Power Control of a Polymer Electrolyte Membrane Fuel Cell

Ind. Eng. Chem. Res. 2006, 45, 4661-4670

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Power Control of a Polymer Electrolyte Membrane Fuel Cell Kevin C. Lauzze and Donald J. Chmielewski* Center for Electrochemical Science and Engineering, Department of Chemical & EnVironmental Engineering, Illinois Institute of Technology, Chicago, Illinois 60616

The polymer electrolyte membrane fuel cell (PEMFC) has been projected to be the fuel cell of choice for future automotive applications. Among the most challenging aspects of the this application is the occurrence of severe and frequent changes in power demand. However, set-point tracking in a PEMFC is complicated by the need to regulate many additional operating variables. In this work, a simplistic PEMFC model is used to illustrate the operational goals and challenges associated with power set-point tracking. Among the measures of performance, we find tracking response time, available power range, and energy conversion efficiency. Challenges range from stack cooling and oxygen starvation to membrane flooding and dehydration. In this work, a feedback structure is proposed to address these many facets of the PEMFC control problem. 1. Introduction Toward transportation propulsion applications, the polymer electrolyte membrane fuel cell (PEMFC) holds significant advantages over other fuel cell types. The PEMFC is light and compact and runs at a low temperature, which allows for relatively quick start-up. Even with these potential advantages, a number of operational hurdles exist. Proper cooling, humidification, and hydrogen supply are all important issues being investigated.1,2 The overall objective of the envisioned PEMFC control system is to deliver power at levels equal to that requested by a command signal (presumably coming from the cell user or a higher level controller). While this objective appears to be straightforward, further analysis reveals a number of additional more specific goals. Clearly, the response-time to set-point changes will be a measure of controller quality. However, achieving a wide range of possible power conditions is also important. In particular, physical limitations of the fuel cell components suggest that maximum and minimum power conditions exist. Thus, the second goal of the fuel cell controller is to expand the range of power conditions available to the user. Combining the first and second objectives, we note the cell’s ability to respond to large changes in power output. That is, if the change is too large (or too abrupt), the cell may be pushed out of its stability region and, thus, lose ignition or overheat. This challenge will manifest as a limitation on the rate of change one can send to the power set-point command signal. The final dimension of the PEMFC control problem is the important issue of energy conversion efficiency. Since power delivery output will specify only a single degree of freedom in the system’s operating space, the remaining degrees of freedom can be used to improve efficiency throughout the operating power range. As one would expect, the literature on PEMFC systems is large and rather diverse. On the subject of steady-state stack models, Dutta et al.3 present a comprehensive 3-dimensional, multicomponent model with fairly sophisticated electrochemical aspects. Of particular interest is the membrane hydration and conductivity aspects of the model, which appear to be derived from the foundational efforts of Springer et al.4 and Mann et al.5 Turning to dynamic type models, the effort by Pukrushpan et al.6 presents a dynamic PEMFC system model intended to aid in control system design studies. It is interesting to note * To whom correspondence should be addressed. Phone: 312-5673537. Fax: 312-567-8874. E-mail: [email protected].

that the stack portion of the model in ref 6 is very similar to that of Dutta et al.3 and could be characterized as a lumped parameter dynamic version. However, the Pukrushpan et al.6 effort extends the model to include auxiliary components such as compressors, coolers, humidifiers, and gas flow manifolds. A similar dynamic modeling effort proposed by Tang et al.7 is distinguished by its inclusion of the charge double layer phenomena. Toward control system design, there are a number of efforts currently in the literature. Most of these advocate using cathode gas flow as a means to regulate power output. In the work of Mufford et al.,8 a proportional-integral-derivative (PID) controller manipulates cathode air flow to regulate power output. Similarly, Nguyen et al.9 propose a cathode gas purge cycle scheme, again with the aim of regulating power output. In the work of Stueber et al.,10 a fuzzy logic based controller is used to regulate cell impedance. In addition to using measured values of cell impedance, the fuzzy control law incorporates measured cell voltage. Moving away from the cathode flow manipulation, Golbert and Lewin11 propose adaptive and model based controllers to regulate power output via a manipulation of stack voltage and the coolant inlet temperature. Pukrushpan et al.6 propose a feed-forward/feedback scheme for power tracking based on manipulations of cathode flow as well as fuel cell current output. The feedback scheme also employs a state observer driven by a system model and measurements of compressor flow rate, manifold pressure, and stack voltage. This controller also regulates the cathode oxygen ratio, motivated by the authors observation that this variable will impact system efficiency. In the next section, we present the model to be used in our analysis, which includes material and energy balances, electrochemistry, and finally a discussion of expected operating characteristics. The remaining sections will focus on building a control structure. Initially, the scheme will focus on load tracking via a power controller. Then, to this initial configuration, we appended a feedback loop aimed at temperature/ humidity regulation. Finally, we add a third loop that employs cathode air flow to regulate oxygen content in the cathode. 2. Model From a global perspective, we assume that the PEMFC stack is of sufficient size that air cooling is a requirement (approximately 10 kWe). However, the actual model presented reflects the volume and surface areas of a single flow channel,

