intermediate Power Systems

Gauss-Seidel vs Newton-Raphson

Compare Gauss-Seidel and Newton-Raphson power flow solvers side-by-side. Understand why NR's quadratic convergence makes it the industry standard despite requiring Jacobian computation at each step.

Click 'Run Both Solvers'. The convergence chart plots mismatch (||ΔP||, ||ΔQ||) against iteration count on a log scale for both Gauss-Seidel (orange) and Newton-Raphson (blue). Observe how many more iterations GS requires than NR.

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A straight line on a log-scale mismatch chart indicates linear convergence. A curve that bends steeply downward indicates quadratic (or super-linear) convergence.
Newton-Raphson converges in typically 4–6 iterations. Each iteration solves a full linear system J·Δx = f (expensive). Gauss-Seidel converges in 30–100 iterations for simple networks and may diverge for ill-conditioned networks. Each GS iteration is cheap but many are needed.

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For n-bus systems, building and factoring the Jacobian costs O(n³). But NR's quadratic convergence means far fewer total iterations — so for large n, NR wins overwhelmingly.
The convergence rate of NR can be observed on the chart: after the first few 'burn-in' iterations, the mismatch approximately squares each iteration (e.g. 1e-2 → 1e-4 → 1e-8 → 1e-16). This is the hallmark of quadratic convergence.

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Quadratic convergence means: if ε_k is the error at iteration k, then ε_{k+1} ≈ C·ε_k². Doubling the digits of accuracy each iteration.
The Fast Decoupled Load Flow (FDLF) is a popular middle ground. It approximates the Jacobian by exploiting the decoupling between P-θ and Q-V in high X/R ratio networks. FDLF is faster per iteration than NR and converges faster than GS. It was the industry standard before NR implementations became efficient.

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The Fast Decoupled approximation assumes: ∂P/∂|V| ≈ 0 and ∂Q/∂θ ≈ 0. This is accurate for high-voltage transmission networks where X >> R.