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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 3, MAY 2008 1309

Inter-Harmonic Identification Using Group-HarmonicWeighting Approach Based on the FFT

Hsiung Cheng Lin

AbstractThefast Fourier transform (FFT) is still a widely-usedtool for analyzing and measuring both stationary and transientsignals with power system harmonics in power systems. However,the misapplications of FFT can lead to incorrect results caused bysome problems such as aliasing effect, spectral leakage and picket-fence effect. A strategy of group-harmonicweightingdistribution isproposed for system-wide inter-harmonic evaluation in power sys-tems. The proposed algorithm can restore the dispersing spectralleakage energy caused by the FFT, and calculate the power dis-tribution proportion around the adjacent frequencies at each har-monic to determine the inter-harmonic frequency. Therefore, notonly high-precision in integer harmonic measurement by the FFTcan be retained, but also the inter-harmonics can be identified ac-curately, particularly undersystem frequency drift. The numericalexamples are presented to verify the performance of the proposedalgorithm.

Index TermsDiscrete Fourier transform (DFT), fast Fouriertransform (FFT), group-harmonic, inter-harmonics.

I. INTRODUCTION

POWER system harmonics have been of great concern since

the early 1900s when alternating current was first widely

applied. Some of power electronic devices and industrial con-trollers, for instance, the cycloconverters, induction motors, arc

furnaces, etc., produce inter-harmonics. Excess use of the elec-

tronic controlled equipment in power systems has caused an in-

creasing pollution on both power line current and voltage. As

a matter of fact, such devices can inject current harmonics into

the power line that, in turn, produce voltage harmonics because

of the mains impedance [1]. Inter-harmonics is a type of wave-

form distortion that may severely degrade the performance of

a power system. The resulting symptoms include over heating,

torsional oscillations, CRT flicker, overload of conventional fil-

ters, interference in telecommunication, and so on. The health

state of power network must be therefore closely monitored.Conventionally, Discrete Fourier transform (DFT) method is

efficient for signal spectrum evaluation because of the simplicity

and easy implementation. The use of the fast Fourier transform

(FFT) can reduce the computational time required for DFT by

several orders of magnitude. An improper use of DFT (or FFT)

based algorithms can, however, lead to multiple interpretations

of spectrum [2][4]. For example, if the periodicity of DFT data

Manuscript received March 1, 2007; revised April 16, 2007. Recommendedfor publication by Associate Editor V. Staudt.

The author is with the Department of Automation Engineering, ChienkuoTechnology University, Taiwan, R.O.C. (e-mail: [email protected]).

Digital Object Identifier 10.1109/TPEL.2008.921067

set does not match the periodicity of signal waveforms, the spec-

tral leakage and picket-fence effect will occur. Since the power

system frequency is subject to small random deviations, some

degree of spectral leakage can not be avoided. A number of al-

gorithms, e.g., short time Fourier transform [5], least-square ap-

proach [6][8], Kalman filtering [9], [10], artificial neural net-

works [4], [11], have been proposed to extract harmonics. The

approaches may either suffer from low solution accuracy or less

computational efficiency. None is reported to perform well in

subharmonic identification under system frequency variations

though each demonstrates its specific advantages.Recent techniques for subharmonic estimation are based

on Wavelet transform theory, which exploits time-frequency

characterization of input signal to identify particular harmonics

within subbands of interest. However, this technique requires

a complex procedure, i.e., a calculation in the discrete wavelet

packet transform (DWPT) for the decomposition of waveforms,

and also the analysis of nonzero decomposed components

by continuous wavelet transform (CWT) [12][14]. These

algorithms are complicated and require expensive computation.

Also, only the low frequency bands are subdivided stepwise

to achieve a high resolution in time, whereas low resolution in

high frequency bands.For all above algorithms, the dilemma has not been resolved

to reach the satisfactory solution in a practical application.

That is why the improved FFT-based approaches are still called

for as an important research field even now [15][18]. IEC

61000-4-7 established a well disciplined measurement method

for harmonics. This standard recently has been revised to add

methodology for measuring inter-harmonics [19]. The key

to the measurement of both harmonics and inter-harmonics

in the standard is the utilization of a 10 or 12 cycle sample

window upon which to perform the Fourier transform. How-

ever, the spectrum resolution with 5 Hz is not sufficiently

precise to reflect the practical inter-harmonic locations for both50-and 60-Hz systems. This paper presents inter-harmonic

identification using FFT-based group-harmonic weighting ap-

proach which retains the merits of FFT analysis and extends to

inter-harmonic identification under system frequency variation

environments. This paper is organized as follows. Section II

gives a background of signal analysis computations using

Fourier transform as well as the concept of group-harmonic.

