Review of Load Flow Calculation Methods

916

PROCEFSMGS OF TEE IEEE, VOL.

62, NO. 7, JULY 1974

Review of Load-Flow Calculation Methods BRIAN

smm, MEMBER, IEEE In&d

Abstract-A m e y is presented on the currently availabIe numerical techniques for power-ayetem load-flow calculation using the digital computer. The review deals with methods that have receivedwidespreadpracticalapplication,recentattractivedevelopments, and other methodsthat have interesting or useful characteristics.The analytical bases, computational requirements, and comparative n u m d d performances of the methods are discuesed.Attention is given to the problems and techuiquea of adjustments in load-flowsolutions, a d the suitabilities of variousmethodsfor modern applications mch as s e a u i t y monitoring and optimal load flow are examined.

Paper

TABLE I L ~ A D ~ O CALCULATIONS-TYPES W AND REQLTIPEYBN~S

Typea of Solution ACCUIate Unadjusted OfT-Line Single-Caae

Approximate Adjusted On-Line Multiple Cases

Properties Required of Lcrad-Flow Solution Method High speed

eepedallyfor:

large systems

I, INTRODUCTION real-time applications multiple cases OAD FLOW (or power flow) is thesolution for the interactive applications static operating condition of an electric-power transLow storage espedally for: large systems computers with small core-atore availmission system, and is the most frequently performed ability of routine digital-computer power network calculations. Over Reliability espedally for: ill-conditioned problems the last 20 years an enormous amount of effort has been exoutage studies ml-time applications pended in research and development on the numerical calcu- Versatility ability to handle conventional and special features lation process. For instance, [ 11 gives a large but byno means (adjustments, representation of power-system apparatuej ; suitability for incorporation into more compliexhaustive bibliography on the subject, comprising more than cated procesaea 200 “respectable” papers in the English language alone. Out Simpliaty ease (and Coet) of coding, maintaining, and enhancing of these, 134 haveappeared as publications of the IEEE b a d on it. the algorithm and computer program Power Engineering Society and its predecessor. This review will clearly be unable to cover every aspect of the problem and every proposed solution algorithm, nor is i t some of the main types of load-flow solutions currently in intended to compete as a catalog of references with other application, and the requirements imposed on the numerical recent sources [l], [2]. T h e aim is to present the underlying processes. Eachapplicationrequires a combination of the principles andtechniques of thepopularlyacceptedaptypes shown, e.g., some forms of security assessment call for proaches, those more recent methods that seem to offer par- approximate, unadjusted, on-line, multiple-case solutions. ticular promise, and a selection of other methods that contain ideas of practical or theoretical interest. A . Brief History of Load Flow Perhaps the most recurrent question arising in the loadPrior to, and for some time after, the advent of digital flow field is-which is the best method to choose for a given computers, load-flow solutions were obtained using network application? The answer is rarely easy. The relative properanalyzers. The first really practical automatic digital solution ties and performances of different load-flow methods can be methods appeared in the literature in 1956 and subsequently influenced substantially by the types and sizes of problems to [lo]-[12]. TheseY-matrixiterativemethods werewellbe solved, by the computing facilities available, and by the of computerssincethey resuitetotheearlygenerations precise details of implementation. Any final choice is almost quireminimalcomputerstorage.Althoughtheyperform invariably a compromise between the various criteriaof goodsatisfactorily on many problems, they converge slowly, and ness bywhichload-flow methods are to be compared with too often notat all. The incentive to overcome thisdeficiency each other. Every such criterion is directly or indirectly asled tothedevelopment of 2-matrixmethods [19]-[21], sociated with financial cost, in the actual calculation itself,in which converge more reliably but sacrifice some of the adthe engineering application, or in the computer hardware and vantages of Y-matrix iterative methods, notably storage and software requirements. speed when applied to large systems. Around the same time, Load-flowcalculationsareperformedinpower-system the Newton(-Raphson)methodwasshown to havevery planning, operational planning, and operation/control. They powerful convergence properties [24], [25], but was computaare increasingly being used to solve very large systems, to tionally uncompetitive. Major breakthroughs in power-system solvemultiple cases forpurposessuch as outagesecurity network computation came in the mid-l960’s, with the deassessment, and within more complicated calculations such as velopment by Tinney and others of very efficient sparsityoptimization and stability. Table I gives a brief summary of programmed ordered elimination [SI. One of its earliest successes was in dramatically improving the computing speed Manusaipt receivedDecember 14, 1973; revisedJanuary 21, 1974. and storage requirementsof Newton’s method, which hasnow The author is with theUniversity of Waterloo, Waterloo, Ont., come to be widely regarded as the preeminent general-purpose Canada, on leave from the Universityof Manchester Institute of Science load-flow approach [26], and has been adopted by much of and Technology, Manchester, U. IC

L

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S l V T l ' : LOAD-FLOW CALCULATION METHODS

industry.Currently,withthestimulus of increasingproblem sizes, on-lineapplications,andsystemoptimization, newer methods are emerging which are also expected to find wide application.

11. NOTATIONS A . Complex Quantities at Bus i Ei = Vi/& - = ei+jfi Nodal voltage. Nodal injected current. Ii Si =Pi+jQi Net nodal injected power. Power mismatch. ASi =AP,+jAQi A Ii mismatch. Current Correction to nodal voltage. AE i =A V i/Mi

Siep = pim +_jQiap _

-

Y=G+jB

J

specified valuebyreactive-powerinjection. The busconstraints are P p = PQ~W P~iep= Re (EJi*) (3)

-

*

admittance Nodal matrix. matrix. impedance Nodal matrix. Jacobian

C. General Number of buses system. in Reactance of branch between buses and k.

?8

Xik

i

Bp =8i -ek S

index.

bus

(2)

A P V bus i is one at which the total injected activepower is specified, and the voltage magnitude is maintained at a

B. Matrices

z

-

= PQ~"P P ~ i ' p + j ( Q ~ i m Q L P ) = EJi*.

Slack

Superscripts ' P I and * denote specified value, calculated value, and complex conjugate, respectively. respecSubscripts Q and L denote generation and load, tively. All unsubscripted upper-case ,quantities denote vectors or matrices. h i denotes a bus k directly connected to bus i. kpi denotes a PQ-type bus directly connected t o bus i. kEiis thesameas R w i , except t h a t i includes t the case k = i . OF 111. BASICANALYTICAL FORMULATION LOAD-FLOWPROBLEM

The basicformulation of the load-flowproblemis now weli-known [4]-[7], and is presented only briefly in this section. .Virtually all load-flow calculation methods have been devisedinitially for the solutionof this basic problem. Extensions to it areusually 'grafted on" afterwards, and SectionI X discusses some of them.

v i m = (e