10.1021/ie050985z CCC: $33.50 © 2006 American Chemical Society Published on Web 05/25/2006

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Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006

to the stack volume (10 kWe stack assumed). This effective surface area is then applied to the volume of our unit cell. Using the above description, the following ordinary differential equation model was derived using standard material and energy balance methods around the cathode and cooling jacket chambers as well as the solid material. Due to our assumption of a pure hydrogen feed, an anode material balance is not required. However, in the case of a reformed and/or humidified hydrogen feed in which an exit stream is required, inclusion of such balances is straightforward. We further note that the density and heat capacity of air have been assumed constant with respect to the temperature and water content.

Vcat

dCH2O dt

Vcat Figure 1. Schematic of the unit cell of the PEMFC model.

under the assumption that macroscopic stack values for power, current, and flow rate can be obtained by appropriate multiplication of this modeling unit. Additionally, the spacial aspect of the single flow channel will be ignored in favor of the simplicity of a continuous stirred tank reactor (CSTR) form. Despite this neglect of spacial dependence, we have found this model to exhibit sufficient richness to illustrate the main challenges associated with the control problem. (This is smilar to the work of Moxley et al.,12 in which a CSTR model is used to illustrate the possibility of multiple steady states in a PEMFC.) 2.1. Material and Energy Balances. The unit cell of the proposed model consists of two gas chambers separated by a membrane electrode assembly (MEA), see Figure 1. On the surface of the two sides of the MEA, all electrochemical reactions are assumed to take place. On the anode side, we assume pure H2 is fed to the gas chamber. At the anode surface, electrons are disassociated from the hydrogen molecules. Since the polymer membrane is electrically nonconductive, the electrons will move through the anode material to a current collector and then on to the electric load. The membrane is, however, conductive to hydrogen ions and thus will pass these to the cathode side of the MEA (note that our assumption of pure hydrogen suggests that no exit stream is needed to purge unreacted species). Returning from the load, electrons travel through the cathode material to combine with hydrogen and oxygen to form water. Through the cathode gas chamber, humidified air is passed, at which point oxygen is consumed and the water vapor product is added before being exhausted to the environment. At the cathode, we also find air flow to be an important mechanism for the removal of reaction generated heat. As an auxiliary cooling mechanism, we assume a third chamber, through which a cooling fluid (either air or water) may be passed. In both the cathode and cooling jacket cases, the surface area available for heat transfer to the gas is much more than just the membrane surface area. This area is mostly due to the current collecting/gas chamber walls of the stack. For thermal purposes, we assume the temperature of all this solid material to be a single lump and, thus, arrive at a solid energy balance with significantly increased surface area and thermal mass as compared to the membrane alone. As a final mechanism for heat removal, we have included losses to the environment from the stack edges. In this case, we assumed an effective surface area based on the expected ratio of the insulation surface area

dCO2 dt

in out ) Fin catCH2O - Fcat CH2O + rH2OAmem

(1)

1 in out ) Fin catCO2 - Fcat CO2 - rH2OAmem 2

(2)

Vcat

dCN2 dt

out in ) Fin catCN2 - Fcat CN2

1 in CFout cat ) CFcat + rH2OAmem 2 Vcat

(3) (4)

UAcat dTcat in out ) Fin (T - Tcat) (5) catTcat - Fcat Tcat + dt (FCp)air sol

(FCp)solVsol

dTsol ) UAcat(Tcat - Tsol) + UAjac(Tjac - Tsol) + dt UAeff(Tamb - Tsol) + QgenAmem (6)

(FCp)jacVjac

dTjac ) (FCp)jacFjac(Tin jac - Tjac) + dt UAjac(Tsol - Tjac) (7)

As suggested above, the PEMFC has the overall reaction H2 + 1/2O2 f H2O. The rate of this reaction (generation of water per unit area of membrane) is given by the following expression.

rH2O )

j nF

(8)

where j is the current density, F is Faraday’s constant, and n is the number of electrons being dissociated. It is noted that we have assumed all generated water will arrive in the vapor phase. Clearly, other approaches exist for this aspect of the model (see, for example, refs 2, 3, 6, and 11). The heat generation term, Qgen, is the amount of heat produced by the electrochemical reaction. It is a combination of reversible and irreversible losses and is calculated as the energy left over after the production of electric power (Qgen and Pe are also with respect to the membrane surface area).

