Power Quality Analysis Using Bilinear Time-Frequency Distributions

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Hindawi Publishing CorporationEURASIP Journal on Advances in Signal ProcessingVolume 2010, Article ID 837360,18pagesdoi:10.1155/2010/837360

Research ArticlePower Quality Analysis Using BilinearTime-Frequency Distributions

Abdul Rahim Abdullah1 and Ahmad Zuri Shaameri2

1 Faculty of Electrical Engineering, Technical university of Malaysia Malacca, 76100 Malacca, Malaysia2 Faculty of Electrical Engineering, Technical university of Malaysia, 81310 Johor, Malaysia

Correspondence should be addressed to Abdul Rahim Abdullah,[email protected]

Received 8 January 2010; Revised 4 June 2010; Accepted 8 December 2010

Academic Editor: Ulrich Heute

Copyright 2010 A. R. Abdullah and A. Z. Shaameri. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Bilinear time-frequency distributions (TFDs) are powerful techniques that offer good time and frequency resolution of time-frequency representation (TFR). It is very appropriate to analyze power quality signals which consist of nonstationary and multi-frequency components. However, the TFDs suffer from interference because of cross-terms. Many TFDs have been implemented,and there is no fixed window or kernel that can remove the cross-terms for all types of signals. In this paper, the bilinear TFDsare implemented to analyze power quality signals such as smooth-windowed Wigner-Ville distribution (SWWVD), Choi-Williamsdistribution (CWD), B-distribution (BD), and modified B-distribution (MBD). The power quality signals focused are swell, sag,interruption, harmonic, interharmonic, and transient based on IEEE Std, 1159-1995. A set of performance measures is defined andused to compare the TFRs. It shows that SWWVD presents the best performance and is selected for power quality signal analysis.Thus, an adaptive optimal kernel SWWVD is designed to determine the separable kernel automatically from the input signal.

1. Introduction

Power quality is an issue that is becoming increasinglyimportant to electricity consumers at all levels of usage [1].Poor power quality can cause very serious problems likereduction of lifetime of the load, the ineffective performanceof protection devices, and instabilities and interruptions in

manufacturing operation. For example, voltage sags to 80%of the nominal voltage with durations of 40 ms or greaterwould shut down the control electronics of production lineof an industrial plant[2]. Thus, there is a need for heightenedawareness of power quality among electricity users thatrequire ultrahigh availability of service and precision man-ufacturing systems. Accordingly, an automated monitoringsystem is required to provide adequate coverage of the entiresystem, understand the causes of these disturbances, resolveexisting problems, and predict future problems [1]. Promptand accurate diagnosis of problems will ensure quality ofpower line signal, reduce diagnostic time in the presence ofpower disturbance, and rectify failures.

In the current research trend, short-time Fourier trans-form (STFT) [3] is a popular technique for power qualitysignals analysis. The technique presents the signals jointly intime-frequency representation (TFR) which provides tempo-ral and spectral information. However, it has the limitationof a fixed window width that results is a compromisebetween time and frequency resolution. The greater temporal

resolution required, the worse frequency resolution will beand vice versa. To overcome the limitation of the fixedresolution of STFT, wavelet transform (WT) was proposedby various researchers[4]. WT offers high time resolutionfor high frequency component and high frequency resolutionfor low frequency component. Consequently, the techniqueis suitable to detect the duration of high frequency signalsuch as transient. For low frequency signal, typically sag,swell, and interruption, it does not produce reliable results[5]. In addition, WT also exhibits some disadvantages suchas its computation burden, sensitivity to noise level, andthe dependency of its accuracy on the chosen basis wavelet[6].
mailto:[email protected]:[email protected]


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2 EURASIP Journal on Advances in Signal Processing

Bilinear time-frequency distributions (TFDs) [7] havebeen intensively used to characterize and analyze non-stationary signals. The bilinear TFDs offer a good timeand frequency resolution and are successfully applied tovarious real-life problems such as radar, sonar, seismic dataanalysis, biomedical engineering, and automatic emission.

However, the TFDs suffer from the presence of cross-terms interferences because of its bilinear structure. Thisinhibits interpretation of its TFR, especially when signalhas multiple frequency components. Some members ofthe bilinear TFDs are Wigner-Ville distribution (WVD),windowed Wigner-Ville distribution (WWVD), smooth-windowed Wigner-Ville distribution (SWWVD), Choi-Williams distribution (CWD), B-distribution (BD), mod-ified B-distribution (MBD), and Born-Jordan distribution(BJD). An analysis of the autoterms presentation using thereduced interference distributions (RID) has been discussedin [8]. A procedure for designing a kernel that will producethe desired autoterm shape and an optimal kernel with

respect to the autoterm quality and cross-term were demon-strated.

In this paper, bilinear TFDs are implemented to analyzepower quality signals. The popular bilinear TFDs are chosensuch as SWWVD, CWD, BD, and MBD. To verify theperformance of the TFDs, a set of performance measures isdefined to compare the TFRs in terms of main-lobe width(MLW), peak-to-side lobe ratio (PSLR), absolute percentageerror (APE),and signal-to-cross-terms ratio (SCR).From thecomparison, the best bilinear TFD is chosen, and its adaptiveoptimal kernel system is designed. The adaptive system isto determine the optimal kernel parameters, automaticallyfrom the input signal, without prior knowledge of the signal.

The optimal kernel is capable of removing the cross-terms,preserving the autoterms, and maintaining accurate TFR.

2. Power Quality Signal

According to the IEEE Standards 1159, electromagneticphenomena are classified into several groups as shown inTable 1[9]. This paper focuses on six types of power qualitysignals: swell, sag, interruption, harmonic, interharmonic,and transient.

3. Signal Model

This paper divides the power quality signals into threeclasses. They are voltage variation for swell, sag, and in-terruption signal, waveform distortion for harmonic andinterharmonic signal, and transient for transient signal. Thesignal models of the classes are formed as a complex expo-nential signal, and defined as

zvv(t) = ej2 f1 t3

k=1Akk(t tk1), (1)

zwd(t) = ej2 f1t +Aej2 f2t, (2)

ztrans(t) = ej2 f1 t3

k=1k(t tk1)

+Ae1.25(tt1)/(t2t1) ej2 f2(tt1)2(t t1),(3)

k(t) =1, for 0 t tk tk1,

0, elsewhere,(4)

where zvv(t), zwd(t), and ztrans(t) are the voltage variation,waveform distortion, and transient signal, respectively.k isthe signal component sequence,Ak is the signal componentamplitude, f1 and f2 are the signal frequency,tis the time,and (t) is a box function of the signal. In this analysis,f1, t0, and t3 are set at 50 Hz, 0 ms, and 200 ms, and otherparameters are defined as follows:

(1) swell: A1 = A3 = 1, A2 = 1.2, t1 = 100 ms, t2 =140 ms,

(2) sag:A1=A3= 1,A2= 0.8,t1= 100 ms,t2= 140ms,(3) interruption:A1 = A3 = 1, A2 = 0, t1 = 100ms,

t2= 140 ms,(4) harmonic:A = 0.25, f2= 250 Hz,(5) interharmonic:A = 0.25, f2= 275 Hz,(6) transient:A= 0.5, f2= 1000 Hz,t1= 100ms,t2=

115 ms.

