Forecasting Solar Power Using Wavelet Transform Framework Based on ELM

Abstract

Forecasting solar power with good precision is necessary for ensuring the reliable and economic operation of electricity grid. In this paper, we consider the task of predicting a given day photovoltaic power (PV power) outputs in 30 min intervals from previous solar power and weather data. We proposed a method combining extreme learning machine and wavelet transform (WT-ELM), which build a separation prediction model for every moment using weather data and corresponding PV power, the weather characteristics are treated as input features and the PV power data are the corresponding ground truth. In addition, we also compared our method with K Nearest Neighbour (K-NN) and support vector machine (SVM) based on clustering using the same data. Then we evaluated the performance of our approach for all data with different time interval. The results show that our result performs much more better than KNN based on clustering.

Y. Yu—This work is supported by National Natural Science Foundation of China (NSFC) under grant 61473089.

Appendix: Wavelet Transform

In multi-resolution analysis (MRA), a scaling function is used to create a series of approximations of a function or image [20]. Consider the set of expansion functions composed of integer translation and binary scaling of the real, square-integrable function $\varphi (x)$; this is the set $\varphi _{j,k}(x)$, where

$$\begin{aligned} \varphi _{j,k}(x)=2^{j/2}\varphi (2^jx-k) \end{aligned}$$

(8)

for all j,k$\in$Z and $\varphi (x) \in L^2$(R) [6]. Here, k determines the position of $\varphi _{j,k}(x)$ along the x-axis, and j determines the width of $\varphi _{j,k}(x)$. Generally, we denote the subspace spanned over k for any j as

$$\begin{aligned} V_j=\overline{Span_k\{\varphi _{jk}(x)\}} \end{aligned}$$

(9)

$$\begin{aligned} if \ f(x) \in V_j,\ f(x)=\sum _k{\alpha _k\varphi _{jk}(x)} \end{aligned}$$

(10)

MRA has four requirements as following.

Requirement 1: the scaling function is orthogonal to its integer translates:

$$\begin{aligned} \langle \varphi _{j,k}(x),\varphi _{j,k'}(x)\rangle = \int \varphi _{j,k}^*(x)\varphi _{j,k'}(x)dx=0,k\ne k' \end{aligned}$$

(11)

Requirement 2: The subspaces spanned by the scaling function at low scales are nested within those spanned at higher scales. As shown in Fig. 9. That is,

$$\begin{aligned} V_{-\infty }\subset ...\subset V_{-1}\subset V_0\subset V_1 \subset V_2\subset ... V_\infty \end{aligned}$$

(12)

Requirement 3: The only function that is common to all V$_j$ is f(x) = 0. i.e. $V_{-\infty }$ = 0.

Fig. 9.

The nested function spaces spanned by a scaling function

Requirement 4: Any function can be represented with arbitrary precision. That is,

$$\begin{aligned} V_\infty = \{L_2(R)\} \end{aligned}$$

(13)

Under these conditions, the expansion functions of subspace V$_j$ can be expressed as a weighted sum of the expansion function of subspace V$_{j+1}$. Using Eq. 10, we let

$$\begin{aligned} \varphi _{j,k}(x) = \sum _n{\alpha _n\varphi _{j+1,n}(x)} \end{aligned}$$

(14)

combine Eqs. 14 and 8 we have

$$\begin{aligned} \varphi _{j,k}(x) = \sum _n{h_\varphi (x)2^{(j+1)/2}\varphi (2^{j+1}x-n)} \end{aligned}$$

(15)

because $\varphi =\varphi _{0,0}(x)$, Eq. 15 can be expressed as

$$\begin{aligned} \varphi (x) = \sum _n{h_\varphi (n)\sqrt{2}\varphi (2x-n)} \end{aligned}$$

(16)

The $h_\varphi (n)$ coefficients in Eq. 16 are called scaling function coefficients; $h_\varphi$ is called the scaling vector.

