IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 4, OCTOBER 2009 1789
A PLL-Based Multirate Structure for Time-Varying
Power Systems Harmonic/Interharmonic Estimation
Janison Rodrigues de Carvalho, Carlos A. Duque, Member, IEEE, Moisés V. Ribeiro, Member, IEEE,
Augusto S. Cerqueira, Thomas L. Baldwin, Senior Member, IEEE, and Paulo F. Ribeiro, Fellow, IEEE
Abstract—This paper describes a phase-locked-loop (PLL)-
based power systems harmonic estimation algorithm, which uses
an analysis filter bank and multirate processing. The filter bank
is composed of bandpass filters. The initial center frequency of
each filter is purposely chosen to be equal to harmonic frequen-
cies. However, an adaptation strategy makes it possible to track
time-varying frequencies as well as interharmonic components.
A downsampler device follows the filtering stage, reducing the
computational burden, especially because undersampling opera-
tions are performed. Finally, the last stage is composed of a PLL
estimator which provides estimates for amplitude, phase, and
frequency of the input signal. The proposed method improves
the accuracy, computational effort, and convergence time of the
previous harmonic estimator based on cascade PLL configuration.
Index Terms—Multirate signal processing, phase-locked loop
(PLL), time-varying harmonic/interharmonic estimation.
I. INTRODUCTION
W
ITH the increased application of power electronics,
controllers, motor drives, inverters, and flexible ac
transmission systems (FACTS) devices in modern power
systems, distortions in line voltage and current have been
increasing significantly. These distortions have affected the
power quality (PQ) of the power system and to maintain it
under control, the monitoring of harmonic and interharmonic
distortion is an important issue [1]–[3].
The discrete Fourier transform (DFT) is a suitable approach
to estimate the spectral content of a stationary signal. However,
it loses accuracy under time-varying conditions [4]. As a result,
other algorithms must be used. The short-time Fourier transform
(STFT) can partly deal with time-varying conditions but it has
the limitation of a fixed window width chosen
a priori and this
imposes limitation for the analysis of low-frequency and high-
frequency nonstationary signals at the same time [5].
Manuscript received November 12, 2007; revised September 07, 2008. Cur-
rent version published September 23, 2009. This work was supported in part by
CAPES and in part by CNPq from Brazil. Paper no. TPWRD-00682-2007.
J. R. de Carvalho, M. V. Ribeiro, and A. S. Cerqueira are with the Elec-
trical Engineering Department, Juiz de Fora MG 36036-330, Brazil (e-mail:
janison@sel.eesc.usp.br).
C. A. Duque is with the Engineering Faculty at Federal University of Juiz
de Fora (UFJF), Juiz de Fora, Brazil 36036-330, and also with the Center for
Advanced Power Systems/Florida State University, Tallahassee, FL 32306 USA
(e-mail: carlos.duque@ufjf.edu.br).
T. L. Baldwin is with the Center for Advanced Power Systems/Florida State
University, Tallahassee, FL 32306 USA (e-mail: tom.baldwin@ieee.org).
P. F. Ribeiro is with the Center for Advanced Power Systems/Florida State
University, Tallahassee, FL 32306 USA, and also with Calvin College, Grand
Rapids, MI 49546 USA (e-mail: pfribeiro@ieee.org).
Digital Object Identifier 10.1109/TPWRD.2009.2027474
The IEC standard drafts [6] have specified signal-processing
recommendations and definitions to harmonic and interhar-
monic measurement. These recommendations utilize the DFT
over a rectangular window of exactly 12 cycles for 60 Hz (10
cycles for 50 Hz) and a frequency resolution of 5 Hz. However,
different authors [7], [8] have shown that the detection and
measurement of interharmonics, with acceptable accuracy, are
difficult to obtain by using the IEC specification.
Unlike the previous methods that follow the IEC standard,
other techniques based on the Kalman filter, adaptive notch
filter, or PLL approaches have been applied in harmonic
and interharmonic estimation. The main disadvantage of the
Kalman filter is the higher order model required to estimate
several components [9]. In [3], the enhanced phase-locked
loop (EPLL) [2] is used as the basic structure for harmonic
and interharmonic estimation, and several of these sections are
arranged together. Each one is adjusted to estimate a single
sinusoid waveform. The convergence takes about 18 cycles,
but it can take more than 100 cycles for higher harmonic fre-
quencies, mostly if there is a fundamental frequency deviation.
