# A study of the AESOPS/PEALS algorithms

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A study of the AESOPS/PEALS algorithms E Z Zhou Power System Research Group, Department of Electrical Engineering, University of Saskatchewan, Saskatoon, Sask 57N 0W0, USA

The basic idea of the AESOPS/PEALS algorithms has been re-examined theoretically and mathematically. The algorithms have been.formulated in rigorous mathematical language, so that the future development of the alqorithms may be based on sound theoretic' (7round. The reason for the success of the AESOPS/PEALS alyorithms in power system swin9 mode computations has been explained. The convergence of the algorithms and the appropriate way to choose the disturbed machine has been studied numericalO' in three example systems. Based on the theoretical analysis and numerical examples some su99estions have been made which may lead to improve the convergence of the alyorithms.

Keywords: power system stability, swin9 oscillations, eiyen-analysis, the AESOPS/PEALS alyorithms

I. I n t roduct ion The existence of insufficiently-damped swing oscillations has posed serious problems to the secure operation of modern interconnected power systems. Analysis of the problem and finding possible means to improve system damping have attracted much attention for many years 1-5. In general, the damping of swing oscillation modes does not depend on the size of system disturbances and hence can be analysed by considering a system linearized about a steady state operation point 5"6. This allows the use of powerful linear analytical methods, such as eigenvalue methods, to determine the damping and assist the design of corrective controls. However con- ventional eigenvalue methods, such as the QR trans- formation method 7, do not exploit the sparsity structure of the system matrix and require a large amount of computer storage and CPU time. As a result, it is restricted to small or medium size systems having fewer than about 500 dynamic states and is not directly applicable to the analysis of interarea oscillations of very large interconnected power systems s.

Received January 1992; revised May 1992

To overcome the size limitation problem many alternate approaches, which allow one to find an eigenvalue or a set of eigenvalues at one time and take the advantage of the sparse structure of power systems, have been reported in literature 3'4s-13. Among these approaches the AESOPS algorithm in Reference 3 is particularly interesting because of its potential appli- cation in the study of very large power systems. The AESOPS algorithm was further developed to the PEALS algorithm in Reference 8. The AESOPS/PEALS algorithms have been tested on several very large systems in References 5 and 8. The test results show the good performances of the algorithms in finding interarea modes of very large systems.

The eigenvalue estimation formula in the AESOPS/ PEALS algorithm has originally been derived, by intuition and experimentation, from the equations of a classical one-machine model 3. The formula was later explained in Reference 8 in a one-machine system in terms of the return difference and the Newton Raphson root finding scheme. The one-machine eigenvalue estimation formula was extended to multimachine system situations by using the equivalent machine inertia to substitute the inertia of the disturbed machine in the formula, also by intuition and experimentation 3, without any rigorous mathematical explanation. This experimental or, in some sense, mysterious development of the algorithms makes the theoretical analysis of their convergence and the future development of the algorithms very difficult. Some explanation of the AESOPS/PEALS algorithms was given in Reference 14 and an alternate solution scheme for the AESOPS algorithm was proposed in Reference 10.

This paper tries to use rigorous mathematical language to re-examine the AESOPS/PEALS algorithms and find the reasons mysteriously hidden behind the success of the algorithms in finding the swing modes of power systems. In Section II, as a preparation for the discussion of multimachine system situations, the problem of finding the swing mode of a one-machine system by iterative methods is investigated. Two algorithms: a simple iterative algorithm and a Newton-like algorithm are pre- sented and the one-machine AESOPS/PEALS algorithm is re-examined. The AESOPS/PEALS algorithm is ex- plained in a much simpler, and yet rigorous, way than was explained in References 3 and 8. The convergence speed

402 0142-0615/92/60402-09 1992 Butterworth-Heinemann Ltd Electrical Power & Energy Systems

of these three algorithms is compared numerically by applying them to a one-machine power system to compute the swing mode.

The multimachine AESOPS/PEALS algorithms are re-examined in detail in Section III. First the problem of using AESOPS-like iterative algorithms to find an eigen- value of a multimachine power system is mathematically formulated and the right/left eigenvector concept defined. Then a multimachine eigenvalue estimation formula is derived, and an iterative algorithm, named LR algorithm, is constructed based on the estimation formula. It is found that the only difference between the LR algorithm and the AESOPS/PEALS algorithms is the way in which the equivalent machine inertia is calculated. The situa- tions and the assumptions where the LR algorithm (ancestor) may be simplified to the AESOPS/PEALS algorithms (descendents), and the reasons why the iterative AESOPS/PEALS algorithms could be used to compute the swing modes and it is very likely the iteration process would converge to a swing mode, to the swing mode of interest, are discussed.

