# Statistical and Artificial Neural Networks Models for Electricity Consumption Forecasting in the Brazilian Industrial Sector

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Statistical Models

#### 2.1.1. Holt–Winters Method

#### 2.1.2. SARIMA

#### 2.1.3. Dynamic Linear Model

#### 2.1.4. Trignometric Box–Cox Transform, ARMA Errors, Trend, and Seasonal Components (TBATS)

#### 2.2. Artificial Neural Networks Approach

#### 2.2.1. Autoregressive Neural Networks (NNAR)

#### 2.2.2. Multilayer Perceptron (MLP)

#### 2.3. Mean Absolute Percentage Error (MAPE)

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Year | Mean | Variance | St. Dev. | Amplitude | Min. | Max. |
---|---|---|---|---|---|---|

1979 | 4616.83 | 70,024.88 | 264.62 | 725.00 | 4215.00 | 4940.00 |

1980 | 5123.83 | 65,639.24 | 256.20 | 699.00 | 4806.00 | 5505.00 |

1981 | 5095.83 | 11,991.61 | 109.51 | 329.00 | 4906.00 | 5235.00 |

1982 | 5324.08 | 74,528.27 | 273.00 | 809.00 | 4836.00 | 5645.00 |

1983 | 5669.92 | 142,872.81 | 377.99 | 1076.00 | 4994.00 | 6070.00 |

1984 | 6704.58 | 254,915.17 | 504.89 | 1578.00 | 5834.00 | 7412.00 |

1985 | 7570.00 | 100,401.82 | 316.86 | 890.00 | 7084.00 | 7974.00 |

1986 | 8094.83 | 286,936.52 | 535.66 | 1529.00 | 7190.00 | 8719.00 |

1987 | 8116.92 | 67,643.36 | 260.08 | 797.00 | 7749.00 | 8546.00 |

1988 | 8377.25 | 45,443.66 | 213.18 | 695.00 | 8073.00 | 8768.00 |

1989 | 8583.08 | 246,853.36 | 496.84 | 1705.00 | 7595.00 | 9300.00 |

1990 | 8322.58 | 262,576.99 | 512.42 | 1949.00 | 7145.00 | 9094.00 |

1991 | 8550.08 | 391,295.36 | 625.54 | 1814.00 | 7466.00 | 9280.00 |

1992 | 8610.58 | 93,964.45 | 306.54 | 1139.00 | 7953.00 | 9092.00 |

1993 | 8915.08 | 145,204.81 | 381.06 | 1083.00 | 8290.00 | 9373.00 |

1994 | 8921.92 | 136,503.90 | 369.46 | 1082.00 | 8368.00 | 9450.00 |

1995 | 9305.50 | 44,580.09 | 211.14 | 577.00 | 9070.00 | 9647.00 |

1996 | 9709.67 | 217,869.70 | 466.77 | 1637.00 | 8753.00 | 10,390.00 |

1997 | 10,143.08 | 185,066.81 | 430.19 | 1272.00 | 9455.00 | 10,727.00 |

1998 | 10,164.83 | 152,053.79 | 389.94 | 1178.00 | 9545.00 | 10,723.00 |

1999 | 10,324.33 | 300,464.79 | 548.15 | 1607.00 | 9257.00 | 10,864.00 |

2000 | 10,940.00 | 163,874.18 | 404.81 | 1398.00 | 10,024.00 | 11,422.00 |

2001 | 10,211.50 | 609,427.18 | 780.66 | 2160.00 | 9178.00 | 11,338.00 |

2002 | 10,635.67 | 247,841.33 | 497.84 | 1609.00 | 9431.00 | 11,040.00 |

2003 | 10,852.67 | 114,157.70 | 337.87 | 1186.00 | 10,345.00 | 11,531.00 |

2004 | 12,846.83 | 291,560.70 | 539.96 | 1585.00 | 11,829.00 | 13,414.00 |

2005 | 13,217.33 | 133,968.42 | 366.02 | 1105.00 | 12,496.00 | 13,601.00 |

