1
Sensitivity Analysis of the Modeling Parameters
Used in Simulation of Proton Exchange
Membrane Fuel Cells
J. M. Corrêa* (Student Member, IEEE), F. A. Farret**,
V. A. Popov** , M. Godoy Simões* (Senior Member, IEEE),
Abstract -- The simulation of Proton Exchange Membrane
Fuel Cells (PEMFC) may work as a powerful tool in the
development and widespread testing of alternative energy
sources. In order to obtain an adequate PEMFC model, (which
could be used in the analysis of fuel cell generation systems), it is
necessary to define the values for a specific group of modeling
parameters. The simulation results are strongly affected by the
choice of such modeling parameters.
Multi-Parametric
Sensitivity Analysis (MPSA) is a tool that can be used to define
the relative importance of the factors related to the model,
because it encompasses the entire parameter space. This paper
presents a sensitivity investigation of PEMFC electrochemical
models, and aims to determine the relative importance of each
parameter on the model results.
Index Terms — Fuel cells, modeling and simulation, sensitivity
analysis.
T
I. INTRODUCTION
HE utilization of fossil fuels has increased, as has the
concentration of toxic and greenhouse gases in the
environment, such as SOx, NOx, CO and CO2. In addition, the
world is strongly dependent on these energy sources, which
are becoming scarce and more expensive. Research on
alternative and renewable energy sources is an international
matter. Such sources should be friendlier to the environment,
cleaner, and more efficient than the conventional sources.
Among the energy sources considered, (such as wind power,
photovoltaic power and small hydropower), the Fuel Cell (FC)
stacks have received heightened attention in the last few years.
This is particularly due to their high electrical and overall
efficiency (up to 80% for combined heat and power), low
aggression to the environment, excellent dynamic response,
and superior reliability and durability.
Among the various FC models actually available, the Proton
Exchange Membrane Fuel Cell (PEMFC) seems a promising
source to be used in residences, industries, and small- and
large-scale distributed generation systems.
The main
characteristics of PEMFC stacks are: (i) they produce water as
This work was supported in part by the Brazilian Agencies CAPES, AESSul, FAPERGS, and CNPq and the American Agency NSF, under grant ECS #
0134130.
*
**
Colorado School of Mines, United States (jcorrea, msimoes@mines.edu)
Federal University of Santa Maria, Brazil (ffarret, vpopov@ct.ufsm.br)
a residue; (ii) they have high efficiency when compared to
thermal generation; (iii) they operate at low temperatures (up
to 90 oC), which allows a fast start-up; and, (iv) they use a
solid polymer as the electrolyte, which reduces concerns
related to construction, transportation, and safety.
From the electrical engineering viewpoint, the importance
of a good fuel cell model is related to: the facility of tests of
the fuel cell controllers; evaluation of the available power and
energy for a certain load profile; and, evaluation of the needs
for hydrogen and additional storage systems. Also, in power
generation systems, the dynamic response is of extreme
importance for the control planner and system management,
especially when there is injection of energy into the network;
the dynamic response must be included in the fuel cell model.
However, the current high costs of the FC stacks make both
development and the widespread usage of these systems
difficult, especially in developing countries. To remedy this,
the Group of Micropower Plants Development (NUDEMI) of
the Federal University of Santa Maria (UFSM - Brazil) and the
Engineering Division of Colorado School of Mines (CSM –
USA) have been working with the application of renewable
energy sources, mathematical modeling, and physical
simulation of power sources particularly with PEMFC stacks
[1]. These groups have recently developed an FC power
system simulator (FC-SIM) that uses a computer-controlled,
ac-to-dc power converter to supply power to electrical loads,
in a way similar to the simulated stack. The FC-SIM can be
used effectively with its electrochemical PEMFC model in the
improvement of PEMFC energy systems, including its
automation and integration with electrical networks [1].
The main difficulty in obtaining an accurate PEMFC
dynamical model is the lack of information about the exact
values that should be used for the modeling parameters. The
choice of parameters may significantly affect the voltage,
power, and temperature characteristics of the simulated stack.
There are several papers [1]-[6] dealing with modeling and
simulation of PEMFC, some of which discuss the dynamical
behavior of unit cells and stacks [1], [3], [4] and which also
present some aspects related to the modeling parameters [8].
The values used for the parameters are primarily based on
manufacturing data and laboratory experiments. However,
some aspects of PEMFC operation are still difficult to model
accurately and some processes are property of the
manufacturers. For a detailed description of the processes
2
involved in fuel cell manufacturing and operation, see [4]. The
disagreements between measurements and calculations arise
because of uncertainties stemming not only from experimental
measurements, but also from ill-defined parameters [10], [11].
