Paralel bağlı senkron generatörlerin öz değerler metodu ile dinamik kararlılık incelemesi

Abstract

Genel anlamda güç sistemlerinin kararlılığı, bütün enterkonnekte üretici ve tüketici sistemlerin paralel olarak senkron işletimlerinin sağlanmasıyla ilgilidir. Paralel çalışmadaki aksaklık, küçük veya büyük bozucu etkiler tarafından oluşabilir. Enterkonnekte güç sistemlerinin lineer dinamik performansı, normal işletim koşullarında güç sisteminde oluşan küçük bozucu etkiler altında sistem makinelerinin davranışlarıyla ilgilidir. İlk birkaç salınımda açık olarak gözlenmemesine karşılık osilasyonlar sistemin işletim karakteristiklerine bağlı belirli bir değere erişince, dinamik kararsızlık ortaya çıkar. Bir enterkonnekte güç sisteminin küçük bozucu etkilere karşı dinamik davranışı, temel altsistem modellerinin kullanımıyla incelenmektedir. Küçük bozucu etkiler altında genel kararlılık inceleme metodu, sistemi tanımlayan diferansiyel ve cebrik denklem takımının lineerleştirilmesi olmuştur. Güç sisteminin dinamiğini gösteren eşitliklerin bir çalışma noktası civarında lineerleştirilmesi ile toplam sistem, dış girişlerden arınmış bir özerk sistem olarak elde edilebilir. Dinamik kararlılık analizleri, altsistemlerin oluşturduğu toplam sistem matrisinin özdeğerlerinin bulunmasıyla yapılır. Asimptotik kararlılık için toplam sistem matrisinin özdeğerlerinin reel kısımları negatif işaretli olmalıdır. Dinamik kararlılığın iyileştirilmesi de özdeğer duyarlılığı ve lineer programlama kullanılarak yapılmaktadır. Lineer programlama teknikleri ile sistemin kritik özdeğerleri sol yarı düzleme kaydırılarak sistemin kararlı çalışması sağlanabilir. Lineer programlamayla birleştirilmiş özdeğer duyarlılığından yararlanarak dinamik kararlılığın oluşturulması, farklı dinamik karakteristiklere sahip güç sistemlerine başarıyla uygulanabilecek bir yöntem olarak görünmektedir.

Parallel operation of synchronous generators is commonly used in limited independent power stations. This paper introduces an approach to the determination of dynamic stability of such electrical power stations. The role and influence of the automatic voltage regulators in improving the quality of the parallel operation is discussed. A state-space representation of a power system including two synchronous generators connected in parallel and supplying loads has been accomplished. The eigenvalues method is applied for the determination of the system dynamic stability. The method presented here has the advantage that it provides a technique for evaluating the dynamic stability limit of interconnected-synchronous generators systems. The stable loading conditions for the limited independent power station can also be found by defining the stable regions of operation. Stable-regions charts have been derived for the cases of a single generator station and a power station that consists of two parallel-connected generators. These chrats are drawn for a typical independent power station supplying the electric loads of a gas plant. The comparison of the charts for a single and two parallel generators has shown that parallel operation increases the stable regions at the same loading conditions. An induction motor load of a comparable size to that of the generating unit is applied. Its influence on the dynamic stability of the system is investigated. The dynamic stability of electric power systems has been a subject of major theoretical and practical Interest since the advent of interconnection of large electric power systems, and it continues to grow in importance as the control requirements of the power plants become more sophisticated and demanding. VJ COMPUTATION OF EIGENVECTORS IN LARGE SVSTEMS For a particular eigenvalue, say Xi, the corresponding eigenvector is determined by solving the equation (A -^1) M1 =0 (I) and the reciprocal eigenvector is determined by solving the equation (AT-Xil)Wi=0 (2) Since |A - Xjl | is zero, the n algebraic equation implied by (1) have an infinite number of solutions. This difficulty can be avoided by arbitrarily assigning the value one to an element of the eigenvector, say the n-th, and dropping the last equation from the set. This results in a non-homogeneous set of equation from the set. This results in a non-hemogeneous set of equations with a unique solution which can be found presumably by one of the well-known procedures for solution of algebraic equations. a 11 "^ 1 a12 e21 s22"x 1... a1,n-l... a2,rH 3n-1,1 a n-1,n-1"*1 m21 m"_ n-1,1 -a- "a2n 3n-1,n (3) A similar set of equations must be solved for each eigenvalue and th M matrix is built up column by column. All of this presupposes that the eigenvalues are known exactly. Although the QR transform is a good algorithm, it Is not realistic to assume that the eigenvalues can be computed exactly by this method. Since the accuracy when finding eigenvectors by some methods depends on the accuracy of the eigenvalues, Wilkinson's