Energy Conversion and Management 171 (2018) 787–806
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Variational mode decomposition based low rank robust kernel extreme learning machine for solar irradiation forecasting
T
⁎
Irani Majumder, P.K. Dash , Ranjeeta Bisoi Siksha O Anusandhan Deemed to be University, Bhubaneswar, India
A R T I C LE I N FO
A B S T R A C T
Keywords: Solar irradiation prediction Variational mode decomposition Empirical mode decomposition Low rank robust kernel extreme learning machine Morlet Wavelet Kernel
In this paper a new hybrid method has been implemented by combining Variational Mode Decomposition (VMD) and a new low rank robust kernel based Extreme Learning Machine (RKELM) for solar irradiation forecasting. This hybrid model presents an efficient and effective short term solar irradiation prediction approach using the historical solar irradiation data. The original non-stationary time series data is decomposed into various modes using VMD approach. The proposed VMD-RMWK (VMD based reduced Morlet Wavelet Kernel extreme learning machine) method is used to predict the solar irradiation of an experimental 1 MW solar power plant in Odisha, India. Different time intervals of 15 min, 1 h and 1 day ahead in different weather conditions are considered for forecasting purpose. The VMD technique decomposes the original nonlinear irradiation into a set of Variational Mode Functions (VMFs), and the extracted VMFs are used to train the kernel based robust ELM. Comparison with empirical mode decomposition (EMD) based low rank kernel is also presented in this paper. As a new contribution to the previously performed literature survey this paper presents a more accurate solar irradiation prediction paradigm for distinctive weather conditions, and different time intervals varying from very short duration of 15 min to one day ahead. Also to improve the reliability of the KELM and to make it robust under noisy conditions and the presence of outliers in the data, a weight loss matrix has been derived using a nonparametric kernel density estimation method and incorporated in the new formulation (RKELM). A typical solar power experimental solar power station in India has been taken for detailed study showing clearly the accuracy and robustness of the proposed approach. For cross validation of the proposed model, solar irradiation data from a solar power plant located in the state of Florida has been implemented.
1. Introduction Due to increased demand in renewable energy, photovoltaic generation systems have gained more importance. PV power penetration into the grid led to precious solar irradiation prediction for stable functioning of the power plant. Effective solar power prediction is necessary mostly for high energy integration [1,2]. The intermittent nature of the solar irradiation gives way to variable problems like stability, power quality issues and voltage fluctuations [3]. The solar output is not easily predicted and differs based on variable weather conditions and location of the solar plant as this affects the irradiation. Previously many researchers have applied different forecasting methods for solar irradiation or power forecasting. These methods can be broadly categorised into two different types namely linear and nonlinear forecasting techniques, predominantly implemented linear methods are ARMA, ARIMA and physical model like (NWP) [4–10]. The numerical weather prediction method (NWP) previously used requires more detailed information of the input data accompanied with greater ⁎
Corresponding author. E-mail address:
[email protected] (P.K. Dash).
https://doi.org/10.1016/j.enconman.2018.06.021 Received 1 March 2018; Received in revised form 20 May 2018; Accepted 6 June 2018 0196-8904/ © 2018 Elsevier Ltd. All rights reserved.
computational complexity. The ARMA and ARIMA models are very popular for solar irradiation forecasting but suffer from inaccuracies due to inability to handle highly nonlinear and fluctuating data. On the other hand the latter method includes different artificial intelligence and machine learning techniques. This category includes fuzzy logic system [11–13], artificial neural network (ANN) [14–17], support vector machine (SVM) [18–22], etc. Although the ANN based methods for irradiation forecasting are known for their widespread implementations, but they suffer from problems like generalization, over fitting, local minima, and lower convergence speed. The ANN techniques proved to be unstable in case of small variation in time series data and selection of different attributes, contributing to inaccurate solar irradiation prediction. A number of hybrid methods were introduced to predict the solar irradiation time series data which is a combination of two or more methods. The hybrid models imparted better prediction accuracy and hence are considered to be superior when compared to other methods. The different hybrid models initially used for solar irradiation prediction include hybrid ANN like ARMA-
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RMWK), and also other RKELM variants. The rest of the paper is organised as follows: Section 2 describes the RKELM (robust kernel extreme learning machine) model along with its reduced or low rank version. In Section 3 VMD and EMD techniques are described for the decomposition of solar irradiation time series data into different modes which are used as inputs to the forecasting kernel models. In Section 4 the detailed numerical experimentation results are outlined for both VMD based RKELM and EMD based RKELM prediction models in different atmospheric conditions followed by conclusion in Section 5.
ANN [10,18], etc. In order to overcome the limitations of ANN based models, many researchers have proposed a new learning machine algorithm known as the Extreme Learning Machine (ELM) [23–28]. This is a single layer feed forward neural network (SLFNs) which structurally shows a resemblance with random vector functional link (RVFL) network and where the input weights are randomly selected. The randomly chosen weights cause variations in the output for different trial runs and hence making the system non-robust. A new incremental ELM (IELM) was proposed by Huang et al. [26], and in this case the accuracy of the training samples was improved. But the major drawback of ELM and its variants are their dependence on the correct choice of neurons in the hidden layer and the right activation function, which are still problems for larger input data set. In order to mitigate the problem of hidden layer selection, kernel functions can be used. The kernel function improves the stability of the prediction system when applied in ELM technique and is termed as Kernel based Extreme Learning Machine (KELM) [29,30]. Different types of kernel functions namely Gaussian kernel, Polynomial Kernel, Sigmoid kernel and Wavelet Kernel are implemented for prediction purpose. The kernel based formulation provides a consolidated framework and generalized model for forecasting time series data. In spite of the afore mentioned advantages, the kernel technique suffers from a disadvantage of larger training time while dealing with large data set. A recent literature [31] shows that the training time and computational overhead can be substantially decreased by reducing the size of the kernel matrix. In this paper the row rank kernel method is described for reducing the execution time without affecting the accuracy of the system to a greater extent. Further the KELM is made robust against the presence of noise and outliers in the fluctuating solar irradiation data during weather changes by incorporating a variable weight for the residual error using a non-parametric kernel density technique [35] and this model will be known as RKELM. The nonlinearity of the time series data can be handled by decomposition techniques. Empirical mode decomposition (EMD) technique is used to decompose the data into intrinsic mode functions and a residue [32]. The basic disadvantage of this technique is that it lacks mathematical foundation. In recent time, a decomposition technique known as the Variational mode decomposition (VMD) [33,34] has been implemented as a better alternative to the previously used EMD technique. Unlike the EMD model VMD processes an exact mathematical model