Gauss-Seidel Method

The Gauss-Seidel method is one of the earliest iterative techniques for solving the nonlinear power flow equations. It updates the voltage at each bus sequentially using the latest available values from neighboring buses, repeating the sweep across all buses until voltages stabilize within a specified tolerance.

Key Aspects of the Gauss-Seidel Method:

• Algorithm: For each bus i, the voltage is recalculated from the power flow equation using the known power injection and the most recently computed voltages of all connected buses. One full sweep through every bus constitutes a single iteration, and the process repeats until convergence.
• Linear Convergence: The method converges linearly, meaning the error decreases by a roughly constant factor each iteration. This is much slower than the quadratic convergence of Newton-Raphson, often requiring 50–200 iterations where Newton-Raphson needs 3–5.
• Acceleration Factor: An acceleration factor (typically 1.4–1.6) can be applied to the voltage correction at each step, effectively over-correcting to speed up convergence. Choosing the optimal acceleration factor is problem-dependent and can significantly reduce the number of iterations needed.
• Low Memory Requirement: Because the method processes one bus at a time without forming or factoring large matrices, its memory footprint is very small. This was a decisive advantage in the 1950s–1960s when computer memory was severely limited, and explains why it was the first computerized power flow algorithm.
• Modern Relevance: Although Newton-Raphson has largely replaced Gauss-Seidel in production-grade software, the method remains valuable in education as an intuitive introduction to iterative power flow, and it is occasionally used as a pre-conditioner or initial solver to provide a starting point for Newton-Raphson in difficult cases.