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Distributed Optimal Power Flow Algorithm for Balanced Radial Distribution Networks
Qiuyu Peng, Steven H. Low
TL;DR
OPF is difficult to solve quickly and distributively in large renewable-rich distribution networks because it is nonconvex and involves global power-flow constraints. The paper develops an ADMM-based algorithm for balanced radial networks using the SOCP relaxation and closed-form local subproblem solutions. Simulations on a real-world 2,065-bus network demonstrate scalability and substantially faster subproblem computation than CVX.
Problem
Large-scale renewable penetration creates a need for fast, distributed OPF solutions, while centralized approaches face computation and communication overhead and OPF is nonconvex.
Method
The paper decomposes the SOCP-relaxed OPF problem for balanced radial distribution networks using ADMM, with local closed-form subproblem solutions and adjacent-bus communication.
Results
The algorithm converges within 0.6s on a 2,065-bus system, while its ADMM subproblems are solved 1,000x faster than with CVX.
Takeaways & Limitations
The proposed distributed algorithm is scalable to a real-world 2,065-bus distribution network and reduces computation time through closed-form subproblem solutions.
Abstract
from arXiv · showhide
The optimal power flow (OPF) problem is fundamental in power system operations and planning. Large-scale renewable penetration in distribution networks calls for real-time feedback control, and hence the need for fast and distributed solutions for OPF. This is difficult because OPF is nonconvex and Kirchhoff's laws are global. In this paper we propose a solution for balanced radial distribution networks. It exploits recent results that suggest solving for a globally optimal solution of OPF over a radial network through the second-order cone program (SOCP) relaxation. Our distributed algorithm is based on alternating direction method of multiplier (ADMM), but unlike standard ADMM algorithms that often require iteratively solving optimization subproblems in each ADMM iteration, our decomposition allows us to derive closed form solutions for these subproblems, greatly speeding up each ADMM iteration. We present simulations on a real-world 2,065-bus distribution network to illustrate the scalability and optimality of the proposed algorithm.
I. INTRODUCTION
OPF is important for power-system operations and planning but is challenging because of nonconvexity, global power-flow constraints, and the scaling demands of future distribution networks. The paper addresses these challenges with a distributed ADMM algorithm whose closed-form subproblems and adjacent-bus communication improve scalability and computation time.
- Motivation: OPF optimizes objectives such as power loss or generation cost subject to power-flow equations and operational constraints.It supports operations and planning tasks including economic dispatch, unit commitment, state estimation, and volt/var control.
- Challenges: Nonconvexity makes OPF difficult to solve, motivating convex relaxations including SDP and SOCP formulations.The paper focuses on the SOCP relaxation for balanced radial distribution networks.
- Challenges: Centralized OPF algorithms may not scale to networks with many intelligent endpoints because of computation and communication overhead.The paper therefore proposes a distributed algorithm for the SOCP-relaxed OPF problem.
- Contribution: Time of convergence equals the number of iterations multiplied by the computation time for solving subproblems in each iteration.The paper targets this metric by reducing subproblem computation time.
- Contribution: The proposed ADMM decomposition provides closed-form solutions for optimization subproblems and requires communication only between adjacent buses.These design choices eliminate iterative SDP/SOCP solves within each ADMM iteration.
- Results: The algorithm converges within 0.6s on a real-life 2,065-bus distribution network.The paper also reports a 1,000x subproblem-solving speedup over CVX.
III. DISTRIBUTED ALGORITHM FOR OPF
The paper applies ADMM to a convexified OPF formulation under an exact SOCP-relaxation assumption. Its convergence framework decomposes the problem into local updates, enabling distributed solution of the resulting convex subproblems.
- Algorithmic framework: The algorithm assumes the SOCP relaxation is exact and uses ADMM to solve the resulting relaxed OPF problem.The method first reviews standard ADMM and then derives closed-form expressions for its OPF subproblems.
- Preliminary: ADMM: ADMM combines the decomposability of dual decomposition with the convergence properties of the method of multipliers.The paper applies it to optimization problems with convex feasible sets Kx and Kz.
- Preliminary: ADMM: ADMM alternates updates of the decomposed variables and uses an augmented Lagrangian with penalty parameter ρ.When ρ = 0, the augmented Lagrangian reduces to the standard Lagrangian.
- Convergence: Under closed, proper, convex objective functions and a saddle point of the unaugmented Lagrangian, ADMM converges to an optimal solution.The paper states these conditions as assumptions A1 and A2.
B. Apply ADMM to OPF problem
The paper applies ADMM to decompose the convexified OPF problem into local subproblems that agents can solve simultaneously. Closed-form subproblem solutions and adjacent-bus communication reduce per-iteration computation and communication requirements.
