1607IEEE Transactions on Power Apparatus and Systems, Vol. PAS-102, No. 6, June 1983

OPTIMAL CHOICE OF FIXED AND SWITCHED SHUNT CAPACITORS ON RADIAL DISTRIBUTORS BY

THE METHOD OF LOCAL VARIATIONS

m . Ponnavaikko, Student Member IEEE

System TInprovement CellRural Electrification Corporation Ltd.

New Delhi 110 019India

Abstract - This paper presents a method of opti-mally choosing fixed and switched shunt cap'citors onradial distribution feeders, considering load growth,growth in load factor and increase in cost of energy.Mathematical models to represent cost saving due toenergy loss reduction taking the growth factors intoaccount, cost saving due to release in systam capac-ities, capacitor cost and voltage rise during off-peekhours, as a function of capacitive current flows inthe feeder sections have been formulated. Cost func-tions have been defined for optimizing the choice ofboth fixed and switched capacitors. A direct searchtechnique known as the Method of Local Viariations hasbeen employed for solving the resulting discrete vari-ational problem. The problem has also been solved us-ing Dynamic Programming Approach for comparison. Theproposed method has been illustrated through some ac-tual cases of radial feeders existing in an Indian di-stribution network. The results highlighting the in-fluence of the growth factors have also been discussedin this paper.

INTRODUCTION

The expansion of rural power distribution systemf or farm electrification means addition of more andmore inductive loads to the system, in an agriculturalcountry like India, which result in the reduction ofthe system power factor considerably. Thus, the highconcentration of the low power factor inductive loadsin a rural system normally keeps the system power fac-tor some times as low as 0.6. The low power factor ofthe system leads to increased power and energy losses,voltege regulation problems and loss of system capaci-ties such as generation, substation and feeder capaci-ties. Further, the reduction in the power factor callsfor creation of additional system facilities to mainta-in the customer voltage within permissible limits. Itis, therefore, necessary to explore the possibilitiesof improving the system power factor.

One of the easier ways of improving the systempower factor is the application of shunt capacitors.The a,pplication of shunt capacitor results in a heavyreduction of system losses, appreciable release in sy-stem capacities and, in addition, produces, though notlarge, a uniform voltage boost alonq the feeder. Theresulting improvement in system voltage leads to in-creased revenue, postponement of investments on new

system facilities, earning the goodwill of the custo-mers etc.

82 SM 478-6 A paper recommended and approved by theIEEE Transmission and Distribution Committee of theIEEE Power Engineering Society for presentation at theIEEE PES 1982 Summer Meeting, San Francisco, California,July 18-23, .1982. Manuscript submitted February 2, 1982;made available for printing May 5, 1982.

K.S. Prakasa Rao, Sr.Member IEEE

Department of Electrical EngineeringIndian Institute of Technology

New Delhi 110 016India

The probable places of location of shunt capacitorsare the customer load points, along the feeder mains(secondary and/or primary distribution feeders) and ateither secondary or primary distribution substations.Though the application of LT capacitors at the customerterminals yields the maximum benefit, it may sometimeslead to self-excitation and resonance problems in theinduction motors, resulting in the overheating of motorwindings and damage to the winding insulation. Also,from the practical utilization point of view, from theexperience it has been found that the capacitors instal-led at the customer terminals are rarely maintained tobe in service. A survey in an Indian power system hasindicated that more than 50% of the capacitors, instal-led at the customer load points were found to be eitherdefective or disconnected from service, thus resultingin a heavy loss of investment. Application of capacit-ors at the substations does not help the feeders. Undersuch circumstances, application of on-line capacitorsalong the feeder mains appears to be the best alterna-tive for improving the system power factor.

In thp rural distribution system, the off-peak topeak load ratio is very low and the off-peak load mos-tly consists of resistive loads. Thus, the power factorduring the off-peak hours is normally high. Therefore,heavy capacitor compensation may lead to over-voltageproblRms during off-peak hours. This situation posescertain limitations to the extent of capacitor compen-sation and hence leads to the concept of fixed andswitched type capacitor applications. The fixed capa-citors, capacity being limited by the over-voltage con-straints, remain in service continuously to yield themaximum benefit. The switched capacitors, costing moredue to the switcthing control equipment, remain in serv-ice only during peak hours and give comparatively lessreturns. The problem is thus to optimize the number,location and size of fixed and switched capacitors fora given feeder.

A distribution feeder is normally designed and con-structed to meet certain load growth in the system. Theload growth in any system is always followed by a growthin load factor and increase in the cost of power andenergy. These growth factors of the system certainlyinfluence the optimal choice of capacitor banks alongthe feeder mains when the present worth of the net cashflow during the life of the capacitor banks is maximi-zed. Hence the models developed for optimizing theallocation of capacitor banks should necesserily consi-der these growth factors to achieve the overall economy.

