# Forecasting Electricity Prices: A Machine Learning Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}coupons from the largest EU energy market, we show that the proposed model provides more accurate predictions of future electricity prices compared to extant prediction methods.

## 2. Challenges of Electricity Price Forecasting

## 3. Method

_{1}, T

_{2}: R

^{n}→ R, the geometric semantic crossover returns the real function T

_{XO}= (T

_{1}· T

_{R}) + ((1 − T

_{R}) · T

_{2}), where T

_{R}is a random function such that T

_{R}: R

^{n}→ [0; 1].

_{R}when producing values in [0; 1], we make use of the sigmoid function TR = 1 / (1 + e

^{-Trand}), where T

_{rand}is a random tree with no constraints on the output values.

^{n}→ R, the geometric semantic mutation with mutation step ms returns the real function T

_{M}= T + ms · (T

_{R1}− T

_{R2}), where T

_{R1}and T

_{R2}are random real functions. Henceforth, we refer to the GP algorithm that uses GSOs as Geometric Semantic Genetic Programming (GSGP).

Algorithm 1: The GP algorithm. |

1: Population ß InitializePopulation(population_size, problem_variables) |

2: EvaluatePopulation(Population) |

3: S_best ß GetBestSolution(Population) |

4: num_iterations ß 0 |

5: While (num_iteration < num_generations) |

6: Children_pop ß Ø |

7: Children_pop ß Children_pop U S_best |

8: While (Size(Children_pop) < population_size) |

9: Operator ß SelectGeneticOperator(Crossover_Rate, Mutation_Rate) |

10: If (Operator == CrossoverOperator) |

11: Parent_1 ß SelectParent(Population) |

12: Parent_2 ß SelectParent(Population) |

13: Child = GSC(Parent_1,Parent_2) |

14: Children_pop = Children_pop U Child |

15: ElseIf (Operator == MutationOperator) |

16: Parent_1 ß SelectParent(Population) |

17: Child = LSGSM(Parent_1) |

18: Children_pop = Children_pop U child |

19: Else |

20: Parent_1 ß SelectParent(Population) |

21: Children_pop = Children_pop U parent_1 |

22: End |

23: End |

24: EvaluatePopulation(Children_pop) |

25: S_bset ß GetBestSolution(Children_pop) |

27: Population ß Children_pop |

28: num_iteration ß num_iteration + 1 |

29: End |

30: Return (S_best) |

## 4. Experimental Phase

#### 4.1. Data

_{0}), the crude oil spot price in USD per barrel for EU was used [49]. Apart from alternative energy sources, the EEX EU Emission Allowances have also been taken into consideration (variable X

_{1}) as the spot EU emissions allowance in price per EUR/tCO

_{2}[50]. Variables X

_{2}through X

_{15}relate to the weather conditions and include air temperature, air density, ground temperature, air pressure, relative humidity, wind speed, maximum intraday air temperature, minimum intraday air temperature, minimum intraday air temperature at ground level, maximum intraday wind speed, next-day precipitation of rain to cover a horizontal ground surface, next day precipitation of snow to cover a horizontal ground surface, sunshine duration, and snow depth. The data for the weather conditions in Munich, Germany were obtained from the online data source “Deutscher Wetterdienst” [51]. All the variables were normalized and are dimensionless. Thus, it is possible to combine all variables without considering their different units of measurement. Table 1 provides a list of all variables used in this study.

#### 4.2. Experimental Settings and Results for GP-Based Systems

^{−11}for the training MAE and 8.6632 × 10

^{−7}for the test MAE) confirm that LSGS is able to outperform GSGP by producing solutions with a significantly smaller MAE than the latter system.

#### 4.3. Comparison with Other State-of-the-Art Machine Learning Techniques

#### 4.4. Interpretability of the Models

## 5. Discussion

_{0}, x

_{1}, x

_{4}, x

_{6}, and x

_{7}are the main variables used by the system to forecast the spot energy price. Our model shows that the spot electricity price depends on the crude oil spot price (x

_{0}), the settlement price EUR/t CO

_{2}(x

_{1}), ground temperature (x

_{4}), relative humidity (x

_{6}), and wind speed (x

_{7}). Interestingly, variables x

_{0}, x

_{1}, and x

_{4}are used in the primary factor of all the models obtained at the end of the experimental phase. On the other hand, the variables from x

_{8}to x

_{13}are never used.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## List of Acronyms

AI | Artificial Intelligence |

GP | Genetic Programming |

GSC | Geometric Semantic Crossover |

GSGP | Geometric Semantic Genetic Programming |

GSM | Geometric Semantic Mutation |

GSOs | Geometric Semantic Operators |

ISO | Isotonic Regression |

LIN | Linear Regression |

LSGP | Geometric Semantic Genetic Programming with Local Search |

LSGSM | Geometric Semantic Mutation with Local Search |

MAE | Mean Absolute Error |

NN | Multilayer Perceptron trained with Backpropagation |

OLS | Ordinary Least Square regression |

RBF | Radial Basis Function Network |

SQ | Least Square Regression |

STGP | Standard Genetic Programming |

SVM | Support Vector Machine (with polynomial kernel) |

SVM2 | Support Vector Machine (with polynomial kernel of second degree) |

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**Figure 1.**Comparison between the execution time of different GP systems. GSGP stands for Geometric Semantic Genetic Programming, the GP system that uses the GSC and GSM operators; LSGP is the GSGP version in which the GSM operator is combined with the local search operator; STGP refers to the traditional GP algorithm that makes use of standard syntax-based genetic operators.

