With the increasing share of renewable energy, concerns regarding ensuring power system stability are ever more relevant and have been accompanied by discussions to address this yet unsolved issue. Nonetheless, enhancing sparsity and increasing generation capacity by overplanting wind turbines not only mitigates the stability problem but also accelerates the transition from fossil fuel to renewable energy sources. With the high penetration of wind energy, there will be a paradigm shift from maximizing energy extraction to generating energy on demand. In this panorama, a cooperative wind farm control may strengthen the stability of the wind power plant through compensation strategies. Still, large-scale farms raise relevant control issues regarding computation effort and information sharing, such as topology constraints and communication overhead. Here, we contribute by presenting a multi-rate distributed control strategy based on average consensus. This strategy involves estimating the power-tracking errors at a fast sampling rate and executing local control actions that collaboratively mitigate these errors over an extended sampling period. This approach achieves performance comparable to that of the resource-intensive centralized approach. The reliability is therefore enhanced by improving the power regulation while reaching modularity and sparsity inside the farm.