Voltage Stability Kernel: A Cofactor Theory of Voltage Stability in Lossy Power Systems

Takayuki Ishizaki, Jigen Koizumi, Hiroo Yashiba, Boqiang Sun

This paper introduces the voltage stability kernel (VSK), a cofactor-based bus-wise representation of voltage stability in lossy power systems. The VSK is defined as the vector of principal cofactors of the voltage stability Laplacian (VSL), a reduced Jacobian that retains voltage source internal angles while eliminating the other variables. We show that the VSK constitutes the left kernel of the VSL, which is typically nonsymmetric in lossy power systems. We also define the voltage stability margin (VSM) as the sum of all VSK components and show that it is equal to the product of all eigenvalues of the VSL except the trivial zero eigenvalue due to phase-shift symmetry. Thus, the VSK provides a bus-wise decomposition of the VSM. Furthermore, the VSK offers an algebraic interpretation of CPF calculations with a fixed slack bus. The singularity of the Jacobian in CPF calculations obtained by deleting the slack-bus row and column is characterized by the vanishing of the VSK component selected by the slack bus. In contrast, the static bifurcation is characterized by the vanishing of the VSM. Since these two conditions are generally different, our theory explains why a CPF nose point does not necessarily correspond to a static bifurcation in lossy cases.