Wavelet Transform Application in Active Power Filter Used for Slip Energy Recovery Drives H.T.Yalazan, T. Sürgevil, and E. Akpinar Department of Electrical and Electronics Engineering, Dokuz Eylül University, Kaynaklar Campus 35160 Buca, Izmir, TURKEY tel.: 9+ 0 232 4127145, 9+ 0 232 4127163, fax: 9+ 0 232 4531085, e-mail:
[email protected],
[email protected],
[email protected] Abstract-- In this paper, discrete wavelet transform (DWT) and symmetrical component (SC) analysis are used together for extracting the fundamental component and oscillations in supply current drawn by slip energy recovery drive (SERD). Discrete Meyer wavelet is used as mother wavelet due to its orthogonal and biorthogonal properties to generate reference signal for active power filter (APF) in order to compensate current harmonics. Simulations are performed using Matlab Simulink and Wavelet Toolbox. Simulation results show that the DWT algorithm can abstract the fundamental component of supply currents effectively and the oscillations in the supply currents are considerably reduced when the DWT is combined with SC analysis for generating the reference current of the APF.
I. INTRODUCTION Wavelet transform is used in many power-engineering applications. Some of these are power disturbance detection and localization, power disturbance data compression and storage, power disturbance identification and classification, power devices protection and power disturbance network analysis [1]. The time -frequency localization property of wavelets makes them suitable for the analysis of signals encountered in power engineering problems and electrical machines’ noise analysis. The motor drives generate harmonics distortion and this affects power quality. The slip energy recovery drive (SERD), which has a rectifier and inverter connected between the rotor and power system, also creates harmonics in the supply currents. The active power filters (APFs) have been widely used to control harmonic distortion [2]. The APFs use power
converters in order to inject harmonic components that cancel out the harmonics in the source currents. Detection and processing of the harmonics are one part of the APF, while the generation of them by means of switching is the other part of APF. The first and most important step in compensating harmonics distortion is to generate the control reference signals quickly and accurately; i.e. harmonics detection [3]. In this study, wavelet decomposition and reconstruction algorithm is used as well as applying symmetrical component (SC) analysis for detecting the harmonics and oscillations in supply current drawn by slip energy recovery drive. Discrete Meyer wavelet is used as mother wavelet due to its orthogonal and biorthogonal properties to generate reference signal for APF. Simulation results show that the wavelet algorithm can abstract the fundamental component of supply currents effectively. The extracted fundamental components of the supply currents are processed via SC analysis and the fundamental positive sequence component is obtained. II. WAVELET TRANSFORM In this section some properties of wavelet transform will be examined. Let’s consider a real or complex-value continuous time function ?(t). This function is a mother wavelet or wavelet if it integrates to zero (1), has finite energy (2) and satisfies the admissibility condition given in (3). ¥
òy ( t) dt = 0
-¥ ¥
ò y (t)
2
dt < ¥
(2)
-¥
¥
Cº
ò
-¥
1-4244-0891-1/07/$20.00 2007 IEEE
(1)
y ( w) w
2
dw
(3)
The continuous wavelet transform of a signal f(t) that is defined in L2(R) is given in (4), where the asterisk denotes complex conjugate, and a, b Î R, a ? 0, are the scale parameter and translation parameter, respectively. A given function f(t) can be synthesized from translates and dilates of the mother wavelet by using (5) if the value of C given in (3) is such that 0 < C < 8 [4]. ¥
W ( a, b) º
ò f (t)
-¥ ¥
æt -b ö y *ç ÷ dt a è a ø
1
(4)
¥
1 1 f (t ) = W ( a, b)y a,b (t ) da db ò ò C a= -¥ b=-¥ a 2
(5)
waveform is extended by a number of periods [9]. In Fig.2, the output of the DWT is shown when the original signal is extended by 6 periods. The extraction error is reduced in following virtual cycles. Hence, the corresponding sampled set of output is chosen from the middle of the produced output and the accuracy of the DWT increases in that way. The response of DWT against the rapid amplitude variations is shown in Fig.3, when real time application is considered with 6 periods signal extension. The extracted signal is obtained with approximately one cycle delay. In order to reduce this delay, a least squares approximation based method was proposed in [9].
where
y a ,b (t ) =
æt - bö yç ÷ a è a ø
1
(6)
The discrete wavelet transform (DWT) can be obtained by using discrete values for the scale parameter a, and the translation parameter b. In the case of dyadic transform a=2k and b=2kl, where k and l are integers. DWT provides a decomposition of a signal into subbands, where each spectral band is approximately one octave wide [1]. The sequence f(n) obtained after digitizing original signal f(t), can be decomposed into an approximation signal a j and a detail signal dj . Approximation signal aj is a low-resolution representation of the original signal; on the other hand, detail signal dj is the difference between two successive low-resolution representations. The original signal can be expressed as: f ( n ) = d 1( n ) + a1 ( n ) M
= d1 ( n ) + d 2 ( n ) + a 2 ( n ) =
åd
j (n )+aM ( n )
Fig.1. Reconstruction of a sampled set of sine signal via DWT.
