Ref3 - s t o c h a s t i c l o a d Flows

8/8/2019 Ref3 - s t o c h a s t i c l o a d Flows

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IEEE Transactions onPowerApparatusand Systems, vol . PAS-94, no. 2 , March/April 1975

J. F. DopazoS T O C H A S T I CO A D FLOWS

0. A. Klit inAmerican Electric Power Service Cow.

N. Y., N. Y.

A. M. Sasson

Abstraci:The oa d flow study has been at he center of stud ies made fordesig ning and opera ting power systems or many years. I t i s w e llknown hat orecasted data used in oad f low studies contain errorsthat affect the solution, as can be evidenced by running many casespertur bing the nput data.Th is paper presents a method for calculat ing he effect of hepropagation of data naccuracies through the oad flow calculation s,thus obtaining a range of values or each output quantity hat, to ahigh degree of probability, encloses he operating conditions of hesystem.The method is ef f ic ien t and can be added to any exist ing oadflow program. Results of cases run on the AE P system are included.

IntroductionThe oad flow study has been at the center of analysis made todesign power systems and delineateoperatingpractices. From thestudies performed in network nalyzer days to the ophist icateddigital computer programs of today, conceptuallypeaking,otmuch alterationhas taken place to thede f in i t ionof the oad lowproblem, The ast methods availab lenow ,wi th he i re f f ic ien tut i-lization of computer core and theircapabilityofaccess to largedata file s, have made possible for the engineer to run a argenumberofcases,starting from some basesystemcondition, to study h isdesign or his proposedplanofoperation under many altern atives .

Important decisions evolve from these studies.For each load low case hat is solved, i t i s necessary or heengineer to provide he equired data hat define he conditions orthe case, This datausually nclude eal and reactive oadingsatso-called oadbusses and realpowergeneration and voltage m a gnitude at generator busses, as well as the electrical charact eristic sof the system model.The formula tion of the oad flow problem assumes that the dataprovidedisabsolutelypreciseand prov idesresu l ts to ta l ly compat ib lewith the given data apart from round-off errors. However, in practice,it can be readily appreciated hat oad low data can only be knownwithin some finite preci sion, his being more the case as the studyrepresentsconditions hat are more dist ant nto the future. As a

normal screening process, the engineerlooks at the range of poss ib levalues or a part icu lar p iece of data and setects an average valueas the number to be used in the o ad flow study.Given the importanceof hedecisions hatevolve from loadflowstudies, i t appears that i t is important to know the possib le

ranges of result quantities corresponding to the known range of dataquantities. nother words, it is o f n teres t to determine he eff ecton load low results of our ignorance of nput data values.Th is paper wi l l address i ts e l f to the problem of processing theexpected errors in the data of he oad low problem, assuming hatthe electricalcharacter ist icsof he model are known suff ic ientlywell that the effect of their naccuracies i s neglig ib le.Esse ntially , the method converts he oad low problem formu-lation from adeterminist icon e to a tocha stic one. The esultsobtained can be considered to define the range of variat ion of resu ltquantities, husdetermining heworstcond itions for each. Sim ilarresults to those obtained with the stochastic load f low can be arrived

at by repeatinga arge number ofconventionaldeterminist ic oadflow cases in which or each case he data i s perturbed, such hatthevarious cases represent possib le sets of data with in the precis ionthat the data is known. Once al l these stu dies have been carried out,the range of values for a specified result quantity can be identif ied.What the stochastic oad low does is to obtain hese ranges in onedirecta lculat ion. An importantract icalharacterist ic fhemethod is that i t can be added to any exis ting oad low programsince he process requires f ir st the solut ion of a conventional oadflow, ollowed then by aseriesofsteps hatdetermines the prop-agation o fthe errors of input data. The method i s not only ap plicableto the oad low problem bu t to any problem where the model i s asystem of a sufficient number of equations.MATHEMATICAL BACKGROUND

The method is based on the r inc ip les f ta t is t ica le a s tsquares estimat ion or inear systems. Some of the ini ti al idea s ofth estochastic oad low came about from theexperienceobtainedwith s imulat ion of and test ing on the AE P monitor ing project [ 1-61.Th ederivationof some of the equations to be presented can befound in these references or in advanced st ati sti cal texts.Given a non-linear set of equations,

y ' = f ( x ' ) + Ea Taylor's series expansion can be used to l inearize ( l) ,

y = J x + Ewhere

y = y ' - f ( x ) I ox = x ' - x bJ = Jacob ian o f f ( x f )E = vector o f error random varia bles

Equation (1) i s in terpreted in thefo l lowing way.There are dataquantitiesywhich epresent the average value o fthe range of possib le values the p iece of datamay have according tosome statisticald istr ibutiondefin ed from ourphysical nowledgeof the problem and the methods that were used to forecast the data.There are problem ariables x ' from which allquan tit ies can bePap e r T 74 308-3, reco-nded and approved by t h e IEEE Power Syste m computed. These ariableshavea true, although unknown, value

at the Summer Resources Conf,, Anaheim, which epresentconditions as they w il l ex is t at some future date.JUIY 14-19, 1974. Manuscript submitted n gust 29, 197 3; made vailable for The Vector E represents the error between y ' and fsEngineering Committee of the BEE PowerEngineering S o c i e t y for presentationprinting April 3 , 1 9 7 4 . wi l lo t be known until the futureondit ionsre encountered, E can

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only be describe d statist ica lly as a random varia ble hat has some deviat ion of G rom Xt. Equation (13) presents the variance of $3)mean and variance hat epresentou rexpectationsof the way E and Equation (14) says hat hisvariance i s zero ifJhere are ascould vary. T he meanings of y and X are simi lar to those of y ' and many equations as unknows. This simply means thaty i s equal toTh e fol lo win g wil l assume the inearization presented n Equa- y, or hat he actual solution of Equa tion (2) has tieen obtained, ation 2. fact also ef lected n hat F (3 = 0.The stat ist ics of the vector E can thus be speci f ied as Another stat ist ical property is that

E ( ) = OE ( E E ~ ) = V

F(x) = (y-Jx)tV-l (y-Jx) (4) Assuming hat herear eotherquantit ies z' related to x ' asand thevalueof x thatminimizes F(x) i s ca l l ed 4 and i s givenby ollows,[6 1 Z ' = g ( x ' ) (17)4 = ( J k l J ) - l J N - l Y (5 ) a aylor'seriesxpansionf g l inearizes (17) intowhich reduces to z = K x (18)2 = J - l y (6) from which?canbeobtained from 2,fo rheas ehat i s a square matrix,mply ing the same number of Aequations as unknowns. F(Q) i s equal to zero for thi s case. z = K 2 (19)

As he original Equat ion (1) wa snonlinear, 12 should be inter-preted as the converged valueafter some iterations nwhichJ i srecomputed in eac hte ra tio n. Z t = K x tand z-tfrom X t

Corresponding to $ t he re i s a 3 computed directly from EquationThe tatist icalpropert iesof can be ummirize das ol lows:E(?) = Z t (21)9 = J ; (7 )

and corresponding to the true, but unknown valueof X, ca l led X tthere i s a true, and also unknown, valu e f,al led yt, Equation (22) presents the variance f 2Yt = J X t (*) Statistical Properties: Confidence Limits

(a,

E ( ( k t ) (z*-zt)b = K ( J ~ v - ~ J )1 ~ t (22)

