Analysis of electrical and thermal models and modeling techniques for polymer electrolyte membrane fuel cells

*Co esponding au ho : anciscoja ie [email protected]
Analysis o elec ical and he mal models and modeling echniques o polyme
elec oly e memb ane uel cells
F.J. Asensioa*, J.I. San Ma ína, I. Zamo ab, G. Saldañaa, O. Oñede ab
aElec ical Enginee ing Depa men , Enginee ing School o Gipuzkoa (Sec ion o Eiba ), Uni e si y o he Basque Coun y UPV-EHU, Eiba 20600, Spain
bElec ical Enginee ing Depa men , Enginee ing School o Bilbao, Uni e si y o he Basque Coun y UPV-EHU, Bilbao 48013, Spain
A B S T R A C T
Polyme ic Elec oly e Memb ane Fuel Cell (PEMFC) modeling conside ing he mal and elec ical beha io in a coupled manne is a key aspec when
e alua ing new designs, ma e ials, physical phenomena o con ol s a egies. Depending on he beha io o be emula ed, i is impo an o choose he
modeling echnique ha bes sui s he needs equi ed. In his sense, his pape desc ibes he mos commonly used PEMFC modeling echniques in he
con ex o analy ical-mechanis ic app oach, semi-empi ical app oach based on heo e ical o mula ion and empi ical co ela ions, as well as empi ical
app oach based on expe imen a ion wi h a eal sys em. In addi ion, an in-dep h analysis o PEMFC models a he cell and s ack le el ha emula e he
he mal and elec ical beha io o hese sys ems in a coupled manne is ca ied ou . A ch onological classi ica ion o he mos ele an models has been
made based on he modeling echnique used, pu pose o he model, s a e and dimension o he model, and he eal sys em, o he de eloped models o
expe imen al esul s ha ha e been used o alida e he p oposed new model. Addi ionally, guidelines o imp o e he ene gy e iciency o PEMFC
sys ems h ough he de elopmen o new models a e gi en.
Key wo ds
Polyme Elec oly e Memb ane Fuel Cell (PEMFC), Modeling echniques, Elec ical and he mal beha io , Model De elopmen , Re iew.
1. In oduc ion
In he cu en li e a u e, he easibili y o PEMFC (P o on Exchange Memb ane Fuel Cell) sys ems o be used in he ield o
s a iona y cogene a ion, au omo i e applica ions and po able gene a ion has been shown [1], [2], [3]. Howe e , he e is s ill
oom o imp o emen in he de elopmen o new memb anes, ca alys s, bipola pla es, collec o s, e c., as well as in he
managemen o hese sys ems o be op imally in eg a ed and o inc ease hei ene gy e iciency and educe he cos s o he sys em
in which hey a e in eg a ed.
On he one hand, o e i y he e ec i eness o new ma e ials and designs o he elemen s ha o m a PEM ype cell, i is
necessa y o de elop models ha ep oduce wi h high accu acy he beha io o he desi ed objec i e (imp o emen o mass
anspo , ac i a ion losses, ohmic losses, concen a ion losses, hea ans e , memb ane humidi ica ion, e c.) when ope a ing
wi h he new design o ma e ial. On he o he hand, o op imally ope a e a PEMFC, i is essen ial o de elop accu a e models
acco ding o he op imiza ion o ope a ion s a egy ha will be implemen ed. In his sense, he modeling o PEMFC sys ems ha
This is he accep ed manusc ip o he a icle ha appea ed in inal o m in Renewable and Sus ainable Ene gy Re iews 113 : (2019) //
A icle ID 109283, which has been published in inal o m a h ps://doi.o g/10.1016/j. se .2019.109283. © 2019 Else ie unde CC BY-NC-
ND license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/)
exac ly emula e i s ope a ion is a key aspec when s udying ope a ion s a egies, i s in eg a ion wi h o he sys ems o he
e alua ion o con ol algo i hms aimed a maximizing he ene gy e iciency o he sys em. To do his, he model mus be able o
p edic accu a ely he e iciency o he sys em based on he ene gy p oduc ion o he PEMFC. Besides, he choice o he
app op ia e modeling echnique is a undamen al aspec when modeling a PEM cell o a comple e s ack. In his sense, o e i y
he e ec i eness o new componen s, ma e ials o phenomena, i usually in ol es he use o mo e complex models ha equi e
a highe compu a ional cos . Howe e , i he pu pose o he model is o e alua e con ol s a egies a sys em le el o o analyze
he beha io o he PEMFC when i ope a es simul aneously wi h o he gene a ion o s o age echnologies, he modeling
echnique will be comple ely di e en , so ha he use can abs ac om he p ocesses ha a e no ela ed o he a iables ha
ake pa in he p ocess o be con olled o analyzed.
A p esen , he e a e a la ge numbe o models ha ep oduce wi h su icien p ecision he elec ical beha io o PEMFC sys ems
in dynamic egime, as well as in s eady s a e [4]-[5]. On he o he hand, he e is also a wide a ie y o models ha allow s udying
he he mal beha io o PEMFC sys ems [6]-[7]. Howe e , since he elec ochemical and he modynamic eac ions ha occu in
a PEMFC a e s ongly coupled, o emula e he global beha io o a PEMFC, he model necessa ily has o con empla e bo h
aspec s in a coupled manne . In his con ex , he e a e se e al esea ch wo ks ha p esen non-iso he mal models, which in eg a e
he elec ical and he mal beha io oge he and which can be aken as a e e ence so ha he scien i ic communi y could imp o e
wha has been done so a in e ms o e alua ing new designs, ma e ials o con ol s a egies.
Taking in o accoun all men ioned abo e, sec ion 3 o his pape desc ibes he modeling echniques used o de elop PEMFC
models. Nex , sec ion 4 desc ibes he cu en li e a u e o he PEMFC models ha conside coupled elec ical and he mal
beha io , which a e classi ied in ch onological o de conside ing he modeling echnique used, he pu pose o he model, and
he eal sys em o published da a used o i s alida ion o compa ison, among o he s. Addi ionally, sec ion 5 desc ibes he
oppo uni ies o imp o ing he ene gy e iciency o PEMFC sys ems by de eloping new models aimed a de eloping new
op imiza ion s a egies.
P io o he desc ip ion o he models men ioned abo e, he ollowing sec ion 2 in oduces gene al aspec s o PEMFC modeling
necessa y o acili a e he unde s anding o he desc ip ions made in he ollowing sec ions.
2. Gene al aspec s o modelling PEMFC sys ems
Rega ding he modeling o PEMFC sys ems, he e a e many cha ac e is ics ha allow di e en ia ing some models om o he s.
In gene al, models o PEMFC sys ems can be classi ied acco ding o he o mula ion used (semi-empi ical, empi ical o
analy ical-mechanis ic), size o he model (ze o, one, wo o h ee dimensions), s a e o he model (s a iona y, ansien ) o he
limi o he model (cell, comple e s ack, e c.), among o he cha ac e is ics. Table 1 shows he gene al modeling cha ac e is ics
o PEMFC sys ems.
Table 1. Gene al modeling cha ac e is ics o PEMFC sys ems
Key aspec s Op ions / Cha ac e is ics
App oach
Sys emic app oach (semi-empi ical, empi ical) o analy ical-mechanis ic app oach (mac oscopic /
mic oscopic)
S a e S a iona y; dynamic ( ansien ); eal ime
Limi A componen o a cell; a comple e cell; s ack; s ack including auxilia y sys ems; In eg a ed PEMFC.
Compu a ional domain Single domain; mul idomain
Spa ial dimensions Ze o (0D); one (1D); wo (2D); h ee (3D). I is also possible o combine he abo e.
Pu pose
The mal analysis, e iciency analysis, s uc u al analysis, pa ame ic s udies, elemen design (gas and
cooling channels, GDL, ca alys , collec o s), e c.
In eg a ion in con ol sys ems, in eg a ion in cogene a ion and igene a ion sys ems, hyb idiza ion wi h
o he echnologies, e c.
Complexi y
