lEEE Transactions on Power Systems, Vol. 12, No. 3, August 1997 1393

A SENSITIVITY ANALYSIS OF EIGENSTRUCTURES

E. E. S. Lima, Member, IEEE. Department of Electrical Engineering

Universidade de Brasilia 70910 - 090, Brasilia, DF, Brazil

Abstract - The concepts of condition inumber, taken from numerical linear algebra combined with logarithmic sensitivity are applied to analyze the robustness for model uncertainty of power systems eigensets. Sensitivity matirices to facilitate the estimations of the eigenvalue sensitivities in relation to ele- ments of the state matrix are proposed. Eigenvector derivatives are analyzed and it is evidenced the trend of power system ei- genvectors to present large sensitivities. A case study of a real- istic single machine with excitation system connected to an infinite bus example is presented: Lack of stability robustness and also of eigenvector robustness are demonstrated.

Keywords - Eigenanalysis, Selective Modal Analysis, Sensi- tivity Function, Robustness, Ill-Conditioned Problem.

I. INTRODUCTION

In dynamic stability studies, non-limnear equations that represent an actual electric power system are linearized around some operating point and a system of first order dif- ferential equations is established, as described below:

x(t) = Ax# . (1) Eigenvalues and eigenvectors have been utilized to ex-

amine the properties and solutions of (1). In fact, i fA ER-has distinct eigenvalues, ?(A)= (Al, ..., 1.1, the state variable vector is given by

n n x(t) = c c k v k E h k t = ~ $ x ( 0 ) v k Ehkt (2)

k=I k=l

where C, are constants that depend on the initial conditions,

v k are the right column eigenvectors, yf are the transposed

PE-604-PWRS-0-11-1996 A paper recomrnended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for publication in the IEEE Transactions on Power Systems. Manuscript submitted August 5, 1996; made available for printing November 25, 1996.

conjugated left eigenvectors corresponding to h~ , (A-1) and 40) is the initial state vector [1,2].

Furthermore, the partial derivative of a simple eigenvalue h k with respect to a parameter a [3],

(3)

has been used to place h(A) properly in the complex plane. If the subsystems have eigenvalues that are close to each

other, even if they are weakly coupled electrically [2], they may exhibit considerable interaction. Some methods to study these troublesome couplings, based on the variations of their related eigenvectors as one or more parameters of the system are varied, were given in 14).

The existence of several types of machines and the variety of their operating conditions motivated the authors of [5] to try to adjust the stabilizing signal to accommodate a wide range of situations. However, stabilizers, designed to im- prove damping of one mode, can produce unexpected effects on other modes [6].

Many sensitivity functions have been used to measure the effects of parameter changes on system performance [7-81. In particular, the rate of change of the roots of the charac- teristic equation has been expressed by the logarithmic sen- sitivil'y (LS) given by:

sh

(4) - k

where k is any system parameter. Values so that 5': 1 1 indicate increasing relative errors.

In the care of matrix computations specialists has warned [9-131 that the eigenvahe of A(6) E Rm ,

[o 1 0 e.. 01

10 0 1 *.* 01 . . . . A(S)=j : : *. -. : I (5)

0885-8950/97/$10.00 0 1997 IEEE

1394

differs in absolute value by lo-''), if 6 = 0 or 6 = lo-'' However, this last variation is so small that it may not be distinguished on a computer. Consequently, problems whose results are very sensitive to small changes in the data have challenged specialists in numerical linear algebra.

When dealing with power systems some characteristics must be taken into account:

Numerical analysts were faced with problems because they verdied that estimations upon ill-conditioned problems could give rise to results that diverge com- pletely from the exact solutions. In electric power sys- tem engineering comparisons between estimations and

Severe lack of robustness for model uncertainty are also de- tected on two eigenvectors.

