Theoretical and Practical Aspects for Study and Optimization of the Aircrafts Electro Energetic Systems

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1 Ncola Jula, Cosn Cpsca, Lungu Mha, Cpran Racucu, Tudor Ursu, Dan Raducanu Thorcal and Praccal Aspcs for Sudy and Opmzaon of h Arcrafs Elcro Enrgc Sysms NICOLAE JULA COSTIN CEPIŞCĂ Mlary Tchncal Acadmy SC Elcos Proc SRL -3 G.Cosbuc Blv.,Scor Buchars Ianculu 4, s., Buchars ncola.jula@gmal.com cosn@wng.ro LUNGU MIHAI Unvrsy of Craova, Faculy of Elcrochncs Blv. Dcbal, No.07, Craova, Dolj Lma3@yahoo.com CIPRIAN RĂCUCIU Mlary Tchncal Acadmy -3 G.Cosbuc Blv.,Scor Buchars cpran.racucu@gmal.com TUDOR URSU DAN RĂDUCANU Mlary Tchncal Acadmy Mlary Tchncal Acadmy -3 G.Cosbuc Blv.,Scor Buchars -3 G.Cosbuc Blv.,Scor Buchars ursuudor9@yahoo.com dan.raducanu@gmal.com Absrac: - Th papr prsns a concr cas of an acual arcraf lcrc powr sysm analyss. Usng h Boolan logcal srucurs w dfn a concpual faul r. Th faul r wll xprss all h combnaon of facors ha can lad o sysm falur n h onboard lcrc sysm. Th furhr on analyss rly on AND OR logc lmns, and h goal s o mprov h faul-olranc bhavor of h sysm. Th xampls and numrc fgurs ar for a c.c. lcrc powr sysm of an opraonal arcraf. Ky-Words: - rlably analyss, Boolan logc, arcraf, lcrc powr, nrgc sysm. Inroducon Larg-scal sysms rlably analyss s basd on h quanfcaon of h falur procss a h srucural lvl. Thus, any sysm falur s a rsul of a quanfd squnc of sas of h falur procss. Th quanfcaon lvl s chosn n accordanc wh h dsrd goal and prcson, down vn o h sngular componns. Th mor dald h quanfcaon lvl gs, h mor accura ar h rsuls [], [], []. Th concpual rprsnaon of an mrgn falur sa s a srs of prmary vns, nrconncd hrough a Boolan logcal srucur, whch ndcas h possbl combnaon of hos lmns havng h rsul of a sysm falur. Th arcraf lcrc sysm rlably drmnaon, usng h Boolan algbra, consss n h calculus of h probably of h falur vn. From h srucural pon of vw, for h rlably analyss, w wll us h rms: - Prmary lmns componns or blocks a h bas lvl of h quanfcaon; - Prmary falurs prmary lmns falurs; - Unwand vn sysm falur sa; - Falur mod h s of prmary lmns ha whn smulanously n falur mod, drvs o a sysm falur; - Mnmal falur mod h smalls s of prmary componns ha whn smulanously n falur mod, drv o a sysm falur; - Hrarchc lvl all lmns ha ar srucurally quvaln and havng quvaln posons n h sysm falur rprsnaon. Th analyss mhod s basd on bnary logc [3], [4], []. Thus, a sysm funcon s quvaln wh a bnary funcon, whch varabls ar h vns (h falurs). Ths bnary funcon: Y = f ( X, X,..., X n ) () s synhszd wh logcal lmns AND/OR, usng h followng symbols and sas: U (runon) for h funcon OR; ISSN: Issu, Volum 7, Dcmbr 00

