**6. Optimization and identification**

#### **6.1 Optimization**

out in order to explore mass transport phenomena occurring in DMFCs for portable applications as well as to reveal an interplay between the local current density and methanol crossover rate. In [37], 3-D modeling is then extended to transient conditions. The authors note that cathode processes, for example oxygen and water transport coupled to electrochemical reaction, are inherently transient so that an unsteady-state model gives more accurate prediction than a steady-state model. Numerical simulations indicate that the cathode catalyst layer porosity has major effects on oxygen transfer and water removal. A three-dimensional multiphase model of DMFC is developed in [38], in which the Eulerian-Eulerian model is adopted to treat the gas and liquid two-phase flow in channel. By 3-D simulation, cell performance is found to be severely affected by accumulation of carbon dioxide mainly at the anode channel and by high-temperature operations. Ref. [39] shows that three-dimensional models are suitable for analyzing DMFC stacks with flowing electrolyte. A multiscale approach is therefore proposed in order the reduce the computational cost arising from 3-D modeling of the entire stack geometry. By this

*/s) [30] (courtesy of Electrochimica Acta).*

*Methanol concentration in the anode and membrane (unit: m<sup>3</sup>*

*Thermodynamics and Energy Engineering*

solution strategy, fully 3-D flow fields, backing layers, and membranes are numerically solved, whereas electrochemical reactions are analytically

It should be finally noted that multiphysics models coupling electrochemical reactions, methanol, water, and heat transport are still under investigation due to

Three different designs are used in planar SOFCs depending on their operating

multiphysics behavior is indispensable in identifying optimal design and operation of such SOFCs and a multiphysics numerical model is required at this purpose. Both

temperatures. High-temperature SOFCs (around 1000°C) usually present an electrolyte-supported structure, with thin electrodes (e.g., 50 μm) supported by a thick electrolyte (above 100 μm) [1]. The high temperature ensures so high a conductivity that the electrolyte resistance remains within acceptable values. In low-temperature SOFCs (though not less than 600°C), thinner electrolytes are used (e.g., 20 μm) and the cell is supported by either anode or cathode (300–1500 μm)

with the other electrode being thinner (e.g., 50 μm). Understanding the

simulated.

**40**

**Figure 6.**

their high complexity.

**5.3 Solid oxide fuel cells**

A number of nonlinear deterministic optimization methods (DOMs) have been applied to PEMFCs in the last decade, proving successful in dealing with specific tasks. Least squares methods have been applied to the estimation of single material parameters as well as parameters evolution under degradation events [2]. The gradient method has been exploited in the search for optimal designs and parameters evolution, such as cathode configuration optimization, geometric optimization, and flow field serpentine optimization [45]. A review of deterministic optimization methods used for identification problems in PEMFCs is given in [46].

Deterministic methods are known for their efficiency, that is, speed of convergence, but their applicability may be hindered, depending on the specific algorithm, by lack of flexibility in handling arbitrary constraints, sensitivity to noise in the objective function, possible need of function derivatives, and premature convergence to local minima. On the other hand, stochastic optimization methods (SOMs), in spite of their comparatively low efficiency, typically overcome the above-mentioned shortcomings of deterministic methods. Although the convergence to the global optimum for SOMs is only asymptotically guaranteed, there is abundant numerical evidence that very good solutions can be obtained for many problems, including FC design problems. It is worth noticing that a crucial feature of FC circuit models is that they avoid partial differential equations, thus resulting in numerical formulations with relatively low computational costs, which make them ideally suited for SOMs. It should also be noted that since optimization problems related to PEMFCs are characterized by highly nonlinear device models, the resulting objective functions subject to minimization have many local minima, which need to be all identified in the search for the global one. Therefore, for this type of problems, stochastic optimizers may end up being almost as efficient as deterministic ones. A further advantage, which however is also shared by some deterministic methods, is that stochastic optimizers can also deal with nondifferentiable or fully discrete, optimization problems. In recent years, the application of stochastic methods for the solution of FC optimization problems has been constantly increasing, and interesting results have been reported, for example, with genetic algorithms (GA) [47], particle swarm optimization (PSO) [48], and differential evolution (DE) [49]. The above-mentioned methods are all population-based, that is, they explore several candidate solutions concurrently, which makes them ideally suited for parallelization. SOMs can also be combined among them or hybridized with DOMs in order to tailor their behavior to the specific optimization problem. Multiobjective stochastic approaches have also been recently investigated [50].

identification problem is a constrained one, that is, the domain *A* where *x* values are defined is supplemented with a number of constraints and the problem is also typically burdened by model nonlinearity, as is the case of an FC model, which results in the non-convexity of ƒ and consequent local minima. Moreover, large problems lead to high computational cost. Given the problems of "curse of dimensionality," presence of local minima, and computational costs, smart strategies are needed to find good solutions, if not the absolute best one, which actually may be

*Distributed and Lumped Parameter Models for Fuel Cells*

*DOI: http://dx.doi.org/10.5772/intechopen.89048*

In the last decade, stochastic methods have been applied to the study of FC parameter identification problems and their use has been strongly increasing in the last 3 years. Studies reported in literature typically aim at using stochastic methods in order to obtain a good fit of PEMFC polarization curves and usually resort to simplified empirical PEMFC models. Such models tend to be interpolatory in nature and contain enough parameters (e.g., 5–7) to allow for a good fit. Given their nature, fitting a set of empirical parameters to match a given polarization curve is a relatively easy task for most optimization procedures, but the usefulness of the obtained results is rather limited. The relatively small number of parameters also helps in avoiding so-called duplicity problems, that is, multiple distinct solutions achieving the same minimal values of ƒ. However, duality is not crucial, because empirical parameters have no direct physical meaning. A more ambitious challenge consists in identifying several physical parameters of the materials of a PEMFC by means of an optimization approach. An algorithm of this kind, built over an early investigation on the capability of stochastic methods to deal with FCs [52], uses a detailed multiphysical performance model that employs such parameters and takes into account some physical control quantities [53]. A straightforward use of a stochastic optimizer with a large number of unknowns and a weakly constrained nonlinear problem can result in duplicity. In order to overcome it, a possible strategy consists in splitting the overall identification problem into a sequence of distinct identification sub-problems, each having a lower number of unknowns, and thus suffering less from the "curse of dimensionality." This approach basically relies on isolating a group of equations, dominated by some of the unknowns only, and has already been applied successfully to fuel-cell problems [54]. This behavior emerges, for example, since the parameters related to the activation losses and to the concentration losses, which are strongly nonlinear, prevail over the ohmic losses, which are linear at given hydration and temperature, and their identified values tend to vanish [55]. This behavior can be exploited by separately considering the typical three parts of the polarization curves. The experimental data obtained at low current density can be used to identify the parameters related with activation losses, while those obtained at high current density can been used to identify the parameters related with the concentration losses. Finally, experimental data at intermediate current density values can be used to identify the parameters related with the ohmic losses, which dominate in the central part of the polarization curve (**Figure 7**). Some final considerations emerge: first of all, the accuracy and reproducibility the experimental data and the experimental conditions must increase with growing number of unknown parameters. This may be hard to obtain in the case of the polarization curves of PEMFCs, which depend on many factors related to both the samples under test and the experimental conditions, some of which are hard to control. Such difficulties can be mitigated by collecting more curves in the same nominal operating conditions and performing a statistical selection of the data. Moreover, enriched experimental plans may allow to identify also some parameters that have a minor effect on the polarization curve, for example, a set of experimental data obtained at different back pressures may allow the identification of the anodic exchange current density, which is otherwise masked by the larger effect of

impossible to identify.

**43**

## **6.2 Identification**

Since fuel cells present a stratified structure of thin layers made of different materials, analyzing their behavior requires the full characterization of these materials, that is, the determination of their chemical, physical, thermal, and electrical parameters. The identification of these parameters is crucial for guiding the research for advanced functionalized materials. These parameters are also needed in FC models, used in the fast exploration of different operating scenarios and in the research of optimized structural design and operating conditions [51]. The systems of equations involved (Nernst equation, Butler-Volmer equation, Darcy's equation, Fourier's law, Ohm's law, etc.) are strongly nonlinear, making the models extremely sensitive to parameter variations and uncertainties.

Unfortunately, they are hard to measure in real operating conditions and their identification still remains challenging when dealing with direct measurements. Careful ex situ measurements can be performed by means of a number of diagnostic techniques; however, the transferability of results to operative fuel cells raises a number of issues. Conversely, in situ measurements can provide meaningful operational values, but very few, often complicated and cumbersome, techniques are available to determine a limited number of parameters. A different approach consists in tackling the identification of multiple parameters by using a very large body of experimental data collected at different of physical conditions (e.g., temperature, pressure, and humidity). However, this approach suffers from the well-known "curse of dimensionality," that is, the problem becomes exponentially harder to solve as the number of parameters increases. This challenging problem can be approached with an optimization procedure (i.e., the search of the minimum of a function ƒ(*x*)). When using optimization algorithms for model parameter identification, *x* is the *n*-dimensional vector of the unknown parameters to be identified and ƒ(*x*) consists in an error function that assesses the difference between the output of the parameter-based model and the measurements. The parameter

#### *Distributed and Lumped Parameter Models for Fuel Cells DOI: http://dx.doi.org/10.5772/intechopen.89048*

the above-mentioned shortcomings of deterministic methods. Although the convergence to the global optimum for SOMs is only asymptotically guaranteed, there is abundant numerical evidence that very good solutions can be obtained for many problems, including FC design problems. It is worth noticing that a crucial feature of FC circuit models is that they avoid partial differential equations, thus resulting in numerical formulations with relatively low computational costs, which make them ideally suited for SOMs. It should also be noted that since optimization problems related to PEMFCs are characterized by highly nonlinear device models, the resulting objective functions subject to minimization have many local minima, which need to be all identified in the search for the global one. Therefore, for this type of problems, stochastic optimizers may end up being almost as efficient as deterministic ones. A further advantage, which however is also shared by some deterministic methods, is that stochastic optimizers can also deal with nondifferentiable or fully discrete, optimization problems. In recent years, the

application of stochastic methods for the solution of FC optimization problems has been constantly increasing, and interesting results have been reported, for example, with genetic algorithms (GA) [47], particle swarm optimization (PSO) [48], and differential evolution (DE) [49]. The above-mentioned methods are all population-based, that is, they explore several candidate solutions concurrently, which makes them ideally suited for parallelization. SOMs can also be combined among them or hybridized with DOMs in order to tailor their behavior to the specific optimization problem. Multiobjective stochastic approaches have also been

Since fuel cells present a stratified structure of thin layers made of different materials, analyzing their behavior requires the full characterization of these materials, that is, the determination of their chemical, physical, thermal, and electrical parameters. The identification of these parameters is crucial for guiding the

research for advanced functionalized materials. These parameters are also needed in FC models, used in the fast exploration of different operating scenarios and in the research of optimized structural design and operating conditions [51]. The systems of equations involved (Nernst equation, Butler-Volmer equation, Darcy's equation, Fourier's law, Ohm's law, etc.) are strongly nonlinear, making the models extremely

Unfortunately, they are hard to measure in real operating conditions and their identification still remains challenging when dealing with direct measurements. Careful ex situ measurements can be performed by means of a number of diagnostic techniques; however, the transferability of results to operative fuel cells raises a number of issues. Conversely, in situ measurements can provide meaningful operational values, but very few, often complicated and cumbersome, techniques are available to determine a limited number of parameters. A different approach consists in tackling the identification of multiple parameters by using a very large body of experimental data collected at different of physical conditions (e.g., temperature, pressure, and humidity). However, this approach suffers from the well-known "curse of dimensionality," that is, the problem becomes exponentially harder to solve as the number of parameters increases. This challenging problem can be approached with an optimization procedure (i.e., the search of the minimum of a function ƒ(*x*)). When using optimization algorithms for model parameter identification, *x* is the *n*-dimensional vector of the unknown parameters to be identified and ƒ(*x*) consists in an error function that assesses the difference between the output of the parameter-based model and the measurements. The parameter

sensitive to parameter variations and uncertainties.

recently investigated [50].

*Thermodynamics and Energy Engineering*

**6.2 Identification**

**42**

identification problem is a constrained one, that is, the domain *A* where *x* values are defined is supplemented with a number of constraints and the problem is also typically burdened by model nonlinearity, as is the case of an FC model, which results in the non-convexity of ƒ and consequent local minima. Moreover, large problems lead to high computational cost. Given the problems of "curse of dimensionality," presence of local minima, and computational costs, smart strategies are needed to find good solutions, if not the absolute best one, which actually may be impossible to identify.

