Computation of bifurcation boundaries for power systems: a new Δ-plane method

536 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 47, NO. 4, APRIL 2000

Computation of Bifurcation Boundaries for PowerSystems: A New�-Plane Method

Yuri V. Makarov, Senior Member, IEEE, David J. Hill, Fellow, IEEE, and Zhao-Yang Dong, Member, IEEE

Abstract—This paper is devoted to the problems of finding theload flow feasibility, saddle node, and Hopf bifurcation boundariesin the space of power system parameters. The first part contains areview of the existing relevant approaches including not-so-well-known contributions from Russia. The second part presents a newrobust method for finding the power system load flow feasibilityboundary on the plane defined by any three vectors of dependentvariables (nodal voltages), called the� plane. The method exploitssome quadratic and linear properties of the load flow equationsand state matrices written in rectangular coordinates. An advan-tage of the method is that it does not require an iterative solutionof nonlinear equations (except the eigenvalue problem). In addi-tion to benefits for visualization, the method is a useful tool fortopological studies of power system multiple solution structuresand stability domains. Although the power system application isdeveloped, the method can be equally efficient for any quadraticalgebraic problem.

Index Terms—Bifurcation, load flow analysis, power systems.

I. INTRODUCTION

POWER system transient behavior is described by differen-tial and algebraic equations (see [1]–[3], for example) of

the form

(1)

(2)

wherevector of dynamic state variables;vector of algebraic variables;vector of power system parameters, includingany parameters of generators, control system,loads, and network which can be varied in plan-ning, tuning, and control.

The full state vector is .In small-signal analysis of system (1), (2), the steady-state

stability of equilibrium points satisfying algebraic equations

(3)

is studied. In principle, (3) may denote the set (1) and (2) whereall derivatives are set to zero. If so, the functionin (3)

Manuscript received May 13, 1998; revised July 22, 1999. This work wassupported in part by the Australian Electricity Supply Industry Research Boardunder Grant “Voltage Collapse Analysis and Control.” This paper was recom-mended by Associate Editor M. K. Kazimierczkuk.

Y. V. Makarov was with the Department of Electrical Engineering, the Uni-versity of Sydney, Sydney, NSW 2006 Australia. He is with Southern CompanyServices, Inc., Birmingham, AL 35202 USA.

D. J. Hill and Z-Y. Dong are with the School of Electrical and InformationEngineering, University of Sydney, Sydney, NSW 2006, Australia.

Publisher Item Identifier S 1057-7122(00)02920-2.

represents and in (1) and (2). However, in many cases (3)is the load flow equation and implies particular simple modelsof generators, their control systems, and loads. Thenusuallyconsists of bus voltages (in polar or rectangular form). For theseload flow equations, the mismatch functiondoes not exactlystand for the functions and in (1) and (2). It defines thesum of power flows or currents into each bus from the rest of thenetwork and fixes the generator terminal voltages. The vectorusually represents the specified independent parameters, suchas active and reactive powers of loads and generators or fixedvoltages. By solving (3) for a given, the unknown nodal volt-ages can be found, and the equilibrium point and can bethen unambiguously derived to keep the right side of (1) and (2)equal to zero. In view of this fact, we will not make a distinctionbetween in (1) and (2) and in (3) for the sequel.

Let us go back to the system (1) and (2). If the algebraic Ja-cobian matrix is nonsingular, the algebraic constraintmanifold whereis regular, i.e., the rank of equals the number of alge-braic equations – and there exists a locally defined func-tion that satisfies [1],[2]. Under these conditions, the system (1) can be locally repre-sented as follows:

(4)

Under these conditions, the possibility of singularity inducedbifurcations is avoided. The state matrix of (4) calculated at theequilibrium point is expressed as follows:

(5)

Together with the state matrix , we will use the load flowJacobian matrix . To simplify expressions, we willuse the common notationfor the vectors and defined for(3) and for and . The meaning ofand will be evident from each concrete context.

Due to nonlinearity and physical structure, a power systemoperating state is subjected to limitations of different origins.The power flow feasibility and stability constraints play an im-portant role among them. Operating conditions are to be keptinside the feasibility and stability domains which are defined byconstraint hypersurfacessatisfying one of the following gen-eral conditions:

(6)

(7)

1057–7122/00$10.00 © 2000 IEEE

MAKAROV et al.: COMPUTATION OF BIFURCATION BOUNDARIES FOR POWER SYSTEMS: A NEW-PLANE METHOD 537

whereor ;

vector of load flow variables or state variables;

identity matrix;imaginary part of an eigenvalue with zero realpart.