2 Ncola Jula, Cosn Cpsca, Lungu Mha, Cpran Racucu, Tudor Ursu, Dan Raducanu I (nrscon) for h funcon AND; X s f h prmary lmn s good and 0 ohrws, and Y s f h sysm s good and 0 ohrws. Thus, h mhod rprsnaon s dpcd n Fgur. Fg.. a) Th gnral concp of h mhod basd on Boolan algbra (,,..., n ar ndpndn prmary vns); b) h schmacs of h logc funcon AND; c) h schmacs of h logc funcon OR For h rlably funcon ndcaors calculus, n h hypohss of h falur nnsy havng an xponnal dsrbuon, w us h rlaons: n R( ) = xp ( = xp ) () = whr: n =. = R( ) = xp n = [ ( )] (3) Rlaon () s usd for h sral conncon and h rlaon (3) s usd for h paralll conncon of h lmns. Th Analyss Mhod Applcaon In h purpos of xmplfyng h mhod for h rlably ndcaors drmnaon w wll focus on h c.c. lcrc powr supply sysm of an arcraf. Fgur dpcs h lcrc powr supply sysm for h arcraf [0], [], []. In prncpl, hs lcrc powr supply sysm qups (as h man lcrc powr supply sysm) a larg numbr of mlary arcraf from h MG famly (MG-, MG-3, MG-7, c.). Th xampl rfrs only a c.c. lcrc powr supply sysm, bu h mhod can b usd also for h alrnav currn, mxd and complx sysms [7], [], [9], [3], [4]. In Fgur : E sarr-gnraor [], [] sarup m of svral sconds (as sarr), afr a succssful sar (hr amps prmd) gos o a gnraor rgm, supplyng a V c.c. volag; 4E accumulaor swch; E nvrs polary procon dod; 3E accumulaor; 4E accumulaor o c.c. bar swch; 4E gnraor o c.c. bar couplr / d-couplr; 47E fus; 7E volag rgulaor. Th mrgng falur sa schmacs usng AND/OR lmns s dpcd n Fgur 3. Th falur vn s h loss of volag a h V bar. For h falur nnsy of h componns w us h rlaon: = k 0 (4) whr: k mannanc and way of us coffcn (for arcraf componns h coffcn vars bwn 0 and 0 []); 0 falur nnsy manufacurr spcfc daa. Th daa rlav o h lcrc powr supply sysm ar prsnd n Tabl. In hs condons, h Boolan funcon assocad o h logc srucur dpcd n Fgur 3 has h followng form: Y = X I X = ( X U X U X U X U X U X ) I ( X U X U X U X ) 9 0 I () To ransform h logc xprsson no algbrac form [3], [4], [0] w us h followng rlaons: X I X = X X, X U X = X X X X, n n () U X = ( X ) = = Thus, w hav: Y = X X = 7 = [ ( X )( X )( X )( X )( X )( X )] (7) 3 4 [ ( X )( )( )( )] X 9 X 0 X whch s smlar o Y = X X = ( X ) ( 7 X ) k () = k = Consdrng h falur nnsy as xponnal dsrbuon, h sysm falur probably: F( ) = xp { [ ( ) ]} [ xp( ) ] 9 0 = xp = (9) xp xp k p. k= p= p 7 Thus, R( ) = F( ) = xp xp k = k= (0) xp, p p= p ISSN: Issu, Volum 7, Dcmbr 00

3 Ncola Jula, Cosn Cpsca, Lungu Mha, Cpran Racucu, Tudor Ursu, Dan Raducanu Th man m bwn falur (MTBF) s [], [3] MTBF = R( )d = = 0 = = ( ) 0 ( ) k k = ( ) On rsuls MTBF = 09,79. p p= p () Thus, man m bwn falur for h non mprovd sysm may b approxmad as follows MTBF 070 hours. Fg.. Th lcrc powr supply schmacs for a c.c. man lcrc supply sysm arcraf (fragmn) Symbol Dscrpon [ ] 0 h Numbr k = nk 0 [ h ] 4E Swch E Dod E Accumulaor E Couplr E Fus Conacs E Sarr- gnraor 0 0 4E Couplr / D-couplr E Volag rgulaor Conacs F = Tabl = F = = F = = F3 = = F4 = = F = = 0 0 F = = F = = F9 = = F0 = = 0 0 F = ISSN: Issu, Volum 7, Dcmbr 00

