# Application of Discrete-Interval Moving Seasonalities to Spanish Electricity Demand Forecasting during Easter

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## Abstract

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## 1. Introduction

## 2. Forecasting for Special Events

## 3. Materials and Methods

#### 3.1. Holt–Winters Models with Discrete Interval Moving Seasonalities

#### 3.1.1. Obtaining Seeds of the Model

#### 3.1.2. Parameter Optimisation and Forecast

- The first consists in determining all parameters simultaneously. Chatfield [39] studied the optimisation of the parameters separately and jointly. He concluded that the best solution is to accomplish the joint optimisation of the parameters.
- The second consists in accomplishing an optimisation of the regular model and subsequently optimising the parameters of the DIMS. Although this contradicts the previous point of view, no differences in precision are found, but there are differences in optimisation time.

#### 3.2. A Case Study: Easter

## 4. Results and Discussion

#### 4.1. Analysis of Results

#### 4.2. Comparison

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Comparison of the effect produced by a special event (Easter in this case) on the hourly time series of electricity demand in Spain.

**Figure 2.**Flow chart for the utilisation of the discrete-interval moving seasonality (DIMS). Initialisation is found at the left side of the figure. The right side explains the parameter optimisation and forecasts.

**Figure 3.**Hourly electricity demand in Spain in the weeks around Easter. It can be seen that the dates of Easter vary every year and that a decrease in consumption always takes place during the Easter week. Furthermore, this descent has a similar behaviour every year.

**Figure 4.**Comparison of 24-h-ahead forecasts obtained by the regular method and the DIMS method. The abscissa represents the lead time in hours from the start of the forecast (SoF) to 24 h ahead. The black line represents the real (observed values) with which the forecasts are compared. The blue line is the forecast obtained by the regular method; the red line is the forecast obtained by the new DIMS method.

**Figure 6.**Comparison of MAPE for all methods. ANN is the artificial neural network; TBATS stands for the general Exponential Smoothing State Space Model with Box-Cox Transformation, ARMA Errors, Trend and Seasonal Components (TBATS) method, and TBATS JASA stands for TBATS with regressors. $AM{C}_{24,168}$ is the regular nHWT method, and $AM{C}_{24,168,Easter}$ stands for nHWT with DIMS.

**Table 1.**Estimate of the weights of the values of the series on the rebuilt time series. In the columns each appearance of the discrete-interval moving seasonality (DIMS) is shown, and in the rows every time instant within the discrete seasonality.

1 | 2 | 3 | … | n | |
---|---|---|---|---|---|

1 | $\frac{{X}_{{p}_{1}}}{{R}_{{p}_{1}}}$ | $\frac{{X}_{{p}_{2}}}{{R}_{{p}_{2}}}$ | $\frac{{X}_{{p}_{3}}}{{R}_{{p}_{3}}}$ | … | $\frac{{X}_{{p}_{n}}}{{R}_{{p}_{n}}}$ |

2 | $\frac{{X}_{{p}_{1}+1}}{{R}_{{p}_{1}+1}}$ | $\frac{{X}_{{p}_{2}+1}}{{R}_{{p}_{2}+1}}$ | $\frac{{X}_{{p}_{3}+1}}{{R}_{{p}_{3}+1}}$ | … | $\frac{{X}_{{p}_{n}+1}}{{R}_{{p}_{n}+1}}$ |

… | … | … | … | … | |

${s}_{h}$ | $\frac{{X}_{{p}_{1}+{s}_{h}-1}}{{R}_{{p}_{1}+{s}_{h}-1}}$ | $\frac{{X}_{{p}_{2}+{s}_{h}-1}}{{R}_{{p}_{2}+{s}_{h}-1}}$ | $\frac{{X}_{{p}_{3}+{s}_{h}-1}}{{R}_{{p}_{3}+{s}_{h}-1}}$ | … | $\frac{{X}_{{p}_{n}+{s}_{h}-1}}{{R}_{{p}_{n}+{s}_{h}-1}}$ |

Year | Palm Sunday | Resurrection | ${\mathit{t}}_{\mathit{Easter}}^{*}$ | ${\mathit{s}}_{\mathit{Easter}}^{*}$ | |
---|---|---|---|---|---|