Qgen ) (∆Hf,H2O)rH2O - Pe

(9)

Pe ) jEcell

(10)

As with all electrical systems, the cell current and voltage are intimately related. As suggested above, the total current is determined as the product of current density and the membrane surface area, Amem, and must be the same for both the fuel cell and the load. The voltage Ecell is the voltage observed by both the cell and the load. In general, this current-voltage operating

Ind. Eng. Chem. Res., Vol. 45, No. 13, 2006 4663

point is not a fixed value but is rather arrived at through negotiations between the cell and the load. Toward determining the current-voltage operating point, one should note that the load is typically not a passive component of the circuit. In fact, manipulation of load characteristics is one of the primary vehicles for modifying the amount of power drawn from the cell. Thus, in this work, we assume that the load characteristics are easily modified. However, to avoid going into too much detail concerning electrical loads, we assume that the load is able to specify either the cell voltage or the current density. Then, the remaining unspecified variable will be determined by the electrochemical model discussed next. 2.2. Electrochemistry. To determine the voltage-current relationship of the cell, we start by defining Ecell as the difference between an ideal, Nernst voltage and a number of voltage losses,1,2,13

Ecell ) Ener - Eact - Eohm - Emt

(11)

The Nernst potential, Ener ) E° + (RTsol/nF ) ln(PH2PO21/2/PH2O), represents the equilibrium voltage resulting from the bulk concentration of reactants and products in the gas chambers. As these concentrations change (due to eqs 1-7), Ener will change accordingly. The first loss, Eact, is defined as the activation loss and represents the kinetics of the electrochemical reaction. This irreversible loss is usually characterized by the Tafel equation, Eact ) (1/R)(RT/nF ) ln(j/jo). The exchange current density, jo, is calculated with respect to a reference exchange current density at a reference oxygen concentration, jo ) joo(CO(s)2/COo 2)γ. It is also important to note that the Tafel equation is only valid if j > jo. If this does not hold, then Eact should be set to zero. The third loss, Emt, is due to mass transfer effects and can be defined as Emt ) -1/2(RT/nF ) ln(CO(s)2/CO2). In both the activation and mass transfer cases, the surface concentration of oxygen, CO(s)2, is calculated with the aid of a mass transfer coefficient, K.

1 KAmem(CO2 - CO(s)2) ) rH2OAmem 2

(12)

In this model, the mass transfer coefficient is not only used to define the ratio of surface to bulk oxygen concentration, but it is also used to describe the phenomena of flooding. Flooding occurs as the relative humidity (RH) in the cathode approaches 100% and results in a liquid water covering of the cathode pores. However, due to the extremely small pores of the cathode, condensation will actually to occur at humidity levels lower than 100%. The Kelvin effect suggests that this critical humidity will be a function of pore diameter.14 Combining this phenomenon with the fact that the cathode contains a distribution of pore diameters, we conclude that the mass transfer coefficient should decrease gradually as the RH approaches 100%, rather than the sharp drop one would expect from a poreless surface. It should also be noted that hydrophobic antiflooding agents can be added to the cathode. While these agents can only partially prevent flooding, the result is typically enough to allow the cell to continue operating, although in a very inefficient state. To capture the above characteristics of a flooding scenario, we propose the following relative humidity dependent mass transfer coefficient:

K(RH) ) Ko[1 - Fo exp{(RH - 1)/ψ}]

(13)

where Ko is the nominal, unflooded mass transfer coefficient and ψ and Fo are newly defined porosity and antiflooding

Figure 2. (a) Influence of the porosity coefficient on mass transfer. (b) Review of Antoine’s equation, xH2O ) 0.35. (c) Influence of RH on ionic conductivity.

coefficients. Figure 2a illustrates how changing ψ and RH will effect the mass transfer coefficient. (In Figure 2a and in the simulations to follow, we assumed no anitflooding agents and, thus, Fo is set to unity.) We calculate the relative humidity via Raoult’s law, RH ) xH2OP/P Hsat2O, where P Hsat2O is determined from Antoine’s equation (see Figure 2b). Concerning the transport of water from the membrane surface to the bulk of the cathode gas, the present model assumes no mass transfer resistance. However, inclusion of a relation similar to 12 is straightforward. In this case, the appropriate RH to be used in 13 would be that based on surface concentration. Returning to the voltage loss terms of eq 11, the second loss is termed the ohmic loss, Eohm ) jR. In this model, we will focus on the ionic resistance of the membrane. If the electrical

