A Wirtinger Power Flow Jacobian Singularity Condition for Voltage Stability in Converter-Rich Power Systems

The progression of modern power systems towards converter-rich operations calls for new models and analytics in steady-state voltage stability assessment. The classic modeling assumption of the generators as stiff voltage sources no longer holds. Instead, the voltage- and current-limited behaviors of converters need to be considered. In this paper, we develop a Wirtinger derivative-based formulation for the power flow Jacobian and derive an explicit sufficient condition for its singularity. Compared to existing works, we extend the explicit sufficient singularity condition to incorporate all bus types instead of only slack and PQ types. We prove that the singularity of the alternative Jacobian coincides with that of the conventional one. A bus-wise voltage stability index, denoted $C_{\mathrm{W}}$, is derived from diagonal dominance conditions. The condition $\min_i C_{W,i}$ being greater than one certifies the nonsingularity of the Jacobian and provides a fast, non-iterative stability margin. Case studies in standard IEEE test systems show that the proposed index yields less conservative and more localized assessments than classical indices such as the L-index, the $K_{\mathrm{R}}$ index, and the SCR index.