4. Bilinear Time-Frequency Distribution

Bilinear TFDs are powerful tools in the analysis of non-

stationary and multicomponent signals. Many of these TFDsare invariant to time and frequency translations and can beconsidered as energy distribution in time-frequency domain[10]. From the TFR, characteristics of the signals can becalculated and used as input for signals classification. Thesignal characteristics are duration of swell, sag, interruption,and transient and average of total waveform distortion,total harmonic distortion, and total nonharmonic distortion.Further discussion of the signal characteristics can be foundin [11].

In general, the bilinear TFDs can be formulated as

Pzt,f =

G(t, )

(t)Kz(t, ) expj2 f d, (5)

whereG(t, ) is the time-lag kernel function, Kz(t, ) is thebilinear product, and the asterisk with tdenotes the timeconvolution of the signals. The bilinear product is furtherdefined as

Kz(t, ) = z

t+

2

z

t 2

, (6)

where z(t) is the analytic signal of interest. Smooth-windowed Wigner-Ville distribution (SWWVD) has a sep-arable kernel [12] which is capable of reducing the effects ofthe interferences or cross-terms and at the same time, having


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The lag window,w(), has a cutofflag at = Tg. The Dopplerrepresentation of the TS function,H(t), that is obtained fromthe Fourier transform with respect to time is

h(v) = sin(vTsm)vTsm

+1

2

sin((v 1/2Tsm))(v 1/2Tsm)

+12

sin((v+ 1/2Tsm))(v+ 1/2Tsm)

,

(11)

where it is a low-pass filter in the Doppler domain, and thecutoffDoppler frequency is

c= 32Tsm

. (12)

The Choi-Williams distribution (CWD) kernel is devel-oped to reduce interference in TFDs [7] and can be definedas

G(t, ) =

|| e2t2/2 , (13)

whereis a real parameter that can control the resolutionand the cross-terms reduction [10]. This kernel has showngood performance in reducing cross-terms while keepinghigh resolution with a compromise between these tworequirements.

The B-distribution (BD) kernel [10] is defined in thetime-lag plane and can be expressed as

G(t, ) = ||cosh2t, (14)where is a positive real parameter that controls the degreeof smoothing, and its value is between zero and unity.This kernel is a low-pass filter in the Doppler domain butnot in the lag domain.

To improve the time resolution, the B-distributionwas modified by making the lag-dependent factor exactlyconstant [10]. The resulting modified B-distribution (MBD)had a lag-independent kernel and can be defined as

G(t, ) = cosh2t

cosh2 d

. (15)

5. Time-Lag Signal Characteristic

Generally, bilinear product of the signal interest is rep-resented in time-lag representation. The bilinear productconsists of autoterms and cross-terms and can be defined as

Kz(t, ) = Kz,auto (t, )+Kz,cross(t, ). (16)In the time-lag representation, normally, the autoterms

are concentrated along the time axis and centered at =0, while the cross-terms are located away from the axis.The autoterms must be preserved, while the cross-termsare suppressed by choosing appropriate kernel parameters.For SWWVD, TS function is used to remove Dopplerfrequency component existing in cross-terms, while lagwindow suppresses cross-terms that lie away from the originof the lag axis.Detail derivation of the autoterms and cross-terms for all signals that are shown in(17) to(32) is derivedinAppendix A.

t1

t1

t2

t2

t3

t3

t1

t2

t3

A1 A2 A3

A1,A2

A2,A1 A3,A2

A3,A1

(t2 t1)

(t2 t1)t

10

2 3

1, 2

A1,A3

A2,A3

1, 3

2, 3

2, 1

3, 1

3, 2

Figure 1: Bilinear product of the voltage variation signal. Theautoterms are highlighted in green, while the cross-terms aredensely dotted.

5.1. Bilinear Product of Voltage Variation Signal. Voltagevariation signal in (1) has a variation in the root mean square(RMS) value from nominal voltage [9]. The autoterms andcross-terms of this signal can be expressed as

Kauto,vv(t, ) =3

k=1A2ke

j2 f1Kk,k (t, ),

Kcross,vv(t, ) =3

k=1

3l=1k/= l

AkAlej2 f1 Kk,l(t, ),

(17)

wherek andlrepresent the signal component sequence,AkandAlare the signal components amplitude, and the bilinearproduct of the box function, (t), is defined as

Kk,l(t, ) = k

t+

2 tk

l

t

2 tl

. (18)

From (17), it is observed that the autoterms lie along thetime axis and are centered at= 0, while the cross-terms areelsewhere as shown inFigure 1.

For example, autoterm whenk = 1 andl= 1 is expressedas

Kauto,vv

tt12, =A21ej2 f1K1,1t t12, . (19)

This autoterm is located at t= t1/2 and is centered at theorigin of the lag axis. It has a single lag-frequency componentwhich is at f = f1. Similar result is observed for autotermwhen k = 2 and l = 2. This autoterm which is at t =t1/2 is also centered at the origin of the lag axis and has asingle lag-frequency component at f= f1 . It can be definedas

Kauto,vv

t t1+t2

2 ,

=A22ej2 f1K2,2

t t1+t2

2 ,

.

(20)


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EURASIP Journal on Advances in Signal Processing 5

Meanwhile, for cross-term whenk = 1 and l= 2, it islocated att= (t2 + 2t1)/4 and= t2/2. The cross-term is dueto the interaction between 1st and 2nd signal component andhas only a single lag-frequency component at f= f1. It canbe expressed as

Kcross,vv

tt2+ 2t1

4 , t22

=A1A2ej2 f1 K1,2

t t2+ 2t14

, t22

.

(21)

Another example is cross-term which is due to theinteraction between 2nd and 1st signal component. Thiscross-term is centered att= (t2+ 2t1)/4 and= t2/2 andalso has only a single lag-frequency component at f= f1 asexpressed in the following equation:

Kcross,vv

t t2+ 2t1

4 , +

t22

=A1A2ej2 f1 K1,2

t t2+ 2t14

, + t22

.

(22)

The examples above prove that the autoterms are cen-tered at= 0 and lie along the time axis, while the cross-terms are elsewhere. Since the signal has only frequencycomponent at f = f1, it results that the autoterms andcross-terms have a lag-frequency component at f = f1and zero Doppler frequency. Thus, in order to suppressthe cross-terms and to preserve the autoterms, lag windowshould cover all autoterms while removing the cross-termsas much as possible. The lag window width, Tg, can be setas

Tg t2 t1. (23)By using this limit, cross-terms such as whenk = 1,l= 2

andk= 2,l= 3 are preserved, since they are adjacent to theautoterms as shown inFigure 1.The remaining cross-termscan be reduced by using smaller Tg, but it will compromisethe concentration of the autoterms. This results in smearingin frequency domain that reduces frequency resolution. Inaddition, the lag window width should contain at least onecycle of fundamental signal such that Tg 1/2f1. Theactual effect of this setting will be discussed in the nextsection.