Considerable attempts have bemn made in the past foroptimally allocating shunt capacitors along the feedersl-5,11] . J.11. Schmill's 1i] work has been abeginning in this direction. He has proposed a modeland a search procedure to optimize the size and locat-ion of a given number of capacitor tbanks along the fee-der. R.F. Cook t2) developed an accurate formule todetermine the exact energy loss reduction achieved byshunt capacitors in radial distribution feeders. Basedon Cook's formula, H. Duran t3] formulated a goodnumber of theorems on the economics of capacitor allo-cations and suggested a dynamic programming approach

0018-9510/83/0600-1607$01.00 © 1983 IEEE

1608

for optimizing the number, location and size of shuntcapacitors in radial distribution feeders. J.J.Graingerand S.A4. Lee C4] defined three sub-problems for thethree aspects, location, sizing and switching time ,

treating one aspect as a variable and the other twoaspects as fixed in each subproblem and suggested aniterative procedure for obtaining the optimal results.They also developed [s5 an equal area criterion foroptimally sizing and locating fixed capacitors on radialfeeders. Y.G. Basa l] gave a simple mathematical equ-ation for calculatin,g the loss reduction and suggesteda procedure for optimizing the capacitor allocation.

8ut none of, the above approaches have accountedfor the benefits due to either capacity release orthe effects of the growth factors such as load growth,growth in load factor and change in cost of energy.But these aspects have direct and more pronouncedinfluence on the optimal capacitor allocation alonqthe radial feeders. Also, no model suggested so farconsiders voltage rise problems during off-peak periodes a constraint to limit the extent of compensation.

Mathematical models to represent cost saving dueto energy loss reduction taking the growth factors in-to account, cost saving due to release in system capa-cities, capacitor cost and voltage rise during off-peak hours, as a function of capacitive current flowsin thp feeder sections have been formulated in thispaper based on the models developed in t2,3,6,9p,10Oand an attempt has been made in this work to suggestsolution procedures for obtaining optimal number, loc-ation and size of fixed as well as switcied capacitorsalong the radial distribution feeders using the methodof local variations L7,83 . The technique, presen-ted does not depend on the circuit voltage, load distr-ibution and other system and load ch,-racteristics andthus, is applicable to all types of radial feeders inpractice.

FORMIULATION OF MATHEM~ATICAL MOJDELS

Cost saving due to energy loss reduction in the feeders

With fixed capacitors S The annual energy loss reduc-tion, that can be achieved by fixed shunt capacitors ina three-phase line can be obtained [2| as

n 2S= 8.76 E 3 Ri( 2 LU 'c 1 ) (1 )

The annual cost saving due to energy loss reductionusing (1) can be written as

designed and constructed in a system on a long termbasis can accept additional loads to the extent it

can.,satisfying the associated voltage regulation andthermal constraints. Once' the load increases beyondthe feeder capacity, limited by either voltage regulat-ion or thermal constraints, new system facilities arecreated and the system configuration gets changed.It isfurther assumed that the feeder load grows at a pre-determined annual rate, g, in proportion to the loadsconnected at the tapping points, implying that the fes-der load profile, on per unit basis, remains unchangedwith load growth. This assumption is made for went ofsufficient information on the exact pattprn of loadgrowth along the feeder main. However, the effect ofthis assumption on optimal capacitor allocation is ne-gligible, as experienced by the authors. As in £6] theeffect of load growth on the cost saving can be intro-duced by multiplying (3) by a factor (1+g)2k, k = 1,..,M, where Ni is the plan p-riod up to which the fepdercan take load growth, subject to a maximum period limi-ted by the capacitor life.

rffect of growth in load factor s As in t6] the sys-tem experiences a continuous growth in load factor withtime due to various reasons such as increase in loaddivprsity, increase in the energy consumption per kWconnected load, measures taken by the utilities to curbtie irotith in peak demand etc. The new load factor atany year, k, can be computed from (4) as in tio3.if =LF-% LF -LF) (4)k u - Yk u p

where Y (0.)k/16 (5)This variable load factor, LF , k = 1,.., n, can be in-troduced in (3) for considerig the effect of growth inload factor on the cost saving due to loss reduction.

Effect octfn y The? cost of equipment, mate-rials, construction, operation and maintenance in a sys-tpen increase with time. This results in a continuousincrease in the cost of enprgy in the system. The effe-ct of increase in the cost of enorgy on cost saving dueto loss reduction can be considered in the model by de-fining the term 'c' in (3) as a time dependent variable,ck, k - 1, ..., N.

Thus, considering the effect of the growth factors,discussed above, the present worth of the cost saving dueto ennrgy loss reduction during the life of the capaci-tors can be written as el

n

SN i=tSNi

(2)C = S.c8

where c is the cost of energy in Rupees per kWh.This cost saving is a recurrinq income annually to theutilities throughout the life of the capacitors. Thesum of the present worth of the income through costsaving in the life period of N years of the capacitorsat a discount rate of r, can be obtained £6] as

C =26.28 c Rf(2LF Ic 'I cI2 k= (1 (3)so ~i=1 i i i i k=1 (l*r)

As can be seen from (3), the cost saving is a functionof load current, load factor and cost of energy. There-fore (3) should be suitably modified to account for theeffect of load growth, growth in lond factor and in-crease in the cost of energy.

