Jiefeng XIONG1,2,Bolin WANG1
Hohai University (1), Nanjing University of Information Science and Technology (2)
Measuring power system harmonics and interharmonics by
envelope spectrum analysis
Abstract. An envelope spectrum analysis-based algorithm for harmonics and interharmonics estimation is proposed. First the envelope is extracted
with narrow band Hilbert transform. Then the spectrum of the envelope and the harmonic components are calculated with windowing and
interpolation method. Finally the interharmonic parameters are restored according to amplitude modulation equation. The proposed method has two
distinguish features, first, it is able to confirm if the calculated interharmonic components do exist, and second, it is not affected by the spectral
leakage caused from the harmonics. Several simulation examples are given to demonstrate the precision, effectiveness, and feasibility.
Abstract. Zaproponowano algorytm analizy widmowej umożliwiający określanie składowych harmonicznych. Najpierw wydzielana jest obwiednia
przy wykorzystaniu wąskopasmowej transformaty Hilberta. Następnie obliczane jest widomo obwiedni i składowych harmonicznych metodami
interpolacyjnymi. Wreszcie parametry interharmonicznych są odtwarzane na podstawie równania modulacji amplitudowej. Zaproponowana metoda
ma dwie istotne zalety  umożliwia obliczanie składowych interharmonicznych i nie jest obciążona wpływem przecieku od harmonicznych.
Zaprezentowano kilka przykładów symulacji potwierdzających skuteczność metody. (Pomiary harmonicznych i interharmonicznych w
systemach mocy metodÄ… analizy spektralnej obwiedni).
Keywords: harmonic analysis, interharmonic, envelope extraction, discrete Fourier transforms, spectral leakage.
SÅ‚owa kluczowe: analiza widmowa, wydzielanie obwiedni, Dyskretna Transformata Fouriera.
Introduction processing technique based on advanced spectrum
Accurate harmonic/interharmoincs analysis and estimation were also used for harmonic and interharmonic
measurement in electrical power systems are of particular analysis, which theoretically has an infinitely frequency
importance, since a true and exact spectrum of a waveform resolution, and their improvements can be found in [15-17].
provides a clear understanding of the causes and effects of Whereas, spectrum estimation methods operate effectively
waveform distortion. only on the narrow-band signal in frequency domain which
The most popular and effective algorithm for harmonics has limited components. Moreover, the computational
and interharmonics measurement is windowed discrete burden may result sensibly increased when high accuracy
Fourier transform (DFT). When interharmonics are present, is required.
the direct application of the DFT with a constant sampling In this paper an envelope spectrum analysis-based
rate may lead to inaccurate measurement results due to the method is proposed for interharmonics estimation of
spectral leakage and picket fence effects[1-3]. These signals. The proposed method focuses on the point that the
effects strongly increase difficulties in measuring envelope of the power system signal contains information
interharmonics, which can even  create new interharmonic for interharmonic estimation. The method extracts the
components (fake interharmonics) in the spectrum that do envelope of the signal, calculates its spectrum, and then
not exist at all[4]. restores the interharmonic parameters according to
Various methods have been proposed to overcome these amplitude modulation equation. It is shown that the effects
effects, especially the spectral leakage effect to obtain (or fake interharmonics), caused by the spectral leakage
better estimates of the power harmonics or interharmonics. from harmonics, can be avoided with the proposed method.
References [5-8] put forward methods based on windowing The new method benefits higher computing speed and
and interpolation in the frequency domain, in which the more stable than advanced spectrum estimation-based
errors created by leakage are eliminated by windowing method, and more accurate than traditional DFT-based
technique, and the errors by picket effects are reduced by method.
the interpolation algorithm. A desynchronized processing The organization of this paper is as follows. The
technique was employed for harmonic and interharmonic relationship between interharmonics, voltage fluctuation and
analysis, in which harmonics are filtered out from the signal voltage flicker are recalled in section II. The interharmonics
to obtain better estimates of the interharmonics [9]. An measurement method based on envelope spectrum
adaptive window width method based on correlation analysis is proposed in section III. Then, simulation results
calculation can be found in [10], and claimed suffering no to demonstrate the precision, feasibility and robustness of
leakage effect. In [11], the time-domain averaging was used the algorithm are presented in section IV. At last the
for harmonic processing, and then a difference filter for the conclusions are given in section V.
improved detection of interharmonics was proposed.
