Power Optimization of Wind Turbines Subject to Navier-Stokes Equations

In this thesis, we first develop a second-order corrected-explicit-implicit domain decomposition scheme (SCEIDD) for the parallel approximation of convection-diffusion equations over multi-block sub-domains. The stability and convergence properties of the SCEIDD scheme is analyzed, and it is proved that this scheme is unconditionally stable. Moreover, it is proved that the SCEIDD scheme is second-order accurate in time and space. Furthermore, three different numerical experiments are performed to verify the theoretical results. In all the experiments the SCEIDD scheme is compared with the EIPCMU2D scheme which is first-order in time. Then, we focus on the application of numerical PDEs in wind farm power optimization. We develop a model for wind farm power optimization while considering the wake interaction among wind turbines. The proposed model is a PDE-constrained optimization model with the objective of maximizing the total power of the wind turbines where the three-dimensional Navier-Stokes equations are among the constraints. Moreover, we develop an efficient numerical algorithm to solve the model. This numerical algorithm is based on the pattern search method, the actuator line method and a numerical scheme which is used to solve the Navier-Stokes equations. Furthermore, the proposed numerical algorithm is used to investigate the wake structures. Numerical results are consistent with the field-tested results. Moreover, we find that by optimizing the turbines operation while considering the wake effect, we can gain an additional 8% in the total power. Finally, we relax the deterministic assumption for the incoming wind speed. The developed model is ultimately a PDE-constrained stochastic optimization model. Moreover, we develop an efficient numerical algorithm to solve this model. This numerical algorithm is based on the Monte Carlo simulation method, the pattern search method, the actuator line method and the corrected-explicit-implicit domain decomposition scheme which we develop for the parallel approximation of three-dimensional Navier-Stokes equations. The developed numerical algorithm, the parallel scheme, and the model are validated by a benchmark used in the literature and the experimental data. We find that by optimizing the turbines operation and considering the randomness of incoming wind speed, we can gain an additional 9% in total power.