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This paper investigates the electricity storage requirements to support the transition towards a high renewable energy source (RES) penetration in a cost-optimal manner. The achieved reduction of renewable energy curtailments and the decrease in the total generation cost of the system are quantified against a counterfactual scenario without storage. A methodology is presented to determine the optimum mix of short- and medium-duration storage needed to support system operation at increased RES penetration levels, using the mixed integer linear programming mathematical optimization. The Greek power system serves as a realistic study case, in its planned development for the year 2030, with a targeted annual RES energy penetration in the order of 60%. Li-ion batteries and pumped-hydro are selected as the representative technologies to include in the storage mix, assuming energy-to-power ratios of up to 6 hours for the former and 10 hours for the latter. It is shown that the introduction of a suitable mixture of storage facilities may improve renewable energy integration and, at the same time, reduce system cost to the extent that entirely compensates for the full cost of storage, thus allowing for a net economic benefit for the system. The optimum storage portfolio for the study case system and the targeted RES penetration level combines 2-h batteries and 6-h pumped-hydro stations, with an aggregate capacity of new facilities between 1250 MW and 1750 MW, on top of the existing 700 MW of open-loop pumped hydro plants. The optimum storage requirements vary with the targeted RES penetration and with the balance of RES technologies in the generation mix, particularly the level of PV integration.