p-q Theory Power Components Calculations

Transcription

1 ISIE 23 - IEEE Interntionl Symposium on Industril Eletronis Rio de Jneiro, Brsil, 9-11 Junho de 23, ISBN: p-q Theory Power Components Clultions João L. Afonso, Memer, IEEE, M. J. Sepúlved Freits, nd Júlio S. Mrtins, Memer, IEEE DEI, University of Minho, Cmpus de Azurém, Guimrães, Portugl e-mil: jl@dei.uminho.pt, mjs@dei.uminho.pt, jmrtins@dei.uminho.pt Astrt The Generlized Theory of the Instntneous Retive Power in Three-Phse Ciruits", proposed y Akgi et l., nd lso known s the p-q theory, is n interesting tool to pply to the ontrol of tive power filters, or even to nlyze three-phse power systems in order to detet prolems relted to hrmonis, retive power nd unlne. In this pper it will e shown tht in three phse eletril systems the instntneous power wveform presents symmetries of 1/6, 1/3, 1/2 or 1 yle of the power system fundmentl frequeny, depending on the system eing lned or not, nd hving or not even hrmonis (interhrmonis nd suhrmonis re not onsidered in this nlysis). These symmetries n e exploited to elerte the lultions for tive filters ontrollers sed on the p-q theory. In the se of the onventionl retive power or zero-sequene ompenstion, it is shown tht the theoretil ontrol system dynmi response dely is zero. Index Terms p-q Theory, Ative Power Filters, Digitl Controller, Sliding Window, Power Qulity. I. INTRODUCTION In 1983 Akgi et l. [1, 2] proposed new theory for the ontrol of tive filters in three-phse power systems lled Generlized Theory of the Instntneous Retive Power in Three-Phse Ciruits", lso known s Theory of Instntneous Rel Power nd Imginry Power, or Theory of Instntneous Ative Power nd Retive Power, or Theory of Instntneous Power, or simply s p-q Theory. The theory ws initilly developed for three-phse threewire systems, with rief mention to systems with neutrl wire. Lter, Wtne et l. [3] nd Aredes et l. [4] extended it to three-phse four-wire systems (systems with phses,, nd neutrl wire). Sine the p-q theory is sed on the time domin, it is vlid oth for stedy-stte nd trnsient opertion, s well s for generi voltge nd urrent wveforms, llowing the ontrol of the tive filters in rel-time. Another dvntge of this theory is the simpliity of its lultions, sine only lgeri opertions re required. The only exeption is in the seprtion of some power omponents in their men nd lternting vlues. However, s it will e shown in this pper, it is possile to exploit the symmetries of the instntneous power wveform for eh speifi power system, hieving lultion dely tht n e s smll s 1/6 nd never greter thn 1 yle of the power system frequeny. It is lso shown tht lultions for retive power nd zero-sequene ompenstion do not introdue ny dely. Furthermore, it is possile to ssoite physil mening to the p-q theory power omponents, whih eses the understnding of the opertion of ny three-phse power system, lned or unlned, with or without hrmonis. II. p-q THEORY POWER COMPONENTS The p-q theory implements trnsformtion from sttionry referene system in -- oordintes, to system with oordintes α-β-. It orresponds to n lgeri trnsformtion, known s Clrke trnsformtion [5], whih lso produes sttionry referene system, where oordintes α-β re orthogonl to eh other, nd oordinte orresponds to the zero-sequene omponent. The zerosequene omponent lulted here differs from the one otined y the symmetril omponents trnsformtion, or Fortesue trnsformtion [6], y 3 ftor. The voltges nd urrents in α-β- oordintes re lulted s follows: v v vα = T v vβ v where, T = 1 3 i i iα = T i iβ i The p-q theory power omponents re then lulted from voltges nd urrents in the α-β- oordintes. Eh omponent n e seprted in its men nd lternting vlues (see Fig. 1), whih present physil menings: A. Instntneous Zero-Sequene Power ( p ) p = v i = p + p (2) p Men vlue of the instntneous zero-sequene power. It orresponds to the energy per time unity tht is trnsferred from the power soure to the lod through the zero-sequene omponents of voltge nd urrent. p Alternting vlue of the instntneous zero-sequene power. It mens the energy per time unity tht is exhnged etween the power soure nd the lod through the zero-sequene omponents of voltge nd urrent. The zero-sequene power exists only in three-phse systems with neutrl wire. Moreover, the systems must hve (1)

2 oth unlned voltges nd urrents, or the sme third order hrmonis, in oth voltge nd urrent, for t lest one phse. It is importnt to notie tht p nnot exist in power system without the presene of p [3]. Sine p is lerly n undesired power omponent (it only exhnges energy with the lod, nd does not trnsfer ny energy to the lod), oth p nd p must e ompensted. B. Instntneous Rel Power ( p ) p = v i + v i = p + α α β β p (3) p Men vlue of the instntneous rel power. It orresponds to the energy per time unity tht is trnsferred from the power soure to the lod, in lned wy, through the -- oordintes (it is, indeed, the only desired power omponent to e supplied y the power soure). p Alternting vlue of the instntneous rel power. It is the energy per time unity tht is exhnged etween the power soure nd the lod, through the -- oordintes. Sine p does not involve ny energy trnsferene from the power soure to lod, it must e ompensted. C. Instntneous Imginry Power ( q ) q = v i v i = q + q β α α β (4) q Men vlue of instntneous imginry power. q Alternting vlue of instntneous imginry power. The instntneous imginry power, q, hs to do with power (nd orresponding undesirle urrents) tht is exhnged etween the system phses, nd whih does not imply ny trnsferene or exhnge of energy etween the power soure nd the lod. Rewriting eqution (4) in -- oordintes the following expression is otined: [( v v ) i + ( v v ) i + ( v v ) i ] q = (5) 3 This is well known expression used in onventionl retive power meters, in power systems without hrmonis nd with lned sinusoidl voltges. These instruments, of the eletrodynmi type, disply the men vlue of eqution (5). The instntneous imginry power differs from the onventionl retive power, euse in the first se ll the hrmonis in voltge nd urrent re onsidered. In the speil se of lned sinusoidl voltge supply nd lned lod, with or without hrmonis, q is equl to the onventionl retive power ( = 3 V I1 sinφ1 q ). POWER SYSTEM SOURCE N p p q p p POWER SYSTEM LOADS Fig. 1 p-q theory power omponents It is lso importnt to note tht the three-phse instntneous power ( p 3 ) n e written in oth oordintes systems, -- nd α-β-, ssuming the sme vlue: 3 = v i + v i + v i = p + p p (6) 3 = vα iα + vβ iβ + v i = p p (7) p + p + Thus, to mke the three-phse instntneous power onstnt, it is neessry to ompenste the lternting power omponents p nd p. Sine, s seen efore, it is not possile to ompenste only p, ll zero-sequene instntneous power must e ompensted. Moreover, to minimize the power system urrents, the instntneous imginry power, q, must lso e ompensted. The ompenstion of the p-q theory undesired power omponents ( p, p nd q ) n e omplished with the use of n tive power filter. The dynmi response of this tive filter will depend on the time intervl required y its ontrol system to lulte these vlues. III. EXPLOITING POWER SYMMETRIES One the p-q theory power omponents re lulted it is importnt to deompose the instntneous rel power ( p ) in its men vlue ( p ) nd lternting vlue ( p ), sine only the seond quntity must e ompensted [7, 8]. The dynmi response of n tive power filter ontroller depends minly on the time required to seprte p nd p. With digitl ontroller it is possile to exploit the instntneous rel power wveform symmetries, so tht only smples of frtion of the period of the mesured voltge nd urrent wveforms re required to seprte the referred omponents. In ft, only 1/6, 1/3, 1/2 or 1 yle is enough, depending on the system eing lned or not, nd hving or not even hrmonis, s shown next. These symmetries n e exploited y digitl ontrol system tht mkes use of sliding window with numer of smples orresponding to the symmetry period. A. Symmetry of 1/6 Cyle In power systems with lned voltges nd urrents, without even hrmonis, the rel instntneous power presents symmetry equl to 1/6 of the fundmentl period (3.33 ms to fundmentl frequeny of 5 Hz). This mens tht the frequeny of the instntneous rel power wveform is 6 times the fundmentl frequeny. B. Symmetry of 1/3 Cyle In power systems with lned voltges nd urrents, ut with even hrmonis in voltges nd/or urrents, the rel instntneous power presents symmetry equl to 1/3 of fundmentl frequeny (6.66 ms to fundmentl frequeny of 5 Hz). This mens tht the frequeny of instntneous rel power wveform is 3 times the fundmentl frequeny.

