The Power Flow ProblemJames D. McCalley, Iowa State UniversityT7.0 IntroductionT7.1 Generator Reactive LimitsIt 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 capability 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.Fig T7.1: Generator Capability Curve and Approximate Reactive LimitsT7.2 TerminologyT7.3 The Admittance MatrixT7.4 The power flow equationsFig. T7.8: Three Bus System for Example T7.4The Power Flow ProblemThe Power Flow ProblemJames D. McCalley, Iowa State UniversityT7.0 IntroductionThe power flow problem is a very well known problem in the field of power systems engineering, wherevoltage magnitudes and angles for one set of buses are desired, given that voltage magnitudes and powerlevels for another set of buses are known and that a model of the network configuration (unit commitmentand circuit topology) is available. A power flow solution procedure is a numerical method that is employedto solve the power flow problem. A power flow program is a computer code that implements a power flowsolution procedure. The power flow solution contains the voltages and angles at all buses, and from thisinformation, we may compute the real and reactive generation and load levels at all buses and the real andreactive 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 engineersconsidered different network configurations necessary to serve an expected future load. Later, it became anoperational problem as operators and operating engineers were required to monitor the real-time status ofthe network in terms of voltage magnitudes and circuit flows. Today, the power flow problem is widelyrecognized as a fundamental problem for power system analysis, and there are many advanced, commercialpower flow programs to address it. Most of these programs are capable of solving the power flow programfor tens of thousands of interconnected buses. Engineers that understand the power flow problem, itsformulation, and corresponding solution procedures are in high demand, particularly if they also haveexperience with commercial grade power flow programs. The power flow problem is fundamentally a network analysis problem, and as such, the study of it providesinsight 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 smallerphysical size, are quite similar otherwise to the power flow problem. For example, references [1,2] arewell-known network analysis texts in VLSI design that also provide good insight into the numericalanalysis needed by the power flow program designer. Similarly, there are numerous classical power systemengineering texts, [3-11] are a representative sample, that provide advanced network analysis methodsapplicable to VLSI design and analysis problems.Section T7.1 identifies a feature of power generators important to the power flow problem – real andreactive power limits. Section T7.2 defines some additional terminology necessary to understand the powerflow problem and its solution procedure. Section T7.3 introduces the so-called network “Y-bus,” otherwiseknown 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 wereintroduced. Section T7.5 provides an analytical statement of the power flow problem. Section T7.6 uses asimple example to introduce the Newton-Raphson algorithm for solving systems of non-linear algebraicequations. Section T7.7 illustrates application of the Newton-Raphson algorithm to the power flowproblem. 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 associatedwith a commercial power flow program. T7.1 Generator Reactive LimitsIt is well known that generators have maximum and minimum real power capabilities. In addition, they alsohave maximum and minimum reactive power capabilities. The maximum reactive power capability1The Power Flow Problemcorresponds to the maximum reactive power that the generator may produce when operating with a laggingpower factor. The minimum reactive power capability corresponds to the maximum reactive power thegenerator may absorb when operating with a leading power factor. These limitations are a function of thereal power output of the generator, that is, as the real power increases, the reactive power limitations movecloser to zero. The solid curve in Figure T7.1 is a typical generator capability curve, which shows thelagging and leading reactive limitations (the ordinate) as real power is varied (the abscissa). Most powerflow programs model the generator reactive capabilities by assuming a somewhat conservative value forPmax (perhaps 95% of the actual value), and then fixing the reactive limits Qmax (for the lagging limit) andQming (for the leading