Stability analysis of large‐scale dynamical systems by sub‐Gramian approach

SUMMARYIn this paper, we consider two methods for solving differential and algebraic Lyapunov equations in the time and frequency domains. Solutions of these equations are finite and infinite Gramians. In the first approach, we use the Laplace transform to solve the equations, and we apply the expansion of the matrix resolvent of the dynamical system. The expansions are bilinear and quadratic forms of the Faddeev matrices generated by resolvents of the original matrices. The second method allows computation of an infinite Gramian of a stable system as a sum of sub‐Gramians, which characterize the contribution of eigenmodes to the asymptotic variation of the total system energy over an infinite time interval. Because each sub‐Gramian is associated with a particular eigenvector, the potential sources of instability can easily be localized and tracked in real time. When solutions of Lyapunov equations have low‐rank structure typical of large‐scale applications, sub‐Gramians can be represented in low‐rank factored form, which makes them convenient in the stability analysis of large systems. Our numerical tests for Kundur's four‐machine two‐area system confirm the suitability of using Gramians and sub‐Gramians for small‐signal stability analyses of electric power systems. Copyright © 2013 John Wiley & Sons, Ltd.