3 C. Symmetry of 1/2 Cyle In power systems with unlned voltges nd/or urrents, without even hrmonis, the rel instntneous power presents symmetry equl to 1/2 of fundmentl frequeny (1 ms to fundmentl frequeny of 5 Hz). This mens tht the frequeny of instntneous rel power wveform is 2 times the fundmentl frequeny. D. Symmetry of 1 Cyle In power systems with unlned voltges nd/or urrents, nd with even hrmonis in voltges nd/or urrents, the rel instntneous power presents symmetry equl to the fundmentl frequeny (2 ms to fundmentl frequeny of 5 Hz). Fig. 2 shows the results of the instntneous rel power men vlue lultions exploiting the four different types of symmetry previously desried, nd the using of sliding window pproh.. Assuming tht the power system does not presents interhrmonis or suhrmonis (s result of fliker prolems, for instne), one yle is enough to lulte the vlue of p for ll ses. For some tive filters ontrollers it is importnt to otin the men vlue of the instntneous zero-sequene power ( p ) [4]. It n e oserved tht, in the ses where p exists, it presents symmetry equl or smller thn the symmetry of the instntneous rel power ( p ). Therefore, the sme sliding window time intervl used to determine p n e pplied to otin p. IV. COMPARISON WITH AN ANALOG CONTROLLER Fig. 3 () shows the time neessry to otin the vlue of p when using 4 th order Butterworth low-pss filter, with ut-off frequeny of 5 Hz, when the power system operting onditions re the sme of those presented in Fig. 2 () (lned system without even hrmonis). It n e seen tht the digitl solution, using sliding window nd exploiting the 1/6 yle power symmetry, is muh fster (3.33 ms) thn the Butterworth (tht tkes out 4 ms to otin the orret vlue of p ). Fig. 3() illustrtes tht this sme Butterworth filter would not e suitle to get the vlue of p in n unlned power system without even hrmonis, like the one of Fig. 2(). Instntneous Rel Power Men Vlue Clultion yle yle yle yle () Blned system Without even hrmonis Instntneous Rel Power Men Vlue Clultion yle yle yle yle () Blned system With even hrmonis Instntneous Rel Power Men Vlue Clultion yle yle yle yle () Unlned system Without even hrmonis Instntneous Rel Power Men Vlue Clultion yle yle yle yle (d) Unlned system With even hrmonis Fig. 2 Clultion of p exploiting symmetries with the use of sliding window for different power system onditions

4 Instntneous Rel Power Men Vlue Instntneous Rel Power Men Vlue () Blned system Without even hrmonis Butterworth 4 th order 5 Hz () Unlned system Without even hrmonis Butterworth 4 th order 5 Hz Fig. 3 Otining of p with Butterworth filter An tive power filter with ontrol system tht presents fst response, like the one sed on the p-q theory, hs silly two dvntges: It presents good dynmi response, produing the orret vlues of ompensting voltge or urrent in short time fter vritions of the power system operting onditions. If this response is good enough, the tive filter n e used in hrsh eletril environments, where the lods or the voltge system suffer intense nd numerous vritions. An tive