Dynamic characteristics of PEM fuel cellsP.J.H. Wingelaar, J.L. Duarte and M.A.M. Hendrix

Eindhoven University of TechnologyElectrical Engineering – EPE group –

Den Dolech 2, – el 1.02P.O.Box 513

5600 MB EindhovenThe Netherlands

Email: P.J.H.Wingelaar@tue.nl

Abstract— Fuel cell applications have become increasinglyattractive. Therefore, comprehensive models, simulation andanalysis tools are required to characterize fuel cell behavior.For years, the focus has been on describing the steady statecharacteristics of fuel cells, which is an important applicationbut certainly not the only one. This paper presents methods tomeasure steady state and transient behavior of Polymer Elec-trolyte Membrane fuel cells (PEMFC). An active load that enablesall required measurements is described. Furthermore, a simplelarge-signal dynamic PEM fuel cell model is introduced of whichthe parameters can be found with an interrupted current method.This method estimates the parameters of the dynamic model.However, there was doubt whether the influence of small-signaldisturbances in the fuel cell operation could be analyzed with theinterrupted current method. Therefore, impedance spectroscopymeasurements were performed on the fuel cell stack to determineif the large-scale model was applicable to small-signal excitations.Practical measurements show that an additional small-signalmodel should be introduced to characterize a fuel cell stack whenswitching ripple is present in the output current.

I. INTRODUCTION

Although the principles of fuel cells have been known formore than 150 years, these green power generators have onlybeen used commercially for a few decades [1]. The energyof a fuel cell is released by reducing oxygen at the cathodeside of the membrane Eq. (2) and oxidizing hydrogen at theanode side Eq. (1). The waste product of the redox reactionsis water:

H2 → 2H+ + 2e−, (1)

O2 + 4H+ + 4e− → 2H2O, (2)

2H2 + O2 → 2H2O + electricity + heat. (3)

Because redox reaction (3) is exothermic, heat is produced.Figure 1 shows a schematically drawn PEM fuel cell. Thenegative terminal of the fuel cell, the anode, takes in hydrogenand the positive terminal, the cathode, takes in oxygen or airand functions as water outlet. The diffusion layers are madefrom carbon with a thin platinum coating. Platinum is thecatalyst for the reaction. The membrane is embedded into arubber gasket, to reduce leakage of the gasses.

Fuel cells are efficient, low voltage, high current DC powergenerators, which have a highly nonlinear voltage-currentcharacteristic, shown in Fig. 2 [2]. This paper focusses on the

aeb

de

Fig. 1. Schematically drawn PEM fuel cell, with a) a metal plate (anode),which is the negative terminal of the fuel cell, b) is the diffusion layer, madefrom carbon with a small layer of platinum on it, c) is the membrane insidea rubber gasket, d) is a diffusion layer made from carbon with a platinumcatalyst layer and e) is a metal plate (cathode), which is the positive terminalof the fuel cell. Hydrogen is fed into the cell at the anode side, air or oxygenon the cathode side. The only waste product of the redox reaction is water,which is blown out of the cell at the cathode side.

Polymer Electrolyte Membrane (PEM) fuel cell. The fuel cells’membrane is made from thin plastic, often Nafion. The maintransport mechanism of the protons through the membrane isdiffusion. The temperature of these fuel cells is low (±60 to80C) in comparison to other types of fuel cells [3].

Many papers have been written about (steady-state) fuel cellmodels, most of them based on a chemical or an electrochem-ical approach. In general, two fuel cell models are describedwell. First, the steady-state model describing almost everyelectrochemical and physicochemical aspect related to fuelcells [4–7]. This model is referred to as the “more theoreticalmodel”.

The second model is based on empirical measurementsof current and voltage, which is coupled to the same the-oretical background of the first model [8–10]. Because thecharacteristics of the fuel cell play a significant role in themodel, this method is referred to as the “more empiricalmodel”. The main advantage of the second method overthe first is that all variables of that model are obtained byelectrical measurements. Instead of disassembling the stackafter measurements, the fuel cell can be characterized bymeasuring currents and voltages.

