Wind Turbine Energy Yield Calculator

The reference-height annual mean is extrapolated with the power law $\bar v_h=\bar v_r(h_h/h_r)^\alpha$ (NOAA wind-shear guidance). Equal heights or $\alpha=0$ bypass the extrapolation. The Weibull shape $k$ is unchanged and its scale is $c=\bar v_h/\Gamma(1+1/k)$ (UL Windographer). The survival probability is $S(v)=\exp[-(v/c)^k]$.

The general model uses $A=\pi D^2/4$ for a horizontal-axis rotor or $A=DH$ for a vertical-axis rotor, then $P_e(v)=\min[P_r,\rho A C_p\eta v^3/2]$ between cut-in and cut-out and zero outside. The kinetic-power relation follows the DOE Small Wind Guidebook (https://www.energy.gov/cmei/systems/windexchange/small-wind-guidebook). Constant Cp, constant efficiency and the hard rating/shutdown limits are explicit modeling assumptions, not a fitted manufacturer curve.

The Excel 10 option linearly interpolates the measured AC power bins in the SWCC report (https://bergey.com/wp-content/uploads/excel-10-swcc-summary-report-2016.pdf), attached source s1 p. 4. Below 0.5 m/s it holds the measured −12 W standby value; above 20.5 m/s it assigns zero output. Positive power scales by $\rho/1.225$; standby does not. It has no artificial 10 kW cap because the measured peak exceeds the nominal rating.

The Cp model splits expected power at the rating speed. With $v_t=\operatorname{clamp}(v_{\mathrm{rated}},v_{\mathrm{in}},v_{\mathrm{out}})$, the cubic contribution is $Kc^3\int_{(v_{\mathrm{in}}/c)^k}^{(v_t/c)^k}x^{3/k}e^{-x}\,dx$, where $K=\rho A C_p\eta/2$. The rated plateau contributes $P_r[S(v_t)-S(v_{\mathrm{out}})]$. This follows directly by substituting $x=(v/c)^k$ in the Weibull expectation. The library's adaptive Gauss–Kronrod rule targets relative accuracy $10^{-6}$. The dimensionless integration stops at $x=50$; the omitted mean power is bounded by $P_r e^{-50}$ (under $4\times10^{-14}$ kWh/year even at the maximum 20 MW rating). The kernel is multiplied by $10^{12}$ and the integral divided by the same factor to make the solver's absolute tolerance negligible in annual-energy units.

The Cp result is recomputed with four subintervals for a convergence diagnostic. Relative differences are compared on the rated-power scale with a $10^{-12}$ floor; only this diagnostic is rounded to $10^{-10}$ to remove floating-point cancellation. Property checks require agreement within $10^{-5}$ across narrow distributions and wide operating ranges. The independently reviewed low-wind case verifies 137.478 kWh net annual energy.

The measured preset retains 820 midpoint samples weighted by exact probabilities $S(v_{lo})-S(v_{hi})$. Those samples also illustrate the general power curve, but they no longer determine Cp-model energy. Neither method renormalizes away calm or storm hours. Wind power density is $\rho c^3\Gamma(1+3/k)/2$ over the whole Weibull distribution. Annual positive generation and standby consumption are accumulated separately over 8760 h. Net energy is $a[(1-L)E_{gen}-E_{idle}]$; site losses apply to generation and availability assumes disconnection during downtime. Capacity factor is net energy divided by rated power and 8760 h. Generating hours use the exact probability of positive-output speeds times availability.

Independent reference cases reproduce s1 p. 2 annual-energy entries at hub means of 4, 5 and 6 m/s under Rayleigh winds, sea-level density and no site losses. The reference tolerances cover interpolation of rounded bins and tail omission, and are much smaller than the report's stated measurement uncertainties.

Rotor Cp curves and ideal control illustration

The separate horizontal-axis illustration follows Slootweg et al. (2003), p. 146, equations (4)–(5), DOI 10.1109/TPWRS.2002.807113 (https://doi.org/10.1109/TPWRS.2002.807113). With numerical pitch $\beta$ in degrees, define $q=1/(\lambda-0.02\beta)-0.003/(\beta^3+1)$ and $C_p=0.73(151q-0.58\beta-0.002\beta^{2.14}-13.2)e^{-18.4q}$. Negative fitted values and the non-generating branch $\lambda\le0.02\beta$ are displayed as zero; the fit is not scaled to the entered yield coefficient. The ideal unshrouded-rotor Betz line is $16/27$.

