Covariance Analysis of Voltage Waveform

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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006

Covariance Analysis of Voltage Waveform Signature
for Power-Quality Event Classification
Ömer Nezih Gerek, Doğan Gökhan Ece, and Atalay Barkana

Abstract—In this paper, covariance behavior of several features
(signature identifiers) that are determined from the voltage waveform within a time window for power-quality (PQ) event detection
and classification is analyzed. A feature vector using selected signature identifiers such as local wavelet transform extrema at various
decomposition levels, spectral harmonic ratios, and local extrema
of higher order statistical parameters, is constructed. It is observed
that the feature vectors corresponding to power quality event instances can be efficiently classified according to the event type using
a covariance based classifier known as the common vector classifier. Arcing fault (high impedance fault) type events are successfully classified and distinguished from motor startup events under
various load conditions. It is also observed that the proposed approach is even able to discriminate the loading conditions within
the same class of events at a success rate of 70%. In addition, the
common vector approach provides a redundancy and usefulness
information about the feature vector elements. Implication of this
information is experimentally justified with the fact that some of
the signature identifiers are more important than others for the

discrimination of PQ event types.
Index Terms—Covariance analysis, event classification, higher
order statistics, power-quality (PQ) analysis.

I. INTRODUCTION
HERE is an ever increasing interest in detection and classification of power-quality (PQ) events due to costly investments on delicate electronic devices. One of the most common
PQ event types of voltage sags, due to the distribution system
faults or switching on large loads, may shut down processes run
by susceptible electronic devices. There has been several system
proposals for the detection and classification of PQ events. The
first step in these proposals is the monitoring and acquisition of
large amount of waveform data from a distribution system under
consideration. The next step is the detection and classification
of PQ events using automated, on or offline systems from acquired data. The focus of this paper is related with the second
step. Since the acquired waveform is usually in digital form,
the available analysis tools are digital signal processing (DSP)
and computer-aided statistical analysis. The DSP tools are relatively well established, and they mostly include spectral domain analysis, correlative methods, and wavelets. The survey in
[1] cites most of the work in this area. The DSP-based methods
such as wavelet and Fourier transforms provide the quantitative

T

Manuscript received March 7, 2005; revised February 28, 2006. This work
was supported by the Anadolu University Research Fund under Contract
06000212. Paper no. TPWRD-00124–2005.
The authors are with Anadolu University, School of Engineering and Architecture, Electrical and Electronics Engineering Department, Eskişehir 26470,
Turkey.
Digital Object Identifier 10.1109/TPWRD.2006.877102

data (or the feature vector) for the automated detection and classification. These data are then fed to a classifier that is designed
using statistical analysis tools, ranging from simple thresholders
[14] to rather sophisticated inference methods [2]. In this multidisciplinary area of PQ event analysis, even algorithms used
for speech recognition such as “Dynamic Time Warping” are
employed as a classification tool [4]. No matter how powerful
and efficient a proposed classifier may be, a poorly selected feature vector would yield inaccurate classification results. For example, consider two different voltage sag events, due to the same
type of a fault in a distribution grid. Although both events are
caused by the same type of a fault, the recorded voltage waveform properties may be quite different in terms of sag magnitude, duration, and harmonic content of transient imposed upon
voltage waveform. Therefore, prior to the automated classification, a well defined feature vector, which accurately describes
the signature of an event, is needed.
In this work, the system voltage is acquired as described in

Section II, and a multidimensional feature vector is constructed
from a time window that contains not only the event instant but
also pre- and post-event voltage waveform data. This feature
vector development is based on a combination of modern signal
processing methods including local wavelet extrema, short-time
spectral harmonics, and local higher order statistical parameter
extrema. The details of the considered feature vector including
its extraction from waveform data and element-wise characteristic behavior are explained in Section III.
Although the selection of signature identifiers for a well
defined feature vector is crucial, another critical step is the
design of a classifier which uses the feature vector. In Section IV, a novel covariance-based subspace classifier, known
as the “common vector classifier” (CVC), which was recently
used for speech recognition [9], is illustrated. Since it is a
covariance-based classifier, the CVC structurally resembles
the Karhunen Loeve Transform (KLT). However, the discriminative feature vector selection of the CVC is inherited from
a different idea, and results in a different transform [8]. The
reason of adopting the CVC as a classifier is two-fold. First,
it is a successful classifier which eliminates the weak points
of KLT or principle component analysis (PCA) by incorporating different subspaces to each class. Second, it provides
information and insight about classification strength of each

feature vector element according to their distances from selection boundaries. Therefore, using this classifier, it is shown
that some of the proposed feature elements are more effective
in classification process. Other celebrated classifiers such as
Bayesian classifiers, support vector machines (SVMs), and
neural networks, are also more or less capable of exhibiting a

