# An Improved Whale Algorithm for Support Vector Machine Prediction of Photovoltaic Power Generation

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## Abstract

**:**

## 1. Introduction

## 2. Prediction Model and Its Improvement

#### 2.1. Support Vector Machine

#### 2.2. Whale Optimization Algorithm

#### 2.2.1. Surround the Prey

#### 2.2.2. Bubble Net Predation

#### 2.2.3. Hunting for Prey

#### 2.3. Improved Whale Optimization Algorithm

#### 2.3.1. Tent Mapping Initialization

- (1)
- Generate ${x}_{0}$ randomly, which represents the initial values of chaotic variables, and make ${x}_{0}$ not equal to 0.5.
- (2)
- Generate the chaotic mapping sequence iteratively according to the method of Formula (17), and if the chaotic variable enters a cycle, proceed to step (4).
- (3)
- Judge the end condition. When the end condition is satisfied, proceed to step (5).
- (4)
- Disturb ${x}_{0}$ and regenerate chaotic sequences.
- (5)
- Map the obtained chaotic value into the solution space of the optimization problem to form the initial IMWOA population.

#### 2.3.2. Variation Disturbance of Optimal Position

#### 2.3.3. Differential Evolution Algorithm

- Mutation: Select one individual in the population as the current individual ${X}^{*}$, and then select three individuals except ${X}^{*}$ randomly. First, two randomly selected individuals are subjected to a vector difference operation, and then the result is subjected to a vector sum operation with the third individual to generate a mutant individual. The Formula is shown as follows [47].$${X}^{mut}=random(Minsc,Maxsc)\times ({X}_{1}(t)-{X}_{2}(t))+{X}_{3}(t)$$
- Crossover: Crossover individual is composed of some elements of the current individual and mutation individual. The crossover Formula is shown in (20).$${U}_{i,j}=\{\begin{array}{ll}{X}_{i,j}^{*}& ran{d}_{i,j}\le CRorj={I}_{rand}\\ {X}_{i,j}^{mut}& otherwise\end{array}$$
- Selection: Compare the fitness values of ${X}^{*}$ and ${U}_{i}$, and retain the individuals with high fitness into the next generation population. The selection Formula is shown in (21).$${X}_{i}(t+1)=\{\begin{array}{ll}{X}_{i}(t)& iffobj({U}_{i}(t))\le fobj({X}_{i}(t))\\ {U}_{i}(t)& otherwise\end{array}$$

- (1)
- Initialize IMWOA parameters, including iteration number, solution dimension, population size, decision variable matrix size, cross probability, etc.
- (2)
- Generate the initial population of IMWOA based on the chaotic tent map, and calculate the fitness value.
- (3)
- Update the position and fitness of the whale population. The renewal process includes mutation disturbance to the optimal individual.
- (4)
- Carry the mutation and crossover operations of differential evolution algorithm to generate crossover individuals.
- (5)
- Select the individuals between the cross individuals and the original individuals according to the fitness to form the next generation.
- (6)
- Judge the end condition. If the end condition is met, output the optimal solution. Else, proceed to step (3).

#### 2.4. IMWOA Performance Test

## 3. Input Data Preprocessing and Experimental Arrangement

#### 3.1. Selection of Input Data

#### 3.2. Denoising and Normalization of Input Data

- (1)
- Decompose the original signal and obtain the wavelet coefficients.
- (2)
- Set threshold and threshold function.
- (3)
- Denoise the wavelet coefficients by threshold to filter the noise information in the signal.
- (4)
- Reconstruct the processed wavelet coefficients to obtain the denoised signal.

#### 3.3. Experimental Arrangement

- (1)
- Select training data and testing data in sunny and cloudy weather, respectively.
- (2)
- Preprocess training input data and testing input data by wavelet soft threshold denoising.
- (3)
- Normalize training and testing data.
- (4)
- Initialize the parameters of the IMWOA-SVM photovoltaic output power prediction model.
- (5)
- Train the prediction model by training data. Apply the IMWOA to optimize the SVM. Test the prediction model by the test data.
- (6)
- Obtain the optimal prediction model of PV power. Predict the PV power.
- (7)
- Normalize the prediction output power inversely and output the experimental results.

