# Residential Demand Response Scheduling with Consideration of Consumer Preferences
## 2 Proposed Model
### Mixed Integer Programming (MIP) Model
The second and third objectives of interest are related to the preference satisfaction of households. The satisfaction level of an individual household can be defined as follows:  
\[n_{h}=\sum_{a\in\hat{A}}a_{ha} (h\in\hat{H}) \tag{6}\] \[d_{h}=\frac{1}{n_{h}}\sum_{t\in\hat{T},a\in\hat{A}}\frac{1}{t_{a }}c_{hat}u_{aht} (h\in\hat{H}) \tag{7}\]  
Equation (7) states that the satisfaction of a household will be equal to the average satisfaction of all of the used appliances in the household \(a\). To be exact, the satisfaction level of household \(h\) is equal to the average satisfaction level of all of the used appliances \(a\) employed in \(h\). The satisfaction level for each appliance \(a\) in household \(h\) is equal to the sum of satisfaction levels \(c_{hat}\) over all pf the time periods \(u_{aht}\) in which it is divided by the total time running time of that appliance. We will now define the objective functions related to household preference satisfaction. The first function represents the minimization of the average dissatisfaction for all of the households. In the used notation, a low satisfaction value indicates a low level of dissatisfaction, so we are minimizing the average satisfaction value. This can be done using the following equation:  
\[\mathit{AvgDissat}=\frac{1}{H}\sum_{h\in\hat{H}}d_{h} \tag{8}\]  
The second objective function is to minimize the maximal level of dissatisfaction of all of the households. Since a high satisfaction corresponds to a low satisfaction value, the goal is to minimize the maximal satisfaction value for households. This type of min-max objective can easily be defined as an auxiliary variable \(\mathit{MaxDissat}\) (the approach used in [22]) satisfying the following constraint:  
\[\mathit{MaxDissat}\geq d_{h} (h\in\hat{H}) \tag{9}\]Finally, to full specify the proposed MIP, we need to include constraints for the proper use of appliances as follows:  
\[\sum_{t\in\hat{T}}u_{aht}=t_{a}a_{ha} (a\in\hat{A},h\in\hat{H}:a_{ha}=1) \tag{10}\] \[\hat{l}_{t}=b_{t}+\sum_{a\in\hat{A}}\sum_{h\in\hat{H}}l_{a}u_{aht} (t\in\hat{T}) \tag{2}\]0  
The constraint given in Equation (10) guarantees that the total working time of the appliance \(a\) used in household \(h\) is equal to the operating time of the corresponding type \(t_{a}\). The next constraint given in Equation (11) is used to secure that an appliance can only operate in time periods that have a comfort level greater than zero. Defining the constraints that provide that the execution time for appliances is uninterrupted is slightly more complicated and needs the introduction of an additional binary variable \(s_{tah}\), which indicates if appliance \(a\) in household \(h\) starts working at time period \(t\). Using \(s_{tah}\), we define the following constraints:  
\[\hat{l}_{t}=b_{t}+\sum_{a\in\hat{A}}\sum_{h\in\hat{H}}l_{a}u_{aht} (t\in\hat{T}) \tag{2}\]1 \[\hat{l}_{t}=b_{t}+\sum_{a\in\hat{A}}\sum_{h\in\hat{H}}l_{a}u_{aht} (t\in\hat{T}) \tag{2}\]2 (13) \[\hat{l}_{t}=b_{t}+\sum_{a\in\hat{A}}\sum_{h\in\hat{H}}l_{a}u_{aht} (t\in\hat{T}) \tag{2}\]3 (14) \[\hat{l}_{t}=b_{t}+\sum_{a\in\hat{A}}\sum_{h\in\hat{H}}l_{a}u_{aht} (t\in\hat{T}) \tag{2}\]4  
Equation (13) is used to secure that an appliance has only onestart. Equations (14) and (15) provide that \(s_{aht}\) has a value of one if an appliance started working at time \(t\). As a summary, we will be solving the multi-objective minimization problem given in Equation (1), for one of production cost functions given in Equations (4) and (5). The minimization will be conducted over the set of decision variables satisfying the constraints given in Equations (2), (3), (6), (7) and (9)-(15).