Uncertainty, Geometry, and Phase Transitions in Energy Networks: A Unified Framework for Cascades and Optimal Control

Emerging economies increasingly face critical energy bottlenecks as highly stochastic industrial demand interacts with grid infrastructures designed under deterministic assumptions. This paper develops a unified mathematical framework that integrates robust optimization, network dynamics, and stochastic optimal control to characterize and manage this structural vulnerability. First, we introduce the concept of robust capacity geometry , showing that demand uncertainty induces an anisotropic contraction of the feasible operating region, governed by the topology-dependent covariance of network flows. Second, we establish an endogenous stabilization bound , proving the existence of a spectral phase transition: when demand variance exceeds a critical threshold relative to the network’s algebraic connectivity, decentralized stabilization mechanisms fail and cascading failures emerge endogenously. Third, we formulate a stochastic control problem and solve the associated Hamilton–Jacobi–Bellman equation, deriving a dynamically optimal load-shedding policy that allocates deficits according to marginal system value. Numerical simulations on a stylized regional grid validate the theoretical results, demonstrating (i) a substantial contraction of the feasible region under stochastic demand, (ii) a spectral collapse of network connectivity during cascades, and (iii) a reduction of aggregate economic loss exceeding 50% under the optimal policy relative to conventional rolling blackout strategies. Collectively, the results provide a rigorous operational framework for managing energy systems under uncertainty, with direct implications for infrastructure planning and real-time control in rapidly industrializing economies.