Section III presents the proposed group-harmonic weighting

approach. In Section IV, the model validation with a numerical

example is demonstrated. Performance results under system

frequency drift is also included and discussed. Conclusions are

given in Section V.

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1310 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 3, MAY 2008

II. BACKGROUND OF POWER SYSTEM HARMONIC

MEASUREMENT

A. Signal Analysis Using Fourier Transform

By Fourier theory, any repetitive waveform can be expressed

as a series of various sinusoidal frequencies. Harmonics are

defined as components of a waveform which are multiples of

the fundamental frequency. Using Fourier series expansion, the

distorted (nonsinusoidal) source current (or voltage) waveform

can be expressed as a series of harmonics; therefore, the

response to each harmonic can be determined by the following

equations [20]:

(1)

where

and is the dc component.

In symmetrical systems, one finds symmetry usually so that, and can be expressed as

(2)

where

.

The root mean square (RMS) value of source current is de-fined as shown in (3) and (4) at the bottom of the page, which is

the RMS value of input harmonic current.

The total harmonic distortion (THD) is well-known as the

most important index to evaluate the power system quality. The

THD factor is defined as the ratio of the RMS value of all the

harmonic components and the RMS value of the fundamental

component, shown as follows.

(5)

B. The Concept of Group-HarmonicThe measurement of inter-harmonics is difficult with results

depending on many factors. IEC 61000-4-7 suggests a method

of inter-harmonics measurement based on the concept of the

so-called group [19]. Therefore, initially the concept of

group-harmonic is introduced as follows.

Suppose the waveform is sampled as discrete points

using the sampling rate , i.e., the truncation interval

(second). With the digital signal processing (DSP) tech-

nology, the continuous signal can be converted to a discrete

signal , and then can be transformed by DFT as

(6)

where denotes the discrete Fourier transform of at

frequency , i.e., , and .

(3)

(4)

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LI N: I NTE R- HARMO NIC I DEN TI FI CAT ION US IN G GRO UP- HAR MONI C WEI GHTI NG APP ROAC H BASE D ON THE FF T 1 31 1

The inverse DFT, which allows us to recover the signal from

its spectrum, is given by

(7)

By the Parseval relation in its discrete form, the power of the

waveform, , can be expressed as [21], [22]

(8)

As above, both positive and negative values of spectral com-

ponents are considered to transform the frequency domain sam-

pled signal into a periodic time domain signal. In the case of

actual signals spectral component relevant to symmetrical fre-

quencies are complex conjugates each other. However, mostreal-world frequency analysis instruments display only the pos-

itive half of the frequency spectrum because the spectrum of a

real-world signal is symmetrical around dc. Thus, the negative

frequency information is redundant.

Therefore, the power at the discrete frequency can be ex-

pressed as [22]

(9)

where .

The RMS value of the harmonic amplitude at the discrete

frequency is

(10)

The power of the harmonic at may disperse over a fre-

quency band around the due to the spectral leakage. Hence,

the total power of harmonics within the adjacent frequencies

around can berestored into a group power [3]. Each group

power, i.e., , is collected between and as

the following equation:

(11)

where is an integer number and denotes the group bandwidth.

Consequently, each harmonic amplitude can be estimated as

(12)

Indeed, amplitudes of the spectral components in the DFT

(or FFT) analysis are related to the DFT algorithm. For this

reason there is a correlation between spectral amplitudes

and frequency of the actual spectral component. Applying

the group power identification will solve the problems in

dispersing spectral leakage energy, arising from measuringinter-harmonics or drifted system frequency in power systems.

An interesting way to view this phenomenon is to observe the

FFT results, for details in Section III-B. Most leakages can be

collected into one group and are considered as though they

were all at the dominant harmonic frequency. The amplitude of

inter-harmonics (and/or subharmonics) can be thus identified.

Additionally, the deviation of harmonic frequency adjacent to

the centre frequency is proportional to the group-harmonicpower distribution that creates a group-harmonic weighing

method as the following section.

