Optimization and equilibrium methods in power systems

Efficient operation and planning of electric power systems promise huge gains to society, from reduction of global poverty to avoidance of drastic climate change. There is a great need for mathematical and economic modeling to support these efficiency efforts. This thesis explores optimization methods to directly improve planning and operation and equilibrium methods to understand how markets composed of multiple economic agents carry out these optimal plans or in some cases thwart them. The challenges faced by this thesis are rooted not only in economic behavior and engineered systems but also in mathematical complexity.;We characterize the mode of divergence of agent-decomposable iterative algorithms for equilibrium problems. These problems pose a challenge for agent-decomposable algorithms because the applications where equilibrium seems to be most necessary for a faithful model have strong interactions between agents. We introduce an equilibrium model for understanding the effect of risk aversion in investment in power grid components.;We develop a novel global optimization technique for a nonconvex problem arising from coal mine quality planning. Our technique relies on isolating the nonconvexity to a low dimensional structure, which is then approximated by a discrete grid. The low dimensionality keeps the computational cost manageable.;We model the effect of unit commitment on the behavior of strategic power suppliers under various market rules by computing discrete approximations of mixed Nash equilibria in continuous spaces. We are able consider rules such as a single price, a price with an uplift, and pay-as-bid.;We also document our contributions to collaborative work on semidefinite programming relaxations of nonconvex power flow problems and on the social benefit of expanded bidding structures in wholesale power markets.