IEEE Transactions on Power Apparatus and Systems, Vol. PAS-102, No. 10, October 1983

OPTIMAL CAPACITOR PLACEMENT ON THREE-PHASE PRIMARY FEEDERS:LOAD AND FEEDER UNBALANCE EFFECTS

J. J. Grainger, Senior Member, IEEE A. A. El-Kib, Member, IEEEElectrical Enqineering DepartmentNorth Carol ina State University

Raleigh, NC 27650

Abstract: Generali,zed procedures are presentedfor optimizing the net dollar savings resulting fromreduction of power losses through shunt capacitorplacement on three-phase, four-wire, multigroundedprimary distribution feeders. These procedures arebased on a voltage-dependent model which is developedfrom results of a three-phase a, b, c phase-frame ofreference load flow. Power-invariant transformationsare introduced to decouple the actual three separatephases and thereby yield three equivalent singlephases with common definition of feeder length. Theeffects of ground-wire and earth-return paths forbalanced and unbalanced system and load conditions areaccommodated within the models presented.

INTRODUCTION

Usually the bulk-power system is analyzed by con-sidering the system to be balanced- and simplifying thesystem representation to a single-phase model. Assump-tions are made regarding line transpositions and reli-ance is placed on diversity of loads at bulk-powerstations to produce an overall "smoothing" or balanc-ing effect on the transmission system. The distribu-tion system, however, is inherently unbalanced owingto feeder configurations, unbalanced transformer con-nections, existence of single-phase and unbalancedthree-phase loads, multigrounded connections and earth-return paths. Fortunately, the distribution system isgeographically localized and, geherally, radial innature. Accordingly, it can be logically subdividedfor analysis; this affords the opportunity to considerthe system in more realistic detail and allows a com-plete three-phase representation in which actual phase(i.e., a, b, c phase-frame of reference) quantitiesappear. Syinmetrical component networks, which de-couple for a balanced system and thereby allow decom-position to per-phase analysis, lose this advantagefor synthesis problems involving unbalanced primarydistribution feeders. Hence,, phase-frame reference(i.e.,. a, b, c) modeling is appropriate and necessary.

The new iiethodology presented here, for power-lossreduction via fixed shunt capacitor-bank placement onradial feeders, is based on realistic modeling proce-dures (see Reference [3]) which account for:

(a) Mutual coupling between phases andground wires (multi grounded)'.

(b) Concentrated single-phase and un-balanced three-phase loads.

*Now with Carolina Power and Light CompanyRaleigh, NC 27602

83 WM 160-9 A paper recommended and approved bythe IEEE Transmission and Distribution Committeeof the IEEE Power Engineering Society for presen-tation at the IEEE/PES 1983 Winter Meeting, NewYork, New York, January 30-February 4, 1983.Manuscript submitted September 8, 1982; madeavailable for printing December 6, 1982.

S. H. Lee*, Member, IEEE

(c) Current earth-return paths.(d) Feeder tapering to handle different wire-

sizes in the various sections along thefeeder length.

An a, b, c phase-frame of reference load flow analysisis unde'rtaken for the balanced and unbalanced three-phase loads on the feeder. This yields a voltage pro-file along each phase of the system and provides thereal and reactive current flows in each section of thefeeder. The peak kilowatt loss due to reactive currentflows in each section of the feeder is expressible ina real, quadratic form. Via power-invariant (i.e.,orthogonal) transformations, mathematically equivalentfeeders are developed which are devoid of mutual cou-pling and non-uniform resistance in each section. Thegeneralized voltage-dependent procedures of References[1] and [2) are then modified and extended for use inan iterative manner to find the optimal locations andindividual phase capacitors to be placed on the actualfeeder so as to minimize the peak kilowatt loss.

DECOUPLED NORMALIZED THREE-PHASE FEEDER

The concept of uniform normalized feeder intro-duced in Reference [1] allows introduction of thenotion of distance and, thereby, permits calculationof optimal locations along the feeder for shuntcapacitors. However, the three-phase distribution pri-mary feeder, by its very construction,- is an asymmetric(i.e., unbal anced) el ement. The conductors of thephases may differ and may not be spaced equilaterallywith the result that the self-inductance of each phaseand the mutual inductance between phases are not equal,in general. Even when balanced voltages are consideredapplied at the substation, unbalanced currents willflow in each phase whether the loads supplied arebalanced or unbalanced. The presence of ground wiresand earth-return paths further complicate the calcu-lations of feeder parameters.

To exploit the uniform normalized feeder conceptfor representative feeder configurations actuallyfound on existing distribution systems, it is requiredto replace the physical three-phase feeder by an equi-valent feeder of three-phases each of which has a uni-form, yet equal, resistance per unit length. This canbe accomplished via a two-step procedure now brieflydescribed. Full detgils appear in the APPENDIX.

Consider the jtn section of the actual three-phase feeder shown in Figure 1; the concentrated P,Qloads appear at the end of each section as shown.Ground-wires and earth-return paths are not shown butare accounted for in the values of the impedanceparameters.

When the reactive currents in the ith section ofthe feedt; are modified as in Eq. (14), the power lossin the i section can be written as follows:

pLi = [Iqabct [Rabc] [Iqabc] (1)

0018-9510/83/1000-3296$01.00 (C 1983 IEEE

3296

3297

resistances Rd, Re, R1 in Eq. (6) are not equal. There-

fore, it is not directly possible to develop the normal-ized equivalent feeder of uniform resistance for eachof the phases d, e, and f; to do so directly for eachseparate phase in Figure 2 would cause the distances ofthe load buses from the substation to be distorted withrespect to one another. Therefore, we require a second

current transformation,[Tl] for the th section of the

feeder, with the following properties:

(i) The power loss PLi in the ith sectionof the feeder must remain invariant.

(ii) The resistance per unit length of eachof the resultant phases x, y, and zmust be identical.

