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Apparent Power in MultiphaseSystemswithUnequal NeutralHector Arango J. PolicarpoG.AbreuDepartment ofElectricalEngineeringFederalSchool of Engineeringat ItajuM - EFEI

Itajuba,MG BrazilAbstract: This paper addresses and old question of scientific aswell as practical significance:"How to define apparent power fordistatedsystems while retainingthe basicPropertiesand meaningof the orthodox definition?".Keywords: power &tixitiom, apparent power, u n e c e dSy-.

And the powerlost in the supplying lineon the eft side offigure 1 givena:Ap+ ai) (2)

wherethematrix4 as he form-I. INTRODUCTION = -=U+p 11' (3 )

Apparent power "Sa s here ocused- for any multi-phasesystem with arbitrary voltage and current - as being the"maxi" ctive power which could be transferred underthe same voltage and feeder losses". Suitably to thisapproach expressions are devised both in the time and thefrequency domains. Also, a linear transformation of thephase current for which "S"ispmerly given as the productof the R.M.S values of the image vectors is also presented.Thus it regains the basic feature found in the orthodoxdefinition.

11.BASIC DEFINITIONSIN TIME DOMATNRegarchng to them-phase system with neutral depicted in

Fig. 1, let be the vector (column-matrix ) of phase-toneutral instantaneous voltages, and the one for phasecur".he nstantaneouspower flowingintothe load willbe given as:p = E'.! (1)

U s the unit matr ix and L stands for the vector of onesand p is tbe resistance elation between neutral and phases,supposedly indqmdent of frequency.

tA Qt-dwvb-+*

P+AP+FEEDER LOAD

Fig. 1.SystanarranganadThe average power supplied to the toad during someinterval of ti meTwill be denoted as:

P= AKU(p)re E' stands or the transpose( row-matrix d. . (4)And the average losses are:

III.APPAREN"POWERASAMAXIMUM-Paper accepted for p resentation at the 8" InternationalConference on Harmonics and Qualio of PowerICHQP '98, ointly organized by IEEEPES and NTUA,Athens, Greece, October 1416,1998

EFFICIENCYCONCEPTNow, within the general contextdescribed in the previoussection - i.e. or any number of phases, any non-sinusoidal,

0-7803-5105-3/98/$10.00 0 1998 EEE and any "heed non-periodid phenomena - the

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challenge arises of how to pmperly &fine the apparentpower "S"which isbemgconveyed o he load.hs p in t~ on s taken from the conclusive studies in 121which state, among he basic properties of apparent power -that it is "the maximum avexage power that can betransferred through the f&g line with the same averagelosses AP, nder the same iastantaneousvalues 2 (t)duringT"This property canbe turnedout o be the very definitionof"S". In fad, there exists an univocal value"g uchpmperty, as can be p" y solving the optimizationproblem hphed. In faact the associatedLagrangian functionwill be:

ar-=v-2rA AI =Oai _ - (J

IV. PERIODICALWAVEFORMS THE REQUENCYDOMAIN

This V&ES be associ ated [35 ] with the " B u c ~ ~ o ~ ~ z -Gdhue-Emanuel Equivalent Values"V,' , f ,orwhichtheapparent power isgiven as:( BGE values)(8)

the definitionadopted, t comes: It is to be noticed thaty or thisutmost general conditions,VGE"values are not the RMS valuesof the instantaneousIn bet, the so called "FBD theory" I6-81, another

1AP=--AVR(x'R r --' E) ( 9 ) phasevoltages and currents.

5 (A& Here it wil l be proposed a somewhat dffnt approach,i, =r . I (/-'E) (10) seekingT " e d Phase Vectors"ofdimensionin whoseRMS values d d oincide with the BGE's equwalentvalues.The followingSection dealswith such approach.(Am(y'A-' x))?And the maximum power "S"comes inally to have thefollowingform:A . How to express the apparent power as a product of

( ' 1 complex-vector-m values.Thisisan expression of the apparentpower which is validover the most gamd conditionsCotlCejvaMe for electricalIt i s worth to remark, "en passant",that it c h a m "the behavior of the most efficient load regarded to the

praportionbetweenpower t r a n s f e r r e d to itand power lost inthefeeder.A s o m e w h a t ~ E a c t a b ~ u t ~ ~ c o m e s t o b e i t s

non-parallelsm with the i nam oltagevedor 2 ,unless p = O or y o = O .

energytransference.While not bemg a matterof too greatpractical valuey t ishoweverof speculativeinterest to attain an expression of theappamt power able to retain the origmal form of the

sinusoida- case, hat is to sayya form in which " Scan be given as the product of RMS voltage and current.Such a goal can be reached by means of a lineartransfonaation of phasevectors,whose form is:

Where

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Observe that this tt;"ation is p o w q " h g , thatis to say:

Io such conditions,

whete,obviously:

RMs(E, )=(. R E: E.))RMs(i* )=(AYR(f: .))and

These resultscanbecomparedwith the apptoachoutlinedin j9-111.

V.ANILLUS"ExAMpLEConsider a threeyphase system with sinusoidal voltagesand aments whose phasor veztors are as f ol l ows and are

also representedinFig. . .

1 + O j- I + j

then

Fig 2. ?breeph.scvoltagleandcumnt.

I + j1 - j1 + 3 j

Zi=6 Zf = 8

and1 + i2 + 4 j

P = R e ( y ;4 = 5 y : E: =-9 j * I * = 32,s = v, I , =45 4 -*-*

Showing that the proposed transformation efktivelyallows e x jm s h g the apparent power in terms of RM.S.values.It is alsoof interest tocalculak the "maximum4ciencyN

ament 1,.starting from (IO), it can be shown that:

Fig. 3 illustmtes it.

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transformation of the space of phaecoodiuates into afictitious space where apparent power can be indeedexpmsed in he stme fonnat asinthe orthodox notion.

The resultsofthe paper raise a host of questionsand newdevelopments to be searched in the future, such as: thehandling of series and shunt feeder losses; the dualityinvolved either in the formulation of theoptimal problem orin the electrical system;matters related with power factordefinitionand "swements, revenue issues, et cetera.

5 + O jL p = K I = ! ? [P 5 - 6 j ] = ~ co57 - 7 + 6 j

The d t tresses the fact that the --images of thevoltage and maximum-efliciency currents are patallel, asopposed o their anti-images in the original complex hyper-space. ~bviously,the active pawer f lowing for f, is thesameasinthetraasfonnationdomain:

VI.CONCLUSIONSAND "TREEXTENSIONS

111

131VIP I161

Vn. EFERENCES

[lo] DIN 40 110 Teil 2 Mcbr;Leitakst"krasc - V o l t a g e f i t r ~(drift)hiatdl 1995;[l 11 J.G.MAYORDOM0, J. USAOLA, "ApparmtPow= andP o w a FactorD c b i t i a ~a Po l y p he N m-h ea r toads whm Supply Condu&orsRaid DiffgartRe&", TEP, v 3,n 6,Nov/Dec 1993.

VIII. BIOGWHIES

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