Statistical Properties: Expected Values nd Variances It is now necessary to ment ionherobabil i tyistributioncharac terist ics of he various quantit ies. The nput quantity y canThequant i t ies 2, X , Y, F and Yt haveknownStat ist ical Pr op have m y probabi l itydistribution unction,which is known from theertieshat are ofnteresf,nowledgefhehysicalroblemt hand. It s stat ist icalropert iesgiven in Equation (3) can then be computed. The output quant i t iesE($) = x t (9) x or z are l inearombinat ionsf y and byheentral Li m i t Theoremcan be aken as norm2lly distributed random variables. It can thenE (x^-xt) (c-t)t } = (J tV-1J) -1 (10) be saidhat and z are N( q, ~ ~ 2 )nd N(zt, oZq espectively.(11) Aormal lyistributed random variable can alwayserans-formed int o a unit normal, thus

which reduces to the fo l low ing f J i s a square matrix, Az - Z t (24)- N(0, 1)E U E Y ) (?-v)t> = 0 (14)Equations (9) and (11) imply that f E h 3 h e s ta t is t ic a l p ro p-ert iesiven in Equation (31, then, i f 3 and y had been detyminedx2 = d i ag { (JW- l J ) - l } (25)many times from differen t value s of y , the average of :and y end sto the rue values of x andy. In stat ist ic al terms this s cal l ed an uz 2 = d i agK(JtV- lJ ) - lK t ) (26)unbiased process. Although X t and z are not known, a statement can be made asEquation (10) presentshe ariance of :which represen ts,he to a range which ncloses them with some prob abil i tyo fbeing

UZwhere

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Withina range of. f 3 imes he unit standard deviation ofdef ined by (23) and (24), it can be said hate is ap proximately a 99% probabi l i ty ofenclosing thetrue values.i t can be stated hat

X t = x^ * 3 u x (27)Z t = z f 3 U Z (28)A

99% prob abi l i ty of being correct .THE LO AD FLO W PRO BLEM

Th eprevioussectionpresented he general theory equired orstoch ast ic oad low. This sect ion wi l l relate ha t theory to theoad flow problem. The various quantit ies referred to in the prev iousect ion w i l l now be def ined i n terms of the variables fami l iar to th eload f low.refers to load f low data, that s , oad bus P and Q and generatorbus P and E.refers to bus state variables E and s

refers to me variancesof heerrors assumed on hey nputdata.refers to the Jacobianof he oad lowequat ion, hat i s thepar t ia ls o f P, Q and E nput quant i ties y wi th respect to E ands load f low v ariables X.refers to output quant i t iesof he oad low computed from X ,such as, l ine low P and Q and generator bus Q.refers to the Jacobianof output quant i t ies z wi th espect toload f low variables X.

torization echn iques can then be used to t r iangularize his matrixand repeated ack ubst i tut ions wi l l ie l d the lements fhecovariance matrix. It i s on lynecessary to save hose terms thatstructurally are in the same posit ions as theoriginalJtV-lJ andthenonlyasymmetricalpart . Th is s an importantcharacterist icsince, n general, the Cov (x ) is a ful l matrix.Step 3 - Computation of Covarian ce Mat rix of 2

The covariance matrix of ;was given in Equation (22) asCOY($ = K ( J ~ v - ~ J ) - ~ K ~ (22)

Tocalcu late he Cov(?) theJacobianofoutputquant i t ies zmust becalculated.Th estructureof hismatrix i s also dependenton thatof headmittancematrix and is thus elated to thatof J.Matrix K is thus a sparse m atrix and should be calcu lated as such.Because of ts st ructural elat ionship to J, in the triple product^indica ed inEquat ion (22) onlytV-lJtructured lements of( J V - t J) -l are needed. Th is s the reason for saving only hoseelements in Step 2 . In the case of Equation (22), onlydiagonalelements need be computed and saved.Step 4 - Computation of Confiden ce L i m i t sand (28) asConf idence Limi ts or 2 and ? were given nEquat ions (27)

X t =;" + 3 u xz t = z * 3 U Z

The u z vector are the diagonals of the Cov(9whi le the C$vector are l h e diagonals of the Cov(?). Thoseelements of thesetwo matrices have been already alculated and saved in Steps 2and 3.AREACONSTRAINTSHaving elated hevariablesused n hePreviousSect ion to

problem, heyshall be used nterchangeably The stochastic oad f low, as presented in the previous section,n whatollows.The arious steps of the process wi l l now be amounts to summarizing the results o f many load lows, as far asthe quantit ies btainedn the confidence l im its are concerned.These oad f lows could have beenrun perturbing al l n put quant i t iesaccording to th esta t i s t i csof theuncertaintyof hesevaluesasderived from the nowledge of thephysic al problem. The esultsThe f i rs t step i s to compute as 'Iwwn in generation decreased, forcinghelack generator to haveidewould eflectcases in which otal oad was increasedwith maybe6) wi t h an i terat iveprocess to take nto account henon-l inear limits. This situationmight be impved if there is additionalof he oad low equations, Equation (1). Equation (6) shows informationvailable. In addit ion to th e forecastmade on individualfo r the O f number O f equations as unknowns' the loads and generators, i t i s often he case hat otal oad and gener-propert ies of theerrorsof Y drop ou t of heequations.ationofgiven areas of he system are known to agreaterprecision.

converts Equation 6) o imply the solutionf a loadlow I f convent ionaloa dlows were run perturbing the data, area loadtit ies y. Any exist ing oad low program can be used inva l ues ass i f l ed to the load low and generationconstraintswould be used to restrict the data.step, thus making his step equivalent to the solut ion of Area load and generationconstraints can be include d n theeproblem as i s done in Practice, with average valuesor the stochast ic load f low. An area cons traint cts as an addit ionalequation inEquat ions (1) and (2). Th eaddit ionalequat ion is ob

Step 2 - Computation of Covarian ce Matr ix of 2 tained by adding some of the other equationsalready ncluded in(1) and (2).The ovariancematrixof x" wa spresented in Equation (10) as For example, i f theoadsof nodes 3, 5 and 7 are to be con-strain ed to a finer degree, new equations are formed,

@ 1 A Load F l ow

C O V ( ~ = ( J ~ v - ~ J )1 (10)P n t lP 3 + P 5 + 4c p , n t l (29)oca lcu la te he Cov(;), the Ja co bia nof he oad lowequat ionust be f i rst formed and evaluatedtheolut ionointfhen+ 1 = Q3 + Q 5 + Q 7 + E q , n t 1oad lowof Step 1. Sparsity echniquesshould be used as both and J tV - lJ ae s arsematrices. An ordering scheme should be Th is new equation appears as the equation of a new imaginary node,used to order J !- J. In the sameway that he structure of he connected to 3,5 and 7 and to al l nodes to which hese are connect-

admittancematrix Y can beused to order J i n aewton's oad low, ed to. However, the new node does notcontribu te any new variablethe structureof Y tY can beused to order JtV-Y . Triangular ac- x to the problem, thusmakingJa non-square matrix. It is important301


