wind farm planners

Abstract: Wind Farm Layout Optimization places turbines to maximize the expected energy under a wind rose, each turbine lowering the wind downwind. In the EndoWake model the wind field is a variable, propagated by linear inequalities, and layouts are certified by branch-and-check. On a ladder of grids it certifies the $10\times10$ grid and leaves the larger grids open after one hour. We tighten the relaxation by a convexification computed rather than derived: every pattern of turbines inside a small tile of the grid is enumerated once with its power and enters the master as a column. The device is valid because the wind field is antitone, and exact, for the cubic power curve, with a Benders cut at integer layouts. Reduced-cost fixing and propagation between overlapping tiles shrink the master before the tree search, and the tree repeats the fixing at every improving incumbent. The incumbent comes from a tile heuristic or an LP dive. On the same computer as the branch-and-check, the $12\times12$ and $14\times14$ grids close in 48 and 395 seconds. The $16\times16$ grid is certified on five seeds from random starts, one job per seed, the heuristic on a GPU, in 1413 seconds under the uniform rose and 6484 seconds under a non-uniform one. Times are medians. A pre-specified test set of 36 unseen instances with forbidden cells confirms the advantage in the axial setting. Certified means closed at the solver's gap of $10^{-4}$: the optimum lies within it plus a residual of at most 0.00042%.

Abstract: Wind farm layout optimization decides where to place turbines to maximize annual energy production, and wake effects decide how much of that energy is produced. Accurate wake models are too slow for a search. We build on EndoWake, an endogenous wake model in which the wind speed at every grid cell is a variable linked to its upwind neighbors by linear constraints, so that wake field, siting decisions and project constraints share one mixed-integer model. Its four parameters are calibrated once against the single-wake field of a Reynolds-averaged Navier-Stokes (RANS) simulation, and a GPU scores over a million layouts per second per wind direction. Comparing EndoWake-guided searches with a PyWake-based pipeline at Lillgrund, under a RANS judge that no search calls, revealed that the fidelity of a wake model is not the quality of the layouts optimized under it. A search is drawn to the gaps between sampled wind directions, the sirens; a diagnostic of the field finds a second artifact, an optimistic band off each wake edge, the mermaid cells, not reported before to our knowledge. Judged at random directions, the two pipelines are statistically indistinguishable. Our main result puts this speed to use: EndoWake screens the candidate moves of a local search on the GPU, and PyWake confirms every accepted move. On twenty unseen starts, with one GPU and thirty CPU processes, this screen-and-confirm search matches the PyWake value of a search on PyWake alone 7.9 times faster, and 14.9 times faster on the annual energy.