In Section IV, a 3-machine power system and a 13-machine power system are used as numerical examples to compare the convergence of the LR algorithm, a RR algorithm which is a simplified version of the LR algorithm and the AESOPS/PEALS algorithms. The validity of the assumptions, based on which the AESOPS/PEALS algorithms are derived from their ancestor, the LR algorithm, is examined in the 3-machine system. The question of how to choose the appropriate disturbed machine is discussed.

II. Finding the swing mode in one-machine systems In a one-machine to infinite bus system there is an oscillation mode, called swing mode, which has the closest relationship to the swing loop of the machine. In this section the problem of finding the swing mode of a one-machine power system by iterative methods will be investigated. The basic idea of the AESOPS algorithm will be re-examined. The purpose of this analysis in one-machine systems is to establish the bases for the discussion of multimachine systems in the next section.

Based on the Heffron-Phillips model 1'16, a block diagram describing the small-disturbance dynamic be- haviour of a one-machine system is shown in Figure 1. Here K(s), D(s) are transfer functions, AT is an external disturbance torque introduced for the purpose of analysis. The system equation in terms of the input AT and the output Ato is as follows:

[Ms+D(s )+ ls tooK(s ) lA to=AT (l)

and the system characteristic equation ~9 is:

Ms+D(s)+ 1- tooK(s) = 0 S

(2)

or

Ms 2 d- sD(s) + tooK(s ) = 0 (2)'

The swing mode is an eigenvalue or characteristic root of the system, and therefore a solution to equation (2) or equation (2)'.

Figure 1. A block diagram for one-machine systems

I1.1 A simple iterative algorithm Based on equation (2)' a simple iteration formula for swing mode computation may be constructed as follows:

)'k l =.J ~ 2kD()k)+tOK(2k) + M (k =0, 1 . . . . ) (3)

The iteration is repeated until the difference in successive steps is sufficiently small, [2k+l--2k[

The derivation of equation (7) from equation (6) is based on some approximations. The obtained A). and ).k+l-=-)~kq-Af~. are not 100% accurate, but usually will reduce the error AT. Therefore an iteration process is needed in order to find the accurate solution. The iterative AESOPS algorithm for one-machine systems may be summarized as follows:

(1) Choose an initial value 2 o, usually 2o=j6.28; (2) With 2k (k = 0, 1 . . . . ) known, calculate A T in equation

(4); (3) Find the eigenvalue increment A2 in equation (7),

2k+l=2k+A2; (4) Check the convergence IA).I < e. If not converged, go

back to step 2.

11.3 A Newton-like method ~3 In order to investigate the convergence speed of the AESOPS algorithm, Newton's method of solving non- linear equations is used to derive an algorithm to solve the eigenvalue problem. In equation (5), the following equations may be used:

('A

)~k+lD(ftk+l)~--2kD()~k)+ D('~k)nt-2k ~2 J !

(8)'

Substituting the above equations into equation (5), we have:

M2k+ 1 +b2k+ 1 -[- C = 0 (9)

where

b=D(Xk)+ Xk ~D(')'k) +09 0 ~g(2k)__

( (?K(')'k) + )k(?D!')'k)l (_'=(~OoK(f'~k)--)~ k (0 0 (-))~ (?)~ /

By solving the above 2nd-order complex coefficient equation, a new eigenvalue estimation 2k+~ may be obtained. This Newton-like algorithm will have better

Pe - - . l i ra -

Xe

Figure 2. A one-machine power system

convergence because the derivatives arc used in the eigenvalue estimation (equation (9)).

11.4 A one-machine system example A one machine system is shown in Figure 2. The study results are shown in Table 1. There are five cases studied. The differences between these cases are the transmission

k line reactance Xe, the generatmn Pe, and with a power system stabilizer PSS (K~ss= 10) or without PSS (Kps s = 0) on the machine. The data are in per unit based on 100 MVA. 2 in the table is the obtained swing mode, p is the swing mode participation ratio defined in Reference 13. The number of iterations needed to find the solution for the simple iterative algorithm (SI), the AESOPS algorithm (AE) and the Newton-like algorithm (NL) are shown in the table. The convergence criterion is IA2I < 10 6. By examining the results it can be seen that:

(1) The AESOPS algorithm has nearly the same con- vergence speed as the simple iterative algorithm, which has been proved to have a linear convergence speed 4'1s. The Newton-like algorithm has much faster convergence speed when the participation ratio p is approaching 1.0.

(2) As explained in References 4 and 18, the convergence speed of the simple iteration algorithm is depending on the participation ratio p. As can be seen in the table, the larger the p, the faster the convergence of the simple iterative algorithm and the AESOPS algorithm. As for the Newton-like algorithm, it seems that the convergence is not affected very much by the decrease of p.