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to realize that even if J is non-square in Steps 2, 3 and 4, t h is doe s 1. Area constraintsdeteriorate hesparsity of thestate n-notalter the fact hataconv entiona l oad low Program canbe verse covariance matrix as their ef fect i s s imi lar to the ntroduct ionused in Step 1. Th is comes f rom the condi t ion hat the solut ion Of of a node connected to a ll other nodes in the area. In the limit,th eoriginalnequat ions in (2) auhmat ical lysolve headdi t ional considering heentire systemas one sing le area, this matr ix be-equations ntroduced by the area constraints. comes ful l .The in clu sio n of area cons traints as presented in Eq. (29) con-siders he n put covariance matrix to be diag onal making he addi-t ional equations independent from the original set. Considering an

area of k nodes, Eqs. (1) becomeY = 4 i Iy) + Ei , i = I, ... k (30)

whic h Eq. 29) has he form

The relation between Eqs. (30) and (31) is the fol lowing,k kf n t I (X ) = 5 +i x ) = f (Yi - c i )I (32)Sub stituting n Eq. 31),

(33)

2 . Confidence limit resu lts would, in general, be somewhatsens i t i ve to slack bus location.The above l imi tat ionscan be eas i ly overcome by mode l l ingthe oad low slack bus as any other generator bus, with P and Eequat ionsbu tal lowingonlyE to be astatevariable,keeping 6f ixed. The orecasted error in total system oad plus osses can bedistributed among generator Pequations, thus making all generatorsCDmpensate for orec astingerrors. It isno tnecessary to d iv ideth eotalxpe cted error eve nly among generators. No areaconstraints are used,Thechangingofposi t ion or heangle eferencebus has noeffec t on confidence l imi ts other than those of voltage angles them-selves. As angles are rea lly angle differences between busses andthe reference bus, these alues and their onfiden ce imitsaredependent on the eferencebus ocation. However, theconfidencel imi tsof angulardifferencesbetween any twobusses are inde-pendent of referencebus ocation.

(341It is of interest to compare Eqs. (31) and 34). The area con strain tformulation considers

However, had yn t 1 been defined as in Eq. 34), the statis t ics ofEn t 1 would have been the fol lowing,assuming independence,

E ( , t I E n t l ! V i t V n t I (38)Eqs. (37) and (38) show that here is a distin ct differ enc e betweenmode l l ing the area co nstrain t as afunct ion ofasta t ionary t rue value

kI

and as the sum of random varia bles.CONSTRAINT ON TOTAL SYSTEM LOAD

The previous sect ions have shown that the stochast ic load f lowrepresents the combined results of many deterministic oad lows.Two ypes of load low unshave been modelled:

1. As et of load f lo ws n which each piece of data is randomlyvaried. It was discussed hat his should produce large conf idencelim its for the sla ck generation and for ines near that bus due t osome cases representing changes in tota l load and in generator outputs.

As n the case of area constraints, he reference bus equ ationis considered to be an independentequat ion.Th emodell ing ofthe referencebu sequation is madeas a unction of the systemstate ariables and not by adding a l l other inputs. This modelalso reflects the nature of the distrib ution of unc erta inties through-out the system to be one of non-simultaneity of occurrence of indi-vidualunce rtainties and not one ofcorrelation.Of negligible effec t s the con dit ion of having one extra equa-tion,producing an overdeterm ined et of input quations and a

f i l tering ef fect .SOLUTION OF PRAC TI CAL CASES

Several testcases are presented and summarized in Ta ble 1.Th e AEP system data, as described n Reference 7, was used in al lthe cases. Reference 7 presented results of 50 runs of a convention-al load low n which he oad f low datawas randomly perturbedassuming it behavedin a rectangular dist r ibuted manner w i thin someerror bounds that, in general, were in the order of 10 to 20 percentof he mean data. T hat same reference presented he maximum errorof each result quantity. The maximum error of one qua ntity does notoccur in he same load low run as the maximum errorof anotherquantity, so i n t ha tsense hey are no t simultaneous. However, asit i s not known what are the true nputs n a given run, any of hemaximums could occur. Table I summarizes the average values(resul ts of the base case oad f low or Step 1 of he stochast ic oadflow) together with the maximum errors as determined from Reference7.Thevariances of the input data ref lect the ectangu lardist r i -buted error assumed in the relatively few cases of Reference7 bycomputingV from,

2. Aetf load f lowsnhich each piece of data is ran- V = (Refer ence error bound/2.5)2domly variedbu tconstruing hevariationssuch hat the load andamounts. from theentral Limit Theorem. Thus,eference 7 resu l ts areconsidered to be approximately three sigma valu es and are compared

The area constraints would reduce slackus f low, It i s to be consideredhat whi le the 50 casesun in ReferenceinTable I with he orrespon ding esultsof the stochast ic oadconf idence l imi ts but has twoas icim i ta t ions:retat ist ical lyign i f i canthey are les s so than the res ultsf

generation of areas within thesystem are allowed to vary by smallThe oubut Wantit ies Can be considered to be normallydist r ibuted

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stochast ic oad low sumofagreater number ofcases than the 50 ofTable I presents esults f some typical utput uantit ies.part icular, i t i s o f n terest to look at he lows n ines near he1. These are presented i n the f i rs t 8 rows inI. Th enext wo groups of 3 rows each refer to l i nes n twosectionsof hesystem,both removed from the sla ck bus.fol low ing group of 3 rows efer to generat ion resul tsat the

bus and two other generators. Since he ealpowers of thetw o generators are data quantit ies hey are not nc lud ed nI. The ast group of 3 row spresents hewltage magni tudeat three di f ferentbusse s, bus 21beinga generator bus.The six cases presented n Table I are now discussed:1

Case 1 is referred to as the FixedVol tagescase nTable I.his case generator bus voltagemagnitudes were considered toxed, al l other quant i t ies having he bounds given i n ReferenceA comparison with the results of Refere nce 7 shows hat he reallow Conf idence Lim i ts areof he same order of magnitude,ally sl igh tly larger. Re active low for l in es near the slack busn Case 1, the same being nre or he esultso f r egeneration.

2Case 2 is referred o as the Variable Vol tages case in Tab le I.case generator voltage magnitudes were considered both as1 - STOCHASTIC OAD FLOW TEST CASES

inputquant i t ies and as variables Of the stochast ic oad low, Acomparison with Case 1 show s hat esults are very similar exceptfor react ive generat ion and react ive f low of l lnes near the sla ck bus.An error bound of 1.2 percent was assumed at a l l generator busses,this being theonly error bound data not bein g den tical to that ofReference 7, although it i s o f the same order of magnitude.Cases 1 and 2 appear t o draw the fol lowing wo conc lusions:

1. The inearizat ion mpl ied ngoing from Equation (I) to Equa-tion (2) is valid for the level of error bounds considered.

