Research ArticleSources, Effects, and Modelling of Interharmonics

Hsiung-Cheng Lin

Department of Electronic Engineering, National Chin-Yi University of Technology, No. 57, Sec. 2, Zhongshan Road,Taiping District, Taichung 41170, Taiwan

Correspondence should be addressed to Hsiung-Cheng Lin; hclin@ncut.edu.tw

Received 21 February 2014; Accepted 29 March 2014; Published 5 May 2014

Academic Editor: Her-Terng Yau

Copyright © 2014 Hsiung-Cheng Lin. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Recently, the increasing use of power electronic systems and time-variant nonlinear loads has brought number of powerharmonics/interharmonics, and the power supply quality is therefore seriously threatened.The presence of interharmonics stronglyposes more difficulties in modelling and measuring the distorted waveforms.Therefore, this paper reviews the sources, effects, andmodelling of interharmonics. It provides a variety of crucial phenomena caused by interharmonics.More importantly, it also gives apossible solution for engineers/researchers to use appropriate tools tomeasure interharmonics. Somemethodswith implementationresults are introduced and discussed for details.

1. Introduction

The presence of power system interharmonics has not onlybrought many problems as harmonics but also producedadditional problems. For instance, there are thermal effects,low frequency oscillation of mechanical system, light andCRT flicker, interference of control and protection signals,high frequency overload of passive parallel filter, telecom-munication interference, acoustic disturbance, saturation ofcurrent transformer, subsynchronous oscillations, voltagefluctuations, malfunctioning of remote control system, erro-neous firing of thyristor apparatus, the loss of useful life ofinduction motors, and so forth. These phenomena may evenhappen under low amplitude [1–5].

The leading methods referred to in the relevant literaturefor harmonic penetration studies can be classified as (a) directcurrent injection; (b) harmonic power flow; (c) iterativeharmonic analysis; (d) experimental analog modeling; (e)time-domainmodeling. In principle, it is very easy to includeinterharmonics in the classical model by the main concept ofthe Fourier fundamental periods [1]. Unfortunately, in prac-tice, some concerns may still apply, being briefly concludedas follows. (1) The extension of low-power analog modelsand of time-domain models does have practical difficultieslimit in the use of these models to cases of small system size.(2) The extension of the direct injection method is easy to

obtain inaccurate results. (3)The extension of the harmonicpower flow is very difficult inmodeling nonlinear loads in thefrequency domain when interharmonics are present. (4)Theextension of the iterative harmonic analysis is quite complexin modelling the nonlinear loads [14].

2. Sources of Interharmonics

Basically, interharmonics types can be classified into twocategories. The first type is that interharmonics are locatedaround sidebands of the system frequency and harmonicsdue to a change of amplitude and/or phase caused by arapid current variation of facilities. Actually, it is also thesources of power supply voltage fluctuation. The secondtype is asynchronous switching in static converters usingsemiconductor devices. It is the case that the switchingfrequency is not synchronised with the power system. Somemajor sources are discussed as follows.

(1) Variable-Load Electric Drives. Induction motors usingstator and rotor sloting are one of the sources of interhar-monics. When the motor is working at a constant speed,the interference frequency would occur between 500Hzand 2000Hz. If it is during the acceleration period, therange would be larger. The motor which has nonsymmetricalcharacteristics, for example, rotor without accurate aim, will

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 730362, 10 pageshttp://dx.doi.org/10.1155/2014/730362

2 Mathematical Problems in Engineering

140

120

100

80

60

40

20

040 45 50 55 60

Frequency (Hz)

Curr

ent (

A)

(a) Current spectrum

40 45 50 55 60

25

20

15

10

5

0

Frequency (Hz)

Curr

ent (

A)

(b) Current spectrum

120

100

80

60

40

20

040 45 50 55 60

Volta

ge (V

)

Frequency (Hz)

(c) Voltage spectrum

40 45 50 55 60

Volta

ge (V

)

Frequency (Hz)

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

(d) Voltage spectrum

Figure 1: Spectrum of motor current and voltage: ((a), (c)) complete signal spectrum and ((b), (d)) spectrum removing fundamentalcomponent [6].

cause interharmonics. Figure 1 shows the spectrum of motorcurrent and voltage [6]. Other cases are the drives addingvariable-torque loads such as forge drives, forging hammers,automatic stamp machines, and electric saws.