**Figure 2.**Training Mean Absolute Error (MAE). The plots show the median over 100 runs. The training MAE is expressed in Euros. The light gray line represents Geometric Semantic Genetic Programming, while the black line represents Geometric Semantic Genetic Programming with Local Search.

**Figure 3.**Test MAE. The plots show the median over 100 runs. The Test MAE is expressed in Euros. The light gray line represents Geometric Semantic Genetic Programming, while the black line represents Geometric Semantic Genetic Programming with Local Search.

**Figure 4.**Boxplots of the training MAE calculated over 100 runs. On each box, the central mark is the median, the edges of the box are the 25th and 75th percentiles, and the whiskers extend to the most extreme data points not considered outliers. The MAE is expressed in Euros. LIN stands for linear regression, SQ stands for ordinary least square regression, ISO stands for isotonic regression, RBF stands for radial basis function network, NN stands for multilayer perceptron trained with backpropagation, and SVM stands for support vector machine with a polynomial kernel (of second degree for SVM2).

**Figure 5.**Boxplots of the test MAE calculated over 100 runs. On each box, the central mark is the median, the edges of the box are the 25th and 75th percentiles, and the whiskers extend to the most extreme data points not considered outliers. The MAE is expressed in Euros. LIN stands for linear regression, SQ stands for ordinary least square regression, ISO stands for isotonic regression, RBF stands for radial basis function network, NN stands for multilayer perceptron trained with backpropagation, and SVM stands for support vector machine with a polynomial kernel (of second degree for SVM2).

**Table 1.**Model variables available in the considered dataset. Observations collected on a daily basis; the task is to predict the target variable in the next 24 h.

Variable | Description |
---|---|

X_{0} | Crude oil spot price (Euro) |

X_{1} | Settlement price EUR/t CO_{2} (Euro) |

X_{2} | Air temperature (degrees Celsius) |

X_{3} | Air density (Kg/m^{3}) |

X_{4} | Ground temperature (degrees Celsius) |

X_{5} | Air pressure (Pascal) |

X_{6} | Relative humidity (expressed as a percentage) |

X_{7} | Wind speed (meters/second) |

X_{8} | Maximum intraday air temperature (degrees Celsius) |

X_{9} | Minimum intraday air temperature (degrees Celsius) |

X_{10} | Minimum intraday air temperature at ground level (degrees Celsius) |

X_{11} | Maximum intraday wind speed (meters/second) |

X_{12} | Next-day rain precipitation forecast to cover a horizontal ground surface (millimeters) |

X_{13} | Next-day snow precipitation forecast to cover a horizontal ground surface (centimeters) |

X_{14} | Sunshine duration (hours) |

X_{15} | Snow depth (centimeters) |

TARGET | Electricity price in EUR in the following day (24 hour prediction) |

**Table 2.**Median and standard deviation of the running time (in seconds) for the two GP based systems.

Running Time(s) | ||
---|---|---|

Median | Standard Deviation | |

GSGP | 35.73 | 0.59 |

LSGP | 36.18 | 0.81 |

**Table 3.**P-values returned by the Mann–Whitney test under the alternative hypothesis that the samples do not have equal medians. In particular, the test compared the median MAE obtained by the considered techniques over the 100 runs we performed.

**Bold**denotes values indicating a statistical difference among the considered techniques.

Training | |||||||

LSGP | LIN | SQ | ISO | RBF | NN | SVM | SVM2 |

1.21 × 10^{−}^{12} | 1.21 × 10^{−}^{12} | 1.21 × 10^{−}^{12} | 1.21 × 10^{−}^{12} | 1.21 × 10^{−}^{12} | 1.21 × 10^{−}^{12} | 3.83 × 10^{−}^{10} | |

Test | |||||||

LSGP | LIN | SQ | ISO | RBF | NN | SVM | SVM2 |

4.72 × 10^{−}^{8} | 4.72 × 10^{−}^{8} | 4.72 × 10^{−}^{8} | 4.72 × 10^{−}^{8} | 4.72 × 10^{−}^{8} | 4.72 × 10^{−}^{8} | 2.59 × 10^{−}^{17} |

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**MDPI and ACS Style**

Castelli, M.; Groznik, A.; Popovič, A.
Forecasting Electricity Prices: A Machine Learning Approach. *Algorithms* **2020**, *13*, 119.
https://doi.org/10.3390/a13050119

**AMA Style**

Castelli M, Groznik A, Popovič A.
Forecasting Electricity Prices: A Machine Learning Approach. *Algorithms*. 2020; 13(5):119.
https://doi.org/10.3390/a13050119

**Chicago/Turabian Style**

Castelli, Mauro, Aleš Groznik, and Aleš Popovič.
2020. "Forecasting Electricity Prices: A Machine Learning Approach" *Algorithms* 13, no. 5: 119.
https://doi.org/10.3390/a13050119