Study of the local univariate effects

In this section, we propose a study of the models estimated at the level of the substations. The goal is twofold. First a quick look at the distribution of the coefficients for different substations proves that the local loads do not have exactly the same behaviors as the national load and are not perfectly homogeneous. The parameters of the model should consequently be adapted for this level. Secondly, we see that a common structure is nevertheless encountered in these models and it justifies the ambition of models learned within a multi-task framework in Chapter4.

Since the loads of the substations have different amplitudes, the graphs that we present in this section represent the models estimated to forecast the loads normal-ized as in Equation (3.33).

Cyclic inputs The remarks about the effects of the hour of the week in the national model are also valid for the substations, namely that the estimated effect of the hour of the week and the residuals are largest on Mondays in Figure 3.17.

Marginal quantiles q10 and q90 are relatively close and a similar weekly cycle can be observed for most of the substations.

Similarly, the quantiles of the effect of the day of the year, presented in Fig-ure F.35, show that the quantiles q10 and q90 have a similar shapes, even if this input has a minor effect on the forecasts.

Note that although these illustrations are convenient because they summarize in a single graph the effects estimated for all the substations, they are still limited because they do not guarantee that the effects do not oscillate between quantiles q0 and q100. However, we did observe that for most of the substations, the estimated effects have the same shape as the quantiles and the curves do not oscillate. We do not represent those individual effects to be concise.

Acyclic inputs The quantiles of the estimated effects of the temperature, its de-layed and its extremal values over24hours are illustrated in Figures3.18,3.19,F.36 and F.37. The analogous graphs for a delay of 48 hours are presented in Fig-ures F.38, F.39 and F.40. Again, they exhibit a relationship similar to what is encountered in the national load forecasting model.

Unlike the hour of the week and the day of the year, the temperatures are not cyclic and the fact that quantiles are wide for low and high temperatures is at least partially due to a low density of the data near the boundaries and not only to the fact that load forecasting is more problematic under these weather conditions.

Finally, the quantiles of the effects of the past loads are given in Figure3.20with a delay of 24 hours and in FigureF.41for a delay of 48 hours. Between quantiles q10 and q90, the effect is almost linear in Figure3.20, like in the national model. Besides, the number of 3 knots has been selected empirically. As explained in Section3.6.6, the modeling of this input only requires a couple degrees of freedom.

Christmas period In Figure 3.21, we present a histogram of the coefficients learned to model the effect of the Christmas period. They spread around a small negative values. We indeed expect a slight decrease of the loads during this period.

However, some substations consistently exhibit an augmentation of the electricity

0 25 50 75 100 125 150 175 h (hours)

−0.25 0.00 0.25 0.50 0.75

f3(h)

q0 q10

q50 q90

q100 average

FIGURE 3.17: Local effects of the hour of the week

Quantiles for the estimated univariate effect of the hour of the week at each of the 168 knots. Since the loads at the different substations have different amplitudes the quantiles correspond to the forecasting models for the normalized load, like in Equation (3.33). The mean and the quantile q50 are almost superimposed. Note also that the quantiles q0 and q100 that correspond to the smallest and largest coefficients encountered over the substations are not especially robust.

demand during this period. We believe that this is due to large crowds going to hol-idays resorts in particular in the Alps, as is illustrated in Figure3.22. Note however that low temperatures are observed during this period in the Alps and our causal interpretation for the effect of the Christmas period is not certain.

Timestamp A histogram of the coefficients for the timestamp is presented in Figure 3.23. The coefficients spread around zero with both positive and negative values : although the national load seems to be slightly decreasing over time, there might be variations in both directions at a local level due to different local economic and demographic contexts. Note also that the database is quite short to estimate precisely this effect and the significance of the estimated coefficients has not been tested. The apparent decrease could in particular be due to the fact that the elec-tricity demand is smaller during the second half of the year and the database starts in January and ends in December, 5 years later. The introduction of the timestamp in the models is discussed in Section3.6.5 with Table 3.11.

0.0 0.2 0.4 0.6 0.8 1.0 T^s

−0.2 0.0 0.2 0.4 0.6 0.8

fs 5(Ts )

q0 q10

q50 q90

q100 average

FIGURE 3.18: Local effects of the temperature

Quantiles of the univariate effect associated to the temperature at each of the 9 knots. The mean and the median are almost superimposed. The substations have different amplitudes so the coefficients correspond to the forecasting model of the normalized load. Also, since the substations are associated to different weather stations that have different ranges of possible values, the temperatures that are affinely transformed to lie in [0,1] have not been transformed back to their original values.

0.0 0.2 0.4 0.6 0.8 1.0

FIGURE 3.19: Local effects of the 24 h-delayed temperatures Quantiles of the univariate effect associated to the 24 hours-delayed de-layed temperatures at each of the 9 knots. The substations have different amplitudes so the coefficients correspond to the forecasting model of the nor-malized load. Also, since the substations are associated to different weather stations that have different ranges of possible values, the temperatures that are affinely transformed to lie in [0,1] have not been transformed back to their original values.

FIGURE 3.20: Local effects of the 24 h-delayed loads

Quantiles of the univariate effects associated to the 24 hours-delayed load at each of the 3 knots. Since the substations have different amplitudes, the coefficients correspond to the forecasting models of the normalized loads.

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 Normalized coefficient α1 in front of 1xmas

0 100 200

#substations

FIGURE 3.21: Local effects of the Christmas period

Distribution of the coefficients corresponding to the Christmas period.

0.00 0.05 0.10 0.15 0.20

FIGURE 3.22: Positive coefficients for the Christmas period Positive part max(α1,0) of the coefficients in front of the indicator of the Christmas period for the different substations. Only the positive coefficients were colored. These coefficients are particularly large in the Alps, where many people spend their Christmas holidays.

−0.04 −0.02 0.00 0.02 0.04 0.06 Normalized coefficient γ2 in front oft

0 100 200

#substations

FIGURE 3.23: Local effects of the timestamp

Distribution of the coefficients in the local models corresponding to the timestamp.