(2) Double Conversion Systems. In power electronics facilities,it may be a source of interharmonics if the system contains aDC-link connecting two different AC systems. The classicalexamples are variable-speed motor, HVDC, other staticfrequency converters, and so forth. These systems generallycontain both AC/DC rectifier and DC/AC inverter, and DC-link using reactor or capacitor is coupled between two sides.Ideal rectifier with infinite reactor or capacitor only generatescharacteristic harmonics as follows:

𝑓ℎ = (𝑝1𝑛 ± 1) 𝑓, (1)

where 𝑝1 is the pulse number of rectifier, n is integer, and 𝑓is the system frequency.

In practice, the reactor or capacitor is finite so that DC-link will consist of a series component, depending on the typeof inverter, that is, current source inverter (CSI) and voltagesource inverter (VSI) [7].

(a) CSI: the pulse number is 𝑝2, and output frequencyis 𝑓𝑜. The DC ripple will contain the followingfrequency:

𝑓𝑟 = 𝑛𝑝2𝑓𝑜, (2)

where 𝑛 is integer.(b) VSI: using synchronous PWM modulation, the DC

ripple will contain the following frequency:

𝑓𝑟 (𝑚𝑓, 𝑗, 𝑟) =𝑚𝑓𝑗 ± 𝑟

⋅ 𝑓𝑜, (3)

Mathematical Problems in Engineering 3

Inte

rhar

mon

ic am

plitu

de(%

)

0 100 200 300 400 500 600 700 800 900 1000

Interharmonic frequency (Hz)

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

fo = 40Hz

(a) CSI

Inte

rhar

mon

ic am

plitu

de(%

)

1.4

1.2

1.0

0.8

0.6

0.4

0.2

00 100 200 300 400 500 600 700 800 900 1000

Interharmonic frequency (Hz)

fo = 40Hz, mf = 9

(b) PWMmodulated VSI

Figure 2: Current interharmonics [7].

where 𝑗 and 𝑟 depend on themodulation ratio and𝑚𝑓is related to the switching method. In conclusion, themodulated source frequency is

𝑓𝑖 = 𝑓ℎ ± 𝑓𝑟. (4)

Figure 2 indicates interharmonics visible from (a) CSIand (b) VSI, where 𝑓 = 50Hz and 𝑓𝑜 = 40Hz [7].

(3) Cycloconverters. Cycloconverters are one of the majorinterharmonics sources. They are widely applied in rollingmill, linear motor, drives, static-var generator, and so forth.The frequency of generated characteristic interharmonics is

𝑓𝑖 = (𝑝1 ⋅ 𝑚 ± 1) 𝑓1 ± 𝑝2 ⋅ 𝑛 ⋅ 𝑓𝑜, (5)

where 𝑝1 = pulse number of rectifier and 𝑝2 = output pulsenumber; 𝑚, 𝑛 = 0, 1, 2, 3,. . . (integer number); 𝑓𝑜 = outputfrequency of cycloconverters.

Figure 3 shows the current interharmonics of typicalcycloconverters with 6 pulses. Its output frequency is 5Hz [8].

(4) Time-Varying Loads. Time-varying loads can gener-ate interharmonics, including regular or irregular fluctuat-ing loads. Typical regular time-varying loads are Weldermachines, laser printer, integral cycle control instruments,and so forth. The produced interharmonics are dependenton the load frequency. Assume that the system voltage isV(𝑡) = sin(2𝜋𝑓𝑡) and load is 𝑅(𝑡) = 1 − 𝑟 cos 2𝜋𝑓𝑜𝑡, where

0.1

0.2

0.40.60.8

1.0

2

4

6810

20

40

6080100

10 20 30405060 80 100

200

300

500700

1000

Har

mon

ic cu

rren

t (%

)

Harmonic frequency (Hz)

Figure 3: Current spectrum of typical cycloconverters (60Hz) [8].

𝑟 < 1 and𝑓𝑜 is the load frequency.Therefore, the load currentis

𝑖 (𝑡) =V (𝑡)𝑅 (𝑡)

=sin (2𝜋𝑓𝑡)

1 − 𝑟 cos 2𝜋𝑓𝑜𝑡

= sin (2𝜋𝑓𝑡)

× (1 + 𝑟 cos 2𝜋𝑓𝑜𝑡 + 𝑟2cos22𝜋𝑓𝑜𝑡 + 𝑟

3cos32𝜋𝑓𝑜𝑡 + ⋅ ⋅ ⋅ ) .(6)

Extend (6), and it is found that interharmonics areinvolved such as 𝑓 ± 𝑓𝑜, 𝑓 ± 2𝑓𝑜, 𝑓 ± 3𝑓𝑜,. . .. As can beseen, the current interharmonics are unvoidable if 𝑓𝑜 is notsynchronized with 𝑓. The spectrum obtained from the laserprinter is shown in Figure 4 [9].

