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FAST DECOUPLED LOAD FLOWB. STOTT 0. A L S A qPower Systems LaboratoryUniversity of Manchester Institute of Science and Technology, Manchester, U.K.

ABSTRACTThis paper describes a simp le, very reliable andextremely fast load-flow solution method with a widerange of practical application. It i s attractive foraccurate or approximate off- and on-line routine andcontingency calculations for networks of any size,andcan be implemented efficiently on computers with res-trictive core-store capacities. The method i s a dev-elopment on other recent work employing the MW-(/MVAR-V decoupling principle,and its precise algorith-mic form has been determined by extensive numericalstudies. The paper gives details of the method's per-formance on a series of practical problems of up to1080 buses. A solution to within 0.01 MW/MVAR maxi-mum bus mismatches i s normally obtained in 4 to 7iterations, each iteration being eq ual in speed to 1 1Gauss-Seidel iterations or 1/5th of a Newton itera-t i o n . Correlations of general interest between the

power-mismatch convergence criterion and actual sol-ution accuracy are o b t a i n e d .INTRODUCT ION

Load-flow calculations are performed in systemp l a n n i n g , operational planning and operation/control.The choice of a solution method for practical appli-cation i s frequently difficult. It r e q u i r e s a care-f u l a n a l y s i s of the comparative merits and demeritsof the many available methods1 in such respects asstorage, speed and convergence characteristics, toname but t h e most o b v i o u s , and to relate these to therequirements of the s pe c i fi c a p p li c a t i o n and comput-i n g f a c i l i t i e s . The difficulties a r i s e from the factthat no one method possesses al l the desirable fea-tures of the others. For routine solutions Newton'sm e t h o d 2 has now gained widespread p o p u l a r i t y . Howeveri t i s limited for small-core applications where theweakly-convergent Gauss-Seidel-type method i s themost economical, and it is not as fast as newermethods for a p p r o x i m a t e r e p e t i t i v e solutions such asi n AC s e c u r i t y m o n i t o r i n g 4 ' 5 .

Numerical methods are generally at their mostefficient when they take advantage of the p h y s i c a lp r o p e r t i e s of the system being solved. Hence, fore x a m p l e , the exploitation of network sparsity b yordered elimination and skilful programming in t h eNewton and other methods has been a very importanta d v a n c e 3 . More recently, attention has b een givento the e x p l o i t a t i o n of the loose physical interac-

P a p e r T 7 3 4 6 3 - 7 . r e c o m m e n d e d a n d a p pr o ve d b y t h e I E E E P o w e r S y s t e mE n g i n e e r i n g C o m m i t t e e o f t h e I E E E P o w e r E n g i n e e r i n g S o c i e t y f o r p r e s e n t a t i o n a tt h e I E E E P E S S u m m e r M e e t i n g & EHV/UHV C o n f e r en c e , V a n c o u v e r , B . C . C a n a d a ,J u l y 1 5 - 2 0 , 1 9 7 3 . M a n u sc r i pt s u bm i tt e d F e b r u a r y 1 2 , 1 9 7 3 ; m a d e a v a i l a b l e f o rp r i n t i n g May 1 5 , 1 9 7 3 .

tion between MW and MVAR fl ows i n a power system,by m a t h e m a t i c a l l y d e c o u p l i n g the MW-F and MVAR-V4-11calculationsThe method described i n this paper i s a rationalintegration of some of these i d e a s . It combines manyof the advantages of the e x i s t i n g " g o o d " methods. Thealgorithm i s s i m p l e r , faster and more reliable thanNewton's method, and has lower storage requirementsfor entirely in-core solutions. Using a small numberof core-disk block transfers i t s core requirements aresimilar to those of the Gauss-Seidel method. Themethod i s equally suitable for routine accurate loadflows as for outage-contingency evaluation studiesperformed on- or off-line.

NOTATIONAPk + jAQk = comp lex power mismatch at bus k , where

A P = P 5 s - V ~V Gcs Bkik V k m ( G k m c o s k m + B k m s i n O k m )mek= 9 s V k - V m ( G s i n - B c o s A Q k k V k X m km km- km k mmE k

( 1 )( 2 )

P k p + i k p = scheduled c o m p l e x power at bus kOk ' V k = v o l t a g e a n g l e , m a g n i t u d e at bus k( km ( 3 k mG km + j B k m ( k , m ) t h element of bus admittancematrix [ G ] + j [ B IA O , AV = voltage angle, magnitude correctionsmek signifies that bus m i s connected to bus k ,including the case m = k , and [ ] signifies vector o rmatrix.max A P ,max I Q | = largest absolute elemnt of [ A P I J A Q ]m a x l e v , m a x i e S i = largest absolute bus-voltage magni-tude error, branch MVA-flow error, r e s p e c t i v e l y .

DERIVATION OF BASIC ALGORITHMThe well-known polar power-mismatch Newtonm e t h o d 2 i s taken as a convenient and meaningful star-t i n g point for the derivation. The Newton method i sthe formal a p p l i c a t i o n of a g e n e r a l a l g o r i t h m forsolving nonlinear e q u a t i o n s , and constitutes success-ive solutions of the sparse real j a c o b i a n - m a t r i x e q t a -tion L APiHA

AQ L AV/V I ( 3 )The first step in applying the MW-O/MVAR-V decou-p l i n g principle i s to neglect the c o u p l i n g submatrices[ N ] and [ J ] i n ( 3 ) , g i v i n g tw o separated equations

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[ A P ] = [ H ] [ A ) ] ( 4 )[ A Q ] = [ L I [ A V / V I ( 5 )

where H = L = V k V ( G s i n k m - B cos ) for mOk1m Lkm VkVm km km km km2 2H k k = BkkVk-Qk and Lkk = -BkkVk+Qk

Equations ( 4 ) and ( 5 ) may be solved alternately as adecoupled Newton m e t h o d 1 1 , r e - e v a l u a t i n g and retriang-ulating [ H ] and [ L I at each iteration,but furtherphysically-justifiable simplifications may be made.Inpractical power systems the f o l l o w i n g a s s u m p t i o n s arealmost a l w a y s valid:csOk 1 ; G k m s i n k m k m ; Q k BkkVk J

so that good approximations to ( 4 ) and ( 5 ) are:[ A P ] = [ V . B ' . V ] [ A O ] ( 6 )[ A Q ] = [ V . B " . V ] [ A V / V ] ( 7 )

At this stage of the derivation t h e elements ofthe matrices [ B ' ] and [ B " ] are s t r i c t l y elements of [ - B IThe decoupling process and the final algorithmic formsare now completed b y :( a ) omitting from [ B ' I the representation of those net-work elements that predominantly affect MVARflows,i . e . shunt reactances and off-nominal in-phasetransformer taps( b ) omitting from [ B " I the angle-shifting effects ofphase shifters( c ) t a k i n g t h e left-hand V terms i n ( 6 ) and ( 7 ) onto the left-hand s i d e s of the e q u a t i o n s , and theni n ( 6 ) removing the influence of MVAR flows on t h ecalculation of [ A O ] by s e t t i n g all the right-handV terms to 1 p.u. Note that the V terms on the

left-hand sides of ( 6 ) and ( 7 ) affect the behav-iours of the d e f i n i n g functions and not the coup-ling( d ) neglecting series resistances i n calculating theelements of [ B ' ] , which then becomes t h e DC-approx-imation load-flow matrix.This i s of minor i m p o r t -ance, but i s found e x p e r i m e n t a l l y to g i v e s l i g h t l yi m p r o v e d results.

