Contents lists available at ScienceDirect Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes Prediction intervals for electricity demand and price using functional data Juan Vilar ⁎ , Germán Aneiros, Paula Raña Departamento de Matemáticas, Facultad de Informática, Universidad de A Coruña, Campus de Elviña, s/n, 15071 A Coruña, Spain ARTICLE INFO Keywords: Load and price Electricity markets Functional data Functional time series forecasting Prediction intervals ABSTRACT This paper provides two procedures to obtain prediction intervals for electricity demand and price based on functional data. The proposed procedures are related to one day ahead pointwise forecast. In particular, the first method uses a nonparametric autoregressive model and the second one uses a partial linear semi-parametric model, in which exogenous scalar covariates are incorporated in a linear way. In both cases, the proposed procedures for the construction of the prediction intervals use residual-based bootstrap algorithms, which allows also to obtain estimates of the prediction density. Applications to the Spanish Electricity Market, in year 2012, are reported. This work extends and complements the results of Aneiros et al. (2016), focused on pointwise forecasts of next-day electricity demand and price daily curves. 1. Introduction Nowadays, in many countries all over the world, the production and sale of electricity is traded under competitive rules in free markets. Thus, the electricity demand and price forecasting is a main target for the agents and companies involved in the electricity markets. In par- ticular, short term forecast, which is the one day ahead hourly forecast, has been extensively studied in the literature. A wide range of meth- odologies and models for forecasting have been proposed and studied. These methods can be classified into two large groups: the first one based on statistical approaches including time series, dynamic regres- sion, exponential smoothing, regression analysis, etc. and the second one based on artificial intelligence techniques such as neuronal net- works, fuzzy neural networks, support vector machines, etc. A nice monograph on electricity demand and price forecasting can be found in [1]. Also [2,3] contain reviews on electricity demand forecasting and [4,5] on electricity price forecasting. In recent years, one can find some studies addressing this problem from a functional perspective, this is, defining the daily curves of electricity demand or price by some func- tional form (functional data). The books [6,7] are comprehensive re- ferences for functional data analysis. See also [8] for a recent mono- graph on inference for functional data. Statistical models for functional data were also used to predict electricity demand and price. The reader will find in [9] a parametric model to predict electricity consumption curves; [10] introduced a novel functional time series methodology that is applied to historical daily curves of load; [11] proposed a hybrid approach which was applied to French demand curves; [12] proposed a new methodology to obtain probabilistic forecasts of electricity load that is based on functional data analysis of generalized quantile curves; [13] studied the short term forecasting of household-level intra-day electricity load curve (they proposed a nonparametric functional ap- proach based on functional kernel regression estimator with the use of an unsupervised clustering step of the historical segments); [14] used three approaches, two of them of functional type, to forecast the France’s daily electricity load consumption; [15] proposed an adaptive functional autoregressive (AFAR) forecast model to predict electricity price curves; finally, the case of residual demand curves in Spain was analysed in [16]. When the interest is to forecast scalar values (not curves) from functional data, the reader can see [17] (nonparametric and semiparametric models), [18] (functional factor model) or [19] (censored response). The case of forecasting curves (as well as scalar values) of demand and price from functional data are studied in [20]. In particular, nonparametric and semi-functional partial linear models are analysed within the Spanish Electricity Market. There are many methods and studies about short-term point forecast of electricity demand and price. However, methods for obtaining pre- diction intervals (PI) or prediction densities (PD) have not been studied extensively up to date. The importance of the PIs and PDs relies on the information that they provide, related to the evolution and variability of future demand or price, which allows to plan different strategies of action. Some papers on this topic are the following: [21–23] obtained PIs in the problem of electricity price prediction; [24–26] computed PIs in forecast of demand, and [27,28] studied methods of estimation of PD in this context. This paper proposes two algorithms to obtain PIs and PDs in the problem of the next-day forecasting of electricity demand and price. http://dx.doi.org/10.1016/j.ijepes.2017.10.010 Received 7 April 2017; Received in revised form 21 September 2017; Accepted 7 October 2017 ⁎ Corresponding author. E-mail addresses: juan.vilar@udc.es (J. Vilar), german.aneiros@udc.es (G. Aneiros), paula.rana@udc.es (P. Raña). Electrical Power and Energy Systems 96 (2018) 457–472 0142-0615/ © 2017 Elsevier Ltd. All rights reserved. MARK