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Progressive Integrality Outer-Inner Approximation for AC Unit Commitment with Conic Formulation

Yongzheng Dai

Abstract

The alternating-current unit commitment (AC-UC) problem provides a realistic representation of power system operations, which is a nonconvex mixed-integer nonlinear programming problem and hence is computationally intractable. A common relaxation to the AC-UC is based on the second-order cone (SOC), which results in a mixed-integer second-order cone program and remains computationally challenging. In this paper, we propose an outer-inner approximation framework that alternatively solves a mixed-integer linear programming (MILP) as an outer approximation and a convex second-order cone programming as an inner approximation to find a (near-)optimal solution to the SOC-based AC-UC. To improve computational efficiency, we introduce a progressive integrality strategy that gradually enforces integrality, reducing the reliance on expensive MILP solutions in early iterations. In addition, time-block Benders cuts are incorporated to strengthen the outer approximation and accelerate convergence. Computational experiments on large-scale test systems, including 200-bus and 500-bus networks, demonstrate that the proposed framework significantly improves both efficiency and robustness compared to state-of-the-art commercial solvers. The results show faster convergence, higher-quality solutions, and improved scalability under different formulations and perturbed load scenarios.

Progressive Integrality Outer-Inner Approximation for AC Unit Commitment with Conic Formulation

Abstract

The alternating-current unit commitment (AC-UC) problem provides a realistic representation of power system operations, which is a nonconvex mixed-integer nonlinear programming problem and hence is computationally intractable. A common relaxation to the AC-UC is based on the second-order cone (SOC), which results in a mixed-integer second-order cone program and remains computationally challenging. In this paper, we propose an outer-inner approximation framework that alternatively solves a mixed-integer linear programming (MILP) as an outer approximation and a convex second-order cone programming as an inner approximation to find a (near-)optimal solution to the SOC-based AC-UC. To improve computational efficiency, we introduce a progressive integrality strategy that gradually enforces integrality, reducing the reliance on expensive MILP solutions in early iterations. In addition, time-block Benders cuts are incorporated to strengthen the outer approximation and accelerate convergence. Computational experiments on large-scale test systems, including 200-bus and 500-bus networks, demonstrate that the proposed framework significantly improves both efficiency and robustness compared to state-of-the-art commercial solvers. The results show faster convergence, higher-quality solutions, and improved scalability under different formulations and perturbed load scenarios.
Paper Structure (44 sections, 14 equations, 2 figures, 5 tables, 4 algorithms)

This paper contains 44 sections, 14 equations, 2 figures, 5 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Contribution and Organization
  3. Formulation
  4. Model Formulation
  5. Objective function: minimize expected cost
  6. Constraints for commitment variables
  7. Logic constraints
  8. Minimum up/down constraints
  9. Constraints for active and reactive power output
  10. Range for power outputs
  11. Ramping Limits
  12. Area reserve requirement
  13. Constraints for power flow and network
  14. Constraints for power flows
  15. Active and reactive power balance
  16. ...and 29 more sections

Figures (2)

  • Figure 1: Relative gaps (left graph) and relative optimality gaps (right graph) for different methods
  • Figure 2: Performance of Gurobi and M3 over 10 random perturbed instances. The solid line shows the mean, and the shaded region represents the min–max envelope across samples.

Theorems & Definitions (2)

  • Remark 1
  • Remark 2