Chapter 26
Time-Varying Power Harmonic
Decomposition using Sliding-Window
DFT
P.M. Silveira, C. Duque, T. Baldwin, P. F. Ribeiro
26. 1 Introduction
While estimation techniques are concerned with the process used to extract useful
information from a signal, such as amplitude, phase and frequency; signal decomposition is
concerned with the way that the original signal can be split in individual components, such as
harmonics, interharmonics, sub-harmonics, etc.
This chapter presents a time-varying harmonic decomposition using sliding-window Discrete
Fourier Transform. Despite the fact that the frequency response of the method is similar to the
Short Time Fourier Transform, with the same inherent limitation for asynchronous sampling
rate and interharmonic presence, the proposed implementation is very efficient and helpful to
track time-varying power harmonic. The new tool allows 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 has also the potential to assist with control and
protection applications.
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 individual harmonic components from the
input signal can be achieved, the estimation task becomes easier; (b) the decomposition is
carried out in the time-domain, such that the time-varying behavior of each harmonic
component is observable.
Several techniques can be used to separate frequency components. For example, Short Time
Fourier Transform (STFT) and Wavelets Transforms [1], are two well-known decomposition
techniques. Both can be seen as a particular case of filter bank theory [2].
When the fundamental frequency is time varying and the sampling frequency is not
synchronous other techniques must be used in order to obtain a good harmonic
decomposition. For example, the adaptive notch filter and the Phase-Locked Loop (PLL) [3]-[4]
have been used for extracting time-varying harmonics components. In [5] the authors
presented a similar approach to [3] and [4], where an adaptive filter-bank, labeled as resonatorin-a-loop filter bank, was used to track and estimate voltage or current harmonic signal in the
power system. In [6] the authors presented a technique based on multistage implementation of
narrow low-pass digital filters to extract stationary harmonic components. The technique
utilizes multirate approach for the filter implementation and needs to know the system
frequency for the modulation stage.
Attempts to visualize time-varying harmonics using Wavelet Transform have been proposed
in [7] and [8]. However, the structure was not able to decouple the frequencies completely. In
[9] the authors presented a new methodology to separate the harmonic components using
multirate and filter bank approach. The method is able to track time-varying power harmonic
frequencies without frequency spillover.
However if the fundamental frequency is supposed to be constant and synchronous sampling
frequency is assumed, and only harmonic component is present in the signal then the STFT is
better fitted for harmonic separation.
The STFT uses filters with coefficients that are complex numbers, which generates a complex
output signal, whose magnitude corresponds to the amplitude of the harmonic component into
the band. If a rectangular window is used in the STFT a low computation burden recursive
algorithm can be employed. This algorithm is well known as Recursive DFT or as Sliding Window
DFT [10], [11].
This chapter presents a new structure for time varying harmonic decomposition using the
Sliding window DFT and an efficient digital sinusoidal generator to reconstruct each harmonic.
The advantage of this method compared to a previous one [9] is the low computational effort,
no phase delay and a single cycle of transient time.
26. 2 Discrete STFT
Given a signal x(n) , the discrete STFT for harmonic h at time n is defined as [2],
X STFT (e jwh , n) 
 x(k ). (n  k )e
k
 jwh k
(26.1)
where,  (n) is a suitably chosen window function (e.g., a rectangular window) of size L and
wh 
2h
,
N
h  0,1, 2,
N -1
(26.2)
is the digital harmonic frequency in radian, and N is the total number of harmonics. The digital
harmonic frequency is related with the real frequency (rad/sec) by the following expression,
h 
wh . f s
2
where fs is the sampling frequency.
(26.3)
Equation (26.1) can be rewrite according to the following formula, for which a graphical
representation can be seen in Fig. 26.1,
X STFT (e jwh , n)  e  jwhn
 x(k ). (n  k )e
jhwh ( nk )
k
(26.4)
Figure 26.1- Filtering interpretation of the STFT
According to the Fig. 26.1, the STFT can be interpreted as a convolution of the input signal
with the impulse response hh(n) of a hth complex bandpass filter followed by a modulation,
which is accomplished by an exponential signal. If the window  (n) is a real function, the
impulse response of the filter will be a complex number. The modulating signal, after the filter
stage, shifts the resultant spectrum to the left side by an amount of wk. Figure 26.2 illustrates
how the STFT works when a signal x(n)  A cos(w1n   ) is injected at the filter input. Figure 26.2.a
jw
shows the spectrum of x(n) and the filter H1 (e ) . It is important to remark that the filter is not
symmetric, since its coefficients are not real. Figure 26.2.b shows the filtered signal, which is a
 jnwh
complex exponential. Then the modulation by e
shift left the spectrum in Fig.26.2b,
translating it to zero frequency. The amplitude and phase can be computed from the
modulated signal taking the module and the angle of it.
