Distributed power generation, micro-grids, and networks working in islanding mode have strong deviations in voltage quantities. These deviations can be divided into amplitude and frequency. Amplitude deviations are well-known and studied, as they are common in small and big grids. However, deviations on the ac mains frequency have not been widely studied. The literature shows control schemes capable of bearing these variations, but no systematic analysis has been performed to ensure stability. As the majority of power converters are designed for big grids, their analysis and design neglect frequency disturbances, therefore those devices allow a very small frequency operating window. For instance, in power converters that need to be synchronized to the grid, the standard deviation does not go beyond 0.5 Hz, and for grid-tied inverters it does not go beyond 1 Hz, whereas variations of around 8 Hz can be expected in micro-grids. This work presents a comprehensive analysis of the control system's stability, where two different control schemes for a back-to-back static converter topology are implemented and studied under a wide variable grid frequency. Because the behavior of power converters is nonlinear and coupled, dynamic and static decouplers are usually introduced in the controller, being a key element on the scheme according to the findings. The results show that using just a static decoupler does not guarantee stability under frequency variations; meanwhile, when a dynamic decoupler is used, the operating window can be greatly extended. The procedure shown in this paper can also be extended to other control algorithms, making it possible to carefully choose the control system for a variable frequency condition. Simulated and experimental results confirm the theoretical approach.