Comparing different solutions for forecasting the energy production of a wind farm Darío Baptista1-2, Joao Paulo Carvalho2, F. Morgado-Dias1-3 1
M-ITI, Madeira Interactive Technologies Institute, Funchal, Portugal 2 INESC-ID, Instituto Superior Técnico, Lisboa, Portugal 3 UMa, Universidade da Madeira, Funchal, Portugal
[email protected];
[email protected];
[email protected] Corresponding author contact information:
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Abstract — The production of different renewable and nonrenewable energies sources can be coordinated efficiently to avoid costly overproduction. For that, it is important to develop models for accurate energy production forecasting. The energy production of wind farms is extremely dependent on the meteorological conditions. In this paper, computational intelligence techniques were used to predict the production of energy in a wind farm. This study is held on publicly accessible climacteric and energy data for a wind farm in Galicia, Spain, with 24 turbines of 9 different models. Data pre-processing was performed in order to delete outliers caused by maintenance and technical problems. Models of the following types were developed: Artificial Neural Networks (ANN), Support Vector Machines (SVM) and Adaptive NeuroFuzzy Inference System (ANFIS) models. Furthermore, the persistence method was used as a time series forecast baseline model. Overall, the developed computational intelligence models perform better than the baseline model, being ANFIS the model with the best results: a ~5% performance improvement over the baseline model. Keywords — support vector machine; artificial neural network; adaptive neuro fuzzy inference system; persistence method; wind farm; time series forecast; wind power; eolic energy
I. INTRODUCTION Currently, there are several ways to get energy from different natural resources. Energy production of wind farms is rapidly expanding into a large-scale industry. This kind of resource is non-polluting and has the advantage of being installed in remote places that were previously not covered by the electrical grid. For that reason, wind farms have been increasing in many European countries, and as an example, 4% of the energy consumption in Spain already originates from wind farms [1].
integration into the electrical grid. Therefore, accurate forecasting models able to predict the energy production from wind, are a very important means to improve the management of electricity grid. In the literature there are some interesting solutions. However, as most wind farms are recent, it is normal that their performance has not been satisfactorily studied. Barbounis et al. [2] developed a Recurrent Neural Network (RNN) based on the climatological data. Li et al. [3] developed an Artificial Neural Network (ANN) for wind turbine energy estimation. Torres et al. [4] developed a times series Auto-Regressive Moving Average (ARMA) model after transformation and standardization. Damousis et al. [5] used a fuzzy logic model trained using genetic algorithm. Tande et al. [6] examined 10 seconds forecasts for the 1s average output of a wind turbine and verified that an ANN did perform only marginally better than persistence method. Fernandes et al. [7] developed an ANN to predict the energy produced in a wind farm with 6 VESTAS turbines. Bechrakis et al. [8] used an ANN to utilize information from upwind direction. Sfetsos et al. [9] developed AutoRegressive Integrated Moving Average (ARIMA) and FeedForward Neural Network (FFNN) models to wind speed timeseries data from United Kingdom and Greece. Zeng et al. [10] proposed a wavelet support vector machine (SVM) for a shortterm wind forecast. In Kusiak et al. [11], the wind speed was predicted fairly accurately based on its historical values examines time series models for forecasting the energy produced by a wind farm with 100 turbines. Zhang et al. [12] developed a two-layered ANN for predicting the wind speed to determine the wind energy.
The energy production through wind is determined by wind speed. In turn, the wind speed is easily influenced by obstacles and terrain. Thus, the energy generated from wind is uncertain and, as consequence, the operation for its integration in the electrical grid is affected. In order to minimize this problem, detailed schedule plans should be made by the grid regulators. So, an accurate forecasting of energy wind production is needed.
More recently, Ma et al.[13] developed a system which combines a de-noising method with a dynamic fuzzy neural network. Santamaría-Bonfil et al. [14] developed a SVR (Support Vector Regression) to predict wind speed, whose parameters are tuned by a genetic algorithm. The proposed method is compared against the persistence model, and autoregressive models. Yang et al.[15] proposed a hybrid forecasting model, combining the decomposition and ensemble strategy and fuzzy time series forecasting algorithm.
