21 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July 2, 21 FrC1.6 Control of a PEM Fuel Cell Base on a Distribute Moel Michael Mangol Abstract To perform loa changes in proton exchange membrane PEM fuel cells is a challenging task, because PEM fuel cells typically show a strongly nonlinear behavior. Especially the water management of low temperature PEM fuel cells is challenging. In this work, a nonlinear controller is presente that is able to keep the water content an the temperature of a PEM fuel cell on constant levels even uner strong changes of the electrical loa. A passive controller esign base on a spatially istribute fuel cell moel is use. The controller is teste in simulations. It is compare with a conventional linear control approach. I. INTRODUCTION Fuel cells are a promising technology for the prouction of electrical energy. The control of fuel cell systems is challenging ue to their intrinsic nonlinear behavior an ue to strong interactions between electrochemical reaction, heat, mass an charge transfer insie a fuel cell stack. These properties make it ifficult to perform loa changes an to keep the water content of low temperature fuel cells on a esire level. More avance control concepts are esirable that fully exploit the potential of fuel cell systems. The focus of this work is on proton exchange membrane PEM fuel cells, as this is the most wiely use fuel cell type. In open literature, the majority of publication on control of PEM fuel cells treats linear control methos. The number of publications on nonlinear control of fuel cells is rather small. Na an Gou suggest exact inputoutput linearisation [1]. Nonlinear moel preictive control approaches were evelope in [2], [3]. Danzer et al. [4] applie flatness base control to the air supply system of a PEMFC. Most of the work on fuel cell control is base on lumpe system moels. A small number of publications present linear control laws for spatially istribute fuel cell moels [5], [6], [7]. The objective of this work is to provie a control concept that is applicable to fuel cells with spatial epenencies, that guarantees stability over a wie range of operation conitions an allows fast loa changes. Passivity base control [8], [9] seems to be a suitable approach for this kin of problem. Current applications of passivity base control are mainly in the area of mechanical an electrical systems, because in these areas it is comparatively easy to fin Lyapunov functions. To fin a Lyapunov function for thermoynamic systems is much harer. This problem was stuie in etail by Ystie, Alonso an coworkers, who mae various suggestions on how to choose Lyapunov functions for chemical processes [1], [11], [12]. These techniques M. Mangol is with Max Planck Institute for Dynamics of Complex Technical Systems, Santorstraße 1, 3916 Mageburg, Germany, mangol@mpi-mageburg.mpg.e are applie here to a spatially istribute fuel cell moel. The moel is presente in the next section. In Section III an Section IV a nonlinear passive feeback controller an a conventional linear controller are esigne. The two controllers are compare in simulations in Section V. II. SIMULATION MODEL This section briefly summarizes the main features of a spatially istribute fuel cell moel that will be use as the basis for controller esign. More etails on the moel can be foun in [13]. The moel escribes a fuel cell as shown in Figure 1. Spatial epenencies are consiere in the irection of the gas flows along the z coorinate. The moel contains the gas channels, the gas iffusion layers, the catalyst layers an the membrane. It is a one-phase moel, i.e. it is vali only as along as no liqui water formation occurs. Fig. 1. Scheme of the one-imensional fuel cell moel A. Equations for the gas channels The mass balances rea for the gas channels with convective transport in z-irection: c j i = vj c j i jj i z δ j 1 v j c j i = j j i,in 2,t The variable c is a gas concentration, v is the flow velocity. Superscript j stans either for the anoe sie j = A or for the cathoe sie j = C. The subscript i is the component mass inex, where i = H 2, H 2 O on anoe sie, an i = O 2, N 2, H 2 O on cathoe sie; δ j is the height of the gas channel in y-irection; j j i are the mass flux ensities between gas iffusion layers an gas channels. 978-1-4244-7425-7/1/$26. 21 AACC 663
