In this paper, we develop the game-theoretic electricity market model ElStorM that includes the possibility of strategic electricity storage. We apply the model to the German electricity market and analyze different realistic and counterfactual cases of strategic and non-strategic pumped hydro storage utilization by different players. We find that the utilization of storage capacities depends on the operator and its ability to exert market power both regarding storage and conventional generation capacities. The distribution of storage capacities among players also matters. A general finding is that strategic operators tend to under-utilize their storage capacities. This affects generation patterns of conventional technologies and market outcomes. Strategic under-utilization of storage capacities might also diminish their potential for renewable energy integration. Accordingly, economic regulation of existing and future storage capacities may be necessary, depending on policy objectives. We also find that the introduction of electricity storage generally increases overall welfare, while outcomes vary between different cases. Strategic storage utilization decreases consumer rent compared to non-strategic storage utilization. However, this effect is less pronounced if storage capacities are distributed among several players.