# A Novel Hybrid Feature Selection Method for Day-Ahead Electricity Price Forecasting

^{1}

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## Abstract

**:**

## 1. Introduction

- A novel hybrid method based on elitist GA and tree based method for input FS in price prediction,
- Fixing the error margins during price prediction by applying the confidence interval, and
- Season-wise optimize FS to have a better forecasting accuracy.

## 2. Proposed Methodology

## 3. Sequential Minimal Optimization (SMO) Regression Algorithm

#### 3.1. Methodology

_{1}and µ

_{2}).

_{1}and µ

_{2}keeping other µ’s fixed.

_{1}and d

_{2}in the vector. 0 defines the lower bound C is higher bound or upper bound in the constraint.

#### 3.2. Calculation of SMO Regression

## 4. Input Feature Selection Using Proposed Algorithm

## 5. Result and Discussion

_{1}) was selected in all the runs i.e., 36 times. At the same time, the price of the previous day (Pr

_{4}) was selected 22 times and the hour type (Ht

_{o}) 24 times, which shows their relative importance accordingly. The load of immediate hour (Lo

_{1}) and previous day load (Lo

_{4}) was an important feature and selected 20 and 23 times respectively. The temperatures of the present day (Te

_{1}) selected more times than on the previous day. Wind speed of the previous day (Wi

_{6}) was selected more number of times than the same day. The humidity of the previous day (Hu

_{6}) was selected more times than on the same day. Effects of features can also be analyzed according to seasons. Table 2 also indicates the top 10 features selected most times.

_{6}and Lo

_{1}etc. are features which are assumed to have more importance during the winter season. Hu

_{6}and Pr

_{5}etc. are features which are assumed to have more importance during the spring season. Lo

_{4}and Te

_{5}etc. are features which are assumed to have more importance during the summer season. Pr

_{1}is seem to be feature regardless of the season. These analysis indicated the relative importance of feature in terms of seasonal variations.

_{p}and F

_{p}are actual and forecasted value of electricity price at time p and K is the length of forecast horizon and ${A}_{week}=\frac{1}{K}{\displaystyle \sum _{p=1}^{K}{A}_{p}}$.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FS | Feature Selection |

GA | Genetic Algorithm |

MCP | Market Clearing Prices |

ANN | Artificial Neural Network |

SDA | Stacked Denoising Autoencoders |

FFNN | Feed Forward Neural Networks |

WT | Wavelet Transform: |

FA-PSO | Fuzzy Adaptive Particle Swarm Optimization: |

RF | Random Forest |

SVM | Support Vector Machine |

DE | Differential Evolution |

MI | Mutual Information |

SMO | Sequential Minimization Optimization |

QP | Quadratic Programming |

WoFS | Without Feature Selection |

KKT | Karush–Kuhan–Tucker |

10-FCV | 10-Fold Cross Validation |

AEMO | Australian Energy Market Operator |

RMSE | Root-Mean Square Error |

MAE | Mean Absolute Error |

MAPE | Mean Absolute Percentage Error |

EV | Error Variance |

## Appendix A

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Name of Variable | Name of Input | Timing of Variable |
---|---|---|

Load (Lo) | Lo_{6} | Lo_{(D-23:00)} |

Lo_{5} | Lo_{(D-23:30)} | |

Lo_{4} | Lo_{(D-24:00)} | |

Lo_{3} | Lo_{(D-01:30)} | |

Lo_{2} | Lo_{(D-01:00)} | |

Lo_{1} | Lo_{(D-00.30)} | |

Price (Pr) | Pr_{6} | Pr_{(D-23:00)} |

Pr_{5} | Pr_{(D-23:30)} | |

Pr_{4} | Pr_{(D-24:00)} | |

Pr_{3} | Pr_{(D-01:30)} | |

Pr_{2} | Pr_{(D-01:00)} | |

Pr_{1} | Pr_{(D-00.30)} | |

Wind Speed (Wi) | Wi_{6} | Wi_{(D-23:00)} |

Wi_{5} | Wi_{(D-23:30)} | |

Wi_{4} | Wi_{(D-24:00)} | |

Wi_{3} | Wi_{(D-01:30)} | |

Wi_{2} | Wi_{(D-01:00)} | |

Wi_{1} | Wi_{(D-00.30)} | |

Temperature (Te) | Te_{6} | Te_{(D-23:00)} |

Te_{5} | Te_{(D-23:30)} | |

Te_{4} | Te_{(D-24:00)} | |

Te_{3} | Te_{(D-01:30)} | |

Te_{2} | Te_{(D-01:00)} | |

Te_{1} | Te_{(D-00.30)} | |

Humidity (Hu) | Hu_{6} | Hu_{(D-23:00)} |

Hu_{5} | Hu_{(D-23:30)} | |

Hu_{4} | Hu_{(D-24:00)} | |

Hu_{3} | Hu_{(D-01:30)} | |

Hu_{2} | Hu_{(D-01:00)} | |

Hu_{1} | Hu_{(D-00.30)} | |

Day Timing (Hto) | Ht_{o} | Ht_{o(D-00.00)} |

**Table 2.**Number of feature selected year wise highlighted on the top 10 features selected most times.

Feature Name | Number of Time Selected | Feature Name | Number of Time Selected |
---|---|---|---|

