observed and predicted wind speed time series in the ... · system due to wind energy and the...
TRANSCRIPT
ECN-C--06-007
Observed and predicted wind speedtime series in the Netherlands and
the North SeaWork Package 2 We@Sea project
System Integration and Balance Control
A.J. Brand
2 ECN-C--06-007
Acknowledgement This work was performed in the framework of Work Packages 2a, 2b and 2c of the We@Sea project "Systeemintegratie en balans-handhaving bij grootschalige windenergie op zee (Si&Bh)" (System Integration and Balance Control in the case of Large-scale Offshore Wind Energy); We@Sea/BSIK 2004-010; ECN 7.9428.
Abstract This report presents the wind data for a study on the power variations in the Dutch electricity system due to wind energy and the measures to mitigate these variations. These wind data con-sist of:
• A representative offshore wind regime and an estimate of wind speed forecasting accu-racy, and
• Simultaneous time series of 15-minute averaged observed and predicted wind speed in 5 locations in the North Sea and in 2 locations in the Netherlands.
Also presented are the transformations from sensor height to evaluation height and from 10-minute periods to 15-minute periods that were used to create the observed wind data from the measured wind data. The time series are applied in an analysis of wind speed variations on different scales (and the resulting wind power variations) and an analysis of wind speed forecasting errors.
ECN-C--06-007 3
Contents
List of tables 5
List of figures 5
1. Introduction 7 1.1 Project objective and expected results 7 1.2 Work breakdown and ECN tasks 7 1.3 Objectives and methodology of this study 8 1.4 Overview of this report 9 1.5 Related reports 10
2. Representative wind regime and accuracy of wind speed forecasts 11 2.1 Motivation and outline 11 2.2 Representative wind regime 11 2.3 Accuracy of wind speed forecasts 11
3. Observed and predicted wind speed time series 13 3.1 Motivation and outline 13 3.2 Observed wind speed 13
3.2.1 Measured wind data 13 3.2.2 Estimated wind data 13
3.3 Predicted wind speed 14 3.4 Simultaneous wind speed time series 14
3.4.1 Created and delivered time series 14 3.4.2 Partial load sub-sets 14 3.4.3 Storm sub-sets 16
4. Wind speed variations on different scales and wind speed forecasting errors 19 4.1 Outline and context 19 4.2 Wind speed variation on different scales 19
4.2.1 Wind speed variation in 15-minute periods 19 4.2.2 Wind speed variation between 15-minute periods 19 4.2.3 Wind speed persistence 21 4.2.4 Conclusions on wind speed variations on different scales 23
4.3 Wind speed forecasting error 24 4.3.1 Systematic and stochastic forecasting error per location 24 4.3.2 Systematic and stochastic forecasting error for the ensemble 24 4.3.3 Cross-correlation between locations 26 4.3.4 Auto-correlation per location 26 4.3.5 Impact of wind speed 27 4.3.6 Impact of lead-time 29 4.3.7 Conclusions on wind speed forecasting error 30
5. Recapitulation 31
References 32
Appendix A Transformation of wind speed from sensor height to evaluation height 33 A.1 Method 33 A.2 Approach 34
A.2.1 Average wind speed profile 34 A.2.2 Wind speed standard deviation 35 A.2.3 Accuracy of the average wind speed 37 A.2.4 Accuracy of the wind speed standard deviation 39 A.2.5 Comparison of the two height transformation methods 39
4 ECN-C--06-007
Appendix B Transformation of wind speed from 10-minute periods to 15-minute periods 42
B.1 Method 42 B.2 Approach 42
B.2.1 Averages 42 B.2.2 Standard deviations 43
B.3 Accuracy 44 B.3.1 Averages 44 B.3.2 Standard deviations 45
ECN-C--06-007 5
List of tables
Table 1.1 Work breakdown 8 Table 2.1 Representative wind regime in a Dutch offshore location 11 Table 2.2 Stochastic forecasting error as a function of wind speed interval and lead-time;
indicative values. Bold values may be used in a Normal distribution. Underlined values are below the minimal lead-time of a weather prediction model and are valid for persistence based forecasts only 12
Table 3.1 Characteristics of the wind mast locations in the North Sea and the Netherlands 13 Table 4.1 Indicative occurrence of wind power regimes with a duration of 15 and 240
minutes in the case of a solitary turbine or farm and a cluster of turbines/farms 22 Table 4.2 Systematic and stochastic wind speed forecasting errors per location 24 Table 4.3 Systematic and stochastic wind speed forecasting errors of the ensemble
averaged wind speed 24 Table A.1 The relative magnitude of the terms in the height transformation of the average
wind speed, and the magnitude of three typical errors in the transformed average wind speed if zh = 70 m 36
Table A.2 The differences in the outcome of the two methods 41
List of figures
Figure 3.1 The wind mast locations 15 Figure 3.2 Wind speed in location F3, sub-period Storm Ramp-up 17 Figure 3.3 Ensemble averaged wind speed for the 7 locations, sub-period Storm Ramp-up 17 Figure 3.4 Wind speed in location F3, sub-period Storm Ramp-down 18 Figure 3.5 Ensemble averaged wind speed for the 7 locations, sub-period Storm Ramp-
down 18 Figure 4.1 Probability of the wind speed standard deviation in 15-minute periods 20 Figure 4.2 Probability of an absolute wind speed change between two consecutive 15-
minute periods 20 Figure 4.3 Probability of an absolute wind speed change between two 15-minute periods
separated by a 2-hour interval 21 Figure 4.4 Probability of the wind speed in single location being larger than a given value
during an indicated period 22 Figure 4.5 Probability of the wind speed in an ensemble of locations being larger than a
given value during an indicated period 23 Figure 4.6 Observed versus forecasted ensemble averaged wind speed 25 Figure 4.7 Observed versus forecasted wind speed in location F3 25 Figure 4.8 Cross-correlation between the wind speed forecasting error as a function of the
distance between locations 26 Figure 4.9 Auto-correlation of the wind speed forecasting error for the 7 locations 27 Figure 4.10 Systematic and stochastic wind speed forecasting error in location F3 as a
function of forecasted wind speed 28 Figure 4.11 Systematic and stochastic wind speed forecasting error in location EWTW1 as a
function of forecasted wind speed 28 Figure 4.12 Systematic and stochastic wind speed forecasting error in location F3 as a
function of lead time 29 Figure 4.13 Systematic and stochastic wind speed forecasting error in location EWTW1 as a
function of lead time 30
6 ECN-C--06-007
Figure A.1 The friction velocity as determined by the two methods 40 Figure A.2 The wind speed average at 70 meter as determined by the two methods 40 Figure A.3 The wind speed standard deviation at 70 meter as determined by the two
methods 41
ECN-C--06-007 7
1. Introduction
1.1 Project objective and expected results The objective of the We@Sea project1 "Systeemintegratie en balanshandhaving bij grootschalige windenergie op zee (Si&Bh)" (System Integration and Balance Control in the case of Large-scale Offshore Wind Energy) is to contribute options for an optimal design of an im-balance market in the Netherlands which deals with large-scale wind energy. Embarking from scenarios for the development of wind power and other plant (Work Package 1), wind power variations on different scales and their impact on balancing are addressed (WP 2). Next, mitigat-ing measures (fast responding production units, wind energy storage, short-term prediction of wind power, optimal wind farm layout and wind farm control strategies) are assessed and for-eign experience is reported (WP 3). Finally, the power scenarios, the potential impact of wind power on balancing and the mitigating measures are integrated in a matching analysis and in an analysis of the cost of imbalance (WP 4). The outcome of these analyses will indicate if and to what extent the current design of the imbalance market in the Netherlands has to be modified. The expected results of the project Si&Bh are:
• Insight in the flexibility and the variability of electricity production on basis of different scenarios for the development of (wind) power plant in the Netherlands until the year 2020.
• An assessment of the impact of large-scale wind power on the flexibility of the other plant.
• Identification of mitigating measures, their potential to reach the required flexibility and the co-lateral effects (e.g. cost, environment, ...).
• An overview of international experience with a high penetration of wind energy. • Insight in the cost of imbalance due to wind power. • Methods for risk (e.g. cost) management in system balancing. • Recommendations and arguments to be used in a discussion on the design of an imbal-
ance market in the Netherlands which deals with large-scale wind power.
1.2 Work breakdown and ECN tasks The work in the project Si&Bh is broken down into various work sub-packages and is distrib-uted over Ecofys (applicant and project coordinator), Tennet (applicant), Delft University, ECN and KEMA; see table 1.1. In contrast to what this breakdown suggests, ECN's work in WP 2a, 2b and 2c does not consist of determining wind power variations but consists of creating wind input for the simulations (to be performed by KEMA) that will give the wind power variations. Since the wind data are defined on basis of the research request by KEMA, these include obser-vations as well as forecasts. As a consequence the data to be employed in WP 2e (postponed to 2006) and WP 3c (scheduled for 2006) became available by December 2005.