With t h e above modifications the f i n a l fast de-coupled load-flow equations become[ A P / V ] = [ B ' ] [ A C ] ( 8 )[ A Q / V I = [ B " I [ A V ]

when [ A P / V I and [ A Q / V ] are zero. A brief account ofthe alternative decoupled algorithms investigated isgiven in Appendix 1 .APPLICATION TO PRACTICAL LOAD FLOW SOLUTIONS

Iteration schemeOf the iteration strategies tried, undoubtedlythe best scheme for al l applications i s to solve ( 8 )

and ( 9 ) alternately, always using the most recentvoltage values. Each iteration cycle comprises onesolution for [ A O @ to update [ 0 1 and then one s o l u t i a nfor [ A V ] to update [V],termed here the ( l 1 , l V ) s c h e m e .Separate convergence tests are used for ( 8 ) and ( 9 )with t h e criteria:max AP < c p , m a x i A Q i < c q ( 1 0 )

The fl ow diagram of the process i s given in Fig.1 . The convergence-testing logic permits the calcul-ation to terminate after a [ ( A ] solution ( c a l l e d a 2iteration). I t is also possible, though unusual inp r a c t i c e , to terminate after more than one consecutive[ A O ] or [ A V ] solution if [ A Q ] or [ A P I respectively donot need converging further.

K P = K Q = 1 l

CALCULATE [ A P / V ]

?es ~ ~ - , yeL K S O L V E ( 8 ) & UPDATE o l

| C A L C U L A T E [ A Q / 7 V ] | OUT

|SOLVE ( 9 ) & UPDATE ( V I YesI K P = 1 I to

Both [ B I ] and [ B " ] are real,sparse and have the stru-ctures of [ H I and [ L ] r e s p e c t i v e l y . Since they con-tain only network admittances they are constant andneed to be triangulated once only at the beginning ofthe s t u d y . [ B " ] i s symmetrical s o that only i t s uppertriangular factor i s stored,and i f phase shifters areabsent or accounted f o r by alternative means [ B ' I ] i salso symmetrical.

F i g . 1 Flow d i a g r a m of the iteration scheme

Convergence characteristicsThe i m m e d i a t e appeal of ( 8 ) and ( 9 ) i s that veryfast repeat solutions for [ A ] and [ A V ] can be c b t a i i e du s i n g the constant triangular factors of [ B ' ] and [ B " ] .These solutions may be iterated with each other insome defined manner towards t h e exact solution, i . e .

A typical convergence pattern for the method i sshown i n F i g . 2 , i n terms of the largest mismatches atevery half i t e r a t i o n . Each solution of ( 8 ) and ( 9 )produces r a p i d reduction of [ A P ] and [ A Q ] respectivelytowards the exact solution of the load-flow problem.86 0

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The effect of 0-changes on the MVAR flows i s sho wn b ythe increase produced in m a x [ A Q [ after each solutionof ( 8 ) . In t h i s and most other problems the effectof V-changes on MW flows i s less pronounced, becausethe active l osses due to MVAR flows are normallysmal l er than the reactive losses due to MW flows.Although this suggests that faster overall convergencecould be achieved by adopting a ( 2 , 1 V ) iterationscheme,this idea ( a m o n g others) w as not successful inany of the systems s t u d i e d . Overconverging [ A P ] at anystage produces a severe increase in [ A Q ] , therebyslowing down the overall convergence rate.

X 4

: Ru zU)

m

u

E - 4U ;

5NUMBER OF ITERATIONS

F i g . 2 Typical convergence pattern of themethod ( I E E E 118-bus system)

Algorithms ( 8 ) and ( 9 ) each have geometric con-vergence, a s explained more fully in Appendix 2 . Thisi s not a s fast as Newton's quadratic rate,but i s morethan compensated for by the much-faster iterations p e e d . Unlike Newton's method, the fast decoupledmethod has not failed on any feasible problem.Accuracy requirements

The question as to what convergence tolerancesc and cq i n ( 1 0 ) give an acceptably-accurate load-flow solution i s relevant not only to the presentmethod but more generally. Overconvergence, andtherefore unnecessary extra iterations,can be avoidedby correlating the solution errors with the b us MW andMVAR mismatches. Since power f l o w s , voltage magni-tudes and losses are the primary objectives of a load-flow calculation,an investigation of their errors wasincluded in the numerical tests performed.Numerical results

The studies carried out on the fast decoupledmethod included u n a d j u s t e d , adjusted and sequential-outage solutions. In a l l cases m a x i A P | and m a x | A Q |

were both monitored at every half iteration. In eachcase the process was iterated to the "exact" (veryaccurate) solution. By storing the voltages, branchMVA flows and losses at each stage of the calculationa post-solution history of the largest errors i n thesquantities during the iterations was constructed.The load-flow problems used were chosen as del-iberately-severe tests. With the exception of theIEEE Standard Test Systems, the problems are prac-tical systems from seven countries supplied b y ind-ustry because they have presented convergence diff-iculties. The systems contain a wide variety offeatures, and cover the range of voltage levels fromE HV to distribution. They include long EHV line andcable circuits, capacitive series branches, shuntcapacitors and reactors, and very small and verylarge series impedances and X/R r a t i o s . Some net-works are highly radial and others are highly m e s h e d ,and t h e proportion of P V buses varies from 0 - 4 5 % .In some cases the voltage regulation i s nearly 0.3p .u.Table I gives results for unadjusted base-casesolutions of the test problems using the normal flatvoltage start. Details of the errors are shown forthe first three iterations. The largest bus voltage-magnitude error m a x i e v i at each iteration i s giveni n percent. The largest branch-flow error m a x i e sat each iteration i s given both in MV A and as a per-centage of the branch f l o w . Except while the solu-tion i s still very inaccurate,the errors i n the totalsystem MW and MVAR losses are i n s i g n i f i c a n t , and arenot presented. A l s o , the m a x i e s i occurs on one ofthe most heavily-loaded lines or transformers, andi s nearly all MW-flow error.The cl ose correspondence b etw een max | A P I andm a x f e s | gives a good guide to the specification ofc in ( 1 0 ) . M a x | A Q I and m a x l e v | do not correlate soclosely, but by iteration 3 they are seen to be ofthe same order as each other. Depending on the pur-pose of the load-flow s t u d y , most of the solutionsshown i n Table I are s u f f i c i e n t l y accurate after 3

i t e r a t i o n s , and even after 2 iterations i n sewralcases. The number of iterations r e q u i r e d i s seen notto be a function of problem s i z e .Ap art f ro m the results shown, the fast decoupledprogram has been used on other problems. Noteworthyamong these i s a very difficult 43-bus system whichfails with Newton's method and takes 2 2 ( l 0 , l V ) iter-ations for an accurate solution. Otherwise, the 27-bus African system of Table I takes the largest num-ber of iterations of any problems t r i e d . The methodhas also solved a 1080-bus system ( s e e the section onSolution Speed and S t o r a g e ) .

ADJUJSTED SOLUTIONS

The most common adjustments in routine load-flowsolutions are s i n g l e - c r i t e r i o n controls such as on-load transformer taps, phase-shifters and area inter-changes,and generator Q limits and load-bus V l i m i t s .Conventionally, adjustments are made in between orduring iteration c y c l e s by some s i m p l e e a s i l y - p r o -grammable parameter- or i n j e c t i o n - c h a n g i n g error-feedback l o g i c or by b u s - t y p e s w i t c h i n g 1 2 . Theseschemes usually work satisfactorily with the s l o w l y -c o n v e r g i n g load-flow methods when the system i s suf-ficiently well-conditioned. With a powerful methodsuch as Newton the number of extra iterations due tothe adjustments can be large relative to t h e 3 or 4

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Two approaches are available. In the first, eachviolating bus i s explicitly converted to P Q t y p e sothat the MVAR output is held at the limiting valUe.The b us remains a P Q type during the rest of the sol-ution unless at some stage it can be re-converted toa P V type at t h e original voltage magnitude withoutthe violation. In the fast decoupled program, con-verting any number o f bus types at one time involvesretrianulating [ B " ] .The second approach i s to correct the voltage ofeach violating P V bus k by an amoUnt A V k at e a cfollowing iteration to reduce the error A Q k = ( Q k

- Q k ) to zero. The convergence of this process israpid when an approximate sensitivity factor S k rela-t i n g A V k and A Q k i s used thus:AVk = S k A k / v k ( 1 2 )

I f S k is defined according to ( 9 ) it i s the diagonalelement corresponding to bus k in the inverse ofmatrix [ B " I augmented by the previously-absent rowand column for bus k . Appendix 3 shows that S k canbe calculated easily and very rapidly without retri-angulating [ B " ] . For an operational network each S kmay be stored permanently and updated only for s i g -nificant system configuration c h a n g e s . The correc-tion ( 1 2 ) ceases to be a p p l i e d i f at some s t a g e inthe solution t h e value of Vk i s restored to or goesbeyond i t s original value.

