Considerations on the non-active power using geometric algebra

Seediscussions,s a s,andau ho p o iles o  hispublica iona :h ps://www. esea chga e.ne /publica ion/252049908 Conside a ionson henon-ac i epowe using geome icalgeb a A icle·May2011 DOI:10.1109/Powe Eng.2011.6036438 CITATIONS 2 READS 79 5au ho s,including: Someo  heau ho so  hispublica iona ealsowo kingon hese ela edp ojec s: libe aconlapapayaViewp ojec MEDIDACONTINUADELAFRECUENCIAENREDESELÉCTRICASCONPERTURBACIONES ELECTROMAGNÉTICASCONDUCIDASDEBAJAFRECUENCIA.Viewp ojec JuanCa losB a o Uni e sidaddeSe illa 27PUBLICATIONS109CITATIONS SEEPROFILE Juan-Ca losMon año Uni e sidaddeSe illa 70PUBLICATIONS573CITATIONS SEEPROFILE ManuelO doñez Uni e sidaddeSe illa 28PUBLICATIONS101CITATIONS SEEPROFILE Allin- ex  e e encesunde linedinbluea elinked opublica ionsonResea chGa e, le ingyouaccessand ead hemimmedia ely. A ailable om:JuanCa losB a o Re ie edon:09No embe 2016 1 Conside a ions on he Non-Ac i e Powe Using Geome ic Algeb a M. Cas illa, Membe IEEE, J. C. B a o, J. C. Mon año, Senio Membe IEEE, M. O dóñez and A. López. Abs ac —Se e al app oaches ha e been de eloped o de ine he non-ac i e powe concep unde nonsinusoidal si ua ions in elec ical sys ems. Ne e heless, hese con ibu ions do no p o ide a comple e and sa is ac o y solu ion o he non-ac i e powe e e sibili y be ween equency domain and ime domain. This pape p esen s a non-ac i e powe mul i ec o concep , based on an o iginal ec o space equency-domain app oach ha b idges he gap be ween bo h domains. The sugges ed co espondence can p o ide a con enien desc ip i e language o econcile F yze’s ins an aneous non-ac i e powe wi h Budeanu´s deac i e-powe . To cla i y his co espondence, a basis example is conside ed. Index Te ms—Non-ac i e powe , appa en powe , ins an aneous powe , mul i ec o , geome ic algeb a. I. INTRODUCTION A. Mo i a ion ne o he mos con o e sial issues in elec ical enginee ing is he uni e sal ep esen a ion o he powe equa ion o he elec ical ci cui s in n-sinusoidal ope a ion. Thus, while he concep o ac i e powe is ully accep ed, he de ini ion o powe ela ed o “ eac i e” and “ha monic” phenomena is s ill unde discussion. This is o absolu e impo ance, since i is he ounda ion o he non-ac i e powe concep ha pe mi s he design o sui able de ices o he compensa ion o his powe componen . This pape gi es a new ision on he ep esen a ion o he non-ac i e powe ( Σ ) and sugges s a mul i ec o ial in e p e a ion in a simple ci cui wi h n-sinusoidal wa e o ms. F om a heo e ical s andpoin , he app oach p esen ed he e di e s om he equency domain app oaches desc ibed in he li e a u e by i s emphasis on complex Geome ic Algeb a ( . In his amewo k, he non-ac i e powe is ep esen ed by a mul i ec o ha egis e s i s magni ude, di ec ion and sense. Mo eo e , in o de o examine he ecip oci y be ween equency-domain and ime- domain, he e a e some doub s which should be cla i ied. A igo ous ea men o his issue can lead o new a enues and pe spec i es in powe