- Closed-form updates: Each bus has a closed-form solution for its local optimization subproblem, avoiding iterative SDP/SOCP solves during ADMM iterations.For one subproblem, the efficient procedure requires solving zeros of three polynomials with degree at most 4.
- Communication: Communication is required only between adjacent buses.Agents exchange parent and child variables needed for the x-update and z-update message-passing steps.
- ADMM decomposition: ADMM decomposes the modified ROPF problem into x- and z-updates that each agent can solve simultaneously through small local subproblems.Consensus constraints are relaxed using Lagrangian multipliers, while each bus maintains local variables, copies, and associated multipliers.
- x-update: The x-update collects parent and child information, stacks local variables into a real vector, and solves a linearly constrained quadratic subproblem.The resulting subproblem has a positive diagonal quadratic matrix, a full-row-rank constraint matrix, and a closed-form solution.
- Initialization and stopping: The algorithm initializes z from a feasible point and stops when both primal and dual residuals fall below 10^-4√|N|.The initialization is interpreted as solving the branch-flow equation under zero line impedance.
IV. CASE STUDY
The case study evaluates the proposed distributed algorithm on a 2,065-bus distribution circuit in the Southern California Edison service territory. It examines scalability and compares closed-form subproblem computation with an off-the-shelf solver.
- Case-study design: The evaluation uses a model of a 2,065-bus distribution circuit in the Southern California Edison service territory.The study tests scalability on a real-world distribution network.
- Case-study design: The study compares subproblem computation time between the proposed closed-form procedure and the off-the-shelf solver CVX.This comparison is intended to show the advantage of deriving closed-form expressions.
- Case-study design: The study also simulates the proposed algorithm on networks of different sizes.
A. Simulation on a 2,065 bus circuit
The algorithm is evaluated on a real-world 2,065-bus distribution circuit with decoupled phases, voltage constraints, rooftop PV controls, and distributed bus agents. It converges quickly, while closed-form subproblem solutions substantially reduce computation time relative to CVX.
- The test network contains 1,409 household loads, 142 commercial loads, and 135 rooftop PV panels distributed across the houses.
- Each bus acts as an agent in the distributed OPF, with rooftop PV real and reactive injections controlled to minimize network power loss.The voltage magnitude at each load bus is constrained to [0.95, 1.05] pu.
- 1,114 iterations satisfy the stopping criteria, requiring 1,153s on one computer and an estimated 0.56s distributed ToC excluding communication overhead.
- Each ADMM iteration takes 6.8 × 10−4s with the proposed method versus 0.5s with CVX, yielding a 1,000x speedup.The x-update averages 1.7 × 10−4s versus 0.2s, while the z-update averages 5.1 × 10−4s versus 0.3s.
B. Rate of Convergence
The convergence-rate study examines how network size and diameter affect iteration counts. Simulations indicate that diameter has a stronger effect than network size, especially when comparing line and fat-tree topologies.
- The study analyzes convergence rate using network size N and diameter D, alongside the computation time per ADMM iteration.
- Fitting T = aN + bD to subnetworks gives a = 0.34 and b = 5.53, indicating stronger impact from network diameter.
- Line networks have the largest diameter for a given size, whereas fat-tree networks have the smallest diameter, equal to 2.
- Iteration counts increase notably with size for line networks, while the trend is less obvious for fat-tree networks.
- The conclusion reports scalability to a 2,065-bus system and a 1,000x speedup for solving each ADMM optimization subproblem.
APPENDIX A
Appendix A derives solutions for local optimization subproblems using KKT conditions and case analysis. The procedure recovers primal variables and multipliers by solving reduced polynomial equations under constraint-activity cases.
- Boundary cases set z*_3 to a lower or upper bound before recovering the remaining variables and checking the multipliers.
- The appendix introduces Lagrange multipliers for the constraints and derives the KKT optimality conditions for the local problem.
- Strict convexity of the objective and constraints gives a unique solution to P1 and its KKT conditions.
- The solution procedure enumerates multiplier cases according to whether constraints are inactive or active, then checks feasibility and multiplier signs.
- For the interior case, variables are reduced through substitutions involving p and recovered sequentially from the resulting equations.
- The appendix considers potentially multiple real solutions for p* and requires checking them before selecting a solution satisfying the KKT conditions.
APPENDIX B
Appendix B derives a closed-form solution for a local quadratic optimization problem with a second-order cone constraint. The solution reduces to a quartic equation with a unique strictly positive multiplier.
- When I_i has the specified form, the local optimization problem is written as a quadratic program with variables p and q.
- The derivation separates solutions into cases based on the signs of b_1 and b_2.
- The multiplier λ is obtained by solving a polynomial equation of degree 4, which has a closed-form expression.
- Although four solutions exist for the polynomial, KKT conditions establish that only one λ* is strictly positive.
- The optimal p* and q* are recovered from λ* using the stated rational expressions.