Arc furnaces are examples of irregular time-varyingloads. They have time-variant and nonlinear characteristics,covering both harmonics and interharmonics. Accordingly,they are very difficult to be modelled using mathematicsequation. Figure 5 shows the current spectrum of arc furnace,where Figure 5(a) is AC-60Hz system, and its interharmonicsare concentrated on the surrounding of the system frequency.On the other hand, Figure 5(b) indicates that the mostdistinct interharmonics are located on the neighboring ofharmonics for DC-50Hz system [10].

(5) Wind Turbines. Wind turbines play a critical role inthe interharmonics source due to the mechanical operation.During continuous operation with a constant speed, thevariation of wind speed and tower shadow effect can causea power line voltage fluctuation. Figure 6 shows the voltagespectrum (logarithm). The outcome is obtained from the

4 Mathematical Problems in Engineering

−3

−2

−1

0

1

2

3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Curr

ent(

A)

Time (s)

(a)

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200

Frequency (Hz)

Curr

ent(

A)

(b)

Figure 4: Laser printer current: (a) waveform, (b) spectrum [9].

analysis of time-domain model (solid line) and frequency-domain model (dotted line). The phenomenon comes fromthe mechanical torque oscillations. It is obvious that majorinterharmonics are focused on the fundamental frequency[11].

(6) Unexpected Sources. Nonlinear load itself cannot generateinterharmonics. Some current interharmonics have the samefrequency as the system voltage, but some of them resultfrom interharmonics voltage modulation. Consequently,when interharmonics voltages appear, these componentswill become the source of interharmonics. Figure 7 revealsthe rectifier current spectrum that is from 50Hz voltagemodulated with 1% 120Hz interharmonic [9]. The outcomealso shows special frequency couples (voltage and current),for example, (20Hz, 20Hz), (20Hz, 120Hz), (120Hz, 20Hz),and (120Hz, 120Hz). It exhibits that 20Hz (or 120Hz)interharmonic voltage can produce very high interharmoniccurrent.

3. Effects of Interharmonics

Themost effect by interharmonics is the light flicker problem.The system voltage may be disturbed as interharmonic volt-age is beyond the limit tolerance, and thus light flicker mayoccur. Figure 8 shows the sensitive area of incandescent lamp.Note that the system frequency is 50Hz. This figure provesthat themost sensitive area is located around the fundamentalfrequency, particularly at a lower frequency range [12].

The influence of RMS voltage of interharmonic on thevoltage fluctuation is briefly discussed as follows.

Consider the power supply contains interharmonic com-ponents:

V (𝑡) = sin (2𝜋𝑓1𝑡) + 𝑎 sin (2𝜋𝑓𝑖𝑡) , (7)

where 𝑓1 is the power supply frequency, 𝑓𝑖 is the interhar-monic frequency, and 𝑎 is the amplitude (p.u.) of interhar-monic.

RMS voltage is defined as

𝑉 = √1

𝑇∫𝑇

0

V2 (𝑡) 𝑑𝑡, (8)

where 𝑇 (= 1/𝑓1) is the period.Assume that interharmonic voltage has 0.2% deviation

over the fundamental voltage. According to (7) and (8), wecan obtain the result shown in Figure 9 [12]. It is foundthat the influence by the interharmonics of high frequencybeyond twice of fundamental frequency is small. However,the interharmonics cannot be ignored for lower frequencycomponents. Note that the system frequency is 50Hz.

4. Modelling of Interharmonics

4.1. DFTMethod. By Fourier theory, any repetitive waveformcan be extended to a series of sine waveforms in differentfrequencies. Harmonic is defined as a component of thesewaveforms. Its frequency is a multiple of fundamental fre-quency. For a distorted waveform 𝑖𝑠(𝑡), it can be representedas

𝑖𝑠 (𝑡) =

∞

∑𝑘=−∞

𝐼𝑠 (𝑘𝜔0) 𝑒𝑗𝑘𝜔0𝑡,

𝐼𝑠 (𝑘𝜔0) =1

𝑇∫𝑡+𝑇

𝑡

𝑥 (𝑡) 𝑒−𝑗𝑘𝜔0𝑡𝑑𝑡,

(9)

where 𝜔0(= 2𝜋/𝑇 = 2𝜋𝑓) is the fundamental angularfrequency and 𝐼𝑠(𝑘𝜔0) is the 𝑘th coefficient.