16350-7803-9033-4/05/$20.00 ©2005 IEEE.

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

IFC

(A)

VF

C (

V)

Region ofActivation Polarization

Region ofOhmic Polarization

Region ofConcentration Polarization

Fig. 2. PEM fuel cell voltage-current characteristic showing a highly non-linear voltage drop in the activation polarization region and in the region ofthe concentration polarization.

Lately, the interest in the dynamic response of a fuel cellincreases. The transient behavior can be a function of thetemperature [6], but this type of transient behavior is outof scope of this paper. Nevertheless, the dynamics can alsobe characterized by applying step responses [10] or smallcurrent harmonics at the outputs of the fuel cell [11]. Althoughboth methods describe dynamic behavior, no qualitative norquantitative comparison between both approaches can befound in the literature. However, it is essential to powerelectronic engineers to fully understand the dynamics of apower source. Therefore, this paper presents the large andthe small signal response of the fuel cell. Moreover, differentequivalent models for large and small scale excitations areintroduced and compared as well.

II. BASIC ELECTROCHEMICAL AND ELECTRICAL FUEL

CELL MODELS

A. Steady-state model

Figure 2 shows the steady state fuel cell characteristic. Thiscan be described by means of a reversible voltage, Erev ,combined with irreversible loss voltages, or overpotentials.The loss voltage responsible for the activation polarization isthe activation overpotential, ηact; the ohmic polarization is re-lated to the ohmic overpotential, ηohmic; and the concentrationpolarization is caused by the concentration overpotential, ηl.The steady state fuel cell voltage, VFC , is then calculated using

VFC = Erev − ηact − ηohmic − ηl. (4)

The reversible voltage Erev is related to the change in Gibbsfree energy ∆G and to pressure-effects, being expressed as [3]

Erev =−∆G

2F+

RT

2Fln(

aH2

√aO2

aH2O

), (5)

where R is the universal gas constant, T is the stack tem-perature, F is Faraday’s constant and a is the activity of thereactants. If the reactant gasses act like ideal gasses, then the

activity can be calculated by dividing the partial pressure ofthe reactant by the standard gas pressure (p0 = 100 kPa).

The activation losses ηact are caused by the slowness ofthe reaction at the surface of the electrodes. The rate of theseelectrochemical reactions are expressed by the Tafel equation[12]. A simplified expression for the activation overpotentialis

ηact =2.3RT

αnFln(

IFC

I0

), (6)

where IFC is the fuel cell output current, α is the transfercoefficient, n is the number of electrons involved in thereaction and I0 is the exchange current.

The ohmic overvoltage is calculated using

ηohmic = IFCRint, (7)

where Rint is defined as the sum of electric and protonicresistance.

The concentration losses ηl are generated by the depletionof the reactants at the surface of the electrodes as the fuel isconsumed. If a mixture of gases is used to supply the fuelcell, the consumption of one reactant causes a small reductionof the partial pressure. This is directly linked to a change involtage (5). The limiting current Il is defined as the current atwhich the fuel is consumed at a rate equal to the maximumsupply flow. Assuming that the partial pressure of the fuel gasfalls down linearly to zero when the load current increaseslinearly to the limiting current, the concentration losses areexpressed as [3]

ηl = −RT

nFln(

1− IFC

Il

). (8)

The basic fuel cell equation (4) combined with (6), (7) and(8) can be reduced to

VFC = Erev −A ln(

IFC

I0

)−

−RintIFC + B ln(

1− IFC

Il

), (9)

where the numerical values of A, the activation polarizationconstant, and B, the concentration polarization constant [3],can be acquired by means of electrical test measurements [10].

B. Large-signal model

Concerning the transient behavior, the dynamics of a fuelcell can be predicted by a phenomenon called the chargedouble layer. One can imagine from Fig. 1 that two ideallypolarized metal plates separated by a thin plastic sheet act asa capacitor. This capacitor is called the charge double layercapacitor, CDL. In case of the PEM fuel cell, the dynamicsof the system can be modeled using this charge double layercapacitor in combination with a parallel and a series resistor,RP resp. RS , as shown in Fig. 3 [3].