E ec s o empe a u e, wo-phase low, po osi y modeling, CO kine ics and poisoning, ca alys deg ada ion,
con ac esis ances, g a i a ional e ec s, e c.
In he ollowing subsec ions, he gene al aspec s shown in Table 1 a e in oduced in o de o acili a e he unde s anding o he
desc ip ion o he models ha can be ound in he cu en li e a u e.
2.1. App oach o he model
Depending on he equi emen s and he expec ed unc ionali y, a sys emic model (semi-empi ical o empi ical) o an analy ical-
mechanis ic model can be used.
Models based on an analy ical-mechanis ic app oach a e usually e y accu a e and p o ide minu e de ails o he ope a ion o he
uel cell a he mic oscopic o mac oscopic le el [8]. I s o mula ion is based on a se ies o elemen a y elec ochemical and
he modynamic ela ionships ha desc ibe he p ocesses ha occu wi hin a PEM (P o on Exchange Memb ane) uel cell. This
ype o o mula ion, commonly, add esses h ee main p ocesses: he elec ochemical eac ions in he ca aly ic laye s, he
mig a ion o p o ons in he polyme ic memb ane and he anspo o hea and mass in all egions o he cell. The complexi y o
hese models depends on he desi ed pu pose and a e gene ally e y di icul o implemen due o he highly non-linea na u e
ha desc ibes he beha io o a PEMFC. In his sense, in mos cases, due o he highly complex equa ions on which hey a e
based and he high compu a ional cos equi ed o sol e hem, hese models a e no e y sui able o use in eal- ime con ol
sys ems. Consequen ly, in mos cases hey a e used as suppo o he design o componen s.
Models based on a sys emic app oach ocus on he gene al beha io o he uel cell, wi hou going in o de ail in he p ocesses
ha occu a he a omic le el. Depending on whe he pa o all o he o mula ion ha desc ibes he beha io o he model is
based on empi ical ela ionships, a dis inc ion is made be ween semi-empi ical o empi ical models, espec i ely. This ype o
o mula ion is widely used when he e a e pa ame e s ha a e no known a p io i, such as he di usi i y o conduc i i y o he
memb ane, which a e desc ibed by semi-empi ical exp essions. The empi ical o mula ion co e s a e y ex ensi e ield, since i
allows o iden i y he beha io o he sys em h ough a g ea a ie y o iden i ica ion echniques, such as spec oscopy,
ol amme y, in e up ion o he cu en , es and e o me hods, as well as implemen a ion me hods, such as he elec ical ci cui s
o he black-boxes [4], [9]. These ypes o models a e mo e sui able o use in ope a ion, con ol o op imiza ion s a egies.
2.2. S a e o he model
The s a e o a model di e s be ween s eady s a e and dynamic ( ansien ) s a e. In he s eady s a e i is assumed ha he a iables
do no change wi h ime, ha is, he olume ic lows, en opies, en halpies, e c., a e no a unc ion o ime. Commonly, hese
ypes o models a e used o cha ac e ize he pola iza ion cu e o a uel cell, including losses in he ac i a ion, ohmic and
concen a ion egion. They a e also used o s udy in de ail he beha io o he di e en pa s o a PEM cell, such as he GDL
(Gas Di usion Laye ), he CL (Ca alys Laye ) o he memb ane unde di e en ope a ing condi ions.
T ansien s a e models include de i a i es as a unc ion o ime o cha ac e ize he beha io o he sys em dynamically. Gene ally,
ime cons an s a e used o cha ac e ize he elec ochemical beha io o he double laye , he global ans e o hea and mass and
he dehyd a ion o he memb ane, which usually a y se e al o de s o magni ude [10], [11]. No mally, hese ypes o models
a e used o analyze he beha io o he sys em in he e en o changes in ope a ing condi ions.
2.3. Limi o he model
Depending on he applica ion, he model can be a componen o a PEM cell (GDL, CL, memb ane, e c.), a comple e cell including
he MEA (Memb ane Elec ode Assembly), a s ack composed o se e al cells, o a s ack including he auxilia y sys ems (cooling
ci cui , coolan pump, ai comp esso , e c.).
In gene al, he models de eloped using an analy ical-mechanis ic app oach ocus on componen s o on a single cell, since
simula ing a comple e s ack would equi e a e y high compu a ional cos . On he con a y, models de eloped wi h a sys emic
app oach a e mo e sui able o modeling comple e s acks.
2.4. Compu a ional domain
A PEMFC model can be based on a single domain o on mul iple domains (mul i-domain). When a single domain is used, only
he sou ce e ms (oxygen and hyd ogen low, p essu es, eloci ies and inle empe a u es, e c.) o sink (cu en densi y gene a ed,
cell ol age, e c.) a y acco ding o he posi ion inside he cell (wi hou in e nal bounda y condi ions). In his sense, all he
equa ions a e w i en in he o m o a gene ic con ec ion-di usion equa ion, and all he e ms ha do no con o m o ha o ma
a e coupled wi h he sou ce o sink e m [12].
Mul idomain models use di e en modeling equa ions in each domain (GDL, CL, memb ane, e c.) and equi e ca e ul
managemen wi h bo de bounda y condi ions, ini ial and in e nal ( o example, con inui y) and ex e nal (p essu e, empe a u e,
e c.).
2.5. Spa ial dimension
A model o a cell o a s ack can be ze o (0D), one (1D), wo (2D) and h ee (3D) dimensions. In his sense, a ze o-dimensional
model is one in which i is assumed ha all he magni udes ( empe a u e, lows, p essu es, e c.) a e homogeneous in space,
conside ing only he a e age alues o he inpu s and ou pu s o he sys em. 1D models con empla e a ia ions o physical
magni udes in only one di ec ion, o example, he a ia ion o he empe a u e h ough he GDL, he CL and memb ane.
Following he same line, 2D and 3D models, con empla e a ia ions o magni udes in a su ace and space, espec i ely.
In he ea ly s ages o uel cell modeling, esea che s used 1D models, wi h a ying deg ees o complexi y, in c oss di ec ion
assuming he cell a anged as a "sandwich". Wi h his ype o models, lows, concen a ions, empe a u es and elec ic po en ials
we e analyzed o de e mine he limi condi ions o he cells. These ypes o models p o ide eno mous in o ma ion, especially
when modeling CLs.
2D models a e p esen ed as an imp o emen o 1D models. In his sense, hey o e a mo e ealis ic iew o ce ain phenomena
since spa ial a ia ions a e conside ed. These models a e usually implemen ed wi h he cell in a sandwich a angemen in he x-
y di ec ion o in a domain along he channel (design known as along- he-channel) in he y-z di ec ion. The sandwich models a e
mainly used o low analysis, hea and mass ans e and concen a ions ha include he e ec s o bipola pla e and gas channels.
The models wi h domains along he channel a e used, mainly, o analyze he concen a ions o he eac an s along he gas
channels. 1D and 2D models can include he same conse a ion equa ions as he 3D models, so hey p o ide a lo o in o ma ion
wi h su icien accu acy i he bounda y condi ions and ini ial condi ions a e ca e ully selec ed.
3D models (x-y-z di ec ions) a e he mos app op ia e when i is equi ed o s udy he gene al beha io o a PEMFC. In p ac ice,
i can be in e p e ed ha a 3D model is ob ained om he combina ion o 2D domains, which allows s udying he blocking e ec
o bipola pla es, he de ailed dis ibu ion o cu en densi y, empe a u e o e iciency o a ield low design.
As an example, Figu e 1 shows a schema ic illus a ion o he di e en dimensions ha a e con empla ed when modeling a PEM
cell, based on he selec ed coo dina e axes (1D, di ec ion y; 2D, di ec ion x-y and y-z; 3D, di ec ion x-y-z).