III. PROPOSED GROUP-HARMONIC WEIGHTING APPROACH

Inter-harmonics in voltage and current waveforms are

frequency components that are not integer multiples of the

fundamental frequency. Subharmonics are a special case of

inter-harmonics for frequency components less than the power

system frequency. Subharmonics or inter-harmonics analysis

using FFT can not achieve an accurate outcome with the spec-

tral leakages. Nevertheless, the linearity relationship betweensubharmonic (inter-harmonic) frequency and group-harmonic

power distribution is found to be proportional according the

induction of empirical observation, for some examples refer-

ring to Section III-B. This scientific background of deduction

leads to the concept of the proposed group-harmonic weighting

(GHW) method.

A. Model of the Group-Harmonic Weighting Approach

In this section, for the determination of inter-harmonics

(and/or subharmonics) components, i.e., frequency and ampli-

tude, the model of the group-harmonic weighting algorithm

is developed with a deduction rule based on the empiricaloutcomes using FFT. This model also extends the basic idea of

group concept that has been mentioned by IEC 61000-4-7

and some papers [3], [19], [21], [22]. For further illustrations,

some typical examples are demonstrated on Section III-B.

Initially, this model classifies the frequency of inter-har-

monics in the decimal point into two situations, i.e., in small

frequency deviation and large frequency deviation. Small fre-

quency deviation includes 0.1 to 0.5 Hz in the decimal point,

e.g., 37.1 to 37.5 Hz. On the other hand, the large frequency

deviation is the frequency that is larger more than 0.5 Hz in

the decimal point, e.g., 37.6 to 37.9 Hz. Inter-harmonics in

small and large frequency deviation are shown in Figs. 1 and 2,respectively. According to the effect of spectrum analysis using

FFT, the second stronger amplitude is found to be located at the

right side of the dominant amplitude, i.e., ,

for the small frequency deviation (less than or equal to 0.5 Hz).

For the large frequency deviation (more than 0.5 Hz), the

second stronger amplitude is located at the left side of the

dominant amplitude, i.e., .

The frequency of inter-harmonic can be defined as the centre

frequency plus frequency deviation, i.e., , where

denotes F.D.R., as shown in (13), at the bottom of the next

page.

Generally, the system frequency may be at a drift situation,i.e., the system frequency is not exact 50 Hz. As a result, the

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Fig. 1. Frequency spectrum of inter-harmonics for small frequency deviation.

Fig. 2. Frequency spectrum of inter-harmonics for large frequency deviation.

restored amplitude (R.A.) that is in fact a recovered amplitude

of inter-harmonic is defined as

(14a)

where denotes the centre frequency of inter-harmonic.

The RA at the system frequency (R.A.S.F.) is defined as

(14b)

where it is a special case at Hz. Note that using FFT with

normalization, R.A.S.F. is n ot equal to in c ase

of system frequency drift, but R.A.S.F. is equal to

with no system frequency drift. Additionally, normalization

in this context is to unify the amplitude of system (fundamental)

frequency.

Consider a particular circumstance with no system frequency

drift, i.e., the system frequency is exact 50 Hz. The R.A. can be

therefore simplified as

(14c)

B. Observation of SubHarmonic Frequency and Amplitude

Analysis Using FFT

Different cases of subharmonic frequency and amplitude

analysis using FFT are investigated and discussed in this sub-

section. For easy demonstration, initially only one harmonic

component is discussed as the following equation:

(15)

where , and is a noninteger number. is the

amplitude, and is the phase.

The frequency lines occur at interval as

(16)

There is strong correlation between the sampling rate, sampling

point and the accuracy of the FFT analysis. For the following

discussions, the sampling rate is set as 1 kHz, and

so that Hz. In deed, should be chosen as a highersampling frequency, e.g., 4 kHz, to satisfy the Nyquist theorem

according to IEC 60160 if the analysis of a power system signal

should be performed up to the 40th harmonic.

1) Consider no System Frequency Drift: The system (funda-

mental) frequency, i.e., 50 Hz, is assumed as an ideal case, and

its amplitude is normalized as 1.0 in the following case exam-

ples.

Case 1: , and Hz, for small

frequency deviation case.

The results of spectrum analysis using FFT is shown in Fig. 3.

Based on these results, frequency deviation ratio (F.D.R.) be-

yond the 23 Hz can be calculated as (17), shown at the bottomof the next page.

The frequency of subharmonic in this case is thus equal to

23 Hz plus 0.1 Hz, i.e., 23.1 Hz, with consistency as the actual

frequency.

(13)

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Fig. 3. Frequency spectrum at 23.1 Hz with no system frequency drift.

R.A. in subharmonic can be calculated as (18), shown at the

bottom of the page.

