1876-6102 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the organizing committee of the General Assembly of the European Geosciences Union (EGU) doi: 10.1016/j.egypro.2016.10.029
Energy Procedia 97 ( 2016 ) 100 – 107
ScienceDirect
European Geosciences Union General Assembly 2016, EGU
Division Energy, Resources & Environment, ERE
Spatial Modelling of Extreme Wind Speed Distributions in
Switzerland
Mohamed Laib∗, Mikhail Kanevski
Institute of Earth Surface Dynamics, Faculty of Geosciences and Environment, University of Lausanne, CH1015 Lausanne, Switzerland
Abstract
The understanding of wind phenomena is of primary importance, especially regarding the renewable energy. One traditional way of wind modelling concerns the use of parametric probability distributions. This work presents a method in two steps to model this phenomena. The first of which studies and compares different extreme distributions by modelling data collected from the Swiss meteorological service. This comparison should serve as a basis for the second step, which applies spatial modelling of distribution parameters throughout the country. The modelling is performed in a high dimensional input space with the help of extreme learning machine. The knowledge of probability distributions allows us to give a comprehensive information about a wind speed distribution.
c
2016 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of the organizing committee of the General Assembly of the European Geosciences Union (EGU).
Keywords: Wind speed, Extreme values, Machine learning algorithms, Spatial modelling, Switzerland.
1. Introduction
Wind can be regarded as either a positive or negative phenomena. The positive side is the renewable energy it pro-duces, and the negative side is the losses caused by wind-storms. The Swiss Federation considers that electricity from wind power is an environmentally friendly source, and in its Energy Strategy 2050, plans to expand the proportion of wind power [1]. On the other hand there are, however, enormous losses that have been caused by extremely violent wind-storms in Switzerland [2] [3], an excellent catalog of which has been produced by Stucki et al. 2014 [4], and Usbeck et al.[5].
Recently wind data were efficiently treated using parametric approaches, in order to learn more about wind speed distributions. Extreme value theories (EVT) has been the most frequently applied modelling approach. The main purpose is to estimate parameters of a given data distribution [6][7][8], which helps to understand the behaviour of the tail of the distribution on this data. There are many areas where extreme value theory plays an indispensable role for modelling rare event, such as environmental risk (wind, temperature, rainfall, etc.)[9][10][11][12].
∗Corresponding author. Tel.:+41 21 692 35 41.
E-mail address: [email protected] (M. Laib)
© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Nowadays, machine learning algorithms (MLA)[13][14] are rapidly gaining popularity in modelling environmental phenomena[15][16][17]. Machine learning is a part of artificial intelligence that allows us to find non-linear depen-dencies observed in data, and to understand better the structure between the input and the output. This paper uses one of the recently developed algorithm : The Extreme Learning Machine (ELM) proposed by Huang et al., 2006. ELM has the same structure as a classical multilayer perceptron (MLP), but in ELM the weights and biases between input and hidden layer are generated randomly, and then the linear equation from hidden to output layer is solved. ELM has one parameter to be tune which is the number of hidden nodes. Moreover this parameter allows us to control the complexity of the phenomena under study.
The main advantage of this algorithm is the speed of the training step, and the capacity to learn complex data. The techniques of cross-validation and data splitting, as usual, are used to avoid overfitting and to test the accuracy of the model [14].
Modelling extreme values with MLA is a challenging task, and in order to make use of the performance of MLA, several papers use more information which can explain the phenomena [18], such as environmental data to model extreme wind speed [19].
The purpose of this paper is to find, using a high dimensional topographical input space, a flexible approach to under-stand the behaviour of extreme wind speed in Switzerland. This was achieved by using the three most known extreme distributions : Weibull, Gamma, and Generalized Extreme Values (GEV). The approach starts with an estimation of distribution parameters using maximum likelihood. And then a spatial modelling of the distribution parameters is carried out employing ELM, and using a methodology to validate results to avoid over-fitting [16].
The study was carried out by using extRemes and elmNN packages of the R language [20].
The paper is organized as follows: Section 2 describes how the data were prepared, and gives a brief description of the methods used. Section 3 deals with an estimation of extreme distribution parameters, and the spatial modelling of these parameters. In section 4, the main results as well as the discussion are presented, and in the last section the conclusions are given with suggestion for challenging future research.
