In this paper, we study a generalization of reflected backward doubly stochastic differential equations (RBDSDEs) and present a link to a general mean field game. In our case, the RBDSDEs are associated with a lower optional not right continuous barrier. First, we establish the existence and uniqueness of a solution of such RBDSDEs. We then study a mean field game with a new type of common noise related to an electricity grid with storage allowing jumps and prove the existence of a mean field Nash equilibrium.