Convergence of Backward/Forward Sweep for Power Flow Solution in Radial Networks

Solving power flow is a fundamental problem in power systems. The normally radial (tree) topology of a distribution network induces a spatially recursive structure in ac power flow, which enables a class of efficient solution methods—backward/forward sweep (BFS). In this article, we revisit BFS from the perspective of its convergence, which was rarely addressed before. We introduce three variants of BFS algorithms: the first one calculates voltages and line currents in a single-phase network model; the second algorithm extends the first one to an unbalanced three-phase network with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$Y$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Delta$</tex-math></inline-formula> configurations; the third one calculates voltages and line power flows in the classical dist-flow model. We prove a sufficient condition, under which the first algorithm is a contraction mapping on a closed set of voltages and thus converges geometrically to a unique solution. This proof is extended to the second algorithm for three-phase networks. We then use the monotone convergence theorem to prove convergence of the third algorithm. We verify the convergence conditions, solution accuracy, and computational efficiency of BFS algorithms through simulations in IEEE test systems.