A PEM Fuel-Cell Model Featuring Oxygen-Excess-Ratio Estimation and Power-Electronics Interaction

1914 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 6, JUNE 2010

A PEM Fuel-Cell Model FeaturingOxygen-Excess-Ratio Estimation and

Power-Electronics InteractionCarlos Andrés Ramos-Paja, Student Member, IEEE, Roberto Giral, Member, IEEE,

Luis Martinez-Salamero, Member, IEEE, Jenny Romano, Alfonso Romero, Member, IEEE, andGiovanni Spagnuolo, Senior Member, IEEE

Abstract—In this paper, a polymer-electrolyte-membrane fuel-cell (FC) model that is useful for simulation and control purposesis presented. The model uses both electrical-circuit componentsand functional blocks to reproduce both static and dynamic FCbehaviors. Its main feature is in the reproduction of the oxygen-excess-ratio behavior, but it is also able to interact with anyelectrical device connected at the FC terminals, e.g., a load ora switching converter. Consequently, the proposed model can beused to develop new control strategies aimed at avoiding theoxygen-starvation effect and/or minimizing the fuel consumption.The model has been customized for a Ballard Nexa 1.2-kW powersystem, and this has allowed an experimental validation by meansof measurements performed on a real FC device.

Index Terms—Oxygen excess ratio, polymer-electrolyte-membrane (PEM) fuel-cell (FC) modeling, power-electronicssimulation.

I. INTRODUCTION

IN THE literature, there are many models that describea polymer-electrolyte-membrane (PEM) fuel cell (FC) in

terms of static and dynamic behaviors. These models can beclassified into electrical equivalents and electrochemical mod-els [1]. Among the latter, the dynamic model proposed byLarminie [2] is widely accepted: It represents each electrode byusing a capacitor in parallel with a resistor and a voltage source.The resistors and the capacitors reproduce the correspondingFaraday effects at the electrodes, and an independent voltagegenerator gives the FC open-circuit voltage. In [3], a morecomplex model that reproduces the different regions of the FC

Manuscript received February 26, 2009; revised April 30, 2009; acceptedJune 11, 2009. Date of publication June 30, 2009; date of current versionMay 12, 2010. This work was supported in part by the Spanish Ministeriode Ciencia e Innovación under Projects ENE2005-06934 and TEC2009-13172,by the Formación Personal Investigador (FPI) scholarship BES-2006-11637,and by the University of Salerno.

C. A. Ramos-Paja is with the Facultad de Minas, Universidad Nacional deColombia, Medellin, Colombia (e-mail: [email protected]).

R. Giral, L. Martinez-Salamero, and A. Romero are with the Departamentod’Enginyeria Electrònica, Elèctrica i Automàtica, Escola Tècnica Superiord’Enginyeria, Universitat Rovira i Virgili de Tarragona, 43007 Tarragona, Spain(e-mail: [email protected]).

J. Romano and G. Spagnuolo are with the Department of Information andElectrical Engineering, University of Salerno, 84084 Fisciano, Italy (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2009.2026363

polarization curve is presented. In that model, a diode, twotransistors, and a resistor give the static response. The diodereproduces the activation losses, and its parasitic resistancemodels the ohmic effect in the cell. The transistors and theresistor reproduce the concentration losses, and a capacitorand an inductor together with the transistors account for thedynamic behavior.

In [4], Capel et al. present a model using the electricaleffects of the semiconductor devices to reproduce the differentregions of the FC polarization curve. The activation effect ismodeled by a diode, the concentration effect by an electricalnet with a diode and a voltage source, and a resistor models theohmic effect.

A different approach based on an FC electrical equivalentwas presented by Famouri and Gemmen [5], with the fuel-flow dynamics that were taken into account in the output-powercalculation. The circuit models the transient effects caused bythe hydrogen-flow variation, the reactants’ partial pressures,and the mechanical losses inside the FC. This model is a cir-cuital implementation of the electrochemical equations wherethe FC and the humidifier dynamics are taken into accountthrough mass-conservation equations. The model calculates thecell voltage as a function of partial pressures, load current,and mechanical losses, but considers the stack-temperature con-stant (77 ◦C).

Electrochemical models use physical equations, buildinglinear or nonlinear models in state-space representations andblock diagrams, without identifying an equivalent electricalcircuit. In the literature, there are different models proposedand validated: detailed nonlinear models in state space [6],empirical polynomial-based approximations [7], models of effi-ciency based on interpolation and experimental characterization[8], control-oriented models [9]–[12], models that consider thestack with its auxiliary systems [13], [14], and also models thatinclude a power converter for specific applications [15].

Another interesting model is the one proposed byCorrea et al. [16], where the analysis of the electrochemicalreactions giving the stack voltage as output is presented. Theanalysis is focused on the activation, concentration, and ohmiclosses, and the FC dynamics are modeled by a first-orderdelay with a time constant τ = R · C, where C and R arethe equivalent capacitance and resistance of the system. Thismodel requires low computational resources, and it is useful for

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RAMOS-PAJA et al.: PEM FUEL-CELL MODEL FEATURING OXYGEN-EXCESS-RATIO ESTIMATION 1915

simulations involving also switching converters that manage thepower produced by the FC system, thus interfacing it with aload or the energy grid.