## 4. Experimental Results and Discussion

#### 4.1. Prediction Results in Sunny Weather

#### 4.1.1. Comparison with Other SVM Models

#### 4.1.2. Comparison with BP Neural Network and ELM

#### 4.2. Prediction Results in Cloudy Weather

#### 4.2.1. Comparison with Other SVM Models

#### 4.2.2. Comparison with BP Neural Network and ELM

## 5. Conclusions

- (1)
- The PV power is determined by some meteorological factors, and it has significantly different characteristics in different weather conditions. Through the correlation analysis, it is found that PV power has the strongest correlation with the meteorological factors including light intensity, ambient temperature and humidity. Furthermore, these meteorological factors can be used to accurately predict PV power.
- (2)
- The wavelet soft threshold denoising can be applied for the pretreatment of PV input data. It can effectively eliminate the noise contained in input data and improve the coherence of the input data, which is beneficial to remove the adverse impact of noise and enhance the stability of the prediction model in complex weather conditions.
- (3)
- The BP neural network and ELM have large demand for training data. When the training data are not sufficient, the ideal prediction accuracy cannot be achieved. SVM has less demand for training data, and can achieve ideal prediction accuracy when there is less training data, which is suitable for PV output power prediction models with less training data.
- (4)
- The optimization performance of WOA can be effectively improved through combination with the hybrid improved method. By combining the original WOA with tent chaos initialization, mutation disturbance of the optimal individual and DE algorithm, the comprehensive performance of the IMWOA is significantly enhanced.
- (5)
- The IMWOA-SVM photovoltaic output power prediction model applies wavelet denoising to process the predicted input data, and applies the hybrid improved whale algorithm to optimize the SVM, which significantly improves comprehensive prediction performance in different weather conditions.
- (6)
- The proposed IMWOA-SVM photovoltaic output power prediction model can symmetrically achieve the accurate prediction for PV power in different weather conditions, especially in complex weather conditions. It can provide the operation and scheduling department with reliable reference, help to improve the utilization rate of renewable energy power generation and maintain the security of renewable energy power systems. It is of great significance to the application of clean energy.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notations

Acronyms list | |

ALO | Ant lion optimization algorithm |

DE | Differential evolution algorithm |

IMWOA | Improved whale optimization algorithm |

MAPE | Mean absolute percent error |

MFO | Moth to fire algorithm |

MSE | Mean square error |

PSO | Particle swarm optimization algorithm |

RMSE | Root mean square error |

SVM | Support vector machine |

SVR | Support vector regression |

WOA | Whale optimization algorithm |

Nomenclature variables | |

a, ${a}^{*}$, u, ${u}^{*}$ | Iteration variables of whale algorithm |

A, C_{w} | Iteration variables of whale algorithm |

b | Spiral motion constant of whale or Regression bias |

C | Penalty factor |

C_{1}, C_{2} | Learning factor |

CR | Crossover probability |

$Maxsc$ | Maximum scale factor |

$Max\_t$ | Maximum number of iterations |

$Minsc$ | Minimum scale factor |

t | Current iterations |

U | New individuals obtained by crossing |

X | Individual whale population |

${X}_{best}$ | Optimal individual of whale population |

${X}^{n}$ | New individuals obtained by mutation |

${X}_{rand}$ | Random individuals of whale population |

${X}^{*}$ | Current individuals |

${x}_{t}$ | Tent chaotic mapping sequence |

$\phi (x)$ | Nonlinear mapping function |

$\epsilon $ | Permissible deviation |

$\rho $ | Pearson correlation coefficient |

$\xi ,{\xi}^{*}$ | Relaxation variable |

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Expression | Dim | Range | Optimum |
---|---|---|---|

${F}_{1}(x)={\displaystyle {\sum}_{i=1}^{n}{x}_{i}^{2}}$ | 10 | [−100, 100] | 0 |

${F}_{{}_{2}}(x)={\displaystyle {\sum}_{i=1}^{n}\left|{x}_{i}\right|}+{\displaystyle {\prod}_{i=1}^{n}\left|{x}_{i}\right|}$ | 10 | [−10, 10] | 0 |

${F}_{3}(x)={\displaystyle {\sum}_{i=1}^{\mathrm{dim}}{({\displaystyle {\sum}_{j=1}^{i}{x}_{j}})}^{2}}$ | 10 | [−100, 100] | 0 |

${F}_{4}(x)={\displaystyle \sum ([1:\mathrm{dim}].\times [x.^4])+rand}$ | 10 | [−1.28, 1.28] | 0 |

${F}_{5}(x)=-20\mathrm{exp}(-0.2\sqrt{\frac{1}{n}{\displaystyle {\sum}_{i=1}^{n}{x}_{i}^{2}}})-\mathrm{exp}(\frac{1}{n}{\displaystyle {\sum}_{i=1}^{n}\mathrm{cos}(2{\displaystyle \prod {x}_{i}})})+20+e$ | 10 | [−32, 32] | 0 |

Algorithm | Parameter | Value |
---|---|---|

ALL | Number of iterations | 300 |

Number of search agent | 50 | |

IMWOA | Minsc | 0.20 |

Maxsc | 0.80 | |

CR | 0.20 | |

PSO | C_{1} | 1.49 |

C_{2} | 1.49 |

Function | Index | IMWOA | WOA | ALO | MVO | PSO | MFO |
---|---|---|---|---|---|---|---|

