217IEEE Transactions on Power Systems, Vol. PWRS-1, No. 1, February 1986EFFICIENT EIGENVALUE AND FREQUENCY RESPONSE METHODS APPLIED

TO POWER SYSTEM SMALL-SIGNAL STABILITY STUDIES

Nelson Martins

Member IEEE

Centro de Pesquisas de Energia Eletrica (CEPEL)Rio de Janeiro, Brazil

Abstract - Frequency response and eigenvalue techniques are fundamental tools in the analysis of smallsignal stability of multimachine power systems. Thispaper describes two highly efficient algorithms whichare expected to enhance the practical application ofthese techniques. Oiie algorithm calculates exact ei-genvalues and eigenvectors for a large power system,while the other produces the frequency response of thetransfer functions between any two variables in thesystem. This paper also presents alternative comput-ing procedures for the AESOPS eigenvalue estimationalgorithm which are simpler and at least as efficientas those described in El].

I. INTRODUCTIONLow damped electromechanical oscillations have

become a common phenomenon in modern electric powersystems. The damping of these oscillations, which insome circumstances may even be negative, is dependenton the system structure, its operating conditions, andthe effects of automatic-controller action. AnrM ef-ficient way to combat these problems is through theinstallation of additional signals to the generatorexcitation systems, which need be properly tuned. Thisarea of work has received continuous attention in re-cent years, leading to the development of methods andcomputer programs to determine the source of theseproblems and obtain solutions by means of control [2].

Among the present requirements in this field,there is the need for simple, reliable and efficientalgorithms for the computation of eigenvalues and fre-quency response of transfer functions for large powersystems. This paper describes two efficient algo-rithms for the calculation of eigenvalues, eigenvectorsand the frequency response of transfer functions be-tween any two specified variables in a multimachinesystem. These algorithms exploit the sparse nature ofthe power system Jacobian matrix and therefore producefast solutions at low computational cost. The powersystem Jacobian matrix formulation caters for variousmodels of generators and associated controllers, in-duction motors, non-linear loads of different charac-teristics, and static VAR compensators.

The first successful attempt to calculate thedominant eigenvalues of large power systems has beenthe AESOPS program [1], which has quite complex computing procedures. This paper presents alternative computing procedures for the AESOPS algorithm which aresimpler and at least as efficient as those describedin [1].

This paper was sponsored by the IEEE Power Engineer-ing Society for presentation at the IEEE Power IndustryComputer Application Conference, San Francisco,California, May 6-10, 1985. Manuscript was publishedin the 1985 PICA Conference Record.

used.The notations adopted in the paper are defined as

II. THE POWER SYSTEM JACOBIAN MATRIXThe power system stability problem can be repre-

sented by a set of differential equations together witha set of algebraic equations, to be solved simultane-ously with each other. Equation (1) shows the Jacobianmatrix of the entire set of equations, evaluated at anoperating point (o , yo):

_ J3 IJ4I(1)

The power system state matrix can be obtained byeliminating the equations for the algebraic variablesin the Jacobian matrix:

Ax = (J-lJ J- I- l2 4 J3) Ax = A Ax (2)

The symbol A is used to represent the systemstate matrix, whose eigenvalues determine the singularpoint stability of the non-linear system.

The most reliable method to date for the computa-tion of the full set of eigenvalues of an assymetricalmatrix is the QR method developed by Francis [3], whichdoes not exploit the sparsity of the given matrix. Re-search work has been quite intensive on methods whichproduce full eigensolutions by exploiting the sparsenature of assymetrical state matrices. These methodswill, unfortunately, not have application in the analy-sis of low-damped electromechanical oscillations, sincethe associated state matrices are themselves not sparse.

Consider, to illustrate this nonsparsity, thecase where every generator in the system is represent-ed by the state vector A(Ef XE,Ws,6SVfd), where thestate AVfd is associated with a first order model of anexcitation control system. Reference 4 discusses thisparticular example, and shows that, even as the numberof generators in the system approaches infinity, theratio of non-zero elements of the state matrix does notfall below 48 percent. Research efforts should, therefore,be directed towards methods of eigenvalue calcu-lation which do not require the explicit formation ofthe power system state matrix. The eigenvalue calcu-lation algorithm, presented in this paper, is appliedto the power system Jacobian matrix, which is verysparse.

The Jacobian matrix formulation used in this workis an improved version of that developed by Vorley [5]for solving the power system transient stabilityproblem via the "simultaneous solution" approach [6].It caters for various models of synchronous generatorsand associated controllers, induction motors,non-linearloads of different characteristics and static VAR com-pensators. Models for HVDC links and associated con-trols can also be easily incorporated.

0885-8950/86/0001-0217$01.00O1986 IEEE

218

-As methods of assembling linear equations for powersystems have been described by many authors, this sec-tion is intentionally brief and aims at givIng 1thereader some basic information on the Jacobian matrixformulation. The computer program developed is quitegeneral, and every bus in the system may have any com-bination of elements (generators, induction motors,loads) connected to it. A synchronous generator andassociated controls can be represented by up to 21incremental variables, which for explanatory purposesare grouped below in the following sub-vectors:

x, = A(Erj, El', El, W, 6)t generator statevariables.

