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Group-Delay-Controlled Multiple-Resonator-Based
Harmonic Analysis
Miodrag D. Kušljević
Termoelektro Enel AD
11060 Belgrade, Uralska 9, Serbia
e-mail: Miodrag.Kusljevic@te-enel.rs
Josif J. Tomić
Faculty of Technical Sciences
University of Novi Sad
21000 Novi Sad, Trg Dositeja Obradovića 6, Serbia
e-mail: tomicj@uns.ac.rs
Predrag D. Poljak
Department of Microelectronic Technologies
Institute of Chemistry, Technology and Metallurgy
11000 Beograd, Njegoševa 12, Serbia
e-mail: Predrag.Poljak@nanosys.ihtm.bg.ac.rs
Abstract— In this paper, a modified and improved approach
to the recently proposed multiple-resonator-based observer
structure for harmonic estimation has been proposed. In the
previous papers, two inherent particular cases have been
considered. In the first case, estimation is performed in the point
located in the middle of the observation interval, and exhibits
good noises and unwanted harmonics attenuation but possesses a
large response delay. In the second case, the estimation point is at
the end of the observation window. In this case, the filters are
able to form a zero-flat phase response about the operation
frequency and hence able to provide instantaneous estimates, but
with large overshoots caused by resonant frequencies at the edges
of the pass band, and the high level of the sidelobs. In this paper,
the estimation point is shifted along the observation interval
reshaping frequency responses to tradeoff between those opposite
requirements. The effectiveness of the proposed algorithm is
shown through simulations.
Keywords—component; Discrete Fourier transform (DFT);
group delay; harmonic analysis; finite-impulse-response (FIR)
filter; maximally flat (MF) filter; multiple-resonator (MR);
recursive algorithm; Taylor-Fourier transform (TFT).
NOMENCLATURE
kth derivative of the mth harmonic phasor at the nth
time instant.
D Prescribed group delay.
Fundamental component frequency (in hertz).
Modulating signal frequency (in hertz).
Sample rate (in hertz).
Complex gain assigned to the resonator of pole .
Phase modulation depth.
Maximum differentiator order in the cascade.
M Maximum harmonic order .
N Number of harmonic cascades (.
Coefficient assigned to the output of the kth resonator
in the cascade of the mth harmonic component.
The transfer function corresponding to the kth
differentiator ), related to the mth
harmonic component (
.
Total feedback transfer function of the harmonic
component (
!
"
#$
).
$
% Reshaped transfer function of the mth harmonic
component ($
%$
%
.
& Time (in seconds).
Sampling period (in seconds).
' Input signal to the estimator at the nth time instant.
' Output of the kth resonator in the mth cascade at the
nth time instant ('()*+.
'
Feedback signal related to the mth harmonic
component '
!'
"
#$ .
'$
% Reshaped signal related to the mth harmonic
component ,--.(-.-/0(/12-,1-'$
%
'$!'
"
# .
kth pole in the cascade of the mth harmonic
component.
Multiple pole (with K multiplicity) in the mth
harmonic component (()*+3.
Angular frequency of the fundamental component (in
rad/s).
I. INTRODUCTION
A huge volume of papers has been written on harmonics
tracking in power systems. Parametric and nonparametric
methods that commonly have been used by the community of
signal processing have been applied to power system harmonic
estimation. The challenge is producing a harmonic estimator
with high convergence ratio, high accuracy, low computational
burden, the ability to track non-stationary signals, resistance to
noise and immunity to the presence of interharmonics:
conditions that are not easy to simultaneously deal with. Good
surveys of some techniques are presented in [1] and [2]. It is
This work has been supported in part by the Ministry of Educational,
Science and Technological Development, Republic of Serbia (No. TR32019)
.
19th International Symposium POWER ELECTRONICS Ee2017, October 19-21, 2017, Novi Sad, Serbia
9781538635025/17/$31.00 ©2017 IEEE
1
important to highlight that, in practice, the harmonic analysis is
computed using a sample window that includes the reference
time. Thus, each harmonic component is computed with a
certain delay with respect to the reference instant. Many
algorithms refer the calculated value to the center of the
window, introducing thus a delay equal to a half of the length
of the time window. This delay can be a critical aspect in real-
time applications [3].