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EFC

RS

RP

CDLIFC

VFC

+

-

VFC

EFC

R∞

RR

CRIFC

RA

CA

C∞

+

-

Fig. 3. Equivalent dynamic circuit model of a PEM fuel cell. The dynamicsof the model are caused by the double layer capacitor (the capacitance of theanode and cathode separated by the membrane).

R∞

CDL

EFC

RS

RP

CDLIFC

VFC

+

-

VFC

EFC

IFC

C∞

+

-

RDL Z

ZRA

LA

Z

RA

LA

RR

CR

Fig. 4. Small signal equivalent circuit model of a PEM fuel cell. Thedynamics of the model can be characterized by the bulk resistor, R∞,the double layer capacitor, CDL, the reaction resistor, RDL and additonalimpedances modeled with Z.

C. Small-signal model

The small-signal model will have a double layer capacitordominating a part of the frequency domain. In Fig. 4 the chargedouble layer capacitor is represented by CDL. Possible addi-tional time constants can be expected due to the adsorption re-actions or other electrode-electrolyte interface reactions. Theseimpedances can be modeled with the unspecified impedance,Z [13–16]. All electrode-material systems have a geometricalcapacitance, C∞ and a bulk resistor R∞ in parallel, leading tothe dielectric relaxation time, τD of the basic material. Becausethe τD is normally very small, it is not likely that this time-constant can be identified [13].

III. MATERIALS

The uncontrolled fuel cell stack is one of the cartridges fromthe commercially available Avista Labs SR-12 fuel cell system.The cartridge contains four membranes stacked in series of two[17], [18].

Theoretically, the cartridge is capable of delivering 42Woutput power, but the air-supply used in the test-setup limitsthe maximum output power to 7W. The oxygen usage of aPEM fuel cell can be calculated using [3]

O2,usage = 8.29 · 10−8 · Pe

Vcell

[kg

s

], (10)

Airusage = 3.57 · 10−7 · λ · Pe

Vcell

[kg

s

], (11)

where Pe is the electrical output power, Vcell is the cellvoltage, and λ is the stoichiometry. The maximum outputpower is reached when the individual cell voltage of the stack

Control circuit

Fuel cell system

MOSFET L (see text)

dSpace

Shunt (see text)

(a)

Ref.

VGS

VFC

Kp

Ki

++

+-

1s

IFC

ActiveLoad

(b)

Fig. 5. The test-setup topology (a) and control implementation (b) of theuncontrolled fuel cell stack. The linear regulator topology is coupled withSimulink / dSpace for measurement and control purposes.

is about 0.5 V [18]. The air-supply in the test setup delivers0.13 l

min , when the stoichiometry is assumed to be one [3].The reversible cell voltage per cell is calculated using for

the Gibbs free energy ∆G = −228.2 kJmole and 80C for the

temperature T . The pressure of the pure hydrogen supply isPH2 = 0.5P 0. If the water product of the reaction is assumedto be in liquid form, then the activity of the water is aH2O = 1[3]. The reversible cell voltage (5) is obtained as being

Erev,cell =228.2 · 103

2 · 96485

+8.314 · 3532 · 96485

ln

(0.5√

0.211

)= 1.16V. (12)

The fuel cell current IFC is one of the major controlparameters in (9). Therefore, it would be convenient to im-plement an active load as a computer controlled, continuouslyadjustable current source. The linear regulator topology shownin Fig. 5(a) has been implemented for this goal. The loadcurrent is continuously adjustable and the advantages of thistopology over switched regulators are: there are no switchingcomponents in the load, which can influence the steady-statemeasurements; and the topology is simple and easy to control.

