slack bus selection

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 2, MAY 2004 987

Slack Bus Selection to Minimize the SystemPower Imbalance in Load-Flow Studies

Antonio Gómez Expósito, Senior Member, IEEE, José Luis Martínez Ramos, Member, IEEE, andJesús Riquelme Santos

Abstract—This paper reconsiders the notion of slack bus inload-flow studies. Instead of determining a priori which bus playsthe role of slack bus, it is selected on the fly during the load-flowiterative process in such a way that the system power imbalance isminimized. The problem of selecting the best slack bus, or buses,is first formulated as a nonlinear optimization problem. Then, theresults obtained are justified on the basis of the involved equalityconstraint being a quasilinear function, leading to an LP problemwith trivial solution. It turns out that the optimal solution canbe easily found from the results of a conventional load flow at amoderate cost. The proposed heuristic procedure is tested on theIEEE test systems.

Index Terms—Incremental transmission losses, linear program-ming, load flow, slack bus.

I. INTRODUCTION

SOLVING the load-flow problem requires that total gen-erated power matches the total demand plus transmission

losses. However, as such losses cannot be determined before-hand, total generation needed to supply a known demand cannotbe exactly specified a priori. In consequence, it is necessary tohave at least one bus (the slack bus) whose real power genera-tion can be rescheduled to supply the difference between totalsystem load plus losses and the sum of active powers specifiedat generation buses [1]. Following [9] and [10], this differencewill be named system power imbalance. Furthermore, as phaseangles of bus voltage phasors must be referred to some arbitraryreference, the voltage phasor of the slack bus is usually taken asreference and, hence, its phase angle becomes zero [1], [2].

Existing approaches for selecting the slack bus can be broadlyclassified as follows:

A. Single Slack Bus

Most textbooks fall within this category [3], [4]. The slack busis considered a mathematical artifact created by the load-flowanalyst, without any direct link with the physical system [5]. Ac-tive power is specified at generation buses, most likely includingan estimation of ohmic losses. Hence, the difference betweencomputed and specified active power at the slack bus representsthe error in the prior estimate of system losses. Only in rare,typically small tutorial cases does the system imbalance repre-sent total power losses, which may exceed the rating of certaingenerators for realistic systems. Usually, the largest generator is

Manuscript received July 31, 2003. This work was supported in part by theSpanish MCYT and in part by Junta de Andalucía under Grants DPI2001-2612and ACC-1021-TIC-2002, respectively.

The authors are with the Department of Electrical Engineering, University ofSeville, Seville, Spain (e-mail: [email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/TPWRS.2004.825871

arbitrarily proposed as slack in absence of better criteria, whichis a good choice in case the total imbalance (i.e., the loss esti-mation error) is relatively large.

Earlier load-flow algorithms were mainly concerned withconvergence problems when selecting the slack bus. The firstsystematic study about this issue was presented in [6], whereit is concluded that the optimal choice from this point ofview is the bus with largest short-circuit current (i.e., the onewhose diagonal element is smallest). Those concernsvirtually vanished when the Newton–Raphson method wasfully developed, as it proved to be much more tolerant of slackbus location than older methods [7].

Other suggested criteria for single slack bus selection are [6]:a) have a large number of lines connected to it; b) have a voltageleading all other voltages of the system.

B. Distributed Slack Bus

The distributed slack bus concept is implemented in somepower flow programs, particularly those used in EMS applica-tions. Under this approach, total system imbalance is frequentlyinterpreted as a deviation of the load-frequency control mecha-nism. Therefore, it is assumed that the set of generators involvedin the automatic generation control (AGC) contribute to balancethe system in proportion to the so-called participation factors[8], [9]. Such coefficients can be determined based on [8]: a) ma-chine inertias; b) governor droop characteristics; c) frequencycontrol participation factors; d) economic dispatch [9], [10].

However, the units used for steady-state loss compensationneed not be the same as those participating in AGC [9].In [11], for instance, the vector of participation factors ismade colinear with the specified generation vector (i.e., eachfactor is proportional to the respective generation scheduling).For localized power-flow solutions, a geographical criterionis suggested in [12], where it is shown that distributing thesystem imbalance among several nearby generators improvesthe performance with respect to the use of a single, probablyremote slack.

