# Probabilistic Forecasting of German Electricity Imbalance Prices

## Abstract

**:**

## 1. Introduction and Motivation

- It is the first work on direct probabilistic forecasting of electricity imbalance prices;
- The imbalance market is inevitable for any market player, and thus this paper may contribute also to electricity trading literature;
- Various probabilistic models are compared in an exhaustive forecasting study;
- We contribute to the scarce electricity balancing literature by drawing researchers’ attention to the German electricity balancing market;
- The paper provides evidence of the efficiency between the intraday and balancing markets and concludes that the traders should rather avoid participating in the balancing market;

## 2. Electricity Balancing Market

#### 2.1. Balancing Market in Germany

#### 2.2. Imbalance Price

- Price cap in the case of a small GCC balance;
- Additional price cap in the case of a small GCC balance;
- Price comparison with the intraday market and setting a minimum price distance to it in such direction that it is less profitable to contribute to the imbalance;
- Surcharge/discount on the imbalance price in the event of GCC reaching 80% of the positive/negative balancing capacity.

#### 2.3. Data

## 3. Models and Estimation

#### 3.1. Input Features

- Corresponding EPEX price indices: DA${}^{d,h}$, IA${}^{d,qh}$, ID${}_{1}^{d,i}$, ID${}_{3}^{d,i}$, and ID-Index${}^{d,i}$ for $i=h,qh$ (8 regressors) (The ID-Index is a volume-weighted average price of all corresponding ID transactions);
- Most recent 15-minute intraday prices ${}_{x}{\mathrm{ID}}_{15\mathrm{min}}^{d,h}$ for $h=1,\dots ,24$ and ${}_{x}{\mathrm{ID}}_{15\mathrm{min}}^{d,qh}$ for $qh=1,\dots ,96$ ($24+96=120$ regressors);
- Corresponding intraday price differences $\Delta {}_{x}^{}{\mathrm{ID}}_{5\mathrm{min}}^{d,qh}={}_{x}^{}{\mathrm{ID}}_{5\mathrm{min}}^{d,qh}-{}_{x+5\mathrm{min}}^{}{\mathrm{ID}}_{5\mathrm{min}}^{d,qh}$ with $x=30,35,\dots ,55$ min (6 regressors);
- DA forecasts of load, wind onshore, wind offshore and solar generation: ${\mathrm{Load}}^{d,qh}$, ${\mathrm{WiOn}}^{d,qh}$, ${\mathrm{WiOff}}^{d,qh}$, ${\mathrm{Solar}}^{d,qh}$ for $qh=1,\dots ,96$ ($96\times 4=384$ regressors);
- DA forecasts mentioned above for the previous day: ${\mathrm{Load}}^{d-1,qh}$, ${\mathrm{WiOn}}^{d-1,qh}$, ${\mathrm{WiOff}}^{d-1,qh}$, ${\mathrm{Solar}}^{d-1,qh}$ for $qh=1,\dots ,96$ ($96\times 4=384$ regressors);
- Most recent available estimation (which is provided by the TSOs around 15 min after delivery) of the imbalance volumes ${\mathrm{Imb}}^{d,qh-i}$ for $i=4,\dots ,7$ (4 regressors);
- aFRR prices: ${\mathrm{aFRR}}_{i,j,k}^{d,qh}$ for $i=\mathrm{POS},\mathrm{NEG}$ indicating the positive or negative balancing side, $j=\mathrm{CAP},\mathrm{EN}$ indicating the capacity or energy price, and $k=\mathrm{min},\mathrm{avg},\mathrm{max}$ indicating the minimum, average or maximum price ($6\times 2$ regressors);
- mFRR prices: ${\mathrm{mFRR}}_{i,j,k}^{d,qh}$ with $i,j,k$ as above ($6\times 2$ regressors);
- Previous day coal, gas, oil and EUA prices: ${\mathrm{Coal}}^{d-1}$, ${\mathrm{Gas}}^{d-1}$, ${\mathrm{Oil}}^{d-1}$, ${\mathrm{EUA}}^{d-1}$ (4 regressors);
- Weekday dummies ${\mathrm{DoW}}_{i}^{d}$ for $i=1,\dots ,7$ (7 regressors);
- Cubic periodic B-splines ${S}_{i}^{d}$ for $i=1,\dots ,6$ constructed as in Ziel et al. [44] (6 regressors).

