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Review

A Review on solid oxide fuel cell models

K. Wang a,*, D. Hissel a, M.C. Pera a, N. Steiner b, D. Marra c, M. Sorrentino c, C. Pianese c,M. Monteverde d, P. Cardone d, J. Saarinen e

aUniversity of Franche-Comte, FEMTO-ST (UMR CNRS 6174), FCLAB, 90000 Belfort, FrancebEIFER, European Institute For Energy Research, Emmy-Noether Strasse 11, 76131 Karlsruhe, GermanycDepartment of Industrial Engineering, University of Salerno, 84084 Fisciano, SA, Italyd Faculty of Engineering, University of Genoa, 16145 Genoa, ItalyeVTT Technical Research Centre of Finland, Biologinkuja, P.O. Box 1000, FI-02044 VTT, Finland

a r t i c l e i n f o

Article history:

Received 10 December 2010

Received in revised form

6 March 2011

Accepted 10 March 2011

Available online 13 April 2011

Keywords:

SOFC

Modelling

Artificial intelligent

Neural network

Electrochemical impedance

spectroscopy

Model-based diagnosis

* Corresponding author.E-mail address: kun.wang@utbm.fr (K. W

0360-3199/$ e see front matter Copyright ªdoi:10.1016/j.ijhydene.2011.03.051

a b s t r a c t

Since the model plays an important role in diagnosing solid oxide fuel cell (SOFC) system,

this paper proposes a review of existing SOFC models for model-based diagnosis of SOFC

stack and system. Three categories of modelling based on the white-, the black- and the

grey-box approaches are introduced. The white-box model includes two types, i.e. physical

model and equivalent circuit model based on EIS technique. The black-box model is based

on artificial intelligence and its realisation relies mainly on experimental data. The grey-

box model is more flexible: it is a physical representation but with some parts being

modelled empirically. Validation of models is discussed and a hierarchical modelling

approach involving all of three modelling methods is briefly mentioned, which gives an

overview of the design for implementing a generic diagnostic tool on SOFC system.

Copyright ª 2011, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights

reserved.

1. Introduction cells at altitude, the hybrid SOFC/gas turbine cycle is

Fuel cell systems are considered as an alternative to

conventional fuel combustion power generation, thanks to

their lower emissions and higher efficiency. Amongst various

types of fuel cell, solid oxide fuel cell (SOFC) at high

temperature operation allows systems design that well uses

the fuel cell thermal output, which leads to higher system

efficiency than other fuel cell systems such as comparable

proton exchange membrane (PEM) fuel cell systems [1]. Due

to the importance of efficiency and the need to operate fuel

ang).2011, Hydrogen Energy P

a potentially attractive option for applications of auxiliary

power unit ofaircraft [1,2] and vehicle as well as for industrial

power supply, in stationary and even non-stationary elec-

tricity generation applications [3,4]. Besides, SOFCs possess

other advantages, i.e.

1. Due to its high temperature operating condition, internal

reforming (IR) can be realised;

2. Its insensitivity to gas contaminants enables utilisation of

unconventional fuels such as biomass or coal gas;

ublications, LLC. Published by Elsevier Ltd. All rights reserved.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 6 ( 2 0 1 1 ) 7 2 1 2e7 2 2 8 7213

Additionally, developers expect commercial SOFCs to have

lifetimes of 10e20 years, two to four times longer than other

fuel cells [5]. However, the capabilities of IR and gas insensi-

tivity lead to more complex electrochemical reactions inside

SOFCs. Moreover, carbon formation thermodynamically can

take place on the anodematerial in hydrocarbon-fuelled case.

These disadvantages canmake SOFC system suffer from a low

reliability. In order to avoid catastrophic system failures, an

online diagnostic tool for assessing and tracking the state of

health of SOFC stack or/and system is very necessary.

Nowadays, the diagnosis technique mainly relies on model-

based method [6], by analysing the residuals/deviations of the

measured system response from the simulated one by model

[7]. Following this diagnostic concept, a reliable and general

model which is capable of predicting the normal performance

of SOFC is required.

In the past decades, a great number of researchers had

investigated in SOFC modelling and the internal process

simulation based on physical principles. By using physical and

analytical equations, they translated successfully the elec-

trochemical reactions, the electronic and ionic properties of

materials as well as gas flow process to detailed physical

models. These models range from zero-dimensional (0-D) to

three-dimensional (3-D) with different features and point to

different research objectives. From the viewpoint of model

function, 2-D and 3-D modelling is typically concerned with

the cell and stack design issueswhile 0-D and 1-Dmodelling is

aimed at control purposes (on system-level) such as predic-

tion of both the transient and steady-state performance of

fuel cell/stack and establishing the optimal operating condi-

tions [5]. For the research target of setting up an online diag-

nostic tool, low dimensional models (0- and 1-D) are more

appropriate due to the less computational time in comparison

with the high dimensional ones (2- and 3-D). Moreover, high

dimensional models require information about material

properties or electrochemical parameters that are not always

available or might be difficult to determine. Even so, high

dimensional models are still helpful to learn the operation

behaviour of fuel cells of different geometry design and very

useful for creating training data for black-box modelling

which will be introduced in the fifth section.

Another method is AC impendence modelling. It is based

on electrochemical impedance spectroscopy (EIS) measure-

ments. The electrochemical information on an operating fuel

cell system can be obtained from the measured EIS data and

interpreted by fitting this data to an impedance model.

Recently, specific applications of EIS in SOFCs have appeared

frequently in the literature. The obtained results demonstrate

that this technique is an effective modelling approach. It is

worth noting that EIS is a tool used to acquire electro-

chemical parameters. It is also known as AC impedance

technique. When a perturbation signal (voltage or current) is

imposed on a SOFC, a corresponding output signal (current or

voltage) can be obtained. This signal is the reaction of the

SOFC to the perturbation. Comparing these two signals can

give a characteristic impedance Z(u). In EIS measurement,

a series of Z(u) in various frequencies are collected. They are

supposed to exhibit the SOFC characteristics and should give

information on physical behaviours inside the operating

fuel cell.

In fact, both physical and equivalent circuit fuel cell

models are mainly based on the knowledge of physicochem-

ical characteristics (electrically, chemically and kinemati-

cally), thus also called as “white”models. They presents a high

generalisability level that enables modelling SOFC stacks of

different geometric features, but require a high computational

effort. In contrast, there is another approach only based on

experimental database (no requirement for any physical

property), known as the black-box modelling. Black-box

models are developed particularly for control-oriented appli-

cations, i.e. system monitoring, online control and diagnosis.

This approach is appropriate for complex fuel cell system.

Nevertheless, the high dependency upon experimental data

makes it less generalisable and the fourth approach is thus

developed. It falls in between white and black-box

approaches, named grey-box modelling. Models based on this

method are partially physical and partially empirical.