Interharmonics, voltage fluctuation and voltage flicker
Interharmonic-subgroups were recommend by the IEC
Interharmonics, voltage fluctuation and voltage flicker
group to reduce the spectral leakage problem, which aims
have an inherent relationship. At steady state without any
at standardization, simplification and unification, more
disturbance, the voltage waveform in a power system is
details can be found in [12-13].
sinusoidal with constant amplitude. When a voltage
Anyway, it is well known that a through solution for the
waveform contains interharmonics (generated from the
problems due to the DFT spectral leakage is to select
operation of fluctuating loads), the peak and RMS
window width as an exact multiple of all signal periods,
which is called the synchronization of the sampling magnitudes of the waveform will fluctuate. This is because
procedure. However synchronizing to interharmoics is the periods of the interharmonics components are not
practically infeasible because their frequencies are usually
synchronous with the fundamental frequency cycle. Figure
unpredictable or the necessary window width is too large.
1 shows the waveform of
Reference [14] uses wavelets for spectral estimation to
reduce the spectral leakage problem. Modern signal
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M
y(t) =ð sin 2pð 50t -ð 0.05sin 2pð 45t -ð pð / 2
(ð )ð (ð )ð
+ðD2 sin 4pðhðt +ðqð2 Am sin 2mpð f1t +ðfðm +ð
(1) (ð)ð(ð )ð
åð
+ð 0.05sin 2pð 55t -ð pð / 2 m=ð1
(ð)ð
M
It contains a fundamental component (50Hz) and two
LðLð +ð DL sin 2Lpðhðt +ðqðL Am sin 2mpð f1t +ðfðm
(ð)ð (ð )ð
åð
interharmonics (45Hz and 55Hz), and its envelope appears
m=ð1
with a noticeable 5Hz fluctuation.
M
=ð Am sin 2mpð f1t +ðfðm +ð
(ð)ð
åð
m=ð1
1
+ð DA1 éðcos 2pð f1 -ðhð t +ðfð1 -ðqð1 Å‚ð
(ð )ð
1
ëð ûð
2
1
-ð DA1 éðcos 2pð f1 +ðhð t +ðfð1 +ðqð1Å‚ð +ð
(ð )ð
1
ëðûð
2
1
+ð D2 A1 éðcos 2pð f1 -ð 2hð t +ðfð1 -ðqð2 Å‚ð
(ð )ð
ëð ûð
2
Fig. 1. Amplitude modulation voltage waveform caused by
1
interharmonics
-ð D2 A1 éðcos 2pð f1 +ð 2hð t +ðfð1 +ðqð2 Å‚ð +ð
(ð )ð
ëð ûð
2
If the fluctuation magnitude is sufficiently large and the
+ðLðLð+ð
fluctuation frequency is in a range perceptible by human
eyes (0.5 to 30Hz), a light flicker will occur. Consequently, a
1
+ð DL A1 éðcos 2pð f1 -ð Lhð t +ðfð1 -ðqðL Å‚ð
(ð )ð
conclusion can be drawn that if there are interharmincs in ëð ûð
2
the signal, a voltage flicker or a modulation of the voltage
1
waveform will occur, and vice versa.
