ISSN Article. Analysis of the Coupling Behavior of PEM Fuel Cells and DC-DC Converters

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1 Enrgis 29, 2, 71-96; doi:1.339/n2171 OPEN ACCESS nrgis ISSN Articl Analysis of th Coupling Bhavior of PEM Ful Clls and DC-DC Convrtrs Markus Grötsch 1,, Michal Mangold 1 and Achim Kinl 1,2 1 Max Planck Institut for Dynamics of Complx Tchnical Systms, Procss Synthsis and Procss Dynamics Group, Sandtorstraß 1, 3916 Magdburg, Grmany 2 Otto-von-Gurick-Univrsität Magdburg, Lhrstuhl für Automatisirungstchnik / Modllbildung, Univrsitätsplatz 2, 3916 Magdburg, Grmany Author to whom corrspondnc should b addrssd; grotsch@mpi-magdburg.mpg.d. Rcivd: 8 January 29 / Accptd: 19 Fbruary 29 / Publishd: 4 March 29 Abstract: Th connction btwn PEM ful clls and common DC-DC convrtrs is xamind. Th analysis is modl-basd and don for boost, buck and buck-boost convrtrs. In a first stp, th ffct of th convrtr rippls upon th PEM ful cll is shown. Thy introduc oscillations in th ful cll. Thir apparanc is xplaind, discussd and possibilitis for thir supprssion ar givn. Aftr that, th ovrall bhaviors of th coupld ful cll-convrtr systms ar analyzd. It is shown, that nithr stationary multiplicitis nor oscillations can b introducd by th couplings and thrfor sparat control approachs for both th PEMFC and th DC-DC convrtrs ar applicabl. Kywords: PEM ful cll; DC-DC convrtrs; analysis; rippl; oscillations. 1. Introduction Ful clls ar a promising tchnology for lctrical powr gnration. Thy ar abl to convrt chmical nrgy dirctly into lctrical nrgy, avoiding an intrmdiat stp of producing mchanical nrgy. Thrfor, th lctrical fficincy of ful clls is considrably highr than that of most convntional procsss for lctrical powr gnration. For mobil applications, Polymr-Elctrolyt-Mmbran ful clls (PEMFCs) ar most suitabl. Ths clls can b charactrizd by a high powr dnsity, an asy production, a low-tmpratur opration and a fast rspons to load changs. A ful cll is lctrically connctd to its load via a powr conditioning unit (PCU) [1]. This is don

2 Enrgis 29, 2 72 for th purpos of powr transfr and powr convrsion. A PCU is gnrally mad up from storag units and/or convrsion units and is typically dsignd and opratd according to rquirmnts and charactristics of th load. Whil storag units buffr lctrical nrgy, convrsion units or convrtrs ar usd to adapt th DC lctricity from th ful cll to th load s dmands. Two typs of convrtrs ar suitabl in ful cll opration: DC-DC and DC-AC convrtrs. If a ful cll is connctd to a load via a PCU a complx dynamic systm is cratd. Such a connction might lad to phnomna lik multiplicitis or oscillations, which ar not prsnt in th singl systms. Ths phnomna can contribut to th prformanc of th whol systm ithr in a positiv or in a ngativ way. Thrfor, a dtaild invstigation of th coupling is ncssary to adapt and improv th dsign and opration of th whol systm, spcially if th original dsign was basd on singl sparat subsystms. Th coupling bhavior of PEMFCs and PCUs is a currnt fild of rsarch. First rsults hav bn obtaind for th coupling of PEMFCs and DC-AC convrtrs [1 3]. Th coupling of ths systms lads to a rippl in th ful cll currnt at a frquncy that is twic th output frquncy of th convrtr. This ffct was analyzd in [1 3] and may contribut to ful cll dgradation. In [4] th control of a PEMFC connctd to a buck-boost convrtr was invstigatd. Th mphasis was on convrtr control and th coupling phnomna wr hardly considrd. This work tris to xtnd th rsults in litratur by analyzing th coupling phnomna btwn PEMFCs and DC-DC convrtrs. Th contribution is dividd in 5 sctions. In th nxt sction th usd PEMFC modl is prsntd. Aftr that th DC-DC convrtr modls ar statd. In th following sction th rsults of th analysis ar shown. Th contribution nds with a conclusion. 2. Modling of th PEM ful cll In this contribution w us a dynamic, lumpd on-phas ful cll modl. Th main modl charactristics ar: It is assumd that thr is no liquid watr in th gas bulks and th diffusion layrs. Only th cathod of th ful cll is considrd. Th anodic raction is assumd to b in quilibrium and th anodic ovrpotntial is qual to zro. Th cathodic gas bulk and gas diffusion layr ar modld as on prfctly mixd phas (Figur 1). Th cathodic catalyst layr and th mmbran ar modld by th quivalnt lctrical circuit in Figur 1, as was suggstd in [5]. Th dynamic bhavior of th mmbran s watr houshold is nglctd. Th modl is isothrmal and th gas phass ar isobaric and bhav lik an idal gass. Th Tafl approach is usd for th cathod kintics. In th following th modl quations ar prsntd. For a drivation of th modl quations plas s Appndix A. Th apparing quantitis and thir valus and units ar listd in th nomnclatur at th nd of this articl. Th paramtrs for th ful cll modl ar mainly takn from [6], whil th paramtrs for

3 Enrgis 29, 2 73 Figur 1. Modling approach for th PEM ful cll. V c U cll + i cll x in o 2 x in h 2 o V in x o2 x h2 o x o2 x h2 o V out C dl i c i r (η c, x o2 ) η c U cll lctrod 2H O 2 H 2 O(g) mmbran U cll r m (x h2 o) H + th DC-DC convrtrs ar chosn according to guidlins in [7, 8]. First of all, th dynamic quations of th modl ar spcifid: V c ẋ o2 = ˆq(1 + x o2 ) i r + (x in o 2 x o2 ) V in, (1) V c ẋ h2 o = ˆq(2 x h2 o) i r + (x in h 2 o x h2 o) V in, (2) C dl η c = i r i cll with i r := i r x o2 xp( ˆαη c ) (3) and ˆq := A g /2c t nf, ˆα := (1 α)nf/rθ. (4) Th diffrntial quations (1) and (2) ar usd to calculat th contnt of oxygn x o2 and watr vapor x h2 o in th PEMFC. Th symbol η c dnots th ovrpotntial at th cathodic catalyst and is calculatd from Eqn. (3). Th lctrical currnt dnsity in th ful cll is givn by i cll. V in dscribs th volum flow rat of humidifid air that ntrs th cll. Th othr trms ar constant modl paramtrs. An additional algbraic quation is usd to calculat th cll voltag U cll : U cll = U cll + η c r m (x h2 o) i cll. (5) Equation (5) includs activation losss of th catalyst via η c and ohmic losss in th cll voltag via th rsistanc r m of th mmbran for proton transport. Th mmbran s rsistanc is calculatd from its proton conductivity σ p : r m (x h2 o) := d m /σ p (x h2 o) whr th following dpndncy from [9] is usd: σ p (x h2 o) = σ p xp(14 (x h2 o p g /p sat ).2 ). In summary, th ful cll modl is mad up by a systm of 3 nonlinar ordinary diffrntial quations and an additional algbraic quation. Th PEMFC modl is opratd at on-phas conditions with rspct to th watr houshold and at a common oprating tmpratur of Θ = 353K [1]. It is fd with air and will b opratd in rhostatic mod du to th coupling with th DC-DC convrtrs. Although this ful cll modl is rlativly simpl and consists only of a singl cll with a cross-sctional ara of on squar cntimtr [6, 11], th modling approach is complx nough to analyz and outlin th ssntial qualitativ ffcts du to th coupling btwn on-phas or two-phas PEM ful clls and DC-DC convrtrs as w will show during th analysis.

