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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 3, JULY 2007
Prony-Based Optimal Bayes Fault Classification
of Overcurrent Protection
Jawad Faiz, Senior Member, IEEE, Saeed Lotfi-fard, Student Member, IEEE, and Saied Haidarian Shahri
Abstract—The development of deregulation and demand for
high-quality electrical energy has lead to a new requirement
in different fields of power systems. In the protection field, this
means that high sensitivity and fast operation during the fault are
required while maltripping of relay protection is not acceptable.
One case that may lead to a maltrip of the high-sensitive overcurrent relay is the starting current of the induction motor or inrush
current of the transformer. This transient current has the potential to affect the correct operation of protection relays close to the
component being switched. In the case of switching events, such
transients must not lead to overcurrent relay operation; therefore,
a reliable and secure relay response becomes a critical matter.
Meanwhile, proper techniques must be used to prevent maltripping of such relays, due to transient currents in the network. In
this paper, the optimal Bayes classifier is utilized to develop a
method for discriminating the fault from nonfault events. The
proposed method has been designed based on extracting the modal
parameters of the current waveform using the Prony method. By
feeding the fundamental frequency damping and ratio of the 2nd
harmonic amplitude over the fundamental harmonic amplitude
to the classifier, the fault case is discriminated from the switching
case. The suitable performance of this algorithm is demonstrated
by simulation of different faults and switching conditions on a
power system using PSCAD/EMTDC software.
Index Terms—Fault, optimal Bayes classifier, overcurrent relay,
prony method, switching.
I. INTRODUCTION
T
RANSIENT analysis of the induction motor starting and
transformer energizing and their effects upon the performance of other power system components have been proposed
as a major research topic for many years, such as studying the
voltage sag due to motor starting and transformer energizing [1]
and proposing a solution to prevent the maltripping of differential relays due to transformer energizing at startup [2], [3].
The development of deregulation in power systems leads to
a higher requirement on power quality and introduces new requirements in various power system fields. In the area of relay
protection, this means that faster protection is needed, while the
maltripping of relay protection is not acceptable. Faster protection can guarantee that when an abnormal operation mode occurs somewhere in a power system, such as the voltage sag due
to faults, it could be quarantined quickly, so as not to propagate to the rest of the system and cause instability. To accomManuscript received January 10, 2006; May 2, 2006. This work was supported by the University of Tehran, Tehran, Iran. Paper no. TPWRD-000022006.
The authors are with the School of Electrical and Computer Engineering, University of Tehran, Tehran 1439957131, Iran (e-mail: jfaiz@ut.ac.ir).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPWRD.2006.886794
plish this, a relay protection should be sensitive enough. Unfortunately, high sensitivity sometimes causes maltripping of relay
protection when there is no fault in the system.
In a deregulated power market, this directly leads to penalty
compensation to the users that suffer from the blackout [4].
Therefore, identification of the factors that produce these maltrips and the introduction of procedures to discriminate them
from the real fault cases are very important. In [4], the factors
producing these maltrips in the viewpoint of overcurrent relays
have been investigated. Power system switching, such as induction motor starting and transformer energizing, is the most important source of maltrips. In [5], a method has been recommended to study the overcurrents due to the switching of the
operation of overcurrent relays. But [4] and [5] have not introduced a method that could discriminate these nonfaulty cases
from the faulty ones.
Normally, in order to prevent maltripping of overcurrent relays due to transients, a longer delay is initiated for relay tripping; therefore, as time passes and the amplitude of the transient
current diminishes, the maltripping of the relays is prevented.
However, this imposed delay slows down the relay operation
during the fault and, consequently, reduces its sensitivity.
However, there are different ways for reducing the starting
current, such as using autotransformers to step down the terminal voltage and star–delta connection for the stator windings,
but high sensitivity of relays can influence the operation of overcurrent relays, particularly when many switchings are considered simultaneously. This may occur in energizing a feeder after
it has been disconnected for a long time, which may lead to high
starting currents that affect the operation of relays.
Also in the case of controlled switching, the starting currents
can be diminished theoretically; however, in practice, there are
some factors that make it impossible to achieve this goal, such
as deviations in circuit-breaker (CB) mechanical closing time,
effects of CB prestrike, errors in the measurement of residual
flux and transformer core or winding configurations which prevents optimal switching [6], [7]. Therefore, it is necessary to
introduce a method to discriminate the switching case from the
faulty case. Although an offline method has been presented in
[8] for the aforementioned objective, it is most convenient for
classification of the recorded data in power-quality applications
and not protection purposes.
