IGEC-1 Proceedings of the International Green Energy Conference 12-16 June 2005, Waterloo, Ontario, Canada Paper No. 126
OPTIMIZATION OF A FUEL CELL SYSTEM BASED ON EMPIRICAL DATA OF A PEM FUEL CELL STACK AND THE GENERALIZED ELECTROCHEMICAL MODEL J. Wishart, M. Secanell, and Z. Dong* Department of Mechanical Engineering and Institute for Integrated Energy Systems (IESVic) University of Victoria Victoria, BC, Canada V8W 3P6 ∗ Email:
[email protected] G. Wang School of Mechanical Science and Engineering Jilin University, Changchun, China
ABSTRACT A fuel cell system model is implemented in MATLAB in order to optimize the system operating conditions. The implemented fuel cell model is a modified version of the semi-empirical model introduced by researchers at the Royal Military College of Canada. In addition, in order to model the whole fuel cell system, heat transfer and gas flow considerations and the associated Balance of Plant (BOP) components are incorporated into the model. System design optimizations are carried out using three different methods, including the sequential quadratic programming (SQP) local optimization algorithm and simulated annealing (SA) and genetic algorithm (GA) global optimization algorithms. Using the operating conditions of the fuel cell system as the design variables, the net output power of the system is optimized. The three methods are used in order to gain some insight into the nature of the objective function and the performance of the different algorithms. The optimization results show a good agreement and provide useful information on the design optimization problem. This study prepares us for more complex modeling and system optimization research. INTRODUCTION As a promising technology that may successfully supersede the combustion of fossil fuels as the dominant method of energy-generation, hydrogen fuel cells are studied worldwide with an aim to improve the power output and lower the cost for wide-spread applications. Among various types of fuel cells, the Proton Exchange Membrane Fuel Cell (PEMFC) is arguably the fastestgrowing type and the fuel cell that is most likely to be widely used in the near future. The modeling and optimization of PEMFC system, carried out in this work, is aimed at achieving better fuel cell system designs. Modeling of real-world applications has been seen as a useful tool for decades. Fuel cell modeling is in its relative infancy, but already a significant amount of effort has been put forth to understand the parameters and issues affecting the performance of the fuel cell.
Fuel cell modeling requires a broad skill base, as electrochemical, thermodynamic and fluid dynamic relationships must be combined with heat transfer and mass and energy balance equations to construct an appropriate model (Heraldsson and Wipke, 2004). A balance must be found between simplifying assumptions that compromise the accuracy of the model and increasing complexity that makes the model an unworkable behemoth. Given the dubious credibility of the current fuel cell models, contemporary fuel cell optimization results must be met with skepticism. Nevertheless, some laudable attempts at fuel cell optimization can be found in the literature. Xue and Dong (1998) used a semi-empirical model of the Ballard Mark IV fuel cell and models for the auxiliary systems to create a model of the fuel cell system. The optimal active stack intersection area and air stoichiometric ratio to maximize net power output is achieved, and, at the same time, minimize production costs. Grujicic and Chittajallu (2004) used a 2D computational fuel cell dynamics model to optimize the electric current per fuel-cell width at a cell voltage of 0.7V. In the optimization sequential quadratic programming was used to obtain the operational and geometric parameters for achieving the maximum electric current, including air inlet pressures and cathode thickness, cathode length for each shoulder segment of flow channel, and fraction of cathode length associated with the flow channel. In general, optimization of fuel cell systems is still a challenge not only because of the inaccuracy of the models but because the optimization is a highly non-linear problem where the objective function is obtained using a numerical model of the fuel cell and fuel cell system. Nonlinear optimization involves the search for a minimum of a non-linear objective function subject to non-linear constraints. It is common for these optimization problems to have multiple optima. Due to this difficulty, two different approaches have emerged in the area of non-linear optimization: local methods and global methods. Local methods aim to obtain a local minimum, and they cannot guarantee that the minimum obtained is the 1
absolute minimum for a non-unimodal objective function and/or a non-convex feasible region. These methods are usually first-order methods. Some of the most popular local methods for optimization include the conjugate gradient algorithms and the quasi-Newton methods for unconstrained optimization and the sequential linear and quadratic programming methods for constrained optimization. Although local methods do not aim for the global optima, several approaches can be used to continue searching once a local minimum has been obtained to obtain the global optima, such as the stochastic based approaches of random multi-start methods (Schoen, 1991) (He and Polak 1993) and ant colony searches (Dorigo et al, 1996). Global methods aim to obtain the absolute minimum of the function. Mostly based on stochastic procedures, these methods do not need any information about the gradient. Some of the most