Currents Physical Components (CPC) in systems with semi-periodic voltages and currents

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1 JUNE 5-8, 05, ŁAÓW, POLAND Currens Physical Componens (CPC) in sysems wih semi-periodic volages and currens Lesze S. Czarneci School of Elecrical Engineering and Compuer Science Louisiana Sae Universiy Baon Rouge, USA Absrac here are siuaions in elecrical disribuion sysems where volages and currens canno be regarded as periodic quaniies. hey could be non-periodic. Noneheless, properies of elecrical sysems confine his non-periodiciy of volages and currens o a paricular sub-se of non-periodic quaniies, referred o as semi-periodic quaniies in his paper. he paper presens he concep of semi-periodic quaniies and defines major funcionals, such as he running acive power, he running rms value, he scalar produc and he running complex rms (crms) value of quasi-harmonics, needed for describing elecrical sysems wih such volages and currens in power erms. A recursive approach o calculaion of hese funcionals was presened as well. he paper presens fundamenals of he Currens Physical Componens (CPC) based power heory of sysems wih semi-periodic volages and currens. An applicaion of he semi-periodic concep o a load curren decomposiion in single-phase circuis wih linear loads and wih harmonics generaing loads is presened as well. he paper presens also a concep of exrapolaion of CPC ino he closes fuure, which enables quasi-insananeous generaion of conrol signals for swiching compensaors.. eywords Power definiions, power heory, non-periodic quaniies I. INRODUCION I loos lie he power heory can evenually provide he answer o Seinmez s old quesion, formulaed in 89, []: why can he apparen power S in a resisive circui wih an elecric arc be higher han he acive power P? he answer, in he frame of he Currens Physical Componens (CPC) based power heory [5], applies o single- and hree-phase sysems on he condiion ha volages and currens are periodic. here are siuaions in disribuion sysems where his condiion is no saisfied, however. Volages and currens of some loads are non-periodic. An example of such a nonperiodic volage and curren a a erminal of a hree-phase energy pump is shown in Fig.. Waveform non-periodiciy is mainly due o power elecronics devices, which provide versaile echnical means for fas conrol of energy flow, disurbing waveform periodiciy. Adjusable speed drives driven by power elecronics converers, are common sources of non-periodic phenomena. Also some loads are non-periodic by naure. Fig.. Waveforms of he load curren and he supply volage of energy pump. Pulsing loads are examples of loads ha draw nonperiodic currens from a supply source, such as spo welders, generaors of beams of X-rays, or elecromagneic guns. hese loads release energy in he form of pulses of curren of he order of hundreds of A and only a few milliseconds duraion. he energy of he pulse is sored, by a sor of energy pump, over several periods of he supply volage in a capacior, o be released nex in a shor inerval of ime, and he process is repeaed. An example of such an energy pump is shown in Fig.. Fig.. Example of energy pump for a pulsing load. Waveforms of he supply curren and he volage of such an energy pump are shown in Fig.. he supply curren of he pump increases over several periods of he supply volage and declines o zero when he capacior volage approaches is maximum value, so ha he energy needed for he pulsing load is sored in he capacior. Due o he volage drop on he supply source impedance, here is also a variaion of he pump volage. I causes he volage non-periodiciy. Some of he mos remarable devices wih nonperiodic currens are arc furnaces in uneasy phase of meling. aing ino accoun ha he power of an arc furnace could be in he range of hundreds MVA, non /0/$ IEEE

