pq-direction of harmonics

7/31/2019 PQ-Direction of Harmonics

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214 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 1, JANUARY 2003

An Investigation on the Validity of Power-DirectionMethod for Harmonic Source Determination

Wilsun Xu, Senior Member, IEEE, Xian Liu, Member, IEEE, and Yilu Liu, Member, IEEE

AbstractThe power-direction method has been used widelyto identify the locations of harmonic sources in a power system.A number of utility-customer disputes over who is responsible forharmonic distortions have been settled with the help of the method.A closer examination of the method, however, reveals that it is un-able to fulfill the task of harmonic source detection. Case studiescan easily show that the method yields incorrect results. In thispaper, problems associated with the method are investigated usingcase studies and mathematical analysis. The results show that thepower direction method is theoretically incorrect and should notbe used to determine harmonic source locations. The main cause oftheproblemis that thedirection of activepower flow is a function ofthe phase-angle difference between the two sources. The directionof reactive power flow, on the other hand, has a better correlationwith the source magnitudes.

Index TermsHarmonics, harmonic source detection, harmonicsources.

I. INTRODUCTION

WHENEVER significant harmonic voltage or current dis-

tortions are observed in a power system, it is always

useful to find the sources of the distortions. Correct identifica-

tion of harmonic source locations is essential for designing ef-

fective harmonic mitigation means and for determining the re-

sponsibility of the parties involved. The most common situation

that needs harmonic source detection is to resolve the disputesover who is responsible for harmonic distortions at the point of

common coupling between a utility and a customer or between

two customers.

The most common method for harmonic source detection is

the power direction method. This method checks the directionofharmonic powerflow. The side that generates harmonic power is

considered to contain the dominant harmonic source or to have

a larger contribution to the harmonic distortions observed at the

measurement point. This method is apparently sound and has

been used in industry as a tool for many years [1], [2]. A number

of power quality monitors have included it as a key product fea-

ture. However, we have found concrete proofs that the method is

not suitable forharmonic sourcedetection. The method has beenfound theoretically incorrect. In this paper, problems associated

with the method are investigated using case studies and mathe-

Manuscript received April 25, 2001; revised March 2, 2002. This work wassupported by a Grant from the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC).

W. Xu and X. Liu are with the Department of Electrical and Computer En-gineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail:[email protected]).

Y. Liu is with the Department of Ele ctrical Engineering, Virginia PolytechnicInstitute and State University, Bla cksburg, VA 24061-0111 USA.

Digital Object Identifier 10.1109/TPWRD.2002.803842

Fig. 1. Problem of harmonic source detection.

matical analysis. Causes of the problem are identified. In addi-

tion, this paper proposes superposition-based indices to quantify

the contributions of harmonic sources.

II. POWER DIRECTION METHOD

The harmonic source detection problem can be explained

with the help of Fig. 1. In this figure, the disturbance sources

are the customer harmonic source and the utility harmonic

source . and are the harmonic impedances of the re-

spective parties. The circuit is applicable to different harmonic

frequencies (the values will be different). The task of harmonic

source detection is to determine which side contributes moreto the harmonic distortion at the PCC, subject to the constraint

that measurements can only be taken at the PCC.

To determine which side causes more harmonic distortion at

the harmonic order , the power direction method first measures

voltage and current at the PCC and then calculates the following

harmonic power index:

(1)

where and are the harmonic voltage andcurrent atthe

PCC for a particular harmonic number. Since this paper deals

with one harmonic at a time, the subscript that represents har-

monic number will be omitted throughout the paper to avoid

excessive subscripts. The direction of is defined as from

side to the side. Conclusion of the power direction method is

the following.

If , the side causes more th harmonic distortion.

If , the side causes more th harmonic distortion.

III. QUANTITATIVE HARMONIC CONTRIBUTION INDICES

While the power direction method could be intuitively sound,

the term causing more harmonic distortions is vaguely de-

0885-8977/03$17.00 2003 IEEE


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XU et al.: VALIDITY OF POWER DIRECTION METHOD FOR HARMONIC SOURCE DETERMINATION 215

Fig. 2. Determination of harmonic source contributions for th harmonic.

