In modern societies, electricity is ubiquitous and its availability and reliability are often taken for granted. However, the reality is that these properties rely on the proper operation of one of man's most complex creations: electric power grids. These systems transport electric energy across wide geographical regions from places where it is produced to places where it is consumed. They do so in many cases with virtually no storage. To ensure an efficient and reliable operation of these complex machines, numerical optimization algorithms are applied at almost every level of the operations and planning spectrum. For example, they are used for planning network expansions, scheduling generators, adjusting control devices, determining system state, computing security margins, and for many other crucial tasks. However, many of the algorithms currently used by system operators and planners are based on heuristics and have severe limitations. For example, common power flow algorithms, which are used for scenario analysis, typically fail to provide useful system information in the absence of accurate estimates of system state. Optimal power flow algorithms, which are used for planning system adjustments, typically use inadequate techniques for handling discrete variables and identifying key control actions. Security assessment algorithms typically require expensive system simulations that prevent frequent security analyses. These deficiencies may compromise system performance as grids become more complex, variable and unpredictable, and are operated closer to their limits. In this work, modeling and numerical optimization techniques are explored for overcoming many of the limitations associated with the algorithms used for scenario analysis, planning and control, and online security assessment. The proposed techniques have been tested on mid to large-scale real power networks obtained from South American, North American, and European electric institutions. The results demonstrate the potential benefits of the proposed techniques for improving the tools used by operators and planners for maintaining system reliability and efficiency.