Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Activate the Dual Cones: A Tight Reformulation of Conic ACOPF Constraints

Saba Rafiei, Samuel Chevalier

Abstract

By exploiting the observed tightness of dual rotated second-order cone (RSOC) constraints, this paper transforms the dual of a conic ACOPF relaxation into an equivalent, non-conic problem where dual constraints are implicitly enforced through eliminated dual RSOC variables. To accomplish this, we apply the RSOC-based Jabr relaxation of ACOPF, pose its dual, and then show that all dual RSOC constraints must be tight (i.e., active) at optimality. We then construct a reduced dual maximization problem with only non-negativity constraints, avoiding the explicit RSOC inequality constraints. Numerical experiments confirm that the tight formulation recovers the same dual objective values as a mature conic solver (e.g., MOSEK via PowerModels) on various PGLib benchmark test systems (ranging from 3- to 1354-buses). The proposed formulation has useful performance benefits, compared with its conic counterpart, and it allows us to define a bounding function which provides a guaranteed lower bound on system cost. While this paper focuses on demonstrating the correctness and validity of the proposed structural simplification, it lays the groundwork for future GPU-accelerated first-order optimization methods which can exploit the unconstrained nature of the proposed formulation.

Activate the Dual Cones: A Tight Reformulation of Conic ACOPF Constraints

Abstract

By exploiting the observed tightness of dual rotated second-order cone (RSOC) constraints, this paper transforms the dual of a conic ACOPF relaxation into an equivalent, non-conic problem where dual constraints are implicitly enforced through eliminated dual RSOC variables. To accomplish this, we apply the RSOC-based Jabr relaxation of ACOPF, pose its dual, and then show that all dual RSOC constraints must be tight (i.e., active) at optimality. We then construct a reduced dual maximization problem with only non-negativity constraints, avoiding the explicit RSOC inequality constraints. Numerical experiments confirm that the tight formulation recovers the same dual objective values as a mature conic solver (e.g., MOSEK via PowerModels) on various PGLib benchmark test systems (ranging from 3- to 1354-buses). The proposed formulation has useful performance benefits, compared with its conic counterpart, and it allows us to define a bounding function which provides a guaranteed lower bound on system cost. While this paper focuses on demonstrating the correctness and validity of the proposed structural simplification, it lays the groundwork for future GPU-accelerated first-order optimization methods which can exploit the unconstrained nature of the proposed formulation.
Paper Structure (14 sections, 3 theorems, 36 equations, 1 figure, 8 tables)

This paper contains 14 sections, 3 theorems, 36 equations, 1 figure, 8 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. ACOPF Formulation
  4. Implementation of Jabr's SOCP Relaxation
  5. Canonicalization of Jabr's Relaxation
  6. Formulating the Dual
  7. Dual RSOC Inequality Constraints are Tight
  8. Explicitly enforcing dual RSOC constraints
  9. Handling non-differentiability via $\epsilon$-stabilization and dual postprocessing
  10. Numerical Experiments
  11. Conclusion
  12. AI Usage Disclosure
  13. Affine function for RSOC Dualization
  14. Additional results

Key Result

Lemma 1

When eq: conic_soln is solved to optimality, the conic inequalities eq: dual_conic_eqs2-eq: dual_conic_eqs4 are tight, i.e., they hold with equality.

Figures (1)

  • Figure 1: Change of dual objective as a function of $\epsilon$. As the stability parameter is swept (log scale), the dual objective value for the 14-bus case increases monotonically.

Theorems & Definitions (7)

  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof