# Energy Consumption Forecasting for Smart Meters Using Extreme Learning Machine Ensemble

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## Abstract

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## 1. Introduction

- The diversity of the ensemble is introduced by the employment of different forecasting methods such as autoregressive (AR), multilayer perceptron (MLP), extreme learning machine (ELM), radial basis function (RBF), and echo state network (ESN).
- The combination step employs an ELM model in order to map nonlinear relations between forecasts and to perform more accurate combinations.
- The proposed method is versatile, since different forecasting methods can be used in the pool, and then combined by the ELM.

## 2. Related Work

## 3. Proposed Ensemble Method

#### 3.1. Single Model: Autoregressive Model

#### 3.2. Single Model: Multilayer Perceptron (MLP)

#### 3.3. Single Model: Echo State Networks (ESN)

#### 3.4. Single Model: Radial Basis Function Network (RBF)

#### 3.5. Single Model: Extreme Learning Machine (ELM)

## 4. Experimental Evaluation

#### 4.1. Data Description

#### 4.2. Preprocessing and Postprocessing Stages

#### 4.3. Experimental Setup

- The coefficients of the AR model were calculated using the Yule–Walker equations, a closed-form solution [54];
- All artificial neural networks used hyperbolic tangent as activation function of the hidden neurons [59];
- The number of neurons in the hidden layer was determined by previous empirical tests, considering a range of [3:500].
- All models were implemented in Matlab
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#### 4.4. Error Metrics

#### 4.5. Results

#### 4.6. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Stages of preprocessing and postprocessing employed in the modeling of the forecasting method.

Set | Number of Samples | Mean (kWh) | Standard Deviation |
---|---|---|---|

Whole Series | 2880 | 0.20077 | 0.10115 |

Training | 1824 | 0.20794 | 0.10238 |

Validation | 384 | 0.19789 | 0.10065 |

Test | 672 | 0.18296 | 0.09579 |

**Table 2.**The performance results in terms of the MSE, MAE, MAPE, RMSE, and IA metrics of the proposed Ensemble and literature models for each day of the week. The best values are highlighted in bold.

Model | Measure | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday | Max | Min | |
---|---|---|---|---|---|---|---|---|---|---|---|

Single | SARIMA | MSE ($\times {10}^{-3}$ kWh) | 2.8720 | 2.7614 | 3.3822 | 2.9381 | 1.7513 | 2.0152 | 2.2561 | 3.3822 | 1.7513 |

MAE (kWh) | 0.0325 | 0.0332 | 0.0359 | 0.0353 | 0.0217 | 0.0288 | 0.0246 | 0.0359 | 0.0217 | ||

MAPE (%) | 15.8363 | 14.2251 | 17.1321 | 19.2733 | 13.2665 | 16.0358 | 14.9922 | 19.2733 | 13.2665 | ||

RMSE (kWh) | 0.0535 | 0.0525 | 0.0581 | 0.0542 | 0.0418 | 0.0448 | 0.0474 | 0.0581 | 0.0418 | ||

IA | 0.8400 | 0.8991 | 0.9272 | 0.8162 | 0.8970 | 0.9296 | 0.9419 | 0.9419 | 0.8162 | ||

AR | MSE ($\times {10}^{-3}$ kWh) | 1.1708 | 1.8802 | 1.9812 | 1.8478 | 2.2187 | 3.4516 | 2.2879 | 3.4516 | 1.1708 | |

MAE (kWh) | 0.0225 | 0.0297 | 0.0234 | 0.0300 | 0.0281 | 0.0355 | 0.0337 | 0.0355 | 0.0225 | ||

MAPE (%) | 14.3075 | 17.0181 | 14.9965 | 16.1628 | 11.8905 | 15.8668 | 19.7286 | 19.7286 | 11.8905 | ||

RMSE (kWh) | 0.0342 | 0.0434 | 0.0445 | 0.0430 | 0.0471 | 0.0588 | 0.0478 | 0.0588 | 0.0342 | ||

IA | 0.9355 | 0.9304 | 0.9568 | 0.9027 | 0.9202 | 0.9285 | 0.8827 | 0.9568 | 0.8827 | ||

MLP | MSE ($\times {10}^{-3}$ kWh) | 1.1413 | 1.7036 | 1.8264 | 1.7168 | 2.1059 | 3.2103 | 1.9979 | 3.2103 | 1.1413 | |