III Re-examine the multimachine AESOPS/PEALS algorithms In an n-machine system there are (n - 1) swing modes 2. The swing modes may be found by first linearizing the system and then computing the eigenvalues of the state matrix, for instance with the QR method. The QR method does not exploit the sparsity structure of the system matrix and is restricted to small or medium size systems. To overcome the size limitation problem the AESOPS/PEALS algorithms were developed. The algorithms have been tested on several very large systems and the test results show the good performance of the algorithms. Yet, until now the algorithms have not been clearly explained in terms of rigorous mathematical language. In this section the idea behind the AESOPS/ PEALS multimachine algorithms will be investigated. It should be pointed out that the power system model used in the following investigation is inefficient from the computation point of view, because the model does not

Table 1. Study results of the one-machine system

Cases

(1) X=0.06, Pc= 10, Kpss=0 (2) Xe=0.06, Pc = 10, Kpss= 10 (3) Xe=0.06, Pe=15, Kvss=10 (4) Xc=0. l , Pe= 10, Kpss = 10 (5) Xe=0.1 , Pe--- 10.5, Kpss= 10

Iterations

). p SI AE NL

0.0937 +j6.0251 43.7 5 5 5 - 0.1083 +j5.7677 4.5 17 l 7 8 - 0.7301 +j4.4851 3.5 30 30 8 -0.2731 +j3.2418 5.6 19 18 7 - 0.1264 +j2.7015 4.8 20 20 7

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Aw AG

Figure 3. A block diagram for multimachine systems

preserve the sparse structure. However, it is very helpful when the discussion of concept is concerned.

II1.1 Problem descriptions Based on the multimachine system model in Reference 13, after some derivations, a block diagram describing the small-disturbance dynamic behaviour of a multi- machine power system may be obtained as shown in Figure 3, where K(s), D(s) are (n n) transfer function matrices (see Appendix A in Reference 13 for the details of K(s), D(s)); A~, A& AT are (n x 1) vectors; M is a diagonal matrix of machine inertia coefficients; ! is an identity matrix. The multimachine system equation in terms of the input vector AT and the output vector Ae~ is as follows:

[Ms + ! A(s)lAto = AT (10)

where

A(s) = sD(s) + cooK(s).

The eigenvalue problem of equation (10) may be formulated as to find the solutions to the following equations:

IM2 + ~A(2)]Ato=0 (11)

v~[M2+ ~A(2) ]=0 (12)

where Ato is the right eigenvector and v the left eigenvector. The same symbol Act is used for the machine speed deviation and the right eigenvector. The reason is that the right eigenvector measures the relative magni- tudes of the machine speed deviations participating in the oscillations of mode 24 . It should be noted that the definitions of the right eigenvector Art and the left eigenvector v are the solutions to equations (11), (12), and differ slightly from the standard definitions of the right/left eigenvectors in terms of a state matrix in text books 19. Eigenanalysis of equations (1l) and (12) is a non-linear problem. Therefore, iterative methods are needed to solve the problem ~3.

111.2 A multimachine eigenvalue estimation formula Supposing that the approximate estimations of an eigenvalue 2 k and the corresponding right eigenvector AoJ k are known, the error vector AT of equation (11) is as follows:

AT=IM2k +IA(2k)IAto (13)

To find the solutions 2, Act to equation (11), let 2 ---- 2 k d- A,,]., ACt = At ) k -+- A(Aco). Multiplying equation (13) by the left eigenvector v, we have:

F 1 (2A(2k)_).kA(2))]AtO k (14) vTAT = vT[_ - MA~ + 2.2 By assuming A(2k) ~ A(2), 2k2 ~ 2 2 and A2A(ACO) ~ 0, the eigenvalue increment A2 may be estimated by the following equation:

vT AT A;: - (15)

2vrMAto

Based on the above eigenvalue estimation formula an iterative algorithm for computing a swing mode may be formulated as follows.

LR algorithm (1) Choose an initial value 20, for example, 2o =j6.28. (2) Set all elements of the error vector AT equal to zero

except the one corresponding to the disturbed machine, where it is set to be 1. Therefore, AT = [0 . . . . 1 . . . . 0] r.

(3) With 2k (k=0, 1 . . . . ) known and AT set according to step 2, solve equation (13) for the right eigenvector A~ok. Normalize Aca~ such that its element corre- sponding to the disturbed machine is (1 +j0). Also AT is multiplied by the same factor as AoJ k to keep the balance of equation (13).

(4) Find the left eigenvector Vk and normalize it in the same way as the right engenvector.

(5) Estimate A2 by equation (15) and 2k+ 1 =2k+A2. (6) Check IA21 < e. If not converged, go back to step 2.

For the convenience of the discussion the above algorithm is named as LR algorithm. L stands for left eigenvector and R right eigenvector. LR implies that both the left and right eigenvectors are used to calculate A2 in equation (l 5).

