A decomposition-clustering-ensemble learning approach for solar radiation forecasting

Solar Energy 163 (2018) 189–199

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A decomposition-clustering-ensemble learning approach for solar radiation forecasting

T

⁎

Shaolong Suna,b,c, Shouyang Wanga,b,d, , Guowei Zhanga,b, Jiali Zhenga,b a

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China School of Economics and Management, University of Chinese Academy of Sciences, Beijing 100190, China c Department of Systems Engineering and Engineering Management, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong d Center for Forecasting Science, Chinese Academy of Sciences, Beijing 100190, China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Solar radiation forecasting Decomposition-clustering-ensemble learning approach Ensemble empirical mode decomposition Least square support vector regression

A decomposition-clustering-ensemble (DCE) learning approach is proposed for solar radiation forecasting in this paper. In the proposed DCE learning approach, (1) ensemble empirical mode decomposition (EEMD) is used to decompose the original solar radiation data into several intrinsic mode functions (IMFs) and a residual component; (2) least square support vector regression (LSSVR) is performed to forecast IMFs and residual component respectively with parameters optimized by gravitational search algorithm (GSA); (3) Kmeans method is adopted to cluster all component forecasting results; (4) another GSA-LSSVR method is applied to ensemble the component forecasts of each cluster and the final forecasting results are obtained by means of corresponding cluster’s ensemble weights. To verify the performance of the proposed DCE learning approach, solar radiation data in Beijing is introduced for empirical analysis. The results of out-of-sample forecasting power show that the DCE learning approach produces smaller NRMSE, MAPE and better directional forecasts than all other benchmark models, reaching up to accuracy rate of 2.96%, 2.83% and 88.24% respectively in the one-day-ahead forecasting. This indicates that the proposed DCE learning approach is a relatively promising framework for forecasting solar radiation by means of level accuracy, directional accuracy and robustness.

1. Introduction With the continuous consumption of fossil fuels, energy and environmental problems become increasingly severe. It is urgent to find a solution to solve energy and environmental problems and achieve sustainable development. Therefore, renewable energy has attracted much attention all over the world and been rapidly developed in recent years. The solar energy is considered to be one of the cleanest and most promising renewable energy sources, which makes solar power an important direction of exploring renewable energy. The solar power generation is widely used in developed countries and considerable developing countries at present, and has partially replaced the traditional power generation. In recent years, China has gained more than 25% growth rate every year in the development and utilization of renewable energy. The solar radiation plays an important role in solar photovoltaic power generation. With the development of solar technologies, demands for solar radiation data with high accuracy are increasing. Consequently, the solar radiation forecasting has become one of core contents of solar photovoltaic power generation. ⁎

The scholars and researchers so far have conducted in-depth research for solar radiation forecasting theory and put forward a number of feasible and efficient methods to predict actual solar radiation intensity, which has achieved satisfactory results. These methods can be divided into three categories: traditional mathematical statistics, numerical weather forecasting and machine learning. The traditional mathematical statistics includes: regression analysis (Trapero et al., 2015), time series analysis (Huang et al., 2013; Voyant et al., 2013), gray theory (Fidan et al., 2014), fuzzy theory (Chen et al., 2013), wavelet analysis (Mellit et al., 2006; Monjoly et al., 2017) and Kalman filter (Akarslan et al., 2014), etc.; numerical weather forecasting solves thermal fluid dynamics equations with weather evolution by computers with high performance to predict the solar radiation intensity of the next period mainly based on the actual situation of the atmosphere (Chow et al., 2011; Mathiesen and Kleissl, 2011; Mathiesen et al., 2013), which is complicated and time-consuming. With the rise of big data mining, machine learning techniques currently have attracted much attention, for example, artificial neural networks (ANN) (Amrouche and Le Pivert, 2014; Benmouiza and Cheknane, 2013; Niu et al., 2015; Paoli et al., 2010), support vector machines (SVM) (Gala

Corresponding author at: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. E-mail address: [email protected] (S. Wang).

https://doi.org/10.1016/j.solener.2018.02.006 Received 28 July 2017; Received in revised form 26 December 2017; Accepted 1 February 2018 0038-092X/ © 2018 Elsevier Ltd. All rights reserved.

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conclusions and indicates the direction of further research.

