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.

REFERENCES

Folland, G. B., 1992. Fourier Analysis and its

Applications. Wadsworth and Brooks..

Charnes A., Cooper, W. W. and, Rhodes, E., 1978.

Measuring the efficiency of decision-making units. In

European Journal of Operational Research, vol. 2, pp.

429-444.

Pessanha, J. F. M., Souza, R. C. and, Laurencel, L. D. C.,

2007. Um modelo de análise envoltória de dados para

o estabelecimento de metas de continuidade do

fornecimento de energia elétrica. In scielo vol. 27,

pp. 51-83.

Angulo-Meza, L., Soares de Mello, J. C. C. B., Gomes, E.

G. and, Fernandes, A. J., 2007. Selecção de variáveis

em DEA aplicada a uma análise do mercado de

energia elétrica. In Investigação Operacional, vol. 27,

pp. 21-36.

Soares de Mello, J. C. C. B., Angulo-Meza, L., Gomes, E.

G., Fernandes, A. J. S.and, Biondi Neto, L. 2008.

Estudo não paramétrico da relação entre consumo de

energia, renda e temperatura. In IEEE Latin America

Transactions, vol. 6, pp. 153-161.

Lins, M. P. E., Sollero, M. K. V., Caloba, G. M. and,

Silva, A. C. M., 2007. Integrating the regulatory and

utility firm perspectives, when measuring the

efficiency of electricity distribution, In European

Journal of Operational Research, vol. 181, pp. 1413-

1424.

Lima, A., Ferro, F., Martins, J., Roncolatto, R., and,

Santos, N., 1994. Custos da Qualidade de Energia em

Grandes Consumidores Industriais." In XII Seminário

Nacional de Distribuição de Energia Elétrica, pp. 16-

21.

Angulo-Meza, L., Biondi Neto, L., Soares de Mello, J. C.

C. B., and, Gomes, E. G., 2005. ISYDS - Integrated

System for Decision Support (SIAD Sistema Integrado

ICEIS 2011 - 13th International Conference on Enterprise Information Systems

44

de Apoio a Decisão): A Software Package for Data

Envelopment Analysis Model. In Pesquisa

Operacional, vol. 25, pp. 493-503.

Biondi Neto, L., Soares de Mello, J. C. C. B. and, Gomes,

E. G., 2003. Método Fourier-DEA na medição de um

componente da qualidade de energia elétrica. In XXIII

Encontro Nacional de Engenharia de Produção Ouro

Preto.

Biondi Neto, L., 2001. Neuro-DEA: Nova metodologia

para determinação de eficiência relativa de unidades

tomadoras de decisão. In Engenharia de Produção.

vol. D.Sc. Rio de Janeiro: COPPE/UFRJ.

Sexton, T. R., 1986. Measuring Efficiency: An assessment

of Data Envelopment Analysis. In New Directions For

Program Evaluation. Jossey-Bass. San Francisco.

J. Doyle and R. H. Green, "Efficiency and cross-efficiency

in DEA derivations, meanings and uses," Journal of

the Operational Research Society, vol. 45, pp. 567-

578, 1994.

Yamada, Y., Matui, T. and, Sugiyama, M., 1994. New

analysis of efficiency based on DEA. In Journal of the

Operations Research Society of Japan, vol. 37, pp.

158-167.

Entani, T., Maeda, Y. and, Tanaka, H., 2002. Dual models

of interval DEA and its extensions to interval data. In

European Journal of Operational Research, vol. 136,

pp. 32-45.

Lins, M. P. E., Novaes, L. F. D. and, Legey, L. F. L.,

2005. Real estate appraisal: A double perspective data

envelopment analysis approach. In Annals of

Operations Research, vol. 138, pp. 79-96.

Anderson, T. R., Hollingworth, K. and, Inman, L., 2002.

The Fixed Weighting Nature of A Cross-Evaluation

Model. In Journal of Productivity Analysis, vol. 17,

pp. 249-255.

Soares de Mello, J. C. C. B., Gomes, E. G., Angulo-Meza,

L.and, Leta, F. R., 2008. DEA Advanced Models for

Geometric Evaluation of used Lathes. In WSEAS

Transactions on Systems, vol. 7, pp. 500-520.

Allen, R., Athanassopoulos, A., Dyson, R. G. and,

Thanassoulis, E., 1997. Weights restrictions and value

judgements in data envelopment analysis: evolution,

development and future directions. In Annals of

Operations Research, vol. 73, pp. 13-34.

Lins, M. P. E. and, Moreira, M. C. B., 1999. Método I-O

Stepwise para Seleção de Variáveis em Modelos de

Análise Envoltória de Dados. In Pesquisa

Operacional, vol. 19, pp. 39-50.

Senra, L. F. A. D. C., Nanci, L. C., Soares de Mello, J. C.

C. B. and, Angulo-Meza, L., 2007. Estudo sobre

métodos de seleção de variáveis em DEA. In Pesquisa

Operacional, vol. 27, pp. 191-207.

Biondi Neto, L., Lins, M. P. E., Gomes, E. G., Soares de

Mello, J. C. C. B. and, Oliveira, F. S., 2004. Neural

data envelopment analysis: A simulation. In

International Journal of Industrial Engineering:

Theory Applications and Practice, vol. 11, pp. 14-24.

Soares de Mello, J. C. C. B., Lins, M. P. E. and, Gomes, E.

G., 2002. Construction of a smoothed DEA frontier. In

Pesquisa Operacional, vol. 28, pp. 183-201.

Soares de Mello, J. C. C. B., Gomes, E. G., Biondi Neto,

L. and Lins, M. P. E., 2004. Suavização da Fronteira

DEA: O caso BCC tridimensional. In Investigação

Operacional, vol. 24, pp. 89-107.

TOWARDS AN INTELLIGENT SYSTEM FOR MONITORING ELECTRICAL ENERGY QUALITY - Foundations and

Motivations

45