When a distributed slack bus is adopted, all active powermismatches are retained in the unknown vector while thestate vector gets augmented with the system power imbalancein order to compensate for the missing phase reference [9],[10], [13]. Nonzero elements of the extra Jacobian column aresimply the respective participation factors.

In addition to the stand-alone load-flow problem, partici-pation factors frequently arise in other applications where theload-flow problem constitutes the core, like economic dispatch[10], [11], deregulated markets [13], [14], sensitivity analysis[15], etc.

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988 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 2, MAY 2004

It is worth noting that, by definition, participation factors areall positive, which means that all involved generators will in-crease (decrease) its scheduled power when the net power im-balance is positive (negative).

Somewhat between both schemes, the multiple slack bus con-cept can be used to achieve net interchange adjustments be-tween control areas in multiarea load-flow algorithms [16]. Inthis case, each control area may have one of its generating unitsdesignated as “area slack,” and net interchange is adjusted bychanging the power output of all area slacks except the systemslack bus, which must account for total system imbalance. Thisscheme can be also considered a particular case of distributedslack bus in which “participation factors” are determined on thefly to match the desired interchanges.

It may be argued that a distributed slack bus better resemblesthe way power systems are operated, provided participation fac-tors are properly chosen. However, this depends on the contextin which the load flow is used (EMS, planning, deregulated mar-kets, etc.).

In this paper, no assumption is made a priori about the slackbus being unique or distributed. Any generation bus, or com-bination of buses, can play the slack bus role but, instead ofresorting to a given set of participation factors, total power in-jected by each generator is decomposed into a major constantterm, specified beforehand in the input file, and a slack powerrepresenting its unknown contribution to the net power imbal-ance. As the need for the slack bus is a direct consequence ofthe existence of the power imbalance, it makes sense to selectas slack, the bus or set of buses that minimize such an imbal-ance, using the slack powers as control variables. This criterioncan be shown to be equivalent to that of minimizing active powerlosses, even though the net system imbalance will differ, in gen-eral, from actual losses. Conditions leading to the choice of asingle or distributed slack are also presented.

The ideas presented in this paper might find application insome of the market-related issues still open, like that of loss al-location, but this is out of the scope of the paper which is focusedon ordinary load flows. The proposed criterion to select the slackbus(es) is original and easy to implement in existing load flows.Unlike most slack bus selection criteria, which require the in-tervention of a skilled user, familiarized with the power systembeing solved, the proposed scheme provides an automatic pro-cedure that may be useful in those cases where the user has nopreliminary idea about the right candidate for slack bus.

The paper is organized as follows: First, the problem ofselecting the slack generator(s) in load-flow studies so as tominimize the power imbalance is presented, and the math-ematical formulation as an optimization problem is derived.Then, some test results are presented and discussed. Finally,a heuristic approach to automatically find the optimal slackbus from the results of a conventional load flow is presentedand tested on some IEEE systems.

II. MOTIVATION

In load-flow studies, it is customary to choose as slack busthe same bus whose phase angle is arbitrarily set to zero. How-ever, while it is irrelevant for the load-flow solution which busis taken as phase origin, the total system imbalance, and hence,

power losses, will be affected to a certain extent by the slack busselection.

Consider, as an example, the three-bus network shown inFig. 1, where generation scheduling is determined in such a waythat the system imbalance exactly matches ohmic losses. Whenbus 1 is the slack bus, total losses amount to 2.385 MW, whilethese losses are 2.412 if bus 3 is selected as slack bus. There-fore, it is preferable, from the point of view of reducing losses,to select bus 1 as slack bus in this case. Note that further reduc-tion of losses could be achieved by increasing (decreasing) thenet power injected at bus 1 (bus 3), as a consequence of branch1–2 resistance being much smaller than the other, but this goesbeyond the load-flow scope in which generation powers are ex-ternally determined and the only degree of freedom is the slackbus choice.