4 Enrgis 29, Modling of th DC-DC convrtrs Th purpos of DC-DC convrtrs is th transformation of dirct lctricity. Thy ar built up from powr lctronic dvics and opratd as switchd systms. Du to th switchd opration th output quantitis of ths systms show an unavoidabl rippl which should b small. Thr DC-DC convrtrs ar considrd in this contribution: boost, buck and buck-boost convrtr [7]. Th convrtrs ar assumd to b losslss and ar modld with rsistiv loads. In ordr to xamin th coupling ffcts btwn PEMFC and convrtrs du to switching, th DC-DC convrtrs ar modld via switchd diffrntial quations. In th cas of th boost convrtr (Figur 2) thy rad: L I l = U (1 s) U c (6) C U c = (1 s) I l U c /R L (7) whr I l is th inductor currnt, U c th capacitor voltag and s is th switching function shown in Figur Figur 2. Losslss boost convrtr with rsistiv load. I l S s = L U s = 1 C U c RL 3. Th load rsistanc R L is assumd to b constant ovr on switching priod T. Not that th input currnt of th convrtr, dnotd with I is qual to th inductor currnt: I = I l. Anothr widly usd Figur 3. Duty cycl of th DC-DC convrtrs. s 1 t on T t t + t on t + T t DC-DC convrtr is th buck convrtr (Figur 4). It can b modld with th following quations: L I l = s U U c (8) C U c = I l U c /R L (9) whr I l is dnots th convrtr s inductor currnt and U c its capacitor voltag. Th switching function s is th sam as for th boost convrtr (Figur 3). Th input currnt I to th convrtr in this cas is qual to: I = s I l. Th buck-boost convrtr (Figur 5) is th last considrd convrtr and can also b

5 Enrgis 29, 2 75 Figur 4. Losslss buck convrtr with rsistiv load. I s = 1 S I l L s = U C U c R L modld by switchd diffrntial quations: L I l = s U + (1 s) U c (1) C U c = (1 s) I l U c /R L. (11) Again, th inductor currnt of th convrtr is dnotd by I l, th capacitor voltag by U c and th switching function s is givn by Figur 3. For th input currnt I th sam statmnt as for th buck convrtr is tru: I = si l. Figur 5. Losslss buck-boost convrtr with rsistiv load. s = 1 s = I S I l U C U c RL L DC-DC convrtrs can also b dscribd by avragd modl quations [7] if th intrinsic rippl is ngligibl. Th structur of ths quations is th sam as for th switchd modls, only th timdpndnt quantitis ar substitutd with thir avragd countrparts, whr th avrag is takn ovr on duty cycl T, i.. [ a(t ), I l (t ), U c (t ), U (t ) ] := 1 T t +T t [s(t), I l (t), U c (t), U (t)] dt (12) and th input currnt I is also intgratd to b I (t ) = I l (t ) for th boost and I (t ) = a(t ) I l (t ) for th buck and buck-boost convrtr. Th avragd modl quations allow a simpl charactrization of th spcifid DC-DC convrtrs in trms of thir input/output bhavior. In Figur 6a th stationary output voltags U c of th thr convrtrs ar shown. On can s, that a boost (buck) convrtr can b usd to produc an output voltag U c that is gratr (smallr) in magnitud than th input voltag U. Th buck-boost convrtr is a mixd form and is usd to invrt th output voltag U c and dcras or incras its magnitud with rspct to th input voltag U. In Figur 6b th stationary input rsistancs of th convrtrs R := U /I ar dpictd.

6 Enrgis 29, 2 76 Figur 6. Stationary and avragd output voltags U c (a)and input rsistancs R (b) of boost, buck and buck-boost convrtrs with rspct to thir duty ratio a. a) b) boost buck buck boost U c / U [-] R / RL [-] boost 1 4 buck buck boost duty ratio a [-] duty ratio a [-] 4. Intrconnction analysis of PEMFC and DC-DC convrtrs In this sction th connction btwn PEMFC and DC-DC convrtrs is analyzd. First of all, th coupling conditions ar spcifid. For th coupling btwn th PEM ful cll (Eqns. (1-5)) and th convrtrs (Eqns. (6-11)) th following conditions apply: U cll U and i cll I /A g. (13) With th abov quation th ful cll and th convrtrs ar connctd and a fdback of th convrtr s input currnt I to itslf via th cll voltag U cll (Eqn. (5)) is stablishd. It is th aim of this contribution to analyz th ffct and xtnt of this fdback. This is don for ach connction in thr stps. In a first stp, th ffct of th convrtr rippl upon th PEMFC is shown and xplaind, whil in th scond stp th found ffct is discussd. Finally, th ovrall bhavior of th connctd PEMFC - convrtr systm is xamind in ordr to chck for th apparanc of stationary multiplicitis and oscillations du to th coupling PEMFC and Boost-convrtr In this sction th coupling btwn th PEMFC and th boost convrtr is xamind. Effct of th convrtr rippl upon th PEMFC In a first stp th ffct of th convrtr rippl upon th PEMFC is shown. For this purpos th PEMFC (Eqns. (1-3)) and th boost convrtr modl (Eqns. (6,7)) ar coupld via Eqn. (13) and form a systm of switchd diffrntial quations. This systm is dynamically simulatd using stp changs of th load rsistanc R L dpictd in Figur 7. Th convrtr s duty ratio a is st to.2 and th othr modl paramtrs ar kpt constant at thir nominal valus. Thr simulations I, II, III ar prformd with th sam initial valu RL I. In simulation I th load rsistanc is kpt constant at RL I whras in simulations II and III th load is stppd to RII L and RIII L rspctivly.

