Wind methods

Implemented from public descriptions of IEC 61400-26-1/-2 and IEC 61400-12-1 and from the reliability literature cited below; not a certified implementation of the standards.

Availability by information category

Every interval of every turbine carries one information category. The list follows the top levels of the standard's information model (its tree is finer: "out of environmental specification" alone covers low wind, high wind, icing and temperature limits):

category	meaning
full_performance	generating without limits
partial_performance	generating with a technical derating
technical_standby	ready, waiting (e.g. low wind, cable untwisting)
environmental	out of environmental specification
requested	requested shutdown or curtailment (grid operator or owner)
electrical	out of electrical specification (grid)
scheduled_maintenance	scheduled maintenance
planned_corrective	planned corrective action
forced_outage	forced outage
no_info	information unavailable

A contract says for each category whether its time is available, unavailable or excluded:

\[ A_\mathrm{time} = \frac{T_\mathrm{available}}{T_\mathrm{available} + T_\mathrm{unavailable}} \qquad A_\mathrm{energy} = 1 - \frac{E_\mathrm{lost,\,unavailable}}{E_\mathrm{actual} + E_\mathrm{lost}} \]

with the energy sums taken over the categories that are not excluded. A maintenance allowance (hours per turbine and year) moves scheduled maintenance from unavailable to excluded until it is used up; its lost energy is excused in the same proportion. Two illustrative presets ship with the library: iec_operational (every operative state available, missing information excluded) and oem_warranty (grid, environment and requested stops excused, scheduled maintenance allowance 60 h, missing information counted as available). wind.remap re-labels categories, which shows what a misclassification costs: counting grid stops as turbine faults lowers a warranty availability by as much as the stops last.

Power curve

Air density of dry air from temperature and pressure (the ISO standard atmosphere at the site elevation when the pressure is not measured):

\[ \rho = \frac{p}{R_d\, T}, \qquad R_d = 287.05\ \mathrm{J\,kg^{-1}\,K^{-1}}, \qquad p(z) = 101325\,(1 - 2.25577 \cdot 10^{-5} z)^{5.25588} \]

Normalisation of the wind speed to the reference density \(\rho_0 = 1.225\ \mathrm{kg/m^3}\) for pitch-regulated turbines:

\[ V_n = V \left(\frac{\rho}{\rho_0}\right)^{1/3} \]

Method of bins. Bins of 0.5 m/s centred on multiples of 0.5 m/s; in each bin the mean normalised wind speed and mean power, and the statistical (category A) uncertainty of the bin mean \(s_i / \sqrt{N_i}\). Bins with fewer than three 10-minute records are dropped. A data set is complete when every bin from 1 m/s below cut-in to 1.5 times the wind speed at 85 % of rated power is present and the records cover at least 180 hours.

Annual energy production for a Rayleigh distribution of mean \(\bar V\), the curve held at its last value up to cut-out:

\[ \mathrm{AEP} = N_h \sum_i \left[F(V_i) - F(V_{i-1})\right] \frac{P_{i-1} + P_i}{2}, \qquad F(V) = 1 - \exp\!\left(-\frac{\pi}{4}\left(\frac{V}{\bar V}\right)^2\right) \]

with \(N_h = 8760\) hours. The nacelle anemometer of a SCADA data set is not the met mast of the standard, so the absolute curve carries the anemometer's bias; comparisons between turbines and years remain valid when the anemometer and its transfer function do not change (see the Kelmarsh example).

Reliability

Operating time is the observation time minus every stop: a failure cannot start while the turbine stands.

MTBF \(= T / n\) for \(T\) operating hours and \(n\) failures, with the two-sided chi-square interval of a time-truncated observation at confidence \(1 - a\):

\[ \mathrm{MTBF}_\mathrm{lo} = \frac{2T}{\chi^2_{1-a/2,\;2n+2}} \qquad \mathrm{MTBF}_\mathrm{hi} = \frac{2T}{\chi^2_{a/2,\;2n}} \]

MTTR is the mean time to restore, from the stop to the restart: it includes waiting, travel and logistics, not only hands-on repair time. Its median and 90th percentile are reported too, since restore times are heavily skewed.

Crow-AMSAA (power-law non-homogeneous Poisson process, Crow 1974). The failure intensity at unit age \(t\) is \(u(t) = \lambda \beta t^{\beta - 1}\). Over units observed in windows \((S_j, T_j]\) (late starts and gaps are handled exactly) with failure ages \(t_i\), the log-likelihood is

\[ \ln L = n \ln\lambda + n \ln\beta + (\beta - 1) \sum_i \ln t_i - \lambda \sum_j \left(T_j^\beta - S_j^\beta\right). \]

For fixed \(\beta\) the maximum in \(\lambda\) is \(n / \sum_j (T_j^\beta - S_j^\beta)\); \(\hat\beta\) solves the profile score equation. Bounds on \(\beta\) come from the profile likelihood, the verdict from the likelihood-ratio test of \(\beta = 1\): \(\beta < 1\) failures thin out (reliability grows), \(\beta > 1\) they become more frequent. For one window starting at age 0 the estimate reduces to the textbook \(\hat\beta = n / \sum_i \ln(T / t_i)\).

Laplace test (model-free): under a constant rate each failure time is uniform in its window, and

\[ U = \frac{\sum_i \left(t_i - S_i - L_i / 2\right)}{\sqrt{\sum_i L_i^2 / 12}} \]

is standard normal (\(L_i\) the window length); \(U > 0\) means failures cluster late. It is applied on each unit's observed-time clock (its windows joined end to end), so that it does not depend on the age origin.

References

• Crow, L. H. (1974). Reliability analysis for complex, repairable systems. In Reliability and Biometry, SIAM, 379-410.
• Rausand, M., Høyland, A. (2004). System Reliability Theory, 2nd ed. Wiley (chi-square bounds, Laplace test).
• Tavner, P. (2012). Offshore Wind Turbines: Reliability, Availability and Maintenance. IET.