Ef fect of lo Load growth is a natural pheno-menon in a system. The growth in feeder load may be dueto the incremental additions to the existing loads or

due to the addition of new lo3ds to the feeder. A feeder

Uith switcd.,penergy loss reducdial feeders can

n ) l= 26.28, jj7R4li ck 12

1=1l Lk= l

c .1 +r

i( I ) ( LFk)

N F1 I gM( )211I

+ iIIck 21 (1+ ) ( 1 +rk=P1M+1 L Ii;

v.(6)ecitors : As derived in Appendix A, thetinn due to switched capacitors in ra-be obtained as

_Tn

< ; f2 )i1 1 1

(7)

where T is the total duration during which the capa-citor will rpmain in service over tne annual lond cycleand f is a ratio between the average inductive currentduring the period T and the maximum inductive current,recorded during the year. .pplying the effect of growthfactors, thp present worth of the cost saving due to

switclhed capacitors during its life period can be

1609

obteined from (6) by replacing the constent 26.28 andthe term LFk in (6) by 3T/1000 and f respectively, as

i=1 SN 1000 =1 k k 1(i )kfI I

(1-) ++ Ck 12II(1+g) f Iilc1r k=M'l~l 'i C 1

(-+r~)3(8)cost savin due to n s

f ormers

The energy loSs in the transformers consists oftwo components, one corresponding to the core lossesand the other corresponding to the copper losses. Whilethe core loss is constant during the load cycle,the co-pper loss is dependent on the loading level of the tra-nsformer and the load factor of the load cycle which isgiven by

ELcu 8.76 cu (LLF) (UF) (9)

where UF 8tk= (10)

The capacitor current flow in the system reduces theutilization factor of the transformer due to reducedloading which in turn reduces the copper loss componen-ts of the energy losses. If 0 is the reduction in thekVJA londing of the trensfocmer, the new utilization fa-ctor UF'' can be obtained as

UF' = tkVA (11)

Thus, the reduction in anerqy loss in tzhe transformersdue to fixed capacitors can be obtained using (9), (10)and (1i) as

2St8.76 c ( LLF) 2(UF) D - (-kV) (12)

The corresponding cost saving can be obtained by multi-plying (12) by the variable cost of energy, ck and thepresent worth factor as

cSt = St t ( )k (13)k=I (1+r)

However, this component of cost saving and the loss re-

duction in the transformer is very insignificant- Thisis due to the fact that the copper loss component ofthe energy losscs is much less compared to the coreloss component in the transformer, because of very lowloss load factors. Also the reduction in UF achievedby shunt capacitor compensation i3 quite small, the ma-

ximum reduction with 100 percent compensation of theinductive current being only 20 percent. As the optim-al compensation is normally of thu. order of LF timesthe inductive current, the actual reduction in UF willbe much lesse. -fence this componrant has practically noinfluence on the choice of cipacitor allocation and maybe ignorpd in thp optimization model.

Systm C t_lseThe capacitor current, flowinq in the system to-

wards the source, compensates the inductive current,demanded by the loads, thus reducing the net currentflow in thp system. This reduction in the current flowin the feeders, transformers, transmission lines andin the generators results in an alppreciable quantum ofrelease in their capacities which were otherwise uti-liz'ed because of the inductive current flow. Thisrelease in capacity, U, in any component with a loadedcapacity of P (in kVIA), and a capacitor current flowof IC through the component can be obtained as

(Appendix 8),

C k'I 2 k' I l 2)D=P 1- L1+( p c) - p c sin 11/23 (14)

The cost saving due to capacity release can now be wri-tten as

C = D c (15)r p

where c is the marginal cost required to supply eve-ry additional kVA demand through the component, underconsideration.In case of the distribution feeders to which the capac-itor banks are allocated optimally, the kk/A capacityand the capacitor current flow in different feeder sec-tions will be different. Hence the release in feedercapacities and the corresponding cost saving in thedistribution feeders can be obtained using (14) and (15)for each feeder section and the total cost saving in thedistribution feeders due to capacity release will be

n n L kh IC,2 2k'Irf:

i=1i=Pi i

(16)The cost saving due to the capacity release in the sub-station trinsf'rrmers, trannomission lines and in thegeneration capacities can be combined together, sincethe k/A loading due to the feeder loads and the capac-itor currant due to the capacitors allocated on thedistribution feeder will be the same in all these sys-tem components. Hence, the cost saving, due to theralease in the generation, transmission and substationcapacities can be obtained as 1/2

r k' I 2 2k'I /t) c

Cro c Pm 1I- i+( in (17)ro [ m m

where cp is the marginal cost reljuirerd to invest in

the substations, tr-nsmission lines and generation ca-pacities to meet every addlitional kVIA demand in thesystemt Pm is the maximum kVIA demand in the feeder and

Ic is the capacitor current due to all the capacitors

located on the feeder

Capctor cost

The capital cost oF the cap-icitor banks in a sin-gle install1tion bears a linear relationship to thecapacitor current as