Since the cross-terms do not have Doppler frequency, theuse of the TS function will not contribute to the cross-termssuppression. Thus, the resulting TFD that uses a lag windowand an impulse function as TS function is also known aswindowed Wigner-Ville distribution (WWVD). It can beexpressed as

z,wwvd

t,f =

Kz(t, )w()ej2 f d, (24)

wherew() is the lag window.

5.2. Bilinear Product of Waveform Distortion Signal. Wave-form distortion signal is a steady-state signal which consists

of multiple frequency components [13]. The autoterm andcross-term of the signal in (2) can be defined as

Kauto,zwd(t, ) = ej2 f1 +A2ej2 f2, (25)

Kcross,zwd(t, )=

2Aej2((f2 +f1)/2)cos2f2 f1t. (26)As shown in (25), the autoterm is centered at the origin

of the lag axis and has two lag-frequency components whichare f1 and f2 . For the cross-term as shown in (26), it is alsocentered at the origin of the lag axis. However, the cross-termconsists of a lag-frequency component at f= (f2+f1)/2 anda Doppler frequency component at = (f2 f1).

Based on the observation, the Doppler frequency com-ponent only exists in the cross-term. TS function whichis a low-pass filter in Doppler frequency domain can beused to remove the cross-term. Since the Doppler frequencycomponent is at= (f2 f1), the Doppler cutofffrequencyshould be set atc

|f2

f1

|. It can be achieved by setting

the TS function parameter,Tsm, as

Tsm 3

2f2 f1 . (27)

Forsignal that has more than two frequency components,|f2 f1| is set as the smallest frequency deviation among thesignal frequency components. When Tsmis set lower than thelimit in(27), the cutofffrequency will bec >|f2 f1| thatwill include the cross-term. Thus, the TS function will not beable to remove the cross-term. Besides that, any higher Tsmvalue would result in a small cutoffDoppler frequency but

would cause the autoterm to smear in time. Thus,Tsmshould

be set at an appropriate value to remove the cross-term andavoid the smearing of autoterm in time.

Besides using the TS function to remove the cross-terms, lag window is also used to obtain desirable lag-

frequency resolution in TFR. The lag-frequency resolutionis set such that f f1/2 to d ifferentiate harmonicand interharmonic frequency components. Therefore, thelag window width should be set at Tg 1/2f. HigherTg offers higher lag-frequency resolution, but it increasescomputation complexity and memory size to calculate TFR.Thus,Tgshould be set at a sufficient value to obtain desirablelag-frequency resolution and avoid higher computation

complexity and memory size used. In this analysis, sincethe signal fundamental frequency chosen is at f1= 50Hz,Tg is set at minimum value which is 20 ms to reduce thecomputation complexity and memory size of the analysis. Itresults in the fact that the lag-frequency resolution of the TFRis f= 25 Hz. In addition, the setting is also applicable forall waveform distortion signals.

5.3. Bilinear Product of Transient Signal. Transient signal isa sudden signal which changes in steady-state condition atnonfundamental frequency [9]. As indicated in transientsignal model in (3), there are three signal components. Thefirst and third components consist of fundamental signal,


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f1, while the second component has additional frequencycomponent which is transient frequency, f2. Thus, thebilinear product of this signal produces three autoterms and

seven cross-terms in time-lag representation as shown inFigure 2.The autoterms and cross-terms can be expressed as

Kauto,trans(t, ) = ej2 f1 K1,1 (t, )

+

ej2 f1 +A2e2.5(tt1)ej2 f2

K2,2 (t, )

+ej2 f1K3,3 (t, ),(28)

Ktrans,cross(t, ) =3

k=1

3l=1k/= l

ej2 f1Kk,l(t, )

+

3l=1

Ae1.25(t+/2t1)ej2(f2f1)tf2t1

ej2(f2 +f1)/2K2,l(t, )

+3

k=1Ae1.25(t+/2t1)ej2(f2 f1 )tf2t1

ej2(f2 +f1)/2Kk,2 (t, ).

(29)

Similar to the voltage variation signal, the autoterms in

(28) are centered at = 0 and lie along the time axis ascolored in Figure 2. For example, autoterm atk = 1 is locatedatt= t1/2 and the origin of the lag axis. It has a lag frequencyat f= f1and can be defined as

Kauto,trans

t t1

2,

= ej2 f1 K1,1

t t12

,

. (30)

The location of the cross-terms in (29) is densely dottedin Figure 2. The figure shows that cross-terms that aregenerated by different signal components (k/= l) are locatedaway from the time axis, /= 0. As example, a cross-termdefined in (31) is produced because of the interactionbetween the first and second signal components (k= 1 andl= 2). It is centered att= (t2 + 2t1)/4 and= t2/2 andhas a Doppler frequency component at = (f2 f1) and twolag-frequency components at f= f1and f= (f2+ f1)/2:

Ktrans,cross

t t2+ 2t1

4 , t2

2

=

ej2 f1 +Ae1.25(t+/2t1)ej2(f2f1 )tf2t1 ej2(f2 +f1)/2

K1,2

t t2+ 2t14

, t22

.

(31)

t1

t1

t2

t2

t3

t3

t1

t2

t3

(t2 t1)

(t2 t1)

t10 2 3

1, 2 2, 3

2, 1

3, 1

3, 2

f1

f1 f1

f1

f1,f2

f1,f2

f1,f2f1,f2

f1,f2

1, 3

Figure2: Bilinear product of the transient signal.

The second signal component has two different frequen-cies which are f1 and f2. Its bilinear product can be definedas

Kz2 (t, )

=

ej2 f1 +A2e2.5(tt1)ej2 f2 + 2Ae1.25(t+/2t1)

cosj2f2 f1tf2t1ej2(f2+f1 )/2K2,2 (t, ).(32)

This bilinear product introduces a cross-term which islocated att

= (t2+ t1)/2 and also centered at

=0, where

it is similar to the autoterms. The cross-term has a Dopplerfrequency component at = (f2 f1) and a lag frequencycomponent at f= (f2+ f1)/2 and can be defined as

Ktrans,cross

t t2+t1

2 ,

= 2Ae1.25(t+/2t1)ej2(f2 +f1)/2 cosj2f2 f1t f2t1

K2,2

t t2+t12

,

.