4 Ncola Jula, Cosn Cpsca, Lungu Mha, Cpran Racucu, Tudor Ursu, Dan Raducanu Fg.3. Th logc srucur ha drvs o h sysm falur saus. Fg.4. Elcrc powr supply sysm ncludng h back-up subsysm (fragmn) Symbol Dscrpon [ h ] 0 Numbr k = nk 0 [ h ] F = Tabl 0E E Couplr Swch = F = 0 0 = F = - Conacs = F3 = ISSN: Issu, Volum 7, Dcmbr 00

5 Ncola Jula, Cosn Cpsca, Lungu Mha, Cpran Racucu, Tudor Ursu, Dan Raducanu 3 Elcrc Powr Supply Rlably Opmzaon W can mprov h lcrc powr supply sysm rlably usng a rdundan (rsrv) subsysm. Th proposd mprovd lcrc powr supply sysm, ncludng h back-up subsysm (dod lns) s dpcd n Fgur 4. Furhr on w wll analyz h mprovd lcrc powr supply sysm rlably, usng h sam mhod. Ths analyss also allows a drmnaon of a rlaon bwn h sysm rlably and h sysm wgh. Such a rlaon s ncssary o mphasz h varaon of h sysm rlably wh h oal wgh of sysm componns. Through a compard analyss of dffrn rlably mprovng varans, mposng as mnmum condon h componn wgh, w can oban an opmal soluon. Th logc srucur ha drvs o h sysm falur saus (for h mprovd sysm schmacs) s dpcd n Fgur. Tabl prsns h valus of h falur nnsy for h supplmnary componns from h back-up sysm, n h xponnal dsrbuon hypohss. Fg.. Th logc srucur ha drvs o h sysm falur saus (mprovd sysm). Th Boolan funcon n hs cas s: Y = ( X I X ) I X = ( X U X U X ) I I ( X U X U X U X U X U X ) I 3 4 () I ( X U X U X U X ). 9 0 Transformng n algbrac form, w hav: Y = [ ( X )( X )( X )] 3 4 [ ( X )( X )( X 3)( X 4 )( X )( X )] (3) [ ( X )( X )( X )( X )]. 9 0 Y = ( X) ( Xk) ( X ) p = 3 k= p= (4) From (4) w can drmn h sysm falur probably F() : F( ) = xp xp k = 3 k= xp p= xp xp xp k p () k= p= = 7 xp xp = = 3 p xp = 7 = 7,,9,0,, xp. = F () and R() ar complmnary funcons, hus, for h lcrc powr supply sysm rlably R() w wll hav h followng rlaon: ISSN: Issu, Volum 7, Dcmbr 00

. () MTBF 070 ( ) 0 valu k = 0. For hs valu on calculad MTBF boh for nal sysm (fg.) and mprovd sysm (fg.4).

6 Ncola Jula, Cosn Cpsca, Lungu Mha, Cpran Racucu, Tudor Ursu, Dan Raducanu R( ) = xp xp k = 3 k = xp p= xp xp = 7 xp = 7 = 7,,9,0,, xp. = For h MTBF, w wll hav [0]: MTBF = R( )d = 0 = 7 p = 7,,9,0,, = 3 = 7 k= = k p= p 9. () (7) Thus, usng h back-up subsysm, w ncrasd h sysm rlably. Th rsrvaon ffcncy [], [] w hav: ( ) 9 γ = MTBF r =.. () MTBF 070 ( ) 0 valu k = 0. For hs valu on calculad MTBF boh for nal sysm (fg.) and mprovd sysm (fg.4). Malab program maks a complx analyss of h nflunc of coffcn k on sysm falur probably, sysm s rlably and MTBF. In fg. and fg.7 m characrscs F() and R(), for dffrn valus of coffcn k, ar prsnd: k = 0 (blu), k = 30 (rd), k =40 (black), k =0 (magna) and k =0 (grn). As can b sn, h ncras of k s drc proporonal wh funcon F() and nvrs proporonal wh rlably R(). 4 Influnc of k Coffcn on MTBF Takng no accoun h sysm falur probably s xprssons - F () and rlably R () for schms from fg. and fg.4 on maks h Malab program prsnd n Appndx. Usng hs program on obans m voluons of varabls F and R. Fg.7. Sysm s rlably for dffrn valus of k (nal sysm) Fg.. MTBF for dffrn valus of k (nal sysm) Fg.. Sysm falur probably for dffrn valus of k (nal sysm) Coffcn k from quaon (4) has h sarng Man m bwn falur (MTBF) of h sysm s bggr for small valus of coffcn k. Thr s an nvrs proporonaly rlaonshp bwn h wo varabls (fg.). Th oband valus boh for nal ISSN: Issu, Volum 7, Dcmbr 00