1 Project We@Sea/BSIK 2004-010
8 ECN-C--06-007
Table 1.1 Work breakdown
1.3 Objectives and methodology of this study The objective of this study is to obtain wind input for the simulation of onshore and offshore wind power in the Netherlands2. These wind input include general as well as specific informa-tion. The general wind input consists of a representative wind regime and an estimate of wind speed forecasting accuracy. Both the wind regime and the accuracy estimate are obtained from avail-able information, viz. the Offshore Wind Atlas, own experience with wind speed forecasts and literature. The specific wind input consists of time series that have to be created for this purpose. Since the simulations will be used in order to investigate wind power variations on different scales and their impact on balancing, the requirements are as follows:
• Time series with observed wind speed averages, which subsequently will be employed to determine observed wind power averages.
• Time series with predicted wind speed averages, which subsequently will be used in or-der to determine predicted wind power and, together with the observed wind power, the imbalance.
• Averaging period 15-minutes; corresponding to the time period of the electricity sector (the so-called Programme Time Unit).
• Total period such that it contains zero-load events at low as well as high wind speed, partial load events and full load events.
• Various locations covering the spatial distribution of wind turbines and wind farms.
2 De Boer, 2005a
W P 1 : S c e n a r i o ’ s e l e k t r i c i t e i t s s e c to ra . i n k a a r t b r e n g e n h u id ig p r o d u c t i e p a r k K E M Ab . o n tw ik k e l e n s c e n a r io ’ s t a v p r o d u c t i e t o t 2 0 2 0 K E M A , T U D e lf tc . o n t w ik k e le n s c e n a r io ’ s t a v o n t w ik k e l i n g w in d v e r m o g e n t o t 2 0 2 0 K E M A , T U D e lf t
W P 2 : V a r i a t i e s a l s g e v o lg v a n in p a s s in g w i n d e n e r g i ea . I n k a a r t b r e n g e n w in d v e r m o g e n -v a r i a t i e s b in n e n P T E E C N , T e n n e Tb . I n k a a r t b r e n g e n b in n e n u u r s e w in d v e r m o g e n -v a r i a t i e s E C N , E c o f y s , T U D e lf tc . I n k a a r t b r e n g e n l a n g e r e - t e r m i j n w in d v e r m o g e n -v a r i a t i e s E c o fy s , E C N , T U D e lf td . E f f e c t o p b a l a n s h u i s h o u d in g K E M Ae . O n d e r z o e k n u t v e r m o g e n s v o o r s p e l l e r v o o r l a n d e l i j k e n e tb e h e e r d e r ( o p t i e ) E C N , T U D e lf t
W P 3 : M id d e le n v o o r b a l a n s h a n d h a v i n ga . S n e l i n z e t b a r e g e n e r a t i e -u n i t s K E M A , T U D e lf tb . O p s l a g w i n d e n e r g i e K E M A , E c o f y s , T U D e lf tc . K o r t e - t e r m i j n w in d v o o r s p e l l i n g e n . E c o fy s , E C N , T U D e lf td . O p t i m a l i s a t i e v a n p a r k l a y -o u t E C Ne . R e g e l s t r a t e g i e ë n v o o r w in d p a r k e n ( in c l . k o s t e n e f f e c t e n ) E C N , E c o f y s , K E M A , T U D e lf tf . I n t e r n a t io n a l e e r v a r in g e n E c o fy s , K E M A
W P 4 : A n a ly s e e n o p lo s s in g s r i c h t i n g e na . M a te v a n m a tc h i n g A lle nb . O n b a la n s k o s t e n A lle nc . A a n p a s s e n s p e l r e g e l s ? A lle nd . O p lo s s in g s r i c h t in g e n A lle ne . A a n b e v e l i n g A lle n
M a n a g e m e n t e n c o ö r d in a t i e E c o fy s
ECN-C--06-007 9
There are 3 methods to create the "observed" dataset, viz:
1) By using measured wind time series, 2) By using synthesized data wind time series (e.g. Doherty, 2004), and 3) By using a combination of measured and synthesized wind time series (e.g. Giebel
2000, Holltinen, 2004 or Nørgård, 2004). In order to correctly account for the random nature (in particular the spatial cross-correlations) of the wind in the area under consideration, it was decided to employ measured wind time series in this study3. Unlike the alternatives this method does not require models (and hence assump-tions) that treat the regional averaging and the cross-correlation of the wind speed. In addition, to resolve the Programme Time Unit, 10-minute averaged wind data is required4. There are 2 methods to create the "predicted" dataset, viz:
1) By using real wind forecasts, and 2) By using synthesized wind forecasts (e.g. Nørgård, 2004).
In order to correctly account for the limitations and the uncertainties in predicted wind speed it was decided to use real wind forecasts in this study, in this case originating from the ECN wind power forecasting method AVDE5. Unlike the alternative this method does not require assump-tions on the distribution, correlation and aging of wind speed forecasting errors. In order to cover all load events the total period must contain wind speeds in the range from 0 to about 30 m/s, which warrants a total period of at least one year. The period from 1st June 2004 thru 31st May 2005 was selected because by the time this study was started it was the most re-cent period. The area where wind turbines and wind farms may be placed consists of the Netherlands and the Netherlands Exclusive Economic Zone. The locations with wind data evidently must cover this area. A scan of potential locations with 10-minute measured wind data yielded 5 offshore loca-tions (NSW, FINO1, Europlatform, K13 and F3) and 2 onshore locations (EWTW1 and Ca-bauw).
1.4 Overview of this report This report presents the wind data prepared by ECN under the umbrella of WP2abc, subse-quently to be employed in the WP2 studies on the power variations due to wind energy in the Dutch electricity system and in the WP3 studies on the measures to mitigate these variations.
• Chapter 2 contains the descriptions of a representative wind regime and an estimate of wind speed forecasting accuracy.
• Chapter 3 introduces the simultaneous time series of observed and predicted wind speed in 7 locations in the North Sea and the Netherlands.
• Chapter 4 contains analyses of wind speed variation on different scales (including the resulting wind power variations) and wind speed forecasting errors.
3 Note these wind data must be transformed from the measurement height (or sensor height) to the evaluation height (or wind turbine hub height) 4 Note these 10-minute data have to be transformed into 15-minute data 5 Brand, 2003
10 ECN-C--06-007
The appendices describe the transformations that were developed in order to create observed wind data from measured wind data:
• Appendix A presents the transformation from sensor height to evaluation height, and • Appendix B the transformation from 10-minute periods to 15-minute periods.
1.5 Related reports Separate reports will present the subsequent application of the wind data, which includes the studies on:
• The variation of wind power on different scales (KEMA, WP 2abc)6, • The impact of wind power on balancing (KEMA, WP 2d)7, • Short-term wind power prediction as a tool to be used by the Transmission System Op-
erator (ECN, WP 2e)8, and • Wind power prediction as one of the measures to mitigate power variations (ECN, WP
3c)9.
6 de Boer, 2005b 7 de Boer, 2005b 8 To appear in 2006 9 To appear in 2006
ECN-C--06-007 11
2. Representative wind regime and accuracy of wind speed fore-casts
2.1 Motivation and outline In this chapter a representative wind regime (section 2.2) and an estimate of the accuracy of wind speed forecasts (section 2.3) are presented. Guideline to this information was the request for wind data by KEMA10.
2.2 Representative wind regime The representative wind regime in an offshore location is presented in table 2.1 in the form of the occurrence (number of hours per year) of 6 wind speed intervals and 12 wind directions sec-tors.
Table 2.1 Representative wind regime in a Dutch offshore location Regime Wind
speed [m/s]
Hours per wind regime [m/s] and per wind direction sector [deg]
-15 ... +15
15 ... 45
45 ... 75
75 ... 105
105 ... 135
135 ... 165
165 ... 195
195 ... 225
225 ... 255
255 ... 285
285 ... 315
315 ... 345 omni
Zero load, low wind speed ws < 3 37 31 45 55 53 58 61 34 37 40 38 51 540 Cut-in 3 <=
ws < 4 29 31 37 37 49 43 57 37 33 30 28 41 452 Partial load 4 <=
ws < 14 391 427 374 449 591 431 802 917 675 453 375 455 6339
Full load 14 <= ws < 24 93 28 26 133 35 37 78 288 299 201 114 40 1371
Cut-out 24 <= ws < 25 1 0 0 3 0 0 0 2 5 6 2 0 18
Zero load, high wind speed
25 <= ws 1 0 0 6 0 0 0 2 11 16 4 0 40
552 517 482 683 727 569 999 1279 1060 745 561 587 8760
These data originate from the wind distribution at 60 meter above mean sea level of Location C (Longitude 3o40' East and Latitude 53o12' North) in the Wind Atlas of the Netherlands Exclu-sive Economic Zone11.