In the following 4 sections (from the 2nd to the 5th section),

four modelling approaches for SOFC will be introduced by

presenting the models available in literature. It is worth

noting that all models reviewed in this paper are with the aim

of proposing a state-of-the-art of existing models which may

be useful for model-based SOFC system diagnosis. In addition,

whichever modelling approach to be used, it should be kept in

mind that since phenomena occurring in nature are too

complex to be completely described by mathematical equa-

tions, the required details to be described by the model must

be goal-driven, i.e. the complexity of the model, and the

related results, must be strictly connected to the main goal of

the analysis itself [8]. In the 6th section, the functions of the

reviewedmodels have been summed up and their application

on SOFC stack and system diagnosis is proposed. The valida-

tion of models is discussed, too.

2. Physical models

A great number of papers can be found on SOFC physical

modelling. Some were aimed at cell design modification or

material development. In this case, the models involve

simulations for the temperature distribution, the heat gener-

ation, and the flow diffusion. Others focus on predicting cell

performance which is expressed either in term of output

current density at fixed potential or in term of potential at

given applied current. Research objective determines the

complexity and the dimension of model [9,10]. In this paper,

the focus is put on general models which depict cell perfor-

mance for system analysis. The physical models covered in

this section are classified into 4 categories on the basis of

model dimensionality and will be introduced in the order of

dimension decreasing from 3-D to 0-D.

Multi-dimensional (MD) models are set up in the consid-

eration of spatial variation in the physical and chemical

variables such as gas concentration, temperature, pressure

and current density, for example [8]. In this review, the

covered MD models have an identical assumption that the

stack is made of repeating single cells stacked together thus

a single cell is simulated and its outcome is multiplied by the

number of cells to obtain stack results. Such a cell model

usually consists of three sub-models, i.e. thermal model, fluid

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model and electrochemical model. The former two (combined

together as a thermo-fluid model in some literature) are used

for the evaluation of temperature profile andmolar flux of one

or more dimensions inside the fuel cell: the first calculates

temperatures of each component (electrodes and electrolyte);

the second calculates gas flow rates as well as their partial

pressures at each electrode. The third is an electrical model

for predicting cell voltage. It is based on the following relation

at a given current density:

Ucell ¼ Uocv � ðiRþ hact þ hconcÞ (1)

where Ucell is the cell voltage, Uocv is the open circuit voltage

(OCV), iR is the ohmic drop (or the ohmic polarisation) and

hx are the activation and concentration polarisations. The

three polarisations contribute to irreversible losses in an

operating fuel cell. The OCV equals theoretically to the Nernst

potential, related with the local temperature and gas partial

pressures. However, in practise, it deviates from this ideal

value during cell operation due to concentration drop of

reactants. For an overall cell reaction, the cell potential

increases with an increase in the activity (concentration) of

reactants and a decrease in the activity of products [11].

Zero-dimensional (ZD) models are often used when fuel

cell is regarded as a single component of a bigger system, for

capturing the general operating behaviour/performance of

fuel cell and meeting the requirement of fast computation.

When themain purpose of this type of model is to analyse the

whole system, the physical-chemical variables variations are

not relevant, however, the performances, in terms of power,

heat and input requirements are important [12]. Therefore,

spatial variation of the parameters considered in MD models

can be not taken into account in ZD ones.

2.1. 3-D models

In a SOFC system, the fuel utilisation and the average cell

temperature can be controlled by the delivery rate and the

temperature of the gases into the cell. If the fuel concentration

is high at a cell area, the local electrochemical reaction is

active, leading to increased local temperatures and thereby

yielding faster reaction rate; in reverse, for a case of fuel

depletion, the reaction is inactive, thus a decreased local

temperature and a slower reaction rate [13]. Although

increased fuel flow tends to increase uniformity of the reac-

tion rates across the active area, it decreases fuel utilisation

[13]. Therefore, management of the flow and the inlet

temperature of gases is critical to stable cell operation. A 3-D

model allows simulating fuel cell internal behaviours and

giving information about the temperature and the fuel

distributions on three physical dimensions. The finite-volume

method is usually employed to separate a unit cell into several

parts and thus to simplify the calculation.

Fergusonetal. (1996) [14]presenteda3-DSOFCmodelwhich

could predict the voltage, the mass and electrical distribution

at cell-level. The heat source consisted of two terms, i.e. the

ohmic heat and the heat from shift and reforming reactants.

Since discontinuities of the potential and heat flux at the

electrode/electrolyte interfaces exist due to the surface elec-

trochemical reactions, the heat andmass transfer betweengas

channels and solid parts were taken into account. A potential

drop at the electrode/electrolyte interfaces due to electro-

chemical reactionswas considered. Thepotential at solid parts

was equal to Nernst potential. Faraday’s law depicts the

correlation between the electric current and the mass flux at

these interfaces. This model could be used as a design tool to

analyse the cell efficiency in different geometries (tubular and

planar) and configurations (co-, counter-flow and cross-flow

designs for a planar geometry). According to the analysis of

Ferguson et al., the counter-flow design was considered to be

the most efficient for planer geometry which showed less

ohmic loss in comparisons with the tubular geometry.

However, the effect of radiation was not considered in the

model.

Yakabe et al. (2001) [15] took into account the radiation

mechanism which was regarded as an essential effect on the

heat exchange inside the channels while the stack operating

at 900e1000 �C. At such high temperatures, the excess thermal

stress would lead to the non-homogeneous temperature

distributions and themismatch on cell components due to the

different thermal expansion coefficients. Therefore, the con-

ducted model in [15] was used to estimate the thermal

stresses in the cell components, so as to optimise operating

conditions to decrease the temperature gradients of PEN

(positive-electrolyte-negative). Paying attention on electrical

performance simulation, in the electrochemical model, the

concentration polarisation was included in the Nernst

potential; the activation polarisation was divided into

a constant term and a current-dependent term. The later was

integrated into a formula of ohmic resistance. The cell resis-

tance, with the contact resistances included, was estimated

from the experimentally measured IeV data of a unit cell. The

electric current density was expressed by the Faraday’s law

like in [14]. This model is applicable for SOFCs of counter- and

co-flow geometries.

Recknagle et al. (2003) [13] proposed a 3-D model for planar

SOFC, aiming at investigating the effects of cell flow configu-

ration on the distribution of temperature, current density and

fuel distribution. The model can predict the fuel utilisation,

the electric current density, and the temperature distribution.

In the thermal model, the radiant heat exchange was small

and neglected due to the large aspect ratios of the flow

channels in the studied stack (length-to-height, roughly 100:1

on the cathode side and 200:1 on the anode side). The elec-

trochemical model could predict local electrical responses to

changes in fuel flow rate, local fuel composition, and local

temperature. Three geometric configurations were examined

on the model and it was concluded that the co-flow case had

the most uniform temperature distribution and the smallest

thermal gradients. The same investigation was performed by

Wang et al. (2007) [16] and the relevant modelling equations

can be found in [16].

2.2. 2-D models

The 2-D model is simplified one from the 3-D model by

neglecting one dimension. It prompts to some assumptions

and simplifications, causing a reduction in the resulting

information [17]. There are two ways for choosing a 2-D

section and they are shown in the following figure (see Fig. 1).

Fig. 1 e 2-D cross-sections representation of an SOFC unit.

Fig. 2 e SOFC equivalent circuit [18].

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The first case takes the x-z plane, assuming that all

parameters (such as the temperature, the gases concentra-

tions and pressures, etc) are uniform in y direction. In the

second case, the current collectors are separated from PEN by

gas channels; thereby, the electrical potential on electrode

boundary is not constant. The ohmic resistance variation

along the neglected� direction needs to be taken into account

in the model formulation [17].

