2.3.6. Predicting increases in usage: 12 pm-5 pm (5-hour) weekday peak demands

Next, we used similar methods to analyze and forecast the weekday 5-hour-peak-demand (12 pm – 5 pm) as a function of the two factors (WBT_12pm-5pm and DCC_Avg7Day). The increase was defined as follows:

where ypeakincior(ii) denotes the increases of the 5-hour-peak-demand from 2019 to 2020, each determined as the difference between the observed peak demand in 2020 ypeak2020ior(ii) and the modeled peak demand in 2019 y^peak2019ior(ii) (modeled for the respective WBT_12pm-5pm observed in 2020; see section 2.3.2). Again, the superscripts “(i)” or “(ii)” represent the two cases of no cooling required (N = 105 observations) and cooling required (N = 69 observations), respectively.

For DCC_Avg7Day, Fig. 11 (a) reveals an approximately logarithmic trend, similar to the one for increases in 8-hour-electricity-use in Fig. 9 (a). The relationship with WBT_12pm-5pm shown in Fig. 11 (b) is similar to a step function, as above. Therefore, we chose again to set the dependence of the increases in 5-hour-peak-demand on WBT_12pm-5pm to zero. The final model is as follows:

(a) Increase in weekday 5-hour-peak-demand (12 pm – 5 pm) vs. CDD_Avg7Day. (b) Same vs. WBT_12pm-5pm. Data points are for times when cooling is not required. The red line in (a) shows the increase in weekday 5-hour-peak-demand as predicted by the regression in Eq. (12a), and the R² in (a) represents the evaluation result of the intermdiate regression in Eq. (12a), not however the prediction accuracy of the final model 3. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Model 3:

where y^peakinc(i) denotes the predicted increase in the 5-hour-peak-demand, and y^peak2020(i) denotes the predicted 5-hour-peak-demand in 2020. β3.1 and β3.2 are the two coefficients of the logarithmic regression model, and DCCAvg7Day is as above. The corresponding statistical metrics and modeling performance are shown in Table 4, Table 6 , respectively.

Coefficients for Model 3 (prediction of the 5-hour-peak-demand when cooling is not required.). 95% confidence intervals of the coefficients are reported in parentheses. N denotes the number of data points in the regressions.

The relationships for increases in 5-hour-peak-demand in Fig. 12 are similar to what we described for the increases in 8-hour-electricity-use: (i) When cooling is required, only considering DCCAvg7Day is not sufficient to predict the increases. Instead, the impact of WBT_12pm-5pm must be considered as well; (ii) a logarithmic and exponential transformation can be used to maximize the forecasting accuracy of the subsequent linear regression model. The factor transformations were as follows:

where DCCAvg7Day, DCCAvg7Daylog, WBT12pm-5pm and WBT12pm-5pmexp are as defined above. c1, c2, d1, and d2 are the coefficients of the log and exponential transformations. The maximum operator in Eq. (13) sets a zero floor for the transformation so that the subsequent regression model does not yield negative predicted values.

(a) Increase in weekday 5-hour-peak-demand (12 pm – 5 pm) vs. CDD_Avg7Day. (b) Same vs. WBT_12pm-5pm. Data points are for times when cooling is required. The data points in the black dashed circle represent weekdays with relatively high WBT (approximately 24 °C) and relatively low daily case numbers. The datapoints in the black solid circle represent weekdays with relatively low WBT (about 20 °C) but relatively high daily case numbers. The red line in (a) [(b)] shows the prediction of a single factor model using the transformed DCC_Avg7Day [WBT_12pm-5pm]. The R² in (a) [(b)] represents the evaluation result of the intermediate regression using the transformed DCC_Avg7Day [WBT_9am-5pm], not however the prediction accuracy of the final model 4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Next, the two transformed variables DCCAvg7Daylog and WBT12pm-5pmexp were used as independent variables in a two-factor linear regression model to forecast the increase in 5-hour-peak-demand, as follows:

Model 4:

where, y^peakinc(ii) denotes the predicted increase in the 5-hour-peak-demand, and y^peak2020(ii) denotes the predicted 5-hour-peak-demand in 2020. β4.1, β4.2, and β4.3 are the three coefficients of the 2-factor linear regression model, whose statistical metrics and modeling performance are shown in Table 5, Table 6 , respectively.

Coefficients for Model 4 (prediction of the 5-hour-peak-demand when cooling is required.). 95% confidence intervals of the coefficients are reported in parentheses. N denotes the number of data points in the regressions.