OPTIMAL SIZING OF CAPACITORS PLACED ON A RADIAL DISTRIBUTION SYSTEM

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IEEE Transactions on Power Delivery, Vol. 4, No. 1, January 1989 735

OPTIMAL SIZING OF CAPACITORS PLACED O N A RA DIAL DISTRIBUTION SYSTEM

Mesut E. Baran Felix F. WuDepartment of E lectrical Engineering and Computer SciencesUniversity of Califomia, Berkeley

Berkeley, CA 94720

Abstract - The capacitor sizing problem is a special case of the generalcapacito r placement problem. The problem is to determine the optimalsize of capacitors placed on the nodes of a radial distribution system sothat the real power losses will be minimized for a given load profile.This problem is formulated as a nonliiear programming problem. Theac power flow model of the system, constraints on the node voltagemagnitudes, and the cost of capacitors are explicitly incorporated in theformu lation A solution algorithm, based on a Phase I - Phase II feasi-,ble directions approach, is proposed. Also presented are a new formula-tion of the power flow equations in a radial distribution network and anumerically robust, computationally efficient solution scheme. This solu-tion scheme is used as a subroutine in the optimization algorithm. Testresults for both the capacitor sizing problem and the power flow solutionschemes are presented.

1. INTRODUCTIONCapacitors are widely used in distribution systems for reactivepower com pensation to achieve pow er and energy loss reduction, voltageregulation, and system capacity release. The extent of these benefits willdepend upon how the capacitors are placed on the system, i.e., (ii) thenumber and location, (i) the type (fixed or switched), (iii) the size , and(iv) the control scheme of the capacitors.Therefore, the capacitor placement problem in distribution feedersconsists of determining the place ( number and location), type, size, andcontrol scheme of capacitors to be installed such that the benefits men-tioned above be weighted against the fixed and running costs of thecapacitors and their accessories.The early analytical methods for capacitor placement are mainlydeveloped by Neagle and Samson [l] . The problem is defined as deter-mining the location and size of a given number of fixed type capacitorsto minimize the power loss for a given load level. The results, the sim-plest of which is the celebrated 2/3 rule, were based on a simplified, vol-tage independent system model obtained by: (i) considering only themain feeder with uniform load distribution and wire size, (ii) takig intoaccount only the loss reduction due to the change in the reactive com-ponent of the branch cumnts, (iii) ignoring the changes in the node vol-tages, and (iv) neglecting the cost of capacitors. Later, a morecomprehensive derivation of these rules was given by Bae, [ 5 ] . Fawziet. al. [ l l] incorporated the released substation W A and the voltageraise at light load conditions into the m odel.These analytical methods are also integrated with some heuristics

to obtain computer oriented solution methodologies and thus to over-come some of their shortcomings. Cook [2] extended the formulation ofthe problem to include peak power and energy loss reduction and pro-

88 W M 065-5 .4 paper recommended and approvedby the I E E E Transmiss ion and Dis t r i bu t ion Commit teeof the IEEE Power Engineering Society for p r e s e n t a t -i o n a t t he IEEEIPES 1988 Winter Meeting, Ne w York,Ne w York, January 31 - February 5 , 1988. Manuscriptsubmitted September 1, 1987; made ava i lab le orprinting November 13, 1987.

posed such a solution method to determine the number as well as loca-tion and the size of capacitors. Chang [4 ] considered voltage limits inhis solution scheme.

Duran [3], by exploiting the discrete nature of the capacitor sizes,used a more realistic model Tor the reeder with many sections ofdifferent wire sizes and concentrated loads and proposed a dynamic pro-gramming solution method. Ponnavaiko and Prakaso [12] consideredload growth as well as system capacity release and voltagc raise at lightload conditions and used a local optimization technique called themethod of local variables by treating capacitor sizes as discrete variables.Recently, interests in Distribution Automation and Control (DAC)have generated quite a lot of new studies in capacitor placement prob-lem. Grainger and Lee pioneered in refoimulating the problem as a non-h e a r programming problem and in developing computationally efficientsolution methods. In [6], they generalized Cooks formulation for aradial feeder, regarded the capacitor sizes as continuous variables, anddeveloped a computationally simple, iterative solution scheme. Theylater introduced switched type. capacitors with simultaneous switching [7]and a voltage dependent model for loss reduction [8]. In [9], togetherwith El-Kib, they proposed. a solution method to determine the optimal

design and real-time control scheme for switched capacitors with non-simultaneous switching under certain assumptions. Grainger and Civan-lar considered the optimal design and control scheme for continuouscapacitive compensation case [lo] and introduced a voltage dependentsolution scheme [14]. El-Kib et. al., [16] extended the methodologiesdeveloped in [6]-[9] to encompass unbalanced three-phase feeders.Kaplan [13] extended the formulation for multi lateral feeders and pro-posed the use of heuristics for the solution. Graing er and Civanla r [151combined capacitor placement and voltage regulator problems for a gen-eral distribution system and proposed a decoupled solution methodology.

The basic idea in our formulation of the general problem is similarto that of Grainger et. al., i.e., formulating it as a nonlinear programmingproblem. We will consider a capacitor placement problem that (i) incor-porates directly the ac power flows as system model, (i) enforces thevoltage constraints. We develop a solution methodology by usingdecomposition techniques. At the lowest level of the decomposition, wehave special capacitor placement problcms, callcd the buse problem s. Abase problem is a sizing type problem which is used to find the optimumsizes of a set of selected capacitors placed on the nodes of the distribu-tion system and thcrefore, it is of practical intercst in its own right. Thesolution algorithm for the sizing problem is a special Phase I - Phase I1Feasible Directions method which takes into account the structure of theproblem. We present the formulation of the sizing problem and thesolution algorithm developed for it in this paper. The solution of thegeneral problcm is presented in another papcr [17].Also in this paper, a new formulation of the ac power flow equa-tions for the radial distribution systems is introduced and a computation-ally efficient and numerically robust solution methodology, callcd Dist-

Flow, is presented. Such a solution methodology is needed because ofits repeated use in the optimization algorithm.In section 2, the new power flow equations for radial distributionsystem s are presented. The sizing problem is stated in section 3. Sec-tion 4 describes the general approach for the solution. The structure ofthe problem is studied in section 5 and the proposed solution algorithmis prcsented in section 6. Section 7 contains the tcst runs and conclu-sions are given in section 8.

