TOWARDS AN INTELLIGENT SYSTEM FOR MONITORING
ELECTRICAL ENERGY QUALITY
Foundations and Motivations
Luiz Biondi Neto, Pedro Henrique Gouvêa Coelho
State University of Rio de Janeiro, R. S. Francisco Xavier, 524 Sala 5006E, Maracanã, R.J., 20550-900, Brazil
João Carlos C. B. Soares de Mello, Lidia Angulo-Meza
Flum. Federal University, Prod. Eng. Dep.
R. Passo da Pátria 156, S. Domingos, Niterói, R. J., 20210-240, Brazil
Eliane G. Gomes
EMBRAPA, S. G. E., Parque Estação Biológica, W3 Norte final, Asa Norte,70770-901, Brasília, D.F., Brazil
Keywords: Harmonic analysis, Optimization methods, Fourier series, Intelligent systems.
Abstract: Electrical energy must be supplied in enough amount but with adequate quality. One of the components of
electrical energy quality is the harmonic distortion. In this paper, we show an alternative way to measure
distortion, mixing Data Envelopment Analysis (DEA) and Fourier Analysis. The technique here presented is
specially useful for comparative analysis and is intended to be the basis for an intelligent system for
monitoring electrical energy quality.
1 INTRODUCTION
One of the components of electrical energy quality is
the proportion of harmonics in the electric signal. On
the other hand, electronic equipments plugged in
power lines generate harmonics. In real-life
situations where a large number of such devices are
plugged in power lines, the electric signal affected
by harmonic distortion should be classified
according to some degree of quality. Such
classification helps to check whether electrical
power facilities are acceptable and also to decide
which devices have to used with suitable filters.
Data Envelope Analysis is (DEA) is a technique
based on Linear Programming that is used to
calculate the performance of operational units also
known as Decision Maker Units (DMUs) in
scenarios involving several inputs and outputs in a
such a way that comparisons are difficult to be
established. Such approach defines an index known
as the relative efficiency for each DMU that results
in a relative classification for each unit among all
others comprising the investigated group.
The method compares the DMU efficiencies in
terms of capability of transforming inputs into
outputs by means of a ratio between the output due
to a particular input. In the end of the analysis, the
method is able to decide which units are relatively
efficients or inefficients.
In this paper, results of traditional theoretical
studies in Power Electric waveforms are compared
with those of DEA. The DEA modeling is carried
out in a nonconventional way, once the defined
inputs (Fourier coefficients) are considered the same
for all DMUs (waveforms).
2 BASIC FOUNDATIONS
Mathematical tools used in this paper are shown in
this section as well as the relationship between the
series expansion of a periodic function and the
harmonic distortion, and also the basis of Data
Envelopment Analysis.
40
Biondi Neto L., Gouvêa Coelho P., C. B. Soares de Mello J., Angulo-Meza L. and G. Gomes E..
TOWARDS AN INTELLIGENT SYSTEM FOR MONITORING ELECTRICAL ENERGY QUALITY - Foundations and Motivations.
DOI: 10.5220/0003436200400045
In Proceedings of the 13th International Conference on Enterprise Information Systems (ICEIS-2011), pages 40-45
ISBN: 978-989-8425-54-6
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
2.1 Harmonic Analysis
Harmonic analysis is based on the Fourier Series
(FS), that for a function
xF , defined on an interval
0,20 TTx , is given by equations (1), where
dx
T
nππ
f(x)
L
A
T
n
cos
1
2
0
dx
T
nππ
f(x)
L
B
T
n
sen
1
2
0
.
1
0
sencos
2
n
nn
T
nππ
B
T
nππ
A
A
(1)
If

xf and

xf ' are continuous and

xfTxf 2 , then the series converges to
xF .
It is possible to determine the magnitude of the
Harmonics (
A
n
and B
n
) in terms of its order n
(Folland, 1992).
2.2 Harmonic Distortion
Harmonic distortion is a paramount index indicating
the quality of electrical energy. High harmonic
distortion waveforms differ somewhat from those
having a sinosoidal shape and moreover are danger
to electric appliances. Usually harmonic distortion is
defined in terms of the total harmonic distortion
percentage (THD) that compares amplitudes of high
frequency harmonics with that of the fundamental
frequency that is given by the term for n equals 1 in
the Fourier series expansion.
