Journal of Power Sources 256 (2014) 470e478

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Journal of Power Sources

journal homepage: www.elsevier .com/locate/ jpowsour

Temperature dynamics and control of a water-cooled fuel cell stack

Daniel O’Keefe a, M.Y. El-Sharkh b, John C. Telotte c, Srinivas Palanki a,*aDepartment of Chemical & Biomolecular Engineering, University of South Alabama, 150 Jaguar Drive, Mobile, AL 36688-0002, USAbDepartment of Electrical & Computer Engineering, University of South Alabama, 150 Jaguar Drive, Mobile, AL 36688-0002, USAcDepartment of Chemical & Biomedical Engineering, FAMU-FSU College of Engineering, Tallahassee, FL 32310-6046, USA

h i g h l i g h t s

� A model for a water-cooled PEM fuel cell stack was developed.� A time-varying PID controller was implemented on this model.� The controller was tested on an experimental power profile from a 3-bedroom house.

a r t i c l e i n f o

Article history:Received 13 September 2013Received in revised form17 December 2013Accepted 19 December 2013Available online 30 December 2013

Keywords:Water-cooled fuel cell stackThermal modelingPI controllerStationary power

* Corresponding author.E-mail address: spalanki@southalabama.edu (S. Pa

0378-7753/$ e see front matter � 2014 Elsevier B.V.http://dx.doi.org/10.1016/j.jpowsour.2013.12.100

a b s t r a c t

In this paper, a time-varying proportional-integral (PI) controller is designed for controlling the tem-perature of a water-cooled 5 kW hydrogen fuel cell stack. This controller is designed using a mathe-matical model for the stack temperature, which is derived using basic chemical engineering material andenergy balances. The controller affects the stack temperature by changing the flow rate of cooling waterthat passes across the stack. The model is then analyzed using a number of power demand profiles todetermine the effectiveness of the controller. The results show that a time-varying PI controller isadequate for maintaining the stack temperature within 5 K of the target point.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

In the search for alternative fuel sources, hydrogen has become apopular option primarily due to the advantages offered byhydrogen fuel cells over the modern internal combustion engine[1]. Fuel cells have multiple advantages over conventional internalcombustion engines such as improved efficiency, lack of movingparts, reduced noise during operation and the ability to store en-ergy directly in batteries without any additional steps. Althoughfuel cells are already readily available for use as backup generatorsand other stationary power sources, the idea of using a fuel cell for adynamic power demand is relatively new and the development ofsuch a system is still being researched. One of the biggest concernsin this situation is the ability to keep the fuel cell stack at optimalconditions to maintain the highest efficiency in the face of fluctu-ating power demand. In the case of a constant power demand, it ismuch easier to maintain the stack at optimal operating conditions,

lanki).

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but a dynamic power demand requires some reaction to thechanges so that the efficiency of the stack can bemaintained. In thispaper, we focus on the dynamic analysis and control of a watercooled fuel cell stack, which is being used as a backup generator in astationary environment (e.g. a residential house).

While there have been a large number of papers in the area ofmodeling fuel cells for different applications, very few of thesepapers have focused on the issue of temperature dynamics of thefuel cell and its effect on performance. Some of the earliest researchinto the thermal characteristics of a PEM fuel cell was actuallyexperimental. Amphlett et. al. [2] conducted numerous physicalexperiments on aworking fuel cell stack and determined propertiessuch as the effective mass and surface area of the fuel cell stack.These values compared well with estimates based on bulk stackmeasurements. The fuel cell stack was also run at various experi-mental conditions to determine the stack response. These resultswere then used to experimentally determine further characteristicsof the stack including a prediction for the cell voltage, the heattransfer coefficient, and the thermal capacity of the stack. A tran-sient model was also constructed to determine the stack temper-ature with time, but relies on a large number of inputs that are, at

D. O’Keefe et al. / Journal of Power Sources 256 (2014) 470e478 471

this point, experimentally determined. Grasser and Rufer [6]developed a thermal model for liquid cooling of a PEM fuel cell intheir dynamic model. With their experimental results, they wereable to design the thermal model using simple energy balances toexpress the stack temperature in a first-order differential equation,similar to how Amphlett et al. performed their research.

