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 752 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 754 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. 756 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 References [1] H. Douglas, P. Pillay, P. 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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

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