2006 | 13,598.42 | 149,381.17 | 386.50 | 1313.00 | 12,851.00 | 14,164.00 |

2007 | 14,530.67 | 222,010.42 | 471.18 | 1433.00 | 13,592.00 | 15,025.00 |

2008 | 14,652.83 | 404,200.70 | 635.77 | 1995.00 | 13,417.00 | 15,412.00 |

2009 | 13,483.17 | 710,177.42 | 842.72 | 2628.00 | 11,924.00 | 14,552.00 |

2010 | 14,956.58 | 380,298.81 | 616.68 | 2031.00 | 13,425.00 | 15,456.00 |

2011 | 15,298.00 | 187,278.18 | 432.76 | 1386.00 | 14,467.00 | 15,853.00 |

2012 | 15,285.42 | 112,891.36 | 335.99 | 1061.00 | 14,567.00 | 15,628.00 |

2013 | 15,390.25 | 197,482.39 | 444.39 | 1516.00 | 14,370.00 | 15,886.00 |

2014 | 14,925.42 | 68,056.63 | 260.88 | 723.00 | 14,537.00 | 15,260.00 |

2015 | 14,071.50 | 127,615.73 | 357.23 | 1238.00 | 13,327.00 | 14,565.00 |

2016 | 13,687.75 | 174,712.57 | 417.99 | 1598.00 | 12,538.00 | 14,136.00 |

2017 | 13,903.92 | 138,837.36 | 372.61 | 1211.00 | 13,105.00 | 14,316.00 |

2018 | 14,121.92 | 119,368.63 | 345.50 | 1014.00 | 13,525.00 | 14,539.00 |

2019 | 13,858.17 | 71,473.42 | 267.35 | 864.00 | 13,442.00 | 14,306.00 |

2020 | 13,802.42 | 1,019,275.90 | 1009.59 | 2936.00 | 12,173.00 | 15,109.00 |

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**Figure 1.**Electricity consumption (

**a**) in GWh and (

**b**) box-plots for the Brazilian industry. Source: Central Bank of Brazil.

**Figure 3.**Model fit for the six considered models applied to the Brazilian industrial electricity consumption.

**Figure 4.**Model forecasting for the six considered models applied to the Brazilian industrialelectricity consumption.

**Figure 5.**Mean absolute percentage error, considering the six models, for $h=1,\dots ,24$ steps-ahead out-of-sample forecasts applied to the electricity consumption of the Brazilian industry.

Equations | Additive Method | Multiplicative Method |
---|---|---|

Level (${\ell}_{t}$) | $\alpha \left({y}_{t}-{s}_{t-m}\right)+(1-\alpha )\left({\ell}_{t-1}+{b}_{t-1}\right)$ | $\alpha \frac{{y}_{t}}{{s}_{t-m}}+(1-\alpha )\left({\ell}_{t-1}+{b}_{t-1}\right)$ |

Trend (${b}_{t}$) | $\beta \left({\ell}_{t}-{\ell}_{t-1}\right)+(1-\beta ){b}_{t-1}$ | $\beta \left({\ell}_{t}-{\ell}_{t-1}\right)+(1-\beta ){b}_{t-1}$ |

Seasonal (${s}_{t}$) | $\gamma \left({y}_{t}-{\ell}_{t-1}-{b}_{t-1}\right)+(1-\gamma ){s}_{t-m}$ | $\gamma \frac{{y}_{t}}{\left({\ell}_{t-1}+{b}_{t-1}\right)}+(1-\gamma ){s}_{t-m}$ |

Forecast (${\widehat{y}}_{t+h\mid t}$) | ${\ell}_{t}+h{b}_{t}+{s}_{t+h-m(k+1)}$ | $\left({\ell}_{t}+h{b}_{t}\right){s}_{t+h-m(k+1)}$ |

**Table 2.**Mean absolute percentage error for the six models under consideration for model fit and model forecasting considering the training and testing data, respectively.

