Critical Evaluation of FBD, PQ and Generalized Non-Active Power Theories

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1 Crcal Ealuaon of FBD, PQ and Generalzed Non-Ace Power Theores Keywords Fan Xu 1, Leon M. Tolber 1,, Yan Xu 1 The Unersy of Tennessee, Knoxlle, TN , USA Oak Rdge Naonal Laboraory, Oak Rdge, TN 37831, USA E-Mal: fxu6@uk.edu FBD heory, PQ heory, Generalzed non-ace ower heory, Curren decomoson, Non-ace ower comensaon. Absrac Due o he wdesread use of non-snusodal, non-erodc loads and he exsence of dsored olages, many defnons of non-ace ower for non-snusodal and non-erodc waeforms hae been formulaed. Ths aer nesgaes he major smlares and dscreances of hree non-ace ower heores whch are wdely used n conrol algorhms for shun comensaon sysems. The hree aroaches are FBD, PQ and generalzed non-ace ower heores. The ealuaon and comarson of hese hree heores as he non-ace ower comensaon sraeges for sngle-hase sysem, hreehase sysem wh unbalanced load, sub-harmonc load, and dsored sysem olage wll be ncluded n hs aer. The conclusons are based on boh he smulaon and he exermenal resuls of dfferen comensaon objeces. I. Inroducon One of he man ons n he deelomen of alernang curren (ac) ransmsson and dsrbuon ower sysems a he end of he 19h cenury was based on snusodal olage a consan-frequency generaon. Hence, he conenonal ower heory, based on ace, non-ace and aaren ower defnons, was suffcen for desgn and analyss of ower sysems. Howeer, wh he wdesread use of nonlnear loads and elecronc ower conerers, non-snusodal and non-erodc loads are becomng more common n oday s elecrcal sysems. These requre he mroemen and adaaon of non-ace comensaor echnology and meerng echnques, whch are based on he new nonace ower defnons and curren decomoson mehods, under such condons. Two moran aroaches o ower defnons under non-snusodal condons were frs formulaed by Budeanu, n 197, and Fryze, n 193. Fryze defned ower n me doman, whereas Budeanu dd n frequency doman [1]. Oher non-ace ower heores, whch sasfy he major requremen of non-ace comensaor echnology and meerng echnques under such condons, were dered from hem n a few decades. In general, ower defnons n he me doman offer a more robus bass for he use n a conroller for ower elecronc deces, because hey are also ald durng ransens []. Deenbrock exended Fryze s defnon of non-ace ower and curren decomoson mehod for a sngle hase o a mulhase sysem, known as FBD heory [3]. In 1983, Akag, Kanazawa and Nabae nroduced a noel conce, he PQ heory, by alyng Park ransformaon, for he conrol of ace flers conneced o hree-hase hree-wre sysems ncludng some noes for four-wre sysems [4-6]. Boh FBD heory and PQ heory are wdely used n he comensaon echnologes nowadays. More recenly, a generalzed non-ace ower defnon was roosed [7-9]. Seeral aers hae deal wh he comarson of dfferen heores. Howeer, some of hem focus on heorecal analyss [1-1]; some do no nclude he generalzed non-ace ower defnon [13-15]. Ths aer nesgaes he major smlares and dscreances of FBD, PQ, and generalzed non-ace ower heores used for non-ace ower comensaon n dfferen cases, by smulaon and exermenal resuls. Ths aer sars wh he bref nroducon of FBD, PQ, and generalzed non-ace ower heores n Secon II. The analyss and comarson of hese heores n a sngle-hase sysem, a hree-hase sysem wh unbalanced load, sub-harmonc load, and dsored sysem olage, based on smulaon

2 resuls, are gen n Secon III. The exermenal resuls are roded n Secon IV. The fnal conclusons are dscussed n he las secon. II. Non-ace ower heores A. FBD heory Deenbrock exended Fryze s defnon of non-ace ower and curren decomoson mehod for a sngle hase o a mulhase sysem wh m hases. He ncororaed some of he aaren ower heory ublshed by Buchholz [16]. Muldmensonal olage and curren ecors are,. Buchholz defned he nsananeous collece alue of olages ( Σ ) and currens ( Σ ) n a sysem wh m hases as Σ 1, m = m μ μ = 1 =, 1 m = Σ m μ μ= 1 (1) = () Deenbrock defned collece nsananeous ower Σ and nsananeous collece ower curren Σ as m m Σ μ μ μ, Σ μ= 1 μ= 1 Σ = = Σ = (3) He hen defned nsananeous hase ower curren μ and hase zero ower curren μ z as Σ μ = μ, μz = μ μ ( μ = 1,,, m) (4) Σ Deenbrock also decomosed he ower curren μ no ace curren μ a and araon curren μ, whch deend on he mean alues of Σ and Σ oer one sysem olage erod, Σ and Σ. Σ μa = μ, μ = μ μa (5) Σ Then he non-ace curren μ n could be wren as μn = μ μa = μz + μ (6) So he hase curren μ could be decomosed as = + = + + = + (7) B. PQ heory μ μ μz μa μ μz μa μn The PQ heory s based on he Park ransformaon of olages and currens n hree-hase sysems (a, b, c) no ( α, β,) orhogonal coordnaes. The hase olages n he (a, b, c) coordnaes hae he form a a α = C 1 b, C 1 α = b (8) β c β c Σ

