  # Long-term Correlations and Extreme Wind Speed Estimations

Fund Project:

National Key R&D Program of China (Grant No. 2016YFC0208802) and the National Natural Science Foundation of China (Grant Nos. 41675012 and 11472272)

• In this paper, we use fluctuation analysis to study statistical correlations in wind speed time series. Each time series used here was recorded hourly over 40 years. The fluctuation functions of wind speed time series were found to scale with a universal exponent approximating to 0.7, which means that the wind speed time series are long-term correlated. In the classical method of extreme estimations, data are commonly assumed to be independent (without correlations). This assumption will lead to an overestimation if data are long-term correlated. We thus propose a simple method to improve extreme wind speed estimations based on correlation analysis. In our method, extreme wind speeds are obtained by simply scaling the mean return period in the classical method. The scaling ratio is an analytic function of the scaling exponent in the fluctuation analysis.
摘要: 风速的极值估计在风工程结构设计中具有广泛的应用。现有的极值估计方法假设风速时间序列中不存在统计相关特征。然而，真实的风速往往具有“聚团”现象，即较大的风速值往往聚集出现。“聚团”现象不存在于统计独立的时间序列中，表明风速时间序列存在明显的统计相关特征。本文利用起伏分析方法，分析了持续时间长达40年的9组小时风速历史记录。研究结果表明，风速时间序列的起伏指数均约等于0.7，大于统计独立时间序列的起伏指数0.5，证明了风速时间序列普遍具有长程统计相关的特征。因此，基于统计独立假设的传统方法对风速极值的估计必然存在偏差，而这一偏差体现在伴随长程相关特征出现的“聚团”现象，引起了平均回归时间与极值发生概率的关系出现偏差。本文通过数据分析，提出了长程相关风速时间序列极值的平均回归时间与极值发生概率的关系，并基于这一关系，提出了风速极值估计的新方法。这一方法可以通过一个简单的变换关系修正传统方法对风速极值的高估。
• • Figure 1.  Fluctuation analysis of wind speed time series listed in Table 1. Lines are linear fittings in log-log plots, and the fitted scaling exponent of the fluctuation function $\alpha$ is also shown in each plot.

Figure 2.  Probability density functions of return periods for wind speeds (the Schiphol data) greater than different thresholds of $v$ . Note that the return period is divided by its mean. For comparison, the exponential distribution (dashed line) and the stretched exponential distribution (line) are also shown in this plot.

Figure 3.  The T−P relation for wind speed time series listed in Table 1. Note that the mean return period is divided by ${C_\kappa }{\rm{\Delta }}t$ , where the coefficient $C_\kappa$ is shown in Eq. (9) and $\Delta t$ is the sampling time. The line denotes a reciprocal function of the probability of extreme occurrences (see Eq. (10)).

Figure 4.  (a) The wind speed time series measured at Schiphol from 1 January 2005 to 31 December 2005. (b) The time series in (a) having been randomly shuffled. An arbitrarily chosen threshold is denoted by a dashed line in each panel.

Figure 5.  Diagnostic plots of the GPD fittings to the wind speed time series listed in Table 1. Points show the empirical conditional probability of extreme wind speeds. Lines show the GPD with their parameters estimated by the maximum likelihood method (see Table 2)

Figure 6.  The maximum likelihood estimator of T-year return level ${\hat z_T}$ as a function of the mean return period T. Lines show the estimated return levels and dashed-dotted lines show the 95% confidence intervals. For an illustration of our method, the 50-year return level ${\hat z_{50}}$ without correlations and the corresponding 50-year return level $\hat z_{50}^*$ with long-term correlations are marked by circles in the plot.

Table 1.  The KNMI HYDRA project data used in this paper. ${T_0}$ is the start time of data records and ${N_{\rm{m}}}$ is the number of missing data. All the data end at 31/12/2006.

 Station T0 (day/month/year) Nm Schiphol 1/3/1950 0 De Bilt 1/1/1961 1 Soesterberg 1/3/1958 4 Leeuwarden 1/4/1961 0 Eelde 1/1/1961 1 Vlissingen 2/1/1961 0 Zestienhoven 1/10/1961 4 Eindhoven 1/1/1960 1 Beek 1/1/1962 1

Table 2.  Thresholds $v$ and maximum likelihood estimations of the GPD parameters $\xi$ and $\sigma$ . CI denotes the 95% confidence interval for the parameter estimated.

 Station $v\;\left( {{\rm{m}}\;{{\rm{s}}^{ - 1}}} \right)$ $\xi$ ${\rm{CI}}\left( \xi \right)$ $\sigma$ ${\rm{CI}}\left( \sigma \right)$ Schiphol 9.5 −0.0893 (−0.0967, −0.0819) 2.4566 (2.4282, 2.4854) De Bilt 7.2 −0.0799 (−0.0876, −0.0722) 1.8141 (1.7913, 1.8372) Soesterberg 7.6 −0.0406 (−0.0491, −0.0320) 1.7858 (1.7630, 1.8089) Leeuwarden 9.1 −0.0921 (−0.0994, −0.0848) 2.2890 (2.2609, 2.3175) Eelde 8.3 −0.0831 (−0.0914, −0.0748) 2.0926 (2.0658, 2.1198) Vlissingen 9.5 −0.1142 (−0.1219, −0.1066) 2.3182 (2.2893, 2.3474) Zestienhoven 9.3 −0.1195 (−0.1249, −0.1141) 2.3018 (2.2759, 2.3281) Eindhoven 8.0 −0.0876 (−0.0957, −0.0796) 2.1138 (2.0869, 2.1410) Beek 8.1 −0.1065 (−0.1143, −0.0987) 2.0573 (2.0314, 2.0836)
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

## Long-term Correlations and Extreme Wind Speed Estimations

###### Corresponding author: Fei HU, hufei@mail.iap.ac.cn
• 1. State Key Laboratory of Atmospheric Boundary Layer Physics and Atmospheric Chemistry, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
• 2. University of Chinese Academy of Sciences, Beijing 100049, China

Abstract: In this paper, we use fluctuation analysis to study statistical correlations in wind speed time series. Each time series used here was recorded hourly over 40 years. The fluctuation functions of wind speed time series were found to scale with a universal exponent approximating to 0.7, which means that the wind speed time series are long-term correlated. In the classical method of extreme estimations, data are commonly assumed to be independent (without correlations). This assumption will lead to an overestimation if data are long-term correlated. We thus propose a simple method to improve extreme wind speed estimations based on correlation analysis. In our method, extreme wind speeds are obtained by simply scaling the mean return period in the classical method. The scaling ratio is an analytic function of the scaling exponent in the fluctuation analysis. followshare  DownLoad:  Full-Size Img  PowerPoint