The amplitude of subharmonic is almost equal to 0.5, with

consistency as the actual amplitude.

Case 2: , and Hz, for middle



Based on these results, F.D.R. beyond the 23 Hz can be calcu-

lated as (19), shown at the bottom of the page.




frequency, shown in (20) at the bottom of the page.



Case 3: , and Hz, for large




lated as (21), shown at the bottom of the next page.

F : D : R :

=

p

0 : 0 5 6

2

+ 0 : 0 2 7

2

+ 0 : 0 1 8

2

+ 0 : 0 1 4

2

+ 0 : 0 1 1

2

+ 0 : 0 0 9

2

p

0 : 0 0 7

2

+ 0 : 0 0 8 6

2

+ 0 : 0 1 1

2

+ 0 : 0 1 5

2

+ 0 : 0 2 3

2

+ 0 : 0 4 4

2

+ 0 : 4 9

2

+

p

0 : 0 5 6

2

+ 0 : 0 2 7

2

+ 0 : 0 1 8

2

+ 0 : 0 1 4

2

+ 0 : 0 1 1

2

+ 0 : 0 0 9

2

=

0 : 0 6 7

0 : 4 9 3 + 0 : 0 6 7

= 0 : 1 2 0

=

0 : 1 (17)

R : A : =

p

0 : 0 0 7

2

+ 0 : 0 0 8 6

2

+ 0 : 0 1 1

2

+ 0 : 0 1 5

2

+ 0 : 0 2 3

2

+ 0 : 0 4 4

2

+ 0 : 4 9

2

+ 0 : 0 5 6

2

+ 0 : 0 2 7

2

+ 0 : 0 1 8

2

+ 0 : 0 1 4

2

+ 0 : 0 1 1

2

+ 0 : 0 0 9

2

= 0 : 4 9 7

=

0 : 5 (18)

F : D : R : =

p

0 : 3 2

2

+ 0 : 1

2

+ 0 : 0 6

2

+ 0 : 0 4 2

2

+ 0 : 0 3 2

2

+ 0 : 0 2 6

2

p

0 : 0 2 8

2

+ 0 : 0 3 3

2

+ 0 : 0 3 9

2

+ 0 : 0 4 9

2

+ 0 : 0 6 7

2

+ 0 : 1 1

2

+ 0 : 3 2

2

+

p

+ 0 : 3 2

2

+ 0 : 1

2

+ 0 : 0 6

2

+ 0 : 0 4 2

2

+ 0 : 0 3 2

2

+ 0 : 0 2 6

2

=

0 : 3 4 5

0 : 3 4 9 + 0 : 3 4 5

= 0 : 4 9 7

=

0 : 5 (19)

R : A : =

p

0 : 0 2 8

2

+ 0 : 0 3 3

2

+ 0 : 0 3 9

2

+ 0 : 0 4 9

2

+ 0 : 0 6 7

2

+ 0 : 1 1

2

+ 0 : 3 2

2

+ 0 : 3 2

2

+ 0 : 1

2

+ 0 : 0 6

2

+ 0 : 0 4 2

2

+ 0 : 0 3 2

2

+ 0 : 0 2 6

2

= 0 : 4 9 1

=

0 : 5 (20)

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The frequency of subharmonic in this case is equal to 23 Hz

plus 0.9 Hz, i.e., 23.9 Hz, with consistency as the actual fre-quency, shown in (22) at the bottom of the the page.



2) Consider the System Frequency Drift: The system (fun-

damental) frequency is assumed with a small drift to 50.2 Hz,

and its amplitude is normalized as 1.0. The results of spectrum

analysis using FFT at this fundamental frequency is shown in

Fig. 6.

R.A. in system frequency (R.A.S.F.) can be calculated as (23)

and (24), shown at the bottom of the page.

The system frequency in this case is found to be 50 Hz plus

0.2 Hz, i.e., 50.2 Hz, with consistency as the actual frequency.

Fig. 6. Frequency spectrum at 50.2 Hz.

As can be seen, the R.A.S.F increases 0.06, compared with the

actual amplitude. Accordingly, the R.A. needs to be modified.

On the other hand, the calculation of F.D.R. still remains ac-curate without modification. This effect is closely investigated

using the same case examples as above, for more details as fol-

lows.

Case 1: , and Hz, for small


The results of spectrum analysis using FFT that is influenced

by the system frequency drift is shown in Fig. 7. Based on these

results, F.D.R. beyond the 23 Hz can be calculated as (25),

shown at the bottom of the next page.



frequency.

