Basic data
The IEEE 33 [27] and 119bus systems [28] are used as test cases to verify the effectiveness of the proposed model and solution algorithm. In the DS, the load demand of each node is satisfied by the utility grid under normal operation conditions, in which the tie switches are open and the sectionalizing switches are closed. There is one access point for MPS in each node to guarantee the electricity supply after the fault. In terms of the TN, there are multiple roads between any two nodes in the DS. The parameters in the model are listed in Table 1. The test case is implemented in a computer with Intel Core i58250 CPU 1.60 GHz, 16 G memory, and MATLAB 2018a is used as the testing environment for the solution.
Case I: IEEE 33bus system
Results of the IEEE 33bus system
Figure 2 illustrates the topology of the IEEE 33bus system with the coupled TN. The load distribution of each node is shown in Fig. 3. The fault is considered to be in branches {8–9, 15–16, 19–20, 22–23, 30–31}, and the depots (1, 2, and 3) are set around nodes 5, 11, and 28, respectively. There are three MPSs with an upper limit of power generation of 300 kW in each depot. It is clear that there is always a road for MPS from the depot to the access point of the electrical island.
Through the resilient outage recovery method, optimal strategies can be obtained and are shown in Fig. 4. In terms of the DS, the tie switches in branches {7–20, 17–32, 24–28} are closed, leading to the formation of two electrical islands. From Fig. 4, the electrical island 1 consists of nodes 9, 10, 11, 12, 13, 14, 15, while the electrical island 2 is formed by closing the tie switch in branch {17–32}, connecting branch {16–17} and branch {31–32}. The two islands and other nodes are separately supplied by the MPSs and utility grid. It is clear that the DS apart from the islands satisfies the radial operation condition. The power outages in nodes 20, 21, 23 and 24 caused by the fault are recovered by performing NR, and another power outage load is recovered with the help of MPS. The optimized node voltage is presented in Fig. 5, which shows that the voltage of every node is higher than 0.95 and meets the voltage constraints. Due to the access of MPS, the voltage of nodes 11 and 31 is 1 p.u..
In terms of the TN, MPSs in depots 2 and 3 are dispatched to the electrical island 1 and 2, respectively. The access point in electrical island 1 is in node 11, and there is a depot (i.e., depot 2) near node 11. Therefore, the corresponding routing distance and power outage time are both the shortest, and the routing cost of MPS is the smallest. The optimal access point is node 31 for the electrical island 2. This can obtain the shortest power outage time and smallest routing cost from depot 3. Considering the power generation upper limit of the MPS, two MPSs in depot 2 and 3 are separately dispatched to satisfy the load demands of 465 kW and 420 kW in electrical islands 1 and 2, respectively. The load demand can be met in nodes 16 and 17 without MPS accessing the nodes, because the tie switch in branch {17–32} is closed by NR.
Comparison between the resilient outage recovery method and different scenarios

(1)
Comparison with the scenario of only dispatching MPS
The scenarios of only dispatching MPS, only performing NR, and without considering power losses are analyzed, and objective function values are shown in Table 2. Compared with the scenario of only dispatching MPS, it is evident that the power outage time, routing cost of MPS and power generation cost of the resilient outage recovery method are lower, but the power losses are higher. The reason is that MPS will be dispatched to fewer electrical islands to restore power supply after a fault if NR is considered. This decreases the power outage time, routing cost and power generation cost, while improving the resilience of the DS. However, in the scenario of only dispatching MPS, since the DS operates in the form of a multiisland and the voltage of load nodes connected by the MPS increases, the power loss of a single line decreases, and then the total power loss of DS decreases. Therefore, NR can effectively reduce power outage time and dispatching cost of MPS while slightly increasing power losses.

