# A Thermal Discomfort Index for Demand Response Control in Residential Water Heaters
## 6 Application of the Proposed Model: TDI Calculation
### Scenario #3: Setting \(\rho\) to Tune the Interruption Priority of a Special Water Heater  
In several circumstances, an electric water heater is installed with specific objectives and should not be considered to be interrupted in a demand-management event. An example of this situation is a nursing home, where hot water is used for sanitary purposes. A simple approach to deal with this scenario is excluding the electric water heater from the list of appliances to be interrupted by the energy utility. This approach is not flexible because it only allows inserting or removing a device from the interruptible list and does not allow prioritizing special appliances. However, using the proposed TDI definition, the parameter \(\rho\) is used to independently weight those _special_ electric water heaters. In this experiment, the case presented in Section 6.3 is extended by adding a household (\(H_{E}\)) with an special water heater. The hypothesis of the experiment is that the special water heater in household \(H_{E}\) has a water use profile almost constant throughout the day, and it cannot be interrupted in demand-response events. Figure 10 shows in orange the hourly probability vector of this special water heater. Green and blue dotted lines correspond to the profiles of households with water use concentrated in the morning and in the evening, respectively. The experiment consists of adding the device with the same value of the parameter \(\rho=1\) used in the base experiment and calculating _Ranking\({}_{m}\)_ and _Ranking\({}_{a}\)_. Then, the value of the parameter \(\rho\) for the special device is set to a very large value (\(\rho=1000\)), and differences are analyzed.  
The proposed scenario was studied via simulations. Two special household values were included, \(H_{E}^{(1)}\) using \(\rho=1\) and \(H_{E}^{(1000)}\) using \(\rho=1000\). Table 5 reports the TDI values obtained in the simulations, considering the two special households \(H_{E}^{(1)}\) (highlighted in red) and \(H_{E}^{(1000)}\) (highlighted in orange). Table 6 reports the order defined by \(Ranking_{m}\) and \(Ranking_{a}\) considering \(H_{E}^{(1)}\), and Table 7 reports the order defined by \(Ranking_{m}\) and \(Ranking_{a}\) considering \(H_{E}^{(1)}\)\(H_{E}^{(1000)}\).  
Figure 10: Water use profiles for a day in the special household.  
Results reported in Table 6 demonstrate that the special electric water heater \(H_{E}^{(1)}\) is not ranked at the end of either \(Ranking_{m}\) or \(Ranking_{a}\). A different situation is observed in the results reported in Table 7, which indicate that using the value \(\rho=1000\) only for the special electric water heater \(H_{E}^{(1000)}\), this appliance appears last in both rankings \(Ranking_{m}\) and \(Ranking_{a}\), as expected.  
Overall, results obtained for scenario #3 validate the proposed approach and confirm the effect of modifying the \(\rho\) parameter for some water heaters to allow flexible demand-response strategies to be designed. For instance, a possible strategy is to define a classification of the complete set of households in three classes and associate each class with a different \(\rho\) value to define water heaters that are _not interruptible_, _interruptible if necessary_, and _no restrictions for interruption_. The flexibility of the proposed TDI using different values  
of \(\rho\) facilitates, in a real scenario that is continually changing in structure, performing active demand management to quickly adapt to the needs of the electrical system.