F : D : R :

=

p

0 : 4 9

2

+ 0 : 0 4 5

2

+ 0 : 0 2 4

2

+ 0 : 0 1 7

2

+ 0 : 0 1 3

2

+ 0 : 0 1

2

p

0 : 0 0 6 3

2

+ 0 : 0 0 7 5

2

+ 0 : 0 0 9 2

2

+ 0 : 0 1 2

2

+ 0 : 0 1 6

2

+ 0 : 0 2 5

2

+ 0 : 0 5 4

2

+

p

0 : 4 9

2

+ 0 : 0 4 5

2

+ 0 : 0 2 4

2

+ 0 : 0 1 7

2

+ 0 : 0 1 3

2

+ 0 : 0 1

2

=

0 : 4 9 3

0 : 0 6 3 + 0 : 4 9 3

= 0 : 8 8 7

=

0 : 9 (21)

R : A : =

p

0 : 0 0 6 3

2

+ 0 : 0 0 7 5

2

+ 0 : 0 0 9 2

2

+ 0 : 0 1 2

2

+ 0 : 0 1 6

2

+ 0 : 0 2 5

2

+ 0 : 0 5 4

2

+ 0 : 4 9

2

+ 0 : 0 4 5

2

+ 0 : 0 2 4

2

+ 0 : 0 1 7

2

+ 0 : 0 1 3

2

+ 0 : 0 1

2

= 0 : 4 9 7

=

0 : 5 (22)

R : A : S : F : =

p

0 : 0 2 8

2

+ 0 : 0 3 5

2

+ 0 : 0 4 4

2

+ 0 : 0 5 9

2

+ 0 : 0 8 7

2

+ 0 : 1 6

2

+ 1 : 0

2

+ 0 : 2 5

2

+ 0 : 1 2

2

+ 0 : 0 7 5

2

+ 0 : 0 5 6

2

+ 0 : 0 4 5

2

+ 0 : 0 3 8

2

= 1 : 0 6 (23)

F : D : R : =

p

0 : 2 5

2

+ 0 : 1 2

2

+ 0 : 0 7 5

2

+ 0 : 0 5 6

2

+ 0 : 0 4 5

2

+ 0 : 0 3 8

2

p

0 : 0 2 8

2

+ 0 : 0 3 5

2

+ 0 : 0 4 4

2

+ 0 : 0 5 9

2

+ 0 : 0 8 7

2

+ 0 : 1 6

2

+ 1 : 0

2

+

p

0 : 2 5

2

+ 0 : 1 2

2

+ 0 : 0 7 5

2

+ 0 : 0 5 6

2

+ 0 : 0 4 5

2

+ 0 : 0 3 8

2

=

0 : 3 0 0

1 : 1 9 2 + 0 : 3 0 0

= 0 : 2 0 1

=

0 : 2 (24)

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Fig. 7. Frequency spectrum at 23.1 Hz under system frequency drift.

R.A in subharmonic can be modified, i.e., divided by

R.A.S.F., as (26), shown at the bottom of the page.

The amplitude of subharmonic is thus obtained as 0.5, with


Case 2: , and Hz, for middle








frequency.

R.A in subharmonic can be modified, i.e., divided by

R.A.S.F., as (28), shown at the bottom of the page.



Case 3: , and Hz, for large


F : D : R : =

p

0 : 0 5 4

2

+ 0 : 0 2 3

2

+ 0 : 0 1 4

2

+ 0 : 0 0 9

2

+ 0 : 0 0 6 4

2

+ 0 : 0 0 5

2

p

0 : 0 1 2

2

+ 0 : 0 1 4

2

+ 0 : 0 1 6

2

+ 0 : 0 2 1

2

+ 0 : 0 2 9

2

+ 0 : 0 5 2

2

+ 0 : 5 3

2

+

p

0 : 0 5 4

2

+ 0 : 0 2 3

2

+ 0 : 0 1 4

2

+ 0 : 0 0 9

2

+ 0 : 0 0 6 4

2

+ 0 : 0 0 5

2

=

0 : 0 6 2

0 : 5 3 3 + 0 : 0 6 2

= 0 : 1 0 4

=

0 : 1 (25)

R : A :

=

p

0 : 0 1 2

2

+ 0 : 0 1 4

2

+ 0 : 0 1 6

2

+ 0 : 0 2 1

2

+ 0 : 0 2 9

2

+ 0 : 0 5 2

2

+ 0 : 5 3

2

+ 0 : 0 5 4

2

+ 0 : 0 2 3

2

+ 0 : 0 1 4

2

+ 0 : 0 0 9

2

+ 0 : 0 0 6 4

2

+ 0 : 0 0 5

2

R : A : S : F :

