  In this paper, we formulate the Load Flow (LF) problem in radial electricity
distribution networks as an unconstrained Riemannian optimization problem,
consisting of two manifolds, and we consider alternative retractions and
initialization options. Our contribution is a novel LF solution method, which
we show belongs to the family of Riemannian approximate Newton methods
guaranteeing monotonic descent and local superlinear convergence rate. To the
best of our knowledge, this is the first exact LF solution method employing
Riemannian optimization. Extensive numerical comparisons on several test
networks illustrate that the proposed method outperforms other Riemannian
optimization methods (Gradient Descent, Newton's), and achieves comparable
performance with the traditional Newton-Raphson method, albeit besting it by a
guarantee to convergence. We also consider an approximate LF solution obtained
by the first iteration of the proposed method, and we show that it
significantly outperforms other approximants in the LF literature. Lastly, we
derive an interesting comparison with the well-known Backward-Forward Sweep
method.
