# Multi-Objective Electric Vehicles Scheduling Using Elitist Non-Dominated Sorting Genetic Algorithm
## 3 Case Study
### Best Compromise Solution  
After determining the Pareto front by the NSGA-II algorithm, the fuzzy-based method was applied to determine the best compromise solutions taking into account the decision maker's perspective. Before applying this fuzzy method, it is required to set the maximum and minimum limits that VPP admits for each objective function (presented on Table 1). These parameters depend on the VPP strategy for the ORS problem. Therefore, a scenario was defined based on a balanced solution between the economic and environmental perspectives. This results in the limits shown in Table 1.  
Afterwards, the fuzzy-based method was applied for the proposed scenario and the five best compromise solutions are depicted in Table 2. \begin{table}
\begin{tabular}{c c c} \hline
**Objective Function** & **Limits** & **Scenario** \\ \hline Operation Cost & Max-\(F_{1}\)\({}^{max}\) & 26,500 \\  & Min-\(F_{1}\)\({}^{min}\) & 24,500 \\ GHG emissions & Max-\(F_{1}\)\({}^{max}\) & 3400 \\  & Min-\(F_{1}\)\({}^{min}\) & 2800 \\ \hline \end{tabular}
\end{table} 
Table 1: Limits in the objective functions allowing the virtual power plants (VPP) to select the best solution. \begin{table}
\begin{tabular}{c c c} \hline
**Best Solution** & **Cost (m.u)** & **GHG (ton)** \\ \hline
1 & 25,649.69 & 2808.86 \\
2 & 25,689.80 & 2774.97 \\
3 & 25,580.27 & 2832.88 \\
4 & 25,297.04 & 2925.31 \\
5 & 25,265.53 & 2947.30 \\ \hline \end{tabular}
\end{table} 
Table 2: Best compromise solutions. It can be seen that the best solutions are around the minimum GHG of 2800 tons established for this scenario, and they represent a middle-term solution between the cost and GHG emissions for the VPP. The first three best solutions show close results in terms of cost and GHG emissions. The fourth and fifth-best solutions show a lower cost than the first three best solutions but achieve GHG emissions around 2900 tons. One of the first three best solutions could be used if the VPP is more interested in minimal GHG emissions, while the fourth and fifth solutions could be adopted for reaching a lower cost. For this scenario, the energy resource scheduling of the best compromise solution is shown in Figure 6.  
The scheduling, presented in Figure 6 only uses CHP units in the peak hours when the other resources are more expensive, trying to find a compromise solution between costs and emissions. The EVs discharge is also scheduled in these periods reaching a total energy scheduling of 3.2 MWh. It is important to mention that the cost and emissions of energy supplied by external suppliers are also dependent on the requested power. Looking at the total energy resources' availability (optimization research space), it is possible to conclude that in peak hours only the demand response is not scheduled and the CHP is only partially scheduled. The compromise solution presented in Figure 6 is now compared to the solution considering only the single objective functions (extremes points in Figure 5). The obtained diagrams are presented in Figure 7 (minimization of only cost objective function) and Figure 8 (minimization of only GHG emissions objective function). Figure 9 presents the total difference in energy scheduling considering the different objective functions. nondominated sorting genetic algorithm (NSGA-II). The objective function of the optimal resource scheduling minimizes the operation cost and the greenhouse gas emissions of the available resources, including a high number of electric vehicles. These objective functions are strategic for the aggregators as well as for society in general. The main contribution of this paper lies in the modifications of traditional NSGA-II so that it can produce better solutions compared to the traditional NSGA-II and show significantly lower computational time compared to the deterministic mixed-integer nonlinear programming approach. The significant reduction of the execution time allows wider use of the proposed methodology, allowing its integration in industrial decision support tools.