As shown in the APPENDIX, these properties are satisfiedby the current transformation

Figure 1. Model for actual three-phase feeder usedin three-phase load flow. Load points andsections numbered as shown; ground wiresand earth-return not shown.

where

[R a] =

R 1 R 1 R 1

E i i i -

Rba Rbb Rbc

I i i_- RCa Rcb Rc c

where

(2)

[Idef] = [T [I xyz]

-1 0

[Ti] -= 0

0

(7)

0

0

Rf R

(8)

The power loss in the ith section of the feeder is thengiven by

andPL = [ I xyz [ Rxyz] [I xyz

i i i t[qab,c = 'qa Tqb Tqc' (3)

The resistance matrix in Eq. (2) is real and symmetricwhile the superscript t in Eq. (3) denotes transposeof the vector of phase7currents. In order to decouplethe a, b and c phases (i.e., to diagonalize the matrix

in Eq. (2)) a real, orthogonal transformation, [Ti], is

introduced which preserves invariant the power loss ineach section of the feeder. Accordingly for the ithsection, we have the current transformation

[Iqabc] = [Ti] [Idef]

from which follows

PLi = [Idef ] [Rdef ] [ Idef ]

where

i itT iI[Rdef]I = [T1] [Rabc] [T1] =

0 Oi I

Re O

0 RI_

The power invariant transformation of Eq. (4) leads toa mathematically equivalent feeder, with d, e, f de-coupled as shown in Figure 2. It is noteJ that thepower loss within each section of the actual feeder is'preserved and the load buses are unchanged at the samedistances from the substation. In general, the

where

[Tyz ]=[T 1 [IqaIc] = [T2] [Tilt [Iqabc]

and

[Rxz =yz

(4)

0R

0 R

0

0

0 0 Rd

(10)

(11)

The physical interpretation of the transformation ma-

(5) trix [Ti] in Eq. (7) is shown in Figure 2 in which it

is seen that each of the phases x, y, and z is de-coupled, section by section, andThas the same sectionlength and identical resistance as the original. For

the j h Rd.tei section, the resistance isRdRemark: In Eq. (8), it is apparent that the transfor-mation is based upon the arbitrary choice of maintain-

ing the resistance Rd as the uniform resistance in theth section of the x, y, z phases. Obviously, Re ore

Ri could have been chosen with corresponding changesin the matrices of Eqs. (8) and (11).

The normalization procedure. for a single phaseintroduced in Reference [1] can now be applied to each

(9)

Rd

0

0

3298

of the phases of the mathematically equivalent x, y, z

feeder shown in Figure 2. While the current flows in

each section of the separate phases will differ, theycan be used to construct normalized reactive currentdistribution functions F (x), F (x), and F (x) where

y z

x represents distance from the substation measuredalong three identical phases of unity length havinguniform resistance y ohms per phase [1].

EQUIVALENT CURRENT DISTRIBUTION FUNCTIONS

A three-phase load flow solution, using the a, b,c phase-frame of reference, can be found for the actual-feeder parameters and (balanced or- unbalanced) loadingconditions. The a, b, c voltages for each bus alongthe feeder is the7n knowni. Using the procedures setforth in Reference [2], we can construct a voltage-modified representation of the reactive currents foreach phase in the following manner.

Consider phase a of the physical feeder. The loadcurrent flowing' in the section of the phase between'bus i and bus (i+1) is given by

N N N

a da jI qa k=.ak kI=dak

where subscripts d and q respectively denote the d-axisand q-axis components of the complex currents of sec-

tion i, Il, and of the load currents at bus k, Iak.

Note that the ith section and bus load currents onphase a are related by

N

Ia-a~. Iqk'( 13)

In the manner explained in Reference [2], we now definefor'the it section of phase a a modified current forsection i by

N

Iqa = Z(iqakk=i

where

Iqak

Nsin

* (1/N) j=1sin 6.ak

.L)a, (14)

(15)Qakcos 6ak- Pak sin 6ak

Vak

In Eq. (15), Pak and Qak represent the respective kil-

owatt and kilovar loads at bus k and a while Va and- ak6ak represent the voltage magnitude and phase-angle at

the same bus. Equations similar to those in Eqs. (12)-(15) can be developed for phases b and c.

In order to establish current dist7ribution func-tions for the decoupled uniform phases x, y, and z ofFigure 2, we use the transformation matrlices of Eq. (10)as follows:

[Iqa Iqb Iti 1 i i i t

y z,(16)

The transformed section currents I', I and I con-y' z

stitute the reactive load levels which define the cor-responding current distribution functions F (x), F (x),

x y

and F (x) which are staircase in nature [1].z

LOSS REDUCTION AND SAVINGS FUNCTION

It is required to locate and size n fixed three-phase capacitor banks'so as to minimize the power losson the actual three-phase feeder. Let i' representthe bus at which the jth capacitor is located onphases a, b, and c. It then follows from Reference [2]that the correspon-ding q-axis capacitor currents onphase a are given by

ICa i

A

V cos si (i=1 92 l. .. ,n)Ci= Qai 'ai-os ai~(17)

where Qcai is the nominal rating of the ith capacitor

connected phase-to-ground. These a, b, c capacitorcurrents are'related to the.x', y, z frame of referencecurrents by

(FZ ICabcj]I C

[T 1][I]I,Ix z

j=

(18)

where

i i i i

and the left hand vector current of Eq. (18) is

similarly defined.The peak power loss reduction effected by the n

shunt capacitors in the x, y, z frame of refe'rence is

given byn

LP = LPIi=1

(20)

qal0 I R IN N 0

x i-1 R I1NNRacl;g lacai aN d I-i d d Id N 0S R Ii i N

I I i - ~~~~~~~~~~~~~dA-M

i N N 0e -i1 R I T I N Iy d Iy N

4i---- l I t iebb----ee e e e y---~ ~ ~~~~~1

'qbl V qbiIqbN If Iftz i-l Rd Ii Ij NqbN 0 if ~~~~~R'f If N

~bc NJ+--if -I- ~0 I~ I IN N° c c c

'"V c 1 klqc i IqcN

Figure 2

Decoupling power-invariant transformations relating reactive-current flows and feeder-section resistances.