inverse iteration method has the significant advantage of being relatively insensitive to the accuracy of the eigenvalue. The algorithm which will be described is a modified version of Wilkinson's scheme developed by Van Ness. VII The basic inverse iteration scheme can be described by the two equations. [A -pl]Y will be n yCk) = I _i_ Mj j=l UJ-P)k (II) (The cj coefficients will actually have changed due to the normalizing of Y but they will still be scalers and their ratios will not have changed so that (\\) still holds with proper interpretation of cj). If p is close to Xj Uj-p) will be small and the coefficient Cj will be by far arpk) the largest term in the summation of (11) Hence, zM will be very nearly proportional to Mj, the i-th eigenvector of A. If another eigenvalue, sayL, is closer to o then X\ then the iterative scheme will converge to Mn instead of fy (This eigenvector is correct but is associated with the wrong eigenvalue and this error will probably be corrected later if the Rayleigh quatient is used to correct the eigenvalue). If two or more eigenvalues are almost equally close to p, then convergence will be slow. For many systems the QR transform will give values of p which differ from X\ only in the fifth or sixth figure and zM changes only in the seventh and eighth figure from the second to third iteration. Systems wtih p off by as much as 10 % have converged in seven or eight iterations. Significant computing effort in solving (4) each iteration is saved by factoring (A - pi) into upper and lower triangular matrices which do not change during the Iterative process. Thus, a back substitution process is all that is required to solve equation (4) for each new value of Z. Since most of the computational effort in solving for the eigenvectors will be spent in factoring (A-pl) or some other equivalent operation, the inverse iteration method Is not appreciably slower than other methods for finding eigenvectors. It has a considerable advantage in accuracy as a result of its insensitivity to small errors in the eigenvalues. The computation steps «re ; 1. Set all elements of G and H Initially to unity, 2. Evaluate [BG^ + (A + I) TH^] and back substitute to find D. 3. Solve (10 for [(pT) -Mk+1>] by forward substitution. 4. Reorder the vector found in step 3 to correct for row interchanges and obtain T^+1^ 5. Substitute in (2 6. Normalize (S. + JT as in equation (2® to obtain the new value of (G+JH) 7. Compare the new (6 + JH) with the previous value. If the change is sufficiently small, this is the required eigenvector; if not, repeat steps 2 through 7 as often as necessary. A difficulty arises in factoring if the original A matrix contains decoupled subsystems (that is, A can be partitioned into block diagonal form). This can be recognized when the matrix is converted to Hessenberg form, preparatory to use of the QR transform. However, Van Ness describes an alternative approach which avoids the need to make reference to the Hessenberg subroutine. USE OF THE RAVLEIGH QUOTIENT TO EMPROVE THE ACCURACV EIGENVALUES Having found the eigenvectors, it is possible to correct the eigenvalues of A. The Rayleigh quotient Is described by the equation p(k+1)=p(k) + (29) WTiM1 where pM is the current estimate of the i-th eigenvalues and p(k+D is the corrected value. Mj and Wj are the i-th eigenvectors. The equation can be used with both real and complex eigenvalues. The product (A - p^OMj should be accumulated in double precision a.s considerable cancellation will occur in it. To tit that this result simplifies, it is noted that Cj is a scalar and that WTj and Mj are orthogonal. Hence, yd) = 2 _JL_ Mj j=1 JlJ-p (I°) After k Iterations, Z will be n yCk) = I _i_ Mj j=l UJ-P)k (II) (The cj coefficients will actually have changed due to the normalizing of Y but they will still be scalers and their ratios will not have changed so that (\\) still holds with proper interpretation of cj). If p is close to Xj Uj-p) will be small and the coefficient Cj will be by far arpk) the largest term in the summation of (11) Hence, zM will be very nearly proportional to Mj, the i-th eigenvector of A. If another eigenvalue, sayL, is closer to o then X\ then the iterative scheme will converge to Mn instead of fy (This eigenvector is correct but is associated with the wrong eigenvalue and this error will probably be corrected later if the Rayleigh quatient is used to correct the eigenvalue). If two or more eigenvalues are almost equally close to p, then convergence will be slow. For many systems the QR transform will give values of p which differ from X\ only in the fifth or sixth figure and zM changes only in the seventh and eighth figure from the second to third iteration. Systems wtih p off by as much as 10 % have converged in seven or eight iterations. Significant computing effort in solving (4) each iteration is saved by factoring (A - pi) into upper and lower triangular matrices which do not change during the Iterative process. Thus, a back substitution process is all that is required to solve equation (4) for each new value of Z. Since most of the computational effort in solving for the eigenvectors will be spent in factoring (A-pl) or some other equivalent operation, the inverse iteration method Is not appreciably slower than other methods for finding eigenvectors. It has a considerable advantage in accuracy as a result of its insensitivity to small errors in the eigenvalues. The computation steps «re ; 1. Set all elements of G and H Initially to unity, 2. Evaluate [BG^ + (A + I) TH^] and back substitute to find D. 3. Solve (10 for [(pT) -Mk+1>] by forward substitution. 