# Strong SOCP Relaxations for the Optimal Power Flow Problem

arXiv: https://arxiv.org/abs/1504.06770 · HTML: https://mlsanity.com/paper/1504.06770/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Apr 2015.

## Built on by (most cited first, 9 of 9)

- [Unsupervised Optimal Power Flow Using Graph Neural Networks](https://mlsanity.com/paper/2210.09277/) (2022)
- [Dual Conic Proxies for AC Optimal Power Flow](https://mlsanity.com/paper/2310.02969/) (2023)
- [A Review of Safe Reinforcement Learning Methods for Modern Power Systems](https://mlsanity.com/paper/2407.00304/) (2024)
- [Scalable Learning for Optimal Load Shedding Under Power Grid Emergency Operations](https://mlsanity.com/paper/2111.11980/) (2021)
- [Escaping Local Minima Provably in Non-convex Matrix Sensing: A Deterministic Framework via Simulated Lifting](https://mlsanity.com/paper/2602.05887/) (2026)
- [Machine Learning for Scalable and Optimal Load Shedding Under Power System Contingency](https://mlsanity.com/paper/2405.05521/) (2024)
- [Dual Conic Proxy for Semidefinite Relaxation of AC Optimal Power Flow](https://mlsanity.com/paper/2502.06978/) (2025)
- [Online Planning of Power Flows for Power Systems Against Bushfires Using Spatial Context](https://mlsanity.com/paper/2404.13391/) (2024)
- [Learning Samples Importance: Parameterizing Dual Variables in Everywhere Learning](https://mlsanity.com/paper/2609.36310/) (2026)

## Abstract (arXiv)

This paper proposes three strong second order cone programming (SOCP) relaxations for the AC optimal power flow (OPF) problem. These three relaxations are incomparable to each other and two of them are incomparable to the standard SDP relaxation of OPF. Extensive computational experiments show that these relaxations have numerous advantages over existing convex relaxations in the literature: (i) their solution quality is extremely close to that of the SDP relaxations (the best one is within 99.96% of the SDP relaxation on average for all the IEEE test cases) and consistently outperforms previously proposed convex quadratic relaxations of the OPF problem, (ii) the solutions from the strong SOCP relaxations can be directly used as a warm start in a local solver such as IPOPT to obtain a high quality feasible OPF solution, and (iii) in terms of computation times, the strong SOCP relaxations can be solved an order of magnitude faster than standard SDP relaxations. For example, one of the proposed SOCP relaxations together with IPOPT produces a feasible solution for the largest instance in the IEEE test cases (the 3375-bus system) and also certifies that this solution is within 0.13% of global optimality, all this computed within 157.20 seconds on a modest personal computer. Overall, the proposed strong SOCP relaxations provide a practical approach to obtain feasible OPF solutions with extremely good quality within a time framework that is compatible with the real-time operation in the current industry practice.