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Register a new userA Novel Riemannian Optimization Approach to the Radial Distribution Network Load Flow Problem
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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.
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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 symplectic Stiefel manifold. We first investigate various theoretical aspects of this optimization problem such as characterizing the sets of critical points, saddle points, and global minimizers as well as proving that non-global local minimizers do not exist. Based on our recent results on constructing Riemannian structures on the symplectic Stiefel manifold and the associated optimization algorithms, we then propose solving the symplectic eigenvalue problem in the framework of Riemannian optimization. Moreover, a connection of the sought solution with the eigenvalues of a special class of Hamiltonian matrices is discussed. Numerical examples are presented.
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 linearized approximation errors originating from the currently-used PWL approximation function can be very large and thus affect the applicability of the PWL based PF constraints. This letter analyzes the approximation error self-optimal (ESO) condition of the PWL approximation function, refines the PWL function formulas, and proposes the self-optimal (SO)-PWL based PF constraints in EDS optimization which can ensure the minimum approximation errors. Numerical results demonstrate the effectiveness of the proposed method.
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 semi-Riemannian manifold allows the metric tensor to be indefinite on each tangent space, i.e., possessing both positive and negative definite subspaces; differential geometric objects such as geodesics and parallel-transport can be defined on non-degenerate semi-Riemannian manifolds as well, and can be carefully leveraged to adapt Riemannian optimization algorithms to the semi-Riemannian setting. In particular, we discuss the metric independence of manifold optimization algorithms, and illustrate that the weaker but more general semi-Riemannian geometry often suffices for the purpose of optimizing smooth functions on smooth manifolds in practice.
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 approach to improve the solve time of the LP relaxation of a Mixed Integer Programming (MIP) model that encodes such an instance is to aggregate all commodities that have the same origin or the same destination. However, the bound of the resulting LP relaxation can significantly worsen, which tempers the efficiency of aggregating techniques. In this paper, we introduce the concept of partial aggregation of commodities that aggregates commodities over a subset of the network instead of the conventional aggregation over the entire underlying network. This offers a high level of control on the trade-off between size of the aggregated MIP model and quality of its LP bound. We apply the concept of partial aggregation to two different MIP models for the multicommodity network design problem. Our computational study on benchmark instances confirms that the trade-off between solve time and LP bound can be controlled by the level of aggregation, and that choosing a good trade-off can allow us to solve the original large-scale problems faster than without aggregation or with full aggregation.
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 around the extreme point than the conventional GM. The property of asymptotic convergence of conventional GM and FOGM is also discussed. To achieve both a super convergence capability and an even faster convergence rate, a novel switching FOGM is proposed. Moreover, we extend the obtained conclusion to a more general case by introducing the concept of p-order Lipschitz continuous gradient and p-order strong convex. Numerous simulation examples are provided to validate the effectiveness of proposed methods.
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