Chapter 22
Novel Method for Tracking TimeVarying Power Harmonic Distortions
without Frequency Spillover
C. A. Duque, P.M. Silveira, T. Baldwin, P. F. Ribeiro
22.1 Introduction
Although it is well known that Fourier analysis is in reality only accurately applicable to
steady state waveforms, it is a widely used tool to study and monitor time-varying signals,
which are commonplace in electrical power systems. The disadvantages of Fourier analysis,
such as frequency spillover or problems due to sampling (data window) truncation can often be
minimized by various windowing techniques, but they nevertheless exist. This chapter
demonstrates that it is possible to track and visualize amplitude and time-varying power
systems harmonics, without frequency spillover caused by time-frequency techniques. This new
tool allows for a clear visualization of time-varying harmonics which can lead to better ways of
tracking harmonic distortion and understanding time-dependent power quality parameters. It
also has the potential to assist with control and protection applications.
While estimation technique is concerned with the process used to extract useful
information from the signal, such as amplitude, phase and frequency; signal decomposition is
concerned with the way that the original signal can be split in other components, such as
harmonic, inter-harmonics, sub-harmonics, etc..
Harmonic decomposition of a power system signal (voltage or current) is an
important issue in power quality analysis. There are at least two reasons to focus on harmonic
decomposition instead of harmonic estimation: (a) if separation of the individual harmonic
component from the input signal can be achieved, the estimation problem becomes easier; (b)
the decomposition is carried out in the time-domain, such that the time-varying behaviour of
each harmonic component can be observed.
Some existing techniques can be used to separate frequency components. For example,
STFT (Short Time Fourier Transform) and Wavelets Transforms [1], are two well known
decomposition techniques. Both can be seen as a particular case of filter bank theory [2]. The
STFT coefficients filters are complex numbers which generates a complex output signal whose
magnitude corresponds to the amplitude of the harmonic component being estimated [2]. The
main disadvantage of this method is to set up an efficient band-pass filter with lower frequency
spillover. On the other hand, although wavelet transform utilizes filters with real coefficients,
the common wavelet mothers do not have good magnitude response in order to prevent
frequency spillover. Besides, the traditional binary tree structure is not able to divide the
spectrum conveniently for harmonic decomposition.
Adaptive notch filter and PLL (Phase-Locked Loop) [3]-[4] have been used for extracting
time-varying harmonics components. However, these methods work well only if a few
harmonics components are present at the input signal. In the other case the energy of adjacent
harmonics spills over each other and the decomposed signal becomes contaminated. In [5] the
authors presented a technique based on multistage implementation of narrow low-pass digital
filters to extract stationary harmonic components. Thus, no digital analytical technique has
been proposed or used to track time-varying power systems harmonics without frequency
spillover. Consequently, the method proposed in this chapter adequately utilizes filters banks to
avoid the frequency mixture, particularly associated with time-varying signals.
Attempts to visualize time-varying harmonics using Wavelet Transform have been
proposed in [6] and [7]. However, the structures were not able to decouple the frequencies
completely.
The methodology proposed in this chapter is constructed to separate the odd and the
even harmonic components, until the 15th. It uses selected digital filters and down-sampling to
obtain the equivalent band-pass filters centered at each harmonic. After the signal is
decomposed by the analysis bank, each harmonic is reconstructed using a non-conventional
synthesis bank structure. This structure is composed of filters and up-sampling that
reconstructs each harmonic to its original sampling rate.
The immediate use for this method is the monitoring of time-varying individual power
systems harmonics. Future use may include control and protection applications, as well as
inter-harmonic measurements.
The chapter is divided into method description, odd/even harmonic extraction and
simulation results.
22.2 Method description
Multirate systems employ a bank of filters with either a common input or summed
output. The first structure is known as analysis filter bank [8] as it divides the input signal in
different sub-bands in order to facilitate the analysis or the processing of the signal. The second
structure is known as synthesis filter bank and is used if the signal needs to be reconstructed.
Together with the filters the multirate systems must include the sampling rate alteration
operator (up and down-sampling). Figure 22.1 shows two basic structures used in a multirate
system, where the up and down-sampling factor, M and L, equal 2. Figure 22.1-a shows a
decimator structure composed by a filter following by the down sampler and Fig. 22.1-b the
interpolator structure composed by a up-sampler followed by a filter. The decimator structure
is responsible to reduce the sampling rate while the interpolator structure to increase it. The
filters H(z) and F(z) are typically band-pass filters.
