# A Trading-Based Evaluation of Density Forecasts in a Real-Time Electricity Market

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## Abstract

**:**

## 1. Introduction

- Firstly, the value of forecasting with densities, compared to mean-values, is shown, through back-testing real-time trading strategies on the Austrian balancing zone of the German/Austrian electricity market,
- Secondly, a new density modeling technique, previously only applied to prices, is extended successfully to forecasting the imbalance volumes at 15 min resolution, and outperforms a more conventional benchmark.

## 2. The Austrian Balancing Market

^{BA}for these imbalances is determined from a “basis price” and a “transfer function”. The basis price p

^{Basis}is

_{max}= 40 €/MWh and U

_{min}= 3 €/MWh, being the fixed maximum and minimum parameter values of the transfer function $T\text{}$for the monthly data in our analysis. These values are set by the Energy Regulatory Authority (ERA) and are adapted from time to time. The ex post balancing price is then:

## 3. Data Analysis and Predictive Methodology

_{U}(in its alternative parametrization as in reference [27], JSU), the sinh-arcsinh (as in reference [28], see SHASHo and SHASHo2), the skew-t (as in references [29,30,31], respectively ST1, ST2 and ST5). The second class is a 3-parameter family represented by the skew-normal distributions, specifically the skew normal ‘type 1’ (SN1), which is a special case of the skew exponential power with τ = 2. Thirdly, selected as a baseline, the 2-parameter normal distribution (NO) as this is often used for simplicity in operational models. Figure 2 presents the density fits for best fitting distributions, after the time series for imbalances have been seasonally adjusted for daily frequency (by using dummy variables for days of the week, from Monday to Saturdays, and holidays).

_{U}(JSU) distributions seem to be the most appropriate to the series of imbalances. For this purpose, three measures for assessing the goodness-of-fit have been considered, specifically: the Kolmogorov-Smirnov (KS), the Cramér–von Mises (CVM), and the Anderson–Darling (AD). The Anderson–Darling and Cramér–von Mises statistics belong to the class of quadratic statistics, using the squared and the weighted squared differences between the empirical distribution function and the cumulative distribution function of the supposed reference distribution. Hence, both statistics place more weight on observations in the tails of the distribution, with weights being larger for the AD than for the CVM. On the contrary, the Kolmogorov–Smirnov statistic quantifies the absolute maximum distance between the empirical distribution function and the cumulative distribution function of the supposed reference distribution. Therefore, given the observed statistical properties of imbalances series with almost zero asymmetry and moderate kurtosis (especially compared to electricity prices), the latter measure should be preferred. According to results reported in Table 2, the general superiority of the JSU and the skew student-t distributions (specifically ST2) is observed (note that computational difficulties can emerge, as with infinite values for the SHASHo distribution), consistent with Hagfors et al. [8] for hourly electricity prices.

_{U}distribution for Californian and Italian electricity price densities, both distributions have been retained to test their forecasting performances.

- $im{b}_{t-2}$ is the imbalance variable with a time lag of 2,
- $fwin{d}_{t-2}$ is the wind forecast error, calculated as the difference between the day-ahead forecast and the latest value measured at (t − 2),
- $floa{d}_{t-2}$ is the load forecast error, calculated as the difference between the day-ahead forecast and the latest value measured at (t − 2), and
- $fsola{r}_{t-2}$ is the solar forecast error, calculated as the difference between the day-ahead forecast and the latest value measured at (t − 2).

_{i}= 1, 5, 10, …, 99 for i = 1, …, 21. Then, the values of these series have been used in the Diebold and Mariano (DM) test with the null hypothesis of equal performance versus the alternative one that JSU is less accurate than ST2; adjusting for missing values and with the differential loss function defined as ${\mathsf{\Delta}}_{ST2,JSU,t,{q}_{i}}={\widehat{\epsilon}}_{ST2,t,{q}_{i}}-{\widehat{\epsilon}}_{JSU,t,{q}_{i}}$; in practice, nominal values instead of absolute ones are used for the estimated forecasting errors, given that the pinball scores are always positive. Results of the DM test show that the null of equal performance is always rejected (in favor of the alternative of JSU being less precise than ST2 at the 1%, and also at the more common 5%, level of significance). Altogether, these results show the forecasting superiority of the ST2 distribution over the JSU.

