DECOUPLED NEWTON LOAD FLOW

Brian StottPower Systemnb ory

University of Manchester Institute of Scienc and technology, Manchester, U.K.

Abstract-In Newton load-flow solutions the mathematical decouplingof busbar-voltage angle and magnitude calculations has several computa-tional and conceptual attractions. A hybrid Newton formulation ex-ploiting this principle has been developed and well tested. For moder-ately-accurate solutions the method has advantages over the formalNewton approach in terms of computer storage and speed, particularlyin adjusted solutions, and is at least as reliably convergent.

INTRODUCTION

This paper presents a new and relatively simple variant of theNewton-type a.c. load flow method for digital-computer applications.The method takes advantage of the special characteristic of electricalpower-system transmission networks that MW flows/voltage angles andMVAR flows/voltage magnitudes are only loosely interdependent, byseparating the iterative re-evaluation of the angles and magnitudes.This principle is of course not new, but the algorithm developed appearsto be the first demonstrably satisfactory such load-flow scheme to haveemerged. The method is an extension of previous workl in which thereliability and speed of the formal Newton method were improved by anon-iterative starting process.

THE FORMAL NEWTON APPROACH

In conventional Newton methods the equations of load flow arewritten as a single set F (X) = 0 and solved by the formal application ofthe generalised Newton (-Raphson) algorithm:

F(Xk) = _ r( xk ).AXk+l (1)

The most popular and successful formulation is that in which F isthe set of busbar active and reactive power mismatches and the solutionvariables are the unknown busbar-voltage angles and magnitudes. In thispolar power-mismatch version, (1) becomes

1APk Hk |Nk | Ak

k3 k kkL 0 E:::1 (2)

The details of this method are well-documented in the literature.2 Thecorrection term A\Vk+l is usually divided by Vk, but this does notaffect the numerical performai e of the algorithm, except to simplifythe calculation of some of the Jacobian-matrix elements. The squareJacobian matrix in (2) is highly sparse, and (2) is solved at each iterationby sparsity-programmed ordered elimination. The convergence of theprocess is fast, compared with alternative load-flow algorithms, and isquadratic, so that the convergence rate increases rapidly as the exactsolution is approached, giving high-accuracy solutions in say 3-6 itera-tions. Its reliability is comparatively good, but it fails on some ill-con-ditioned problems. Its storage requirements are moderate, and likesolution time, increase roughly linearly with problem size providedthat the programming is sufficiently skillful.

Paper T 72 135-7, recommended and approved by the Power System Engi-neering Committee of the IEEE Power Engineering Society for presentation at theIEEE Winter Meeting, New York, N.Y., January 30-February 4, 1972. Manuscriptsubmitted September 13, 1971; made available for printing November 19, 1971.

DEVELOPMENT OF THE DECOUPLED METHOD

It has often been suggested that the angle and magnitude calcula-tions may be decoupled by neglecting submatrices N and J in (2), sincetheir elements are comparatively small, leaving two separated sets

APk = Hk AOktl (3)ati

%tJ p AQk = Lk AVk1 (4)

The obvious attraction of this is that (3) and (4) may be solved inde-pendently, reducing the storage requirements and possibly the compu-tation per iteration.

When (3) and (4) are iterated in a simultaneous-displacementsmode, convergence is weak and by no means competitive with (2) forreliability or total solution time, as confirmed by the author's experi-ments. It would appear to be much better to iterate (3) and (4)successively, using the newly-calculated 0 k+l from (3) in establishingAQk and Lk in (4). This is also found not to be very satisfactory formost problems, and the reason is that (4) is a relatively unstablealgorithm at some distance from the exact solution.

In the method of this paper, (4) is replaced by the polar current-mismatch Newton formulation in terms of the voltage-magnitudecorrections, and the algorithm takes the form

APk = Hk ABek+l

AIm = Dm AV+l

(3)

(5)

The derivations of (3) and (5) are given in the Appendix. The startingvalues are 0 =0, V 1, unless better estimates are available.