111.3 The AESOPS/PEALS algorithms 8 In the AESOPS/PEALS algorithms a machine, designated the disturbed machine, has a torque disturbance of complex frequency /~k applied to it. The amplitude and the phase of the torque disturbance are chosen such that the speed change of the disturbed machine is (1 +j0). Mathematically this process is exactly the same as solving equation (13) in steps 2 and 3 in the LR algorithm to compute the right eigenvector, where the torque disturb- ance is termed as the error vector and the speed change vector as the right eigenvector. The complex frequency 2 k is an approximation to a system eigenvalue and a closer approximation )~k + t =)ok + A2 is calculated by

ATdis A2 . . . . . . (16) 2Me

where ATai~ is the torque disturbance, Me is the equivalent machine inertia coefficient.

AESOPS algorithm: Me=E, Mi]ACOil (17) PEALS algorithm: M~ = Z MilAcoil 2 (18)

Equation (17) was used in the original AESOPS algorithm in Reference 3. The later version of the AESOPS algorithm in Reference 15 used equation (18), which was also used in the PEALS algorithm 5'8. For the

Vol 14 No 6 December 1992 405

convenience of discussion, the author gives equation (17) and equation (18) different names, Obviously equation (16) is an extension of equation (7) to multimachine system situations.

111.4 The equivalent inertia Mo There is a very important question remaining un- answered, that is, why the equivalent inertia M~ could be used in the multimachine situations and result in good convergence. In the LR algorithm the left eigenvector v is normalized such that its element corresponding to the disturbed machine is (1 +j0), and AT only has one non- zero element corresponding to the disturbed machine. Therefore if equation (15) is expressed in terms of the equivalent inertia, M~ may be calculated as follows:

LR algorithm: M e = Y MiviAo9 i (19)

where, v i, A~o~ are respectively the ith elements of the left and right eigenvectors.

The right eigenvector Ao9 and the left eigenvector v are defined as the solutions to equations (11) and (12). When A(2) in the equations is symmetric (A().)T=A(2)), the right and left eigenvectors are equal (Ae~=v). For the cases where Aorta v, another way to calculate the equivalent inertia Me may be used as follows:

RR algorithm: M~ = Y.M~A(~ (20)

The approach using the above equation to calculate M~ is named RR algorithm, where R stands for right eigenvector. RR implies that M~ is calculated by Ae~rMAo~.

111.5 Re-examination of the AESOPS/PEALS algorithms Our goal is to compute the swing modes in multimachine power systems. Four similar algorithms have been discussed so far. They use exactly the same right eigenvector computation procedure. In the AESOPS/ PEALS algorithms the procedure is explained in terms of the torque disturbance and the machine speed changes. It is known from field experience that if an external torque of a frequency around 1.0 HZ is applied to the shaft of one of the machines in a power system, it is very likely that the excited dynamics are dominated by the swing oscillations between machine rotors. The mode of the oscillations is likely to be very similar to the mode shape of the swing mode in which the disturbed machine participates significantly.

The author believes that the key to the success of the AESOPS/PEALS algorithms is that the above field experience or the above physical property of power systems has been translated to a very efficient right eigenvector computation procedure. By using equation (13), if the complex frequency 2k of the exciting torque is close to a swing mode in which the disturbed machine participates significantly, the resulting right eigenvector A~o k is likely to be very close to the mode shape of the swing mode of interest. Another very important feature of the right eigenvector computations procedure is that it can exploit the sparse structure of power systems, which has been fully demonstrated in References 5 and 8. This makes it very attractive for the application to very large power systems.

The differences between the four algorithms are the ways in which the equivalent inertia M~ in equation (t6)

is calculated. In the AESOPS algorithm thc machinc inertias are weighted by the absolute values of the machine speed changes IAtoil as ira equation (17). ill the PEALS algorithm by ]A(,hl 2 as in equation {18), in the RR algorithm by A(,)~ as in equation (20), and in tile LR algorithm by the participation factors 4 v~A(,)~ as in equation (19).

Swing modes represent the dynamics of machine swing loops, which is mainly dependent on machine inertias and the synchronous torque coefficient matrix K~~~ Therefore for a swing mode or at a swing frequency, A(s) in equation (10) is dominated by ~%K~. For a lossless power network, K~ is symmetric 2t. As it will be seen in the next section, for a power network with losses, K 1 is not far from symmetric. Hence it is not unreasonable to assume that there are many cases where the A().) matrix is almost symmetric. For these cases the LR algorithm and the RR algorithm will be similar because of the symmetry and AeJ ~ v.

Swing modes of power systems are mostly either local modes or interarea modes. A local mode represents a machine or a group of machines swinging against the rest of the system, which may be abstracted to be a one-machine to infinite bus system, as shown in Figure 4a. An interarea mode represents two groups of machines swinging against each other, which may be abstracted to be a two machine system as in Figure 4b, where the two machines are identical. The swing oscillation mode shapes are shown in the figures. For these two typical cases M~ given by Y-.MiAo,)zi, ZMilA(.oil or EMilAeoil z is identical, indicating that the RR algorithm, the AESOPS algorithm and the PEALS algorithm make no difference for these cases.

