Control and Analysis of Air, Water, and Thermal Systems 
for a Polymer Electrolyte Membrane Fuel Cell 
 
by  
 
Jong-Woo Ahn 
 
 
 
 
A dissertation submitted to the Graduate Faculty of 
Auburn University 
in partial fulfillment of the 
requirements for the Degree of 
Doctor of Philosophy 
 
Auburn, Alabama 
August 6, 2011  
 
 
 
 
Keywords: PEM fuel cell, air, water, thermal management. 
 
 
 
Copyright 2011 by Jong-Woo Ahn 
 
 
Approved by 
 
Song-Yul Choe, Chair, Associate Professor of Mechanical Engineering 
David Beale, Professor of Mechanical Engineering 
Jay Khodadadi, Alumni Professor of Mechanical Engineering 
ii 
 
Abstract 
 
 
The polymer electrolyte membrane (PEM) fuel cell is a power source that can potentially 
replace the internal combustion engine in vehicles of the future. When hydrogen stored in a tank 
and oxygen from the air chemically react in a PEM fuel cell, electricity is generated and water 
and heat are produced as by-products. The management of fuel, water, and heat are crucial issues 
in order to ensure reliable operation and to maintain high efficiency with continuously changing 
loads.  
During the operation of a fuel cell, oxygen from the air is supplied to the cathode side of the 
fuel cell. Insufficient oxygen supply during dynamic loads causes oxygen starvation, which 
forces protons transported through the membrane to reduce the amount of hydrogen on the 
cathode catalysts. Reduced hydrogen could chemically generate heat on the platinum particles 
and result in local hot spots in the membrane electrode assembly (MEA), which could lead to 
failure of the fuel cells. Conversely, an excessive oxygen supply increases the parasitic power 
dissipated by an air supplier such as a blower, which leads to low efficiency of the overall 
system. Therefore, an optimal supply of oxygen is one of challenging issues addressed by 
researchers.   
When a chemical reaction takes places, water is produced in the cathode catalyst. A fraction 
of the water moves from the catalyst though the membrane and to the anode. In contrast, protons 
crossing the membrane take up water from the anode to the cathode. The water content in 
iii 
 
membranes directly affects proton conductivity. Insufficient water causes dehydration that 
decreases proton conductivity and increases voltage loss, which leads to a reduction in output 
power. However, the water produced by chemical reactions causes flooding in gas flow channels, 
gas diffusion layers, and catalyst layers, which blocks oxygen transport to the catalysts and 
reduces the catalyst activation areas. As a result, the degradation of components is accelerated. 
Therefore, excessive water in the cell should be removed to prevent flooding, and at the same 
time the membrane should be kept fully humidified.  
When operating, heat produced in the stack continuously changes as the load current varies. 
Consequently, the temperature inside the cells also varies. Variation of the temperature directly 
affects rates of chemical reaction and the phase change of water, and finally the water transport. 
Improper rejection of the heat might produce local hotspots and degrade the thin layers of the 
cell. Low temperatures decrease the rate of the chemical reaction and reduce efficiency.  
Conversely, elevated temperatures increase the reaction rate, ease the removal of water, and 
increase the mobility of water vapor in the membrane, which alleviates over-potential losses. In 
addition, parasitic power necessary for operating a coolant pump should be reduced in 
conjunction with the heat rejection strategy, which contributes to an increase in system 
efficiency. Therefore, thermal management is another challenging issue for reliable and efficient 
operation.  
The research conducted for my doctoral program focused on the development of fuel cell 
models for air, water, and thermal systems, and on the related components. The control-oriented 
models were used in the design of control strategies and in the analysis of integrated systems. 
The air and water supply systems consist of a blower, a humidifier, and inlet and outlet 
manifolds. The thermal management system is composed of a bypass valve, a liquid-to-liquid 
iv 
 
heat exchanger, a radiator with a fan, reservoirs, and pumps. The objectives of the proposed 
control strategies are to prevent oxygen starvation, maintain a proper water balance in the cells, 
reject excessive heat in the stack, and at the same time the parasitic powers are minimized. 
Considering the aforementioned system configurations and objectives, state feedbacks with 
integral controls were designed and optimized. The entire system was simulated and analyzed, 
and the resulting static and dynamic behavior obtained by experiments is described.  
v 
 
Acknowledgements 
 
 
I would like to thank Dr. Song-Yul Choe for his constructive advice and considerable 
support during this research. Thanks are given to my parents, Hong-Kyu Ahn and Ok-Jae You, 
for their support. In particular, I would like to thank Jinglin He for his advice and discussion, and 
other faculty and graduate students who have supported this research. 
vi 
 
Table of Contents 
 
 
Abstract ........................................................................................................................................... ii 
Acknowledgements ......................................................................................................................... v 
List of Tables .................................................................................................................................. x 
List of Figures ................................................................................................................................ xi 
Nomenclature ............................................................................................................................... xiv 
Chapter 1 Introduction .................................................................................................................... 1 
1.1. Fuel cell technology ........................................................................................................... 1 
1.2. The PEM fuel cell system .................................................................................................. 4 
1.2.1. A PEM fuel cell ........................................................................................................ 4 
1.2.2. Balance-of-plant ........................................................................................................ 8 
1.3. Literature review .............................................................................................................. 10 
1.3.1. Air supply system ................................................................................................... 10 
1.3.1.1. The effects of oxygen starvation on a fuel cell .............................................. 10 
1.3.1.2. Strategies for air supply system ..................................................................... 10 
1.3.2. Water management ................................................................................................. 11 
1.3.2.1. The effects of water management on a fuel cell ............................................ 11 
1.3.2.2. Strategies for water management ................................................................... 14 
1.3.3. Thermal management.............................................................................................. 18 
1.3.3.1. Effects of thermal management on a fuel cell ............................................... 18 
vii 
 
1.3.3.2. Strategies for thermal management ............................................................... 20 
1.4. Objectives of this dissertation .......................................................................................... 21 
Chapter 2 Model Development of PEM Fuel Cell ........................................................................ 22 
2.1. Dynamic fuel cell model .................................................................................................. 23 
2.1.1. Model setup ............................................................................................................. 23 
2.1.2. Governing equations ............................................................................................... 25 
2.2. Non-isothermal dynamic fuel cell model ......................................................................... 30 
2.2.1. Model setup ............................................................................................................. 30 
2.2.2. Governing equations ............................................................................................... 32 
2.3. Non-isothermal and two-phase dynamic fuel cell model ................................................ 34 
2.3.1. Two-phase model for GDL ..................................................................................... 35 
2.4. Model for a stack and experimental validation ................................................................ 38 
2.5. Summary .......................................................................................................................... 40 
Chapter 3 Air and Water Management ......................................................................................... 41 
3.1. System configurations ...................................................................................................... 41 
3.2. BOP models ..................................................................................................................... 43 
3.2.1. Model for air supplier ............................................................................................. 43 
3.2.1.1. Modeling a blower ......................................................................................... 44 
3.2.2. Model for humidifier............................................................................................... 46 
3.2.2.1. Humidifier model ........................................................................................... 48 
3.2.3. Model for inlet and outlet manifolds ...................................................................... 50 
3.2.4. Model for a bypass valve ........................................................................................ 50 
3.3. Control strategies ............................................................................................................. 51 
3.3.1. Air ........................................................................................................................... 51 
viii 
 
3.3.2. Air and water management (AWM) ....................................................................... 57 
3.4. Simulation and analysis ................................................................................................... 61 
3.4.1. Oxygen excess ratio ................................................................................................ 62 
3.4.2. Membrane water content and cell voltage .............................................................. 64 
3.5. Experimental validation ................................................................................................... 68 
3.5.1. Model validation for a gas-to-gas humidifier ......................................................... 68 
3.5.1.1. Experimental setup......................................................................................... 68 
3.5.1.2. Uncertainty analysis ....................................................................................... 70 
3.5.1.3. Experimental procedure ................................................................................. 71 
3.5.1.4. Experimental results and analysis .................................................................. 71 
3.5.2. Experiments for controls of BOP ............................................................................ 75 
3.5.2.1. Experimental setup......................................................................................... 75 
3.5.2.2. Uncertainty analysis ....................................................................................... 81 
3.5.2.3. Experimental procedure ................................................................................. 82 
3.5.2.4. Experimental results and analysis .................................................................. 83 
3.6. Summary .......................................................................................................................... 86 
Chapter 4 Thermal Management .................................................................................................. 88 
4.1. System configurations ...................................................................................................... 88 
4.2. Models for system components ....................................................................................... 90 
4.2.1. Bypass valve ........................................................................................................... 90 
4.2.2. Radiator and fan ...................................................................................................... 91 
4.2.3. Reservoir ................................................................................................................. 92 
4.2.4. Pump ....................................................................................................................... 92 
4.2.5. Heat exchanger........................................................................................................ 93 
ix 
 
4.2.5.1. Model for a plate heat exchanger ................................................................... 94 
4.3. Control strategies ............................................................................................................. 96 
4.3.1. Temperature controls .............................................................................................. 96 
4.3.2. Temperature controls based on a reduced-order observer ...................................... 99 
4.3.3. Temperature controls for two thermal circuits ...................................................... 103 
4.4. Simulation and analysis ................................................................................................. 105 
4.4.1. Temperature distribution in two cells ................................................................... 106 
4.4.2. Effect of temperature on membrane water content ............................................... 107 
4.4.3. Effect of temperature on oxygen excess ratio ....................................................... 108 
4.4.4. Response of temperatures with a step current profile ........................................... 109 
4.4.5. Response of the accumulated total parasitic power with a step current profile .... 111 
4.4.6. Responses of controllers and temperatures for the coolant leak ........................... 112 
4.4.7. Responses of temperatures with FUDS ................................................................ 114 
4.5. Summary ........................................................................................................................ 115 
Chapter 5 Conclusion .................................................................................................................. 117 
References ................................................................................................................................... 119 
Appendix A Matrices for air supply system ............................................................................... 126 
Appendix B Linear Quadratic Regulator method ....................................................................... 127 
Appendix C Matrices for thermal management .......................................................................... 130 
 
x 
 
List of Tables 
 
 
Table 3-1. Parameters of fuel cell model. ..................................................................................... 62 
 