Xue et al. (2005) [18] developed a dynamic quasi-2D model

for a tubular counter-flow SOFC. It was capable of character-

ising the transient/time- and spatial-dependent properties of

critical state variables. The studied section belonged to the

second case and was separated, by applying the control

volume (CV) method, into four CVs, i.e. anode channel, cell,

cathode channel and thermal insulator. the section. Physical

properties, within each CV, were assumed to be uniform, but

they changed in different CVs. The mass/species balance

equation was used to describe the mass/species conservation

in each CV. The momentum effect on the main flow stream

caused by electrochemical reaction was neglected. The radi-

ation heat transfer between the gas CVs (anode and cathode

channel) and the solid CVs (cell and thermal insulator) was

not considered but that between the cell CV and its adjacent

thermal insulator CV was taken into account in the thermal

model. As considering the case that the external load potential

was higher than the cell Nernst potential, the fuel cell in this

research was regarded as a combination of a Nernst potential

source and a capacitor as shown in Fig. 2. Moreover, the three

polarisation resistances were calculated based on the instant

conditions. As a result, a dynamic model of a tubular SOFC

was implemented and it was useful for studying both the

steady-state and the transient cell behaviours. It concluded

that this model could be used in system optimisation and

dynamic controlling.

In [19], amodel-based study focused on transient operation

was carried out based on a dynamic 2-Dmodel. The possibility

for improving the cell/stack performance by proper gas flow

configurationwas confirmed. The possible degradation effects

due to increased thermal stresses were pointed out in the

study. Themodel was validated against empirical data as well

as another 1-D model in [20]. Furthermore, the transient

behaviour of thismodel was compared against that of another

two-dimensional dynamic model with tubular cell configura-

tion in [21] and the planar stack configuration was found to

adapt faster to the operating condition changes.

Chnani (2007) [23,22] also took the second case for model-

ling a planar SOFC with co-flow channels. He developed the

thermal and the fluidic sub-models through electrical

analogy. This method allowed exhibiting gas flows and ther-

modynamic behaviours in term of equivalent circuit so that

multiple identical models could be connected together to be

a stack-level model. For a stack module, therefore, this

advantage is obvious: the thermal circuit can describe

temperature gradient along cells. Fig. 3 shows the schematic

diagram for overall cell modelling. The transient thermal

model was used to compute the solid and the gas tempera-

tures. The fluidic model calculated the partial pressures of

chemical species. The electric (electrochemical) model

computed the stack voltage and the polarisations with the

parameters from these two sub-models.

In thermal behaviour modelling [23], the cell was firstly

divided into 7 isothermal volumes (as shown in Fig. 4),

including anode interconnect, anode channel, electrolyte/

anode interface, electrolyte, electrolyte/cathode interface,

cathode channel and cathode interconnect. Fig. 5 shows the

2-D thermal equivalent circuits for each volume. They were

connected by temperature nodes to be a nodal network. In

order to capture thermodynamic behaviours inside the cell,

both heat generation and heat transfer were considered. The

former referred to chemical reactions and ohmic losses; it was

represented by thermal source symbols. The later was due to

three fundamental heat transfer mechanisms (convection,

conduction and thermal radiation) as well as mass transfer

(heated gas transportation in channels); these four mecha-

nisms were expressed by temperature drops on the corre-

sponding thermal resistances (Rx_conv, Rx_cond, Rx_ray and Rin/out_x

illustrated respectively by blue, green, red and black resis-

tance symbols in Fig. 5). The heat capacities at particular

nodes indicated thermal energy stored between and inside

volumes. In thismodel, the heat source of chemical reaction is

only located at the interface anode/electrolyte.

For the fluidic model, an equivalent circuit based on the

electric fluid analogy (as shown in Fig. 6) was built to depict

the fluidic behaviour in gas channels. The gas flow was

homologous to the electrical current while the pressure to the

Fig. 3 e Schematic of stack modelling [22].

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voltage. The fluidic resistance was regarded as the electrical

resistance. It was considered that the pressure drop between

the air and the fuel sides was linear with the gas flow rate.

In the electric model, three modes of polarisation were

considered but the resistance of contact was ignored due to its

little contribution to the ohmic resistance, when compared

with the electrolyte resistance. Finally, a group of the cell-level

model could be easily connected in parallel or in series to

obtain a stack-level model.

2.3. 1-D models

In 1-D model, the fuel cell is usually treated as a set of layers

including interconnects, air channel, electrodes, electrolyte

and fuel channel [8], just like the case shown in Fig. 4 but

neglecting physical variations at vertical axis. Both gas

composition and flow rate in each channel are assumed to be

constant and theirmean values are used in the simulation. For

Fig. 4 e Heat transfer and heat

planar SOFC, the dimension is following along the gas channel

and the direction is determined by the gas flow. It is necessary

to note that the fuel cell of cross-flow design cannot be

simulated by 1-D models. For tubular SOFC, the kept dimen-

sion is usually the tube axis which coincides with the direc-

tion of the fuel and oxidant flow [17].

Magistri et al. (2004) [24] built a one-dimensional model for

tubular SOFC, where the cell coordinate x is the axis of the tube

and its origin corresponds to the bottom of the cell. The main

hypotheses of the single cellmodel are: 1) the cell is adiabatic, 2)

the cell voltage is uniformand all the chemical reactionswithin

the anodic stream are at equilibrium, 3) the electrochemical

reaction of H2 is taken into consideration; the electrochemical

reaction with CO is neglected. The cell model includes: electro-

chemical performance, equilibrium of reforming and shifting

chemical reactions, mass balances of anodic and cathodic,

energy balances of gaseous flows, energy balance of the tube

and of the solid PEN structure. In the paper, the 1-D model was

sources in 7 volumes [22].

Fig. 5 e Cell-level thermal model [23].

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described and the results were compared to the 0-D model

simulation proposed in [25]. In both twomodels, the input data

are geometrical characteristics, operating conditions, inlet flow

conditions and gas and material properties. The 1-D model

represented the cells as a plug-flow reactor, so it integrated

along the cell coordinate the values of the operating. The 0-D

model considered the tubular cell as a continuous stirred-tank

Fig. 6 e Electrode fluidic model [22].

reactor; and the thermodynamic and electrochemical parame-

ters were thus uniform along the cell coordinate. Both the

models were integrated through a relaxation method for the

evaluation of the cell performance and were included into

a whole system model. The model comparison showed how

discriminating the simulation accuracy could be in studied

cases. Under some operating condition, it was possible that not

great difference was there between the average simulation

results from the detailed and the simplified SOFC models; at

a first glance, the results seemed reasonable and compatible

with the technological limits of the components of the hybrid

system. But a deep analysis of the results from the detailed

model revealed that the temperature inside the stack was not

uniform and, although the average value was acceptable, the

maximum values were too high. The comparison of two

different SOFC models is very important for investigating how

the studyof thewhole system isaffectedby the approximations

of SOFCmodels. Indeed, detailed fuel cell models require a long

Fig. 7 e Schematic of gas flows in a tubular SOFC [30].

