# 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

## References

- Nowotarski, J.; Weron, R. Recent advances in electricity price forecasting: A review of probabilistic forecasting. Renew. Sustain. Energy Rev.
**2018**, 81, 1548–1568. [Google Scholar] [CrossRef] - Weron, R. Electricity price forecasting: A review of the state-of-the-art with a look into the future. Int. J. Forecast.
**2014**, 30, 1030–1081. [Google Scholar] [CrossRef] - Jónsson, T.; Pinson, P.; Nielsen, H.A.; Madsen, H. Exponential smoothing approaches for prediction in real-time electricity markets. Energies
**2014**, 7, 3710–3732. [Google Scholar] [CrossRef][Green Version] - Bello, A.; Bunn, D.; Reneses, J.; Muñoz, A. Parametric Density Recalibration of a Fundamental Market Model to Forecast Electricity Prices. Energies
**2016**, 9. [Google Scholar] [CrossRef] - Chan, J.C.C.; Grant, A.L. Modeling energy price dynamics: GARCH versus stochastic volatility. Energy Econ.
**2016**, 54, 182–189. [Google Scholar] [CrossRef][Green Version] - Jiang, P.; Liu, F.; Song, Y. A hybrid multi-step model for forecasting day-ahead electricity price based on optimization, fuzzy logic and model selection. Energies
**2016**, 9. [Google Scholar] [CrossRef] - Uniejewski, B.; Weron, R.; Ziel, F. Variance Stabilizing Transformations for Electricity Spot Price Forecasting. IEEE Trans. Power Syst.
**2018**, 33, 2219–2229. [Google Scholar] [CrossRef] - Hagfors, L.I.; Bunn, D.; Kristoffersen, E.; Staver, T.T.; Westgaard, S. Modeling the UK electricity price distributions using quantile regression. Energy
**2016**, 102, 231–243. [Google Scholar] [CrossRef] - Lago, J.; De Ridder, F.; De Schutter, B. Forecasting spot electricity prices: Deep learning approaches and empirical comparison of traditional algorithms. Appl. Energy
**2018**, 221, 386–405. [Google Scholar] [CrossRef] - Singh, S.; Yassine, A. Big Data Mining of Energy Time Series for Behavioral Analytics and Energy Consumption Forecasting. Energies
**2018**, 11, 452. [Google Scholar] [CrossRef] - Gajowniczek, K.; Ząbkowski, T. Two-Stage Electricity Demand Modeling Using Machine Learning Algorithms. Energies
**2017**, 10, 1547. [Google Scholar] [CrossRef] - Wang, H.-Z.; Li, G.-Q.; Wang, G.-B.; Peng, J.-C.; Jiang, H.; Liu, Y.-T. Deep learning based ensemble approach for probabilistic wind power forecasting. Appl. Energy
**2017**, 188, 56–70. [Google Scholar] [CrossRef] - Yang, Z.; Ce, L.; Lian, L. Electricity price forecasting by a hybrid model, combining wavelet transform, ARMA and kernel-based extreme learning machine methods. Appl. Energy
**2017**, 190, 291–305. [Google Scholar] [CrossRef] - Areekul, P.; Senjyu, T.; Toyama, H.; Yona, A. A hybrid ARIMA and neural network model for short-term price forecasting in deregulated market. IEEE Trans. Power Syst.
**2010**, 25, 524–530. [Google Scholar] [CrossRef] - Maciejowska, K.; Nowotarski, J.; Weron, R. Probabilistic forecasting of electricity spot prices using Factor Quantile Regression Averaging. Int. J. Forecast.
**2016**, 32, 957–965. [Google Scholar] [CrossRef] - Panagiotelis, A.; Smith, M. Bayesian density forecasting of intraday electricity prices using multivariate skew t distributions. Int. J. Forecast.
**2008**, 24, 710–727. [Google Scholar] [CrossRef] - Serinaldi, F. Distributional modeling and short-term forecasting of electricity prices by Generalized Additive Models for Location, Scale and Shape. Energy Econ.
**2011**, 33, 1216–1226. [Google Scholar] [CrossRef] - Gianfreda, A.; Bunn, D. A Stochastic Latent Moment Model for Electricity Price Formation. Oper. Res. Forthcom.
**2018**. [Google Scholar] [CrossRef] - Gianfreda, A.; Ravazzolo, F.; Rossini, L. Comparing the Forecasting Performances of Linear Models for Electricity Prices with High RES Penetration. arXiv, 2018; arXiv:1801.01093. [Google Scholar]
- Kupiec, P.H. Techniques for Verifying the Accuracy of Risk Measurement Models. J. Deriv. Winter
