How this calculator works
A year of wind measurements contains tens of thousands of 10-minute values. To design or compare turbines, wind engineers compress them into a frequency distribution: how many hours the wind blows in each speed band. The Weibull distribution fits that histogram remarkably well with just two numbers, the shape factor k and the scale factor c.
You can start from whatever you know:
- mean speed + k: the usual case, e.g. from the Global Wind Atlas;
- mean + standard deviation: from a logger’s summary statistics; k is solved exactly from σ/v̄;
- k + c: directly from an atlas or a fitting report.
The calculator then draws the probability density and cumulative curves, tabulates hours and share of energy in each speed bin, and reports the operating hours of a turbine with your cut-in, rated and cut-out speeds.
The formulas
mean v̄ = c × Γ(1 + 1/k) σ = c × √(Γ(1 + 2/k) − Γ(1 + 1/k)²) median = c × (ln 2)^(1/k) mode = c × ((k − 1) / k)^(1/k) speed carrying max. energy = c × ((k + 2) / k)^(1/k) hours in bin [a, b] = 8,760 × (F(b) − F(a))
Worked example: mean 6.5 m/s, k = 2
With the defaults (mean speed 6.5 m/s, k = 2):
- scale c = 6.5 / Γ(1.5) = 7.33 m/s, standard deviation 3.40 m/s
- median 6.11 m/s, most probable speed 5.19 m/s
- the speed carrying the most energy is 10.4 m/s, well above the mean
- energy pattern factor 1.91
For a turbine with a 3 m/s cut-in, 11.5 m/s rated speed and 25 m/s cut-out, the wind is in the operating window about 7,400 hours a year (85%), above rated speed about 750 hours, below cut-in about 1,350 hours, and above cut-out for well under one hour. Switch the mode to mean + standard deviation with the default 3.4 m/s and the exact solution returns k = 2.00, while the Justus approximation gives 2.02. In the normal range the two agree closely.
How to read the results
- pdf curve: the relative frequency of each speed; its peak is the most probable speed.
- cdf curve: the share of time below a given speed; read off, for example, the fraction of calm hours below cut-in.
- Hours per bin chart: bars are hours, the line is each bin’s share of the wind’s energy. The two peaks are far apart: time is concentrated at moderate speeds, energy at high ones.
- Operating and rated hours are useful for maintenance planning and for estimating how often a turbine will be clipping at full power.
Typical values
| Climate | Typical k | Mean speed | Character |
|---|---|---|---|
| Sheltered inland, valleys | 1.3–1.7 | 3–5 m/s | many calms, occasional strong winds |
| Continental plains | 1.8–2.2 | 6–8 m/s | close to Rayleigh |
| Western European coasts | 2.0–2.3 | 7–9 m/s | frequent fronts |
| North Sea, offshore | 2.1–2.4 | 9–10.5 m/s | strong and fairly steady |
| Trade-wind islands | 2.8–3.5 | 7–9 m/s | very steady |
Tips
- Fit on the right averaging period. Statistics from hourly means give slightly higher k than 10-minute data, because averaging removes short-term variability.
- k changes with height. It usually increases a little up to around 50–100 m, then decreases. Use parameters from the height you need, or the wind speed at height calculator for the mean.
- Use k to sanity-check data. A fitted k below 1.3 or above 4 at a normal site often signals sensor problems or icing.
- Then compute energy. Plug mean and k into the wind turbine calculator (manual mode) or the wind power density calculator.
Limitations
The Weibull model is a fit, not a law. Sites with two wind regimes (a sea breeze plus synoptic storms, or monsoon and dry seasons) can be bimodal, and a single Weibull curve then misplaces energy between bins. Very calm sites may have more zero-wind hours than the curve predicts. The method-of-moments fit used for mean + σ is dependable but slightly less accurate in the high-speed tail than maximum-likelihood or the WAsP energy-matching method. Always compare the fitted curve with a histogram of your own data when it is available.
Method & assumptions
pdf f(v) = (k/c)(v/c)^(k−1) e^{−(v/c)^k}; cdf F(v) = 1 − e^{−(v/c)^k}. Mean = c·Γ(1+1/k), σ = c·√(Γ(1+2/k) − Γ(1+1/k)²), median = c(ln 2)^{1/k}, mode = c((k−1)/k)^{1/k}, speed carrying maximum energy = c((k+2)/k)^{1/k}. With mean + σ, k is solved exactly from σ/v̄ by bisection (the Justus 1978 approximation k ≈ (σ/v̄)^−1.086 is shown for comparison). Hours in a bin = 8,760 × [F(b) − F(a)].
Frequently asked questions
What is the Weibull distribution in wind energy?
How do I calculate Weibull k and c from wind data?
What is a typical Weibull shape factor?
What is the Rayleigh distribution?
What is the energy pattern factor?
How many hours a year does a wind turbine operate?
Sources & references
- Justus, C.G., Hargraves, W.R., Mikhail, A., Graber, D. (1978). Methods for estimating wind speed frequency distributions. Journal of Applied Meteorology 17(3)
- Troen, I. & Petersen, E.L. (1989). European Wind Atlas. Risø National Laboratory (WAsP Weibull fitting)
- IEC 61400-12-1: Power performance measurements — AEP for Rayleigh distributions
- Global Wind Atlas — Weibull k and A parameters for any location
Embed this calculator
Free for blogs, installers, schools and NGOs. Copy the code below; it updates itself automatically.