Sunday | Begin | End | |||

2008 | 16 March | 23 March | 1897 | 2017 | — |

2009 | 5 April | 12 April | 11,237 | 11,257 | 9220 |

2010 | 28 March | 4 April | 19,705 | 19,825 | 8448 |

2011 | 17 April | 24 April | 28,945 | 29,065 | 9120 |

2012 | 1 April | 8 April | 37,345 | 37,465 | 8280 |

2013 | 24 March | 31 March | 45,913 | 46,033 | 8448 |

2014 | 13 April | 20 April | 55,153 | 55,273 | 9120 |

2015 | 29 March | 5 April | 63,553 | 63,673 | 8280 |

Model | $\mathit{\alpha}$ | $\mathit{\gamma}$ | ${\mathit{\delta}}^{\left(24\right)}$ | ${\mathit{\delta}}^{\left(168\right)}$ | ${\mathit{\delta}}^{\left(\mathit{Easter}\right)}$ | ${\mathit{\phi}}_{\mathit{AR}}$ |
---|---|---|---|---|---|---|

$AM{C}_{24,168,Easter}$ | 0.0502 | 0.0001 | 0.3080 | 0.0614 | 0.0627 | 0.9178 |

$AA{C}_{24,168,Easter}$ | 0.0001 | 0.0071 | 0.2981 | 0.1127 | 0.1167 | 0.9658 |

Period | Starting at | Length |
---|---|---|

0 | Regular Model, no DIMS included | |

1 | Monday after Palm Sunday | 192 h |

2 | Holy Thursday | 120 h |

**Table 5.**Mean Absolute Percentage Error (MAPE) for $AM{C}_{24,168}$, $AM{C}_{24,168,Easter}$, $AA{C}_{24,168}$, and $AA{C}_{24,168,Easter}$ for different periods and optimisation methods.

MODEL | ${\mathit{AMC}}_{24,168}$ | ${\mathit{AMC}}_{24,168,\mathit{Easter}}$ | ${\mathit{AAC}}_{24,168}$ | ${\mathit{AAC}}_{24,168,\mathit{Easter}}$ | |||||
---|---|---|---|---|---|---|---|---|---|

VARIANT | CASE 0 | CASE 1 | CASE 2 | CASE 0 | CASE 1 | CASE 2 | |||

M2 | M1 | M2 | M1 | ||||||

Thursday | 17/04/2014 | 13.12 | 3.05 | 2.73 | 2.79 | 12.25 | 6.42 | 5.90 | 9.18 |

Friday | 18/04/2014 | 11.38 | 1.74 | 2.41 | 3.06 | 14.22 | 1.79 | 2.39 | 2.14 |

Saturday | 19/04/2014 | 10.97 | 1.68 | 1.44 | 1.58 | 11.34 | 1.40 | 1.28 | 1.09 |

Sunday | 20/04/2014 | 3.25 | 1.65 | 1.88 | 4.04 | 5.85 | 4.33 | 4.67 | 4.79 |

Monday | 21/04/2014 | 4.94 | 6.25 | 3.57 | 7.02 | 5.80 | 4.87 | 5.06 | 5.87 |

Thursday | 02/04/2015 | 11.35 | 2.90 | 2.79 | 3.07 | 12.02 | 5.47 | 4.81 | 8.80 |

Friday | 03/04/2015 | 10.66 | 3.44 | 1.55 | 2.62 | 12.82 | 0.97 | 1.13 | 2.44 |

Saturday | 04/04/2015 | 10.51 | 1.10 | 2.58 | 2.04 | 10.32 | 2.64 | 2.45 | 2.28 |

Sunday | 05/04/2015 | 3.60 | 2.56 | 1.53 | 2.47 | 4.86 | 3.11 | 3.44 | 2.88 |

Monday | 06/04/2015 | 5.91 | 1.51 | 3.93 | 4.69 | 7.40 | 2.84 | 3.04 | 2.96 |

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**MDPI and ACS Style**

Trull, Ó.; García-Díaz, J.C.; Troncoso, A.
Application of Discrete-Interval Moving Seasonalities to Spanish Electricity Demand Forecasting during Easter. *Energies* **2019**, *12*, 1083.
https://doi.org/10.3390/en12061083

**AMA Style**

Trull Ó, García-Díaz JC, Troncoso A.
Application of Discrete-Interval Moving Seasonalities to Spanish Electricity Demand Forecasting during Easter. *Energies*. 2019; 12(6):1083.
https://doi.org/10.3390/en12061083

**Chicago/Turabian Style**

Trull, Óscar, J. Carlos García-Díaz, and Alicia Troncoso.
2019. "Application of Discrete-Interval Moving Seasonalities to Spanish Electricity Demand Forecasting during Easter" *Energies* 12, no. 6: 1083.
https://doi.org/10.3390/en12061083