Ij i-I i-i

[ ICxi (Fx (X)- 2E ICxj )+ICy; (F (X)-E Icy)j=j Cyiy j=1Cj

IQ _CoicO=a,b,ccoi v;i,cos 6dj " i=1,... ,n

Subproblem B: Opt-imal Locations

i-1

+ IC s(F (X) E I j dx - yh (I 2+I 2+I 2Cziz j=1Czj i Cxi Cyi Czi

(21

Obviously, because all the reactive currents in eachframe of reference are interrelated by power-invarianttransformations, the power loss reduction, LP., is thesame for the ith feeder section in each referenceframe. The net dollar savings resulting from peakpower loss reduction due to the placement of n capaci-tors is shown in the APPENDIX to be

S KpLP Kc [Pi I Py Icj+

pz Ij=1C Cxj y Cyj z Czj (22)

Where K and K are constants to convert kilowatt sav-p c

ings and capacitor kilovar to dollars while Pi" P3 and

Pi are defined in Eq. (A.20).z

OPTIMUM LOCATIONS AND BANK SIZES

The optimal design problem now is to simultaneous-ly solve for the optimal locations and sizes of the ncapacitor-banks to be installed at buses along thethree-phase primary feeder of Figure 1 so as to maxi-mize the net dollar savings represented by the functionS in Eq. (22). This is done via an iterative procedureinvolving two subproblems in which either the locationsor bank-sizes are known while the other is sought.

In the x, y, z reference frame, it is requiredthat each of th-en capacitors be placed at buses com-mon to each of the phases. This is to ensure that thelocations found for the actual a, b, c phases will beconsistent for three-phase capacitor Eanks. The de-tails leading to solution of the necessary conditionsfor optimal locations are provided in the APPENDIX andlead to the following sequential procedure:

(a) Compute Ml, M and Ml using Eq. (A. 28)y

(b) Weight the current distribution functionsFx(X), Fy(X) and F (X) by Mx,My and M,respectively. Then compute the composite

current distribution function FXyZ (X),representative of all three phases,Eq. (A.30).

(c) Compute the locations hi (i=1,2,. . .n)from the inverse of the functionF1 (h.) using Eq. (A.31).xyz I

(d) Return to step (a) until optimal locationsare found for all n capacitors.

A flow graph for the computer-based iterativeinvolving the transformation subroutines, thephase load flow, Subproblem A and B, etc., isin Figure 3.

scheme,three-shown

Subproblem A: Optimal Bank-Sizes

It is shown in the APPENDIX that if the locationsare specified, then the optimal capacitor sizes foreach of the phases x, y, and z can be determined bysolving the followinig set of Tinear equations for

IC,i (i=1,2,... ,n) where ¢ = x, y, z.

h1 h2 . hn] O -F(X)dX 2: P

h2 h2 . h I Fi (X)dX c- 2

h h."Ihn

hn ... hn

L.ip

I C~ i

ZY p

li

CF, (X)dX Kc pi2yK

p

p

K ncJi(AX)A

0

UNSITUCTI IHE IRANSFOURMA-TION MATRICES [ Ti] FOR ALL

CONSTRUCT THE X, Y AND 2i-lEDER USING EQS. (10) -

INITIALIZATION: BEGINH PROPR INIITIAL VALUES

FOR Q and h., (i = 1,2,.nJ4 = A,9,C)

(23)

The solutions for capacitor currents so found are inthe x, y, z reference frame. It is necessary to trans-form these7currents to the a, b, c physical frame ofreference by use of Eq. (A.13) ofthe APPENDIX. Thenall that remains is to convert the resultant a b ccapacitor- currents to per unit kilovar ratings using

SOLVE THE THREE-PHASELOAD-FLOW

OBTAIN THE MODIFIED SECTIONREACTIVE CURRENTS FOR PHASESA. B AND C (EC. (14)

OBTAIN THE SECTION REACTIVECURRENTS FOR THE X, Y AND ZFEEDER EQ. (10)

ONSTBRUCT FH (X). (=

LUETRROB.EM A:OSING UVDATED LOCATIONS hi,((i=1,2, .. .,In), I DETF t-1;I NE,1CAPACITOR SI/ES ICli; EQ. (23)

SUBPROBLEM B:USING UPDATFO CAPACITOR SIZESIC 4 = X,Y,Z), DETERMINELOCATIONS, hi; EQS. (A.30) -LA.31)

DOES THE S LUTION

YES

BACK TRANSFORM IC AND OBTAINIC U,( = 1,2....n), ( = A,

/ARE lID AND hi, (=1,2/ ..,n) AND ( t = A,B,C) TH

SAME AS THOSE AT THE PRE-OUS ITERATION?

NO YES

FPDATE QCoi (i = 1,2,

...iIn I A BIQiH

CH =V Cos w Hi-

Figure 3. Computer Based Iterative Solution Procedure

where

h

LP1 = 2y0

3299

(24)

1-0

IIConL _j

3300

Remark: Closed-form algebraic expressions can bereadily developed for the transformations

[Ti], [T2] and [Ti].

Accordingly, they need be calculated only once for any,

given feeder and [T i] alone needs to be stored.

NUMERICAL EXAMPLE

The iterative desi§n procedure described abovehas been applied to an elaborate numerical example. Wechose a realistic, multigrounded, three-phase feeder offive wire-sizes; the chbsen ground-wires correspond tothose actually used by an operating utility (seeTable 1).

SECTION FEEDER PHASE WIRE GROUND WIRE

N.LENGTH TYPEQNo. (MILES) SIZE PER SIZE PERMILE MILE

1 0.63 1 300 Cu 0.1966 2 Cu 0.8640

2 0.88 2 336.4 Al 0.2790 1/0 ACSR 0.8880

3 1.70 2 2/0 Cu 0.4440 4 Cu 1.3740

4 0.81 3 2 Cu 0.8640 4 Cu 1.3740

5 2.30 3 2 Cu 0.8640 4 Cu 1.3740

6 1.05 3 2 Cu 0.8640 4 Cu 1.37407 1.50 4 4 Cu 1.374 6 Cu 2.1800

8 3.50 4 4 Cu 1.374 6 Cu 2.1800

9 3.90 4 4 Cu 1.374 6 Cu 2.180

Table 1. Wire sizes-lehgths and per-mile resistancesfor the example feeder; feeder types referto Table 2.

