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Abstract

This short communication shows a case in which instantaneous frequencies are detected on synthetic signals by both Wavelet (WT) and Hilbert Huang Transform while Fourier Transnsform (FT) is not effective in identifying them. The selected synthetic signal is an example of how using FT in electric power systems with time varying frequencies, can mislead the interpretation of the results. The problem of leakeage is showcased in the example. This is confirmed by the results obtained from the analysis of a single-phase microgrid voltage in which Fourier based WT did not detect the presence of a time varying frequency on the microgrid, as this frequency fluctuates faster than the fundamental frequency.
Instantaneous Frequency in Electric Power Systems
Marta Molinas1, Geir Kulia2, and Olav B. Fosso3
1Department of Engineering Cybernetics, NTNU,
2Department of Electronics and Telecommunications, NTNU
3Director of NTNU’s Strategic Research Area Energy, NTNU
Introduction
For almost a century, Fourier-based analyses have been
used to interpret and understand frequency-related phe-
nomena in electric power systems. The notion of In-
stantaneous Frequency has not been previously explored
in electric power systems [1]. The reason for that has
been that the century-old electric power system has been
dominated by large electromechanical generators that
produced an excellent quality voltage with a constant
frequency. Steady and stable frequency has been one
essential attribute of a stable electric power system and
a pre-condition of stable operation. In this short com-
munication, we discuss the notion of instantaneous fre-
quency in electrical systems as first presented in [2] and
use Hilbert Huang Transform [1] to detect its existence
in a single-phase microgrid. This new notion might sig-
nificantly change our understanding of the stability of
electric power systems.
For the sake of demonstration, first a synthetic sig-
nal v(t)given by (1) is analyzed using Fourier Trans-
form (FT), Wavelet Transform (WT) and Hilbert Huang
Transform (HHT). This signal models two generic har-
monic transitions in a power system with an instanta-
neously varying frequency.
v(t) =
v1(t),if t<t0
v2(t),if tt0
(1)
where
v1(t) = v0(t) + 10cos(2π50t) + 3 sin(2π150t)(2a)
v2(t) = v0(t) + 10cos(2π100t) + 3 sin(2π300t)(2b)
v0(t) = 5 cos2π1000t+300sin(2π300t)(2c)
It consists of three components: one with a time-varying
frequency, while the two others change from one con-
stant frequency to another when t=t0. In the demon-
stration, t0=2.5 s.
Wavelet vs Hilbert Huang
Transform
The figure below shows v(t), along with the FT of v1(t),
v2(t), and v(t). The FT of v(t)(Fv(jω)) contains more
harmonics than the sum of Fv1(jω)) and Fv2(jω)). This
is due to the leakage created by the transition at t0=2.5
s.
0 500 1000 1500
frequency [Hz]
-100
-50
0
50
magnitude [dB]
Fv1(jω)
Fv2(jω)
0 500 1000 1500
frequency [Hz]
-100
-50
0
50
Fv(jω)
2.46 2.47 2.48 2.49 2.5 2.51 2.52 2.53
Time [s]
-10
0
10
v(t)
v1(t)
v2(t)
t=t0
Figure 1: Top: the signal v(t)given by (1). Its scope
is focused on the transition area for better visualization.
Bottom left: the Fourier transform of v1(t)(blue) and
v2(t)(red). Bottom right: the Fourier transform of v(t).
The Fourier had a window of 5 seconds.
The Fourier spectra in the figure above do not give
any temporal information about when the frequency is
changing. To better capture the temporal properties of
v(t), its wavelet transform (WT) was processed. The
WT was compared with the Hilbert-Huang transform
(HHT). HHT estimates the instantaneous frequency and
amplitude of a given signal [1] and is, unlike WT, not
based on the FT. It decomposes any signal down to
monocomponents called intrinsic mode function (IMF)
by using the Empirical mode decomposition (EMD), so
that
v(t) = r(t) + ci(t) = r(t) + ai(t)cosθi(t)(3)
where ci(t)is the IMF number i,ai(t)and θi(t)are
ci(t)s instantaneous amplitude and phase respectively.