x-y
Anode supply (H
2
)
Ca hode supply
(O
2
)
y-z
y
Memb ane
(PEM)
Ca alys Maye
(CL)
Gas Di ussion Laye
(GDL)
Bipola Pla es
(BP)
y
x
z
Fig. 1. Schema ic illus a ion o he di e en dimensions ha a e con empla ed when modeling a PEM cell.
In addi ion o he men ioned dimensions, i is also possible o combine he p e ious ones (1 + 1, 2 + 1, 2 + 1/2) when a highe
dimension model is coupled wi h a lowe dimension model o when one o he magni udes o he model is calcula ed in a di e en
dimension om he global dimension o he model.
2.6. Pu pose o he model
The ype o modeling chosen is ela ed o he pu pose ha will be gi en o he esul ing model. In gene al, 1D designs a e used
o analyze he limi a ion o mass and load anspo , while 2D o mo e dimensions a e used o he mal analysis, low analysis,
s uc u al analysis, e c. In addi ion, i he model is o ien ed o he design o elemen s (gas and cooling channels, GDL, ca alys ,
collec o s, e c.), he model is implemen ed wi h an analy ical-mechanis ic app oach.
When he pu pose o he model is o be used o implemen a con ol o op imiza ion s a egy, be in eg a ed in o a cogene a ion
sys em o hyb idized wi h o he echnologies, ze o-dimensional models de eloped wi h a sys emic app oach a e usually used
(semi-empi ical o empi ical), al hough i is also possible o use mo e complex models depending on he le el o de ails desi ed.
Howe e , o cases in which eal- ime compu a ion is equi ed, he models a e limi ed o semi-empi ical o empi ical models o
0D, since he esolu ion o analy ical models o mo e dimensions supposes a lo o compu a ional ime and i s implemen a ion
would no be iable. o ha ype o applica ions [13]. In his con ex , in [14], o emula e he he mal beha io o a 600 W
PEMFC, s a ing om a 3D compu a ional model, a echnique ha has allowed con e ing he 3D model in o a 0D model o be
implemen ed in a eal ime con ol sys em has been de eloped. The ac o de eloping he 3D model and hen con e ing i o a
0D model p o ides he combined ad an age o bo h models. In his sense, he 3D model p o ides high accu acy in he p edic ion
o he empe a u e p o iles o he s ack and he e ige a ion ci cui , and a high speed in compu a ion is ob ained by subsequen ly
educing i o a 0D model.
2.7. Complexi y o he model
Thanks o he cu en compu a ional capaci y, simula ions wi h PEMFC models a e becoming mo e complex and demanding
and include a high le el o de ail. In his sense, depending on he le el o de ail equi ed, compu a ional models can con empla e
he e ec s o empe a u e, e iciency, wo-phase lows, po ous media, kine ics and ca bon monoxide (CO) poisoning, ca alys
deg ada ion, con ac esis ances, g a i a ional e ec s, e c. Howe e , depending on he ope a ing condi ions o he desi ed
pu pose, a se ies o assump ions and simpli ica ions a e usually es ablished ha minimize he complexi y o he model [15].
3. PEMFC modeling echniques
One way o classi y modeling echniques is based on he me hod by which a model is implemen ed. The mos complex models
ha inco po a e mo e de ails a e he models based on he echnique o analy ical-mechanis ic o mula ion. The models ob ained
h ough his echnique, also called whi e-box models, a e o mula ed by means o e y complex equa ions ha desc ibe he
physic-chemical phenomena ha in e ene in he unc ioning o he PEMFC.
When he pu pose o a model is o be used in a con ol s a egy o o be in eg a ed wi h o he sys ems, simpli ied models can be
conside ed, so ha hey emula e only he necessa y a iables ha allow hei con ol o in eg a ion. These simpli ied models,
commonly known as g ay-box models, a e de eloped based on analy ical o mula ion complemen ed by a p io knowledge o
he sys em (expe imen al da a), so ha some e y complex ma hema ical equa ions a e eplaced by empi ical equa ions o
mapping ables.
Finally, he e is ano he modeling echnique in which he ela ionships be ween he inpu s and ou pu s o he sys em a e no
based on equa ions o physical laws, bu a e deduced only h ough physical expe imen a ion wi h he eal sys em o h oughou
expe imen al da abases. Models de eloped using pu ely empi ical echniques can be classi ied in o wo g oups: elec ic models
o models based on a i icial in elligence. The la e a e known by he name o black box models. Fig. 2 shows an ou line o he
al e na i es o modeling app oaches, implemen a ion me hods, as well as esolu ion s a egies ypically used.
Fig. 2. Ou line o he al e na i es o modeling echniques, implemen a ion me hods, as well as esolu ion s a egies o PEMFC modeling.
In summa y, h ee majo g oups a e dis inguished: implemen a ion h ough analy ical-mechanis ic o mula ion, implemen a ion
h ough semi-empi ical o mula ion and implemen a ion h ough pu ely empi ical me hods. Nex , hese modeling echniques
used o model a PEMFC a e desc ibed.
3.1. Theo e ical o analy ical-mechanis ic modeling echniques
Commonly he models de eloped using his echnique a e implemen ed in a mul idimensional domain, conside ing ha physical
magni udes a y in wo o h ee axes o he spa ial domain. The analy ical-mechanis ic o mula ion ypically con empla es he
laws o Fick, Ne ns -Planck and Bu le -Volme o ep oduce he phenomena o cha ge anspo (elec ic and ionic) and mass
ans e [16].
The is law o Fick ela es di usi e low o concen a ion assuming a s eady s a e. I is pos ula ed ha he low goes om
egions o high concen a ion o egions o low concen a ion, wi h a magni ude p opo ional o he concen a ion g adien (spa ial
de i a i e), o in simplis ic e ms he concep ha a solu e will mo e om a egion o high concen a ion o a low concen a ion
egion h ough a concen a ion g adien . Equa ion (1) shows he o mula ion o Fick's law, assuming a spa ial dimension.
𝐽=−𝐷𝑑𝜑
𝑑𝑥 (1)
whe e 𝐽 is he di usion low [mol/(m2·s)] o he subs ance lowing h ough an a ea du ing a ime in e al, 𝐷 is he di usion
coe icien [m2/s], 𝜑 is he concen a ion o he subs ance [mol/m3] and 𝑥 is he posi ion [m].
The law o Ne ns -Planck ex ends he law o di usion o Fick assuming ha he di using pa icles also mo e wi h espec o he
luid due o elec os a ic o ces. I s o mula ion is based on he conse a ion o he mass used o desc ibe he mo emen o a
chemical species cha ged in a luid medium. Equa ion (2) shows he o mula ion o he Ne ns -Planck law.
𝜕𝜑
𝜕𝑡=∇·𝐷∇𝜑−𝑢𝜑+𝐷𝑧𝑒
𝑘𝑇𝜑∇ϕ+∂A
∂  (2)
whe e 𝑡 is he ime [s], ∇ ep esen s he g adien , 𝑧 is he alence o he ionic species, 𝑒 is he elemen a y cha ge [C], 𝑘 is he
Bol zmann cons an [J/K], 𝑇 is he empe a u e [K], 𝑢 is he eloci y ec o o he luid [m/s], ϕ is he elec ic po en ial [V] and
A is he ec o o he magne ic po en ial [V·s/m].
The law o Bu le -Volme is one o he mos undamen al ela ions in elec ochemical kine ics. I desc ibes how he elec ical
cu en in an elec ode depends on he po en ial o he elec ode, conside ing ha in he same elec ode a ca hodic, as well as
anodic eac ion is p oduced. Equa ion (3) shows he o mula ion o he law o Bu le -Volme .
𝑗=𝑗·󰇧𝑒𝑥𝑝𝛼𝓏𝐹
𝑅𝑇𝐸−𝐸−𝑒𝑥𝑝𝛼𝓏𝐹
𝑅𝑇𝐸−𝐸󰇨 (3)
whe e 𝑗 is he cu en densi y a he elec ode [A/m2], 𝑗 is he exchange cu en densi y [A/m2], 𝐸 is he elec ode po en ial [V],
𝐸 is he equilib ium po en ial [V], 𝓏 is he numbe o elec ons in ol ed in he eac ion, 𝐹 is he Fa aday cons an [C/mol], 𝑅
is he uni e sal gas cons an [J/(mol·K)], and 𝛼 and 𝛼 a e he coe icien s o anodic and ca hodic cha ge ans e , espec i ely.
The way o sol e his ype o models based on he o mula ion desc ibed abo e, in mul idimensional domains, is based on he
use o ad anced nume ical me hods. The i s compu e p og ams based on ad anced nume ical me hods a e known as
Compu a ional Fluid Dynamics (CFD) p og ams, which is a compu e ized ool o simula e he beha io o sys ems ha
con empla e luid low, hea ans e and o he ela ed physical p ocesses. This so wa e wo ks by sol ing he equa ions ha
desc ibe he low o he luid o e a egion o in e es , wi h bounda y condi ions p e iously speci ied in he limi o ha egion.
Cu en ly, he e is so wa e ha is based on ad anced nume ical me hods and combines he simula ion o mechanical, he mal,
elec ical and luid-mechanical p ope ies, allowing o comple ely model i ually any sys em, including all he pa s in ol ed
in a PEMFC. Two o he mos comple e comme cial so wa e packages ha include mul iphysics packages a e ANSYS and
COMSOL Mul iphysics.
The me hodology o calcula ion by means o his ype o me hods consis s o he ollowing poin s:
 P e-p ocessing. Du ing p e-p ocessing, he geome y o he p oblem o be sol ed is de ined by CAD (Compu e -Aided
Design) so wa e. Once he geome y is de eloped, he domains o each elemen a e es ablished. Fo example, in he
case o an MEA, he gas channels, he GDL, he CL, he PEM, e c.
X
XX
X
X X
Ø
Op imal
hype plane
Ø ( ) Ø ( )
Ø ( ) Ø ( )
Ø ( )
X
Ø ( )
X
Ø ( )
X
Ø ( )
X
Ø ( )
X
Ø ( )
X
Ø ( )
Fig. 7. Illus a ion o he mapping o a nonlinea unc ion o a linea ly sepa able unc ion.
In he exp ession (9) he gene al o mula ion o a classi ie based on SVM is shown.
𝑓(𝑥)=𝑤Φ(𝑥) + 𝑏 (9)
whe e pa ame e s 𝑤 and 𝑏 ep esen he weigh ec o and he bias, espec i ely, ha a e de e mined du ing he aining p ocess
by minimizing a cos unc ion, and Φ(·) ep esen s he non-linea mapping unc ion o map he inpu ec o 𝑥 in a space o highe
dimension, in o de o be able o easily sepa a e he da a con ained in 𝑥 by a linea hype plane (Fig. 8).
S a ing om he ac ha a aining sample (𝑥,𝑦) is a suppo ec o i i sa is ies 𝑦𝑓(𝑥)≤1, and deno ing he suppo
ec o s ex ac ed by 𝑠∈[1,𝐾], he unc ion o he SVM can be ep esen ed acco ding o he exp ession (10).
𝑓(𝑥)=𝐾(𝑥,𝑠)+𝑏


