Critical Evaluation of FBD, PQ and CPT Power Theories for Three-Wire Power Systems

Transcription

1 Critil Evlution of FBD, PQ nd CPT Power Theories for Three-Wire Power Systems Fernndo MARAFÃO Group of Automtion nd Integrting Systems, São Pulo Stte University Soro, São Pulo, 17-1, Brzil Helmo MORALES nd Luis DA SILVA Deprtment of Energy Systems nd Control, University of Cmpins Cmpins, São Pulo, 133-5, Brzil ABSTRACT Considering unlned nd non liner three-phse three-wire power systems, this pper disusses the reltionship nd disrepnies mong three modern power theories. The onsidered pprohes were the so lled FBD Theory (Fryze- Buhholz-Depenrok), the pq-theory nd the Conservtive Power Theory (CPT). Suh theories nd relted definitions my influene not only the energy ontrol nd regultion, mostly in terms of power mesurement tehniques, revenue metering, instrumenttion tehnology or eletroni power onditioning, ut lso some onepts of power engineering edution. The three methods hve een summrized, disussed nd ompred y mens of omputtionl simultion. Although the three methods re sed on different onepts, the results otined under idel onditions re very similr. The min differenes pper in the presene of unlned nd non liner lod onditions. Under liner unlned onditions, oth FBD nd pq-theory suggest tht the some urrent suomponents ontin third-order hrmoni. Besides, neither pq-theory nor FBD method re le to provide urte informtion for retive urrent under unlned nd distorted onditions, wht seems to e done y mens of the CPT. The pper tries to explin the uses of these differenes in terms of the foundtions of the power nd urrent s deompositions nd the resulting wveforms nd spetr. Keywords: CPT, Current Deomposition, FBD Theory, Hrmonis, Homo-vriles, pq-theory, Power Theory. 1. INTRODUCTION The enormous efforts to find generlized power theory, pplile for power systems under non-sinusoidl nd/or unlned onditions, hs mostly een motivted y the ever inresing of power eletroni onverters utiliztion. It point out the mjor requirement of improvement nd dpttion of retive/hrmoni ompenstors tehnology nd revenue metering tehniques under suh onditions. In this sense, numerous new power theories hve een defined. A numer of them re sed on the frequeny domin to desrie suitle power nd urrent terms under non-sinusoidl nd unlned onditions [1-], while others re sed on instntneous quntities in order to propose prtiulr urrent deompositions nd power definitions [7-13]. Thus, onsidering just the time domin pprohes, one ould ll ttention to the proposls of Depenrok (FBD) [9], Akgi et l. (pq-theory) [1] nd Tenti et l. (CPT) [11], whih re strongly relted to energy onditioning pplitions. Given tht the pq-theory is very well-known nd epted y the power eletronis ommunity, some uthors tend to onsider it theoretil tool not only for tive filters ontrol [1,1,15], ut lso for energy properties definitions nd understnding [1,17], regrdless of ll the misunderstnding out physil phenomen under non sinusoidl nd unlned onditions [-]. In [1], Willems hd lredy verified tht the pq-theory fes some oneptul prolems. More reently, Depenrok et l. [19] hve investigted the originl nd modified pq-theory for three-phse four-wire systems, ut it ws Czrneki [,1], who investigted how the properties of three-phse systems re desried y mens of the pq-theory nd disuss why suh theory should not e used for understnding the power properties of the lod instntneously. Considering three-phse three-wire iruits, this pper will demonstrte tht the FBD method extly mthes the pq- Theory, in se of sinusoidl nd lned PCC voltges, nd oth n jeoprdize the power properties nlysis under unlned or non liner lods. These two theories re ompred to the proposl of Tenti et l. nd it will e demonstrted tht this lst one seems to e more onsistent thn the other two, in terms of the urrent deompositions nd power phenomen explntion.. THE FBD THEORY The FBD (Fryze-Buhholz-Depenrok) method is n extension of the Fryze [7] nd Buhholz [] theories, on whih Prof. Depenrok mkes use of Fryze s urrent deomposition nd Buhholz s instntneous nd RMS olletive vlues for the definition of new urrent deompositions. Suh urrents were lso pplied to the lultion of novel power omponents nd for the proposition of ompenstion strtegies [9]. With the orret onsidertions, the FBD method n e pplied in ny multiphse power iruit, whih n e represented y uniform iruit on whih none of the ondutors is treted s n espeil ondutor. In this uniform iruit, the voltges in the m-terminls re referred to virtul str point *. The single prerequisite is tht Kirhhoff s lws must e vlid for the voltges nd urrents t the terminls. Considering multiphse power iruits, the FBD-method uses multidimensionl voltge nd urrent vetors (,) nd their instntneous olletive vlues ( Σ, Σ ), defined respetively s: ; (1) Σ ; Σ, ()