2. Dataset and methods
2.1. dataset and study area
Data used in this work were downloaded from the website of the Federal Office of Meteorology and Climatology of Switzerland (IDAWEB, Meteoswiss). They present measurements at weather stations distributed uniformly in all Switzerland (fig.1 elevations from 203 m to 3302 m). In total there are 127 stations. However, some stations were eliminated because they contain an important number of missing values. The final dataset contains wind speed measurement of 111 station for a time period between 2009 and 2013, taken at 10 minutes intervals. This high frequency allows us to obtain good approximation of wind speed distribution, even the behaviour of extremes wind speed.
Distribution parameters are estimated for each station using maximum likelihood. And in order to choose the suitable one, a comparison between the different probability distributions is carried out by using a graphical method called Quantile plot, with a visual statistical protocol, as it is proposed in [21].
To predict the estimated parameters in all Switzerland, a high dimensional input space (independent variables) com-posed of thirteen topographical features was constructed (Table 1, see details and description in [22]). Furthermore, a feature selection approach is used to choose the best set of variables which helps to model better each parameter. It was performed by considering all possible combinations of input spaces ([213− 1] = 8191) and selecting the subset
with a minimum validation error of ELM.
2.2. Statistical modelling
Traditional statistical methods are efficiently used in environmental data analysis. Wind speed is well known by the presence of extremes, and therefore, EVT is an appropriate approach. Several papers have proposed Weibull distribution for wind data [23]. This work compares three extreme distributions and then chooses the best one. These distributions are the following:
1000 2000 3000 4000
Altitude (m)
Fig. 1: Locations of MeteoSwiss stations.
Table 1: Input space features generated from digital elevation model.
ID Name of the variable Scale
X X coordinate
Y Y coordinate
Z Z (elevation)
dogs Diff. of Gauss. at small scale σ1= 0.25 km σ2= 0.5 km
dogm Diff. of Gauss. at medium scale σ1= 1.75 km σ2= 2.25 km
dogl Diff. of Gauss. at large scale σ1= 3.75 km σ2= 5 km
S s Slopes at small scale σ = 0.2 km
S m Slopes at medium scale σ = 1.75 km
S l Slopes at large scale σ = 3.75 km
dns Dir. deriv. in South–North dir. at small scale σ = 0.25 km
dws Dir. deriv. in East–West dir. at small scale σ = 0.25 km
dnm Dir. deriv. in South–North dir. at medium scale σ = 1.75 km
dwm Dir. deriv. in East–West dir. at medium scale σ = 1.75 km
2.2.1. Weibull distribution
It is a two parameters distribution, with the following density function [24]:
f (x;λ, k) =⎧⎪⎪⎪⎨⎪⎪⎪⎩ k λ(xλ)k−1e−( x λ)k x≥ 0 0 x< 0 (1)
where k is the shape parameter andλ is the scale. This function is a continuous probability distribution and mostly used to describe wind data.
2.2.2. Gamma distribution
Gamma distribution is also a two parameter distribution. It used to present several phenomena especially those that varies over time, and it is defined by the following density function:
f (x;α, β) =β
αxα−1e−xβ
Γ(α) f or x≥ 0 and α, β > 0 (2)
Whereα is the shape and β is the rate [25].
2.2.3. Generalized Extreme Values
It combines three families of distributions: Gumbel, Fr´echet and Weibull. The GEV distribution is used to model the maxima of long sequences of random variables, and the treatment of risk [26]. It is defined as follow:
F(x;μ, σ, ξ) = exp{−[1 + ξ(x− μ
σ )]
−1
ξ} (3)
For 1+ ξ(x − μ)/σ > 0, where μ is the location parameter, σ is the scale parameter and ξ the shape. This latter indicates the tail behaviour of the distribution, and the sub-families are defined as following:
• Gumbel distribution or type I when ξ = 0. • Fr´echet or type II when ξ > 0.
• Weibull distribution or type III when ξ < 0.
2.3. Extreme Learning Machine
This paper uses an extreme learning machine as algorithm to model wind speed data. Proposed by Huang et al. 2006, ELM is inspired by the multi-layer perceptron (MLP) with one hidden layer. For a fixed number of hidden nodes
N, ELM generates randomly the weight and the biases of each node. Then the result passes through a differentiable
activation function g which gives the matrix H where each row corresponds to the output of hidden layer for one input data vector:
Hi j= g(xi.wj+ bj) (4)
where xi = x1i, x
2
i, . . . , xdi (i= 1, . . . , n) are the input data, wj( j= 1, . . . , N) are the vector of weights, and bjare the
biases of each node.
Finally, to get the connection vectorβ between the hidden layer and the output layer, ELM uses the Moore-Penrose generalized inverse of the matrix H:
β = H†y
At the end of these steps, new data points can be predicted as well as the validation and testing errors.
