# Simplified Energy Model and Multi-Objective Energy Consumption Optimization of a Residential House
## 2 Materials and Methods
### Energy Demand Optimization
These bounds were considered to be fixed, but it is possible that the user's preferences or willingness to accept discomfort might change during the day. Such changes in the preferences might be reflected in changes to the bounds as well as the weights of the weighted sum (13). The weighted sum (13) acts as the utility function, which is supposed to approximate the user's preference function. Such a function would represent the user's satisfaction with changes to the value of each objective function. As mentioned in [41], the utility function used in the form of a weighted sum is a linear approximation of the preference function. However, the preference function might be non-linear, meaning the preferences might change in the criterion space, for example, if one of the objective functions reaches a certain value. As the objective function is calculated using an iterative algorithm with non-linearities (i.e.,: constraints on the value of u), an iterative method is used for the solution of the optimization problem. The function fminbnd in MATLAB has been used for the optimization [43]. The algorithm is based on a golden section search and parabolic interpolation [43]. This function is a one-dimensional minimizer that finds a minimum for a problem specified by  
\[\mathrm{min\ f(x)},\,\mathrm{such\ that\ x_{1}<x<x_{2}} \tag{20}\]  
where  
\(\mathrm{x_{1}}\) is the lower bound;  
\(\mathrm{x_{2}}\) is the higher bound. The main idea behind the optimization in this paper was to show how the predicted energy price changes with respect to different values of the desired indoor temperature. This reflects a compromise between comfort and price. The pricing range can best be shown by calculating the optimization for different values of the weight of comfort. The resulting pricing range can be seen in Figure 16:0 It is divided into three parts. The first graph shows how the price changes with regard to the value of the comfort weight. The second graph represents a distance from the ideal comfort for different values of comfort weight. It is calculated as the sum of the difference between the desired temperature originally set by the user and the optimized desired temperature. Moreover, the third graph shows how each of the three optimized desired temperature deltas changes for different values of the comfort weight. It can be seen that there is a compromise between the price paid and comfort. By increasing the value of the weight, comfort gains more importance, and the price rises. These graphs also show that there are 4 or 5 main areas the optimization leads to. The desired temperature deltas in the last graph were defined as three deltas used at different times of the day. The different timings represent the times when the user is at home, and when the user is not present, and the last timing represents night-time. The temperature delta, when the user is not present, was used during the day from 11.00 to 17.00. Its range was between \(-4\) and +4 \({}^{\circ}\)C. The delta did not change very much, which was caused by the high temperatures and sunshine during the day. So, the optimization found that changing this temperature delta did not lead to a significant change in the power consumed by cooling since when increasing the desired temperature, the cooling still ran at the maximal level. The temperature delta when the user was present was used at hours from 9.00 to 10.00 and then from 18.00 to 23.00. Its range was set to boundaries at \(-2\) and +2 \({}^{\circ}\)C. When the weight was low, indicating that comfort was not important, this delta was set to \(-2\)\({}^{\circ}\)C. This means that when the desired temperature set by the user was lowered to \(-2\)\({}^{\circ}\)C, that would be 20 \({}^{\circ}\)C in the model scenario. As the weight of comfort increased, this delta lowered until it was 0 \({}^{\circ}\)C. In this case, the desired temperature was 22 \({}^{\circ}\)C, which was the same as defined by the user. The third temperature delta was in the night-time frame and was used between 00.00 and 8.00.