2 where, m indites the numer of ondutors (wires), old letters indite vetor representtion nd T represents the trnspose opertion. Thus, the olletive instntneous power results from the inner (dot) produt: Σ. (3) Under periodi onditions the olletive RMS vlue of the urrents nd of the voltges n e lulted s: Σ 1 Σ ; Σ 1 Σ, () nd the olletive tive power results from: Σ 1 Σ. (5) Thus, ording FBD method, the instntneous urrent in eh phse of the system ( ) is deomposed in some omponents, proportionl nd orthogonl to the voltges: Ative urrents ( ): responsile for the trnsferene of verge energy to the lod. These definitions re generlly vlid under periodil ondition nd is responsile for the sme tive power s the urrent. Σ. () Σ Nontive urrents ( ): ssoited to ny type of disturne nd osilltions tht ffet the instntneous power, ut do not trnsfer verge energy to the lod.. (7) Power urrents ( ): responsile for the instntneous power, inluding possile osilltions relted with hrmoni nd unlnes. Σ. () Σ Powerless urrents ( ): they do not ontriute for the energy onveyne nd n e ompensted instntneously without the neessity of energy storge elements.. (9) Vrition urrents ( ): responsile for the osilltion of the instntneous equivlent ondutne round its verge vlue, or lso, vritions of Σ round Σ.. (1) Thus, the orthogonl urrents deompositions proposed y FBD results tht:.(11) 3. THE PQ-THEORY The instntneous power theory proposed y Akgi et l. [1] is usully known s pq-theory. This theory is sed on the Clrke trnsformtion of voltges nd urrents in three-phse systems (,,) into (α, β,) orthogonl oordintes. The pq-theory desries the power properties of three-phse three-wire systems y mens of two min instntneous power omponents: the instntneous rel power p, nd the instntneous imginry power q. This theory ws originlly proposed s mthemtil tool, speilly direted for the ontrol of tive power filters nd this ws one of the most importnt motivtions for its dissemintion. It is worth mention tht Akgi et l. spred the onept of retive ompenstion without energy storge elements. Thus, using the Clrke s Trnsformtion, the phse voltges in the α, β nd oordintes hs the form of (1). Similrly, the instntneous urrents (, nd ) n e trnsformed to the α, β nd oordintes. Note tht in the se of three-phse three-wire systems, the mesured phse voltges re referred to virtul str point (s in the FBD) nd with four-wire systems, the voltges re referred to the return ondutor. 1. (1) Therefore, onsidering three or four-wire iruits, the uthors define three instntneous power omponents: the zerosequene power (just in se of four-wire system), the rel power nd the imginry power s:. (13) The sum of nd results in the trditionl instntneous power of three-phse systems:, (1) whih n e simplified in se of three-wire systems s:. (15) Then, the uthors introdued the onept of the instntneous imginry power, defined s:. (1) Using these two instntneous power, nd, the orthogonl urrents nd n e deomposed into instntneous tive ( nd ) nd retive ( nd ) urrents, s follows: ;, (17) ;, (1) where. Aordingly, the instntneous zero-phse sequene, tive nd retive urrents n e lulted in the originl oordintes, y mens of the inverse Clrke trnsformtion: 1, (19) nd. So, the instntneous three-phse urrents (, nd ) n e deomposed on the following omponents:. () Aording to [1,13], the powers of (13) ould lso e deomposed into:,, (1) where nd represent the verge nd osillting omponent of ; nd where nd represent the verge nd osillting omponent of. It ws ssumed tht the osillting prts of nd re relted to the ourrene of unlned nd/or distorted voltges nd urrents. Consequently, the instntneous tive urrent n lso e deomposed into verge () nd osillting () omponent. ; ()