The output voltage of the fuel cell ranges theoreticallybetween 0V and 4.64V, with a maximum power of 7W at 2V.For the MOSFET the Philips BUK436-50A has been selected,which has a low RDS,on resistance [19]. The control circuitconverts the fuel cell output voltage to a range between –10Vand 10 V and is connected to dSpace. The control of the linear

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regulator is performed in Simulink using a PI feedback loop,Fig. 5(b).

The shunt-resistor in the topology is not included, becausethe current measurement is accomplished by a LEM LA 50-Psensor. The sensor is positioned between the positive terminalof the fuel cell and the drain of the FET’s.

The maximum sample frequency of dSpace is limited to10 kHz. As a consequence, the maximum control frequencywithout aliasing, the Nyquist frequency, is 5 kHz, which israther low for controlling power electronic circuits. Therefore,high frequency characteristics are not measurable with thistest-setup. The accuracy of the digital measurements is limitedby the bit error of the used DA and AD converters. The ADconverter used for the measurements of the drain and thesource voltages is 12 bits wide. The maximum voltage rangeis 20 V (–10 V to +10 V) resulting in a bit-error of 5 mV.

IV. TESTS AND RESULTS

There are two methods known to characterize dynamicfuel cell models. The easiest to implement is the Interruptedcurrent method. This method steps the load-current instan-taneously to a lower or higher value. The fuel cell voltagechanges dynamically to the new steady state value. The modelshown in Fig. 3 is fitted to the data.

The second method is Impedance spectroscopy. The outputcurrent of the fuel cell is excited with a small sinusoidalsignal superposed on a DC-current, while the fuel cell voltageamplitude and phase shift are monitored [13]. The fuel cellimpedance can be determined and fitted to an equivalentcircuit, for instance, the model presented in Fig. 4. In thefollowing, both methods are used to characterize the fuel cellstack dynamics.

A. Steady state characteristic

The steady state characteristic of the fuel cell is determinedbefore the dynamics of it is considered. To that purpose, theactive load is adjusted to sink a certain current after whichthe fuel cell stack voltage is measured with dSpace. In Fig. 6the measurement points are represented by the ‘o’ marks. Theparameters from (9) are calculated from the data, resulting inA = 0.16V , I0 = 1.4 · 10−5A, Rint = 0.34Ω, B = −0.063Vand Il = 7.7A. The fitted characteristic, the full-line in Fig.6, is derived from the obtained parameters.

B. Interrupted current method

As described before, the interrupted current method stepsthe load current of the fuel cell suddenly from one value toanother. This method can be very useful when studying the fuelcell behavior to switched mode converters, because the inputof these converters excites the fuel cell in the same way. Thelarge-signal model shown in Fig. 3 is fitted to the obtaineddata. Figure 7 shows the result of the step-response of thefuel cell stack to a load current-change from 1A to 5A. Theparameter values are Rs = 0.236Ω, Rp = 0.140Ω and Cdl =0.221F .

The value of Rint found from steady state measurementsis much lower than the series and parallel resistor obtained in

0 2 4 60

1

2

3

4

IFC (A)

VFC

(V)

A = 0.16 VIo = 1.4⋅10-5 A

Rint = 0.34 ΩB = -0.063 VIl = 7.7 A

Fig. 6. Steady state characteristic measured at the fuel cell stack. The ‘o’are the measured points, the full line is the fitted characteristic (9) using forA = 0.16V , I0 = 1.4 · 10−5A, Rint = 0.34Ω, B = −0.063V andIl = 7.7A.

43 43.5 44 44.5 45 45.5 46

1

1.5

2

2.5

t (s)

VF

C (

V)

Fig. 7. Step-response of the small uncontrolled fuel cell stack. The loadcurrent is changed from 1A to 5A. The gray dotted line is the fitted fuelcell model on the data with for Rs = 0.236Ω, Rp = 0.140Ω and CDL =0.221F .

this test. A possible explanation to this behavior is the effectof temperature. If the voltage is monitored for several moreseconds after a load step-change, an additional rise in the fuelcell voltage is seen, Fig. 8. The voltage rise can be relatedto an increasing temperature caused by the higher currentdrawn from the fuel cell. A higher current leads to a higherproduction of water on the membrane, which increases theproton-conductivity and lowers the internal resistance.

