Inter-Annual Variability and Uncertainty in Wind Farm Annual Energy Production Estimates

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Martina Reeves
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1,2, Jim Salmon 2, 1 York University, 2 Zephyr North

1 Inter-Annual Variability and Uncertainty in Wind Farm Annual Energy Production Estimates Peter Taylor 1,2, Jim Salmon 2, 1 York University, 2 Zephyr North Canada Contact: pat@yorku.ca ICEM, Toulouse, June 2013

2 Uncertainties in the estimated Annual Energy Production (AEP) are crucial factors in assessing the financial viability of a wind farm project. A good estimate of the mean (P50) value is the starting point but finance agencies give at least as much weight to the P90, P95 or P99 estimates of future production, both annual and averaged over multiple years. These are based on uncertainty estimations and the Inter- Annual Variability (IAV) of the wind resource plays an important role. When acquisition of a portfolio of spatially distributed wind farms is under consideration then correlations between the wind speeds at the different farm locations play a role in assessing the overall IAV for the portfolio.

3 A structure is presented for uncertainties in the estimation of annual energy production for wind farms. The typical industry practice of assuming 6% variability in wind IAV is too conservative for many locations. IAV at hub height appears to be lower than the IAV at 10m. At hub height a value of 3% appears to be appropriate but 4% may represent a conservative compromise for most North American locations. For investment in a portfolio of multiple wind farm projects, a less than 100% correlation between winds at different sites will reduce the uncertainties in AEP. A structure for assessing the benefits of the "portfolio effect" is presented and some sample Canadian cases discussed.

4 Fixed Uncertainties a) Initial AEP estimate errors - wind measurement, and flow and wake modelling. Hardware uncertainties (e.g. power curves). Uncertainties that can vary Year to Year b) Operational uncertainties (turbine failures, maintenance, grid connection availability etc.) c) Meteorological factors, icing and inter-annual variability (IAV). Uncertainty is central in the prediction of the secure (P90) multi-year return on investments.

5 Single Wind Farm We denote the long term AEP (P50) by P50. Perturbations from P50 in any given year are denoted by p' with a long term mean of zero. If the p' are normally distributed and if we denote the root mean square of p' over many years as σ, then the 1-year P90 is P(90-1 year) = P σ (1-year) The same relationship would hold for multi-year P90 (Y-year) values with a reduced multi-year standard deviation (assuming each year is independent) as, σ (Y-year) = σ (1-year)/ Y But not all uncertainties will vary from year to year - e.g. the fixed errors such as possible errors in the resource assessment, so...

6 Denoting uncertainties in AEP as u = f + g where the f are fixed uncertainties associated with pre-construction estimates and g are uncertainties which will vary from year to year with zero long term mean, the multi-year uncertainties are assumed to be given as u (Y )= f 2 +g 2 /Y where f and g are not correlated. As an example, consider a case with yearly wind IAV and random technical variability giving a relative production uncertainty (g/p50) of 8% and the relative fixed uncertainties (f/p50), associated with original measurement errors, modelling errors, turbine variability, etc., of 8%. Combined, these give a 1-year relative AEP uncertainty (square root of sum of squares) of 11.3% and a 10- year average AEP uncertainty of 8.4% mostly associated with the potential errors in the pre-construction wind monitoring, MCP analysis, wind farm modelling and turbine performance. IAV impacts will be smoothed out by averaging over 10 years but the fixed uncertainty remains at 8%.

7 Inter-Annual Variability (IAV) Many groups assume wind speed IAV = 6%, based in part on an analysis of 22+ years of annual wind speeds at 11 UK met stations. (6.2% IAV) Raftery A., Tindal J. and Garrad A. (1997), Understanding the risks of financing windfarms, Proc. EWEA Wind Energy Conference, Dublin. and on the EWEA web site, or Fig in the 2009 EWEA book Wind Energy - The Facts ( %)

