## 2 Mathematical Model  
In this section, the formulation of distribution network operation with reconfiguration and DGs [23; 24; 25] is expressed. It is an optimal AC power flow of the distribution system considering network loss minimization. Therefore, its formulation is written by:  
\[\min\sum_{l\in\varphi_{l}}365\times C_{p}  
\[\left(V_{i}^{\min}\right)^{2}\leq\left(V_{i,t}^{re+}\right)^{2}+\left(V_{i,t}^{ im+}\right)^{2}\leq\left(V_{i}^{\max}\right)^{2}\ \ \ \ \ \forall i,t \tag{11}\]  
\[-\overline{w}_{l}^{re}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\leq w_{l,t}^{re} \leq\overline{w}_{l}^{re}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\ \ \ \ \ \forall l,t \tag{12}\]  
\[-\overline{w}_{l}^{im}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\leq w_{l,t}^{im} \leq\overline{w}_{l}^{im}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\ \ \ \ \ \forall l,t \tag{13}\]  
\[0\leq IL_{l,t}^{re+}\leq IL_{l}^{\max}y_{l,t}^{+}\ \ \ \ \ \forall l,t \tag{14}\]  
\[0\leq IL_{l,t}^{re-}\leq IL_{l}^{\max}y_{l,t}^{-}\ \ \ \ \forall l,t \tag{15}\]  
\[y_{l,t}^{+}+y_{l,t}^{-}\leq 1\ \ \ \ \ \forall l,t \tag{16}\]  
\[\sum_{l\in\varphi_{l}}\left(y_{l,t}^{+}+y_{l,t}^{-}\right)=N_{bus}-1\ \ \ \ \ \forall t \tag{17}\]  
\[0\leq IL_{l,t}^{im+},IL_{l,t}^{im-}\ \ \ \ \ \forall l,t \tag{18}\]  
\[y_{l,t}^{+},y_{l,t}^{-}\in\left\{0,1\right\}\ \ \ \ \ \forall l,t \tag{19}\]  
\[\sum_{l\in\varphi_{l}}\left(y_{l,t}^{+}+y_{l,t}^{-}\right)\leq N_{ch}\ \ \ \ \ \forall l \tag{20}\]  
\[-\overline{w}_{l}^{re}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\leq w_{l,t}^{re} \leq\overline{w}_{l}^{re}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\ \ \ \ \ \forall l,t \tag{12}\]0  
\[-\overline{w}_{l}^{re}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\leq w_{l,t}^{re} \leq\overline{w}_{l}^{re}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\ \ \ \ \ \forall l,t \tag{12}\]1  
\[-\overline{w}_{l}^{re}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\leq w_{l,t}^{re} \leq\overline{w}_{l}^{re}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\ \ \ \ \ \forall l,t \tag{12}\]2  
\[-\overline{w}_{l}^{re}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\leq w_{l,t}^{re} \leq\overline{w}_{l}^{re}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\ \ \ \ \ \forall l,t \tag{12}\]3  
\[-\overline{w}_{l}^{re}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\leq w_{l,t}^{re} \leq\overline{w}_{l}^{re}\left(1-y_{l,t}^{+}-y_{l,t}^{-}\right)\ \ \ \ \ \forall l,t \tag{12}\]4  
Equation (1) describes the objective function that deals with the minimization of annual energy losses. In Equation (1), the term \(\sum\limits_{l\in\varphi_{l}}\sum\limits_{l\in\varphi_{l}}R_{l}\left(\left( IL_{l,t}^{re+}+IL_{l,t}^{re-}\right)^{2}+\left(IL_{l,t}^{im+}+IL_{l,t}^{im-} \right)^{2}\right)\) for calculating the network energy loss in a day or the operation horizon is 24 h. However, this term is multiplied by \(365\times C_{p}\). The number 365 denotes the number of days of a year [26]. Additionally, as the consumption behavior of all days of a year in the distribution network is not the same, the term \(C_{p}\) has been used, which is known as the coincidence factor, and it is around 70% for distribution level [27]. Moreover, real and imaginary terms of the current flowing through lines include two positive and negative components, each of which determines the direction of current in the distribution lines [28]. The reduction of losses is proportional to the reduction of the current passing through the distribution lines. The reduction in the current of the distribution lines also leads to a reduction in the voltage drop of the lines, which results in a favorable voltage profile. In other words, the minimization of energy losses and the minimization of the voltage deviations function are in the same direction, that is, the reduction of losses corresponds to the reduction of the voltage deviations function. However, in multi-objective optimization, it is necessary that the changes of different objective functions in terms of increase and decrease are in the same direction.