Assessing Eigen Value Sensitivities

7/30/2019 Assessing Eigen Value Sensitivities

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Asses sing Eigenvalue SensitivitiesE v a n d r o E. Souza Lima, Membec IEEE, and Luis Filomeno de Jesus Femandes

Abstract-The motivation for this pape r is the fact that inpractice, the parameters of a p o w r system are only approximatelyknown. The pa per d iscusses the sensitivityof eigenvalues n termsof state matrix entry changes (model uncertainty) and param eterchanges (parameter uncertainty). The cnnccpts of asymptoticstability robustness for model and para me ter tincertainty arc pre-sented. b o ndexes derived from eigenvahie sensitivity matricesare suggested to measure eigenvalne sensitivity and asymptoticsma ll-signa l stability roliustness. As an e xam ple, the sensitivities ofthc clectrome clianical modes of a 9 -machine system arc analyze d.The results show lack of asymptotic stability robustness for bo thmodel and param eter uiicertainties. Th e paper shows that lack ofasymptotic stability robustness is a tre nd of actual multimachinepower systems because too small stability margins and largeeigenvalue sensitivitiesoccur simultaneously.Index Terms-&igenanalysis, Ill-Conditioned Problem, Modal

Analysis, Robustness, S ensitivity Function, Small-Signal Stability.1. INTRODUCTION

N PRACTICE, full information on the parameters of an en-I .ire system may be dilficult, if not impos sible, to obtain [ I I.Only approximations to the actual system, machine, and reg-ulator parameters are known, and they may vary during thesystem operation [Z]. Therefore, there arc always differencesbetween the model on which the design i s based and the ac-tual power system. The se differences are usually referred to asmodel u n c e r tu i n t i e s 131.

For small-signal stability stndies, non-linear equations de-scribing actual powcr systems are linearized around operatingpoints, and plant models of first order differential equations areimplicitly or explicitly established as

Since it is assumed that (I ) depicts the esscnc c of the actualplant dynamics, the 7 1 x n matrix A is called system or statematrix. If A has distinct eigenvalnes, X(A) = {XI , . ., A n } ,the n x I state vector z(1) i s given by

z(t) = C b U k ~ =c ; r : (0 ) l )khh*l (2 )k = 1 b = l

where Ch are c onstants that depend on the initial conditions, v gare the right column eigenvectors, yf are the transposed con-jugated left eigeuvectors corresponding to A y , an d z (0) s theinitial state vector [4], .51. Thus, significant efforts [4], 161-1 101have bccn made for the calculation of the eigenvalues; particu-larly those associated with electromechanical mode s.

Manuscript receivul Augus t 6, 19%.The a ~ l l i ~ r src with the Ueptn de Hog. Elelrim, UnB. CEP. 70910-900,Publisher llcm ldcnlilicr S 0885-8~50(00)01887-3.Brasilia, DF, Brazil (c-mail: e . l imaLasi .cnm.br) .

The problem of the reliability of eigcnvalue estimation beeneventually disciisscd 141,161,hu t no systematic study eigenvaluesensitivities bad been proposed until 15.1.That work has demon-strated through an example that some eigenvalues, associatedwith voltage oscillations of a single machine with an exc itationsystem connected to an infinite 11 11, have very large sensitivi-ties in relation to so me entries of the state matrix.

This pape r extends the eigenvalue sensitivity analysis given in[5] and points out that the main cause f or eigenvalue misestima-tion is their sensitivities in relation to data. Mos t of small-signalstability studies are not based on mea sured models but on cal-culated models, i.e., the state matrix entries n i j are obtainedthrough algebraic expressions relating parameter values. Be-cause of this, a distinction between eigenvalue sensitivity inrelation to any u i j an d to any system parameter was made atthe beginning of the paper. Two indexes derived fxom the loga-rithmic sensitivity matrix (LS-m atrix, [ 5 ] ) re suggested to mea-sure eigenvalue sensitivity for model uncertainty and to assessasymptotic small-signal stability robustness.