8 EWEA/GLGH: Wind Map of Europe Inter annual variations. Shown as Standard Deviation as a percentage of mean

9 Inter-Annual Variability (IAV) Effects of trend. Suppose X(t) = X 0 + at + x' for -T/2 < t < T/2 where <x'> = 0, <x' 2 > = s T 2 and averages <..> are over the interval from -T/2 to T/2. Then <X(t)> = X 0, but <(X(t) - X 0 ) 2 > = <(at) 2 > + 2a <tx'> + <x' 2 > We can note that <(at) 2 > = a 2 T 2 /12,for the continuous case. Assuming <tx'> = 0, and using the continuous case result, the IAV computed would be σ T 2 = (s T 2 + a 2 T 2 /12) 1/2

10 Results from Cabauw Values computed from annual mean winds at the Cabauw tower 100m (interpolated) 10m Year Range Mean σ a s s/mean Mean σ a s s/mean N Table 1 Inter-Annual Variability computed from yearly-averaged data from the Cabauw tower. Wind speeds and standard deviations are in m/s. σ is the standard deviation computed directly from the annual averages over the year range indicated. A linear fit to the data gives slopes, a (m/s/year) and s is the standard deviation after trend removal. N is the number of annual data values used. Note that no data were available Data kindly provided by Fred Bosveld (KNMI). Note reductions in s/mean at 100m versus 10m.

11 Inter-Annual Variability (IAV) Length of record: If there is no trend, or trend has been ignored, and if individual annual averages are considered as independent and identically distributed (IID) with a normal distribution then an average over T values would have a variance of σ2/t, where σ is the standard deviation of the complete data set of IID annual averages. If this is regarded as an error relative to the true mean then the computed variance from the limited data set (σt 2 ) would, on average, be reduced relative to the variance (σ2) relative to the long term mean by a factor (1-1/T) since, with sums over i = 1 to T, σ 2 = (1/T) (U i - U ) 2 = (1/T) (U i - U T + U T - U ) 2 = (1/T) {( U i - U T ) 2 + (U T - U ) 2 } since (U i - U T ) = 0

12 If we average over many cases, σ 2 = σ T 2 + σ 2 /T, This implies that, on average, σ = σ T (1-1/T) -1/2 So for an 11 year sample, assuming no trend, or after trend removal, we could increase the computed IAV by a factor For Cabauw 100m data we would have long term IAV estimates of 3.6 and 4.0% over the two 11-year periods for which data are available.

13 IAV at Canadian Surface Stations MSC station wind Speeds m/s Broadview, Weyburn, Regina Indian Hd, Egbert Location Mean SK 4.17 SK 4.91 Apt, SK 5.22 SK 4.85 CS, ON 3.23 σ T σ T /Mean Hamilton, Goderich, Windsor, Wiarton, Mt Forest, Location Mean ON 4.21 ON 4.70 ON 4.27 ON 3.62 ON 3.06 σ T σ T /Mean

14 Wind Speed (m/s) Broadview, SK Weyburn, SK Regina Airport, SK Indian Head, SK Egbert, ON Hamilton Airport, ON Goderich, ON Windsor Airport, ON Wiarton, ON Mount Forest, ON Average (8 stations) Fit 1: Linear Calendar Year

15 IAV in reanalysis data for Canadian locations No Canadian long term tall tower data were available. Alternative sources of wind information are provided by the "re-analysis" data sets from the National Centre for Environmental Prediction (NCEP) or the European Centre for Medium-Range Weather Forecasting (ECMWF). At least two commercial organizations can supply down-scaled, site-specific data sets for potential wind farm sites. The analysis to be presented here uses hourly SERIES data from the Vortex company (Vortex Factoria de Calculs S.L. located in Barcelona, Spain). We use their results based on down-scaling to 3 km resolution using site-centred Weather Research and Forecasting Model (WRF) simulations driven by NCAR-NCEP CFSR reanalysis data. The down-scaled site-specific data cover eleven complete years from January 1, 2001 to Dec 31, 2011, and can be interpolated to hub heights which we will take as 100m.

16 Typical comparison between hourly VORTEX series and tower data. We find that they generally track well but there may be a scaling factor and some slight seasonal variation - which would not significantly impact IAV.

