# Hybrid LSTM-Based Fractional-Order Neural Network for Jeju Island’s Wind Farm Power Forecasting

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## Abstract

**:**

^{2}values.

## 1. Introduction

- The LSTM model is designed to predict missing input parameters, including wind speed and direction. Its performance is evaluated through root mean squared error (RMSE) assessment.
- The FONN model predicts wind power using the LSTM’s forecast data and evaluates performance with a coefficient of determination (R
^{2}) and mean squared error (MSE). - The models developed were evaluated in two case studies involving missing data scenarios for specific parameters.

## 2. Dataset Description

#### 2.1. Correlation Analysis of Wind Speed Parameter with Missing Data

#### 2.2. Correlation Analysis of Wind Direction Parameter with Missing Data

## 3. Proposed Methodology

^{2}) and MSE, with comparisons made against a conventional neural network model. This comprehensive methodology offers a structured approach to predicting wind power and leveraging predictive modelling techniques for renewable energy applications. As shown in Figure 6 and as explained earlier, the first part of the methodology is the LSTM model’s development for forecasting missing input data of wind speed and direction. The next part presents the FONN model that is used to make predictions of the generated wind power.

#### 3.1. LSTM Model

#### 3.2. FONN Model

#### 3.3. Fractional-Order Tangential Activation Functions

#### 3.4. Performance Metrics

^{2}, is frequently used to show the predictive capability of forecasting methods in fitting actual data (${Y}_{i}$), calculated as [52]

^{2}yields values ranging from 0 (indicating a poor match) to 1 (representing a perfect fit).

## 4. Results and Discussion

#### 4.1. Performance of LSTM Model

- The LSTM model exhibits the lowest RMSE values compared to the NAR and ARIMA models for forecasting missing wind speed data across all sites.
- At Site A, the LSTM model achieved the lowest RMSE value of 0.16, followed by NAR with an RMSE of 0.353 and ARIMA with an RMSE of 0.583.
- Similarly, the LSTM model at Site B outperformed the other models with an RMSE of 0.185, while the NAR and ARIMA models showed higher RMSE values of 0.297 and 0.458, respectively.
- Finally, at Site C, the LSTM model exhibited the lowest RMSE of 0.112, followed by ARIMA with an RMSE of 0.387 and NAR with the highest RMSE of 0.457.
- The following analysis is related to missing wind direction data forecasting, where the performance of the models varies across different sites.
- At Site A, the LSTM model had the lowest RMSE of 0.18, followed by ARIMA with an RMSE of 0.386 and NAR with the highest RMSE of 0.442.
- Similarly, at Site B, the LSTM model performed best with an RMSE of 0.425, followed by NAR with an RMSE of 0.185, and ARIMA with the highest RMSE of 0.572.
- Finally, at Site C, the NAR model had the lowest RMSE of 0.395, followed by LSTM with an RMSE of 0.126, and ARIMA with the highest RMSE of 0.454.

#### 4.2. Performance of FONN Model

#### 4.2.1. Case Study 1

^{2}and MSE. The analysis of the performance of the activation function is given in Table 3, based on the results obtained, as follows.

^{2}values of 0.9749 and 0.9831 during the training and testing phases, respectively, with MSE values of 0.0205 and 0.0142. Similarly, the fractional hard tansig function performed the best, with R

^{2}values of 0.9263 and 0.9369 and MSE values of 0.0424 and 0.0397 during the training and testing phases, respectively. The hard tansig function also performed well, with R

^{2}values of 0.8954 and 0.9075 in the training and testing phases, respectively, and an MSE value of 0.0521 in both phases. On the other hand, the conventional tansig function had the lowest accuracy, with R

^{2}values of 0.8578 and 0.8642 during the training and testing phases, respectively, and MSE values of 0.0753 and 0.0764, respectively.