The description of the FC behavior in terms of chemical andphysical relations was also performed by del Real et al. [17];this model was also experimentally validated using a Ballard1.2-kW Nexa power module, which is representative of the stateof the art in PEM FC prototypes.

The model introduced in [18] and [19] considers the fullsystem, the electrochemical and physical reactions of the FCtogether with the dynamic behavior of the auxiliary devices.Regulation strategies of the reactants aimed at having an ap-propriate relation between the anode and cathode pressures areintroduced. The model is used to create control strategies for theoverall system, considering the fuel flow, the current, and thevoltage of the cell as control variables. Many electrochemicaleffects at the FC anode and cathode are modeled by using theequations of mass conservation and water transport through themembrane. Finally, the mathematical descriptions of the polar-ization curve and of the different voltage drops are documented.

The model proposed in this paper reproduces the behaviorof a FC system, and it is experimentally validated using a1.2-kW Nexa power module shown in Fig. 1(a). The maincharacteristic of the model proposed is its simplicity comparedwith complex electrochemical models, but it still allows pre-diction of the oxygen excess ratio [18], which is critical in FCcontrol design [20], [21]. The model combines electrochemicalequations and experimentally identified relations and considersthe thermal effects to achieve a general behavior reproduction.Such peculiarities of the proposed model make it suitable fordeveloping control strategies aimed at ensuring the FC safetyby means of an accurate reproduction of the current value ofthe oxygen excess ratio; moreover, it is proposed as a goodcandidate for testing power-electronic circuits for FC applica-tions [22]–[27] by simulations in circuit-oriented environments.The model accuracy has been validated by means of experimen-tal data.

This paper is organized as follows. Section II outlines thefeatures of the model. Sections III–V describe the detailsof the models for the oxygen excess ratio, the thermal effects,the compressor dynamics, and losses. Section VI describesthe nonlinear circuit simulating the FC voltage–current char-acteristic. Section VII proposes the model validation by meansof experimental results, and Section VIII gives an applicationexample of the model in FC control-system evaluation. Finally,the conclusions of this paper are given in Section IX.

II. MODEL OVERVIEW

The model proposed is control oriented and considers exper-imentally measurable inputs and outputs to adjust it to the realprototype. The interaction of the model with the FC electricalload (dc/dc converter, battery charger, inverter, and so on) takesplace through an electrical circuit that models the FC outputimpedance. The thermal effects that are having an impact on thestack voltage are also taken into account, and the main internalstate predicted in the model is the oxygen excess ratio λO2 ,which is an important parameter in FC control and safety [18].

Fig. 1. Ballard 1.2-kW Nexa power module. (a) Nexa power module. (b) Nexapower module diagram.

The modeling procedure is supported by experimental datathat allow identification of physical relations and simplifycomplex equations derived from involved models [17]. In thisway, the model includes physical and electrochemical equa-tions as well as behavioral relations obtained by interpolat-ing experimental data. The experimental setup used in thispaper is the Ballard 1.2-kW Nexa power module shown inFig. 1(a), which is composed of a stack of 46 cells with 110-cm2

area membranes. The physical configuration of the FC Nexamodule is shown in Fig. 1(b), where the interaction between thestack, the air compressor, the humidifier, the cooling system, thehydrogen supply, and the anode purge valve is shown.

In addition, the Nexa module has a control-boardimplementing strategies aimed at regulating the anode–cathodepressure ratio and the stack temperature and humidity to ensuresafe operation conditions. Similarly, the control board executessafe start-up, load connection, and shutdown sequences andallows command and monitoring procedures in a PC throughthe Nexa module control interface. Finally, the control boardincludes regulation strategies for the anode purge valve and theair-compressor voltage that are intended to avoid undesiredphenomena like flooding and oxygen starvation. The floodingphenomenon occurs when the water generated and the inertgases supplied with the hydrogen get stuck in the anode,producing a decrease in the stack voltage and power [28].The oxygen-starvation phenomenon occurs when the oxygensupplied to the FC is not enough to react in agreement with thedemand of stack current, causing degradation of the FC and


Fig. 2. Proposed model structure.

decrement of the output power and frequently requiring to shutdown the FC [29].

The oxygen-starvation phenomenon implies a steep de-crease in cell voltage, potentially leading to hot spots or evenburn through on the surface of a membrane. To prevent thiscatastrophic event, the FC system controller must regulate thecathode oxygen excess ratio by controlling the compressor-motor voltage according to changes in the current drawn fromthe FC. As pointed out in [18], [20], [21], and [30], the FCair-flow control is a challenging task, since, on one hand, itmust ensure that the partial pressure of oxygen does not fallbelow a critical level at the cathode and, on the other hand,it must also minimize the parasitic losses of the air compres-sor. One of the main difficulties in implementing this con-trol algorithm is related to measuring the oxygen-excess-ratiovalue.

The model inputs are the load current Inet and the ambienttemperature Tca,in, and the main outputs are the oxygen excessratio λO2 and the stack voltage Vst. The stack temperature Tst

is calculated in a thermal model used to estimate the voltagedeviation caused by stack-temperature changes, but it can alsobe used for prediction purposes.

The FC dynamics, included in λO2 and thermal models, havealso been considered. The FC load interaction is performedby a nonlinear circuit, being the voltage at the FC terminalsdependent on the oxygen excess ratio and load current. Finally,the output voltage deviation dVT caused by changes in the stacktemperature is also modeled, reproducing in this way the non-linear FC electrical impedance. The proposed model structureis given in Fig. 2, which shows the compressor dynamics, itslosses, and Nexa board-control structure, where the effectivestack current depends on the load current and on the compressorconsumption.

