Submodular Optimization of Dynamic Thermal Rating for Cascading Failure Risk Mitigation Considering Braess Paradox

Cascading failure poses a significant risk to society. One approach to mitigate failure risk is through dynamic thermal rating (DTR) sensor, placed in transmission lines to achieve both risk mitigation and investment postponement of new lines. Sensor placement, as the basis of DTR analysis, is intrinsically a combinatorial optimization problem, while traditional solving methods cannot provide optimality guarantee and suffer easily from dimensionality curse. Besides, the risk mitigation may result in Braess paradox, a counterintuitive phenomenon that line update inversely increases failure risk. This paper proposes a submodular optimization-based DTR placement model for risk mitigation considering Braess paradox. First, a model based on Markov probability and important sampling weight techniques is utilized to efficiently quantify the failure risk before and after DTR placement. Then the risk model is applied to analytically reveal the Braess paradox condition which can invalidate the submodular formulation. For this invalidation issue, a novel submodular optimization approach is established to reformulate the risk mitigation model containing estimation error. Finally, a computationally efficient solving algorithm is designed to address this nonmonotone submodular optimization, which provides a provable approximation guarantee. The benefits brough from DTR and performance of the proposed algorithms are verified by case results.