3 . Resulting in the (,,) oordintes:. (3) Finlly, the instntneous phse urrents yield: (). However, neither the interprettion of these urrent omponents, nor their ssoition with speifi physil phenomen ws suggested y the uthors of the pq-theory.. THE CPT FRAMEWORK The third onsidered pproh (Conservtive Power Theory, CPT) ws reently proposed in [11] nd it is sed on the definition of instntneous omplex power under non-sinusoidl onditions nd it represents n extension of the usul omplex power, defined for sinusoidl onditions. Even though detiled disussion hs een direted to single-phse systems, this theory is lso esily extended to multiphse systems []. The uthors hd introdued the so-lled homo-vriles (integrl nd derivte) whih n e defined under periodi onditions nd re homogeneous to the urrent, voltge nd power terms. Sine homo-voltges nd homo-urrents stisfy the Kirhhoff s Lws, the orresponding homo-powers re onservtive in ny eletri network, wht llows introduing the onept of onservtion of the omplex power under nonsinusoidl onditions. In ddition, urrent deomposition ws proposed, on whih every term is relted to speifi physil phenomenon (power sorption, energy storge, voltge nd urrent distortion ). Assuming multidimensionl systems, the following definitions mke use of the sme symols pplied to the FBD method, it mens, old vriles to vetor representtion nd the index µ for eh m-phse vrile. Regrding to the voltge referentil, the uthors suggest using the return ondutor in se of its existene (in different wy from the FBD pproh) nd the virtul point unless []. Thus, the homo-integrls of the voltges nd urrents re defined s: () where:, integrl of voltges nd urrents, nd,, re time re the verge vlue of eh nd, over period T. Note tht nd re dimensionlly homogeneous to voltge nd urrent respetively. This mens tht the opertion of integrtion does not influene the mplitude of the resultnt signls, sine they re multiplied y the ngulr frequeny. Likewise, the homo-derivtives of the voltges nd urrents re given y: 1 ; 1 () As well s nd the vriles nd re lso dimensionlly homogeneous to the originl voltges nd urrents, respetively. In this se the derivtive is multiplied y the inverse of the ngulr frequeny. Considering periodil quntities, with period nd fundmentl ngulr frequeny, it is well-known tht the internl produt of voltge nd urrent vetors is defined s:,, 1. (7) nd in the sme wy, the voltge nd urrent norms re:,, (),, whih results equl to the RMS olletive vlues from the FBD method ( Σ, Σ ). For single-phse iruits or onsidering eh phse of n m-dimensionl system:,, yields the RMS phse voltges nd urrents. Under the ssumption of periodi ehvior, the following quntities hve een defined, whih re vlid for oth sinusoidl nd distorted, lned or unlned onditions: Ative power: tht represents the onveyed verge power. This definition is identil to the onventionl tive power (Steinmetz, Budenu, Fryze, Buhholz)., 1.. (9) Retive power: tht represents the verge energy stored in the network nd ws defined s:, 1.. (3) The physil mening of this nd other power terms re widely disussed in [11]. Following, the originl urrents re split into some prels, regrding to their ssoition to the power terms. The tive urrent is the minimum urrent (i.e., with minimum norm) onveying tive power P to the lod nd it is defined s:... (31) The retive urrent is the minimum urrent trnsferring retive power, nd it is relted to the verge energy eing exhnged through the iruit:. (3) Both the tive nd retive urrents hve n expliit physil mening. They re ssoited with the presene of the tive nd retive powers, nd, nd re relted to the lod verge equivlent ondutne, nd suseptne,. The void urrent is the remining urrent (residul term), sine it does not onvey tive nor retive power:. (33) Aording to the uthors [11], the void urrents my exist only in presene of urrent distortion, however it will e demonstrted in the following simultions tht it is lso influened y urrent unlnes. Further detils of eh urrent omponent nd its physil mening n e found in [11]. By definition, ll urrent terms re orthogonl:. (3) 5. COMPUTATIONAL SIMULATION RESULTS: COMPARISON AND DISCUSSION In view of demonstrting the min similrities nd differenes mong the proposls, three different onditions were simulted nd nlyzed my mens of the PSIM softwre. In order to mke esier the omprisons, the following under sripts were pplied: FBD, pq, CPT.