2. Thedecoupling of P-s and Q-E quan t i t ies n the oad low isv is ib le n thecomparison of the wocases.Th eaddit ion ofvoltagemagnitudevariations had aconsiderableeffect on re-active low and generation, and a neg ligible oneon real powerflow and angles.Case 3

Case 3 includesaddi t ionalconstraints, inbot h oad and gen-eration areas, sl lch hat he otal oad or generation of each areahas an error bound of 5% of he average tota l oad or genera tion orthe area. A l l other nputquantit ies have ind ivid ua l bounds as inCase 2.The results of Case 3 show a gene ral attenu ating effect on theConf idence Lim i ts of al l quant i t ies, espec ial ly those near the slac kbus. Th is s as expected. In reference 7, and in heequivalentresults of Cases 1 and 2 , thedata used for he oad f lows mplie dsituatio ns where the total oad was increased or decreased from theaverage value, he sla ck bus having to adjust he differences. As a

C O N F I D E N C EI M I T SFLOW BETWEENBUS TO BUS

1 381 21 338 3938403 77 1216 2015153332 3737GENERATIONAT BU S

12140VOLTAGEAT BU S

102139

AVERAGEVALUES-118-J 685 1 5 t J5 1237-J 0680 -J6951 J380733 -J130- 1 O l t J 3 9- 1 5 8 t J 8- 96-J 63-102-J150- 1 8 4 t J5 1-260-J 34-222-J 15229-J 28

634-J 17- 3 5 t J 5 9800 -J2009 m1 . o u9 u

REFERENCE7411 tJ2457 8 t J 9 11 6 2 t J 6 31 6 3 t J 5 1272 +J 174208 tJ1631 7 5 t J 9 9152 tJ16 38 1 t J 3 55 0 t J3 139+J 3342 t J 2 347 t J 195 3 t J 28

609 t J334J377J 540

0.8514.91

CASE 1FIXEDVOLTAGES1 3 0 t J 1 91 9 1 t J 42 1 8 t J 483 2 1 t J 1 4250 tJ1 022 0 5 t J 2 52 1 9 t J5 17 6 t J 5 05 5 t J 445 3 t J 426 2 t J 246O t J 136 1 t J 22

5 0 5 t J 5

773 t J 21J117J105

3 . 6 0 u0.3016.36L m

CASE 2VARIAELEVOLTAGES?!I!191 J 48218 tJ 56321 J 166249 tJ20 1206 +J 102219 tJ1387 6 t J5 156 +J 4453 +J 4362 +J 2860 +J 196 1 +J 26

173 tJ200J347J5D6

3.8015.351 . 4 0 w. 9 O W

CASE a5% AREACO"lNTS(1)180 tJ14 86 2 t J 637 2 t J 4097.tJ 32131 J 139109 tJ1489 4 t J 8 3110 t J11340 t J 2940 +J 3634 +J 3250+J 2148+J 1449+J 19

264 tJ166J272J413

1 . 7 0 u1 . 2 o w0.7012.48

CASE 4 CASE 52XAREA CASE65% AREA 3% TOTAL LOADCosmwTs COWSTRAINTS(2) c&srrWwT9f tJ14814tJ148 54 tJ20 65 t J 63 4 9 t J 6 2 55 t J 884 3 t J 407 6 t J 2 999 t J13894 +J 1477 9 t J 8 398 t J1133 4 t J 283 9 t J 363 2 t J 3 250+J 214 7 t J 1 44 9 t J 1 9

117 t J 166J272J413

L 6 0 w1 . 2 o u0 . l O j ~

iootJ411 7 8 t J 46222 t J 39198tJ1471 8 6 t J 8 5225tJ112

8 1 t J 326 1 t J 455 9 t J 288 7 t J 1 88 6 t J 148 9 t J 1 9

290 t J 166J272J413

1 . 8 U1.201J&0 . 7 0 m

23 t J 5676 t J 3 076 tJ13868 tJ16764 t J 8 377 tJ11453 t J 4140 t J 3636 t J 3617 t J 2 29 t J 159 t J 20

15 tJ295J289J420

0.70 LL611.200.70(1) Individual data errors are as in Re ference 7.(2) Individ ual data error s are 40% of Average Values.

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consequence, not on ly the slac k generator but also he ine s nearthe slack bus were found to have large Conf idence Lim i ts . By con-straining he oad and generation of areas of he system to smallervariation s han the individua l oads and generations, he total oadvariat ion mpl ied is reduc ed, producing the at tenuat ing ef fect .Case 4

Case 4 is simi la r to Case 3, the area constraints reduc ed from5% to 2%. Some furthe r attenua tion occu rs, espe cially to the slackbusgene ration However, most outp utquant i t iesd idnotexhibi tsigni f icantConf idence Limit variat ions, ndicat ing hat hey weremore due tq the ndividu al data error bounds than to the area erroreffect.Case 5

Th is .case presents heef fectof ncreasing heerror boundsof the ndividua ldataQua ntit ies beyond thoseofReference 7. Anormal lydist r ibutederr or bound of 40% of theaveragedatavaluewas assumed foreachquantity,eonstrainingat he same time heareas to wi t h i n 5% of total average values.A comparison of Cases 3 and 5,both wi th 5% area constraints,shows hatwhile heConf idence Li mi ts of eal power f lows and

vol tage angles ncrease, hose of react ive low and vol tagemagni-tudes are negl igibly f fected. hempl icat ion fheseesul tsappear to be that eal power is a muchmore sens i t ive quan t i ty tovar iatio ns n load and generationdata error bounds than eactivepower. Reactive power produced by ransm ission ines appears tobe the cause.The same voltag e magnitude error bounds were usedin Cases3 and 5. However,Cases 1 and 2 showed that eac tivepower was very sen sit ive to voltage magnitude errors. Thus, argervoltage magnitude errors n Case 5 would have increased he Con-f idence Limi ts of react ive power.Case 6

This caseuses he f inal ormulat ion of thestochast ic oadflow mod ell ing he slac k bus equ ation as any other generator andconvert ing i ts role only to that of voltage angle reference. A otalsystem oadplus osses naccurac y of about 3% resulte d n a 210MW error bound which was distributedeven ly among the 14 gen-erators. Thus, the differe nce between Refere nce and Case 6data are on ly n generator real power error bounds. Apa rt from theconf idence l im i ts of vol tage ngles, al l other conf idence imitswere invariant as the reference bus posit ion was altered.

It i s in terest ing o note that the research program used through-ou t th is nvest igat ion ca lcu la ted theconfidence limit of the refer-ence bus eal power as an output quantity. I n Case 6, however,the error bound of P, was given a value of 15 MW as an input quan-tity . Th is same numerical value was produced by the program as acalculatedquant i ty,conf i rming hat here wasno f i l teringef fectcaused by the redundancy of the slack real power equation.

CONCLUSIONSThis paper hasdiscussed heextension of theconventionalload f low problem to include hecalculat ion of th eeffectsof n-accuraciesnnput data on a l l output uant i t ies.The dhorshave rec ently become aware of a paper [81addressing he sameproblem but with a different solution approach.

1.2.3.

Three models were presented in the paper:A model inwhic h a l l input data was assigned error bounds.A second model includingconstraints on load and generationof areas within a system.A f inal model inwhichacons traint on totalsystem oadpluslosses was placed b y including a eal power equat ion or theslack bus.As a conclusion of the theoret ical discussions and numericalsolut ionsusing he hree models, the hird model is recommended

for mpleme ntation or being conserva tive n case requiremen ts, forbeing independent ofvoltageangle eferencebus ocation and forhandl ing he mportantpract icalcons traint on totalsystem oadplus osses. An attrac tivecharacterist ic of the method is that itinvolvesaseriesofnon-i terat ivecalculat ions t o be carried outafter the solution of a onventional oad low by any method.ACKNOWLEDGEMENTS

The uthorswish to acknowledgehe ugge stion made byMr.A. F.Gab riel le, Head of he Computer Ap plic atio nDiv is ionof AEP, to study he ef fect of area constraints n he ormulat ionof the tochas ticoa dlow. he uthors' ppreciat ion to Mr.F . Aboytes o f ImR r ia l C o l lege, Lmdon for h is pr i va te d iscuss ionson the stat ist icals ign i f i canceof some of the models consideredis

1.