The model is intended to test power-electronic circuits andcontrol strategies, which are being used in power-electronicsimulation environments. Consequently, the model implemen-tation has been performed using the power-electronics simu-lation software PSIM [31], which allows the use of electricalcircuits for the polarization and thermal models, and continuousand discrete transfer functions for the λO2 and compressormodels.

The next sections describe λO2 , thermal, compressor, andpolarization models and their implementations.

III. OXYGEN EXCESS RATIO

A PEM FC generates electricity from the chemical reac-tion between hydrogen (H2) and oxygen (O2) [19]. In theanode, the hydrogen is dissociated in protons and electronsfollowing

2H2 ⇒ 4H+ + 4e−. (1)

The protons flow through the PEM and arrive to the cathode.The electrons flow to the cathode through an external electriccircuit connected between the electrodes, thus generating elec-tric current. The protons and electrons react with the oxygen inthe cathode as

O2 + 4H+ + 4e− ⇒ 2H2O + heat (2)

and the overall reaction in the FC is

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

To prevent membrane stress, the anode–cathode pressureratio is regulated in a safe relation by modifying the hydrogencontrol valve, which regulates the anode pressure following thecathode pressure [17], [18]. This implies that hydrogen andoxygen flows are correlated, the oxygen flow being the mainvariable for control objectives [32].

The pressure pca,in in the input of the cathode depends on thepartial pressures of water vapor pv,ca,in and dry air pa,ca,in ofthe inlet air

pca,in = pv,ca,in + pa,ca,in (4)

and the vapor pressure can be calculated as (5) from theinlet-air humidity φca,in, temperature Tca,in, and saturationpressure psat(T ), which depends on the temperature as givenin (6) [18], [19]

pv,ca,in = φca,in · psat(Tca,in) (5)

log10(psat) = (−1.69 × 10−10) · T 4 + (3.85 × 10−7) · T 3

+ (−3.39 × 10−4) · T 2 + 0.143 · T − 20.92.

(6)

Similarly, the molar mass of the dry air Ma,ca,in can beexpressed as a function of the oxygen mole fraction yO2,ca,in ofthe inlet air (0.21 for atmospheric air) and of the oxygen MO2

and nitrogen MN2 molar masses

Ma,ca,in = yO2,ca,in · MO2 + (1 − yO2,ca,in) · MN2 . (7)

The relation between the masses of water vapor and dry airin the inlet air, called humidity ratio ωca,in, can be calculatedfrom the ideal gas law as [18]

ωca,in =Mv

Ma,ca,in· pv,ca,in

pa,ca,in

=Mv

Ma,ca,in· φca,in · psat(Tca,in)pca,in − φca,in · psat(Tca,in)

(8)


Fig. 3. Cathode-inlet-pressure experimental data.

where Mv is the water-vapor molar mass. Equation (8) is usedfor calculating the dry-air flow Wa,ca,in available in the inlet-airflow Wca,in as

Wca,in =Wa,ca,in + ωca,in · Wa,ca,in (9)

Wa,ca,in =(

11 + ωca,in

)· Wca,in. (10)

The oxygen flow available in the inlet-air flow is calculatedby means of the molar mass relation rO2,a between the oxygenand dry air

rO2,a =yO2,ca,in · MO2

Ma,ca,in. (11)

Finally, the oxygen flow that arrives to the cathode dependson the mass relation rO2,a and on the dry-air flow Wa,ca,in

available in the inlet-air flow

WO2,ca,in = rO2,a · Wa,ca,in. (12)

A set of experimental measurements taken on the BallardNexa 1.2-kW system has allowed obtaining the cathode inlet-flow pressure, which has been modeled by the linear regressiongiven in (13). The experiments have been performed by mea-suring the pressure in the cathode input pipe under differentair-compressor-flow and stack-current conditions. The resultsof the experiments are summarized in Fig. 3, where the air-compressor flow is given in standard liters per minute (SLPM),the stack current is given in amperes, and the cathode inletpressure is given in bars. The figure also shows the {Ist,Wcp}region where the experiments were performed, avoiding operat-ing conditions with high probability of oxygen starvation, i.e.,high currents at low cathode air flow

pca,in = 1.0033 + (2.1 × 10−3) · Wcp − (475.7 × 10−6) · Ist.(13)

Wca,in in (10) has to be set in kilograms per second, but theoutput of the air compressor Wcp is expressed in SLPM. Toconvert SLPM into kilograms per second, the inlet-air molar

mass Mam (14) is calculated: This depends on the dry airMa,ca,in and water-vapor Mv molar masses, as well as on thewater-vapor fraction in the inlet air xv (15). The latter, in turn,depends on the stack temperature, by means of the saturationpressure, and on the inlet-air humidity [18]

Mam = (1 − xv) · Ma,ca,in + xv · Mv (14)

xv =φca,in · psat(Tst)

1 − φca,in · psat(Tst). (15)

Finally, the molar flow (moles per second) corresponding tothe SLPM representation is calculated by using the ideal gaslaw and mol definition (mol volume is equal to 22.4 L) andalso by converting minutes into seconds. The inlet-air flow inkilograms per second is obtained by multiplying the molar flowby the inlet-air molar mass Mam as follows:

Wca,in =Wcp

22.4 × 60· Mam. (16)

The oxygen and hydrogen flows (WO2,reac and WH2,reac,respectively) consumed in the reaction depend on the stackcurrent, and they are defined by electrochemistry principles asfollows [17], [18]:

WO2,reac =MO2

n · Ist

4F(17)

WH2,reac =MH2

n · Ist

2F. (18)

The parameters MH2 , n, and F are the hydrogen molar mass,the number of cells in the stack, and the Faraday constant,respectively.

