مدل سازی توزیعی و پیش بینی کوتاه مدت قیمت برق توسط مدل های افزودنی تعمیم یافته برای محل، مقیاس و شکل

In the context of the liberalized and deregulated electricity markets, price forecasting has become increasingly important for energy company's plans and market strategies. Within the class of the time series models that are used to perform price forecasting, the subclasses of methods based on stochastic time series and causal models commonly provide point forecasts, whereas the corresponding uncertainty is quantified by approximate or simulation-based confidence intervals. Aiming to improve the uncertainty assessment, this study introduces the Generalized Additive Models for Location, Scale and Shape (GAMLSS) to model the dynamically varying distribution of prices. The GAMLSS allow fitting a variety of distributions whose parameters change according to covariates via a number of linear and nonlinear relationships. In this way, price periodicities, trends and abrupt changes characterizing both the position parameter (linked to the expected value of prices), and the scale and shape parameters (related to price volatility, skewness, and kurtosis) can be explicitly incorporated in the model setup. Relying on the past behavior of the prices and exogenous variables, the GAMLSS enable the short-term (one-day ahead) forecast of the entire distribution of prices. The approach was tested on two datasets from the widely studied California Power Exchange (CalPX) market, and the less mature Italian Power Exchange (IPEX). CalPX data allow comparing the GAMLSS forecasting performance with published results obtained by different models. The study points out that the GAMLSS framework can be a flexible alternative to several linear and nonlinear stochastic models.

In the liberalized electricity markets, the market clearing price is established through one-sided auctions (power pool-type market) or two-sided auctions (power exchange-type market) as the intersection of the supply curve and the estimated demand or demand curve, respectively. Usually, the clearing price is determined the day before delivery by means of 24 or 48 auctions, one for each hour or half an hour of the following day. Therefore, a reliable price forecast plays an important role in the bidding strategies of generator firms as well as of distribution companies, traders and large consumers. According to the planning horizon, price forecasts are classified as long-term (years), medium-term (monthly), and short-term (from few hours to few days), which are particularly useful in the day-to-day operations carried out in the auction-based day-ahead spot market (Bunn, 2000 and Weron and Misiorek, 2006). As pointed out by Aggarwal et al. (2009), electricity price time series exhibit patterns more complex than load sequences, and some characteristics (such as non constant mean and variance, multiple strong seasonality, and calendar effect) that can be attributed to the features of the electricity market (non-storability of electricity, inelasticity of the short-term demand, wide spectrum of costs, oligopolistic behavior of the generators) and distinguish electricity from other types of commodities (see also Blumsack et al., 2002, Blumsack et al., 2006 and Bosco et al., 2007). Weron and Misiorek, 2006 and Weron, 2006 classified the models available in the literature for the electricity price forecasting in six broad classes and focus on statistical methods as the best suited for the short-term forecasting. More recently, Aggarwal et al. (2009) classified price forecasting methods in three wide groups, namely, game theory, simulation, and time series models. The first group aims at mimicking the bidding strategies of the market participants, the second one simulates the physical phenomenon related to the actual dispatch with system requirements and constraints, whereas the last group describes the historical pattern of the prices involving sometimes exogenous variables, such as loads and weather variables. In their review, Aggarwal et al. (2009) considered 47 works, whose results are grouped according to several criteria, namely: (1) type of model, (2) time horizon for prediction, (3) input variables used, (4) output variables, (5) analysis of results, (6) sample size used for the analysis, (7) preprocessing procedures, and (8) model architecture. Focusing on time series models, there is no evidence that one approach clearly outperforms the others. The performance of the models depends on the properties of the data on hand (market maturity, volatility, regulation, time span, etc.), and is usually assessed by comparing point forecasts (i.e., price profiles, which are the most common output of the short-term forecasting models) with the actual clearing prices. Aggarwal et al. (2009) recognized that only four papers out of 29 dealing with time series models provide confidence intervals or the probability density function associated with price profiles (see Tables 3 and 5 of their paper). Weron (2006) also pointed out the lack of extensive literature on interval forecasting of electricity prices. From 2006, only few authors have complemented the stochastic point forecasts with confidence intervals (e.g., Misiorek and Weron, 2006, Misiorek et al., 2006, Nogales and Conejo, 2006 and Weron and Misiorek, 2008). However, the interval forecasting should always be performed to provide a picture of the estimation uncertainty and reliability. Interval forecasting is usually carried out by assuming that prediction errors (i.e., the differences between point forecasts and actual clearing prices) are normally distributed. As an alternative, quantiles with a given probability can be extracted from the empirical distribution of the prediction errors. In other words, the expected price forecast resulting from the point estimation is commonly complemented by confidence intervals deduced from the distribution of the forecasting residuals. In this study, we assume that the elements of an electricity price time series are realizations of a nonstationary distribution function, the parameters of which change dynamically according to intra-day, weekly, annual seasonality and other possible explanatory variables. Since the position, scale, and shape parameters of a suitable multi-parameter distribution are linked to its expectation, variance, skewness and kurtosis coefficients, a proper choice of the rules driving the change of the parameters' values from a time instant to the following one allows obtaining a price distribution whose expected value represents the time-varying point estimate (forecast), and the time-varying scale and shape parameters can account for heteroskedastic and leptokurtic (or platykurtic) behavior. Generalized Additive Models for Location, Scale and Shape (GAMLSS) proposed by Rigby and Stasinopoulos (2005) are a well-suited framework to perform this type of analysis because they allow the forecasting of the entire day-ahead distribution and not just the expectation and approximate confidence intervals. The remainder of the paper is organized as follows. In Section 2, the GAMLSS rationale, inferential aspects and model setup are introduced. Section 3 briefly describes structure and setup of the benchmark models used to compare the GAMLSS performance. Section 4 introduces the criteria used to assess the performence of the forecasting methods. In Section 5, two datasets are analyzed: the California Power Exchange (CalPX) market prices for which results from a number of models are well documented in the literature, and the data from the Italian Power Exchange (IPEX) market, which are of specific interest in the present study. Finally, conclusions close the study.

In this study, GAMLSS models proposed by Rigby and Stasinopoulos (2005) have been introduced as an alternative stochastic method to carry out one-day ahead forecasting of electricity market prices. While traditional approaches for time series analysis try to model the expected price pattern and the residuals, GAMLSS consider the prices as realizations of a variable whose distribution function parameters vary dynamically with time and other explanatory variables. Therefore, the complete instantaneous distribution function of price is available without performing separate analyses of the expected price and model residuals. Since the distribution parameters are linked to the statistical moments, and complex dependence functions between parameters and explanatory variables can be introduced in the modeling, GAMLSS allows accounting for nonstationarity of high order moments, say > 2, in a coherent and flexible framework. Nevertheless, this approach involves some subjectivity, common to all stochastic methods, which can influence the results. For example, even though the choice of the most appropriate distribution, link and dependence functions, and explanatory variables can be carried out through performance indices related to the maximum likelihood, such as AIC and SBC, the model structure has to be refined by a trial-and-error approach based on suitable performance measures according to forecasting goal. As shown in the applications, in some cases GAMLSS can yield results better than the other reference models; however, all methods can be improved by modifying the model setup appropriately. The advantage of GAMLSS is that these setup changes can be applied to expectation, variance, skewness, and kurtosis of price distribution within quite a simple framework similar to Generalized Linear Models and Generalized Additive Models. Moreover, since the comparison of different methods is fundamental to assess the reliability of the results, GAMLSS provide a further tool for the cross-validation of other forecasting procedures. The following are the supplementary materials related to this article.