interaction between proton exchange membrane fuel cells and power converters for ac integration

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Published in IET Renewable Power GenerationReceived on 27th June 2007Revised on 24th November 2007doi: 10.1049/iet-rpg:20070057

ISSN 1752-1416

Interaction between proton exchangemembrane fuel cells and powerconverters for AC integrationD.D. Marquezini D.B. Ramos R.Q. Machado F.A. FarretCentre of Studies in Energy and Environment, Electrical Post-graduation Program, Federal University of Santa Maria,Santa Maria, BrazilE-mail: [email protected]

Abstract: New aspects about steady-state and transient behaviours are considered in the relationships among themodel variables for fuel cell (FC) stacks that are of great interest in the scientific environment. In this way, themost important variables are voltage, current, power, heat from cooling system, membrane temperature andhydrogen pressure. The transient model aims at reproducing FC variations of its internal resistance underdistinct current levels. This current effect modifies the time response during load turning on/off conditions.Results from a modified FC model are presented and comparisons with real data are made. Additionally, theseresults are included in an analysis about the electrical interaction between FCs and converters as a cause ofgreat concern among power electronics designers. As the number of such converters has significantlyincreased in the last few years, FC generation systems are steadily calling attention for operational problemsrelated to their efficiency, stability and durability when DC–DC converters are connected across its terminals.Therefore design proceedings of a DC–DC converter associated to ‘T’ filters to avoid fast current transitionscaused by converter connection across the FC stack terminals are included. To deliver the energy produced bythe FC system to the grid, it is presented, also, an analysis of a DC–AC converter used to improve powerquality when the FC is, simultaneously, supplying load and grid.

1 IntroductionApplications of FC stacks include essential activities such asbank system operations, military defence, industry processingand medical care areas [1, 2]. However, this kind ofrenewable energy source produces low DC voltage,meaning that it must be boosted to a minimal level, whichis usually converted to AC voltage in residential andindustrial applications.

Similar to galvanic cells, FCs are electrochemical-baseddevices that convert energy from a hydrogen-rich fueldirectly into electricity and operate as long as they aresupplied with fuel. FCs also have many advantages sincethey have low level emissions, and almost none of sulphurand nitrogen compounds are released into the atmosphere.Natural gas, coal-derived gas, landfill gas, bio-gas or

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alcohols are some of the fuels to be used in theseequipments. According to [1, 3–5], proton exchangemembrane FCs (PEMFCs) are specially attractive forautomotive and static applications because of their higher-power density and lower temperature compared with otherkinds of FCs. Beyond environmentally friendly, this highlyefficient energy source is promising new trends in energygeneration. However, this technology is not mature yetleading customers to doubts about some feasibility points[6] such as the high installation costs of an FC power planthindering so its widespread deployment [7]. FC technologystill demands for a lot more research in spite of its growinguse either for stationary or mobile applications [8].

Currently, the FC production costs are decreasing [1], andthey have nearly reached the ordinary energy costs, thereforecompetitive with some local utility rates. As further assistance

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to its costs reduction, the price of ancillary devices such as airand water compressors, refrigerant equipment and powerconverters for FC systems is also decreasing, while,simultaneously, it is expected increased efficiency, reliabilityand suitable power quality levels [3] as energy sources. Alow-cost approach in inverter designs would be able toexpand the commercialisation of small-scale FC systems.Consequently, it would encourage the development offlexible distributed power systems. This paper focuses alsoon the FC behaviour in the presence of converters reflectedon their static characteristics, dynamical behaviour underdifferent load surges and effects of power convertersconnected across the FC terminals. Additionally, a DC–AC converter for connection to the grid together with theFC system and its DC–DC converter are presented.

This paper is organised as follows. Section 1 provides abrief revision of the modern FCs status. Section 2 presentsa PEMFC modified model suitable for power systems.Section 3 gives static simulations from the proposed model.Section 4 shows the dynamic behaviour of the modifiedalgorithm. Section 5 explains about DC–DC and DC–AC interfaces. Section 6 demonstrates the dynamicbehaviour of an FC interacting with a DC–DC converter,Section 7 provides discussions over the main results andSection 8 concludes.