Both approaches have been programmed and tested,and are equally e f f e c t i v e . The s e n s i t i v i t y methodusually takes more i t e r a t i o n s , but a g a i n s t this i trequires n o retriangulations for limit enforcementand back-off, and i s s i m p l e r p r o g r a m m i n g - w i s e . Usinga 0 . 0 1 MW/MVAR tolerance on bus-mismatches and Q - l i m i tenforcement, t y p i c a l results for the IEEE 30-bus and1 18 -b us systems are quoted. With tw o buses in t h e30-bus system v i o l a t i n g their Q - l i m i t s by 17 % and 6%after i n i t i a l convergence,the solution takes 7 2 iter-ations by a p p r o a c h 1 , and 9 iterations b y a p p r o a c h 2 .For the 118-bus system, with 1 2 buses i n i t i a l l y v i o -lating their l i m i t s by an average of 12% and a maximumof 2 0 % , the solutions take 7 2 and 1 2 1 iterations res-p e c t i v e l y .V-limits on P Q buses are handled i n the inversemanner, using either of the tw o a p p r o a c h e s .

APPLICATION TO OUTAGE STUDIESS e q u e n t i a l b ranch- and generator-outage load-flow calculations are p e r f o r m e d off-line for p l a n n i n gand o p e r a t i o n a l p l a n n i n g studies and on-line or semi-on-line f o r ' o p e r a t i o n / c o n t r o l , to evaluate systemperformance and p a r t i c u l a r l y s e cu r it y f o ll o w i n g cred-ible outage c o n t i n g e n c i e s . In al l these a p p l i c a t i o h sa l a r g e and time-consuming number of load flows are

solved c o n s e c u t i v e l y and a u t o m a t i c a l l y after a s o l u -tion of the base-case problem, and branch f l o w s , busvoltages and g e n e r a t o r M V A R outputs are monitored todetect insecure and unsatisfactory conditions. Forthese p u r p o s e s very accurate s o l u t i o n s are not norm-a l l y r eq u i r e d.The a p p r o x i m a t e decoupled approach has alreadybeen developed successfully for the outage-studya p p l i c a t i o n 4 . The present method can also beemployed e f f i c i e n t l y without r e t r i a n g u l a t i n g the base-case [ B ' ] or [ B " ] at any s t a g e .

Transmission network outagesLine and transformer outages are simulated by anadaptation of the inverse-matrix modification t e c h -nique applied to [ B ' ] and [ B " ] . The experiments haveshown that it i s only necessary to simul ate the re-moval of series transmission elements from thesematrices. Shunt capacitors and reactors, line charg-ing capacitance, and the shunt branches of off-nominal-tap transformer equivalent circuits can remain

in [ B " ] without affecting convergence noticeably. Al loutages must of course be reflected correctly i n thecalculation of [ A P / V I and [ t x Q / V ] .For the outage of a series branch t w o n o n - s p a r s evectors [ X ' ] and [ X " must be calculated,each requir-ing one repeat solution using t h e factors of [ B ' ] and[ B " ] , respectively. After each solution of ( 8 ) , [ A O ]i s corrected b y an amount

- c ' [ X ' ] [ M ' [ ( A B ] ( 1 3 )where c ' i s a scalar and [ M ' ] is a highly-sparse rowvector containing one or two elements. This corre-tion requires about n multiplication-additions. Sim-i l a r l y , after each solution of ( 9 ) , [ A V ] i s correctedby an amount

-c" [ X " [ M " ] [V] ( 1 4 )The details of this technique are given in Appendix 4 ,including the case of multiple outages. The schemecan equally b e used for b r a n c h switch-in operationsif desired.Generator outages

A generator outage requires as input data ther e d i s t r i b u t i o n of scheduled MW generation t o compen-sate for the loss of s u p p l y . It i s usual to caterfor the partial l o s s of generation at a plant, i nwhich case i t i s only necessary to reflect the changesi n scheduled MW i n [ A P / V ] . The complete loss of aplant means that the net MVAR of the rel evant busmust b e fix ed b y either of the methods used i n enforc-ing generator Q l i m i t s .

Since only one bus is involved, an alternativei s t o augment t h e upper-triangular f a c t o r s of [ B ' ]and [ B " ] by an extra column each. This i s e x a c t l yt h e same calculation as performed i n obtaining thes e n s i t i v i t y factor S k used i n t h e 9 - l i m i t adjustments,and i s d e s c r i b e d i n A p p e n d i x 3 .Experience with outages

Tests were c o n d u c t e d o n several s y s t e m s , i n c l u d -i n g a s e r i e s of 3 5 line and transformer s i n g l e - o u t a g ecases o n the heavily-loaded 180 bus problem which i spart of the main British transmission system. Ten ofthe outaged branches were chosen as the leAst-loadedones in the base-case, and 2 5 were chosen as the mostloaded ones, with base-case branch power fl ows of upto 2370 MW and 240 M V A R .

It was found that the ( l , l V ) iteration schemeremains the best i n this a p p l i c a t i o n , and it was con-firmed that u s i n g t h e base-case solution to s t a r t theo u t ag e c al cu la ti o n s is on the average d i s t i n c t l ybetter than using the normal flat voltage start ( e a c ht i m e . T h e worst branch MVA-flow error at each itera-

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TABLE II RESULTS FO R 3 5 SINGLE-BRANCH OUTAGE CASES ON 180-BUS SYSTEM

N o . o f ( l 0 , l V ) No. of ERRORS I N GROUPSiterations for outage mmax. branch-flow cases i n v serror of 3 % groups largest average largest average largest average l argest averageGroup 1 , 1 iteration 2 1 4 9 1 3 1. 8 0.3 0.13 0.05 2 3 8Group 2 , 2 iterations 1 2 10 4 0 .8 0.3 0.62 0.18 1 7 5Group 3 , 3 iterations 2 1 0.7 1.3 0.7 0 .0 9 0.05 0.1 0.06

tion was obtained a s before. Since this always seemsto occur on one of the most heavily-loaded branches,the performance of the method for security-monitoringcan be meaningfully interpreted even though branchMVA-rating data w as not available.In the tests, one ( 1 0 , 1 V ) iteration was adequatein the majority of cases to g i v e a maximum branch-flowerror of an a r b i t r a r i l y - c h o s e n 3 % , and in two casesthree iterations were n e e d e d . Table II gives a break-down of the 3 5 cases.

SOLUTION SPEED AND STORAGECentral to the fast decoupled method i s the useof efficient sparsity-programmed routines for the tri-a n g u l a t i o n of the sparse, real, symmetric matrices[ B ' ] and [ B " ] by Crout elimination and the subsequentsolutions of ( 8 ) and ( 9 ) by forward and backward sub-stitutions 3 . I f , as i s becoming more widespread, suchroutines are available as standardpackages,the methodi s not difficult to program efficiently.At the beginning of the load-flow study the sys-tem buses are re-ordered to avoid excessive fill-up i nthe table of factors d u r i n g the t r i a n g u l a t i o n of [ B ' ] .It can be shown that this bus ordering remains equallysuitable for [ B " ] with the P V buses by-passed i n theo r d e r i n g l i s t . I f the system i s very large or if con-s e c u t i v e load-flow solutions are to be calculated thenit i s advantageous to minimise fill-up as far as poss-ible by using a good o r d e r i n g scheme 6.Since [ B ' ] and[ B " ] remain unchanged, triangulation routines that per-form the o r d e r i n g d u r i n g elimination are at no greatdisadvantage.The triangulation routine is programmed to takeaccount of matrix symmetry so that only the upper-tri-angular factors of [ B ' ] and [ B " ] are stored for use i nthe solution r o u t i n e . Using dynamic re-ordering accor-d i n g to row s p a r s i t y , the o r d e r i n g , t r i a n g u l a t i o n andsolution times each vary roughly linearly with networks i z e 3 . Fo r a t y p i c a l well-developed network with nbuses and b branches the numb er of elements i n the