In this paper, it will be assumed, like in the former example,that active power at all generation buses is specified. This is thecase also when a distributed slack bus is adopted. Total specifiedgeneration may, or may not, as in the example, take into accountan estimation of network losses, leading to positive or negativesystem imbalances. The important thing, as shown above, is thatthe resulting imbalance depends upon which particular bus ischosen as slack. Consequently, there is a margin to obtain im-proved load-flow solutions by carefully selecting the slack bus.Furthermore, there is no reason a priori to believe that choosinga single slack bus is better than distributing the total imbalanceamong several generators. In fact, based mainly on intuition, it isgenerally believed that distributing the system imbalance amongseveral generators leads to lower losses [9]. The above observa-tions provide the foundation for the work reported in this paper.

III. PROBLEM FORMULATION

Mathematically, the choice of a slack bus to minimize thesystem power imbalance, including the possibility for it tobe distributed, can be exactly formulated as an optimizationproblem.

Let denote the contribution of generator to match thesystem imbalance (in the sequel, will be referred to as theslack power of bus ). Then, the load-flow equations can bewritten as follows:

(1)

(2)

(3)

where, assuming bus 1 is the reference bus, the augmented statevector is given by

(4)

(5)

(6)

Note that the sum of slack powers must be equal to the systempower imbalance, that is

(7)


EXPÓSITO et al.: SLACK BUS SELECTION TO MINIMIZE THE SYSTEM POWER IMBALANCE IN LOAD-FLOW STUDIES 989

Fig. 1. Three-bus system and associated data (S = 100MVA).

In compact form, the set (1)–(3) can be expressed as

(8)

The above system comprises equations inunknowns, which means that there are degrees of

freedom. Conventionally, values of coefficients areset to zero, and that of the slack bus is determined after the loadflow is solved, but there are other possibilities to obtain suchcoefficients keeping in mind a certain merit function.

The objective function proposed in this paper is to minimizethe net power imbalance

(9)

which, according to (7), amounts to minimizing the powerlosses.

In case each slack power is assigned a certain cost , the slackbus(es) could be alternatively selected in such a way that thescalar

(10)

is minimized. Minimizing the cost associated with the systemimbalance is just a matter of scaling the slack powers, and theprocedure developed below for is directly applicable to .

Two types of inequality constraints could be enforced whensolving the equality-constrained optimization problem formu-lated above, as follows.

1) Equipment physical limits, such as maximum powerflows, generator ratings, etc. However, except for re-active power limits, the remaining constraints are outof the scope of a conventional load flow and should behandled by ad hoc congestion management procedures.Furthermore, as the system imbalance is usually quitesmall, it is very unlikely that the slack bus choice drivesany power flow beyond the respective branch rating.Anyway, simple constraints like

(11)

could be easily added as a safeguard against largedeviations.

2) If no constraints are imposed on the slack variables ,it may well happen that certain generators strategicallylocated significantly increase their power share, whileothers get their scheduled power virtually cancelled bya negative (i.e., ). For instance, in the

three-bus system of Fig. 1, the unconstrained minimum ofis achieved when and .

This is not a desirable feature for a conventional load flow,which should not be allowed to redistribute power in thisway when selecting the slack bus. Therefore, the slackpowers should be further constrained, in addition tothe upper limits given by (11). In the same way as partic-ipation factors are all positive when a distributed slack isadopted, it is reasonable to force each to be positiveif the system imbalance is positive, in order that no gen-erator reduces its scheduled power when there is a powerdeficit. Similarly, all ’s should be negative when thereis a surplus of generated power. The problem is that thereis no way to determine a priori which is the case, unlessa load-flow solution exists. Considering that each term

(or ) is null only when(or ), the requirement that all ’s are simulta-neously positive or negative can be fulfilled by enforcingthe following constraint:

(12)

Consequently, ignoring for simplicity the upper bounds (11),the problem of choosing the slack powers in an optimal waycan be formulated as follows:

min

(13)

In order to remove the operator from the model, it iscustomary to split each as follows:

(14)

where and are both positive. This way, the equivalentproblem that must be solved in practice is

min

(15)

It should be noted that the solution to this problem minimizesthe extra power that generators must deliver when the powerimbalance is positive, or maximizes the reduction of generatedpower when the imbalance is negative.

IV. TEST RESULTS

To begin with, the five-bus system of Fig. 2 will be tested. Ac-cording to the data presented therein, total generation is equal



Fig. 2. Five-bus system and associated data.

in this case to total demand, which means that the system im-balance amounts to power losses.

Table I collects the results of solving the optimizationproblem (15) in three cases. Case 0 refers to the base case;case 1 differs from case 0 in the way generators 4 and 5 sharethe total load, while in case 2, the load at bus 1 is significantlyincreased and the generation scheduling is also changed. Themain conclusion is that, in all cases, the optimum is achievedwith a single, rather than distributed, slack bus. The results alsoshow that the optimal slack bus location is a function of bothload level and generator scheduling, contrary to the systematicchoice of the same slack bus conventionally performed inload-flow studies.

The IEEE 57-bus system, comprising four generators, hasbeen also tested. Table II presents the results obtained in fourcases. Case 0 refers to the standard base case, in case 1, the totalload is equally split among the generators, case 2 differs fromthe base case in the fact that the scheduled powers for generators3 and 8 are exchanged, and the power of bus 12 is increased incase 3 so that the system imbalance is negative. The lowest rowshows, for comparison, the losses that would take place by run-ning the load flow with the standard slack bus (bus 1). For thissystem, this particular slack bus is actually the worst in mostcases, which shows the importance of carefully choosing thebest slack bus in accordance with network topology, load level,etc. This is clearly seen in case 3, where there is an excess ofgenerated power and bus 1 is selected to maximize the power re-duction. Like in the five-bus system, it turns out that the systemimbalance (power losses except for case 3) is optimally providedby a single bus, which means that losses can be further reducedonly if the slack powers ’s are allowed to become positiveand negative simultaneously.

V. EXPLANATION OF RESULTS

The results presented above, and many more obtained in otherexperiments, lead to the conclusion that the system power im-balance is minimized when a single generator accounts for it,

TABLE ITEST RESULTS FOR THE 5-BUS SYSTEM (POWERS IN MEGAWATTS)

TABLE IITEST RESULTS FOR THE 57-BUS SYSTEM (POWERS IN MEGAWATTS)

Losses using bus 1 as slack bus.

provided negative reschedulings are not allowed when there isa power deficit (and vice versa). This section is devoted to jus-tify this conclusion and to discuss possible situations in whicha distributed slack may arise.

As stated in Section III, when the slack powers areadded to the load-flow model, degrees of freedom arise.This means that, if it was possible for the conventional statevariables to be eliminated from (1)–(3), then all would belinked by a single nonlinear equation such as

(16)

Of course, because of the nonlinearity of the load-flow equa-tions, there is no way to explicitly write (16). However, it is easyto obtain its linearized counterpart around a load-flow solutionpoint ( , , ). Let denote the column vectorwhen the reference bus is omitted. Then, linearization of (1)–(3)around ( , , ) yields

(17)

(18)

where is the conventional load-flow Jacobian, is the Jaco-bian row corresponding to the reference bus (missing in ordinaryload flows), and the column vector has been split for conve-nience into its generator and demand components and ,respectively. From (17) and (18), the following relationship isobtained:

(19)



or, in more compact form

(20)

where refers to the columns of corresponding to .Note that the row vector can be obtained from the LUfactors of , without computing explicit inverses.

Therefore, for small deviations around the load-flow solution,(16) can be linearized as

(21)

Note that the right-hand side of the above equation reduces toin conventional load-flow solutions, where the slack and

reference buses are the same. Rearranging terms in (21), thehyperplane that approximates (16) around ( , , ) canbe written in standard form as

(22)

It is important to realize that the coefficients depend onthe solution point. The more constant these coefficients are, themore linear (16) is, and vice versa. Table III presents these co-efficients for the three networks tested above (case 0), when theload-flow equations are linearized around every possible solu-tion with single slack bus.

