
not have any important variation. This is due to the fact that at low
load level (under 5% of the nominal torque) the fault frequencies
are practically superimposed to the fundamental and not significant. On the other hand during deceleration (t = 1.8–2.9 s), a8 shows
particular magnitude escalation proportional to the progressive
1450
1400
1450
1400
0
2
4
6
8
10
0
2
4
6
8
10
0
2
4
6
Time (s)
8
10
d
Current (A)
c
Current (A)
1500
20
0
-20
0
2
4
6
Time (s)
8
10
20
0
-20
Fig. 16. Instantaneous values of speed ((a) simulation and (b) experimental) and stator current ((a) simulation and (b) experimental).
762
Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
b
70
(1+4s)f
60
(1+2s)f
50
(1-2s)f
40
30
70
Frequency (Hz)
Frequency (Hz)
a
60
2
4
6
8
(1+2s)f
50
(1-2s)f
40
(1-4s)f
0
(1+4s)f
30
10
(1-4s)f
0
2
4
Time (s)
d
12.5
(1-2s)f
6.25
0
Frequency (Hz)
Frequency (Hz)
c
2
4
8
10
6
8
10
12.5
6
8
(1-2s)f
6.25
(1-4s)f
0
6
Time (s)
0
10
(1-4s)f
0
2
4
Time (s)
Time (s)
Fig. 17. IFFEs in stator current under rotor unbalance. Before FS (a) simulation and (b) experimental results. After FS of 40 Hz (c) simulation and (d) experimental results.
deceleration until reaching a quasi steady-state magnitude (at the
nominal torque).
The experimental results, under healthy (Fig. 18c) and faulty
condition (Fig. 18d), corroborate simulations, although the magnitude evolutions, in some tests, are even bigger than those registered
in simulation. The 8th approximation obtained from the experimental results, show the sensitivity and the effectiveness of this
particular approximation signal to reproduce the contribution of
the frequency component (1 − 2s)f under rotor unbalance. This new
approach has very interesting advantages in avoiding successive
frequency sampling and confusions with other harmonics around
the frequency of interest. The proposed technique reduces considerably the latency and the memory required.
7. Quantitative fault evaluation in DFIM under
time-varying conditions
Once the state of the machine has been qualitatively diagnosed,
a quantitative evaluation of the fault degree is a necessary step to
where sj (n) is the approximation or the detail signal of interest. As
depicted in Fig. 19 the fault indicator is periodically calculated (each
ın = 400 samples) using a window of n samples (n = 6400 samples). The n sequences are indexed by a time interval number
(TIN). These parameters (ın and n) were regulated experimentally to reduce variations that can lead to false interpretation.
Once the fault occurs, the energy distribution of the signal is
changed at the resolution levels related to the characteristic frequency bands of the default. Hence, the energy excess confined in
the frequency band of interest is considered as the beginning of
an anomaly. All fault events were quantified using the parameters
3
2
2
1
1
0
(14)
n=1
c
3
2
1 sj (n)
n
N
mPsj =
a8
a8
a
decide for the operating continuity of the machine. For this purpose
a dynamic multiresolution mean power indicator mPsj at different
resolution levels j was introduced as a diagnostic index to quantify
the extent of the fault as in (14):
0
-1
-1
-2
-2
-3
-3
0
2
4
6
8
0
10
2
4
Time (s)
d
3
2
1
1
0
8
10
8
10
3
2
a8
a8
b
6
Time (s)
0
-1
-1
-2
-2
-3
-3
0
2
4
6
Time (s)
8
10
0
2
4
6
Time (s)
Fig. 18. Wavelet decomposition of the signal (Isli )r : simulation results under (a) healthy and (b) faulty condition (Radd = Rr ). Experimental results under (c) healthy and (d)
faulty condition (Radd = Rr ) during speed transient.
''aj'' signal
sequences
''aj'' signal
Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
tions (Radd = Rs ) the calculated mPa8 indicator shows significant
increase meeting the criterion for stator fault detection. The large
energy deviation observed in faulty conditions prove the effectiveness of the proposed approach, to extract the energy excess related
to the negative sequence −f contribution under stator unbalance.
The results that have been obtained experimentally (Fig. 21c and d)
corroborate simulations thus proving the effectiveness of this new
approach.
Time
Δn
TIN=1
TIN=2
δn
TIN=3
7.2. Stator fault quantification under fault-varying conditions
Samples
Fig. 19. Principle of time interval calculation (TIN, time interval number).
mentioned above except for the intermittent stator fault conditions
where a reduced sequence (n = 1600 samples) was adopted to
improve the dynamic sensitivity of the fault indicator.