popular methods for global optimization are Genetic Algorithms (GA), Simulated Annealing (SA), Tabu Search and Stochastic Programming. In this work, one local optimization algorithm and two global optimization algorithms will be coupled to a fuel cell system model. The optimization algorithm will use only operating parameters such as the temperature, pressure ratios and the stoichiometries of the reactants as design variables. A dedicated mathematical model is used to assess the performance of the fuel cell system. The optimization routines are used to search for the best operating parameters that lead to peak fuel cell system performance. Three optimization techniques are used: Simulated Annealing (SA), Genetic Algorithms (GA) and Sequential Quadratic Programming (SQP). A description of the optimization setup is given and the results for SA, GA and SQP optimizations are discussed. MODEL OF THE FUEL CELL SYSTEM The fuel cell model implemented in this work is based on the work at the Royal Military College of Canada by Amphlett, et al (1995), Mann, et al (2000) and Fowler, et al (2002). The model, known as the Generalized SteadyState Electrochemical Model (GSSEM), is zerodimensional, semi-empirical and static in nature, thus the parameters of the equations are determined experimentally to provide the time-independent polarisation curves, power curves and system efficiencies at various operating conditions, and the model is applicable to an entire fuel cell stack. The voltage of the fuel cell is modeled as (Fowler, et al, 2002):
VCell = ENernst + ηact + ηohmic + ηconc
(1)
where ENernst is the Nernst equation, which is an expression for the electromotive force (EMF) for given product and reactant activities; ηact is the activation overvoltage, which is the amount of voltage used to drive the reaction; ηohmic is the ohmic overvoltage, which is the amount of voltage lost to the resistance to electron flow in the electrodes and the resistance to ion flow in the
electrolyte; ηconc is the concentration overvoltage, which is the voltage lost when the concentration of reactant at the electrode is diminished. The terms of Eq.(1) are discussed in the following sections. Nernst Voltage The Nernst equation for the reaction described above is given by Mann, et al (2000): ENernst = 1.229 − 0.85 ⋅10−3 (T − 298.15 ) + 4.3085 ⋅10−5 ⋅ (2) interface T ln pHinterface + 0.5ln p O2 2
( (
)
))
(
where T is the stack temperature (in K), and
pOinterface 2
and pHinterface 2
are the hydrogen and oxygen partial gas
pressures (in atm) at the surface of the catalyst at the anode and cathode, respectively. It should be noted that there is no term for the partial pressure of the water product in Eq.(2). The assumption has been made that the water product is in pure liquid form, and that a thin film of liquid water covers the catalyst and allows the reactants to diffuse through the water film. Future research will be required to elucidate the effects that the vapour portion of the product may have on the Nernst voltage equation. It is further noted that the expression in Eq.(2) for the Nernst voltage incorporates the voltage loss due to fuel crossover (where H2 passes through the electrolyte without reacting) and internal current (where electrons pass through the membrane rather than through the electrodes). The partial pressures at the catalyst surface are assumed to be the same across the entire cell, and are given by Dong (2004a):
p
Interface H2
(
= 0.5 p
sat H 2O
)
⎡ ⎤ 1 ⎢ ⎥ − 1⎥ ⎢ 1.653i ⎢ e T 1.334 ⋅ xHchannel ⎥ 2O ⎣ ⎦
(3)
and 0.291i ⎡ ⎤ sat channel T 0.832 pOInterface P 1 x x e = − − ⋅ ⎢ ⎥ H 2O other gases 2 ⎣ ⎦
where
xHsat2O
(4)
is the molar fraction of water in a gas stream
at saturation for a given temperature,
channel xother gases
is the
molar fraction of other gases (apart from oxygen) in the air stream,
pHsat2O
is the saturation pressure of water vapour
at a given temperature, and i is the current density. The molar fraction of water at saturation in a gas stream for a given temperature is given by Dong (2004a):
xHsat2O =
pHsat2O
(5)
P
where P is the gas pressure.
The
pHsat2O
term is
determined in a fuel cell by the following empirical equation (Dong, 2004a):
2
7362.6981 + T 0.006952085 ⋅ T − 9.0000 ⋅ ln T .
(
)
ln pHsat2O = 70.434643 −
(6)
channel xother gases =
where
x
−x
out , hum other gases
in , hum ⎛ xother gases ln ⎜ out , hum ⎜x ⎝ other gases
(
⎞ ⎟⎟ ⎠
)
in , hum sat xother gases = 1 − x H 2 O ⋅ 0.78
(7)
(8)
and out , hum xother gases =
1 − xHsat2O ⎛ λ − 1 ⎞ ⎛ 0.21 ⎞ 1 + ⎜ air ⎟⎜ ⎟ ⎝ λair ⎠ ⎝ 0.79 ⎠
(9)
In Eqs. (8) and (9), the 0.79 term refers to the dry molar fraction of other gases in air, while in Eq.(9), the 0.21 term refers to the dry molar fraction of oxygen in air. The λair term denotes the stoichiometry of the air stream. Activation Overvoltage Some of the voltage created by the reaction of Figure 1 is lost to driving the reactions at the electrodes and in driving the flow of electrons either to or from the electrode. This voltage loss is known as activation overvoltage. This loss can be decreased if steps are taken to increase the reaction rates, i.e. by increasing the temperature or pressure in the fuel cell. For a PEMFC, however, which operates in a temperature range of approximately 6585°C, the activation overvoltage irreversibility is significant. The semi-empirical equation for the activation overvoltage is given by Dong (2004a): (10) η act = β1 + β 2T + β 3T ln(COInterface ) + β 4T ln( I ) 2
where
= COInterface 2
pOInterface 2 ⎛ −498 ⎞ ⎜ ⎟ ⎝ T ⎠
(12)
This results in an activation overvoltage term of:
It should be noted that the expression for saturation pressure is a function of cell temperature only. The molar fraction of other gases (mostly nitrogen gas) in the air stream is given by a log mean average between the molar fraction of nitrogen in a humidified stream of air at the inlet and the molar fraction at the outlet (Dong, 2004a): in , hum other gases
β1 = −0.95140 β 2 = 0.003120 β 3 = 0.000074 β 4 = −0.000187
(11)
5.08 × 106 ⋅ e Interface and where the expression for pO is given in Eq.(4) 2 and the β coefficients are empirically determined for each individual fuel cell stack. The coefficients were determined for the Ballard Mark V stack by Xue and Dong (1998). The improvement upon the current methodology for coefficient determination is another potential future area of research. The coefficients used in this work are:
ηact = −0.9514 + 0.00312T + 0.000074T ln(COInterface ) 2
−0.000187T ln( I )
(13)
Ohmic Overvoltage Another portion of the voltage generated by the electrochemical reaction is lost due to the resistance to electron flow in the electrodes and graphite collector plates and resistance to ion flow in the electrolyte. The latter resistance is usually the dominant irreversibility. Choosing MEA materials that have high conductivities can reduce the ohmic overvoltage. However, most PEMFCs use a polymeric electrolyte manufactured by Dupont that is known as Nafion, and thus there is little flexibility in the electrolyte material choice to improve on the ohmic overvoltage significantly. Decreasing the thickness of the electrolyte does reduce its ohmic resistance to ion flow, but the electrolyte must maintain structural integrity and thus cannot be made too thin. The ohmic overvoltage can be expressed in accordance with Ohm’s Law as (Mann, et al 2000): electronic protonic ηohmic = ηohmic + ηohmic
= −i ( R electronic + R protonic )
(14)
where Relectronic is assumed to be a constant over the operation temperature of the PEMFC, but since it is difficult to measure or predict, it is further assumed to be inconsequential in comparison to the Rprotonic, and is thus ignored. Further research may be conducted at a later date to confirm this assumption. The term Rprotonic is known to be a complex function of water content and distribution in the membrane, which in turn is a function of the cell temperature and current. A general expression for the ohmic resistance of the electrolyte is given by (Mann, et al 2000):
R protonic =
rM l A
(15)
where rM is the membrane-specific resistivity for the flow of hydrated protons (in ohm⋅cm), l is the thickness of membrane (in cm) and A is the active cell area (in cm2). The active cell area is given by Dong (2004a): (16) A = 0.56 × At where At is the total stack cross-sectional area. While the other two terms in Eq.(15) are known parameters of a specific cell, the term rM is difficult to describe phenomenologically, and thus the following semi-empirical expression has been derived (Mann, et al 2000): 2 ⎡ ⎛ T ⎞ 2.5 ⎤ 181.6 ⎢1 + 0.03 ( i ) + 0.062 ⎜ ⎟ i ⎥ ⎝ 303 ⎠ ⎢⎣ ⎥⎦ rM = ⎛ T −303 ⎞
[λ − 0.634 − 3i ] ⋅ e
4.18⎜ ⎝
T
(17)
⎟ ⎠
3
where again i is the current density and λ is in this case an adjustable fitting parameter, influenced by the method of manufacture of the membrane and a function of the relative humidity and stoichiometric ratios at the anode and cathode and of the age and use of the membrane. The parameter is usually assigned a value between 10 and 20 (Fowler, et al 2000). Concentration Overvoltage As the fuel and oxygen are extracted from the gas streams at the anode and cathode, respectively, the initial concentration that satisfies the electrochemical reaction is diminished, and there is a voltage loss as a result of the pressure drop of the reactant gases. The overvoltage depends on the amount of current drawn from the cell, as well as the physical characteristics of the gas supply systems. The expression used in the model for the concentration overvoltage is given by Dong (2004a):
⎛ RT I⎞ ⋅ ln ⎜1 − ⎟ 2F ⎝ Il ⎠
ηconc =
(18)
Fuel Cell Gross Output The main outputs of the fuel cell operation are power, water and heat production. The power output of a fuel cell is an important measure of its performance. Much of the current research in the area of fuel cells is focused on attempts to increase the power output while decreasing the manufacturing costs. The gross output of the fuel cell stack (in W) is given by:
Wgross , stack = I × vcell × ncell
(19)
where ncell is the number of cells in the stack. The net power output of the fuel cell stack is a more important consideration when assessing its performance, and is given by:
Wnet output = Wgross , stack × η net
(20)
where ηnet is the electrical efficiency, and is given by:
where
M H 2 is
Wgross , stack ⋅ M H 2 I ⋅ m& H 2 ⋅ HHVH 2
(21)
the molar mass of H2 gas (which has a
value of 2.016×10
-3
kg/mole),
m& H 2 is
m& H 2O , produced = where
M H 2O is
I ⋅ ncell ⋅ M H 2O 2F
(22)
the molar mass of water (which has a
value of 18.018×10-3 kg/mole).
where R is the universal gas constant (which has a value of 8.3145 J/(mole⋅K)), T is the cell temperature, F is Faraday’s constant (which has a value of 96485.34 C/(gmole⋅electron)), I is the total current in the fuel cell and IL is the current at which the fuel is used up at a rate that is equal to its supply rate, hence the current I can never be greater than IL. Eq.(18) has several problems, including the fact that the ηconc term alters the polarization curves in a way that does not match experimental values for PEMFCs that use air as a reactant, which is the case for the model. The development of an accurate concentration overvoltage term that is phenomenologically based is another possible area of research.