The relative importance of the physical and electrochemical
processes occurring in a fuel cell can be evaluated using a
generalized Multi-Parametric Sensitivity Analysis (MPSA),
which encompasses the entire model parameter space. Such
analysis is conducted in this paper using a fuel cell
electrochemical model and data from a 500 W PEMFC stack
manufactured by the company BCS Technology [9].
II. PEM FUEL CELL ELECTROCHEMICAL MODEL
This section presents an electrochemical model that can be
used to predict the dynamic behavior of PEMFC stacks. This
mathematical model uses a group of parameters whose
definition is essential for the best simulation results. The
output voltage of a single cell VFC, can be defined as follows:
VFC = E Nernst − Vact − Vohmic − Vcon
(1)
And, for n cells connected in series and forming a stack, the
voltage Vs, can be calculated by:
Vs = n ⋅ VFC
(2)
In (1), ENernst is the thermodynamic potential of each unit
cell and represents its reversible voltage; Vact is the voltage
drop associated with the activation of the anode and of the
cathode; Vohmic is the ohmic voltage drop (a measure of the
voltage drop associated with the conduction of protons and
electrons); and Vcon represents the voltage drop resulting from
the decrease in the concentration of oxygen and hydrogen [4].
The first term of (1) represents the FC open circuit voltage,
while the last three terms represent reduction in this voltage.
The resulting voltage VFC, is the FC useful voltage for a
certain operating condition. In addition to the three terms
representing voltage drops, there is another term involving the
PEMFC operation. This additional voltage drop results from
the circulation of electronic currents through the electrolyte or,
similarly, from the fuel crossover through the electrolyte [4].
This voltage drop is modeled considering a permanent FC
current density (Jn) that is added to the main FC current
density, even when the FC is operated without any load. Each
individual term of (1) is defined by [1], [2], [4]:
E Nernst = 1.229 − 0.85.10 −3.(T − 298.15) +
( )
( )
1


4.31.10 −5.T .ln PH 2 + ln PO2 
2


[
( )
Vact = − ξ1 + ξ 2 .T + ξ 3 .T . ln cO2 +
ξ 4 .T . ln(i FC )]
(3)
(4)
Vohmic = iFC .(RM + RC )
(5)

J 

Vcon = − B. ln1 −

 J max 
(6)
PO2
cO2 =
6
5.08.10 .e
(7)
 498 
−

 T 
where PH2 and PO2 are the partial pressures (atm) of hydrogen
and oxygen, respectively. T is the cell absolute temperature
(K). iFC is the cell operating current (A). cO2 is the
concentration of oxygen in the catalytic interface of the
cathode (mol/cm3). The ξi (i = 1...4) represent the parametric
coefficients for each cell model [2]. RM is the equivalent
membrane resistance to proton conduction [2]. RC is the
equivalent contact resistance to electron conduction. Jmax is
the maximum current density. B (V) is a constant dependent
on the cell type and its operation state [4]. And, J is the actual
cell current density (A/cm2) including the permanent current
density Jn.
The equivalent membrane resistance (RM) can be calculated
by [2]:
RM =
ρ M .l
(8)
A
where ρM is the membrane specific resistivity (Ω.cm) obtained
by:
ρM =
2 .5
2

i 
 T   i FC  
181.6.1 + 0.03. FC  + 0.062.
 
 .
 303   A  
 A 



 i FC 
 T − 303 
. exp 4.18.

ψ − 0.634 − 3.
A
 T





(9)
where the term 181.6/(ψ-0.634) is the specific resistivity
(Ω.cm) at no current (iFC = 0) and at temperature of 30oC (T =
303 K). The exponential term in the denominator is the
temperature factor correction if the cell is not at 30oC. The
parametric coefficient ψ is considered an adjustable parameter
with a possible minimum value of 14 and a maximum value of
23 [2].
Most variables in (3) to (9) are dependent on the cell
temperature and pressure operating conditions. Variations in
these operating conditions directly affect the fuel cell
performance; a higher operating temperature and pressure will
increase the fuel cell voltage and efficiency, for a certain
current. The operating temperature of a fuel cell can be
obtained from the following equation:
 dT 
M ⋅Cs 
 = ∆Q&
 dt 
(10)
where M is the cell mass (kg); Cs is the equivalent average
specific heat coefficient (J.K-1.kg-1); and, ∆Q& is the rate of
heat variation (J/s).
The value for the product M.Cs must be obtained by the
summation of the individual masses and heat coefficients of all
cell components, (as with graphite and iron, for example).
