Evaluation of automatic power quality classification in

microgrids operating in islanded mode

Raul Igual, Carlos Medrano Franz Schubert
EduQTech group, Electrical Engineering Department / Department für Informations- und Elektrotechnik
Electronics Engineering and Communications Department Fakultät Technik und Informatik, HAW Hamburg
EUPT/IIS, Universidad de Zaragoza Hamburg, Germany
Teruel, Spain Franz.Schubert@haw-hamburg.de
rigual@unizar.es, ctmedra@unizar.es

Abstract—Microgrids operating in islanded mode are more prone On the other hand, microgrids can cause power quality
to fundamental frequency variations and power quality disturbances. This is especially due to the high presence of
distortions. To mitigate power quality problems, it is essential to power electronics interfaces, which are sources of distortion.
first identify the type of distortion using a proper classifier. There Power quality issues are undesirable in electrical grids and may
are many classifiers in the literature. However, they are tested lead to several problems such as incorrect operation of grid
assuming that there is no fundamental frequency variation. In equipment, damage of sensitive loads or erratic operation of
this paper, we investigate the effect of fundamental frequency electronic controls, among many others [5], [6].
variations on classification accuracy. For that purpose, a well-
known classifier is tested with data sets of different fundamental To mitigate the effects of power quality distortions, it is first
frequencies. Then, accuracies are compared using statistical tests. necessary to detect when an electrical signal is distorted and to
To the best of our knowledge, this is the first work adopting this know the type of distortion. For that, an automatic classifier
approach. The results of the comparison show that changes in should be used. As this is a major, well-documented problem,
fundamental frequency greatly affect classification accuracy. A there are many studies in the literature on this subject [7]. The
large decrease occurs even with moderate frequency deviations. structure of power quality classifiers is as follows: first, signals
Therefore, future studies should consider this effect, since non- are processed by applying an appropriate technique and features
adapted classifiers may perform poorly in weak microgrids are extracted; second, a classifier is designed and fed with the
operating in islanded mode. selected features; and third, its classification performance is
Index Terms—classification, fundamental frequency, islanded
obtained. As an example, the work of Khokar et al. [8] used the
mode, microgrids, power quality. multiresolution analysis (MRA) of the Discrete Wavelet
Transform (DWT) as a signal processing technique. Then, 9
I. INTRODUCTION features were extracted from the coefficients of the processed
signals. Previously they had been selected as the most
Microgrids are entities that coordinate distributed energy discriminative ones. Those features were then used in a
sources (typically microturbines, PV panels, and fuel cells), a probabilistic neural network (PNN) to classify the different
cluster of loads and energy storage units in a consistently more power quality distortions. Wang et al. [9] also implemented a
decentralized way [1]-[3]. Their components usually include PNN to perform the classification. Borges et al. [10] used the
power electronic interfaces [2]. Microgrids can operate in either signals in the time domain and the signals processed with the
grid-connected or stand-alone modes [3]. In this last situation, Fourier Transform to extract the features. Then, artificial neural
they are also expected to remain operational as an autonomous networks and a decision tree were used to classify the power
(islanded) entity [1]. The most challenging operation of quality distortions. Similar classifiers were considered by
microgrids occurs when they operate in stand-alone mode [2]. Kumar et al. [11], processing the signals by means of the
There are two main concerns: the deviation of the grid Stockwell-transform (ST). Rodriguez et al. [12] also used the
frequency and power quality issues. On the first hand, the ST. They adopted a decision tree to classify the distortions.
deviation of the grid frequency is caused by an imbalance Other authors such as Huang et al. [13], Ray et al. [14], He et
between generation and consumption. Regarding the grid- al. [15] or Li et al. [16] considered a variation of the ST. The
connected mode of operation, the frequency of the microgrid is work of Kumar et al. [17] used symmetrical components to
dominantly determined by the host grid [4]. In this case, the process the data and extract 5 features. The classifier used was
high number of synchronous generators ensures a relatively based on decision rules. Decision trees are popular classifiers
large inertia. However, if microgrids operate in stand-alone widely adopted in this field. Other authors that used them were
mode, the low inertia can lead to frequency deviations. This is Singh et al. [18], Kubendran et al. [19] or Deokar et al. [6],
particularly serious if there is a significant share of power among many others.
electronic-interfaced distributed generation units [4].

‹,(((
 ௡ ௡ିଵ ȁ
All these studies have in common that signals at the grid ȁܸோெௌ െ ܸோெௌ
fundamental frequency are used to test the classification ‫ܤ‬௡ ൌ (3)
ο‫ݐ‬
systems. Therefore, frequency deviations are not considered.
௡
This approach can be appropriate for most grid-connected Where ܸோெௌ is the RMS value of the nth cycle and ο‫ ݐ‬is the
systems or strong islanded microgrids. However, in weak time period of the 50-Hz signal. If any of the elements of ‫ܤ‬௡ is
islanded microgrids with low inertia, there may also be above 0.3 (noise threshold), then F2 is 1, else F2 is 0.
significant changes in fundamental frequency. Therefore, Feature 3 (F3): number of oscillations (zero crossings) of
existing studies are not considering this scenario. The objective the RMS values of the signal.
of this work is to determine whether existing automatic
classifiers can also be applied in microgrids operating in ‫ ܥ‬ൌ ‫ݐ݋݋ݎ‬൫ܸோெௌ െ ݉݁ܽ݊ሺܸோெௌ ሻ൯ (4)
islanded mode. In other words, this paper studies whether
classification results are also appropriate when there are Where ܸோெௌ is a vector that contains the RMS values of
variations in the fundamental frequency. This is a novelty with each of the n cycles considered, root is a function that calculates
respect to traditional approaches adopted in power quality the number of zero crossings of a given vector ൫ܸோெௌ െ
studies. Additionally, statistical tests were used to compare the
results. ݉݁ܽ݊ሺܸோெௌ ሻ൯. F3 is taken as 1 if ‫ ܥ‬൒ ͵, else F3 is 0. To avoid
noise effects, if an element of the vector ൫ܸோெௌ െ
The rest of this paper is organized as follows: Section II ݉݁ܽ݊ሺܸோெௌ ሻ൯ is in the range ሺെͲǤͲͳǡͲǤͲͳሻ, it is set to 0.
presents the classification method, the data set generation
process and the structure of the experiments carried out in this Feature 4 (F4): total harmonic distortion (THD) factor.
study; Section III shows the results of those experiments;
ே
Section IV discusses the results and limitations of this study; ටσ௜௡௧ሺ ଶ ሻȁܸ ௡ ሾ݇ሿȁଶ
and Section V draws some conclusions and outlines future ௞ୀଶ (5)
ܶ‫ܦܪ‬௡ ൌ
research efforts. ܸ ௡ ሾͳሿ
II. METHODS
Where ܸ ௡ ሾ݇ሿ is the magnitude of the coefficients of the
A. Feature Extraction and Classification decomposition level k obtained as a result of applying the DFT
The classification method used to study the effect of to the samples of the nth cycle. If any of the elements of ܶ‫ܦܪ‬௡
fundamental frequency variation on classification accuracy is is above 0.05, then F4 is 1, else F4 is 0.
similar to that published by Zhang et al. [20]. In that study, only
Feature 5 (F5): low harmonic distortion (LHD) factor.
the accuracy rates for the traditional case (without fundamental
frequency variation) were considered. That system was selected ඥσ଺௞ୀଶȁܸ ௡ ሾ݇ሿȁଶ
due to its high classification accuracy, high execution speed and ‫ܦܪܮ‬௡ ൌ (6)
ease of implementation. Five features were taken to feed the ܸ ௡ ሾͳሿ
classifier. Two of them were extracted directly from the
electrical signals in the time domain, while the remaining three If any of the elements of ‫ܦܪܮ‬௡ is greater than 0.07 and is
were obtained from the signals processed with the Discrete equal to or greater than ܶ‫ܦܪ‬௡ െ ‫ܦܪܮ‬௡ , then F5 is 1, else F5 is
Fourier Transform (DFT). Both the features and the classifier 0.
were designed to operate with signals of a fixed fundamental These features are used in a rule-based decision tree [20]. It
frequency (50 Hz in this case), similar to most studies in this was built from expert knowledge. Fig. 1 shows the structure of
field. In this case, it is assumed that distortions are composed of the decision tree.
10 cycles. Next, the different features are presented.
B. Data Set
Feature 1 (F1): RMS value of the 50-Hz fundamental
component in p.u. (per unit). To test the effect of fundamental frequency variations,
signals with power quality distortions of different fundamental
ܸ௡ frequencies are required. These signals were synthetically
‫ͳܨ‬௡ ൌ (1)
ܸோெௌ generated using the integral mathematical model published in
[21], which can be freely accessed and used. That model
ξʹȁܸ ௡ ሾͳሿȁ (2) generates signals with power quality distortions whose
ܸ௡ ൌ
ܰ parameters are selected at random. As an example, considering
the case of a sag event, this model randomly selects its starting
Where ܸோெௌ is the RMS value of the normal undistorted point, its ending point and its amplitude. Nine types of power
signal, ܸ ௡ ሾͳሿ is the value of the DFT-coefficient corresponding quality distortions were considered in this study: normal
to the 50-Hz fundamental frequency, N is the number of signal, sag, swell, interruption, flicker, oscillatory transients,
samples in one cycle and n is the number of the cycle that is harmonics, harmonics with sag and harmonics with swell. The
being computed. sampling frequency was set at 16 kHz, which is a common
value in this field [21]. Regarding the fundamental frequency,
Feature 2 (F2): variation of the RMS values of two 50 Hz was considered as the reference value. Then, the 40
consecutive signal cycles. closest possible values around 50 Hz were taken. It should be
noted that the sampling frequency (16 kHz in this case)
determines the resolution and, therefore, the closest possible

