Optimum filter design for distribution feeders with multiple harmonic sources

Electric Power Systems Research, 23 (1992) 103-113

103

Optimum filter design for distribution feeders with multiple harmonic sources R. K. Hartana Power Systems Engineering Department, General Electric Company, Schenectady, N Y 12345 (USA)

G. G. Richards Department of Electrical Engineering, University of New Orleans, New Orleans, LA 70148 (USA) (Received July 30, 1991)

Abstract A procedure is developed for selection and placement of optimum filters on d i s t r i b u t i o n feeders t h a t have d i s t r i b u t e d and b a l a n c e d or u n b a l a n c e d h a r m o n i c sources and v a r i a b l e c o m p e n s a t i n g capacitors. The objective is to minimize voltage harmonic distortion for all buses on the feeder. Analysis of an example feeder system shows t h a t optimum filters chosen and placed in this m a n n e r may produce results superior to the filter designed to reduce distortion at a single bus and other multibus filter design procedures. Keywords: filter design, d i s t r i b u t i o n system, power system harmonics.

Introduction

Harmonic voltage distortion at distribution buses is caused by nonlinear loads such as power semiconductor devices, devices with saturated magnetic circuits, and arcing devices. These harmonic sources often cannot be eliminated, so that the frequency response of the distribution system must be altered to reduce distortion by adding filters. The usual approach is to reduce voltage distortion and/or improve the power factor at a particular bus by selecting a filter or LC compensator and placing it at that bus [1, 2]. This type of filtering is applicable when there is only a single bus to be corrected. However, harmonic filtering which eliminates harmonic distortion at only one bus may actually cause worse distortion at another bus on the same distribution feeder, especially when there are several harmonic sources. In many cases, particularly when several buses have harmonic sources, it is desirable to minimize voltage distortion for an entire feeder. Techniques for such feeder filtering are discussed in refs. 3-5, but the solution approach is qualitative and intuitive, rather than quantitative. The subject of distortion caused by distributed harmonic sources on feeders is also discussed in ref. 6, which introduced a method of analyzing harmonics from multiple harmonic sources by considering the 0378-7796/92/$5.00

nonlinear load to be a distributed current source. This assumption suggests that distortion can be associated with an entire distribution feeder, rather than a single bus. However, quantitative minimization of harmonic distortion for all feeder buses by optimizing filter admittances at harmonic frequencies has not been done. In finding the optimum filters for a feeder, the problem of shifting system response caused by variable compensating capacitors must be taken into account. In refs. 3-5, it is suggested that the filters be selected with switched capacitor banks at their 'worst-case' values. The worst case is found by an exhaustive search for all the switched capacitor combinations for all possible resonant conditions. After placing the filter, the search is done again to assure that there is no new resonance. This approach requires iteration and may not converge. Another question that arises in distribution feeder filtering, in addition to filter admittance optimization, is filter placement. Best results are obtained with properly placed filters. Filter components for all harmonic frequencies need not be at the same bus. Filter placement is discussed in refs. 3 5 which suggest the simple rule of placing the filter at the capacitor bank farthest from the substation. However, this rule is not always best for the case of distributed harmonic sources. ~$ 1992 - - Elsevier Sequoia. All rights reserved

104

In this paper, the three issues r e l a t i n g to minimizing h a r m o n i c voltage distortion for the entire feeder buses with multiple and b a l a n c e d or unb a l a n c e d h a r m o n i c sources are addressed. First, a m e a s u r e of combined bus distortion for the entire feeder is i n t r o d u c e d and a method of finding o p t i m u m filter a d m i t t a n c e s with v a r y i n g c o m p e n s a t i n g capacitors is presented. Symmetrical components are used for the case of an unbalanced h a r m o n i c source. Then, the problem of filter placement is explored. Finally, filter realization with the lowest cost (size), including sensitivity analysis, is discussed.

c o m p e n s a t i n g capacitors. To create a "worst case', these capacitors will be considered to be variable from zero to their m a x i m u m values, t h e r e b y insuring t h a t the worst case is included. The possibility of u n b a l a n c e d h a r m o n i c c u r r e n t sources will also be considered. Using this minimization, the problem of filter placement will be explored and some conclusions reached.

Objective function For a balanced and symmetrical n-bus distribution system, the h a r m o n i c voltages along the feeder for h a r m o n i c order h can be c a l c u l a t e d as

Feeder voltage harmonic reduction

h

~'Zh

lh

(2)

V b u s = LJbUS.L bu s

The d i s t r i b u t i o n feeder u n d e r s t u d y is considered to serve a large n u m b e r of n o n l i n e a r loads. Therefore, h a r m o n i c c u r r e n t sources and comp e n s a t i n g capacitors are n u m e r o u s and widely d i s t r i b u t e d t h r o u g h o u t the feeder. The distribution feeder parameters, h a r m o n i c sources and capacitors (fixed or variable) are assumed to be known a priori. The objective is to select and place filters to minimize the combined t o t a l h a r m o n i c distortion (CTHD) of all buses on an n-bus d i s t r i b u t i o n feeder, such t h a t the t o t a l h a r m o n i c distortion (THD) of each individual bus is w i t h i n the allowable THD limit. The CTHD is defined as a w e i g h t e d root m e a n square of the individual bus THDs: CTHD =

,., WiTHDi

where vL.

= [

.....