The inethodology is illustrated by the study of electro-mechanical mode sensitivities of a nine-machine systemstabilized by a pole-placement PSS design 1121. Large sensitiv-ities to model a nd to parameter uncertainties are shown, lack ofasymptotic stability robustness of the design is demonstrated.Usually, non-oscillatory modes arc not the primary focus ofsmall-signal stability studies. In the nine-machinc example,it is shown that there are four non-oscillatory modes that caneasily cause non-oscillatory instability. These eigenvalues areassociated mainly with m achine fields, excitation systems, andPSSS.The pa per shows that large eigenvalue sensitivities inherent tomultimachine pow er systems and that power systems work withtoo small stability margins. Hence, negative damping modes c aneasily occur in widespread manner in the system.

At the end, the authors refer to a siiniilation in whic h a singlemachine system with a rule-based fuzzy PSS rcmains stable forsevere loading and parameter changes, whilc the same systemwith ii conventional PSS becomes unstable.

11. PRRLIMINARIESIn setting out a mathematical model we should have in mindthat all of the factors that influence in some way the dynamic

behavior of a powcr system can not be taken into account endno factor can remain absolutely constant during the system op-eration.Hence, it is natural to consider that on establishing ( l ) , er -rors on parameter values always happen. Therefore crisp models

only have a meaning when small variations of the parameters d onot substantially vary the character of the system pe rformance.


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A. Matrix Comoutation Framework: Stable Algorithms and Actual inaccuracies on the data can cause significant. .Condition of a Problem changes on ,some elements of A, shifing the eigenvalues muchmore than the ruanding effect of usual numerical algorithms.These inaccuracies m ake the positions o eigenvalues doubtful,i some eigenvalues are a ctually on the lefr-half complex plane( L H P ) or right-halfcomplex plane (RHP ).

The subject of rounding error was placed in its proper per-spective in matrix computation, with the distinction betweenthe concepts o l ll-conditioned problem and unstable algorithm1151-1 171.

The groundwork was the demonstration that the efSect.s orounding errors could be accounted o r by a perturbation in theoriginalproblem, Thus, it is required that a good algorithm yielda solution that is near the exact solution of a slightly perturbedproblem. Algorithms w ith this property are called stable [15].For the eigenvalue problem, it is well known that methodsbased o n the QR algorithm are stable, i.e., they yield a set of ap -proximate eigenvalues @ the ma trix .4 that are the eigenvaluescif a slightlyperturbeclmatrix A +E , where E i s a n n x 11 smallrandom matrix [151.

However, if the eigenvalues arc very sensitive with respectto data, even eigenvalues computed by the QR algorithm canhe far from the exact solution. Such eigenvalues are said to heill-conditioned. Thus, the accuracy of eigenvalue calculationsdepend on the sharpness qf the numerical algorithm, as well ason the degree o eigenvalue sensitivities in relation to data.

To assess the effects of the perturbation E on the eigenvaluesof .4 , a condit ion number (CN) has been associated with eacheigenvaluc to have, a priori, an upper bound on the eigenvalueabsolute sensitivity -CN(Xk) = l/(yi'vl:), [5], [15]-[18]. I thas been roughly estimated [ I 61 that if order 6 perturbations a remade , i n A , then an eigenvalue X may be perturbed by /AX156 x CN(X). Also, it has been stated [lX] that large CN s implythat the rnatrix is near a matrix with multiple eigenvalues.H. Condition of the Eigenvalue Problem in the Power SysternFramework

Occasionally, some comments about the ill-conditioning ofpower system eigenvalues and the rounding effect of numericalalgorithms havc bcen m ade, especially in paper discussions [4],161. However, cven when sensitivities were adm itted, it has bee nargued that sensitivities to coefficients are not meaningfiil-re-ferring to charac teristic polynom ial [4]-but sensitivities withrespect to system paramelers are. This very argument was one ofthe causes of the little importance given to investigations aboutthe reliability of outcomes of small-signal stability studies, untilnow [51. In order to address this matter, in the next section, be-sides the concept of sensitivity to mod el uncertainty (SMU) [3],the concept oi sensitivity to parameter uncertainty (SPU) is pre-sented.