C. Impedance spectroscopy

The dynamic model from Fig. 3 is a rather basic modeldescribing only one double layer capacitor. The model is basedon large-signal changes. However, two electrode-electrolyteinterfaces are present in a fuel cell, resulting in two double-layer capacitors. To compare the large-signal model with thesmall-signal behavior, impedance spectroscopy measurementswere performed on the fuel cell.

Figure 9 shows the measurement results from the impedance

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45 50 550.8

0.9

1

1.1

t (s)

VF

C (

V)

Fig. 8. Extension of Fig. 7: cell voltage effect, probably related to the heatingof the fuel cell, that causes higher resistor values in the large-signal model(Rs and Rp) than the internal resistor of the steady state model has (Rint).

0.3 0.4 0.5 0.6

-0.2

-0.15

-0.1

-0.05

0

0.05

Z' (Ω)

Z'' (Ω

)

0.01

0.07 0.1

0.5

1 1.8 2

3 4

5 6

10 15

22

Fig. 9. Impedance plot of the four membrane fuel cell stack, measured witha DC-current of 1A, and a frequency ranging from 0.01Hz (right) to 22Hz(left, near the real axis). The measured error-band is included in the figure.

spectroscopy carried out on the fuel cell stack. The frequencyranges from 0.01 Hz to 22 Hz, and the DC-current was setto 1A. To stabilize the control with dSpace, a small inductor(L = 12µH) is placed in series with the source of the linearregulator MOSFETs, shown in Fig. 5(a).

The figure shows the averaged data-points obtained fromthe measurements. The test is repeated at least ten times forall frequencies. The data is converted to the frequency-domainby using a fast fourier transform. The maximum deviation ofthe data-points is marked with the gray, dashed lines.

Because Fig. 7 shows the step-response from 1A to 5A, theimpedance plot of 5A DC-current setting is also depicted, Fig.10. A difference in the measurement method is the removal ofthe series inductor from the linear regulator. To compare theimportant values of the step-response with the small-signalresponse measured with impedance spectroscopy, figures Fig.9 and Fig. 10 are fitted on a ladder network, shown in Fig. 4.

Both figures show an inductive behavior in the low-frequency range. Literature indicates that this behavior is

Fig. 10. Impedance plot of the four membrane fuel cell stack, measured witha DC-current of 5A, and a frequency ranging from 0.01Hz (right) to 45Hz(left, near the real axis). The measured error-band is included in the figure.The inset shows the inductive behavior in the low-frequency region of theimpedance plot.

TABLE I

FITTED RESULTS OF THE IMPEDANCE SPECTROSCOPY TO A SECOND

ORDER AND THIRD ORDER EQUIVALENT CIRCUIT

Impedance-response2nd order 3rd order1A 5A 1A 5A

R∞ (Ω) 0.291 0.270 0.283 0.266CDL (F) 0.281 0.309 0.221 0.229RDL (mΩ) 284 84.9 202 65.7LA (mH) 49.6 56.4 36.0 38.5RA (mΩ) 14.7 9.15 19.1 9.37CR (F) - - 0.450 0.566RR (mΩ) - - 92.3 24.2

probably caused by adsorption [14], [16].As the results of the measurements indicate inductive be-

havior in the low-frequency response, and the mid-frequencyrange shows capacitative behavior, the model presented in Fig.3 will not deliver an adequate fit. The additional impedance Z,introduced in the model of Fig. 4 is first substituted with theimpedance shown in Fig. 11(a). LA, the adsorption induction,and RA, the adsorption resistance, model the low frequencyimpedance behavior.

Figure 11(b) shows the best-fit solutions (gray lines), forthe 1A DC-current set-point measurement, ‘x’-marks, and forthe 5A DC-current set-point measurement, ‘+’-marks. The fitis performed with a Nonlinear Least Squares method.

