**6. Conclusions**

This article analyses the stability of the smoothing parameters in the multiple seasonal Holt-Winters models. This is crucial for the proper appliance of these models to provide accurate and trusted forecasts. Although most of the time, an automatic forecasting algorithm can provide good results, forecasters need to understand the behaviour of these parameters before submitting the forecasts. We use the time series of the hourly Spanish electricity demand.

In this work, we analyze the behaviour of the smoothing parameters of the multiple seasonal Holt-Winters models applied to the series of hourly electricity demand in Spain. There are many variables that affect the parameters within the same time series, including the computation of the initial values, in addition to other factors, such as the calendar or climate conditions. This variability of the parameters is subsequently reflected in the accuracy of the forecasts since they depend greatly on the calculation of the parameters.

The variation of the parameters is analyzed when different seasonal and trend methods are used, as well as different climatic situations of the series. Additionally, it is analyzed how the size of the data set used in order to adjust the model influences on the parameters, and, thus, in the forecasts. It was observed that the seasonal parameters are strongly dependent on the period of the year, making autumn a really difficult period to deal with. This effect is more pronounced in the intra-weekly seasonality than the daily one. However, with the increase in the size of the fitting set, these parameter values stabilise around a value, which depends on the time series. When the size of the set is bigger than 5000 hours, the values are stable. Triple seasonal models have much more stable parameters, although a much larger set of data is necessary to adjust the model. It was found that the main source of variability of the parameters is the calendar effect, which strongly influences the accuracy of the forecasts, not being so much the climatic effect on the series. The accuracy of the forecasts is also stabilised with a larger set of data, it is not necessary to have more than 5000 observations. It is also observed how double seasonal models provide better predictions than triple seasonal ones. The results of the current analysis are limited to being used with Spanish electricity demand. Of course, similar results are expected when faced with load forecasting in other countries or systems. Future works will be addressed toward the development of new models able to include calendar effect in the own model as well as models for the prediction of holidays through the inclusion of discrete seasonalities.

**Author Contributions:** Conceptualization, Ó.T. and J.C.G.-D.; methodology, J.C.G.-D.; software, Ó.T.; validation, Ó.T., A.T. and J.C.G.-D.; writing–original draft preparation, Ó.T. and J.C.G.-D.; writing–review and editing, A.T.; supervision, A.T. and J.C.G.-D.; project management: A.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding. **Acknowledgments:** The authors would like to thank the Spanish Ministry of Economy and Competitiveness for the support under project TIN2017-8888209C2-1-R.

**Conflicts of Interest:** The authors declare no conflict of interest.