By extensive use of the digital relays, the application of intelligent methods, such as neural networks and fuzzy logic as powerful tools for classification purposes, is being increased. However, a large number of required patterns, retraining requirement
subsequent to the addition of new classes and a great deal of
computational effort to minimize overfitting is considered as
disadvantageous.
0885-8977/$25.00 © 2007 IEEE
FAIZ et al.: PRONY-BASED OPTIMAL BAYES FAULT CLASSIFICATION OF OVERCURRENT PROTECTION
To reduce the required patterns for the training phase which
has a direct relationship with the number of inputs, feature
extraction (extracting the feature that could represent different
cases as abstract as possible) is the mainstream problem.
In this paper, the modal parameters of the current waveform
including frequency, damping, amplitude, and phase shift are
extracted using the Prony method. By choosing the fundamental
frequency damping and ratio of the second harmonic amplitude
over the fundamental harmonic amplitude as input, a large reduction in the number of inputs can be obtained.
An optimal Bayes classifier as a type of generative classifier
is used because of its inherent modular architecture and fewer
samples are required compared to discriminative classifiers.
The proper performance of the proposed scheme is studied
by the simulation of various faults and switching cases (motor
starting and transformer energizing) using PSCAD/EMTDC
software. Since the recommended algorithm is based on the
data obtained from the sampled current, which is normally
obtained in the overcurrent relays, it prevents the maltripping
of relays without extra cost.
II. EFFECT OF SWITCHINGS ON RELAY RESPONSE
It should be considered that in the power system when a fault
occurs, there are harmonics, interharmonics, and dc components
in the current waveform. On the other hand, there are some nonfault events in the system that seem to distort the waveform in
a similar way [9]. In this section, the effect of some nonfault
switching conditions on the current waveform is studied.
A. Transformer Energizing
High inrush current is expected when transformers are involved. The inrush current generates a large flux linkage which
pushes the transformer magnetizing core deeply into saturation.
This results in an extremely large current in the primary side
winding, abundant in harmonics. The initial energizing inrush
current can reach values as high as 25 times full-load current
and will decay with time until the normal magnetizing current
value is reached. The decay of the inrush current may vary from
as short as 20 cycles to as long as minutes for highly inductive
circuits [10].
Another concern about transformer energizing is transient
propagation. This normally occurs during transformer energizing, which causes a considerable amount of even harmonics
and dc component in the voltage. These disturbances may
propagate through transformers to the rest of the system, and
be magnified due to the resonance effect. Because of this, the
load currents at other busbars can be severely distorted, which
might have a detrimental impact on the locally installed current
relays.
B. Motor Starting
Motor starting that occurs in medium-voltage and
low-voltage systems is another subject to be considered.
The starting of a large induction motor leads to a current, which
is typically 5 to 6 times the normal operating current. In fact, the
starting current has a very high initial peak which is damped out
after a few cycles, normally no more than two cycles depending
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Fig. 1. Induction motor direct starting (V = 1 p.u.).
of the circuit time constant [11], and after that, drops rapidly to
a multiple value of its nominal level, and is maintained during
most of the acceleration process. The current is then smoothly
reduced to the nominal value that depends on the steady-state
motor mechanical load. This trend has been shown in Fig. 1
[12] which indicates the direct starting of a three-phase motor
connected to the supply at the worst switching angle. The
motor has the following data: 380 V, 7.5 kW, 50 Hz, 1500
r/min
, and
, where
and
are the stator resistance and reactance, respectively. Generally,
the starting time of an induction motor is less then 5 or 6 s.
However, it can be as high as 20 to 30 s for motors having
loads with high inertia. This puts a severe strain on overcurrent
protection. In addition, undervoltage protection is potentially
affected [13].
III. PRONY METHOD
The prony method is a tool that has been used in signal
processing applications and recently has been introduced to the
power system protection field [14]–[16]. Prony is considered
as a powerful tool to analyze a signal and extract its modal
information. This method can be used to analyze time-independent signal and damped signals. The fact that Prony can handle
damped signals and estimate the damping coefficient makes
it suitable for applications based on power system transients.
