Robust Digital Redesign of Continuous PID Controller for Power system Using Plant-Input-Mapping

Transcription

1 Rcnt Avancs in lctrical nginring Robust Digital Rsign of Continuous PID Controllr for Powr systm Using Plant-Input-Mapping. Shabib, sam H. Ab-lham,. Magy Dpartmnt of lctrical Powr nginring, Faculty of nrgy nginring, Aswan Univrsity, Aswan, 858, YP. -mail: Abstract: - th igital rsign tchniqu is on of th most popular approachs to th sign of igital controllrs in inustris. Which convrting a goo-sign continuous tim controllr to a igital controllr suitabl for igital implmntation. In this papr, th Plant-Input-Mapping algorithm (PIM) is us for convrting th S-omain mol of th PID controllr to a Z-omain mol countrpart. h propos igital PID controllr is us to nhanc th amping of a singl machin powr systm. h propos mtho is bas on a transfr function from th rfrnc input to th plant input, which call continuous tim plant input transfr function C-PIF. All th pols of th transfr function that n to b controll must appar in th C-PIF. h rsults obtain from th propos igital PID controllr mor convrgnc to th C-PID controllr spcially for longr sampling prio whr ustin's mtho is violat. h propos algorithm is stabl for any sampling rat, as wll as it taks th clos loop charactristic into consiration. h computation algorithm is simpl an can b implmnt asily. h propos igital PID controllr is succssfully appli to th linariz mol of a singl machin infinit bus systm an th prformancs of th analog PID controllr, ustin's controllr an th propos igital PID controllr ar compar an thir rsults ar prsnt. y-wors:- Dynamic stability, Digital rsign, Discrtization, Plant-Input-Mapping, Discrt systms.. Introuction hr ar many iffrnt approachs to signing iscrt-tim controllrs for a continuous-tim systm in a fback configuration. hr ar two igital sign approachs for igital control systms []. h first approach, call th irct igital sign approach, is to iscrtiz th analog plant an thn trmin a igital controllr for th iscrtiz plant. h scon approach, call th igital rsigns approach [, ], is to sign a goo analog controllr for th analog plant an thn carry out th igital rsign for th goo sign analog controllr. Many igital vics hav bn put into practical us in powr systm such as igital PID, igital PSS an igital AR. h analog PID controllr is wily us in powr systm to gnrat supplmntary control signal for th xcitation systm in orr to amp th low frquncy oscillations. In th igital rsign tchniqu, a goo-sign continuous tim controllr is convrt to a igital controllr countrpart. It is bas on an optimal matching of continuous-tim clos loop stp rsponss of both continuous-tim an iscrtiz systms. Diffrnt tchniqus ar us to convrt continuous systms into iscrt systms. Howvr, it is to b not that continuous systm can only b approximat an th iscrt systm can nvr b xactly quivalnt. On of most popular igital rsign mtho is th bilinar transformation (ustin mtho []. his mtho is consir as local iscrtization an it proucs satisfactory rsults whn th sampling prio is sufficintly low. In rcnt yars, applications of iscrt tim controllrs to powr systms wr rport in a numbr of publications [4, 5, 6, 7, 8, an 9]. It solv th transint stability problm arss by analog controllr, xcpt that iscrt tim controllr is just a mattr of rprogramming a softwar program. []. In [5] th sign of a iscrt powr systm stabilizr PSS, which has bn prsnt by linar approximation for singl-machin infinit-bus systm, was rprsnt by nonlinar iffrntial quations, th transfr function of th PSS was iscrtiz using ustin s