upper-triangular factor of [ B ' ] i s a b o u t 3 ( n + b ) / 2 .D e p e n d i n g on the number a n d location of P V buses, thetriangulation and s o l u t i o n processes for [ B " ] arefaster and r e q u i r e less storage than fo r [ B ' ] , in somecases c o n s i d e r a b l y s o .The calculations of t h e vectors [ A P / V I and [ A Q / V ]can each be performed r a p i d l y i n a s i n g l e sweep of theb r a n c h admittance l i s t . F o r each series-branch connec-t i n g buses k and m , i t can be seen from ( 1 ) and ( 2 )that where a p p r o p r i a t e , t h e terms VkVmGkm and VkVmBkmcan be used directly in accumulating both A P k and A P m ,

or A Q k and A Q m . Also, it is noted that s i n i k m = - s i n O m l kand c o s k m = c o s O m k . These trigonometrical functionsare calculated using the approximations sinO % O - C 3 / 6and cosO t 1 _ 0 2 / 2 + 0 4 / 2 4 , which have been found to givefinal load-flow results indistinguishable from thoseobtained with the accurate function evaluations. Ifs t o r a g e i s not critical, the sine and c o s i n e termscalculated during the construction of [ A Q / V ] can bestored for use i n the next construction of [AP/V].Froma flat voltage start, each sine and cosine term can beinitialised to 0 and 1 respectively for the first sol-ution of ( 8 ) .

During the experiments the constituent parts ofthe solutions were timed on the CDC 7600 computer.Results are given for a 1080 bus, 1862 branch systemwith 4 1 P V buses,and the IEEE 118 bus, 18 6 branch sys-tem with 53 P V buses,using non-optimised FORTRAN com-pilation.The inferior ' s t a t i c ' bus ordering scheme wasused and the sine and cosine terms were not stored.Time, seconds

10 80 - bus 118-busBus orderingformation & triangulation of [ B ' ]formation & triangulation of [ B " ]calculation of [ A P / V I & conver-gence testingsolution of ( 8 ) & updating [ 0 ]calculation of [ A Q / V ] & conver-gence testingsolution of ( 9 ) & updating [ V ]

0 . 0 6 20 . 6 3 50.5150.0070.0280 . 0 0 7

0.059 0.0060.06 3 0.0050 . 0 5 3 0.0050.050 0.001

Fo r the 1 0 8 0 - b u s problem, the solution to within 0 . 0 1 M Wand MV AR bus-mismatch tolerances from a flat voltagestart required 7 ( 1 0 , l V ) iterations plus tw o additional( 1 0 ) iterations i n the final stages. Excluding i n p u t /output, t h e total time was 3 . 2 seconds of which 1.98seconds were for the iterative process. For the 118-bus problem these times were 0.13 and 0.09 secondsrespectively. The iteration times coul d be reducedc o n s i d e r a b l y by using a better ordering scheme and morerealistic mismatch tolerances.The storage requirements of the fast decoupledmethod are about 4 0 % less than t h o s e c f Newton's m e t h o d .This saving i s reduced somewhat i f the sine and cosineterms are s t o r e d , and for outage cases where vector s[ X ' ] and [ X " ] are needed.Each of the parts of the so l-ution g i v e n in the above t i m i n g listis c o n v e n i e r t l y per-formed i n a separate program subroutine. Apart froms i m p l e subroutine overlaying,core storage can be econ-omised by reading certain selected vectors from diskwhen t h e y are required and writing them back after therelevant s u b r o u t i n e . These block transfers are per-formed a limited number of times during a solution,and

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should not degrade the method's speed too severely.Using this scheme,the core storage r e q u i r e m e n t s of themethod are about the same as those of the Gauss-Seidelmethod.Network tearing fo r load-flow calculations i s notusually economical in computing speed when orderedelimination i s used for solving the sparse networkequations. However, if a piecewise approach i s desir-able for other reasons, e.g. storage,any of the avail-

able Y-matrix d e c o m p o s i t i o n or d i a k o p t i c a l methods canbe applied to give a s e r i e s of smaller constant [ B ' ]and [ B " ] matrices. Note that since branches connectedto P V buses are absent from [ B " ] , a certain amount ofnetwork tearing i s already inherent in this matrix.CONCLUSIONS

The fast decoupled method offers a uniquely attra-ctive combination of advantages over the establishedmethods, including N e w t o n ' s , in terms of speed,reliab-i l i t y , simplicity and storage, for conventional loadflow s o l u t i o n s . The basic algorithm remains unchangedfor a variety of different a p p l i c a t i o n s . Given a seto f good ordered elimination routines,the basic programi s easy to code e f f i c i e n t l y , and its s p e e d and storagerequirements are roughly p r o p o r t i o n a l to system s i z e .Auseful feature o f the method i s the ability to reducei t s core-storage r e q u i r e m e n t s to a p p r o x i m a t e l y those ofThe Gauss-Seidel method with a small r L u n b e r o f core- diskblock transfers. The method performs wel l with conven-tional adjustment a l g o r i t h m s , a n d solves network o u t a g esecurity-check cases usually i n one or tw o iteratiors.It i s computationally suitable for o p t i m a l load-flowc a l c u l a t i o n s , and d e v e l o p m e n t s in this area are to bereported separately.

ACKNOWLEDGEMENTSThe authors are grateful to the U n i v e r s i t y of Man-chester R e g i o n a l C o m p u t i n g Centre for r u n n i n g the pro-grams. The Middle East Technical U n i v e r s i t y , A n k a r a ,Turkey, is acknowledged b y 0 . A l s a g for g r a n t i n g leaveof absence.

APPENDIX 1Alternative d e c o u p l e d versions i n v e s t i g a t e d

This appendix g i v e s a very brief outline of thelarge number of e x p e r i m e n t a l studies conducted on var-iants of the decoupled a p p r o a c h . Each version w as usedon all or a selected group of the load-flow test p r o b -l e m s of Table I . In nearly all cases the p r o b l e m s weresolved successfully by the different v e r s i o n s , demon-strating the general effectiveness of d e c o u p l i n g thea n g l e and m a g n i t u d e calculations. The o b j e c t of thestudies was then to find the version that best combinesconsistent and r a p i d success with other c o m p u t a t i o n a ladvantages.Initially,the methods of r e f s . 6 , 7 and 11 were com-p a r e d 1 7 . The D e s p o t o v i c and D e c o u p l e d Newton methodsconverge well but r e q u i r e r e - t r i a n g u l a t i o n s . Uemura'smethod i s not very s a t i s f a c t o r y on some p r a c t i c a l sys-tems because MVAR conditions are not excluded from the" P - 0 " matrix. The idea i n r e f . 1 8 of u s i n g the currentmismatch [ A I ] in ( 9 )

[ A I ] = [ B " ] [ A V ] , ( 1 5 )and a variant in which each V k is divided b y c o s O k ,were less successful than a n t i c i p a t e d . The other ideain r e f . 18 of u s i n g the constant matrix [ B ' ] in thea n g l e calculation of refs. 4 and 6 g i v e s g o o d conver-gence.

Having established this latter point, attentionwas devoted to the decoupled Newton a p p r c z c i m a t i o n s ( 6 )and ( 7 ) which can be re-written[ A P / V ] = [ B ' ] [ V . A ][ A Q / V ] = [ B " ] [ A V ]

( 1 6 )( 1 7 )

A question at this stage was whether the V terms onthe left-hand sides of ( 1 6 ) and ( 1 7 ) and on the rightof ( 1 6 ) should be omitted. The seven resulting combi-nations of 0 and V algorithms were compaied with eachother on al l the test systems,and the combination ( 8 )with ( 9 ) emerged a s the best, although one or two o fthe others w ere not much worse. ( 8 ) and ( 9 ) and otherversions can be rearranged to give direct solutionsfor [ 0 ] and [ V I instead of [ A ] and [ A V ] ,but there isno convergence difference u n l e s s further approximationsare made. The [ A ] and [ A V ] forms were p r e f e r r e d , inspite of a few extra arithmetic operations, becausethis form directly retains the familiar AP and AQ m i s -match quantities for convergence testing.

