Semi-implicit Continuous Newton Method for Power Flow Analysis

TL;DR

The paper tackles ill-conditioned power-flow analysis by addressing the stability-convergence trade-offs of existing continuous Newton methods. It introduces a semi-implicit CNM (SICNM) that reformulates the power-flow equations as differential-algebraic equations and solves them with a stiffly accurate Rosenbrock-type method, avoiding nonlinear inner iterations while achieving high damping. A new 4-stage, 3rd-order SARM named Rodas3d is constructed with a carefully chosen damping parameter $\gamma$ and an embedded error estimator for adaptive step sizing; its stability function $R(z)$ ensures hyper-stability and effective damping of both stable and unstable modes. Case studies on ill-conditioned benchmarks demonstrate that the Rodas3d-based SICNM (M8) converges faster and more robustly than alternative methods, with only a single LU factorization per step reducing computational overhead. The authors also provide a MATPOWER extension on GitHub for benchmarking and cross-validation in practical power-flow analyses.

Abstract

As an effective emulator of ill-conditioned power flow, continuous Newton methods (CNMs) have been extensively investigated using explicit and implicit numerical integration algorithms. Explicit CNMs are prone to non-convergence issues due to their limited stable region, while implicit CNMs introduce additional iteration-loops of nonlinear equations. Faced with this, we propose a semi-implicit version of CNM. We formulate the power flow equations as a set of differential algebraic equations (DAEs), and solve the DAEs with the stiffly accurate Rosenbrock type method (SARM). The proposed method succeeds the numerical robustness from the implicit CNM framework while prevents the iterative solution of nonlinear systems, hence revealing higher convergence speed and computation efficiency. A new 4-stage 3rd-order hyper-stable SARM, together with a 2nd-order embedded formula to control the step size, is constructed to further accelerate convergence by tuning the damping factor. Case studies on ill-conditioned systems verified the alleged performance. An algorithm extension for MATPOWER is made available on Github for benchmarking.

Figures (3)

• Figure 1: The stability region indicated by shading.
• Figure 2: Convergence errors versus iteration times in Case 9241pegase.
• Figure 3: Step size versus iteration times in Case 9241pegase.