3
The rate of heat variation ∆Q& is the difference between the
rate of heat generated by the cell operation and the rate of heat
removed by the cell cooling system:
∆Q& = Q& ger − Q& rem
(11)
The rate of heat generation by the fuel cell is obtained from:
1 
Q& ger = PFC ⋅  − 1
η 
(12)
where PFC is the fuel cell electrical output power and η is the
fuel cell efficiency.
Part of this heat must be removed from the fuel cell to limit
and control the temperature, as presented in (11). This heat
can be removed using one or more of the following cooling
methods: (i) using part of the reaction air; (ii) using a
separated cooling air; and, (iii) using cooling water.
In the following analysis, membrane temperature and
reactant pressure are assumed to be constant. As presented
above, the temperature can be controlled by the rate of
removed heat, as shown in (11). The hydrogen is assumed to
come from a high-pressure bottle (and is reduced by a pressure
controller). In this way, the hydrogen pressure is assumed to
be constant. In addition, the oxygen comes from the air where
it has a constant pressure. An analysis of the influence of
these operating conditions is interesting, so a thermodynamical
model should be used in this case.
Equations (1) - (9) represent the fuel cell stack static
electrochemical behavior. An electrical circuit can be used to
model the FC dynamical behavior [1], [4], as represented in
Fig. 1.
+
vohmic
RΩ
+
Ract
_
C
Rcon
+
ENernst_
+
dvd 1
1
= iFC − vd
dt
C
τ
(13)
where vd represents the dynamical voltage across the
equivalent capacitor (associated with Vact and Vcon); C is the
equivalent electrical capacitance; and, τ is the FC electrical
time constant defined as:
 V act + Vcon
i FC

τ = C ⋅ R a = C ⋅ (R act + Rcon ) = C ⋅ 




(14)
where Ra is an equivalent resistance.
Including the dynamic behavior represented by (13), the
resulting fuel cell voltage is then defined by:
VFC = E Nernst − Vohmic − vd
(15)
III. PARAMETRIC SENSITIVITY ANALYSIS
iFC
_
near the electrolyte/electrode interface. Then, when there is an
increase (decrease) in the FC current, there is a delay until the
FC voltage decreases (increases). The ohmic overpotential is
not affected by the charge double layer effect as it is directly
related to the current (represented by the resistance RΩ).
On the equivalent circuit of Fig. 1, the capacitor is
positioned in parallel with the activation and concentration
voltages (represented by their equivalent resistor), to take into
account the dynamic effect of these voltage drops. This
resulting loop is then connected in a series with the Nernst
potential (thermodynamic potential) and with the ohmic
voltage drop (represented by its equivalent resistance).
The dynamical equation of the model presented in Fig. 1 is
represented by:
VFC
vd
LOAD
_
Fig. 1. FC Dynamical Model - Electrical Equivalent Circuit
In the equivalent circuit in Fig. 1, there is a first order delay
in the activation and the concentration voltage components
(represented by the resistances Ract and Rcon, respectively).
This delay is caused by the charge double layer effect [4].
Such phenomenon normally exists, on every contact between
two different materials, due to a charge accumulation on the
opposite surfaces or a load transfer from one to the other. The
charge layer on both electrode/electrolyte interfaces (or close
to the interface) is storage of electrical charges and energy; in
this way, it behaves as an electrical capacitor. This effect
causes a retardation in the dissipation of the electrical charges
The model presented in Section II requires the definition of
several parameters prior to computer simulation. In order to
investigate the influence of such parameters in PEMFC
analysis, a 500 W BCS stack was simulated. The base
parameter set is presented in Table I. These parameters are
based on literature data for similar stacks [2], [4], in addition
to manufacturer data [9].
The parameters presented in Table I have the following
meanings:
• n: number of cells used in the stack
• A: cell active area (cm2)
• l: membrane thickness (µm)
• T: cell operating temperature (K)
• PO2; PH2: oxygen and hydrogen partial pressures (atm)
• RC: contact resistance (Ω)
• ξi and ψ: parametric coefficients
• Jn: no-load current density (A/cm2)
• Jmax: maximum current density (A/cm2)
• C: equivalent electrical capacitance (F)
TABLE I
PARAMETERS SET OF A 500 W BSC STACK
Param.
Value
Param.