All of the above tests w ere carried out using a( 1 , 1 V ) iteration scheme,flat voltage starts, and noadjustments. Having finalised on the algorithms ( 8 )and ( 9 ) , the problems of Tab le I were solved us ingeach of the following schemes: ( a ) ( 2 O , 1 V ) , ( b ) ( 2 O , 1 V )once and subsequently ( l 1 , l V ) , ( c ) ( 2 , 2 V ) once andsubsequently ( 1 0 , l V ) . , ( d ) ( l V , l 1 ) , and ( e ) ( 3 , 2 V ) .None of these schemes was as good a s t h e s i r p l e ( l , l V )approach, which i s also best for transformer-tap a d j u s -tment solutions, and branch outage solutions.

Several different attempts at acceleration, inc-luding block successive over-relaxation a r d a heuristicapproach based on testing sign-changes in the bus mis-matches, proved to be unrewarding.For al l the branch-outage studies,the removal ofline-charging and transformer n r - e q u i v a l e n t shunt ele-ments from [ B " ] was found to affect the convergenceonly minimally, provided that they are correctly re-moved in the Calculations of [ A P / V I and [ A Q / V ] , and

similarly for shunt capacitor or inductor outages.However,failing to remove the outaged series elementsfrom [ B " ] and [ B ' ] i s totally unsuccessful.Appendix 4shows how these outages are carried out,and also ' t h a tthere i s no extra effort involved in removing trans-former T r - c i r c u i t shunts.APPENDIX 2

N otes o n the convergence of the fast decoupled methodThe iteration process of the f a s t d e c o u p l e d methodhas three distinct components, each with i t s ow n con-vergence characteristics: ( a ) the solution of [ A P / V ] =0 for [ 0 ] using algorithm ( 8 ) , ( b ) the solution of[ A Q / V ] = 0 for [ V I using algorithm ( 9 ) , and ( c ) theinteractive effects of V-changes and 0-changes on the

defining functions [ A P / V I and [ A Q / V ] respectively.Algorithms ( 8 ) and ( 9 ) are both Newton-like,except that instead of re-evaluating t h e t r u e Jacobian-matrix tangent-slopes to the left-hand functions ateach i t e r a t i o n , fixed approximated t a n g e r t - s l o p e s [ B ' ]and [ B " ] are used. The algorithms therefore corres-pond to t h e generalised ' f i x e d - t a n g e n t ' or ' c o n s t a n t -s l o p e ' m e t h o d 1 9 which has geometric convergence. Forreasonably-behaved functions, this method i s very re-liable and i f , as i n the present a p p l i c a t i o n , t h e fixedtangent-slopes correspond closely to the Jacobianmatrix at the initial point ( f o r a flat voltage s t a r t ) ,the initial convergence i s very rapid.The process doesnot ' h o m e i n ' as f a s t as the quadratic Newton method as

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the exact solution i s approached, but load-flow solu-tions are rarely required to very high accuracy.The fixed-tangent method i s not thrown off-coursewhen i t encounters ' h u m n p s ' in the defining functions,whereas Newton's method tends to be mis-directed evento t h e extent of divergence from the desired solutionin such cases. In the decoupled load-fbw problem,thechanging V-values during the solution have the effectof causing sometimes-oscillatory shifts i n the shapesof the multi-dimensional surfaces [ A P / V I a s functionsof [ 0 1 ] ,and likewise for 0 - v a l u e s on [ A Q / V I a s functionsof [ V I . Since [ B ' ] does not represent MVAR conditionsi t corresponds to fixed tangent-directions that take rcognisance of t h e s e shifts and a r e t h e r e f o r e not affec-ted by V-oscillations. Likewise for [ B " ] i n relationto MW c o n d i t i o n s .

APPENDIX 3Calculation of sensitivity factor

The matrix factors of [ B " ] are expressed in thesymmetric f o r m , so that ( 9 ) i s[ A Q / V ] = [ U ] [ D ] [ U ] [ A V ] ( 1 8 )

where [ D ] 1 i s a diagonal matrix and [ U ] i s upper-triangular with unit diagonal elements. [ U l t i s notstored. The solution of ( 9 ) i s then performed accord-i n g to the following three steps: ( a ) a forward-sub-stitution process on [ A Q / V ] and [ U ] t , g i v i n g an i n t e r -mediate vector [ F ] ; ( b ) the trivial matrix multipli-cation [ G ] = [ D ] [ F ] ; ( c ) a backward-substituti o nprocess on [ G ] and [ U ] to g i v e [ A V ] .I f a new PQ-type bus i s created i n the network,an extra row and column must be added to [ B " ] .Lettingt h e order of [ B " ] be ( n - l ) , then the new P Q bus i sordered n , i . e . l a s t , and the enlarged matrix i s

the high sp arsity of [ C 1 ] and [ G I , the calculation be-comes extremely fast.The process can be expressed i n more formal matrixnotation using the 'multi-product of inverse' approachand is easily generalised for any non-symmetric matrix.In the present case, the process can be used for theoriginal triangular factorisation of [ B ' ] and [ B " ] ,adding one column at a time.

APPENDIX 4Branch outage calculation

Let either ( 8 ) or ( 9 ) be represented in the base-case problem as the equation[ R I = [B] [E]0 .0 ( 2 2 )

for which a solution

[ E ] = [ B ] [ R ]0 0 ( 2 3 )can be obtained using t h e factors of [ B ] . In the most0g e n e r a l case, the outage of a line ( n e gl ec t i n g c h ar g -i n g c a p a c i t a n c e ) or transformer can be reflected in[ B o ] by modifying two elements i n row k and two in ro wm . The new o u t a g e matrix i s then

[ B 1 ] = [ B 0 ] + b [ M ] t [ M ] ( 2 4 )where:

b = line or nominal transformer series admittance[ M ] = row vector which i s null except for Mk = a ,

and M - - 1ma = off-nominal turns ratio referred to the buscorresponding to ro w m , for a transformer= 1 for a l i n e .

1 2To obtain the matrix factors of [ B " ] i t i s only nec-essary to e n l a r g e the e x i s t i n g factors by a d d i n g ann t h column to [ U ] and a d i a g o n a l element to [ D ] .

R e p l a c i n g [ A Q / V ] i n step ( a ) above by t h e sparse[ C 1 ] , [ G I i n step ( b ) gives the new sparse column of[ U ]( 2 0 )G I = [ U l n U 2 * . . . U ] t

where U = 1 . The new element D i s given bnn nnD = ( C - E Uu / D . )nn 2 j=l j n i i

= the s e n s i t i v i t y factor S used fkl i m i t s and generator outages.U s i n g t h e normal solution r o u t i n e , this calction takes about the same time as a solution ofBy reprogramming steps ( a ) and ( b ) to take accoun

Depending on t h e types of the( 1 9 ) row, k or m , might be presentcase either Mk or Mm above i sboth the connected buses arerequires no modification.

connected buses,only onein [ B ' ] or [ B " ] , i n whichzero, as a p p r o p r i a t e . I fP V or slack, then [ B " ]It can be shown that

-l - l - l[ B 1 ] = [ B ] - c [ X ] [ M ] [ B I1 ~ 0 0w h e r e :

c = ( l / b + [ M l [ X ] ) and [ X ] = [ B o [ M ]0

( 2 5 )

( 2 6 )The solution vector [ E 1 ] to the outage problem i s :

- 1[ E 1 ] = [ B 1 ] [ R I ( 2 7 )( 2 1 ) and from ( 2 3 ) , ( 2 5 ) and ( 2 7 ) we have

[ E 1 ] = [ E o ] - c [ X ] [ M ] [ E 0 ] ( 2 8 )Hence, the solution to the base-case problem i s e a s i l yc o r r e c t e d , as in ( 1 3 ) and ( 1 4 ) . [ X ] i s calculated atthe b e g i n n i n g of the outage case b y a repeat solutionu s i n g the factors of [ B 0 ] with [ M ] t as the i n d e p e n d e n tvector.