,(((0LODQ3RZHU7HFK
fundamental frequencies around 50 Hz. For the 50-Hz simulations were carried out (42 different fundamental
fundamental frequency, 10 signal cycles were generated in frequencies multiplied by 3 SNRs multiplied by 10 simulations
each distortion. Thus, each distortion was composed of 320 per case).
points.
9 types of X 100 signals/type
One hundred distortions of each type were created for the distortions
900 signals
42 different fundamental frequencies considered in this study
(50 Hz, the 40 closest possible values around 50 Hz and a X 3 (no. SNRs
considered)
combination of those frequencies—see Section II.C for
details). Three sets of distortions were obtained considering 320
signal-to-noise ratios (SNRs) of 50 dB, 40 dB and 30 dB. For points 2700 signals
each fundamental frequency and SNR, the data set generation each
process repeated 10 times. Therefore, a total of 1,134,000 X 42 (no.
fundamental
signals with power quality distortions were obtained. Fig. 2
frequencies)
shows a graphic representation of the data set generation.
X 10 repetitions 113,400 signals
1,134,000 signals
F1 0.99<F1<1.01
F1<0.99 F2 Figure 2. Structure of the generation of power quality distortions.
0 0 Normal
F1>1.01 F4
1 signal
1 Then, to check whether there are differences between the
1 F3 classification accuracies obtained with data at 50-Hz
0
fundamental frequency and those obtained with data at other
Flicker F5 0
1 0.91<F1<1.09
fundamental frequencies, statistical tests were performed on
Oscillatory
F1
transients the mean accuracies. In total, 41 statistical tests were carried
0 F2 F1<0.91 out comparing two sets of accuracies in each test (the
F1>1.09 Harmonics
Harmonics 1 accuracies obtained with distortions at 50 Hz were compared
with sag
F4 >2 0.1<F1<0.9 with each of the forty-one remaining accuracies obtained with
<=2 signals at other fundamental frequencies). As simulations were
F5
0.1<F1<0.9 F1 1 F1 done independently, (a new set of distortions was generated for
0 Harmonics
>1.1 each one) the t-test for independent samples can be used as a
Sag >=1.1 Swell with swell
>=1.1 statistical test [22]. In this test, the null-hypothesis is that “the
Interruption
means of the two data sets are equal”. Therefore, if a p-value
Figure 1. Decision tree used to classify power quality distortions. obtained as a result of the t-test is above the significance value
(usually 0.05), the null hypothesis cannot be rejected and the
C. Experiments mean accuracies of the two sets can be considered equal with
To test the effect of frequency variation on classification high probability. Otherwise, the accuracies of the two sets
accuracy, the following experiments were performed: cannot be considered equal, which implies that the value of the
fundamental frequency can affect the classification accuracy.
• In the first one, the classification system was tested This test assumes that data follow a normal distribution. To
with distortions of 50-Hz fundamental frequency. check this point, the Shapiro-Wilk normality test was
This led to reference accuracy values. This is the performed for each set of accuracies (in total, 42 times). The
typical experiment performed by existing studies in null-hypothesis of this test is that “the set of samples is
this field. normally distributed”. If a p-value obtained as a result of the
test is above the significance level (0.05), the null hypothesis
• In the second one, distortions at other fundamental probably cannot be rejected and it can be assumed that the data
frequencies (20 frequencies above 50 Hz and 20 set comes from a normally distributed population. Therefore,
frequencies bellow 50 Hz) were generated using the the data would meet one of the conditions for the t-test.
same mathematical model. The classification system
was tested with those signals. The 40 fundamental The t-test also presents differences depending on whether
frequencies were in the range of 46.92 Hz to 53.16 the variances of the two data sets are equal or unequal.
Hz. Therefore, this point must be determined for each comparison.
For that, the variances of the classification accuracies for the
• In the third one, distortions with fundamental 50-Hz fundamental frequency distortions were contrasted with
frequency variations (randomly generated in the the variances of the classification accuracies for the remaining
range of 46.92 Hz to 53.16 Hz) were obtained and fundamental frequencies (in total, 41 comparisons). For this
used to test the classification system. This situation purpose, the Bartlett’s test was used. In this test, the null-
simulates the behavior of a weak islanded microgrid hypothesis is that “population variances are equal”. Thus, if a
subject to fluctuations in demand or generation. p-value obtained in the test is above the significance level
As a result of these experiments, the classification accuracy (0.05), the null hypothesis cannot be rejected, and variances
was obtained for each fundamental frequency. The are assumed to be equal. Otherwise, they cannot be considered
experiments were repeated 10 times per frequency with three equal. Based on the results of the Bartlett’s test, the appropriate
different SNRs (50 dB, 40 dB and 30 dB). In total 1,260 t-test (equal or unequal variances [22]) is applied.

,(((0LODQ3RZHU7HFK
 If the results of the t-test suggest statistical differences TABLE II. ACCURACIES OF THE CLASSIFIER FOR DISTORTIONS WITH
DIFFERENT FUNDAMENTAL FREQUENCIES (SNRS FROM 30 DB TO 50 DB)
between mean accuracies, the effect size can be quantified. For
that, the Cohen’s d is calculated. A widely accepted “rule of Frequency Accuracy: Mean (std)
thumb” for Cohen’s d values is the following: 0.2 means (Hz) SNR 50 dB SNR 40 dB SNR 30 dB
“small” effect, 0.5 “medium” effect, 0.8 “large” effect, 1.2 46.92 33.08 (0.58) 33.32 (0.52) 32.42 (0.42)
47.06 33.81 (0.34) 33.90 (0.72) 33.06 (0.39)
“very large” effect and 2 “huge” effect [23].
47.20 34.04 (0.36) 33.74 (0.43) 33.04 (0.47)
The structure of the experiments is represented 47.34 34.50 (0.54) 34.64 (0.42) 33.22 (0.38)
schematically in Fig. 3. 47.48 34.76 (0.54) 34.34 (0.56) 34.04 (0.32)
47.62 34.89 (0.52) 34.89 (0.62) 34.24 (0.51)
47.76 36.27 (0.51) 36.11 (0.71) 35.59 (0.44)
47.90 44.66 (0.78) 44.09 (0.54) 41.89 (0.37)
Power quality 48.05 46.74 (0.53) 46.47 (0.45) 45.48 (0.48)
model 48.19 47.71 (0.72) 47.84 (0.74) 45.91 (0.3)
48.34 49.29 (0.73) 49.61 (0.76) 46.72 (0.74)
48.48 54.81 (0.95) 53.76 (0.80) 48.97 (0.72)
50 20 lower 20 higher Frequency 48.63 64.18 (1.23) 62.69 (0.89) 53.53 (0.65)
Hz frequencies frequencies variations 48.78 75.04 (0.94) 75.57 (1.06) 61.51 (0.94)
48.93 83.22 (0.83) 81.89 (0.87) 71.14 (0.42)
49.08 94.33 (0.44) 92.06 (0.72) 78.88 (0.83)
49.23 97.73 (0.51) 96.98 (0.43) 87.33 (0.63)
49.38 98.09 (0.42) 97.89 (0.64) 93.43 (1.03)
Accuracies X 3 SNRs
49.54 98.43 (0.51) 98.17 (0.33) 96.24 (0.35)
X 10 repetitions
49.69 98.62 (0.47) 98.60 (0.34) 97.27 (0.47)
Mean 10 repetitions 49.84 98.83 (0.28) 98.59 (0.48) 97.03 (0.31)
Cohen’s Statistical tests 50.16 98.89 (0.36) 98.82 (0.41) 97.20 (0.64)
d 50.31 98.26 (0.47) 98.42 (0.47) 97.26 (0.6)
50.47 98.16 (0.43) 98.16 (0.60) 95.84 (0.71)
Each lower and 50.63 97.97 (0.59) 97.89 (0.49) 91.79 (1.04)
higher frequency 50.79 96.83 (0.54) 96.64 (0.50) 86.63 (0.87)
t-test 50 Hz VS 3 SNRs 50.96 91.80 (0.71) 90.64 (0.96) 77.02 (0.79)
Frequency 51.12 79.88 (0.51) 79.29 (0.60) 69.14 (1.2)
Shapiro- Bartlett variations 51.28 73.53 (0.86) 71.37 (0.65) 56.43 (1.12)
Wilk test 51.45 56.30 (0.66) 55.71 (1.02) 50.00 (0.93)
norma. 51.61 49.56 (0.82) 48.94 (0.72) 47.02 (0.39)
test 51.78 46.69 (0.48) 46.37 (0.51) 45.63 (0.42)
51.95 45.49 (0.63) 45.42 (0.65) 44.32 (0.54)
Figure 3. Structure of the experiments. 52.12 35.24 (0.47) 35.58 (0.63) 35.18 (0.48)
52.29 34.06 (0.39) 33.93 (0.29) 33.89 (0.36)
52.46 33.66 (0.33) 33.73 (0.29) 33.60 (0.31)
III. RESULTS 52.63 33.68 (0.41) 33.42 (0.33) 33.30 (0.25)
Table I presents the mean classification accuracies and their 52.81 33.40 (0.41) 33.21 (0.25) 32.94 (0.46)
standard deviations (10 repetitions used) for distortions at 50- 52.98 32.87 (0.43) 32.89 (0.27) 32.91 (0.34)
53.16 32.69 (0.2) 32.76 (0.27) 32.82 (0.43)
Hz fundamental frequency with SNRs of 50, 40 and 30 dB. The
accuracies presented in this table are the reference values with To perform a rigorous analysis of the results obtained,
which the rest of the accuracies are compared. statistical tests are required. In particular, t-tests are necessary
Table II presents the same results for the different to determine the upper and lower fundamental frequencies from
fundamental frequencies considered in this study. This table which the mean classification accuracies can be considered
shows how the classification accuracy decreases when signals different from the mean accuracies of the classifier tested with
deviate from the 50-Hz fundamental frequency. In this sense, distortions at 50-Hz fundamental frequency. To perform the t-
Table III presents the mean accuracies (10 repetitions) for the tests, data must follow a normal distribution. In this regard,
third part of the study: distortions with random fundamental Table IV and Table VI (first column) show the results of the
frequency variations around 50 Hz (included). In this case, Shapiro-Wilk normality test. All p-values are above the
accuracy values are much lower than those of the reference case significance level (0.05). Therefore, it can probably be assumed
(50 Hz) without fundamental frequency variations. that all sets of accuracy values follow a normal distribution.
This allows performing the appropriate t-test. For that, the
TABLE I. MEAN CLASSIFICATION ACCURACIES AND THEIR STANDARD results of the Bartlett’s test are needed (Table V, second column
DEVIATIONS (IN BRACKETS) FOR DISTORTIONS AT 50-HZ FUNDAMENTAL and Table VI, third column).
FREQUENCY (SNRS FROM 30 DB TO 50 DB)

