[Paper Review] Sufficient Conditions for Exact Semidefinite Relaxation of Optimal Power Flow in Unbalanced Multiphase Radial Networks

Fengyu Zhou, Yue Chen+1 more|arXiv (Cornell University)|Oct 18, 2019

Optimal Power Flow Distribution24 references4 citations

TL;DR

This paper establishes sufficient conditions for exact semidefinite programming (SDP) relaxation in unbalanced multiphase radial distribution networks by proving that exact relaxation holds when critical buses—those contributing to the cost or binding injection constraints—are non-adjacent. The result generalizes single-phase tree network conditions to multi-phase systems using voltage and injection relationships in complex vector form.

ABSTRACT

This paper proves that in an unbalanced multi-phase network with a tree topology, the semidefinite programming relaxation of optimal power flow problems is exact when critical buses are not adjacent to each other. Here a critical bus either contributes directly to the cost function or is where an injection constraint is tight at optimality. Our result generalizes a sufficient condition for exact relaxation in single-phase tree networks to tree networks with arbitrary number of phases.

Motivation & Objective

• To address the lack of sufficient conditions for exact SDP relaxation in unbalanced multiphase radial distribution networks.
• To generalize existing sufficient conditions from single-phase radial networks to multi-phase systems with arbitrary phase counts.
• To identify structural network properties—specifically, non-adjacency of critical buses—that guarantee exact relaxation.
• To provide a priori verifiable conditions for exact relaxation without requiring knowledge of the optimal solution.
• To extend theoretical guarantees to cases with unique or multiple optimal solutions, and to discuss implications for nonlinear cost functions.

Proposed method

• Model the unbalanced multiphase radial network using a complex admittance matrix Y ∈ ℂ^(mn×mn), where m is the number of phases and n the number of buses.
• Define phase-specific voltage and injection vectors V_j^ϕ and s_j^ϕ, and use projection matrices E_j^ϕ to extract phase-specific components of the network admittance.
• Express real and reactive power injections at each bus-phase as quadratic forms: Re(s_j^ϕ) = V^H Φ_j^ϕ V and Im(s_j^ϕ) = V^H Ψ_j^ϕ V, where Φ_j^ϕ and Ψ_j^ϕ are Hermitian matrices.
• Formulate the optimal power flow (OPF) problem as a minimization of a linear combination of real and reactive injections, subject to voltage magnitude and injection bounds.
• Apply semidefinite relaxation by lifting the voltage vector V to a positive semidefinite matrix W = V V^H, transforming the non-convex problem into a convex SDP.
• Establish sufficient conditions for exact relaxation by analyzing the structure of critical buses (those with active cost or binding constraints) and proving that non-adjacency ensures rank-1 solution recovery.

Experimental results

Research questions

• RQ1 Under what conditions is the semidefinite relaxation of the optimal power flow problem exact in unbalanced multiphase radial networks?
• RQ2 Can the sufficient conditions for exact relaxation in single-phase radial networks be generalized to multi-phase systems with arbitrary phase counts?
• RQ3 How does the spatial distribution of critical buses—those with active cost contributions or binding constraints—affect the exactness of the SDP relaxation?
• RQ4 Can the exactness of the relaxation be verified a priori without solving the original OPF problem?
• RQ5 What happens to the exactness guarantee when the SDP relaxation has multiple optimal solutions rather than a unique one?

Key findings

• The SDP relaxation is exact when critical buses—those contributing to the cost or with tight injection constraints—are not adjacent in the radial network.
• A priori verifiable sufficient conditions (Corollary 1) exist that do not require knowledge of the optimal solution, relying only on network topology and cost function signs.
• Theoretical results are proven for unique optimal solutions; for multiple solutions, the conditions must be replaced by linear separability, as in prior work.
• In a numerical example with an 11-bus network, the optimal SDP solution had eigenvalues 36.90 and 1.44×10^−10, confirming rank-1 recovery up to numerical precision.
• The proposed conditions generalize prior single-phase results to multi-phase systems, and simulation results confirm exact relaxation in IEEE 13, 37, 123-bus, and a 2065-bus real-world network—even when all buses are critical, violating the non-adjacency condition.
• The method applies to convex, monotonic, and additively separable cost functions, and the core argument extends to nonlinear cases based on cost function sign structure.