1880 IEEE Transactions on Power Systems. Vol. 9. No. 4, November 1%

AN EFFICIENT IMPROVEMENT OF THE AESOPS ALGORITHM FOR POWER SYSTEM EIGENVALUE CALCULATION

D.M. Lam H. Yee B. Campbell The University of Sydney, Australia

ABSTRACT

This paper dpcribes a simple but efficient modification to im- prove the well known PEALS/AESOPS algorithm for power system eigenvalue calculation. The moQified algorithm is a N e w t o n - R a p h ikration scheme which converges signifi- cantly faster than AESOPS. The algorithm is linked to the operational trader matrix equation method proposed by Okubo et id pad Chats et al. An dficient opurtional m& trix based fotrmtla is suggested to calculate the sensitivity of an electrdechamical eigenvalue rrith respect to the tr-fer function of a power system s t a b h r . Results for a 21 gener- ator pawer system are described.

keywords: small signal stability, eigenvalues

INTRODUCTION

Eigenvalue calculation is universally used in the quantitative analysis of power system small signal stability. The QR al- gorithm, which ia conventionally used for eigenvalue calcula- tion, has been found u n d t a b k 6or very large syskms, for which special analysis methods have been dwaloped [1E[12]. The methods take full advantage of the nature of ~ m d signal stability and of the network s p s i t y of large power systems. They are designed to calculate a small number of eigenvalues, particularly those awociated with generator rotor oscillations. This is adequate in most small signal stab&ty rtudies. In typ- ical systems, interest is centered on only a small number of poorly damped rotor oscillation modes.

One of the early and best known methods is the AESOPS algorithm of [3] which calculates rotor oscillation eigenval- ues 'one at a time. Sparsity based implementations of the AESOPS algorithm are reported in [5], [9] and [ll]. The A E SOPS algorithm requires an initial guess of the eigenvalue and choice of a generator at w h d a disturbing torque is applied. An appropriate disturbed generator is one which participates significantly in the oscillatory mode described by the eigenvalue. AESOPS has the disadvantage of requiring

94 Wn 250-1 PWRS A paper recommended and aRRrovad

repeated factorization of large matrices, unlike more recent methods such as the modiied Arnoldi method [lo, 111. The latter can also find several eigenvalues a t a time. Despite the higher computatrion cost, there is a distinct advantage in using AESOPS became it always converges to the dominant elec- tromechanical modes of the disturbed generator. AESOPS is actually a mon+variable quasi Newton-Raphson method with the Jacobian, the derivative of the torque disturbance, approximated heuristically.

A more general algorithm is proposed in [12] and related to AESOPS. The algorithm is a mono-variable Newton-Raphson iteration d e m t &hen cakulating rotor oscillation eigenval- ues. Reference [13] describes another improvement to A E SOPS, formulating it as a problem in finding zeros of a trans- fer function and solving iteratively by the Newton-Raphson method. Both algorithms use the augmented system equa- tion approach to take advantage of sparsity. However, [12] does not give details on the way sparsity is used to improve computation efficiency, and [13] is a multi-variable iteration scheme which requires that more variables be initialized and solved in each iteration. Moreover, the derivations of both improvements are rather complicated.

The major objective of this paper is to show that the actual derivative of the torque disturbance of the disturbed genera- tor of AESOPS can be computed efficiently and directly from the AESOPS algorithm itself. This is achieved by a sim- ple modification to the equations of the PEALS/AESOPS algorithm[ll] which transforms AESOPS into an efficient Newton-Raphson algorithm. The modified AESOPS algo- rithm resembles the original PEALS/AESOPS algorithm. There is one additional step in each iteration but the extra computation time incurred is not great.

We also show the underlying relationship between PEALS/AESOPS (and the modified AESOPS) and the oper- ational transfer matrix equation approach p e a l l y proposed by [l] and [2]. The same operational transfer matrix equ& tion can be derived from the AESOPS equations, with the following differences: (1) generators are modelled in the more flexible state space form and (2) network sparsity is preserved. Use of the operational transfer matrix equation has the ad- vantage that a reasonable initial guess of the rotor oscillation eigenvalues and eigenvectors (rotor speeds only) is available.

by the IEEE Power Sys& Engimering Comittei- of the Furthermore, the operational transfer matrix equation is the IEEE Power Engineering Society for presentation at Starting Point for the derivation of an efficient formula, a P the IEEE/PES 1994 Winter Meeting, New York, New York, plicable to large systems, for calculating the sensitivity of an January 30 - February 3 . 1994. Manuscript submitted oscillation eigenvalue with respect to the transfer function of November 2, 1992; made available for printing a power system stabilizer(PSS). December 7, 1993.