𝐾(𝑥,𝑠)=Φ(𝑥)Φ(𝑠) (10)
whe e 𝐾(·,·) ep esen s he ke nel unc ion o be implemen ed o ep esen he non-linea mapping o Φ(·). The mos commonly
used ke nel unc ions a e linea , polynomial and RBF (Radial Base Func ion) [38]. Some examples o PEMFC models based on
SVM a e hose p esen ed in [39], [40].
B) Fuzzy Logic sys ems
Fuzzy logic sys ems ocus on ixed and app oxima e easoning as opposed o ixed and exac easoning. A a iable in uzzy logic
can ake a ange o ue alues be ween 0 and 1, ins ead o aking " ue" o " alse" alues as in adi ional bina y se s. Since he
ue alue is a ange o alues o he o al se , uzzy logic sys ems can only pa ially handle he u h.
A model based on uzzy logic maps inpu s o ou pu s combining h ee componen s: i - hen ules, membe ship unc ions and
logical ope a o s, ha is, AND and OR [41]. Nume ical da a a e con e ed in o linguis ic a iables h ough membe ship unc ions
ha de ine how well a a iable belongs o he ou pu , ha is, an e alua ion be ween 0 and 1 [41]. Fo he co ec de ini ion o

hese unc ions o ules, unlike he ANNs, p io knowledge o he use is equi ed, which does no allow o abs ac om he
physical beha io o he p oblem o be modeled [9]. Some examples o PEMFC modeling based on uzzy logic a e hose
p esen ed in [42], [43].
C) A i icial Neu al Ne wo ks (ANN)
Like SVMs and sys ems based on uzzy logic, ANNs a e sys ems ha can be used o sol e classi ica ion p oblems. ANNs a e
inspi ed by biological neu al ne wo ks, and ha e p o ed o be a powe ul ool o he modeling o non-linea sys ems [44]. Fig.
6 shows he s anda d model o an a i icial neu on, which was desc ibed by D.E. Rumelha and J.L. McClelland [45], [46].
Xn
Xj
X2
X1
∑ ()
wi1
wi2
wij
win
-1
Synapse
Cellula body
yi
Ou pu
bias
Inpu s
dend i es
axon
i Neu on
Fig. 6. S anda d model o an a i icial neu on.
A biological neu onal sys em is composed o millions o neu ons o ganized in laye s. In he emula ion o said biological neu al
sys em, a hie a chical s uc u e simila o ha exis ing in he b ain can be es ablished by means o an a i icial neu al sys em.
The essen ial elemen will be he a i icial neu on, which will be o ganized in laye s. Se e al laye s will cons i u e a neu al
ne wo k. Finally, a neu al ne wo k oge he wi h he inpu and ou pu in e aces will cons i u e he neu onal sys em (Fig. 7).
∑
F ()
O
U
T
P
U
T
S
Algo i hm
I
N
P
U
T
S
Neu on Laye Ne wo k Neu onal sys em
Fig. 7. Global s uc u e o a neu onal sys em.
The e a e se e al models o neu al ne wo k (linea associa o , simple pe cep on, Adaline, mul ilaye pe cep on, e c.),
a chi ec u es (unidi ec ional, eedback, monolaye , mul ilaye , e c.) and lea ning algo i hms (Hebbian algo i hm, Rosenbla
algo i hm, Wid ow-Ho algo i hm, backp opaga ion algo i hm, e c.). Howe e , due o he limi a ions o some o he neu al
ne wo k models and a chi ec u es when emula ing e y complex non-linea sys ems, wi hin he PEMFC con ex , he mos used
model and a chi ec u e co esponds o he MLP (Mul i-Laye Pe cep on) wi h eedback. The a chi ec u e o he MLP has
become so popula because, wi h a single hidden laye , his neu al ne wo k model can app oxima e any con inuous unc ion in
a ange up o he desi ed le el [47].
The MLP model is usually ained h ough he BP (Back P opaga ion) algo i hm. Tha is why in he cu en li e a u e, his
a chi ec u e can commonly be ound unde he name o backp opaga ion ne wo k. Fig. 8 shows he s uc u e o he MLP wi h a
single hidden laye , as well as he ac i a ion unc ion used in he mul ilaye pe cep on, which is a sigmoid unc ion.
(x)
x
Inpu Hidden Ou pu
j

