Cluster Optimization for Takagi & Sugeno Fuzzy Models and Its Application to a Combined Cycle Power Plant Boiler

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1 Clut Optmzaton o Takag & Sugno Fuzzy Modl It Applcaton to a Combnd Cycl Pow Plant Bol Do Sáz, Mmb IEEE, Robto Zuñga, Studnt Mmb IEEE Abtact- In th pap, a nw mthod o clut numb optmzaton o Takag & Sugno modl popod. Alo, a gnal dntcaton mthodology dcbd, ncludng a ntvty analy o nput vaabl lcton. Th nw mthod xmpld ung a bnchmak poblm,.., Chn. At that, th uzzy modl o a combnd cycl pow plant bol, ung th popod mthodology, a dvd. C I. INTRODUCTION OMBINED cycl pow plant a o gat ntt o many count du to th hgh cnc th low nvtmnt cot []. Alo, du to th hghly non lna bhavo o thmal pow plant bol, non lna modlng ncay to pnt th poc opaton. Patculaly, th non lna multvaabl uzzy modl dvlopd n th wok wll b ud o a utu contol dgn o upvoy contoll n od to conomc optmz th plant pomanc []. Idntcaton o uzzy modl a complx poblm gvn by many tp. A lvant tp dtmnng th optmal numb o clut [3]. In that aa th a om pvou wok. Rnc [4] popo th clut valdty mau bad on th pomanc o th obt patton ung cta lk th wthn clut dtanc, th patton dnty, th ntopy, tc. Th appoach mpl a hgh computatonal ot a clutng mut b patd val tm. Khnapuam & Fg [5] dcb a Compatbl Clut Mgng (CCM) o ndng th numb o lna o plana clut. Th algothm tat wth a maxmum numb o clut, thn th numb o clut ducd untl om thhold achd no mo clut can b mgd. Kaymak & Babuka, [6] popo a modd CCM algothm bad on l convatv cta. Th wok wa uppotd n pat by th Facultad d Cnca Fíca y Matmátca, Unvdad d Chl und pojct Dgn o upvoy optmal contol tatg o multvaabl non lna ytm, by FONDECYT -- Chl und gant no Th autho a wth th Elctcal Engnng Dpatmnt, Unvdad d Chl, Tupp 007, Santago, Chl (ax: ; -mal: daz@ng.uchl.cl). In th wok, w popo a nw mthod o dtmnng th optmal numb o th clut. Alo, a gnal mthod o non lna ytm bad on uzzy modl dcbd, ncludng a bnchmak xampl. At that, a combnd cycl pow plant bol dcbd. Thn, uzzy modl o a thmal pow plant bol a dvlopd om mulaton data, ung th popod mthod. Fnally, th wok concluon a pntd. II. NON LINEAR IDENTIFICATION METHODOLOGY BASED ON FUZZY MODELS A. Non lna multvaabl uzzy modl In th wok, th u o th Non Lna Autogv wth Xognou vaabl (NARX) modl condd, wh t tuctu gvn by th ollowng quaton: y(k) = (y(k ), y(k ),..., y(k na), u(k nk),..., u(k nb nk + )) () wh y(k) a output vcto o modl, th non lna uncton to b tmatd u(k) copond to a vcto o manpulatd vaabl. Th Takag&Sugno uzzy modl ha th ollowng non-lna uncton: y(k -) u(k - nb - nk +) A thn y (k) = g + g + g na A L y(k na) + g na + nb 0 na +nb- + g y(k ) + g y(k ) +... na + u(k nb nk + ) u(k nk) +... wh A th uzzy t o vaabl o ul, g conqunc paamt o ul y th output o ul. Th output o th uzzy modl : yt ()= N wy = N w = () (3)

