Estimation of Weibull parameters for wind energy analysis across the UK

Harvesting wind energy resources is a major part of the UK strategy to diversify the power supply portfolio and mitigate environmental degradation. Based on wind speed data for the period 1981–2018, collected at 38 surface observation stations, this study presents a comprehensive assessment of wind speed characteristics by means of statistical analysis using the Weibull distribution function. The estimated Weibull parameters are used to evaluate wind power density at both station and regional levels and important, turbine-specific wind energy assessment parameters. It is shown that the Weibull distribution function provides satisfactory modeling of the probability distribution of daily mean wind speeds, with the correlation coefficient generally exceeding 0.9. Site-to-site variability in wind power density and other essential parameters is apparent. The Weibull scale parameter lies in the range between 4.96 m/s and 12.06 m/s, and the shape parameter ranges from 1.63 to 2.97. The estimated wind power density ranges from 125 W/m2 to 1407 W/m2. Statistically significant long-term trends in annual mean wind speed are identified for only 15 of the 38 stations and three of the 11 geographical regions. The seasonal variability of Weibull parameters and wind power density is confirmed and discussed.

. 41 While the benefits of harnessing wind energy are evident, the implementation may 42 be subject to a number of practical difficulties and uncertainties, one of which is the 43 intermittent and unsteady nature of wind. The theoretical energy carrying by wind (P) 44 is linked to the third power of wind speed, as shown in Eq.(1), where is the air density, 45 represents the area swept out by the rotor blades perpendicular to the prevailing 46 direction of the wind and is the wind speed [7]. Hence, accurate understanding of 47 wind speed characteristics is imperative in different aspects of wind energy 48 development, ranging from identification of desirable sites to prediction of the 49 economic viability of wind farm to structural design of wind turbines. 50 = 1 2 3 (1) However, precise prediction of wind is not an easy task since wind, like many other 51 meteorological parameters [8], often exhibits significant variability over a range of 52 scales, both spatially and temporally [9] [10]. In the view of wind power development, 53 the variation of wind speed at a given location is generally characterized by a 54 probability distribution [11] which indicates the likelihood that a given wind speed will 55 occur. Most commonly used for wind energy assessments is the two-parameter Weibull 56 distribution, which has been shown to accurately capture the skewness of the wind In the UK, estimation of Weibull parameters for wind energy analysis has been 63 carried out previously by Earl et al.,[21] and Früh [22]. Based on 2-year surface wind 64 observation at 72 stations, Früh [22] concluded that the shape parameter ranges from 65 1.43 to 2.23, and the scale parameter at 10m height ranges from 4.76m/s to 8.71 m/s. 66 Given the assertion of Gross et al. [24] show that at least 7 years of wind speed data is 67 required due to year-to-year variability (this variability has been estimated as about 4% 68 [25]) the 2-year period seems short, but a similar range of shape parameter is also 69 reported by Earl  (1 ≤ ≤ 10) (3) where is the gamma function. 119 Once the shape parameter, k, is estimated based on Eq.
(3), an alternative, empirical 120 method was also proposed by Lysen [35] in which is the wind speed data measured at the time interval i, and n is the number 126 of non-zero data. 127 The power density method (PDM), originally proposed by Akdag and Dinler [36], 128 calculates the shape parameter using: where 3 ̅̅̅ is the mean of the cubed wind speed. The scale parameter in PDM is 130 estimated in a the same manner as in the EMJ, as shown in (4).

131
Once these Weibull parameters are determined, they can be applied to estimate a 132 number of parameters that are important to wind power assessment. Each model of 133 wind turbine has several characteristic wind speeds: the cut-in wind speed, , the cut-134 off wind speed, , and the rated wind speed, . Below or above the turbine will 135 not operate, while energy production is maximal at . The probability that a turbine 136 will be in operation can therefore be calculated based on the cumulative Weibull where is the density of ambient air (often adopted as 1.225 kg/m 3 ). This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0038001
analysis can also be found in several previous studies [16] [43]- [45]. A further 195 discussion on the use of daily wind data will be given hereinafter in Section 4. 196 In addition, UK is one of the countries that most frequently affected by the where is the probability determined from the wind speed histogram for wind speed 339 , is the probability predicted by the Weibull distribution function for , and This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

SWT-2.3-93 wind turbine. Coloured version is available online 403 404
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0038001 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0038001 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0038001
The link between the scale parameter and the mean wind speed is clear from 484 comparison of the gradients (Table 3 and Table 1 (Table 3 and Figure 13). 490 The implications of these changes for wind power production can be seen from the 491 WPD and the variation of its seven-year value with time (  [66].Correspondingly, as can be seen 531 in Figure 15, the seasonal variation of Weibull distribution fit is clearly distinguishable, 532 where the wind speed distribution during the summer months of June, July and August 533 tends to be more peaked with smaller scale parameter (i.e, abscissa of the distribution 534 peak), whereas those during the winter months of December, January and February 535 appears to be much wider with lower peaks. Figure 16  This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0038001 [4] that there exists a positive relationship between the wind power output and the 551 electricity demand in the UK, i.e., the availability of wind power during times of peak 552 electricity demand is higher than that at times of low electricity demand. Overall, the 553 broad similarities in the seasonal pattern of wind power and electricity demand is 554 encouraging. 2) The lack of consistent trends over all stations in a region implies the importance 577 of local topographical effects. 578 3) South-East England has a statistically significant increase in annual mean wind 579 speed, but this amounts to less than 0.5 −1 over the entire period. 580 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0038001 4) The probability distributions are modelled well using a Weibull distribution. 581 The scale parameter follows trends which are similar to those of the annual mean wind 582 speed, though with a greater proportion of statistical significance; the trends in the 583 shape parameter are significant for all regions.