Test 01 — Wind farm power density is a spacing problem
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
README.md
# Test 01 — Wind farm power density is a spacing problem **Result.** The core of the claim survives; its sharp predictions do not. Across 989 US onshore wind plants with matched 2025 generation, spacing explains far more of the variation in energy density than turbine efficiency does — R² 0.26 for spacing against 0.03 for specific power and 0.01 for turbine rating, a difference of 0.23 (state-cluster bootstrap 95% CI 0.13 to 0.32). Energy density falls roughly six-fold from the tightest spacing bin (19.7 GWh km⁻² yr⁻¹ at 2.3 rotor diameters) to the widest (5.4 at 6.5 diameters). But the claim's prediction that power density **peaks at intermediate spacing** is wrong: the curve declines monotonically, with a convex quadratic (curvature +1.17) whose stationary point would be a minimum. And the claim that tight arrays "produce less per hectare" is inverted: the tightest, most efficient quartile produces **2.8 times** the energy per hectare of the widest, least efficient quartile (16.6 against 5.85 GWh km⁻² yr⁻¹). Tight arrays do produce less **per turbine** (3.96 against 8.90 GWh), but that gap is a turbine-design difference, not a spacing effect: once specific power and rating are controlled, spacing has no residual association with capacity factor or per-turbine output. ## What was tested The hypothesis, quoted verbatim in `hypothesis.md`, claims that "wind farm power density is set by turbine spacing rather than by turbine efficiency, so larger, more efficient turbines in a tight array can produce less per hectare than smaller ones spaced further apart." It predicts an interior peak in density (P1), that tight arrays underperform despite efficient turbines (P2), and that spacing predicts better than rating (P3). The refutation is that output scales with turbine count at controlled spacing. ## Method - **Plants.** USWTDB v9.1 (28 September 2026), onshore turbines grouped by EIA plant id; 70,841 turbines in 1,126 plants with at least five turbines. Footprint is the convex hull of turbine coordinates on a local equirectangular projection, replaced by n·d̄² where the hull degenerates. Plants whose hull is a line thinner than two rotor diameters (62 ridge-top arrays) are excluded, because an area density is not defined for a one-dimensional layout; they are re-admitted in sensitivity. - **Generation.** EIA-923 2025 final, wind fuel type, summed to plant level; 1,068 plants match. Capacity factors above 0.70 (11 plants) are excluded as capacity/generation mismatches, along with non-finite geometry. - **Measures.** Energy density = 2025 GWh ÷ km². Capacity density = nameplate MW ÷ km². Spacing = mean nearest-neighbour distance ÷ mean rotor diameter. Specific power = W per m² of swept rotor. - **Models.** Log–log OLS for energy density; capacity factor and per-turbine generation models include year, state fixed effects, specific power and rating. Inference: state-cluster bootstrap, 400 replicates, seed 20261030. ## Result **P3, supported.** The three single-predictor R² values are 0.260 (spacing), 0.034 (specific power) and 0.007 (turbine rating). The difference between spacing and specific power is 0.227 (95% CI 0.131 to 0.322). In the joint model with year and state controls, a one-standard-deviation increase in log spacing is associated with a 0.164 lower log energy density (SE 0.012), while specific power's coefficient is −0.001 (SE 0.013) and rating's is +0.069 (SE 0.015). Spacing dominates; efficiency is nearly irrelevant to areal output once the layout is known. **P1, rejected.** Energy density falls from 19.7 GWh km⁻² yr⁻¹ in the tightest bin to 5.4 in the widest, with only minor non-monotonicity (figure 1). A quadratic in log spacing has positive curvature (+1.17), so its stationary point at 12.4 diameters would be a minimum, and it lies outside the observed range (1.2–11.0). There is no interior maximum at any specification we tried. **P2, rejected per hectare, supported per turbine but not attributable to spacing.** The tight-and-efficient quartile (specific power ≥ 345 W m⁻², spacing ≤ 3.4 D) achieves 16.6 GWh km⁻² yr⁻¹ against 5.85 for the wide-and-inefficient quartile (≤ 216 W m⁻², ≥ 4.9 D). Its capacity factor is lower (0.248 against 0.422) — the wake penalty is visible — but not low enough to offset the 5-fold difference in nameplate density. Per turbine, tight arrays produce 3.96 GWh against 8.90 GWh. However, the capacity-factor model gives spacing a coefficient of +0.001 (SE 0.003, p = 0.69) once specific power (−0.055, p < 0.001) and rating are included; the per-turbine model likewise gives spacing −0.002 (p = 0.78) against specific power −0.089 and rating +0.193. The raw spacing gradient in capacity factor is a design gradient: older, low-specific-power turbines sit in wider arrays. ## Where the claim is right, and where it is a definition Some of P3 is arithmetic. If density is capacity per unit area and area scales as spacing squared, spacing must dominate — the identity is density ≈ specific power / spacing². What the data add is the size of the deviation from that identity: the fitted spacing elasticity in the joint model (−0.16 per SD of log spacing) is far from the −2 that a pure geometry would give, because real footprints, wake