Matrix Minor Reformulation and SOCP-based Spatial Branch-and-Cut Method for the AC Optimal Power Flow Problem

Alternating current optimal power flow (AC OPF) is one of the most fundamental optimization problems in electrical power systems. It can be formulated as a semidefinite program (SDP) with rank constraints. Solving AC OPF, that is, obtaining near optimal primal solutions as well as high quality dual bounds for this non-convex program, presents a major computational challenge to today's power industry for the real-time operation of large-scale power grids. In this paper, we propose a new technique for reformulation of the rank constraints using both principal and non-principal 2-by-2 minors of the involved Hermitian matrix variable and characterize all such minors into three types. We show the equivalence of these minor constraints to the physical constraints of voltage angle differences summing to zero over three- and four-cycles in the power network. We study second-order conic programming (SOCP) relaxations of this minor reformulation and propose strong cutting planes, convex envelopes, and bound tightening techniques to strengthen the resulting SOCP relaxations. We then propose an SOCP-based spatial branch-and-cut method to obtain the global optimum of AC OPF. Extensive computational experiments show that the proposed algorithm significantly outperforms the state-of-the-art SDP-based OPF solver and on a simple personal computer is able to obtain on average a 0.71% optimality gap in no more than 720 seconds for the most challenging power system instances in the literature

Oracle-Based Robust Optimization via Online Learning

Robust optimization is a common framework in optimization under uncertainty when the problem parameters are not known, but it is rather known that the parameters belong to some given uncertainty set. In the robust optimization framework the problem solved is a min-max problem where a solution is judged according to its performance on the worst possible realization of the parameters. In many cases, a straightforward solution of the robust optimization problem of a certain type requires solving an optimization problem of a more complicated type, and in some cases even NP-hard. For example, solving a robust conic quadratic program, such as those arising in robust SVM, ellipsoidal uncertainty leads in general to a semidefinite program. In this paper we develop a method for approximately solving a robust optimization problem using tools from online convex optimization, where in every stage a standard (non-robust) optimization program is solved. Our algorithms find an approximate robust solution using a number of calls to an oracle that solves the original (non-robust) problem that is inversely proportional to the square of the target accuracy

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

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. 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

On affine scaling inexact dogleg methods for bound-constrained nonlinear systems

Within the framework of affine scaling trust-region methods for bound constrained problems, we discuss the use of a inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model. Focusing on bound-constrained systems of nonlinear equations, an inexact affine scaling method for large scale problems, employing the inexact dogleg procedure, is described. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given