X2 = A(v1, v2, Vfd, v3)t - excitation control.2 - A (V4, v5, v6, Pm)t - speed-governor and

turbine.

x4= A(v7, v8, v9, VpSS)t - power system stabilizer.A(Id, Iq, Sdd Sq)t - generator algebraic

variables.where the symbol A signifies an incremental change froma steady-state value. The symbol Vfd represents theapplied field voltage, Pm is the prime-mover mechanicalpower and Vpss is the power system stabilizer outputsignal. The symbols vl to vg represent state variablesof the various controllers which do not interface withother controllers or the generator equations. Thesymbols Id and I represent the components of statorcurrent expressea in the generator d-q frame of refer-ence. Two algebraic equations are used to model theiron incremental saturation effects on the direct andquadrature axes of the synchronous generator. Themodelling of generator saturation is similar to thatdescribed in [7] and the symbols Sd and S representalgebraic variables which are functionals ot the gener-ator air-gap voltage.

By keeping in the Jacobian matrix the algebraicequations relating the generator internal and terminalvoltages, expressed in the generator d-q referenceframe, the Jacobian matrix elements can be derived in astraightforward manner. Induction motors are repre-sented by three state variables: the rotor slip S andthe internal voltages Er and EV. These voltages areexpressed in the synchronously rotating frame of refer-ence, which is made coincident with the real andimaginary axes and is represented by the symbol 'r-m'.

A static VAR compensator is represented by afourth(or lower) order transfer function model whose outputvariable is the effective admittance value of thecontrolled reactor. The input variable is the terminalvoltage magnitude of the controlled bus which is compared with a reference value. There is also provision fora fourth order stabilizing signal, sensitive to busfrequency or power flow in a transmission line, whichshould be tuned to provide additional damping to electromechanical oscillations.

The Jacobian matrix is formed by first enteringthe blocks of equations for every system generator.After this the induction motor blocks are built and arefollowed by the static VAR compensator blocks. Thenetwork equations come last in the Jacobian matrix for-mation. Submatrix JD, shown in Figure 1, is practicallyequal to the nodal admittance matrix of the network,expanded into its real and imaginary parts. The onlydifference lies in the fact that the (2x2) diagonalblocks of JD corresponding to the nodes which containlinear, non-linear or induction motor loads, have extrapartial deriv ative tertms added to them.

Figure 1: The Power System Jacobian Matrix.(The hatched blocks contain non-zeroelements and have sparse stucture)

The injected current into bus 'i' due to a staticload (PI + jQ') is given by:

mj

I4

Ii 1I: Vuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu12uuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuPL

IQLand a voltage-dependent load can be represented eitherin a polynomial or exponential form:

PL miVila ot Pl = A +B Vii + C IVQ1 = Vi1b i 2L= njV1 or QL =Aq + Bq Vi- + Cq Vl

(4)

The derivative of equatiog (3) with respect tothe voltage components V] and VI is added to the (2x2)diagonal block associates with bus 'i' in submatrix J.Power system loads which are functionals of b-is fre-quency can also be modelled by using a first order de-rivative block to obtain instantaneous bus frequencyfrom the bus absolute angle deviation (see equation(17)).

The power system Jacobian matrix is extremelysparse and of very high order, and the full use of sparsity techniques becomes imperative. As an example, consider a power system model having 10 generators withspeed-governor and excitation control systems, 3 induc-tion motors, 40 buses and 80 lines. The Jacobian ofthis system will be of order n = 299 but the percentageof non-zero elements is below 3 percent.This percentageis rapidly reduced as the system size increases.

III. THE IMPLICIT INVERSE ITERATION ALGORITHMInverse iteration is one of the most powerful

methods for calculating an eigenvector associated withan eigenvalue which is known only approximately. It isalso efficient in finding the eigenvalue closest to a

219

specified point in the complex plane and its respectiveeigenvector. The basic inverse iteration algorithm [3]can be described by:

(i) Solve for W. l

(A - qI)W = Z (5a)-kc+l -k

(ii) Compute the vector ik+l for the next iteration:

_k+l-k+l max(W+l) (Sb)

and return to (i). Convergence occurs when the changein Z at any iteration is less than some specified tolerance. In this algorithm the subscript 'k' is the iteration number, 'I' is the identity matrix,'q' the ap-proximation of the desired eigenvalue 'Xi'and max(k+1)is the element of largest magnitude in this vector. Thevector Sk has arbitrary initial value, and correspondsto the desired eigenvector at convergence. After con-vergence, the factor l/(Xi-q) will be dominant in theelement max(Wk+l) and the correct eigenvalue Xi isgivenby:

Xi = q +l/max(W ) (6)

The rate of convergence of the inverse iterationalgorithm depends on the ratio of the factor (Xi-q)- tothe next larger factor (Xj-q) [3]. Provided a good ei-genvalue estimate 'q' is given this method converges in2 or 3 iterations. A variation of this method, whichaccording to [3] is more suitable when the given ei-genvalue estimate is not good, involves refactorizingthe matrix (A-qI) at every iteration,using as the newestimate 'q' a corrected value obtained from theRayleigh quotient.