2 Modified FC modelA PEMFC converts the chemical energy of a fuel, commonlyhydrogen (H2), and an oxidiser, oxygen (O2) or air, intoelectrical energy. On one side of the cell, referred to asanode, the fuel is supplied under certain pressure. Fuel forthis model is a pure gas H2, although other compositionsof gases can be used. In these cases, the hydrogenconcentration should be determined in the mixture. Thefuel spreads through the electrode until it reaches thecatalytic layer of the anode where it reacts to form protonsand electrons, as shown below in the reaction [1, 3, 4, 9–11]

H2 ! 2Hþþ 2e� (1)

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Protons are transferred through the electrolyte (solidmembrane) to the catalytic layer of the cathode. On theother side of the cell, the oxidiser flows through the flowfield channels of the plate, and it spreads through theelectrode until it reaches the catalytic layer of the cathode.The oxidiser used in this model is air or pure O2. Theoxygen is consumed with the protons and electrons and theproduct, liquid water, is produced with residual heat on thesurface of the catalytic particles. The electrochemicalreaction that happens in the cathode is

2Hþþ 2e� þ 1

2O2 ! H2Oþ heat(2)

Then, the full physical–chemical FC reaction is

H2 þ 1

2O2 ! þH2Oþ heatþ electricalenergy(3)

According to previous studies, feeding the FC with H2 andO2 a quantity of energy is produced [1, 3, 5, 9, 12, 13]. Inthe equivalent electrical circuit of Fig. 1a representing theFC chemical formulation, the activation voltage (Vact) isdefined via Tafel equation and the concentration voltage(Vcon) expresses the mass transport effects, representing thenonlinear behaviour of the current through the FCterminals by (4) and (5) [3, 10]. Voltage Vact is obtainedwith constant parametric coefficients ji for i ¼ 1, . . . , 4 [3,5, 9, 13], the FC current (IFC), oxygen concentration onthe anode (C�

O2) and temperature (T ). The ji coefficients

give an additional degree of freedom to the FC model,since they are based on fitting procedures over measuredvalues obtained from the polarisation curve of the FC stack[3, 9]. In this way, using (5), it is possible to determine thevoltage because of the reactant concentration, where ln, Ia,J, Jmax and B are, respectively, the natural logarithm, currentthat flows through the Ra resistance, actual FC current

Figure 1 FC representation

a Electrical model of an FC stackb Dynamic FC flow chart based on the electrochemical model

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density, maximum current density and the Tafel slope.

Vact ¼ �bj1 þ j2 � T þ j3

� T � ln (C�O2)þ j4 � T � ln (Ia)c (4)

Vcon ¼ �B � ln 1�J

Jmax

� �(5)

As (4) and (5) represent the nonlinear effects caused by thechemical process, it is possible to obtain the FC parameterstaking into account the nonlinear characteristic representedby the previous equations and thus, it is possible to write Ra

in function of Vact, Vcon and Ia differently that what it ispresented in literature [1, 3, 5, 9, 14] considering theresistance Ra as a constant term. As Ra is a function of (Vcon,Vact and Ia) and as Vact is time-variant, the Ra resistanceshould present the same characteristic. In this way, the Ra

value must be updated whenever the load changes its value (6).

Ra ¼Vcon þ Vact

Ia¼ Rcon þ Ract (6)

To explain (6) more clearly, Fig. 1a shows the conventionaldynamic model of a PEMFC whose correct calculationflowchart is given in Fig. 1b. This diagram allows theestablishment of the state space electrochemical model of anFC stack, as given in (7) [4]

Eo ¼ IFC � (Ra þ Rohm þ Rload)� Ca � Ra �dVd

dt(7)

where Rohm and Ra represent the ohmic and activation dropsplus concentration losses, and Ca is the capacitance because ofthe charge double layer (CDL), respectively [1, 3, 5, 9, 10, 12,13]. By representing (7) as a state space model, the result is (8).

dVd

dt¼

IFC � (Ra þ Rohm þ Rload)

Ca � Ra

�Eo

Ca � Ra

(8)

To understand exactly what this modified calculation algorithmproposes it is necessary to observe references [3, 9, 15] and verifytheir proposed block diagram. In those references, it was used a

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series representation of the FC dynamic circuit (Rohm, Ra andCa), meaning that the current circulating through Ra and Ract

is IFC. However, the calculation algorithm proposed in thispaper takes more appropriately the solution of the parallelcircuit, exactly as presented in Fig. 1b. Splitting the IFCcurrent is used to make more precise the algorithmrepresenting the FC dynamic behaviour (6).

