Abstract
Hourly data usually exhibit complex seasonal variations characterized by yearly, monthly, weekly or daily seasonal patterns. Each seasonal variation is modelled by using an evolving spline function in such a way that a seasonal effect at a proportion of the seasonal period is defined as a non-fixed parametric formulation of this proportion. Subsequently, the areas under the splines are proposed as a useful tool to measure the changes in the magnitude of seasonal variations over time, and to compare the relevance of seasonal variations with different seasonal periods. Furthermore, two indexes are suggested to compare seasonal variations with the same seasonal period in different time series: a dissimilarity index accounts for the area between the splines corresponding to the seasonal variation for each series, whereas a complementarity index accounts for this area when seasonal effects have opposite signs. The proposal is illustrated by applying it to hourly series of energy demand in Canary Islands.
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Notes
It means that \(m\) is the number of days, weeks, months or years over the sample period when the seasonal cycle is, respectively, the daily, weekly, monthly or yearly pattern.
Note that there are \(4k\) unknown parameters in the original spline function, but the formulation of \(3k+1\) constraints allow us to express the original function in terms of \(k-1\) free parameters.
An illustration of this procedure is shown in Fig. 10 in the electronic supplementary material.
An illustration of this procedure is shown in Fig. 11 in the electronic supplementary material.
The idea is similar to the proposal in Martin-Rodriguez and Caceres-Hernandez (2013) to measure the evolution of yearly seasonal patterns in weekly series. However, to compare the magnitude of seasonal variations with different seasonal periods, the length of these seasonal periods must be taken into account. Note that the areas calculated are proportional to the length of the segment identifying the seasonal period and the value in this segment moves always between 0 and 1. However, the lengths of the weekly and yearly seasonal periods are, respectively, 7 and 365 or 366 times the length of the daily seasonal period. Therefore, the areas calculated according to Eq. (16) to measure the relevance of weekly and yearly seasonal variations in a day should be multiplied by 7 and 365 or 366 to be compared with the relevance of the daily seasonal variation in this day.
The consumption level in MW h is measured from the hourly programmed electricity production by the power generation plants (operational hourly schedule), drawn up by Red Eléctrica de España (REE) depending on the evolution of demand. These data, provided by the System Operators’ Information System (SIOS), are available from htttps://www.esios.ree.es/es/analisis.
It is assumed that the amplitude of seasonal variations in the original series increases as the underlying level rises according to a multiplicative model. However, the log transformation provides an additive combination. Although the model proposed is flexible enough to capture changing seasonal patterns corresponding to an additive model for the original series, it seems to be more appropriate to assume that the magnitude of seasonal variation depends on the underlying level. Furthermore, a less parsimonious model would probably be required to model the effects of public holidays for series in levels. On the other hand, we have observed the log transformation clearly stabilizes the variance of the series, and, according to Lutkepohl and Xu (2012), improvements in forecasting accuracy are expected.
Unobserved components model decomposes a time series into components such as trend or seasonal, and also allows for exogenous variables to represent the most salient features of the series under study. The most elementary structural time series model deals with a series whose underlying level changes over time according to a local level model. This flexible stochastic specification of a time-varying trend component implies that the level at time \(t\) is equal to the level at time \(t-1\) plus a white noise disturbance (see Harvey 1989: 18–19).
Some of these anomalous observations are related to unexpected interruptions in power supply from electrical grid.
Daylight-saving time is the period between spring and autumn when clocks are moved forward to have an extra hour of daylight in the evening. All EU member states move clocks forward one hour on the last Sunday in March and back one hour on the last Sunday in October.
In the common power system for Fuerteventura and Lanzarote, some public holidays are not the same on both islands (Virgin La Peña is only celebrated in Fuerteventura and Virgin Los Dolores is only observed in Lanzarote). For hourly observations during these public holidays, the corresponding dummy variable is assumed to be equal to the relative weight of Fuerteventura or Lanzarote population compared with the total population covered by this power system.
Chida et al. (2015) propose a similar procedure to filter seasonal variations in hourly data, but another method is applied in that paper to define the window of moving averages. Zhang and Wang (2018) introduce a nonparametric technique to identify and extract trend, periodic components or noise in time series analysis.
Six-segment cubic splines with equally spaced break points were chosen.
Estimating these models is computationally hard due to the high number of observations and the co-existence of several seasonal cycles. However, the RESM model is very useful to reduce the number of parameters needed to capture this type of seasonal component. This formulation of the seasonal model is included as a set of explanatory variables into a structural time series model to estimate conjointly the seasonal components and the remaining components in the original series. Estimates by maximum likelihood are obtained by using the Stamp 8.30 module of Ox-Metrics 6.20 package.
