This paper proposes a novel control approach for optimizing wind farm energy capture with a nested-loop scheme of extremum seeking control (ESC). Similar to Bellman’s Principle of Optimality, it has been shown in earlier work that the axial induction factors of individual wind turbines can be optimized from downstream to upstream units in a sequential manner, i.e. the turbine operation can be optimized based on the power of the immediate turbine and its downstream units. In this study, this scheme is illustrated for wind turbine array with variable-speed turbines for which torque gain is controlled to vary axial induction factors. The proposed nested-loop ESC is demonstrated with a 3-turbine wind farm using the SimWindFarm simulation platform. Simulation under smooth and turbulent winds show the effectiveness of the proposed scheme. Analysis shows that the optimal torque gain of each turbine in a cascade of turbines is invariant with wind speed if the wind direction does not change, which is supported by simulation results for smooth wind inputs. As changes of upstream turbine operation affects the downstream turbines with significant delays due to wind propagation, a cross-covariance based delay estimate is proposed as adaptive phase compensation between the dither and demodulation signals.

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