Modeling and simulation of PEM fuel cell power converter system

Transcription

1 Modelng and smulaton of PEM fuel cell power conerter system M. Petrnć and Ž. Jakopoć Department of electrc machnes, dres and automaton Faculty of Electrcal Engneerng and omputng Unska 3, Zagreb, roata Telephone: , Fax: Emal: Abstr Parastc capactances of Proton Exchange Membrane (PEM) fuel cell are causng electrcal effects resultng wth change of dynamc behaor of fuel cell stack output oltage. Ths paper shows PEM fuel cell dynamc model wth capablty of easy ntegraton of humdty, temperature and pressures dynamcs, as well as ther control. Fuel cell stack dynamc model was lnked wth boost conerter aeraged dynamc model, obtaned usng state space method, contanng controller that keeps conerter output oltage constant. A arety of step load changes was smulated on ths structure, resultng effects were obsered on stack current, stack oltage and conerter output oltage responses. Obtaned results are showng dfference between correspondng responses of ths model and the one whch neglects effects of parastc capactances of PEM fuel cell. I. INTRODUTION Drect electrochemcal transformaton wth effcency of up to 45 %, hgh energy densty out of small dmensons (up to W/cm ), slent operaton and zero gas emssons are attrbutes that cause large nterest n PEM fuel cells and are reasons that applcaton of ths technology s consdered n transport, statc producton of electrc energy and power supply for wreless electrcal deces. But for desgnng a good power source based on ths technology, fuel cell behaor, ts charerstcs, connected power conerter and ts control, ther nteron, as well as mathematcal models used to descrbe and smulate dfferent operaton modes, hae to be well known. Only then, satsfory system behaor and energy qualty can be acheed. Today a large number of mathematcal PEM fuel cell models exsts, whose purpose spreads from fuel cell desgnng, across descrbng statc and dynamc behaor, up to operaton analyss n complex systems, where fuel cells hae to meet specfc work condtons. Most of dynamc models are dealng wth temperature and flud pressures, because these arables are known to hae tme constants that can last een few seconds, whle nfluence of pararastc capactances on output cell oltage s often neglected. Models that take nto account ths effect [4] usually use hghly eoled fuel cell equalent electrcal crcuts composed of resstancecapactance parallels connected n seres. These complex structures are descrbng hgh frequency electrc effects ery precsely, but determnaton of ther capactance alues demands addtonal electrc measurement on a real fuel cell, and presents seere problem. That s the reason why smplfed model, that has only one tme constant, was used n ths case, to descrbe electrc dynamc of PEM. In ths model, all parastc capactances are represented wth one connected n parallel wth aton and entraton resstances of fuel cell. As far as power conerters are erned, papers can be found that focus on power conerters whose charerstcs and regulaton technques hae been customzed to sut PEM fuel cells. Some of them deal wth desgns that allow cheap producton [5], whle others, based on good knowledge of PEM charerstcs, descrbe complex structures resultng wth hgh energy qualty, by usng hybrd PEMbattery power source [6]. As these models are often customzed for certan purpose (for nstance automote applcatons), they are not sutable for studyng of fuel cell operaton effects on power conerter operaton, nether for analyzng effects caused by dfferent controllers. For those reasons, approprate boost power conerter dynamc model s used here, that when connected to a fuel cell model, ges stack current, stack oltage and conerter output oltage responses n short smulaton executon tme. In that way, a system sutable for nteron between PEM fuel cell and power conerter was obtaned. II. FUEL ELL MODEL PEM fuel cell electrochemcal process starts on the anode sde (Fg.) where H molecules are brought by flow plate channels. Anode catalyst ddes hydrogen on protons H that trael to cathode through membrane and electrons e that trael to cathode oer external electrcal crcut. At the cathode hydrogen protons H and electrons e combne wth oxygen O by use of catalyst, to form water H O and heat. Descrbed reons can be expressed usng equatons: H H e (Anode), () O H e 0 (athode). () H Amount of chemcal energy released n these reons depends on hydrogen pressure, oxygen pressure and fuel cell temperature. Usng change n Gbbs free energy, ths amount can be expressed as: 0 Δ g f = Δg f RT ln( ph ) ln( po), (3)

