## 3 Case Study  
_Applying the Methodology to Real Data Profile Disaggregation_  
This methodology will be applied to real cases, using actual consumption data provided by the electricity company. Typically, they provide aggregated consumption data from the transformation centers to which the consumers are connected. Therefore, it is necessary to select the specific dates of interest for analysis. To assess the amount of flexible consumption, it is necessary to know the consumption trends for the end uses of interest: washing machines, dishwashers, dryers, heating, air conditioning, and hot water. Since the patterns of these specific consumptions are not known beforehand, they must be estimated based on the total consumption patterns and the probability values obtained from statistical analysis. Next, the procedure used to reconstruct disaggregated consumption profiles by end-use is presented, followed by the results obtained from the flexibility analysis. The method can be described with the following points:  
1. Construction of the probability matrix: For each end use, a probability matrix \(P_{end\_use}(h,i)\) is constructed, where  
* \(h=1,2,\ldots,24\) are the hours of the day (rows of the matrix). * \(i=1,2,\ldots,\mathbb{C}\) represents the consumers for each transformation center (columns of the matrix). 2. Distinction between modified and non-modified end uses: The distinction is made between modified end uses (washing machine, dishwasher, dryer, and hot water) and non-modified end uses (all others). The probability matrices for non-modified end use remain as initially estimated. However, the probability matrices for the modified end uses are adjusted based on the difference between the real measured data and the sum of the non-modified consumption. This ensures that the consumption trends for the modified end uses align with the measured data, providing a more accurate reflection of their actual behavior. 3. Adjustment based on the difference between total consumption and non-modified end uses: First, the total consumption \(E_{metoral}(h)\) for each hour is taken from measured data. Then, the sum of the consumption for all non-modified end uses (i.e., all except washing machine, dishwasher, dryer, and hot water) is subtracted from the total real measured consumption to isolate the portion related to the modified end uses. The non-modified consumption, \(E_{total\_nan\_mad}(h)\), is represented as follows: \[E_{total\_nan\_mod}(h)=E_{metoral}(h)-E_{washing\_machine}(h)+E_{dishwasher}(h)+E_{ dryer}(h)+E_{hot\_water}(h)\] (17) The remaining consumption for the modified end uses is then: \[E_{total\_mad}(h)=E_{metoral}(h)-E_{total\_nan\_mad}(h)\] (18)
4. Calculation of new consumption ratios: Since the non-modified consumption has been subtracted, the consumption ratios for the modified end uses are recalculated. The new consumption ratios (\(c.ratio_{ond\_use\_new}\)), for each end-use, are: \[c.ratio_{ond\_use\_new}=\frac{c.ratio_{ond\_use\_}{\_}{\_}{\_}{\_}{\_}{\_}{10} \] (19)
5. Calculation of kWh for modified end uses: Once the new ratios for modified end uses are determined, the kWh \(E_{end\_use}(h)\) for each modified end-use is calculated as: \[E_{end\_use}(h)=E_{total\_mad}(h)\cdot c.ratio_{ond\_use\_new}\] (20)Calculation of \(P_{sum}\) for modified end uses: The sum of probabilities for each hour \(P_{sum}(h)\) for modified end uses is calculated using the following formula: \[P_{sum}(h)=\sum P_{end\_use}(h)\cdot c.ratio_{end\_use\_new}\] (21)
7.