11. ASSESSMENT OF ABSOLUTE AND RELATIVE SENSITIVITIES

A Resultspom Numerical Linear Algebra A condition number (CN) of a problem is a bound on the

error induced in the solution of the problem by a perturba- tion in the original matrix. CN's for eigensets were esti- mated by means of first-order perturbation approaches [9 - 131 The following bounds were set up [ll]:

Ifh is a simple eigenvalue of a matrix A and its right and left eigenvectors scaled (A-1) so that 1 1 ~ 1 1 ~ = y H v = 1, then

actual results are much more complex. Ih - h'l EllYll2 (6) Many parameters are not known accurately. Some of them can hardly be estimated within relative errors of lo-' Not quite as many people consider the fact that time- domain performances depend on the scalar products

where h'is an eigenvalue of a slightly perturbed matrix (A+@ with E = IIE112, the Euclidean norm of E. Hence,

lbllz is a CN for 31. and, if it is large, then A. is ill condi- tioned. Following , it was settled that

(7) yFx(0). The expression at the far right in (2) shows E

that the left eigenvectors weight the excitation of the I '* I

system modes by the initial state vector. Without the guarantee that the eigenvalues are robust in relation to the data, much effort deployed for the as-

where V I is an eigenvector of (A+@ and h k an eigenvalue Of A, Other than h . Stewart [ 111 still stated that: "lf h is

sessment of small signal and voltage stability tends to be irrelevant, ( A design is said to be robust if it performs well for substantial variations of the system parameters from the designed values).

Therefore, the objective of this paper is to present evi- dence of the need to assess the reliability of the calculated eigensets and to suggest some handy sensitivity matrices.

At the beginning of section 11, a brief review of condition numbers of eigenvalues and eigenvectors is given based on [ 1 I]. It is demonstrated that the condition number of an ei- genvalue is an upper bound of its derivatives in relation to the elements of the state matrix. The derivative of an eigen- vector in relation to any element of state matrix is set and an useful insight of eigenvector sensitivity is given. Finally, for easier robustness tests, some matrices, generalizations of eigenvalue derivatives and LN's are set forth.

The application of these tools is illustrated for a single machine with a W Low rE Brushless exciter connected to a infinite bus system [14]: First, overall evaluations of absolute and relative sensitivity of eigenvalues are given. . Next, LN's of dominant and worst damping eigenvalues and of their eigenvectors are exarmned in detail. It is shown that t b s small system does not have stability robustness for model uncertainty [7]: A 1% variation in only one element of the state matrix turns the system unstable. The symbolic expres- sions of this and others A-matrix elements in relation to which the eigenvalues have larger sensitivities are given.

near another eigenvalue of A then its eigenvector will be ill conditioned. However, he also remarked that: It should not thought that the only ill-conditioned eigenvectors are those corresponding to poorly separated eigenvalues. Indeed, it will be shown that well separated but ill-conditioned eigen- values can also give rise to ill-conditioned eigenvectors.

B. Absolute and Relative Sensitivities of Eigenvalues.

If all eigenvectors are scaled so that Ikll, = Ild12 = 1, than

Golub [E] shows that (8) is an upper bound on the de- rivative of A(&) at E = 0, by differentiating

( A + EF)v( E) = A( E).( E)

with respect to E , with 11F11, = land ~\v(E)/\, = 1. This ap-

proach leads us to look for the relation between th~s CN(L)

and the derivatives - . a . day

If the derivative of h k is taken in relation to a(i j ) instead of a system parameter a, then (3) can be rewritten as:

1395

* where y k ( i ) is the complex conjugate of the i& coordinate

of the left eigenvector and vk (j) the j th coordinate of the right eigenvector, both corresponding to h k , The numerator and denominator of (9) are homogenous functions of v and y , that is, F(cy, m) = c 2 F ( y , v ) . Therefore, if‘ we wish all the range of the derivative of h k , it is siacient to examine

it on the set lbk 112 = 1l.k 112 = L The upper bound for the

eigenvalue derivative (9) is attained when Iyk(i)l= Ivk( j)l = 1. Thus, CN (1) is an upper bound on m. - for ( i =1, ... n) and ( j = 1 ,..., n ) .

dav

A matrix whose elements are the derivatives of h in relation to the correspondent entries of A , D i E CW, can be built from (9):

*

where the numerator is the outer product of the conjugate of its left eigenvector y and its right eigenvector v .

From (4) and (IO), a matrix of the logarithmic sensitivities of h , S,h E Cnm , can be built:

*

where the symbol .* denotes element-by-element, multiplication.

C. Eigenvectors ’ Sensitivities

Any variation on the magnitude of an eigenvector corre- sponds to another variation on its associated left eigenvector, insofar as their scalar product remains unchanged (A-1) and consequently their participation in the state (2). Thus, only the changes in the eigenvector direction are of interest.

