# Simplified Energy Model and Multi-Objective Energy Consumption Optimization of a Residential House
## 2 Materials and Methods
### Energy Demand Optimization  
The biggest part of the overall energy consumption of the building is caused by heating and cooling, which correspond with the desired indoor temperature that the controller tries to achieve. An optimization strategy based on the compromise between comfort and price was implemented into the model. The cost function is composed of two terms, namely, the deviation from the comfort temperature and the price paid for the electricity. It is expected that the owner would define the desired indoor temperature. One constant desired temperature was set for the whole horizon in the model. This defined desired temperature was then considered as the best achievable comfort. This comfort has some value for the owner, and deviating from it has a 'price' weighted by parameter r.  
When the main aim of the optimization is to lower the price paid for the electricity consumed by the HVAC, the best result is obtained by turning off the HVAC. When comfort is of the utmost importance, then the price paid for the HVAC is defined only by the circumstances such as outdoor temperature, solar irradiance and the quality of control by the controller. If the owner is willing to make a compromise with the comfort, then the optimization finds the balance between saving money and keeping the comfort within some bounds. These bounds are defined by the user as an acceptable deviation from the desired temperature. Such an optimization approach falls into the category of multi-objective optimization [41]. The problem of multi-objective optimization can be described as finding an optimum for a set of objective functions that are to be optimized simultaneously. This optimization problem can be generally described as  
\[\text{min}\ (\text{f}_{1}(\text{x}),\text{f}_{2}(\text{x}),\ \dots\ \text{f}_{\text{k}}(\text{x})),\ \text{subject to:}\ \text{x}\in\text{X} \tag{12}\]  
where k is the number of objective functions and X is the search space or solution space defined by constraint functions such as inequalities or equalities. These objective functions represent conflicting objectives because improving one objective leads to degrading the other. This means that there is no single solution that would be optimal for all objective functions. A solution then consists of some trade-offs. Figure 16: Box plot for non-absolute pricing error. Such a solution is called the Pareto optimal, when moving away from this solution might lead to improvement in one objective function but degrade the other [41]. It is thus clear that there is no single solution to multi-objective optimization, but rather a set of equally optimal solutions. Such a set is called the Pareto front [42]. Finding a solution to a multi-objective optimization problem can be achieved in different ways as overviewed in [41]. One possibility is to have no information about the preferences among the objectives. In such cases, the method is about finding a set of neutral compromises between the objectives. Another possibility is to subject the solutions to preferences, which can be set by the user or the decision maker before the optimization runs. These methods are called a priori. The optimization runs first and finds a set of Pareto optimal solutions from which the user or the decision maker then chooses one with regard to the preferences. This method is called a posteriori. This work uses a weighted sum method that transforms the formulation (8) into a composite objective function [41]:  
\[\mathrm{U}=\sum\
olimits_{\mathrm{i}=1}^{\mathrm{k}}\mathrm{w_{i}}\cdot \mathrm{f_{i}(x)} \tag{13}\]  
where  
\(\mathrm{w_{i}}\) is the weights;  
\(\mathrm{f_{i}}\) is the objective functions;  
\(\mathrm{k}\) is the number of objective functions;  
\(\mathrm{x}\) is the vector of variables.