AÑOS TERMINADOS EN DICIEMBRE 31, 2018 Y 2017
NOTA 2 PRINCIPIOS Y PRACTICAS CONTABLES
2.1 Condition monitoring criterion
When a synchronous machine is operated as a generator, the prime mover drives the generator at synchronous speed, vs, as
shown inFig. 1. In steady state, the mechanical torque Tpmof the prime mover balances with the electromagnetic torque Tem produced by the generator and the electro-mechanical loss torque Tloss, that is
Tpm¼TemþTloss (1)
For a well designed, large synchronous generator, with an efficiency in excess of 96%, the loss torque Tlosswill be small
and can be neglected. Accordingly, (1) can be rewritten as
Tpm’ Tem (2)
The electromagnetic torque produced by a three-phase non- salient pole synchronous machine is[9]
Tem¼3 Ea Va vsXa ¼3EaVa vsXa sin d (3)
where Ea refers to the phasor per-phase induced emf in the winding, Va the phasor per-phase terminal voltage, Xa the synchronous reactance and d the load angle between Vaand
Ea. FromFig. 1, the circuit equation can be derived as
Va¼ Ea jXaþRa
Ia (4)
where Iais the phase current. As the winding resistance Rain
large synchronous generators is much smaller than the synchronous reactance Xa, (4) can be approximately
expressed as
Va’ EajXaIa (5)
Substituting (5) into (3), gives
Tem¼3 Ea EajXaIa vsXa ¼3Ea EajXaIa vsXa sin d (6)
For a phase winding with N turns in series, the induced emf from Faraday’s law is
Ea¼ Nkw df
dt (7)
where kwis the winding factor, which includes the distribution and pitch factors; f ¼ ^fsin vset
is the flux per pole of a round rotor revolving at angular velocity vs,where vse refers to the electrical supply frequency and vs ¼vse=p where p is the number of pole pairs. As the flux revolves, the emf in the phase winding can be expressed as
Ea¼ Nkwvsef^ cos vset
(8) Figure 1 Synchronous machine operated as a generator
2 IET Renew. Power Gener., 2009, Vol. 3, No. 1, pp. 1 – 11
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The Institution of Engineering and Technology 2008 doi: 10.1049/iet-rpg:20080006and Ea EajXaIa / E2 a¼N 2 k2wv2sef^2 cos2 vset (9) Further substitution of (9) into (6), gives
Temð Þ /t 3 N2k2wv 2 sef^ 2 cos2vset vsXa sin d ¼3N 2 k2wvsf^ 2 cos2vset p2X a sin d (10)
when load, d, and flux, f, are constant, then Tem/
vs Xa
(11) In steady-state conditions Tpmis approximately balanced with Tem; see (2), then
Tpm/ vs Xa
(12) This is why the mechanical torque Tpmcan be regarded as a valid indicator of the running condition of synchronous machines. However, as large wind turbines operate at variable rotational speed, the effects of mechanical/electrical faults in Tpm are hidden and difficult to distinguish. Therefore to remove the effect of the variable speed, a condition monitoring criterion C is proposed
C ¼Tpm
vr (13)
where vr ¼‘vsrefers to the rotational speed of the shaft, l a positive constant representing the gear ratio of a gearbox if fitted. All drive train mechanical faults of the wind turbine, for example in the shaft or gearbox, will have a response in C, and owing to C / 1=Xa all electrical faults, for example stator and rotor winding faults, will also have response in C. The criterion C is therefore versatile for machine condition monitoring and can be used to monitor drive train mechanical and generator electrical faults.
2.2 Test rig
The proposed criterion was demonstrated by experiments performed on a wind turbine condition monitoring test rig with a synchronous generator, shown in Fig. 2 [9]. The test rig comprised a 50 kW DC drive motor for speed control, a two-stage gearbox and a three-phase generator. The synchronous generator used was slightly unusual, being a permanent magnet machine, the small-scale prototype of a lightweight synchronous machine originally designed for application on large direct-drive wind turbines [10]. The machine had 84 coils on the stator and 108 permanent magnets on the rotor and the output of each coil was rectified, then coupled to a DC bus and fed to a resistance
load bank. The load was therefore fixed and did not represent the variable condition of an infinite bus. However, the machine had advantages in that it was a synchronous machine that allowed easy application of electrical and mechanical faults.
