Comparative analysis of the convergence characteristics of second-order Ioadflow methods B J Cory Department of Electrical Engineering. Imperial College. London SW7 2BT. UK A 0 Ekwue National Grid Research and Development Centre, Kelvin Avenue, Leatherhead KT 7ST. UK R B I Johnson Pacific Gas and Electric Company, 77 Beale Street, San Francisco, CA 941 06. USA The second-order loadflow technique defined in rectangular coordinates has become oltractive because of the large amount of computation required in the evaluation of polar trigonometric functions in conventional loadflow methods'. Seven versions of the second-order loadflow technique are developed and their convergence characteristics and computin 9 times determined with tests carried out on the IEEE 14 and 30 bus networks. It is" suggested that the version based on the CRIC approach warrants further developments into optimal power flows and automatic' contingency selection algorithms Jbr real-time control. Keywords: network analysis, loadflow, optimal power ./lows I. Introduction The early developments in the formulation of the loadflow problem for digital computer solution were based on the admittance matrix or the Z-matrix using Gauss-Seidel 1'2 and Newton Raphson methods 3. With the increasing complexity in the power networks and the application of efficient numerical structures such as optimal ordering, sparsity, etc. the decoupled Newton method 4 was introduced. This sacrificed the true quadratic convergence behaviour of the Newton-Raphson method, with some computational advantages. Simplifi- cations were further made giving rise to the fast- decoupled loadflow method (FDLF) 5 which has been widely accepted by the industry because it is fast, simple to implement and has reduced computer storage requirements. Several refinements have been made such as the CRIC modifications 6 for the reactive model and a hybrid model by Behnam-Guilani 7 for improved convergence characteristics. Figure 1 shows the ioadflow Received 22 October 1 987, revised 1 3 March 1990 development. The loadflow problem consists essentially of the solution for a set of nonlinear equations with the Jacobian matrix for Newton's technique formed by considering only the first 2 terms of the Taylor's series. Because the 3rd term of the Taylor's series may be significant, its inclusion in the problem formulation resulted in second-order loadflows (SOLF) being developed. Sachdev and Medicherla s developed a SOLF procedure defined in polar coordinates, but the cartesian coordinates approach has been developed and reported 9 16. This latter approach is more attractive as it is exact since no trigonometric functions are involved which make power calculations during the iterative process time consuming and third- and higher-order terms of the Taylor series expansion are zero. A comparative analysis of the SOLF technique with the FDLF 16 shows that they are comparable in speed, both being about 5 times faster than the Newton-Raphson method. The SOLF offers the opportunity to perform very fast multiple-case solutions as the Jacobian is only triangulated once during a solution process. Reviews on loadflow methods can be found elsewhere 17--19 In this paper, a critical evaluation of the SOLF problem defined in rectangular coordinates is made. Seven refinements of an earlier development 16 are formulated and their convergence characteristics com- pared for various tolerance limits. The objective of the paper is to recommend a better approach for further development into optimal power flows, real-time control, etc. Notation Vii = ei + Jfi nodal voltage Aei nodal voltage real-component correction Vol 1 2 No 4 October 1 990 0142-0615/90/040251-06 (£') Butterworth-Heinemann Ltd 251