7 Enrgis 29, 2 77 Figur 7. Tim plot of load rsistanc R L and stationary voltag-currnt profil of th PEM ful cll with assignd oprating points. 1 1 stationary: R I I I L RL [Ohm] U [V ] O P I I I R I I I R I O P I R I I O P I I I [A] 1 RL I 2 R L I I t [sc] Th simulation scnario can b furthr illustratd with th stationary voltag-currnt profil of th PEMFC togthr with th considrd oprating points OP I, OP II and OP III shown insid of Figur 7. Th oprating points ar dtrmind by th load rsistanc R L. A rlationship btwn R and R i L can b drivd from th avragd vrsion of th boost convrtr modl in Eqns. (6,7): L İl(t ) = U (t ) (1 a(t )) U c (t ) C U c (t ) = (1 a(t )) I l (t ) U c (t )/R L For th stationary opration of th convrtr on obtains: U = (1 a)u c and I l = U c /(1 a)r L. Th avrag input rsistanc of th boost convrtr is givn by R = U /I and with th dpndncy I = I l on obtains from th prvious statmnts R = R L (1 a) 2. Th simulation rsults ar shown in Figurs 8 and 9. Th diagrams ar split into two parts. Th first part shows th tim plots from th simulation start to th sttlmnt of th lctrical transints of th coupld systm. Th scond part shows stationary simulations aftr th transints for mass transport of oxygn and watr vapor hav sttld. It can b sn, that during simulations I and II no significant impact from th PEMFC to th boost convrtr or vic vrsa can b found. Aftr th applid stp th simulation sttls and finally rachs a stady stat. Th oscillations of th quantitis ar small and can b nglctd. In contrast, if simulation III is considrd a clar intraction of PEMFC and convrtr can b obsrvd. Th ovrpotntial η c and th cll voltag U in Figur 8 show rlativly larg oscillations compard to th cass I, II. Th oscillations ar prsnt immdiatly aftr th applid stp and also at stady stat. This is not th cas for th convrtr input currnt I and th capacitor voltag U c in Figur 9. Both of thm show only small oscillations in cas III, similar to th simulation cass I and II. Th givn intraction is thrfor on-sidd in dirction from boost convrtr to PEMFC and is locatd at small cll currnts in th activation polarization rgion of th ful cll (Figur 7). Th rason for this intraction can b found from th modl quation of th ovrpotntial in Eqn. (3). A linarization of this quation at an avragd and stationary oprating point (x s o 2, x s h 2 o, η s c, I s, U s c ) of

8 Enrgis 29, 2 78 Figur 8. a) Stp rspons of ovrpotntial η c and b) cll (=convrtr input) voltag U. a) b).6 η I I c 1 ηc [V ] I I I I I I.2.6 η I c η I I I c u [V ] I I I I I I I I I I I I t [sc] t [sc] Figur 9. a) Stp rspons of convrtr input (=inductor) currnt I and b) capacitor (=convrtr output) voltag U c. a) b) I [A] I I I I I I I I I I I I I I I t [sc] t [sc] Uc [V ] I I I I I I I I I I I I

9 Enrgis 29, 2 79 th coupld systm lads to C dl ˆα I s δ η c + δη c = δi /A }{{ g ˆα I s } τ:= (14) with I s = A g i rx s o 2 xp( ˆαη s c ) (15) whr δη c and δi ar th variations of th ovrpotntial and th convrtr input currnt around th oprating point rspctivly. It is assumd, that th variation of th oxygn contnt x o2 du to th convrtr switching can b nglctd. Th variation δi of th convrtr input currnt is considrd as an input quantity in Eqn. (14), which is indpndnt from δη c bcaus of th obsrvd on-sidd intraction from convrtr to PEMFC. Equation (14) is thrfor a linar ordinary diffrntial quation of first ordr with constant cofficints whos transint bhavior is dtrmind by its tim constant τ. If th tim constant τ is small compard to th givn tim intrval of th duty cycl T thn th variation δη c can approximatly b calculatd by δη c 1 ˆαI s δi = ( η c s ) I s δi. (16) This rlationship is dtrmind from th Tafl kintic in Eqn. (15) whr ( ηc s )/ I s is th snsitivity of th ovrpotntial ηc s with rspct to th cll currnt I s. It can b sn, that th snsitivity incrass with dcrasing cll currnt. If th variation δi dos not chang vry much at diffrnt cll currnts I s, th chang of variation δη c can approximatly b dtrmind by th changd snsitivity. This is th cas for th thr simulation xprimnts abov. Th oscillations I, i i = I, II, III in th cll currnt (Figur 9a) ar narly th sam for all thr simulation cass, but th avrag valus ar clarly diffrnt. For simulation cas III th avrag cll currnt is th smallst rsulting in th largst snsitivity of th thr cass. Th larg oscillations in th ovrpotntial for cas III (Figur 8a) ar th consqunc. In Figur 1 th abov statmnts ar illustratd. Th Tafl quation (15), th oscillation amplituds of th cll currnt I i (Figur 9a) and th corrsponding amplituds of th ovrpotntial ηc i (Figur 8a) ar shown for th thr simulation cass I, II and III. W hav sn, that th rason for th larg oscillation ηc III is a too small tim constant τ with rspct to th duty priod T. From Eqn. (14) w can s, that th tim constant τ is proportional to th doubl layr capacitanc C dl and (with Eqn. (16)) to th snsitivity ( ηc s )/ I s. Th snsitivity in simulation III is th largst, so th doubl layr capacitanc C dl of th ful cll is rsponsibl for th small τ. Thrfor, th oscillations in th activation polarization rgion of th PEMFC in III ar causd by an insufficint adaption of th doubl layr capacitanc C dl and th duty priod T to ach othr. Discussion of th ffct W hav shown and xplaind in th prvious sction that th convrtr rippl introducs oscillations in th activation polarization rgion of th ful cll. This statmnt is tru for th usd doubl layr capacitanc C dl and duty priod T, but it is also in gnral valid as long as th ratio btwn C dl and T is lss or qual to th givn on. This mans for xampl, that w cannot incras T in ordr to dcras switching losss in th boost convrtr, bcaus this will rsult in largr oscillations in th ful cll. This also mans for xampl, that if th ful cll owns a largr doubl layr capacitanc C dl and w us th sam duty priod T th oscillations will vanish. If w incras th duty priod T (and

10 Enrgis 29, 2 8 Figur 1. Tafl quation for simulation cass I, II and III. I s [A] I I I I I I I I I η I I I c I I I ηc s [V ] I η I c I I η I I c th boost convrtr s inductivity L, capacitanc C to stay at th sam rippl in th output voltag U c ) th oscillations will roccur. Th impact of convrtr introducd oscillations in th ovrpotntial is currntly undr rsarch and up to now it is not clar, if thy lad to cll dgradation, as long as no ractant dpltion appars [12]. Anyway, in ordr to avoid oscillations in th ful cll w hav to tak suitabl masurs which ar prsntd in th following. For th abov simulations w usd a small doubl layr capacitanc in th ordr of magnitud as in [13, 14]. In othr publications lik in [15, 16] a largr doubl layr capacitanc in th PEMFC is obsrvd. For th lattr cas, th oscillations in th ful cll vanish for th givn duty priod and no furthr ffort has to b takn to avoid thm. In th first cas thr ar two simpl possibilitis to avoid oscillations. Th first is to dcras th duty priod T. This has a smallr variation I of th cll currnt and a largr impact of th tim constant τ within th tim intrval T as a consqunc, but can also lad to largr switching losss in th convrtr. Th scond altrnativ is to incras th doubl layr capacitanc C dl of th PEMFC and thrfor th tim constant τ. Th first point can b achivd via th control of th boost convrtr. Aftr th boost convrtr has bn dsignd [7] and th duty priod T has bn adjustd to mt th boost convrtr s rquirmnts, a minimal cll currnt I s,min I s has to b spcifid. This puts an uppr bound on th snsitivity in Eqn. (16). If th doubl layr capacitanc and all othr ncssary paramtrs ar roughly known thn th rlvant tim constant C dl A g /ˆαI s,min can b stimatd. If T C dl A g /ˆαI s,min oscillations ar xpctd to appar if th ful cll is opratd in th activation polarization rgion. In ordr to avoid ths oscillations th duty priod T can b dcrasd,.g. T C dl A g /ˆαI s,min. Th scond possibility can b implmntd for xampl by insrting a capacitor btwn PEMFC and boost convrtr. This lads to an incrasd doubl layr capacitanc as is shown in appndix B. Ovrall bhavior of th coupld systm With th abov suggstions th impact of th convrtr rippl can b supprssd and w can dscrib th connction btwn th PEMFC and th boost convrtr with avragd modl quations and chck th ovrall bhavior of this coupld systm for th occurrnc of