C = (e iC + ad) (la)j=1 c

where i =0 and i = IC1 ~~cn cn+1 n

ic =tIc - IC. * j=2,..*, n (19)i J-i j

a =1 for ic > 0

-0 for i 0

The capacitors generally do not require any maint-enance. However, to account for the occasional inspec-tion charges, replacement costs of the fuses and theother control accessories and for the power and enargylosses wlithin the capacitors, a cost component in termsof annual expenditure can be defined as

N e1(ei + ad)

k=1 J=1 (1 + r)k 20)

where h is the annual expenditure represented as aper unit of the capital cost of the capa.citor bank.Combining (13) and (f20), the gneral cost model forthe capacitor bank can be written as

1610

n n

C = c Ze I -I )j+ ad'com i=1 I i=1 i i+1

whe re eP = e [1 +! r1

d' = d E + j h1

Voltage rise

As discussed in the introduction, it is essentialto consider a constraint in the optimization model on

the voltage rise during lean hours. An approximateformula for the voltag' rise at any receiving node can

be written as (Appendix C), - 1/2

_( -I sin g'

(CIc-Io sin 9') X-I I I (22)

where Io = I + Ic - I (23)[0~~~~~~

and I 1 I sinlQ'0

OBJECTIVE FUNCTIJN

(24)

The objective function, in this case, is defined

as

F = Cost saving due to feeder energy loss reduction+ Cost saving due to transformer energy loss

reduction+ Cost saving due to release in feeder capacity+ Cost saving due to release in other system capa-cities (transformers, transmission lines andgeneration)

- Cost of c-oacitor Danks (25)

The energy loss reduction in the transformers bp-

ing neqligible the second term in (25) can be dropped.

Though the cost saving due to release in feeder

capacity is quite considerable and influences the allo-

cation of capacitor banks alonq the feeder, this gainin cost saving is only theoratical. There is no

scope of utilizing the capacity released, under the

prevailing conditions in the rural distribution syste-ms. Actually, in the distribution systems, thei capec-ity,limited by the voltage regulation is reached muchearlier than the thermal capacity of the feeders. Inmost of the r'ural distribution systems in the develop-ing countries, the thermAl limit of the feeder capaci-ties is never reached. Also, the feaders when desig-ned and con3tructed on long time basis, adequate pro-vision is normally made in the feeder capacities for

meeting the expected load growth in the area.Thus, as

tne feeder capacity is already available for meetingthe load development in the area, the capacity relea-

sed by the capacitors will be of no avail and hence

will give no monetary benefit in practice. Therefore,if this theoretical unrealizable benef it is considered

in the model, it may lead to a result wherein the oth-

er benefits actually realizable may be off-set by this

theoretical gain and the capacitor allocated may beco-

me a liability to the system .conomy. Hence,the third

term in (25) can also be dropped.

Oce

The objective function for determining the optimalnumber, location and size of fixed capacitors, alongthe radial feeders can be expressed using (6),(17) and

(21) as a function of capacitor currsnt flows in the

fpeeder sections as

n nl(21) f E SNi ci r C c c

(26)The main constraint is that the voltage rise, G givenby (22) during off-peak hours at any node, should notexceed a given limit. If G is kapt at zero or less,then the voltage level at all the nodes will be lessthan or equal to the source voltage. This constraintcan be stated as

G (Ic ) 0, i = 1,..., n

ci(27)

TThe objactive function for determining the switch-ed capacitor banks in addition to the fixed capacitorsalnng the feeder main can be expressed in a similar wayusing (8), (17) and (21) as

nFs ~ cSSi(Ii) (I,+1 1)- r(I )

i=1 Ni IC'i) C0' 1 1 r0 1I

et (It _ I I ) + ad]3 (28)

where tie IP is the capacitor current due to switchedci

capacitors alone. The capacitor current due to f ixedcapacitors, Ic' which is known in advance has to be

adjusted with the inductive current flows in the feedersections and the n't inductive current flows alone have

to be substituted in (8). That is, the term

kIl (l+g) in (8) has to be replaced by a term

[I(ig)k IC where IC is the capacitor current

curresponding to the fixed capacitors already decided.

Since the switchetd capacitors will not be in ser-

vice during the off-peak hours, the voltage rise con-

straint need not be considered in this case.

SOLUTIJN APPROACHI

A close examination will, reveal that the present

problem of the choice of shunt capacitors along a radialfeeder is a simple process of determinino the capacitive

current flows in the different feeder sections to maxi-

mize the net cost saving, (26) or (28), by varying the

flow of the capacitive currents in descrete steps, deci-ded by the minimum size of the capacitor bank, availablesubject to voltage constraint. Thus, it is a descrete-

variational problem with a separable objective functionand constraint equation. In this paper the dynamic pro-

gramming (DP) approach, and the Mlethod of Local Varia-tions (MLu) were employed to solve this problem.

