# The Short-Term Forecasting of Asymmetry Photovoltaic Power Based on the Feature Extraction of PV Power and SVM Algorithm

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The effective decomposition of the original PV power output.

- (2)
- The reasonable identification of specified IMFs.

- (3)
- The parameter optimization of the SVM.

- (4)
- The reconstruction of the forecasting result.

## 2. Related Work

#### 2.1. CEEMDAN Algorithm

- (1)
- Obtain the noisy signal ${x}^{i}$; ${x}^{i}=x+{\omega}_{0}{\xi}^{i}\left(i=1,\cdots ,N\right)$, where ${\xi}^{i}$ is the added white noise with unit variance, ${\omega}_{0}$ is the corresponding coefficient, and $x$ is the original signal;
- (2)
- Extract the first IMF (${c}_{1}^{i}$) from each noisy signal ${x}^{i}$ with EMD;
- (3)
- Obtain the first IMF (${c}_{1}$) by taking the average of each ${c}_{1}^{i}$;$${c}_{1}=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{c}_{1}^{i}$$
- (4)
- Obtain the first residue: ${r}_{1}=x-{c}_{1}$;
- (5)
- Decompose ${r}_{1}+{w}_{1}{E}_{1}\left({\epsilon}^{i}\right)$ with the EMD algorithm, and extract the first IMF to obtain the second IMF (${c}_{2}$), where ${w}_{1}$ means the coefficient of the added white noise, and the operator ${E}_{m}(\cdot )$ indicates the $m$-th IMF with EMD;$${c}_{2}=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{E}_{1}\left({r}_{1}+{w}_{1}{E}_{1}\left({\epsilon}^{i}\right)\right)$$
- (6)
- Compute the $m$-th residue mode ($m=2,\dots ,K$), and extract the first IMF to generate the $\left(m+1\right)$-th IMF with Equation (3);$${c}_{m+1}=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{E}_{1}\left({r}_{m}+{w}_{m}{E}_{m}\left({\epsilon}^{i}\right)\right)$$
- (7)
- Repeat the above steps until the residue $\mathrm{R}$ contains fewer than two extrema;$$\mathrm{R}=\mathrm{x}-{\displaystyle \sum}_{m=1}^{K}{c}_{k}$$

#### 2.2. SVM Algorithm

#### 2.3. PSO Algorithm

## 3. Proposed Work

#### 3.1. Effective Decomposition for PV Power

#### 3.2. Selection of Related Modes

_{N}contains similar fluctuation components. If another fluctuation component is introduced into MIMF

_{(N+1)}, the waveform between the MIMF

_{N}and MIMF

_{(N+1)}is different, obviously. It is illustrated that if a kind of fluctuation component is mixed into that of another component, the waveform of the adjacent MIMF must be changed.

#### 3.3. Establishment of the Forecasting Sub-Model for PV Power Output

- (1)
- Determine the training sample for each ${\mathrm{Group}}_{k}\text{}\left(k=1,2,\dots \right)$ and the corresponding input variables:$$\left(\mathit{x},\mathit{y}\right)=\left\{\left({\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{i}}\right)|i=1\cdots n\right\}$$
- (2)
- Determine the parameter range of the SVM.
- (3)
- Optimize the parameters of the SVM with the modified PSO (MPSO).
- (4)
- Train the SVM with the training sample and calculate the fitness value of each particle. The fitness value is compared with the global best position ${\mathit{p}}_{\mathit{g}}$. If the fitness value is superior to ${\mathit{p}}_{\mathit{i}}$, the fitness value is considered as ${\mathit{p}}_{\mathit{i}}$. Here, the mean square error (MSE) is used to perform the fitness function:$$\mathrm{fitness}=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{\left({y}_{i}-{y}_{i}^{\prime}\right)}^{2}}$$
- (5)
- Update the position and velocity with Equations (8) and (9).
- (6)
- Look for the optimal solution until the end condition is satisfied; otherwise, go back to Step 3.

