Empirical avelet Transform based Single Phase Quality ndices T. Karthi Dept. of Electrical Engg. T ndore ndore, ndia phd300004@iiti.ac.in Amod C. Umariar Dept. of Electrical Engg. T ndore ndore, ndia Trapti Jain Dept. of Electrical Engg. T ndore ndore, ndia Abstract This paper presents an application of Empirical avelet Transform (ET) to the power signals for estimation of quality ndices (PQs). This technique first estimates the frequency components present in the distorted signal, computes the boundaries and extracts mono components based on the boundaries computed. Several stationary and non-stationary signals have been analyzed to show the effectiveness of this technique. The PQs obtained with the proposed technique has been compared with Discrete avelet Technique (DT) and the proposed technique is found to be effective and superior over DT. The results confirm that the ET efficiently extracts the mono component signals from the time varying distorted signal and hence this technique would be suitable for real time PQ indices estimation. Keywords Empirical avelet Transform (ET), Harmonics, EEE Std. 459-000, Quality ndices (PQs).. TRODUCTO Recently, Quality has gained significant importance with the deregulation of System and has become more critical due to the emergence of nonlinear device based new technologies. The power quality disturbances can be either stationary or non-stationary. Since most of the power quality disturbances are non-stationary in nature, an advanced signal processing technique is required to accurately decompose the non-stationary power signal into mono components and determine the instantaneous frequency. EEE std. 459-000 [], [] is a standard that provides definitions for calculating the traditional PQs. The fast computing Fourier Transform (FT), which is still being used, limits its application in analyzing the non-stationary signals because of the spectral leaage [3]. Moreover, FFT does not give time information i.e., where those spectral components appear in time. To overcome this limitation, the Short Time Fourier Transform (STFT) based PQs for aperiodic signals are suggested in [4]. The recent signal processing techniques for estimation of power quality indices and algorithms for classifying power disturbances are well explained in [5], [6]. The existing PQs have been reformulated and applied to the power signals using the Discrete avelet Transform (DT) [7], [8]. The DT provides non-uniform frequency bands, moreover, if suitable sampling frequency and adequate 978--4799-54-3/4/$3.00 04 EEE decomposition levels are not chosen, then two or more frequency components will be present in the single sub band. avelet Pacet Transform (PT) is an extension of the avelet Transform, which is effective in analyzing nonstationary signal and can accurately measure the electrical quantities [9]-[]. Even though the PT shows better accuracy, the frequency subdivision scheme is based on a constant prescribed ratio, limiting its adaptability. The suitable selection of sampling frequency and adequate choice of mother wavelet [] can minimize the leaage errors between wavelet levels [3], [4]. Therefore, it is revealed that the analysis of a signal with the DT or the PT requires a proper selection of mother wavelet, decomposition levels and sampling frequency. Moreover, the selection of these parameters differs for the signals containing different frequency components and this limits the application of DT and PT to analyze real time non-stationary signals. n this paper, a fast and adaptive wavelet technique, Empirical avelet Transform (ET) proposed recently [5], is considered to extract the individual frequency components and thereby compute the PQs. This method has an advantage of adaptability according to the analyzed signal and can isolate the different modes of the signal. The performance of this method has been investigated on ten different signals by taing time varying frequencies and phase angles into account. The accurate frequency estimation and adaptive wavelets maes this technique well suited to analyze the stationary and highly distorted non-stationary signals. The organization of this paper is as follows, section presents a brief review of ET technique required to analyze the signal. n section, the ET based PQ indices of single phase system are listed. Section considers several test signals and the PQ ndices are calculated for the coefficients obtained from the ET. This section also provides results which exhibit the effectiveness of this technique. Finally, conclusions are drawn in section.. EMPRCAL AELET TRASFORM n [5], Jerome Gilles proposed a new time frequency technique to decompose a multicomponent signal using the adaptive wavelets. This method wors as follows. i) Firstly, determine the frequency components of the input signal using FFT. ii) Then, different modes are extracted by proper