The degree of parallelism between two different hyperplanescan be assessed by performing the scalar product of the orthog-onal vectors associated with them, whose coordinates are deter-mined by the coefficients . This leads to the angles shownin the bottom rows of Table III, the largest one being 0.22 .Therefore, it can be concluded that (16) is almost a linear func-tion if the slack powers are restricted to the region(or when the imbalance is negative). Fig. 3 representsthe subspace (16) for the three-bus and five-bus networks. Asthe visual representation suggests, both subspaces fit pretty wellthe respective linear variety (the curvature has been exagger-ated). For the five-bus system, for instance, a set of 35 load flowswas run by choosing different slack powers within the feasible“triangular” region. A linear regression provides the best hyper-plane coefficients shown in Table IV, along with the upper andlower hyperplanes defining the 95% confidence interval. Thesefigures confirm that the subspace (16) is nearly a linear variety.

Assuming the coefficients are positive and ignoring the non-linearity of (16), the problem of choosing the slack bus that min-imizes the system imbalance can then be formulated as follows:

min

(23)

It is easy to show (see, for instance, [17]) that the above LPproblem converges to the vertex where

min (24)

TABLE IIIHYPERPLANE COEFFICIENTS AT DIFFERENT SOLUTION POINTS

Original slack bus.

Fig. 3. Relationship among the coefficients P for two- andthree-dimensional spaces.

TABLE IVHYPERPLANES OBTAINED FOR THE 5-BUS SYSTEM BY LINEAR

REGRESSION ON 35 POINTS

When the coefficients ’s are negative (i.e., negative imbal-ance), the inequality constraints to be enforced are but(24) still applies (the slack bus corresponds in this case to thatwith largest ).

For very large system imbalances (positive or negative),however, the validity of the linear model (23) could be ques-tioned, and the LP theory can no longer be applied to concludethat the optimum lies at a vertex. Fig. 4 shows, in addition tothe linear problem, two nonlinear functions with increasingcurvature for the two-dimensional case (the thin parallel linesrepresent constant values of the objective function). As theo-retically predicted, in the linear case, the solution point “ ”lies at the vertex with smallest coefficient ( in theexample). This can also be the case for slightly nonlinear func-tions, like in Fig. 4(b), but not necessarily, like in Fig. 4(c),where “ ” lies at an intermediate point.



Fig. 4. Two-dimensional optimization problem: (a) linear; (b) and (c)nonlinear equality constraint.

As proved in the Appendix, for every pair of coordinatesand , the following two conditions must be simultaneouslysatisfied for the optimum to lie at an intermediate point:

(25)

where refers to the coefficient of in the tangent sub-space (22) computed at the vertex given by and

for .Since the function (16) is so linear, it is very unlikely that

the two conditions (25) are satisfied, which explains the test re-sults presented in Section IV. However, such a possibility exists,particularly when the weighted function (10) is adopted in orderto minimize the cost rather than the system imbalance. Definingthe scaled variables , it is straightforward to provethat (25) is equivalent to

(26)

For instance, applying (26) to the three-bus system yields

which is indeed a very narrow interval. The intermediate pointand is reached when

.The Appendix also shows that the hyperplane coefficients

are related with the incremental transmission loss coefficientsthrough

which means that a given slack bus is optimal, from the point ofview of reducing the system imbalance, when the resulting ITLcoefficients are all positive. Otherwise, the bus with lowest ITLshould be chosen as slack.

VI. HEURISTIC SELECTION OF THE SLACK BUS

From the above discussion, it can be concluded that there isno need to worry in practice about the possibility for the systemimbalance to be distributed among several generating buses, un-less it is very large and the two smallest hyperplane coefficientsare very similar. Furthermore, it would be quite awkward to re-sort to an optimization package every time a load flow must besolved.

Based on the above findings, the following heuristic proce-dure is proposed that should lead, except for very pathologicalcases, to the same solution as if (15) was actually solved.

1) Obtain the load-flow solution by provisionally using thereference bus as slack bus. At this stage, there is no needto adopt a very tight convergence criterion.

2) Compute the coefficients of the tangent hyperplane (22)based on the LU factors of the last Jacobian.

3) Select as slack bus the one whose coefficient satisfies(24).

4) Using the new slack, perform extra iterations startingfrom the solution of step 1, until full convergence. Thisrequires that the rows corresponding to the old and newslack buses be exchanged (phase angles should be shiftedas well so that the new slack bus is also the referencebus).

