DETERMI3ATION OF TIE DESIGN PARAME'i'EiCj OF PASSIVIS

HARMONIC FILTERS USIIVC; NONLilW EAR 0 P'1' IhIlZ A T 10 N.

S. D. Upadhye. Senior Systems Analyst,

Prof. Y . R. Alrc Dcpt. Of KlcctricaI Ex;.

Tata Infotech t t d . Walclratld Collcn,c 0: Punc, India I!iiginccring, Sang!i.

400096. Iiidia. JI64IS.

INTRODUCi?ON:

Power systcm harmonics are a mcnacc xitli disastrous consequences. A tiumber of ways tvcrc tried out io tackle the harmonic problem poscd by nonliiicar electric devices, but $+ far. ihe most popular and quiet crfective onc is the use of passivt filters. This provides a low impcdzncc path to a harmonic or a group of iiarnionics as tliown in Fig. 1. Usually the AC ' m o n i c s arc odd harnionics. So ntrnimlly the filler combination selected is such that for thc predominant harmonics, one or more single tuned filters arc used, and for the rest of the harmoiiics, a high pm filtcr is used. An altcmte :ombination is that a damped filter is uscil for the hvo pdominant harmonics with the filter being tuned at a frequency in behveen the hvo harmonics and another damped filter Tor ihe higher harmonics. If all thc harmonics arc not of v q high magnitudc thcn usually one high j x s s film is suficient ;'or all rlic Iiariuoriics. [ 1 1 Instcad of tiic conventional Jes:y of thesc filters. wliicii is bnsctl OII experience cf thc dcsicncr. a bcttcr way :S to usc thc optimization 1iScc:hms io oht3in niaximrrlli pcrforinsricc from these 5i::x.

0-7803-4509-6/98/$10.00 1998 EEE

/ . \

Po:w j?Zt=m harmonics are a major problein :vi& disastrou :orsequenceS. These a& usually reduced using s k t ysjsive filters. This naoer discusses he applicaiicn o i 3ctimization a l g o r i b to the dcsign of passive 5iz: :o obtain optimal dejign of these R i m s . To judgc d e ;,t:fomance of the optiml design, it is brcaal:: i ivicci into three categories. The first is for maximuxi xlcc$on in Total Harmonic Distortion of pow: qis;ex aehvork. Under th'is, magnitude of hnrinonics p ~ c d to tlic systcni is optimizcd. l l i c sccund is for ociinal :css at fuaidamciital frequenc:i. Under Ais the losscs si :;le Siter bank and hcnce curient drawn at fiindanezai 5e;uency is optimized. T i e third involves oprimizxon sf Cost of the filter, vhich optimizes

package zwz.cTed on this algorithm is tested using hamonit 5 2 s collected from field. Resuits are enumerucd mi21 magnitude of harmonies bypassed by the filter; azG -;le magniiude passing to :he powcr system ntnvork. ?!CS of the impedance characteristics of &e rcspective Ira :ho enlisted.

running x s : 2:

whcre 0 = 2n f bcing tuning frequcncy At the resonance frequency, imaginav part IS equal to zero and the current through the filter is limited by R. Since the two reactances are equal at tuning frequency,

X, = n2 X, From this it follows that,

1

This frequency f is an integer multiple of the system fundamental' frequency. This integer multiple, designated by *n' is called harmonic number.

There is another factor that influences the dcsign of a single tuned fiiter. It is called Dctuning factor and is represented by 5. I t represents the variations in the tuning frequcncy caused by following factors.

