Control of Chaotic Behaviour in Parallel-Connected DC-DC Boost Converters

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1 Aalable ole at Eergy Proceda 8 (0 ) Cotrol of Chaotc Behaour Parallel-Coected DC-DC Boost Coerters G.N. Aldahm a, A.N. Natsheh b, H. Orkozak a, a* a Departmet of Power Egeerg, Damascus Uersty, Syra b Faculty of Egeerg, Al-Ahlyya Amma Uersty, Post Code 938 Amma, Jorda Abstract Chaos cotrol meas to desg a cotroller that s able to elmatg the chaos behaour of olear systems that experecg such pheomeo. ths paper, a delayed feedback cotrol mode s preseted. The paper descrbes the cotrol of the bfurcato behaour of a modular peak curret-mode cotrolled DC-DC boost coerter whch s used to prode a terface betwee eergy storage batteres ad photooltac (PV) arrays reewable eergy sources. The parallel-put/parallel-output coerter comprses two detcal boost crcuts ad operates the cotuous-curret coducto mode. A comparso s made betwee waeforms obtaed from a MATLAB/SMULNK model for ope-loop coerter ad the cotrolled oe. The study shows the effecteess of the desged delayed feedback cotrol mode. 0 0 Publshed by by Elseer Elseer Ltd. Selecto Ltd. Selecto ad/or peer ad/or reew peer-reew uder resposblty uder resposblty of The TerraGree of TerraGree Socety. Socety. Keywords: Bfurcato ad Chaos Theory; Boost Coerter; Delayed Feedback Cotrol Mode.. troducto Curret-cotrolled DC-DC swtch-mode coerters are o-lear electroc deces ad t has bee obsered that they may behae a chaotc maer. The desg of such coerters, wth a low harmoc cotet ad low oltage rpple s a costat challege; ths has geerated much terest, partcularly sgle-stage topologes such as the buck, boost, ad cuk coerters [-7]. Ths paper estgates the cotrol of chaotc behaour a two-module parallel put/parallel-output boost coerter operatg uder a peak curret-mode cotrol scheme usg a delayed feedback cotrol * Correspodg author. Tel.: E-mal address: eg.ghadad@hotmal.com Publshed by Elseer Ltd. Selecto ad/or peer reew uder resposblty of The TerraGree Socety. do:0.06/j.egypro

2 76 G.N. Aldahm et al. / Eergy Proceda 8 ( 0 ) strategy, whch s based o the dea of stablzato of ustable perodc orbts that exst the chaotc attractor [8], where each module has ts ow curret feedback loop. The coerter cossts of two smlar boost crcuts operatg the cotuous-curret coducto mode. A o-lear map for the coerter was dered usg dscrete tme modellg ad umercal terato of the map produces bfurcato dagrams whch dcate the chaotc operato. order to check the aldty of the cotroller, a MATLAB/SMULNK model for the cotrolled coerter was deeloped. The waeforms for the ope-loop model already produced by the MATLAB/SMULNK model whch tegrates umercally the system state-arable equatos for the coerter. The results for the cotrolled model are compared wth the ope-loop model.. Bfurcato ad Chaos Theory Bfurcato theory, orgally deeloped by Pocare, s used to dcate the qualtate chage behaour of a system terms of the umber ad the type of solutos, uder the arato of oe or more parameters o whch the system depeds [9-0]. The system state arables ad the cotrol parameters are defed ad the relatoshps betwee ay of these cotrol parameters ad a state arable s called the state-cotrol space. ths space, locatos at whch bfurcatos occur are called bfurcato pots. Bfurcatos of a equlbrum or fxed-pot soluto are classfed as ether statc bfurcatos, such as saddle-ode, ptch fork, or trascrtcal bfurcatos; or as dyamc bfurcatos whch are also kow as the Hopf bfurcato that exhbts perodc solutos. For the fxed-pot solutos, the local stablty of the system s determed from the egealues of the Jacoba matrx of the learzed system. O the other had, wth perodc-solutos, the system stablty depeds o what s kow as the Floquet theory ad the egealues of the Moodromy matrx that are kow the lterature as Floquet or characterstc multplers. The types of bfurcato are determed from the maer whch the Floquet multplers leae the ut crcle. There are three possble ways for ths to happe [9-0]: ) f the Floquet multpler leaes the ut crcle through +, the three possble bfurcatos may occur: trascrtcal, symmetry-breakg, or cyclc-fold bfurcato. ) f the Floquet multpler leaes through -, the perod-doublg (Flp bfurcato) occurs. ) f the Floquet multplers are complex cojugate ad leae the ut crcle from the real axs, the system exhbts secodary Hopf bfurcato. To obsere the system dyamcs uder all the aboe possble bfurcatos, a complete dagram kow as a bfurcato dagram s costructed. A bfurcato dagram shows the arato of oe of the state arables wth oe of the system parameters, otherwse kow as a cotrol parameter. 3. Coerter Operato A crcut dagram for the module boost coerter s show Fg.. A smplfed dagram for the proposed coerter cossts of two peak curret-mode cotrolled DC-DC boost coerters whose outputs are coected parallel to feed a commo resste load. Each coerter has ts ow curret feedback loop comprsg a comparator ad a flp-flop. Each comparator compares ts respecte peak ductor curret wth a referece alue, to determe the o-tme of the swtch.