The QR algorithm has been used to validate the eigenvaluecalculations of each new method developed for the analysis oflarge power system [4], LC]-[lO], [13]. Generally, the c ompar-isons have showed results agreeing to about 2 significant fig-ures, therefore with relative errors of about 10V2, enough ac-curacy for cngineering pinposes. Based on such comparisons,one can say that these numerical methods like the Q R algorithmdo n ot introduce significant perturbations on the A-matrices ofactual large power systems. How cver, simulations of actual in-accuracies in some parameters (v i., example, Section V and [ 5 ] )show that:

111. SENSlTIVlTlES TO MODEL AND TO PARAMETERUNCERTAINTIESSM U A N D SPU)When the elements of a state-matrix are obtained by mea ns ofmeasurements, it is natural that eigenvalue sensitivities be e sti-mated w ith respect to these eleme nts. Sensitivities estimated in

such way are called sensitivity to model uncertainty (SMU) [3].However, usually the n i j 's are explicitly obtained from systemparameters as

with ( i , =1, . ', n), or through im plicit algebraic relationstaken from matrix equations. Therefore, an eigenvalue A k ca nbe considered either as a function of the elements of the statematrix or as a function of the system parameters, that is

X I , = X a ( w 1 , " ' , a , , n ) = h ( e l , . . . , m p ) , (4 )w i t h ( k = l , . . . , n ).

Wh en the sensitivity is calculated in relation to parameters,as described in (4), it will be called sensitivity to parameter un-certainty (SPU).A. Absolute Sensitivir);a i j can b e obtained [5] y

An estimate of the absolute sensitivity of Ai, in relation to an y

where yr. i ) s the conjugate of the i th coordinate of yl:with respect to any system parameter mi is given by

If explicit algebraic equations bold (3), the sensitivity of X I:

where the first expression was originally used in [19].B. Relative Sensitivity

Relative sensitivity to model uncertainty may be obtained bythe reciprocal of Bode'sfunction [3] or simply the logarithmicsensitivity (LS) of the eigenvalue [20], given by

By replacing ai j by a I in (7),we obtain the relative sensitivityto parameter uncertainty


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When the e lement a i j of the state matrix (7) or the parametera! 8) changes by a certain percentage, S;! or ,S$ magnifiesthis percentage in the Xi, change. Thus, values o LS greater thanon e indicate increased relative ermrs or decreased precision.C . Sensitiviry Matrices,for Model Uncertainty [SI

Matrices whose entries are eigcnvalue derivatives or cigen-value logarithmic sensitivities in relation to the u i j ' s can be builtfrom (5) and (7):a) A matrix of derivatives of X with respect to the corre-

spondent entries of A , D i CfLX" ,s built from (5)(9)

b) A matrix of the logarithmic sensitivities of Xk. (LS-ma-trix) S p t C n x n ,s constmcted from (7) an d (9).

ES I indicates how much the relative error of the state matrix en-tries is magnified on the estimated e igenvalue. Like the LS (S),values of ES I (A ) >1 indicate that there is a loss of precision.B. The Asymptotic Stability Robustness Index (A SRIJ

The AS RI establishes ii hound for the relative error on statematrix entries so that this eigenvalue does not migrate to theRHP. To guess how easily an eigenvalue migrates to the righthalf complex plan e (KHP), under the influence of n i j uncertain-ties, the relative variation of th e real part of Ax can be evaluatedinstcad of its magnitude, and a bound for the ai j relative errorsi s a t t a in ed( l3 )wh en IdRe(Xx) l = 1 Re(X s)/ .Hen ce, as tabil i tyrobustness index (A SRI) may be defincd as

'I',Sp=- AA* A =?/iiviL0 , This index is adequate to be applied for dominant modes, whichusually are in neighborhoods of the im aginary axis. If X k i s r e dASRI(Xi,) i s simply the inverse ~ ~ E S I ( X I . ) .C . Comments o n the Proposed Indexes

Xi, X k y f 7 j k (I0)The symbol "0'enotes element-by-element multiplication.