11 Enrgis 29, 2 81 stationary multiplicitis and oscillations. At first, w considr th stationary opration of PEMFC and boost convrtr. Thrfor, th stationary and avragd rlationship in Figur 6 for th boost convrtr is valid. Du to th coupling in Eqn. (13) th input rsistanc of th convrtr R srvs as load rsistanc of th PEMFC V cll /I cll = U A g /I = R A g and forcs a rhostatic opration of th cll. Morovr, th mapping btwn th convrtr s input rsistanc R and th load rsistanc R L is uniqu as is indicatd in Figur 6b. Thrfor, no furthr stationary multiplicitis ar addd by th coupling PEMFC and boost convrtr to th ons that ar alrady prsnt in a rhostatic opratd PEM ful cll [17 19]. Howvr, oscillations inducd by th coupling ar still possibl. Thy appar if a Hopf bifurcation occurs du to th coupling. A Hopf bifurcation appars in a nonlinar systm ż = f(z, p) if a pur imaginary pair of ignvalus of th Jacobian matrix J = f/ z, valuatd at th stady stat (z, p ) ariss at th paramtr p. Thrfor, in ordr to sarch for th onst of oscillations, w hav to chck th Jacobian matrix of th coupld systm. For this purpos w start with th following avragd modl of PEMFC and boost convrtr: V c ẋ o2 = ˆq(1 + x o2 ) i r + (x in o 2 x o2 ) V in, (17) V c ẋ h2 o = ˆq(2 x h2 o) i r + (x in h 2 o x h2 o) V in, (18) C dl η c = i r I l /A g with i r = i r x o2 xp( ˆαη c ), (19) L İl = U cll + η c r m (x h2 o) I l /A g (1 a) U c, (2) C U c = (1 a) I l U c /R L. (21) It is drivd by coupling Eqns. (1-7) using Eqn. (13) and avraging, lik in Eqn. (12), th rsulting modl ovr on duty cycl. For this purpos it is assumd that th stats x o2, x h2 o, η c, I l, U c ar approximatly constant during on duty cycl. This is a valid assumption du to th ngligibl impact of th convrtr rippl. Th abov systm of quations includs avragd modl quations for oxygn and watr transport (Eqn. (17,18)). This mass transport typically shows transint tims in th ordr of magnitud of sconds whil th rsonant bhavior of th convrtr is in th ordr of magnitud of milli sconds and smallr. Du to this w considr th mass transport Eqns. (17,18) as static and us only th quations (19-21) to sarch for th apparanc of a Hopf bifurcation. Th first stp in ordr to dtct a Hopf bifurcation is th calculation of th Jacobian matrix. If w calculat th Jacobian matrix of quations (19-21) at th stady stat (x ss o 2, x ss h 2 o, ηss c, I ss l, U ss c, RL ss, a ss) w gt J := b 11 b 12 b 21 b 22 b 23 b 32 b 33 = ˆαI ss l /A g C dl 1/A g C dl 1/L r m (x ss h 2 o )/A gl (1 a ss )/L (1 a ss )/C 1/R ss L C. (22) Th duty ratio a ss of th boost convrtr is typically btwn a ss < 1 and thrfor, th cofficints b ij of J ar always gratr than zro. In th nxt stp w hav to chck th location of th ignvalus of th Jacobian matrix J. Th ignvalus of J ar th roots of th charactristic polynomial P (λ) =

12 Enrgis 29, 2 82 dt(λi J) which is givn by P (λ) = λ 3 + c 2 λ 2 + c 1 λ + c (23) with c 2 := b 11 + b 22 + b 33, (24) c 1 := b 11 b 22 + b 11 b 33 + b 22 b 33 + b 23 b 32 + b 12 b 21, (25) c := b 11 b 22 b 33 + b 11 b 32 b 23 + b 21 b 12 b 33. (26) Th location of th roots of P (λ) can b dtrmind with th critrion of LIÉNARD-CHIPART [2]. Th polynomial has only roots with ngativ ral parts if th following ncssary and sufficint conditions ar fulfilld: c >, c 2 > and c 2 c 1 c >. Th first two conditions ar fulfilld through Eqns. (24,26), bcaus th cofficints c 2, c 1 and c of th polynomial ar always positiv. Th third condition is also fulfilld, bcaus of c 2 c 1 c = b 2 11b 22 + b 2 11b 33 + b 11 b 12 b 21 + b 11 b b 11 b 22 b 33 + b 2 22b 33 + b 22 b 23 b 32 + b 12 b 21 b 22 + b 11 b 22 b 33 + b 11 b b 22 b b 23 b 32 b 33 >. (27) Thrfor, th charactristic polynomial P (λ) (Jacobian matrix J) has always roots (ignvalus) with ngativ ral parts and bcaus of this th connction btwn a PEMFC and a boost convrtr cannot lad to a Hopf bifurcation in th coupld PEMFC - boost convrtr systm PEMFC and Buck-convrtr In this sction th coupling btwn th PEMFC and th buck convrtr is xamind. Effct of th convrtr rippl upon th PEMFC In th first stp w considr th ffct of th buck convrtr rippl upon th ful cll. For this purpos w coupl th PEMFC (Eqns. (1-3)) and th buck convrtr modl (Eqns. (8,9)) with Eqn. (13) and apply stp changs of th duty ratio a (Figur 11a) to this systm. Th load rsistanc R L is chosn such that th ful cll is opratd clos to th maximum powr point. Th othr modl paramtrs ar at thir nominal valus. Two simulations I, II ar carrid out. In simulation I th duty ratio a is kpt constant at a I and in II th duty ratio is stppd to a II. Th simulation scnario can b furthr illustratd by th stationary voltag currnt profil of th ful cll and th considrd oprating points OP I and OP II. It is shown insid of Figur 11a. Th oprating points ar dtrmind by th buck convrtr s input rsistanc R via th convrtr s duty ratio a from th following dpndncy R i = R L /a 2 i with i = I, II. This rlationship can b drivd in an analogous mannr from an avragd and stationary vrsion of th buck convrtr modl lik it was don for th boost convrtr in sction Th simulation rsults ar shown in Figurs 11b and Figur 12. Th diagrams ar split into two parts. As in sction 4.1., th first part shows th fast dynamics du to lctrical ffcts, th scond part shows th long trm bhavior, aftr th transints of th mass balancs hav sttld. In simulation I th duty ratio is qual to a = a I = 1. This mans that th switch S of th buck convrtr is always in position s = 1 and no oscillations occur. In contrast, if simulation II is considrd, rlativ larg oscillations in th ovrpotntial ηc II (Figur 11b), cll currnt I II (Figur 12a) and th cll voltag U II (Figur 12b) ar apparing. Th larg oscillations ar prsnt immdiatly aftr th applid stp and also at th stady stat. This is not th cas for th inductor currnt Il II (Figur 12a) and th capacitor voltag Uc II (Figur