4 A. Cse I: Unlned resistive lod smll line impedne Figure 1 shows the power iruit for Cse I, while the Tle 1 presents the vlues of grid voltges, line impedne nd lod phse resistnes. Figure shows the mesured PCC voltges (v) nd urrents (i). Tle 1 Voltges nd impednes for Cse I. Soure Line Lod (Y) V = 17 º Vrms R L = 1mΩ L L= 1 μh R = 9,35Ω V = 17-1º Vrms R L = 1mΩ L L =1μH R =,7Ω V = 17 1º Vrms R L = 1mΩ L L =1μH R =3,1135Ω Figure 1: Power iruit for Cse I unlned resistive lod. v [V] i Time [s] Figure : PCC voltges nd urrents for Cse I. Note they re not extly in-phse, sine the voltges re mesured to virtul point, whih in prtie represents tht the voltges re referred to the power soure entrl point, insted of the lod entrl point. Figure 3 shows the urrent deomposition y mens of the FBD method (), the pq-theory () nd CPT (). In this se, the tive urrents sed on the FBD nd CPT re extly the sme, sine they re sed on the Fryze s definition. Besides, they mth with the verge prt of the Akgi s tive urrent ( ) nd re in-phse with the respetive voltges, sine equtions (), (31) nd the left side of () re equivlents under sinusoidl nd lned voltges ( ). However, some onfusing results n e pointed out, e.g.: the omponents nd results distorted, even without ny voltge distortion or non liner lod, wht ould indite tht they re not good representtion of the power phenomen in suh ondition. In ddition, the osillting prt of the Akgi s tive urrent ( ) is lso distorted, wht mens tht the totl tive urrent is not sinusoidl nd, even onsidering liner resistive lod, with no pitors or indutors, the deomposetion indites the existene of retive urrent (Fig. 3-). Finlly, Fig. 3- shows the urrent deomposition sed on the CPT. Note tht the resulting retive urrents re equl to zero ( ), inditing the sene of energy storge elements, nd the void urrents re sinusoidl ( ) nd represents the resistive lod unlnes. Moreover, onsidering the distorted urrent omponents, the omprisons of their spetr nd wveform (Fig. ) llow identifying the following reltions: Czrneki [,1] hs demonstrted tht the totl tive nd retive urrents from the pq-theory present third hrmoni ontent, originted y the deomposition itself. From Figs. 3 nd, one n oserve tht the hrmoni ontent omes from the osillting tive urrent ( ) nd the retive urrent ( ). On the other hnd, it hs never een disussed in literture the equivlene with the FBD method, under suh onditions. Fig. shows tht either the powerless urrent or the vrition urrent lso presents the sme third hrmoni ontent. In terms of physil phenomen interprettion, suh results suggest tht the Tenti s pproh ould e more pproprite nd onsistent, if onsidering the trditionl definitions for sinusoidl systems. The lod unlne is represented y mens of the sinusoidl void urrent omponents. Note tht the phse sequene of the deomposed urrents is negtive if ompred to the tive urrent. Aording to [11], suh urrent ( ) does not ontriute either to the tive or to the retive power phenomen. Thus, if ( ), it is proved tht the hrmoni omponents re introdued y the FBD nd pq deompositions nd even though the differenes in nmes or in the deomposition proess, their results re very similr in this first se. i i i z Tempo[s] ) FBD-deomposition ) pq-deomposition i i 5 i i p- i p~ Tempo[s] Tempo[s] ) CPT-deomposition Figure 3: Current deompositions for Cse I.

5 i z F [Hz] F [Hz] 3 3 ) FBD-deomposition ) pq-deomposition Figure : Spetr of some FBD nd pq urrent omponents - Cse I. B. Cse II Two non liner nd one liner lod (smll line impedne) Considering the sme power soure nd line impednes, Figure 5 shows the iruit for Cse II, while Tle presents the different lod impednes. Over gin, the results from FBD nd pq-theory re equivlent: Figure 5: Power iruit for Cse II unlned non liner lod. Tle Lod impednes for Cse II. RL Retifier RC Retifier RL L CC = mh R CC = 5Ω C CC = mω R CC = Ω L = mh R = 5Ω In this se, the differenes mong the CPT nd the other two methods re still more signifint, sine