Power Flow Analysis

7.1

INTRODUCTION

Power ﬂow analysis is concerned with describing the operating state of an entire power system, by which we mean a network of generators, transmission lines, and loads that could represent an area as small as a municipality or as large as several states. Given certain known quantities—typically, the amount of power generated and consumed at different locations—power ﬂow analysis allows one to determine other quantities. The most important of these quantities are the voltages at locations throughout the transmission system, which, for alternating current (a.c.), consist of both a magnitude and a time element or phase angle. Once the voltages are known, the currents ﬂowing through every transmission link can be easily calculated. Thus the name power ﬂow or load ﬂow, as it is often called in the industry: given the amount of power delivered and where it comes from, power ﬂow analysis tells us how it ﬂows to its destination.

Owing mainly to the peculiarities of a.c., but also to the sheer size and complexity of a real power system—its elaborate topology with many nodes and links, and the large number of generators and loads—it turns out to be no mean feat to deduce what is happening in one part of the system from what is happening elsewhere, despite the fact that these happenings are intimately related through well-understood, deterministic laws of physics. Although we can readily calculate voltages and currents through the branches of small direct current (d.c.) circuits in terms of each other (as seen in Chapter 2), even a small network of a handful of a.c. power sources and loads deﬁes our ability to write down formulas for the relationships among all the variables: as a mathematician would say, the system cannot be solved analytically; there is no closed-form solution. We can only get at a numerical answer through a process of successive approximation or iteration. In order to ﬁnd out what the voltage or current at any given point will be, we must in effect simulate the entire system.

Historically, such simulations were accomplished through an actual miniature d.c. model of the power system in use. Generators were represented by small d.c. Electric Power Systems: A Conceptual Introduction, by Alexandra von Meier Copyright # 2006 John Wiley & Sons, Inc.

195

196

POWER FLOW ANALYSIS

power supplies, loads by resistors, and transmission lines by appropriately sized wires. The voltages and currents could be found empirically by direct measurement. To ﬁnd out how much the current on line A would increase, for example, due to Generator X taking over power production from Generator Y, one would simply adjust the values on X and Y and go read the ammeter on line A. The d.c. model does not exactly match the behavior of the a.c. system, but it gives an approximation that is close enough for most practical purposes. In the age of computers, we no longer need to physically build such models, but can create them mathematically. With plenty of computational power, we can not only represent a d.c. system, but the a.c. system itself in a way that accounts for the subtleties of a.c. Such a simulation constitutes power ﬂow analysis.

Power ﬂow answers the question, What is the present operating state of the system, given certain known quantities? To do this, it uses a mathematical algorithm of successive approximation by iteration, or the repeated application of calculation steps. These steps represent a process of trial and error that starts with assuming one array of numbers for the entire system, comparing the relationships among the numbers to the laws of physics, and then repeatedly adjusting the numbers until the entire array is consistent with both physical law and the conditions stipulated by the user. In practice, this looks like a computer program to which the operator gives certain input information about the power system, and which then provides output that...