Distributionally Robust Optimal Power Flow with Strengthened Ambiguity Sets

Abstract

Uncertainties that result from renewable generation and load consumption can complicate the optimal power flow problem. These uncertainties normally influence the physical constraints stochastically and require special methodologies to solve. Hence, a variety of stochastic optimal power flow formulations using chance constraints have been proposed to reduce the risk of physical constraint violations and ensure a reliable dispatch solution under uncertainty. The true uncertainty distribution is required to exactly reformulate the problem, but it is generally difficult to obtain. Conventional approaches include randomized techniques (such as scenario-based methods) that provide a priori guarantees of the probability of constraint violations but generally require many scenarios and produce high-cost solutions. Another approach is to use an analytical reformulation, which assumes that the uncertainties follow specific distributions such as Gaussian distributions. However, if the actual uncertainty distributions do not follow the assumed distributions, the results often suffer from case-dependent reliability.

Recently, researchers have also explored distributionally robust optimization, which requires probabilistic constraints to be satisfied at chosen probability levels for any uncertainty distributions within a pre-defined ambiguity set. The set is constructed based on the statistical information that is extracted from historical data. Existing literature applying distributionally robust optimization to the optimal power flow problem indicates that the approach has promising performance with low objective costs as well as high reliability compared with the randomized techniques and analytical reformulation. In this dissertation, we aim to analyze the conventional approaches and further improve the current distributionally robust methods.

In Chapter II, we derive the analytical reformulation of a multi-period optimal power flow problem with uncertain renewable generation and load-based reserve. It is assumed that the capacities of the load-based reserves are affected by outdoor temperatures through non-linear relationships. Case studies compare the analytical reformulation with the scenario-based method and demonstrate that the scenario-based method generates overly-conservative results and the analytical reformulation results in lower cost solutions but it suffers from reliability issues.

In Chapters III, IV, and V, we develop new methodologies in distributionally robust optimization by strengthening the moment-based ambiguity set by including a combination of the moment, support, and structural property information. Specifically, we consider unimodality and log-concavity as most practical uncertainties exhibit these properties. The strengthened ambiguity sets are used to develop tractable reformulations, approximations, and efficient algorithms for the optimal power flow problem. Case studies indicate that these strengthened ambiguity sets reduce the conservativeness of the solutions and result in sufficiently reliable solutions.

In Chapter VI, we compare the performance of the conventional approaches and distributionally robust approaches including moment and unimodality information on large-scale systems with high uncertainty dimensions. Through case studies, we evaluate each approach's performance by exploring its objective cost, computational scalability, and reliability. Simulation results suggest that distributionally robust optimal power flow including unimodality information produces solutions with better trade-offs between objective cost and reliability as compared to the conventional approaches or the distributionally robust approaches that do not include unimodality assumptions. However, considering unimodality also leads to longer computational times.