# Sample-based estimation of mean electricity consumption curves for small domains Section 1. Introduction and context

Many studies conducted by the French electric company EDF are based on the analysis of the mean curves of electricity consumption by groups of customers who share common characteristics (e.g., similar electrical equipment or a common rate). In this text, these groups will be called domains. These mean consumption curves, also called demand curves, are estimated using a sample of several thousand curves measured at half-hourly intervals over long periods (often years).

In the literature, estimation of a total or mean demand curve for various sampling plans and the construction of confidence intervals has been examined in the recent work of Cardot, Dessertaine, Goga, Josserand and Lardin (2013), Cardot, Degras and Josserand (2013), and Cardot, Goga and Lardin (2013). The estimation of totals or means for functional data raises specific problems regarding the sample estimate of the finite population, as the strong time dependencies of the data must be exploited and preserved.

Here, we will focus on the problem of estimating mean curves for small domains, i.e., cases where we look simultaneously at several subpopulations, which may be small in size. With the advent of smart meters, it will become increasingly easy and less and less costly to create and maintain large samples of demand curves. It will therefore be possible to produce estimates of mean curves not only throughout France, but also for small geographic areas such as regions, departments and even cities. For example, these estimates could be used to propose services based on an analysis of consumption curves in territorial communities or for publication as part of an open data process.

This issue of small domains is frequently addressed in sampling theory outside the framework of functional data. The recent book by Rao and Molina (2015) proposes a state-of-the-art report on existing methods. When the domain is small, direct estimators (i.e., constructed solely from individuals in the sample within the domain) are not very effective. To improve the quality of estimates, auxiliary information is used and estimators are constructed based on implicit or explicit modelling of the link between quantity of interest and auxiliary information, common to all domains. In the context of EDF, this auxiliary information can, for example, be from known billing data (rate, contract power, total consumption in the previous year in particular) for each individual in the population, but also from open data proposed by the INSEE for small geographic aggregates (IRIS).

In the literature, there are estimation methods for small domains specific to temporal series. For example, Pfefferman and Burck (1990) and Rao and Yu (1994) superimpose temporal series models on series of variables and/or coefficients of the various instants to take into consideration temporal dependencies. However, those space-state-type models were developed for relatively short temporal series (a few dozen points). They are estimated using Kalman filters, which require a lot of calculation time, which would present a problem in our context, in which the number of domains studied can vary widely.

To our knowledge, the estimation of small domains in surveys for functional data has not yet been examined in the literature. To address this problem, we propose two types of methods. First, we apply parametric methods such as linear mixed models and functional linear regressions to the coordinates of the projected curves in a finite base, e.g., the base of principal components of a principal components analysis. We also propose two non-parametric methods based on regression trees and random forests adapted to the curves, respectively. All these methods are part of the model-based survey approach.

In Section 2, we formalize the problem and introduce a few notations. In Section 3, we present two direct estimators (the Horvitz-Thompson estimator and the calibration estimator for functional data) that will be the references to which we will compare ourselves to evaluate the performance of our methods. In Section 4, we propose two parametric methods based on unit-level linear functional and mixed models, adapted to the context of the functional data, and two non-parametric methods based on regression trees and random forests. For each method, we also propose a procedure for approximating the bootstrap variance. Finally, in Section 5, all estimation methods proposed in this article are tested and compared to a data set from actual electricity consumption curves for households in France. The conclusions and perspectives are presented in Section 6. In particular, the respective benefits and drawbacks of the various methods are compared in Subsection 5.4.

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