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Figure 3. Typical polarization and power output curves.

resistance of the electrolytes and current collectors are determined to be important, then inclusion in the model is straightforward. The ionic resistance of the membrane with respect to the membrane surface area is defined as R ) tmem/σ, where tmem is the membrane thickness and σ is the membrane conductivity, defined as3

{ (

σ ) (0.514λ - 0.326) exp 1268

1 1 303 Tsol

)}

(14)

where λ is the water content in the membrane. As one would expect, λ is heavily dependent on relative humidity at the cathode. Assuming 0 < RH e 1, λ is modeled as,3 (see Figure 2c)

λ(RH) ) 0.043 + 17.81RH - 39.85RH2 + 36.0RH3 (15) It should be noted that the conductivity model, given in eqs 14 and 15, is significantly reduced from those commonly found in the literature. In particular, we have based the water content of the membrane solely upon the bulk portion of the cathode gas. Clearly, it would be more appropriate to use a surface humidity value as well as include the impact of anode side humidity (potentially averaging over the two sides). Alternatively, one could track membrane water content directly, through the inclusion of a separate material balance. This would have the additional advantage of capturing the dynamics of membrane hydration. Certainly, such knowledge of how long it takes for the membrane to dry out or rehydrate will be critical to designing the fuel cell controller. The reader is referred to refs 2 and 15 for additional details concerning conductivity models. 2.3. Stack Operation. Collecting the above electrochemical notions into eq 11, we arrive at a voltage-current relationship dependent upon a variety of stack operating conditions. In simplified form, this relationship (denoted as the polarization curve) is given as

Ecell ) f(j; CH2, CH2O, CO2, CO(s)2, K, σ)

(16)

where K depends on RH, σ depends on RH and Tsol, and RH depends on CH2O, P, and Tsol. A typical polarization curve is given in Figure 3. This plot also indicates the amount of power that would be delivered to the load at a given current density (simply calculated via eq 10). It is important to note that this example polarization curve is with respect to fixed operating conditions and thus is not representative of actual stack

operation. Clearly, if stack operation moves along this curve, and, for example, decreases current density, then the amount of water produced at the cathode will decrease and eventually result in a decrease in relative humidity. This decrease in relative humidity will then result in increased membrane resistance and, ultimately, a change in the polarization curve. It should be noted that the simulation does not calculate the entire polarization curve at each time step. In fact, the solver sees eq 16 as a nonlinear algebraic equation that is used to calculate current density, given a specified cell voltage. In this work, fuel cell efficiency is defined as the ratio between the actual cell voltage and the highest attainable voltage, η ) Ecell/Eeq, where Eeq ) ∆Hf,H2O/nF. 1 Using the lower heating value for the enthalpy of formation, the equilibrium potential is 1.2 V at 25 °C and unit activity and defines cell voltage at 100% efficiency. Equation 9 shows heat production to be defined as the total power available (due to enthalpy of reaction) minus the electrical power delivered to the load. Consider Figure 3, at 400 mA/cm2, the fuel cell has a voltage of about 0.62 V and a power output of 0.25 watts/cm2. If we then determine that the lossless cell voltage is 1.2 V, we then find that 0.49 watts/cm2 of power would be available. Thus, 0.24 watts/cm2 is generated as heat, and the resulting efficiency is 50%. The above definition suggests that high efficiency is attained by keeping the voltage as high as possible. However, the voltage at a given (or set-point) power output is defined by the polarization curve. Thus, to increase voltage (at a fixed power output), the polarization curve must be modified by changing the cell operating conditions. A number of possible manipulated variables (MVs) exist to alter the operating conditions of the PEMFC as well as influence the control objectives. Possible MVs include the coolant flow, cathode gas flow, cell voltage (or current), inlet humidification and temperature, and operating pressure. Certain variables, like the inlet humidification, would make for poor MVs due to the amount of time needed to change the variable as well as the cost of adding the required hardware. However, air flow rates (coolant and cathode) are easily manipulated and quickly changed if necessary. The cell voltage/ current is also easily changed and responds almost instantaneously. Conditions which are relatively easy to measure include temperature and relative humidity, although a time lag will likely exist for each. Oxygen content, on the other hand, is not as readily measured, but it may be possible to infer measurements via a soft sensor configuration. Finally, electrical values such as power, voltage, and current density are clearly easily measured. 3. Power Control Toward achieving the desired power output, there are a number of options. One is to fix the cell operating voltage at a constant value and then modify the power curve by changing the cell operating conditions. For example, increasing the gas flow rate through the cathode will typically increase the power output. Thus, it is possible to use cathode flow as the MV to control power output.8-10 Unfortunately, increasing cathode flow will have additional impacts, such as an increased oxygen concentration and decreased relative humidity. Furthermore, once the flow is changed, the increase in power results only after a new polarization curve is attained. The delay associated with shifting the polarization curve will clearly decrease the closed-loop performance of this configuration. Given these drawbacks, we focus on controlling power by other means. The power control scheme advocated by Golbert and Lewin11 is to simply modify the cell voltage until the desired power