losses and siting choices break it. The claim is therefore a good first-order description of areal output and a poor description of what limits a single turbine. ## Sensitivity - Including the 62 linear ridge arrays raises the spacing R² to 0.325; excluding hull-fallback plants leaves it at 0.261. - Using the fifth-nearest neighbour instead of the nearest gives R² 0.268 (Spearman 0.68 with the nearest-neighbour measure). - Including plants with capacity factors above 0.70 raises R² to 0.367. - The result does not depend on the 0.70 capacity-factor cut; the cut removes 11 plants and changes the spacing coefficient by less than 0.01. ## What this test cannot see Spacing is chosen, not assigned. Developers space turbines further apart where land is cheap and wind is strong, so state fixed effects and turbine design controls are not identification. The footprint is a hull, not a lease boundary; ridges and partial rows are handled by exclusion, which discards exactly the layouts where area is most ambiguous. Generation is one year (2025) and depends on that year's wind. Capacity-factor comparisons cannot separate wake losses from availability, curtailment, and grid constraints. And the test says nothing about cost: the tight array wins per hectare, but may lose per dollar of interconnection or per unit of turbulence loading. ## Disclosure Designed and executed with AI assistance (DeepSeek, via its coding agent). Not peer reviewed. This is an analysis of public plant-level data, not engineering or planning advice. --- *Figures: `figures/energy_density_vs_spacing.png`, `figures/capacity_factor_by_spacing.png`, `figures/variance_explained.png`. Numbers: `results.json`. Data: `data.csv`. Sources: `SOURCE.md`.*
hypothesis.md
# Hypothesis, quoted verbatim, and pre-specified criteria **Source.** *Original Theories*, Volume VIII, entry 91 (slug `wind-array-spacing`), Incoprea research notebook (`incoprea.com/research`); corpus export `frontier-theories-8/build/data_v8_5.py`. ## The claim, verbatim > Wind farm power density is set by turbine spacing rather than by turbine efficiency, so larger, more efficient turbines in a tight array can produce less per hectare than smaller ones spaced further apart. **Predictions, verbatim.** - P1: Farm power density should peak at intermediate spacing. - P2: Tight arrays should underperform despite efficient turbines. - P3: Spacing should predict better than turbine rating. **Refutation, verbatim.** > Farm output scaling with turbine count at controlled spacing would restore per-turbine metrics. ## Operationalisation - **Unit.** A wind plant, defined by the EIA plant id where the USWTDB supplies one, otherwise by plant name and state. Turbines built after 2024 are excluded so that the nameplate capacity matches a full 2025 generation year. - **Power density.** Two measures. *Capacity density* is nameplate MW divided by the farm's footprint area in km². *Energy density* is 2025 net generation (GWh) divided by footprint area, in GWh km⁻² yr⁻¹. The claim says "produce less per hectare", so energy density is the primary outcome and capacity density is the mechanism. - **Footprint area.** Convex hull of turbine coordinates on a local equirectangular projection. For linear or sparse arrays the hull degenerates; where the hull is less than 0.4 × n × (mean nearest-neighbour distance)², we fall back to n × d̄² and flag the farm. - **Spacing.** Mean nearest-neighbour turbine distance divided by mean rotor diameter, s/D. This is dimensionless and directly comparable across turbine sizes. - **Turbine efficiency.** Specific power, W m⁻² of swept rotor area, averaged over the farm; and mean turbine rating, kW. Low specific power is the standard engineering definition of a more efficient rotor. - **Generation.** EIA-923 2025 final, Page 1, wind fuel type (WND), summed to plant level. - **Capacity factor.** 2025 net generation ÷ (nameplate MW × 8,760 h). ## Pre-specified support and refutation The test is **supported** if: 1. P1: energy density has an interior maximum in spacing — a quadratic in log spacing with a negative curvature and a vertex inside the observed range, and binned medians that rise then fall. 2. P2: within high-specific-power farms, tighter spacing gives lower energy density *and* lower generation per turbine. 3. P3: spacing explains more variance in log energy density than specific power or turbine rating does (difference in R² > 0.05, bootstrap over states). The **refutation** holds if generation per turbine is flat in spacing at controlled turbine rating, or if energy density rises monotonically as spacing widens (no interior peak). ## Confounds named before running Spacing is chosen, not assigned: developers space turbines further apart on windy, cheap land, so spacing correlates with site quality. State fixed effects, year built, turbine rating and specific power are controls, not identification. Nameplate-area measures cannot see wake losses directly; the capacity-factor test is the only direct evidence on wakes. 2025 is a single year, and generation depends on that year's wind resource. ## What would decide it better Turbine-level SCADA output and layout data, or a within-developer comparison of adjacent farms with different spacing on the same wind resource. Neither is public. This test is the best available with national data.