Effectiveness of the modified algorithm for a 500 W SR-12 FC stack was evaluated comparing simulations andexperimental results to observe the FC behaviour when aload step is applied across its terminals. Table 1 presentsthe estimated values of an SR-12 FC stack using intrinsicparameters in the simulation test. More accurateapproximations to j1, . . . , i parameters were obtained byfitting it on the results from previous simulations.Additional equations are presented in Appendix 1.

3 FC steady-state behaviour model(model validation)With the algorithm presented in Section 2, several tests wereperformed using the FC steady-state behaviour. First, acomparative study between experimental and simulatedresults using this modified calculation algorithm as explainedabove is presented, Fig. 2a. To perform this test, the inputdata required for model implementation are given in Table 1.

In all tests, the maximum measured error between theexperimental and simulated results was lower than 3.0%.However, more evident difference is observed when themodified calculation algorithm is compared with the resultsgiven in [3]. Usually, these models present a significant errorwhen the concentration region of the output characteristic isreached, but a minimal error is observed by using themodified algorithm when IFC reaches 35 A (concentrationregion). The discrepancy between the modified algorithm andthe technique presented in [3] is because of the calculationmethod used. In [3] the FC current is not split through thecapacitance and the activation resistance. In this way, all IFC

Table 1 Parameters of the 500 w SR-12 FC

Parameter Value Parameter Value

T, K 333.15 j1 20.948

A, cm2 62.50 j2 0.0002 � ln Aþ (4.3 � 1025) � ln C�H2þ 0.00286

l, m 250 � 1026 j3 7.60 � 1025

PO2

�, Pa 26344.50 j4 21.930 � 1024

PH2

� , Pa 131723.0 c 16.0

RC, V 0.002 J, mA/cm2 22.0

B, V 0.20 Jmax, mA/cm2 672.0

n 48.0 Ca, F 0.0072

The pressure values must be converted from Pa to atm in the equations of the FC model.

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Figure 2 Static behaviour of an SR-12

a SR-12 polarisation curveb Simulated output power of an SR-12 stack

values were used to calculate Vd, whereas in the modifiedalgorithm only the proper fraction of the IFC current was used.

Following this idea, it is possible to plot the FC outputpower as in Fig. 2b.

4 FC dynamic behaviour modelBecause of the argument presented in Section 3, a load stepwas used to analyse the electrical behaviour across theFC terminals. Experimental and simulated tests were usedas terms of comparison and were performed in a MatLab/SIMULINKTM environment.

4.1 Circuital analysis

As the FC has a nonlinear characteristic, distinct time responsesin terms of IFC, VFC and consequently, PFC behaviour areobserved when the FC is submitted to a load step. In thisway, it is possible to observe two different circuits: one whencharging the equivalent capacitance and the other whendischarging it. Because of the chemical process, when the loadis inserted, the effective value of Ra changes from a high to alow value and when the load is disconnected, the effective Ra

changes from a low to a high value leading the rated value ofRa to reach more than a 1000.0% variation. These transitionscan be represented by a first-order behaviour and allow betterunderstanding of the electrochemical process through adetailed circuital analysis (Fig. 3). When the load is connectedacross the FC terminals, there is a fast reduction on the Ra

resistance which is caused by the gas combination. In thesame way, Rohmþ Rload changes also from a high to a lowvalue when the load is connected. This effect turns theequivalent resistance a much lower value than Ra so chargingthe capacitor Ca more rapidly (9).

Req ¼ (Rohm þ Rload)==Ra (9)

After that, the load is removed and a new value of the equivalentresistance to discharge Ca found as in (10). As the equivalentresistance, in this case, is determined exclusively by Ra andbecause of the recombination process inside the FC, a slower


time response is observed once Ra changes from a lower to ahigher value, Fig. 3.

