Solution Characteristics of Quadratic Power Flow Problems

KEYWORDS Load ow analysis, numerical techniques. ABSTRACT A number of facts about quadratic power ow problems f(x) = y + g(x) = 0, x 2 R n x , y 2 R n y and applied Newton-Raphson methods with optimal multipliers are presented. The main results are about solution structure, singular points and Newton-Raphson solutions. Comments and discussions are given. Although we address here to the power ow problems, there are many areas where presented results may be eeectively used. Actually they are valid for any problem described by an algebraic system of quadratic equations. INTRODUCTION The Newton-Raphson (NR) method and its various modiications are the most popular numerical techniques used in load ow problems-see for instance 1]-5]. There are two main forms used for the load ow equations: the polar and rectangular forms. Both of them have some advantages. The polar form provides signiicant reduction of computations. For instance, the method, which uses P-Q decomposition of the load ow problem 3], is widely used on practice. The rectangular form of load ow equations can be eeectively used as well-see 6]-12]. The most important feature of that form is that the power mismatch function can be exactly expressed using linear and second order terms of the Taylor series. It is well known that the NR method has good quadratic convergence if the initial estimates are close to a solution point. However, if they are far from a solution or the load ow problem is an ill-conditioned one, convergence of the NR method may be slow or not at all. To overcome this problem, a number of numerical techniques were proposed-see 4]-8]. The general idea behind all of them is to apply corrections to each step of the methods in such a way that iterative processes do not oscillate or diverge. Due to its nonlinearity, the load ow problem may have a number of distinct solutions. Studies of the multiple solutions of the load ow problem play a role in determining proximity to voltage collapse 9, 13]. In order to obtain multiple load ow solutions, Tamura et al used a set of quadratic load ow equations and the NR optimal multiplier method 11]. Iba et al used Tamura's approach and some newly discovered convergence peculiarities of the NR method to nd a pair of closest multiple solutions 12]. It is observed from experimental results 12] that if a point x comes …