Internal-energy regulation in grid-forming MMC-HVDC: A factorial study across weak-to-strong AC grids

Grid-forming (GFM) control of modular multilevel converters (MMCs) for HVDC links is governed by three coupled design choices: the inner AC loop (current control, CC, or direct voltage control, VC), the outer synchronization law (power-synchronization PI or virtual synchronous machine, VSM), and the inner energy loop (energy-based, EB, or non-energy-based, Non-EB). This paper evaluates these choices with a 2 × 2 × 2 × 5 factorial campaign: 40 active-power step scenarios and 72 large-disturbance cases covering low-voltage ride-through, bolted three-phase faults, and phase jumps over S C R = 1 –5. The simulations use an average-value MMC model that resolves arm-energy and circulating-current dynamics; it is validated in real time on an OPAL-RT OP5600 simulator against a switching-function model of the same converter, which it tracks to within 0 . 2 ms in rise time and 2 . 5 × 1 0 − 4 pu in steady state. The main result is that EB is required for large-disturbance energy containment within the tested framework: during a 1 5 0 ms, 0 . 3 pu voltage sag at ( C C , S C R = 1 . 3 ) , removing EB increases the stored-energy excursion from 1 . 0 6 % to 9 3 % , while EB keeps the excursion near 2 % across controller families and disturbances. The step campaign shows two additional effects. Under CC, the current loop masks the energy-loop choice on active-power overshoot ( < 0 . 2 percentage points, pp), supporting CC-PI as the simplest default when intrinsic current limitation is required. Under VC, the EB effect depends on the synchronization law: enabling EB raises VC-PI overshoot by 4 . 4 pp but reduces VC-VSM overshoot by 2 0 . 1 pp at S C R = 1 . These results condense into a compact, SCR-aware design map for selecting the inner loop, the synchronization law and the energy regulation of grid-forming MMC-HVDC converters.