An Analytical Expression For Electromechanical Oscillation...

ï€   An Analytical Expression for Electromechanical Oscillation Frequency Yongli Zhu, Student Member, IEEE, Kai Sun, Senior Member, IEEE Abstract—In this letter, a mathematic proof and improvement of a classic result for multi-machine systems modal analysis is given considering non-uniform damping and line resistance. An analytical formula is derived for the estimation of electromechanical oscillation frequency. Matrix theory based proof is conducted via determinant simplification and matrix triangularization. A half size shrunken system matrix is obtained, which brings an extra benefit of faster eigenvalue solving. The accuracy of the formula is validated against public available toolbox on two large power systems. the results highlight the†¦show more content†¦1. With only the generator internal buses left, the system model is (under common system MVA base): Fig. 1. Reduced network of n-machine system where: Mi=2Hi and D = diag (D1, D2, †¦, Dn). Pei is expressed by: where, Gij+jBij=|Yij|ïÆ' Ã¯  ¡ij is the element of the reduced nodal admittance matrix. Then, linearizing the above equations around the equilibrium point gives: where, the n-by-n Jacobian matrix J is: ÃŽ ´ij0= ÃŽ ´i0−Î ´j0, is the steady state rotor angle difference between the ith and jth generator. B. Analytic Eigenvalue Solution Step-1: Determinant simplification From above section, the 2n-by-2n system matrix A is: the eigenvalues ÃŽ » are the roots of the determinant equation: where, I is the identity matrix and the property det(XT) = det(X) for any square matrix X is utilized. Then the following matrix lemma will be utilized to obtain the analytic eigenvalue solution [3]: Lemma: For matrix , if , then: Obviously, the condition holds for the bottom two block matrices in (6). (6) becomes (use det(XT) = det(X) again): Step-2: Approximate Matrix Triangularization Theorem (Simultaneous Triangularization): A matrix set {A1, A2, †¦An} can be simultaneously (upper) triangularized by a nonsingular matrix P if and only if each commutator of the form AiAj−AjAi is nilpotent, i.e. ï€ ¤ kïÆ'ŽZ+, such that (AiAj−AjAi)k =O [4]. For set {I, M-1D, M-1J}, the above condition holds as long as T=(M-1D)(M-1J) − (M-1J)(M-1D)=[tij]