A variety of theoretical models have been developed to studyand model PEM fuel cells. Pukrushpan et. al. [12] developed anonlinear dynamic model that is suitable for control study. Xue et.al. [14] developed a dynamic lumped-parameter model of a PEMfuel cell and discovered that hydrogen diffusion through the elec-trode affects the stack temperature. Pathapati et. al. [11] furtherdeveloped the model set forth by Xue et. al. [14] specificallyfocusing on the dynamics and mechanisms that are of particularimportance for automotive purposes. Chu et. al. [3] also developeda lumped parameter model, and investigated the changes in stacktemperature for various inputs, but control of the stack tempera-ture was not in the scope of their research. Meyer and Yao [9]presented a Multiple Input Single Output (MISO) cooling systemmodel using linearization and m-synthesis, but this model is againrather complicated and integrated into an overall dynamic modelfor the stack. Other research has been performed to analyze howthe fuel cell stack with perform under adverse conditions, such asstarting the stack in freezing temperatures [13].

While various theoretical models have been developed to studythe PEM fuel cell stack, they tend to have thermal models devel-oped for a fuel cell stack open to the atmosphere. This air-cooledstack is of interest in the laboratory, but it does not realisticallymodel what the temperature profile of the stack would begin tolook like when used in an automobile. When in operation, the at-mosphere under the hood of a vehicle can get much hotter than thesurrounding environment outside of the vehicle, making the air inthe engine compartment unsuitable for adequate cooling of the fuelcell stack. If an air-cooled system is not possible, the only otherways to reliably affect the stack temperature are to modify the inletconditions or to use a liquid cooling system around the outside ofthe stack. Since modifications of the inlet conditions will obviouslyhave an adverse effect on the desired power output, a liquid coolingsystem is the only other feasible option. Significantly less theoret-ical research has been done on liquid cooling of a fuel cell stack.Recently, Fang et. al. [5] developed a dynamic model using a liquidcooling thermal model, but control of the stack temperature wasonce again not in the scope of their research.

In this research, a theoretical model is developed to model thethermal characteristics of a hydrogen fuel cell stack, as well as itsresponse to changes in parameters that can be used in temperaturecontrol. A schematic of the system under consideration is shown inFig. 1.

The specific objectives are (1) to develop a time-varying pro-portional-integral (PI) controller for coolingwater flow rate across ajacketed PEM fuel cell stack with the intent of controlling the stack

Fig. 1. Basic schematic of fuel cell system.

temperature and (2) test the controller using a variety of powerprofiles.

Fuel cell stacks currently offer one of the best, cleanest alterna-tives to the fossil fuels used inmodern engines, thereby reducing theamount of harmful greenhouse gases released into the atmosphere.Improvement in the design and efficiency of fuel cell stacks will alsomake them more affordable for stationary power applications.

2. Development of thermal model

A fundamental premise of the approach taken here is that theelectrical characteristics of a fuel cell change much more rapidlythan the temperature. A time-scale analysis of this system indicatedthat the electrical time-scale was of the order of milliseconds, themechanical flow process time-scale was of the order of seconds andthe thermal time-scale was of the order of minutes [8]. Thus we canexamine the dynamics of the stack temperature assuming that thevoltage adjusts nearly instantaneously when current draw changes.This does not mean that the voltageecurrent relationship does notdepend on temperature, just that the changes occur so rapidly thatthis part of the overall model can be represented by an algebraicmodel.

If the fuel cell stack and its contents are chosen as the system,the energy balance can be written as:

dUdt

¼Xi

_Ni;inHi;in �Xi

_Ni;outHi;out � _Q � _W (1)

The material balance for any species i is the following:

dNi

dt¼ _Ni;in � _Ni;out þ ni

_x (2)

Even though there are separate reactions occurring at the anodeand cathode, we combine them into an overall water formationreaction. The accumulation of the species i is very small comparedto the mass of the stack. It is therefore safe to assume steady-statefor all gaseous and liquid species. This assumption allows theaccumulation term in the material balance to be set equal to zero,resulting in:

_Ni;out ¼ _Ni;in þ ni_x (3)

The material and energy balances may now be combined togive:

dUdt

¼Xi

_Ni;in�Hi;in � Hi;out

�� DHrxnðToutÞ _x� _Q � _W (4)

where the heat of reaction for the overall reaction, the formation ofwater, will be evaluated at the actual cell outlet temperature. Notethat since the _Ni;out term can be written in terms of the _Ni;in termfrom Eq. (3), the combined equation presented in Eq. (4) is afunction of _Ni;in only.

If the hydrogen for the fuel cell is generated by the reforming of ahydrocarbon then the anode inlet flow has hydrogen (H2), carbondioxide (CO2), carbon monoxide (CO) and water (H2O). The cathodeinlet flow ismade up of oxygen (O2), nitrogen (N2), andwater (H2O).All required species flows can be related to the inlet hydrogen flow.The mole fraction of hydrogen in the inlet is given by:

yH2¼

_NH2;in_NH2;in þ _Nother;in

(5)

Then all of the other gas flows (carbon dioxide, carbon mon-oxide, and water) can then be expressed by

D. O’Keefe et al. / Journal of Power Sources 256 (2014) 470e478472

_Nother;in ¼�1� yH2

�_NH2;in (6)

yH2

The cathode inlets can similarly be expressed as a function of thehydrogen inlet flow.

_NO2;in ¼ ð1þ XsÞ _NH2;in (7)

_NN2;in ¼ 7921

ð1þ XsÞ _NH2;in (8)

_NH2O;in ¼ fsð1þ XsÞ _NH2;in (9)

where fs represents the humidity ratio of the entering air. Theprevious equations also take into account any excess of oxygen inthe system and a cathode stream that is saturated with water. Thesaturation calculationmay be solved or estimated as necessary. Thisinformation can now be used to simplify the inlet flow terms of thecombined material and energy balance.

Xi

_Ni;in�Hi;in � Hi;out

� ¼_x

XðTin � TstackÞCp;m (10)

where

Cp;m ¼ Cp;H2þ�1� yH2

yH2

�Cp;CO;CO2;H2O þ ð1þ XsÞCp;O2

þ 7921

ð1þ XsÞCp;N2þ fsð1þ XsÞCp;H2O (11)

and X represents the fractional conversion of hydrogen in the celland Cp,m is an effective heat capacity for the entering gases. Theinlet flow term can further be simplified by setting a constanta ¼ Cp,m/X. The inlet flow rate terms can now be expressed as:

Xi

_Ni;in�Hi;in � Hi;out

� ¼ a _xðTin � TstÞ (12)

The change in extensive energy term in the combined energyand material balance can also be simplified. It is known that theextensive energy of the system is equal to the sum of the extensiveenergy of the stack, and the extensive energy of the contents of thestack. It can again be assumed that the mass of the stack makes anychange in the contents relatively insignificant when it comes to thestack temperature. The differential term can therefore be approxi-mated as the change in the extensive energy of the stack [2].

dUdt

¼ mstCp;stdTstdt

(13)

wheremst and Cp,st represent the mass and specific heat of the fuelcell stack. The work term of the combine energy and materialbalance must be defined. Since the work done by the stack iselectrical power, it is only logical to express it as such. The poweroutput for each fuel cell is the product of current draw and voltage.The voltage is a function of current draw, current density, and stacktemperature. The total power output for the system is therefore thepower output from each cell multiplied by the number of fuel cellsin the stack.

_W ¼ ncellIVcell (14)

where the cell voltage Vcell is a function of the current density (i¼ I/Acell) and the stack temperature (Tst) that is represented by thepolarization curve. Thus, we can express

Vcell ¼ f ðI; TstÞ (15)

where f is a functional form representing the polarization curve. Ina fuel cell stack, the polarization curve captures the reduction involtage with increasing current density due to ohmic and transportresistances and is a function of temperature, pressure and stackmaterial properties. For maximum efficiency, it is desirable tooperate at a point in the polarization curve where the power pro-duced is maximum. Thus, a change in stack temperature changesthe operating point on the polarization curve. Combining all ofthese equations, the overall energy and material balance now be-comes the following.

mstCp;stdTstdt

¼_x

XðTin�TstÞCp;m�DHrxnðToutÞ _x� _Q�ncellsIVcellðI;TstÞ

(16)

The heat of reaction term shown in Eq. (16) must be calculatedat the actual operating conditions for the fuel cell stack. This meansthat it must be evaluated at the stack operating temperature and forthe appropriate state (liquid or vapor) of the reaction products.Because we can be in a situation where some of the water formedcan be in liquid state, the actual heat of reaction can be calculatedfrom

