Power-Circle Vulnerability Assessment and Two-Stage IPSO-Based Multi-STATCOM Allocation for Fine Voltage Regulation in Distribution Networks

A two-stage improved particle swarm optimization (IPSO) framework is proposed for multi-STATCOM siting and sizing in radial distribution networks, together with a power-circle external-tangency margin for independent electrical-vulnerability assessment. The power-circle margin is formulated from branch active- and reactive-power circle relationships, aggregated by the arithmetic mean of incident-branch margins to avoid topological-degree bias, and cross-checked against Q–V continuation-power-flow results and load-level sensitivity. STATCOM planning minimizes the annual cost of active-power losses, annualized device investment, and operation and maintenance under hard voltage constraints. In the first optimization stage, two installation locations and coarse capacities are searched jointly; the second stage fixes the selected locations and continuously refines the capacities. On the IEEE 33-bus system, the power-circle and Q–V rankings show a Spearman correlation of 0.8644 and 90% Top-10 overlap. Deterministic enumeration identifies Bus 7 with 3.6689 Mvar as the best single-STATCOM solution, whereas the optimal dual configuration is 0.5583 Mvar at Bus 16 and 1.1571 Mvar at Bus 30. The dual solution reduces annual cost by 38.86% relative to the optimal single-device solution and reduces active-power loss to 0.1402 MW while satisfying all normal-operation voltage limits. Across 30 independent runs on the dual-device problem, the proposed IPSO attains the highest location-pair and global-solution success rates (33.33%) and the lowest mean objective value among six compared algorithms. Under restored N − 1 contingencies, the normal-operation economic optimum improves post-contingency voltage performance but is capacity-limited; with the locations fixed, a total rating of 6.5693 Mvar is required to satisfy all 31 restorable contingencies, revealing the trade-off between normal-operation economy and post-contingency voltage compliance.