A Cycle-Based Solvability Condition for Real Power Flow Equations

Certifying power flow solvability is important for reliable power system operations under volatile operating conditions, but solving power flow equations repeatedly can be costly and may encounter convergence issues. In this paper, we develop an explicit cycle-based solvability condition for the lossless real power flow equations on meshed networks. We decompose every feasible nodal balance solution into a particular flow plus a cycle flow correction vector. The power flow problem is then reduced to enforcing edge-wise feasibility and cycle consistency. We show that the cycle consistency function is strongly monotone, and is the gradient of a strongly convex energy function. By exploiting these properties, we derive an explicit condition on the existence and uniqueness of power flow solution with bounded angle difference. The resulting condition is invariant under the choice of cycle basis and can be verified through simple algebraic computations. Numerical results on standard test systems show that the proposed condition is significantly less conservative than existing sufficient conditions and closely approximates true loading limits.