Prony calculates the modal information, such as frequency,
amplitude, damping, and phase shift; these can be used to
reconstruct the original signal or to make inferences about
system conditions. The fact that Prony can be used for system
stability and protection applications makes it a good candidate
for the modern concept of wide-area protection and emergency
control [17].
Compared to other oscillatory signal analysis techniques,
such as Fourier and wavelet transform, Prony analysis has
the advantage of estimating damping coefficients apart from
frequency, phase, and amplitude. In addition, it fits the best
reduced-order model to a high-order system, both in time and
frequency domains.
The Prony analysis directly estimates the parameters of the
signal by fitting a sum of complex damped sinusoids to evenly
spaced sample (in time) values of the output
(1)
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 3, JULY 2007
Fig. 2. Modular architecture of optimal Bayes classifier with parzen-window
pdf estimates for ! classes.
amplitude of component ;
damping coefficient of component ;
phase of component ;
frequency of component ;
total number of damped exponential components.
The Prony problem is formulated by knowing the value of
in the form of a series of time samples. The
the signal
problem is solved to estimate the value of parameters of the
Prony method such that the squared errors of fitting are minimized. The mathematical details have been reported in [18].
IV. OPTIMAL BAYES CLASSIFIER WITH
NONPARAMETRIC ESTIMATION
Classification methods, from a general point of view, can be
divided into two broad categories, which are the discriminative and generative models. Discriminative classifiers model the
directly, or learn a direct map from inputs
posterior
to the class labels, while generative classifiers learn a model
of the inputs and the label
of the joint probability
and make their predictions by using the Bayes rules to calculate
, and then picking the most likely label . Contrary to a
widely held belief that discriminative models have lower asymptotic error, a generative model was favored in this work. The
reason for this choice is two fold. First, although discriminative
models have lower asymptotic error, a generative classifier may
reach its (higher) asymptotic error much faster and with fewer
samples [19]. The second reason is that the generative classifier
inherently admits a modular architecture as shown in Fig. 2, in
which to add any other class is as simple as gathering some new
data and estimating the class conditional probability distribution
functions (pdfs).
Density estimation can be approached in three ways depending on the problem specifications [20]. When the underlying data are generated from a known distribution or its form
approximately resembles so, one can evidently gain the most
advantage from imposing some constraints on the pdf form.
This class of algorithms is known as parametric density estimation. Although it makes the estimation problem much simpler,
it would not be suitable for models where the constraints do
not hold. All parametric densities are unimodal, whereas many
practical problems involve multimodal densities. To overcome
this problem one can either use mixture density models, which
are somewhat semiparametric or nonparametric forms. Nonparametric procedures can be used with arbitrary distributions
and without the assumption that the forms of the underlying
densities are known. Also, one drawback of nonparametric
forms is that one has to preserve all of the training data in order
to estimate the posterior for each sample. However, since there
were relatively few data in this work, nonparametric density
estimation was the method that has been chosen. Because of the
strong features extracted via the Prony method, the posteriors
are not so complex and, henceforth, the burden of carrying
large amounts of data is alleviated. The details of Parzen and
nearest neighbor nonparametric density estimations used in this
paper are elaborated in the Appendix.
After obtaining the class conditional pdfs, the optimal Bayes
classifier is readily available for use. Considering a uniform
prior, the maximum a posteriori estimate is equal to the maximum likelihood estimate of class conditional pdfs.
V. PROPOSED ALGORITHM
Considering the formerly mentioned themes, the starting current of induction motor and inrush current of the transformer can
influence the proper operation of the overcurrent relays. Therefore, it is necessary to introduce a procedure for discrimination
of the faulty case from the switching case.
Among the basic differences between fault case and induction motor starting and transformer energizing are their magnetic cores which results in completely different behavior with
the faulty case, as noted below.
• Since there is core saturation over initial instants of motor
starting and transformer energizing, this generates the 2nd
harmonic amplitude; therefore, one component that can be
used to discriminate the faulty case from the switching is
to take the 2nd harmonic into account. In fact, this idea is
traditionally utilized in differential protection of the power
transformers to discriminate the internal fault from the inrush current [21].