2 Fg.. Fuel cell mage Fg.. Fuel cell losses where Δg 0 f s change n Gbbs free energy at standard pressure, R unersal gas constant, T PEM temperature and p O and P H are gas pressures. Because electrcal work done by fuel cell s equalent to released chemcal energy, alue of open crcut fuel cell oltage E meets equaton: Δg f E =, (4) F where F s Faraday's constant. To attan ual cell oltage (on electrcal couplngs), oltage drops caused by aton, entraton and c losses hae to be deducted from open crcut oltage (Fg ). athode and anode aton losses are result of breakng and formng electronproton chemcal bonds, and parastc electrochemcal reons caused from hydrogen proton mgraton through membrane at zero current. Ther oltage drop was calculated usng formula: ( e ) c 0 a =, (5) where aton oltage drop at zero current densty 0 depends on fuel cell temperature, cathode pressure and water saturaton pressure 0 = f ( T, pca, psat ). Voltage drop a nserts n (5) correlaton wth current densty and depends on fuel cell temperature, oxygen pressure and water saturaton pressure a = f ( T, po, psat ), and c s aton oltage constant. oncentraton losses are caused by drop n reant entraton due to dynamc flow problems between water and oxygen on cathode sde, and also electroosmotc water drag that occurs when protons trael through membrane. Voltage drop caused by these losses s descrbed wth equaton: c3 = c, (6) max where max represents current densty that causes steep PEM oltage drop, parameter c s functon of temperature, oxygen pressure and water saturaton pressure, c = f ( T, po, psat ), and c 3 s entraton oltage constant. Ohmc losses are dered from membrane resstance R whose alue depends of membrane thckness t m, fuel cell temperature T and membrane water content degree λ m : R t m =, (7) σ m ( ) σ m = b λm b exp b. (8) 303 T Value σ m n (8) represents specfc membrane conductty and b, b and b are membrane conductty constants. Voltage drop of c losses s expressed as: = R. (9) Usng calculated oltage drops and open crcut cell oltage, alue of ual cell oltage n statc condton can be attaned usng equaton: = E. (0) Dynamc electrc model was ganed by mplementng nfluence of parastc capactance n preously descrbed statc model. Fuel cell equalent electrc crcut wth capactance s shown n fgure 3. On t R s resstance that corresponds to aton losses, and R resstance that represents entraton losses: R R ( e ) c 0 a = =, () c3 = = c. () max

3 Fg. 3. Fuel cell equalent electrc crcut Based on electrcal crcut from fgure 3, followng equatons that show currentoltage relatons can be wrtten: d =, (3) dt R R = E R. (4) Usng (3), (4), and Laplace transformatons, transfer functon (5) was obtaned, n whch s represents Laplace operator: R R = E s R R ( R R ) c R R, (5) Usng expressed equatons and MatLab Smulnk, PEM fuel cell dynamc model was created (fgure 6), that allows change of membrane water content degree (lambdam), alue of pressures (ph, po, pca), temperature (T) and current densty (). Because of ths feature easy mplementaton of pressures, humdty and temperature dynamcs as well as ther regulaton can be acheed usng external blocks. III. DYNAMI BOOST ONVERTER MODEL Boost conerter, shown n fgure 4, was desgned for 500 W electrc power, nput oltage n between V and 8 V and output oltage out of 48 V. Takng nto account aboe mentoned nput oltage span and f swtchng frequency of 0 khz, nductor L was desgned to keep nput current L rpple under 5 % (L =0,405 H), and capactor to keep output oltage out rpple under 0,5 % ( =,68 mf). Vn L L ESR T Fg. 4. Boost conerter electrc crcut T D D E R Vout To descrbe ths conerter usng state space method, followng ectors had to be defned: state space ector: L x =, nput ector: u = [ n ], output ector: y = [ out ]. Usng these ector equatons, conerter behaor s expressed: x & = Ax Bu, (6) y = Ex Fu. (7) where A, B, E and F represent data matrces. Parameters of these matrces were separately determnated for both (turn on & off) condtons of T transstor swtch. By mergng them, aeraged conerter model s obtaned (8, 9): ESR & L L L L L, (8) = [ n ] D & D R L [ out ] = [ 0 ] [ 0] [ n 0 ]. (9) In these equatons ESR represents nductor resstance, R load resstance, D swtch dutyrato and capactor oltage. Whle desgnng PI controller, each tme dependant parameter was consdered as a sum of one statc and one ~ tme dependant part (for example: ). Usng L = I L L (8), (9) and Laplace transformatons, transfer functons that show relatons between conerter output oltage and dutyrato ( ~ ~ d ( s) ), conerter output and d = out nput oltage ( ~ ~ ) and conerter output n = out n oltage and Pulse Wdth Modulator (PWM) output ( ~ ~ ( s) ) were obtaned. c = out c Usng frequency analyss on an open crcut transfer functon: T ( s) = β, (0) of a boost conerter control crcut (fgure 5), proportonal K p gan and ntegral K I gan of PI regulator were obtaned. Equaton () shows PI regulator transfer functon: G c K Pωz K I ( s) = K P = K, () P s s ca ca Wth regulator parameter alues K p = 0,05 and K I = 8,85, gan margn of 7,7 db, and phase margn of 9.7, was acheed, whch s n accordance wth recommendatons (6 db < G.M. < 0 db, 45 < P.M.).

