Weibull Distribution for Wind Energy: What k and c Actually Mean

The Weibull distribution describes how often wind blows at each speed throughout the year, using two parameters that control your turbine's energy production more than any marketing spec sheet. Shape parameter k determines whether you have steady trade winds (k > 2.5) or gusty continental weather (k < 1.8), while scale parameter c sets your site's average wind speed in meters per second. A Bergey Excel 10 at a site with k=2.1 and c=6.5 m/s delivers 40% more annual kWh than the same turbine at k=1.7 and c=6.5 m/s because consistent moderate winds beat occasional gales for small turbine economics.

Why Average Wind Speed Lies to Homeowners

Wind developers stopped using simple averages decades ago because a 5 m/s average from constant 5 m/s winds produces triple the energy of a 5 m/s average split between 2 m/s calms and 8 m/s gusts. Power output follows the cube of wind speed—doubling speed multiplies power eightfold—so the distribution of speeds matters more than the mean.

The Weibull probability density function captures this reality:

f(v) = (k/c) × (v/c)^(k-1) × e^(-(v/c)^k)

Where v is wind speed, k is dimensionless shape, and c carries units of m/s. Most wind software (HOMER, Windographer, RETScreen) fits Weibull curves to your anemometer data automatically, but understanding what the numbers reveal separates realistic projects from expensive lawn ornaments.

Shape Parameter k: The Consistency Indicator

Shape k quantifies how tightly wind speeds cluster around the average. Coastal Maine sites with steady onshore flow produce k values of 2.4-2.8. Kansas plains with thermal turbulence and frontal passages drop to k=1.6-1.9. The National Renewable Energy Laboratory's distributed wind resource characterization tools classify sites by k bands because turbine selection changes dramatically across the range.

k < 1.5: Extremely variable. Common in complex terrain, urban canyon microclimates, and sites with strong diurnal heating cycles. A Primus Air 40 rated at 10 kW will rarely reach rated output because winds spike briefly then die. Annual capacity factors stay below 8% even with favorable average speeds. These sites need battery storage to smooth output or should reconsider wind entirely.

k = 1.5-2.0: Moderate variability. Typical of continental interiors, ridge tops with mixed exposure, and suburban residential areas. The "standard" Weibull distribution (k=2.0) appears in IEC small wind turbine certification test procedures. Most 5-10 kW turbines deliver manufacturer-projected output within 15% at these sites if c exceeds 5.5 m/s at hub height.

k = 2.0-2.5: Good consistency. Found on Great Plains open farmland, coastal bluffs, and offshore platforms. Turbines spend more hours near rated output. A Southwest Windpower Skystream 3.7 with 2.4 kW nameplate capacity at a k=2.3 site generates 450-550 kWh monthly where k=1.7 produces 320-400 kWh at identical c values.

k > 2.5: Exceptional steadiness. Trade wind belts, coastal headlands with geographic funneling, mountain gap flows. Rare in the lower 48 states outside Hawaii and select Pacific Northwest gorge locations. Permits downsizing turbine rated capacity by 20-30% while maintaining target kWh output, reducing upfront capital cost.

image: Weibull probability curves showing k=1.5, k=2.0, and k=2.5 distributions with identical c=6 m/s scale parameters, demonstrating how higher k values concentrate probability around the mean wind speed

## Scale Parameter c: The Speed Baseline

Scale c approximates 1.13 times your site's mean wind speed for the k=2.0 case, but the relationship shifts as k changes. A site with 5.0 m/s average and k=1.8 has c=5.65 m/s; the same 5.0 m/s average with k=2.4 gives c=5.45 m/s. Wind energy software handles this automatically, but manual checks prevent data entry errors that compound through project financing.

For small turbine siting, c at hub height determines feasibility:

• c < 4.5 m/s: No small turbine achieves positive return on investment at residential electricity rates. The 30% federal tax credit (IRC §25D through 2034) and state incentives from DSIRE databases cannot overcome physics. A 5 kW Aeolos-H turbine at c=4.2 m/s generates $180-240 annual value against $15,000-18,000 installed cost.