The system was instrumented using LabVIEW such that a variety of wind speed inputs could be applied and the relevant signals could be collected from the drive train and terminals of the generator. In the experiments, the speed of the DC motor was controlled by an external model, in which the properties of both natural wind and the turbine rotor aerodynamic behaviour were incorporated [11]. Both the generator electrical and wind turbine mechanical faults could be simulated on the test rig. A number of transducers had been fitted to the rig to measure the shaft rotational speed, torque, vibration and the load DC current and voltage of the generator, as illustrated inFig. 3.
Shorted-coil faults in the stator winding could be applied to the generator with the aid of remote relays, installed on the stator, to enable either one or more coils to be shorted or connected normally; the arrangement is shown inFig. 4.
2.3 Validity of the criterion
The effectiveness of criterion C in monitoring the running condition of the wind turbine generator is demonstrated in
Fig. 5, by measuring the torque-speed characteristic of the generator at half load and full-load before and after the three coils were shorted. The results from the half-load test with and without faults are compared inFig. 5.
From Fig. 5, it is confirmed that for a direct-drive wind turbine synchronous generator:
i. there is a linear positive proportional relationship between mechanical torque Tpm and shaft speed vs and the relationship depicted in (12) is correct;
Figure 2 Test rig of the wind turbine with a synchronous generator
ii. the criterion C is sensitive for monitoring changes in wind turbine running condition.
The transient load and dynamic torque conditions experienced by a wind turbine indicate that Tpm is not always balanced with Tem and transient changes in Tloss must also be taken into account. However, observation of the torque-speed characteristic under transient conditions shows that the quantity C can still be considered a valid condition monitoring criterion, a full transient analysis being beyond the scope of this paper.
Further experiments will reveal that criterion C is also effective in detecting the presence of drive train mechanical faults.
2.4 Experiments demonstrating use of
the criterion
In the first experiments, the generator was partially loaded with an aerodynamic load, representing a real wind driving condition, over a long period of 900 s and three stator coils were successively shorted and unshorted by operating the remote relays. The corresponding shaft rotational speed and torque signals were monitored at a data rate of 1 kHz as
shown in Fig. 5. The shaft speed was derived from a DC tachometer, which had significant intrinsic noise, and was converted to the generator speed using the gear ratio. In
Fig. 6, the upper subplot shows the converted generator shaft speed; the lower subplot shows the torque.
2.5 Wavelet theory
FromFig. 6, the variations because of wind driving are clearly visible and the noise originating from various sources is present in both speed and torque signals, regardless of generator condition. This significantly increases the difficulty of condition monitoring the wind turbine. To reduce the noise WTs were employed. Two basic approaches are possible to reduce noise, the first based on the singularity information analysis with wavelets [12] and the second on thresholding wavelet coefficients [13]; the second was adopted in this work, in view of its much simpler computational algorithm. Wavelets are families of functions obtained by the dilation and translation of a mother wavelet c(t). The daughter wavelets ca,b(t) at scale a and translation b may be expressed mathematically as
ca,b(t) ¼ 1ffiffiffi a p c t b a (14) Figure 3 Locations of the transducers fitted to the test rig and circuit arrangement of the synchronous generator
4 IET Renew. Power Gener., 2009, Vol. 3, No. 1, pp. 1 – 11
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The Institution of Engineering and Technology 2008 doi: 10.1049/iet-rpg:20080006The CWT of a signal x(t) is implemented by the following equation CWTa,b(t) ¼ ð1 1 x(t)ca,b(t) dt (15)
Wavelet-based noise reduction is accomplished by using the DWTs. The dyadic DWT is a special form of the CWT with dilation a ¼ 2j and translation b ¼ 2jn, that is
DWTj,n(t) ¼ CWT2j,2jn(t) ¼ 1 ffiffiffiffi 2j p ð1 1 x(t)c t 2jn dt, ( j, n [ Z) (16)
The practical calculation of the DWT is conducted through a digital filter tree combined with decimation blocks [12], as shown inFig. 7.