Dyna.imic programming approach was successfully appl-

ied by Ouran r3] for optimizing the choice of fixedcapacitors along the radial feeders, considering only

the loss saving aspect of the problem and with no con-

straint on voltage rise. The modified objective fun-

ction, considering the benefits due to both loss savin-gs and capacity release, subject to voltage rise con-

straint was solved first by using dynamic programming

approach for both fixed and switched capacitor applica-tions, in this endeavour. The Method of Local Variation

7,8I , was then employed to solve the above problem.M V has pertain well established advantages [7 ] over

the DP approach in terms of computational requirementsin addition to the special feature that at any stage inthe solution processt using MILV, there is always a feasi-

ble solution that can be considered as suboptimal.

and IC %1 -°

1611

Method of Local Variations

The problem for the fixed capacitor case can nowbe stated under the f'rame work of the MU/L, as follows:

max r(k) C (I)+ Cr (IcI,V..Ii i=1 SNi ( c r 1

c cI n

subject to

((c i)I ;ci -

n .I t-_j7 e I - I + ad'

i=51 ci ci+l

0, i = 1,..., n

I , i = 1, **..,n-1)i+1

a = 1 for I - I ))0 and IC

0 (32)c ci+1 n

- U for (I - ) 0 and I = 0ci cii Cni = 1,..., n (33)

The nodes are numbered from source to far end.It may be noted thet (29) will be replaced by an appro-priate cost function using (28) for the switched capa-citor case.Descr4tn the Solution A_grthm

The terminals of the line segments are identifiedas the different stages of the problem and at each sta-ge, the net capacitiv;e current flowing into the nodewhich is the current due to the capacitor connected tothe node, t.k.is considered as state variable. The

cumulative current flow in each feeder segment is thedecision variable. A change in the decision variableis achieved by varying the capacitor size connected tothe nodes in steps of the minimum capacitor size avai-lable.

To stert with an initial trajectory of capacitorunits connected to all the nodes is chosen by settingany valup for k within NC which results in a currentflow of Ici (i = 1,..., n), in all the feeder sectionsthat would satisfy the constraints (30) to (33). Thenstarting from the ith stage (i=1,..,n+l), change thenominal state trajectory by one unit on either side ofthe number of capacitors connected to the ith nodewhich would result in a change in Icj, (J-=,..,i-1) by

a value of Aice As a first step, add one capacitor

unit to the ith node by setting k at k+1, keeping allothpr state values unchanged. The resulting Icj isgiven by

k-1, thus removing one capacitor unit from the ith node.This chanqe in the state value results in

cj = I ic ' 3 = 1, *i., (i-i) (35)

If this proves successful, the state at the ith stageis changed from the nominal trajectory value to thisnew state. OthPrwiset the old value is retained andthe local variation operation is performed at the(i+1)th stage. In this way the local variations opera-tion is applied successively up to the (n+1)th stage tocomplete one iteration. The resulting trajectory atthe end of one iteration forms the nominal trajectoryfor the next iteration. This process is continued aslong as it ensures an increase in the value of the obj-ective function in each iteration. The iterativeprocess is terminated when a satisfactory convergenceto an optimum value has been achieved. If an improve-ment in the objective function is achieved through achange in the state value at a particular stage, thevariation process is immediately shifted to the nextstage retaining the state at this new value* If thevalu,e of the objpctive function remains unaltered in aparticular iteration, when the variation procedure hasbeen tried at all the stages, the convergence is saidto have been achieved.

RESULTS ANO CONCLUSION

The proposqd method has been tested on a number of11 kV radial distribution feeders in an Indian distri-bution system. The results of one such typical feederwith non-uniform loading, studied in (6) and of atheoretical feeder with a uniform loading of 1QQ kidtapped at an equal length of 1 km are presented herefor discussion. Fig.1 gives the non-uniformly loadedfeeder. In the presentation of the method, the nodeshave bean niJhbered from the source to the far and. How-ever, in the results tme nodes are numbered in thereverse order as shown in Figv1 as per common practice.14 13 12 11 10 9 8 7 6 5 4

1.0 12.3 10.8 0.8I3.0 11.0 j0.8O.8 j1i9 -.9 j 0.8

30 128 90 30 60 30 30 15 30 301 2 3

1.8 2.4Legend Loads in kW IF

Lengths in km 20 61 20Conductors : Between nodes I and 8 squirrel

8 and 10 weasel10 and 14 rabDit

Fig.1 A typical radial distribution feeder

Table 1

Data(34)Tc = Ic + b,ic p t-1,-***, (i-i)c c

Tests are then performed to see whether

(i) this change in the state value Satisfios the con-straints (30) to (33) in the case of fixed capa-citors and (31) to (33) in the case of switchedcapacitors and

(ii) the objective function value F(k+1) with the

changed state is more than the value F'(k) withthe old state at the ith stage. In computing the

change in the objective functions sectionsj 1,..., (i-1) (in wnich the capacitive currentflow changes) only need be considered.