xi 
 
List of Figures 
 
 
Figure 1-1. U.S. primary energy flow by source and demand sectors, 2009 [1]. ........................... 2 
Figure 1-2. Applications of fuel cells. ............................................................................................ 3 
Figure 1-3. Schematic diagram of a PEM fuel cell. ........................................................................ 5 
Figure 1-4. Construction of a fuel cell stack. .................................................................................. 8 
Figure 1-5. Schematic diagram of a PFM fuel cell system. ............................................................ 9 
Figure 1-6. Schematic diagram of mechanism of water transport. ............................................... 12 
Figure 1-7. Flow field designs: (a) interdigitated and (b) serpentine flow fields. ........................ 15 
Figure 2-1. Schematic diagram of a fuel cell. ............................................................................... 24 
Figure 2-2. Block diagram for a fuel cell stack at constant temperature. ..................................... 25 
Figure 2-3. Schematic diagram of a single cell stack. .................................................................. 31 
Figure 2-4. Energy balance model. ............................................................................................... 34 
Figure 2-5 Schematic diagram for two-phase GDL. ..................................................................... 36 
Figure 2-6. Schematic diagrams of (a) 2-cell stack, and (b) 20-cell stack. ................................... 38 
Figure 2-7. Experimental apparatus. ............................................................................................. 39 
Figure 3-1. Air supply system with humidifier [51]. .................................................................... 42 
Figure 3-2. Air supply system with gas-to-gas humidifier and three-way valves. ....................... 43 
Figure 3-3 Structures of (a) centrifugal blower and (b) regenerative blower [53]. ...................... 44 
Figure 3-4. Block diagram of the blower model. .......................................................................... 46 
Figure 3-5. Diagram of (a) an enthalpy wheel [56], and (b) a plate-type humidifier [57]. .......... 47 
Figure 3-6. Structure of a gas-to-gas humidifier [58]. .................................................................. 48 
xii 
 
Figure 3-7. Principle of a gas-to-gas humidifier ........................................................................... 48 
Figure 3-8. Air supply system using a blower. ............................................................................. 51 
Figure 3-9. Control configurations of (a) a sFF, and (b) a sFB with a sFF. ................................. 52 
Figure 3-10. Optimal blower motor voltage for static feed-forward controller. ........................... 53 
Figure 3-11. Block diagram for feedback control with a feed-forward control. ........................... 54 
Figure 3-12. Effect of weighting factor QI on the recovery behavior of oxygen excess. ............. 56 
Figure 3-13. Effect of weighting factor Qz on the oxygen excess ratio. ....................................... 56 
Figure 3-14. Block diagram for control of the air flow rate with a humidifier. ............................ 58 
Figure 3-15. Control configuration of air and water management system. .................................. 59 
Figure 3-16. Block diagram of the proposed integrated system model. ....................................... 61 
Figure 3-17. Simulated oxygen excess ratio for two different controllers. .................................. 63 
Figure 3-18. Comparison of (a) membrane water content and (b) cell voltages at 0.2 A cm-2. ... 65 
Figure 3-19. Comparison of (a) membrane water content and (b) cell voltages at 1.0 A cm-2. ... 66 
Figure 3-20. Membrane water content change with respect to the step current profile. ............... 68 
Figure 3-21. Layout of the test station for humidifier experiments [58]. ..................................... 69 
Figure 3-22. Temperatures for variable exhaust flow rates (dry air flow rate = 140 slpm). ........ 72 
Figure 3-23. Humidity for variable exhaust flow rates (dry air flow rate = 140 slpm). ............... 73 
Figure 3-24. Temperatures for step exhaust flow rate. ................................................................. 74 
Figure 3-25. Humidity for step exhaust flow rate. ........................................................................ 75 
Figure 3-26. Fuel cell test station. ................................................................................................. 76 
Figure 3-27. Structure of air supply and humidification system. ................................................. 78 
Figure 3-28. Structure of hydrogen supply system. ...................................................................... 79 
Figure 3-29. Structure of thermal circuit. ..................................................................................... 80 
Figure 3-30. Stack voltages with different three-way valve opening factors at 10 A. ................. 83 
xiii 
 
Figure 3-31. Stack voltages with different three-way valve opening factors at 60 A. ................. 84 
Figure 3-32. I-V characteristics of stack between different humidification. ................................ 85 
Figure 3-33. Dynamic behavior of the stack voltage for step valve opening factor at 10 A. ....... 86 
Figure 4-1. Configuration of thermal management using a bypass valve. ................................... 89 
Figure 4-2. Configurations of thermal management with two thermal circuits. ........................... 90 
Figure 4-3. Block diagram of the coolant pump. .......................................................................... 96 
Figure 4-4. Block diagram of a state feedback controller with an integral controller. ................. 98 
Figure 4-5. Block diagram for coolant control with a sFB and a reduced-order observer. ........ 100 
Figure 4-6. Example of the order reduction: cathode. ................................................................ 101 
Figure 4-7. Block diagram for temperature control with a state feedback controller ................. 103 
Figure 4-8. Block diagram of the proposed integrated system model for TMS. ........................ 105 
Figure 4-9. Temperature variation in the two cells depending on current density. .................... 106 
Figure 4-10. Water content in the membrane. ............................................................................ 108 
Figure 4-11. Comparison of oxygen excess ratio at constant and varying temperatures. .......... 109 
Figure 4-12. Comparison of temperatures of the catalyst layer and coolant channel. ................ 110 
Figure 4-13. Comparison of accumulated total parasitic power between bang-bang and 
state feedback controller. .............................................................................................................111 
Figure 4-14. (a) Coolant flow rate, (b) bypass valve opening factor, and (c) temperatures 
of the catalyst layer and coolant channel for the coolant leak. ....................................................113 
Figure 4-15. (a) FUDS and a current profile with a base load of 120 A, (b) temperature of 
the catalysts and coolants for control only, (c) temperature of the catalysts and coolants 
for control with the reduced-order observer, and (d) the oxygen excess ratio. ...........................115 
 
