energies

Article

Short-Term Wind Speed Forecasting Based onSignal Decomposing Algorithm and HybridLinear/Nonlinear Models

Qinkai Han 1, Hao Wu 2, Tao Hu 2,* and Fulei Chu 1

1 The State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China;hanqinkai@mail.tsinghua.edu.cn (Q.H.); chufl@mail.tsinghua.edu.cn (F.C.)

2 School of Mathematical Sciences, Capital Normal University, Beijing 100048, China; 2160502002@cnu.edu.cn* Correspondence: hutao@cnu.edu.cn

Received: 9 October 2018; Accepted: 21 October 2018; Published: 1 November 2018�����������������

Abstract: Accurate wind speed forecasting is a significant factor in grid load management andsystem operation. The aim of this study is to propose a framework for more precise short-term windspeed forecasting based on empirical mode decomposition (EMD) and hybrid linear/nonlinearmodels. Original wind speed series is decomposed into a finite number of intrinsic modefunctions (IMFs) and residuals by using the EMD. Several popular linear and nonlinear models,including autoregressive integrated moving average (ARIMA), support vector machine (SVM),random forest (RF), artificial neural network with back propagation (BP), extreme learning machines(ELM) and convolutional neural network (CNN), are utilized to study IMFs and residuals, respectively.An ensemble forecast for the original wind speed series is then obtained. Various experiments wereconducted on real wind speed series at four wind sites in China. The performance and robustnessof various hybrid linear/nonlinear models at two time intervals (10 min and 1 h) are comparedcomprehensively. It is shown that the EMD based hybrid linear/nonlinear models have betteraccuracy and more robust performance than the single models with/without EMD. Among the fivehybrid models, EMD-ARIMA-RF has the best accuracy on the whole for 10 min data, and the meanabsolute percentage error (MAPE) is less than 0.04. However, for the 1 h data, no model can alwaysperform well on the four datasets, and the MAPE is around 0.15.

Keywords: wind speed forecasting; hybrid modeling; EMD; ARIMA; machine learning models

1. Introduction

Wind energy has been growing fast in recent years. By the end of 2017, the worldwide totalcapacity of wind turbines reached 539 GW (52.6 GW added in 2017) [1]. Because of the stochasticvariation of wind speed, wind energy behaves in a more unstable and volatile manner than traditionalenergy sources. Direct integration of unstable wind power will have a serious impact on the wholegrid, especially for the areas with high levels of wind power penetration [2–4]. If the wind speedcould be predicted accurately, the dispatching plan of the power system could be adjusted to reducethe adverse impact of the wind power on the whole grid. It is also beneficial for the improvement ofthe power limit of the wind power penetration. Therefore, accurate wind speed forecasting is veryimportant for grid load management and system operation [5–9].

Many efforts have been made on developing accurate wind speed forecasting models.The autoregressive moving average (ARMA) or autoregressive integrated moving average (ARIMA)models are considered to be the most widespread models in wind speed forecasting. Torres et al. [10]first carried out the transformation and standardization of the original wind speed series to allow

Energies 2018, 11, 2976; doi:10.3390/en11112976 www.mdpi.com/journal/energies

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the use of ARMA models. Shi and Erdem [11] comprehensively evaluated the effectiveness ofARMA-generalized autoregressive conditional heteroscedastic (GARCH) approaches for modelingthe mean and volatility of wind speed. Lydia et al. [12] built various ARMA models with andwithout external variables to predict wind speed at 10-min intervals up to 1 h. Note that theARIMA models are considered as inherent linear models, although curvilinear (or nonlinear) relationscould be incorporated in these models. Besides ARIMA models, machining learning (ML) models,such as artificial neural network (ANN) with back propagation (BP) and radial basis function(RBF) [13,14], support vector machine (SVM) [15], extreme learning machines (ELM) [16], and deeplearning networks (DLN) [17,18], have shown excellent potentials in accurate wind speed predictions.Usually, ML models are considered as nonlinear models as they are more flexible compared withlinear models, and thus could better deal with nonlinear relations. Despite showing superiority overlinear models, the ML models also have their own shortcomings, e.g., dilemma of local minima,the over-fitting problem, poor efficiency with fewer samples. Many studies have shown that singlelinear or nonlinear models could not give satisfactory results for all situations due to their drawbacks[19]. Therefore, hybrid models, by synthesizing the advantages of single models, are the trend ofadvanced wind speed forecasting.