The water-vapor flow Wv,gen generated in the electrochem-ical reaction also depends on the stack current, and it isdefined by

Wv,gen = Mvn · Ist

2F. (19)

The relation between the oxygen flow provided to the stackand the one required to supply the current demand is normallyexpressed by the oxygen excess ratio λO2 [18]

λO2 =WO2,ca,in

WO2,reac. (20)

A high oxygen excess ratio, and thus high oxygen partialpressure, improves the power of the stack; however, after anoptimum value is reached, a further increase of its value causesan excessive increase in air-compressor losses, thus degradingthe system efficiency.

The control of the oxygen flow is critical because a loweroxygen concentration than the one required to supply the stackcurrent generates the oxygen-starvation effect, which leads tothe FC degradation [29]. Therefore, the oxygen excess ra-tio must be regulated to λO2 ≥ 1 to prevent the starvationphenomenon. In [33] and [34], the authors propose to trackλO2 = 2 because this value prevents oxygen-starvation effectand ensures a high efficiency in its experimental system.


The evaluation of λO2 , together with thermal effects and thereproduction of the electric power generated by the FC, givesa useful model to evaluate the efficiency and safety of powerelectronics and control systems that are interacting with the FC.

IV. THERMAL MODEL

The thermal model can be obtained by an energy balance: bydefining Hreac as the energy produced in the chemical reactionof water formation, Pelec as the electric power supplied, andQrad,B2amb and Qconv,B2amb as the amount of heat evacuatedby radiation and both natural and forced convection, respec-tively; the energy balance is the following [17]:

mst·CstdTst

dt= Hreac − Pelec − Qrad,B2amb − Qconv,B2amb

(21)Hreac = mH2,reac · ΔhH2 + mO2,reac · ΔhO2

− WH2O,gen(g)

(h0

f,H2O(g) + ΔhH2O(g)

)ΔhH2 = cp,H2 · (Tanch,in − T 0)ΔhO2 = cp,O2 · (Tcach,in − T 0)ΔhH2O(g) = cp,H2O(g) · (Tst − T 0)

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(22)

Pelec = Pst = Vst · Ist (23)

where h0f,H2O(g) is the mass specific enthalpy of formation of

water steam and cp,H2 , cp,O2 , and cp,H2O(g) are the specificheats of hydrogen, oxygen, and water steam, respectively. T 0 isthe reference temperature for the enthalpy, Tanch,in and Tcach,in

are the inlet temperatures of the anode and the cathode, respec-tively, Tst is the stack temperature, mst is the mass of the FCstack, and Cst is its heat capacity. mO2,reac and mH2,reac are themass-flow rate reaction of oxygen and hydrogen, respectively,and WH2O,gen(g) is the water mass-flow rate. Finally, Vst andIst are the voltage and the current of the FC stack, respectively.

The amount of radiated heat depends on the exchange areaAB2amb,rad and the emissivity ε as given in (24), where σ is theStefan–Boltzmann constant. Qrad,B2amb can also be approxi-mated by (25) obtained by means of a polynomial regression

Qrad,B2amb = σ · ε · AB2amb,rad ·(T 4

st − T 4cach,in

)(24)

Qrad,B2amb ≈ 0.81249 · T 4st − 0.81262 · T 4

cach,in. (25)

The heat extracted by means of natural and forced convection(26) includes two terms: Qconv,B2amb,nat that corresponds tonatural convection (27) and Qconv,B2amb,forc that correspondsto forced convection (28) [17]

Qconv,B2amb = Qconv,B2amb,nat + Qconv,B2amb,forc

(26)

Qconv,B2amb,nat = (hB2amb,nat · AB2amb,conv) · ΔTca,in

(27)

Qconv,B2amb,forc = (hB2amb,forc · AB2cool,conv) · ΔTca,in

(28)

ΔTca,in =Tst − Tcach,in (29)

hB2amb,forc =Kh1 · (Wcool)Kh2 . (30)

Fig. 4. PSIM schematic subsystem simulating the thermal model.

TABLE IBLOCKS AND THE CORRESPONDING EQUATIONS IN THE SUBSYSTEM

SIMULATING THE THERMAL MODEL (FIG. 4)

In each term, the convective heat-transfer coefficients(hB2amb,nat and hB2amb,forc) are different, just as the exchangeareas (AB2amb,conv and AB2cool,conv) are; this is because nat-ural convection takes place in the FC lateral walls, and forcedconvection occurs across the internal walls of the cells, whichare constructed as a radiator. Kh1 and Kh2 are the heat-transfercoefficients. In the Nexa system, the air flow supplied by thecooling fan Wcool has been identified as follows [17]:

Wcool = 36 · ucool (31)

where the input ucool is the control signal of the fan. The newfinal energy balance expressed by means of the polynomialregression is

dTst

dt=

15500

[57.64 · Ist + 0.0024 · Ist · (Tcach,in − 298)

− 8.13807 · (Tst − Tcach,in)

− (0.81249 · Tst − 0.81262 · Tcach,in)

−Vst · Ist] (32)

where the stack current is in amperes, the temperatures are inkelvin, and the stack voltage is in volts.