• Since current tends to the nonlinear region of the core (saturation region), the amplitude of the fundamental component increases and as time passes and the current amplitude is damped, the fundamental component is also decayed. In fact, at the time of transformer energizing and
motor starting, the amplitude of the fundamental compo, but
nent has a descending form as in
in the faulty case, the equivalent circuit is an RL circuit and
[22]. Therethe fundamental component is
fore, in the faulty case, damping (á) of the fundamental harmonic is very small (almost zero) compared to the damping
ratio of the switching case. Another reason for this trend
is confronting with the variable impedance in the above
switching cases. It means that as time passes, back emf in
the induction motor is generated and the transformer magnetizes amplitude of the current decreases which concludes
the increase of the visualized impedance. However, in the
faulty case, there is constant impedance.
This has been clearly shown in Fig. 3. The horizontal axis indicates the damping of the fundamental current and the vertical
axis denotes the ratio of the 2nd harmonic to the amplitude of
the fundamental component. As seen in this figure, the ratio
of the harmonics amplitude and also damping is much larger
than that of the faulty case. In fact, they form two independent
regions. So, in order to discriminate the faulty case from the
FAIZ et al.: PRONY-BASED OPTIMAL BAYES FAULT CLASSIFICATION OF OVERCURRENT PROTECTION
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Fig. 4. Block diagram of the proposed method.
Fig. 3. Ratio of the amplitude of the 2nd harmonic to the amplitude of the fundamental harmonic versus damping of the fundamental harmonic: (3) induction
motor starting, ( ) transformer energizing and (+) faulty case.
O
switching case, first the current signal components are obtained
by Prony method over 1.5 cycle after an increase in current (due
to switching or fault), then the ratio of the 2nd harmonic amplitude to the fundamental harmonic and also the damping of the
fundamental harmonic are given to the optimal Bayes classifier
as input. Although three preliminary classes are assumed for the
three cases (i.e., motor starting, transformer energizing and fault
states), the two first states are announced as nonfault, since their
discrimination is of no importance. However, this scheme assists in making a modular architecture for the classifier, since
any other state can be added without major modification in the
classifier’s architecture.
A simple block diagram of the suggested algorithm has been
shown in Fig. 4. First the analog input signal is converted to
the digital signal by data acquisition unit and goes to the relay
characteristic and detector. If amplitude of the current becomes
larger than the relay setting, the start command is applied to the
detector unit, and the proposed parameters are evaluated by the
Prony method after 1.5 cycles. It is then passed to the classifier
for decision making. In such a case, if a fault occurs, the output
of the detector or pin 2 of the AND gate becomes one. Now, if
pin 1 of the AND gate, considering the relay characteristic, becomes one, then the tripping command is issued. In a nonfaulty
case, the output of the detector or pin 2 of the AND gate becomes zero. In this case, if pin 1 becomes one by mistake (i.e.,
the overcurrent of the switching case leads to a mistake with the
fault case, the tripping command is not issued because one of
the inputs of the AND gate is zero). In order to use the idea that
the switching currents are transient and damped after a while,
the Delay block is considered. Initially its output is zero, when
the start command is issued, after the predefined time its output
and consequently the pin 2 of AND gate becomes 1. In this instant, if pin 1 of the AND gate is still equal to 1, it means that this
is a fault case (because if the switching case had occurred, after
time passes and the current amplitude reduces, pin 1 became 0).
Therefore, the inopportune blocking of the relay is prevented
during the fault. In fact, as time passes, the transient currents
damp and the relay does not trip the relay, this has been considered as a factor for enhancement of the dependability of the
relay operation, while does not cause any delay for relay operation during the fault. It means that on the contrary to the case in
which a delay was applied to the relay for preventing the maltrip that leads to the delay for relay operation during the fault, in
the suggested algorithm there is no such delay in the faulty case
and this delay is applied as a Delay block. This is done to prevent blocking the relay only in the rare cases that the faulty case
may be detected as switching case. In such highly rare cases,
the delay in the relay operation is equal to the delay in the operation of the present relays. Of course, taking into account the
high precision of the suggested algorithm, such a case is very
rare and this block is only considered for enhancement of the
reliability.
VI. SIMULATION RESULTS
To investigate the merit of the proposed algorithm, a part
of a distribution system shown in Fig. 5 [23] is modeled
using PSCAD/EMTDC software. The network parameters of
the 13-bus distribution system are illustrated in this figure.
Several nonfault events are applied to this system along with
some short circuit events at different times. The simulation
results show that how the proposed algorithm could help the
over-current relay to discriminate fault from nonfault situations.