Some signal de-noising measures need to be implemented to make the wind speed series lessnoisy and more stable. Empirical mode decomposition (EMD) [20] is a de-noising method basedon local characteristics of the signal. EMD absorbs the advantages of multi-resolution of waveletdecomposition, overcomes the difficulty of determining the wavelet base and decomposition scale,and is more suitable for nonlinear and non-stationary signal sequences [21]. Thus, EMD is usuallyused to decompose the original wind speed series into a small number of intrinsic mode functions(IMFs) and residuals, which will be respectively studied by various models to formulate a hybridforecast for original wind speed series. The ANN family is the most commonly used prediction tool inEMD based hybrid forecasting models [22–32]. Guo et al. [22] first proposed a modified EMD-ANNmodel for multi-step wind speed forecasting. After simulations on the monthly and daily windspeed data in Zhangye of China, the proposed model showed the best accuracy comparing with thesingle ANN and unmodified EMD-ANN model. By introducing the latest decomposing algorithm,fast ensemble EMD (FEEMD), Multilayer perceptron (MLP) neural networks and Adaptive neuro fuzzyinference system (ANFIS) neural networks, two new hybrid models (FEEMD-MLP, FEEMD-ANFIS)were proposed by Liu et al. [23]. After comparisons with the wavelet packet decomposition (WPD)based hybrid models (WPD-MLP and WPD-ANFIS), they found that the FEEMD-MLP hybrid modelhas the best performance in the three-step predictions [24]. Liu et al. [25] also used the FEEMD for asecondary decomposition, and found that the FEEMD could further improve the forecasting accuracyof WPD based hybrid models. Xiao et al. [26] and Sun and Wang [27], respectively, adopted the batalgorithm and phase space reconstruction to improve the accuracy of BP models, and then developednew forecasting architectures based on FEEMD and modified BP. Santhosh et al. [28] utilized theadaptive wavelet neural network (WNN) to regress each signal decomposed by EMD. The proposedEMD based hybrid approach was subsequently investigated with respect to the wind farm of southIndia. In recent years, Wang and his collaborators [29–32] developed a series of powerful EMD basedhybrid forecasting systems. Several ANN models, including Elman neutral network (ENN) [29],WNN [30,32] and generalized regression neural network (GRNN) [31], have been adopted in thesystems. Some popular parameter optimization algorithms, such as the multi-objective ant lionoptimization algorithm [29], meta-heuristic optimization algorithm [31] and multi-objective sine cosinealgorithm [32], were respectively developed to ensure the hybrid forecast models are in the optimalstate. The experimental results indicated that the average values of the mean absolute percent errors ofthe developed model utilizing 10-min, 30-min and 60-min interval data are lower than 8% [29]. ELM isanother popular prediction tool in EMD based hybrid forecasting models [33–37]. Liu et al. [33] firstintroduced the ELM to the hybrid forecasting architecture using EMD and FEEMD as wind signaldecomposing algorithms. Their experiments indicated that by using the EMD and FEEMD, all the

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hybrid algorithms have better performance than the single ELMs. Then, Liu et al. [34] used the outliercorrection method to guarantee the robustness of ELM during the forecasting computation. They alsodeveloped a multi-decomposing strategy based on the combination of EMD and WPD, and showedthat the WPD-EMD-ELM hybrid model has the best predicting performance [35]. Two improved ELMmodels, named by regularized ELM [36] and composite quantile regression outlier-robust ELM [37],were introduced to model each EMD decomposed sub-series. Besides the ANN and ELM, the SVMalso received the attention of some scholars in EMD based hybrid wind speed forecasting [38,39].

Most of the above studies adopted only single nonlinear model (ANN [22–32], ELM [33–37]or SVM [38,39]) to study the IMFs decomposed by the EMD. Because the characteristics of IMFsand residuals are not the same, hybrid strategies with one more different types of forecast modelsmight get even better results. More recently, some scholars have made meaningful attempts in thisarea. Zhang et al. [40] employed a recent type of EMD to divide the original wind speed data into afinite set of IMFs and residuals, and then utilized five neural networks (including BP, RBF, GRNN,WNN and ENN) to forecast each IMFs and residuals. Experimental results of their study showed thatthe proposed hybid model can take advantages of individual models and has the best performance.Li et al. [41] developed a novel hybrid forecasting model, which combines EMD and several singlemodels (BP and ENN). A modified SVM was then used to integrate all the results to obtain thefinal forecasting results. Experimental studies on real 10-min wind speed series showed that thedeveloped hybrid model outperforms other benchmark models. Among the EMD decomposed IMFsand residuals, the low frequency dominated residuals behave more stable and might be more suitablefor forecasting with linear models (ARIMA). For the IMFs with higher frequency components, it isbetter to be modeled by nonlinear prediction models (such as ANN, ELM, SVM, etc.). Thus, combiningboth linear and nonlinear models for predicting IMFs and residuals with various frequency bandcomponents might be a good choice for precise wind speed forecasting. However, such idea has notbeen realized in current studies.

The above literature review indicates that it is rare to see that two (or more) prediction toolsare used in EMD based hybrid wind speed forecasting; this is particularly true for the hybridlinear/nonlinear modeling and prediction of wind speed series. In addition, few efforts have beenpaid to comprehensively compare the performance and robustness of both linear and nonlinear modelsin short-term wind speed forecasting. Therefore, the novelty and contributions of this study can besummarised as follows:

� A framework for short-term wind speed forecasting is introduced based on EMD and hybridlinear/nonlinear models. The EMD is adopted to decompose the original wind speed series into afinite number of IMFs and residuals, i.e., low-frequency residuals (LFR), medium-frequency IMF(MIMF) and high-frequency IMF (HIMF).

� Several popular linear models (ARIMA) and nonlinear models (SVM, random forest (RF), BP,ELM and convolutional neural network (CNN)) are, respectively, utilized to study each IMFs andresiduals. An ensemble forecast for the original wind speed series is then obtained.

� Various experiments are conducted on the real wind speed data at four wind sites in China.The performance and robustness of various hybrid linear/nonlinear models at two time intervals(10 min and 1 h) are compared comprehensively. The forecasting model with the best performanceis then recommended for real applications.

The remainder of this paper is organized as follows. Section 2 explains the structures andprocedures of EMD based hybrid models, Section 3 introduces the measurement of wind speeddata at four sites in China, Section 4 presents the evaluation of model parameters and compares theperformances of various models, and Section 5 summarises the conclusions of the study.