heo y. ) n CG This wo k was suppo ed by he Minis y o Educa ion and Science as pa o a esea ch h ough p ojec DPI-2006-17467-CO2-01. M. Cas illa and J. C. B a o a e wi h he Elec ical Enginee ing Depa men o he Uni e si y o Se illa, C/Vi gen de Á ica 7, (41011) - Se illa, Spain. (phone:+34 954 55 28 47; ax: +34 954 55 16 88; e-mail: [email p o ec ed]; [email p o ec ed]). J. C. Mon año is wi h he Spanish Resea ch Council (CSIC), Reina Me cedes Campus, POB 1052, 41080-Se illa, Spain. (email: mon ano@i nas.csic.es). M. O dóñez is wi h he Depa men o Applied Ma hema ics II. Escuela, Poli écnica Supe io , Se illa, Spain. (e-mail:m[email p o ec ed]). A. López is wi h he Depa men o Elec onic Technology, Escuela Poli écnica Supe io , Se illa, Spain. (email: [email p o ec ed]). B. Li e a u e e iew Budeanu and F yze o mula ed he i s powe equa ions o n-sinusoidal ope a ion be ween 1920 and 1930. The o me , [1], add essed he ques ion in he equency domain, whils he la e , [2], add essed he p oblem in he ime domain. Mo i a ed by hese heo ies, nume ous aluable a icles ha e appea ed [3-13] al hough none conside s he mul i ec o ial cha ac e o appa en powe componen s. On he o he hand, he mul idimensionali y o he powe in n-sinusoidal ope a ion sys ems is he unde lying obs acle ha complica es he p oblem a hand. The powe concep is be e unde s ood i a ec o space is used o ep esen a ion o he powe equa ion. In his sense [14], [15] a e wo impo an and o iginal con ibu ions o he powe analysis in linea /non- linea ope a ion. The la ge numbe o pape s published on he physical and/o ma hema ical na u e o hese e e sibili y, sugges s ha wo k on his opic emains un inished. C. Con ibu ions The one- o-one co espondence be ween he non-ac i e mul i ec o e ms and he ins an aneous powe componen s a e conside ed and cla i ied. These objec i es canno be eached in a Complex Algeb a amewo k. II. NON-ACTIVE POWER MULTIVECTOR Suppose ha a nonsinusoidal ol age ( ) () p p pN u 2 U sinp ω α ∈ =+ ∑ (1) is applied o a linea load, Fig.1, whe e p is he ha monic o de o u( ). The esul ing cu en has an ins an aneous alue gi en by ( ) ( q qN i 2 Isinq ) q ω β ∈ =+ ∑ (2) whe e q is he ha monic o de o i( ). Fo cla i y o p esen a ion and wi hou loss o gene ali y, he phases angles o he ha monic ol ages a e . p α=0 By applying he Cli o d-Fou ie ans o m [15] o he O P oceedings o he 2011 In e na ional Con e ence on Powe Enginee ing, Ene gy and Elec ical D i es To emolinos (Málaga), Spain. May 2011 978-1-4244-9843-7/11/$26.00 ©2011 IEEE 2 ol age (1) and cu en (2), he ins an aneous quan i ies can be exp essed as a linea combina ion o ha monic geome ic phaso s o complex- ec o s in n CG () () 2 sin p pp pp u U p U U ω σ =⇒= ∑∑  (3) () () 2sin q j qq q qq i I q I Ie ϕ q ωϕ σ =+⇒= ∑∑  (4) In his amewo k, he powe mul i ec o [15] en e ing he one-po in Fig. 1, is n CG S  N SU I UI UI •∧ ∗∗ ΩΩ ⎧⎫ ⎪ ==⋅⊕∧ ⎨ ⎪⎪ ⎩⎭     : ∗ ⎪ ⎬ (5) whe e pp pN UU σ ∈ =∑  and q j q qN IIe ϕ q σ ∗ ∈ =∑ a e he geome ic phaso s o ol age and