#### 3.4. Final Forecasting Model

#### 3.5. Evaluation of Forecasting Result

- (1)
- Determination coefficient (${R}^{2}$).$${R}^{2}=1-\frac{RSS}{TSS}$$$$RSS={\displaystyle \sum _{i=1}^{n}{\left({x}_{i}-{y}_{i}\right)}^{2}}$$$$TSS={\displaystyle \sum _{i=1}^{n}{\left({x}_{i}-\overline{x}\right)}^{2}}$$

- (2)
- Mean absolute error (MRE).$$\mathrm{MRE}=\frac{1}{n}\frac{{\displaystyle \sum _{i=1}^{n}\left|{x}_{i}-{\widehat{x}}_{i}\right|}}{\overline{x}}$$

- (3)
- Root-mean-square error (RMSE).$$\mathrm{RMSE}=\sqrt{\frac{{\displaystyle {\sum}_{i=1}^{n}{\left({x}_{i}-{\widehat{x}}_{i}\right)}^{2}}}{n}}$$

- (1)
- Decompose the PV power output into the IMFs with physical meaning with the improved CEEMDAN (ICEEMDAN). Here, the two critical parameters (the ensemble size and amplitude of the added white noise) are determined by introducing the noise level.
- (2)
- Classify IMFs into the individual feature groups including the most relevant fluctuation components by introducing the comprehensive factor adaptively. It may avoid the drawbacks of threshold determination depending on the empirical method and is an adaptive way.
- (3)
- Optimize the parameters in the SVM for individual feature groups with the modified PSO (MPSO), and obtain the corresponding sub-forecasting model. The MPSO can enhance the global search ability to make the particle traverse all the space and also local search performance to increase the speed of convergence.
- (4)
- Reconstruct the sub-forecasting model to obtain the final forecasting model.
- (5)
- Evaluate the effectiveness of the proposed method (ICEEMDAN-IF-MPSO-SVM), which comes from Step 1 to Step 4.

## 4. Results and Discussion

_{opt}). Here, the noise level is set from 10

^{0}to 10

^{−10}, with the step 10

^{−1}. The determined index of the noise level is plotted in Figure 4. The findings are that the determination index is the smallest at the noise level of 10

^{−3}. Thus, the optimal critical parameters are two trails (ensemble size) and ${10}^{-3}\sigma $ (the amplitude of the added white noise), respectively. The $\sigma $ means the standard deviation of the PV power output.

^{0}, 10

^{−1}and 10

^{−2}. This is shown in Figure 5.

^{0}, 10

^{−1}, 10

^{−2}and 10

^{−3}, respectively. The decomposition number with the proposed method is the smallest. This indicates that the proposed method (ICEEMDAN) can overcome the spurious mode. It also proves that the proposed method can be more suitable for decomposing the PV power output. In the decomposition results under the other noise levels, because the amplitude of the added white noise is relatively large, the white noise is difficult to eliminate. It follows that the interaction of white noise and PV power output introduces a spurious mode (no real physical meaning) and puts the added noise in the IMFs. Finally, the forecasting model is influenced due to the spurious mode and extra noise.

^{2}are used to evaluate the forecasting effectiveness for each forecasting method, respectively. Table 4 shows the compared results. The findings are that the MRE and RMSE of the proposed method are the smallest among those of all the methods, and the corresponding R

^{2}is the biggest. It indicates that the forecasting effectiveness of the proposed method outperforms the other forecasting methods. Additionally, the findings in Figure 13b,c show that the forecasting accuracy for a high amplitude in PV output is better than that for low amplitude.

## 5. Conclusions

- (1)
- The ICEEMDAN method is introduced to decompose the PV power output into IMFs with physical meaning. The two critical parameters, which include the ensemble size (EN) and amplitude of the added white noise (NA), can be determined by setting the ensemble size as two trails and introducing the noise level. This method can avoid the interference of a spurious mode for the forecasting of PV power output.
- (2)
- The adaptive identified method of the relative mode is proposed to classify the IMFs into the corresponding feature groups. The method can separate the complex fluctuating components in PV power output into single components to enhance the forecasting accuracy.
- (3)
- The modified PSO (MSPO) is proposed to optimize the hyper-parameters in the SVM. The MPSO applies the piecewise inertial weight to enhance the global search ability to make the particle traverse all the space and also local search performance to increase the convergence speed.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

PV | photovoltaic |

IF | selection method of relative modes |

GA | genetic algorithm |

SVM | support vector machine |

EMD | empirical mode decomposition |

EEMD | ensemble EMD |

NA | amplitude of added white noise |

IMFs | intrinsic mode functions |

ICEEMDAN | improved CEEMDAN |

Cr | crest factor |

${R}^{2}$ | determination coefficient |

RMSE | root-mean-square error |

PAF | partial autocorrelation function |

PSO | particle swarm optimization |

MPSO | modified particle swarm optimization |

GS | grid search |

ANN | artificial neural network |

VMD | variational mode decomposition |

CEEMDAN | complementary EEMD with adaptive noise (CEEMDAN) |

EN | ensemble trials |

IIMFs | identified IMFs |

Sh | shape factor |

Ku | kurtosis |

CF | comprehensive factor |

MRE | mean absolute error |

AF | autocorrelation function |

BP | back propagation neural network |

RF | random forest |

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**Figure 3.**Analysis of autocorrelation function (AF) and partial autocorrelation function (PAF) of PV power.

**Figure 8.**Combined IMF with proposed method. (

**a**) plots the combined result from IMF1 to IMF3, and the (

**b**) plots the combined result from IMF4 to IMF6, and the (

**c**) plots the combined result from IMF7 to IMF9.

Number | Statistical Parameter | Expression |
---|---|---|

1 | Shape factor | $SH=\sqrt{\left(1/N\right){\displaystyle \sum _{i=1}^{N}x{\left(i\right)}^{2}}}/\left(1/N\right){\displaystyle \sum _{i=1}^{N}\left|x\left(i\right)\right|}$ |

2 | Crest factor | $CR=max\left(\left|x\left(i\right)\right|\right)/\sqrt{\left(1/N\right){\displaystyle \sum _{i=1}^{N}x{\left(i\right)}^{2}}}$ |

3 | Kurtosis | $KU={\displaystyle \sum _{i=1}^{N}{\left(x\left(i\right)-\overline{x}\right)}^{4}}/N{\left({x}_{\sigma}\right)}^{4}$ |

Group Order | First Group | Second Group | Three Group |
---|---|---|---|

Combined IMFs | IMF1 + IMF2 + IMF3 | IMF4 + IMF5 + IMF6 | IMF7 + IMF8 + IMF9 |

Order | Name | Value | Order | Name | Value |
---|---|---|---|---|---|

1 | Maximum iteration number | 100 | 6 | Iteration number ${t}_{1}$ | 45 |

2 | Start weight | 0.9 | 7 | Iteration number ${t}_{2}$ | 90 |

3 | End weight | 0.4 | 8 | Parameter C range | 0–100 |

4 | Slope ${a}_{1}$ | −10^{−4} | 9 | Parameter $\gamma $ range | 0–100 |

5 | Slope ${a}_{2}$ | −10^{−5} | 10 | Population number | 20 |

Model | MRE | RMSE | R^{2} |
---|---|---|---|

BP | 0.3764 | 0.1919 | 0.9279 |

PSO-SVM | 0.4817 | 0.2417 | 0.8856 |

RF | 0.4194 | 0.2157 | 0.9089 |

EMD-IF-MPSO-SVM | 0.3076 | 0.1596 | 0.9501 |

EEMD-IF-MPSO-SVM | 0.3829 | 0.1984 | 0.9299 |

CEEMDAN-IF-MPSO-SVM | 0.3422 | 0.1813 | 0.9356 |

ICEEMDAN-IF-PSO-SVM | 0.3453 | 0.1741 | 0.9406 |

ICEEMDAN-IF-MPSO-SVM | 0.2607 | 0.1329 | 0.9654 |

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**MDPI and ACS Style**

Wang, L.; Liu, Y.; Li, T.; Xie, X.; Chang, C.
The Short-Term Forecasting of Asymmetry Photovoltaic Power Based on the Feature Extraction of PV Power and SVM Algorithm. *Symmetry* **2020**, *12*, 1777.
https://doi.org/10.3390/sym12111777

**AMA Style**

Wang L, Liu Y, Li T, Xie X, Chang C.
The Short-Term Forecasting of Asymmetry Photovoltaic Power Based on the Feature Extraction of PV Power and SVM Algorithm. *Symmetry*. 2020; 12(11):1777.
https://doi.org/10.3390/sym12111777

**Chicago/Turabian Style**

Wang, Lishu, Yanhui Liu, Tianshu Li, Xinze Xie, and Chengming Chang.
2020. "The Short-Term Forecasting of Asymmetry Photovoltaic Power Based on the Feature Extraction of PV Power and SVM Algorithm" *Symmetry* 12, no. 11: 1777.
https://doi.org/10.3390/sym12111777