2 wh N th ul numb w th actvaton dg o ul gvn by: = µ L µ Lµ (4) w na+ nb wth µ µ th mmbhp dg o uzzy t A : ( a ( x b ) = xp 0.5 (5) wh a b a mmbhp paamt x on o th ollowng nput vaabl: ( yk ( ),..., y( k na), uk ( nk),..., o u( k nb nk + )). B. Idntcaton pocdu Th man tp o th non lna dntcaton mthodology a pntd n Fgu. Fo modl dntcaton t ncay to lct al data om th poc. Tho data mut nclud nough nomaton to pnt th dnt nomal opaton condton o th poc. Nxt, th lvant nput vaabl o th non lna modl a lctd. At that, a tuctual optmzaton mad. Thn, non lna modl paamt ung only lvant nput vaabl optmal tuctu a calculatd. Fnally, th non lna modl valdatd. In th nxt cton, th man tp o th mthod a dcbd n dtal. Fg.. Flow dagam Data Slcton Slcton o Rlvant Vaabl Stuctual Optmzaton Paamt Idntcaton Modl Valdaton ) Data lcton Ncay data t o non lna uzzy modlng a: Tanng t. Fom th data, th uzzy modl tuctu modl paamt a obt. Tt t. An addtonal tt t dnd. Th t not dctly ud n th tanng algothm; howv, t allow to valuat th modl gnalzaton capacty gvn by th uzzy modl bhavo und a nw data t. Valdaton t. Ncay nw data to valuat th appopat bhavo o adjutd modl. ) Slcton o lvant nput vaabl Fo any poc modlng, on o th mot mpotant pont th appopat lcton o th lvant nput vaabl (x : yk ( ),..., y( k na), uk ( nk),..., u( k nb nk + )) that mut b ncludd n th modl. To olv th poblm th ntvty analy mthod condd [7]. Th mthodology cont n adjutng an ntal modl wth th maxmum pobl nput vaabl, lookng o lmtng th poblm complxty. Thn, th nlunc o nblt o ach nput vaabl a dtmnd. Fnally, th optmum modl that u only th nput vaabl wth bggt aocatd nblt obt. Th nput vaabl ntvty ξ o th NARX modl (quaton ()) dnd by: ξ = (6) wh th non lna uncton x an nput vaabl. Th nblt (ξ ) dpnd on nput vaabl x, thy a valuatd ung tanng t. Thn, t ncay to calculat th man o nblt at that, th nput vaabl wth low man valu a lmnatd. In th ca, th nput vaabl ntvty o uzzy modl (quaton () to (5)) gvn by th ollowng quaton []): y(t) = w µ w y w N N N N y + w w x x = = = = N w = µ = µ L µ µ + L µ = µ c ( ( ) ) c = a x b a y = g Nx w y 3) Paamt dntcaton In gnal, th paamt o th non-lna modl a obt mnmzng th tanng t man quad o. Sugno & Yaukawa [8] popod a mthod that mnmz th numb o ul makng a patton o th output vaabl unv that pojctd to th nput unv ndng th optmal uzzy t ul. Th patton bad on a uzzy clutng mthod. Ung th (7)

