Detection of rotor faults under transient operating ...

SDEMPED 2005 Symposium on Diagnostics for Electric Machines, Power Electronics and Drives Vienna, Austria, 7–9 September 2005

Detection of rotor faults under transient operating conditions by means of the Vienna Monitoring Method C. Kral, H. Kapeller, F. Pirker, G. Pascoli Arsenal Research Faradaygasse 3 1030 Vienna, Austria fax: +43–50550–6595, e-mail: [email protected]

Abstract— This contribution investigates the Vienna Monitoring Method during transient operation. The Vienna Monitoring Method is a rotor fault detection technique based on two space phasor machine models. These models derive quantities such as flux and torque. An induction machine with rotor asymmetries gives rise to double slip frequency oscillations of shaft torque. These oscillations are also reflected in the modelled torques. Accordingly, the fault indicator of the Vienna Monitoring Method is derived from the difference of the computed torques to compensate load effects. This paper presents measurement results regarding the Vienna Monitoring Method during stationary and transient operating conditions. The transient conditions include varying voltage and frequency as well as varying torque. The two transient cases are a challenge for any rotor fault detection technique, since the magnitudes and frequencies of fault-specific signatures are highly load and speed dependent.

(double slip frequency). Slip dependency also indicates load dependency. Besides frequencies, the magnitudes of these harmonics are also load dependent. If the induction machine is not loaded (s → 0), rotor currents vanish and the reactions of the rotor fault on the stator currents as well as power and torque disappear. Therefore, the magnitudes of the side band currents merge with the fundamental and their magnitudes diminish. The frequency components of the torque and power (3) merge with the dc components (average torque and real power respectively) and their magnitudes diminish, as well. For a loaded machine, the fault-specific harmonics in the Fourier spectrum arise sharply if the total data acquisition time, T , is long enough in order to achieve a sufficient frequency resolution, ∆f = T1 :

I. Introduction

Such a relation refers to steady state operating conditions. Clearly, the determination of any fault-specific harmonic requires sufficient load torque (slip) and measurement time (frequency resolution). The magnitudes of the fault-specific harmonics are also dependent on the total inertia of the drive. Increasing inertia shifts the magnitudes of the side band currents from the upper to the lower side and also increases the magnitudes of the fault-specific torque and power harmonics [1]. Variable speed drives either have an open loop (Volts per Hertz) or a closed loop control. The latter control application is for examaple, direct or indirect field oriented control [2]. In this case, currents may be controlled in such way that side band harmonics in the current spectrum do not arise. Therefore, the side band harmonics arise in the voltage spectrum instead. It is thus obvious, that the actual type of drive control has a major impact on the appearance of the side band harmonics in either the voltages or currents. Thus, for a variable speed drive the careful selection of a suitable fault detection technique is of high importance.

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OTOR faults such as broken rotor bars and end rings disturb the mainly sinusoidal distribution of rotor currents and react therefore, on the magnetic field of the air gap. These reactions give rise to faultspecific signatures in the current, power, torque and speed of an induction machine. For a mains supplied machine there arise two sidebands next to the fundamental of the current spectrum. Their respective frequency distance from the fundamental is twice the slip frequency. Consequently, they are called the lower and the upper side band harmonics of the current: fi,1 = fs (1 ± 2s)

(1)

Besides this first order harmonic component (index 1), also higher order harmonics arise (index k): fi,k = fs (1 ± 2ks)

(2)

In (1) and (2) fs represents stator frequency and s is slip. The interaction of the current and flux harmonics give rise to torque and power oscillations with the following frequencies: ft,k = 2kfs

(3)

From (1)–(3) it is obvious, that the frequencies of the fault-specific harmonics are slip dependent. The most distinct harmonic belongs to the first order, k = 1

∆f  2sfs

(4)

II. Common fault detection techniques Many fault detection techniques are based on the evaluation of the side bands currents (1). Such techniques are therefore called current signature analysis (CSA) [3], [4], [5], [6]. These techniques usually determine the Fourier spectrum of usually one of the line