xiv 
 
Nomenclature 
 
 
Alphabet:  Units 
a Water activity  
A Area cm2 
b Membrane expansion coefficient  
C Mass concentration kg m-3 
Cp Specific heat W m-2 oC-1  
 D Diffusion coefficient m2 s-1 
F Faraday number  
FM Flow meter  
i or I Current density A cm-2 
HS Humidity sensor  
h Heat transfer coefficient kW m-2 oC-1 
J Rotational inertia kg m2 
K Permeability m2 
m Mass kg 
M Molar mass kg mol-1 
n Number of cells  
N Molar flux mol m-2 
p (Partial) pressure  Pa 
Q Heat J 
R Universal gas constant or resistance (-) or ? 
xv 
 
s Entropy or liquid water saturation ratio J mol-1 K-1 or (-) 
T 
t 
Temperature K 
TC Thermocouple  
t Thickness m 
W Mass flow rate kg s-1 
z Ratio of the species molar flux  
   
Greek:   
? Volumetric condensation coefficient s-1 
? Porosity   
? Efficiency  
? Water content or ratio  
? Density kg m-3 
? Viscosity kg m-1 s-1 
? Membrane conductivity S m 
? Torque N m 
? Relative humidity or flux linkage (-) or V s rad-1 
? Angular velocity rad s-1 
   
Superscripts, subscripts:   
amb Ambient  
an Anode  
bl Blower  
c Coolant  
ca Cathode  
xvi 
 
catl Catalyst layer  
cond Conduction  
conv Convection  
cv Control volume  
diff Diffusion  
elec Electro-  
evap Evaporation  
fc Fuel cell  
g Gas  
i Index  
im Immobile  
m Motor  
memb Membrane  
pl Plumbing  
rad Radiator  
res Reservoir  
s Stator  
sou Source  
st Stack  
 
 
 
1 
 
Chapter 1 Introduction 
 
1.1. Fuel cell technology 
We require energy for electricity, heating, cooling, and fueling vehicles.  Figure 1-1 shows a 
diagram representing the sources of energy and energy demand in the U.S. According to the 
survey [1], the transportation sector is heavily dependent upon fossil fuels: 97% of our energy is 
from petroleum and natural gas, while only 3% is from renewable energy sources. Our 
dependence on fossil fuel results in two challenging issues. First, an imbalance between supply 
and demand, and depletion in near future causes fluctuations in oil prices and makes the global 
economy vulnerable. Second, emissions due to combustion processes include chemicals that 
adversely affect health. In particular, the carbon dioxides have been identified as chemicals that 
cause global warming. Many universities, industries, and governments are seeking solutions to 
replace fossil fuels with renewable, sustainable, and economical fuels and related devices that 
convert the new fuels into electric, thermal, or mechanical energy.  
 
2 
 
 
Figure 1-1. U.S. primary energy flow by source and demand sectors, 2009 [1]. 
 
A fuel cell is an electrochemical device that converts chemical energy into electrical energy. 
Fuel cell technology is considered to be a power source that can potentially replace traditional 
internal combustion engines (ICEs) due to its high efficiency, high power density, and low 
emissions. Fuel cell technology can be categorized by the type of electrolyte, fuel, and operating 
temperature as follows: polymer electrolyte membrane fuel cells (PEMFCs), direct methanol fuel 
cells (DMFCs), phosphoric acid fuel cells (PAFCs), molten carbonate fuel cells (MCFCs), and 
solid oxide fuel cells (SOFCs) [2]. These technologies have been demonstrated in a variety of 
applications as shown in Figure 1-2. 
3 
 
 
Figure 1-2. Applications of fuel cells. 
 
Among these fuel cell technologies, PEM fuel cells have attracted attention for transportation 
applications because of their eco-friendliness, quite operation, low operating temperature, and 
high efficiency [2], [3]. For example, PEM fuel cells are very advantageous for electric vehicles 
because the driving range can be extended. As a result, the size of the required rechargeable 
batteries can be minimized [4]. In order to commercialize fuel cell technology for automotive 
applications, technical barriers must be overcome, and infrastructure such as refueling gas 
stations must be developed. The two barriers are durability and price. The lifespan of a fuel cell 
should be at least 5000 hrs, and the price must be comparable to that of ICEs (in the range of 
$30/kW) [5]. 
 
 
4 
 
1.2. The PEM fuel cell system 
A PEM fuel cell generates electricity and produces heat and water though an electrochemical 
reaction between oxygen and hydrogen. Since the electrical energy generated in a single fuel cell 
is very insufficient to power a vehicle, multiple cells are assembled into a stack. A PEM fuel cell 
system consists of a fuel cell stack and a balance-of-plant (BOP). The BOP manages reactant 
gases and removes byproducts such as heat and water. The subsystems of the BOP include air 
and hydrogen supply systems, thermal systems, and water management systems; these strongly 
affect the performance and durability of the fuel cell system. In particular, these subsystems play 
a significant role in ensuring highly efficient and reliable operations by maintaining proper 
operating conditions.  
 
