Powerline faults are responsible for major bushfires around the world where arcs provoked from ground faults are common causes for igniting such fires. The grounding techniques (mainly resonant grounding) used in distribution substations and arc suppression devices play a crucial role for compensating fault currents in order to extinguish arcs so that the likelihoods of powerline bushfires are significantly reduced. Though passive arc suppression devices (e.g. Petersen coils) are extensively used for compensating the reactive component of the fault current, the active component of this current is still large enough to ignite the fire in bushfire prone areas for which active arc suppression devices are recently used. These active arc suppression devices incorporate residual current compensation inverters and the full compensation of the fault current rely on the control scheme of these inverters. This paper comprehensively reviews different control schemes that are used for compensating the fault current in resonant grounded power distribution systems. The existing control schemes are discussed in terms of the model used during the controller design process, loop structures, control block diagrams, and performance analysis frameworks (i.e. the type of fault impedances). It is worth mentioning that faults on resonant grounded power distribution networks exhibit the characteristics of high impedance faults and it important to consider this aspect for performance analysis of the control scheme. This paper also covers a brief overview of different grounding techniques used for mitigating fault currents and finally, the challenges with the existing controllers are identified in terms of extinguishing powerline bushfires. The comprehensive review motivates and guides future research activities on developing more efficient fault compensation techniques.