Figure 4. Feedback structure for the power controller.

level is achieved. Looking back to Figure 3, we see that a decrease in voltage will increase current density. At low current densities, an increase in current will result in increased power. However, after 670 mA/cm2, this response will switch to decreased power. Golbert and Lewin11 suggest that this sign change in the (linearized) transfer function gain is an important aspect that must be included in control system design. However, it should be noted that operation to the right of this peak power condition is never desired from an efficiency standpoint. This is due to the fact that an equal amount of power can always be

Figure 5. Performance of the power controller.

achieved using less current, which translates to lower fuel consumption. (Alternatively, recall our definition of η which indicates more efficient operation at higher operating voltages.) It should also be noted that a capital cost perspective of the problem seems to indicate an advantage to operating at the peak power condition. However, one should recall that this maximum in the power curve is only with respect to the electrical conditions. That is, for different thermal and chemical operating conditions, the value and location of the peak power condition will change. Thus, if one would like to find the true maximum power for a given stack, then an optimization based search over all operating variables will be required. Returning to the question of operation at or near the peak power condition, our position is that such an operating point should be avoided, if possible. From the previous paragraph, it is clear that one should be able to do so in all cases except for those approaching the overall maximum power output of the stack. If successful in avoiding this difficult to control operating point, then one should be able to avoid having to use a sophisticated controller for this low level feedback loop. Unfortunately, this approach will place additional burden on

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higher level controllers. Ideally, before sending a command to the low level power loop, one of these controllers should verify that a crossing of the peak power condition is not expected. Since the current effort does not consider the design of these higher level controllers, we will simply limit the local size of power request increases. This will allow the nonelectrical operating conditions to catch-up and establish a new polarization curve (recall section 2.3). Hopefully, this new operating point will have a larger peak power condition and, thus, will allow for further power request increases. Another issue in power control is the impact on the cell voltage. In particular, if the cell voltage becomes too low, then the electrode catalyst becomes exposed to possible damage. This issue, in particular, is one advantage of using cathode flow to manipulate power (in which case cell voltage is held constant for all time). To address this issue, we propose the feedback structure illustrated in Figure 4. Here, we see an inner loop using voltage to manipulate current density. Then, the current density set-point is used to manipulate power. In addition to providing additional tuning parameters (and thus increasing our ability to avoid voltage undershoots), this configuration is perceived to be an advantage for future predictive type controllers. That is, if one would like to constrain cell voltage, then these limits will be more reliably observed if voltage is an input rather than a model dependent output. Using the feedback structure of Figure 4 and assuming all other possible inputs are held constant (including pressure, inlet conditions, and coolant and cathode gas flows), we simulated the response to increasing and decreasing step changes in the power set-point. The scenario of Figure 5 is that the command is considered as an external disturbance that signal P(sp) e undergoes step changes at 10, 20, and 30 s. This will cause the controller of Figure 4 to act on Ecell in an effort to achieve the desired power output. This will have the additional impact of causing the current density, temperature, and relative humidity to change. We start by considering the 10-20 s interval of the left-hand plots of Figure 5. At 10 s, the requested power increase is achieved by reducing the cell voltage, which increases current density, the heat generation rate, and ultimately the temperature of the cathode gas. Additionally, the production and mole fraction of water in the cathode will increase. However, from the perspective of RH, this change in water content is small as compared to the impact of the gas temperature. The resulting RH decrease will cause a drop in membrane conductivity. Normally, this loss of conductivity would cause a drop in power output. However, the power controller will sense the onset of such power losses and correct by further decreasing the voltage. This is most easily observed in the current density curve, where an initial jump in current is followed by a slow rise (i.e., initially responding to the set-point change and then regulating in the face of conductivity changes). In the 20-30 s interval of Figure 5, left, we observe similar behavior for most of the interval. That is, as the membrane conductivity drops (due to decreasing RH), the controller is dropping the cell voltage in an effort to maintain power output at the set-point level. However, at 29 s, this loss of conductivity becomes great enough that the peak power of the associated polarization curve is less than the set-point power. At this point, the closed-loop system becomes unstable, due to a sign change in the open-loop gain (recall the peak power discussion above). In particular, a decrease in the cell voltage will now decrease the power output, which is contrary to previous operation (i.e., earlier in the interval). The resulting drop in power causes the

Figure 6. Feedback structure for the power/temperature controller.

