Problem settings

Having presented the database and the behaviors of the different time series, we now describe more precisely the forecasting problems considered in this work. A load forecasting problem is defined by a perimeter, a time horizon and the available information. In this manuscript, we are only interested in day-ahead load fore-casting but we distinguish 2 possible settings for the available data and 5 different aggregation levels.

2.5.1 Middle-term and short-term models

The shorter the time horizon, the more recent information is available to forecast electricity loads. While more settings can be considered, we only introduce two problems that seem particularly relevant for the TSO. In short, they differ on the availability of the recent loads among the inputs and have consequently distinct use cases.

Middle-term forecasts The middle-term model is only based on exogenous in-puts : the forecasts depend on the calendar variables and the meteorological condi-tions but do not depend on the past loads. With the adequate weather scenarios, a middle-term model can reasonably be used in practice to forecast the load several days or weeks ahead, the main limitation being the accuracy of weather forecasts for long time horizons. Nevertheless, a middle-term model can also serve as an estima-tor of the average load during this period if average weather conditions are available for a future time period of the year. More generally, it provides a simple tool to analyze the relationships between the weather, the time and the loads.

Short-term forecasts In addition to the information available to a middle-term model, a short-term model also has access to endogenous information through the recent values of the target time series. It constitutes a significant advantage over the middle-term model and clearly impacts the performances.

While the relationships of the loads with the weather or the economic activity are relatively well-understood, the introduction of the past loads in the inputs has a less clear interpretation. Certainly, it provides information on the economic activity that are not contained in the calendar data but we could equally conjecture that it informs the model of a sensitivity to other weather conditions. Improvements with the short-term models could also indicate that the expressiveness of the middle-term model is inappropriately limited. We do not go further in this interpretation and use the model with the best performances, considering that the past loads provide complementary information.

2.5.2 Aggregation levels

Aggregated Loads Because the priority of the TSO is to ensure the supply and demand equilibrium, the first load forecasting problem to be addressed was for the national demand. A few years ago, this work was extended to forecast the load of the administrative regions and more recently, the load forecasting at the substations

level was considered. In this manuscript, we are especially interested in the latter setting but introduce two more intermediate aggregation levels, respectively defined by the organization of the network and by the position of the weather stations.

Their goal is to enrich the set of possible problems.

An aggregation level of the K substations in the database is characterized by a partition Z := (Zk)k∈[[1,K]] of the K substations into K zones. Given κ∈[[1,K]], we denoterκ the load of substation κ and for k ∈[[1, K]], we denote :

`k := X

κ∈Zk

r_κ, (2.6)

the sum of the loads in zone Zk. For simplicity and because we do not consider different aggregation levels simultaneously, we omit the partitionZ in the notation.

Weather information Given a zoneZk in a partitionZ, the weather information extracted from the W = 32 weather stations and injected in the modeling of the electricity load ofZk should obviously depend on Zk. We consider two possibilities.

First, to model the load in the zoneZk, we can consider a linear combination of all the weather stations with weightsα∈[0,1]^W such thatP_W

s=1α_s = 1, we denote the mean of the temperatures weighted by the vectorα :

T^α:=

XW s=1

αsT^s, (2.7)

whereT^s is the temperature at the weather station s∈[[1,W]]. The same notation is used for the cloud covers :

c^α:=

XW s=1

αsc^s. (2.8)

Such linear combinations are used for instance for the operational forecasting model of the national load.

In the second case, the zone Z_k is associated with a subset Wk ⊂[[1,W]] of the weather stations and the forecasts of the aggregated load in zoneZkonly rely on the weather information extracted from the weather stations in Wk, the temperatures at the different substations being injected in the models as distinct inputs.

National setting In the national setting, the partition is made of a single set that contains all the substations and the goal is to forecast the sum of all the loads :

`national :=

XK κ=1

rκ (2.9)

In the historical national model, RTE has decided of the linear combination given in Table F.1 of the 32 weather stations [RTE, 2011]. Thereby, a single fictive tem-perature obtained as a weighted average of the 32 weather stations is used for the national setting.

Substations level The local forecasting problem corresponds to the prediction of the loads(r_κ)_κ=1,...,K, at each individual substation illustrated in Figure2.26. Conse-quently, we set for the substations levelK =Kand for eachk ∈[[1,K]],`k :=rk. It is not relevant to consider for each substation all the weather stations or the previously defined weighted mean of the weather stations. Instead, we consider for a substation k ∈ [[1, K]] the two weather stations s^k₁, s^k₂ ∈ [[1,W]] that are geographically closest to the substationk. This decision is discussed in Section 3.6.5.

FIGURE 2.26: Voronoi diagram of the substations

Note that the size of each area is not proportional to the load. Large cities correspond to regions with a high density of substations that have consequently small areas on the map.

The time series at the level of the substations are much noisier. They may also present heterogeneous behaviors that make the local load forecasting problem sig-nificantly different from the national problem. For this reason, we introduce inter-mediary settings.

RTE regions Based on the topology of the high-voltage network, the TSO parti-tioned the country in 7regions that we denote N₁, . . . , N₇. In order to forecast the aggregated loads in one of these regions k∈[[1,7]], we use the two weather stations

that are closest to the center of the region. Thus we obtain 7 aggregated loads (`_k)_k=1,...,7 as presented in Figure 2.27 and Table2.1.

FIGURE 2.27: Map of the 7 RTE regions

Map of the 7 Regions defined by the high-voltage network. First we have computed the Voronoi diagram of the substations and secondly painted each area with the color of the corresponding regions.

Administrative regions The 12 administrative metropolitan regions of France that we denoteA1, . . . , A12 form a slightly thinner partition of the country and have consequently slightly noisier load time series (`k)k=1,...,12. They are described in Figure 2.28 and Table 2.1. The weather stations associated to each administrative region are also the two closest to the center of the region.

FIGURE 2.28: Map of the12metropolitan administrative regions

Districts We introduce32districtsD1, . . . , D32 that form a partition of the whole set of substations, defined by the Voronoi diagram of the weather stations presented

in Figure2.29. As explained in Table2.1, this setting lies between the administrative setting and the local setting. Although the corresponding time series(`_k)_k=1,...,32are more sensitive to the local weather conditions than for the coarser regions, noise is reduced compared with the loads of the individual substations. To simplify, we consider that each district only has access to the temperature of the associated weather station.

FIGURE 2.29: Map of the 32 districts

National RTEregions Administrative

regions Districts Local

K 1 7 12 32 1751

K/K 1751 250 146 55 1

`¯(MWh) 39 000 5 500 3 200 1 200 21

|S| 1 2 2 1 2

TABLE 2.1: Characteristics of the different aggregation levels The number of zones for an aggregation level is denotedK and the average number of substations per zone is K/K. The average hourly load of the zones is denoted `¯and the number of weather stations that we use to model the load in each zone within an aggregation level is denoted |S|. Note that the unique station used at the national level is fictive and obtained with the linear combination of Equations (2.7) and (2.8). Besides, the choice of the number of weather stations at the local level is discussed in Section 3.6.5.