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The above procedure can be applied recursively formultiple simultaneous outages.The solution vector [ E li s corrected successively as the effect o f each branchoutage i s introduced one at a t i m e . For a set of out-a g e d branches 1 ..... m , the recursion i s[ E . ] [ E 1 i - 1 c i [ X . l [ M i l [ E . i l I for i=l...m ( 2 9 )

where [ M . ] i s defined as before for the outagedbranzhintroduced ith in the sequence.The scalars c i and thevectors [ X i 1 are calculated at the beginning o f thecase from the recursive a l g o r i t h m :

M-1 t[ X i ] = [ B 0 I []i[ X i ] = [ X i ] - c. X ] [ M ] [ X i ] for j=l...i-l1 1 J J J ( 3 0 )c . = ( l / b - [ M . ] x 1 ) - 11 1 1 [i)

- 1all for i = 1 . . . . m . I f any ci= 0 a s p l i t network i sindicated.Although this scheme avoids the retriangulation oft h e network matrices, it i s faster for at most three

simultaneous ly-outaged branches.REFERENCES

1 . D . A . Conner, " R e p r e s e n t a t i v e bibliography on l o a d -flow analysis and related topics",paper C73-104-7presented at the 1 97 3 IEEE Winter Power Meeting,N ew Y ork , N.Y., January 28-February 2 , 1973.2 . W.F. Tinney and C.E.Hart, "Power flow solution byNewton's method", IEEE Trans. ( P o w e r Apparatus andSystems), Vol.PAS-86, p p . 1 4 4 9 - 1 4 6 0 , November 1 9 6 7 .3 . W.F. Tinney and J.W. Walker, "Direct solution ofsparse network e q u a t i o n s by o p t i m a l l y ordered tri-angular f a c t o r i z a t i o n " , Proc. I EE E , V o l . 5 5 ,pp . 1801-1809, November 1 9 6 7 .4 . N.M.Peterson, W.F.Tinney and D.W.Bree, "Iterativelinear AC power flow solution for fast approximateoutage s t u d i e s " , IEEE Trans. ( P o w e r Apparatus andSystems), Vol.PAS-91,pp. 2048-2053, September/Oct-o b e r , 1 9 7 2 .5 . W.F. Tinney, N.M.Petersen, " S t e a d y state securitym o n i t o r i n g " , Proc. Symposium on real time controlof e l e c . power systems, Brown, Boveri & C o m p 6 L t d . ,Baden, Switzerland, 1 9 7 1 .6 . S . T . D e s p o t o v i c , B.S.Babic and V.P.Mastilovic, " Arapid and reliable method for solving load flow

problems", IEEE Trans. ( P o w e r Apparatus and Systems),Vol. PAS-90, pp . 123-130, January/February 1971.7 . K . Uemura, "Power flow solutions by a Z - m & 3 r i x t y p emethod and its application to contingency evalua-t i o n " , PICA Conference Record, p . 3 8 6 , May 1971.8 . K . Uemura, "Approximated Jacobians i n N e w t o n ' s pcwerflow method", Proc. Power System Computation C o n -f e r e n c e , paper 1 . 3 / 2 , September 1 9 7 2 .9 . C.H. Jolissaint, N.V.Arvanitidis & D.G.Luenberger"Decomposition of real and reactive power flows: Amethod suited for on-line a p p l i c a t i o n s " , I E E E T r a r s( P o w e r Apparatus and Systems) ,Vol.P AS-91, pp.6 61-670, March/April 1972.

1 0 . B . Stott, "Effective starting process for Newton-Raphson load flows", Proc. IEE, Vol. 118, pp.983-9 8 7 , August 1 9 7 1 .1 1 . B . Stott, "Decoupled New ton l oad flows",IEEETrars.( P o w e r Apparatus and Systems) , V o l . P A S - 9 1 , p p . 1955-1957, September/October 1972.1 2 . G.W. Stagg and A.H. El-Abiad,"Computer methods i npower system a n a l y s i s " , M c G r a w - H i l l , 1 9 6 8 .1 3 . N.M.Peterson and W.S.Meyer, "Automatic adjustmentof transformer and phase shifter t a p s i n the Newtonpower f l o w " ,IEEE Trans. ( P o w e r Apparatus and Syst-e m s ) , Vol. PAS-90, pp . 103-106, January 1 9 7 1 .1 4 . J.P. Britton, "Improved l oad fl ow performancethrough a more general equation f o r m " , I E E E Trans.( P o w e r Apparatus and S y s t e m s ) ,Vol.PAS-90, pp.109-1 1 4 , January 1 9 7 1 .1 5 . H.W. D o m m e l , W.F.Tinney and W.L. P o w e l l , "Furtherdevelopments in Newton's method for power systema p p l i c a t i o n s " , p a p e r 70CP-161-PWR presented at the1970 IEEE Winter Power M e e t i n g , New York, N . Y . ,January 25-30, 1 9 7 0 .1 6 . B . Stott and E . Hobson, "Solution of large power-system networks by ordered elimination: a c o m p a r i -son of ordering s c h e m e s " , Proc. I E E , Vol. 1 1 8 , p p .12 5- 13 5 , January 1 9 7 1 .1 7 . J . Hassall, "Comparison of modern l o a f l o w m e 1 h o d s "B.Sc. Project Report, UMIST, April 1 9 7 2 .1 8 . B . S t o t t , discussion of reference ( 4 ) .1 9 . C . - E . Froberg, " I n t r o d u c t i o n to Numerical A n a l y s i s " ,Addison-Wesley P u b l i s h i n g Comp., I n c . , p p . 19-261 9 6 6 .

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D iscu ssi o nE . H o b s o n ( T h e C a p r i c o r n i a I n s t i t u t e o f A d v a n c e d E d u c a t i o n ; R o c k -h a m p t o n 4 7 0 0 ; QLD, A u s t r a l i a ) : T h e a u t h o r s h a v e made s i g n i f i c a n tc h a n g e s t o p r e v i o u s l y d e s c r i b e d d e c o u p l e d load f l o w m e t h o d s i n o r d e rt o i m p r o v e s o l u t i o n s p e e d a n d t o - p e r m i t r a p i d r e l i a b i l i t y a s s e s s m e n t s t ob e m a d e . I am g r a t e f u l t o t h e m f o r g i v i n g me t he o p p o r t u n i t y t o e x -p e r i m e n t w i t h t h e i r m e t h o d s b e f o r e p u b l i c a t i o n .My c o m m e n t s r e f e r t o e x p e r i e n c e g a i n e d wh il st i n co r po r a t i n gs o m e o f t h e a u t h o r s ' t e c h n i q u e s i n t o a n o p e r a t i o n a l i n t e r a c t i v e m i n i -c o m p u t e r l o a d f l o w p r o g r a m w r i t t e n i n B a s i c a n d e m p l o y i n g t h e G a u s sS e i d e l a l g o r i t h m l 1 ] . T h i s p ro gr am w a s d e s i g n e d pri m ari ly f o r s m a l l d i s -t r i b u t i o n a u t h o r i t i e s w h e r e v e r s a t i l i t y w a s c o n s i d e r e d m o r e i m p o r t a n tt h a n sp e e d .T h e cla im t h a t c o r e r e q u i r e m e n t s a r e s i m i l a r t o t ho se o f t he G a u s sS e i d e l m e t h o d w a s f o u n d t o b e a p p r o x i m a t e l y t r u e . I n c r e a s e d c o m -p l e x i t y o f p r o g r a m m i n g a n d t h e n e c e s s i t y t o s t o r e t he f a c t o r i s e d f o r m so f b o t h B ' a n d B " m a t r ice s o f e q u a t i o n s ( 8 ) a n d ( 9 ) l e d t o a n i n c r e a s ei n c o r e r e q u i r e m e n t s o f a p p r o x i m a t e l y 2 : 1 . T h i s i n c r e a s e p ro ve d e m -b a r r a s s i n g o n a n 8 k PDP8 c o m p u t e r when a t t e m p t i n g t o r e p l a c e t h eG a u s s S e i d e l a l g o r i t h m w i t h t h e new d e c o u p l e d m e t h o d . I n o r d e r t of i t t h e n e w m e t h o d o n t o t he P D P 8 , c e r t a i n c h a n g e s w e r e made t o t hea u t h o r s ' a p p r o a c h . T h e s e m o d i f i c a t i o n s p e r m i t t e d t h e r e p l a c i n g o fm a t r i c e s B ' a n d B " w i t h a s i n g l e m a t r i x B , c o m p r i s i n g t he n e g a t e d r e -a c t a n c e s o f t h e n o d a l a d m i t t a n c e m atri x . F o r P V b u s e s , t h e r e a c t i v em i s m a t c h e s o f e q u a t i o n ( 9 ) w e r e p u t t o z e r o , a n d n o a d j u s t m e n t s madet o V . S i m p l e pre - orderi ng w a s co n s id e r e d a d e q u a t e f o r t h e s i z e o f n e t -w o r k s c o n t e m p l a t e d , h e n c e n o r e - n u m b e r i n g s c h e m e w a s e m p l o y e d .W i t h t h e a b o v e m o d i f i c a t i o n s , s o l u t i o n s w e r e o b t a i n e d f o r a r a n g eo f s m a l l d i s t r i b u t i o n n e t w o r k s . I n g e n e r a l t h e a u t h o r s ' c l a i m s w e r e s u b -s t a n t i a t e d . E a c h i t e r a t i o n t o o k a p p r o x i m a t e l y 1 . 5 t i m e s a G a u s s S e i d e l

