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In large wind farms, the vertical interaction of the farm 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. The current dissertation investigates the use of optimal control techniques in large-eddy simulations of wind farm boundary-layer interaction with the aim of increasing the total energy extraction in wind farms. The individual wind turbines are considered as flow actuators and their energy extraction is dynamically regulated in time so as to optimally influence the flow field and the vertical turbulent energy transport. The dissertation focuses on the development of a framework for a gradient-and adjoint-based scheme for wind-farm power optimization. To this end, a receding-horizon optimal control approach is employed in combination with the non-linear Polak-Ribière conjugate gradient method and the Brent line search algorithm. The gradient of the cost functional required by the conjugate-gradient method is determined using a continuous adjoint-based approach. The adjoint equations for the standard Navier-Stokes equations are extended to include the adjoints for the subgrid-scale model and wall-stress model, and the adjoint of the wind-turbine model. In the first optimization studies, the optimal control of an infinite wind farm is investigated. The first control case focuses on the direct maximization of the energy extraction. It is found that the energy extraction increases by 16% compared to the uncontrolled reference. This is directly related to an increase in 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. It is further observed 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 the penalization of turbulent dissipation. For the current wind-farm geometry, it is found that the ratio between the wind-farm energy extraction and turbulent boundary-layer dissipation remains roughly around 70%, but can be slightly increased by a few percent by penalizing the dissipation in the optimization objective. For a pressure-driven boundary layer in equilibrium, it is estimated that such a shift can lead to an increase in wind-farm energy extraction of 6%. The second optimization study investigates the application of the optimal control to a finite-sized wind farm. A fringe region is employed to impose non-periodicity to the domain and the adjoint for the fringe forcing term is added to the original adjoint LES equations. It is found that the energy extraction increases by 7.3% compared to the uncontrolled case. The value is significantly lower when compared to the optimization of the infinite wind farm. One possible reason for this could be that the turbines in the front row - which contribute 16.5% of the whole farm power in the current case - are already operating close to the optimal condition, and hence, their performance cannot be improved much further by coordinated control. However, even the 7.3% gain achieved in this dissertation can be beneficial, especially for large wind farms. nrpages: 132 status: published |
