Our study is motivated by the impact of uncertainty, nonlinearity, and hierarchy on critical long-term planning (such as transmission expansion) and operational problems (such as unit commitment problems (UCPs) and transmission switching) in power systems and markets. The optimal resolution of such problems is of significant relevance. For instance, algorithms for UCPs have saved billions of dollars annually while analogous schemes for transmission switching may have similar potential. Yet, much of the available technology can only cope with deterministic linear problems. But nonlinear generalizations are assuming increasing relevance and emerge from incorporating reactive power and voltage management through AC load flows, while uncertainty in problem data, such as demand and availability, leads to a massive growth in problem complexity. Equally challenging are the hierarchical problems, arising from optimizing the expansion of transmission assets subject to subsequent energy market behavior, recognizing long run regulatory, economic, and technology uncertainties. Both the planning and operational problems lead to inordinately challenging optimization problems and no general purpose algorithms exist for the scalable resolution of such problems, motivating the proposed research. Through relationships with independent system operators for two US markets, our research will help inform stakeholder discussions concerning the design of markets for electric energy and capacity. More efficient short-run and long-run markets lower the cost and enhance the environmental sustainability of the power sector. In particular, our collaboration with PJM Interconnection will examine commitment, switching, and expansion problems under uncertainty, reliability pricing models for ensuring generation adequacy, and the design of reliability premiums in the context of integrating renewables. The proposal is equipped with an educational plan that includes the organization of research workshops and professional course development.
Our goals are twofold: (a) Stochastic optimization models: We consider development of: (i) Operating models for stochastic unit commitment problems emphasizing reactive power management, transmission switching, and stochasticity; and (ii) Planning models for transmission expansion in uncertain settings as well as the pricing of reliability, in particular the adequacy of generation and transmission resources. (b) Scalable algorithms: These models lead to mixed-binary stochastic optimization problems possibly complicated by nonlinearity. Unfortunately, existing decomposition schemes are ill-equipped to address such problems and we consider two broad directions. (i) A stochastic barrier-cut scheme that utilizes a combination of interior point methods, cutting plane techniques, and Schur-complement methods to develop scalable interior-point methods for contending with mixed-binary stochastic nonlinear programs; and (ii) Stochastic mixed-integer quadratic programs, including potential use of semidefinite programming relaxations, both in terms of developing cutting-plane schemes as well as approximate solutions. If successful, this research will lead to scalable computational tools for resolving a range of stochastic optimization problems, complicated by hierarchy, discreteness, and nonlinearity.