This thermal model has been implemented in the PSIM envi-ronment by using electrical equivalents [5], so that a capacitorvoltage models the stack temperature. Fig. 4 shows the modelscheme, and Table I helps the reader in relating the differentblocks with the equations discussed.

V. MODELING THE AIR-COMPRESSOR

DYNAMICS AND LOSSES

The air compressor is usually modeled by using mechani-cal and electrical relations: An example is proposed in [18].


Nevertheless, a simplified model that allows a low computa-tional effort and, at the same time, good accuracy can be basedon a Laplace representation of its dynamic behavior. The fol-lowing transfer function has been identified from experimentaldata taken on the Ballard Nexa 1.2-kW system following thereaction-curve method [35]

Wcp = Gcm(s) · Vcp − 45Gcm(s) = 0.1437s2+2.217s+8.544

s3+3.45s2+7.324s+5.745

}(33)

where Vcp is the compressor control signal (0%–100%) andWcp is the air mass flow supplied to the FC stack.

The compressor control law implemented in the Nexa controlboard has been identified experimentally

Vcp = 0.99873 · Ist + 46.015. (34)

To account for the power consumption due to FC systemancillaries, particularly to the air compressor, the stack currentIst must be obtained by the sum of the net current Inet requestedby the load and the compressor current Icm. The parasiticconsumption and losses of the ancillaries have been identifiedexperimentally from the prototype to obtain the following rela-tion between the air mass flow Wcp and Icm:

Icm = −3.231 × 10−5 · W 2cp + 0.018 · Wcp + 0.616. (35)

VI. POLARIZATION-CURVE CIRCUIT EQUATIONS

To obtain a reproduction of the FC polarization curve byusing an equivalent circuit, including linear and nonlinear com-ponents, the electrochemical processes that take place at eachelectrode and between them have been studied. They includethe following.

1) The activation of both electrodes: The voltage contribu-tions of each electrode depend on species and electrodematerials. According to the Butler and Volmer law [4], therelationship between the voltage vA and the cell current ifor a given electrode is

i = ia + ic = io

[e

(αanFvA

RT

)− e

(−αcnFvART

)](36)

where ia and ic are the currents of the electrode involvedin the oxidation or the reduction processes when theyoperate as an anode or as a cathode. The current i0 is thecell current at steady state, and the parameters αa andαc are the charge-transfer coefficients of the generatedspecies at the electrode, in the oxidation (FC area oranode operation) and in the reduction (water electrolysisor cathode operation) processes, respectively; n is thenumber of electrons, F is the Faraday constant, R is thegas constant, and T is the FC temperature.

2) The charge transfer from electrode to electrode accordingto Fick’s first law of diffusion, related to the carrierconcentration, which states that the voltage drop vD dueto the current i is given by [4]

vD =RT

nF· ln

(1 − i

iL

)(37)

Fig. 5. Equivalent electrical circuit of the FC.

where iL is the maximum current produced by the FCfor a given flow of hydrogen. This limit condition corre-sponds to the short-circuit point of the FC characteristic.

3) The voltage drop vR introduced by the Ohm’s law appliedto all resistive parts of the cell, i.e., the ohmic losses dueto the circulation of current i through the connectors,the electrodes, and the electrolyte, representing a totalresistance rc

rc =vR

i. (38)

Modeling the electrical characteristic of a cell consists inrepresenting these three effects to define an electrical equivalentcircuit with the same behavior. This circuit exhibits the samev(i) characteristic of the cell, i.e., at any time t

v(i, t) = ΔE0 + vA(i, t) + vD(i, t) + vR(i, t) (39)

where ΔE0 is the open-circuit voltage, i.e., the differencebetween the standard potentials of the electrodes.

The electrical circuit, which represents an electrochemicalsystem and behaving as an FC, has to assemble the three dif-ferent contributions of the chemical reaction on each electrode.As shown in [4], the activation process is equivalent to a diodeconnected in series with a voltage source that represents thestandard potential of the electrode; the diffusion is representedby a current source with a diode in series with a negativevoltage source, and the ohmic effect is represented by a seriesresistance.

The equivalent electrical circuit is shown in Fig. 5. It iscomposed of a current source iL feeding, at the same time,two branches: one constituted by a diode DD in series witha negative voltage source vaD and the second consisting ofthe series connection of a resistance rc, one diode DA, andtwo voltage sources E0

2 and −E01 . This branch is connected

to the external network (Load) and could represent the pos-itive terminal of the cell. The relationships of the activa-tion and diffusion processes imply three voltage sources E0

2 ,−E0

1 , and vaD, and one current source iL. These voltagesources can be suppressed by fulfilling the following con-dition, which imposes that the difference between the stan-dard reference potentials sets the open circuit voltage of thecell, i.e.,

vaD = E02 − E0

1 . (40)

The cell characteristic v(i) becomes

v = vL − vA − vR (41)


that is

v=ADkT

q· ln

(1+

iL−i

iRD

)−AAkT

q· ln

(1+

i

iRA

)−rc · i.