et al., 2016; Lauret et al., 2015) and heuristic intelligent optimization algorithms (Jiang et al., 2015; Niu et al., 2017; Niu et al., 2016a; Wang et al., 2015) have been widely used in solar radiation forecasting. According to the literature above, Trapero et al. (2015) applied dynamic harmonic regression model (DHR) to predict short-term direct solar radiation and scattered solar radiation in Spain for the first time in 2015. Huang et al. (2013) took autoregressive model to forecast the solar radiation in the framework of meteorological factors-dynamic adjustment system in 2013, which increased the accuracy by 30% than general neural network or random models. Fidan et al. (2014) predicted hourly solar radiation in Izmir, Turkey by integration of Fourier transform and neural network. Mellit et al. (2006) applied infinite impulse response function to filter time series of the total solar radiation in Algeria from 1981 to 2000, and then substituted the filtered data into adaptive wavelet neural network model to forecast the total solar radiation in 2001, where the error percentage is less than 6% performing better than conventional neural network models and classical statistical methods. However, Akarslan et al. (2014) first used the multidimensional linear predictive filtering model to forecast the solar radiation, which surpasses the two-dimensional linear predictive filtering model and the traditional statistical forecasting method by means of empirical analysis. Amrouche and Le Pivert (2014) adopted the idea of spatial modeling and artificial neural networks (ANNs) to forecast the daily solar total radiation of four sites in the United States, in which the empirical results show that the forecasting results of this proposed model satisfy the expected accuracy requirements. Benmouiza and Cheknane (2013) classified the input data by K-means, then modeled different class by using nonlinear autoregressive neural network, and finally predicted the solar radiation of test data by the corresponding model. In recent years, the integrated model has grown rapidly. Paoli et al. (2010) used the integrated model to predict the total solar radiation in three sites in France. First, they pretreated the original total solar radiation sequence by using the seasonal index adjustment method. And then they used the multi-layer perceptron neural network (MLPNN) for daily solar radiation prediction. The results show that the mean absolute percentage error of the multi-layer perceptual neural network model is about 6%, which is superior to the ARIMA, the Bayesian, Markov chain model and K-nearest neighbor algorithm. Lauret et al. (2015) used three different methods which are artificial neural network (ANN), Gaussian process (GP) and support vector machine (SVM) to forecast global horizontal solar irradiance (GHI). Through analyzing the actual data in three different places in France, the three machine learning algorithms proposed in this paper are found to be better than the linear autoregressive (AR) and persistence model. Gala et al. (2016) used a combination of support vector regression (SVR), gradient boosted regression (GBR), and random forest regression (RFR) to predict three-hour accumulated solar radiation of 11 regions in Spain. Wang et al. (2015) proposed a new integrated model to predict hourly solar radiation in 2015. First, they used the network structure of multiresponse sparse regression (MRSR), leave-one-out cross validation, and extreme learning machine (ELM), then used the cuckoo search (CS) to optimize its weight and threshold, and finally analyzed six sites in the US. The prediction results show that this combination model is stronger than the ARIMA, BPNN and optimally pruned extreme learning machine (OP-ELM). The main innovation of this paper is to propose a novel decomposition clustering ensemble (DCE) learning approach integrating EEMD, K-means and LSSVR to improve the performance of solar radiation forecasting by means of forecasting accuracy and robustness, and to compare its forecasting power with some popular existing forecasting models. The rest of the paper is organized as follows. Section 2 describes the formulation process of the proposed DCE learning approach in detail. Related methodologies are illustrated in Section 3. The empirical results and effectiveness of the proposed approach are discussed in Section 4. Finally, Section 5 provides some

2. Decomposition-Clustering-Ensemble (DCE) learning approach According to the work of TEI@I methodology (Wang et al., 2005; Wang, 2004), which is based on integration (@ Integration) of text mining (T) plus econometrics (E) plus intelligence techniques (I). Yu et al. (2008) proposed a decomposition ensemble learning approach for crude oil price forecasting. Recently, this learning approach has been applied in many fields, including financial time series forecasting (Yu et al., 2009), nuclear energy consumption forecasting (Tang et al., 2012; Wang et al., 2011), hydropower consumption forecasting (Wang et al., 2011), crude oil price forecasting (He et al., 2012; Yu et al., 2014; Yu et al., 2015; Yu et al., 2016), etc. By analyzing the literature above in detail, there are three main steps involved in the decomposition and ensemble learning approach, i.e. decomposition, single forecasting and ensemble forecasting. Firstly, some decomposition algorithms can be used to decompose the original time series into a number of meaningful components. Secondly, some forecasting methods are applied to forecast all components respectively. Finally, these forecasting results of each component are combined to generate an aggregated output as the final forecasting result using some ensemble methods. It can be concluded that ensemble learning is critical to the final forecasting results. Sub-component forecasting has different attributes in different time. If ensemble weights are the same all the time, different attributes cannot be captured. Therefore, DCE learning approach is proposed in this paper, which employs clustering scheme to cluster the sub-component forecasting results. By using different ensemble weights in different forecasting time, a better performance can be obtained by compared with the fixed ensemble weights. The framework of DCE learning approach is shown in Fig. 1. It can be seen from Fig. 1, DCE learning approach contains four steps: (1) Decomposition. Decomposition method is introduced to decompose the original time series into some relatively simple and meaningful component series. (2) Individual forecasting. Various forecasting models are employed to forecast each component series. (3) Clustering. A clustering method is used to cluster the individual forecasting results. (4) Ensemble forecasting. An ensemble method is applied to calculate the ensemble weights of different cluster. Then the corresponding clusters’ ensemble weights are used for component forecasts to obtain the ensemble forecasting results. 3. Related methodologies 3.1. Ensemble empirical mode decomposition The empirical mode decomposition (EMD) is an adaptive signal decomposing method which is proposed by Huang et al. (1998). It can be used to decompose the signal into several intrinsic mode functions (IMFs) and a residual series. Each IMF has a zero local mean value and its number of extreme values and zero crossings are equal or differ at most by one. Different IMFs have different frequency ranges and represents different kinds of natural oscillation modes embedded in the original signal. Thus, different IMFs may highlight different details of the signal. Compared with other methods, the key advantage of EMD is that the basis function is derived directly from the original signal rather than a priori fixed basis function. It is especially suitable for analyzing non-linear and non-stationary signals. The ensemble empirical mode decomposition (EEMD) is proposed as an improved version of EMD which overcomes the mode mixing problem of EMD. The main idea of EEMD is to add white noise into the original signal with many trials. Then the EMD is applied to the noisy 190

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Fig. 1. General framework of decomposition-clustering-ensemble (DCE) learning approach.