(33)

The purpose of using lag window in this signal is similar

to the voltage variation signal. The lag window width is setsuch that |Tg| (t2 t1) to remove the cross-terms locatedaway from the time axis and to preserve the autoterms lyingalong the time axis. In addition, similar to the waveformdistortion signal, TS function is also employed to remove theDoppler frequency component of the remaining cross-terms.Since the Doppler frequency component is = (f2 f1),the cutoffDoppler frequency of the TS function is set atc |f2 f1| by setting the TS function parameter, Tsm,as in (27). An appropriate value ofTsm and Tg should bechosen to optimize the cross-terms suppression as well asto minimize the smearing of the autoterms in time andfrequency domain.


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Table2: Limit of the kernel parameters.

Signal Tg,min (ms) Tsm,min(ms)

Swell 10 0

Sag 10 0

Interruption 10 0

Harmonic 20 7.5Interharmonic 20 6.67

Transient 10 1.578

5.4. Kernel Parameters. The analysis of bilinear productin time-lag representation to determine kernel parametersfor all power quality signals is discussed in the previoussubsections. Based on the analysis, the limits of the kernelparameters as defined in (23) and (27) are summarizedinTable 2. The smallest lag window width, Tg,min, and TSfunction parameter, Tsm,min, can be set in (9) and (10),respectively, to obtain sufficient cross-terms suppression

with minimal autoterms bias as well as to reduce thecomputation complexity and memory size of the analysis.

6. Performance Comparison ofKernel Parameters

Several performance measures are created and used to verifythe TFR of the bilinear TFDs. They are main-lobe width(MLW), peak-to-side lobe ratio (PSLR), signal-to-cross-terms ratio (SCR), and absolute percentage error (APE).These measurements are adopted to evaluate concentration,accuracy, interference minimization, and resolution of TFRs[12].

6.1. Performance Measurements. MLW and PSLR are calcu-lated from the power spectrum which is obtained from thefrequency marginal of the TFR [12] as shown inFigure 3.MLW is the width at 3 dB below the peak of the powerspectrum, while PSLR is the power ratio between the peakand the highest side lobe calculated in dB. Low MLWindicates good frequency resolution, and it gives the abilityto resolve closely spaced sinusoids. PSLR should be as highas possible to resolve signal of various magnitudes.

SCR is a ratio of signal to cross-terms power in dB. HighSCR indicates high cross-terms suppression in the TFR andis defined as

SCR= 10log

signal power

cross terms power

. (34)

Besides the MLW, PSLR, and SCR, APE is also appliedto quantify the accuracy of signal characteristics that arecalculated from the TFR. This measurement has beendiscussed in [11] and is expressed as

APE = xi xmxi

100%, (35)

where xi is actual value and xm is measured value. LowAPE shows high accuracy of the measurement. In general,

100

80

60

40

20

0

20

Power(dB)

0 500 1000 1500 2000

Frequency (Hz)

PSLR

3 dB

MLW

Figure3: Performance measures used in the analysis.

an optimal kernel of TFD should have low MLW and APEwhile high PSLR and SCR.

6.2. Performance Comparison of Smooth-Windowed Wigner-Ville Distribution. The performance of SWWVD with vari-ous kernel parameters for power quality signals is shown inTable 3. In this table, the kernel parameters are chosen basedon the observation made in Section 5. The bold values inTable 3presents the parameters that give optimal TFR foreach type of signal. Even though, the discussion of the tablewill focus on transient signal and similar observation can bemade for voltage variation and waveform distortion signal.

As shown in the table, the optimal kernel parameters forthe transient signal are atTg= 10 ms andTsm= 1.578ms.To observe the performance response corresponding to thekernel parameters, the performance measures of this signal

at optimalTsmwith variousTgand optimalTgwith variousTsmare plotted in Figure 4.Figure 4(a)shows that, at optimalTsmand whenTgis set higher, the MLW is smaller indicatinga higher frequency resolution of the TFR. However, it suffersfrom the reduction of the cross-terms suppression whichresults in smaller SCR. This is because higherTgcovers moreadjacent cross-terms in lag axis in the bilinear product. As aresult, the APE is higher which presents lower accuracy of thesignal characteristic measurement. Since the fundamentalfrequency, f1, is set at 50 Hz, the minimum Tg should beset at 10 ms to cover at least one cycle of the fundamentalsignal.

At the optimal Tg and when Tsm is set higher than its

optimal value, the SCR increases, while the MLW remainsconstant as shown inFigure 4(b).It indicates that higherTsmimproves the cross-terms suppression and does not give anyeffect to the frequency resolution. However, the APE is alsohigher which shows that the time resolution of the TFR islower. This is because the application of TS function withhigherTsm increases the smearing of the autoterms in timedomain. Thus, there is a compromise between cross-termssuppression and time resolution to obtain optimal TFR.

The optimal kernel parameters for voltage variation sig-nal are atTg= 10 ms andTsm= 0 ms. For this signal, the useof the TS function does not introduce any improvement inthe cross-terms suppression because all cross-terms have zero


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0

5

10

15

20

25

30

MLW(Hz),A

PE(%),SCR(dB)

MLW

APE

SCR

PSLR

81

82

83

84

85

86

87

88

89

PSLR(dB)

10 15 20 30 40

OptimalTg

(a) MLW, APE, PSLR, and SCR at optimal Tsmwith variousTg

0

20

40

60

80

100

120

140

160

180

MLW(Hz),A

PE(%),PSLR(dB)

12.212.412.612.813

13.213.413.613.81414.214.4

SC

R(dB)

0 1.578 6.67 7.5 10

MLW

APE

PSLR

SCR

OptimalTsm

(b) MLW, APE, PSLR, and SCR at optimalTgwith variousTsm

Figure4: Performance of the TFR using SWWVD with various kernel parameters for transient signal. (The kernel parameters chosen mustgive low MLW and APE but high PSLR and SCR.)

Table3: Performance comparison of SWWVD with various kernel parameters.