ηnet =
gas (in kg/s), and HHV is the Higher Heating Value of enthalpy (which for H2 gas is 285812 J/mole). The production of water is both a benefit and a hindrance to the operation of the fuel cell, as a delicate balance must be struck between having enough moisture to maintain the humidity of the membrane while also ensuring that the MEA is not flooded with excess water. Either situation will result in a reduction in the efficacy of the fuel cell operation. The model used in this work assumes that the membrane is fully hydrated without any water management issues. Water management of PEMFCs is another possible future avenue of research. The water is produced at a rate of two moles for every four electrons released at the anode. The equation for water production is given by (Larminie and Dicks 2003):
the flow rate of H2
Heat Considerations Heat will be generated by the operation of the fuel cell since the enthalpy that is not converted to electrical energy will instead be converted to thermal energy. This will raise the temperature of the fuel cell beyond its operating temperature range, and must be addressed using a cooling system. The model in this work assumes that the passage of reactant air through the cell will account for some of the heat transfer, while the cooling system will account for the remaining excess heat to maintain the proper cell temperature. A fuel cell stack will generate the following amount of heat during operation (Dong, 2004a):
Q& total = ( Emax − Vcell , gross ) I ⋅ ncell
(23)
where Emax is the maximum EMF of the fuel cell, or 1.48V if the HHV is used; Vcell,gross is the voltage of the cell calculated by Eq.(1); and I is as usual the cell current. Heat will be lost to the surroundings and to the air stream through three heat transfer processes: radiation, convection and evaporation. Thus, the amount of heat lost to the air stream can be calculated using the following expression:
Q& air ,total = Q& rad + Q& conv ,total + Q& evap
(24)
This heat quantity must be transferred away from the cell in its entirety in order to maintain the operating temperature of the fuel cell. Even a minute amount of heat that is not transferred away for each reaction will eventually cause the cell to fail, due to the drying out of the membrane. The equation for radiation is given by: 4 4 Q& = σ A (Tbody − Tsurrounding )
(25)
where σ is the Stefan-Boltzmann constant (which has a value of 5.67×10-8 W/(m2·K4)), A is the area of the radiative body, Tbody is the temperature of the radiative body and
4
Tsurrounding is the ambient air temperature. For the Mark V fuel cell, the area is given by Dong (2004a): (26) Acell = 2 ⋅ AreaEnds ⋅ 0.85 + AreaSides ⋅ 0.70 ⋅1.3 where the decimal terms denote empirical parameters that illustrate the internal shape of the fuel cell. The equation for heat due to radiation is then (Dong, et al 2004b): 4 4 Q& rad = σ Acell (Tcell , average − Tair ,in )
(27)
The heat loss due to convection will occur differently for the various parts of the fuel cell interior. The convection lost by the vertical walls of the cell is given by:
Q& conv , walls = 1.15 ⋅ AreaSides ⋅ hair (Tcell ,average − Tair ,in ) (28)
where hair is the heat transfer coefficient of the cell channel sides (in W/(m2·K)), and is calculated by the following (Dong, et al 2004b):
hair =
k air ⋅ Nu l
(29)
where kair is the thermal conductivity of air (in W/(m·K)), Nu is the Nusselt number (which is dimensionless) and l is the stack length. The heat transfer due to convection on the top and bottom of the channel is given by:
Q& conv , top + bottom = ( htop + hbottom ) ⋅
(T
stack , average
− Tair , in ) ⋅ Area Ends
(30)
(31)
The model calculates the amount of heat lost to evaporation, using an assumption that 90% of the water produced is evaporated. At present it is believed that this is a reasonable value; however, further research to confirm this assumption is required. In any event, the amount of heat required to convert a mass of liquid water to vapour water is equal to the product of the latent heat of vaporization lvap and the mass of the water. Incorporating the 90% evaporation assumption into this equation, an expression for the amount of heat lost to evaporation is given as (Dong, et al 2004b):
Q& evap = lvap ⋅ 0.90m& H2O
W parasitic , total = Wcooling system + Wreactant pump , total + Wcompressor
(32)
Parasitic Power Considerations The fuel cell is the core of a fuel cell system; however, it is not the only part. In order to provide a higher voltage fuel cells are stacked together in series, creating what is known as a fuel cell stack. The fuel cell stack will not be able to operate if the reactants are not provided to the cell; furthermore, the cell temperature must be within a certain range to ensure successful operation. Therefore, several auxiliary systems, known as the Balance of Plant (BOP), are necessary for the correct operation of a fuel cell system. The most important auxiliary systems are the air