MAE (kWh) | 0.0217 | 0.0282 | 0.0216 | 0.0286 | 0.0289 | 0.0332 | 0.0304 | 0.0332 | 0.0216 | ||

MAPE (%) | 13.5270 | 15.7165 | 12.6500 | 15.4307 | 12.3867 | 14.9471 | 17.9853 | 17.9853 | 12.3867 | ||

RMSE (kWh) | 0.0338 | 0.0413 | 0.0427 | 0.0414 | 0.0459 | 0.0567 | 0.0447 | 0.0567 | 0.0338 | ||

IA | 0.9367 | 0.9375 | 0.9583 | 0.9100 | 0.9260 | 0.9325 | 0.9000 | 0.9583 | 0.9000 | ||

ELM | MSE ($\times {10}^{-3}$ kWh) | 1.1526 | 1.6701 | 1.7890 | 1.7343 | 2.0591 | 3.1423 | 1.8294 | 3.1423 | 1.1526 | |

MAE (kWh) | 0.0209 | 0.0271 | 0.0222 | 0.0285 | 0.0292 | 0.0329 | 0.0287 | 0.0329 | 0.0209 | ||

MAPE (%) | 12.7378 | 15.2818 | 13.9267 | 15.4566 | 12.6752 | 14.6467 | 17.0593 | 17.0593 | 12.6752 | ||

RMSE (kWh) | 0.0340 | 0.0409 | 0.0423 | 0.0416 | 0.0454 | 0.0561 | 0.0428 | 0.0561 | 0.0340 | ||

IA | 0.9356 | 0.9383 | 0.9594 | 0.9091 | 0.9284 | 0.9322 | 0.9079 | 0.9594 | 0.9079 | ||

ESN | MSE ($\times {10}^{-3}$ kWh) | 1.1806 | 1.5424 | 1.7851 | 1.7948 | 1.9928 | 3.3204 | 2.0896 | 3.3204 | 1.1806 | |

MAE (kWh) | 0.0213 | 0.0269 | 0.0220 | 0.0293 | 0.0279 | 0.0336 | 0.0305 | 0.0336 | 0.0213 | ||

MAPE (%) | 12.9235 | 15.4666 | 13.9666 | 15.9643 | 11.9069 | 14.9907 | 18.2586 | 18.2586 | 11.9069 | ||

RMSE (kWh) | 0.0344 | 0.0393 | 0.0423 | 0.0424 | 0.0446 | 0.0576 | 0.0457 | 0.0576 | 0.0344 | ||

IA | 0.9345 | 0.9460 | 0.9605 | 0.9044 | 0.9320 | 0.9320 | 0.8964 | 0.9605 | 0.8964 | ||

RBF | MSE ($\times {10}^{-3}$ kWh) | 1.7832 | 1.7691 | 3.0669 | 2.1245 | 2.2380 | 3.4564 | 2.5668 | 3.4564 | 1.7691 | |

MAE (kWh) | 0.0261 | 0.0287 | 0.0326 | 0.0313 | 0.0313 | 0.0368 | 0.0322 | 0.0368 | 0.0261 | ||

MAPE (%) | 15.2994 | 15.4352 | 22.0301 | 17.4932 | 13.7047 | 17.3610 | 20.1081 | 22.0301 | 13.7047 | ||

RMSE (kWh) | 0.0422 | 0.0421 | 0.0554 | 0.0461 | 0.0473 | 0.0588 | 0.0507 | 0.0588 | 0.0421 | ||

IA | 0.8957 | 0.9364 | 0.9243 | 0.8828 | 0.9245 | 0.9242 | 0.8633 | 0.9364 | 0.8633 | ||

Ensemble | Ensemble Mean | MSE ($\times {10}^{-3}$ kWh) | 1.1632 | 1.6345 | 1.8150 | 1.7466 | 2.0379 | 3.1300 | 1.9460 | 3.1300 | 1.1632 |