^{2}values of 0.9428 and 0.9497 during training and testing phases, respectively, and MSE values of 0.0534 and 0.0529. The hard tansig function also performed well, with R

^{2}values of 0.9489 and 0.9517 during the training and testing phases, respectively, and MSE values of 0.0583 and 0.0578, respectively. On the other hand, the worst-performing function for Site B was the conventional tansig function, with R

^{2}values of 0.9328 and 0.9436 in the training and testing phases, respectively, and MSE values of 0.0662 and 0.0652, respectively. Moreover, the fractional hard tansig function performed better during the training and testing phases, with R

^{2}values of 0.9543 and 0.9609 and MSE values of 0.0464 and 0.0432, respectively. The conventional LiSHT function had R

^{2}values of 0.9532 and 0.9584 and MSE values of 0.0428 and 0.0414 during the training and testing phases, respectively. The fractional LiSHT function performed better, with R

^{2}values of 0.9572 and 0.9621 and MSE values of 0.0399 and 0.0386 during the training and testing phases, respectively. For the highest accuracy, the fractional arctan function proved to be the best option, with R

^{2}values of 0.9929 and 0.9952 in the training and testing phases, respectively, and MSE values of 0.0046 and 0.0032, respectively. Similarly, the conventional arctan function also performed well, with R

^{2}values of 0.9901 and 0.9948 during the training and testing phases, respectively, and MSE values of 0.0063 and 0.0035, respectively.

^{2}values of 0.8931 and 0.9026 and MSE values of 0.0742 and 0.0629 during the training and testing phases, respectively. Similarly, the fractional hard tansig function performed better than the conventional hard tansig function, with R

^{2}values of 0.9035 and 0.9163 and MSE values of 0.0598 and 0.0586 during the training and testing phases, respectively. For the LiSHT function, the fractional LiSHT function performed slightly better than the conventional LiSHT function, with R

^{2}values of 0.8864 and 0.8973, and MSE values of 0.0752 and 0.0745 during the training and testing phases, respectively. The highest accuracy was achieved using the fractional arctan function, with R

^{2}values of 0.9573 and 0.9635 and MSE values of 0.0123 and 0.0115 during the training and testing phases, respectively. The conventional arctan function also performed well, with R

^{2}values of 0.9469 and 0.9529 and MSE values of 0.0158 and 0.0134 during the training and testing phases, respectively.

^{2}values and lower MSE values for training and testing across different sites.

#### 4.2.2. Case Study 2

^{2}and MSE values during both training and testing phases. On the other hand, the fractional function performed better overall, with higher R

^{2}values and lower MSE values than the corresponding conventional functions. For instance, for Site A, arctan had the highest R

^{2}value of 0.9898 and the lowest MSE value of 0.0081 for training, while for testing, it had the highest R

^{2}value of 0.9931 and the lowest MSE value of 0.0059. Similarly, the fractional arctan function had the highest R

^{2}value of 0.9899 and the lowest MSE value of 0.0081 for training, while for testing, it had the highest R

^{2}value of 0.9946 and the lowest MSE value of 0.0048. These values indicate that arctan and its corresponding fractional function were the best-performing functions for Site A. The second-best performing function for Site A was hard tansig and its corresponding fractional function. For training, hard tansig had the highest R

^{2}value of 0.9264 and the lowest MSE value of 0.0372, while for testing, it had the highest R

^{2}value of 0.9378 and the lowest MSE value of 0.0346. Similarly, the fractional hard tansig function had the highest R

^{2}value of 0.9726 and the lowest MSE value of 0.0218 for training, while for testing, it had the highest R

^{2}value of 0.9832 and the lowest MSE value of 0.0169. On the other hand, the tansig function had the lowest R

^{2}value of 0.8973 and the highest MSE value of 0.0621 during training. Similarly, during testing, it had the lowest R

^{2}value of 0.9043 and the highest MSE value of 0.0594. Similarly, the fractional tansig function had the lowest R

^{2}value of 0.9264 and the highest MSE value of 0.0519 during training, while it had the lowest R

^{2}value of 0.9329 and the highest MSE value of 0.0497 when tested. These values indicate that tansig and its corresponding fractional function were the worst-performing functions at Site A.

^{2}and MSE values demonstrate the consistent and dependable performance of both functions for predicting generated wind power using the provided input data.

^{2}values and lower MSE values than the other functions, indicating better predictive capabilities of the FONN model. These findings have important implications for fields that rely on predictive modelling, such as finance, economics, and engineering.

