Hybrid modular multilevel converter control and design for pumped hydro storage plants is presented, addressing application-specific shortcoming of conventional back-to-back half-bridge modular multilevel converter: high common-mode-voltage machine stress. Common-mode-voltage-free operation in the entire or partial frequency range is enabled by variable DC link voltage control, through introduction of minimal full-bridge submodule share in hybrid active front-end converter stage. The developed generalized converter design approach for arbitrary DC link voltage range operation and additional internal energy balancing control layers enable down-to-zero DC voltage control. The results are verified through high-fidelity switched-model simulations of 6 kV converter, with 10/6 ratio of full-bridge to half-bridge submodules in active front-end. The analyzed hybrid active front-end stage benefits from lower converter losses for equal machine operation flexibility compared to full-bridge design, while trade-off between grid-side power factor range and full-bridge submodule share is offered within the design stage. Compared to the state-of-the-art, zero-to-rated DC link voltage operation is possible at lower full-bridge submodule share (62% against 75 %), at the penalty of reduced grid-side power factor. Alternatively, operation at higher full-bridge submodule share (62% against 50% existing solution) enables grid-side reactive power support over wide speed range, without branch current overload.