3.3. Least square support vector machine

signal with additional white noise rather than the original signal to obtain the IMFs. For further details on these methods, please refer to (Huang et al., 1998; Wu and Huang, 2009; Zhang et al., 2008).

Support vector machine (SVM) is firstly proposed by Vapnik based on statistical learning theory and the principle of structural risk minimization which possess good performance even for small samples (Vapnik, 2013). However, it is time-consuming and leads to a high computational cost when deal with large-scale problem. Therefore, least square support vector regression (LSSVR) is proposed by Suykens and Vandewalle (1999). The basic idea of support vector regression is to map the original data into a high-dimensional feature space and make a linear regression in the space. It can be formulated into:

3.2. Kmeans clustering Kmeans is an unsupervised learning algorithm for clustering, which was proposed by MacQueen (1967). The basic idea of Kmeans algorithm is to split the historical dataset into several subsets where each subset has its own unique characteristics. Assuming that training set is X = {x1,x2,…,x n} , the detail steps of Kmeans clustering are as follows:

f (x ) = wT φ (x ) + b

(1) Initialize cluster centroids (c1,c2,…,ck ) randomly. (2) If ‖x i−ck ‖ ⩽ ‖x i−cp ‖,p = 1,2,…,K ,k ≠ p, x i is assigned to cluster k. N 1 (3) Update cluster centroids by ci = N ∑ j =i 1 x ij,i = 1,2,…,k .

where φ (x ) is a nonlinear mapping function, f (x ) is the estimation value, wT and b are the weights. It can be transformed into an optimization problem:

i

k

(1)

N

(4) Compute the least square error of clustering J = ∑i = 1 ∑ j =i 1 ‖x ij−ci ‖2 . (5) Repeat step 2–4 until J convergence.

T

min

1 T w w 2

s. t .

T ⎧ w φ (x t ) + b−yt ⩽ ε + ξt ,(t = 1,2,…,T ) ⎪ y −(wT φ (x t ) + b) ⩽ ε + ξt∗,(t = 1,2,…,T ) ⎨ t ∗ ⎪ ξt ,ξt ⩾ 0 ⎩

+ C ∑ (ξt + ξt∗) t=1

From the above processes, it can be seen that the number of clusters K and the election of the initial cluster center are critical to the final results of clustering. For further details on this method, please refer to (MacQueen, 1967).

where C is the penalty parameter, ξt and 191

ξt∗

(2)

are the nonnegative slack

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Fig. 2. The framework of EEMD-LSSVR-K-LSSVR learning approach.

residual series. (2) Each component series is modeled by using optimized LSSVR respectively. (3) Kmeans is introduced to cluster the sub-series forecasting results of the in-sample to obtain cluster centers. (4) Another optimized LSSVR is employed to calculate each cluster’s ensemble weights. By using corresponding clusters’ ensemble weights, final forecasting results are obtained.

variables. It is time-consuming to solve the above problem and LSSVR is proposed to transform the problem into: T

min

1 T w w 2

s. t .

yt = wT φ (x t ) + b + et ,(t = 1,2,…,T )

+ C ∑ et2 t=1

(3)

where et is the slack variable. Generally speaking, the parameters of SVR and LSSVR have a great influence on the accuracy of the regression estimation. Thereby, gravitational search algorithm (GSA) is employed to automatically choose the optimal parameters of SVR and LSSVR in this paper. For more information about GSA, please refer to (Niu et al., 2016b; Rashedi et al., 2009).

4. Empirical analysis In this section, Beijing daily solar radiation data are carried out to evaluate the performance of the proposed DCE approach. The data description and evaluation criteria are first introduced in Section 4.1. Then empirical results are further discussed in Section 4.2.

3.4. EEMD-LSSVR-K-LSSVR learning approach 4.1. Data description and evaluation criteria An EEMD-LSSVR-K-LSSVR learning approach is obtained by using EEMD as the decomposition method, Kmeans as the clustering method, LSSVR as the forecasting and ensemble method. The framework of EEMD-LSSVR-K-LSSVR learning approach is illustrated in Fig. 2. In Fig. 2, the proposed EEMD-LSSVR-K-LSSVR learning approach consists of four steps:

The daily solar radiation data used in this paper are obtained from a meteorological station of Beijing with 3103 observations ranging from January 1, 2009 to June 30, 2017, which is shown in Fig. 3. The datasets are divided into in-sample subset and out-of-sample subset. The in-sample subset is used for model training with data from January 1, 2009 to June 30, 2015, while out-of-sample subset is used for model testing with data from July 1, 2015 to June 30, 2017. The detailed data

(1) EEMD is applied to decompose the original series into the IMFs and 192

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Fig. 3. The original daily solar radiation series in Beijing.

