CONTROL-ORIENTED MODELING AND ANALYSIS FOR AUTOMOTIVE FUEL CELL SYSTEMS

Transcription

1 Abstract CONTROL-ORIENTED MODELING AND ANALYSIS FOR AUTOMOTIVE FUEL CELL SYSTEMS Jay T. Pukrushan Huei Peng 1 Anna G. Stefanooulou Automotive Research Center Deartment of Mechanical Engineering University of Michigan Ann Arbor, Michigan ukrush@umich.edu Fuel Cells are electrochemical devices that convert the chemical energy of a gaseous fuel directly into electricity. They are widely regarded as a otential future stationary and mobile ower source. The resonse of a fuel cell system deends on the air and hydrogen feed, flow and ressure regulation, and heat and water management. In this aer, we develo a dynamic model suitable for the control study of fuel cell systems. The transient henomena catured in the model include the flow and inertia dynamics of the comressor, the manifold filling dynamics (both anode and cathode), reactant artial ressures, and membrane humidity. It is imortant to note, however, that the fuel cell stack temerature is treated as a arameter rather than a state variable of this model because of its long time constant. Limitations and several ossible alications of this model are resented. 1 Introduction Fuel cell stack systems are under intensive develoment for mobile and stationary ower alications. In articular, Proton Exchange Membrane (PEM) Fuel Cells (also known as Polymer Electrolyte Membrane Fuel Cells) are currently in a relatively more mature stage for ground vehicle alications. Recent announcements of GM AUTOnomy concet and federal rogram FreedomCAR are examles of major interest from both the government and automobile manufacturers regarding this alternative energy conversion concet. To comete with existing internal combustion engines, fuel cell systems must oerate at similar levels of erformance. Transient behavior is one of the key requirements for the success of fuel cell vehicles. Efficient fuel cell system ower roduction deends on roer air and hydrogen feed, and heat and water management. During transients, the fuel cell stack breathing control system is required to maintain roer temerature, membrane hydration, and artial ressure of the reactants across the membrane to avoid degradation of the stack voltage, and to maintain high efficiency and long stack life [1]. Creating a control- 1 Corresonding author, Associate Professor, Deartment of Mechanical Engineering, University of Michigan, Ann Arbor, MI , , heng@umich.edu DS Author: Peng, H. 1

2 oriented model is a critical first ste for the understanding of the system behavior, and the subsequent design and analysis of model-based control systems. Models suitable for control studies have certain attributes. Imortant characteristics such as dynamic (transient) effects are included while effects such as satial variation of arameters or dynamic variables are discretized, lumed, or ignored. In this aer, only dynamic effects that are related to automobile oerations are included in the model. The extremely fast electrochemical reactions and electrical dynamics have minimal effects on automobile alications and thus are neglected. The transient behavior due to manifold filling dynamics, membrane water content, suercharging devices, and temerature may imact the behavior of the vehicle [2], and should be included in the model. However, since the stack temerature is much slower comared with other dynamic henomena, it could be simulated and regulated with its own (slower) controller. The temerature is thus treated as a arameter in the model. Desite a large number of ublications on fuel cell modeling, relatively few are suitable for control studies. Many ublications target the fuel cell erformance rediction with the main urose of designing cell comonents and choosing fuel cell oerating oints [3-6]. These models are mostly steady-state, analyzed at the cell level, and include satial variations of fuel cell arameters. They usually focus on electrochemistry, thermodynamics and fluid mechanics. These models are not suitable for control studies. However, they do rovide useful knowledge about the oeration of fuel cell stacks. On the other end of the sectrum, many steady-state system models were develoed for comonent sizing [7,8], and cumulative fuel consumtion or hybridization studies [9-11]. Here, the comressor, heat exchanger and fuel cell stack voltage are reresented by look-u tables or efficiency mas. Usually, the only dynamics considered in this tye of models is the vehicle inertia, and sometimes fuel cell stack temerature. The temerature dynamic is the focus of several ublications [12-14]. Many of these aers focus on the startu eriod, during which the stack oerating temerature needs to be reached quickly. A few ublications [2,15,16] include the dynamics of the air suly system and their influence on the fuel cell system behavior. In this aer, a dynamic fuel cell system model suitable for control studies is resented. The transient henomena catured in the model include the flow and inertia dynamics of the comressor, the manifold filling dynamics (both anode and cathode), and membrane humidity. These variables affect the fuel cell stack voltage, and thus fuel cell efficiency and ower. Unlike other models in the literature where a single olarization curve or a set of olarization curves under different cathode ressure is used, the fuel cell olarization curve used in this aer is a function of oxygen and hydrogen artial ressure, stack temerature, and membrane water content. This allows us to assess the effects of varying oxygen concentration and DS Author: Peng, H. 2

3 membrane humidity on the fuel cell voltage, which is necessary for control develoment during transient oeration. The current status of the fuel cell industry and research is highly secretive. Therefore, we are not able to obtain test data to comletely verify our model. The main contribution of this aer is thus in comiling the scattered information in the literature, and constructing a model temlate to reflect the state of the art. The obtained model is useful in showing the behavior of fuel cell systems qualitatively, rather than quantitatively. 2 Nomenclature A fc Fuel cell active area (cm 2 ) A T Valve oening area (m 2 ) C Throttle discharge coefficient D C Secific heat (J kg -1 K -1 ) D w Membrane diffusion coefficient (cm 2 /sec) E Fuel cell oen circuit voltage (V) F Faraday s number (Coulombs) I Stack current (A) J Rotational inertia (kg m 2 ) M Molecular Mass (kg/mol) P Power (Watt) R Gas constant or electrical resistance (Ω) T Temerature (K) V Volume (m 3 ) W Mass flow rate (kg/sec) a Water activity c Water concentration (mol/cm 3 ) d c Comressor diameter (m) i Current density (A/cm 2 ) m Mass (kg) n Number of cells n d Electro-osmotic drag coefficient Pressure (Pa) t Time (sec) t m Membrane thickness (cm) u System inut v Voltage (V) x Mass fraction or system state vector y Mole fraction or system measurements γ Ratio of the secific heats of air η Efficiency λ Excess ratio or water content ρ Density (kg/cm 3 ) τ Torque (N-m) φ Relative humidity ω Rotational seed (rad/sec) Subscrits act Activation Loss air Air an Anode ca Cathode conc Concentration Loss c Comressor fc Fuel cell gen Generated in Inlet m Membrane membr Across membrane ohm Ohmic loss out Outlet rm Return manifold sm Suly manifold st Stack v Vaor w Water 3 Fuel Cell Proulsion System for Automobiles A fuel cell stack needs to be integrated with several auxiliary comonents to form a comlete fuel cell system. The diagram in Figure 1 shows an examle fuel cell system. The fuel cell stack is augmented by four auxiliary DS Author: Peng, H. 3

