The Power Flow Problem 1 The Power Flow Problem James D. McCalley, Iowa State University T7.0 Introduction The power flow problem is a very well known problem in the field of power systems engineering, where voltage magnitudes and angles for one set of buses are desired, given that voltage magnitudes and power levels for another set of buses are known and that a model of the network configuration (unit commitment and circuit topology) is available. A power flow solution procedure is a numerical method that is employed to solve the power flow problem. A power flow program is a computer code that implements a power flow solution procedure. The power flow solution contains the voltages and angles at all buses, and from this information, we may compute the real and reactive generation and load levels at all buses and the real and reactive flows across all circuits. The above terminology is often used with the word “load” substituted for “power,” i.e., load flow problem, load flow solution procedure, load flow program, and load flow solution. However, the former terminology is preferred as one normally does not think of “load” as something that “flows.” The power flow problem was originally motivated within planning environments where engineers considered different network configurations necessary to serve an expected future load. Later, it became an operational problem as operators and operating engineers were required to monitor the real-time status of the network in terms of voltage magnitudes and circuit flows. Today, the power flow problem is widely recognized as a fundamental problem for power system analysis, and there are many advanced, commercial power flow programs to address it. Most of these programs are capable of solving the power flow program for tens of thousands of interconnected buses. Engineers that understand the power flow problem, its formulation, and corresponding solution procedures are in high demand, particularly if they also have experience with commercial grade power flow programs. The power flow problem is fundamentally a network analysis problem, and as such, the study of it provides insight into solutions for similar problems that occur in other areas of electrical engineering. For example, integrated circuit designers also encounter network analysis problems, although of significantly smaller physical size, are quite similar otherwise to the power flow problem. For example, references [1,2] are well-known network analysis texts in VLSI design that also provide good insight into the numerical analysis needed by the power flow program designer. Similarly, there are numerous classical power system engineering texts, [3-11] are a representative sample, that provide advanced network analysis methods applicable to VLSI design and analysis problems. Section T7.1 identifies a feature of power generators important to the power flow problem – real and reactive power limits. Section T7.2 defines some additional terminology necessary to understand the power flow problem and its solution procedure. Section T7.3 introduces the so-called network “Y-bus,” otherwise known more generally as the network admittance matrix. Section T7.4 develops the power flow equations, building from module T1 where equations for real and reactive power flow across a transmission line were introduced. Section T7.5 provides an analytical statement of the power flow problem. Section T7.6 uses a simple example to introduce the Newton-Raphson algorithm for solving systems of non-linear algebraic equations. Section T7.7 illustrates application of the Newton-Raphson algorithm to the power flow problem. Section T7.8 provides an overview of several interesting and advanced attributes of the problem. Section T7.9 summarizes basic power flow input and output quantities and provides an example associated with a commercial power flow program. T7.1 Generator Reactive Limits It is well known that generators have maximum and minimum real power capabilities. In addition, they also have maximum and minimum reactive power capabilities. The maximum reactive power capabilityThe Power Flow Problem 2 corresponds to the maximum reactive power that the generator may produce when operating with a lagging power factor. The minimum reactive power capability corresponds to the maximum reactive power the generator may absorb when operating with a leading power factor. These limitations are a function of the real power output of the generator, that is, as the real power increases, the reactive power limitations move closer to zero. The solid curve in Figure T7.1 is a typical generator capability curve, which shows the lagging and leading reactive limitations (the ordinate) as real power is varied (the abscissa). Most power flow programs model the generator reactive capabilities by assuming a somewhat conservative value for Pmax (perhaps 95% of the actual value), and then fixing the reactive limits Qmax (for the lagging limit) and Qming (for the leading limit) according to the dotted lines shown in Fig. T2.1. Pmax Qmax Qmin P Q leading operation lagging operation Fig T7.1: Generator Capability Curve and Approximate Reactive Limits T7.2 Terminology Bulk high voltage transmission systems are always comprised of three phase circuits. However, under balanced conditions (the currents in all three phases are equal in magnitude and phase separated by 120), we may analyze the three phase system using a per-phase equivalent circuit consisting of the a-phase and the neutral conductor. Per-unitization of a per-phase equivalent of a three phase, balanced system results in the per-unit circuit. It is the per-unitized, per-phase equivalent circuit of the power system that we use to formulate and solve the power flow problem. For the remainder of this module, we will assume that all quantities are in per-unit. The reader unfamiliar with per-phase equivalent circuits or the per-unit system should refer to modules B3 and B4, respectively. It is convenient to represent power system networks using the so-called one-line diagram, which can be thought of as the circuit diagram of the per-phase equivalent, but without the neutral conductor (module B3 also provides additional background on the one-line diagram). Figure T7.2 illustrates the one-line diagram of a