3 patton, th pm paamt a obt. Nxt, th conqunc paamt a obt ung th Takag & Sugno mthod bad on lat qua, dcbd n [9]. 4) Stuctual optmzaton In gnal, th tuctual optmzaton o a non lna multvaabl modl a achng pocdu contng n popong dnt achtctu, ncang complxty. Thn, o ach popod tuctu, th paamtc optmzaton mad mnmzng th tanng t o valuatng th tt t o. Fnally, both optmzaton conclud whn th tt o th ncad o tabld. Dtmnng th optmal numb o clut. In od to dtmn th optmal numb o clut, a tuctual optmzaton o uzzy modl popod. Thn, th nw mthod bad on a achng pocdu contng n popong dnt numb o clut, ncang complxty. Thn, o ach clut numb popod, th paamtc optmzaton mad mnmzng th tanng t o valuatng th tt t o. Fnally, both optmzaton conclud whn th tt o th ncad o tabld. Th oot man qua o condd a ndx o o tanng tt data t, gvn by: N ( y() ŷ() ) = RMS = N (8) wh y() th ytm output, ŷ () th tmatd output by uzzy modl N th data numb. 5) Modl valdaton Th adjutd uzzy modl valuatd ung a valdaton t. Thn, th adjutd modl valuaton appopat, th modl dntcaton pocdu nh; othw t convnnt to vw th pvou tp. C. Exampl: Chn Th xampl o dynamc ytm dntcaton wa pntd by Chn n [0]. Th xampl gvn by th ollowng quaton: ( ( y(k) = xp y (k ) ) ( xp( y (k ) ) y(k y(k ) ) + u(k ) + 0.u(k ) + 0.u(k )u(k ) + ε(k) wh y(k) th output vaabl, u(k) th nput vaabl gvn by unom dtbuton ( µ = 0, σ = ) ε (k) wht no ( µ = 0, σ = 0. ). 50 tanng data, 50 tt data 50 valdaton data a condd. Alo, th pm conqunc paamt a dtmnd ung th mthod dcbd n (9) cton..3. Fo lcton o gncant nput vaabl, th ntvty mthod, dcbd n cton.., ud. Dtmnng th optmal numb o clut Fo th lcton o th optmal numb o clut, th popod mthod condd. In Fgu, th tanng tt o o dnt numb o clut hown. Th tanng o dca, by ncang th clut numb. Othw, th tt o ha a mnmum bo, n od to gnalz th ytm dynamc o Chn. Thn, a how Fgu, th optmal numb o clut o th uzzy modl ou. Fg.. Tanng tt o o Chn III. FUZZY MODELS OF A COMBINED CYCLE POWER PLANT BOILER A. Poc dcpton Th combnd cycl pow plant cont n a ga tubn, a bol a tam tubn to gnat lctcty []. In Fgu 3, th bol conguaton pntd. Th dwat uppld to th dum, wh th thmal ngy o combuton poduct t to b condnd. Thn, th dwat nt th, wh th unac hat ud to nca th wat tmpatu vntually t cau t vapoaton. Thu, th cculaton o wat, tam wat tam mxtu tak plac n th dum. Stam gnatd n th paatd n th dum, om wh t low though th uphat to th hgh pu tubn. Thn, th tam cycld to th bol n th hat wh t ngy contnt ncad.

Th phnomnologcal mulato wa dvlopd o bol o a combnd cycl pow plant (50 MW). Th bol mulato bad on th phnomnologcal quaton, th paamt a dtmnatd adaptd om [].

4 A Econom Rhat Suphat Ful R Combuton Chamb Fg. 3. Bol conguaton Fdwat Stam om HP Tubn Rhatd Stam Suphatd Stam Dum ) Paamt dntcaton Th paamt o th uzzy modl popod o p a dtmnat ung th uzzy clutng lat man qua mthod. ( cton II.B.3). Th phnomnologcal mulato wa dvlopd o bol o a combnd cycl pow plant (50 MW). Th bol mulato bad on th phnomnologcal quaton, th paamt a dtmnatd adaptd om []. Th mulato cont on 5 non lna dntal quaton 40 non lna algbac quaton apox. B. Fuzzy modlng A an xampl o uzzy modlng popod, a non lna uzzy multvaabl modl o uphatd tam pu (p ) dvlopd, ung ul low (w ) dwat low (w ) a manpulatd vaabl. In Fgu 4, th p modl dntcaton data a pntd. Th xctaton gnal (w ) (w ) a dct wht no. Fo modl adjutng valuaton, th o ndx (RMS), dnd by quaton (7), condd. Fg. 4. Idntcaton data ) Data lcton On thou data o tanng, tt valdaton t, ung 30 cond amplng pod, a condd. Fgu 5 pnt th tanng, tt valdaton data t o p. Fg.5. Tanng, tt valdaton data t. 3) Slcton o lvant nput vaabl Du to th computatonal qumnt gvn by adjutng an ntal modl wth th hght od, th ollowng ntal uzzy modl tuctu popod: p (k -) A L p w (k -) A w (k -) A thn p (k) = g 0 + g p (k ) + L+ g p (k 5) + g w (k -) + L+ g + g 6 6 L w (k - 5) A L w (k - 5) A w (k ) + L+ g 5 (k - 5) 0 5 A w (k 5) w (k 5) (0) Fgu 6 pnt th man nblt gaphc o th popod ntal modl, wth 5 nput vaabl. In th gaphc, th vaabl wth th lat tattcal valu o th nblt a cdat to b lmnatd. Thn, t pobl to conclud that th vaabl mut not b ncludd n th poc modl. By th way, th ollowng optmal tuctu obt wth only modl lvant nput vaabl: p(k -) A p(k - ) A p(k - 3) A3 p(k - 4) A4 p(k - 5) A5 w (k -) A6 w (k - ) A7 w (k - 4) A8 w (k - 5) A9 w(k -) A0 thn p(k) = g0 + gp (k ) + gp(k ) + g3p(k 3) + g4p(k 4) + g5p(k 5) + g6w (k ) + g7w (k ) + g8w (k - 4) + g w (k 5) + g w (k ) 9 0 ()

