Modeling and Numerical Solution of a 17-Bus Electrical System using RK4 and Newton- Raphson Methods

This article presents an integrated numerical framework for the analysis of electromechanical dynamics and steady‑state behavior in a 17‑bus subset of the Spanish peninsular transmission system. Motivated by the severe blackout that occurred in April 2025, which revealed the vulnerability of low‑inertia, renewable‑dominated grids, the study couples the fourth‑order Runge–Kutta method with the Newton–Raphson method, as a power‑flow algorithm, to provide a unified representation of transient frequency evolution and static system equilibrium. The selected network subset includes a Slack bus, several PV buses, and multiple PQ buses with significant load concentrations and strong reactive interactions. The Newton–Raphson procedure exposes high sensitivity in nodes with intense reactive demand. This is the case around branches featuring large susceptance values, and when we use branch parameters and shunt contributions. Iterative behavior shows notable mismatch magnitudes in the flat start, substantial angle corrections, and conditioning challenges associated with reactive stiffening. At the same time, the dynamic model captures the influence of inertia, damping, and primary control on frequency trajectories following large disturbances. The coupled approach ensures consistency between electrical power evaluations and dynamic states at each integration step, enabling detailed assessment of electromechanical responses under stressed operating conditions. Overall, the study demonstrates that combining high‑order time integration with robust nonlinear algebraic solvers provides a reliable tool for understanding system sensitivity, evaluating operational risks, and supporting the design of protection and control strategies in increasingly renewable electrical networks.