4. Reorder the vector found in step 3 to correct for row interchanges and obtain T^+1^ 5. Substitute in (2 6. Normalize (S. + JT as in equation (2® to obtain the new value of (G+JH) 7. Compare the new (6 + JH) with the previous value. If the change is sufficiently small, this is the required eigenvector; if not, repeat steps 2 through 7 as often as necessary. A difficulty arises in factoring if the original A matrix contains decoupled subsystems (that is, A can be partitioned into block diagonal form). This can be recognized when the matrix is converted to Hessenberg form, preparatory to use of the QR transform. However, Van Ness describes an alternative approach which avoids the need to make reference to the Hessenberg subroutine. USE OF THE RAVLEIGH QUOTIENT TO EMPROVE THE ACCURACV EIGENVALUES Having found the eigenvectors, it is possible to correct the eigenvalues of A. The Rayleigh quotient Is described by the equation p(k+1)=p(k) + (29) WTiM1 where pM is the current estimate of the i-th eigenvalues and p(k+D is the corrected value. Mj and Wj are the i-th eigenvectors. The equation can be used with both real and complex eigenvalues. The product (A - p^OMj should be accumulated in double precision a.s considerable cancellation will occur in it. The computation steps «re ; 1. Set all elements of G and H Initially to unity, 2. Evaluate [BG^ + (A + I) TH^] and back substitute to find D. 3. Solve (10 for [(pT) -Mk+1>] by forward substitution. 4. Reorder the vector found in step 3 to correct for row interchanges and obtain T^+1^ 5. Substitute in (2 6. Normalize (S. + JT as in equation (2® to obtain the new value of (G+JH) 7. Compare the new (6 + JH) with the previous value. If the change is sufficiently small, this is the required eigenvector; if not, repeat steps 2 through 7 as often as necessary. A difficulty arises in factoring if the original A matrix contains decoupled subsystems (that is, A can be partitioned into block diagonal form). This can be recognized when the matrix is converted to Hessenberg form, preparatory to use of the QR transform. However, Van Ness describes an alternative approach which avoids the need to make reference to the Hessenberg subroutine. USE OF THE RAVLEIGH QUOTIENT TO EMPROVE THE ACCURACV EIGENVALUES Having found the eigenvectors, it is possible to correct the eigenvalues of A. The Rayleigh quotient Is described by the equation p(k+1)=p(k) + (29) WTiM1 where pM is the current estimate of the i-th eigenvalues and p(k+D is the corrected value. Mj and Wj are the i-th eigenvectors. The equation can be used with both real and complex eigenvalues. The product (A - p^OMj should be accumulated in double precision a.s considerable cancellation will occur in it. The computation steps «re ; 1. Set all elements of G and H Initially to unity, 2. Evaluate [BG^ + (A + I) TH^] and back substitute to find D. 3. Solve (10 for [(pT) -Mk+1>] by forward substitution. 4. Reorder the vector found in step 3 to correct for row interchanges and obtain T^+1^ 5. Substitute in (2 6. Normalize (S. + JT as in equation (2® to obtain the new value of (G+JH) 7. Compare the new (6 + JH) with the previous value. If the change is sufficiently small, this is the required eigenvector; if not, repeat steps 2 through 7 as often as necessary. A difficulty arises in factoring if the original A matrix contains decoupled subsystems (that is, A can be partitioned into block diagonal form). This can be recognized when the matrix is converted to Hessenberg form, preparatory to use of the QR transform. However, Van Ness describes an alternative approach which avoids the need to make reference to the Hessenberg subroutine. USE OF THE RAVLEIGH QUOTIENT TO EMPROVE THE ACCURACV EIGENVALUES Having found the eigenvectors, it is possible to correct the eigenvalues of A. The Rayleigh quotient Is described by the equation p(k+1)=p(k) + (29) WTiM1 where pM is the current estimate of the i-th eigenvalues and p(k+D is the corrected value. Mj and Wj are the i-th eigenvectors. The equation can be used with both real and complex eigenvalues. The product (A - p^OMj should be accumulated in double precision a.s considerable cancellation will occur in it.