# Multi-Objective Electric Vehicles Scheduling Using Elitist Non-Dominated Sorting Genetic Algorithm
## 2 Optimization Problem and Methodology
### Objective Functions  
The considered objective functions intend to optimize the use of DER, in concerning cost optimization and GHG emissions minimization and, therefore, the overall distribution system operation. The most traditional objective function intends to reduce the operation costs considering the costs associated with the use of all types of DER, for instance, distributed generation (DG), the energy provided by external suppliers (ES), the costs associated with the charge and discharge of electric vehicles (EV) and electric storage systems (ESS). The costs associated with the activation of demand response programs were also considered for each generic load connected to the network. When the load does not have demand response (DR) contracts, the activation is limited to zero. Additionally, a penalization cost related to the service not supplied (SNS) is included. This service means that the aggregator is unable to provide services for grid users such as supply energy to the consumers and battery charging, as well as absorbing the energy injected by generators and battery discharge. These costs are depicted in Equation (1).  
In addition to this objective function traditional approach, we considered a quadratic function to determine the GHG emission cost [12] (Equation (2)). It is important to mention that in the present study only the GHGs produced during the use of the resources are considered. The GHG emissions function is directly related to the fuel consumption by the generators. Therefore, we can build a graphic relating the GHC emissions and the output power of the plant, as shown in Figure 1.  
The building and decommissioning emissions are not considered since these emissions are not dependent on the use and, consequently, should not be considered on the optimal resource scheduling (ORS) problem. Taking this approach in mind, only the emissions generated by the DG aggregated to a VPP and the emissions of the external suppliers are considered. The GHG emissions of EVs and  
Figure 1: Greenhouse gas (GHG) emissions quadratic function example.  
storage systems are not considered in Equation (2), neither the DG based on natural resources, such as wind and sun.  
\[F_{\text{Cost}}=\sum\limits_{t=1}^{T}\left(\begin{array}{l}\sum\limits_{DG=1}^{ N_{DG}}f\Big{(}c_{(DG,t)}\times P_{(DG,t)}\Big{)}+\sum\limits_{EV=1}^{N_{EV}}f \Big{(}c_{(EV,t)}\times P_{(EV,t)}\Big{)}+\\ \sum\limits_{ES=1}^{N_{ES}}f\Big{(}c_{(ESS,t)}\times P_{(ESS,t)}\Big{)}+\sum \limits_{L=1}^{N_{U}}f\Big{(}c_{(DR,t)}\times P_{(DR,t)}\Big{)}+\\ \sum\limits_{ES=1}^{N_{ES}}f\Big{(}c_{(ES,t)}\times P_{(ES,t)}\Big{)}+\sum \limits_{SNS=1}^{N_{ES}}f\Big{(}c_{(SNS,t)}\times P_{(SNS,t)}\Big{)}\end{array}\right) \tag{1}\]  
\[F_{\text{GHG}}=\sum\limits_{t=1}^{T}\left(\begin{array}{l}\sum\limits_{DG=1}^ {N_{DG}}\alpha_{(DG)}X_{(DG,t)}+\beta_{(DG)}P_{(DG,t)}+\gamma_{(DG)}P_{(DG,t)} ^{2}\\ \qquad\qquad\qquad\qquad\qquad+\sum\limits_{ES=1}^{N_{ES}}\alpha_{(ES)}X_{(ES, t)}+\beta_{(ES)}P_{(ES,t)}+\gamma_{(ES)}P_{(ES,t)}^{2}\end{array}\right) \tag{2}\]