## 3 Methods
### Objective Function  
Followed by the constraints for the proposed eLCVRP-CS-PDS presented in (1)-(37), (54), we seek to minimize \(F\) in (55), which is the objective function incorporated through the summation of the equations shown in (56)-(61):  
\[\min F=F_{1}+F_{2}+F_{3}+F_{4}+F_{5}+F_{6} \tag{55}\]  
\[F_{1}=\left[T\cdot[C_{r}+C_{m}]\sum_{i\in V}\sum_{j\in V}\sum_{k\in K}x_{i,j, k}\cdot d_{i,j}\right]\cdot AF \tag{56}\]  
\[F_{2}=\left[T\cdot[C_{r}+C_{m}]\sum_{m\in V}\sum_{n\in V}\sum_{l\in L}y_{m,n,l }\cdot d_{m,n}\right]\cdot AF \tag{57}\]  
\[F_{3}=C_{c}\cdot\sum_{g\in V}U_{g} \tag{58}\]  
\[F_{4}=\left[T\cdot t_{c}\cdot P_{b}\cdot C_{e}\cdot\sum_{i\in V}Y_{i}^{crrp} \right]\cdot AF \tag{59}\]  
\[F_{5}=\left[T\cdot t_{c}\cdot P_{b}\cdot C_{e}\cdot\sum_{n\in V}\sum_{l\in L} Y_{n,l}^{last}\right]\cdot AF \tag{60}\]  
\[F_{6}=\left[T\cdot t_{c}\cdot C_{e}\cdot\left[P_{loss}-\overline{P_{loss}} \right]\right]\cdot AF \tag{61}\]  
Equations (56) and (57) are the corresponding routing costs of the EVs that follow the CVRP and LMDP, respectively. Notice that routing and maintenance are costed per kilometer traversed \((C_{r}+C_{m})\), and it is assumed that the vehicles repeat the routes on a daily basis over the course of the year, e.g., \(T=365\). Expression in (58) denotes the cost of the charging stations installed. Expressions (59) and (60) are the operation costs related to EV charging operation. Besides parameter \(T\) that relates the operation along the year, other parameters are found in these two expressions, i.e., \(t_{c}\), \(P_{b}\) that correspond to the period of time that the EV is connected while charging and the maximum power drawn from the distribution network, respectively. Additionally, the cost of energy is also considered in the parameter \(C_{e}\). Equation (61) is the difference between the energy losses with charging stations sited and the energy losses in the benchmark case, i.e., when no charging stations are installed in the distribution system. Be advised that the expressions above are in terms of energy cost except for expression in (58) that lies on the cost for charging stations installed. In addition, the discussed expressions are affected by the annualization factor \(AF\) for comparison purposes between the initial investment of charging stations, and operation costs translated into energy costs distributed over the future years. By the other hand, the term \(P_{loss}\) in Equation (61) is computed in (62) for the three-phase system case:  
\[P_{loss}=V_{R}^{T}G_{Bus}V_{R}+V_{I}^{T}G_{Bus}V_{I} \tag{62}\]  
\(V_{R}\) is the real part of the voltage at slack node \(V_{RS}\) and the rest of the nodes \(V_{RW}\), and \(V_{I}\) is the imaginary part of the voltage at slack node \(V_{IS}\) and the rest of the nodes \(V_{IW}\), as shown by (63) and (64). \(G_{BUS}\) is denoted in (65) and represents the real part of the admittance matrix:  
\[V_{R}= \begin{bmatrix}V_{RS}\\ V_{RW}\end{bmatrix}  \tag{63}\]  
\[V_{I}= \begin{bmatrix}V_{IS}\\ V_{IW}\end{bmatrix}  \tag{64}\]  
\[G_{BUS}= \begin{bmatrix}G_{SS}&G_{SW}\\ G_{WS}&G_{WW}\end{bmatrix}  \tag{65}\]  
From (62), it is observed that \(P_{loss}\) can be deployed as demonstrated in (66):  
\[P_{loss}= \begin{array}{c}V_{RS}G_{SS}V_{RS}+V_{RS}G_{SW}V_{RW}+V_{RW}G_{WS} V_{RS}+V_{RW}G_{WW}V_{RW}+\\ V_{IS}G_{SS}V_{IS}+V_{IS}G_{SW}V_{IW}+V_{IW}G_{SS}V_{IS}+V_{IW}G_{WW}V_{IW}\end{array}  \tag{66}\]  
Notice that (66) contains nonlinear terms where \(V_{RW}\) and \(V_{IW}\) are involved in a quadratic context, i.e., the fourth and last term. In this sense, a linearization is proposed using Taylor series to make linear Equation (66), considering that a linear mathematical model can be solved in less computational time than a nonlinear approach and that the optimal solution can be warranted. Then, expression (66) is replaced by (67)-(70) that stands for the linearization of \(P_{loss}\) using Taylor series at an operation point.