4 systems: (i) hydrogen suly system, (ii) air suly system, (iii) cooling system, and (iv) humidification system. In Figure 1, we assume that a comressed hydrogen tank is used. The control of hydrogen flow is thus achieved simly by controlling the hydrogen suly valve to reach the desired flow or ressure. The air is assumed to be sulied by an air comressor, which is used to increase the ower density of the overall system. Figure 1 shows an external humidification system for both anode and cathode gases. PEM fuel cells without any external humidification have also been studied (e.g., [4]). Secial membranes can be used in self-humidification designs [17]. In contrast to these other methods, external humidification usually rovides higher authority and better erformance, albeit at higher system comlexity and cost. Figure 1 Automotive fuel cell roulsion system The ower of the fuel cell stack is a function of the current drawn from the stack and the resulting stack voltage. The cell voltage is a function of the stack current, reactant artial ressure inside each cell, cell temerature and membrane humidity. In this aer, we assume that the stack is well designed so that all cells erform similarly and can be lumed as a stack. For examle, all the cell temeratures are identical, and thus we only need to kee track of the stack temerature; if starvation or membrane dehydration exists, it occurs simultaneously in every cell and thus all cells are reresented by the same set of olarization curves. This assumtion of invariable cell-to-cell erformance is necessary for low-order system models. As electric current is drawn from the stack, oxygen and hydrogen are consumed, and water and heat are generated. To maintain the desired hydrogen artial ressure, the hydrogen needs to be relenished by its suly system, which includes the ressurized hydrogen tank and a suly servo valve. Similarly, the air suly system needs to relenish the air to maintain the oxygen artial ressure. The air suly system consists of an air comressor, an electric motor and ies or manifolds between the comonents. The comressor not only achieves desired air flow but also increases air ressure which significantly imroves the DS Author: Peng, H. 4

5 reaction rate at the membranes, and thus the overall efficiency and ower density. Since the ressurized air flow leaving the comressor is at a higher temerature, an air cooler may be needed to reduce the temerature of the air entering the stack. A humidifier is used to revent dehydration of the fuel cell membrane. The water used in the humidifier is sulied from the water tank. Water level in the tank is maintained by collecting water generated in the stack, which is carried out with the air flow. The excessive heat released in the fuel cell reaction is removed by the cooling system, which circulates de-ionized water through the fuel cell stack and removes the excess heat via a heat exchanger. Power conditioning is usually needed since the voltage of fuel cell stack varies significantly. 4 Fuel Cell System Model In this aer, we will not resent a model that includes all sub-systems shown in Figure 1. Rather, the roblem is simlified by assuming that the stack temerature is constant. This assumtion is justified because the stack temerature changes relatively slowly, comared with the ~100ms transient dynamics included in the model to be develoed. Additionally, it is also assumed that the temerature and humidity of the inlet reactant flows are erfectly controlled, e.g., by well designed humidity and cooling sub-systems. The system studied in this aer is shown in Figure 2. It is assumed that the cathode and anode volumes of the multile fuel cells are lumed as a single stack cathode and anode volumes. The anode suly and return manifold volumes are small, which allows us to lum these volumes to one anode volume. We denote all the variables associated with the lumed anode volume with a subscrit (an). The cathode suly manifold (sm) lums all the volumes associated with ies and connection between the comressor and the stack cathode (ca) flow field. The cathode return manifold (rm) reresents the lumed volume of ies downstream of the stack cathode. In this aer, an exander is not included; however, one will be added in future models. It is assumed that the roerties of the flow exiting a volume are the same as those of the gas inside the volume. Subscrits (c) and (cm) denote variables associated with the comressor and comressor motor, resectively. The rotational dynamics and a flow ma are used to model the comressor. The law of conservation of mass is used to track the gas secies in each volume. The rincile of mass conservation is alied to calculate the roerties of the combined gas in the suly and return manifolds. The law of conservation of energy is alied to the air in the suly manifold to account for the effect of temerature variations. The model is develoed rimarily based on hysics. However, several henomena are described in emirical equations. In the following sections, models for the fuel cell stack, comressor, manifolds, air cooler and humidifier are resented. DS Author: Peng, H. 5

6 Figure 2 Simlified fuel cell reactant suly system 4.1 Fuel Cell Stack Model The electrochemical reaction at the membranes is assumed to occur instantaneously. The fuel cell stack (st) model contains four interacting sub-models: the stack voltage model, the anode flow model, the cathode flow model, and the membrane hydration model (Figure 3). We assume that the stack temerature is constant at 0 80 C. The voltage model contains an equation to calculate stack voltage based on fuel cell ressure, temerature, reactant gas artial ressures and membrane humidity. The dynamically varying ressure and relative humidity of the reactant gas flow inside the stack flow channels are calculated in the cathode and the anode flow models. The rocess of water transfer across the membrane is governed by the membrane hydration model. These sub-system models are discussed in the following sub-sections. Figure 3 Fuel cell stack block diagram DS Author: Peng, H. 6

