Power curve evaluation from tests on multiple turbinesAxel Albers, Jochen Cleve

Vindkraftnet, 9. April 2018

Agenda

▪ The problem of averaging multiple power curves and annex R of IEC 61400-12-1

▪ Three methods to average power curves

▪ Examples from power curve tests using nacelle lidars

▪ Conclusion

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Annex R of IEC 61400-12-1 Ed. 2 is ‘underdeveloped’

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Annex R states:

➢ Average AEP as average over individual test AEPs

➢ Reference to IEC 61400-12-2 (nacelle power curves) Annex I&J

➢ Uncertainty as simple average over individual uncertaintiesOR by means of error propagation (but method not specified).

➢ Correlation between uncertainty components mentioned and table given, but …

Shortcomings of Annex R

– No clear recommendation which method to use for averaging power curves

– Texts mentions ‘… with minor adjustments in the handling of correlation’ but doesn’t explain these adjustments

– Table R.1 of correlation is incomplete

– Editorial errors

=> Ørsted asked Deutsche Windguard to work out a correct way to calculate the uncertainties of multiple power curve tests and include the use of nacelle lidars.

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Method 1: simple average of AEP & error propagation for uncertainties

𝐴𝐸𝑃AVG =

𝑚=1

𝐿1

𝐿𝐴𝐸𝑃𝑚

𝑢AEP,𝑘AVG =1

𝐿2

𝑚=1

𝐿

𝑛=1

𝐿

𝑢AEP,𝑚,𝑘𝑢AEP,𝑛,𝑘𝜌𝑚,𝑛,𝑘

𝑢𝐴𝐸𝑃AVE =

𝑘=1

𝑀

𝑢AEP,𝑘AVE2

These equations are basically the sophisticated option described in IEC – correlations 𝜌𝑚,𝑛,𝑘 have to be worked

out properly.

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M is the total number of uncertainty

components, containing both

category A and B uncertainties.

Method 2: optimal weighting of power curves in terms of AEP

More accurate measurements should get higher weight on average!

𝐴𝐸𝑃weighted =

𝑚=1

𝐿

𝑡𝑚 𝐴𝐸𝑃𝑚

𝑚=1

𝐿

𝑡𝑚 = 1

𝐴𝐸𝑃weighted: weighted average annual energy production

𝐴𝐸𝑃𝑚: measured annual energy production of the m-th test turbine

𝑡𝑚: weighting factor of the m-th test turbine

L: total number of power curve tests

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Uncertainty of optimally weighted AEP

The uncertainty is calculated by error propagation of the relative uncertainties of the single power curve tests.

𝑢AEP,𝑘weighted =

𝑚=1

𝐿

𝑛=1

𝐿

𝑡𝑚𝑢AEP,𝑚,𝑘𝑡𝑛𝑢AEP,𝑛,𝑘𝜌𝑚,𝑛,𝑘

𝑢𝐴𝐸𝑃weighted =

𝑘=1

𝑀

𝑢AEP,𝑘weighted2

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Weighting factors 𝑡𝑚 are determined from minimising the total uncertainty in AEP

– 𝑢𝐴𝐸𝑃weighted 𝑡1, … , 𝑡𝐿 = 𝑢𝐴𝐸𝑃weighted = σ𝑘=1𝑀 𝑢AEP,𝑘weighted

2

– minimise 𝑢𝐴𝐸𝑃weighted with boundary condition σ𝑚=1𝐿 𝑡𝑚 = 1

– minimisation problem is robust, the results converge quickly.

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Method 3: direct power curve averaging and weighting

Weighting and averaging of measured power and wind speed is performed in each wind speed bin:

𝑃weighted,i =

𝑚=1

𝐿

𝑡𝑚,𝑖 𝑃𝑚,𝑖

vweighted,i =

𝑚=1

𝐿

𝑡𝑚,𝑖 v𝑚,𝑖

𝑢𝑘,𝑖weighted =

𝑚=1

𝐿

𝑛=1

𝐿

𝑡𝑚,𝑖𝑢𝑚,𝑘,𝑖𝑡𝑛,𝑖𝑢𝑛,𝑘,𝑖𝜌𝑚,𝑛,𝑘,𝑖

𝑢𝑖weighted =

𝑘=1

𝑀

𝑢𝑘,𝑖weighted2

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𝑚=1

𝐿

𝑡𝑚,𝑖 = 1

Uncertainty component k:

Uncertainty in wind speed bin i:

Example of combining three power curve tests

Overview of power curve campaigns

– Three power curve tests have been conducted at undisclosed locations on an undisclosed turbine type

– Offshore wind farms

– Wind measured with nacelle lidars

– Measured power curves are close to each other (no outliers)

– Only changes in uncertainty will be shown here

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Reduction of uncertainty by combining three power curves

Mean annual wind speed

(Rayleigh distr.)

Reduction weighted

uncertainty compared to

average uncertainty

Reduction uncertainty optimal

weighting compared to equal

weighting

[m/s] [%] [%]

6.0 -11.2 -0.6

7.5 -10.5 -0.5

9.0 -9.8 -0.3

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• Error propagation of uncertainties leads to ~10% reduction in uncertainty compared to average uncertainty

• Optimal weighting yields minor improvement compared to equal weighting

Comparison of uncertainties for direct power curve averaging

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– Direct weighting of power curvesenables to weigh individually in each wind speed bin.

– Additional advantage: problem of different wind range coverageis avoided.

Reduction of uncertainty with direct weighting of power curves

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– Uncertainty reduction is also in the orderof 10%.

– At lower wind speeds a higheruncertainty reduction is achieved.

– Optimal weighting compared to equalweighting gives minor improvement of uncertainty.

Reduction in uncertainty by direct power curve weighting

Mean annual wind speed

(Rayleigh distr.)

Reduction in uncertainty direct

power curve weighting

compared to AEP weighting

[m/s] [%]

6.0 -0.2

7.5 -0.8

9.0 -1.7

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Direct power curve weighting is a bit more complicated but leads to a (small) reduction in total uncertainty compared to AEP weighting.

Topics not covered by this presentation

– Problem of different wind speed ranges covered by different tests

– Discussion of individual uncertainty components and their correlation

– Sampling uncertainty which is mentioned in Annex I&J of IEC 61400-12-2

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Conclusions

– With three tests a reduction of about 10% in uncertainty can be achieved.

– A weighted average of the power curve / AEP yields a slightly more accurate mean AEP / power curve.

– Direct power curve weighting offers only slight benefits over weighted AEP averaging but it is the most consistent approach.

– IEC should not allow to use a simple average of the uncertainty because the chief benefit from doing multiple tests would not be realised

– … and should state more clear recommendation, e.g. to always use weighted power curve averaging.

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