In summary, the AESOPS/PEALS algorithms are based on a very efficient right eigenvector computation procedure. With a disturbance torque at a complex frequency close to a swing mode of interest applied to a disturbed machine, the procedure is likely to result in a right eigenvector which is very close to the mode shape of the swing mode of interest. The key to understand the procedure is that AT physically represents the torque disturbance, its mathematical meaning is an error vector: Am represents physically the machine speed changes, mathematically the right eigenvector. The eigenvalue estimation formula equation (16) is obtained from equation (15) of the LR algorithm based on the following assumptions.

(1) For a swing mode )~, the complex matrix A(Z) in equations (11) and (12)is likely to be not far from symmetric. This results in the assumption Am ~ v and the RR algorithm.

(2) For most cases swing oscillations in a power system are similar to either a one-machine local mode as in Figure 4a or a two-machine interarea mode in Figure 4b. For these two typical cases the equivalent inertia M~ obtained by equations (I 7) and (18) of the AESOPS/PEALS algorithms and by equation (20) of the RR algorithm are the same.

AcU=IZO ALo=O L~w=IZO ALu=IZ 180

a b Figure 4. Mode shapes of (a) the local mode and (b) the interarea mode

406 Electrical Power & Energy Systems

The LR algorithm is based on sound mathematical principles. But from the computation point of view the RR algorithm may be better. First, there is no need for left eigenvector computation in the RR algorithm as required in the LR algorithm. Secondly, an efficient left eigenvector computation procedure has yet to be found. The author believes that the move from the RR algorithm ot the AESOPS/PEALS algorithms is unnecessary because no speed up of the whole computation process could be achieved. It should be realized that the development of the AESOPS/PEALS algorithms is by intuition and experimentation, the move from M=T-,,MIAe~o 2 to Me=Y_,MilA6oil or Me=Y-,MilAe)il 2 is not deliberate.

IV. Numerical application examples Four algorithms: AESOPS, PEALS, RR and LR for computing a swing mode of a multimachine power system have been discussed. It would be very interesting to find out how these algorithms perform numerically when they are used to compute the swing modes of power systems. The potential of the application of the AESOPS/PEALS algorithms to very large systems has been demonstrated fully in References 5 and 8. The emphasis of the investigation reported in this paper is on the under- standing of the problem and the convergence of the algorithms rather than the applicability of the algorithm to large power systems. Therefore efficiency related issues (storage, CPU time) will not be discussed. The LR algorithm requires computing the left eigenvector in the iteration process (v~ in step 4). The author has not yet found an efficient left eigenvector computation procedure for the algorithm. For the purpose of comparison a shortcut has been taken where the accurate left eigen- vector is 'borrowed' from the eigen-analysis results by the QR method, and used in the LR algorithm.

IV.1 A 3-machine system example A 3-machine power system from Reference 20 is shown in Figure 5, where machine capacities in MW and machine inertia in seconds are shown. L1, L2 are bus loads. Each machine is represented by a 3rd-order E'q machine model 17, equipped with a voltage regulator and a speed governor, both represented by l st-order models. There are two swing modes in the study system, one is an interarea mode (about 0.4HZ) representing the oscillation between machine l, 2 as a group and machine 3; the other is a local mode (about 1.6HZ) representing the oscillation of machine 1 against the rest of the system.

@

@1 IOOMW M=5.6s

1 IO00MW M: 56s

@

L2

Li

Figure 5. A three-machine power system

1000MW M =56s

Two cases have been studied, case 1 is without any PSS in the system, case 2 is with PSS on machine 1 and 2.

Case 1." In this case there is not any PSS in the system. The eigen-analysis results of the case by the QR method are shown in Table 2, where pf~ (=viA~oi) is the participation factor 4, which may be used to measure the participation of a machine in the oscillations of a mode. The results in the table show that machine 3 is the most significant machine participating in the oscillations of the interarea mode 21, according to the relative value of the participation factor. Also machine 2 participates in mode )-1 significantly. In considering the local mode )~2, machine 1 is the significant machine. The disturbed machine number, the initial value 2 o and the iterations needed by the four different algorithms to find the swing modes are shown in Table 3. When machine 1 is chosen as the disturbed machine the iteration process converges to the local mode "~2 because machine 1 dominates the oscillations of mode 2 2. When machine 2 or 3 is the disturbed machine the iteration process converges to the interarea mode 21. For the computation of the local mode 22 the PEALS algorithm and the RR algorithm are faster than the AESOPS algorithm and the LR algorithm. For mode )41 the four algorithms have nearly the same convergence speed.