3. Modelling extreme wind speed
3.1. Estimation of distribution parameters
In the present work the maximum likelihood is used to estimate distribution parameters.
It is a very simple tool to find estimators, it chooses the value of the parameter which maximizes the likelihood of seeing the sample data. The likelihood function is [26]:
L(θ) =
n
i=1
where xiare independent realizations of a random variable with a probability density function f (xi;θ). As is known, it is more convenient to work with the log-likelihood function:
log L(θ) =
n
i=1
log f (xi;θ) (6)
The log-likelihood takes its maximum at the same point as the likelihood function, and it is found by differentiating the log-likelihood and equating to zero.
For each measuring station, the parameters for the three proposed probability distributions are estimated.
0 2 4 6 8 10
02468
10
12
Q-Q plot for Weibull 1-1 line 95% confidence bands 0 2 4 6 8 10 02468 10 12
Q-Q plot for Gamma 1-1 line 95% confidence bands 0 2 4 6 8 10 02468 10 12
Q-Q plot for GEV 1-1 line 95% confidence bands 0 2 4 6 8 10 12 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Histogram of Magadino-Cadenazzo Wind Speed D ens ity Weibull Gamma GEV
Fig. 2: Examples Magadino Cadenazzo 203 meters.
the QQplot is used to choose the best probability distribution regarding extremes (Fig.2, example of the QQplot). In our case, the used data are well-modelled by the GEV. For this reason The remain work is based on the Generalized extreme values distribution.
3.2. Spatial modelling
For the purpose of spatial modelling, extreme parameters are predicted in all Switzerland using extreme learning machine. Which was applied on the thirteen dimensional input space based on topographical information. The methodology used can be described as follows:
• The data are projected into the interval [0,1], and then are split it into training and testing set, in total 31 measuring stations are assigned as testing set and 80 as training set.
• The remain data are used to train ELM with N number of hidden nodes where N ∈ {1, . . . , 100}
• The optimal number of nodes N is selected by using k-fold cross-validation with respect to the mean square error.
• Generate the optimal model and evaluate the mean square error for the testing set.
• This process of learning is repeated 20 times, with random splitting of the data, and at the end the mean of repetition is taken.
The same process is carried out, For the three parameters of GEV.
0 2 4 6 8 10 05 10 15 4í4SORWfRU*(9 íOLQH FRQILGHQFHEDQGV 0 5 10 15 0.0 0.1 0.2 0.3 0.4 +LVWRJUDPRI0RQWH5RVDí3ODWWMH :LQG6SHHG 'HQVLW\ (VWLPDWHGE\0/( 3UHGLFWHGE\(/0 (VWLPDWHGE\0/( 3UHGLFWHGE\(/0
Fig. 3: Comparison between predicted and estimated distribution for a testing measuring station.
4. Results and discussions
The results presented allow us to visualize better each parameter. (fig. 4 variability of location (μ) parameters of the GEV) distribution. Moreover, the produced map is coherent with the topographical information provided by the input space. 2 3 4 5 6 7 8
Fig. 4: Visualisation of the location parameter of GEV distribution.
Furthermore, the advantage of knowing the extreme distribution allows us to predict the probability for a given wind speed and vice versa, fig.5 presents probability map that a wind speed exceeds 20 m/s.
0.00 0.05 0.10 0.15 0.20
Fig. 5: Probability map for wind speed greater than or equal to 20 m/s.
One of the most important results in this work is the possibility to predict extreme wind speed in new places without measuring stations. And also the comparison between the three proposed distributions offers more precision to the predicted parameters.
5. Conclusion
Wind energy remains the most attractive resource for providing sustainable power. The research presented here, gives an initial exploration of wind speed data, and a global idea about extremes of this phenomena in Switzerland. Such modelling can be useful in developing intelligent decisions for wind-powered electrical generators. Spatial modelling of distributions can be used to optimize existing network and to propose new places for the aeolian energy production. Combination of parametric models and MLA, especially ELM, is a practical approach to model complex phenomena like extremes, which is important for natural hazards and risk assessments.
This research is an example of application of machine learning algorithms for extreme wind speed, and it can be applied to other extreme environmental phenomena, e.g. precipitation.
Future developments can be in adaptation of the methodology for spatio-temporal environmental data and quantifica-tion of the uncertainties.
Acknowledgements
The authors are grateful to Jean Golay and Michael Leuenberger for many fruitful discussions about machine learning and extreme values.
The research was partly supported by the Swiss Government Excellence Scholarships for Foreign Scholars. The authors thank MeteoSuisse for giving access to the data via IDAWEB server.
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