ij
w
kj
w´
k
´

Ta ge

i
x

j
y

k
z

k
º
Fig. 8. S uc u e (le ) and ac i a ion unc ion ( igh ) o a MLP.
The ope a ion pe o med by a mul ilaye pe cep on wi h a single hidden laye and ac i a ion unc ions o he hidden laye and
ou pu laye o sigmoid and linea ype, espec i ely, can be de ined acco ding o he exp ession (11).
𝓏=𝑤
󰆒𝑦−𝜃󰆒

 =𝑤
󰆒𝑓𝑤𝑥−𝜃

 −𝜃󰆒

 (11)
whe e 𝑥 a e he 𝑛 inpu s o he ne wo k, 𝑦 a e he 𝑜 ou pu s o he hidden laye and 𝓏 a e he 𝑠 ou pu s o he ou pu laye
( hose ha ha e o be compa ed wi h he a ge s 𝑡). 𝑤 and 𝜃 ep esen he weigh s and biases o he hidden laye , espec i ely,
and 𝑤
󰆒 and 𝜃󰆒 ep esen he weigh s and biases o he ou pu laye , espec i ely. 𝑓 ep esen s a unc ion o sigmoid ype.
Conside ing a h ee-laye MLP (Fig. 4), ha is, including a single hidden laye , and wi h he inpu s, ou pu s, weigh s and biases
o he neu ons de ined abo e, gi en an inpu pa e n 𝑥(𝑟=1,…,𝑁), he global ope a ion o his a chi ec u e o each o he 𝑘
ou pu neu ons wi h (𝑘=1,…,𝑠) can be de ined by means o (12).
𝓏=𝑤
󰆒𝑦−𝜃󰆒

 =𝑤
󰆒𝑓𝑤𝑥−𝜃

 −𝜃󰆒

 (12)
The commonly used cos unc ion is he MSE (Mean Squa e E o ), being o he case o he MLP he exp essed in (13) and
(14).
𝐸(𝑤,…,𝑤,…,𝑤
󰆒,…,𝑤
󰆒,𝜃,…,𝜃,𝜃󰆒,…,𝜃󰆒) 𝑤ℎ𝑒𝑟𝑒 𝐸∈ℜ(×)(×) (13)
𝐸(𝑤,𝑤󰆒,𝜃,𝜃󰆒)=12(𝑡−𝓏)



 (14)
The e a e se e al me hods o he minimiza ion o (14), al hough he mos e ec i e a e he Le enbe g-Ma qua d , Bayesian
Regula iza ion and Conjuga e G adien me hods [48]. Some examples o PEMFC modeling based on ANNs a e hose p esen ed
in [44], [49].
4. PEMFC sys ems modeling: elec ical and he mal beha io
This sec ion includes a compila ion o he models ha can be ound in he cu en li e a u e, which emula e bo h elec ical and
he mal beha io o PEMFC sys ems. Models ha a leas emula e he ol age and empe a u e a ia ion as a unc ion o cu en
densi y a e conside ed. In his sense, i should be no ed ha only non-iso he mal models ha conside he empe a u e as an
ou pu a iable and no as an inpu a iable a e conside ed. Besides, only s ack o cell domain models a e conside ed,
As seen abo e, he e a e se e al ways o classi y a model (app oach, egime, domain, e c.). To classi y all he models
con empla ed in his sec ion, a classi ica ion based on he dimension o he model has been conside ed.
4.1. Ze o-dimensional models (0D)
Among he i s esea ch wo ks con empla ed in his sec ion ha p esen ze o-dimensional models is he model p oposed by S.
Busque e al. [50]. The model p esen ed is a no el empi ical model o accu a ely calcula ing he V-I cha ac e is ic o a PEM
uel cell, an elec olyze o a e e sible uel cell. The empi ical adjus men is made h ough expe imen a ion wi h a PEMFC o
4 kW. Addi ionally, he new me hodology p esen ed allows o adjus he model h ough expe imen a ion o any PEMFC o
elec olyze . The model is ma hema ically cohe en and con e gen nea ze o, allowing o in e pola e he expe imen al esul s o
he PEMFC when he model is ope a ed in egions whe e he cu en densi y does no c oss he mass ans e limi a ions. The
same yea , A. Kazim [51], p esen s a comp ehensi e exe gy analysis o a 10 kW PEM uel cell a a iable ope a ing empe a u es,
p essu es, cell ol ages and ai s oichiome y. The analysis is ca ied ou by a ying he empe a u e and p essu e, cell ol age
and s oichiome y o he ai , in o de o de e mine he e ec o hese a ia ions on he e iciency o he uel cell. The esul s
ob ained highligh ed he impo ance o he ope a ing empe a u e, p essu e, cell ol age and ai s oichiome y on he exe gy
e iciency o he uel cell. Howe e , hey ecommend o ope a e he uel cell a s oichiome ic p opo ions below 4 o main ain
he RH (Rela i e Humidi y) le el in he ai and o p e en he memb ane om d ying ou a high ope a ing empe a u es.
One o he i s ze o-dimensional models implemen ed by elec ical ci cui s is ound in he wo k o X. Kong e al. [52], in which
a uel cell model ha is able o cha ac e ize he s eady s a e beha io o he uel cell, as well as he ansien beha io is p esen ed.
To make he empi ical adjus men o he model, a Balla d Nexa o 1.2 kW o elec ic powe is used. The p oposed model shows
good beha io (93% accu acy) when alida ing i wi h he expe imen a ion esul s in s eady and dynamic s a e. La e , in [53], an
upda e o he model implemen ed wi h elec ical ci cui s in [52] is p oposed, in such a way ha he dynamics o he sys em using
an ANN based on he MLP is calcula ed. The ANN is composed o wo hidden laye s wi h 30 neu ons pe laye . By means o
his echnique i is managed o educe o hal he MSE o he model based on he Nexa Balla d equi alen ci cui . In [54], K.C.S.
Wang e al. p opose a model based on elec ical ci cui s de eloped in Pspice en i onmen o model he dynamic beha io o a
Nexa Balla d o 1.2 kW o elec ical powe . In he s udy, i is pu special emphasis on he dynamic beha io du ing he cold s a
o he s ack, as well as on he empe a u e e olu ion in ime. The simula ion esul s show consis ency wi h he expe imen al
esul s.
In [55], Z. Zhang e al. p esen a dynamic model o an equi alen elec ical ci cui o he Nexa Balla d o 1.2 kW, conside ing
he cha ac e is ics o he empe a u e and he equi alen in e nal esis ance. The esul s o he model show ha he de eloped
model can accu a ely ep esen he expe imen al esul s in a wide ange o load condi ions. M. Miansa i e al. [56], de elop a
model o a PEM cell o s udy he e ec o di e en ope a ing condi ions, such as empe a u e, p essu e and ai s oichiome y on
he exe ge ic e iciencies and he i e e sibili ies o he cell. The e ec o he dep h o he anode and ca hode channels on
e iciency is also calcula ed. In his sense, he highes e iciencies o a channel dep h o 1.5 mm o he anode and 1 mm o he
ca hode a e ob ained. ME. Yousse e al. [57], p opose a model o ze o dimensions, o ien ed o he s udy o he e ec o
empe a u e, p essu e, s oichiome y, hickness o he memb ane and hickness o he gas di usion laye on he beha io o he
cell. The esul s ob ained wi h he model a e compa ed wi h he esul s p esen ed by A.R. Mahe e al. [58].
K. Hyun-il e al. [59], p esen a model o cha ac e ize he slow ansien esponse o PEMFC. In he wo k, he es ima ion o he
necessa y pa ame e s o ob ain he beha io in s a ic, as well as in dynamic egime is p esen ed. The model is alida ed wi h
expe imen al esul s ca ied ou wi h a Nexa Balla d o 1.2 kW. R. da Fonseca e al. [60], p esen a s ack-le el model, o ien ed
o applying a con ol s a egy using he heo y o di e en ial la ness. The model esponds o he con ol signals ha egula e he
mos impo an a iables in he ai supply subsys em: oxygen s oichiome y and ca hode p essu e. The model is based on a s ack
o 5 kW o elec ical powe composed o 80 cells.
In [61], I. San Ma ín e al. de elop a model o he Balla d Nexa 1.2 kW in he MATLAB/Simulink en i onmen , ob aining he
pa ame e s ha emula e he beha io o he elec ochemical and he modynamic phenomena empi ically. The model is alida ed
in s a ic and dynamic egime. The beha io o he model is alida ed o ming a mic o-g id wi h 4 uel cells o he same ype. R.
Salim e al. [62], p esen a echnique wi h a heu is ic app oach o es ima e up o 18 pa ame e s o model a Nexa Balla d o 1.2
kW. The iden i ica ion algo i hm is based on PSO (Pa icle Swa m Op imiza ion). The esul ing model shows good accu acy
and equi es ew ma hema ical ela ionships. M.M. Ba zega i e al. p esen in [63] a s ack le el PEMFC model o in es iga e he
empe a u e e ec on pe o mance o dead-end cascade PEMFC s ack wi h an in eg a ed humidi ie and sepa a o . The equa ions
a e posed using a semi-empi ical app oach and a e sol ed using a ou h-o de Runge-Ku a me hod. The model can p edic he
bulk humidi ie and PEMFC empe a u es and he s ack ol age. Besides, cascade PEMFC ope a ion in a dead-end mode is
compa ed wi h an open-end mode. Au ho s p opose he ob ained model o sys em iden i ica ion and con ol pu poses.
F.J. Asensio e al. de elop in [64] a 0D s ack le el model o a 600 W PEMFC o e alua e he elec ical and he mal e iciency
o he sys em, including powe elec onics. The model is implemen ed using ANNs and is de eloped in MATLAB/Simulink
en i onmen . The model p o ides he empe a u e and he hyd ogen consump ion as a unc ion o elec ical and he mal demand
wi h good accu acy. La e , same au ho s p esen in [65] an imp o emen o he model by adding he cooling luid low a e as
an inpu a iable o he model. The empe a u e o he cooling luid is calcula ed using a 3D model in COMSOL Mul iphysics
en i onmen and a dynamic look-up able is de eloped o couple he he modynamic model o he p e iously de eloped ANN-
based model, esul ing in a 0D model mo e accu a e. Au ho s p opose he model o de elop eal- ime con ol, op imiza ion
s a egies and o op imally manage he cooling sys em o he PEMFC.
In [66], X. Chen e al. show a he modynamic model o a PEMFC ha includes he main auxilia y componen s. In his sense,
he model con empla es a hea exchange , a wa e ank, a cooling pump, as well as he inpu gas p ocessing componen s
(humidi ie and comp esso ). A pa ame ic s udy is ca ied ou o s udy he elec ical and he mal e iciency o he PEMFC and
he e iciency o he o al sys em. The PEMFC is con olled wi h MOEA/D (Mul i-Objec i e E olu iona y Algo i hm based on
Decomposi ion) o op imize he ope a ing pa ame e s o he sys em, aimed a maximizing he e iciency and powe o he sys em.
The model is ob ained by o mula ing semi-empi ical ela ionships. In [67], A. Khei andish e al. p opose an AI-based model
using FCMs (Fuzzy Cogni i e Map) o desc ibe he beha io o a uel cell o 250 W o a powe elec ic bicycle sys em. Au ho s
use uzzy ules o explain he cause and e ec be ween concep s. C. Ziogou e al. p esen in [68] a PEM cell model aimed o
apply MPC (Mul i a iable P edic i e Con ol) s a egies. The model is de eloped using semi-empi ical o mula ion and
gPROMS so wa e. Au ho s show how using he model and s a egy implemen ed uel cell sys em ope a es economically and a
a s able en i onmen ega dless o he a ying ope a ing condi ions.
J. Chen e al. p opose in [69] a dynamic scalable model o PEMFC sys ems conside ing wo-phase wa e low. The model is
de eloped in MATLAB/Simulink en i onmen using he oolbox Simscape. Simula ion esul s show easonable dis ibu ions o
cu en densi y, empe a u e, p essu e, and wo-phase wa e low a bo h he s eady s a e and dynamic ope a ions. Au ho s
highligh ha he pa e n econ igu abili y and he segmen a ion scalabili y o he p oposed model mee he equi emen s o
bo h con olle design and sys em analysis o uel cells. K. Sanka e al. p esen in [70] a PEMFC sys em le el 0D model aimed
a e alua ing a nonlinea MSMC (Mul i a iable Sliding Mode Con ol) s a egy. In addi ion o he elec ochemical and
he modynamic beha io o he PEMFC, he model de eloped is capable o emula ing he beha io o he ai comp esso , ai
coole , p ima y mani old, supply mani old, humidi ie and e u n mani old. All equa ions in ol ed a e implemen ed in
MATLAB/Simulink en i onmen .

