Towards managing of the financial risk of electricity load predictions. A product of linear time series
Electricity is a special commodity with very limited storage possibilities. Hence, electricity supply and demand must be constantly balanced, making short-term operational planning a crucial issue for electricity trading companies. Production or cost management is usually based on electricity load or demand predictions, published on the day-ahead manner by the transmission system operators (TSO). However, these forecasts are burdened with prediction errors, which might cause a large deviations of the actual energy cost from its projections. Hence, the proper evaluation of the risk associated with load or demand uncertainties is essential for market strategies planning.
The financial cost of the load or demand prediction errors is equal to the product of electricity price and the difference between the actual and forecasted load or demand. Both variables, electricity prices and load or demand, are usually modelled using linear time series based on common explanatory variables, like, e.g., weather or business conditions. A proper modeling of their product, thus, leads to nonlinear effects in the resulting time series for the financial cost of predictions errors. Mathematically it is definitely more challenging task than an analysis of separate linear times series. However, a joint common approach for all those market variables leads to a model that is consistent with real market relations between the variables.
In our research, see , we have analyzed data from the Danish electricity market. We have shown that the bi-dimensional vector autoregressive time series of order 1 (called VAR(1) model) with Student’s t distribution is well fitted for both variables, the prices and load prediction errors. This observation allows for modeling the total financial cost of the TSO load forecast errors, being the product of both variables. In Fig. 1. we demonstrate the analyzed data related to the weekly means of the day-ahead electricity prices and the weekly means of the TSO load forecast (top and middle panel, respectively). In the bottom panel, we demonstrate the product of two considered time series that is the total financial cost of the load forecast errors.
As it was mentioned, from a mathematical point of view, one considers the product of two components of the bi-dimensional time series, where the dependence between them may have a crucial impact on their product. Moreover, the non-Gaussian behavior of the data also influences the product characteristics. In  we have analyzed the theoretical properties of one-dimensional time series that arises as the product of the components of bi-dimensional VAR(1) model. The main attention was paid to the auto-dependence structure of the considered time series (expressed in the manner of the auto-covariance function, ACVF) and its relation with the correlation coefficient between the bi-dimensional model components. We have demonstrated that a seemingly simple operation (product of two components) applied to a simple linear model may give interesting properties of the resulting time series. What is more, the proposed methodology, can be considered as universal. It is dedicated to finite-variance VAR(1) models without an assumption of residuals’ distribution form. We have demonstrated that the correlation between VAR(1) model components has different influence on the final product characteristics in the case when the residuals are Gaussian than in the case when they have Student’s t distribution. This makes the considered problem also interesting from the theoretical point of view. On the other hand, the theoretical properties of the product of the VAR(1) model components, presented in , describe the properties of the analyzed energy data well, what was demonstrated, e.g., by the comparison between the sample and theoretical ACVFs, see Fig. 2.
We have also conducted a similar research in the case when the analyzed bi-dimensional data are correlated, however their auto-dependence structure is not taken into account. In  we considered transactions data from the continuous German electricity market. Each transaction is characterized by two values: the volume and the price. Hence, the final transaction value, being the amount of the total profit for the energy seller or a total cost for energy buyer, is the product of these two variables.
From the mathematical point of view, one considers here the product of two random variables that may be related. In  we have demonstrated that the probabilistic properties of separate random variables have a strong influence on the product random variable. We have paid the main attention on the sensitivity of the final product to the correlation coefficient between the variables. In case the random variables have the same distribution, we could analyze the cases when they are dependent and not dependent. In the Gaussian case, the independence is equivalent to the zero correlation. For other considered cases this situation does not occur. We have also examined the cases when the considered random variables come from different classes of distributions and analyzed the influence of the heavy-tail behavior of one of the random variable on the final product’s characteristics. The theoretical results were applied to the transaction data from the German continuous energy market. We have shown that the distribution of transaction values resembles the product of log-normal and Student’s t distributions.
A product variable occurs naturally in many real life processes. Another possible application of the obtained results is modelling of commodity prices in the currency of the country, where a company operates. Commodities prices are fixed at main commodities exchanges, where they are quoted usually in USD. Then the price of a commodity in local currency is just the product of the price in USD and the currency exchange rate. On the other hand, the relation between commodity price in USD and local currency exchange rate throughout the time is extremely weighty as it has a significant impact on entity’s profitability and can be modelled using VAR model. A recent analysis of such modelling approach for a mining company was conducted in . Dynamic situation in financial markets, triggered mainly by the outbreak of a pandemic, causes that a lot of previously existing patterns and relations between commodities and local currencies has been changing. This makes the analysis of the price of commodities in various currencies very important.
 J. Janczura, A. Puć, Ł. Bielak, A. Wyłomańska (2022) Dependence structure for the product of bi-dimensional finite-variance VAR(1) model components. An application to the cost of electricity load prediction errors, submitted, working paper version: https://arxiv.org/abs/2203.02249.
 J. Adamska, L. Bielak, J. Janczura, A. Wyłomańska: On the distribution of the product of two continuous random variables with an application to electricity market transactions. Finite and infinite-variance case, submitted, working paper version: arXiv:2111.13487, 2021
 Ł. Bielak, A. Grzesiek, J. Janczura, A. Wyłomańska (2021) Market risk factors analysis for an international mining company. Multi-dimensional, heavy-tailed-based modelling, Resources Policy 74, 102308.
The work is conducted under the project “Probabilistic forecasting as a tool for the optimization of decision processes in electricity markets”, financed by the National Science Center, Poland, under the grant nr 2019/35/D/HS4/00369 and the project “Market risk model identification and validation using novel statistical, probabilistic, and machine learning tools” financed by National Center of Science under Opus Grant No. 2020/37/B/HS4/00120.
The research is conducted in the cooperation between Faculty of Pure and Applied Mathematics (WUST, Poland) and the KGHM PM SA. KGHM is one of the biggest copper and silver producer in the world. Company’s mines and smelters located in Poland are one of the biggest energy consumer in the country.
By Joanna Janczura (firstname.lastname@example.org), Agnieszka Wyłomańska, Hugo Steinhaus Center, Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology (WUST), and Łukasz Bielak, KGHM, PM SA, Poland