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Abstract--This paper gives an overview of techniques to

determine the harmonic content of the currents of three-phase

power systems. The different methods, such as the Fourier

transformation, the FBD-method, the instantaneous reactive

power theory, the dq-transformation and the multiple rotating

transformation method are described in detail. Advantages and

disadvantages are given and the most favorable methods are

compared.

As an example, these methods are used for the control of a

Shunt Active Power Filter to investigate them in detail. For this

purpose, the filter is simulated in the PSpice environment. Two

different methods to calculate the harmonic content in three-

phase power systems are implemented in a special DSP-model to

simulate the time discrete behavior of processors. The simulation

results are presented in this paper.

Index Terms - Harmonic Measurement, Harmonics, Active

Filters.

I. INTRODUCTION

With the increasing use of non-linear loads such as diode-

bridge rectifiers, adjustable speed-drives and cycloconverters

the generation of harmonic currents has steadily increased. Thedisadvantages [1] of these harmonic currents are well known.

Among others, additional heat in power cables, transformers,

electrical machines and capacitors is generated due to the

increased RMS current. In three-phase four-wire systems, the

triplen harmonics all add up in the neutral conductor which

can lead to an unacceptable high current in this conductor. The

power factor is typically reduced, and harmonics accelerate the

system’s aging process causing extra costs.

In the past, different passive, active or a combination of

both, the so-called hybrid filters have been investigated [2, 3].

Especially the Shunt Active Power Filter (SAPF) has proven to

be very effective and is therefore available on the commercialmarket and has been installed in industrial power systems.

The control of these filters requires the calculation of the

harmonic content of the load currents. In this paper different

methods such as the Fourier analysis, FBD-Method, multiple

rotating transformation method, the instantaneous reactive

power theory, and the dq-transformation are described in detail

and are discussed. Based on this discussion two methods are

selected for comparison with the help of the simulation tool

PSpice.

II. SHUNT ACTIVE POWER FILTER

The single phase diagram of a Shunt Active Power Filter is

depicted in Fig. 1. The source supplies a harmonic generating

load. A SAPF is installed in parallel to the load to compensate

the harmonics making the source current is sinusoidal. In this

case, the active filter current iAF has to be equal to the

harmonic content of the load current iL. Therefore, depending

on the control, the load current iL or the source current iS can

be measured and its harmonic content can be calculated. In all

examples in this paper the load current is measured. The

calculated harmonic content is used as the current command

signal for the SAPF.

In this paper, the authors mention that the goal of an activefilter is to generate sinusoidal source currents. The other

option is to control the SAPF in a way that the source currents

become proportional to the source voltage, i.e. the filter tries

to model a resistive load. Opinions vary about which control is

the preferred one. It is the authors' opinion that from the utility

customers view-point the sinusoidal source current generation

is the most favorable because in this case active power is

drawn only from the fundamental voltage.

iS i

L

iAF

Fig. 1: Single-phase diagram of the Shunt Active Power Filter installed in a

power system.

For the investigations in this paper we assume a three-

phase 50 Hz / 400 V voltage source with a source inductance

of 50 µH. This voltage source feeds a non-linear load whichconsists of a B6 diode bridge rectifier with a series connected

resistor (R = 1 Ω) and inductor (L = 8.3 mH). Capacitive loads

are not considered in this paper because it is known that series

active filters are better suited for compensating harmonics

generated by voltage inducing loads.

The SAPF consists of a current controlled three-phase

IGBT voltage source inverter, filter inductors ( Lf = 500 µH) to

smooth the inverter's output current and a 800 V dc-link. In the

simulations, the dc-link is assumed to be ideal. The filter

current is controlled with a hysteresis controller.

The control (including the calculation of the harmonic

content of the load current) of the SAPF is implemented in aDSP-model to simulate the time discrete behavior of

microprocessors or DSPs [11]. This model allows to write the

An Overview of Methods to Determine the

Harmonics in Three-Phase SystemsJoep Jacobs, Dirk O. Detjen, Rik W. De Doncker

Institute for Power Electronics and Electrical Drives

Aachen University of Technology, Germany

[email protected]

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complete control algorithm in C -code. The sample frequency

of the controller is f S = 12.8 kHz which results in a number of

256 cycles per fundamental period.

