Vertical wind speed extrapolation: Modelling using a response surface methodology (RSM) based on unconventional designs

Abstract

The main objective of this paper is to develop a predictive model of vertical wind speed profile. Response surface methodology (RSM) is used for this purpose. RSM is a set of statistical and mathematical techniques useful for the development, improvement and optimisation of processes. It is mainly used in industrial processes and is successfully applied in this paper to model the wind speed at the hub height of the wind turbine. An unconventional model is adopted due to the nature of the input parameters which cannot be controlled or modified. The model validation indicators, namely correlation coefficient ( $R^{2} = 0.99$ ) and root mean square error (RMSE = 1.02), give excellent results when comparing predicted and measured wind speeds. For the same data, the RSM model gives a better RMSE compared to the conventional power law and the artificial neural network.

Keywords

Wind profile wind speed extrapolation RSM unconventional designs

Introduction

The world is increasingly moving towards the use of renewable energy, including wind power. Wind energy is really cost-effective because wind is clean, free, domestic and readily available. Every day, taller and taller wind turbines are being installed to capture wind’s power and convert it into electricity. For investors, the correct estimation of site’s wind resource is of great importance and the only way to assess wind correctly is in-situ measurement. Therefore, extrapolation of wind speed data to different hub heights is a crucial and important step prior to the installation of wind farms (Akdag et al., 2013). Extrapolation techniques with the most accurate models possible are proving to be a good alternative to the costly installation of taller towers. Two laws are widely used in the wind industry: the power law (Emeis, 2013), empirical but which gives good results while being very simple, and the logarithm law (Emeis, 2013), resulting of the boundary layer theory. Although simple, the accuracy limitations of both methods have been discussed in various studies (Lubitz, 2009; Optis et al., 2016). Recently, significant researches have been carried out to overcome the limitations of conventional methods. These are methods involving artificial intelligence techniques such as machine learning (Alao and Konneh, 2009; Bodini and Optis, 2020; Cheggaga, 2018; Mohandes et al., 1998; Saiful Islam et al., 2017). In most of them, the models are trained on historical observation data to adjust the model parameters based on wind speeds only. However, the addition of other variables such as air temperature could improve the accuracy of the models (Adjiri et al., 2018).

In this paper, a simple and effective approach, based on the response surface methodology (RSM), is proposed for extrapolation of wind speed from 10 to 50 m. Wind speed and air temperature are considered as model parameters. The resulting model is used to predict the vertical wind speed profile. The results of the proposed approach are then compared with those of the conventional power law technique and the artificial neural network approach.

RSM is a powerful technique initially developed by Box and Wilson in chemical processing (Box and Wilson, 1951). Its field of application is constantly expanding, addressing problems in physics (Charles et al., 2014; Hannane et al., 2013; Kirouani et al., 2018) mechanics (Alao and Konneh, 2009), food engineering (Khuri, 2017) and even human sciences (Humberg et al., 2019; Meyer, 1963). RSM consists of a collection of mathematical and statistical techniques allowing the studying of the behaviour of a system. It is used for modelling, analysing and optimising problems in which a response is influenced by several input variables and the objective is to optimise this response. The input variables are sometimes called independent variables, and they are subject to the control of the engineer or scientist, at least for purposes of a test or an experiment (Montgomery, 2013). A multiple regression model of the empirical data is determined by solving multivariate equations simultaneously (Myers et al., 2009). Once a satisfactory model is obtained, it can be used for predicting future responses for given settings of the input variables or for determining the optimum settings of input variables that result in the optimum of the response over a certain region of interest (Khuri and Mukhopadhyay, 2010). The advantage of this technique lies in the fact that it disregards the studied system and considers only the inputs, the outputs and the relationship between them, making it a powerful tool for reflection and analysis.

There are several types of designs of experiments in RSM (Alao and Konneh, 2009; Goupy and Creighton, 2007; Montgomery, 2013). The choice of a design depends on the needs of the study, and experiments are conducted to maximise the information that can be extracted in a limited number of experiments.

The experiments are usually presented in tables that indicate the levels of the different factors, and the corresponding designs are optimal or near-optimal designs. In many situations, the experimental constraints do not allow to be in ideal conditions and real experiments can be so different from textbook designs. In such case, (Goupy, 1996) has demonstrated that all kinds of experiments can be treated using experimental design methodology employing some caution. Thus, a set of experimental results that have not been obtained according to a conventional design can be used if the experimental points are appropriately placed. In other words, even if the levels of the factors cannot be imposed and manipulated according to a pre-selected design, they can be taken as close as possible.

