Study of the national bivariate effects

The addition of interactionsgenerates more complex models that are more suit-able to describe the relationship between the loads and the inputs. In this section, we illustrate the estimation of these relationships in thenationalshort-term model, although the interpretations that we can make are less obvious than for the univari-ate effects.

Past loads and hour of the week It is essential that the effects of the past loads in the model depend on the day of the week since the relationships between the loads of two consecutive days are clearly different if these days are working or non-working days. We consequently included this interaction and it is illustrated for the national model in Figure3.13. It modifies the effect of the past loads, especially on Mondays (h∈[[0,23]]), Fridays and Sundays (h∈[[120,167]]).

Although it is not intuitive why this bivariate effect is small on Sundays and Mondays, what matters to analyze this interaction is the sum f3(h) +f15,924(`924) + g16,924(h, `924) which is, on the contrary larger on weekdays than on weekends. This sum is represented in Figure3.14.

The analogous graphs for the load with a delay of 48 hours are given in Fig-ure F.29 and FigureF.30.

Cloud cover and daylight The estimated effect of the interaction between the cloud cover and the indicator of the daylight is presented in Figure 3.15. Roughly, the more clouds during the day, the larger the electricity demand while this term has no effect during the night (1sun = 0). Note that this effect is relatively small compared with the effects of the temperatures or the hour of the week.

Holidays and hour of the week The impact of the holidays depend obviously on the day of the week. The interactions between the hour of the week and the indicators of holidays are presented in Figures 3.16, F.31and F.32.

We observe on the learned effect in Figure3.16that the electricity demand during working hours from Monday to Friday is reduced on holidays. During the weekend, the estimated effect is almost constant as expected.

In Figure F.31, it appears on the learned effect that the electricity demand is lower on a Monday if it precedes a holiday, which makes sense given that it often leads to a 4-day-long weekend. Whether we see the inverse relation for Fridays in Figure F.32 is debatable. This might be due to the little number of non-working weekdays in the database.

In fact, the effect1_hld⁺h₁₄(h)in FigureF.32leads to larger forecasts on days that follow a holiday. We believe that it is because the potential decrease of the economic activity on days that follow a holiday is already introduced in the model through the effects related to the past loads.

Day of the year with temperature The interaction in the national model between the temperature and the day of the year is presented in FigureF.33. Clearly, the modification induced by the interaction g^s₁₀(T^s,d) could not be obtained with a sum of univariate functions. However, note that the regions of the input space

FIGURE 3.13: Interaction between past loads and week hours ( top left ) Interaction g_16,924(h, `₉₂₄) between the hour of the week

and the past load.

( bottom left ) Estimated marginal densityµtest(h, `924)of the inputs in the test set.

( top right ) Average residuals in the test set E^test[|`−`ˆ||h, `924].

whereg₁₀^s (T^s,d) is small correspond to a low-density and their interpretation seems difficult.

Day of the year with hour of the week Finally, the estimated interaction in the national model between the hour of the week and the day of the year is presented in FigureF.34. Although this bivariate effect clearly increases the estimated loads on Mondays during summer and reduces the estimated loads on Saturdays and Sundays during the same period, the coefficients are small compared with other effects.

Also, we can see in Figure F.34 a continuous shift during the year of the peak in the morning from Mondays to Fridays. It occurs around 8 a.m. in summer and one or two hours later in winter. This may be partially due to the Daylight Sav-ing Time (DST). However, there is no abrupt transition at the end of March and October, when the shift occurs.

For information, the time in the model and in the plots is always the UTC time and the DST is not explicitly taken into account, any more than with this interaction. Looking closer at electricity load curves near the switch between winter and summer times permits to see a transition but we did not take it into account in the models. We discuss this in more details in Section3.7.3.

10 20 30 40 50 60 ℓ₉₂₄ (GWh)

0 20 40 60 80 100 120 140 160

h(hours)

11 70

(GWh)

FIGURE 3.14: Total effect of the 24 h-delayed loads

Sum of the univariate and bivariate effects β0 + f3(h) + f15,924(`924) + g16,924(h, `924) in the national short-term model.

FIGURE 3.15: Estimated effect of the cloud cover during the day

( 1^st row ) Effect β0 + 1sunh^s₁₁(c^s) of the cloud cover, when the sun is down (1sun = 0) and up (1sun = 1).

( 2^nd row ) Target loadsE^train[`|1sun,c<sup>s</sup>]andE<sup>test</sup>[`|1sun,c^s], with the av-erage forecasts E^train[ˆ`|1sun,c<sup>s</sup>] and E<sup>test</sup>[ˆ`|1sun,c^s].

( 3^rd row ) Marginal norm of the residuals E^train[|` − `ˆ||1sun,c^s] and E^test[|`−`ˆ||1sun,c^s].

( 4^th row ) Density of the cloud cover in the training and the test sets.

25

FIGURE 3.16: Interaction of holidays with hours of the week (1^st row) Interaction β0 + 1hldh12(h) between the indicator of holidays

and the hour of the week.

(2^nd row) Target loads E^train[`|1hld,h] and E<sup>test</sup>[`|1hld,h] with the fore-casts E^train[ˆ`|1hld,h] and E<sup>test</sup>[ˆ`|1hld,h].

(3^rd row) Marginal norm of the residuals E^train[|` − `ˆ| |1hld,h] and E^test[|`−`ˆ| |1hld,h].

The marginal loads, forecasts and residuals are incomplete in the column 1hld = 1 because there was no holiday on Wednesdays and Saturdays in 2016.