## 5. Conclusions

^{2}and low MSE values, around 0.97 and 0.003, respectively, during training and testing. Similarly, both activation functions exhibited strong predictive capabilities in predicting wind power using wind speed. During training and testing, the forecast wind direction in the second case achieved high R

^{2}and low MSE values, around 0.98 and 0.004, respectively. The results highlight the potential of the developed arctan arctan function, which consistently proved its effectiveness in enhancing predictive capabilities compared to the conventional arctan function and among all the tangential functions in both case studies. The study provides valuable insights into predicting generated wind power and fills gaps in missing data, demonstrating the potential of advanced neural networks in renewable energy applications. The developed arctan tangential activation functions have improved predictive capabilities compared to the conventional tangential functions, but their increased complexity may limit their practical implementation. In future work, there is a possibility of expanding the analysis carried out on fractional activation functions at $\alpha $ = 0.1 to determine the optimal $\alpha $ value. This extension of $\alpha $ could potentially increase the predictive accuracy of power in wind farms.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 13.**Performance of conventional neural network model during training and testing with forecast missing wind speed.

**Figure 15.**Performance of conventional neural network model during training and testing with forecast missing wind direction.

Data Aspect | Site A | Site B | Site C |
---|---|---|---|

Data Collection Period | 11 January 2014–25 January 2014 | 11 January 2014–20 January 2014 | 11 January 2014–25 January 2014 |

Collection Time Interval | 10 min | 10 min | 10 min |

Wind Turbine Specifications | |||

Model | U88 | U50 | U50 |

Output | 2000 kW | 750 kW | 750 kW |

Wind Speed | Up to 12 m/s | Up to 12.5 m/s | Up to 12.5 m/s |

Rotor Speed Range | 6–17.5 rpm | 9–28 rpm | 9–28 rpm |

Voltage and Frequency | 690 V/60 Hz | 690 V/60 Hz | 690 V/60 Hz |

Rotor Diameter | 88 m | 50 m | 50 m |

Hub Height | 80 m | 50 m | 50 m |

Power Control | Pitch Regulation | Pitch Regulation | Pitch Regulation |

**Table 2.**Performance comparison of various forecasting models for missing data of wind speed and direction at different sites.

Model | Site | Wind Speed (m/s) | Wind Direction (deg) |
---|---|---|---|

RMSE | RMSE | ||

LSTM | Site A | 0.18 | 0.16 |

Site B | 0.425 | 0.185 | |

Site C | 0.112 | 0.126 | |

NAR | Site A | 0.353 | 0.442 |

Site B | 0.297 | 0.185 | |

Site C | 0.457 | 0.395 | |

ARIMA | Site A | 0.583 | 0.386 |

Site B | 0.458 | 0.572 | |

Site C | 0.387 | 0.454 |

**Table 3.**Performance comparison of different functions in training and testing phases for various sites under case study 1.

Site | Conventional Function | Training | Testing | Fractional Function | Training | Testing | ||||
---|---|---|---|---|---|---|---|---|---|---|

R^{2} | MSE |
R^{2} | MSE |
R^{2} | MSE |
R^{2} | MSE | |||

Site A | Tansig | 0.8578 | 0.0753 | 0.8642 | 0.0764 | Tansig | 0.8739 | 0.0628 | 0.8864 | 0.0612 |

Hard tansig | 0.8954 | 0.0521 | 0.9075 | 0.0516 | Hard tansig | 0.9263 | 0.0424 | 0.9369 | 0.0397 | |

LiSHT | 0.8749 | 0.0683 | 0.8873 | 0.0621 | LiSHT | 0.9025 | 0.0612 | 0.9173 | 0.0598 | |

Arctan | 0.9727 | 0.0227 | 0.9733 | 0.0207 | Arctan | 0.9749 | 0.0205 | 0.9831 | 0.0142 | |

Site B | Tansig | 0.9328 | 0.0662 | 0.9436 | 0.0652 | Tansig | 0.9428 | 0.0534 | 0.9497 | 0.0529 |

Hard tansig | 0.9489 | 0.0583 | 0.9517 | 0.0578 | Hard tansig | 0.9543 | 0.0464 | 0.9609 | 0.0432 | |

LiSHT | 0.9532 | 0.0428 | 0.9584 | 0.0414 | LiSHT | 0.9572 | 0.0399 | 0.9621 | 0.0386 | |

Arctan | 0.9901 | 0.0063 | 0.9948 | 0.0035 | Arctan | 0.9929 | 0.0046 | 0.9952 | 0.0032 | |

