# Stability of Multiple Seasonal Holt-Winters Models Applied to Hourly Electricity Demand in Spain
## 4 Results
### Double Seasonal Models  
Table 2 shows the results of the forecasts made by double seasonal models. In particular, the average of the MAPE when predicting the 250 random samples of two weeks. Only models with the correction of the first order autocorrelation error setting are shown, since they offer better results. Removing the year 2009, the year in which the crisis was emphasized in the Spanish industrial sector until that uncertain moment, the rest of the years there is a clear stability in the forecasts. Figure 1: Demand split in years and seasons for stability analysis. Holt-Winters models are very robust to small variations. The forecast accuracy values are around 2.6% in terms of MAPE for 24 hours-ahead forecasts. In the triple seasonal case these values are around 7.9% of MAPE. The \(NMC_{24,168}\) is selected as it offers the best forecast accuracy. As the MAPE provided is an average each year, a split including the months is graphed in Figure 2 as weather conditions and months may influence on the results. Clearly the winter and autumn months have higher levels of MAPE, this time the winter months when worse forecasts occur. Table 3 shows the distribution of the parameter values according to the year and seasonal period for the model \(NMC_{24,168}\). These values were obtained using the sets for adjustment composed of the 250 random samples of 8 weeks. The total row is the mean of the parameters if they are calculated without dividing into seasons. It can be seen that there was a shift in the parameter values from autumn to winter. The values for the daily seasonality are in the order of 0.2 to 0.3 with the exception of winter, where it increases to the value of 0.4. For the intra-weekly seasonality it has a value of 0.2 and it becomes 0.5 for the winter season. \begin{table}
\begin{tabular}{l l l l l l l l l l l} \hline \hline
**Year** & **2008** & **2009** & **2010** & **2011** & **2012** & **2013** & **2014** & **2015** & **2016** & **2017** & **Mean** \\ \hline \(AAC_{24,168}\) & 2.4755 & 3.3322 & 2.8705 & 2.5367 & 2.6761 & 2.2657 & 2.5447 & 2.4527 & 2.6741 & 2.6238 & 2.6963 \\ \(AMC_{24,168}\) & 2.4644 & 3.2576 & 2.8389 & 2.4974 & 2.5697 & 2.2284 & 2.4225 & 2.4131 & 2.7129 & 2.6972 & 2.6521 \\ \(dAAC_{24,168}\) & 2.4402 & 3.3052 & 2.8227 & 2.5120 & 2.5927 & 2.2272 & 2.5114 & 2.4397 & 2.6361 & 2.6189 & 2.6571 \\ \(dAMC_{24,168}\) & 2.4028 & 3.1723 & 2.8047 & 2.4720 & 2.5209 & 2.2190 & 2.3866 & 2.4164 & 2.6447 & 2.5585 & 2.6034 \\ \(DAMC_{24,168}\) & 2.4483 & 3.3217 & 2.8034 & 2.5183 & 2.6054 & 2.2351 & 2.4846 & 2.4279 & 2.6399 & 2.6296 & 2.6601 \\ \(DAMC_{24,168}\) & 2.4119 & 3.1572 & 2.8087 & 2.4519 & 2.5267 & 2.2045 & 2.4077 & 2.4117 & 2.6330 & 2.6184 & 2.6026 \\ \(MAC_{24,168}\) & 2.4722 & 3.3320 & 2.8427 & 2.5260 & 2.6059 & 2.2430 & 2.5031 & 2.4420 & 2.6520 & 2.6325 & 2.6746 \\ \(MMC_{24,168}\) & 2.4462 & 3.1844 & 2.8538 & 2.4670 & 2.5681 & 2.2444 & 2.4787 & 2.5781 & 2.7370 & 2.7615 & 2.6531 \\ \(NAC_{24,168}\) & 2.4434 & 3.3208 & 2.8131 & 2.4970 & 2.5932 & 2.2258 & 2.4869 & 2.4256 & 2.6497 & 2.6216 & 2.6552 \\ \(NMC_{24,168}\) & 2.4061 & 3.1339 & 2.8043 & 2.4196 & 2.5126 & 2.1720 & 2.4407 & 2.3999 & 2.6366 & 2.5810 & 2.5872 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the 24-hour ahead forecasting MAPE of double seasonal models, split by years. Figure 2: MAPE radar diagram of the forecasts for the \(NMC_{24,168}\) model using the demand time series from 2008 to 2017. Figure 3 shows a representation of the parameters over time. It can be observed how the values follow a pattern according to the period of the year. The level and AR(1) adjustment values are slightly different from those of the previous analysis. The level values are higher while the AR(1) adjustment is lower. This is because when the trend is removed, possible long-term variations are supplied by the level. The MAPE in winter season worsens while the smoothing parameters increase their variability in autumn.