2 JUNE 5-8, 05, ŁAÓW, POLAND periodic phenomena caused by such a furnace could be remarable. he presence of non-periodic volages and currens in disribuion sysems raises wo issues. Firs, a cogniive quesion occurs: how can power relaed phenomena be explained and described in a siuaion where he conceps developed for sysems wih periodic quaniies are no longer valid? he second quesion is pracical: how can loads wih non-periodic volages and curren be compensaed, and in paricular, how should reference signals for swiching compensaor conrol be generaed? his paper is focused on he firs of hese wo quesions, namely, on describing power properies of loads wih semi-periodic volages and currens. he sudy is confined o singlephase sysems, bu he approach and resuls can be generalized easily o hree-phase sysems. Volages and currens in siuaions described above are non-periodic and consequenly, well esablished definiions in elecrical engineering such as he acive power P = u () i() d = U I cosϕ, () he volage and curren rms values 0 u = u ( ) d, i = i ( ) d, () 0 0 as well as he complex rms (crms) value of harmonics jα ω X n n= X ne = x() e d, (3) i.e., he main funcionals in he CPC based power heory, canno be defined, because non-periodic quaniies simply do no have he period. o calculae powers, non-periodic quaniies are someimes approximaed by periodic ones. Noneheless, when powers in sysems wih non-periodic quaniies are no defined, he error of such an approximaion canno even be evaluaed. Non-periodic volages and currens in elecrical power sysems usually have some paricular properies ha enable us o caegorize hem as semi-periodic quaniies and he CPC-based power heory can be exended o sysems wih such quaniies. II. SEMI-PERIODIC VOLAES AND CURRENS he main sources of elecric energy in power sysems, synchronous generaors, creae a volage which is wih high accuracy sinusoidal and jus his generaed volage is he driving force for he energy delivery o he power sysem. he period of his generaed volage can be deeced by filering and be used as a sor of ime-frame for he energy flow analysis. Due o ime variance of he load parameers, he load curren can be non-periodic. his non-periodiciy can be permanen or ransien, so ha afer some ime, he load curren could be periodic wih he period again. he same applies o volages ha conain a response o he load currens. he mechanism of semi-periodic disorion is illusraed for a single-phase circui in Fig. 3. Volages and currens in sysems wih ime-varying loads, bu wih he energy delivered by generaors of sinu- 0 soidal volage, are referred o in his paper as semi-periodic volages and currens. Fig. 3. he mechanism of semi-periodic disorion. In mahemaics here exiss [] a concep of quasiperiodic funcions. he meaning of a semi-periodic quaniy or funcion, as defined in his paper, differs from he meaning of a quasi-periodic funcion, however. he raio of he supply source impedance o he load impedance is usually confined in such a way ha he load volage rms value U does no decline a he maximum load power by more han 5%. herefore, he semi-periodic componen of he load volage should no be higher han a few per cens of he periodic componen. he load curren does no have his sor of limiaion and consequenly, semi-periodiciy can be much more visible in currens han in volages. Fundamenal harmonic u of he volage periodic componen, separaed by a filer, can be used for deecing he period. A ime inerval of duraion will be referred o as an observaion window. I is shown in Fig. 4. Fig. 4. Observaion window. Such a window sars a some insan of ime = and ends a insan. If he quaniy x() as observed in he window, shown in Fig. 4, is reproduced every period from minus o plus infiniy, hen a periodic exension n= x () = x( n) (4) n= is creaed, shown in Fig. 5, which is a ficiious periodic quaniy, idenical wih quaniy x() only in he observaion window, bu no ouside of i. Fig. 5. Semi-periodic quaniy x() and periodic exension x ().