The iteration indices in (3) and (5) are different to signify that0 and V might not be iterated strictly consecutively with each other.The best iteration strategy is evidently that which produces the fastestreduction of both AP and Al to within the prescribed accuracytolerances.

PERFORMANCE OF THE METHOD

The values 01 and V1 after the first iteration differ little fromthose calculated by the starting process in [ 1] Each 0 is usually withinone or two degrees of the solution, but may be in error by five degreesor so at some busbars with large angles. The special feature of (5) isthat each value VI is remarkably accurate, averaging less than 0.2%error, and no individual error VI - V i larger than about 2%* has everbeen found in numerous studies on a wide range of power systems,including many ill-conditioned ones. These 01 and V1 thus have theproperty not always possessed by formal Newton methods of beingreliably-good first estimates of the solution. If the problem is physicallyunstable, or marginally stable, this is indicated by V1 and possibly by01, which may be used as feasibility monitors in the solution, as indi-cated in [1 ].

The rate of convergence in the early stages of the solution is fasterthan that of the formal Newton approach, although the decoupledmethod does not subsequently have the quadratic property needed forrapid high-accurate solutions. However, in the routine solution of loadflows, particularly for large well-developed systems, busbar power-

*Two such cases cited in [I ] were later found to be erroneous.

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mismatch tolerances of 1 MW/MVAR or more are regarded as adequatefor many purposes. The reliability of the method is high. On a rangeof specially selected test problems that are difficult or impossible tosolve with any of the popular non-Newtonian methods, it performs atleast as reliably as (2), and oscillates without converging on only oneextremely ill-conditioned problem which can only be solved by themethod described in [ 1] .

The decoupled method performs much better than the formalcurrent-mismatch methods. These have been found by the author to beless satisfactory in general than (2) with the possible exception oflight-load cases as described in [3]. The decoupled method also per-forms well on light- or no-load, since (5) becomes an almost linear set.

The convergence of the method has the oscillatory characteristicof Newton methods with the peculiarity that the MW or MVAR mis-matches are capable of increasing at certain stages of the process, de-pending upon the relative rates of convergence of the voltage angles andmagnitudes. This is because at each application of (3) and (5) theNewton algorithm is attempting to solve an incorrect problem.

In the tests performed, (3) and (5) were iterated consecutivelyfrom a normal flat start for the solution of ordinary unadjusted loadflows, with a convergence criterion of 1 MW/MVAR for the busbarmismatches. As shown in the Appendix, the mismatches may be com-puted most conveniently during the construction of (3), so thatTable 1 gives whole numbers of iterations. In fact, for more than half

TABLE 1

RESULTS FOR DECOUPLED METHOD

Largest No, of LargestNo. of initial iterations mismatchesbusbars mismatches to satisfy in solutiontessse MWcriterion oftest system MW MVAR 1 MW/MVAR MW MVAR

6 50 53 2 0.14 0.02

13 538 1015 3 0.62 0.09

14* 92 62 3 0.4 0.1

19 253 193 3 0.3 0.03

22 68 231 3 0.49 0.23

27 548 1562 3 1.0 0.15

30* 93 72 3 0.3 0.09

38 123 111 3 0.85 0.06

43 290 561 5+ 47 16

57* 287 173 5 0.53 0.15

107 1935 1317 >5 1.97 0.3

118* 589 323 3 1.0 0.02

125 1879 2658 3 0.18 0.12

134 2563 2839 3 0.28 0.62

180 1802 1762 3 0.65 0.72

205/ 700 52 4 0.9 0.11

* IEEE Test Systems,+ not converging, also not soluble by (2).+ not soluble by formal rectangular power-mismatch Newton Method.

of the test problems, the accuracy criterion has been satisfied half aniteration previously, i.e. before the final solution of (5).

ADJUSTED SOLUTIONS AND OTHER APPLICATIONS

Adjustments of system quantities in the Newton load flow tend todegrade its basic solution speed considerably, because the extra numberof iterations required is a large proportion of the total, much more so

than with slowly-converging load-flow processes. Common examples ofsuch adjustments are in the control of transformer taps, phase shifts,and inter-area transfers, and the enforcement of Q and V limits by PVand PQ busbar conversions or vice versa.