III. HARMONIC CURRENT CALCULATION

In literature, different methods to calculate the harmonic

content of voltages and currents have been presented during

the past 25 years [4 - 11]. In this paper the authors will

describe the following methods:

• Fourier Transformation (FFT[5], DFT [6])

• Fryze-Bucholz-Depenbrock (FBD)-Method [7]

• Instantaneous Reactive Power Theory [4]

• dq-Transformation [9, 10]

• Multiple dq-Transformations [8]

A. Fourier Transformation

The result of a Fourier Transformation is a continuous

frequency spectrum of a function in the time domain. The

discrete equivalent is called the Discrete FourierTransformation (DFT). For active filter applications the source

or load currents are measured. The DFT of a time discrete

current i[n] sampled with frequency f s is determined as

∑−

−=

−⋅=

1 2 j

][][a

N an

N

kn

enik I (1)

k = 0, 1, 2, ..., N -1 (2)

The spectrum ][k I represents all harmonic values of the

fundamental frequency f 0 up to ( N -1)th. The number of samples

taken during one fundamental period is

0

S

f

f N = (3)

With the Euler-equation we obtain for the spectrum

∑∑−

−=

−

−=

⋅⋅−

⋅=

11 2sin][ j

2cos][][

a

N an

a

N an N

knni

N

knnik I (4)

With the help of this equation the harmonics that need to

be compensated can be determined. The inverse DFT is used

to generate the inverter’s reference signals in the time domain.

An advantage of the DFT is that all harmonics can be

identified individually. Hence, harmonics that need to be

compensated can be selected and compensated. Furthermore,the DFT can be implemented in single-phase systems as well

as multi-phase-systems.

Mayor disadvantage, however, is that a large amount of

computing power is required. Especially, in multi-phase

systems the frequency spectrum has to be calculated for every

phase. On the other hand, if all harmonics need to be

compensated only the amplitude of the fundamental has to be

determined, so saving calculation time. Although reducing the

required computing power drastically the DFT method still

requires powerfull processors. Another drawback is that in

transient conditions the harmonics can not be accurately

detected rapidly, because the spectrum is calculated from thesamples of the previous fundamental cycle. In commercial

harmonic filter products the DFT is not applied due to these

disadvantages.

B. Fryze-Bucholz-Depenbrock (FBD)-Method

The FBD-Method [7] tries to shape the source currents iSa,

iSb and iSc in such a way that they become proportional to the

source voltages eSa, eSb and eSc. Therefore, the source voltagesand the load currents are measured. With the help of these

quantities the equivalent conductance can be calculated:

SiSi

LiSi

i

ee

ieG

⋅⋅

= , i = a, b or c (5)

The index i indicates the three phases a, b or c. The terms

in the formula can be calculated with

Pt t it eT

ie

t

T t

=⋅⋅=⋅ ∫ −

d)()(1

LiSiLiSi(6)

2d)()(

1 2

SiSiSiSi

E t t et e

T ee

t

T t

=⋅⋅=⋅ ∫ −

(7)

This means that the minimum current required by the load

can be determined:

)(Sii

*

imin,,Lt eGi ⋅= (8)

To achieve unity power factor the current difference has to

be compensated. Therefore, the active filter has to inject a

current:

)()(SiiiL,

*

iAF,t eGt ii ⋅−= (9)

The advantages of this method are that it is very easy to

implement and that it requires little computing power. This

method can be used in single- as well as in multi-phase

systems. Especially, in stationary systems the damping

performance is quite good.

On the other hand this method can not distinguish between

reactive and distortion power, i.e. it is not possible to

compensate the harmonic load currents only. Single harmonics

such as the 5th and the 7th can not be detected and

compensated, too. A second disadvantage is that source

voltage harmonics result in proportional source current

harmonics, causing additional power flow. Lastly, the

equivalent conductance G is gained with voltage and currentsamples of the last fundamental period, which means that this

method requires one fundamental period to respond to

transient conditions making the active filter rather slow.

C. The instantaneous reactive power theory

The instantaneous reactive power theory [2 - 4] or pq-

theory (Fig. 2) is already used in a lot of filter applications.

The phase voltages ea, eb, and ec and the load currents iLa,

iLb and iLc are measured and transformed into α-β orthogonal

coordinates.

⋅ −−−⋅=c

b

a

e

e

e

e

e

330

1

21

2

1

21

2

1

32

β

α (10)

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⋅

−−−

⋅=

Lc

Lb

La

L

L

i

i

i

i

i

330

1

21

21

21

21

32

β

α (11)

According to [4], the instantaneous real power pL and the

instantaneous imaginary power qL

on the load side can be

calculated with the help of these equations.