and sensitive to both noise and sampling. VMD is a non-recursive variational model that explores for a number of modes and there central frequencies in a band limited manner which reconstructs the original signal in the least square sense. As compared to EMD VMD has a superior denoising property and ability to separate tones of similar frequencies. In this paper the VMD is used to decompose the original solar irradiation data in order to obtain the fundamental variational mode functions (VMF) which combine together to form the original data set. The VMFs act as the predictor for KELM to forecast the future data. Although various VMD based hybrid models have been implemented for wind power and price forecasting, they are still not implemented for solar irradiation or power forecasting. The main contribution of this paper is that it presents a hybrid technique implementing VMD and different low rank robust kernel extreme learning machines (RKELM) namely VMD-RMWK, VMD-RMHWK, VMD-RGK, VMD-RPK and VMDRSK) for solar irradiation prediction under different weather conditions like sunny, rainy and foggy at various time horizon (15 min, 1 h and 1 day). The RKELM reduces the effects of the outliers and hence increasing the reliability of the samples, and this process decreases the execution time without affecting the accuracy of prediction to a greater extent. The decomposition technique (VMD) divides the main nonlinear data into different modes. This reconstructs the original signal with the help of least square technique which removes the nonlinearity of the original signal. Further to validate the superiority of the proposed prediction model VMD-RMWK, its performance is compared with the basic model (VMD-MWK) and decomposition based RKELM (EMD-
2. Robust kernel based extreme learning (RKELM) The output function of the basic ELM with L hidden nodes can be represented by L
fL (x ) =
∑ βi hi (x ) = h (x ) β
(1)
i=1
where the output vector β = [β1, β2, ...........,βL] between the L hidden neurons and the output neuron and the ELM feature mapping functionh (x ) = [h1 (x ), h2 (x ), .............,hL (x )]; the number of input samples x = [x1, x2 , ...............,xN ], and N represents the number of patterns. The initial weights between the input layer consisting m inputs for each pattern and hidden layer with L neurons need not be tuned and the activation function of the hidden neurons could comprise almost all nonlinear piecewise continuous functions. Thus using the tanh function as the activation function
hi (x ) = tanh(wi0 + wi1 x i1 + wi2 x i2 + ...............+wim x im)
(2)
Thus the hidden layer randomized matrix is written as
⎡ h (x1) ⎤ ⎡ h (x1)⋯hL (x1) ⎤ ⋮⋮⋮ H=⎢ ⋮ ⎥=⎢ ⎥ ⎢ h (x ) ⎥ ⎢ h (x )⋯h (x ) ⎥ N N L N ⎦ ⎣ ⎣ ⎦
(3)
and the target vector is given by
t ⎡ 1⎤ t2 ⎥ ⎢ T=⎢.⎥ . ⎢t ⎥ ⎣ N⎦
(4)
Eq. (1) is expressed in a matrix form as
Hβ = T
(5)
To solve for β , a constrained optimization problem needs to be solved that overcomes the over fitting problem and provides better generalization ability in comparison to the original ELM. This is similar to the structural risk minimization of the statistical learning theory and is expressed as N
LCELM =
1 1 ∥β∥2 + C ∑ wi ξi2 2 2 i=1
(6)
Subject to h (x i ) β = ti−ξi, i = 1, 2, ...............,N where the error vector ξ is obtained as: ξ = [ξ1, ξ2, ...............,ξN ], C is the regularization parameter. However, if the data samples are corrupted by noise or outliers, the reliability of the regularized ELM is not robust. Therefore, in Eq. (6) a weighting factor wi on a sample to sample basis is introduced using the nonparametric kernel density estimation [35] to the output error. This provides better reliability of the samples by unevenly weighting each sample, so that the effect of noise or outliers in the data is eliminated. This makes the ELM robust. Using KKT theorem the constrained optimization problem in Eq. (6) is converted to a dual optimization problem as N
L DELM = 788
N
1 1 ∥β∥2 + C ∑ wi ξi 2− ∑ αi (h (x i ) β−ti + ξi ) 2 2 i=1 i=1
(7)
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where α = [α1, α2, ...........,αN ] is the vector of Lagrange multipliers; and W = diag [w1, w2, ...............,wN ] is the weighting matrix for the error vector ξ . Now taking the partial derivatives Eq. (7) with respect to β , ξ , and α and equating them to zero for optimality condition the following equations are obtained:
β = H T α, α = CWξ , Hβ−T + ξ = 0
Table 1a Summary of the experimental data site. Location Coordinates Size Data recorded Data resolution Data period
(8)
Tangi, Odisha, India 19.924°N, 85.3966°E 1 MW Irradiation, ambient temperature, current and power 15 min 1h 1 day 1st Jan 2015-31st Dec 2015
Table 1b Components specification of the experimental solar power station.
Thus the value of β is obtained from Eq. (8) in the following way:
WHβ−WT + Wξ = 0
Components
Specification
Number
−1 I ⎛ + HH T ⎞ α = WT ⎠ ⎝C
Solar module Solar inverter Step up transformer SCADA system
250 Wp 500 kW 1250kVA –
4400 2 1 1
and thus −1 I β = H T ⎛ + WHH T ⎞ WT C ⎝ ⎠
Table 1c Summary of the Florida data site.
(9)
where I is an unit matrix of appropriate dimension. From the above equation the output for a sample x comprising m inputs from the RELM is obtained as −1 I f (x ) = h (x ) H T ⎛ + WHH T ⎞ WT ⎝C ⎠
(10)
Thus for N number of patterns the total output vector becomes equal
Location
Miami, Florida
Coordinates Data recorded Data resolution Data period Plant ID
25.7617° N , 80.1918° W Irradiation, ambient temperature, sunny glow 1h 1st Jan 2010-31st Dec 2010 722020
to average irradiation
−1 I O = HH T ⎛ + WH T H ⎞ WT ⎝C ⎠
solar irradiation (kWh/m2)
(11)
The learning performance of the regularized ELM like stability and generalization is influenced by the choice of the number of neurons in the hidden layer and a proper selection of activation function which are still an unsolved problem. Thus when feature mapping function is unknown, Kernel functions can be used for the ELM to provide better stability and generalization and the ELM will be designated as Kernel Extreme Learning Machine or simply as KELM. Thus the Kernel matrix for the KELM based on Mercer theorem is given by
KELM = HH T , and KELM (x i , x j ) = h (x i ) h (x j )
max power 950
8 7 6 5 4 3 2 1 0
900 850 800 750 JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC months
(12)
Thus the output function is obtained as
Fig. 1. Variation in solar irradiation: monthly average irradiation data.
−1
I f (x ) = h (x ) H T ⎛WKELM + ⎞ WT C⎠ ⎝
condition and are convenient to use. These Kernel functions are listed as: Polynomial kernel, Gaussian Kernel, Hyperbolic tangent kernel (Sigmoid kernel), Wavelet Kernel, etc., whereas the Multi- kernel functions is a weighted combination of the above mentioned kernels. The various kernel functions are described as follows:
(13)
Eq. (13) is written in an expanded form as
I −1 f (x ) = [(x , x1), (x , x2), .....................,(x , xN )] ⎛WKELM + ⎞ WT ‵ C⎠ ⎝
(14)
For obtaining the probability distribution of the error vector ξ , initially W is chosen as W = I, an identity matrix of size N × N . Thus
ξ = Hβ−T , ξi = h (x i ) β−ti
(1) Polynomial kernel:
K (x i , x j ) = (1 + μx iT x j )2
(15)
Using the approach given in [35] the probability density function of the residual error ξi is obtained as
f (ξi ) =
1 2π γN
N
(2) Gaussian kernel:
K (x i , x j ) = exp (− ∥(x i−x j )∥2 )/(2b2)
2
⎡ ξ −ξ j ⎤ ∑ exp ⎢ ⎜⎛ γ ⎟⎞ ⎥ j=1 ⎠⎦ ⎣⎝
(16)
(19)
(3) Sigmoid kernel:
where γ = width of the estimated window and is obtained as: γ = 1.06σ / N 0.2 , and σ is the standard deviation of the residual errors. Thus the greater the value of f (ξi ) , the greater is the reliability of the data samples. Therefore the final weight matrix is obtained as
W = I + δWi and δWi = f (ξi ), i = 1, 2, ................,N
(18)
K (x i , x j ) = tanh(cx iT x j + d )
(20)
(4) Morlet Wavelet Kernel:
(17)
K (x i , x j ) = cos ⎛ ⎝
⎜
There are various types of Kernel functions that satisfy the Mercer 789
1.675 ∥(x i−x j ) ∥ ⎞ e
exp ⎜⎛− ⎠ ⎝
⎟
∥(x i−x j ) ∥2 ⎞ 2e 2
⎟
⎠
(21)
Energy Conversion and Management 171 (2018) 787–806
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(a)
(b)
(c)
Fig. 2. Original solar irradiation for sunny weather: (a) 15 min interval solar irradiation, (b) 1 h interval solar irradiation, (c) 1 day interval solar irradiation.
