# Stability of Multiple Seasonal Holt-Winters Models Applied to Hourly Electricity Demand in Spain  
[PERSON]  
1 Department of Applied Statistics, Operational Research and Quality, Universitat Politecnica de Valencia, 46022 Valencia, Spain; [EMAIL_ADDRESS] (O.T. ); [EMAIL_ADDRESS] (J.C.G.-D.)  
[PERSON]  
1 Department of Applied Statistics, Operational Research and Quality, Universitat Politecnica de Valencia, 46022 Valencia, Spain; [EMAIL_ADDRESS] (O.T. ); [EMAIL_ADDRESS] (J.C.G.-D.)  
[PERSON]  
2 Division of Computer Science, Universidad Pablo de Olavide, ES-41013 Seville, Spain  
22  
\[S_{t} = \alpha\left(\frac{X_{t}}{I_{t-s}}\right)+(1-\alpha)(S_{t-1}+T_{t-1}) \tag{1}\] \[T_{t} = \gamma(S_{t}-S_{t-1})+(1-\gamma)T_{t-1}\] (2) \[I_{t} = \delta\left(\frac{X_{t}}{S_{t}}\right)+(1-\delta)I_{t-s}\] (3) \[\widehat{X}_{t+h} = (S_{t}+hT_{t})I_{t-s+h} \tag{4}\]  
where \(S_{t}\), and \(T_{t}\) are the equations for the level and additive trend, with smoothing parameters \(\alpha\) and \(\gamma\). \(I_{t}\) are the seasonal indices of length \(s\), with smoothing parameters \(\delta\). \(X_{t}\) is the observed data. Finally, \(\widehat{X}_{t+h}\) is the equation to forecast h time instants ahead. It collects the information contained in the model and makes predictions. The introduction of double and triple seasonal models (HWT), described in [9; 10], improved the accuracy of these methods, and their use is being generalized. Additionally, these methods behave very accurate for such type of series, with a strong seasonal effect [11; 12]. The generalization to include up to \(n\) seasonal patterns is known as Multiple Seasonal Holt-Winters (nHWT), developed in [13]. The model is defined as in Equations (5)-(8),  
\[S_{t} = \alpha\left(\frac{X_{t}}{\prod_{i=1}^{n_{s}}I_{t-s_{i}}^{(i)}} \right)+(1-\alpha)(S_{t-1}+T_{t-1}) \tag{5}\] \[T_{t} = \gamma(S_{t}-S_{t-1})+(1-\gamma)T_{t-1}\] (6) \[I_{t}^{(i)} = \delta^{(i)}\left(\frac{X_{t}}{S_{t}\prod_{j=1,j\
eq i}^{n_{s}}I_ {t-s_{j}}^{(j)}}\right)+(1-\delta^{(i)})I_{t-s_{i}}^{(i)}\quad i=1,\ldots,n_{s}\] (7) \[\widehat{X}_{t+h} = (S_{t}+hT_{t})\prod_{i=1}^{n_{s}}I_{t-s_{i}+h}^{(i)}+\varphi_{AR} \varepsilon_{t} \tag{8}\]  
where \(I_{t}^{(i)}\) are the seasonal indices of length \(s_{i}\), with smoothing parameters \(\delta^{(i)}\). There are as many equations as seasonal patterns allocated in the time series, \(n_{s}\). The \(\varphi_{AR}\) parameter is entered to correct the model including effectively the first order autocorrelation error (\(\varepsilon_{t}\)), known as AR(1) adjustment. Depending on the way the equations are defined, additive or multiplicative methods can be used. The method of combining the smoothing equations will determine the Holt-Winters model to be used. The combination provides 30 different models, according to the trend and seasonality combination, as well as AR(1) adjustment, as described in [13]. The nomenclature for these models is as follows: Three letters to describe the combination, where the first one describes the trend method, the second one the seasonality method and the last one whether the adjustment was applied. This combination is shown in Table 1. Furthermore, the model is described by adding as subscript the different seasonal patterns considered, using the length of the cycle. As an example, the \(AMC_{24,168}\) one is a model with additive trend and multiplicative seasonal, the model is adjusted using first order autocorrelation error and it has two seasonalities, one daily (subscript 24) and one weekly (subscript 168). The smoothing parameters are obtained by adjusting the model to data observed in the past, and try to reproduce the same pattern for the near future. The adjustment is done by solving a non-linear problem, in which the 1-hour-ahead forecasting error is minimised. This topic is better developed in Section 3. The smoothing parameters obtained are assumed to be optimal, and the model is then exploited. The optimality of these parameters is locally obtained, valid only for the dataset and the model selected.