1.2.1. A PEM fuel cell 
A typical PEM fuel cell structure, shown in Figure 1-3, is constructed of various components. 
These include gas flow channels (GFCs) that provide pathways for hydrogen and oxygen to gas 
diffusion layers (GDLs); gas diffusion layers that distribute the reactants evenly to catalyst layers 
(CLs) where electricity and water are generated on the cathode side; and a membrane located in 
the middle of the fuel cell that separates the anode and cathode while conducting hydrogen 
protons and blocking electrons. 
5 
 
 
Figure 1-3. Schematic diagram of a PEM fuel cell. 
 
During fuel cell operation, oxygen or air, and pure hydrogen from hydrogen storage, are 
supplied as reactant gases into the gas flow channels. Parts of the reactants diffuse into the gas 
diffusion layers. When hydrogen reaches the catalyst layer on the anode side, the hydrogen 
becomes ionized, releasing electrons and creating protons. This reaction occurring at the anode is 
called a hydrogen oxidation reaction (HOR). Electrons released by the HOR travel along an 
external circuit to the cathode side while the protons travel though membranes and arrive at the 
catalyst of the cathode side. Here, oxygen molecules react with protons permeating though the 
membrane, and with electrons arriving though the external circuit, thereby producing water 
molecules and heat. This reaction occurring in the cathode is called an oxygen reduction reaction 
(ORR), which can be written as follows:  
6 
 
OHeHO
eHH
22
2 244 442 ??? ??
??
??
         (1- 1) 
 
Water and heat are generated as byproducts of the electrochemical reactions. Water 
generated at the catalyst layer is transported though the gas diffusion layer and gas flow channel 
at the cathode side, and though the membrane to the gas diffusion layer and gas flow channel on 
the anode side.  
 
Since protons take up water, there is a water concentration gradient in the membrane from 
the anode to the cathode. On the other hand, a difference in water concentration between the 
cathode and anode causes back diffusions. Therefore, the overall balance in the membrane is 
determined by these two phenomena. Depending upon operating conditions, the membrane could 
be dehydrated and thus require proper humidification. As the membrane is drying, the ohmic 
resistance for protons increases and potential losses increase. As a result, fuel cell performance 
decreases. On the other hand, when the water removal rate is lower than the water generation rate 
at the cathode, excessive water vapor in the GDL and CL can condense into liquid water. The 
accumulated liquid water blocks the pores, providing pathways for oxygen transport in the GDL 
and covers up the active sites in the CL. This results in degradation of the fuel cell?s performance 
[6]. 
Therefore, water management in fuel cells is a challenging issue. The membrane should be 
properly humidified for all operating conditions, and at the same time water flooding at the 
cathode should be prevented to ensure pathways for the reactants. However, it has to be noted 
that the use of the highly pressurized air flow supplied to the fuel cell to remove water can 
increase the parasitic power of the air supply system.  
7 
 
The other byproduct resulting from electrochemical reactions is heat. Generated heat 
determines the temperature profiles inside the fuel cells, which directly influence the rates of 
chemical reaction and water transport. For example, increased temperature can ease the removal 
of water produced in the catalysts and increase the mobility of water in the membrane, thereby 
alleviating over-potentials. Conversely, a low operating temperature (less than 60 oC) could 
increase the rates of water condensation and flooding in the electrodes, which subsequently 
increases over-potentials because of increased flow resistance for reactant mass transport. In 
addition, a low operating temperature results in a low electrochemical reaction rate [6]. 
Therefore, thermal management is another challenging issue. However, improper rejection of 
heat might locally produce hotspots and destroy thin layers of cells 
  
A fuel cell stack consists of multiple cells with end plates as shown in Figure 1-4. A unit cell 
is constructed with five components: a membrane electrode assembly (MEA), two GDLs, and 
two bipolar plates (BPs) at the cathode and anode sides. The MEA is referred to as the heart of 
the PEM fuel cell, and is usually made of a polymer electrolyte membrane sandwiched between 
two catalyst-coated carbon papers. The gas diffusion layer is made of porous media such as 
carbon fiber/cloth, and is used uniformly to diffuse reactant gases reaching the catalyst layer for 
chemical reaction. The bipolar plate located between cells performs as a current conductor 
between cells and provides flow channels for reactant gases and coolant. The current collector 
plate is made of highly conductive materials and is connected to an external circuit. The end 
plate of the fuel cell stack functions as the supporting structure for the fuel cells.  
8 
 
 
Figure 1-4. Construction of a fuel cell stack. 
 
1.2.2. Balance-of-plant  
A typical configuration of a PEM fuel cell system, including a fuel cell stack and the BOP, is 
shown in Figure 1-5. The BOP system consists of an air and water management system (blue 
line), a thermal management system (green line), and a hydrogen delivery system (red line).   
9 
 
 
Figure 1-5. Schematic diagram of a PFM fuel cell system. 
 
The main components of the air and water management system are a blower and a 
humidifier. Air supplied to the fuel cell stack by the blower is humidified by a humidifier. The 
thermal circuit includes a radiator to exchange heat with the ambient media, a fan to increase the 
effectiveness of heat convection, and a reservoir to store and thermally insulate the coolants. A 
coolant pump is used to supply coolant to the heat source. 
The fuel delivery system consists of a hydrogen tank to store hydrogen, a low pressure 
regulator to control pressure, a supply solenoid valve to adjust flow rate, and a purge valve to 
remove excessive liquid water and impurities on the anodic side. 
 