The application of the inverse iteration methodin the form described in equations (5a) and (5b) isimpractical for large power systems since the statematrix A is not sparse. Equation (5a) may however bewritten in implicit form [5] by deLining an extravector x

|Jj-qI| J2 | + and U. are again ofno direct interest in the solution.

The partitioned solution [6] of equation (9) canbe described as:

(i) Compute JD = i-J (JA) JBD D C A B (lOa)Where J* has the same sparsity structure as J

D DS(ii) Solve forQ

J* Q =- J (t)lZ eD -k+l C (JA)(iii) Solve for Tie:

3AH.k+l < B +1

(lOb)

(lOc)

The submatrix JD has the sparsity structure ofthe nodal admittance matrix and should be ordered so asto minimize fill-in during factorization. The matrixproduct JC (J) 1.JB is block diagonal and each (2x2)diagonal block can be obtained from the productJC.(JkiYl.JA where J' J' and Jl are blocks associatedB A'l B Cwith the "i-th" system component (generator, inductionmotor or static compensator). Note that each JA can befactorized separately, which allows savings in corerequirements and solution time. Note also that thetranspose eigenvector, needed for eigenvalue sensitivity and Rayleigh quotient calculations, can be obtainedusing the same LU factors of JA and J* by noting thatAt =Ut.Lt.A D

IV. EFFICIE'NT CALCULATION OF FREECNcY RESPONSEPLOTS FOR LARGE POWER SYSTEMS

Frequency response methods allow a deeper insightinto small-signal dynamics than eigenvalue methods, andhave widespread use in the design of power system con-trollers. The transfer function matrix F(s) may be

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obtained from the Laplace transformation of the dynamicsystem state space equations [8], and is given by:

-1F(s) = C(sI - A) B + D (II)

where A is the system state matrix, B is the input orcontrol matrix, C is the output matrix and D is usuallya null matrix for most physical systems. In the appli-cation considered in this paper matrix D is zero.

The frequency response analysis of a dynamicsystem model can be performed by replacing the Laplacevariable 's' by 'jw' in equation (11) and numericallycalculating matrix F(jw) for discrete values of 'jw'within the frequency range of interest. The use ofequation (11) becomes prohibitive for large order sys-tems due to excessive computational time and memoryrequirements.

A variation of the proposed implicit inverse iteration algorithm can be used to efficiently obtain thefrequency response of transfer functions between anytwo variables of large multimachine power systems. Inthis method, neither matrix A or matrix C, which are notsparse, need to be formed.

In the following analysis, a single input dis-turbance is considered so that the matrix B reduces tothe vector b. The next equation shows an intermediatestep in the derivation of equation (11):

(sI - A) X(s) =- b u(s) (12)Note that X(s) and u(s) are the Laplace transforms

of the system state vector and the applied input, re-spectively. Replacing 's' by 'jw' in (12):

(jwI - A) X(jw) b u(jw) (13)The solution for X(jw) can be obtained through the

LU factorization of matrix (jwI - A) followed byforward and backward substitution on vector b. Equation(13) is equivalent to equation (14) which is based onthe power system Jacobian matrix:

.u(1w)(14)

where matrix L has been defined1 previously and vector his contained in be. The vector Xe(jw) contains thephasor values, at the applied frequency 'w', of all thestate variables for the generators, induction motorsand static VAR compensators in the system, apart fromthe phasors Id(jw), Iq(jw), Sd(jw) and Sq(jw) for everysynchronous machine represented. Vector V(jw) containsthe phasor values for the voltage components, expressedin the synchronously rotating (r-m) frame of reference,for all the buses in the network.Therefore,the frequencyresponse of the transfer function between the inputsignal u(jw) and any of the variables containied invectors Xe(jw) and V(jw) can be obtained simultaneously.

Normally, generator terminal voltage magnitude andactive power are among the output variables of interesLtThe linearized expressions for these variables in thefrequency domain are similar to those in the time do-main, and are given by:

LV(jw) o AV(jw) + n AV (jw) (15)Vto VtoAPt(jw) VroAIr(w) + VMO AIm(w) + Iro AV (1w) +

+ I AV (jw)Mo m (16)where the subscript "o" signifies a ste.ady-state value

for the variable.The phasor values for the active and reactLve power

flows in any transmission line may also be obtainedthrough sinmle relationships. The relation arctan(VM/Vr) provides the starting pointc for the derivationof the expression in the frequency domain for the phaseangle deviations observed at any systoic bus:

V 'V (w} - V (w)ro m Mo rAd(jw) - Vro2 + \ o2 (7 )

and the bus frequency deviation phasor s obtainedmultiplyin, the expression for AO(jw) by jw'.

by

The use of expressions of the type shown in equa-tions (15)-(17) obviate the need to calculate the systemoutput matrix C, which is non-sparse, with large savingsis computation.