Equation (2) calculates THD where it is assumed
that the distortion is evaluated for an electric
waveform current (I) and the indexes in the sum are
related to the harmonic number.
1
2
2
I
I
THD
n
n
(2)
2.3 Data Envelopment Analysis
Data Envelopment Analysis (DEA) is a linear
programming based technique for measuring
performances of predefined Decision Making Units
(DMUs) in presence of multiple inputs and outputs
where comparison is difficult. Such method
calculates the relative efficiency of each of the
DMUs among all others comprising the group in
evaluation.
Data Envelopment Analysis (DEA) is a linear
programming based technique for measuring
performances of predefined Decision Making Units
(DMUs) in presence of multiple inputs and outputs
where comparison is difficult. Such method
calculates the relative efficiency of each of the
DMUs among all others comprising the group in
evaluation.
The model used in (Chames, Cooper, and
Rhodes, 1978) known as CCR, defines a piecewise
linear, non parametric surface on the data and
evaluates the DMUs efficiency over such surface.
Such model is input oriented i.e. it minimizes the
input number keeping the output values and assumes
Constant Return to Scales (CRS) so that every input
variation yields a proportional variation in the
outputs. The problem is then to calculate the weights
u
j
and v
i
such that the ratio of the outputs weighted
sum over the inputs weighted sum is maximized.
The method should be repeated for each of the n
DMUs so that the relative efficiency for each DMU
is determined using the weights. The DMU model is
shown in equation (3).
1
1
1
1
subject to
11
0
s
jjO
j
O
r
iiO
i
s
jjk
j
r
iik
i
ji
uY
Max h
vX
uY
, k ,...,n
vX
u and v j,i


(3)
efficiency of DMU
number of
number of
number of DMUs
produced by DMU
- used by DMU
weight for
weight for
O
jk
ik
j
i
where
h o
r inputs
s outputs
n
Youtput j k
Xinputi k
u output j
v input i
The problem defined in equation (3) is known as
fractional programming and can be linearized as
shown in equation (4) and it is called Multiplier
Problem.
TOWARDS AN INTELLIGENT SYSTEM FOR MONITORING ELECTRICAL ENERGY QUALITY - Foundations and
Motivations
41
1
1
11
subject to
1
01
0
s
OjjO
j
r
iik
i
sr
jjk i ik
ji
ji
Max h u Y
vX
u Y v X , k ,...,n
u and v j,i




(4)
The Dual of the Multiplier Problem is known as
Envelope Formulation. The solution of such problem
yields, besides the values for the efficiencies,
benchmarks for the inefficient DMUs. Although in
management decision applications that is important
information, in this paper application there is no
such interest once the defined DMUs are
mathematical entities that make no decision.
For the case in study, the model compares the
DMUs efficiencies by their ability of transforming
inputs into outputs, measuring the ratio between the
produced product (output) and the input. The
analysis results make possible to identify which
units are relatively efficients and inefficients.
DEA has also been used to aid in defining
Electric Energy Distribution targets (Pessanha et al.,
2007), in evaluating the energy efficiency of cities
and areas (Angulo-Meza et al., 2007 and, Soares de
Mello et al., 2008) or in evaluating the performance
of companies using a double perspective (Lins et al.,
2007).
3 HARMONIC DISTORTION
Electrical energy have aspects that make it different
from the other industrial products. It must be
generated as it is used, it can not be stored by the
users, nor carried through usual means of
transportation and last but not least, its quality
depends on both the user and the producer.
Moreover, current electrical energy systems use
more electronic than mechanic devices which make
them easier to generate higher order harmonics due
to their nonlinearity characteristics.
Currently, over 50 % of electrical energy runs on
electronic devices. Although that contributed to an
increase of industrial productivity and a more
efficient use of electrical energy, it also changed the
electrical energy quality requirements. While
electromechanical systems are insensible to energy
supply interruptions to the order of magnitude in
seconds, electronic systems are insensible to such
interruptions to the order of magnitude in
milliseconds, apart from showing sensitivity to
voltage variations. Due to that larger sensitivity
property, even typical procedures in electrical
systems may cause interruptions in large automatic
industrial unities, turning inefficients the standard
evaluation indexes, as far as quality assessment is
concerned.