2.

3:

4 .

5.

6.

7.

a.

also gratefully acknowledged.REFERENCES

J. F. Dopazo,.O. A. Klit in, G. W. Stagg and L. S. Va nSlyck,"State Calc ulatio n of Power Systems from Line Flo w Measure-ments," IE EE PAS-89, pp. 1698-1708, September/October, 1970.J. F. Dopazo, 0. A. K l i t i n , and L. S. Van Slyck "State Calcu-lation of Power Systems from L ine Flo w Measurement, Part I \ "IEE E PAS-91, pp. 145-151, JanuaryFebruary, 1972.J. Dopazo, 0. Klit in, A. Sasson, L. S. Van Slyck, Real-TimeLoadFlow for the AE P System," Paper No. 3.3/8, 4thPowerSystems Computation Conference Proceedings, Grenoble, France,September, 1972.J. F. Dopazo, S. T. Ehrmann, 0. A. K l i t i n and A . M. Sasson,"Just i f i ca t ion f the AEPReal -T ime oad lowProject , "IEEE Paper No. T 73 108-8, Winter Power Meeting, New York,1973.J. F. Dopazo and A. M. Sasson, "AEP Real-TimeMonitoringComputer System," Symposium on Implementation o f Real-T imePower System Contr ol by Digita l Computer, Imper ial College ofScience and Technology , London , September, 1973.J. F. Dopazo, 0. A. K l i t i n andA. M. Sasson, "State Estim ationfor Power Systems: Detection and Identif ication of Gross Measurement Errors,"Proceed ings of the 8th EEEPICAConference,June, 1973.L. S. Van Slyck and J. F. Dopazo, "Convent ional Load Flow NotSuited or Real Time Power System Monitoring," Proceed ings ofthe 8th IEE E PIC A Conference, June, 1973.B,Borkowska, P robabi l ist icLoadFlow," EEEPaper No, T73 485-0, presentedat he Summer Meeting ,Vancouver, 1973.

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DiscussionA. Semlyen (University of T oronto, T oronto, O ntario, Canada): This isa very timely paper. Probabilistic methods are of increasing significancein many power system studies due to the prohibitively large computerrequirements f or handling a huge nu mb er of individual deterministicproblems resulting from many different comb inations and magnitudesof the nput variables. Fortuna tely, linear fun ctions of ndependentmndom variables, with Gaussian distribution, are also Gaussian andtheir statistical characteristics are easy to correlate w ith those of the in-put variables. The merit of the paper consists in applying this fact t o theload flow problem where the forecasted data are only estimated.I would appreciate clarifications on the following details.1) The load flow problem is basically non-linear an d its lineariza-tionproduc es some inaccuracy in the simple relationshipbetweenstatistical characteristics. This may be insignificant in many cases butmay have importance in long range planning. Some variables may bemore affected than others. Would the authors comment on the effect oflinearizing the load flow solution?2) Some input variables, like ,P and Q at he same bus, areand wou ld it significantly alte r the esults based on the approachapparently not uncorrelated. How would this affect the general theoryadopted in the paper? I feel that even if the matrix V is not strictlydiagonal (say, blockdia gonal) and the inputs close to Gaussian (which,probably, is in general a reasonable assumption) the Central LimitTheorem will still apply for practical evaluations.My belief is hat he application of stochastic method s to thisimportant power system problem will prove to be stimulating toengineers engaged in research in some other areas of power systemswhere direct statistical results are more meaningful and practical thanvery large numb ers of determ inistic calculations.

Manuscript received August 5, 19 7 4.

H. Duran (University of he Andes, Bogota): The auth ors should becommended on their effort to formulate and solve a problem whoseimportance has not yeteen fully recognized.As described in the introduc tion, the stochastic load flow problemis concerned with finding the probability distribution , and in particularthe expec ted value and he variance of the solutionofa load flowproblem. As such, itis a problem o f probability calculus and not one ofstatistical estimation. Hence the approach that the auth ors ake to solvethe problem using statisticalprinciplesappears to be misleading andunnecessarily complicated.A more direct approach would be as follows. Let 7 and V be theexpecte d value and he covariance matrix, respectively, of the nputdata. Let i be the solution of the load flow problem using 7 as data,that is

y = f ( x ,Using a Taylors series expansion around 2, and neglecting second andabove order termsgives:

Finally,

paper relates to he accuracy of the formulas. It should be borne inOne important question tha t the auth ors do not consider in theirmind that du e to the linearization introduced in the Taylors expansionj , is not the actual expectedvalue of x, and equation ( 1 0) does not givethe exact value of cov (x). To see this,consider the following two-node example. A generator supplies a load of P MW at unity powerfactor nd nit voltage magnitude through a transmission line ofresistance R and negligible capacitance. Th e Generation G is then givenby 3

Let us assume that P has a normal distribution withmean P and varianceu 2. What is the distribu tion, m ean and variance of G? It can easiiy beJ o w n t ha t:

and

while, the fo rmulas in the paper give instead,-8 = H ? RF2an dHence the errors in the expected value and the variance of G, are Rop2an d 2 R2up2 respectively. Are these errors significant? and, if they are,how can they be calculated or estimated in the general case? If the errorintroduced in calculating the expectedvalue of a variable is of th e sameorder o f mag nitude of its standard dev iation, w hat degree of confidencecould we have in the Confidence Limits? Regarding the probability dis-tribution of G, one thing that can be said is that t does not have anormal distribution since the P2 erm gives rise to a X2 pattern. I couldno t follow the authors use of the Central Limit Theorem to concludethat the output quantities can be taken as normally distributed. Wouldthey like to comm ent on the assumptions underlying its use here?As a final remark I would like to compliment the authors again forpointing a t a very impo rtant, interesting and difficult problem. Theyhave given one of the fust bites to a hard bone. I hope this discussionwill encourage them to co ntinue their work since the problem has notbeen solved yet.

R N.Allan and C. H.Crigg (Univ. of Manchester Inst. of Science andTechnology,Manchester,England): We would fust like to state howmuch we agree with the authors for the need t o treat the power flowproblem probabilistically. We also feel that, because the variables in-volved vary statisticallyand are forecastedstatistically,deterministiccalculations can lead t o erroneous planning and operating co nditionsand are, at best, only subjective assessments.We would, however, like to com men t on the authors use of th eCentral Limit Theorem to assume that f 30 predicts satisfactory con-fidence limits. We, in a mutual collaborative effort between UMIST inEngland and t he nsti tute of Power in Warsaw, have been currentlyinvestigating the same problem. We, however, not only characterise thepower lows by expected values, standarddeviationsan dconfidencelimits similar to hat proposed by the au thors, but also calculate thecomplete probability density curves of the power flows of interest. Itwas in order t o calculate these densitycurves that the nitial formulation,as described by Borkow skia,l. was limited to the d.c. case since thesecalculations are inherently complex.in Fig. 1. This is clearly not a normal distribution and suggests tha t theOn e of ou r typical calculations produced the density curve shownCentral Limit Theorem does no t apply. The difficulties arise becausewhch is not necessarily the case when analysing typical power systems.the Theorem is only applicable for a large numb er of random variablesOne consequence is illustrated in Fig. 1 where the value of the densityfunction at a pow er level of E + u is nearly three times greater than at apower level of E.limits enclose more than 99% of the probabilities of occurrenceasWe can c o n f m that all our results to date indicate that the k 30assumed by th e authors. We feel, however, tha t this could stiU lead toerroneous decisions if power flows of the type shown in Fig. 1 occur. Inthis example, the f 30 confidence limits enclose power levels between100 and 1100 MW. Our results show, however, that the probability ofpower flows greater han 800 MW is neghgible. Therefore,with heauthors confidence limits, the line could be almost 40 % overdesigned.obj ectio ns may be raised concerning their accuracy. Preliminary workWe accept our results were obtained using a d.c. representation andusing an a.c. model, however, indicates that the same trend prevails.In ending we would like to add our weight to the authors beliefthat probabilistic assessment of system behaviour is necessary to enable