The forecasts for the level component were obtained from a six-segment non-periodic cubic spline adjusted to the estimates for the sample period.
The forecasts for the seasonal cycles were obtained according to Eq. (14) as explained in the methodological section.
The forecasts for the level component were also obtained from a six-segment non-periodic cubic spline adjusted to the estimates for the sample period.
Forecasts of electricity demand are obtained from these forecasts for each seasonal component and also for the level component (by using a non-periodic spline adjusted to the estimates of the stochastic level from 2009 to 2017). To illustrate these forecasts, averages of observed and forecast values for any hour in a day are shown in Fig. 13 in the electronic supplementary material.
The identification of roots of the spline function for each seasonal cycle (daily, weekly or yearly) into the sample (2009–2017) and out of the sample (2018–2019) is a computationally demanding task. For each one of the six time series, there are 11 yearly cycles, 575 weekly cycles and 4.017 daily cycles. Fortunately, the RESM model imposes smooth changes between consecutive seasonal cycles and the roots of the spline function corresponding to a seasonal cycle are usually similar to the corresponding ones to the following seasonal cycle.
To obtain estimates of the area corresponding to the days in 2018 and 2019, the spline parameters of the last segment were applied to the proportions higher than 1 corresponding to each hourly observation.
References
Adeoye O, Spataru C (2019) Modelling and forecasting hourly electricity demand in West African countries. Appl Energy 241:311–333. https://doi.org/10.1016/j.apenergy.2019.03.057
Alberini A, Prettico G, Shen C, Torriti J (2019) Hot weather and residential hourly electricity demand in Italy. Energy 177:44–56. https://doi.org/10.1016/j.energy.2019.04.051
Alonso-Blanco E, Gómez-Moreno FJ, Artíñano B, Iglesias-Samitier S, Juncal-Bello V, Piñeiro-Iglesias M, López-Mahía P, Pérez N, Brines M, Alastuey A, García MI, Rodríguez S, Sorribas M, del Águila A, Titos G, Lyamani H, Alados-Arboledas L (2019) Temporal and spatial variability of atmospheric particle number size distributions across Spain. Atmos Environ 190:146–160. https://doi.org/10.1016/j.atmosenv.2018.06.046
Amara F, Agbossou K, Dubé Y, Kelouwani S, Cardenas A, Bouchard J (2017) Household electricity demand forecasting using adaptative conditional density estimation. Energ Buildings 156:271–280. https://doi.org/10.1016/j.enbuild.2017.09.082
Andersen FM, Larsen HV, Gaardestrup RB (2013) Long term forecasting of hourly electricity consumption in local areas in Denmark. Appl Energy 110:147–162. https://doi.org/10.1016/j.apenergy.2013.04.046
Andersen FM, Baldini M, Hansen LG, Jensen CL (2017) Households’ hourly electricity consumption and peak demand in Denmark. Appl Energy 208:607–619. https://doi.org/10.1016/j.apenergy.2017.09.094
Andersen FM, Henningsen G, Moller NF, Larsen HV (2019) Long-term projections of the hourly electricity consumption in Danish municipalities. Energy 186:115890. https://doi.org/10.1016/j.energy.2019.115890
Birt BJ, Newsham GR, Beausoleil-Morrison I, Armstrong MM, Saldanha N, Rowlands IH (2012) Disaggregating categories of electrical energy end-use from whole-house hourly data. Energ Buildings 50:93–102. https://doi.org/10.1016/j.enbuild.2012.03.025
Caceres-Hernandez JJ, Martin-Rodriguez G (2017) Evolving splines and seasonal unit roots in weekly agricultural prices. Aust J Agric Resour Econ 61:304–323. https://doi.org/10.1111/1467-8489.12205
Cancelo JR, Espasa A, Grafe R (2008) Forecasting the electricity load from one day to one week ahead for the Spanish system operator. Int J Forecast 24:588–602. https://doi.org/10.1016/j.ijforecast.2008.07.005
Chen A, He X, Guan H, Zhang X (2018) Variability of seasonal precipitation extremes over China and their associations with large-scale ocean-atmosphere oscillations. Int J Climatol 39:613–628. https://doi.org/10.1002/joc.5830