In this study, the data come from in-situ climate measurements at a meteorological station. An unconventional two-level factorial design is used to model the wind speed at the hub of the turbine since the parameters involved cannot be controlled or imposed.

Wind speed is considered as a response (explained variable) of the system. It is also included in the input variables (explanatory variables) with a difference in height. Therefore, there is an unavoidable dependency between the input variables (wind speed and temperature). However, this does not prevent the use of multiple regression, and thus RSM, since multiple regression is not only applicable to independent variables (Dette et al., 2013; Erkel-Rousse, 1995). Moreover, when validating the model, the RMSE obtained is an element that justifies the use of the RSM approach.

This paper is structured as follows:

In section 2, the study site, the data set and the wind speed prediction model are described,

Section 3 explains the validation of the methodology, the results of the evaluation and the discussion, and

In Section 4, the conclusion is discussed.

Materials and methods

Data

The study site is Ksar Chellala, a semi-arid region of the Algerian highlands in the western part of the wilaya of Tiaret. A meteorological mast was installed on relatively neutral ground (35.21°N latitude, 2.32°E longitude and 839 m altitude). The data used for this study were provided by the ONM (Office National de Météorologie). They were recorded in-situ in the period January to December 2007 using a measuring mast equipped with an ENERCO system (Table 1). Anemometers are placed at 10 and 50 m for wind speed measurements, and a thermometer at 10 m for temperature measurements.

Table 1.

Characteristics of sensors used for the data collection.

Parameter	Sensor type	Sensor height (m)	Operating range	Accuracy
Wind speed	Pulse sensor wind speed	10 50	0.3–50 m/s	±0.1 m/s
Temperature	Platinum resistance Thermometer (PT100)	10	−40°C to +60°C	±0.1°C

Hourly data are used to build the RSM model. Part of the data is used to build the predictive model while the other part is used to validate the resulting model. In addition, they have been processed to eliminate any errors and missing data on wind speed or temperature have been removed. A summary of the monthly data is presented in Table 2.

Table 2.

Summary of used data.

	Temperature at 10 m (°C)			Wind speed at 10 m (m/s)			Wind speed at 50 m (m/s)
	Min	Max	Average	Min	Max	Average	Min	Max	Average
Jan.	−9.900	21.000	2.831	0.480	15.020	5.553	0.440	17.010	7.372
Feb.	−8.400	26.500	7.336	0.540	17.820	6.798	0.410	19.720	8.551
Mar.	−1.500	28.100	10.022	0.410	20.180	6.294	0.360	22.850	7.917
Apr.	−1.600	31.500	14.316	0.720	16.580	6.659	0.890	18.550	8.275
May	7.500	39.100	7.500	0.360	15.150	6.500	0.360	17.370	8.075
June	9.300	38.700	22.828	0.360	15.510	5.944	0.400	17.460	7.230
July	15.000	41.000	26.973	0.400	12.970	5.243	0.360	15.720	6.539
Aug.	16.000	44.400	30.178	1.160	11.350	5.898	1.530	13.170	7.509
Sep.	4.800	45.000	23.454	0.360	14.840	6.365	0.360	16.450	7.942
Oct.	−3.900	36.600	16.786	0.470	14.490	5.837	0.650	16.210	7.618
Nov.	−10.000	26.400	5.070	0.360	15.040	5.633	0.390	16.920	7.259
Dec.	−17.100	18.100	−3.312	0.360	19.170	6.418	0.360	21.650	7.947

Building the RSM model

One of the main objectives of RSM is to establish an approximate relationship, between the response and the input variables or factors, which can be used to predict response values for given settings of the input variables. To build the RSM model, a methodological approach must be followed:

Definition of the domain of each factor (low level and high level);

Choice of experimental design;

Experimentation;

Interpretation of the results (i.e. analysis and validation);

Decision to stop or continue the study.

Experimental design

Wind speed at 50 m is taken to be the response variable ( $y$ ). Wind speed ( $X_{1}$ ) and air temperature ( $X_{2}$ ) both at 10 m height are the two input variables or factors (Table 3). With two factors each taking two levels (low commonly denoted by (−1) and high by (+1)), the total number of runs is 4 (i.e. 2² as shown in Tables 4 –6).