Therefore, in order to verify the forecasting performance of the proposed DCE learning approach, four single forecasting models, i.e., single ARIMA, ANN, SVR and LSSVR models and five hybrid ensemble learning approaches, i.e., EMD-LSSVR-ADD, EMD-LSSVR-LSSVR, EMDLSSVR-K-LSSVR, EEMD-LSSVR-ADD and EEMD-LSSVR-LSSVR are considered as benchmark models. For consistency, all LSSVR models, as both component forecasting and nonlinear ensemble forecasting, are the same as LSSVR of single benchmark model in the determination method, as mentioned above. Generally speaking, the parameter specification is crucial for model performance. For single SVR and LSSVR models, the Gaussian RBF kernel function is selected and GSA algorithm is used to determine values of optimal parameters γ , σ 2 and ε in terms of the smallest error in the in-sample subset (Niu et al., 2016b; Rashedi et al., 2009). For single ANN forecasting technique, input neurons is set to 6 chosen by autocorrelation function (ACF) and partial correlation function (PCF), hidden neuron is set to 7 and output neuron is set to 1 (Niu et al., 2016b). ANN models are iteratively run 10,000 times to train the model (Yu et al., 2009). In ARIMA (p-d-q) model, the optimal form is estimated by Schwarz Criterion (SC) minimization (Yu et al., 2009). Six decomposition ensemble learning approaches are performed. For EEMD decomposition, the ensemble member is set to 100 and the standard deviation of added white noise is set to 0.2. Fig. 4 shows the decomposition results of solar daily radiation data in Beijing. All IMF components are listed from the highest frequency to the lowest frequency and the last one is the residual term. Therefore, four steps are involved in empirical results analysis: (1) four single models are performed to forecast solar radiation in Beijing to select the best single model; (2) the forecasting performance of the proposed EEMD-LSSVR-K-LSSVR approach is compared with other ensemble benchmark approaches to ensure that the proposed DCE approach can be statistically superior to all other benchmark approaches; (3) the robustness of the DCE approach is analyzed; (4) some conclusions are summarized from the empirical analysis.

are not listed here, but can be obtained from the authors. The m-step-ahead forecasting horizons are employed to evaluate the performance of the proposed DCE learning approach. Given a time ̂ m: series x t ,(t = 1,2,…,T ) , we make m-step-ahead forecasting for x t +

x t +̂ m = f (x t ,x t − 1,…,x t − (l − 1) ),t = 1,2,…,T .

(4)

̂ m is the m-step-ahead forecast value at time t , x t is the actual where x t + value at time t , and l denotes the lag orders which is chosen by autocorrelation and partial correlation analysis (Lewis, 1982). In addition, in order to assess the level forecasting accuracy and the directional forecasting accuracy of the proposed DCE learning approach with some other benchmark models. Three main evaluation criteria are employed to compare the in-sample and out-of-sample forecasting performance (Coimbra et al., 2013). For level forecasting accuracy evaluation, the normalized root mean squared error (NRMSE) and the mean absolute percentage error (MAPE) are selected: NRMSE =

100 x

1 T

T

∑t=1 (xt −xt ̂)2 ,MAPE =

1 T

T

∑ t=1

x t −x t ̂ × 100% xt

(5)

where T is the number of observations in the out-of-sample subset. The performance to forecast direction of movement can be measured by a directional symmetry (DS) as follows:

DS =

1 T

T

∑ t=2

1 if (x t −x t − 1)(x t ̂−x t − 1) > 0 dt × 100%, where dt = ⎧ ⎨ ⎩ 0 otherwise

(6)

Furthermore, in order to further evaluate the level forecasting performance and the directional forecasting performance from the statistical view, the Diebold-Mariano (DM) statistic and PesaranTimmermann (PM) statistic are adopted to test the statistical significance of all models. The main process of the DM test and the PM test can be found in (Diebold and Mariano, 2002; Pesaran and Timmermann, 1992; Sun et al., 2017). 4.2. Empirical results

4.2.1. Performance comparison of single models The performances of four single forecasting models are analyzed by means of the NRMSE, MAPE and DS evaluation criteria as shown in Figs. 5–7. Tables 1–4 demonstrate the results of DM test and PT test with respect to different forecasting horizons. From Figs. 5–7, it can be concluded that: (1) the performance of LSSVR is the best single benchmark model, followed by other AI techniques, and ARIMA ranks the last, by means of forecasting accuracy and statistical test; (2) it may be difficult for the ARIMA, as a traditional linear model, to capture the complexity and nonlinearity of solar radiation data, while AI techniques are more appropriate for exploring the nonlinear patterns; (3) the forecasting performances of SVR and ANN are quite similar in different forecasting horizons. The DM test and PT test are employed to test the difference of four single benchmark models in terms of level accuracy and directional accuracy, and the results are shown in Tables 1–4. According to the above empirical results, it can be obtained that (1) all the single models are almost ineffective in solar radiation forecasting; (2) the level

For comparison purposes, some other traditional forecasting methods are regarded as benchmark models to be compared with the proposed DCE learning approach in terms of forecasting performance. According to previous literature review, the most popular univariate forecasting models are introduced for solar radiation series, i.e., the typical ARIMA model and the most popular AI techniques of ANN and SVR. In particular, these three tools of ARIMA, ANN and SVR might be the most popular time series forecasting models, which have widely been used as single forecasting models and component forecasting model for decomposition ensemble learning approach. Therefore, these three typical univariate forecasting tools are introduced as single benchmark models in this study. Furthermore, we also introduce LSSVR as the single benchmark model since LSSVR is selected as the component forecasting tool for extracted components in formulating the proposed DCE learning approach. In the decomposition stage, EMD, the original form of EEMD, is regarded as benchmark decomposition method. 193

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Fig. 4. The IMFs and a residual for Beijing solar radiation data decomposed by EEMD. LSSVR

SVR

ANN

ARIMA

LSSVR

25

22.36

SVR

12.34

13.25

12.38

14.03

14.88

15.19 15.07

MAPE (%)

11.26

ARIMA 19.61

19.25

18.03

15

15

ANN

20.09

19.18

20 NRMSE (%)

20

13.15

10

10

12.89 9.69

10.54 11.17

10.83

14.35 14.23

13.63 11.76

5

5

0

0

One-step-ahead forecasting Three-step-ahead forecasting Six-step-ahead forecasting

One-step-ahead forecasting Three-step-ahead forecasting Six-step-ahead forecasting

Fig. 6. Performance comparison of different single models in terms of MAPE criteria.