Case 2." In this case there are two PSS on machines 1 and 2. The eigen-analysis results by the QR method are shown in Table 4. Comparing Tables 2 and 4 it can be seen that the presence of PSS in the system does not change the participation pattern of the local mode 22 very much, it does change the participation pattern of the interarea mode 21 significantly. The participation factor of machine 2 changes from 0.737 in case 1 to 0.167 in this case, indicating that machine 2 does not participate significantly in the interarea oscillations with PSS in the system. The disturbed machine number, the initial value 2 o and the iterations are shown in Table 5. As shown in Table 4, machine 1 dominates the oscillation of the local mode 22 and machine 3 the interarea mode 2~. When these two machines are chosen as the disturbed machines

Table 2. Eigen-analysis results of case 1 by the QR method

)- 1 = 0.0004 +j2.6010 22 = 0.0061 +j 10.2102

IzXcoil Iril Ipf/I IAcoll Iril IPfil

0.813 0.091 0.074 1.000 0.999 0.999 0.811 0.909 0.737 0.100 1.000 0.100 1.000 1.000 1.000 0.000 0.001 0.000

Table 3. Iterations required by the four algorithms, case I

Disturbed machine )~o

Iterations

AESOPS PEALS RR LR

# 1 j6.28 15, 2 z 4, "~2 4, )'2 15, )~z # 2 j6.28 6, 22 6, 21 6, 21 5, 21 # 3 j6.28 6, '~1 5,/~1 5, AI 5, 21

Vol 14 No 6 December 1992 407

there is no difficulty to find 21 and '/'2" Machine 2 does not participate significantly in either mode 21 or mode 22 . When it is the disturbed machine the iteration process does not converge within 30 iterations to any eigenvalue when the AESOPS/PEALS algorithms are used. When an initial value 2 o =j2.0 very close to mode 21(-0.329+j2.08) is used, the RR algorithm converges to mode 2i. As mentioned before, the accurate left eigenvector by the QR method is used in the LR

Table 4. Eigen-analysis results of case 2 by the QR method

21 = - 0.3287 +j2.0748 22 = - 1.6976 +j10.4655

[A(Di[ Ivil Ipfi[ IA~il b'~l IPf/I

0.387 0.043 0.017 1.000 0.998 0.998 0.386 0.433 0.167 0.101 1.000 0.101 1.000 1.000 1.000 0.000 0.002 0.000

Table 5. Iterations required by the four algorithms, case 2

Disturbed machine

#1 #2 #2 #3

Iterations

2 o AESOPS PEALS RR LR

j6.28 15, )~2 4, 2 2 7, 2 2 15, '/~2 j6.28 x x x 13, 21 j2.00 x x 10, 21 6, 21 j6.28 11, 21 9, 21 8, 21 8, 21

x : Computation does not converge within 30 iterations

algorithm, this ensures the convergence of the iteration process.

The synchronous coefficient matrix K 1 of the study system for the operating condition studied is as follows:

K1

[ 1.372061 -1.334863 -0.0371981

= ~-1.342748 1.721076 -0.378327 / L-0 .050424-0.509287 0.559712J

As can be seen, the K 1 matrix is not far from symmetric, although the transmission line resistance in the system is not equal to zero, and the E'q machine model (salient) instead of the E' machine model (non-salient) is used 21. This indicates that the symmetric assumption which leads to the simplification of the LR algorithm to the RR algorithm is not unreasonable for the study system. The equivalent inertia M e by equations (17) to (20) for case 1, case 2 is shown in Table 6. The figures in the table show some difference, but not very much. They may be the reason that the performances of the four different algorithms are not far from each other for the 3-machine study system.

IV.2 A 13-machine system example A 13-machine system from References 13 and 17 is studied to compare the performances of the four algorithms. Each machine in the system is represented by the E'q model and equipped with a voltage regulator. There are 12 swing modes in the system, as shown in Table 7 (2 in the table). The swing modes are obtained by the QR method, together with the right and left eigenvectors to investigate the selection of the disturbed machine. The disturbed machine in the table is chosen according to the magnitude of the right eigenvector elements (see more discussions

Table 6. The equivalent inertia Me by different algorithms

Cases Mode ZMilAoJit

Case 1 2, 105.94 22 11.22

Case 2 )- 1 79.78 2 2 11.26

~MilAfoil 2

96.48 6.16

65.18 6.17

Y~MiAfo 2 Y, Mivi~9 i

96.48 -j0.04 97.68 +j0.02 6.16 --j0.00 11.21 -j0.01

48.85 +j5.76 48.57 +j5.82 6.17 +j0.07 11.24 +j0.52

Table 7. Results of the 13-machine system

Disturbed No. 2 machine

1 -0.108 +j2.285 # 13 2 -0.096+j4.042 # 6 3 -0.169 +j5.002 # 5 4 -0.967 +j5.700 # 5 5 -0.246 +j6.015 # 7 6 -- 0.194 +j6.286 # 13 7 -0.113 +j6.378 # 9 8 -0.156 +j7.329 # 10 9 -0.068 +j7.313 # 3