conjuga e cu en espec i ely. In he expanded o m o (5), he wo componen s (scala complex) and (bi ec o complex) can be exp essed • Ω ∧ Ω  { } { } 0 00 Re Im ( cos sin ) ()() pp p pp p P pp P jUIjUI PjQ PjQ ϕ ϕσ σσ •• • Ω= Ω + Ω = + =+ =+ ∑ ∑   (6) ( q j p q pq pq pq pq pq pq UI e j ϕ ) σ σ ∧ ≠≠ Ω= = Δ +Λ ∑∑  (7) whe e {} Re Ω• is he ac i e powe P and { } Im Ω•  is associa ed o Budeanu and Slonim’s eac i e powe Q. Combining (6) and (7), we ob ain he non-ac i e powe mul i ec o ha is de ined as Σ  ( • ) p qpq pq Im + j pq σ ≠ ⎡⎤ ΣΩ Δ+Λ ⎣⎦ ∑  = (8) whe e he e ms ep esen s he in e ac ion o ol age and cu en ha monics wi h di e en equencies. The sugges ed ep esen a ion ( pq pq jΔ+Λ ) (8) can be ex ended in a s aigh o wa d manne o he exis ing powe equa ions in he equency domain. In his way, he mul i ec o enables sepa a e ea men o he smalles non-ac i e powe e m which conside s only he undamen al componen , Emanuel app oach [11], and he la ges one which includes he imagina y pa o he complex scala and he o al complex bi ec o , F yze app oach [2]. I is ema kable ha he quan i ies p oposed in Σ  (5) ha e h ee basic a ibu es: magni ude, di ec ion and sense. III. POWERS IN FREQUENCY-DOMAIN AND TIME -DOMAIN The inne o scala p oduc in (5) is symme ic bu , on he o he hand, he ou e p oduc is an isymme ic ( ) p qqp σσ =− . Thus he appa en powe (5) and i s no m can be exp essed as ( )( ) ( ) ΔΔ ΛΛ p p 0 pq qp pq qp pq Ppq S(PjQ) j σ σ ≠ =+ + −+− ∑∑ (9) ()( ΔΔ ΛΛ 22 222 pq qp pq qp pq pq SPQ ≠≠ ) ⎡ ⎤⎡ ⎤ =++ − + − ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ ∑∑  (10) and he non-ac i e powe mul i ec o is now Load i( ) u( ) + Fig. 1. Linea load wi h n-sinusoidal wa e o ms ( )( ) ( ) ΔΔ ΛΛ p 0pqqppqqp Ppq Scala pa Bi ec o pa Σ=j Q + j pq σ σ ≠ −+ − ∑∑    (11) which a e exp essed as a linea combina ion o he mul i ec o basis. In ime domain, he ins an aneous powe s ( ) en e ing he gene ic ci cui o Fig.1 is gi en by s ( )= u( )i( ) (12) Now i is seen om (3),(4) and (12) ha i may be w i en in he o m () ()() ()() () ()() p 2 p sin sin sin cos cos sin sin sin sin cos p p p pq pq pq pp p pp p pq q pq q P p( ) Q q( ) p( ) s( )= U I cos 2 pω UI 2 pω pω +U I 2 pω qω UI 2 pω qω Δ Λ ⎡⎤ ϕ⎣⎦ −ϕ ⎡ ⎤ ⎣ ⎦ ϕ ⎡ ⎤ ⎣ ⎦ −ϕ⎡ ⎣ ∑             pq q( ) ⎤ ⎦   (13) whe e is he ins an aneous ha monic ac i e powe e m, is he ins an aneous ha monic eac i e powe e m, and and a e he ins an aneous c oss- ha monic powe e ms. This analysis can eadily be ex ended o mo e gene al si ua ions in ol ing any numbe o ha monic componen s, as long as he sui able p p( ) p q( ) pq p( ) pq q( ) n CG s uc u e is chosen. The e o e, (13) consis o wo pa s, whe e he i s pa con ains a g oup o ins an aneous powe e ms in ol ing like- equency and he second pa con ains hose in ol ing c oss- equency. Then, he o al ins an aneous powe s ( ) , (13), can be seen as o med by wo powe componen s, ac i e , and non ac i e , gi en by a p( ) na p( ) p( (14) a p p )= p( ) ∑ na p pq pq ppqpq p( )= q( ) p q( ) ≠≠ ++ ∑ ∑∑ (15) In (9), each