The main objectives of this research are (a) to develop advanced simulation methods for voltage-recovery phenomena using improved, realistic system models and accurate solution techniques and (b) to develop methods for the mitigation of problems related to slow voltage recovery. Therefore, this work concentrates on the areas of voltage-recovery analysis in electric power systems, dynamic load modeling with emphasis on induction-motor models, dynamic simulation with emphasis on the numerical integration methods, and optimal allocation and operation of static and dynamic VAr resources. In the first part of this work, a general framework for power-system analysis is presented the main characteristics of which are (a) the utilization of full three-phase models and (b) the use of a "quadratized" mathematical formulation, which models the system under study as a set of mathematical equations of order no more than two. The modeling approach is essentially the same for steady-state, quasi-steady-state, and dynamic analysis. Furthermore, a new approach for time-domain transient simulation of electric power systems and dynamical systems, in general, is introduced in this research. The new methodology has been named quadratic integration method. The method is based on a numerical integration scheme that assumes that the system states vary quadraticaly within an integration time step. Accurate modeling and simulation of voltage-recovery phenomena allows the development of mitigation methodologies via the optimal allocation and operation of static and dynamic VAr resources over the planning horizon. This problem is solved with successive dynamic programming techniques with the following two innovations: (a) the states at each stage (candidate solutions) are obtained with static and dynamic (trajectory) sensitivity analysis and (b) each candidate solution is evaluated by considering the optimal operation of installed static and dynamic VAr sources utilizing concepts from the theory of applied optimal control and trajectory optimization.