# An Effective Hybrid Symbolic Regression–Deep Multilayer Perceptron Technique for PV Power Forecasting

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## Abstract

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## 1. Introduction

- A novel feature selection technique was employed to investigate the feature patterns;
- A novel hybrid algorithm was explored for PV power forecasting;
- A fair evaluation was presented by showing the numerical and graphical performances of the proposed hybrid model.

## 2. Background and Proposed Architecture

#### 2.1. Symbolic Regression

#### 2.2. Deep Multi-Layer Perceptron

#### 2.3. Genetic Programming

#### 2.4. Problem Formulation

## 3. Hybrid Model

## 4. Case Study

#### 4.1. Features Selection

#### 4.2. Training and Simulation Results

#### 4.3. Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Binary genetic tree programming representation: $f(x,y)=sin\left(\frac{\pi x+0.5+y}{z}\right),z\in {\mathbb{R}}^{*}$.

**Figure 6.**Coefficients of the nonlinear correlation between the PV power and related system attributes.

Model | Reference | Score Metrics | Lowest Score | Dataset |
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XGBF ${}^{1}$-DNN ${}^{2}$ | [24] | RMSE, MBE ${}^{3}$, FS ${}^{4}$ | RMSE = 51.35 W | PV data in Limberg, Belgium |

SR-FFNN | [25] | RMSE, MBE ${}^{2}$, MAE, ${R}^{2}$ | ${R}^{2}$ = 0.932 | Solar power in Flanders, Belgium |

LSTM | [26] | NMAE, RMSE | RMSE = 38.13 kWh | 1 MW PV site in Goheung, Korea |

Modified LSTM | [27] | MAE, RMSE | RMSE = 0.55 kW | Ansan, Gyeonggi-do, Korea |

LSTM-EMA ${}^{5}$ | [28] | RMSE, ${R}^{2}$, MAPE | ${R}^{2}$ = 0.96 | Yeonseong-gun, Gyeonggi-do, South Korea |

ENS ${}^{6}$ | [29] | NRMSE, nMBE, MAE, nMAE | MAE = 74.1 kW | 32 PV plants installed at different latitudes in Italy |

GA-PSO-ANFIS | [30] | RMSE, MAE, NMAE, FS ${}^{4}$ | RMSE = 2.08 kW | Goldwind microgrid system found in Beijing |

SOM ${}^{7}$, LVQ ${}^{8}$, SVR ${}^{9}$ | [31] | MRE ${}^{10}$ and RMSE | MRE = 1.79% | Taiwan Central Weather Bureau |

PFLRM ${}^{11}$ | [32] | RMSE, MAD ${}^{12}$, MAPE | RMSE = 59.38 kW | Coloane island of Macau |

ANN | [33] | RMSE, ${R}^{2}$ | ${R}^{2}$ = 0.999 | Solar power plant in Dhaka |

LSH ${}^{13}$ | [34] | RMSE, MRE, QR ${}^{14}$ | RMSE = 4.23 kW | PV power station in Ashland |

AE ${}^{15}$-LSTM | [35] | MAPE, RMSE, MAE | RMSE = 0.14 kW | PV inverter installed in Haenam, South Korea |

SFLA ${}^{16}$-ANN | [36] | MAPE | MAPE = 5.38% | PV sites in Florida |

PCPOW ${}^{17}$ | [37] | ${R}^{2}$ | ${R}^{2}$ = 0.938 | Yunnan Electric Power Research Institute |

^{1}Extreme Gradient Boosting Forest.

^{2}Deep Neural Network.

^{3}Mean Bias Error.

^{4}Forecast Skill.

^{5}Exponential Moving Average.

^{6}Ensemble of Methods.

^{7}Self-Organization Map.

^{8}Learning Vector Quantization.

^{9}Support Vector Regression.

^{10}Mean Relative Error.

^{11}Partial Functional Linear Regression Model.

^{12}Mean Absolute Deviation.

^{13}Local Sensitive Hashing.

^{14}QR pass rate.

^{15}Auto-Encoder.

^{16}Shuffled Frog Leaping Algorithm.

^{17}PSO-based sky images cloud motion speed calculation method for PV power.

Errors | SR | MLP | SR-MLP |
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RMSE (kW) | 7.21 | 6.48 | 5.58 |

MAE (kW) | 4.92 | 3.81 | 3.3 |

${R}^{2}$ | 0.988 | 0.990 | 0.993 |

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**MDPI and ACS Style**

Trabelsi, M.; Massaoudi, M.; Chihi, I.; Sidhom, L.; Refaat, S.S.; Huang, T.; Oueslati, F.S.
An Effective Hybrid Symbolic Regression–Deep Multilayer Perceptron Technique for PV Power Forecasting. *Energies* **2022**, *15*, 9008.
https://doi.org/10.3390/en15239008

**AMA Style**

Trabelsi M, Massaoudi M, Chihi I, Sidhom L, Refaat SS, Huang T, Oueslati FS.
An Effective Hybrid Symbolic Regression–Deep Multilayer Perceptron Technique for PV Power Forecasting. *Energies*. 2022; 15(23):9008.
https://doi.org/10.3390/en15239008

**Chicago/Turabian Style**

Trabelsi, Mohamed, Mohamed Massaoudi, Ines Chihi, Lilia Sidhom, Shady S. Refaat, Tingwen Huang, and Fakhreddine S. Oueslati.
2022. "An Effective Hybrid Symbolic Regression–Deep Multilayer Perceptron Technique for PV Power Forecasting" *Energies* 15, no. 23: 9008.
https://doi.org/10.3390/en15239008