7 4.1.1 Stack Voltage Model The stack voltage is calculated as a function of stack current, cathode ressure, reactant artial ressures, fuel cell temerature and membrane humidity. The current-voltage relationshi is commonly given in the form of the olarization curve, which is lotted as cell voltage, v fc, versus cell current density, i fc (see Figure 4 for an examle). Since the fuel cell stack consists of multile fuel cells connected in series, the stack voltage, v st, is obtained as the sum of the individual cell voltages; and the stack current, I st, is equal to the cell current. The current density is then defined as stack current er unit of cell active area, i fc = I st A fc. Under the assumtion that all cells are identical, the stack voltage can be calculated by multilying the cell voltage, v fc, by the number of cells,, of the stack ( v n st n vf c = ). The fuel cell voltage is calculated using a combination of hysical and emirical relationshis, and is given by [18] vfc = E vact vohm vconc (1) where E is the oen circuit voltage and v, v and v are activation, ohmic and concentration act ohm overvoltages, which reresent losses due to various hysical or chemical factors. The oen circuit voltage is calculated from the energy balance between the reactants and roducts, and the Faraday Constant, and is [3] conc E = ( T ) T ln( ) + ln( ) (Volts) 2 where the fuel cell temerature exressed in atm fc fc H2 O2 T fc is exressed in Kelvin, and reactant artial ressures H 2 and O 2 (2) are The activation overvoltage, v act, arises from the need to move electrons and to break and form chemical bonds at the anode and cathode [19]. The relationshi between the activation overvoltage and the current density is described by the Tafel equation, which is aroximated by v = v + v e act ci 1 0 a (1 ) The activation overvoltage deends on temerature and oxygen artial ressure [3,20]. The values of, va and c1 and their deendency on oxygen artial ressure and temerature can be determined from nonlinear regression of exerimental data. (3) v 0 DS Author: Peng, H. 7

8 Figure 4 Fuel cell olarization curve fitting results at 80 degree Celcius The ohmic overvoltage, v ohm, arises from the resistance of the olymer membrane to the transfer of rotons and the resistance of the electrodes and collector lates to the transfer of electrons. The voltage dro is thus roortional to the stack current v ohm = i R (4) ohm The resistance, R ohm, deends strongly on membrane humidity [21] and cell temerature [22]. The ohmic resistance is roortional to membrane thickness t and inversely roortional to the membrane m conductivity, σ ( λ, T fc ) ( Ω cm) m m 1 [6,23], i.e., R ohm t σ m = (5) m where λ m reresents the membrane water content. The membrane conductivity is a function of membrane humidity and temerature in the form [6] 1 1 σm = ( b11λm b12)ex b2 ( ) (6) 303 Tfc m b where the value of t, and b for the Nafion 117 membrane [6] are used, and b is adjusted to fit our fuel cell data. The calculation of λ m is given in Section Its value varies between 0 and 14 [6], which corresond to relative humidity (RH) of 0% and 100%, resectively. The concentration overvoltage, v conc, results from the increased loss at high current density, e.g., a significant dro in reactant concentration due to both high reactant consumtion and head loss at high flow rate. This DS Author: Peng, H. 8

9 term is ignored in some models, e.g. [24], erhas because it is not desirable to oerate the stack at regions where v conc is high (efficiency is low). If the stack will oerate at high current density, however, this term needs to be included. An equation to aroximate the concentration losses is given by [2] v conc i c2 i max c3 i = (7) where c, c and i are constants that deend on temerature and reactant artial ressure and can be 2 3 max determined emirically. The coefficients in Equations (3) and (7) are determined using nonlinear regression with olarization data from an automotive roulsion-sized PEM fuel cell stack [1]. By assuming that the data is obtained from the fuel cell stack oerating under a well-controlled environment, where cathode gas is fully humidified and oxygen excess ratio (ratio of oxygen sulied to oxygen reacted) is regulated at 2, the ressure terms in the activation and concentration overvoltage terms can be related to oxygen artial ressure, O 2, and vaor saturation ressure, sat. The regression results are 4 5 ca sat ( ca sat ) v0 = ( Tfc ) Tfc ln( ) + ln( ) O ( )( ) 2 ( O va = Tfc + + sat + Tfc 0.166)( + sat ) ( T ) c = 10, c = 2, i = fc max b = , b = , b = 350 c O 2 if + sat < 2 atm, O2 3 ( Tfc 0.622)( + sat ) + ( Tfc ) = else 5 O2 4 ( Tfc 0.068)( + sat ) + ( Tfc ) The redicted voltage and exerimental data for T = 80 C and various cathode ressure is shown in Figure fc 4. Results at other temeratures (between 40 C and 100 C ) all have similar accuracy. Based on the stack model develoed above, the effect of membrane water content on cell voltage is illustrated in Figure 5, which shows fuel cell olarization curves for a membrane water content at 100% and 50%. The model redicts significant reduction in fuel cell voltage due to change in the membrane water content, which illustrates the (8) DS Author: Peng, H. 9

10 imortance of humidity control. It should be noted that over-saturated (flooding) conditions will cause condensation and liquid formation inside the anode or the cathode, which leads to voltage degradation [25]. This effect is currently not catured in our model. Note also that the coefficients in Equation (8) are derived from exerimental data for a secific stack. To model a different stack, the same basis functions (2)-(7) may be used but the coefficients need to be determined using data from the new stack. Figure 5 Effect of membrane water content (100 o C and 2.5 bar air ressure) Cathode Flow Model This model catures the cathode air flow behavior, and is develoed using the mass conservation rincile and the thermodynamic and sychrometric roerties of air. Several assumtions are made: (1) All gases obey the ideal gas law; (2) The temerature of the air inside the cathode is equal to the stack temerature; (3) The roerties of the flow exiting the cathode such as temerature, ressure, and humidity are assumed to be the same as those inside the cathode; (4) When the relative humidity of the gas exceeds 100%, vaor condenses into the liquid form. The liquid water does not leave the stack and will either evaorate when the humidity dros below 100% or accumulate in the cathode; (5) finally, the flow channel and cathode backing layer are lumed into one volume, i.e. the satial variations are ignored. The mass continuity is used to balance the mass of the three elements oxygen, nitrogen and water, inside the cathode volume. dm O2 dt dm N2 dt dm wca, dt = W W W O2, in O2, out O2, reacted = W W N2, in N2, out = W W + W + W v, ca, in v, ca, out v, ca, gen v, membr Using the masses of oxygen,, nitrogen, m, and water, m, and the stack temerature, m O2 N2 w (9) T st, we use the ideal gas law and thermodynamic roerties to calculate oxygen, nitrogen and vaor artial ressure, O 2,,, N 2 vca, cathode total ressure, = + +, relative humidity, φ ca, and dry air oxygen mole ca O2 N2 v, ca DS Author: Peng, H. 10

11 fraction, y O 2, ca, of the cathode flow. If the calculated water mass is more than that of the saturated state, the extra amount is assumed to condense to liquid form instantaneously. Figure 6 illustrates the cathode model in a block diagram format. The inlet (in) and outlet (out) mass flow rate of oxygen, nitrogen and vaor in Equation (9) are calculated from the inlet and outlet cathode flow conditions using thermodynamic roerties. The detailed calculations are given in Aendices A and B. The cathode inlet flow rate and condition are calculated in the humidifier model (Section 4.5). A linearized nozzle equation (A4) is used to calculate the cathode exit flow rate, W. Electrochemistry rinciles are used to calculate the rates of oxygen consumtion, W roduction, W vc, agen,, from the stack current W I st : O2, reacted O2 W vcagen,, O2, reacted ca, out, and water nist = M (10) 4F ni st = Mv (11) 2F where F is the Faraday Constant (=96485 Coulombs) and M O and M 2 v are the molar masses of oxygen and water, resectively. The water flow rate across the membrane, W membrane hydration model in Section vmembr,, in (9) is calculated from the Figure 6 Cathode flow model DS Author: Peng, H. 11

12 4.1.3 Anode Flow Model This model is quite similar to the cathode flow model. Hydrogen artial ressure and anode flow humidity are determined by balancing the mass of hydrogen, m, and water in the anode,. H 2 m wan, dm dt dm H2 wan, dt = W W W (12) H2, in H2, out H2, reacted = W W W (13) v, an, in v, an, out v, membr In this model, ure hydrogen gas is assumed to be sulied to the anode from a hydrogen tank. It is assumed that the hydrogen flow rate can be instantaneously adjusted by a valve while maintaining a minimum ressure difference across the membrane. This has been achieved by using a high gain roortional controller, discussed in Section 4.7, to control the hydrogen flow rate such that the anode ressure, cathode ressure, an, tracks the ca. The inlet hydrogen flow is assumed to have 100% relative humidity. The anode outlet flow reresents ossible hydrogen urge and is currently assumed to be zero (W an, out = 0 ). The anode hydrogen temerature is assumed to be equal to the stack temerature. The rate of hydrogen consumed in the reaction, W H 2, reacted, is a function of the stack current where M H 2 is hydrogen molar mass. W H2, reacted H2 nist = M (14) 2F Membrane Hydration Model The membrane hydration model catures the effect of water transort across the membrane. Both water content and mass flow are assumed to be uniform over the surface area of the membrane, and are functions of stack current and relative humidity of the gas in the anode and cathode. The water transort across the membrane is achieved through two distinct henomena [6,23]. First, the electro-osmotic drag henomenon is due to the water molecules dragged across the membrane from anode to cathode by the rotons. The amount of water transorted is roortional to the electro-osmotic drag coefficient, n d, which is defined as the number of water molecules carried by each roton. Secondly, the gradient of water concentration across the membrane results in back-diffusion of water, usually from cathode to anode. The water concentration, c v, is assumed to change linearly over the membrane thickness, t m. Combining the two water transort mechanisms, the water flow across the membrane from anode to cathode is DS Author: Peng, H. 12

13 ni W = M A n D The coefficients n and D vary with membrane water content, d w the water content at the anode ( λ an ) and cathode ( λ ca ). c c d st vca, van, vmembr, v fc w F tm (15) = = [ ] λ m, which is calculated from the average of λ an and λ ca are calculated from the membrane water activity, a i yv, i i / sat. i v, i / sat. i : i an, c a, from the following equation: + a a + a < a λi = ( ai 1), 1< ai i i 36 i,0 i 1 The electro-osmotic and diffusion coefficients are calculated from [26] and n λ λ (16) 2 19 d = m m (17) D w 1 1 = Dλ ex 2416( ) 303 T fc (18) where and fc D λ 6 10, λm < (1 + 2( λm 2)),2 λm < 3 = 6 10 (3 1.67( λ 3)),3 λm 4.5 m < , λm 4.5 T is the fuel cell temerature. The water concentration at the membrane surfaces, and, are functions of water content on the surface, λ ca and λ an. Secifically, c, ρ, λ / M, vi mdry i mdry c vca, c van, (19) =, i [ an, ca] where ρ mdry, (kg/cm 3 ) is the dry membrane density and M mdry, (kg/mol) is the membrane dry equivalent weight. These equations are develoed based on exerimental results measured for Nafion 117 membrane in [6]. During the last decade, PEM membranes have evolved tremendously and the emirical equations (16)- (19) may no longer be a good reresentation of the new membrane roerties. However, information about newer membranes is not available in the literature at the time this aer is written. 4.2 Comressor Model A lumed rotational model is used to reresent the dynamic behavior of the comressor, J dω c c τ cm τ c dt = (20) where τ cm ( v cm, ω c ) is the comressor motor (CM) torque and τ c is the load torque to be exlained below. The comressor motor torque is calculated using a static motor equation DS Author: Peng, H. 13

14 where kt, τ kt = η ( v k ω ) (21) R cm cm cm v c cm R cm and kv are motor constants and η cm is the motor mechanical efficiency. The torque required to drive the comressor is calculated using the thermodynamic equation γ 1 C T atm γ sm τ c = 1 W ωc η c atm where γ is the ratio of the secific heats of air (= 1.4), C c (22) is the constant-ressure secific heat caacity of 1 1 air (= 1004 Jkg K ), η c is comressor efficiency, sm is the ressure inside the suly manifold and atm and T atm are the atmosheric ressure and temerature, resectively. Figure 7 Comressor ma A static comressor ma is used to determine the air flow rate through the comressor, W. The comressor flow characteristic sm Wc (, ω c ) is modeled by the Jensen and Kristensen nonlinear curve atm fitting method [27], which reresents the comressor data very well as shown in Figure 7. The comressor model used here is for an Allied Signal comressor [28]. Thermodynamic equations are used to calculate the exit air temerature. c T c γ 1 T atm γ sm = Tatm + 1 η c atm (23) DS Author: Peng, H. 14

15 4.3 Suly Manifold Model The cathode suly manifold (SM) includes ie and stack manifold volumes between the comressor and the fuel cells. The suly manifold ressure, equations where dm dt sm, is governed by mass continuity and energy conservation sm = W W (24) c sm, out dsm γ Ra = ( W c T c W sm, out T sm) (25) dt V sm R a is the air gas constant, V sm is the suly manifold volume, and T sm is the temerature of the flow inside the manifold calculated from the ideal gas law. The suly manifold exit flow, W sm, out, is calculated as a function of sm and ca using the linearized nozzle flow equation shown in Aendix A. 4.4 Static Air Cooler Model The air temerature leaving the comressor is usually high due to the increased ressure. To revent the fuel cell membrane from damaging, the air may need to be cooled down before it is sent to the stack. In this study, we assume that an ideal air cooler (CL) maintains the temerature of the air entering the stack at T cl = 80 C. Further, it is assumed that the ressure dro across the cooler is negligible, cl = sm. The humidity of the gas exiting the cooler is then calculated from where 0.5 atm vcl, clvatm, clφatm sat ( Tatm) φ cl = = = (26) ( T ) ( T ) ( T ) sat cl atm sat cl atm sat cl φ = is the assumed ambient air relative humidity and ( ) sat i is the vaor saturation ressure. 4.5 Static Humidifier Model Since we assume that the inlet air is humidified to the desired relative humidity before entering the stack, a static humidifier model is needed to calculate the required water that needs to be injected into the air. Additionally, the model also determines the changes in total flow rate and ressure due to the added water. The temerature of the flow is assumed constant. The water injected is assumed to be in the form of vaor. The amount of vaor injected is calculated from the vaor flow at the cooler outlet and the required vaor flow for the desired humidity, cl sm, out cl cl cl φ des. Based on the condition of the flow exiting the cooler W ( W,, T, φ ), the dry air mass flow rate, W, the vaor mass flow rate, W, and the dry air acl, vcl, DS Author: Peng, H. 15

16 ressure, acl,, are calculated using equations (B1)-(B5). The flow rate of vaor injected is then calculated from where M and v W M φ ( T ) = W W (27) v des sat cl vinj, acl, vcl, Ma a, cl M a are molar mass of vaor and dry air, resectively. The cathode inlet flow rate and ressure are Wca, in = W cl +Wv, inj and ca, in = a, cl + φdes sat ( Tcl ). 4.6 Return Manifold Model Unlike the suly manifold, where temerature changes need to be considered, the temerature in the return manifold, T rm, is assumed constant and equal to the temerature of the flow leaving the cathode. The return manifold ressure, assumtions. rm, is governed by the mass conservation and the ideal gas law through isothermal d dt R T = ( Wca, out Wrm, out ) (28) V rm a rm rm The nonlinear nozzle equations (A2)-(A3) are used to calculate the return manifold exit air flow rate, W, A Trm, as a function of the return manifold ressure and back-ressure valve oening area,. rm, out 4.7 Hydrogen Flow Since ressurized hydrogen is used, the anode hydrogen flow can be regulated by a servo valve to achieve very high loo bandwidth. The goal of the hydrogen flow control is to minimize the ressure difference across the membrane, i.e., the difference between anode and cathode ressures. Since the valve is fast, it is assumed that the flow rate of hydrogen can be directly controlled based on the feedback of the ressure difference. ( ) W = K K (29) an, in 1 2 sm an kg/s where K 1 = 2.1 kpa is the roortional gain and K 2 = 0.94 takes into account a nominal ressure dro between the suly manifold and the cathode. A urge valve is commonly installed at the anode exit to remove water drolets. The urge valve can be used to reduce the anode ressure quickly if necessary (e.g., when anode flow calculated from Equation (29) is negative). 4.8 Model Summary The fuel cell system model develoed above contains nine states. The comressor has one state: rotor seed. The suly manifold has two states: air mass and air ressure. The return manifold has one state: air ressure. The stack has five states: O 2, N 2, and vaor masses in the cathode, and H 2 and vaor masses in the anode. DS Author: Peng, H. 16

17 These states then determine the voltage outut of the stack. Under the assumtions of a erfect humidifier and air cooler, and the use of roortional control of the hydrogen valve, the only inuts to the model are the stack current, I st, and the comressor motor voltage, v. The arameters used in the model are given in Table 1. Most of the arameters are based on the 75 kw stacks used in the FORD P2000 fuel cell rototye vehicle [29]. The active area of the fuel cell is calculated from the eak ower of the stack. The values of the volumes are aroximated from the dimensions of the P2000 fuel cell system. Table 1 Model arameters for vehicle-size fuel cell system Symbol Variable Value ρ mdry, Membrane dry density kg/cm 3 M mdry, Membrane dry equivalent weight 1.1 kg/mol t m Membrane thickness cm n Number of cells in stack 381 A fc Fuel cell active area 280 cm 2 d c Comressor diameter m J Comressor and motor inertia kg m 2 c V an Anode volume m 3 V ca Cathode volume 0.01 m 3 V sm Suly manifold volume 0.02 m 3 V rm Return manifold volume m 3 C D, rm Return manifold throttle discharge coefficient A Trm, Return manifold throttle area m 2 k Suly manifold outlet orifice constant kg/(s Pa) sm, out k Cathode outlet orifice constant kg/(s Pa) ca, out k v Motor electric constant V/(rad/sec) k t Motor torque constant N-m/A R cm Comressor Motor circuit resistance Ω η Comressor Motor efficiency 98% cm cm 5 Steady-State Analysis The model we have develoed will be used to conduct a few examle analyses imortant to fuel cell system engineers. In this section, the otimal steady-state oerating oint for the air comressor is studied. The net ower of the fuel cell system, Pnet, which is the difference between the ower roduced by the stack, and the arasitic ower, should be maximized. The majority of the arasitic ower for an automotive fuel cell system is sent on the air comressor, thus, it is imortant to determine the roer air flow. The air flow excess is reflected by the term oxygen excess ratio, used in the cathode, i.e., λ O 2 P st,, defined as the ratio of oxygen sulied to oxygen DS Author: Peng, H. 17

18 WO 2, in λ O = (30) 2 W, O2 react High oxygen excess ratio, and thus high oxygen artial ressure, imroves P st and value is reached, however, further increase in λo 2 P net. After an otimum will result in an increase in comressor ower and only marginal increase in P st. Therefore, P net decreases. To identify the otimal value for λ O 2, we run the model reeatedly, and then lot steady-state values of λ and P, at different stack current I (see Figure 8). It O 2 net st can be seen that the otimal oxygen excess ratio varies between 2.0 and 2.4, and decreases slowly when the stack current increases. Note that the results may also be influenced by factors not included in the model, such as stack flooding. Figure 8 System net ower at different stack current and oxygen excess ratios 6 Dynamic Simulation A series of ste changes in stack current (Figure 9(a)) and comressor motor inut voltage (Figure 9(b)) are alied to the stack at a nominal stack oerating temerature of 80 C. During the first four stes, the comressor voltage is controlled so that the otimal oxygen excess ratio (~2.0) is maintained. This is achieved with the simle static feedforward controller as shown in Figure 10. The remaining stes are then alied indeendently, resulting in different levels of oxygen excess ratios (Figure 9(e)). During a ositive current ste, the oxygen excess ratio dros due to the deletion of oxygen (Figure 9(e)), which causes a significant dro in the stack voltage (Figure 9(c)). When the comressor voltage is controlled by the feedforward algorithm, there is still a noticeable transient effect on the stack voltage (Figure 9(c)), and oxygen artial ressure at cathode exit (Figure 9(f)). The ste at t = 18 seconds shows the resonse of giving a ste increase in the comressor inut while keeing constant stack current. An oosite case is shown at t = 22 seconds. The resonse between 18 and 22 seconds shows the effect of running the system at an DS Author: Peng, H. 18

19 excess ratio higher than the otimum value. It can be seen that even though the stack voltage (ower) increases, the net ower actually decreases due to the increased arasitic loss. Figure 9 Simulation results of the fuel cell system model for a series of inut ste changes Figure 10 Static feedforward using steady-state ma Figure 11 shows the fuel cell resonse described above lotted on the olarization ma. The results are qualitatively similar to the exerimental results resented in [21]. The comressor resonse for this simulation is shown in Figure 12. The lot shows that the comressor resonse does not follow the oerating line (dashed line) during transient. It is also clear from Figure 12 that large and raid reductions of the comressor voltage should be avoided as the comressor may be ushed into the surge (instability) region [30]. Figure 11 Fuel cell resonse on olarization curve. Solid line assumes fully humidified membrane; dashed line reresents drying membrane. DS Author: Peng, H. 19

20 Figure 12 Comressor transient resonse on comressor ma 7 Observability Analysis Simulation results in the revious section show that a static controller is not good enough in rejecting the adverse effect of disturbances (stack load current). The design of an advanced control algorithm is beyond the scoe of this aer; however, we would like to resent examle analysis of the observability of different measurements, a critical ste for multivariable control develoment. In this section, a linearized model will be used to study the system observability. Three measurements are investigated: comressor air flow rate, y = W, suly manifold ressure, y2 = sm, and fuel cell stack voltage, y3 = Vst. These signals are usually 1 c available because they are easy to measure and are useful for other uroses. For examle, the comressor flow rate is tyically measured for the internal feedback of the comressor. The stack voltage is monitored for diagnostics and fault detection uroses. The LTI system analysis in MATLAB/SIMULINK control system toolbox is used to linearize the model. The nominal oerating oint is chosen to be P net = 40 kw and λ = 2, which corresond to nominal inuts 2 O of I = 191 Am and v = 164 Volt. The linear model is st cm x = Ax + Bu y = Cx+ Du (31) where x m m m ω m m, = u [ v I ] T O2 H2 N2 c sm sm w, an rm T = cm st and y = Wc sm vst. Here, the units of states and oututs are selected so that all variables have comarable magnitudes, and are as follows: mass in grams, ressure in bar, rotational seed in krpm, mass flow rate in g/sec, ower in kw, voltage in V, and current in A. The matrices of the linearized model (31) are given in Aendix C. DS Author: Peng, H. 20

21 The linear model has eight states while the nonlinear model has nine states. The state removed is the mass of water in the cathode. The reason is that the arameters of the membrane water flow available in the literature always redicts excessive water flow from anode to cathode which results in fully humidified cathode gas under all nominal conditions. Additionally, our nonlinear model does not include the effects of liquid condensation, also known as flooding, on the fuel cell voltage resonse. Table 2 Eigenvalues, eigenvectors, and observability The analysis results on system observability are shown in Table 2. The table shows system eigenvalues, λ i, eigenvectors, and corresonding rank and condition number of the matrix λ I A i C for three different cases: 1) measuring only W c (y 1 ), 2) measuring W c and sm ( and y ), and 3) measuring all W, c y1 2 sm, and v st (all three oututs). The eigenvalue is unobservable if the corresonding matrix (32) loses rank [31]. A large condition number of a matrix imlies that the matrix is almost rank deficient, i.e., the corresonding eigenvalue is weakly observable. (32) Table 2 shows that with only the comressor air flow rate W, the system is not observable, and adding c sm measurement does not change the observability. This is because ressure and flow are related by an integrator. The eigenvalues and are not observable with measurements W and c sm. The eigenvectors associated with these eigenvalues reveal that the unobservable mode is associated with the DS Author: Peng, H. 21

22 dynamics at the anode side. This analysis is consistent with intuition since these two measurements are on the air suly side and the only connection between them to the anode is through a weak membrane water transort. These two unobservable eigenvalues are, however, stable and fast, and thus they may only have a small effect on the estimation of other states. On the other hand, the slow eigenvalues at and can degrade estimator erformance because they are weakly observable, as indicated by large condition number at and , resectively. Adding the stack voltage measurement substantially imroves the state observability, as can be seen from the rank and the condition number for Case 3. Stack voltage is currently used for monitoring, diagnostic, and emergency stack shut-down rocedures. The observability analysis above suggests that the stack voltage should be used for state estimation uroses. 8 Conclusion A control-oriented fuel cell system model has been develoed using hysical rinciles and stack olarization data. The inertia dynamics of the comressor, manifold filling dynamics and time-evolving reactant mass, humidity and artial ressure, and membrane water content are catured. This model has not been fully validated but it reflects extensive work to consolidate the oen-literature information currently available. Transient exerimental data, once available, can be used to calibrate the model arameters such as membrane diffusion and osmotic drag coefficients to obtain a high fidelity model. Additionally, stack flooding effects needs to be integrated into the model. Examles of alication of the current model on the analysis and simulation of fuel cell systems are rovided selection of otimal system excess ratio, transient effect of ste inuts, and analysis of system observability. Acknowledgment The authors wish to acknowledge the Automotive Research Center at the University of Michigan and the National Science Foundation CMS and CMS for funding suort, and the Ford Motor Comany for roviding us with valuable fuel cell data. References [1] W-C Yang and B. Bates and N. Fletcher and R. Pow, Control Challenges and Methodologies in Fuel Cell Vehicle Develoment, SAE Paer 98C054. [2] L. Guzzella, Control Oriented Modelling of Fuel-Cell Based Vehicles, Presentation in NSF Worksho on the Integration of Modeling and Control for Automotive Systems, [3] J.C. Amhlett, R.M. Baumert, R.F. Mann, B.A. Peley and P.R. Roberge, Performance modeling of the Ballard Mark IV solid olymer electrolyte fuel cell, Journal of Electrochemical Society, v.142, n.1,.9-15, [4] D.M. Bernardi and M.W. Verbrugge, A Mathematical model of the solid olymer-electrolyte fuel cell, Journal of the Electrochemical Society, v.139, n.9, , [5] J.H. Lee and T.R. Lalk, Modeling fuel cell stack systems, Journal of Power Sources, v.73, , DS Author: Peng, H. 22

23 [6] T.E. Sringer, T.A. Zawodzinski and S. Gottesfeld, Polymer Electrolyte Fuel Cell Model, Journal of Electrochemical Society, v.138, n.8, , [7] F. Barbir, B. Balasubramanian and J. Neutzler, Trade-off design analysis of oerating ressure and temerature in PEM fuel cell systems, Proceedings of the ASME Advanced Energy Systems Division, v.39, , [8] D.J. Friedman, A. Egghert, P. Badrinarayanan and J. Cunningham, Balancing stack, air suly and water/thermal management demands for an indirect methanol PEM fuel cell system, SAE Paer [9] S. Akella, N. Sivashankar and S. Goalswamy, Model-based systems analysis of a hybrid fuel cell vehicle configuration, Proceedings of 2001 American Control Conference, [10] P. Atwood, S. Gurski, D.J. Nelson, K.B. Wike and T. Markel, Degree of hybridiztion ADVISOR modeling of a fuel cell hybrid electric sort utility vehicle, Proceedings of 2001 Joint ADVISOR/PSAT vehicle systems modeling user conference, , [11] D.D. Boettner, G. Paganelli, Y.G. Guezennec, G. Rizzoni and M.J. Moran, Comonent ower sizing and limits of oeration for roton exchange membrane (PEM) fuel cell/battery hybrid automotive alications, Proceedings of 2001 ASME International Mechanical Engineering Congress and Exosition, [12] W. Turner, M. Parten, D. Vines, J. Jones and T. Maxwell, Modeling a PEM fuel cell for use in a hybrid electric vehicle, Proceedings of the 1999 IEEE 49th Vehicular Technology Conference, v.2, , [13] D.D. Boettner, G. Paganelli, Y.G. Guezennec, G. Rizzoni and M.J. Moran, Proton exchange membrane (PEM) fuel cell system model for automotive vehicle simulation and control, Proceedings of 2001 ASME International Mechanical Engineering Congress and Exosition, [14] K-H Hauer, D.J. Friedmann, R.M. Moore, S. Ramaswamy, A. Eggert and P. Badrinarayana, Dynamic Resonse of an Indirect-Methanol Fuel Cell Vehicle, SAE Paer [15] J. Padulles, G.W. Ault, C.A. Smith and J.R. McDonald, Fuel cell lant dynamic modeling for ower systems simulation, Proceedings of 34th universities ower engineering conference, v.34, n.1,.21-25, [16] S. Pischinger, C. Schönfelder, W. Bornscheuer, H. Kindl and A. Wiartalla, Integrated Air Suly and Humidification Concets for Fuel Cell Systems, SAE Paer [17] M. Watanabe, H. Uchida, M. Emori, Analyses of Self-Humidification and Suression of Gas Crossover in Pt- Disersed Polymer Electrolyte Membranes for Fuel Cells, Journal of the Electrochemical Society, Volume 145, Number 4, , Aril [18] J. Larminie and A. Dicks, Fuel Cell Systems Exlained, West Sussex, England, John Wiley & Sons Inc, [19] J.H. Lee, T.R. Lalk and A.J. Aleby, Modeling electrochemical erformance in large scale roton exchange membrane fuel cell stacks, Journal of Power Sources, v.70, , [20] K. Kordesch and G. Simader, Fuel Cells and Their Alications, Weinheim, Germany, VCH, [21] F. Laurencelle, R. Chahine, J. Hamelin, K. Agbossou, M. Fournier, T.K. Bose and A. Laerriere, Characterization of a Ballard MK5-E roton exchange membrane fuel cell stack, Fuel Cells Journal, v.1, n.1,.66-71, [22] J.C. Amhlett, R.M. Baumert, R.F. Mann, B.A. Peley, P.R. Roberge and A. Rodrigues, Parametric modelling of the erformance of a 5-kW rotonexchange membrane fuel cell stack, Journal of Power Sources, v.49, , [23] T.V. Nguyen and R.E. White, A Water and Heat Management Model for Proton-Exchange-Membrane Fuel Cells, Journal of Electrochemical Society, v.140, n.8, , [24] R.F. Mann et al., Develoment and alication of a generalized steady-state electrochemical model for a PEM fuel cell, Journal of Power Sources, Vol.86, 2000, [25] J.J. Baschuk and X. Li, Modeling of olymer electrolyte membrane fuel cells with variable degreees of water flooding, Journal of Power Sources, v.86, , [26] S. Dutta, S. Shimalee and J.W. Van Zee, Numerical rediction of mass-exchange between cathode and anode channels in a PEM fuel cell, International Journal of Heat and Mass Transfer, v.44, , 2001 [27] P. Moraal and I. Kolmanovsky, Turbocharger Modeling for Automotive Control Alications, SAE Paer [28] J.M. Cunningham, M.A Hoffman, R.M Moore and D.J. Friedman, Requirements for a Flexible and Realistic Air Suly Model for Incororation into a Fuel Cell Vehicle (FCV) System Simulation, SAE Paer DS Author: Peng, H. 23