(7)
j =1
where
a j (n) = a j +1 ( n) + d j +1 ( n)
(8)
In real-time applications of DWT, it is necessary to obtain a sampled-based output at corresponding time of input signal. The DWT transforms a sampled set of signal at its input to another sample set of signal in the frequency band of interest at its output. In order to obtain the DWT of the measured signal, the sampled set of measured signal is stored in a circulating buffer. The buffer length is N, which corresponds to the number of samples in one period of the signal. When the signal data in buffer is processed via DWT, an error in extraction of the signal is occurred as shown in Fig.1. In order to solve this problem, the sampled set of original
Fig.2. Reconstruction of a sampled set of sine signal via DWT with extending the original signal by 6 similar periods.
é ù 1 3 S 90 ( f b - f c )ú ê fa - ( fb + fc )+ 2 2 é f a+ ù ê ú ê -ú 1ê ú 1 3 ê fa ú = ê f a - ( fb + fc )S 90 ( f b - f c )ú 2 2 ê 0ú 3ê ú êë f a úû ê ú f a + fb + f c ê ú úû ëê
(10)
where S90 stands for 90o phase-shift operation. The fundamental symmetrical components of other phases can be obtained in the same way as shown in (10). Fig.3. Performance of DWT against the stepped amplitude variations.
III. CALCULATION OF SYMMETRICAL COMPONENTS Any unbalanced system of n set of signals can be resolved into three balanced set of signals, which are called symmetrical components and have been widely used in the analysis of unbalanced faults and systems as well as other purposes such as protection, design, control and compensation [6-8]. The instantaneous symmetrical components of three-phase quantities (voltages or currents) can be expressed as follows [7-8]:
é f a+ (t)ù é1 a ê - ú 1ê 2 ê f a (t)ú = ê1 a 3 ê f 0 (t) ú ê1 1 ë ë a û
f a 2 ù é f a (t) ù ú úê a ú ê f b f (t) ú 1 úû ê f c f (t) ú ë û
(9) j1200
where a = e corresponds to 120o phase shift operation in the time domain and superscript f identifies the fundamental component. Equation (9) can be also expressed in terms of to 90o phase-shift operator for easier realization and can be implemented in various ways as discussed in [6]. The authors of [7] and [8] suggest an enhanced phase-locked loop (EPLL) circuit in order to obtain fundamental components and their 90o phase-shifted signals. In this study, the fundamental components are obtained via wavelet transform and then their fundamental symmetrical components are calculated using the following formula [7]:
IV. SIMULATION RESULTS The slip energy recovery drive is simulated in MATLAB Simulink using the model parameters given in [5]. The load torque is set to 19.1Nm. The source current of phase-a is sampled using 4800 samples per second. Discrete Meyer wavelet is used as the mother wavelet. Six-level wavelet decomposition is applied to source currents using MATLAB Wavelet Toolbox, which have positive and negative sequence harmonics. The fundamental component of the SERD current signals are obtained in the frequency range of interest during the reconstruction stage of DWT. The frequency bands obtained by using six-level wavelet decomposition are shown in Table I. The upper frequency band limits of the approximate signal are determined by the sampling frequency (fs) and the number of wavelet decomposition level (k), and can be expressed as f s / 2k +1 . The detail component d 6 contains fundamental, while the approximate component a6 contains subharmonics. The higher order harmonics are contained by the detail parts of reconstruction from d 1 to d 5. In simulations, the currents drawn by SERD system was recorded and processed via DWT offline. T ABLE I. FREQUENCY BANDS FOR SIX-L EVEL DECOMPOSITION
Decomposition coefficient
Frequency (Hz)
d1 d2 d3
1200 – 2400 600 – 1200 300 – 600
d4 d5
150 – 300 75 – 150
d6 a6
37.5 – 75 0 – 37.5
In Figure 4a, the phase current of the SERD at steady-state is shown. The fundamental phase currents and subharmonic components (having a frequency band of 0-75Hz, which is the sum of a6 and d 6) are extracted via 6-level wavelet decomposition. These current waveforms have oscillations, which yields to oscillating positive and negative sequence components as shown in Figure 4b. Here it must be noted that no zero sequence component exists since the system has no neutral line. When the fundamental phase current waveform is obtained from the detail signal d6 within 37.5-75Hz frequency band via 6-level wavelet decomposition, the oscillations in the fundamental current waveform are observed to be reduced under this operating condition of SERD and as a result a very small amount of negative sequence current is calculated as shown in Figure 4c. Simulations show that the low frequency oscillations in the supply currents can be treated as subharmonics caused by the torque and speed oscillations. When conventional FFT method is used for obtaining fundamental symmetrical components, it is observed that the oscillations in the supply currents are treated as current imbalance in the system and a fundamental negative sequence component exists. As long as it is aimed to obtain fundamental positive sequence component and subtract it from the actual supply currents, the oscillations can be suppressed via an active power filter using both FFT and DWT methods.
Figure 4. (a) Phase current (b) calculated instantaneous symmetrical components via 6-level wavelet decomposition within 0-75Hz. (c) calculated instantaneous symmetrical components via 6-level wavelet decomposition within 37.5-75Hz.