Differentiating the zero-pitch fit gives $q_*=13.2/151+1/18.4$, $\lambda_*=1/(q_*+0.003)$ and $C_{p,*}\approx0.4412$. The independent published peak is Melício et al. (2011), p. 467, equation (6), DOI 10.5772/14597 (https://doi.org/10.5772/14597). Its equation (7) reports a different optimum tip-speed ratio; the calculator retains the derivative of the stated equations.

For $r=v/v_r$, ideal below-rated control uses $\omega/\omega_r=r$, $\lambda=\lambda_*$ and zero pitch. Above rated, $\omega/\omega_r=1$, $\lambda=\lambda_*/r$ and pitch is solved by bisection so $C_p=C_{p,*}/r^3$. Thus $P/P_r=r^3C_p/C_{p,*}$ equals $r^3$ below rated and unity above. This steady equilibrium is an explanatory simplification of the speed and pitch control described on pp. 147–148, not the paper's dynamic controller. The play input changes only this small calculation and drawing; Weibull integration and annual yield do not depend on it.

• A stationary two-parameter Weibull distribution represents a complete non-leap year; no directional, seasonal or chronological wind model is inferred from the mean.
• Power-law shear is uniform over the site and applied to the mean. Rotor shear, atmospheric stability, displacement height, veer and complex terrain are not resolved. One hub-height wind represents the entire rotor.
• General-model Cp is an aerodynamic coefficient and efficiency covers the generator/converter once. Generic standby consumption is assumed zero. No claim of actual turbine performance follows from these defaults.
• The preset is the tested Excel 10 / Powersync II configuration, not every turbine sold with a similar name. Its nominal rating is 10 kW; measured peak is 12.555 kW, so capacity factor on the nominal basis can exceed 100% in unusual wind distributions.
• Availability is independent of wind speed. Extra losses are constant fractions of positive energy. Downtime standby and restart hysteresis are omitted. Negative net yield is retained as net consumption.
• The schematic is a layout illustration, not a dimensioned tower or rotor design. Input/result units use the catalog's metric/US conversion; chart axes and schematic annotations are explicitly SI.
• The Slootweg Cp family and animation are a separate generic horizontal-axis rotor illustration. They do not define the annual-yield power curve, represent vertical-axis aerodynamics, or reproduce the Excel 10 furling/controller. Constant generator efficiency would cancel in the illustrated power ratio.
• The ideal control illustration neglects inertia, minimum rotor speed, torque limits, pitch rate limits, cut-in, cut-out and transient overshoot. Its wind cycle is prescribed for teaching, not a wind record.

Use for preliminary single-turbine energy estimates and teaching, with annual mean wind 0.5–20 m/s, Weibull shape 1–5, air density 0.6–1.5 kg/m³ and site-specific power-law shear where possible. Large height extrapolations and means above 20 m/s at hub height are flagged. The rotor must remain above ground, and general-model cut-out must exceed cut-in.

The Cp curve is a simplified operating envelope: it has a finite jump at cut-in and assumes ideal regulation at rating. The measured preset's high-speed tail is incomplete; a warning appears if at least 1% of the wind probability lies above 20.5 m/s. Density scaling of a measured curve is an approximation, especially near furling. The histogram shows 0–40 m/s only; its omitted probability is calculated separately, while yield integration uses the selected turbine range.

This is an estimate of long-term expected energy, not a P50/P90 uncertainty assessment or a guarantee. It does not establish structural strength, turbine class, electrical protection, permits, grid compliance or financial viability. SWCC test context is AWEA Standard 9.1–2009; this tool does not certify compliance. Where certification or installation requirements matter, check against the standard in force. Installation work and electrical connection belong to qualified persons.

The separate Cp illustration plots tip-speed ratio 0–15 and pitch slices 0–20°. The animation covers wind/rated ratios 0.55–1.50, assuming the turbine is already operating; the equilibrium pitch remains in this plotted range. The empirical Cp fit is not valid as a universal turbine performance map. Slootweg et al., p. 146, explicitly distinguish dynamic-model approximation from turbine-specific energy-yield accuracy.