## 3 Optimal Island Partition Strategy  
### Island Partition Model  
The goal of island operation lies in maximizing the power supply capacity of the distribution system, isolating vulnerable nodes for protection, and reducing a secondary fault caused by the transient fluctuation of vulnerable nodes upon power supply restoration in the case of a fault of the distribution system. In ensuring the power supply, the loads of different grades have different influences upon personal safety and economic damage, so it is necessary to grade the load of the distribution system to ensure the supply reliability of important loads in priority and lower the influence of blackout upon social activities. According to constraints, like generator operation condition, power supply and demand balance, and controllable load proportion when the distribution system with multiple distributed generators operates in multiple islands, the island partition model is constructed as follows:  
\[\max f=\omega_{1}\sum_{i\in V_{1}}\lambda_{i}CT_{i}P_{i}^{\text{load}}+\omega_{ 2}\sum_{i\in V_{2}}\lambda_{i}CT_{i}P_{i}^{\text{load}}+\omega_{3}\sum_{i\in V _{3}}\lambda_{i}CT_{i}P_{i}^{\text{load}} \tag{9}\]  
s.t. :  
\[\lambda_{i}\in\{0,1\},\text{if}:\;i\in\Omega_{l}^{\text{ucc}} \tag{9a}\] \[\lambda_{i}\in[0,1],\text{if}:\;i\in\Omega_{l}^{\text{c}}\] (9b) \[\lambda_{j}\geq 0,j\in\Omega_{\text{load}}\cap\Omega_{is}^{s} \cap\Omega_{i\to g^{*}}\exists g\in\Omega_{dg}\cap\Omega_{is}^{s}\] (9c) \[\sum_{i}P_{i}^{\text{dg}}-\sum_{i}P_{i}^{\text{load}}>0,i\in \Omega_{is}^{s}\] (9d) \[P_{i}^{\text{dg,min}}\leq P_{i}^{\text{dg}}\leq P_{i}^{\text{dg,max}}\] (9e) \[\sum_{i=1}^{m}P_{i}^{\text{dg}}>0,i\in\Omega_{is}^{s}\] (9f) \[(1+\sigma)\sum_{i=1}^{m}P_{i}^{dgc}\leq\sum_{i=1}^{m}P_{i}^{dgc \_{\text{max}}}\] (9g) \[U_{i\to j}^{\text{min}}\leq U_{i\to j}\leq U_{i\to j}^{\text{max}},i,j\in\Omega_{is}^{s}\] (9h) \[P_{i\to j}^{\text{min}}\leq P_{i\to j}\leq P_{i\to j}^{\text{max}},i,j\in\Omega_{is}^{s} \tag{9i}\]  
where \(\omega_{1}\), \(\omega_{2}\), \(\omega_{3}\) are primary, secondary, and tertiary load weights, respectively; \(V_{1}\), \(V_{2}\), \(V_{3}\) are primary, secondary, and tertiary load sets; \(CT_{i}\) is the vulnerability index of node \(i\); and \(P_{i}^{load}\) is the load active power of node \(i\). Due to the different types of loads, the load nodes can be divided into two categories: an uncontrollable load set (\(\Omega_{l}^{uc}\)) and a controllable load set (\(\Omega_{l}^{c}\)). \(\lambda_{i}\) is a variable that defines the connection state of a load. When the load node \(i\) is uncontrollable, i.e., \(i\in\Omega_{l}^{uc}\), \(\lambda_{i}\) can only equal zero or one, \(\lambda_{i}=0\) means the uncontrollable load node \(i\) is disconnected to any island, and \(\lambda_{i}=1\) means the uncontrollable load is connected to some island. For a controllable load node, i.e., \(i\in\Omega_{l}^{c}\), the value of \(\lambda_{i}\), which dependents to how much load is restored, is alterable. Assuming that node \(j\) is on the route (\(\Omega_{i\to g}\)) connecting node \(i\) and node \(g\), and node \(g\) is a generator belonging to the node set \(\Omega_{is}^{s}\) in the \(s^{th}\) island. Constraint (9c) ensures that there is always a route connecting to the generators inside the island for any load node therein, namely the connectivity constraint of the island node. \(P_{i}^{dg}\) is the power output value of the generation at node \(i\), and constraint (9d) ensures power supply and demand balance between the generator and the load after island partition. \(P_{i}^{dg\_min}\) and \(P_{i}^{dg\_max}\) represent the minimal and maximal values of the generation output power. Given that uncontrollable generators (PV, wind generator) cannot independently supply power for loads, constraint (9f) ensures that there is at least one controllable generation unit in any island.