## 3 Proposed Service Restoration Scheme  
### Island Partitioning Method  
Intentional islanding operation of DGs can provide a distribution system in an emergency with the self-sufficient capability by supplying power to important loads in outage areas. To restore the outage loads and achieve minimal generation-load imbalance, outage parts within the distribution system isolated from the main source should be divided into multiple islands making them self-sufficient areas. Figure 3 shows the overall procedure of the island partitioning method used in this paper. As indicated in Figure 3, firstly, a topology matrix containing information on connection relationships between all buses (nodes) in a distribution system is generated according to the current switch status. Elements in the matrix are encoded with 1 for 'connected' and 0 for 'disconnected' so that network searches can be performed by tracking all the 'connected' paths. Once the topology matrix is built, then a tree is created where the root is a DG-connected bus, and an initial island is built according to the path in the tree. Three quantities in the island are calculated to evaluate whether the initial island is feasible considering the power balance: (1) the total DG capacity (SDG), (2) the total  
Figure 3: The overall procedure of the island partitioning method. amount of loads (SL), and (3) the total amount of loads that can be disconnected for load sheddings (CL). For the initial island, the power balance constraint is satisfied if the SDG is greater than the SL. Therefore, the information on the initial island can be stored in a list without any changes since the island can be considered a feasible one satisfying the power balance constraint. Otherwise, controllable loads to be disconnected should be selected for meeting the constraint. To select which controllable loads in the island should be disconnected, all possible combinations considering the controllable loads are examined, and the one resulting in the minimum generation-load imbalance is chosen as the best one. The procedure of island partitioning is conducted until all DGs in the system are exploited to make islands. The islanding partitioning is performed by the number of DGs because the proposed scheme does not allow parallel operation of DGs meaning that there is only one DG within an island. Figure 4 illustrates an example of island partitioning for a simple system with 11 buses, two DGs, and two controllable loads. As shown in Figure 4a, two branches are disconnected to make islands in the system, and the corresponding system topology matrix is created by Equation (5). Based on the matrix, a subtree with one DG as a root node is generated as indicated in Figure 4b. By using the tree search and load information shown in Table 1, the SDG and SL for the DG1 are calculated as 100 kW and 90 kW. Since the SDG is greater than the SL, the island with DG1 is stored without any load shedding by controllable loads. For the DG2, the SDG and SL are derived as 200 kW and 240 kW and it needs to make use of load sheddings to restore more important loads in the island. Three combinations are available because there are two controllable loads, and the load at bus 8 is chosen to be disconnected to make the least generation-load imbalance. For the sample system, two islands are generated with one load shedding, and the island with DG1 contains buses 4, 5, 6, and 7 whereas the island with DG2 includes buses 2, 3, 8, 9, 10, and 11. \[\textit{topology matrix}= \begin{bmatrix}1&0&0&0&0&0&0&0&0&0&0&0\\ 0&1&1&0&0&0&0&0&0&0&0\\ 0&1&1&0&0&0&0&1&0&0&0\\ 0&0&0&1&1&0&0&0&0&0\\ 0&0&0&0&1&1&1&0&0&0&0\\ 0&0&0&0&0&1&1&0&0&0\\ 0&0&1&0&0&0&0&1&1&0&0\\ 0&0&0&0&0&0&0&1&1&1&0\\ 0&0&0&0&0&0&0&1&1&1\\ 0&0&0&0&0&0&0&0&1&1\\ \end{bmatrix}  \tag{5}\]