Stability of power systems and other systems of second order differential equations
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Analytic criteria have been developed to predict the stability or instability of equilibrium points of systems of second order differential equations. The results were applied to the swing equations for electric power systems;It was shown that the trivial solution is exponentially stable if and only if the linearization matrix is stable. This rapid rate of convergence ensures that the asymptotic stability of a certain subsystem will imply the stability of the whole system. In the case of uniform damping, stability results proved by linearization allow one to classify equilibrium points as either stable or unstable, except for a certain critical case. To help implement these and related results, an algorithm was developed to determine whether all eigenvalues of a matrix are real and non-positive. Applied to the swing equations with non-trivial transfer conductances, these results demonstrate that an equilibrium solution may not be stable even though (VBAR)(delta)(,i) - (delta)(,j) - (alpha)(,ij)(VBAR) < (pi)/2 for all rotor angle pairs (delta)(,i), (delta)(,j) where (alpha)(,ij) is the complement of the phase of the transfer impedance between machines i and j;Other stability criteria were proved using invariance theory and an extension of the concept of Hamiltonian systems. These results may be applied to the swing equations with negligible transfer conductances. It was shown that equilibrium states for this system may be stable even though most pairs of rotors are more than 90(DEGREES) out of phase; some may be 180(DEGREES) out of phase;A sufficient condition for a non-linear map being one-to-one in a convex region was proved. An application to the swing equations showed that there can be no more than one equilibrium satisfying (VBAR)(delta)(,i) - (delta)(,j) - (alpha)(,ij)(VBAR) < (pi)/2 for all i, j. A simple analytic criterion for the existence of stable equilibrium states was proved for the swing equations with negligible transfer conductances.