Note that only one sparse LU factorization of thzeJacobian is needed for each discrete value of frequency,in order to calculate the frequency response of thetransfer functions between the specified input variableand all the other system variables. The specified inputvariable may be changed by simply changing appropriate-ly the vector be of equation (14).

Use can be made of fast-forward substitution [91,since generally only one non-zero element will exist invector be. The use of fast-backward substitution aswell would make it possible to obtain only the desiredfrequency response plots, with significant gains incalculation speed.

The frequency response of the transfer functionbetween any two specified system variables, vl(s) andv2(s), can be obtained from the solution vectors Xe(jw)and V(jw) by making use of the following relationship.The ratio between the frequency responses of vl(s)/VRl(s)and v2(s)/VRI(s), where VRl is a system input variable,will yield the frequency response of the desired trans-fer function vl(s)/v2(s). The use of this relation canbe quite useful in studies of the type described inReference 10.

V. AN ALTERNATIVE SOLUTION SCHEMEFOR THE AESOPS PROGRAM

AESOPS is a recently developed computer program[171, based on a one-eigenvalue-at-a-time solution me-thod, which can determine practically all eigenvaluesassociated with synchronizing power flow in largeelectric power systems.

In this section it is assumed that the AESOPS for-mulation and its notation [1] are familiar to thereader. Although the power system state matrix A isnot actually built in the AESOPS program, it is con-venient when explaining its methodology to wrn te theeigenvalue estimation algorithm as:(i) Solve for X(zk), given an initial estimate of zk:

(zk I -A) X(zk) = b . TX(zk)Note that (18) is similar to (12).

(ii) Compute the eigenvalue estimate (a + j kiteration k:

(Cl j+)k = zk - I (U + IV)k O +2TI

(18)

at

(19)(iii) Define the complex frequency zk+l of the mechani-

cal torque input Tx for the next iteration:k+1 k

z =(Ca+j ) (20)and return to (i). Convergence occurs when thedifference between two successive eigenvalue esti-mates is below a specified tolerance. At conver-gence, the vector X(zk) is equal to the eigenvectorassociated with the system eigenvalue zk.

221

The following definitions are necessary regardingthe above algorithm:

Tx(zk)

E 2 H.1w. (z )Ii=1 3

= external torque phasor applied at aspecified generator, at iteration k,which yields a rotor speed phasorw(zk)=(1. + jO.0) at this generator.

= momentum function involving all gener-ator inertia constants and the absolutevalues of the rotor speed phasors.

(U + SV)k = Tx(z )/ ( E 2 H ilui(z )i=l

(01 + jo5) = acceleration factorThe rotor speed phasors wi(zk) for all generators

are a subset of X(zk). The calculation of these rotorspeed phasors is the only large computational task re-quired by the AESOPS algorithm. It is now described analternative method to calculate these rotor speed pha-sors which is much simpler and at least as efficient asthat used in the AESOPS program [1]. Noting that equa-tion (18) is similar to (12) we can express the formerin implicit form, yielding an equation similar to (14).The alternative solution scheme for the AESOPS algo-rithm, here proposed, is as described in steps (i),(ii) and (iii) of this section, but vector X(zk) issolved for by using the implicit form of equation (18).

It was stated in [1] that frequency dependent loadswere not considered in the AESOPS program since theywould increase the computing costs enormously. As seenin section II of this paper, a state variable is included in the power system Jacobian matrix for every frequency dependent load represented. Therefore, when usingthe alternative solution scheme for the AESOPS algo-rithm, the computing costs are only marginally increas-ed if frequency dependent loads are considered.

VI. RESULTS OF EIGENVALUE CALCULATIONS

Small System StudyThe New England test system [11], used in several

other dynamic stability-related papers, was chosen todemonstrate the performance of the proposed algorithms.This test system comprises 10 generators, 39 buses and46 lines and its complete data is given in Appendix Fof [11].

The alternative solution scheme for the AESOPSalgorithm, described in section V of this paper, wasimplemented by adding an extra routine of about 40FORTRAN statements to the implicit inverse iterationprogram. Table I summarizes the results obtained withthe AESOPS algorithm. The system generators were dis-turbed with different initial eigenvalue estimates inorder to evaluate the performance of the algorithm. Theconverged eigenvalues shown in Table 1 are those associated with the 9 electromechanical modes of oscillationof the New England test system. They are very close tothose presented in Table 4-5 of [11], a fact which givesconfidence to the digital program developed.

The AESOPS algorithm tends to converge to the dominant eigenvalues of the driven generator. As a clearexample of this fact, note in Table 1 that even bydriving the generator at bus 39 with an initial complexfrequency (-0.111 + j7.09) which is practically a cor-rect eigenvalue, the algorithm converges to the dominant eigenvalue of that generator: -.248 + j3.68. Incases where the driven generator participates signifi-cantly in several modes of oscillation, the convergedeigenvalue will depend on the given initial estimate.The results showed in Table 1 were obtained without theuse of accelleration factors, which are said to hasten

convergence[l] in the neighbourhood of an eigenvalue.The first computer implementation of the implicit

inverse iteration method incorporated refactorizationof the implicit form of matrix (A-qI) at every iteration,and the updatedeigenvalue estimate was calculated fromthe Rayleigh quotient (see Section III). The resultsobtained were disappointing as the converged eigen-values, in a good number of cases, were not the closestto the given initial estimate. Accordingly, the use ofthe Rayleigh quotient for eigenvalue estimation wasdismissed.