IV. EIGENVALUE SENSITIVITY AN D ASYMPTOTIC STABILITY a ) The CN, j" Se ct io n I[-/,, ha s heen usecl to concisely es-ROBUSTNESSSSESSMENTSTwo indexes, dcrived from a matrix norm, are suggested.

timate an uppcr bound for the absolute eigenvalue sensi-tivity. The ESI is being proposed to give an upper boundfor the relative cigenvaluc sensitivity. Like the CN, com-bining iufoimation about a set of numbers into only a. The Eicenvalue Sensitivitv Index (ES I),

Th e ESI measures the relative sensitivity of its magnitude inrelation to the relative error of the state matrix entries. It is de-fined as the summation of thc absolute values of the entries ofDerivatives ( 5 )of simple cigcnvalues in relation to any u i j al -

ways exist in a neighborhood of the point ((11 1 , . . , ann).From(4) an d (7). he relationship between the total differential (1x1.an d A s can be written as

S? (10).

ndXi, 1 dX d o . .n i j- - = $2.1 1 )X k Z , J = I. dai j i , j = l

Taking the absolute values on both sides of ( 1 I ) , we have

Assuming the same relative error to al l a i j , then

where the symbol 1 1 . 1 1 represcnts the matrix n orm defined as th esummation of the absolute values of all entries 1151, [21j . T he

single number also leads to thc ESI (13). It is not sur-prising that there is loss of information in the process.b) In [SI it was stated that som e parameters co uld hardly heestimated within relative errors of 10%.It is importaul

to notice that ASK1 was calculated under the assum ptionthat the relative errors for each u i j are equal. Thus, i t isreasonable to assume that the value of 5% is like an av-erage value. Therefore, there might be higher and lowererror values, and, to be safe, it would he recommendedthat if ASRI(X) 5 3% for a d ominant eigenvalue, a thor-ough sensitivity analysis in relation to thc system param-eters must be done.

D. ensitivities for System Parameter Uncertainty (SPU JIn order to comp ute the absolute (6 ) and relative sensitivities

(8) or any system parameter q , A/&[ must be calculated. I fonly a numerical form of A is available, this matrix derivativecan be numerically estimated as dA/&r = AA/Aml whereAA H'"x" is a matrix increment in A an d Anr s a scalarincrement in or.v. A N E X A M P LE F SENSITIVITY AN D ASYMPTOTIC TABILITYROBUSTNESS ASSESSMENTA. Model Description

The co uccpts oregoing presente d are now illustrated by testscarried out on a 48th-order, 9-machine, 21-bus system takenfrom 1121. Fig. 1 shows its one-line diagram where bus 10 isconsidered to be a n infinite bus. Each machine is described bya fourth ordcr model in the following order of state-variables


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132

r 1.

Fig. I. A stabilized nine-machinc system [I216, U , E; an d Kt:f.1. he open-loop system (without PSS's) wasunstable, with four eigenvalue pairs in the RHP. According to[121,"the best damping found fo r the entire system was oblaiuedby assigning a set of non-uniform dampings of 0.3,0.8,0.7 and0.4, to machines 3 , 7 , 8, and 9"-actually the autho rs are re-ferring to damping ratios. Duc to the introdnction of th e PSSs,three other state variables were added to the machine model.

In the present work, a test ofeigenvalue robustness was per-formed, by modifying only one m achine parameter. Another testwas done modifying six machine parameters and tw o PSS pa-rameters. In Appendices A and B, some generator and PS S pa-rameters are copied in Table V and Table VI. Remaining data, toconstluct the closed loop state matrix (with PSSs), wesc takcnfrom [12].B. Eigenvalue Calculations

Using a program written in FORTRAN, the matrix A of thestabilized system [I21 was obtained 1221 and its eigenvaluescalculated using E ISPACK routines [231. Th e same eigenvalueswere calculated using a Computer Aided System Design(CASD) software package [18], but using an approximationwith four decimal figures of this matrix A. The s et of eigen-values, their CNs and ESIs calculated for this last matrix arcshown in Table 1.This last set of eigenvalues is nearly identical to those fromthe EISPACK routines. The relative errors among th e correspon-dent eigenvalues are about I OW3, except for a few cases in whichthe relative errors are about I 0V2. This coincidence was ex-pected because the QR algorithm is the basis for both softwarepackages.

