Estabilidad Aroldi Método PS Feb 07

324 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 1, FEBRUARY 2007

Critical Eigenvalues Tracing for Power SystemAnalysis via Continuation of Invariant Subspaces

and Projected Arnoldi MethodDan Yang, Student Member, IEEE, and Venkataramana Ajjarapu, Senior Member, IEEE

AbstractThe continuation of the invariant subspaces algo-rithm combined with the projected Arnoldi method is presented totrace the critical (right-most or least damping ratio) eigenvaluesfor the power system stability analysis. A predictor-correctormethod is applied to calculate the critical eigenvalues trajectoriesas power system parameters change. The method can handleeigenvalues with any multiplicity and very close eigenvaluesduring the tracing processes. The subspace dimension inflationand deflation is applied to deal with eigenvalue overlap. The sub-space update is proposed by an efficient projected Arnoldi methodthat can utilize known information from the traced eigenvaluesand subspaces.

Index TermsArnoldi method, continuation of invariant sub-spaces, critical eigenvalues, eigenvalue tracing, power systems,stability.

I. INTRODUCTION

E IGENVALUES play an important role in power systemstability analysis. Eigenvalues can be used to determinethe small signal stability of the power systems. They are alsoa good indication for the bifurcation points such as saddle-nodebifurcation and Hopf bifurcation. Eigenvalues with the largestreal parts and eigenvalues with the least damping ratios are ofgreat importance for power system applications.

From a mathematical point of view, eigenvalue calculationsare special root-finding problems where the polynomial func-tion is for matrix A. There is no direct methodto find the eigenvalues from square matrices whose dimensionis greater than 5 because no analytical formula exists for the rootof a general polynomial function. Thus, all the existing eigen-value algorithms are essentially iterative methods. Among allthe eigenvalue solver methods for general nonsymmetrical ma-trices, the QR algorithm is regarded as the most efficient methodup to now. However, in most of the applications, there is noneed to calculate the whole set of the eigenvalues. In general,the group of the critical eigenvalues is usually a small subset inthe whole spectrum. Several dominant subspectrum algorithmsexist to serve this purpose such as the power method, subspaceiteration, and the Arnoldi method [1], [2].

Manuscript received October 25, 2005; revised April 29, 2006. Paper no.TPWRS-00673-2005.

D. Yang is with the Department of Market Monitoring, California Indepen-dent System Operator, Folsom, CA 95630 USA (e-mail: [email protected]).

V. Ajjarapu is with the Department of Electrical and Computer Engineering,Iowa State University, Ames, IA 50011 USA (e-mail: [email protected]).

Color versions of Figs. 58 are available online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRS.2006.887966

Among all the dominant eigenvalue algorithms, the Arnoldimethod is believed to be the most efficient approach. To obtaingood convergence properties, the Arnoldi method is heavily in-fluenced by the selection of the number of guard vectors [3]. Apractical question is how to decide the number of guard vectorsso as to have the best computational efficiency. The only way toexplore this is by numerical tests [4]. Furthermore, it is pointedout in [5] that very close eigenvalues or eigenvalues with multi-plicity may cause algorithms to terminate prematurely.

As eigenvalues are closely related to power system stability,eigenvalue calculation is a very active area in the power systemliterature. The efficiency of dominant eigenvalue calculationmethods are discussed and compared in [3] and [6]. The matrixtransformation for the dominant eigenvalues is described in[7]. The algorithm to efficiently calculate dominant poles oftransfer functions is proposed in [8] and [9]. The methods forsmall-signal and oscillatory stability boundary are proposed in[10] and [11]. The sparse eigenvalue techniques are shown in[4]. In the power system application, eigenvalues are widelyused to study system uncertainty, estimate stability boundary,and enhance system stability performance. The robust analysisof system uncertainty effect on eigenvalues is presented in [12].Eigenvalues are used to estimate voltage stability sensitivity in[13]. Eigenvalue-based control is applied in [14] to enhancesmall signal stability transfer capability. Parameter tuning andcontroller design techniques are shown in [15] and [16] by usingeigen-analysis. In [17], saddle-node-related eigen-analysis isemployed to avoid voltage collapse.