Req ¼ Ra (10)

4.2 Comparison between simulated andexperimental results

Now, a discussion between simulated and experimental tests iscarried out. Time ranges from 0.0 to 2.0 s, which is sufficient toovercome the initial transient state representing the FCwarming-up period. Fig. 4a shows that the current responsegoes to about 43.0% above the steady-state current throughthe FC terminals when the rated load is connected.Regarding the VFC time response (Fig. 4b), it took about60.0 ms to achieve the steady-state value after the rated loadwas applied. For this reason, a power surge of 198.0%appears across the FC terminals (Fig. 4c). Power surges likethis may cause heavy stress on the membrane electrodeassemblies, Fig. 4c. On the other hand, when the load wasremoved, the voltage VFC took about 150.0 ms to achieve theopen-circuit voltage. These phenomena can be explained byusing an equivalent circuit to represent the electrochemicalprocess occurring when load transitions are imposed acrossthe FC terminals, illustrated in Fig. 3.

Figure 3 Percentage of the resistance Ra


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Figure 4 Dynamic behaviour of an SR-12

a FC output currentb FC output voltagec FC output power

To verify the modified algorithm, a comparison betweensimulated and experimental (Figs 4 and 5) results forcurrent, voltage, power and time errors was made. Timeerrors are understood as the time difference betweensimulated and experimental results. When the load isturned on the time error was 9.6%, whereas 4.0% of timeerror was obtained when the load is turned off.

The error of voltage across the FC terminal is lower than2.0% in both tests. The same happens to the FC currentand its error is lower than 4.0%. However, the worst caseof stack power mismatch is an 8.5% error.

5 DC–DC interfaceA step-up converter using a proportional–integral (PI)current controller in continuous conduction mode waschosen as the DC–DC interface and control technique,respectively, because of the wide variation of the FCparameters. Current mode was used to allow the selectionof an optimal point of power delivery for a given set ofoperating conditions by adjusting only the FC current.

In Fig. 6 it is presented the current control block diagramwhere it is used a simplified boost converter transfer function(11). Variables IL2

and d represent the oscillations aroundthe middle point of the boost current and duty cycle,respectively.

~I L2

~dffi

VDC

s � L2

(11)

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where, PIIL2 corresponds to the PI subroutine to control theboost converter current and PWMIL2

is the pulse wavemodulation routine, respectively. According to (12) and(13) phase margin (mfFC) and cut-off frequency (vFCLFC)were used to obtain the PI parameters (kpropFC and kintFC)[16–20].

kpropFC �GOLFC

vFCLFC

¼ 1 (12)

kintFC ¼ kpropFC �vFCLFC

tan(mfFC)(13)

where GOLFC is the open loop gain.

Figure 5 Experimental results when the stack is submittedto a load variation From top to bottom: M, FC outputpower (200.0 W/div.); CH1, FC output voltage (20.0 V/div.); CH2, FC output current (10.0 A/div.)Horizontal:20.0 ms/div

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Figure 6 Block diagram of the system

6 Interaction between the FCstack and converterTaking into account the previous sections, a DC–DCconverter working in continuous conduction mode waschosen as the interface between the FC stack and the DClink of the DC–AC converter. To obtain a low-voltageripple across the FC terminals, a high-order L � Cnetwork (‘T’ filter) was inserted. Fig. 6 shows theconnection between the FC stack and the boost converter.In this circuit Eo, Rohm, Ra and Ca are the FC parameterswhereas, L1, L2, CFC (boost or ‘T’ filter parameters) wereused to minimise the current ripple through the FC stackterminals whereas, RL2

, RL1, RFC are the most important

parasitic components of the ‘T’ filter.

6.1 ‘T’ filter design criteria

Design of the passive components for the ‘T’ filter isaccording to the parameters showed in Table 2. Theseparameters are selected from the FC characteristics andusing a simplified transfer function for the step-upconverter, the inductance L2 may be represented by (14) [21].

L2 ¼VFC

DIL2� fs

�V 2FC

DIL2� VDC � fs

(14)

where, DIL2is the current ripple.

