Electrical energy storage (EES) systems allow shifting the time of electric power generation from that of consumption, and they are expected to play a major role in future electric grids where the share of intermittent renewable energy systems (RES), and especially solar and wind power plants, is planned to increase. No commercially available technology complies with all the required specifications for an efficient and reliable EES system. Reversible solid oxide cells (ReSOC) working in both fuel cell and electrolysis modes could be a cost effective and highly efficient EES, but are not yet ready for the market. In fact, using the system in fuel cell mode produces high temperature heat that can be recovered during electrolysis, when a heat source is necessary. Before ReSOCs can be used as EES systems, many problems have to be solved. This paper presents a new ReSOC concept, where the thermal energy produced during fuel cell mode is stored as sensible or latent heat, respectively, in a high density and high specific heat material and in a phase change material (PCM) and used during electrolysis operation. The study of two different storage concepts is performed using a lumped parameters ReSOC stack model coupled with a suitable balance of plant. The optimal roundtrip efficiency calculated for both of the configurations studied is not far from 70% and results from a trade-off between the stack roundtrip efficiency and the energy consumed by the auxiliary power systems.