Moreover, relative errors of the same magnitude are vel.i-fied, if the set of eigenvalues in Table I is also compared withthat in 112, Table 1111. In the 2 nd an d 3 rd col um ns of Ta ble 11,the mechanical modes of the nine machines are copied from[12, Table IV]. T he 4th column is composed of the m odes cal-culated using the CASD package [18 ].

These results briefly described above evidence that roundingerrors of different implementations of stable algorithms donot introduce a significant perturbation on any eigenvalue,In other words, the'product E x CN(X), see Section 11-A, issmall if compared with any eigenvalue. The CNs range is10 7 5 CN(X) 5 1860, but the E ' S are insignificant. This is afortunate con sequ enc e of the large relative accuracy, of approx-imately 16 decimal digits, of the modern microproce ssors 1161,[MI. It is reasonable to suppose that parameter inaccuracies

TABLE 1EIGENVALUES,N'S AND BSI's OF THE C L OS ~D - L OOPIN S MACHINC TCSTSYSTEMI2 1

ESl(h) C N Q Eig-"a'"e ESI(1) CN(1) Eigenvalues[rad I SI [rad I SI1.16 214 -41.181 0.94 154 -0.2244fi8.64951.22 370 -39.189 11.88 691 - .9029fj7.54221.15 163 -36.599 1.94 232 -0.1112fj7.15681.16 149 -35.118 6.55 516 - .8465 fj7.37203.41 279 - a i m 3.68 316 -0.2688fj6.13356.71 901 -3.5SISfj l3.7S38 10.24 788 - .8404ij5.84935.01 406 - 19.523 56.06 1710 -3.8283 fj3.90826.51 752 -9.W55fj11.850 29.88 1860 -4.2768fj3.21052.45 184 -17.546 67.12 1728 -2.6691 fj3.44858.38 882 -17.012 54.39 762 - .1237fj2.56243.32 159 -15.859 8.77 120 - .60607.76 675 -9.2730fj10.018 11.59 194 - .656311.88 68 2 -14.402 250.88 212 -0.22114.34 469 -10.284 fj6.4307 lS0.27 107 -0.2032930 556 -10.221 ij2 .91 56 108.49 114 -0.20740 9 3 187 -0.3342fj9.1231 151.06 139 -0.2047

TABLE IIRIFPNvAI.UES OP'rLln MECIIANICALODES RADISJ, FROM [IZ, ableIV1 AND FROM TABLE I

0.22483 j6.127310.50830 L j5.323540.52595 fj5.446220.09219 fj2.80204-0.09469 i 7.2452-0.33543 fj912669-0.19702 f j630435-0.2188 fj8.64231- .06273f 7.0242

-1.83941j5.8490- .2782 tj3.208 7- .8301 tj3.9075i- .1214kj2.5695i- .1118 fj7.1568i- .3342 fj9.1235i-0.269fj6.1339i- .2249 fj8.6497i- .845f 7.3728i

- 1.8404fj5.8493- .2768 fj3.21 05- .8283 fj3.90 82- 3.1237fj2.5624- 0.1112fj7.1568- .3342 fj9.1231-0,2688 fj6.1335-0.2244fj8.6495-0.8465 fj7.3720

andlor parameter variations cause, at least, uncertainties onthe A-matrix such as E 2 0.01 . As the CNs range is a s men-tioned above, these errors may be far more unimportant thanthosc caused by rounding errors introduced by the numericalalgorithms. This land of perturbations may mislead someeigenvalue estimates.C. Eigenvalue Sensitivity fo r Model Uncertainty