0885-8977/89/0190-0735$01 WO1989 IEEE


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136II. DISTRIBUTION SYSTEM POWER FLOW

Power flows in a distribution system obey physical laws (Kirchofflaws and Ohm law), which become part of the constraints in the capaci-tor placement problem. In our proposed solution algorithm for the capa-citor placement problem, the distribution system power Row solution isto be used as a subroutine in every iteration. Therefore, it is essential tohave a computationally efficient and numerically robust method for so lv-ing the distribution system power flow.In this section , we present the new power flow equations for radial

distribution systems. The formulation is conducive to efficient solutionmethods. For pedagogic convenience, we first consider a special casewhere there is only one main feeder. The general case for any radialdistribution system is considered next. To simplify the presentation, thesystem is assumed to be balanced 3-phase system.Special Case :Radial Main Feeder

Consider a distribution system consists of a radial main feederonly. The one-line diagram of such a feeder comprising n branches /nodes is shown in Fig.1.V l vi

Fig.1 :One line diagram of a main distribution feederIn the figure, V o represents the substation bus voltage magnitude and isassumed to be constant. Lines are represented by a series impedancez, =r, + j x l and loads are treated as constant power sinks,S, =PL + Q L. Shunt capacitors to be placed at the nodes of the systemwill be represented as reactive power injections.

With this representation, the network becomes a ladder networkwith nonlinear shunt loads. If the power supplied from the substation,So =P o + Q o is known, then the power and the voltage at the receivingend of the first branch can be calculated as follows.s, =so- , - ', =so- , lS,I*/V,2 - s 1V , B , =v, - 11, =VO- , s ; / v ,

Repeating the same process yields the following recursive formula foreach branch on the feeder.pi+,=pi - ri+,(p:+e:)/v: - pf i+ , (1.i)Q i + l =Qi - Xi+l(P?+Q?YV? - Q ~ i + t+Qci+l (1 ii)Vi:, =V: - 2(ri+,Pi+xi+,Qi)+( r z , + x ~ , ) ( P : + Q ~ ) / V ~l.iii)

Where,P i , Qi : real and reactive power flows into the sending end ofbranch i+ l connecting node i and node i+ l,Vi :bus voltage magnitude at node i ,Q , : reactive power injection from capacitor at node i.

Eq.( ). called the branch flow equation, has the following formXoi+l =foi+l(Xoi*ui+l) (2 )

Where, xgi =[Pi i V:lT andNote that if there is no capacitor at node i, then ui does not appear inEq.(2). By abusing notation , we will simply use U as an nc dimensionalvector c ontainin g the nc Capacitors to be sized i.e.,

=Qci+l .

Note also that we have the following terminal conditions:(i) at the substation; let the given substation voltage be V sp , hen% =v; =v sP= (3.i)

(ii) at the end of the main feeder;X k , =P , =0 . x k l = Q , = O (3.ii)

The 3n branch flow equations of (2) together with the 3 boundaryconditions of (3) constitute the system equations and will be referred to

as DirtFlow Equa tions. They are of the formG(xo,u) =0 (4)

Where, 4 [ x& . . . IT is the branch variables and U is the capacitorsizes. For a given load profile (i.e., set of demands P f i , Q f i :i =1 , . . . n ) and capacitor sizes (connof variables). U , 3(n+I) Dist-Flow equations can be used to determine the operating point, x,, of thesystem. Such a solution algorithm is developed in the appendix.General Case

The DistFlow equations can be generalized to include laterals.Consider a lateral branching out from the main feeder as shown in Fig.2.For notational simplicity, the lateral branching out from node k will bereferred to as the lateral k and the node k will be referred to as thebranching node.V"

Y----?Figure 2 :One line diagram of a section of a feeder

For lateral k with nk branches, the same process of formulationapplied to the main feeder can be repeated for the lateral by using theline flow equations of (1) and the new terminal conditions,V k o = V , , P,,, = O , Q,,, =0 . Where, we have used the dummyvariable V ~ Oor notational simplicity. As a result, we have the follow-ing 3(nk +l) equations.

i =0, . . . , k-1L+, =fb+l(xkiUL+,)

Hence. in general, for a distribution network of n branches and 1laterals, there are 3(n+1+1) DistFlow equations (ac power flow equa-tions) comprising of Eq.(5) for each lateral k =0, . . . , , including themain fee der as the O'th lateral. Thcy are of the form( 6 )(x,u) =0

DistFlow equations can be used to determine the operating point. xof the system if the capacitor sizes, U are given. We prefer to use Dist-Flow equations over conventional ac power flow equations because thespecial structure of the DistFlow equations can be utilized to develop acomputationally efficient and numerically robust solution algorithm. Thedetails of the derivation of the solution algorithm are presented in theappendix. Other distribution systcm load flow methods have been pro-posed, see [18] and the references cited therein.III. SIZING PROBLEM IN CAPACITOR PLACEMENTThe capacitor placement problem is to determine the location, type,and size of the capacitors to be placed on a distribution system. The

objectives are to reduce the losses on the system and to maintain thedesined voltage profile while keeping the cost of the capacitor addition ata minimum. In this general form, the problem includes the followingfactors: (i) the locati on of the capacitors, (ii) the type of capacitors -fixedor sw itched, (iii) load variations, and (iv) the cost of capacitors. A solu-tion scheme for this general problem is proposed in another paper [17].The proposed approach calls for the solution of a Sizing Problem as asubroutine.The sizing problem is a special case of the capacitor placementproblem; it assumes that

the capacitors are placed (thcir number and places aregiven),there is only one load profile,


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131the cost of a capacitor is a differentiable function of its size.