The superior performance of the proposed modified A E SOPS algorithm compared with the original algorithm, and

0885-8950E94/so4.00 43 1994 IEEE

1881

the method of eigenvalue sensitivity calculation, are demon- strated by analysis of a 21-generator power system.

From ( 5 ) and (6 ) , we have, in the Laplace domain

Ai = Gl(p)Av+Gz(p)Aw

ALGORITHM In the above equations, p is the complex frequency. Gi(p), Gz(P), G ~ ( P ) and G ~ P ) are:

GI(P) =Cr(pI -Arr ) - 'Br + D g (9) We assume that there are n nodes and m generators, that the nodes are numbered from 1 to n with the first m nodes WO WO

P P

G ~ ( P ) = -M [bl + air(pZ - Arr)-lBr]

Gz(p) = c1 + -cZ + ~ ~ ( ~ 1 - Arr)-'(ar1 + -ar2) (10) being the generator nodes, and that, apart from generators, there is no other type of dynamic plant in the system. Loads (11) are represented by static voltage dependent models which, 7

after linearization, are absorbed into the network admittance G ~ ( ~ ) = M WO 1z - a l r ( p ~ - Arr)-l(arl + yOarz)] matrix Y. P

The AESOPS algorithm [3, 5, 9, 111 calculates a rotor oscil- (12) lation eigenvalue p using the iteration scheme Calculation is simplified by diagonalizing A,, in terms of its

eigenvalues [ll]. In (7) and (e ) , if the relevant generator is the disturbed generator the speed Aw equals 1; for all other generators, the disturbing torque A T is 0.

pi+l = pi - AT(pi)/2Me(pi) ( l )

where m The equivalent admittance Yq@) of a non-disturbed genera-

Me(pi) = Mr lAwr(Pi)lz (2) tor obtained from (7) and ( 8 ) with AT zero is r=l

is an equivalent inertia defined in terms of the generator iner- = -Gl(p) + Gz(P)G3(P)/(MP + G4(p)) (13) tias M-and perturbed speeds Aw, and AT(pi) is the torque

is kept constant at 1 P.u.. disturbance of a disturbed generator whose perturbed speed For the disturbed generator, with nw set to in (7), the

admittance is equivalent to -Gl(p) in parallel with a current source Gz(p). .. ~

AESOPS is a quasi-Newton method with the torque deriva- tive The calculation steps of the PEALS/AESOPS program

arehll: of the disturbed generator approximated by 2M.. L .

The PEALS/AESOPS program of [ll] is an efficient imple- mentation of the AESOPS algorithm. In this section, we show that this implementation can be easily modified, using the true value of the torque derivative, to become an efficient New ton-Raphson met hod.

1. With an estimate of the eigenvalue p, solve the equivalent network equation (14) for the first 2m elements of the bus voltage vector [AV] by a sparsity method. Details are given in [9].

The linearized state space equation of a generator with a torque disturbance input AT 4111

x = A , x + B , A v + e , A T (3) Ai = C,x+D,Av (4)

where x is the state vector, Ai and AV are the perturbed stator current and voltage vectors, and A,, etc. are matrices and vectors of appropriate dimensions.

Equations (3) and (4) may be partitioned as

(Y + P'q(P)I)[AVI = LI (14)

p,(p)] is a block diagonal matrix of the generator equiv- alent admittances and [i,] is the current source vector having a non zero entry only for the disturbed generator equal to G z @ ) .

2. Calculate AT for the disturbed generator and Aw for

3. Update p using (1) and (2).

4. Repeat the above steps until p converges.

other generators using (8 ) .

The description of PEALS/AESOPS to this point basically follows [ll] apart from the expression for Yg(p) in (13).

