Electricity generation via renewable energy systems can be fed into the electrical grid when it is needed or stored when it is excessive. This is important for demand management. In this regard, power to gas technology (PTG) is recognized as a potential option for an efficiently, environmentally friendly and long-term storage solution for renewable energy systems. An integrated wind power system comprised of wind turbine, proton exchange membrane (PEM) electrolyzer and a methanation unit is considered in this study for thermodynamic analyses. The energy and exergy efficiencies of the overall developed system are found to be 44% and 45%, respectively. The methanation unit works based on the Sabatier reaction for synthetic natural gas (SNG) production. A steam perm-selective membrane is considered for methanation and the products are integrated with other parts of the system for heat recovery. An increase in the wind speed results in a decrease in the exergetic efficiency and an increase in hydrogen and methane production. Hence, multi-objective optimization method based on genetic algorithm is employed to determine the optimal values for the decision variables. The exergetic efficiency of the overall system is 41%, and hence, the CH₄ production by the present system is 1.68 kg/h which are found to be the optimal values where the wind velocity is 4.33 m/s, and the power coefficient becomes 0.57, respectively.