(561b) Control of a Proton Exchange Membrane Fuel Cell, DC-DC Power Converter, an Ultracapacitor System Using Sliding Mode Controls

In this paper we consider control of a power system that comprises three
controlled components: Proton Exchange Membrane fuel cell (PEMFC), ultracapacitor, and a boost DC-DC power converter, using
sliding mode control techniques.  Control of fuel cells using sliding mode control (SMC)
technique (Levant, A., (1993),
Fridman, L. and Levant, A., (2002) and Shtessel, Y., et. al., (2011)) that is robust to the fuel cell model
uncertainties and the external disturbances is studied in Sabanovic, A., et. al., (2004). Sliding mode
control design for power systems, which consists of fuel cell and boosts DC-DC
power converter, is presented in (Spurgeon
S. and Davies, R., (1993), Shtessel, Y. and Ashok, R.S., (2011)).  The use of
ultracapacitor was addressed in the work (T. Azib, et. al.,
2010). 
All three subsystems are controlled to provide a required power
system's performance.

I. PHYSICAL DESCRIPTION OF PEMFC

The schematic of a Proton Exchange Membrane (PEM) fuel
cell has two electrodes (anode and cathode) isolated by a membrane acting as an
electrolyte as shown in Figure 1. (Matraji, I. 2010).

Figure 1. Schematic of a Fuel Cell Operation

The performance of a PEM fuel cell increases, when its
operating temperature is around 70-80⁰C at a partial reactant
pressure of 3-5atm. The chemical reactions in PEMFC can be described by the
following equations (Larminie J. and Dicks, A. (2003),
Kunusch, C. (2006)).

 QUOTE
 
                                                                                                      (1)

II.
MATHEMATICAL MODELS

The
mathematical model of the fuel cell electric power unit is derived dynamically,
by combining the thermodynamic flow rate, boost DC-DC power converter and the
ultra-capacitor. A typical PEMFC has the following V-I characteristic or
polarization curve as shown in Figure 2.

Figure 2. V-I Characteristics of
the Fuel Cell

The voltage across the fuel cell (
) is a function of the stack current, partial pressure
of reactants, temperature and humidity of the membrane.

                                                                                                             (2)

where
 the number of
cells in the stack,
 the total
voltage across the stack,
  the
thermodynamic cell potential,
activation overvoltage,
 ohmic voltage drop, and
 the mass
transport voltage drop or the concentration.

A.
Reversible Cell Voltage 

The
open circuit voltage
, (T. Azib, et. al., 2010.), is calculated using

                                                                                          (3)

where
 is a change in
entropy,
 Faraday's
constant,
 universal gas
constant,
,
 partial
pressure of hydrogen in the cell,
 partial pressure f oxygen in the cell,
  partial pressure of water and
 the
standard  temperature (
).  The operating
temperature
is in
. The sum of the partial pressures equals the total pressure
.

B. Activation Over Potential

Slow transfer of charge at the electrodes
causes this loss.  A part of the
electrode potential is used for electron transfer to match the current
demand.  Hence the voltage at the
fuel cell drops by;

                                                                                                                            (4)

where
 is a
double layer capacitance, and
 is the
equivalent resistor to activation.

The double layer of charge (dl) is needed to understand the dynamics
of the fuel cell. Whenever two materials of opposite charge come in contact,
there is an accumulation of the charge on the surface. The
charge layer that is deposited on the interface of the electrode or electrolyte
acts as storage of the electrical charges and acts as an electrical capacitor
 (Correa, J., et. al., (2004)).

C. Ohmic Over Potential

The ohmic over
potential
 results from the
resistance to electron transfer in the electrolyte and is modeled by;

                                                                                                                                         (5)

       
 is the equivalent resistor to the ohmic
over potential, and
 is the current
through the fuel cell.

D. Concentration over potential

The electrical current and the physical
characteristics of the system are directly proportional to the pressures of the
oxygen and hydrogen. Maximum current density is used to determine the voltage
drop, under which the fuel is being used at the same rate.  Thus, the drop in voltage is caused due
to the mass transport,

                                                                                                                                  (6)

where coefficients
vary with the temperature and are given to by;

.

A
stack of 300 fuel cell units are used, along with an Ultracapacitor, are the
primary source of electrical energy for the boost DC-DC power converter.  The equivalent circuit (Larminie J. and Dicks,
A. (2003), Kunusch,
C. (2006)) is presented in Figure 3 below;

Figure 3. Circuit Diagram of Fuel Cell System

     
→   current
of the fuel cell, same as
, the inductance current,

 →  resistance of the load,

 → inductance of the load,

 →   charge build up due to diffusion
and reactions between the electrons in the electrode and protons in the    electrolyte,

→ resistance that
causes activation loss (
) due to slow rate of the reaction,

→ resistance that causes the ohmic
over potential (
)

→ resistance that causes the drop in concentration of
reactants at the reaction sites

 → total voltage across the fuel cell,

  →  output load voltage of the DC/DC
convertor,

 QUOTE
 
→    current through the load
inductance.
III. SIMULATIONS

The
system of electrical energy supply consists of the boost DC-DC power converter,
which receives electrical energy from a stack of 300 fuel cell elements and is
controlled by SMC and ASTW control is simulated. The parameters of the model
are presented in Table 1.

Table 1. Model Parameters

The load
voltage command profiles were selected in accordance with (Sabanovic, A., et.
al., (2004), Spurgeon S. and Davies, R., (1993) and Utkin, V., et.
al., (1999)):

                                                                                                  (7)

The
load resistor was changed at
:

                                                                                                                 (8)

The
fuel cell varying resistor changed its value at

                                                                                                          (9)

The simulation plots are presented in Figures
4-10.  The plots on Figure 4
illustrate high accuracy direct tracking of the load voltage command profile
via classical sliding mode control (Figure 8) in the presence of unknown load
resistor while the voltage command profile is changing twice during simulation
time. High accuracy tracking of the fuel cell current command profile that is
generated on line is confirmed via plots in Figure 5.  The ultracapacitor current
is shown in Figure 6. 
The time history of adaptive control gains is demonstrated in Figures 7
and 8.  The control functions of
the PEMFC current and DC-DC boost converter are shown in Figures 9 and 10.

IV  CONCLUSIONS

 Two types of sliding mode control
(traditional SMC and adaptive-gain 2-SMC, continuous super-twisting control)
are used at the same time for controlling PEMFC power system. It is observed
that a three-fold SMC feedback control structure (adaptive 2-SMC PEMFC current
and traditional SMC voltage and ultracapacitor current controllers) avoids
non-minimum phase property of boost DC-DC power converter, which output voltage
is controlled directly and simplifies the controller design.  The ultracapacitor not only acts as an
auxiliary power source when there is an interruption in power supply but also
responds rapidly to a fast load demands. 
The energy produced from a fuel cell is managed efficiently using
appropriate Sliding mode control techniques.  The efficacy and robustness of the SMC and 2-SMC controllers
are confirmed via computer simulations.

V REFERENCES

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