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In the line of research on inclusion of Energy Storage Systems (ESS) into HVDC stations, this paper presents the integration of partially rated ESS into Modular Multilevel Converter (MMC) substations using parallel branches to the arm inductors. The proposed ESS-MMC topology keeps the design of the original MMC unmodified, as it just adds partially rated ESS with electrical connections outside the valve hall. The system utilizes another frequency component in the arm circulating current to exchange power between the energy storage branches and the original MMC stacks. This third frequency component allows decoupling to some extent the power on the dc and ac sides of the substation. Analysis of the circuit performance led to the development of control algorithms to maximize the power flow capabilities from the ESS within the operational limits of the existing MMC design. This allows the addition of up to 37% power from the ESS considering similarly rated power semiconductors in a simulated 1 GW MMC substation. Analysis shows that extra device losses are kept under 1% for an addition of $\pm$10% of ESS power to the nominal substation power. Therefore, the proposal is suitable for adding services such as fast frequency response or black start capability to existing MMC designs.