(33)
As mentioned previously, 80% of the total heat produced is assumed to be carried away by the cooling system. The model allows the user to choose between water and air cooling systems. For the case of a water cooling system, the mass of water in kg/s required to perform the heat transfer to the water coolant is thus (Dong, 2004a):
m& H 2O =
c pH O 2
(
Q& total − Q& air , total
c pH O Tstack , average − TH 2O in , average 2
where
where htop and hbottom are the heat transfer coefficients of the top and bottom of the fuel cell channel, respectively. The total heat lost to convection is then given by the sum of Eq.(28) and Eq.(30):
Q& conv ,total = Q& conv , walls + Q& conv ,top +bottom
compressor necessary to pressurize the incoming air to the correct fuel cell operating pressure, the humidifier to guarantee that the fuel cell has enough water for optimal performance and the cooling system to maintain the cell temperature. All of these auxiliary systems are described and modeled in the next sections. The BOP components will draw power from that produced by the fuel cell operation itself, thereby reducing the overall power output. The total parasitic power of the fuel cell stack is assumed to be modeled by the equation:
)
(34)
is the specific heat of water at constant
pressure. The model assumes that the efficiency of the pump and pump motor are 60% and 80%, respectively. Also, the model assumes the pressure drop of the water through the system to be 1×104 Pa. The power (in W) required pumping the water coolant is then given as (Dong, 2004a):
Wcooling system =
m& H 2O
ρH O
⋅ Pdrop ⋅
2
FoS η pump ⋅ηmotor
(35)
where FoS is a factor of safety (assigned a value of 1.5) to account for any pressure losses that are not considered explicitly. If the cooling system is to be air, Eq.(34) is modified to be:
m& air , cooling =
Q& total − Q& air , total
c pair (Tstack , average − Tair in , average )
(36)
and Eq.(35) becomes:
Wcooling system =
m& air , cooling
ρ air
⋅ Pdrop ⋅
FoS
η pump ⋅ηmotor
(37)
The pressure drop is assumed to be the same for both cooling system types. To compute the power loss at the humidifiers, an existing model from (Dong, et al 2004b) is used. The total humidifier power is related to the amount of water needed to fully humidify the reactants, a humidifier pressure drop and the air density by the equation:
W waterpump,total = Pdrop ⋅
FoS (m H 2O,inH 2 + m H 2O,inAir )
ρ airη pumpηmotor
(38)
where the amount of water necessary to fully humidify the
5
hydrogen is (Dong, et al 2004b):
m& H 2O , in H 2 =
8.937 ⋅ pHsat2O P − pHsat2O
⋅ m& H 2
(39)
Similarly, the amount of water in the air stream is given by:
m& H 2O , in air =
0.6219 ⋅ pHsat2O P − pHsat2O
⋅ m& air
(40)
The model then assumes that the pressure drop in the reactant gases to be 1×105 Pa, and that the pump and motor efficiencies are the same as for the cooling system. Proof of the connections to phenomenologically based theory for Eqs.(38)-(40) has not yet been fully established, and the equations should be studied in detail in the future to determine their accuracies. However, use of these equations is allowed since the ultimate goal of the study is design optimization. A compressor is required only for the cathode side of the fuel cell since the hydrogen gas is assumed to be stored in a pressurized container. The compressor is assumed to operate isentropically. The compressor efficiency is assigned a value of 60% while the compression ratio is calculated from the ratio of operation pressure P to atmospheric pressure. The expression for the compressor power is given as (Larminie and Dicks, 2003):
Wcompressor
γ −1 ⎛ ⎞ Tin ⎜ ⎛ P ⎞ γ ⎟ = c pair m& air ⎜ ⎟ − 1⎟ ⎜ ηcomp ⎜ ⎝ Patm ⎠ ⎟ ⎝ ⎠
(41)
where γ is the isentropic compression ratio, or Cp/Cv. The net power is thus the net output power of the stack, given in Eq.(20), subtracted by the total parasitic power, calculated in Eq.(33):
Wnet = Wnet output − W parasitic ,total
(42)
The main output of the model computer program is to provide the polarization curve depending on the various constants and parameters that the programmer has input. Additionally, the output includes the net power and parasitic power. The final step of the model is a calculation of the system efficiency and exergetic efficiency. The efficiency is calculated from the expression (Dong, et al 2004b):
η system =
Wnet V I
(43)
and the exergetic efficiency is given as (Dong, et al 2004b):
ηexergetic
V ⋅M H2 I = m& H 2 ⋅ HHVH 2
(44)
The model equations have now all been described, and the optimization can be implemented.