In Table I, the CN's range is 107 5 CN(X) 5 1860. TheseCNs are always significantly larger than the C Ns of any 48 x 48matrix with random entries chosen from a uniform distributionand with the same Euclidean norm of the state miltrix IlAll2 =2331.7 . Onc such matrix was simulated and the CNs' rangewas just 1.66 5 CN(X) 5 8.30. However, the eigenvaluesof the random matrix were widespread on the complex plane,-1078.5 5 Re(X) 5 -1101.9 and -1172.9 5 Im(X) 5-1 172 .9. This tcst confirms the statement in S ection 11-A, thatthe closer the eigenvalues, the larger the C N s .


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MAQNFJDESW LCG.SEN3l l "eS W ReAL[-O.1112+7.1571)

Pig.2.7.1668i (radis) in relittion to thc entries of matrix A .Plotof milgnitudesolLSsofthereelp:atuftheeigenvaliic -U.I I l a t

The rated angular velocity (w , ~= 377 rad/s) is constant,thus the corrcsponding A-entries were not considered in ESIcalculations, nevertheless most of ES Is are bigger than I .In Table I, the ESIs of the real eigenvalues

-0.221 1 , -0.2032, -0.2074 and -0.221 I are too large:250.9, 150.3, 108.5 and 151.1, respectively. Thus, their ASRAare only 0.4,0.7 , 0.9 and 0.7%, respectively. These four modesare associated mainly with the fields, excitation systems andPSS's of machines 3, 7, 8, and 9.

Themechanical mode, -[1.224&j8.G495, ESI =0.94, has asmallerrealpartthan-ll.2688f~6.1535,butitas alargerESI,equal to 3 . 68 . Hence, this eigenvaluc and -0.1112 f 7.1568,ES I =1.94, are selected as examples for a coniprehensivc sen-sitivity analysis.Fig. 2 shows the plot of the LS matrix of the real part of-0.1112 f j7 .15 68 , the electromechanical mode with thesmallest real part. It is associated mainly with the mechanicaloscillations of machines I , 2, 4 an d 5. Fig. 3 is the plot forto -0.2688 f G.1335 (radls). It is relatcd mainly with themachines 4, 5 , 6 and 7 , this last on e with a PSS.

Table 111presents the five biggest LS's observed in Fig. 2 andFig. 3. The LSs associated with 71111 are not included.

Fig. 4shows the 9 electromechanical modes of the examplcand their corresponding ASRls (l4),with the eigenvalue sensi-tivities in relation to tu11 excluded. However, small ASRIs fornon-dominant eigenvalues only show that the real parts of thesceigenvalues have large sensitivities. For instancc, the ASR l's of2 .1 a n d 4 . 6 o f t h e - . 1 . 8 2 8 3 i j 3 . 9 0 8 2 a n d - 4 . 2 7 6 8 ~ j 3 . 2 1 0 : ,inodes just indicate that the relative errors of the real parts ofthese modes would be about 47, 6 and 21, 7 times the averagerelative error of the aij 's.The electromechanical mode -0 . I I1 2 i 7.15 68 has a smallASRI = I .7, indicating that it can migrate to th e RBP under theinfluence of small variations on some A-entries. By the way, ifthe absolute values of ail,, mi, a6 5are increased by 10% andthoseofu61, a1716aredecreasedbythesamcamount, this modewill chan ge to +0.0081 j7 .2484. The entry a8l was chosen,

MAONlTUDESOF LOO.SENSllMrlES OFRUL 4.2E3at6.133 )

Pig. 3.D.13.3Ri (radls) in relation to the eiiliies of A.Plot ofmagnitudes olLSs of the rc d part of thecigenvalue -0.2688+

TABLE II ILARGESTNTRIPSP THR 1.S MAIKICYS O r RE(-U. l I 1 2* j7,I668)AN D RI?.-0.2688 *,j6.1336)RADIS I

0, ~Re(-O.2688+/6.1335) puaga4 1 - 1.40 a17 16 - .19a6 I 1.69 a25 24 3.21a6 5 -3.99 32 1.63

(11716 3.60 a34 30 1.44a 2 1 2 0 - 1.44 a34 32 1.93

4.d.

o.*a.+ 1

Fig. 4.inodes [I21ancl their ASRls, i n S.Eigenvalues with psitivc imaginary puts of the 9 electromechanical

in spite of its LS not being one of the five largest LS , as thosein Table 111.


