Journal of Power and Energy Engineering, 2015, 3, 262-268
Published Online April 2015 in SciRes.
How to cite this paper: Peng, J.C., Zhou, J. and Jiang, H. (2015) A Liability Division Method for Harmonic Pollution Based on
Line-Transferred Power Components. Journal of Power and Energy Engineering, 3, 262-268.
A Liability Division Method for Harmonic
Pollution Based on Line-Transferred
Power Components
Jianchun Peng*, Jun Zhou, Hui Jiang
College of Mechatronics and Control Engineering, Shenzhen Univer si ty , Shenzhen, China
Email: *
Received February 2015
The existing l iability d ivis ion me th ods f or h armonic pollution are either inexp licit or incompl ete
in physic al mea ning. To comp ensate th ese d efects , two new method s are p roposed b ased on
line-transferred p ower co mponen ts in this p ap er. At first , al l h armo nic sou rces are rep resented by
ideal equ ivalent curr ent s ou rce, line cur ren t c ompo nents an d bu s vol tag e c omp onents of a source
are determ ined by stimul ati on of this s ou rce with all other sou rces d isabled . Then, the line -tr an s-
ferred po wer comp onen t owin g to a source under all sou rce s act i on tog ethe r is determi ned by the
theor y of line-tra nsferr ed power compo nents, and called source’s line-transferred power compo-
nent. At last , th e li abili ty of a sourc e f or line-end h armonic pollution is divided by t wo meth ods:
the r ati on of th e s ou rce ’s line-transfe rred ac tive powe r comp onent to the to tal l ine-t ransfe rred
power, and the ratio n of projection of the source’s line-tran sfe rred complex power compon ent to
absolute value of the total line-transferred complex power. These two metho d s are taken into ac-
count no t only h ar mon i c v ol tage but als o ha rmonic c u rren t in the li abili ty di vision. Simulation re-
sults show tha t the pr oposed liability division meth od based on active power c ompon ent is the
most eff ective and id e al one.
Harmonic Pollution, Liability Division, Line-T r ansfer red Powe r, P ow er C ompon ent, Grid
1. Introduction
With the developme nt of mode rn indust ry and the use of a large number of power electro nic devices, the har-
monic sources have been increasing in p ower networ k. They seriously influence the no rmal wo rk of sensitive
loads and precision devices in the grid [1]-[4]. For the power quality pollution caused by harmo ni c waves, a
mana gement sc heme of reward s and punishments is suggested in [5]. Therefore, a comprehensive, reasonable
and quantitative liability division method of a source for harmonic pollution is crucial under the coexistence of
several harmonic sources.
*Corresponding author.
J. C. Peng et al.
Cons i der i ng t he volt age sensi tivity of electrical equipment, many researchers proposed a liability division
met hod for harmonic pollution based on harmonic voltage. A non-invasive liability division met hod was pro-
posed in [6] for ha rmo ni c pollutio n based on the fa st compone nt s of harmo ni c cur rent injected by harmonic
sources and the statistical features of indep endent rand om vectors. A liability division method was proposed in
[7] for assessi ng harmonic pollution based on the improvement of traditional least squares method and M-esti-
mation robust regres si on. It overcomes the d i sadvanta ges of the traditional least squares method, such as the lo w
accuracy, sensiti vit y to singular data. A liabilit y divisio n method was proposed in [8] for q ua ntify the har monic
pollution, wh ich is based on the estimated harmonic impedance and b ack ground harmonic volt a ge using com-
plex least squares method. It overcomes the short a ge of the linear re gr ess ion metho d by using the real and im-
aginary components respectively to estimate the harmo ni c i mpedance and quantify the har moni c liability.
The re are some other researchers proposing a liability division method fo r harmonic po llution based on har-
monic currents. A liability divisio n me thod was proposed in [9] for harmonic pollutio n from the perspective of
harmonic c urrents based on QR-RLS ( Recursive Least-Squar e s Method based on QR decomposition). A liabilit y
division method was proposed in [10] for harmonic pollut io n, whic h apportions the voltage and curr ent harmon-
ic step by step considering the filter, initial p hase angles of har monic voltage and cur rent as well as the number
of harmonic so ur c e s.
In addition, from the viewpoint of harmonic p ower, a qualit a tive liabil it y division method for harmonic pollu-
tion was proposed in [11]. The method evaluated the liabilit y of each harmoni c sour c e qualitative l y according to
the power r e sp onse produced by the actio n alone of this source. Compared to the har monic voltage or har monic
curr ent based meth od in which the V and I characteristics are not invol ved completely, t h is qualitative liabilit y
division me thod co mpensates this drawback. However, it is a challenging problem that how to quantitati vel y di-
vide the harmonic po llution lia bility among sources based on harmo nic power, because the power does not meet
the superposition the orem.
The electrical characteristics of harmonics are reflected by not only the volt age (or electric field) but also t he
curr ent (or magnetic field). It is the power (there is no any other qua nt i ty) that invol ves both of them (vol t a ge
and current). So only the q uan titative l iab ilit y division method for harmonic pollutio n based on harmonic power
is the most complete and explicit one. In this paper, e mployi ng o ur achievements, two q u antitati ve liabilit y divi-
sion methods for harmo ni c po llution based on the ha rmonic power are proposed. Simulation and anal ysi s show
that the quantitat ive liab ilit y division me thod for harmonic po llution based on line-transferred active power
component is the most desirable o ne .
2. Concept of Har monic Po llu tio n Liabil ity
In a power gri d, there ofte n coexist several harmonic sources. Fig ure 1 sh ows a point of common coupling
(PCC) connected to several lines. The harmonic voltage and current as well as complex power o verline end-
denoted by
, are respo nse s under all harmonic sources action together . T he level of harmoni c
pollution liabilit y of a source should be decided by the ration of the sou rce’s component contrib ution to t he sum
of component contributions over a ll sour c e s.
For a harmonic wave o ve r line end-
(or the PCC), the liability of a source for harmonic pollution is defined
as the ration of the source’s line -transferred compo nent con trib ution to the sum of compone nt contributio ns over
all sources, and br iefly called thi s source’s harmonic p o llutio n liability.
Electrical quantities representing harmonics are harmonic c urrent, volt age and power. Currently, there are
onl y two q ua ntita tive l iab ilit y division methods for harmonic pollution, one is based on har moni c cur rent and the
other is based on harmo ni c volta ge. As s ume that there are n harmonic sources in a grid. For line end-
, a
source’s harmo ni c pollutio n li ab ilit y based on harmo ni c volt age is calculated by
... ...
Fig ure 1. A PCC connected to several lines.
J. C. Peng et al.
= ×
 