Let Avk = hkvk. zf we take partial derivatives on both sides of this expression with respect to a(ij), we get,

The first term on the right side is a vector with all compo- nents equal to zero, except the i fh component, vk( j) . Since

the eigenvectors ofA span R” we can write

Thus, considering (9), (13), (Al) and after some manipula- tions on (12),

*

l+k

where a k is the component of the derivative along V k and does not have any influence over the state vector. However, looking at (14) it can be seen that the derivative of an eigen- vector depends on the distance of its associated eigenvalue porn the others and on the CN(h)’s of these. Golub stated in [ 121, when the eigenvalues coalesce, the eigenvectors asso- ciated with nearby eigenvalue are wobbly .

MSO, a vector of logarithmic sensitivity of vk , svk E C” OZJ

can be built from (4) and (14),

\I#k

where the symbol -/ denotes element-by element quotient.

D. Infrences fiom, the derivative of an eigenvector

Unfortunately, power systems tend to have some wobbly eigenvectors: Indeed, when the machine parameters are in per unit, it is realized that they are approximately equal, in spite of the maclune size differences. Conse- quently, the eigenvalues fall in groups, forming clusters in the complex plane, according to their similar oscilla- tory modes, [ 16 J . Equation (14) gives insights into the “resonance” phe- nomena studied in (21 and [4]. Eigenvectors associated

0

TABLE I

EIGENVALUE CHARACTERISTICS OF THE CASE STUDY

Eigenvalues Damping CAT@) m4wlD-%i’j)l m m l s 2 ( j > j)l h(A) x pu Ratios, 5 - 35.9 f 999i 0.036 12.41 4.95 6.29 - 26.525 - 3.79 2.01 1 .oo - 98.6 - 21.31 15.43 16.59 - 121.7 - 30.46 20.65 9.36

-167.2 f 8661 0.19 57.45 30.09 0.49 4 . 4 9 4 1.2i 0.39 41.26 18.30 557.22 -1.6 i 28.9i 0.057 60.10 38.53 31.27

1396

with nearby eigenvalues even from hstanced machines have large sensitivities. Theoretically, their derivatives (14) become infinite when the eigenvalues coincide.

IV. A CASE STUDY

A single machine with a W Low ZE Brushless excitation system connected to an infinite-bus ( B 1 ), is examined, [ 141

Table I shows its eigenvalues, damping ratios, condition numbers, magnitudes of the largest magmtude elements of D i and S i .The data used was taken from [ 141. a) As expected from (12), the CN’s are upper bounds for the

magnitudes of __ .

b) The L S s of eight eigenvalues have magnitudes larger than 1. The one with the largest magnitude, for the fre- quency O.O72Hz, asserts a relative error of 557% in the magnitudes of eigenvalues for an eventual relative error of 1% in a17. (shown in Table Cl). In Table XI, which de- scribes the coordinates of its eigenvector shows that this mode manifests itself mainly in regulator and field output voltages. The real part of these complex eigenvalues are the smallest, thus they contribute to the bulk of the settling time. But, with such LS’s values, their actual positions on the complex plane can not be determined. c) Table I shows that the 59.94 J3.z oscillatory mode has the worst damping ratio and its LS is 6.29. An eventual relative error of 10% in a41, will cause an eigenvalue magmtude error of 62.9%. In the 2nd column of Table 11, it can be seen that it excites mainly the direct and quadrature axis compo- nents of stator current, quadxature axis component of damper current and regulator output voltage.

dh

da,

TABLE I1

EIGENVECTORS AND STATE VARIABLES OF THE CASE STUDY ~~

State PU PU vector

Eigwvecior of Eigenvector of ( - 0.49 + 1.2 i ) x

- 0.0163 + 0.0069i

(- 35.9 + 999 i ) x

Id 0.1580 + 0.25271 0.0403 - 0.0130i 0.0004 - 0.0042i

- 0.0101 + 0.0040i - 0.0002 + 0.0005i

0.0000 - 0.OOOOi - 0.0207 + 0.0088i

0.0125 - 0.0053i - 0.0132 + 0.0036i

0.3035 + 0.67671 0.2771 + 0.60831

- 0.0675 - 0.07991 - 0.0860 - 0.16771 0.2628 - 0.13721 - 0.2613 + 0.12481

0.0001 + 0.0001i 0.0001 - 0.0001i - 0.0050 - 0.0019i - 0.0040 - 0.0107i 0.7099 - 0.4102i - 0.051 1 - 0.13391

MAGNKUDES OF THE LOG. SENSlTNlTlES OF (-0.00049+0.0012i)

MAlRIX ROWS MAlRIX COLUMNS

Fig. 1. Plot ofmagnitudes oftheLSofthe eigenvalue- 0.00049 + 0.0012i in relation to the entries of A.