Table 3.

Experimental domain.

Factor	Unit	Range
Factor	Unit	−1	1
Wind speed at 10 m ( $X_{1}$ )	m/s	00.36	20.18
Air temperature ( $X_{2}$ )	°C	−15.40	45.00

Table 4.

Data of the model with 1 month of data.

Run	Coded values		Real values		Response
Run	$x_{1}$	$x_{2}$	$X_{1}$	$X_{2}$	$y_{mesured}$	$y_{predicted}$
1(A’)	−0.96	−1	0.77	−15.40	2.890	2.8996
2(B’)	0.41	−0.63	14.36	−4.30	16.200	16.2274
3(C’)	−0.91	0.09	1.23	17.60	1.800	1.7717
4(D’)	0.41	0.1	14.37	17.70	16.340	16.3803
−1 level	00.36	−15.40
+1 level	20.18	45.00

Table 5.

Data of the model with 3 months data.

Run	Coded values		Real values		Response
Run	$x_{1}$	$x_{2}$	$X_{1}$	$X_{2}$	$y_{mesured}$	$y_{predicted}$
1(A’)	−0.96	−1	0.77	−15.40	2.890	2.8996
2(B’)	0.41	−0.63	14.36	−4.30	16.20	16.2274
3(C’)	−0.74	0.25	2.98	22.50	3.440	3.4815
4(D’)	0.76	0.2	17.82	20.90	19.720	19.7400
−1 level	00.36	−15.40
+1 level	20.18	45.00

Table 6.

Data of the model with 6 months data.

Run	Coded values		Real values		Response
Run	$x_{1}$	$x_{2}$	$X_{1}$	$X_{2}$	$y_{mesured}$	$y_{predicted}$
1(A’)	−0.96	−1.00	0.77	−15.40	2.890	2.8996
2(B’)	0.41	−0.63	14.36	−4.30	16.200	16.2274
3(C’)	−0.86	0.80	1.73	39.10	1.940	1.9155
4(D’)	0.53	0.59	15.51	32.60	17.420	17.4064
−1 level	00.36	−15.40
+1 level	20.18	45.00

As $X_{1}$ and $X_{2}$ are meteorological parameters that cannot be controlled but can nevertheless be observed and measured, it is practically impossible to set them to −1 or +1 (according to a 2² factorial experimental design). Therefore, an unconventional factorial design is adopted to model the response $y$ as a function of $X_{1}$ and $X_{2}$ .

The choice of the number and the placement of experimental points is the fundamental problem of designs of experiments (Goupy and Creighton, 2007). In an unconventional design, there can be as many experimental points as one wants and the factors can take any value (Goupy, 1996). For wind modelling, the unconventional factorial design 2² has 4 experimental points that differ from the corners of the experimental domain but are chosen to be as close as possible (Figure 1). The levels of the factors are therefore not always exactly −1 or +1 (Table 3), unlike a standard or a conventional 2² factorial design where each factor is measured at two levels, coded to take the values −1 and +1.

Wind speed prediction model

There are two important models in the RSM: the first-order model (equation (1)) and the second-order model (equation (2)) (Khuri and Mukhopadhyay, 2010)

y = a_{0} + \sum_{i = 1}^{n} a_{i} x_{i} + e

(1)

y = a_{0} + \sum_{i = 1}^{n} a_{i} x_{i} + \sum_{i < j} \sum a_{ij} x_{i} x_{j} + \sum_{i = 1}^{n} a_{ii} x_{i}^{2} + e

(2)

where $y$ is the observed response; $x_{i}$ represent the coded variables; $a_{0}$ , $a_{i}$ , $a_{ij}$ and $a_{ii}$ are the unknown coefficients to be determined; $n$ is the number of studied parameters; and $e$ is the approximation error.

The first model (equation (1)) is usually used in the initial phase of the experiment, as part of an exploratory process to assess important factors. The second model (equation (2)) is then developed for the identification of important factors to be taken into account in the experiment. It is used to determine the significance of the model parameters, to arrive at optimal operating conditions on the control variables.

