# Electricity Price Forecasting in the Danish Day-Ahead Market Using the TBATS, ANN and ARIMA Methods

^{*}

## Abstract

**:**

## 1. Introduction

#### Nordpool

## 2. Research

#### 2.1. Research Goals

#### 2.2. Methodology

#### 2.3. Forecasting Methodology

_{L}is the amount of water in reservoirs in Norway, S

_{L}is the amount of water in reservoirs in Sweden and F

_{L}is the amount of water in Finish reservoirs. N

_{C}, S

_{C}, and F

_{C}correspond to the total available water capacities in respective countries. Using the equation a value between 0 and 1 is reached, where 1 means that the reservoirs are full and 0 means that they are empty:

#### 2.4. Methodology and Tools Used

#### 2.4.1. ARIMA Model

#### 2.4.2. TBATS Model

#### 2.4.3. ANN Models

## 3. Preliminary Data Analysis

#### 3.1. Autocorrelation of the Time Series

#### 3.2. Decomposition of the Data

_{t}is the main data, S

_{t}is the seasonal component, T

_{t}is the trend component and R

_{t}is the remainder component [17]:

_{t}is the Daily seasonality, W

_{t}is the Weekly seasonality and Y

_{t}is the Yearly seasonality. The data are relatively of high frequency and the decomposition model should be able to handle this:

#### 3.3. Explanatory Variables

#### 3.3.1. Temperature

_{PT}= 0.0567. It seems insignificant however looking at Figure 8, which is a scatterplot, it is seen that prices below 0 degrees has a linear effect on prices. This is consistent with our previous comments and it suggests that temperature should be used as a variable with a knot at 0 degrees.

#### 3.3.2. Consumption Prognosis

_{PCP}= 0.43, which is significant and much higher than temperature.

#### 3.3.3. Production Prognosis

_{PPP}= −0.0448, it is even lower than temperature. Inspecting the scatterplot the only meaningful pattern is seen when production prognosis is below 1000 MWh, where very low prices can be observed. Production is actually driven by the consumption and prices and not the other way around, however this data might be more useful with a knot at 1000 MWh.

#### 3.3.4. Wind Production Prognosis

_{PWP}= −0.368 which is almost 8 times of production prognosis. Wind is very important for Denmark due to high penetration rate.

#### 3.3.5. Oil Prices

#### 3.3.6. Natural Gas Prices

#### 3.3.7. Hydro Reservoir Level

## 4. Forecasts and Results

#### 4.1. TBATS

#### 4.2. Non-Seasonal ARIMA with External Regressors

#### 4.3. Artificial Neural Network-Nnetar

#### 4.4. Seasonal Naïve (Benchmark)

#### 4.5. Evaluation and Comparison of Forecasts

_{h}is the actual value and $\widehat{{P}_{h}}$ is the forecasted value. In our calculations “accuracy()” function from the forecast package is utilized. This function calculates the mean error, MAE, mean absolute percentage error, mean absolute seasonal error, mean percentage error and RMSE automatically. Then the mean minimum and maximum values of the related errors were taken, to reach to the values at Table 3:

#### 4.6. Backwards Feature Elimination

#### 4.7. Combination Forecasts

## 5. Conclusions

- (1)
- Determine the best forecasting method out of the three individual methods
- (2)
- Suggest and implement ways for improving the individual forecasts
- (3)
- Demonstrate a solid forecasting methodology

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**DK west day ahead electricity prices (

**a**), monthly mean, coefficient of variation, and standard deviation (

**b**).

**Figure 3.**DK west day ahead electricity prices daily means, coefficient of variation, and standard deviation.

**Figure 4.**Hourly means for each weekday with coefficient of variation, and the standard deviation bars.

Explanatory Variable | Correlation Coefficient | Actual Contribution | Suggested Knot | Knot Contribution |
---|---|---|---|---|