(42)

In the case of a stack consisting of the series connectionof m cells, some parameters in the equivalent circuit multiplyby m. The final equation of the proposed stack-impedancemodel is

Vst = m · ADkT

q· ln

(1 +

Isc − Ist

iRD

)

− m · AAkT

q· ln

(1 +

Ist

iRA

)− RC · Ist (43)

where m = 46 for the Ballard Nexa is the number ofindividual cells connected in series, RC = m · rc is the overallstack resistance, and α = ADkT/q and β = AAkT/q are theproducts of the diodes’ ideal factors AD and AA, the Boltzmannconstant k, the electron charge q, and the temperature T inkelvin that is equal to 35 ◦C, i.e., the reference temperature.The experimental data of different polarization curves fordifferent λO2 were taken, regulating the stack to the referencetemperature. In this circuit, some nonlinear elements have beenused to model the diodes to define the following completenonlinear equations:

iDd = iRD

[e(

vm·α ) − 1

](44)

iDa = iRA

[e(

vm·β ) − 1

](45)

where iRA and iRD are the diode saturation currents. Thevoltage-controlled current source gives the short-circuit currentIsc that corresponds to a given λO2 condition. The circuit hasbeen parameterized by using experimental polarization curveswith λO2 values between 3.0 and 6.5, obtaining the identifiedrelation given as

Isc = −0.45 · λ2O2

+ 8.5 · λO2 + 35. (46)

The polarization model and its correspondingvoltage–current characteristic have been identified by means ofexperimental measurements, thus obtaining iRA = 0.0039 A,iRD = 0.7908 A, α = 0.1999 V, β = 0.0069 V, andRC = 0.0926 Ω.

In the experimental setup, the stack current was limited tothe desired operating range of Ist ≥ 6 A, where the compressordynamics are linear. The fitting was performed using a newcurrent axis (Ist−6) to reproduce the behavior of the diodes.This was implemented in the model circuit using a currentsource of 6 A (I_sum) to obtain the new stack-current axis.

The stack voltage predicted by the circuit shown in Fig. 6 isvalid for the modeling reference temperature. To consider theeffect of the temperature on the stack voltage, the deviationof the polarization curve dVT depending on the changes ofthe stack temperature from the reference temperature has beenmodeled. In the experimental prototype, the reference temper-

Fig. 6. Equivalent electrical circuit to obtain the polarization curve.

Fig. 7. Polarization curves for different λO2 values obtained by fitting andexperimental data.

ature is set to T0 = 35 ◦C, and the following deviation law hasbeen experimentally identified [17]:

dVT = kdV · (Tst − T0)

kdV ={

Tst > T0 → kdV = 0.138Tst ≤ T0 → kdV = 0.250.

(47)

The polarization curves obtained by means of the modelpresented in this paper have been compared with the experi-mental data: Fig. 7 shows the model prediction as well as theexperimental measurement data for different λO2 values.

VII. EXPERIMENTAL VALIDATION

The FC model previously described has been implementedin the PSIM environment as shown in Fig. 8. Table II helpsthe reader in relating the different blocks with the equationsdiscussed in the previous sections. The inputs of the model arethe load current Inet and the ambient temperature Tca,in, and theoutputs are the oxygen excess ratio λO2 and the stack voltageVst. The stack temperature Tst is calculated in the thermalmodel and can also be used for prediction purposes, but it isnot a main output variable.

Numerical results obtained by means of the developed modelhave been validated by means of experimental measurementson a 1.2-kW Ballard Nexa FC system. The model parametersused in this validation process are given in Table III.

The comparison between the experimental data and themodel response is shown in Fig. 9. The physical variablesexternally imposed to the real system and to the model havebeen the load current Inet and ambient temperature Tca,in.


Fig. 8. PSIM schematic of the Ballard FC.

TABLE IIBLOCKS AND THE CORRESPONDING EQUATIONS

IN THE FC MODEL (FIG. 8)

The applied Inet profile shown in Fig. 9(a) exhibits high- andlow-frequency transients that allow a realistic validation of themodel. The Tca,in behavior measured during the experiments isshown in Fig. 9(b).

The air-compressor dynamic and loss models have beencompared with the experimental measurements: Fig. 9(c) showsthe comparison between the losses predicted by the modeland the measured current consumption of the auxiliary sys-tems, with the model reproducing the main consumption only.Fig. 9(d) shows that the compressor dynamic model reproducesthe experimental air mass flow in a satisfactory way.

The good agreement between the model and the experimentalresults also holds for the oxygen excess ratio λO2 [Fig. 9(e)].Experiments have also been allowed to identify the λO2 profilefollowed by the control law implemented in the Nexa controlboard (48), which presents a static relation between the desiredλO2,Nexa and the stack current Ist. This relation gives a safeλO2 reference against oxygen-starvation phenomenon only insteady-state conditions and not during transient behavior. TheNexa board controller parameters are given in Table IV

λO2,Nexa =a3 · I3

st + a2 · I2st + a1 · Ist + a0

b1 · Ist + b0. (48)

Similarly, Fig. 9(f) shows the comparison between the stack-voltage values obtained from both simulations and experimentalmeasurements: A satisfactory fitting of both results is evident.

To provide an estimation of the model accuracy, an erroranalysis has been carried out by using the mean relative error(MRE) criterion [36] given in the following equation:

MRE(%) = 100 × 1N

N∑j=1

∣∣∣∣mj − fj

mj

∣∣∣∣ (49)

where mj and fj represent the experimental and estimated datasets, respectively, and N represents the number of samples.

The MRE criterion, applied to the data referred to the po-larization curves shown in Fig. 7, gives a relative mean error

TABLE IIIMODEL PARAMETERS

equal to 0.34%. The application of the same criterion to thedynamical responses shown in Fig. 9 gives the relative meanerrors shown in Table V. Such results confirm the accuracyof the model. The errors in the ancillary system consumptionand losses, compressor air flow, oxygen excess ratio, and stackvoltage are mainly due to the nonmodeled ripple componentaffecting those signals.

VIII. APPLICATION EXAMPLE

In addition to the virtual monitoring of the oxygen excessratio, the model proposed in this paper also allows performingevaluations of control strategies and dc/dc power-converterdesign. To enforce this feature, an example involving a linearcontroller regulating the oxygen excess ratio, designed accord-ing to the procedure proposed in [30], has been simulated. It al-lows defining a control strategy with a feedforward componentdepending on the stack current, thus giving a fast compensationof the net-current transients. This is specified in terms of thecompressor time constants, with the desired value being equalto ten. The controller also includes a feedback component toachieve null steady-state error

Vap(s) =2.674 · s + 6.077

s + 0.039· (λO2,ref − λO2(s))

+ 1.2 · Ist(s) + 40. (50)


Fig. 9. FC model experimental validation. (a) Load-current profile. (b) Ambient temperature. (c) Parasitic losses. (d) Air mass low profile. (e) λO2 profile.(f) Stack-voltage profile.

TABLE IVλO2,Nexa CONTROLLER IDENTIFIED PARAMETERS

TABLE VDYNAMICAL VALIDATION ERROR ANALYSIS (MRE)

The oxygen-excess-ratio controller, given in (50), defines thecompressor control signal Vap, depending on the stack currentand on the error in the oxygen excess ratio. In the experimentalmeasurements given in [30], it was also evidenced that themaximum oxygen excess ratio that can be set constant is λO2 =3.29; this is due to the saturation limit of the air compressor.This λO2 value is far from the oxygen-starvation limit λO2 ≥1 and also close to the high-efficiency λO2 trajectory. TheλO2 controller (50) was tested with the proposed model toregulate the oxygen excess ratio to 3.29 with the nonlinearnet-current profile shown in Fig. 10(a), which is typical ofinductive loads like the one represented by a switching dc/dcconverter. The regulated λO2 profile is shown in Fig. 10(c),where a satisfactory response to the net-current transients isobserved. In addition, Fig. 10(b) and (d) shows the stack

Fig. 10. λO2 control-strategy simulation. (a) Net current. (b) Stack voltage.(c) Oxygen excess ratio. (d) Compressor control signal.

voltage and compressor control signal profiles generated by thecontroller.

This application example shows the usefulness of the pro-posed model in the design and evaluation of FC control strate-gies, by using a standard power-electronics-oriented softwareplatform.

IX. CONCLUSION

In this paper, a new PEM FC dynamic model has beenintroduced. It is useful to design new strategies for controllingthe oxygen excess ratio, and it is also suitable for simulating


power-electronic circuits dedicated to FC systems. The modelincludes experimentally derived behavioral relations, electro-chemical equations, and a nonlinear circuit reproducing the FCpolarization curve. The model has been implemented in PSIMenvironment and also validated by means of experimental data,referred to a Ballard Nexa 1.2-kW FC power system.

The main characteristic of the proposed model is the interac-tion of electrochemical equations and circuital-based models;this is to provide an estimation of the oxygen excess ratiotogether with an electrical interaction with the load based onthe modification of the output impedance of the model.

The PSIM implementation of the FC model (Fig. 8) hasalso put into evidence the circuital connection between themodel and any electrical load, thus allowing the performanceof different analyses and simulations aimed at studying theinteraction between the FC system, the power-electronics inter-faces, and the controller of both of them. For example, it mightbe useful for the evaluation of the λO2 profiles generated bypower-electronics systems operation. Moreover, it might allowanalysis of the effects of the 120-Hz component, generatedin grid-connected applications, on the stack voltage and evenon λO2 that leads to oxygen-starvation (safety) considerations.Furthermore, the developed model might help in evaluating theeffects of the load dynamics on the FC control system andimproving the system efficiency by using optimal operationanalyses based on the λO2 trajectory reported in the literature[30]. The model can also be parameterized to take into accountthe aging effects into the FC impedance to predict such long-term deviations [37].

The model is also useful for online estimation of the oxygenexcess ratio, since it is possible to adjust the model parametersto real prototypes and run it in real-time systems, measuring theFC net current. In addition, using other variables like stack volt-age and temperature, the veracity of the oxygen-excess-ratioestimation can be evaluated, allowing, in this way, the design ofan oxygen-excess-ratio virtual sensor, which can be very usefulfor an FC safe experimentation and optimal operation.

Finally, the model evaluation shows a satisfactory agreementwith the experimental measurements, and the given applicationexample illustrates one of the main advantages of the model:the oxygen-excess-ratio estimation in power-electronics design,control, and simulation.

