**2. Stability of Periodic Solutions**

Power grid dynamics with initial values *t*0 are shown as follows:

$$
\dot{\mathbf{x}} = f(t, \mathbf{x}), \mathbf{x}(t\_0) = \mathbf{x}\_0 \tag{1}
$$

System stability can be determined by the real value of eigenvalue as system is linear. However, for a nonlinear system, there are plenty of ways to determine system stability. The maximal Lyapunov exponent is an indicator that gives information on stability of time-dependent solution from Equation (1). The fluctuation of solution can be decreased as maximal Lyapunov exponent is negative and vice versa. However, different approaches are required to analyze behavior as the solution is oscillating. A monodromy matrix gives the characteristics of the oscillatory solution of the nonlinear system (1).
