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Uncertainty is a major factor in power system operations. In recent years, with the emergence of the smart grid, uncertainty level has been further elevated in both the generation and demand side of power systems. Increasing uncertainty exposes the electric grid to potential safety issues and economic loss, thus posing significant challenges to the grid operations. Traditionally, power system operations use certainty equivalent approach to deal with uncertainty, i.e., replacing random variables by their expected values. With this simplification, the original stochastic optimization is reduced to a deterministic problem. However, the certainty equivalent method is inadequate for the modern electric grid with deep penetration of distributed energy resources. Due to increasing uncertainty, operations and decision makings need to incorporate system dynamics over a broad range of temporal and spatial horizons. To this end, this thesis provides a new paradigm for operation under uncertainty and computationally efficient algorithms based on multiparametric programming theory. Under this new paradigm, uncertainty is characterized by conditional distributions and decisions are made by incorporating such probabilistic descriptions. To illustrate the new paradigm, we consider two specific problems. For characterization of system uncertainty, we develop a formal methodology for probabilistic forecasting of real-time operations and locational marginal prices. Conditioning on the current system state, we provide a full distribution of future operations and prices. For operational decision making, we propose an optimal stochastic approach to interchange scheduling in multi-area systems. By incorporating the conditional distribution of load and generation, the optimal interchange is obtained through an iterative process.

The Parametric Programming method is investigated to consider its applicability to structural optimization problems. It is used to solve optimization problems that have design variables as implicit functions of some independent input parameter(s). It provides optimal solutions as a parametric function of the input parameter(s) for the entire parameter space of interest. It does not require the detailed discrete optimizations needed at a large number of parameter values as in traditional non-parametric optimization. Parametric programming is widely used in optimal controls, model predictive control, scheduling, process synthesis and material design under uncertainty due to the above mentioned benefits. Its benefits are however, still unexplored in the field of structural optimization. Parametric programming could for instance, be used to aid designers in identifying and optimizing for uncertain loading conditions in complex systems. The first objective of this thesis is to identify a suitable multi-parametric programming algorithm among the many available ones in the literature to solve structural optimization problems. Once selected, the second goal is to implement the chosen algorithm and solve single parametric and multi-parametric sizing optimization problems, shape optimization problems, and use multi-parametric programming as a multi-objective optimization tool in structural optimization. In this regard, sizing optimization of truss structures and shape optimization of beams for load magnitude and load directions as varying parameters are solved for single and multi-parameter static and/or dynamic load cases. Parametric programming is also used to solve the multi-objective optimization of a honeycomb panel and the results are compared with those from non-parametric optimization conducted using commercial optimization software. Accuracy of results, and computational time are considered. From these studies, inferences are drawn about the issues and benefits of using parametric programming in structural optimization.