• c = 4.5-5.5 m/s: Marginal economics. Works only with net metering at $0.15+/kWh retail rates, excellent k values above 2.2, and owner-installed systems avoiding contractor markups. Payback stretches to 18-25 years before equipment replacement.

• c = 5.5-6.5 m/s: Viable residential zone. Professional installation of certified turbines (Small Wind Certification Council standards) achieves 12-18 year simple payback. NEC Article 705 interconnection requirements add $2,000-3,500 to project cost through required disconnect switches and anti-islanding protection.

• c > 6.5 m/s: Strong commercial opportunity. Farms, ranches, and light industrial sites see 8-14 year payback on 10-50 kW turbines. Equipment must meet FAA Part 77 notice requirements if total structure height exceeds 200 feet above ground level.

Converting Between k, c, and Familiar Metrics

Engineering literature lists wind resources in multiple formats. Converting maintains consistency across feasibility studies:

Mean wind speed v̄ relates to k and c through the gamma function: v̄ = c × Γ(1 + 1/k)

For k=2.0, this simplifies to v̄ ≈ 0.886c. For k=1.8, v̄ ≈ 0.897c. For k=2.5, v̄ ≈ 0.863c.

Most probable wind speed occurs at: v_mp = c × [(k-1)/k]^(1/k) for k > 1

This is the peak of the Weibull curve—the speed you'll observe most frequently. At k=2.0 and c=6.0 m/s, the most probable speed is 4.24 m/s, explaining why your anemometer display rarely shows the calculated average.

Wind power density (W/m²) integrates the cube of the Weibull distribution: WPD = (ρ/2) × c³ × Γ(1 + 3/k)

Where ρ is air density (1.225 kg/m³ at sea level, 15°C). A site with k=2.0 and c=6.0 m/s delivers 267 W/m², placing it in NREL Wind Class 2. The same c=6.0 m/s with k=1.5 produces 371 W/m² because the higher probability of strong gusts contributes disproportionately to the cubic relationship.

*Estimated for 5 kW turbine, 2.4 m swept diameter, 30m hub height

image: Bar chart comparing annual kWh output from the same 5 kW turbine at three sites with identical 6.0 m/s scale parameter but varying shape parameters k=1.5, 2.0, and 2.5

## Collecting Site Data That Fits Weibull Models

The National Renewable Energy Laboratory's WindWatts project provides computational wind resource estimates for distributed generation, but on-site measurement remains essential for projects above $10,000 investment. A $400-800 NRG Systems or Renewable NRG data logger with cup anemometer at proposed hub height for 12 months captures seasonal variation that annual k and c parameters summarize.

Mount the anemometer atop a telescoping mast or building corner at the planned turbine hub height, not on a roof ridge where flow separation creates false readings. Log 10-minute average wind speed and direction. Software like Windographer ($500-1,000 annual license) or free NREL tools processes the time series into k and c values with 95% confidence intervals.

If your measured k confidence interval spans 1.7-2.2, the site exhibits moderate uncertainty. Consider extending measurement to 18 months or comparing against nearby airport ASOS stations with 20+ year records. If confidence intervals remain wide after extended sampling, the site genuinely experiences high variability—proceed with conservative energy projections.

Some installers skip measurement and substitute wind atlas data or airport observations 15-30 miles distant. This works for preliminary screening but introduces 25-40% uncertainty in projected turbine output. For a $22,000 Bergey Excel 10 installation, that uncertainty represents $5,500-8,800 in expected lifetime value—more than the cost of proper measurement equipment and analysis.

Matching Turbine Power Curves to Your k and c

Small wind turbine manufacturers publish power curves showing kW output at each wind speed, but annual energy production depends on how often each speed occurs. The calculation integrates the power curve against the Weibull distribution:

AEP = 8760 × ∫[0 to ∞] P(v) × f(v) dv

Where AEP is annual energy production in kWh, 8760 is hours per year, P(v) is turbine power output at speed v, and f(v) is the Weibull probability density function. Wind software automates this, but checking the math by hand for 2-3 wind speed bins confirms the software's assumptions match your site.

