h29 y7b ffb fs13 fc0 sc0 ls2e ws2c"> 
. (3)
kk k
SP jQ
′′ ′
= +
 
is the so urce-k-driven line-e nd-
-transferred complex power component.
k
P
and
k
Q
are the real and imagina ry parts of
k
S
.
k
I
is the conjugatio n of
k
I
. 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 -
.
1
n
k
k
SS
′′
=
=

. (4)
SP jQ
′′ ′
= +
 
is the total complex power over line -end-
under all sources action together .
P
and
Q
are the real and imagi nar y parts of
S
.
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.
265
100%
pkk
L PP
′′
= ×
 
. (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
component.
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
k
S
on phasor
S
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
100%
skk
LSS
′′
= ×
 
. (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.
k
S
is the proj ec tio n of phaso r
k
S
on phaso r
S
.
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
Harmonic
source H2
Harmonic
source H3
Harmonic
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.
266
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.
267
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.
268
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.
Funding
This work is supported by National Natural Science Foundation of China under grant 51177102.
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