Chapter 3 Real data analysis: Power Consumption in National
3.3 Phase I and Phase II control scheme
3.3.4 Example of Semester with AC data (WAC data)
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3.3.4 Example of Semester with AC data (WAC data)
I. Control bands under each hour
In Phase I, we calculate the control bands by equations (2.7) and (2.8). Figure 3.16 part (a) shows the control bands under each time unit and the electricity consumption in weekday32 from SE data. All the observations from NE data fall in the bands, which means that the subgroups are in-control. We can use this chart to monitor the data.
In phase II, we plot the SE data together with control bands (2.7) and (2.8). Table 3.11 shows the time points that electricity consumption are out of the control bands. It represents that six of seven weekdays are detected to be out-of-control due to the special events. Only weekday 35 can’t be detected.
To monitor the variance, we calculate the control bands of MS by equations (2.9).
Figure 3.16 part (b) shows the control bands of MS under each time unit and the MS of weekday32 from SE data. All the MS from NE data fall in the bands, so we can use this chart to monitor. In phase II, we plot the SE data into control bands. It represents that six of seven weekdays are detected to be out-of-control due to the special events. Only weekday 35 can’t be detected as Table 3.11 shows.
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Table 3.11 Table of the OOC time point by control bands under each hour for WAC
Weekday Out-of-control time point (t)
Weekday14 24, 35
Weekday15 67
Weekday31 15, 22, 23, 24, 47, 50, 86, 87, 108~112 Weekday32 11, 15~17, 33~43, 58~65, 83~89, 107~112
Weekday33 9~23
Weekday34 24, 105~118
Weekday35 None
Figure 3.16 Plot of control bands under each hour in Phase I for WAC data
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II. Confidence bands based on the Kalman Filter approach
To construct the chart in Phase I scheme, we use the model (3.2) to apply to NE data. To the Kalman Filter method, the State-Space equation for each weekday can be expressed by equations (2.11) and (2.12) where
Φi = functions “Kfilter0” and “Ksmooth0”, we get the estimate of the state variable 𝛼̂𝑡𝑖|𝑇 and 𝑃̂𝑡𝑖|𝑇, i=1,2,…,7, t=1,2,…,120. Confidence bands can be calculated by equations (2.13) and (2.14). All the fitted values fall in the confidence bands based on Kalman Filter approach.
Figure 3.17 shows the confidence bands based on Kalman Filter approach in Phase I and the electricity consumption in weekday32 from SE data.
In phase II, we apply model (3.2) to the SE data, and plot the fitted values of a profile into equations (2.13) and (2.14). According to Table 3.5, the data of weekday14, weekday31, weekday33, weekday35 can’t be applied to model (3.2), so we plot the observations into the confidence bands. Table 3.12 shows the time points that fitted value of a profile are out of the bands. It represents that six of seven weekdays are detected to be out-of-control due to some special events. Only weekday35 can’t be detected.
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Figure 3.17 Plot of CB based on Kalman Filter approach in Phase I for WAC data
Table 3.12 Table of the OOC time point by CB based on Kalman Filter approach for WAC
Weekday Out-of-control time point (t)
Weekday14 10, 13~19, 34~45 Weekday15 60, 67
Weekday31 14~ 22, 45, 46, 50, 59, 63, 64 86, 87, 107~112 Weekday32 11, 14~18, 33~44, 58~66, 83~90, 107~113
Weekday33 8~23
Weekday34 105~118
Weekday35 None
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III. Confidence bands based on Bootstrap approach
To construct the confidence bands in Phase I scheme, we use the model (3.2) to apply to NE data. We calculate the confidence bands by equations (2.15) and (2.16). Figure 3.18 shows the confidence bands based on Bootstrap approach and the electricity consumption in weekday32 from SE data. All the fitted values of profile for NE data fall in the bands. It means that the subgroups are in-control. We can use this confidence bands to monitor the future data of subgroups. In phase II, as we do in Kalman Filter approach, we plot the fitted value of profile for SE data into equations (2.15) and (2.16). Table 3.13 shows the time points that the electricity consumption are out of the confidence bands. It represents that six of seven weekdays are detected to be out-of-control due to some special events. Only weekday35 can’t be detected.
