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This paper aims to develop a combination method for the evaluation of power quality disturbances. First, we apply the Fast Fourier Transform, Wavelet Transform and Hilbert Huang Transform on a synthetic signal that represents typical behavior in a power system with high penetration of Renewable Energies. Then, we combine the methods to extract the best of each of these and achieve a better signal decomposition. The paper seeks to generate decision criteria on the method of analysis of signals to be used according to the application.
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Hybrid Technique for the Analysis of Non-Linear
and Non-Stationary Signals focused on Power
Quality
Mauricio Sanabria-Villamizar , Maximiliano Bueno-L´
opez, Marta Molinas and Efrain Bernal
Department of Electrical Engineering, Universidad de la Salle, Bogot´
a, Colombia
Email: jsanabria16@unisalle.edu.co, maxbueno@unisalle.edu.co
Department of Engineering Cybernetics
Norwegian University of Science and Technology
Email: marta.molinas@ntnu.no
Department of Automation Engineering, Universidad de la Salle, Bogot´
a, Colombia
Email: ebernalal@unisalle.edu.co
Resumen—This paper aims to develop a combination method
for the evaluation of power quality disturbances. First, we apply
the Fast Fourier Transform, Wavelet Transform and Hilbert
Huang Transform on a synthetic signal that represents typical
behavior in a power system with high penetration of Renewable
Energies. Then, we combine the methods to extract the best of
each of these and achieve a better signal decomposition. The
paper seeks to generate decision criteria on the method of analysis
of signals to be used according to the application.
Index Terms—Hilbert-Huang Transform, Hilbert Spectrum,
Power Quality, Fourier Fast Transform, Wavelet Transform,
Instantaneous frequency.
I. INTRODUCTION
Increased energy demand has given the need to generate
energy from unconventional methods, producing high varia-
tions in the network due to its nonlinear nature, caused by
nonlinear devices [1]. Quality in power systems is focused on
ensuring a voltage supply with an excellent waveform accor-
ding to the parameters set by the IEC 61000, to protect the
operation of the equipment, which handles critical processes of
great importance to customers [2] [3]. Due to the penetration
of the new technologies, such as non-conventional sources
of renewable energies (NCSRE) in distribution networks,
the harmonic distortion of current and voltage waveform is
becoming an important issue [4]. Therefore, it is necessary
to develop new methodologies for the analysis of signals that
have a wide content of harmonics and noise from NCSRE
signals [5].
Alterations in the quality of the power can be classified into
two categories, these are: variations and events. Variations are
disturbances that influence each cycle, such as harmonics or
voltage imbalances. Events are disturbances that last a while,
from a fraction of a cycle to several of these, and then they
may not be repeated for several hours or days [6]. The most
common energy quality problems are: voltage drop, harmonics
and switching transients. For the most part, these are caused
by the intermittent nature of unconventional power generation
and large load feeders [7].
The basic idea of a time-frequency analysis is the design
of a strategy that can describe the characteristics of a signal
in both domains, so that it is not only possible to detect the
presence of any frequency component but also the moment of
time in which it is presented [8].
In this paper, we present an analysis methodology capable
of extract the instantaneous frequency of a typical signal that
appears in a power system with high penetration of renewable
energy. Based on the combination of Hilbert Huang Transfor
(HHT), Wavelet Transform (WT) and Fast Fourier Transform
(FFT), the proposed methodology not only can display the
estimation of the instantaneous frequency-amplitude, but also
the fundamental frequency. Furthermore, with the advantage
of requiring less computational time with greater efficiency
compared to the other conventional methods of analysis for
this type of signals.
II. NO N-LINEAR AND NO N-STATIONA RY SIG NAL
ANALYSIS
II-A. Hilbert-Huang Transform (HHT)
In order to solve the time-frequency-energy paradigm of
non-stationary and non-linear signals, Norden E. Huang in
[9], presented the Hilbert-Huang Transform. The method is
based on the local characteristic timescale of the signal x(t),
helped by the Empirical Mode Decomposition (EMD), finding
the Intrinsic Mode Functions (IMFs). The IMF should be a
function that fulfils two conditions:
1. The number of ends and the number of crosses by zero
should be different from just one.
2. Its local medial is zero.
For a signal x(t), the analytic signal z(t)is defined as:
z(t) = x(t) + iy (t) = a(t)e(t)(1)
where y(t)is the Hilbert Transform of x(t), so that:
y(t) = 1
πPZ
−∞
x(τ)
tτ(2)
Where P is the Cauchy Principal Value.a(t)and θ(t)are
defined in (3) and (4) respectively.
a(t) = px2+y2(3)
θ(t) = arctan y(t)
x(t)(4)
The instantaneous frequency is defined by (5).
f(t) = 1
2π
(t)
dt (5)
II-A1. Masking Signal: In order to find a solution to Mode
Mixing problem, Deering and Kaiser [10] defines the Masking
Signal method, supported by the EMD. The algorithm is
defined as:
Algorithm 1 Masking Signal Method
Result: Descomposici´
on Emp´
ırica en Modo con Masking
Se construye una se˜
nal de m´
ascara s(t)basada en G.
Rilling y Fosso, respaldada por la informaci´
on de se˜
nal
original, x(t).
El EM D se obtiene de las se ˜
nales: x(t)ys(t).
x+(t) = x(t) + s(t)
x(t) = x(t)s(t)
Luego, se obtienen IM F +yI M F .
Calcule el valor principal, IM F se define como:
IMF (t) = IM F +(t) + IM F (t)
2
Realizando el proceso para cada residuo.
Some recent variations to the method can be seen in [11]
and [12].
II-B. Wavelet Transform (WT)
In 1980 was developed the wavelet (WT) transformed by
Morlet and Grossmam, defined as:
W(u, s) = 1
sZ
−∞
x(t)ψtu
sdt (6)
The WT breaks down a signal in frequency bands, which
are generated by means of a function called wavelets mother
(7) and function translation operations in the time domain:
ψus (t) = 1
sψtu
s(7)
So, it responds to the optimal resolution needs in a time-
frequency analysis for all signal ranges [13]. Currently the
WT is a popular method in the analysis of sound signals and
vibration in engineering [14].
II-C. Fast Fourier Transform (FFT)
The Fast Fourier transform (FFT) advance an efficient
computational algorithm for converting signals from the time
domain into the frequency domain [15]. One variation of the
FFT is the Discrete Transfer Function (DTF) and is defined
according to the equation (8).
X(k) =
N1
X
n=0
x(n)ej2πk n
N(8)
II-D. Comparison between methods
The FFT has the inability to break down a non-stationary
and non-linear signal into a monotonic signal, this issue is
called aliasing. The above-mentioned issue is not only present
in the FFT, but also in the WT and WT, but with different
consequences. The most important weakness of the WT is
the selection of the type of mother wavelet (window), this
should have coupling characteristics and adaptation to the
signal to be analyzed. The WT has better resolution in different
applications, but just like the FFT has the issue of aliasing,
therefore, it does not generate an adequate frequency spectrum.
However, HHT is not free of problems, its weakness is in
the heuristic way of selecting EMD type, like the WT it
depends on a mother function; therefore, a hybrid technique
is performed in this document using the above-mentioned
analysis methods in order to extract the advantages of each
and generate a robust method of analysis.
III. CAS E STU DY
III-A. Synthetic Signal
To verify the methodologies presented above, the analysis
based on the EMD Masking, EMD Masking+FFT, EMD
Masking+WT and EMD Masking+WT+FFT for a synthetic
signal is presented. The signals present a phenomenon called
Mode Mixing, to solve this applies the methodology of the
signal mask in [10], [11] and [16]. The Mode Mixing Problem
appears frequently in modern power systems where different
frequency components are detected. The synthetic signal is
described in equation (9) and (10) and also shown in Fig. 2,
with its corresponding frequency modes.
The following conditions are considered for the design
and selection of the synthetic signal: High harmonic and
inter-harmonic content and a significant amount of noise,
characteristic of modern unconventional energy sources, as
the following shows Fig. 1 and Fig. 2. This signal simulates
the ensuing phenomena: voltage dips, voltage swells, voltage
fluctuations, rapid voltage changes and transients [17].
x(t) = 3 ·sen (2π·21 ·t)
+ 4 ·sen (2π·5·t)
+ 3 ·sen (2π·2·t)
+S3
+NOISE
(9)
S3=
2·sen(2π·12 ·t)1<t<2
0other wise
(10)
Figura 1. Synthetic signal to be analyzed.
Figura 2. Synthetic signal with their respective frequencies.
By knowing the characteristics of the signal to be analyzed,
it is expected that the frequency separation will be visible
when applying EMD Masking to the signal. For this reason,
the mask signal is applied with the form s(t) = A0·sen(2π·
f·t), where a0and fare obtained according [11], [11],
[16] and is explained as follows: the mask signal parameters
are chosen from a map of the ratio between amplitudes and
frequencies shown in Fig. 3, where the blue region represents
the separation of the mixed modes from the original signal.
In terms of frequency, this occurs when the ratio ¯
f/f < 0,67
(this number may change depending on the resolution used in
the map construction), where ¯
fis the highest frequency of the
signal. For the analysis of this case, the ratio is 0.3 and the
amplitude ratio is -0.398. In this case the mask signal used
was s(t) = 11 ·sen(2π·70 ·t).
IV. RES ULT S
IV-A. Masking signal
The EMD with masking signal is shown in Fig. 4, where
the first two IMFs show the characteristic noise of the signal,
Figura 3. Separation performance measurement for two-mode signals, a 2-D
representation.
The other two IMFs show a severe problem of Mode Mixing.
Similarly, Fig. 5 shows the above-mentioned, evidenced in its
instantaneous frequencies.
Figura 4. Decomposition using EMD with Masking Signal.
Figura 5. Instantaneous frequency using EMD with Masking Signal.
IV-B. Masking signal + FFT
Looking to find an efficient technique for frequency se-
paration, avoiding what is shown in Fig. 4 and Fig. 5, the
combination of FFT and HHT is proposed. Fig. 6 shows the
procedure performed, in the first part appears the original
signal, the following shows the signal after applying FFT and
the frequency spectrum.
Figura 6. Original signal, signal after applying FFT and frequency spectrum.
The final result after applying making signal and FFT is
shown in Fig. 7. The first IMF show any remaining noise after
applying the FFT. The second IMF presents a Mode Mixing in
t= 1sand t= 2s, mixed frequencies correspond to 21Hz and
12Hz. The third and fourth IMFs show frequencies of 5Hz
and 2Hz, respectively. Fig. 8 corresponds to the instantaneous
frequency of this method, with the frequencies said above. In
addition, an acceptable accuracy in the separation of modal
frequencies is observed.
Figura 7. Decomposition using EMD with Masking Signal+FFT
IV-C. Masking signal + WT
In order to join the WT with the HHT, the selection of
the Wavelet (mother’s function) most suitable for the original
signal had to be carried out. The selection of this function is
done heuristically, due to the different characteristics of the
signal to be analyzed. For this case, the Wavelet Daubechies
6 (db6) was selected. When this process is carried out at the
Figura 8. Instantaneous frequency using EMD with Masking Signal+FFT
original signal, the approximation of the WT is obtained Fig.
9.
Figura 9. Original signal, signal after applying WT
The result of applying this methodology is shown in Fig.
10, frequency separation is similar to that obtained when
performing Masking+FFT, for example, the form of frequency
separation performs in the same way, a visible advantage of
this strategy is that the noise is more accurately separated.
The other characteristics continue to behave the same, the first
IMF show any remaining noise after applying the FFT. The
second IMF present a Mode Mixing in t= 1sand t= 2s,
the third and fourth IMFs show frequencies of 5Hz and 2H z,
respectively, shown in Fig.11.
IV-D. Masking signal + WT + FFT
In order to seek greater accuracy in the decomposition of
frequencies, we propose the combination of EMD with mas-
king signal + WT + FFT. First, we applied WT (Daubechies 6,
db6), and then perform the FFT. In Fig. 12 appears the result
of this process.
The execution of this methodology produces an excellent
decomposition, shows in Fig. 13 and Fig. 14. Fig. 13 in its first
part has an IMF only with noise components, the second IMF
presents a Mode Mixing between the frequencies of 21Hz
Figura 10. Decomposition using EMD with Masking Signal+WT
Figura 11. Instantaneous frequency using EMD with Masking Signal+WT
Figura 12. Original signal, signal after applying WT, finally the signal with
WT+FFT
and 12Hz, in this case, in less proportion and greater stability.
Finally, the latest IMF shows the frequency of 5Hz
V. DISCUSSION
In order to better understand the qualities of each met-
hodology, table 1 brief the advantages and disadvantages of
the different transforms analyzed. The masking signal does
not allow adequate noise extraction, making impossible to
efficiently extract signal characteristics. On the other hand,
Figura 13. Decomposition using EMD with Masking Signal+WT+FFT
Figura 14. Instantaneous frequency using EMD with Masking Sig-
nal+WT+FFT
Cuadro I
COMPARISON OF TIME-FREQUENCY ANALYSIS METHODOLOGIES
Masking Masking+
FFT
Masking+
WT
Masking+
FFT+WT
Noise
Reduction
X≈ ≈ X
Sampling
Frequency
X X XX
Feature
Extraction
X≈ ≈ X
the Masking+FFT and Masking+WT have a similar result
but with different difficulties: both present difficulties in ef-
ficient feature extraction proportional to their partial noise
isolation capacity, in-contrast the sampling rate is acceptable
in Masking-FFT and in Masking-WT this frequency is reduced
losing details of signal behavior. However, these problems are
solved with Masking+FFT+WT, breaking the paradigms that
were had for signal analysis. This technique has the ability
to separate noise completely with the dexterity of keeping
the sampling rate, improving signal details. For this reason,
this novel technique has an excellent extraction of the signal
characteristics.
VI. CONCLUSIONS
In this paper we have discussed different methods of sig-
nal analysis, in order to find the most efficient strategy in
terms of frequency separation for signals with high noise
content, characteristic of signals with high penetration of Non-
Conventional Sources of Renewable Energies [5]. In Fig. 4,
it is possible to see the IMFs obtained with EMD Masking,
which do not adequately break down the signal, this is caused
by the high noise of the signal; this being evident in Fig. 5.
To solve this problem, three strategies have been proposed:
1) EMD with Masking+FFT, 2) EMD with Masking+WT, 3)
EMD with Masking+WT+FFT. The results of applying EMD
with Masking+FFT and EMD with Masking+WT are similar
but very efficient, this process is a shown in Fig. 12 and Fig. 4,
both have the same characteristics: the first IMF represents the
excess noise after each process (FFT or WT), the second IMFs
present a Mode Mixing in t= 1sand t= 2s, the third and
fourth IMFs show frequencies of 5Hz and 2H z, respectively,
shown in Fig. 7 and Fig. 11. Achieving more accurate and
efficient frequency separation.
The limitation of applying the FFT together with EMD and
Masking signal is the choice of the fundamental frequency of
operation of the system, in this case, knowing beforehand the
characteristics of the signal the method is expected to work
better. Applying the WT loses resolution due to the reduction
of the original sample rate of the signal. In addition, it has the
limitation of the choice of the Wavelet signal, this must have
peculiar characteristics depending on the signal to be analyzed.
These defects mentioned can be counteracted by joining the
two methodologies, executing the EMD+WT+FFT results in
an excellent and accurate mode decomposition, and a smaller
magnitude of Mode Mixing, reducing the number of IMFs
needed to demonstrate better time-frequency resolution.
The proposed strategies emerge as an alternative to imple-
menting HHT aided by different methods of analysis, taking
advantage of the qualities and strengths of each of them. This
methodology results in a discovery of a robust and efficient
analysis technique for those signals where not only the ampli-
tude and frequency values are known previously, if not also,
those that have a high noise content. Common characteristics
of signals from unconventional energy sources. These results,
give an indication to the emergence of a technique of global
analysis capable of breaking the limitations described, being
a disruptive technique for the analysis of amplitude-time-
frequency, nonlinear and non-stationary signals.
VII. ACKNOWLEDGMENT
This paper is part of the project number 111077657914 and
contract number 031-2018, funded by the Colombian Admi-
nistrative Department of Science, Technology and Innovation
(COLCIENCIAS) and developed by the ICE3 Research Group
at Universidad Tecnologica de Pereira (UTP) and CALPOSA-
LLE Group at Universidad de La Salle.
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