Fig. 5. Performance comparison of different single models in terms of NRMSE criteria.

194

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SVR

ANN

ARIMA

EEMD-LSSVR-K-LSSVR EMD-LSSVR-LSSVR

70 60

59.51 59.09 58.14 45.83

50

44.32

EEMD-LSSVR-LSSVR EMD-LSSVR-ADD

10

55.68 54.99 54.04

30 20

7.74

7.65

8

43.78

40

NRMSE (%)

DS (%)

56.91 56.49 54.86

EMD-LSSVR-K-LSSVR EEMD-LSSVR-ADD

5.89 5.73

6

6.54

8.96

7.02 7.09

6.24 6.41

5.86

5.14

4.53 4

8.46

8.14

4.87

3.15

2.96

2

10 0

0

One-step-ahead forecasting Three-step-ahead forecasting Six-step-ahead forecasting

One-step-ahead forecasting Three-step-ahead forecasting Six-step-ahead forecasting

Fig. 7. Performance comparison of different single models in terms of DS criteria.

Fig. 8. Performance comparison of different hybrid ensemble approaches in terms of NRMSE criteria.

Table 1 DM test results for single models in one-step-ahead forecasting. Benchmark model SVR LSSVR SVR ANN

ANN

−0.4121 (0.3401)

EEMD-LSSVR-LSSVR

EMD-LSSVR-LSSVR

EMD-LSSVR-ADD

EEMD-LSSVR-ADD

10

ARIMA

−0.5129 (0.3040) −0.1182 (0.4530)

8

−3.3627 (0.0004) −3.5628 (0.0001) −3.7143 (0.0001)

MAPE (%)

Tested model

EEMD-LSSVR-K-LSSVR EMD-LSSVR-K-LSSVR

6.91

6.89

6.07

6

4.68

4.01 2.83

6.23

5.89 6.06

5.11 5.26

4

7.58 7.86

7.54 6.76

5.04 4.12

3.06

2

Table 2 DM test results for single models in three-step-ahead forecasting.

0 One-step-ahead forecasting Three-step-ahead forecasting Six-step-ahead forecasting

Tested model

LSSVR SVR ANN

Benchmark model Fig. 9. Performance comparison of different hybrid ensemble approaches in terms of MAPE criteria.

SVR

ANN

ARIMA

−0.4017 (0.3440)

−0.3892 (0.3486) −0.2053 (0.4187)

−4.1026 (0.0000) −3.8968 (0.0000) −4.1236 (0.0000)

EEMD-LSSVR-K-LSSVR

EMD-LSSVR-K-LSSVR

EEMD-LSSVR-LSSVR

EMD-LSSVR-LSSVR

EEMD-LSSVR-ADD

EMD-LSSVR-ADD

100

LSSVR SVR ANN

DS (%)

Tested model

88.24 80.31 80.03 81.94 82.08 75.51 75.11 72.23 71.55 68.39 69.91 64.52 65.12 64.71 60.05 62.11 58.82 55.81 60

80

Table 3 DM test results for single models in six-step-ahead forecasting. Benchmark model SVR

ANN

ARIMA

−0.4173 (0.3382)

−0.3156 (0.3762) −0.3319 (0.3700)

−4.6027 (0.0000) −5.5361 (0.0000) −4.9586 (0.0000)

40 20 0 One-step-ahead forecasting Three-step-ahead forecasting Six-step-ahead forecasting

Fig. 10. Performance comparison of different hybrid ensemble approaches in terms of DS criteria. Table 4 PT test results for single models. Horizon

One-step-ahead forecasting Three-step-ahead forecasting Six-step-ahead forecasting

Value

SPT p-Value SPT p-Value SPT p-Value

4.2.2. Performance comparison of hybrid ensemble learning approaches The forecasting performances of six hybrid ensemble approaches are discussed in this section. Figs. 8–10 show the comparison results of NRMSE, MAPE and DS evaluation criteria. The results of DM test and PT test with different forecasting horizons are reported in Tables 6–9. The cluster numbers of the proposed DCE learning approach are set to 3 which can be interpreted as the rainy, cloudy and fine day respectively (Benmouiza and Cheknane, 2013). Table 5 describes the cluster centers and member numbers of component forecasting results by using Kmeans method. According to above comparison results, it can be concluded that the proposed DCE learning approach significantly outperforms all other hybrid ensemble approaches in both level accuracy and directional accuracy for solar radiation forecasting. The results of DM test indicate that the proposed DCE learning approach statistically confirms its superiority at the 95% confidence level. Furthermore, the results of PT test reveal that the DCE learning approach is more powerful than all

Single model ARIMA

ANN

SVR

LSSVR

0.3126 0.7546 0.3125 0.7547 −0.4027 0.6872

−0.8926 0.3721 −1.6017 0.1092 −1.5129 0.1303

−0.5923 0.5536 −0.9137 0.3609 −0.6674 0.5045

1.4018 0.1610 1.9896 0.0466 1.7123 0.0868

accuracy and directional accuracy of LSSVR forecasting model are the optimal among all the single models. Hence, the LSSVR is considered as the individual forecasting model and ensemble forecasting method in decomposition ensemble learning approach and the proposed DCE learning approach.

