## 3 Modeling of the Cost Function: A DER-CAM Case Study
### Proposed two-objective Pareto-based Optimization
\tag{7}\]  
where:  
\(\gamma_{1}\), \(\gamma_{2}\), \(\gamma_{7}\) = shifting constants, \(d\) = day of the week, \(\delta_{t}\), \(\rho_{t}\) = shifting parameters  
Constraints for \(f_{1}\)(\(x\),\(t\)) include:  
\[\theta_{1,h}=\theta_{k,24}\ (\text{battery constraint}) \tag{8}\]  
\[f_{c}\leq gen_{max,der}\text{-}c_{cap,der}\text{-}F_{a}+c_{fix}\ (\text{ DER investment constraint}) \tag{9}\]  
\[P_{gen,pv}+P_{max,pv}\leq P_{cap,pv}\text{-}I_{pv}\ (\text{PV module generation capacity constraint}) \tag{10}\]  
\[P_{gen,der}+P_{max,der}\leq c_{inv,der}\text{-}P_{maxgen,der}\ (\text{ DER maximum power gen. constraint}) \tag{11}\]  
\[P_{gen,der}+P_{max,der}\leq c_{inv,der}\text{-}P_{maxgen,der}\text{-}t_{op,der} \ (\text{ToU constraint}) \tag{12}\]  
\[I_{max,der}=\ P_{gen,der}+P_{sup,der}\text{-}COP_{der}\ (\text{heat-pump operating constraint}) \tag{13}\]  
Constraints for \(f_{2}\)(\(x\),\(t\)) include:  
\[S_{min}\leq S_{op,h,24}\leq S_{max}\ (\text{battery constraint}) \tag{14}\]  
\[e_{d,h,24}\leq\ P_{conv}\ (\text{battery constraint}) \tag{15}\]  
\[e_{c,h,24}\leq P_{conv}\ (\text{battery constraint}) \tag{16}\]  
\[V_{pv}=V_{occ}+V_{m,T}\ (\text{PV module operating constraint}) \tag{17}\]  
\[f_{c}\leq gen_{max,der}\text{-}c_{cap,der}\text{-}F_{a}+c_{fix}\ (\text{ DER investment constraint}) \tag{9}\]0  
\[f_{c}\leq gen_{max,der}\text{-}c_{cap,der}\text{-}F_{a}+c_{fix}\ (\text{ DER investment constraint}) \tag{9}\]1  
\[f_{c}\leq gen_{max,der}\text{-}c_{cap,der}\text{-}F_{a}+c_{fix}\ (\text{ DER investment constraint}) \tag{9}\]2  
\(f_{1}\) represents the constructed cost function based on distributed energy resources (similar to that used in DER-CAM optimization model). \(f_{2}\) specifies the variables for the considered energy sources (PV, battery, and air source heat pump). The two-objective optimization space has been used in order to validate the optimization capability of DER-CAM. Three test cases are considered that have been selected from the DER-CAM optimization by [PERSON] et al. in [PERSON] et al. (2017) for the following cases:  
**Case 1**: Investment is made in solar PV panels with the capability to export excess energy, and a battery, air source heat pump, and load shifting are also utilized. **Case 2**: Investment is made in solar PV panels for self-consumption only, and a battery and an air source heat pump, with load shifting, are also utilized. **Case 3**: Investment is made in solar PV panels with the capability to export excess energy, and a battery and air source heat pump are utilized. **Case 4**: This is the base case without the DER strategy, and all of the supply and demand are met by distribution utility. The three test cases (Case 1-Case 3) have been considered in this paper because they are the optimal cases selected by DER-CAM based on the analysis by [PERSON] et al. in [PERSON] et al. (2017). Parameter variation for the above investment scenarios are compared with the base case in which no investment is made in DER, with all of the supply and demand being made by the energy distribution utility. For the above-mentioned cases, the effect of the utilization of PV panels on the Pareto set for both self-consumption and energy export will also be considered. The Pareto-based genetic algorithm used to optimize the dynamic bi-objective problem specified in Equations (1) and (2) is detailed in Algorithm 1. All of the simulations are done using PlatEMO open source MATLAB-based platform [29]. ```
Input: OF parameters, optimization parameters Output: Pareto optimal set, MIGD Start Specifyf\({}_{2}\)(x,t) based on case (n) Activate input matrices for f\({}_{2}\)(x,t) Perform genetic mutation, crossover, and selection based on specified crossover rate.