i t e r a t i o n , a n d o v e r a l l s o l u t i o n t i m e f o r ' d i f f i c u l t ' n e t w o rk s i m p r o v e d b y1 0 : 1 o r m o r e . T h e n u m b e r o f i t e r a t i o n s w a s g r e a t e r t h a n f o r p u r eN e w t o n R a p h s o n , b u t n o s p e e d c o m p a r i s o n s w e r e p o s s i b l e s i n c e d i f -f e r e n t c o m p u t e r s w e r e i n v o l v e d . C o n v e r g e n c e w a s a ppr e cia b ly s l o w e da n d d i v e r g e n c e p r o b l e m s w e r e s o m e t i m e s i n t r o d u c e d i n t h e s i m p l i f i e dm e t h o d i f i n - p h a s e t r a n s f o r m e r t a p s d e p a r t e d m o r e t h a n a b o u t 5%f r o m n o m i n a l s e t t i n g s . T h i s w a s p r e s u m a b l y b e c a u s e t h e e f f e c t s o f t a pc ha ng in g w e r e b e in g i n c o r p o r a t e d i n t o t he P - 0 c o u p l i n g . A f u r t h e rd iv e r ge n ce p r o b l e m ar o s e w i t h a v e r y w e a k 9 b u s r a d i a l s y s t e m i n c l u d -i n g a l i n e w i t h R / X a p p r o a c h i n g 2 . C o n v e r g e n c e w a s a ch ie v e d b y m ak i ngR / X c l o s e r t o u n i t y . R e m o v a l o f s e r i e s r e s i s t i v e e l e m e n t s f r o m B ' , a sd e s c r i b e d b y t h e a u t h o r s , e lim i n a t e d t h e c o n v e r g e n c e p r o b l e m b u t d idn o t s i g n i f i c a n t l y r e d u c e t h e n u m b e r o f i t e r a t i o n s . C o n v e r g e n c e c o u l dn o t b e a ch ie v e d b y r e m o v i n g s e r i e s r e s i s t i v e e l e m e n t s f r o m t h e s i n g l em a t r i x B .A f t e r e x p e r i m e n t i n g w i t h t he a u t h o r s ' d e c o u p l e d m e t h o d s I c o n -clu d e th a t t h e y h a v e much t o r e c o m m e n d t h e m i n t e r m s o f s p e e d ,s t o r a g e a n d s i m p l i c i t y i n c o m p a r i s o n w i t h e x i s t i n g m e t h o d s .REFERENCE:[ 1 ] E . H o b s o n " U s i n g t h e P D P 8 / e a s a N e t w o r k A n a ly se r " . P r o c e e d -i n g s o f t h e 1 9 7 2 D e c u s A u s t r a l i a S y m p o s i u m

M a n u s c r i p t r e c e i v e d Au g u s t 1 , 1 9 7 3 .

T . E . D y L i a c c o a n d K . A . R a m a r a o ( C l e v e l a n d E l e c t r i c I l l u m i n a t i n gC o m p a n y , C l e v e l a n d , O h i o 4 4 1 0 1 ) : T h i s p a p e r , w h i c h i s a m o d e l o fc l e a r e x p o s i t i o n , r e f l e c t s t he t h o r o u g h n e s s a n d e x t e n t o f t h e e f f o r t smade b y D r . S t o t t a n d h i s c o l l e a g u e s i n d e v e l o p i n g a n d t e s t i n g d e -c o u p l e d load f l o w m e t h o d s . F o r t he p a s t t w o y e a r s we h a v e b e e n p l a n -n i n g o n u s i n g a d e c o u p l e d l o a d f l o w f o r b o t h s y s t e m o p e r a t i o n a n ds y s t e m p l a n n i n g b ut h av e n o t h a d t i m e t o t r y o u t v a r i o u s te ch n iq u e s.W e d i d t r y D r . S t o t t ' s f i r s t m e t h o d 1 1 o n o u r s y s t e m w i t h v e r y g o o d r e -s u l t s b ut c ou ld n o t p u r s u e f u r t h e r e x p e r i m e n t a t i o n w i t h i t . T h e a u t h o r sh a v e d o n e u s a n d c e r t a i n l y many others i n t h e i n d u s t r y a g r e a t s e r v i c eb y d o i n g much o f t h e t e s t i n g f o r u s .T h e i m p o r t a n t r e s u l t o f a l l t h i s work i s t h e d e m o n s t r a t i o n t h a t ad e c o u p l e d lo a d fl ow u s i n g a f i x e d s u s c e p t a n c e m a t r i x r e q u i r i n g o n l y o n ei n i t i a l t r i a n g u l a r i z a t i o n i s m o r e e f f i c i e n t a n d r e l i a b l e t h a n o t h e r e x i s t i n gm e t h o d s . T h i s i d e a h a d a lw ay s b ee n a p p e a l i n g t o u s s i n c e i t w a s f i r s tp r o p o s e d b y D r . U e m u r a 7 . W e b e l i e v e t h a t t h e i n d u s t r y s h o u l d t r y o u tt h i s a p p r o a c h e x t e n s i v e l y i n o r d e r t o g a i n a s w i d e a n e x p e r i e n c e a sp o s s i b l e . T h e a u t h o r ' s r e f i n e m e n t o f t he d e c o u p l i n g proc es s b y r e -m o v i n g a l l s h u n t r e a cta n ce s f r o m t he B ' m a t r i x i s a r e m a r k a b l e i d e a .H a v e t h e a u t h o r s t r i e d U e m u r a ' s m e t h o d w i t h t h e m o d i f i e d B ' ? D o e st h e i n c l u s i o n o f t he v o l t a g e m a g n i t u d e s i n t h e f o r c i n g f u n c t i o n s makea s i g n i f i c a n t d i f f e r e n c e i n o v e r a l l s p e e d o r c o n v e r g e n c e c h a r a c t e r i s t i c s

M a n u s c r i p t r e c e i v e d J u l y 3 0 , 1 9 7 3 .

