## 2 Framework of the Proposed Self-Healing Recovery Strategy  
In this section, the framework of the self-healing recovery strategy proposed in this paper, which is as shown in Figure 1, is illustrated. Once a fault of SDN is detected by distribution automation system (DAS), the self-healing recovery strategy is put forward as follows. (1) The distribution network after a fault occurrence can be divided into several trouble-free subnets, each of which is called an initial island. At the first step, the initial island division is carried out based on the connectivity analysis of the distribution network and each initial island is the minimum application object of optimization model of self-healing, which consists of load-shedding optimization and network reconfiguration optimization; the following two points are important for island division. Keep the normal operation of widely distributed high-priority loads as normal as possible. The island with larger capacity and more adjustable equipment can achieve better power quality indexes. Based on the above understandings, the following initial island division strategy is adopted with the consideration of tie-switches. If the network is connected after the fault, the network is operated as a whole island. If the network is no longer connected after the fault but the connectivity can be recovered by the tie-switches, the network is still operated as a whole island. If the network is no longer connected after the fault and the connectivity cannot be recovered by the tie-switches, the network shall be divided into independent small islands. In the proposed strategy, the method in Reference [17] is used to judge the network connectivity. The algorithm to judge the network connectivity is explained as follows. (a) \(G=(A,E)\) is a network diagram, \(A=(A_{1},A_{2},\ldots,A_{n})\) is a set of the nodes, and E is a set of the lines. (b) \(Z\) is an adjacency matrix, \(Z[i,j]=1\) if \((A_{i},A_{j})eE\); otherwise, \(Z[i,j]=0\). (c) Referring to the short connection in the circuit, the network diagram can be simplified in a similar way. If two nodes are directly connected, they can be merged into one node to form a new network graph. Figure 1: Framework of the proposed self-healing recovery strategy. For example, if node \(i\) is connected to node \(j\), which means \(Z[i,j]=1\), the simplification of the network diagram can be operated as follows. In detail, for a row operation, apply the \"OR\" operation to the corresponding element of row \(i\) and row \(j\), and the result is placed in row \(j\). For a column operation, apply the \"OR\" operation to the corresponding element of column \(i\) and column \(j\), and the result can be placed in column \(j\). After the operations, the diagonal element can be set to zero. Since node \(i\) has been merged into node \(j\), row \(i\) and column \(i\) are removed. The above simplification decreases the dimension of adjacency matrix \(Z\) from \(n\) to \(n\)-1. (d) Repeat the simplification. The network diagram is not connected if there is a row in which all the elements are zeros during the process. Otherwise, the network diagram is connected, and when the dimension decreases to 2, all nondiagonal elements of adjacency matrix \(Z\) are 1. Through the above simplification, each connected part of the network diagram is considered as an initial island. After the initial island division, the proposed load-shedding optimization and network reconstruction optimization are applied for each initial island. In practical application, this procedure is in parallel for each island, thus the increase of the number of initial islands may lead to the growth of the total amount of computational burden, but it will not lead to the delay of strategy implementation for each initial island by taking advantage of parallel computation. In consequence, there is no specific restrictions on the number of islands. (2) For each initial island, the load-shedding optimization and network reconfiguration optimization are carried out as follows, respectively. 1.