1. Variations in the system frequency. 2. Temperature variations in the values of

Another important factor that influences the dcsign of a single iuned filter is the-Filter Quality facfor Q . It is calculated always at the tuning frequency and can be given as

capacitance and inductance

\Wicrc X, is tlic renctance of inductor or capacitor at thc tuning frcqucncy. I-lcncc

0

1 and c=- QR

The value of Q usually lies in some range. The circuit resisiance R IS usually the DC resistance of the inductor L. Tile filter impedance can be expressed in tci-ms of Q and 5 as [ 11

I

DESIGN PARAMETERS OF DAMPED FILTERS: Duc to its good Zltering pcrforniaiicc at mocicratc losscs. die second order damped filtcr as shown i n Fig. 4 , is an ideal choice for harmonic supprcssioii. Ainsworth[ I ] lias introduced two factor m and f for this particular filter. These factors detemiine its frcqiicncy rcspoiisc. Tlic filter impedance can be given as

whcrc 0- 2x1

Here it is to be noted that j is not tlic tuning frequency of this filter but this is tlic frcqwiicy at which the filter impedance is iiiiiiiinum and nbovc wliicli i t offcrs low impedance. 'Iliis I1-cqucncyfis given by

1 f=- 2x cx

The other important factor be given as

L 111 = -

R'C Fig. S(2) illustrates the

is desiznated by 111. 1 I ] i t can

( 4 )

effect o f variations in 111, on the characteristics of the second order danipcd filtcr for various valucs of ni.

Tlic filtcr quality factor Q is givcn by

Where X, is the reactancc of thc filter. at frequency f. Fig. 6[2] shows how a typical second ortier damped filter, having a minimum inipctlniicc frequency at 1 0.7Ih harmonic, behaves with variations in the valuc of R and thereby in Q. X high Q means the filtering action will be more pronounced and a low Q nieaiis very little filterins action of the filter.

The chi-t design constraints Tor a Iiish pass filler are mainly Q, f and m. If the capacitance m i bc dctcrmincd from 111c oihcr pnramcicrs, f and hcncc 111 w i l l be dctcrmiilcd i n such :I \vny 3s to ol,ln!n n lllqll admittance over tlic requirctl I'Icc~uci~cy I niigc. 'i\'hcii expressed in temis of in and R. the contlrictaiice and susceptance :enis for a sccontl ortlcr high pass :iltcr nre given as

156

x( n? x - IlfX + 1)

J. f o

Whcrc X = -

f is the frequency at which the filter has minimum impedance.

OPTIMIZA TION A L GOiUTHMS USED D UIUiVG FILTER DESIGN:

Optimization is a process of finding a global or true maximum value or true minimum value of a variable using various optimization algorithms. It is distinct from the conventional maximization or minimization methods that obtain only nearby maxima or minima.

Mainly in the dcvelopmcnt process, the Interior Penalty Function Method [3][4] is used. The unconstrained function Q(x) formed thereby is optimized using Modified Cauchys Steepest Descent Gradient Partan method. [3![4] Both thc algorithms arc briefly dcscribcd bellow.

Ipitcrii>r. Pciinliy Fttrtctiori Mcrhotl: This is a constrained optimization nicthod whose unconstraincd optimization equation for a constrained functionf(x) with constraints g, (x) on x is,

@(.) = f ( x ) + rkgG, (g,(.x)) ( G ) j = I

Where the second term is called Pena!ty term. The penalty term is used to represent the constraints on x. [4]

Cauchy in 1 S47, this unconstraincd optimization method is used 70 optimize the unconstrained equation formed by Penalty Cmction method as above.

Tie negated gadient of 4 is used for determining the scat direction S .

Modified Steepest D~scerif Mefhod: Formulated by

s = - v + Froiii an initial starting point, D step sf optimal step

length i s takcn along the search direction. Thc next point is @en 3s follows.

Whcrc S is search dircctinii iit~tl h is opi i i : i \ sicp Icngtli. (41

OPTIMAL DESIGN OF FILTERS

Due to distinction of application. tlic design i. tliwtlccl into three headings or schemes. Tlicsc arc

1. Minimum level of harmonics ill ilic systcin. 2. Mininium liltcr losscs at fwidaiiiciiial l icc l i~c~icy 3. kliniiiiuiii cost of tlic riiiilc I I I I C I 11:iiik.