Site C | Tansig | 0.8216 | 0.0853 | 0.8362 | 0.0817 | Tansig | 0.8931 | 0.0742 | 0.9026 | 0.0629 |

Hard tansig | 0.8453 | 0.0732 | 0.8564 | 0.0695 | Hard tansig | 0.9035 | 0.0598 | 0.9163 | 0.0586 | |

LiSHT | 0.8762 | 0.0789 | 0.8758 | 0.0778 | LiSHT | 0.8864 | 0.0752 | 0.8973 | 0.0745 | |

Arctan | 0.9469 | 0.0158 | 0.9529 | 0.0134 | Arctan | 0.9573 | 0.0123 | 0.9635 | 0.0115 |

**Table 4.**Performance comparison of different functions in training and testing phases for various sites under case study 2.

Site | Conventional Function | Training | Testing | Fractional Function | Training | Testing | ||||
---|---|---|---|---|---|---|---|---|---|---|

R^{2} | MSE |
R^{2} | MSE |
R^{2} | MSE |
R^{2} | MSE | |||

Site A | Tansig | 0.8973 | 0.0621 | 0.9043 | 0.0594 | Tansig | 0.9264 | 0.0519 | 0.9329 | 0.0497 |

Hard tansig | 0.9264 | 0.0372 | 0.9378 | 0.0346 | Hard tansig | 0.9726 | 0.0218 | 0.9832 | 0.0169 | |

LiSHT | 0.9163 | 0.0583 | 0.9289 | 0.0542 | LiSHT | 0.9517 | 0.0487 | 0.9619 | 0.0453 | |

Arctan | 0.9898 | 0.0081 | 0.9931 | 0.0059 | Arctan | 0.9899 | 0.0081 | 0.9946 | 0.0048 | |

Site B | Tansig | 0.8245 | 0.0982 | 0.8463 | 0.0968 | Tansig | 0.8562 | 0.0841 | 0.8678 | 0.0832 |

Hard tansig | 0.8674 | 0.0721 | 0.8689 | 0.0708 | Hard tansig | 0.8864 | 0.0682 | 0.8949 | 0.0617 | |

LiSHT | 0.8462 | 0.0819 | 0.8573 | 0.0798 | LiSHT | 0.8693 | 0.0739 | 0.8715 | 0.0716 | |

Arctan | 0.9826 | 0.0129 | 0.9875 | 0.0094 | Arctan | 0.9835 | 0.0124 | 0.9867 | 0.0094 | |

Site C | Tansig | 0.9041 | 0.0528 | 0.9146 | 0.0512 | Tansig | 0.9317 | 0.0425 | 0.9462 | 0.0419 |

Hard tansig | 0.9089 | 0.0481 | 0.9163 | 0.0479 | Hard tansig | 0.9273 | 0.0341 | 0.9526 | 0.0252 | |

LiSHT | 0.8932 | 0.0514 | 0.9023 | 0.0506 | LiSHT | 0.9172 | 0.0459 | 0.9251 | 0.0445 | |

Arctan | 0.9793 | 0.0085 | 0.9866 | 0.0054 | Arctan | 0.9816 | 0.0076 | 0.9865 | 0.0052 |

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## Share and Cite

**MDPI and ACS Style**

Ramadevi, B.; Kasi, V.R.; Bingi, K.
Hybrid LSTM-Based Fractional-Order Neural Network for Jeju Island’s Wind Farm Power Forecasting. *Fractal Fract.* **2024**, *8*, 149.
https://doi.org/10.3390/fractalfract8030149

**AMA Style**

Ramadevi B, Kasi VR, Bingi K.
Hybrid LSTM-Based Fractional-Order Neural Network for Jeju Island’s Wind Farm Power Forecasting. *Fractal and Fractional*. 2024; 8(3):149.
https://doi.org/10.3390/fractalfract8030149

**Chicago/Turabian Style**

Ramadevi, Bhukya, Venkata Ramana Kasi, and Kishore Bingi.
2024. "Hybrid LSTM-Based Fractional-Order Neural Network for Jeju Island’s Wind Farm Power Forecasting" *Fractal and Fractional* 8, no. 3: 149.
https://doi.org/10.3390/fractalfract8030149