Figs. 1 and 2 [ 141 show plots of the S,h matrices of these

two modes. In Fig. 1, four entries are greater than 100. In Table C-1 are the biggest twelve entries of each of these

S i matrices. It is followed by the symbolic expressions of the respective a y . It can be seen that they are functions of self and mutual inductances. But these parameters change with the operating conditions of the machme. Moreover, the terms ~ 9 , ~ 1 and al 1,1 are functions of SE that represent the effect of saturation in the exciter. These large values of LS, for both modes, and the nature of the corresponding a y ,

allow to state that in practice, the actual values of these ei- genvalues are unpredictable.

For instance, if 1770.2 is used instead of 1752.7 at a17 in @1), a perturbation of only 1%, seven eigenvalues in Tab. I rest practically unchanged. But, the complex pairs -0,0016 t- 0.0289 i and - 0.0005 * 0.0012 i pu are replaced by +

I I

MAGNKUDES OF ME LOG. SENSmVlllES OF (-0.0359+0.9991)

MATRIX COLUMNS

Fig. 2. Plot of magnitudes ofLS’s ofthe eigenvalue - 0.0359 + 0.999i in rela- tion to the entries of A

1397

expression of the state matrix is needed, in order to accom- plish and accurate determination of which system parame- ters are involved and their estimated precision.

c; 2000 lo00

0 0

15

MATRIXROWS 15 ' 0

Fig. 3. %Norms of the vectors of the LS 's ofthe eigmvector corresponding to the eigenvalue - 0.00049 + 0.0012 i pu, m relation to the erRries of A

0.0042 f 0.0309, -0.0123 and - 0.0003 pu, and the system becomes unstable! Thus, this system has not stability ro- bustness for model uncertainty.

Fig. 3 and 4 show the 2-norms of the vectors of the eigen- vectors' logarithmic sensitivities in relation to the aij's ,

corresponding to the modes -0.00049 % 0.0012i pu and - 0.00359 f 0.999i pu, respectively. It can be seen that some LS 's of both eigenvectors are extremely large.

E. Practical Procedures.

Each entry &, in matrix A is a combination of several sys- tem parameters and conversely a single ]parameter may cause several entries. Thus, if an eigenvalue ox eigenvector has too large sensitivity to changes on any ai j , it likely has also large sensitivity in relation to some system parameters. Thus, if large LS in relation to any aij is detected, a symbolic

2501 200 A f 150

? 100 N

50

0 0

15

MAWCOLUMNS 1 :ig. 4.2-Nams ofthe vectors ofthe LS ofthe eigmvedm correspondmgtothe

eigenvalue - 0.00359+0.9991 pu, m relatica to the entries of A

MA= ROWS

V. CONCLUSIONS

This paper applies the concepts of condition number from numerical linear algebra, and logarithmic sensitivity to ana- lyze power system eigenset robustness for model uncertainty. Matrices of derivatives and logarithmic sensitivities of ei- genvalues, in relation to the elements of the state matrix facilitate the assessment of their sensitivities. By the expres- sion of the derivative of an eigenvector it is evidenced that as the eigenvalues of actual electric power systems have the tendency to fall closely in groups on the complex plane, wobbly eigenvectors must be expected. A case study of a realistic single machine, with a W Low zE Brushless exciter, connected to an infinite bus system is developed. It is dem- onstrated that this small system does not have stability ro- bustness for model uncertainty: A 1% variation of only one element of the state matrix turns it unstable. Severe non robustness of two eigenvectors was also ascertained.