An important line of research in the smart grids context is to
identify and estimate time-varying harmonics that may appear
in the current and voltage signals, and from this information,
correct and adjust the digital algorithms that are part of
protection equipment, power quality monitors and intelligent
electronic devices (IEDs). The smart phasor measurement unit
for control (PMU/C) which from the substation can feed
accurate fundamental and low order harmonics phasors is key
component for advanced wide-area control and monitoring
applications [4]. In particular, it has recently emerged in the
literature the need for phasor measurements that can be
accurate even under dynamic conditions. The focus of recent
literature has been on preprocessing and postprocessing
methods for fixed-sample-rate algorithms surrounding a core
DFT (or similar) analysis with a fixed number of samples, e.g.,
in [5]-[7]. Idea of considering a dynamic model to better
estimate phasors/harmonics has been emerging in the last years
[8], and its importance for harmonic and/or phasor estimation
has been pointed out in [3]. In [8], the discrete Taylor–Fourier
transform (TFT), was proposed as an extension of the full DFT
and its polynomial implementation is given in [9]. It can be
seen as a FIR filter bank formed by a set of maximally flat
filters with less distortion and less interharmonic interference
than the DFT-based estimator. Each filter has a maximally flat
differentiator gain around its harmonic frequency and almost
perfect rejection (zero-flat gain) about the remaining harmonic
frequencies. When harmonics are narrow band pass signals
with spectral density confined into the flat-gain harmonic
intervals, the coefficients of the TFT provide good estimates of
the first derivatives of their complex envelopes.
The new standard C37.118.1 for synchrophasor
measurements [10] has pointed out the importance of dynamic
conditions for synchrophasor estimation. A new dynamic
definition for phasors has been adopted, considering phasor
amplitude and phase as functions of time. The dynamic phasor
can thus be seen as a narrow-band signal around the
fundamental frequency, thus keeping into account amplitude,
phase and frequency variations that can occur in power
systems. The standard gives accuracy limits for the estimation
algorithms under different test conditions. In [11],
synchrophasor algorithms approximating C37.118.1 filtering
requirements are extended to provide phasor measurement units
(PMUs) with the capability of accurately tracking single-phase
harmonic phasors subject to varying nominal frequency and
out-of-band interharmonic interference.
A new approach to the harmonic estimation in dynamic
conditions based on the multiple-resonator (MR) structure
previously having been introduced in [12], has been proposed
in [13]. With an interior grid of resonators output taps, this
structure represents itself as an excellent platform for
estimation of not only the complex transfer values but also,
according to the actual resonator multiplicity, of their first,
second, etc. derivatives at the corresponding frequencies. In
[14], the proposed algorithm for the harmonic analysis uses a
quasi- instead of the true MR-based computation technique.
This approximation significantly eases design by utilizing of
Fig. 1 Block diagram of the K-type MR-based harmonic analyzer.
19th International Symposium POWER ELECTRONICS Ee2017, October 19-21, 2017, Novi Sad, Serbia
2
the classical Lagrange instead of Hermite interpolation
technique, while the frequency responses of the estimators FIR
filters are insignificantly reshaped compared to the true MR-
based ones. The closed-form solution for resonator gains for a
general case is given.
In the proposed estimation structure, two inherent particular
cases have been considered. The first case, which uses an
output of the last resonator in the cascade, exhibits both good
noises and unwanted harmonics attenuation but possesses a
large response delay. For the second one, which is formed as a
sum of the outputs from all resonators in the cascade and used
for a feedback, the filters are able to form a zero-flat phase
response about the operation frequency and hence they are able
to provide instantaneous estimates but with large overshoots
caused by resonant frequencies at the edges of the passband.
In order to avoid aforementioned drawbacks, a linear
combination of the resonator outputs, instead of their sum, is
used in [15]. The maximally flat (MF) conditions are imposed
to the filter polynomial transfer functions to reshape frequency
responses avoiding resonant frequency peaks. The estimation
point is placed in the middle of observation interval; a group
delay is present and the zero–phase–flatness is not more
preserved.
