Measurement 42 (2009) 478–484

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Measurement

journal homepage: www.elsevier .com/ locate /measurement

Higher-order characterization of power quality transientsand their classification using competitive layers q

Juan José González de la Rosa a,b,*, Antonio Moreno Muñoz a,c, Antolino Gallego d,Rosa Piotrkowski d,e, Enrique Castro d

a Research Group PAIDI-TIC-168: Computational Instrumentation and Industrial Electronicsb University of Cádiz, Area of Electronics, EPSA, Av. Ramón Puyol S/N, E-11202, Algeciras-Cádiz, Spainc University of Córdoba, Area of Electronics, Campus de Rabanales, Ed. Leonardo da Vinci, E-14.071, Córdoba, Spaind University of Granada, Department of Applied Physics, Research Group, PAIDI-TEP-232, Numerical Methods and Signal Processing in Dynamical Systemse School of Science and Technology, San Martin University, M. de Irigoyen 3100 (1650) San Martin, Buenos Aires, Argentina

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 July 2008Received in revised form 12 September 2008Accepted 15 September 2008Available online 22 September 2008

Keywords:Higher-order statistics (HOS)Neural classifiersPower-quality analysisTransient categorization

0263-2241/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.measurement.2008.09.006

q This work is funded and supported by the Spanisand Education (projects DPI2003-00878, PTR95-0802970) and by the Andalusian Government (excelleTIC00155).

* Corresponding author. Address: University of Cáics, EPSA, Av. Ramón Puyol S/N, E-11202, Algeciras-C956028020; fax: +34 956028020/01.

E-mail address: juanjose.delarosa@uca.es (J.J. Go

This paper deals with power-quality (PQ) event detection, classification and characteriza-tion using higher-order sliding cumulants (which are calculated over high-pass filtered sig-nals to avoid the low-frequency 50-Hz sinusoid), whose maxima and minima are thecoordinates of two-dimensional feature vectors. The classification strategy is based in com-petitive layers. We focus on the problem of differentiating two types of transients: short-duration (impulsive transients) and long-duration (oscillatory transients). The results showthat the measured vectors are classified into clearly differentiated clusters in the featurespace. The experience aims to set the foundations of an automatic procedure for PQ eventdetection.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Power Quality (PQ) or power disturbances are con-cerned with deviations of the voltage or current from theideal single-frequency sine wave of constant amplitudeand frequency. PQ problems commonly faced by facilitiesoperations include transients, sags, swells, surges, outages,harmonics, and flickers that vary in quantity or magnitudeof the voltage. A consistent set of definitions can be foundin [1].

. All rights reserved.

h Ministry of Science24-OP and DPI2006-ncy project PAI2005-

diz, Area of Electron-ádiz, Spain. Tel.: +34

nzález de la Rosa).

PQ event detection and classification is gaining impor-tance due to worldwide use of delicate electronic devices.Things like lightning, large switching loads, non-linear loadstresses, inadequate or incorrect wiring and grounding oraccidents involving electric lines, can create problems tosensitive equipment, if it is designed to operate within nar-row voltage limits, or if it does not incorporate the capabil-ity of filtering fluctuations in the electrical supply [2,3].

The solution for a PQ problem implies the acquisitionand monitoring of long data records from the energy distri-bution system, along with an automated detection andclassification strategy, which allows the identification ofthe cause of these voltage anomalies. The goal of the signalprocessing analysis is to get a feature vector from the datarecord under study, which constitute the input to the com-putational intelligence modulus, with the task of classifica-tion. Signal processing for this purpose are mainly based inspectral analysis and wavelet transforms. Second-ordermethods are based on the independence of the spectral

J.J. González de la Rosa et al. / Measurement 42 (2009) 478–484 479

components and the evolution of the spectrum in the timedomain. Other tools are threshold-based functions, linearclassifiers and Bayesian networks.

Some recent works bring a different strategy, based inhigher-order statistics (HOS), in dealing with the analysisof transients within PQ analysis [4,5], and other fields ofScience [6–8]. Without perturbation, the 50-Hz of the volt-age waveform exhibits a Gaussian behavior. Deviationsfrom Gaussianity can be detected and characterized viaHOS; non-Gaussian processes need third and fourth-orderstatistical characterization in order to be recognized (com-pletely characterized), because 2nd-order moments andcumulants could be not capable of differentiate non-Gaussian events.

The problem of differentiating between a transient oflong duration named oscillatory (within a signal period)and a short duration transient, or impulsive transient(25% of a cycle), falls into the set of HOS applicability.The short transient could also bring the 50-Hz voltage tozero instantly and, generally affects the sinusoid dramati-cally. By the contrary, the long-duration transient couldbe considered as a modulating signal (the 50-Hz signal isthe carrier), and is associated to load charges [2]. Thesetransients are intrinsically non-stationary, so it is neces-sary a battery of observations (sample registers) to obtaina reliable characterization.

The contribution of this paper consists of the applica-tion of higher-order central cumulants to characterize PQevents in the time-domain (measuring maxima and min-ima values of higher-order cumulant sequences), alongwith the use of a competitive layer as the classificationtool. Results reveal that two different clusters, associatedto both types of transients, can be recognized in the 2Dgraph, attending to third and fourth-order characteriza-tion. The successful results convey the idea that the phys-ical underlying processes associated to the transients,generate different types of deviations from the typical ef-fects that the noise cause in the 50-Hz sinusoid voltagewaveform.

The paper is structured as follows. The following Sec-tion 2 explains the fundamentals of the importance forpower quality monitoring. Higher-order statistics are out-lined in Section 3. Competitive layers are summarized inSection 4. Results are presented in Section 5. Finally, con-clusions are drawn in Section 6.

2. The importance of power-quality characterization

As more and more electronic equipment enter the resi-dential and business environment, the subjects related toPower Quality (PQ) and its relationship to vulnerability ofinstallations is becoming an increasing concern to theusers. Particularly has increased the need to protect sensi-tive electronic equipment from damaging over-voltages[9,10].

Over-voltage is an RMS increase in the AC voltage, at thepower frequency, for durations greater than a few seconds[11]. Over-voltage can be the result of a programmedutility operation, or the effect of an external eventuality.Under normal operating conditions, the steady-state

voltage is regulated by the utility within a limits band ac-cepted by the EN-50160. Deviations from these limits arerare, and the utility can actuate readily to correct them, ifknown their occurrence, by acting on conventional distri-bution technologies, such as tap-changing transformers[12].

However, under the typical operating conditions of apower system it is always possible the presence of damag-ing momentary excess of voltage. Although by themselvesthey would be described as ‘‘abnormal”, it is possible todistinguish between surge and swell. A surge is an over-voltage that can reach thousands of volts, lasting less thanone cycle of the power frequency, that is, less than 16 ms. Aswell is longer, up to a few seconds, but does not exceedabout twice the normal line voltage.

Power system surges, based on waveform shapes, canbe classified into ‘‘oscillatory transients” and ‘‘impulsivetransients” [1,3] and they are the goal of this researchwork. Oscillatory transient surges show a damped oscilla-tion with a frequency range from 400 Hz to 5 kHz or more.Impulsive transient surges present a fast rise time in theorder of 1 ns–10 ls over the steady-state condition of volt-age, current or both, that is unidirectional in polarity (pri-marily either positive or negative), reaching hardly twicethe peak amplitude of the signal. They are damped quickly,presenting a frequency range from 4 kHz to 5 MHz, occa-sionally reaching 30 MHz.

Categorization of electrical transients based on wave-form shapes and their underlying causes (or events) hasbeen studied in [2], and a few previous studies [4,5] usingHOS for feature extraction of electrical signals have shownthe possibility of distinguish transients based on detailsbeyond the second-order.

In the following Section 3 we present higher-order sta-tistics in the time-domain in order to present the signalprocessing tool, along with a basic example which showsthe performance of the statistical estimators which havebeen used in the computation of the cumulants. Thisexample also motivates the use of HOS in time-seriescharacterization.

3. Higher-order statistics

3.1. Mathematical foundations

Higher-order cumulants are used to infer new proper-ties about the data of non-Gaussian processes [6,13]. Inmultiple-signal processing it is very common to definethe combinational relationship among the cumulants of rstochastic signals, fxigi2½1;r�, and their moments of orderp; p 6 r, given by using the Leonov-Shiryaev formula[14,15]

Cumðx1; � � � ; xrÞ ¼Pð�1Þp�1 � ðp� 1Þ! � Ef

Qi2s1

xig

�EfQi2s2

xjg � � � EfQi2sp

xkg;ð1Þ

where the addition operator is extended over all the parti-tions, like one of the form ðs1; s2; . . . ; spÞ, p ¼ 1;2; � � � ; r; and

480 J.J. González de la Rosa et al. / Measurement 42 (2009) 478–484

ð1 6 i 6 p 6 rÞ; being si a set belonging to a partition of or-der p, of the set of integers 1,. . .,r.

Let fxðtÞg be an rth-order stationary random real-valuedprocess. The rth-order cumulant is defined as the joint rth-order cumulant of the random variables x(t), x(t + s1),. . .,x(t + sr�1),

Cr;xðs1; s2; . . . ; sr�1Þ

¼ Cum½xðtÞ; xðt þ s1Þ; . . . ; xðt þ sr�1Þ�ð2Þ

The second-, third- and fourth-order cumulants of zero-mean x(t) can be expressed via [16]:

C2;xðsÞ ¼ EfxðtÞ � xðt þ sÞg ð3aÞ

C3;xðs1; s2Þ ¼ EfxðtÞ � xðt þ s1Þ � xðt þ s2Þg

C4;xðs1; s2; s3Þð3bÞ

¼ EfxðtÞ � xðt þ s1Þ � xðt þ s2Þ � xðt þ s3Þg

�C2;xðs1ÞC2;xðs2 � s3Þ � C2;xðs2ÞC2;xðs3 � s1Þ

�C2;xðs3ÞC2;xðs1 � s2Þð3cÞ

By putting s1 ¼ s2 ¼ s3 ¼ 0 in Eq. (3a), we obtain

c2;x ¼ Efx2ðtÞg ¼ C2;xð0Þ ð4aÞ

c3;x ¼ Efx3ðtÞg ¼ C3;xð0;0Þ ð4bÞ

c4;x ¼ Efx4ðtÞg � 3ðc2;xÞ2 ¼ C4;xð0;0;0Þ ð4cÞ

The expressions in Eq. (4c) are measurements of the vari-ance, skewness and kurtosis of the distribution in termsof cumulants at zero lags (the central cumulants).

Normalized kurtosis and skewness are defined asc4;x=ðc2;xÞ

2 and c3;x=ðc2;xÞ3=2, respectively. We will use and

refer to normalized quantities because they are shift andscale invariant. If xðtÞ is symmetrically distributed, itsskewness is necessarily zero (but not vice versa); if x(t) isGaussian distributed, its kurtosis is necessarily zero (butnot vice versa). In the experimental section, results areobtained by using sliding cumulants, i.d. a moving windowin the time domain over which to compute the eachcumulant.

In practice, the computation of the cumulants and thepoly-spectra is based in estimates. For example, given anN-sample signal vector xðnÞ; n ¼ 0; � � � ;N � 1, the followingexpressions, Eq. (5)1, Eq. (6) and Eq. (7) describe three esti-mates for the second, third and fourth-order cumulants,respectively.

C2;xðsÞ ¼ Rx;unbiasedðsÞ ¼1

N � sXN�s

i¼1

½xðiÞ � �x�½xðiþ sÞ � �x�;

ð5Þ

C3;xðk; lÞ ¼ ^Cum½xðnÞ; xðnþ kÞ; xðnþ lÞ�

¼ 1N

XN�1

n¼0

xðnÞxðnþ kÞxðnþ lÞ; ð6Þ

1 Unbiased autocorrelation.

C4;xðk; l;mÞ ¼ ^Cum½xðnÞ; xðnþ kÞ; xðnþ lÞ; xðnþmÞ�

¼ 1N

XN�1

n¼0

xðnÞ � xðnþ kÞ� � xðnþ lÞ� � xðnþmÞ�

� 1N2

XN�1

n¼0

xðnÞ � xðnþ kÞ�" # XN�1

n¼0

xðnþ lÞ� � xðnþmÞ�" #

� 1N2

XN�1

n¼0

xðnÞ � xðnþ lÞ�" # XN�1

n¼0

xðnþ kÞ� � xðnþmÞ�" #

� 1N2

XN�1

n¼0

xðnÞ � xðnþmÞ�" # XN�1

n¼0

xðnþ kÞ� � xðnþ lÞ�" #

;

ð7Þ

where s is a generic time-lag, k; l; m 2 ½�v; :::; �1;0;1; :::; þv�, and n ¼ 0;1; :::; N � 1; v is the index of themaximum time shift (lag) between samples of a record.

Eq. (5) represents an unbiased estimator, while Eqs. (6)and (7) represent biased estimates. If the total number ofterms averaged are considered in the denominator (e.g.N � s), the estimator is unbiased. By the contrary, if thenumber of sampled data (N) are considered, the estimatoris biased.

If different signals are involved Eq. (6) turns into Eq. (8):

Cx;y;zðk; lÞ ¼1

N3

XN2

n¼N1

xðnÞyðnþ kÞzðnþ lÞ; ð8Þ

where N1 and N2 are chosen such that the summations in-volve only signal components with n 2 [0, N)-1], and N isthe number of samples; unbiased estimates are obtainedif N3 is set equal to the actual number of terms whichare averaged.

3.2. An introductory example

To show the relevance of HOS a previous example is pre-pared. Four noise processes: Gaussian; uniform; exponen-tial and Laplacian, previously catalogued in [13], andindistinguishable from the second-order perspective (auto-correlation sequence), are presented in this subsection in or-der to illustrate the importance of introducing higher-ordercumulants. The 4th-order cumulants are computed accord-ing to the estimate given in Eq. (7). We consider a 2048-point sample register for each random set of data. The fouridentical autocorrelation sequences are drawn in Fig. 1.

If we look into the fourth-order sequences, substantialdifferences are observed, specially those corresponding tozero time lags. This can be seen in Fig. 2, where thefourth-order cumulant sequences are depicted. The theo-retical values of the cumulants at zero time-lag are: 0(Gaussian), -1 (uniform), 6 (Exponential), 12 (Laplacian),according to [13]. The difference between the theoreticaland the experimental value is due to the lack of averaging(only one sample register is consider). The convergency ofthe estimate is assured.

4. Competitive layers

The neurons in a competitive layer distribute them-selves to recognize frequently presented input vectors.The competitive transfer function accepts a net input

500 1000 1500 2000

-2

0

2

Gau

ssia

n

Realizations of the processes

-20 -10 0 10 20-0.5

0

0.5

1

Autocorrelation sequences

500 1000 1500 2000

-1

0

1U

nifo

rm

-20 -10 0 10 20-0.5

0

0.5

1

500 1000 1500 200002468

Expo

nent

ial

-20 -10 0 10 20-0.5

0

0.5

1

500 1000 1500 2000-5

0

5

10

Sample number

Lapl

acia

n

-20 -10 0 10 20-0.5

0

0.5

1

Lag index

Fig. 1. The four second-order cumulants corresponding to four sample registers, each of which is a realization of a different noise process (left column ofsub-figures). They exhibit the same autocorrelation sequence (sub-graphs in the right column).

-20 -15 -10 -5 0 5 10 15 20-0.1

00.1

Gau

ssia

n

c4,x(τ1,0,0); τ2=τ3=0

-20 -15 -10 -5 0 5 10 15 20-1.5

-1-0.5

0

Uni

form

-20 -15 -10 -5 0 5 10 15 2002468

Expo

nent

ial

-20 -15 -10 -5 0 5 10 15 20-505

1015

τ1

Lapl

acia

n

-0.075

-1.16

5.55

10.78

Fig. 2. Fourth-order cumulant sequences for the four noise processes.Sample values at zero time lag are included in each sub-figure.

J.J. González de la Rosa et al. / Measurement 42 (2009) 478–484 481

vector p for a layer (each neuron competes to respond to p)and returns neuron outputs of 0 for all neurons except forthe winner, the one associated with the most positive ele-ment of net input. If all biases are 0, then the neuron whoseweight vector is closest to the input vector has the leastnegative net input and, therefore, wins the competitionto output a 1.