Ncola Jula, Cosn Cpsca, Lungu Mha, Cpran Racucu, Tudor Ursu, Dan Raducanu sysm and mprovd sysm ar prsnd n Tabl 3. On also maks h sam analyss of h mprovd sysm (fg.4). Th conclusons ar h sam.

. Comparav analyss of h wo sysms rlably for dffrn valus of k Fg.0. Sysm s rlably for dffrn valus of k (mprovd sysm) Comparav analyss of h wo sysms rlably for dffrn valus of k s mad n fg.

corrspondng o h nal sysm (fg.3). In fg.3 h voluon of MTBF for h nal sysm s rprsnd wh dashd ln, whl h voluon of MTBF for h mprovd sysm s rprsnd wh connuous ln.

7 Ncola Jula, Cosn Cpsca, Lungu Mha, Cpran Racucu, Tudor Ursu, Dan Raducanu sysm and mprovd sysm ar prsnd n Tabl 3. On also maks h sam analyss of h mprovd sysm (fg.4). Th conclusons ar h sam. Th graphc characrscs ar h ons from fg. 9-. Fg.. MTBF for dffrn valus of k (mprovd sysm) Fg.9. Sysm falur probably for dffrn valus of k (mprovd sysm) Fg.. Comparav analyss of h wo sysms rlably for dffrn valus of k Fg.0. Sysm s rlably for dffrn valus of k (mprovd sysm) Comparav analyss of h wo sysms rlably for dffrn valus of k s mad n fg. (for nal sysm on uss h blu color and rd color for h mprovd sysm) For h fv valus of coffcn k, h mprovd sysm usng a rdundan (rsrv) subsysm s characrzd by supror valus of MTBF n rappor wh h valus corrspondng o h nal sysm (fg.3). In fg.3 h voluon of MTBF for h nal sysm s rprsnd wh dashd ln, whl h voluon of MTBF for h mprovd sysm s rprsnd wh connuous ln. MTBF for dffrn k k = 0 k = 30 k = 40 k = 0 k = 0 Inal sysm (fg.3) 4.4 hours 3.7 hours. hours 4. hours 09. hours Improvd sysm (fg.4) 9.34 hours.0 hours 7.90 hours 7.33 hours.9 hours ( MTBF γ = ) r MTBF ( ) 0 Tabl 3 ISSN: Issu, Volum 7, Dcmbr 00