The self-and mutual-inductances were calculatedfor four different spatial configurations which aredescribed in Table 2. The parameter-calculations ac-count for an earth-retuF'n path with resistivity of 100ohm-meter [3]. The optimal design problem is now tolocate ahd size three shunt capacitor-banks along thefeeder so as to reduce the peak kilowatt losses whileaccounting for the cost of capacitors. The parametersKp = $120/three-phase kW/year. and Kc = $0.5/three-phasep

kvar/year were chosen. A comparative analysis of thecapacitor design schemeS was undertaken for differentfeeder mnodels and diffetent levels of load bala-nce.

1\'Y5 | 2 3 4

TYPE *a\ * b *b *b

*c a * *c as c ao ec *b

StAC NG(inches) ow * w ow w

Dab 42 32 28.4 42

IDbe 42 32 28.4 42

D 84 36.5 36 84ca

D 142 60.8 43.86 62.3aw

Dbw 100 84 53 50

Dcw 58 60.8 43.86 62.3

Deq. 52.92 33.44 30.74 52.92

Table 2. The configuration types used along the sec-tions of the example feeder; interconductorspacings are indicated. Deq is the equia-lent equilateral spacing between phases.

The P,Q loading conditions at each bus of the radialfeeder are shown in Table 3. The tabulation of resultsis set forth in Table 4 from which it is apparent thatthree feeder representations were used. The symmetricfeedor corresponds to an equivalent equilateral spac-ing of D = D"T a . The asymmetric feeder

eq a ccmodel uses parameters with and without ground wires i'n-cluded. The numbers in Table 4 have been rounded-offto the nearest integer values; hence, some totals willnot be the exact sum of their components.

TYPE OF BALANCED UNBALANCEDLOADING

PHASE

A,B,C A B C

S.S. 0.0, 0.0 0.0., 0.0 0.0 0.0 0.0, 0.0

1 613, 153 .613, 153 490, 123 736, 184

2 326, 113 326, 113 261 91 392, 1363 597 149 597, 149 447 119 716, 1784 533, 613 533, 613 326, 491 639 7365 537 200 537 200 429, 100 644 240

6 260, 37 260, 37 208, 29 312, 44

7 383 ,20 383 ,20 307, 16 460 ,24

8 327, 43 327, 43 261, 35 392, 52

9 547, 67 547, 67 437, 53 656, 80

Table 3. Balanced and Unbalanced loads (kW, kvar) atbuses of example feeder. S.S. meansSubstation.

DISCUSSION AND CONCLUSIONS

Since Table 4 show-s results for a specific ex-ample it is inadvisable to base general statementsthereon. However, it is observed that for

(1) The total feeder compensation level for peakpower loss reduction altays exceeds thetotal kilovar load. It is easily showinfrom Table 4 that the total compensationlevel varies from 124% to 142%.

(2) The capacitive compensation level on an in-dividual phase oasis varies from 93% to 155%.

(3) The symmetric feeder, which is the basis ofmost previous studies (involving only sin-gle-phase analysis) is a reasonably goodmodel for balanced loads. Asymmetry of thefeeder configurations appears not to be avery significant factor in capacitive compen-sation schemes when all loads are balanced.

(4) Unbalanced load conditions require the use ofa three-phase load-flow program. Capacitivecompensation schemes based on the more real-istic models (those involving asymmetry andwith/without ground-wires) appear to moreproperly reflect the kilowatt loss reductionand dollar savings. Accordingly, the pro-cedures developed herein provide a meansnecessary for design of capacitive compensa-tion schemes for unbalanced three-phase sys-tems.

(5) In the case of unbalanced three-phiase loads,it is not sufficient to separately treateach phiase of the feeder for capacitor loca-tion and sizing. In Table 4 it is noted thatlocations do not differ between cases stud-ied; this is considered to be due to the load

3301

TABLE 4. OPTIMAL RESULTS FOR DIFFERENT FEEDER MODELS AND LOAD REPRESENTATION

distribution chosen and not a general indicationof insensitivity of locations to model-variationsor degrees of unbalance in loads.

While other observations may be made, the overall con-clusion is that the procedures presented here providethe distribution engineer with means to optimally de-

sign three-phase capacitive-compensation schemes forrealistic feeders with commonly encountered unbalancedloads. To the authors' knowledge, this is the firstreported methodology for this purpose.

It is important to note that the design procedurepresented here incorporates, at each iteration, the re-sults from a full three-phase a.c. load flow. Accord-ingly, the real and reactive power and current flows,as well as voltage variation along the feeder, areused to update the feeder reactive current distribution.It is already demonstrated in Table 3 of [2] that thisvoltage-dependent methodology (which is extended andgeneralized here for three-phase systems) yields kW

savings which are indeed larger than those found bytreating reactive current flows separately from totalcurrent flows. Therefore, we emphasize that-the treat-ment of capacitor currents and reactive current flows,as presented here, more properly reflects the-true kWloss savings without overstatement. This is readilyseen by comparing results from a full three-phase loadflow solution with those from the model which excludesthe load flow in the iterative procedure.

The design methodology which is set forth aboveapplies general'ly to all three-phase feeders and there-fore offers a basis for cost/benefit analyses by thedistribution engineer. The authors are currently work-ing with a large utility on feasibility tests involvingthe new methods. The authors suggest that the inter-ested reader consult references [1] and [21 to assisteasier assimilation of this paper.

ACKNOWLEDGEMENT

One of the authors (J.J.G.) wishes to acknowledgethe cooperation provided by Professor W. H. Kersting,New Mexico State University, Las Cruces, New Mexico.This research was supported by the- Energy Division ofthe Union Carbide Corporation, Nuclear Division on be-half of the U.S. Department of Energy under SubcontractNo. 7955 and Purchase Order No. 19X-22294V.