Most adjusted quantities predominantly affect either MW flows orMVAR flows but not both. The decoupled nature of the new methodenables sub-iteration to be carried out on either (3) and (5). For in-stance, an on-load in-phase transformer-tap study with a conventionaladjustment algorithm that ordinarily takes 10 iterations of (2) can beconverged with adequate accuracy in the following manner:

(3) + (5) to give quite good initial estimates of V; tapadjustment-+ (5);

+ (3) for correction of 0+; (5) + (5) + (5) with tapadjustments each time;

+ (3) for correction of 0; + (5) + (5) + (5) with tapadjustments; + (3) to finalise the solution.

This represents a speed advantage at least in the ratio 3:5 comparedwith the use of (2).

Controlled devices may alternatively be incorporated as primaryvariables in the Newton solution,3'5'6 and the above tapping studycould be undertaken in this way, with (5) modified appropriately. Inmany cases the formalised adjustment approach does not appear to bequite so attractive in practice as it does in theory. The author's experi-ence of this with in-phase transformers is that they often weaken theconditioning of the problem so that the solution takes more iterationsto converge than normal, or tends to diverge. Devices are required tocounteract this, such as the imposition of limits on the tap correctionsand sometimes on AV, particularly in the early iterations, which slowthe solution down even if successful. If the taps, which are treated ascontinuously variable, reach their limiting positions and/or are finallycorrected to their nearest actual settings, usually one or two iterationsmore are required. The average total number of iterations for a widerange of systems studied was 7-8.

With some controls, zero-valued diagonal elements appear in theJacobian matrix, and modifications have to be made to the eliminationprocess, whose solution efficiency decreases.

For optimal power-flow solutions, the decoupling of voltageangles and magnitudes has some obvious drawbacks, e.g. for limits onMVA loadings. In many power-system optimisation studies, however,active- and reactive-powers can be and are treated substantially sepa-rately. MW flows are optimised for minimum generation cost assumingfixed voltage-regulating-device settings, while the latter are optimisedalone assuming specified MW generations. Separated optimisations ofthese kinds can be built around the decoupled load-flow approach witheconomy. Matrix D in (5) is a linearising approximation to the voltagemagnitude/MVAR problem which is valid over a much wider range ofvariation than matrix L in (4).

SOLUTION EFFICIENCY ASPECTS

The sparse matrix equations (3) and (5) are constructed compactlyrow-wise and are reduced to upper-triangular form by sparsity-pro-grammed ordered Crout elimination, for which routines in FORTRANwere developed by the author. The elimination routine must be pro-grammed to accept matrix rows with no off-diagonal elements to caterfor PQ busbars not connected to other PQ busbars, in the solution of(5). The conventional pre-iteration ordering of the network nodes isperformed according to initial row sparsity or dynamic row sparsityas selected by an input parameter, and the same ordering is used for (3)and (5), since in networks with a low proportion of PV busbars, thestructures of (3) and (5) are similar. The computer times for orderingand elimination are roughly linearly proportional to network size. Onthe Atlas Mark I computer and a non-optimising FORTRAN compiler,the respective ordering schemes take 0.08 sec and 1.04 sec for a 705-busbar 1073-line network. The solution of (3) takes 1.3 sec and 1.05sec respectively for the two ordering methods.

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The time for constructing and solving (3) and (5) for 0 and V isabout the same as that using (2). Although more terms have to be calcu-lated and addressed in constructing (2), this is counterbalanced by thefact that the opportunity of using some calculated terms in more thanone expression is lost by constructing (3) and (5) separately.

The storage required by the elimination routine for the solutionof (3) and (5) is 3n+2b real words and n+2b+60 integer words, wheren = no. of busbars, and b = no. of lines. This storage could be reducedslightly at the expense of speed. The corresponding storage for thesolution of (2) is 8n + 8b real words and 4n + 8b + 60 integer words. Forb= 1.5n the decoupled solution requires less than one third of thestorage of the formal Newton approach. A little less storage is neededif the compact rows of the Jacobian equations are generated and elimi-nated one at a time, but with a similar proportional saving. One ofseveral advantages of constructing all the equations at once is that anextra complete iteration to detect convergence to the specified accuracyis avoided.

