Solid Oxide Fuel Cell Modeling: IEEE Transactions On Industrial Electronics January 2009
Solid Oxide Fuel Cell Modeling: IEEE Transactions On Industrial Electronics January 2009
Solid Oxide Fuel Cell Modeling: IEEE Transactions On Industrial Electronics January 2009
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Abstract—This paper discusses the modeling of a solid oxide the effects of heat/mass transfer and electrochemical reactions
fuel cell using both lumped and distributed modeling approaches. in each CV.
In particular, the focus of this paper is on the development of a Detailed analyses of interconnecting a fuel cell to a grid
computationally efficient lumped-parameter model for real-time
emulation and control. The performance of this model is compared are presented in [11]–[15]. References [11]–[14] focus on the
with a detailed distributed model and experimental results. The design and control algorithms of the power conditioning system
fundamental relations that govern a fuel cell operation are utilized only. Reference [15] includes the fuel cell model based on
in both approaches. However, the partial pressure of the species empirical equations derived from experimental data of an actual
(fuel, air, and water) in the distributed model is assumed to vary fuel cell.
through the length of the fuel cell. The lumped model approach
uses the partial pressure of the species at the exit point of the However, the complexity and computation time associated
fuel cell. The partial pressure of the species is represented by an with these models are the drawbacks for real-time emula-
equivalent RC circuit in the lumped model. tion and control applications. This paper therefore focuses
Index Terms—Fuel cells, modeling, simulation. on a lumped-modeling approach using electrical components
in PSpice and/or Matlab/Simulink for real-time applications,
and parameter estimation. It also allows the study of the
I. I NTRODUCTION performance and reliability of the SOFC under various flow
rates and load conditions. Moreover, the modeling takes into
A SOLID OXIDE fuel cell (SOFC) converts chemi-
cal energy into electrical energy at high temperature
(800 ◦ C–1000 ◦ C), in contrast to a PEM fuel cell that op-
consideration the effects of reactant and product concentrations,
polarization losses, and the effects of internal (inherent) resis-
erates at a lower temperature (80 ◦ C–100 ◦ C). The SOFC is tances. Steady-state simulation results of the SOFC model were
a promising technology for distributed power generation with compared with experimental data to validate the model.
high efficiency and no moving parts. The transient and static This paper is organized as follows. Section II is a background
model of an SOFC, which takes into account the effect of study of fuel cell operation, in particular SOFC, followed by
electrochemical, thermal, and mass flow, is proposed in [1]. In two modeling approaches—a distributed model incorporating
[2]–[4], dynamic models of SOFCs are developed for analyzing detailed calculation of all phenomena and a lumped model us-
power system performance and fuel cells. A more compre- ing electrical components in Section III. The simulation results
hensive mathematical model of an SOFC is conducted in [5]. and experimental data are presented in Section IV. Finally,
It estimates the parameters for a 1-D cathode microstructure Section V discusses the conclusions.
SOFC model, such as the composition and particle size. A zero-
dimensional model is presented in [6] to show the limitation II. SOFC O PERATION
of the empirical assumptions derived from observation and
measurements of physical process. The Butler–Volmer equation A fuel cell (SOFC) generates electrical power by contin-
analysis for approximating the activation losses in SOFC mod- uously converting chemical energy of a fuel into electrical
els is presented in [7]. These papers do not include a detailed energy through an electrochemical reaction. The fuel cell itself
analysis of all the losses in a fuel cell. A dynamic fuel cell has no moving parts, making it quiet and reliable. Fuel cells
model, which uses a similar approach to that in [2], is proposed typically utilize hydrogen as the fuel and oxygen (usually
in [8]. Reference [9] focuses only on the effect of polarization from air) as the oxidant in the electrochemical reaction. It
losses of an anode-supported SOFC for various cell parameters generates electricity, and its by-products are water and heat.
such as the geometry of the cell. A dynamic model of the SOFC, The system has higher efficiency compared to conventional
where a single cell is divided into small control volumes (CVs), combustion engines [16], because it is not limited by Carnot
is presented in [10]. It is a detailed model that accounts for both efficiencies. The electrochemical reactions that occur in an
SOFC that utilize fuel (hydrogen) and air (oxygen) [1]–[10] are
as follows.
