Invertibility Conditions for the Admittance Matrices of Balanced Power Systems
Daniel Turizo, Daniel K. Molzahn
TL;DR
This work addresses when balanced power-system admittance matrices are invertible, a property essential for Kron reduction and fault analysis. It corrects a technical flaw in prior invertibility conditions by introducing a complex-valued, generalized-incidence framework with $Y_N = A_{L,N}^H Y_L A_{L,N} + Y_T$ that handles lossless branches and transformers with off-nominal taps. The authors prove a generalized invertibility theorem, develop a reactive-component reduction, and provide practical sufficiency conditions for common network topologies, accompanied by an algorithm and numerical validation on realistic grids. The results enable efficient, condition-based certification of invertibility in large-scale power systems and point to directions for extending the theory to polyphase networks and broader transformer models.
Abstract
The admittance matrix encodes the network topology and electrical parameters of a power system in order to relate the current injection and voltage phasors. Since admittance matrices are central to many power engineering analyses, their characteristics are important subjects of theoretical studies. This paper focuses on the key characteristic of \emph{invertibility}. Previous literature has presented an invertibility condition for admittance matrices. This paper first identifies and fixes a technical issue in the proof of this previously presented invertibility condition. This paper then extends this previous work by deriving new conditions that are applicable to a broader class of systems with lossless branches and transformers with off-nominal tap ratios.