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Methodology

Problem Setup

Target Wind Farm

  • Size: 16 turbines in a 16D × 16D square area
  • Rotor diameter (D): 200 m
  • Hub height: 120 m
  • Rated power: 10 MW per turbine
  • Minimum spacing: 4D (800 m)

Neighboring Farm Representation

Potential neighboring farms are represented as "blobs" - morphable shapes defined by B-spline boundaries with 4 control points. For each analysis run, we randomly sample multiple blob configurations to explore how different neighbor geometries affect design tradeoffs.

Important: The blob shapes are randomly sampled, not optimized to find worst-case configurations. This Monte Carlo approach provides a distribution of possible regret values across different neighbor geometries, but does not guarantee finding the absolute worst-case scenario.

Blob Example Example blob (coral/red region) positioned upwind of the target farm (dashed rectangle). Turbines within the blob create wakes that affect the target farm.

Blob Sampling Parameters

  • Position: Center sampled within (-10D to -4D, 0.2L to 0.8L) where L = target size
  • Size: Radius sampled between 5D and 10D
  • Shape: Aspect ratio sampled between 0.6 and 1.6
  • Number of blobs: 20 random configurations per wind rose type

Wake Model

We use the Bastankhah Gaussian deficit model with:

  • Turbulence intensity factor: k = 0.04
  • Superposition: Root-sum-of-squares (SquaredSum) by default; LinearSum available via --superposition linearsum

The Annual Energy Production (AEP) is computed as:

\[\text{AEP} = \sum_{d} \sum_{i} P_i(U_{i,d}) \cdot w_d \cdot 8760\]

where \(P_i\) is the power curve, \(U_{i,d}\) is the effective wind speed at turbine \(i\) for direction \(d\), and \(w_d\) is the probability weight.

Wind Rose Types

Von Mises Distribution

The Von Mises distribution is the circular analog of the normal distribution:

\[f(\theta; \mu, \kappa) = \frac{e^{\kappa \cos(\theta - \mu)}}{2\pi I_0(\kappa)}\]

where: - \(\mu\) = mean direction (270° = West) - \(\kappa\) = concentration parameter - \(I_0\) = modified Bessel function of order 0

κ value Interpretation
0 Uniform distribution
1 Mild concentration
2 Moderate (typical offshore)
4+ High concentration

Wind Rose Configurations Tested

Type Description Parameters
Single Unidirectional 270° only
Uniform Omnidirectional 24 directions, equal weights
Von Mises κ=1 Diffuse μ=270°, 24 directions
Von Mises κ=2 Moderate μ=270°, 24 directions
Von Mises κ=4 Concentrated μ=270°, 24 directions
Bimodal Two peaks 270° (70%) + 90° (30%)

Optimization Methodology

What Is Optimized vs. Sampled

Component Method Description
Blob shape Random sampling B-spline control points sampled from bounded distributions
Neighbor positions Fixed grid 25 potential positions on a 5×5 grid, masked by blob
Target layout SGD optimization Turbine positions optimized via gradient descent

Pooled Multi-Start Approach

For each randomly sampled blob configuration, we run a pooled multi-start optimization on the target farm layout only:

  1. Liberal Strategy (20 starts): Optimize target layout assuming neighbors are absent
  2. Objective: Maximize AEP_absent

  3. Gradient-based optimization with random initialization

  4. Conservative Strategy (20 starts): Optimize target layout accounting for neighbor wakes

  5. Objective: Maximize AEP_present

  6. Same optimization procedure

  7. Cross-Evaluation: All 40 layouts are evaluated under both scenarios

  8. Pareto Analysis: Identify non-dominated layouts

SGD Optimization Settings

SGDSettings(
    max_iter=3000,      # For single direction
    # max_iter=2000,    # For 24 directions
    learning_rate=D/5,  # 40 m step size
)

Constraint Handling

  • Boundary constraints: Soft penalty for turbines outside target area
  • Spacing constraints: Soft penalty for turbines closer than 4D
  • Gradient projection: Ensures feasibility during optimization

Regret Computation

Pareto Frontier

A layout is Pareto-optimal if no other layout achieves both: - Higher AEP when neighbors are absent, AND - Higher AEP when neighbors are present

Regret Definition

\[\text{Regret} = \text{AEP}_{\text{present}}(\text{conservative-opt}) - \text{AEP}_{\text{present}}(\text{liberal-opt})\]
  • Liberal-optimal: Layout optimized to maximize AEP without neighbors (i.e., designed in isolation), then evaluated with neighbors present
  • Conservative-optimal: Layout optimized to maximize AEP with neighbors present

Both terms are evaluated under the same conditions (neighbors present). The regret isolates the design effect: how much AEP is lost by having designed in ignorance of the neighbors, not the total AEP impact of the neighbors themselves.

If regret > 0, there exists a fundamental tradeoff between the two objectives.

Convergence Verification

To ensure results are not artifacts of insufficient optimization, we verified convergence by varying the number of random starts:

Configuration n=5 n=10 n=20 n=40
Single direction 53.70 38.62 41.15 38.62
Uniform 24.27 24.27 20.29 20.29
Von Mises κ=4 16.69 17.75 9.77 9.77

Convergence Plot

Results stabilize by n=20 starts per strategy. The full analysis uses n=20.