When parametersare varied slowly, either conditions (6) or (7)can be achieved, and this means that the system is on the edge ofits transfer capability, i.e., and ,or small signal stability, i.e., . In the lastcase, once the conditions (6) or (7) are satisfied, the saddle node

or Hopf bifurcations (instabilities) occurin the system, respectively. These hyperspaces can be in eitherstate-space or parameter space.

Among the requirements of power system control and plan-ning is the provision of adequate stability and loadability mar-gins. Knowledge of the stability boundary geometrical configu-ration gives guidance for development of control strategies andproper decision making. Hence, it follows that study of the bi-furcation surfaces is an important task. This paper contains abrief review of the existing and relevant approaches. However,as the conditions (6) and (7) correspond to very complicated sur-faces, visualization of their geometry is a difficult problem. Thisis usually done in planes defined by a pair of chosen parameters

or . However, some important features of the bifurcation sur-faces and multiple equilibrium points may not be observed inthis way. For example, on a plane of two nodal voltages, bifur-cation surfaces and certain equilibrium points may not be repre-sented completely; these planes generically do not contain anyequilibria. This paper contributes to development of techniquesfor such visualization. The introduction of aplane defined byany three linearly independent vectors, , and enablesviewing of the bifurcation surface as it intersects this plane, i.e.,a tool is provided for fully flexible observation. For instance,any three equilibria can be chosen to form theplane and therelative positions of these points and bifurcation surfaces willbe reflected on the plane in all the details which are required.

The structure of the paper is as follows. Section II givesa review and discussion of some ideas associated withcomputing the feasibility and stability boundaries includingnot-so-well-known contributions from Russia. Section III con-tains theoretical results about the quadratic algebraic problemswhich form the basis of the -plane method. Section IV isdevoted to the proposed-plane method itself. The methodcomputes the power system load flow feasibility boundaryon the plane defined by any three vectors of dependentvariables (nodal voltages). An advantage of this new reliablemethod is that it does not require an iterative solution ofnonlinear equations (except the eigenvalue problem). Resultsof numerical studies for the New England Test System aregiven in Section V.

II. THE CURRENT STATE-OF-ART

The study of nonlinear power system behavior associatedwith multiple equilibrium points (load flow solutions) andtheir relationship with bifurcations has attracted considerable

attention recently (see [3] for a recent collection of reviewpapers). Among these, [2] contains an extended review ofmodern local bifurcation theory, its application to powersystem analysis and control, and some relevant computationalaspects. An important reference is [4] which is devoted tomathematical and computational issues of bifurcation analysis,namely, to finding the closest saddle node, Hopf, transcritical,and pitchfork bifurcation in parameter space. Many of the ideasreviewed and developed in [2]–[4] can be used to study thebifurcation boundary geometry. Here we will consider some ofthese and other approaches. It should be noted that, in some ofthe reviewed work, the application of the ideas to explorationof the bifurcation boundaries is not addressed directly. In thesecases, we will attempt to show the obvious developments.

In the pioneering works [6]–[10], condition (6) is substitutedby the following equivalent equations:

(8)

where and are the right and left eigenvectors ofcorresponding to the zero eigenvalue. The vectorsand haveremarkable properties [4] which are very useful in stability anal-ysis. The right eigenvector indicates the direction of the ini-tial dynamics of the unstable behavior (voltage collapse, for ex-ample), and the extent to which a variable collapses is given bythe relative magnitude of the corresponding components of.The left eigenvector may be interpreted as the normal vectorto or, more precisely, to the tangent hyperplane spanned bythe columns of . In the case of parameter space, forgeneral parameters, is a normal vector to the hy-perplane.

