## 4 Outage Optimization Strategy Considering the Importance of Users  
The outage sequence optimization problem for LVDNs aims to minimize the importance of load curtailment and the impact of outages while satisfying the number of outages. The objective function is shown in Equation (37):  
\[\min\sum_{t=t_{0}}^{t_{\text{end}}}\sum_{i=1}^{n}\text{S}_{i}\cdot\text{P}_{i, t}^{\text{D}} \tag{37}\]  
where t\({}_{0}\) and t\({}_{\text{end}}\) are the starting and ending moments of the outage, respectively; \(n\) is the number of users of all LVDNs; and \(S_{i}\) is the importance of the \(i\)th LV user. \(\text{P}_{i,t}^{\text{D}}\) is the outage load of the \(i\)th LV customer at time \(t\). To ensure the safe and stable operation of the LVDNs, the system should satisfy the following constraints:  
(1) Power Limits  
\[\text{P}_{i,t}^{\text{D}}<\text{P}_{i,t}^{\text{L}} \tag{38}\]  
\[P_{pv,t}<\text{P}_{\text{pv,max}} \tag{39}\]  
\[P_{\text{wt},t}<\text{P}_{\text{wt,max}} \tag{40}\]  
\[P_{pv,t}+P_{wt,t}=\sum_{i=1}^{n}\left(\text{P}_{i,t}^{\text{L}}-\text{P}_{i, t}^{\text{D}}\right) \tag{41}\]  
Equation (38) constrains the magnitude of the reduced load, and \(\text{P}_{i,t}^{\text{L}}\) is the original load at node \(i\). Equations (39) and (40) constrain the magnitude of the output of the WT and PV unit of DG. \(P_{pv,t}\) and \(P_{wt,t}\) are the magnitudes of the PV and WT outputs at time t, respectively; \(\text{P}_{\text{pv,max}}\) and \(\text{P}_{\text{wt,max}}\) are the maximum values of the PV and WT outputs, respectively. Equation (41) represents that the sum of the output values of PV and WT is equal to the sum of the original load minus the sum of the reduced load. (2) Power Flow Equations  
\[\left\{ \begin{array}{l}P_{pv,i,t}+P_{wt,i,t}-(P_{i,t}^{L}-P_{i,t}^{D})=P_{i,t}\\ P_{i,t}=U_{i,t}\sum\limits_{j=1}^{m}U_{j,t}(G_{ij}\cos\theta_{ij,t}+B_{ij}\sin \theta_{ij,t})\\ \\ Q_{pv,i,t}+Q_{wt,i,t}-(Q_{i,t}^{L}-Q_{i,t}^{D})=Q_{i,t}\\ Q_{i,t}=U_{i,t}\sum\limits_{j=1}^{m}U_{j,t}(G_{ij}\cos\theta_{ij,t}-B_{ij}\sin \theta_{ij,t})\\ \end{array} \right. \tag{42}\]  
Constraints (42) and (43) are active and reactive power node balance constraints. Here, \(P_{i,t}\) and \(Q_{i,t}\) are the active and reactive net power injected into node \(i\) at time \(t\); \(P_{pv,i,t}\), \(Q_{pv,i,t}\), \(P_{wt,i,t}\), and \(Q_{wt,i,t}\) are the active and reactive power emitted by the connected PV and WT at node \(i\) at time \(t\); \(U_{i,t}\) is the voltage amplitude of node \(i\) at time \(t\); G\({}_{ij}\) and B\({}_{ij}\) are the conductance and conductance between lines at nodes \(i\) and \(j\); and \(\theta_{ij,t}\) is the phase angle difference in voltages at the ends of nodes \(i\) and \(j\) at time \(t\). (3) Voltage constraints  
The permissible deviation of the 220 V single-phase supply voltage is \(+5\%\sim-10\%\) of the rated voltage. \[0.9\text{U}_{N}\leq U_{i,t}^{\varphi}\leq 1.05\text{U}_{N} \tag{44}\]  
where U\({}_{N}\) is the rated phase voltage; \(U_{i,t}^{\varphi}\) is the actual phase voltage of node \(i\) at time \(t\). (4) Branch current constraints  
\[\left|I_{ij}(t)\right|\leq I_{ij}^{\max} \tag{45}\]  
where \(I_{ij}(t)\) is the branch current between nodes \(i\) and \(j\) at moment \(t\), and \(I_{ij}^{\max}\) is the maximum allowable value of the branch circuit current. (5) Island constraint  
Equation (46) represents the island connectivity constraint. \[y_{i,u,t}\leqslant\sum\limits_{k\in\
abla_{k,u,t}}y_{k,u,t} \tag{46}\]  
where \(y_{i,u,t}\) is a 0-1 variable that indicates whether the node belongs to an island at time t: 1 if it belongs, and 0 if it does not. \(\
abla_{k,u,t}\) is the set of parents of the nodes in the island at time \(t\). This constraint indicates that if there is a node \(i\) in island \(u\), then there must be at least one parent node \(k\) of node \(i\) belonging to island \(u\) to form a pathway from the power node to node \(i\) in that island.