13 Enrgis 29, 2 83 Figur 11. a) Tim plot of duty ratio a and (insid) th stationary voltag-currnt profil of th ful cll with considrd oprating points. In b) th stp rspons of th ovrpotntial η c is shown. a) a I b).6 η I c η I c a [ ].5 U [V ] O P I I R I I.1.2 R I.2.3 I [A].4 O P I t [sc] a I I ηc [V ] η I I c.3.4 t [sc] η I I c Figur 12. a) Stp rspons of inductor currnt I l and cll (=convrtr input) currnt I. In b) th stp rsponss of cll (=convrtr input) voltag U and capacitor (=convrtr output) voltag U c ar shown. a) b).5 I, Il [A] I I I I I I l I I = I I l I I = I I l I I I l I I I U, Uc [V ] t [sc] t [sc] U I I U I = U c I U I I c U I I U I = U c I U I I c

14 Enrgis 29, b) of th convrtr. Both of thm show only small oscillations. Th givn intraction is thrfor onsidd in dirction from th buck convrtr to th PEMFC. Th rason for th rlativly larg oscillations in th PEMFC is du to th prsnc of th switching function s in th coupling of ful cll and buck convrtr: i cll = I /A g = s I l /A g. This lads to a switchd ODE for th ovrpotntial: C dl η c = i rx o2 xp( ˆαη c ) s I l /A g (28) and causs th larg oscillations in th ovrpotntial and in th cll voltag. Discussion of th ffct Th abov quation can b usd to furthr discuss th oscillation amplituds of th ovrpotntial. With th abov obsrvation that th intraction is on-sidd from buck convrtr to th ful cll and th assumptions that th changs in th inductor currnt I l and th oxygn contnt x o2 ar small ovr on duty priod T and can b approximatly dscribd by thir avrag valus I l and x o2, th following formula can b drivd for th stationary oscillation amplituds η c of th ovrpotntial: ( ) γ γ(1 a)t β η c = ln 1 + 1ˆα xp( βat ) 1 ( ) ( ) (29) γ γ xp( βat ) 1 γ + xp( βat ) γ(1 a)t β β β with γ = ˆα i r x o2 / C dl, β = ˆα I l / A g C dl and (1 a)t β > 1. Th drivation of this quation is givn in Appndix C. Th quation can b usd to furthr invstigat th oscillations in th ful cll. Figur 13a shows th oscillation amplituds η c calculatd with Eqn. (29) at diffrnt duty ratios a. Th ratio of duty priod and doubl layr capacitanc T/C dl is usd as paramtr and th othr quantitis rmain constant at thir nominal valus. It can b sn, that a Figur 13. a) Stationary oscillation amplituds η c of th ovrpotntial with rspct to th buck convrtr s duty ratio a at diffrnt ratios of duty priod and doubl layr capacitanc T/C dl and b) stationary simulations of th ovrpotntial for th cass i to iii at a duty ratio a =.5. a) b) ηc [V ] ηc i i : T.41 C dl η.37.1 ii :.1 T c ii.35 C dl ηc iii i iii :.1 C T dl a [ ] t [sc] 1 3 ηc [V ] iii ii η c iii η c ii η c i dcrasing ratio of T/C dl lads to smallr oscillations in th ovrpotntial and vic vrsa. Thrfor, in ordr to rduc oscillations in th ful cll ithr th duty priod T of th buck convrtr has to b

15 Enrgis 29, 2 85 dcrasd or th doubl layr capacitanc C dl of th PEMFC has to b incrasd or both things hav to b don. As was discussd in sction 4.1., this can b achivd ithr by adjusting th switching priod of th convrtr or by adding a capacitor. In Figur 13b an incrasd doubl layr capacitanc is usd. Stationary simulation rsults of th ovrpotntial η c for th coupld PEMFC and buck convrtr modl at a duty ratio a =.5 ar shown. Th duty priod T is hld constant and th doubl layr capacitanc is incrasd from its nominal valu in cas i to 1C dl in cas ii and 1C dl in cas iii. On can s, that th oscillation amplituds of th ovrpotntial dcras ( η i c > η ii c > η iii c ) as it is prdictd in Figur 13a. Th oscillations in th ovrpotntial du to th coupling of PEMFC and buck convrtr may also b usd to stimat paramtrs of th ful cll. This may b usful for monitoring or control purposs of th PEMFC. Rathr xpnsiv to obtain ar th paramtrs dscribing th raction kintics of th ful cll. Thir idntification is usually don in xprimnts using th impdanc spctroscopy, th currnt intrrupt tchniqu [21] and th lctrochmical paramtr idntification [22]. Equation (29) may also b usd for this purpos. For an stimation of th ful cll s raction kintics th xchang currnt dnsity togthr with th cll s oxygn contnt i r x o2, th xponnt in th Tafl quation ˆα and th doubl layr capacitanc C dl hav to b dtrmind. If w want to idntify ths paramtrs from Eqn. (29) w nd to know th avrag inductor currnt I l and th oscillation amplitud η c of th ovrpotntial whil th othr quantitis ar rathr wll known. Th quantity I l can b obtaind by masuring and avraging th inductor currnt. Th oscillation amplitud η c can b obtaind by masuring th oscillation amplitud U of th cll voltag. If th ful cll is wll humidifid th ohmic and concntration losss ar ngligibl and w hav η c U. If I l and η c ar known w hav to analyz th snsitivity of ths masurmnts with rspct to th unknown paramtrs in Eqn. (29) to gt an indication about th quality of th obtainabl stimation rsults. For th doubl layr capacitanc C dl w can us Figur 13 for this purpos. If w dfin th changs of th oscillation amplitud η c with rspct to changs in C dl at som fixd duty ratio a as snsitivity ( η c )/ C dl a w can s from Figur 13a that this snsitivity should b larg nough for all duty ratios to gt accptabl stimation rsults for C dl. Th snsitivity with rspct to th xchang currnt dnsity i r is analyzd in Figur 14. If w considr th changs of th oscillation amplitud η c with rspct to th changs in i r at som duty ratio a as snsitivity ( η c )/ i r a it can b sn, that this snsitivity is rathr small. Thrfor, w cannot xpct to gt accptabl stimation rsults for i r x o2 from Eqn. (29). Finally, in Figur 15 th snsitivity with rgard to th paramtr ˆα (Eqn. (4)) is xamind. If w considr th changs of th stationary oscillation amplitud η c with rspct to th changs in ˆα at a duty ratio a as snsitivity ( η c )/ ˆα a it can b sn from Figur 15, that this snsitivity should b larg nough for duty ratios btwn.1 and.9 to gt accptabl stimation rsults for th paramtr ˆα. Th stimation rsults for ˆα can b usd to dtrmin th transfr cofficint α from Eqn. (4), sinc th rlativ chang of ±.4 in ˆα corrsponds to a rlativ chang of.2 in th transfr cofficint. To dtrmin α from Eqn. (4) th cll tmpratur Θ has to b roughly known,.g. from masurmnts. In summary, th snsitivity analysis rvals that accptabl stimation rsults can b xpctd for th doubl layr capacitanc C dl and th paramtr ˆα. Th xchang currnt dnsity cannot b stimatd du to its small snsitivity. It should b notd, that du to this fact th prcis valu of th xchang currnt dnsity as wll as th prcis valu of th oxygn contnt in th cathodic catalyst is not ncssary