𝑖𝑠(𝑡) can be converted into a discrete signal 𝑖𝑠[𝑛]. Withdiscrete Fourier transform (DFT), it can be

𝐼𝑠 [𝑘] =1

𝑁

𝑁−1

∑𝑛=0

𝑖𝑠 [𝑛]𝑊𝑘𝑛

𝑁, (10)

where 𝐼𝑠[𝑘] is the magnitude of 𝑖𝑠[𝑛] at 𝑓𝑘 and 𝑓𝑘 = 𝑘/𝑇 and𝑊𝑁 = exp(𝑗2𝜋/𝑁).

Inverting DFT can recover the original signal as

𝑖𝑠 [𝑛] =

𝑁/2−1

∑𝑘=0

𝐼𝑠 [𝑘]𝑊−𝑘𝑛

𝑁. (11)

Assume that 𝑖𝑠[𝑛] is periodical and its period is𝑇. Fourierfundamental angular frequency (Δ𝜔) is defined as

Δ𝜔 =2𝜋

𝑇. (12)

Mathematical Problems in Engineering 5

Frequency (Hz)

0.400

0.350

0.300

0.250

0.200

0.150

0.100

0.050

0.000

(p.u

.)

0 50 100 150 200 250 300 350 400 450 500

(a) AC-60Hz system

I(p

.u.)

0 500 1000 1500 2000 25000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency (Hz)

(b) DC-50Hz system

Figure 5: Current spectrum of arc furnaces [10].

100

10−1

10−2

10−3

10−4

10−5

10−6

40 50 60

Frequency (Hz)

Line

vol

tage

(p.u

.)

Figure 6: Voltage spectrum [11].

200 300 400 500 600 700020

100120

0

0.5

1

1.5

Frequency (Hz)

Figure 7: Rectifier current spectrum [9].

The sampling length is selected as 𝑝 periods (𝑝 > 1 withpositive integer); Δ𝜔 can be rewritten as

Δ𝜔 =2𝜋

𝑝𝑇=𝜔0

𝑝. (13)

20 40 60 80 100

0

0

0.1

0.2

0.3

0.4

0.5

Interharmonic frequency

Inte

rhar

mon

ic v

olta

ge (%

)

Envelope

Flicker sensation area

Figure 8: Sensitive flicker area of incandescent lamp [12].

Interharmonic frequency (Hz)0 50 100 150 200 250 300 350

0

0.05

0.1

0.15

0.2

0.25

Dev

iatio

n (%

)

Figure 9: RMS voltage distortion with 0.2% deviation [12].

Therefore, Fourier fundamental frequency (Δ𝑓) can berepresented as

Δ𝑓 =1

𝑝𝑇=

1

𝑝𝑁𝑠𝑇𝑠=

1

𝑁𝑇𝑠=𝑓𝑠

𝑁, (14)

where𝑁𝑠 ≜ 𝑁/𝑝 and 𝑇𝑠 ≜ 1/𝑓𝑠.

6 Mathematical Problems in Engineering

0 50 100 150 200 250 300 350 400 450 500

0

0.2

0.4

0.6

0.8

1

1.2

Frequency (Hz)

Am

plitu

de

Figure 10: Spectrum analysis using DFT.

Figure 10 shows the waveform spectrum using DFT,where both 281Hz and 353Hz interharmonics are containedbased on (15). Obviously, the DFT may obtain accurateharmonics spectrum but result in incorrect results for inter-harmonics analysis:

𝑠 (𝑡) = sin (2𝜋 ⋅ 60 ⋅ 𝑡) + 0.3 sin (2𝜋 ⋅ 180 ⋅ 𝑡)

+ 0.15 sin (2𝜋 ⋅ 300 ⋅ 𝑡) + 0.2 sin (2𝜋 ⋅ 281 ⋅ 𝑡)

+ 0.1 sin (2𝜋 ⋅ 353 ⋅ 𝑡) .

(15)

4.2. IEC Grouping Method. IEC Standard suggested theinterharmonics measurement method using the concept of“grouping” [13]. The principle is illustrated as the schematicdiagram shown in Figure 11. Based on the Fourier analysis,the sampling window time is 10 periods for 50Hz system. For60Hz system, 12 periods are required.

