CONVEX OPTIMIZATION METHODS FOR ROBUST AND DETERMINISTIC OPTIMAL POWER FLOW PROBLEMS
Today's energy sector faces a trilemma of challenges: keeping electricity reliable, affordable, and clean. Providing affordable and reliable service entails the determination of an operating point for the power system, which minimizes the total cost of generation while respecting the physical and operational constraints of the system. At the core of reliability and affordability challenges is the AC optimal power flow problem (AC-OPF). In its most general form, the AC-OPF problem is a high-dimensional optimization problem that is nonconvex and NP hard in general. In addition to reliability and affordability challenges, in recent years there has been a growing need to develop optimization methods, which enable the reliable and efficient operation of power systems that have a large fraction of their power supplied from intermittent renewable energy resources, like wind and solar. The need to accommodate the intrinsic uncertainty in the power supply of such resources will require the development of robust optimization methods for the AC-OPF problem. In its most general formulation, the robust AC optimal power flow(RAC-OPF) problem amounts to a two-stage robust optimization problem, in which the system operator must determine a day-ahead generation schedule that minimizes the expected cost of dispatch, given an opportunity for recourse to adjust its day-ahead schedule in real-time when the uncertain system variables have been realized. In addition to being nonconvex, the RAC-OPF problem is an infinite-dimensional optimization problem due to the need to optimize over an infinite-dimensional recourse policy space. The central topic of this thesis is the development of computationally tractable convex inner and outer approximations (relaxations) for the AC-OPF problem and the RAC-OPF problem. In the first part of the thesis, we focus on the AC-OPF problem, its equivalent reformulation as a rank one constrained semidefinite program, and its semidefinite programming relaxation. First, we study an a priori sufficient condition developed in the literature, which guarantees the exactness of the relaxation and then we develop an posteriori sufficient condition, which can be used to verify the inexactness of the relaxation. For AC-OPF problems that do not satisfy the sufficient condition for exactness, we investigate the extent to which it is possible to apply a structured perturbation to the problem data to obtain a problem, which satisfies said sufficient condition and which yields an optimal solution that is feasible for the original problem. An explicit bound on the performance of the feasible solution is also derived. In addition to perturbation-based inner approximations, we also propose an inner approximation scheme, which is based on an equivalent representation of the rank one constraint as the difference of two convex functions. Using this representation, we develop an algorithm, which is guaranteed to generate a sequence of feasible solutions with nonincreasing costs. Lastly, we propose an iterative linearization-minimization algorithm to uncover rank one optimal solutions for the semidefinite relaxation when the relaxation is exact, but its optimal solution set contains both rank one and high rank optimal solutions. A simple bisection method is also proposed to address problems for which the linearization-minimization procedure fails to return a rank-one optimal solution. In the second part of the thesis, we focus on the RAC-OPF problem. By restricting the space of recourse policies to those which are affine in the uncertain problem data, we propose a method to approximate RAC-OPF from within by a finite-dimensional semidefinite program. The solution of this optimization problem gives rise to an affine recourse policy for the RAC-OPF problem. In addition to the inner approximation, we develop a method for constructing a second-order cone outer approximation to the RAC-OPF problem. The crux of our approach centers on the reformulation of RAC-OPF as a robust rank one constrained semidefinite program, which is then relaxed to a robust linear program. The relaxation is obtained by eliminating the rank one constraint and by approximating the cone of positive semidefinite matrices from without by a polyhedral cone. A recursive method is also developed, which refines said polyhedral cone in those regions that are important for optimization\ to improve the performance of the relaxation. The practical value of our method is that one can obtain a feasible solution to the RAC-OPF problem by solving a finite-dimensional semidefinite program whose suboptimality can be bounded by solving a second-order cone program. In addition, if the gap between the optimal values of the outer and inner approximations is small, we have a certificate of near optimality of the feasible solution obtained. To the best of our knowledge, our approach is the first to provide a systematic method for computing a feasible solution to the RAC-OPF problem and a nontrivial global lower bound on its optimal value via convex optimization. As the final contribution, we develop computationally tractable inner and outer approximations to robust semidefinite programs, a class of optimization problems that is NP-hard in general. The proposed method relies on approximating the positive semidefinite cone from within and without by appropriate polyhedral cones. In addition, we develop a recursive method, which refines these cones to sharpen the approximations. In particular our method, eliminates the optimal solution of the approximation at the current iteration step from the feasible set of the approximation at the next iteration step.
AC Optimal Power Flow; Convex Relaxations; Robust AC Optimal Power Flow; Robust Semidefinite Programs; Semidefinite Relaxations; Electrical engineering; Operations research
Bitar, Eilyan Yamen
Tong, Lang; Lewis, Adrian S.
Electrical and Computer Engineering
Ph. D., Electrical and Computer Engineering
Doctor of Philosophy
dissertation or thesis