After testing different algorithm tunings it wasdecided to implement an automatic refactorization ofthe implicit form of matrix (A-qI) every time a certainnumber of iterations was reached or when the change inZ (see equation (5)) at any iteration was below ar"soft" convergence tolerance (taken as 10-4). The valueof the estimate 'q' used in the next factorization wastaken from the residual correction formula shown in (6).

Table 2 summarizes the results obtained with theimplicit inverse iteration algorithm as described above.The convergence tolerance used was the same as for theAESOPS algorithm : C = 10-6.

DRIVEN INITIAL ITERA-GENERATOR EIGENVALUE CONVERGED EIGENVALUE TIONSAT BUS ESTIMATE

30 0.0 + j4.0 -.1117 + j7.094 2430 0.0 + 57.5 -.1117 + j7.094 1930 0.0 + j9.0 -.1117 + j7.094 2231 0.0 + j4.0 -.2834 + j6.282 1731 0.0 + j7.5 -.2817 + j7.537 931 0.0 + j9.0 -.2817 + j7.537 932 0.0 + j4.0 -.2834 + j6.282 1532 0.0 + 57.5 -.2817 + j7.537 832 0.0 + j9.0 -.2817 + j7.537 1033 0.0 + j4.0 -.2489 + j3.687 1233 0.0 + 57.5 -.3707 + j8.613 1733 0.0 + j9.0 -.3707 + j8.613 1734 0.0 + j4.0 -.2489 + j3.687 2834 0.0 + 57.0 -.2834 + j6.282 1934 0.0 + 59.0 -.2968 + j6.956 1435 0.0 + j4.0 -.2489 + j3.687 1435 0.0 + 57.0 -.2968 + j6.956 1735 0.0 + 59.0 -.4670 + j8.963 736 0.0 + j4.0 -.2489 + j3.687 1336 0.0 + 57.0 -.2968 + j6.956 1636 0.0 + j9.0 -.4670 + j8.963 1137 0.0 + j4.0 -.4117 + j8.778 2837 0.0 + 57.0 -.1117 + j7.094 4537 0.0 + j9.0 -.4117 + j8.778 1838 0.0 + 54.0 -.2489 + j3.687 2838 0.0 + j5.0 -.3008 + j5.793 1738 0.0 + 57.5 -.2834 + j6.282 1539 0.0 + 54.0 -.2489 + j3.687 1239 0.0 + j9.0 -.2489 + j3.687 1339 -.111 + 57.09 -.2489 + j3.687 15

Table 1: Eigenvalues for the New England Test Systemobtained with the AESOPS Algorithm(Convergence Tolerance = 10-6).

It is seen from Table 2 that the implicit inverseiteration algorithm performs very satisfactorily andfinds all 9 eigenvalues associated with the electro-mechanical modes of oscillation. The initial eigenvalue

222

estimates given were purely imaginary numbers close toexpected values for the electromechanical oscillationfrequencies. The generators which participate signifi-cantly in the various modes of oscillation can be readily identified from the contents of the eigenvectorsassociated with the converged eigenvalues.

INITIAL CONVERGED ITERA- FACTOR-ESTIMATE EIGENVALUE TIONS IZATIONS|0.0 + 14.0 -.2489 + j3.687 9 20.0 + j4.5 -.2489 + j3.687 11 20.0 + j5.0 -.3008 + j5.793 19 30.0 + j5.5 -.3008 + j5.793 9 20.0 + j6.0 -.3008 + 55.793 14 30.0 + j6.5 -.2834 + j6.282 16 30.0 + j6.7 -.2968 + j6.956 23 30.0 + j7.0 -.1117 + j7.094 10 20.0 + j7.5 -.1117 + j7.094 12 20.0 + j7.9 -.2817 + j7.537 15 20.0 + j8.0 -.3707 + j8.613 20 30.0 + j8.5 -.3707 + j8.613 12 20.0 + j9.0 -.4117 + j8.778 18 30.0 + j9.5 -.4670 + j8.963 19 3

Table 2: Eigenvalues for the New England TestSystem obtained with the ImplicitInverse Iteration Algorithm_(Convergence Tolerance = 10 6)

Large System StudyFor the purpose of algorithm testing,the data for

a fictitious power system of large size was generatedas follows. The computer program developed is able tomodel power systems having many generators connected tothe same system bus. By using this feature of thecomputer program, each one of the generators of the NewEngland system was replaced by 10 smaller generators.These smaller generators had the same parameters buttheir loading and MVA rating were exactly one tenth ofthose of the original generator. This fictitious systemhad then 100 generators and a subset of its eigenvalueswere exactly equal to those of the original 10 gener-ator New England system. All the remaining eigenvalueswere associated with intra-plantmodes of oscillation[2].

