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Hypergraph-Based Fast Distributed AC Power Flow Optimization

Xinliang Dai, Yingzhao Lian, Yuning Jiang, Colin N. Jones, Veit Hagenmeyer

TL;DR

This paper rigorously establishes the convergence guarantee of the proposed algorithm with a locally quadratic rate and the one-step convergence of the inner algorithm when using the Levenberg-Marquardt regularization and demonstrates that the computational complexity of the proposed algorithm is much lower than the state-of-art distributed algorithm.

Abstract

This paper presents a novel distributed approach for solving AC power flow (PF) problems. The optimization problem is reformulated into a distributed form using a communication structure corresponding to a hypergraph, by which complex relationships between subgrids can be expressed as hyperedges. Then, a hypergraph-based distributed sequential quadratic programming (HDQ) approach is proposed to handle the reformulated problems, and the hypergraph-based distributed sequential quadratic programming (HDSQP) is used as the inner algorithm to solve the corresponding QP subproblems, which are respectively condensed using Schur complements with respect to coupling variables defined by hyperedges. Furthermore, we rigorously establish the convergence guarantee of the proposed algorithm with a locally quadratic rate and the one-step convergence of the inner algorithm when using the Levenberg-Marquardt regularization. Our analysis also demonstrates that the computational complexity of the proposed algorithm is much lower than the state-of-art distributed algorithm. We implement the proposed algorithm in an open-source toolbox, i.e., rapidPF, and conduct numerical tests that validate the proof and demonstrate the great potential of the proposed distributed algorithm in terms of communication effort and computational speed.

Hypergraph-Based Fast Distributed AC Power Flow Optimization

TL;DR

This paper rigorously establishes the convergence guarantee of the proposed algorithm with a locally quadratic rate and the one-step convergence of the inner algorithm when using the Levenberg-Marquardt regularization and demonstrates that the computational complexity of the proposed algorithm is much lower than the state-of-art distributed algorithm.

Abstract

This paper presents a novel distributed approach for solving AC power flow (PF) problems. The optimization problem is reformulated into a distributed form using a communication structure corresponding to a hypergraph, by which complex relationships between subgrids can be expressed as hyperedges. Then, a hypergraph-based distributed sequential quadratic programming (HDQ) approach is proposed to handle the reformulated problems, and the hypergraph-based distributed sequential quadratic programming (HDSQP) is used as the inner algorithm to solve the corresponding QP subproblems, which are respectively condensed using Schur complements with respect to coupling variables defined by hyperedges. Furthermore, we rigorously establish the convergence guarantee of the proposed algorithm with a locally quadratic rate and the one-step convergence of the inner algorithm when using the Levenberg-Marquardt regularization. Our analysis also demonstrates that the computational complexity of the proposed algorithm is much lower than the state-of-art distributed algorithm. We implement the proposed algorithm in an open-source toolbox, i.e., rapidPF, and conduct numerical tests that validate the proof and demonstrate the great potential of the proposed distributed algorithm in terms of communication effort and computational speed.
Paper Structure (16 sections, 2 theorems, 42 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 16 sections, 2 theorems, 42 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. Hyergraph-based modeling
  4. Hypergraph-based AC Power Flow Problem
  5. Distributed Optimization Algorithm
  6. Preliminaries
  7. Hypergraph-Based SQP
  8. Local Convergence Analysis
  9. Numerical Case Study
  10. Implementation
  11. Case Studies
  12. HDSQP vs. ALADIN
  13. Conclusion and Outlook
  14. Proof of Proposition \ref{['thm::convergence::hdq']}
  15. Proof of Theorem \ref{['thm::gn::rate']}
  16. ...and 1 more sections

Key Result

Proposition 1

Let the Levenberg-Marquardt method be used to evaluate $B_{\ell}^k$ such that the condensed qp subproblem eq::dsqp::subqp::reduced is strongly convex, then given by Algorithm alg solves eq::dsqp::subqp::reduced at iteration $k$.

Figures (4)

  • Figure 1: Decomposition by sharing components for a two-region system
  • Figure 2: The open-source toolbox rapidpf
  • Figure 3: Convergence behavior of hdsqp
  • Figure 4: Comparison of different algorithms for Case 3

Theorems & Definitions (6)

  • Remark 1
  • Proposition 1
  • Definition 1: Regular KKT point of \ref{['eq::pf::ls::hypergraph']}
  • Remark 2
  • Theorem 1
  • Remark 3: Initialization