Alleviating Fractal and Ill-Conditioning Problems of the AC Power Flow Using a Polynomial Form

Power system resilience assessment, and mitigation, which deals with high-impact, low probability events, is gaining a great deal of attention among power researchers. To perform such an analysis, the availability of tools able to analyze modern power grids under normal and extreme disturbances is imperative. One such tool is the AC power flow algorithm. The latter should exhibit a numerical stability and an excellent convergence property under a broad range of operating conditions of a power system. One major difficulty that needs to be overcome stems from the nonlinearity and non-convexity of the AC power flow equations. As shown by Thorp and Naqavi, the Newton-Raphson (NR) method suffers from a fractal behavior, and the power flow Jacobian matrix becomes singular at points located on the fractal hypersurface, preventing that algorithm to converge. Furthermore, the Jacobian matrix becomes ill-conditioned for power systems that have relatively short lines or are heavy loaded. To address these problems, we propose a transformation of the AC power flow equations from a sinusoidal form to a polynomial form, which is solved using a modified Newton's method. The new polynomial form, whose order is equal to the number of solutions, encompasses node-based equations, Pythagorean equations, and loop-based equations. Furthermore, our approach is quadratic in three cases. Simulation results performed on several power systems reveal the good performance of the proposed method.