In Section II, the filters frequency responses of the MR-
based estimator are reshaped to avoid large gains in the
subharmonic frequencies at the edges of the passband, and to
provide simultaneously a minimized group delay. The idea is,
likewise in [15], to use a linear combination of the outputs of
the resonators instead of their sum. However, in this paper, a
different design criteria is considered. In [15], only an
amplitude frequency response is considered and MF properties
are provided. In this paper, both amplitude and phase frequency
responses in the passband area are considered. The objective is
to control the group delay of the transfer function in the center
of the passband subject to the controlled values of resonant
peaks at the edges of the passband. Consequently, the resulting
performances of the estimation algorithm are between to the
best and worst case obtained by other solutions. In addition to
that, it is possible to obtain a filter bank for a set of different
linear combinations corresponding to different signal dynamics.
II. RESHAPING OF THE FREQUENCY RESPONSES
The block diagram of the K-type MR-based harmonic
analyzer is shown in Fig. 1. The cascade in the mth channel
consists of K+1 resonators with poles distributed closely (not
necessarily equidistantly in a general case) on the unit circle
around the pole related to the mth harmonic frequency. In this
paper, a true MR-based estimator is considered (that means
that all poles in the cascade are placed in the single point,
3 ). In this section, reshaping of the
filters for the zeroth- and the first order derivatives of the
harmonics for estimator (the third-order resonator
structure) is demonstrated. The overall system order is 4,
where ;
, is an angular
frequency of the fundamental component and M is number of
harmonics to be analyzed. The 4 estimators with four-
order resonators are also investigated, but developed equations
are not given due to the shortage of the place. The design for
the higher multiplicity cases is straightforward, only difference
is that the higher order derivatives are also to be considered.
Frequency responses of zeroth-, first- and second-order
discrete FIR filters corresponding to the transfer functions
related to the fundamental
component and the fifth harmonic (3 and 5,
3), for (the third-order resonator structure) are
shown in Fig. 2.
It can be shown that the first and second derivatives of the
transfer function
$
6 in the pole ()*+ are equal to zero, which
means that
is MF, (see Fig. 3). The main feature of the
filter
is in its phase response. It has null phase in the
interval around the mth harmonic frequency which means that
the harmonic phasor estimates of this filter are instantaneous,
i.e. without any delay. On the other hand, the high gains at
resonance frequencies at the edges of the passband and
interharmonic gains of the sidelobes are notable in the
amplitude frequency response.
In order to adapt the achieved digital differentiators to their
ideal frequency responses around the harmonic frequencies, a
linear combination of the resonator outputs $
%
$66, instead of their sum
, is used.
Fig. 2 Magnitude f
requency responses of the zeroth, first and second order
derivatives (transfer functions $,, and 6,
respectively)
related to the fundamental component (3
) and the fifth harmonic
(5) for 578 and 978.
Fig. 3 Frequency responses of
$
,
and
related to the fundamental
component, for 578 and 978, for estimator (the third-
order resonator structure).
19th International Symposium POWER ELECTRONICS Ee2017, October 19-21, 2017, Novi Sad, Serbia
3
For dead-beat estimators we have:
$$
:;6<
;= >?:;6<
;=>?
>?@$
(1)
:6A<
;=
>?:6A<
;=>? >?@
(2)
66
:A6<
;=
>?:A6<
;=>? >?@6
(3)
where
@$:;6<
;=>?
@:6A<
;=>?
@6:A6<
;=>? B
(4)
<C AD;
E
D#>E
DF D()*+D, DG.
: 6, :6 6 and :; $6.
It follows that
$
%$66>?@$
%
(5)
where
@$
%@$@6@6
H:;6:6A6:A6I<
;=>?
(6)
Transfer function @$
% is not causal which means that
can not be implemented real time since needs advance J
samples which are not available in the estimation time instant.
However, it can be used for estimation of the harmonic
coefficients referred to the time instant AJ) postpone in
the instant , with a latency of J samples. It is obvious that
K$
%KK@$
%K. The amplitude frequency responses of
the transfer functions are equal but the phase responses are
different. Item >? introduces a group delay of J samples and
the zero–phase–flatness is not more preserved. Hence, while
@$
% has a total flatness of both amplitude and phase
frequency responses, $
% preserves the flatness of the
amplitude response only.
