High-Order Approximate Power Flow Solutions and Circular Arithmetic Applications

Recent research has presented a sufficient condition for the existence of a practical power flow solution in distribution networks, and a fixed-point linear approximation in complex coordinates. By employing Wirtinger calculus, this paper shows that the fixed-point linear approximation can be obtained via a complex-variable first-order Taylor series expansion around the no-load operating point; it then presents new simplified formulas for second- and third-order Taylor series expansions. The high-order power flow solutions are shown to be highly accurate on standard distribution networks available in the literature, and significantly more precise than the existing linear approximation. The proposed power flow approximations are useful in circular arithmetic methods, which extend the modeling of parameter uncertainty from the real line to the complex plane. The application of circular arithmetic is finally illustrated for the solution of power flow based Volt/VAr control under power injection uncertainty. The results show the capacity of the approach in mitigating voltage violations whenever injections deviate from their nominal values.