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Table 5 Cluster centers and member numbers of EEMD-LSSVR-K-LSSVR ensemble approach. Horizon

Cluster

Cluster center

Number

One-step-ahead forecasting

1 2 3 1 2 3 1 2 3

(0.228, −0.180, −0.178, −0.194, −0.173, −0.001, −0.815, −0.122, 0.037, −0.010, 8.445) (0.010, 0.137, 0.028, −0.001, 0.052, −0.394, −0.348, −0.016, 0.061, 0.010, 8.609) (−0.087, −0.260, 0.206, −0.074, −0.290, −0.369, 0.478, 0.086, −0.008, −0.010, 8.701) (−0.147, −0.0.088, 0.028, 0.146, −0.204, 0.144, −0.465, −0.093, 0.034, 0.008, 8.446) (−0.449, 0.209, 0.217, 0.001, −0.006, 0.217, −1.050, 0.006, 0.041, 0.009, 8.560) (0.535, −0.208, 0.317, −0.004, −0.317, −0.350, 0.494, 0.086, −0.008, −0.010, 8.700) (0.096, 0.002, −0.078, −0.045, −0.167, −0.366, 0.272, −0.026, 0.029, −0.008, 8.450) (0.385, 0.038, −0.226, −0.091, 0.064, −0.050, 0.097, −0.017, 0.067, 0.010, 8.612) (−0.118, −0.058, 0.306, 0.048, −0.402, −0.218, 0.596, 0.087, −0.011, −0.009, 8.703)

715 989 662 726 982 656 695 987 679

Three-step-ahead forecasting

Six-step-ahead forecasting

Table 6 DM test results for hybrid ensemble approaches in one-step-ahead forecasting. Tested model

EEMD-LSSVR-K-LSSVR EMD-LSSVR-K-LSSVR EEMD-LSSVR-LSSVR EMD-LSSVR-LSSVR EEMD-LSSVR-ADD

Benchmark model EMD-LSSVR-K-LSSVR

EEMD-LSSVR-LSSVR

EMD-LSSVR-LSSVR

EEMD-LSSVR-ADD

EMD-LSSVR-ADD

−1.9124 (0.0279)

−2.0125 (0.0221) −1.7029 (0.0443)

−2.0964 (0.0180) −1.2385 (0.1078) −1.3346 (0.0910)

−2.9974 −1.6951 −1.0263 −1.5968

−3.8961 (0.0000) −3.5264 (0.0002) −1.5788 (0.0572) −2.9816 (0.0014) 0.8463 (0.1987)

(0.0014) (0.0450) (0.1524) (0.0552)

Table 7 DM test results for hybrid ensemble approaches in three-step-ahead forecasting. Tested model

EEMD-LSSVR-K-LSSVR EMD-LSSVR-K-LSSVR EEMD-LSSVR-LSSVR EMD-LSSVR-LSSVR EEMD-LSSVR-ADD

Benchmark model EMD-LSSVR-K-LSSVR

EEMD-LSSVR-LSSVR

EMD-LSSVR-LSSVR

EEMD-LSSVR-ADD

EMD-LSSVR-ADD

−2.2354 (0.0127)

−2.9025 (0.0019) −1.8974 (0.0289)

−3.4458 (0.0002) −2.5961 (0.0047) 0.7842 (0.2165)

−4.5543 −3.2594 −1.0268 −1.1269

−4.0056 (0.0000) −4.2674 (0.0000) −1.3642 (0.0863) −1.6874 (0.0458) 0.4419 (0.3293)

(0.0000) (0.0005) (0.1523) (0.1299)

Table 8 DM test results for hybrid ensemble approaches in six-step-ahead forecasting. Tested model

EEMD-LSSVR-K-LSSVR EMD-LSSVR-K-LSSVR EEMD-LSSVR-LSSVR EMD-LSSVR-LSSVR EEMD-LSSVR-ADD

Benchmark model EMD-LSSVR-K-LSSVR

EEMD-LSSVR-LSSVR

EMD-LSSVR-LSSVR

EEMD-LSSVR-ADD

EMD-LSSVR-ADD

−2.9156 (0.0018)

−2.6671 (0.0038) −1.7849 (0.0371)

−3.1146 (0.0009) −2.5963 (0.0047) 1.0284 (0.1519)

−6.1573 −4.5267 −1.6157 −1.8816

−6.2105 −4.5249 −6.4892 −3.1257 −0.3674

(0.0000) (0.0000) (0.0531) (0.0299)

(0.0000) (0.0000) (0.0000) (0.0008) (0.3567)

Table 9 PT test results for hybrid ensemble approaches. Forecasting horizons

Value

Hybrid ensemble approach EMD-LSSVR

One-step-ahead forecasting Three-step-ahead forecasting Six-step-ahead forecasting

