Transient Stability Analysis of Power Systems via Occupation Measures

Abstract

We propose the application of occupation measure theory to the classical problem of transient stability analysis for power systems. This enables the computation of certified inner and outer approximations for the region of attraction of a nominal operating point. In order to determine whether a post-disturbance point requires corrective actions to ensure stability, one would then simply need to check the sign of a polynomial evaluated at that point. Thus, computationally expensive dynamical simulations are only required for postdisturbance points in the region between the inner and outer approximations. We focus on the nonlinear swing equations but voltage dynamics could also be included. The proposed approach is formulated as a hierarchy of semidefinite programs stemming from an infinite-dimensional linear program in a measure space, with a natural dual sum-of-squares perspective. On the theoretical side, this paper lays the groundwork for exploiting the oscillatory structure of power systems by using Hermitian (instead of real) sums-of-squares and connects the proposed approach to recent results from algebraic geometry.

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

“…The original motivation behind our current work is the study of transient phenomena in large-scale electrical power systems, see [4] and references therein. Our objective is to design a hierarchy of approximations of the MPI set for largescale systems described by non-linear differential equations.…”

“…where the infimum is with respect to u ≥ 0, v ∈ C 1 (X ) and w ∈ C 0 (X ). (5) is very similar to the initial problem (2), but with an additional slack variable u related to the constraint µ(X ) ≤ T λ (X ) in the primal (4). For u = 0, (2) and (5) are equivalent.…”

Review on set-theoretic methods for safety verification and control of power system

“…In [40 ], a control policy is designed and certified using barrier certificates such that the voltage limits during transients are respected under generated active and reactive power set points. Closely related works are the stability analysis based on Lyapunov functions [41–45 ].…”

“…In [43], a control policy is designed and certified using barrier certificates such that the voltage limits during transients are respected under generated active and reactive power setpoints. Closely related works are the stability analysis based on Lyapunov functions [44]- [48]. One advantage of passivity-based methods compared with the set operation-based methods is that the certificate is a function of system states.…”

Stability of Multi-Microgrids: New Certificates, Distributed Control, and Braess's Paradox

“…This is partly because there is no global energy function for such systems [12], and therefore, some basic techniques (e.g., the energy function method) for studying these equations cannot be directly applied. Nonetheless, several workarounds are developed over the years [9], [10], [13]- [18]. For instance, the method in [15] constructs Lyapunov functions for lossy swing equations using dissipative systems theory.…”