## 2 Service Restoration Problem Statements
### Objectives and Constraints  
The SR can be regarded as a nonlinear multi-objective problem since SR plans are established by considering various objectives such as the outage loads, power losses, and the number of switching operations required to achieve a new topology. In this paper, minimizing outage loads in a distribution system is considered an objective since it is a more important factor than others, especially in emergencies that do not last for a long time. In order to emphasize load importance in an emergency, load importance coefficients are applied to ensure that the SR scheme restores as many critical loads as possible despite the disconnection of controllable loads within an island. The objective function considering a load importance index can be expressed as Equation (1).  
\[\text{min}F=\sum_{i=1}^{I}\left(\alpha_{\text{C1}}\left(1-\frac{PRE_{\text{C1}i }}{P_{\text{C1}}}\right)+\alpha_{\text{C2}}\left(1-\frac{PRE_{\text{C2}i}}{P_{ \text{C2}}}\right)+\alpha_{\text{C3}}\left(1-\frac{PRE_{\text{C3}}}{P_{\text{ C3}}}\right)\right) \tag{1}\]  
where, \(l\) is the number of islands in a distribution system; \(\alpha_{\text{C1}}\), \(\alpha_{\text{C2}}\), and \(\alpha_{\text{C3}}\) are coefficients of load importance for load classes 1, 2, and 3, respectively; \(PRE\) and \(P\) are the sum of loads restored in an island and the total load in a distribution system, respectively.  
Figure 1: General service restoration process in distribution systems.  
The topology derived by the SR scheme needs to satisfy distribution system operating constraints. The following constraints are applied to check whether an SR plan derived is feasible or not.  
* Power balance;
* Bus voltage;
* Branch capacity;
* Anti-parallel operation.  
The DG capacity in an island should be greater than all loads in the island to meet the power balance constraint, and bus voltages and branch capacities should be within the allowable range. The constraints applied to the SR problem can be expressed by Equations (2)-(4).  
\[P_{DG}\geq\sum_{i=1}^{N}P_{Li} \tag{2}\]  
\[S_{max,i}\geq S_{i} \tag{3}\]  
\[V_{lower}\leq V_{i}\leq V_{upper} \tag{4}\]  
Figure 2: An example of the service restoration process: (**a**) Before the fault section isolation; (**b**) After the fault section isolation.  
where, \(N\) is the number of loads in an island; \(P_{DG}\), \(P_{L}\) are active power capacity of DG and load, respectively; and \(S_{max}\) is the branch capacity limit; \(V_{lower}\) and \(V_{upper}\) are the lower and upper limits for bus voltages, respectively.  
It is noted that the anti-parallel operation is introduced as a constraint in this paper. An island with more than two DGs imposes complexity in system operation since the parallel operation requires an improved control system that reflects the different dynamics of DGs. Protection-related challenges are also expected due to multiple fault current sources and the resulting reverse power flows. Furthermore, a tool or control center for the automatic management of various DGs should be introduced, which causes additional equipment and costs for reliable control strategies [25, 26]. For example, in the case of inverter-based DGs, a communication-based control scheme collecting the operating information of each DG should be introduced to deal with the parallel-connected mode control [28].