(2)
Comparison with the scenario of only performing network reconfiguration
Under the scenario of only performing NR, the optimal reconfiguration strategy is to close the tie switch in branches {8–14, 7–20, 24–28}, and the power loss is 504.83 kW. The routing cost and power generation cost of MPS are 0, because no MPS is dispatched. From Fig. 2, it is clear that there are two electrical islands in the reconfiguration strategy, i.e., nodes 16 and 17, and nodes 31 and 32. Since there are no MPSs, the power supply of the two electrical islands cannot be restored. Therefore, the power outage time is infinity, as shown in Table 2. Compared with the scenario only performing NR, although the MPSs’ routing cost and power generation cost obtained by the resilient outage recovery method are higher, the power outage time and power losses are smaller. The reason for smaller power losses is that loads of nodes 9–15 are not restored by the main grid, which decreases the power losses on branches {0–1, 1–2, 2–3, 3–4, 4–5} and {23–24, 5–25, 25–26, 26–27, 27–28, 28–29, 29–30}. To sum up, the MPS dispatching model and NR are cooptimized in the resilient outage recovery method, ensuring the power supply being restored and the cost resulting from the fault being reduced.

(3)
Comparison with the scenario of without considering the power losses in the objective function
The optimal schedule of the scenario without considering the power losses in the objective function is closing the tie switches in branches {8–14, 11–21, 17–32, 24–28}, and dispatching two MPSs from depot 3 to node 31. In this scenario, there is only one electrical island, i.e., island 2 in Fig. 4. It can be seen from Table 2 that the power outage time and routing cost of MPS are the same, both from depot 3 to node 31. It is noted that the routing distance from depot 2 to node 11 is nearly 0 as the result of the outage management framework. However, the MPS power generation cost of the resilient outage management framework is higher, because there are more electrical islands. The number of switch actions is decreased by one, which increases the switch life. The power losses of the scenario with the resilient outage recovery method are greatly decreased by 77.18%, compared to the scenario without considering power losses in the objective function. The reason is that the loads at nodes 9–15 are not restored by the main grid in the scenario with the resilient outage recovery method. This decreases the power losses of branches {0–1, 1–2, 2–3, 3–4, 4–5} by reducing the injecting power of nodes 1, 2, 3, 4, and 5. In addition, the current flowing through branches {23–24, 5–25, 25–26, 26–27, 27–28, 28–29, 29–30} is less resulting from loads at node 9–15 being restored by MPS, which reduces the power losses on these branches. In conclusion, the proposed resilient outage recovery method simultaneously minimizes power outage time, dispatching cost of the MPS and power losses by comprehensively considering multiple objectives and it enhances the DS’s ability to deal with faults.
Through the comparison between the resilient outage recovery method and different scenarios, it is clear that the power supply can be quickly restored and costs resulting from the fault can be minimized simultaneously by this method. Thus, it can be concluded that the proposed method has better performance in power outage recovery for the IEEE 33bus system.
Case II: IEEE 119bus system
Results of the IEEE 119bus system
The test in the IEEE 119bus system is used to verify the scalability of the proposed resilient outage recovery method and solution algorithm. The IEEE 119bus system and coupled TN are shown in Fig. 6. There are 15 loops and 118 nodes in the IEEE 119bus system. It is evident that there is one access point for MPS near the node. 7 depots are built in the traffic network, and are respectively set near nodes 11, 36, 42, 67, 76, 100, 110. Each depot has 7 MPSs with an upper limit of the power generation of 300 kW. There is at least one road for MPS from the depot to the access point. The load distribution of each node in the IEEE 119bus system is shown in Fig. 7. The electrical faults happen in branches {7–8, 24–25, 31–32, 42–43, 52–53, 62–63, 73–74, 81–82, 93–94, 101–102, 109–110}. In the initial operational condition, all the tie switches are open to satisfy the radial structure.
The optimized results are obtained by the resilient outage recovery method and are shown in Fig. 6. Through closing the tie switches in branches {8–24, 17–27, 27–48, 45–56, 38–65, 51–65, 61–100, 76–95, 78–91, 86–113}, the power supply of corresponding outage load resulting from the fault in branches {7–8, 24–25, 31–32, 42–43, 52–53, 62–63, 73–74, 81–82, 93–94, 109–110} is restored. There is no loop structure in the optimized DS, but an electrical island is formed after switch actions. The electrical island consists of two load nodes, i.e., nodes 102 and 103, whose power demand is supplied by the MPS from the nearest depot. In terms of the TN, the MPS is dispatched from depot 6 located near node 100 to the access point in node 102, as shown in Fig. 6b. The load demand of the electrical island is 72.38 kW. Thus only one MPS is needed to supply the outage load. The number of MPS needed is reduced by NR, which greatly reduces the dispatching cost of MPS. Because performing NR is quicker than dispatching MPS, the power outage time is decreased and quick outage recovery is realized.
Comparison between the resilient outage recovery method and different scenarios