Therefo re, the sizing problem determines the optimal sizes of the capaci-tors once they are placed. This special case was considere d by Lee andGrainger in [8]. However, in this proposed formulation, the voltageprojile of the system (voltages at all nodes) is required to lie within theacceptable limits.To formulate the sizing problem as a nonlinear programming prob-lcm, consider a radial distribution system with n branches, J laterals, andnc capacitors placed at the nodes of the system and a load profile,

Pli ,Q, , i =1, . . . ,n . The objective comprises two terms; the sav-ings due to real power loss reduction in the system as a result of capaci-tor placement, and the cost of capacitors. The real power losses in thenetwork can be calculated as the sum of the i2 r loss on each branch,i.e.,(7)

The other term in the objective function, namely the capacitor cost func-tion, is to be assumed linear with a constant marginal cost r,$/kvur forsimplicity. Then the objective can be written as

Where, kp is the weighting factor for the losses.There are three sets of constraints.

(i) The power flow equations :G (x ,u)=0(ii) Voltage limits :v Y 2I xk 3 =V: I vkw2(iii) Capacitor Limits :0


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738(9.i)

(9.ii)(9.iii)

Note that the real power flow is decoupled and can be dropped from cal-culations. This simplified DistFlow model is similar to the currentmodel adopted in literature [ 6 ] . The appealing properties of these equa-tions are that they are linear and can be solved directly. The generalform of the voltage equation is as follows.It

k = lV =(VjY +2C j k Qck (10)Where, el k is the total line reactance of the section of the network whichcorresponds to intersection of two p aths extending between the nodes 0-j

Therefore, the upper and the lower voltage constraints becomehyperplanes and the feasible region defined by these constraints will be apolytope. To illustrate such a feasible region, consider the small systemshown in Fig.4.

and 0-k.

VO V 1 vz v3 v4 v5I I+QO

Figure 4 :A small main feederSome of the voltage equations areV? =WP)' +b i Q c z +b i Q c 4V =(V,O)' +2 ( ~ 1 +Z)QCz+W 1 + xz +x3)Qc4V,Z =(V,O)' +2 ( ~ 1 + d Q c z+2 ( ~ 1 x2 +x3 +x4,QC4The corresponding feasibility region is shown in Fig.5.

Figure 5 :Feasible region defined by vo ltage constraintsNow consider the hyperplanes that corresponds to the lower boundconstraints. For the example, they are stacked on top of each other inalmost parallel fashion as depicte d in Fig.5. There fore, the intersectionof two hyperplanes of the lower bound constraints will not create sharpcomers. A sharp comer may be created by h e hyperplanes of con-straints V =V" and V? =Vmm2 as shown in the figure.The foregoing observation on the example leads us to assume thatin general:

the constraints of the same type (either the lower voltage limit con-straints or the upper voltage limits constraints) will not create sharpcomers in the feasible region,sharp comers may be created by the constraints of different types.calculations involved in a Phase I - Phase I1 type solution algorithm.This assumption will be used in the next section to simplify the

VI. SOLUTION ALGORITHMIn this section, we will prcscnt an efficient solution algorithm for theSizing Probl em. The algorithm involves modification of the Phase I -Phase I1 feasible directions method specifically for the radial distributionsystem cap acitor placement problem.From the analysis of Sec.V, it is reasonable to assume that: (i) theconstraints of the same type do not crcate sharp comers in the feasible

region, and (ii) sharp comers are created only when constraints ofdifferent types are binding. Therefore, we modify the feasible directionsby including only one or two gradients from the violated constraints. Ifonly lower voltage limits are violated, we include the gradient of themost violated constraint (corresponding to the lowest bus voltage) in thefeasible direction. If both lower voltage limit and upper voltage limitviolations are present, we include the gradient of the most violated lowervoltage constraint and the gradient of the most violated upper voltageconstraint in the feasible direction.To be more specific, the selection of the feasible direction h is as fol-lows.

1. Phase I (to get a feasible point)Case 1:There are only lower voltage limit violations. Then, we cpn-sider only the smallest voltage V, below the lower bound limit V"" ,i.e.,

'

V, S V , -F j and VI


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if phase =2 thenI

600

300

step 4 : pdateUi+l =U, +hi h

Step 5 : onvergence checkIf h, Ih l >Step size calculations in stcp 3 involvcs the solution of system equa-tions, G(xj+ l,u,+h, h) =0 every time Dk is updated to get thecorresponding state variables x,+I(ui+l).Therefore, computationally it i simportant to have a quick search. One way is to limit the search intcrval

by putting an upper limit, AQch on the maximum allowable change inconml variables, Aii hk I-. This will speed up the search in Phase 11since the step size will get smaller as solution approaches to the optimalpoint. Also an upper bound K has to be defined on the number ofsearches when implem enting the search.Another improvement usually made on the above algorithm, I201involves the m odification of feasible direction for Phase I by combiningphase I direction h, and the phase I1 direction Vf, as follows.

then go to step 1 else stop

--

,-

This way the solution will be pushed towards the optimal point in thefeasible region and hence the convergence will improve in phase 11The algorithm can start from any initial point, h. owever, a goodinitial point can easily be obtained by employing the approximatemethod of [6] with simplified DistFlow equations of (9).