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GEREK et al.: COVARIANCE ANALYSIS OF VOLTAGE WAVEFORM SIGNATURE

Fig. 1. Experimental low-voltage system.

similar performance to the CVC; however, they do not provide
the information regarding the discriminative features of the
event waveform signature.
In Section V, experimental classification results obtained by
the proposed feature vector using the CVC method are presented
and compared to those obtained from two other popular classifiers; SVMs and Bayesian. To test the detection and classification performance of the proposed scheme, a real-life experimental system is constructed. Since all the acquired data are obtained from real life experimental system, the data naturally contains parametrical randomness and added noise due to system
loads and environmental conditions. The use of such real life
data justifies the effectiveness and generality of the proposed

method as opposed to methods tested by using computer generated data. Nevertheless, the proposed system parameters can be
further tuned and improved by collecting more voltage waveform data under numerous loading and environmental conditions. Using the experimental data, good classification results
are obtained for the discrimination of arcing fault-type events
from motor startup events. Interestingly, it was also observed
that the proposed feature vector—classifier combination is even
capable of providing an information for the load condition of the
experimental system within the same class of PQ events. Since
the CVC method is capable of indicating which elements of the
feature vector are more discriminative (or useful) for the classification of certain types of events, by eliminating the less useful
signature identifiers from the feature vector, successful classification results were also obtained.
II. EXPERIMENTAL SETUP
Using the experimental setup shown in Fig. 1, statistically
sufficient number of PQ experiments were conducted under
available variation of loading conditions. Voltage waveforms
of real-life PQ events were captured at 20 kHz from the experimental system. The system is composed of a three-phase
wye-connected 400-V, 50-Hz, 25-kVA, five-wire supply loaded
with RL load bank and three-phase induction motors coupled
with varying mechanical loads. The system also includes
adjustable speed drives controlling the induction motors for
studying load generated harmonics. Experimental voltage sag

events were obtained by starting mechanically loaded induction
motors in a controlled way. During the motor starting experiments, mechanical loading was changed between 50%–100%

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of rated load. Also, the instantaneous value of the supply
voltage at the time of motor starting was naturally random. As
a result, various voltage sag levels were obtained and acquired
to be used in the proposed algorithm.
Arcing fault events were staged between a phase wire and
the ground wire by stripping their insulation for a few millimeters, aligning the stripped parts, and placing several strands of
AWG 12 electrical wire between the stripped portions of the
phase and ground wires. The arcing faults were initiated in a
controlled fashion by turning on the switch connected in series
to the phase wire. Once initiated, arcing fault experiments were
recorded until the fault clears itself. Due to the randomness of
the physical way of preparing the wire samples and randomness of the instantaneous value of the voltage supplying the fault
at the instant of fault initiation, a wide variety of fault sample
records was obtained. As an example to the physical randomness of the process, it was observed that while some of the arcing
faults restriked several times before clearing itself, others striked

once and cleared quickly.
The data-acquisition unit consists of an analog-to-digital converter (ADC) unit that was set to perform signal sampling at 20
kHz at each channel simultaneously from four different channels. The data-acquisition system also includes programmable
digital filters that can be adjusted to perform sharp frequency
selective filtering operations in real time.
III. PQ EVENTS AND FEATURE VECTOR CONSTRUCTION
The feature vector used in this work consists of scalar numbers obtained by three major methods; wavelets, spectrum analysis, and higher order statistical parameters. The instrumentation in our experimental system acquires the voltage and current
waveforms and their 50-Hz notch-filtered versions at a sampling
rate of 20 kHz for each waveform. The local estimation window
size was selected as 800 samples which corresponds to twice the
fundamental period length.
The feature vector contains 19 signature identifiers. The
first eight correspond to the wavelet transform extrema for
the four-level decomposition of voltage waveform using the
Daubechies-4 (db4) orthogonal wavelet as follows:
for
for
(1)
where corresponds to the th detail decomposition level of the
wavelet transform, corresponds to the time instance of the PQ

event, and corresponds to a time window size that depicts the
time vicinity of the PQ event.
These four levels depict time-frequency-localized signatures
at different frequency resolutions. At decomposition levels
higher than four, the time resolution is decreased beyond a
factor of 32, which is below the desired time-resolution level for
the acquired waveform with a sampling frequency of 20 kHz.
The extrema correspond to the maximum and the minimum
transform values around the instance of a PQ event. It was
previously shown by several authors that the transform-domain
values exhibit high energy around PQ event instances. Usually,
simple thresholding of these coefficient magnitudes is enough
to detect the existence of a PQ event. However, for the detection