SNR 50 dB SNR 40 dB SNR 30 dB TABLE III. MEAN CLASSIFICATION ACCURACIES AND THEIR STANDARD
DEVIATIONS (IN BRACKETS) WHEN USING DATA WITH RANDOM
99.03 (0.28) 98.49 (0.23) 97.07 (0.73) FUNDAMENTAL FREQUENCY VARIATIONS

SNR 50 dB SNR 40 dB SNR 30 dB

63.27 (1.26) 62.67 (0.92) 58.47 (1.46)

,(((0LODQ3RZHU7HFK
TABLE IV. P-VALUES OF THE SHAPIRO-WILK NORMALITY TEST FOR THE 51.95 0.025 <2.2e-16 109.52 < 2.2e-16 <2.2e-16
ACCURACIES OF THE CLASSIFIERS OBTAINED WITH DISTORTIONS OF DIFFERENT 52.12 0.144 <2.2e-16 164.33 < 2.2e-16 <2.2e-16
FUNDAMENTAL FREQUENCIES
52.29 0.352 <2.2e-16 191.08 < 2.2e-16 <2.2e-16
Freq. Shapiro- Freq. Shapiro- Freq. Shapiro- 52.46 0.630 <2.2e-16 211.74 < 2.2e-16 <2.2e-16
(Hz) Wilk (Hz) Wilk (Hz) Wilk 52.63 0.290 <2.2e-16 186.52 < 2.2e-16 <2.2e-16
p-value p-value p-value 52.81 0.272 <2.2e-16 185.51 < 2.2e-16 <2.2e-16
46.92 0.4568 48.93 0.5479 51.12 0.4698 52.98 0.223 <2.2e-16 181.53 < 2.2e-16 <2.2e-16
47.06 0.1424 49.08 0.09175 51.28 0.963 53.16 0.329 <2.2e-16 270.51 < 2.2e-16 <2.2e-16
47.20 0.4379 49.23 0.1838 51.45 0.4377 a. The results of the Bartlett’s test and Cohen’s d are only presented for the case of 50-dB SNR
47.34 0.9607 49.38 0.7367 51.61 0.3308 due to space restrictions.

47.48 0.3 49.54 0.6311 51.78 0.4615

47.62 0.4208 49.69 0.5322 51.95 0.5751 TABLE VI. STATISTICAL ANALYSIS OF THE ACCURACIES OBTAINED
WITH DISTORTIONS AT DIFFERENT FUNDAMENTAL FREQUENCIES RANDOMLY
47.76 0.3054 49.84 0.703 52.12 0.3248
SELECTED
47.90 0.1613 50 0.2404 52.29 0.7177
48.05 0.7844 50.16 0.48 52.46 0.8816 SNR Shapiro-Wilk Bartlett test t-test Cohen’s d
48.19 0.2851 50.31 0.06567 52.63 0.1222 normality test p-value p-value
48.34 0.5937 50.47 0.8699 52.81 0.1154 p-value
48.48 0.1707 50.63 0.7528 52.98 0.3681 50 dB 0.656 0.0001 1.2e-15 39.22
48.63 0.8343 50.79 0.829 53.16 0.7322 40 dB 0.871 0.0003 <2.2e-16 53.38
48.78 0.2868 50.96 0.2291 30 dB 0.621 0.0491 <2.2e-16 33.45

