The optimal online control of the instantaneous power and the multiphase source s current
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1 BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 65, No. 6, 27 DOI:.55/bpasts-27-9 The optimal olie cotrol of the istataeous power ad the multiphase source s curret M. SIWCZYŃSKI ad K. HAWRON* Faculty of Electrical ad Computer Egieerig, Istitute of Electrical Egieerig ad Computer Sciece, Cracow Uiersity of Techology, 24 Warszawska St., 3-55 Cracow, Polad Abstract. The paper presets the ew optimal real-time cotrol algorithm of the power source. The miimum of the square-istataeous curret was assumed as a optimal criterio, with a additioal costrait o source istataeous power. The mathematical model of a multiphase source was applied as a oltage-curret coolutio i the discrete time domai. The resultig cotrol algorithm was the recursie digital filter with ifiite recursio. Key words: power source, real-time cotrol, optimum, operators.. Itroductio The paper attempts to sole ew problems of the electrical source optimal real-time cotrol. I preious papers, optimal solutios were achieed i.e. for istataeous power exceed criterio [, 2], for miimum eergy losses [3, 4], with usage of the similarity priciple [5], etc. This article is a sigificat geeralizatio of preious studies for sigle-phase systems [6 8] ad for three-phase systems [9, ]. Those studies hae bee usuccessful i the research for mathematical expressios for the so-called iactie powers, while i this paper, a much more efficiet approach based o optimizatio methods is applied. I this paper, the two alteratie cost fuctios were assumed as the optimal criterio: the achieemet of the maximum istataeous power or the achieemet of the miimum istataeous source s curret ABS alue with the gie istataeous source power. A mathematical model of multiphase power source i the discrete time domai (Fig. ) was applied. The source s oltage-curret equatio i this model is: * kohawpk@gmail.com Mauscript submitted , reised , iitially accepted for publicatio , published i December 27. u = e z m i m () m= where: u = col α [u α ], e = col α [e α ], i = col α [i α ] the colum ectors of: the termial phase oltage, source oltage ad source output curret; α =,,, M the lie (phase) umber; =,, 2, the oltage ad curret sample s umber; z m = mat α, β [z m α, β ] the square matrix of the impulse resposes of source s iteral impedace operator; α, β =,,, M e z Fig.. A multiphase electrical power source the umber of phase. This meas that the source model assumed is a liear, time-iariat, i.e. the coolutioal ier impedace operator. I particular, it is the discrete time coolutio. I the olie optimal cotrol system, the trasitio from the state to the state takes place so as to meet the gie quality criteria. Two alterate criteria are assumed: the maximum of istataeous power criteria (called p algorithm), i.e.: i T u where T meas fuctio traspositio, the miimum of istataeous curret ABS alue at a gie istataeous source power p, called ji j MIN algorithm: i T i MIN i T u p =. I order to sole these tasks, () is trasformed i the form of recursie digital filter with ifiite recursio: u = z i (2) i u Bull. Pol. Ac.: Tech. 65(6) 27 Dowload Date 7/22/8 4:56 AM 827
2 M. Siwczyński ad K. Hawro where: = e z m i m. (3) m= I expressios (2) ad (3) the ector of sample s sequece { } ad the matrix z will be treated as gie alues, while the ector {i } of sample s sequece will be sought. Assumed cotrol algorithms take the followig forms: p algorithm: where: i T u = i T ( z i ) = i T i T z i = i T u = i T i T r i or equialetly: i T r i i T MIN (4) r = 2 (z + z T ) is positie defied, time-iariat matrix; ji j MIN algorithm: i T i MIN or usig Lagrage s factor: i T ( r i ) p =, i T ( + λr )i λ T i MIN (5) where is the idetity matrix. The solutio of both optimizatio tasks, at a gie e ad z for =,, 2, is the i sequece as a fuctio of, this i tur is fuctio of i, i 2,, i, i. Thus, the solutios of those tasks should hae form of a theoretically ifiite recursie digital filter. 