Value
n
32
ξ1
-0.948
64 cm2
A
ξ2
PH2
(4.3.10-5).ln (cH )
2
178 µm
ξ3
7.6.10-5
T
333 K
ξ4
-1.93.10-4
0.2095 atm
ψ
23.0
1 atm
Jn
3 mA/cm2
RC
0.0003 Ω
Jmax
469 mA/cm2
B
0.016 V
C
3F
PO
PH
2
2
FUEL CELL
0.00286+0.0002.ln (A)+
l
4
T, iFC
Using the parameters presented in Table I, Fig. 2 shows the
simulated polarization curve obtained with the electrochemical
model (Section II). Fig. 2 also shows the polarization curve
presented in the manufacturer data [9]. As shown in Fig. 2,
the simulated results agree with the real data, except at the
very beginning and very end of the polarization curve.
Fig. 2. 500 W BCS Stack Polarization Curve
The discrepancies observed in Fig. 2 are mainly caused by
the difficulty in obtaining exact parameter values. As shown
in Table I, there are various parameters to be defined before an
accurate simulation can be obtained. In addition, though there
was agreement between observed and simulated results, it is
not possible to identify the relative importance of each
parameter used in the model. To investigate this, the
parametric sensitivity of the fuel cell electrochemical model
can be tested using a Multi-Parametric Sensitivity Analysis
(MPSA) approach [10]. Fig. 3 presents a block diagram of the
model inputs, outputs, and feedback signals for application of
the MPSA to the fuel cell electrochemical model. The input
parameters shown in Fig. 3 will be evaluated, in order to
define their relative importance on the model results. Only the
number of cells (n) will not be evaluated, because it is a
constant number with 100% certainty.
Vs
T
PO2
iFC
LOAD
MODEL
Parameters:
n, A, l, ξi, ψ, Jn,
Jmax, RC, B, C
Fig. 3. Block Diagram of the Fuel Cell Model
To apply MPSA, the following steps may be followed for a
certain set of parameters [10]:
1. Select the parameters to be tested.
2. Set the range of each parameter.
3. For each parameter, generate a series of independent,
random numbers with a uniform distribution within
the defined range.
4. Run the model using the selected series and calculate
the objective function using (16), for each value of
cell current.
5. Determine the relative importance of each parameter,
for each value of current using (17).
6. Evaluate parametric sensitivity, (to define the
sensitive and insensitive parameters), using (18).
The objective function values of the sensitivity analysis
are usually calculated from the sum of square errors between
observed and modeled values [10]:
k
[
]
f h = ∑ x 0,h − x c,h (i ) 2
i =1
(16)
where fh is the objective function value for a certain fuel cell
current h. x0,h is the observed value at this current h. xc,h(i) is
the calculated value xc at current h for each series element i
(where i represents an element in the random series). And, k is
the number of elements contained in the random series (Step
3). The observed values used in this analysis are obtained
from a simulation using the base value for each parameter
(presented in Fig. 2). The range used for each parameter to be
evaluated is presented in Table II for the 500 W BCS stack,
and the base values corresponding to the parameters are
presented in Table I.
The following can be used to evaluate the relative
importance of each parameter independently on the stack
voltage:
δh =
fh
x 0 ,h
(17)
5
where h represents each current point. By applying the
described procedure to the PEMFC model (using a series of
500 data for each current value), the results presented in Fig. 4
were obtained for each evaluated parameter. These results
were obtained using (17). For easier reference, the results
presented in Fig. 4 are grouped under: insensitive parameters,
sensitive parameters, and highly insensitive parameters.
Parameters B, ξ4 and ψ have more influence on the stack
voltage for high current values. However, their effect is less
accentuated than Jmax. The parameter B defines the form of the
polarization curve, especially in its final portion (near the
maximum stack current). The final portion of the polarization
curve is characterized by a fast decrease in the voltage, as
shown in Fig. 2.
TABLE II
RANGE OF PARAMETER USED IN MPSA
TABLE III
SENSITIVITY CLASSIFICATION OF THE MODELING PARAMETERS
Param.
Test range
Param.
Test range
Param.
64 ±5% [cm2]
ξ1
Index γ
A
-0.948 ±10%
A
7.917.10-3
l
178 ±5% [µm]
ξ3
7.6.10-5 ±10%
RC
0.1605
RC
0.0003 ±15% [Ω]
ξ4
-1.93.10-4 ±10%
l
0.5583
B
0.016 ±15% [V]
ψ
15 – 24
Jn
1.7008
Jn
3 ±25% [mA/cm ]
1 – 5 [F]
B
4.7754
Jmax
469 ±10% [mA/cm2]
ψ
52.9100
ξ4
78.3325
Jmax
173.8125
ξ3
423.6830
ξ1
2338.4800
2
C
The following was used to summarize the data obtained
from the MSPA, and to define the relative importance of each
parameter:
γ =
Insensitive
Sensitive
Highly Sensitive
i FC ,max
∑δ
h
(18)
h =0
where the fuel cell current was evaluated from no-current (h =
0) to the maximum value (iFC,max), which is equal to 30 A for
this stack.