its deomposition identifies the tive urrent (, relted to the tive power onsumption (resistors or equivlent verge ondutne), the retive urrent ( ), relted to the pitors nd indutors present in the power iruit (equivlent verge suseptne) nd the void urrent, relted to the lod unlnes nd nonlinerities ( ). In this se, ( ) nd ( ) re lned nd prtilly sinusoidl, sine the line impednes re very smll. Further, the following similrities n e drwn:. C. Cse III - Two non liner nd one liner lod (high line impedne) Assuming the sme iruit of Fig. 5, ut hnging the line i p ~ F [Hz] F [Hz] 3 3 impedne to R L = 1mΩ nd L L = mh, whih represents PCC with worse short-iruit level, Fig. indites the PCC voltges nd urrents. Note tht, the voltges re distorted nd unlned, euse of the lrger line impedne. This is ritil se. From Fig. 7, their spetr nd using mthemtil mnipultion, it is possile to onlude the following reltions: ;. Oserve tht the verge prt of the Akgi s tive urrent ( ) does not gree with FBD nd CPT s tive urrent. It hppens sine in se of nonsinusoidl or unlned voltges, the denomintor of () nd (31) re equl, ut they do not orrespond to ( + ) in (). It mens tht even if the numertors from the three equtions re equl, the rtios re not. i i i z v [V] i Time [s] Figure : PCC voltges nd urrents for Cse III Time[s] ) FBD-deomposition ) pq-deomposition i i i i p- i p~ ) CPT-deomposition Figure 7: Current deompositions for se III. -5 Time[s] -5 Time[s]

6 Indeed, the rtio in () nd (31) re onstnt over fundmentl period (verge equivlent ondutne), while the rtio in () is not (nd does not represent ny physil phenomen). Another interesting oservtion is tht the Tenti s retive urrents re slightly distorted (see Fig. 7). Sine they represent the portion of urrent reltive to energy storge elements nd their lultion is sed on n integrl (3) funtion, it is expeted some ttenution in the hrmoni ontent if ompred to the voltges or to the tive urrents. These lst two oservtions my e vlid even for Cse II, however in tht onditions, the voltges were prtilly sinusoidl nd lned (smll line impedne).. CONCLUSIONS Notwithstnding the sis of the FBD method seems to e more suitle to the understnding of different power phenomen [3], if ompred to the pq-theory, espeilly euse it does not uses ny xis trnsformtion nd sine it is sed on the extension of well-epted Fryze s definitions, their results re very similr in the se of three-phse three-wire iruits. In [,1], the uthor s nlysis were direted to the instntneous tive nd retive urrent omponents from pq- Theory. In this pper, ses I nd II show tht under some onditions, the interprettion of either the FBD nd pq-theory n led one to erroneous onlusions, e.g., onsidering third hrmoni ontent of liner resistive lod iruit. Cse III shows tht under signifint voltge deteriortion (or high PCC impednes), even the verge tive urrent from pqdeomposition is useless for the interprettion of the iruit. Besides, it hs een demonstrted tht the tive urrent from FBD nd CPT methods mth for three-wire iruits. Considering tht suh urrent deompositions nd relted power omponents my onern to severl power system pplitions, suh s regultion, revenue metering, power onditioning, power qulity monitoring nd so on, the results suggest tht the proposl from Tenti et l. seems to e n interesting lterntive to the nlysis of distint modern power iruits. 7. REFERENCES [1] C. I. Budenu, Puissnes Retives et Fitives Instytut Romin de l'energie, Pu. No., Buhrest, Rumni. 