(5) Mexican Hat Wavelet Kernel:
∥WKN × L β−T∥2 < ε
2 x −y ⎞2⎤ 1 x −y ⎞ ⎤ K (x , y ) = ⎡1−⎛ exp ⎡− ⎛ ⎢ ⎝ α ⎠⎥ ⎢ 2⎝ α ⎠ ⎥ ⎦ ⎣ ⎦ ⎣
where ε is non zero training error and L is always less than N . Using ℓ2 norm minimization and sample number constraint, the optimization problem using the weight loss matrix (for robustness) is formulated as 1 1 Minimize 2 ∥β∥2 + 2 CW ∥ (KN × L β−T ) ∥2 −α (WKN × L β−T + ξ ) and using the KKT theorem the solution for the output vector is obtained as
(22)
The various parameters like μ , b c, d, e, and α are to be chosen appropriately to enhance the performance of the kernel based KELM forecasting model.
−1 I β = ⎡ + WL × L KNT × L KN × L⎤ WL × L KNT × L T ⎣C ⎦
2.1. Low rank robust KELM
T
⎡ K (x , x1) ⎤ ⎢ K (x , x2) ⎥ −1 I . ⎥ ⎡ + WL × L KNT × L KN × L⎤ WL × L KNT × L T O (x i ) = ⎢ . ⎥ ⎣C ⎢ ⎦ . ⎥ ⎢ ( , ) K x x L ⎦ ⎣
(26)
3. Data pre-processing The original solar data (irradiation) is decomposed and these decomposed time series data is givenas the input to the RKELM technique to improve the forecasting ability of this new method. The non-linear and non-stationary solar irradiation signals are analyzed in an improved manner with the help of decomposition technique. An improvedexploration of the input signals upgrades the prediction efficiency and effectiveness of the proposed RKELM model. Two decomposition techniques implemented in this paper for data pre-processing are Variational mode decomposition (VMD) and Empirical mode decomposition (EMD). A brief description of these methods is given in the Section 3.1.
L p=1
(25)
Thus the predicted output Oi for the sample x i is obtained as
Although the Kernel based formulations provide a unified learning frame work and generalization for prediction and classification it suffers from larger training time while dealing with large data sets. In a recent literature [31] it has been shown that the training time and computational overhead can be substantially decreased by reducing size of the Kernel matrix (N×N in Eq. (12)) to N × L, where L denotes the randomly selected subset of training data that serve as support vectors. However, the reduced KELM or suffers from a slight loss of prediction accuracy while reducing the training time and computational complexity substantially. For the robust KELM with kernel function K (x i , x j ) with L arbitrarily chosen support vectors, the following equation is written:
∑ WK (xi , xj) βp = T (i), i = 1, 2, ..............,N , j = 1, 2, .............,L
(24)
(23)
or in matrix form WL × L KN × L β = T , and β = [β1, β2, ................,βL] Further it is well known that for distinct N patterns or samples
∥WKN × N β−T∥2 = 0 However, with L number of random samples 790
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(a)
(c)
(b)
Fig. 3. Original solar irradiation for rainy weather: (a) 15 min interval solar irradiation, (b) 1 h interval solar irradiation, (c) 1 day interval solar irradiation.
After the bandwidth estimation, the resulting constrained variational problem is expressed as
3.1. Variational mode decomposition (VMD) The basic principle of VMD is non-recursive decomposition of a time varying signal into different modes. It decomposes the original signal X into a number of modes with limited bandwidth; the modes can be called as XK. It is mandatory for each mode K to be compact about a centre pulsation WK along with the decomposition. A specific sparsity is maintained by each mode, this sparsity is nothing but the bandwidth in its spectral domain. The bandwidth of each mode is obtained with the help of following algorithm. 1. Hilbert transform is applied to the mode XK to attain the unilateral frequency spectrum. 2. The frequency spectrum of the mode is shifted to a baseband region by mixing an exponential tuned to its respective estimated centre frequencies. 3. Estimation of the bandwidth of each mode is obtained through H1 Gaussian smoothness of the demodulated signal, which is squared L2 i.e. norm of the gradient. The Variational Mode Decomposition is then carried out by solving the following problem [33,34]. For each mode XK(t) an associated analytical signal is computed by means of Hilbert transformation,
K
⎧ ∑ {XK }·{WK } ⎨ ⎩ K =1 min
j ⎞ ∗XK (t ) ⎤ e−jwK t ∂t ⎡ ⎛δ (t ) + ⎢ ⎥ π t⎠ ⎣⎝ ⎦
⎫ ⎬
(29)
2⎭
Such that K
∑ XK
=X (30)
K =1
where X is the original signal to be decomposed {XK} is the set of modes {X1, X2,…,XK} and WK is the set centre pulsation i.e. {W1, W2, …, WK}. K being the number of modes, * is the convolution and δ is the Dirac distribution. The above mentioned constrained problem can be transferred to unconstrained problem by combining the quadratic penalty term and the lagrangian multipliers and hence can be discussed as below. K
∑
L ({XK }, {WK }, λ ): =α
K = 1‵
j ⎞ ∗XK (t ) ⎤ e−jwK t ∂t ⎡ ⎛δ (t ) + ⎥ ⎢ πt ⎠ ⎦ ⎣⎝ K
+
X (t )−
2
2
2
∑ XK (t ) K =1
j ⎛δ (t ) + ⎞ ∗XK (t ) πt ⎠ ⎝
2
+
λ (t ),
2
K
(27)
X (t )−
K =1
The frequency spectrum of each mode is shifted to its respective estimated central frequency
(31)
Eq. (29) is obtained in a sequence of iterative measures known as the alternate direction method of multipliers (ADMM), ADMM is applied to obtain the saddle point of the previously derived augmented Lagrange expression
−jωK t
⎡ ⎛δ (t ) + j ⎞ ∗XK (t ) ⎤ ⎢⎝ ⎥e πt ⎠ ⎣ ⎦
∑ XK (t )
(28) 791
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(a)
Fig. 4. Original solar irradiation for foggy weather: (a) 15 min interval solar irradiation, (b) 1 h interval solar irradiation, (c) 1 day interval solar irradiation.