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computational time and knowledge about geometrical data,

materials and lay out of the fuel cell, which are rarely available.

For these reasons, simplified fuel cellmodels are generally used

inHSsimulations, but in thiswaycriticalaspect suchashot spot

temperature of the SOFC cannot be investigated, and the

calculation could give results very different from the real

performance of the fuel cell and whole system. On the other

hand, there are several caseswhere the results of the two types

of simulation coincide, and it is difficult to have information

a priori on the range of operating conditions where this occurs.

In [26,27], a so-called dynamic behaviour model of an SOFC

was developed and verified. Themodel was capable of solving

the I-V-behaviour and the temperature distribution in the gas

flow direction inside a cell operating under either co- or

counter-flow mode and it was found to be sufficiently accu-

rate for rapid system simulation [28]. The model enabled, e.g.,

designing the gas flow rates accordingly in respect to the

maximum drawn current density and, thereby, to prevent

overheating of cell.

Aguiar et al. (2004) [29] developed a 1-D dynamic anode-

supported intermediate temperature planar SOFC with direct

internal reforming. This model predicted the SOFC charac-

teristics both in the steady and the transient states. It con-

sisted of mass and energy balances, and an electrochemical

model. For the mass balance the molar flux in the gas chan-

nels was considered convective in the flow direction. It was

assumed that only hydrogen was electrochemically oxidised

and that all of CO was converted through the shift reaction,

considered to be at equilibrium. In the fuel channel, three

reactions are taken into account: 1)methane steam reforming;

2) water gas-shift; 3) and hydrogen electrochemical oxidation.

In the air channel, only the reduction reaction of O2 was

considered. Faraday’s law related the flux of reactants and

products to the electric current arising from an electro-

chemical reaction. In the energy balance were included the

released heat from electrochemical reactions and ohmic los-

ses; the convective heat transfer between cell components

and gas streams; and the in-plane heat conduction through

cell components. The thermal fluxes were supposed to be

conductive and radiate between the PEN and the interconnect

components. However, in the gas channels, they were

assumed to be convective in the gas flow direction and from

the gas channels to the solid parts. In the electrochemical

model the OCVwas calculated by the Nernst equation and the

SOFC stack was considered isopotential.

Costamagna et al. (2004) [31] studied an innovative fuel cell

concept, the Integrated Planar Solid Oxide Fuel Cell (IP-SOFC)

which was substantially a cross between tubular and planar

geometries, seeking to borrow thermal compliance properties

from the former and low cost component fabrication and

short current paths from the latter. In this new concept,

several cells of small dimensionwere deposited over a vertical

porous substrate and electrically connected in series, in order

to obtain high voltages and low electrical currents. The scope

of IP-SOFCmodelling was multi-fold: to better understand the

physical-chemical phenomena occurring in the electrodes, in

the fuel cell and in the stack, to predict the local behaviour of

the cell and to identify dangerous effects (for example, hot

spots) whichmight lead to damage. This model could serve as

the basis for planning experimental campaigns and provide

a useful tool for optimisation of fuel cell systems, in aspects of

both operating conditions and design parameters. The model

involved different levels of simulation: electrode, single cell,

tube, bundle, stack and block. At the electrode level, the main

phenomena taken into account were the electrochemical

reaction, the charge conduction and the mass transfer. At the

tube and bundle level, the models included themass balances

of the gaseous streams and energy balances of the gaseous

streams and of the solid.

Jiang et al. (2006) [30] set up a 1-D dynamic model for

a tubular SOFC with external reforming. The cell was divided

into elements along the flow direction, like shown in Fig. 7. For

each element, therewere 4 CVs separated along perpendicular

axis (see Fig. 8): the fuel, the solid, the reaction air and the pre-

heated air CVs. Several assumptions were made for the

thermal model: 1) for every element, the temperature within

each CV was uniform; 2) the radiation and the conduction

heat transfer were not taken into account; 3) the convection

heat transfer was assumed as the only reason for the

temperature gradient of gas streams in the flow. The heat

generation due to the reactions (shifting, reforming and

electrochemical) and the ohmic losseswas calculated. The cell

voltage at each element was uniform. An equivalent circuit

(see Fig. 9) was built to evaluate the influence of the current

path length to the ohmic loss. This model was capable of

predicting SOFC characteristics in both the steady and the

transient states and showed a good reliability. Results from

the model showed that elevated pressure could improve the

cell performance whereas higher operating temperature

decreased both theNernst potential and the irreversible losses

(ohmic, activation and concentration losses).

Zhang et al. (2006) [32] developed a 1-D non-linear, control-

oriented dynamic model for planar SOFC. Two kinds of fitting

function, namely the exponent decay function and the expo-

nent associate function were introduced to fit the distribution

characteristics of the gaseous molar fractions and the

temperature along the streamwise direction. The spatial

effect was lumped into the dynamic model by fitting the three

parameters of the used function. These parameters were

determined through numerical simulations.

Sorrentino (2006) [33] developed a 1-D steady-state model

for co-flow planar SOFC. The model was divided into three

sub-models: 1) mass balance model; 2) energy balance model;

3) voltage model. The model was based on the control volume

(CV) approach, according to which the cell was discredited in

CVs in the flow direction and divided into three layers: anode

channel, cathode channel and cell (solid layer). The cell was

assumed to be isopotential and the pressure drop across the

fuel and air channels was neglected. The radiant heat transfer

Fig. 8 e Control volume definitions for one element [30].

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and the heat conduction in the solid layer were neglected. The

stack was assumed to be adiabatic. The heat convection

between solid layer and gas streams and the energy transfer

due to the reactants and products were considered dominant

in the energy balance. The model showed a good accuracy in

the simulation of SOFC states and variables. It was adopted to

generate SOFC stack data to be used in a hierarchic modelling

approach to implement a control-oriented model [34].

Cheddie et al. (2007) [35] upgraded a 0-D real time model to

a dynamic 1-D model in order to predict more accurately the

temperature and pressure variations along the gas flow

direction. The real time capability was maintained by setting

up several simplifications: the current density distribution

was considered uniform and there was no need of calculating

the cell current iteratively, therefore resulting in reduction of

computational effort. The overpotentials at each node were

replaced by the average one across the cell. It was assumed

that the voltage immediately responded to changes in current

so the transient states were not taken into account. The gas

concentration was considered dependent only on partial

pressure rather than both pressure and temperature. In

thermal model, all heat generations were assumed to occur in

Fig. 9 e Equivalent circuit for the tubular SOFC cross [30].

the PEN. The heat conduction was negligible in the fluid phase

due to the fact that the thermal conductivity ismuch higher in

the solid regions than in the fluid phases. The 1-D model with

21 nodes was proven to require 3.8 ms of computational time

for each iteration. The model validation showed that the

limiting assumptions did not lead to the significant simulating

difference when comparing with a more comprehensive 1-D

model without these assumptions. Moreover, the proposed

model was capable of predicting more accurately the trans-

port phenomena.

Kang et al. (2009) [36] modified a 1-D dynamic model for

a planar internal reforming SOFC also by integrating two

simplifications: 1) the PEN, interconnects and gas channels

were integrated together along the perpendicular direction,

that is, the SOFC is considered to have only one temperature

layer; 2) the current density distribution is considered to be

uniform within the SOFC, and the cell voltage is determined

by the average gas molar fractions and cell temperature.

