Q4Power System 2

Comparison Between Gauss-Seidel and Newton-Raphson Methods

• Convergence characteristic: Gauss-Seidel (GS) exhibits linear convergence, meaning the error reduces by only a roughly constant fraction each iteration, while Newton-Raphson (NR) exhibits quadratic convergence near the solution, meaning the number of correct digits in the solution roughly doubles with each successive iteration, giving NR a dramatically faster rate of convergence once it is reasonably close to the true solution.
• Number of iterations required: GS typically requires a number of iterations that increases roughly in proportion to the number of buses in the system (and can become very large, sometimes hundreds of iterations, for bigger systems), whereas NR typically converges in only a handful of iterations (commonly 3 to 5) essentially independent of system size, since its iteration count is governed mainly by the quality of the initial guess and the desired convergence tolerance rather than by the number of buses.
• Memory and computational requirement per iteration: GS requires comparatively little memory (only the admittance matrix and bus voltage values need to be stored) and each iteration is computationally cheap and simple; NR requires forming and repeatedly solving (or updating) a large Jacobian matrix at every iteration, which is both more memory-intensive and computationally more expensive per individual iteration, though sparse-matrix and matrix-factorization techniques are used in practice to make this per-iteration cost manageable even for very large systems.
• Sensitivity to system size: GS's convergence tends to worsen noticeably as system size grows (and can even fail to converge for some larger or more heavily-loaded systems), while NR's convergence behavior (given a reasonable starting point) remains comparatively robust and largely insensitive to system size, which is precisely why NR became the preferred method for practical load flow studies as computer memory and processing capability improved over time, despite its higher per-iteration cost.
• Handling of reactive power/voltage-controlled buses and system ill-conditioning: NR handles PV (voltage-controlled) buses and reactive power limits, as well as generally ill-conditioned systems (heavily loaded, or with wide disparities in line impedances), considerably more robustly than GS, which can converge very slowly or fail altogether under such conditions.

Suitability for large power systems: based on this comparison, the Newton-Raphson method is clearly the more suitable choice for load flow calculations of a large power system, since its convergence characteristics remain essentially independent of system size (converging reliably in only a few iterations even for networks with many hundreds or thousands of buses), whereas the Gauss-Seidel method's iteration count and convergence reliability both deteriorate significantly as the number of buses increases, making GS increasingly impractical for large-scale systems despite its simplicity. In practice, GS today survives mainly as a teaching tool for illustrating the basic iterative load flow concept (as in the accompanying three-bus numerical problem elsewhere in this paper) and occasionally as a means of generating an initial voltage estimate before switching to NR for final, rapid convergence, while virtually all modern production-grade load flow software for real power systems relies on the Newton-Raphson method (or its computationally-optimized variant, the Fast Decoupled Load Flow method).