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Fig. 5.z i , =0 .1 5 p o i n s t e a d o f . r ; , = 0 . 1 2 p u [ 1 2 ] .Th e 9 cleclroincchmical modes and their ASRI's of Fig. 4but wilh

TABLE IVORlOlNAL AN D P!3RTlJRDED MACI I I NEND Pss PARAMRTPRS, REACTANCESNPR R UNIT, AN D H , T'S IN SECOND

D. igenvalue Sensitivity for Pnrnmeter Uncertainty (ESPU )The exam ination of LS-matrix and the symbolic form of thematrix A [22] allows the straightforwa rd determina tion of thelargest eigenvalue SP U's. For instance, it was verified that just

a change in the unsaturated & axis subtransient reactance xL1 ofthe machine I , from = 0.12-0.15 pu. Table V, brings themode -0.1 1 1 5 2 ~ t j 7 . 1 5 6 8rad/s) into the RHP, more prec iselyto +0. 009* 7.03 11, as shown in Fig. 5 .Furthermo re, it was noticed that many com binations of a fewparam eter perturbations of about 10%or less also destabilize theoriginal system. For exam ple, Table IV shows a total of six ma-chine parameters and twu PSS parameters pcrturbed by about10%. For these perturbed parameter values the electromechan-ical mode -0.2688 f S . 1335 becomes +0.002 & jG.2317 pu ,Fig. 6.

VI. LARGEIGENVALUEENSITIVITIES AN D TO O SMALLSTABILITY ARGINSIn [Z], control design technique based on the concepts of

phase and gain margins showed that if a damping ratio of C=0 .5 were a ssumed for the mechanical dominant mode, a singlemach ine connected to an infinite bus would work with reason-able phase and gain margins, Cor a variety of machine parame-ters and loading conditions. In [24], for a simp le multimach ine

EMNVALUESOFORGMLANDPERTURBED DATA

4 I tI-2 -1 0 1REAL PARTFig. 6 . Eigenvalues with positive imaginary parts of 9 originalalectramechanical modes "+" and aitcr data perturbation "0" given onTable IV.

system, the same phase and gain margin technique would pro-vide a damping ratio of 6=0.26 for a dominant mode, only ifall the ma chines were provided w ith PSSs.

The analyzed multimachine PSS design i s based on apole-placement technique [121. However, the two electro-mechanical modes more carefully examined, Figs. 2 an d 3,-0.1112 i 57.15638 and -0.2688 f 6.1335, have dampingratios of only 0.016 and 0.043. The robustness tests aboveconfirm the suspicion that such small damping ratios do notcorrcspond to reasonab le stability margins. Also, a recent w orkbased on an optimal control method does not get for somedominant modes, damping ratios larger than 0.07, even for asimple ten-machine power system [25].It was mentioned in Section 11-A that matrix computationspecialists consider that large eigenvalue sensitivities ( C N s )are tightly related with eigenvaluc closeness. Also, it is wellknown that the eigenvalues of every multimachine system fallin clusters, on the com plex plane. H ence, roughly speaking, ifthe eigenvalues of the power system remain close to the imag-inary axis, negative dam ping mo des can easily occur even in asimple multimachine power system.The general procedure of robustness test suggested in thispaper may be sum ma rked as follows:

1. First, the easier SM U analysis should be performed, de-termining the ESI's (13) and ASRIs (14).2. If large ESIs, or small ASR Is are verified, a SPU analysis

must be done , even if a symbolic expression for the statematrix is not y et available.The authors would like to see these tests being applied toa large number of actual power systems. These studies should

help to se e the small-stability problem in the proper perspective;the au thors think that a fuz zy rule-based approac h is an attractivealternative for power system controller designs.