Manuscript received November 9, 2006; revised May 1, 2008. First pub-
lished November 18, 2008; current version published December 30, 2008. This Anode:
work was supported by NanoDynamics, Inc.
A. Gebregergis and P. Pillay are with the Department of Electrical and H2 + O2− − > H2 O + 2e− . (1)
Computer Engineering, Clarkson University, Potsdam, NY 13699-5720 USA
(e-mail: natiabraham@gmail.com; pillayp@clarkson.edu).
D. Bhattacharyya and R. Rengaswemy are with the Department of
Cathode:
Chemical Engineering, Clarkson University, Potsdam, NY 13699-5707 USA
1
(e-mail: debangsu@clarkson.edu; raghu@clarkson.edu). O2 + 2e − > O2− . (2)
Digital Object Identifier 10.1109/TIE.2008.2009516 2
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140 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 1, JANUARY 2009
Overall reaction:
1
H2 + O2 − > H2 O. (3)
2
The stoichiometric ratio of hydrogen to oxygen is 2 : 1.
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GEBREGERGIS et al.: SOLID OXIDE FUEL CELL MODELING 141
has been modeled is an anode-supported counterflow tubular is the effective binary diffusivity of x into y, Rout,PEN is the
SOFC. The modeling considers not only the reactants and outside radius of PEN (porous cathode electrode, electrolyte,
product composition but also the physical shape (geometry) of and porous anode electrode assembly), Rin,INS is the inside
the cell. The modeling is done based on a CV approach, as radius of the insulator, and subscripts ac, an, cc, and ca de-
shown in Fig. 2 [6], [8], [10]. The fuel (hydrogen) enters into note the anode channel, anode, cathode channel, and cathode,
the anode channel, and air goes to the cathode channel. The fuel respectively.
diffuses into the anode tpb through the porous anode, where it The momentum conservation equations for the distributed
reacts with the incoming oxygen ions to produce water. In the SOFC model are as follows [10], [22].
same manner, oxygen diffuses into the cathode tpb through the Anode channel:
porous cathode. The oxygen ions which are generated in the tpb
of the cathode travel through the solid oxide electrolyte to the ∂(ρac uz,ac ) ∂ ρac u2z,ac ∂(ρac ur,ac uz,ac ) ∂pcc
=− − − .
anode tpb to react with the fuel. The unutilized air and hydrogen ∂t ∂z ∂r ∂z
are exhausted to the balance of the plant facilities. The fuel and (15)
air enters each CV at different concentrations from the cathode Cathode channel:
channel and the anode channel, respectively. Hence, the partial
pressure measured at each CV is different and is a function of ∂(Ccc uz,cc ) ∂ ρcc u2z,cc ∂(ρcc ur,cc uz,cc ) ∂pcc
=− + + .
time and space. This partial pressure governs the dynamics of ∂t ∂z ∂r ∂z
the fuel cell. (16)
The species conservation equations in the channels for the
ρx is the density in the x channel and px is the pressure
distributed SOFC model are as follows [10], [22].
in the x channel. These equations are developed based on the
Hydrogen conservation: assumption that u is a function of z and t only. The radial
∂CH2 (∂CH2 uz,ac ) 2DH2 −H2 Oeff ∂CH2 component of the velocity vector is ignored. Since the Reynolds
=− + . (11) number, the ratio of the rate of the convective transport of
∂t ∂z Rac ∂r
momentum to the rate of the molecular transport of momentum,
Water conservation: is greater than one in both the anode and cathode channels
in the operating range of the cell, the molecular transport of
∂CH2 O (∂CH2 O uz,ac ) DH2 O−H2eff ∂CH2 O
=− + . (12) momentum in the main flow direction is ignored.
∂t ∂z Rac ∂r The species conservation equations in the electrode are as
Oxygen conservation: follows [22].