A balance of the internal energy u gives ρu j = v j c j i z h it j i + α 1 δ j T S T j i + λ j 2 T j z 2 j j i δ j h it j 3 The terms on the right-han sie escribe energy transport in z-irection ue to convective flow, heat conuction accoring to Fourier s law, heat transfer between soli fuel cell parts an gas channels, an enthalpy transport cause by mass flow from the gas channels to the soli. The corresponing bounary conitions are j j i,in h it j in = v j i cj i h it j λ j T j z 4 i,t,t i λ j T j z = 5 L,t The temperature in the gas channels is given implicitly by the caloric equation of state: B. Gas iffusion layers ρu j + p j = i c j i h it j 6 The purpose of the gas iffusion layer GDL moel is to introuce a mass transport limitation between gas channels an catalyst layers. Stefan-Maxwell iffusion in an ieal gas is assume to escribe the mass fluxes perpenicular to the z-coorinate: x j i = k x j k jj i xj i jj k c j D eff i,k j = A, C; i = H 2 for j = A; i = O 2, N 2 for j = C C. Catalyst layers The catalyst layers are assume to have no mass storage capacity an no graients in y irection. Quasi-stationary component mass balances lea to 7 = jh2 A r A 8 = jh2o A jh2o AM 9 = j C O2 1 2 rc 1 = jn2 C 11 = jh2o C jh2o CM + r C, 12 where r A is the rate of the anoe reaction an r C is the rate of the cathoe reaction. D. Membrane moel To capture the humiity epenence of the membrane s properties correctly, a rather etaile moel is use in this work. The membrane moel taken from [14]. It is assume that water is only transporte along the y coorinate perpenicular to the gas flow. Neglecting the swelling of the membrane, one obtains the following equation for the water concentration c M H2O from a mass balance: The fluxes j AM H2O c M H2O = 1 j AM δ M H2O + jh2o CM an jcm H2O 13 stan for the water exchange between membrane an anoe gas bulk an cathoe gas bulk, respectively. Usually the humiity epenence of the membrane is not formulate as a function of the water concentration c M H2O, but as a function of the water content Λ, which is efine as the ratio between moles of water in the membrane an moles of polymer in the membrane. Both quantities are relate by: c M H2O = Λρ M ΛX M Λ 14 The fluxes of water an protons through the membrane are assume to be riven by graients of the chemical potentials of water an protons with humiity epenent transport number t W an self iffusion coefficient D W j = AM, CM: j j H2O = t W ΛκΛ F 2 j H+ = κλ F 2 µ j H+ D W Λc M H2O RT S µ j H2O 15 µ H+ t W ΛκΛ F 2 µ H2O 16 The electrical current through the membrane i M is irectly couple to the proton flow j H+ by E. Energy balance for the soli part i M = F j H+ 17 A pseuo-homogeneous energy balance for the soli parts of the fuel cell reas: δ S ρes = i,j T S z j j i h it j + j=a,c α 1 T j T S +α 2 T cool T S + λ S δ S 2 T S z 2 Φ C Φ A i M 18 = T S,t z = 19 L,t The energy fluxes on the right-han sie of 18 are cause by mass exchange between gas bulks an soli, by heat exchange between gas bulks an soli, by heat exchange between coolant an soli, by Fourier heat conuction, an by electrical work one to lift charge from the lower potential Φ A to the higher potential Φ C. The total energy ρe S comprises the internal energy an the electrical energy in the ouble layers: 2 δ S ρe S = δ S ρu S + C A AC ΦA2 δ + C C CC ΦC δ 2 2 2 Equation 2 can be use to etermine the temperature of the soli T S. 6631
F. Charge balances It is assume that charge is transporte through the electroes in the irection of the z-coorinate, an through the membrane perpenicular to the z-coorinate. Charge balances at the anoe ouble layer an at the cathoe ouble layer give: C A AC ΦA δ = i M 2F r A 21 C C CC ΦC δ = i M + 2F r C 22 An integral charge balance over the length of the membrane reas: I Cell = L x i M z 23 The potential ifferences Φ A z, t an Φ C z, t can be compute from 21 an 22. If the total cell voltage U Cell t is known potentiostatic operation, then the potential rop in the membrane follows from U Cell t = Φ C z, t Φ M z, t Φ A z, t 24 an 23 is an explicit relation for the cell current I Cell t. If I Cell is fixe galvanostatic operation, 23 serves as an implicit relation for U Cell. In the case consiere here neither the cell voltage nor the cell current are specifie externally, but the cell power. Therefore, the aitional equation P Cell = U Cell I Cell 25 is neee to complete the set of moel equations. III. DESIGN OF A PASSIVE CONTROLLER The objective of fuel cell control to be evelope is the following: For a varying cell power, keep the internal state of the fuel cell stack in a esire range. The thermoynamic state of the stack is etermine by the composition an the temperatures of the gas bulks, the electrical charge store in the ouble layers, the temperature of the soli parts, the water store in the membrane. Not all of these quantities are of equal importance for the operation of a fuel cell. Usually there is no nee to keep the gas concentrations an gas temperatures exactly