These two simplifications are similar to the assumptions in

Cheddie’s modelling. In fact, by introducing them, the SOFC

model was greatly simplified in form. This model contained

100 nodes and its computational time was decreased

comparing with the lumped one. Moreover, it showed an

improvement with regard to accuracy because it took into

account the spatially distributed nature of SOFCs to a certain

extent.

2.4. 0-D models

The 0-D model is the simplest one. No dimension is deter-

mined; and thereby spatial variations are not taken into

account. The transformations are considered to define output

variables from input ones. 0-D models are simplified based on

assumptions and practical information. They can be used for

numerical analysis of fuel cells in energy systems such as

SOFC/gas turbine hybrid system. In such a system, the single

elements, for instance, compressors, heat exchangers, fuel

reformer, partial oxidisers, and contaminant removal appa-

ratus are simulated through independent box models [17].

Furthermore, they allow being easily calibrated and modified

for new developed materials.

Costamagna et al. (2000) [25] described a hybrid system

where the SOFC was simulated with the 0-D model approach.

The balance equations were written as macroscopic balances,

in form of finite equations. Those equations expressed

a balance between inlet and outlet flows ofmass and energy in

each component of the group; under suitable assumptions,

they allowed the evaluation of the average values of the

physical-chemical variables of each components and the

electrochemical performance of the group itself.

In Campanari’s 0-D SOFCmodel (2001) [37], the cell voltage

was a function of the current density, the operating temper-

ature and pressure as well as the reactants and product

composition. Bove et al. (2005) [12] built a macro model in

which the ohmic polarisation depended only upon the mate-

rial properties. The open circuit voltage and the activation

polarisation were related to gas concentration while the

concentration polarisation was ignored. The mean current

density was regarded as an input variable. Three different

modes of gas composition, i.e., inlet gases composition, outlet

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gases composition and a mean value between the previous

two, were respectively used in the simulation for estimating

the cell voltage. The simulated results showed that if the first

mode was considered, the effect of fuel utilisation variation

could not be estimated; on the contrary, the cell voltagemight

be underestimated if with the second mode. Therefore, as

a conclusion, the third mode seemed as the best, which could

get a compromise between these two aspects.

Magistri, Ferrari et al. [38e40] made a transient analysis of

hybrid system based on SOFC. This system was mainly

composed of three parts: the stack, the anodic recirculation

systemwith fuel feeding and the cathodic side (air side) where

turbo-machinery and heat exchangers were installed. These

transient researches allowed a deep investigation of the Fuel

Cell Stack complete with reformer and post-combustor

models. In this way it was possible to define the start-up and

shut-down procedures, avoiding risks for thewhole plant. The

tubular SOFC was considered as a stand-alone unit in order to

understand its behaviour without taking into account the

influence of the other components of the plant. This model

was developed in the MATLAB-Simulink environment with

the TRANSEO [41] tool and was successfully verified at design

and off-design conditions. Moreover a special time charac-

terisation of the transient phenomena was introduced in

order to automatically suggest to the user the proper inte-

gration time step (Dt) to employ. Analysing the results of the

model, they observed, during transients condition, unex-

pected fluctuations could occur and need to be carefully

monitored, in order to avoid the system running into

“forbidden” or “dangerous” areas.

Modelling the dynamic behaviour of SOFCs is nowadays

a highly strategic research area to well address safe operation

as well as degradation prevention of SOFC stacks. Bhatta-

charyya and Rengaswamy, in their extended literature [3],

reviewed SOFC dynamic models, highlighted the importance

of simulating SOFCs in transient conditions in order to opti-

mise design, control and diagnosis of SOFC systems. With

particular regard to these latter goals, on field of performance

monitoring, as well as the management of energy and mass

flows during system start-up and load changes, it is required

that the development of modelling tools meeting the

compromise between satisfactory accuracy and affordable

computational burden.

The above compromise can be easily achieved by 0-D (i.e.

lumped)modelling approaches, thus explaining the increasing

number of such scientific contributions that recently appeared

in the SOFC literature [3]. An Interesting lumped approachwas

followed by Sedghisigarchi and Feliachi [42] for control and

stability enhancement of SOFC-based distributed generators

[43]. Nevertheless, in [42] average cell temperature was

assumed as state variable, thus not allowing to provide some

basic information for balance of plant analysis, such as

temperature of exhaust gases (i.e. outlet SOFC temperature).

Sorrentinoet al. [34]proposedhierarchicalmodellingapproach

to derive a lumped, control-oriented model of planar SOFC

capable of accurately simulating temperature and voltage

dynamics as function of the main operating variables (i.e.

current density, fuel and air utilisations, inlet and outlet

temperatures). Thecontribution [34], differently than [42], does

take into account temperature variation across the channels,

thus being suitable to perform, at low computational cost,

accurate balance of plant analyses, including heat exchangers

sizing [3]. Thus, Sorrentino andPianese [44] proposed to extend

the lumped approach presented in [34] to the modelling of

a fully integrated SOFC-APU (i.e. auxiliary power unit). This

latter contribution was also proven to be valid for the devel-

opment of model-based diagnostics tools for mobile SOFC-

APUs [45]. The 0-D approach was also applied to transient

modelling of tubular SOFC by Hajimoliana [46], to develop

suited strategies aimed at controlling voltage and cell-tube

temperature by properly acting on both temperature and

pressure of the inlet air flow.

3. Equivalent circuit models with EIStechnique

EIS is a powerful technique which is usually used to assess the

data on the internal resistance, degradation and failure within

an electrochemical system. The approach of EIS relies on

measuring the impedance of a system at different frequencies

by superimposing a small voltage/current AC perturbation

onto the voltage/current operating point. The measured

impedances are a function of frequency. This data is then

used for impedance modelling, including model structure and

parameter identification. For fuel cell systems, the measured

AC impedances Z(u) are usually displayed through the

impedance plot method such as Nyquist plot or Bode plot and

three-dimensional perspective plot like in [47].

Impedance modelling is a critical procedure because it is

the basis of data interpretation. After data fitting, the param-

eters in the model will be identified. In some cases, the

parameters can intuitively provide the information about the

processes inside the system; in other cases, however, an

interpreting action should be carried out. This phase has

a function equal to that of mapping, i.e. transferring from the

identified model parameters to the investigated characteris-

tics of the system [48]. In SOFC performance diagnosis,

equivalent circuitmodels are often constructedwith electrical

elements such as a parallel ReC circuit in the simplest case

[49]. Each discrete element is supposed to have corresponding

physical meanings and thereby, the interpreting action is not

necessary. Once the values of these elements (parameters) are

identified in various operating condition, they can be used as

an estimated data set for fitting physical models. This model-

ling method is explicated in detail in [50].

Takano et al. (2004) [51] investigated specially the imped-

ance corresponding to mass transfer, usually called gas-

diffusion impedance (GDI). In their work, it was assumed that

the solid electrolyte, the electrodes, and the current collectors

were all of homogeneous composition with uniform thick-

ness. The pressure and the temperature inside the cell were

uniform. The ECM with GDI shown in Fig. 10 consists of the

resistance (Rwg) and capacitance (Cwg) for GDI, the reaction

resistance (Rr), the double layer capacitance (Cd), the elec-

tronic resistance (Re), the ionic resistance (Ri) and the ohmic

resistance (Rs). The simulation results showed that the GDI

was significantly dependent on the fuel utilisation. The GDI

became larger as the fuel utilisation approached both 0 and

Fig. 10 e ECM with GDI [51].

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100% and had the inversive relationship with the gas flow

rates.

Lang et al. (2008) [52] used EIS to measure the different

kinds of resistances which contribute to the area specific

resistance (ASR) value (or the slope the I-V curve), due to the

fact that it is not possible to distinguish them via the IeV

curves. Their essential idea is varying the gas composition so

that the different processes can be ordered to the different

frequency ranges. The measured resistances were fitted into

the equivalent circuit (see Fig. 11) so as to simulate the elec-

trochemical behaviours of the cells. Among, the impedance of

the current collector wires is taken into account by the

inductive element ZL; the three terms combining a resistance

and a constant phase element (CPE) represent the polarisation

of electrodes and the gas conversion impedance; the ohmic

resistance is connected in series with these terms.

4. Grey-box models

The aim of the grey-box modelling is to develop model-based

tools aiming at optimal design, management and diagnosis of

SOFC units destined to a wide application area [45].

The grey-box method is based on a priori knowledge con-

cerning the process and on the mathematical relations which

describe the behaviour of the system. This means that the

starting point is a specific model structure based on physical

relations. The construction procedure of a grey-box model

based on mathematical relations can be divided into different

sub-procedures: basic modelling, conduct experiment of the

process, calibration and validation [53]. The flexibility of

Fig. 11 e ECM for a SOFC stack [52].

a grey-box model allows us to extract rules that describe the

behaviour of a device.

Sorrentino and Pianese [45] presented a grey-box model of

a SOFC unit. The core part of the model is the fuel cell stack,

made of planar co-flow SOFCs and surrounded by a number of

auxiliary devices, namely air compressor/blower, regulating

pressure valves, heat exchangers, pre-reformer and post-

burner. As a consequence of low thermal dynamics charac-

terising SOFCs, a lumped-capacity model is proposed to

describe the response of fuel cell and heat exchangers to load

change.

5. Black-box models

The black-box model is a behavioural model that is derived

through statistical data-driven approach. Contrary to the

physical models, they are not based on explicit physical

equation definitions but the measured database which is

capable of reflecting the relationship between inputs and

outputs.

As stated in the previous sections, SOFC is a non-linear,

dynamic system with multiple inputs and outputs. So far,

most of existing models are based on physical conversion

laws and governing equations. Although being useful for

analysis and optimisation of SOFC, they are too complex for

model-based control system. This drawback impelled some

researchers [54e65] to attempt black-box method which is

based on mapping inputs to the appropriate outputs. The

black-boxmodel is constructed without any physical laws but

only a set of input-output pairs for training procedure. It has

been verified that the black-box models based artificial intel-

ligent are very suitable for non-linear dynamic systems [6].

However, such a model requires an amount of database/

experimental data which should well represent a specific

feature of the system. Therefore, the experimental time for

collecting data is very long.

The following paragraphs will give a summary of this kind

of SOFC models most of which aim to predict the cell perfor-

mance in terms of voltage and/or electric power.

5.1. ANN (artificial neural network)

Artificial neural network is a statistical data-driven approach.

It is inspired by the central nervous system, exploiting features

such as high connectivity and parallel information processing,

exactly like in the human brain. An ANN is capable of

producing a response to a specific combination of input data.

With a great number ofmeasured data froma system, theANN

can be trained to learn the internal relationships that govern

the system, and then to predict its behaviour at a given input.

Across the whole modelling, no relevant physical equation is

used. Moreover, the highly parallel connectivity of ANN can

reduce the computational time.

Arriagada et al. (2002) [54] proposed a non-linear fuel cell

model by utilising ANN for evaluating SOFC performance. This

model is a two-layer feed-forward network (see Fig. 12). The

outputs are air flow, current density, air outlet T (tempera-

ture), fuel outlet T, mean solid T, fuel outlet T, mean solid

T and the reversible voltage. The model was trained with

Fig. 12 e Feed-forward 2-layer neural network.

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a reduced amount of input and correct output data pairs

generated by a physical cell model. The BP (backpropagation)

algorithmwas used to modify the weights, which requires the

use of differentiable transfer functions (logistic-sigmoid and

tanh-sigmoid). Comparing the outputs of the ANNmodel with

that of the physical model, the average values of the errors are

well below 1% and the maximum below 4%. Besides the

numerical accuracy, the ANNmodel is much faster and easier

to use, which makes it suitable for the generation of perfor-

mance maps.

Milewski et al. (2009) [61] applied an ANN with the same

configuration to simulate the SOFC behaviour, using experi-

mental data for training and testing process. This ANN-based

SOFC model had 9 input parameters (current density,

cathode inlet O2 and N2 flow densities, anode H2 and He flow

density, anode thickness, anode porosity, electrolyte thick-

ness and electrolyte temperature) according to which cell

voltage could be predicted. A hyperbolic tangent sigmoid

transfer function was used as the neuron activation function

in the first layer, whereas a linear transfer function was used

in the output layer. The testing results show that ANN can be

successfully used in modelling of the singular solid oxide fuel

cell. However, its practical design suffers from some draw-

backs such as the existence of local minima and over-fitting

Fig. 13 e RBF neural network.

as well as the determination of the number of hidden layer

nodes, etc.

5.2. RBFNN (radial basis function neural network)

Compared with the general form of ANN as stated above,

radial basis function neural network has a number of advan-

tages, such as better approximation properties, simpler

network structures and faster learning algorithms. The RBF

neural network is a feed-forward neural network and can

uniformly approximate any continuous function to a pre-

specified accuracy (Warwick, 1996) [63]. It consists of an input

layer, a non-linear hidden layer and a linear output layer (see

Fig. 13). The input variables are each assigned to nodes in the

input layer and connected directly to the hidden layer without

weights. The hidden layer nodes (RBF units) calculate the

Euclidean distances between the centres and the network

input vector, and pass the results through a non-linear func-

tion. The output layer nodes areweighted linear combinations

of the RBF in hidden layer.

The most used non-linear function for RBF units is

Gaussian activation function. For realising the RBF algorithm,

it is very important that how to choose the optimum initial

values of the three parameters: the output weights, the

centres and the widths of the Gaussian function. If these

parameters are not appropriately chosen, the RBF neural

network may degrade validity and accuracy of modelling [64].

The standard training method determines the hidden centres

by clustering approach which usually results in a large

number of selected centres. In addition, this kind of training

method is time-consuming, since it requires examining many

different network structures by using a trial and error proce-

dure. Chakraborty built RBF neural networks with 3, 4, 5 and

10 hidden neurons, respectively, in order to find the optimal

network structure for the SOFC modelling [55].

