On DC-link voltage stabilization of shunt active filters

Transcription

1 On DC-lnk voltage stablzaton of shunt actve flters F. Ronch, A. Tll Dept. of Electroncs, Computer Scence and Systems (DEIS) Unversty of Bologna Vale Rsorgmento, 6 Bologna ITALY Abstract: - In ths paper the control of DC-lnk voltage n shunt actve flters s studed. The man control objectve n ths knd of actve power flters s to nject a proper current n the electrc mans n order to compensate for harmonc dstorton generated by nonlnear loads. Accordng to ths purpose, the proposed stablzng algorthm allows the DC-lnk voltage to properly oscllate wthn the admssble range when load currents are compatble wth the energy stored n the capactor. Dfferently, when a transent dsturb pushes the DC-lnk voltage near the bounds, the controller ncreases ts acton n order to avod unnecessary turn-off by hardware/software protecton. Ths soluton leads to an mproved relablty of the shunt actve flter Key-words: - Actve power flters, AC/DC converters, DC-lnk voltage regulaton Introducton The use of electrc nonlnear devces, generates harmoncs n voltage and current mans spectra. Ths phenomenon can determne adonal power losses and the rsk of equpment damage or malfunctonng. Hence some countermeasures have to be taken to reduce ths harmonc dstorton. Current harmoncs have been traonally compensated wth passve flters, whch have several drawbacks: ther operaton depends on the network mpedance, they have to be tuned on fxed frequences, etc. In the last decades, the fast development of power electroncs components and control processors has led to the ntroducton of the so-called Actve Power Flters (APF) [],[6]. These devces can be more expensve than the passve flters, but they are less network-dependant and can be tuned on dfferent frequences changng a few software parameters. A partcular knd of APF are the Shunt Actve Flters, whose purpose s to nject nto the mans wres a proper current n order to compensate partally or totally for the harmonc current generated by nonlnear loads. Shunt actve flters are based on AC/DC boost converter topology and ther performances are determned by: (a) the converter parameters, (b) the method to compute current references and (c) the control algorthm adopted. The generaton of the current reference s typcally based on the detecton of the harmonc content of load currents by means of dfferent technques as the nstantaneous power theory [], the tme-doman correlaton technques [8], the FFT, etc. The control of the current produced by the shunt actve flter s deeply analyzed n lterature, whle an aspect usually not well hghlghted s the DC-lnk voltage control. Focusng the control objectve only on current harmoncs compensaton, the DC-lnk voltage dynamcs becomes a stable but oscllatng nternal dynamcs for the system. A proper selecton of the DC-lnk capactor guarantees voltage oscllaton nsde an admssble nterval for a gven range of load harmoncs [7]. However, load current transents and parastc effects can make DC - lnk voltage to leave ts allowed nterval, requrng the software/hardware protectons to stop the flter operatons. The objectve of ths paper s to provde a voltage stablzng method whch s compatble wth the current compensaton objectve and can face conons as load current transents avodng halts of the power flter. In order not to mpar the harmoncs compensaton, the voltage controller mnmzes ts acton n

2 normal load current conons; whle, when overload occurs, the controller sgnfcantly changes the flter current n order to keep the DC-lnk voltage nto ts admssble range. Ths behavor has been acheved applyng Input to State Stablty (ISS) approach [] wth a sutable shapng of the voltage controller asymptotc gan. The paper s organzed as follows. In the frst secton the model of the shunt actve flters shown, n the second one the nomnal behavor and the admssble range of the DC-lnk voltage are descrbed. In secton the proposed control algorthm s presented. Ths secton s focused on voltage control, whle the current control s just sketched. Smulaton results are reported n secton [5]. Model The scheme of the actve flter consdered n ths artcle s presented n Fg.. It s a three-phase ag replacements AC/DC N v ma v mb v mc K ma mb mc a b c L L L C v la lb lc Fgure : Shunt actve flter scheme LOAD boost converter, where the capactor s the man energy storage element and the nductors are used for the control of the flter currents by means of the converter voltages. In fgure (): v ma, v mb, v mc are the mans voltages, ma, mb, mc are the mans currents, la, lb, lc are the load currents, a, b, c are the flter currents, v s the capactor voltage, L s the value of the nductances, C s the value of the DC-bus capactor. The mans voltages are co-snusodal of frequency ω m π f m, f m 5 Hz or 6 Hz, balanced and equlbrated. v ma t V m cos ω m t π v mb t V m cos ω m t v mc t v ma t v mb t The load currents la lb lc are balanced and perodc of frequency f m : l j t I l j M j a b lc t la t lb t I l jk cos ω m t θ jk k The flter model can be defned, startng from nductor dynamcs and neglectng parastc resstances v m t L d t u xyz t v t v NK t where v m t v ma t v mb t v mc t T, t a t b t c t T, u xyz t u x t u y t u z T t are arrays representng, respectvely: mans voltages, flter currents, control nputs of the sx-swtches-brdge. In partcular, a PWM control [] s assumed for the swtches, hence u x u y u z. From the sum of the three scalar equatons above, t follows that Defnng v NK t u abc t u xyz t u x t u y t u z t v t u x t u y t u z t and consderng the dynamcs of capactor C, the complete flter model n the a-b-c reference frame can be wrtten as follows: d v t L u abc t L v m t () C ut abc t t () where all the arrays are balanced (.e. for each vector x x ). Ths property allows to represent equatons () and () n a classcal two-phase Space