The definition of harmonic/interharmonic grouping isillustrated as follows.

(a) RMS Value of a Harmonic Group. Within the obser-vation window, it is the square root of the sum ofthe amplitude squares of a harmonic and the spectralcomponents adjacent to it.The energy contents of theneighbouring lines with that of the harmonic properare summed as follows:

𝐺2

𝑔,𝑛+1=𝐶2𝑘−5

2+

4

∑𝑘=−4

𝐶2

𝑘+𝑖+𝐶2𝑘+5

2. (16)

(b) RMS Value of a Harmonic Subgroup. It is square rootof the sumof the amplitude squares of a harmonic andthe two spectral components immediately adjacentto it. The subgroup of output components of theDFT is to sum the energy contents of the frequency

component directly adjacent to a harmonic on theharmonic proper:

𝐺2

𝑠𝑔,𝑛=

1

∑𝑘=−1

𝐶2

𝑘+𝑖,

𝐺2

𝑠𝑔,𝑛+2=

1

∑𝑘=−1

𝐶2

𝑘+𝑖.

(17)

(c) RMS Value of an Interharmonic Group. It is squareroot of the sum of all interharmonics amplitudesquares in the interval between two consecutiveharmonic frequencies:

𝐶9

𝑖𝑔,𝑛+1=

9

∑𝑖=1

𝐶2

𝑘+𝑖. (18)

(d) RMS Value of an Interharmonic Centred Subgroup. Itis the RMS value of the squares of all interharmonicsamplitudes in the interval between two consecutiveharmonic frequencies, excluding frequency compo-nents directly adjacent to the harmonic frequencies:

𝐶9

𝑖𝑔,𝑛=

8

∑𝑖=2

𝐶2

𝑘+𝑖. (19)

It is known that IECmethod is a practical tool to measureinterharmonics because it can reduce the effect of spectrumleakage, also disclosing the level of interharmonics. Onlyshort sampling time (200ms, Δ𝑓 = 5Hz) required is anotheradvantage. However, it has two limitations. (1) Spectrumleakage still exists. (2) It is unable to identify individualcomponent (frequency and amplitude) so it is not suitable insystem diagnostic purpose.

4.3. A Strategy of Leakage Energy Allocation (LEA) Method[15]. The power of the waveform, P, can be expressed by theParseval relation in its discrete form [16] as

𝑃 =1

𝑁

𝑁/2−1

∑𝑛=0

𝑖𝑠[𝑛]2=

𝑁/2−1

∑𝑘=0

𝐼𝑠[𝑘]2. (20)

The power at the discrete frequency 𝑓𝑘 can be expressedas

𝑃 [𝑓𝑘] = 𝐼𝑠[𝑘]2+ 𝐼𝑠[𝑁 − 𝑘]

2= 2𝐼𝑠[𝑘]

2, (21)

where 𝑘 = 0, 1, 2, . . . , 𝑁/2 − 1.Therefore, the 𝑚th harmonic amplitude at the frequency

𝑓𝑘 is expressed as

𝐴𝑚 [𝑓𝑘] = √𝑃 [𝑓𝑘] = √2𝐼𝑠 [𝑘] , (22)

where𝑚 = 1, 2, . . . ,𝑀.The power of the 𝑚th harmonic at 𝑓𝑘 may disperse over

around the 𝑓𝑘 caused by the spectral leakage. By the conceptof grouping, all spilled power within adjacent frequencies of

Mathematical Problems in Engineering 7

Output DFT

Output DFT

Harmonic order

Harmonic order

Harmonic groupHarmonic subgroup Harmonic subgroup

Interharmonic groupInterharmonic group

n + 1 n + 2n

n + 1 n + 2n

G2sg,n =1

∑k=−1

G2g,n+1 = +4

∑k=−4

+ G2sg,n+2 =1

∑k=−1

C2isg,n =8

∑i =2

C2ig,n+1 =9

∑i =1

C2k+5

C2k+i

C2k+iC2k+i

C2k+i

C2k+i

2

C2k−52

Figure 11: Depiction of harmonics/interharmonics [13].

harmonics can be restored into a “group power” [13]. In otherwords, each “group power,” that is, 𝑃∗

𝑚[𝑓𝑘], can be collected

between 𝑓𝑘−Δ𝑘 and 𝑓𝑘+Δ𝑘 as follows:

𝑃∗

𝑚[𝑓𝑘] =

+𝜏

∑Δ𝑘=−𝜏

(𝐴𝑚 [𝑓𝑘+Δ𝑘])2, (23)

where 𝜏 is an integer number that denotes the group band-width.

