Topology-adaptive Estimation of Maximum Voltage Unbalance for Distribution Networks with Finite-sample Guarantee

High penetration of single-phase distributed power production and frequent topology changes can significantly increase voltage unbalance in distribution networks. Reliable probabilistic power flow (PPF) assessment for the network, therefore, must account for topology-dependent power-flow equations, numerical error in the voltage unbalance factor (VUF), and finite-sample uncertainty in power measurements. In this paper, we propose Topology-Adaptive Certified Quantile Estimation (TACQE) to estimate the maximum VUF across monitored electric buses. Following a local network change, TACQE updates the cross-couplings among power-flow sequences, and certifies candidate PPF solutions against the updated nonlinear equations. The resulted power-flow state enclosures provide rigorous bounds on the maximum VUF. Furthermore, we derive independent reference-solver evaluations to calibrate discrepancies from the reference PPF solution. This leads to a uniform sampling bound that yields simultaneous upper-tail probabilities and quantiles, which provides a finite-sample guarantee for the estimated maximum-VUF distribution over an entire range of VUF thresholds. From our numerical simulations across 101 fixed-topology quantiles, it demonstrates that the designed TACQE can provide a fast candidate post-change power flow solution where it reuses most of the previous network model and modifies only the affected bus blocks.