The critical eigenvalues are calculated for a specified powersystem operating state in normal applications. As the operatingstates often vary with respect to control and load changes,critical eigenvalues may also change. For the cases where theeigenvalue movements are of interest (one of the potentialapplications is the coordinated control in a large operatingrange), eigenvalue trajectories can be obtained from continu-ation methods. The continuation methods can be categorizedas eigenvalue derivative-based methods and invariant sub-space-based methods.

The derivative-based continuation algorithms are proposed in[18] to update the spectrum of a continuously parameterizedfamily of sparse matrices. Numerical integration is used to com-pute the eigenvalues through eigenvalue derivatives. The deriva-tive-based method is applied to power system oscillatory sta-bility analysis in [19].

The other continuation category is based on the concept ofinvariant subspaces. The column vectors in Q matrix of Schur

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YANG AND AJJARAPU: CRITICAL EIGENVALUES TRACING FOR POWER SYSTEM ANALYSIS 325

decomposition provide the basis of the eigen-spacewhere a subset of column vectors in Q form an invariant sub-space for A. As the matrix A changes with respect to the param-eter, the corresponding invariant subspace also changes and canbe used to calculate the eigenvalues. The simplest case of theinvariant subspace is the single eigen-pair whose dimension is1. Several homotopy methods for single eigen-pair problem arepresented in [20][22].

A generalized continuation method for the calculation of theinvariant subspace with any dimension is presented in [23]. Itapplies the Newton method to the associated Riccati equationand solves a Sylvester equation in each step. The method wasapplied in [24][26].

Another scheme for the continuation of invariant subspace isproposed in [27] and [28]. Compared with [23], it improves thecomputational efficiency by solving a Sylvester equation with aBartelsStewart algorithm.

One advantage of the invariant subspace method over thederivative-based method is the treatment of eigenvalues withmultiplicity and very close eigenvalues. In the cases of mul-tiple eigenvalues, or at the point of double real eigenvalues split-ting into a pair of complex eigenvalues, the derivatives of theeigenvalues and eigenvectors vanish. In addition, when com-puting the derivatives numerically, a simple eigenvalue that isclose to others may behave like a defective one [29]. The eigen-vectors may be nearly linearly dependent, making some numer-ical schemes for the computation of derivatives badly behaved[30]. When a cluster of eigenvalues occurs, it is better to con-sider these nearby eigenvalues together by the invariant sub-space method.

In this paper, the continuation of the invariant subspacemethod in combination with the projected Arnoldi method isproposed to trace the critical eigenvalue set in power systemstability analysis. The accompanying projected Arnoldi methodcan efficiently update new critical eigenvalues without recalcu-lating the traced eigenvalues by utilizing the traced subspaceinformation.

The remaining parts of this paper are organized as follows.Section II demonstrates the algorithm for the continuation ofinvariant subspaces, and the projected Arnoldi method is givenin Section III. The method is demonstrated via the New England39-bus system and the IEEE 118-bus system in Section IV. Theconclusions are presented in Section V.

II. CONTINUATION OF INVARIANT SUBSPACE

A. Eigenvalues and Invariant SubspacesEquation (1) represents equilibrium of nonlinear dynamical

systems

(1)

where is a smooth nonlinear functiondepending on a real parameter. All of the eigenvalues of theJacobian matrix should have negativereal parts to maintain the stability.

The dimension of the above system corresponds to thenumber of the system variables. Usually only a small number of

eigenvalues may approach near or close to the imaginary axisthat is connected with a low-dimensional invariant subspace.The idea of the dominant eigenvalue calculation algorithms is tofind out the dominant subspace and thus dominant eigenvalues.For example, the Arnoldi method is based on a reduction tech-nique in which a large matrix is reduced to an upper Hessenbergform. In general, spectral decomposition can be represented bySchur decomposition, which has the form

(2)

In (2), is the original matrix, is anmatrix, and is an matrix. The column vectors in

are the basis in the invariant subspace, and the eigenvaluesof are a small portion of the corresponding eigenvalues inthe full matrix .