The uncertain behavior of wind is the major obstacle for forecasting wind farm energy production and its further
In this paper, three time series 1-hour ahead forecast models were developed to predict the energy produced by a wind farm
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composed of turbines with different features. The models’ inputs were optimised based on a model order study, and as a result, only 4 wind speed lags and 1 wind farm energy lag are used. A baseline model was developed in order to have a proper a performance comparison. II. WIND FARM DESCRIPTION The wind farm used as a case-study for this work is located in the south-west of Europe, in Galicia, Spain, in “A Serra da Loba”. Its altitude varies between 600 and 700 meters above sea level. It has an installed capacity of 17.56 MW, consisting of 24 wind turbines of 9 different models (the features are presented in table I) [16] . TABLE I.
WIND TURBINE GENERAL FEATURES
Fig. 2. Wind turbines distributions in “Serra da Loba”.
When the wind speed sensors (anemometers) detect wind from any direction, the controller sends the following commands to the turbine:
Features Turbine Type
Turbine ID
Neg Micon NM48 750
2-8-13-20
48
45
4
750
Gamesa G-47
3-7-14-19
47
45
4
660
Made AE-46
6-10-17-23
46
45
4
660
Izar-Bonus MKIV
5-9-18-22
44
40
4
600
Ecotecnia 44/640 4-11-15-21
44
46
4
640
Neg Micon NM52 900
12
48
45
1
900
The wind speed exceeds a margin limit. i.e., when the wind speed is higher than 25m/s or 90Km/h.
Made AE-52
16
52
50
1
800
Malfunction detection.
Made AE-61
24
61
60
1
1320
Supervision routines interventions.
Izar-Bonus 1.3 MW
1
62
49
1
1300
Maintenance interventions.
Emergency reasons: This occurs when faced with turbine’s safety risk or to avoid risk to people.
Rotor Tower Unit Mounted diameter height Power Units (m) (m) (kW)
An image of the wind farm can be seen in figure 1. Figure 2 shows the turbine distribution inside the wind farm in Serra da Loba.
When the wind speed is between 2m/s to 3m/s, the controller sends an order to position itself against the wind. This command is called turbine orientation.
When the speed is higher than 3m/s, the controller disengages the brakes to allow the rotation of the turbine.
The controller sends the blade position command from 90º - 0º.
Turbine stoppage may occur due to several reasons such as, for example:
III. DATABASE PREPARATION The process of data preparation is an important modeling step since the acquired data may contain errors that hinder the modeling process. For that reason, in this section a statistic description and a process of inspecting and cleansing are presented. A. Data pre-processing The power installed in a wind farm is equal to the addition of the power of all turbines existing in the farm. For each turbine, the power density (W/m2), P, is directly proportional to the wind speed, v, and it is given by [17]: 1
𝑃 = 𝜌𝐴𝑣 3 2
(1)
Fig. 1. Image of wind farm in “A Serra da Loba”.
2
Where, A is the swept area of a wind turbines blades in m2, ρ is the density of air under normal conditions for temperature and pressure (1.225Kg/m3 - according to UNE 61400-12). Figure 3 shows the relation between the measured power produced by the wind farm and measured wind speed with a sampling time, T, equal to 1 hour.
As consequence, there are some interruptions in the time series. So, it was necessary to select sequences without interruptions to train and to test the models. The data sequence used to train and test the models will be discussed afterwards. Figure 5 presents the effect of applying the condition to the data. The first illustration shows the data in blue dots. The second illustration shows the data in blue dots and the power curve in a red dashed line. Finally, the third illustration represents the data after applying the filtering procedure described in figure 4.
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Fig. 3. Relation between wind speed and power produced by the wind farm.
As can be seen in figure 3, a large quantity of data was collected. This data contains some errors produced by measurement errors, malfunctioning of the sensor, interruptions by supervision or maintenance and other technical operations. These errors were revealed by lack of data or by data whose value was out of the expected range. The pre-processing of data involves removing the data which is distant from the power curve. For that, the procedure shown in figure 4 was applied.
Fig. 5. Top: Data measured in the wind farm (blues dots); Middle: Data measured (blue dots) and the power curve (red dashed line); Down: Data after applying the filtering procedure (blue dots).