⎧ XKn + 1 = arg min α ⎨ XK ∈ X ⎩
j ⎞ ∗XK (t ) ⎤ e−jω K t ∂ (t ) ⎡ ⎛δ (t ) + ⎢⎝ ⎥ πt ⎠ ⎣ ⎦
X (t )− ∑ Xi (t ) +
+
i
λ (t ) 2
2
3.2. Empirical mode decomposition (EMD)
2
EMD technique decomposes the main signal into small and finite number of intrinsic mode functions (IMF) and a residual by using Hilbert –Huang transform (HHT) [32]. IMF can be best characterised as a hidden oscillation mode satisfying the following conditions:
2
⎫ ⎬ 2⎭
(32)
Perceval Fourier isometry is applied to Eq. (30) and WK and ΣiXi(t) corresponds to Wn+1 and ΣiKXi(t)n+1 ,respectively. k
XKn + 1 =
arg
⎧
XK̂ , XK ∈ X ⎨
(1) In the entire data set the number of extrema and the number of zero crossings must be equal or the difference must be at max by one. (2) At any point, the mean value of the envelope defined by the local maxima and minima is zero. The IMF decomposition method is implemented in the following manner: (a) Pinpoint every local extrema (b) The local maxima is connected with a cubic spline line for the upper envelope and rerun the procedure for the production of lower envelope as the upper and lower envelope should cover the entire data between them. There mean might be designated as mn (t ) , and the difference between the signal x n (t ) and mn (t ) is the first component hn (t ) hence
α ∥jω [(1 + sgn(ω + ωK )) XK̂ (ω + ωK )] ∥22
⎩
X ̂(ω)− ∑ Xi ̂ (ω) +
+
i
λ (ω) 2
2
⎫ ⎬ 2⎭
(33)
the new solution obtained for XK and WK are as follows:
ωKn + 1 = arg min ωK
⎧ ⎨ ⎩
∞
∫ (ω−ωK )|XK̂ (ω)|2 dω⎫⎬ ⎭
0
(34)
hn (t ) = x n (t )−mn (t )
Central frequency ωK can be converted to frequency domain as
X ̂(W )− ∑ Xi ̂ (W ) + n+1
XK̂
(W ) =
i≠K
1 + 2α (W −WK )2
In ideal condition hn (t ) must satisfy the IMF definition, however that does not occur in most of the cases as changing local zero from a rectangular to a curvilinear system introduces a new extrema and hence adjustments are required and hence repetition and shifting of the procedure is necessary [31]. In the subsequence repetition of the process hn (t ) is treated as a proto- IMF and data for the next iteration
λ ̂(W ) 2
(35)
∞
∫ W |XK̂ (W )|2 dw WKn + 1 =
hn1 (t ) = hn (t )−mn1 (t )
0 ∞
∫ |XK̂ (W )|2 dw 0
(37)
(38)
The shifting process is repeated for k times until and unless hnk becomes the true IMF that is
(36)
n+1 Here X ̂(W ) , Xi ̂ (W ) , λ ̂(W ) and XK̂ (W ) are the Fourier transforms n+1 of X (t ) , Xi (t ) , λ (t ) and XK (t ) , respectively and n is the total number of iterations.
hnk (t ) = hn (k − 1) (t )−mnk (t ) And finally it is designated as 792
(39)
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MF10 MF9 MF8
MF7
MF6
MF5
MF4 MF3 MF2 MF1
(a) 15min ahead solar irradiation 0.8 0.7 0 0.2 0 -0.2 0 0.2 0 -0.2 0 0.2 0 -0.2 0 0.1 0 -0.1 0.10 0 -0.1
0 0.1 0 -0.1 0 0.1 0 -0.1 0 0.1 0 -0.1 0 5 0
0
20
40
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120
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number of samples
(b)1 hr ahead solar irradiation
(c) 1day ahead solar irradiation
Fig. 5. Various modes decomposition (VMD) of irradiation for sunny weather (a) VMF of 15 min ahead solar irradiation, (b) VMF of 1 h ahead solar irradiation, (c) VMF of 1 day ahead solar irradiation (all in per unit).
cn (t ) = hnk (t )
(40)
Prediction of solar irradiation principally the short term prediction (few minutes to one-day ahead) is a challenging task and the output solar power alters largely with correct prediction of irradiation and other external conditions like weather, and temperature, etc. The next section describes the detailed numerical experimentation results of solar irradiation prediction using the measured historical values from an experimental solar power station in India.
The stopping criterion for this shifting process was suggested by Huang et al., a normalized squared difference between the two successive shifting operations is required. T
S1 Dk =
∑ t=0
|hn (k − 1) (t )−hnk (t )| 2 hn2(k − 1) (t )
(41)
In general cn (t ) must contain a component that has the finest scale and a shortest signal. cn (t ) and the rest of the data can be separated as
rn (t ) = x n (t )−cn (t )
4. Numerical experimentation results
(42)
4.1. Model description
The residue obtained is rn (t ) containing a component with a longer period than the previous component and considering it to be a new signal and repetition of the process and hence the second IMF and residue cn2 (t ) and rn2 (t ) . This procedure can be repeated with all the subsequent r j (t ) and the result becomes
rni (t ) = rn (i − 1) (t )−cni (t )
This section describes the background of the employed data set based on the real time data available from a solar power plant of 1 MW capacity situated at the geographical location of Tangi, Odisha, India (19.92° N and 85.396° E). Table 1a gives the test site values and description where the total solar power generated over 12 months is taken (January 2015 to December 2015). Further the whole year has been divided into three weather conditions namely sunny weather (MarchJune), rainy weather (July-August) and foggy weather (SeptemberFebruary). Daily power and irradiation data collection are performed between 6 am and 7 pm; a pyranometer is used for measuring the daily solar irradiation. The average temperature for sunny weather is 36 °C, while that of rainy weather is 30 °C and foggy weather has an average temperature of 16 °C. The 1 MW solar power plant situated over an area
(43)
where i = 1, 2, 3, …, m. The shifting process is stopped by any predetermined criteria, by summing all the IMFs and the final residue the signal x n (t ) can be reconstructed as m
x n (t ) =
∑ cni (t ) + rnm (t ) i=1
(44) 793
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(a) 15min ahead solar irradiation
(b) 1hr ahead solar irradiation
(c) 1day ahead solar irradiation
Fig. 6. Various modes decomposition (VMD) of solar irradiation for rainy weather (a) VMF of 15 min ahead solar irradiation, (b) VMF of 1 h ahead solar irradiation, (c) VMF of 1 day ahead solar irradiation (all in per unit).
The proposed model predicts the solar irradiation using only the measured historical solar irradiation data samples between 6 am and 7 pm only assuming the irradiation is zero during the night. For execution of the experiment MATLAB R2015a is used in Windows-7 with 64 bit operating system. As mentioned before the model consists of different kernel functions based RKELM hybridized with either VMD or EMD techniques to predict the solar irradiation with measured irradiation as input for multi-horizon forecasting. A comparative study with different kernel functions is performed under different weather conditions and time horizon. Comparison of results shows the best model and the best weather for obtaining the lowest values of the performance metrics. Total number of samples used for 15 min interval prediction in sunny weather is 6200, 5000 for rainy weather and 4300 in foggy weather.
of 5 acres operates at 11 kV and is constructed to be a three phase, three wire, 50 Hz system with 4400 solar modules, 200 strings, and 23 SCB’s. Maximum DC current obtained is 1120 Amps with current and voltage harmonics of less than 3%. The supervisory control and data acquisition (SCADA) system has provision for local and remote display. The SCADA system collects all the related data and hence the performance is monitored and recorded which can be implemented for future studies. In this paper the SCADA data collected over a time span of 12 months is integrated for forecasting purpose. The key components of the solar plant are as follows: 1. 2. 3. 4. 5. 6. 7. 8. 9.