(confused) controller to drop the voltage further, and thus, the unstable scenario is observed in the plot. Turning to the right-hand plots of Figure 5, we observe expected operation during the 10-20 s interval. However, during the 20-30 s interval, the current and voltage responses are distinctly different. In particular, the voltage initially increases but then drops below the steady-state voltage of the previous interval. Clearly, this behavior is due to the controller’s successful attempt to regulate power in the face of a dropping polarization curve. However, in contrast to the left-hand plots where a drying out of the membrane occurred, this drop is due to the cell’s approach to a flooded condition (as evidenced by RH values of about 99.5%). Then, as the power output is further decreased (at 30 s), the cell becomes fully flooded, and this results in the unstable oscillations observed in the figure. (These oscillations seem to be a result of the highly sensitive nature of the polarization curve at RHs close to 100%.) 4. Temperature and Humidity Control The previous section clearly illustrates the set of available power outputs for the control scheme of Figure 4 (from Figure 5, the range is from 0.165 to 0.205 watts/cm2). In the high power case, greater heat production results in a drying out of the membrane (due to low RH), while at low power the increased RH causes flooding to occur. To address these issues, and hopefully increase the range of available power conditions, it is clear that a temperature and/or relative humidity controller is desired. The first step in designing such a controller is to select a suitable set of manipulated variables. Initial inspection of the model suggests that manipulation of the inlet conditions (temperature and humidity) would be a good choice. However, as stated earlier, the slow response of these auxiliary units is a concern. Alternatively, we could look to the cathode gas flow rate. Clearly, this variable will quickly impact both the cathode gas temperature and water vapor content. While increasing the flow rate will decrease both, it is somewhat difficult to predict the impact on relative humidity (one will decrease it while the other will increase it), especially at various operating conditions. Another shortcoming of manipulating cathode flow is the unpredicted impact on inlet conditions (due to new flow conditions in the auxiliary units) as well as changes to the oxygen content in the cathode chamber. The next possible manipulation is coolant flow rate. Clearly, this input will only impact solid temperature and then indirectly cathode gas temperature. While this delay is undesirable, we are willing to accept it in favor of the decoupled impact of this input. Using the feedback structure of Figure 6, we again simulated the response to increasing and decreasing step changes in the power set-point. As with Figure 5, P(sp) is the external disture bance, which changes values at 10 s intervals. In these simulations, the temperature controller was tuned aggressively which required a scaling back of the power controller tuning, leading to slower power responses as compared to Figure 5.


Figure 7. Performance of the power/temperature controller.

Toward analyzing the actions of the controller, consider the 1020 s interval of the left-hand plots of Figure 7. At 10 s, the increase in power output causes the cathode temperature to increase. Then, the temperature controller uses the jacket flow rate to bring this temperature back to the temperature set-point fairly quickly. Additionally, it is noted that the mole fractions of water and oxygen in the cathode find new steady-state values at the end of the interval. Similar responses are found in the other intervals as well as in the right-hand plots. Using the control structure of Figure 6, the range of available power conditions is increased (the new range is from 0.03 to 0.31 watts/cm2). However, an analysis of the failures at the two extremes indicates somewhat different sources, as compared to the power controller only cases. In particular, the high power limit is no longer due to a drying out of the membrane. It is now due to a lack of oxygen at the surface of the cathode, which causes the polarization curve to drop substantially (i.e., such that the peak power is below the set-point and the controller again becomes unstable). It should be noted that other failure scenarios are possible for this case. For example, if the inlet RH was higher or the set-point temperature was lower, then

flooding would have likely occurred prior to oxygen starvation. At the low power limit, we find coolant flow to be saturated at a value of zero (i.e., the condition of the least heat removal). The result is that the cell cannot maintain the desired operating temperature. This drop in temperature and eventual rise in relative humidity (potentially to the point of flooding) will cause the cell to fail. Returning to the high power condition, it is important to note that limits on maximum coolant flow (neglected here) would certainly reduce the maximum achievable power. To address these issues, we propose a relative humidity controller as illustrated in Figure 8, which uses the cathode temperature set-point signal as the manipulated variable. Consider the 10-40 s interval of the left plots of Figure 9. At 10 s, the increase in power output causes the cathode temperature to increase. This in turn causes the RH to drop, which is seen by the RH controller. However, the MV of the RH controller is the set-point to the temperature controller. Thus, the call from the RH controller is to bring the temperature down, which it eventually does (at around 20 s). From that point to the end of the interval, the cathode temperature and its set-

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cases indicates an improvement at all power conditions, especially in the low power cases (see Figure 12). It should be noted that this increase in efficiency (about 2.5%) is most likely due to our selection of 95% as the RH set-point. That is, selecting a lower set-point is expected to result in less efficient operation. 5. Oxygen Control

Figure 8. Feedback structure for power/temperature/RH controller.

point are about equal but both growing so as to kill the RH increase resulting from a greater production of water vapor at the cathode. Similar responses are observed in the other intervals as well as in the right-hand plots. Using the RH controller, we find that the range of available power conditions is almost unchanged (0.03-0.295 watts/cm2). However, a comparison of the efficiency curves of these two

Figure 9. Performance of the power/temperature/RH controller.