Given a scaling function that meet the MRA requirements, we have a wavelet function $\psi (x)$, together with its integer translates and binary scalings functions $\psi _{j,k}(x)$, which spans the difference between any two adjacent scaling subspaces V$_j$ and V$_{j+1}$. The set $\{\psi _{j,k}(x)\}$ of wavelet was defined as:

$$\begin{aligned} \psi _{j,k}(x) = 2^{j/2}\psi ({2^{j}x-k)} \end{aligned}$$

(17)

for all k $\in$ Z that span the W$_j$ spaces in the Fig. 10. The same as scaling functions, wavelet subspaces can be write

$$\begin{aligned} W_j=\overline{Span_k\{\psi _{jk}(x)\}} \end{aligned}$$

(18)

$$\begin{aligned} i\!f \ f(x) \in W_j, \ f(x)=\sum _k{\alpha _k\psi _{j,k}(x)} \end{aligned}$$

(19)

The relationship between scaling and wavelet function subspaces are as

$$\begin{aligned} V_{j+1}=V_j\oplus W_j \end{aligned}$$

(20)

where $\oplus$ denotes the union of spaces. And W$_j$ is the orthogonal complement of V$_j$ in V$_j+1$. That is all members of V$_j$ are orthogonal to the members of W$_j$.

$$\begin{aligned} \langle \varphi _{j,k}(x),\psi _{j,k'}(x)\rangle = 0 \end{aligned}$$

(21)

We can now express the space of all measurable, square-integrable functions as

$$\begin{aligned} L^2(\mathbf R ) = V_0 \oplus W_0 \oplus W_1 \oplus ... \end{aligned}$$

(22)

or

$$\begin{aligned} L^2(\mathbf R ) = V_{j_0} \oplus W_{j_0} \oplus W_{j_0+1} \oplus ... \end{aligned}$$

(23)

where $j_0$ is an arbitrary starting scale.

According to the relationship between scaling and wavelet subspace and Eqs. 9 and 16, we have that any wavelet function can be expressed as a weighted sum of shifted, double-resolution scaling functions (Eq. 1).

Similarly, h$_\psi$(n) are called the wavelet function coefficients and $h_\psi$ is the wavelet vector. Using the condition that shown in Fig. 10 and that integer wavelet translates are orthogonal, the relationship between h$_\varphi$ and h$_\psi$ can be write as Eq. 2.

Consider Eq. 16 again, scaling x by $2^j$, translating it by k, and letting m = 2k + n gives

$$\begin{aligned} \begin{aligned} \varphi (2^i-k)&=\sum _n{h_\varphi (n)\sqrt{2}\varphi (2(2^jx-k)-n)}\\&= \sum _n{h_\varphi (n)\sqrt{2}\varphi (2^{j+1}x-2k-n)} \\&= \sum _n{h_\varphi (m-2k)\sqrt{2}\varphi (2^{j+1}x-2m)} \end{aligned} \end{aligned}$$

(24)

Equation 24 can be written as Eq. 25, similarly, we could get Eq. 26.

$$\begin{aligned} W_\varphi (j,k) = \sum _m{h_\varphi (m-2k)W_\varphi (j+1,m)} \end{aligned}$$

(25)

$$\begin{aligned} W_\psi (j,k) = \sum _m{h_\psi (m-2k)W_\varphi (j+1,m)} \end{aligned}$$

(26)

Therefore, the coefficients expression of FWT can be translated into Eqs. 27 and 28. The structure of FWT are shown as Fig. 2.

$$\begin{aligned} W_\varphi (j,k) = h_\varphi (-n)*W_\varphi (j+1,n)\mid _{n=2k,k\ge 0} \end{aligned}$$

(27)

$$\begin{aligned} W_\psi (j,k) = h_\psi (-n)*W_\varphi (j+1,n)\mid _{n=2k,k\ge 0} \end{aligned}$$

(28)

Fig. 10.

The relationship between scaling and wavelet function spaces