It follows that for the first derivative is
L@$
%
L
MH:;:6A6:AI<
H:;6:6A6:A6INO
;=>?
(7)
where
NL<
L A4AJ<
From L@$
%L
KP#PQ, it follows that:
A;RN
<S
(8)
Further, it follows for the second derivative:
L6@$
%
L6
TH:;:66:I<
H:;:6A6:AIN
H:;6:6A6:A6IUV
4AJ
(9)
where
ULN
L A4AJN
LN
L L6<
L6A4AJL<LA<6
From L6@$
%L6
KP#PQ, the following closed-form
solution is derived:
6 A6
W
X
X
Y
Z;N
;U[
<;
\
]
]
^
(10)
Similarly, a MF first differentiator is
@
%_@_6@6
H_:6A_6:A6I<
;=>?
(11)
From L@
%L
KP#PQ3 and L6@
%L6
KP#PQ it
follows that:
_ ;=>?
:6<;>?
(12)
_6 A_6R3N
<S
(13)
19th International Symposium POWER ELECTRONICS Ee2017, October 19-21, 2017, Novi Sad, Serbia
4
III. FREQUENCY VARIATIONS TRACKING
For frequency excursions outside the ideal bands, the
relocation of the MF bands of the filters at each iteration can be
necessary. If an extern reference signal is available to directly
control the resonator pole positions, then center frequencies of
the measuring channels can be matched with that of the signal
components. If there is no available reference signal, since the
TFT filters provide instantaneous frequency estimates [9], they
can be used to relocate the filter bands. Then, the concept of the
Adaptive Fourier Analyzer (AFA) can be used and generalized
[12]. The convergence analysis of this AFA, even for single-
resonator structure, is extremely difficult, and until now,
theoretical results have not been available. In [16], a modified
version of the original AFA, the concept of a so-called block
AFA (BAFA), is introduced, which avoids the transient-
induced disturbance and therefore makes the convergence
property investigation much easier. For the case of the BAFA,
the conditions of a global convergence have been derived. Its
only inconvenience is the long-lasting transient stage, which is
unacceptable for some real-time applications.
Since the aforementioned convergence problems in the
recursive adaptation of a resonator-based structure are present,
an external decoupled module for frequency estimation has
been applied in [17] and has shown good dynamic properties.
Good properties are achieved by cascade connection of the
antialiasing low-pass filters with the cutoff frequency that is
high enough to ensure a fast response (up to 8–16 harmonics)
and the reduced-order FIR comb filter that is applied on the
decimated signal. This way, both coefficient sensitivity
problems are avoided, and the frequency estimation accuracy is
improved. The method proposed in [9] can also be successfully
applied to the extern frequency estimation module
implementation.
IV. DESIGN EXAMPLE
Table I shows the coefficients values for the third order of
resonator multiplicity () for J4
, J4 `
, J
4 a
and J4 9
, with 578 and 978. It is
important to mention that in the case of the synchronous
(coherent) sampling conditions, i.e., when the sampling
frequency is a multiple of the fundamental frequency, it is
enough to perform the calculation of parameters for only one
harmonic. It means that parameters having been designed for
one harmonic are valid also for other ones.
Fig. 4 shows the amplitude and phase responses of the
fundamental component transfer function $
% for different
values of D. Note that a maximally flatness of the amplitude
responses around the actual harmonic (with KL$
%L
K
and KL6$
%L6
K for ) is present in all cases,
but gains in the resonant frequencies at the edges of the
bandpass and a level of sidelobes increase with decreasing of
the delay D. On the other hand, from the phase responses it is
visible that the group delay decreases with the decrease of D.
This way, $
% trades-off between high resonant gains and
small group delays around the actual harmonic frequency. For
J4 `
, there are no resonant peaks of the amplitude
response, similarly as for J4
, but the group delay is a
half of one obtained by J4
. This feature can be
important in the control applications.
In view of brevity, only the responses for fundamental
component (3) are given. Responses for other harmonics
have the same shapes.