10 
 
1.3. Literature review 
1.3.1. Air supply system 
1.3.1.1. The effects of oxygen starvation on a fuel cell 
During operation, a sudden change of the oxygen demand such as start-up and load change, 
and water flooding can cause oxygen starvation [7]. Taniguchi et al. [8] observed that the 
decrease of the activation area in oxygen starved cell is related with the increase of the size of the 
catalyst particles, and disappearing of the small particles. A consequence of oxygen starvation is 
a rapid cell reversal [8]. When cell reversal occurs in the cathode catalyst, the proton and the 
electron are reduced to the hydrogen. During cell reversal, the heat is generated and results in the 
local hot spots, which give the damage to the fuel cells.  
 
1.3.1.2. Strategies for air supply system 
Currently, the control strategies for air supply system were proposed by many researchers. 
Rodatz et al. [10] proposed a dynamic model for a compressor that accounts the dynamics of the 
compressor and inlet and outlet manifolds filling processes. Based on the models, a regulator is 
designed by using the Linear Quadratic Gaussian (LQG) means and compared with a PI 
controller. The result shows that the use of the LGQ significantly improves the response time of 
the air supply.  
Pukrushpan et al. [11] proposed different control strategies for an air supply system that 
optimally maintain the oxygen excess ratio that refers to a ratio between the oxygen consumed 
and supplied. A Static Feed-Forward control (sFF) solely based on a map date and a feedback 
control (sFB) based on observer are employed to increase robustness of the ratio at any load 
11 
 
applied and improve recovery dynamics. The optimization of the advanced controller gains is 
carried out by using the Linear Quadratic Gaussian (LQG). 
Vahidi et al. [12] addressed phenomena of oxygen starvation that the air cannot be quickly 
delivered to the fuel cell at instant change of current because of the air flow rate limited by the 
dynamics of inlet manifold and operational constraints of compressor. Hybridization with a 
battery is proposed to minimize the oxygen starvation, which levels a peak load. Optimal balance 
of the two sources is carried out by the Model Predictive Control (MPC) 
Zhang et al. [13] proposed an adaptive control algorithm to stabilize the oxygen excess ratio 
around an optimal level. The adaptive control includes the estimation of time-varying parameters 
in a stack and the pole assignment of the closed-loop system. Finally, the proposed adaptive 
control was evaluated on a test bench. 
 
1.3.2. Water management 
1.3.2.1. The effects of water management on a fuel cell 
A schematic diagram for the mechanism of water transport in a cell is shown in Figure 1-6. 
During operation, there are two source terms for water transport mechanism, which are ones 
supplied by humidification of reactant gases and generated internally by the ORR. Water 
transport in membranes occurs through two forces: an electro-osmotic driving force and back 
diffusion. The former moves water from the anode to the cathode along with protons. The latter 
is caused by a concentration gradient of water across the membrane that moves water from the 
cathode to the anode. In addition, water at the catalyst layer is transported though the gas 
diffusion layer into the gas flow channels, and is then exhausted at the fuel cell outlet by 
diffusion and capillary forces combined with pressure in the gas flow channels.  
12 
 
 
Figure 1-6. Schematic diagram of mechanism of water transport.  
 
If the water removal rate is higher than the water generation rate, the membrane water 
content decreases, which increases ohmic resistance for proton transport. Consequently, potential 
losses in the membrane become higher, fuel cell performance is degraded, and the membrane 
may be permanently damaged under severe drying conditions [6]. 
Conversely, when the water generation rate by electrochemical reaction exceeds the water 
removal rate by diffusion of water vapor and capillary transport of liquid water from the GDL 
and the CL, the water vapor is oversaturated and condensed. Liquid water that has accumulated 
in the porous gas diffusion and catalyst layers blocks the gas pores used for oxygen transport to 
the catalyst layer [14], [15], and covers up active areas of the catalyst layer. This leads to fewer 
oxygen reduction reactions. In addition, excessive water transport leaches impurities within the 
cell and deposits them on catalysts or the membrane. This results in increasing ohmic and mass 
transport loss [16]. Flooding occurs under certain operating conditions such as low temperature, 
13 
 
high relative humidity of fuel cell entering gases, and a high current density region where the 
water vapor inside the fuel cell becomes oversaturated and condensed on the cathode side. 
Flooding occurs depending on interaction of the properties of the components and operating 
conditions. The effects of the catalyst layer on water management have rarely been studied 
experimentally or numerically, in contrast to numerous studies on the effects of the GDL and 
GFC on water management. Eikerling and Kornyshev [17] considered the effects of the cathode 
catalyst layer relating to water handling ability and cell performance. The results show that the 
cathode catalyst layer is a prime component for the conversion of liquid water to vapor, and 
regulates the water flux balance between the GDL and the membrane. 
The gas diffusion layer plays the most significant role in the water flow that maintains the 
sensitive balance between membrane hydration and water removal. To help repel water from 
catalysts, the GDL is often treated with hydrophobic materials (e.g., polytetrafluoroethylene 
(PTFE)) to change its wetting characteristics. In addition, a micro-porous layer (MPL) is often 
added between the GDL and the CL, which boosts water removal and assists in the distribution 
of reactant gas flows to catalyst surfaces. 
The water transported from the GDL to the GFC depends on water management along the 
GFC. Two different and widely used flow patterns are the inter-digitated and serpentine flow 
fields. The capability of water removal depends on the operating conditions. Therefore, an 
understanding of the effects of operating conditions on water transport is important. 
The key operating conditions for the PEM fuel cell are the flow rates of the reactants with 
respect to the stoichiometric number and back pressure; the inlet humidity of the reactant gases; 
and coolant flows for maintaining cell temperature. Generally, the humidified reactant gases 
supplied to the fuel cell ensure membrane hydration in the low current density region. However, 
14 
 