DHrxn ¼ flDHrxn:l þ ð1� flÞDHrxn;v ¼ DHrxn;v � flDHvap (17)

where fl is the fraction of the water formed that exits as a liquid, thesubscripts on the heat of reaction terms refer to the values calcu-lated for the water produced as either a vapor or a liquid, and theterm DHvap is the heat of vaporization of water. To justify neglectingthe last term, we should realize that the ratio of the heat ofvaporization to the heat of reaction of the vapor is given by:

DHvap

DHrxn;v¼ 44

241:8¼ 0:182 (18)

This if even as much as 10% of the water exits as a liquid (thiswould be a very high value), the possible error in using only theheat of reaction to form water vapor (and neglecting the flDHvapterm in Eq. (17)) is less than 2%. A more detailed model couldincorporate the phase equilibrium calculation that would allow forthe calculation of the fraction that exits as a liquid. We hope toaddress this point in a future publication.

Eq. (16) can be further simplified, however, since the extent ofreaction can be related to current draw.

I ¼ b _x (19)

where b denotes the conversion frommaterial flow to current. Thiscan be calculated using Faradays constant, F, and the number ofelectrons involved in the reaction, ne� . When substituted into thematerial and energy balance and combining all of the constantparameters into singular constants, the inlet and outlet flow termscan be set as a function of current draw.

mstCp;stdTstdt

¼ C1IðTin � TstÞ � C2I � _Q � ncellsIVcell (20)

The constants C1 and C2 can be calculated as follows

C1 ¼ Cp;mXne�F

(21)

C2 ¼ DHrxn

ne�F(22)

Table 1Parameters from a Ballard Mark V 5 kW fuel cell stack.

Cooling water specific heat, CPcw 4.1796 J g�1 K�1

Cooling water density, rcw 1000 kg m�3

Inlet temperature of cooling water, Tin 300 KHeat transfer parameter for stack, UAst 35.55 þ 0.025I W K�1

Number of cells, ncell 35Stack length, L 0.38 mStack width, W 0.21 mCooling water channel thickness, h 0.02 mProduct of stack mass and specific heat, mCst 35 kJ K�1

Lumped parameter CPgas=Xne� F 1.544 W K�1 A�1

Lumped parameter DHrxn=ne� F 626.4 W A�1

Active fuel cell area, Acell 232 cm2

D. O’Keefe et al. / Journal of Power Sources 256 (2014) 470e478 473

Just like the fuel cell stack itself, the cooling water used toregulate the stack temperature has its own dynamics. However, thecooling water temperature is also dependent on position. If thecooling water jacket is divided into n zones, where the temperaturewithin each zone, i, is assumed to be uniform, the energy balancefor each zone can be defined as:

dTidt

¼ _mcw

rViðTi�1 � TiÞ þ

_qirViCPcw

(23)

where

_qi ¼ UAðTst � TiÞ (24)

This is analogous to considering a plug-flow reactor as a numberof CSTRs in series. This simplification results in a model for thecooling water temperature dynamics that is a series of ordinarydifferential equations. This simplification essentially implies thatthe stack itself is at a uniform temperature and with this assump-tion, there is no effect of co-current or counter-current flow. Whilethis “lumped model” approach might not be accurate for a verylarge stack (e.g. one that supplies 100 kW of power), for a smallstack that produces up to 5 kW of power, this is a reasonableapproach as shown from the experimental data of Khan and Iqbal[7].

The heat transfer term for the fuel cell stack can be shown to be:

_Q ¼ �Xni¼1

_qi ¼ �Xni¼1

UAðTst � TiÞ (25)

This results in the following stack temperature dynamics:

mstCPstdTstdt

¼ CPgasXne�F

IðTin � TstÞ � DHrxn

ne�FI � UA

Xni¼1

ðTst � TiÞ

� ncellIVcell

(26)

A proportional-integral (PI) controller is modeled in state spaceas:

_mcw ¼ _mcw;ss þ kc�Tsp � Tst

�þ kcsI

Zt

0

�Tsp � Tst

�dt (27)

where the first term _mcw;ss is the flow rate of cooling water atsteady-state, the second term represents the proportional gainterm in the control law and the third term represents the integralterm in the control law.

Fig. 2. Cooling water flow rate to achieve stack temperature of 350 K.