16 Enrgis 29, 2 86 Figur 14. Stationary oscillation amplituds η c of th ovrpotntial with rspct to th buck convrtr s duty ratio a at diffrnt xchang currnt dnsitis i r i ii iii iv ηc [V ] i :.1 i r ii :.1 i r iii : i r iv : 1 i r a [ ] Figur 15. Stationary oscillation amplituds η c of th ovrpotntial with rspct to th buck convrtr s duty ratio a at diffrnt valus of ˆα. ηc [V ] iii:.6 ˆα ii: ˆα i: 1.4 ˆα a [ ]

17 Enrgis 29, 2 87 for an stimation of C dl and ˆα. Th stimation rquirs th masurmnt of th avrag inductor currnt and th oscillation amplitud of th cll voltag at a highly humidifid ful cll. It should not b carrid out at too small oscillation amplituds η c to rduc th impact of th nglctd inductor currnt rippl in Eqn. (29). Ovrall bhavior of th coupld systm If w supprss th oscillations in th ful cll and nglct th impact of th buck convrtr rippl, w can dscrib and analyz th coupling btwn th PEMFC and th buck convrtr with avragd modl quations in ordr to chck th ovrall bhavior of th coupling for th apparanc of stationary multiplicitis and oscillations. First of all, th stationary opration of PEMFC and buck convrtr is considrd. Thrfor, th stationary and avragd rlationship in Figur 6 for th buck convrtr is valid. Lik in th cas of th PEMFC and th boost convrtr, th sam rasoning is tru and thrfor th coupling btwn PEMFC and buck convrtr cannot introduc furthr stationary multiplicitis as ar alrady prsnt in th PEMFC. Howvr, oscillations inducd by th coupling ar still possibl. In ordr to analyz th onst of oscillations w start with th following avragd modl of PEMFC and buck convrtr: V c ẋ o2 = ˆq(1 + x o2 ) i r + (x in o 2 x o2 ) V in, (3) V c ẋ h2 o = ˆq(2 x h2 o) i r + (x in h 2 o x h2 o) V in, (31) C dl η c = i r a I l /A g with i r = i r x o2 xp( ˆαη c ), (32) L İl = a ( U cll + η c r m (x h2 o) a I l /A g ) U c, (33) C U c = I l U c /R L. (34) It is drivd by coupling th modl quations of th PEMFC (1-3) and th buck convrtr (8,9) via Eqn. (13) and avraging th rsulting quations ovr on duty cycl. This is don in th sam way as for th boost convrtr abov. Lik thr, w considr th quations for th mass transport (3,31) as static and us only th avragd modl quations (32-34). Th Jacobian matrix of ths quations valuatd at th stady stat (x ss o 2, x ss h 2 o, ηss c, I ss l, U ss c, RL ss, a ss) is givn by J := b 11 b 12 b 21 b 22 b 23 b 32 b 33 = ˆαa ss I ss l /A g C dl a ss /A g C dl a ss /L a 2 ssr m (x ss h 2 o )/A gl 1/L 1/C 1/R ss L C. (35) Th duty ratio for a buck convrtr is typically btwn < a ss 1. With this, th cofficints b ij in J ar always positiv and thrfor th sam rasoning as in th prvious analysis of PEMFC and boost convrtr is tru: Th connction btwn a PEMFC and a buck convrtr cannot introduc a Hopf bifurcation in th coupld PEMFC - buck convrtr systm PEMFC and Buck-Boost-Convrtr In this sction th coupling btwn th PEMFC and th buck-boost convrtr is xamind. Effct of th convrtr rippl upon th PEMFC In a first stp th ffct of th convrtr rippl upon th ful cll is considrd by analyzing th coupld systm of switchd diffrntial quations mad up

18 Enrgis 29, 2 88 from th PEMFC (Eqns. (1-3)) and th switchd buck-boost convrtr modl (Eqns. (1,11)). Th analysis rvals that th buck-boost convrtr introducs oscillations in th ful cll in th sam way as th buck convrtr. As in this prvious cas, th rason is du to th prsnc of th switching function s in th coupling of th ful cll and th buck-boost convrtr: i cll = s I l /A g. This lads to th sam switchd ODE for th ovrpotntial (28) and causs th oscillations in th ful cll. Discussion of th ffct Th formula for th oscillation amplitud η c in Eqn. (29) can also b usd to dscrib th stationary oscillations introducd by a buck-boost convrtr. Figur 16 shows th oscillation amplitud of th ovrpotntial calculatd with this formula at diffrnt duty ratios. Th ratio of th duty priod and th doubl layr capacitanc T/C dl is usd as paramtr. Th load rsistanc R L is st to 7 Ω whil th othr quantitis rmain constant at thir nominal valus. It can b sn from Figur 16, Figur 16. Stationary oscillation amplituds of th ovrpotntial with rspct to th buckboost convrtr s duty ratio a at diffrnt ratios of duty priod and doubl layr capacitanc T/C dl. Th quantity a(p max ) dnots th duty ratio at th maximal cll powr P max a(p max ) ηc [V ] i: T C dl ii:.1 T C dl iii:.1 T C dl a [ ] that a dcrasing ratio of T/C dl lads to smallr oscillations in th ovrpotntial and vic vrsa. This is th sam qualitativ bhavior as in th cas of th buck convrtr in sction Thrfor, th sam possibilitis to rduc th oscillations ar applicabl. Th connction btwn th PEMFC and th buck-boost convrtr can also b usd to stimat paramtrs of th ful cll. W can us Eqn. (29) for this purpos again. In dtail, th doubl layr capacitanc C dl and th xponnt of th Tafl kintics ˆα can b stimatd. In th cas of th doubl layr capacitanc this can b sn from Figur 16. Th snsitivity ( η c )/ C dl a of th oscillation amplitud with rspct to th doubl layr capacitanc should b larg nough to gt accptabl stimation rsults for C dl. In th cas of th paramtr ˆα w can us Figur 17. W s, that th snsitivity ( η c )/ ˆα a of th oscillation amplitud η c with rspct to ˆα should b larg nough to gt accptabl stimation rsults for ˆα too.