in the high current density region where more water is generated, highly humidified reactant 
gases may cause water flooding at the cathode.  
Back pressure and stoichiometric numbers affect the pressure drop of two-phase flow, which 
is directly related to water movement in the GDL and along the flow channels. The operating 
temperature determines the saturation levels of the reactant gas flows in the GDL and the CL, 
which in turn are related to evaporation and condensation rates. When the cell temperature 
increases, more liquid water is evaporated and the water flooding level becomes low [18].  
Cell performance strongly depends on the maintenance of membrane hydration and on the 
prevention of water flooding. To achieve these objectives, considerable research has been 
conducted in developing design and operating strategies for proper water management. The main 
strategies developed thus far for humidification of the membrane and/or the mitigation of water 
flooding are summarized as follows. 
 
1.3.2.2. Strategies for water management 
a. Design of the flow field and MEA 
An appropriate design of gas flow channels is the most successful approach to the prevention 
of water flooding. Two typical flow patterns are the inter-digitated and serpentine flow fields, as 
shown in Figure 1-7. The inter-digitated flow field proposed by Nguyen [19] consists of dead-
end inlet and outlet channels. This design provides a higher transport rate of gases than a 
conventional flow field with straight parallel channels because incoming reactant gases are 
forced to flow though the GDL to the catalyst layer. As a result, shear force exerted by the gas 
flow flushes liquid water out of the cathode. Water flooding is reduced, and fuel cell 
performance is significantly improved. However, forcing reactant gases to flow into the GDL 
15 
 
requires high pressure, which results in significant parasitic power loss. Furthermore, highly 
enhanced mass transport in the GDL could lead to membrane dehydration at low current density 
[20]. 
In contrast, the serpentine flow field is the most widely known and used because it ensures 
better performance and durability [21]. The gas flows along the main channels in this design, and 
there are cross-sectional small channels that create a high pressure drop between adjacent main 
channels. This high pressure drop drives a cross-leakage flow between adjacent main channels 
and through the gas diffusion layer, which makes water removal easier. However, it requires 
significant parasitic power loss because of the high pressure drop. Moreover, this design often 
causes membrane dehydration near the channel inlet while water flooding occurs near the 
channel outlet because of excessive water removal by the gas stream. For a dry reactant stream 
or for cases in which operations occur at less than 100% relative humidity, a single serpentine 
flow field can result in membrane dehydration that leads to degradation of cell performance in 
the low current density region [22]. 
 
Figure 1-7. Flow field designs: (a) interdigitated and (b) serpentine flow fields. 
16 
 
 
In addition to a channel design strategy, incorporation of special hydrophilic wicking 
structures into cathode flow channels can help redistribute liquid water. For example, Ge et al. 
[23] proposed an internally humidified polymer electrolyte fuel cell (PEFC) mounted with two 
strips of absorbent polyvinyl alcohol sponge. Their results show that the mounted sponge wicks 
were sufficient to humidify dry air flow entering the fuel cell, and to remove water produced in a 
cell under correct operating conditions. However, this humidification strategy did not work under 
abnormal conditions such as high temperature, a high gas stoichiometric ratio, and low pressure.  
Ihonen et al. [14] proposed treating the GDL with hydrophobic materials such as 
polytetrafluoroethylene (PTFE); this would allow separate paths for gas and liquid transport. 
Water removal improves using this approach.  Spernjak et al. [24] investigated liquid water 
formation and transport in the PEM fuel cell using untreated carbon cloth, untreated carbon 
paper, and 5 wt.% PTFE-treated carbon fiber. Experimental results show that water emerged as 
droplets over the surface of the flow channels for PTFE-treated carbon fiber, whereas the water 
moved along the sidewall of the flow channels in the form of films and slugs for the untreated 
carbon cloth and paper. Under the same operating conditions, the gas flow channel with PTFE-
treated GDL has a higher quantity of liquid water than that of the untreated GDL, which 
indicates that the water is efficiently removed from the catalyst layer.  
Qi and Kaufman [25] experimentally studied the performance of single cells with and 
without micro-porous layers (MPLs), which are added to conventional GDLs to improve water 
removal from the catalyst layer.  
Freire and Gonzalez [26] investigated the effects of membrane characteristics and 
humidification conditions on the impedance response of cells. Their results showed that thinner 
17 
 
membranes shortened the water back-diffusion process, reduced the anode to be humidified, and 
lowered the ionic resistance or ohmic losses though the membrane. However, thinner membranes 
often cause poor durability and high gas crossover rates, which limits the practical membrane 
thickness for fuel cell applications. 
The internal humidification systems have advantages such as compact and lightweight 
structures. However, their performance in a fuel cell system with internal humidification was 
lower than that of external humidification, and the operating conditions were limited. 
 
b. Operating conditions  
Water balance in a cell can be operationally manipulated by removing the water present on 
the anode side and by using an electro-osmotic pump. Buchi and Srinivasan [27] investigated 
operation of a PEM fuel cell without external humidification of the reactant gases. The dry 
reactant gases were humidified by the water produced in a cell. Since back diffusion is the 
dominant factor for water flow though a membrane in the low current density region, hydrogen 
on the anode side is internally humidified. However, the operation of fuel cells with an internal 
humidification system is limited for certain gas flow rates and cell temperatures. The 
performance of a cell without humidification is 20 to 40% lower than for the case using both 
humidified gases. 
 