In the figure, g(t) is a low-pass filter and h(t) is a high-pass filter. At each level of the filter tree, the input is decomposed into two frequency bands; the high-frequency band shows the ‘details’ and the low-frequency band the ‘smoothed approximations’ of the input signal. Finally, the signal x(t) is expressed in the following discrete form
x(t) ¼ X 1 n¼1 a0,n(t)g0,n(t) þ X1 j¼0 X1 n¼1 dj,n(t)hj,n(t) (17) Figure 4 Remote relay for simulating short-coil fault and
circuit arrangement
Figure 5 Generator torque-speed curves before and after the coils were shorted
Figure 6 Shaft rotational speed and mechanical torque signals
where a0,n(t) ¼ Ð1 1x(t)g0,n(t) dt dj,n(t) ¼ Ð1 1x(t)hj,n(t) dt ( (18) with gj,n(t) ¼ 2 j=2 g 2 jt n hj,n(t) ¼ 2j=2h 2jt n ( , j [ Zþ, n [ Z (19) After applying the thresholding technique, the reconstructed de-noised signal, ^x(t) can be obtained by using the same pyramidal scheme as that used for the DWT. In practice, two thresholding approaches are possible. They are
i. Hard thresholding ^dj,n(t) ¼ dj,n(t) 0 if dj,n(t) . u if dj,n(t) u (20)
ii. Soft thresholding
^dj,n(t) ¼ dj,n(t) u 0 dj,n(t) þ u 8 < : if dj,n(t) . u if dj,n(t) u if dj,n(t) , u (21)
In the equations, ^dj,n(t) are the thresholded values of dj,n(t). The threshold u can be estimated as follows[13]
u ¼ spffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 log (L) (22) Where s is an estimate of the noise level, L the number of wavelet coefficients in the current level. Then, the de- noised signal ^x(t) may be constructed by using the following equation ^x(t) ¼ X 1 n¼1 ^a0,n(t)g0,n(t) þX 1 j¼0 X1 n¼1 ^dj,n(t)hj,n(t) (23) The calculation of ^a0,n(t) is similar to that of ^dj,n(t). The hard thresholding approach is computationally simpler, but the use of a fixed threshold value inevitably leads to discontinuities in the de-noised signal. By contrast, soft thresholding solves the problem by using a variable threshold; for this reason, it was used so that the smoothness of the de-noised signal could be guaranteed. The noise cancellation operation may be fulfilled using the MATLAB functions ‘wavedec.m’ and ‘waverec.m’, the former for multilevel wavelet decomposition and the latter for multilevel wavelet reconstruction. The de-noised torque signal, obtained using Daubechies mother wavelet ‘db16’, is also shown in Fig. 6, where the noise in the original signals has been dramatically removed. As ‘db16’ is
a well defined mother wavelet and its coefficients can be accessed by running MATLAB function ‘WFILTERS.m’, its detailed description is not repeated here. The criterion C is then calculated by using the de-noised signals and the results are shown in Fig. 8, in which C derived from the original signals are also shown for comparison. The signal in Fig. 8 reduces with time because of the rising temperature of the generator windings during the test, which increases the generator winding resistance.
From Fig. 8, it is found that when the three coils are shorted, the criterion C gives increased sensitivity, which is explicable through (12) and (13) that the shorted coils reduce the generator synchronous reactance Xa, which
therefore results in a larger mechanical torque Tpm during the fault to achieve the same shaft rotational speed vr. The comparison of the results shown inFigs. 5 and8show that the criterion C becomes more sensitive to changes in machine condition when the generator is fully loaded.