The confusion and confidence matrices of the classifier with
Parzen window (constant ) and k-nearest neighbors (constant
) estimates show that both nonparametric methods separate
extremely well with high confidence (Table I). The confusion
matrix shows the probability with which the classifier discriminates different classes, and the confidence matrix reveals the
confidence of the classifiers decisions. For example the 1.0 in
the first row and column of confusion matrix represents the
probability with which real fault examples recognized as fault
and the zero to its right shows the probability of real fault
examples recognized as nonfault, and the 1.0 in the first row
and column of the confidence matrix shows the probability of
the classifiers decision in the fault–fault region being correct.
As seen in both cases, the suggested algorithm enables discriminating the fault from the nonfault case very precisely.
Some simulated cases are given in detail in the following section.
The following cases are presented here:
transformer energizing;
motor starting;
fault;
simultaneous transformer energizing and fault.
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 3, JULY 2007
Fig. 6. Current of phase A due to transformer energizing.
memory (RAM) 256 MB. In addition, parameters computation
takes 1.5 cycles; therefore, the total time will be shorter than
2 cycles (33.8 ms). This is so fast for over-current relays. It
is noted that this period of time has been spent for running
the Matlab on the PC, but the utilized hardware on the relay
is much faster. In addition, this computation is carried out in
parallel with the time taken by the relay characteristic; it does
not lead to any delay and is shorter than it. In fact, the position
of pin 2 of the AND gate is always determined faster than pin 1
which is related to the relay characteristic.
A. Transformer Energizing
Fig. 5. Diagram of the 34.5-kV simulated distributed system.
TABLE I
CONFUSION AND CONFIDENCE MATRICES FOR THE OPTIMAL BAYES
CLASSIFIER WITH TWO ESTIMATION METHODS
The performance of the suggested algorithm has been also compared with that of the conventional overcurrent relays. In all
cases, the characteristic of the overcurrent relay is the CO-6
type.
The decision making time by optimal Bayes classifier takes
3.8 ms on computer Pentium 3, 1.2-GHz, random-access
In order to study a transformer energizing, various inrush current conditions were simulated at different parts of the network.
Various parameters which have considerable effect on the characteristic of the current signal (e.g., core residual magnetization,
nonlinearity of transformer core, and switching instant) were
changed and the current signal was analyzed by the proposed
method. In all cases, correctness of the proposed algorithm has
been proved. A detailed study of a typical case is presented
below. In this case, the transformer at busbar 12 is switched at
and the currents are measured at busbar 7. Fig. 6 shows
the current of phase A. In this case, the damping of the fundamental harmonic (50 Hz) is 2.7 and the ratio of the 2nd harmonic
to the amplitude of the fundamental harmonic is 0.35. In this situation, the probability of the case being faulty is 2.7182
and being nonfaulty is 0.99997, so the occurring case is diagnosed as a nonfaulty case correctly and the maltrip of the relay
is prevented.
Fig. 7 shows the output of the present relays. In this case, the
pickup current relay is set at 1.3 and time dial setting is 0.2 (it
means that the relay is very sensitive). As seen, energizing the
transformer produces overcurrent and this leads to the maltripping of the relay at 0.78 s. To overcome this, the pickup current
must be increased; however, this delays the relay operation when
the fault occurs (this will be studied in Section C).
Fig. 8 presents the performance of the relay suggested in this
paper. The output of the relay characteristic or pin 1 of the AND
FAIZ et al.: PRONY-BASED OPTIMAL BAYES FAULT CLASSIFICATION OF OVERCURRENT PROTECTION
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Fig. 7. Output of the conventional relays due to transformer energizing.
Fig. 10. Output of the conventional relays due to motor starting.
Fig. 8. (a) Pin 1 of the AND gate. (b) Pin 2 of the AND gate. (c) Output of the
proposed relay due to transformer energizing.
Fig. 11. (a) Pin 1 of the AND gate. (b) Pin 2 of the AND gate. (c) Output of the
proposed relay due to motor starting.
B. Induction Motor Starting
Fig. 9. Current of phase A due to motor starting.
gate of Fig. 4 leads to the maltrip at 0.78 s, but pin2 of the AND
gate (output of the suggested algorithm) detects the nonfault
case less than 40 ms (at 0.5338 s) and keeps pin2 of the AND
gate equal to zero. Finally, the output of the relay becomes zero
and prevents the maltrip of the relay.