Chida T, Saitoh H, Toyoda J (2015) Analysis of hourly demand data before and after the 2011 Tohoku earthquake. Electr Eng Jpn 192:46–53. https://doi.org/10.1002/eej.22737
Darbellay GA, Slama M (2000) Forecasting the short-term demand for electricity. Do neural networks stand a better chance? Int J Forecast 16:71–83. https://doi.org/10.1016/S0169-2070(99)00045-X
De Livera AM, Hyndman RJ, Snyder RD (2011) Forecasting time series with complex seasonal patterns using exponential smoothing. J Am Stat Assoc 106:1513–1527. https://doi.org/10.1198/jasa.2011.tm09771
Dordonnat V, Koopman SJ, Ooms M, Dessertaine A, Collet J (2008) An hourly periodic state space model for modelling French national electricity load. Int J Forecast 24:566–587. https://doi.org/10.1016/j.ijforecast.2008.08.010
Elamin N, Fukushige M (2018) Modelling and forecasting hourly electricity demand by SARIMAX with interactions. Energy 165:257–268. https://doi.org/10.1016/j.energy.2018.09.157
Filik UB, Gerek ON, Kurban M (2011) A novel modeling approach for hourly forecasting of long-term electric energy demand. Energ Convers Manage 52:199–211. https://doi.org/10.1016/j.enconman.2010.06.059
Gould PG, Koehler AB, Ord JK, Snyder RD, Hyndman RJ, Vahid-Araghi F (2008) Forecasting time series with multiple seasonal patterns. Eur J Oper Res 191:207–222. https://doi.org/10.1016/j.ejor.2007.08.024
Harvey AC (1989) Forecasting, structural time series models and the Kalman filter. Cambridge University Press, Cambridge
Harvey AC, Koopman SJ (1993) Forecasting hourly electricity demand using time-varying splines. J Am Stat Assoc 88:1228–1236. https://doi.org/10.2307/2291261
Harvey AC, Koopman SJ, Riani M (1997) The modelling and seasonal adjustment of weekly observations. J Bus Econ Stat 15:354–368. https://doi.org/10.1080/07350015.1997.10524713
Huai S, Zio E, Zhang J, Xu M, Li X, Zhang Z (2019) A hybrid hourly natural gas demand forecasting method based on the integration of wavelet transform and enhanced Deep-RNN model. Energy 178:585–597. https://doi.org/10.1016/j.energy.2019.04.167
Jelica D, Taljegard M, Thorson L, Johnsson F (2018) Hourly electricity demand from an electric road system. A Swedish case study. Appl Energy 228:141–148. https://doi.org/10.1016/j.apenergy.2018.06.047
Keppler JH, Meunier W (2018) Determining optimal interconnection capacity on the basis of hourly demand and supply functions of electricity. Energ J 39(3):117–139. https://doi.org/10.5547/01956574.39.3.jkep
Kipping A, Tromborg E (2016) Modeling and disaggregating hourly electricity consumption in Norwegian dwellings based on smart meter data. Energ Build 118:350–369. https://doi.org/10.1016/j.enbuild.2016.02.042
Koopman SJ (1992) Diagnostic checking and intra-daily effects in time series models. Thesis Publishers Tinbergen Institute Research Series, Amsterdam, p 27
Liu YL, Ge YE, Gao HO (2014) Improving estimates of transportation emissions: modelling hourly truck traffic using period-based car volume data. Transp Res D 26:32–41. https://doi.org/10.1016/j.trd.2013.10.007
Lobato E, Sigrist L, Rouco L (2017) Value of electric interconnection links in remote island power systems: the Spanish Canary and Balearic archipelago cases. Int J Electr Power Energy Syst 91:192–200. https://doi.org/10.1016/j.ijepes.2017.03.014
Lutkepohl H, Xu F (2012) The role of the log transformation in forecasting economic variables. Empir Econ 42:619–638. https://doi.org/10.1007/s00181-010-0440-1
Ma Y, Xu W, Zhao X, Li Y (2017) Modeling the hourly distribution of population at a high spatiotemporal resolution using subway Smart card data: a case study in the central area of Beijing. Int J GeoInf 6:128. https://doi.org/10.3390/ijgi6050128
Martin-Rodriguez G, Caceres-Hernandez JJ (2005) Modelling the hourly Spanish electricity demand. Econ Model 22:551–569. https://doi.org/10.1016/j.econmod.2004.09.003
Martin-Rodriguez G, Caceres-Hernandez JJ (2010) Splines and the proportion of the seasonal period as a season index. Econ Model 27:83–88. https://doi.org/10.1016/j.econmod.2009.07.021