10 -0.299 +j7.737 # 4 11 -0.799 +j8.433 # 11 12 - 0.680 +j8.936 # 1

Iterations

2o AESOPS

j2.50 5 j6.28 15, )~3 j5.00 29, 24 j6.28 26 j6.28 18 j6.28 6 j6.28 11 j6.28 11, )~7 j6.28 15 j6.28 25, 29 j6.28 15 j6.28 18

PEALS RR LR

8 9 6 16, 'J~3 8 7 9, 24 9, 2,, 50 8 7 7 4 6 8 6 5 5 7 5 5

10, 27 5, 27 7 9 7 8

12 8 14 6 6 9 8, 27 9 5

408 Electrical Power & Energy Systems

on this subject late in this section). For the most cases the iterations converge to the corresponding swing modes of interest, for example, at the first row in the table, with machine 13, whose right eigenvector (Ato 1 corresponding to 21) entry has relatively the largest magnitude, as the disturbed machine, the iteration converges to mode 21. For the cases where the iteration does not converge to the corresponding mode of interest, the mode which the iteration converges to is shown, for example, at the second row the AESOPS/PEALS algorithms converge to mode 23 instead of mode 2 2 .

From the results in the table the unrealistic LR algorithm (because the accurate left eigenvector by the QR method is used) has the best overall performance among the four tested algorithms. The accurate left eigenvector by the QR method ensures the iterations of the LR algorithm converge to the corresponding modes of interest. The AESOPS algorithm has the worst performance. The RR algorithm has better performance than the PEALS algorithm in eight out of ten cases. It seems that the RR algorithm converges faster (six cases) and is more reliable (two cases) than the PEALS algorithm.

The disturbed machine The success of the AESOPS/PEALS algorithms lies in the very efficient right eigenvector computation pro- cedure. To solve equation (13) for Ato k a machine must be chosen as the disturbed machine. In a multimachine power system how to choose an appropriate machine to be the disturbed machine is decisive to the convergence of the iteration process to the swing mode of interest. In References 3 and 8 it was suggested conceptually that the machine having the greatest participation in the mode of interest should be disturbed. Mathematically there are two possible ways to choose the disturbed machine: one is according to the participation factor, the machine with the largest participation factor be disturbed; the other is according to the right eigenvector, the machine with the largest oscillation magnitude be disturbed. For the 3-machine system, as shown in Tables 2 and 4, in both cases the machine with the largest qAog~( is the machine

with the largest Ipf~l, Therefore there is no difference if the disturbed machine is chosen according to the right eigenvector or the participation factor. The disturbed machines according to the right eigenvector entries (mfDmax) and the participation factors (Pfmax) for the 12 swing modes of the 13 machine system are shown in Table 8. Out of 12 cases there are only two differences: 21 and 2 3 .

A comparison of the two different approaches to select disturbed machine is shown in Table 9. The results indicate that the participation factor is more suitable for the selection of the disturbed machine. For mode 23 when machine 5 is chosen as the disturbed machine according to IA~oil the iteration does not converge to the mode of interest (23). When machine 7 is as the disturbed machine according to Ipfil the iteration process converges to mode 23 without any difficulty. The right eigenvector informa- tion may be obtained by the time-domain simulation of a power system 3, while the participation factor informa- tion is not available until the right and left eigenvectors are found. Therefore from the practical application point of view the.feasible disturbed machine selection approach is according to the right eigenvector. One important thing must be borne in mind, that the selection of the disturbed machine according to the right eigenvector may result in the convergence of the iteration process to a mode other than the mode of interest.

V. Conclusions The basic idea of the one-machine system and multimachine system AESOPS/PEALS algorithms has been re-examined theoretically and mathematically. The algorithms have been formulated and explained in rigorous mathematical language. This may allow the future development of the algorithms to be based on sound theoretic ground. The success of the AESOPS/ PEALS algorithms in power system swing mode com- putations lies in a very efficient right eigenvector computation procedure.

The basic idea of the one-machine AESOPS/PEALS algorithm is that based on the error AT of the system

Table 8. The disturbed machines according to AoJ and pf

Mode mO)ma x Pfmax Mode AfDma x Pfmax Mode A(Dma x Pfmax

21 #13 #12 2 s # 7 # 7 2 9 # 3 # 3 2 2 # 6 # 6 )'6 #13 #13 210 # 4 # 4 23 # 5 # 7 27 # 9 # 9 211 #11 #11 24 # 5 # 5 28 #10 #10 212 # 1 # 1

Table 9. Comparisons of the different disturbed machine selections

Iterations Disturbed

2 machine IA~oil IPf~l 2o AESOPS PEALS RR LR

21 # 13 1.000 0.772 j2.5 5 8 9 6 # 12 0.971 1.000 j2.5 5 6 7 12

2 z # 5 1.000 0.197 j5.0 29, 24 9, 24 9, 2 4 50 # 7 0.951 1.000 j5,0 15 9 6 12

Vol 14 No 6 December 1992 409

characteristic equation at a complex frequency )~k, a new eigenvalue 2k+ 1 = 2k + A2 is estimated such that the error may be reduced to zero. The numerical example indicates that the one-machine AESOPS/PEALS algorithm has linear convergence speed, and its convergence is mainly dependent on the swing loop participation ratio p.