filter with suh ontroller hs lso the pity of performing well in power systems with some types of suhrmonis, like fliker. It my hppen euse fliker usully hs period (not neessrily onstnt) of some or even mny power system yles, nd the proposed ontroller n respond in time intervl lwys inferior or equl to one yle. If the tive filter hs fst ontrol system, its energy storge element will suffer less to ompenste the power system prmeters vrition. Fig. 4 shows the voltge t the pitor used in the DC side of the inverter of shunt tive power filter, like the one presented in Fig. 5, when lod hnge ours. Voltge t Shunt Ative Power Filter DC side Cpitor Lod Chnge Fig. 4 Ative filter pitor voltge when lod hnges: () When p is otined with Butterworth filter; () When p is lulted with sliding window Power Soure N v v i i i v v v v i* i* i* in* Vd + Shunt Ative Power Filter Cpitor is is is isn Controller Inverter Vd i i i i i i i n Fig. 5 Blok digrm of shunt tive power filter in Lods When there is lod hnge, the shunt tive filter n t inorretly, delivering (s shown in this se) or umulting energy. If it delivers energy, the pitor voltge flls. This ehviour must e onsidered, nd is one of the ftors tht influene the sizing of the pitor. Fig. 4() refers to the ehviour of the pitor voltge when ertin new lod is dded to the power system, nd the ontrol system uses the lredy desried Butterworth filter to otin the vlue of p. Fig 4 () refers to the sme sitution, ut with digitl ontrol system tht lultes p with 1/6 yle symmetry sliding window. The seond solution presents dvntges: the tive filter n ompenste greter lod vritions, or, under the sme operting onditions, the pitor n e downsized. V. SIMULATED EXAMPLE OF COMPENSATION Fig. 6 shows n exmple of ompenstion performed y shunt tive power filter with ontrol system sed on the p-q theory. The simultion ws mde with Mtl/Simulink The voltges of this power system re lned nd sinusoidl. The lods re unlned nd present 2 nd order hrmoni urrents, so sliding window with symmetry of 1 yle ws hosen to dynmilly lulte the vlue of p. Initilly there re only two single-phse lods: two hlfwve retifiers with resistive lod. Then three-phse full-wve retifier with RL lod is turned-on (Fig. 6 ()). Fig. 6 () shows tht, y the tion of the shunt tive filter the soure urrents eome sinusoidl nd lned. The tive filter ontroller ompenstes instntneously the neutrl wire urrent, nd this urrent is kept lwys equl to zero in the soure, s seen in Fig. 6 (). The ontroller lso ompenstes the instntneous imginry power immeditely, mintining null vlue for q t the soure (Fig. 6 (e)). However, the instntneous rel power (p) must e seprted in its men nd lternting vlue, nd the sliding window tkes 1 yle (2 ms) to perform this tsk with this power system operting onditions. It n e seen in Fig. 6 (f) tht, fter the lod hnge, while the lultion of the new orret vlue of p is not finished, the soure delivers less energy to the lods thn it should do. So, during this time intervl, the pitor of the tive filter must supply energy to the lod, s lredy explined.