In [10], a new multirate filter bank structure for harmonic and
interharmonic extraction is presented. The method uses EPLL
as an estimation tool in combination with sharp bandpass filters
and downsampler devices. As a result, an enhanced and low
computational complexity method for parameters estimation of
time-varying frequency signals is obtained.
This paper presents a new version of the proposed method
in [10]. The main contribution of this new version is concerned
with the new concept of apparent frequency and the undersam-
pling. The undersampling is deliberately used in order to re-
duce the computational effort of the overall estimator, while the
signal reconstruction remains possible. Furthermore, the con-
vergence time of the parameter estimator is reduced. Finally, but
not least, the new approach leads to better structures to be imple-
mented in the fixed-point arithmetic processor. This is because
the downsampler feeds the PLL estimator with a fake signal
whose frequency is close to 60 Hz. As the PLL parameters are a
function of the signal frequency to be estimated, all PLL cores
will have a similar structure that can be optimized to be imple-
mented in fixed-point processors. Implementation in fixed-point
processors means lower cost of the final product.
This paper is organized as follows. Section II presents some
concepts about the digital filter bank. Section III describes the
multirate processing, the concepts of undersampling, and ap-
parent frequency. Section IV describes the proposed structure
and presents the recursive equations of the EPLL method [2].
Section V presents two structures of harmonic estimation for
0885-8977/$26.00 © 2009 IEEE
1790 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 4, OCTOBER 2009
Fig. 1. Typical analysis filter bank.
comparison. The simulation results are shown and discussed in
Section VI. Finally, conclusions are stated in Section VII.
II. D
IGITAL
FILTER BANK
A digital filter bank [11] is a collection of digital bandpass fil-
ters with either a common input (the analysis bank) or a summed
output (the synthesis bank). The analysis bank is the focus of
this section.
The analysis filter bank decomposes the input signal
into
a set of
subband signals , each one
occupying a portion of the original frequency band. Fig. 1 shows
a typical analysis filter bank.
In this paper, the filter bank differs from traditional filter
banks found in the literature [11], [12]. This is due to the filters
employed here. Basically, they are conventional parametric IIR
bandpass filters [12] given by
(1)
The main objective of using the bandpass filters consists of
extracting each of the components, fundamental and harmonics
(or interharmonics), normally found in power systems. Another
objective is to reduce the noise level at the input of the estimation
stage. This filter was chosen, in this design, for the following
reasons:
• it is a parametric filter, which means that the central fre-
quency may be adjusted easily by changing the
factor in
(1); treating the
factor as an input parameter provides a
quick way to set the central frequency in an adaptive algo-
rithm;
• at the central frequency, the phase shift induced by the filter
is zero; by tuning the central frequency of the IIR filter to
the desired bandpass frequency, the signal of interest will
not be shifted in phase;
• despite the low order of the IIR bandpass filter (it can used
in cascade configuration), the frequency response has an
appropriate characteristic.
The magnitude responses of (1) are plotted in Fig. 2(a) for the
bandpass centered at a fundamental frequency (60 Hz) and some
of its odd harmonics. Solid lines are for the single second-order
filter of (1) while dashed lines show the response for a cascade
structure of two second-order filters. The phase response of the
filter centered at the fundamental component filter is detailed in
Fig. 2(b). Notice the zero-phase response at the frequency 60
Hz mentioned previously.
The previous transfer function has a narrow bandwidth when
the poles are a pair of complex conjugates near the unit circle.
Fig. 2. Frequency response: (a) Magnitude responses of parametric bandpass
filters of (1). (b) Phase response for the filter centered at the fundamental fre-
quency.
Fig. 3. Block diagram representations of sampling rate alteration devices. (a)
The downsampler. (b) The upsampler.
The parameter controls this proximity defining the 3 dB-band-
with of the filter. The maximum magnitude value occurs at dis-
crete frequency
which is related to by the expression
.