If the answer is in the affirmative for the above tests,the state value at the ith stage is set at the new

value. If the answer is in the negative, k is set at

1. I linimum capacity of a capacitor unit2. Variable cost of the capacitor bank3. rixed cost per location of capacitor

installation4. Annual ch-rrgs on capacitors

(p.u. on capital)5. Cost of substation capacity6. Cost of energy7. Rate of growth of cost of energy8. Discount rate9. Power factor10. Present load factor11. Ultimate load factor12. Rate of load growth13. Load growth period14. Off-peak to peak load ratio

: 50 kVAr: Rs.100/kV/Ar

: Rs. 500

: 0.053 Rs.60.38/kVIA:s. 0 .3/k Wh: 1%: 1%: 0.7: 0.2: 0.45: lo%s 7 years

: 0.2

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The results outained through the proposed method forthe two cases with the above data are given in Tabl e2

Table 2Ontimal results by the proposed method

Allocation of fixed Voltage magnitudecapacitor banks Cost during off-peak

Cases -Saving loadLocetion Capacity (Rs.) Maximum Node(Node ) (WVAr) number

Feeder 2 50in Fig.1 S 50 46 023 11.0 14

9 50

Uniformly 3 100loaded 73 089 11.0 14feeder 8 100

To highlight the effects of load growth, growth in loadfactor, the increase in cost of energy and change insystem capacity cost, they are varied from a ininimumvalue to a maximum value, independently, and the resultsare tabulated in Tables 3 to 6.

Table 3

Results showing the effect of load growth

Optimal fixed capacitor banksalilocatedRate of alct

load ~~~~~~~~~~~Costgrowth , Capacity Total savingwith loca- toaTotal (Rs.)

tion in number cityp aranthesis

0 50(3) 1 50 10 468

S 50(2),50(7) 2 100 23 253

10 50(2),50(5),50(9) 3 150 46 023

15 50(2),50(4),QO0(8)3 200 86 285

20 100(2), 50(5),100(8),I00(11) 4 350 156 890

Table 4

Results showing the effect of change in present loadfactor with and without growth in future

Optimal fixed capacitor banksPresent allocated

L sie9 ot savingNumber Location (kVA)

With no growth in LF

0.10 1 5 50 1 2930.15 1 3 50 8 6880.20 2 2,8 100 19 5310.25 3 2,6,9 150 33 9600.30 3 2,5,8 150 53 8090.35 3 2,4,9 200 77 6230.40 5 2,4,6,9,10 250 105 130

With growth in LF as in C930.10 2 2,7 100 23 0750.15 3 2,6,9 150 32 7130.20 3 2,5,9 150 46 0230.25 4 2,A,8,10 200 59 8680.30 3 2,4 ,9 200 76 8020.35 4 2,4,8,11 250 94 8360.40 4 2,3,6,9 250 115 050

Table 9Results showing the effect of rate of increase in cost

of energy

Optimal fixed capacitor banksRate of allocated Costgrowth % N >r X ti * b - Saving

growth ~ Number Location Size S(Node) (kIAr)

0 3 2,5,9 150 38 4981 3 2,05,9 150 46 0232 3 2,5,9 150 54 3803 3 2,5,9 150 64 0384 4 2,5,8,11 200 76 1055 4 2,5,8,10 200 90 312

Tablpe 6

Results snowing the effect of change in system capacitycost

Optimal fixed capacitor banksSystem allocated Costcapacitycost Capacity with location Total Total ng

in paranthesis Number (kVAr)

0 50(2), 5o(5), 50(9) 3 150 40 02050 50(2), 50(5), 50(9) 3 150 44 991

100 50(2), 50(5), So(9) 3 150 49 962500 50(2), 50(5), 50(9),

100(u1), 150(13) 5 400 115 2801000 50(2), 50(5), 50(9),

100(11), 200(13) 5 450 227 840

1500 50(2), 50(5), 50(9),100(11), 250(13) 5 500 347 110

2000 50(2), 50(5), 50(9),100(11), 250(13) 5 500 466 800

As can be seen from the results tabulated in Tables3 to 6, the effects of the growth factors on the optimalchoice of number, location, and Size of fixed capacitorsis well establishpd. The effect of load growth is morepronounced than the obther factors. As per the results,presented, the effect of increase in the cost of energyduring the life of the capacitor on the optimal alloc-tion of the same is no.t pronounced; it may be due tothe fact that the increase in cost of energy is verylow.

From Table 6 it can be seen that the increase inthe cOst of system capacity results in more allocationof capacitor banks towards the source, not influencingmuch on the size or location of the capacitors allocatedat the nodes away f rom the source, since the release infeeder capacity is of no practical value as discussedearlier.

It has been further observed from the study that ifthe benefit of the feeder caoscity release is not accou-nted for in the cost savinq function, the switched capa-citors,due to their high cost, become uneconomical.Hance, for the feeders studiRd, ignoring this part ofthe cost saving function,no allocation of switchedcapacitors has been made. Thus, switched capacitors foron-line application in the low load density rural dist-rioution areas will be uneconomical and hence should bediscouraged. In the rural distribution systems switchedcapacitors can be installed at the substations. However,switched capacitors may be economical for high load den-sity areas wnere the release in feeder capacity could beusefully utilized for meeting further load growth in thearea.

The optimal results of the feeder in Fig. 1 thro-ugh the proposed algorithm,have been obtained for onecase in about 7 seconds of cpu time on ICL 2960 computersystem. The same problem when solIved by DP approach, ittook about 9 seconds of cpu time on the same computersystem. The memory core required for IILI is only about13 k.bytas while that required by the OP problem isabout 34 k.bytes.