Different motor starting cases for motors with different ratings have been applied at different parts of the power system.
For all cases the correctness of the proposed algorithm has been
proved. A detailed study of a typical case is presented below. In
this case a 2.5–MVA induction motor at busbar 13 is switched at
and currents are measured at busbar 7. Fig. 9 shows the
current of phase A. In this case, the damping of the fundamental
harmonic (50 Hz) is 1.1 and the ratio of the 2nd harmonic amplitude to the fundamental harmonic amplitude is 0.02. In this
situation, the probability of the case being faulty is 0.00012153
and being nonfaulty is 0.99987, therefore the occurred case is
diagnosed as a nonfaulty case correctly and the maltrip of the
relay is prevented. Fig. 10 shows the output of the present relays. In this case, the pickup current of relay sets on 1.3 and
time dials setting at 0.2. As seen, induction motor starting leads
to the maltrip of the relay at 0.8 s. To overcome this, the pickup
current must be increased; however this delays the relay operation when fault occurs (this will be studied in section C).
Fig. 11 presents the performance of the relay suggested in
this paper. Output of the relay characteristic or pin 1 of the AND
gate of Fig. 4 leads to the maltrip at 0.8 s, but pin2 of the AND
gate (output of the suggested algorithm) detects the nonfault
case shorter than 40 ms (at 0.5338 s) and keeps pin2 of the AND
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Fig. 12. Current of phase A due to fault (A–G).
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 3, JULY 2007
Fig. 14. (a) Pin 1 of the AND gate. (b) Pin 2 of the AND gate. (c) Output of the
proposed relay due to fault (A–G).
of Fig. 4, there is no need to consider factor 1.5. So, in this case,
the pickup current of relay is set at 1.3 and time dial setting at
0.2 s. As seen, the relay characteristic, pin 1 of the AND gate of
Fig. 4, becomes 1 at 0.65 s and pin 2 of the AND gate (output
of the suggested algorithm) detects the fault case after 40 ms
(at 0.5338 s) and pin 2 of the AND gate becomes 1. Finally, the
output of the relay issues the trip command at 0.65 s. As seen in
this case, the suggested relay operates 0.11 s quicker (5 cycles).
It is noted that this time difference is longer for further relays
taking into account the relays coordination.
Fig. 13. Output of the conventional relays due to fault (A–G).
gate equal to zero. Finally, the output of relay becomes zero and
prevents the maltrip of the relay.
C. Fault
In this case, a phase–ground fault (A-G) with
is
0.5 s and currents are measured at
applied at busbar 13 at
busbar 7. Fig. 12 shows the current of phase A. In this case,
the damping of the fundamental harmonics (50 Hz) is 0.0044
and the ratio of the 2nd harmonic amplitude to the fundamental
harmonic amplitude is zero. In this situation, the probability of
;
the case being faulty is 1 and being nonfaulty is 3.9037
therefore, the occurred case is diagnosed as a faulty case correctly and the trip command is issued.
Fig. 13 shows the performance of the conventional relays.
As mentioned in the introduction, in order to prevent maltrip
of present overcurrent relays due to transients, a longer delay is
initiated for relay tripping; This delay is generated by increasing
the pickup current of the relay, such that factor 1.3–2 times the
set value is considered, here factor of 1.5 has been assigned.
Therefore, in this case, the pickup current of relay is set at 2 and
time dial setting at 0.2. As seen, the relay issues trip command
at 0.76 s.
Fig. 14 shows the performance of the suggested relay. Since
preventing the maltrip of the relay is done by the detector part
D. Simultaneous Transformer Energizing and Fault
One of the rare cases that may occur and influence the performance of the suggested algorithm is the Simultaneous occurrence of the fault and switching. The reason for its rareness
is that the suggested algorithm diagnoses the faulty case from
no-faulty case shorter than 2 cycles (33.8 ms), therefore if their
time intervals are shorter than 33.8 ms, the Simultaneous case
occurs. This happens if a faulty transformer or motor is switched
on. Since these apparatus have own protections that operate in
a very short time (for example, 0.25 cycles [24]) which cannot
excite the overcurrent relay. Despite this, in order to verify the
correctness of the performance of the suggested relay in this
case, consider the case that the overcurrent due to the fault is
low and the fault is located after transformer which leads to the
error in the detector (in other cases, the detector operates correctly, because if the fault current becomes very large or fault
occurs before the transformer, the ratio of I /I and damping is
very low and detector operates properly)
Therefore, in this case, a phase–phase fault (A-B) with
is applied at busbar 13 after transformer at busbar 12 at
and currents are measured at busbar 7. Fig. 15 shows
the current of phase A. In this case, the damping of the fundamental harmonics (50 Hz) is 0.88 and the ratio of the 2nd
harmonic amplitude to the fundamental harmonic amplitude is
0.13. In this situation, the probability of the case being faulty is
and being nonfaulty is 1; therefore, the occurring
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Fig. 17. (a) Pin 1 of the AND gate. (b) Pin 2 of the AND gate. (c) Output of the
proposed due to the simultaneous fault (A–B) and transformer energizing.
