## 4 Case Study
### Island Partition Verification  
Branch 2-3 in the PG and E 69-node system was set to have a fault of the three-phase grounding, with the branch switch disconnected through fault isolation, leading to the failure of the power supply to node 3 and its downstream nodes through the public distribution system. The primary, secondary, and tertiary load weights were set to be \(\omega_{1}=100\), \(\omega_{2}=10\), \(\omega_{3}=1\), respectively [21]. The proposed algorithm, according to node vulnerability, position of distributed generators, and its capacity, was adopted to conduct island partition in the case of a fault of the distribution system with multiple distributed generators, as shown in Figure 6.  
As shown in Figure 6, the distribution system with multiple distributed generators was divided into two islands for independent operation after a fault, with a total load restoration rate of 62.04%. Among them, the restoration rate was 100% for primary loads, 59.23% for secondary loads, and 47.04% for tertiary loads. Specifically, for island-2 of Figure 6, all the primary loads and uncontrollable loads in the power transmission line are restored at the first stage. In this period, only controllable non-primary loads (secondary loads and tertiary loads) are waiting to be selected to be restored. To restore the secondary loads most, all the controllable tertiary load nodes in island-2 of Figure 6 are not fully restored and the controllable secondary load is partly restored according to the remaining power supply ability of DGs at the same time. In this case, there are still some controllable secondary load nodes are not under a fully-restored state. In other words, the DGs in island-2 of Figure 6 have already supplied their maximum power to the loads. Thus, it is meaningless to put node 53 into island-2, and node 53 may be under a no-power state even if it is connected to island-2. In addition, in order to ensure dynamic power balance in each island's independent operation after island partition, the standby power in one island should be the sum of 3% of the local load and 5% of uncontrollable generator power. Therefore, some controllable generators are not at maximum power for balancing the power fluctuation from the local load and the uncontrollable generators. Detailed load restoration and power output is shown in Tables 4 and 5. Figure 6: Island partition scheme of the distribution system with multiple distributed generators. The load restoration amount and rate of three methods are as shown in Table 6.  
Table 6 presents the statistical results of load restoration after island partition through the three methods. Reference [8] divided the distribution system into four islands for independent operation; but due to the calculation not considering that uncontrollable units, like photovoltaics and wind power, cannot independently supply power to the local load, it could not form stable and effective islands. Therefore, the statistics abandoned the load contained by ineffective islands so as to obtain a total load restoration rate of 53.44% in [8]. Both the method in this paper and [22] combined uncontrollable and controllable power generators, prescribing that every island should contain at least one controllable power unit and dividing the distribution system into two effective islands for independent power supply. According to Table 6, the total load restoration of the method proposed in this paper was close to the method in [22], being 62.04% and 62.55%, respectively. After the fault of branch 2-3 is fixed, the distribution system should close all the brakes which were open before for power restoration. During this power restore process, the voltage of the power grid will fluctuate, caused by the sudden change of load power and the switch of brakes. Therefore, two kinds of island partition were set in PSCAD. Assuming that branch 2-3 is fixed, the power restoration is, respectively, executed under two different island partition configurations.