The proposed method is a simple generalized appro-ach and is applicable to any type of feeder with anytype of loading pattern, with same or different loadpower factors. The proposed model is also very usefulto carryout sensitivity analysis of the dependent costcomponents on capacitor compensation. The algorithm,suggested considers the system growth factors, maximi-zing the life time benefit, thus giving the overal'loptimal solution to the problem. The method has a spe-cial advantage that the optimal capacitor allocation islimited by the lean pariod voltage rise constraint andthus avoiding over-voltage probloms during the off-peakhours. Thp solution technique MLV has been proved tobe a better method than the OP approach from the pointof view of computational requirements. The proposedmethod is simple, easy to program, more efficient andin authors' opinion is quite promising.

NOMENCLATURE

Il maximum inductive phase current in the ithfeeder segment

Ri per-phase resistance in ohms of the ith feedersegment

Xi per-phase reactance in ohms of the ith feedersegment

Ic capecitive current flow in the ith feederi segment

LF load factorLFD present load factor

LFu

LFknrNck

cu

LLFUFptkVAk I

kVIc

Pic

Piic '

d e d I

Io09

ultimate- load f actor

load eactor in the kth year

number of feeder seqmenntsannual discount rate in p.u.life of the shunt capacitorscost of energy in the kth year

copper loss of the transformer in kW

loss load factorutilization factor of the transformer capacitypeak load on the transformer in kVAtransformer capacity in kVAa constant given by k' = / (kV)circuit voltage in kilo voltscapacitive currant flow

peak load flow in kVA in the ith feeder segment

cost of feeder capacity at the ith feeder seg-mentcapacitor current of the capacitor banksconnected to node ivariable cost components of the capacitor costmodelfixed cost component of the capacitor costmodeloff-peak load current in the feeder

annual rate of load growth in p.u.

NC maximum number of capacitor units connected toa node

m plan period of the feeder up to which it can

take load growth

T duration of switched capacitor3 in service

over the load cycle.

1613

RE FRFNCES1. J.1I. Schmill, "Optimum size and location of shunt

capacitors on distribution feeders," IEFY Transac-to oP oweApr-tusandSs! ms, 7& tAb-84,pp. 825-832, September 1965.

2. R.F. Cook, 'Calculating loss reduction afforded byshunt capacitor application," IEEF Transactions onPower Apparatus and Systems, vol. PAS-83, pp.1227-1230, December, 1964.

3. Hernando Duran, "$Optimum number, location, and sizeof shunt capacitors in rural distribution feeders-A dynamic programming approach"t IEEE TransactionsonPwerpatus and Systems, vol.PAS-87,pp.1769-1774, September 1968.

4. S.H. Lee and J.J . Grainger, "Optimum placement offixed and switched capacitors on primary distribu-tion feeders," IEE T c o

an Sstem, vol. PAS-1130, pp.345-352, January,1981.

5. J.J. Crainger and S.H. Lee, "Optimum size and Loca-tion of shunt capacitors for reductian of losses ondistribution feeders," IEFE rransactions on PowerALusan stems, vol.PAS-100, pp.1105-1118,March 19 81 .

6. M'. Ponnavaikko and K.S. Prakasa Rao, "An approachto optimal distribution system planning throughconductor gradation," Presented at the IEF PFS1991 Transmission and Distribution conference andExposition, Minneapolis, Minnesota, paper No.TD 662-6, September 20-25, 1931.

7. K.S. Prakasa Rao, S.S. Prabhu and R.P. Aggarwal,"Optimal scheduling in hydro-thermal powver systemsby the method of local variations, t presented atthe IEFE PrS winter meeting, New York, N.Y.January 27-February 1, 1974.

8. F.L. Chernous Ko, "A local variation method fornumerical solution of variatiinal problem,"USSR Corn.putatinal Mathmmatics and MathematicalP c, vol. 5, No.4, pp. 234-242, 1965.

9g. C.G3 Scheer, "Future Power Prediction", ergyInternational, pp. 14-16, February 1966.

10. M. Ponnavaikko and K.S. Prakasa Rao, "Optimaldistribution system planning," IEEE Transactions onPower Apparatus atd tems, vol. PAS-100,pp.2969-29 77, June 1981.

11. Y.G. Bae, "Analytical method of capacitor allocationon distribution primary feeders," IEEE Transactionsot vol. PAS-97,pp. 1232-1238, July/August 1978.

APPENDIX A

Consider the -load cycle shown in Fig. a

kW

II

iIIi

ti '1 :

W* T -4 s_T _'

1 -* Time 2

Fig.a Load cycle of a feeder section

Let T1 and T be the Periods during which the capa-citor is switcAed on in service.The enrgy loss during the two periods Ti and T2 due tothe inductive current floWS in tne feader segments with-out capacitors can be obtained as

1614

EL0= [1I 12 t2+ 13 t3 I4 t4 +I5

6 6 (

The energy loss with capacitors can be obtained as

ELC = R [(I1IC)2t1 + ( )-I2)2t + ( I)2t

+ (I4-I)2t4 +( I5_I) 5+ (iI )2t] (A

APPENDIX C

a.1) Consider the phasor diagram shwon in Fig. c which showsthe voltage and current vectors of a feeder sectionsbetween the nodes S and R.