SPT p-Value SPT p-Value SPT p-Value

EEMD-LSSVR

ADD

LSSVR

K-LSSVR

ADD

LSSVR

K-LSSVR

1.6587 0.0972 1.5543 0.1201 1.5021 0.1331

2.2206 0.0264 2.0017 0.0453 1.8954 0.0580

3.4106 0.0006 3.0146 0.0026 2.8914 0.0038

1.9687 0.0490 1.9217 0.0546 1.7416 0.0816

2.5861 0.0097 2.6349 0.0084 2.4417 0.0146

4.0129 0.0001 3.8874 0.0001 3.6253 0.0002

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4.2.4. Discussion To evaluate the forecasting performance of our proposed DCE approach against other multiscale hybrid models we applied to this problem the models proposed in Monjoly et al. (2017). In that paper the authors proposed an EMD-hybrid model, an EEMD-hybrid model and a WD-hybrid model to forecast hourly solar radiation. Table 12 lists the results obtained when we applied those models to our problem. The results show that the proposed DCE approach obtained better results than the multiscale hybrid models in Monjoly et al. (2017). The main reason why DCE approach performs better is that the clustering leverage improves the final forecasting performance of the decomposition ensemble learning approach. Meanwhile, according to above comparisons and statistical test results, the following six main conclusions can be summarized:

other hybrid ensemble approaches in terms of directional accuracy. From Figs. 8–10, certain interesting findings can be summarized: (1) the proposed EEMD-LSSVR-K-LSSVR outperforms all other benchmark models in different forecasting horizons, which implies that the EEMDLSSVR-K-LSSVR is a powerful learning approach for solar radiation forecasting in both level accuracy and directional accuracy. In contrast, the forecasting performances of EEMD-LSSVR-ADD and EMD-LSSVRADD are poor. The main reason may be that the addition ensemble is very difficult to identify the characteristic of component forecasting results; (2) it clearly shows that the hybrid ensemble approach with EEMD is much better than the one with EMD by means of level accuracy and directional accuracy, which reveals that EEMD is a more effective decomposition algorithm; (3) according to Kmeans clustering method, it can be found that the hybrid ensemble approach with clustering outperform the one without clustering in both level accuracy and directional accuracy, which indicates that the clustering scheme is a simple and effective technique for hybrid ensemble learning approach; (4) the forecasting performance of hybrid ensemble approach is significantly better than single model. The possible reason is that the decomposition can dramatically improve the forecasting performance of single models. Similarly, the DM test and PT test are performed to statistically evaluate the difference of hybrid ensemble approaches from both level accuracy and directional accuracy. The comparison results are shown in Tables 6–9, which demonstrates the same conclusions as above. Therefore, the empirical results show that the proposed DCE learning approach significantly improves the performance of decomposition ensemble learning approach, which also improves the forecasting performance of single models.

(1) The proposed DCE learning approach significantly outperforms all other benchmark models by means of level accuracy and directional accuracy, which indicates that the clustering strategy can effectively improve the performance of hybrid ensemble learning approach. (2) The performance of LSSVR is the best in comparison with other single models. (3) The hybrid ensemble learning approaches perform better than the single models in terms of different forecasting horizons, which imply that the scheme of decomposition-ensemble can effectively improve the forecasting performance of solar radiation. (4) The proposed EEMD-LSSVR-K-LSSVR learning approach is a stable and effective forecasting framework with respect to robustness analysis. (5) Due to the intrinsic complexity of solar radiation data, nonlinear AI techniques with powerful self-learning ability are much more appropriate for solar radiation forecasting than linear ARIMA model. (6) The proposed DCE learning approach is a promising framework for forecasting solar radiation in terms of level accuracy, directional accuracy and robustness.

4.2.3. Robustness analysis The robustness of single models and hybrid ensemble learning approaches are evaluated in this section. The ARIMA, ANN, SVR and LSSVR models produce different forecasting results with different initial settings. Therefore, we run all single models and hybrid ensemble learning approaches for twenty times, and analyze their robustness in terms of standard deviation of NRMSE, MAPE and DS. The results are shown in Tables 10 and 11, and some interesting findings can be summarized: (1) it is obvious that LSSVR is the most stable model in all single models, since its standard deviations of NRMSE, MAPE and DS are far less than other single models; (2) EEMD-LSSVR-K-LSSVR is the most robust hybrid ensemble learning approach, and its standard deviations of NRMSE, MAPE and DS are far less than 0.01; (3) ADD-based ensemble forecasting are most unstable among all the hybrid ensemble learning approaches.

5. Conclusions This paper proposes a decomposition-clustering-ensemble (DCE) learning approach for solar radiation forecasting. The DCE learning approach employs EEMD for decomposition, Kmeans for clustering analysis and LSSVR for ensemble learning. For further verification, the proposed EEMD-LSSVR-K-LSSVR learning approach is applied to forecast solar radiation. The empirical results show that the EEMD-LSSVRK-LSSVR learning approach can significantly improve the forecasting performance and outperform some other popular forecasting methods in terms of forecasting accuracy and robustness analysis, which indicates that the EEMD-LSSVR-K-LSSVR learning approach is a relatively promising approach for solar radiation forecasting problems. Besides solar radiation forecasting, the proposed CNE learning approach can be applied to solve other complex and difficult forecasting problems, including wind speed forecasting, container throughput forecasting, crude oil price forecasting, traffic flow forecasting, etc. Furthermore, this paper only focuses on univariate time series analysis, while other factors affecting solar radiation are not taken into consideration. If those factors can be introduced into the proposed DCE learning approach, the forecasting performance may be better. In addition, the kernel function of LSSVR is selected only by means of the experience and the selected kernel function may not be optimal for the specific problem. Hence, the optimal selection of kernel function for a specific problem is left for a further study.