Wu et al. (2007) [64] built a 2-3-1 RBF neural network and

utilised a genetic algorithm (GA) to optimise the parameters of

the network. The optimum values are regarded as the initial

values of the RBFNN parameters and the gradient descent

learning algorithms were used to adjust them. GA is a kind of

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self-adaptive global searching optimisation algorithm based

on the mechanics of natural selection and natural genetics

[57]. Different from conventional optimisation algorithms,

GA is based on population, in which each individual is evolved

parallel, and the ultimate result is included in the last

population [64].

The similar work was done by Huo et al. (2008) [59] for

realising a Hammerstein model of the SOFC in which the non-

linear static part was approximated by an RBFNN and the

linear dynamic part was modelled by an autoregressive with

exogenous input model. Such a model aimed at controlling

fuel utilisation and output voltage of a SOFC stack so that the

stack could be protected and the voltage demand of DC type

loads could be meted. The natural gas input flow, the oxygen

flow, the operating temperature and the stack current were

chosen as the model inputs. The fuel utilisation was kept

constant. The gas input flow was controlled according to the

stack current which was proportional with the load. Through

a large number of tests, an RBFNN with 6 hidden nodes was

proven for obtaining a better performance.

According to Chakraborty [55], the number of hidden layer

neurons in RBFNNcan be determined from the training data by

a learning algorithmwhile optimising the number of Gaussian

neurons with a global search algorithm, such as genetic

algorithm or differential evolution, would cause more

computational time to be spent.

5.3. LS-SVM (least squares support vector machine)

LS-SVM was proposed by Suykens and Vandewalle [62] as

a modification of the standard SVM. It possesses prominent

advantages over ANN, such as few occurrence of over-fitting

through the structural risk minimisation principle, and the

capability to get the global optimal solution by solving a set of

linear equations [65].

A non-linear model of SOFC was established in [58], based

on LS-SVM. Fuel utilisation and cell current were chosen as

the two inputs and cell voltage as the output. The training data

was generated by a mathematical cell model operating at

steady-state regime. The RBF function was used as the kernel

function of LS-SVM, in which the two important parameters,

regularisation parameter and kernel width, were tuned

rapidly with a 10-fold cross-validation procedure and a grid

search mechanism by LS-SVM toolbox. In comparison with

the RBFNN approach, simulation results in this research

showed that the LS-SVM yielded higher prediction accuracy.

5.4. ANFIS (adaptive neural-fuzzy inference system)

ANFIS is a fuzzy inference system (FIS) implemented in the

framework of adaptive networks. It was put forward by

Dr. Jang while various combinations of methodologies in

“soft” computing emerged. It integrates the advantage of both

neural networks and fuzzy system, which not only has good

learning capability but can also be interpreted easily [65].

The architecture of ANFIS and the methods to update

parameters in membership functions during learning process

have been introduced in detail in [60]. The FIS is composed of

five functional blocks (see Fig. 14):

1. A rule base containing a number of fuzzy if-then rules;

2. A database which defines the membership functions of the

fuzzy sets used in the fuzzy rules;

3. A decision-making unit which performs the inference

operations on the rules;

4. A fuzzification interface which transforms the crisp inputs

into degrees of match with linguistic values;

5. A defuzzification interface which transform the fuzzy

results of the inference into a crisp output [60].

Entchev et al. (2007) [56] applied the Fuzzy Logic Toolbox of

MATLAB to build an ANFIS model which could predict SOFC

stack current and voltage. This model initialled the parame-

ters in membership function and then they were adjusted by

applying a combination of the least squares estimate (LSE)

method and the back propagation (BP) gradient decent

method. Similarly, Wu et al. (2008) [65] applied ANFIS to build

a dynamic model of SOFC stacks for predicting stack voltage.

Note that at MATLAB, the ANFIS usually applies a hybrid

learning algorithm in which the consequent parameters are

identified by the LSE and the antecedent (premise) parameters

by the BP.

6. Validation and application of models

6.1. Validation of models

Model validation is the last step of model development that

should always be carried out only after the modelled physical

set up is fixed. It is usually done by comparing the simulated

results with the actual measurement data which were never

presented in model parameterisation and tuning. What

results to be selected for comparing is dependent on model-

ling objectives. For system simulation and optimisation, the

following parameters may be chosen to be verified:

1. the polarisation curves (IV curves) or/and the impedance

spectrum (only for AC impedance modelling approach)

where steady-state operating point (such as gas flow rates

and compositions, operating temperature, fuel inlet and

outlet temperatures) are required;

2. fuel cell performance (such as electrical power or efficiency,

operating voltage and temperature profile) at different load

and operating conditions;

3. transient behaviour of the stack (stack temperature evolu-

tion against time, voltage or current profile) during load

changes.

If a model can reproduce the same results (at a given level

of confidence) with the experimental ones, this model is

considered valid.

However, some degree of uncertainty in experiments and

also in calculations always exists [8] which might result in

large deviation between the computed and the measured

values. In practise the data obtained from even repeated

experiments are never identical while in simulation themodel

should always produce the same output when given the same

input. Due to this fact, the uncertainty of measurement data

should be considered when validating a model. In [18], the

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conducted model was validated at steady-state condition by

verifying the correlation of the experimental polarisation and

power density curveswith the simulated ones. Themodel was

finally considered valid because the simulation results had

a consistent trend when compared to experimental data.

Furthermore, due to high temperature operating, there is

a large chance of having some variation in materials resulting

in defects in the SOFC structure. Hence, a lower experimental

performancemust be expected. In [31], the authors considered

during model validation the effects of micro-cracks which

lead to 1) a not perfect adherence between cell components

and 2) some cross-over of the reactant gases between two

electrode sides. These effects were simulated respectively

1) by introducing an additional constant resistance into the

model and 2) by considering occurrence of a chemical reaction

between oxygen and hydrogen which causes the presence of

water in both the anodic and the cathodic flow rates. The

polarisation curves obtained from the simulations and the

experiments are compared to verify the agreement of these

results.

Besides the uncertainty of experiment, another difficulty of

SOFC model validation is due to limited resources and

measurement techniques such as the difficulties ofmeasuring

variables like local current density, temperature or gas

composition, especially for multi-dimensional SOFC models.

In this case, an indirect validation can be performed by

comparing the predictions of two or more independent

models for an identical test case [8]. Achenbach (1994) [66]

compared eight independent models for a predefined bench-

mark test. The values of three outputs from these models, i.e.

the maximum solid temperature, minimum solid tempera-

ture and the air exit temperature, were analysed by statistical

method. The 2-D model in [19] was validated against a 1-D

model as well as empirical data. The validation of the 2-D

model in [67] was carried out with measurements and a 3-D

computational fluid dynamics model. The 1-D model in [24]

was validated by being compared with a 0-D model. In [12],

the authors compared three 0-D models with different gas

composition modes as an input for simulating SOFC perfor-

mance evaluation. Through analysing the results with phys-

ical and empirical knowledge on SOFC system, it was

concluded that using the average composition rather than

inlet or outlet gases composition when the fuel utilisation is

low, the SOFC behaviour could be approximated well.

For black-box models, before modelling, experimental

data are divided into three parts the third of which is for

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Fig. 14 e Fuzzy inference system [60].

model validation. It should be ensured on one hand that none

of points in such a data set is involved in the training data set

and on the other hand that this validation data set should be

involved in the mathematical space of the training one.