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2. Methods

The solution process of EMD is explained in this section. After introduction of both linear(ARIMA) and nonlinear (SVM, RF, BP, ELM and CNN) single models, the structures and proceduresfor EMD based hybrid linear/nonlinear models are proposed in detail. Several metrics for measuringthe forecast accuracy are also introduced briefly.

2.1. EMD

The original wind speed data is processed by EMD, and a finite number of IMFs with differentscales or trends are obtained. Compared with the original data series, each IMF has better stabilityand regularity. The decomposed IMFs should satisfy two conditions [21]: (a) The number of extremaand the number of zero crossings are equal or differ at most by one; (b) At any point, the mean valueof the envelope defined by the local maxima and the envelope defined by the local minima is zero.The computation of the EMD can be given as follows [21]:

(1) For wind speed series X(t), all of the local maximal and minimal data points are foundand located;.

(2) Two cubic spline lines are used to connect all of the local maximal and minimal points, respectively.Then, upper and lower envelopes XH(t) and XL(t) are gained accordingly.

(3) Mean values M(t) = (XH(t) + XL(t))/2 are calculated.(4) New time series C(t) = X(t)−M(t) is defined. If C(t) satisfies condition (a) and (b), then the

C(t) could be considered as a IMF; otherwise, replace X(t) by C(t), and repeat step (1)–(4) untilthe condition (a) and (b) are simultaneously obeyed.

(5) Residuals R(t) = X(t)− C(t) are then calculated. Replace X(t) by R(t), and repeat step (1)–(5)until all the IMFs and residuals are found.

In our study, two IMFs (HIMF and MIMF) and residuals (LFR) are used to represent the originalwind speed series, and are respectively studied by both linear and nonlinear models to obtain anensemble forecast.

2.2. Single Linear Models (ARIMA)

The model is known as ARIMA(p, d, q), where p is the order of the autoregressive part, q isthe order of the moving average part, d is the degree of first difference involved. The AugmentedDickey-Fuller (ADF) [42] is used to determine whether the wind speed series is stable or not. If itis unstable, first difference should be applied, i.e., d > 0. The autocorrelation function (ACF) andpartial autocorrelation function (PACF) [42] plots are then utilized to determine the order of ARIMAmodel. The maximum likelihood method is utilized to estimate the model parameters. In principle,the ARIMA model with estimated parameters should generate the lowest residuals.

2.3. Single Nonlinear Models

Five nonlinear models (SVM, RF, BP, ELM and CNN) are used in this study. They are brieflyintroduced as follows:

(1) The SVM model has recently been used in a range of applications such as regression and timeseries forecasting. The basic idea of SVM for regression is to use a nonlinear mapping model totransform the data into a high-dimensional feature space, and then perform a linear regressionin the feature space. Optimal weight and bias values are obtained by solving the quadraticoptimization problem [43].

(2) The RF model, which was suggested by Breiman [44], is an ensemble learning method forclassification and regression. It is operated by constructing a multitude of decision tress atthe training stage and outputting the mean prediction of individual trees. Classification and

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regression trees (CARTs) [45] in the RF model use the binary rules to divide data samples.CARTs could correct decision trees’ habit of overfitting to their original dataset.

(3) Among many available learning algorithms, BP has been the most popular implemented learningalgorithm for all ANN models [46]. The time series data is introduced by the input layer, and theforecast value is produced by the output layer. The layer between the input and output layers iscalled the hidden layer, where data are processed. The procedure of BP is repeated by adjustingthe weights of the connection in the network using the gradient descent. Ref. [47] presented thedetailed algorithm.

(4) The ELM’s structure is similar to the single hidden layer feed-forward neural network [48].The main idea of the ELM model is to randomly set the network weights and then obtain theinverse output matrix of the hidden layer. This concept makes the ELM model operate extremelyfast and maintain better accuracy compared with other learning models. The number of hiddennodes, which is the key parameter of ELM model, should be carefully estimated in order to obtaingood results [16].

(5) CNN is a class of deep, feed-forward ANNs, most commonly applied to image classification andthen generalized for time series prediction. In order to simply the preprocessing, CNN utilizes avariation of multilayer perceptrons. Except that the filter weights need to be shared, there are noother connections between the neurons. Thus, CNN could be trained more efficiently and havereliable abilities to extract the hidden features [49].

2.4. Hybrid Linear/Nonlinear Models

The idea of hybrid linear/nonlinear models is shown in Figure 1. The EMD is first adopted todecompose the linear and nonlinear characteristics of original wind speed data into a finite number ofIMFs and residuals (HIMF, MIMF and LFR) as

yt = HIMF + MIMF + LFR (1)

Usually, the LFR retains more linear characteristics, while the nonlinear characteristics aredominant in HIMF and MIMF. The linear (ARIMA) model is utilized to fit LFR as

Lt = linear(LFR) (2)

By applying the nonlinear models on both HIMF and MIMF, the nonlinear characteristics of windspeed data are extracted as follows

Nt = nonlinear(MIMF) + nonlinear(HIMF) (3)

From Equation (1), we can obtain the ensemble prediction results of the hybridlinear/nonlinear model

yt = Lt + Nt (4)

As five nonlinear models (SVM, RF, BP, ELM and CNN) are adopted, so the hybridlinear/nonlinear models are named by EMD-ARIMA-SVM, EMD-ARIMA-RF, EMD-ARIMA-BP,EMD-ARIMA-ELM and EMD-ARIMA-CNN. In order for comparisons, the obtained HIMF, MIMF andLFR are studied by single linear or nonliear models, which are named by EMD-ARIMA, EMD-SVM,EMD-RF, EMD-BP, EMD-ELM and EMD-CNN. Moreover, single linear or nonlinear models are alsodirectly applied on the original wind speed data to show the effect of EMD on the forecasting accuracy.