Fig. 3. Decomposition of of the th harmonic into two components.

fined. There is a need to be more precise on this term so that the

effectiveness of any harmonic source detection methods can beassessed. To this end, the principle of superposition is applied to

the th system shown in Fig. 1 [3]. According to the principle,

the contribution of each source to the th harmonic current

can be determined according to Fig. 2 and the following equa-

tions:

(2a)

(2b)

(2c)

where and are the contributions of respective

sources to the PCC current. Again, all quantities in Fig. 2 and (2)

refer to a particular harmonic, the th harmonic. The equations

shown before are phasor equations. There are still some ambi-

guities to define harmonic contributions since (2c) is a phasor

summation. A more precise method is to decompose into

two scalar components as shown in Fig. 3 and (3) [4]

(3)

where is the projection of onto , and is the

projection of onto . Thus, is the algebraic sum-mation of twoscalar components, one dueto customer harmonic

source and the other due to utility source. For example, if

is equal to 3 A and is equal to 7 A, the current is 10 A.

One can say that the customers contribution is 70%.

Both and are scalars and can have opposite signs. If

they have the same sign, the customer and utility harmonics add

up toform . Ifthey haveopposite signs, the negativeone has

the effect of reducing the harmonic flow at PCC. We believe that

the and indices accurately characterize the contributions

of respective harmonic sources to the PCC current. They are

used in this paper to refute the validity of the power direction

method.

A similar index can be defined from the voltage distortion

perspective. The contribution to voltage distortions at PCC by

each harmonic source can be determined as follows

(4a)

(4b)

(4c)

Again, the phasors and need to be projected

onto the axis to provide scalar harmonic contribution in-

dices. The process is similar to what has been applied to the

current indices.

IV. SAMPLE TEST RESULTS

The validity of the power direction method can be tested with

the following simple experiment:

1) Select a set of practical , , and data that cor-

respond to any particular harmonic number. In this paper, wechose the following case:

p.u. p.u.

angle p.u. p.u.

Let the phase angle of vary from 0 to 360 . Phase angle of

is set to 0 as a reference.

2) Check the correlation between the power direction index

and the superposition indices. If the power direction method was

correct, the following condition should always hold:

If then (5)

Fig. 4 shows the results. The figure depicts the variation of

normalized harmonic power , current , and its components

and withrespectto the phase angle of . The normaliza-

tion divides thecurrentsand powerby their respective maximum

values sothat theyreside between the range of 1 to 1 and can

be easily plotted in one chart for comparison. It can be clearly

seen from the figure that the condition shown before (5) does

not always hold true. For example, is always greater than

and the side should be considered as the main harmonic

current contributor. The harmonic power, on the other hand, can

flow in either direction. When the angle is greater than 190 ,

the power direction contradicts to the principle of superposition.

There is an approximate 50% chance that the contradiction canoccur. Fig. 5 is a practical case that can create a situation where

the phase angle between and is far apart. Since there is

no guarantee that the phase angle between and resides in a

certain range, one can conclude that the power direction method

is unsuitable for harmonic source detection.

Test results are also obtained for the voltage distortion index

defined in (4). The results are shown in Fig. 6. It is noted that,

according to the voltage superposition index, the side con-

tributes more to the PCC voltage distortion. This conclusion ap-

plies to all customer current angles. The power direction index,

however, changes sign when the current angle is between 190

and 340 . The contradiction between the two indices is obvious.


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Fig. 4. Correlation between the power direction and current superpositionindices.

Fig. 5. A practical case that can cause large phase-angle difference betweenand .

Fig. 6. Correlation between the power direction and voltage superpositionindices.

V. MATHEMATICAL ANALYSIS

Figs. 4 and 6 can be produced easily for many test cases. The

evidence against the power direction method is overwhelming.

This finding is also supported by the results shown in [5] and[6]. In this section, we try to demonstrate mathematically that

the contradiction does exist. The sources of the contradiction are

also identified and explained.

A. DC Circuit Case

The first case is a general dc circuit shown in Figs. 7 and 8. Al-

though the actual harmonic source detection problem involves

ac circuits, the dc circuit case can reveal key characteristics of

the power direction method. A dc circuit is much simpler to an-

alyze since there is no phasor involved. The dc case can also be

considered a special ac case where the system has resistances

only and the harmonic sources have the same phase angle.

Fig. 7. DC circuit to demonstrate the power direction method ( andoppose each other).

Fig. 8. DC circuit to demonstrate the power direction method ( and addup).

The PCC voltage and current of the circuit can be determined

as follows:

(6a)

(6b)

Since the voltage is always positive, the condition for the

power ( ) to flow from side to side is or

(7)

On the other hand, the superposition current index shows that

if

(8)

or

(9)

holds, the sidecontributes more current than the side. Equa-

tions (7) and (9) are therefore consistent, which implies that the

power direction method works well in this case.

If the customer current changes polarity as shown in Fig. 8,

however, the current becomes positive all of the time. The

condition for the power to flow from side to side is ,

or

(10)

[based on Equation (6b)]. The condition for is still

the same as that shown in (9). This equation does not alwaysagree with (10), depending on the relative size of and .

