# Multi-Objective Electric Vehicles Scheduling Using Elitist Non-Dominated Sorting Genetic Algorithm
## 3 Case Study
### Pareto front Result  
In this simulation, the external suppliers and CHP units presented in Figure 4 are the ones producing GHG emissions. Table A2 of Appendix A contains the coefficients used in Equation (2). The NSGA-II algorithm executed 1000 run trials and the hypervolume measure was used to select the trial with the most suitable Pareto front. This measure determines the volume of the objective space dominated by a Pareto front of a certain trial. The trial that presents the highest hypervolume is selected as the best one. For the WMINLP, the weighted sum method solved the problem 21 times, and in each run, the weight factor changed from 0 to 1 with a step of 0.05. Figure 5 depicts the Pareto front obtained by both algorithms. In the case of NSGA-II, the Pareto front (i.e., individuals with rank equal to 1) of the initial population is represented by a green line with triangles. The individuals that only minimize the operation cost or the GHG emissions are highlighted with purple and light blue circles, respectively. Each solution presented in the Pareto Front represents the scheduling (active and reactive power setpoints) of all the energy resources. The respective costs and CO2 emissions are obtained through the objective functions. The WMINLP algorithm obtained a Pareto front containing 19 nondominated solutions, while the NSGA-II obtained a final Pareto front with 30 nondominated solutions. In this case, the initial population only had 13 nondominated solutions and the remaining individuals had a rank higher than 1. The fact of obtaining more solutions with a rank equal to 1 proves the effectiveness of the crossover and mutation operators used in the study. These operators helped to obtain a Pareto front well spread along with the objective functions space, as compared to the initial population. More precisely, the proposed NSGA-II method was able to find solutions close to the region near to the minimum GHG point. Figure 4: 37-Bus distribution network with distributed energy resources (DERs) penetration for 2050. Figure 5: Pareto front of min(Cost, GHG). As expected, the WMINLP found better solutions than the NSGA-II algorithm in most of the objective functions space. A small number of the WMINLP solutions are dominated by the Pareto front of NSGA-II; in particular, the ones close to the minimum GHG point. For this region, a small step of 0.01 was tested to find more nondominated solutions. This smaller step enabled the WMINLP to find nondominated solutions in a region where the WMINLP algorithm with 0.05 step was not able to find (but NSGA-II was). However, it increased the number of runs and implied more computation time. Moreover, the solution that minimizes GHG emissions is dominated by the last nondominated solution of the Pareto front obtained by the WMINLP algorithm (counting from left to right in the operation cost axis). This solution was obtained with a weight factor equal to 0.1, which obtains the same minimum GHG, but with a lower operation cost. In terms of execution time, the NSGA-II achieved a mean time of around 779.39 s (i.e., around 13 min), and the initial population was obtained with a mean time around 624.75 s. On the other hand, the WMINLP algorithm solved the problem considering just the linear coefficient for the energy cost (1) and emissions (2) to have a faster response in each run. However, the WMINLP algorithm took a huge time (around 1,018,243.46 s), corresponding to 282.85 h (i.e., around 12 days). The mean time obtained for each run of the WMINLP method was around 48,487.78 s (13.47 h). A single run was made with the quadratic functions in (1) and (2), which resulted in a time around 26 h. This allows the conclusion that the proposed hybrid NSGA-II is the most suitable algorithm for this type of multi-objective problems. From these results, it is also possible to observe that the weighted sum method (WMINLP) needs an unrealistic execution time to produce a result for mixed-integer nonlinear programming problems in general.