3. Simulation studies

The model developed earlier was simulated in MATLAB. Themodel parameters used are for a Ballard Mark V 5 kW Fuel Cellstack and are given in Table 1 [7]. It is necessary to parameterize thevoltage versus current density relationship in 16 depending onthermodynamic state of the system. This polarization curve isparameterized using the approach proposed by Pukrushpan et al.[12]. A MATLAB programwas written to determine the voltage as afunction of current, gaseous partial pressures and the temperatureat each instance. More details can be found in Ref. [10].

Before performing the calculations, it is first necessary todetermine the number of zones that the cooling water channelshould be broken into. The number of zones was determined byperforming the calculations using various numbers of zones anddetermining at what point the temperature of the final zone was

consistent. There is practically no change in the temperature of thefinal zone when more than 15 zones are considered, so 15 zoneswere used in this simulation study.

The first term in the PI controller design equation is the coolingwater flow rate needed to keep the stack at the optimal tempera-ture. The literature shows that the target operating temperature fora PEM fuel cell is about 350 K [1]. The difficulty arises, however, inconsidering the fact that the needed cooling water flow rate isdependent on the current draw; greater current drawswill result ingreater power output, but also greater heat produced. This addi-tional heat must be removed from the stack to maintain its tem-perature at the target value.

The first task is therefore to determine the relationship betweenthe current draw and the cooling water flow rate to maintain thestack temperature at 350 K. This was done by considering the samemodel used for calculating the stack dynamics and setting the stacktemperature as constant. It is then possible to perform the calcu-lations for a series of current draws and determine the necessarycooling water flow rate. The needed cooling water flow rates for aninlet cooling water temperature of 300 K are plotted in Fig. 2.

A time-varying PI controller was designed with the goal ofminimizing the change in stack temperature from the optimalpoint. In quantitative terms, the goal was to develop a controllerthat would keep the stack temperature within 5 K of the targetpoint. The model was given a number of current draw profiles totest the controllers ability to react.

The parameters used to define the PI controller are given inTable 2. These parameters were determined by first consideringtheir proper order of magnitude such that the first term was themajor contributor to the steady-state value of the cooling water

Table 2PI controller parameters.

Set point, Tsp 350 KSteady state water flow rate, _mcw;ss 2e36 g s�1

Controller gain, kc �0.03 g s�1 K�1 I < 25 A�0.02 g s�1 K�1 I � 25 A

Integral time constant, sI 75 s I < 25 A125 s I � 25 A

D. O’Keefe et al. / Journal of Power Sources 256 (2014) 470e478474

flow rate. The constants were then adjusted using a guess-and-check method until the results of the model had been optimized.The values for these parameters were then used to attempt tocontrol the stack temperature for other current draw profiles.

It was determined that a different set of controller constants wasmore effective for low current drawswhere the stack is operating ina non-linear portion of the polarization curve. This is due to acombination of the polarization curve being non-linear and thesmall changes needed in the cooling water flow rates compared tothe changes for larger current draws. The needed cooling waterflow rate for current draws below 20 A is very small compared tothe flow rate needed for larger current draws, meaning that whenthe current draw changes, the controller makes much smallerchanges to the flow rate than it does when the current draw is

Fig. 3. Stack dynamics for

Fig. 4. Cooling water dynamic

greater. The controller constants are therefore adjusted so that thesecond and third terms are more effective at adjusting the steady-state flow rate.

Fig. 3 shows a linear current draw (the simplest current drawprofile considered) and the effect of the PI controller on the stacktemperature for this current draw profile for a constant coolingwater flow rate of 5 g s�1, for comparison. Fig. 3 shows that thecontroller is extremely effective at maintaining the stack temper-ature under such simple demands, with virtually no change in thestack temperature from the optimal value. The plots of the corre-sponding cooling water flow rate and temperature are shown Fig. 4.

Further analysis of the results does reveal that the stack tem-perature begins to stray at the high end of the current draw profile,suggesting that the controller has difficulty in properly adjustingthe cooling water flow rate. Considering the controller model andthe correlation between the current draw and necessary coolingwater flow rate, it would be expected that the controller perfor-mance would degrade when the second and third terms were toosmall. If the adjustments made to the flow rate were too small, thenthe stack temperature would be expected to rise, which is exactlythe result seen. This breakdown begins to occur at a current draw ofabout 280 A, which is beyond the applicability of the polarizationcurve correlation.

linear current draw.

s for linear current draw.

