Modeling and control of fuel cell-battery hybrid power systems for portable electronics

Abstract

Hybrid power sources can combine the high power density of batteries with the high energy density of fuel cells, in order to improve runtime of portable electronic devices. In this work a non-linear coupled electro-chemical model for predicting the steady-state and dynamic performances of a micro fuel cell is presented. This model is implemented in Matlab/Simulink^reg in order to simulate a hybrid system (Li-ion cell+direct methanol fuel cell) under variable load and supply conditions.

Full text

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Modeling and Control of Fuel Cell-Battery Hybrid
Power Systems for Portable Electronics

Piergiorgio Alotto, Massimo Guarnieri, Federico Moro
Dipartimento di Ingegneria Elettrica, Università di Padova, Italy
moro@die.unipd.it


Abstract-Hybrid power sources can combine the high power
density of batteries with the high energy density of fuel cells, in
order to improve runtime of portable electronic devices. In this
work a non-linear coupled electro-chemical model for predicting
the steady-state and dynamic performances of a micro fuel cell is
presented. This model is implemented in Matlab/Simulink® in
order to simulate a hybrid system (Li-ion cell+direct methanol
fuel cell) under variable load and supply conditions.
I.
INTRODUCTION
Hybrid power sources composed of fuel cells (FC) and
secondary batteries (Bh) can combine the high energy density
of fuel cells with the high power density of batteries, in order
to improve operational lifetime of portable electronic devices
such as laptops, mobile phones, or PDAs [1][2]. Small-scale
direct methanol fuel cells (μ-DMFCs) are particularly suited
in that case, since they exhibit a very simple structure, and do
not require any additional external device for fuel reforming
or conditioning. Moreover, μ-DMFCs can operate at room
temperature, and (in principle) with 4-6 times the energy
density of Li-ion cells [3]. A limited operating voltage and
load following capability are the main drawbacks, which can
be overcome by coupling DMFCs and Li-ion cells. The latter
can provide a good dynamics, and can supply peak loads.
A non-linear equivalent circuit for modeling steady state
and transient dynamics of μ-DMFCs is presented (Section II).
The circuit model is implemented in Matlab/Simulink® for
simulating the dynamic performance over the short and long
time scales of a FC/Bh hybrid system (Section III).

II. DIRECT METHANOL FUEL CELL MODELING
Direct methanol fuel cells generate electric energy from
methanol oxidation. Fig. 1 shows that, inside DMFCs, a
polymeric proton exchange membrane (PEM) is sandwiched
between anode and cathode electrodes. In passive DMFCs
fuel is stored in a tank, while oxygen is taken from the air.
Reactants flow to catalyst layers through diffusion layers,
where electro-chemical energy conversion occurs. Electric
power is generated by the electron flow, which is driven to
the external circuit by metallic current collectors.
The multiphysics model takes into account the following
phenomena: electrochemical reactions and electric current
generation inside catalyst layers, electronic and protonic
conduction, methanol crossover through the PEM, diffusion
of reactants inside the diffusion layers.

Fig.1. DMFC schematic (a=anode, c=cathode, pem=proton exchange
membrane, dl=diffusion layer, cl=catalyst layer).
A. Static modeling of a μ-DMFC
The electric steady-state external characteristic of DMFC
is obtained by coupling mass transport and electro-chemical
relations under the assumption of one-dimensional geometry.
The external circuit is connected to the fuel cell by the
following generalized continuity equations:

+
+⋅∇
J
−
g
−−
+
g
+
∂=∂+⋅∇
∂=∂
tt
tt
ρρ
ρρ
J
(1)

where superscripts indicate protons and electrons,
stored/ generated charge densities and J the current densities.
The anodic and cathodic current densities are computed by
means of Butler-Volmer’s equations, written as [4]:

/(
aaa
CJJ
=

where concentrations
c
C , and over-voltages
variables, and the other quantities are constant.
Reactant concentrations at both catalyst layers depend on
the diffusion rate across diffusion layers by Fick’s law:
g
ρρ,
the
)exp()/(
)exp()
**
**
cccccc
aaa
vfC
vfCCJJ
α
α
γ
γ
=
(2)
aC
cavv , are

CD∇−=
N
(3)

where N is the reactant molar flow, and D is the diffusivity.