III. Vienna Monitoring Method The Vienna Monitoring Method (VMM) uses two mathematical space phasor models. Measured quantities are three voltages v1 , v2 and v3 , three currents i1 , i2 and i3 and rotor position γ, which is computed from measurement data of a rotor position encoder and the number of pole pairs. For some applications the VMM is also applicable without rotor position sensor (sensorless). In the sensorless case rotor position has

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currents and assess the magnitudes of the side band harmonics. This is a challenging task, since the obtained fault indicator is dependent on inertia. However, without any additional measures this type of CSA is only applicable for steady state operation [7]. Some fault detection techniques evaluate the double slip frequency harmonics of the electrical power signature [8]. Such techniques are therefore called power signature analysis (PSA). The main difference between CSA and PSA is that three (or two, if no neutral is present) currents and voltages have to be measured in order to determine instantaneous power. The drawback of the additional and therefore costly measurement is compensated by the fact, that the magnitude of only the first order spectral component at twice the slip frequency has to be assessed – according to (3). Due to the acquisition of voltages and currents the actual operating condition can be determined, too, which is advantageous if load dependencies are intended to be taken into account. The Vienna Monitoring Method (VMM) is a model based technique, which evaluates the calculated torques of two machine models with different model structure [9], [10]. If the induction machine is symmetric both models calculate the same torque and torque difference equals zero. In case of any electrical rotor asymmetry, typical double slip frequency power and torque oscillations arise. These oscillations are also reflected in the modelled torques. Due to the different model structure and different input quantities, magnitudes and phase angles of the respective oscillations are different. Therefore, the difference of the computed torques (torque difference) also shows a double slip frequency oscillation. Earlier investigations showed, that the magnitude of the torque difference is linearly proportional to the actual load torque [11], [12]. If the torque difference is divided by load torque, a load independent magnitude is obtained. For steady state operation it has been shown, that the fault indicator derived by the VMM is independent of inertia [1] and machine supply [2]. It also turned out that the high sensitivity of the VMM allows under certain conditions the detection of small rotor faults such as cracked rotor bars [13]. For steady state operation either CSA or PSA can be applied to detect electrical rotor asymmetries in squirrel cage induction machines. Time varying load or speed commands causes the fault-specific components (2) and (3) to be time dependent, too. For example, CSA techniques have to take into account that both the fundamental and slip are not constant under transient conditions. Therefore, the usual fast Fourier analysis cannot be performed. Higher sophisticated techniques such as time frequency approaches (shorttime Fourier analysis, STFT) have to be performed.

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to be estimated from modelled state variables, which is currently usable only for steady state operation [14], [15]. Transient operating conditions cannot be handled by the sensorless approach, at the moment. Another drawback of the sensorless VMM is the less rotor fault sensitivity due to given deviations of the estimated and actual rotor position. From the measured voltages and currents, per unit (p.u.) voltage and current space phasor are determined. 2 (v1 + ej 2π/3 v2 + e− j 2π/3 v3 ) (5) v ss = 3Vref 2 iss = (i1 + ej 2π/3 i2 + e− j 2π/3 i3 ) (6) 3Iref Reference voltage Vref and reference current Iref are the peak values of the rated phase voltage and current, respectively. The subscript index s refers to the stator side of the induction machine, whereas the superscripted s represents affiliation to the stator fixed reference frame. The real axis of the stator fixed reference is aligned with the first winding axis of the stator winding. This reference frame is certainly not rotating. The second frame is the rotor fixed reference frame. The phase angle between real axis of the rotor fixed reference frame and the real axis of the stator fixed reference frame is the angle, γ (fig. 1). Any space phasor can be transformed from the stator to the rotor reference frame and vice versa: irs = iss e− j γ

(7)