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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006

and classification of different classes of PQ events, more waveform signature identifiers as well as sophisticated classifiers are
required. Furthermore, since wavelet subspaces require signal

changes with high-frequency components, if the transition from
the steady-state condition to a PQ event (specifically a voltage
sag or swell) is a smooth transition, then wavelets naturally fail
even to detect the event.
The ninth coefficient of the feature vector was selected according to classical spectral analysis. The signal energy was calculated exactly at the line frequency (50 Hz), and proportioned
to the remaining spectral energy at all other frequencies, then
the reciprocal of this quantity was taken. Analytically, this expression can be expressed as

(2)
where is the feature vector and
corresponds to its 9th elis the power spectral density of the voltage
ement, and
waveform
.
The remaining ten coefficients were obtained from the statistical parameters of the 50-Hz notch-filtered voltage waveforms. The low-variance Gaussian behavior of the 50 Hz-removed voltage waveform under normal conditions without any
event is the reason for selecting higher order statistical parameters as part of the signature identifiers in the feature vector
because each PQ event can be modeled as a noise contribution over the voltage waveform. In fact, in [3], it was shown
that the power system voltage waveform can be modeled as
a combination of a pure sinusoid and noise components imposed upon that sinusoid. The sinusoidal component is not the
informative part in terms of an event detection or classification, but its existence greatly perturbs local statistical parameters. On the other hand, the noise component contains valuable

information in case of PQ events and event-driven transients.
For that purpose, during the data acquisition, the 50-Hz sinusoid of the voltage waveform was removed using Frequency
Devices’ ASC-50 programmable filter adjusted to a very sharp
(20th-order Elliptic) 50-Hz notch filter. Under event-free operation conditions, the output waveform of the filter can be modeled as Gaussian. This model is pretty accurate, because the
voltage waveform may be noise-corrupted due to the ambient
conditions such as electromagnetic-interference (EMI)-generating loads running on or near the system and from the central
limit theorem, the combination of independent random sources
add up to a Gaussian process as the number of sources grows.
In this specific work, in addition to the local central cumulants
of order 2, 3, and 4, skewness and kurtosis were also used as
statistical parameters. The Gaussian model observation justifies
the use of higher order cumulants together with skewness and
kurtosis which specifically provides a measure for the amount
of deviation from Gaussianity. The use of such parameters also
makes the classifier robust to noisy conditions since the noise is
generally in the form of an additive Gaussian process.
Classical waveform noise detection methods, on the other
hand, usually refer to sample means and sample variances (cor-

responding to lowest two statistical orders) of the waveform.

However, higher order statistics are known to be effective tools
to detect and discriminate deviations from Gaussianity [5], [6].
The mean and variance only carry information related to the
“spectra,” which is only the mono-spectra. This much information may be sufficient to detect and identify Gaussian-type processes. However, complicated fluctuations cannot be visualized
or differentiated in such low orders. Poly-spectra is a commonly
used method to detect and identify non-Gaussian processes or
phase nonlinearities [7] as is the case in this work.
Calculated local extrema of these five parameters add up to
the last ten coefficients of our feature vector. Definitions of
central cumulants and their relation to moments are briefly described in the Appendix. In order to better discriminate PQ event
types, cumulants of order higher than 4 can also be used. However, the meanings of such very high order cumulants are physically and statistically unclear. Therefore, they are hard to encounter in any application in the literature. It is reasonable to
keep the parameter estimation window size a small integer multiple of the fundamental period length. The selected window
size of twice the fundamental period length is statistically long
enough to accurately estimate statistical parameters, and short
enough to accurately resolve the time localization. When using
such a window size, it is not statistically meaningful to approximate and use cumulants of order higher than 4. Therefore, they
are avoided.
Consequently, the last ten coefficients of the feature vector
are formed as

and

and

and
and
and
and

(3)

The total feature vector, finally, consists of three main types of
signatures. The first two types are sample values of power spectral density and wavelet decomposition extrema which are calculated using voltage waveforms. The third signature type consists of statistical parameters which are calculated from 50-Hz
notch-filtered versions of these waveforms. The resulting feature vectors are used for discriminating the event classes as
follows.
1) Class 1: Arcing fault with resistive, inductive, and ASD
load.
2) Class 2: Arcing fault with resistive and inductive load.
3) Class 3: Motor startup with resistive and inductive load.
4) Class 4: Motor start-up with resistive, inductive, and ASD
load.
Events were staged under several loading conditions. For both
arcing faults and motor starting events, the experimental system
was first loaded with inductive and resistive loads only. Next,
adjustable speed drives (ASDs) were also connected in order
to observe the performance of the proposed PQ event detection
and classification method in the presence of harmonics and EMI