The p-values of the t-tests are shown in Tables V and VI. IV. DISCUSSION
The p-values that reject the null-hypothesis are highlighted in In view of the p-values of the t-tests in tables V and VI, it is
bold. In those cases, the alternative hypothesis can probably be possible to conclude that the fundamental frequency of the
considered: there are differences in mean accuracies. The effect power quality distortions greatly affects the classification
sizes are quantified in the same two tables for cases in which performance. For example, in the case of a SNR of 50 dB, the
differences were identified. classification performance begins to decrease in the second step
of fundamental frequency variation. Therefore, the
TABLE V. P-VALUES FOR THE T-TESTS THAT COMPARE THE MEAN
classification performance worsens for fundamental
ACCURACIES FOR THE DISTORTIONS AT 50-HZ FUNDAMENTAL FREQUENCY
WITH THOSE OBTAINED AT OTHER FREQUENCIES frequencies below 49.69 Hz or above 50.31 Hz. When
considering random frequency variations (Table III),
Frequency SNR 50 dBa SNR SNR
(Hz) 40 dB 30 dB
classification performance decreases dramatically. This has
Bartlett t-test Cohen’s d t-test t-test important implications in existing studies on power quality
test p-value p-value p-value classification, since most studies do not take into account the
p-value possible fundamental frequency variations when calculating
46.92 0.043 <2.2e-16 144.39 <2.2e-16 <2.2e-16 classification performances. The consequence of this is that
47.06 0.590 <2.2e-16 208.85 <2.2e-16 <2.2e-16 many classifiers that perform well in grids with a very stable
47.20 0.500 <2.2e-16 202.48 <2.2e-16 <2.2e-16
47.34 0.065 <2.2e-16 149.25 <2.2e-16 <2.2e-16
fundamental frequency could not work equally well in grids
47.48 0.064 <2.2e-16 148.36 <2.2e-16 <2.2e-16 with moderate frequency variations. We have demonstrated this
47.62 0.080 <2.2e-16 152.46 <2.2e-16 <2.2e-16 effect in a well-known classifier of power quality distortions
47.76 0.095 <2.2e-16 152.65 <2.2e-16 <2.2e-16 [20], which was slightly modified.
47.90 0.006 <2.2e-16 92.76 <2.2e-16 <2.2e-16
48.05 0.071 <2.2e-16 122.37 <2.2e-16 <2.2e-16 If the p-values of the t-tests for the different SNRs are
48.19 0.010 <2.2e-16 93.58 <2.2e-16 <2.2e-16 analyzed, it is possible to conclude that the decrease in
48.34 0.009 <2.2e-16 89.89 <2.2e-16 <2.2e-16 performance occurs in all of them. As for SNRs of 40 dB and
48.48 0.001 <2.2e-16 63.362 <2.2e-16 <2.2e-16 30 dB (Table V), the decrease in performance begins in the
48.63 2e-4 1.06e-15 39.170 <2.2e-16 <2.2e-16 third, or even fourth, step of frequency deviation. Therefore,
48.78 0.001 6.63e-16 34.413 2.0-14 <2.2e-16 classification accuracy seems a bit more robust to fundamental
48.93 0.004 4.81e-15 25.601 2.99e-14 <2.2e-16 frequency variations in noisy conditions. This can be explained
49.08 0.192 2.35e-16 12.624 3.0e-11 <2.2e-16
49.23 0.089 1.52e-6 3.136 1.37e-08 <2.2e-16
by the fact that accuracies for the reference case in these
49.38 0.266 1.25e-5 2.661 0.0177 3.53e-08 scenarios (more noise) are a bit lower than for the 50-dB SNR
49.54 0.091 0.0045 1.452 0.0213 0.0066 scenario (less noise) and, therefore, the comparison is made
49.69 0.146 0.0289 1.062 0.4069 0.4754 with slightly worse accuracy values.
49.84 0.968 0.1281 - 0.5646 0.8962
50.16 0.504 0.3274 - 0.0876 0.6676 Regarding the comparison of accuracies, Cohen’s d results
50.31 0.142 0.0003 2.001 0.6938 0.5351 show a “large” effect size from the first frequency deviations.
50.47 0.236 3.70e-5 2.428 0.1277 0.0013 This is a logical result, since differences in performance are
50.63 0.038 0.0002 2.298 0.0039 1.15e-10 large even for moderate frequency deviations. For example, if
50.79 0.065 1.19e-9 5.088 1.07e-07 <2.2e-16 the fundamental frequency increases by 0.96 Hz from the
50.96 0.011 1.87e-12 13.355 2.31e-10 <2.2e-16 reference case, the classification accuracy decreases in a range
51.12 0.097 <2.2e-16 46.729 < 2.2e-16 <2.2e-16
of 7.23 % to 20.05 % (depending on the SNR), which are very
51.28 0.003 <2.2e-16 40.028 < 2.2e-16 <2.2e-16
51.45 0.019 <2.2e-16 84.352 < 2.2e-16 <2.2e-16
large values in the field of automatic power quality
51.61 0.004 <2.2e-16 80.993 < 2.2e-16 <2.2e-16 classification. Therefore, this effect cannot be ignored, since
51.78 0.135 <2.2e-16 133.63 < 2.2e-16 <2.2e-16 classifiers with a high performance under simulated or

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de España - Estancias de movilidad en el extranjero José complex power quality disturbances using S-transform amplitude
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