Use of bottom-up techniques at the aggregated level

An important issue for bottom-up techniques, and one which has not been tackled by the literature reviewed, is the relation between the percentage of appliances monitored and the

6_{l og (400) ≈ 6} 7_{l og (0.1) ≈ −2}

accuracy obtained in the estimation of the aggregated energy consumed by the small power loads. To address this, an extrapolation method that provides accurate estimations of the total energy consumed for each of the possible permutations of an individual meter has been proposed. The method is implemented in four stages:

1. In the first stage, the aggregated energy collected by each of the individual meters during the monitoring period is calculated, creating a set of energy values 4.3:

e =©ei

ª

(4.3)

where:

i is the meter counter (i=1,2,....n);

n is the total number of individual meters considered;

and ei is the total aggregated energy, in kWh, monitored by meter i during the period t.

2. In order to consider all the possibilities for the metering granularity, in the second stage of the method all of the possible permutations of the n elements from set e are obtained and the values of their elements summed up, resulting in n-sets of aggregated permuted energy values: pk=©pj ª k= © mk X 1 ei ª k (4.4) where:

k represents the numbers of meters considered in the permutations (k=1,2....n);

• mk= n!/ ((n − i )) i !; for i<n

• mn= n; for i=n

and j is the value counter for the elements of the sets (j=1,2....mk).

3. In the third stage, each of the previous n-sets are extrapolated for estimating the energy that would be consumed if all the 12 meters were considered, this obtained n-sets of estimated total energy values:

Ek=©Ej

ª

k=©pj∗ (n/k)

ª

k (4.5)

4. In the fourth and final stage, the uncertainty8is expressed as a standard deviation of the dispersion of each set Ek, in accordance with Equation 4.6:

σk= s 1 mk mk X 1 (Ej− µk)2 (4.6)

whereµkis the mean of the the set Ek, and is given by Equation 4.7;

µk=

Pmk

1 Ej mk

(4.7)

and the relative standard uncertainty percentage (RUP) is given by Equation 4.8.

RU Pk=σk

µk

∗ 100 (4.8)

8_{In accordance with the EURACHEM/CITAC Guide Quantifying Uncertainty in Analytical Measurement. [142],}

when the uncertainty component is evaluated experimentally, it can be expressed as a standard deviation of the dispersion of repeated measurements.

Once defined, the new proposed extrapolation method is tested by its implementation on the 12 individual desk profiles from Database I (Table 4.2)9. Figure 4.8 presents the RU Pk obtained for

the permutations of 11 meters10.

Figure 4.8: RUP of the aggregated desk energy estimation depending on the number k of meter used

In Figure 4.8, each RU Pkvalue is represented by a blue star and the exponential regression fitting

line in red (i.e., given by the equation y = ba1x

1 ). The fitting line shows relatively small squared errors of prediction (SSE), which means a good fit of the model to the data. The graph presents an exponential decay shape where the RUP values decay exponentially with k, a large initial value is followed by an abrupt collapse which approaches the RUP=0 value asymptotically. Plotting the data on a logarithmic scale rearranged them into a linear regression line, as shown in Figure 4.9.

9_{The shared appliance profiles (fridge, printers, and shredder) are considered separately.}

10_{Note that k=12 is excluded from the implementation of the extrapolation method because to include it would}

Figure 4.9: RUP logarithmic values of the aggregated desk energy estimation depending on the number k

of meter used

In Figure 4.9, each RU Pklogarithmic value is represented by a blue star and the new regression

fitting line is shown in yellow (i.e., given by the equation l og (y) = a2x + b2), with a similar SSE value to the exponential one. Both equations maintain the exponential-logarithmic relationship (i.e., a1≈ a2and b1≈ l og (b2)). This relationship between RUP percentage and the number of meters used means that the variation between consecutive values of RUP is greater for low values of k and decreases dramatically when the number of meters used increases. Translated to the present case study, this means that, for instance, increasing the number of meters from k=1 to k=6 will achieve a reduction in the RUP of nearly 25%, but the addition of more meters would only improve the prediction by a maximum of 10%.

The study of the range in values for the different numbers of monitored appliances also provides significant additional information relevant to the statistical analysis. In Figure 4.10 each blue box represents the variation in samples of one set Ek, and offers a visual presentation of the degree of

dispersion of the different sets which, as shown in the graph, decreases with the increase in the number of meters. On each box, the central red mark indicates the normalised median (Mk)11

for each of the correspondent number of desks monitored, given by the middle value separating the greater and lesser halves of the data set, the bottom and top edges of the box indicate the 25t hand 75t h percentiles, respectively, and the whiskers extend to the most extreme data points without considering outliers, which are plotted individually using the ’+’ symbol. The last ”box” appears as a single line because the set E12contains only one energy estimation value.

The confidence interval limits for each set Ekare defined by Equation 4.9.

C Ik= µk± t1−α/2,mk−1

σk

p mk

(4.9)

whereµkis the mean (considered as the ”true” value by the method),σkthe standard deviation

and mkthe size of each set Ek; andα is the desired significance level, that defines the confidence

coefficient 1 − α. The upper and lower limits for each mean µk provided by a 95% two-sided

confidence interval are also represented in Figure 4.10 by a green and pink asterisk respectively.

Figure 4.10: Box plot graphical of the normalised energy distribution against the number of desks moni-

tored.

The confidence interval is an indication of how much uncertainty there is in the method esti- mation of the ”true”’ value (defined as the mean of the distribution). A low number of desks monitored in Figure 4.10 provides wider confidence intervals, meaning more uncertainty in the method estimation.

The skewness of the Ek set distribution can be also analysed, as a measure of the asymmetry,

or more precisely, the lack of symmetry of the data around the sample mean. This magnitude contains relevant information, indicating the degree of dispersion and existing outliner values in the data, which are values behaving outside general. If the skewness is negative, the data are spread out more to the left of the mean than to the right. If it is positive, the data are spread out more to the right, zero representing a normal distribution. The skewness percentage Skof each

Sk= 1 mk Pmk 1 (Ej− µk)3 ³q 1 mk Pmk 1 (Ej− µk)2 ´3∗ 100 (4.10)

Figure 4.11 represents the skewness percentages for each Ekagainst the number of meters used.

Figure 4.11: Skewness percentage of the energy distribution set against the number of meter.

In accordance with Figure 4.11, only the set E6has a normal distribution, i.e. the possibilities of over-estimated or lower-estimated the results are 50%-50%. The use of more meters will result in the probability of over-estimate the total energy consumption values up to nearly 60% for k=1 (right skewness of the distribution); and the use of fewer meters will result in the probability of an under-estimation of these values down to nearly 60% for k=11 (left skewness of the distribution).