SYSTEM OPTIMIZATION Concurrent engineering design is an extremely valuable tool for reducing costs and design lead time. An important step in this process is to use optimization techniques to determine the optimal design without the need to construct expensive and time-consuming physical prototypes. In the case of fuel cell system design, the mechanisms governing the operation of a fuel cell are of such a complex nature that the objective function will likely have a plethora of local minimums, and therefore, optimization of such systems can be quite challenging. Many different algorithms have been developed by researchers in the field of optimization. Three popular and successful techniques have been chosen: (1) simulated annealing (SA), (2) genetic algorithm (GA), and (3) sequential quadratic programming (SQP). The first two are global optimization algorithms, while the third is a local optimization algorithm. The local optimization algorithm is chosen so that the form of the objective function can be examined. Thus, if the SQP algorithm does not produce results similar to those of the global optimization methods, then it will be clear that the objective function is not unimodal and smooth, but in fact contains local minima which “trapped” the local optimization algorithm. In order to be able to use an optimization technique, a problem with several conflicting objectives must be identified. For single cell optimization, where the output power is to be maximized, the conflicting goals are mainly due to heat and water management issues. For example, if the air stoichiometry is small, the air will not be able to eliminate all the water produced in the cathode of the cell and flooding will occur at the gas diffusion layer (GDL), which will result in a large reduction of the generated power. On the other hand, if the stoichiometry is too large the membrane will dry out, reducing the proton conductivity and therefore reducing the output power. There exists an optimum value of the air stoichiometry for which there is enough water for the membrane to be humidified without flooding occurring in the gas diffusion layer; it is at this value where the maximum output power occurs. As mentioned previously, the model assumes that the membrane is fully hydrated; therefore, the aforementioned conflict of objectives does not occur. Similarly, heat management is not considered. For these reasons, if optimization of the output power of a single cell or a fuel cell stack is performed, it yields unrealistic results. Increasingly complex models are necessary to perform stack and single fuel cell optimization. In this work, to handle conflicting objectives the fuel cell system is optimized as a whole. In this case, the power consumed by the auxiliary systems creates several conflicts of interest when the net output power is to be maximized. The operation of the fuel cell is ameliorated significantly at a high temperature and pressure; however, these increases would result in higher power losses due to the auxiliary systems, in particular due to the cooling system and the air compressor, respectively. It is because of these conflicting interests that the design of a fuel cell 6
system is a good candidate for optimization. To solve the optimization problem the fuel cell system model described in the previous section is coupled to the optimization algorithms described above as shown in Figure 1. The coupling between the optimization algorithm and the model is achieved through the design variables and the objective function and constraints. First, the optimization algorithm or the user selects the initial value for the design variables. In the case of global methods, the initial value for the design variables is selected randomly. In the case of a local method, the user gives the initial value for the design variables. The value of the design variables is then given to the fuel cell system model. The fuel cell system model then computes the performance of the system and, from those results computes the value of the objective function subject to the design constraints. With this information, the optimization algorithm chooses a new set of design variables that can potentially increase the system performance. This process is repeated until a convergence criterion is satisfied or the maximum number of iterations is achieved, depending on the optimization algorithm used. Finally, if a local optimization algorithm is used for optimization, the gradients are computed using adaptive forward differences. In this case, the optimization algorithm automatically calls the fuel cell system model with a small perturbation in each one of the design variables and computes the numerical gradient.
Fuel cell system model Objective function (average net power) Design variables (p, T, airStoich)
Optimization algorithm Design variables (p, T, airStoich) NO Convergence ?
Optimal operating conditions (p, T, airStoich)
YES
Figure 1. Flow diagram of the optimization program used to compute the optimal fuel cell operating conditions
DISCUSSION AND RESULTS The fuel cell system model and the optimization methods described above are used to solve two optimization problems where the optimal operating conditions of the full cell system are obtained. In both optimization problems, the maximization of the net power is the main objective. For the first optimization problem, it is assumed that the fuel cell will operate over a wide range of power
requirements, for example in a vehicular application. Therefore, it is desired to obtain the fuel cell operating conditions that yield the maximum net power over a range of power demands. The optimization problem is solved as a multi-objective optimization problem: N
Minimize: f ( x, i ) = (−1) ⋅ ∑ wi Pi ( x, ii ) w.r.t. x
i =1
subject to: x max ≥ x ≥ x min , where Pi (x,ii ) is the net power obtained from the fuel cell system at a current density ii , w i are the weighting variables and x is the vector of design variables. The net power output of the fuel cell system is obtained using the computational model described above. The design variables include the operating temperature, the air stoichiometry, and the operating pressure of the fuel cell stack. The upper and lower limits of these design variables are x max = [358, 5, 15] and x min = [338, 1.5, 1] , respectively. The multi-objective optimization problem with different weights will yield different optimal operating conditions. If only two objectives were present, it would be possible to create a Pareto curve to see the trade-off between objectives. In this case, the multiple objectives are the maximization of the net power for current densities of i = [0.25, 0.5, 0.75, 1] A/cm2. The weights are set to have equal value for all objectives since the assumption is made that adequate performance is desired over the whole range of current densities of the fuel cell. The aforementioned problem is solved using the three optimization algorithms described above and the results are provided in Table 1. The solution of the three methods is almost identical. Only the solution achieved using the GA has a small error and increasing the number of generations in the algorithm can probably reduce this error. The SA algorithm stopped when a convergence criterion of three successive iterations with a difference in the objective function smaller than 1×10-6 was satisfied. The GA algorithm stopped after running 200 generations; the convergence criterion was not specified. Finally, the SQP algorithm stopped when the module of the gradient of the objective function reached a value less than 1×10-8. The solution of the SQP is dependant on the initial point; therefore, the problem was solved from a variety of different initial points and the same solution was achieved. Due to the different nature of the methods and the different convergence criteria it is difficult to do a thorough comparison of the computational expense of these methods. Looking at the number of function evaluations and the required computational time in Table 1 of each one of these methods we can observe that the local method (SQP) converges to the solution extremely quickly. From the assessment, it can be observed that the shape of the objective function is likely to be smooth and convex since the local method converges to the same solution independently of the initial point.