VI. APPENDIX A

A . Right-Le3 Eigenvector Scalar Product If the matrix A E R - has simple eigenvalues, its n line-

arly independent eigenvectors can form a non singular ma- trix v=[vl v2 ..- v , ] . n u s , a matrix Y =[VI y2

- - a y n ] such as Y H =V-', can be defined. Also, A V =

VA withA=diag(h1,h2 ,..., h,) and Y H A = A Y H . The vector columns y j are called the left eigenvectors of A. More formally, it can be written,

Y y v J = ~ ~ ~ ~ ~ 2 ~ ~ v j ~ ~ 2 cosL(YlyvJ)=6! , (All

where 6,, = 1 for i =j or 6,, = 0 for i f j . Usually, the

eigenvectors are scaled so that 11 v /I2 = 1 .

VII. APPENDIX B

A. Single-Machine and Znjnite Bus System [14].

State-space representation for a synchronous machine, with a W Low ZE Brushless exciter, connect to a infinite bus [14, pages 290-2911 are shown in (E%-1). The A-Matrix was obtained by A = - M ' K and the numerical expressions of these matrices are on page 290, [ 141. However some entries have to be calculated with numerical values from Table 7.8, page 291, for the case of the W Low zE Brushless exciter.

B. Nomenclature.

- - - id -36.080 0.4361 14.069 -3488.3 -2548 -2445.2 1752.7 0 0 0 -0.0489 id 'F 12.434 -4.9519 76.613 1202 878.10 842.74 -604.1 0 0 0 0.5548 iF iD 22.835 4.3611 -95.674 2207.7 16125 1547.6 -1109.3 0 0 0 -0,4886 io

'9 3589 26488 2648.9 -36.059 90.107 1775.6 2387.4 0 0 0 0 I.1 -3504 -2586.4 -2586.4 35.208 -123.37 -1734 -2331.1 0 0 0 0 iQ 'Q

w -0.GiI78 -0.2026 -0.2026 -0.7994 -0.4365 0 0 0 0 0 0 W

s 0 0 0 0 0 1000 0 0 0 0 0 6 v; 12252 12589 12537 9676.3 10268 24458 -5696.32 -26525 0 0 0.1887 V, v; 0 0 0 0 0 0 0 0 -5.3050 14.147 -15.735 V, VR 0 0 0 0 0 0 0 -53050 -53050 -132.63 0 VR

-Em)- - 0 0 0 0 0 0 0 0 0 176.84 -1%.69 - E m -

A. Entries of S with the biggest magnitudes. I $ 1 Table C.1 shows the twelve entries of S i , with the

TABLE C-I

THE TWELVE ENTIUES OF THE LS MATRICES WITH THE LARGEST MAGNITUDES OF THE EIGENVALUES - 0.0359 & 0.999 i

AND - 0.0005 -I 0.0012 i

~-(0.49+1.21)E-3 PU s-0.0359+0.999z pu

a41

a51

a42

a52

a43

a53

al 4

a24

a34

a15

a25

a35

6.29 - 0.32 i

- 4.49 + 0.28 i

- 1.6 + 0.32 i

1.14- 0.24 i

- 2.93 - 0.1 i

2.1 + 0.05 i

6.25 + 0.16 i

- 1.59 - 0.03 i

- 2.92 - 0.05 i

- 4.46 - 0.27 i

1.13 + 0.06 i

2.08 + 0.11 i

a910

a1 110

a42

a17

a27

a37

al 4

a24

a34

a15

091 1

a1111

39.5 + 63.0 i

- 39.6 - 63.1 i

12.6 + 6.77 i

77.7 + 555 i

- 28.6 - 203 i

- 49.3 - 352 i

- 62.9 - 542 i

23 + 198 i

39.9 + 343 i

- 13.6 - 4.48 i

- 40.02 - 62.9 i

40.0 + 63.0 i

KF . 1 a910 = - ' a l l l o =-

ZFTE ZE

1399

, where ;all11 = --- ('E + K E ) KF(sE + K E ) a911 =-

ZFZE "E

C. Nomenclature [14]:

do stator voltage, direct axis component

k=&T KE exciter constant related to self-excited field KF regulator stabilizing circuit gain Ld , L, unsatured d and q axis synchronous inductances

LF rotor self-inductance LD, LQ unsatured damper self-inductances, &red and quadrature axis windings M D , M Q mutual inductances between stator and damper

windings, direct and quadrature axis windings M F mutual inductance between stator and rotor windings MR mutual inductance between field and damper windings SE exciter saturation function z E exciter time constant ZF regulator stabilizing circuit time constant wo = 1, generator rated speed V, infinite bus voltage

E. ACKNOWLEDGES

The author wishes to thank Dr. Alqumdar Pedroso of COPPE - UFRJ at Rio de Janeiro, Brazil, for his several fiuuitful discussions when there were only intuitions. His demands for pradical criteria of condition estimations were the origins ofthis work.