Although the parameter
near the unit produces a sharper
magnitude response, it increases the transient time response.
This fact is very important because the convergence time of the
estimator is proportional to the duration of the transient response
caused by the filter. For example, using (1) with
0.98 and
an input signal of 60 Hz with 128 points per cycle, the transient
virtually decays at about four cycles.
III. M
ULTIRATE PROCESSING
The two basic components in sampling rate modification are:
1) the downsampler, to reduce the sampling rate; and 2) the up-
sampler, to increase the sampling rate [12]. The block diagram
representation of these two components is shown in Fig. 3. The
downsampler will be described in this section.
A downsampler with a downsampling factor
, where is
a positive integer, creates an output sequence
with a sam-
pling rate
times smaller than the sampling rate of the input
sequence
.
The input–output relationship can be written as
(2)
In frequency domain, it can be shown that [12]
DE CARVALHO et al.: PLL-BASED MULTIRATE STRUCTURE 1791
Fig. 4. Alternative interpretation of the downsample effect in a single sinu-
soidal signal. (a) Original position of the component with frequency
f
. (b) Po-
sition of the component after downsampling
<
. (c) Position of the com-
ponent after downsampling
<
<
2
. (d) Position of the component after
downsampling
2
<
<
3
.
(3)
Equation (3) implies that the DTFT of the downsampled
output signal
is a sum of uniformly shifted and stretched
versions of DTFT of input
, scaled by a factor .
The direct application of (3) is sometimes unnecessary, es-
pecially with the assumption of a single sinusoidal component
at the downsampler input. A simple method can be applied al-
ternatively, as shown in Fig. 4. Fig. 4(a) shows a circle where
a sinusoidal component of frequency
is correctly placed with
the angle
(in radians) given by
(4)
where
is the sampling frequency. After downsampling with
a factor
, the output component exhibits a new angle position
, as shown in Fig. 4(b)–(d).
There are some singularities to be considered when a sinu-
soid is downsampled. It is clear that undersampling can be per-
formed. As a result, the frequency of the downsampled sequence
can be different (lower) from the frequency of the input se-
quence. This new frequency will be referred to as the apparent
frequency. We can resume singularities as:
Case 1) The new angle
is lower than radians. This case
is shown in Fig. 4(b). The output signal represents
the original signal with frequency
, that is, no un-
dersampling was performed.
Case 2) The new angle
falls into the lower semicircle, as
shown in Fig. 4(c). The output signal was obtained
by undersampling the input and the mirror compo-
nent, in the dashed line, has to be considered. The
apparent frequency
is obtained by analyzing the
angle
(5)
Case 3) The new angle
falls into the upper semicircle
after one revolution, as shown in Fig. 4(d). Under-
sampling was performed again. The apparent fre-
quency
is obtained by analyzing the angle
(6)
If the number of revolutions is greater than 1, they must be
discounted and then case 2 or case 3 can be applied.
The common sense indicates that undersampling (when the
sampling frequency is lower than the Nyquist rate) is a pro-
hibitive operation in signal processing, because the aliasing phe-
nomena [12] deforms the signal frequency spectrum. However,
there are at least two applications where undersampling is ap-
plied and the original signal can be reconstructed without error:
1) in filter bank theory and 2) single sinusoid signal. In the first
case, the aliasing phenomena in the filter bank stage is avoided
by the proper design of the analysis and synthesis filter banks
[11]. The second case is easier to understand and correct, as
discussed previously. If the signal is composed by a single sinu-
soid, the aliasing has no effect in superimposing other compo-
nents with the component of interest. Thus, the undersampled
signal is a representation of the sinusoidal component but with
a different frequency—the defined apparent frequency.
IV. P
ROPOSED STRUCTURE
The proposed structure for harmonic estimation is shown in
Fig. 5. Fifteen bandpass filters compose the filter bank. The
th
bandpass filter is previously designed with its central frequency
being equal to
, where . Although this
design is specific to the analysis of harmonic content up to the
15th component, it can be expanded to analyze harmonics of
higher order. Basically, the main idea of this research is to show
that undersampling can be used in benefit to achieve an efficient
and low-cost structure, incorporating the concepts of multirate
processing with an estimation tool—the PLL.