Kernel Parameters Performance

measures

Signal

Voltage variation signal Waveform distortion signal Transient signal

Swell Sag Interruption Harmonic Interharmonic Transient

Tg= 10msTsm= 0 ms

MLW (Hz) 25 25 25 25 25 25

PSLR (dB) 614.82 614.82 614.82 623.12 50.795 19.102

SCR (dB) 15.641 17.799 55.446 4.4785 4.5758 13.764

APE (%) 0.2083 0.625 0.625 0.3755 100 55

Tg= 40msTsm= 0 ms

MLW (Hz) 6.25 6.25 6.25 6.25 6.25 6.25

PSLR (dB) 117.60 117.55 89.657 644.84 652.71 86.259

SCR (dB) 8.9462 11.216 48.903 9.0262 9.0721 9.4476

APE (%) 1.4583 1.875 1.0417 141.27 25.031 19.444

Tg= 10msTsm= 1.578ms

MLW (Hz) 25 25 25 25 25 25

PSLR (dB) 218.09 216.68 198.78 18.835 49.889 86.145

SCR (dB) 13.491 15.038 35.107 5.6064 5.7679 14.171

APE (%) 3.125 12.708 100 55.544 100 1.6667

Tg= 15msTsm= 1.578ms

MLW (Hz) 13.333 13.333 13.333 13.333 13.333 13.333

PSLR (dB) 130.99 130.68 127.89 17.623 17.712 84.025

SCR (dB) 11.751 13.297 33.514 10.959 11.289 12.567

APE (%) 69.791 117.08 100 100 100 6.6667

Tg= 20msTsm= 6.67ms

MLW (Hz) 12.5 12.5 12.5 12.5 12.5 12.5

PSLR (dB) 229.98 228.53 210.04 51.168 638.24 154.53

SCR (dB) 9.8581 11.485 31.371 27.934 23.752 10.442

APE (%) 14.583 29.375 100 20.142 0.125 107.78

Tg= 40msTsm= 6.67ms

MLW (Hz) 6.25 6.25 6.25 6.25 6.25 6.25

PSLR (dB) 120.03 100.92 81.315 51.168 655.78 129.74

SCR (dB) 6.7221 8.4121 28.701 28.570 24.256 7.7198

APE (%) 20.416 35.417 100 20.142 0.125 90.556

Tg= 20msTsm= 7.5 ms

MLW (Hz) 12.5 12.5 12.5 12.5 12.5 12.5

PSLR (dB) 229.97 228.53 209.88 643.28 68.843 156.11

SCR (dB) 9.8073 11.439 31.289 41.007 31.001 10.248

APE (%) 16.666 31.667 100 0.125 1.4268 115

Tg= 40msTsm= 7.5 ms

MLW (Hz) 6.25 6.25 6.25 6.25 6.25 6.25

PSLR (dB) 122.91 102.56 111.22 664.29 651.32 131.79

SCR (dB) 6.6568 8.3551 28.600 41.739 32.356 7.4996

APE (%) 21.458 36.667 100 0.125 0.125 97.778


9/18


Table4: Performance comparison of Choi-Williams distribution.

Signal

Kernel Parameters Performance

measures Voltage variation signal Waveform distortion signal Transient signal


= 1.0MLW (Hz) 2.34375 2.34375 2.34375 2.34375 2.34375 2.34375PSLR (dB) 66.6589 66.6589 66.6589 45.5389 47.4593 67.5132

SCR (dB) 5.61185 7.81444 28.6804 21.4932 22.3112 7.0376

APE (%) 2.70833 3.54166 81.8750 59.5457 59.0395 21.6667

= 0.5

MLW (Hz) 2.34375 2.34375 2.34375 2.34375 2.34375 2.34375

PSLR (dB) 78.8655 78.8655 78.8655 45.1447 47.3009 81.7881

SCR (dB) 5.51383 7.68141 27.6884 22.9633 23.9232 6.75077

APE (%) 1.45833 2.29166 85.2083 53.8937 55.8645 34.4444

= 0.1

MLW (Hz) 2.34375 2.34375 2.34375 2.34375 2.34375 2.34375

PSLR (dB) 51.309 51.309 51.309 45.3051 52.875 51.9201

SCR (dB) 6.0203 8.1977 27.1346 25.3153 26.4656 6.22225

APE (%) 0.20833 0.62500 63.7500 49.3376 49.6516 49.4444

= 0.05

MLW (Hz) 2.34375 2.34375 2.34375 2.34375 2.34375 2.34375

PSLR (dB) 52.0779 52.0779 52.0779 47.666 49.1118 53.6057

SCR (dB) 6.53223 8.73298 27.4525 26.7083 27.7299 6.30179

APE (%) 0.20833 0.20833 56.8750 44.1498 43.7457 52.2222

= 0.01

MLW (Hz) 5.85938 5.85938 5.85938 4.6875 4.6875 4.6875

PSLR (dB) 57.8314 57.8314 57.8314 59.4144 61.5886 67.2973

SCR (dB) 8.18709 10.4501 28.9207 30.6248 31.5286 7.20599

APE (%) 0.62500 0.41666 75.0000 15.5563 28.2073 51.6667

= 0.005

MLW (Hz) 5.85938 5.85938 5.85938 5.85938 5.85938 5.85938

PSLR (dB) 45.6061 45.6061 45.6061 62.4682 54.3076 57.5039

SCR (dB) 9.03793 11.3288 29.758 30.8003 32.5905 7.8471

APE (%) 1.04166 0.62500 78.3333 13.1708 24.7269 48.8889

= 0.001

MLW (Hz) 9.375 9.375 9.375 9.375 9.375 9.375

PSLR (dB) 53.3157 53.3157 53.3157 60.5077 56.1806 58.7548

SCR (dB) 11.2395 13.5999 32.0283 29.0768 30.1635 9.93482

APE (%) 1.45833 1.25000 83.3333 2.36060 12.7563 49.4444

Doppler frequency. In addition, higherTsmreduces the timeresolution of the TFR. Therefore, as theTsmis set higher, theSCR is lower, and APE is higher. For waveform distortionsignal, the optimal kernel parameters for harmonic signal are

at Tg= 20 ms andTsm = 7.5 ms while for interharmonicsignal are at Tg = 20 ms and Tsm = 6.67 ms. All cross-terms of these signals have Doppler frequency and can beremoved by using the TS function at the optimalTsm. HigherTgdoes not improve the cross-terms suppression. However,it is still used to set the frequency resolution of the TFRthat can differentiate between harmonic and interharmonicfrequency components.

6.3. Performance Comparison of Choi-Williams Distribution.Performance of the CWD is also compared with variouskernel parameters. The kernel parameter, , is set at 1.0,0.05, 0.1, 0.05, 0.01, 0.005, and 0.001 as shown in Table 4.

This table shows that the optimal parameter for voltagevariation, waveform distortion, and transient signal is at=0.05,0.001, and 1.0, respectively.

All signals present similar performance response when

is set higher or smaller than their optimal value. As example,the performance measures of sagsignal using various isshown graphically in Figure 5. The graph illustrates that,whenis set higher than its optimal kernel, the MLW andSCR are smaller. Higherincreases frequency resolution ofthe TFR, but it reduces cross-terms suppression. As a result,the APE is higher. As is set smaller, the SCR is higherbecause smallerremoves more cross-terms. However, thefrequency and time resolution get worse, resulting in higherMLW and APE. Thus,should be chosen based on the signalparameters, and a compromise between time and frequencyresolution and cross-terms suppression is required to obtainoptimal TFR.


10/18


0

24

6

8

10

1214

16

MLW(H

z),

APE(%),SCR(dB)

01020

30405060708090

PSLR(dB)

0.001 0.005 0.01 0.05 0.1 0.5 1

Optimal

MLW

APE

SCR

PSLR

Figure5: Performance comparison of the TFR using CWD with variousfor sag signal.

Table5: Performance comparison of B-distribution.