-ð DL A1 éðcos 2pð f1 +ð Lhð t +ðfðL +ðqðL Å‚ð +ð
(ð )ð
ëð ûð
It should be mentioned that the summations of one or
2
more small interharmonics to fundmental frequency can +ðLðLðLðLðLðLðLðLðLðLðLðLðLðLðLðLð+ð
always be interpreted in terms of amplitude modulation and
1
phase modulation [18]. However, the most traditional
+ð DAM éðcos2pð Mf1 -ðhð t +ðfðM -ðqð1Å‚ð
(ð )ð
1
ëðûð
2
approach to study the voltage flicker (voltage fluctuation) is
based on amplitude modulation [19] and a voltage with an
1
-ð DAM éðcos2pð Mf1 +ðhð t +ðfðM +ðqð1Å‚ð +ð
(ð )ð
amplitude modulation can be described as 1
ëðûð
2
(2) y(t) =ð 1+ð d (t) A1 sin 2pð f1t +ð fð1
[ð ]ð (ð)ð
1
+ð DAM éðcos2pð Mf1 -ð2hð t +ðfðM -ðqð2Å‚ð
(ð )ð
where d(t) is the  envelope (modulating signal), A1, f1,and 2
ëðûð
2
Åš1 are the fundamental amplitude, frequency, and phase of
1
the system individually.
-ð DAM éðcos2pð Mf1 +ð 2hð t +ðfðM +ðqð2Å‚ð +ð
(ð )ð
2
ëðûð
Due to the load characteristics, d(t) can be cyclic, such
2
as operation of a reciprocation pump. And it also can be
1
+ðLðLð+ð DAM éðcos2pð Mf1 -ð Lhð t +ðfðM -ðqðL Å‚ð
stochastic, such as operating electric arc furnaces [20]. (ð )ð
L
ëð ûð
2
With no additional explanation, only periodic modulating
1
signal is considered in this paper, and d(t) can be
-ð DAM éðcos2pð Mf1 +ð Lhð t +ðfðM +ðqðL Å‚ð
(ð )ð
L
ëðûð
expressed as
2
L
From (5), it can be known that if · is not integral multiple
(3) d (t) =ð Dl sin 2lpðhðt +ð qðl
(ð)ð
åð
of f1, components with frequency (f1Ä…·), (f1Ä…2·),& , (f1Ä…L·),
l =ð1
(2f1Ä…·), (2f1Ä…2·),& (2f1Ä…L·),& (Mf1Ä…·) (Mf1Ä…2·), & (Mf1Ä…L·)
where Dl and ¸l are amplitude and phase of the lth
are all interharmonics. We can also find that the envelope
harmonic component of d(t), · is the fundamental frequency
signal d(t) never changes the amplitude and phase of
of d(t). It should be noted that although the flicker model (2)
harmonic components , and it only  produces
is still needs to be improved, many field measurement
interharmonics. Note that if L·< f1/2 is also satisfied, each
results demonstrate its effectiveness in calculating voltage
interharmoic component will appear once in (5), and great
flicker and identifying the interharmonic polluters [20-23].
When harmonics are considered, a more complex simplification of interharmonic measurement can be
model can be expressed as: achieved.
M
(4) y(t) =ð 1+ð d (t) Am sin 2mpð f1t +ð fðm Envelope spectrum analysis-based method
[ð ]ð (ð)ð
åð
m=ð1
Based on the above discussion, the new measurement
If we substitute (3) into (4), this will yield
algorithm for power system harmonics and interharmonics
M
is completely presented with the help of the flowchart
y(t) =ð Am sin 2mpð f1t +ðfðm
(ð)ð
åð
(Figure 2) in this section. The phases of ¸l and Åš1 are
m=ð1
(5)
assumed to be zero, and this will simplify the calculations
LM
+ð sin 2lpðhðt +ðqðl Am sin 2mpð f1t +ðfðm
(ð)ð (ð )ðwithout affecting the interpretation of the algorithm.
åðD åð
l
The signal y(t) is digitized with equally sampling space Ts
l =ð1 m=ð1
in the sampling block, thus the output of this block is
M
LM
=ð Am sin 2mpð f1t +ðfðm
(ð)ð éð
åð
(6)
y(n) =ð Dl sin 2lpðhðnTs Å›ð Am sin 2mpð f1nTs
(ð )ðÅ‚ð (ð )ð
åðåð
m=ð1 Ä™ð1+ð
ëð l =ð1 ûð m=ð1
M
+ðD1 sin 2pðhðt +ðqð1 Am sin 2mpð f1t +ðfðm The low pass filtering block is composed of a sixth order
(ð)ð(ð )ð
åð
Butterworth low-pass filter with an 85Hz cut-off frequency.
m=ð1
The amplitude response of this filter is shown in Figure 3.