The residual r(t)is a monotone function. For v(t),
|r(t)|<1014. The IMFs of v(t)are shown in the figure
below.
2.46 2.47 2.48 2.49 2.5 2.51 2.52 2.53
Time [s]
Intrinsic mode functions of v(t)
Figure 2: The Intrinsic Mode Functions of v(t). The
scope of the IMFs are focused on the transition area for
better visualization.
The instantaneous frequency for each IMF is defined by
fi(t)def
=== 1
2π·dθi(t)
dt (4)
where fi(t)is the instantaneous frequency of ci(t). A
Hilbert Spectrum (HS) is a 3D plot of the instantaneous
amplitude and frequency as a function of time for each
IMF. The figure below shows the Wavelet Spectrum
(WS) and HS of v(t).
Figure 3: Top: WS of v(t)using a 8 level WT. Bottom:
HS of v(t)using a median filter with a length of 2 % of
the sampling frequency to remove artifacts.
It is possible to recognize the functions from (1) in both
the WS and HS. However, WS’s resolution is lower than
that of HS’s and WS’s computation time is more than 28
% longer than HS’s. The WS has also what seems to be
mirrors of the time varying frequency of v0(t).
WS and HS of Microgrid Voltage
A voltage waveform measured at the output of the dc/ac
inverter of a single phase stand-alone microgrid was an-
alyzed using both WT and HHT. The WS and HS are
shown in the figure below.
Figure 4: Top: WS of v(t)using a 14 level Wavelet trans-
form. Bottom: HS of v(t).
A time varying frequency, similar to the above syntheti-
cally created signal is detected with HS. Analytical and
simulation investigations of the microgrid also detected
a time varying frequency with a 10 ms cycle. This phe-
nomena, visible in the HS, was not detected in the WS.
The fact that the HHT can identify the time varying fre-
quency hidden in the WS shows one of the strengths
of the HHT over Fourier-based techniques in detecting
instantaneous frequencies. In addition, the HS shows
greater temporal and frequency resolution and has lower
computation time than the WS.
Discussion
This letter shows a case in which instantaneous frequen-
cies are detected on synthetic signals by both WT and
HHT while FT is not effective in identifying them. The
selected synthetic signal is an example of how using FT
in electric power systems with time varying frequen-
cies, can mislead the interpretation of the results. This
is confirmed by the results obtained from the analysis
of a single-phase microgrid voltage in which Fourier-
based WT did not detect the presence of a time varying
frequency on the microgrid, as this frequency fluctuates
faster than the fundamental frequency.
References
[1] Norden E. Huang et. al.
The empirical mode decomposition and the hilbert spectrum
for nonlinear and non-stationary time series analysis.
Proceedings: Mathematical, Physical and Engineering
Sciences, 454(1971):903â ˘
A¸S995, 1998.
[2] Geir Kulia, Marta Molinas, Lars Lundheim, and
Bjørn B. Larsen.
Towards a real-time measurement platform for microgrids in
isolated communities.
Humanitarian Technology: Science, Systems and Global
Impact 2016, HumTech2016, 2016.
Department of Engineering Cybernetics, NTNU, Trondheim, Norway - July 2016
... measurements from a stand-alone PV microgrid in Bhutan is shown in a Hilbert spectrum in fig. 5. The instantaneous frequency results from applying the HHT approach to the voltage signal [8]. ...
... 6b) and to visualize the frequency components ( fig. 6c) of the originally generated signal [8]. ...
... Their magnitude and frequencies must therefore (c) Hilbert spectrum of vt using a median filter with a length of 2% of the sampling frequency to remove artifacts. Fig. 6: Conceptual proof of the HHT capability [8]. ...
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