The multimachine AESOPS/PEALS algorithms are derived from their ancestor, the LR algorithm, based on some assumptions. Theoretically the equivalent machine inertia M r should be calculated by weighting the machine inertias with the participation factors (LR algorithm: Me=ZMiviAe)i). When the system transfer function matrix A(s) in equation (10) is symmetric (v=Ao), the equivalent machine inertia may be calculated by weighting the machine inertias only with the right eigenvector entries (RR algorithm: ZMiAeh2). For the typical local mode or interarea mode, 5"MilAehl z and ZN/~JAco~[ used in the AESOPS/PEALS algorithms for calculating M e are equal to Z M~Aco{ of the RR algorithm. The numerical results in Section IV indicate that the RR algorithm outperforms the AESOPS/PEALS algorithms. The numerical results suggest that the participation factor is the best index for the disturbed machine selection. Practically only the right eigenvector information is available. If the disturbed machine is selected according to the right eigenvector, the iteration process may sometimes converge to a mode other than the mode of interest.

VI. References 1 de Mello, F P and Concordia, C 'Concept of synchronous

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2 Larsen, E V and Swann, D A 'Applying power system stabilizers', IEEE T-PAS, Vol 100 (1981) pp3010 3046

3 Byerly, R T, Bernon, R J and Sherman, D E 'Eigenvalue analysis of synchronizing power flow oscillations in large electric power systems', IEEE T-PAS, Vol 101, No 1 (January 1982) pp235-243

4 Perez-Arriaga, I J, Verghese, G C and Schweppe, F C 'Selective modal analysis with application to electric power systems', IEEE T-PAS, Vol 101 (September 1982) pp 311 7 3134

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6 Barbier, C 'Questionnaire on electromechanical oscillation damping in power systems', Electra, No 64 (1979) pp59 90

7 Wilkinson, J H The algebraic eigenvalue problem, Clarendon Press, Oxford (1 965)

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9 Obbata, Y, Takeda, S and Suzuki, H 'An efficient eigen- value estimation technique for multimachine power system dynamic stability analysis', /EEE T-PAS, Vol 100 (January 1981) pp259-263

10 Martins, N 'Efficient eigenvalue and frequency response methods applied to power system small-signal studies', IEEE T-PWRS, Vol 1 (February 1986) pp217 226

11 Uchida, N and Nagao, T 'A new eigen-anatysis method of steady-state stability studies for large power system: S matrix method', IEEE T-PWRS, Vol 3, No 2 (May 1 988) pp 706 714

12 Wang, L and Semlyen, A 'Application of sparse eigenvalue techniques to the small signal stability analysis of large power systems', Proc. of the 16th PICA Conf., Seattle, Washington (May 1989) pp358 365

13 Zhou, E Z, Malik, O P and Hope, G S 'A reduced-order iterative method for swing mode computation',/EEE T-PWRS, Vol 6, No 3 (August 1991) pp1224 1230

14 Sauer, P W, Rajaopalan, C and Pal, M A 'An explanation and generation of the AESOPS and PEALS algorithms', IEEE T-PWRS, Vol 6, No 1 (February 1991) pp293-299

15 EPRI EL-724, Contract 747-1, Frequency domain analysis of low frequency oscillations in large electric power systems, Final Report, Part 1 (April 1 982)

16 Heffron, W G and Phillips, R A 'Effect of a modern amplidyne voltage regulator on underexcited operation of large turbine generators', A/EE T-PAS, Vol 71, No 1 (August 1952) pp 692 697

17 Yu, Y N Electric power system dynamics, Academic Press (1 983)

18 Semlyen, A and Wong, L Discussion to Reference 8 1 9 Franklin, G F, Powell, J D and Emami-Naeini, A Feedback

congro/ of dynamic systems (2nd edn) Addison-Wesley Publishing Company (1 991 )

20 Chen, S J, Malik, O P and Hope, G S 'An adaptive synchronous machine stabilizer', IEEE T-PWRS, Vol 1, No 3 (1986) ppl01 109

21 Winkelman, J R, Chow, J H, Bowler, B C, Avramovic, B and Kototovic, P V 'An analysis of interarea dynamics of multimachine systems', IEEE T-PAS, Vo1100, No 2 (February 1 981 ) pp 754 763

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