## 3 Optimal Island Partition Strategy
### Island Partition Model
Considering the fluctuation and randomness of uncontrollable generators and loads after island partition, the power margin coefficient of controllable generators is set as \(\sigma\) in constraint (9g), to allow controllable generators to conduct real-time tracking compensation for power fluctuation in the island. \(P_{i}^{dgc}\) is the output value of controllable generation at node \(i\), and \(P_{i}^{dgc,\max}\) is its maximal value of generation output power. Constraints (9h) and (9i) are, respectively, the branch voltage constraint and capacity constraint in an island. \(U_{i\to j}^{\min}\) and \(U_{i\to j}^{\max}\) are the minimal and maximal values of the branch voltage between nodes \(i\) and \(j\). \(P_{i\to j}^{\min}\) and \(P_{i\to j}^{\max}\) are the minimal and maximal values for branch power transmission capacity. According to [17], standby power in an island is the sum of 3% of the local load and 5% of the uncontrollable generator power:  
\[\lambda_{i}\in\{0,1\},\text{if}:\;i\in\Omega_{l}^{\text{ucc}} \tag{9a}\]0  
where \(P_{i}^{dguc}\) is the output power of the uncontrollable generators, \(i\in\Omega_{\rm is}^{\rm s}\). Equation (10) constrains the relationship between all generation unit power and load power at the time of island partition. Given that the right side is identically greater than, or equal to, 0, it also plays the role of ensuring power supply and demand balance within the island. Converting Equation (10) to obtain the power constraint of controllable generators:  
\[\lambda_{i}\in\{0,1\},\text{if}:\;i\in\Omega_{l}^{\text{ucc}} \tag{9a}\]1  
Equation (11), in considering the dynamic changes of power for load and uncontrollable generation units within an island, ensures that there will be no insufficient power supply-caused transient fluctuations in the island operation through superposing the power gain coefficient before islanding the load and uncontrollable generators (making an increase of 3% in load power and a decrease of 5% in uncontrollable generation output as the ultimate state of system operation to calculate the minimal standby power margin for controllable generators). In addition, because the operation mode of the distribution system with multiple distributed generators after island partitioning is similar to a microgrid, it can be considered to adopt a microgrid island operation generation control mode (such as _V-f_ control) to realize the transient stability of loads within the island. Combining the above factors, equipment parameters, like line capacity, are not considered. Then the island partition model can be simplified as:  
\[\lambda_{i}\in\{0,1\},\text{if}:\;i\in\Omega_{l}^{\text{ucc}} \tag{9a}\]2  
In Equation (12), the generation constraint conditions in Equation (9) are simplified through calculating the power supply and demand relationship between controllable generation, uncontrollable generation and load, as well as power margin conditions. Through further setting \(1.03\sum\limits_{i\in\Omega_{is}^{c}}P_{i}^{\text{load}}-0.95\sum\limits_{i\in \Omega_{is}^{c}}P_{i}^{\text{dgc}}\geq 0\), it can be seen that \(\sum\limits_{i\in\Omega_{is}^{c}}P_{i}^{\text{dgc}}>0\). Therefore, while ensuring dynamic power balance in independent operation after island partitioning, constraint (12d) contains at least one controllable generation unit used for dynamically balancing the power supply and demand relationship.