at preefine values. It is sufficient to prevent epletion of hyrogen on the anoe sie an of oxygen on the cathoe sie. This can be achieve by a simple fee forwar control of the gas flows that ajusts the flows to the eman of electrical current. The charge in the ouble layers is responsible for the anoic an cathoic overpotentials an hence affects the cell voltage. It shoul be controlle if one wants to keep the cell voltage on a given level. However, in most cases this is not necessary, because the cell stack is connecte to a electronic converter that ensures a constant voltage on the loa sie, inepenent of the cell voltage. Therefore, it is enough to avoi strong overpotentials that may amage the catalyst. There remain two quantities whose control seems to be crucial for the operation of a cell stack: the soli temperature an the water content. The soli temperature has an impact on the electrochemical reaction rates, as they are temperature epenent; further, the soli temperature etermines the physical properties of the materials an shoul remain below a certain level in orer not to amage the membrane. The membrane humiity is important, because it strongly affects the internal resistance of the fuel cell. On the other han, the water concentration in the cell shoul not be too high, as this woul lea to the formation of liqui water blocking the gas transport to the electroes. In the following a passive controller is evelope for the internal energy of the soli, which is irectly relate to the soli temperature, an for the water content of the membrane. A variant of the propose approach that also controls the gas compositions an the overpotentials was presente in [13]. The control concept is base on integral storages or inventories, similar to the approach in [15]. Suitable inventories an their corresponing balance equations will be given in the next section. A. Inventories As state above, the control aims at setting the soli temperature an the membrane humiity. Uner some aitional simplifying assumptions these control objectives can be relate to two inventories that etermine the thermoynamic state of the soli part of the cell: The total amount of energy v e := {δ A ρu A + δ C ρu C + δ S ρe S }z 26 store in the cell consists of the internal energies of the gas bulks, the internal energy of the soli, an the electrical energy in the ouble layers. Compare to the internal energy of the soli, which is mainly a function of the soli temperature, the contributions of the other energies are very small. Therefore, it is possible to control the soli temperature profile via the inventory v e. The amount of water store in the gas phases is small compare to the amount of water store in the membrane. Therefore, one can keep the membrane humiity on a esire level by controlling the total amount of water in the cell, which is given by v H2O := {δ A c A H2O + δ M c M H2O + δ C c C H2O}z. 27 Differential equations for the inventories can be erive from the mass, energy an charge balance equations in Section II. An equation for v e follows from the energy balances for the gas channels 3, 4, 5 an the global charge balance 23: v e = } δ {j j j i,in t h it j in vj c j i h it j L,t i,j + α 2 Tcool T S z P cell L x =: f e 28 Integrating the water mass balances of the gas channels an of the membrane over space an inserting a quasi-stationary 6632
cathoe charge balance 22 gives an approximate equation for v H2O : v H2O = { } δ j j j H2O,in t δj v j c j H2O + I Cell L,t 2F L j x =: f H2O 29 B. Lyapunov function The control objective is to let the system go towars a esire steay state or towars a given esire trajectory. The inventories at the esire states are given by vet an vh2o. As suggeste in [15], the functional V = 1 2 i=e,h2o v i v i 2 3 is chosen as a caniate Lyapunov control function. The time erivative of the Lyapunov function reas V = i If one efines the fictitious output vector v i v i v i v i 31 y = v e v e, v H2O v H2O T 32 an the fictitious input vector u = v e v e, v H2O v H2O T, 33 then the time erivative of the Lyapunov function can be written as V = y T u. 34 Therefore, the passivity conition V y T u is fulfille. C. Feeback law The feeback law has to be chosen in such a way that V <. A simple way to this is to require or u i = 1 τ i y i, i = e, H 2 O 35 τ i v i v i + v i v i = 36 because then V = i v i vi 2 /τ i. The time constants τ i are esign parameters for the controller. The fictitious feeback law 35 has to be translate back into a feeback law for the physical manipulate variables. Here, two manipulate variables are neee to control the two inventories. A possible choice is the water flow on the cathoe