W. EST RESULTSThe proposed algorithm has been implemented in Fortran-77 onboth VAX 11/750 and IBM PC-AT. In the implemented algorithm, theoptimization parameters are set as follows:

u = O A , p=O . 6 , ~ = 0 . 3 , , = 0 . 6 ,~ . = 0 . 0 4 , = 5 , AQch=3.0We will present three capacitor sizing test runs and sample power flowsolutions for two test systems.

The first test system, TSl is a 9-branch main feeder developed byGrainger et. al. [8]. We also follow Grainge r et. al. in settingkp =168 $lkW, r, =4.9 $lkvvar and placing three capacitors Q , b , a ~ ,QCl n the nodes 6, 5, 1 of the feedcr respectively. Th e second testsystem, TS2 is a 69 branch, 9-lateral feeder derived from a poltion ofthe PG&E distribution system. The network data of this system is givenin [17]. There are three capacitors placed on the system; one on themain feeder, Qcla nd two on a lateral. Qc47 , Qc52.

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-igure 7 : olution Trajectory for TS1 Test A

739A sum mary of the power flow solutions for the test systems beforethe capacitor placement is presented in Table 2. The solutions areobtained by the DistFlow solution scheme mentioned in the appendix.They converge in 4 and 3 iterations for TS1 and TS2 respectively.These results indicate that the convergence of DistFlow is fast. Consid-ering the fact that the feeders have branch cs with diverse r/x ratio (ratiovaries between 3 to 0.02). and TS1 has a low voltage and high lossprofile, the results also indicate the method's num erical robustness. Therun time figures shown in Table 2 are obtained by mnning the p r o g mon a VAX ll n5 0.Table 2 :Power Flow results for the test systems by DistFlow

Test results of the proposed method for capacitor sizing are presentedbelow.Test System 1 TS1)Test run A : n this test run, we (i) neglect the cost of capacitors. (ii)ignore the voltage constraints. Therefore, the solution is an uncon-strainted optimal point and gives the maximum loss reduction. As theinitial starting point, the approximate solution given in [ 8 ] is used. Theresulting solution trajectory is given in Fig.7. A summary of the solu-tion and the computation time are presented in Table 3. The followingobservations are made from Fig.7 and the comparison of Table 2 andTable 3.

The convergence is achieved in about 7 iterations (2 times thenumber of capacitors to be placed).The approximate solution is not close to the actual solution aspointed out in [8].At the unconstrainted optimal pint. the power loss reduction ismaximum (100 kW and 4890 kvar) and the voltage profile is better(the minimum voltage raises form .837 p.u. to .882 p.u. and all thenode voltages are still below the substation voltage).

Test run B : n this test run, cost of capacitors are considered and thelower and upper lim its on all bus voltages are set to 0.9 p.u. and 1.1 p.u.respectively. The resulting solution trajectory is shown in Fig.8. Thetrajectory starts with Phase I since the initial point is infeasible (lowervoltage limit violation). In the forth iteration a feasible point is obtainedand the program switches to Phase 11. The convergence is obtained inthe 12 'th iteration. A summary of the results and the computation timeare presented in Table 3. We have the following observations about thetest"

Two test N~IS ere conducted for TSl .

27------00

0 1 2 4 5 i + rFigure 8 :Solution Trajectory for TSl Test B Figure 9 :Solution Trajectory for TS2


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740

Test ATest B

The convergence in Phase I is fast; the number of iterations isroughly equal to the number of capacitors.The convergence in Phase I1 gets slower since the solution trajec-tory moves along the boundary of a lower limit voltage limit (thetotal number of iterations is roughly equal to 2 to 3 times thenumber of capacitors.The incremental changes of capacitors gets smaller as iterationincreases.The results show the feasibility of capacitor placement for bothloss reduction and voltage regulation (minimum voltage is raiscd to.901 p.u. while still achieving 64.32 kW loss reduction).Table 3 :Summary of test results for TSl.

I Test Ru n I Substation I Loss Reduction 1 V-:- I Ru n Time IP o ( k W ) Qo(kvar) APo(kW) AQo(kvar) V CPU I/O

13 050.87 329.62 100.90 4 892.85 .882 4.38 1.313 087.45 279.43 64.32 4 943.04 .901 6.08 1.9

Substation3 959.55 1778.38Po(kW) Q o W r )

Test System 2 (TS2)In this test run, c apacito r costs and the voltage constraints are con-sidered also. The initial point is obtained by the application of theapproximate method of [6] with the simplified DistFlow equationspresented in section 5. The corresponding solution trajectory is given inFig.9. Since the the system without any capacitors is feasible, the testrun uses Phase I1 only. A summ ary of the solution and the computationtime are presented in Table 4. The following observations are made.

The convergence is obtained in 4 iterations (less than 2 times thenumber of capacitors).The test result. =0 indicates that the capacitor on bus 15 isnot ebno mi cal to install.Considerable loss reduction is obtained through optimal capacitorplacemcnt (67.55 kW and 1018.39 kvar).

LossReduction V,,,,,,/Vo Run Time67.55 1018.39 .925 10.91 0.92

@ o ( ~ W ) A Q o ( ~ ~ ) CPU VO

Table 4 :Summ ary of test results for TS2.