⋅

−

=

L

L

L

L

i

i

ee

ee

q

p(12)

According to [2], the instantaneous real power pL and the

instantaneous imaginary power qL are divided into three

components.

L L L L

L L L L

qqqq

p p p p

~ˆ

~ˆ

++=++=

(13)

In the case the source voltage is undistorted, the dc

component L p corresponds to the fundamental active power

and L

q corresponds to the fundamental reactive power. The

high frequency (≥ 150Hz) components L

p~ and L

q~ correspond

to the harmonics and the low frequency (0.9 Hz < f < 150 Hz)

components L

p and L

q correspond to the subharmonics and

to negative sequence components of the three-phase system.

For example, with a high pass filter with a cut-off frequency of

150 Hz the harmonic components L

p~ and L

q~ can be

determined.

=

∗

∗

L

L

q

p

q

p~

~

(14)

By an inverse transformation, the current command signals

can be calculated.

⋅

−

−

⋅

−−−⋅=

∗

∗

∗

∗

∗

q

p

ee

ee

i

i

i 1

3

3

01

2

1

21

2

1

21

32

AF3

AF2

AF1

(15)

The instantaneous power theory is implemented in the

majority of today’s harmonic filters. Its easy implementation

and relatively small computing power are the driving forces

for this. Furthermore, it can respond rapidly in transient

conditions mainly depending on the high- or band-pass filter.The disadvantage of this method is that the source of the

harmonics, the voltage source or the load current, can not be

determined and that in the case of voltage and current

harmonics not all harmonics are detected. In fact this method

tries to modulate a resistive load, which could be the preferred

control strategy. Minor disadvantages of this method is that it

is only applicable in three-phase power systems and that the

harmonics can not be identified separately.

Fig. 2: Instantaneous Reactive Power Theory

D. dq-Transformation

Another method to generate the harmonic content of the

currents is the dq-transformation [9, 10]. This method is based

on a space vector transformation. The time dependent load

currents iLa, iLb, and iLc are measured in a stationary reference

frame and subsequently transformed into the rotating dq-

reference frame. The dq-system rotates with the fundamental

frequency f 0 of the ac supply. The phase angle θ is provided by

a PLL as mentioned in the next paragraph. The principle of

this control is depicted in Fig. 3

Fig. 3: Calculation of the harmonic content with dq-transformation

⋅

+

−

+

−

=

Lc

Lb

La

Lq

Ld

3

2sin

3

2sinsin

3

2cos

3

2coscos

3

2

i

i

i

i

i (16)

The dq-signals of the load currents can be divided into dc-

and ac-components.

LqLqLq

LdLdLd

~

~

iii

iii

+=

+=(17)

The active dc-component Ldi represents the positivesequence fundamental active power of the load currents, the

reactive dc-componentLq

i represents the positive sequence

fundamental of the reactive power of the load currents, and the

ac-componentsLd

~i and

Lq

~i represent the total harmonic content

of the load current.

If the dq-components are high pass filtered, the dc-

components are eliminated, and one obtains the total harmonic

content.

Lq

*

Lq

Ld

*

Ld

~

~

ii

ii

=

=(18)

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The reference signals *

AFai , *

AFbi , and *

AFci for the current

controller are calculated by multiplying the dq-components *

Ldi

and *

Lqi with the inverse transformation matrix.

⋅

+

+

−

−=

*

Lq

*

Ld

*

AFc

*

AFb

*

AFa

3

2sin

3

2cos

3

2sin

3

2cos

sincos

i

i

i

ii (19)

The phase angle θ of the source voltage eSa has to be

determined to synchronize the reference signals with the

fundamental components of the phase voltages. This phase

angle and the fundamental frequency f 0 are provided by a

Phase Locked Loop (PLL).

The three phase voltages eSa, eSb, and eSc are measured and

the reactive or q-component eq of these voltages is calculated

[9, 10].

⋅

+

−=

Sc

Sb

Sa

q3

2sin

3

2sinsin

e

e

e

e (20)

When the angle θ is equal to the phase angle of the voltage

vector, the q-component of the fundamental is zero. Hence, by

adjusting the angle θ the q-component can be controlled to

zero, realizing a tracking of the phase angle of the phase

voltages.