= [r;,

.....

zL+ =

(3) z

,

z,'i2

...

z ,,:,,

V~u~ is an n × 1 bus h a r m o n i c voltage vector, [~,.~ is an n × 1 bus h a r m o n i c c u r r e n t injection vector, Z~u.~is an n x n bus impedance m a t r i x (cons t r u c t e d with fixed and variable capacitors), and superscript T indicates transposition. The THD for voltage at bus i is defined as

(1)

i=l

where THD~ is the total h a r m o n i c distortion at bus i a v e r a g e d for all three phases, and Wi is the weight. W e i g h t i n g factors are used to emphasize significant THDs at critical buses. The CTHD would appear to be a reasonable figure of merit for e v a l u a t i n g filter performance in multiple-source situations. It is a generalization of the THD to two or more buses. Buses with n o n c r i t i c a l loads which nonetheless must be kept w i t h i n reasonable d i s t o r t i o n limits may be assigned r e l a t i v e l y small weights W~. If the minimization of the CTHD results in decreasing general d i s t o r t i o n at the expense of a single, u n a c c e p t a b l y distorted bus, the problem can be addressed by assigning a h i g h e r weight to the bus in question. The CTHD will be minimized u n d e r a condition of k n o w n distributed h a r m o n i c c u r r e n t sources, t a k i n g into a c c o u n t several variable

r,:],

THDi

:

~

,(

h ~ ' I V : + I+

(4)

where V~ and V~' are the f u n d a m e n t a l and h t h h a r m o n i c voltages at bus i. Assuming f u n d a m e n t a l frequency voltages at all buses are c o n s t a n t and equal to 1.0 p.u., the CTHD is obtained by s u b s t i t u t i n g (4) into (1):

CTHD =

~ [Vh [ C b u s ]1,w~,zl, vv " b u s

(5)

12 > 1

where superscript * indicates c o n j u g a t i o n of complex variables, and W is the d i a g o n a l m a t r i x of weights Wi. In a distribution system with c o m p e n s a t i n g capacitors, if there are j variable capacitors with a d m i t t a n c e s v a r y i n g from 0 to Yc, ..... (l = 1, 2 . . . . . j), the bus h a r m o n i c voltages in eqn. (5) are averaged to include all variable capacitor value ranges. The CTHD can t h e n be r e w r i t t e n as

105 I

nml

CTHD=

1

ml

~

E

"mjh>~k,=o

"'"

mj

E

hj=o

[ vhu~(kl A Y c , , . . . , k~ AYc)] *w 1/2

× W[Y~u~(kl AYc, . . . . . kj AYcfl

(6)

w h e r e AYc = Ycjmax/mj is the a d m i t t a n c e increment for c~pacitor j, m r is the n u m b e r of equal switching steps for the c a p a c i t o r j, and V~u~(klAYc, . . . . . k~ AYc~) is the bus harmonic voltage v e c t o r as a function of (k~ AYc~ . . . . . kj AYc), the possible combinations of the capacitor admittances. For the case of u n b a l a n c e d harmonic c u r r e n t sources, the equations shown above are still valid, except the scalar n o t a t i o n s of voltage, current and impedance for each bus on the righth a n d side of eqn. (3) are replaced by their corresponding v e c t o r n o t a t i o n s for representing the positive-, negative- and zero-sequence components (using the F o r t e s c u e transformation). The bus harmonic voltage and c u r r e n t vectors for harmonic order h are now 3n × 1 vectors: h " ' " V hs, n] T Y s ,h bus ~- [Ysh, 1 " ' " V s,i

I s h bus :

[/shl

•

"

" I

s,i

h

"

"

"

I h. . . . IT

(7a~

(7b)

w h e r e subscript s denotes sequence components, Vsh, i :

[ Yhi,

Ishi = [ I +hi

y h ._t, V h i ]

. .I.h. Ihoi]

and + , - , 0 r e p r e s e n t positive-, negative- and zero-sequence components, respectively. The bus impedance matrix is a 3n × 3n matrix F zsh'll Z h s, bus =

Z sh, 21

L Z sh, nl

Z sh, 12 h 22 Zs,

""" . ""

Z sh, In Zs,h 2n

Z sh, n2

...

Z hs, nn

O p t i m u m filter a d m i t t a n c e To minimize the CTHD with r e s p e c t to filter admittances, the steepest descent m e t h o d is used. Filter a d m i t t a n c e s at r e l e v a n t harmonic frequencies are i n d e p e n d e n t variables. The a c t u a l filter components are then realized from optimum filter admittances. Minimization of the CTHD is obtained by minimizing eqn. (6) s e p a r a t e l y for each harmonic frequency. The superscript h is omitted hereafter for clarity.

One filter W h e n a three-phase filter with a d m i t t a n c e Yp¢((p = a, b, c) is placed at bus p, the bus voltages are: V~,~ = V ° ~ - Z~,~pYf~,oV~.o

(8)

where Zsh nn ----diag[Zhnn

impedances [3-6]. The ideal harmonic c u r r e n t source model is a d e q u a t e for a wide range of circuit conditions w h e n the voltage distortion is n o t greater t h a n 10%. W h e n the distortion is high (over 10%), the harmonic c u r r e n t source model in parallel with load impedance is necessary. Linear loads such as induction motors are represented as an equivalent RL in series. For loads which are not well defined, a parallel RL repres e n t a t i o n is used. Transformers are r e p r e s e n t e d as an equivalent l e a k a g e i n d u c t a n c e or an equivalent leakage resistance and i n d u c t a n c e in series. For analysis of low frequency h a r m o n i c s (up to 17th harmonic), o v e r h e a d distribution lines are r e p r e s e n t e d as nominal x-equivalent circuits c o n n e c t e d in cascade. The other s h u n t capacitances t h a t are included in the analysis are the c o m p e n s a t i n g capacitors. The exact x-equivalent is used for higher frequencies.

i--1,2,...,n

i#p V~, p = [YP] -1 V 0s , p

(9a) (9b)

where

zhn~ Zhnn]

[YP] = I + Zs,~ Yfs,v

Then, replacing V~us in eqn. (6) by Vs.h bu~ in eqn. (7), the objective function of eqn. (6) is now expressed in terms of symmetrical components.

and

H a r m o n i c distribution s y s t e m model

where

H a r m o n i c - p r o d u c i n g (nonlinear) loads can be modeled as ideal harmonic c u r r e n t sources or as harmonic c u r r e n t sources in parallel with load

:[!!