A Pikasola 5000W turbine with cut-in at 2.5 m/s, rated at 12.5 m/s, and cut-out at 25 m/s performs differently under different k scenarios at c=6.5 m/s:

• k=1.5: Low-end winds below 4 m/s occur 32% of the year (Pikasola produces 100-600W during these hours). High winds above 10 m/s occur 21% of the year (near rated 5 kW). Annual output: 9,200-10,400 kWh.

• k=2.0: Winds below 4 m/s drop to 18% frequency. Winds above 10 m/s drop to 14%. Annual output: 8,600-9,800 kWh despite identical c value.

• k=2.5: Winds cluster tightly around 5-7 m/s. Below 4 m/s falls to 9%, above 10 m/s to 8%. Annual output: 8,000-9,200 kWh.

This counterintuitive result—lower k producing more energy at same c—stems from the power curve's cubic relationship below rated output. The Pikasola generates 1,800W at 8 m/s and 600W at 5 m/s. Variable winds that swing between these speeds accumulate more kWh than steady 6 m/s winds that never reach the high-output range.

However, higher k sites permit selecting smaller, less expensive turbines. Where k=1.5 demands a 5 kW nameplate for target 700 kWh/month, k=2.5 achieves the same with a 3 kW unit costing $7,000-9,000 less installed.

Real Installations: How k and c Predicted Performance

A 2019 Vermont project installed an ARE442 10 kW turbine after 14 months of wind measurement at 21 meters (proposed hub height) revealed k=2.18 and c=5.92 m/s. Windographer projected 14,600 kWh annual output. Actual metered production through four years averaged 14,180 kWh (97.1% of projection). The k value above 2.0 meant steady output without inverter cycling stress, and the turbine has required only scheduled maintenance.

A contrasting 2021 Oklahoma installation used six months of summer-fall data showing k=1.63 and c=6.41 m/s to project 18,500 kWh from a Bergey Excel 10. The site sits in rolling prairie with a creek valley 400 meters west that channels night drainage flows. Winter-spring months—missing from the abbreviated measurement period—experience thermal inversions and lower wind speeds, dropping actual k to 1.52 and c to 5.87 m/s over the full annual cycle. Two-year average output: 13,900 kWh (75% of projection). The low k also means the turbine frequently operates in partial-load conditions where mechanical efficiency suffers.

Both sites meet NEC Article 705 interconnection standards with IEEE 1547-compliant inverters. The Vermont installation qualifies for 30% federal tax credit ($6,600 on $22,000 system cost) plus Vermont's net metering at $0.19/kWh retail rate. Oklahoma has no state incentive beyond the federal credit, and the utility's net metering caps export at avoided cost ($0.04/kWh), making the actual vs. projected shortfall especially costly.

image: Side-by-side comparison photos of Vermont and Oklahoma installations with data overlays showing measured vs projected annual kWh output and the k and c parameters for each site

## When Standard Weibull Models Break Down

The two-parameter Weibull assumes a single wind regime throughout the year. Sites with distinct seasonal patterns—coastal areas with winter nor'easters and summer calms, or mountain locations with monsoon season wind reversals—need bi-modal distributions or separate k and c values per season.

A California Central Valley site measured k=1.82, c=7.1 m/s from April-October (crop irrigation season with strong thermal gradients), but k=2.41, c=4.6 m/s from November-March (winter high-pressure dominance). A single annual Weibull fit produced k=2.05, c=6.2 m/s that overestimated winter output and underestimated summer. The installation required separate seasonal models to accurately predict battery charging patterns for off-grid operation.

Urban and suburban sites with nearby buildings create turbulence that violates Weibull assumptions. A residential development in suburban Denver measured k=1.24, c=5.8 m/s at 15 meters above a two-story roofline. Windographer flagged the fit as poor (R² = 0.78) due to calm periods during building wake effects. A Rayleigh distribution (k fixed at 2.0) actually matched observations better by ignoring the site's true but problematic characteristics. The project proceeded with conservative estimates assuming the 15-meter measurement overstated actual turbine performance.

Complex terrain—ridges, valleys, forest clearings—exhibits wind shear that changes k and c with height faster than the standard power-law profile predicts. Always measure at hub height, not ground level with extrapolation formulas.

