## 5 Load Segmentation and Aggregation
### The Customer Demand Baseline  
Aggregation is also important in evaluating DR&EE performance according to the objectives that are pursued. DR simulation or real measurements after DR (e.g., through a more effective NIALM [30], considering a higher granularity of data provided by the deployment of smart devices [4]), must be compared to its CBL, which is the theoretical demand of an aggregation of uncontrolled loads. The actual demand can involve penalties if DR resources are consuming less than the defined CBL during the availability periods [31].  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Index** & **Zone** & **Day 1** & **Day 2** & **Day 3** & **Day 4** & **Day 5** & **Day 6** & **Day 7** \\ \hline \multirow{3}{*}{RMSE} & Hot-WH1 & 0.997 & 1.133 & 0.904 & 0.874 & 1.093 & 1.200 & 1.556 \\  & Mixed-WH2 & 1.277 & 1.338 & 1.403 & 1.870 & 1.902 & 2.054 & 0.956 \\  & Cold-WH3 & 1.058 & 1.082 & 0.773 & 0.904 & 0.885 & 0.950 & 0.427 \\ \hline \multirow{3}{*}{MPE} & Hot-WH1 & \(-\)1.135 & \(-\)0.931 & 0.125 & \(-\)0.367 & \(-\)0.573 & \(-\)1.252 & \(-\)2.020 \\  & Mixed-WH2 & \(-\)0.196 & \(-\)0.994 & \(-\)0.850 & \(-\)2.062 & \(-\)1.921 & \(-\)2.319 & \(-\)0.076 \\  & Cold-WH3 & 1.365 & 0.757 & 0.328 & 0.220 & 0.072 & 0.069 & 0.616 \\ \hline \multirow{3}{*}{MAPE} & Hot-WH1 & 1.488 & 1.697 & 1.121 & 1.206 & 1.524 & 1.827 & 2.347 \\  & Mixed-WH2 & 1.864 & 1.468 & 1.611 & 2.273 & 2.260 & 2.652 & 1.088 \\ \cline{1-1}  & Cold-WH3 & 1.448 & 1.932 & 1.476 & 1.698 & 1.776 & 1.945 & 0.719 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Metrics for the evaluation of the model performance (state variables X\({}_{1}\), X\({}_{2}\), X\({}_{3}\)).  
Figure 8: Predicted and real profiles for the temperatures; water draw and power measurements for the tested HPWH (**a**) Prediction with the model. (**b**) Comparison of all WH zones.  
PBLM can improve the evaluation of CBL, especially before and after DR is applied [30]. Some feedback from smart devices to the aggregator (demand, temperatures from some appliances) can also help in this task, such as the evaluation of offline flexible devices into the control groups.  
For the simulation, a QHCG group of 80-110 L HPWH has been defined. It should be remembered that the average size of EWH tanks in Spain is around 90 L (Figure 1). Moreover, the WH manufacturer being considered (ARISTON [32]) cells and distributes residential HPWHs in this range of capacity. Aside from the tank capacity, standard patterns for the use of hot water have been considered, enabling a dispersion in the time of use of water demand. In the literature, other researchers have considered monitored DHW heat use, national and international standard profiles (e.g., [33] for Norway), and the estimated water draws according to specific WH demand profiles to suit specific tariffs [7]. In this paper, the authors have applied profiles based on EN 12831 European Standard to calculate heat loads in dwellings [34]. Figure 9 shows one of these profiles according to the size of the WHs being considered. Notice that the average HPWH load profile roughly represents a proportional reduction in the load shape of ERWH due to load shape diversity (see also Figure 2b).  
This load profile explains a water temperature of at least 55 \({}^{\circ}\)C, which is the usual temperature of HPWH recommended by manufacturers to maintain efficiency (a maximum COP; see Table 2). Obviously, the demand profile presented in Figure 9 also depends on water inlet temperature (in this case, it is assumed to be 20 \({}^{\circ}\)C based on weather data). It depends on the week or season selected for the simulation. This can be done by changing PBLM inputs. Moreover, some efficiency measures can also be analyzed through PBLM (COP improvement, the insulation of the tank, change of patterns, etc.).