Energy and Power Engineering, 2013, 5, 1126-1133
doi:10.4236/epe.2013.54B215 Published Online July 2013 (http://www.scirp.org/journal/epe)
The Use of LMS and RLS Adaptive Algorithms for an
Adaptive Co ntrol Method o f Active Pow e r Filter
Radek Martinek1, Jan Zidek1, Petr Bilik1, Ja kub M anas 1, Jiri Koziorek1,
Zhaosheng Teng2, He Wen2
1Faculty of Electrical Engineering and Computer Science, VSB – Technical University of Ostrava, Czech Republic
2College of Electrical and Information Engineering, Hunan University, Changsha, Hunan Province, China
Received March, 2013
ABSTRACT
This paper deals with the adaptive control mechanism management meant for shunt active power filters (SAPF). Sys-
tems driven this way are designed to improve the quality of electric power (power quality) in industrial networks. The
authors have focused on the implementation of two basic representatives of adaptive algorithms, first, the algorithm
with a stochastic LMS (least mean square) gradient adaptation and then an algorithm with recursive RLS (recursive
least square) optimal adaptation. The system examined by the authors can be used for non-linear loads for appliances
with rapid fluctuations of the reactive and active power consumption. The proposed system adaptively reduces distor-
tion, falls (dip) and changes in a sup p ly voltage (flicker). Real signals for measurement were obtained at a sophisticated,
three-phase experimental workplace. The results of executed experiments indicate that, with use of the certain adaptive
algorithms, the examined AHC system shows very good dynamics, resulting in a much faster transition during the AHC
connection-disconnection or during a change in harmonic load on the network. The actual experiments are evaluated
from several points of view, mainly according to a time convergence (convergence time) and mistakes in a stable state
error (steady state error) of the investigated adaptive algorithms and finally as a total harmonic distortion (THD). The
article presents a comparison of the most frequ ently used adaptiv e algorithms.
Keywords: Active Power Filters; Total Harmonic Distortion; Least Mean Square; Adap tive Filtering; Recursive Least
Square
1. Introduction
Currently, the active harmonic compensation AHC [1],
which is intended to compensate the harmonic flow gen-
erated by the powerful electronics, is being used more
and more often in the industrial applications [2]. Modern
industrial corporations usually have numerous types of
electrical equipment and the load [2] which can be di-
vided into the linear load [24] (regenerative power unit,
active loads, standing on alternating current) and
non-linear loads [23] (variable speed drives, arc units,
etc.).
The authors of this article have primarily focused on
adaptive filters [20] from the viewpoint of ad aptive algo-
rithms utilized [13] in order to improve efficiency of the
Active Power Filters - APF [23]. This article concen-
trates on the adaptive control method - ACM [7] de-
signed for the APF, especially from perspective of the
examining the performance data of adaptive algorithms
used. The authors do not deeper examine their own APF
project as this topic is very broad and it has been covered
in a number of vari ous pu bl i cat ions s uc h as: [ 4, 1 4, 2 2].
In the framework of the experiment, the investigated
algorithms least mean squares – the LMS [5] based adap-
tive FIR filters [5] and recursive least squares – the RLS
[5] based adaptive FIR filters [5 ] were tested on real sig-
nals. Real data were obtained at the earlier mentioned
three-phase experimental workplace, see Picture 3.
2. The Situation in 2013
At present, we can see notable development in the indu s-
trial APF supply networks [2]. The role of active com-
pensation is increase of its importance alongside with the
passive filters [22], which are traditionally used to im-
prove the quality of a power supply. However these tra-
ditional approaches in the form of a passive compensa-
tion have, from the perspective of modern power net-
works, certain disadvantages [22]. We are talking espe-
cially about situations where the proposed filter has to
operate in an unfamiliar environment, where preliminary
identification is difficult. Here, it is hard to set the filter
parameters [9] to solve the given task in th e best possible
way. Another case is where there is an environment with
a variable (where there are various, unspecified loads

R. MARTINEK ET AL. 1127
connected), whose development in the future cannot be
predicted. In this case, the optimal filter setting values
change over time and also it would appear very appropri-
ate to use the adaptive filtering, where the optimal filter
coefficients are altered over time with the use of the
adaptive algorithm.