. (1)
is the harmonic pollutio n liab ilit y of source-k
( )
over line end-
is the projection
of phasor
(response of harmonic voltage c ompone nt over line end-
under sour ce-k action alone) on
(total re sponse of harmonic volt age over line end-
A source’s harmonic po llu tion liab ilit y based on harmonic curr ent is calculated by
= ×
 
. (2)
is the har monic po llu tion lia b ilit y of source-k over line end-
is the projection of phasor
sponse of har monic curre nt compone nt o ver line end-
under source-k action alone) on phasor
(total re-
sponse of har monic c urre nt o v er line end -
As the voltage is an important power qualit y index, the research of voltage-based liability division method for
harmonic po llution is relati ve l y deeper than that of current-based one. The level of harmonic vo l t age (current)
will increase sharply when series (p ar alle l) harmonic impedances mat ch in a grid. So the harmoni c vo ltage- or
curr ent-based method can’t reflect completely t he electrical characteristics of harmo ni cs. In addition, the calcu-
lation of voltage - or current-based harmonic po llut ion liability needs the projec tion of component on the total,
whi ch ma kes no t only the ph ysical meaning inexplic it but also the dive rge nce in the level of harmonic pollution
liabilities big [12]. Thus, it wo u ld be more complete and reasonable to take into account both harmonic voltage
and current , or take into account harmonic power, in determining har monic po llutio n liabil ity. However, the re is
no power-based qua ntitat ive liab ilit y divisio n meth od fo r harmonic pollution till now. Thi s is because “power
does not meet the superposition theorem”. In this paper, employing our achievements, t wo liabilit y division me-
thods for ha rmonic po llut ion based on harmonic power are proposed.
3. Liabil ity D ivis ion M ethod for Ha r m o n i c Pollution Based on Power
3.1. Theory of Line-Transferred Power Component
By circuit theory, p ower does not meet the superposition theore m. In order to quantitatively divide the harmo ni c
pollution liabilit y a mong so urces based on power, source-k-driven line-transferred complex power compo nent
under coexistence of all harmonic sources must be determined at fir st. A formula for determining source-k-driven
line-transferred complex power component is derived and proved in [13] [14] based on the additive, effe ct ive-
ness and symme t ry, wh ich complies wi th t he circuit l aws wit hout any assumptio n.
0.5 0.5
kkk kk
′′′ ′′
= =
= +
 