AfI8Ii%liJhl L E VEL OF I I A Itil I O N I C S As the nanic suggests, this sclicnic tlcsigiis tlic filtc~s 20

that the total harmonic coiitciit prcsciit in thc systciii will be reduced as much as possible. As showii iii Fig. I , liltcr current has to be increased as far as possible, to reduce the harmonic current entering the system, to niiuinium; wliicli means, Z,:he total impcdancc of lil(cr kink as a wliolc, has to be optimized. Hence it is nccessary to optimize I I I C individual filter impedances. This entails ihat thc filtcrs sliould be optimized in such a way that they will ofrer minimum impedance at their rcspcctive tuning frcqwticics miis sclicnie 3s coniparcd to othor sclicincs givcs lowczt value of TOTAL I-lARMONICS I)ISfORIION. [ 11[3 I

Sir@ Ticried Filter: As cliscusscd carltci. IIIC impcdancc of thc singlc tuncd fiitcr as ;I function o f ilic filter quality factor (Q), 31 any frcqucncy is givcn as,

22 (e) = Z2(Q) = R2(1 + KQ)

22 (Q) is the optimization fuiiction for tlic singlc tuned filter and Q is called tlic optiiiiizatioii variable. Q usually lies in some range, say il

This ;3 is optimized using Steepest Descent Ngor i t lm . During this optimization process, rk remains constant as i t varies during the constrained optimization. Hence Q is thc only variable.

If the system network angle 4%" is known, then according to Amllaga [ I ] , the optimal value of Q is given by

1 1 1 wliicli case tlic optiiiiizatioii is a11 optional addctl fcaturc.

Damped F i l m : l l i e impedance of thc damped filter at its tuning frequency is given by

( 1 1 )

As C can be detenxined from other factors, the independent variable here is m alone. According to Ainsworth [l], m normally lies in a range specified as say a < iii < b. Thcrcforc thc aini is to find such a value of m in the specified rangc that 2 2 and hence Z will bc at its minimum. The penalty term can be given by,

T h i s (I 'formed above is optimized using Steepcst Descent Algorithm. During this process, tk remains constant as it varies in the constrained optimization loop. Thus m is the only variable.

For any filter, the magnitude of the current flowing through it is generally maximum at fundamental frcqucncy. Hence the filter o h m i c losscs will coricspondingly bc largcst at fui1danicnta1 frcqucncy. To reduce thesc losscs, the filtcr will h a w to bc cspccially designed so that i t will offer comparativcly optimal dc resistance at fundamental frequency. This means the fundamental impedancc will have to be optimized. For a single tuncd Slter, this resistance is mostly equal to the o h m i c resistance of the inductor. For a damped filter, the resistance at fumdainental frequency is largely

dependent upon all the compnncnts of tlic danipcti fiitcr viz. R, L and C.

SINGLE TUNED FILTER: As seen nhovc, tlicsc is not much scope for optimization of 1.k rcsisiancc of this filtcr as the parametcrs it dcpcnds 011, arc dctermincd by the design constraints of tlic iiidwtor that leavc littlc scope. All that can be donc to rcducc I< is to design tlic filter such that the value of I, is minimum. (1][37

DAA4PED Fll,7El?: ' I h icsislaiicc a l fundaiiicntal frcqucncy of a tlaiiipc'l I'iltcr is givcii hy.

,112(2111? - 21,r -I- I ) (12)

Where 01, i s that frequency a t wliicli tlic filter has minimum irnpcdancc. Tl i i i s 111 is the oiily vnrial>lc. ns C can be determined froni other factois.

As discussed earlier, in nornially lies in soiiie range. ni < b. Therefore the aini is to find such n value

Tlic say a of m in this range, R, will be at its miniinrim. penalty tenii can bc givcii by,

j = I

T i u s (T, is given by.

73;s (T, is optimized using Stccpcst I ) C S C C I I ~ Algoritliiii. K, rcmains constant as it varics i n the constrained loop; thus ni is the only variable.