17 1.08 Normalized annual wind speed BC1 AB1 SK 1 SK2 SK3 ON1 ON3 ON5 Average Linear Fit Year Annual Average 100m winds, Vortex SERIES Vortex SERIES annual average wind speeds at 8 Canadian locations. Note expanded scale.

18 Note Standard Deviations = IAV Normalized speeds Location BC1 AB1 SK1 SK2 SK3 ON4 ON1 ON3 ON5 ON2 Hub Height 100m 100m 100m 100m 100m 100m 100m 100m 100m 100m Year Mean Std Dev So IAV < 3% except for BC1 (3.7%)

19 Portfolios If we had production data for N wind farms with individual long term average productions Pi and perturbations in any given year p'i, we would let The long term variance of the total portfolio perturbation could be written as, N p ' 2 =[( i=1 p ' i ) 2 N ]= i=1 N P= i =1 N [ p ' 2 i ]+2 i=1 where [... ] indicates a long term average. In the simple case where all <p'i 2 > are equal to σ 2, the overall standard deviation, p', could vary from N 1/2 σ with no correlation to Nσ if there is 100% correlation. P i N [ p ' i p ' j ] j=i+1

20 Portfolios In terms of estimated uncertainties in AEP, let the uncertainty for each wind farm be u i = f i + w i + t i where the f i are fixed uncertainties, w i are AEP uncertainties caused by the wind IAV and t i are uncertainties expected to vary from year to year associated with technical and similar problems.[... ] indicates a long term average. Then the total estimated, annual AEP uncertainty for the portfolio of N wind farms becomes, N [( i=1 u i ) 2 N ]= i=1 N ([ f 2 i ]+[w 2 i ]+[t 2 i ])+2 i=1 N [w i w j ] j=i+1 assuming independence of all except the wind variability. So we need to look at wind induced AEP uncertainties and their correlations.

21 A few portfolio results To illustrate the correlations we have computed gross AEP for six imaginary wind farm projects at locations across Canada based on Vortex SERIES wind data. The same idealized 2 MW, 100 m hub height turbine is used at each location and each farm is assumed to have the same number of turbines. The locations are organized from West (BC1) to East (ON5). The correlations decrease with increasing separation with a slightly negative correlation between ON5 and BC1. For a portfolio of all 6 wind farms the (normalized) double sum N N < i= 1 j= i+ 1 w i w j 6.34 can be compared to a sum of 15 (from (N-i) summed from 1 to N) if there were correlations of 1.0 between each farm. > =

22 AEP cross-correlations CROSS CORRELATIONS Location BC1 AB1 SK1 ON1 ON3 ON5 BC AB SK ON ON ON Cross-correlations of AEP perturbations from six imaginary wind farm projects across Canada. All calculations based on Vortex SERIES hourly wind data at locations indicated with a single,representative wind turbine and a generic 2MW power curve. Locations are from West to East. Orange and Yellow highlights indicate smaller portfolios.

23 Conclusions A structure is presented for uncertainties in the estimation of annual energy production for wind farms. Inter-annual variability in the average hub-height wind speed plays a significant part in the uncertainty and it is argued that the typical industry practice of assuming 6% variability is too conservative for many locations. IAV at hub height appears to be less than at 10 m. A hub height value of 3% may be more realistic - at least for Canada. Site specific estimation based on reanalysis data is also a viable method. For investment in a portfolio of multiple wind farm projects, a less than 100% correlation between winds at different sites will reduce the uncertainties in total AEP. A structure for assessing the benefits of the "portfolio effect" is presented and some sample cases discussed. Acknowledgements We are grateful to a number of Zephyr North clients for permission to make use of their Vortex SERIES data for part of this research.

A PRESENTATION BY THE AMERICAN ACADEMY OF ACTUARIES TO THE NAIC S CLIMATE CHANGE AND GLOBAL WARMING (C) WORKING GROUP

A PRESENTATION BY THE AMERICAN ACADEMY OF ACTUARIES TO THE NAIC S CLIMATE CHANGE AND GLOBAL WARMING (C) WORKING GROUP MARCH 24, 2018 MILWAUKEE, WISCONSIN COPYRIGHT 2018 2018 American Academy of Actuaries.