3 !! Vector [] reference frame. In partcular, t s useful to consder a d-q synchronous reference frame, algned to the mans voltages. The matrx that allows to descrbe the three-phase balanced sgnals of (), () n ths b-dmensonal reference frame s: dq T abc t k c C ωm t C ωm t π C ωm t π S ωm t S ωm t π S ωm t π where C x cos x, S x sn x, k c s a sutable postve constant. In the new coordnates, the flter equatons (), () become d ω m t v t L u v t m L () where v m ω m k c C ut t t () k c V m l t ld t lq t u t u d t u q T t t d t q T t V md I ld ldh t lq t Remark: accordng to the admssble range for u x u y u z, the vector u dq T abc u abc must be ncluded nto the hexagon reported n Fg b c Fgure : Hexagon of feasble u abc Nomnal behavor of the DC-lnk voltage The man control objectve of a shunt actve flter s to cancel all or a part of the reactve and harmonc a current of the load. Hence the reference for the flter current s a sutably defned functon of the load current L. For nstance, f a full harmonc and reactve compensaton s needed, then d q Ld I Ld Lq where I Ld s the mean value of the actve component Ld. In order to fully acheve ths purpose, the DClnk voltage v t has to oscllate. The necessty of ths behavor s clear consderng the energy balance equaton C d v k c v m t L d T and takng nto account the perodc behavor mposed to the flter currents. On the other hand, the nstantaneous value of v t must be upper and lower bounded, as deeply dscussed n [7]. The upper bound s related to the maxmum admssble voltage for the capactor C, whle the lower bound depends on current controllablty constrants, as brefly reported n the followng. Consderng the () and substtutng the current wth ts reference, t follows that u t L v t # L V mn ω m ω m ω m ω m t t v m L v m L d d In order to ensure that ths control vector s ncluded nto the hexagon of Fg.(), the voltage bound V mn must be suffcently hgh. Moreover, from a practcal pont of vew, V mn must be also greater than a mnmum value to guarantee the correct behavor of all the electronc devces, as the system control board whose supply s usually derved from the DClnk capactor. As reported n [7], the belongng of voltage v to the allowed regon s manly related to the correct selecton of capactor value C wth respect to the nomnal harmonc load to be compensated. Hence, the DC-lnk voltage control acton must be focused on the satsfacton of voltage constrants when transtory overload conons occur.

4 # # Control algorthm Consder the model (),(). In order to make lnear and ndependent the current dynamcs, the followng control u s chosen u v t %$ ω m L ω m L t v m ξ t '& (5) where ξ s an auxlary control varable. Defnng the current error vector ĩ t t and consderng (5),the current error dynamcs s where dĩ ξ t d t (6) L d d (7) The auxlary control ξ t must be a functon of ĩ sutably chosen n order to stablze ĩ and to reject the dsturb d t. Snce ths paper manly concerns the DC-lnk voltage stablzaton, the detaled analyss of the current controller s omtted and t s assumed that the auxlary control ξ t s fast enough to produce a neglgble ĩ wth respect to the voltage dynamcs (ths assumpton s usually admssble owng to the large value of C). Hence consderng (),(5), the voltage dynamcs s the followng: d v k c C ( v d t d t v q t q t *) (8) where v d t v q t T v t u are the voltage mposed to control d t q t The control objectve for the DC-lnk voltage s twofold. To guarantee the voltage boundness # V mn v t V max for a gven maxmum current. To perform a low control acton when the voltage v t s nsde the admssble range V mn V max. In ths way, the man current control objectve s not mpared. The strategy adopted to obtan voltage control objectve s based on the ntroducton of an adonal part n the reference currents: where dl ql d q dl ql are the orgnal reference components derved from the load currents. In order to desgn the voltage control, let defne the varable ẽ v t e wth e + 5 V max, V mn, constant value around whch e t has to oscllate. The resultng ẽ dynamcs s the followng: where d v has been defned as ẽ - v d t t d v t (9) d v t v d t dl t v q t q t.+ The frst step to fulfll the control objectves s to guarantee the Input-to-State Stablty of (9) wth respect to the dsturbance d v t. Choosng the control acton as t sgn ẽ f / ẽ / f / ẽ / and defnng, accordng to (5): v d V md v d t () the followng dynamcs s derved: ẽ t k C V md v d t sgn ẽ f / ẽ / d v t Let consder the followng Lyapunov canddate functon W ẽ It follows that Ẇ k C V md v d t f / ẽ / / ẽ / d v t ẽ Assumng that the current control voltage v d s bounded as follows / v d t / V d V md

