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A Hybrid Sequential Convex Programming Framework for Unbalanced Three-Phase AC OPF

Sary Yehia, Alessandra Parisio

TL;DR

This work tackles unbalanced three-phase AC OPF in distribution networks by introducing a hybrid sequential convex programming framework that couples McCormick outer relaxations with first-order linearisations under an adaptive trust-region. Each iteration solves a convex subproblem that remains faithful to the nonlinear physics, and convergence to a KKT-consistent stationary point is established through alignment-based constraints and a shrinking trust region. Empirical validation on IEEE feeders and a real Cyprus LV network shows near-IPOPT accuracy (<=0.1% optimality gap) with substantial runtime speedups (up to ~2x), demonstrating scalability to large, unbalanced networks. The approach offers a practical, fast, and robust alternative for real-time distribution-network OPF, with potential extensions to dynamic and multi-energy scenarios.

Abstract

This paper presents a hybrid Sequential Convex Programming (SCP) framework for solving the unbalanced three-phase AC Optimal Power Flow (OPF) problem. The method combines a fixed McCormick outer approximation of bilinear voltage-current terms, first-order Taylor linearisations, and an adaptive trust-region constraint to preserve feasibility and promote convergence. The resulting formulation remains convex at each iteration and ensures convergence to a stationary point that satisfies the first-order Karush-Kuhn-Tucker (KKT) conditions of the nonlinear OPF. Case studies on standard IEEE feeders and a real low-voltage (LV) network in Cyprus demonstrate high numerical accuracy with optimality gap below 0.1% and up to 2x faster runtimes compared to IPOPT. These results confirm that the method is accurate and computationally efficient for large-scale unbalanced distribution networks.

A Hybrid Sequential Convex Programming Framework for Unbalanced Three-Phase AC OPF

TL;DR

This work tackles unbalanced three-phase AC OPF in distribution networks by introducing a hybrid sequential convex programming framework that couples McCormick outer relaxations with first-order linearisations under an adaptive trust-region. Each iteration solves a convex subproblem that remains faithful to the nonlinear physics, and convergence to a KKT-consistent stationary point is established through alignment-based constraints and a shrinking trust region. Empirical validation on IEEE feeders and a real Cyprus LV network shows near-IPOPT accuracy (<=0.1% optimality gap) with substantial runtime speedups (up to ~2x), demonstrating scalability to large, unbalanced networks. The approach offers a practical, fast, and robust alternative for real-time distribution-network OPF, with potential extensions to dynamic and multi-energy scenarios.

Abstract

This paper presents a hybrid Sequential Convex Programming (SCP) framework for solving the unbalanced three-phase AC Optimal Power Flow (OPF) problem. The method combines a fixed McCormick outer approximation of bilinear voltage-current terms, first-order Taylor linearisations, and an adaptive trust-region constraint to preserve feasibility and promote convergence. The resulting formulation remains convex at each iteration and ensures convergence to a stationary point that satisfies the first-order Karush-Kuhn-Tucker (KKT) conditions of the nonlinear OPF. Case studies on standard IEEE feeders and a real low-voltage (LV) network in Cyprus demonstrate high numerical accuracy with optimality gap below 0.1% and up to 2x faster runtimes compared to IPOPT. These results confirm that the method is accurate and computationally efficient for large-scale unbalanced distribution networks.

Paper Structure

This paper contains 21 sections, 21 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Network Model
  4. AC OPF Formulation
  5. Hybrid SCP Model
  6. Outer Approximation of Bilinear Terms
  7. First-Order Linearisation of Bilinear Terms
  8. Trust-Region Constraint
  9. Alignment-Based SOC Constraint
  10. Adaptive Adjustment Strategy
  11. Convergence Analysis
  12. Relaxation Deviation
  13. Linearisation Error
  14. Numerical Evaluation
  15. Simulation Setup
  16. ...and 6 more sections

Figures (4)

  • Figure 1: Proposed Hybrid SCP framework for unbalanced three-phase AC OPF.
  • Figure 2: McCormick relaxation of the bilinear term $z = x \cdot y$ showing the true surface, convex envelope, and optimal solution.
  • Figure 3: IEEE distribution test systems used for evaluation.
  • Figure 4: Single-line diagram of the Cyprus LV distribution feeder.