Matters as what should be such new quality
energy indexes and which actions should be taken in
order to improve the quality of energy were not
satisfactorily answered yet. The pressure for finding
quick and efficient answers for such questions
requires the use of automatic management systems
and protection and filter equipments (Lima et al.,
1994).
Harmonic voltage and current distortion is one of
the problems usually found in electrical systems
caused by highly nonlinear loads such as electronic
converters, rectifiers, inverters, controllers, etc. Such
high distortion indexes may cause problems such as
capacitor bank faults, fuse and thyristor burn outs or
even malfunctioning in electronic devices,
particularly computers and electronic sensitive
devices, fed by power lines degraded by the
presence of harmonic distortion.
Simple solutions include the reduction of the
operation time for such nonlinear loads or the use of
filters. The former means production losses for the
consumer whereas the latter equipment expenses. An
accurate measure of the harmonic distortion allows a
selective use of filters for reducing the expenses.
This paper deals only with harmonic distortion,
but the quality of energy is concerned with other
issues such as voltage drops and surges, frequency
variations and blackouts, to mention a few.
3.1 Modeling by Harmonic Analysis
Table 1 shows the THD for typical functions used in
engineering (Folland, 1992). Waveforms rank in
terms of THD percentage will be later used as a
basis of comparison to validate the obtained results
using the DEA approach.
Table 1: THD (%) for some waveforms.
Waveform (Wav.) % of THD
Triangular (T) 12.05
Half-wave rectification (HW) 21.75
Full-wave rectification (FW) 22.48
Square-wave (SW) 42.88
Sawtooth wave (S) 74.15
Controlled rectification (CR) 107.60
ICEIS 2011 - 13th International Conference on Enterprise Information Systems
42
3.2 DEA based Calculations
DEA - CRS model with the ISYDS (Angulo-Meza
et al., 2005) software yielded initially the results
shown in Table 2. In such a model, the considered
output is the RMS value for the fundamental
frequency (F1) and the inputs, the RMS values for
the harmonics up to order ten (H2 to H10) ( Biondi
Neto et al., 2003, and, Biondi Neto, 2001).
Table 2a: Initial results for DEA-CCR model.
DMUs Inputs x 10
-2
Eff.
(%)
Wav. H2 H3 H4 H5 H6
HW 15.0 0 3.0 0 1.28 100
FW 0 0 6.0 0 2.57 100
SW 0 30.0 0 18.0 0 99.97
S 70.7 47.1 35.3 28.2 25.3 16.38
CW 16.7 11.2 6.10 3.70 3.90 11.29
T 0 10.0 0 3.6 0 100
Table 2b: Initial results for DEA-CCR model.
DMUs Inputs x 10
-2
Out.
Wav. H7 H8 H9 H10 F1
HW
0.71 0 0 0.45 70.7
FW
0 1.42 0 0.9 30.0
SW
12.8 0 10.0 0.0 90.0
S
20.2 17.6 15.7 14.14 141.4
CW
3.70 2.80 2.20 2.20 20.90
T
1.83 0 1.11 0 90.03
Initial results show the presence of a large
number of inputs for quite a few DMUs and also
several null harmonics due to the properties of some
waveforms, e.g. parity, which may mask the results.
Moreover, there are many null inputs and so all the
weights affect the units having the largest ratio
output over input that is due to the fact that DEA is a
benevolent model for the evaluated units.
There are several methods in the DEA literature
to overcome those problems that avoid a proper
waveform classification regarding their harmonic
distortion, such as cross-evaluation (Sexton, 1986,
and Doyle et al., 1994), inverted frontier (Yamada et
al., 1994, Entani et al., 2002, Lins et al. 2005,
Anderson et al., 2002, and, Soares de Mello et al.,
2008), weights restrictions approach (Allen et al.,
1997) and, selection of part of the set of variables to
be considered in the model ( Lins et al., 1999, and,
Senra et al. , 2007).
A selection of variables may seem, at first sight,
incoherent to traditional DEA modeling. Actually,
DEA considers that all the DMU’s use the same set
of inputs to produce the same set of outputs.
However, one should consider that, the waveforms
are not DMU’s in the normal meaning of a DMU.
Moreover, in this paper context, the inputs can be
seen as the price one has to pay in terms of
harmonics RMS values, in order to obtain the
output, i.e. the fundamental RMS value. So, it makes
sense consider, for each waveform, the harmonics
that really cause distortion in particular those of least
order. On most cases of convergent series, those
harmonics show the largest values.