For a certain parameter, the higher the value of the index γ
the more sensitive the fuel cell model is to this parameter.
With the index γ, the following criteria was used to define the
relative sensitivity of the fuel cell model to a certain
parameter:
• γ ≤ 1 ⇒ model parameter insensitive
•
Sensitivity
Regarding the parameter Jmax, the model results are also
more affected by high current values. This can be explained
by the logarithm term in (6). When the current density is close
to the maximum value, the logarithm term tends to zero, as
does the concentration voltage. This then changes the
resulting stack voltage.
However, for parameters ξ1 and ξ3, the model results are
affected by all current values in a same high order. Their
electrochemical exact definition is [2]:
ξ1 = −
1 < γ ≤ 100 ⇒ model parameter sensitive
• γ > 100 ⇒ model parameter highly sensitive
As a result, the parameters can be grouped as such:
• Insensitive: A, RC, l
• Sensitive: Jn, B, ψ, ξ4
• Highly Sensitive: Jmax, ξ3, ξ1
The results are summarized in Table III.
The insensitive parameters are basically those related to the
cell construction. Their influence on the model accuracy is not
critical, and it is not necessary to know their exact values to
have a good response.
Parameter Jn only affects the simulation results at low
current values, because its value will define the resulting opencircuit voltage considering the internal current and crossover
effect [4].
ξ3 =
∆G a
∆G c
−
2 ⋅ F αc ⋅ n ⋅ F
R ⋅ (1 − α c )
αc ⋅ F
(19)
(20)
where:
∆Ga: free activation energy for the standard state (J/mol)
referred to the anode;
∆Gc: free activation energy for the standard state (J/mol)
referred to the cathode;
αc: parameter for the anode chemical activity;
F: Faraday Constant;
R: gases universal constant;
A: cell active area (cm2);
cH2: hydrogen concentration (mol/cm3); and
cH2O: water concentration (mol/cm3).
0.012
0.04
0.0006
0.009
0.03
0.0004
0.0002
Sensitivity [l ]
0.0008
Sensitivity [Rc ]
Sensitivity [A]
6
0.006
0.003
0.01
0.000
0
0
5
10
15
20
0.00
0
25
5
10
15
20
25
0
(i) Cell Active Area
1.20
1.80
Sensitivity [B]
2.40
0.80
25
0.40
1.20
0.60
0.00
5
10
15
20
25
0
5
10
Current (A)
15
20
25
20
25
Current (A)
(i) Internal Current Density
(ii) Parameter B
4.00
3.20
3.00
2.40
Sensitivity [ξ
[ 44]]
Sensitivity [ψ]
[ ]
20
(iii) Membrane Thickness
1.60
0
2.00
1.00
1.60
0.80
0.00
0.00
0
5
10
15
20
0
25
5
15
Current (A)
(iii) Parameter ψ
(iv) Parameter ξ 4
Parameters Sensitive to the Model
8.8269
Sensitivity [ξ
[ 33]]
75.00
50.00
25.00
0.00
48.720
Sensitivity [ξ
[ 11]]
100.00
8.8268
8.8267
15
20
25
0
Current (A)
10
15
20
25
(ii) Parameter ξ 3
(c)
Fig. 4. Sensitivity Analysis of the Modeling Parameters
5
Current (A)
(i) Maximum Current Density
48.718
48.715
48.713
48.710
8.8266
10
10
Current (A)
(b)
Sensitivity [Jmax ]
15
Parameters Insensitive to the Model
0.00
5
10
Current (A)
(ii) Contact Resistance
(a)
0
5
Current (A)
Current (A)
Sensitivity [Jn]
0.02
Parameters Highly Sensitive to the Model
0
5
10
15
Current (A)
(iii) Parameter ξ 1
20
25
7
Stack voltage (V)
All these elements are related to the electrochemical
process needed for electrodes activation, and they are difficult
to determine with great accuracy. The values used in the
presented model are based on calculation and measured results
[2].
Taking into account the results presented in Fig. 4, the
process to define the fuel cell stack parameters is not a simple
task; and, once the parameter set is defined, it is only valid for
that specific stack. To know if the parameter values are good
enough, it is necessary to compare the simulation results with
real data from similar test conditions.
Considering the relative importance of ξ1 and ξ3 in the
simulation results, further analysis was carried out for these
parameters. Fig. 5 presents the stack polarization curve
considering parameter ξ1 changing on the range presented in
Table II. Fig. 6 presents the stack polarization curve assuming
changes in parameter ξ3 according to Table II. The other
parameters are assumed to be the base parameters, as
presented in Table I. These figures show that the stack
voltage changes considerably depending on these parameter
values.
