New Formulation and Strong MISOCP Relaxations for AC Optimal Transmission Switching Problem

Abstract

As the modern transmission control and relay technologies evolve, transmission line switching has become an important option in power system operators' toolkits to reduce operational cost and improve system reliability. Most recent research has relied on the DC approximation of the power flow model in the optimal transmission switching problem. However, it is known that DC approximation may lead to inaccurate flow solutions and also overlook stability issues. In this paper, we focus on the optimal transmission switching problem with the full AC power flow model, abbreviated as AC OTS. We propose a new exact formulation for AC OTS and its mixed-integer second-order conic programming (MISOCP) relaxation. We improve this relaxation via several types of strong valid inequalities inspired by the recent development for the closely related AC Optimal Power Flow (AC OPF) problem [27]. We also propose a practical algorithm to obtain high quality feasible solutions for the AC OTS problem.Extensive computational experiments show that the proposed formulation and algorithms efficiently solve IEEE standard and congested instances and lead to significant cost benefits with provably tight bounds.

A low-rank coordinate-descent algorithm for semidefinite programming relaxations of optimal power flow

“…Our approach seems to bring the use of SDP relaxations of ACOPF closer to the engineering practice. A number of authors have recently explored elaborately constructed linear programming [5] and second-order cone programming [27,42,46,24,26] relaxations, many [27,26] of which are not comparable to the SDP relaxations in the sense that they may be stronger or weaker, depending on the instance, but aiming to be solvable faster. Algorithm 1 suggests that there are simple first-order algorithms, which can solve SDP relaxations of ACOPF faster than previously, at least on some instances.…”

“…constraints(21),(24),(27), and (31) are box constraints, while the remainder of (20-31) are linear equalities O 2 : using elementary linear algebra:…”

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

“…Different techniques have been proposed in the literature to obtain tighter relaxations of OPF [70]- [74]. Several papers have developed SDP-based approximations or reformulations for a wide range of power problems, such as state estimation [75], [76], unit commitment [17], and transmission switching [16], [77]. The recent findings in this area show the significant potential of conic relaxation for power optimization problems.…”

On Subadditive Duality for Conic Mixed-integer Programs

“…For instance, the Karush-Kuhn-Tucker (KKT) optimality conditions are generalized for MINLPs in [3], Lagrangian-based methods are used in [10,14], and other geometric [4] or algebraic [27] approaches are utilized to obtain strong duality results in particular cases.Conic MIPs: Conic MIP problems generalize MILPs and have significantly more expressive power in terms of modeling. To name just a few application areas, conic MIPs are used in options pricing [22], power distribution systems [17], Euclidean k-center problems [8] and engineering design [12]. We note here that all the conic MIPs used in these applications include binary variables, and that this feature, rather than being the exception, is a general rule when modeling real life problems.In spite of the growing interest in conic MIP applications, conic MIP solvers are not as mature as their MILP counterparts.…”

“…Conic MIPs: Conic MIP problems generalize MILPs and have significantly more expressive power in terms of modeling. To name just a few application areas, conic MIPs are used in options pricing [22], power distribution systems [17], Euclidean k-center problems [8] and engineering design [12]. We note here that all the conic MIPs used in these applications include binary variables, and that this feature, rather than being the exception, is a general rule when modeling real life problems.…”