In the works [5]–[7] based on quadratic representation of theload flow problem , the feasibility re-gion is a cone whose vertexis at the origin of . The supporting (tangent) hyperplanesimply the condition where is the normal vector to

and, if the cone is convex, all feasible independent parame-ters satisfy the inequality (see Fig. 1). In [5]–[7] itwas conjectured that the feasibility region is convex. Examples[8] show that this is not true in general. Though, under what sit-uation it is true remains to be proven for further research. Forvectors within the feasibility region, the following optimiza-tion problem gives the tangent planes:

or (9)

(10)

where is a vector corresponding to a positive definite(where is a linear matrix function of [5]–[7]) andis a vector not parallel to and is a nonzero constant. Theconstraint (10) defines a plane which intersects the cone (Fig. 1).The problem (9), (10) can be expressed as follows [5]–[7]:

or (11)

(12)


Fig. 1. A supporting hyperplane in [5]–[7].

By means of differentiation of the Lagrange functionwith respect to , the following

generalized eigenvalue problem can be stated:

(13)

To get the supporting planes , the minimal andmaximal eigenvalues of the problem (13) are used.The set of tangent hyperplanes for differentand gives anapproximation of the feasibility region boundary . Difficul-ties may arise for nonconvex feasibility regions.

Similar ideas were developed in Russia [9]–[12]. In [9], it isshown that the load flow feasibility domain boundary is a setof points where a linear combination of the mismatch functions

reaches its extreme values, i.e.,

or (14)

where , are direction cosines of a normal vectorto . By optimizing (14) for distinct nonzero vectors, thefeasibility boundary can be traced. It is obvious that (8) directlyfollows from (14). In [11] and [12], (8) is used to reveal . Bysolution of (8) for distinct or , points of are obtained.Note that all dependent variablesand independent parameters

in (nodal powers and terminal generator voltages) arefree to vary. In polar coordinates, the equality constraints such asfixed generator voltages may be taken into account by fixing andeliminating them from and withdrawing the correspondingfunctions from .

Further developments are given by Kontorovichet al.in[10]–[12] and in the work by Dobson [13]. The authors presentmethods for finding the load flow feasibility and saddle nodebifurcation points by solving a system which can be generalizedas follows:

(15)

(16)

(17)

where (15) and (17) are the load flow and conditions, re-spectively, is the vector of independent parameters at a givenoperating point, is a direction of variation of , is a scalarunknown parameter, andis a nonzero vector. In [4] and [13]

this system is used to obtain the locally closest bifurcation point,but it can give sequences of the saddle node bifurcation pointsas well. By successive rotations of in a plane it gives theintersection of ( ) or by this plane.For simpler models the load flow feasibility and saddle nodeboundary can be shown to coincide, however, this is not true ingeneral [14].

There are some other questions regarding the technique forgenerating bifurcation boundaries.

i) What is a reliable way to get an initial estimate ofon ?

ii) How reliable is the procedure in view of the geometricpeculiarities and nonsmoothness of?

iii) Whether is it possible to obtain all the parts of?iv) How many points are to be found for an accurate assess-

ment of ?In [15], all the questions above are addressed. A high-order

numerical method is used to get the initial point and providefurther motion along . This method may be considered as ei-ther a generalization of the Newton–Raphson method, includingnonlinear terms of the Taylor series, or a parameter continuationtechnique providing more reliable solution properties. The ideaof the method is the following. A nonlinear system ,which is the power flow problem or set (15)–(17), can be ex-pressed as

(18)

wheregiven nonzero vector;scalar parameter.

The equation (18) implicitly defines the function .If , a solution of (18) corresponds to the point where

. At , becomes a solution of the originalproblem . The dependence can be expressed asthe Taylor series

(19)

Formula (19), where the upper infinite sum limit is replaced bya given finite number , is used at each iteration of the method.The correction vectors are defined recurrently through thevalues of the mismatch function computed at certain pointsand is chosen within the range to get reliable conver-gence of (19) and good accuracy of the method [15]. The methodprovides many desirable features: reliable solution of nonlinearalgebraic problems up to points of singularity; convergence toa singular point if it occurs on the way of the iterative process;almost straight line motion of the iterative process in the spaceof mismatches; and retention of zero mismatches of (18).

