[Paper Review] Optimal Power Flow with Step-Voltage Regulators in Multi-Phase Distribution Networks

Mohammadhafez Bazrafshan, Nikolaos Gatsis+1 more|arXiv (Cornell University)|Jan 14, 2019

Optimal Power Flow Distribution25 references4 citations

TL;DR

This paper proposes a convex relaxation-based optimal power flow (OPF) framework for three-phase distribution networks that jointly optimizes tap settings of wye, closed-delta, and open-delta step-voltage regulators (SVRs). By leveraging a phase-separation assumption on SVR secondary voltages, it applies novel McCormick relaxations to trilinear voltage-tap terms and approximates rank-1 constraints, achieving a tight convex relaxation with small optimality gaps even in large networks.

ABSTRACT

This paper develops a branch-flow based optimal power flow (OPF) problem for multi-phase distribution networks that allows for tap selection of wye, closed-delta, and open-delta step-voltage regulators (SVRs). SVRs are assumed ideal and their taps are represented by continuous decision variables. To tackle the non-linearity, the branch-flow semidefinite programming framework of traditional OPF is expanded to accommodate SVR edges. Three types of non-convexity are addressed: (a) rank-1 constraints on non-SVR edges, (b) nonlinear equality constraints on SVR power flows and taps, and (c) trilinear equalities on SVR voltages and taps. Leveraging a practical phase-separation assumption on the SVR secondary voltage, novel McCormick relaxations are provided for (c) and certain rank-1 constraints of (a), while dropping the rest. A linear relaxation based on conservation of power is used in place of (b). Numerical simulations on standard distribution test feeders corroborate the merits of the proposed convex formulation.

Motivation & Objective

• To develop a unified optimal power flow (OPF) formulation that incorporates tap selection for wye, closed-delta, and open-delta step-voltage regulators (SVRs) in three-phase distribution networks.
• To address the non-convexity introduced by SVR modeling, including trilinear equalities between voltages and taps, and rank-1 constraints on voltage matrices.
• To improve computational tractability and solution quality by leveraging a practical phase-separation assumption on SVR secondary voltages.
• To provide a tight convex relaxation of the non-convex OPF problem that enables efficient solution with small optimality gaps.

Proposed method

• Expands the branch-flow semidefinite programming (SDP) framework to include SVR edges with continuous tap variables.
• Applies novel McCormick relaxations to trilinear equality constraints linking SVR voltages and taps, enabled by a phase-separation assumption on secondary voltages.
• Approximates certain rank-1 constraints on SVR secondary voltage matrices using the phase-separation assumption to improve relaxation quality.
• Uses a linear relaxation based on power conservation to handle nonlinear equality constraints on SVR power flows.
• Employs a tap recovery scheme to reconstruct integer tap settings from the relaxed solution, minimizing optimality gap.
• Leverages radial network topology and chordal SDP relaxation to enhance computational efficiency.

Experimental results

Research questions

• RQ1 Can a unified OPF formulation be developed to jointly optimize tap settings of wye, closed-delta, and open-delta SVRs in three-phase distribution networks?
• RQ2 How can the trilinear non-convexities arising from SVR voltage-tap relationships be effectively relaxed while preserving solution quality?
• RQ3 To what extent does the phase-separation assumption on SVR secondary voltages enable tighter convex relaxations of rank-1 constraints and trilinear terms?
• RQ4 What is the computational and optimality gap performance of the proposed convex relaxation compared to standard MINLP or SDP approaches?
• RQ5 Can the proposed method achieve near-optimal solutions with low optimality gaps in large-scale distribution feeders?

Key findings

• The proposed convex relaxation achieves very small optimality gaps—often less than 1%—even in large-scale three-phase distribution networks.
• The phase-separation assumption on SVR secondary voltages enables effective McCormick relaxations of trilinear voltage-tap terms and improves rank-1 constraint approximation.
• Numerical simulations on standard IEEE test feeders confirm the method’s ability to handle all three SVR types (wye, closed-delta, open-delta) within a single unified OPF framework.
• The relaxation quality is significantly improved by approximating rank-1 constraints on SVR secondary voltage matrices using the phase-separation assumption.
• The method outperforms existing approaches in computational efficiency while maintaining high solution accuracy, as evidenced by low optimality gaps.
• The tap recovery scheme successfully reconstructs feasible integer tap settings from the continuous relaxation, preserving solution quality.