Hydrogen conservation:
∂CO2 (∂CO2 uz,cc ) 2DO2 −N2eff ∂CO2
=− + . (13) ∂(CH2 ,an ) 1 ∂ ∂CH2 ,an
∂t ∂z Rcc ∂r an = rDH2 −H2 Oeff . (17)
∂t r ∂r ∂r
Nitrogen conservation:
Oxygen conservation:
∂CN2 (∂CN2 uz,cc ) 2DN2 −O2eff ∂Cn2
=− + . (14)
∂t ∂z Rcc ∂r ∂(CO2 ,ca ) 1 ∂ ∂CO2 ,ca
ca = rDO2 −N2 eff . (18)
∂t r ∂r ∂r
2 2 2
R = Rin,INS
+ Rout,PEN
Cx is the molar concentration
,
of species x, uz is the velocity in the main flow direction z, an and ca are the porosities of the anode and cathode,
Rac is the radius of the anode channel, t is the time, Dx−y respectively. Similar equations are also written for water and
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142 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 1, JANUARY 2009
TABLE I
CHEMICAL-TO-ELECTRICAL EQUIVALENCE
B. Lumped Model
1) Species Conservation: In the case of the distributed-
modeling approach, the flow rate and partial pressure at each
CV are different. The lumped model considers the fuel cell
as a single lumped system, which reduces the complexity and
computation time. The species enter the cell through one end
and leaves through the other end, as shown in Fig. 3. The mass
conservation is applied at one end of the fuel cell. The partial
pressure at the exit point of the fuel cell is calculated using (9).
The relation between the reactant/product flow rate and the cell
current of the fuel cells is given by [1]–[10]
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GEBREGERGIS et al.: SOLID OXIDE FUEL CELL MODELING 143
current densities, both equations have the same results. How- It is necessary to have experimental data for determination
ever, the error of the Tafel equation becomes large if Ifc < 4I0 , of the limiting current density. The limiting current density
and the activation voltage calculated using (26) is negative for depends upon a number of things like reactant concentration,
Ifc < I0 , as shown in Fig. 5. Therefore, the activation loss for temperature, material properties, etc. Equation (31) can be
all current densities is calculated using the inverse hyperbolic alternatively expressed as [24]
sine of (27)
ln(1 − pre )
RT Ifc IL = KL (34)
Vact = ln (26) T
F I0
where pre is the reactant concentration, T is the operating
RT −1 Ifc
Vact = sinh . (27) temperature of the cell, and KL = 69 is a constant factor.
F 2I0
Alternatively, (27) can be written as (28), since it is easy to
C. Implementation of the SOFC Model Using PSpice
implement the equation in PSpice or Matlab/Simulink
All the equations and the equivalent circuit of the lumped
RT
SOFC model discussed in the previous section are implemented
Vact = z + (1 + z 2 ) (28)
F in PSpice. The implemented model of the SOFC in PSpice
is shown in Fig. 6. The PSpice model contains a circuit
where z = (Ifc /2I0 ).
that represents the output voltage, as shown in Fig. 6(a), and
It is evident from the graph that the exchange current I0 of
equations that calculate the polarization losses and equivalent
the two Tafel lines is not the same. This difference may occur
circuit of the flow rate conservation. Fig. 6(b) and (c) shows the
for a number of reasons such as the cell temperature, reactant
activation and concentration losses, respectively, while the flow
and product concentration, electrode material, its geometry, and
rate equivalent electrical circuit is shown in Fig. 6(d), where
so on [3], [6], [8], [10], [16]. The exchange current density can
n = 2 for the hydrogen and water circuits and n = 1 for the
be expressed as [6], [23]
oxygen circuit. The dynamics of the SOFC is embedded in the
equivalent RC circuit. Moreover, the SOFC model is imple-
I0 = Ae−Eact /RT (29)
mented in Matlab/Simulink for the development of emulator
and control application. The parameters used in the simulation
where A = 101.2 kA/cm2 is a preexponential factor ob-
are given in Table II, where CH2 , CH2 O , and CO2 are the
tained by curve fitting with the distributed model and
capacitances of hydrogen, water, and oxygen, respectively.
Eact = 120 kJ/mol is the activation energy of the electrochem-
ical reaction.