This procedure is based on the empirical observation thatthe bus whose hyperplane coefficient is the smallest seldomchanges with the slack bus, because of the parallelism of thedifferent hyperplanes. In case of doubt, step 2 can be performedagain to check that the slack bus remains the same.

Furthermore, let be the chosen slack bus and the buswhose coefficient is closest to that of . In case ,the possibility of a distributed slack can be confirmed or dis-carded by obtaining a new load flow with as slack bus andthen checking (25). This extra work is seldom justified because,even if the system imbalance is distributed between and , thepower reduction achieved will be very small.

The need for a distributed slack bus may arise, however, whenupper limits are imposed upon slack powers. For instance, if

, then and

is the optimal solution provided .

VII. RESULTS PROVIDED BY THE HEURISTIC APPROACH

The procedure presented in the former section has beenapplied to the IEEE 118-bus system (data taken from [18]).Table V presents the three buses , , and with smallesttangent hyperplane coefficients. The first row corresponds tothe standard slack bus of the test case, the second refers to theslack bus deemed as optimal , and the third is obtained withthe best competitor of the chosen slack bus .

The following comments are in order:

• As happened with the 57-bus system, the original slackbus is not the best choice to reduce the system imbalance(in this case, it is third in the ranking).

• The ranking of candidate slack buses is not altered by theparticular slack bus adopted to obtain the tangent hyper-plane. This may not be true in all cases.

• The hyperplane coefficients are negative this time,which means that the generated power must be reduced.Choosing bus 89 as slack, instead of bus 69, leads to anextra 8.6% reduction, which is not negligible.

• Computing the coefficients when bus 87 is the slack busis not actually needed. They are included in Table V just



TABLE VSMALLEST HYPERPLANE COEFFICIENTS AT DIFFERENT VERTICES

(118-BUS SYSTEM)

to check that the two conditions (25) are not satisfied and,consequently, to confirm that a single slack bus is the op-timal choice.

VIII. DISCUSSION AND REMARKS

The following issues deserve further clarification.

• OPF versus load flow. Selecting one among several slackbus candidates, keeping in mind a certain merit function,constitutes an optimization problem and, as such, can beconsidered a particular case of the so-called OPF. One ofthe goals of the paper is to show that, for this purpose,there is no need to resort to such a complex tool, as the re-quired information is a byproduct of the load-flow iterativeprocess. The proposed procedure can be easily embeddedwithin existing load flows, and the user will simply noticethat the selected slack bus sometimes differ from that ofthe input file when this option is activated.

On the other hand, the OPF problem formulated in thispaper to theoretically found the proposed method is orig-inal in the way control variables, objective function, andconstraints are combined. Conventionally, active powersare used as control variables when cost is minimized,while reactive powers are rescheduled to minimize losses[19]. In this paper, a preliminary loss minimization isachieved by carefully selecting the bus that takes care ofthe system imbalance.

• Constraining the slack power signs. From the OPF per-spective, constraining the slack powers to be all positive(negative) when the power imbalance is positive (negative)might seem artificial. However, this is required to preventthe generation scheduling, determined beforehand, frombeing redistributed merely because of a small power im-balance. The reader should be aware that this constraint isalso implicit when a distributed slack, based on conven-tional participation factors, is employed. This is clearlyseen from the expression

(27)

where is the system imbalance and the respectiveparticipation factor [10]. As by definition, it fol-lows that the sign of is the same as that of the powerimbalance.

Note that this constraint is crucial to reach one of the mainconclusions of the paper, namely the fact that the system imbal-ance is minimized virtually in all cases when the slack poweris provided by a single generator, which is somewhat contraryto intuition and general belief. For instance, in [9] a five-bus

system is solved twice, first with a single slack bus and then witha distributed one. The authors claim that, in the second scenario,“power losses are less than in the case of a single slack bus, asexpected.” However, [9, Tables 1 and 6] actually show that theopposite is true, confirming the theory presented in this paper.