To find the actual torque derivative, we now differentiate (14) and ( 8 ) for the disturbed generator with respect to p, noting that Aw = 1, to yield

0 ari arz Arr

+ [ z r ] A v + [ % ] A T

'AT -- aG3(p) AV + G3 (p) - 'AV + M + - 'G4(P) ( l q

Aw

'P 'P 'P A i = [ CI cz Cr ] [ +DgAv ( 6 ) F -

where A6, WO and D are the perturbed rotor angle , base an- gular frequency and mechanical damping factor respectively.

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In (15) and (16), the partial derivatives of the operational matrices can be analytically determined from (9)-(12). The calculation is very simple because @I - Arr)-' in these equa- tions is in diagonal form. The vector [AV] is obtained by solving (14). Only the first 2m elements of [AV] are needed because the last 2n-2m columns of 9, in (16) are zero.

Equation (14) can thus be solved in the same way as for A E SOPS. The torque derivative is readily calculated from

after solving (16) for the corresponding two elements of

the coefficient matrix of (14) and (16) are the same, they can be solved with just one sparse matrix factorization.

Replacing ZM, in (1) by 9, we transform AESOPS to a Newton iteration scheme. The steps in this modified AESOPS algorithm can be summarized as follows:

[ I p2; -1 of the disturbed generator by a sparsity method. Since

1. Select an estimate for the eigenvalue p. Solve (14) by a sparsity method for the first 2m elements of AV.

2. Solve (16) for the pair of elements of corresponding to the disturbed generator by again applying a sparsity method making use of the same matrix factors of (Y + [Yg]) obtained in Step 1.

of the disturbed generator using (8) and (15).

4. Compute an improved estimate of the eigenvalue p using the Newton method, i.e.

3. Calculate AT and

5. Check for convergence. If p has not converged, go to step 1.

Following convergence, the perturbed speeds A w of the non- disturbed generators are computed using (8) which are the elements of the relevant converged system eigenvector.

As can be seen, the modified AESOPS algorithm is very sim- ilar to the original AESOPS with just the addition of cal- culation step 2. Though the modified AESOPS needs some more time for each iteration, it generally requires far fewer iterations than AESOPS and has better overall efficiency.

and [Awl is the generator speed vector. [Gl(p)], [G2(p)] and [G3(p)] are block diagonal matrices with some redundant zero columns or rows appended, and [Mp+D+G&)] is a diagonal matrix.

Equations (7) , (8), (18) and (19) have the same basic form as in [I] and [2] except that the more flexible state space equation model instead of the operational impedance model is used for generators and that the network admittance matrix Y is not reduced.

Reference [2] proposes a technique to find a reasonable es- timate of the electromechanical eigenvalues and the associ- ated eigenvectors from (18). This is used in the work of this paper to initialize the AESOPS and the modified A E SOPS algorithms. We take the generator which corresponds to the largest component of an estimated eigenvector as the disturbed generator for the associated eigenvalue. The ini- tial eigenvalues and disturbed generators determined by this method usually give satisfactory result but we recognize that, in some cases, further engineering judgment may be needed in choosing an effective disturbed generator because an eigen- vector is only an indication of the observability of the swing mode.

An electromechanical eigenvalue X and its eigenvector U sat- isfy (18)

We define the corresponding left eigenvector w for F(X), which will be needed in the next section, as the vector which satisfies the transposed operational matrix equation of (18)

F(X)u = 0 (20)

F'(X)W = 0 (21)

w is different from the left eigenvector v of X for the system state matrix taking only the corresponding components. It can, in fact, be shown that they are related by

v = [M]w (22)

w(and hence v) can be quickly calculated by applying either AESOPS or the modified AESOPS, with the correct eigen- value as initial value, to the same system model but with Y and Gl(p) of the generators transposed, and G2(p) and G3(p) of every generator transposed and swapped(see (19)).

RELATION T O OPERATIONAL TRANSFER MATRIX EQUATION METHOD

CALCULATION OF EIGENVALUE SENSITIVITY

The approach used in [ll] and this paper is closely related to the operational transfer matrix equation method of [l] and

With AT set to zero, and combining (7) and (8) for all gen- erators with the network equation [Ai] = Y[Av], we arrive at the following operational transfer matrix equation.

PI.

J W [ A w l = 0 (18)

F(P) = W P + D + G 4 P ) l + [WP)I(Y - [Gl(P)l)-'[GdP)l (19)

where

Eigenvalue sensitivity is a very useful concept in system anal- ysis and in ranking locations and designs of power system sta- bilizers. A number of formulae have been proposed in the lit- erature to calculate eigenvalue sensitivities, particularly those in relation to the effect of PSS. Recently, [14] and [15] present two improved methods of calculation. [14] uses the technique of transfer function residue and augmented eigenvector to cal- culate the eigenvalue sensitivity. The augmented eigenvectors are obtained by a sparsity technique, which makes it applica- ble to large systems. Reference [15] calculates the sensitivity of an eigenvalue with respect to the transfer function of a PSS based on a reduced order power system model. However, the reduced order model is a very crude model, using third order representation of generators, and network reduction, which

1883

is actually a restricted form of (18). It is neither accurate enough in general nor suitable for large systems.

Here, we present an alternative efficient expression for com- puting the sensitivity of an oscillation eigenvalue with respect to a PSS transfer function. The expression is derived using the operational matrix approach discussed above.

Suppose that an extra signal AV, derived from a speed type PSS with transfer function k,(p) is injected into the reference junction of the automatic voltage regulator of a generator. (3) becomes

where fg in partitioned form is [ 0 (7) and (8) are rewritten as

x = Agx + B,Av + f,Av, (23)

0 f: 1'. Equations

A i = G l ( p ) A v + (Gz(p) + Gs(p)k,(p))Aw (24)

0 = GJ(P)Av + (Mp + D + Gi(p) + G~(p)ks(p))Aw (25) where

Gs(p) = Cr(p1- Arr)-'fr (26)

(27) G6(p) = -Mair(pl - Arr)-'fr

In deriving (24) and (25), we have used the relation AV, = k,(p)Aw. As is the case for Gl(p) to G*(p), the calculation of Gs(p) and G6(p) is simplified because A,, has been diag- onalized.

With (24) and (25) for all generators, (18) then becomes

(F(P) + L(P)[ks(P)l)[A4 = 0 (28)

where

L(P) = [G6(p)l + [G~(P)I (Y - [Gl(p)l)-1[G5(P)I (29)

[k.(p)] and [G6(p)] are diagonal matrices, and [Gs(p)] is a block diagonal matrix with some redundant rows appended.

Let A, U and w be an oscillation eigenvalue and the associated right and left eigenvectors respectively satisfying (28) and its transpose. Consider the case where [k,(p)] is identically zero. In this case, (28) and its transpose are reduced to (20) and (21) respectively. By linearizing (28) and making use of (20) and (21), we can obtain the following expression for the sensi- tivity of the eigenvalue X with respect to the transfer function k,,(A) of a speed type PSS on the ith generator :

where ut is the z t h element of U and L;(A) is the z t h column of L(X). The product *U is

where q 1 and q 2 are solutions of

In (30) to (33), A, U and w are efficiently calculated using the proposed modified AESOPS algorithm. L;(A), ql and

q 2 are sparsely solved from (29), (32) and (33), using matrix factors of (Y - [Gl(A)]) common to the three equations. The computation effort for the factorization of (Y - [Gl(A)]) is same as that for the AESOPS and modified AESOPS a lge rithms. As before, the differentiation of Gl(X), etc. is made straightforward by the diagonalization of A?,.

The expression for the sensitivity of eigenvalue A with respect to the transfer function k p i ( A ) of an electric power type PSS on the ith generator is obtained by multiplying (30) with a factor -(MA + 0)[14].

The eigenvalue sensitivity formula (30) given here is similar to that of [15] but are much superior in that there is no limitation in generator modelling and network sparsity can be utilized. It is efficient and suitable for large systems.

If the aim of sensitivity analysis is simply to rank the effec- tiveness of possible PSS locations, it is sufficient to calculate only the numerator of (30).

ILLUSTRATIVE EXAMPLE

The 21-generator 80-bus power system of [16] is used here to illustrate the proposed modified AESOPS algorithm and the method of calculating eigenvalue sensitivities, and to provide a comparison with the AESOPS algorithm.

The mathematical models adopted for this example are basi- cally the same as in [16]. Generators., excitation control sys- tems and PSS are represented in detail. The effect of gover- nors is ignored. Loads are modelled as constant impedances. There are 5 PSS in the system.