A. Equivalent circuits It is assumed that the resistive and reactive parameters of an induction machines are known. These parameters are rs (stator resistance), x0sσ (stator leakage reactance), x0m (magnetizing reactance), x0rσ (rotor leakage reactance), rr0 (rotor resistance). The stationary equivalent circuit associated to these parameters is shown in fig. 2(a). It is shown in [16], that the machine parameters x0sσ , x0m , x0rσ and rr0 can be transformed into a set of equivalent parameters xσ , xm and rr which are associated to the equivalent circuit of fig. 2(b). Either of the equivalent circuits of fig. 2 has the same coefficient of leakage x02 m (x0m + x0sσ )(x0m + x0rσ ) xm , =1− xm + xσ

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(8)



 


















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However, such transformation from one to the other equivalent circuit is invariant with respect to the stator current space phasor, power and torque. The following transformations are valid without any restrictions and can be applied in order to simplify machine equations: rr = rr0 xm



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(10)

(11) (12)

(13)

The voltage model utilizes both the measured voltage space phasor, v ss , and the current space phasor, iss , in order to determine the stator flux linkage phasor, λss , in the stator fixed reference frame (fig. 3): dτ

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can be used to substitute the rotor current space phasor, irr , by the stator current space phasor, irs . This leads to a differential equation of the rotor flux phasor, λrr , where the only input quantity is the stator current space phasor with respect to the rotor fixed reference frame (fig. 3): dλrr xr r 1 (17) = i − λr dτ τr s τr r The parameters of the current model are the rotor reactance, xr , and the rotor time constant, τr . D. Torque difference

B. Voltage model

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The current model utilizes the rotor voltage equation in the rotor fixed reference frame. dλrr = −rr irr (15) dτ The applied equation refers to the rotor fixed reference frame. If it is assumed that the total leakage inductance is assigned to the stator side according to fig. 2(b), the rotor voltage equation can be simplified. Rotor flux linkage equation λrr = xm (irs + irr )

For the equivalent circuit of fig. 2(b) the transformed rotor current phasor is: ir = i0r

Fig. 3. Voltage and current model of the Vienna Monitoring Method

(14)

This equation is derived from the stator voltage equation of a symmetrical machine with respect to the stator fixed reference frame. In this equation τ is p.u. time, which is characterized by a period of 2π at rated frequency. The only required parameter of the voltage model is the stator resistance, rs . This model does not evaluate any inductances. Therefore, (14) can be used to determine the stator flux linkage only by integrating the inner voltage, v ss −rs iss . In practice some measures have to be taken in order to stabilize the open integration of (14).

Torque is determined by the cross product of the flux and current vector. The corresponding space phasor equation derives the negative imaginary part of a flux and a conjugate complex current space phasor. The torques associated with the voltage model (superscript v) and the current model (superscript i) are: tv = − Im(λss is∗ s )

(18)

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(19)

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Relative torque difference is the difference of both the determined torques (18) and (19), divided by the estimated (averaged) load torque, which can be estimated from (18), too: tv − ti (20) tload If the rotor cage is symmetrical, both calculated torques are equal and torque difference vanishes. However, for low load conditions (less than approximately 40% of rated load), both the numerator and the denominator of (20) are small numeric values, which ∆t =