MAE (kWh) | 0.0216 | 0.0277 | 0.0220 | 0.0289 | 0.0282 | 0.0325 | 0.0300 | 0.0325 | 0.0216 | ||

MAPE (%) | 13.0506 | 15.4767 | 13.5962 | 15.7273 | 12.1594 | 14.2690 | 17.7973 | 17.7973 | 12.1594 | ||

RMSE (kWh) | 0.0341 | 0.0404 | 0.0426 | 0.0418 | 0.0451 | 0.0559 | 0.0441 | 0.0559 | 0.0341 | ||

IA | 0.9342 | 0.9404 | 0.9583 | 0.9061 | 0.9289 | 0.9336 | 0.8992 | 0.9583 | 0.8992 | ||

Ensemble Median | MSE ($\times {10}^{-3}$ kWh) | 1.1378 | 1.6290 | 1.8093 | 1.7949 | 2.0108 | 3.1865 | 1.9856 | 3.1865 | 1.1378 | |

MAE (kWh) | 0.0215 | 0.0279 | 0.0214 | 0.0294 | 0.0281 | 0.0332 | 0.0303 | 0.0332 | 0.0214 | ||

MAPE (%) | 13.2260 | 15.7893 | 13.1504 | 15.9486 | 12.1569 | 14.8732 | 17.9705 | 17.9705 | 12.1569 | ||

RMSE (kWh) | 0.0337 | 0.0404 | 0.0425 | 0.0424 | 0.0448 | 0.0564 | 0.0446 | 0.0564 | 0.0337 | ||

IA | 0.9365 | 0.9409 | 0.9593 | 0.9046 | 0.9291 | 0.9329 | 0.8992 | 0.9593 | 0.8992 | ||

Ensemble MLP | MSE ($\times {10}^{-3}$ kWh) | 1.1588 | 1.5856 | 1.7507 | 1.7038 | 1.9816 | 3.0892 | 1.7540 | 3.0892 | 1.1588 | |

MAE (kWh) | 0.0210 | 0.0266 | 0.0219 | 0.0278 | 0.0292 | 0.0319 | 0.0275 | 0.0319 | 0.0210 | ||

MAPE (%) | 12.7435 | 14.9182 | 13.5774 | 15.2293 | 12.6736 | 14.1799 | 16.2368 | 16.2368 | 12.6736 | ||

RMSE (kWh) | 0.0340 | 0.0398 | 0.0418 | 0.0413 | 0.0445 | 0.0556 | 0.0419 | 0.0556 | 0.0340 | ||

IA | 0.9357 | 0.9428 | 0.9592 | 0.9095 | 0.9300 | 0.9328 | 0.9117 | 0.9592 | 0.9095 | ||

Ensemble ELM | MSE ($\times {10}^{-3}$ kWh) | 1.2162 | 1.7898 | 1.5592 | 1.7071 | 1.9356 | 3.4034 | 1.5598 | 3.4034 | 1.2162 | |

MAE (kWh) | 0.0217 | 0.0261 | 0.0203 | 0.0278 | 0.0303 | 0.0296 | 0.0243 | 0.0303 | 0.0203 | ||

MAPE (%) | 12.5109 | 15.0994 | 12.1248 | 14.8702 | 13.2287 | 13.0881 | 13.8786 | 15.0994 | 12.1248 | ||

RMSE (kWh) | 0.0349 | 0.0423 | 0.0395 | 0.0413 | 0.0440 | 0.0583 | 0.0395 | 0.0583 | 0.0349 | ||

IA | 0.9321 | 0.9382 | 0.9624 | 0.9124 | 0.9300 | 0.9249 | 0.9263 | 0.9624 | 0.9124 |

**Table 3.**MSE, MAE, MAPE, RMSE, and IA values for the evaluated models. The number of neurons used by each neural network is shown in the NN column. The performance corresponds to the whole test set of the energy consumption series. The best value for each metric is highlighted in bold.