The AESOPS and the modified AESOPS algorithms were used to calculate the rotor oscillation eigenvalues of the sys- tem, of which there are a total of 20, starting with the same initial guesses and disturbed generators. The initial eigenval- ues and disturbed generators were determined by the method discussed above. The calculation was carried out on a 80486 CPU personal computer. Results were obtained for error tol- erances c of lo-' and

The eigenvalues computed by the two algorithms agree with each other to within the error tolerances. Table 1 shows the eigenvalues calculated and the magnitudes of the disturbance torques AT at convergence with an error tolerance of Table 2 shows the number of iterations required by each al- gorithm. lATl should ideally be zero at convergence, and is therefore a measure of the accuracy of the calculation results.

Of the two algorithms, the modified AESOPS needs far fewer iterations, especially for inter-area oscillation eigenvalues, and converges in most cases in 3 to 4 iterations. Eigenvalue 5, the slowest to converge, represents a poorly damped inter- area mode. When the error tolerance is tightened from lo-' to the increase in iterations for the modified AESOPS algorithm is also less than for AESOPS and is very small. The values of lATl at convergence for the modified AESOPS are generally much small than that for AESOPS. This indicates that the modified AESOPS is more accurate. These results

*The model for generators is different from [16] though it is of the same order.

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mode

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

initial &genvalue

-5.17+j15.84 -1.87+j10.68 -1.48+j10.90 -0.54+j10.42

0.22+j2.33 -0.90+jS.S9 -0.38+j4.49 -0.36+j5.06 -1.64+j5.59 -1.36+j5.19 -0.43+j9.27 -1.34+j9.14 -0.80+j8.67 -0.26+j6.15 -0.64+j6.99 -0.33+j6.36 -0.70+j8.24 -0.5l+j7.94 -0.64+j7.79 -0.61+j7.78

convaed eigenvalue

10-6

10

10 25 12 17 23 3 5 3 11

19

45

of 6 A &-MA -1.81106+j18.38791 AE!

10-4 8 4

5 8 4

3 6 4 4 4

3 4 3

4 7 3

2 9 3 5 2 7 3 5 3 6 3 6 3

-1.42097+j11.57642 -0.66358+j9.94274

-0.44239+j 10.57412 -0.03338+j2.62707

-0.69192+j10.09013 -0.31552+j4.47087 -0.37768+j5.26019 -1.59297+j4.47889 -1.86326+j5.08933 -0.41985+j9.32918 -1.04598+j9.55471 -1.06721+j8.81666

-0.65037+j6.95602 -0.32627+j6.36915 -0.61144+j8.37304 -0.46698+j7.92561 -0.75155+j7.83352 -0.59469+j7.89807

-0.25660fj6.15216

-l OA le-5 3e-5 2-4 Se-6 le-3 2e-5 4e-4 2e-4 3.3-4 7e-5 8e-5 6e-5 706 2-4 6e-5 le-4 7e-5 2e-5 3e-5 2e-5

-

-

rl -MA 2e-10 7e-7 9 4

3e-10 2-7 le-5 le-7 7 e 5 8-7

9-11 2e-6 107

4-11 7-6 2e-5 2e-8

2-10 2-6

7-10 2e-11

-

-

Table 1: Eigenvalues calculated with error tolerance of ( O A and M A stand for original and modified AE-

SOPS respectively)

verify the fast convergence property of the modified AESOPS algorithm. It is notable that the AESOPS algorithm at times needs a considerable number of iteration to converge, even when a very good initial estimate of the eigenvalue is made. It is also very sensitive to a change of error tolerance.

In terms of computer time, the modified AESOPS algorithm is again faster, taking only 65 seconds and 74 seconds to com- plete the entire calculation with error tolerances of lo-' and

respectively. In contrast, the calculation time for AE- SOPS is 112 seconds with error tolerance of lo-', increasing to 170 seconds when the error tolerance is tightened to lo-'. Reference [2] presents another algorithm to compute the os- cillation eigenvalues from (18). The modified AESOPS was compared with this algorithm using the 21-generator system, and found to have much superior performance. Due to limi- tation of space, details of the comparison are omitted.