The deployment of the AHC brings a number of ad-
vantages in comparison with the passive compensation:
 It is easy to design it, according to a given applica-
tion, since it is independ ent from the line impedance [2].
 It does not generate any resonance [2].
 It actively restricts and controls the harmonic and
reactive currents [2].
Despite considerable benefits of the AHC, this solu-
tion had a limited market share during last several years.
This was mainly caused by particularly high costs. Now,
at the beginning of 2013, it can be said that change is
about to set in. Mass deployment of the AHC is evoked
by the new trends in design of systems for the industrial
power networks, more detail in [6, 11, 25]. Another key
factor is the economy. Currently, there is a rise in inex-
pensive mass production of semiconductor devices that
are used in the AHC. On the other hand, the costs of the
passive filter are increasing because of the large volume
of copper and steel used in its production as a primary
material. These production costs are having a greater
impact on the passive solutions than on the active solu-
tions.
3. Basic Principles of APF
APF, using adaptive filtering, works similarly to the
adaptive filtering systems techniques for noise reduction
[21], which are used in various applications (ECG filter-
ing [8], equalization [13], the suppression of noise in
communication [5], etc.). The authors have dealt with
various areas of the use of adaptive filtering in their own
publications [5, 8, 13, 17, 18], where individual tech-
niques of adaptive filtering are de scribed into detail.
3.1. Description of APF Activity
An active filter monitors the input current (measured
through a monitoring current transformer), including its
distortion. From this signal, the controlling system de-
rives the required compensation and forms a switching
sequence. In this manner a conduc tive path with low im-
pedance is created in the filter and harmonic currents
flow through the filter instead of continuing towards the
power supply. By suppression, the current distortion is
almost completely eliminated. The filter performs con-
tinuous evaluation of the current harmonic suppression [3,
10, 15, 16] so the instant or daily changes of load on the
device appear to be a no problem for the filter. More de-
tailed description of the APF can be found in, [1, 12, 15].
Figure 1 shows a simplified schematic diagram of the
APF.
In modern supply systems, non-linear appliances are
being used more often and there is no sinusoidal wave-
form in the network. This waveform will consist of a
sinusoidal current whose frequency corresponds to the
frequency of network and the current of high-
er harmonics, whose frequencies are multiples of the
network frequency.
Figure 1. Simplified scheme of the connected active harmonic filter for experiments with adaptive algorithms connected to an

R. MARTINEK ET AL.
1128
i
ndustrial plant.
In a power system, the current can be described as
following
(1)
The fundamental current is denoted by i1(t) and the
harmonic current as
(2)
With passage of a non-harmonic current to the imped-
ance networks higher harmonic voltage is created, which
causes an additional torque in motors and generators,
cancelling out zero crossing detectors, etc. Th e frequency
of higher harmonics has been marked with a serial num-
ber "n" or "v". (For example the network frequency = 50
Hz, n = 3 is the 3rd harmonic = 150 Hz, n = 5, the 5th
harmonic = 250 Hz, etc.).
Similar to the LC filter circuits, the active filter is
connected parallel before the source of harmonic currents
(see Figure 1), but it works on a completely different
principle. The development of the microprocessor and
semiconductor technology has allowed us to construct
an electronic device that can produce (generate) such a
current (minus sinusoidal current drawn from the real),
which is injected into the access points, added together
with the drawn current and creates a curve that ap-
proaches a sinusoid.
4. Adaptive Filtering
Since the principles of adaptive filtering are used in this
work this chapter is devoted to the progressive process-
ing of digital signals. Figure 2 illustrates the basic block
diagram of an adaptive filter [5].
