Forecasting functional time series in electricity markets with a seasonal ARIMAX Hilbertian model

Abstract

A functional time series is the realization of a stochastic process where each observation is a continuous function defined in a finite interval. In order to forecast these functions, a seasonal ARIMAX Hilbertian model is presented. It extends the structure of ARIMA models to functional data using integral operators in the L2 space. The kernels of operators are modeled as linear combinations of sigmoid functions, where the parameters of each sigmoid are estimated using a Quasi-Newton algorithm which minimizes the sum of squared errors. This functional model allows forecasting functional time series taking into account time dependencies (autoregressive and moving average terms), seasonality as well as scalar and functional exogenous variables. An empirical study is presented for the time series of hourly residual demand curves in the Spanish day-ahead electricity market. The residual demand curves model the competitive behavior of the agents bidding in an electricity market. For every auction, the residual demand is defined as the clearing price of the market expressed as a function of the amount of energy an agent is able to buy or sell. Being able to forecast these curves is the first and essential step in the design of optimal bidding strategies.