2. The solutio of the optimal tasks for the sigle phase source For the sigle phase source its oltage-curret equatio takes the simpler (scalar) form: u = e z m i m = e z m i m z i = m= m= (6) u = z i where: = e z m i m. m= The p task has the form: i ( z i ) (7) ad the [(i ) 2 ] MIN task has the form: (i ) 2 MIN i ( z i ) = p (8) or yet aother form usig Lagrage s factor: ( + λz )(i ) 2 i MIN. (9) I order to sole these tasks the discrete time idex is skipped i (7) ad (8), therefore the p takes form: i r i 2 () where r = z, ad the (i 2 ) MIN takes form: ( + λr )i 2 i MIN. () The solig equatios for () ad () tasks take form: thus: ad: thus: i = i i d = ( + λr )i = 2 λ i i opt = Λ where Λ is a udetermied real scalar. The obtaied currets i d ad i opt will be called adjustmet curret ad optimal curret respectiely. Usig these currets i the source s balace coditio: r i 2 i + p = the maximum power is obtaied: p = 2, ad the equatio which allows to calculate the Λ factor takes form: r 2 Λ 2 2 Λ + p =. This quadratic equatio ca be formed as follows: r Λ 2 Λ + x =, p where x = is the so called fractio of the source s load. p The oe ad oly allowed solutio for the quadratic equatio is: i = i opt = ( x ) = i d ( x ). 828 Bull. Pol. Ac.: Tech. 65(6) 27 Dowload Date 7/22/8 4:56 AM
3 The optimal olie cotrol of the istataeous power ad the multiphase source s curret The same result ca be obtaied usig the source s balace coditio: r i 2 i + p = which takes form: r i2 i + x =, ad solig it with respect to i yields: i = ( x ) = i d ( x ). Thus, the followig theorem of the sigle phase source has bee proed: the curret that deliers the gie istataeous power from the source has the miimum or the maximum istataeous-square alue. The [(i ) 2 ] MIN algorithm works recursiely, i.e. (i, i, i 2,, i ) i accordig to the scheme: (i ) 2 MIN i u = p ad is soled by the followig sequeces of operatios: where r = z, i = u = e z m i m, m= i d = p = ( ) 2 x =, p p, ( i d x for x < i d for x > (2) where =,, 2, is the discrete time idex. The sequece of the gie istataeous power p is limited by the (2) coditio. It meas that the source caot proide curret that trasfers greater power tha the source s maximum power p. I such case the source is delierig the adjustmet curret i d, which trasfers the maximum power p at the momet. It will be show i a example i Sectio The solutio of the optimal tasks for the multiphase source Skippig the discrete time idex i (4), the p task takes the form: i T r i i T MIN. (3) Doig the same with (5), the jij MIN task takes the form: i T ( + λr )i λ T i MIN. (4) These miimum tasks were cosidered i paper [] ad its solig equatios are: for the p task, ad: r i d = 2 (5) ( + λr )i λ = λr i d (6) for the jij MIN task. Solig (5) ad (6) the multiphase ector of adjustmet curret is obtaied: i d = 2 r ad the λ based curret ectors: i λ = 2 (λ + r ). (7) Simultaeously the source s power balace coditio is met: i T ( r i) p = which allows to calculate the maximum power: p = 4 T r. The Lagrage s factor λ i (7) ca be calculated from the so called power equatio: where F(λ) = P. F(λ) = (i λ ) T ( r i λ ) is the so called source s eergy fuctio. That fuctio ca be approximated by expressio []: where F(λ) = r = 2 + λr ( + λr) 2 λrp, T T r is the so-called source ormatie resistace. Solig the eergy equatio by usig the approximatio fuctio, the Lagrage s factor is obtaied: λ x = r x, p where x = is fractio of the source s load. A istace p of the optimal curret is defied by (7) which ow takes the form: Bull. Pol. Ac.: Tech. 65(6) 27 Dowload Date 7/22/8 4:56 AM 829
4 M. Siwczyński ad K. Hawro i opt = 2 (λ + r ). Itroducig agai the discrete time idex causes appropriate alues to take the sese of the istataeous alues. I this way, the sequetial algorithm for determiig the curret s istataeous-alue ector is obtaied: (i, i 2, i 3, ) i, which operates as follows: the istataeous alue of is calculated: = e z m i m, (8) m= the the maximum istataeous power: p = 4 T r, ext the istataeous fractio of the source s load: x = p p, ext the source s ormatie-istataeous resistace: Also, i both cases, the costat sequece of the istataeous power is assumed: { p } ={2, 2, 2, }. Accordig to the iteral impedace operator, oltage will take the form: = e z i. I Example the source s o-siusoidal oltage sequece is assumed as follows (Fig. 2): {e } ( ) 8, 9,, 9, 8, 8, 9,, {e } =. 9, 8, 8, 9,, 9, 8, 8, r = T T r Fig. 2. The sequece of the source s oltage (Example ) ext the istataeous Lagrage s factor: x λ r for x < = x, for x > ad fially the istataeous optimal curret: i opt = 2 (λ + r ). 4. Examples for the sigle phase source Two similar examples are used for the sigle phase source i the discrete time domai. I both of them the source s iteral impedace was assumed as follows: Z(s) = R + sl s = f( z) 4 f = / τ R + X L X L z, where: R = 4, X L = fl = 2 (so-called digital reactace); from which the iteral impedace sequece is obtaied: {z } = {R + X L ; X L ; ; ; } = {6, 2,,, }, thus: z = r = 6, z = 2. The optimal olie cotrol algorithm begis with sample = : the sample = : = e z i = e = 8, i d = 2r = 6,666667, p = ( ) 2 = 266,666667, p x = p =,75, i = i d ( x ) = 3,333333, = e z i = 96,666667, i d = 2r = 8,55556, p = ( ) 2 = 389,35852, p x = p =,53674, i = i d ( x ) = 2,437848, 83 Bull. Pol. Ac.: Tech. 65(6) 27 Dowload Date 7/22/8 4:56 AM