Fig. 5. Stack dynamics for pulse function current draw.

Fig. 6. Cooling water dynamics for pulse function current draw.

Fig. 7. Stack dynamics for high current draw.

D. O’Keefe et al. / Journal of Power Sources 256 (2014) 470e478 475

Fig. 8. Cooling water dynamics for high current draw.

Fig. 9. Stack dynamics for low current draw.

D. O’Keefe et al. / Journal of Power Sources 256 (2014) 470e478476

The next simple current profile considered is that of a pulsefunction. In this case, it is possible to examine the ability of thecontroller to react to large, sudden changes in the current draw.Sudden changes such as this are fairly common in many real worldapplications, so it is vital that the controller be able to quickly andeffectively respond. The results of this model are given in Figs. 5 and6, and demonstrate that the designed controller is very effective at

Fig. 10. Cooling water dynam

handling isolated step changes. The stack temperature shows avisible controller response within about 50 s of each disturbance,allowing the stack temperature to change by only about 2.5 Kbefore returning to the target temperature in each case. That cor-responds to a 97% reduction in the change in stack temperaturefrom the reference case of constant cooling water flow. Fig. 6 alsoprovides the best example of the PI controller at work. At t¼ 4000 s,

ics for low current draw.

D. O’Keefe et al. / Journal of Power Sources 256 (2014) 470e478 477

there is a step change in the current draw. This step change causesthe controller to suddenly change the cooling water flow rate fortwo reasons; the first reason is simply the difference in the steady-state cooling water flow, which is the first term in Eq. (26). Thecontroller also accounts for the deviation in the stack temperaturefrom the set point, which explains the small changes in the coolingwater flow rate for about 500 s following the step change in currentdraw.

The model was tested using two current profiles obtained fromexperimental data from a 3 bedroom house [4]. The experimentaldata was broken into two current profiles, each with a differenttrend. The first profile considers the current draw on a home duringpeak hours, so the profile has multiple drastic changes in the powerdemand. The second profile considers the power demand in non-peak hours, featuring sections of time with little change in thecurrent draw. The results for the first current profile based onexperimental data are given in Figs. 7 and 8.

The results show that the controller greatly reduces the resultingchanges in stack temperature. The greatest deviation in stack tem-perature is only about 1.5 K; bycomparison, the greatest deviation instack temperature from the average temperature in the modelwithout the controller is approximately 41 K. That means if themodel without the controller was designed to operate at is averagestack temperature (368 K), then the maximum deviation from theideal temperature would be about 41 K. Comparing the controllerdeviation in stack temperature to this best case scenario modelwithout a controller gives a 96.5% reduction in the variation of stacktemperature. The average outlet temperature of the coolingwater isaround 320 K, with a maximum outlet temperature of 333 K. Thismaximum change in the cooling water temperature could proveproblematic for a system with recycled cooling water, but not for asystemwhere the water is disposed or used for other purposes.

The second current profile from experimental data is designedusing the same procedure as the first. Figs. 9 and 10 show thecurrent profile and the results from the calculations. The resultsshow that the controller is notably less effective, mainly becausethe average current draw for the second profile is much lower than20 A. The maximum deviation from 350 K is just under 5 K and thedeviations average about 2 K. Using the same comparison as in theprevious current draw profile, the average operating temperaturefor the reference model without the controller is approximately335 K, and the maximum deviation from this temperature is 48 K.This means that the controller still reduces the deviation in stacktemperature by as much as 96%. Although the controller is certainlyless effective at lower current draws, it still achieves the desiredresult within the given restraints. The required cooling flow ratesare not excessive in any of the simulations and it is not necessary touse chilled water.

4. Conclusions

A dynamicmodel of awater cooled fuel cell stack was developedand simulated in conjunction with a state space model of a PIcontroller. The stack temperature dynamics and cooling watertemperature dynamics models were linked and solved simulta-neously, while all other dynamics inside the fuel cell were consid-ered to operate at pseudo-steady-state. A time-varying PI controllerwas designed for the theoretical model of a PEM fuel cell stackjacketed by cooling water. The controller adjusts the cooling waterflow rate through the jacket tomaintain the stack temperature at anoptimal operating point. The model was tested using multiple po-wer profiles, including two power profiles obtained experimentallyfrom a residential house. It was determined that the controller iscapable of maintaining the stack temperature within 2 K for mostcurrent draws, and within 5 K for current draws below 20 A.