. (3)
kk k
′′ ′
= +
 
is the so urce-k-driven line-e nd-
-transferred complex power component.
are the real and imagina ry parts of
is the conjugatio n of
. Obviously, the sum of line-end-
transferred complex power components over all so urces is equal to the total complex powe r ove r line-end -
. (4)
′′ ′
= +
 
is the total complex power over line -end-
under all sources action together .
are the real and imagi nar y parts of
With formul a (3), the ha rmonic po llution liab il it y can be quantitativel y divid ed among harmo nic sources
based on power.
3.2. Liability Division Method for Harmonic Pollution Based on Line-Trans f e rre d Active
Power Component
The active power is the physical quantity really reflecting the capability of power transmission in a grid, while the
reactive power is just the amplitude of the power travelling back and forth in the grid. A source’s harmonic po l l u-
tion liability based on line-transfer red active power component can be determined by the following fo rmul a.
J. C. Peng et al.
= ×
 
. (5)
is the harmo ni c p o llution li ab ilit y of source-k ove r line end-
based on line-transferred active power
3.3. Liability Division Method for Harmonic Pollution Based on Line-Trans f e rre d Complex
Power Component
Similar to the quantitative liabili ty division metho ds for harmonic pollutio n ba se d on voltage or c ur re nt , the
projection of phasor
on phasor
can be used to quantitatively divide the harmonic po llut ion lia bil-
ity of sour ce-k over line end -
. As a res ul t , a source’s harmonic pollutio n liab ilit y ba sed on line-transferred
complex powe r component can be determined by
= ×
 