Model | NN | MSE ($\times {10}^{-3}$ kWh) | MAE (kWh) | MAPE (%) | RMSE (kWh) | IA | |
---|---|---|---|---|---|---|---|

Single | SARIMA | - | 2.5675 | 0.0303 | 15.4004 | 0.0506 | 0.9129 |

AR | - | 2.1195 | 0.0290 | 15.7090 | 0.0460 | 0.9318 | |

MLP | 200 | 1.9574 | 0.0275 | 14.5376 | 0.0442 | 0.9391 | |

ELM | 120 | 1.9110 | 0.0271 | 14.5393 | 0.0437 | 0.9405 | |

ESN | 40 | 1.9579 | 0.0274 | 14.7819 | 0.0442 | 0.9402 | |

RBF | 60 | 2.4292 | 0.0310 | 17.1017 | 0.0493 | 0.9226 | |

Ensemble | Ensemble Mean | - | 1.9247 | 0.0273 | 14.5826 | 0.0439 | 0.9373 |

Ensemble Median | - | 1.9363 | 0.0274 | 14.7307 | 0.0440 | 0.9375 | |

Ensemble MLP | 40 | 2.1671 | 0.0284 | 14.2228 | 0.0466 | 0.9358 | |

Ensemble ELM | 60 | 1.8817 | 0.0257 | 13.5424 | 0.0434 | 0.9410 |

**Table 4.**p-values of the Wilcoxon statistical test comparing the Ensemble ELM and ELM with the other forecasting models.

Models | p-Value (Ensemble ELM) | p-Value (ELM) |
---|---|---|

Ensemble ELM | — | 0.0013 |

ELM | 0.0013 | — |

SARIMA | 1.21$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 1.21$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ |

AR | 7.47$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ | 0.0045 |

MLP | 0.0241 | 0.0323 |

ESN | 1.72$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | 0.0478 |

RBF | 3.01$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 3.01$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ |

Ensemble Mean | 1.91$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 3.35$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ |

Ensemble Median | 2.05$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.35$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ |

Ensemble MLP | 8.48$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 1.35$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

Model | MSE (kWh) | MAE (kWh) | MAPE (%) | RMSE (kWh) | IA | Mean | Rank | |
---|---|---|---|---|---|---|---|---|

Single | SARIMA | 10 | 9 | 8 | 10 | 10 | 9.4 | 9 |

AR | 7 | 8 | 9 | 7 | 8 | 7.6 | 8 | |

MLP | 5 | 6 | 3 | 5 | 4 | 4.6 | 5 | |

ELM | 2 | 2 | 4 | 2 | 2 | 2.4 | 2 | |

ESN | 6 | 4 | 7 | 6 | 3 | 5.2 | 6 | |

RBF | 9 | 10 | 10 | 9 | 9 | 9.4 | 9 | |

Ensemble | Ensemble Mean | 3 | 3 | 5 | 3 | 6 | 4 | 3 |

Ensemble Median | 4 | 5 | 6 | 4 | 5 | 4.8 | 4 | |

Ensemble MLP | 8 | 7 | 2 | 8 | 7 | 6.4 | 7 | |

Ensemble ELM | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

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## Share and Cite

**MDPI and ACS Style**

de Mattos Neto, P.S.G.; de Oliveira, J.F.L.; Bassetto, P.; Siqueira, H.V.; Barbosa, L.; Alves, E.P.; Marinho, M.H.N.; Rissi, G.F.; Li, F.
Energy Consumption Forecasting for Smart Meters Using Extreme Learning Machine Ensemble. *Sensors* **2021**, *21*, 8096.
https://doi.org/10.3390/s21238096

**AMA Style**

de Mattos Neto PSG, de Oliveira JFL, Bassetto P, Siqueira HV, Barbosa L, Alves EP, Marinho MHN, Rissi GF, Li F.
Energy Consumption Forecasting for Smart Meters Using Extreme Learning Machine Ensemble. *Sensors*. 2021; 21(23):8096.
https://doi.org/10.3390/s21238096

**Chicago/Turabian Style**

de Mattos Neto, Paulo S. G., João F. L. de Oliveira, Priscilla Bassetto, Hugo Valadares Siqueira, Luciano Barbosa, Emilly Pereira Alves, Manoel H. N. Marinho, Guilherme Ferretti Rissi, and Fu Li.
2021. "Energy Consumption Forecasting for Smart Meters Using Extreme Learning Machine Ensemble" *Sensors* 21, no. 23: 8096.
https://doi.org/10.3390/s21238096