References

[1] Fuel Cell Handbook, fifth ed., U.S. Department of Energy, Morgantown, WV,2000.

[2] J. Amphlett, R. Mann, B. Peppley, P. Roberge, A. Rodrigues, J. Power Sources 61(1996) 183e188.

[3] K. Chu, J. Ryu, M. Sunwoo, J. Power Sources 171 (2007) 412e423.[4] M. El-Sharkh, A. Rahman, M. Alam, P. Byrne, A. Sakla, T. Thomas, J. Power

Sources 138 (2004) 199e204.[5] L. Fang, D. Li, R. Yang, A Dynamic Model of PEM Fuel Cell Stack System for Real

Time Simulation, School of Mechanical Engineering, South China University ofTechnology, Guanzhou, China, 2009.

[6] Felix Grasser, Alfred C. Rufer, A Fully Analytical PEM Fuel Cell System Modelfor Control Applications, Laboratoire dElectronique Industrielle, Ecole Poly-technique Fdrale de Lausanne, Switzerland, 2007.

[7] M. Khan, M. Iqbal, Fuel Cells 4 (2005) 463e475.[8] P. Kolavennu, J.C. Telotte, S. Palanki, Int. J. Hydrogen Energy 34 (2009) 380e

387.[9] R.T. Meyer, B. Yao, in: 2006 ASME International Mechanical Engineering

Congress and Exposition. IMECE2006e14151, ASME, Chicago, IL, 2006.[10] D.B. O’Keefe, Temperature Dynamics and Control of Fuel Cell Systems, M.S.

thesis, University of South Alabama, 2012.[11] P. Pathapati, X. Xue, J. Tang, Renew. Energy 30 (2005) 1e22.[12] J.T. Pukrushpan, A.G. Stefanopoulou, H. Peng, Proc. Am. Control Conf. (2002).[13] M. Sundaresan, R. Moore, J. Power Sources 145 (2005) 534e545.[14] X. Xue, J. Tang, A. Smirnova, R. England, N. Sammes, J. Power Sources 133

(2004) 188e204.

Notation

English letters

Acell: active fuel cell area, m2

Acs: reactor cross sectional area, m3

C1: lumped parameter, W K�1 A�1

C2: lumped parameter, W A�1

Cp: specific heat, J mol�1 K�1

CV: conversion factorDp: catalyst particle diameter, mfs: saturation calculationF: Faraday’s constant, C mol�1

G: superficial mass velocity, kg m�2 s�1

h: cooling water channel height, mH: molar enthalpy, J mol�1

DH+298: standard heat of reaction, kJ mol�1

DHrxn: molar heat of reaction, J mol�1

i: current density, A m�2

I: current, Akc: controller proportional constant, kg s�1 K�1

L: fuel cell stack length, mm: mass, kg_m: mass flow rate, kg s�1

ncell: number of cells in stackne� : number of electrons_N: molar flow rate, fuel cell stack, mol s�1

p: partial pressure, kPaP: pressure, kPaq: molar flow rate, reactor, mol s�1

Q: heat flow into reactor per unit volume, W m�3

_Q: heat flow, Wr: reaction rate, mol s�1 m�3

R: universal gas constant, J mol�1 K�1

T: temperature, KU: internal energy, JUA: heat transfer parameter, W K�1

V: reactor volume, m3

Vcell: cell voltage, VVn: volume per cooling water zone, m3

W: fuel cell stack width, m_W: work flow, WX: fractional conversion of primary reactantXs: fractional excess oxygen fed to fuel cell stacky: gaseous mole fraction

Greek letters

ε: efficiency factorm: viscosity, kg m�1 s�1

m: stoichiometric ratio_x: extent of reaction, mol s�1

r: density, kg m�3

D. O’Keefe et al. / Journal of Power Sources 256 (2014) 470e478478

s1: transfer function dimensionless parameters2: transfer function dimensionless parametersI: controller integral constant, sf: void fraction of catalyst in reformer

Subscripts

c: catalystcw: cooling water

CO2: carbon dioxideC2H5OH: ethanolH2: hydrogenH2O: waterin: inlet conditionout: outlet conditionsp: set pointst: stackT: total sum of chemical components in reactor