Size Matters: Estimation Sample Length and Electricity Price Forecasting Accuracy.

• Author: Fezzi, Carlo

1. INTRODUCTION

   During the past 30 years, the widespread liberalization of the energy sector has entrusted the electricity price formation process to the law of supply and demand. In most developed economies, electricity is now traded in high-frequency (hourly or half-hourly) wholesale markets, where power companies sell directly to retailers and large consumers. In this relatively new environment, developing effective short-term, day-ahead, Electricity Price Forecasting (EPF) tools can be of tremendous value (e.g. Hong, 2015).

   EPF has proven to be particularly challenging. Electricity prices are characterized by a level of variability that is unobserved in any other commodity or financial asset, and peculiar dynamics such as abrupt and short-lived spikes, heteroscedasticity, and pronounced daily, weekly and yearly seasonality (e.g. Weron, 2014), which follows the dynamics of demand, often referred as "load" in this literature. Given this background, it is not surprising the recent proliferation of EPF techniques, which include statistical models (e.g. linear and non-linear regressions, time series models), machine-learning algorithms (e.g. neural networks) and various hybrid methods. Despite this significant effort, the comprehensive review by Weron (2014) concludes that a leading, best-performing methodology is yet to emerge.

   Why is EPF so difficult? Arguably, the main cause is the substantial instability that characterizes the process of price formation. The physical laws governing the electric grid always require production and consumption to be perfectly balanced, making economically-sound storage virtually impossible. For this reason, minor changes in demand, which frequently go unnoticed, can sometimes have tremendous repercussions on prices, particularly when margin (i.e. the additional generation capacity available for production) is low. In such cases, even relatively small utilities can exercise a significant amount of market power, and influence prices substantially (e.g. Fabra and Toro, 2005; Hortacsu and Puller, 2008; Ito and Reguant, 2016). Although margin can sometimes be predicted, unobservable determinants, such strategic behavior and asymmetric information, create an unstable price formation process that continuously evolves through time.

   In forecasting, a common approach to handle time-instability is to estimate parameters using moving windows that include only the most recent observations: the so-called "rolling estimation" method (e.g. Inoue et al., 2017). While rolling estimation is the standard approach also in EPF (e.g. Dudek, 2016; Nowotarski and Weron, 2016; Steinert and Ziel, 2019), perhaps surprisingly, there is no established strategy to guide the window-size selection process. The typical approach is to simply set a relatively large window (usually between 180 and 365 observations, i.e. from 6 months to 1 year of data) a priori for all models and markets. (1,2)

   This paper demonstrates that such stylized approach produces subpar results. Window size dramatically affect EPF models' performance, and selecting the optimal rolling sample prior to estimation significantly reduces forecasting errors. To the best of our knowledge, there are only two contributions studying this issue in EPF: Hubicka et al. (2018) and Marcjasz et al. (2018a). Both articles explore the performance of weighting schemes constructed by averaging predictions across models estimated on different window sizes. The first work analyzes the performance of a regression model and an artificial neural network under different weighting schemes, while the second article focuses on regression models and explores a larger selection of weighting techniques. Both papers conclude that using appropriate weights improves upon selecting a single window size when averaging prediction errors across all 24 trading hours of the day.

   This paper develops a different and highly complementary approach. We propose a simple, two-step strategy to select both best performing models and window sizes. Our analysis includes a wide selection of models including time series, regressions and computational intelligence methods for a total of eleven different approaches. In addition, rather than evaluating the best performing model using average measures across all 24 hourly predictions within a day, we evaluate predictions for each hourly trading period separately. This finer analysis reveals that both optimal window size and best performing model change greatly across hours, with the stable off-peak hours favoring long windows and complex (i.e. with a large number of parameters) models, and the more volatile peak hours selecting short samples and relatively simple (i.e. with only a few parameters) specifications. We conclude that different models and window sizes should be used for different hours. Furthermore, our simple two-step strategy significantly outperforms the standard fixed rolling window approach in the majority of the trading periods we investigate.