Each harmonic amplitude can be calculated as

𝐴∗

𝑚[𝑓𝑘] = √𝑃

∗𝑚[𝑓𝑘]. (24)

Figure 12 indicates the energy dispersing around thedominant component. Based on the empirical observation,the relation between sampling length and harmonic dis-persed energy can be classified into two cases.

Case 1. The second stronger amplitude is found to belocated at the right side of the dominant component, that is,𝐴𝑚[𝑓𝑘+1] > 𝐴𝑚[𝑓𝑘−1], due to overlong truncated-windowlength.

Case 2. The second stronger amplitude is located at theleft side of the dominant component, that is, 𝐴𝑚[𝑓𝑘+1] <𝐴𝑚[𝑓𝑘−1], due to insufficient truncated-window length.Based on the inductive method from empirical results, it isfound that the frequency deviation amount has a relationin dispersed energy distribution [15]. Accordingly, the truefrequency of interharmonic can be represented by the domi-nant frequency (𝑓𝑘) plus “frequency deviation” (Δ𝑓𝑘), that is,𝑓𝑘 + Δ𝑓𝑘.

Am[fk]

Am

fk

Frequency (Hz)

Am

plitu

deA

m[f

k−𝜏]

Am[f

k−2]

Am[f

k−1]

fk−1

fk−2

fk+𝜏

fk+2

fk+1

fk−𝜏

Am[f

k+1]

Am[f

k+2]

· · ·· · ·

[fk+𝜏]

Figure 12: Energy dispersing around the dominant component.

The frequency deviation range (FDR) is defined as

Δ𝑓𝑘 =√∑+𝜏

Δ𝑘=1𝐴𝑚[𝑓𝑘+Δ𝑘]

2

√∑0

Δ𝑘=−𝜏𝐴𝑚[𝑓𝑘+Δ𝑘]

2+ √∑

+𝜏

Δ𝑘=1𝐴𝑚[𝑓𝑘+Δ𝑘]

2

⋅ Δ𝑓,

(25)

where Δ𝑓 is a factor of fundamental frequency, that is, 50Hz.It means that Δ𝑓must be 1, 2, 5, 10, 25, and 50.

8 Mathematical Problems in Engineering

According to the analysis of group-harmonic frequencydeviation, the restored amplitude (RA) can be used forretrieving dispersed amplitude, defined as

RA = √+𝜏

∑Δ𝑘=−𝜏

𝐴𝑚[𝑓𝑘+Δ𝑘]2, (26)

where 𝜏 = 0, 1, 2, 3, . . ..The flowchart of the proposed LEA algorithm shown in

Figure 13 is described briefly as follows.

(1) Set 𝑓𝑠,𝑁, 𝜏.

(2) Sample power line waveform 𝑖𝑠(𝑡) using the samplingtime 𝑇𝑓.

(3) Implement DFT.

(4) Set 𝑚 = 1 and determine the number (𝑀) of majorharmonics/interharmonics.

(5) Determine the location of major frequency.

(6) Calculate Δ𝑓𝑘, RA.

(7) Find the frequency and amplitude of the 𝑚th har-monic/interharmonic, that is, 𝑓

𝑚, 𝐴𝑚.

(8) Let𝑚 = 𝑚 + 1,𝑀 = 𝑀 − 1.

(9) Go back to Step (6) until 𝑀 = 0. In other words,the procedure will continue until all major harmon-ics/interharmonics (𝑓

𝑚, 𝐴𝑚) are found.

(10) Go back to Step (2) until the system is requested tostop.

Firstly, a waveform with fundamental frequency drift isconsidered [7, 8]. Equation (27) indicates that 𝑖𝑎 has 0.2Hzdrift at the fundamental frequency (𝑓1). It also contains twointerharmonics (𝑓2 and 𝑓3):

𝑖𝑎 = 𝑎1 sin (2𝜋𝑓1𝑡 + 𝜑1) + 𝑎2 sin (2𝜋𝑓2𝑡 + 𝜑2)

+ 𝑎3 sin (2𝜋𝑓3𝑡 + 𝜑3) ,(27)

where 𝑓1 = 49.6Hz is the fundamental (system) frequencywith 0.2Hz drift. The 𝑓2 and 𝑓3 are interharmonics that are123Hz and 327Hz, respectively. The 𝑎1 = 1.0, 𝑎2 = 0.3, and𝑎3 = 0.15 are their respective amplitudes. The 𝜑1 = 0

∘, 𝜑2 =12∘, and 𝜑3 = 35

∘ are their respective degrees.In order to demonstrate the LEA method, an example

using Δ𝑓 = 1Hz (𝑓𝑠 = 1 kHz, 𝑁 = 1000, 𝑇𝑓 = 1 sec) and𝜆 = 4 is given as follows.

