# Multi-Objective Electric Vehicles Scheduling Using Elitist Non-Dominated Sorting Genetic Algorithm
## 2 Optimization Problem and Methodology
### Constraints  
The problem constraints are modelled according to the characteristics of the distribution system and technologies. Taking this principle into account, the constraints are considered regardless of the multi-objective functions used in this study. A detailed description of these constraints can be seen in [10]. First, the multi-objective ORS problems consider an AC OPF to ensure that the results cause no technical constraints to the grid. The OPF allows congestion verification of balanced distribution grids, like voltage limits and line thermal limits. This means that the use of a full AC-OPF also improves the quality of the technical objective functions described in the previous section. For a VPP in centralized management, it is not enough to have a scheduling solution since the VPP also has control over the network in the future paradigm. Thus, incorporation of the AC OPF is fundamental to ensure a feasible solution in terms of network operation. It can also represent an economical saving when compared to the coordination of DERs in the current operation that separates the ORS problem from the OPF [13; 14]. For instance, a resource can be selected by the OPF to solve a constraint that was previously not scheduled, causing an extra cost. On the other hand, the proposed approach can optimize the cost of scheduling extra resources to solve congestion problems. The active and reactive power flow in each branch, and voltage magnitude and angle in each node are determined by the AC OPF. The application of [PERSON]'s current law in each bus leads to the following equations for the active (3) and reactive (4) power:  
\[ \begin{array}{l}P_{G(i,t)}-P_{D(i,t)}=G_{ii}\times V_{i(t)}^{2}+V_{i(t)} \times\sum\limits_{j\in TI}V_{j(t)}\times\Big{(}G_{ij}\cos\theta_{ij(t)}+B_{ij }\sin\theta_{ij(t)}\Big{)}\\ P_{G(i,t)}=\sum\limits_{DG=1}^{N_{i,C}^{i}}\Big{(}P_{(DG,t)}^{i}-P_{(SNS,t)}^{i }\Big{)}+\sum\limits_{ES=1}^{N_{ES}}P_{(ES,t)}^{i}+\sum\limits_{EV=1}^{N_{EV} ^{i}}P_{(EV,t)}^{i}+\sum\limits_{ESS=1}^{N_{ES}^{i}}P_{(ESS,t)}^{i}\\ P_{D(i,t)}=\sum\limits_{L=1}^{N_{L}^{i}}\Big{(}P_{(L,t)}^{i}-P_{(DR,t)}^{i} \Big{)}\\ \forall t\in\{1,\ldots,T\};\forall i\in\{1,\ldots,N_{Bus}\};\theta_{ij(t)}= \theta_{i(t)}-\theta_{j(t)}\\ \end{array}  \tag{3}\]  
\[ \begin{array}{l}\sum\limits_{DG=1}^{N_{loc}^{i}}\Big{(}Q_{(DG,t)}^{i}-Q_{ SNS(L,t)}^{i}\Big{)}+\sum\limits_{ES=1}^{N_{loc}^{i}}Q_{(ES,ES(ES,t)}^{i}- \sum\limits_{L=1}^{N_{U}^{i}}Q_{(L,t)}^{i}=V_{i(t)}\times\sum\limits_{j\in TI _{U}}V_{j(t)}\times\Big{(}G_{ij}\sin\theta_{ij(t)}+B_{ij}\cos\theta_{ij(t)} \Big{)}-B_{ii}\times V_{i(t)}^{2}\\ \forall t\in\{1,\ldots,T\};\forall i\in\{1,\ldots,N_{Bus}\};\theta_{ij(t)}= \theta_{i(t)}-\theta_{j(t)}\\ \end{array}  \tag{4}\]  
For a feasible solution of AC power flow, it is necessary to keep the voltage magnitude (5) and angle (6) between a maximum and a minimum limit. \[V_{Min}^{i}\leq V_{i(t)}\leq V_{Max}^{i};\forall t\in\{1,\ldots,T\} \tag{5}\]\[\theta^{i}_{Min}\leq\theta_{i(t)}\leq\theta^{i}_{Max};\forall t\in\{1,\ldots,T\} \tag{6}\]  
A slack bus was previously selected, and both the reference voltage magnitude and angle are specified for it, typically 1 and 0, respectively. In the distribution network, the upper and lower bounds for voltage magnitude are typically 0.95 and 1.05, and for voltage angle are typically \(-\pi/2\) and \(\pi/2\). A constraint regarding the power flow in the line (connecting bus \(i\) to bus \(j\)), that must be lower than a maximum limit (line thermal limit), is shown in (7). \[ \begin{array}{l}\left|\overline{V_{i(t)}}\times\left[\overline{y_{ij}} \times\left(\overline{V_{i(t)}}-\overline{V_{j(t)}}\right)+\overline{y_{sh(i) }}\times\overline{V_{i(t)}}\right]^{*}\right|\leq S^{max}_{TL}\\ \forall t\in\{1,\ldots,T\};\forall i,j\in\{1,\ldots,N_{Bus}\};i\
eq j;\forall TL \in\{1,\ldots,N_{TL}\}\end{array}  \tag{7}\]  
The power generation can be obtained from DG units, external suppliers, electric storage systems and electric vehicles.