2. DATA

   We analyze the day-ahead time series of two large but very different wholesale electricity markets: the Nord Pool (NP) and the Italian Power Exchange (IPEX). For both markets, we consider hourly prices and forecasted load (published on the day-ahead by the system operator and, therefore, available to all market participants for price forecasting) for three years covering the period from January 1, 2014 to December 31, 2016. These data are accessible from the system operator websites (www.nordpoolspot.com and www.mercatoelettrico.org) and from the webpage of the corresponding Author of this paper. (3) In both the NP and the IPEX (as in most electricity markets) the clearing prices and quantiles for each hour of the day are generated via 24 simultaneous auctions taking place on the day before the delivery.

   The two markets are of similar size (a peak of about 70GWh for the NP and one of 55GWh for the IPEX) but have very different histories, generation mixes, demand patterns and resulting price dynamics. The NP was inaugurated in 1991 and it is now one of the oldest liberalized power markets in the world, encompassing Denmark, Finland, Norway and Sweden. Its price and load dynamics have been subject to extensive research (e.g. Haldrup and Nielsen, 2006; Weron and Misiorek, 2008; Nowotarski and Weron, 2016; Marcjasz et al., 2018a). Figure 1 presents the NP hourly day-ahead price and forecasted load time series.

   Load displays a strong yearly seasonality with peaks in the winter months that, however, are not always mirrored by high prices. In fact, the NP is characterized by a large share of hydro-power (Norway, for example, is almost entirely reliant on this type of generation) which generates prices that are typically lower than in other European markets. However, significant spikes are present when demand is high and water storage is low (e.g. during early 2016). Figure A1 in the Appendix illustrates the daily seasonality. Both load and prices are characterized by two daily peaks (around hour 10 and hour 20), which also present the highest volatility and, therefore, are the most challenging for forecasting. This daily seasonality is essentially the same in both weekdays and weekends, while, of course, weekdays peaks are generally much higher for both load and price.

   The IPEX opened in year 2004, which makes it one of the youngest power markets in Europe. Consumption is met with a mix of fossil fuels (about 65%), renewables (21%) and direct imports (14%). This mixture of relatively expensive generation and large share of imports makes the IPEX price rather high and subject to frequent spikes, as highlighted in Figure 2. Yearly seasonality is not very strong, since in Italy the main source of heating is natural gas and, therefore, there is not a winter demand peak. On the other hand, as show in Figure A2 in the Appendix, daily seasonality is quite pronounced. Such seasonality is characterized by two distinctive peaks, with the highest being the one in the evening, around hour 20. Volatility is highest when power plants are increasing production in order to reach this peak. Recent analyses of IPEX prices are Bigerna and Bollino (2015), Grossi and Nan (2015), Gianfreda et al. (2016, 2019), Lisi and Edoli (2018).

3. METHODOLOGY

   As mentioned in the previous section, the NP and IPEX generate clearing prices and quantities via 24 simultaneous day-ahead auctions. This mechanism breaks down the temporal structure of the time series, since the information available to traders is updated every day, and not every hour (Huisman et al., 2007). Acknowledging this feature, it has become standard practice to model the prices of the 24 hours of the day as separate series. This approach is also superior from a purely forecasting perspective, since it recognizes that electricity generators faces very different constraints throughout the daily cycle (e.g. Weron, 2014). Our analysis follows this convention, generating 24 different sets of estimates and predictions for each one of the models we analyze. We focus on day-ahead predictions, i.e. one-step ahead, which is the most common short-term EPF exercise. Section 3.1 presents the different forecasting models in detail, while Section 3.2 illustrates how we determine the optimal sample length for each model and evaluate forecasting performance.

   3.1 Forecasting models

   The literature proposes a truly extensive collection of short-term EPF techniques. According to the review by Weron (2014), the two most popular classes of methods are: a) statistical models, which include both time series (e.g. Conejo et al., 2005; Weron and Misiorek, 2008; Koopman et al., 2009) and multiple regression models (e.g. Maciejowska and Nowotarski, 2016; Nowotarski and Weron, 2016; Marcjasz et al., 2018a; Steinert and Ziel, 2019) and b) computational intelligence algorithms, in particular neural networks (e.g. Dudek, 2016; Hubicka et al., 2018; Marcjasz et al., 2019)...