At a particular operating point, there exists a dominant sub-space associated with dominant eigenvalues. When the oper-ating point changes with respect to the parameter, the dominantinvariant subspace also changes. The new subspace and eigen-values at the new operating points can be updated directly fromthe previous results. In such case, there exists a family of ma-trices smoothly parameterized by satisfying (2). In (2), matrixelements of and are unknown. There are

unknowns, but only equations exist. Hence, additionalequations must be added to remove the under-determina-

tion. Let be an orthonormal matrix. Then, where is the m-dimension identity matrix, could be a

choice of the additional conditions. However, the usual normal-ization may not be differentiable [21]. Thus, it is generally pre-ferred to consider a linearized constraint

(3)

where is a fixed matrix with rank .Now the equation set for the invariant subspace is

(4)

The equation set can be solved by the predictor-corrector tech-niques along the parameter path.

B. Predictor-Corrector StepThe predictor-corrector methods are applied to (4) in the con-

tinuation method. Let and be the so-lutions of (4) at , and the tangent . Bydifferentiating (4), the following linear system can be derived:

(5)

After solving (5), the predicted values are given as

(6)

In the corrector step, (4) is solved via Newtons methodwith as the initial guess. The step


approximation in the Newton iteration is obtained from stepby solving

(7)

The unknowns in (5) and (7) are matrices. The equations canbe further expanded into standard forms with vector unknowns,and the total unknown elements are . Thus,the dimension of the Jacobian matrix is also ,which is roughly times the size of the system states. It isinefficient to solve such a large-dimensional system directly. Asan alternative, the system can be decomposed into several smallsubsystems to improve the efficiency.

C. Solve Sylvester Equation by BartelsStewart AlgorithmEquations (5) and (7) in the predictor-corrector step require

solving the similar systems

(8)

which is the so-called bordered Sylvester equation in which un-knowns , , , and are the right-handside of (5) and (7).

The BartelsStewart algorithm is a sequential approach thatcan be applied to solve the bordered Sylvester equation effec-tively. It consists of orthogonal transformation of to an uppertriangular matrix . For any matrix , it can be done by com-plex Schur decomposition

Since is an dimension matrix where is small, theSchur decomposition can be done efficiently.

By right-multiplying Q to both sides of (8), we get

(9)

Since , we can get . Define newvariables as

Equation (9) can be rewritten as

(10)

Since is an upper triangular matrix, the equations can besolved in column wise and the column vectors and of

and can be computed sequentially. At each step, the linearequation set is

(11)After solving (11) sequentially for , the orig-

inal matrix and can be reversely computed, and the eigen-

values in the subspace can be easily obtained from small squarematrix .

As in each step of the BartelsStewart algorithm, a relativelysmall system whose dimension is only is solved. There-fore, the predictor and corrector equations can be calculated rel-atively efficiently.

The eigenvector of the matrix A is given as , where isthe eigenvector of . Let , be the eigenvalue and eigenvectorof ; thus, , and . Finally,

, and denote ; thus, .

D. Continuation of Invariant Subspaces for DifferentialAlgebraic Equations

Power system dynamic analysis includes both differentialequations and algebraic equations. The differential algebraicsystem generally has the form

(12)

The total system Jacobian matrix is

The equivalent system matrix is whoseeigenvalues can determine the system stability.

For DAE systems, the extended eigenvector can be definedas to preserve the sparsity, and the equationsbecome

(13)

The continuation algorithm can be directly applied to thewhole DAE system.

Denote , and the invariant subspace equationbecomes

(14)

The predictor equation is

(15)

The corrector equation is

(16)

Both (15) and (16) can be solved by the BartelsStewart al-gorithm effectively.