B. Descriptive Statistic
Initial Data
After the data is pre-processed, the central tendency and their distribution was computed. Table II presents the information about the data after pre-processing. TABLE II. DESCRIPTIVE STATISTICAL MEASURES
Power reject > Curve +σ and Power reject < Curve –σ
Data Filtered
Speed (m/s)
Power (kW)
Maximum
17.7800
16 178.0
Minimum
0.3500
0
Median
5.6300
1 759.4
Mean
5.8641
3 164.5
Standard Deviation
2.5763
3 582.5
Fig. 4. The data filtering procedure
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The probability–probability plot test (P-P plot) and the Kolmogorov–Smirnov test (K-S test) with Lilliefors Significance correction was used to analyze the data distribution [18]. Figures 6 and 7 present the P–P plot test of wind speed and wind farm energy. These plots show the cumulative probability of a variable against the cumulative probability of a normal distribution. If the data has a normal distribution, it will result in a straight diagonal line.
significance value, p, is determined by the comparison of the value of D and the critical values for testing normality [20]. If p > 0.05, the data distribution is not significantly different from normal distribution. Otherwise, the data distribution is significantly different from a normal distribution. Table III shows the K–S test results for wind speed and wind farm energy. Analyzing both data, it can be verified clearly that it presents a non-normal distribution (nonparametric data) because the p values are lower than 0.05 for both tests. TABLE III. THE KOLMOGOROV-SMIRNOV TEST WITH LILLIEFORS SIGNIFICANCE CORRECTION
Fig. 6. Left: P–P plot test of Speed; Right: The data deviation from the normal distribution.
Statistic (D)
Sig. (p)
Wind Speed
0,037
0,000
Wind Farm Energy
0,197
0,000
Thus, analyzing all the tests, it is concluded that the best solution is to use a non-linear model to forecast the energy production. Also, the correlation between the wind speed and the energy production by the wind farm was calculated. The coefficient of the correlation has values between [–1,1]. According Nangolo et al. [21], the correlation values can be grouped in ranges to define the strength of the correlation between variables. Table IV shows the classifications of these ranges. TABLE IV. THE CORRELATION COEFFICIENT AND THEIR RESPECTIVE SCALE.
Fig. 7. Left: P–P plot test of Energy; Right: The data deviation from the normal distribution.
Looking at the P–P plots, the first thing to notice is that the wind energy data presents values with deviation from the ideal straight diagonal line and, consequently, its distributions are not normal. In figure 6, it can be observed some values with a little deviation from the ideal diagonal line (the maximum deviation is smaller than 0.04). So, the following question arises: Is this little deviation enough to declare that the distribution of wind speed values has a non-normal distribution? To clarify this problem the K-S test is used. The K-S statistic value, D, verify whether the dataset is a normal distribution or not [19]. The KS test statistic is defined by [19]: 𝐷 = 𝑚𝑎𝑥𝑥∈ℝ |𝐹 ∗ (𝑥)– 𝑆𝑁 (𝑥)|
(2)
Modulus of coefficient of correlation ( | r | or | ρ | )
Correlation
[ 0.0 ; 0.3 ]
Weak
] 0.3 ; 0.5 ]
Moderate
] 0.5 ; 0.8 ]
Strong
] 0.8 ; 1.0 ]
Very Strong
The Spearman's correlation coefficient, 𝜌, is a test for nonparametric data. It is given by the following equation [22]:
𝜌=
2 1– 6 ∑𝑁 𝑖=1(𝑥𝑖 – 𝑦𝑖 ) 𝑁 3– 𝑁
(3)
Where 𝑁 is the number of samples and (𝑥𝑖 – 𝑦𝑖 ) is the difference between a pairs of scores. In this test, it was concluded that the wind speed was significantly correlated with the wind farm energy, r=0.9656. IV. DEVELOPMENT OF THE TIME SERIES PREDICTION MODEL
Where 𝑆𝑁 (𝑥) is the sample cumulative distribution function and 𝐹 ∗ (𝑥) is the cumulative Gaussian distribution function whose mean and variance are estimated from the sample. The
Time series prediction consist in determining future measures based on known measurements at consecutive time
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intervals. A time series prediction model is the nonlinear AutoRegressive with eXogenous signal (nonlinear ARX), defined as follows:
Following this step, the times series prediction models that appeared to be the most promising were developed.
y^(t) = 𝐹(𝑦(𝑡 − 1), 𝑦(𝑡 − 2), 𝑦(𝑡 − 3), … , 𝑢(𝑡), 𝑢(𝑡 − 1), 𝑢(𝑡 − 2), … ) (4)
The persistence method assumes non-changing values from one sample to the next, that is, it assumes that the conditions at the time of the prediction will be the last observed ones. For that reason, this method is the simplest way of making a one step ahead prediction [24]. If the measured wind speed and wind energy at t are 𝑣(𝑡) and 𝐸(𝑡), then the forecasting wind speed and wind energy at 𝑡 + 𝛥𝑡 can be expressed as the following term:
Where y(t−1),y(t−2),y(t−3),...,u(t),u(t−1),u(t−2),... delayed input and output variables (lags).