2.3 Accuracy of wind speed forecasts The accuracy of local wind speed forecasts is expressed in terms of the systematic forecasting error m V and the stochastic forecasting error s V:
10 de Boer, 2005a 11 Brand, 2004
12 ECN-C--06-007
[ ]∑=
=N
1iV
iVN1
m and [ ]( )2
∑=
−=N
1iV
2V miV
N1
s .
Here V[i] = Vrea[i] – Vfrc[i] is the wind speed forecasting error12 at time i, that is the difference between the observed and the forecasted value. For wind speeds between 4 m/s and 14 m/s m V and s V are the parameters of a Normal distribution. For other wind speeds the forecasting error is asymmetric about the mean. Usually the absolute value of the systematic forecasting error is less than 0.5 m/s (which value may be obtained after correlating observations to forecasts) and may be neglected13. Usually the stochastic forecasting error is larger than 1 m/s but smaller than 3 m/s, and depends on the lead-time14. (Lead-time is the period looked ahead.) Indicative values are presented in ta-ble 2.2.
Table 2.2 Stochastic forecasting error as a function of wind speed interval and lead-time; indicative values. Bold values may be used in a Normal distribution. Underlined values are below the minimal lead-time of a weather prediction model and are valid for persistence based forecasts15 only
Regime Wind speed
[m/s] Stochastic forecasting error per wind regime [m/s] and per lead time [h]
+24 +12 +8 +4 +2 +1 Zero load, low wind speed ws < 3
- - - - - -
Cut-in 3 <= ws < 4 - - - - - - Partial load 4 <= ws < 14 2.1 1.9 1.8 1.75 1.725 1.7 Full load 14 <= ws < 24 2.1 1.9 1.8 1.75 1.725 1.7 Cut-out 24 <= ws < 25 2.1 1.9 1.8 1.75 1.725 1.7 Zero load, high wind speed 25 <= ws
2.1 1.9 1.8 1.75 1.725 1.7
These numbers originate from ECN's past experience with wind speed forecasts and from the literature16, and are collaborated by the analysis in section 4.2. The forecasting errors in an ensemble of locations are correlated. The analysis in section 4.2 shows that the decay constant is about 165 km if an exponential decay of correlation with dis-tance between two locations is assumed.
12 As a rule of thumb, for wind speeds between cut-in (~ 3 m/s) and nominal (~ 13 m/s) a wind speed forecasting er-ror of 1 m/s yields a wind power forecasting error of 10% of the nominal power 13 Lange, 2004 14 Lange, 2004 15 In a persistence based forecast the predicted value is equal to the observed value 16 Lange, 2004
ECN-C--06-007 13
3. Observed and predicted wind speed time series
3.1 Motivation and outline This chapter presents the simultaneous time series of observed and predicted wind speed in 7 locations that were created on basis of KEMA's request for wind data. First, it is explained how the observed 15-minute data at hub height is created from measured 10-minute data at sensor height (section 3.2). Next follows a brief description of the creation of the predicted 15-minute wind data (section 3.3). Finally an overview is presented of the created series and sub-series and to whom these were delivered (section 3.4).
3.2 Observed wind speed
3.2.1 Measured wind data Wind data measured at 7 locations (figure 3.1) are available for the period 1st June 2004 thru 31st May 2005 Universal Time. For 4 sites these data include continuous time series of 10-minute average μu(zs) and 10-minute standard deviation σu(zs) of the wind speed, where zs is the sensor height. For 3 sites μu(zs) only is available. Table 3.1 presents the characteristics of the lo-cations.
Table 3.1 Characteristics of the wind mast locations in the North Sea and the Netherlands Coordinates Sensor
height zs Average stability length Lesti
Ten-minute wind speed av-erage μu
Ten-minute wind speed std. dev. σu
Longitude E Latitude N m m m/s m/s NSW 04o25' 52o37' 70 1.042 yes yes FINO1 06o35' 54o00' 60 809 yes no Europlatform 03o17' 52o00' 29 945 yes no K13 03o13' 53o13' 74 742 yes no F3 04o44' 54o51' 59 712 yes yes EWTW1 05o05' 52o49' 70 1.200 yes yes Cabauw 04o56' 51o58' 60 1.200 yes yes
3.2.2 Estimated wind data Continuous 15-minute average μu(zh) and 15-minute standard deviation σu(zh) of the wind speed at evaluation height zh are created from the 10-minute values at sensor height. This warrants two transformations of the measured wind data:
• From sensor height to evaluation height, in this case 80 and 100 meter17, and • From 10-minute periods to 15 minute periods, for data at sensor and evaluation height.
17 Rule of thumb: Offshore hub height H = 0.5 D + 20, where D is the rotor diameter. Diameters ranging from 90 m (e.g. Vestas V90) to 120 m (e.g. Vestas V120) will require hub heights between 65 and 80 m
14 ECN-C--06-007
These transformations are described in appendix A and appendix B, respectively.
3.3 Predicted wind speed Predicted wind data in the 7 locations are available for the period June 2004 thru May 2005 Universal Time. These data consist of 15-minute average μu(zs) and 15-minute standard devia-tion σu(zs) of the wind speed at sensor height zs, as created with the ECN wind power forecast-ing method AVDE18. The forecasts are day-ahead forecasts where the wind forecast in a given day originates from the Hirlam run 06 of the preceding day. This implies a forecast contains lead-times between 16 and 41 hours. Since usually run 06 is issued well before noon, a forecast fits to an Electricity-programme19.
3.4 Simultaneous wind speed time series
3.4.1 Created and delivered time series The time series with simultaneous observed and predicted wind speed starts 1st June 2004 pe-riod 2:00-2:15 hour and ends 1st June 2005 period 1:45-2:00 hour Local Time. The full time se-ries was delivered to Delft University. A sub-set was delivered to KEMA, containing the sub-series Partial Load #1 (19th March 2005, 4 days), Partial Load #2 (30th September 2004, 4 days), Storm Ramp-up (10th November 2004, 8 days) and Storm Ramp-down (24th October 2004, 4 days) for all locations except NSW1.
3.4.2 Partial load sub-sets The selection criterion is: 4 m/s < μu[i] < 13 m/s (15-minute averaged wind speed at time i) si-multaneously in at least 6 locations, and 4 m/s < μu,ensemble[i] < 13 m/s (15-minute ensemble av-eraged wind speed at time i). The following periods, sorted on duration, were identified: Partial load #1: Begin: i = 28016; date 2005/3/19 period 19:45-20:00 hour Local Time End: i = 28186; date 2005/3/21 period 14:15-14:30 h LT Duration: 42.75 hours Partial load #2: Begin: i = 11714; date 2004/10/1 period 0:15-0:30 h LT End: i = 11862; date 2004/10/2 period 13:15-13:30 h LT Duration: 37 hours Partial load #3: Begin: i = 18036; date 2004/12/5 period 02:45-21:00 h LT End: i = 18183; date 2004/12/7 period 9:30-9:45 h LT Duration: 36.75 hours
18 Brand, 2003 19 An Electricity-programme (E-programme) contains the expected demand and supply per Program Time Unit of 15 minutes, to be submitted to the TSO Tennet by a Programme Responsible Party a day in advance at noon
ECN-C--06-007 15
Figure 3.1 The wind mast locations
16 ECN-C--06-007
Partial load #4: Begin: i = 31575; date 2005/4/26 period 0:00-0:15 h LT End: i = 31718; date 2005/4/27 period 3:15-3:30 h LT Duration: 36 hours Partial load #5: Begin: i = 3757oftewel datum 2004/7/10 period 3:00-3:15 h LT End: i = 3893; date 2004/7/11 period 13:00-13:15 h LT Duration: 34.25 hours Partial load #6: Begin: i = 18401; date 2004/12/9 period 16:00-16:15 h LT End: i = 18527; date 2004/12/10 period 23:30-23:45 h LT Duration: 31.75 hours Partial load #7: Begin: i = 11402; date 2004/9/27 period 18:15-18:30 h LT End: i = 11514; date 2004/9/28 period 22:15-22:30 h LT Duration: 28.25 hours An actual sub-set is created by adding 12 hours of data before the first and after the last record meeting the criterion, and expanding to a whole number of days.
3.4.3 Storm sub-sets Storm Ramp-up: Begin: i = 15670; date 2004/11/11 period 04:30-04:45 h LT End: i = 16206; date 2004/11/16 period 19:15-19:30 h LT The wind speed in location F3 and the ensemble averaged wind speed are shown in figure 3.2 and 3.3, respectively. In location F3 the wind speed first increases from 3 m/s to 20 m/s, next decreases (but remains greater than 8 m/s) again increases (to 20 m/s), and eventually decreases to 1 m/s. In some quar-ters the wind speed is larger than 20 m/s. The same patterns occur in the ensemble averaged wind speed, though with a smaller amplitude. In location F3 the ramp-up occurs early in the period and consists of a wind speed increase with 15 m/s in 14 hours. During this period the difference from one 15-minute period to the other may be as large as 0.8 m/s. During the ramp-up the ensemble averaged wind speed increases from 4 m/s to 15 m/s in 20 hours.