**1995**, 3, 73–84. [Google Scholar] [CrossRef] - Christoffersen, P. Evaluating Interval Forecasts. Int. Econ. Rev. (Phila.)
**1998**, 39, 841–862. [Google Scholar] [CrossRef] - Hong, T.; Pinson, P.; Fan, S.; Zareipour, H.; Troccoli, A.; Hyndman, R.J. Probabilistic energy forecasting: Global Energy Forecasting Competition 2014 and beyond. Int. J. Forecast.
**2016**, 32, 896–913. [Google Scholar] [CrossRef][Green Version] - Kraas, B.; Schroedter-Homscheidt, M.; Madlener, R. Economic merits of a state-of-the-art concentrating solar power forecasting system for participation in the Spanish electricity market. Sol. Energy
**2013**, 93, 244–255. [Google Scholar] [CrossRef] - Barthelmie, R.J.; Murray, F.; Pryor, S.C. The economic benefit of short-term forecasting for wind energy in the UK electricity market. Energy Policy
**2008**, 36, 1687–1696. [Google Scholar] [CrossRef] - Zareipour, H.; Canizares, C.A.; Bhattacharya, K. Economic Impact of Electricity Market Price Forecasting Errors: A Demand-Side Analysis. IEEE Trans. Power Syst.
**2010**, 25, 254–262. [Google Scholar] [CrossRef] - Bunn, D.W.; Kermer, S. Statistical Arbitrage and Information Flow in an Electricity Balancing Market. SSRN Electron. J.
**2018**. [Google Scholar] [CrossRef] - Johnson, N.L. Systems of frequency curves derived from the first law of Laplace. Trab. Estad.
**1954**, 5, 283–291. [Google Scholar] [CrossRef] - Jones, M.C.; Pewsey, A. Sinh-arcsinh distributions. Biometrika
**2009**, 96, 761–780. [Google Scholar] [CrossRef][Green Version] - Azzalini, A.; Capitanio, A. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. J. R. Stat. Soc. Ser. B Stat. Methodol.
**2003**, 65, 367–389. [Google Scholar] [CrossRef][Green Version] - Jones, M.C.; Faddy, M. A skew extension of the t-distribution, with applications. J. R. Stat. Soc. Ser. B Stat. Methodol.
**2003**, 65, 159–174. [Google Scholar] [CrossRef] - Azzalini, A. Further results on a class of distributions which includes the normal ones. Statistica
**1986**, 46, 1973–2201. [Google Scholar] - Ziel, F.; Weron, R. Day-ahead electricity price forecasting with high-dimensional structures: Univariate vs. multivariate modeling frameworks. Energy Econ.
**2018**, 70, 396–420. [Google Scholar] [CrossRef][Green Version] - Weber, C.; Just, S. Strategic Behavior in the German Balancing Energy Mechanism: Incentives, Evidence, Costs and Solutions. J. Regul. Econ.
**2012**, 48. [Google Scholar] [CrossRef] - Ding, H.; Pinson, P.; Hu, Z.; Wang, J.; Song, Y. Optimal Offering and Operating Strategy for a Large Wind-Storage System as a Price Maker. IEEE Trans. Power Syst.
**2017**, 32, 4904–4913. [Google Scholar] [CrossRef] - Krishnamurthy, D.; Uckun, C.; Zhou, Z.; Thimmapuram, P.R.; Botterud, A. Energy Storage Arbitrage Under Day-Ahead and Real-Time Price Uncertainty. IEEE Trans. Power Syst.
**2018**, 33, 84–93. [Google Scholar] [CrossRef] - Browell, J. Risk Constrained Trading Strategies for Stochastic Generation with a Single-Price Balancing Market. Energies
**2018**, 11. [Google Scholar] [CrossRef] - Pinson, P.; Chevallier, C.; Kariniotakis, G.N. Trading wind generation from short-term probabilistic forecasts of wind power. IEEE Trans. Power Syst.
**2007**, 22, 1148–1156. [Google Scholar] [CrossRef][Green Version]

**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