are
The Lipschitz criterion was used to determine the lags. This method is based on the continuity property of nonlinear functions, representing the continuous dynamic input-output models [23]. Evaluating the modification of an index, the appropriate order can be determined. Figure 8 presents a surface containing the order index for each combination of 10 lags for wind speed and 10 lags for energy production from wind farm. To determine the order, it is necessary to find the point where the slope of the surface changes from decreasing in a pronounced way to decreasing in a less pronounced way. This methodology is used to find a balance between the ease to establishing the model and its accuracy. Analyzing figure 8, it is verified that the best order is 4 wind speed lags and 1 wind farm energy lag.
A. Persistence Method
𝑣𝑡(+Δ𝑡) = 𝑣(𝑡)
(5)
𝐸(𝑡 + Δ𝑡) = 𝐸(𝑡) (6) This method works pretty well when the changes from step to step are relatively small, and it can indeed outperform many forecast techniques in difficult models [25]. As, such it can be used as a good baseline in forecasting performance. However, it should be noted that whenever the inputs have a large change from sample to sample, the persistence method is not the best forecasting method to use. The main goal of using a baseline is to compare it with more advanced models. If an advanced model achieves performance at or below the baseline model, the technique should be fixed or, in some cases, abandoned. B. Support Vector Machine SVMs are popular machine learning model for classification and regression. SVM regression with a non-linear kernel function is considered a nonparametric technique. The basic idea behind this machine learning technique is mapping the input data into a nonlinear higher dimensional feature space.
Fig. 8. Surface containing the order index for each combination of 10 wind speed lags and 10 wind farm energy lags.
Thus, the 1-hour ahead prediction of wind farm energy, E^(t), is computed using 4 lags for wind speed (S(t),S(t-T),S(t2T) ,S(t-3T)) and 1 lag energy farm wind, E(t-1). Figure 9 shows a schematic of the time series prediction model. Time
E(t-4)
E(t-3)
E(t-2)
E(t-1)
E^(t)
The SVM depends only on a subset of the train data which is essentially the support vectors [26]. To solve SVM problems, the Sequential Minimal Optimization (SMO) was used. SMO performs a series of two-point optimizations. In each iteration, a working set of two points are chosen based on a selection rule that uses second-order information. Then, the Lagrange multipliers are solved analytically using the method described in [27] and [28]. The SVM developed in this work has 635 Lagrange multipliers. Additionally, some regression problems cannot adequately be described using a linear model. The SVM developed to predict the energy produced from wind farm uses a Gaussian function as Kernel. The SVM developed to predict the energy produced has a scale factor equal to 1. C. Artificial Neural Network The ANN employed is a MLP (Multilayer Perceptron) with three layers: input, hidden and output layer. Each of these layers, except for the input layer, contains elements which process the information called artificial neurons [29].
Model
S(t-3)
S(t-2)
S(t-1)
S(t) Time
Fig. 9. Block diagram of the time series prediction model (1-hour ahead prediction).
In this work a hyperbolic tangent and a linear function were used in the hidden and output layer, respectively. The number of neurons in the input, ni, and output layer, no, depends on the
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complexity of the problem to model and the available data. The main difficulty is to know how many neurons should be used in the hidden layer, nh, without unnecessarily increasing the complexity (and based on having enough training data). A decision is often made based on experimentation testing the equilibrium between the convergence and the generalization of the ANN [30] . ANNs with 2 to 14 hidden neurons (20 of each kind) were tested. The superior limit was found according to [31], being the number of neurons required to obtain 90% of accuracy. Figure 11 shows the best ANN according to the number of hidden neurons. The blue line represent the train dataset performance, the green line represent the validation dataset performance and the red line represent the test dataset performance.
V. COMPARISON BETWEEN THE MODELS As has been previously mentioned, some data had to be removed. As a consequence, there are some interruptions in the time series. So, it was necessary to select a sequence without interruptions to train and test the model. For this motive, the largest sequences were chosen to train the SVM, the ANN and the ANFIS. Table V shows the number of samples of each dataset. TABLE V.
