## 2 Methodology
### Identification of Consumption Patterns
This reduction reflects the lower heating or air conditioning (AC) use during those hours. Essentially, it adjusts the probability model to account for the fact that heating and AC are typically used less at night, resulting in a lower overall probability of demand for these systems in those periods. This factor takes values from 0 and 1. In this case, an average value for Fnight of 0.6 has been obtained from 1 AM to 7 AM for heating, while this value was 0.3 from 1 AM to 8 AM in the winter. This function was chosen because it represents a classic exponential model, allowing the probability to be expressed as a function of the temperature difference (\(\Delta T\)). The approach is justified because heating usage tends to be a non-linear response to the difference between the ambient temperature and a reference temperature, increasing rapidly as the temperature drops below a certain level. The \(\Delta T\) for heating can be defined as follows:  
\[\Delta T=T_{ref}-T_{amb} \tag{10}\]  
where  
* \(T_{ref}\) is the reference temperature (the temperature at which the respective heating or air conditioning systems are activated). * \(T_{amb}\) is the ambient temperature measured at a specific time. The greater the difference \(\Delta T\), the higher the probability of heating usage. If \(T_{amb}>T_{ref}\), meaning when the ambient temperature exceeds the reference temperature, \(\Delta T=0\) is set, and thus the probability of heating usage becomes zero. The approach used for air conditioning is similar to that of heating, with the difference that the calculation of \(\Delta T\) follows the inverse relation:  
\[\Delta T=T_{amb}-T_{ref} \tag{11}\]  
In this case, the probability of air conditioning usage increases as the ambient temperature \(T_{amb}\) exceeds the reference temperature \(T_{ref}\). If \(T_{amb}<T_{ref}\), which means that the ambient temperature is lower than the reference temperature, \(\Delta T=0\) is set, nullifying the probability of the use of air conditioning. In the exponential function, the parameter \(\alpha\) plays a crucial role in determining the sensitivity of the probability with respect to the temperature difference \(\Delta\)T. In this case, after testing other options from 0 to 1.5 with a step of 0.1, \(\alpha=0.5\) was chosen. When \(\alpha\) was closer to zero, the sensitivity the demanded power to \(\Delta T\) was very low, so that flatten load curve for this process was obtained and so, the underestimation of the real response of consumers with temperature. On the other side, when \(\alpha\) was closer to 1 and higher, the probability to increase or reduce the power of this process was very sensitive to \(\Delta T\) even for small temperature variations, quickly modifying the value of power. Therefore, an intermediate value of \(\alpha\) was chosen, reflecting a moderate sensitivity to the increase in \(\Delta T\), which is consistent with the idea that the use of heating or air conditioning does not increase too abruptly but follows a more gradual pattern. After calculating the probability of heating usage for each hour of the day in January and air conditioning usage for the month of July, as described previously, variability was introduced for all the households. For each hour, instead of using a single calculated probability value P, a random probability value was generated within 90% to 110% of the initially calculated probability, meaning within the range [0, 9 \(\cdot\) P; 1, 1 \(\cdot\) P]. This allows the probability for each household to vary within this range, simulating diversity in consumption behavior across different houses. After calculating the probability for each hour, the values were normalized. This means that all the probabilities for each hour were summed, and then each individual hour's probability was adjusted by dividing it by this sum. This process ensures that the total sum of all hourly probabilities equals 1. Normalizing the values creates a comparable scale, allowing heating or air conditioning usage patterns to be easily compared with other end uses.