Adopting the initial estimates shown in Table 1,the results obtained for the 100 generator system withthe implicit inverse iteration algorithm were exactlylike those of the 10 generator system. The only difference was in the computation time which rises for thelarge system. The implicit inverse iteration algorithmwould converge to those eigenvalues associated with theintra-plantmodes of oscillation depending on the initialeigenvalue estimate.

Table 3 shows the results for the fictitious 100generator system obtained with the AESOPS algorithm. Itcan be seen that the AESOPS algorithm tends to convergeto eigenvalues associated with intra-plant modes ofoscillation. This is an expected result since the me-chanical torque disturbance Tx applied to one out ofthe 10 small generators of a plant excites the intra-plant modes of oscillation to a greater extent than theinter-plant modes. Convergence also occurs towardseigenvalues associated with the inter-plant modes,depending on the initial estimate.

The AESOPS algorithm takes about 4.8 seconds periteration when solving the 100 generator system, whichis modelled here by 800 state variables. Each iter-ation basically involves one factorization of the as-symetrical Jacobian matrix and one repeat solution. Theimplicit inverse iteration algorithm, for solving the

same system, requires 3.9 seconds for the Jacobianmatrix factorization and about 1.3 seconds per iter-ation. Each iteration of this algorithm inivolves arepeat solution, one vector normalization and con-vergence checking on the calculated eigenvector. Whenthe same 100 generator system is modelled by 1400 statevariables, the implicit inverse iteration algorithm re-quires 9 seconds for the matrix factorization and about3 seconds per iteration.

DRIVEN INITIALGENERATOR FSTIMATEAT BUS

313333

353535

363639

0.0 + j4.0-0.3 + j8.0-0.4 + j9.00.0 + j9.0

-0.4 + j9.0-.46 + j8.90.00.00. 0

+ j4.0+ j9.0+ j4.0

CONVERGEDEIGENVALUE

-. 2834-. 3707-. 467 4-. 8863-. 8863-. 4670

-. 2968-. 4517-. 9037

+

+

+

+

+

ITERA-TIONS

j 6. 282*j8. 613*j l0.74**j 10. 23**j 10. 23-*j 8. 9 63*-j6.956*j10. 93**j 6. 87 6**

181622

23265

202324

Table 3 Eigenvalues for the 100 Generator SystemObtained with the AESOPS Algorithm byDriving a Single Generator at a PowerPlant.(Convergence Tolerance = 10-)

Key * - Eigenvalue associated with inter-plant modei* - Eigenvalue associated with intra-plant mode

The C.P.U. times given are for a VAX-11/780 com-puter and it is believed that they could be signifi-cantly reduced if the computer program was further optimized.

A Comparison Between the Two Eigenvalue SolutionAlgorithms

Some comments are now presented on the implicitinverse iteration and AESOPS algorithms based on theresults described in this section and on the experienceaccumulated in the analysis of other power systems.

The convergence criterion adopted in this paperwas much stricter than that of [11] so as to make theresults obtained with the AESOPS algorithm match thoseof the implicit inverse iteration. On normal use a lessstrict criterion may be adopted with considerable re-duction in computation.

Both implicit inverse iteration and AESOPS algo-rithms pay a price for their relatively small computermemory requirements since these one-eigenvalue-at-a-time solution methods do not allow a full stability as-sessment of the dynamic system as does the QR method,for instance. These algorithms may also have diffi-culties to locate eigenvalues which are tightly group-ed, and this problem occurs to a greater extent withthe implicit inverse iteration algorithm. Cases of non-convergence may occur with the two algorithms but arevery rare. Convergence may however occur towards eigenvalues which are not the desired ones.

The AESOPS algorithm tends to converge to thoseeigenvalues associated with the dominant electromecha-nical oscillations of the driven generator. This is avery good characteristic since in this area of workthese eigenvalues are of main concern. If the drivengenerator is equipped with a power system stabilizer,one dominant eigenvalue may be associated with the wellknown "exciter mode" [2] of this generator. As powerplants are generally modelled by one equivalent gener-

223

ator in large system studies, the fact that the AESOPSalgorithm converges to eigenvalues associated with intra-plant electromechanical modes of oscillation is not aserious drawback.

The inverse iteration algorithm converges to thesystem eigenvalue which is nearest to the given esti-mate. Provided the desired eigenvalue is not within acluster of eigenvalues, the better the estimate thefaster the rate of convergence of the algorithm. TheAESOPS algorithm does not always present this charac-teristic and may require an acceleration factor to hastenconvergence in the neighbourhood of an eigenvalue.

It can be seen from Table 1 that in one case theAESOPS algorithm converged after 45 iterations. However,if the non-converged eigenvalue estimate, obtained fromthe 10th iteration of the AESOPS algorithm, is used asthe initial approximation for the implicit inverse iteration algorithm,convergence is reached after 6 iterations. In most of the eigenvalue searches shown inTable 1 it would be computationally advantageous if theimplicit inverse iteration algorithm replaced theAESOPS algorithm once a "soft" convergence criterionwas satisfied.