MT. CONCLUSIONSIn this paper, a capacitor sizing problem for capacitors placed on aradial distribution system is formulated as a non-linear programmingproblem and a solution algorithm is developed. The problem finds theoptimal size of the capacitors so that the power losses will be minimizedfor a given load profile while considering the cost of Qe capacitors. Theformu lation also incorpo rates the ac power Row mo del for the systemand the voltage constraints.The solution algorithm developed for the capacitor sizing problemis based on a Phase I - Phase I1 feasible directions approach. This gen-eral method is made computationally more efficient by exploiting thespecial structure of the problem.Another contribution of the paper is the introduction of new powerBow equations and a solution method, called DistFlow , or radial distri-bution systems. The method is com putationally efficicnt and numericallyrobust, especially for distribution systems with l arge r/x ratio branches.The DistFlow is used repeatedly as a subroutine in the optimization algo-rithm for the capacitor sizing problem.The test results of the proposed algorithm for the capacitor sizingproblem are also presented. They indicate that the method is computa-tionally efficient and has good convergence characteristics. Theydemonstrate the feasibility of capacitor placement for both loss reductionand voltage regulation purposes.The cap acitor sizing problem considered in this paper is a specialcase of the general capacitor placement problem. The general problem

determines the place and the type of capacitors to be installed as well astheir sizes. The load variations as a function of time are also incor-porated in the general problem in the calculation of the the energylosses. The general problem is considered in another paper [17]. Thesolution algorithm developed in [17] uses the algorithm developed hereas a subroutine.

AcknowledgementsWe thank Mr. Wayne Hong and Dr. D ariush Shirmohammadi ofPacific Gas and Electric for providing the data for the test system andhelpful discussions. This research is supported by TUBITAK-TURKEYand by National Science Foundation under grant ECS-8715132.

REFERENCESN. M. Neagle and D. R. Samson, "LossReduction from CapacitorsInstalled on Primary Feeders", AIEE Transactions part I l l , vol. 75,R. F. Cook, "O ptimizing the Application of Shunt Capacitors forReactive Volt-Ampere Control and Loss Reduction" ,AIEE Trun-suctions part I l l , vol. 80, pp. 430-444, August 1961.H.Duran, "Optimum Num ber, Location, and Size of Shunt Capaci-tors in Radial Distribution Feeders: A Dynamic ProgrammingApproach, IEEE Trans. on Power Apparatus and Systems, vol.87, pp. 1769-1774, Sept. 1968.N. E. Chang, "Locating Shunt Capacitors on Primary Feeders forVoltage Control and Loss Reduction", IEEE Trans. on PowerApparatus and Systems, vol. 88 , pp. 1 574-1577. Oct. 1 969.Y.G. Bae, "Analytical Method of Capacitor Application on Distri-bution Primary Feeders", IEEE Trans. on Paver Apparatus andJ. J. Grainger, and S. H. Lee, "Optimum Size and Location ofShunt Capacitors for Reduction of Losses on Distribution Feeders",IEEE Trans. on Power Apparatus and Systems, vol. 100, pp.S. H. Lee and J. J. Grainger, "Optimum Placement of Fixed andSwitched Capacitors on Primary Distribution Feeders", IEEETrans. on Power Apparatus and Systems, vol. 10 0, pp. 345-352,January 1981.1. J. Grainger, and S.H. Lee, "Capacity Release by Shunt Capaci-tor Placcment on Distribution Fecdcrs: a Ncw Voltage DependentModel", IEEE Trans. on Power Apparatus and Systems, vol. 101,pp. 1236-1244, May 1982.

pp. 950-959, Oct. 1956.

SyStems, vol. 9 7, pp. 1232-1 237, JUly/AUg. 1978 .

1105-1118, March 1981.


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74 1the beginning of the lateral, xk o=[PbQkoV,'O]; since then the rest of thevariables, xy can be calculated by using the branch flow equations succes-sively. T o be more specific, let us consider two cases again, namely firstthe main feeder and then the general case.Special Case :Main Feeder Only

Sin ce the s ubstation voltag e, Vo is give n, the only Variables need to bedetermined arez,= P oQolT ,.e., the power supplied from the substation .Therefore,z, onstitutes the "state" of the system. To eliminate the othervariables, xgi from the Eq.(a.l), we use the terminal conditions at the endof the feed er, i.e.,

p n =Po,,xo,,-l) =0 ; Q, =40. (xh-1) =0 (a.2)and eliminate % by substituting the branch flow equations, Eq.(a.l.i)recursively fori = n - l , n - 2 , . . ,1. A s a result, we get two equations ofthe following form.

Po,, (zo0.u)=0 ; (foe (%.U) =0 (a.3)We will use these two equations to solveGeneral Case : eeder with Laterals

For sim plicity, it will be assumed that the system consists only of amain feeder and 1 primary laterals (laterals branching out from the main).We generalize the procedure of reducing the system equations as follows.For lateral k,we choose two new variables, Pk o and Qko the real and thereactive power flows into the ateral respectively - as the extra state vari-ables zb = PbQbIT .Then the same process of reduction applied to themain feeder can be repeated for the late ral by using the branch equationsof (a.1) and the associated terminal conditions. Ph =0 ,Qh =0 . Thiswill give two new equations of the following form.Ph 210, . 2 ,zo0,u)=0 (a.4.i)ff h 210. f .., b z,.u) =0 (a.4.ii)

Hence, in general, for a distribution network of n branches and 1laterals, reduced DistFlow equations comprise Q(a.4) for tach lateralk =1, . . ,I together with the equations for the main feeder, Eq(a.3)which can now be rewritten as,Pol, (z10. . . 210,%*U) =0 (a.5 .i)dol ,(210, . . lo .%J4 =0 (a.55)

for a given U .