It is an iiidisputable fact that cost of tlic filtcr hniik is a critical element in determining thc installation of ihc bank itself. Tie cost of a filter consists of two parts V I Z . Fixed capital cost and the vnriablc runniiig cost. l'lic capital cost largely depends upon tlic capacitor valiic ( h a n k s i x ) nccdcd. As i t also tlcpcntls o i l ollicr h c ~ o i < iliat niay or may not bc coii~i~oi1:iI~le~ tlic liltcr riiliiiy sliould h e optiiiiizcd s o t1i:it tlini riiiiiiiiig c r ~ t ivi l l itc iriinirnuni. :\Ithough the cost lijr at1diiioii;iI iiniis rcciiiirc'l for higher ratings. is iiot cxactly liiiear, for siinplicity i t i s assunicd that tlic cost of n capacitor is pro1ioclioiial IO llic rating. Tlic cost of a11 inductor is matlc tip of two parts .i*iz. one constant part arid the other variable part proportional to tlic ratins o f tlic inductor. [ I J

158

SINGLE TUNED FIL7ER: For a singlc tuncd filxr, tlic sizc or rating in kvar can bc givcii as

2 I1 2 Var. A' = - xc 4, x, 11 - 1 rc

Whcrc X,- and XL are respective reactances at thc fundameiital frequency. As the aim is to optimize the ninning cost of the filter, i t is necessary to optimize the nmiiing losscs of thc filtcr. For this tlic individrinl componcnt losses, which arc depcndent upon the total loading of tlie componcnts, will have to bc optimized.

Cnpacitor. losses: For calculating the total losses occurring in the capacitor, total loading consisting of fundamental as well as harmonic losses is to be calculated. The total loss in the capacitor is

= K C L ( - g +

(13)

Where bL is the capacitor loss factor and I, is the current passing through the capacitor at the harmonic.

Resisfnirce losses: Urdike the capacitor, an inductor does not have intrinsic losscs hut rathcr has extrinsic lnsscs designated by tlie resistancc R in thc circuit. Because for n siriglc tuncd fil:cr, Ilcticc the loss ill thc rcsistancc will be,

Tlius the total filter loss from Eqs.13 and 14 is given by,

Tliis I;,,, is optimized for such a value of k, that the loss will be minimum. As k, is an unconstrained variable. F,,, can be optimized usins Steepest Descent A I go r i t h in.

IMMPEU FIL7ER: 11 tlninpctl filler Iins wiially vcry small valuc of Q as.coniparcd to tlic siiiglc tunctl filtcr. As Q is given by

I< s

Q = -

This means, tlic filfcr rcsistaiicc I< i s w r y siii;lil compared to X . hciicc ncglcctiiig I < . tlic filtcr rating is,

r:' 11; x 11; - 111 K,, =T- Wlicrc

Twiirig jk,wcticj* ?/ tiic ,fiiter Fitnciantentcil ji-ajiwrcy

- 1l0 =

Cnpncitor losscs: For calculatiiig tlic total losses occurring in the capacitor, losscs consisting of fundamental as vel1 as harmonic losscs will linvc to bc caicuiatcd. I Ihcsc are

~essismrice io.ssc?s: Wic iosscs 111 thc rcsistancc I< can be given as or

I-ieiicc the total filfcr loss from Eq5. 15.1 6 i s givcti by

kr,(Ill - 1 ) y I l ~ - I n ) t l ? + i- 112-11 1 1 2 - 1

This F,,, is oytiiiiizcd li)r siicli a vnlrie 0 1 kr(: tlinf tlic loss \vi11 be niiniiiiiiiii. A s kj,. is nn uiicoiistrniiictl variable, Floss can be optimized using Stcclmt 1 ) c s c o i t uncor~strair~cd optiniization iiictl~nd.

159

A sofhvare package was designed to implement the thrcc schemes discusscd above. With ihis soltwarc package, some field trials were carried out. For these trials, daia regarding harmonic content was collected from bulk power consumers and the results obtained were analyzed. One such case is described bellow.

