Lyapunov modal analysis and participation factors applied to small-signal stability of power systems

Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms

“…The use of canonical forms of controllability allow to propose a pioneering approach to computation of Gramians based on the use of Routh tables and Xiao matrices. This paper proposes to improve this approach by using spectral decompositions of the Gramians and the representation of the resolvent of the dynamics matrix by extending the scope to multivariable linear and bilinear control systems [11,[26][27][28][29].…”

“…into Formulae (11) and ( 13), we obtain new formulas for the expansion of the controllability and observability Gramians…”

“…Thus, the system of linear algebraic Equations ( 24)-( 28) implements an alternative method to determine elements of the Gramian control matrices in a simpler way than solving the linear algebraic equation in the form of Kronecker products and applying vectorization method (8) [16]. In addition, Formulas (10) and (11) give analytical solution of Lyapunov equation based on Gramian method. If a linear system is stable and the eigen numbers of its dynamics matrix are different, the solution of the Lyapunov equation is singular.…”

“…Let us discuss properties of the matrix and solution elements on the basis of analysis of solutions to the system of Equations ( 24)- (28) and Formulas (10) and (11).…”

“…New mathematical modelling techniques, such as generation and load forecasting, optimal control of energy storage systems, and new methods for energy systems control will play a key role in 21st century power engineering and in realizing the Internet of Energy concept [9]. The development of new innovative technologies requires the involvement of a modern mathematical control theory, in which the study of the structural properties and computational methods of energy systems Gramians is an important task [10,11].…”