In this case, DIL2represents 20.0% of the nominal FC

current. The capacitance CFC (15) is defined using the ‘T’

Table 2 Parametersof the ‘T’ filter design

Po, W 2.0 � 103

fs, Hz 12.0 � 103

VFC, V 20.0–40.0

VDC, V 300.0

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filter cut-off frequency ( f ), for at least one decade belowthe switching frequency. In this case, f is 500.0 Hz.

CFC ¼(1=(2� p� f ))2

L2

(15)

The next step is to design the inductance L1. This passivecomponent needs to follow an important rule since itreduces most of the current ripple from its original 20.0%with L2 to 2.0% when including the branch L1 � CFC.The design was based on an adapted equation of thecascade filter as in (16) [22].

L1 ¼

ffiffiffi24

p

ffiffiffi3

p �1ffiffi

rp

� 16� p 2 � f 2 � CFC

(16)

where r represents the percentage current ripple through theFC terminals.

Based on (14)–(16) and taking into account theparameters presented in Table 2, it is possible to obtain the‘T’ filter parameters as the ones presented in Table 3.

6.2 Transfer function including boostconverter, FC source and ‘T’ filter

To attend the FC steady-state features, the proposedconverter is controlled in current mode. This is necessarybecause the FC stack will have its best performance and itwill reach its selected generation power when it works inthe linear region of the V–I polarisation curve. A typicalapplication to this system is its connection to the grid. The

Table 3 Calculated ‘T’ filter parameters and its parasiticvalues

Parameter Value Parameter Value

L1, H 400.0 � 1026 L2, H 1.0 � 1023

RL1, V 0.15 RL2, V 0.25

RFC, V 0.50 CFC, F 100.0 � 1026


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Figure 7 Root locus of the system

a Open loop transfer function (at rated load)b Open loop transfer function (at no-load)

common bus voltage (VDC) is considered constant, and itmust be controlled by a DC–AC interface [23, 24].

The state–space averaging was used to obtain the smallsignal transfer function (17) current/duty cycle (IL2

/d )when the converter operates in the continuous conductionmode. Variables N3, . . . , N1 and D3, . . . , D0 are thetransfer function coefficients used in (17) and defined inAppendix 2.

~I L2

~d¼

VDC � [N3 � s3 þ N2 � s2 þ N1 � s þ 1]

[D4 � s4 þD3 � s3 þD2 � s2 þD1 � s þD0]

(17)

6.3 Root locus of a PEM stack andDC–DC converter plus ‘T’ filter

According to Fig. 4, it is possible to observe the widevariation of Ra. This situation makes poles and zeros of thetransfer function to change their position enough to causesome troubles when adjusting the controller parameters ofthe DC–DC converter.


To understand this behaviour, a root locus from the no loadto the full load condition was plotted, as in Figs 7a and 7b.Despite that the transfer function presents high-order terms(Appendix 2), only a dominant zero is close to theimaginary axis and its variation does not cause problemsonce exclusively the zero moves nearer to the instable regionbehaviour. As expected, it is possible to simplify (17) totransform it in a first-order transfer function making easierthe application of the design criteria for the current controller.

6.4 DC–AC control diagram

The simplified transfer function of the proposed DC–ACconverter is defined as (1/s � Lconv). This approximation ispossible when the system is connected to the grid and itsimpedance is higher than the load and the capacitive filterimpedance [4]. To control the VDC voltage a power-balancing method was used. The current reference may beprovided either from the utility voltage waveform orthrough a phase-locked loop-based synchronism algorithm.The block diagram of Fig. 8 allows a definition of the PIparameters (kpropx and kintx) based on the open loop gainGOLx, the closed loop cut-off frequency FCLx and the

Figure 8 AC control block

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phase margin mfx, given respectively by equations (18) and(19) [16, 17, 19, 20]. The sub-index ‘x’ represents the ACcurrent (i) and the control voltage VDC [4].

kpropx ¼FCLx

GOLx

� 2� p (18)

kintx ¼kpropx � 2p� FCLx

tan(mfx)(19)

7 ResultsSimulated and experimental results were performed to shownthe controller effectiveness. A sinusoidal PWM control at12.0 kHz was used as the switching technique for bothconverters. The experimental set-up was built up using an

Figure 9 AC circuit


ADSP 21992 from the analogue devices as its digitalcontrol interface. Fig. 9 shows the system diagram wherethe FC is represented by a current source and the DC–ACconverter is connected to the grid, and to the AC linear load.