(1)
Comparison with the scenario of only dispatching MPS
Table 3 shows objective function values of four scenarios. The values are power outage time, routing cost of MPS, power generation cost, the number of switch actions, and power losses. In the resilient outage recovery method, the power outage time and routing cost of MPS are only a quarter of those in the scenario of only dispatching MPS, and the power generation cost of MPS is much lower than that of that scenario. There is only one electrical island as the result of the outage management framework, but 11 electrical islands exist if only dispatching MPS to recover the power supply of electrical islands. The more electrical islands the longer the routing distance and there is more generation power required of the MPS. Also, the formation of multiple electrical islands leads to the reduction of the radical network scale, which decreases the power losses of DS. However, considering the stability of the operation, the island operation mode only runs for a short time and the resilience of DS is extremely low in the island operation mode. Therefore, the optimal recovery strategies obtained by the proposed resilient outage recovery methods are superior to the scenario of only dispatching MPS.

(2)
Comparison with the scenario of only performing NR
The objective function values of the scenario of only performing NR are shown in Table 3. The routing cost and power generation cost of MPS are 0 due to no MPS being dispatched. The number of faults is small and branches where faults occur are scattered, thus power supply can be restored simply by NR. The reason for power outage time being 0 is that NR can be performed instantaneously with the help of a highspeed switching device [23]. However, the power losses obtained by the scenario of only performing NR are higher than that obtained by the resilient outage recovery method. Because the scale of series load in the radial network is larger in the scenario of only performing NR, it results in a larger voltage drop and power losses on each branch. Therefore, the reliability of the DS will be decreased. In conclusion, although the power outage time, MPSs’ routing cost and power generation cost obtained by the resilient outage recovery method are larger than those obtained by only performing NR, the optimal schedule of the resilient outage recovery method is better.

(3)
Comparison with the scenario without considering the power losses in the objective function
In the scenario without considering power losses in the objective function, the optimal strategy only performs NR by closing the tie switches in branches {8–24, 17–27, 27–48, 45–56, 38–65, 51–65, 76–95, 78–91, 80–103, 86–113,115–123}, but not dispatching the MPS. The reason is that the number of faults is fewer than the number of loops, so the power supply can be recovered only through NR without forming electrical islands. Accordingly, the power outage time, routing cost and generation cost of MPS is 0. However, this recovery strategy results in larger power losses due to the larger scale of the series load. It is clear from Table 2 that the power losses of the resilient outage recovery method are only half that of the scenario without considering power losses. Moreover, the power line heating phenomenon arises because of the large power losses. This can potentially cause new faults and harm the operation of the utility grid. Therefore, even if the power outage time, MPSs’ routing cost and power generation cost of the resilient outage recovery method are slightly higher, its recovery strategy is the optimal one on the whole.
Compared with other scenarios, the resilient outage recovery method considers multiple objectives to ensure the operational reliability of the DS and reduce dispatching costs while minimizing power outage time. Therefore, the outage recovery problem can be better solved by the proposed method.
Analysis of practical feasibility
The proposed algorithm is a centralized algorithm, which is implemented by using a computer with Intel Core i58250 CPU 1.60 GHz, 16 G memory, in MATLAB 2018a. There is no interaction nor communication requirement for realizing the algorithm. Thus, the algorithm can be easily realized in the existing power grid infrastructure. In this environment, the computation time of the algorithm is 1.56 min for the IEEE 33bus system and 5.40 min for the IEEE 119bus system. The computation time shows the optimal scheduling strategy can be obtained quickly to reduce the power outage time and cost resulting from the fault.
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