CONCLUSIONS

The decoupled method is a competitive alternative to formalNewton methods for routine load flows with moderate accuracy re-quirements. The storage for the solution of the Jacobian equation isless than one third of that for the formal method. The reliability ofconvergence appears to be at least equally good, noting that most ofthe practical test problems used are difficult ones. For unadjusted loadflows, the solution speed of the decoupled method is comparable withthat of the formal methods, where very efficient elimination routinesare available. It appears that a variety of elimination routines used inpractice increase in solution time rather more than linearly with systemsize, and in these cases, the decoupled method gains in the comparisonof speeds. For adjusted solutions, computing time can be saved bysub-iteration of the voltage angles or magnitudes, and this applies tocontrollable devices using either conventional adjustment algorithmsor where the controlled parameter is incorporated as a primary variablein the Newton solution. For optimal solutions, the opportunity ofseparating the MW and MVARproblems with computational economyis available.

APPENDIX

Derivation of eqs. (3) and (5)

The active-power mismatch equation at busbar i is

APi = Ps Vi I Vk (Gik cosOik + Bik sinOik) (6)

By partial differentiation with respect to 0, for k 7 i

i/ k Hik Vi Vk Bik CoS0ik- Gik sin eik (7)and

-aWP./ae. = H.. = -H. (8)1 1 11 k.i l

If busbar i is a PQ type, the reactive-power mismatch, used for con-vergence testing, is

-DAI./av = D = G.. sine. + B.. cose.1 1 11 11 1 11 1

-i-(pi~ sineO.- QSP cose.)/V.2 (12)As with (2), the above trigonometrical expressions can be replaced byrectangular-coordinate terms if desired.

ACKNOWLEDGEMENTS

Grateful acknowledgement is made to Prof. L. M. Wedepohl forencouragement and for the facilities provided in the Power SystemsLaboratory, UMIST, and to the University of Manchester RegionalComputing Centre for running the programs.

REFERENCES

[1] B. Stott, "Effective starting process for Newton-Raphson loadflows," Proc. IEE, vol. 118, p. 983, August 1971.

[21 W. F. Tinney and C. E. Hart, "Power flow solution by Newton'sMethod," IEEE Trans. Power Apparatus & Systems, vol. PAS-86,p. 1449, November 1967.

[3] H. W. Dommel, W. F. Tinney and W. L. Powell, "Further develop-ments in Newton's Method for power system applications," Paperno. 70 CP 161-PWR, presented at the IEEE Winter Power Meeting,N.Y., January 25-30, 1970.

[4] W. F. Tinney and J. W. Walker, "Direct solution of sparse networkequations by optimally ordered triangular factorisation," Proc.IEEE, vol. 55, p. 1801, November 1967.

[5] N. M. Peterson and W. S. Meyer, "Automatic adjustment of trans-former and phase shifter taps in the Newton power flow," IEEETrans. Power Apparatus & Systems, vol. PAS-90, p. 103, January1971.

[6] J. P. Britton, "Improved load flow performance through a moregeneral equation form," IEEE Trans. Power Apparatus & Systems,vol. PAS-90, p. 109, January 1971.

Discussion

R. H. Jordan (Indianapolis Power and Light Company, Indianapolis,Ind. 46206): The author is to be complimented for an ingenious methodwhich appears to reduce storage requirements without sacrificing speedof solution. In solving the load flow problem, there is always the trade-off of speed and storage. The Gauss-Seidel method has always beenattractive for its low storage requirement. The Newton-Raphson methodoffers some advantage in speed and accuracy at a tremendous increasein storage and programming complexity.

One major advantage of the Newton-Raphson method is its abilityto handle negative impedances. The author's paper states that the de-coupled method will handle difficult convergence problems. Does thisinclude negative impedances, or have systems with negative impedancesbeen attempted?