3 JUNE 5-8, 05, ŁAÓW, POLAND he energy delivered o a load during he observaion window preceding he insan is equal o W ( ) = uid () (). (5) When he volage and curren a he load erminals are periodic wih period, his energy is consan, meaning independen on he beginning of inegraion. I is no consan when he volage and curren are semi-periodic. Alhough inerval is no a period of semi-periodic quaniies, i can be used for calculaing he value of he average rae of energy flow over he observaion window. A he end of his flow observaion, a insan, his average value is equal o W ( ) = u() i() d = P ( ). (6) he power calculaed in such a way can be regarded as he acive power of a load wih a semi-periodic volage and curren. When he volage and curren are no periodic, hen he acive power defined in such a way is no consan, bu i is a funcion of ime. his is emphasized wih he wave symbol. I will be referred o as a running acive power. he periodic exension x () of wha is observed in he observaion window x() has he rms value, which can be calculaed a he insan =, namely x = x ( ) d. (7) If he quaniy is no periodic, hen his value is no consan, bu changes wih he observaion window. I will be referred o a running rms value of a semi-periodic quaniy. When a semi-periodic curren i() flows hrough a resisor of resisance R, hen he running acive power of his resisor a he insan =, is equal o () () (). (8) P = u i d = Ri d = R i hus, he running rms value has exacly he same meaning as he convenional rms value. I enables calculaion of he energy loss on he resisor in he preceding observaion window. I does no allow calculaion ouside of his window, however. For wo semi-periodic quaniies x() and y(), which have periodic exensions wih he same period, a scalar produc can be defined and calculaed a he end of observaion window, namely ( x, y ) = x() y() d. (9) he running rms value of he sum of wo semi-periodic quaniies x() and y() is equal o x + y = + ) d ( x, y ) +. x y = x + y ( (0) rms value of he sum componens, as on he condiion ha = x + y x + y, () ( x, y ) = 0, () he number of samples has o obey he Nyquis his rms value can be expressed in erms of only running crierion. If n max is he highes harmonic order of he /0/$ IEEE i.e., when hese quaniies are muually orhogonal. Propery (), a condiion (), applies only o he observaion window of duraion jus preceding insan =. he periodic exension x () can be expressed, moreover, as a Fourier series ( ) 0 Re Xn n= 0 ω x = X + e, (3) wih crms values of is harmonics X n n ω = X ( ) = xe () d, (4) calculaed a insan of ime. I means ha quaniy x() can reconsruced from is harmonics, assuming ha i does no have poins of disconinuiy, wih full accuracy. his reconsrucion applies only o he observaion window of duraion, however, bu no beyond i. III. DISCREE IDENIFICAION OF SEMI- PERIODIC QUANIIES Alhough he running acive power and rms value of semi-periodic volages and currens, as well as he crms values of heir harmonics were defined above as funcionnals of coninuous quaniies, discree mehods implemened in digial meers are needed for heir measuremen. Such a meer performs arihmeic operaions on digial samples of he load volage and curren, provided by analog o digial (A/D) converers. he sequence of curren samples is shown in Fig. 6. Fig. 6. Curren samples in observaion window. o avoid confusion wih symbols of harmonics, denoed usually by u n and i n, hese samples will be denoed by u m and i m in his paper. he las sample, i.e., a he insan of observaion, has index. Assuming ha here are samples of he volage and curren in he observaion window, as shown in Fig. 6, hen such a digial meer can calculae a he insan = he running acive power according o formula: m= m m m= + P ( ) = P = u i. (5)

4 JUNE 5-8, 05, ŁAÓW, POLAND periodic exension, hen he minimum number of samples in he period has o be higher han he double value of n max, i.e., > n max. his is he condiion for preserving full informaion on he waveform of a periodic quaniy. A discree formula for running rms value of semiperiodic quaniy has he form x = while for he scalar produc m= x m m= + m= ( x, y ) xm y m m= +, (6) =. (7) wo semi-periodic quaniies are orhogonal on he condiion ha m= xm y m = 0. m= + (8) he complex rms (crms) value of he n h order harmonic periodic exension of a semi-periodic quaniy in observaion window x() can be obained from a discree formula Xn = m= m x me m= +. (9) o implemen a Fas Fourier ransform (FF) algorihm, an