TABLE I. CO EFFICIENTS VALUES FOR A DIFFERENT D
J
6
J
4
24 -0.0213 -0.1250
J
4
`
12 0.4894 0.1562
J
4
a
8 0.6596 0.3750
J
4
9
6 0.7447 0.5078
V. SIMULATION RESULTS
The estimates of the amplitude, the phase, and the total
vector error (TVE) in percentage, obtained by the proposed
estimation structure for the third order of resonator
multiplicity () for J4
, J4 `
, J4 a
and J4 9
, with 578 and 978, are shown
in Figs. 5 and 6, only for illustration purposes. Similar results
are obtained in cases of amplitude changes. In view of brevity,
only the estimates for fundamental component (3) are
given. Responses for other harmonics have the same shapes.
At first, the modulation test is applied to reflect the
variations in phase angle of the voltage waveforms that occur
during a stable power swing. The test signal proposed by the
standard is:
b&cde2H&de2&AI (14)
where cis the signal basic amplitude, is the phase
modulation depth, and is the modulating signal frequency.
Fig. 5 Estimation of a) amplitude, b) phase, and c) TVE in percentage,
corresponding to the modulated signal with 5f, obtained by
type estimator for J4
, J4 `
, J4 a
and J4 9
with
578 and 9Hz.
Fig. 4 Frequency responses of the zeroth-order
type differentiato
r
related to the fundamental component (3) for 578 and
9
78, for J4
, J4 `
, J4 a
and J4 9
.
19th International Symposium POWER ELECTRONICS Ee2017, October 19-21, 2017, Novi Sad, Serbia
5
The considered phase angle modulation depth is B3g,h,
the modulation frequency is 578, c3_BiB.
In Fig. 5, it is notable that the estimates obtained by J
4
are unable to suppress the systematic delay. In this case
the TVE value is about 10%. It is notable that the limit of 3%
for TVE is respected when the estimate is obtained by $
%
for J4 a
. For J4 `
, the maximum TVE is about
5%, a half of J4
case. It is also visible from the upper
diagram that the response delay is a half of one obtained by
J4
.
In the second example, a signal with the 10° phase angle
step is fed to the algorithm. The test signal can be represented
as follows:
b&cde2H&i&A&I (15)
where
( )
u t
is the unit step function and
s
t
is the step instant.
The phase step is generated with 39
(10° phase angle
step). The response times of filters are three cycles. In
addition, it should be noticed that the overshoots of the
estimates obtained for J4 a
and J4 9
, caused by
large peaks at the resonant frequencies, are very large and do
not satisfy the standard requirements. Estimate obtained in the
case J4 `
provides a much lower overshoot of about
10%, Fig. 6.
It is obvious that the decrease of J provides faster estimates
and a lower group delay. However, the obtained overshoots for
step changed input responses are much larger. And vice versa,
if J increases, the lower overshoots for the step changed input
responses are provided, but with the higher response time and
group delay. Jjk3 `
can be considered as an
optimal (compromised) solution. Alternatively, it is possible to
obtain a filter bank, surrounding the core MR structure, for a
set of different linear combinations corresponding to different
signal dynamics.
VI. CONCLUSION
The proposed algorithm for harmonic analysis uses a
modified recursive MR-based computation technique which
provides MF frequency responses. For this purpose, the
frequency responses around the harmonic frequencies are
reshaped providing the resonant frequencies at the passband to
be avoided and the interharmonic gains to be reduced. The
simulations results have demonstrated that the resulting
performances of this estimation algorithm are between the best
and worst results obtained by two particular cases having been
investigated in the previous papers. Thus, estimates are
simultaneously enough accurate and resistant to interharmonics
and noise presence, and still fast to allow tracking of the
nonstationary signal content. Due to its parallel form, recursive
calculation, and high accuracy, the proposed algorithm can be
very useful for real-time digital systems. The proposed method
has been investigated for up to 64 harmonics, under different
conditions, and found to be a valuable and efficient tool for
detection of signal components.
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Fig. 6
Estimation of a) amplitude, b) phase, and c) TVE in percentage,
corresponding to the phase angle step signal, obtained by for J
4
, J4 `
, J4 a
and J4 9
with 578 and
9
78.
19th International Symposium POWER ELECTRONICS Ee2017, October 19-21, 2017, Novi Sad, Serbia
6