NUMBER OF SAMPLES OF EACH DATASET
Data set
Number of samples
Test Dataset
730
Train Dataset
998
Table VI presents the values of mean absolute error for the SVM, ANN, ANFIS and Persistence method using two different data sets. TABLE VI. MEAN ABSOLUTE ERROR OF ALL THE MODELS DEVELOPED.
Fig. 10. The best network according to the number of hidden neurons.
Taking into consideration the equilibrium between the complexity (number of hidden neurons) and the accuracy, the best ANN to forecast the energy produced is the one that has 12 hidden neurons.
Model
Test Dataset
Train Dataset
Total
SVM
744.86
707.32
726.09
ANN
737.39
705.48
721.43
ANFIS
697.94
689.54
693.74
Persistence Method
750.25
710.50
730.37
D. Adaptive Neuro-Fuzzy Inference System An Adaptive Neuron-Fuzzy Inference System (ANFIS) is a kind of artificial neural network based on Takagi–Sugeno fuzzy inference system. The model combines the principles of both artificial neural networks and fuzzy logic, and captures the potential of both in a single model [32]. The ANFIS architecture contains five layers. The first layer generates the membership grades. The firing strengths are produced by the second layer through the multiplication of incoming signals and the outputs of the t-norm operator results. The third layer normalizes the firing strengths. The next layer computes the first-order Takagi-Sugeno rules for each fuzzy rule. The fifth layer calculates the weighted output and the summation of incoming signals [33]. Obtaining an ANFIS that performs well requires taking into consideration the number of parameters, the number of inputs and fuzzy rules. The parameters are trained using a hybrid learning algorithm. This algorithm applies a combination between the least-squares and the back-propagation gradient descent method for emulating a given training data set. The main advantage of this learning algorithm that it converges much faster.
Analyzing table VI, it can be verified that the ANN, SVM and ANFIS models introduce an improvement when compared with the Persistence Method. In order to get a better notion of the improvement introduced by ANN, SVM and ANFIS, the percentage of error between each developed model and the persistence method is presented in table VI. To compute this percentage, the equation 7 was used. 𝐸% = |
𝐸𝑟𝑟𝑜𝑟𝑚𝑜𝑑𝑒𝑙 – 𝐸𝑟𝑟𝑜𝑟𝑏𝑎𝑠𝑒𝑙𝑖𝑛𝑒 𝑚𝑜𝑑𝑒𝑙 (𝑃𝑒𝑟𝑠𝑖𝑠𝑡𝑒𝑛𝑐𝑒) | × 100 (7) 𝐸𝑟𝑟𝑜𝑟𝑏𝑎𝑠𝑒𝑙𝑖𝑛𝑒 𝑚𝑜𝑑𝑒𝑙 (𝑃𝑒𝑟𝑠𝑖𝑠𝑡𝑒𝑛𝑐𝑒)
TABLE VII. IMPROVEMENT IN PERCENTAGE BETWEEN ANN, SVM AND ANFIS COMPARING TO THE PERSISTENCE METHOD.
Model
Test Dataset
Train Dataset
Total
SVM
0,75%
0.50%
0.625%
ANN
1.72%
0.71%
1.22%
ANFIS
6.97%
3.04%
5.01%
6
According to the results presented in table VII, the ANFIS model is the best performing when predicting the produced energy 1 hour step ahead. Figure 13 presents the comparison between the measured data and the predicted wind farm energy (1 hour ahead) using SVM, as well its absolute error. Figure 14 presents the comparison between the measured data and predicted wind farm power (1 hour ahead) using ANN, as well its absolute error. Figure 15 presents the comparison between the measured data and predicted wind farm power (1 hour ahead) using ANFIS, as well its absolute error. Figures 16 presents the comparison between the measured data and predicted wind farm power (1 hour ahead) using persistence method, as well its absolute error.
(b) Fig. 12. Measured and predicted wind farm energy using artificial neural network and its absolute error. (a) Test Dataset; (b) Train Dataset;
(a) (a)
(b) Fig. 11. Measured and predicted wind farm energy using support vector machine and its absolute error. (a) Test Dataset; (b) Train Dataset;
(b) Fig. 13. Measured and predicted wind farm energy using adaptive neuro-fuzzy inference system and its absolute error. (a) Test Dataset; (b) Train Dataset;
(a) (a)
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REFERENCES
(b)
[1]
I. Sánchez, “Short-term prediction of wind energy production,” Int. J. Forecast., vol. 22, no. 1, pp. 43–56, 2006.
[2]
T. G. Barbounis, J. B. Theocharis, M. C. Alexiadis, and P. S. Dokopoulos, “Long-term wind speed and power forecasting using local recurrent neural network models,” IEEE Trans. Energy Convers., vol. 21, no. 1, pp. 273–284, 2006.