Temperature | 0.0567 | No | 0 Degree | No |

Consumption Prognosis | 0.43 | Yes | - | - |

Production Prognosis | 0.0448 | Yes | 1000 MWh | No |

Wind Prognosis | 0.368 | Yes | - | - |

Oil Prices | 0.259 | Yes | - | - |

Natural Gas Prices | 0.293 | Yes | - | - |

Hydro Reservoir Levels | 0.313 | Yes | - | - |

Year | Mean | Min | Max | Standard Dev. |
---|---|---|---|---|

2016 | 175.82 | −100.26 | 692.12 | 57.02 |

2017 | 219.31 | −112.30 | 870.75 | 63.90 |

Model | RMSE | MAE | ||||||
---|---|---|---|---|---|---|---|---|

Mean | Min | Max | Std Dev | Mean | Min | Max | Std Dev | |

TBATS | 45.01 | 11.87 | 210.13 | 29.56 | 37.51 | 9.86 | 177.77 | 24.71 |

Non-Seasonal ARIMA | 39.53 | 7.95 | 215.22 | 26.85 | 33.24 | 6.01 | 212.11 | 23.91 |

ANN-nnetar | 48.41 | 7.27 | 152.07 | 28.70 | 41.41 | 5.53 | 132.42 | 26.02 |

Seasonal Naïve (Benchmark) | 65.95 | 3.43 | 305.62 | 50.20 | 52.53 | 3.01 | 295.19 | 43.45 |

Excluded Variable | RMSE | MAE | ||||||
---|---|---|---|---|---|---|---|---|

Mean | Min | Max | Std Dev | Mean | Min | Max | Std Dev | |

Benchmark ARIMA | 39.53 | 7.95 | 215.22 | 26.85 | 33.24 | 6.01 | 212.11 | 23.91 |

Temperature | 38.88 | 7.15 | 153.32 | 20.61 | 31.96 | 5.97 | 120.29 | 17.35 |

Consumption Prognosis | 45.41 | 5.44 | 215.35 | 27.82 | 38.49 | 4.46 | 211.09 | 25.04 |

Production Prognosis | 42.04 | 7.61 | 236.63 | 30.10 | 34.71 | 6.10 | 233.27 | 26.55 |

Wind Prognosis | 44.88 | 9.29 | 233.06 | 33.28 | 37.19 | 8.14 | 229.41 | 29.41 |

Oil Price | 41.04 | 4.85 | 215.49 | 26.77 | 34.52 | 3.81 | 212.38 | 23.88 |

Natural Gas Price | 41.06 | 4.85 | 215.59 | 26.80 | 34.56 | 3.80 | 212.49 | 23.90 |

Hydro Res. | 41.00 | 4.86 | 215.58 | 26.75 | 34.50 | 3.81 | 212.47 | 23.84 |

Model | RMSE | MAE | ||||||
---|---|---|---|---|---|---|---|---|

Mean | Min | Max | Std Dev | Mean | Min | Max | Std Dev | |

Benchmark ARIMA | 39.53 | 7.95 | 215.22 | 26.85 | 33.24 | 6.01 | 212.11 | 23.91 |

Temperature Excluded | 38.88 | 7.15 | 153.32 | 20.61 | 31.96 | 5.97 | 120.29 | 17.35 |

Temperature with Knot | 39.52 | 8.03 | 215.31 | 26.87 | 33.22 | 6.04 | 212.18 | 23.93 |

Production Prognosis with Knot | 40.96 | 9.58 | 235.27 | 30.20 | 34.04 | 7.91 | 232.13 | 26.49 |

Model | RMSE | MAE | ||||||
---|---|---|---|---|---|---|---|---|

Mean | Min | Max | Std Dev | Mean | Min | Max | Std Dev | |

Benchmark TBATS | 45.01 | 11.87 | 210.13 | 29.56 | 37.51 | 9.86 | 177.77 | 24.71 |

Benchmark ARIMA | 39.53 | 7.95 | 215.22 | 26.85 | 33.24 | 6.01 | 212.11 | 23.91 |

Benchmark ANN | 48.41 | 7.27 | 152.07 | 28.70 | 41.41 | 5.53 | 132.42 | 26.02 |

ARIMA (Temp. Excluded) | 38.88 | 7.15 | 153.32 | 20.61 | 31.96 | 5.97 | 120.29 | 17.35 |

ANN (Temp. Excluded) | 47.57 | 8.33 | 151.39 | 29.29 | 40.44 | 7.10 | 132.27 | 26.49 |