7.1. Stator fault quantification under speed-varying conditions
The mean power of the detail d5 resulting from MRA applied
to rotor currents, which have been obtained from simulations in
Figs. 4 and 5 are depicted in Fig. 20a and b. In healthy conditions
(Radd = 0) and under large range of speed variations (acceleration, deceleration), the calculated mPd5 indicator doesn’t show
any variation being equal to zero. In faulty conditions (Radd = Rs )
the calculated mPd5 indicator shows significant increase meeting
the criterion for stator fault detection. The large energy deviation
observed in faulty conditions prove the effectiveness of the proposed approach, since the variable speed motor operations do not
disturb the assessment compared with the healthy case. The corresponding experimental results under healthy and faulty conditions
are depicted respectively in Fig. 20c and d. The results that have
been obtained experimentally corroborate simulations thus proving the effectiveness of the proposed approach.
The mean power of the approximation a8 resulting from MRA
applied to the shifted stator current space vector, which have been
obtained from simulations in Figs. 8 and 9 are depicted in Fig. 21a
and b. In healthy conditions (Radd = 0) and for the considered acceleration and deceleration transients, the calculated mPa8 indicator
doesn’t show any variation being equal to zero. In faulty condi-
a
7.3. Rotor fault quantification under speed-varying condition
The mean power of the approximation a8 resulting from MRA
applied to one stator current after the frequency sliding process and
obtained from simulation and experiments are depicted in Fig. 23.
In healthy conditions (Radd = 0) and under large range of speed variation (deceleration), the calculated mPa8 indicator doesn’t show
any variation and is practically zero. In faulty conditions (Radd = Rr )
the indicator shows significant increase meeting the criterion for
rotor fault detection. The large energy deviation observed in faulty
Radd=0
Radd=Rs
1.5
1
0.5
0
2.5
Radd=0
Radd=Rs
2
mPd5
mPd5
The mean power of the detail d5 resulting from the MRA applied
to rotor currents, which have been obtained under progressive
(Fig. 12) and intermittent (Fig. 14) stator unbalance are depicted
respectively in Fig. 22a and b. When the stator asymmetry occurs,
under progressive or intermittent stator fault-varying conditions,
this fact is directly reproduced by the fault indicator mPd5 , which
shows proportional evolution relatively to the fault severity propagation. Experimental results are in total agreement with simulation
results although the magnitude indicator evolutions are bigger than
in simulation. The mean power of the approximation a8 resulting
from the MRA applied to the shifted stator current space vector,
which have been obtained under the same fault-varying conditions (Figs. 13 and 15) are depicted respectively in Fig. 22c and
d. Also in this case the fault indicator mPa8 shows proportional
evolution relatively to the fault severity propagation allowing its
detection. Experimental results corroborate simulations thus proving the effectiveness of this new approach for the diagnosis of stator
faults in DFIM.
b
2.5
2
1.5
1
0.5
0
10
20
0
30
0
10
TIN
c
d
Radd=0
Radd=Rs
2
30
20
30
2.5
Radd=0
Radd=Rs
2
1.5
mPd5
mPd5
20
TIN
2.5
1
0.5
0
763
1.5
1
0.5
0
10
20
TIN
30
0
0
10
TIN
Fig. 20. Cyclic values of the mPd5 fault indicator calculation issued from the details d5 of a rotor phase current analysis. Simulation results under speed: (a) acceleration and
(b) deceleration. Experimental results under speed: (c) acceleration and (d) deceleration.
764
Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
a
b
2.5
Radd=0
Radd=Rs
2
1.5
mPa8
mPa8
2
1
0.5
0
2.5
Radd=0
Radd=Rs
1.5
1
0.5
0
10
20
0
30
0
10
TIN
c
d
2.5
20
30
Radd=0
Radd=Rs
2
1.5
mPa8
mPa8
30
2.5
Radd=0
Radd=Rs
2
1
0.5
0
20
TIN
1.5
1
0.5
0
10
20
0
30
0
10
TIN
TIN
Fig. 21. Cyclic values of the mPd5 fault indicator calculation issued from the approximation a8 of the signal Isli . Simulation results under speed: (a) acceleration and (b)
deceleration. Experimental results under speed: (c) acceleration and (d) deceleration.
a
2
Simulation
2
1
1
0
Experimental
3
Simulation
mPa8
mPd5
c
Experimental
3
0
10
20
30
40
50
60
0
70
0
10
20
30
TIN
b
d
5
mPa8
mPd5
60
70
Experimental
4
Simulation
3
2
Simulation
3
2
1
1
0
50
5
Experimental
4
40
TIN
0
0
10
20
30
40
50
60
70
80
0
10
20
30
40
50
60
70
80
TIN
TIN
Fig. 22. Cyclic values of the mPsj fault indicators, issued from the wavelet signals d5 and a8 under (a and c) progressive and (b and d) intermittent stator fault condition
respectively.
