Online optimization addresses problems whose input is incomplete. To solve online optimization problems, the decision maker needs to invent algorithms or mechanisms under uncertainty and achieve a performance comparable to when the information is completely given. The power system, non-surprisingly, is an area rich in online problems. In particular, the expanding use of renewable energy, including wind and solar, has imposed great uncertainty and new challenges on the power grid. Recent large-scale power blackouts also raise discussions about improving stability and fairness in the scheduling of electricity generation. To address these emerging issues, novel online optimization tools catered for the power grid must be considered.

In this Ph.D. dissertation, we aim to use online optimization techniques to solve decision-making problems in power systems, when different kinds of uncertainty are present. We choose three emerging online problems in the power systems that can greatly benefit from the proposed optimization frameworks, either in promoting equity among electricity consumers or enhancing the resilience of the existing power infrastructure. Moreover, the invented online algorithms and mechanisms have utilities beyond the power systems and can be used for general online problems in other application areas.

The first problem is related to the inequity in the net metering billing scheme that is currently adopted in the electricity market. The growing number of energy prosumers (e.g., solar panel owners), whose gross power consumption is hidden from the public, has been shifting the grid costs to traditional energy consumers, causing great unfairness. We study a penalty mechanism that incentivizes prosumers to report their true consumption, which results in a fairer pricing scheme than net metering. We model the problem as a repeated game with one or multiple players and provide the minimum penalty rates such that players voluntarily report their private value of electricity consumption.

The second problem is to address the uncertainty in energy procurement where utility companies need to secure contracts with generators to fulfill the demand. The contracts arrive one by one during a time period and the utility company must decide irrevocably whether to accept the current contract or not. This is an application of the well-known prophet inequality problem. In the literature, the distribution of the value of the contract is known to the utility companies, which is hardly the case in practice. In this work, we consider the setting where the utility company can make a handful of queries to a distribution oracle and has to design algorithms to select the contract such that the value obtained is maximized. We give competitive ratios when two to five queries can be made. When only one query is allowed, we propose an alternative algorithm whose competitive ratio is strictly better than the one obtained from the single-threshold algorithm in the past literature.

Finally, for the last problem, we study the classic unit commitment (UC) problem, the scheduling and dispatch of power generators subject to operating constraints so that the total cost is minimized. The UC problem is a special case of the more general location-allocation problems that have been widely adopted in application areas including disaster preparedness and supply chain management. We design a two-stage data-driven robust optimization framework to address general location-allocation problems with two types of uncertainties: the binary network uncertainty and the continuous unknown demand. Moreover, we add fairness constraints to the formulation such that the ratio of the allocated resources to demand is consistent across all regions. We provide tractable reformulations of the original problem and conduct numerical studies for an IEEE test system and the 2010 Yushu earthquake data set. Our simulation results show that the addition of network uncertainties and fairness constraints effectively improves equity and reduces costs.