Lo_{6} | 16 | Wi_{2} | 20 |

Lo_{5} | 10 | Wi_{1} | 17 |

Lo_{4} | 23 | Te_{6} | 14 |

Lo_{3} | 16 | Te_{5} | 19 |

Lo_{2} | 18 | Te_{4} | 19 |

Lo_{1} | 20 | Te_{3} | 11 |

Pr_{6} | 15 | Te_{2} | 16 |

Pr_{5} | 18 | Te_{1} | 21 |

Pr_{4} | 22 | Hu_{6} | 23 |

Pr_{3} | 17 | Hu_{5} | 20 |

Pr_{2} | 18 | Hu_{4} | 15 |

Pr_{1} | 36 | Hu_{3} | 10 |

Wi_{6} | 21 | Hu_{2} | 18 |

Wi_{5} | 13 | Hu_{1} | 18 |

Wi_{4} | 18 | Ht_{o} | 24 |

Wi_{3} | 15 |

Feature Name | Winter | Spring | Summer |
---|---|---|---|

Lo_{6} | 8 | 4 | 4 |

Lo_{5} | 3 | 4 | 3 |

Lo_{4} | 5 | 8 | 10 |

Lo_{3} | 4 | 6 | 6 |

Lo_{2} | 3 | 7 | 8 |

Lo_{1} | 9 | 5 | 6 |

Pr_{6} | 3 | 6 | 6 |

Pr_{5} | 4 | 8 | 6 |

Pr_{4} | 9 | 7 | 6 |

Pr_{3} | 7 | 5 | 5 |

Pr_{2} | 5 | 7 | 6 |

Pr_{1} | 12 | 12 | 12 |

Wi_{6} | 7 | 7 | 7 |

Wi_{5} | 4 | 5 | 4 |

Wi_{4} | 7 | 6 | 5 |

Wi_{3} | 6 | 4 | 5 |

Wi_{2} | 5 | 7 | 8 |

Wi_{1} | 5 | 7 | 5 |

Te_{6} | 3 | 7 | 4 |

Te_{5} | 6 | 5 | 8 |

Te_{4} | 6 | 6 | 7 |

Te_{3} | 4 | 2 | 5 |

Te_{2} | 6 | 7 | 3 |

Te_{1} | 6 | 7 | 8 |

Hu_{6} | 6 | 10 | 7 |

Hu_{5} | 6 | 7 | 6 |

Hu_{4} | 6 | 3 | 6 |

Hu_{3} | 4 | 3 | 3 |

Hu_{2} | 6 | 5 | 7 |

Hu_{1} | 7 | 6 | 5 |

Ht_{o} | 7 | 9 | 8 |

Sr. No. | Error Measures | Methods | Season | Average | ||
---|---|---|---|---|---|---|

Winter (22–28 August 2015) | Spring (22–28 October 2015) | Summer (22–28 January 2016) | ||||

