This paper investigates the controllability of modular multilevel converters for high voltage direct current applications. The paper provides an in-depth analysis of a computationally efficient method to calculate the required amount of triplen harmonic injection (THI) and to devise an overmodulation strategy based on a three-dimensional representation of the converter control voltages in the vector space. The proposed method allows simultaneous converter operation and power quality control in an extended ac voltage range without degrading the submodules (SM's) capacitor voltage ripple or spectra of generated waveforms with respect to well-known THI methods. The advantages of the presented technique with respect to traditional THI techniques are: first, the converter controllability is extended during ac-side voltage disturbances; second, the SM's capacitor voltage ripple is upper bounded; and third, the insulation stress due to the common-mode voltage at the transformer's secondary windings is minimized. The performance of the claimed features is supported by simulation results for both steady-state and transient operating conditions.