Experimental (Radd=Rr)
Experimental (Radd=0)
Simulation (Radd=Rr)
Simulation (Radd=0)
mPa8
1.5
1
0.5
0
0
10
20
30
40
50
60
70
TIN
Fig. 23. Mean power of the approximation a8 resulting from the wavelet decomposition of stator current under rotor fault condition. Simulation and experimental results.
Y. Gritli et al. / Electric Power Systems Research 81 (2011) 751–766
conditions proves the effectiveness of the proposed approach, since
the motor variable speed operations do not disturb the assessment
with respect to the healthy case. The corresponding experimental
results, depicted in the same figure corroborate simulations thus
proving the effectiveness of the proposed approach under rotor
fault conditions too.
8. Discussion and recommendations for future work
Under rotor fault conditions, some fluctuations on the mPa8
related to the (1 − 2s)f contribution can be noticed (Fig. 23)
when the fault index is computed from simulation data. These
fluctuations are not observed when the mPa8 is retrieved from
experimental results (Fig. 17c and d). In fact, these small fluctuations are due to the natural wavelet overlapping effect, described in
section III-2, when the IFFE is very close to the frequency band limits considered [0:6.25] Hz. Eventually, the use of high order mother
wavelet, can reduce considerably these fluctuations, but in expense
of a higher computation time [23,30].
With regard to the dynamic fault indicator parameters (ın and
n) it’s evident that the conditions n > ın must always be verified and that the more their values are reduced, the more the
dynamic sensitivity of the mPsj is increased. Anyway the proposed
approach is very effective in discriminating healthy from several
faulty condition degrees of the machine.
The proposed technique could be easily implemented in general
purpose software designed for monitoring and diagnosis of various types of faults in wind generator equipped with DFIM. Once
the speed limits corresponding to a maximum power tracking of
wind power is defined, the fault components can be shifted in a
frequency band of interest, with cyclic acquisitions of the currents
being analyzed. Thanks to the cyclic quantification methodology
developed in Section 7, the simplest decision making technique can
be obtained by placing appropriate thresholds on the fault indicators magnitude (mPsj). In this way, faults can be detected when
the value of an indicator is higher than the related threshold value,
leading to an effective diagnosis procedure.
9. Conclusion
The aim of this paper was to validate the effectiveness of a
new and reliable approach for the characterization of stator and
rotor faults in DFIM under time-varying conditions. The proposed
method, based on frequency sliding and HMRA capabilities of the
DWT, was tested under speed-varying conditions and fault-varying
conditions.
Once the state of the machine has been qualitatively diagnosed,
a dynamic multiresolution mean power indicator at different resolution levels was introduced as a diagnostic index to quantify the
extent of the fault over time. Simulation and experimental results
carried out, demonstrate the effectiveness of this new approach
that can be applied to any type of machine and extended for diagnosing other types of faults under time-varying conditions.
Appendix A. List of symbols
isa , isb , isc
ira , irb , irc
f
s
Te
p
Rs
Rr
stator phase currents
rotor phase currents
stator frequency
slip
electromagnetic torque
number of pole pairs
stator resistance
rotor resistance
˛, ˇ
Radd
ϕJ,p
j,p
fsam
fksa
fkra
is
fsli
CaJ,p
Cdj,p
765
initial phases for stator and rotor currents
angular displacement between stator and rotor references
additive resistance used in the tests
scaling function
mother wavelet function
sampling frequency
stator frequency components due to a stator fault
rotor frequency components due to a rotor fault
stator current space vector
sliding frequency
approximation coefficients
detail coefficients
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Glossary
DFIM: doubly fed induction machine
MCSA: motor current signature analysis
FSA: flux signature analysis
RMSA: rotor modulation signature analysis
FA: Fourier analysis
SD: signal demodulation
STFT: short-time Fourier transform
FFT: fast Fourier transform
WT: wavelet transform
DWT: and discrete wavelet transform
FS: frequency sliding
MRA: multi-resolution analysis
HMRA: high multi-resolution analysis
IFFE: instantaneous fault frequency evolution