In much RSM work it is convenient to transform the real variables in coded variables which are usually defined to be dimensionless (i.e. without their natural units such as degrees Celsius (°C), metres per second (m/s), or grammes (g)). Each variable $x_{i}$ is coded according to the following equation:

x_{i} = \frac{2 X_{i} - (X_{iMax} + X_{iMin})}{X_{iMax} - X_{iMin}}

(3)

with $X_{iMax}$ and $X_{iMin}$ respectively the greatest and the smallest value that can take the real variable $X_{i}$ .

The mathematical model can be approximated by fitting by the polynomial regression technique (Box and Wilson, 1951) as following:

y = a_{0} + a_{1} x_{1} + a_{2} x_{2} + a_{12} x_{1} x_{2}

(4)

with $y$ the predicted response (wind speed at 50 m) and $x_{1}$ , $x_{2}$ the coded variables (wind speed and air temperature at 10 m).

In this study, three models are formulated using 1, 3 and 6 month’s data (Tables 4 –6). Figure 1 shows, in a, b and c, the scatter plot (Goupy and Creighton, 2007; Montgomery, 2013) of data wind speed and temperature measured at 10 m in the coded values for 1, 3 and 6 months respectively.

The three models gives the same value for the coefficient of correlation $R^{2}$ (Table 7). The model with 6 months of data gives the better RMSE (Root Mean Square Error). It is also the one whose experimental points A’, B’, C’ and D’ are closest to the corners of the experimental domain (Figure 1). The closer the experimental points are to the corners of the experimental domain, the better the model built (Goupy, 1996). It is easy to see that as the number of data increases, the experimental points become closer and closer to the corners of the experimental domain (bounded by the values (−1) and (+1) that the factors take).

Table 7.

Results of the three models.

Model	$R^{2}$	RMSE
1 month data	0.99	1.15
3 months data	0.99	1.25
6 months data	0.99	1.02

Results and discussion

The model used for estimating the vertical profile of wind speed is the one with 6 months of data. With only four runs (points), the resulting model takes the following polynomial form:

y = 11, 83846 + 10, 51569 x_{1} - 0, 39887 x_{2} + 0, 77888 x_{1} x_{2}

(5)

As the wind speed is included in both the system response and the input variables, with different heights of course, this implies that the input variables (wind speed and temperature) are dependent. This seems to be a problem for the application of the RSM method where the input variables are generally assumed to be independent. However, the multiple regression technique used to build the model required by RSM can be used for both dependent (Dette et al., 2013) and independent variables.

In Figure 2 (Goupy and Creighton, 2007; Montgomery, 2013), the data used to build the predictive model is analysed to check the relationship between the measured and predicted response. It is shown that the data points used for building the model are distributed close to a straight line, which suggests an excellent correlation between the measured and predicted response value. The value of the coefficient $R^{2} = 0.99$ indicates that a simple variation of the wind speed at 50 m is 99% explained by the input variables for the four selected points. Moreover, this high value suggests a high significance of the model (equation (5)) and a good fit to the measured data. It also indicates that the position of the four experimental points in the experimental domain was adequate.

Wind speeds measured at 50 m height and those predicted by the RSM model are shown. Figure 3 represents 1 day. Figure 4 represents a month, and Figure 5 represents a year.

In order to check the adequacy of the model, RMSE is calculated. It indicates the average deviation. Table 8 shows a comparison, in terms of RMSE, between different schemes: RSM, artificial neural network (ANN) (Cheggaga and Ettoumi, 2011) and power law (PL) (Kasbadji-Merzouk et al., 2007). ANN was chosen because it is one of the recently used machine learning approaches for wind extrapolation and PL was chosen because it is one of the mostly used approaches in wind industry. The results obtained in Table 8 justify very well the application of RSM approach since it gives the butter RSME, and even if the input variables are dependent.

Table 8.

Root mean square error for different methods.

Model building period	Method
Model building period	RSM	ANN	PL
1 month	1.15	4.31	45.21
3 months	1.25	–	45.21
6 months	1.02	–	45.21

Conclusion

The purpose of this work is using RSM technique in modelling and predicting wind speed at different high levels (here 50 m) for the improvement of estimation of wind energy producible.

Since it is not possible to control the climatic parameters involved (input parameters, i.e. wind and temperature), an unconventional design is applied. Data collected in-situ by the meteorological station of Ksar chellala are used for the construction and validation of the model. For the same data, the error indicators give good results in comparison with the other methods, namely the power law and the ANN. The expected objective is therefore achieved.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

F. Adli

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