After the filtering stage, the downsampler devices re-
duce the sampling rate, performing an undersampling for
, according to Table I. Regarding this table,
from
5to 15, the difference between the true frequency
values, placed in column
, and the apparent frequency values,
placed in column
can be noticed.
Finally, the last stage is the estimator stage. This is composed
by the Enhanced-PLL (EPLL) system [2], which is responsible
for extracting three parameters from its input signal
. These
parameters are the magnitude, the frequency, and the total phase,
that is
(7)
1792 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 4, OCTOBER 2009
Fig. 5. Proposed structure for harmonic estimation.
TABLE I
T
YPICAL
VALUES FOR THE
DOWNSAMPLING
FACTOR AND ITS
EFFECTS IN
CHANGING FREQUENCY OF THE
INPUT
SIGNAL
The EPLL discrete-time recursive equations are
(8)
where
, , and are constants that determine the speed of
convergence and steady-state error,
is the sampling period,
and
is the error signal given by
(9)
From Fig. 5, it can be seen that the estimated frequency of
each EPLL block is used to update its respective bandpass filter.
There are many strategies to do this. Here, we chose to use an
instantaneous update by using the output of a moving average
filter (MAF) attached to the EPLL block output.
A. Theoretical Analysis of the Multirate Structure
Signal enhancement is a common preprocessing approach
used in parameter estimation. It consists of extracting an un-
wanted signal component or noise from the source signal that
would degrade the estimation performance. When used as an
estimator, the EPLL shows improvements when signal enhance-
ment is performed first. The supporting theoretical argument
comes from the periodic orbit theorem associated with the EPLL
dynamic equations [13].
First, assume an electrical signal is composed of the funda-
mental, several harmonics components, and noise. This signal
can be written as
(10)
where
is the fundamental component, is the sum of
harmonics, and
is the noise. Let the objective be the ex-
traction of the
component. The theorem concludes that
the response of dynamic equations of EPLL will converge to a
periodic solution in the neighborhood of
. This occurs for
a proper choice of parameters
and the neigh-
borhood is determined by these parameters and the component
. In other words, if the fundamental component of
a highly polluted electrical signal is to be extracted, the estima-
tion will present oscillations due to the presence of harmonics
and noise. When using analysis filter banks (for signal enhance-
ment), the energy
is reduced, so the periodic orbit
becomes close to
.
This theorem can be extended to the estimation of each har-
monic and to the case when the frequency is time varying as
stated below.
Assume that the
th bandpass filter was designed with a cen-
tral frequency equal to
and there is a frequency deviation
. Despite the sharp magnitude response of the
bandpass filter (Fig. 2), the component close to
is not com-
pletely attenuated and the value of the frequency, phase, and
amplitude of it will be estimated. The capability of EPLL of
tracking the actual frequency of the attenuated component is
used to update the central frequency of the bandpass filter and,
after the transient response, all parameters are correctly esti-
mated, since the phase response of the filter is zero at the cen-
tral frequency. The smaller the frequency deviation is (such as
in electric power systems), the faster the transient response is.
Note that all other frequency components than the central
frequency are attenuated by the filter, which reduces the term
in (10). Furthermore, the bandpass filter increases the
signal-to-noise ratio (SNR) (i.e., it reduces the noise variance)
and consequently
, by a factor that depends on the param-
eter
. The theoretical relationship between the output variance
and input variance is given by [12]
(11)
For example, for
0.98, the SNR is increased by 17 dB.
Thus, the analysis filters are used to split the components,
improving PLL performance by providing (“theoretically”) a
single sinusoidal signal at the input of each PLL.