III. SUBSPACE UPDATE BY PROJECTED ARNOLDI METHODIn the previous section, the continuation procedure is de-

scribed to trace a subspace of the eigenvalues that satisfy a


given criterion. In this section, a projected Arnoldi method isused to combine with the continuation method for efficientupdate. The subspace update is required because there maybe cases such as subspace inflation and deflation during thetracing process. In the case of subspace deflation, some of thetraced eigenvalues may be outside the critical region and haveto be removed from the set. In the case of subspace inflation,some new eigenvalues that previously are not in the subspacemay enter into the critical region of interest. In the first case,the subspace is deflated and the dimension of the subspace isreduced, while in the second case, the subspace is inflated toinclude the new ones.

Since the eigenvalue movement is nonlinear, it is generallydifficult to predict when some new eigenvalues would enter thecritical region. Thus, the subspace dimension update is accom-plished by recalculating the critical eigenvalues periodically forthe check. The problem with recalculation is that since the setof the traced eigenvalues is already known, it is desirable not torecalculate these eigenvalues again to save computational time.The conventional methods do not utilize the known informationabout the traced eigenvalues, and the remedy proposed here is aprojected Arnoldi method. The projected Arnoldi method aimsto find out only the critical eigenvalues outside the traced sub-spectrum.

In the following section, the traditional method to identify therightmost eigenvalues is given first, and the eigenvalues withleast damping ratio can be calculated with slight modification;then the projected Arnoldi method is shown to directly identifythe remaining critical eigenvalues outside the existing ones.

A. Arnoldi Method and Cayley Transform1) Arnoldi Method: The Arnoldi method is an orthogonal

projection method onto Krylov subspace for general matrices.The method was introduced in 1951, and it was later discoveredthat it leads to an efficient technique for approximating eigen-values. Several variants of the Arnoldi method exist to improveconvergence properties such as the deflated Arnoldi method andthe implicitly restarted Arnoldi method [1][3].

Basic steps involved in the Arnoldi method are given in thefollowing.

Start: Choose an initial vector with unit norm.

Iterate: for compute

.

The Arnoldi method converges when .2) Cayley Transform for the Eigenvalues With Largest Real

Part: The results of the basic Arnoldi method are the eigen-values with the largest moduli. To obtain the rightmost eigen-values, it is necessary to map the rightmost eigenvalues of A tothe largest moduli eigenvalues of a transformed matrix. Cayley

Fig. 1. Cayley transformation for rightmost eigenvalues.

Fig. 2. Cayley transformation for least damping ratio eigenvalues.

transform [3], [31] is a general tool to achieve the transforma-tion. Cayley transform is defined as

It is shown in Fig. 1 that the half plane to the right ofis mapped outside the unit circle, and the farthest eigen-

value of the matrix corresponds to the rightmost eigen-value of A.

3) Cayley Transform for the Eigenvalues With LeastDamping Ratio: The standard Cayley transform only mapsthe half plane outside a unit circle. As a result, it will providethe eigenvalues parallel close to the imaginary axis. In powersystem analysis, it is also important to find out the eigenvalueswith least damping ratio. This can be done via some modifiedforms of Cayley transform such as semi-complex or coordi-nation transforms. Since the eigenvalues are symmetrical withrespect to the real axis for a real matrix, only the eigenvalueswith positive imaginary parts are considered. Suppose all theeigenvalues whose damping ratio is less than a given number Dare needed. To map the eigenvalues in the left shaded area intothe outside of a unit circle in the right diagram in Fig. 2, it ispreferred to do the coordinate transform first. By rotating thereal and imaginary axis (90 -DampingAngle) in counterclock-wise direction, the eigenvalues with least damping ratio will


become the eigenvalues with largest real part. Then under thenew coordinates, the standard Cayley transform can be appliedto find out the eigenvalues outside unit circle, which are exactlythe eigenvalues whose damping ratios are less than D.