Manuscript received May 6 ,1 97 4 .anuscript received August 7 ,1 97 4305


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-30I

id-0

IwJ

L

EIII ,30III

1

oopl I 00 * 800I I 200power flow, MT

Fig. 1 - Typical probability density curveaccurate security prediction of a present system and a balance betweencost a nd security of a future planned system to be treated objectivelyand not, as at present, subjectively. We consider, however, that orprobabilistic methods to supercede presently sed deterministic metho ds,the full potential of probab ility assesanent must be utilised as we arecurrently pursuing.

REFERENCE[ 11 Borkowska, B. Probabilistic Load Flow ,EE Eapero.173 8 5 - 0 , presented a t Summer Power Meeting, Vancouver, 1 973.

Barbara Borkowska (Inst ytu t Energetyki, Warsaw, Poland ): I regard themethod presented by J. F. Dopazo, 0. A. Klitin an d A. M. Sasson asvery interesting. The me thod seems to be useful and fficient innumerical calculations. The uncerta inty of the input data can be con-sidered from two points of view: a posteriori [uncertainty of the pastor present state of system] and a priori[uncertainty of the future stateof system]. The uncertain ty aposteriori has given the impulse toelaborating he various methodsknow n as state stimation. Theformulation of the load flow problem used in a posteriori study wastaken by the auth ors as the starting poi nt to elaborate the meth od of apriori analysis. This is right since the redundancy of data takes place ina priori study too. The use of global forecast for areas in order to limitthe variance of outpu t quan tities is an nteresting idea. Howev er, theuncertainty of a posteriori data is at least in tw o points different fromthis of a priori.1. Thecorrelationsbetween the measurement errorsoften arenegligible while the errors of the forecasted input data may be strongrandomparameter [temp erature, he configurationof lower voltagecorrelated because of nodal inpu t powers being the functio n of the somenetwork etc.] .Therefore the matrix V d ef ie d by the equation 3 is not

Manuscript received August 13, 197 4.

diagonal. It complicates the com putatio ns but not he mathematicalmodel. It should be not ed hat taking intoaccou nt he correlationdiminishes the variance of th e out put quantities.2. The variance of the inpu t data a posteriori is constant while thevariance a priori is variable according to th e ime horizon. The u rther infuture the forecasted mom ent [ time horizon] thegreater is the varianceof inp ut data. I t leads to great uncertainty of the results. Takingadvantage of the additional information makes the space of the possiblesystem states limited. This result was achieved by the auth ors by intro-ducing additional Equation [2 9] which is to be combined with hese[3 5] to [3 7] . Particularly nteresting seems to be the Equation [361which means t hat the errors of individual data are not correlated t o theerror of global forecast. I regard such assumptio n as fully acceptablebecause the m ethod s and the data used for individual and global fore-casting are different.

As o Central Limit Theorem some com ment seems to be necessary.Since Jacobian J is sparse, the variances of the in put data are different,and the density curves of the errors of the inpu t data are not defined,then it is not obvious that the CLT holds. It is known that, especiallyfor the furth er ime horizon, the density curves of the power generation(of the power stations with few large generatingunits)differs sub-stantially rom the Gaussian curve. The somemight pply to hedensity curves of the power flows in branches connected to generationnodes. In such a case the Eq. [ 2 7 ] , [ 2 8 ] are not true. However incases in which the density curves of errorsof input data can be properlydefined [a.g. as a normal] the equ ations hold.linearisation applied n going from Eq uation [ 1 ] to Equation [21 is valid.It would be interesting to know to what level of error bounds theoperational problems and in some problems of power system planning.Concluding the metho d seems to be very useful in power system

G.T.Hey dt (Purdue U niversity, West LaFay ette, Indiana): This paperpresents a new formulation of the pow er flow problem in stochasticterms in which th e load/generation schedule is a random vector. Thesolutionvector to the power flow problem is written in a inearapproximation to the nonlinearpower flow problem, and n such aformulation, the solution is a linear transformation of the input (wherethe input is interprete d as t he load/generation schedule). The mathe-matical formulation involves the usual decoupling process between bu svoltage m agnitudes and angles, and therefore in the Newton-Raphsonsolution,only real num bers appear. The central imit theorem pre-m ib es that the distribution of the solutio n variables will be Gaussiansince these variables are linear combinations of a large number of ran-dom variables. Messrs. Dopazo, Klitin and Sasson have made a valuablecontribution in theirobservationsof the various intricacies of hestochastic power flow formulation. I have worked o n thisproblem froma different point o f view, and I would like to briefly present this alter-nate approach in order to elucidate certain generalizations of the meth-esting questio ns which I would like t o pose to the authors.o d s used in this paper. Th e alternate approach also r a i s e s some inter-An alternate formulation is obtain ed by ignoring bus voltages asproblem variables; instead, consider line complex power flows as out-lation, the complex factors relating these variables are know n[ 1 andput variables and bus demand/generation as inputs. In linear formu-the ou tpu t is simply a linear transformation of the input. Unlike theformulation in this paper, the variables so formulated are not real butcomplex. Data used for load and generation schedules are randomcomplex vectors of know n distribut ion (or at least known mean andcovariance). The covariance matrix should not be considered diagonalsince there is no reason to believe that bus demands a t several differentbusses are independent. The covariance matiix is Hermitian, however,as may be observed from the definition of element i, j of the co variancematrix V,

where the p entries are the means of the random variables, x, an d (.)*deno tes complex conjugation. In vector notatio n,v - E( (X -M)(x -x)H) (2 )

where M is the mean vector and (.penotes the H ermitian operationof his discussion with Eq. (3) of the paper shows the generalization(complex conjugation followed by transposition). Comparison of Eq. (2 )necessary to approach this problem as I suggest. When the problem isformulated w ith line power flows rather than bus voltages, not only does

Manuscript received August 13, 1974 ,306


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E,q. (5 ) of the paper

= (JH V- J)l JH V-l Y.also the nform ation derived changes. Line loading is obtainedy, and many of the power flow statistical nferences carefullyin this paper are also available. Th e advantage of th e m ethodf Messrs. Dopazo, Klitin and Sasson primarily lies in the availability ofwhich

certain other comp utational factors.suggest primarily lie in the speed of comp utation (no inversion isI would like to raise some questions of the authors in order tois covariance matrix V,tV - E@ , E )

E are available, evens where independence does not hold could be handled. Secondly, Io ask a difficult question concerning the accuracy of theof the line flows (labelled z in the paper) by the metho din the paper is used,zation is required to obtain bus voltage information @. (1) ofin order to obtain the(Eq.18 ) of the paper). Would more accurate statistics of z beby formulating the load flow problem directly with z as theWhen this s done, onlyone lineari-is needed.This paper presents a well written accoun t of stochastic calcula-I have no doubt that the results and tech-will be used by others to obtain many o ther useful statistical in-

REFERENCE11 G . T. Heydt, B. M. Katz, A Stochastic Model for SimultaneousInterchange Capacity Calculations, to be published, IEEE Trans.on Power App aratus and Systems.