ineger power of wo is usually seleced for value. he crms values (9) are no consan, bu hey change wih ime. hey specify harmonics of only a single periodic exension x (), which changes wih he change of he observaion insan o a new periodic exension. herefore, semi-periodic quaniies, as non-periodic, canno be described in erms of harmonics, bu by eniies referred in his paper as quasi-harmonics, wih varying ampliude and phase. Only inside of he observaion window quasi-harmonics are idenical wih he common harmonics. Siuaions when wo semi-periodic quaniies are orhogonal do no differ from hose for periodic quaniies. here are five such siuaions. Four of hem apply o single-phase quaniies, and he las one applies o hreephase quaniies. wo semi-periodic quaniies are orhogonal, if: - for each pair of samples x m and y m in he observaion window, heir produc x m y m = 0, meaning, only one sample in he pair is differen from zero, - one of hese quaniies is a derivaive or inegral of he oher quaniy, 3 - hese quaniies are quasi-harmonics of differen order, 4 - hese quaniies are quasi-harmonics of he same order, bu hey are shifed muually by π/, 5 hese quaniies are hree-phase symmerical quaniies of a differen sequence. When a semi-periodic quaniy is composed of a sum of muually orhogonal componens, i.e., componens ha saisfy any of hese five condiions, hen he running rms value of such a quaniy can be calculaed as a roo of sum of squares of rms values of is componens. Formulae (5) (9) along wih he concep of orhogonaliy provide a basic mahemaical ool for exension of he concep of he Currens Physical Componens (CPC) o elecrical sysems wih semi-periodic volages and currens. A he insan of ime =, daa on volages and curren a he load erminals, colleced over he observaion window of duraion, are sufficien for he load curren decomposiion ino he Physical Componens ha uniquely idenify physical phenomena in he load over ha window. he only concern ha could arise is he amoun of calculaion needed for ha beween he preceding observaion insan of ime = - and he curren insan =. his compuaional burden can be, as shown in he following secion, only apparen, however. IV. RECURSIVE CALCULAION OF RUNNIN QUANIIES he running acive power a insan = can be calculaed [7] as follows m= m= = m m= m= + m= + m m+ m= P u i ( u i u i u i ) = = umim ( ui u i ) = + (0) m= M = P + ( ui u i ) i means ha i can be calculaed by updaing is value calculaed previously, a insan = -. When he volage and curren are periodic, hen u = u, i = i and updaing is no needed, since P = P. Similarly, he running rms value of a semi-periodic x() quaniy is m= m= xm xm x x m= + m= + x = = + = m= ( xm x x ) m= = + = () = x + ( x x ). hus i is enough o updae he previously calculaed value. he same applies o he complex rms values of quasiharmonics, namely m= m X = n xme = m= + m= m ( ) = [ x ( ) me x x e ] + = m= + m= π m π = x ( ) me x x e + m= and evenually X = + ( ) n Xn x x e (). (3)

5 JUNE 5-8, 05, ŁAÓW, POLAND hus, only he difference of he laes sample x and ha, aen he period earlier x -, has o be aen ino accoun when updaing he crms value of he quasi-harmonic. Recursive calculaion reduces drasically he amoun of calculaions, bu heir resuls can be affeced by accumulaion of rounding error. herefore, rounding should be as random as possible, and his should be aen ino accoun in he calculaion algorihm consrucion. Moreover, special procedures, such as periodic rese of calculaed funcionals migh be considered. he heoreical frame presened above maes descripion and inerpreaion of power properies of disribuion sysems wih semi-periodic volages and currens possible. his can apply o all disribuion sysem srucures and load properies, bu equaions and definiions developed for semi-periodic volages and currens are valid only in he -long observaion window, jus a he insan =, when he las samples of volages and currens in ha window are provided for calculaion. Semi-periodic currens occur in disribuion sysems due o semi-periodic supply volage, and/or due o load parameers variabiliy. his variabiliy could be fas, causing curren waveform disorion or slow, observed raher as a sor of ampliude or phase modulaion. he load curren disorion due o periodic variabiliy of he load parameers can be inerpreed as caused by generaion of quasiharmonics in he load. A slow variabiliy of hese parameers i can be assumed ha load properies can be approximaed and inerpreed as properies of linear loads. V. CPC