5 Fg. 6. Sntvty analy 4) Stuctual Optmzaton: Dtmnng th optmal numb o clut. In od to dtmn th optmal numb o clut, uzzy modl wth dnt numb o clut a adjutd. A hown Fgu 7, th tanng o dca by ncang th clut numb th tt o ha a mnmum. Tho, th optmal o clut numb two o th uzzy modl obt n quaton (). Fg. 8. Pdcton o p ung optmal uzzy modl. TABLE. RMS ERRORS Tanng Tt Valdato n Lna Modl Intal Fuzzy Modl Optmal uzzy modl Fg.7. Tanng tt o o uzzy modl 5) Modl Valdaton Fgu 8 pnt th pdcton o uphatd tam pu p ung th uzzy modl dnd by quaton () th optmal numb o clut obt n cton III.B.4. Tabl how th RMS o o th tanng, tt valdaton data t o a lna modl, th ntal uzzy modl (quaton (0)) optmal uzzy modl (quaton () clut). Th lat RMS o obt, ung th optmal uzzy modl, o tt valdaton t. IV. CONCLUSIONS Th pap pnt a nw mthod o dtmnng th optmal numb o clut o uzzy modl. Alo, a complt dntcaton mthodology o non lna uzzy modl dcbd, ncludng a ntvty analy. Fo th dtmnng th optmal numb o clut, a Chn xampl pntd. Th popod dntcaton mthodology wa appld to th modlng o a combnd cycl pow plant bol. Th optmal uzzy modl o uphatd tam pu wa avoably compad vu an ntal uzzy modl a lna modl. Th non lna multvaabl uzzy modl o a thmal pow plant dvlopd n th wok wll b ud o a utu contol dgn o upvoy contoll n od to conomc optmz th plant pomanc. REFERENCES [] Ody, A., Pk, A., Johnon, M., Katb, R., Gmbl, M. (994). Modllng mulaton o pow gnaton plant. Spng- Vlag. [] Sáz, D., Cpano, A. (00). A nw mthod o tuctu dntcaton o uzzy modl t applcaton to a combnd cycl pow plant. Engnng Intllgnt Sytm o Elctcal Engnng Communcaton, Vol. 9, Nº, pp [3] Babuka, R. (998). Fuzzy Modlng o Contol. Kluw Acadmc. [4] Gath, I., Gva, A. (989). Unupvd optmal uzzy clutng. IEEE Tanacton on Pattn Analy Machn Intllgnc, Vol. 7, pp

6 [5] Khnapuam, R., Fg, C. (99). Fttng an unknown numb o ln plan to mag data thought compatbl clut mgng. Pattn Rcogntaton, Vol 5, Nº 4, pp [6] Kaymak, U., Babuka, R. (995). Compatbl clut mgng o uzzy modlng. Pocdng o FUZZ-IEEE/IFES 95, Yokohama, Japan, pp [7] Cznchow, T. (996). Appot d éaux à la pévon d é tmpoll, applcaton à la pvon d conommaton d élctcté. Thè d Dotoat d Unvté Pa 6. [8] Sugno, M. Yaukawa, T. (993). A uzzy-logc-bad appoach to qualtatv modlng. IEEE Tanacton on Fuzzy Sytm, Vol., Nº, Fbuay, pp [9] Takag, T., Sugno, M. (985). Fuzzy dntcaton o ytm t applcaton to modlng contol. IEEE Tanacton on Sytm, Man Cybntc, Vol. SMC-5, pp [0] Chn, S. Bllng, S. (989). Rpntaton o non-lna ytm: Th NARMAX modl. Intnatonal Jounal Contol, Vol. 49, Nº 3, pp

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