sie an the coolant temperature. By inserting the efinitions for u an v into 35, one obtains implicit equations for the physical manipulate variables: f i v i = 1 T i v i v i, i = e, H 2 O 37 IV. DESIGN OF A LINEAR CONTROLLER In orer to get a better assessment of the passive controller s properties, it is compare to a more conventional linear controller. To esign the linear controller, the spatially iscretize moel equations from Section II are linearize at a steay state setpoint. A linear ifferential algebraic system of the following structure results: E ẋ = ẋ a A A a + A a A aa x x SP x a x SP a B B a u u SP 38 The superscript SP stans for the values at the set point. In accorance with the assumptions unerlying the passive controller esign, only the ynamics of the water concentration in the membrane an of the soli internal energy are consiere, the remaining equations are set quasistationary. So x contains the vector of spatially iscretize membrane water concentrations an the vector of spatially iscretize soli internal energies. The input vector u consists of hyrogen, oxygen, an water gas flows, of the coolant temperature an of the cell current: u = j A H2,in, j C O2,in, j C H2O,in, T Cool, I Cell T 39 Because A aa an E are invertible, the algebraic variables x a can be eliminate, an 38 can be transforme into an ODE system for the ynamic states x : with ẋ = A x x SP + B u u SP 4 A = E 1 A A a A 1 aa A a B = E 1 B A a A 1 aa B A 41 42 In analogy with the passive controller esign, only the water flow an the coolant temperature are use as manipulate variables for the linear controller, the other elements of u are seen as isturbances. With u = jh2o,in C, I T Cell, 4 is rewritten as + B u u SP ẋ = A x x SP +B H2 jh2,in A j A,SP H2,in +B O2 jo2,in C j C,SP +B I Icell I SP cell O2,in 43 The manipulate input u is splitte up into a feeback part u F B an into a feeforwar part u F F : u = u SP + u F B + u F F 44 The feeback part is proportional to the eviation of the states from the setpoint: u F B = K x SP x 45 6633
The observer gain K, which has a imension of 2 1 for the chosen spatial iscretization, is etermine by LQ optimal esign using the Matlab comman lqr. The feeforwar part is to compensate the isturbances ue to varying gas flow rates an electrical current. It is chosen as u F F = B + B H2 jh2,in A j A,SP H2,in +B O2 jo2,in C j C,SP where B + = O2,in +B I Icell Icell SP, 46 B T B 1 B T is the pseuo-inverse of B. V. SIMULATION RESULTS First, the behavior of the close loop system with passive control is stuie. Figure 2 shows steay state results, when the cell current is varie an the reference values for v H2O an v e are kept constant. While the current-voltage plot has a quite conventional shape, the interesting point is that along this curve the average membrane humiity an the average soli temperature o not change. Therefore one can jump between arbitrary steay state points on the i-v-curve without changing the state of the water storage an the internal energy storage, which both have slow ynamics. The consequence is a very fast response of the fuel cell stack to changes of the electrical loa that is only limite by the ynamics of the manipulators gas blowers an cooling system, but not by the stack ynamics. This is illustrate average quantities, the spatial profile of the membrane water content an of the soli temperature vary slightly uner loa changes. This is shown in Figure 5 for a time interval after the first strong power rop. But this small rift in the spatial profiles is of minor importance for the operation of the stack. Actually, the close loop system is able to cope with much faster loa changes than those shown in Figure 3. The long intervals of constant loa were only chosen in orer to show that the system is able to return to the exact setpoint an to make the results of the passive controller comparable to the results of the linear controller presente in the following. Fig. 3. Varying power ensity use as a test signal for the simulation of the close loop system shown in Figure 4, 5 an 6 Fig. 2. Steay state behavior of the close loop system when using the passive controller: voltage an electrical power output uner constant water an internal energy content of the cell. by the ynamic simulation epicte in Figure 4 an 5, which show a response of the close loop system to the power changes in Figure 3. One can see from Figure 4 that the internal energy storage v e is kept perfectly constant even uner large loa changes an that the water content is isturbe only slightly; this can be explaine by the simplifying assumptions mae when esigning the passive controller. Because the controller