Inexactness of SDP Relaxation and Valid Inequalities for Optimal Power Flow

By Burak Kocuk, Santanu S. Dey, Xu Andy Sun. IEEE Transactions on Power Systems, 2014

138 citations recorded by the source corpus.

It has been recently proven that the semidefinite programming (SDP) relaxation of the optimal power flow problem over radial networks is exact under technical conditions such as not including generation lower bounds or allowing load over-satisfaction. In this paper, we investigate the situation where generation lower bounds are present. We show that even for a 2-bus 1-generator system, the SDP relaxation can have all possible approximation outcomes, that is 1) SDP relaxation may be exact, 2) SDP relaxation may be inexact, or 3) SDP relaxation may be feasible while the optimal power flow (OPF) instance may be infeasible. We provide a complete characterization of when these three approximation outcomes occur and an analytical expression of the resulting optimality gap for this 2-bus system. In order to facilitate further research, we design a library of instances over radial networks in which the SDP relaxation has positive optimality gap. Finally, we propose valid inequalities and variable bound tightening techniques that significantly improve the computational performance of a global optimization solver. Our work demonstrates the need of developing efficient global optimization methods for the solution of OPF even in the simple but fundamental case of radial networks.

Subfields: Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/cd27b1a255a5a3eec3814e717a5e9adfed738365

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In Electrical and Electronic Engineering

1. Vijay Vittal
2. Qifeng Li
3. Xu Andy Sun
4. Santanu S. Dey
5. Burak Kocuk

Citations in the corpus

• Spectral Algorithms for Computing Fair Support Vector Machines (2017, background): “Despite this result, we believe that additional convexification techniques [Kocuk et al., 2016, Madani et al., 2017] can be used to strengthen the quality of the SDP relaxation; we leave the problem of how to design a strengthened SDP relaxation for future work.”
• Non-Iterative Enhanced SDP Relaxations for Optimal Scheduling of Distributed Energy Storage in Distribution Systems (2017, methodology): “Illustrative examples are provided in [13], [14] and [17] to demonstrate the impact of some bounds/constraints on the exactness of the SOCP relaxation for OPF problems with a unique objective function.”
• Strong SOCP Relaxations for the Optimal Power Flow Problem (2015, methodology): “A straightforward approach is to optimize cij and sij over the feasible region of the SOCP relaxation as is proposed in Kocuk et al. (2015). However, this procedure can be expensive because we need to solve 4|L| SOCPs, each of the size of the classic SOCP relaxation.”
• Convex Hull of the Quadratic Branch AC Power Flow Equations and Its Application in Radial Distribution Networks (2017, methodology): “Based on the BFM in radial networks, reference [11] proposes a set of non-iterative linear cuts to obtain a tight root node for solving the OPF problem in BARON [12]; valid linear cuts are generated iteratively for tightening the SOCP relaxation of OPFs in [13]; Several convex constraints which can ”
• New Formulation and Strong MISOCP Relaxations for AC Optimal Transmission Switching Problem (2015, unknown): “AC OPF) for brevity, unless stated otherwise.”