F_{1} | Mean | 0 | 0 | 0 | 0.0145 | 0 | 0 |

RMSE | 0 | 0 | 0 | 0.0145 | 0 | 0 | |

F_{2} | Mean | 0 | 0 | 1.54 | 0.0428 | 0.136 | 0 |

RMSE | 0 | 0 | 2.14 | 0.0429 | 0.0912 | 0 | |

F_{3} | Mean | 0 | 227 | 0.458 | 0.209 | 0.00801 | 0.591 |

RMSE | 0 | 228 | 0.675 | 0.238 | 0.00941 | 0.828 | |

F_{4} | Mean | 0 | 0 | 0.0222 | 0.00211 | 0.0126 | 0.00717 |

RMSE | 0 | 0 | 0.0245 | 0.0223 | 0.0160 | 0.00753 | |

F_{5} | Mean | 0 | 0 | 0.385 | 0.964 | 0.0450 | 0 |

RMSE | 0 | 0 | 0.667 | 1.18 | 0.0632 | 0 |

${\mathit{\rho}}_{\mathit{X},\mathit{Y}}$ | Degree of Relevance |
---|---|

[0.8,1.0] | Extremely strong |

[0.6,0.8] | Strong |

[0.4,0.6] | Moderate |

[0.2,0.4] | Weak |

[0.0,0.2] | Extremely weak |

Month | Light Intensity | Diffuse | Temperature | Wind Speed | Humidity |
---|---|---|---|---|---|

1 | 0.992 | 0.350 | 0.521 | 0.199 | −0.590 |

2 | 0.994 | 0.449 | 0.521 | 0.262 | −0.563 |

3 | 0.989 | 0.321 | 0.426 | −0.053 | −0.373 |

4 | 0.996 | 0.212 | 0.627 | 0.428 | −0.596 |

5 | 0.995 | 0.256 | 0.696 | 0.580 | −0.698 |

6 | 0.990 | 0.326 | 0.500 | 0.512 | −0.731 |

Mean | 0.993 | 0.319 | 0.549 | 0.321 | −0.591 |

Relevance | Extremely strong | Weak | Moderate | Weak | Moderate |

Parameter | IMWOA | ALO | MFO | MVO | PSO | WOA | BP | ELM |
---|---|---|---|---|---|---|---|---|

MSE | 0.069 | 0.859 | 0.970 | 0.069 | 0.072 | 0.962 | 0.587 | 8.887 |

RMSE | 0.263 | 0.927 | 0.985 | 0.263 | 0.268 | 0.981 | 0.766 | 2.981 |

MAE | 0.212 | 0.783 | 0.923 | 0.212 | 0.203 | 0.870 | 0.632 | 2.578 |

MAPE | 0.047 | 0.103 | 0.130 | 0.048 | 0.057 | 0.161 | 0.179 | 0.506 |

R2 | 0.995 | 0.933 | 0.924 | 0.995 | 0.994 | 0.925 | 0.954 | 0.305 |

Parameter | IMWOA | ALO | MFO | MVO | PSO | WOA | BP | ELM |
---|---|---|---|---|---|---|---|---|

MSE | 0.257 | 1.297 | 4.987 | 0.783 | 0.655 | 0.809 | 2.717 | 2.756 |

RMSE | 0.507 | 1.139 | 2.233 | 0.885 | 0.809 | 0.899 | 1.648 | 1.660 |

MAE | 0.331 | 0.996 | 2.012 | 0.671 | 0.630 | 0.748 | 1.338 | 1.283 |

R2 | 0.979 | 0.893 | 0.588 | 0.935 | 0.946 | 0.933 | 0.775 | 0.772 |

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## Share and Cite

**MDPI and ACS Style**

Liu, Y.-W.; Feng, H.; Li, H.-Y.; Li, L.-L.
An Improved Whale Algorithm for Support Vector Machine Prediction of Photovoltaic Power Generation. *Symmetry* **2021**, *13*, 212.
https://doi.org/10.3390/sym13020212

**AMA Style**

Liu Y-W, Feng H, Li H-Y, Li L-L.
An Improved Whale Algorithm for Support Vector Machine Prediction of Photovoltaic Power Generation. *Symmetry*. 2021; 13(2):212.
https://doi.org/10.3390/sym13020212

**Chicago/Turabian Style**

Liu, Yu-Wei, Huan Feng, Heng-Yi Li, and Ling-Ling Li.
2021. "An Improved Whale Algorithm for Support Vector Machine Prediction of Photovoltaic Power Generation" *Symmetry* 13, no. 2: 212.
https://doi.org/10.3390/sym13020212