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Carlos Andrés Ramos-Paja (S’08) received theB.S. degree (with honors) in ingeniero electrónicoand the M.S. degree in ingeniería automática fromthe Universidad del Valle, Cali, Colombia, in 2002and 2005, respectively.

He was with the Departamento d’EnginyeriaElectrònica, Elèctrica i Automàtica, Escola TècnicaSuperior d’Enginyeria, Universitat Rovira i Virgilide Tarragona, Tarragona, Spain. Since August 2009,he has been with the Facultad de Minas, UniversidadNacional de Colombia, Medellin, Colombia, where

he is an Assistant Professor. His main research interests include design andcontrol of power systems for renewable energy sources.

Roberto Giral (S’94–M’02) received the B.S. de-gree in ingeniero técnico de telecomunicación, theM.S. degree in ingeniero de telecomunicación, andthe Ph.D. (with honors) degree from the UniversitatPolitécnica de Catalunya, Barcelona, Spain, in 1991,1994, and 1999, respectively.

He is currently an Associate Professor with theDepartamento d’Enginyeria Electrònica, Elèctrica iAutomàtica, Escola Tècnica Superior d’Enginyeria,Universitat Rovira i Virgili de Tarragona, Tarragona,Spain, where he works in power electronics.

Luis Martinez-Salamero (S’77–M’85) received theB.S. degree in ingeniero de telecomunicación andthe Ph.D. degree from the Universidad Politécnicade Catalunya, Barcelona, Spain, in 1978 and 1984,respectively.

From 1978 to 1992, he taught circuit theory, ana-log electronics, and power processing at the EscuelaTécnica Superior de Ingenieros de Telecomunicaciónde Barcelona. During the academic year 1992–1993,he was a Visiting Professor with the Center for SolidState Power Conditioning and Control, Department

of Electrical Engineering, Duke University, Durham, NC. During the academicyear 2003–2004, he was a Visiting Scholar with the Laboratoire d’Architectureet d’Analyse des Systémes (L.A.A.S) of the Research National Center (CNRS),Toulouse, France. He is currently a full Professor with the Departamentod’Enginyeria Electrònica, Elèctrica i Automàtica, Escola Tècnica Superiord’Enginyeria, Universitat Rovira i Virgili de Tarragona, Tarragona, Spain. Hehas published a great number of papers in scientific journals and conferenceproceedings and is a holder of a U.S. patent on the electric energy distributionin vehicles by means of a bidirectional dc–dc switching converter. His researchinterests are in the field of structure and control of power-conditioning systemsfor autonomous systems.

Dr. Martinez-Salamero was a Guest Editor of the IEEE TRANSACTIONS ON

CIRCUITS AND SYSTEMS (August 1997) for the Special Issue on Simulation,Theory and Design of Switched-Analog Networks. He was a DistinguishedLecturer of the IEEE Circuits and Systems Society in the period 2001–2002.He served as a President of the Joint Spanish Chapter of the IEEE PowerElectronics and IEEE Industrial Electronics Societies in the period 2005–2008.He is the Director of the Grupo de Automàtica y Electrònica Industrial, aresearch group on industrial electronics and automatic control, whose mainresearch fields are in power conditioning for vehicles, satellites, and renewableenergy.

Jenny Romano was born in Salerno, Italy, in 1983.She received the B.S. and M.S. degrees in me-chanical engineering from the University of Salerno,in 2005 and 2008, respectively.

From May to September 2008, she was on hertraining activity with the Erasmus Placement Pro-gramme in the Departamento d’Enginyeria Elec-trònica, Elèctrica i Automàtica, Escola TècnicaSuperior d’Enginyeria, Universitat Rovira i Virgilide Tarragona, Tarragona, Spain, where she workedon fuel-cell systems, control, and power electronics.

She is currently with the Department of Information and Electrical Engineering,University of Salerno, Fisciano, Italy.

Alfonso Romero (S’94–M’02) received the B.S.degree in ingeniero de telecomunicación and thePh.D. degree from the Universitat Politécnica deCatalunya, Barcelona, Spain, in 1994, and 2001,respectively.

He is currently an Associate Professor with theDepartamento d’Enginyeria Electrònica, Elèctrica iAutomàtica, Escola Tècnica Superior d’Enginyeria,Universitat Rovira i Virgili de Tarragona, Tarragona,Spain, where he is working in the fields of instrumen-tation and power electronics.

Giovanni Spagnuolo (M’98–SM’10) was born inSalerno, Italy, in 1967. He received the “Laurea”degree in electronic engineering from the Universityof Salerno, in 1993 and the Ph.D. degree in electricalengineering from the University of Naples FedericoII, Naples, Italy, in 1997.

Since 1993, he has been with the Dipartimentodi Ingegneria dell’Informazione ed Ingegneria Elet-trica, University of Salerno, where he was a Post-doctoral Fellow from 1998 to 1999, an AssistantProfessor of Electrotechnics from 1999 to 2003, and

has been an Associate Professor since January 2004. His main research interestsinclude the analysis and simulation of switching converters, circuit and systemsfor renewable energy sources, and tolerance analysis and design of electroniccircuits.

Dr. Spagnuolo is an Associate Editor of the IEEE TRANSACTIONS ON

INDUSTRIAL ELECTRONICS.

Documents

A PEM Fuel-Cell Model Featuring Oxygen-Excess-Ratio Estimation and Power-Electronics Interaction