[Paper Review] Distributed Algorithm for Optimal Power Flow on Unbalanced Multiphase Distribution Networks

Qiuyu Peng, Steven H. Low|arXiv (Cornell University)|Dec 21, 2015

Optimal Power Flow Distribution11 references13 citations

TL;DR

This paper proposes a distributed ADMM-based algorithm for optimal power flow (OPF) in unbalanced multiphase distribution networks, leveraging problem-specific structure to replace computationally heavy semidefinite programming (SDP) subproblems with closed-form solutions or 6×6 Hermitian matrix eigen-decompositions. The method achieves sub-second convergence (under 2.5 seconds) across IEEE 13-, 34-, 37-, and 123-bus systems, with a 100x speedup over CVX-based SDP solvers in subproblem resolution.

ABSTRACT

The optimal power flow (OPF) problem is funda- mental in power distribution networks control and operation that underlies many important applications such as volt/var control and demand response, etc.. Large-scale highly volatile renewable penetration in the distribution networks calls for real-time feed- back control, and hence the need for distributed solutions for the OPF problem. Distribution networks are inherently unbalanced and most of the existing distributed solutions for balanced networks do not apply. In this paper we propose a solution for unbalanced distribution networks. Our distributed algorithm is based on alternating direction method of multiplier (ADMM). Unlike existing approaches that require to solve semidefinite programming problems in each ADMM macro-iteration, we exploit the problem structures and decompose the OPF problem in such a way that the subproblems in each ADMM macro- iteration reduce to either a closed form solution or eigen-decomposition of a 6x6 hermitian matrix, which significantly reduce the convergence time. We present simulations on IEEE 13, 34, 37 and 123 bus unbalanced distribution networks to illustrate the scalability and optimality of the proposed algorithm.

Motivation & Objective

• To address the lack of scalable, distributed OPF solutions for unbalanced multiphase distribution networks with high renewable penetration.
• To overcome the computational bottleneck of existing ADMM-based methods that require solving SDP subproblems in each iteration.
• To enable real-time, distributed control in distribution systems by reducing per-iteration computation time while preserving convergence and optimality.
• To incorporate practical constraints such as voltage magnitude limits and general objective functions for broader applicability.
• To analyze the impact of network topology—particularly diameter and size—on convergence rate in distributed OPF algorithms.

Proposed method

• The algorithm uses the alternating direction method of multipliers (ADMM) to decompose the OPF problem into local subproblems per bus, enabling distributed computation.
• It exploits the structural properties of unbalanced multiphase networks to reformulate subproblems as either closed-form solutions or eigen-decomposition of 6×6 Hermitian matrices.
• The method avoids solving semidefinite programs (SDPs) in each ADMM iteration by leveraging problem-specific algebraic structure and constraints.
• Voltage magnitude constraints are explicitly included in the formulation, enhancing practical relevance for distribution system operation.
• The algorithm requires only local communication between adjacent buses, minimizing communication overhead.
• A sufficient condition for the existence of efficient subproblem solutions is derived, ensuring applicability to practical networks.

Experimental results

Research questions

• RQ1 Can a distributed ADMM-based OPF algorithm be designed for unbalanced multiphase distribution networks that avoids solving SDP subproblems?
• RQ2 How does the proposed method reduce computation time per ADMM iteration compared to off-the-shelf SDP solvers?
• RQ3 What is the impact of network topology—specifically diameter and size—on the convergence rate of the distributed OPF algorithm?
• RQ4 Can the algorithm handle general objective functions and practical constraints such as voltage magnitude limits?
• RQ5 Does the algorithm maintain optimality and convergence guarantees while achieving significant speedups?

Key findings

• The proposed algorithm converges in under 2.5 seconds when distributed across multiple nodes for all tested IEEE 13-, 34-, 37-, and 123-bus unbalanced networks.
• Each ADMM iteration takes an average of 3.8×10⁻³ seconds using the proposed method, compared to 0.58 seconds using the CVX SDP solver—achieving over 100x speedup.
• The average time to solve a subproblem in the x-update is 9.8×10⁻⁵ s (proposed) vs. 0.13 s (CVX), and in the y-update is 3.7×10⁻³ s (proposed) vs. 0.45 s (CVX).
• Network diameter has a stronger impact on convergence rate than network size, with line networks (high diameter) requiring significantly more iterations than fat tree networks (low diameter).
• For a 50-bus line network, convergence required 6,061 iterations, while the same-size fat tree network required only 389 iterations.
• The algorithm maintains optimality and convergence under realistic constraints, including voltage magnitude limits and general objective functions.