Time-Varying Optimization and Its Application to Power System Operation

The main topic of this thesis is time-varying optimization, which studies algorithms that can track optimal trajectories of optimization problems that evolve with time. A typical time-varying optimization algorithm is implemented in a running fashion in the sense that the underlying optimization problem is updated during the iterations of the algorithm, and is especially suitable for optimizing large-scale fast varying systems. Motivated by applications in power system operation, we propose and analyze first-order and second-order running algorithms for time-varying nonconvex optimization problems. The first-order algorithm we propose is the regularized proximal primal-dual gradient algorithm, and we develop a comprehensive theory on its tracking performance. Specifically, we provide analytical results in terms of tracking a KKT point, and derive bounds for the tracking error defined as the distance between the algorithmic iterates and a KKT trajectory. We then provide sufficient conditions under which there exists a set of algorithmic parameters that guarantee that the tracking error bound holds. Qualitatively, the sufficient conditions for the existence of feasible parameters suggest that the problem should be "sufficiently convex" around a KKT trajectory to overcome the nonlinearity of the nonconvex constraints. The study of feasible algorithmic parameters motivates us to analyze the continuous-time limit of the discrete-time algorithm, which we formulate as a system of differential inclusions; results on its tracking performance as well as feasible and optimal algorithmic parameters are also derived. Finally, we derive conditions under which the KKT points for a given time instant will always be isolated so that bifurcations or merging of KKT trajectories do not happen. The second-order algorithms we develop are approximate Newton methods that incorporate second-order information. We first propose the approximate Newton method for a special case where there are no explicit inequality or equality constraints. It is shown that good estimation of second-order information is important for achieving satisfactory tracking performance. We also propose a specific version of the approximate Newton method based on L-BFGS-B that handles box constraints. Then, we propose two variants of the approximate Newton method that handle explicit inequality and equality constraints. The first variant employs penalty functions to obtain a modified version of the original problem, so that the approximate Newton method for the special case can be applied. The second variant can be viewed as an extension of the sequential quadratic program in the time-varying setting. Finally, we discuss application of the proposed algorithms to power system operation. We formulate the time-varying optimal power flow problem, and introduce partition of the decision variables that enables us to model the power system by an implicit power flow map. The implicit power flow map allows us to incorporate real-time feedback measurements naturally in the algorithm. The use of real-time feedback measurement is a central idea in real-time optimal power flow algorithms, as it helps reduce the computation burden and potentially improve robustness against model mismatch. We then present in detail two real-time optimal power flow algorithms, one based on the regularized proximal primal-dual gradient algorithm, and the other based on the approximate Newton method with the penalty approach