Another advantage of the Newton-Raphson method is its accuracy.Has the author made direct comparisons of systems solved by the de-coupled method and the Newton-Raphson method to determine theeffect of mis-match upon the line flows?

Finally, I would like to compliment the author on a very wellwritten paper, clear and concise.

AQi = QSP + Hii + Bii Vi (9) Manuscript received February 16, 1972.

The imaginary part of the current-mismatch equation at busbar i is

Ai. = (PPp sinG. - QSP cose.)/V-TG sinO +B cose )V1 i ~ 1 1 1 i~(Gksinkikcsk)Vk

Partially differentiating with respect to V, for k f i

i/avk= Dik = Gik sinek + Bik cosek

(10) Norris M. Peterson and W. Scott Meyer (Systems Control, Inc., PaloAlto, Calif. 94306): The author is to be complimented for an excellentpaper. Based on the Table I results, an algorithm rivaling Newton's

( 11) Manuscript received February 16, 1972.

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method in both speed and storage requirements is indicated. Others willno doubt further test the idea in this country, thereby enlarging theexperience upon which comparisons can be made. In the meantime, wewould be grateful for the author's comments on the several pointswhich follow.

First, regarding the behavior of procedures for the inclusion ofarea-interchange control and local transformer-tap adjustment, theauthor's experience seems contrary to that of references 3, 5, and 6. Ithas been our experience that convergence for a wide range of problemshas been essentially identical with that of the associated fixed system(system without such adjustments), and that transformer taps and trans-former type-switching is a very well-behaved function. We would ap-preciate the author's comments on this. Also, does the author have anynew quantitative data indicating how the possible existence of a zerodiagonal term degrades the computational efficiency of the elimination?

The author has chosen to replace the usual reactive equation bythe imaginary part of the current equation. These relations wouldappear to be closely related when all voltage phase angles are small(near zero). However, the coupling between the imaginary current in-jection and the voltage magnitude approaches zero as the phase angleapproaches 90 degrees. Has the author experienced such a difficulty asthis might suggest? In this country it is generally the case for largepower flow studies that phase angles on the periphery are 60 or 70degrees out of phase with the slack bus.

What has been the author's experience with modifying the itera-tion cycle? i.e., has the author tried multiple solutions of (3), and thenmultiple solutions of (5)?

*Convergence speed in the vicinity of the solution is an importantfactor. Realizing that the author's scheme cannot actually convergequadratically due to the decoupling, does the convergence rate remainconstant as the solution is approached? For a typical system, how is thenumber of required iterations related to the largest bus mismatch, fortolerances much smaller than those of Table I?

It is felt that the author's storage-requirement comparison withthe formal Newton algorithm deserves scrutiny. The stated figure ofmore than a 3 to 1 improvement does not imply that his decoupledprogram will generally run in one third of the memory partition used bya Newton power flow - not without added use of auxiliary storage,beyond that used by a typical Newton power flow.2 Assuming 1.5branches per node, a typical subroutine for solution by Newton'smethod uses about 24n real and 1 On integer words of core storage.Similar assumptions applied to the author's algorithm indicate a figureof 13n real and 1On integer words. As a percentage of the Newtonrequirement, this is 68% on a fixed-word-length machine, or 58% on abyte-oriented machine (or when typical integer packing is used).This,however, still represents a significant storage reduction.

An examination of the time per iteration reveals that1. The matrix formation of the decoupled formulation is, for allpractical purposes, identical to that for the full ac power flow.2. The decoupled solution requires about twice the number of logicaloperations per iteration as does the Newton factorization.3. The required floating-point operations per iteration are signifi-cantly less for the decoupled solution, which should more than offsetpoint 2.Thus there appears to be both a storage and a time saving per iterationfor the author's decoupled algorithm, and its final acceptance as a stand-ard solution technique should depend heavily on its convergencecharacteristics and solution reliability for a wide range of problems.

