The envisaged decarbonisation of electricity systems has attracted significant interest around the role and value of energy storage systems (ESSs). In the deregulated electricity market, there is a need to investigate the complex impacts of ESSs, considering the potential exercise of market power by strategic players. This study aims at comprehensively analysing the impacts of both price-taking and price-making storage behaviours on energy market efficiency, corresponding to potential settings with small and large storage players, respectively. In order to achieve this and in contrast to previous papers, this work develops a multi-period equilibrium programming market model to determine market equilibrium stemming from the interactions of independent strategic producers and ESSs, while capturing the time-coupling operational constraints of ESSs as well as network constraints. The results of case studies on a test market capturing the general conditions of the GB electricity system demonstrate that the introduction of ESSs mitigates market power exercise and improves market efficiency, with this beneficial impact being higher when ESSs act as price takers. When the electricity network is congested, the location of ESSs also affects the market outcome, with their beneficial impact on market efficiency being higher when they are located in higher-priced areas.