t o make i t w o r t h w h i l e ?T h e a u t h o r s s t a t e t hat t h e c u r r e n t m i s m a t c h i d e a l 8 d i d n o t w o r ko u t t o o w e l l . S i n c e we h a d more t h a n j u s t a pas s i ng i n t e r e s t i n t he c u r -r e n t m i s m a t c h f o r m u l a t i o n , w h a t s p e c i f i c a l l y w a s i t s w e a k n e s s ?W e c o m pli m en t a n d t h a n k t he a u t h o r s f o r a n e x ce l l e n t p a p e r .A . K . L a h a a n d K . E . B o l l i n g e r ( U n i v e r s i t y o f S a s k a t c h e w a n , S a s k a t o o n ,S7N OWO C a n a d a ) : T h e a u t h o r s a r e t o b e c o m p l i m e n t e d f o r an i n t e r -e s t i n g paper. I n s o l v i n g t he l o a d f l o w p r o b l e m t h e r e i s a l w a y s a t r a d e - o f fo f s p e e d a n d s t o r a g e . T h e f o r m a l Newton m e t h o d h a s d e m o n s t r a t e dd i s t i n c t a d v a n t a g e s o v e r o t h e r i t e r a t i v e s c h e m e s . T h i s p a p e r i s a w e l c o m ea d d i t i o n t o t h e d i f f e r e n t m o d i f i c a t i o n s o f N e w t o n ' s m e t h o d , t o a c -c e l e r a t e t h e s p e e d o f c o n v e r g e n c e , w h i c h a r e a l r e a d y publi s hed i n t h el i t e r a t u r e . O t h e r s w i l l no d o u b t further t e s t t he i dea o u t l i n e d i n t h i sp a p e r t h e r e b y c o n f i r m i n g t h e a u th or s' c l a i m a b o u t i t s r e l a t i v e s u p e r i o r i t yo v e r o t h e r m e t h o d s . I n t h e m e a n t i m e we w o u l d l i k e t o comment onsome o f t h e s a l i e n t p o i n t s o f t h e p a p e r .A l t h o u g h t he i de a o f n e g l e c t i n g t h e o f f - d i a g o n a l m a t r i c e s N a n d Jwas f i r s t s u g g e s t e d b y C a r p e n t i e r as e a r l y as 1 9 6 3 , t h i s was l a t e r f o u n dt o s u f f e r f r o m weak c o n v e r g e n c e a n d p o o r r e l i a b i l i t y . A s we u n d e r -s t a n d , t h e a u t h o r s ' d e c o u p l e d method i s a m o d i f i c a t i o n o f e q u a t i o n s ( 4 )a n d ( 5 ) i n t h e s e n s e t h a t t h e y i n t r o d u c e d t he ( 1 0 , l v ) s u c c e s s i v e i t e r a -t i o n s c h e m e , w h i c h i n t u r n , i n c r e a s e s t h e s t a b i l i t y p r o p e r t y o f t h es o l u t i o n . The m o d i f i e d Newton m e t h o d 1 w h i c h we a r e d e v e l o p i n g a tp r e s e n t , a l t h o u g h c o m p l e t e l y m a t h e m a t i c a l l y d i f f e r e n t f r o m t h e a u -t h o r s ' m e t h o d , f ol lo w s t h e id e a o f a s u c c e s s i v e i t e r a t i o n s c h e m e . Them o d i f i c a t i o n s s u g g e s t e d i n e q u a t i o n s ( 6 ) - ( 9 ) o f t h e a u t h o r s ' p a p e re s s e n t i a l l y i n c r e a s e t h e s p e e d o f c o m p u t a t i o n t i m e , b u t d o e s n o t a f f e c tt he c o n v e r g e n c e p r o p e r t i e s o f t h e a l g o r i t h m n o r w i l l i t a f f e c t t h e t o t a ln u m b e r o f ' i t e r a t i o n s . T h u s t h i s a l g o r i t h m a p p a r e n t l y c ontradi c ts o ne o ft he a u t h o r s ' e a r l i e r comments r e g a r d i n g t h e a l g o r i t h m 2 t h a t " t h i s i sa l s o f o u n d n o t t o b e v e r y s a t i s f a c t o r y f o r m o s t p r ob le m s a n d t h e r e a s o ni s t h a t e q u a t i o n ( 4 ) ( w h i c h i s same a s e q u a t i o n ( 5 ) o f t h i s pa pe r ) i s ar e l a t i v e u n s t a b l e alg orith m a t some d ista n ce f r o m t h e e x a c t s o l u t i o n . "I t may b e i n t e r e s t i n g t o know w h e t h e r t he authors ' comment r e g a r d i n gt h e f a s t c o n v e r g e n c e p r o p e r t y i s b a s e d s o l e l y o n t h e r e s u l t s a s s h o w n i nT a b l e I a n d f i g u r e 2 o f t h i s more r e c e n t p a p e r o r i s t h e r e a n y m a t h e -m a t i c a l j u s t i f i c a t i o n b e h i n d i t .I n D r . S t o t t ' s e a r l i e r p a p e r 2 , h e i n t r o d u c e d t h e id e a o f c u r r e n tm i s m a t c h . S i n c e t hat method a n d t h e one o u t l i n e d i n t h i s p a p e r a r ec o m p e t i t i v e t o e a c h o t h e r , we w o u l d apprec i ate i t i f t h e a u t h o r s c o u l dg i v e u s a r e l a t i v e c o m p a r i s o n o f b o t h o f t he m e t h o d s f r o m t h e p o i n t o fv i e w o f s p e e d , c o n v e r g e n c e c h a r a c t e r i s t i c s , s t o r a g e r e q u i r e m e n t e t c . a n dw h i c h o n e , t h e a u t h o r s t h i n k , i s s u p e r i o r f r o m a n o v e r a l l p o i n t o f v i e w .One may n o t e t h a t t he t he 2 7 t h b u s s y s t e m , t he t w o m e t h o d s t a k e 3 a n d1 0 / 2 i t e r a t i o n s r e s p e c t i v e l y t o a r r i v e a t t h e c o r r e c t s o l u t i o n .W e f u l l y a g r e e w i t h t h e a u t h o r s ' s u g g e s t i o n t o a p p r o x i m a t e C o s i n ea n d S i n e t r i g o n m e t r i c f u n c t i o n s b y t h e f i r s t f e w t e r m s o f t h e S e r i e s , a si t d e f i n i t e l y i n c r e a s e s t he s p e e d o f c o m p ut a t io n . One s hould, h o w e v e r ,u s e t h i s a p p r o x i m a t i o n w i t h c a u t i o n as t h i s may g i v e e r r o r s i f t h e a n g l ei s q u i t e l a r g e . T h e s e a re c a s e s w h e r e t he p h a s e a n g l e o f some b u s e s a r e6 0 0 o r s o o u t o f p h a s e w i t h t h a t o f t h e s l a c k b u s .A g a i n we w o u l d l i k e t o commend t h e a u t h o r o n a w e l l w r it t en i m -p o r t a n t p a p e r .

REFERENCES1 . R. B i l l i n t o n , K . E . B o l l i n g e r a n d S . B . Dhar, "A New M o d i f i e dN e w t o n ' s M e t h o d F o r L o a d - F l o w A n a ly sis " , p r e s e n t e d a t t he IEEEW i n t e r P o w e r M e e t i n g , 1 9 7 2 .2 . R e f e r e n c e ( 1 1 ) o f t h i s p a p e r .

M a n u s c r i p t r e c e i v e d A u g u st 6 , 1 9 7 3 .

W . 0 . S t a d l i n a n d B . F . W o l l e n b e r g ( L e e d s & N o r t h r u p C o m p a n y ,N o r t h Wa l e s, P e n n s y l v a n i a ) : The a u t h o r s h a v e p e r f o r m e d a n in v a l u a b l es e r v i c e i n t h e i r e v a l u a t i o n o f a l t e r n a t e a p p r o x i m a t e f o r m s o f t h eNewton m e t h o d . I n o r d e r t o f ur t he r c o n s er v e c o m p u t e r memory wew o u l d w e l c o m e t h e a u t h o r s comments a s t o t h e f e a s i b i l i t y o f s e t t i n g[ B " ] e q u a l t o [ B ' ] . S h u n t r e a c t a n c e s a n d l i n e c h a r g i n g w o u l d t h e n b ec o n s i d e r e d a s p a r t o f t h e Q l o a d a n d t he c o r r e s p o n d i n g MVAR v a l u e o ft h e se s h u n t e l e m e n t s w o u l d b e u p d a t e d a s p a r t o f t he Q m i s m a t c hc a l c u l a t i o n b a s e d o n t h e previ ous ly ca l cu l a te d v o l t a g e . S e c o n d l y , i f s u c ha l o a d f l o w i s p a r t o f a n o p t i m a l p o w e r f l o w p a c k a g e c a n a p p r o x i m a t ei n c r e m e n t a l l o s s e s b e d e r ive d u s i n g s i m i l a r d e c o u p l i n g t e c h n i q u e s a su s e d i n t he sol u t io n - o r m u s t o n e r e v e r t t o t h e n o r m a l J a c o b i a n ?M a n u s c r i p t r e c e i v e d A u g u s t 2 , 1 9 7 3 .

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