iscrtization mtho. h mtho in [6] analyz th asymptotical stability of th igital controls of powr systms with a spcial mphasis on th igital PSS. It trat th powr systms as nonlinar hybri ynamical systms so th powr systms can b analyz in a mor xact way. In [4] a tchniqu bas on sampl-ata control was propos for optimal iscrtization of analog controllrs whil taking into account both clos-loop an intrs ampl bhavior. In [7] a iscrt fuzzy PID xcitation controllr utilizing th bilinar transforms (ustin' was implmnt. his controllr was vlop by first signing iscrt tim linar PID control law an thn ISBN:

2 Rcnt Avancs in lctrical nginring progrssivly riving th stps ncssary to incorporat a fuzzy logic control mchanism into th moification of th PID structur. h mtho in [9] prsnt a igital rsign mtho for iscrtizing a continuous-tim powr systm stabilizr PSS for a singl machin powr systm using Plant-Input-Mapping PIM mtho. his tchniqu guarant th stability for any sampling rat as wll as it took clos-loop charactristics into consiration In this papr th Plant Input-Mapping (PIM) is appli to rsign an analog PID controllr. his analog PID must hav goo prformancs controllr. Our goal in this papr is to vlop a high prformanc igital PID controllr for singl machin powr systm that taks into consiration th clos loop prformanc, which cannot b attain whn using th traitional igital rsign mtho. h PIM mtho is a iscrtization schm that can guarant th stability for virtually any sampling rats (non-pathological sampling rat an that has goo prformancs vn for larg sampling intrvals [,,, ]. Ovrall, th PIM mtho pav th way to th igital r-sign of a gnral analog controllr with guarant stability an continuous-tim prformanc rcovry. Such gnrality an stability ar not availabl by any othr mthos. On th othr han th major isavantag of this algorithm is pning on plant mol. his sign tchniqu provis th signr a usful altrnativ to xisting igital r-sign mthos as wll as to possibly a wi class of irct igital sign mthos incluing th mol rfrnc control as xplain in this rsarch. his articl is organiz as follows. In sction (), scribs th systm configuration that consists of two subsctions, which ar riving a powr systm mol an xplains th continuous tim proportional, intgral an rivativ controllr (PID Controllr) mol. Sction (), scribs th stanar PIM igital rsign mtho is consir. h iscrtization of PID controllr by using ustin s mtho scribs in sction (4). h application of th PIM mtho to a singl-machin powr systm is consir in sction (5). Sction (6) analysis th simulation rsults. Finally th conclusions ar givn in sction (7).. h Systm Configuration. Powr systm mol Fig. shows Schmatic of th stui systm, which a singl machin infinit bus (SMIB) powr systm is consir. h SMIB systm, call th plant which consists of a synchronous gnrator connct through transmission lin to a vry larg powr ntwork approximat by an infinit bus. h synchronous gnrator is rivn by a turbin with a govrnor an xcit by an xtrnal xcitation systm. h xcitation systm is controll by an automatic voltag rgulator (AR) an a PID controllr. h powr systm consir in this stuy is th fourth orr linariz on-machin an infinit bus systm []. Fig. Schmatic of a singl-machin infinit-bus (SMIB) powr systm. Fig. shows a block iagram of transfr functions scribing th iffrnt subsystms of th on machin infinit bus powr systm. h iffrnt subsystms blocks ar givn as [9]; A. xcitation systm () + s whr is th gain of xcitr an is tim constant of xcitr. B. Fil flux cay () + s whr is th -axis transint opn circuit tim constant. C. Machin mchanical ynamics loop: () Hs + whr H is th inrtia constant an is amping cofficint. Paramtrs.6 ar th constant of linariz mol of synchronous machin. From th block iagram shown in Fig., an using qs. (, an) th following fourth orr linariz on machin infinit bus systm can b riv as scrib in [9,]. h quations which scrib th systm ar (qs: 4:7):- H H H H ω ω δ ψ f + () f + P + m B f f m () δ ω ω + ( ) δ + () ψ + () + () P + () ψ f () ω ψ 4 δ f + f + () Pm + o o o 5 6 ( ) ω δ ψ f () f f + () Pm + rf rf rf rf ISBN:

