A study of the AESOPS/PEALS algorithms

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<ul><li><p>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 </p><p>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. </p><p>Keywords: power system stability, swin9 oscillations, eiyen-analysis, the AESOPS/PEALS alyorithms </p><p>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. </p><p>Received January 1992; revised May 1992 </p><p>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. </p><p>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. </p><p>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 </p><p>402 0142-0615/92/60402-09 1992 Butterworth-Heinemann Ltd Electrical Power &amp; Energy Systems </p></li><li><p>of these three algorithms is compared numerically by applying them to a one-machine power system to compute the swing mode. </p><p>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. </p><p>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. </p><p>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. </p><p>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: </p><p>[Ms+D(s )+ ls tooK(s ) lA to=AT (l) </p><p>and the system characteristic equation ~9 is: </p><p>Ms+D(s)+ 1- tooK(s) = 0 S </p><p>(2) </p><p>or </p><p>Ms 2 d- sD(s) + tooK(s ) = 0 (2)' </p><p>The swing mode is an eigenvalue or characteristic root of the system, and therefore a solution to equation (2) or equation (2)'. </p><p>Figure 1. A block diagram for one-machine systems </p><p>I1.1 A simple iterative algorithm Based on equation (2)' a simple iteration formula for swing mode computation may be constructed as follows: </p><p>)'k l =.J ~ 2kD()k)+tOK(2k) + M (k =0, 1 . . . . ) (3) </p><p>The iteration is repeated until the difference in successive steps is sufficiently small, [2k+l--2k[</p></li><li><p>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: </p><p>(1) Choose an initial value 2 o, usually 2o=j6.28; (2) With 2k (k = 0, 1 . . . . ) known, calculate A T in equation </p><p>(4); (3) Find the eigenvalue increment A2 in equation (7), </p><p>2k+l=2k+A2; (4) Check the convergence IA).I &lt; e. If not converged, go </p><p>back to step 2. </p><p>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: </p><p>('A </p><p>)~k+lD(ftk+l)~--2kD()~k)+ D('~k)nt-2k ~2 J ! </p><p>(8)' </p><p>Substituting the above equations into equation (5), we have: </p><p>M2k+ 1 +b2k+ 1 -[- C = 0 (9) </p><p>where </p><p>b=D(Xk)+ Xk ~D(')'k) +09 0 ~g(2k)__ </p><p>( (?K(')'k) + )k(?D!')'k)l (_'=(~OoK(f'~k)--)~ k (0 0 (-))~ (?)~ / </p><p>By solving the above 2nd-order complex coefficient equation, a new eigenvalue estimation 2k+~ may be obtained. This Newton-like algorithm will have better </p><p>Pe - - . l i ra - </p><p>Xe </p><p>Figure 2. A one-machine power system </p><p>convergence because the derivatives arc used in the eigenvalue estimation (equation (9)). </p><p>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 </p><p>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 &lt; 10 6. By examining the results it can be seen that: </p><p>(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. </p><p>(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. </p><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 </p><p>Table 1. Study results of the one-machine system </p><p>Cases </p><p>(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 </p><p>Iterations </p><p>). p SI AE NL </p><p>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 </p><p>404 Electrical Power &amp; Energy Systems </p></li><li><p>Aw AG </p><p>Figure 3. A block diagram for multimachine systems </p><p>preserve the sparse structure. However, it is very helpful when the discussion of concept is concerned. </p><p>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&amp; 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: </p><p>[Ms + ! A(s)lAto = AT (10) </p><p>where </p><p>A(s) = sD(s) + cooK(s). </p><p>The eigenvalue problem of equation (10) may be formulated as to find the solutions to the following equations: </p><p>IM2 + ~A(2)]Ato=0 (11) </p><p>v~[M2+ ~A(2) ]=0 (12) </p><p>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. </p><p>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: </p><p>AT=IM2k +IA(2k)IAto (13) </p><p>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: </p><p>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: </p><p>vT AT A;: - (15) </p><p>2vrMAto </p><p>Based on the above eigenvalue estimation formula an iterative algorithm for computing a swing mode may be formulated as follows. </p><p>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 </p><p>except the one corresponding to the disturbed machine, where it is set to be 1. Therefore, AT = [0 . . . . 1 . . . . 0] r. </p><p>(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). </p><p>(4) Find the left eigenvector Vk and normalize it in the same way as the right engenvector. </p><p>(5) Estima...</p></li></ul>