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Table 1. Solution of the multi-objective problem with equal weights Optimization Algorithm
Objective Function: Average Net Power [W]
Simulated Annealing
539.2532
Genetic Algorithms
539.2414
SQP
539.2532
Solution: [T[K], AirStoich[-], P[bar]] [356.848, 1.50000, 2.96721] [357.065, 1.50000, 2.95062] [356.848, 1.50000, 2.96721]
Number of function evaluations
CPU time [s]
3301
8.690
4000
13.60
70
0.395
Performances of the fuel cell system operating at the optimal operating condition and at a base operating condition are compared. The base operation conditions are at temperature of 343K, hydrogen and air stoichiometry ratios of 1.5 and 2.5, respectively, and a pressure of 3 bar. Figure 2 shows the polarization curve of the fuel cell at the base and optimal conditions. The curves are very similar with a small voltage increase in the optimal case at high current densities due to an increase in the operating temperature. The difference in gross power output of the fuel cell due to the changes of the polarization curve will be small.
Figure 2. Polarization curve of the fuel cell under optimal and base operating conditions
Figure 3 depicts the net output power of the fuel cell system at the optimal and base operating conditions. A large increase on the net output power can be observed. Figure 4 shows the gross power output of the fuel cell at the optimal and base operating conditions. The increase in net power is due to a decrease in the necessary power of the auxiliary devices since the gross output power between the two operating conditions is similar. At the base operating conditions and at a current density of 0.75A/cm2, the consumed power in the cooling pump,
humidifier pump and air compressor are 70W, 50W and 300W, respectively. At the optimal operating conditions and identical current density, the cooling pump, humidifier pump and air compressor consume powers of 60W, 50W and 200W, respectively. The large increase in net power is mainly due to the reduction in the power consumed by the air compressor, caused by a large reduction in the air stoichiometry ratio and a small reduction in the operating pressure. The compressor power is reduced at all current densities, as shown in Figure 5.
Figure 3. Net output power of the fuel cell system vs. current density under optimal and base operating conditions
Figure 4. Gross power output of the fuel cell system vs. current density when operating at the base and optimal conditions
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Table 2. Optimal solution of the single objective problem
Figure 5. Consumed power by the air compressor vs. current density when operating at the base and optimal conditions
In the second optimization problem it is assumed that the fuel cell is to be operated at maximum net power. This will be the case for a power generation PEM fuel cell. Therefore, the fuel cell operating conditions and the current that yields the maximum net power are to be obtained. The optimization problem can be formulated as:
Minimize : f (x) = (−1) ⋅ P(x) w.r.t. x
subject to : x max ≥ x ≥ x min where P(x) is the net power and x is the design variables that include operating temperature, air stoichiometry, operating pressure of the fuel cell stack, and current density at which the fuel cell system will operate to achieve maximum net power. The upper and lower limits of the design variables are defined as: x max = [358, 5, 15, 1.2 ] and x min = [338, 1.5, 1, 0.1] . This problem is also solved using the three optimization algorithms described and the solutions are presented in Table 2.
Optimization Algorithm
Objective Function: Maximum Net Power [W]
Simulated Annealing
659.805
Genetic Algorithms
659.185
SQP
659.80
Solution: [ T [K], AirStoich[-], P [bar], i [A/cm2] ] [358.000, 1.50000, 3.05960, 0.85002] [357.588, 1.50011, 3.13007, 0.829284] [358.000, 1.50000, 3.05960, 0.85002]
Number of function evaluations
CPU time [s]
4401
6.85
4000
9.74
98
0.29
Again, the solutions of the three optimization algorithms are extremely similar and only a small discrepancy is observed between the solution obtained using the GA optimization method and the other two optimization algorithms. As in the previous case, an increase in the number of generations or the population would likely result in a better performance at the expense of a larger computational time. In this second problem, the temperature and pressure are slightly higher than in the previous optimal solution. This shows the compromise achieved in the previous case between the performances at high and low current densities. The temperature reached its upper bound. A new optimization run was undertaken with an upper bound of 368K, instead of 358K in order to understand to what extent the temperature boundary value was limiting the maximum output net power. The new optimization showed that the optimal temperature was 358.990K, only slightly higher than the previous temperature and this results in an almost negligible increase in the maximum net power output from 659.805 to 659.865 W. Figure 6 shows a curve of the maximum net output power with respect to the current density at the base and optimal operating conditions. We can easily observe that the current density obtained in the solution matches the maximum of the net output power curve, denoted by the small circle on the peak of the optimal net power curve on the graph. We can also observe that the optimal operating conditions not only increase the maximum net output power, but also move the point at which the maximum net power occurs to higher current densities.