[3]
S. Li, D. C. Wunsch, E. O’Hair, and M. G. Giesselmann, “Comparative analysis of regression and artificial neural network models for wind turbine power curve estimation,” Journal of Solar Energy Engineering, Transactions of the ASME, vol. 123, no. 4. pp. 327–332, 2001.
[4]
J. L. Torres, A. García, M. De Blas, and A. De Francisco, “Forecast of hourly average wind speed with ARMA models in Navarre (Spain),” Sol. Energy, vol. 79, no. 1, pp. 65–77, 2005.
[5]
I. G. Damousis, M. C. Alexiadis, J. B. Theocharis, and P. S. Dokopoulos, “A Fuzzy Model for Wind Speed Prediction and Power Generation in Wind Parks Using Spatial Correlation,” IEEE Trans. Energy Convers., vol. 19, no. 2, pp. 352–361, 2004.
[6]
J. O. . L. L. Tande, “A 10 sec. Forecast of Wind Turbine Output with Neural,” in Proceedings of the 1993 ECWEC, 1993, pp. 774–777.
[7]
F. Fernandes, D; Baptista, D; Jardim, B; Figueira, A, Morgado-Dias, “10th Portuguese Conference on Automatic Control – CONTROLO’12,” in LOIRAL wind park modelling using artificial neural network, 2012.
[8]
D. A. Bechrakis and P. D. Sparis, “Wind speed prediction using Artificial Neural Networks,” Wind Eng., vol. 22, no. 6, pp. 287–295, 1998.
[9]
A. Sfetsos, “A novel approach for the forecasting of mean hourly wind speed time series,” Renew. Energy, vol. 27, no. 2, pp. 163–174, 2002.
[10]
J. Zeng and W. Qiao, “Short-term wind power prediction using a wavelet support vector machine,” IEEE Trans. Sustain. Energy, vol. 3, no. 2, pp. 255– 264, 2012.
[11]
A. Kusiak, H. Zheng, and Z. Song, “Short-term prediction of wind farm power: A data mining approach,” IEEE Trans. Energy Convers., vol. 24, no. 1, pp. 125–136, 2009.
[12]
L. Wu, J. Park, J. Choi, J. Cha, and K. Y. Lee, “A study on wind speed prediction using artificial neural network at Jeju Island in Korea,” in 2009 Transmission &
Fig. 14. Measured and predicted wind farm energy using persistence method and its absolute error. (a) Test Dataset; (b) Train Dataset;
VI. CONCLUSIONS In this paper, time series models were built for predicting the energy of a wind farm with a horizon of 1 hour. The models are appropriate to the electricity market management and its predictive control. The data used in this work is from a wind farm in Galicia. This wind farm presents 24 turbines of 9 different models. A large quantity of data has been collected which contains errors and interruptions. Thus the data was preprocessed by removing the data that is too far away from energy curve. The obtained results show that all developed models outperform the baseline model, being ANFIS the time series model with the best accuracy to predict the power produced by the wind farm. This model presents a 5.01% improvement when compared with the baseline model. The complexity of building the models for this particular farm is increased by the fact that it is composed of wind turbines of different models and types spread out through a large area. Nevertheless the developed models were able to capture the main properties of the system without requiring an individual model for each of the wind towers. It was concluded that it is possible to predict the energy production by the wind farm with a good performance. ACKNOWLEDGMENT Agradecimentos à ARDITI - Agência Regional para o Desenvolvimento da Investigação Tecnologia e Inovação através do apoio concedido no âmbito do Projeto M1420 - 09-5369-FSE-000001- Bolsa de Doutoramento. Acknowledgments to the Portuguese Foundation for Science and Technology for their support through Projeto Estratégico LA 9 - UID/EEA/50009/2013. Conflicts of Interest: The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.
