Maximal Positively Invariant Set Determination for Transient Stability Assessment in Power Systems

Abstract

This paper assesses the transient stability of a synchronous machine connected to an infinite bus through the notion of invariant sets. The problem of computing a conservative approximation of the maximal positive invariant set is formulated as a semi-definitive program based on occupation measures and Lasserre's relaxation. An extension of the proposed method into a robust formulation allows us to handle Taylor approximation errors for non-polynomial systems. Results show the potential of this approach to limit the use of extensive time domain simulations provided that scalability issues are addressed.

Inner Approximations of the Maximal Positively Invariant Set for Polynomial Dynamical Systems

“…Our objective is to design a hierarchy of approximations of the MPI set for largescale systems described by non-linear differential equations. A first step towards non-polynomial dynamics can be found in [12]. Since the initial work [5] relied on the mathematical technology behind the approximation of the volume of semi-algebraic sets, we already studied in [14] the problem of approximating the volume of a large-scale sparse semi-algebraic set.…”

“…One can also find numerical applications of this method to actual eletrical engineering problems in [12] with very promising results.…”

Transient Stability Analysis of Power Grids with Admissible and Maximal Robust Positively Invariant Sets

“…Let G and L be the set of indices of all generator and load nodes, respectively, with |G| = p G , |L| = p L , and let p = p G + p L . For each generator, the dynamics is described by the swing equation, which is often used for transient stability analysis [8]- [12], [15], [18], [19], defined by…”

“…Hybrid approaches combining direct methods and timedomain simulations [13] and set-based methods, such as [14]- [16], where forward reachable sets are found for the "post-fault" dynamics, and the references [17], [18] where backwards reachable sets are found that constitute regions from where the state is guaranteed to reach a neighbourhood of an equilibrium point. Another set-based method is presented in [19] where invariant sets for synchronous machines are computed using the idea of occupation measures, and it is shown that inner approximations may be found through the solution of semi-definite programs.…”

Constrained control for microgrids with constant power loads

“…In particular, the ideas of [4] regarding the boundary structure of invariant sets of nonlinear systems characterise the structure of these sets. Several new advances have applied set invariance concepts, such as the maximal positive invariant set, to power system transients, see [11]. These set invariance methods provide the necessary tools to obtain certificates of stability and constraint satisfaction that are safety critical.…”