Individual Invariance and Functions Approaches to Transient Stability in Power Systems

Student thesis: Doctoral Thesis


Power systems represents one of the most advanced and most active fields in nonlinear systems' stability analysis. Due to the critical nature of power systems combined with high degrees of uncertainty, Power system operators always seek methods to evaluate and maintain system's stability. Lyapunov and Energy function approaches dominated the field of stability assessment in power systems due to their ability to provide quantitative measure of the degree of system stability and an insight about the system behavior at the stability boundary without the need of numerical integrations. Lyapunov functions in power system applications require two steps, the construction of a Lyapunov function to assert the stability of the equilibrium point, and the enlargement of its sub level set to estimate the relevant region of attraction. Hence, the challenges of Lyapunov functions in power systems are twofold. First, the lack of systematic method to construct a Lyapunov function. Second, the nonlinear nature of the vector field in power system lead to nonlinear (non-convex) Lyapunov functions which makes the task of estimating the region of attraction computationally challenging. In this work, we represent a general autonomous vector field as a summation of individual vector fields and examines the correlation between each individual vector and the original vector. This redefinition of nonlinear systems led to new results in nonlinear theory, mainly individual invariance theorem and individual functions theorem. Individual invariance relate invariant sets of individual systems to invariant sets of the original system leading to significant simplification in estimating regions of stability. In context of individual vectors, a Lyapunov-like individual functions theory was derived to assert the stability and asymptotic stability of general autonomous systems through conditions on individual vector fields. As a corollary to this theorem, it is shown that level sets of individual functions are contained in the region of attraction. Results were derived for convex quadratic individual functions which allows the formulation of stability problem as a sequence of semi definite programming problems for functions construction and region of attraction estimation. The proposed method shows substantial improvement against traditional Lyapunov and Energy function methods with test cases including: standard Lure control systems as well as single machine and multi-machine power systems.
Date of AwardDec 2017
Original languageAmerican English


  • Power Systems
  • Stability Assessment
  • Lyapunov Functions Energy Functions
  • Single Machine Power Systems
  • Multi-machine Power Systems.

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