F. Schenk and K. Singh (University of Ottawa,Ottawa, Ontario,This paper is another one ofa series ofpapers which thesystematic efforts to develop practicalof tatistical notions t o the analysis ofpower ystemThis is anaturaldevelopmentwhen one recognizes thatmodels and solutions donot it well with real-typedeterministic esults have agreaterappeal thanta and models. T h i s s, in our view, a step forward in the rightion. T he authors are to be congratulated for their timely efforts.E, may be considered to be normally distributed as a conse-As is pointed out in the paper, th e vector of error randomof the Central Limit Theorem. his being the case, the estimatorsz and uZ2)may be obtained by maximizing theThis is, of course, a well know n fact, but it ives an

are unbiased (not all of themonal insight into the formulation of th e problem. Moreover, maxi-) also exhibit some other properties which make them very desirableinimum variance, consistent, efficient, sufficient, com pleteindepend ent. Further more, under uite general regularity condition sthe density function[ 1 1 , any function u ( 2 ) of a m.!Z.e. 2 of i is theful results can be e xtracted from the metho d if it i s properlyIn addition, t may be worthwhile to po int out that usefulandOnly those cases which are considered normal with small dataoperating poin t will give useful results. Extrem eions in the input data (which are allowed by a normal distribution)

REFERENCE11 F. A. Graybill, A n Introduction to Linear Statistical Models,Volume 1 , McGraw-Hill, 1961.

Manuscript received July 2, 1974.

S.T. p o t o v i c (Research Institute Nikola Tesla, Beograd, Yugoslavia):The authors have presented an interesting, useful and simple method forcalculating the effect of the propagation of forecasted data inaccuraciesthrough the conventional load flow. The method, using the principles ofstatistical least squares estimation for linear systems and addressing itselfto the problem of processing the expected errors n the inp ut data, thatis, load bus P an d Q, and generator bus P and E, converts the load flowproblem formulation from a deterministic one t o a stochastic one. Theproposed m ethod , through the extension of the conventional load flowproblem , has included the calculation of the effects of inaccuracies ininput data on all ou tpu t q uantities, nam ely o n all bus state variables Ean d S or on line flow P and Q and generator bus Q. The stochastic loadflow represents, in this way, the combined results of many deterministicload flows in which f or each flow the data are perturbed, such that thevarious flows represent possible sets of data within the precision thatthe input data is known.The model of the load flow in which a constraint on total systemload plus losses has been placed by includin g a real power equ atio n forthe slack bus can be considered as a realistic one, since the forecastederrors are then distributedamonggeneratorP equations. It s to bebelieved, that further experience in using the developed me thod willmake it simpler and more attractive.The authors should be complimented for this nice paper.

Manuscriptreceived August 13,1974.

A. Petroianu (National Center of Roma nian Power System, Bucharest,Romania): The authors are to be commended for a thought provokingpaper related to an interdisciplinary .in character problem.The paper is devoted to the research problemof nfluenceofinitial data random errors upon the results of load flow calculation.Such an nvestigation is very timely for the comparison of differentmathematical models suitable for off and on line load flow calcula-tions.The authors have aprobabilistic pproach t o the problem ofestimation of th e results of calculation. From the probabilistic point ofview the initial data should be considered as random variables withmultidimensional distribution law.We can underline two distinct direction of research in the frame ofthe probabilistic approach :1 Initial dat a errors are of stochastically definite natureIn this case the initial data are given by their mean value (mathe-matical expectatio ns) an d by the ir extre me limits of deviations (errors)from mean values. The law of distributionof nitial dataerrors issupposed to be known.It seems to the discusser that such an approach was used in[ 11.2-Initial data errorsare of stochastically i nde f~ t e atureIn this case only th e extrem e values of initialdata errors are known.I believe that considering a model in which all input data wasassigned error bounds t he auth ors are approaching a more practicalway since the law of distribution of errors is often nknown.In the frame of the probabilistic approach, considering that hedeformation of the initial data is small, the knowledge of the expecta-distribution law of the calculated values which constitutesour aim.tions andof the variance-covariance matrix sufficiently characterizes heI would like to remark that a geometrical approac h could introducesome fresh point of view that corresponds to the necessity of reducingnumb er of pertinent, synthetic and intuitive information [ 2 + 51 .the great outpu t data streams, generated by the comp uter, to a smallThe geometrical theory , which can be conceived in parallel to theprobabilistic approach, permits us to find and to describe an area ofpossible errors and the guaranteed region to which the calculated valuesbelong.

wh ere x, y are column vectors in m- and n- dimensional Euclideanspacesy = y + i ; y (2 )

where Sy- small random vector with variancecovariance m atrix Ay.In accordance with the principle of linearization:-= t (y ) + a x (316~ = J L y.- (4)

Manuscript received July 30,1974.307


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where J Jacobian matrix for p taken at the pointy.accordingly,Th evariancecovariance matrix Ax fo r S x an be calculated

L

and characterizes the errors in the solution vector x du e to the non-exactness in giving initial vector data - y.quadratic form: XT 45Formulae (5),corresponds to the general matrix expression of aIn the realm of an n-dimensional space with which is our presentconcern this quadratic form can be geometrically interpreted as repre-senting the area of the erro r yperellipsoid.Even under the simplifying assumptions that the admissible erro rsof x and y are sufficiently small to be related by the linear transforma-tion (4), the small region in which S y is contained is stretched in onedirection and compressed in the other.If the estimate of errors ismade b the ctral vector norm, henthe admissible regions for Sy will be an Ky per zps oid .If in tur n we estimate 6x by the spectral norm we shall be forcedto take as admissible the hypersphere with radius equal to the largesthalf-axis of the hyperellipsoid. This hypersphere will be larger than thehyperellipsoid; this replacement of the hyperellipsoid by the hyper-sphere involves the lossof information due to the uncertaintynd errorsin input data.Formulae (5) expresses a hyperellipsoid which is in a skew positionwith respect to the oblique coordinate axes; by means of a suitabletransformation to the principal axes of the hyperellipsoid the equationcan be reduced t o the canonic form

where &l .. An are the eigenvalues.That is equivalent to consider only the variance of co mpone nts(i.e. ignoring the correlation coefficient); in this casewe shall obtain theconfidence yperellipsoidswithhe xes being directed along thecoordinate axes.The degree in whichhe uncertainty of basic informatio n isconditioning the uncertainty of the steady state solution is expressedby the square root of the ratio of the eigenvalue of maximal modulusto the eigenvalue of minimal modulus.Formulae ( 7) which gives the elliptic norm (conditionality number),

can be geometrically int erpre ted as the atio of he hyperellipsoidsemiaxes.By virtue of the above form ulae it is evident that the precision,convergence and stabilityproperties of the load flow solut ion areoptimum orH = 1,

that corresponds to the perfect symmetry of a hypersphere.The discussers comments, regarding the geometrical approach,were intende d to underline the importance of the authors contributionto a new emerging research problem - the probabilistic load flowvarious existent schools of tho ught.problem - an d to add some bits of information in an effort to relate

REFERENCES[ 1 Power System Com pute r Feasibility Stud y - 1968 - vol. I , IBM[21 C. Lanczos, Applied Analysis Prentice Hall, 1956.[3 A. Petroianu A geometrical approach to the steady state problemof electrical networks. Rev. Roum. Sci. Techn.-Electrotechn. et[41 V. I. Idelchik About t k error influence of initial data upon theEnerg., 14 ,4 p. 623-630, Bucharest, 1969 .result of power flows calculations in power systems Izvestia, theUSSR A.S. Energetics and Transport, 1968, No. 2, p. 9-15 .[ 5 1 D. K. Faddeev, V. N. Faddeeva Stability in Linear AlgebraProblems A.F.I.P.S. Congr. 1972. Ljubliana.