OF SLOWLY VARYIN SINLE-PHASE LINEAR LOADS Le us assume ha a linear load is supplied wih a semiperiodic volage ω u ( ) = U0 Re n e + U. (4) If his load is no a source of curren quasi-harmonics, hen i draws he curren he raio ω i ( ) = I0 Re n e In Un + I. (5) Yn n jbn = = + (6) specifies he running admiance of he load for harmonic frequencies. Wih respec o he running acive power a insan, such a load is equivalen o a resisive load of conducance P e u =. (7) he curren of such a resisive load a insan = a e e 0 Re e Un i = u = U + e (8) is he acive curren of he load, as defined by Fryze [3]. he remaining par of he load curren a = ( 0 e) 0+ Re ( Yn e) Un i i U e (9) /0/$ IEEE can be decomposed ino he scaered curren 0 e 0+ n e Un = s ( ) U Re ( ) e i (30) and ino he reacive curren n Un = r Re jb e i. (3) hus, he load semi-periodic curren can be decomposed exacly as in he case of periodic currens ino hree componens such ha a insan = = a + s + r (3) i i i i which are Physical Componens of his curren. Inside of he observaion window i.e., for < <, his decomposiion of he semi-periodic curren is idenical wih he common decomposiion i() = i () + i () + i (), (33) a s r bu no ouside of ha window. Decomposiion (3) canno be found before he insan =, i.e., a he insan when componen of he decomposiion (3) can be calculaed. hese componens are muually orhogonal, so ha i = ia + is + ir. (34) Muliplying (34) by he square of he supply volage running rms value, he power equaion is obained wih S = P + Ds + Q, (35) S = u i Ds = u is Q = u ir. (36) Power equaion (35) describes he relaionship beween powers of single-phase loads supplied wih a semi-periodic volage, however confined o loads ha do no generae quasi-harmonics. I is valid in he inerval of duraion preceding insan =. I could be wrien, for convenience, in he radiional form = + s + S P D Q, (37) bu i should be remembered, ha hese powers may no have a consan value and his equaion is valid only in he window of duraion preceding ime insan =, bu no ouside of ha ime window. VI. CPC OF HARMONICS ENERAIN SINLE- PHASE LOADS Nonlinear loads or loads wih fas varying parameers or quasi-periodic swiches can generae quasi-harmonics. he running acive power of such quasi-harmonics could be negaive, which means ha hey convey he energy from he load bac o he supply source. his phenomenon, revealed in sysems wih periodic volages and currens [6], is a disincive physical phenomenon, which has o be

6 JUNE 5-8, 05, ŁAÓW, POLAND aen ino accoun when he power properies of elecrical circuis are analyzed. Le us describe such loads in erms of CPC when he volage and curren are semi-periodic. Le us consider he equivalen circui shown in Fig. 7. Decomposiion (39) of quasi-harmonic orders and he volage and curren according o (40) and (4), mean ha he sysem, as presened in Fig. 7, can be described as superposiion of wo sysems. he firs, shown in Fig. 8a, has an LI load and he second, shown in Fig. 8b, has only a curren source on he cusomer side while he disribuion sysem is a passive energy receiver. Fig. 7. Equivalen circui of a load and he supply source. he running acive power of he n h order quasi-harmonic Pn = Un In cosϕn (38) depending on he phase-shif, can be posiive, negaive or zero. he se N of quasi-harmonics orders n can be decomposed ino wo sub-ses. When he energy flow is caused by a quasi-harmonic of he n h order in he disribuion volage, i.e., he power P n is posiive, his quasi-harmonic order belongs o sub-se N C. When his flow occurs because he quasi-harmonic is generaed in he load, i.e., he power P n is negaive, is order belongs o sub-se N. he sign of he acive power depends on he phase-shif beween he volage and curren quasi harmonic, hus if ϕn π/, hen C if ϕn > π/, hen. (39) I enables he volage and curren decomposiion ino componens wih harmonics from sub-ses N C and N. A he insan =, (40) = n = n + n = C + C i i i i i i. (4) = n = n + n = C C u u u u u u he volage u is defined as he negaive sum of volage quasi-harmonics because, as a supply source response o load generaed curren, i, i has he opposie sign as compared o he sign of he disribuion sysem originaed volage quasi-harmonics. he same applies o quasiharmonic acive power, hus. (4) = n = n + n = C C P P P P P P Sub-ses N C and N do no conain common quasi-harmonic orders n, hus currens i C and i are muually orhogonal. Hence heir running rms values saisfy he relaionship i = ic + i. (43) he same applies o he