4.2. One-dimensional models (1D)
One o he mos ele an one-dimensional models can be ound in he wo k p esen ed by J.C. Amplhe e al. [71], in which a
PEMFC model designed o p edic elec ical and he mal beha io , bo h in s eady s a e and in ansi o y egime is p esen ed. To
do his, he ansien cha ac e is ics o hea and mass ans e a e inco po a ed in o an elec ochemical model o o m a gene al
model ha p edic s he ansien esponses o a PEMFC. The de eloped model is based on he Balla d Ma k V sys em, which is
a PEMFC sys em o 5 kW o elec ical powe , o med by 35 uel cells. Th ough expe imen a ion wi h his equipmen , a he mal
model o he s ack based on he conse a ion o mass and ene gy balance was de eloped. The he mal cha ac e iza ion o he
s ack includes he de e mina ion o sensible hea changes in anode, ca hode and wa e ci cula ion lows, he heo e ical ene gy
de i ed om he eac ion, he elec ical ene gy p oduced by he uel cell, and he hea eleased h ough he su ace o he s ack.
The he mal model is coupled o an elec ochemical model, ela ing he powe p oduced by he s ack and he empe a u e o he
s ack wi h he amoun o hea ha mus be ex ac ed om he s ack. The elec ochemical model calcula es he elec ical powe
p oduced by he s ack by p edic ing he cell ol age based on a complex exp ession in ol ing he ope a ing cu en , he
empe a u e o he s ack, and pa ial low a es and p essu es o hyd ogen and oxygen.
A. Rowe e al. [72], p esen a one-dimensional model o a PEM cell on which he e ec o designing and ope a ing condi ions
on cell e iciency, he mal esponse and wa e managemen is esea ched. I is shown how he wa e phase change in he
elec odes a ec s he empe a u e p o ile, especially o unsa u a ed eac an s and a low ope a ing empe a u es. The simula ion
esul s ob ained a e compa ed wi h he expe imen al esul s p esen ed by E.A. Ticianelli e al. in [73] and [74]. N. Djilali e al.
[75], p esen a wo k in which a heo e ical model o anspo phenomena is o mula ed o a PEM uel cell. The model conside s
he di usion o humidi ied uels and oxidizing gases h ough he po ous elec odes, he anspo o wa e h ough he elec odes
and he memb ane, as well as he g adien s o hea ans e and gas p essu e in he uel cell. The mic o-hyd odynamic phenomena
associa ed wi h he low pe meabili y o he elec odes a e also conside ed. The model is implemen ed in a one-dimensional code,
and a pa ame ic s udy is pe o med, compa ing he esul s ob ained wi h hose p esen ed in he wo k ca ied ou by D.M.
Be na di e al. [76] and E.A. Ticianelli e al. [77]. In his sense, i is e i ied ha , unlike he iso he mal and isoba ic models, he
non-uni o m empe a u e and p essu e dis ibu ions ha e a g ea impac on he lows o liquid wa e and in he o m o simula ed
apo in he anode and ca hode di usion laye s. In pa icula , he esul s indica e ha wa e managemen equi emen s (i.e.,
humidi ica ion o emo al o wa e ) o p e en possible dehyd a ion o he memb ane o looding o elec odes a e much mo e
conse a i e han when assuming iso he mal condi ions. I is shown ha , in he pe meabili y ange o he po ous elec odes used
in he PEMFC (10-16 – 10-17 m2), he Knudsen di usion mus be conside ed when modeling he gas anspo .
In he wo k p esen ed by X. Xue e al. [78], a dynamic model a PEMFC sys em le el ha emula es he empe a u e, he gas
low h ough he channels and he capaci ance o med by he double cha ge laye in he MEA is p esen ed. To quan i y he
dynamic in e ac ions, he PEMFC sys em is di ided in o h ee con ol olumes: he anode channel, he ca hode channel and he
uel cell body; de eloping he espec i e dynamic models wi h g ouped pa ame e s. The esul ing model is simula ed in Simulink
en i onmen and alida ed by compa ing he esul s ob ained wi h hose de i ed om [71]. I is concluded ha he model is
use ul o be used in he op imiza ion and eal- ime con ol o PEM uel cells ins alled in au omo i e o s a iona y applica ions.
Y. Shan e al. [79], p opose a model ha is cons uc ed based on he laye s o a PEM cell, conside ing he ollowing ac o s:
dynamics o he empe a u e g adien ac oss he cell, dynamics in he edis ibu ion o wa e concen a ion in he memb ane,
dynamics o he concen a ion o p o ons in he ca alys laye , and dynamics in he edis ibu ion o he concen a ion o eac an s
in he GDL o he ca hode. Fo he cons uc ion o he model, hey a e based on he pa ame e s p esen ed in [80], [81]. In he
wo k, he esul s ob ained in ansi o y egime, du ing he s a o he PEMFC and in s eady s a e a e shown.
C. Wang e al. [82], p esen he de elopmen o a dynamic model o PEM uel cells using elec ical ci cui s implemen ed in
MATLAB/Simulink and Pspice en i onmen s. The model con empla es he e ec o double laye cha ge and he he modynamic
cha ac e is ic wi hin he cell. The model esponses ob ained in s eady s a e and ansien condi ions a e alida ed by expe imen al
da a acqui ed om he PEMFC A is a Labs SR-12 o 500 W o elec ic powe . The au ho s p opose he model o be used in
PEM uel cell con ol s udies. In [83], S. Kjels up e al. p esen a model o a PEM cell aimed a s udying he a e o p oduc ion
o local en opy in a ious pa s o he uel cell. Au ho s p esen i e se s o anspo equa ions o a one-dimensional
he e ogeneous s eady-s a e cell (compa ible wi h he second law o he modynamics) and sol e hem by an i e a i e p ocess. Fo
he implemen a ion o he model, da a ound in he li e a u e on cells ha use he memb ane based on Na ion 115 is used.
In [84], A.Z. Webe e al. p esen a one-dimensional model o a PEM cell based on he memb ane Na ion 112, which is used o
compa e he beha io o i in an iso he mal and non-iso he mal si ua ion. P. Sang-Kyun e al. [85], p esen a PEMFC model o
s udying he e ec o wa e a ia ion (con empla ing one and wo phases) and he empe a u e dis ibu ion along he s ack a
a iable loads on he beha io o he s ack. Au ho s include he cooling ci cui in he model. Fo he cons uc ion and alida ion
o he model, an PEMFC o 80 W composed o wo cells o 140 cm2 is used. F om he de eloped model, se e al s a ing s a egies
o a PEMFC composed o 20 cells a e shown. A.A. Shah e al. p opose in [86] a wo-phase model ha includes a complex
kine ic mechanism o desc ibe he elec ode eac ions. The model is aimed a s udying he sul ide poisoning in PEMFCs and is
de eloped in COMSOL Mul iphysics en i onmen . Ob ained esul s a e compa ed wi h expe imen al da a published in [87].
Au ho s conclude ha he kine ic mechanism in he anode is in ima ely linked wi h he empe a u e and he wa e ac i i y, which
yield a wide in luence on pe o mance, h ough he o m o he eac ion a es. These eac ions cause a educ ion on he wa e
le els in he anode, which educe he cu en densi y and es ic s back di usion o wa e ia p o on mig a ion.
P. Hu e al. [88], p esen an ANN-based model aimed a cha ac e izing he non-linea dynamic beha io o a PEMFC o 1.5 kW
o elec ical powe . Fo he implemen a ion o he ANN au ho s use a hyb id algo i hm based on PSO and LM (Le enbe g-
Ma qua d ). The a chi ec u e o he ANN is based on MLP wi h eedback, and consis s o 3 neu ons in he inpu laye , 11 in he
hidden laye and 3 in he ou pu laye . The model shows good accu acy compa ed o he eal sys em. S. M. Sha i i e al. [89]
p esen a model aimed a emula ing he dynamic esponse o a PEMFC o a ia ions in he load. The inno a ion o he model is
ha i calcula es he wa e con en in he memb ane and conside s he p esence o wa e apo in he ca hode channel. The model
is alida ed wi h expe imen al esul s om se e al eal sys ems (SR-12, Balla d Ma k V and BCS 500), ob aining good
co ela ion wi h hem. In [90], F. Tiss e al. p opose a non-iso he mal model ha akes in o accoun he double laye e ec , he
geome ic capaci y and he empe a u e g adien . The model is de eloped o ope a e in a dynamic egime and he esul s ob ained
a e compa ed wi h he iso he mal model p esen ed in he wo k o A. Haddad e al. [91]. Au ho s conclude ha he e ec o he
empe a u e dis ibu ion signi ican ly in luences he cell ol age and he gas low a e.
N. Nogue e al. [92], p esen a one-dimensional and wo-phase model, aimed a de eloping a me hod o e alua e he eliabili y
o a PEMFC. The me hod combines physical modeling wi h s a is ical analysis. The model is de eloped wi h he Modelica-
Dymola so wa e and allows analysis in ansi o y egime wi h ime cons an s g ea e han 0.1s. In [93], J.A. Sal a e al. de elop
a one-dimensional, wo-phase model o a 50 cm2 PEM cell o emula e he cell ol age and wa e con en in he memb ane. Fo