Figure 22.1- Basic structures used in multirate filter bank and its equivalent
representation for L=M=2. (a) Decimator; (b) Interpolator
The direct way to build an analysis filter bank, in order to divide the input signal in its
odd harmonic component, is represented in Fig. 22.2. In this structure the filter H k (z ) is a bandpass filter centered in the kth harmonic and must be projected to have 3dB bandwidth lower
than 2f0, where f0 is the fundamental frequency. If only odd harmonics are supposed to be
present in the input signal, the 3dB bandwidth can be relaxed to be lower than 5f0. Note that
Fig. 22.2 is not a multirate system, because the structure does not include sampling rate
alternation, which means that there is only one sampling rate in the whole system.
The practical problem concerning the structure shown in Fig. 22.2 is the difficulty to
design each individual band-pass filter. This problem becomes more challenging when a high
sampling rate must be used to handle the signal and the consequent abrupt transition band.
In this situation the best way to construct an equivalent filter bank is to utilize the
multirate technique. Figure 22.3 shows how an equivalent structure to Fig. 22.2 can be
obtained using the multirate approach. The filters H0(z) and H1(z) are orthogonal filters [6], with
the first one as a low-pass filter and the second one as a high-pass filter.
Figure 22.2 – Analysis bank filter to decompose the input signal in its harmonics
components.
.
Figure 22.3- Multirate equivalent structure to the filter bank in Figure 22.2
Figure 22.4 shows the amplitude response for the analysis bank. This figure was
obtained using sampling rate equal to 226 samples by cycle and FIR filters of 69th order. These
filters correspond to so-called orthogonal filter banks also known as power-symmetric filter
banks [8].
Analysis Filter Bank Frequency Response
0
H1
-5
H3
H5
H7
H9
H11 H13 H15
Magnitude (dB)
-10
-15
-20
-25
-30
-35
-40
0
2
4
6
8
10
12
14
frequency (harmonic number)
16
18
Figure 22.4- Amplitude response
The main difference between the structure shown in Fig. 22.2, and the other one
obtained using the multirate technique, is the decimator at the output of the bank, as shown in
Fig. 22.5. In fact, Fig.22.5 is equivalent to Fig. 22.3. It was obtained using the multirate nobles
identities [2] and moving the downsampler factor inside Fig. 22.2 to the right side. The
decimated signal at the output of each filter has a sampling rate 64 times lower then the input
signal. To reconstruct each harmonic into its original sampling rate it is necessary to use the
synthesis filter bank structure.
H1(z)
xd1(n)
64
xd3(n)
x(n)
H3(z)
64
Hk(z)
64
xdk(n)
Figure 22.5- Equivalent multirate analysis filter bank
22.3 The synthesis filter banks
The synthesis filter bank used here is a different implementation of the conventional
one. As the interest is to reconstruct each harmonic instead of the original signal, the filter bank
must be divided in order to obtain the corresponding harmonic. Figure 22.6 shows the filter
bank utilized to reconstruct the harmonic components.
Figure 22.6- Modified synthesis bank
It is important to remark that the frequency response of the synthesis filter bank is
similar to Fig. 22.4.
The analysis of Fig. 22.4 reveals that the filter bank in this configuration is unable to
filter even harmonics appropriately, which means that these components will appear with the
adjacent odd harmonics. To overcome this problem an additional filter stage is included. This
filter is composed by a second order Infinite Impulse Response (IIR) notch filter.
22.4 Extracting even harmonics
To extract the even harmonics the same bank can be used together with a
preprocessing of the input signal. Hilbert transform is then used to implement a technique
known as Single Side Band (SSB) modulation [8]. The SSB modulation moves to the right all
frequencies in the input signal by f0 Hz. In this way, even harmonics are changed to odd
harmonics and vice-versa.
Figure 22.7 shows the whole system for extracting odd and even harmonics.
Figure 22.7 – Whole structure to harmonic extraction
22.5 Simulation results
This section presents two examples: the first is a synthetic signal, which has been
generated in Matlab using a mathematical model, and the second is a signal obtained from the
Electromagnetic Transient Program including DC systems (EMTDC) with its graphical interface
Power Systems Computer Aided Design (PSCAD). This program can simulate any kind of
power system with high fidelity and the resulting signals of interest are very close to physical
reality.
Synthetic Signal
The synthetic signal utilized can be represented by:
N
x(t )   Ah sin(h 0t). f (t )  g (t )
h 1
(22.1)
Where h is the order (1 up to 15th) and A is the magnitude of the component, 0 is the
fundamental frequency, and finally, f(t) and g(t) are exponential functions (crescent, decrescent or alternated one) or simply a constant value  . Besides, x(t) is partitioned in four
different segments in such way that the generated signal is a distorted one with some
harmonics in steady-state and others varying in time, including abrupt and modulated change
of magnitude and phase, as well as a DC component. Figure 22.8 illustrates the synthetic signal.
The structure shown in Fig. 22.7 has been used to decompose the signal into sixteen
different harmonic orders, including the fundamental (60 Hz) and the DC component.