A rule-base d fuzzy controller is a non-linear controller, eas yto design, and thanks to fuzzy chips, prod uces responses nearlya hundred time s shorter than tho se of a conventional PID c on-troller [26]. Moreover, using an adaptive rule-bas ed fuzzy con-trol scheme, one can design control actions that can improve


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both sm all-signal stability and transient stability, in an indepen- TABLE VGENERATOR DATA AR R I N PBR-UNIT,Excrrr I I ANI) Ti , IN SECONDI21dcnl way.

VII. CONC1,USIONSThis paper defines separately the concepts of eigenvalue sen-sitivity with respect to the elements or the state inalrix, SMU,aud to parameter uncertaiuty, SPU. I n small-signal stabilitystudies the SMU must be always estimated: the eigenvalue

sensitivity index (E SI), the logarithmic sensitivily (LS) matrixand the asymptotic stability robustness index ( A S R I ) proposedhere are useful tools to accomplish SMU estimates. If an LSand/or EST bigger than one or an ASRI about 5% are detectedFor dominant eigenvalues, the SP U of these modes must beinvestigated.These procedures were carried out for two electromechanicaloscillatory inodcs of a nine-machine system: ESl's of I .Y4 and3.68 show (hat the natural frequencies have large sensitivitieswith rcspccl lo some state matrix entries; ASRI's of 1.7 and

2.0% evidence lack of asym ptotic stahility robustness for modeluncertainty of the system.A change of the iinsalurated d-axis transient reactance of justone machine from z&= I . I2 pu o o:(=0.15 pu caused loss ofthe asymp totic stability. Also, a variation of 10% in six machine

parameters and in two PSS parameters displaced to the RHP anclcctroinechanical mode with a damping ratio of 0.043. Hence,lack of asymptotic stability robustness Cor parameter uncertaintywas also vcrificd.

Four non-oscillatory inodes, usually not considered on small-signal studies, have the smallest ASRl ' s , 0.4,0.7,0.Y and 0.7%.Such extremely low ASRls w e a l that these modes can he theearliest causes of instability. These eigenvalues are mainly as-sociated with niachine fields, excilation system s and PSS's.

It is shown that in generel, the eigenvalues of the inullima-chine s ystem s tend to have large sensitivity. Thu s, poor dam pingratiw of about 0.06 are evidences of lack of asym ptotic stability.The authors hope that the robustucss tests proposed stimu-late the research for new types of power system controllers.We think that inodein adaptive non-linear controllers, like adap-

tive rule-bascd fuzzy controllers, could, under severe and small-signal transient disturbances, have a better performance thancontrollers based on linear control techniques.

APPENDIXATable V presents the generator data. The second columnshows the subindices of the stale variables zi concerned toeach machine. Reactances are in pu, I-I an d ' I;, are io second.Each generator is equipped with a static excitation systemconsisting of a first order transfer function with a gain IC*an d a time constant 7'".Machines 1-6 have I


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306 IERE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. I. EBRUARY 2000

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EvandroE. SouzaLima (M81) receivedhisB.S.E.E. degreelramITA(Brari1)in 1966 and his D.E.A. an d Dr.hg. degrees in automation from the UniversitePaul Sabatier, Toulouse.France, in 1969 and 1972 respectively. Since 1996 hehas been a retired Professor of the Deparlment of Electrical Enginecring of th eUniversity of Brasilia and healso worked as B visiting Senior Researcher of thesame institution until February 1998. His main interests are in convo l methodsnpplied to power systems, mainly rule-basedfuzzy controllers,

Luis Filnmeno de Jesus Fernandes wa s born in 1961, in L uanda, Angohi. Hereceived the B.S.E.E. degree fram the U niversity of Angola, in 1990 and theM.Sc. degree in electrical engineering fram lhe University of Brasilia, Brazil,in 1996. His main inlercals arc concorned with power system dynamics.

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