Continuous adjoint formulation for wind farm layout optimization: A 2D implementation

Highlights: • The developed adjoint method can effectively be used for WFLO gradient computations. • The adjoint method significantly reduces the cost for the gradient computation. • The AEP is improved on average by 15% with gradient-based layout optimization. Current methodologies to optimize wind farm layouts to maximize the farm energy production rely on simple analytical models for wake loss estimations. In this paper, we present an innovative continuous adjoint formulation for gradient calculations within the framework of a gradient-based wind farm layout optimization. The developed optimization methodology integrates high-fidelity CFD models and, thanks to the adjoint method, overcomes the computationally high costs of a CFD-based optimization. The proposed continuous adjoint formulation allows for a derivation of the general adjoint equations, before any discretization is being applied, and therefore allows for a more flexible implementation in CFD software packages. Adjoint formulations for different conditions in the flow equations, namely, laminar, frozen-turbulence and turbulent flows are presented. The proposed formulation was implemented in a 2D domain and verified by comparing the calculated gradients with finite-difference approximations. Gradient calculations using the developed adjoint method were implemented in a gradient-based optimization methodology with open source software libraries, and were used to solve a 2D wind farm layout optimization problem under a wide array of wind resource scenarios. Our results showed that the annual energy production (AEP) of a given wind farm layout can be effectively improved within 30–60 iterations, depending on the initial layout and wind resource distribution. Improvements in AEP were found to be in the range of 7–37%, with an average of 15%.