. (6)
is the harmonic pollution liability of sourc e-k ove r l i ne e nd -
based on line -transferred complex pow-
er component.
is the proj ec tio n of phaso r
on phaso r
4. Case Study
The two ne wl y proposed methods for qua ntit ativ e div isio n of har mo ni c pol lutio n liability, represented by (5)
and (6), are based on line-transferred ac ti ve -a nd complex -power components, and called P-method and
S- method, respectively. A ca se st ud y is performed to show the effectiveness of the two methods. And the
simulation res ult s by the volt a ge - and current-based methods (respectively called V-method and I-method)
are also given for compariso n.
The IEEE 6-bus system shown in Figu re 2 is used for t he te st. T he system contains thre e ge ne r at o r s, six
bus, and seven transmi ssio n lin es. Li ne impeda nces (in p.u.) at rated fre q ue ncy are also s hown in the fig ur e.
Assume tha t the re are three harmo ni c sour c e s (their freq uenc y is 5 time s of the rated) located at busse s 2
and 3 as well as 5, and denot ed by H2 and H3 as well as H5, respectively. In the test, all har moni c so urc e s
are represented by ideal eq ui vale nt cur re nt sourc e, all loads in the harmo n i c do mai n are represented by im-
pedance model II as in [15]. The har mon ic curre nt s of the t hr ee har moni c so urc e s are 0.3780 + j0.2823,
0.8193 + j0.6608 and 0.9450 + j1.0318 p.u., respectively.
In order to obviously sho w the features of the four harm oni c poll utio n liability division methods (P-me-
thod, S-method, V-method and I-method), only the simulation result s of the six line s, 2-1, 2-3, 3-6, 4-2, 5-6,
5-4, are selec ted a nd li sted in Table 1-3.
For intuitiveness, the bar graphs of one sourc e’s harmonic pollution liabilities over individual lines by different
methods are sh own in Figure 3-5.
Look at t he H3 ’s har mo n i c p oll ution liab ilit ies s h own in Figure 3 and Ta bl e 1: By I- method, the ran ge of
the har mo ni c pol lutio n liab ili tiesi s [–58.681%, 176.646%], the b i gg es t of all. T he stand a rd de viatio n of the m
Fig ure 2. The IEEE 6-bus system.
Bus 1Bus 2Bus 3
Bus 4Bus 5Bus 6
source H2
source H3
source H5
0.08 +j0.36 0.02 +j0.12
0.28 +j0.64
0.12 +j0.52
0.07 +j0.30
0.03 +j0.22
J. C. Peng et al.
Table 1. Liabilities of H3 for line-end harmonic pollution by different methods.
Li n e P-method (%) S-method (%) V-method (%) I-method (%)
2-1 10.158 9.672 38.15018.806
2-3 106.932 107.398 38.150 176.646
3-6 –8.3218.321 42.03758.681
4-2 9.744 4.805 28.41518.806
5-6 –10.05720.036 18.60858.681
5-4 9.5450.099 18.60818.806
Table 2. Liabilities of H2 for line-end harmonic pollution by different methods.
Li n e P-method (%) S-method (%) V-method (%) I-method (%)
2-1 0.734 0.754 18.05416.545
2-3 –18.81018.730 18.05455.514
3-6 4.609 4.030 16.6748.615
4-2 0.4861.377 13.79116.545
5-6 4.1530.026 8.5638.615
5-4 0.734 0.754 18.05416.545
Table 3. Liabilities of H5 for line-end harmonic pollution by different methods.
Li n e P-method (%) S-method (%) V-method (%) I-method (%)
2-1 89.107 89.573 43.796 135.351
2-3 11.878 11.332 43.79621.132
3-6 103.712 104.292 41.288 167.296
4-2 89.770 96.572 57.794 135.351
5-6 105.904 120.063 72.829 167.296
5-4 90.088 104.090 72.829 135.351
Figure 3. Bar graphs of H3’s harmonic pollution liabilities versus lines.
J. C. Peng et al.
Figure 4. Bar graphs of H2’s harmonic pollution liabilities versus lines.
Figure 5. Bar graphs of H5’s harmonic pollution liabilities versus lines.
is 0.885, also the biggest of all. It indicat es that the I-method is the most unre a so na b l e and extreme method.
By V-method, the ra n ge of the har monic pol lutio n liabil itie si s [18.608%, 42.037%], the smallest of all. The
sta nda rd de viatio n of the m is 0.104, also the smallest of all. Thus V-method goes to a not he r extreme opposite
to the I-method. By P -me tho d and S- method, the ra n ge of the har moni c pollu tio n liabi litie s are respectively
[10.057%, 106.932%] and [20.036%, 107.398%], t he mediu m a mo n g al l. The standard deviations of them
are respectively 0.437 and 0.462, also t he medi u m a mon g all. Both P- method a nd S -method are reasonable
viewing fro m the ranges of thei r harmonic po llut io n liabil itie s.
The real part of a complex po wer mea n s the average power delivered by the gr id , wh il e the ima ginary p art
of a c o mp l e x p ower is the amplitude of powe r tr avell i ng back and fo r t h in the grid . T he ir physical meanings
are quite different. In addition, the S-method needs pro jectio n of compl ex p owe r co mp o nent on total com-
plex power. T he se mak e the S -method inexp licit in physical meaning.
In conc l us ion, the P-method (liab ili ty division met ho d for harmo ni c poll utio n ba sed on line-transferred
active power co mp o nent) is the mos t ide al met ho d, wh i ch is no t only explicit in physical meaning but also
comple te a nd reasonable .
For harmon ic sour c e H2 or H5, the sa me co nclu sion can be made fro m F igur e 4 and Table 2, or Fig ure 5
and Ta ble 3.
J. C. Peng et al.
5. Conclusions
The range and standard deviatio n of liabilities for harmonic pollution by the c urrent-based meth od are the big-
gest of all. Those by the voltage-based method are the smallest of all. The two methods go to opposite extremes
and are unreasonable. The newly proposed two methods, which respectively based on line-transferred active
power compo nent and line-transferred complex power component, take all factors into account (complete) and
give reasonable levels of liabilities for harmoni c p o ll ution.
However, the met hod based on line-transferred complex power component needs projection of the co mplex
power component, thus ine xp lic it in physical meaning. As a result , the liab ili t y divisio n metho d for har monic
pollution based on line-transferred active power compone nts is not only complete and explicit but also reasona-
ble, and it is worth of recommendation.
This work is supported by National Natural Science Foundation of China under grant 51177102.
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