he esolu ion o non-linea equa ions, hey use he so wa e EES 9.705-3D. The model is alida ed wi h a no el echnique based
on neu on images. Subsequen ly, in [94], au ho s use he model o emula e he beha io o a s ack composed o 3, 5 and 7 PEM
cells.
4.3. Two-dimensional models (2D)
The i s modeling based on wo dimensions, is ound in he wo k p esen ed by T.V. Nguyen e al. [95], in which a wo-
dimensional and single-phase model o hea and mass ans e o a PEM uel cell is p esen ed. To de eloped he model, he
elec o-osmo ic coe icien is assumed o be cons an . The model is de eloped as a designing ool o he de elopmen o PEM-
ype cell humidi ica ion sys ems. Subsequen ly, au ho s p esen in [96] an upda e o he model wi h upda ed da a on he
comme cial Na ion memb anes o ha ime, in o de o in es iga e se e al echniques o memb ane humidi ica ion. In he same
line, in [97] T.F. Fulle e al. de elop a model ha allows quan i ying he amoun o hea o be ex ac ed om he cell.
J.H. Lee e al. de elop in [98] a echnique o nume ically model a MEA in wo dimensions, in o de o be in eg a ed as pa o a
PEMFC dynamic model. The MEA model includes p ocesses, losses and elec ical cha ac e is ics. The equa ions used o he
cons uc ion o he model a e based on he p e ious wo ks de eloped by J. Kim e al. [99] and J.H. Lee e al. [100]. Fo he
de elopmen o he nume ical models, au ho s use a MEA o 350 cm2 and a s ack composed o 125 cells. The simula ion esul s
show ha he model de eloped using he p oposed nume ical echnique is especially use ul o s udy he e ec o empe a u e,
p essu e, humidi y, and a ia ions in oxygen concen a ion on he e iciency o he MEA.
Based on a semi-empi ical o mula ion echnique, V. Gu au e al. p esen in [101] a model o a wo-dimensional PEM cell wi h
wo-phase low, o s udy he dis ibu ion o oxygen and wa e apo in he GDL o a ious cu en densi ies. Au ho s also s udy
he wa e con en in he memb ane and se e al aspec s ha in luence he e iciency o he cell. The equa ions a e sol ed by means
o SIMPLE algo i hm o he esolu ion o CFD sys ems, de eloped by S.V. Pa anka [21]. The simula ion esul s o he
implemen ed model a e compa ed wi h he esul s o he wo k p esen ed by E.A. Ticianelli e al. [73]. In [102], M. Noponen e
al. p esen a wo-dimensional model in which he cu en densi y in he ac i e laye o he ca hode is modeled assuming an
agglome a ed ma e ial. The model is de eloped in he en i onmen o COMSOL (FEMLAB 2.3) and alida ed using a segmen ed
PEM cell.
E. Bie ge sson e al. p opose in [103] a wo-phase wo-dimensional model de eloped in COMSOL (FEMLAB 2.5), wi h which
he e ec o con ac esis ances be ween cell componen s and he e ec o di e en capilla y p essu es a e s udied. The ca alys
laye is ea ed as a eac i e limi . Au ho s conclude ha hea ans e by con ec ion is negligible unde he gi en ope a ing
condi ions. S. Li s e e al. p esen in [104] a model o s udying hea and mass ans e on he ca hode side o a PEM cell. The
model, which includes he cooling sys em, is sol ed using he CFX so wa e and he SIMPLEC algo i hm. The compu a ional
domain consis s o mo e han 30,000 mesh elemen s o he ai domain and 580 mesh elemen s o bo h elec odes.
In [105], J.J. Hwang de elops a wo-dimensional model o cha ac e ize he elec ochemical beha io and hea ans e o a PEM
cell in a coupled manne . The model con empla es a single phase o wa e s a us. The esul ing mesh consis s o 8,789 elemen s
and he coupled equa ions a e sol ed by he New on-Rapshon algo i hm. The pola iza ion cu e o he model is alida ed wi h
he wo k p esen ed in [106].
M. Acos a e al. p esen in [107] a wo-dimensional wo-phase model o s udy he pe o mance o he low ield. In he wo k
emphasis is placed on physical pa ame e s and capilla y sa u a ion p essu e, and he esul s a e compa ed wi h expe imen al
in es iga ions. A sa u a ion le el o liquid wa e o 60% is p edic ed o low cell ol ages. The compu a ional domain consis s
o he GDL and he CL ea ed as a hin laye (6,400 elemen s o equal size). To sol e i , au ho s use he so wa e MUFTE_UG.
In [108], H. Wu e al. p esen a wo-dimensional model designed o emula e he dynamics o wa e anspo (in a single phase)
in PEM cells unde non-iso he mal empe a u e condi ions. In he wo k done, au ho s ocus on s udying he e ec o memb ane
wid h (Na ion 117) on he e iciency o he cell. Au ho s conclude ha he he mal e ec has a g ea impac on he ansien
beha io o he cell.
In [109], Y. Zhang e al. de elop a model in wo dimensions o s udy he a mosphe ic ai in ake in a PEM cell wi h an ac i e
a ea o 6 cm2. The e ec s o he o ien a ion o he cell, he ope a ing condi ions and he geome ical pa ame e s a e analyzed.
Fluen so wa e is used o sol e he model oge he wi h use -de ined sub ou ines, consis ing o 5,748 elemen s. The esul s o
he model a e compa ed wi h he expe imen al esul s p esen ed in he echnical epo p esen ed in [110]. In he same esea ch
line, B.P.M. Rajani e al. p esen in [111] a wo-dimensional model aimed a s udying he cell espi a ion. Despi e being a wo-
dimensional model, he ca alys laye is conside ed e y hin and is ea ed as a single dimension. Au ho s also use Fluen and
use -de ined sub ou ines o s udy a ious e ec s. In he wo k au ho s show ha mos o he dynamic esponse p ocesses a e
wi hin ew seconds.
[54] 2005 0D
Empi ical
Elec ical ci cui
S a iona y/
T ansien
Cha ac e ize he dynamic beha io du ing he cold s a
and he e olu ion o he empe a u e.
Pspice
Nexa Balla d 1,2
kW
[83] 2005 1D
Theo e ical
Non-linea equa ions
S a iona y De e mine he local en opy a e in a PEM cell MATLAB 6.0.088
Na ion 115-based
memb ane cell
[127] 2006 3D
Semi-empi ical
FEM
S a iona y
S udy he phenomena o anspo , sa u a ion o he liquid
phase o wa e , ields o eloci ies and empe a u e
g adien s.
Ansys Fluen 6.0.1.2
SIMPLE Algo i hm
Valida ion
conside ing
balance o species
[104] 2006 2D
Theo e ical
FEM
S a iona y
S udy hea and mass ans e on he ca hode side o a PEM
cell
CFX
SIMPLEC
Algo i hm
Valida ion
conside ing local
Nussel numbe s
[128] 2006 3D
Semi-empi ical
FEM
S a iona y La ge-scale s udy o a 200 cm2 MEA Ansys Fluen 6.0.1.2
SIMPLE Algo i hm
200 cm2 MEA
[84] 2006 1D
Semi-empi ical
Non-linea equa ions
S a iona y
Compa e he he mal and non-iso he mal e ec s on he
beha io o wa e anspo in he cell
Na ion 112-based
PEM cell
[105] 2006 2D
Theo e ical
FEM
S a iona y
Cha ac e ize elec ochemical beha io and hea ans e in
a coupled manne
New on Raphson
Algo i hm and
Gaussian elimina ion
Compa ison wi h
[106]
[53] 2006 0D
Empi ical
ANN
S a iona y/
T ansien
Upda e he p oposed model in [52] implemen ing he
dynamics o he PEMFC by using ANNs
DSpace
Nexa Balla d 1,2
kW
(Compa ison wi h
[52])
[55] 2006 0D
Empi ical
Elec ical ci cui
S a iona y/
T ansien
Emula e he dynamic beha io o he Nexa Balla d 1.2 kW
by means o he in e nal equi alen esis ance
Nexa Balla d 1,2
kW
[107] 2006 2D
Semi-empi ical
FEM
S a iona y
O ien ed o simula e PEM cells wi h con en ional and
in e digi a ed gas dis ibu o s
MUFTE_UG
Célula PEM con
elec odo ELAT-
DS de E-TEK inc.
[129] 2007 3D
Theo e ical
FEM
T ansien
Desc ibe he ansien p ocess and he dynamic
cha ac e is ics o a PEM cell wi h a luid channel in he
o m o a coil.
Ansys Fluen and
SIMPLE Algo i hm
21,32 cm2
PEM cell
[130] 2007
~
3D
Semi-empi ical
Non-linea equa ions
S a iona y/
T ansien
O ien ed o apply con ol s a egies in PEM-based s acks
MATLAB/
Simulink
Na ion 112-based
25 cm2 PEM cell
[58] 2007 3D
Semi-empi ical
FEM
S a iona y
Use as a compu e -assis ed ool o op imize uel cells wi h
high powe densi y and lowe cos
So wa e CFD
Compa ison wi h
[124]
[108] 2007 2D
Semi-empi ical
FEM
S a iona y/
T ansien
S udy he dynamic cha ac e is ics o he cell and he
in luence o he wid h o he memb ane on he e iciency
COMSOL
Mul iphysics
Na ion 117-based
PEM cell
[109] 2007 2D
Semi-empi ical
FEM
S a iona y
In es iga e he beha io o a PEM cell ha uns on
hyd ogen ed a he anode and ai supplied by na u al
con ec ion a he ca hode
Ansys Fluen
Compa ison wi h
[110]
[111] 2007 2D
Semi-empi ical
FEM
S a iona y
S udy he espi a ion o a PEM cell unde a mosphe ic
condi ions o 23 °C and 20% o RH.
Ansys Fluen
6 cm
2
PEM cell
[112] 2007 2D
Theo e ical
FEM
T ansien
S udy he e ec s o liquid wa e anspo and hea ans e
phenomena on he ansien esponses o a PEM cell
du ing a change in cell ol age
Ansys Fluen PEM cell
[131] 2007
2D
3D
Semi-empi ical
FEM
S a iona y/
T ansien
S udy o he dynamics o GDL dehyd a ion and i s impac
on he e iciency o a PEM cell
Ansys Fluen 6.0.12
PISO Algo i hm
Compa ison wi h
[104]