coe icien p ppqpqqpqp P,Q, , , ,ΔΛΔΛo a co esponding mul i ec o is a he same ime a p opo ional ms alue o a co esponding ins an aneous powe (13). In ac , he equa ion (13) gene a es he keys one o he 3 b idge be ween he magni udes in equency domain and hei associa ed oscilla ions in ime domain. Thus, he classic appa en powe and non-ac i e powe alues can also be in e p e ed in e ms o Cli o d coe icien s and no ms o co esponding oscilla ing componen s 222 p p pq pq p=q p=q p q p q 2 S S= p( )+2 q( )+ p( ) q( ) 3≠≠ =+ ∑∑∑∑ 2 (16) 222 Σ=S -P   (17) No e ha he abo e men ioned powe mul i ec o s, and a e he complemen s o he me ely alues S  Σ SS= and Σ= Σ . Bo h quan i ies (mul i ec o and magni ude) lead o alid powe equa ion ep esen a ion, bu i depends on he si ua ion a hand o see which app oach is mo e app op ia e. Fu he mo e, in mos cases di ec ion and sense may no be equi ed, bu e en in his case, he mul i ec o can handle he p oblems equally o e en be e han he magni ude S . Howe e , he e a e cases whe e di ec ion and sense a e necessa y, whe eby ailing o dis inguish he powe componen s in e ms o di ec ion and sense may lead o e oneous esul s. The e o e, he Gene alized Geome ic Algeb a [15] p o ides he necessa y ools o ex end he concep s o ins an aneous and complex geome ic analysis o highe dimensions. IV. NUMERICAL EXAMPLE In his Sec ion, a nume ical example is p oposed o illus a e he sugges ed non-ac i e mul i ec o concep and i s na u al capabili y o a one o one co espondence be ween i s componen s and he e ms o ins an aneous non- ac i e powe . Uni s o physical quan i ies a e hose s anda ds o he MKSA sys em and hus can be omi ed. Le ol age and cu en o a non-linea one-po ci cui be () u( ) 2 100sin 100sin 3 =+ ⎡⎤ ⎣⎦ (18) i( ) 2 50 2 sin( 45º) 50 2 sin(3 ) ⎡ =−− ⎣⎤ ⎦ (19) and he co esponding geome ic phaso s a e hen 13 100 50 111.8UU σσ =+ ⇒=  (20) 45 180 13 70.71 20 73.48 jj Iee I σσ ∗=+⇒=  ⎤ ⎦ ω ⎣    ) (21) The ins an aneous powe equa ion and he appa en powe mul i ec o may be w i en P () ( ) ( ) ( ) () () ( ) () ( ) () ()() 3 1 13 13 1 113 31 3 P P 22 p ( ) p ( ) Q q( ) p ( ) p s( ) 5000 2sin 1000 2sin 3 5000 2sin cos 2000 2sin sin 3 2500 2sin 3 sin Δ Δ ⎡⎤⎡ ⎤ =ω+− ω+ ⎣⎦⎣ ⎦ − ω ω+− ω ω+⎡⎤⎡ ⎣⎦⎣ +ωω ⎡⎤ ⎣⎦              () ()() 31 131 ( ) q ( ) 2500 2sin 3 cos Λ −ω ⎡⎤ ⎦ (22) ()()( 13 1 3 31 03 5000 1000 5000 0 4500 2500 PP P QQ Q Sj j 1 σ σ ∧ =+ =+ Ω ⎡⎤ ⎢⎥ =−++++ ⎢⎥ ⎣⎦    (23) and he con en ional appa en powe is gi en by S S 8216==  The o al ins an aneous powe is gi en by (22) and i s wa e o m is depic ed in Fig.2 Acco ding o (14) and (15) he ins an aneous ac i e and 0 3.14 6.28 7899.08 2202.02 s () ω ⋅ Fig.2. To al ins an aneous powe s( ) non-ac i e powe s a e gi en by a13 p ( )= p ( )+ p ( ) (24) na 1 13 31 31 p ( )= q ( )+ p ( )+ p ( )+q ( ) (25) 0 3.14 6.28 8447.45 265.83 8979.1 pa () pna () ω ⋅ Fig.3. Ac i e (pa( )) and non-ac i e (pn( )) ins an aneous powe s and hese wa e o ms a e shown in Fig.3. Ob iously, (22) and (23) can egis e he bidi ec ional