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This low-pass filter gives a strong attenuation at frequencies accomplished by filter, Wavelet transform and Hilbert-
higher than 85Hz. Consequently, the terms of (5) that Huang Transaction, and the narrow band Hilbert
include frequencies of (2f1Ä…·), (2f1Ä…2·),& (2f1Ä…L·),& (Mf1Ä…·) transform is employed in the paper.
(Mf1Ä…2·), & (Mf1Ä…L·) are suppressed. The elimination in (5) 4. calculate the spectrum of d(t) with DFT based algorithm
of all these components leads to the following expression: when d(t) is periodic, and Hanning window and
interpolation are recommended again to obtain better
results.
5. calculate the frequencies and amplitudes of each
interharmonic using (5).
6. calculate the harmonic and interharmonic sub-groups
according to IEC standards.
Simulation results
According to IEC standard, instrument precision for
interharmonic analysis is tested with the input signal which
contains fundamental component and only one
interharmonic component. It should be mentioned that, if
this interharmonic locates far from the fundamental
component in the spectrum, spectral leakage effect from the
fundamental component can always be negligible for
interharmonic measurement. Whereas, this tested signal is
quite different from the practical waveform in power system,
in which multiple harmonic and interharmonic components
always exist.
The proposed algorithm based on envelope spectrum
Fig. 2. Flowchart of the envelope spectrum analysis-based method
analysis extracts the envelope (modulating) signal,
L
éð
(7) calculates the spectrum of modulating signal, and then
y1(n) =ð Dl sin 2lpðhðnTs Å›ð A1 sin 2pð f1nTs
(ð )ðÅ‚ð (ð )ð
åð
Ä™ð1+ð
ëð l =ð1 ûð
interharmonic parameters can be restored according to (5).
In this way the effects (or fake interharmonics), caused by
If Hilbert transform is applied to (7), it yields
L the spectral leakage from harmonics, can be eliminated, it is
éð
(8)
w
1(n) =ð Dl sin 2lpðhðnTs Å›ð A1 cos 2pð f1nTs
(ð )ðÅ‚ð (ð )ð more accurate for interharmonic analysis than traditional
åð
Ä™ð1+ð
ëð l =ð1 ûð
methods in the real world.
The analytic signal constructed by (7) and (8) is
Four simulations are performed in Matlab6.5 to
expressed as
demonstrate the effectiveness of the proposed algorithm.
(9) z(n) =ð y1(n) +ð 5
y1(n) The sampling frequency for all the experiments is 10KHz.
And its amplitude takes the following form
A) Waveforms with only Harmonics
L
1/ 2
éð
(10) z(n) =ð y12(n) +ð w
12(n) =ð A1 Ä™ð1+ð l sin 2lpðhðnTs Å›ð Synchronization characteristics of both the proposed
(ð )ðÅ‚ð
åðD
method and the IEC technique are studied in this section.
ëð l =ð1 ûð
The signal x(t)=220×"2sin(2Ä„ft)+220×"2sin(6Ä„ft) is
considered in the case, which consists of the fundamental
component and the 3rd harmonic. The ideal fundamental
frequency f is assumed to be 50 Hz and it is assumed to be
varying from 49.5Hz to 50.5Hz. The harmonic and
interharmonic subgroup evaluated with both the new
method and the IEC technique are given in Table 1.
