Optimal control of energy extraction in wind-farm boundary layers

Abstract

In very large wind farms, the vertical interaction with the atmospheric boundary layer plays an important role, i.e. the total energy extraction is governed by the vertical transport of kinetic energy from higher regions in the boundary layer towards the turbine level. In the current study, we investigate optimal control of wind-farm boundary layers, considering the individual wind turbines as flow actuators, whose energy extraction can be dynamically regulated in time so as to optimally influence the flow field and the vertical energy transport. To this end, we use large-eddy simulations of a fully developed pressure-driven wind-farm boundary layer in a receding-horizon optimal control framework. For the optimization of the wind-turbine controls, a conjugate-gradient optimization method is used in combination with adjoint large-eddy simulations for the determination of the gradients of the cost functional. In a first control study, wind-farm energy extraction is optimized in an aligned wind farm. Results are accumulated over one hour of operation. We find that the energy extraction is increased by 16 % compared to the uncontrolled reference. This is directly related to an increase of the vertical fluxes of energy towards the wind turbines, and vertical shear stresses increase considerably. A further analysis, decomposing the total stresses into dispersive and Reynolds stresses, shows that the dispersive stresses increase drastically, and that the Reynolds stresses decrease on average, but increase in the wake region, leading to better wake recovery. We further observe also that turbulent dissipation levels in the boundary layer increase, and overall the outer layer of the boundary layer enters into a transient decelerating regime, while the inner layer and the turbine region attain a new statistically steady equilibrium within approximately one wind-farm through-flow time. Two additional optimal control cases study penalization of turbulent dissipation. For the current wind-farm geometry, it is found that the ratio between wind-farm energy extraction and turbulent boundary-layer dissipation remains roughly around 70 %, but can be slightly increased by a few per cent by penalizing the dissipation in the optimization objective. For a pressure-driven boundary layer in equilibrium, we estimate that such a shift can lead to an increase in wind-farm energy extraction of 6 %.

Boundary-layer development and gravity waves in conventionally neutral wind farms

“…Examples are studies of fully developed wind farms by Calaf et al (2010), Calaf, Parlange & Meneveau (2011), Yang, Kang & Sotiropoulos (2012, Goit & Meyers (2015) and of wind farms with entrance effects and a developing boundary layer by Porté-Agel, Wu & Chen (2013), , Stevens, Gayme & Meneveau (2014a), Stevens, Graham & Meneveau (2014b), Stevens, Gayme & Meneveau (2015), Nilsson et al (2015), Goit, Munters & Meyers (2016), Munters, Meneveau & Meyers (2016). The main working assumption in these studies is that the wind turbines are located in the inner region of the ABL (i.e.…”

Online model calibration for a simplified LES model in pursuit of real-time closed-loop wind farm control

“…The turbine forcing term in (1), f, at spatial location s = x y T ∈ R 2 is expressed as with position t i ∈ R 2 . The scalar C Ti is a variation of the non-dimensional thrust coefficient of turbine i which can be related to physical turbine parameters such as the generator torque and blade pitch angles (Goit and Meyers, 2015). The scalar γ i is the yaw angle of turbine i with respect to the incoming wind, and U i is the average flow velocity over the rotor of turbine i.…”

“…Note that C T has a direct mapping to the turbine power P turb,i , and thus replaces the usual non-dimensional 15 power coefficient, following the example of Goit and Meyers (2015). …”

“…predictive optimization method for dynamical models (e.g., Goit and Meyers, 2015;Vali et al, 2017;Siniscalchi-Minna et al, 2018). A more in-depth discussion on optimization algorithms for the framework of Fig.…”

Instantaneous Response and Mutual Interaction between Wind Turbine and Flow

“…Gebraad et al [8] and Fleming et al [9] The aforementioned studies have generally focussed on the changes to the flow, often from steady state or at least static operating conditions. Dynamic control of large wind farms has recently been investigated in detail by Goit and Meyers [10] and subsequently a derived simpler and practical control strategy by Munters and Meyers [11], although the dynamic analysis does not include the effect on the turbine load.…”