Multiple seasonal cycles forecasting model: the Italian electricity demand

Abstract

Forecasting energy load demand data based on high frequency time series has become of primary importance for energy suppliers in nowadays competitive electricity markets. In this work, we model the time series of Italian electricity consumption from 2004 to 2014 using an exponential smoothing approach. Data are observed hourly showing strong seasonal patterns at different frequencies as well as some calendar effects. We combine a parsimonious model representation of the intraday and intraweek cycles with an additional seasonal term that captures the monthly variability of the series. Irregular days, such as public holidays, are modelled separately by adding a specific exponential smoothing seasonal term. An additive ARMA error term is then introduced to lower the volatility of the estimated trend component and the residuals’ autocorrelation. The forecasting exercise demonstrates that the proposed model performs remarkably well, in terms of lower root mean squared error and mean absolute percentage error criteria, in both short term and medium term forecasting horizons.

Appendix: Competing models

In this “Appendix”, we shortly describe the alternative model formulations we consider as benchmark to test the forecasting ability of the proposed model. The double seasonal exponential smoothing model of Taylor (2003), named ${\mathcal {M}}_{\mathsf {DES}}^{\mathsf {AR}}$, is specified as follows:

$$\begin{aligned} y_t&=\mu _{t-1}+s_{1,t-m_1}+s_{2,t-m_2}+\varepsilon _t,\qquad t=1,2,\dots ,n\\ \mu _t&= \mu _{t-1}+\alpha \varepsilon _t,\\ s_{1,t}&=s_{1,t-m_1}+\gamma _{11}\varepsilon _t\\ s_{2,t}&=s_{2,t-m_2}+\gamma _{22}\varepsilon _t\\ \varepsilon _t&=\phi \varepsilon _{t-1}+\zeta _t, \end{aligned}$$

where all the parameters $\left( \alpha ,\gamma _1,\gamma _2,\phi \right)$ are restricted to belong to the unit interval, i.e. $\left( 0,1\right)$, and, for homogeneity we assume $\zeta _t$ to be independently and identically distributed drawn from a standardised Gaussian random variable, i.e. $\zeta _t\sim {\mathcal {N}}\left( 0,1\right)$. Here $m_1=24$ is the daily seasonal period in hours, while $m_2=24\times 7$ denotes the weekly seasonal period.

The triple seasonal exponential smoothing model of Taylor (2010a), named ${\mathcal {M}}_{\mathsf {TES}}^{\mathsf {AR}}$, include an additional seasonal term accounting for the annual cycle:

$$\begin{aligned} y_t&=\mu _{t-1}+s_{1,t-m_1}+s_{2,t-m_2}+\varepsilon _t,\qquad t=1,2,\dots ,n\\ \mu _t&= \mu _{t-1}+\alpha \varepsilon _t,\\ s_{1,t}&=s_{1,t-m_1}+\gamma _{11}\varepsilon _t\\ s_{2,t}&=s_{2,t-m_2}+\gamma _{22}\varepsilon _t\\ s_{3,t}&=s_{3,t-m_3}+\gamma _{\mathsf {Y}}\varepsilon _t\\ \varepsilon _t&=\phi \varepsilon _{t-1}+\zeta _t, \end{aligned}$$

where the additional smoothing parameter $\gamma _\mathsf {Y}\in \left( 0,1\right)$ account for the annual cycle. The period of the annual cycle is set to $m_3=24\times 7 \times 52$, about 52 weeks of data. As before, the stochastic term $\zeta _t$ are independently and identically distributed drawn from a standardised Gaussian distribution, i.e. $\zeta _t\sim {\mathcal {N}}\left( 0,1\right)$. Finally, the parsimonious multiple seasonal patterns exponential smoothing model of Taylor and Snyder (2012), denoted ${\mathcal {M}}_{\mathsf {MSPT}}^{\mathsf {AR}}$, consists of the following set of equations

$$\begin{aligned} y_t&=\mu _{t-1}+\sum _{i=1}^kx_{it}s_{it-m_1}+\varepsilon _t,\qquad t=1,2,\dots ,n\\ \mu _t&= \mu _{t-1}+\alpha \varepsilon _t,\\ s_{it}&=s_{it-m_1} + \left( \sum _{j=1}^k\gamma _{ij}x_{jt}\right) \varepsilon _t,\qquad i=1,2,\dots ,k,\\ s_{k+1,t}&=s_{k+1,t-m_3}+\gamma _{\mathsf {Y}}\varepsilon _t\\ \varepsilon _t&=\phi \varepsilon _{t-1}+\zeta _t, \end{aligned}$$

where $x_{it}$, for $i=1,2,\dots ,k$ have been previously defined and $\zeta _t\sim {\mathcal {N}}\left( 0,1\right)$, independently and identically distributed. Of course, $s_{k+1,t}$ denotes the yearly seasonal cycle of period $m_3=24\times 7 \times 52$ h, while $m_1$ and $m_2$ are the daly and weekly seasonal period, respectively. All the alternative model are estimated by minimising the Gaussian log-likelihood function over the same in sample period used for the proposed model. In particular we employed the same two step procedure consisting of running the simulated annealing procedure whose optimal values are subsequently introduced as starting values for a standard quasi Newton–Raphson algorithm. Concerning the specification of the seasonal patterns for the different days of the week in the multiple seasonal pattern model of Taylor and Snyder (2012), we used the same parsimonious representation of the daily seasonal cycles identified for our model. This means that we only consider 5 different daily patterns and we do not consider a specific cycle for the irregular days