YY+ 1 Yo YY÷ Yo

Y+ = (Ypa + aYpb + ~2Ypc)13 Y- =

(Y.. +

~2Ypb + aYpc)/3

Yo = (Ypa + Y,b + Ypo)/3

(10)

106

= 1/120 ~ a n d I is a 3 × 3 i d e n t i t y m a t r i x . V~.~ a n d V°p are t h e v o l t a g e s at buses i a n d p before p l a c i n g t h e filter. N o t e t h a t Y+ a n d Y in eqn. (10) are d e r i v e d for p o s i t i v e - s e q u e n c e h a r m o n i c c u r r e n t s (h = 7 a n d 13). T h e y are i n t e r c h a n g e d for n e g a t i v e - s e q u e n c e h a r m o n i c c u r r e n t s (h = 5 a n d 11). By s u b s t i t u t i n g eqn. (9) i n t o eqn. (7) a n d t h e n r e p l a c i n g Vbu~ in eqn. (6) by V~.~.... in eqn. (7), the o b j e c t i v e f u n c t i o n , CTHD, b e c o m e s a f u n c t i o n of t h r e e v a r i a b l e s (Y~.... Y,~,, a n d Y,,,,). F o r t h e b a l a n c e d h a r m o n i c s o u r c e case, the t h r e e - p h a s e filter a d m i t t a n c e s (Y~,~,, Yp,,, a n d Yp,.) are i d e n t i c a l , b u t for the u n b a l a n c e d case t h e y are n o t i d e n t i c a l .

Two filters M a n y d i s t r i b u t i o n s y s t e m s h a v e m o r e t h a n one m a j o r feeder b r a n c h . In t h i s case, a single filteri n g s c h e m e m i g h t n o t be sufficient to r e d u c e the h a r m o n i c d i s t o r t i o n at all buses on the feeder, b e c a u s e t h e filter will t e n d to be m o s t effective at the buses on the feeder b r a n c h w h e r e it is l o c a t e d a n d leave t h e d i s t o r t i o n at o t h e r buses quite high. T h e r e f o r e , two or m o r e filters m a y be required, one at e a c h m a j o r feeder b r a n c h [4, 5]. F o r t h e case of a d i s t r i b u t i o n s y s t e m w i t h two m a j o r b r a n c h e s , t h e bus v o l t a g e e q u a t i o n s for t w o filters are d e r i v e d below. F o r a d i s t r i b u t i o n s y s t e m w i t h t h r e e or m o r e m a j o r b r a n c h e s , equat i o n s c a n be e x t e n d e d to t h r e e or more filters. F o r two t h r e e - p h a s e filters, one w i t h a d m i t t a n c e Yp,p (~ = a, b, c) p l a c e d at bus p a n d the o t h e r w i t h a d m i t t a n c e Y~,p (~ = a, b, c) p l a c e d at bus q, t h e bus v o l t a g e s a l o n g t h e feeder buses are:

V~.~ = V ° , i - Z~,~,,Y~.pV~.p - Z~.~,,Y~.,,V~.,, i = 1 , 2 . . . . . n, iv~p,

i#q

'Z~.,,~Y~,,, } ~a

V~,p = {[YP] - Z ~ , p q Y ~ , q [ Y Q ]

(11a) V~.q = {[YQ]

-

Z~,pq Y~.p[YP] 'Z~,pq Y~, u } 'B (llb)

The steepest d e s c e n t a l g o r i t h m u s e d for minim i z a t i o n is o u t l i n e d in A p p e n d i x A. The comb i n e d t o t a l h a r m o n i c d i s t o r t i o n is m i n i m i z e d w i t h r e s p e c t to six v a r i a b l e s (two t h r e e - p h a s e filter a d m i t t a n c e s ) . O n l y h a r m o n i c f r e q u e n c i e s of c o n c e r n n e e d to be i n c l u d e d in the m i n i m i z a t i o n .