ECN-C--06-007 17
Storm single location (F3)
0
5
10
15
20
25
30
0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528
15-minute period from 11 Nov 2004 04:30 LT
Win
d sp
eed
[m/s
]
Figure 3.2 Wind speed in location F3, sub-period Storm Ramp-up
Storm ensemble (7 locations)
0
5
10
15
20
25
30
0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528
15-minute period from 11 Nov 2004 04:30 LT
Win
d sp
eed
[m/s
]
Figure 3.3 Ensemble averaged wind speed for the 7 locations, sub-period Storm Ramp-up
Storm Ramp-down: Begin: i = 14030; date 2004/10/25 period 03:15-03:30 h LT End: i = 14190; date 2004/10/26 period 19:15-19:30 h LT The wind speed in location F3 and the ensemble averaged wind speed are shown in figure 3.4 and 3.5, respectively. In location F3 the wind speed decreases from 25+ m/s to 2 m/s; in a sub-period even with 12 m/s in 6 hours. Note that just before start of the ramp-down in several quarters the wind speed is lar-ger than the common cut-out wind speed 25 m/s; and that wind turbines may restart when the wind speed has been below 20 m/s. During the ramp-down the ensemble averaged wind speed decreases from 19 m/s tot 4 m/s. (Gaps originate from missing data from 1 location.) Note the gradual decrease in comparison with the decrease in F3.
18 ECN-C--06-007
Storm single location (F3)
0
5
10
15
20
25
30
0 24 48 72 96 120 144
15-minute period from 25 Oct 2004 03:15 LT
Win
d sp
eed
[m/s
]
Figure 3.4 Wind speed in location F3, sub-period Storm Ramp-down
Storm ensemble (7 locations)
0
5
10
15
20
25
30
0 24 48 72 96 120 144
15-minute period from 25 Oct 2004 03:15 LT
Win
d sp
eed
[m/s
]
Figure 3.5 Ensemble averaged wind speed for the 7 locations, sub-period Storm Ramp-down
Although these storms contain large local wind speed changes between consecutive 15-minute periods, these changes are not the largest possible20. So general conclusions may not be drawn from these storms.
20 For example wind speed changes of 3 ... 4 m/s from one 5-minute period to the other may be inferred from wind power changes during a thunderstorm; Kristoffersen, 2005
ECN-C--06-007 19
4. Wind speed variations on different scales and wind speed forecasting errors
4.1 Outline and context This chapter presents an analysis of the observed and predicted wind data in the 7 locations con-sidered in this studies. In particular wind speed variation on different scales (section 4.2) and wind speed forecasting error (section 4.3) are addressed. The context for these analyses is as fol-lows:
• Wind power variations on different scales mainly originate from wind speed variations on different scales. Although the actual wind power variations (WP2abc) and their im-pact on balancing (WP2d) are studied on basis of simulations21, conclusions on wind power variations may be drawn from an analysis of wind speed variations only.
• Wind speed forecasting error is needed in three studies, namely on the effect of wind power on balancing (WP2d), on short-term wind power prediction as a tool to be used by the system operator (WP 2e), and on wind power prediction as one of the measures to mitigate power variations (WP 3c).
4.2 Wind speed variation on different scales
4.2.1 Wind speed variation in 15-minute periods Figure 4.1 shows the probability of the 15-minute averaged wind speed standard deviation, where for the ensemble it is assumed that the wind speeds in the 7 locations are uncorrelated (all cross-correlation coefficients equal to 0)22. Wind speed variation on sub 15-minute scale is rele-vant because the corresponding wind power variation is to be handled by the system operator. As a rule of thumb, for wind speeds between cut-in and nominal a wind speed standard devia-tion of 1 m/s gives a wind power standard deviation of 10% of the nominal wind power. The figure shows that the probability of small wind speed standard deviations (< 0.4 m/s here) is increased whereas the probability of large wind speed standard deviations ( 0.4 m/s) is de-creased in the case of an ensemble of locations. This implies that the wind power standard de-viation will be reduced in the case of a cluster of wind turbines and/or farms. On basis of the data in this figure23 about 67% of the wind power standard deviations would be smaller than 8% of the nominal wind power for a solitary turbine or farm whereas smaller than 3% for a cluster.
4.2.2 Wind speed variation between 15-minute periods Figure 4.2 shows the probability of an absolute change of the 15-minute averaged wind speed between two consecutive periods. Wind speed changes on this time scale are relevant because the corresponding power variations are the fastest to be handled by the electricity market and the associated power plant. As a rule of thumb, for wind speeds between cut-in and nominal a wind speed change of 1 m/s yields a change of 10% of the nominal wind power.
21 See notes 2 and 3 22 The standard deviation is substantially larger if the wind speed forecasting errors are correlated (all cross-correlation coefficients equal to 1), and substantially smaller if these are anti-correlated (all cross-correlation coeffi-cients equal to -1) 23 These numbers may not be generalized
20 ECN-C--06-007
Wind speed variation in 15 minute periods
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.0 0.5 1.0 1.5 2.0
Wind speed standard deviation [m/s]
Prob
abili
ty [-
]
single location (F3)
ensemble (7 locations)
Figure 4.1 Probability of the wind speed standard deviation in 15-minute periods
Wind speed variation betweenconsecutive 15 minute periods
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.0 0.5 1.0 1.5 2.0 2.5
Wind speed change [m/s]
Prob
abili
ty [-
]
single location (F3)
ensemble (7 locations)
Figure 4.2 Probability of an absolute wind speed change between two consecutive 15-minute
periods
ECN-C--06-007 21
The figure 4.2 shows that the probability of small wind speed changes (< 0.3 m/s here) increases whereas the probability of large wind speed changes ( 0.3 m/s) decreases when an ensemble rather than a single is considered. This implies that the number of small wind power changes will be increased whereas the number of large wind power changes will be decreased in the case of a cluster of wind turbines and/or farms. On basis of the data in this figure about 75% of the wind power variations would be smaller than 7% of the nominal wind power for a solitary tur-bine or farm but smaller than 3% for a cluster. Figure 4.3 shows the probability of an absolute change in the 15-minute averaged wind speed for two periods separated by a 2-hour interval. The corresponding power variations can be fol-lowed by most of the power plant.
Wind speed variation between15 minute periods 2 hours apart
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.0 0.5 1.0 1.5 2.0 2.5
Wind speed change [m/s]
Prob
abili
ty [-
]
single location (F3)
ensemble (7 locations)
Figure 4.3 Probability of an absolute wind speed change between two 15-minute periods
separated by a 2-hour interval
The figure 4.3 shows that the probability of small wind speed changes (< 0.4 ... 0.8 m/s here) decreases whereas the probability of large wind speed changes ( 0.4 ... 0.8 m/s) increases if the separation between two 15-minute periods increases. This implies that the number of small wind power changes will decrease whereas the number of large wind power changes will increase if the separation between two 15-minute periods increases. On basis of the data in this figure 75% of the wind power variations would be smaller than 18% of the nominal wind power for a soli-tary turbine or farm and 11% for a cluster.
4.2.3 Wind speed persistence Figure 4.4 shows the probability of the average wind speed in a single location being larger than a given value during periods of 15, 30, 60, 120 and 240 minutes. The wind speed persistence is relevant to wind power persistence, which indicates the probability that the wind power is greater than a given value during an indicated period. As a rule of thumb, wind power is zero for wind speeds between 0 and 3 m/s, increases from zero to the nominal power for wind speeds between 3 and 13 m/s, is equal to the nominal power for wind speeds between 13 and 25 m/s, and is zero for wind speeds beyond 25 m/s.
22 ECN-C--06-007
Wind speed persistence single location (F3)
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20
Minimal wind speed [m/s]
Prob
abili
ty [-
]
15 minutes30 minutes60 minutes120 minutes240 minutes
Figure 4.4 Probability of the wind speed in single location being larger than a given value
during an indicated period
The figure shows that in the case of a single location (1) the higher the wind speed the lower the probability that it is exceeded, and (2) the longer the period the lower the probability that a given wind speed is exceeded. Implication: (1) the higher the wind power the lower the prob-ability that it is exceeded, and (2) the longer the period the lower the probability that a given power level is exceeded. Table 4.1 presents indicative values for the percentages of zero load, partial load and full load 15-minute periods as following from the data in this figure.