4 V ref V c V out G ca(s) G c(s) In presented aeraged model of regulated boost conerter, realzed usng MATLAB Smulnk (fgure 7), PI regulator conerter load alue can be changed by nput (R), and nput oltage can be changed usng nput (Vn), whle output β oltage and nput (nductor) current can be obsered usng outputs (Vout) and (L) respectely. Fg. 5. Boost conerter control crcut block representaton Fg. 6. MATLAB/Smulnk dynamc fuel cell model Fg. 7. MATLAB/Smulnk dynamc model of controlled boost conert

5 VI. SYSTEM RESPONSE DUE TO INFLUENE OF PARASITI APAITANES Wth presumpton of the same condtons n each cell of a fuel cell stack, fuel cell model was modfed nto a PEM stack model by multplyng fuel cell output oltage wth number of cells N: = N. () stack Also because stack current I s used as an nput arable of PEM stack, current densty was calculated usng (3), where A represents a fuel cell e area: I =. (3) A ontrolled boost conerter model was lnked wth descrbed PEM model and a number of step load changes was smulated usng nput port (R) (fgure 8). Fuel cell stack output oltage, fuel cell stack current and boost conerter output oltage responses (shown n fgures 9,0 and ) were obsered for cases of ncluded and excluded PEM dynamc caused by parastc capactance. They show that effects caused by parastc capactances decrease system stablty, resultng wth the ncrease of transent tme duraton and related oscllatons. These effects are n accordance wth responses of a real PEM fuel stack expermentally obtaned n [7]. It s nterestng to note that n case of fuel cell stack output oltage V, parastc effects are dmnshng the sze of transent response peak alue. That means that PEM fuel cell model wth ncluded capactances ges more realstc results than common smple PEM fuel cell model. Fg. 8. MATLAB/Smulnk model of system PEM regulated boost conerter Fg. 9. Fuel cell stack current response I

6 Fg. 0. Fuel cell stack output oltage response V V. ONLUSION Fg.. Boost conerter output oltage response V out Smulaton results are showng that applyng a model structure of fuel cell and boost conerter, takng parastc capactances nto consderaton, results wth better understandng of behaor of controlled system based on PEM fuel cell as power source. Ths allows new possbltes n research and can result wth better system component dmensonng and controller desgn for such power systems. REFERENES [] J. T. Pukrushpan, A. G. Stefanopoulou, H. Peng, ontrol of Fuel ell Power Systems, nd prntng, SprngerVerlag, London Lmted, 005. ISBN: [] G. Hoogers, Fuel cell technology handbook, R Press LL, 003. ISBN [3] Z. Lemeš, A. Vath, Th. Hartkopf, H. Mäncher, Dynamc fuel cell models and ther applcaton n hardware n the loop smulaton, Journal of Power Sources 54, , 006. [4] J. Garner, M.. Pera, D. Hssel, F. Harel, D. andusso, N. Glandut, J. P. Dard, A. De Bernardns, J. M. Kauffmann, G. oquery, Dynamc PEM Fuel ell Modelng for automote applcatons, IEEE /03, 003. [5] P. T. Kren, R. Balog, Low ost Inerter Sutable for MedumPower Fuel ell Sources, IEEE X/0, 00. [6] D. K. ho, B. K. Lee, S.W..,. Y. Won, D. W. Yoo, A noel power conerson crcut for costeffecte batteryfuel cell hybrd systems, Journal of Power Sources 5, 45 55, 005. [7] G. Fontes,. Turpn, R. Sasset, T. Meynard, S. Aster, Interons between fuel cells and power conerters Influence of current harmoncs on a fuel cell stack, IEEE /04, 004. [8] J. Golbert, D. R. Lewn, Modelbased control of fuel cells: () Regulatory control, Journal of Power Sources 35, 35 5, 004. [9] D. Žarko, Boost conerter desgnng, FER, Zagreb, 006. (In roatan)