segmentation of the Fourier spectrum and iii) Apply scaling and wavelet functions corresponding to each detected support. Segmentation of the Fourier spectrum is the most important step that provides the adaptability to this technique according to the analyzed signal. Consider a discrete signal x() which is sampled at a frequency of f s. First, apply the FFT to the signal x(), obtain the frequency spectrum and then find the set of maxima,.., in the Fourier spectrum and deduce their corresponding frequency. Here M is the total number of frequency components present in the real signal. ow, with this set of frequencies,.., corresponding to maxima, obtain the boundaries of each segment (the dotted line in Fig. ) as the center of two consecutive maxima. ωi + ω i Ω + i = () where,, are the frequencies and Ω is their Ω= Ω corresponding boundary and the set is { } i i=,,..., M Fig. illustrates the detection of boundaries from the Fourier spectrum, where, the red colored triangles shown correspond to the maxima and the dotted lines represent the boundaries computed. t can be clearly seen from the figure that the algorithm is able to isolate the different modes. After obtaining the set of boundaries, a ban of M wavelet filters comprising of one low pass filter and M- band pass filters are defined based on the well detected boundaries. The expressions for Fourier transform of the empirical wavelets and scaling function are given by [5] if ( + γ) Ωi ω ( γ) Ωi+ π cos β ( γ, Ωi+ ) if ( γ ) Ωi+ ω ( + γ ) Ω i+ ψi ( ω) = π sin β ( γ, Ωi) if ( γ ) Ωi ω ( + γ ) Ω i 0 otherwise and if ω ( γ) Ω π φ( ω) = cos β( γ, Ω) if ( γ) Ω ω ( + γ) Ω (3) 0 otherwise where β ( γ, Ω ) = β ( ω ( γ) ) i Ωi γωi where, γ is a parameter to ensure no overlap between the two consecutive transitions areas and is an arbitrary function defined as 0 if x 0 β ( x) = if x β( x) β( x) if x 0, + = Equipped with this set of filters, the ET can be defined in the similar way as the normal wavelet transform. The approxi- () (4) Fig.. Fourier Spectrum with detected boundaries mate coefficients are obtained by the inner product of the applied signal x, with the empirical scaling function as given below x (, t) = x, φ = x( τ) φ( τ t) dτ (5) The detail coefficients are obtained by the inner product of applied signal x, with empirical wavelets as given below x ( i, t) = x, ψ i = x( τ) ψi( τ t) dτ (6) This shows that this technique can accurately extract the mono frequency components. Since the basis function is generated according to the information contained in the analyzed signal, this approach is more suitable for decomposing the real time signals. The steps required to analyze the signal and estimate the PQs using the ET are as follows.. Apply the FFT to the discrete signal x().. Estimate the frequency components of the applied signal.,, and then detect the boundaries Ω,Ω, Ω of each frequency,. 3. Perform filtering with the scaling function and empirical wavelets to extract the components of different modes. 4. Compute the single phase PQs using the formulas listed in section.. ET BASED SGLE PHASE POER QUALTY DCES This section presents the brief review of Single Phase PQ indices recommended in [] and [] and the ET based reformulated indices. Consider the single phase system having non sinusoidal periodic voltage and current containing the fundamental and harmonic components as and Hmax n ( π n θn) (7) vt () = sin ft n= Hmax n ( π n αn) (8) it () = i sin ft n= here, t is the time and H max is the maximum frequency