19 Enrgis 29, 2 89 Figur 17. Stationary oscillation amplituds η c of th ovrpotntial with rspct to th buck-boost convrtr s duty ratio a at diffrnt valus of ˆα. Th quantity a(p max ) dnots th duty ratio at th maximal cll powr P max iii:.6 ˆα a(p max ) ηc [V ].15.1 ii: ˆα i: 1.4 ˆα a [ ] Ovrall bhavior of th coupld systm If w rduc th oscillations and ar abl to nglct th impact of th convrtr rippl w can finally analyz th ovrall bhavior of th coupld PEMFC and buck-boost convrtr with avragd modl quations. First of all, th stationary opration of PEMFC and buck-boost convrtr is considrd. Lik for th prvious two convrtrs th sam rasoning is tru and thrfor th coupling btwn PEMFC and buckboost convrtr cannot introduc furthr stationary multiplicitis as ar alrady prsnt in th PEMFC. Howvr, oscillations ar still possibl and thir apparanc has to b analyzd. This is don by coupling and avraging th modl quations of th PEMFC (1-3) and th buck-boost convrtr (1,11) in th sam way lik in th prvious two cass. W obtain th sam mass transport quations for oxygn and watr vapor lik in th cas of th buck convrtr (Eqns. (3,31)). Lik thr, w assum thm as static and us only th modl quations for th ovrpotntial and th buck-boost convrtr s inductor currnt and capacitor voltag: C dl η c = i r x o2 xp( ˆαη c ) a I l /A g, (36) L İl = a ( U cll + η c r m (x h2 o) a I l /A g ) + (1 a)u c, (37) C U c = (1 a)i l U c /R L. (38) If w calculat th Jacobian matrix of th abov systm at th stady stat (x ss o 2, x ss h 2 o, ηss c, I ss l, U ss c, RL ss, a ss) w gt J := b 11 b 12 b 21 b 22 b 23 b32 b 33 = ˆαa ss I ss l /A g C dl a ss /A g C dl a ss /L a 2 ssr m (x ss h 2 o )/A gl (1 a ss )/L (1 a ss )/C 1/R ss L C. (39) Th duty ratio of th buck-boost convrtr is typically btwn < a ss < 1. With this, th cofficints b ij of J ar always positiv and th cofficints b 23, b 32 ar always ngativ. If w calculat th charactristic

20 Enrgis 29, 2 9 polynomial of J w gt th quations (23-26) with ngativ quantitis b 23 = b 23 and b 32 = b 32. Dspit this diffrnc, th cofficints c, c 1, c 2 of th charactristic polynomial and th condition c 2 c 1 c ar still positiv du to th fact that only th positiv product b 23 b 32 = b 23 b32 > ntrs th dtrmining quations (24-27). Thrfor, th sam conclusion as in th cas of th PEMFC and th boost convrtr applys: Th connction of a PEMFC and a buck-boost convrtr cannot induc a Hopf bifurcation in th coupld systm. 5. Conclusion Th connction of PEM ful clls and DC-DC convrtrs is analyzd in this contribution. Th analysis is don for common DC-DC convrtrs lik th boost, buck and buck-boost convrtrs. First of all, th ffct of th convrtr rippls ar shown. Thy introduc oscillations in th ful cll. Thir origin is xplaind, discussd and possibilitis for thir supprssion ar givn. In th cas of th coupling btwn PEMFC and buck and buck-boost convrtr it is shown, that th oscillations may also b usful to stimat paramtr of th ful cll s Tafl kintic. Finally, th ovrall bhaviors of th coupld systms ar xamind. This is a ncssary stp, bcaus PEM ful clls can show a complx nonlinar bhavior lik multiplicitis, instabilitis and oscillations [18, 19, 23 25] and th coupling with DC-DC convrtrs might hav introducd additional nonlinar ffcts. W hav shown mathmatically, that this is not th cas: Th connction btwn PEM ful clls and boost, buck and buck-boost convrtrs can nithr lad to stationary multiplicitis nor to oscillations in th coupld systms. As a consqunc, it is not ncssary to dvlop intgratd control approachs for th couplings. Instad, w can concntrat ourslvs on th dvlopmnt of control stratgis considring only th PEMFC and w can us xisting control approachs for th DC-DC convrtrs [7] in ordr to control both subsystms. Although th PEMFC modl usd in this contribution is quit simpl, th obtaind rsults ar also valid for PEMFC stacks in gnral. This du to th fact, that in PEMFC stacks th lctrochmical ractions in th catalysts can also b dscribd by th modling approach w us in our analysis. Othr transint ffcts that appar in PEMFC stacks, lik th mass transport of gass and liquid watr through th gas diffusion layrs, th catalyst layrs or th mmbran as wll as th transint bhavior of th oprating tmpratur of th ful cll ar ordrs of magnituds slowr than th lctrochmical procsss and can thrfor b nglctd. Acknowldgmnt Th authors thank Prof. Lindmann for hlpful discussions. A Drivation of th PEMFC modl In this sction th quations for th ful cll modl ar drivd. At first, this is don for th diffrntial quation of th ovrpotntial from th quivalnt lctrical circuit in Figur 1. Th charg balanc at th doubl layr capacitor C dl rads dq/dt = i c = i cll i r. Th charg Q can b xprssd in trms of th ovrpotntial to b Q = C dl η c whr η c <. If th doubl layr capacitanc is assumd to b constant

21 Enrgis 29, 2 91 and th currnt i r is xprssd using th Tafl quation i r = i rx o2 xp( ˆαη c ) with ˆα := (1 α)nf / RΘ, (4) th diffrntial quation for th ovrpotntial in Eqn. (3) rsults. Th algbraic quation for th cll voltag in Eqn. (5) can b obtaind using Kirchhoff s voltag law. Th modl quations for th oxygn and watr contnt in th ful cll ar drivd from th CSTR in Figur 1. Mass balancs for th spcis k {O 2, N 2, H 2 O} rad ṅ k = V in c in k V out c k + ν k ra g (41) whr n k (c k ) dscribs th molar amount (concntration) of spcis k in th CSTR. Th symbol ν k dnots th stoichiomtric cofficints of th cathodic raction with ν o2 =.5, ν h2 o = 1 and ν n2 = and th quantity r rfrs to th raction rat and is givn by Faraday s law r = i r /nf. Th volum flow rat of air at th inlt V in is kpt constant. Th volum flow rat at th output V out can b dtrmind if th gas phas is assumd to b isothrm, isobaric and idal. Th idal gas law rads p g V c = n all RΘ. Th symbol n all = k n k dnots th whol amount of gas insid th CSTR and is constant if isobaric, isothrmic conditions as wll as a constant CSTR volum V c is assumd. This mans ṅ all = k ṅk =. If th mass balancs from Eqn. (41) ar insrtd an algbraic quation for th output flow rat V out rsults: V in k cin k V out k c k + k ν kra g =. This quation can b simplifid if th total concntration in th CSTR c t := n all /V c = k n k/v c = k c k = p g /RΘ = const. is dfind. Th total concntration in th CSTR and th inlt is th sam: c in t = c t bcaus w assum an idal and isobaric, isothrmic gas phas in th inlt as wll. With ths simplifications th output flow rat can b writtn as V out = V in + k ν k A g r/c t = V in + A g i r /2c t nf. (42) If w insrt th abov quation in Eqn. (41) and us n k = V c c k and molar fractions x k := n k /n all = c k /c t, x in k = cin k /c t th modl quations (1,2) rsult for k = O 2, H 2 O. B Enlargmnt of th doubl layr capacitanc In this sction it is brifly shown that a capacitor connctd in paralll to a PEMFC can b usd to incras th doubl layr capacitanc of th cll. Th capacitor with capacitanc C II is assumd to b losslss and is connctd to th ports of th quivalnt lctrical circuit in Figur 1. Th capacitor can b usd to supprss oscillations in th ful cll inducd by th duty cycl of a connctd DC/DC-convrtr. A tim intrval of on duty priod T is considrd for th following quations. For th ovrpotntial η c and th cll voltag U cll th Eqns. (3,5) ar still valid. Th voltag at th nw capacitor is idntical to th cll voltag and is calculatd by C II du cll dt = i cll i, (43) whr i dnots th nw output currnt of th ful cll that is diffrnt from i cll. If Eqn. (5) is diffrntiatd with rspct to tim and insrtd in Eqn. (43) on gts ( ) dηc C II dt r di cll m = i cll i, (44) dt