GEREK et al.: COVARIANCE ANALYSIS OF VOLTAGE WAVEFORM SIGNATURE

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Fig. 2. Phase-to-ground arcing fault event.

generating loads. In Fig. 2, first column depicts voltage waveform and parameters obtained from this waveform for an arcing
fault event when the system is loaded with inductive and resistive load banks and ASDs. Similarly, the second column gives a
50-Hz notch-filtered voltage waveform for the same event and
statistical parameters derived from the filtered waveform. Fig. 3
shows the waveforms and obtained feature vector parameters for
a voltage sag event due to an induction motor starting.
Due to the general behavior of an energy change in the feature
vector parameters, the event detection becomes mainly a thresholding problem. The thresholding can be carried out both along
samples of wavelet decomposition, and/or samples of statistical
parameters. However, thresholding should be applied to the parameter(s) that shows definite change in energy at the instant of
a PQ event. Notice that in Fig. 3, wavelet decomposition levels
do not show any significant change in energy during the voltage
sag event. The wavelet samples would normally become large if
there were high-pass discontinuities incorporated to the voltage
waveform during a PQ event. We purposely selected the event
shown in Fig. 3 as an example for the case of a smooth transition
from normal system voltage to voltage sag without any high-frequency transients. Therefore, wavelet decomposition levels do
not show any significant change in energy. As a result, if one
selects feature vector parameters composed of only wavelet decomposition level, this particular event can neither be detected
nor classified. On the other hand, local statistical parameters
yield variations in magnitude around regions of the voltage sag

event, indicating a strong deviation from the steady-state distributions and providing a tool for detection and classification.
Experimental observations show that the variance of the
50-Hz notch-filtered voltage waveform is the most robust
indicator of a PQ event. Therefore, the event detection criterion
was selected as thresholding the variance. The threshold is
automatically determined by calculating the variance of the
first ten cycles of the steady-state waveform and setting it as
twice the steady-state variance.
In general, the changes in signature identifiers for the cases
of arcing faults and motor startup events are visually similar.
Most of the signature identifiers exhibit variations in magnitude during a PQ event. However, in the next section, it will be
shown that feature vectors combined with the common vector
based classifier is capable of discriminating arcing faults from
motor startup events. The possibility of such a classification
arises from the fact that, the structure of the energy change in the
signature identifiers are different for the two classes. Following
the stage of event detection, the feature vector is generated according to the rule described above. The classifier is used only
when an event is detected.
IV. PQ EVENT CLASSIFIER
Apart from the judicious selection of signature identifiers as
elements of feature vector for PQ event classification, the significance of this work is the adoption of a novel classifier based
on the common vector approach. The CVC is a covariance-

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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006

Fig. 3. Motor starting event.

based technique which has a theoretical basis similar to the
Karhunen–Loeve Transform (KLT) or the PCA. However, the
derivation of the transform and its application for classification
is unique. The common vector analysis suggests that, one should
select eigenvectors corresponding to the zero or smallest eigenvalues of covariance matrices of each class for discrimination.
On the other hand, the KLT requires the selection of eigenvectors corresponding to the largest eigenvalues for the best (or
minimum error) representation of a signal. This implies that the
best representation subspace does not correspond to the best discriminating subspace. This situation is quite clear from a covariance illustration described in Fig. 4. In this figure, the principal
components of the covariance scatter contours corresponding to
two separate classes are along the direction of the major axis of
. However, along this direction, the separability
the ellipses
of the two classes is very poor. Conversely, the direction of the
is the correct (Fisher) choice for the separation.
minor axis
A single transform matrix is enough to separate classes as
long as the major and minor axes of the covariance scatters are
parallel for both cases. On the other hand, separate transforms
are needed if the covariance scatters of classes exhibit different
principle angles. In order to remedy this situation, the common
vector classifier treats each class separately (within-class approach). In this way, covariance matrices of each class are calculated, and an eigen-analysis is carried out for each class. The
,
covariance within a class is ideally represented as
where
is the expectation operator, corresponds to the
class index, and corresponds to the feature vector element
index. This theoretical expectation can approximated using the

Fig. 4. Covariance contours and principal components for a sample two class
scatter.

available data. A feature vector consists of 19 scalar elements
in a column form. For example,
corresponds to the feature vector obtained from the th experiment for a given class
of PQ event. First, these vectors are stacked in columns of a
corresponds to the number of data
matrix whose width
. Next,
within the class, and height is 19:
the mean vector corresponding to the same class is calculated:
and it is extended to a matrix form as:
to make the matrix sizes compatible. A
. Fimean removed data matrix corresponds to:
nally, the intraclass covariance matrix is simply

GEREK et al.: COVARIANCE ANALYSIS OF VOLTAGE WAVEFORM SIGNATURE

for class . The common vector classifier for this class proand eigenvalues
ceeds as follows; First, eigenvectors
are evaluated from the covariance matrix. Next, the eigenvectors corresponding to the zero eigenvalues are taken. Practieigenvalues among
available are concally, the smallest
sidered. These eigenvectors are stacked into a matrix as
, and the common vector transform matrix corresponding to class is created:
. This transform
matrix is applied to the mean-removed versions of the original
feature vectors of class . The result of this transformation produces a vector with elements close to zero if eigenvalues are
, where
small
is the common vector.
Ideally, the transform vector becomes completely zero if true
expectations were used. It was shown in [8] that the transform
vector corresponds to the projection of the input vectors onto
the zero space of the covariance scatter. The zero output situation is valid as long as the input feature vector belongs to the
considered event class. If the input belongs to another class, the
transform operation gives a vector which is large in magnitude.
Due to this property, the within-class common vector transform
constitutes a distinguishing signature for the class. For the multiple-class problem, common vector transforms must be constructed for each class of training data, and the feature vector,
which is considered for classification, should be applied to each
of these common vector transforms corresponding to different
classes. Each transform gives a separate output vector. The classification rule selects the vector with the smallest magnitude,
and assigns the vector to the class corresponding to this situation. This is a novel approach which was recently applied to
speech and face recognition [9], [10]. Due to the analysis made
in [8], it is known that the common vector approach yields the
optimum discrimination space from the covariance of the input
data. This fact is also justified by our experimental results.
In this study, four different transforms were obtained corresponding to feature vectors derived from the four considered PQ
events explained in Section III. The first two cases correspond
to the PQ event type of arcing faults, and the last two cases correspond to PQ event type of motor startup events. For different
load conditions, even for the same class of event, separate feature vectors were obtained in order to monitor the classification
performance of the selected signature identifiers and classifier
combination. The experimental results for the classification of
the two fundamental PQ event types, as well as results for the
resolution of load conditions are presented in Section V.
A particularly useful aspect of the common vector approach
is that it also provides a usefulness measure for each signature
identifier used in the feature vector. As seen from the above
explanations, the ideal transform output in case of a correct class
should correspond to a zero vector. Conversely, if an incorrect
class transform is used, the output should be a vector large in
magnitude. Experimentally, it was observed that some of the
signature identifiers satisfy this property better than others. In
order to reduce the set of parameters used for the feature vector
construction, the elements of the eigenvectors were analyzed.
Consequently, the more useful signature identifiers were kept
and the rest were discarded in the common vector transform
matrix construction. The results for parameter space reduction
is presented in Section V-A.

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V. EXPERIMENTAL CLASSIFICATION RESULTS
The voltage waveform data is acquired during PQ events corresponding to arcing faults and motor startup events with different loading conditions, as described in Section III. For both
arcing faults and motor starting events, a total of 60 experiments
were carried out with inductive and resistive load conditions; 30
of them for arcing faults, 30 of them for motor startup events.
Next, ASDs are also connected and 30 set of experiments were
carried out for both arcing faults and induction motor startup
events. The detection of the events was obtained by monitoring
the change in local variance, and comparing it with a threshold
which was set according to the steady state variance measured
from the first ten cycles of the voltage waveform. All of the experimentally generated events in this work were successfully detected using this method.
Once an event is detected, wavelet extrema, spectral harmonic
ratio, and HOS parameter extrema are calculated within the
local time window corresponding to the detected event location
to form a 19-dimensional feature vector. Consequently, a total of
120 feature vectors were ensembled corresponding to four different classes for the experimental studies. In this section, we
present results for
1) fundamental PQ event-type classification that separates the
first two classes from the last two, namely discriminating
arcing faults from induction motor starting events; and
2) separation of the four classes which implies a method to
identify loading conditions.
Due to the structure of the common vector classifier, a
projection operator (consisting of a transform matrix and a
common mean vector) is constructed separately for each of the
four classes using available training data. The classification
performance obtained by using the training data as the test
data would not be a fair indication about the success of the
method. In order to efficiently use the restricted number of
experimental data, the celebrated leave-one-out technique is
used. In this technique, for each acquired data, one of the
calculated feature vectors belonging to class is taken as the
test data, and the rest of the feature vectors (29 out of a total of
30) from class constitutes the training data. Using this training
data, a projection operator is constructed for class . For the
other three classes, all of the data is used as the training data,
and three more projection operators are constructed. Next, the
test data is projected using the four projection operators, and
the test data is classified to a class whose projection operation
gives the minimum norm projection vector. This operation is
carried out for each of the experimental data belonging to any
of the four classes. In each case, one of the class projections is
determined using 29 elements while the other class projections
are determined using 30 elements. This is, effectively, a method
to obtain 120 test data and 120 training data out of a total of
120 elements.
The classification results for four classes are cross-tabulated
in Table I. Since Class 1 and 2 are never mixed with Class 3
and 4, it can be concluded that the PQ event type classification
accuracy is 100%. The feature vectors also carry some information regarding the load conditions during the same type of PQ
events. From the same table, it can be observed that the load