3) Concentration Voltage Loss: According to the polariza- IV. S IMULATION AND E XPERIMENTAL R ESULTS
tion curve, this voltage drop appears when the current density
of the fuel cell approaches the maximum current density of A. Experimental Test Setup
the cell and is dependent on the flow rate [10], [19]. The The experiments were carried out on an anode-supported
concentration loss is determined using (7). The cell current Ifc tubular SOFC manufactured by NanoDynamics, Inc., Buffalo,
produced is related to the concentration of the species using the NY. The anode material is nickel–yttria-stabilized zirconia, and
following: the cathode is lanthanum strontium manganite. Pure hydrogen
was supplied to the anode gas flow channel from a bank of
Ifc = K · (C∞ − Cb ). (30) cylinders through a flow controller. Compressed air regulated
by a flow controller was provided through an electrical furnace
The limiting current IL is the cell current evaluated at Cb = 0. to the cathode channel. The entire cell was kept well insulated
That is to minimize the radiation heat loss from the system. The cell
was heated up slowly by the hot air. When the cell temper-
IL = K · C∞ . (31) ature reached about 200 ◦ C, hydrogen was introduced, and
the temperature was brought to 700 ◦ C. Once a steady open-
Hence, the ratio of the concentration in (7) can be written circuit potential was obtained, the data collection was started.
alternatively as the ratio of the cell current to the limiting DC polarization of the cell was maintained by an electronic
current of the fuel cell load controller run in constant-voltage mode. The V –I data
were collected while decreasing the voltage gradually to 0.5 V.
Cb Ifc
=1− . (32) While keeping the temperature constant, the data were collected
C∞ IL at hydrogen flow rates of 31, 36, 41, and 51 mL/min. Then,
Therefore, substituting (32) into (7), the concentration loss is the temperature was increased to 750 ◦ C, and the V –I data
written as were collected at all flow rates mentioned earlier. The same
procedure was repeated at 800 ◦ C and 850 ◦ C. At every flow
RT Ifc and temperature, the data were collected after a steady-state
Vcon = ln 1 − . (33)
nF IL condition was reached.
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144 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 1, JANUARY 2009
Fig. 6. PSpice simulation circuit diagram of an SOFC. (a) SOFC circuit diagram. (b) Activation loss. (c) Concentration loss. (d) Partial pressure circuit.
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GEBREGERGIS et al.: SOLID OXIDE FUEL CELL MODELING 145
Fig. 8. V –I curves for different input flow rates at 850 ◦ C. (a) Polarization
Fig. 7. V –I curves for different input flow rates at 800 ◦ C. (a) Polarization
curves at 31 mL/min. (b) Polarization curves at 36 mL/min. (c) Polarization
curves at 31 mL/min. (b) Polarization curves at 36 mL/min. (c) Polarization
curves at 41 mL/min. (d) Polarization curves at 51 mL/min.
curves at 41 mL/min. (d) Polarization curves at 51 mL/min.
is set as a reference and then compared with the simulation 750-◦ C, 800-◦ C, and 850-◦ C cell temperatures. The cell voltage
results. The maximum relative error occurs at 31-mL/min flow increases with an increase in the operating cell temperature.
rate and 800-◦ C cell temperature. The error becomes more Fig. 10(b) is an enlarged portion of Fig. 10(a) at a cell current
severe for lower flow rate and cell temperature. Ifc = 2.5 A. The cell voltage increases almost linearly as a
Simulation results at 41-mL/min flow rate were considered function of temperature at this particular cell current and its
to study the effect of cell temperature on the cell voltage. vicinity, while the cell voltage is almost the same in the activa-
Fig. 10(a) is a plot of the V –I polarization curve at tion and concentration regions of the polarization curve.
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146 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 1, JANUARY 2009
TABLE III
RELATIVE ERRORS OF THE LUMPED SOFC MODEL
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GEBREGERGIS et al.: SOLID OXIDE FUEL CELL MODELING 147
A PPENDIX B
B OUNDARY C ONDITIONS FOR THE L UMPED M ODEL
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148 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 1, JANUARY 2009
fuel cell power plant,” IEEE Trans. Energy Convers., vol. 14, no. 4, Debangsu Bhattacharyya received the B.S. de-
pp. 1651–1657, Dec. 1999. gree from Regional Engineering College (currently
[21] M. D. Lukas, K. Y. Lee, and H. Ghezel-Alagh, “An explicit dynamic National Institute of Technology), Durgapur, India,
model for direct reforming carbonate fuel cell stack,” IEEE Trans. Energy in 1993. He is currently working toward the Ph.D.
