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On the Tightness of the Second-Order Cone Relaxation of the Optimal Power Flow with Angles Recovery in Meshed Networks

Ginevra Larroux, Matthieu Jacobs, Mario Paolone

Abstract

This letter investigates properties of the second-order cone relaxation of the optimal power flow (OPF) problem, with emphasis on relaxation tightness, nodal voltage angles recovery, and alternating-current-OPF feasibility in meshed networks. The theoretical discussion is supported by numerical experiments on standard IEEE test cases. Implications for power system planning are briefly outlined.

On the Tightness of the Second-Order Cone Relaxation of the Optimal Power Flow with Angles Recovery in Meshed Networks

Abstract

This letter investigates properties of the second-order cone relaxation of the optimal power flow (OPF) problem, with emphasis on relaxation tightness, nodal voltage angles recovery, and alternating-current-OPF feasibility in meshed networks. The theoretical discussion is supported by numerical experiments on standard IEEE test cases. Implications for power system planning are briefly outlined.
Paper Structure (10 sections, 3 theorems, 12 equations, 1 figure)

This paper contains 10 sections, 3 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. SOC-OPF formulation by Low_PartI
  4. SOC-OPF formulation by Zhao
  5. Discussion
  6. Theoretical considerations
  7. Numerical counterexample
  8. Step 1: verify feasibility in the non-relaxed (o-ACOPF) model
  9. Step 2: test cycle consistency of the voltage angle differences across branches implied by the (SOC-ACOPF) model
  10. Conclusions

Key Result

Theorem 1

Assume $(\theta_l^{\min}, \theta_l^{\max}) = \left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ and $(v_n^{\min}, v_n^{\max})=(0.9,1.1)$, replacing eq:KVL_phase with eq:soc_eq2d relaxes the (o-ACOPF) problem.

Figures (1)

  • Figure 1: Residuals when solving the linear systems $(A x \approx b)$ and $(C\theta_n \approx d)$, for the radial IEEE 33-bus (red) and meshed IEEE 39-bus (blue) networks.

Theorems & Definitions (4)

  • Theorem 1: Theorem 1 in Zhao
  • Theorem 2: Theorem 4 in Zhao
  • Lemma 1
  • proof