Optimum filter location If a feeder is to h a v e o n l y one o p t i m u m filter for e a c h h a r m o n i c , t h e best l o c a t i o n c a n be f o u n d by s i m u l a t i n g C T H D w i t h an o p t i m u m filter p l a c e d at e a c h bus u n t i l the l o c a t i o n w i t h m i n i m u m C T H D is found. H o w e v e r , this a p p r o a c h is t i m e consuming if t h e r e are m a n y buses on t h e feeder, or if it is n e c e s s a r y to l o c a t e two filters on t h e feeder. In t h e case of a single feeder b r a n c h w i t h m a n y buses i n j e c t i n g h a r m o n i c c u r r e n t s , a d i r e c t e d search, s u c h as the g o l d e n s e c t i o n m e t h o d , m a y be useful b e c a u s e m i n i m u m v a l u e s of C T H D t e n d to form a u n i m o d a l f u n c t i o n of filter d i s t a n c e from t h e end of the feeder. S u c h a s e a r c h m u s t be done over the l e n g t h of e a c h b r a n c h of the feeder. The g o l d e n s e c t i o n s t r a t e g y u s e d for the s i m u l a t i o n s t u d y is given below. The n o t a t i o n is: Pu,P[. u p p e r a n d lower b o u n d s of i n t e r v a l , meas u r e d in t e r m s of l e n g t h bu, b,. n e a r e s t buses to Pu a n d p,., r e s p e c t i v e l y p~,p~ p o i n t s w i t h i n i n t e r v a l , m e a s u r e d in t e r m s of l e n g t h b~, b2 n e a r e s t buses to Pl a n d P2, r e s p e c t i v e l y g(b~ ) m e a s u r e d v a l u e of bus i in t e r m s of l e n g t h f(bi) m i n i m u m v a l u e of C T H D w h e n filter is p l a c e d at bus i G i v e n the l e n g t h of a feeder b r a n c h e q u a l to d. p,j = 0, Pu = d, the f o l l o w i n g steps i l l u s t r a t e t h e algorithm.

Step 1. C a l c u l a t e Pl = P L + 0.38(ptj - p , , ) find bl, c a l c u l a t e

where

A = VO B=V

Minimization technique

Zs,p q

f Y~,,,[YQ]

1

0 V~,,,

°s , q - Z ~ ~,p q Y f ~, p [ Y p ] - 1 V 0s , p

[YQ] = I + z~. q~ Y~. q a n d y r q is t h e s a m e as in eqn. (10) w i t h s u b s c r i p t p c h a n g e d to s u b s c r i p t q. By s u b s t i t u t i n g eqn. (11) i n t o eqn. (7) a n d t h e n r e p l a c i n g Vb,s in eqn. (6) by V~. bu~ in eqn. (7), t h e o b j e c t i v e f u n c t i o n is n o w a f u n c t i o n of six v a r i a b l e s (Ypa, Ypb, Ypc, Yq~, Yq~, a n d Yq~.).

P2 = Pu - 0.38(pu - p,.) find be, e v a l u a t e f(b~) a n d f(b2). Step 2. If f(bl) <<.f(b2), go to step 6. Step 3. O t h e r w i s e , set b,. = bl, p~. = g(b~), a n d

f(bl.) = f(b, ). Step 4. Set bl = be, pa =g(b2), a n d f(b,) = f(b~). Step 5. C a l c u l a t e P2 = Pu - 0.38(pu - PL) a n d find b2; e v a l u a t e f(b2) a n d go to step 9.

107 Step 6. Set bv = b2, Pu = g(b2) and f(bu) = f(b2). Step 7. Set b2 = bl, P2 =g(bl) and f(b2) =f(b~). Step 8. Calculate

S1 = •

.

S; =

°

1

S 1h

Sih 2 _ h2 +

SMh • • • + PM2

h2

(12) where

Opl~numBus

L

1 co2LiCi

/)2

1

_

(.oL M

(D2LMCM

Pi is the pole order (or tuned frequency) for filter branch i. The total filter cost is

M

M

COST = UL ~ VARL, + Uc ~ VARc, i=l

(13)

i=l

where UL and Uc are, respectively, the unit cost of the reactor and capacitor per kVAR, and VARLi, VARc, are the voltampere ratings for reactor L / a n d capacitor C~. In the following examples, it will be assumed that UL = Uc. The voltampere ratings required can be found from [7]

h

p2_h2j J h~

A one-port filter with multiple branches of L and/or C circuits can be constructed at the optimum bus location using filter admittances obtained from minimization of eqn. (6). The values of Ls and Cs in the filter are selected such that the total filter cost is minimized• For M harmonics (with harmonic order h > 1), the one-port filter consists of M branches, each composed of L C in series, as shown in Fig. 1. The admittance at harmonic order h is

p12 _ h2 + • • • -4- p i

.

P~ =

1

SM =

Filter realization

-

•

(gL i

VARLi

Yf(h)

1

t92L1C1

P12

,

Pl = PL + 0.38(pu --PL) and find bl; evaluate f(b~). Step 9. If (bL, bu) or (bL, bl, b2, bv) are adjacent buses, then find the minimum of f(b* ); b* is the optimum filter location and stop. Otherwise, go to step 2. For the two-filter case in an n-bus distribution system with feeder branches, the suggestion from refs. 4 and 5 of placing one filter per feeder branch is followed. One filter is placed in the m-bus feeder branch at the bus with highest distortion level, and the other filter is then placed at the bus in the branch with n - m buses• In total, there are m(n - m) combinations of locations to be searched for the best two locations• The golden section search can be applied to the filter being placed among the n - m buses, each to achieve minimum CTHD. This result is then used in the directed search in the m-bus branch.

1 caLl

~CTi L..... LM~_j_

Fig. 1. A one-port filter configuration.

and VARc,

pi 2 - h 2} J ip2_h21]

kVAR

(14b)

The optimum filter realization is achieved by selecting Pi s such that the optimum filter admittances at harmonic frequencies of interest are satisfied by eqn. (12), and the total filter cost defined by eqn. (13) is minimized. The filter admittance defined by eqn. (12) is always positive (capacitive) at fundamental frequency, which is desirable for reactive power compensation. However, a desired fundamental frequency admittance may be included in eqn. (12) as a constraint.