Solar PV module Inverter LT panel Lightning arrestor Step up transformer LT panel Energy meter SCADA monitoring system Battery bank and UPS
4.2. Performance metrics The robustness of the proposed hybrid method is validated with various performance metrics like the Mean Absolute Percentage Error (MAPE) between the absolute and forecasted values, the Mean Absolute error value (MAE), and the Root Mean Square Error (RMSE). Another performance metric like correlation coefficient (CC2) provides the amount of correlation between the measured values and the targeted values and thus can be used as an indicator for choosing the most efficient model for solar irradiation prediction.
Specifications of the above mentioned components used at the solar power plant are mentioned in Table 1b. The data used for cross validation of the technique is collected from the NREL solar irradiation data [36]. The geographical location and solar plant information’s are mentioned in Table 1c.
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(a) 15min ahead solar irradiation
(c) 1day ahead solar irradiation
(b) 1hr ahead solar irradiation
Fig. 7. Various modes decomposition (VMD) of solar irradiation for foggy weather (a) VMF of 15 min ahead solar irradiation, (b) VMF of 1 h ahead solar irradiation, (c) VMF of 1 day ahead solar irradiation (all in per unit).
⎛1 MAPE = ⎜ N ⎝
N
∑ i=1
x′ =
|Ac (i)−Fc (i)| ⎞ ⎟ ∗100 Ac (i) ⎠
(45)
∑ |Ac (i)−Fc (i)|
RMSE =
1 N
(46)
i=1 N
∑ |Ac (i)−Fc (i)|2
(47)
i=1 N
N
2
n0
⎛N ∑ O (x ) T (x )− ∑ O (x ) ∑ T (x ) ⎞⎟ i i i i i=1 i=1 ⎝ i=1 ⎠
⎜
CC 2 =
N
N
2
N
N
2
⎛ 2 ⎛ ⎞ ⎞⎛ ⎞⎞ 2 ⎛ ⎜N ∑ O (xi) −⎜ ∑ (O (xi)) ⎟ ⎟ ⎜N ∑ T (xi) −⎜ ∑ T (xi) ⎟ ⎟ ⎝i = 1 ⎠ ⎠ ⎝ i=1 ⎝i = 1 ⎠⎠ ⎝ i=1
(49)
where the normalized value of the whole data is x ′; x is the total data x min and x max are the minimum and maximum generated solar irradiation values in the entire data set considered for forecasting. The scaled solar power data is first processed with the help of either VMD or EMD technique. The time series data is decomposed into several modes which are then fed as the input to the learning technique. In this paper we considered the irradiation data of three different seasons of the year 2015 for forecasting purpose for every 15 min, 1 h and 1 day intervals ahead. The performance analysis of the proposed VMD-RMWK (VMD-Morlet Wavelet Kernel) is performed by comparing it with other hybrid methods like VMD-Polynomial Kernel (VMD-PK), VMD-Sigmoid Kernel (VMD-SK), VMD-Gaussian Kernel (VMD-GK), and VMD-Mexican Hat Wavelet Kernel (VMD-MHWK). The irradiation data is fed as input to get solar irradiation for a particular weather condition with time intervals varying from 15 min to 1 day ahead. Different parameters selected for computation purpose are basically based on trial and error method. Suitable values of these parameters are found as μ = 1, b = 0.85, c = 3, d = 2, e = 0.7, α = 0.9 for optimal solar irradiation predictions using kernel variants. The regularization parameter C is chosen as 4 for best results. Fig. 1 gives the average original solar irradiation and measured power data for different months for 1 year and the maximum power generated was around 0.85–0.9 MW in the month of May. Figs. 2–4 show the variations of one month original irradiation data for three different time intervals of 15 min, 30 min, and 1 h during the sunny, rainy, and foggy weather conditions, respectively. From the figures it is quite evident that there are huge fluctuations in solar
N
1 MAE = N
x −x min x max −x min
(48)
where N is the total number of samples, Ac (i) is the actual value and Fc (i) is the forecasted value and T is the target; O is output. 4.3. Result analysis The solar data originally obtained is of 15 min interval which is further subdivided into, 1 h and 1 day interval, out of this individual time series data 70% of the data is used for training purpose, while the rest 30% is for testing purpose. For numerical experimentation the whole data with different time interval is normalized and scaled between 0 and 1 using Eq. (48) this improves the computational complexity of the forecasting process. 795
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(a) 15min ahead solar irradiation
(b) 1hr ahead solar irradiation
(c) 1day ahead solar irradiation
Fig. 8. Empirical modes decomposition(EMD) of solar irradiation for sunny weather (a) IMF of 15 min ahead solar irradiation, (b) IMF of 1 h ahead solar irradiation, (c) IMF of 1 day ahead solar irradiation (all in per unit). Table 2a Different random training sample data for best corresponding MAPE for irradiation forecasting for 15 min ahead in sunny weather using the proposed VMDRMWK prediction model. (N, Tn and Tr are the total number of data, randomly selected training data and training time, respectively). N X Tn 6200 6200 6200 6200 6200
X X X X X
6200 3500 3000 2500 2000
Table 2c Different random training sample data for best corresponding MAPE for irradiation forecasting for 15 min ahead in foggy weather using the proposed VMDRMWK prediction model. (N, Tn and Tr are the total number of data, randomly selected training data and training time, respectively).
MAPE
MAE
RMSE
CC2
Tr (s)
N X Tn
1.162 1.244 1.932 1.960 2.009
0.008 0.008 0.014 0.014 0.014
0.011 0.011 0.032 0.033 0.033
0.996 0.996 0.986 0.978 0.975
196.39 53.96 50.44 47.39 44.87
4300 4300 4300 4300 4300
5000 5000 5000 5000 5000
X X X X X
5000 2500 2300 1500 1200
MAPE
MAE
RMSE
CC2
Tr (s)
1.461 1.421 1.695 1.932 1.993
0.012 0.011 0.015 0.016 0.016
0.024 0.023 0.028 0.030 0.030
0.991 0.991 0.987 0.985 0.984
98.14 31.91 28.22 25.83 23.58
4300 2500 2300 1200 1000
MAE
RMSE
CC2
Tr (s)
1.523 1.537 1.646 1.750 1.768
0.007 0.008 0.008 0.009 0.009
0.009 0.009 0.010 0.010 0.010
0.996 0.996 0.995 0.994 0.994
83.70 37.43 32.43 17.15 14.15
irradiation magnitudes. Figs. 5–7 show different modes of the irradiation for sunny, rainy and foggy weathers conditions, respectively using VMD technique, while in Fig. 8 different IMFs obtained from EMD based decomposition of solar irradiation for sunny weather are displayed for illustration. These solar irradiation decomposed modes in VMD and EMD are quite different and exhibit a number of frequency fluctuations depending on the seasonal conditions and the choice of collected data samples for different intervals of time varying from 15 min to 1 h, respectively.
