Wind Weibull Distribution, Annual Energy, and Capacity Factor

Annual energy does not come from average wind speed, it comes from the DISTRIBUTION, and using a mean instead understates the resource by nearly half. A site at Weibull k = 2.0 and c = 18 mph has a mean speed of 15.95 mph -- but an energy pattern factor of 1.910, because the mean of the cubes is that much larger than the cube of the mean. Every serious wind calculation is an integral of the power curve against the distribution for exactly that reason, and the shape parameter controls how pronounced the effect is: a low k means a broad distribution with more very windy hours and a higher pattern factor. THE HOURS MATTER AS MUCH AS THE ENERGY. That same site spends 1,230 hours of the year below a 7 mph cut-in making nothing at all, runs 7,530 hours, and reaches rated power in only 923 of them. A mean speed hides all of it. Integrating a 2,500 kW machine cut in at 7, rated at 27 and cut out at 55 against this distribution gives 6,772 MWh gross, 5,756 MWh after 15% of wake, availability and electrical losses -- a capacity factor of 26.3%, below the 35 to 45% band a modern onshore machine reaches at a decent site. CAPACITY FACTOR IS THE COMPARABLE FORM. It is the honest measure of a site and a machine together and it is what makes a large rotor on a modest tower comparable with a small rotor on a tall one; a figure claimed above about 60% onshore deserves the same scepticism as a power coefficient above Betz. Fitting Weibull parameters requires a year or more of measured data at or near hub height; parameters from a wind atlas or a nearby station are indicative only and are routinely wrong for a specific site, especially in complex terrain. The curve used here is the standard idealisation -- cubic between cut-in and rated, flat at rated to cut-out -- and a real machine curve is not that; the manufacturer warranted curve governs. The loss factor stands for wake losses within an array, availability, electrical and transformer losses, blade soiling and icing, curtailment, and high-wind hysteresis, typically 10 to 20% combined. A bankable energy assessment to IEC 61400-12 by an independent assessor governs.

the Weibull pdf f(v) = (k/c)(v/c) raised to (k-1) times exp of minus (v/c) raised to k; mean speed = c x Gamma(1 + 1/k); energy pattern factor = Gamma(1 + 3/k) / Gamma(1 + 1/k) cubed; annual energy is that distribution integrated against a curve cubic from cut-in to rated and flat to cut-out; capacity factor = annual energy / (rated power x 8,760 hours).

The two-parameter Weibull distribution, the energy pattern factor, and the capacity-factor definition by name, with IEC 61400-12 named for the measurement campaign that fits the parameters. An independent energy assessor and the manufacturer's warranted power curve govern a bankable estimate.

An integral over the user's own Weibull parameters and machine ratings; no wind atlas, measured dataset, or manufacturer power curve is reproduced.

Estimate. AHJ and licensed professional govern.

Field names used by the API: weibull_k, weibull_c_mph, rated_power_kw, cut_in_mph, rated_speed_mph, cut_out_mph, loss_factor_pct, mean_speed_mph, energy_pattern_factor, hours_below_cut_in, gross_aep_mwh, net_aep_mwh, capacity_factor

• The distribution, not the mean an average-speed estimate is low by roughly the energy pattern factorfirst principles
• The curve here is an idealisation cubic to rated then flat; a real machine's curve is not thatmanufacturer power curve
• Losses are 10 to 20% combined wake, availability, electrical, soiling, icing, curtailmentenergy assessment practice