## 3 Problem Formulation  
### Bi-Level Optimization Model  
In general, a bi-level optimization model is a decision model with a two-level structure and multiple participants [27]. The upper-level decision regulates the lower-level behavior while the optimal strategy of the lower-level influences the upper-level decision-making. Mathematically, the bi-level optimization model can be expressed as a pre-defined objective function subject to a set of static physical and operating limits, with its compact form being as follows:  
\[\begin{array}{l}Upper-level:\ \ \ \begin{cases}&\underset{U}{Min}\ \omega_{u}F(Z)\\ s.t.&G(U,\ X)\leq 0\\ &\underset{X}{Max}\ \omega_{l}f(Z)\\ s.t.&g(U,X)\leq 0\end{cases}\end{array} \tag{7}\]  
where \(F(Z)\) and \(f(Z)\) are the objective functions of the upper and lower levels, respectively; \(G(U,X)\) and \(g(U,X)\) are the vector functions representing the equality constraints of the upper and lower levels, respectively; \(U\) and \(X\) are the decision variables of the upper and lower levels, respectively. \(\omega_{u}\) and \(\omega_{l}\) are weighting factors of upper-level and lower-level, respectively.  
The bi-level decisions influence and constrain one another, because the optimization model can express the hierarchical relationship between the two levels. Figure 2 presents the overall scheme of the bi-level optimization model which is designed to incorporate the energy band during the problem formulation. The MG is integrated into the DN, and the DSO provides power to the MGO to balance the load. Furthermore, the WT, PV panel, MT, and battery energy storage system (BESS) are connected to different load nodes of the MG. In this case, the power flow between the DN and MG is bidirectional. The MG can not only purchase energy from the DSO but also sell energy to it. As shown in Figure 2, the DSO incentivizes the MGO to lower the cost of the energy it supplies using the optimal scheduling information for the tie-line exchange, and the MGO operates economically and safely by following the scheduling requirements. On the upper-level, the DSO who has the responsibility of operating the DN in an optimal fashion, optimizes the power exchanged between the DSO and MGO, so that its operating cost is minimized. In response, the lower-level determines the rescheduled tie-line flow with the energy band as feedback information. The exchanged power of the tie-line between the DN and MG is considered as coupling variables between the DSO and MGO, which determine the amount of the tie-line purchased power by the DSO at the upper-level in consideration of the tie-line exchange power for MGO's profit for stable operation of the DN, and it forces the decision maker to consider a multi-objective optimization problem. The DSO wants to buy less transaction levels from the MGO to minimize operating costs for optimal operation of the DN. Conversely, the MGO want to sell them to the DSO to maximize profits in the DN. These relationships are trade-off (i.e., minimized operating cost for DSO and maximized profit for MGO), and the solutions must be solved by a multi-objective problem in order to obtain a Pareto-front between the two operators. Thus, the proposed approach provides the optimal solution for cooperation between the DSO and MGO to help in decision-making, whenever there is a trade-off between operating cost of the DSO and profit of the MGO in the DN.