Figure 22.9 shows the decomposed signal which its corresponding components from DC
up to 11th harmonic. The left column represents the original components and the right column
the corresponding components obtained through the filter bank.
For simplicity and space limitation the higher components are not shown. However, it is
important to remark that all waveforms of the time-varying harmonics are extracted with
efficiency along the time.
Naturally, intrinsic delays will be present during the transitions from previous to the
new state. Figure 22.10 shows some components of both the original and decomposed signal in
a short time scale interval.
5
4
3
Magnitude
2
1
0
-1
-2
-3
0
0.5
1
1.5
2
Time (s)
2.5
3
3.5
4
Figure 22.8 – Synthetic signal used
Simulated Signal
It is well known that during energization a transformer can draw a large current
from the supply system, normally called inrush current, whose harmonic content is high.
Although today’s power transformers have lower harmonic content Table 22.1, shows
the typical harmonic components present in the inrush currents [9]. These values are normally
used as reference for protection reasons, but they do not take into account the time-varying
nature of this phenomenon.
Table 22.1- typical harmonic content of the inrush current
Order
Content %
Dc
55
2
63
3
26.8
4
5.1
5
4.1
6
3.7
7
2.4
In recent years, improvements in materials and transformer design have lead to inrush
currents with lower distortion content [10]. The magnitude of the second harmonic, for
example, has dropped to approximately 7% depending on the design [11]. But, independent of
these new improvements, it is always important to emphasize the time-varying nature of the
inrush currents.
In being so, a transformer energization case was simulated using EMTDC/PSCAD and the
result is shown in Fig. 22.11.
Using the methodology proposed to visualize the harmonic content of the inrush
current, Fig. 22.12 reveals the rarely seen time-varying behaviour of the waveform of each
harmonic component, where the left column shows the DC and even components and the right
column the odd ones. This could be used to understand other physical aspects not observed
previously.
DC component
DC component
2
2
1
1
0
0
1
2
1st
3
4
1
0
0
1
2
1st
3
4
0
1
2
2nd
3
4
0
1
2
3rd
3
4
0
1
2
Time s
3
4
1
0
0
-1
-1
0
1
2
2nd
3
4
1
1
0
0
-1
-1
0
1
2
3rd
3
4
1
1
0
0
-1
-1
0
1
2
Time s
3
4
4th
4th
1
1
0
0
-1
-1
0
1
2
5th
3
4
0.5
0.5
0
0
-0.5
0
1
2
6th
3
4
0.2
0
-0.2
-0.5
0
1
2
5th
3
4
0
1
2
6th
3
4
0
1
2
7th
3
4
0
1
2
Time s
3
4
0.2
0
-0.2
0
1
2
7th
3
4
0.5
0.5
0
0
-0.5
-0.5
0
1
2
Time s
3
4
8th
8th
1
1
0
0
-1
-1
0
1
2
9st
3
4
0.2
0
-0.2
1
2
9st
3
4
0
1
2
10th
3
4
0
1
2
11rd
3
4
0
1
2
Time s
3
4
0.2
0
-0.2
0
1
2
10th
3
4
1
1
0
0
-1
0
0
1
2
11rd
3
4
0.5
-1
0.5
0
0
-0.5
-0.5
0
1
2
Time s
3
4
Figure 22.9 – First column: original components, second column: decomposed signals.
DC component
2
1
0
2.99
3
3.01
3.02
2nd
3.03
3.04
3.05
3
3.01
3.02
3rd
3.03
3.04
3.05
3
3.01
3.02
7th
3.03
3.04
3.05
3
3.01
3.02
Time s
3.03
3.04
3.05
1
0
-1
2.99
1
0
-1
2.99
0.5
0
-0.5
2.99
Figure 22.10 – Comparing original and decomposed component
14
12
Magnitude kA
10
8
6
4
2
0
-2
0
0.1
0.2
0.3
0.4
0.5
Time (s)
0.6
0.7
0.8
0.9
1
Figure 22.11 – Inrush current in phase A.
Table 22.1- typical harmonic content of the inrush current
Order
Content %
Dc
55
2
63
3
26.8
4
5.1
5
4.1
6
3.7
7
2.4
22.6 Final consideration and future work
The methodology proposed has some intrinsic limitations associated with the
analysis of inter-harmonics as well as with real time applications. The Inter-harmonics (if they
exist) are not filtered by the bank, so it would corrupt the nearby harmonic components. The
limitation regarding with real time applications is related to the transient time response
produced by the filter bank. For example, in the presence of abrupt change in the input signal,
such as in inrush currents, the transient response can last more than 5 cycles, which is not
appropriate for applications whose time delay must be as short as possible. The computational
effort for real time implementation is another challenge that the authors are investigating. In
fact the high order filter (69th) used in the bank structure demand high computational effort. By
using multirate techniques to implement the structure it is possible to show that the number of
multiplication by second to implement one branch of the analysis filter bank is about one
million. Some low price DSPr (Digital Signal Processors) available in the market are able to
execute 300 Million Float Point operations per second. This shows that, despite the higher
computational effort of the structure, it is feasible to be implemented in hardware. However,
new opportunities exist for overcoming the limitations and the development of improved and
alternative algorithms. For example, the authors are investigating the possibility of extracting
inter-harmonic components and developing a similar methodology based on Discrete Fourier
Transform (DFT). The first results show similar visualization capabilities with the promise to
provide a reduced transient time response and lower computational burden. This process has
the potential of addressing specific protective relaying needs such as detecting a high
impedance fault during transformer energization and detection of ferro-resonance.