component present in the signal.,, are the amplitude, frequency and phase angle of the n th component of voltage signal, respectively and,, are the amplitude, frequency and phase angle of the n th component of current signal, respectively. ow, for this set of voltage and current, ET is applied and the corresponding mono component signal coefficients of voltage and current are obtained. The approximate coefficients are obtained by inner product of the input signal with the scaling function, and the detail coefficients are obtained by inner product of the input signal with the empirical wavelets ψ as explained in detail in section of this paper., = v( ), Φ,,, = v( ), Ψ,,,, i( ),, = Φ (9), i( ),, = Ψ (0) where, is the harmonic level. A. RMS calculations The RMS value of the non-sinusoidal voltage [] is defined as rms H > = + = + () where =, and =, The RMS value of the non-sinusoidal current [] is defined as rms H > = + = + () where =, and =, Here, and are the RMS values of the fundamental frequency components of the voltage and current, respectively. and are the set of RMS values of the harmonic voltage and harmonic current. Also, and, are the ET coefficients of voltage and current at level one and sample while, and, are the ET coefficients of voltage and current at any level higher than the level one and sample. is the total length of the signal. B. Total Harmonic Distortion (THD) The total harmonic distortion of voltage and current is as follows thd = = H > and thd H > = = C. Active The fundamental active power P is defined as =,, (3) P (4) The harmonic active power is defined as where, P H = P (5) > P M M =,, The total active power is defined as the sum of the fundamental active power and the harmonic active power P = P + PH (6) D. Apparent The fundamental apparent power is defined as S = (7) The current distortion power can be defined as D = (8) H The voltage distortion power can be defined as D = (9) v H The harmonic apparent power can be defined as S = (0) H H H The harmonic distortion power is defined as H H H The total apparent power is D = S P () rms rms v H S= = S + D + D + S () The non-fundamental apparent power is defined as v H S = D + D + S (3) The non-active power is defined as = S P (4) E. Reactive The fundamental reactive power is defined as Q = S P (5) The reactive power at each wavelet level can simply be computed from the corresponding active and apparent power at each wavelet level thus applying the time-domain concept at each wavelet level. The harmonic reactive power at each wavelet level is Q = S P (6) The harmonic reactive power can be define as Q H = Q (7) >
So the total reactive power also called Budeanu s reactive power can be defined as QB = Q + QH (8) F. Factors The displacement power factor is defined as P dpf = (9) S The total power factor is the ratio of total active power to total apparent power PF = P (30) S n order to measure the quality of the transmitted power especially the oscillation behavior, illems proposed the use of the oscillation power factor PF osc P PF = = P + S + PF (3) G. Harmonic Pollution The harmonic pollution is defined as the ratio of the nonfundamental apparent power to the fundamental apparent power S HP = (3) S. SMULATO RESULTS AD DSCUSSOS The ET technique has been verified using 0 test signals considering a variety of stationary and non-stationary conditions. However, due to the space constraints, simulation results of only signals are presented in the paper. The signals are simulated in the MATLAB environment with sampling frequency of 0 Hz i.e., 00 samples/cycle. First, the true values are computed as per the EEE standard definitions and then the PQs are calculated for the mono frequency components obtained from the ET. The percentage difference has been calculated for all the indices using the mathematical expression shown below by ndicestrue ndicescalculated % difference = *00 (33) ndicestrue To show the effectiveness of the ET-, DT-based PQ ndices are also calculated considering the same parameters proposed in [6] and the percentage difference obtained with these techniques has been compared with the ET technique. The sampling frequency, mother wavelet and decomposition levels of the DT are considered same for all the case studies to compare the performance of the DT and the ET. A. Case study-: Stationary signal n the first case, the algorithm is tested for stationary signals containing three frequency components each. The frequency, amplitude and phase angle of voltage and current signal are Fig.. TABLE. PARAMETERS OF OLTAGE SGAL Time (S) Frequency, Amplitude, 0.0-0. 50 50 350 0 3. TABLE. PARAMETERS OF CURRET SGAL Time (S) Frequency, Amplitude, Phase angle 0.0-0. 