# Simplified Energy Model and Multi-Objective Energy Consumption Optimization of a Residential House
## 2 Materials and Methods
### Energy Demand Optimization
In this paper, the composite objective function is thus defined by a weighted sum formulation:  
\[\mathrm{F}=\mathrm{r}\cdot\sum\
olimits_{\mathrm{k}=1}^{\mathrm{N}}\left( \mathrm{T_{desired\_by\_user}(k)}-\mathrm{T_{desired\_delta}(k)}\right)+\sum \
olimits_{\mathrm{k}=1}^{\mathrm{N}}\mathrm{P(k)} \tag{14}\]  
where  
\(\mathrm{r}\) is the weight of comfort;  
\(\mathrm{N}\) is the number of samples from the weather forecast;  
\(\mathrm{T_{desired\_by\_user}(k)}\) is the desired temperature defined by the user, \({}^{\circ}\mathrm{C}\), in the k-th time step;  
\(\mathrm{T_{desired\_delta}(k)}\) is the optimized desired temperature, \({}^{\circ}\mathrm{C}\) in the k-th time step;  
\(\mathrm{P(k)}\) is the price paid for heating and cooling, in EUR, in the k-th time step. In this case, the variable is the real-valued vector containing the differences of the desired temperature, \(\mathrm{T_{desired\_delta}}\), in the time steps of the forecasting horizon:  
\[\mathrm{T_{desired\_delta}}=[\mathrm{T_{desired\_delta}}\mathrm{_1},\mathrm{ T_{desired\_delta}}\mathrm{_2},\mathrm{T_{desired\_delta}}\mathrm{_3}] \tag{15}\]  
The price for heating and cooling is calculated by:  
\[\mathrm{P(k)}=\mathrm{P_{consumed}}\cdot\mathrm{P_{kWs}}\cdot\mathrm{T_{s}} \tag{16}\]  
where  
\(\mathrm{P(k)}\) is the price paid for heating and cooling, in EUR, in the k-th time step;  
\(\mathrm{P_{consumed}}\) is the power consumed by heating and cooling, kW;  
\(\mathrm{P_{kWs}}\) is the price per kWs consumed, EUR/kWs;  
\(\mathrm{T_{s}}\) is the sampling period, s.  
The price paid for heating and cooling does not have a corresponding weight, or rather, the weight is considered to be equal to 1. This approach would fall into the category of ranking methods, a subset of rating methods, where one of the objectives is given a fixed weight of 1 [41]. As also mentioned in [41], the absolute values of the weights might not represent the actual preferences of the user because the magnitudes of each objective function might vary greatly, thus affecting the resulting composite objective function differently. In such cases, the weights might compensate for these different magnitudes of each objective function more than reflecting the preferences of the user. The optimization calculates \(\mathrm{T_{desired\_delta}}\) so that the objective function in (13) is minimized. Two scenarios have been considered: one in which the value of \(\mathrm{T_{desired\_delta}}\) is constant for the whole forecast horizon (one variable for the optimization problem), and another in which there are three values of \(\mathrm{T_{desired\_delta}}\) used at different times each day (three variables for the optimization problem). These times reflect the presence of the residents inside the home. The value of f is calculated for the whole forecast horizon. If r is equal to 0 then the resulting change in the desired temperature corresponds to minimizing the price on the forecast horizon, so the optimization becomes a parametric optimization instead of a multi-objective optimization. Such a case shows the maximal potential for a decrease in price on the forecast horizon while the optimization is constrained to finding one or three values of \(\mathrm{T_{desired\_delta}}\). Each \(\mathrm{T_{desired\_delta}}\) is considered to be bounded. In this model scenario, they are bounded differently. \[-4\ ^{\circ}\mathrm{C} \leq\mathrm{T_{desired\_delta}}\mathrm{_1} \leq 4\ ^{\circ}\mathrm{C} \tag{17}\] \[-2\ ^{\circ}\mathrm{C} \leq\mathrm{T_{desired\_delta}}\mathrm{_2} \leq 2\ ^{\circ}\mathrm{C}\] (18) \[-3\ ^{\circ}\mathrm{C} \leq\mathrm{T_{desired\_delta}}\mathrm{_3} \leq 3\ ^{\circ}\mathrm{C} \tag{19}\]  
These bounds are used at different times during the days of the forecast. One is used when the user is present, another when he is not, and the third is used at night. The bounds, as well as the timings, can be set by the user and reflect the maximal distance from the desired temperature that the user is willing to accept.