CASE 1:

The industrial installation has 2 to 3 local gcncrating ririits with capacitics of about 6 to 10 M V A at (3.6 kV. Tlic installation has a n H.T. liiic from state electricity supply utility at the same voltage with a provision to co- generate or draw power. l l i e load consistj of mainly six pulse rectifiers and other loads. The total load is 3415 kVA or 2800 kW at 0.82 power factor lagging. The short circuit level is 28.4 MVA or equivalent reactance of 1.5333 n. The power factor is to be improved to 0.95 lagging. This requires about 1050 kvar of leading reactive power. [ 6 ]

It is assumed that the power supply source (namely the state supply network) is free of harmonic sources. Table I illustratcs the original and filtered harmonic niagiiitude for minimum harmonic inagnirlide.

17

10

10.0537 + j0.76

-0.054 - j0.76 10.0584 + j0.78

-0.058 - j0.78 9.0963 i j l . 4 1

-0.004 - j 1.42

1 larm-onic

Filter bank currcnt

Filter bank current

19

7

7.02 + j0.48

-0.02 - j0.48

7.024 + j0.49

-0.024 - j0.49

6 998 +j0.87

-0.002 - j0.87

5

70

i0.779+ 14.59

-0.779 - j4.59

70.779+ j4.59

J4S9 -0.777 -

70.046- J1.71

-0.046 - jl.71

7

45

45.026 + j0.76

-0.0267 - j0.76

45.026 + j0.76

-0.0267 - 1076

44.996 + jO.84

4.0043 - j 0.S-l

I I

25

25.763 -+ j3.32

-0.763 - j3.32

25.GS8 4 j3.44

-0.657 -j3.444

24.2624 jG.39

-0.734 - j 6 3

MINIMU,?-! iiARMONICS DESIGN: LVlicn filtcrs arc designed using thc niinimuni harmonic

dcsign scheme, the filter data obtained is summarized bellow.

For ihc single tuncd filter, for SIh hamonic , L = I ? . 8385 in1-I C = 3 I 3 G Y &ti:

For tlic siiiglc !..;:ic(l Iiilci.. rCll 7"' I i : i r i i i o i i i ~ : ,

For the dampec 5Itcr, for tiic I I"' Iial-iiioiiic a i d aliovc. R = 2.21574 5 (1 = 130.303

L = 3.203GS 3111 ( 1 - (14.4243 }'I:

L = 0.SX21 I O tiill

The magnimics of various Iiamotiics cntcriiip tlic Power System Network aftcr usiiig tlicsc filters arc sivcii in TABLE I.

111:

Thc ncgali-Lt magiiitridcs iiitlicatc tiic rriii:iiiiiiii: capacity to by11nis thc lnxriiioiiics. of tlic filtcr Iwi ik :I$ :I \vliolc. 'Ilic TI ID or clirrciit tlislcii t i o i l . riiitlci ~ l i c v , circuriistarices iq rctlucccl to 0.00 I ! < , . (;r:iplis iii I.'ig. 7 show tlic charac:rristics of tlicsc filtcis.

MlNlMVM LOSSES AT FUNDAMENTAL FREQUENCY: When filtcrs are designcd usiiig thc niiniiiiiitni loss desiyi scheme for the above-nieiitioiicd data, ilic Lltcr parameters obtained are summarizcd bcllow.

For die single tuned filter, for 5'" harmonic. C = 31.5679 pl:

For the sin9le tuncd filter, for 7Ih harmonic, C = 04.4243 ikr

For the dampcd filtcr, for [lie I 1 ' " harnioinic aiitl ahovc. R = 2.21834 Q

L = 12.8385 mH

I, = 3.2096; inl-I

L = 0.934952 nil1 C = 130.393 { I F

ThC magnitutlcs of various Iiariiioiiics cntcriiil: tlnc I'on.rr Systcni Nctivork aftcr using tlicsc filters are as givcii iii TABLE I. Tlic TIID for cui-rciit distortion riiiilcr thcsc circumstanccs is reduced to 0.00 Yo. Cirqilis o f Ihc characteristics of tlic filters arc similar to thosc i n 1,'ig. 7.

