CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 52, 2016 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Petar Sabev Varbanov, Peng-Yen Liew, Jun-Yow Yong, Jiří Jaromír Klemeš, Hon Loong Lam 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-42-6; ISSN 2283-9216 
 

Performance Analysis and Control Structure Design for 

Proton Exchange Membrane Fuel Cell 

Narissara Chatrattanaweta, Thanaphorn Hakhena, Dang Saebeab,   

Amornchai Arpornwichanop*a 

aComputational Process Engineering Research Unit, Department of Chemical Engineering, Faculty of Engineering, 

Chulalongkorn University, Bangkok 10330, Thailand 
bDepartment of Chemical Engineering, Faculty of Engineering, Burapha University, Chonburi 20131, Thailand 

amornchai.a@chula.ac.th 

A proton exchange membrane fuel cell (PEMFC) is regarded as a highly efficient power generation device. 

The performance of PEMFC depends on various key operating parameters, such as pressure, flow rate and 

humidity of reactant gases, and cell temperature (60 - 100 °C). Generally, the cell temperature is known to 

have an importance on transportation within the PEMFC and its performance. The fuel cell temperature should 

be limited under a working temperature that does not affect the material properties of the components. 

Therefore, the aim of this study is to perform the control structure design of the PEMFC. The steady-state 

analysis is done to find the suitable operating conditions of the PEMFC. The effect of input parameters on cell 

voltage and cell temperature is analyzed to investigate the dynamic behavior of PEMFC that is important for 

control design. To obtain an efficient control system, the suitable controlled and manipulated variables of the 

PEMFC are determined via the control structure design approach. The selection of input-output pairings is 

based on the relative gain array (RGA) as a controllability index. The results show that the cell voltage and cell 

temperature depend on the inlet molar flow rates and temperatures of hydrogen and air, and operating current 

density. According to the RGA, the inlet molar flow rates of air and hydrogen are manipulated variables to 

regulate the cell temperature and partial pressure of hydrogen, respectively. 

1. Introduction 

Fuel cells have been received great attention due to the promising alternative to traditional energy conversion 

technology. Among the various fuel cell types, the polymer exchange membrane fuel cell (PEMFC) has 

attracted considerable interest because of its high energy efficiency, low emission of pollutants to the 

environment, quick start-up and low-temperature operation (60 - 100 °C). It is also a promising fuel cell type 

for automotive, residential and portable applications due to its high power density and rapid response to 

changes in power demand (Arpornwichanop et al., 2014).  

Regarding the control design of the PEMFC, the dynamic behavior is critical and the control system is needed 

to provide that the flow rates and temperatures of fuel and air can be stabilized during normal operation at 

variable loads, as well as during system start-up and shut-down. Fuel and air fed into the fuel cell have to be 

controlled in order to keep the cell temperature, hydration of the membrane and the reactants in a good level 

to avoid membrane degradation and to maintain efficiency of the PEMFC system (Gruber et al., 2008). 

Moreover, the cell temperature involves the rate of the electrochemical reaction and the transport of protons in 

the membrane, leading to the enhanced PEMFC performance. However, the cell temperature can affect the 

material properties of components when it is too high.   

In the past, various control strategies have been proposed for PEMFC, but less attention has been given to 

the control structures design. To obtain the efficient control of the PEMFC system, the control structure design 

including the selection of manipulated variables, controlled variables, control configuration, and controller type 
is necessary (Chatrattanawet et al., 2014). 

                                

 
 

 

 
   

                                                  
DOI: 10.3303/CET1652167 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Chatrattanawet N., Hakhen T., Saebea D., Arpornwichanop A., 2016, Performance analysis and control structure 
design for proton exchange membrane fuel cell, Chemical Engineering Transactions, 52, 997-1002  DOI:10.3303/CET1652167   

997



 

Figure 1: The structure of PEMFC.  

The purpose of this study is to implement the control structure design to determine the good controlled and 

manipulated variables for PEMFC. The performance of PEMFC is first analyzed under steady-state and 

dynamic behaviors to find the optimal operating points. The analysis of the steady-state relative gain array 

(RGA) is used for pairing of the controlled and manipulated variables. The control structure is step-by-step 

designed using the method proposed by Skogestad (2004).  

2. Proton exchange membrane fuel cell 

PEMFC can convert the chemical energy of a fuel and an oxidant into electrical energy. The PEMFC structure 

consists of seven components: anode flow channel, gas diffusion layer (GDL) of anode, anode catalytic layer, 

membrane, cathode catalytic layer, GDL of cathode, and cathode flow channel, as can be seen in Figure 1. 

The membrane electrode assembly (MEA) where the electrochemical reaction occurs is the critical interface 

between the membrane and the electrodes. At the anode side, the fuel or hydrogen is continuously supplied 

while the oxidizer or oxygen in air is supplied at the cathode side. Hydrogen flows through gas channel on the 

anode side and diffuses through diffusion layer to the catalytic layer where it is oxidized to form proton and 

electrons. Proton ions are transferred through the membrane to the catalytic layer of cathode while electrons 

pass through the current collector and external circuit to the cathode. At the cathode side, oxygen is reacted 

with protons ions and electrons to form water. 

3. PEMFC model 

The dynamic model of PEMFC includes the mass balances for describing gas variation, the energy balance, 

and the electrochemical model (Khan and Iqbal, 2004). Several assumptions are made as: (i) the fuel cell is 

described by a lumped model; (ii) all gases behave as the ideal gas; (iii) pure hydrogen is fed to the anode 

and air is supplied to the cathode; (iv) the outlet temperatures of hydrogen and oxygen are equal to the cell 

temperature and (v) the membrane is fully saturated with water. 

3.1 Mass balances 
The transient dynamics in the anode and cathode channels are studied. In the anode channel, it is simplified 

by assuming pure hydrogen (99.99%) flows into the channel. Then, hydrogen diffuses through the anode 

electrode into the anode active layer where it is consumed by the electrochemical reaction. Therefore, the 

dynamic change in the partial pressure of hydrogen (pH2) at the anode is as: 

 2
2 2

H

H ,in an H atm

an
2

 
    

 

dp RT jA
m k p p

dt V F
 (1) 

The dynamic of the partial pressure of oxygen (pO2) at the cathode when oxygen in air flows into the channel 

can be described as: 

 2
2 2

O

O ,in ca O atm

ca
4

dp RT jA
m k p p

dt V F

 
    

 
 (2) 

998



where 
2H ,in

m and 
2O ,in

m denote the inlet molar flow rates of hydrogen and oxygen (mol/s), respectively, T is the 

cell temperature (K), j is the current density (A/cm2), patm is the ambient pressure (atm), R is the gas constant 
(J/mol K), F is the Faraday constant (C/mol), A is the active area (cm2), kan and kca are the flow constant for 
anode and cathode (mol/s atm), and Van and Vca are the anode and cathode volume (m3), respectively. 

3.2 Thermal model 
The heat produced from the electrochemical reaction causes the PEMFC temperature increase and can affect 

a membrane degradation. In this study, the temperature is defined as one of the state variables and a control 

strategy is developed to manage the transient thermal behavior of the PEMFC. The energy balance of the 

PEMFC can be written as: 

i tot elec in out loss

dT
C P Q Q Q Q

dt
      (3) 

The total energy (Ptot) is calculated by the product of the reaction heat (H) and the molar flow rate of reacted 

hydrogen (
2H ,react

m ). The electrical power output ( elecQ ) generated by the fuel cell can be expressed in Eq(5). 

Heat loss ( lossQ ) at the stack surface is described in Eq(6) where Tamb is the ambient temperature and Rt is the 

thermal resistance of the cell stack. 

2H ,react 2
tot

jA
P m H H

F
     (4) 

elec
Q jVA  (5) 

 amb
loss

t

T T
Q

R


  (6) 

The input heat flow rate ( inQ ) and the output heat flow rate ( outQ ) by the reactant can be described as: 

     
2 2 2 2 2 2 2 2 2 2 2H ,in p,H H O,an,in p,H O H ,in 0 O ,in p,O N ,in p,N H O,ca,in p,H O air,in 0

g g

in
Q m c m c T T m c m c m c T T        (7) 

  
2 2 2 2 2 2 2 2 2 2 2 2H ,out p,H H O,an,out p,H O O ,out p,O N ,out p,N H O,ca,out p,H O H O,ca,out p,H O 0

g g gen liq

out
Q m c m c m c m c m c m c T T        (8) 

where cp,H2, cp,O2, cp,N2, and cp,H2O denote the specific heat of hydrogen, oxygen, nitrogen, and water, g and liq 

denote gas and liquid states, respectively, and TH2,in and Tair,in are the inlet temperatures of hydrogen and air, 

respectively. 

The input molar flow rate of vapor is expressed as follow: 

 
 

 
 2 2 2

an an,in ca ca,in

H O,an,in H ,in H O,ca,in air,in

an an an,in ca an ca,in

,     
sat sat

sat sat

p T p T
m m m m

p p T p p T

 

 

   
    

       

 (9) 

The water generation rate and the output molar flow rate of vapor can be described by Eqs(10) and (11).   

2H O,ca,out 2

gen jA
m

F
  (10) 

 

 

 

 2 2 2 2 2 2
an ca

H O,an,out H O,an,in H ,react H O,ca,out H O,ca,in O ,react

an an ca an

,    
sat sat

sat sat

p T p T
m m m m m m

p p T p p T

 

 

   
      

       

 (11) 

3.3 Electrochemical model 
The electrochemical model for PEMFC is shown in Table 1. The cell voltage (Vcell) decreases from its open-

circuit voltage (E) that is the maximum voltage due to activation overpotentials (Vact), ohmic losses (VOhm), and 

concentration overpotentials (Vcon). The activation overpotentials are caused by the slowness of the reactions 

taking place on the surface of the electrodes. The ohmic losses arise from the resistance of membrane to the 

transfer of proton and the resistance of the electrode and collector plate to the transfer of electron. Moreover, 

the concentration overpotentials are the result of the mass transport of gases to the reaction. 

999



Table 1: Electrochemical models for PEMFC 

Parameters Equations*  

Output voltage 
act ohm concell

V E V V V     (12) 

Open circuit voltage  
2 2

4 0.5

H O
1.229 8.5 10 298.15 ln

2

RT
E T p p

F

        
 (13) 

Activation overpotentials  
21 2 3 O 4

ln ln
act

V T T c T jA          
(14) 

 

 
2

1

5

2 H

5

3

4

4

0.948

0.00286 0.0002 ln 4.3 10 ln

7.6 10

1.93 10

A c















 

   

 

  

 

 

Ohmic losses 
intohm

V jR  (15) 

 

 

2

2.5

mem

int

181.6 1 0.03 0.062
303

303
0.634 3 exp 4.18

T
l j j

R
T

A j
T



  
   

   


  
    

  

 

 

Concentration overpotentials 
max

ln 1
2

con

RT j
V

F j

 
  

 
 (16) 

*It is noted that cO2 and cH2 are the concentrations of oxygen and hydrogen at the catalytic interface of the cathode and 

anode, respectively. lmem is the membrane thickness and  denotes the function of membrane humidity and stoichiometric 

ratio (usually from 10 to 23). 

4. PEMFC performance 

4.1 Model validation 
To confirm that the PEMFC model used in this study can reliably predict the performance of the PEMFC. The 

simulated polarization curve is compared with the experimental data (Mueller et al., 2007) under the range of 

current densities from 0 to 1 A/cm2 as shown in Figure 2. It is observed that the simulated polarization curve 

shows a good agreement with the experimental polarization curve. The minor errors at the low and high 

current densities are caused by the estimated parameters, especially in the electrochemical model.  

 

 

Figure 2: Comparison between the model prediction and experimental data from Mueller et al. (2007).  

4.2 Steady state analysis and dynamic behavior 
The performance analysis of the PEMFC at steady state conditions is performed and used to select its optimal 

operating conditions. Figure 3 shows the cell voltage, power density, and temperature as a function of the 

current density. It is revealed that the current density increases with decreasing cell voltage due to the 

concentration losses. The cell temperature increases at high current density because the electrochemical 

reactions are more pronounced. In this work, the steady-state operating points are selected at current density, 

j = 0.51 A/cm2, cell voltage, V = 0.59 V, power density, P = 0.30 W/cm2, and cell temperature, T = 332 K. 

 

 

 

1000



 

Figure 3: The cell voltage, power density, and the cell temperature as a function of the current density. 

 

Figure 4: The responses of the cell voltage and cell temperature due to step changes of current density and 

flow rates of hydrogen and air 

Next, the dynamic responses of the cell voltage and cell temperature are discussed. The dynamic responses 

are analyzed under a variation of various inputs, such as current density and flow rates of hydrogen and air. 

Figure 4 shows the effects of these inputs on the cell temperature and voltage. When the current density 

increases, the voltage rapidly decreases due to large concentration overpotentials, whereas the cell 

temperature increases because of an increase in the hydrogen consumption by the electrochemical reaction 

(exothermic reaction). The increased inlet flow rates of hydrogen and air can improve cell performance in 

terms of the cell voltage and temperature because they can increase the partial pressures of hydrogen, 

oxygen and membrane conductivity and excess water can be removed. 

5. Control structure design 

The stepwise procedure of Skogestad (2004) for selecting an active constraint and a self-optimizing variable is 

presented. Moreover, the relative gain array (RGA) considered as a controllability index for the selection of 

input-output pairings is also implemented. 

5.1 Self-Optimizing control 

The self-optimizing control is used to alter the controlled variables for PEMFC. 

Step 1: Define the cost J and constraints. With a given load (current density), an economic cost function J is 

defined to minimize the cost of the hydrogen feed minus the power value, subject to the constraints on the cell 

temperature kept constant at 332 K.  

Step 2: Identify the degree of freedom. At steady state, there are two operational degrees of freedom; the inlet 

flow rates of hydrogen and air. We have to optimize the system in detail, the fuel cell temperature constraint of 

T = 332 K is assumed to always be active.  

Step 3: Identify importance disturbances. The main disturbances are the current density and the temperatures 

of hydrogen and air. 

Step 4: Solve the optimization problems when the PEMFC is operated with/without disturbances.  

Step 5: Identify candidate controlled variables. The cell temperature is active constraint. The following 

candidates for the controlled variable are considered three alternatives that are partial pressure of hydrogen 

(pH2), partial pressure of oxygen (pO2), and cell voltage (V).  

1001



Step 6: Evaluation of loss. The loss, L(u,d) = J(u,d) - Jopt(d) is calculated with each of the candidate variables 

kept constant at their nominal optimal value. The results of the loss for alternative controlled variables show 

that the partial pressure of hydrogen is selected as self-optimizing variable because it results in a small 

economic loss. 

Step 7: Evaluation and selection. This is the pairing issue which is discussed on “controllability analysis”.  

5.2 Controllability analysis 
Relative gain array (RGA) is chosen as the controllability index for the selection of control pairing. Generally, 

the RGA formed in the gain matrix close to the unity matrix is preferred but control structures with high RGA 

elements should be avoided. A change in manipulated variables impacts all of the controlled variables; 

therefore, pairing of the controlled and manipulated variable is important for minimizing the interactions in 

PEMFC system. In this work, the cell temperature (y1) and partial pressure of hydrogen (y2) as the controlled 

variables are needed to be regulated by manipulating inlet flow rates of hydrogen (u1) and air (u2) as the 

manipulated variables. The result of the RGA is 1 for the diagonal pairings (u1-y2, u2-y1). This implies that the 

partial pressure of hydrogen is controlled by using the inlet flow rate of hydrogen as manipulated variable. The 

cell temperature is controlled by using the inlet flow rate of air as manipulated variable.  

6. Conclusions 

In this work, the performance analysis and control structure design for PEMFC are performed. The steady 

state analysis is used to select the optimal operating point of the PEMFC. The dynamic responses of PEMFC 

in terms of the cell voltage and temperature are investigated with respect to a step change in current density 

and inlet molar flow rates of air and fuel. Moreover, the self-optimizing control is implemented to find good 

controlled variables and RGA as the controllability index is also used to pairing suitable input-output. The 

results show that the cell temperature and fuel cell voltage are dependent on the fuel and air inlet flow rates as 

well as current density. The cell temperature is selected as an active constraint that should be maintained at 

its setpoint. Furthermore, the partial pressure of hydrogen is selected as good controlled variable based on the 

self-optimizing concept. According to the RGA, the inlet molar flow rates of hydrogen and air are the 

manipulated variables to control the partial pressure of hydrogen and the cell temperature, respectively. 

Acknowledgments 

Support from the Ratchadaphiseksomphot Endowment Fund of Chulalongkorn University and the Thailand 

Research Fund is gratefully acknowledged. N. Chatrattanawet would also like to acknowledge the 

Postdoctoral fellowship, Chulalongkorn University. 

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Proton Exchange Membrane Fuel Cell System Fuelled by a Mixture of Bio-Ethanol and Methane, 

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Gruber J.K., Bordons C., Dorado F., 2008, Nonlinear control of the air feed of a fuel cell, American Control 

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Khan M.J., Iqbal M.V., 2004, Modelling and Analysis of Electrochemical, Thermal, and Reactant Flow 

Dynamics for a PEM Fuel Cell System, Fuel Cells, 5, 463-475. 

Mueller F., Brouwer J., Kang S., Kim H.S., Min K., 2007, Quasi-three dimensional dynamic model of a proton 

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Skogestad S., 2004, Control structure design for complete chemical plants, Computers and Chemical 

Engineering, 28, 219-234. 

 

 

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