Research Division,San Jose, California.

B.EWollenberg (Leeds and Northru p Company, North Wales, Pa.): Thisis a very im porta nt paper. The techniques used in calculating the covari-ance matricesof system statesan dquantitiescomputable from thestates are definitely useful in all analytical studies requiring a load flowsolution. I agree that any time a load flow is run, wheth er for eal timestudies or planning studies, a recognition of the data inaccuracies is al-ways useful - nd necessary.The autho rs have been careful to develop a technique which is usedin conj unct ion with a conventional load flow solution and not a leastsquares state estimator type solution. The primary contribution of thepaper is in showing how to compu te he covariance matrices whenadjustments such as area load or system total load, known to somespecified accuracy, are placed on he solution.The authors correctlypoint out that uch adjustmen ts d o not imply anything more than a con-ventional load flow solution. The covariance matrix calculations used tomodel the effects of the area or system load adjustments are correct fora least squares solution to th e load flow with the area or system loadvalues acting as redundant nformation. When a load flow is solved withthe loads preadjusted to meet area or system conditions, the numb er ofdegrees of freedom remains the same and ther e is no filtering effect. Aleast squaressolutionwitharea or system load values as edun dantinformation will produce filtering. The statem ent in the description ofthe authors Case 6 that I . . . here was no iltering effect caused by theredundancy of the slfck real power equation. is somewhat misleading.It should have read . ..noapparent filtering. . ., that is, there was nofiltering which could be measured. In part th is is due to the fac t tha t theerror bound of 210 MW was allocated to the 14 generators by dividing2 10 MW by 14 to give a 15 MW error bound. This calculation shouldhave divided 210 MW by the square root of 14which would have givena 56.1248 MW error bound for the generators. The 15 MW used by theauthors was so much smaller than the 40%error band on the loads thatthe 56.1248 MW error bound also and therefore shows that the covari-any filtering effect would be minimal. This result is probably true forance matrix calculation used is quite adequate.

Manuscriptreceived August 2, 1974.

J . F.Dopazo, 0.A. Klitin, and A. M. asson: We are very pleased t o theresponse of so many discussers to our paper and acknowledge theircontributions to the subject. We are encouraged tha t in all cases the dis-cussers emphasize the need for treating the load flow stochastically.which they calculate actual density curves with simplified models. WeMessrs. Allan and Grigg and Ms. Borkowska present approaches inconsider tha t the distribution of inpu t quantities, which is needed intheir method, is generally not known. For instance, consider thementions. Over a period of a day the outpu t of the plant can vary sub-generationofa large plan t with several units which Ms. Borkowskastantially and a non-normal distribution could be determined. However,th edistribution of plant ou tpu t at peak system condition s at somefuture year can be considered as anormaldistributionwitha smallvariance that depends significantly on what thedistributionof heupon he applicability of the Central Limit Theorem. T he reasoningsystem peak is. The same discussors and Dr. Duran, express reservationsgiven for the distribution of in put quan tities also apply here. Line flowsnear generating lants, which is the examplementioned by Ms.Borkowska, can thus be considered as normal. Dr. Duran finds someWe agree that results are only pproximatelynormal in th e sensechi-squared co mpon ent in the output when no inearizations are made.approaches a no rmal as the degrees of freed om increases which justifiesDr. Duran pointsout. However, the chi-squared distribution apidlyreally after the bound s of o utput quan tities an d not their distribution.ou r use of the CLT for practical purposes. On the other hand, we areare fur ther justified.The unknown but bounded theory (9) applies here and our proceduresThe argumentsgiven above apply t o a large extent t o the com mentsmade by Messs. Heydt and Semlyen and Ms.Borkowska on the use ofa diagonal input convariance m atrix. Even in the case of P, Q values at thesame bus which Dr. Semlyen refers to, while the variations during a dayfor system peak cond itions are considerably less so.are strongly correlated our expectation of P and Q at some future yearMessrs. Schenk, Sigh,Semlyen and D uran an dMs. Borkowska askon the accuracy of results for large input variances, given the lineariza-tions involved in our method . Obviously, one can not expectto increaseinpu t variances indefinitely wit hout serious deterioration in t he accuracy.We would advise tha t t ests similar to the ones we performed be made toanswer this question for a specific system. If Monte Carlo load flowsresults can be reasonably duplicated by the stochastic load flow, the nthe latterapplies for that evel of input variances.

Manuscript received November 4,1974.308


11/11

We were pleased to find the comm ents made by Ms. Borkowskad Dr.Despotovic on our differentiation between global and individualand he nclusion of total oaderror as variances over al ltors thus elimin ating the need for defining a slack bus. On hi sour calculation of individual generatoris in error. Considering a total load erro r bound of 210 to be14 gen erators he consid ers th at th e sum of the variances ofbe equal to that of the total. Thus,

is to divide the 210 evenly among the 14 generators. WithWollenbergs sugg estio n we wo uld be considering load flow situa-in which generation excee ds the bou nds of total system load thusDr. Heydt briefly presents a method he has developed in which eare some com putatio nal advantages. We understan d that hebe presenting his metho d in a paper that will be presented shortlyike to reserve our comm ents until we read his forth -paper and unde rstand his appro ach better. We will only point

ou t that the use of sparsity techniques makes our method computation-ally attractive.We appreciate Dr. Petroianus commentsonour paperan d hisgeometric nterpretation of the mapping of npu terrors ooutputquantities.Dr. Durans derivation of the papers equat ions are he same asours. We com plicated m atters by trying t o show the relation betwe enthe heory of stateestimationand hat of stochastic oad flows toemphasize that historically the idea of the latter came from the former.We fmd Dr. Durans error analysis quite enlightn ing Howev er th elast term of the exact formula for the varianceof G should be 2 RZup4and not 2 R2o 2. This makes the errors in the mean and variance to beRup2, and 2 Rs 4 w hich are second order effects in their respectiveequations. For ins%nce, consider as n example that per unit Ran d P areequal o 1 and an error bound for P equa l o 0.3. This makes O p 2 =(0.3/3)2 = 0.0 1. Then , the error n the mean and variance of G are 0.01and 0.0002 while our computed mean andvariance are 2 and0.9respectively.

REFERENCE[9 ] F. C. Schweppe U ncertain Dynam ic System s, Prentice-Hall Inc.,Englewood Cliffs, N. J ., 1973.

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