=

0 : 5 3 7

1 : 0 6 4

= 0 : 5 0 4

=

0 : 5 (26)

F : D : R :

=

p

0 : 3 3

2

+ 0 : 1 1

2

+ 0 : 0 6

2

+ 0 : 0 4

2

+ 0 : 0 2 9

2

+ 0 : 0 2 3

2

p

0 : 0 3 5

2

+ 0 : 0 4

2

+ 0 : 0 4 6

2

+ 0 : 0 5 7

2

+ 0 : 0 7 7

2

+ 0 : 1 2

2

+ 0 : 3 5

2

+

p

0 : 3 3

2

+ 0 : 1 1

2

+ 0 : 0 6

2

+ 0 : 0 4

2

+ 0 : 0 2 9

2

+ 0 : 0 2 3

2

=

0 : 3 5 6

0 : 3 8 6 + 0 : 3 5 6

= 0 : 4 8 0

=

0 : 5 (27)

R : A : =

p

0 : 0 3 5

2

+ 0 : 0 4

2

+ 0 : 0 4 6

2

+ 0 : 0 5 7

2

+ 0 : 0 7 7

2

+ 0 : 1 2

2

+ 0 : 3 5

2

+ 0 : 3 3

2

+ 0 : 1 1

2

+ 0 : 0 6

2

+ 0 : 0 4

2

+ 0 : 0 2 9

2

+ 0 : 0 2 3

2

R : A : S : F :

=

0 : 5 2 6

1 : 0 6 4

= 0 : 4 9 4

=

0 : 5

(28)

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The resultsof spectrum analysis using FFT is shown in Fig. 9.





frequency.

R.A. in subharmonic can be modified, i.e., divided byR.A.S.F., as (30), shown at the bottom of the page.



For all above examples, similar outcomes are obtained using

a variety of phase angles for F.D.R. and R.A. This effect

confirms that the phase angle isinsensitiveto thisproposed

GHW model.

IV. MODEL VALIDATION WITH A NUMERICAL EXAMPLE

The proposed GHW algorithm has been tested by the synthe-

sized line signal (voltage/current) to verify the effectiveness of

inter-harmonic analysis. The following example is used to illus-

trate the harmonic analysis of a distorted waveform [23][25].

(31)

where and are 19.2, 50.0,98.7, 250, 350,

450 Hz, respectively.

The line signal has a fundamental frequency of 50 Hz and ascaled amplitude of 1 V. A noninteger subharmonic (19.2 Hz)

below the 50 Hz is to be considered, reflecting a possible subhar-

monic case. Another inter-harmonic (98.7 Hz) beyond the 50 Hz

is also included. The 250, 350, and 450 Hz components are cov-

ered in the synthesized waveform so that all possible worse sit-

uations in general power systems can be represented.

Generally, the system frequency drift is another concern in

power systems because it may vary slightly from time to time

due to the change of system loads. This effect, in deed, influ-


ences the traditional FFT-based spectrum analysis. The pro-

posed GHW method is thus developed to extract not only inter-harmonic frequencybut also its amplitude accurately even under

system frequency drift. Note that the inter-harmonic phase angle

was not involved in the estimation in this study. In this section,

note that kHz, , i.e., Hz, as discussed

in Section III.

A. Spectrum Analysis With no System Frequency Drift

Initially, the subharmonic, Hz, and

, was tested by varying at Hz. The results,

as shown in Table I, are found very close to the actual value,

no matter in frequency and amplitude. However, as the group

bandwidth increases, the amplitude identification is moreaccurate, but the extracted frequency may be slightly apart from

the actual value with a larger . Furthermore, inter-harmonics,

Hz, and , have been also tested for

the case beyond the system frequency (50 Hz). Clearly, the re-

sults, shown in Table II, indicate a full consistency with Table I.

Actually, a variety of situations covering all low and high fre-

quency inter-harmonics have been examined for the evaluation

of the proposed GHW algorithm. All performance results reveal

highly accurate outcome that is similar as above.