## 4. Optimal Imbalance Positions

## 5. Backtesting

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Balancing price formation (source: APCS, 2018, [26]).

Whole Year | Winter | Spring | Summer | Autumn | |
---|---|---|---|---|---|

(Dec.–Feb.) | (Mar.–May) | (Jun.–Aug.) | (Sept.–Nov.) | ||

Average | 0.131 | .629 | −4.877 | 0.316 | 5.753 |

Maximum | 151.146 | 131.480 | 138.899 | 151.146 | 140.798 |

Minimum | −320.021 | −205.058 | −197.641 | −320.021 | −230.439 |

Standard Deviation | 33.525 | 31.638 | 36.503 | 33.325 | 31.482 |

Skewness | −0.834 | −0.637 | −0.566 | −1.631 | −0.387 |

Kurtosis | 6.904 | 5.151 | 4.644 | 12.186 | 5.757 |

JB statistics | 26311 | 2250 | 1466 | 34967 | 2986 |

SHASHo | SHASHo2 | JSU | ST1 | ST2 | ST5 | SN1 | NO | |
---|---|---|---|---|---|---|---|---|

AD | Infinity | 4208.7000 | 0.2900 | 315.9700 | 297.4300 | 6654.8900 | 244.5600 | 244.5600 |

CVM | 7474.8600 | 860.8500 | 0.0307 | 51.7100 | 45.8900 | 1296.4100 | 40.5800 | 40.5800 |

KS | 0.6600 | 0.2700 | 0.0020 | 0.0590 | 0.0520 | 0.3570 | 0.0550 | 0.0550 |

Percentiles | |||||||||||

1 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | |

JSU | 1.1105 | 3.1324 | 4.9126 | 6.2800 | 7.3589 | 8.2189 | 8.8851 | 9.3779 | 9.7175 | 9.9177 | 9.9832 |

ST2 | 0.9817 | 2.9732 | 4.7045 | 6.0299 | 7.0642 | 7.8750 | 8.4990 | 8.9629 | 9.2725 | 9.4502 | 9.4893 |

Percentiles | |||||||||||

55 | 60 | 65 | 70 | 75 | 80 | 85 | 90 | 95 | 99 | Average | |

JSU | 9.9163 | 9.6987 | 9.3224 | 8.7776 | 8.0552 | 7.1444 | 6.0174 | 4.6109 | 2.8202 | 0.8550 | 6.9577 |

ST2 | 9.3875 | 9.1468 | 8.7631 | 8.2270 | 7.5387 | 6.6794 | 5.6191 | 4.3038 | 2.6368 | 0.7693 | 6.5892 |

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**MDPI and ACS Style**

Bunn, D.W.; Gianfreda, A.; Kermer, S.
A Trading-Based Evaluation of Density Forecasts in a Real-Time Electricity Market. *Energies* **2018**, *11*, 2658.
https://doi.org/10.3390/en11102658

**AMA Style**

Bunn DW, Gianfreda A, Kermer S.
A Trading-Based Evaluation of Density Forecasts in a Real-Time Electricity Market. *Energies*. 2018; 11(10):2658.
https://doi.org/10.3390/en11102658

**Chicago/Turabian Style**

Bunn, Derek W., Angelica Gianfreda, and Stefan Kermer.
2018. "A Trading-Based Evaluation of Density Forecasts in a Real-Time Electricity Market" *Energies* 11, no. 10: 2658.
https://doi.org/10.3390/en11102658