Spectrum of 𝑖𝑎 using Δ𝑓 = 1Hz is shown in Figure 14. Ascan be seen, the spectrum has serious spilled energy aroundthe system frequency due to the frequency drift. On the otherhand, the spectrums of 𝑓2 and 𝑓3 are correct.

(a) System Frequency 𝑓1. According to (25), the FDRbeyond 49Hz is calculated as

Δ𝑓𝑘 =√0.752 + 0.212 + 0.122 + 0.0862

× (√0.0872 + 0.122 + 0.192 + 0.512

+√0.752 + 0.212 + 0.122 + 0.0862)−1

⋅ 1

≈0.799

0.564 + 0.799⋅ 1 ≈ 0.586 ≅ 0.6.

(28)

As a result, 49Hz (𝑓𝑘) plus 0.6Hz (Δ𝑓𝑘) is equal to49.6Hz, matching the real value (𝑓1 = 49.6Hz).

According to (26), the restored amplitude is calculated as

RA = (0.0872 + 0.122 + 0.192 + 0.512

+0.752+ 0.21

2+ 0.12

2+ 0.086

2)1/2

≈ 1.01.

(29)

Consequently, the RA is almost equal to 1.0, matching thereal value (𝑎1 = 1.0).

(b) Interharmonic (𝑓2). The 𝑓2 = 123Hz and 𝑎2 = 0.3can be obtained directly from DFT, matching the realvalues. In this case, no dispersed energy is foundaround the interharmonic (𝑓2), and therefore it is notnecessary for further process.

(c) Interharmonic (𝑓3). Similarly, the 𝑓3 = 327Hzand 𝑎3 = 0.15 can be obtained directly from DFT,matching the real values. No dispersed energy isfound around the interharmonic (𝑓3), and thereforeit is not necessary for further process.

Using different Δ𝑓 with 𝜆 = 4, it can be concludedin Table 1. Obviously, DFT cannot give an accurate solutionexcept 𝑓2 and 𝑓3 components using Δ𝑓 = 1Hz. For theproposed LEA scheme, all components identification usingΔ𝑓 = 1, 5, and 10 can achieve a very correct value foreither frequency or amplitude. On the other hand, Δ𝑓 using25Hz is unable to obtain a satisfactory result due to noremarkable adjacent dispersed energy. Actually, it is clear thatthe sampling time (𝑇𝑓) will be reduced if a large Δ𝑓 up to10Hz is chosen, not paying the cost of accuracy. However, inview of general practice, the risk of reciprocal interferencebetween surrounding harmonics/interharmonics of spilledenergy may arise once a larger Δ𝑓 is used. To reach acompromise, Δ𝑓 = 5Hz is taken into account in this study.

5. Conclusions

In recent years, the increasing generated power harmonicsand interharmonics have caused serious power line pollution.For this reason, the research on harmonics and interhar-

Mathematical Problems in Engineering 9

Start

DFT

Determine Mm = 1

Determine majorfrequency fk

Findfm, Am

m = m + 1

Stop system?

Stop

No

No

Yes

Yes

Set fs, N, 𝜏

M = M− 1

M = 0?

Sample is(t)

Calculate Δfk, R.A.

Figure 13: Flowchart of the LEA scheme.

monics is still a crucial task. Accordingly, this paper hasreviewed major sources and effects of interharmonics. Ithas also illustrated how to apply DFT and IEC groupingmethods in industry. A new leakage energy allocation (LEA)

method is given for an alternative solution. It can recover allspilled leakage energy to obtain its original interharmonicsamplitude. Also, every individual frequency component canbe calculated using the principle of the distribution state of

10 Mathematical Problems in Engineering

Table 1: Result comparison between DFT and LEA.