J. J. Grainger, S.H. Lee. and A. A. El-Kib. "Design of a Real-TimeSwitching Control Scheme for Capacitive Compensation of D istribu-tion Feeders", IEEE Trans. on Power Apparatus and Systems. vol.101. pp. 2420-2428. August 1982.J. J. Grainger. S.Civanlar, and S.H. Lee, "Optimal Design and Con-trol Scheme for Continuous Capacitive Compensation of Distribu-tion Feeders", IEEE T rans. on Power Apparatus and System s, vol.102, pp. 3271-3278, October 1983.T. H. Fawzi, S. M. El-Sobki, and M. A. Abdel-Halim, "A NewApproach for the Application of Shunt Capacitors to the PrimaryDistribution feeders", IEEE Trans. on Power Apparatus and Sys-tems, vol. 102. pp.10-13, Jan. 1983.M. Ponnavaikko and K. S. hak as a Rao, "Optimal Choice of Fixedand Switched Shunt Capacitors on Radial Distribution Feeders bythe Method of Local Variations", IEEE Trans. on Power Apparatusandsys tems, vol. 102, pp.1607-1614, June 1983.M. Kaplan, "Optimization of Number, Location, Size, Control Type,and Control Setting of Shunt Capacitors on Radial DistributionFeeders", IEEE Tran s. on Powe r Apparatus and Systems, vol. 103,J. J. Grainger. S.Civanlar, and K. N. Clinard, L. J. Gale, "OptimalVoltage Dependent Continuous Time Control of Reactive Power onPrimary Distribution Feeders", IEEE Trans. on Power ApparatusandSystems. vol. 103, pp. 2714-2723, Sept. 1984.S . Civanlar and J. J. Grainger, "VolWar Control on DistributionSystems with Lateral Branches Using Shunt Capacitors and VoltageRegulators: Part 1 Part 11 Part III", IEEE Trans. on PowerApparafus andsysKems, vol. 104, pp. 3278-3297, Nov. 1985.A. A. El-Kib, J. J. Grainger, and K.N. Clinad, L. J. Gale, "Place-ment of Fixed and/or Non-Simultanmusly Switched Capacitors onUnbalanced Three-phase Feeders Involving Laterals", IEEE Trans.on Pa ver Apparatus and Systems, vol. 104, pp. 3298-3305. NOV.1985.M. E. Baran and F. F. W u, "Optimal Capacitor Placement on RadialDistribution Systems". submitted toIEEE PES winter meeting, 1988.D. Shirmo hamm adi. H. W. Hong. and A. Semlyen , G. X. Luo. "ACompansation Based Power Flow Method for Weakly Meshed Dis-tribution and Transmission Networks" , EEE PICA meeting, June1987. Montreal.F. F. Wu. G. Gross. nd J. F. Luini, P. M. Look. "A Two StageApproach to Solving Large Scale Optimal Power Flows", PICAProceedin gs. Cleveland, May 15-18 1979. pp. 126-136.E. Polak, R. Trahan, andD. Q. Mayne, "Combined Phase I - Phase I1Methods of Feasible Directions", Mathematical Programming,E. Polak, Computational Methodr in Optimization, Academic Press.New York, 971D. G.Luenberger, Introduction to Linear and Nonlinear Progrm-ming, Second Edition, Addison-Wesley Publishing Co., Mas-sachusetts, 1984.P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization.Academic Press, New York, 1981.

pp.2659-2665, Sept. 1984.

~01 .17 , 0.1, pp.61-73, 1979.

Appendix : olution of DistFlow EquationsThe DistFlow equations of a radial distribution system, as shown insection 2. comprises branch flow equations and the associated terminalconditions for each lateral including the main feeder which is treated as heO'th lateral. They are of the ollowing form.

(a.1.i)X , ~ = Q ~ , xh = x h , = O i =0,1,. &- I (al.ii)

We can use these equations to determine the operating point,xy = PYQYVz]of the system for a given, U. But rather then using thea b ve equations to solve for xY's directly, we are going to conceptuallyreduce the number of equations first and then propose an efficient solution

k =0,1 , . . IX y+ 1 =ki+l(XY .UY+l)

algorithm.ReductionofDistFlow Equations

Note that for a given lateral k. we need only to know the variables at

Solution ofDistFlow EquationsThe educed DistFlow equations are of the form

H(z,u)=0 (a.6)and can be used to solve for the "state variables" z =[zL . . . ZL41.Let's consider solving (a6) for z by N e,wton-Raphson Method (NR) or agiven U. From an estimated value of 9, n iteration step of NR nvolvesthreesteps:Step 1 : alculation of the mismatches H(d)Step 2 : onstruction of the system Jacobian matrix

(a.7)aH '(z') =-a2 lZ=dStep 3 : solution of the following system of equations to update the states2.

J(z')AzJ =-H(d) (a.8)We consider the special case - the main feeder only - first. Themismatches for t h i s case can be calculated as follows. At iteration j, thesubstation bus voltage Vo is given and the updated values of the substationpowers Po, o are available. Therefore, the recursive branch Row equa-

tions of (a.1.i) c an be emplo yed to update the variables~i =[PiQi .V?lT i =1, .. , successively along the feeder startingfrom the substation and proceeding towards the end of the feeder. Whenwe reach the end of the feeder, the updated variables P,, ,Q, , will be themismatches as indicated by (a.2). We'll r efer to t h i s procedure as afor-ward sweep.The elements of system Jacobian can be calculated by using the ChainRule. For the main feed er case, the Jacobian is a 2x2 matrix of the follow-ing form.


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742

(a.9)

By using the branch flow equations, (a.1.i) and applying the Chain Rule,J(z,-,,) can be constructed as follows.

Where, 1i-1 Qi-1 ( P A+QA1-2ri- -2ri- riI Vi?, v;2,Ji is the Jacobian of branch flow equation i and will be referred as thebrunch Jaco bian. Therefore, calculation of the 2x2 Jacobian in Eq.(a.9)can easily be conducted by the multiplication of 2x3 and 3x3 matrices in(a.lO).

The third step of the solution procedure involves the solution of a2x2 matrix equation.This calculation procedure can be generalized for general radial distri-bution feeders. Again, we consider a system having a main feeder and Iprimary latera ls only.The mismatches can be calculated by generalizing the forward sweepprocedure as follows. For the lateral k, given the estimated powerinjected into lateral, Pko.Qro.and the voltage at the branching node (subs-tation for the main), V k , he forward sweep method of the main feedercase can be applied to update the variables xk =[Pk.Q,,V,] alonglateral and get the mismatches corresponding to this lateral: P , and Q h .