IX. CONCLUSION

In this paper, it is assumed that when solving the load-flowproblem, all generated powers are specified and that any bus,or combination of buses, can play the role of the slack bus. Forthis purpose, a slack power is introduced at each generation buswhose value is determined in such a way that the system powerimbalance is minimized. When the slack powers are all con-strained to be positive (or negative if there is a power surplus),the resulting optimization problem becomes, in practice, an LPproblem with a single equality constraint, whose solution is triv-ially obtained at one of the vertices (single slack bus). Condi-tions for the unlikely possibility of a distributed slack bus toarise are also deduced from linearization around a load-flow so-lution. A simple modification to existing load-flow tools is pro-posed that should detect in nearly all cases the slack bus whichminimizes the net power imbalance. It is also shown how theproposed method can be easily adapted to the case in which thecost of the system imbalance is to be minimized.

Major contributions of this paper are: 1) an original andsimple criterion to select the slack bus in load-flow studieswithout user intervention; (2) a theoretical development pro-viding further insight into the implications of choosing a singleor distributed slack bus from the point of view of ohmic losses;and (3) an algebraic and geometrical justification about theremote possibility of a distributed slack bus being preferable toa single slack when the power imbalance is a concern.

APPENDIX AREQUIREMENTS FOR DISTRIBUTED SLACK BUS

Let and be a couple of candidate slack buses. When theremaining generators are discarded, the objective function (9)becomes

(28)

Also, in the resulting two-dimensional subspace, the tangenthyperplanes computed at the vertices and are

(29)

(30)

For small perturbations, the objective function can be linearizedaround both vertices as follows:

(31)

(32)



The optimal solution will lie within the two vertices when theobjective function decreases for perturbations in that direction,that is, when the following two conditions are satisfied:

and

or, equivalently

(33)

Otherwise, the optimal solution will be the vertexwhen (and vice versa). Geometrically, (33) can beinterpreted as both angles and in Fig. 5 being larger than45 .

APPENDIX BRELATIONSHIP OF TANGENT HYPERPLANE COEFFICIENTS WITH

INCREMENTAL TRANSMISSION LOSSES

In this section, it will be assumed that the system imbalanceamounts to the power losses.

For a given slack bus , the tangent hyperplane is defined by

(34)

Assuming this linear variety is a good approximation of thenonlinear function (16), losses can be expressed as follows:

(35)

where the variable has been eliminated by means of (34).Consequently, the incremental transmission loss coefficient ofbus , , can be obtained from

(36)

This shows that, as expected, the ITL information is embeddedin the hyperplane definition. The opposite is not true, as eachindividual cannot be obtained from the ITL coefficientsunless is specified. From the load-flow solution ( ,

), this coefficient can be computed as . Notethat is the optimal slack bus if and only if for all

.The following expression, relating the ITL coefficients for

two different slack buses and can be found in the literature[20]

(37)

Fig. 5. Geometrical interpretation of conditions for distributed slack bus.

However, according to (36), this is only true if

(38)

which is not exactly the case, as shown by Tables III and V (thehyperplanes are not perfectly parallel).

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Antonio Gómez Expósito (SM’95) was born inSpain in 1957. He received the electrical engineeringand doctor engineering degrees from the Universityof Seville, Seville, Spain.

Currently, he is a Professor and Head of theDepartment of Electrical Engineering, Universityof Seville, where has been since 1982. His primaryareas of interest are state estimation and optimizationtechniques.

José Luis Martínez Ramos (M’99) was born in DosHermanas, Spain, in 1964. He received the Ph.D. de-gree in electrical engineering from the University ofSeville, Seville, Spain.

Currently, he is an Associate Professor with theDepartment of Electrical Engineering at the Univer-sity of Seville, where he has been since 1990. His pri-mary areas of interest are active and reactive poweroptimization and control, power system analysis, andpower quality.

Jesús Riquelme Santos was born in Las Palmas deGran Canarias, Spain, in 1967. He received the Ph.D.degree in electrical engineering from the Universityof Seville, Seville, Spain.

Currently, he is an Associate Professor with theDepartment of Electrical Engineering at the Univer-sity of Seville, where he has been since 1994. Hisprimary areas of interest are active power optimiza-tion and control, power system analysis, and powerquality.


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