Forecasting system with sub-model selection strategy for photovoltaic power output forecasting

Abstract

Photovoltaic power output forecasting has been focused on worldwide due to its environmental benefits and soaring load demand of the electricity market. Many forecasting technologies have been developed to increase photovoltaic power output forecasting performance. However, due to the various characteristics of different photovoltaic power output time series, no commonly used technology can always reach satisfactory prediction performance. To solve this dilemma and further improve photovoltaic power output forecasting accuracy and stability, a novel photovoltaic power output forecasting system is developed, where the data preprocessing method is first used to capture the primary characteristic of photovoltaic power output time series. Then, six forecasting models are employed to predict the preprocessed data. Sub-model selection strategy is introduced to select the best three forecasting models for obtaining good prediction results under different circumstances. Finally, the forecasting results of three forecasting models are combined based on a multi-objective grey wolf optimizer. The developed system is proved to be effective in terms of prediction accuracy and stability in three simulation experiments. Thus, the proposed system can be widely used to improve photovoltaic power output prediction performance in practical applications and it will provide valuable technical support for the operation and management of power systems.

Appendices

Appendix 1. SSA

• Step 1: Embedding

Given the PV power output time series $\mathbf{P}=\left[{p}_{1},{p}_{2},\cdots ,{p}_{N}\right]$, it can be projected into a lagged vector $\mathbf{Z}=\left[{z}_{1},{z}_{2},\cdots ,{z}_{K}\right]$, where ${z}_{i}={\left[{p}_{i},{p}_{i+1},\cdots ,{p}_{i+L-1}\right]}^{\mathrm{T}}\in {R}^{L},i=1,\dots ,K$, $K=N-L+1$, and $2\le L\le N$. Thus, the trajectory matrix can also be expressed as:

$$\mathbf Z=\left[z_1,z_2,\cdot\cdot\cdot,z_k\right]=\begin{pmatrix}p_1&p_2&\cdots&p_k\\p_2&p_3&\cdots&p_{k+1}\\\cdots&\cdots&\cdots&\cdots\\p_L&p_{L+1}&\cdots&p_N\end{pmatrix}$$

(16)

• Step 2: SVD

The decomposition in terms of the trajectory matrix is completed by SVD process. Firstly, construct a covariance matrix ${\mathbf{Z}\mathbf{Z}}^{T}$, whose eigenvalues and eigenvectors are ${\lambda }_{1}\ge {\lambda }_{2}\ge \cdots \ge {\lambda }_{d}\ge 0$ and ${u}_{1},\cdots ,{u}_{d}$, respectively. Then, the trajectory matrix Z is presented as follows:

$$\mathbf{Z}={e}_{1}+{e}_{2}+\cdots +{e}_{d}$$

(17)

where ${e}_{i}=\sqrt{{\lambda }_{i}}{u}_{i}{{\mathbf{v}}_{i}}^{T}$, ${\mathbf{v}}_{i}={\mathbf{Z}}^{T}{u}_{i}/\sqrt{{\lambda }_{i}}$ denotes the principal components.

• Step 3: Grouping

In this process, the interval $\left\{1,2,\dots ,d\right\}$ can be grouped into several disjointed subsets $\left\{{K}_{1},{K}_{2},\cdots ,{K}_{m}\right\}$. The matrix Z can now be presented by Eq. (18).

$$\mathbf{Z}={z}_{{K}_{1}}+{z}_{{K}_{2}}+\cdots +{z}_{{K}_{m}}$$

(18)

• Step 4: Diagonal averaging

In this step, every matrix ${z}_{{K}_{i}}\left(1\le i\le m\right)$, will be switched into a new series $\mathbf{C}=\left[{c}_{1},{c}_{2},\cdots ,{c}_{N}\right]$ with length of N. Suppose that $\mathbf{C}=\left[{c}_{1},{c}_{2},\cdots ,{c}_{N}\right]$ is the switched one-dimensional series, ${c}_{k}\left(k=1,\dots ,N\right)$ can be described as follows:

$$c_k=\left\{\begin{array}{lc}\sum\limits_{j=1}^kc_{j,k-j+1}^\ast/k,&1\leq k<L\\\textstyle{\displaystyle\sum\limits_{j=1}^{L^\ast}}c_{j,k-j+1}^\ast/L^\ast,&L^\ast\leq k\leq K^\ast\\\textstyle{\displaystyle\sum\limits_{j=k-K^\ast+1}^{N-K^\ast+1}}c_{j,k-j+1}^\ast/\left(N-k=1\right),&K^\ast\leq k\leq N\end{array}\right.$$

(19)

where $L^\ast=min\left(L,K\right),K^\ast=max\left(L,K\right)$. Moreover, when $L<K$, ${c}_{j,k-j+1}^{*}={c}_{j,k-j+1}$, else ${c}_{j,k-j+1}^{*}={c}_{k-j+1,j}$.

Appendix 2. MOGWO

Algorithm 1:

MOGWO

Appendix 3. Parameter Settings

Tables 12 and 13.

Table 12 Parameter values of data preprocessing technology and six forecasting models

Table 13 Parameters of optimization algorithms and data preprocessing techniques in Experiment II

Appendix 4. Distribution fitting performance

Tables 14, 15, 16, 17 and 18.

Table 14 Distribution fitting performance for the prediction error of SSA-MODA-CM

Table 15 Distribution fitting performance for the prediction error of SSA-MOGOA-CM

Table 16 Distribution fitting performance for the prediction error of EMD-MOGWO-CM

Table 17 Distribution fitting performance for the prediction error of VMD-MOGWO-CM

Table 18 Distribution fitting performance for the prediction error of the proposed SSA-MOGWO-CM

Appendix 5. Introduction of Evaluation Index

Tables 19 and 20.

Tables

Table 19 Evaluation indexes of prediction sensitivity

Table 20 Improvement percentages of five indexes involved in PF and IP