Martin-Rodriguez G, Caceres-Hernandez JJ (2012) Forecasting pseudo-periodic seasonal patterns in agricultural prices. Agric Econ 43:531–543. https://doi.org/10.1111/j.1574-0862.2012.00601.x
Martin-Rodriguez G, Caceres-Hernandez JJ (2013) Canary tomato export prices: comparison and relationships between daily seasonal patterns. Span J Agric Res 11:882–893. https://doi.org/10.5424/sjar/2013114-4063
Mushin M, Sunilkumar SV, Ratnam MV, Murthy BVK (2017) Seasonal and diurnal variations of tropical tropopause layer (TTL) over the Indian Peninsula. J Geophys Res Atmos 122:672–687. https://doi.org/10.1002/2017JD027056
Pielow A, Sioshansi R, Roberts MC (2012) Modeling short-run electricity demand with long-term growth rates and consumer price elasticity in commercial and industrial sector. Energy 46:533–540. https://doi.org/10.1016/j.energy.2012.07.059
Pina A, Silva C, Ferrao P (2011) Modeling hourly electricity dynamics for policy making in long-term scenarios. Energy Policy 39:4692–4702. https://doi.org/10.1016/j.enpol.2011.06.062
Ramos-Real FJ, Barrera-Santana J, Ramírez-Díaz A, Pérez Y (2018) Interconnecting isolated electrical systems. The case of Canary Islands. Energy Strategy Rev 22:37–46. https://doi.org/10.1016/j.esr.2018.08.004
REE (2015) Planificación Energética. Plan de Desarrollo de la Red de Transporte de Energía Eléctrica 2015–2020. Red Eléctrica de España. Centro de Publicaciones del Ministerio de Industria, Energía y Turismo.
Siddiquee MSA, Hoque S (2017) Predicting the daily traffic volume from hourly traffic data using artificial neural network. Neural Netw World 3:283–294. https://doi.org/10.14311/NNW.2017.27.015
Soares LJ, Medeiros MC (2008) Modeling and forecasting short-term electricity load: A comparison of methods with and application to Brazilian data. Int J Forecast 24:630–644. https://doi.org/10.1016/j.ijforecast.2008.08.003
Taylor JW (2003) Short-term electricity demand forecasting using double seasonal exponential smoothing. J Oper Res Soc 54:799–805. https://doi.org/10.1057/palgrave.jors.2601589
Taylor JW (2010) Triple seasonal methods for short-term electricity demand forecasting. Eur J Oper Res 204:139–152. https://doi.org/10.1016/j.ejor.2009.10.003
Wu X, Zhang X, Xiang X, Zhang K, Jin H, Chen X, Wang C, Shao Q, Hua W (2018) Changing runoff generation in the source area of the Yellow River: mechanisms, seasonal patterns and trends. Cold Reg Sci Technol 155:58–68. https://doi.org/10.1016/j.coldregions.2018.06.014
Yuan J, Farnham C, Azuma C, Emura K (2018) Predictive artificial neural network models to forecast the seasonal hourly electricity consumption for a University campus. Sustain Cities Soc 42:82–92. https://doi.org/10.1016/j.scs.2018.06.019
Yukseltan E, Yucekaya A, Bilge AH (2017) Forecasting electricity demand for Turkey: modeling periodic variations and demand segregation. Appl Energy 193:287–296. https://doi.org/10.1016/j.apenergy.2017.02.054
Zhang X, Wang J (2018) A novel decomposition-ensemble model for forecasting short-term load-time series with multiple seasonal patterns. Appl Soft Comput 65:478–494. https://doi.org/10.1016/j.asoc.2018.01.017
Acknowledgements
We would like to thank Alfredo Ramirez Diaz and Francisco Ramos Real for their help to collect data. We also thank anonymous reviewers for their useful suggestions to improve and clarify this manuscript.
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All authors contributed to the study conception and design. Data collection was performed by Jonay Hernandez-Martin. Statistical analysis was performed by Jose Juan Caceres-Hernandez and Gloria Martin-Rodriguez. The first draft of the manuscript was written by Jose Juan Caceres-Hernandez and Gloria Martin-Rodriguez and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
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Caceres-Hernandez, J.J., Martin-Rodriguez, G. & Hernandez-Martin, J. A proposal for measuring and comparing seasonal variations in hourly economic time series. Empir Econ 62, 1995–2021 (2022). https://doi.org/10.1007/s00181-021-02079-3
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DOI: https://doi.org/10.1007/s00181-021-02079-3