#### 3.2. Naive

#### 3.3. Lasso with Bootstrap

#### 3.4. Gamlss with Lasso

#### 3.5. Probabilistic Neural Networks

- Input feature selection as described in Section 3.1 (20 hyperparameters);
- Dropout layer—whether to use the dropout layer after the input layer, and if yes at what rate. The rate parameter is drawn from $(0,1)$ interval (up to 2 hyperparameters);
- Size of the network—either two or three hidden layers (1 hyperparameter);
- Activation functions in the hidden layers. The possible functions are: elu, relu, sigmoid, softmax, softplus, and tanh (1 hyperparameter per layer);
- Number of neurons in the hidden layers. The values are drawn from $[24,1024]$ interval (1 hyperparameter per layer);
- ${L}_{1}$ regularization–whether to use the ${L}_{1}$ regularization on the hidden layers and their weights and if yes at what rate. The rate is drawn from $({10}^{-5},10)$ interval (up to four hyperparameters per layer);
- Learning rate for the Adam algorithm drawn from $({10}^{-5},{10}^{-1})$ interval (one hyperparameter).

## 4. Application Study

#### 4.1. Setting

#### 4.2. Evaluation

#### 4.3. Results

## 5. Conclusions

## Funding

## Conflicts of Interest

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**Figure 1.**The daily routine of the German electricity spot (

**top**) and balancing (

**bottom**) markets. $d,h$ correspond to the day and hour of the delivery, respectively.

**Figure 4.**Time series plots of selected external regressors. POS and NEG stand for positive and negative, respectively.

**Figure 7.**Pinball score (

**left**) and its ratio to the naive (

**right**) over quantiles $\tau \in r$. The right graph shows selected models for better clarity.

**Figure 8.**CRPS (

**left**) and its ratio to the naive (

**right**) over quarter-hours $qh\in QH$. The right graph shows selected models for better clarity.

**Figure 9.**Results of the Diebold–Mariano test. The plots present p-values for the CRPS${}^{d,qh}$ loss—the closer they are to zero (→ dark green), the more significant the difference is between forecasts of the x-axis model (better) and forecasts of the y-axis model (worse).

**Table 1.**Error measures of the considered models. Colour indicates the performance columnwise (the greener, the better). With bold, we depicted the best values in each column.

CRPS | MAE | RMSE | 50%-Cov | 90%-Cov | 98%-Cov | |
---|---|---|---|---|---|---|

Naive | 23.04 | 61.22 | 115.2 | 0.2847 | 0.8011 | 0.9525 |

Lasso | 24.91 | 65.06 | 124.5 | 0.2784 | 0.7825 | 0.9417 |

gamlss.N | 36.09 | 88.70 | 154.5 | 0.3294 | 0.6158 | 0.7067 |

gamlss.t | 24.24 | 65.70 | 124.2 | 0.4074 | 0.8851 | 0.9739 |

probNN.N | 35.67 | 92.84 | 368.5 | 0.5037 | 0.8460 | 0.9249 |

probNN.t | 25.85 | 68.74 | 129.2 | 0.3912 | 0.8733 | 0.9645 |

Combination | 22.94 | 62.84 | 117.8 | 0.3480 | 0.8749 | 0.9787 |

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**MDPI and ACS Style**

Narajewski, M.
Probabilistic Forecasting of German Electricity Imbalance Prices. *Energies* **2022**, *15*, 4976.
https://doi.org/10.3390/en15144976

**AMA Style**

Narajewski M.
Probabilistic Forecasting of German Electricity Imbalance Prices. *Energies*. 2022; 15(14):4976.
https://doi.org/10.3390/en15144976

**Chicago/Turabian Style**

Narajewski, Michał.
2022. "Probabilistic Forecasting of German Electricity Imbalance Prices" *Energies* 15, no. 14: 4976.
https://doi.org/10.3390/en15144976