Therefore, validation of black-box model permits to assess

how well the model can explain the significant information

in the training data and how it will generalise to an inde-

pendent data set with the same or similar information

involved in.

6.2. Application of models for SOFC system diagnosis

As stated in the introduction, this review is aimed to propose

a state-of-the-art of existing SOFC models for system model-

based diagnosis. The design for a fault-diagnosis system

begins frequently withmathematicalmodelling of the process

[68], following the idea of comparing the measured response

of system with the simulated normal one to determine

whether or not a fault has occurred during the real process. In

our case, the diagnosed object will be the FC system with

potential failures in operation that yield a sudden drop of

performance. The stack will be treated as a sensor. The output

deviation from the model will be analysed in order to assess

system performance. Setting up such a generic diagnostic tool

requires an intensive model use for example by combining

fast models with a statistical representation of both operative

and state variables, and in this case black-box and grey-box

models are more appropriate than physical ones [45].

Black-box model is very suited for interpreting and pre-

dicting the performance of SOFC at both normal and abnormal

operating conditions and of different geometric designs,

avoiding using complicated differential equations to describe

the stack [64]. As long as the input-output data for depicting

the studied process is available, an accuratemodel of this type

can be achieved relying upon optimisation on training algo-

rithm or/and model architecture. Unfortunately, the avail-

ability of experimental data that should be enough

representative for themodelled phenomena is always amajor

problem of black-boxmodelling, especially in SOFCmodelling

for diagnostic application due to the high risk of fuel cell

damage and experimental failure when a system fault occurs

(for example a low flow rate of input or a too high operating

temperature which may be catastrophic for fuel cell and lead

to system shut-down). Overcoming this problem needs to

borrow validate white-box models to supplement the experi-

mental data matrix. In [54,55,58,59,64,65], the black-box

models were trained and validated by using pure data

produced from physical models. Thesemodels showed a good

congruence with the physical ones and to be generic to

various operational conditions as well. The performances of

them are, however, limited by the accuracy of the applied

physical models and should be thus further validated and

modified with experimental data.

It is worth noting that all the reviewed black-boxmodels in

this paper are straightforward type aiming at simulating static

SOFC processes. However, the application of neural networks

to themodelling or fault diagnosis of control systems requires

taking into account the dynamics of processes or system

considered [6]. Hence, SOFC recurrent neural network models

that include feedback loops from output would be very

Fig. 15 e Interactions and Applications of different model types in online SOFC system diagnosis.

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necessary to be considered and developed. In fact, this type of

model had been developed for proton exchange membrane

(PEM) fuel cell dynamic modelling by Jemei et al. (2004) [69]

and Puranik et al. (2010) [70] but at present, there is no such

model for SOFC dynamics in the available literature. Accord-

ingly, recurrent neural network for SOFC dynamic modelling

will carry more weight in the future work.

Different from black-box model, grey-box model requires

both the knowledge on the process or studied system (repre-

sented by mathematical relations) and empirical data. More-

over, the more the model is detailed, the more data/

information is required, especially with respect to geometric

design and material properties. However, a reliable grey-box

model for fuel cell can be realised also with a general knowl-

edge of the geometry and an empirical definition of the losses

as a function of stack operating temperature, e.g. area specific

resistance. The existing dynamic 2-D and 1-D models can be

used as a starting point in the development of a grey-box

model. For diagnosis application, it can be assumed in such

a model that SOFC behaves as a first-order system and

thermal dynamics is much slower and thus dominant with

respect to the dynamics of electrochemistry and mass trans-

fer. In this case, SOFC is simulated by applying the conserva-

tion of energy principle (heat balance) for a lumped control

volume, which includes air and fuel channels, as well as

interconnect and solid tri-layer (i.e. electrolyte and electrodes)

[33,44,34].

7. Conclusion & prospective

In this paper four modelling approaches for SOFC are pre-

sented. The physical models have been greatly developed in

recent decade. They are constructed based on the physical

laws in thermo-fluid and electrochemical courses, which can

describe the internal physical and chemical behaviours inside

a fuel cell unit. The 3-D and 2-D models are complicated and

time-consuming in spite of their considerable accuracy. The

improved 1-D and 0-D models by being integrated several

simplifications are regarded as the most pertinent for real

time simulation applications. These simplifications (or

assumptions) are required to not only reduce the computa-

tional time but improve the model reliability as well. They are

suitable for those applications requiring a satisfactory

compromise between accuracy and computational time, such

as model-based control and diagnostics. The equivalent

circuit models can be used to simulate the AC impedance

spectrum of SOFCs, providing information about the indi-

vidual behaviours inside a cell or a stack. However, due to the

complexity and coupling of physicochemical processes in

SOFCs, it remains technical gaps in the AC impedance

modelling and simulation in terms of fundamental under-

standing [48]. The grey-boxmethod is based on a combination

of a priori knowledge concerning the process and the math-

ematical relations which describe the behaviour of a SOFC

system. Its flexibility allows us to extract rules that describe or

interpret the behaviours. The black-box models can predict

the fuel cell performance without knowledge of numerous

physical, chemical and electrochemical parameters. The

underlying self-learning process ensures adapting the model

to new situations. The black-box models based on artificial

intelligence are flexible and pertinent for the non-linear

dynamic FC system [6]. The disadvantages are that 1) the

collection of experimental database should be perform in

a long time and 2) a raw data processing is necessary in order

to reduce the training time and improve the fidelity of the

model.

The validation of models of each type has been discussed.

With the aim of setting up a generic model-based diagnostic

tool for SOFC systems, the roles of models within the diag-

nostic algorithm development have been pointed out.

Dynamical simulation of system process is required for

i n t e rn a t i o n a l j o u r n a l o f h y d r o g e n en e r g y 3 6 ( 2 0 1 1 ) 7 2 1 2e7 2 2 87226

realising online fault diagnosis, which may lead to selecting

recurrent neural network for SOFCmodelling. Considering the

limited information level retained by the experimental data, it

will be taken into account the extension of the available data

set by means of hierarchical modelling approach in future

work. An overall prospective for real time model-based diag-

nosis of SOFC system is sketched in Fig. 15. The black-box

model is expected for static and especially dynamic simula-

tion of SOFC responses to inputs. The grey-box model devel-

oped from 1-D or 2-D physical models is aimed to describe

thermal dynamics of fuel cell and to represent the balance of

plant, providing details in physical sense. 3-D and 0-D models

are two extremes in physical models and they will be used to

supplement experimental data matrix (set up based on design

of experiment (DOE)) for reinforcing the black-box model

applicability. The former aims at various geometries of SOFCs

and the later at different operating conditions. In addition,

these data may also be useful for knowledge/feature extrac-

tion of considered faults, serving for fault identification and

localisation (refer to [68]) in the last diagnosis step. The

equivalent circuit models based on EIS technique will be used

for stack degradation analysis in frequency domain to know in

what frequency range a given fault can be observed. These

results can be used for signal-based diagnosis (refer to [68]).

Acknowledgements

The financial support of the European Commission for the

GENIUS Collaborative Project is gratefully acknowledged.

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