5 () Lod phse urrents () Soure phse urrents () Neutrl wire urrents Lod (d) Compenstion phse urrents Soure (e) Instntneous imginry power ( q ) Lod (f) Instntneous rel power ( p ) Lod Soure Soure Fig. 6 Opertion Exmple of shunt tive filter with ontrol system sed on the p-q theory nd with sliding window VI. EXPERIMENTAL RESULTS A shunt tive filter ontrol system sed on the p-q theory, where p is lulted using sliding window exploiting 1/6 yle symmetry, ws implemented with n Intel 8296SA miroontroller. The numer of points per yle is 3. So, the sliding window hs 5 points (3/6). Fig. 7 shows three mesured wveforms for phse : the ompenstion urrent referene (i *) in the upper window, phse to neutrl voltge (v ), nd soure urrent (i s ). Initilly only three-phse lned RL lod is swithedon. Lter three-phse retifier with RL lod is lso onneted to the power system. It n e seen tht i * is modified immeditely fter the lod hnges. This hppens e-

6 use the instntneous imginry power vrition n e ompensted without ny dely. It is lso possile to notie tht fter short time intervl, neessry to lulte the new vlue for p, the ompenstion urrent referene wveform rehes stedy stte gin. The wveforms in Fig. 7 were otined with the tive filter inverter turned-off. VII. CONCLUSIONS The pper riefly desried the lultions nd physil mening of the p-q theory power omponents. In three-phse eletril systems the instntneous rel power presents symmetries of 1/6, 1/3, 1/2 or 1 yle of the power system fundmentl frequeny, depending on the system eing lned or not, nd hving or not even hrmonis. Sine the instntneous zero-sequene power presents n equl or smller symmetry period, the instntneous three-phse power, whih is the sum of p nd p (eq. 7), presents the some kind of symmetries. This pper suggests the utiliztion of sliding window, in digitl ontrol system, to lulte the men vlue of the instntneous rel power, exploiting the symmetries desried ove. The importne of fst response for the tive filter ontrol system is ommented. Besides the improvement of the dynmi response of the filter, it n lso ontriute to redution in its ost, sine the tive filter pitor n e smller. Experimentl results prove the good dynmi response of the tive filter ontroller sed on the p-q theory, nd using sliding window to lulte p. This pper did not intend to disuss prtil prolems relted with the implementtion of digitl ontrol systems, like the delys imposed y nlog-to-digitl onversion times nd proessing times. However, it is ertin tht these kinds of prolems re eoming eh time less importnt, sine nowdys there re lredy ville fst ADCs (with onversion times of 8 ns, for instne) nd miroontrollers/dsps operting t frequenies s high s 15 MHz. v Lod Chnge i * i s Fig. 7 Experimentl results with lod hnging VIII. ACKNOWLEDGEMENT The uthors re grteful to FCT (Fundção pr Ciêni e Tenologi), projet funding POCTI/ESE/4117/21. IX. REFERENCES [1] H. Akgi, Y. Knzw, A. Ne, Generlized Theory of the Instntneous Retive Power in Three-Phse Ciruits, IPEC'83 - Int. Power Eletronis Conf., Tokyo, Jpn, 1983, pp [2] H. Akgi, Y. Knzw, A. Ne, Instntneous Retive Power Compenstor Comprising Swithing Devies without Energy Storge Compenents, IEEE Trns. Industry Appli., vol. 2, My/June [3] E. H. Wtne, R. M. Stephn, M. Aredes, New Conepts of Instntneous Ative nd Retive Powers in Eletril Systems with Generi Lods, IEEE Trns. Power Delivery, vol. 8, no. 2, April 1993, pp [4] M. Aredes, E. H. Wtne, New Control Algorithms for Series nd Shunt Three-Phse Four-Wire Ative Power Filters, IEEE Trns. Power Delivery, vol 1, no. 3, July 1995, pp [5] E. Clrke, Ciruit Anlysis of A-C Power Systems, Vol I Symetril nd Relted Componentes, John Wiley nd Sons, [6] C. L. Fortesue, Method of Symetril Co-ordintes Applied to the Solution of Polyphse Networks, A.I.E.E. Trns., vol. 37, June 1918, pp [7] João Afonso, Crlos Couto, Júlio Mrtins, Ative Filters with Control Bsed on the p-q Theory, IEEE Industril Eletronis Soiety Newsletter, vol. 47, nº 3, Sept. 2, pp [8] J. L. Afonso, H. R. Silv, J. S. Mrtins, Ative Filters for Power Qulity Improvement, IEEE Power Teh 21, Porto, Portugl, 1-13 Sept. 21.