In this paper, we address the analysis of resilience properties related to electric power grids modeled as a (large) dynamical system. To this end, we introduce the notion of p-robustness as the capability of ensuring the proper functioning of the electric power grids, in the sense of guaranteeing generic controllability of the associated dynamical system, under arbitrary p transmission line failures. Then, we provide conditions under which the electric power grid is p-robust, and an algorithm that determines the minimum number of transmission lines in the electric power grid that is need to add in order to transform a non-robust (0-robust) electric power grid into a 1-robust electric power grid. Further, we discuss how the methodology can be extended to ensure p-robustness with a relatively small number of additional transmission lines. We present an illustrative example of the proposed analysis and methodology using the IEEE 39-bus system, whose dynamical model is described by 127 state variables.