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Polynomial SDP Cuts for Optimal Power Flow

Hassan Hijazi(hassan.hijazi***at***anu.edu.au)
Carleton Coffrin(carleton.coffrin***at***nicta.com.au)
Pascal Van Hentenryck(pvanhent***at***umich.edu)

Abstract: The use of convex relaxations has lately gained considerable interest in Power Systems. These relaxations play a major role in providing quality guarantees for non-convex optimization problems. For the Optimal Power Flow (OPF) prob- lem, the semidefinite programming (SDP) relaxation is known to produce tight lower bounds. Unfortunately, SDP solvers still suffer from a lack of scalability. In this work, we introduce an exact reformulation of the SDP relaxation, formed by a set of polynomial constraints defined in the space of real vari- ables. The new constraints can be seen as “cuts”, strengthening weaker second-order cone relaxations, and can be generated in a lazy iterative fashion. The new formulation can be handled by standard nonlinear programming solvers, enjoying better stability and computational efficiency. This new approach benefits from recent results on tree-decomposition methods, reducing the dimension of the underlying SDP matrices. As a side result, we present a formulation of Kirchhoff’s Voltage Law in the SDP space and reveal the existing link between these cycle constraints and the original SDP relaxation for three dimensional matrices. Preliminary results show a significant gain in computational efficiency compared to a standard SDP solver approach.

Keywords: OPF, SDP relaxation, SOCP relaxation, Polynomial Constraints, Nonlinear Programming.

Category 1: Applications -- OR and Management Sciences

Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 3: Nonlinear Optimization (Constrained Nonlinear Optimization )

Citation: @article{Hijazi:15, year={2015}, month={October}, journal={{ANU} Technical Report}, institution={{The Australian National University}}, title={Polynomial SDP Cuts for Optimal Power Flow}, author={Hijazi, Hassan and Coffrin, Carleton and Van Hentenryck, Pascal}, }

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Entry Submitted: 10/07/2015
Entry Accepted: 10/08/2015
Entry Last Modified: 10/07/2015

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