Mr. John P. Stovall is the projett director andDr. T. W. Reddoch is the program manager. Theirassistance and cooperation are greatly appreciated.

REFERENC ES

[1] J. J. Grainger and S. H. Lee, "Optimum Size and Lo-cation of Shunt Capacitors for Reduction of Losseson Distribution Feeders," IEEE Thans. Powier Appa-ratus and Systems,, vol. 100, pp. 1105-1118, March,1981.

[2] J. J. Grainger and S. H. Lee, "Capacity release byshunt capacitor placement on distribution feeders:a new voltage-dependent model," IEEE Trans. PowerApparatus and Systems, vol. 101, pp. 1012-1020,

[3]

reay, 1982.

P. !1. Anderson, Analysis of Faulted Power SystemsIAmes, Iowa: The Iowa State University Press, 1976.

[4] B. Noble and J. W. Daniel, Applied Linear Algebra,Englewood Cliffs, New Jersey: Prentice-Hall, Inc.,1977.

FEEDER ODEL SMMETRICFEEDER ASYMMETRIC FEEDER ASYMMETRIC FEEDER SYMTI EDRI ASYMMETRIC FEEDER IASYMMETRIC FEEDERFEEDER MODEL SYMMETRIC FEEDER |WITHOUT GROUND WIREI WITH GROUND WIRE MWITHOUT GROUND WIRE WITH GROUND WIRETYPE OF LOADING BALANCED LOADS UNBALANCED LOADS

PHASE A B C A 8 C A B C A B C A 8 C Ar B CKILOWATT LOAD PER PHASE 4123 4123 4 412 3 4123 4123 4123 4123 [4123 4123 [3268 4947 4123 3268 4947 4123 3268 4947THREE PHASE KILOWATT LOAD 12368 12368 12368 12368 12368 12368KILOVAR LOAD PER PHASE 1396 139 1396 139 1396 1396 1396 1396 1116 75 1396 111611675 1396 [1116 j1675THREE PHASE KILOVAR LOAD 4187 4187 4187 4187 4187 4187KILOWATT LOSS BEFORE 287 287 287 309 280 271 315 276 269 287 171 444 97 217 630 171 173 609CAPAC TOS INSTALLED _OPTIMAL SIZETOF 147 147 147 155 149 149 161 154 150 147 97 215 147 196 314 147 89 268

OPTIMAL SIZE OF 365 365 365 358 348 349 361 347 348 365 265 486 342 116 603 343 227 521CAPACITOR 2 (kvar)OPTIMAL SIZE Of 1255 1255 1255 1237 1221 1222 1437 1436 1422 1255 961 1584 1402 729 1708 1393 1143 1807

CAPACITIVE COMPENaTIOLEVEL PER PHASE (kvar) 767 1767 1767 1750 1719 1719 1959 1937 1920 1767 1323 2284 1891 1041 2025 1883 1459 2595COMPENSATION LEVEL 5300 5188 5815 5373 5557 5938

OPTIMAL LOCATION OF CAPACI- 16.27 16.27 16.27 16.27 16.27 16.27TOR 1 (MILES FROM SUBSTATION)

OPTIMAL LOCATION OF CAPACI-TOR2 (MILES FROM SUBSTATION) 6.32 6.32 6.32 6.32 6.32 6.32OPTIMAL LOCATION OF CAPACI- 4.02 4.02 4.02 4.02 4.02 4.02TOR 3 (MILES FROM SUBSTATION) . . 4 4KILOWATT LOSS REDUCTION |3 6 333 36 40 34 3 66 15 (34 71 14 6 126PER PHASEj j j f4j11 jj~j3-PHASE KILOWATT LOSS REDUC- 109 109 108 121 152 146TION: CAPACITORS INSTALLEDDOLLARS SAVINGS PER ANNUM $10449 $10490 $10025 $11862 $15416 $14518

APPENDIX

A.1 Three-phase Power Loss Reduction

The three-phase power loss in section i (i=1,2,...,N) of the actual feeder is given by

= Re abc abc abc

a[ aa[ Zab zac jFia

e Jb Lba Zbb Zbc bL

where Re me.ans "real part of" and-

Ip d,p jIq4,*t ab,bc

(A.1)

(A.2)

Equation (A.1) may be rewritten as follows:

i 2 i 2 i [ j-i , + ( i 2Li = { aa[(Ida) + 'qa) ] bb'db' (qb) ]

I i 2 i 1i+ Rcc[dc) + (Iqc) ]+ 2[Rab(IdaIdb+ IqaIqb)+ R (111I1 + I1 I1 + R11(11 I1 + 1i 1

ac da dc qa qc bc db dc qb qc

(A.3)

Before placing capacitors, define the section icurrents by

df= I- q = a,b,c (A.4)

Assuming the capacitors affect only the q-axiscomponents of the complex currents, define the ithsection current by

Iif= ~Id 1Iqf (A.5)

Using Eqs. (A.3) - (A.5), we express the three-phase power loss reduction in section i as follows:

LP Ri [(I )2 _ (I 2] + Ri [(I1 )2 - 2aa qa) qa' bbl(qb qb

i )2 -i 2 Ri 'z i - 1iRcc[(Iqc) '(Iqc) 1+ 2 Rab(iqaIqb - Iqaqb)

ac qa qc - qb qc

+ 2 R' (Ibl -I -I 1 ) (A.6)

The normalization procedure in Reference [1] mustbe modified since the self and the mutual resistancescannot be made uniform simultaneously. It is requiredto express the power loss expression so that the quad-ratic form is diagonal. Using the modified quadratureaxis components of the currents in Eq. (14) the powerloss is expressed as

PL. = [Ij ]t[Rl ][I, ]qabc abc qabc

(A.7)

where symbolic definitions are given in Eqs. (2) and(3).

Using Eq. (4), we may rewrite Eq. (A.7) asfollows:

(A.8)PLj = [I I][T I [R Ic[T' ][I Ifdef 1 abc 1 def

or, as in Eq. (5),

PLj = [Idef[Rle][I'ef (A.9)

The matrix in Eq. (A.8) is.diagonal with elementsequal to the eigen values of [Rlbcl. Since [RAbc] isreal and symmetric, a real orthogonal transformation[TI] is readily found [4].