If a rectangular window is chosen to perform the STFT, the computational effort can be
drastically reduced by using the recursive formulas to calculate the DFT [10] and [11].
In the theory of Fourier series, it is well known the equation (26.5), for real periodic signals.
But another commonly encountered form is the rectangular one, given by (26.6).
x(n)  a0  2 Ah .cos(wh n  h )
x(n)  a0  2YCh (n).cos(wh n)  YSh (n).sin( wh n)
(26.5)
(26.6)
Figure26.2- The complex exponential generated in the STFT.
In being so, the rectangular (quadrature) terms YCh(n) and YSh(n) can be obtained by using
the structure shown in Fig. 26.3. The factor cos(wh n) is used to obtain the term YCh(n) and
 sin(wh n) to the term YSh(n).
Figure 26.3- Recursive filter to compute the quadrature term YCh(n)
Based on this structure, the recursive equation can be easily written as:
YCh (n)  YCh (n 1)   x(n)  x(n  N )  .cos(whn)
(26.7.a)
YSh (n)  YSh (n 1)   x(n)  x(n  N )  .sin(wh n)
(26.7.b)
26.3 The decomposition structure
Normally, the DFT recursive algorithm is used to extract and compute the amplitude and phase
of the fundamental and other harmonics components [10], but not the waveform.
Nevertheless, the main point of this work is, in fact, to obtain the fundamental waveform, as
well as each individual harmonic. This task can be performed using the own equation (26.6). For
real time implementation this can be obtained by two methods: (a) Table searching or (b) using
a digital sine-cosine generator. The second approach is more effective and has been used to
decompose and analyze some signals from power systems events.
A digital sine-cosine generator is presented in [11]. It can be implemented using the following
state equations:
cos(wh )  1  s1 (n  1) 
 s1 (n)   cos(wh )
s (n)  cos(w )  1 cos(w ) .s (n  1)
h
h
 2  

 2
(26.8)
where s1(n) is a sine function and s2(n) is a cosine function, and the initial states must be
correctly set.
The advantage of this approach is that the sine and cosine signal can be used at the same
time in both, decomposition and reconstruction tasks.
Figure 26.4 presents the block diagram of the core structure that extracts the hth harmonic.
For extracting N harmonics it is necessary to employ N cores as shown in Fig.26.4. The
advantage of this structure can be summarized as following:
i) low computational effort, suitable for real time decomposition implementation;
ii) no phase delay;
iii) transient time of only one cycle.
The disadvantage of this structure comes from the limitation of the DFT:
i)
ii)
a synchronous sampling is needed;
interharmonics produce estimation error of the quadrature components.
Figure26.4- The core structure for extracting the hth harmonic.
26. 4 Simulation results
In order to present some results of tracking time-varying harmonics using the structure
shown in Fig. 26.4, two examples are considered: (a) a synthetic signal, which has been
generated in Matlab using a mathematical model, and (b) a signal obtained from the
“Electromagnetic Transient Program including DC systems” (EMTDC) with its graphical interface
known as “Power Systems Computer Aided Design” (PSCAD). This program can simulate
power systems with high fidelity and the resulting signals of interest are very close to physical
reality.
26.4.1 Synthetic Signal
The synthetic signal utilized can be represented by:
N
x(t )   Ah sin(h 0t). f (t )  g (t )
h 1
(26.8)
Where h is the order (1st 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 portioned 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 26.5 illustrates the synthetic signal.
Figure. 26.5 – Synthetic signal used.
The structure composes of 16 cores like Fig. 26.4 has been used to decompose the signal into
sixteen different harmonic orders, including the fundamental (60 Hz) and the DC component.
Figure 26.6 shows the decomposed signal from 4th up to 7th harmonic components. 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 other
components are not shown. However, it is important to remark that all waveforms of the timevarying harmonics are extracted with efficiency along the time axis.
Naturally, intrinsic transients will be present during the transitions from previous to the new
state. Figure 26.7 shows both the original and the estimated components (DC to 3rd harmonic)
in a short time scale interval. The transient of one cycle is shown in each output.