5 The optimal olie cotrol of the istataeous power ad the multiphase source s curret {i } 4, 3, 2,,,, 2, 3, 4, Fig. 3. The sequece of the optimal curret (Example ) ad so o. The optimal curret (i 2 ) MIN is show i Fig. 3. I this example, all samples of curret are optimal, which meas that i each sample the required power p is smaller tha the maximum power p. I Example 2, the source s oltage sequece is assumed as follows (Fig. 4): {e } ( ) 4, 8,, 5,, 4, 8,, {e } =. 5,, 4, 8,, 5,, Fig. 4. The sequece of the source s oltage (Example 2) The optimal olie cotrol algorithm for = : ad for = : = e z i = e = 4, i d = 2r = 3,333333, p = ( ) 2 = 66,666667, p x = p = 3, i = i d = 3,333333, = e z i = 86,666667, i d = 2r = 7,222222, p = ( ) 2 = 32,962963, 6, 4, 2,, 2, 4, 6, {i } x = p p =,63953, i = i d ( x ) = 2,88395, ad so o. The optimal curret (i 2 ) MIN is show i Fig Fig. 5. The sequece of the optimal curret (Example 2) I sample = it ca be see that the source s maximum power is lower tha required p, therefore the optimal curret is limited to the adjustmet curret (delierig the maximum power from the source) (2). 5. Coclusio A ew curret cotrol algorithm for the real-oltage source was formulated i the paper. The discrete time coolutio oltage-curret model with the iteral impedace operator was used. First, the sigle-phase model was cosidered, the the multiphase oe. The optimal source s curret cotrol brigs to searchig such a curret sigal as to proide both its miimum square alue all the time ad the gie istataeous power alue. The cotrol algorithm is recursie, i.e. the actual samples are obtaied from the precedig oes. For the sigle-phase source, the problem of seekig curret sigal that proides the gie istataeous power has the uequiocal solutio ad thus, the obtaied curret sigal has miimum square alue all the time. The multiphase source works i a differet way. The ector space of currets which delier the gie istataeous power hae ifiite dimesio ad oly oe of them has the miimum RMS alue. It may be cocluded that the obtaied optimal curret cotrol algorithm is a recursie digital filter with ifiite recursio. Howeer, i practice the ifiite recursio ca be approximately replaced by a fiite oe. Due to the stability of ier impedace operator, it ca be assumed that z m = (the zero matrix) for m > M >. The the recursio (8) takes the fiite form: = e z m i m. m= Bull. Pol. Ac.: Tech. 65(6) 27 Dowload Date 7/22/8 4:56 AM 83
6 M. Siwczyński ad K. Hawro Refereces [] M. Siwczyński ad M. Jaraczewski, The source optimal curret calculatio accordig to the istat power criterio, Przegląd Elektrotechiczy 85 (6), (29). [2] M. Siwczyński ad M. Jaraczewski, The three-phase source optimal curret calculatio accordig to the istataeous power exceed criterio, Przegląd Elektrotechiczy 85 (), 7 75 (29). [3] M. Siwczyński ad M. Jaraczewski, Fractioally matched load to power source, Przegląd Elektrotechiczy 86 (4), (2) [i Polish]. [4] M. Siwczyński, S. Żaba, ad A. Drwal, Eergy-optimal curret distributio i a complex liear electrical etwork with pulse or periodic oltage ad curret sigals. Optimal cotrol, Bull. Pol. Ac.: Tech. 64 (), 45 5 (26). [5] M. Siwczyński ad M. Jaraczewski, Priciple of similar equatios for optimizatio of theory of electric power ad eergy, Przegląd Elektrotechiczy 86 (a), (2) [i Polish]. [6] H. Akagi, Y. Kaazawa, ad A. Nabae, Geeralized theory of the istataeous reactie power i three phase circuits, IPEC 83, (983). [7] H. Akagi, E.H. Wataabe, ad M. Aredes, Istataeous Power Theory ad Applicatios to Power Coditioig, Wiley Itersciece, A Joh Wiley ad Sos, New Jersey, 27. [8] S. Su ad S. Huag, O the meaig of osiusoidal actie currets, IEEE Trasactios o Istrumetatio ad Measuremet 4 (), (99). [9] J.L. Willems, A ew iterpretatio of the Akagi-Nabae power compoets for osiusoidal three-phase situatios, IEEE Trasactios o Istrumetatio ad Measuremet 4 (4), (992). [] F.Z. Peg ad J.S. Lai, Geeralized istataeous reactie power theory for three phase power systems, IEEE Trasactios o Istrumetatio ad Measuremet 45 (), (996). [] M. Siwczyński ad K. Hawro, Power theory optimizatio tasks algorithms reiew, (to be published). 832 Bull. Pol. Ac.: Tech. 65(6) 27 Dowload Date 7/22/8 4:56 AM
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