ac i e powe low occu ing in his case ( ) and o al ac i e powe is 5 31000P=− 000 1000 4000 − =. This alue coincides wi h he a e age powe o (22). Mo eo e , equa ion (22) also e eals a non ac i e e m ( ) ()()( ) ()() () ( ) 13 31 13 31 p ( ) p ( ) 2000 2sin sin 3 2500 2sin 3 sin 500 2sin sin 3 ΔΔ −ωω+ ωω ⎡ ⎤⎡ ⎤ ⎣ ⎦⎣ =ωω ⎡⎤ ⎣⎦ ⎦       (26) which ampli ude alue is ( ) 2000 2500 500−+ = VA. This is no he same alue o ha one deduced om (23), since 13 31 31 2000 2500 4500 σ σσ − +=. Al hough (22) and (23) a e ma hema ically co ec , he wo e ms ha appea ed in ins an aneous powe equa ion (22), and , ha e he same na u e despi e hei opposi e signs. These possible e oneous esul s om he ins an aneous powe () 13 p () 31 p (22) occu due 4 o he need o in o ma ion conce ning di ec ion and sense o he componen s. The non-ac i e mul i ec o e ms (8) a e always applicable o any powe heo y. Thus, in he Example, he 5000 • Im Ω=  alue is equal o Budeanu and Slonim’s eac i e powe . The magni ude Σ= 7176  coincides wi h F yze’s eac i e powe and he esul gi en by {} {} 5590 22 • 13 Im Ω+ImΩ= ∧  is he same ha he eac i e powe p oposed by Shephe d [4], Sha on [5], and Cza necki [9] on linea ope a ion. The sca e ed powe de ined in [8] coincides wi h {} 4500 13 Re Ω= ∧  and a he same ime wi h he complemen a y eac i e powe o Sha on [5] and he ac i e dis o ion o Slonim [6]. Thus, he p oposed ep esen a ion is uni ied and in e nally consis en wi h exis ing powe equa ions. Howe e , ou equa ions a e no de i ed om pu ely algeb aic manipula ions o he appa en powe componen s in equency domain, bu a he hey a e a consequence o he powe mul i ec o . Today's accep ed heo ies canno explain he esul s ob ained he e. V. CONCLUSION In his pape , a new concep o he non-ac i e powe mul i ec o unde pe iodic n-sinusoidal linea /non-linea ope a ion has been p esen ed. Mo eo e , powe o mulas ha e been gi en i s in Cli o d equency domain and hen in ime domain. Fu he , a non-ac i e powe mul i ec o has been de eloped, which condenses all powe in o ma ion and obeys he usual conse a ion law [16]. In his sense, i b idges he gap be ween equency domain and ime domain. The new non-ac i e powe mul i ec o concep plays a simila ole o he eac i e powe in he S einme z phaso model o he sinusoidal case; his mul i ec o ha e a simple and compac exp ession and can iden i y he mos ele an exis ing powe equa ions. Σ  Finally, om o he poin o iew, he sugges ed ep esen a ion can p o ide a new language o he design o compensa o ci cui s, and op imiza ion algo i hms. The s udy o hese applica ions is a ask ha dese es u he esea ch. LIST OF SYMBOLS 1...k σ = basis o Cli o d algeb a : = geome ic p oduc ab⋅ = inne p oduc ab∧ = ou e p oduc o bi ec o p Z , p Z = no m o p- h geome ic-phaso () Z ∗  = conjuga e elemen p U  = p- h ol age geome ic-phaso p I  = p- h cu en geome ic- phaso S  = powe mul i ec o () s = ins an aneous powe Σ  = non-ac i e powe mul i ec o • Ω  = complex-scala Ω ∧  = complex-bi ec o q ϕ = q- h impedance phase angle p q