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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, Newtons), 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.
We address the problem of computing the smallest symplectic eigenvalues and the corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamsons theorem. It is formulated as minimizing a trace cost function over the symp
As a representative mathematical expression of power flow (PF) constraints in electrical distribution system (EDS), the piecewise linearization (PWL) based PF constraints have been widely used in different EDS optimization scenarios. However, the lin
We introduce in this paper a manifold optimization framework that utilizes semi-Riemannian structures on the underlying smooth manifolds. Unlike in Riemannian geometry, where each tangent space is equipped with a positive definite inner product, a se
When solving hard multicommodity network flow problems using an LP-based approach, the number of commodities is a driving factor in the speed at which the LP can be solved, as it is linear in the number of constraints and variables. The conventional
In this paper, we give some new thoughts about the classical gradient method (GM) and recall the proposed fractional order gradient method (FOGM). It is proven that the proposed FOGM holds a super convergence capacity and a faster convergence rate ar