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Figure 1. Solution process for the empirical mode decomposition (EMD) based hybrid linear/nonlinearmodels. Intrinsic mode functions (IMF), high-frequency IMF (HIMF), medium-frequency IMF (MIMF),low-frequency residuals (LFR), support vector machine (SVM), random forest (RF), back propagation(BP), extreme learning machines (ELM), convolutional neural network (CNN), autoregressive integratedmoving average (ARIMA).

2.5. Forecasting Performance Metrics

Three metrics, including root mean square error (RMSE), mean absolute error (MAE) andmean absolute percentage error (MAPE), are utilized to evaluate the forecasting performance.Their mathematical definitions are given as follows

RMSE =

√∑n

t=1(yt − yt)2

n(5)

MAE =1n

n

∑t=1|yt − yt| (6)

MAPE =1n

n

∑t=1

∣∣yt − yt

yt

∣∣× 100% (7)

where yt, yt denote the actual and predicted values, n is the sample number. Using the above threemetrics, various models, including the single models (ARIMA, SVM, RF, BP, ELM, CNN) and EMDbased single models (EMD-ARIMA, EMD-SVM, EMD-RF, EMD-BP, EMD-ELM, EMD-CNN) andEMD based hybrid linear/nonlinear models (EMD-ARIMA-SVM, EMD-ARIMA-RF, EMD-ARIMA-BP,EMD-ARIMA-ELM, EMD-ARIMA-CNN), will be evaluated for their forecast performance on differentwind speed datasets, respectively.

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3. Data Descriptions

At a wind observation site, the wind speed data is continuously measured using the anemometers.Four wind sites in China, listed in Table 1, were selected for this study. The wind speed data with twotime intervals (10 min and 1 h) are considered, and shown in Figures 2 and 3, respectively. The lengthfor every wind speed series is 1000 points. The first 800 points of data are used for constructingand training the forecasting models, while the remaining 200 points of data are utilized for tests.Descriptive statistics of wind speed data are also given in Table 2.

Table 1. Locations of four observation stations in China.

Observation Station Longitude Latitude Altitude (m)

AnHui (AH) 117◦17′ E 31◦52′ N 20GuangDong (GD) 113◦17′ E 23◦8′ N 11

GanSu (GS) 103◦44′ E 36◦2′ N 1500HeiLongJiang (HLJ) 126◦38′ E 45◦45′ N 128

Figure 2. Wind speed series with 10 min interval at four sites in China: (a) AnHui (AH); (b) GuangDong(GD); (c) GanSu (GS) and (d) HeiLongJiang (HLJ).

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Figure 3. Wind speed series with 1 h interval at four sites in China: (a) AH; (b) GD; (c) GanSu (GS) and(d) HLJ.

Table 2. Descriptive statistics of wind speed data at four sites in China.

Max-Min Values (m/s) Mean (m/s) Standard Deviation (m/s) Skewness Kurtosis

AH-10 min 9.42-0 4.8756 1.8278 0.1328 2.5460AH-1 h 10.41-0 4.6009 1.6494 0.1648 2.9031

GD-10 min 11.9-3.0 7.4867 1.7079 0.1357 2.6211GD-1 h 13.05-0 6.7162 2.3287 −0.0621 2.5962

GS-10 min 19.15-0 7.2653 4.1582 0.6476 2.9146GS-1 h 18.54-0 5.8787 3.3801 0.9460 3.8657

HLJ-10 min 15.97-0 9.4065 2.7275 −0.9519 4.8296HLJ-1 h 16.56-0 8.5214 2.8915 −0.3391 3.0986

4. Results and Discussions

Model parameters of various single models should be determined first (i.e., ARIMA, SVM, RF,BP, ELM and CNN). Also, after the EMD, the obtained HIMF, MIMF and LFR are, respectively,studied by the single models (EMD-ARIMA, EMD-SVM, EMD-RF, EMD-BP, EMD-ELM andEMD-CNN). The parameters of the EMD based hybrid linear/nonlinear models (EMD-ARIMA-SVM,EMD-ARIMA-RF, EMD-ARIMA-BP, EMD-ARIMA-ELM, EMD-ARIMA-CNN) are then identified.Finally, short-term predictions are made based on the constructed models.

4.1. Single Models without EMD

4.1.1. ARIMA

In this study, we utilize the “auto.arima” function in R to fit the best ARIMA model to theunivariate time series. For instance, ARIMA(1,1,1) is obtained as the best model for the 10 min data ofthe AnHui (AH) site. The maximum likelihood method is applied to estimate the model parameters,whose values are given in Table 3. In order to verify the obtained model, the parameter significant testis carried out and shown in Figure 4. Obviously, the fitting residuals are random series and the p valueis found to be significantly greater than 0.05, indicating that the obtained ARIMA model is reasonable.By applying the “auto.arima” function, the single ARIMA model parameters for both 10 min and 1 hdata of four wind speed datasets are determined and given in Table 3.

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Table 3. Single ARIMA model parameters of various wind speed datasets.