Table 2b Different random training sample data for best corresponding MAPE for irradiation forecasting for 15 min ahead in rainy weather using the proposed VMDRMWK prediction model. (N, Tn and Tr are the total number of data, randomly selected training data and training time, respectively). N X Tn
X X X X X
MAPE
4.3.1. Reduction of execution time by using low rank kernel matrix for changing weather conditions After mode decomposition a diagonal weighting matrix given in Eq. 796
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(b) MAPE vs training time during rainy weather
(a) MAPE vs training time during sunny weather
(c) MAPE vs training time during foggy weather Fig. 9. Different training time corresponding to its respective MAPE values at different weather condition (a) describes the MAPE value in sunny weather (b) describes the MAPE value in rainy weather and (c) describes the MAPE value in foggy weather.
Table 3 Performance evaluation of the proposed VMD non-robust low rank KELM technique for solar irradiation prediction in sunny weather (Tr is the training time). Kernel functions
VMD-RMWK VMD-MHWK VMD-GK VMD-PK VMD-SK
MAPE (%)
MAE(p.u)
CC2
RMSE (p.u)
Tr (s)
15 min
1h
1 day
15 min
1h
1 day
15 min
1h
1 day
15 min
1h
1 day
15 min
1h
1 day
1.494 2.202 2.526 2.839 3.228
2.028 2.339 2.995 3.036 3.389
2.464 2.805 3.102 3.319 4.075
0.009 0.014 0.017 0.018 0.018
0.013 0.015 0.020 0.019 0.022
0.053 0.060 0.066 0.071 0.087
0.013 0.017 0.024 0.027 0.022
0.016 0.019 0.028 0.024 0.031
0.067 0.085 0.072 0.083 0.096
0.994 0.987 0.984 0.978 0.902
0.991 0.986 0.985 0.976 0.871
0.987 0.979 0.963 0.954 0.889
50.96 48.92 42.65 38.72 32.61
26.24 20.41 18.68 17.49 16.99
0.32 0.35 0.36 0.37 0.32
Table 4 Performance evaluation of the proposed VMD-RKELM technique for solar irradiation prediction in sunny weather (Tr is the training time). Kernel functions
VMD-RMWK VMD-RMHWK VMD-RGK VMD-RPK VMD-RSK
MAPE (%)
MAE(p.u)
CC2
RMSE (p.u)
Tr (s)
15 min
1h
1 day
15 min
1h
1 day
15 min
1h
1 day
15 min
1h
1 day
15 min
1h
1 day
1.244 1.543 2.079 2.265 2.862
1.546 1.825 2.275 2.344 3.036
1.823 1.970 2.404 2.612 3.331
0.008 0.010 0.014 0.014 0.018
0.010 0.011 0.015 0.015 0.019
0.039 0.042 0.051 0.056 0.071
0.011 0.012 0.020 0.022 0.023
0.012 0.014 0.024 0.020 0.024
0.050 0.064 0.058 0.071 0.081
0.996 0.996 0.987 0.986 0.972
0.991 0.986 0.985 0.974 0.971
0.973 0.978 0.979 0.984 0.890
53.96 51.59 37.96 46.74 37.31
28.72 24.45 21.96 28.41 18.94
0.37 0.37 0.38 0.37 0.34
(17) is multiplied to the low rank kernel matrix (Eq. (26)) using only a subset of training data in a random manner to reduce the execution time substantially and simultaneously improve the performance metrics
during solar irradiation forecasting. This has been demonstrated by showing the prediction results in Tables 2 for three different weather conditions, namely the sunny, rainy, and foggy using the proposed 797
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Table 5 Performance evaluation of the proposed VMD-RKELM technique for solar irradiation prediction in rainy weather (Tr is the training time). Kernel functions
VMD-RMWK VMD-RMHWK VMD-RGK VMD-RPK VMD-RSK
MAPE (%)
MAE (p. u)
CC2
RMSE (p. u)
Tr (s)
15 min
1h
1 day
15 min
1h
1 day
15 min
1h
1 day
15 min
1h
1 day
15 min
1h
1 day
1.421 1.642 2.338 2.664 2.949
1.777 1.943 2.498 2.706 3.140
2.484 2.827 3.092 3.237 3.485
0.011 0.007 0.011 0.020 0.022
0.010 0.016 0.017 0.022 0.025
0.015 0.017 0.016 0.023 0.025
0.023 0.010 0.014 0.036 0.040
0.014 0.020 0.022 0.028 0.032
0.022 0.023 0.020 0.033 0.034
0.991 0.995 0.998 0.984 0.973
0.991 0.988 0.993 0.978 0.969
0.986 0.984 0.992 0.969 0.965
31.91 26.88 36.33 39.94 39.33
8.81 6.00 5.16 5.05 6.59
0.23 0.24 0.17 0.17 0.19
Table 6 Performance evaluation of the proposed VMD-RKELM technique for solar irradiation prediction in foggy weather (Tr is the training time). Kernel functions
VMD-RMWK VMD-RMHWK VMD-RGK VMD-RPK VMD-RSK
MAPE (%)
MAE(p.u) 1h
1 day
15 min
1h
1 day
15 min
1h
1 day
15 min
1h
1 day
15 min
1h
1 day
1.537 1.748 2.365 2.607 3.156
1.860 2.102 2.589 2.954 3.250
2.553 2.871 3.192 3.367 3.573
0.008 0.009 0.010 0.013 0.014
0.011 0.011 0.014 0.015 0.017
0.195 0.022 0.024 0.026 0.027
0.009 0.010 0.014 0.016 0.018
0.011 0.012 0.014 0.018 0.021
0.030 0.026 0.038 0.040 0.041
0.996 0.994 0.997 0.988 0.997
0.995 0.993 0.996 0.983 0.980
0.985 0.988 0.985 0.984 0.976
37.43 31.20 33.05 32.13 29.59
3.98 4.03 3.51 3.07 2.66
0.16 0.18 0.28 0.22 0.16
Hybrid model
Time horizon
MAPE (%)
CC2
VMD-RMWK
15 min 1h 1 day 15 min 1h 1 day
1.244 1.546 1.823 2.403 3.566 4.016
0.996 0.991 0.973 0.983 0.975 0.868
where there is an increase in MAPE and decrease in execution time by reducing the size of the kernel matrix appropriately. For rainy and foggy weather conditions given in Tables 2b and 2c, a similar trend of lower execution time and a slight increase of MAPE is observed for all the prediction intervals. Therefore, it can be inferred that by reducing the size of the data matrix with a random choice of lesser number of support vectors from the data subset, thereby creating a low rank KELM, the forecasting accuracy decreases slightly, whereas the reduction in execution time is found to be significantly large. Also a comparative graph between the MAPE and execution time for the sunny, rainy and foggy weather conditions is shown in Fig. 9 using the low rank RMWK extreme learning machine model that indicates a lower MAPE and higher execution time with full rank kernel matrix, whereas a slightly lesser MAPE and lesser execution time occur with a reduced size kernel matrix.