Table 4.1 Indicative occurrence of wind power regimes with a duration of 15 and 240 minutes in the case of a solitary turbine or farm and a cluster of turbines/farms
Solitary (F3) Cluster (7 locations) 15 minutes 240 minutes 15 minutes 240 minutes Zero load, LWS 5% 10% 1% 2% Partial load 65% 70% 83% 88% Full load 27% 18% 13% 9% Zero load, HWS 3% 2% 2% 1% Figure 4.5 shows the wind speed persistence for an ensemble of locations.
ECN-C--06-007 23
Wind speed persistence ensemble (7 locations)
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20
Minimal wind speed [m/s]
Prob
abili
ty [-
]
15 minutes30 minutes60 minutes120 minutes240 minutes
Figure 4.5 Probability of the wind speed in an ensemble of locations being larger than a given
value during an indicated period
The figure shows that in the case of an ensemble of locations, with respect to the single location the probability that a low wind speed (< 6 ... 7 m/s here) is exceeded is increased whereas the probability that a higher wind speed (> 6 ...7 m/s) is exceeded is decreased. This implies that the number of partial load 15-minute periods will increase, whereas the number of zero load and full load periods will decrease in the case of a cluster of wind turbines and/or farms. Also note that the number of cut-outs at full load will be reduced. Table 4.1 contains indicative values for the percentages of zero load, partial load and full load 15-minute periods as following from the data in this figure.
4.2.4 Conclusions on wind speed variations on different scales Clustering of wind turbines and/or wind farms will results in:
• A reduction of sub 15-minute wind power variations. • A reduction of the variation in wind power from one 15-minute period to the other. • An increase of the variation in wind power between 15-minute periods separated longer. • A reduction of the number of 15-minute periods with zero load and with full load, and
an increase of the number of partial load periods. In a qualitative sense this is in agreement with the effect of clustering as reported in the litera-ture (e.g. Giebel, 2000).
24 ECN-C--06-007
4.3 Wind speed forecasting error
4.3.1 Systematic and stochastic forecasting error per location The systematic and stochastic wind speed forecasting error per location are presented in table 4.2.
Table 4.2 Systematic and stochastic wind speed forecasting errors per location F3 K13 Europl. EWTW1 Cabauw NSW1 FINO1 Systematic forecasting error [m/s]
1.255 1.256 1.103 0.374 0.758 0.183 0.745
Stochastic forecasting error [m/s]
2.687 2.611 2.312 2.039 1.889 2.519 2.742
The systematic forecasting errors, in the range between 0.2 m/s and 1.3 m/s, are large given the reported absolute values of the systematic error of 0.5 m/s or less24. The stochastic forecasting errors are larger than 1.9 m/s but smaller than 2.7 m/s, and are in agreement with the reported range between 1 m/s and 3 m/s25. In a sub-period of 4 to 8 days the systematic error may be different from the values mentioned above, which should be no surprise. The stochastic error, on the other hand, has the same order of magnitude. For this reason the wind speed forecasting error in a sub-period has not been determined; the systematic error could have a different but not representative value whereas the stochastic errors would be of the same order of magnitude.
4.3.2 Systematic and stochastic forecasting error for the ensemble Table 4.3 presents the forecasting errors of the ensemble averaged wind speed. The ensemble consists of 7 locations.
Table 4.3 Systematic and stochastic wind speed forecasting errors of the ensemble averaged wind speed
Ensemble Systematic forecasting error [m/s]
0.821
Stochastic forecasting error [m/s]
1.609
These data show that the predictability of the ensemble averaged wind speed is better than the predictability of a local wind speed. This is illustrated in the scatter plots of the observed and the forecasted wind speed (figure 4.6 and 4.7).
24 Lange, 2004 25 Tambke, 2006
ECN-C--06-007 25
Wind speed observations and forecasts ensemble (7 locations)
Jun'04 - May'05
0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30
Forecasted wind speed [m/s]
Obs
erve
d w
ind
spee
d [m
/s]
Figure 4.6 Observed versus forecasted ensemble averaged wind speed
Wind speed observations and forecastssingle location (F3)
Jun'04 - May'05
0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30
Forecasted wind speed [m/s]
Obs
erve
d w
ind
spee
d [m
/s]
Figure 4.7 Observed versus forecasted wind speed in location F3
26 ECN-C--06-007
4.3.3 Cross-correlation between locations Figure 4.8 shows the cross-correlation between the forecasting errors in 2 locations as a function of their separation. Recall the lead-time ranges from 16 to 41 hours.
Wind speed forecasting error Cross-correlation between locations
0.00.10.20.30.40.50.60.70.80.91.0
0 50 100 150 200 250 300 350Separation [km]
Cro
ss-c
orr.
coef
f. [-]
Figure 4.8 Cross-correlation between the wind speed forecasting error as a function of the
distance between locations
From the figure it is clear that the larger the distance, the lower the cross-correlation. If a loga-rithmic decay is assumed the decay constant is about 165 km. This value is large given the re-ported decrease of the decay constant from 160 km for 48-hours forecasts to 85 km for 12-hours forecasts26.
4.3.4 Auto-correlation per location The auto-correlation of the forecasting error in the 7 locations is presented in figure 4.9.
26 Tambke, 2006
ECN-C--06-007 27
Wind speed forecasting error Auto-correlation per location
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0 12 24 36 48
Time shift [hr]
Aut
o-co
rr. c
oeff.
[-]
Figure 4.9 Auto-correlation of the wind speed forecasting error for the 7 locations
The auto-correlation of the forecasting errors decreases rapidly with time: after 6 and maybe even after 3 hours correlation is non-existent. If a logarithmic decay is assumed the decay con-stant is 4 ... 9 hours.
4.3.5 Impact of wind speed The figures 4.10 and 4.11 display the systematic and the stochastic error as a function of the forecasted wind speed for the locations F3 and EWTW1. (The other locations show similar pic-tures.) The conditional numerical values are to be compared to the bulk values presented in sec-tion 4.3.1.
28 ECN-C--06-007
Wind speed forecasting error single location (F3)
-2
-1
0
1
2
3
4
5
6
0 5 10 15 20 25 30
Forecasted wind speed [m/s]
Fore
cast
ing
erro
r [m
/s]
Syst. ErrorStoch. Error
Figure 4.10 Systematic and stochastic wind speed forecasting error in location F3 as a function
of forecasted wind speed
Wind speed forecasting error single location (EWTW1)
-2
-1
0
1
2
3
4
5
6
0 5 10 15 20 25 30
Forecasted wind speed [m/s]
Fore
casi
ng e
rror
[m/s
]
Syst. ErrorStoch. Error
Figure 4.11 Systematic and stochastic wind speed forecasting error in location EWTW1 as a
function of forecasted wind speed
ECN-C--06-007 29
Three regions are identified in these figures: (1) forecasted wind speed smaller than 4 m/s, (2) between 4 m/s and 16 m/s, and (3) beyond 16 m/s. In the central region the systematic and the stochastic error have opposite trends: the systematic error gradually decreases whereas the sto-chastic error gradually increases with the forecasted wind speed. In both cases a linear location dependent trend would fit to the data. Such a trend could be used in order to compensate the forecasted wind speed.
4.3.6 Impact of lead-time The figures 4.12 and 4.13 display the systematic and the stochastic error as a function of the lead-time for the locations F3 and EWTW1. (The figures for the other locations are similar.) The conditional numerical values are to be compared to the bulk values presented in section 4.3.1.
Wind speed forecasting error single location (F3)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0 5 10 15 20 25 30 35 40 45
Lead time [h]
Fore
cast
ing
erro
r [m
/s]
Syst. ErrorStoch. Error
Figure 4.12 Systematic and stochastic wind speed forecasting error in location F3 as a function
of lead time
30 ECN-C--06-007
Wind speed forecasting error single location (EWTW1)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0 5 10 15 20 25 30 35 40 45
Lead time [h]
Fore
cast
ing
erro
r [m
/s]
Syst. ErrorStoch. Error
Figure 4.13 Systematic and stochastic wind speed forecasting error in location EWTW1 as a
function of lead time
The systematic errors (1.3±0.3 m/s for F3 and 0.4±0.2 m/s for EWTW1) may considered to be constant, whereas the stochastic error increases with the lead-time. The increase of the stochas-tic error with a factor 1.3 for lead times between 17 and 41 hours is smaller than the factor 2 re-ported in literature27.
4.3.7 Conclusions on wind speed forecasting error The systematic wind speed forecasting error:
• Is 0.2 ... 1.3 m/s in a single location, and 0.8 m/s for an ensemble of locations, • Decreases with the forecasted wind speed, and • Does not depend on lead-time.
The stochastic error:
• Is 1.9 ... 2.7 m/s in a single location, and 1.6 m/s for an ensemble of locations, • Increases with the forecasted wind speed, and • Increases with lead-time.
The decay constant of the correlation coefficient is:
• 165 km for the cross-correlation between locations, and • 4 ... 9 h for the auto-correlation in a location.