To get an initial point on , the permanent loading tech-nique can be used [15], [17]. In this application,andare dependent variablesand load flow mismatch functions

respectively. For a specified direction in(15) and parameter which gives a point outside the fea-sibility region, the method follows this direction unless a sin-


Fig. 2. Application of the technique [15] to explore� .

gular point has been met (see Fig. 2, Step 1). Thesystem (15)–(17) is used after that to refine the initial estimateof and get proper initial values of and (seeFig. 2, Step 2). To get an initial guess of for Step 2, the factthat when can be effectively used [15].For further exploring the intersection of by a plane goingthrough the points and , the vector is rotatedin the plane through the angle increment(see Fig. 2, Step 3).Again, the high-order method is applied at the second stage. Asmismatches of (16) and (17) are initially zeros, they are pre-served close to zeros afterward and, due to this, the high-ordermethod follows the feasibility domain boundary. Note that theseries (19) taken as a function ofprovides an analytical ap-proximation of .

Equations similar to (15)–(17), i.e.,

(20)

(21)

(22)

where is now a vector of any two independent parameters, andthe predictor-corrector method [19] are used in [20] to explorethe power flow solution space boundary. The system (20)–(22)is presented as . The unit predictorvector that is tangent to the curve (20) is givenby —see Fig. 3. Predictions of the next point

are found as where is the step length.Corrections of a step are computed by simultaneous solutionof equations describing hyperplanes perpendicular to, that is,

and . The initial point on is found byapplication of the same predictor-corrector technique to solutionof the problem . The predictor-corrector techniqueis considered as a very reliable parameter continuation method[19]. Nevertheless, the above questions ii)–iv) remain relevant.

For the Hopf bifurcation boundary , the state matrixmust satisfy the following conditions [4], [16]:

(23)

In (23), and are the left and righteigenvectors corresponding to eigenvalue . By using

Fig. 3. The predictor-corrector procedure [19], [20].

(23) instead of (8), the following set can be proposed to get aHopf bifurcation point in the direction of :

(24)

(25)

(26)

(27)

(28)

From the dimensions for (1) and (2), the set (24)–(28) consistsof equations and depends upon the same numberof unknown variables , , , , and , where and arethe number of load flow and dynamic state variables, respec-tively. Both the Hopf and saddle node bi-furcation boundaries can be plotted by means of successive ro-tation of . Many difficulties arise regarding the solution of(24)–(28): absence of bifurcation points on the line(see previous comments to the system (15)–(17)) large dimen-sion, complicated topology of the bifurcation surface, and com-putational problems [16]. Among the computational issues, oneof the most important tasks is obtaining the initial guesses of

, , , , and on . One way is proposed in [16] for aspecial case whendoes not affect the load flow conditions,linearly appears in and the operating point lies insidethe stability domain ( , ). Then (24) iseliminated from the system andis a fixed vector, i.e., .These conditions can be satisfied for example when some gen-erator control gains (for example, the PSS gains) are taken as

[16]. Suppose we are interested in finding a point whereisclose to a given value . The value of can be chosen fromphysical considerations. For example, it should be taken from0.1 to 1 Hz for interarea oscillations. The desired point satisfiesthe condition

(29)

By variation of along the gradient which is com-puted quite easily (see [16]) it is possible to get a pointclose to solving (29). For the initial guesses ,

, , and being a solution of thelinear system (23), system (25)–(28) gives a point on. Fur-ther exploration of the bifurcation boundary is done by using thehigh-order method described above.


An alternative way for computing the Hopf bifurcations onthe ray has been proposed by the authors of [21] and[22]. It consists of the use of eigenvalue sensitivity factors to getthe zero real part for a certain oscillatory mode. The procedureimplies the following steps.

1) Set and .2) Take as a solution of the load flow problem

. Find the vectors and from.

3) Compute the state matrix using (5).4) Find the eigenvalue of interest for the

matrix , and corresponding eigenvectorsand. If is small enough, stop the procedure.

5) Evaluate the derivative at the current pointnumerically.

6) Compute the sensitivity factor

7) Find the increment of as and.

8) Go to Step 2 with .The procedure can be used to get the initial guesses forin-cluding the eigenvectors and. It allows choice of a particularoscillatory mode (for example, an interarea mode) to get its bi-furcation point. Difficulties may be caused by the nonlinear de-pendence and by tracing the selected mode between steps.

In Russia, approximations of the stability region boundariesare used for fast decision making in realistic centralized emer-gency control systems [23]–[25]. The general idea behind thesetechniques is to represent the stability domain by its approxima-tion in the form

(30)

where, , and experimental coefficients;

nodal power injections and power transfersthrough certain cut sets.