Therefore, a mathematical proof has been found to show thatthere is a contradiction between the power direction method and

the superposition method. Another conclusion drawn from the

analysis is that the polarity (or phase) difference between the

two sources has more influence on the direction of power flow

than that caused by their magnitude difference.

B. AC Circuit With Reactance Only

The second case is an ac circuit shown in Fig. 1. To sim-

plify the analysis, the impedances are assumed to contain imagi-

nary parts only. The circuit corresponds to a particular harmonic


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frequency. Following the classic power-angle equation for two

source ac circuits, the power flowing from side to side canbe determined as:

(11)

where and are the open circuit th

harmonic voltages of the and sides, respectively. is thephase-angle difference between utility and customer side th

harmonic current sources. The significance of this equation is

the following: the direction of power is controlled by , the

phase-angle difference between two harmonic sources.

For the superposition-based current index, we want to show

at first that condition implies .

According to Fig. 3 and the well known triangular formula, the

superposition currents and can be determined as

follows

(12a)

(12b)

Therefore

(13a)

(13b)

Subtracting the above two equations yields

(14)

This equation shows that if is satisfied,will be true. Note that this is a general conclusion. It

does not rely on the assumption of the impedances having imag-

inary parts only. In the following step, the relationship between

the superposition currents and the source currents is determined.

The principle of superposition shows

(15a)

(15b)

The condition for or to hold

becomes

(16)

The condition just mentioned is not related to phase-angle ,

while (11) is. The powerdirection index is therefore inconsistent

with the current superposition index. Hence, the invalidity of the

power direction index has been demonstrated analytically. The

same process can be used to show the inconsistency between the

power index and voltage-based superposition index.

It is common knowledge for power engineers that the phase

angles of bus voltages mainly affect the flow of active power

while the magnitudes of bus voltages mainly affect the flow of

reactive power. One would, therefore, wonder if the direction of

reactive power could indicate the relative size (i.e., magnitudes)

of two harmonic sources. This question can be analyzed by ex-

amining the reactive power flowing out of source

(17)

where is the phase-angle difference between the two sources.

Since is always less than 1 and if , condition

automatically implies that the direction of reactive

power is from side to the side. In other words, the direction

of reactive power can be used as an (necessary but not sufficient)

indicator to determine which side has a larger voltage source.

The reason that the indicator is not a sufficient one is because

the reactive power can still flow from side to the side if

is greater than but is less than . The phase-angle

, therefore, plays an important role in this case as well. Despite

this restriction, the direction of reactive power is still a more

reliable indicator than the direction of active power in this case.Another important prerequisite for using the direction of reac-

tive power as a harmonic source locator is . While

this condition is generally true at the fundamental frequency, it

may not be true at the harmonic frequencies. This is the main

problem associated with the reactive power direction method.

C. AC Circuit With Resistances Only

It is interesting and important to examine the hypothetical

case where the system and customer impedances are entirely re-

sistive, namely, and . In this case, the branch

resistance consumes power. The amount and direction of the ac-

tive power flowing on the branch are dependent on the location

of measurement. To simplify the problem, we first consider thepower generated or absorbed by harmonic source , as shown

in Fig. 9. For this case, the power flowing out of source is

(18)

where is the phase-angle difference between the two sources.

It can be seen that (18) is very similar to (17). It means that the

direction of active power can be used as an (necessary but not

sufficient) indicator to determine which side has a larger voltage

source for the resistive circuit. The reactive power flow for this

case takes the following form:

(19)

This equation is very similar to (11). Comparing (11) to (19),

and (17) to (18), one can conclude that the characteristics of the

circuit impedance determine which power, or , has

more bearing on the source magnitudes instead of source phase

angles. If the circuit impedance is dominated by reactance, the

direction of reactive power is a better indicator on the relative

magnitude of the two sources. If the impedance is dominated

by resistance, however, the direction of active power is a better

indicator.


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Fig. 9. AC circuit with resistive elements.

The second analysis deals with a general case where the me-

tering point is the PCC. As shown in Fig. 9, the PCC voltage

and current can be determined as follows

(20a)

(20b)

The active power flowing from side to side can be de-

termined as

(21)

The equation just shown reveals that the resistances and

could havea large impacton the signof , orthedirection

of active power. We can quantify the impact by considering the

special case of . For this case, Equation (21) can be

simplified as

(22)

The equation just shown demonstrates clearly that the direc-

tion of active power is affected by the relative size of and

or the location of the meter. If , the power flows

from side to side, even if the two sources have the same

magnitude. This analysis has revealed another impact factor, the

point of measurement, which can influence the reliability of the

power direction method.