26.4. 2 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 26.1 shows the
typical harmonic components present in the inrush currents [12]. 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 26.1- typical harmonic content of the inrush current
Order
Dc
2
3
4
5
6
7
Content
%
55
63
26.8
5.1
4.1
3.7
2.4
DC component
DC component
2
2
1
1
0
0
1
2
1st
3
4
1
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
0
0
1
2
3rd
3
4
1
-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
1
2
7th
3
4
0.5
0.5
0
0
0
1
2
Time s
8th
3
4
1
-0.5
0
0
-1
0
1
2
9th
3
4
0.2
0
-0.2
3
4
0
1
2
6th
3
4
0
1
2
7th
3
4
0
1
2
Time s
8th
3
4
0
1
2
9th
3
4
0
1
2
10th
3
4
0
1
2
11th
3
4
0
1
2
Time s
3
4
0.2
0
-0.2
0
1
2
10th
3
4
1
1
0
0
0
1
2
11th
3
4
-1
0.5
0.5
0
0
-0.5
2
5th
1
-1
-1
1
0.2
0
-0.2
0
-0.5
0
0
1
2
Time s
3
4
-0.5
Figure 26.6 – 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 26.7 – Comparing original and decomposed component.
In recent years, improvements in materials and transformer design have lead to inrush
currents with lower distortion content [13]. The magnitude of the second harmonic, for
example, has dropped to approximately 7% depending on the design [14]. 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.26.8.
Figure 26.8 – Inrush current in phase A.
Using the methodology proposed to visualize the inrush current, the Fig.26.9 reveals the
rarely seen time-varying behavior of the waveform of each harmonic component. This could be
used to understand other physical aspects not observed previously. In Fig. 26.9, the left column
shows the DC and even components and the right column the odd components.
Figure 26.9 – Decomposition of the simulated transformer inrush current.
26.5 Conclusions
This chapter presents a method for time-varying harmonic decomposition based on sliding
window DFT. The technique is able to extract each harmonic in the time domain. The
methodology has the advantages of low computation burden, no phase delay and a short
transient time and can be used as a useful tool for real time applications. The disadvantages are
inherent to all DTF based algorithms, i.e., they need synchronous sampling and are influenced
by the presence of interharmonics. However some strategies can be used to guarantee
synchronization, such as the use of the Phase Looked Loop (PLL) to estimate the fundamental
frequency. The influence of the interharmonics can be minimized through the choice of other
windows, with smaller Main Lobe width and higher Side-Lobe Attenuation. The cost to be paid
is the increasing of the computational effort.
26. 6 References
[1] Yuhua Gu, M. H. J. Bollen, “Time-Frequency and Time-Scale Domain Analysis,” IEEE Trans.
on Power Delivery, Vol. 15, No. 4, Oct. 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 TimeVarying Harmonic based on an Adaptive Notch Filter,” Proc. of IEEE Conference on Control
Applications, Toronto, Canada, Aug. 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] H. Sun, G. H. Allen, and G. D. Cain, “A new filter-bank configuration for harmonic
measurement,” IEEE Trans. on Instrumentation and Measurement, Vol. 45, No. 3, June
1996, pp. 739-744.
[6] C.-L. Lu, “Application of DFT filter bank to power frequency harmonic measurement,” IEE
Proc. Gener. , Transm,. Distrib., Vol 152, No. 1, Jan. 2005, pp. 132-136.
[7] 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, Aug.
2007, Brazil.
[8] V.L. Pham and K. P. Wong, “Antidistortion method for wavelet transform filter banks and
nonstationay power system waveform harmonic analysis,” IEE Proc. Gener., Transm.,
Distrib., Vol 148, No. 2, March 2001, pp. 117-122.
[9] C. A. Duque, P. M. Silveira, T. Baldwin, and P. F. Ribeiro, “Novel method for tracking timevarying power power harmonic distortion without frequency spillover,” IEEE 2008 PES, July
2008, Pittsburgh, PA, USA.
[10] Sanjit K. Mitra, Digital Signal Processing – A computer-based approach, Mc-Graw Hill 2006,
3ª Edition.
[11] R. Hartley, K. Welles, “Recursive Computation of the Fourier Transform", IEEE Int.
Symposium on Circuits and Systems, Vol.3, 1990. pp. 1792 -1795.
[12] C.R. Mason, The Art and Science of Protective Relaying, John Wiley & Sons, Inc. New York,
1956.
[13] B. Gradstone, “Magnetic Solutions, Solving Inrush at the Source”, Power Electronics
Technology, April 2004, pp 14-26.
[14] F. Mekic, R. Girgis, Z. Gajic, E. teNyenhuis, “Power Transformer Characteristics and Their
Effect on Protective Relays”, 33rd Western Protective Relay Conference, Oct. 2006.