The Cayley transform for the least damping ratio eigenvaluesis defined as

DampingAngle

B. Projected Arnoldi Method With Cayley TransformDuring the continuation process, the eigenvalues in the in-

variant subspace spanned by are already known. To avoid re-calculating these eigenvalues, the projected Arnoldi method isproposed here. Let be the orthonormal basis of the subspacespanned by , and the projector can cut off the com-ponents in the invariant subspace and only keep the remainingeigenvalues. The following proposition gives the connection be-tween original matrix A and projected matrix .

Proposition: Suppose an real matrix has nor-malized linearly independent eigenvectors withrespect to eigenvalues . Let bethe orthonormal basis, andfor ; then, are all the eigen-values for the matrix , and the linearly in-dependent eigenvectors corresponding to the zero eigenvaluesmay be chosen as .

The proof is given in the Appendix.Basic steps involved in the projected Arnoldi method are

given in the following.1) Start: Choose an initial vector with unit norm.2) Iterate: for compute

.

The only difference between the projected Arnoldi methodand the basic Arnoldi method is that a precondition matrix ismultiplied to the original matrix. The orthonormal basis in theprecondition matrix may be calculated by the GramSchmidtmethod or by the modified GramSchmidt method [2] and othersophisticated variants [32] for improved robustness. Since mostof the procedure remains unchanged, it is easy to implement thenew algorithm.

Fig. 3 shows the concept of the projected Arnoldi method.By mapping the known eigenvalues into zero, only a few newcritical eigenvalues are left in the critical region. It is pointedout in [6] that the sequential approach is usually used to buildup all the critical eigenvalues. In such cases, the Arnoldi methodwill have to recalculate all the existing eigenvalues again to listall the critical eigenvalues (see the left plot of Fig. 3), while theprojected Arnoldi method only needs to calculate the new ones(see the right plot of Fig. 3). Therefore, the projected Arnoldi

Fig. 3. Projected Arnoldi method.

method is more efficient to update the critical subspace than theArnoldi method.

In the projected Arnoldi method, the eigenvalues in the crit-ical region are guaranteed to be the new entrants. The parame-ters may be chosen as

criticalvalue criticalvalue

where is a positive number. These values are substituted in theprojected Arnoldi iterative algorithm method to test the new en-trants. Once the output of the projected Arnoldi method locatesan eigenvalue outside the unit circle, this eigenvalue should beincluded in the critical set to inflate the subspace.

By combining the continuation algorithm and the projectedArnoldi algorithm, the methodology can identify and efficientlyupdate all the critical eigenvalue trajectories with smoothsystem change. A flowchart of the proposed method is given inFig. 4 to show the general procedure.

In certain power system applications, if the system changescannot be described by a continuous function, such as the casesof contingencies or controllers hitting limits, modifications areneeded to apply the proposed method. If the system changes arediscrete such as contingency, the discrete change can be mod-eled as the results from a continuous function. For example, aline with admittance has two statuses: 0 for line connectionand 1 for line trip. Line admittance can be modeled as a contin-uous function , where is the changing param-eter. When , the line admittance is . When , theline admittance is 0, that is, line trip occurs. By changing param-eter gradually from 0 to 1, discrete change can be smoothlymodeled, and eigenvalue movement can also be traced. Com-pared with contingency, controllers limit hitting is associatedwith much more complex physical phenomenon. For example,before the generator excitation voltage limit is hit, the excitationvoltage is governed by the dynamic equation. However, once thevoltage reaches the upper or lower limit, the differential equa-tion will become an algebraic equation. In such case, the numberof system differential states will be reduced, and so does thenumber of eigenvalues. The change in structure of the differen-tial algebraic system may not only change eigenvalue numberbut also suddenly cause eigenvalue jump, as shown in [33]. Todeal with the structure change associated with limit hitting, the


Fig. 4. Flowchart for critical eigenvalue tracing.

tracing method needs to restart to path-follow the new systemeigenvalue trajectories. In such case, piecewise continuous tra-jectories are given instead of smoothly continuous trajectories.

