  The integration of the frequency dynamics into Micro-Grid (MG) investment and
operational planning problems is vital in ensuring the security of the system
in the post-contingency states. However, the task of including transient
security constraints in MG planning problems is non-trivial. This is due to the
highly non-linear and non-convex nature of the analytical closed form of the
frequency metrics (e.g., frequency nadir) and power flow constraints. To handle
this issue, this paper presents two algorithms for decomposing the MG
investment planning problem into multiple levels to enhance computational
tractability and optimality. Furthermore, the sensitivity of the decisions made
at each level is captured by corresponding dual cutting planes to model
feasible secure regions. This, in turn, ensures both the optimal determination
and placement of inertia services and accelerates the convergence of the
proposed decomposition algorithms. The efficient and effective performance of
the proposed algorithms is tested and verified on an 18-bus Low Voltage (LV)
network and a 30-bus Medium Voltage (MV) network under various operating
scenarios.