A.2)

The reduction in energy losses due to capacitors can beobtained from (A.1) and (A.2) as

S = R 2 Ic IIt1 +12t2 + I3t3 + I4t4 + 555 6 6

i2 Cr r)'(A.3)-Ic (T1 + T2) A3

Defining f as the averaging factor for the combined pe-riods T1 and T2 considerpd, similpr to the load factor

for the entire load cycle, and Im as the maximum induc-tive current during the periods T1 and T2 (A.3) can bewritten as

IC

0

Fiq.c Phasor diagramI

S = RT (2 Im f I )

where T = T + T

(A.4)( A5) The approximate gain in the voltage manitude between

(A*S the nodes S and R from the sending end, S to the recei-ving end R can be written f rom Fig. c as

APPENDIX 8

Consider Fig. b which shows the phasor representationof the real and reactive power flows in a feeder seg-ment or in any other system component, say, transfor-mer, transmission line or generator.

Fig.b Power phasor diagram

kW

OkRiA1

G = AC - 8C

- I ' X sino( - I ' R cose'a o

Also,

ckVAr

( Ic - I0 sin 8')sinK=3 Ic0

coso( = (1 - sin2 o( )1/

From the Fig. b,

Releasein capacity, D = kVA - kURA

kVA - kW kVA cos 0I Cos 1 Cos 1

ckVJAr = kVJA sin 8 - k'JA1 sin 4X

cos 9 = (1 - sin ) 1/2

(-.1)

( B.2)

(8.3)

( 8.4)

Substitutinq for sin c4 and cos o( from (C.2) end (2".3)respectively, C can be written as I

I - I:,sin Q1]C = I X [ ,0

0L ~~I I

1/2

(C.4)- Ia1R [l ( c *

From Fig. c,

From (B.?), (B.3) and (3.4), we get _ 1/2

kVA1 = [ (kA)2 + (ckUAr) - 2 (kVA)(c!dAr)sin 11/ I (I os9 ) + (IC I0 sin QJ)

(8.5) On simplification (C.5) becomesSubstituting for kVA1 from (B.5) in (8.1), 0 can beobtained as

~~~~2

° = k [1 + ( kVUAr 2(-2kAr) sin a]Q (8.6)Replacing the term, ckVlAr, in terms of tne capacitivecurrent c1, (B.6) can be written as

2 2 k' I 1/2D =k~ kVAkI+/A c

sin

where I1

2 2 1/2

Io = (O0 + Ic 2 c I )0

L I0 C '

n

v C *6)

(C.7)

Thus, (C.4) and (C.6) define the voltage rise at the

receiving and R from the sending end S due to thecapacitive current flow Ic.

(C.1)

(C.2)

(C.3)

(C.5)

1615

M. Ponnavaikko (S 4472254) wasborn in 5engamedu, S5outh Arcot,

Tamilnadu, India, on March 7,1946.

He received his S.E.(Elec.) and

X M.Sc.(Engg.) in Power System Eng-ineering from the College of Engi-neering, Guindy, Madras, IndiA in1969 end 1972 respectively. He ispresently working for his Ph.D.degree at the Indian Institute ofTechnology, Delhi.

He joined the Planning Direct-orats of Tamilnadu rlectricity Board in 1972 and was on

deputation at the Southern Regional Electricity Board,Bangalore, from 1974-1976 on special duty for conducti-ng power system planning studies. During 1976 and 1977he was with the Bharat Heavy Electricals Ltd. He is nowat the Rural Electrification Corporetion Ltd. as SpecialOfficer responsible for System Improvement projects andtraining of engineering personnel on computer ad aideddistribution system planning. He has been consultantto many electrical utilities in India. His fields ofinterest are Transmission and Distribution System Plan-ning, Unit Commitment for Load Despatch and GeneratorMiaintenaince Scheduling.

K.S. Prakasa Rao (5'70-M'74-SM'81) was born in Prakkilanke,Andhra Pradesh, India,in 1942.He received his B.E.(Elec.)andM.E.(Power Systems Engg.)degreesfrom Osmania University, Hyde-rabad in 1964 and 1966 respect-ively. In 1974, hp obtainedhis Ph.D. degree from theIndian Institute of TechnologyKanpur, India.

From 1966 to 1970, he wason the faculty of the Electrical Engineering Departmentof the Regional Engineering College, Warangal. From1974 to 1975, he was with the University of Roorkee,Roorkee, and from 1575 to 1978 he was a Professor inPower Systems Engineering at the National Institute ofEngineering, Mysore. Since 1978, he has been with theDepartment of Electrical Engineering, Indian Instituteof Technology, Delhi, India.

His fields of interest are Load Flow Studies,SteadyState Security Analysis, Planning of Distribution Syste-ms and Optimal Operation of Interconnected Power Syst-ems.