Source-linked AI summary
Offshore wind farm layout optimization using mathematical programming techniques
Beatriz Perez, Roberto Minguez, Raul Guanche
TL;DR
Offshore wind’s favorable resource is offset by higher installation and maintenance costs, motivating layout optimization to reduce wake effects. The paper combines heuristic initial layouts with nonlinear mathematical programming for global optimization, and its Alpha Ventus case study reports a 3.758% increase in expected annual power production over the actual configuration. The authors also identify scope boundaries related to farm size, turbine mobility, and unresolved physical and mathematical modeling issues.
Problem
Offshore wind farms must compensate for higher installation and maintenance costs, while layout optimization is challenged by wake effects and non-convexity.
Method
The method generates heuristic initial layouts and refines them with nonlinear mathematical programming under layout constraints and an AEP objective.
Results
3.758% increase in expected annual power production for Alpha Ventus compared with the actual configuration.
Takeaways & Limitations
The procedure is most effective for smaller wind farms, whereas dense farms with limited turbine movement produce layouts similar to grid-like layouts.
Takeaways & Limitations
The study notes that wake modeling, foundations, cabling, visual effects, area restrictions, and optimization approaches still require further detailed review.
Abstract
from arXiv · showhide
Offshore wind power is a renewable energy of growing relevance in current electric energy systems, presenting favorable wind conditions in comparison with the sites on land. However, the higher energy yield has to compensate the increment in installation and maintenance costs, thus the importance of optimizing resources. One relevant aspect to increase profitability is the wind farm layout. The aim of this paper is to propose a new method to maximize the expected power production of offshore wind farms by setting the appropriate layout, i.e. minimizing the wake effects. The method uses a sequential procedure for global optimization consisting of two steps: i) an heuristic method to set an initial random layout configuration, and ii) the use of nonlinear mathematical programming techniques for local optimization, which use the random layout as an initial solution. The method takes full advantage of the most up-to-date mathematical programming techniques while performing a global optimization approach, which can be easily parallelized. The performance of the proposed procedure is tested using the German offshore wind farm Alpha Ventus, located in the North Sea, yielding an increment of expected annual power production of 3.52% with respect to the actual configuration. According to current electricity prices in Germany, this constitutes an expected profit increment of almost 1 M per year.
1. Introduction
Offshore wind offers stronger, steadier winds than land-based sites, but higher installation and maintenance costs make layout optimization important. The paper combines heuristic initialization with nonlinear mathematical programming to address wake losses and non-convex layout optimization.
- Motivation: Offshore sites provide stronger and steadier winds than land sites, but their higher installation and maintenance costs require optimized project designs.This section focuses specifically on layout optimization among several offshore wind-farm design aspects.
- Motivation: Wake effects reduce downstream wind speed and increase turbulence, lowering energy production for turbines within the wake’s influence.Strategic turbine positioning can minimize expected wake-effect losses and maximize expected efficient energy production.
- Existing approaches: Existing heuristic approaches use simplifying assumptions, may miss local or global optima, often fail to satisfy Karush-Kuhn-Tucker conditions, and restrict turbine locations through predefined grids.The introduction identifies these limitations as central motivations for the proposed approach.
- Existing approaches: Gradient-based mathematical programming methods can be computationally prohibitive and may return local solutions that skip the global optimum depending on initialization.The layout problem is non-convex, so the starting solution affects which solution these methods find.
- Proposed method: The paper combines heuristic initial-layout generation with nonlinear mathematical programming to search for an offshore layout over a pre-specified area.The procedure uses heuristic restarts to identify convex subregions and applies nonlinear solvers to refine layouts.
- Proposed method: The methodology supports alternative constraints and objectives, parallelization, KKT-optimal final solutions, and continuous feasible locations without gridding.The authors also characterize the nonlinear solvers as numerically robust and computationally efficient.
2. Wake models
The paper models turbine wakes as downstream flow disturbances that reduce power at downwind turbines. It selects the N.O. Jensen model for energy prediction while recognizing uncertainty across wake models.
- Wake disturbances reduce wind speed and increase turbulence behind turbines, lowering energy production at downwind turbines.
- The study selects the N.O. Jensen wake model because prior comparisons recommend it as a balance between positive and negative prediction errors.
- For downstream locations, the model computes wake velocity deficit from projected distance, thrust coefficient, decay factor, and rotor radius.
- The decay factor represents wake-width growth downstream and depends on ambient turbulence, turbine-induced turbulence, and atmospheric stability.
- The selected wake model has been used in commercial software including WAsP, Garrad Hassan WindFarmer, and WindPRO.
3. Layout problem definition
The layout problem defines turbine positions within a feasible wind-farm area and maximizes expected Annual Energy Production by accounting for wake effects and directional wind variability.