8 Ncola Jula, Cosn Cpsca, Lungu Mha, Cpran Racucu, Tudor Ursu, Dan Raducanu Fg.3. Evoluons of MTBF for h wo sysms Thus, γ c. ndffrn of k' s valus. Appndx clos all; clar all; % BEFORE OPTIMIZING =:000; %Tm [hours] landa0=[ * *0]/000000; SUM=0; for =:lngh(landa0) SUM=SUMlanda0(); nd k=[ ]; for =:lngh(k) landa(,:)=landa0*k(); S()=SUM*k(); nd for =:lngh(k) F(,:)=-xp(-(landa(,7)landa(,)landa(,9) landa(,0))*)-xp(-(landa(,)landa(,) landa(,3)landa(,4)landa(,)landa(,))*) xp(-s()*); R(,:)=-F(,:); MTBF()=/(landa(,7)landa(,)landa(,9) landa(,0))/(landa(,)landa(,)landa(,3) landa(,4)landa(,)landa(,))-/s(); nd plo(,f(,:),'b'); grd; xlabl ('Tm [hours]'); ylabl ('Damag probably of h sysm'); hold on; plo(,f(,:),'r');hold on; plo(,f(3,:),'k');hold on; plo(,f(4,:),'m');hold on; plo(,f(,:),'g'); l('damag probably of h sysm for dffrn valus of k') h=fgur; plo(,r(,:),'b');grd; xlabl('tm [hours]'); ylabl('fably of h sysm'); hold on;plo(,r(,:),'r'); hold on;plo(,r(3,:),'k'); hold on;plo(,r(4,:),'m'); hold on;plo(,r(,:),'g'); l('fably of h sysm for dffrn valus of k') % AFTER OPTIMIZING landa0p=[ * * *4]/000000; SUMp=0; for =:lngh(landa0p) SUMp=SUMplanda0p(); nd kp=k; for =:lngh(kp) landap(,:)=landa0p*kp(); Sp()=SUMp*kp(); nd for =:lngh(kp) Fp(,:)=-xp(-(landap(,)landap(,) andap(,3))*)-xp(-landap(,)landap(,) andap(,3)landap(,4)landap(,) andap(,))*)-xp(-(landap(,7)landap(,) andap(,9)landap(,0))*)xp(-s()*) xp(-(sp()-landap(,7)-landap(,)-andap(,9)- -andap(,0))*)--xp(-sp()*)xp(-andap(,7) andap(,)landap(,9)landap(,0) andap(,)landap(,)landap(,3))*); Rp(,:)=-Fp(,:); MTBF()=/(landap(,)landap(,) andap(,3))/(landap(,)landap(,) landap(,3)landap(,4)landap(,) andap(,))/(landap(,7)landap(,) landap(,9)landap(,0))-/s()-/(sp()- -landap(,7)-landap(,)-landap(,9)- -landap(,0))-/sp()- /(landap(,7) landap(,)landap(,9)landap(,0) landap(,)landap(,)landap(,3)); nd h=fgur; plo(,fp(,:),'b');grd; xlabl('tm [hours]'); ylabl('damag probably of h sysm'); hold on; plo(,fp(,:),'r'); hold on;plo(,fp(3,:),'k'); hold on;plo(,fp(4,:),'m'); hold on;plo(,fp(,:),'g'); l('damag probably of h sysm for dffrn valus of k') h=fgur; ISSN: Issu, Volum 7, Dcmbr 00