Table 10 Robustness analysis for single models. Horizon

One-step-ahead forecasting

Three-step-ahead forecasting

Six-step-ahead forecasting

Std.

Std. of NRMSE Std. of MAPE Std. of DS Std. of NRMSE Std. of MAPE Std. of DS Std. of NRMSE Std. of MAPE Std. of DS

Single model ARIMA

ANN

SVR

LSSVR

0.5123

0.2035

0.1147

0.0876

0.1022 0.6756 0.5374

0.0875 0.2074 0.2194

0.0625 0.1169 0.1873

0.0421 0.0876 0.0967

0.1124 0.6811 0.5486

0.0954 0.0975 0.2367

0.0714 0.1008 0.1982

0.0461 0.0926 0.0923

0.1326 0.6938

0.1059 0.1041

0.0829 0.1073

0.0562 0.0964

Acknowledgement This research was supported by the National Natural Science Foundation of China (Project No: 71373262). The Authors would like to

The bold values denotes the best performance of robustness analysis.

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Table 11 Robustness analysis for hybrid ensemble approaches. Horizon

Std.

Ensemble approach EMD-LSSVR

One-step-ahead forecasting

Std. Std. Std. Std. Std. Std. Std. Std. Std.

Three-step-ahead forecasting

Six-step-ahead forecasting

of of of of of of of of of

NRMSE MAPE DS NRMSE MAPE DS NRMSE MAPE DS

EEMD-LSSVR

ADD

LSSVR

K-LSSVR

ADD

LSSVR

K-LSSVR

0.2049 0.0106 0.1956 0.2149 0.0153 0.2011 0.2218 0.0163 0.2217

0.0509 0.0076 0.0791 0.0583 0.0107 0.0831 0.0623 0.0125 0.0892

0.0512 0.0029 0.0113 0.0664 0.0059 0.0139 0.0704 0.0071 0.0149

0.3671 0.0087 0.1652 0.3792 0.0114 0.1798 0.3884 0.0182 0.1859

0.1327 0.0095 0.0592 0.1428 0.0137 0.0625 0.1568 0.0152 0.0782

0.0256 0.0032 0.0098 0.0329 0.0065 0.0119 0.0395 0.0085 0.0128

The bold values denotes the best performance of robustness analysis. Table 12 Forecasting performance using the hybrid model proposed in Monjoly et al. (2017). Evaluation criteria

Models

One-step-ahead forecasting

Three-step-ahead forecasting

Six-step-ahead forecasting

NRMSE (%)

EMD-hybrid model EEMD-hybrid model WD-hybrid model EMD-hybrid model EEMD-hybrid model WD-hybrid model EMD-hybrid model EEMD-hybrid model WD-hybrid model

3.27 4.38 4.76 3.09 3.56 4.42 86.46 82.76 79.07

3.45 4.13 5.08 3.19 3.74 4.38 87.55 80.57 79.34

4.65 5.28 5.92 4.23 5.15 5.69 78.66 77.98 73.87

MAPE (%)

DS (%)

method, LSSVR as component forecasting method and LSSVR as ensemble method EEMD-LSSVR-K-LSSVR – the model using EEMD as decomposition method, LSSVR as component forecasting method, Kmeans as clustering method and LSSVR as ensemble method ELM – extreme learning machine EMD – empirical mode decomposition EMD-LSSVR-ADD – the model using EMD as decomposition method, LSSVR as component forecasting method and ADD as ensemble method EMD-LSSVR-LSSVR – the model using EMD as decomposition method, LSSVR as component forecasting method and LSSVR as ensemble method EMD-LSSVR-K-LSSVR – the model using EMD as decomposition method, LSSVR as component forecasting method, Kmeans as clustering method and LSSVR as ensemble method GBR – gradient boosted regression GHI – global horizontal solar irradiance GP – Gaussian process GSA – gravitational search algorithm IMFs – intrinsic mode functions LSSVR – least square support vector regression MAPE – mean absolute percentage error MLPNN – multi-layer perceptron neural network MRSR – multi-response sparse regression NRMSE – normalized root mean squared error OP-ELM – optimally pruned extreme learning machine PCF – partial correlation function PM – Pesaran-Timmermann test RFR – random forest regression SC – Schwarz criterion SVM – support vector machines SVR – support vector regression

express their sincere appreciation to Jan Kleissl (the subject editor), Hugo Pedro (the associate editor), and two anonymous referees in making valuable comments and suggestions to this paper. Their comments and suggestions have improved the quality of the paper immensely. Conflict of interest The authors declare that there is no conflict of interests regarding the publication of this paper. Appendix A. List of abbreviations Here, all terms mentioned in this paper and their definitions are listed in alphabetical order: ACF – autocorrelation function AI – artificial intelligence ANN – artificial neural network AR – autoregressive model ARMA – autoregressive moving average ARIMA – autoregressive integrated moving average BPNN – back propagation neural network CS – cuckoo search algorithm DCE – decomposition-clustering-ensemble learning approach DHR – dynamic harmonic regression model DM – Diebold-Mariano test DS – directional symmetry EEMD – ensemble empirical mode decomposition EEMD-LSSVR-ADD – the model using EMD as decomposition method, LSSVR as component forecasting method and ADD as ensemble method EEMD-LSSVR-LSSVR – the model using EMD as decomposition

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