Figure 2 represents x(n) input vector samples y(n) de-
notes the output of the adaptive filter, d(n) is the desired
response, e(n) represents the error signal (estimated er-
ror), wi represents the weighting coefficients vector
transversal FIR filter z-1 represents a delay. In this work,
the input signal in the form of a column vector defined
by the equation:
(3)
Figure 2. Block diagram of an adaptive filter.
Vector weights transversal filter (a FIR adaptive filter
coefficients) will have the form:
(4)
The output signal of the adaptive transversal filter can
be written using the equation as:
(5)
This filter is an N-th order, so indexing the range from
0 to N-1. It can be seen as equivalent to the dot product
[21] between the impulse responses [21] vector and the
input vector, and alternatively as a vector transpose the
filter weights and the input vector, see eq uation
(6)
(7)
The aim of the whole weight adaptation process is the
progressive reduction of the objective function value ξ(n)
[21] to its minimum. ξ(n) depends only on the values of
the error function, therefore, is dependent on the differ-
ence between the desired and actual value (i.e. the dif-
ference between the desired output and the actual output
of the FIR filter):
(8)
In the executed experiments, two basic classes’ repre-
sentatives of adaptive algorithms conducted were exam-
ined. The first class is also known as a mean square error
(MSE) [21] of the adaptive filter (LMS algorithm).
Whole process´s aim is the progressive reduction of ob-
jective function value ξ(n) to its minimum (the smallest
value of mean square error) [5].
(9)
The sec ond class of adap tive algorithms is also known
as a recursive method of least squares (RLS) [21]. The
basic difference, comparing to the family of the LMS
algorithms, is the own statistical concept. Here, we are
working with average values that calculate with variable
developments of the time instead of the sample averages
calculated from several implementations of the same
random process [5].
(10)
(11)
where k =1, 2, 3… n, parameter λ designates the
so-called forgetting factor and is in the range 0 to 1.
A more detailed description of adaptive filtering can
be found for example: [5, 20, 21].
4.1. LMS Implementation
A basic representative of the stochastic gradient algo-
rithms class, based on the theory of Wieners [8], stochas-
tic averaging [8] and the method of least squares [8]. Th e

R. MARTINEK ET AL. 1129
custom LMS algorithm derivation is generally known
and described in many technical publications, such as: [5,
8, 21]. So in this article, there is only a simple mathe-
matical description of the respective algorithm imple-
mentation.
Each items of the LMS algorithm requires three dis-
tinct steps in this order:
1) The output of the FIR filter y(n) is calculated using
an equation 5 or 6 and 7.
2) Value of the estimated error is calculated using
equation 5.
3) FIR weight vector is updated in preparation for the
next iteration of the equation [15]:
(12)
The parameter μ is referred as a step size. It is a small
positive constant which influences the properties of the
algorithm adaptation (filter stability, convergence rate,
etc.).
4.2. RLS Implementation
It's a basic representative of the recursive algorithms
class based on the theory of Kalmans filtering [8], time
averaging [8] and the method of least squares, see [8].
The basic difference comparing to the LMS algorithms
family lies in the respective statistical concepts. We are
working with average values calculated from the time
developments, instead of the sample averages of several
implementations from the same random process. The
structure of the filter remains the same compared to the
LMS algorithm, only the adaptive process is different
because of the averages use. A detailed description and
derivation of the RLS algorithm can be found in [8, 20,
21].
If we want to implement the RLS algorithm it must be
carried out along the following steps in this order:
1) Filter output is calculated using the weights of the
filter from the previous iteration and the current input
vector:
(13)
2) Medium-range vector gain is calculated using the
equation:
(14)
(15)
3) The value of the estimated error is calculated ac-
cording to the equation:
(16)
4) The vector filter weights are u pdated using equ ation
16 and the vector gain is calculated in equation 15:
(17)
5) The inverse matrix is calculated using the fo llowing
equation:
(18)
5. Experimental Workplace
To real voltages and currents generation three-phase ex-
perimental workplace was used with three-phase pro-
grammable source HP 6834B connected as the inputs.
