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The reduction in system inertia under the penetration of power electronic-interfaced generation has fostered various proposals on wind synthetic inertia. Several works in the state of the art propose control strategies to optimize synthetic inertia performance; however, most of the performance metrics indirectly represent frequency control adequacy indicators such as rate of change of frequency or frequency nadir. Also, control definitions assume that there is one variable speed wind turbine and one controller, which limits the applicability of the proposals as power systems normally count with various wind turbines/farms over extended geographical areas with different wind regimes. This work presents a <inline-formula> <tex-math notation="LaTeX">$H_{2}$ </tex-math></inline-formula> optimal control approach for a variable speed wind turbine synthetic inertia controller. First, the objective of the formulation is to explicitly minimize the rate of change of power system frequency. Second, a stability proof for the multi-wind turbine case is proposed, allowing optimal controllers to be independently implemented for an arbitrary number of wind turbines and wind regimes. The effectiveness of the proposed control scheme is demonstrated with a numerical example, considering an empirically-validated, reduced-order model of the Electric Reliability Council of Texas frequency dynamics.

RoCoF-Minimizing H₂ Norm Control Strategy for Multi-Wind Turbine Synthetic Inertia