The values of the components of the equivalent model foundin the second-order fit are summed in Table I. Although thefitted amplitude matches the measured characteristic well, thephase-shift approaches the measured values less accurately.This indicates another time-constant in the data. Therefore,the fit is performed on a equivalent circuit composed of Fig.4 and Fig. 12(a). The reaction capacitor CR and resistor RR

model the second electrode-electrolyte interface.The results of this fit are presented in Table I and graphically

in Fig. 12(b) as the thick gray lines. Not only the amplitude ofthe impedance is matched in this fit, but also the phase-shift.

1639

VFC

EFC

-

RDL Z

ZRA

LA

Z

RA

LA

RR

CR

(a)

0.3 0.4 0.5 0.6

-0.2

-0.15

-0.1

-0.05

0

0.05

Z' (Ω)

Z'' (Ω

)

(b)

Fig. 11. Impedance plots of the 1A (‘x’-mark) and 5A (‘+’-mark) currentsetting with a fit on a second-order equivalent circuit shown in Fig. 4.

The goodness of the fit is graphically shown in Fig. 13. Thetop-figure shows the residuals of the 1A DC-current fit. Thegray line represents the second order fit, while the black linecorresponds to the third order fit. The lower figure shows theresiduals of the 5A DC fit.

D. Analysis of the models

If a comparison between the step-response characteristicsof the fuel cell and the third order impedance model is made,the same double layer capacitor CDL value is found. However,the analysis of the measurements show that the resistor valuesof the models are variable. Therefore, the large-signal modelcannot directly be converted to the small-signal model.

The third order impedance model matches the theory ofthe two double layer capacitors formed from the electrode-electrolyte interfaces of the anode and the cathode. Addi-tionally, one can conclude from the matching double layercapacitor in the small-signal and the large-signal model, thatthe time constant caused by the CDL is dominating the fuelcell transient behavior.

Furthermore, the impedance spectroscopy plots show notransport limitations of the protons through the membrane.Currently, the research focuses on the inductive behavior in thelow-frequency range. It is very well possible that the transportlimitations show up at lower frequencies than 10 mHz.

ZRA

LA

Z

RA

LA

RR

CR

(a)

0.3 0.4 0.5 0.6

-0.2

-0.15

-0.1

-0.05

0

0.05

Z' (Ω)

Z'' (Ω

)(b)

Fig. 12. Impedance plots of the 1A (‘x’-mark) and 5A (‘+’-mark) currentsetting with a fit on a third-order equivalent circuit.

V. CONCLUSIONS

This work shows two methods for identifying dynamicsof a fuel cell stack. The first one is called the interruptedcurrent method, which is easy to implement. With this method,the identification of parameters that describe the large-signalcharacteristics can be performed, resulting in a simple dynamicmodel.

The second method is impedance spectroscopy, which is amore accurate method to identify electrochemical cells. Thelarge-signal model, as described in the interrupted currentmethod, is not able to represent the small-signal behavior ofthe fuel cell. Therefore, in view of power electronic applica-tions, an extended model representation is necessary to includethe small-signal variations.

The impedance spectroscopy has shown that the fuel cellused in this research can best be modeled with a third-orderequivalent circuit. This model includes the dynamics of twoelectrode-electrolyte interfaces and an “inductive” adsorptionreaction.

ACKNOWLEDGMENT

The authors would like to thank Marijn Uyt de Willigen andWim Thirion for their assistance in producing the test-setupand for their efforts during the measurements. Furthermore,the authors thank Marcel Geers, who worked for his practicaltraining on some of the presented measurement results.

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10-2

10-1

100

101

-8

-4

0

4

8x 10

-3

Res

idua

ls (1

A D

C)

10-2

10-1

100

101

-8-4048

x 10-3

f (Hz)

Res

idua

ls (5

A D

C) 2nd order fit

3rd order fit

Fig. 13. The residuals of the fit for the 2nd order (gray lines) and 3rd order(black lines) fit. The residuals for the 1A DC current setting are displayed inthe upper figure, the 5A DC-current setting is depicted in the lower figure.

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