Model | Fitted | Forecast |
---|---|---|

Holt–Winters | 2.51 | 4.09 |

SARIMA | 1.88 | 6.17 |

TBATS | 1.99 | 3.77 |

DLM | 1.87 | 4.09 |

NNAR | 2.40 | 4.77 |

MLP | 1.48 | 3.41 |

**Table 3.**Mean absolute percentage error, considering the six models, for $h=1,\dots ,24$ steps-ahead out-of-sample forecasts applied to the electricity consumption of the Brazilian industry.

Step | HW | SARIMA | TBATS | DLM | NNAR | MLP |
---|---|---|---|---|---|---|

1 | 0.27 | 0.86 | 0.50 | 2.28 | 3.29 | 0.96 |

2 | 0.34 | 1.56 | 0.95 | 2.67 | 3.46 | 1.96 |

3 | 1.00 | 1.97 | 0.95 | 2.62 | 2.96 | 1.82 |

4 | 0.84 | 3.08 | 1.71 | 3.06 | 3.25 | 2.49 |

5 | 1.11 | 2.77 | 1.46 | 2.53 | 2.81 | 2.17 |

6 | 1.43 | 2.86 | 1.76 | 2.65 | 2.91 | 2.03 |

7 | 1.53 | 2.96 | 1.79 | 2.45 | 2.79 | 1.82 |

8 | 1.56 | 3.22 | 1.92 | 2.29 | 2.65 | 1.75 |

9 | 2.03 | 3.65 | 2.29 | 2.36 | 2.85 | 1.81 |

10 | 1.93 | 3.62 | 2.13 | 2.18 | 2.66 | 1.73 |

11 | 2.06 | 3.80 | 2.20 | 2.13 | 2.69 | 1.63 |

12 | 2.44 | 4.12 | 2.37 | 2.33 | 3.10 | 1.62 |

13 | 2.54 | 4.04 | 2.29 | 2.24 | 3.42 | 1.53 |

14 | 2.43 | 4.06 | 2.15 | 2.24 | 3.54 | 1.45 |

15 | 2.37 | 4.10 | 2.04 | 2.18 | 3.48 | 1.41 |

16 | 3.13 | 5.29 | 2.97 | 3.15 | 4.44 | 2.44 |

17 | 3.86 | 6.21 | 3.73 | 3.93 | 5.25 | 3.24 |

18 | 4.52 | 6.86 | 4.28 | 4.48 | 5.83 | 3.55 |

19 | 4.57 | 6.91 | 4.21 | 4.38 | 5.76 | 3.38 |

20 | 4.40 | 6.83 | 4.00 | 4.22 | 5.47 | 3.31 |

21 | 4.28 | 6.69 | 3.85 | 4.15 | 5.25 | 3.26 |

22 | 4.28 | 6.43 | 3.89 | 4.21 | 5.18 | 3.39 |

23 | 4.15 | 6.34 | 3.77 | 4.12 | 4.98 | 3.39 |

24 | 4.00 | 6.17 | 3.78 | 4.09 | 4.77 | 3.42 |

Average | 2.54 | 4.35 | 2.54 | 3.04 | 3.87 | 2.32 |

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## Share and Cite

**MDPI and ACS Style**

Leite Coelho da Silva, F.; da Costa, K.; Canas Rodrigues, P.; Salas, R.; López-Gonzales, J.L. Statistical and Artificial Neural Networks Models for Electricity Consumption Forecasting in the Brazilian Industrial Sector. *Energies* **2022**, *15*, 588.
https://doi.org/10.3390/en15020588

**AMA Style**

Leite Coelho da Silva F, da Costa K, Canas Rodrigues P, Salas R, López-Gonzales JL. Statistical and Artificial Neural Networks Models for Electricity Consumption Forecasting in the Brazilian Industrial Sector. *Energies*. 2022; 15(2):588.
https://doi.org/10.3390/en15020588

**Chicago/Turabian Style**

Leite Coelho da Silva, Felipe, Kleyton da Costa, Paulo Canas Rodrigues, Rodrigo Salas, and Javier Linkolk López-Gonzales. 2022. "Statistical and Artificial Neural Networks Models for Electricity Consumption Forecasting in the Brazilian Industrial Sector" *Energies* 15, no. 2: 588.
https://doi.org/10.3390/en15020588