Simulation studies

A 35-bus radial distribution system taken from ref. 5 was simulated in this study. The system consists of two 13.8 kV feeders fed from a 230 kV source through a 24MVA transformer• The transformer is A - Y connected, with a solidly grounded neutral• A single-line diagram of the distribution system is shown in Fig. 2.

108 9 6

11

230/13.8 kV

I

I

harmonic c u r r e n t s are t a k e n to be 0.0 '~, which is a pessimistic assumption. In the system to be studied, it is assumed t h a t harmonic distortion at every bus is equally important, therefore the weighting matrix W can be omitted from the equation.

110

16 1

12

14

2222

8 17

1200 ~VAR each

1

~

e,ch

One filter

2800 kVAR

. . . .L 32

33

34

25

26

I ~

600 ' kVAR

i, i

1200 kVAR

Fig. 2. Single-line d i a g r a m of a 35-bus radial d i s t r i b u t i o n s y s t e m .

The distribution lines are all o v e r h e a d lines, except some short cables leading from the substation. There are nine capacitors totaling 10.6 M V A R for p o w e r factor correction. At first, only one variable c a p a c i t o r is installed in the system and located at bus 29 with a d m i t t a n c e varying from 0 to 0.05 p.u. (assuming ten switching steps). The line and load impedances t o g e t h e r with c a p a c i t o r r e a c t a n c e s and o t h e r system impedances for f u n d a m e n t a l frequency are given in Appendix B. L o a d s are pessimistically assumed to include typical six-pulse rectifiers, which p r o d u c e h a r m o n i c s of order 5, 7, 11, and 13. The other higher h a r m o n i c s are considered small and ignored in this study. The m a g n i t u d e s of h a r m o n i c c u r r e n t s g e n e r a t e d by this type of load are 0.175, 0.11, 0.045, and 0.029 p.u. of the f u n d a m e n t a l c u r r e n t for the 5th, 7th, l l t h , and 13th harmonics, respectively [7]. The n o n l i n e a r loads simulated in this s t u d y are assumed to total 0.25 p.u. c u r r e n t and therefore c u r r e n t s injected from all loads for each h a r m o n i c order h, I htotal, are 25% of the above magnitudes. By assuming uniformly distributed harmonic c u r r e n t s at all buses, the m a g n i t u d e of the h a r m o n i c c u r r e n t injected at each bus is equal to 1/35 times / t oht a l " The phase angles of all

The results of four simulations with b a l a n c e d harmonic c u r r e n t sources are given in Table 1. The first column of the Table shows the combined total harmonic distortion (CTHD) of the feeder, 4.562%, before placing any filter. The total harmonic distortion (THD) at each bus is higher for the buses near the feeder end, where there are c o m p e n s a t i n g capacitors. The plot of T H D at each bus versus varying capacitive admittance at bus 29 before placing the filter is shown in Fig. 3. In the Figure, the highest THD o c c u r r e d at bus 31, with the T H D ( a v e r a g e d over

Bus 31

% % %

Fig. 3. T h r e e - d i m e n s i o n a l plots of T H D s v e r s u s Y(. at b u s 29 a n d b u s n u m b e r (before p l a c i n g filter).

T A B L E 1. S i m u l a t i o n s w i t h one filter for b a l a n c e d h a r m o n i c c u r r e n t s o u r c e s No filter

C T H D (%) Most distorted bus T H D (%)

4.562 31 8.048

O p t i m u m filter at b u s 22

T u n e d filter at b u s 8

at b u s 27

4.633 31 9.195

2.128 15 4.292

1.481 15 2.775

109

TABLE 2. Optimum filter admittances and realization at bus 22 Harmonic order Admittance (p.u.) Branch

5 2.527

7 2.298

11 1.491

13 1.200

LC

LC

LC

LC

Total filter r a t i n g = 5601.0 kVAR (three-phase) L (mH) 29.469 17.904 26.729 C (pF) 8.967 7.476 2.061 Pole order 5.160 7.250 11.300

6.696 5.423 13.920

Total filter r a t i n g = 4325.0 kVAR (three-phase) L (mH) 66.365 59.770 58.466 C (pF) 4.133 2.352 0.973 Pole order 5.065 7.075 11.120

49.249 0.826 13.155

287.5

the range of variable capacitance) equal to 8.048%. The second simulation shows a filter placed at bus 8, designed to reduce the harmonic distortion at only t h a t bus. The filter is t u n e d exactly to the harmonic frequency. As s h o w n in the second column of Table 1, the CTHD with such a filter and p l a c e m e n t is higher t h a n t h a t with no filter, b e c a u s e the distortion at o t h e r buses, such as bus 31, becomes higher (9.195%). However, when the t u n e d filter is placed at the most distorted bus (bus 27), which is also the placement suggested in refs. 4 and 5, the CTHD is reduced by more t h a n one half, as s h o w n in the third column. The most distorted bus is now bus 15 with a T H D of 4.292%. The last column shows the result when the o p t i m u m filter is found by minimizing CTHD in eqn. (6). The optimum location is bus 22. The THD at each bus is r e d u c e d to less t h a n 2.775%. The o p t i m u m filter a d m i t t a n c e s with two alternative corresponding filter realizations are given in Table 2. Note t h a t the a d m i t t a n c e s are finite and positive, indicating an optimum filter t u n e d above the harmonic frequency. This contrasts with the usual practice of tuning filters slightly below harmonic frequency to avoid a t t r a c t i n g harmonic currents from other buses. In this case, it is desirable to a t t r a c t some harmonic currents as the feeder i n d u c t a n c e acts as part of the filter for the distributed harmonic sources. The first a l t e r n a t i v e for filter realization is selected w h e n the f u n d a m e n t a l frequency filter a d m i t t a n c e has to be zero. The filter consists of five branches: four b r a n c h e s of LC and one b r a n c h of L. If capacitive a d m i t t a n c e at fundamental frequency is desired at bus 22, the second alternative with only four branches of LC is selected. This filter supplies 0.025 p.u. reactive power at bus 22. In practice, the second alterna-

