Optimization plays a central role in the control and operation of electricity power networks. In this thesis we focus on two very important optimization problems in power systems. The first is the optimal power flow problem (OPF). This is an old and well-known nonconvex optimization problem in power system. The existence of local solutions of OPF has been a question of interest for decades. Both local and global solution techniques have been put forward to solve OPF problem but without any documented cases of local solutions. We have produced test cases of power networks with local solutions and have collected these test cases in a publicly available online archive (http://www.maths.ed.ac.uk/optenergy/LocalOpt/), which can be used now by researchers and practitioners to test the robustness of their solution techniques. Also a new nonlinear relaxation of OPF is presented and it is shown that this relaxation in practice gives tight lower bounds of the global solution of OPF. The second problem considered is how to split a network into islands so as to prevent cascading blackouts over wide areas. A mixed integer linear programming (MILP) model for islanding of power system is presented. In recent years, islanding of power networks is attracting attention, because of the increasing occurrence and risk of blackouts. Our proposed approach is quite flexible and incorporates line switching and load shedding. We also give the motivation behind the islanding operation and test our model on variety of test cases. The islanding model uses DC model of power flow equations. We give some of the shortcomings of this model and later improve this model by using piecewise linear approximation of nonlinear terms. The improved model yields good feasible results very quickly and numerical results on large networks show the promising performance of this model.

<p dir="ltr">The safe and reliable delivery of electricity is critical for the functioning of our modern society. However, high-impact, low-probability (HILP) catastrophic events (such as extreme weather caused by climate change, or cyber-physical attacks) pose an ever-growing threat to the power grid. At the same time, modern advancements in computational capabilities, communication infrastructure, and measurement technologies provide opportunities for new operation and control strategies that enhance the resilience of electric power systems to such HILP events. In this work, optimization-based operation and control approaches are proposed to improve resilience in two power systems applications. First, a real-time linearized-trajectory model-predictive controller (LTMPC) is developed for ensuring voltage, frequency, and transient (rotor angle) stability in systems engineered to operate as microgrids. Such microgrids are capable of seamlessly transitioning from grid-connected operation to an islanded mode and thus, enhance system resilience. The proposed LTMPC enables rapid deployment of such systems by reducing engineering costs and development time while maintaining stable operation. On the other hand, some power systems, such as distribution feeders, are not designed to operate as standalone microgrids. For these cases, a method is proposed for forming ad-hoc microgrids from intact sections of the damaged feeder in the aftermath of a HILP event. A feeder operating center-on-a-laptop (FOCAL) is introduced that coordinates the control of possibly hundreds of inverter-interfaced distributed energy resources (e.g., rooftop solar, battery storage) to improve system resilience. Theoretical analysis as well as numerical case studies and simulations of the proposed strategies are presented for both applications.</p>