A.2 The Transformed Currents and Equivalent Feeder

It is required to construct a three-phase normal-ized feeder with phases which are identical in sectionresistances and lengths. This is accomplished bytransforming the currents and leaving the power lossin each section unaltered.

Choosing phase d as a reference, we set

i R IRId

(A.10)

where a = d,e,f has one-to-one correspondence with= x,y,z.

Immediately Eqs. (7-11) follow and thhe peak powerloss reduction which is effected by the t1 shuntcapacitors may now be expressed as in Eq. (21).

A.3 The Savings Function for the xyz Feeder

The net savings function corresponding to peakpower reduction by optimal capacitor placement is givenby [2]:

S = K LP - Kc I r Cai + Cbip c i=1taiICo 6ail VbiCo bi

+ 1Cci 1

Vci.CoS6 J,(A. 11)

As in Eqs. (18) and (19), we now transform thecapacitor ph4 e currents of Eq. (A.11) by the trans-formation [Tl] , applicable to the feeder section ter-minated by bus i', as follows:

ICam I1 Cbm mi ICcm]

[Ti][M=1 TCxm M1ICym m=1ICzml

Expanding and rearranging Eq. (A.12), we get theing lower-triangular matrix equation (A.13)

[Cabc1] rT i]O

CO

[ ICabc i]

_ I ICabcn ]

(A.12)

follow-

:xyZ 1]1

0 0 0

.. IT1 I O 0 [I~XZ

[ICxyzn]

I

3302

(n-1) ]]-[T I... [Tn 1-[T (n-1) ) 1. . I[Tn ,j

k< 0

3303

where

[ Cxyzi 1 A [ Cxi 'Cyi 'Czi]

aLPk

(A. 14) aIcpi

The jth row of Eq. (A.13) is written in the form

[I Cabci I [T I] - [T(i-1) )[ I I'Cxyzm]+ [r1 1~~'C

=

+ [Ti)i [0Cxyz1

Note that [T(i 1) ] = [01 , for i. = 1

S = x

(A.15)

Equation (A.11) may now be rewritten as follows:

<pLP - K {[B ([Ti][I I Ip c mI1 Cxyzm

i-i- [T(')'][ I

m=l ICxyzm])

whe re

[Bi = [Bait Bbil Bci*1and

Bf = (V. , Cos 6+i,))1 for 4 = a,b,c

and i = 1,,2, ... ,N

A simpler form of Eq. (A.16) is

(A.16)

(A. 24)

Substituting from Eqs. (A.22) - (A.24) into Eq.(A.21) we now write

aCsi - K~ naS= K xaIfj P k=l

LPk k'I cCcpi

(A.25)

which yields the desired set of linear equations givenby Eq. (23).

A.5 Derivation of Necessary Conditions for Subproblem B

For optimal capacitor lo ations on the x,y,zfeeder, the location of the ijh capacitor must satisfy

aS 3LP7- Kp .-= 0

1 1(A.26)

It can be easily shown that for i = 1,2,...,n

(A.17) = 2r[I CxiFx(hi) + ICyiFy(hi) + ICziFz(hi)]37i~~~~~~~~Zi-i i-i

- 2r[ICxi 1 ICxm + ICyi m-1 Cym

(A.18)- r[(ICX )2 + (ICyj2 + (I 21 = 0Cxi) yi) Czi)

i-1+ CzmmIi Czm1

(A.27)

S = K LP - K n(P1 I. + PY II+ P Cz)p c .I x Cxi yCyi z Czi

(A.19)

where[k I =[k Ipk Izk

x y z

= [Bn [Tn ] + )[[BJ]I - [B(j+1)']][TJI'

for k = 1,2,...,,N (A.20)

A.4 Derivation of Necessary Condition for Subroblem A

From the necessary conditions for optimal capaci-tor sizes it is required, for i = 1,2,.. .,n and ¢ =

x,y,z, that

Defining

N 2i 21i-1

i 2m 2 IC4j; Lf 2 1CI, EI,; Gf (IC¢5) (A,

for 4 x,y,z. We then write Eq. (A.27) as follows:

M F (hi) + MiF (hi) + M F (hi)i~~~~~~_L +Gi +L+ Gy + L +Gz (A

For further simplicity of notation, we define

F yz(X) = M'FX(X) + My(X) + M (X) (A

whence it follows that Eq. (A.28) can be written as

(A.31)Fi (hi) = L' + Gi + Li + Gi + L1 + Gixyz 1 x x y y z z

a - = K p = 0

For any i, we have:h.

= 2r( fC¢i o

- 2rh I2i)cp

aLPk= - 2rhk I1tk~~~k

i-i

Fg(X) dx - hi ( miICpm)

(A.22)

(A. 23)

.28)

i.29)

.30)

(A.21)

i = 1,92 , . . . ,n

k > i

3304

Discussion

D. J. Ward (General Electric Co., Schenectady, NY): The authors areto be complimented for their analytical work covering capacitor op-timizing with unbalanced impedance and load effects on distributionfeeders.

It should be noted that the authors' "voltage-dependent model" is,in fact, a constant kVA load representation. This representation, inturn, makes the line losses also voltage dependent, with low nodevoltages leading to higher currents and greater line losses. I raise thispoint because the substation voltage in their example-has a value of 1.0per unit and the resulting voltage profile- is well below normal. By in-creasing the substation voltage to 1.05 per unit, the resulting losses aresome 103 kW and 123 kVAR below the base case values.