3 Rcnt Avancs in lctrical nginring h following fourth orr linariz mol of a on machin with infinit bus systm can b givn in stat variabl form as follows X AX + BU Y CX + DU Whr H ωb A H 4 o 5 H 6 o o B [ H ] C [ D [ ] ] h stat variabls compris th gnrator ar sp ω, rotor angl viation δ viation intrnal voltag viation viation f q (8), transint an fil voltag. h viation of th angular vlocity ω is assum to b masur as th output of th systm. h constants of th gnration systm an connct powr systm us for stuy ar givn in appnix I [9, ]. h amping cofficint inclu in th swing quation. h ignvalus of th matrix A shoul li in LHP in th S-plan for th systm is to b stabl. It is to b not that th lmnts of matrix A ar pn on th oprating conition. h valus of : 6 in th matrix A ar to b calculat accoring to th oprating conitions of th gnration systm an connct powr Systm []. Dtails of ths constants ar givn in appnix II. Using th ata givn abov, th transfr function of th powr systm ( givn by Fig. an th stat spac quations givn by q. 8 can b calculat using th MALAB function SSF in th signal procssing toolbox an ar givn by: ( Y.8s +.8s +.s (9) 4 U s +.67s + 47.s + 55s h powr systm transfr function ( pols an zro is givn in abl () abl () powr systm transfr function pols an zros pols zros -.6 ± j ± j ± j Continuous tim PID controllr mol h PID controllr is simpl an asy to implmnt. It is wily appli in inustry to solv various control problms. PID controllrs hav bn us for cas. During this tim, many moifications hav bn prsnt in th litratur [4, 5]. hn th transfr function of th moifi continuous tim PID controllr ISBN:

4 Rcnt Avancs in lctrical nginring [6] is givn by PID( whr P + I + s D s τs + P is th proportional gain, gain, is th rivativ gain an th trm D () I is th intgral acts τs + as an ffctiv low-pass filtr on th D rgulator to attnuat nois in th rivativ block. h iniviual ffcts of ths thr trms on th clos-loop prformanc ar summariz in [6]. PID controllr paramtrs ar trmin from th Matlab tuning givn by p 5.5, I 5., D. 5, τ. whr th sp viation ω is th input to th PID controllr, an th filtr is us to rmov th controllr ffct at stay stat conitions. Utilizing th paramtr of th PID controllr, th transfr function of th PID controllr givn can b calculat as; as + as + a.665s s + 5 PID () b s + b s + b.s + s. PIM Digital Rsign Mtho Fig. shows a SISO systm which consists of plant with a transfr function ( an thr analog controllrs with rational, propr transfr functions A (, B ( an C ( [6, 7]. Fig. Continuous-tim control systm. h continuous-tim plant is linar, tim-invariant, an strictly propr, an is not as n () h plant transfr function ( is now iscrtiz using th stp invariant-mol (SIM), which is a combination of th zro-orr-hol (ZOH), th plant an th samplr as shown in Fig. 4 u( ZOH Samplr y( ( Fig. 4 Stp invariant mol (SIM) of th plant Lt th stp-invariant mol of this plant b xprss as ( n ( r ε + r ε r m m m m () n n ( pnε + pn ε p whr [ n ( ] m, [ ( ] n, an nots th gr of its argumnt. h plant is xprss in ulr oprator [8], which is fin as z ε (4) Whr