Figure 3.18 Plot of CB based on Bootstrap approach in Phase I for WAC data Table 3.13 Table of the OOC time point by CB based on Boostrap approach for WAC
Weekday Out-of-control time point (t)
Weekday14 13~18, 23, 34~45 Weekday15 60, 67
Weekday31 14~ 23,44, 45, 46, 50, 59, 63, 64, 86, 87, 107~112 Weekday32 8, 11, 14~18, 33~44, 58~66, 83~90, 107~113
Weekday33 9~23
Weekday34 8, 23, 105~118
Weekday35 None
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To monitor the variance of residuals, Figure 3.19 part(a) shows the plots of the values of 𝜎̂𝑎𝑖 for each subgroup. The UCL is 27.36. Two of the points exceed the UCL from NE data.
These are false alarm cases in the Phase I control chart. In phase II, from Figure 3.19 part (b), three of seven weekdays’ data exceeds the UCL in s control charts, which indicate that these data are out-of-control. Fitting these data to the in-control model (3.2) leads to violating the stationary condition and reach no estimation results. Due to the parameter estimate problem for the remaining four weekdays’ data (we denote “NA” in the control chart), we can’t fit the same in-control profile model (3.2).
Figure 3.19 Plot of standard deviation of the residuals for WAC data
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the coefficients of a profile and variance of the residuals for NE data. The UCL for log (𝑇12) is 2.925, and the UCL for log ( 22) is 5.163. All the statistics are under the upper limit, except for that the log (𝑇22) of weeday16 (wd16) exceeds UCL. It is a false alarm case in the Phase I control chart. So we can conclude that model (3.2) can be used as an in-control profile data, and six of the seven weekday’s data are in-control data.To compute the values of log (𝑇12) and log (𝑇22) statistics, we use the corresponding
estimates δ̅ and 𝜎̅𝑎2 in the Phase I scheme. From Figure 3.21, three of the seven weekdays’
data exceed the UCL in both log (𝑇12) and log (𝑇22) control charts, which indicate that these data are out-of-control. Fitting these data to the in-control model (3.2) leads to violating the stationary condition and reach no estimation results. Due to the parameter estimate problem for the remaining three weekdays’ data (we denote “NA” in log (𝑇12) and log (𝑇22) control chart), we can’t fit the same in-control profile model (3.2) to get the estimation of coefficients.
It is reasonable to infer that these weekdays are not suitable for model (3.2), so we can’t calculate log (𝑇12) and log (𝑇22) statistics. They have the better time series model which is different from in-control model (see Table 3.14), so they can be identified to be
out-of-control.
Table 3.14 Table of best time series model in Weekday14, 31, 33, 35 for WAC data Weekday Best seasonal ARMA time series model
14 Sarima(3,2)×(1,0)24
31 Sarima(1,1)×(1,0)24
33 Sarima(2,2)×(1,0)24
35 Sarima(2,2)×(1,0)24
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Figure 3.20 Plot of the values of log(T2) statistic for each weekday in Phase I for WAC data (a) log (𝑇12) (b) log (𝑇22)
Figure 3.21 Plot of the values of log(T2) statistic for SE data for WAC data
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In this section, we compare the performance of four methods. In Phase I, as we mention in Section 3.3, for WOAC and WAC data, all the points fall in the bands (or limits) based on four different methods. Therefore, we can conclude that NE data are in-control, so we can use these four charts to detect the special-event weekdays’ data. As one can see, both of the confidence bands based on Kalman Filter approach and Bootstrap approach are similar. We plot out-of-control data together with the confidence bands, if the fitted points fall outside the band at some points, we issue an alarm. As Table (3.15) and Table (3.16) show, all the weekday data can be identified to be out-of-control except weekday6 and weekday35. It indicates that four methods are capable of identifying any change from the model.
The following represents some special-event weekdays’ data to show the capable of identifying to out-of-control. Figure 3.23 represent that some electricity consumption points fall outside the bands due to the abnormal temperature. For weekday8, temperature exceeds 30℃ that rise the electricity consumption. The possible reason to the abnormal temperature is seasonal change in this week. Figure 3.24 shows that when there is a national holiday in a weekday, the electricity consumption becomes lower than the lower band. We can infer that some of students return to their hometown that makes the electricity consumption decrease.
Figure 3.25 part (b) and part (c) represent that the charts can detect two sharp points at time 60 (Wed. 12 am) and time 67 (Wed. 19 pm), while part (a) can only detect one of the two points (time 67 ). From Figure 3.26, weekday 31 is the first weekday in Fall Semester, but the first two days in the chart are during the summer vacation that make the electricity
consumption lower, while third, fourth, fifth peaks are higher than UCB. Because September is an extreme hot month that rises the consumption of the electricity and the use of power is related to temperature, some values are exceed the UCB, which are also shows in part (a) to (c). The first peak in Figure 3.27 is lower than the other peaks due to Super Typhoon Jangmi