35
30
25
20
15
10
0
5
10
15
20
25
35
30
Stack Voltage (V)
Current (A)
ξe1=-0.948
1= −0.948
ξe1=-0.853
1= −0.853
e1=-1.043
ξ1= −1.043
Fig. 5. Effect of the Parametric Coefficient ξ 1 on the Polarization Curve
Stack voltage (V)
The dynamical behavior of a PEMFC stack is modeled as
an equivalent electrical circuit, as shown in Fig. 1. The effects
of parameter variation on the equivalent resistances were
evaluated above, taking into account that the resistance values
actually change with the modeling parameters. However, the
equivalent capacitor needs a different test, in order to evaluate
its affect on the fuel cell response.
As explained in Section II, the charge double layer effect is
responsible for a delay in the FC voltage change, after a
change in its current. The parameter used to describe this
behavior is the equivalent capacitance C whose value, for the
PEMFC, is of a few Farads. This capacitance does not
influence the stack polarization curve, because each point of
this curve is obtained after the voltage has reached its steadystate value. To evaluate its effect, a current interruption test
can be simulated.
Fig. 7 shows the effect of this parameter for a reduction in
the stack current, from 15 A to 0 A (open circuit). The current
reduction occurs after 10 seconds of simulation. As shown in
Fig. 7, the stack voltage presents an instantaneous change
(caused by the ohmic overpotential), followed by a first order
delay until it reaches its new steady-state value (open circuit
voltage). The curves shown in Fig. 7 are from equivalent
capacitances values from 0.5 F to 5 F resulting in a range of
1:10. In practical electronic circuits, the values of the
capacitors are much less. Nevertheless, these values are
representative of the PEMFC dynamical behavior and do not
represent real capacitors.
30
25
20
= 0.5 F
=2F
=3F
=5F
15
0
35
C
C
C
C
C
50
100
150
200
250
300
Time (s)
30
25
Fig. 7. Effect of the Equivalent Capacitance (C) on the Stack Voltage
20
15
10
0
5
10
15
20
25
30
Current (A)
-5
ξe3=6.84E-5
3=6.84×10
-5
ξe3=7.6E-5
3=7.6×10
-5
ξe3=8.36E-5
3=8.36×10
Fig. 6. Effect of the Parametric Coefficient ξ 3 on the Polarization Curve
Even with the strong influence of ξ1 and ξ3 in the
simulation performance, the model is still valid, as long as
values are set for a certain cell or stack. As can be seen from
Figs. 5 and 6, their effects are practically constant over the
whole current range. They introduce a relatively fixed offset
on the stack voltage related to the real data.
The time needed for the stack voltage to reach its new
steady-state value is strongly dependent on the value chosen
for the equivalent capacitance. As can be seen in (13) and
(14), the higher the capacitor value, the higher the time
constant. For a higher value capacitor, the circuit response
takes longer to reach the steady state operation again, as
shown in Fig. 7.
For a PEMFC stack, the time needed to reach the steadystate can run a few milliseconds to tens of seconds. Taking
this into consideration, according to Fig. 7 the equivalent
capacitances should be lower than 3 F.
If there is a possibility of testing the real stack, finding the
exact value of the equivalent capacitance is a straightforward
matter.
It is only necessary to make a similar current
8
Stack Voltage (V)
40
35
30
25
20
15
0.00
0.40
0.80
1.20
1.60
2.00
Time (s)
Fig. 10. Simulated Dynamic Voltage for a 500 W Avista Stack
IV. CONCLUSIONS
15
This paper investigates the influence of the modeling
parameters on the dynamical performance of PEMFC
simulations. To show the effects of some key parameters,
some data from the literature and an electrochemical model
are used to evaluate the stack polarization curve based on the
dynamic behavior of a 500 W BCS stack. Starting with a
basic parameter set, it shows how the choice of the parameters
can influence the initial curve.
The parameters were analyzed using a Multi-Parametric
Sensitivity Analysis (MPSA). As a result, the parameters
were classified according to their influence in the model
results as: insensitive (A, l and RC); sensitive (Jn, B, ξ4 and
ψ.); and, highly sensitive (Jmax, ξ1 and ξ3).
For the most sensitive parameters (ξ1 and ξ3), this paper
shows that the polarization curve can present results that are
not similar to the real data. In addition, the results do not
present a fixed tendency, but are dispersed along the real
curve.