22 Enrgis 29, 2 92 whr w hav assumd, that th chang in watr contnt x h2 o in th cll is ngligibl ovr on duty priod. If Eqn. (3) is diffrntiatd with rspct to tim undr th assumption that th oxygn contnt x o2 is assumd to b constant and solvd for di cll /dt w obtain di cll dt = i r dη c η c dt C d 2 η c dl dt. (45) 2 whr i r dnots th Tafl quation (4). If Eqn. (45) and Eqn. (3) ar insrtd for di cll /dt and i cll in Eqn. (44) a scond ordr ODE for th ovrpotntial rsults: [ ]) d 2 η c r m C dl C II (C dt + i r dηc 2 dl + C II 1 r m η c dt = i r(η c ) i. (46) If th first trm on th lft hand sid is ngligibl compard to th scond trm,.g. if th ful cll is wll humidifid and lads to a fairly small mmbran rsistanc r m, a first ordr ODE for th ovrpotntial follows: (C dl + C II [1 + r m ˆα i r (η c )]) dη c dt = i r(η c ) i. (47) This quation has th sam structur as th original ODE for th ovrpotntial in Eqn. (3) whras th nw quation and thrfor th paralll connction of th ful cll and th capacitor shows an incrasd doubl layr capacitanc. C Formula for stationary oscillations of th ovrpotntial in a PEMFC if connctd to buck or buck-boost convrtrs In this sction th formula in Eqn. (29) is drivd. Th ODE for th ovrpotntial in Eqn. (28) is usd as a starting point. It is assumd that th changs in th inductor currnt I l and th oxygn contnt x o2 ar small ovr on duty priod T of th convrtr and can b approximatly rplacd by thir avrag valus I l and x o2. If th switching function s is dfind by s = { 1, t T on := [t, t + t on [, t T off := [t + t on, t + T [ th following nonlinar ODE in η c rsults: { I l A C dl η c = g + i rx o2 xp( ˆαη c ), t T on (49) i rx o2 xp( ˆαη c ), t T off with initial conditions η c (t ) = η c and η c (t + t on ) = η c1. With th following chang in variabls z := xp(ˆαη c ) and th dfinitions β := ˆα I l /A g C dl and γ := ˆα i r x o2 /C dl a linar ODE in z can b obtaind: { β z + γ, t T on ż = (5) γ, t T off This ODE can b solvd asily and th solution in th original variabls rads ηc on = 1ˆα ( ( γ ln β + xp(ˆα η c ) γ ) ) xp( β(t t )), t T on (51) β η off c = 1ˆα ln (xp(ˆα η c 1 ) + γ(t (t + t on ))), t T off. (52) (48)

23 Enrgis 29, 2 93 With th abov solutions w can now dfin th oscillation amplitud η c of th ovrpotntial. It is givn by η c := lim t t +T ηoff c lim ηc on (53) t t +t on with unknown initial conditions η c and η c1. Th initial condition η c1 can b calculatd by dmanding continuity at t = t + t on btwn both solutions: η c1 lim t t +t on ηc on. W gt from Eqn. (51) xp(ˆα η c1 ) = γ ( β + xp(ˆα η c ) γ ) xp( β t on ). (54) β Th initial condition η c can b obtaind by dmanding η c lim t t +T ηc off. This is valid for th stationary cas and w gt xp(ˆα η c ) = ( γ β )2 xp( β t on ) γ(t t on ) γ β xp( β t on) 1 (55) from calculating this limit from Eqn. (52) aftr insrting Eqn. (54). Th initial conditions in th solutions can now b liminatd by insrting Eqns. (54,55) in Eqns. (51,52). Thrfor, th stationary oscillation amplitud of th ovrpotntial can now b calculatd from Eqn. (53). With th rlation a = t on /T from Eqn. (12) th formula in Eqn. (29) rsults. Not that th oscillation amplitud η c = η c η c1 has to b always gratr than zro, bcaus in ful cll opration w hav η c > η c1. With Eqns. (54,55) and a = t on /T this condition can b rformulatd to (1 a) T β > 1. Nomnclatur α transfr cofficint of cathodic raction,.5 η c σ p σ p ovrvoltag at cathodic catalyst, V proton conductivity of mmbran, Sm 1 min. proton conductivity of mmbran, Sm 1 Θ cll tmpratur, 353 K a duty ratio of DC-DC convrtrs A g cross-sctional ara of ful cll, m 2 C capacitanc of DC-DC convrtrs, F c t C dl d m p g /(RΘ), molm 3 doubl layr capacitanc,.1 F m 2 mmbran thicknss, m F Faraday constant, Cmol 1 I input currnt of DC-DC convrtrs, A

24 Enrgis 29, 2 94 I l i r i cll inductor currnt of DC-DC convrtrs, A xchang currnt dnsity,.1 Am 2 cll currnt dnsity, Am 2 L inductivity of DC-DC convrtrs, H n numbr of lctrons consumd in cathodic raction, 2 p g p sat ovrall gas prssur, P a saturation prssur of H 2 O at Θ, P a R gas constant, 8.314Jmol 1 K 1 R L rsistiv load of DC-DC convrtrs, Ω r m mmbran rsistanc against proton transport, Ωm 2 T duty priod of DC-DC convrtrs, s U c U cll U cll capacitor (=output) voltag of DC-DC convrtrs, V cll voltag, V quilibrium voltag of th cll, 1.17 V V c gas volum of CSTR, m 3 V in x h2 o volum flow rat of air, m 3 s 1 molar fraction of H 2 O x in h 2 o molar fraction of H 2O at inlt,.14 x o2 molar fraction of O 2 x in o 2 molar fraction of O 2 at inlt,.21 Rfrncs and Nots 1. Choi, W.; Howz, J.; Enjti, P. Dvlopmnt of an quivalnt circuit modl of a ful cll to valuat th ffcts of invrtr rippl currnt. J. Powr Sourcs 26, 158, Gmmn, R. S. Analysis for th ffct of invrtr rippl currnt on ful cll oprating condition. Trans. ASME 23, 125, Shirn, W.; Kulkarni, R. A.; Arfn, M. Analysis and minimization of input rippl currnt in pwm invrtrs for dsigning rliabl ful cll powr systms. J. Powr Sourcs 26, 156, Znith, F.; Skogstad, S. Control of ful cll powr output. J. Proc. Control 27, 17,