Table 1. Absolute errors in calculating harmonic-subgroup and
interharmonic-subgroup
Gsg,1ÿV ÿ Gisg,1ÿV ÿ Gsg,3ÿV ÿ Gisg,3ÿV ÿ
Fig. 3. Amplitude response of the low-pass filter
f(Hz)
True value True value True value True value
(220V) (0V) (35V) (0V)
It is important to note that (10) only contains the dc
IEC NEW IEC NEW IEC NEW IEC NEW
component and periodic envelope d(n). With DFT based
50.00 0 0 0 0 0 0 0 0
method, the spectrum of (10) can be obtained, and then the
50.05 0.11 0 1.73 0 0.20 0 1.13 0
spectrum of y(t) can be calculated according to (5) if the
50.50 2.61 0 17.57 0 2.36 0 9.16 0
parameters of harmonics have been obtained.
The following summarizes major steps of the solution for 49.95 0.06 0 1.71 0 0.15 0 1.13 0
harmonic and interharmonics measurement. 49.5 1.85 0 16.00 0 2.30 0 8.99 0
1. digitize the estimated signal y(t) with equally sampling
space for nearly S periods, note that window width
With IEC technique, errors can always be observed on
should cover at least one period of the envelope and
the harmonic/interharmonic sub-groups estimation in the
the synchronous error should be as little as possible.
case of loss of synchronization, and accurate results can
2. calculate the spectrum of integral harmonics of y(n) with
only be obtained under synchronous sampling(f=50Hz).
DFT based algorithm, windowing and interpolation
Special notice should be taken that, under asynchronous
techniques are recommended in order to improve the
sampling, Gisg,1 and Gisg,3 are  fake interharmonic
measurement precision. In this paper, Hanning window
components caused by spectral leakage effect from the
is selected because it is characterized by a relatively
harmonics.
narrow main lobe and fast-decaying side lobes.
Whereas, the proposed method based on demodulation
3. extract modulating signal d(n) from y(n), this can be
spectrum analysis is not affected by asynchronous
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sampling, and so no fake interharmonics are found. For nearly 10Hz component can be found in the spectrum of
simplicity, the calculation procedures with only f=50.5Hz is demodulated signal. This is helpful to confirm the existence
shown in Figure 4. From figure 4.a, we can find that the of the interharmonic.
original waveform and low-pass filtered waveform are not
amplitude modulated due to containing no interharmoinc
component, thus the amplitude of the demodulated signal
(envelope) is almost constant as shown in figure 4.b. And
its detailed waveform is reported in figure 4.c. No frequency
component (0.5 to 30Hz) can be found in the spectrum of
the demodulated signal as shown in figure 4.d, which
demonstrates no interharmonic component existing in the
original signal.
(a)
(a)
(b)
(b)
(c)
(c)
(d)
Fig. 5. Calculation procedures of envelope spectrum analysis-
based method for x1(t).(a) original waveform and low-pass filtered
waveform. (b)envelope of low-pass filtered waveform. (c) detailed
waveform of (b). (d) amplitude spectrum of demodulated signal.
With windowing and interpolation technique, modulation
frequency and amplitude are, respectively, equal to 8.59Hz
and 0.0999 and the parameters of harmonics and
interharmonics can be obtained according to (5), and then
(d)
the harmonic and interharmonic subgroups evaluated with
Fig. 4. Calculation procedures of envelope spectrum analysis-
both the proposed method and the IEC technique are given
based method for x(t).(a) original waveform and low-pass filtered
in Table 2. It is observed that the proposed method enjoys
waveform. (b)envelope of low-pass filtered waveform. (c) detailed
much more accuracy than the IEC technique as expected.
waveform of (b). (d) amplitude spectrum of demodulated signal.
C) Waveforms with multiple Harmonics and
As the harmonics components can be obtained
Interharmonics
accurately with windowing and interpolation technique, the
Practically, waveforms in the power system always
new method leads to a considerable precision improvement
contain multiple harmonics and interharmonics
compared with IEC method.