only works with spatially Fig. 4. Response of the close loop system with passive controller to the test signal in Figure 3; left column: inlet water flow an coolant temperature; right column: water content v H2O an soli internal energy v e. Figure 6 shows the response of the close loop system with linear controller to a power rop at t = 1 s, when the reference values x SP an u SP are kept constant. It can be seen that the linear controller is not able to cope with the large isturbance. The system rifts away to a state with a high soli temperature an a high water content of the membrane. At the en of the simulation, the relative humity in the cathoe iffusion layers reaches 1 %, which means flooing of the cell. An improve behavior of the linear controller is obtaine, if x SP an u SP are ajuste to each 6634
Fig. 5. Response of the close loop system with passive controller to the test signal in Figure 3: Spatial profiles of the water content in the membrane an of the soli temperature for times between 1 s an 4 s Fig. 7. Response of the linear controller to the test signal in Figure 3, if the reference values x ref an u ref are ajuste to the require cell power controller only balances the water an energy fluxes into an out of the stack. In the current work, it is assume that the state vector is completely measurable The more realistic case of limite measurement information is treate in [13]. Fig. 6. Response of the close loop system with linear controller to the test signal in Figure 3, if the reference values x SP an u SP are kept constant change of the electrical loa see Figure 7. However, this is quite inconvenient, as it requires to fin suitable profiles of the membrane water concentration an of the soli internal energy for each esire power output an to store them in a look-up table. VI. CONCLUSIONS A nonlinear controller has been presente that is base on a spatially istribute fuel cell moel an makes use of the concept of passivity. The main avantage of the controller is that it is able to hanle fast loa changes of a fuel cell stack. It is foun to be superior to a linear LQ optimal controller that has been esigne for the purpose of comparison. Furthermore, the nonlinear feeback law is more transparent than the linear feeback law. While the linear controller requires storing a high number of coefficients for the gain matrix an the states at the setpoints, the passive REFERENCES [1] Na, Woon Ki, Gou, Bei, Feeback-linearization-base nonlinear control for PEM fuel cells, IEEE Transactions on Energy Conversion 23 1 28 179 19. [2] J. Golbert, D. Lewin, Moel-base control of fuel cells: 1 regulatory control, Journal of Power Sources 135 24 135 151. [3] M. Danzer, S. Wittmann, E. Hofer, Prevention of fuel cell starvation by moel preictive control of pressure, excess ratio, an current, Journal of Power Sources 19 1 29 86 91. [4] M. Danzer, J. Wilhelm, H. Aschemann, E. Hofer, Moel-base control of cathoe pressure an oxygen excess ratio of a PEM fuel cell system, Journal of Power Sources 176 2 28 515 522. [5] M. Sheng, M. Mangol, A. Kienle, A strategy for the spatial temperature control of a molten carbonate fuel cell system, Journal of Power Sources 162 1 26 1213 1219. [6] R. Methekar, V. Prasa, R. Gui, Dynamic analysis an linear control strategies for proton exchange membrane fuel cell using a istribute parameter moel, Journal of Power Sources 165 27 152 17. [7] B. McCain, A. Stefanopoulou, I. Kolmanovsky, On the ynamics an control of through-plane water istributions in PEM fuel cells, Chemical Engineering Science 63 28 4418 4432. [8] J. Willems, Dissipative ynamical systems, part i: General theory; part ii: Linear systems with quaratic supply rates, Archive for Rational Mechanics an Analysis 45 1972 321 393. [9] H. K. Khalil, Nonlinear Systems, Prentice-Hall, 2. [1] B. Ystie, A. Alonso, Process systems an passivity via the Clausius- Planck inequality, Systems & Control Letters 3 1997 253 264. [11] A. Alonso, B. Ystie, Stabilization of istribute systems using irreversible thermoynamics, Automatica 37 21 1739 1755. [12] M. Ruszkowski, V. Garcia-Osorio, B. Ystie, Passivity base control of transport reaction systems, AIChE Journal 51 12 25 3147 3166. [13] M. Mangol, A. Bück, R. Hanke-Rauschenbach, Passivity base control of a istribute PEM fuel cell moel, Journal of Process Control 2 21 292 313. [14] W. Neubran, Moellbilung un Simulation von Elektromembranverfahren, Logos-Verlag, 1999. [15] M. Ruszkowski, M. Rea, R. Kaiser, P. Richarson, T. Kern, B. Ystie, Passivity base control an optimization of a silicon process, in: Proceeings of 16th IFAC Worl Congress, Elsevier, 25, pp. 345 35. 6635