Equation (30) is applied to calculate the sensitivities of the inter-area oscillation eigenvalue -0.033 +~2.627 with respect to the transfer function of an electric power type PSS for each machine. The result is given in Table 3. The sensitivities for generators 2, 12 and 17 are zero because the excitation control of these machines is on manual. From Table 3, we see that 14, 15, 18 and 20 are the four generators with the largest sensitivities, which means that PSS at these locations can have a strong effect on the eigenvalue. This is consistent with a more detailed analysis. In fact, generators 14 and 15 are the two largest machines in the system and in this example, they are already equipped with PSS. As verification, generators 18 and 20 were individually fitted with an electric power type PSS of 0.7 p.u. static gain(machine base), and the inter- area eigenvalue was re-calculated. The transfer functions of these two PSS are given in [16]. Table 4 shows the result. In Table 4, the actual shift of the eigenvalue is compared with a linear estimation made using (30). It can be seen that the estimated eigenvalue shift is quite close to the actual value.

mode

1 2 3 4

6

8t

10 11 12 13 14 15 16 17 18 19 20

51

71

9+

t :

number of iterstion A S O P S I modified

i i F - 6 7 s 7

16 9

10 14 21 8 3

12 5

23 6 3 5 3 4 4

Iter-area oscillation ma

IPS I0-P

5 5 5 4 7 4 5 4 3 4 3 5 4 3 3 3 4 3 4 4

-

- :s

Table 2: Number of iteration with error tolerances of and loV6

CONCLUSION

An efficient direct refinement of the PEALS/AESOPS algo- rithm is proposed for calculating large power system eigenval- ues associated with generator rotor o d a t i o n . The modified AESOPS algorithm is a Newton-Raphson iteration scheme with fast convergence properties as confirmed by tests on a 21- generator power system . The derivation of the modified A E SOPS algorithm is very simple in contrast with the other two improvements of AESOPS reported in [13] and [12]. More- over, it is shown that AESOPS and the modified AESOPS are closely linked to the operational transfer matrix equation approach, which provides an estimate of the eigenvalues and eigenvectors to initialize calculations. The estimated eigen- vectors can also indicate suitable disturbed generators for the AESOPS and the modified AESOPS algorithms. The oper- ational transfer matrix equation method is further used to derive an efficient expression for calculating eigenvalue sensi- tivities.

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[3] R. Byerly, R. Bennon, and D. Sherman, "Eigenvalue analysis of synchronizing power flow oscillations in large

1885

gener- eigenvalue eigenvalue shift ator with PSS actual estimated 18 -0.10136+j2.54550 -0.068~j0.0816 -0.068S-j0.~50 20 -0.14260+j2.51214 -0.1092-j0.1149 -0.1151-jO.1187

generator 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

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sensitivity 0.30551-1.63

0 0.0472L-30.7 0.3280f 8.8

0.3196L-117.1 0.0006L -162.5 0.0081L-149.1 0.0045L-63.4 0.0568L -137.5 0.78261 -40.9 0.0542L -144.1

0 0.0481L -172.6 2.0172L -65.7 1.9015L-84.6 0.8343f -37.1

0 2.4128L84.6 0.8602f 99.7 4.9249L72.5 1.0738L48.4

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in the analysis of power system dynamics,” IEEE Trons- action on Power Systems, vol. PWRS-6, pp. 1189-1195, Aug. 1991.

[16] A. Fretwell, D.M. Lam, C.Y. Yu, M. Shimomura, S. Tanaka, and M. Itoh, “Design, installation and opera- tion of power system stabilizers on the Hongkong electric power system,“ IEE International Conference on Ad- vances in Power System Control, Operation and Man- agement, Ifong Kong, pp. 534-541, Nov. 1991.

D.M. Lam received his B.Sc.(Eng) degree from the University of Hong Kong in 1980. From 1980 to 1991, he worked as an electrical engineer in the Hongkong Electric Company. He is currently working towards a Ph.D. degree at the University of Sydney in the area of power system dynamic stability analysis and control.

H. Yee(M’70) received a B.Sc. degree in 1965, B.E. degree in 1967 and a Ph.D. degree in 1972, all from the University of Sydney. Dr. Yee is a senior lecturer in the Department of Electrical Engineering at the University of Sydney. He is mainly interested in power system dynamics and control and the use of artificial intelligence techniques.

B. Campbell received his M.E. degree in 1970 from the Uni- versity of Sydney. Mr. Campbell is a senior lecturer in the De- partment of Electrical Engineering at the University of Syd- ney. His main research interest is in power system dynamics and high pressure plasma.