Finding the definition of the values, for fuel cell simulation
parameters, is not a simple task. Moreover, once the
parameter set is defined, it is only valid for a specific cell or
stack. To simulate other fuel cells, almost all the values must
be defined again. Using the data presented in this paper, it is
possible to evaluate the importance of each parameter to the
simulation’s accuracy. To obtain the best results, the
parameter values can then be corrected.
10
V. ACKNOWLEDGMENTS
45
40
35
30
25
20
15
0.00
vd
vohmic
0.40
0.80
1.20
1.60
2.00
Time (s)
Measured
Averaged
Fig. 8. Dynamic Voltage Data from a 500 W Avista Stack
The current used in this test is presented in Fig. 9, and the
simulation results using the equivalent capacitor of 0.01 F are
shown in Fig. 10.
Stack Current (A)
45
Stack Voltage (V)
interruption test, and to record the voltage values and
instantaneous time.
In order to better understand the dynamic behavior of a fuel
cell stack, data of a real dynamic test is presented in Fig. 8.
This example was obtained with a 500 W fuel cell stack
manufactured by the company Avista.
The data acquired for this stack presents some oscillation,
as observed in Fig. 8. These variations originate from the
pressure control system. This stack is a closed system that
does not allow access to the internal controllers. In order to
eliminate the effects of these variations, the data was
averaged; it is presented in Fig. 8. The ohmic voltage drop
and the equivalent dynamic voltage drop are also shown in
Fig. 8. For this specific stack, the average open circuit voltage
is about 40 V, and the average voltage with load current of
10.8 A is about 27.6 V. The time needed to reach steady state,
after the current interruption, is about 250 ms, making this a
very fast fuel cell. In this case, the equivalent capacitor is
very small: around 0.01 F.
5
0
0.00
0.40
0.80
1.20
Time (s)
1.60
The authors recognize and greatly appreciate the Federal
University of Santa Maria and the Colorado School of Mines,
Engineering Division (USA), for allowing all tests to be
conducted in their Laboratories (LHIPAE, NUDEMI and
2.00
NUPEDEE), the Advanced Coatings and Surface Engineering
Laboratory (ACSEL) and the support from National Science
Foundation Grant ECS # 0134130.
Fig. 9. Load Current Data from a 500 W Avista Stack
VI. REFERENCES
[1]
[2]
J. M. Corrêa, F. A. Farret and L. N. Canha, “An analysis of the dynamic
performance of proton exchange membrane fuel cells using an
electromechanical model,” in Proc. of the IEEE - Industrial Electronics
Conference 2001 - IECON’01, pp. 141-146.
R. F. Mann, J.C. Amphlett, M. A. I. Hooper, H. M. Jensen, B. A.
Peppley and P. R. Roberge, “Development and application of a
9
generalized steady-state electrochemical model for a PEM fuel cell,”
Journal of Power Sources 86 (2000), pp. 173-180.
[3] J. J. Baschuck and X. Li, “Modeling of polymer electrolyte membrane
fuel cells with variable degrees of water flooding,” Journal of Power
Sources 86 (2000), pp. 181-196.
[4] J. E. Larminie and A. Dicks, Fuel Cell Systems Explained. John Wiley &
Sons, Chichester, England, 2000, p. 308.
[5] J. C. Amphlett, R.F. Mann, B. A. Peppley, P.R. Roberge and A.
Rodrigues, “A model predicting transient responses of proton exchange
membrane fuel cells,” Journal of Power Sources 61 (1996), pp. 183188.
[6] J. Padullés, G. W. Ault and J. R. McDonald, “An integrated SOFC plant
dynamic model for power systems simulation,” Journal of Power
Sources 86 (2000), pp. 495-500.
[7] D. Chu and R. Jiang, “Performance of polymer electrolyte membrane
fuel cell (PEMFC) stacks - part I. evaluation and simulation of an airbreathing PEMFC stack,” Journal of Power Sources 83 (1999), pp. 128133.
[8] F. Laurencelle, et al, “Characterization of a ballard MK5-E proton
exchange membrane fuel cell stack,” Fuel Cells 2001, Vol. 1; No. 1, pp.
66-71.
[9] BCS Technology Co, Data sheet for a 500 W FC stack; 2001.
[10] J. Choi, J. W. Harvey, and M. H. Conklin, “Use of multi-parameter
sensitivity analysis to determine relative importance of factors
influencing natural attenuation of mining contaminants,” in Proc. of the
U.S. Geological Survey Toxic Substances Hydrology Program Technical Meeting, Charleston, South Carolina, USA, March 8-12,
1999; Vol. 1, Section C, pp. 185 – 192.