In this section, we address the oxygen starvation problem. In addition to degrading performance (from a Nernst potential perspective), this depleted oxygen state could damage the electrocatalyst. Previous efforts to address this problem utilized the notion of an oxygen ratio (defined as the ratio of oxygen fed to that reacted). In particular, the goal is to maintain an oxygen ratio greater than 2. In the work of Pukrushpan et al.16 and Sun and Kolmanovsky,17 manipulations of cathode flow and current density are used to achieve this objective. (In ref 16, a feedback/feed-forward controller is used, while ref 17 employs the load governor concept of ref 18.) In Vahidi et al.,19 hybridization of the PEMFC with a battery (or ultracapacitor)


Figure 10. Feedback structure for the power/temperature/RH/oxygen controller.

is used to augment fuel cell power output, while oxygen ratio limitations are being enforced. Toward the oxygen starvation problem, we propose a simple feedback loop for regulating the oxygen mole fraction within the cathode chamber. Unfortunately, the only remaining variable capable of manipulating this control variable (CV) is cathode gas flow rate. In previous sections, we have avoided this MV due to its coupled impact on multiple operating variables. However, given the previous feedback loops, aimed at regulating the other conditions, we now feel more comfortable using this MV. Another challenge with this configuration is to obtain a measurement of the oxygen mole fraction. In the absence of a sufficient oxygen sensor, we suggest a model based soft sensor (similar to those found in refs 16, 17, and 19). The proposed feedback structure is illustrated in Figure 10. Figure 11 illustrates the new system’s response to power staircases. The mole fraction of oxygen set-point is selected to be 0.095, and the oxygen controller is tuned to react much more slowly than the other loops. However, during the tuning process, we observed poor performance at low power densities (in particular, large oscillations were observed). To address this issue, we turned the oxygen controller off at low power conditions (less than 0.1875 watts/cm2). We additionally note that without the oxygen

Figure 11. Performance of the power/temperature/RH/oxygen controller.

Figure 12. Effectiveness of the proposed control structures.

controller (before 120 s) the temperature set-point is quite low, due to the relative humidity controller. 6. Conclusions In this work, a simple dynamic model of the PEMFC was presented and a set of control schemes were proposed. As a final comparison between the schemes, Figure 12 illustrates both the range of available power conditions as well as the efficiency of each. Clearly, the more complex configurations yield better performance with respect to these measures. However, one should note the increase in oscillatory behavior as well as the slower response time of the power output (as indicated by the previous plots with respect to time). While additional effort toward tuning the PI loops may improve the situation, this task is hindered by the nonlinear and highly coupled nature of the

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PEMFC. An alternative would be to employ a multi-input, multioutput nonlinear controller to exploit the couplings. If this nonlinear controller is of the predictive variety, then one would find great utility for constraint handling, especially with respect to avoiding flooded and oxygen starvation conditions. It is also noted that the set-point values for relative humidity and oxygen content were selected almost arbitrarily (based solely on intuition with respect to efficiency). Clearly, these additional degrees of freedom could be utilized to improve performance. For example, a steady-state optimization scheme could be used to maximize efficiency. Alternatively, these variables could be selected to yield greater controllability (although a new measure, indicating this notion, would need to be developed). Unfortunately, application of these advanced control methods will require the development of more sophisticated models, so as to reduce the model mismatch degradation resulting from the feed-forward characteristics inherent to these controllers. In particular, a more realistic consideration of spatial aspects will be required. However, this effort to improve accuracy will be tempered by the computational burden commonly associated with advanced/ predictive methods. Acknowledgment This work was supported by the Department of Chemical and Environmental Engineering and the Graduate College at the Illinois Institute of Technology. Notation j, Ecell, Pe ) current density, cell voltage, and power to the load Eo ) reversible potential R, F, n ) gas, Faraday’s, and electron disassociation constants R, jo ) charge-transfer coefficient and exchange current density jo, COo 2 ) reference current and concentration Amem, tmem ) membrane surface area and thickness Tcat, Fcat, Vcat ) cathode gas temperature, volumetric flow, and volume Tsol, Vsol ) solid temperature and volume Tjac, Fjac, Vjac ) jacket temperature, volumetric flow, and volume CH2O, CO2, CN2, C ) concentration of water, oxygen, nitrogen, and air CO(s)2 ) concentration at the cathode surface PH2, PO2, PH2O ) partial pressures F, Cp ) density and heat capacity Acat, Ajac, Aeff ) surface areas available for heat transfer U ) heat transfer coefficient rH2O, Qgen ) rate of reaction (water generation) and rate of heat generation ∆Hf,H2O ) enthalpy of formation for water (lower heating value) K,ψ ) mass transfer and porosity coefficients xH2O, xO2 ) mole fractions R, σ ) ionic resistance and conductivity