B. Downsampling Benefit
The advantages of inserting the downsamplers between the
bandpass filter and the EPLL estimators are twofold: 1) it re-
duces the computational complexity of the algorithm and 2) it
allows the definition of a single set of EPLL parameters, ap-
plied uniformly to each frequency band. The first advantage is
DE CARVALHO et al.: PLL-BASED MULTIRATE STRUCTURE 1793
based on exchanging the extra operations from the filtering stage
with the simplification of the estimation stage by using a lower
sampling rate. The second advantage comes from a single set
of EPLL constants. In a single-rate approach (conventional),
these constants must be modified and adjusted for each har-
monic component. In a multirate approach, the EPLL estimates
an undersampled signal whose frequency is between 60–240 Hz
(see Table I) and only one set of constants is necessary. This
strategy is adopted to allow the parameter estimation of any har-
monic in a frequency band where the EPLL constants were ex-
haustively tested. This fact explains the reasons for the choice
of the downsampler factors in Table I.
V. M
ETHODS FOR
COMPARISON
A brief description of two methods used for comparison is
made here. Two parameters are used for comparison with these
methods. The first one is the convergence time, defined here as
the time necessary to attain and stay inside the
error band.
The second one is the mean squared error (MSE) on steady state.
For any parameter
, taken from (8), the MSE is given by
(12)
where
is the number of samples and is the actual value.
A. Short-Time Fourier Transform (STFT)
The STFT is also known as the time-dependent Fourier trans-
form. Let
be the signal of interest. The STFT of is de-
fined as [12]
(13)
where
is a suitably chosen window of length and
is the harmonic order.
Interpreting (13), the STFT of
at time is the DFT
of the sequence
. Shifting from to , the
STFT is now the DFT of the sequence
. It can
be concluded that the STFT provides the spectral contents of the
sliding window in time of the sequence
.
The choice of window sequence depends on desirable char-
acteristics of each one in the frequency domain, such as a re-
duction of leakage effect, etc. [14]. Some examples of windows
are: Rectangular, Hamming, Hann, and Bartlett.
B. EPLL Cascade Structure
This method was proposed in [3]. It employs a cascade con-
figuration of modified units of (8), as is shown in Fig. 6. The
signal to be analyzed is the input of the unit responsible to ex-
tract the fundamental component
. This component is sub-
tracted from the input signal and then the second unit is respon-
sible to extract the second harmonic component. This process
continues until the
th component of interest.
The EPLL blocks on Fig. 6 are said to be modified because of
the use of two first-order lowpass filters on their internal struc-
ture. These filters are included to prevent the error on estimating
Fig. 6. EPLL cascade structure proposed in [3] for harmonic and interharmonic
estimation.
the component of interest caused by the other harmonic and in-
terharmonics components and noise. Besides, first-order low-
pass filters are also included on the output to smooth results.
C. Comments on Asynchronous Sampling
Although the EPLL cascade structure performance does not
depend on the sampling process, the STFT performance is di-
rectly related to this process. If the sampling rate is supposed to
be constant and synchronous with the fundamental component,
and only the harmonic component is presented in the signal, the
STFT becomes the better fit for harmonic estimation. However,
the STFT is not able to track frequency changes, and provides
only the amplitude and phase angle of the
th component by cal-
culating the modulus and phase of the complex number of (13).
With asynchronous sampling, it loses accuracy, as discussed in
the next section.
The STFT can be interpreted as an analysis filter bank com-
posed of complex filters. Assuming a frequency deviation in an
electrical signal under analysis, a possible way of adapting the
STFT is to provide synchronous sampling. Thus, the central fre-
quency of the
th filter of (13) is adapted to match the actual
frequency. We can observe a similar strategy in [15] where a
variable sampling frequency is presented to improve the phasor
estimation for digital relaying.
Fig. 7 shows the magnitude response of the STFT filter, with a
rectangular window centered at 60 Hz. The magnitude response
for the cascade filter consisting of two second-order bandpass
filters (1) is shown in the same figure for comparison.
Note that the STFT has high rejection for the harmonic fre-
quencies. If the sampling rate is not synchronous or if there are
interhamonic components in the input signal, then the estima-
tion error increases significantly. On the other hand, when com-
pared with the STFT, the bandpass filter presents better noise re-
jection over all frequencies, except at the harmonic frequencies.
Furthermore, the update of the central frequency of the bandpass
filter is easily accomplished because the filter is parametric.
In this paper, the STFT was implemented without employing
a synchronization algorithm and using a Hamming window,
being used as an additional comparison method. However, it is
worth mentioning that there is a lack of works in the literature