F : D : R :

=

p

0 : 5 3

2

+ 0 : 0 5 1

2

+ 0 : 0 2 9

2

+ 0 : 0 2 2

2

+ 0 : 0 1 9

2

+ 0 : 0 1 7

2

p

0 : 0 0 8 1

2

+ 0 : 0 0 9

2

+ 0 : 0 1

2

+ 0 : 0 1 3

2

+ 0 : 0 1 7

2

+ 0 : 0 2 6

2

+ 0 : 0 5 6

2

+

p

0 : 5 3

2

+ 0 : 0 5 1

2

+ 0 : 0 2 9

2

+ 0 : 0 2 2

2

+ 0 : 0 1 9

2

+ 0 : 0 1 7

2

=

0 : 5 3 4

0 : 0 6 7 + 0 : 5 3 4

= 0 : 8 8 8

=

0 : 9 (29)

R : A : =

p

0 : 0 0 8 1

2

+ 0 : 0 0 9

2

+ 0 : 0 1

2

+ 0 : 0 1 3

2

+ 0 : 0 1 7

2

+ 0 : 0 2 6

2

+ 0 : 0 5 6

2

+ 0 : 5 3

2

+ 0 : 0 5 1

2

+ 0 : 0 2 9

2

+ 0 : 0 2 2

2

+ 0 : 0 1 9

2

+ 0 : 0 1 7

2

R : A : S : F :

=

0 : 5 3 8

1 : 0 6 4

= 0 : 5 0 5

=

0 : 5

(30)

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TABLE IRESULTS OF SUBHARMONIC SPECTRUM AT f = 5 0 : 0 HZ

TABLE II

RESULTS OF INTER-HARMONIC SPECTRUM ATf = 5 0 : 0

HZ

B. Spectrum Analysis With the System Frequency Drift

To verify the effectiveness of the proposed GHW algorithm

in a real-world industrial application, the system frequency drift

has been tested further in this subsection. Firstly, assume the

worst cases that the system frequency may drift from 49.5 to

50.5 Hz. The results, set , are shown in Table III.

The measured frequency is found very accurate as the actualone. Similarly to Section IV-A, the extracted frequency may

be slightly apart from the actual value with a larger . On the

other hand, R.A.S.F. will increase dramatically as the system

frequency deviates considerably from the normal frequency, i.e.,

Hz. This phenomenon confirms the limitation of FFT

analysis.

R.A. using the R.A.S.F. modification, can achieve very accu-

rate results, as shown in Table IV for the subharmonic frequency

( , and Hz) at Hz. Also, the

inter-harmonic frequency ( , and Hz)

identification beyond the system frequency presents the similar

outcome, shown in Table V. The proposed GHW algorithm, in-deed, has been verified using a wide range of system frequency

TABLE IIISPECTRUM RESULTS OF THE SYSTEM FREQUENCY DRIFT

TABLE IVRESULTS OF SUBHARMONIC SPECTRUM AT

f = 4 9 : 8

HZ

TABLE VRESULTS OF INTER-HARMONIC SPECTRUM AT

f = 4 9 : 8

HZ

drift from 49.5 to 50.5 Hz. All performance results reveal thatthe GHW algorithm still carry out a successful performance.

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C. Selection of Group Bandwidths

The power of the harmonic at may disperse over a fre-

quencyband around the if the FFT isused asa spectrum anal-

ysis tool. Therefore, each group power should be collected

between and to ensure satisfactory results achiev-

able. Obviously, the larger group bandwidth can restore all

leakages and regain the actual amplitude. However, with a large

bandwidth the group power may include considerable har-

monic contents at distant frequencies because neighboring nom-

inal harmonics may be dispersed widely. Additionally, the ex-

tracted frequency may be slightly apart from the actual value

with a larger due to the influence of neighboring harmonic

contents. As a consequence, the group bandwidth should be

chosen as large as possible for obtaining an accurate amplitude

but small enough to avoid the overlap between two neighboring

harmonic groups. Based on the results by this proposed GHW

model, the group bandwidth is suggested to be chosen as

to reach the compromise.

D. Error Comparison With FFT Analysis and the GHW Model

The error comparison between the FFT and the GHW al-

gorithm using the same sampling rate ( Hz) and

sampled point is concluded in Tables VIVIII,

except chosen as 4 in the GHW model. Tables VI and VII

indicate the error comparison of subharmonic and inter-har-

monic spectrum using FFT and GHW at Hz and

Hz, respectively. The measured frequency using

FFT can cause up to 0.5 Hz error. For example, is found as

19.0 Hz using FFT when the actual subharmonic frequency is

19.5 Hz at Hz, shown in Table VI. The identified am-plitude is found as 0.19 using FFT when its actual value

is 0.3. On the other hand, is obtained as 19.51 Hz, and

is 0.29 with the GHW model. Table VII, under system fre-

quency drift at Hz, shows similar results as Table VI.