To voltages and currents acquisition the IED (Intelli-
gent Electronic Device) was built connected to the out-
puts on the workplace
Linear and non-linear loads were connected to work-
place.
Programmable source HP6834B can generate sine
waveforms with maximum amplitude 300V/5A per phase
with 100th harmonics. Source was connected remotely to
IED (GPIB interface) for automation testing.
The IED can measure up to three voltages (directly)
and three currents (directly or using current clamps). For
voltage and current measurement precise modules (un-
certainty ±0,1%) is used with an t aliasing filters [25]. All
inputs were calibrated with NI 9925 modules. Sample
rate was set to 30 kS/Channel. In the IED the application
was developed for PQ (Power Quality) parameters anal-
ysis. The application evaluates harmonics, interharmon-
ics and higher frequency components (complies with IEC
61000-4-7 standard), flicker (IEC 61000-4-15), voltage
phenomena (IEC 61000-4-30) and the voltage character-
istics (EN 50160). Th e app licatio n can stream raw d ata to
TDMS files for offline post-processing.
Figure 3. Three-phase experimental workplace.

R. MARTINEK ET AL.
1130
An application created in IED (Pictures.5, 6) utilizes
the concept of a virtual instrumentation [19], therefore
there is no need for more time and resource expandable
to meet future requirements for the development of adap-
tive filter algorithms.
6. Results of the Experiments Carried Out
In Figure 7, there is a depicted reference waveform cur-
rent to the linear load. For each phase the THD [7] dis-
tortion is less than 0.2%. The length of the time window
of 0.1 s for all the charts is selected for the optimal dis-
play of waveform.
THD is a quantity that defines the distortion sin e wave.
It is defined as a ratio of the all harmonic components
Figure 4. Connec ted inputs and outputs of the experimental
workplace without a connected load.
Figure 5. Developed application graphical interface.
(Waveforms).
total power to the fundamental harmonic performance
The lower the THD is, the more constant is the signal.
THD is a way of assessing the extent of distortion. THD
calculation is based on decomposition of a periodic sig-
nal to harmonic components using Fourier series in the
amplitude-phase entry.
In Figure 8, strongly distorted waveform caused by
nonlinear loads. THD distortion is for phase L1 88.1%,
L2 98, 2% an d L 3 92.3%.
In Figure 9, the resulting current waveform is calcu-
lated using the RLS adaptive filter algorithm. THD dis-
tortion is diminished for phase L1 88.1% to 5.43%, L2
98.2% to 6.43% and L3 92.3% to 4.78%.THD for the
reconstructed current waveform now meets the THD
limits set according to the EN 61000-3-2 standard.
In Figure 10 is the output current waveform calculated
using the LMS adaptive filter algorithm. THD distortion
is diminished for phase L1 88.1% to 13.54%, L2 98.2%
to 16.86% and 92.3% to L3 14.32%.
Figure 6. Developed application graphical interface (Re-
sult).
Figure 7. Reference current waveform to a linear load.

R. MARTINEK ET AL. 1131
Figure 8. Current waveform from non-linear load (time
line).
Figure 9. Reconstructed current waveform for a non-linear
load with the RLS algorithm (time line).
Figure 10. Reconstructed current waveform from a
non-linear load with the LMS algorithm (time line).
The frequency spectra of the source current before and
after the compensation are compared in Figures 11, 12
where the amplitude of each harmonic is represented in a
percentage to the fundamental.
6.1. Evaluating the Experiments Carried Out
Looking at the results of the filtering using the LMS al-
gorithm (Figure 10) it is clearly seen that the algorithm
converges slowly. The LMS algorithm therefore needs a
large number of iterations (repetitions) so that the adap-
tive filter is closer to an optimum condition of filtering.
In Figure 10 you can see that in the beginning of filtra-
tion, the error in filtration achieves significant error val-
ues and, with the adaptive algorithm approaching to
(converges) the optimal value, it decreases significantly.
Upon approaching to the optimal value of the LMS algo-
rithm, it begins to fluctuate [21] around this optimal val-
ue and thus a small error occurs even after filtration op-
timal value.
Figure 11. Current waveform from a non-linear load and
reconstructed current waveform with the RLS algorithm
(amplitude spectrum).
Figure 11. Current waveform from a non-linear load and
reconstructed current waveform with the LMS algorithm
(amplitude spectrum).

R. MARTINEK ET AL.
1132
The second investigated algorithm was the RLS algo-
rithm. Looking at the results of the filtration (Figure 9)
using the RLS algorithm, we can say that the RLS algo-
rithm comes closer to the optimal value very quickly.
From the course, it is obvious that there is a good stabil-
ity of the filtration process.
6.2. Comparing the Mathematical Difficulty
The basic factors determining the properties of adaptive
algorithms in terms of implementation of the DPS are:
the speed of convergence, computational complexity,
memory consumption, the steady state error in more de-
tail in [20]. Properties of the studied LMS and RLS algo-
rithms are summarized in Table 1 below.
If we then compare the mathematical difficulty of both
algorithms, it can be seen that the RLS algorithm, which
achieves better results during th e filtration, pays the price
for this accuracy and speed in the form of extreme
mathematical complexity. This is related to the high re-
quirements on hardware (processor, memory), on which
the RLS algorithm will be used.
7. Conclusions
This paper proposes a novel simple current control
method f o r act ive filter s ba s ed on adaptive filtering.
The examined adaptive LMS algorithm is simple and
mathematically undemanding (Table 1). Unfortunately,
in tests it achieves a lower speed and a higher conver-
gence error during the filtration process (Figure 10). In
contrast, the RLS algorithm is very involved mathemati-
cally, (see Table 1). On the other hand the test results
have shown that it is very accurate. It achieves a low
error rate and has an extremely high rate of convergence
(Figure 9). The use of adaptive filters in the THD elec-
trical engineering appears to be an essential parameter in
comparing of the different algorithms effectiveness (see
Figure 11).
Overall, the results have shown that the adaptive sys-
tem for suppress of harmonics based on the RLS algo-
rithm has better filtration properties, however, at the ex-
pense of higher computational requirements on the algo-
rithm. Although the LMS algorithm did not achieved, as
good results as the RLS algorithm, its use is very inter-
esting if we take into account its mathematical complex-
ity. In practice, of course, it is required to have the lowest
Table 1. Characteristics of the examined lms and rls algo-
rithms.
Examined
Algorithms +/- in 1cycle * in 1cycle Memory ootprint Con. Time
LMS M+1 2M 2M 0,08s
RLS M2+M 2M2+3M+50 M2+3M 0,02s
cost implementation of adaptive systems while main-
taining high quality suppress of unwanted harmonics
From the viewpoint of the cost of producing the DSP it
will be preferable to use on e which is meant for the LMS
algorithm. There are currently far more widespread adap-
tive systems using the LMS adaptive algorithms. In the
future, we can expect a steady increase of performance
and quality in the field of computer technolog y. With the
advent of more powerful DPS, the requirements for low
computational complexity and memory consumption of
the algorithms will drop and it might b e even possible to
implement more complicated and more efficient algo-
rithms. Therefore, it is certainly true that the area of
adaptive filtering is and for a long time will remain
widely opened sphere for scientific research and com-
mercial applications
8. Acknowledgements
This research was supported in part by The Ministry of
Education, Youth and Sports of Czech Republic under
the project KONTAKT II registration number LH12183.
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