To update all the variables and thus obtain all the mismatches, aboveprocedure is applied to all the laterals in an order such that the mainfeeder is updated before the laterals branching out from it.System Jacobian, J for the general case will be a Z(l+l)x(l+l) matrixof the following form.

Where,

J =

J 1 1J Z l J zz. . .... .

J I I JrzJO I Jo z

J I OJu ,

Ji r J hJo r JGQ

(a.12)

and can be calculated by Chain Rule similar to (a.lO).The third step involves the solution of 2(l+l)xZ(l+l) matrix equationof the form (a.8).Note that the mismatches and the Jacobian matrix in the proposedmethod involve only the evaluation of simple algebraic expressions andno trigonometric functiomds opposed to the standard load flow case.Thus computationally the proposed method is efficient. In the followingtow subsections, we show that the method has very good numerical pro-perties and it can be modified to make it computationally even moreefficient.

Numerical PropertiesThe DistFlow method described above is numerically robust and thesolution scheme is insensitive to system parameters, in particular, thebranch r/x ratios. This can be seen as follows.When the elements of the branch jacobians, Ji are calculated in perunit, PA., Ji has the following structure

Ji = E 1 AQ; (a.13)[:Where, IZ;,~ = I E I = 1 ~ ~ : ~Vi21 4: 1, APi =P loss on branch i, andAQi =Q loss on branch i. Therefore, det (J i )=1; indicating that Jisare very well conditioned.As a result of (a.13), the system Jacobian, J is well conditioned forthe main feeder only case.For the general case, it can easily be shown that, the diagonal ele-ments, Je of the system Jacobian has the Same properties as that of themain feeder, J , and the system Jacobian is block diagonally dominant,

i.e., det (Je) = 1 > det ( J k ) -V i #k , k #0 (a. 14)There fore, the system Jacobian, J is well conditioned, i.e.,

det(J)=1 (a.15)and t h i s is true independent of the line parameters.

These results indicate that the NR solution algorithm will be numeri-cally stable and robust. This very desirable property is one of theadvantages of the new formulation over the conventional one where con-vergence problems has been experienced when there are branches withhigh r / x ratios in the system [H I.Furthermore, these numerical properties can be exploited to modifythe DistFlow solution scheme to make it even more efficient computa-tionally.Enhancement of Computational EfficiencyWe use the numerical properties and the special structure of the Jaco-bian matrix, J in simplifying its construction and in solving the updateequations of (a.8).Since by (a.14). the off diagonal terms Jk , #k k =1. . . . l aremuch smaller than the diagonal terms, Jk , k =1 , . . . I , off diagonalterms can be dropped in the Jacobian except the ones in the last row,i.e., d =[Jol . . . ar] . This basically decomposes the update equationsinto I+1 equations as follows.

(a. 16.i)(a.16.ii)

Where, Hk =&, , ( z ) &, ( z ) IT , r, =[AzL . . ,$IT. Therefore, the statesthat correspond to the laterals, z i o , =1... I can be updatedindependently first by Eq(a.16.i). then the state for the main feeder, z mis updated by (a.16.ii).

Furthermore, since Jes are well conditioned and almost constant, asindicated by (a.13), the Jacobian needs to be constructed only once andthen it can be used in al l the iterations. Therefore, updating step of NRreduces to the solution of (a.16).This modified solution scheme brings down the overall cost of com-putation to about 8n+8(1+1) multiplications for each iteration; 8n multi-plications to calculate the mismatches and 8(1+1) multiplications to solvethe update equations of (a.16). This amount of computation is compar-able to 6n multiplications involved in just solving the update equations(like Eq4a.16)) of conventional fast decoupled power flow solution

scheme.

JkAzko =H ~ ( z )J& =H d z ) - r,

R =1, . . . I

Mesut E. Baran received his B.S. and M.S. rom Middle East TechnicalUniversity, Turkey. He is cum ntl y a Ph.D. tudent ai the University ofCalifomia, B erkeley.Felix F. Wu received his B.S. from National Taiwan University, M.S.from the University of Pittsburgh. and W.D. from the University of Cal-ifornia, Berkeley. He is a professor of Electrical Engineering and Com -puter Sciences at the U niversity of California, Berkeley.


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743M. . Baran and F. F. Wu : he authors appreciate the interest shown inthis paper by the discussers and thank them for their comments.

Concerning the first question by Mr. Vempati and Dr. Shoults aboutthe extension of the D istFlow algorithm to solve networks with loops , twopossible approaches seem workable. The first approach would be to putDistFlow equations corresponding to the radial system together with addi-tional equations due to looping branches and to apply Newton Raphsonalgorithm to solve the resulting equations. In this case,he structure of thesystem Jacobian can still be exploited to reduce computational burden andhence retain the robusmess and the speed of the algorithm. However,computations for each iteration will increase as the number of loopsincreases since the system Jacobian will get fuller. An altemativeapproach would be opening of the the loops at certain points of the loopingbranches and putting current injections at these points to simulate theeffect of the loops. An iterative scheme can be developed to obtain thesolution as proposed in [18]. In each iteration, currents passing throughthe opened points on the looping branches are estimated and then Dist-Flow equationsare solved to determine the power flow in the radial systemwith extra injections. The robustness of the method would still be validbecause of the usage of the branch powers as variables. However, thespeed of convergence and hence the computational efficiency of themethod would probably suffer as he number of loops increases.