tive may be preferable. It will be used hereafter for optimum filter realization. In the case of u n b a l a n c e d harmonic c u r r e n t sources, the n e t w o r k sequence c o m p o n e n t s are used for representing the distribution feeder. Positive-sequence impedances are given in Appendix B. Negative-sequence impedances are assumed to be equal to positive-sequence impedances. Zero-sequence rectifier harmonic currents are assumed to be zero. One case of u n b a l a n c e d harmonic c u r r e n t source was simulated with single-phase rectifier loads c o n n e c t e d b e t w e e n phase a and phase b (I, = - I b, Ic = 0). The sequence components of c u r r e n t sources generated by such loads are

Ia+ =(Ia/N//3)/--30 °, Ia_=(Ia/x/-3)[30 °,

Ia0=0

for positive-sequence harmonic c u r r e n t s (h = 7 and 13). For the 5th and l l t h (negative-sequence) harmonic currents, Ia+ and I,_ are interchanged. Furthermore, it is assumed that triplen (h = 3, 9 . . . . ) harmonics are not permitted to flow into the system b e c a u s e of the A- or u n g r o u n d e d Y-connected transformers (at the source side). The results are exactly the same as the b a l a n c e d harmonic source case, except the optimum filters here are only placed at phases a and b. As expected, a filter at phase c is not necessary, since there are no zero-sequence harmonic sources and no distortion at phase c. The filters for phases a and b are shown in Table 2. In practice, u n c h a r a c t e r i s t i c h a r m o n i c s of even and/odd orders (h = 2, 3, 4, 6, 8 . . . . ) are produced by six-pulse rectifiers due to u n b a l a n c e d conditions such as AC bus voltage imbalance, transformer impedance imbalance, and t h y r i s t o r firing angle imbalance or 'jitter'. The magnitudes of these u n c h a r a c t e r i s t i c harmonics are usually much smaller t h a n those of a d j a c e n t characteristic harmonics when the percentage of imbalance is small ( 1 % - 2 % ) , therefore, for practical purposes, t h e y are ignored in the case studied above. However, if the magnitudes of u n c h a r a c t e r i s t i c harmonics are significant due to s u b s t a n t i a l l y u n b a l a n c e d conditions, these harmonics can be included in the optimum filter design procedure. The CTHD with one optimum three-phase filter at bus 22 is still above 1.0% for b o t h b a l a n c e d and u n b a l a n c e d harmonic cases, and the highest THDs, particularly at buses on the upper feeder branch, are still above 2.5%. If the maximum allowable T H D at each individual bus on this two-branch feeder is assumed to be 2%, a n o t h e r filter is required on the upper feeder b r a n c h (one filter at each major feeder branch).

110 TABLE 3. S i m u l a t i o n s with two filters for balanced h a r m o n i c c u r r e n t sources No filter

CTHD (%) Most distorted bus THD (%) Total filter r a t i n g (kVAR)

4.562 31 8.048

Tuned filters at buses

O p t i m u m filters at buses

8 and 25

12 and 27

3 and 27

3~ 12 and 27

1.275 20 2.291 5376

1.426 20 3.412 4000

0.447 3 0.907 4424

0.448 35 1.373 4353.5

Two filters Table 3 shows the CTHD and the individual highest bus T H D ( a v e r a g e d over the variable c a p a c i t o r value range) for several possible placements of two filters. The result w i t h o u t any filter is the same as column 1 of Table 1. The second column of Table 3 shows the effect of t u n e d filters at two a r b i t r a r i l y chosen buses, 8 and 25. The CTHD is r e d u c e d to 1.275%, b u t bus 20 still has 2.291% THD. This w o u l d r e p r e s e n t a situation w h e r e two c u s t o m e r s install filters t u n e d at or below harmonic frequencies w i t h o u t regard for o t h e r buses/customers. W h e n t u n e d filters are moved to the c a p a c i t o r b a n k s near the feeder ends (buses 12 and 27, as suggested in refs. 4 and 5), h a r m o n i c distortion becomes even higher. The fourth column of the Table shows that CTHD, and T H D at the most distorted bus, are reduced to less t h a n 1.0%, with o p t i m u m filters found by minimizing eqn. (6) and using the location search to find buses 3 and 27.

placed at buses 3 and 27, is very sensitive to filter c o m p o n e n t values located at bus 3. If the 5th harmonic filter is moved to bus 12, the CTHD is only slightly increased (to 0.448%, as shown in the last column of Table 3), b u t less sensitive to C or L. Therefore, buses 12 and 27 are selected for placing a 5th harmonic filter, and buses 3 and 27 for placing 7th, 11th and 13th harmonic filters. Figure 4 shows the plot of T H D at each bus after placing these optimum filters versus varying capacitive a d m i t t a n c e at bus 29. From the Figure, it can be seen t h a t the highest THD (1.373%) now occurs at bus 35. With a 2% change of the filter c a p a c i t a n c e or a 1% change of system frequency, the CTHD will only change to a maximum of 0.68%, with the highest THD, 1.6%, at bus 35. The optimum filter a d m i t t a n c e s and realization of these filters are listed in Table 4. The minimized total size of two three-phase filters at buses 3, 12 and 27 as listed in Table 3 is 4353.50 kVAR, which is c o m p a r a b l e to the