X. REFERENCES

[l] A. Semlyen, L. Wang, "Sequential Computation of the Complete Eigensystem for the Study Zone in Small Signal Stability Analysis of Large Power Systems", IEEE Trans. on Power Systems, vol. 3, No. 2, pp. 715-725, May 1988. [2] J. E. Van Ness, F. M. Brash, G. L. Landgren and S. T. Naumann, "Analytical investigation of ctynarmc instability occurring at Powerton station", IEEE Transactions on Power Apparatus and Systems, vol. 4, JdyIAug. 1980, pp. 1386-95. [3] J. E. Van Ness, J. M. Boyle, and F. P. Imad, "Sensi- tivities of Large Multiple-Loop Control Systems," IEEE Trans. Automatic Control, vol. AA-10, July 1965, pp. 308- 315.

[4] D. K. Mugwanya, J. E. Van Ness, "Mode Coupling in Power Systems" IEEE Trans. on Power Systems, vol. PWRS- 2, no. 2, May 1987, pp. 264-270. [5] F. P. de Mello and C. Concordia, "Concepts of Synchro- nous Machine Stability As Affected by Excitation System Control," IEEE Trans. on Power Apparatus and Systems,

[6] F. P. de Mello, P. J. Nolan, T. F. Laslowski, and J. M. Undnll, "Coordinated Application of Stabilizers in Multima- chine Power Systems", IEEE Trans. on Power App. and Systems, vol. 99, No. 3, May 1980, pp. 892-901. [7] G. F. Franklin, J. D. Powell, A. Emami-Naeini, Feed- back Control of Dynamic Systems, Addison-Wesley pub- lishing Company, 1994, pp. 144-145,205,420 - 431. [8]B. C. Kuo, Automatic Control Systems, Englewood Cliffs, New Jersey: Prentice-Hall, Inc, 1991, pp. 423-426. [9] J. M. Wilkinson, "The Algebraic Eigenvalue Problem", Claredon: Oxford Univ. Press, 1965, pp. 62-109, [lo] J. M. Willanson, "Modern error analysis", SLAM Rev.,

[l 11 G. W. Stewart, Introduction to Matrix Computations, New York: Academic Press, 1974, pp. 184-187,289-299. [12] Gene H. Golub, C. F. Van Loan, Matrix Computa- tions, Baltimore: The Johns Hopktns University Press, 1989,

[13] P. Lancaster, M. Tismentsky, The Theory of Matn'ces, Academic Press, Inc, 1985, pp. 383-405. [14] M. Anderson and A. A. Fouad, Power System Control and Stability, IEEE Press, Revised Printing, 1993, pp. 159, 214, 286 -293."Matlab, User's Guide, For Microsoft Win- dows", The Mathworks, Inc., July 1993. [ 151 "Maple V Language Reference Manual", Spring Verlag, 1991. [16] G. Gross, C. F. Imparato, P. M. Look, "A Tool for Comprehensive Analysis of Power System Dynamic Stabili- ty", IEEE Trans. on Power Apparatus and Systems, vol.

vol. 88, April 1969, pp. 189-202.

13, 1971, pp. 548-568.

pp. 79-81, 341-348.

PAS-101, NO. 1, Jan. 1982, pp. 226-234.

E. BIOGRAPHY

f

i f

Evandro Lima ( M 8 1 ) was born on August 23, 1941, m Rio de Janeiro. He received his BS m Electronic Engineering ftm ITA, SHo Jose dos Campos, Brazil, in 1966 and his DEA and Dr. Ing degrees in Automation at the UniversitC Paul Sabatier, Toulouse, France, m 1969 and 1972, respectively. During this period he was assigned to the Eledrical Department of EPUFP, at Campma Grande, Paraiba. He is retired senior ledurer and associated researdm of the E l d c a l Department of the Universidade de Brasilia. His mam interests are in robust and fizzy logic based controllers for power systems.