The coefficients , , and are evaluated by means of theleast square and regression methods [23], [24]. In other works,a set of simple equivalents (chain, ring, and radial networks) areused to approximate parts of the stability domain boundary [23],[25].

III. PROPERTIES OFQUADRATIC PROBLEMS

This section deals with some properties of quadratic load flowproblems, their solutions and singularities, which are used indevelopment of the proposed method.

Let the power flow problem be of the form

(31)

Hereafter is the vector of nodal voltages. If nodalvoltages are expressed in rectangular coordinates, thenis

a quadratic function of and the load flow Jacobian matrixis a linear function of . A detailed considera-

tion of the properties reported below can be found in [26] and[27]. Outline proofs are presented here in the Appendix.

Property 1: For any two points and ,the number and location of singularities of the quadraticproblem on the straight line through is definedby real eigenvalues of the matrix . These singularpoints on the line can be found as where

are computed as for all real eigenvaluesof the matrix .

Property 2: The maximum number of solutions of aquadratic equation on each straight line in thestate–space is two.

Property 3: For quadratic mismatch functions , a varia-tion of along a straight line through a pair of distinct solutionsof the problem results in variation of the mismatchvector along a straight line in .

Property 1 gives a basis for the proposed-plane method.Properties 2 and 3 are of auxiliary importance.

IV. THE -PLANE METHOD

Our task here is to develop a method for plotting an intersec-tion of the power flow feasibility domain boundary by a plane inthe space of dependent variables. It is known that the boundaryitself consists of points where .

A. Obtaining Bifurcation Curves on Plane in

A plane in can be defined by any three distinct points,, and provided the condition

(32)

is satisfied for any scalar parameter. Condition (32) preventsthe points , , and from belonging to a straight line in .Once (32) is true, the points , , and form a triangularwhich defines a plane in and gives the name for the proposedmethod. Any point on the plane can be expressed by meansof scalar parameters and (Fig. 4)

(33)

It is clear that , , and .Suppose that at the point , . Then the followingprocedure can be used to find out all singularities of in the

-plane. The idea is based on rotation of a vectorin the -plane and subsequent computations of all singularitiesof on each line defined by this vector (see pointsinFig. 4). Let be the angle which defines the current orientationof the vector with respect to the vectorin the -plane. The following steps are used.

1) Take the angle .2) Compute and , .3) Define a point as in (33).4) Find eigenvalues of the matrix .5) Compute for all real eigenvalues

from the previous step.


Fig. 4. The�-plane in .

6) For each value of , define the corresponding point inthe plane as

(34)

7) Change , where is an increment, and goto Step 2 unless .

The set of points computed for different forms a cutset of the feasibility domain boundary by theplane. The pro-cedure does not require an iterative solution, except as neededfor the eigenvalue problem. The reliable QR technique is rec-ommended for use in Step 4.

B. Visualization of the -Plane in

Although the bifurcation points in (34) belong to the-plane, they are vectors in the multidimensional space. To

get a visual representation for them, it is convenient to use a newtwo-dimensional (2-D) coordinate system associated with theplane itself. For this purpose, we use the following oblique-an-gled coordinate system (Fig. 5):

(35)

where

In the new 2-D coordinate system, the expression for computingthe power flow singular points becomes the following:

(36)

C. -Plane in

Consider a particular case when, , and are distinctsolutions of (31). It follows from Property 2 in Section III thatthese points can not lie on a straight line in. Property 3 saysthat, from , the straight lines

(37)

(38)

Fig. 5. Visual representation of the�-plane.

in are mapped into the straight lines

(39)

(40)

in , respectively. The last two lines pass through a commonpoint and they define a plane which we call theplanein (see plane in Fig. 6).

D. Visualization of the Plane in

From all singular points computed along the lines(37) and (38) are mapped into points of the lines (39) and (40),respectively. Thus, they lie on the plane in the space .The rest of the points mapped from the plane indo not normally belong to the plane in . The plane in

is mapped by into a surface which is not a plane(Fig. 6, surface ). The only thing which we can do hereis to find out their projections on the plane in the space

. The projection can be found by the following way.Let

(41)

where and are parameters and, are defined as

(42)

(43)

where . The coefficientsand can be found by solving the linear equation

(44)

They are then used in (41) to get singular points in the 2-Doblique-angled coordinate system .