Sensitivity analysis For certain filter placements, the values of CTHD can be very sensitive to changes of component values (L or C) or changes of system frequency. In practice, c a p a c i t a n c e changes more t h a n i n d u c t a n c e b e c a u s e of aging, changes of t e m p e r a t u r e and self-heating, and may a m o u n t to several percent. To check sensitivity, the CTHD change c o r r e s p o n d i n g to a 2% change of the filter c a p a c i t a n c e or a 1% change of system frequency is calculated. If the value of CTHD increases to more t h a n twice the minimum CTHD, the o p t i m u m filters and locations are indeed sensitive and o t h e r o p t i m u m filters and locations have to be found. However, the sensitivity problem can be avoided from the beginning of the location search by finding the CTHD with a filter (shorted to ground) at each bus. Buses which, w h e n grounded, cause a s u b s t a n t i a l increase in CTHD are sensitive, and should be omitted from the search. The CTHD in Table 3, with optimum filters

%]

~

Bus 35

Fig. 4. Three-dimensional plots of T H D s v e r s u s Y(, at bus 29 and bus n u m b e r (after placing two o p t i m u m filters).

111

TABLE 4. O p t i m u m filter a d m i t t a n c e s and realization at buses 3, 12 and 27 H a r m o n i c order Admittance (p.u.) at bus 3 at bus 12 at bus 27

5

3.522 5.092

Branch

LC

7

11

13

3.083

4.177

3.776

6.205

4.093

3.081

LC

LC

LC

At bus 3: filter r a t i n g = 969.50 kVAR (three-phase) L (mH) 44.683 21.784 C (pF) 3.141 2.612 Pole order 7.080 11.120

Conclusions 18.214 2.237 13.140

At bus 12: filter r a t i n g = 1560.0 kVAR (three-phase) L (mH) 16.324 C (~F) 16.137 Pole order 5.168 At bus 27: filter r a t i n g = 1824.0 kVAR (three-phase) L (mH) 40.011 21.880 20.410 C (pF) 6.887 6.424 2.789 Pole order 5.053 7.075 11.118

the l l t h and 13th harmonic filter placements. The CTHD found by minimizing eqn. (6) is 0.491%, with the highest THD of 1.407% at bus 35. With a 2% change of filter capacitance or a 1% change of system frequency, the CTHD increases to a maximum of 0.577%, with the highest THD, 1.646%, at bus 35.

19.399 2.101 13.140

minimized size of the single-filter optimization (4325.0 kVAR) from Table 2. With approximately the same filter size (cost), the optimum two-filter placement is preferable, because it reduces the harmonic distortion at every bus to about 1%. For comparison, the minimized sizes of tuned filters placed at buses 8 and 25 (column 2) and buses 12 and 27 (column 3) are calculated and listed in Table 3. Notice that in order to reduce the CTHD with the single-bus tuned filter, the size of the tuned filter (in column 2) is increased to 5376 kVAR, which is 34.4% higher than the lowest size of tuned filter (4000 kVAR) in column 3. It is also observed that the selected optimum filter (in column 5) achieving the lowest CTHD is not the most expensive choice and its total size (4353.5 kVAR) is only 8.84% higher than the lowest size of tuned filter in column 3. The lower cost of the optimum filter is partially due to the fact that the feeder inductance itself comprises part of the filter. The number of variable compensating capacitors may be increased to two, installed at buses 14 and 29, with both admittances varying from 0 to 0.05 p.u. (using five equal switching steps for each capacitor). The results are similar to those of Table 3. The CTHD before placing any filter is 5.053% and the highest THD (7.591%) occurs at bus 31. Using the location search algorithm, the optimum locations found, including sensitivity checking, are buses 12 and 27 for the 5th and 7th harmonic filter placements and buses 3 and 27 for

A design procedure for selecting and placing optimum filters on distribution feeders with variable compensating capacitors has been presented. The case of distributed, balanced or unbalanced harmonic sources is also considered. A sample procedure on a 35-bus distribution feeder shows that optimum filters chosen and placed in the prescribed manner reduce voltage harmonic distortion at every bus to a lower value than that achieved by a filter designed to reduce distortion at only a single bus. The optimum filters are also more effective than the conventional resonant shunts (tuned filters) placed to minimize total feeder distortion. Best results are obtained when optimum filters are placed at selected buses with one filter at each major feeder branch. The results show that the filter components at each feeder branch for different harmonic frequencies need not be at the same bus. The optimum locations are not always at the capacitor banks farthest from the substation. The optimum filters are tuned slightly above harmonic frequency, as opposed to the usual practice of tuning them slightly below harmonic frequency.

Acknowledgements

The work reported in this paper is part of a multiarea research project sponsored by the LSU-Utilities Consortium including Louisiana Power and Light Co., Gulf States Utilities Co. and Central Louisiana Electric Co.

Appendix

A

The steepest descent algorithm for an objective function with six variables, CTHD(Y~, Yq~), ~0 = a, b, c, can be summarized as follows. Step 1. Select an initial point x(y(o) V(o)~ - pep, --qq~ ], ~0 = a, b, c.