The results obtained using such approach are
shown in Table 3. Now the results are more coherent
with the application having in mind waveform
distortion, since the previous detected problems are
not seen making possible a plausible ranking in
terms of such a distortion
Among the investigated waveforms, one can see
that the triangular is the one who shows the least
distortion. It should be stressed that its 100%
efficiency does not mean that distortion is not
present but shows that it is the best of all
investigated waveforms. As a matter of fact, the sine
waveform is the only one who produces no
distortion. Actually, an eye inspection, indicates that
the triangular waverform is the closest one to the
sine waveform. The half-wave rectification shows a
low distortion and all the others produce a large
distortion or low efficiency particularly the
controlled rectification waveform. If one of such
waveforms is present in a distribution circuit, severe
damages will certainly show up if filters are not
used.
Several of the above methods show
shortcomings. The first method results in a fixed
weight model (Anderson et al., 2002) that disfigure
the DEA method, the inverted frontier method is
undermined by the large number of null variables,
and the weights restrictions involve undesirable
subjective aspects.
Table 3: Data and results for the second DEA-CCR model.
DMUs Inputs Output Eff.(%)
H1 H2 F1
HW 0.15 0.03 0.707 94.24
FW 0.06 0.0275 0.3 55.24
SW 0.3 0.18 0.9 33.32
S 0.707 0.471 1.414 22.21
CW 0.1670 0.12 0.309 13.9
T 0.10 0.036 0.9003 100
TOWARDS AN INTELLIGENT SYSTEM FOR MONITORING ELECTRICAL ENERGY QUALITY - Foundations and
Motivations
43
So the restriction of variables approach was selected.
In this paper, the traditional DEA approach is
substituted in a way that not all DMU’s use the same
input variables. The used inputs are the relevant
harmonics for each investigated waveform. For each
waveform, the first two non-zero harmonics (H1 and
H2) are selected as inputs and the fundamental (F1)
is the output.
Table 4 shows a comparison among the THD
values and the DEA-CCR efficiency. Both results
yield the same DMU ranking validating the
approach used in this paper.
Table 4: THD and DEA-CCR Efficiency in percentage
(%) for some waveforms.
Waveform THD(%) DEA-CCR Eff.
Triangular 12.05 100
Half-wave rectification 21.75 94,24
Full-wave rectification 22.48 55,54
Square-wave 42.88 33,32
Sawtooth wave 74.15 22,21
Controlled rectification 107.60 13,90
4 CONCLUSIONS
The use of DEA as an alternative way to measure
the distortion combined with Fourier analysis
techniques was shown to be very useful particularly
to comparative analyses.
Although the THD and DEA methods produce
nearly the same results, at least in the present
theoretical study, there are reasons that justify the
use of DEA instead of THD. The first reason is that
DEA allows automatic calculations quite quick,
particularly when they are jointly used with neural
networks (Biondi, 2001, and, Biondi et al., 2004).
The second regards the fact that DEA is a
comparative method that can cause the decrease the
number of used filters for compensating the
harmonic distortion, indicating for a time instant
which equipments need filters.
Finally, a mathematical comparison between the
methods is needed. Although DEA can be
considered a trivial linear fractional model, THD is a
nonlinear model since involves a root square of a
sum of squares. From a theoretical point of view, it
should be stressed that the measure functions
capabilities of DEA and THD methods are alike.
First, both show only positive values. On the other
hand, a linear version of DEA is a degree one
homogeneous function. The THD numerator which
comprises a square root of a homogeneous function
of degree two, is also a homogeneous function of
degree one. Thus, for both ways of measure, the
numerators show proportionality among variables
and calculated values.
At last, it can be stressed that the fractional
nonlinear version of DEA is a homogeneous
function of zero degree as the final result of the
THD method. Thus, the methods present a certain
mathematical resemblance, although DEA is a
comparative method and nondifferentiable whereas
THD is an absolute method (noncomparative) and
differentiable. A comparative form of THD would
show several properties of a smoothed DEA frontier
( Soares de Mello et al., 2002, and, Soares de Mello
et al., 2004) for the investigated problem in this
paper.
The results indicate that the ideas here presented
could lead and inspire an intelligent system for
monitoring energy quality more efficiently.
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