3 1 1 1 where 1 1. C1 = Consderng four-wre crcus, hree nsananeous ower comonens are defned as = α β (9) α 3 q β α β In hree-hase hree-wre sysems, here are no zero-sequence comonens. In hree-hase four-wre sysems, he sum of and resuls n he radonal nsananeous ower of hree-hase sysems. The nsananeous currens can be calculaed by means of nerse Park ransformaon. a aq a b = C α, bq = C αq, b = C (1) c β cq β q c 1 1 where C = C 1 = The orhogonal curren can be decomosed no nsananeous ace ( α, β ) and reace ( q, ) α β q currens as α β α β α =, β =, αq = q, β q = q (1) α + β α + β α + β α + β The owers could be decomosed no wo comonens, he aerage ( x ) and oscllang ( x ) comonen as = +, q = q + q (11) The nsananeous ace curren can also be decomosed no aerage and oscllang comonens as α α α = _ + ~ = + α α + + β α β α β = + = + (13) β β _ ~ β β α + β α + β Fnally, he nsananeous hase curren could be decomosed as _ ~ a a aq a a a aq a b b bq b _ ~ bq = + + = b b b c c cq c cq c _ ~ c c C. Generalzed nsananeous non-ace ower heory In generalzed nsananeous non-ace ower heory, whch s also exended from Fryze s heory, curren s decomosed no ace curren and non-ace curren q. (14)

4 The ace curren s defned as 1 T where V () = ( τ) ( τ) T, C TC () = () + () P () () = () V () L q (15) (16) 1 T PL ( ) = ( τ) ( τ) T (17) C TC The () s he reference olage ecor, and deendng on he comensaon objeces, can be chosen as sysem olage, or as he fundamenal comonen of, or as somehng else. T C s he aeragng neral. T C s chosen o aerage he ace ower of he load and does no cause any me delay n he sysem resonse. P L () s he aerage load ace ower oer he neral [-T C, ], and V P () s he rms alue of he reference olage () oer he neral [-T C, ]. I s shown n [8-9], ha hs defnon s flexble o mee many dfferen comensaon objeces; s ald for non-snusodal and non-erodc sysems; s ald for sngle-hase and mulhase sysems. More deals abou he choce of () and T C wll be gen n secon III. III. Smulaon resuls In hs secon, he comensaon resuls based on FBD, PQ and generalzed non-ace ower heores for dfferen cases wll be shown and comared. For a sngle or a mulhase sysem, a shun comensaor can be confgured as n Fg. 1. s s he source curren, L s he load curren, and C s he curren ha he comensaor rodes. The smulaons are accomlshed under deal condons: he comensaor has no losses. A. Sngle-hase sysem wh ulse load Fg. 1. Shun comensaor confguraon. Accordng o he defnons, boh FBD and generalzed non-ace ower heores can be used n mulhase sysems, ncludng a sngle hase sysem. The orgnal PQ heory, roosed by Akag e al., s based on he Park ransformaon, whch ransfer curren and olage n hree-hase sysems (a, b, c) no ( α, β,) orhogonal coordnaes. So canno be used n sngle-hase sysems. Alhough a mehod dered from PQ heory can be used n a sngle-hase sysem [17], he smulaon resuls n hs subsecon are only based on FBD and generalzed non-ace ower heores. Some loads draw a large magnude curren for a shor erod, from half of a cycle o a few cycles, whch may cause olage sag n a weak ower sysem. A ulse load curren L s reresened by a rangle waeform n hs subsecon, as Fg. a shows. Fg. b shows he source curren s comensaed by FBD mehod. I s clear ha he non-ace curren of he ulse load s no comensaed comleely. Tha s because n (5), he ace curren μ a s defned based on he mean alues of Σ and Σ, whch are oer one sysem olage erod T. The ulse load, howeer, s a non-erodc load whose erod can be hough as nfne. As a resul, he FBD mehod can be aled o he mulhase, seady sae, erodcal sysem for comensaon uroses, bu s no suable o comensae he non-erodcal load currens.