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Distribution Conference & Exposition: Asia and Pacific, 2009, pp. 1–4. [13]
[14]
[15]
X. Ma, Y. Jin, and Q. Dong, “A generalized dynamic fuzzy neural network based on singular spectrum analysis optimized by brain storm optimization for short-term wind speed forecasting,” Appl. Soft Comput. J., vol. 54, pp. 296–312, 2017. G. Santamaría-Bonfil, A. Reyes-Ballesteros, and C. Gershenson, “Wind speed forecasting for wind farms: A method based on support vector regression,” Renew. Energy, vol. 85, pp. 790–809, 2016. H. Yang, Z. Jiang, and H. Lu, “A Hybrid Wind Speed Forecasting System Based on a ‘Decomposition and Ensemble’ Strategy and Fuzzy Time Series,” Energies, vol. 10, no. 9, p. 1422, 2017.
[16]
Engasoft, “sotaventogalicia,” 2005. [Online]. Available: http://www.sotaventogalicia.com/. [Accessed: 10-Jun-2016].
[17]
D. A. Spera, Wind Turbine Technology: Fundamental Concepts of Wind Turbine Engineering, vol. 62, no. 2. 2009.
[18]
H. Coolican, Research Methods and Statistics in Psychology, vol. 2. 1999.
[19]
H. W. Lilliefors, “On the Kolmogorov-Smirnov test for normality with mean and variance unkown,” J. Am. Stat. Assoc., vol. 62, no. 318, pp. 399–402, 1967.
[20]
G. E. Dallal and L. Wilkinson, “An Analytic Approximation to the Distribution of Lilliefors’s Test Statistic for Normality,” Am. Stat., vol. 40, no. 4, pp. 294–296, 1986.
[21]
C. Nangolo and C. Musingwini, “Empirical correlation of mineral commodity prices with exchange-traded mining stock prices,” J. South. African Inst. Min. Metall., vol. 111, no. 7, pp. 459–468, 2011.
[22]
D. J. Best and D. E. Roberts, “Algorithm AS 89: The Upper Tail Probabilities of Spearman’s Rho,” Appl. Stat., vol. 24, no. 3, p. 377, 1975.
[23]
X. He and H. Asada, “A new method for identifying orders from input-output models for nonlinear dynamic systems,” American Control Conference, 1993. pp. 2520–2523, 1993.
[24]
W. Chang, “A Literature Review of Wind Forecasting Methods,” J. Power Energy Eng., no. April, pp. 161– 168, 2014.
[25]
J. P. Carvalho and F. V Camelo, “One Day Ahead Stream Flow Forecasting,” in IFSA- EUSFLAT2015, 16th World Congress of the International Fuzzy Systems Association (IFSA) and the 9th Conference of the European Society for Fuzzy Logic and Technology
(EUSFLAT), 2015, vol. 89, pp. 1168–1175. [26]
A. J. Smola and B. Schölkopf, “A tutorial on support vector regression,” Statistics and Computing, vol. 14, no. 3. pp. 199–222, 2004.
[27]
R.-E. Fan, P.-H. Chen, and C.-J. Lin, “Working Set Selection Using Second Order Information for Training Support Vector Machines,” J. Mach. Learn. Res., vol. 6, pp. 1889–1918, 2005.
[28]
F. Granata, R. Gargano, and G. de Marinis, “Support Vector Regression for Rainfall-Runoff Modeling in Urban Drainage: A Comparison with the EPA’s Storm Water Management Model,” Water, vol. 8, no. 3, p. 69, 2016.
[29]
F. M. Dias, A. Antunes, J. Vieira, and A. Mota, “A sliding window solution for the on-line implementation of the Levenberg-Marquardt algorithm,” Eng. Appl. Artif. Intell., vol. 19, no. 1, pp. 1–7, 2006.
[30]
J. Sjoberg and L. Ljung, “Overtraining, regularization and searching for a minimum, with application to neural networks,” Int. J. Control, vol. 62, no. 6, pp. 1391–1407, 1995.
[31]
D. Baptista, S. Abreu, C. Travieso-González, and F. Morgado-Dias, “Hardware implementation of an artificial neural network model to predict the energy production of a photovoltaic system,” Microprocess. Microsyst., vol. 49, 2017.
[32]
J. S. R. Jang, “ANFIS: Adaptive-Network-Based Fuzzy Inference System,” IEEE Trans. Syst. Man Cybern., vol. 23, no. 3, pp. 665–685, 1993.
[33]
a Abraham, “Adaptation of fuzzy inference system using neural learning,” Fuzzy Syst. Eng., vol. 83, pp. 53–83, 2005.
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