W. 0. Stadlin and B. F. Wollenberg (Leeds & Northrup Company,North Wales, Pa. 19454): Success of the quadrature current-mismatchequation (5) apparently is due in part to the way that the diagonalcoefficients are derived. For a flat start, i.e., V = 1 and 0 = 0, equations(4) and (5) reduce to,

AQi = (- z Bik - Bii) AVi - E Bik AVk From (4)k k ti

- AIi = AQi = (QSP - Bii) AVi - I Bik AVk From (5)1 k fi

This difference in the main diagonal is due to dividing the complexinjected node power by the complex node voltage as shown by theauthor's reference 1.

We would appreciate the author's comments with respect to solv-ing equations (3) and (5) independently (i.e., not successively) as alinearized ac load flow. This could lead to "one-iteration" approximatecontingency studies utilizing existing Newton-Raphson Load Flowprograms.

Manuscript received February 17, 1972.

H. H. Happ (General Electric Company, Schenectady, N.Y. 12305):I compliment the author on both an excellent paper and on his con-cise presentation.

A few questions arose in reading the paper and perhaps the authorwould care to comment. I had the impression from reading the paperthat the author has tested the method on larger systems than thosereported in Table 1; any additional results which he may care to presentwould, of course, be appreciated. Dr. Stott mentions that the methodis at least as reliable as the complete N-R method. He may wish toclarify whether the comment concerning reliability is a) based primarilyupon the convergence results obtained'in the test cases; and b) restrictedto the successive solution of Equations 3 and 5 as used in the testcases or whether the convergence statement is more inclusive per hiscomments under Equations 3 & 4 of the text.

Manuscript received February 22, 1972.

Brian Stott: I am very grateful to the discussers for their interest, usefulcomments and kind remarks.

The conclusions drawn in the paper of the method's performanceare based entirely on experience with the test systems listed in Table 1,and it is perhaps appropriate to start this reply by giving more detailsabout the test problems. With the exception of the IEEE and 6-bussystems the problems were contributed by industry because of con-vergence difficulties using nodal iterative and the more-powerfulZ-matrix methods. By choosing such test problems it was hoped thatany conclusic.s on reliability of convergence etc. would be likely tohave a wide range of validity. The systems in Table 1 originate fromeight different countries, and include main-transmission, distributionand mixed networks. Most of the features that traditionally cause ill-conditioning are to be found in the systems. The 13-bus network con-tains two large negative series admittances; other problems have largest/smallest series-admittance ratios of 105 p.u., long e.h.v. overhead linesand cables, no PV buses, and minimal interconnection. The formalNewton polar-power, rectangular-power, and polar-current mismatchmethods perform generally well on these problems, failing on one,three and two of them, respectively. Incidentally, using the startingprocess in [I I no convergence difficulties occur on these or any ofhundreds of other feasible problems encountered to date.

In answer to Dr. Happ, the 705-bus problem for which timingswere quoted has not been solved as a complete a.c. load-flow-problem,since only the d.c. load-flow data was available. The performance of thedecoupled method should, like formal Newton methods, be more orless invariant to problem size when using an adequate computer wordlength.

Answering Dr. Happ and Messrs. Peterson and Meyer, the onlyiteration scheme used for unadjusted solutions has been single succes-sive applications of (3) and (5). This is because an examination of thesolution at each stage does not reveal any obvious discrepancies be-tween the convergence rates of the MW and MVAR mismatches. Forunadjusted solutions, other iteration sequences are unlikely to be profit-able except in particularly difficult cases. The Newton algorithm alwaysconverges extremely rapidly at each solution of (3) and (5), and therewould normally be no point in improving this convergence further,since neither (3) nor (5) is individually a correct representation of theload-flow problem. The overall rate of convergence of the solution islimited more by the rate at which the 0 and V values mutually relaxtowards a common solution point. The very fast overall convergencerate during the first one or two iterations is due mainly to the de-coupling and not to Newton's algorithm, and can be achieved by more-approximate algorithms such as that in [7].

After the first iterations, convergence settles down to an oscillatory-linear rate, and in all cases except the 43-bus problem an 'exact' solu-tion can be reached by continuing the iterations. Typically, the largestmismatch reduces to 0.01 MW/MVAR in five iterations from a flatstart, although this is fairly problem-dependent. The Newton decoupledmethod does not appear to suffer from the problem sometimes experi-enced with more-approximate decoupled formulations of oscillationwithout convergence after a certain stage of accuracy has been reached.

