A new HEAP-inspired method for embedded power flows and path-related holomorphic properties

Power flow problems can be robustly solved by holomorphic embedding (HE) methods which require the holomorphic property of solution functions near a reference state and also along a path bridging this state and the target state, for determining the effectiveness of series expansion and physical realizability (PRL), respectively, where PRL means the complex conjugacy of solution values at the target state. Existing studies generally assume holomorphic properties beforehand or assert them from a convergence of series expansion or implicit function theorem at separate points, however, which can hardly be applied to confirm the holomorphic property along a path and then PRL (because a path has infinitely many points and cannot be numerically inspected point-by-point). To this end, a new approach PHVACR is enlightened by HEAP and employs an expansion in the rectangular form and also Cauchy-Riemann (CR) conditions to numerically verify holomorphic properties around a path in an arc-by-arc manner. Here, 'HE' is short for holomorphic embedding, 'AP' in HEAP indicates an arc-length parametrization, and 'PHVA' in PHVACR refers to the path-related holomorphy validation with 'CR' representing the Cauchy-Riemann conditions. To explain the importance of PHVACR and its performance, counter-intuitive examples are discussed to explain the limitations of existing approaches, and large problems with 10,000+ buses are also simulated to examine the effectiveness and scalability of PHVACR which are compared with other HE methods.