Fig. 15. Current of phase A due to the simultaneous fault (A–B) and transformer energizing.
Fig. 16. Output of the conventional relays due to simultaneous fault (A–B) and
transformer energizing.
case is diagnosed as a nonfaulty case and the trip command is
not issued.
Fig. 16 shows the output of the conventional relays. In this
case, the pickup current of relay sets at 2 and time dial setting
at 0.2. As seen, the relay issues the trip command at 0.74 s.
Fig. 17 shows the performance of the suggested relay. The
output of the relay characteristic or pin 1 of the AND gate of
Fig. 4 issues the trip command at 0.63 s and pin 2 of the AND gate
(output of the suggested algorithm) detects the nonfault case of
shorter than 40 ms (0.5338 s) by mistake and pin 2 of the AND
gate is kept at zero. At 0.74 s, the output of the Delay block,
pin 1 of the OR gate, becomes 1 which causes pine 2 of the AND
gate to become 1. Finally, the output of the relay issues the trip
command at 0.74 s.
The delay of the Delay block is set based on the relay characteristic; such that factor 1.3–2 is applied for preventing the
maltrip of the relay at the switching instance.
Taking into account the above-mentioned points, in the worse
case, the time delay of the suggested relay is equal to the delay
time of the conventional relays. The difference is that this time
can be varied as desired and if the security is more important,
this time can be increased and if the dependability is more important, this time can be decreased while it does not slow down
the relay operation during the fault.
VII. CONCLUSION
In this paper, a method for improving overcurrent relay operation was introduced. The suggested algorithm is based on
the decision made by the optimal Bayes classifier based on the
extracted information from the current signal using the Prony
method. The ability of the proposed method was demonstrated
by simulating various cases on a suitable power system. The
advantages of the suggested method include a low number of
inputs with a reduction in the required number of patterns and
fast diagnosis. Another important advantage is the modular architecture, which means adding a new case to the proposed case
does not need any retraining of the system. The new case can be
proposed as a new independent state. This paper studied some
important factors that influence the operation of relays. The suggested method has a modular architecture and it is possible to
add new cases to the states proposed in this paper and provide a
more comprehensive algorithm.
APPENDIX
The basic idea of nonparametric density estimation is to compute the probability that a vector will fall in region
(1A)
is a smoothed version of the density function
a sample of size n; therefore, the probability that
in is then
if we have
points fall
(2A)
and the expected value for
is
(3A)
Now, the maximum likelihood estimation of
is
(4A)
Therefore, ratio
is a good estimate for probability and,
hence, for density function
. When
is continuous and
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 22, NO. 3, JULY 2007
region is so small that does not vary significantly within it,
can be expressed as follows:
(5A)
where is the point within and is the volume enclosed by
. Combining (1A), (3A), and (4A) yields
(6A)
is the space averaged value of
.
Fraction
is obtained only if
approaches zero. Practically,
cannot
be allowed to become small since the number of samples are
always limited. To estimate the density of , we form a sequence
of regions
containing : the first region contains one
be
sample, the second contains two samples, and so on. Let
the number of samples falling in Rn, and
the volume of Rn,
be the th estimate for
(7A)
Three necessary conditions should apply if we want
converge to
to
(8A)
There are two different ways of obtaining sequences of regions
that satisfy these conditions.
and show that
1) Shrink an initial region where
This is called the “Parzen-window estimation method.”
as a function of , such as
; volume
2) Specify
is grown until it encloses
neighbors of .
This is called the “ -nearest neighbor estimation method.”
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