Table 8 Comparative study of VMD-RMWK and EMD-RMWK for solar irradiation prediction for 15 min ahead in rainy weather. Hybrid model
Time horizon
MAPE (%)
CC2
VMD-RMWK
15 min 1h 1 day 15 min 1h 1 day
1.421 1.779 2.484 2.755 3.367 4.173
0.991 0.991 0.986 0.991 0.976 0.972
EMD-RMWK
4.3.2. Comparison of prediction performance between robust and nonrobust VMD-RMWK and other kernel variants Next to study the effect of the weighting matrix W used for creating a robust low rank VMD-RMWK and other kernel variants, W is chosen as an unit matrix in Eq. (26) (non-robust formulation) and the prediction performance is shown in Tables 3 and 4 for both non-robust and robust formulations during the sunny weather condition, respectively for all prediction intervals. It is observed from the prediction results given in these Tables that the robust formulation produces significantly improved performance metrics in comparison to the non-robust kernel variants. As an example for 15 min prediction interval the proposed VMD-RMWK shows lower MAPE and RMSE of 1.244 and .011 in comparison to 1.494 and 0.013 for the non-robust case during sunny weather condition. Also the value CC2 is higher in the case of robust VMD-RMWK than the non-robust VMD-RMWK clearly demonstrating that the robust model is more accurate. This trend is also exhibited for other two weather conditions namely the rainy and foggy. Therefore only the low rank robust kernel ELM variants are used for solar irradiation forecasting for all prediction intervals and the results are depicted in Tables 5 and 6 using both rainy and foggy weather conditions, respectively. From the Tables it is quite clear that for all the weather conditions and time intervals the VMD-RMWK outperforms all other kernel variants in terms of various performance metrics like MAPE, MAE, RMSE, and CC2.
Table 9 Comparative study of VMD-RMWK and EMD-RMWK for solar irradiation prediction for 15 min ahead in foggy weather. Hybrid model
Time horizon
MAPE (%)
CC2
VMD-RMWK
15 min 1h 1 day 15 min 1h 1 day
1.537 1.860 2.553 3.464 5.559 5.762
0.996 0.995 0.985 0.978 0.955 0.951
EMD-RMWK
Tr (s)
15 min
Table 7 Comparative study of VMD-RMWK and EMD-RMWK for solar irradiation prediction for 15 min ahead in sunny weather.
EMD-RMWK
CC2
RMSE (p. u)
VMD-RMWK. For example considering a full size kernel matrix of 6200 × 6200 in Table 2a for the sunny season it is observed that by reducing its size from 6200 × 6200 to 6200 × 3500 thereby performing an inversion of 3500 × 3500 matrix, the MAPE for 15 min ahead solar irradiation prediction is found to be 1.244 instead of 1.162 and the training time is 53.96 s instead of 196.39 s. A similar trend is observed in the 1 h and 1 day ahead prediction of solar irradiation, 798
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(a) 15min ahead irradiation prediction in sunny weather
(b) 1hr ahead irradiation prediction in sunny weather
(c) 1day ahead irradiation prediction in sunny weather Fig. 10. Solar irradiation prediction for various time horizon in sunny weather using various low rank robust VMD based Kernel functions, (a) 15min ahead prediction, (b) 1 h ahead prediction, (c) 1 day ahead prediction.
conditions showing clearly that the hybrid VMD based robust Morlet Wavelet Kernel outperforms the EMD based Morlet Wavelet Kernel in terms of various performance metrics. Thus judging the prediction performance in different weather conditions and different time
4.3.3. Prediction performance comparison of VMD-RMWK with other prediction models during changing weather conditions Tables 7–9 give a comparative study between VMD-RMWK and EMD-RMWK for 15 min ahead prediction in different weather
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(a) 15min ahead irradiation prediction in rainy weather
(b) 1hr ahead irradiation prediction in rainy weather
(c) 1day ahead irradiation prediction in rainy weather Fig. 11. Solar irradiation for various time horizon in rainy weather using various low rank robust VMD based Kernel functions, (a) 15 min ahead prediction, (b) 1 h ahead prediction, (c) 1 day ahead prediction.
intervals, the hybrid VMD-RMWK extreme learning machine exhibits the least MAPE, RMSE and MAE values and least computational time in comparison to VMD-RMHWK and other kernel variants. The CC2 of the
proposed model is found to be the highest (0.99). Also as expected the best performance is found during sunny weather condition for solar irradiation forecasting. Also it can be seen that the performance metrics 800
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(a) 15min ahead irradiation prediction in foggy weather
(b) 1hr ahead irradiationprediction in foggy weather
(c) 1day ahead irradiationprediction in foggy weather Fig. 12. Solar irradiation prediction for various time horizon in foggy weather using various low rank VMD based Kernel functions, (a) 15 min ahead prediction, (b) 1 h ahead prediction, (c) 1 day ahead prediction.
irradiation data with different VMD based predicting models (RMWK, RMHWK, RPK, RGK and RSK) is exhibited at different time intervals for sunny weather. Figs. 11 and 12 present the comparison of the proposed model along with other techniques for different time interval predictions in rainy and foggy weather conditions, respectively. From these
deteriorate as the time interval increases from 15 min to 1 h and one day, respectively. To validate the prediction performance results given in Tables, graphical representation of various VMD based low rank robust KELM is shown in Figs. 10–13. In Fig. 10, a comparison of original solar 801
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(a) Solar irradiation prediction in sunny weather
(b) Solar irradiation prediction in rainy weather
(c) Solar irradiation prediction in foggy weather Fig. 13. VMD based RMWK and EMD based RMWK for 15 min ahead solar irradiation prediction in different weather (a) solar irradiation prediction in sunny weather (b) solar irradiation prediction in rainy weather (c) solar irradiation prediction in foggy weather.
figures it is demonstrated that the prediction performance of VMDRMWM is close to the original solar irradiation measured values and hence is considered as a more accurate prediction model. Further the weather conditions greatly influence the predicted values in comparison to original measured values. Another interesting comparison is presented in Fig. 13(a)–(c) between the VMD-RMWK and EMD-RMWK prediction models, where the VMD-RMWK exhibits a more accurate solar irradiation prediction for 15 min ahead in comparison to the EMDRMWK during the different weather conditions. However, it is seen that for the rainy season the divergence between these two models is very
small. 4.3.4. Case study during a small duration partial shading condition A case study has been performed to mark the performance of the VMD-RMWK technique when there is a cloud covering or partial shading for certain amount of time. In this paper a time gap of 2 min has been considered for cloud covering time. Fig. 14 clearly shows that when there is a sudden drop of irradiation (represented by ‘X’ axis) the tracking is still maintained and prediction is performed with greater accuracy without being much affected by the sudden change, and this 802
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Fig. 14. Prediction of solar irradiation at cloudy condition for 2 min implementing VMD-RMWK . red line (original) blue line (predicted).
MAPE(%)
Error Comparison
Table 10 Performance evaluation of the proposed VMD-RKELM technique for 1 h interval solar irradiation prediction in summer season at Florida (Tr is the training time).
