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In this paper, we describe an approach to the analysis and design of power grid dynamic performance based on hybrid systems theory. Power grid is a large-scale cyber-physical system for transmission of electrical energy. The joint dynamics of physical processes and cyber elements in power grids are typical of a mixture of continuous and discrete behaviors, that is, hybrid dynamics. We address problems on stability that are basic concerns in the performance of current and future power grids with the hybrid dynamics. Measures for stability of power grids are interpreted as safety specifications in hybrid system models and are translated into restrictions on the systems' reachable sets of states. Algorithmic reachability analysis of hybrid systems enables analysis of safe initial states and hence quantitative estimation of stability regions. Also it contributes to synthesis of safe initial states as well as switching conditions in order to satisfy safety specifications in a power grid. We demonstrate the approach for two problems on transient stability of the single machine/infinite bus (SMIB) system and on fault release control of a multimachine power grid.