components. The case in this simulation is to test the
precision and stability of the new proposed method for
B) Waveforms with harmonics and interharmonics
complex signals. The signal model expressed as (3) and (4)
The aim of this case is to test the accuracy of the
is considered, in which M=21, Am=1/mÿm is odd ÿ, Am=1/40
proposed method compared with the IEC technique when
ÿm is even ÿ, f=50.05Hz, Åšmÿ0, L=1, Dlÿ0.1/l, ·=8.6 Hz,
signal containing both harmonics and interharmonics, which
is expressed as x1(t)=[1+0.1×sin(2Ä„×8.6t)] x(t). Figure 5 ¸l =00
shows the calculation procedures in the case of f=50.5Hz
with the new method. It can be seen that the waveform is
modulated by a low frequency component (8.6Hz), and so a
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Table 2. Absolute errors in calculating harmonic-subgroup and
interharmonic-subgroup
Gsg,1ÿV ÿ Gisg,1ÿV ÿ Gsg,3ÿV ÿ Gisg,3ÿV ÿ
f(Hz)
True value True value True value True value
(220V) (11V) (35V) (1.75V)
IEC NEW IEC NEW IEC NEW IEC NEW
2.68 -0.18 -1.32 -0.04 0.39 -0.03 -0.08 0.00
50.00
2.51 -0.17 -0.20 -0.04 0.17 -0.02 0.66 0.00
50.05
-0.52 -0.06 12.90 -0.04 -2.25 0.00 7.92 0.00
50.50
Fig. 8. Amplitude spectrum with proposed method for 0.4s.
0.80 -0.17 -2.27 -0.04 0.56 -0.02 -0.31 0.00
49.95
-0.49 -0.06 0.63 -0.03 -1.62 0.00 6.18 0.00
49.5
Figure 6 displays the amplitudes spectrums with Hanning
windowing and interpolation technique when the window
width is 0.4s, and the results for 0.6s are reported in figure
7. By comparing the results in these two figures, we can
find that the precision and stability of windowing and
interpolation technique are affected by the sampling window
width. Interharmoinics around high order harmonics can not
Fig. 9. Absolute amplitude errors with interpolation and proposed
be estimated due to their relatively small amplitude in the method for 0.4s.
0.4s case.
(a)
Fig. 6. Amplitude spectrum with interpolation technique for 0.4s.
(b)
Fig. 10. Performance of the proposed algorithm when the input
signal is corrupted with a white Gaussian noise of zero mean and
its SNR value is 40dB. (a) relative amplitude errors. (b) relative
frequency errors.
Fig. 7. Amplitude spectrum with interpolation technique for 0.6s.
Modulation frequency of 8.5907Hz and amplitude of
0.0999 can be obtained by the proposed method when the
window width is 0.4s. Then the interharmonic components
can be calculated according to (5), and the results are
shown in figure 8. Absolute error with these two methods for
0.4s are compared in figure 9, it can be clear seen that new
(a)
method leads to more accurate results.
D) Waveforms with added white Noise
Noise characteristics of the proposed algorithm are
studied in this section through simulation. The signal in the
previous simulation is corrupted with an added zero-mean
Gaussian white noise, and three cases are discussed in
which their SNR values are 20dB, 30dB and 40dB
individually based on the rms value of the signal. As the
(b)
results may change in each simulation, only one test results
Fig. 11. Performance of the proposed algorithm when the input
for each case are reported in figure 10, figure 11, and figure
signal is corrupted with a white Gaussian noise of zero mean and
12. From the figures, we can find that the new method
its SNR value is 30dB. (a) relative amplitude errors. (b) relative
exhibits desirable performance to the noise.
frequency errors.
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[8] Hao Q. , Z. RongxiangÿC. Tong,  Interharmonics analysis
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technique for harmonic and interharmonic analysis, IEEE
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[10] Zhu T. X., Exact Harmonics/Interharmonics Calculation Using
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(a)
harmonics and interharmonics by time-domain averaging,
IEEE Trans. Power Del., vol. 20, no. 4, pp. 2370-2380, Oct
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[12] General Guide on Harmonics and Interharmonics
Measurementsÿ for Power Supply Systems and Equipment
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[13] Power Quality Measurements Methods ÿ Testing and
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6-10, 2002.
Fig. 12. Performance of the proposed algorithm when the input
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