Order Intercept ar1 ma1 ma2 ma3

AH-10 min (1,1,1) 0 0.5383 −0.6908 - -AH-1 h (1,0,1) 4.6518 0.8923 0.1126 - -

GD-10 min (1,1,1) 0 0.8936 −0.9664 - -GD-1 h (1,1,3) 0 0.7755 −0.7402 −0.0913 −0.0837

GS-10 min (0,1,1) 0 - 0.2025 - -GS-1 h (1,1,1) 0 −0.1772 0.4780 - -

HLJ-10 min (0,1,1) 0 - 0.0767 - -HLJ-1 h (1,1,2) 0 0.7807 −0.8125 −0.1378

Figure 4. Parameter significance test results of single ARIMA(1,1,1) model for 10 min data from theAH site. ACF: Autocorrelation function.

4.1.2. SVM, RF, BP, ELM and CNN

In the SVM model, two parameters, namely the bandwidth σ2 and regularization factor γ,should be estimated for each of the wind speed sites. Following the suggestions of Zhou et al. [15],we will consider the values of σ2 as 0.1, 0.25, 1, 4, 16, 100, 1000 and γ as 0.25, 1, 4, 16, 256. SingleSVM models with different combinations of σ2 and γ are utilized to fit the wind speed time seriesat the GuangDong (GD) site, and the obtained MAE values are listed in Table 4. Obviously, whenσ2 = 0.1, γ = 0.25, the MAE value of the 10 min data is minimum, i.e., 0.444. For the 1 h data,the minimum MAE value (0.754) is obtained at σ2 = 0.25, γ = 0.25. For the GD site, the optimalparameters of single SVM model are: σ2 = 0.1, γ = 0.25 (10 min data) and σ2 = 0.25, γ = 0.25(1 h data). Similarly, the optimal parameters of SVM models for other three wind speed sites couldalso be determined and shown in Table 5.

For the RF model, the number of trees and number of extracted features should be determined [44].Here, we consider number of trees to vary from 50 to 700 and number of extracted features to varyfrom 1 to 4, and use the RF model to fit each wind speed time series. The calculated MAE values at theHeiLongJiang (HLJ) site are given in Table 6. When the number of trees is equal to 100 and the numberof extracted features is equal to 2, the MAE of 10 min data reaches the minimum, i.e., 0.8992. For 1 hdata, the minimum MAE is obtained for the number of trees equal to 300 and number of extracted

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features equal to 3. Thus, the optimal parameters of single RF models for the HLJ site are determined.Following the same process, the best RF models for other three wind sites can also be gained andshown in Table 5.

Table 4. Single SVM model selection for wind speed time series at the GD site, mean absoluteerror (MAE).

10 min Data

γσ2

0.1 0.25 1 4 16 100 1000

0.25 0.444 0.445 0.512 1.378 1.384 1.384 1.3841 0.542 0.540 0.651 1.383 1.384 1.384 1.3844 0.744 0.739 0.926 1.384 1.384 1.384 1.384

16 1.074 1.088 1.226 1.385 1.384 1.384 1.384256 1.473 1.466 1.415 1.386 1.384 1.384 1.384

1 h Data

γσ2

0.1 0.25 1 4 16 100 1000

0.25 0.758 0.754 0.780 1.596 1.816 1.816 1.8161 0.995 0.996 1.050 1.769 1.816 1.816 1.8164 1.400 1.407 1.530 1.863 1.816 1.816 1.816

16 1.848 1.849 1.879 1.877 1.816 1.816 1.816256 1.964 1.961 1.957 1.879 1.816 1.816 1.816

Table 5. Parameter selection results for single nonlinear models.

Parameters AH-10 min

AH-1 h

GD-10 min

GD-1 h

GS-10 min

GS-1 h

HLJ-10 min

HLJ-1 h

SVM γ 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25σ2 0.1 0.1 0.1 0.25 0.1 0.25 0.25 0.25

RFTree

number 50 200 300 700 200 200 100 300

Featurenumber 3 4 4 3 3 2 2 3

BP Structures s4 s4 s3 s1 s4 s2 s3 s1

ELM Nneu 200 60 200 1000 40 200 40 100

CNN Channels of convolutional layers are 4, 16 and 32

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Table 6. Single RF model selection for wind speed time series at the HLJ site (MAE).

10 min Data

FeatureNumber

Tree Number

50 100 200 300 400 500 600 700

1 0.9346 0.9226 0.9188 0.927 0.9154 0.9268 0.9266 0.9262 0.9018 0.8992 0.902 0.9122 0.9024 0.9144 0.907 0.90443 0.912 0.9302 0.9192 0.9244 0.9186 0.9316 0.9236 0.92424 0.9774 0.9564 0.9702 0.9664 0.969 0.963 0.9752 0.962

1 h Data

Featurenumber

Tree Number

50 100 200 300 400 500 600 700

1 1.0764 1.0744 1.0598 1.062 1.0632 1.0598 1.0598 1.05682 1.034 1.0264 1.024 1.0244 1.022 1.0236 1.0226 1.0263 1.022 1.023 1.0268 1.0194 1.0206 1.0204 1.0206 1.02264 1.0278 1.0212 1.0258 1.0288 1.026 1.0302 1.025 1.0258

For the single BP model, there is no reliable way to determine the optimal parameters. In thisstudy, we just try four structures, named by s1: 10-10-10-10-10, s2: 10-10-10-10, s3: 10-10-10 and s4:10-20-50, to see which structure has the lowest MAE value. Table 7 shows the MAE values of thesingle BP model with different structures for wind speed time series at the AH site. It is shown thateither 10 min data or 1 h data, the BP model with structure s4 always has the minimum value of MAE,and thus it is selected as the best single BP model. Similarly, other three wind sites are studied toobtain the best structure for the single BP model, and the results are given in Table 5.

Table 7. Single BP model selection for wind speed time series at the AH site (MAE).

10 min Data

Structures s1 s2 s3 s4

0.457 0.4544 0.483 0.4084

1 h Data

Structures s1 s2 s3 s4

1.3316 0.6874 0.6496 0.5652

For the single ELM model, the key parameter is the number of hidden neurons (Nneu). Here, we setNneu = 20, 40, 60, 80, 100, 200, 500, 1000, and take the wind speed data at the GanSu (GS) site as anexample. The calculated MAE results are listed in Table 8. It is shown that for the 10 min data,the minimum MAE is obtained at Nneu = 40. For the 1 h data, the minimum MAE is gained atNneu = 200. Thus, for the wind speed data at the GS site, the best number of hidden neurons shouldbe 40 (10 min data) and 200 (1 h data). In this way, the best single ELM models for other three windsites could also be gained and shown in Table 5.

Table 8. Single ELM model selection for wind speed time series at the GS site (MAE).

10 min Data

Nneu20 40 60 80 100 200 500 1000

0.3790 0.1133 0.4325 0.1629 0.0915 0.1169 0.2867 0.2203

1 h Data

Nneu20 40 60 80 100 200 500 1000

0.3287 0.7767 0.5471 0.5139 0.8574 0.2672 0.7078 0.7046

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For the single CNN model, we just adopt the structure recommended by Liu et al. [50]. In theirstudy, the CNN model consists of three convolutional layers and a fully connected layer. The channelsof the convolutional layers are 4, 16 and 32, respectively. In the following study, the CNN model hasthe same structure.

4.2. Single Models with EMD

4.2.1. EMD-ARIMA

The EMD is adopted to decompose the original wind speed data into a finite number of IMFsand residuals (HIMF, MIMF and LFR). Figure 5 gives the EMD results of 10 min data at the AH site.Compared with the original data (see Figure 2a), one can find that the LFR retains the trend and lowfrequency components. The high frequency components are dominant in HIMF and MIMF. Similar tothe solution process presented in Section 4.1, the orders and parameters of ARIMA models for HIMF,MIMF and LFR could be determined accordingly. Table 9 gives the results of both 10 min data and 1 hdata of four wind sites.

Figure 5. EMD results of 10 min data at the AH site: (a) high-frequency intrinsic mode function (HIMF);(b) medium-frequency intrinsic mode function (MIMF) and (c) low-frequency residuals (LFR).

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Table 9. EMD-ARIMA model parameters of various wind speed datasets.

Order Intercept ar1 ar2 ar3 ar4 ar5 ma1 ma2 ma3 ma4

AH-10 minHIMF (2,0,2) −0.0072 0.2677 −0.3288 - - - −0.6236 0.1867 - -

AH-10 minMIMF (2,0,4) 0.0061 1.0409 −0.5285 - - - 0.7289 −0.436 −0.5381 −0.1572

AH-10 minLFR (5,1,3) 0 1.0418 0.4034 −1.0128 0.3023 0.0501 1.6185 0.7437 0.0824 -

AH-1 hHIMF (2,0,2) 0 0.4791 −0.3692 - - - −0.8025 0.2692 - -

AH-1 hMIMF (5,0,4) 0 1.954 −1.711 0.6172 −0.0029 −0.0971 −0.3561 −0.7501 0.0919 0.3172

AH-1 hLFR (3,0,4) 4.6149 2.2188 −1.7555 0.515 - - 1.2482 0.9488 0.6001 0.2004

GD-10 minHIMF (1,0,2) 0 0.1935 - - - - −0.3955 −0.2374 - -

GD-10 minMIMF (4,0,4) 0 2.2572 −2.2493 1.1458 −0.2854 - −0.5511 −0.8215 0.2857 0.3155

GD-10 minLFR (4,1,1) 0 2.2822 −2.1974 1.0208 −0.2126 - 0.2556 - - -

GD-1 hHIMF (2,0,2) 0 0.9411 −0.5456 - - - −1.1692 0.4633 - -

GD-1 hMIMF (5,0,4) 0 −0.0154 −0.0193 −0.4017 −0.0127 −0.2514 1.9025 1.4405 0.55 0.0407

GD-1 hLFR (4,1,2) 0 2.9386 −3.5466 2.0825 −0.5146 - −0.3667 −0.3251 - -

GS-10 minHIMF (2,0,1) 0 0.3224 −0.3344 - - - −0.4286 - - -

GS-10 minMIMF (2,0,4) 0 1.4583 −0.7165 - - - 0.7456 −0.3988 −0.7145 −0.28

GS-10 minLFR (3,1,2) 0 2.2461 −1.7913 0.494 - - 0.6726 0.0931 - -

GS-1 hHIMF (3,0,3) 0 1.1227 −0.2833 0.0774 - - −1.1755 −0.0924 0.3303 -

GS-1 hMIMF (3,0,4) 0 0.6329 0.0703 −0.4106 - - 0.9912 −0.1967 −0.6348 −0.2965

GS-1 hLFR (4,0,1) 6.0967 3.0507 −3.6816 2.0904 −0.4697 - 0.4522 - - -

HLJ-10 minHIMF (2,0,1) 0 0.1808 −0.2572 - - - −0.4975 - - -

HLJ-10 minMIMF (3,0,3) −0.0176 1.6717 −1.1566 0.2937 - - 0.2927 −0.5431 −0.3069 -