27 Tambke, 2006
ECN-C--06-007 31
5. Recapitulation
This report presents the wind data for a study on the power variations in the Dutch electricity system due to wind energy and the measures to mitigate these variations. These wind data con-sist of:
• General information in the form of a representative offshore wind regime and an esti-mate of wind speed forecasting accuracy, and
• Specific information in the form of simultaneous time series of 15-minute averaged ob-served and predicted wind speed in 5 locations in the North Sea and in 2 locations in the Netherlands.
Also presented are the transformations from sensor height to evaluation height and from 10-minute periods to 15-minute periods that were used to create the observed wind data from the measured wind data. The time series are applied in an analysis of wind speed variations on different scales (and the resulting wind power variations) and an analysis of wind speed forecasting errors. Regarding the power variations it is found that clustering of wind turbines and/or wind farms will results in a reduction of sub 15-minute wind power variations, a reduction of the variation in wind power from one 15-minute period to the other, an increase of the variation in wind power between 15-minute periods separated longer, and a reduction of the number of 15-minute periods with zero load and with full load in combination with an increase of the number of par-tial load periods. As to the wind speed forecasting error it is found that the systematic error is 0.2 ... 1.3 m/s in a single location and 0.8 m/s for an ensemble of locations, decreases with the forecasted wind speed, and does not depend on lead-time. The stochastic error is 1.9 ... 2.7 m/s in a single loca-tion and 1.6 m/s for an ensemble of locations, increases with the forecasted wind speed, and in-creases with lead-time. The decay constant of the correlation coefficient is 165 km for the cross-correlation between locations, and 4 ... 9 h for the auto-correlation in a location.
32 ECN-C--06-007
References
W.W. de Boer, 2005a, Gewenste info van ECN, Mail 27th January 2005 W.W. de Boer e.a., 2005b, Balanshandhaving met 6000 MW aan windenergie, KEMA Rapport 3045-0054-Consulting-Draft A.J. Brand & T. Hegberg, 2004, Offshore Wind Atlas, ECN Report ECN-CX--04-136 A.J. Brand & J.K. Kok, 2003, Aanbodvoorspeller Duurzame Energie - Deel 2: Korte-termijn prognose van windvermogen, ECN Report ECN-C--03-049 R. Doherty e.a., 2004, System operation with a significant wind power penetration, In: Power Engineering Society General Meeting 2004, Denver , June 2004 G. Giebel, 2000, On the benefits of distributed generation of wind energy in Europe, PhD The-sis H. Holttinen, 2005, Hourly wind power variations in the Nordic countries, Wind Energy 2005, 8.2 J.R. Kristoffersen, 2005, The Horns Rev Wind Farm and the operational experience with the Wind Farm Main Controller, In: Copenhagen Offshore Wind Energy 2005, Copenhagen, Octo-ber 2005 M. Lange, 2004, On the uncertainty of wind power predictions, In: EWEA Special Topics Con-ference, Delft, April 2004 P. Nørgård e.a., 2004, Fluctuations and predictability of wind and hydro power, Risø Report Risø-R-1443(EN) J. Tambke e.a., 2006, Advanced forecast systems for the grid integration of 25 GW offhore wind power in Germany, In: EWEC 2006, Athens, March 2006 J. Wieringa, 1984, De atmosferische grenslaag, KNMI
ECN-C--06-007 33
Appendix A Transformation of wind speed from sensor height to evaluation height
A.1 Method Given:
• μu(zs) = 10-minute average of the wind speed, where zs is the sensor height • σu(zs) = 10-minute standard deviation of the wind speed
or
• μu(zs) = 10-minute average of the wind speed, where zs is the sensor height Needed:
• μu(zh) = 10-minute average of the wind speed, where zh is the evaluation height • σu(zh) = 10-minute standard deviation of the wind speed
Estimates: If μu(zs) and σu(zs) are available:
• ( ) ( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −+⎟⎟⎠
⎞⎜⎜⎝
⎛+=
esti
sh
s
hsusuhestiu L
zzzz
zzz 5ln, σμμ
with Lesti the location dependent average stability length; see table 3.1. • ( ) ( )suhestiu zz σσ =,
If only μu(zs) is available and the location is offshore:
• ( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −+⎟⎟⎠
⎞⎜⎜⎝
⎛+=
esti
sh
s
hsuhestiu L
zzzz
uzz 5ln5.2 *, μμ ,
with u* according to
( ) 05ln5.2 2*
* =⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
esti
sssu L
zAu
gzuzμ ;
where g = 9.81 m/s2 (acceleration of gravity) and A = 0.011 (Charnock's constant).
• ( ) *, 5.2 uzhestiu =σ .
If only μu(zs) is available and the location is onshore estimates for μu(zh) and σu(zh) cannot be given.
34 ECN-C--06-007
A.2 Approach
A.2.1 Average wind speed profile The vertical profile of the 10 minute averaged wind speed is28:
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛Ψ−⎟⎟
⎠
⎞⎜⎜⎝
⎛=
Lz
zz
uz mu0
* ln5.2)(μ
(note the value 0.4 for the von Karman constant), with
Lz
Lz
m 5−=⎟⎠⎞
⎜⎝⎛Ψ under stable condition
Lz
Lz
m −=⎟
⎠⎞
⎜⎝⎛Ψ 1.1 under unstable condition (Van Ulden approximation29)
and
z = height above surface level, u* = friction velocity, z0 = surface roughness length, L = Monin-Obukhov or stability length.
According to this profile the difference δμu between the average wind speed at evaluation height zh and at sensor height zs is:
( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛Ψ+⎟
⎠⎞
⎜⎝⎛Ψ−⎟⎟
⎠
⎞⎜⎜⎝
⎛=−≡
Lz
Lz
zz
uzzzz sm
hm
s
hsuhushu ln5.2),( *μμδ .
Averaged over a long period the vertical profile is stable and then can be described with the av-erage stability length Lave
30. In that case the difference between the wind speeds at zh and zs is:
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −+⎟⎟⎠
⎞⎜⎜⎝
⎛=
ave
sh
s
hshaveu L
zzzz
uzz 5ln5.2, *,δ ,
so that the estimate for the wind speed at zs is:
( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −+⎟⎟⎠
⎞⎜⎜⎝
⎛+=
ave
sh
s
hsuhestiu L
zzzz
uzz 5ln5.2 *, μμ . (1)
This estimate warrants the friction velocity u* and the average stability length Lave. The friction velocity is determined from the data as explained in sections A.2.2c and A.2.2d. The average stability length must be determined a priori and the recommended values are presented in table 3.1.
28 Wieringa, 1984, p. 35 29 Wieringa, 1984, p. 36 30 Brand, 2004, p. 14
ECN-C--06-007 35
A.2.2 Wind speed standard deviation A.2.2a Relation between σu and u* The standard deviation of the wind speed depends on the friction velocity:
± = *)2.05.2()( uzuσ ,
independent of height and stability class31. This can be rewritten:
± = *)08.01(5.2)( uzuσ , (2)
where the second term is the uncertainty in the wind speed standard deviation. A.2.2b μu(zs) and σu(zs) are available The friction velocity u* is estimated from the wind speed standard deviation σu(zs) at sensor height by using equation 2:
= )(4.0* su zu σ .
Inserting this into equation 1 gives the estimate for the average wind speed at zh:
( ) ( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −+⎟⎟⎠
⎞⎜⎜⎝
⎛+=
esti
sh
s
hsusuhestiu L
zzzz
zzz 5ln, σμμ ,
where Lesti is the estimate for the average stability length. Table A.1 gives the relative magnitude of the terms in this equation if zh = 70 m32, together with the magnitude of three typical errors which are addressed in section A.2.3. For all locations the stability term is smaller than the logarithmic term, but may not be ne-glected. Given that the wind speed standard deviation does not depend on height, the estimate for the standard deviation of the wind speed at zh is:
( ) ( )suhestiu zz σσ =, .
Since this transformation employs the wind speed standard deviation σu(zs) at sensor height it will be referred to as the wind speed standard deviation method.
31 Wieringa, 1984, p. 41 32 Initial value of the "hub height" which was changed into 80 m and 100 m in a later stage
36 ECN-C--06-007
Table A.1 The relative magnitude of the terms in the height transformation of the average wind speed, and the magnitude of three typical errors in the transformed average wind speed if zh = 70 m
zh = 70 m
sz estiL
s
h
zz
ln esti
sh
Lzz −
5 esti
sh
s
h
Lzz
zz −+ 5ln
u
u
σμ 1,Δ
u
au
σμ 2,Δ
u
bu
σμ 2,Δ
m m - - - - - - very stable ...
neutral very unstable ... neutral
NSW 70 1.042 0 0 0 0 0 0 FINO1 60 809 0.154 0.062 0.216 0.043 0.438 ... –
0.062 –0.130 ... –0.062
Europlatform 29 945 0.881 0.217 1.098 0.220 1.833 ... –0.217
–0.545 ... –0.217
K13 74 742 –0.056 –0.027 –0.083 –0.017 –0.173 ... 0.027
0.053 ... 0.027
F3 59 712 0.171 0.077 0.248 0.050 0.473 ... –0.077
–0.152 ... –0.077
EWTW1 70 1.200 0 0 0 0 0 0 Cabauw 60 1.200 0.154 0.042 0.196 0.039 –0.042 ...
