Inexact Convex Relaxations for AC Optimal Power Flow: Towards AC Feasibility

Convex relaxations of AC optimal power flow (AC-OPF) problems have attracted significant interest as in several instances they provably yield the global optimum to the original non-convex problem. If, however, the relaxation is inexact, the obtained solution is not AC-feasible. The quality of the obtained solution is essential for several practical applications of AC-OPF, but detailed analyses are lacking in existing literature. This paper aims to cover this gap. We provide an in-depth investigation of the solution characteristics when convex relaxations are inexact, we assess the most promising AC feasibility recovery methods for large-scale systems, and we propose two new metrics that lead to a better understanding of the quality of the identified solutions. We perform a comprehensive assessment on 96 different test cases, ranging from 14 to 3120 buses, and we show the following: (i) Despite an optimality gap of less than 1%, several test cases still exhibit substantial distances to both AC feasibility and local optimality and the newly proposed metrics characterize these deviations. (ii) Penalization methods fail to recover an AC-feasible solution in 15 out of 45 cases, and using the proposed metrics, we show that most failed test instances exhibit substantial distances to both AC-feasibility and local optimality. For failed test instances with small distances, we show how our proposed metrics inform a fine-tuning of penalty weights to obtain AC-feasible solutions. (iii) The computational benefits of warm-starting non-convex solvers have significant variation, but a computational speedup exists in over 75% of the cases

Adjoint-based predictor-corrector sequential convex programming for parametric nonlinear optimization

This paper proposes an algorithmic framework for solving parametric optimization problems which we call adjoint-based predictor-corrector sequential convex programming. After presenting the algorithm, we prove a contraction estimate that guarantees the tracking performance of the algorithm. Two variants of this algorithm are investigated. The first one can be used to solve nonlinear programming problems while the second variant is aimed to treat online parametric nonlinear programming problems. The local convergence of these variants is proved. An application to a large-scale benchmark problem that originates from nonlinear model predictive control of a hydro power plant is implemented to examine the performance of the algorithms.Comment: This manuscript consists of 25 pages and 7 figure

Decentralized non-convex optimization via bi-level SQP and ADMM

Non-convex optimization problems arise in many problems of practical relevance-for example in distributed nonlinear MPC or distributed optimal power flow. Only few existing decentralized optimization methods have local convergence guarantees for general nonconvex problems. We present novel convergence results for non-convex problems for a bi-level SQP method that solves the inner quadratic problems via ADMM. A decentralized stopping criterion borrowed from inexact Newton methods allows the early termination of ADMM as an inner algorithm to improve computational efficiency. The method shows competitive numerical performance to existing methods for an optimal power flow problem

A Preconditioned Inexact Active-Set Method for Large-Scale Nonlinear Optimal Control Problems

We provide a global convergence proof of the recently proposed sequential homotopy method with an inexact Krylov--semismooth-Newton method employed as a local solver. The resulting method constitutes an active-set method in function space. After discretization, it allows for efficient application of Krylov-subspace methods. For a certain class of optimal control problems with PDE constraints, in which the control enters the Lagrangian only linearly, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method is faster than using direct linear algebra for the 2D benchmark and allows for the parallel solution of large 3D problems.Comment: 26 page

Distributed Optimization with Application to Power Systems and Control

In many engineering domains, systems are composed of partially independent subsystems—power systems are composed of distribution and transmission systems, teams of robots are composed of individual robots, and chemical process systems are composed of vessels, heat exchangers and reactors. Often, these subsystems should reach a common goal such as satisfying a power demand with minimum cost, flying in a formation, or reaching an optimal set-point. At the same time, limited information exchange is desirable—for confidentiality reasons but also due to communication constraints. Moreover, a fast and reliable decision process is key as applications might be safety-critical. Mathematical optimization techniques are among the most successful tools for controlling systems optimally with feasibility guarantees. Yet, they are often centralized—all data has to be collected in one central and computationally powerful entity. Methods from distributed optimization control the subsystems in a distributed or decentralized fashion, reducing or avoiding central coordination. These methods have a long and successful history. Classical distributed optimization algorithms, however, are typically designed for convex problems. Hence, they are only partially applicable in the above domains since many of them lead to optimization problems with non-convex constraints. This thesis develops one of the first frameworks for distributed and decentralized optimization with non-convex constraints. Based on the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) algorithm, a bi-level distributed ALADIN framework is presented, solving the coordination step of ALADIN in a decentralized fashion. This framework can handle various decentralized inner algorithms, two of which we develop here: a decentralized variant of the Alternating Direction Method of Multipliers (ADMM) and a novel decentralized Conjugate Gradient algorithm. Decentralized conjugate gradient is to the best of our knowledge the first decentralized algorithm with a guarantee of convergence to the exact solution in a finite number of iterates. Sufficient conditions for fast local convergence of bi-level ALADIN are derived. Bi-level ALADIN strongly reduces the communication and coordination effort of ALADIN and preserves its fast convergence guarantees. We illustrate these properties on challenging problems from power systems and control, and compare performance to the widely used ADMM. The developed methods are implemented in the open-source MATLAB toolbox ALADIN-—one of the first toolboxes for decentralized non-convex optimization. ALADIN- comes with a rich set of application examples from different domains showing its broad applicability. As an additional contribution, this thesis provides new insights why state-of-the-art distributed algorithms might encounter issues for constrained problems

Distributed Optimization with Application to Power Systems and Control

Mathematical optimization techniques are among the most successful tools for controlling technical systems optimally with feasibility guarantees. Yet, they are often centralized—all data has to be collected in one central and computationally powerful entity. Methods from distributed optimization overcome this limitation. Classical approaches, however, are often not applicable due to non-convexities. This work develops one of the first frameworks for distributed non-convex optimization

Combining Homotopy Methods and Numerical Optimal Control to Solve Motion Planning Problems

This paper presents a systematic approach for computing local solutions to motion planning problems in non-convex environments using numerical optimal control techniques. It extends the range of use of state-of-the-art numerical optimal control tools to problem classes where these tools have previously not been applicable. Today these problems are typically solved using motion planners based on randomized or graph search. The general principle is to define a homotopy that perturbs, or preferably relaxes, the original problem to an easily solved problem. By combining a Sequential Quadratic Programming (SQP) method with a homotopy approach that gradually transforms the problem from a relaxed one to the original one, practically relevant locally optimal solutions to the motion planning problem can be computed. The approach is demonstrated in motion planning problems in challenging 2D and 3D environments, where the presented method significantly outperforms a state-of-the-art open-source optimizing sampled-based planner commonly used as benchmark