Table VIII shows the error comparison of spectrum analysis

using FFT and GHW with a variety of system frequency drift,

i.e., Hz. As can be seen, the measured ampli-

tude using FFT is more far from actual value as the system fre-

quency drift increases. Also, the identified frequency is found

either or Hz, but the actual frequency

may be located between 49.5 and 50.5 Hz. For example, is

found as 50.0 Hz and using FFT when the systemfrequency is drifted as 50.4 Hz, and its actual amplitude

is equal to 1.0. With the GHW model, it is evident that

the amplitude and frequency identification of inter-harmonic is

very close to the actual value. For instance as the above case,

is extracted as 50.41 Hz, and its amplitude identification is

correct, i.e., .

E. Discussion About Industrial Implementation

The proposed GHW algorithm is an advanced FFT-based

method for inter-harmonic analysis. Accordingly, for the prac-

tical application of the methodology in industry, the GHW al-

gorithm can be easily added to such FFT-based measurementdevices that are still widely used currently. Alternatively, the

TABLE VICOMPARISON OF SUBHARMONIC AND INTER-HARMONIC SPECTRUM

USING FFT AND GHW ATf = 5 0 : 0

HZ

TABLE VII

COMPARISON OF SUBHARMONIC AND INTER-HARMONIC SPECTRUMUSING FFT AND GHW AT f = 4 9 : 8 0 HZ

TABLE VIIICOMPARISON OF SPECTRUM ANALYSIS USING FFT AND GHW

WITH THE SYSTEM FREQUENCY DRIFT

GHW algorithm is suitable for online microprocessed imple-

mentation if the FFT is available with I/O interface capability in

the chip.

Based on the proposed group-harmonic process, the inter-har-

monic amplitude and frequency under different system fre-quency drifts are found to be still within a very low error. This

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discovery indicates that the GHW algorithm is adaptive to any

variation of system frequency in power systems. Consequently,

most measurement devices that have some inherent errors

caused by inter-harmonic leakages can be fixed by using the

GHW algorithm, and the robustness of the algorithm can be

thus guaranteed.

V. CONCLUSION

Although the DFT (or FFT) has certain limitations in the har-

monic analysis, it is still widely used in industry today. The

inter-harmonic identification using FFT-based group-harmonic

weighting approach has been developed to extract the inter-har-

monic amplitude and frequency accurately and efficiently. The

test results confirm that the proposed GHW method can even

adapt to a system frequency variation circumstance, which can

never be done by conventional DFT/FFT. There is no theoret-

ical restriction in the locations of inter-harmonic components

while the group bandwidth of each inter-harmonic should

be chosen appropriately. Moreover, the GHW methodology hasbeen implemented successfully by a LabVIEW programming

so that it can be easily extended to other software packages like

microprocessor for online measurement. Additionally, the pro-

posed GHW can provide an advanced improvement for most

measurement devices with some inherent errors because of the

spectrum leakages caused by inter-harmonics.

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[4] H. C. Lin, Intelligent neural network based adaptive power line con-ditioner for real-time harmonics filtering, Proc. IEE Gener. Transm.

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670675.[13] Z. Wenhui, L. Lili, Y. Xiuqing, and G. Weikang, Wavelet transformbased new methods for voltage flicker signal and harmonic detection,in Proc. 5th Int. Conf. Power Electron. Drive Syst., Nov. 1720, 2003,vol. 1, pp. 805810, PEDS 2003.

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Hsiung Cheng Lin was born in Chang Hua, Taiwan,R.O.C., on September 3, 1962. He received the B.S.degreee from National Taiwan Normal University,Taiwan, R.O.C., in 1986, and the M.S. and Ph.D.degrees from Swinburne University of Technology,Australia, in 1995 and 2002 respectively.

His employment experience included Lecturerand Associate Professor at Chung Chou Institute ofTechnology, Taiwan, R.O.C. He is currently a Pro-

fessor in the Department of Automation Engineeringat Chienkuo Technology University (CTU), Taiwan,

R.O.C. His special fields of interest include power electronics, neural network,network supervisory system and adaptive filter design.

Dr. Lin was honored an excellent teaching award from CTU in 2005, 2006and 2007. He was also nominated and included in the First Edition of Who sWho in Asia 2007 and 10th Whos Who in Science and Engineering 2007.

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