Mr. Jordan revives the problematic issue as to what mismatchaccuracy is needed in load-flo'w solutions. In my experience, a maxi-mum mismatch of 1 MW/MVAR on a large power system usuallyrepresents a maximum line-flow error of the same amount or less, andbus voltage-magnitude errors are insignificant, for most purposes. Thevery high accuracy achieved by formal Newton methods is not neces-sarily a- virtue - merely that with these methods it is difficult not tooverconverge the solution.

Manuscript received April 26, 1972.

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Messrs. Peterson and Meyer have raised several very relevant ques-tions. Firstly, I agree with their observation about the coupling be-tween V and Al in (5) for large angles approaching 90 degrees. Thelargest angles encountered have been 48 degrees, without difficulties.This problem could be minimised by the choice of slack, if it exists inpractice. Secondly, although the automatic transformer-tap adjustmentmethod in [5] seems to be normally very good, the coupling betweenbtuses in the Jacobian matrix is inevitably weakened for all but radialtransformers by the removal of previously-nonzero elements. Pre-stumably this makes a noticeable difference only when the problem isalready ill-conditioned, as in many of the systems that we have beensttudying. As soon as the discontinuities of tap limits (with back-offopporttunities) and correction to the nearest physical tapping are con-sidered, we find that the conventional tap-adjustment approach witha graded damping of the tap increments is competitive, requiring 4-8Newton iterations for an accurate solution. With the automatic method,we have had a lot of trouble with tap values tending to be thrown wildlyotutside limits in the early stages of the solution, as mentioned in [8].

Zero-valued diagonal elements in the Jacobian matrix, producedby the incorporation of certain controls in the Newton formulation,6require either prior restructuring of the matrix or partial pivoting atexecution time. Neither of these is attractive programming-wise. With-out having specific data, my guess is that the solution time could insome cases be increased considerably. For the representation of thesetypes of single-criterion controls, we use the optimal Newton load-flowmethod in the mode described in [9], for which the convergenceproperties and flexibility are quite impeccable, even on the most diffi-cult systems, whenever the basic Newton load flow can be converged.The main disadvantage is that of storage, for the lower triangular factorof the Jacobian matrix.

In the Newton decoupled method, the storage required for (3) or(5) and its solution is reduced by a factor of three or four compared

with the formal Newton method. Messrs. Peterson and Meyer are quitecorrect in pointing out that overall, the percentage storage reduction inthe respective programs is much less than this, because during the con-struction of (3) or (5) the compact nodal admittance matrix and thenode-type vector, at very least, must also be in core. In my programs,the decoupled storage is 62% of the formal Newton storage on thisbasis, with or without integer packing, which agrees with their figures.

Messrs. Stadlin and Wollenberg, to whom I am indebted for point-ing out typographical errors in the appendix of the original of thepaper, refer to the important subject of approximate contingencystudies. Using the base case solution to start with, I believe that oneiteration of the decoupled Newton process, i.e. one solution of (3)followed by one solution of (5), should be better than alternative 'one-iteration' methods. However, from the point of view of computingspeed, the algorithms that avoid repeated matrix triangulation seem tobe-more advantageous7. We have now tested the 0 algorithm suggestedin [101, which approximates the diagonal elements of the coefficientmatrix used in [7] to give a truly constant triangulated matrix forevaluation of 0. This works very well and enables the solution to beiterated to high accuracy, where required, economically. Even severalcomplete iterations of such a method would be faster than one iterationof the decoupled Newton method, as well as giving greater accuracy.

REFERENCES

[7] N. M. Peterson, W. F. Tinney and D. W. Bree, "Iterative LinearAC Power Flow Solution for Fast Approximate Outage Studies",Paper no. T 72 140-7, presented at Winter Power Meeting, N.Y.,January 1972.

[8] W. F. Tinney, discussion of reference 5.[9] W. F. Tinney, discussion of reference 6.[101 B. Stott, discussion of reference 7.

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