Signal

Kernel

parameters

Performance

measures Voltage variation signal Waveform distortion signal

Transient

signalSwell Sag Interruption Harmonic Interharmonic Transient

= 1.0MLW (Hz)PSLR (dB)SCR (dB)APE (%)

2.3437517.98194.0005926.4583

2.3437517.98196.1864330.0000

2.3437517.981927.6292100.000

2.3437517.804817.813823.5287

2.3437517.803720.165133.4150

2.3437517.98196.25955

100


2.3437520.44384.228349.37500

2.3437520.44386.3819411.8750

2.3437520.443826.697

100.000

2.3437520.293820.814316.9564

2.3437520.286924.253121.3484

2.3437520.44386.3561

100


2.3437541.65844.751992.70833

2.3437541.65846.874313.75000

2.3437541.658426.1382100.000

2.3437541.360622.857910.8960

2.3437541.269124.220112.3272

2.3437541.65846.73587

100


2.3437556.19064.88367

2.083333

2.3437556.19067.003923.12500

2.3437556.190626.1603100.000

2.34375

55.0188

23.2079

10.5308

2.34375

55

24.1038

10.1349

2.34375

56.1906

6.8444

2.77778


2.3437582.44365.0083

1.66666

2.3437582.44367.127552.50000

2.3437582.443626.205

100.000

2.3437554.194519.998915.0836

2.3437563.156221.378210.2403

2.3437582.44366.9497285.5556


2.34375100

5.040781.66666

2.34375100

7.144422.5000

2.34375100

26.2125100.000

2.3437554.077520.026615.0483

2.3437563.076421.188610.2398

2.34375100

6.9615490

= 0.001

MLW (Hz)

PSLR (dB)SCR (dB)APE (%)

2.34375

1005.04217

1.666667

2.34375

1007.15816

2.50000

2.34375

10026.2188

100.000

2.34375

53.981620.053813.4147

2.34375

63.00921.164810.2426

2.34375

1006.973292.7778

6.4. Performance Comparison of B-Distribution. BD isanother TFD used in this paper. Table 5 presents theperformance of the TFD with various kernel parameters,, at 1.0, 0.05, 0.1, 0.05, 0.01, 0.005, and 0.001. The resultshows that the optimal kernel for voltage variation signal is at = 0.001 while waveform distortion and transient signal areat=0.05. For all signals, as is set other than the optimalvalue, the MLW is similar, and the SCR is smaller. This

indicates thatdoes not change the frequency resolution andreduce the cross-terms suppression in the TFR. As a result,the APE is higher. The trends of this performance are showninFigure 6which proves that the optimal kernel of harmonicis at = 0.05.

6.5. Performance Comparison of Modified B-Distribution.MBD is also analyzed with various kernel parameters similar


11/18


PSLR(dB)

0.001 0.005 0.01 0.05 0.1 0.5 1

MLW

APE

SCR

PSLR

05

10

15

20

25

MLW(H

z),

APE(%),SCR(dB)

0

10

20

30

40

50

60

Optimal

Figure6: Performance comparison of the TFR using BD with variousfor harmonic signal.

Table6: Performance comparison of modified B-distribution.

Signal

Voltage variation signal Waveform distortion signal Transient

signal

Kernelparameters

Performancemeasures



2.34375100

5.607863.54166

2.34375100

7.889284.79166

2.34375

100

29.1182

91.45833

2.34375

45.1706

18.3383

12.7766

2.34375

48.2892

19.6895

10.3902

2.34375

100

7.54363

100


2.34375100

5.349293.12500

2.34375100

7.573293.95833

2.34375100

27.747796.87500

2.3437545.160819.219724.2547

2.3437548.293920.586431.7169

2.34375100

7.31436100

= 0.1

MLW (Hz)PSLR (dB)SCR (dB)APE (%)

2.34375100

5.095442.08333

2.34375100

7.238782.91666

2.34375100

26.4945100.0000

2.3437555.138419.868532.3782

2.3437562.841

21.156743.5187

2.34375100

7.04573100


2.34375

100

5.0679

1.66666

2.34375

100

7.19925

2.50000

2.34375100

26.3538100.0000

2.3437554.567120.011734.0133

2.3437562.912

21.193844.4557

2.34375100

7.01054100


2.34375100

5.047411.66666

2.34375100

7.168982.50000

2.34375100

26.2465100.0000

2.3437554.082120.025334.2055

2.3437562.975121.178144.5223

2.34375100

6.98279100

= 0.005MLW (Hz)PSLR (dB)SCR (dB)

APE (%)

2.34375100

5.04496

1.66666

2.34375100

7.16529

2.50000

2.34375100

26.2334

100.0000

2.3437554.019920.0315

34.2260

2.3437562.983321.1759

44.5324

2.34375100

6.97936

100


2.34375100

5.043011.66666

2.34375100

7.162362.50000

2.34375100

26.223100.0000

2.3437553.969820.029334.2465

2.3437562.99

21.173944.5407

2.34375100

6.97661100

to BD as shown inTable 6. The table shows that the optimalkernel parameter for swell and sag is = 0.05 while forinterruption, harmonic, interharmonic, and transient signalsis= 1.0. For instance,Figure 7shows the performance ofswell signal using variousand its optimal value is identifiedat = 0.05.

7. Adaptive Optimal Kernel

In the previous section, the performance of SWWVD, CWD,BD, and MBD is analyzed with various kernel parameters.From the analysis, the optimal performance of the distri-butions is identified and compared as shown in Table 7.


12/18


PS

LR(dB)

0.001 0.005 0.01 0.05 0.1 0.5 1

MLW

APE

SCR

PSLR

Optimal

0

1

2

3

4

5

6

MLW(Hz),A

PE(%),SCR(dB)

0

20

40

60

80

100

120

Figure7: Performance comparison of the TFR using MBD with variousfor swell signal.

Table7: Performance comparison between optimal kernel parameters for the TFDs.

Signal SWWVD CWD BD MBD

Voltagevariation signal

Swell


25

614.815

15.6408

0.20833

2.3437552.07796.532230.20833

2.34375100

5.042171.66666

2.34375100

5.06791.66666

Kernel parameter Tg= 10ms

Tsm= 0 ms = 0.05 = 0.001 = 0.05

Sag


25

614.815

17.7996

0.625

2.3437552.07798.732980.20833

2.34375100

7.158162.50000

2.34375100

7.199252.50000

Kernel parameter Tg= 10msTsm= 0 ms = 0.05 = 0.001 = 0.05

Interruption


25

614.815

55.4463

0.625

2.3437552.077927.452556.8750

2.34375100

26.2188100.000

2.34375100

29.118291.458


Tsm= 0 ms = 0.05 = 0.001 = 1.0

Waveformdistortion signal

Harmonic


6.25

664.295

41.7393

0.125

9.37560.507729.07682.36060

2.3437555.018823.207910.5308

2.3437545.170618.338312.7766


Tsm= 7.5 ms

=0.001

=0.05

=1.0

Interharmonic


6.25

655.776

42.256

0.125

9.37556.180630.163525.5125

2.3437555

24.103810.1349

2.3437548.289219.689510.3902


Tsm= 6.67ms = 0.001 = 0.05 = 1.0

Transientsignal

Transient


25

86.1447

14.1705

1.66667

2.3437567.51327.0376

21.6667

2.3437556.19066.8444

2.77778

2.3437556.19066.8444

2.77778


Tsm= 1.58ms = 1.0 = 0.05 = 1.0


13/18


The result shows that the SWWVD is the best distribution forpower quality signal analysis and an adaptive optimal kernelfor SWWVD is designed.

Based on the analysis in Sections5 and6, a guideline todetermine the separable kernel parameters of SWWVD forpower quality signals is given as follows.

(i) For voltage variation signal

Tg= 10ms, Tsm= 0 ms. (36)

(ii) For waveform distortion signal

Tg= 20ms, Tsm= 32f2 f1 . (37)

(iii) For transient

Tg= 10ms, Tsm= 3

2

f2 f1

. (38)

The kernel parameters are different based on the char-acteristics of the signals. Hence, an adaptive kernel systemis required which is capable of setting the kernel parametersautomatically from input signal. In this paper, based on thekernels setting given above, the adaptive kernel system forpower quality signal is designed as shown inFigure 8.

Firstly, bilinear product at= 0 for the input signal iscalculated. It can also be called instantaneous energy of theinterest signal,x(t)[14], and can be expressed as

Kx(t, 0) = x(t)x(t). (39)

For the signal models in (1) to(3), the bilinear product

at= 0 of the voltage variation, waveform distortion, andtransient signal are, respectively, defined as

Kz,vv(t, 0) =3

k=1A2kk(t tk),

Kz,wd(t, 0) = 1 +A2 + 2A cos

2f2 f1

t

,

Kz,trans(t, 0) =3

k=1k(t tk)

+A2e2.5(tt1) + 2Ae1.25(t+/2t1)

cosj2f2 f1t f2t12(t t2).(40)

The above equations show that the voltage variation andtransient signal have a momentary energy variation betweent1 and t2, while the waveform distortion has no momentaryenergy variation. Thus, based on this observation and theguideline given in (36) to(38), the lag window width is setat Tg = 10 ms for the signal that has momentary energyvariation, while, for no momentary energy variation,Tg isset at 20 ms.

In process to estimate TS function parameters, Tsm,ambiguity function of the bilinear at = 0 is employed.

Power quality signal

Calculate bilinearproduct at= 0

Calculate ambiguityfunction at= 0

Estimate time-smoothfunction parameter

Setup time-smoothfunction

Calculate TFR

Identify magnitudepattern

Estimate lag windowwidth

Setup lag -window

Figure8: Process flow of the adaptive optimal SWWVD.

It is calculated by using (41) to present the Doppler frequencycomponent of the bilinear product. From the ambiguityfunction, the lowest Doppler frequency, min, is identifiedand used to calculateTsmas defined in (42):

Az(, 0) =

Kz(t, 0)ej2tdt, (41)

Tsm=

3

2min

. (42)

As indicated in (40), the waveform distortion and tran-sient signal have a Doppler frequency component at = |f2f1| while the voltage variation has zero Doppler frequency.Thus, for waveform distortion and transient signal,min isset at|f2 f1| and is then used in (42) to calculate Tsm.Since the voltage variation has no Doppler frequency, thetime-smooth function parameter is set at Tsm = 0 ms, or,in other words, the TS function used is a delta function. Fornormal signal, it has zero Doppler frequency as well as noenergy variation. Therefore, the kernel parameters used aresimilar to the voltage variation signal which areTg= 10msandTsm= 0 ms. Finally, the setting of the kernels is used tocalculate the SWWVD to represent the signal in TFR.

For example, Figures9 to 11show swell, harmonic, andtransient signals and their bilinear product and ambiguityfunction at = 0, respectively. As shown inFigure 9(a),the magnitude of the swell signal is 1.2 pu starting from 100to 140 ms, whileFigure 9(b)shows that its energy increasesfrom 1 to 1.44pu between 100 and 140 ms. The signalhas zero Doppler-frequency as shown inFigure 9(c). Forharmonic signal inFigure 10(a), it is a constant sinusoidalenergy as shown inFigure 10(b), while Figure 10(c) showsits Doppler frequency is= 200 Hz. It is different from thetransient signal that has a short energy variation between 100and 115 ms, while its Doppler frequency is at= 950 Hz asshown in Figures11(b)and11(c),respectively.


14/18


1

0

1

Amplitude

0 50 100 150 200

Signal

Time (ms)(a)

0 50 100 150 2000.5

1

1.5Bilinear product at= 0

Time (ms)

Energy

(b)

0

0.5

1

Energy

0 2000 4000 6000 8000 10000 12000

Doppler -frequency (Hz)

Ambiguity function at= 0

(c)

Figure 9: (a)Swell signal,(b) its bilinear product, and(c) ambiguityfunction at= 0.

Based on the adaptive system design, since the bilinear

product at = 0 of swell and transient signals has amomentary energy variation, the lag window width is set atTg= 10 ms. For harmonic signal that has no momentaryenergy variation, the lag window width is set at Tg =20 ms. The harmonic and transient signals have a Dopplerfrequency. Thus, the values of the Doppler frequency areused in (42) to calculateTsm. As a result, the setting ofTsmfor the harmonic and transient signals is 7.5 and 1.578 ms,respectively. For the swell signal, it has no Doppler frequencycomponent, andTsmis set at 0 ms.

8. Results

In this section, the results of the power quality analysis usingSWWVD, CWD, BD, and MBD are discussed. The exampleof the signals and their TFRs using the TFDs at optimalkernels is shown in Figures12 to 15. The line graphs showthe signal in time domain, while the contour plots show theTFR. The highest power is represented in red color while thelowest in blue color.

Figure 12shows a sag signal and its TFR using SWWVD.The magnitude of the sag signal is 0.8 pu, while its durationis between 100 and 140ms. The contour plot presentsthat there is a momentary decrease of power at 50 Hz(fundamental frequency) from 100 to 140 ms. InFigure 13,there is a harmonic signal in time domain and its TFR

Amplitude

2

0

2

0 20 40 60 80 100 120 140 160 180 200

Time (ms)

Signal

(a)

0 20 40 60 80 100 120 140 160 180 200

Time (ms)

0

0.5

1

1.5

Energy

Bilinear product at= 0

(b)

Energy

0

0.02

0.040.06

0 50 100 150 200 250 300 350 400 450 500

Doppler -frequency


(c)

Figure 10: (a) Harmonic signal, (b) its bilinear product, and (c)ambiguity function at= 0.

by using CWD. The TFR shows that the harmonic signal

consists of two frequency components which are 50 and250 Hz.