P, PHsat2O ) pressure of cathode gas and saturation pressure at Tcat λ, RH ) membrane water content and cathode chamber relative humidity η ) efficiency Literature Cited (1) Larminie, J.; Dicks, A. Fuel Cell Systems Explained, 2nd ed.; John Wiley & Sons: West Sussex, 2003. (2) O’Hayre, R.; Cha, S.-W.; Colella, W.; Prinz, F. Fuel Cell Fundamentals; John Wiley & Sons: New York, 2006. (3) Dutta, S.; Shimpalee, S.; VanZee, J. Numerical prediction of massexhange between cathode and anode channels in a pem fuel cell. Int. J. Heat Mass Transfer 2001, 44, 2029. (4) Springer, T.; Zawodzinski, T.; Gottesfeld, T. Polymer electrolyte fuel cell model. J. Electrochem. Soc. 1991, 138 (8), 2334. (5) Mann, R.; Amphlett, J.; Hooper, M.; Jensen, H.; Peppley, B.; Roberge, P. Development and application of a generalised steady-state electrochemical model for a pem fuel cell. J. Power Sources 2000, 86, 173. (6) Pukrushpan, J. T.; Stefanopoulou, A. G.; Peng, H. Control of Fuel Cell Power Systems: Principles, Modeling, Analysis and Feedback Design; Springer-Verlag: London, 2004. (7) Pathapati, P.; Xue, X.; Tang, J. A new dynamic model for predicting transient phenomena in a pem fuel cell system. Renewable Energy 2005, 30, 1. (8) Mufford, W.; Strasky, G. Power control system for a fuel cell powered vehicle. US Patent 5,991,670, 1999. (9) Knobbe, M.; He, W.; Chong, P.; Nguyen, T. Active gas management for pem fuel cell stacks. J. Power Sources 2004, 138, 94. (10) Schumacher, J.; Gemmar, P.; Denne, M.; Zedda, M.; Stueber, M. Control of miniature proton exchange membrane fuel cells based on fuzzy logic. J. Power Sources 2004, 129, 143. (11) Golbert, J.; Lewin, D. Model-based control of fuel cells: (1) regulatory control. J. Power Sources 2004, 135, 135. (12) Moxley, J.: Tulyani, S.; Benzigner, J. Steady-state multiplicity in the autohumidification polymer electrolyte membrane fuel cell. Chem. Eng. Sci. 2003, 58, 4705. (13) Bard, A.; Faulkner, L. Electrochemical Methods: Fundamentals and Applications, 2nd ed.; John Wiley & Sons: New York, 2001. (14) Satterfield, C. N. Hetrogeneous catalysis in practice; McGrawHill: New York, 1980. (15) Weber, A. Z.; Newman, J. Modeling transport in polymer-electrolyte fuel cells. Chem. ReV. 2004, 104, 4679. (16) Pukrushpan, J. T.; Stefanopoulou, A. G.; Peng, H. Modeling and control for pem fuel cell stack system. Proceedings of the American Control Conference, Anchorage, AK, 2002; p 3117. (17) Sun, J.; Kolmanovsky, I. A robust load governor for fuel cell oxygen starvation protection. Proceedings of the American Control Conference, Boston, MA, 2004; p 828. (18) Gilbert, E.; Kolmanovsky, I. Nonlinear tracking control in the presence of state and control constraints: a generalized reference governor. Automatica 2002, 38, 2063. (19) Vahidi, A.; Stefanopoulou, A. G.; Peng, H. Model predictive control for starvation prevention in a hybrid fuel cell system. Proceedings of the American Control Conference, Boston, MA, 2004; p 834.

ReceiVed for reView August 31, 2005 ReVised manuscript receiVed March 20, 2006 Accepted April 27, 2006 IE050985Z

Power Control of a Polymer Electrolyte Membrane Fuel Cell

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