Comparison of Load Flow Methods:
Comparison of Load Flow Methods – Gauss Seidel and Newton Raphson methods are compared when both use YBUS as the network model. It is experienced that the Gauss Seidel method works well when programmed using rectangular coordinates, whereas Newton Raphson requires more memory when rectangular coordinates are used. Hence, polar coordinates are preferred for the Newton Raphson method.
The Gauss Seidel method requires the fewest number of arithmetic operations to complete an iteration. This is because of the sparsity of the network matrix and the simplicity of the solution techniques. Consequently, this method requires less time per iteration. With the NR method, the elements of the Jacobian are to be computed in each iteration, so the time is considerably longer. For typical large systems, the time per iteration in the NR method is roughly equivalent to 7 times that of the Gauss Seidel method. The time per iteration in both these methods increases almost directly as the number of buses of the network.
The rate of convergence of the Gauss Seidel method is slow (linear convergence characteristic), requiring a considerably greater number of iterations to obtain a solution than the NR method which has quadratic convergence characteristics and is the best among all methods from the standpoint of convergence. In addition, the number of iterations for the Gauss Seidel method increases directly as the number of buses of the network, whereas the number of iterations for the Newton Raphson method remains practically constant, independent of system size. The Newton Raphson method needs 3 to 5 iterations to reach an acceptable solution for a large system. In the Gauss Seidel method and other methods, convergence is affected by the choice of slack bus and the presence of series capacitor, but the sensitivity of the Newton Raphson method is minimal to these factors which cause poor convergence.
Therefore, for large systems the Newton Raphson method is faster, more accurate and more reliable than the Gauss Seidel method or any other known method. In fact, it works for any size and kind of problem and is able to solve a wider variety of ill-conditioned problems . Its programming logic is considerably more complex and it has the disadvantage of requiring a large computer memory even when a compact storage scheme is used for the Jacobian and admittance matrices. In fact, it can be made even faster by adopting the scheme of optimally renumbered buses. The method is probably best suited for optimal load flow studies (Chapter 7) because of its high accuracy which is restricted only by round-off errors.
The chief advantage of the Gauss Seidel method is the ease of programming and most efficient utilization of core memory. It is, however, restricted in use of small size system because of its doubtful convergence and longer time needed for solution of large power networks.
For FDLF, the convergence is geometric, two to five iterations are normally required for practical accuracies, and it is more reliable than the formal Newton Raphson method. This is due to the fact that the elements of [B’] and [B”] are fixed approximation to the tangents of the defining functions ΔP/|V| and ΔQ/|V|, and are not sensitive to any ‘humps’ in the defining functions.
If ΔP/|V| and ΔQ/|V| are calculated efficiently, then the speed for iterations of the FDLF is nearly five times that of the formal Newton Raphson or about two-thirds that of the Gauss Seidel method. Storage requirements are around 60 percent of the formal Newton Raphson, but slightly more than the decoupled Newton Raphson method.
Changes in system configurations can be easily taken into account and though adjusted solutions take many more iterations, each one of them takes less time and hence the overall solution time is still low.
The FDLF can be employed in optimization studies and is specially used for accurate information of both real and reactive power for multiple load flow studies, as in contingency evaluation for system security assessment and enhancement analysis.
When a series of load flow calculations are performed, the final values of bus voltages in each case are normally used as the initial voltages of the next case. This reduces the number of iterations, particularly when there are minor changes in system conditions.