1st
DC component
5
4
0
2
0
-5
0
0.2
0.4
0.6
0.8
0
1
0.2
0.4
2nd
0.6
0.8
1
0.6
0.8
1
0.6
0.8
1
3rd
1
2
0
-2
0
0
0.2
0.4
0.6
0.8
-1
1
0
0.2
0.4
4th
5th
0.5
0.2
0
-0.2
0
-0.5
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
7th
6th
0.2
0.1
0
-0.1
0
-0.2
0
0.2
0.4
0.6
Time (s)
0.8
1
0
0.5
1
Time s
9st
8th
0.1
0.1
0
0
-0.1
-0.1
0
0.2
0.4
0.6
0.8
0
1
0.2
0.6
0.8
1
0.6
0.8
1
0.2
0.4
0.6
15th
0.8
1
0.2
0.4
0.8
1
10th
0.4
11rd
0.05
0.05
0
-0.05
0
-0.05
0
0.2
0.4
0.6
0.8
1
0
0
0
-0.05
-0.05
0.2
0.4
0.4
0.05
0.05
0
0.2
13th
12th
0.6
0.8
1
14th
0
0.05
0.04
0.02
0
-0.02
-0.04
0
-0.05
0
0.2
0.4
0.6
Time (s)
0.8
1
0
0.6
Time (s)
Figure 22.12 – Decomposition of the simulated transformer inrush current.
22.7 Conclusions
This chapter presents a new method for time-varying harmonic decomposition based on
multirate filter banks theory. The technique is able to extract each harmonic in the time
domain. The composed structure was developed to work with 226 samples per cycle and to
track up to the 15th harmonic. The methodology can be adapted through convenient preprocessing for different sampling rates and higher harmonic orders.
Acknowledgments
The authors would like to thanks Dr. Roger Bergeron and Dr. Mathieu van den Berh for
their useful and constructive suggestions.
22.8 References
[1] Yuhua Gu, M. H. J. Bollen, “Time-Frequency and Time-Scale Domain Analysis,” IEEE
Transaction on Power Delivery, Vol. 15, No. 4, October 2000, pp. 1279-1284.
[2] P.P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993.
[3] M. Karimi-Ghartemani, M. Mojiri and A. R. Bakhsahai, “A Technique for Extracting
Time-Varying Harmonic based on an Adaptive Notch Filter,” Proc. of IEEE Conference on Control
Applications, Toronto, Canada, August 2005.
[4] J. R. Carvalho, P. H. Gomes, C. A. Duque, M. V. Ribeiro, A. S. Cerqueira, and J.
Szczupak, “PLL based harmonic estimation,” IEEE PES conference, Tampa, Florida-USA, 2007
[5] C.-L. Lu, “Application of DFT filter bank to power frequency harmonic measurement,”
IEE Proc. of Generation Transmission and Distribution, Vol 152, No.1, January 2005, pp. 132136.
[6] P. M. Silveira, M. Steurer, .P F. Ribeiro, “Using Wavelet decomposition for
Visualization and Understanding of Time-Varying Waveform Distortion in Power System,” VII
CBQEE, August 2007, Brazil.
[7] V.L. Pham and K. P. Wong, “Antidistortion method for wavelet transform filter banks
and nonstationay power system waveform harmonic analysis,” IEE Proc. of Generation,
Transmission and Distribution, Vol 148, No.2, March 2001, pp. 117-122.
[8] Sanjit K. Mitra, Digital Signal Processing – A computer-based approach, Mc-Graw Hill
2006, 3ª Edition.
[9] C.R. Mason, “The Art and Science of Protective Relaying,” John Wiley&Sons, Inc. New
York, 1956.
[10] B. Gradstone, “Magnetic Solutions, Solving Inrush at the Source”, Power Electronics
Technology, April 2004, pp 14-26.
[11] F. Mekic, R. Girgis, Z. Gajic, E. teNyenhuis, “Power Transformer Characteristics and
Their Effect on Protective Relays”, 33rd Western Protective Relay Conference, Oct 2006.