5 From (16) and (17), n a non-erodc sysem, he nsananeous curren ares wh dfferen aeragng neral T C, whch s dfferen from he erodc cases. Snce he erod of he non-erodcal load can be hough as nfne, choosng TC n generalzed non-ace ower heory, (17) wll be 1 T V() = lm ( τ) ( τ) V 1 T =, PL () = lm ( τ)( τ) PL = (18) and he ace curren n (16) wll be PL() P L () = () () (19) V() V So he generalzed non-ace ower heory can only comensae non-erodcal load curren comleely when choosng aerage neral TC. Howeer, s no raccal o choose TC n a ower sysem, because T C mus be a fne alue. Snce a non-erodc curren s usually a sum of he fundamenal comonen and non-erodc comonen, T C s usually chosen as a few mulles of he fundamenal erod. Fg. c and Fg. d show s afer comensaon by generalzed heory, choosng T C = T and T C = 1T, resecely. I s clear ha larger T C wll make more comonens of non-ace curren be comensaed. Less non-ace curren exsng n s wll reduce s eak alue and mgae s dsoron. In addon, Fg. b and Fg. c are he same, whch demonsraes ha he FBD mehod can be deduced from he generalzed heory wh T C = T n hs case. 1 (V) Vs L (s) (a) s and L. (b) s (FBD) (s). (c) s (generalzed, T C = T). (d) s (generalzed, T C = 1T) Fg.. Sngle-hase sysems wh ulse load. B. Three-hase sysem wh unbalanced RL load (s) (s). In hs case, he sysem olage s s ure sne wae and balanced. The unbalanced load currens hae boh unequal amludes and unbalanced hase-angles. For he erodc sysem, choosng he reference olage n generalzed non-ace ower heory as () = s() = V [cos( ω+ α),cos( ω π /3 + α),cos( ω+ π /3 + α)] T () unbalanced load curren s ( ) = [ I1cos( ω+ β1), Icos( ω π / 3 + β), I3cos( ω+ π / 3 + β3)] T (1) and choosng he aeragng neral TC = kt /, where T s he sysem lne erod, T = π / ω, hen (17) wll be kt T V() = Vs() = s( τ) s( τ) = V kt kt T PL( ) = L( τ) L( τ) PL kt = () Equaon () shows V () and P L () wll be consan alues n erodc sysems when choosng T C = kt/. So he non-ace curren of load q () can be easly comensaed comleely by he

6 comensaor. Snce ace curren μ a from FBD mehod deends on he mean alues of ower and olage oer T, wll be he same wh k= n (). For comlee comensaon, PQ heory requres he dc comonen of he nsananeous alues, whch requres a leas half of he fundamenal cycle o deermne. Therefore, has he same resuls as generalzed heory when T C = T/, as () shows. Fg. 3a shows he unbalanced load curren. Fg. 3b o Fg. 3d show he comensaon resuls by he hree heores n hree-hase hree-wre sysems. Balanced sysem curren s can be acheed by all hese heores. In hree-hase four-wre sysems, unbalanced load curren wll cause large neural curren, as Fg. 3e shows. Fg. 3f o Fg. 3h show afer comensaon by hese hree heores. s zero afer he comensaon whch also demonsraes he comlee comensaon for an unbalanced load (s).15 (a) L (s).15 (b) s (FBD) (s).15 5 (c) s (PQ) (s).15 5 (d) s (generalzed) (s).15 5 (e), before comensaon (s) (s) (s).15 (f) (FBD) (g) (PQ) (h) (generalzed) Fg. 3. Three-hase sysems wh unbalanced load curren. C. Three-hase load wh sub-harmoncs Sub-harmoncs are he comonens n a waeform whose frequences are no an negral mulle of he fundamenal frequency. Usng a sngle-hase sysem o make a smle analyss, he source olage s () s s() = V1cos( ω1+ α) (3) and load curren s L( ) = I1cos( ω1+ β1) + Iscos( ωs+ βs) (4) where ω1 = π f1 s he fundamenal frequency of sysem, and ωs = π fs s he sub-harmonc frequency. The nsananeous ower L () s L() = s() L() = VI 1 1cos( α1 β1) + VI 1 1cos( ω1+ α1+ β1) (5) + VI 1 s cos[( ω1 ωs) + α1 βs] + VI 1 scos[( ω1+ ωs) + α1+ βs) The nsananeous ower conans frequences of f 1, f 1 -f s, f 1 +f s, due o he mullcaon beween he fundamenal and he sub-harmoncs. In generalzed non-ace ower heory, f choosng he reference olage () = s (), hen he rms alue V () = V 1 s consan. If T C s an negral mulle of he erods of all he frequences n L (), he aerage alue P L () wll be consan oo. From (16), he ace curren () wll be ure fundamenal sne wae and n hase wh s (). For FBD mehod, s he same wh choosng T C = T n generalzed heory. So he ace curren obaned from sll conans sub-harmonc comonens. Snce he aerage alues are sll used n PQ heory (, q, and q ), and daa from a leas half of a

7 cycle of a fundamenal erod are used o calculae he aerages alues, he comlee comensaon canno be acheed for sub-harmonc currens whose erods are no negral mulle of half of he fundamenal erod. In he smulaon of a hree-hase sub-harmonc sysem, he sysem olage s s 6 Hz ure sne wae, and he load curren L has 8 Hz sub-harmonc comonen, as Fg. 4a shows. Usng FBD and PQ mehods, he non-ace curren canno be comensaed comleely, as shown n Fg. 4b and Fg. 4c. Accordng o he generalzed heory, f he aeragng neral T C = T, he comensaon resul s he same wh FBD s, as Fg. 4d shows. If choosng T C = 3T, whch s he negral mulle of he erods of all he frequences n L (), comlee comensaon can be acheed, as shown n Fg. 4e (s). (a) L (s). (b) s (FBD) (s). (c) s (PQ) (s) (s). (d) s (generalzed, T C = T) (e) s (generalzed, T C = 3T) Fg. 4. Three-hase load wh sub-harmoncs. D. Dsored sysem olage The dsored sysem olage ncludes he sysem olage wh harmoncs, he unbalanced sysem olage, and asymmercal olage sources. In he smulaon, he sysem olage s conans harmoncs, as Fg. 5a shows, and he load curren L also conans large 3rd and 5h harmoncs, shown n Fg. 5b. Accordng o (5), he ace hase curren μ a obaned from FBD mehod wll conan he same harmonc comonens as he hase olage μ. Fg. 5c shows he source curren s comensaed by FBD mehod, and he shae of s he same wh s. From (8) - (13), he ace ower curren calculaed by PQ heory does no elmnae he harmoncs n s, so PQ heory does no allow comlee comensaon n hs case eher, as Fg. 5d shows. For generalzed heory, f choosng reference olage () = s (), from (16) and (17), he harmonc comonens n s wll sll be n he ace curren. Fg. 5e shows he resul. Howeer, f choosng he reference olage as he fundamenal comonen of source olage, () = f (), he ace curren () wll be ure sne waeform and no harmoncs exs, as Fg. 5f shows. 1 (V) (s).15 (a) s (s).15 5 (b) L (s).15 5 (c) s (FBD) (s) (s) (s).15 (d) s (PQ) (e) s (generalzed, = s ) (f) s (generalzed, = f ) Fg. 5. Harmonc sysem olage wh harmonc load curren.

8 Fg. 6a shows he unbalanced source olage. In he smulaon, he load s a balanced RL load, bu he load curren s sll unbalanced due o he unbalanced source olage, shown n Fg. 6b. The unbalanced olages of hree hases can be decomosed no hree sequences: a ose sequence + (), a negae sequence - (), and a zero sequence (). For FBD mehod, (5) uses hase olage whch conans all of hese hree sequences o calculae ace curren, and hs leads o curren unbalance afer comensaon. Fg. 6c shows he resul. The assumon of PQ heory s ha he olage s a ure sne wae and he hree hases are balanced. For balanced, symmercal olage sources, he non-ace ower n (9) s equal o he hree-hase non-ace ower, bu for dsored olage sources, s no equal o he hree-hase non-ace ower. Zero sequence comonens n PQ heory are comleely arbued o ace ower whou consderng s conrbuon o non-ace ower. PQ heory wll nroduce errors f s aled o a non-ace ower comensaon sysem for dsored source olage sysems. Fg. 6d shows he source curren comensaed by PQ heory n an unbalanced source olage sysem. I s clear ha he source curren afer comensaon s dsored serously. For generalzed non-ace ower heory, f he reference olage () equals o he unbalanced source olage s (), he ace curren calculaed by (16) wll sll be unbalanced, as Fg. 6e shows. So n hs case, when generalzed heory s used, choosng () = s + (), he source curren wll no conan negae and zero sequences and wll be balanced, as shown n Fg. 6f. The aeragng neral s chosen T C = T n he smulaon. (V) (s).15 5 (a) s (s).15 (b) L (s).15 1 (c) s (FBD) (s).15 (d) s (PQ) (s).15 (e) s (generalzed, = s ) Fg. 6. Unbalanced sysem olage (s).15 (f) s (generalzed, = s + ) IV. Exermenal resuls Fg. 7. Exermenal confguraon.

9 The exermens are erformed n a hree-hase sysem wh he olage rang 8 V (lne-o-lne rms alue). An IGBT based hree-hase nerer was used as he nerer for he ace fler (DC lnk olage s 45V). Fg. 7 shows he confguraon. The generalzed non-ace ower heory s used as he comensaon algorhm. Sysem olage s s snusodal and balanced, as Fg. 8a shows. Fg. 8b shows he unbalanced RL load curren L wh he hase a olage. The source curren s afer comensaon s shown n Fg. 8c. s s balanced and n hase wh he olage. When an RL load s conneced beween hase a and hase b, he hree-hase load currens L are shown n Fg. 8d. Fg. 8e shows s afer comensaon. The magnudes of s of hase a and hase b are reduced, and here s a curren n hase c. When he load s a dode recfer, here are harmoncs n L, as shown n Fg. 8f. Afer comensaon he source curren s s nearly snusodal, as Fg. 8g shows. The skes n s are due o he hgh d/d n he load curren. (a) s (b) L (unbalanced RL load) (c) s (unbalanced RL load) (d) L (sngle-hase load) (e) s (sngle-hase load) (f) L (dode recfer load) (g) s (dode recfer load) Fg. 8. Exermenal resuls of hree dfferen yes of loads comensaon by generalzed heory. V. Conclusons Ths aer comares he comensaon resuls based on FBD, PQ and generalzed non-ace ower heores for dfferen cases. Table I summarzes he alcably of hese hree heores from he smulaon and exermenal resuls resened aboe. The able shows f each heory s alcable o sngle-, hree-, or mulle-hase sysems, and wheher he heory can comensae non-ace ower comleely or wheher can arally comensae non-ace ower. The generalzed non-ace ower heory allows comlee comensaon n many cases by secfyng suable and T C. Praccally, he generalzed mehod canno comensae non-erodcal sysem comleely, because T C canno be nfne. Howeer, by choosng T C large enough, he comensaon resuls wll be mroed sgnfcanly. The FBD heory can be deduced from generalzed heory, by choosng T C = T and = s. PQ heory can achee comlee comensaon n hree-hase sysem wh ure sne wae source olage. Bu canno be used n sngle-hase sysems, and wll nroduce errors n dsored sysem olage cases, because s based on he Park ransformaon and assumes he source olage s ure sne waeform.