It appears that their methodology is also suitable- for handlingconstant-current loads. Most of our distribution system studies utilize aconstant-current load model. In this representation, each load is assign-ed a current value (and pf) based on nominal or rated voltage. The loadreal and reactive power values, thus, are affected by the voltage profile;however, the magnitude of line losses are not.The first point raised in the Discussion and Conclusions section, that

"the total feeder compensation level for peak power loss reductionalways exceeds the total kilovar load," requires further explanation. Itappears to be saying that you always overcompensate the feeder to aleading power factor condition. However, in checking the balancedthree-phase symmetric case, I find that the reactive power losses areconsiderable-(about 25% of the connected load). This is due to the com-bination of the constant kVA load representation and the low voltageprofile. It would be of interest to know if the compensation levels ofTable-4 ever exceed the total load plus losses.

While-the authors strive for realism in the circuits which are analyzed,they continue to ignore that discrete nominal capacitor bank sizes exist.Utilities have standardized on certain bank sizes for reasons of economyand overcurrent protection. A more meaningful technique, in my opin-ion, would be to specify the available capacitor bank sizes and solve forthe optimal locations. This will actually- make the computational efforta good deal easier. Perhaps the use of actual capacitor bank sizes wouldlead to fewer different solutions than are indicated in Table 4, but,nevertheless, they would represent the choices available to the distribu-tion engineer.

Manuscript received February 17, 1983.

D. L. Allen (Consolidated Edison Co., New York, NY) and M. A.Wortman (Clemson University, Clemson, SC): The capacitor sizing andplacement methodology presented by the authors is an extension ofearlier work as reported in their References. This present paper reportsa novel effort to remove the coarse assumptions made in previous in-vestigations of balanced phase loading of a balanced multiphase elec-trical network. The use of transformed section currents is an appealingmechanism for introducing distance variables for the indication ofcapacitor positions.

In an effort to more fully-appreciate the implications of the informa-tion presented in the paper, these discussers request the authors' com-mentary on the following points:

(i) The grounding resistance associated with each earth electrode in amulti-grounded system is likely-to be of significant size relative to impe-dance values elsewhere in the distribution system. In the authors' opin-ion, how (if at all> are the effects of earth grounding resistance likely toalter the nature of the comparative results for symmetric/asymmetricfeeders given in Table 4 in the paper?

(ii) The authors depend upon Ref. [2] of their paper (herein called[R2]) for the development of an "equivalent current distribution func-tion" which is a heuristic method of accounting for the effects ofrealistic nonuniform voltage profiles. On reading this paper and referr-ing to the explanations in [R2], it is seen that the substation "a" phasevoltage is chosen as a convenient system voltage reference. Also, it ap-pears that "reactive" and "q-axis" current refers to the component ofa current that is in quadrature to its corresponding substation phasevoltage. That is, a load, line, or capacitor current anywhere on thephase "a" system has a component (which can be zero) that is inquadrature with the phase "a" substation voltage, and this componentof current is given the subscript "q" (see Fig. 2 of [R2]). Except at thesubstation bus, the physical significance of such a q-zxis current is notobvious to these discussers.

Given that power flows or dissipations are defined by a complex cur-rent/voltage pair, an alternate specification of reactive current is thatcomponent of complex current that is in quadrature with a "local"

voltage such that when taken together, the current/voltage pair definesa physically- meaningful power. Reactive currents defined in this man-ner yield an interesting current distribution function as is illu-stratedbelow by way of an example.The system shown in Fig. D.l is based on the example presented in

[R2]. Bus voltages, load powers, and complex currents are given.Voltage and current angles are relative to the substation voltage, Vs.Fig. D.2 shows complex load powers and complex power flows basedupon the conditions of Fig. D. 1. Fig. D.3 shows reactive currents com-puted as the components of complex currents which are in quadraturewith the system voltage reference, Vs. These currents are the imaginaryparts of the currents given in Fig. D. 1.

2 .9871/2. 9942°

=2.9830+jO.1560(Apu)

VS=1.00(

Vg=0.99V8=0. 99

2.5575/6.0770°

=2.5431+jO.2707(Apu)

~00 / 0.0000°(Vpu)1

0671 058210(0pu) 0.4399-jO.1147(Apu)

17/-1.33160(Vpu) 2.5431+jO.2'

S =0.4396+jO. 1099(VApu) S8=2.5151-jO.32

707(Apu)

270(VApu)

Figure D. I Example-Feeder Showing Currents, Voltages,and Load Powers.

2.9830(Wpu) 2.9714 2.5319.~~ ~~ D. I

-jo.1560(VARpu) -jO.1857 -jO.29560.4396

I94 JO. 1099

2 .5151

-jO . 3270

2.5151

-j° * 3270 i t T

f1

Figure D.2 Example-Feeder Power Flows and Loads(reactive powers are indicated withhatched arrows).

©D

Figure D.3 Reactive Currents Computed Relative to theSystem Voltage Reference.

Fig. D. 4 shows reactive currents with each computed as the currentcomponent which is in quadrature with a "local" complex voltage(these current/voltage pairs define the complex powers of Fig. D.2).

jO. 3298

jO.3298

Figure D.4 Reactive Currents Computed Relative to"Local" Voltage References.

The reactive currents of Fig. D.4 lead to a reactive ction function, Pq(x), which is sketched in Fig. D.5.

A

FqCx) tApui)

(lead )]

0.156Q

0 1863.

0.2990.

0.3298,

Figure D.5 Reactive Current Distribution FunCorresponding to Fig. D.4.

The sloping portions of tq(x) between "s" and "9"9" and "8" represent the uniform reactive loadinjfeeder. While-complex current in each feeder section mithe components of these currents that are responsible-foiflow change as the magnitude and sign of reactive powe(The boundary values of these currents are shown in Figreactive currents so defined obey Kirchoff's current lav

This definition of reactive current directly- accountsdependencies of IDqi's. Also the capacitor currents,d-axis components, and therefore, interpedance of ICvanishes.The extent to which the proposed construction of the

distribution function affects the positioning and scapacitors remains to be determined. The possibility oftimal" positions is the more interesting question since t]flow does not allow positions to be decision variables. Ibe reduced to performing exhaustive case studies tovarious combinations of possible- positions in gettingflow verification of optimal sizes.

(iii) Under "Discussions and Conclu-sions" the autthat reactive compensation always exceeds total reactithe authors share any insight which-they might have irsignificance and cause of this condition?