As the singular points belong to a nonlinear surface (see thesurface in Fig. 6) and we use their projections on the

plane, the resulting plot reflects a qualitative shape of thesingular boundary in . Nevertheless, all singular points alongthe lines and are defined accurately.

V. RESULTS FOR THENEW ENGLAND TEST SYSTEM

The proposed method has been tested for the New EnglandTest System [28]. The system consists of 39 buses, 10 gener-


Fig. 6. The�-plane in .

TABLE ITHREE DISTINCT LOAD FLOW

SOLUTIONS

ators, and 18 loads. Bus number 31 is chosen as a slack bus.All other generators are represented by means of constant activepowers and terminal voltages. Loads have fixed active and reac-tive demands. We consider three distinct load flow solutions,

, and given in Table I. These solutions were selected from asolution set obtained by the method for computing multiple so-lutions of quadratic algebraic problems given in [26] and [27].All three points are low-voltage solutions. Solutions, , and

define a plane in , shown in Fig. 7. The plane is ori-ented in such a way that corresponds to zero point andthe vector is directed along the horizontal axis. Solution

is taken as a fixed point for the plot. All straight lines alongwhich the singularities are determined belong to theplane andpass through . The cut set of the feasibility domain boundary

Fig. 7. �-plane in (New England Test System).

Fig. 8. �-plane in (New England Test System).

by the plane is shown by dotted curves (solid parts of thecurves are formed by covering of close dots). A detailed consid-eration of the feasibility boundary topology is not the main pur-pose of this paper. Nevertheles, let us make some observations.First, note that the singular points marked by small circles lieexactly in the middle of the dashed lines connecting the points

, , and . This fact is explained in a number of works (see[26] and [27], for example). Second, the singular curves in Fig. 7have an open shape. This appears to say that, by certain unre-stricted variations of independent parametersin , voltages

in may be indefinitely increased without reaching the fea-sibility boundary. In more realistic situations the limitations for, for example, equality constraints for generator terminal volt-

ages and zero power injections in empty buses, must be takeninto account. Fig. 8 shows the correspondingplane in .Point 0 represents the operating condition ofthe system. The singular boundary is plotted by using the map

. Points and correspond to the pointsand in Fig. 7. All singular points which belong to the straightlines defined by – and – are given accurately. All otherpoints are obtained as projections of multidimensional singularcurves on the plane in .

VI. CONCLUSION

Several approaches have been developed during the last 15years to explore the power system load flow feasibility, saddlenode and Hopf bifurcation boundaries. The main idea behindmost of them is to use matrix eigenvalue conditions derived from


the load flow or dynamic state matrices where the real parts ofeigenvalues are put to zero, and imaginary parts are either zero(for the load flow feasibility and saddle node bifurcation bound-aries) or not (for the Hopf bifurcations). These methods are re-viewed, and some problems which need further investigation areconsidered. A number of techniques have been proposed to findan initial point on the bifurcation boundary. As the reviewedmethods require iterative solutions of nonlinear equations, thequestion of their reliability arises. A new technique which hasbeen developed in this paper does not require iterative solutionsof nonlinear equation sets. For quadratic problems, it producesall load flow feasibility boundary points on a straight line inthe space of dependent variables (nodal voltages given in rect-angular form) by solution of an eigenvalue problem. This ro-bust and accurate method is useful for visualization and topo-logical studies of the multiple solution and feasibility domainstructures.

APPENDIX

A. Proof of Property 1

Define the line through as

(45)

For a quadratic function , its Jacobian matrix consistsof elements which are linear functions of, i.e.,

(46)

where , are constant matrices of Jacobian coef-ficients and is th element of .

Using (46), it is easy to show that

(47)

As and , equality (47) can be written asfollows:

(48)

Therefore, is equal to zero if and only if

(49)

where . It is clear that all singular points on theline (45) can be computed in terms of real eigenvaluesof the matrix

(50)

B. Proof of Property 2

Take the function

(51)

For a quadratic mismatch function

(52)

where is a quadratic term of expansion (52). So

Function equals to zero if and only if .At a solution point , and function (51) is

For any fixed direction , equals to zero in the twofollowing cases: a) and b) . The first casegives us the original solution point . The second casemay correspond to solutions on the straight line directedby . However, as is clear from (51) the function cannotbe negative. Thus, and in the case b) it ispossible to have only one additional solution except, but nottwo or more. So, on the line we get one original root andwe can have only one additional root corresponding to conditionb).