0.458 –0.110 ... –0.042
A.2.2c μu(zs) is available If the location is offshore the friction velocity û* is estimated from the wind speed μu(zs) at sen-sor height:
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=
esti
sssu L
zuAgz
uz 5ˆ
lnˆ5.2 2*
*μ ,
for example by applying the Regula Falsi. Note to derive this equation Charnock's rule for the surface roughness length was used:
gu
Az2*
0 = ,
with g = 9.81 m/s2 (acceleration of gravity) and A = 0.011 (Charnock's constant). The equations 1 and 2 now yield the estimates for the wind speed average and standard devia-tion at zh:
( ) ( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −+⎟⎟⎠
⎞⎜⎜⎝
⎛+=
esti
sh
s
hsuhestiu L
zzzz
uzz 5lnˆ5.2 *, μμ
and ( ) *, ˆ5.2 uzhestiu =σ .
Since this transformation employs the friction velocity it will be referred to as the friction veloc-ity method.
ECN-C--06-007 37
If the location is onshore, the friction velocity can not be obtained from μu(zs) alone33. As a con-sequence for onshore locations the wind speed average and standard deviation at zh cannot be determined if the average wind speed at sensor height only is available.
A.2.3 Accuracy of the average wind speed A.2.3a μu(zs) and σu(zs) are available Error type #1: The uncertainty in the relation between the friction velocity and the wind speed standard devia-tion gives an uncertainty in the wind speed at zh. According to equation 2 the uncertainty u* in the friction velocity is proportional to the wind speed standard deviation σu:
uu σ08.0* ≈Δ .
This uncertainty propagates and yields an uncertainty μu in the estimate of the wind speed at zh:
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+⎟⎟⎠
⎞⎜⎜⎝
⎛≈Δ
∂∂=Δ
esti
sh
s
hu
uu L
zzzz
uu
5ln2.0**
1, σμμ .
Note the subscript esti is omitted. Table A.1 shows that the relative magnitude of the type #1 error is smaller than the sum of the logarithmic and the stability term, but not so much smaller that it could be neglected. Order of magnitude of the type #1 error:
• Onshore location (Cabauw): σu = 3 m/s, zh = 70 m, zs = 60 m, Lesti = 1200 m, Error μu,1 = 0.12 m/s.
• Offshore location (Europlatform): σu = 1 m/s, zh = 70 m, zs = 29 m, Lesti = 945 m, Error μu,1 = 0.22 m/s.
Error type #2a: The actual wind speed profile is stable but the actual value of the stability length differs from the average value. If Lactual is the actual positive but unknown value of the stability length, from equation 1 it fol-lows that the error μu in the estimate of the wind speed at zh is:
( ) ( ) ( )( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−−=−≡Δ
estimateactualshsuhestimateuhactualuau LL
zzzzz11
5,,2, σμμμ .
Table A.1 shows that, unless the condition is very stable, the relative magnitude of the type #2a error is smaller than the sum of the logarithmic and the stability term but cannot be neglected. Order of magnitude of the type #2a error:
33 In the sense that the fixed but direction dependent surface roughness length must be available in order to determine the friction velocity from the wind speed at sensor height
38 ECN-C--06-007
• Onshore location (Cabauw):
σu = 3 m/s, zh = 70 m, zs = 60 m, Lactual = 100 m (very stable) ... ∞ m (neutral), Lestimate = 1200 m, Error μu,2a = 1.4 ... –0.13 m/s.
• Offshore location (Europlatform): σu = 1 m/s, zh = 70 m, zs = 29 m, Lactual = 100 m (very stable) ... ∞ m (neutral), Lestimate = 945 m, Error μu,2a = 1.8 ... –0.22 m/s.
The largest type #2a error in the estimated wind speed at zh occurs if the actual condition is very stable; if so the error in the estimated wind speed is of the order 2 m/s which means that the ac-tual wind speed is underestimated with 2 m/s. Error type #2b: The actual wind speed profile is unstable rather than stable. If Lactual is the actual negative but unknown value of the stability length, from the equation for the vertical wind speed profile it follows that the error μu in the estimate of the wind speed at zh is:
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −+−
−−=Δ
estimate
sh
actual
shsubu L
zz
L
zzz 51.12, σμ .
Table A.1 shows that, unless the condition is very unstable, the relative magnitude of the type #2b error is smaller than the sum of the logarithmic and the stability term but can not be ne-glected. Order of magnitude:
• Onshore location (Cabauw): σu = 3 m/s, zh = 70 m, zs = 60 m, Lactual = –100 m (very unstable) ... –∞ m (neutral), Lesti-
mate = 1200 m, Error μu,2b = –0.33 ... –0.13 m/s.
• Offshore location (Europlatform): σu = 1 m/s, zh = 70 m, zs = 29 m, Lactual = –100 m (very unstable) ... –∞ m (neutral), Lesti-
mate = 945 m, Error: μu,2b = –0.54 ... –0.22 m/s.
The largest type #2b error occurs if the actual condition is very unstable; if so the error in the estimated wind speed is of the order –0.5 m/s which means that the actual wind speed is overes-timated with 0.5 m/s. A.2.3b μu(zs) is available (offshore only) Error type #3: The uncertainty in the friction velocity when determined from the wind speed at sensor height gives an uncertainty in the wind speed at zh. Impact of:
• The actual value of the Charnock constant versus the employed value, and • The actual stability length versus the employed one.
Yet to be established; difficult because of explicit expression of u*.
ECN-C--06-007 39
A.2.4 Accuracy of the wind speed standard deviation A.2.4a μu(zs) and σu(zs) are available An error in σu(zs) may originate from the assumption that σu does not depend on height. Hard to quantify. A.2.4b μu(zs) is available (offshore only) See section A.2.3b. In addition: uncertainty in the relation between σu and u*.
A.2.5 Comparison of the two height transformation methods Wind speed data from the offshore F3 platform comprises 10-minute wind speed averages and standard deviations measured at sensor height. These data allow the two height transformation methods to be compared, in this case from 59 meter to 70 meter. Recalling from section A.2.2b and A.2.2c, these methods are as follows:
• Wind speed standard deviation method: If σu(zs) is available, the wind speed average at zh is obtained by employing σu(zs) in order to determine u*, and the wind speed standard deviation at zh is taken equal to σu(zs).
• Friction velocity method: If σu(zs) is not available and the location is offshore, the wind speed average at zh is obtained by employing μu(zs) in order to determine u*, and the wind speed standard deviation at zh is taken equal to 2.5u*.
The comparison is presented in figure A.1 (friction velocity), figure A.2 (wind speed average at zh), and figure A.3 (wind speed standard deviation at zh). Note figure A.1 and figure A.3 are similar because σu = 2.5 u*. Also note the discrete values in these figures - these originate from the measured wind speed standard deviations which are rounded to integer dm/s. In addition ta-ble A.2 presents the characteristic values of the difference between the two methods, where the mean difference gives the bias between the two methods and the rms difference the spread.
40 ECN-C--06-007
Figure A.1 The friction velocity as determined by the two methods
Figure A.2 The wind speed average at 70 meter as determined by the two methods
ECN-C--06-007 41
Figure A.3 The wind speed standard deviation at 70 meter as determined by the two methods
Table A.2 The differences in the outcome of the two methods F3, zh = 70 m u*,esti μu,esti(zh) σu,esti(zh) m/s m/s m/s Mean difference –0.031 –0.019 –0.078 Rms difference 0.168 0.104 0.421 Smallest difference –0.645 –0.400 –1.612 Largest difference 1.179 0.732 2.947 As to the friction velocity and the wind speed standard deviation the mean difference is small but the rms difference is large. This leads to the conclusion that the two methods yield unbiased but different values for the friction velocity. The same holds for the wind speed standard devia-tion. As to the wind speed average, on the other hand, the mean difference and the rms difference are small. This leads to the conclusion that the two methods yield equal values for the wind speed average.