Initially, the AC system is providing all energy to the feeder.When a high-load power is connected across the feeder

Figure 10 Simulated results

a Current delivered or absorbed from the feeder and the three-phase load currentb Currents through inductance L2 and inductance L1, respectively

Figure 11 Experimental results

a Current through inductance L2b Current through inductance L1Vertical: 1.0 A/div; Horizontal: 25.0 ms/div


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terminals its energy is provided by the FC system and only afraction of the energy is supplied by the grid. In this case, atthe instant of load connection it is observed a reversion in thegrid current direction (Fig. 10a).

The ripple current through the FC stack terminals was lowerwhen the ‘T’ filter is connected, as shown in Fig. 10b. A ‘T’filter not only reduces the current ripple from 20.0 to 2.0%but also avoids the fast current transitions through the FCterminals during the load connections.

A low-power test was performed to observe the capabilityof the step-up converter associated with the ‘T’ filter. In thistest it is possible to observe a high increase in the IL2

(currentthrough L2) ripple of more than 100.0%. In the other side ofthe ‘T’ filter, the IL1

(current through L1) ripple isinsignificant. It demonstrates the effectiveness of the ‘T’filter to reduce the ripple current and transients, as shownin Figs 11a and 11b.

8 ConclusionsThis paper presents an improved calculation algorithm tosimulate the dynamic and static states of the PEM FCoperation. Validation of the FC stack model is presented asa comparison term between the behaviours of the FCsteady and dynamic states using this improved calculationalgorithm. Simulated and experimental results show thevariations of the time coefficients used during loading andunloading the FC stack.

Additionally, it was discussed the interaction between thePEM FC stack and the converter under distinct loadconditions. In the first approach; the FC, the ‘T’ filter andthe boost converter were analysed under the small signalpoint of view. With the electrical model proposed in thispaper, it is possible to verify that the FC parameters arechanging under load variations and mostly that the systemis stable when the FC stack changes from the no-load tothe full-load condition.

9 AcknowledgmentsThe authors would like to acknowledge Dr Ing. YalesR. Novaes for supplying experimental data for the FCtransient states and to CNPq and CAPES for their financialsupport of this project.

10 References

[1] LARMINIE J., DICKS A.: ‘Fuel cell systems explained’ (JohnWiley and Sons, 2000, 2nd edn. edn.)

[2] NOVAES Y.R., ZEPELINI R.R., BARBI I.: ‘Design considerations ofa long-term single-phase uninterruptible powersupply based on fuel cells’. Proc. 36th Power

Renew. Power Gener., 2008, Vol. 2, No. 3, pp. 151–161: 10.1049/iet-rpg:20070057

Electronics Specialist Conf., Recife, Brazil, June 2005,pp. 1628–1634

[3] CORREA J.M., FARRET F.A., CANHA L.N., SIMOES M.G.: ‘Anelectrochemical-based fuel cell model suitable forelectrical engineering automation approach’, IEEE Trans.Ind. Electr., 2004, 51, (5), pp. 1103–1112

[4] HIRSCHENHOFER J.H., STAUFFER D.B., ENGLEMAN R.R., KLETT M.G.:‘Fuel cell handbook’ (Parsons Corporation, 2005,6th edn.)

[5] PATHAPATI P.R., XUE X., TANG J.: ‘A new dynamic model forpredicting transient phenomena in a PEM fuel cellsystem’, Renew. Energy, 2004, 30, (1), pp. 1–22

[6] HELLMAN H.L., VAN DEN HOED R.: ‘Characterising fuel celltechnology: challenges of the commercialisation process’,Int. J. Hydrog. Energy, 2007, 32, (3), pp. 305–315

[7] MILLETT S., MAHADEVAN K.: ‘Commercialization scenariosof polymer electrolyte membrane fuel cell applicationsfor stationary power generation in the United States bythe year 2015’, J. Power Sources, 2005, 150, (1),pp. 187–191