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Figure 6. Fuel cell system net output power at different current densities. The circle shows the current density for maximum net output power
CONCLUSIONS Fuel cell optimization is an interesting area of research. Adequate models need to be used in order to obtain meaningful results for fuel cell and fuel cell systems optimization. We have seen in this study how optimization at the fuel cell level could not be performed using semiempirical models because of the simplifications and assumptions used to obtain the model. Optimization at a fuel cell level needs a model that is able to take into account water management issues and transport phenomena. In order to reach this level of accuracy Computational Fuel Cell Dynamics (CFCD) must be used (Wang, 2004). The development of accurate fuel cell models is the goal of many researchers, and contemporary models will no doubt be continuously improved and new models advanced. On the other hand, semi-empirical fuel cell system models such as the model described in this work enable us to perform fuel cell system optimization. When performing fuel cell system optimization, the main trade-offs appear from the selection of the most appropriate auxiliary devices, operating conditions and geometry of the fuel cell system as well as the system constraints; therefore, the semi-empirical models perform well even though they are obtained by an oversimplification of the fuel cell itself. System geometric constraints are not considered in this study because the design variables used in this work are only related to the operating conditions of the fuel cell. However, if the geometry of the fuel cell becomes part of the design variables, geometrical constraints would be necessary. For example, in vehicular and mobile applications geometrical constraints will play a key role. During the study, global and local optimization algorithms were compared. Similar results are obtained using both methods. This allows us to conclude that the optimization problems investigated are both convex and unimodal. However, the addition of design variables and constraints
may change the nature of the problem. In this particular case, the use of a local optimization technique is preferred since its application results in the same solution as the global methods using less computational time. It is recommended to use a combination of global and local optimization algorithms in the future in order to study the nature of the problem and understand more clearly the characteristics of the objective function. This work reveals a large number of possible avenues of future research. The existing fuel cell system model would benefit from an improved model of the parasitic losses by introducing a method to compute the pressure drop in the humidifier and in the cooling channels. A new optimization could be performed that takes into account not only the operating conditions as design variables but also the geometric parameters of the fuel cell. Also, the objective function can be modified: in the current study, two optimization problems are discussed that assume that the fuel cell would work in the whole operating range and at the maximum power point. A more sophisticated working cycle of the fuel cell can be developed that would define the objective function depending on the application. Finally, if single fuel cells are to be optimized more complex models for the fuel cell need to be implemented that take water management issues into account. ACKNOWLEDGMENTS Financial support from the Natural Science and Engineering Research Council of Canada (NSERC) is gratefully acknowledged. REFERENCES Amphlett, J.C., R. M. Baumert, R. F. Mann, B. A. Peppley and P. R. Roberge, “Performance Modeling of the Ballard Mark IV Solid Polymer Electrolyte Fuel Cell. I. Mechanistic Model Development,” Journal of the Electrochemical Society, Vol. 142, No. 1, January 1995, pp. 1-8. Dong, Z., “Fuel Cell System Modeling Notes” last updated October 18, 2004. Dong, Z., M. Guenther, and G. Iuzzolino, “Mathematical Modelling of PEM Fuel Cells and Its Implementation in MatLab,” University of Victoria, Mechanical Engineering Department, 2003. Dorigo, M. and V. Maniezzo and A. Colorni, “The Ant System: Optimization by Colony of Cooperating Agents,” IEEE Transactions on Systems, Man and Cybernetics, Part B, Vol.26, 1996, pp. 29-41. Fowler, M.W., R.F. Mann, J.C. Amphlett, B.A. Peppley and P.R. Roberge, “Incorporation of Voltage Degradation into a Generalised Steady State Electrochemical Model for a PEM Fuel Cell,” Journal of Power Sources, Vol. 106, 2002, pp.274-283. Grujicic, M. and K.M. Chittajallu, “Design and Optimization of Polymer Electrolyte Membrane (PEM) Fuel Cell,” Applied Surface Science, Vol. 227, 2004, pp. 56-72. Haraldsson, K. and K.Wipke, “Evaluating PEM Fuel Cell System Models,” Journal of Power Sources, Vol. 126, 2004, pp.88-97.
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He, L. and E. Polak, “Multistart Method with Estimation Scheme for Global Satisfying Problems,” Journal of Global Optimization, Vol. 3, 1993, pp. 139-156. Larminie, J. and A. Dicks, Fuel Cell Systems Explained, New York: John Wiley and Sons, 2nd edition, 2003. Mann, R.F., J.C. Amphlett, M.A.I Hooper, H.M. Jensen, B.A. Peppley and P. R. Roberge, “Development and Application of a Generalised Steady State Electrochemical Model for a PEM Fuel Cell,” Journal of Power Sources, Vol. 86, 2000, pp.173-180. Schoen, F. “Stochastic Techniques for Global Optimization: A Survey on Recent Advances,” Journal of Global Optimization, Vol. 1, No. 3, 1991, pp. 207-228. Wang, Chao-Yang, “Fundamental Models for Fuel Cell Engineering,” Chemical Reviews, Vol.204, 2004, pp. 47274766. Xue, D. and Z. Dong, “Optimal Fuel Cell System Design Considering Functional Performance and Production Costs,” Journal of Power Sources, Vol. 76, 1998, pp.69-80.
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