9 Ncola Jula, Cosn Cpsca, Lungu Mha, Cpran Racucu, Tudor Ursu, Dan Raducanu plo(,rp(,:),'b');grd; xlabl('tm [hours]'); ylabl('fably of h sysm'); hold on; plo(,rp(,:),'r'); hold on;plo(,rp(3,:),'k'); hold on;plo(,rp(4,:),'m'); hold on;plo(,rp(,:),'g'); l('fably of h sysm for dffrn valus of k') h=fgur; plo(k,mtbf);grd; xlabl('consan k'); ylabl('man Tm Bwn Falur (MTBF)'); l('man Tm Bwn Falur (MTBF) for dffrn valus of k'); hold on; plo(k,mtbf,'*r'); % COMPARATIVE GRAPHICS (BEFORE AND AFTER OPTIMIZING) h=fgur; subplo(,,); plo(,r(,:),'b',,rp(,:),'r');grd; xlabl('tm [hours]'); ylabl('fably'); l('fably of h sysm bfor and afr opmzng (k=30)'); subplo(,,); plo(,r(3,:),'b',,rp(3,:),'r');grd; xlabl('tm [hours]'); ylabl('fably'); l('fably of h sysm bfor and afr opmzng (k=40)'); subplo(,,3); plo(,r(4,:),'b',,rp(4,:),'r');grd; xlabl('tm [hours]'); ylabl('fably'); l('fably of h sysm bfor and afr opmzng (k=0)'); subplo(,,4); plo(,r(,:),'b',,rp(,:),'r');grd; xlabl('tm [hours]'); ylabl('fably'); l('fably of h sysm bfor and afr opmzng (k=0)'); h=fgur plo(k,mtbf,'*r',kp,mtbf,'or');grd; xlabl('consan k'); ylabl('man Tm Bwn Falur (MTBF)'); l('man Tm Bwn Falur bfor and afr opmzng'); hold on; plo(k,mtbf,'--b',kp,mtbf,'b'); Conclusons From h analyzd xampl, w can conclud ha hs mhod can b usd n h onboard lcrc powr supply rlably drmnaon. Th MTBF nfluncng paramrs n h man sysm pons (powr supply bars and dsrbuon panls) can b drmnd and analyzd. Through h falur rlad logc funcon analyss w can drmn h crcus ha can mprov h sysm rlably. In h concr cas, hrough h nroducon of h componns 0E, E and corrspondng conacs, w oband a subsanal ncras of h rlably (approxmaly ms hghr) for h V c.c. powr supply bar. On also mad a complx analyss of h nflunc of coffcn k on sysm falur probably, sysm s rlably and MTBF. On can also nocd ha h ncras of k s drc proporonal wh funcon F() and nvrs proporonal wh rlably R(). Rfrncs: [] Hohan, I., Fablaa ssmlor mar, E.D.P., Buchars, 9. [] Jula, N., Conrbuţ la opmzara ssmlor lcromagnc d la bordul avoanlor mlar. PhD. Thsys, Buchars, 9. [3] Dns-Papn, M., Malgrang, Y. Exrcţ d calcul boolan cu soluţl lor, Ed. Thncă, Buchars, 970. [4] Mosl, G. Tora algbrcă a mcansmlor auoma. Ed. Thncă, Buchars, 979. [] Rus, I. Traara smbolcă a schmlor d comuaţ. Ed. Acadm R.S.R, Buchars, 97. [] Drujnn, G.V., Sguranţa în funcţonar a ssmlor. Edura Thncă, Buchars, 9. [7] Lungu, M., Lungu, R., Jula, N., Cpsca, C., Calburanu, M. Opmal Conrol of Flyng Objcs Mov Afr Esmad Sa Vcor Usng a Rducd Ordr Obsrvr. WSEAS Transacons on Crcus and SYSTEMS, Issu, Volum 7, Jun 00, pp. 0. [] Lungu, R., Lungu, M., Jula, N., Cpsca, C., Modls dnfcaon and adapv conrol of h flyng objcs usng nural nworks. Procdngs of WSEAS Confrncs, Arcachon, Ocobr, 3-, 007. [9] Al Evcl, Progrssv Falur Analyss of Pn Jons n Compos Lamnas. Procdngs of WSEAS Appld Compung Confrnc (ACC '0), Isanbul, Turky, May 7-30, 00. [0] Amrasu Snha, Algorhms on machns. Procdngs of h WSEAS Inrnaonal Confrnc on Robocs, Conrol and Manufacurng Tchnology (ROCOM '0), Hangzhou, Aprl -, 00. ISSN: Issu, Volum 7, Dcmbr 00

10 Ncola Jula, Cosn Cpsca, Lungu Mha, Cpran Racucu, Tudor Ursu, Dan Raducanu [] Sandhu, K.S., Vadhra, S. Racv Powr Rqurmns of Grd Conncd Inducon Gnraor n a Wak Grd. WSEAS Transacons on Crcus and Sysms, Issu, Volum 7, March, 00. [] Muz, F. Ral-m Volag Conrol o Improv Auomaon and Qualy n Powr Dsrbuon. WSEAS Transacons on Crcus and Sysms, Issu, Volum 7, Aprl, 00. [3] Hahanov, V., Obrzan, V., Lvnova E., Ka Lok Man. Algbra-Logcal Dagnoss Modl for SoC F-IP. WSEAS Transacons on Crcus and Sysms, Issu, Volum 7, July, 00. [4] Wn-Yau Chang. An Acv Islandng Dcon Mhod for Small-Scal Dsrbud Gnraors. WSEAS Transacons on Crcus and Sysms, Issu, Volum 7, Jun, 00. [] Huang Pham. Handbook of Rlably Engnrng, Sprngr Vrlag London Lmd 003. ISSN: Issu, Volum 7, Dcmbr 00

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm