Small-signal stability analysis is about power system stability when subject to small disturbances. If power system oscillations caused by small disturbances can be suppressed, such that the deviations of system state variables remain small for a long time, the power system is stable. On the contrary, if the magnitude of oscillations continues to increase or sustain indefinitely, the power system is unstable. Power system small-signal stability is affected by many factors, including initial operation conditions, strength of electrical connections among components in the power system, characteristics of various control devices, etc. Since it is inevitable that power system operation is subject to small disturbances, any power system that is unstable in terms of small-signal stability cannot operate in practice. In other words, a power system that is able to operate normally must first be stable in terms of small-signal stability. Hence, one of the principal tasks in power system analysis is to carry out small-signal stability analysis to assess the power system under the specified operating conditions.The dynamic response of a power system subject to small disturbances can be studied by using the method introduced in Chap. 7 to determine system stability. However, when we use the method for power system small-signal stability analysis, in addition to slow computational speed, the weakness is that after a conclusion of instability is drawn, we cannot carry out any deeper investigation into the phenomenon and cause of system instability. The Lyapunov linearized method has provided a very useful tool for power system small-signal stability analysis. Based on the fruitful results of eigensolution analysis of linear systems, the Lyapunov linearized method has been widely used in power system small-signal stability analysis. In the following, we shall first introduce the basic mathematics of power system small-signal stability analysis.The Lyapunov linearized method is closely related to the local stability of nonlinear systems. Intuitively speaking, movement of a nonlinear system over a small range should have similar properties to its linearized approximation. To study the stability of the nonlinear system at point x e(1) If the linearized system is asymptotically stable, i.e., all eigenvalues of A have negative real parts, the actual nonlinear system is asymptotically stable at the equilibrium point. (2) If the linearized system is unstable, i.e., at least one of eigenvalues of A has a positive real part, the actual nonlinear system is unstable at the equilibrium point. (3) If the linearized system is critically stable, i.e., real parts of all eigenvalues of A are nonpositive but the real part of at least one of them is zero, no conclusion can be drawn about the stability of the nonlinear system from its linearized approximation.The basic principle of the Lyapunov linearized method is to draw conclusions about the local stability of the nonlinear system around the equilibrium point from the stability of its linear ap...