VI. CONCLUSION

The concern for power system harmonics is mainly on thedistortion of sinusoidal voltage and current waveforms. Even

with little harmonic power, a distorted waveform can trigger the

malfunction of electronic circuits. It is, therefore, important to

define the contribution of each harmonic source based on the

current and/or voltage parameters. This paper proposes superpo-

sition-based current and voltage indices to quantify the contri-

butions of harmonic sources. Using these indices, the validity of

the power direction method for harmonic source determination

is investigated. Both case studies and mathematicaly analyses

have shown that the power direction method is not suitable for

harmonic source detection. Main findings of this work are sum-

marized as follows.

1) The direction of active power is mainly affected by the

relative phase angle between the two harmonic sources. It has

little bearing on the relative magnitude of the sources. Note that

it is the source magnitudes instead of phase angles that are of

main interest for the harmonic source detection problem.

2) The direction of reactive power, on the other hand, has

a closer relationship to the source magnitudes. If the circuit

impedance is purely reactive, the direction of reactive power isactually a necessary (but not sufficient) condition indicating one

source has a larger magnitude than the other.

3) The conclusions shown above are applicable to circuits

dominated by reactances. If a circuit consists of mainly resistive

components, the conclusions mentioned before are reversed.

Namely, the direction of active power is mainly affected by

the source magnitudes and that of the reactive power by the

phase angles. The implication of this conclusion is that the

characteristics of the circuit impedance ( ratio) will affect

the reliability of the active or reactive power-direction-based

harmonic source detection methods.

4) The metering point or the relative size of the source and

the customer impedances will also affect the direction of eitheractive or reactive powers. This is another important factor that

makes the powe-direction-based methods unreliable.

The (active)power direction method has been used frequently

as a practical method for locating harmonic sources. This work

has shown that the method does not work and there is an urgent

need to develop new harmonic source detection methods.

ACKNOWLEDGMENT

The authors wish to express sincere thanks to M. B. Hughes

of BC Hydro for valuable comments during the course of this

work.

REFERENCES

[1] P. H. Swart, M. J. Case, and J. D. Van Wyk, On techniques for local-ization of sources producing distortion in three-phase networks, Eur.Trans. Elect . Power Eng., vol. 6, no. 6, Nov./Dec. 1996.

[2] L. Cristaldi and A. Ferrero, Harmonic power flow analysis for the mea-surement of the electric power quality,IEEE Trans. Instrum. Meas., vol.44, pp. 683685, June 1995.

[3] H. Yang, P. Porotte, and A. Robert, Assessing the harmonic emissionlevel from one particular customer, in Proc., 1994.

[4] W. Xu and Y. Liu, A method for determining customer and utility har-monic contributions at the point of common coupling, IEEE Trans.Power Delivery, vol. 15, pp. 804811, Apr. 2000.

[5] A. E. Emanuel, Onthe assessment of harmonic pollution,IEEE Trans.Power Delivery, vol. 10, pp. 16931698, July 1995.

[6] M. B. Marz, J. F. Witte, D. L. Williams, and P. M. Thompson, Findingand determining the influence of multiple harmonic sources on a utilitysystem using harmonic measurements, in Power Quality 2000 Conf.,Boston, MA, Oct. 2000.

Wilsun Xu (M90SM95) received the Ph.D. degree from the University ofBritish Columbia, Vancouver, BC, Canada, in 1989.

He is presently a Professor at the University of Alberta, Edmonton, AB,Canada. He was with BC Hydro from 1990 to 1996 as an Engineer. His mainresearch interests are power quality and voltage stability.

Dr. Xu chairs the Harmonics Modeling and Simulation Task Force of theIEEE PES.


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Xian Liu (M95) received the Ph.D. degree from the University of British Co-lumbia, Vancouver, BC, Canada, in 1994.

He is presentlyan Assistant Professorat theUniversityof Alberta, Edmonton,AB, Canada. He was with Nortel Networks from 1995 to 1999 as an ElectricalEngineer. His main research interests are electric machines, optimization, andreal-time software.

Yilu Liu (M91) is an Associate Professor of electrical engineering at VirginiaPolytechnic Institute and State University, Blacksburg.

Her current research interests are power system transients, power quality, andpower system equipment modeling and diagnoses.

Dr. Liu is the recipient of the 1993 National Science Foundation Young In-vestigator Award and the 1994 Presidential Faculty Fellow Award.

pq-direction of harmonics

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