Due to the unique structure and special characteristics of electric distribution networks, along with the increase in the number and severity of the natural disasters in recent years, presenting a proper framework and procedure for evaluating the resiliency of these systems is more essential. In addition, it is obligatory to model the different capabilities and features of the smart grids components in order to provide a better knowledge about their expected performance in such circumstances. In response to these challenges, this paper addresses the concept of resiliency and its dimensions in distribution networks. A new model based on mixed-integer linear programming is proposed to properly model and evaluate the resiliency of smart distribution systems. In the proposed model, optimal formation of dynamic microgrids (MGs), their service areas, and the optimal management of different technologies such as energy storage (ES) units, demand side management programs and distributed generations (DGs) units are investigated. in addition, employing a two-stage framework based on stochastic programming, the impact of increasing penetration level of the renewable energy resources and their related uncertainties on system resiliency is examined. The efficiency and applicability of the proposed integrated model is verified by performing multiple simulations on the modified 118-bus test system and a real distribution network.