Note that the AESOPS algorithm requires one matrixfactorization at every iteration while the implicit in-verse iteration algorithm,with the presently adoptedtuning, allows a maximum of 3 factorizations. As a consequence, the implicit inverse iteration algorithm may takeconsiderably less computation time than the AESOPS algorithm to obtain a good part of the desired eigenvaluesof a large power system. These algorithms have beenused to analyse a 57-machine model of a Brazilian in-terconnected power system, to determine the maximumpower interchange between areas and to find bettersettings for power system stabilizers. Depending on theproblem, one or the other algorithm was more appropri-ate.

A direct comparison between these algorithms cannot be made on a definitive basis. The implicit in-verse iteration algorithm is more recent and so,furtherimprovement could be made in connection with its appli-cation to the eigenvalue solution of large powersystems.

It is believed that a production program for eigenvalue estimation of large power systems should incorpo-rate both the implicit inverse iteration and the AESOPSalgorithms. The program should provide for the inde-pendent use of these two algorithms and also for theiruse in a combined mode as suggested in this section.

VII. RESULTS OF FREQUENCY RESPONSE CALCULATIONSThe algorithm presented in this paper is thought

to be more efficient than any other for obtainingfrequency response plots of the transfer functions be-tween any two variables in large power systems. Such analgorithm is highly useful for analytical studies ofpower system stabilizer tuning in a multimachine en-viroment [10] and computer duplication of field measurements [2,10]. Specific engineering objectives are notconsidered in this section, which is intended toprovidean indication of the efficiency of the proposed algo-rithm.

Results are presented for the 100-machine system,already described. Figure 2 shows the frequency res-ponse plot for the transfer function between the termi-nal voltage reference of the excitation system for oneof the generators connected to bus 30 and the power flowin the transmission line connecting buses 4 and 14 ofthe New England network.It can be seen that the frequencyresponse plot shows large variations for frequenciesclose to the imaginary parts of the eigenvalues:-.2489 + j3.687 and -.1117 + j7.094 (see Table l),whichare the dominant poles of the chosen transfer function.

u4

Figure 2: Frequency Response of Transfer FunctionAPline 4-14/Vref(generator at bus 30)

The plot shown in Figure 2 was obtained by evalu-ating the required transfer function for 100 discretevalues of applied frequency and the total C.P.U. timespent in this calculation was 510 seconds. The algo-rithm used is however more efficient when more than onetransfer function plot is simultaneously obtained. Asan example, the C.P.U. time needed to obtain 60 frequency response plots, where 20 outputs were considered foreach one of the 3 specified inputs, was about 740seconds. It is believed that the C.P.U. time couldagain be significantly reduced with a more optimizedprogrammning.

VIII. CONCLUSIONSOf previously published techniques, only the AESOPS

algorithm is capable of providing reliable and economicsolution of the eigenvalue problem for very large powersystems. The implicit inverse iteration method of thispaper now provides an alternative approach. The otheralgorithm presented in this paper is thought to be moreefficient than any other for obtaining frequency re-sponse plots of the transfer functions between any twovariables in large power systems.

The package of computer programs developed in thiswork, uses the routines for the formation of the powersystem Jacobian matrix in the following calculations:1. Exact eigenvalue closest to a specified point in

the complex plane, and associated eigenvector (im-plicit inverse iteration algorithm).

2. Exact eigenvalues associated with the dominantmodes of oscillation in the system (AESOPS algo-rithm).

3. Frequency response plots of transfer functions be-tween any two specified variables in the system.

4. Full eigensolution of state matrix: A = J1-J2J4 J3(In this case, the computation time required by theQR method [3] is prohibitive for large powersystems).This is an important feature regarding software

maintenance since a large part of the coding is commonto all programs.

The Jacobian matrix formulation used is very general and allows for the representation, in different de-

224

grees of detail, of the power system components whichare important to the analysis: synchronous generatorsand associated controllers, induction motors,non-linearloads of different characteristics and static VAR compensators.

The AESOPS and the implicit inverse iteration algorithms are recently developed techniques and, accordingly, a period of evolution is anticipated. Thealgorithm confines itself to the power system stabilityproblem. The implicit inverse iteration algorithm,the other hand, will undoubtedly provide an attractivetechnique for eigensolution in other engineeringcation areas where very large system matricesvolved.

ACKNOWLEDGEMENTThe author is grateful to Dr. Brian Stott

helpful discussions.

REFERENCES

[1] R.T.Byerly, R.J. Bennon and D.E. Sherman, "Eigen-value Analysis of Synchronizing Power Flow Oscil-lations in Large Electric Power Systems",IEEEConf., pp. 134-142, 1981.

[2] K.E. Bollinger, J.Hurley, F. Keay, E. Larsen,D.C. Lee, "Power System Stabilization Via Ex-citation Control", Tutorial Course Text, IEEEblication 81EH0 175-0 PWR, 1981.

[3] J.H. Wilkinson, "The Algebraic Eigenvalue Problem",Clarendon Press, Oxford, 1965.

[4] N. Martins, and R.M. Stephan, "Power Systemnamic Stability Limits Considering the EffectsNon-Linear Loads", 10th IMACS World CongressSystem Simulation and Scientific Computation,Montreal, Canada, pp. 120-122, August 1982.