HLJ-10 minLFR (3,1,1) 0 2.2675 −1.8836 0.5543 - - 0.57 - - -

HLJ-1 hHIMF (2,0,1) 0.0101 0.3971 −0.3318 - - - −0.7131 - - -

HLJ-1 hMIMF (2,0,3) 0 1.2475 −0.7229 - - - 0.4612 −0.4266 −0.3792 -

HLJ-1 hLFR (5,1,2) 0 3.0145 −3.533 1.8385 −0.2716 −0.0681 −0.2828 −0.5816 - -

4.2.2. EMD-SVM, EMD-RF, EMD-BP, EMD-ELM, EMD-CNN

In Section 4.1, the single nonlinear models are studied to estimate their parameters and structures.Here, the HIMF, MIMF and LFR obtained by EMD are also analyzed by single nonlinear modelsfollowing the same solution process with that presented in Section 4.1. The results of both 10 min dataand 1 h data of the four wind sites are shown in Table 10.

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Table 10. Parameter selection results for single nonlinear models with EMD.

EMD-SVM EMD-RF EMD-BP EMD-ELM EMD-CNN

γ σ2 TreeNumber

FeatureNumber Structures Nneu Structures

AH-10 minHIMF 16 0.1 400 4 s4 40

Same withsingle CNN

AH-10 minMIMF 4 0.1 600 4 s1 80

AH-10 minLFR 0.25 0.1 500 4 s1 200

AH-1 hHIMF 4 0.1 200 4 s2 1000

AH-1 hMIMF 1 0.1 700 4 s3 60

AH-1 hLFR 0.25 0.1 400 4 s3 40

GD-10 minHIMF 16 0.1 200 3 s1 60

GD-10 minMIMF 1 0.1 500 4 s3 80

GD-10 minLFR 0.25 0.1 100 4 s1 40

GD-1 hHIMF 4 0.1 200 4 s3 80

GD-1 hMIMF 0.25 0.1 100 4 s2 40

GD-1 hLFR 0.25 0.1 200 4 s4 80

GS-10 minHIMF 1 0.1 400 3 s3 200

GS-10 minMIMF 0.25 0.1 500 4 s3 1000

GS-10 minLFR 0.25 0.1 300 4 s2 60

GS-1 hHIMF 1 0.1 700 3 s1 40

GS-1 hMIMF 0.25 0.1 100 4 s1 60

GS-1 hLFR 0.25 0.1 400 4 s1 80

HLJ-10 minHIMF 4 0.1 100 4 s1 40

HLJ-10 minMIMF 1 0.1 700 4 s2 80

HLJ-10 minLFR 0.25 0.1 100 4 s1 80

HLJ-1 hHIMF 0.25 0.1 50 3 s2 500

HLJ-1 hMIMF 0.25 0.1 300 4 s3 100

HLJ-1 hLFR 0.25 0.1 500 4 s4 100

4.3. Hybrid Models with EMD

As shown in the above section, the idea of hybrid linear/nonlinear models is to use the linear(ARIMA) model to analyze the LFR, while the HIMF and MIMF are studied by nonlinear models.

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Thus, the ARIMA models in EMD based hybrid linear/nonlinear models have the same orders andparameters with EMD-ARIMA models for LFR, as shown in Table 9. The single nonlinear models withEMD, as shown in Table 10, will also be utilized in the hybrid models to get the nonlinear predictionsfrom the HIMF and MIMF. Then, the ensemble prediction results of the hybrid linear/nonlinear modelare obtained according to Equation (1).

4.4. Forecasting Performance Comparisons

Short-term predictions for both 10 min data and 1 h data are made using the estimated singleand EMD based hybrid models, respectively. Values of three metrics (RMSE, MAE and MAPE) arecalculated for the predictions of the four sites in China. The results of the 10 min data and 1 h data arecompiled and shown in Figures 6–13, respectively. We can see that:

(1) Single ARIMA models have better prediction accuracy than the single SVM, RF, BP, ELM and CNNmodels, except for the 1 h data of the GD and HLJ sites. For 1 h data of the GD site (see Figure 11),the RMSE value of RF (1.095) is slightly lower than the ARIMA model (1.128). At the HLJ site(see Figure 13), from the MAPE metric, the RF model (0.2306) is better than the ARIMA model(0.2542).

(2) The introduction of EMD is beneficial to most single models’ prediction accuracy, but there areexceptions. For instance, both RMSE and MAE values of EMD-CNN model on the 10 min dataof the AH site (see Figure 6) seem to show little difference from that of the single CNN model.A similar phenomenon could also be found in the EMD-BP model on the 10 min data of the GDsite (see Figure 7).

(3) EMD based hybrid linear/nonlinear models (EMD-ARIMA-SVM, EMD-ARIMA-RF,EMD-ARIMA-BP, EMD-ARIMA-ELM and EMD-ARIMA-CNN) always have the top rankedmetric (see the minimum value in these figures), which means that the EMD based hybridlinear/nonlinear models generally have better accuracy and more robust performance thanthe single models with/without EMD. Quantitative comparisons between hybrid models andsingle ARIMA models are taken as an example. Maximum accuracy increase of RMSE is foundat the 10 min data of the GD site (see Figure 7), from 0.2569 of the ARIMA model to 0.121 ofthe EMD-ARIMA-RF model. The relative accuracy increase is about 52.9%; the maximumaccuracy increase of MAE is found at the 10 min data of the GD site (see Figure 7), from 0.2012of the ARIMA model to 0.088 of the EMD-ARIMA-RF model. The relative accuracy increaseis about 56.26%; the maximum accuracy increase of MAPE is found at the 1 h data of the HLJsite (see Figure 13), from 0.2542 of the ARIMA model to 0.1021 of the EMD-ARIMA-RF model.The relative accuracy increase is about 59.83%. Thus, EMD based hybrid models could greatlyimprove the forecast accuracy (higher than 50%).

