A Dantzig-Wolfe reformulation of the EndoWake model for optimal wind farm layouts
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 10×10 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 12×12 and 14×14 grids close in 48 and 395 seconds. The 16×16 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%.
Related
- Optimal input design via Frank-Wolfe
- Optimal Control Under Signal Temporal Logic Constraints: A Robust Reformulation via Augmented Dynamics
- Optimal Control of Dynamic Gas Networks Using Gaussian Processes with Chance-Constraint Reformulation
- Simple Strategy-Iteration with Dantzig's Rule is not Strongly Polynomial on Turn-Based Deterministic Games
- EndoWake: Modeling Wind by Linear Programming for Wind Farm Layout Optimization
BibTeX
@misc{fischetti2026dantzigwolfe,
title = {{A Dantzig-Wolfe reformulation of the EndoWake model for optimal wind farm layouts}},
author = {Martina Fischetti and Matteo Fischetti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39503},
eprint = {2610.39503},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39503},
}
RIS
TY - UNPB
TI - A Dantzig-Wolfe reformulation of the EndoWake model for optimal wind farm layouts
AU - Martina Fischetti
AU - Matteo Fischetti
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39503
UR - https://arxiv.org/abs/2610.39503
AB - 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 10×10 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 12×12 and 14×14 grids close in 48 and 395 seconds. The 16×16 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%.
ER -