# Stability of Multiple Seasonal Holt-Winters Models Applied to Hourly Electricity Demand in Spain
## 4 Results
### Triple Seasonal Models  
In this section, an analysis of the parameters of the triple seasonal Holt-Winters models is carried out. Data sets of size 3 years (\(53\times 24\times 7\times 3\) h) were used to adjust the models. As in the case of double seasonal models, 24-hour forecasts were made for two weeks. Although all triple seasonal methods  
Figure 4: MAPE of the 24-hours ahead forecasts according to the size of the datasets used to obtain the model. Figure 5: Evolution of parameters versus size of data set. depending on the seasonality and trend were analyzed, there are no major differences between the models, and the \(AMC_{24,168,8766}\) model turns out to be the easiest to study, as it has 6 parameters, including level, trend, seasonality and fit. The results obtained are shown in Table 4, where the values of the parameters are organized according to the season and year. In particular, the mean and standard deviation of the smoothing parameters obtained using the 80 sets for adjustment composed of 8 random weeks are shown. It can be noticed that the standard deviation for Autumn of the year 2016 is not defined because it was only a random set of 8 weeks into the Autumn. Contrary to what happens in the double seasonal models, and as seen in the previous section, when using sets of size greater than 20,000 h, the values of the parameters are stabilized. The value of the \(\delta^{(24)}\) parameter stabilizes around 0.3 for any period and with a minimum variability. The value of the \(\delta^{(168)}\) parameter stabilizes around 0.2 and the new parameter \(\delta^{(8766)}\) around 0.08 and 0.1. These values follow the pattern shown in Figure 5, although there is a transfer between the \(\delta^{(168)}\) parameter and the \(\delta^{(8766)}\). The introduction of the third seasonality has forced the model to update the intra-weekly seasonality with more recent data over time. On the other hand, values of \(\alpha\) stabilize around practically 0, and values of \(\varphi_{AR}\) around 0.95. The existence of so much data makes the AR(1) adjustment practically responsible for adapting the level and makes the level equation redundant. This would imply the elimination of a parameter to be optimized. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{\(\alpha\)} & \multicolumn{2}{c}{\(\gamma\)} & \multicolumn{2}{c}{\(\delta^{(24)}\)} & \multicolumn{2}{c}{\(\delta^{(168)}\)} & \multicolumn{2}{c}{\(\delta^{(8766)}\)} & \multicolumn{2}{c}{\(\varphi_{AR}\)} \\ \hline  & **mean** & **Dev. ** & **mean** & **Dev. ** & **mean** & **Dev. ** & **mean** & **Dev. ** & **mean** & **Dev. ** & **mean** & **Dev. ** \\ \hline  & 2011 & & & & & & & & & & & \\ Autumn & 0.0008 & 0.0009 & 0.0001 & 0.0000 & 0.3081 & 0.0004 & 0.2296 & 0.0019 & 0.0783 & 0.0306  
In addition, an analysis of the size of the data set and its consequences on the forecasts was carried out in the same way as in the previous case. It was used on the same day, 11 July 2016, where the size of the observed data set was gradually increased and 24-hour forecasts were made over two weeks. The results are shown in Figure 6. An asymptotic evolution towards a MAPE of 2% can be seen. Although the MAPE is triggered and reduced again periodically. The shark fin form responds to an adjusting dataset beginning in the autumn, and finishing in the end of autumn--remember that in autumn the calendar effect was much more important than at other times. One of the main characteristics observed in the triple seasonal models is the variability in predictability, mainly produced by the time of year with more holidays, i.e., autumn. The selection of 11 July 2016 for the forecasts responds to the necessity to avoid many special events nearby. The nearest one is the 1 st May, and the series has enough time to react. As the model has low values for alpha, gamma and \(\delta_{8766}\), near to zero, only the intraday and intraweek seasonal components can deal and react against the irregularities of the series, smoothing the model to newer values.