[85] 2008 1D Semi-empi ical
Non-linea equa ions
S a iona y/
T ansien
S udy he e ec o wa e a ia ion ( wo phases) and hea o
a iable loads on he beha io o he s ack
PEMFC o 80 W
composed o wo
cells o 140 cm2
[86] 2008 1D
Semi-empi ical
FEM
S a iona y
In es iga e he e ec s o hyd ogen sul ide con aminan on
pe o mance o PEMFCs
COMSOL
Mul iphysics
Compa ison wi h
[87]
[56] 2009 0D
Semi-empi ical
Non-linea equa ions
S a iona y
S udy he e ec o s oichiome y, empe a u e, p essu e
and ai on he exe ge ic e iciency and i e e sibili ies o a
PEM cell
25 cm2
PEM cell
[132] 2009 3D
Semi-empi ical
FVM
T ansien S udy he cold s a p ocesses in PEMFCs
Ansys Fluen 6.3
UDFs
PISO Algo i hm
Compa ison wi h
[133]
[88] 2010 1D
Empi ical
ANN
T ansien Cha ac e ize he nonlinea dynamic beha io o a PEMFC
MATLAB
R2008a
PEMFC o 1,5 kW
composed o 28
cells o 232 cm2
[89] 2010 1D
Semi-empi ical
Non-linea equa ions
S a iona y/
T ansien
S udy he dynamic esponse o he PEMFC o a ia ions in
he load
MATLAB/
Simulink
SR-12
Balla d Ma k V
BCS-500
[57] 2010 0D
Semi-empi ical
Non-linea equa ions
S a iona y
S udy o he e ec o empe a u e, p essu e, s oichiome y,
hickness o he memb ane and hickness o he gas
di usion laye on he beha io o a PEM cell
MATLAB
Compa ison wi h
[58]
[59] 2010 0D
Semi-empi ical
Non-linea equa ions
S a iona y/
T ansien
O ien ed o cha ac e ize he slow ansien esponse o a
PEMFC.
Excel
Runge-Ku a o
o de 4
Nexa Balla d 1,2
kW
[135] 2010 3D
Semi-empi ical
FVM
S a iona y/
T ansien
S udy he e ec o non-equilib ium phase ans e Ansys Fluen 6.3.26
Expe imen al
esul s p esen ed
in [136].
[90] 2013 1D
Semi-empi ical
Elec ical ci cui
T ansien
S udy o he beha io o a PEM cell in non-iso he mal and
ansi o y condi ions
Ma hcad
Compa ison wi h
[91]
[60] 2014 0D
Empi ical
Elec ical ci cui
T ansien
O ien ed o apply con ol echniques on he ai subsys em
(oxygen s oichiome y and ca hode p essu e)
MATLAB/
Simulink
PEMFC o 5 kW
composed o 80
cells
[61] 2014 0D
Empi ical
Elec ical ci cui
S a iona y/
T ansien
Emula e he s a iona y and ansien beha io o a mic o-
g id o med by 4 PEMFC
MATLAB/
Simulink
Nexa Balla d 1,2
kW
[62] 2015 0D
Empi ical
Non-linea equa ions
S a iona y/
T ansien
Expe imen al me hodology Iden i y up o 18 pa ame e s o
he model expe imen ally
MATLAB
ou h-o de Runge-
Ku a me hod
Nexa Balla d 1,2
kW
[92] 2015 1D
Semi-empi ical
Non-linea equa ions
S a iona y/
T ansien
O ien ed o de elop a me hod o e alua e he eliabili y o
a PEMFC
Modelica-Dymola
220 cm
2
PEM cell
manu ac u ed by
CEA LITEN
[113] 2015 2D
Semi-empi ical
FVM
T ansien
S udy he e ec o pe meabili y on he dynamic ield in he
PEM uel cell
FVM-based so wa e Gene ic PEM cell
[63] 2016 0D
Semi-empi ical
Non-linea equa ions
T ansien
In es iga e he empe a u e e ec on pe o mance o dead-
end cascade PEMFC s ack wi h in eg a ed humidi ie and
sepa a o . Model sui able o sys em iden i ica ion and
con ol pu poses.
Fou h-o de Runge-
Ku a me hod
PEMFC s ack
composed o 4
cells o 225 cm2
[93] 2016 1D
Semi-empi ical
Non-linea equa ions
S a iona y
Emula e he cell ol age and he wa e con en in he
memb ane (new alida ion echnique based on neu on
images)
EES
9.705-3D
50 cm2
PEM cell
[94] 2016 1D
Semi-empi ical
Non-linea equa ions
S a iona y Emula e s a ic beha io a s ack le el
EES
9.705-3D
50 cm
2
PEM cell
[137] 2017 3D
Semi-empi ical
FVM
S a iona y
In es iga e he e ec o a blockage in he low ield
channel o a PEM uel cell on mass ans e o eac an gas
om he channel in o he ca alys laye
Ansys Fluen
SIMPLE Algo i hm
P e iously
published esul s
in [138]
[64] 2017 0D
Empi ical
ANN
S a iona y/
T ansien
Emula e he hyd ogen consump ion and empe a u e as a
unc ion o he elec ical and he mal load
MATLAB/Simulink PEMFC o 600 W
[139] 2017 3D
Semi-empi ical
FVM
T ansien
In es iga e he wa e emo al p ocesses in a PEMFC
du ing he gas pu ging p io o i s shu down
Ansys Fluen and
UDS equa ions
Valida ed using
expe imen al da a
[140] and
nume ical da a
[141]
[66] 2017 0D
Semi-empi ical
Non-linea equa ions
S a iona y
Apply a mul i-objec i e op imiza ion algo i hm
(MOEA/D) aimed a maximizing he ene gy e iciency o
he sys em
PEMFC o 5 kW
composed o 75
cells
[142] 2017 3D
Theo e ical
FVM
S a iona y
S udy he cooling low ields e ec on PEMFC
pe o mance
Ansys Fluen
PEM cell
p esen ed in
[144]
[67] 2017 0D
Empi ical
Fuzzy logic
T ansien
De e mine de beha io o a uel cell elec ic bicycle
sys em
Fuzzy Cogni i e
Map
Rule-based FCM
22 cell ai -cooled
s ack o 250 W
[143] 2017 3D
Theo e ical
FVM
S a iona y
S udy he cooling low ields e ec on PEMFC
pe o mance
Ansys Fluen
PEM cell
p esen ed in
[144]
[145] 2017 3D
Semi-empi ical
FVM
S a iona y
S udy eac an s dis ibu ion, cu en densi y and inal
powe in PEMFCs o ou squa e ubula con igu a ions
(simple, DPIE, DBIE and TPIE)
Ansys Fluen 14
SIMPLE algo i hm
Valida ed using
da a epo ed in
[124]
[146] 2017 3D
Semi-empi ical
FVM
S a iona y/
T ansien
E alua e he pe o mance o a PEM uel cell s ack wi h
a iable inle lows unde simula ed d i ing cycle
condi ions
Ansys Fluen 15
UDFs
Scala s in C code
PEMFC o 320
cells and 1,600
cm2 o ac i e
ca alys a ea
[65] 2018 0D
Theo e ical
FEM
Empi ical ANN
S a iona y/
T ansien
P o ide a ool o de elop new op imiza ion s a egies in
eal ime applica ions
COMSOL
Mul iphysics
MATLAB/Simulink
PEMFC o 600 W
[68] 2018 0D
Semi-empi ical
Non-linea equa ions
T ansien
Apply a mul i a iable model p edic i e con ol (MPC)
s a egies o PEM uel cells.
gPROMS so wa e 6 W PEM cell
[147] 2018 3D
Semi-empi ical
FVM
S a iona y In es iga e he pe o mance o a PEMFC. Ansys Fluen 16.2
Valida ed using
da a epo ed in
[124]
[148] 2018 3D
Semi-empi ical
FVM
S a iona y
S udy he e ec s o agglome a e model pa ame e s on
anspo cha ac e iza ion and pe o mance o PEM uel
cells
Ansys Fluen and
UDS equa ions
Valida ed using
da a epo ed in
[150]
[114] 2018 2D
Semi-empi ical
FEM
S a iona y
In es iga e he compe i ion be ween cu en collec ion and
oxygen supply. Used o ib/channel design.
COMSOL
Mul iphysics
Compa ison wi h
esul s om
Au oS ack-CORE
p ojec [115]
[69] 2018 0D
Semi-empi ical
Non-linea equa ions
S a iona y/
T ansien
De elop a dynamic scalable segmen ed model o PEMFC
sys ems wi h wo-phase wa e low
MATLAB/Simulink
Simscape
[151] 2018 3D
Semi-empi ical
FVM
S a iona y
In es iga e he s eady pola iza ion cu es and long- e m
s abili y o poisoned PEMFC, as well as e alua e he
pe o mance o a ious comme cial GDLs.
Ansys Fluen and
UDF equa ions
Valida ed using
da a epo ed in
[152] and [153]
and a 25 cm2 PEM
cell
[149] 2018 3D
Semi-empi ical
FVM
S a iona y
De e mine he e ec s o GDL de o ma ion and ob ain he
anspo cha ac e is ics in PEMFCs wi h in e digi a ed
low ields
Ansys Fluen and
UDS equa ions
Valida ed using
da a epo ed in
[150]
[70] 2018 0D
Semi-empi ical
Non-linea equa ions
S a iona y/
T ansien
De elop a Nonlinea mul i a iable sliding mode con ol
o a e e sible PEM uel cell in eg a ed sys em
MATLAB/Simulink
PEMFC
composed o 35
cells
[116] 2018 2D
Semi-empi ical
FEM
S a iona y In es iga e he eac an gas c osso e e ec in a PEM cell F eeMem++ PEM cell
5. Oppo uni ies o imp o e ene gy e iciency h ough new model de elopmen s
An impo an aspec when modeling a PEMFC sys em is ha he de eloped model mus con empla e he e ec ha empe a u e
has on he e iciency o he sys em. In his con ex , i would be in e es ing i models would conside he elec ical and he mal
beha io o he PEMFC emula e as p ecisely as possible he e ec ha empe a u e has on he speed a which he elec ochemical
eac ions occu . Since he ope a ing empe a u e is di ec ly ela ed o he amoun o hea ex ac ed h ough he cooling ci cui , a
key aspec o keep in o accoun is ha he model has o conside he he mal managemen o he sys em. In his sense, among all
esea ch wo ks ha can be ound in he cu en li e a u e, only a small g oup o hem ocuses on models ha include he he mal
managemen o he sys em h ough he e ige a ion ci cui . Howe e , mos models ha include in de ail he e ec o hea
ex ac ion h ough he cooling sys em a e de eloped by a complex o mula ion, which does no allow o use he model o e alua e
eal- ime con ol s a egies o op imiza ion s a egies, o which a e y educed calcula ion ime is equi ed.
Conside ing all men ioned abo e, he need o de elop ze o-dimensional models o PEMFC ha con empla e he egula ion o
he low a e o he cooling luid as an inpu a iable o he model is de ec ed. This will allow o es ablish he op imal e e ence
o elec ical and he mal p oduc ion o he uel cell in o de o maximize he elec ical e iciency o he sys em and o minimize
he p oduc ion cos s. This aspec is especially impo an i i is conside ed ha a PEMFC could ope a e connec ed o he powe
g id and ha o he auxilia y elec ical and / o he mal gene a ion de ices could be in eg a ed wi h he PEMFC in a mic og id.
In his way, he model de eloped could be used o p edic he beha io o he sys em in di e en si ua ions ( o p edic ed elec ical
and he mal load p o iles), which in u n will allow es ablishing he op imum ope a ing e e ence ha maximizes e iciency and
minimizes ope a ing cos s. o he sys em based on he hou ly p ices o elec ici y, p ices o he uels used by o he gene a o s, as
well as all he ope a ion and main enance cos s o all he de ices
6. Conclusions
In his pape he gene al modeling aspec s o he PEMFCs ha e been in oduced. Likewise, a sea ch was made o he s a e o he
a o he PEMFC models ha emula e in a coupled manne he he mal and elec ical beha io , a cell and s ack le els,
con empla ing a o al o 78 esea ch wo ks. I has been possible o e i y how app oxima ely 80% o he models de eloped in
he las 5 yea s (2014-2018) co espond o ze o-dimensional and h ee-dimensional models, p ac ically in he same p opo ion.
Due o he exis ing compu a ional capaci y, i has been p o en how simula ions wi h PEMFC models a e becoming mo e complex
and exigen and include a high le el o de ail. In his sense, he global modeling o PEMFC has e ol ed om s eady o dynamic,
om one-dimensional o complex h ee-dimensional models, om iso he mal o non-iso he mal, om single-phase o mul i-
phase and ecen ly om s aigh channels o ield s uc u es wi h mo e complex low, such as se pen ine o in e digi a ed low
ields.
I has been de ec ed ha , o use a PEMFC model in eal- ime con ol o op imiza ion s a egies, i is usually necessa y o eso
o ze o-dimensional sys em models o a oid he high compu a ional cos in ol ed in he inclusion o de ails and phenomena ha
occu a a mic oscopic le el inside a PEM cell. In his sense, i has been de ec ed ha MATLAB is he mos used enginee ing
ool by he esea ch communi y in e ms o modeling PEMFCs in ze o dimensions. In e ms o mul idimensional modeling,
Ansys Fluen (based on FVM) has been shown as he mos widely used mul iphysics so wa e, ollowed by COMSOL
Mul iphysics (based on FEM).
In he con ex o ze o-dimensional sys emic modeling, he modeling app oach based on empi ical echniques has been mo e
widely used, due o he good ela ionship be ween simplici y and p ecision ha i p o ides. Among hese echniques, he
iden i ica ion o pa ame e s h ough EIS and hei subsequen implemen a ion using equi alen elec ical ci cui s has been shown
as one o he mos sui able al e na i es o emula e he dynamic cha ac e is ics o he elec ical beha io o he PEMFCs. Howe e ,
o con empla e he coupled beha io , elec ical and he mal, i has been ound ha a i icial in elligence based on ANNs show a
g ea po en ial in he modeling o hese elec ochemical de ices o a highly non-linea cha ac e .
Finally, i has been e i ied ha , among all he models ha ha e been iden i ied in he scope o he applica ion o ope a ion and
op imiza ion s a egies, he wo ks ha ha e conside ed he low o he e ige an as a con ol a iable om he poin o iew o
ene gy e iciency a e e y limi ed. Since he ope a ing empe a u e o he PEMFC is di ec ly ela ed o he e iciency o he
s ack, and which in u n, he ope a ing empe a u e is ela ed o he amoun o hea ex ac ed om he PEMFC, i is de ec ed he
need o implemen op imiza ion s a egies o con ol PEMFC sys ems ha will ake he egula ion o he cooling low as a con ol
a iable o maximize he elec ical and he mal e iciency o he sys em. In addi ion o he e iciency o he sys em, he
op imiza ion s a egy could be implemen ed as a mul i-objec i e s a egy ha would con empla e he minimiza ion o ope a ing
and main enance cos s and ha m ul emissions.
Acknowledgemen
The au ho s hank he suppo om he Spanish Minis y o Economy, Indus y and Compe i i eness (p ojec ENE2016-79145-
R AEI/FEDER, UE), he Basque Go e nmen (p ojec ELKARTEK KK-2017/00083 and GISEL esea ch g oup IT1083-16), as
well as om he Uni e si y o he Basque Coun y UPV/EHU (p ojec EHUA15/25 and esea ch g oup unding PPG17/23).
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