50 50 350 0.8 0.5 30 0 45 0 75 0 Extracted mono components of the distorted current signal TABLE. POER QUALTY DCES OBTAED ndices EEE standard ET based % difference Definitions ndices oltage = 7.07 3 =.3 7 = 0.8485 = 7.077 3=.53 7= 0.8497 0.06 0.88 0.44 RMS H =.847 H =.795 0.76 Current RMS THD Active Apparent Reactive Distortion Factor Harmonic Pollution rms= 7.430 =.44 3 = 0.5657 7 = 0.3536 rms= 7.430 =.444 3= 0.565 7= 0.3538 0 0.04 0.06 0.0566 H = 0.667 rms=.5636 H = 0.666 rms=.56365 0.649 0 thd= 0.33 thd = 0.33 0.476 thd= 0.477 thd = 0.473 0.0848 P = 8.660 P = 8.6639 0.047 P 3 = 0.8485 P 3 = 0.844 0.5068 P 7 = 0.0778 P 7 = 0.0783 0.647 P H = 0.963 P H = 0.95 0.40 P= 9.5865 P= 9.5864 S = 9.9999 S = 0.038 0.380 S 3 =.0 S 3 =.953 0.567 S 7 = 0.3006 S 7 = 0.3006 S=.687 S=.695 0.0069 Q = 5.000 Q = 5.00 0.040 Q 3 = 0.8485 Q 3 = 0.846 0.89 Q 7 = 0.898 Q 7 = 0.90 0.380 Q H =.383 Q H =.363 0.757 Q B= 6.383 D = 4.77 D = 3.30 S H =.54 D H =.03 S = 5.97 = 6.5645 dpf= 0.8660 PF = 0.85 PF osc= 0.7593 Q B = 6.375 D = 4.75 D = 3.4 S H =.596 D H =.076 S = 5.907 = 6.5660 dpf = 0.8660 PF = 0.850 PF osc = 0.759 0.030 0.044 0.05 0.953 0.983 0.098 0.09 0.0 0.03 HP= 0.597 HP = 0.5908 0.5
Fig. 3. Percentage difference for the two approaches for case study- TABLE. PARAMETERS OF OLTAGE SGAL Time (S) Amplitude, Frequency and Phase angle 0.0-0. 0,50,0 0-5,50,0 0 -,350, 0 0 0.-0.8 0,50,0 0,00,0 0 5,50,0 0 3.5,50,0 0,350, 0 0 0.8-0.37 0,50,0 0-5,50,0 0 3.5,50,0 0-0.37-0.50 0,50,0 0,00,0 0-3.5,50,0 0,350, 0 0 TABLE. PARAMETERS OF CURRET SGAL Time (S) Amplitude, Frequency and Phase angle 0.0-0. 4,50,30 0 -.8,50, - 0.7,350, 48 0 75 0 0.-0.8 4,50,30 0 0.8,00,.8,50,.5,50, 0.7,350, 60 0 48 0 6 0 75 0 0.8-0.37 4,50,30 0 -.8,50,.5,50, - 48 0 6 0 0.37-0.50 4,50,30 0 0.8,00, -.5,50, 0.7,350, 60 0 6 0 75 0 listed in Table and, respectively. The phase angles for all frequency components of voltage signal are zero. n Fig. the extracted mono frequency components of current using the ET are shown. t can be clearly observed that the ET technique is able to accurately estimate the actual frequency components present in the distorted signal. t can also be noticed that no two frequency components are combined in any mode and it do not overestimate the number of frequency components. Table shows the PQs obtained by the ET and also EEE standards. From the table, it is clear that the ET is efficient in estimating the indices with very small percentage difference. Fig. 3 depicts the percentage difference of PQs calculated by the DT and the ET. Since adequate selection of parameters was made for the DT, the results from DT are also closer to the true values. However, the ET provides better results than the DT. B. Case study-: onstationary signal To test the efficiency of the proposed technique for nonstationary signals, a non-stationary voltage and current signal of 5secs duration are simulated. The amplitude, frequency and phase angle of voltage and current signal are listed in Tables and, respectively. Fig. 4 shows the extracted mono components of the nonstationary current signal. t can be seen that the ET provides Fig. 4. Extracted mono components of the non-stationary current signal TABLE. POER QUALTY DCES OBTAED ndices EEE standard Definitions ET based ndices % difference oltage RMS = 7.07 =.077 3 = 3.044 5 =.576 7 =.806 = 7.070 =.0776 3 = 3.0395 5 =.568 7 =.808 0.07 0.0557 0.065 0.037 0.056 H = 4.087 H = 4.0856 0.039 Current RMS THD Active Apparent Reactive Distortion Factor Harmonic Pollution rms= 8.674 =.884 = 0.4308 3 =.0949 5 = 0.7397 7 = 0.448 rms= 8.673 =.883 = 0.4308 3 =.0950 5 = 0.7394 7 = 0.4485 0.080 0.0035 0 0.009 0.0406 0.0669 H =.4603 rms= 3.83 H =.4603 rms= 3.83 thd = 0.578 thd = 0.5777 0.059 thd = 0.563 thd = 0.563 0.0967 P = 7.304 P = 7.39 0.0087 P = 0.30 P = 0.36 0.586 P 3 =.8 P 3 =.59 0.03 P 5 = 0.3 P 5 = 0.8 0.449 P 7 = 0.486 P 7 = 0.485 0.0673 P H =.70 P H =.787 0.055 P = 0.0405 P = 0.0406 5 S = 9.999 S = 0.003 0.065 S = 0.464 S = 0.464 0.043 S 3 = 3.330 S 3 = 3.384 0.0480 S 5 =.596 S 5 =.5948 0.075 S 7 = 0.574 S 7 = 0.5745 0.087 S = 5.9978 S = 5.9978 Q = Q = 0.408 Q 3 =.4747 Q 5 =.59 Q 7 = 0.5544 Q H = 5.030 Q B = 5.030 D = 0.359 D =.560 S H = 5.9685 D H = 5.37 S = 6.6099 = 6.5608 dpf = 0.8660 PF = 0.7709 PF osc = 0.7369 Q = 0.00 Q = 0.408 Q 3 =.4746 Q 5 =.5909 Q 7 = 0.5550 Q H = 5.0 Q B = 5.045 D = 0.376 D =.5557 S H = 5.9665 D H = 5.3 S = 6.607 = 6.5608 dpf = 0.8659 PF = 0.7708 PF osc = 0.7369 0.0 0.004 0.0754 0.08 0.059 0.065 0.0389 0.0335 0.030 0.069 0 0.05 0.030 HP = 0.8305 HP = 0.830 0.036
