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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)eiθ(t)(1)

where y(t)is the Hilbert Transform of x(t), so that:

y(t) = 1

πPZ∞

−∞

x(τ)

t−τdτ (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

dθ (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)ψt−u

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

√sψt−u

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) =

N−1

X

n=0

x(n)e−j2πk 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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