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makes the computation of (20) problematic. Hence, such load conditions should be excluded from the evaluation. The magnitude of the relative torque difference is independent of load torque. Therefore, the VMM uses the magnitude of the relative torque difference to assess rotor asymmetries. Additional investigations showed that the magnitude of the relative torque difference may vary only slightly with the actual motor design [17]. E. Spatial data clustering technique Two steps are taken to condition the relative torque difference. The first step is making the signal time independent and the second step is to apply a filtering (data clustering) technique to suppress measurement noise and disturbances. In case of a rotor fault, the frequency of the relative torque difference is slip dependent. The determined rotor flux space phasor (17) rotates with slip frequency since this phasor refers to the rotor fixed reference frame. One full revolution of the rotor flux space phasor corresponds with two periods of the fault-specific oscillation (reciprocal of double slip frequency). The spatial angle (21) γλ = arg(λrr ) covers one spatial period (2π) during this time period. Therefore, the torque difference is evaluated versus the angle (21) to make the waveform of (20) time independent. The second step is the data clustering. This means that the angle is discretized into 32 equidistant cluster segments. The data value associated with each (discrete) cluster is the averaged instantaneous relative torque difference for that the angle, γλ , coincides with the actual cluster. The averaged cluster values are determined recursively. This algorithm is applied for an evaluation time of several seconds to achieve a representative discrete pattern of the waveform of the relative torque difference. Essentially, the waveform only contains a second harmonic (fig. 4). Additional multiples (fourth, sixth, harmonic etc.) may arise with a much more deteriorated magnitude, however. The discrete waveform pattern is subject to a discrete Fourier analysis to determine the second harmonic. The magnitude of this component serves as fault indicator of the VMM. The phase angle of the achieved waveform is related with the location of the faulty bar and end ring segment. Physical details about this coherence are presented in [2].

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Fig. 5. Fault indicator for stationary operation as a function of load torque; three broken rotor bars

IV. Measurement results Measurement results are obtained for a four pole, 18.5 kW squirrel cage induction machine with 40 rotor bars. Three adjacent bars were completely cracked to cause a severe rotor fault. First, the VMM was applied to stationary operating conditions. The determined fault indicator versus load torque is depicted in fig. 5. This graph demonstrates the load independence of the fault indicator of the VMM. The second investigation refers to the operation at constant supply frequency. Rotor fault detection was applied for variable load torque. However, reliable fault detection can be performed with sufficient load torque only. Fault detection was therefore idle as long as the p.u. torque was less than 0.4. Measured p.u. voltages, current, power and frequency are depicted in fig. 6. The corresponding fault indicators are shown in fig. 7. The third investigation was performed with an open loop inverter in order to operate the machine at variable speed. Operating conditions are shown in fig. 8, the corresponding fault indicator is presented in fig. 9. The depicted p.u. operating conditions are averaged values with respect to a period of 20 s. The fault evaluation period also equals 20 s. The load profile is therefore smoothed in the shown curves although the actual profile has some slow transients with a slope of 8 Nm s as well as some steps (60 Nm → 120 Nm in a few ms) were performed. Each data point of the fault indicators in fig. 7 and 9 represents the average value of ten successive determined fault indicators. This averaging technique suppresses outliers and does therefore not cause overrating of a rotor fault. Measurement results show that the obtained fault indicator remains within a tolerance band of less than 0.01. V. Conclusions In this contribution the Vienna Monitoring Method (VMM) was applied to stationary and transient operating conditions. The measurement results under transient operating conditions were presented for a mains supplied and an inverter fed induction machines. In both these cases, the machine was loaded with transient load torque. The presented results showed a stable rotor fault indicator for stationary and transient supply conditions. Therefore, the measurement results of this paper prove that the VMM is also able to