MIiVIicIUi\~f COST DESIGN: When filters are dcsigncd using tlic iniiiinium cost design scheme. :he filter paramcters are as suinniarizcd bel low.

For the single xncd filter f o r 5"' hai-nionic, c = 56.1353 I t I ;

For the s ingk xncd filtcr for 7'" harmonic, C = 36.8253 111;

For the dan;r:,i filtcr for thc I 1'" Iinrnionic a n d abox-c. R = 3.52952 2 L = 2.04209 nil I

L = 7.2 19-3 mI-I

L = S.Gli.'? ml-I

C = 8 1 .OS67 !:F

160

V i e magnitudes or various harmonics entering thc Powcr Sysicm Network after using these filters arc as given in table I. the THD for current distortion under thcsc circumstances is rcduccd to 0.14 %. The Graphs of filter Characteristics are similar to those in Fig. 7.

From this case it can be obscrved that, minimum harmonics schemc is best suitcd to reduce the: harmonic content as far as possible with a margin for temporary tluctiiatioiis. But iltlic rtltcr Ixi~ik piicc has to bc justifictl against pcrforniance with somc harmonic coiitcnt rcniaining albcil untlcr limits, thcn niinirnum cost schemc is the riglit schcnie. I f the desire is to have the better of the hvo worlds, then minimum loss scheme can be used.

References: 1. J. ArriiaSa. D. A. l lrntllcy atid I). S. Bndgcr.

Power System llariiioiiics 1985 John Wilcy & Sons Ittl. W. 93-109, 296-324. Daniiin A. G O I I Z ~ ~ C Z , Joliii C, klccall, 'Design of Filters to Rctlucc 11:iriiinnic Dislorlioti 111 Industrial Power Sgs(ciiis l1X13 Tratw. oil Industrial Applicatinns. vol. 1A-23. No. 3,

3. S. S. R n o , Opti i i i im(io~~ (Ilcoc-g acid AppIicali~iiis* Jitlic I O X a l , Wilcy 1i:irlci I I 1111. Sccond Edition. 11. 37-38, 306-3 10. 390-39X. Donald A. Picrrc, Optiiiiizntioii tlieory willi Applications 19G9, Jolin Wilcy LQ Soiis inc.

Dr. U. Gudani A New Scliciiir for llesctivc Power Coiiipcnsitiioii. 1 lirriioriic siipprcssion arid Daiiiliiiig or I

APPENDIX IIA CALCULATIOX OF TIIE CAPACITOR LOSSES IN THE M I ~ I U N COST DESIGN OF A DAMPED

FIJ. TER

Vic total iosses occurring in a capacitor include the losses at fundamental frcqucncy and tliosc at the hamioriic frcqucncies. From cq. 4, it can bc sccn that

Therefore for a second order damped filter if V, and V, are thc respective -;oltages across capacitor and the inductor,

The fundamcntai loading is given by,

Siniilarly tlie harmonic loading as a sum of the loading at individual harmonics is givcn by

Thus if kCL is the loss factor of the capacitor. the total power loss of the capacitor will be

The losses in the resisiance R can hc ol)tniiied as follo\vs.

i.Jow the losses in :hc rcsistancc a t liindariicittai I'rcqucncy can be given as

Siinilorly tlic linrnionic loarliiig cnii bc givcn ns

Thus tlie tot31 loss i n tiic rcsistniice n~ill hc

162

List oifigurc captions:

I . Sonlinea: A. C. Load as a source of source o l hanonicj.

2. Single Tmed Fiitcr 3. Cnarac:er'itics of a single tuned fiIter 4. Second ode: high-pass fitter S. Effect of mriations in Q of a second or&r damped

filtcr 6. Characteristics showing c fkc t of variadon in m on

filter ~CCORTIJ~CC 7. CbractCr;stics of filters designed by t h ~ paCkagC.

4-

Fig. 1 Second order high-pass fdter

I Fig. f Single Tmcd FiItcr

I ! - ;\

\:-.:: :.:

163

Charaearistics of the fifth 16 - harmonic filter

I 14 7.

!I 2