Δ = pq- h { } Re pa o dis o ion powe mul i ec o pq Λ = pq- h { } Im pa o dis o ion powe mul i ec o REFERENCES [1] Budeanu CI. Puisances Reac i es e Fic i es. Ins y u Romain de l´Ene gie 1927 Bucha es , Romania. [2] F yze S. Wik-,Blind, un Scheinlei ung in Elek ischen S omk eisen mi nich sinusoidalem Ve lau on S om und Spanung. Elek . Z. 1932; 53: 596-599,625-627,700-702,. [3] Cza necki LS. Budeanu and F ize:Two amewo k o in e p e ing powe p ope ies o ci cui s wi h nonsinusoidal ol ages and cu en s. Elec ical Enginee ing 1997;80: 359-367. [4] Shephe d W and Zhakikhani P. Sugges ed de ini ion o eac i e powe o nonsinusoidal sys ems. P oc. Ins .Elec . Eng. 1972; 119: 1361-1362. [5] Sha on D. Reac i e powe de ini ions and powe ac o imp o emen in nonlinea sys ems. P oc Ins . Elec . Eng. 1973;120: 704-706. [6] Slonim MA and Van Wyk JD. Powe s componen s in a sys em wi h sinusoidal and nonsinusoidal ol ages and/o cu en s. P oc. Ins . Elec Eng 1988; 135: 76-84. [7] Willems JL and Fe e o A. Is he e a Rela ionship be ween Non Ac i e Cu en s and Fluc ua ions in he T ansmi ed Powe ?. ETEP 1998; 8: 265-270. [8] Cza necki LS.: Sca e ed and Reac i e Cu en , Vol age and Powe in Ci cui s wi h Nonsinusoidal Wa e o ms and Thei Compensa ion. IEEE T ans. On Ins um. and Meas 1991; 40: 563-574. [9] Cza necki LS. Conside a ions on he eac i e powe in non-sinusoidal si ua ions. IEE T ans. 1985; IM-34: 399-404. [10] Cza necki LS. Cu en ´s Physical componen s (CPC) Concep : A undamen al o Powe Theo y. IEE T ans. 1985; IM-34: 399-404. [11] Emanuel AE. Powe s in Nonsinusoidal Si ua ions a Re iew o De ini ions and Physical Meaning. IEEE T ansac ions on Powe Deli e y 1990; 5 :1377-1383. [12] Depenb ock M. The FBD- Me hod a Gene ally Aplicable Tool o Analyzing Powe Rela ions. IEEE T ansac ions on Powe Sys ems 1993; 8: 381-387. [13] Sasdelli R. Mon ana i GC. Compensable Powe o Elec ical Sys ems in Nonsinusoidal Condi ions. IEEE T ans. On Ins . and Meas 1994; 43: 592-598. [14] Somma i a AM. Powe Analysis o One-Po s Unde Pe iodic Mul i- Sinusoidal Ope a ion, IEEE T ans. On Ci cui s and Sys ems.-I: Regula Pape s 2006; 53: 2068-2074. [15] Cas illa M, B a o JC, O doñez M, Mon año JC. Cli o d Theo y: A Geome ical In e p e a ion o Mul i ec o ial Appa en Powe . IEEE T ansac ions On Ci cui and Sys ems I-Regula Pape s 2008, 55: 3358- 3367. [16] Cas illa M, B a o JC, O doñez M. Geome ic Algeb a: A Mul i ec o ial P o . o Tellegen’s Theo em in Mul i e minal ne wo ks. IET Ci cui s, De ices and Sys ems 2008; 2, 383-390. [17] Cas illa M, B a o JC, O doñez M, Mon año JC. The Geome ic Algeb a as a Powe Theo y Analysis Tool. P ezglad Elek o echniczny 2009; 1: 202-208. [18] B a o JC. Rep esen ación mul i ec o ial de la Po encia Apa en e en egímenes pe iódicos n-senoidales aplicando Álgeb as de Cli o d. Ph. D. Dise a ion. Uni . o Se ille (Spain) 2008. [19] Hes enes D, Sobczyk G. Cli o d Algeb a o Geome ic Calculus: a uni ied language o Ma hema ics and Physics. Kluwe Academic 1986, Do d ech/Bos on.