3 G.N. Aldahm et al. / Eergy Proceda 8 ( 0 ) Fg.. Module Boost Coerter Crcut Dagram The terate map descrbg the system takes the form: x f ( x, ) () ref where subscrpt deotes the alue at the begg of the th cycle ad x s the state ector c x () c where:, are the currets through ductors L ad L, respectely. c, c are oltages across capactors C ad C, respectely. So the terate map: kl [ V L e kt kt ref ( ) ( ) e s t ref cos t ] (3) k e kv kt kt ref kt c ( ) c( ) V e [ s t ( e )cos t ] / C V R V (4) A MATLAB program was wrtte to produce bfurcato dagrams for the two-module boost coerter, wth the omal parameters ge Table.

78 G.N. Aldahm et al. / Eergy Proceda 8 ( 0 ) 75 79 Table. System Parameters Fg.

Fg.. Bfurcato dagram for the two-module coerter 4.

us)] dt L d dt L [ L C ( u S )] dc dc C [ L ( us) L ( us )] dt dt C R Where u S ad u S take the alue 0 or depedg o

4 78 G.N. Aldahm et al. / Eergy Proceda 8 ( 0 ) Table. System Parameters Fg. shows the bfurcato dagram for load curret of the proposed system wth the referece curret as the bfurcato parameter. Fg.. Bfurcato dagram for the two-module coerter 4. MATLAB/SMULNK Smulato (Smulated Ope-Loop) MATLAB/SMULNK model was deeloped based o the system equatos: dl [ C ( us)] dt L d dt L [ L C ( u S )] dc dc C [ L ( us) L ( us )] dt dt C R Where u S ad u S take the alue 0 or depedg o whether the swtches or are closed or ope. The result for chaotc regmes at ref 5 A s obtaed for the ductor curret waeform ad s ge Fg. 3. The ductor curret at swtch tur-o ow hae may alues. (5) Fg. 3. Chaotc ductor curret

G.N. Aldahm et al. / Eergy Proceda 8 ( 0 ) 75 79 79 5.

the feedback stregth. The dea behd the scheme reles o the fact that a skeleto of a chaotc attractor s formed by a fte set of ustable perodc orbts wth dfferet perods.

5 G.N. Aldahm et al. / Eergy Proceda 8 ( 0 ) Cotrol of Chaos The feedback cotrol force F(t), appled to the system s the dfferece betwee the curret alue of some system arable y(t), ad ts alue secods preously, multpled by a costat K, where K s the feedback stregth. The dea behd the scheme reles o the fact that a skeleto of a chaotc attractor s formed by a fte set of ustable perodc orbts wth dfferet perods. f the alue of the tme delay s exactly equal to the perod T of oe of the orbts, the at the approprate alues of K, the orbt ca become stable ad chaos wll be elmated. As the cotroller coected to the ope-loop model, the ductor curret at swtch tur-o ow does ot hae may alues ad the perodc regme s stable as show Fg Cocluso Fg. 4. Delayed Feedback Cotrol The smulated tme waeforms for L (t) at ref 5 A wth the coerter coected to the delayed curret feedback (the cotrol crcut) showed the coerter s operatg a "perod-" mode as requred. The study shows the effecteess of the desged delayed feedback cotrol mode. The results dcated that delayed feedback cotrol mode results a greater rage of stable perod- operato, compared wth the tradtoal method. Refereces [] Deae, J. H. B., Hamll, D. C. stablty, Subharmoc ad Chaos Power Electroc Crcuts. EEE Trasactos o Power Electrocs; Vol. 5, 990, pp [] Deae, J. H. B. Chaos a Curret-Mode Cotrolled Boost DC-DC Coerter. EEE Trasactos o Crcuts ad systems- : Fudametal Theory ad Applcatos; Vol. 39. No. 8, August 99, pp [3] Hamll, D. C., Deae, J. H. B., Jefferes D. J. Modelg of Chaotc DC-DC Coerters by terated Nolear Mappg. EEE Trasactos o Power Electrocs; Vol. 7, No., 99, pp [4] Tse, C. K. Flp Bfurcato ad Chaos Three-State Boost Swtchg Regulators. EEE Trasactos o Crcuts ad systems-: Fudametal Theory ad Applcatos; Vol. 4, No., Jauary 994, pp. 6-. [5] Tse, C. K., Cha W. C. Y. Chaos from a curret programmed Cuk coerter. t. J. Crcut Theory Applcatos; Vol. 3, May-Jue 995, 7-5. [6] Marrero, J. L. R., Fot, J. M., Verghese, G. C. Aalyss of the chaotc regme for DC-DC coerters uder curret-mode cotrol. Power Electrocs Specalsts Coferece; PESC 996, pp [7] Baerjee, S. Nolear Modelg ad Bfurcato Boost Coerter. EEE Trasactos o Power Electrocs; Vol. 3, No., 998, pp [8] K. Pyragas. Cotuous cotrol of chaos by self-cotrollg feedback. Physcs Letters; 70, 99, [9] Nayfeh, A. H., Balachadra B. Appled Nolear Dyamcs; Joh Wley, New York, 995. [0] Khall, H., Nolear Systems, Thrd Edto, Pretce Hall, Upper Saddle Rer, New Jersey, 00.

d dt d d dt dt Also recall that by Taylor series, / 2 (enables use of sin instead of cos-see p.27 of A&F) dsin

d dt d d dt dt Also recall that by Taylor series, / 2 (enables use of sin instead of cos-see p.27 of A&F) dsin Learzato of the Swg Equato We wll cover sectos.5.-.6 ad begg of Secto 3.3 these otes. 1. Sgle mache-fte bus case Cosder a sgle mache coected to a fte bus, as show Fg. 1 below. E y1 V=1./_ Fg. 1 The admttace