The winning neuron will move closer to the input, afterthis has been presented. The weights of the winning neu-ron are adjusted with the Kohonen learning rule (0.9 in

the present case). Supposing that the ith-neuron wins,the elements of the ith-row of the input weight matrix(IW) are adjusted as shown in Eq. (9):

IW1;1i ðqÞ ¼ IW1;1

i ðq� 1Þ þ a½pðqÞ � IW1;1i ðq� 1Þ�; ð9Þ

where p is the input vector, q is the time instant, and a isthe learning rate. The neuron whose weight vector wasclosest to the input vector is updated to be even closer.The result is that the winning neuron is more likely towin the competition the next time a similar input is pre-sented. As more and more inputs are presented, each neu-ron in the layer closest to a group of input vectors soonadjusts its weights toward those inputs. Eventually, if thereare enough neurons, every cluster of similar input vectorswill have a neuron that outputs 1 when a vector in thecluster is presented, while outputting a 0 at all other times.Thus, the competitive network learns to categorize the in-put vectors.

5. Experimental results

The aim is to differentiate between two classes of PQevents, named long-duration (oscillatory) and short-dura-tion (impulsive events). The experiment comprises twostages. The feature extraction stage is based on the compu-tation of cumulants. Each vector’s coordinate correspondsto the local maxima and minima of the third and fourth-or-der central cumulants. Secondly, the classification stage isbased on the application of the competitive layer to thefeature vectors. We use a two-neuron competitive layer,which receives two-dimensional input feature vectors dur-ing the network training.

0.02 0.04 0.06 0.08-0.5

00.51

Analysis of an oscillatory transient

Original signal

0.02 0.04 0.06 0.08-0.5

0

0.5 High-passed signal: PQ event

Time(s)

100 200 300 400 5000.020.040.060.08

0.10.12

2nd-order sliding cumulant

100 200 300 400 500-4-202

3rd-order sliding cumulant

100 200 300 400 5000

10

20

Number of segment

4th-order sliding cumulant

0.02 0.04 0.06 0.08 0.1-0.5

0

0.5

1Analysis of an impulsive transient

Original signal

0.02 0.04 0.06 0.08 0.1

-0.5

0

0.5 High-passed signal:PQ event

Time(s)

100 200 300 400 500 600

0.020.040.060.08

2nd-order slidingcumulant

100 200 300 400 500 600

024 3rd-order sliding

cumulant

100 200 300 400 500 6000

102030

Number of segment

4th-order slidingcumulant

Fig. 3. Comparative higher-order analysis of an impulsive and an oscillatory electrical transients.

100 200 300 400 500 600-4

-2

0

2

4

Number of segment

Ampl

itude

Comparison of 3rd-order cumulants

Oscillatory (longduration)Impulsive (shortduration)

100 200 300 400 500 60005

101520253035

Number of segment

Ampl

itude

Comparison of 4th-order cumulants

Oscillatory (longduration)Impulsive (shortduration)

Fig. 4. Third-order vs. fourth-order analysis for the former electricaltransients (zoom of the bottom sub-graphs) in Fig. 3.

482 J.J. González de la Rosa et al. / Measurement 42 (2009) 478–484

We analyze a number of 26,1000-point real-liferegisters during the feature extraction stage. Before thecomputation of the biased cumulants, two pre-processingactions have been performed over the sample signals. First,they have been normalized because they exhibit verydifferent-in-magnitude voltage levels. This disparity ofvoltage levels cannot influence the results of the categori-zation. Secondly, a high-pass digital filter (fifth-order But-terworth model with a characteristic frequency of 150 Hz)eliminates the low frequency components which are notthe targets of the experiment.

After pre-processing, a battery of sliding central biasedcumulants second, third and fourth) is calculated. Eachcumulant is computed over 50 points; this window’s length(50 points) has been selected neither to be so long to coverthe whole signal nor to be very short. The algorithm calcu-lates these three central cumulants over 50 points, and thenit jumps to the following starting point (next 50-point over-lapped group); as a consequence we have 98% overlappingsliding windows (49/50 = 0.98). Each computation over awindow (called a segment) outputs three cumulants (2nd,third and fourth-orders).