24 [29] J.A. Adams,W-C Yang, K.A. Oglesby and K.D. Osborne, The develoment of Ford s P2000 fuel cell vehicle, SAE Paer [30] J.T. Gravdahl and O. Egeland, Comressor Surge and Rotating Stall, Sringer, London, 1999 [31] T. Kailath, Linear Systems, Prentice-Hall, New Jersey, [32] P. Thomas, Simulation of Industrial Processes for Control Engineer, London, Butterworth Heinemann, Aendix A: Flow Calculations In this aendix, we first exlain the calculation of mass flow rate between two volumes using nozzle equations. Then we exlain the calculation of mass flow rates of each secies (O 2, N 2 and vaor) into and out of the cathode channel in Aendix B. The flow rates are used in the mass balance equations (9). The nozzle flow equation [32] is used to calculate the flow between two volumes. The flow rate assing through a nozzle is a function of the ustream ressure, into two regions according to the critical ressure ratio: u, and downstream ressure, r crit d 2 = = u γ + 1 crit γ γ 1 d. The flow characteristic is divided where γ = C / is the ratio of the secific heat caacities of the gas. For sub-critical (normal) flow where C v (A1) ressure dro is less than the critical ressure ratio, r > r, the mass flow rate is calculated from crit where Tu is the ustream gas temerature, 1 1 γ 1 2 γ 2γ γ CDAT u W = ( r) 1 ( r) RT γ 1 u C D is the discharge coefficient of the nozzle, (A2) is the oening area of the nozzle (m2) and R is the universal gas constant. For critical (choked) flow, r r crit, the mass flow rate is given by W choked 1 CDAT u 2 2 = γ RT γ + 1 u If the ressure difference across the nozzle is small, the flow rate can be calculated from the linearized equation γ + 1 2( γ 1) A T (A3) W = k ( ) (A4) nozzle u d Aendix B: Standard Thermodynamic Calculations Tyically, air flow roerties are given in terms of total mass flow rate, W, ressure,, temerature, T, relative humidity (RH), φ, and dry air oxygen mole fraction, y O2. We then use thermodynamic roerties to DS Author: Peng, H. 24

25 calculate the mass flow rate of the individual secies. Although these equations are standard, for comleteness and educational uroses we include here all the detailed calculations associated with converting { W T φ y } to { O, N, v},,,, O 2 W W W. Given the total flow, the humidity ratio is first used to 2 2 searate the total flow rate into the flow rates of vaor and dry air. Then, the dry air flow rate is divided into oxygen and nitrogen flow rates using the definition of y O2. Assuming ideal gases, the vaor ressure is calculated from the definition of the relative humidity where v = φ ( T) (B1) sat ( T ) is the vaor saturation ressure. Since humid air is a mixture of dry air and vaor, the dry air sat artial ressure is the difference between total ressure and vaor ressure a = v. The humidity ratio, ω, defined as a ratio between mass of vaor and mass of dry air in the gas ω M M = v v (B2) a a where where M v and M a are vaor molar mass and dry air molar mass, resectively. a O 2 M N2 M a = y M + (1 y ) M (B3) O2 O2 O2 N2 M is calculated from M and are the molar mass of oxygen and nitrogen, resectively. The oxygen mole fraction y O2 is 0.21 for the inlet air and is lower for the exit air. The flow rate of dry air and vaor are and the oxygen and nitrogen mass flow rates can be calculated from Wa Wv WO 2 1 = W (B4) 1 + ω = W W (B5) = x W (B6) O2 a a W x ) N 2 = (1 O2 W a (B7) where y O2, x m m / 2 2 O O dryair is the oxygen mass fraction which is a function of dry air oxygen mole fraction, x O2 = y y M O2 O2 O M (1 y ) 2 O + 2 O M 2 N 2 The calculation of hydrogen and vaor flow rates into the anode is similar to that of the air into the cathode, and is simler because the anode gas only contains hydrogen and vaor. (B8) DS Author: Peng, H. 25

26 Aendix C: Linearized System Matrices Table 3 Linearized System Matrices DS Author: Peng, H. 26