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TABLE I
CLASSIFICATION RESULTS FOR 4-CLASS EXPERIMENTS. DARK-SHADED BOXES
BELONG TO THE CLASS OF ARCING FAULTS, LIGHT-SHADED BOXES BELONG
TO THE CLASS OF MOTOR STARTUPS

conditions can be separated with an accuracy of 70%. When
ASDs are connected to the experimental system as a part of the
load, they draw current with high harmonic content. This current, therefore, causes high harmonic voltage distortions that effects the signature identifiers associated with spectral content of
the voltage waveform. In this work, real-life data are used and
the magnitude, duration, and harmonic content of each event is
effected by the load composition. For some of the staged events,
the effect of the ASD portion of the load may be more pronounced, enabling us to discriminate experimental system load.
Although the effects caused by different load configurations are
quite small as compared to the waveform distortions due to PQ
events, our classifier managed to discriminate load conditions
of 70% of staged events.
Alternative classification results were also obtained using
two popular classifiers; the support vector machines (SVMs)
and Bayesian classifiers using the same training and test data.
In the SVM method, the Gaussian kernel method was used. It
was observed that 100% classification success rate could be
achieved using at least seven support vectors to distinguish
arcing faults and voltage sags due to motor starting events.
Using the Bayesian classifiers, the classification success rate
was found to be 97%. Apparently, the classification performance of SVM is the same as the method proposed here.
However, as explained and illustrated in the succeeding subsection, the the proposed CVC method has a major advantage
of not only yielding successful classification results, but also
providing information about the usefullness of each of the
parameters in the feature vector. The information about such a
usefulness measure gives insight about the descriptive features
of the voltage waveform signature. In this way, non or poorly
descriptive features and computations to generate them can
be avoided. Furthermore, the computational complexity of
the CVC is less than that of similarly performing methods
including SVM.
A. Feature Accuracy Analysis
The selection of feature vector elements as described in Section III is well justified with the above experimental results.
However, the CVC classifier and its construction from the eigenvectors of the covariance matrices are also capable of giving
1) the amount of dimensionality reduction;
2) hints about which elements among the 19-dimensional feature vector are more meaningful and useful in the discrimination process.
In this subsection, the dimensionality reduction from the covariance data is described and selection of the eigenvectors for

Fig. 5. Eigenvalue magnitudes obtained from covariance matrices.

transform matrix construction is explained. During the construction of transform matrices, the eigenvectors of the covariance
matrices corresponding to the zero or small eigenvalues must
be selected. On the other hand, selection of the number of small
eigenvalues is not deterministic. It is usually a good idea to visualize the eigenvalues corresponding to covariances matrices
belonging to two PQ event types. In Fig. 5, the eigenvalues obtained from the covariance matrices belonging to two PQ event
types are plotted. The figure implies selecting 10 of the 19 eigenvalues as small. Therefore, the experiments are carried out using
transform matrices obtained from eigenvectors corresponding
to 10 of the smallest eigenvalues. In fact, selection of number
of zero eigenvalues other than 10 gave experimentally inferior
classification accuracy.
The second analysis described in this subsection is about the
usefulness of the elements inside the 19-dimensional feature
vector. This analysis can be carried out in two ways.
1) The smaller magnitudes of signature identifiers along a dimension for selected eigenvectors mean that the considered
dimension contributes less to the discriminant.
2) The larger magnitudes of signature identifiers along a dimension for unselected eigenvectors mean that the considered dimension is irrelevant to the discriminant.
The averages of absolute values of each dimension within selected and unselected eigenvectors are presented in Figs. 6(a)
and (b) for all of the four classes. According to the usefulness
analysis explained above, Fig. 6(a) implies that dimensions 9,
11, and 15-to-18 are important. While selecting the important
dimension indexes, the exhibited magnitude must be large for
at least one of the classes. Similarly, Fig. 6(b) implies that dimensions 1-to-8, 10, and 12-to-14 are consistently irrelevant,
whereas dimensions 9, 11, and 15-to-18 are relevant. The relevant dimensions are selected according to the rule that the magnitude must be small for at least one of the classes. Interestingly,
the 10th element which corresponds to the variance maxima was
assigned as “not useful.” During the initial detection of the PQ
event, it was observed that this parameter is the most useful. This