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Using (3) methanol flow from the tank can be expressed as:

KN
=

assuming very thin layers and a one-dimensional mass flow.
Similarly, oxygen flow at the cathode can be expressed as:

(
c cd
KN
=

where
c
K ,are the mass transfer coefficients,
C, 0 the methanol and oxygen concentrations in the tank and
in the ambient, and
cc
C , those at catalyst layers.
Due to electro-osmosis and concentration gradient effects,
part of the methanol passes through PEM, causing significant
voltage loss and waste of fuel. This is the so-called methanol
crossover, which is related to the anode current density as:

6
a
J
=

where F is the Faraday’s constant (96.485 C mol-1). Current
density reacting at the cathode side can be expressed as:

4
c
J
=

The methanol crossover in (6) and (7) can be computed by
means of the following mass balance equation:

N
=

where
m
D is the methanol diffusivity on the membrane, and
d n the electro-osmotic drag coefficient.
Anode activation over-voltage is obtained by combining
(4) and (6) with (2). This can be expressed as:

1
)(
a
f
α

where the anode equivalent concentration and the limiting
current values are:

K
C
,
+δ

where
m
δ is the thickness of the PEM (Fig. 1). Similarly, the
cathode over-voltage can be computed from (5) and (7), as:

1
)(
c
f
α

where the equivalent cathode concentration and the limiting
current values are:
)(
, 0acaaad
CC
−
(4)
)
, 0 ccc
CC
−
(5)
aK
a
C, 0 and
c
acC
)(
m ad
NNF
−
(6)
)(
2
3
m ad
NNF
−
(7)
FnCD
admm
/J
+∇
(8)
)/ 1 (
/
log
lim,,
*
a
*
a
aa eq
a
a
JJC
JJ
−
C
Jv
=
(9)
a
mma
ma
a eq
C
DK
, 0
=
δ
,
F
n
F
aa
a
d
C
+
K
J
=
6
1
, 0
lim,

)/1 (
c
/
log
lim,,
*
c
J
*
c
ceq
c
c
JC
JJ
−
C
Jv
=
(10)
a
c
a
mma
m
+
cc
,
eq
C
K
K
DK
D
CC
, 0 , 0
2
3
−=
δ
,
d
FF
aa
n
eqc
+
eqc
c
CCCK
J
2
3
4
1
, 0,,
lim,
)/(
=


Anode and cathode current densities in (9) and (10) can be
related to load current densityJ on the external circuit, as:

J
mc
a
FNJJ
J
6
+=
=
(11)

which state that the anodic electron flow equates the PEM
proton flow, while the cathodic electron flow balances also
proton flow deriving from methanol crossover oxidation.
Finally, the fuel cell voltage between collectors is obtained
from anode and cathode over-voltages in (9) and (10), as

JV
=
)(

where
()()(vJvEJE
ca eq
−−=
the standard cell voltage,
eq
R
resistance,
m
σ the PEM non-linear conductivity, and
contact resistance of collectors and diffusion layers.
The electric behavior is finally given by the external
characteristic V-J computed from (12) for increasing inlet
methanol concentrations in the reservoir (Fig.2).
JRJE
eqeq
−
)(
(12)
)
=
0
J
is the equivalent fem,
+σδ /
0
E
cmm
R
the equivalent
c
R the
0 20 40 6080100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
current density [mA/cm2]
fuel cell voltage [V]
C0,a = 0.1 - 4 M

Fig.2. External characteristics for increasing inlet methanol concentrations.

B. Dynamic modeling of a μ-DMFC
The fuel cell dynamics on the long time scale–i.e., on a few
hours– is dominated by the methanol consumption in the
tank, which is computed by the mass conservation law [4]:

+⋅∇
N

The state variable model is obtained by assembling (4), (6)
and (13) into the following PDE system:

M
1

0
=∂
C
t
(13)
gx
?
Mx
=+
2
(14)

Page 3

where:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−−
=
ac
ad
a
/
a
mmaa
C
CFJnFJ
KK
DKK
, 0
0
21
0
6/
0
00/
xg
MM
δ
δ


and
0
δ is the tank thickness (Fig. 1).
System (14) is solved numerically by the θ-method,
which consists in the following iterative scheme [6]:

M
M
⎢⎣
⎠⎝
τ

where τ is the time integration step, and parameterθ is set to
2/3 in order to ensure unconditional numerical stability.
Fig. 3 shows voltage discharges for different load current
densities, estimated with an initial concentration
gx
M
τ
Mx
+
⎥⎦
⎤
⎡
+−=
⎟
⎞
⎜
⎛
+
+
kk
θ
(
θ
2
11
2
1
) 1
(15)
a
C, 0= 3 M.
00.51 1.522.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
time [h]
fuel cell voltage [V]


250 A m-2
500 A m-2
750 A m-2
1000 A m-2

Fig.3. Voltage discharge profiles at constant current densities.

According to [5], the electric dynamic of the DMFC on
small time scales-- -a few micro seconds-- -can be assumed to be
dependent mainly on the charge double layer effect, i.e.,
charge distributions with different polarities formed across
the boundary between porous catalyst layers and PEM.
The charge double layer behaves like a super-capacitor,
showing a very large capacitance C due to the high degree of
porosity in substrates. Hence, over-voltages vary under
dynamic load changes according to the following equation:

t
v
CI
ICRICR
v
+
acaa
∂
∂
−=
),(),(
, 0, 0
(16)

where

IvR
aa
/
=
, IvR
cc
/
=
are non-linear resistances.
III.
HYBRID POWER SYSTEM
The operating principle of a hybrid FC-Bh power source
can be described as follows. During periods of low power
demand, the DMFC is controlled to supply the (average) load
power demand, charging the Li-ion battery at the same time.
On the other hand, during periods of high power demand, the
fuel cell generates only the rated power and battery provides
the peak power [7]. Li-ion battery acts like a storage device,
allowing DMFC to be operated at constant discharge current
density (which is important for extending fuel cell lifetime).
Fig. 4 shows the layout of the proposed hybrid system. A
synchronous boost converter steps the output DMFC voltage
up to DC-bus voltage (typically 3.3 V for electronic devices).
In real applications the fuel cell is made up of 4-6 stacks,
connected in series, in order to attain acceptable input voltage
levels on full load (2-2.4 V). On the battery side, an H-bridge
buck-boost converter manages the battery charge/discharge.
H-bridge converters, an emerging solution in analog-device
design, can operate either as a buck converter (when the
Li-ion battery is fully charged and its voltage is greater than
DC-link voltage), or a boost converter (when cell is almost
discharged and its voltage drops below the DC-link voltage).
The controller regulates power flows among the fuel cell, the
battery, and the electronic load, by varying the duty cycle at
MOSFET transistor gates in power converters.
The hybrid system was modeled in Matlab/Simulink®, by
using two different approaches for simulating the dynamics
on long and short time scales respectively.
A. Slow dynamics modeling of the hybrid system
Fig. 5 shows the Simulink® model for simulating the hybrid
system dynamics on long time scales. The electric behavior of
the DMFC is modeled by means of a look-up table, which
provides the output signal voltage (of a single stack) for a
given current density and methanol concentration.

Fig.4. Schematic of the hybrid power system (small-scale DMFC and Li-ion battery).

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Fig.5. Simulink® model of the DMFC for long time-scale dynamics.

Polarization curves, i.e., the input data of the look-up table,
are computed for inlet methanol concentration values ranging
from 0.1 to 4 M, as shown in Fig.2. By using a saturation
block, the output voltage signal for small current loads is
approximated to the open circuit voltage of the DMFC, which
is evaluated experimentally (about 0.6 V for a single stack).
The signal from the look-up table block is converted into a
power signal by using a controlled voltage source generator.
The discharge process is modeled by using a state-space
system, which implements the PDE system described in (14).
The slow dynamics model is fully non-linear as the discharge
rate of the fuel cell is determined by the reactant flow, which
depends on the load current density. This non-linear behavior
can be modeled in Simulink® by using algebraic loops, which
provides a feedback from the power circuit. Current through
DMFC is measured at the controlled voltage generator. This
signal is transformed into a current density signal, which is
used either to evaluate the voltage signal from look-up table,
or the concentration signal from state-space block.
Fig. 6 shows the Simulink® model of the hybrid system,
including the DMFC model presented above. For slow circuit
dynamics, both power converters can be modeled by using
controlled current generators. Both converters are assumed to
be operated with a conversion efficiency of 90% (for rated
loads). Input signals of current generators are calculated by
using the power conservation principle and taking into
account power losses. For the boost converter, linking DMFC
to the electronic load, the output converter current becomes:

o
ii
o
U
IU
I
11
1
9 . 0
=
(17)

where
and
varies during the DMFC discharge, as the input current and
the output voltage are set constant by the control system. The
DC-link is modeled as a constant voltage generator (Fig. 6).
Similar considerations apply for evaluating the gate
signal for the controlled current generator representing the
H-bridge converter. This can be computed by:

1 i
U is the output FC voltage,
o
U the DC-bus voltage,
1 iI the (constant) FC current. The output current (17)
2
2
2
9 . 0
i
oo
i
U
IU
I
=
(18)

where
output current flowing to the load. Differently from (17), the
input converter current, i.e., the current flowing through the
battery, is taken as dependent variable. Therefore, the Li-ion
cell is operated as a storage device, supplying the load power
demand or adsorbing the exceeding power from the DMFC.
The dynamic behavior of rechargeable battery is simulated by
using the model of the SymPower® library in Simulink®.
A pulsed profile is used to simulate the varying load
demand of an electronic device such as a cell phone (Fig. 8).
The average power supply is provided by a 6 stack fuel cell
with 10 cm2 cross section and 0.2 A output current.
2 i
U is the fuel cell voltage, and
2oI is the converter
012345678
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
time [h]
electronic load current [A]

Fig. 8. Pulsed load profile used for simulating the electronic device.

Typical parameters of cell phone battery are considered for
the Li-ion cell (nominal voltage 3.6 V, capacity 750mAh).
Fig. 9 show the DMFC discharge characteristic computed
by the proposed model; the same profile, computed for the
Li-ion battery, shows that the storage device is recharged
when the electronic load is in standby (Fig. 10).

Fig.6. Simulink® model of the hybrid system for long time-scale dynamics.

Page 5

012345678
0
0.5
1
1.5
2
2.5
time [h]
Fuel cell voltage [V]

Fig. 9. Fuel cell voltage profile during the discharge.

4.3
012345678
3.6
3.7
3.8
3.9
4
4.1
4.2
time [h]
Li-ion battery voltage [V]

Fig. 10. Li-ion battery voltage profile during the discharge.

B. Fast dynamics modeling of the hybrid system
The previous model cannot analyze system start-up or fast
load transients. More detailed models of both converters and
power sources must be considered at this aim (Fig. 11).
As an example, the dynamic behavior of the hybrid system
at startup is examined. On short time scales, the Li-ion battery
is assumed to be not exchanging power with the system, so
that buck-boost converter has not to be modeled. The double
layer charging effect of the DMFC is accounted for by an
equivalent capacitance
SC
ε
/
=
section (10 cm2), and thickness
frequency of the boost converter is 1.2 MHz, according to last
component data sheets. The rated input and output voltages
are 2.4 V (corresponding to a 6 stack DMFC), and 3.3 V
respectively. The boost converter is assumed to be operated in
continuous conduction mode (CCM). The average duty cycle
is δ=0.27, for an average conversion ratio MCCM=1.375. Input
m
δ
δ is 180 μm. The switching
, where S is the cell cross
m
and output capacitances are 10 μF in order to limit voltage
ripples at 1 % of nominal voltages. The boost inductance is 8
μH, so that current ripple is 40 %. Output voltage and fuel
cell current are both regulated by a peak-current control mode
(PCCM), with a PI controller. Voltage overshoots are
overridden by precharging capacitors at the initial input
voltage (2.4 V). Fig. 12 shows that a smooth voltage profile at
DC-link is obtained: by using the peak-current controller the
reference voltage is attained after a few micro seconds.

3.6
020406080100
2.2
2.4
2.6
2.8
3
3.2
3.4
time [μs]
DC-link voltage [V]

Fig. 12. Pulsed load profile used for simulating the electronic device.

IV.
CONCLUSION
Dynamic models of FC-Bh hybrid system for simulating
performance over short/long time scales have been presented.
Numerical analyses show that a combined used of DMFC and
Li-ion cell can extend effectively runtime of portable devices.
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Fig. 11. Simulink® model of the hybrid system for short time-scale dynamics.