z is th usual z oprator an is th sampling intrval. h ulr oprator is us hr for bttr numrical proprtis in igital control implmntation an as of rlating iscrt-tim rsults to continuous tim countrparts []. Assum that th analog control systm is intrnally stabl, satisfis all th sign spcifications, an is raliz with propr transfr functions, which givn as; na nb nc A(, B(, C (5) A B C In th PIM mtho, both th clos-loop charactristics an plant information ar us in th iscrtization procss in th nam of th Plant-Input-ransfr Function (PIF). h PIF is th transfr function from th rfrnc input to th plant input an is givn by u A( C M (6) r( + B( C h PIF is iscrtiz in th stanar PIM mtho. his is carri out using th Match-pol-zro (MPZ) mtho [9] an th rsulting iscrt tim mol bcoms th targt PIF. h targt iscrt-tim PIF can b xprss as nm ( ( M ( MPZ( M ) (7) M ( It is foun that th nominator of th SIM of th plant appars in th numrator of D-PIF. Choosing th iscrt-tim controllr blocks [] as; m( β ( λ( A (, B (, C ( (8) λ( λ( α( Onc this iscrt-tim PIF is obtain, this must b raliz in clos-loop configuration, such as on shown in Fig. 5. ISBN:

5 Rcnt Avancs in lctrical nginring r ( r( A( + B( u( ε) ZOH u( ( - Fig. 5 Discrt-tim control systm rsign using th PIM mtho. An λ(ε ) is an arbitrary stabl polynomial of appropriat gr [, 6]. h actual PIF of this control systm is givn by m( ( M ( (9) β ( n ( + α( ( h polynomial n (ε ) an (ε ) ar known from of th plant (s q. ). By quating th targt an th actual PIF, it can b sn that th polynomial m( must b in th numrator of polynomial n M (, whras α(ε ) an β ( must b trmin by solving th following Diophantin quation: α( ( + β ( n ( M ( () If th orr of nominator of M ( is P, whr P n -, an n is th orr of nominator of th plant (, which is not satisfi. h uniqunss of th solution of q. is not assur. As in th cas, a stabl polynomial λ ( of orr q must b multipli in th numrator an nominator of th targt PIF M ( to guarant th solution of q. []. q λ ( ( ε + ), whr q n--p () h Diophantin quation aftr moification bcoms; α( ε) ( ε) + β( ε)n ( ε) λ( ε) M ( ε) () quation () can b solv to fin th unknown trms α( an β ( using for instanc liminant matrix an a stat spac formulation []. Fig. 6 shows th thr controllr block of th PIM mol. r( M( ε) C( ε) A( B( ( - C( y( ε) Fig. 6 PIM sign mtho for a plant. y( y( h PIM sign guarants th intrnal stability for any nonpathological sampling intrval an that th prformanc of th rsulting control systm approachs that of th analog control systm as. 4. Discrtization of PID controllr by using (ustin s Mtho) Discrtization of PID controllrs by using bilinar mtho (ustin s mtho) is invstigat []. By rplacing ach S-omain in analog controllrs to Z-omain, accoring to this rlation. ( ) z, Whr is sampling tim () s z + hn, th transfr function of a igital PID controllr (ustin s mtho) is a PID( z) + a + a z + a + a z + a a + a 4 b + b z b z + b b 4 (4) From q., thn; a. 665, a 5. 55, 5 a, b., b, b.an th sampling intrval slct accoring to th sampling thorm which fin as th sampling frquncy shoul b at last twic th highst frquncy contain in th signal []. hn th sampling tim for igital control is.sc slcting by sampling thory [], thn th transfr function of a igital PID controllr is.77z.z.9 PID ( z) (5).z.z.9 Aftr sign of iscrt-tim PID controllrs for iscrt-tim control systms by using traitional mtho (ustin s mtho) compar it with sign of iscrt-tim control systm by using th propos mtho (PIM) which prsnt in th nxt sction. 5. Application of PIM Digital Rsign Mtho to Powr Systm Mol o apply th sign tchniqu prsnt in sction, th transfr function ( for th powr systm givn by q. 9 an th transfr function for th PID controllr givn by q. ar us in th sign Procur with th blocks ISBN:

6 Rcnt Avancs in lctrical nginring A( an C ( qual to on as shown in Fig. [6]. Simulations rsponss of th powr systm bas on th linar mol givn by stat spac rprsntation ar prsnt. h powr systm is subjct to a stp chang in th mchanical torqu not by. h signal to b Pm controll is th rotor sp not by Δω. h analog PID is plac on th block B( of Fig. of th thr block controllrs PIM igital rsign mtho [9]. For comparison, rsults of th analog PID an th igital PID obtain by th bilinar transformation (ustin s mtho) ar invstigat []. h C-PIF is foun to b ( s + )( s +.45s + 8)( s +.5s + 4.6) M (6) ( s )( s +.65s + 9.6)( s +.85s ) It is clar that all powr systm pols an PID controllr pols ar apparing in th numrator of th C-PIF. h C-PIF pols an zro ar givn in abl (). abl () C-PIF pols an zro pols zros -.4 ± j ± j ± j ± j h SIM mol of th powr systm is givn by.87558ε ( ε ε +.5) ( ( ε ε +.6)( ε + 4.6ε +.9) (7) h SIM of th powr systm contains pols an zros ar givn in abl (4) abl (4) SIM pols an zros pols zros -.54 ± j ± j ± j h MPZ mol of th ZOH typ with its DC gain ajust is us for iscrtizing th C-PIF an is givn as.84888( ε + 5)( ε ε +.6)( ε + 4.6ε +.9) M ( ( ε + 5)( ε + 9.5ε +.66)( ε ε ) (8) h D-PIF (Cas of PIM PID) contains pols an zros ar givn in abl (5) abl (5) D-PIF pols an zros pols zros -.67 j ± j j ± j ±.469 ± 4.76 h sampling intrval slct for igital control is. sc, (any sampling intrval > is nonpathological), which is rasonabl compar with th ynamic of th systm. h conition P n -, whr P is th orr of nominator of M ( an n is th orr of nominator of th plant ( is rquir to assur uniqunss for solving th Diophantin quation q., but in this stuy of PID controllr th conition (P n -) is not satisfi, thn th uniqunss of th solution of q. is not assur. o account for this, C. A. Rabbath [] propos a moification of Diophantin quation to solv for this problm, a stabl polynomial λ ( of orr q must b multipli in th numrator an nominator of th targt PIF M ( to guarant th solution of q.. Accoring to q. th polynomial λ( is slct as λ ( ( ε + ) (9) h polynomial m ( is obtain from th numrator of M ( which is fin m (.84898ε () It is clar that th numrator of M ( inclus th pols of th SIM of th powr systm an th polynomial m (. h moifi Diophantin quation in q. can b solving by th liminant matrix mtho. Using th numrator an nominator of th SIM of powr systm th liminat matrix can b construct as follows: ISBN:

7 Rcnt Avancs in lctrical nginring () Solving th moifi Diophantin quation (q. ) with th ai of th liminat matrix givn by q. (), th polynomial α(ε ) an β ( ar obtain as follows; hir plant inputs ar still clos to that of th analog controllr as shown in Fig. 8. Whn th sampling rat bcoms slow as shown in Fig. 9 th ustin s rspons oscillats violntly an is not satisfactory whil th PIM of PID controllr proucs a iffrnt transint rspons from analog on an it has a small ovrshoot, it sttls in 5sc at th sam tim as th analogu on with no stay stat rror an almost no oscillation at th.5hz sampling rat. At th Hz control sampling rat, th ustin s cas oscillats to such an xtnt that it is not accptabl an osn t sttl vn aftr sc. Although th PIM of PID yils transint rsponss that ar iffrnt from analog cas, thir prformanc is vry goo as shown in Fig.. α (.