Besides, each nth-order cumulant, Cumn;x½i�, associatedto the ith computation segment has been normalized byðCum2;x½i�Þn=2, in order to obtain categorization results asso-ciated to the shape of the sliding cumulants. This gives areal statistical characterization. If the cumulants are notnormalized, the maxima and minima also gather informa-tion regarding the value of the cumulants. The higher-or-der (n > 2) normalized cumulants are the skewness andthe kurtosis.

Fig. 3 shows the comparison between the analysis of anoscillatory event and an impulsive event. The second-order

cumulant sequence corresponds to the variance, whichclearly indicates the presence of an event, due to the excessof power. Both types of transients exhibit an increasingvariance in the neighborhood of the PQ event, that presentsthe same shape, with only one maximum. The magnitudeof this maximum is by the way the only available featurewhich can be used to distinguish different events fromthe second-order point of view. This may suggest the useof additional features in order to distinguish different typesof events. Higher-order results allow to differentiate be-tween the transients.

J.J. González de la Rosa et al. / Measurement 42 (2009) 478–484 483

Fig. 4 shows the comparison of the former events inFig. 3 looking at the third and fourth-order cumulants.Maxima and minima of the sliding higher-order cumulantsare selected as features from each sample register.

The results of the training stage (using the Kohonenrule) are shown in Fig. 5 and in Fig. 6, for the third andfourth-order sets of features, respectively. The horizontal(vertical) axis corresponds to the maxima (minima) values.Each cross in the diagram corresponds to an input vectorand the circles indicate the final location of the weight vec-tor (after learning) for the two neurons of the competitivelayer. Before training, both weight vectors pointed to theasterisk, which is the initializing point.

The separation between classes (inter-class distance) iswell defined in both 2-D feature graphs. Both types of PQevents are clustered. The correct configuration of the

3 3.5 4 4.5 5 5.5 6 6.5-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5Cluster classification: 3rd_order cumulants

Cumulant maxima

Cum

ulan

t min

ima

:measured vector:initial neuron weight vector:final neuron weight vector

Fig. 5. Competitive layer third-order training results over 20 epochs.Upper cluster: Short-duration PQ-events. Down cluster: Long-durationevent.

24 26 28 30 32 34 36 38-2.6

-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8Cluster classification:4th_order cumulants

Cumulant maxima

Cum

ulan

t min

ima

:measured vector:initial neuron weight vector:final neuron weight vector

Fig. 6. Competitive layer fourth-order training results over 20 epochs.Right cluster: Short-duration PQ-events. Left cluster: Long-durationevent.

clusters is corroborated during the simulation of the neuralnetwork, in which we have obtained an approximate clas-sification accuracy of 97%. During the simulation, new sig-nals (randomly selected from our data base) wereprocessed using this methodology. The accuracy of theclassification results increases with the number of data.To evaluate the confidence of the statistics a significancetest has been conducted. As a result, the number of mea-surements is significantly correct.

6. Conclusion

In this paper we have proposed an automatic method todetect and classify two PQ transients, named short andlong-duration. The method comprises two stages. The firstincludes pre-processing (normalizing and filtering) andoutputs the 2-D feature vectors, each of which coordinatecorresponds to the maximum and minimum of the centralhigher-order cumulants. The second stage uses a neuralnetwork to classify the signals into two clusters. This stageis different-in-nature from the one used in [5] consisting ofquadratic classifiers and was previously used in [4], with-out having been normalized the cumulants. The configura-tion of the clusters is assessed during the simulation of thenetwork, in which we have obtained acceptable classifica-tion accuracy.

Acknowledgements

The authors thank the Spanish Ministry of Science andEducation for funding the project DPI2003-00878, wherethe different noise processes have been modeled and con-trasted. Thanks for supporting the PETRI project PTR95-0824-OP dealing with plague detection using higher-orderstatistics; and also the project DPI2006-02970, where non-destructive testing based on acoustic emission signals isperformed. Our unforgettable thanks to the trust we havefrom the Andalusian Government for funding the excellencyproject PAI2005-TIC00155, where higher-order statisticsare modeled and applied to plague detection and powerquality analysis.

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