197 (in Romnin). [] N. L. Kuster nd W. J. More, On the Definition of Retive Power under Non-Sinusoidl Condition, IEEE Trns. on Power Apprtus nd Systems, PAS-99, pp.15-15, 19. [3] W. Shepherd, P. Zkikhni Suggested Definition of Retive Power for Non-sinusoidl Systems. Pro. Inst. Ele. Eng., vol. 119, pp , Sept [] IEEE Tril-Use Stndrd Definitions for the Mesurement of Eletri Power Quntities under Sinusoidl, Non-sinusoidl, Blned or Unlned Conditions, IEEE Stndrd 159- ISBN , June,. Full-Use August. [5] L.S. Czrneki, Currents Physil Components (CPC) in Ciruits with Non-sinusoidl Voltges nd Currents. Prt 1: Single-phse liner iruits, Eletril Power Qulity nd Utiliztion Journl, Vol. XI, No.,, pp [] L.S. Czrneki, Currents Physil Components (CPC) in Ciruits with Non-sinusoidl Voltges nd Currents. Prt : Three-phse three-wire liner iruits, Eletril Power Qulity nd Utiliztion Journl, Vol. XII, No. 1,, pp [7] S. Fryze, Wirk-, Blind- und Sheinleistung in Elektrishen Stromkreisen mit Nihtsinusförmigem Verluf von Strom und Spnnung, Elektrotehnishe Zeitshrift, vol. 53, No., 1933, pp , -7, 7-7. [] F. Buhholz, Ds Begriffsystem Rehtleistung, Wirkleistung, totle Blindleistung, Selstverlg Münhen, 195. [9] M. Depenrok, The FBD-Method, Generlly Applile Tool For Anlyzing Power Reltions, IEEE Trns. on Power Systems, vol., n., 31-37, My [1] H. Akgi, Y. Knzw, A. Ne, Generlized Theory of the Instntneous Retive Power in Three-Phse Ciruits, in Pro. 193 JIEE-IPEC Conf. Power Eletronis. Conf., pp [11] P. Tenti nd P. Mttvelli, A Time-Domin Approh to Power Term Definitions under Non-sinusoidl Conditions, Sixth Interntionl Workshop on Power Definitions nd Mesurements under Non-Sinusoidl Conditions, Miln-Itly, 3. [1] F. Peng, G. Ott, nd D. Adms, Hrmoni nd Retive Power Compenstion Bsed on the Generlized Instntneous Retive Power Theory for 3-phse -wire Systems, in Pro IEEE Power Eletronis Speilists Conferene PESC, vol., pp [13] J. L. Willems, Instntneous Sinusoidl nd Hrmoni Ative nd Detive Currents in Three-phse Power Systems, ETEP Europen Trns. on Eletril Power Engineering, Vol., No. 5, pp [1] M. K. Mishr, A. Joshi, nd A. Ghosh, Unified Shunt Compenstor Algorithm Bsed on Generlized Instntneous Retive Power Theory, IEEE Trns. Gen., Trns., Distri., vol. 1, no., pp , Nov. 1. [15] H. Kim, F. Bljerg, B. Bk-Jensen, nd J. Choi, Instntneous Power Compenstion in Three-phse Systems y Using p-q-r Theory, IEEE Trnstion on Power Eletronis, Vol.17, pp ,. [1] E. H. Wtne, R. M. Stephn, nd M. Aredes, New Conept of Instntneous Ative nd Retive Powers in Eletril Systems with Generi Lods. IEEE Trns. Power Delivery Vol., pp , [17] H. Kim, F. Bljerg, nd B. Bk-Jensen, Spetrl Anlysis of Instntneous Powers in Single-phse nd Three-phse Systems with use of p-q-r Theory. IEEE Trnstion on Power Eletronis Vol. 17, No. 5, pp ,. [1] J. L.Willems, A New Interprettion of the Akgi-Ne Power Components for Nonsinusoidl Three-phse Situtions, IEEE Trns. on Instrumenttion nd Mesurement, vol. 1, no., pp , Aug [19] M. Depenrok, V. Studt nd H. Wrede, Theoretil Investigtion of Originl nd Modified Instntneous Power Theory Applied to Four-Wire Systems. IEEE Trnstion on Industril Applition, Vol. 39, No., pp , 3. [] L. S. Czrneki, On Some Misinterprettions of the Instntneous Retive Power p-q Theory, IEEE Trns. on Power Eletronis Vol. 19, No. 3, pp. 3,. [1] L. S. Czrneki, Instntneous Retive Power p-q Theory nd Power Properties of Three-phse Systems, IEEE Trns. Power Delivery, vol. 1, no. 1, pp. 3 37, Jn.. [] E. Tedeshi nd P. Tenti, Coopertive Design nd Control of Distriuted Hrmoni nd Retive Compenstors, in Pro. Interntionl Shool on Nonsinusoidl Currents nd Compenstion Proeeding of the IX Conferene ISNCC, Lgów Poloni. [3] M. Depenrok, D. A. Mrshl, nd J. D. vn Wyk, Formulting Requirements for Universlly Applile Power Theory s Control Algorithm in Power Compenstors, Seond Interntionl Workshop on Power Definitions nd Mesurements under Non- Sinusoidl Conditions, Stres-Itly, [] W. G. Morsi, nd M. E. El-Hwry, Defining Power Components in Nonsinusoidl Unlned Polyphse Systems: The Issues, IEEE Trns. on Power Delivery, vol., No., pp. -3, Otoer 7. [] R. S. Herrer nd P. Slmerón, Instntneous Retive Power Theory: A Comprtive Evlution of Different Formultions, IEEE Trns. on Power Delivery, vol., No. 1, pp. 595-, Jnury 7. [] M. Mlengret nd C. T. Gunt, Deomposition of Currents in Three- nd Four-Wire Systems, IEEE Trns. on Instrumenttion nd Mesurement, vol. 57, No. 5, pp , My.