time power quality monitoring and can extract relevant characteristics, which can be used as inputs to classify the Quality disturbances. Fig. 5. Percentage difference for the two approaches for case study- an accurate estimates of the even harmonics as well and decomposes the input signal into different frequency components with respect to time. However the DT with the same sampling frequency and decomposition level may fail in this case due to the more number of frequency components present in this signal as compared to the signal in the case study-. Moreover, the DT will result two or more frequency components in a single sub-band which has been overcome by the ET. The PQ indices obtained from the EEE standard definitions and the ET for non-stationary signal are listed in Table. t can be noted that the maximum difference is ust 0.5 percentage, which is considerably less and for rest of the indices the percentage differences is very small. The results clearly indicate that the ET is more efficient in analyzing the signal with highly non-stationary nature and subsequently estimating the indices. t can be observed from Fig. 5, that the percentage differences for the DT based PQ indices are very large. The reason behind this is that the DT with the same parameters cannot isolate the frequency components accurately if more number of harmonics is present in the signal as compared to case-. The values obtained using the ET are close to zero even in the case of non-stationary signal. This is due to the adaptive nature of the basis wavelets used in the ET, which do not require either selection of mother wavelet or decomposition levels. COCLUSO The paper presents an application of a new signal processing technique, Empirical avelet Transform to compute the Quality ndices as defined in EEE Std. 459-000. The algorithm uses adaptive scaling functions and wavelets for accurately extracting the frequency components present in the signal. The proposed method has been tested on several single phase stationary and non-stationary signals, of which two examples are presented. Based on the results, the ET proved to be adaptive and also accurate in estimation of PQs as compared to the DT, which depends on the mother wavelet, number of levels and sampling frequency. n the case of multiple close frequency components occurring in the signal, ET is able to isolate them while the DT will give those close frequency components in the same sub band. The proposed technique is very useful for its application in real REFERECES [] EEE Recommended Practice and Requirements for Harmonic Control in Electric Systems, EEE Standard 59-99, Apr. 993. [] Definitions for the Measurement of Electric Quantities under Sinusoidal, on-sinusoidal, Balanced, or Unbalanced Conditions, EEE Standard 459-00, Jan. 000. [3] Heydt GT, Jewell T, Pitfalls of electric power quality ndices, EEE Trans. Del., vol.3, pp.570-578, 998. [4] SH Jaramillo, GT Heydt, E O eill-carrillo, quality ndices for aperiodic voltages and currents, EEE Trans. Del., vol.5, pp.784-790, 000. [5] D. Granados-Lieberman, R.J. Romero-Troncoso, R.A. Osornio-Rios, A. Garcia-Perez and E. Cabal-Yepez, Techniques and methodologies for power quality analysis and disturbances classification in power systems: a review, ET Gen., Transm. Distrib., vol. 5, no. 4, pp. 59-59, 0. [6] M. Bollen and. Yu-HuaaGu, Signal Processing of Quality Disturbances. Hoboen, J, USA; iley, 006. [7] M.S. Kandil, S.A. Farghal and A. Elmitwally, Refined Quality ndices, Proc. nst. Elect. Eng., Gen., Transm. Distrib., vol. 48, no. 6, pp.590-596, 00. [8].G. Morsi, M.E. El-Hawary, Reformulating Components Definitions Contained in the EEE Standard 459-000 Using Discrete avelet Transform, EEE Trans. Del., vol., no. 3, pp.90-96, Jul. 007. [9] J. Barros and R. Diego, Analysis of harmonics in power systems using the wavelet pacet transform, EEE Trans. nstrum. Meas., vol. 57, no., pp. 63 69, Jan. 008. [0] J. L. illems, Reflections on apparent power and power factor in nonsinusoidal and poly phase situations, EEE Trans. Del., vol. 9, no., pp. 835 840, Apr. 004. [] E. Y. Hamid, R. Mardiana, and Z.. Kawasai, Method for RMS and power measurements based on the avelet Pacet Transform, Proc. nst. Elect. Eng., Sci., Meas. Technol., vol. 49, no., pp. 60-66, Mar. 00. [] R. Flores, Signal processing tools for power quality event classification, Lic. Eng. thesis, School Elect. Eng, Chalmers Univ. Technol, Goteborg, Sweden, 003. [3]. G. Morsi and M. E. El-Hawary, Suitable mother wavelet for harmonics and inter harmonics measurements using wavelet pacet transform, in Proc., Can. Conf. Elect. Comput. Eng, Apr. 007, pp. 748 75. [4] J. Barros and R.. Diego, Application of the wavelet pacet transform to the estimation of harmonic groups in current and voltage waveforms, EEE Trans. Del., vol., no., pp. 533 535, Jan. 006. [5] J. Gilles, Empirical avelet Transform, EEE Trans. Signal Process., vol.6, no.6, pp.3999-400, Aug. 03.