In the last decade, stochastic methods have been applied to the study of FC parameter identification problems and their use has been strongly increasing in the last 3 years. Studies reported in literature typically aim at using stochastic methods in order to obtain a good fit of PEMFC polarization curves and usually resort to simplified empirical PEMFC models. Such models tend to be interpolatory in nature and contain enough parameters (e.g., 5–7) to allow for a good fit. Given their nature, fitting a set of empirical parameters to match a given polarization curve is a relatively easy task for most optimization procedures, but the usefulness of the obtained results is rather limited. The relatively small number of parameters also helps in avoiding so-called duplicity problems, that is, multiple distinct solutions achieving the same minimal values of ƒ. However, duality is not crucial, because empirical parameters have no direct physical meaning. A more ambitious challenge consists in identifying several physical parameters of the materials of a PEMFC by means of an optimization approach. An algorithm of this kind, built over an early investigation on the capability of stochastic methods to deal with FCs [52], uses a detailed multiphysical performance model that employs such parameters and takes into account some physical control quantities [53]. A straightforward use of a stochastic optimizer with a large number of unknowns and a weakly constrained nonlinear problem can result in duplicity. In order to overcome it, a possible strategy consists in splitting the overall identification problem into a sequence of distinct identification sub-problems, each having a lower number of unknowns, and thus suffering less from the "curse of dimensionality." This approach basically relies on isolating a group of equations, dominated by some of the unknowns only, and has already been applied successfully to fuel-cell problems [54]. This behavior emerges, for example, since the parameters related to the activation losses and to the concentration losses, which are strongly nonlinear, prevail over the ohmic losses, which are linear at given hydration and temperature, and their identified values tend to vanish [55]. This behavior can be exploited by separately considering the typical three parts of the polarization curves. The experimental data obtained at low current density can be used to identify the parameters related with activation losses, while those obtained at high current density can been used to identify the parameters related with the concentration losses. Finally, experimental data at intermediate current density values can be used to identify the parameters related with the ohmic losses, which dominate in the central part of the polarization curve (**Figure 7**).

Some final considerations emerge: first of all, the accuracy and reproducibility the experimental data and the experimental conditions must increase with growing number of unknown parameters. This may be hard to obtain in the case of the polarization curves of PEMFCs, which depend on many factors related to both the samples under test and the experimental conditions, some of which are hard to control. Such difficulties can be mitigated by collecting more curves in the same nominal operating conditions and performing a statistical selection of the data. Moreover, enriched experimental plans may allow to identify also some parameters that have a minor effect on the polarization curve, for example, a set of experimental data obtained at different back pressures may allow the identification of the anodic exchange current density, which is otherwise masked by the larger effect of

**Acronyms**

*A* area (m<sup>2</sup>

)

*Cdl* double-layer capacitance (F) *ci* molar concentration (mol m<sup>3</sup>

*D* diffusion coefficient (m<sup>2</sup> s

*dm* membrane thickness (m) *E0* standard potential (V) *F* Faraday constant (A s mol<sup>1</sup>

*cp* specific heat capacity (J kg<sup>1</sup> K<sup>1</sup>

*Distributed and Lumped Parameter Models for Fuel Cells*

*DOI: http://dx.doi.org/10.5772/intechopen.89048*

*j* electric current density (A m<sup>3</sup>

*k* Boltzmann constant (J K<sup>1</sup>

*Pdiss* dissipated power (W) *pi* partial pressure (Pa)

*T* temperature (K) *V* cell voltage (V) *Wa* activation energy (eV) *Zw* Warburg impedance (Ω) *α* charge transfer coefficient () *δ* electrode thickness (m) Δ^*s* molar entropic variation (J K<sup>1</sup>

*η* overpotential (V)

*ρ* mass density (kg m<sup>3</sup>

*φ* electric potential (V) *ω* angular frequency (rad s<sup>1</sup>

**45**

*κ* mass transfer coefficient (\_) *<sup>λ</sup>* Nafion® hydration ()

*σ* electric conductivity (S m<sup>2</sup>

*μ* charged particle mobility (m<sup>2</sup> V<sup>1</sup> s

*ξ* electro-osmotic drag coefficient (\_)

)

*J* electric current density vector (A m<sup>2</sup>

*n* number of electrons in reaction () *N* molar flow density (mol m<sup>2</sup> s

*R* universal gas constant (J mol<sup>1</sup> K<sup>1</sup>

*kT* global thermal exchange coefficient (K W<sup>1</sup>

)

1 )

)

)

)

1 )

)

)

)

)

1 )

)

)

)

*a/c* anode/cathode ()

**Figure 7.**

*Identification by means of a hybrid PSO-DE algorithm run simultaneously on all polarization curves. Dashed lines: experimental, continuous lines: optimized models (courtesy of Journal of Power Sources).*

the cathodic one. On the other hand, a failure of the identification procedure, that is, a poor fit of the polarization curves with varying operating conditions, typically hints at weaknesses in the model.