C. Proof of Property 3

Let be a point on the straight line connecting two distinctsolutions ,

(53)

where is a parameter and . For quadraticmismatch functions

(54)

(55)

where is the quadratic term of the Taylor seriesexpansion (55). At points , , we have .So, from (55)

(56)

Using (56), (54) transforms to

(57)


where , . Thus, the functionvaries along the straight line in .

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[18] I. Dobson, “Observations on the geometry of saddle node bifurcationand voltage collapse in electrical power systems,”IEEE Trans. CircuitsSyst. I, vol. 39, pp. 240–243, Mar. 1992.

[19] G. B. Price, “A generalized circle diagram approach for global analysisof transmission system performance,”IEEE Trans. Power Appl. Syst.,vol. PAS-103, pp. 2881–2890, Oct. 1984.

[20] I. A. Hiskens and R. J. Davy, “A technique for exploring the powerflow solution space boundary,” inProc. Int. Symp. Electric Power En-gineering Stockholm Power Tech., Stockholm, Sweden, June 1995, pp.478–483.

[21] I. A. Gruzdev, E. L. Toroptzev, and S. M. Ustinov, “Optimization ofautomatic control devices tuning for a variety of power system operatingconditions” (in Russian),Elektrichestvo, no. 4, pp. 11–15, 1986.

[22] I. A. Gruzdev, V. A. Maslennikov, and S. M. Ustinov, “Development ofmethods and software for analysis of steady-state stability and dampingof bulk power systems” (in Russian), inIssue: Methods and Softwarefor Power System Oscillatory Stability Computations. St. Petersburg,Russia: Federation of Power and Electro-Technical Societies, 1992, pp.66–88.

[23] S. A. Sovalov and V. A. Semenov,Emergency Control in Power Systems(in Russian). Moscow, Russia: Energoatomizdat, 1988.

[24] “Questions of the power system emergency control” (in Russian), inEnergosetproekt Proceedings, Moscow, Russia, 1982.

[25] “Emergency control and governing in power systems” (in Russian), inProc. NIIPT, Leningrad, Russia, 1982.

[26] Y. V. Makarov, A. M. Kontorovich, D. J. Hill, and I. A. Hiskens, “Solu-tion characteristics of quadratic power flow problems,” inProc. PowerSystem Computation Conf., Dresden, Germany, Aug. 19–23, 1996, pp.460–467.

[27] Y. V. Makarov, D. J. Hill, and I. A. Hiskens, “Properties of quadraticequations and their applications to power system analysis,” Int. J. Elec-trical Power Energy Syst., to be published.

[28] “EPRI EL-726, RP 744–1, Interim Rep.,”, Palo Alto, CA, Apr. 1978.

Yuri V. Makarov (M’97–SM’99) received the M.Sc.degree in computers and the Ph.D. degree in electricalengineering from the Leningrad Polytechnic Institute(now St. Petersburg State Technical University), St.Petersbury, Russia, in 1979 and 1984, respectively.

From 1990 to 1997, he was an Associate Professorat the Department of Electrical Power Systems andNetworks at St. Petersburg State Technical Univer-sity. From 1993 to 1998, he had conducted his re-search at the University of Newcastle, the Universityof Sydney, Australia, and Howard University, USA.

He is currently with the Transmission Planning Department, Southern CompanyServices, Inc., Birmingham, AL. His current research activities are in the areasof various theoretical and applied aspects of power system analysis, planning,and control.

David J. Hill (M’76–SM’91–F’93) currently holdsthe Chair in electrical engineering at the Universityof Sydney. His research interests are mainly innonlinear systems and control, stability theory, andpower system dynamics and security.

Dr. Hill is a Fellow of the Institution of Engineers,Australia and a Foreign Member of the RoyalSwedish Academy of Engineering Sciences.

Zhao-Yang Dong(M’98) received the Ph.D. degreefrom the University of Sydney, Sydney, Australia, in1999.

He is continuing his research at the School of Elec-trical and Information Engineering, the University ofSydney. His research interests include power systemanalysis and control, artificial intelligence and its ap-plication in power engineering, power system expan-sion planning, and security.

Computation of bifurcation boundaries for power systems: a new Δ-plane method

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