[8] AGNOLUCCI P.: ‘Economics and market prospects ofportable fuel cells’, Int. J. Hydrog. Energy, 2007,doi:10.1016/j.ijhydene.2007.03.042

[9] AL-BAGHDADI M.A.R.: ‘Modelling of proton exchangemembrane fuel cell performance based on semi-empirical equations’, Renew. Energy, 2005, 30, (10),pp. 1587–1599

[10] REGGIANI U., SANDROLINI L., GIULIATTINI BURBUI G.L.: ‘Modellinga PEM fuel cell stack with a nonlinear equivalent circuit’,J. Power Sources, 2007, 165, (1), pp. 224–231

[11] XUE X.D., CHENG K.W.E., SUTANTO D.: ‘Unified mathematicalmodelling of steady-state and dynamic voltage-currentcharacteristics of PEM fuel cells’, Electrochim. Acta, 2006,52, (3), pp. 1135–1144

[12] CERAOLO M., MIULLI C., POZIO A.: ‘Modellingstatic and dynamic behaviour of proton exchangemembrane fuel cells on the basis of electro-chemicaldescription’, J. Power Sources, 2003, 113, (1),pp. 131–144

[13] COSTA R.A., CAMACHO J.R.: ‘The dynamic and steadystate behaviour of a PEM fuel cell as an electricenergy source’, J. Power Sources, 2006, 161, (2),pp. 1176–1182

[14] HADDAD A., BOUYEKHF R., MOUDNI A.E., WACK M.: ‘Non-lineardynamic modelling of proton exchange membrane fuelcell’, J. Power Sources, 2006, 163, (1), pp. 420–432

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[15] PAGANO M., PIEGARI L.: ‘Electrical networks fed by fuel-cells for uninterruptible electrical supply’. Proc. 2002 IEEEInt. Symp. Industrial Electronics, L’Aquila, Italy, May 2002,pp. 953–958

[16] MACHADO R.Q., BUSO S., POMILIO J.A.: ‘A line-interactivesingle-phase to three-phase converter system’, IEEE Trans.Power Electr, 2006, 21, (6), pp. 1628–1636

[17] MATTAVELLI P.: ‘A closed-loop selective harmoniccompensation for active filters’, IEEE Trans. Ind. Appl,2001, 37, (1), pp. 81–89

[18] ZENITH F., SKOGESTAD S.: ‘Control of fuel cell power output’,J. Process Control, 2007, 17, (4), pp. 333–347

[19] MARWALI M.N., KEYHANI A.: ‘Control of distributedgeneration systems Part I: voltage and current Control’,IEEE Trans. Power Electr, 2004, 19, (6), pp. 1541–1550

[20] MARWALI M.N., JUNG J.-W., KEYHANI A.: ‘Control of distributedgeneration systems Part II: voltage and current Control’,IEEE Trans. Power Electr, 2004, 19, (6), pp. 1551–1561

[21] MOHAN N., UNDELAND T.M., ROBBINS E.P.: ‘Power electronics:converters, applications and design’ (John Wiley andSons, 1989, 2nd edn.)

[22] MILLMAN J., HALKIAS C.C.: ‘Electronic devices and circuits’(McGraw-Hill Book Company, 1967, 1st edn.)

[23] LEE J., JO J., CHOI S., HAN S.-B.: ‘A 10-kW SOFC low-voltagebattery hybrid power conditioning system forresidential use’, IEEE Trans. Energy Convers, 2006, 21, (2),pp. 575–585

[24] ERICKSON R.W., MAKSIMOVIC D.: ‘Fundamentals of powerelectronics’ (Kluwer Academic Publishers, 2001, 2nd edn.)

[25] BUSO S., FASOLO S., MATTAVELLI P.: ‘Uninterruptible powersupply multiloop control employing digital predictivevoltage and current regulators’, IEEE Trans. Ind. Appl,2001, 37, (6), pp 1846–1854

11 Appendix 1: additionalequations used in the FC modelIn this appendix additional equations used in theimplemented FC software can be found.