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TABLE II
CLASSIFICATION RESULTS FOR FOUR-CLASS EXPERIMENTS USING SEVEN
PARAMETERS FROM THE ORIGINAL FEATURE VECTORS. DARK-SHADED
BOXES BELONG TO THE CLASS OF ARCING FAULTS, LIGHT-SHADED BOXES
BELONG TO THE CLASS OF MOTOR STARTUPS

Fig. 7. Scatter plots of data projected to the operators of Class 1.
Fig. 6. Dimension element magnitudes for (a) selected, and (b) unselected
eigenvectors.

implies that eigenvector elements with more significant eigenvalues (as suggested by PCA) are more robust for the detection
or representation of the existence of a PQ event. Conversely,
such elements are not useful for discriminating the PQ event
types. Nevertheless, since this is a parameter specifically used
for detection, it is included in the list of “to be used elements.”
According to these observations, the set of elements for the feature vector is reduced to the elements 9-to-11 and 15-to-18.
These elements correspond to spectral harmonic energy proportion and HOS parameters. It is noteworthy to observe that the
wavelet coefficients are not significantly relevant in the discrimination.
After skimming the feature vector to seven dimensions, the
CVC analysis was carried out once again. The general result is,
the PQ event-type classification is still 100% accurate. However,
the separation between different loading conditions had deteriorated. Normally, separation of loading conditions is not critical
in PQ event classification; therefore, the general use of spectral harmonics and HOS parameters seem to be perfectly suf-

ficient. As a result, the selected and unselected feature vector
elements propose that the more critical parameters are “spectral harmonics,” “2nd, 3rd, 4th central cumulant extrema,” and
“skewness maxima.” The relatively unimportant parameters are
found as “all of the four wavelet transform extrema,” “skewness minima,” and “kurtosis extrema.” The classification results
using the reduced feature vector are shown in Table II.
Following an analysis similar to the one carried out in Fig.
5, the common vector approach projection matrix for the reduced feature vector was calculated with three eigenvectors corresponding to the smallest three eigenvalues. Since the dimensionality reduction implies a projection dimension of 3, unlike
the previous case of reducing to 10 dimensions, the projection
data can be visualized and displayed. In Fig. 7, the three-dimensional (3-D) scatter plot shows the first three dimensions
of the projected input data using the projector corresponding
to Class 1. Consequently, the data belonging to Class 1 gather
around a compact region, and data from other classes are farther away from the centroid of Class 1. Similarly, in Fig. 8, the
3-D scatter plot shows the first three dimensions of the projected
input data using the projector corresponding to Class 4. The data
belonging to Class 4 gather around a compact region, and data

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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006

Fig. 8. Scatter plots of data projected to operators of Class 4.

from other classes are again farther away from the centroid of
Class 4. Since the projection operator in Fig. 7 is obtained from
arcing fault-type event data, the projected data exhibit a sparse
behavior. Although Class 3 and 4 belong to a PQ event type that
is different from Class 1 and 2, several projection points from
Class 3 and 4 are closer to Class 1 than points from Class 2.
On the other hand, it can be seen that along the single dimension corresponding to the first projection vector element, there
is a well-defined separation. In fact, the Fisher projection along
the first dimension is sufficient to make Classes 1 and 2 come
nearer, and Classes 3 and 4 remain far from them along both
directions. The projection operator in Fig. 8 is obtained from a
more uniformly behaving data of motor startup events. As a result, the scatter plots of Classes 3 and 4 are more compact, and
closer to each other in three dimensions. Similar to the previous
case, a one-dimensional (1-D) Fisher axis is visible along which
the separation is still possible.
VI. CONCLUSION
In this paper, a method of PQ event classification using a
novel covariance-based method; the CVA is described. Prior
to the covariance analysis, a feature set consisting of wavelet
decomposition extrema across four levels, power spectral harmonic ratio, and higher order statistical parameter extrema corresponding to orders 2, 3, and 4 is constructed. The event detection is based on the variance value, which is actually the
second-order central cumulant. Once an event is detected, other
extrema are also recorded to form a feature vector consisting of
19 elements. This feature vector can be considered as a signature of the event. A training set of 60 elements from arcing fault
events and 60 elements from motor starting events is acquired.
Using feature vectors obtained from these events, covariance
matrices corresponding to different event types are constructed.
The CVA provides us with projection operators for each class
type. Classification is done by applying each projection to the

data, and selecting the minimum normed output. The “leave one
out” technique is used to separate the test and training sets. Experimental results imply that the classification accuracy is 100%
between our set of arcing faults and motor startup events. An
interesting observation is that the event driven feature vectors
carry discriminative information for loading conditions up to a
rate of 70%. The CVA also enables us to argue about which
of the feature vector elements are more relevant for discrimination. According to the analysis in Section V-A, higher-order
central cumulant parameter extrema and spectral harmonic information are found to be more relevant for the classification
than the wavelet tranform extrema. It is clear that some types of
PQ events may not be detected using the proposed feature vector
and the classifier. The proposed event detection hence, the classification, depends on the change in the behavior of the voltage
waveform by monitoring the variance. Therefore, if the event is
an ongoing flicker or an inherent harmonic from the beginning
of the monitoring process, no change in variance will be detected and the system will fail. Other transient-type events, such
as voltage swell, would require different training of the classifier with probably different set of feature vectors. The overall results indicate that the CVA is a promising tool for classification
of various types of data. It can also be deduced that construction of the feature vector by selecting parameters using signal
processing techniques constitutes an important aspect of the PQ
event detection and classification from voltage waveform.
APPENDIX
MOMENTS AND CUMULANTS
Before proceeding with the experimental results, a minimal
amount of theoretical background about general concepts of
moments and cumulants is provided [7]. The definition of the
order moment of a single random variable, is given as
(4)
where
corresponds to the characteristic
function, and
denotes the expectation operation. For ex,
ample, the first two moments are
. Using the
and
logarithmic form of the characteristic form, we get the second
characteristic function
(5)
The cumulants (which are basically used in this work) are defined in terms of the second characteristic function
(6)
which correspond to the coefficients of the Taylor expansion of
the second characteristic function. There is a relation between
the moment and cumulant values that can be derived using the
Taylor expansion of the logarithm. For the first four orders, let
,
,
, and
. One can obtain
us define the moments as
that the first four cumulants are
,
,

GEREK et al.: COVARIANCE ANALYSIS OF VOLTAGE WAVEFORM SIGNATURE

, and
. The relation between cumulants and
moments of orders higher than four become more complicated.
and Kurtosis
values are determined from
The Skewness
the central cumulants as

(7)
In practice, these values can be evaluated using sample data as

2031

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Ömer Nezih Gerek was born in Eskisehir, Turkey, in
1969. He received the Engineer, M.Sc., and Ph.D. degrees in electrical engineering from the Bilkent University, Ankara, Turkey, in 1991, 1993, and 1998, respectively.
He spent one year as a Research Associate at
EPFL, Lausanne, Switzerland. Currently he is a
Professor with the Electrical and Electronics Engineering Department, Anadolu University, Eskisehir.
His research areas include signal analysis, signal
compression, wavelets, and sub-band decomposition.

Doğan Gökhan Ece was born in Ankara, Turkey, in
1964. He received the Engineer degree from Istanbul
Technical University, Istanbul, Turkey, in 1986 and
the M.Sc. and Ph.D. degrees in electrical engineering
from Vanderbilt University, Nashville, TN, in 1990
and 1993, respectively.
Currently, he is Professor with the Electrical and
Electronics Engineering Department, Anadolu University, Eskisehir, Turkey. His research areas include
power quality, fault detection, and modeling.

Atalay Barkana received the B.S. degree in electrical engineering from Robert College, Istanbul,
Turkey, in 1969, and the M.S. and Ph.D. degrees
in electrical engineering from the University of
Virginia, Charlottesville, in 1971 and 1974, respectively.
From 1974 to 1986, he worked on linear and nonlinear theory. His current interests include speech
recognition, pattern analysis, neural networks, and
statistical signal processing. From 1994 to 2004, he
was with the Electrical and Electronics Engineering
Department, Osmangazi University, Eskisehir, Turkey. Currently, he is a
Professor of Electrical and Electronics Engineering Department, Anadolu
University, Eskisehir.