A swell signal and its TFR using BD are shown inFigure 14. The magnitude of the swell signal is 1.2 pubetween 100 and 140 ms. The TFR shows that the powerincreases from 120 to 160 ms, and its frequency is 50 Hz. Thelast example is transient signal. This signal and its TFR usingMBD are shown inFigure 15.The transient signal begins at100 ms, and its duration is 15 ms. In the contour plot, thetransient power increases between 118 and 125 ms, and itsfrequency is 1000 Hz.

In the contour plots, the TFRs show some delays com-pared to the input signals. This is because the convolution

process between kerneland signal in the TFDs shifts the TFRsin time domain. For the TFR of sag signal using SWWVD,there is no delay because the kernel parameter used fortime-smooth function is set at Tsm = 0 ms. Generally,this observation clearly shows that the TFRs represent thecharacteristics of the power quality signals.

By assuming perfect knowledge of the power qualitysignals, the performance of the bilinear TFDs with variouskernel parameters has been analyzed as shown in Tables3to6.From those tables, the optimal performance of the TFDs isidentified and summarized inTable 7.

A good TFD should have low APE and MLW while highSCR and PSLR. This table shows that, in overall, SWWVD


15/18


0 20 40 60 80 100 120 140 160 180 200

Time (ms)

Amplitude

Signal

1

0

1

(a)

0 20 40 60 80 100 120 140 160 180 200

Time (ms)

0

1

2

Bilinear product at= 0

Energy

(b)

0

2

4

700 750 800 850 900 950 1000 1050 1100 1150 1200

Doppler -frequency


Energy

104

(c)

Figure 11: (a) Harmonic signal, (b) its bilinear product, and (c)ambiguity function at= 0.

1

0

1

700

600

500

400

300

200

100

0

0.6

0.5

0.4

0.3

0.2

0.1

0 50 100 150 200

Time (ms)

Time (ms)

Frequency(Hz)

Time-frequency representation

0 20 40 60 80 100 120 140 160 180

Signal

Magnitude

(Vpu)

Figure12: The TFR of sag signal using SWWVD atTg= 10 ms andTsm= 0ms.

gives good APE, SCR, and PSLR while MLW is poor. ForCWD, BD, and MBD, they offer good MLW but poor APE,SCR, and PSLR. Thus, it clearly proves that the SWWVD isthe best distribution and very appropriate for power qualityanalysis.

1

01

500

400

300

200

100

0

0.6

0.5

0.4

0.3

0.2

0.1

0.7

0 50 100 150 200

Time (ms)

Time (ms)

Frequency(Hz)


0 20 40 60 80 100 120 140 160 180

Signal

Magnitude

(Vpu)

Figure13: The TFR of harmonic signal using CWD at = 0.001.

1

0

1

0 50 100 150 200

Time (ms)

Time (ms)

Fr

equency(Hz)


0 20 40 60 80 100 120 140 160 180

300

250

200

150

100

50

0

1.2

1

0.8

0.6

0.4

0.2

Signal

Magnitude

(Vpu)

Figure14: The TFR of swell signal using BD at = 0.001.

Frequency(Hz)


Time (ms)

0 20 40 60 80 100 120 140 160 180

0.6

0.5

0.4

0.3

0.2

0.1

0.7

2

0

2

1500

1000

500

0

Magnitude

(Vpu)

0 50 100 150 200

Time (ms)

Signal

Figure15: The TFR of transient signal using MBD at = 1.0.


16/18


17/18


The autoterms are bilinear product of a signal with thesame signal, k= l. Thus, the autoterms for this signal aredefined as

Kauto ,vv(t, ) = ej2 f13

k=1|Ak|2k

t+

2 tk1

k

t 2 tk1

= ej2 f13

k=1|Ak|2Kk,k (t, ),

Kk,l(t, ) = k

t+2 tk1

l

t

2 tk1

.

(A.2)

The cross-terms are bilinear product between differentsignal,k /= l, and expressed as

Kcross,vv(t, ) = ej2 f1

3

k=1

3

l=1k/= l

AkAl

k

t+

2 tk1

l

t 2 tl1

= ej2 f13

k=1

3l=1k/= l

AkAlKk,l(t, ).

(A.3)

B. The Waveform Distortion Signal

The waveform distortion signal in (2) is different from the

previous signal that has one signal and consists of twofrequency components, f1 and f2 . Its bilinear product isexpressed as

Kz,wd(t, ) =

ej2 f1 (t+/2) +Aej2 f2 (t+/2)

ej2 f1 (t/2) +Aej2 f2 (t/2)

= ej2 f1 +A2ej2 f2 + 2Aej2((f2+f1 )/2)

cos2f2 f1t.

(B.1)

The function above shows that the autoterms have two lag-frequency components similar to the signal frequencies, f1and f2 frequencies, while the cross-terms exist in the middlebetween the lag-frequency components, (f2 +f1)/2, and havea Doppler frequency at = f2 f1 . The autoterms and cross-terms are defined as

Kauto,zwd(t, ) = ej2 f1 +A2ej2 f2 ,

Kcross,zwd(t, ) = 2Aej2((f2 +f1)/2) cos

2f2 f1

t

.(B.2)

C. The Transient Signal

The transient signal in (3) has three sequence signals. Thefirst and third signals are normal signal at fundamental

frequency, f1, while the second signal has additional transientfrequency, f2. The bilinear product of this signal is defined as

Kz,trans(t, ) =ej2 f1 (t+/2) 3

k=1k

t+

2 tk1

+Ae1.25(t+/2

t

1)/t

dej2 f

2(t+/2

t

1)

2

t+

2 t1

ej2 f1 (t/2) 3

l=1l

t

2 tk1

+Ae1.25(t/2t1)/tdej2 f2 (t/2t1)

2

t 2 t1

=A2e2.5(tt1)ej2 f2 K2,2 (t, )

+3

k=1

3l=1

ej2 f1Kk,l(t, )

+3

l=1Ae1.25(t+/2t1)ej2(f2 f1)tf2t1

ej2(f2 +f1)/2K2,l(t, )

+3

k=1Ae1.25(t+/2t1)ej2(f2f1)tf2t1

ej2(f2 +f1)/2Kk,2 (t, ).(C.1)

As discussed for the voltage variation and waveform distor-tion signal, autoterms and cross-terms of transient signal canbe defined as

Kauto,trans(t, ) =3

k=1ej2 f1 Kk,k (t, )+A

2e2.5(tt1)ej2 f2

K2,2 (t, ),

Ktrans,cross(t, ) =3

k=1

3l=1

ej2 f1 Kk,l(t, )

+3

l=1Ae1.25(t+/2t1)ej2(f2f1 )tf2t1

ej2(f2 +f1)/2K2,l(t, )

+3

k=1Ae1.25(t+/2t1)ej2(f2 f1)tf2t1

ej2(f2 +f1)/2Kk,2 (t, ).(C.2)


18/18


Acknowledgments

The authors would like to thank Technical university ofMalaysia Malacca (UTeM) for its financial support andTechnical university of Malaysia for providing the resourcesfor this research.

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Power Quality Analysis Using Bilinear Time-Frequency Distributions

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