[5] D.H.Vorley, "Numerical Techniques forStability of Large Power Systems", Ph.D. Thesis,Univ. of Manchester, U.K., 1974.

[6] B. Dembart, A.M. Erisman, E.G.Cate, M,A.and H.W. Dommel, "Power System Dynamic Analysis"

,

RP670-1 Final Report, EPRI EL-484, Palo Alto,California, July 1977.

[7] D.W. Olive, "New Techniques for the CalculationDynamic Stability", IEEE Trans. PowerVol. PAS-85, pp. 767-777, July 1966.

[8] 0.I. Elgerd, "Control System Theory", Mc CrawHill,1967.

[9] 0. Alsa,, B. Stott, W.F. Tinney, 'Sparsity-OrientedCompensation Methods for Modified Networktions", Paper n9 82 SM 324-2, presented atPES 1982 Summer Meeting.

[10] P. Kundur, D.C. Lee and H.M. Zein El-Din,System Stabilizers for Thermal Uni'ts AnalyticalTechniques and On-Site Validation". IEEEPower App.y Vol. PAS--00, pp. 81-95,1981.

[11] R.T. Byerly, D.E. Sherman and R.J. Bennon, "Fre-quency Domain Analysis of Low-Frequency Oscilla-tions in Large Electric Power Systems", RP744-1Interim Report, EPRI EL-726, Palo Alto, California,April 1978.

DiscussionA. J. Calvaer (University of Liege, Belgium) and M. Stubbe (TractionelBelgium): The author is to be congratulated for his paper which presentsthe state of the art on the field of eigenvalues computation for large powersystems. The method that he has implemented (implicit inverse iterationalgorithm) occurs efficient).However, some questions arise as regards practical performances:

1) How to choose the initial eigenvalue estimate? In Table 2 we arelooking for for nine electromechanical eigenvalues. The estimates[0.0 + jO.4 + jk 0.05, k =1...11] converge to seven different eigen-values. How were determined the initial values (0.0 + j 6.7) and0.0 + j 7.9)? Is there no risk to multiply the attempts before gettingthe right estimate?2) In the ficticious large size system, the eigenvalues are well separatein the complex plane. In a real system, some eigenvalues would bevery close to others. In our opinion, this situation could be detrimen-tal to the efficiency of the method, particularly as regards the choiceof initial estimates. Has the author some experiment in this field?3) The "package" of computer programs doesn't contain any unit stepresponse calculation. Can we actually miss it? Has the numericalintegration of the linearized system (1) been imagined and has thecorresponding computation time been compared with the one of theeigenvalues, eigenvectors and requency response?

Hugh Rudnick (Universidad Cat6lica de Chile, Santiago, Chile): Theauthor is to be congratulated for his valuable contribution to power systemsteady state stability analysis. The implicit inverse iteration method andthe frequency response method have a significant potential for the studyof large scale systems.The discusser is aware of the difficulties of analizing instabilities in

large power systems as he performed extensive eigenvalue studies of areal situation occuring in a European system (12) and he suggested theincorporation of system stabilizers which were eventually implemented.The computer programs developed kept storage requirements to aminimum through the use of sparsity techniques and bifactorized matrixinversion methods for the handling of the network equations, andpiecewise techniques for the building of the state matrix. Nevertheless,the final state matrix had to be stored and all eigenvalues had to bedetermined.The need to determine all eigenvalues can become a serious drawback

as the size of the system increases and the modeling complexity grows.The paper is a new contribution in a line of research that has orientedto specific eigenvalue calculations (1, 13, 14). Simplified tools and classicalconcepts to determine basic unstable modes have also been used to tacklethe problem(15, 16). In spite of these developments, many utilities inthe world still analize steady state stability with conventional step by steptransient stability computer programs, or do not analize it at all, unlessoperational problems arise.The author goes into extensive work to adequately model power system

load in the system Jacobian matrix. The question is on the practical needto go into that extra complication, considering the usual lack of dataon the matter. Moreover, induction motor loads, although contributingwith stabilizing torques, have traditionally felt to have a negligible in-fluence in relation to the effect of the excitation loops. Has the authorassessed quantitavely the influence of load system stability? Could thefrequency response method be useful to show its influence? The samequestions apply to the influence of the static var compensators.A drawback of the proposed algorithm is the need to calculate eigenvec-

tors to associate generator participation and, therefore, evaluate stabilizerlocations. How does the author propose to deal with this limitation whichis not present in the AESOPS algorithm?The power frequency response method seems a promising tool and

the author is to be commended to pursue further research on it. It wouldbe valuable to explore its applications into the location and tuning ofpower system stabilizers and the design of complete AVR and governorschemes. The influence on stability of the power transmission throughdetermined system lines could be evaluated with this method, providinga guide in operational procedures. The author's comments would beappreciated.

REFERENCES

[121 H. Rudnick, "Steady state stability of multimachine powersystems," Ph.D. Thesis, The Victoria University of Manchester,January 1982.