determine a repeatable and stable fault indicator for stationary and transient operating conditions. References [1] C. Kral, F. Pirker, and G. Pascoli, “Influence of inertia on general effects of faulty rotor bars and the Vienna Monitoring Method,” Conference Proceedings of the IEEE International Symposium on Diagnostics of Electrical Machines, Power Electronics and Drives, SDEMPED, pp. 447– 452, 2001. [2] R. Wieser, C. Kral, F. Pirker, and M. Schagginger, “The Vienna induction machine monitoring method; on the impact of the field oriented control structure on real operational behavior of a faulty machine,” Annual Conference of the IEEE Industrial Electronics Society, IECON, pp. 1544– 1549, 1998. [3] R. Schoen and T. Habetler, “Evaluation and implementation of a system to eliminate arbitrary load effects in current-based monitoring of induction machines,” Conference Proceedings of the IEEE IAS Annual Meeting, pp. 671– 678, 1996. [4] S. Nandi, R. Bharadwaj, H. Toliyat, and A. Parlos, “Study of three phase induction motors with incipient rotor cage faults under different supply conditions,” 1999. ThirtyFourth IAS Annual Meeting. Conference Record of the 1999 IEEE Industry Applications Conference, vol. 3, pp. 1922– 1928, 1999. [5] W. Thomson and M. Fenger, “Current signature analysis to detect induction motor faults,” IEEE Industry Applications Magazine, vol. 7, pp. 26–34, January 2001. [6] A. Bellini, F. Fillipetti, G. Franceschini, C. Tassoni, R. Passaglia, M. Saottini, and G. Tontini, “ENEL’s experience with on-line diagnosis of large induction motors cage failures,” Conference Record of the IEEE Industry Application Conference, IAS, 2000. [7] C. Kral, T. Habetler, R. Harley, F. Pirker, G. Pascoli, H. Oberguggenberger, and C. Fenz, “A comparison of rotor fault detection techniques with respect to the assessment of fault severity,” Symposium on Diagnostics for Electric Machines, Power Electronics and Drives, SDEMPED, pp. 265– 270, 2003. [8] S. Cruz and A. Cardoso, “Rotor cage fault diagnosis in three-phase induction motors by the total instantaneous power spectral analysis,” 1999. Thirty-Fourth IAS Annual Meeting. Conference Record of the 1999 IEEE Industry Applications Conference, vol. 3, pp. 1929–1934, 1999. [9] R. Wieser, C. Kral, F. Pirker, and M. Schagginger, “On-line rotor cage monitoring of inverter fed induction machines, experimental results,” Conference Proceedings of the First International IEEE Symposium on Diagnostics of Electrical Machines, Power Electronics and Drives, SDEMPED, pp. 15–22, 1997. [10] R. Wieser, C. Kral, F. Pirker, and M. Schagginger, “High sensitive rotor cage monitoring during dynamic load operation the Vienna Monitoring Method,” Conference Proceedings of the International Conference on Electrical Machines, ICEM, pp. 432–437, 1998. [11] R. Wieser, C. Kral, F. Pirker, and M. Schagginger, “On-line rotor cage monitoring of inverter-fed induction machines by means of an improved method,” IEEE Transactions on Power Electronics, vol. 14, pp. 858–865, September 1999. [12] C. Kral, F. Pirker, and G. Pascoli, “Influence of load torque on rotor asymmetry effects in squirrel cage induction machines including detection by means of the Vienna Monitoring Method,” Conference Proceedings EPE, 2001. [13] C. Kral, F. Pirker, G. Pascoli, and H. Oberguggenberger, “On the sensitivity of induction machine rotor cage monitoring—the Vienna Monitoring Method,” Conference Proceedings of the symposium on Power Electronics, Electrical Drives, Automotion and Drives, speedam, pp. B1/13 – B1/18, 2000. [14] C. Kral, F. Pirker, and G. Pascoli, “Model based detection of rotor faults without rotor position sensor—the sensorless Vienna Monitoring Method,” Symposium on Diagnostics for Electric Machines, Power Electronics and Drives, SDEMPED, 2003. [15] C. Kral, F. Pirker, G. Pascoli, and C.-J. Fenz, “Influence of load torque on the detection of rotor faults by means of the sensorless vienna monitoring method,” Conference Proceedings of the Second IEE International Conference on Power Electronics, Machines and Drives, PEMD, vol. 1, pp. 108–113, 2004. [16] H. Kleinrath, “Ersatzschaltbilder f¨ ur Transformatoren und Asynchronmaschinen,” e&i, vol. 110, pp. 68–74, 1993.

[17] C. Kral, F. Pirker, G. Pascoli, and H. Oberguggenberger, “Influence of rotor cage design on rotor fault detection by means of the Vienna Monitoring Method,” International Conference on Electric Machines, ICEM, 2002.

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