Like the aggregated energy demand, the electricity consumption is sensitive to the economic context. In 2016, the French energy intensity was estimated around 0,12 ton of oil equivalent per 1000 euros (1,4 MWh per 1000 euros) and the GDP was2225 billions euros.
It is also sensitive to the weather conditions : the slope of the electrical load with respect to the temperature is around dTd` = −2.4 GW/°C in winter [RTE, 2016a] because of the increased heating demand, which corresponds to a relative variation of 100` dTd` = −2.7 % /°C, and +0.4 GW/°C in summer mainly due to the presence of cooling appliances.
Estimating the coming electricity demand and adapting flow-management ac-cordingly is a key step for RTE to carry out the missions entrusted by the State.
There are for RTE multiple load forecasting problems to consider, each being char-acterized by : an aggregation level, between the whole country and the substations, a time horizon ranging from few minutes ahead to several years in the future, and a temporal granularity, for instance every couple minutes for intra-day forecasts and every couple hours for the loads in a decade.
1.2.1 Day-ahead local load forecasting
The day-ahead national load forecasting problem has been studied by the research community for several decades and a forecasting model has been operational at EDF since the 1980s. In this manuscript, we focus on the day-ahead load forecasting problems, at thelocal level of substations.
More precisely, we consider the problem of day-ahead deterministic hourly forecasts of the local load at every substation, meaning the forecast at 23:59 on dayj of the hourly loads on day j+ 1, with a preference for an interpretable model given that eventually its usage should not be restricted to statisticians. With the 2 000 substations considered, this corresponds to 48 000 values to forecast everyday.
1.2.2 Industrial interest
Facing the increased variability of the supply and the demand, a predictive tool is a prerequisite for the local management of power systems to ensure its stability and its resilience. The penetration of electric vehicles and the installation of renewable energy power plants are only 2 of the major challenges to come in the next decade.
Altogether, we identify 4 main needs for the TSO to set a local load forecasting model.
National supply and demand equilibrium RTE is contractually responsible for the national supply and demand equilibrium. This is part of its public service missions agreed with the French State and monitored by the National Regulation Au-thority (NRA) of the energy sector, that justify the financial compensation known as the TURPE. To this end, RTE has a clear interest in anticipating the load and the fatal renewable energy production i.e. the wind and solar productions. Otherwise, RTE is assisted by the so-calledresponsables d’équilibre (responsible for the
equilib-between the electricity injected on the high-voltage network and the electricity ef-fectively consumed.
Safety of the system flow management The equilibrium must also be satisfied locally to ensure the feasibility, determined by the capacity of the lines, of the production planning and the electricity transfers. The resilience of the network is commonly assessed by its capacity to resist the default of a couple random electrical lines. This requires in particular to estimate the future loads at different crucial nodes of the system including the substations, key interfaces between the high-voltage network and medium or low-high-voltage networks.
Maintenance planning Maintenance is a necessity for RTE to ensure the safety of the network. Either predictive or corrective, it often requires to disconnect elec-trical lines. In such a case, the load forecasts permit to check the feasibility of the energy flow planning and ensure the robustness of the network, in spite of the supposedly offline part of the grid.
Loss reduction Finally, the anticipation of electricity demand is necessary to consider the optimization of energy flows and losses, in terms of distance travelled between the production and the consumption sites illustrated in Figure 1.2. The losses due to the Joule effect are indeed more or less proportional to the distance traveled. Although they are reduced thanks to the high voltage, they oscillate between 0.7 GW and 3 GW. Note that this physical phenomenon is not the only cause for energy losses on the network, the iron losses occurring in transformers being of the order of 0.1 GW.
1.2.3 Original motivation of this work
In addition to the importance of accurate forecasts of the local loads in the decision-making processes of RTE, this work is generally motivated by the current dynamic of the Machine Learning community working on forecasting models [Hahn et al.,2009;Kyriakides and Polycarpou,2007;Muñoz et al.,2010;Weron,2007], the new availability of large datasets [Hong and Fan, 2016; Hong et al., 2014] and the accessibility to more computational power, making precisely possible the considera-tion of these datasets. The important impact of the temperatures on the electricity demand also makes essential the quality of local weather forecasts, constantly im-proved during the last decades.
All the factors above motivate the common idea that the results on existing fore-casting problems can be improved thanks to the development of modern scientific tools. However, we explain in this manuscript that the local load curves can be sig-nificantly different from the national or the regional loads since they do not benefit from the same smoothing effect due to the aggregation. Therefore, their volatility is higher, even though a lot of similarities can be observed between the substations.
They have also been less studied and their relationships with the weather and the calendar information are not as well understood. As a first consequence, the models developed to predict the national load may be inadequate at the local levels.
Sec-ondly, computational power is not the only extra ingredient required for local load forecasting.
Instead, the ambition of this work is to characterize the similarities observed at the level of substations and propose a modeling able to benefit from them, in terms of numerical accuracy and computational time. This manuscript addresses the following question :
Is it possible to leverage the similarities between local loads to improve the forecasts with coupled models ?
Organization of the manuscript In Chapter 2, we present the database pro-vided by RTE and Météo-France in order to specify the problems we are interested in. We also present related work and a preliminary data exploration allows us to justify our approach.
Chapter 3is dedicated to load forecasting models where each time series is dealt with independently from the others. We propose a modeling based on B-splines for univariate effects and products of B-splines for bivariate effects, which leads to a standard bivariate linear model. After casting and solving the optimization problem both at the national and at the local levels, we propose an analysis of the results to highlight the difficulties encountered in the modeling and justify the multi-task approach of Chapter4.
Illustrating the models learned and their residuals in the independent setting lets us relate the local load forecasting problem with different multi-task approaches presented with related work at the beginning of Chapter 4. We study three differ-ent multi-task models. The first one assumes that the coefficidiffer-ent of the models for different substations are close in a geometrical sense. It is based on a clustering method. The second approach is geometrical too but only assumes that the coeffi-cients learned for the different models lie in a low-dimensional space. This leads us to considering an optimization problem with a low-rank constraint, which is studied in details in Chapter 5. The analysis of the convergence to critical points and the linear convergence in a neighborhood of the optimal set has been published at the International Conference on Artificial Intelligence and Statistics in 2019. Finally, the third approach presented in Chapter4has two motivations. It is both an attempt to leverage the correlations between the residuals observed with the independent mod-els and a proposition to have local forecasts consistent with the aggregated forecasts, at the national or regional level, meaning that the sum of the local forecasts must be a reasonable forecast of the aggregated electricity demand.