(4) Among the five hybrid models, EMD-ARIMA-RF has the best accuracy on the whole for 10 mindata. However, for the 1 h data, no model can always perform well on the whole dataset.EMD-ARIMA-CNN seems to be better as it outperforms other hybrid models on the 1 h data ofthe AH site (see Figure 10) and GD site (see Figure 11). EMD-ARIMA-SVM and EMD-ARIMA-RFhave better precision on the 1 h data of the GS site (see Figure 12) and HLJ site (see Figure 13),respectively.

Moreover, we can see that EMD based hybrid linear/nonlinear models perform worse on 1 hdata than on 10 min data. Let us consider the metric of MAPE. For the 10 min data, the MAPE isaround 0.03 and the worst is no more than 0.04 (see 0.0368 in Figure 8). However, for the 1 h data,the MAPE is around 0.10 and the worst is close to 0.15 (see 0.1427 in Figure 12). This is mainly becausethe correlation of 10 min data is stronger than that of 1 h data, and thus the 10 min data is easierfor forecasting.

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Figure 6. Wind speed prediction comparisons for the 10 min data at the AH site. Root mean squareerror (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE).

Figure 7. Wind speed prediction comparisons for the 10 min data at the GD site.

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Figure 8. Wind speed prediction comparisons for the 10 min data at the GS site.

Figure 9. Wind speed prediction comparisons for the 10 min data at the HLJ site.

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Figure 10. Wind speed prediction comparisons for the 1 h data at the AH site.

Figure 11. Wind speed prediction comparisons for the 1 h data at the GD site.

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Figure 12. Wind speed prediction comparisons for the 1 h data at the GS site.

Figure 13. Wind speed prediction comparisons for the 1 h data at the HLJ site.

4.5. Discussions

From the above analysis, we can see that the proposed hybrid model is mainly composed of theexisting mature linear and nonlinear prediction algorithms, so it is very suitable for the promotionof practical engineering applications. The proposed hybrid model performs well in short-term windspeed prediction, and the MAPE can reach 0.03 (10 min data). For the 1 h data, the predictionperformance has decreased significantly, and the MAPE has exceeded 0.10. Predictably, in the medium

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and long term wind speed forecasting, the performance of proposed hybrid model would be evenworse. Therefore, future studies will combine weather numerical prediction results to improve theaccuracy and performance of hybrid models in the medium and long term wind speed forecasting.

5. Conclusions

A framework for short-term wind speed forecasting is introduced based on EMD and hybridlinear/nonlinear models. The EMD is adopted to decompose the original wind speed series into a finitenumber of IMFs and residuals, which are studiedby several linear models (ARIMA) and nonlinearmodels (SVM, RF, BP, ELM and CNN) to obtain the ensemble forecast for the original wind speedseries. Forecasting experiments are conducted on real wind speed series at four wind sites in China.The performance and robustness of various hybrid linear/nonlinear models at two time intervals(10 min and 1 h) are compared comprehensively. It is shown that single ARIMA models have betterprediction accuracy than the single SVM, RF, BP, ELM and CNN models. The introduction of EMD isbeneficial to most single models’ prediction accuracy. The EMD based hybrid linear/nonlinear modelsgenerally have better accuracy and more robust performance than the single models with/withoutEMD. Among the five hybrid models, EMD-ARIMA-RF has the best accuracy on the whole for 10 mindata. However, for the 1h data, no model can always perform well on the whole dataset. As the existingmature linear and nonlinear forecast models are adopted, they will greatly enhance the practical utilityof the proposed hybrid wind speed forecasting model.

Author Contributions: Q.H., H.W., T.H. and F.C. developed the theoretical framework. Q.H. and H.W. performedthe numerical simulation. Q.H. wrote the manuscript with help from H.W. T.H. and F.C. supervised the research.

Acknowledgments: The research work was supported by the NSFC under No. 11472147/51335006, and theSKLT under No. SKLT2015B12. Tao Hu’s work was partly supported by Support Project of High-level Teachersin Beijing Municipal Universities in the Period of 13th Five-year Plan (No. CIT&TCD 201804078) and “CapacityBuilding for Sci-Tech Innovation-Fundamental Scientific Research Funds” (No. 025185305000/204).

Conflicts of Interest: The authors declare no conflict of interest. The founding sponsors had no role in the designof the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in thedecision to publish the results.

Abbreviations

The following abbreviations are used in this manuscript:

EMD empirical mode decompositionIMF intrinsic mode functionsARIMA autoregressive integrated moving averageSVM support vector machineRF random forestANN-BP artificial neural network with back propagationELM extreme learning machinesCNN convolutional neural networkMAPE mean absolute percentage errorGARCH generalized autoregressive conditional heteroscedasticRBF radial basis functionDLN deep learning networksFEEMD fast ensemble EMDMLP Multilayer perceptronANFIS Adaptive neuro fuzzy inference systemWPD wavelet packet decompositionWNN wavelet neural networkENN Elman neutral networkGRNN generalized regression neural network

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LFR low-frequency residualsMIMF medium-frequency IMFHIMF high-frequency IMFADF Augmented Dickey-FullerACF autocorrelation functionPACF partial autocorrelation function

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