C�H2

¼P�H2

1:09� 106 � e(77=T )(20)

C�O2

¼P�O2

5:08� 106 � e(�498=T )(21)

where the parameter B is given in Table 1, and the terms

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J and Jmax are defined in Appendix 3.

rM ¼181:6� [1þ 0:03� J þ 0:062� (T=303)2 � ( J )2:5]

[c� 0:634� 3� J ]� e(4:18�((T�303)=T ))

(22)

RM ¼rM � l

A(23)

Vohmic ¼ IFC � (RM þ RC) (24)

EO ¼ 1:229� 0:85� 10�3� (T � 298:15)

þ 4:31� 10�5� T � ln (P�

H2)þ

1

2� ln (P�

O2)

� �(25)

12 Appendix 2: coefficientsdefinitionThis appendix gives the definitions for the coefficients usedin (17).

R0¼ Rohm þ RL1

N1 ¼ CFC � (RFC þ R0þ Ra)þ Ra � Ca

N2 ¼ CFC � (L1 þ (RFC þ R0)� Ra � Ca)

N3 ¼ CFC � L1 � Ra � Ca

D0 ¼ R0þ Ra þ RL2

D1 ¼ L1 þ L2 þ CFC

� ((Ra þ R0)� (RFC þ RL2)þ RFC � RL2

)

þ Ra � Ca � (R0þ RL2

)

D2 ¼ Ra � Ca � (L1 þ L2 þ CFC

� (RFC � (RL2þ R0)þ R0

� RL2))

þ CFC � (L2 � (Ra þ RFC þ R0)

þ L1 � (RL2þ RFC))

D3 ¼ CFC � (L1 � L2 þ Ra � Ca

� (L2 � (RFC þ R0)þ L1 � (RL2þ RFC)))

D4 ¼ L1 � CFC � L2 � Ra � Ca

13 Appendix 3: list of symbolsA cell active area, cm2

B Tafel slope

Ca equivalent capacitance because of thecharge double layer, F

Cconv capacitance of the output filter, F

CDC DC link capacitance, F

CFC ‘T’ filter capacitance, F


IETdo

www.ietdl.org

CH2

� effective hydrogen concentration atthe anode catalyst sites, mol/cm3

C �O2

effective oxygen concentration at thecathode catalyst sites, mol/cm3

Eo Nernst potential, V

fS switching frequency of the boostconverter, Hz

Ia current that flows through the Ra

resistance, A

iALconv, iCLconv

instantaneous currents that flowthrough the output filter, A

ICacurrent through the charge doublelayer, A

IFC FC output current, A

I �Lconv

,ILconvcurrents that flow through the outputfilter, A

IL1, IL2

currents that flow through the ‘T’ filterinductances, A

J current density, A/cm2

Jmax maximum current density, A/cm2

klem, kvolt constants transformation

l thickness of the membrane, m

Lconv inductance of the output filter, H

L1, L2 ‘T’ filter inductances, H

n number of cells

Po maximum output power of the boostconverter, W

P �H2

partial pressure of hydrogen in theanode channel, Pa

P �O2

partial pressure of oxygen at the water-gas interface in the cathode channel, Pa


PFC total power produced by the FC,including the power spent by theinternal losses, W

Ra equivalent membrane resistance, V

Ract resistance of activation, V

RC equivalent contact resistance toelectron conduction, V

Rcon resistance of concentration, V

Req equivalent resistance to charge/discharge Ca, V

Rload load resistance, V

RM equivalent contact resistance to protonconduction, V

Rohm Ohmic losses, V

RL1, RL2

, RFC ‘T’ filter parasitic resistances, V

T temperature, K

Vact activation voltage, V

vA waveform from the utility grid, V

Vcon concentration voltage, V

Vd dynamic voltage over the capacitanceCa, V

V �DC, VDC, vDC DC link voltages (average and

instantaneous value), V

VFC FC output voltage, V

Vohmic Ohmic voltage, V

Vrms rms value of the utility grid voltage, V

Zload//grid equivalent impedance, V

rM Nafion membrane resistivity,V � cm

j1, j2, j3, j4 semi-empirical coefficients forcalculation of activation voltage

161


interaction between proton exchange membrane fuel cells and power converters for ac integration

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