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anode-supported tubular solid oxide fuel cells,” Chem. Eng. Sci., vol. 62, Potsdam, NY.
no. 16, pp. 4250–4267, Aug. 2007. DOI: 10.1016/j.ces.2007.04.020. He was with the refineries division of Indian Oil
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fuel cell,” in Proc. IEEE Power Eng. Soc. Gen. Meeting, Jul. 2003, vol. 3, search interests mainly include modeling, optimiza-
pp. 1436–1440. tion, and control of fuel cells.
[24] K. Huang, “Gas-diffusion process in a tubular cathode substrate of an
SOFC,” J. Electrochem. Soc., vol. 151, no. 5, pp. A716–A719, 2004.
Raghunathan Rengaswemy received the B.S. de-
gree from the Indian Institute of Technology (IIT),
Abraham Gebregergis (S’07–M’08) received the Madras, India, in 1990, and the Ph.D. degree from
B.S. degree from the University of Asmara, Purdue University, West Lafayette, IN, in 1995.
Asmara, Eritrea, in 2001, and the M.S. degree from He is currently a Professor with the Depart-
Stellenbosch University, Stellenbosch, South Africa, ment of Chemical Engineering, Clarkson University,
in 2004. He is currently working toward the Ph.D. Potsdam, NY. From 1996 to 2000, he was with IIT,
degree in electrical engineering in the Department Bombay, India. He was a Visiting Professor with the
of Electrical and Computer Engineering, Clarkson University of Delaware, Newark, in summer 1999;
University, Potsdam, NY. Purdue University, West Lafayette, IN, in winter
His research interests include power electronics, 2001; and the University of Alberta, Edmonton, AB,
drives and machines, control, and fuel cell modeling. Canada, in summer 2002. His research and teaching interests include modeling,
optimization, diagnostics, and control of proton exchange membrane and
solid oxide fuel cells, chemical process calculations, mathematical methods,
computer-aided design, advanced process control, and artificial intelligence
Pragasen Pillay (S’84–M’87–SM’92–F’05) re- techniques in process engineering.
ceived the B.S. and M.S. degrees from the University Dr. Rengaswemy was the recipient of the Young Engineer Award from the
of KwaZulu-Natal, Durban, South Africa, in 1981 Indian National Academy of Engineering (INAE) in 2000. He was chosen by
and 1983, respectively, and the Ph.D. degree from the the students of the Department of Chemical Engineering, Clarkson University,
Virginia Polytechnic Institute and State University, as the Professor of the Year in 2003.
Blacksburg, in 1987, while funded by a Fulbright
Scholarship.
From January 1988 to August 1990, he was with
the University of Newcastle upon Tyne, Newcastle
upon Tyne, U.K. From August 1990 to August
1995, he was with the University of New Orleans,
New Orleans, LA. He is currently with Clarkson University, Potsdam, NY,
where he is a Professor with the Department of Electrical and Computer
Engineering and holds the Jean Newell Distinguished Professorship in Engi-
neering. He has also been an Adjunct Professor with the University of Cape
Town, Cape Town, South Africa, since 1999. He has organized and taught
short courses on electric drives at the Annual Meeting of the IEEE Industry
Applications Society. His research and teaching interests include modeling,
design, and control of electric motors and drives for industrial and alternative
energy applications.
Dr. Pillay is a Fellow of the Institution of Engineering and Technoloy, U.K.,
and a Chartered Electrical Engineer. He is a member of the Academy of Science
of South Africa and the IEEE Power Engineering, IEEE Industry Applications,
IEEE Industrial Electronics, and IEEE Power Electronics Societies. He is also
a member of the IEEE Industry Applications Society (IAS) Electric Machines
Committee, the Past Chairman of the IEEE IAS Industrial Drives Committee,
and the Past Chairman of the IEEE Power Engineering Society Induction
Machinery Subcommittee. He is currently the Chair of the Awards Committee
of the IEEE IAS Industrial Power Conversion Department.
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