ε ε ε () β ( 8.74ε ε ε () o rlat a iscrt-tim systm to continuous tim countrpart, th following oprator is us z ε (4) Whr is th sampling intrval an z is th usual shift oprator. h thr controllr blocks, A (z), C(z) an B (z) ar calculat using th rsults obtain abov by taking A(z) as a unity thn C (z) an B (z) ar; z.5z +.75z.5 ( z) z.54z +.86z. B (5).56z z) z ( 7.7z z.576.5z +.75z.5 C (6) 6. Simulation Rsults h tst systm has bn mol through Matlab programming. Fig. 7 to fig. show simulations rsults of th propos igital rsign tchniqu PIM mtho by using th control sampling rats of 5Hz, 4Hz,.5Hz, an Hz, rspctivly. It is notic that th PIM controllrs is stabl for any sampling rats an closly match thos of th continuous-tim PID controllr. On othr han, it is foun that ustin s mtho is violat whn sampling intrval bcoms larg. As shown in Fig. 7 th rsponss of ustin s an PIM of PID controllr proucs a smallr ovrshoot than analog controllr whil th prformanc of ustin s an PIM of PID controllr convrg to th analog cas at th control sampling rat of 5Hz. At th 4Hz control sampling rat, th ovrshoot of th ustin s an PIM of PID controllr bcom largr than th corrsponing cas of 5Hz, though thy ar accptabl. Fig. 7 Dynamic rsponss to stp chang in th mchanical torqu (sampling intrval. of PID controllr. Fig. 8 Dynamic rsponss to stp chang in th mchanical torqu (sampling intrval.5 of PID controllr. ISBN:

8 Rcnt Avancs in lctrical nginring nabls us to solv th problm an sign th thr iscrt-tim controllr. A comparison stuy of th propos igital PID controllr is carri out with convntional continuous-tim PID controllr an ustin s PID controllr. h rsults obsrv by simulations show that th propos igital PID controllr convrg to th C-PID controllr spcially for longr sampling prio whr ustin's mtho is violat. Acknowlgmnt h authors ar gratful to Profssor N. Hori, Profssor of Mchanical nginring Dpartmnt, sukuba Univrsity, Japan, for his guianc. An all his fforts an suggstions ar ply apprciat. Fig. 9 Dynamic rsponss to stp chang in th mchanical torqu (sampling intrval.4 of PID controllr. APPNDIX I I. nrator paramtrs: ' H4.6, 4.4, 7.67, ω B 77., X.97 pu, ' x.9 pu, X q.55 pu I. xcitr paramtrs: 5.,.5. I. h s:.5758,.978,.6584, 4.566, , I.4 ransmission lin: R., X.997 pu. I.5 Oprating point: Q.5 pu, t.5 pu, P.75 pu. Fig. Dynamic rsponss to stp chang in th mchanical torqu (sampling intrval.5 of PID controllr. 7. CONCLUSION h prsnt tchniqu in this stuy guarants th stability for any sampling rat as wll as it taks closloop charactristics into consiration an that has goo prformancs vn for larg sampling intrvals, unlik th popular convntional such as ustin s mtho of iscrtization. On th othr han th major isavantag of th prsnt tchniqu is pning on plant mol. h propos igital PID controllr is appli to a singl machin infinit powr systm for stability nhancmnt. Dsign PIM-PID controllr rquir to sign of th thr iscrt-tim controllrs A (Z), B (Z), C(Z) which pn on solution of th Diophantin quation, th conition P n -, whr P is th orr of nominator of M ( an n is th orr of nominator of th plant ( must b satisfi assur uniqunss of th solution of this quation, but in this stuy of PID controllr th conition ( P n -) is not satisfi, thn th uniqunss of th solution of this quation is not assur. o account for this, C. A. Rabbath [] propos a moification of Diophantin quation to solv for this problm. It APPNDIX II h constants : 6 ar valuat with transmission lin rsistanc r an ar givn as follows: k X (X q X I + X ) sin δ (X + X ) X (X X + X + X X ) q 4 sin (X + X ) X sin δ δ qo cos δ + (X + X ) q q 5 cos δ + sin (X + X q ) t (X + X ) t 6 X (X + X ) q t X q δ ISBN:

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