[11] D. G. Cacuci, “Global optimization and sensitivity analysis,” Nuclear
Science Eng.; No. 104 (78), 1990.
VII. BIOGRAPHIES
Jeferson M. Corrêa (S’1995) was born in Augusto
Pestana, RS, Brazil, in 1972. He received the B.Sc. in
Electrical Engineering from the Federal University of Santa
Maria, Brazil, in 1997, as well as the M.Sc. Degree in 2002,
and began his Ph.D. in 2002. Currently, he is doing
research at the Colorado School of Mines, USA, working
with integration of renewable energy sources and with
power quality improvement. He has been supported by the National Science
Foundation (NSF - USA) and Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior (CAPES - Brazil).
His employment experience includes General Motors do Brasil Ltda, Philip
Morris Brasil SA and WEG Automação Ltda. For these companies, he
worked primarily as a Manufacturing and Maintenance Engineer, developing
factory automation systems and predictive and preventive maintenance
coordination.
For his superior performance upon graduation, Correa was honored with
awards from the Brazilian agency CACISM and the Brazilian company CRT.
He has received awards for paper presentations, and has recently received the
IEEE Student Travel Grant to attend the 37th Annual IAS, 2002, Pittsburgh,
USA.
His research interests include control systems, renewable energy sources,
modeling and simulation, industrial automation and power electronics.
Felix A. Farret received his B.E and M.Sc. in Electrical
Engineering from the Federal University of Santa Maria in
1972 and 1986, respectively. He received a M.Sc from the
University of Manchester, UK and his Ph.D. from the
University of London, UK. His educational background is
interdisciplinary and related to power electronics, power
systems, nonlinear controls and renewable energy
conversion. Since 1974, Farnet has taught for the Department of Electronics
and Computation of the Federal University of Santa Maria, Brazil. Currently,
he is committed to undergraduate and graduate teaching and to research
activities. He published his first book in Portuguese: Use of Small Sources of
Electrical Energy, published by the UFSM University Press. He is presently
focused on Energy Engineering Systems for industrial application. Farnat is a
visiting professor at Colorado School of Mines, Engineering Division, USA.
In more recent years, several technological processes in renewable sources
of energy were coordinated by Dr Farret and transferred to Brazilian
enterprises such as AES-South Energy Distributor, Hydro Electrical Power
Plant Generation of Nova Palma and CCE Power Control Engineering Ltd
related to integration of micro power plants from distinct primary sources;
voltage and speed control by the load for induction generators; low power
PEM fuel cell application and its model development. Injection of electrical
power into to the grid is currently his major interest. In Brazil, he has been
developing several intelligent systems for industrial applications related to
injection, location and sizing of renewable sources of energy for distribution
and industrial systems including fuel cells, hydropower, wind power,
photovoltaic applications and other ac-ac and dc-ac links.
Vladimir A. Popov was born in Kiev (USSR), in 1951. He
received his M. Sc. and Ph. D. degrees in Electrical
Engineering from the Kiev Polytechnic Institute in 1977 and
1985, respectively. Since 1985, he has been Assistant
Professor and Associate Professor with the Power Supply
Systems Department at the National Technical University of
Ukraine (Kiev Polytechnic Institute). His research interests
include use of soft computing methods in modelling and control of power
systems. Currently he is Visiting Professor at the Federal University of Santa
Maria (Brazil). V. Popov is the author of more than 150 technical papers.
Marcelo G. Simões received the B.E. degree from University of São Paulo,
Brazil, the M.Sc. degree from University of São Paulo,
Brazil, and the Ph.D. degree from The University of
Tennessee, USA, in 1985, 1990 and 1995, respectively
and his D. Sc. degree (Livre-Docência) from the
University of São Paulo in 1998. He is IEEE Senior
Member. He joined the faculty of Colorado School of
Mines in 2000 and has been working to establish research
and education activities in the development of intelligent control for high
power electronics applications in renewable and distributed energy systems.
Dr. Simões is a recipient of a NSF - Faculty Early Career Development
(CAREER) in 2002. It is the NSF’s most prestigious award for new faculty
members, recognizing activities of teacher-scholars who are considered most
likely to become the academic leaders of the 21st century. Dr. Simões is
serving as IEEE Power Electronics Society Intersociety Chairman, AssociateEditor for Energy Conversion as well as Editor for Intelligent Systems of
IEEE Transactions on Aerospace and Electronic Systems and also serving as
Associate-Editor for Power Electronics in Drives of IEEE Transactions on
Power Electronics. He has been actively involved in the Steering and
Organization Committee of the IEEE / DOE / DOD 2003 International Future
Energy Challenge.