Correlation of Uncertainties in Wind Measurement Analysis

Worked Examples Feedback Power Curve Working Group, 2016-12-13, Glasgow Axel Albers

Contents Cumulating of uncertainties across site calibration and power curve test Cumulating of uncertainties across wind speed bins Order of uncertainty cumulating 1 www.windguard.de www.windguard.de

Problem of Cumulating Uncertainties Across Site Calibration and Power Curve Test FDIS IEC 61400-12-1, Ed.2: Error in equation E.34: Correlation of uVHW,iand uVT,ineglected Correct: VHW i , VT i , VHW i , V 2 u u u + + = + + 2 2 2 VM 2 AD u u u u , VT i , VHW i , VT i , i , , method i , 0 where mostly Some guidance given in Annexes C.6.2 and E.9 how to treat this: but inconsistent (only appropriate in E.9.2 for anemometer calibration) VHW , VT i , 2 www.windguard.de www.windguard.de

Practical Approach Correlation of entire uncertainty of SC uVT,iand of wind speed measurement at PC uVHW,iis difficult to estimate: - Coming up with an entire uncertainty of SC is useless! Uncertainties are correlated across SC and PC, but uncertainties are not correlated across components Solution: Assess effective uncertainty of wind speed component by component for combined process SC and PC - take full uncertainty of wind speed measurement at turbine position into account as wind speed measure- ment at PC is corrected back to this measurement - take only uncorrelated part of uncertainties of reference position measurements of SC and PC into account as the correlated part basically cancels out in the combined process SC and PC (correlation of uncertainty of this measurement across SC and PC per component can be estimated) 3 www.windguard.de www.windguard.de

Theoretical Background for Practical Approach Wind speed finally applied for PC analysis is approximately: V V V = SC: site calibration PC: power curve test R: measurement at reference position T: measurement at turbine position T _ SC final R _ PC V R _ SC 2 2 2 Error propagation: V V V V R _ PC T _ SC R _ V PC T _ SC = + + 2 final 2 T 2 R 2 R u u u u _ SC _ PC _ SC 2 R V V R _ SC R _ SC _ SC V V R _ PC T _ SC + 2 u u T _ SC , R _ PC T _ SC R _ PC V V R _ SC R _ SC 2 R V V _ V PC T _ SC 2 u u T _ SC , R _ SC T _ SC R _ SC 3 R _ SC V 2 T V R _ V PC _ SC 2 u u R _ PC , R _ SC R _ PC R _ SC 3 R _ SC with approximation: V V u , u , R _ PC R _ SC R _ SC R _ PC T _ SC , R _ SC T _ SC , R _ PC 2 V ( 1 ) T _ SC = + 2 final 2 T 2 R u u 2 u _ SC _ PC R _ PC , R _ SC V R _ SC 4 www.windguard.de www.windguard.de

Why you should Use Equation E.4 of FDIS IEC 61400-12-1, Ed.2 Instead of E.5 E.5 is only an approximation of E.4 and always leads to a higher uncertainty than E.4 E.4 is easy to apply (no reason to use E.5) E.4 provides uncertainties in AEP of each component, useful to identify uncertainty drivers and unimportant factors: v-average [m/s] 4 [m/s] 5 [m/s] 6 [m/s] 7 uncertainty in AEP [m/s] 8 [m/s] 9 [m/s] 10 [m/s] 11 component [-] uV1 uV2 uV3 uV4 uP1 uP2 uP3 uPd uT1 uT2 uT3 uTd uB1 uB2 uBd sP total [% of AEP] 3.71 3.20 0.00 0.50 0.17 0.00 1.78 0.00 0.07 0.72 0.04 0.02 0.04 0.00 0.02 0.48 5.3 [% of AEP] 2.88 2.67 0.00 0.41 0.15 0.00 0.99 0.00 0.07 0.72 0.04 0.02 0.04 0.00 0.02 0.32 4.2 [% of AEP] 2.25 2.16 0.00 0.34 0.13 0.00 0.67 0.00 0.07 0.72 0.04 0.02 0.04 0.00 0.02 0.24 3.3 [% of AEP] 1.79 1.75 0.00 0.27 0.13 0.00 0.51 0.00 0.07 0.72 0.04 0.02 0.04 0.00 0.02 0.20 2.7 [% of AEP] 1.46 1.45 0.00 0.22 0.12 0.00 0.42 0.00 0.07 0.72 0.04 0.02 0.04 0.00 0.02 0.16 2.2 [% of AEP] 1.23 1.22 0.00 0.19 0.12 0.00 0.37 0.00 0.07 0.72 0.04 0.02 0.04 0.00 0.02 0.14 1.9 [% of AEP] 1.06 1.07 0.00 0.17 0.12 0.00 0.34 0.00 0.07 0.72 0.04 0.02 0.04 0.00 0.02 0.12 1.7 [% of AEP] 0.94 0.95 0.00 0.15 0.12 0.00 0.32 0.00 0.07 0.72 0.04 0.02 0.04 0.00 0.02 0.11 1.6 5 www.windguard.de www.windguard.de