5 7 ;: :9 ;: :9 t results Ẇ # k C V md V d f / ẽ / / ẽ / / d v t /5/ ẽ / Defnng k 6 and choosng f / ẽ / k V md C k V d / ẽ / f / ẽ / () where f s a class-k functon of / ẽ / t follows that k C V md V d f / ẽ / / ẽ / / d v t /5/ ẽ / k / ẽ / () PSfrag replacements / ẽ /86 f / d v t /. Hence the controlled voltage dynamcs s ISS wth respect to d v t. The second step to acheve the control objectves s In Fg., e max ndcates the maxmum acceptable error ampltude. When / ẽ / exceeds that lmt, the software/hardware protectons act to halt the flter operaton. γ x / d v / ag replacements ẽ f x / ẽ / e th Fgure : f / ẽ / e max devoted to a sutable shapng of f / ẽ /. The result s shown n Fg.. For error ampltude below e th a low control acton s performed: the voltage oscllates as requred for correct current compensaton. When error modulus s greater than e th the control effort s ncreased to save the DC-lnk voltage, even f ths acton wll mpar the flter performances n terms of harmonc compensaton. Note that the maxmum slope of f =< s lmted by the dynamc separaton mposed between current and voltage control and by the bandwh lmts determned by actual dscretetme mplementaton. The effect of f =< n terms of allowed oscllaton s better depcted n Fg. where the asymptotc gan γ =< f =< of the voltage dynamcs s reported. Fgure : Bound on / ẽ / Remark:In ths analyss the effects of nductance parastc resstances and electronc devces supply current are neglected. These are dsspatve components whch produce a slow decrease of the DC-lnk voltage. In actual mplementaton an ntegral acton can be added to the DC-lnk controller to compensate for ths effect. 5 Smulaton results The shunt actve flter parameters adopted n smulatons are: L mh C µf The dstorted load current whch must be compensated s shown n Fg.5, n the d-q reference frame. In the d-component a step transent s consdered: a constant value of A s added at tme t + s and t s removed at tme t + s In order to enlghten ts stablzaton propertes, the proposed soluton has been compared wth a lnear proportonal controller: k mn ẽ where k mn s equal to the mnmum slope of the f / ẽ / descrbed n (). In Fg.6 the performances of the two solutons are reported. Both controllers produce a small control acton n steadystate conon, ths gves a neglgble perturbaton

6 > > > ( ag replacements replacements roposed one ortonal one of compensaton propertes of the flter. Dfferently, durng the load transent, the proportonal controller s not able to guarantee a safe DC-lnk voltage (.e. a v t, nsde the nterval + < 5 + < 5 ) ), whle the proposed one acheves the desred behavor, ncreasng the control acton ampltude. e 8 7 Ld 6 5 Ld x 5 Fgure 5: Load consdered, d-q frame t proposed one proposed one t proportonal one proportonal one Fgure 6: Comparson between proportonal control and the proposed soluton 6 Conclusons Dfferently, t s able to avod unnecessary halts by hardware/software protectons f a transent dsturb occurs. References [] H. Akag. New trends n actve flters for power cononng, IEEE Trans.Ind.Applcat., volume, pp., (Nov./Dec.996). [] J. Holtz Pulsewh modulaton - a survey, IEEE Trans. Ind. Electron., volume 9, pp., (December 99). [] Sontag E.D., Wang Y. On characterzatons of the nput-to-state stablty property, Systems and Control Letters, volume, pp. 5 59, (995). [] H. Akag, A. Nabae. Control strategy of actve power flters usng multple voltage source PWM converters, IEEE Trans.Ind.Applcat., volume IA-, pp. 6 65, (May/June 986). [5] L. Malesan, L. Rossetto, P. Tent. Actve flter power flter wth hybrd energy storage, IEEE Trans.Power Electron., volume 6, pp. 9 97, (July 997). [6] P. Mattavell. A closed-loop selectve harmonc compensaton for actve flters, IEEE Trans.Ind.Applcat., volume 7, pp. 8 89, (January/February ). [7] F. Ronch, A.Tll Desgn methodology for shunt actve flters, EPE conference, to be publshed, (September ). [8] G. L. Van Harmelen, J.H.R. Ensln. Realtme dynamc control of dynamc power flters n supples wth hgh contamnaton, IEEE Trans. Power Electron., volume 8, pp. 8, (July 99). A shunt actve flter DC-lnk voltage regulator has been presented. It does not sgnfcantly affects the compensaton performances when load currents are compatble wth the energy stored nto the capactor.