Model | RMSE | MAE | ||||||
---|---|---|---|---|---|---|---|---|

Mean | Min | Max | Std Dev | Mean | Min | Max | Std Dev | |

Benchmark ARIMA | 39.53 | 7.95 | 215.22 | 26.85 | 33.24 | 6.01 | 212.11 | 23.91 |

Benchmark TBATS | 45.01 | 11.87 | 210.13 | 29.56 | 37.51 | 9.86 | 177.77 | 24.71 |

Benchmark ANN | 48.41 | 7.27 | 152.07 | 28.70 | 41.41 | 5.53 | 132.42 | 26.02 |

TBATS+ANN | 40.21 | 8.88 | 174.22 | 25.71 | 33.46 | 7.53 | 136.30 | 21.50 |

ANN+ARIMA | 38.05 | 8.07 | 168.88 | 24.06 | 31.92 | 5.90 | 165.67 | 21.14 |

TBATS+ARIMA | 37.50 | 9.94 | 176.41 | 26.41 | 31.11 | 8.10 | 162.73 | 21.86 |

TBATS+ANN+ARIMA | 36.44 | 8.04 | 164.26 | 24.34 | 30.06 | 6.63 | 148.23 | 20.34 |

Model | RMSE | MAE | ||||||
---|---|---|---|---|---|---|---|---|

Mean | Min | Max | Std Dev | Mean | Min | Max | Std Dev | |

Benchmark ARIMA | 39.53 | 7.95 | 215.22 | 26.85 | 33.24 | 6.01 | 212.11 | 23.91 |

Benchmark TBATS | 45.01 | 11.87 | 210.13 | 29.56 | 37.51 | 9.86 | 177.77 | 24.71 |

Benchmark ANN | 48.41 | 7.27 | 152.07 | 28.70 | 41.41 | 5.53 | 132.42 | 26.02 |

Combination ANN | 37.37 | 8.07 | 166.31 | 25.04 | 31.10 | 6.38 | 153.93 | 21.40 |

Forecast 1 | Forecast 2 | p (power = 1) | Result |
---|---|---|---|

Arima | Tbats | 0.002476 | Null Hypothesis is rejected |

Arima | ANN | 0.00005125 | Null Hypothesis is rejected |

Tbats | ANN | 0.02377 | Null Hypothesis is rejected |

TBATS+ANN | Arima | 0.5596 | Failed to Reject Null Hypothesis |

ANN+ARIMA | Arima | 0.1057 | Failed to Reject Null Hypothesis |

TBATS+ ARIMA | Arima | 0.006285 | Null Hypothesis is rejected |

TBATS+ANN+ARIMA | Arima | 0.0013 | Null Hypothesis is rejected |

Forecast 1 | Forecast 2 | p (power = 2) | Result |
---|---|---|---|

Arima | Tbats | 0.007324 | Null Hypothesis is rejected |

Arima | ANN | 0.003557 | Null Hypothesis is rejected |

Tbats | ANN | 0.201 | Failed to Reject Null Hypothesis |

TBATS+ANN | Arima | 0.4896 | Failed to Reject Null Hypothesis |

ANN+ARIMA | Arima | 0.03984 | Null Hypothesis is rejected |

TBATS+ ARIMA | Arima | 0.08277 | Failed to Reject Null Hypothesis |

TBATS+ANN+ARIMA | Arima | 0.009754 | Null Hypothesis is rejected |

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## Share and Cite

**MDPI and ACS Style**

Karabiber, O.A.; Xydis, G. Electricity Price Forecasting in the Danish Day-Ahead Market Using the TBATS, ANN and ARIMA Methods. *Energies* **2019**, *12*, 928.
https://doi.org/10.3390/en12050928

**AMA Style**

Karabiber OA, Xydis G. Electricity Price Forecasting in the Danish Day-Ahead Market Using the TBATS, ANN and ARIMA Methods. *Energies*. 2019; 12(5):928.
https://doi.org/10.3390/en12050928

**Chicago/Turabian Style**

Karabiber, Orhan Altuğ, and George Xydis. 2019. "Electricity Price Forecasting in the Danish Day-Ahead Market Using the TBATS, ANN and ARIMA Methods" *Energies* 12, no. 5: 928.
https://doi.org/10.3390/en12050928