Uncertainty Cumulating in General M M = 1 k = 2 total u u u Rules of error propagation: Uncertainties of power curve must be cumulated over: - measurement steps (SC, NTF, PC) - direction bins (site calibration) - measurement heights (REWS) - components = = = = = = = = 1 g 1 h 1 e 1 f 1 c 1 d 1 k 1 l k l l , k = l 1 indices: g,h, number: O indices: f,e, number: P indices: c,d, number: Q indices: k,l, number: R Q Q O O P P R R = 2 i , P u i , k , c , e , g c i , k , c , e , g u i , l , d , f , h c i , l , d , f , h u i , l , k , d , c , f , e , h , g In addition, uncertainty in AEP must be cumulated over: - wind speed bins = = = = = = = = = = 1 g 1 h 1 e 1 f 1 c 1 d 1 k 1 l 1 i 1 j indices: i,j, number: N Q Q O O P P R R N N = 2 AEP 2 h u N f f i , k , c , e , g c i , k , c , e , g u j , l , d , f , h c j , l , d , f , h u i j j , i , l , k , d , c , f , e , h , g Example: O=2, P=6, Q=5, R=20, N=40 gives 48000 sub components and 162729000 correlation coefficients Simplification needed! 6 www.windguard.de www.windguard.de

Uncertainty Cumulating Approach in IEC 61400-12-X Uncertainties in different steps O, different direction bins P and different heights Q are considered as additional components (not explicitly written in IEC 61400-12-X, but implicit assumption): M M = = i , l , k 2 i , P u c u c u (E[F].1) i , k i , k i , l i , l = k 1 l 1 M M N N = = = j , i , l , k 2 AEP 2 h u N f f c u c u (E[F].2) , i j i , k i , k j , l j , l = = k 1 l 1 i 1 j 1 = M R O P Q Not yet simplification! 7 www.windguard.de www.windguard.de

Uncertainty Cumulating Simplifications in IEC 61400-12-X (part 1) Uncertainty components fully uncorrelated across each other 1 , 0 i , k l , k i , k l , k = = = Not correct for same type of component across steps SC, NTF, PC! Special approaches for that in each standard! Only a few components are direction dependent (some SC components), these are considered either fully correlated across direction bins, or fully uncorrelated across direction bins. The correlation of these uncertainties across direction bins is independent on their correlation across wind speed bins and measurement heights. The cumulating of each of these components across direction bins provides a set of directional independent components (but some can be height dependent in case of evaluation of REWS). 8 www.windguard.de www.windguard.de

Uncertainty Cumulating Simplifications in IEC 61400-12-X (part 2) Only a few components are height dependent (from REWS measurement). The correlation of these uncertainties across heights is independent on their correlation across wind direction bins and wind speed bins. The cumulating across heights provides a set of height (and direction) independent components. Aim of simplifications: Set of fully uncorrelated components! M = M R O P Q = 2 i , P 2 i , k 2 i , k u c u (E[F].3 simplified ) k 1 Category A uncertainties fully uncorrelated across wind speed bins, category B uncertainties fully correlated across wind speed bins: M = k = 2 AEP u k , 1 9 www.windguard.de www.windguard.de

Order of Uncertainty Cumulating 1. Over steps SC, NTF, PC (e.g. same wind speed components) - must come first for assessing part cancellation 2. Over directions (is basically independent from rest) 3. Over heights (only in case of REWS) - must come after steps 1. and 2. because of rule given in Annex C.1 of FDIS IEC 61400-12-1, Ed.2 to assess separate site calibration for each height 4. Over wind speeds (only for AEP) 5. Over components 10 www.windguard.de www.windguard.de

Thank you for your attention contact: a.albers@windguard.de 11 www.windguard.de