DistFlow method can al so be generalized to represent the voltagedependent loads and capacitors as shunt susceptances. Modification forvoltage dependent load representation will en tail making the real and eac-tive power components of load, in branch Bow equations (1 .i)and (1.ii) a function of volta ge, (i.e. Pu+l=@fi+l(V,+l), QL+I=&+,(V,+I) ).Similarly, reactive power injections by capacitors, e+ in Eq.(l.ii), can bechanged as QP.+~= ,4 .+ lV~l , here Q,4.+1 corresponds to the nominal capa-city of the capacitor, to represent the capacitors as shunt susceptances. W eintend to incorporate voltage dependent loads and capacitors into our pro-gram.Because of the fact that starting point of the algorithm can beselected as the solution of the approximate method (see comment at thelast paragraph of seC.VI), we have not encountered the problem of slowconvergence and suboptimal solutions that are known to exist in the firstorder OPF methods. The success of second order OPF is due largely to theexploitation of the structure of the OPF equations. We believe thedevelopment of a successful second order solution method for the capaci-tor sizing problem would be an interesting research project. Here, wewant also to point out another additional advantage of themethod which isnot mentioned in the paper. The method used can easily be modified toinclude different load/capacitor models as pointed above.

We thankMr. Ch an dr as ek m and Dr. Broadwater for directing ourattention to reference [Dl]. Theuse of branch losses and voltages as statevariables is an interesting idea. The convergence characteristics of thesame method (e.g. Newton-Raphson) applied to two different formulationsof the same problem may be different. We do not have experience on themethod using losses as state variables to comment on its convergencecharacteristics.

Reference [18] pointed out the convergence problems for distributionpower flows. In addition to convergence, different methods may exhibitdifferent computational efficiency. T he so-called simple methods may beconvergent in most cases, but the convergence is usually slow.optimal sizing of capacitors may result in leading power factorcurrents in thebranches depend iig on the load profile. places of capacitorsand the voltage limits. This is more likely the case when capacitors areused for voltage regulation purposes (like TS l TB in the paper). The per-missible upper and lower voltages are set by the load demand require-ments and system operating requirements. The optimality seems n prac-tice tobe dependent on loca l condition.Representing the cost of a capacitor by a fixed cost and a linear capa-city cost is an approximation. The capacity cost in practice will be a stair-case curve. A linear cost cu rve is the simplest approximation of the stair-case curve as a differentiable function. A quadratic function, or any othernonlinear function may be used for a better approximation. T he increasein computation as result is minimal. However the effect, if any, on theoverall convergence is not known.

Manuscript received May 2, 1988.

,

DiscussionA. Chandrasekaran and R. P. Broadwater (Center for Electric Power,Tennessee Technological University, Cookeville, TN): The authors are tobe complimented for bringing under a single mathematical framework allthe various problems of capacitor placement, sizing, time-varying loads,and other special features of radial power distribution systems. A lot ofrigor has gone into the analysis of transmission systems. How ever, radialdistribution systems have received very little, and mostly sporadic anddesultory, mathematical treatment. The paper under discussion fulftlls along-felt need in distribution analysis. We request the authors to commenton the following observations.1) Recognition of the minimum (state?) variables required to solve thedistribution systems as the branch flows in the source branch and the f xstbranches in the radials in the D istFlow algorithm is of special interest. In[Dl] we proposed a load flow formulation based on branch losses andvoltages and the resultingJacobean tructure s similar.What is the authorsopinion about the convergence characteristics when the variables chosen arethe losses?2) Even though much is talked about the difficulties of convergence inradial systems, most of the practical algorithms in use are based on thesimple ladder technique of Kersting [D2]and appear to work very well. Arethere documented cases that the authors are aware of where convergencewas a problem? Is it possible to construct meaningful examples of radialsystems depicting nonconvergence with simple methods?3) The upper voltage constraints used in Test Run B is 1.1 pu, and if thesource voltage is 1 pu, does t h i s mean leading power factor currents canflow in some parts of the system? Are there optimal limits fo rpermissible upper and lower voltages?4) The cost factors kp nd ru used in (8)are oversimplified. What will bethe effect on the solution if more complex factors are used?We congratulate the authors once again f or a very interesting paper.

References[Dl] A. Chandrasekaran and R. P. Broadwater, A New Form ulation ofLoad Flow Equations in Balanced Radii Distribution Systems,C4nadionElectrical Engineering Journal, Vol. 12, No. 4, October[D2] W. H. Kersting and D. L.Mendive, A n Application of LadderNetwork Theory to the Solution of Three-phase Radial Load FlowProblems, Presented at the IEEE Winter Power Meeting, NewYork, 1976.

1987, p ~ .47-151.

Manuscript received February 22, 1988.

N. Vempati and R. R. Sboults (The University of Texas at Arlington,Arlington, TX): The authors are o be commended for their excellent paper.Their approach of rewriting the branch flow equations n a recursive formatenables an efficient algorithm. This is under the assumption of a radialdistribution system. When loops are encountered, similar equations can beformulated. Would the autho rs comment on the robustness and speed of thealgorithm in the presence of loops.Loads and capacitors are considered to constant power devices in thepaper. It is now well known that reactive power load is not a constant powertype load, rather it is more like a constant susceptance. Would the authorscomment on the impact, if any, on accuracy if voltage dependent loadmodels are not used. It has been our experience with various optimizationprocedures that significant differences result when different load modelingassumptions are used.The authors allude to the fact that second-order methods are alsoavailable in addition to first-order methods for the solution of nonlinearoptimization problems. They also present their rationale for implementingthe first-order method. In optimal power flow (OPF) problems, it has beenshown that first-order methods can give misleading results. The conver-gence rate can be too slow and sometimes results n a suboptimal solution.The second-order Newton method has been found to be more reliable. Itwould be interesting to see comparison cases between their first-ordermethod and a second-order Newton type method for their problemformulation. Would the auth ors please comment.We again congratulate the authors for a very fine pap er, which is verycharacteristic of their proven abilities.Manuscript received February 23, 1988.

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OPTIMAL SIZING OF CAPACITORS PLACED ON A RADIAL DISTRIBUTION SYSTEM