Step 2. Calculate the unit gradient vectors at (V(i) y(i)~ for ~0 = a, b, c:

112

Gpvl,, y ( ip)i p ,

1 -

G

(y(i)

(?(CTHD)

(i)

y(i)'~ =

11611

1

From

0(CTHD)_ ( y . ) Yq~

y(i))

(A-lb)

" - p,p, - - q ~ .

where (0(CTHD) is the total gradient vector. Step 3. If

tlGp (y23,_q ,r
Line impedances (p.u.)

(A-la)

(y(i) y(~)~

and (A-2)

where ~ is a preselected small positive number (~ ~ 1 ) , then terminate the iterative procedure, . output the optimum value (Yp*, Yq~), ~0 = a, b, c, and stop. If the stopping criterion (A-2) is not satisfied, then generate a new point given by Y~j~) = Y(p~,~-6Gp~,(Y~)~,

To

RL( x 10 ~)

XI.( × 10 :~)

X c ( x 10:*)

1 2

2 3

1.764 44. 739

2.647 79.647

5.573 12.519

3 4

4 5

5.671 16.383

10.082 29.238

98.903 34.104

5

6

2.521

3.907

3.715

5

7

32.262

57.467

17.351 197.807

7

8

2.773

5.041

8

9

3.151

4.915

2.972

8 8

10 11

8.444 11.342

24.197 32.262

65.940 49.451

11 12

12 13

9.578 22.684

17.139 40.328

58.178 24,726

13 14

14 15

10.208 9.074

18.147 16.131

54.946 61.815

4 16

16 17

13.611 9.578

24.197 17.139

41.210 58.178

17 17

18 19

2.647 24.323

4.033 43.352

3.570 23.000

l9 1 21

2(1 21 22

5.711 1.008 36.169

10.082 1.638 64.146

98.9(10 8.920 l 5.95(I

Y~,~)

(A-3a)

22

23

16. 383

29.238

34.100

(i) y ( i ) ] = y(i) - - q ~ o -- 6Gq,p(Yp~, __q~.

(A-3b)

23 24

24 25

33.396 19.786

59.48"~ 47.637

16.763 28.300

where 6 ( > O) is the step size which has to be judiciously selected. Then, replace (y(~) - p e p , y(~)) - - q ~ p ," by (Y(i+_ p,p ", _q~Y (~÷~)) and return to step 2.

25 26

26 27

32.892 7.940

78.891 19.030

17.100 70.650

27 28

28 29

11.342 3.403

27.220 8.192

49.451 164.840

29 30 23 25

30 31 32 33

22.054 20.920 39.067 7.940

53,056 50.410 93.888 19.030

25.400 26.730 14.330 70,645

26 1

34 35

9.578 20.794

23.188 445.872

58.200 5.740

y ( i + 1) --q¢

Appendix B Network impedances for a distribution system Source impedance Zs = (0.126 + jl.3863) × 10 -a p.u. Transformer impedance Zt,. = (8.9477 + j134.72) × 10 a p.u.

Load impedances (p.u.) From

To

R1,

XI,

2 3 4

0 0 0

1920.00 155.69 117.56

57.59 4.68 3.53

5 6 7

0 0 0

59.99 164.56 457.14

1.80 4.94 13.71

8 9 10 11 13 14 15 16 17

0 0 0 0 0 0 0 0 0

234.15 101.05 107.88 167.41 218.17 120.00 37.12 118.03 70.42

7.02 3.04 3.24 5.03 6.54 3.60 1.11 3.54 2.12

Capacitor reactances (p.u.) From

To

Xc

12

0

20.00

13 14

0 O

20.00 20.00

15 17 19 27 29 35

0 0 0 0 0 0

20.00 40.00 40.00 40.00 20.00 8.57

Note: Vha~e= 13.8 kV and VAbase

=

24 MVA.

113 18 19 20 22 23 24 25 26 28 29 30 31 32 33 34

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

87.28 296.84 96.01 1797.76 423.64 149.99 121.51 124.13 116.60 488.29 91.42 285.22 164.56 51.90 89.73

2.62 8.91 2.89 53.98 12.70 4.50 3.64 3.73 3.50 14.64 2.75 8.56 4.94 1.56 2.70

References 1 G.G. Richards, P. Klinkhachorn, O. T. Tan and R. K. Hartana, Optimal LC compensators for nonlinear loads with uncertain

nonsinusoidal source and load characteristics, IEEE Trans., PWRS-4 (1) (1989) 30-36. 2 R. K. Hartana and G. G. Richards, Comparing capacitive and LC compensators for power factor correction and voltage harmonic reduction, Electr. Power Syst. Res., 17 (1989) 57 64. 3 M. F. McGranaghan, R. C. Dugan, J. A. King and W. T. Jewell, Distribution feeder harmonic study methodology, IEEE Trans., PAS.103 (1984) 3663 3671. 4 R. C. Dugan and C. D. Ko, Analyzing and controlling harmonic distortion on distribution feeders, Int. Conf. Harmonics in

Power Systems, Worcester Polytech. Inst., Worcester, MA, USA, 1984, pp. 38-44. 5 R. C. Dugan and D. T. Rizy, Harmonic considerations for electrical distribution feeders, Rep. No. ORNL /Sub /81-95011/4, Oak Ridge Nat. Lab., Oak Ridge, TN, March 1988. 6 T. Hiyama, M. S. A. A. Hamman and T. H. Ortmeyer, Distribution system modeling with distributed harmonic sources, IEEE Trans., PWRD-4 (2) (1989) 1297 1304.

7 IEEE Guide for Harmonic Control and Reactive Compensation of Static Power Converters, IEEE Standard 519-1981, IEEE 1981.