volage rms values, namely u = uc + u. (44) Fig. 8 (a) Equivalen circui for harmonics n N C and (b) equivalen circui for harmonics n N. he circui in Fig. 8a, as a sysem wih a linear load, can be described according o he CPC approach. Namely, if he equivalen admiance is Yn n jbn and he equivalen conducance wih In Un = + = (45) PC e uc =, (46) uc = un, (47) C hen he curren i C can be decomposed a insan = ino he acive, scaered and reacive componens, namely where is he acive curren and i = i + i + i, (48) C Ca Cs Cr Ca e C i = u (49) Cs = ( n e) 0 + Re ( n e ) Un C i U e (50) is he scaered curren and icr Re jbn Un e C = (5) is he reacive curren. Evenually, he load curren can be decomposed ino four physical componens, where i = i + i + i + i, (5) Ca Cs Cr Re In e i = (53)

7 JUNE 5-8, 05, ŁAÓW, POLAND is he load generaed curren a insan =. hese currens are muually orhogonal, hence a insan = i = i + i + i + i. (54) Ca Cs Cr A diagram which geomerically illusraes his relaionship is shown in Fig. 9. Fig. 9. Diagram of running rms values of he supply curren physical componens of a HL. Loads wih fas varying parameers or HLs are ofen supplied wih sinusoidal or quasi-sinusoidal volage, or such approximaion is quie sufficien for analysis of he power properies in such siuaions. Fas variance of he load parameers could be he mos dominaing feaure of he sysem. In such a siuaion, he se N C of disribuion sysem originaed harmonic orders n is confined o only n =, i.e., N C = {}, while all oher orders belong o he se N, meaning, he se of he load originaed harmonic orders. he curren i C () in such a case is composed only of he acive and reacive currens, which are quasisinusoidal, while he curren i () is composed of all quasiharmonics. he scaered curren does no occur, of course, in he supply curren and consequenly, i can be decomposed ino he acive, reacive and he load generaed currens. he formula (5), could be reduced in such a siuaion o i = i + i + i. (55) Ca Cr VII. EXRAPOLAION IN HE FUURE he approach presened above enables decomposiion of a semi-periodic curren ino physical componens, which is valid only in he observaion window, which direcly pre-cedes he insan of he presen measuremens =. Alhough i provides full informaion on he power phenomena in he load, i is, unforunaely, o some degree useless for compensaion. Assuming ha a swiching compensaor is used for he power facor improvemen, hen properies of he load a he insan = +, i.e., in a direc fuure have o be nown. Currens ha are o be compensaed, of he value jus a ha insan, = +, have o be injeced by he compensaor ino he sysem. heir value can be prediced by exrapolaion. he mos simple is a linear exrapolaion, based on he assumpion ha changes in he direc fuure, in he ime of a single sampling inerval, are equal o changes in he direc pas. his assumpion means ha if x is a value of a semi-periodic quaniy a he insan =, hen in he direc fuure, a = +, is assumed o be x+ x x x ( x x ) = x x = + = +. (56) he measuremen a he insan = +, can updae his exrapolaed value o a rue one. Such exrapolaion can provide fundamenals for quasiinsananeous generaion of reference signals for swiching compensaor conrol. VIII. CONCLUSIONS he paper demonsraes ha he Currens Physical Componens based power heory, originally developed for sysems wih periodic volages and currens, can be generalized o sysems where hese quaniies are semiperiodic. I means ha also in such condiions, he inerpreaion of power phenomena in elecrical circuis, as provided by he CPC, remains valid. REFERENCES [] Ch.P. Seinmez, Does phase displacemen occur in he cur-ren of elecric arcs? (In erman), (89), EZ, 587. [] H. Bohr, (93), Fasperiodische Funionen, Erg. Mah., Springer, Berlin. [3] S. Fryze, Acive, reacive and apparen power in circus wih nonsinusoidal volages and currens, (in Polish) Przegląd Eleroechniczny, (93), z.7, 93-03, z.8, 5-34, (93), z., pp [4] I. Hacing, (98), Scienific Revoluion, (Oxford Reading in Philosophy), Oxford. [5] L.S. Czarneci, Currens Physical Componens (CPC) concep: a fundamenal for power heory, (008), Przeglad Eleroechniczny, R84, No. 6, pp [6] L.S. Czarneci and. Swielici, "Powers in nonsinusoidal newors, heir analysis, inerpreaion and measuremen," IEEE rans. Insr. and Measur., (990), Vol. IM-39, No., pp [7] L.S. Czarneci, Power heory of elecrical circuis wih quasiperiodic waveforms of volages and currens, European ransacion on Elecrical Power, (996), EEP Vol. 6, No. 5, pp /0/$ IEEE