(iv) Would it not streamline the feeder length norncedure to change the (1,1) term in Eq.(8) to V7R anequal to an identity matrix for all sections?

(v) Could the authors' methodology for representifeeders and loads be extended to optimally position andcapacitors in delta-connected and ungrounded wye-conrsation banks?

(vi) The authors indicate in. the Introduction thatinvariant (i.e., orthogonal) -transformations, mathematifeeders are developed which are devoid of mutual coupliform resistance in each section." It should be recogtransformation matrix [T21] is not orthogonal; -hence, Nuniform resistance for each feeder section, an orthogortion is not performed. Could the authors offer some releregarding the relationship between power invariant atransformations. Note that a non-orthogonal transfornplies that section voltages are not in the same reference fcurrents.

(vii) The authors offer a methodology by which optitions in the a,b,c reference frame can be inferred by exti.strained nonlinear objective function in the x,y,z reGiven that transformation by matrix ITi] does not preserit is not immediately obvious that optimum bank locati(reference frame will -infer the optimum bank locationreference frame by inverse transformation. A short proregarding this point would-be most helpful.

(viii) The authors make a distinction between symmemetric feeder configurations based upon interphasepower loss on a section can be calculated in terms of thewhich is the real part of [Zabcl]. It should be noted thaitained under the multiple grounding assumption using wIthe power systems literature as a "Kron reduction". Th

3305

urrent distribu- entries of IRabci] result from the Kron reduction through which the ef-fects of non-phase conductors are implicitly- represented in [Zabc'].Hence, the non-zero off-diagonal entries of [Rabcl] arise from the Kronreduction alone and not from nonequilateral (or asymmetric) phasespacing. Thus, interphase spacings alone have no effect upon realpower loss on a line section, and there is no need to distinguish betweensymmetric and asymmetric feeder configurations (as they are defined inthe paper).

(ix) This final point concerns the stochastic nature of the reactiveload distribution for the instant in time which coincides with peaksystem load. Given that the values taken by the distributed loads are notdeterministic, it is important to know the extent to which optimal posi-tions and sizes are sensitive to changes in load distribution. Could theauthors comment upon the effects of this sensitivity?

Again, the problem treated in this paper is an important one, and it ishoped that this discussion makes a positive contribution to its exposi-tion and solution.

Manuscript received February 18, 1983.

ctionJ. J. Grainger and A. A. El-Kib (North Carolina State University) andS. H. Lee (Carolina Power and Light Co): The authors wish to thank

I" and between the discussers for their interest and complimentary remarks.g effect of the Mr. Ward's remark concerning the low voltage profile on the feederust be constant, is an important one. With constant kVA load representation (usually-r reactive power used in load-flow analysis), it is shown by our results that capacitors dor flow changes. not appreciably- improve the voltage profile for the example feeder ofD.4. Note that the paper; it is clear that a voltage regulator should be used for voltage

w at nodes.) improvement. It will -be found that with 1.05 per unit voltage at thefor the voltage substation the compensation level is indeed considerable-reduced. Con-ICi's, have no sequently, it appears that "over-compensation" will not be required on

"qi'5 and 'Cdi's a feeder with adequate voltage profile. Mr. Ward suggests a strategy ofnot finding optimal sizes of capacitors but rather to prespecify the capa-

reactive current citor sizes and seek optimal locations appropriate to the prespecifiedsizing of fixed sizes. This strategy is, of course, encompassed within our more generalchanges in "op- methodology and is not ignored. The sequence in which the capacitorshe optimal load of known modular sizes should be installed is quite easily- shown fromThus, one could our procedures.consider all -the Drs. Allen and Wortman raise some interesting modeling questionsoptimal power which are being, or have already been, addressed by our research group

at North Carolina State University: (i) The objective of this paper is tothors point out develop a methodology for capacitive compensation of a general feederive load. Could model. All the different factors which can influence the parameters of aito the physical primary feeder can be readily included in developing such a model.

Once available- the model can be used in the manner shown. (ii) Themalization pro- reactive current distribution function in Fig. D.5 of the discussion waskd make [Rxyzi] long ago considered by the authors but was abandoned since our test

results were unsatisfactory. One obvious reason is that, while- the reac-ing unbalanced tive currents computed based on local voltages seem to appeal intuitive-I size individual ly, they actually-provide little information from the system's viewpoint.nected compen- There are other questionable features of Fig. D.5 representation which

the discussers may wish to consider. (iii) The reason for the over-t "Via power- compensation is obvious; the voltage profile-is low and this may not becally- equivalent acceptable in practice. (iv) We do not know what the discussers have ining and nonuni- mind here. If it is felt some simplifications are possible, then theynized that the should be made. We see no conceptual point here. (v) We see no reasonswhen obtaining why it cannot be extended given that the appropriate ac load flow exists.nal transforma- (vi) The transformation matrices [Tli] (i = 1,2, . . n) are orthogonal.vant discussion The matrices [T2i] are obviously not. The important point being madeind orthogonal is that IPR lo-sses per section remain invariant within our procedure andnation here im- the matrix notation is merely a convenience for so stating. (vii) Therame as section identity of buses does not change from on frame of reference to

another. Therefore, the capacitor locations on the x, y, z feeder are theLmal bank loca- same as those on the a, b, c feeder. (viii) Symmetric and asymmetricremizing a con- feeder configurations clearly affect the load flow results and feederference frame. losses. In the load flow analysis the matrix [Zabc'] for each ill section isve eigenvalues, used; the off-diagonal elements are functions of interphase spacingsons in the x,y,z and hence the discussers may wish to reexamine their statements. (ix)Is in the a,b,c, The practicing distribution designer is only too well -aware of the ran-iof or reference domness of the feeder load and its relative geographical distribution.

The representation of feeder load variability is important to appreciate,tric and asym- and is being addressed.spacings. Real The authors appreciate very much the contributing remarks of thematrix .Rabci] discussion and hope that their work will encourage other research in thist [Zabcl] is ob- area.hat is known inie off-diagonal Manuscript received May 23, 1983.

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