42 ECN-C--06-007
Appendix B Transformation of wind speed from 10-minute periods to 15-minute periods
B.1 Method Given:
• μk, μk+1, μk+2, ... = consecutive 10-minute averages • σk, σk+1, σk+2, ... = consecutive 10-minute standard deviations
Needed:
• mk, mk+1, ... = consecutive 15-minute averages • sk, sk+1, ... = consecutive 15-minute standard deviations
Estimates:
• 3
2 22/)1(312/)1(3,
+−+− += kk
estikmμμ
(first 15-minute period of a 30-minute period) and
3
2 32/)1(322/)1(3,1
+−+−+
+= kk
estikmμμ
(second 15-minute period of a 30-minute period)
• 3
2 222/)1(3
212/)1(32
,+−+− +
= kkestik
σσσ and
3
2 232/)1(3
222/)1(32
,1+−+−
+
+= kk
estik
σσσ , likewise.
B.2 Approach
B.2.1 Averages The continuous wind speed signal w(t) is sampled at a given data rate, wi is i-th sample. Consider consecutive 10-minute averages of the wind speed, μk is the k-th average:
∑=
=N
iik w
N 1
1μ , ∑+=
+ =N
Niik w
N
2
11
1μ , ∑+=
+ =N
Niik w
N
3
122
1μ , etc,
where N is the number of samples in the 10-minute period. Also consider consecutive 15-minute averages mk:
∑=
=2
3
132
N
iik w
Nm , ∑
+=
+ =N
Ni
ik wN
m3
12
31 3
2, etc.
Now introduce the estimate mk,esti of the average in the 1st 15-minute period:
ECN-C--06-007 43
∑∑+==
+−+−
++
+=
++
=N
Nii
N
ii
kkestik w
Ncw
Ncc
cc
m2
1111
1
1
22/)1(312/)1(31,
11
1111
μμ.
The error in this estimate is:
∑∑∑+=+== +
−+−+
++−=−≡Δ
N
Ni
i
N
Nii
N
iiestikkestik w
Ncw
Ncc
wNc
cmmm
2
12
31
23
11
1
11
1,,
11
11)1(3
121)1(3
2.
The terms in the right-hand side are the contributions to the error from the wind speed samples in the 1st 10-minute period (1st and 2nd 5-minute period), from the 3rd 5-minute period and from the 4th 5-minute period. By taking c1 = 2 we eliminate the contribution from the samples in the 1st 10-minute period. This yields
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−=Δ ∑∑
+=+=
N
Ni
i
N
Niiestik w
Nw
Nm
2
12
3
2
3
1,
1131
;
the error in the estimated average in the 1st 15-minute period is proportional to the difference between the averages in the 3rd and the 4th 5-minute period, which averages are unknown. By introducing the estimate mk+1,esti of the average in the 2nd 15-minute period
∑∑+=+=
+−+−+ +
++
=+
+=
N
Nii
N
Nii
kkestik w
Ncc
wNcc
cm
3
123
32
133
32/)1(3322/)1(3,1
11
11
11
μμ (3a)
and taking c3 = 2 we find that the error in this estimate is:
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛+−=Δ ∑∑
+=+=+
N
Ni
i
N
Niiestik ww
Nm
2
12
3
2
3
1,1 3
1. (3b)
The error in the estimated average in the 2nd 15-minute period is proportional to the difference between the averages in the 3rd and the 4th 5-minute period. Note mk,esti + mk+1,esti = 0.
B.2.2 Standard deviations The 10-minute standard deviations σk are given by the variances σ2
k:
2
1
2 )(1 ∑
=−=
N
ikik w
Nμσ , 2
2
11
21 )(
1 ∑+=
++ −=N
Nikik w
Nμσ , 2
3
122
22 )(
1 ∑+=
++ −=N
Nikik w
Nμσ , etc,
and the 15-minute standard deviations sk are given by the variances s2
k:
22
3
1
2 )(32 ∑
=
−=
N
ikik mw
Ns , 2
3
12
31
21 )(
32 ∑
+=
++ −=N
Ni
kik mwN
s , etc.
We introduce the estimate for the variance in the 1st 15-minute period:
44 ECN-C--06-007
11
222/)1(3
212/)1(312
, ++
= +−+−
c
cs kk
estik
σσ
and find that the error in this estimate is:
21
1
2
1
122
12
3
2
1
23
1
2
1
1
1
2
1
12,
22, 1
11
11
11)1(3
121)1(3
2+
+=+== ++
++−
++
+−+
++−=−≡Δ ∑∑∑ kkk
N
Ni
i
N
Nii
N
iiestikkestik cc
cmw
Ncw
Ncc
wNc
csss μμ
We eliminate the contribution to this error by the wind speed samples in the first 10-minute pe-riod by taking c1 = 2, and get:
21
222
12
3
22
3
1
22, 3
132
31
+
+=+=
++−⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−=Δ ∑∑ kkk
N
Ni
i
N
Niiestik mww
Ns μμ (4a)
The error in the variance in the 1st 15-minute period is proportional to the difference between the summed squared wind speeds in the 3rd and the 4th 5-minute period, and to the difference of the averages in the 1st and 2nd 10-minute period with respect to the average in the 1st 15-minute period. In a similar way it can be shown that the estimate
13
232/)1(33
222/)1(32
,1 ++
= +−+−+ c
cs kk
estik
σσ
for the variance in the 2nd 15-minute period with c3 = 2 has the error
22
21
21
2
12
3
22
3
1
22,1 3
231
31
+++
+=+=+ ++−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛+−=Δ ∑∑ kkk
N
Ni
i
N
Niiestik mww
Ns μμ . (4b)
B.3 Accuracy
B.3.1 Averages According to equation 3a the error in the estimate of the 1st 15-minute averaged wind speed is proportional to the difference between the averages of the 3rd and the 4th 5-minute period. Since these averages are unknown, this error cannot be established. The maximum of this error can however be estimated. To this end consider the average μk+1 and the standard deviation σk+1 in the 2nd 10-minute pe-riod. Suppose now all wind speed samples larger than μk+1 occurred in one 5-minute sub-period, and those smaller than μk+1 occurred in the other sub-period. Assuming a normal distribution, the averages in the two sub-periods are
11
2++ + kk σ
πμ and 11
2++ − kk σ
πμ ,
respectively. It follows that the difference between the average in the two periods is
ECN-C--06-007 45
1
22 +± kσ
π ,
so that
1max,
231
+±=Δ kestikm σπ
.
In a similar way, starting with equation 3b it can be shown that the maximum of the error in the 2nd 15-minute average is:
1max,1
231
++ =Δ kestikm σπ
m .
Order of magnitude of the error in the 15-minute averages:
• Onshore location: σu = 3 m/s so that | mk,esti|max = 0.80 m/s.
• Offshore location: σu = 1 m/s so that | mk,esti|max = 0.27 m/s.
B.3.2 Standard deviations According to equation 4a the error in the estimate of the variance of the wind speed in the 1st 15-minute period is proportional to
• The difference between the summed squared wind speeds in the 3rd and the 4th 5-minute period, and
• The difference of averages in the 1st and 2nd 10-minute period with respect to the aver-age in the 1st 15-minute period.
First we treat the contribution by the summed squared wind speeds. Like in section A.2.2a, con-sider the average μk+1 and the standard deviation σk+1 in the 2nd 10-minute period, and suppose all wind speed samples larger than μk+1 occurred in one 5-minute sub-period, and those smaller than μk+1 occurred in the other sub-period. Assuming a normal distribution, the standard devia-tions in the two sub-periods are equal:
πσ 2
11 −+k ,
so that
11
2
12
3
22
3
1
2 232
31
++
+=+=
≤⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛− ∑∑ kk
N
Ni
i
N
Nii ww
Nσμ
π.
Next we treat the contribution by the difference between the averages. To this end we use mk = mk,esti + mk with
46 ECN-C--06-007
1, 31
32
++= kkestikm μμ (section A.3.2a) and 1max,
231
+±=Δ kestikm σπ
(section
A.3.3a), and obtain
( ) ( ) 2111
21
21
22
92
22
92
92
31
32
+++++ −+−≤++− kkkkkkkkkm σπ
σμμπ
μμμμ m .
By taking the two contributions together, the error in the variance in the 1st 15-minute period is obtained:
( ) ( ) 21
2111max
2, 9
2922
94
++++ −−+−±=Δ kkkkkkestiks σπ
μμσμμπ
.
In a similar way, starting with equation 4b, the error in the variance in the 2nd 15-minute period is obtained:
( ) ( ) 21
221121max
2,1 9
2922
94
+++++++ −−+−=Δ kkkkkkestiks σπ
μμσμμπ
m
Order of magnitude of the error in the 15-minute variances:
• Onshore location: | μk+1 – μk+2 | = 2 m/s and σu = 3 m/s so that | s2
k,esti|max = –1.9 ... +2.4 m2/s2. • Offshore location:
| μk+1 – μk+2 | = 1 m/s and σu = 1 m/s so that | s2k,esti|max = –0.20 ... +0.51 m2/s2.
Note a negative value of | s2
k,esti|max means that s2k < s2
k,esti (standard deviation is overestimated), and a positive value means that s2
k > s2k,esti (standard deviation is underestimated).