The dq-transformation method has the same pros and cons

as the instantaneous reactive power theory. Except for the fact

that if the source voltages are tracked properly the sourcevoltage harmonics do not have an impact on the filtering

performance. On the other hand this phase tracking requires

more computing power.

E. Multiple dq-transformations

Fig. 4: Multiple dq-transformations

The algorithms of the previously described dq-

transformation can also be used to transform the time

dependent currents into a system rotating with the harmonic

frequency f h. Now, the harmonic h is represented as a dc-

quantity. This dc-component can be filtered by a low pass

filter and transformed into the stationary reference frame. If

several harmonics have to be compensated severaltransformations have to be made. An example is shown in

Fig. 4.

The advantage is that the harmonics can be determined

separately and precisely. The computing power demand is, of

course, larger, compared to the standard dq-transformation.

IV. SIMULATION RESULTS

The instantaneous reactive power theory and the dq-

transformation have been implemented in the PSpice

simulation environment. The control algorithms are

implemented in a DSP model created with the help of the

Device Equations-option of PSpice [11]. Herein the algorithms

can be written in C-code. The DSP model simulates the timediscrete behavior of DSPs and microprocessors. All values are

chosen as described in chapter II.

800

-800

-400

400

0

iL

/ A

t / ms

20 40

Fig. 5: Load current

Fig. 6: Filter current iF and source current iS with instantaneous reactive

power theory

The load current iL is depicted in Fig. 5. The filter and sourcecurrents in the case the instantaneous reactive power theory is

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used are depicted in Fig. 6. The impact of the source voltage is

shown very clearly. Because there are no in- and output filters

used in the simulation. In practice the measured source

voltages are filtered at the input. Further, the inverter will have

an output LC- or LCL-filter. Fig. 7 shows the results of the dq-

transformation implementation. Again, no in- and output filters

are implemented, but with this method the source currentbecomes nearly sinusoidal. Only the steep current rise can not

be compensated because of the dynamic limits of the SAPF.

For the dq-transformation no in- and output filters are

necessary to obtain sinusoidal source currents.

Fig. 7: Filter current iF and source current iS with dq-transformation

V. CONCLUSION

In this paper different algorithms to calculate the reference

signals for SAPF are reviewed. Advantages and disadvantages

are given. Based on these advantages and disadvantages two

algorithms, the instantaneous power theory and the dq-

transformation, have been selected for implementation in

three-phase power systems.

Although the implementation of the instantaneous power

theory is more simple because it does not require a PLL the

dq-transformation does not require in- and output filters to

generate sinusoidal source currents. In this way the SAPF canbe made more cost effective.

VI. REFERENCES

[1] "IEEE recommended practices and requirements for harmonic control in

electrical power systems", IEEE 519-1992

[2] Akagi H.: "Trends in active power line conditioners", IEEE

Transactions on Power Electronics, Vol.9, 1994

[3] Akagi H.: " New trends in active filters for power line conditioning",

IEEE transactions on industry applications Vol. 32, No. 6, 1996

[4] Akagi H., Kanazawa Y., Nabae A, "Instantaneous reactive power

compensators comprising switching devices without energy storage

components", IEEE transactions on industry applications Vol. 20, No. 3,

1984

[5] Zhang F., Geng Z., Yuan W., "The algorithm of interpolating windowed

FFT for harmonic analysis of electric power systems", IEEE transactions

on power delivery Vol. 16, No. 2, 2001

[6] Lev-Ari H., T h x v ü Ã H Ã Lin S., "Application of staggered

undersampling to power quality monitoring", IEEE transactions on

power delivery Vol. 15, No. 3, 2000

[7] Depenbrock M, Skudelny H.-Ch., "Dynamic Compensation of

Nonactive Power Using the FBD-Method. Basic Properties

Demonstration by Benchmark Examples", IEEE Second International

Workshop on Power Definitions and Measurements under Non-

Sinusoidal Conditions, Stresa, Italy, 1993

[8] Bojrup M., Karlsson P., Alaküla M., Gertmar L., "A multiple rotating

integrator controller for active filters", ABB

[9] United States Patent US5,648,894, Jul. 15, 1997, De Doncker R.W.:

"Active filter control"

[10] Jacobs J., Detjen D., De Doncker R. W.: A new hybrid filter versus a

shunt active power filter, EPE 2001, Graz (Austria), 27-29.8.2001[11] Detjen D., Schröder S., De Doncker R.W.: "Embedding DSP control

algorithms in PSpice", COMPEL, Blacksburg/Virginia, 2000

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