Real valuesΔ𝑓

Δ𝑓 = 1Hz Δ𝑓 = 5Hz Δ𝑓 = 10HzDFT LEA DFT LEA DFT LEA

𝑓1 = 49.6 (Hz) 49 49.6 45 49.5 40 49.5𝑎1 = 1.0 1.0 1.01 0.99 1.00 1.0 1.02𝑓2 = 123 (Hz) 123 123 120 123 120 123𝑎2 = 0.3 0.3 0.3 0.22 0.29 0.26 0.29𝑓3 = 327 (Hz) 327 327 325 327 320 327𝑎3 = 0.15 0.15 0.15 0.076 0.146 0.13 0.148

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency (Hz)

Am

plitu

de

Figure 14: Spectrum of 𝑖𝑎 using Δ𝑓 = 1Hz.

spilled leakage energy. Its implementation results prove thatthe proposed LEA approach is feasible in terms of rapid andhigh-precision performance.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

References

[1] H. C. Lin, “Accurate harmonic/inter-harmonic estimation usingDFT-based Group-harmonics energy diffusion algorithm,”Canadian Journal of Electrical and Computer Engineering, vol.36, no. 4, pp. 158–171, 2014.

[2] D. Gallo, R. Langella, and A. Testa, “Inter-harmonics. Part 1.Aspects related to modeling and simulation,” in Proceedingsof the 6th International Workshop on Power Definitions andMeasurements under Non-Sinusoidal Conditions, pp. 168–173,Milan, Italy, October 2003.

[3] D. Gallo, R. Langella, and A. Testa, “Inter-harmonics. Part2. Aspects related to measurement and limits,” in Proceedingsof the 6th International Workshop on Power Definitions andMeasurements under Non-Sinusoidal Conditions, pp. 174–181,Milano, Italy, October 2003.

[4] J. Barros, E. Pérez, A. Pigazo, and R. I. Diego, “Simultaneousmeasurement of harmonics, interharmonics and flicker in apower system for power quality analysis,” in Proceedings of the

5th International Conference on Power SystemManagement andControl, pp. 100–105, April 2002.

[5] M. Karimi-Ghartemani and M. R. Iravani, “Measurement ofharmonics/inter-harmonics of time-varying frequencies,” IEEETransactions on Power Delivery, vol. 20, no. 1, pp. 23–31, 2005.

[6] Z. Hanzelka and A. Bien, Harmonics, Interharmonics, AGHUniversity of Science and Technology, 2004.

[7] A. Testa, M. F. Akram, R. Burch et al., “Interharmonics: theoryand modeling,” IEEE Transactions on Power Delivery, vol. 22,no. 4, pp. 2335–2348, 2007.

[8] Interharmonics in Power Systems, IEEE Interharmonic TaskForce, Cigré 36. 05/CIRED 2 CC02 Voltage Quality WorkingGroup.

[9] D. Gallo, R. Langella, andA. Testa, “Desynchronized processingtechnique for harmonic and interharmonic analysis,” IEEETransactions on Power Delivery, vol. 19, no. 3, pp. 993–1001,2004.

[10] C. Vilar, J. Usaola, and H. Amaŕıs, “A frequency domainapproach to wind turbines for flicker analysis,” IEEE Transac-tions on Energy Conversion, vol. 18, no. 2, pp. 335–341, 2003.

[11] R. Carbone, A. lo Schiavo, P. Marino, and A. Testa, “Frequencycoupling matrices for multi-stage conversion system analysis,”European Transactions on Electrical Power, vol. 12, no. 1, pp. 17–24, 2002.

[12] Interharmonics in Power Systems, IEEE Interharmonic TaskForce, Cigré 36. 05/CIRED 2 CC02 Voltage Quality WorkingGroup.

[13] Z. Hanzelka and A. Bien, Power Quality Application Guide,AGH University of Science and Technology, 2004.

[14] R. Carbone, D. Menniti, R. E. Morrison, and A. Testa, “Har-monic and interharmonic distortion modeling in multicon-verter systems,” IEEE Transactions on Power Delivery, vol. 10,no. 3, pp. 1685–1692, 1995.

[15] H. C. Lin, “Inter-harmonic identification using group-harmonic weighting approach based on the FFT,” IEEETransactions on Power Electronics, vol. 23, no. 3, pp. 1309–1319,2008.

[16] IEC, 61000-4-7: 2002 Electromagnetic compatibility (EMC)Part 4. Testing and measurement techniques Section 7: generalguide on harmonics and interharmonics measurements andinstrumentation for power supply systems and equipmentconnected thereto.

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