10 Table I. Comarson of hree heores comensaon resuls Phase number Load curren Source Sngle Three Mulle Nonerodharmonc Unbalance Sub- olage Theory (3&4-wre) dsoron FBD Yes Yes Yes Parally Comleely Parally Parally PQ No Yes No Parally Comleely Parally Parally Generalzed Yes Yes Yes Comleely Comleely Comleely Comleely References [1] L. M. Tolber and T. G. Habeler, Comarson of me-based non-ace ower defnons for ace flerng, IEEE Inernaonal Power Elecroncs Congress, Acaulco, Mexco, Ocober 15-19,, [] L. F. C. Monero, J. L. Afonso, J. G. Pno, E. H. Waanabe, M. Aredes, and H. Akag, Comensaon algorhms based n he -q and CPC heores for swchng comensaors n mcro-grds, Brazlan Power Elecroncs Conf., COBEP 9, Ro de Janero, Brazl, Seember 7-Ocober 1, 9, [3] M. Deenbrock, The FBD mehod, a generally alcable ool for analyzng ower relaons, IEEE Trans. Power Sysems, ol. 8, May 1993, [4] H. Akag, Y. Kanazawa, and A. Nabae, Generalzed heory of he nsananeous reace ower n hreehase crcus, IPEC-Tokyo 83 In. Conf. Power Elecroncs, Tokyo, Jaan, 1983, [5] H. Akag, Y. Kanazawa, and A. Nabae, Insananeous reace ower comensaors comrsng swchng deces whou energy sorage comonens, IEEE Trans. Indusry Alcaons, ol. IA-, no. 3, May 1984, [6] H. Akag and A. Nabae, The -q heory n hree-hase sysems under non-snusodal condons, Euroe Trans. on Elecrcal Power Engneerng, ol. 3, no. 1, January 1993, [7] F. Z. Peng and L. M. Tolber, Comensaon of non-ace curren n ower sysems - defnons from comensaon sandon, IEEE Power Engneerng Socey Summer Meeng, Seale, Washngon, USA, July 15-,, [8] F. Z. Peng and L. M. Tolber, "Defnons and comensaon of non-ace curren n ower sysems," IEEE Power Elecroncs Secalss Conference, Carns, Ausrala, June 3-7,, [9] Y. Xu, L. M. Tolber, F. Z. Peng, J. N. Chasson, and J. Chen, Comensaon-based non-ace ower defnon, IEEE Power Elecroncs Leers, ol. 1, no., June 3, [1] Y. Xu, L. M. Tolber, J.N. Chasson, J. B. Cambell, and F. Z. Peng, A generalzed nsananeous nonace ower heory for STATCOM, IET Elecrc Power Alcaons, ol. 54, no. 6, Noember 7, [11] J. L. Wllems, Crcal analyss of dfferen curren decomoson and comensaon schemes, Inernaonal School on Nonsnusodal Curren and Comensaon, ISNCC 8, June , Lagow, Poland, [1] M. Malengre and C. T. Gaun, Decomoson of curren n hree- and four-wre sysems, IEEE Trans. Insrumenaon and Measuremen, ol. 57, no. 5, May 8, [13] R. S. Herrera and P. Salmeron, Insananeous reace ower heory: a comarae ealuaon of dfferen formulaons, IEEE Trans. Power Delery, ol., no. 1, January 7, [14] H. K. M. Paredes, F. P. Marafao, and L. C. P. da Sla, A comarae analyss of FBD, PQ and CPT curren decomosons Par I: Three-hase hree-wre sysems, IEEE PowerTech Conference, June 8 July, 9, Buchares, Romana, [15] F. P. Marafao, H. K. M. Paredes, and L. C. P. da Sla, Crcal ealuaon of FBD, PQ and CPT curren decomosons for four-wre crcus, Brazlan Power Elecroncs Conf., COBEP 9, Ro de Janero, Brazl, Seember 7-Ocober 1, 9, [16] F. Buchholz, Das begrffsysem rechlesung, wrklesung, oale blndlesung, Selbserlag Munchen, 195. [17] M. T. Haque, Sngle-hase PQ heory, IEEE Power Elecroncs Secalss Conference, PESC, ol. 4, June,