1 | Mean Absolute Percentage Error (MAPE) | J48 | 12.45 | 9.30 | 10.42 | 10.72 |

J48+ FS | 11.40 | 8.41 | 8.86 | 9.56 | ||

Bagging | 7.86 | 7.67 | 8.61 | 8.05 | ||

Bagging+ FS | 7.98 | 7.31 | 8.43 | 7.91 | ||

M5P | 6.40 | 6.12 | 7.68 | 6.73 | ||

M5P+ FS | 6.05 | 5.77 | 6.47 | 6.10 | ||

SMO regression | 5.25 | 5.73 | 5.35 | 5.44 | ||

SMO regression+ FS | 4.85 | 5.26 | 5.28 | 5.13 | ||

2 | Mean Absolute Error (MAE) | J48 | 4.09 | 3.34 | 3.88 | 3.77 |

J48+ FS | 3.74 | 3.07 | 3.29 | 3.37 | ||

Bagging | 2.62 | 2.76 | 3.24 | 2.87 | ||

Bagging+ FS | 2.68 | 2.57 | 3.19 | 2.81 | ||

M5P | 2.14 | 2.19 | 3.65 | 2.66 | ||

M5P+ FS | 2.03 | 2.07 | 2.74 | 2.28 | ||

SMO regression | 1.75 | 1.99 | 2.11 | 1.95 | ||

SMO regression+ FS | 1.60 | 1.82 | 2.07 | 1.83 | ||

3 | Root Mean Square Error (RMSE) | J48 | 5.37 | 4.35 | 6.91 | 5.55 |

J48+ FS | 4.96 | 4.05 | 5.20 | 4.74 | ||

Bagging | 3.47 | 3.67 | 5.10 | 4.08 | ||

Bagging+ FS | 3.55 | 3.54 | 5.12 | 4.07 | ||

M5P | 3.56 | 2.96 | 13.06 | 6.53 | ||

M5P+ FS | 3.04 | 2.86 | 6.83 | 4.24 | ||

SMO regression | 2.62 | 2.75 | 4.07 | 3.15 | ||

SMO regression+ FS | 2.39 | 2.49 | 4.00 | 2.96 | ||

4 | Error Variance (EV) | J48 | 0.0102 | 0.0062 | 0.0271 | 0.0145 |

J48+ FS | 0.009 | 0.0055 | 0.0134 | 0.0093 | ||

Bagging | 0.0044 | 0.0046 | 0.0129 | 0.0073 | ||

Bagging+ FS | 0.0046 | 0.0047 | 0.0133 | 0.0075 | ||

M5P | 0.0068 | 0.0032 | 0.1307 | 0.0469 | ||

M5P+ FS | 0.0043 | 0.0031 | 0.0325 | 0.0133 | ||

SMO regression | 0.0032 | 0.0029 | 0.01 | 0.0054 | ||

SMO regression+ FS | 0.0027 | 0.0023 | 0.0097 | 0.0049 |

Sr. No. | Method | Average MAPE | Improvement (%) |

1. | SMO regression+ FS | 5.13 | - |

2. | J48 | 10.72 | 52.16 |

3. | J48+ FS | 9.55 | 46.32 |

4. | Bagging | 8.086 | 36.27 |

5. | Bagging+ FS | 7.86 | 35.15 |

6. | M5P | 6.73 | 23.77 |

7. | M5P+ FS | 6.09 | 15.90 |

8. | SMO regression | 5.44 | 5.75 |

Sr. No. | Method | Average MAE | Improvement (%) |

1. | SMO regression+ FS | 1.83 | - |

2. | J48 | 3.76 | 51.33 |

3. | J48+ FS | 3.36 | 45.54 |

4. | Bagging | 2.87 | 36.24 |

5. | Bagging+ FS | 2.81 | 34.89 |

6. | M5P | 2.66 | 31.20 |

7. | M5P+ FS | 2.28 | 19.74 |

8. | SMO regression | 1.95 | 6.15 |

Sr. No. | Method | Average RMSE | Improvement (%) |

1. | SMO regression+ FS | 2.96 | - |

2. | M5P | 6.53 | 54.68 |

3. | J48 | 5.55 | 46.64 |

4. | J48+ FS | 4.74 | 37.51 |

5. | M5P+ FS | 4.24 | 30.23 |

6. | Bagging | 4.08 | 27.44 |

7. | Bagging+ FS | 4.07 | 27.29 |

8. | SMO regression | 3.15 | 5.97 |

Sr. No. | Days | Winter (22–28 August 2015) | Spring (22–28 October 2015) | Summer (22–28 January 2016) | |||
---|---|---|---|---|---|---|---|

SMO Reg | SMO Reg+ FS | SMO Reg | SMO Reg+ FS | SMO Reg | SMO Reg+ FS | ||

1 | Day1 | 5.49 | 5.32 | 4.60 | 4.18 | 10.96 | 10.72 |

2 | Day2 | 4.32 | 4.21 | 6.24 | 5.69 | 4.86 | 4.85 |

3 | Day3 | 6.74 | 6.62 | 5.30 | 4.58 | 3.1 | 3.01 |

4 | Day4 | 5.18 | 4.36 | 5.94 | 5.72 | 4.37 | 4.20 |

5 | Day5 | 6.51 | 5.80 | 5.85 | 5.50 | 3.3 | 3.29 |

6 | Day6 | 4.48 | 3.82 | 5.92 | 5.47 | 4.64 | 4.62 |

7 | Day7 | 4.01 | 3.80 | 6.27 | 5.66 | 6.27 | 6.24 |

Average | 5.25 | 4.85 | 5.73 | 5.26 | 5.35 | 5.28 |

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## Share and Cite

**MDPI and ACS Style**

Srivastava, A.K.; Pandey, A.S.; Elavarasan, R.M.; Subramaniam, U.; Mekhilef, S.; Mihet-Popa, L.
A Novel Hybrid Feature Selection Method for Day-Ahead Electricity Price Forecasting. *Energies* **2021**, *14*, 8455.
https://doi.org/10.3390/en14248455

**AMA Style**

Srivastava AK, Pandey AS, Elavarasan RM, Subramaniam U, Mekhilef S, Mihet-Popa L.
A Novel Hybrid Feature Selection Method for Day-Ahead Electricity Price Forecasting. *Energies*. 2021; 14(24):8455.
https://doi.org/10.3390/en14248455

**Chicago/Turabian Style**

Srivastava, Ankit Kumar, Ajay Shekhar Pandey, Rajvikram Madurai Elavarasan, Umashankar Subramaniam, Saad Mekhilef, and Lucian Mihet-Popa.
2021. "A Novel Hybrid Feature Selection Method for Day-Ahead Electricity Price Forecasting" *Energies* 14, no. 24: 8455.
https://doi.org/10.3390/en14248455