The proposed method was also tested for a different operating condition under unbalanced and distorted supply voltages. Fig.5 shows the distorted supply voltages, which contains a fundamental negative sequence component with 0.15 pu amplitude and a 5th order negative sequence harmonic component with 0.1 pu amplitude. Under this condition, the currents drawn by the SERD are severely distorted as shown in Fig.6. These currents are processed through the DWT and their symmetrical components are calculated. The extracted fundamental component waveforms, which are obtained within the frequency band of 0-75 Hz via DWT, are shown in Fig.7. In Fig.8, extracted fundamental positive, negative and zero sequence components of phase a current are shown. Fig.9 shows the extracted fundamental positive sequence of three phase supply currents . It is observed that the current imbalance in the supply currents is considerably suppressed. Fig.10 shows the extracted fundamental positive sequence components using d 6. It is clearly observed that the oscillation of the current amplitudes is approximately 12.5Hz. This amplitude oscillation is created by the supply current induced at the frequencies of 37.5Hz and 62.5Hz, which is in the frequency band of interest [10].
25
200
20
150
15 10
100
5
50
0
0
-5 -10
-50
-15
-100
-20
-150
-25 1.6
-200 1.6
1.61
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.7
Fig.5. Unbalanced and distorted supply voltages
1.61
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.7
Fig.8 Fundamental positive, negative and zero sequence components of phase a current. 30
40 20 30 10
20 10
0
0 -10 -10 -20
-20
-30
-30 1.6
-40 1.6
1.61
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.7
Fig.6 Supply currents drawn by the SERD under distorted supply voltages
1.61
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.7
Fig.9 Fundamental positive sequence components of three phase supply currents.(0-75Hz) 30
40 20
30 10
20 10
0
0 -10
-10 -20
-20 -30 -40 1.6
-30 1.6
1.61
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.7
Fig.7 Extracted fundamental supply current waveforms via DWT
1.61
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.7
Fig.10 Fundamental positive sequence components of three phase supply currents.(37.5-75Hz)
V. CONCLUSION A wavelet transform based harmonic detection method is applied to SERD. Simulation results show that this method is
successful in detecting the fundamental positive sequence component of the currents in SERD. However, the extraction of positive sequence fundamental component is obtained within a limited frequency band. This yields the harmonics nearby the fundamental frequency in this band causing low frequency amplitude oscillations. These oscillations cannot be completely suppressed by DWT. The fundamental component positive sequence component is subtracted from the actual load current in order to obtain the reference currents for APF. This type of compensation reduces the current imbalance and voltage flickers caused by these oscillating currents at the common coupling point as well as eliminating the load current harmonics.
VI. ACKNOWLEDGEMENT This work was carried out as a part of project, “Power Quality National Projects”, sponsored by Turkish Scientific and Research Council and Turkish Electrical Power Transmission Co. (TEIAS) under contract 106G012.
VII. REFERENCES [1] C.H. Lee, Y.J. Wang and W.L. Huang, “A literature survey of wavelets in power engineering applications,” in Proc. Natl. Sci. Counc. ROC(A) , Vol. 24, No. 4, pp. 249-258, 2000. [2] H. Akagi, “New trends in active filters for power conditioning,” IEEE Trans on Ind. Appl., Vol. 32 (6), pp. 1312-1322, 1996. [3] H. Liu, G. Liu and Y. Shen, “A novel harmonics detection method based on wavelet algorithm for active power filter,” Proc. 6th World Congress on Intelligent Control and Automation, pp.7617- 7621, Dalian, China, 2006. [4] R.M. Rao and A.S. Bopardikar, Wavelet Transforms Introduction to Theory and Applications, Addison-Wesley, 1998. [5] Akpinar, E., Discussion on the qd/qd Model of Slip Energy Recovery Drive, Journal of Electrical Power System Research, vol: 77, no:1, pp 64 -70, Jan. 2007 [6] Ghartemnai, M.K., and Karimi, H., ‘Analysis of symmetrical components in time-domain’, 48th IEEE Midwe st Syposium on Circuits and Systems, vol.1, 7-10 August 2005, Kentucky USA, pp.28-31. [7] Iravani, M.R., and Ghartemani, M.K., ‘Online estimation of steady state and instantaneous symmetrical components’, IEE Proc.Gener. Transm. Distrib. vol.150, no.5, September 2003, pp.616622. [8] Ghartemani, M.K., Iravani, M.R., and Katiraei, F., ‘Extraction of signals for harmonics, reactive current and network-unbalance compensation’, IEE Proc.-Gener. Transm. Distrib. vol.152, no.1, January 2005, pp.137-143.
[9] Forghani, M. and Afsharnia, S., “Online wavelet transform-based control strategy for UPQC control system”, IEEE Trans. on Power Delivery, vol.22, no.1, January 2007, pp.481-491. [10] Papathanassiou,S.A., Papadopoulos, M.P., “On the harmonics of the Slip Energy Recovery Drive”, IEEE Power Engineering Review, April 2001, pp.55-57.