- Data: The model uses wind, turbine, wake-effect, and wind-farm data to evaluate layouts.Inputs include wind speeds and directions, turbine characteristics, wake parameters, the available area, and the number of turbines.
- Problem variables: The optimization variables are the exact coordinates (x_i, y_i) of each turbine.These coordinates form the decision vector for all allocated turbines.
- Scope: Water depth, foundation costs, and electrical connection costs are outside the paper’s scope, although they could be added to the objective function.The authors identify these factors as possible extensions rather than modeled components.
- Constraints: Turbines must remain inside a predefined quadrilateral area and satisfy a minimum spacing of four rotor diameters.The area constraint keeps turbines within the boundary, while the spacing restriction is d_min = 4D.
- Objective function: The objective is expected Annual Energy Production (AEP), calculated by summing turbine energy yields across wind-direction sectors.The calculation uses sector probabilities, wind-speed distributions, turbine power curves, and wake-perturbed wind speeds.
- Evaluation procedure: The algorithm reorients and renumbers turbines for each directional sector before evaluating wake interactions and AEP.The procedure includes a coordinate base change and turbine renumbering according to vertical position.
4. Layout optimization
The layout optimization formulates turbine placement as a nonlinear mathematical programming problem and combines heuristic initialization with gradient-based local searches. Repeated searches explore multiple local solutions while supporting parallel execution and final refinement of the best candidate.
- 4. Layout optimization: The layout problem is nonlinear mathematical programming with linear inequality constraints and can be solved using constrained nonlinear-programming solvers.Examples include MINOS, CONOPT, fmincon, and ktrlink.
- 4.1. Heuristic initial solution: A stochastic feasible layout is generated first, then used as the starting value for a mathematical-programming local search.The heuristic aims to cover possible convex subregions before local optimization.
- 4.2. Local minima search: The sequential heuristic-local search can be parallelized by running independent repetitions on separate cores while sharing the best solution found.This structure is intended to increase computational efficiency and reduce computational costs.
- 4.1. Heuristic initial solution: The heuristic spreads turbines across the assigned area by uniformly sampling coordinates, transforming them into the farm region, and applying a turbine-widespread procedure.The initial layout must be random, feasible, and able to place turbines throughout the wind-farm area.
- 4.1. Heuristic initial solution: Delaunay triangulation optimization maximizes the sum of triangle areas while preventing triangle overlap and edge crossing.The resulting feasible solution is used as the starting point for the gradient-based algorithm, and the maximum summed area equals the limiting farm area.
- 4.2. Local minima search: The combined search repeats heuristic initialization and local optimization, using ktrlink for faster local maxima searches before fmincon refinement.The figure distinguishes starting points, local maxima, and the global maximum; increasing repetitions raises the chance of finding the true global solution but does not guarantee it.
5. Case study: Alpha Ventus
The Alpha Ventus case study evaluates wake-aware layout optimization for a 12-turbine offshore wind farm. The optimized layout improves annual energy production and reduces wake losses, although efficiency declines as turbine density increases.
- Alpha Ventus is a North Sea offshore wind farm 60 kilometres from the coast, selected to demonstrate the proposed methodology under extreme winds, weather, and tides.
- The case study assumes 12 identical NREL 5 MW turbines with hub height z = 90 m and rotor diameter D = 126 m.
- 293.274 GWh is the actual layout’s expected annual AEP, representing a 4.44% reduction from 306.9 GWh without wake effects.
- 304.809 GWh is the optimized layout’s expected annual AEP, reducing wake losses from 4.44% to 0.682%.
- The optimized layout removes wake effects from turbines affected in the initial simulation for the most likely 75°NE sector and improves the configuration in the 45°NE sector.
- 0.9921 M€ per year is the estimated profit increment under a production cost of 0.064 €/kWh and selling price of 0.15 €/kWh.
- Higher installed power capacity reduces efficiency and the optimization improvement; 23 turbines are infeasible because no layout satisfies minimum-distance constraints.
- Simulation times for different turbine counts are approximate because runs did not begin under identical conditions.
6. Conclusions
The paper proposes a sequential layout-optimization method that combines heuristic initialization with nonlinear mathematical programming to reduce wake effects. Tested on Alpha Ventus, it increases expected annual power production while retaining flexible and parallelizable optimization features.
- The proposed method maximizes expected offshore wind-farm power production by selecting layouts that minimize wake effects and indirectly reduce turbine fatigue effects.
- The sequential procedure first generates a random layout heuristically, then applies nonlinear mathematical programming for local optimization from that initial solution.
- 3.758% is the reported increment in expected annual power production for Alpha Ventus compared with its actual configuration.
- The methodology supports robust nonlinear solvers, alternative constraints or objectives, convex-subregion searches, parallelization, and solutions satisfying Karush-Kuhn-Tucker optimality conditions.
- Further work must address wake-model uncertainty, foundations, cabling, visual effects, area restrictions, and mathematical optimization topics.