Using k and c for Turbine Selection Strategy

Given a measured k and c, the economic optimization balances turbine cost against energy capture:

High k (>2.3), moderate c (5.0-6.5 m/s): Select turbines with lower rated wind speeds (10-13 m/s) since steady winds reach rated output often. A Primus AIR Breeze rated at 12.5 m/s outperforms a competitor rated at 16 m/s by 12-18% in annual kWh at these sites. Pay attention to generator efficiency at partial load, as the turbine operates 40-55% of hours between cut-in and rated wind speed.

Low k (<1.8), high c (6.5-8.0 m/s): Prioritize turbines with strong high-wind performance and wide operational range. The frequent gusts above rated speed contain 30-45% of annual energy potential. Ensure cut-out speed exceeds 25 m/s to avoid lost production during storms. Blade pitch control (Bergey Excel series) handles gusts better than stall control, reducing mechanical stress that compounds over a variable-wind turbine's life.

Low k, low c: Reconsider wind. Solar photovoltaic with the same 30% federal tax credit delivers more predictable output per dollar. Exceptions exist for off-grid applications where generator fuel costs exceed $0.70/kWh equivalent or for hobby/education purposes where energy ROI is secondary.

The Small Wind Certification Council (testing per AWEA 9.1 standards) requires turbines demonstrate performance across a range of k values during certification. Check the certified turbine list at smallwindcertification.org for models that passed real-world Weibull distribution testing, not just steady-state wind tunnel performance.

Frequently Asked Questions

What's a "good" k value for small wind turbines?

Values between 1.8 and 2.4 work well for most residential turbines because power curves optimize for moderate variability. Extremely high k (>2.7) wastes a turbine's rated capacity since winds rarely spike, while very low k (<1.5) causes excessive wear from constant speed cycling and reduces capacity factor below 12% even at sites with adequate average wind speed.

Can I estimate k and c without installing an anemometer?

NREL's WindWatts computational tools provide k and c estimates at 250-meter resolution for U.S. locations, sufficient for initial screening. Airport ASOS/AWOS stations publish hourly wind data that Windographer's free viewer can process into Weibull parameters. However, neither substitutes for hub-height measurement at your specific site for final go/no-go decisions on installations above $8,000. Microscale terrain effects shift c by ±1.5 m/s over distances of 200-400 meters in complex topography.

How do I convert scale parameter c to mph?

Multiply c (in m/s) by 2.237 to get mph. A site with c=6.0 m/s equals c=13.4 mph. Most U.S. wind maps and airport data report mph, while Weibull literature uses m/s, creating conversion confusion. Wind energy software accepts either unit but performs calculations internally in m/s per IEC standards.

Does k change with measurement height?

Shape parameter k typically increases 0.05-0.15 units when moving from 10m to 30m height as surface roughness effects diminish. The relationship is site-specific and nonlinear. Scale parameter c increases more predictably with height following power-law or logarithmic profiles (0.15-0.28 exponent for open terrain). When extrapolating from one height to another, adjust c using established shear models but treat k cautiously—changes of 0.2 units alter energy projections by 8-15%.

What k and c characterize trade winds or mountain gap winds?

Hawaiian trade wind sites measure k=2.6-3.2 and c=7-9 m/s at 30m hub height, among the steadiest regimes globally. Columbia River Gorge gap flows produce k=2.4-2.8 with c=8-11 m/s during summer but drop to k=1.9-2.2, c=4-6 m/s in winter. Tehachapi Pass in California averages k=2.2-2.5 with c=7.5-9.5 m/s year-round. These exceptional sites support utility-scale wind farms but offer lessons for siting distributed turbines in similar geographic features elsewhere—seek funneling effects from surrounding terrain.

Bottom Line

Shape parameter k tells you whether your winds blow steadily (k > 2.2) or gust erratically (k < 1.8), while scale parameter c sets your average wind speed baseline in m/s. Together, they predict small turbine annual output far more accurately than "average wind speed" or power curve charts alone. A site with k=2.3 and c=6.0 m/s delivers better economics than k=1.6, c=7.0 m/s despite lower average wind speed because steady flow matches turbine power curves efficiently. Invest $800 in 12 months of hub-height wind measurement and proper Weibull analysis before committing to a $15,000-40,000 turbine installation—the data will either confirm a viable project or save you from an expensive mistake.