IV. NUMERICAL EXPERIMENTSThe continuation of the invariant subspace method and the

projected Arnoldi method are applied to the New England

Fig. 5. Least damping ratio eigenvalues movement.

39-bus system and the IEEE 118-bus system in MATLABenvironment. The New England system has 39 buses and tengenerators, and the IEEE 118 system has 118 buses and 48generators. The generators are represented by the two-axismodel as in [34], and exciter and governor models are the sameas in [35]. There are nine states for each generator. Both thedamping ratio and the real part of the eigenvalues are used asthe criterion to choose critical eigenvalues in the numericalexperiments. To emphasize the basic idea of smooth parameterchange, the control limits are not considered in the numericalexperiments.

A. New England 39-Bus SystemThe total number of differential states and algebraic states of

the system are 89 and 78, respectively. The system real loadis chosen as the control parameter. The real load level at theinitial state is 614 MW. In the first example, all the eigenvalueswith damping ratio less than 10% are considered as the criticaleigenvalues, while in the second example, all the eigenvalueswith real part greater than 0.15 are taken into account. Theload level is increased by 20% each iteration, and the subspaceupdate is checked at approximate intervals of 1000 MW. Thetracing process will end when one of the eigenvalues crossesthe imaginary axis.

1) Tracing of Eigenvalues With Least Damping Ratio: Theeigenvalues with least damping ratios are traced by the invariantsubspace continuation algorithm in this case. Since eigenvaluesare symmetrical to the real axis, only the eigenvalues with posi-tive imaginary parts are counted. Damping ratios of all the crit-ical eigenvalues with respect to load variation are plotted inFig. 5. Initially, six eigenvalues are located in the critical re-gion, and the number of critical eigenvalues is increased to nineduring the tracing process.

To compare the computational accuracy, Table I shows oneeigenvalue at different load levels. The MATLAB estimatedeigenvalue is calculated by MATLAB function, while the tracedeigenvalue is from the continuation algorithm. The results showthat the invariant subspace continuation algorithm providesvery accurate results for the power system analysis.


TABLE ICOMPARISON BETWEEN TRACED AND MATLAB ESTIMATED EIGENVALUES

Fig. 6. Right-most eigenvalues movement.

2) Tracing of Eigenvalues With Largest Real Part: Fig. 6shows the result by the invariant subspace continuation algo-rithm for all the eigenvalues satisfying real part criterion. In thefigure, solid lines represent real eigenvalues, and dashed linesrepresent complex eigenvalues. Initially, only nine eigenvaluesare located in the critical region. Similarly to the least dampingratio case, four more eigenvalues enter the critical region as loadincreases. The final number is 13 at the end of the process.

At the initial state, there are nine eigenvalues in the critical re-gion (seven real eigenvalues and two complex eigenvalues). InFig. 6, only six real eigenvalues are clearly shown. The reasonis that two real eigenvalues are very close to each other, and it isdifficult to differentiate them from the figure. Table II gives thedetailed information about the two eigenvalues. The small dif-ference between these two eigenvalues may be due to roundingerror, and these two eigenvalues may correspond to a pair ofdouble eigenvalues of the system. The results show that the ex-istence of these two eigenvalues does not affect the convergenceof the continuation algorithm.

In the performance comparison, it is found that in the NewEngland cases, computational time of the tracing method issimilar or slightly more than direct calculation by the Arnoldimethod. The simulation performance of the tracing methodand the direct calculation method is shown in Table III. In the

TABLE IICLOSE EIGENVALUES TRACE

TABLE IIINEW ENGLAND SIMULATION PERFORMANCE

results, the average computational time at the different loadlevel is given for both methods.

B. IEEE 118-Bus SystemThe total numbers of differential states and algebraic states of

the system are 431 and 236, respectively, in apart from referencegenerator angle. The system real load is chosen as the controlparameter. The real load level at the initial state is 3665 MW.In the first example, all the eigenvalues with damping ratio lessthan 5% are considered as the critical eigenvalues, while in thesecond example, all the eigenvalues with real part greater than

0.05 are taken into account. The load level is increased by 4%each iteration, and the subspace update is checked at approxi-mate intervals of 1000 MW.