4 3.5 3 2.5 2 1.5 1 0.5 0 15min 1hr Sunny
1day 15min
1hr
1day 15min
1hr
1day
MAPE (%)
MAE (p.u)
RMSE (p.u)
CC2
Tr (s)
VMD-RMWK VMD-RMHWK VMD-RGK VMD-RPK VMD-RSK
1.760 2.025 2.381 2.518 3.463
0.017 0.020 0.023 0.025 0.034
0.024 0.027 0.027 0.032 0.045
0.998 0.997 0.995 0.994 0.981
28.23 26.86 20.96 28.11 17.04
Rainy Foggy time horizon for different weather
VMD-RMWK
VMD-RGK
VMD-RMHWK
VMD-RPK
Table 11 Performance evaluation of the proposed VMD-RKELM technique for 1 h interval solar irradiation prediction in winter season at Florida (Tr is the training time).
VMD-RSK
Fig. 15. Comparative study of MAPE values of different VMD based methodologies for solar irradiation prediction for different time horizons at different weather conditions.
Error Comparison MAPE(%)
Kernel functions
7 6 5 4 3 2 1 0
Kernel functions
MAPE (%)
MAE (p.u)
RMSE (p.u)
CC2
Tr (s)
VMD-RMWK VMD-RMHWK VMD-RGK VMD-RPK VMD-RSK
1.548 1.738 2.480 2.769 3.273
0.020 0.023 0.033 0.037 0.043
0.026 0.025 0.036 0.048 0.056
0.998 0.997 0.994 0.992 0.988
28.82 23.45 20.98 27.41 18.49
Table 12 Performance evaluation of the proposed VMD-RKELM technique for 1 h interval solar irradiation prediction in spring season at Florida (Tr is the training time).
15min 1hr
1day 15min 1hr
1day 15min 1hr
1day
sunny rainy foggy time horizon for different weather VMD-RMWK
Kernel functions
MAPE (%)
MAE (p.u)
RMSE (p.u)
CC2
Tr (s)
VMD-RMWK VMD-RMHWK VMD-RGK VMD-RPK VMD-RSK
1.812 2.046 2.576 2.527 3.483
0.024 0.027 0.034 0.034 0.046
0.031 0.034 0.049 0.044 0.057
0.996 0.996 0.995 0.994 0.992
8.94 6.83 6.30 5.26 5.09
EMD-RMWK Table 13 Performance evaluation of the proposed VMD-RKELM technique for 1 h interval solar irradiation prediction in autumn season at Florida (Tr is the training time).
Fig. 16. Comparative study of MAPE values of the proposed technique i.e. VMD-RMWK and the EMD based RMWK for solar irradiation prediction at different time horizons and weather conditions.
may be inferred as due to the robust nature of the proposed model. Fig. 15 uses a bar graph to show a comparative study of MAPE values for different VMD based RKELM (VMD-RMWK, VMD-RMHWK, VMDRGK, VMD-RPK and VMD-RSK) models for solar irradiation prediction at different time intervals and weather conditions. From this graph it can be clearly noticed that VMD implemented RMWK gives better result
803
Kernel functions
MAPE (%)
MAE (p.u)
RMSE (p.u)
CC2
Tr (s)
VMD-RMWK VMD-RMHWK VMD-RGK VMD-RPK VMD-RSK
1.963 1.988 2.437 2.649 2.834
0.025 0.025 0.030 0.034 0.036
0.032 0.032 0.042 0.043 0.044
0.998 0.996 0.995 0.992 0.989
7.98 7.06 6.81 6.17 5.72
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(a) Solar irradiation prediction in summer season
(b) Solar irradiation prediction in spring season
Fig. 17. VMD based RMWK and RMHWK for 1 h ahead solar irradiation prediction of Florida at different seasons (a) solar irradiation prediction in summer season (b) solar irradiation prediction in spring season (c) solar irradiation prediction in winter (d) solar irradiation prediction in autumn season.
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Fig. 17. (continued)
MAPE(%)
Error Comparison 4 3.5 3 2.5 2 1.5 1 0.5 0
Summer Spring Winter Autumn VMD-RMWK VMD-RMHWK VMD-RGK VMD-RPK VMD-RKELM
VMD-RSK
Fig. 18. Comparative study of MAPE values of different VMD based methodologies for 1 h interval solar irradiation prediction at different season.
and is shown in Fig. 17, where a comparative study is demonstrated between the original solar irradiation using the proposed technique and the second best one. Thus it can be precisely noted that the proposed model performs better than other variants. Fig. 18 gives a comparative study based on MAPE values of all types of VMD-RKELM models in different seasons. Finally a comparative performance of forecasting accuracy for different prediction models in different countries is given in Table 14 showing clearly the proposed model is found to be superior to some of the existing ones.
Table 14 Comparative analyses of previously used forecasting technique and the proposed technique. Reference
Model type
Location
MAPE (%)
[37] [38] [39] [39] [38]
ANN ARIMA SVR EMD-PSO-SVR NN Proposed
France Portugal China China Portugal India
18.5 11.03 6.02 3.43 6.34 1.244
5. Conclusion than all other VMD based RKELM models. In a similar way Fig. 16 displays a comparative study of MAPE values for two types of decomposition technique integrated with the proposed technique, i.e. VMDRMWK and EMD-RMWK, where it can be noticed that the VMD based technique outperforms the previously implemented EMD technique.
In this paper a novel VMD based low rank robust Morlet Wavelet Kernel (VMD-RMWK) extreme learning machine is proposed for efficient short term solar irradiation prediction for different weather conditions and different time intervals using the measured data from an experimental solar power station in India and cross validated with the solar plant data of Florida. Various performance metrics such as MAPE, MAE, RMSE, CC2 are compared with other VMD based low rank robust kernel formulations and EMD based low rank robust Morlet Wavelet Kernel prediction model. The computational time is given the utmost preference as it is very necessary when large data set is taken into consideration. Also by using a weighted kernel matrix formulation based on the non-parametric kernel density estimation technique, the prediction accuracy improves against chaotic fluctuations in solar irradiation data. The numerical experimentation results clearly reveal that the proposed VMD-RMWK corresponding to a MAPE value of 1.244% for 15 min ahead prediction in sunny weather performs better than the EMD technique combined with RMWK with an error of 2.403%
4.3.5. Cross validation of solar irradiation prediction using data from a solar plant in Florida State, USA Different VMD based RKELM techniques implemented for solar irradiation prediction of the Florida solar plant data shows the uniformity of the proposed technique, as its performance is obtained with greater accuracy as compared to the other VMD based RKELM models. Tables 10–13 show a comparative study of different VMD-RKELM models for 1 h interval solar irradiation prediction at different seasonal conditions (summer, winter, spring and autumn, respectively). From these Tables it can be concluded that the proposed technique performs better prediction of solar irradiation irrespective of the location of the solar power plant. The pictorial verification of the above Table is performed 805
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for the same weather condition and time horizon. When run time is taken into consideration it is observed that the proposed low rank robust VMD-RMWK exhibits an execution time of 53.96 sec in comparison to the full rank robust VMD-RMWK that takes a much longer execution time of 196.4 sec for a 15 min prediction interval during sunny weather condition. Thus with the improved prediction precision in a much lesser time, the proposed model is an efficient tool to solve various power quality and operational issues; it is also beneficial for new solar plant installation where historical power is not available, solar irradiation is used for power prediction. The short term (15 min−1 h) solar irradiation prediction is applicable for real-time dispatch whereas the medium term (1 day−1 week) forecasting helps in maintenance and unit commitment. In future other conditions like cloud cover; tilt angle, etc. can be taken into consideration for forecasting purpose in solar power plants located in other geographical regions.
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