Symbolic Direct Construction of Augmented Systems for Levenberg-Marquardt Power Flow

Overview & Synopsis

Levenberg-Marquardt power flow (LMPF) addresses ill-conditioned equations and provides least-squares stationary points for infeasible operating requests. Conventional implementations form damped normal equations, where J^T J introduces fill, squares the spectral condition number of the undamped part, and adds sparse-product cost. We propose an augmented LMPF formulation with a symbolic-permutation-based Jacobian operator that fills the augmented blocks directly. We prove that the matrix is symmetric quasi-definite for positive damping, enabling fixed-order, pivot-free QDLDL factorization. This property also motivates the solver choice: KLU value-only refactorization can reuse an unsuitable pivot order, forcing fresh full factorizations on difficult LMPF trajectories. We further introduce one-probe adaptive trust-region initialization for the radius and damping. Direct assembly is 7.5-11.9x faster than the fastest coordinate-list path and 18.0-58.8x faster than cached normal-equation sparse-product formation; QDLDL reduces numerical refactorization and triangular-solve time by up to 3.40x on identical matrices. Adaptive initialization solves flat-started RTE cases in 19-25 factorizations, 7.5-11.6x faster than reproduced three-step LM. On case2383wp, our 84-iteration run performs 84 factorizations and 168 resolves in 237 ms, versus published times of 200.780-441.72 s with fewer outer iterations, corresponding to speedups of 847-1860x. These results require complete factorization and resolve accounting and reveal initialization and legacy-parameter effects. DC-initialized NR exhibits fast quadratic convergence throughout every tested solvable loading range, motivating clearer, reproducible definitions of ill-conditioned power-flow benchmarks.