1) Tracing of Eigenvalues With Least Damping Ratio:Damping ratios of all the critical eigenvalues with respect toload variation are plotted in Fig. 7. Initially, four eigenvaluesare located in the critical region. The subspace is updated atthe load level of 4730 MW where a new critical eigenvalue isidentified. As load level increases further, this new eigenvaluequickly crosses the imaginary axis.

2) Tracing of Eigenvalues With Largest Real Part: Fig. 8shows the result by the invariant subspaces continuation al-gorithm for all the eigenvalues satisfying real part criterion,where solid lines represent real eigenvalues and dashed linesrepresent complex eigenvalues. Initially, the invariant subspacecorresponds to five real eigenvalues without much movement.A pair of new complex eigenvalues enters the critical regionas load increases. In the Fig. 8, only two real eigenvalues areclearly shown. The data show that there are four eigenvaluesnear 0.0184. Table IV gives the detailed information aboutthese four eigenvalues.

In the performance comparison between the tracing methodand the direct method, it is found that in the IEEE 118-bus cases,the tracing method takes more time than the direct calculationmethod in one case, while in the other case, the tracing methodneeds significantly less time than the direct calculation method.


Fig. 7. Least damping ratio eigenvalues movement.

Fig. 8. Right-most eigenvalues movement.

TABLE IVCLOSE EIGENVALUES AT INITIAL LOAD

The computational performance of the tracing method does notchange much for different cases, while direct calculation by theArnoldi method demonstrates both fast and slow performance.The computation performance for the 118-bus system is shownin Table V.

V. CONCLUSIONSA new method for tracing the critical eigenvalues for power

system analysis is given in this paper. The procedure combinesthe continuation of the invariant subspace method and the pro-jected Arnoldi method. The continuation of the invariant sub-

TABLE VIEEE 118-BUS SIMULATION PERFORMANCE

space method uses the Newton method to trace selected eigen-values. It can also handle eigenvalues with any multiplicity andvery close eigenvalues. The projected Arnoldi method aims toefficiently update the subspace dimension by only calculatingnew critical eigenvalues. By combining these two methods, ageneralized framework is proposed to efficiently trace all thecritical eigenvalues.

APPENDIXProof of the Proposition: From the GramSchmidt process,

we know that , where is an uppertriangular matrix with . Furthermore

wherewhen ;otherwise.

So , where ,; , .

Since

then we see that is an upper triangular matrix whose diag-onal elements are . So the eigenvalues of

are , which are also the eigenvalues of.

Furthermore, , , whereis the upper left principal minor of . Letting

, we have

So the matrix has zero eigenvalues whosecorresponding linearly independent eigenvectors may be chosenas .

ACKNOWLEDGMENTThe authors would like to thank T. Jin with the Department

of Mathematics at the University of Rochester for the help andvaluable discussions.


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Dan Yang (S03) received the B.S. degree in electrical engineering from WuhanUniversity, Wuhan, China, in 1999 and the M.S. degree in electrical engineeringfrom Tsinghua University, Beijing, China, in 2002. He received the Ph.D. de-gree in electrical engineering and the M.S. degree in economics with a minor inapplied mathematics from Iowa State University, Ames, in 2006.

Currently, he is with the Department of Market Monitoring at the CaliforniaIndependent System Operator. His research interests include mathematical op-timization, dynamical systems, power systems, and financial economics.

Venkataramana Ajjarapu (S86M86SM91) received the Ph.D. degree inelectrical engineering from the University of Waterloo, Waterloo, ON, Canada,in 1986.

Currently, he is a Professor in the Department of Electrical and Computer En-gineering at Iowa State University, Ames. His research is in the area of reactivepower planning, voltage stability analysis, and nonlinear voltage phenomena.

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