Wilkinson et al. [28] proposed a strategy to control cathode water flooding. An appropriately 
designed stack can draw liquid water accumulated in the cathode by a concentration gradient 
across the membrane to the anode, and remove it into the fuel stream without additional parasitic 
power loss. Also, anode water removal can be achieved by high differential gas pressure between 
18 
 
the cathode and the anode, which forces liquid water transport though the membrane from the 
cathode to the anode [29]. However this strategy requires a relatively high operating pressure 
that can result in significant parasitic power loss and membrane rupture during operation. On the 
other hand, Buie et al. [30] proposed the use of an electro-osmotic (EO) pump to prevent water 
flooding. For example, two porous glass EO pumps were placed against the wall of cathode flow 
channels. During operation, liquid water formed in the GDL was forced out of the GDL and 
transformed into water droplets. Liquid water droplets were drawn into a hydrophilic porous 
glass structure of the EO pump. Once the EO pump structure was properly saturated, the EO 
pump actively drove the water though the porous glass structure into the integrated water 
reservoirs in the acrylic top plate. It was reported that an EO pump has the capability of 
preventing cathode flooding as well as enhancing the performance of partially flooded fuel cells 
using a small fraction of the fuel cell?s power. 
 
1.3.3. Thermal management 
1.3.3.1. Effects of thermal management on a fuel cell 
Proper thermal management has been recognized as a crucial issue that must be resolved for 
the commercialization of fuel cell vehicles [31], [32]. The PEM fuel cells developed thus far 
operate in a temperature range of 60 to 80 oC. This temperature range is determined by the 
properties of membrane materials such as Nafion? , whose working temperature is commonly 
limited to 80 oC. When the working temperature is lower than 60 oC, water condensation can be 
accelerated. Subsequently, water flooding occurs, which results in voltage losses due to extra 
flow resistance of reactant mass transport. In addition, a low operating temperature causes low 
proton conductivity and electrochemical reaction rates. Therefore, appropriate thermal 
19 
 
management is required to maintain the working temperature of the cells at the desired level. The 
time dependent temperature profile can be described by an energy equation that includes heat 
source terms and heat transfer to components and the ambient. During operation, heat in a single 
cell is generated by entropic changes from chemical reactions, activation over-potential on the 
anode and cathode, and ohmic over-potentials [31], [33]. Joule heating is caused by ohmic 
resistances caused by the conduction of both protons and electrons. The percentages of heat 
generated by these source terms are approximately 55%, 35%, and 10%, respectively [31].  
 
The heat generated in a stack is rejected by the cooling system, or transferred by convection 
or radiation across the surface of the stack. The heat rejection rate for the stack is determined by 
the thermal properties of the individual components of the stack. Different heat transfer rates and 
mechanisms have been observed in different components. The heat transfer mechanism through 
the membrane is primarily heat conduction, while both conduction and convection have 
significant heat transfer contributions in the GDL, CL, and GFC.  
Insufficient heat rejection induces various thermal stresses on components that can cause loss 
of catalysts, membrane dehydration, and cathode flooding. For example, Bi and Fuller [34] 
analyzed the effects of temperature on cathode Pt/C catalyst durability. High temperature 
accelerates Pt/C degradation, loss of Pt catalyst active area, and deposition of Pt in the 
membrane, which results in a decrease of cell potential with operating time. 
On the other hand, high operating temperature increases water removal from both the cathode 
and anode, and increases the mobility of water in the membrane. As a result, the membrane may 
be exposed to dehydration that causes the high ohmic resistance of the membrane, which leads to 
low conductivity of protons. Moreover, according to Maggio et al. [35], the cathode flooding 
20 
 
phenomenon, which is dominated by condensation/evaporation processes in the cathode, is 
heavily affected by the temperature distribution.  
 
1.3.3.2. Strategies for thermal management 
He et al. [36] proposed a temperature fuzzy control strategy. The PEMFC temperature and 
circulating coolant inlet temperature are controlled by regulating the coolant flux and bypass 
valve opening factor, respectively. The simulation results showed that the fuzzy controller with 
integrator effectively controlled the PEMFC temperature and the inlet coolant temperature within 
the desired working temperature ranges. Moreover, the author claimed that the proposed control 
system can be easily applied in various power classes of PEMFC temperature control in real-time 
due to its concise modeling and control process. 
Wen et al. [37] proposed combined passive and active cooling system for PEM fuel cell. 
While six thermally conductive Pyrolytic Graphite Sheets (PGSs) inserted to six central cathode 
gas channel plates are used for passive cooling as heat spreaders and fins, two small cooling fans 
are used for forced convection. In addition, three other cooling configurations with differently 
sized fans are tested and compared.  
Yu and Jung [38] proposed a thermal management strategy of a PEMFC system with a large 
active area for transportation applications, which is based on the investigation and the numerical 
experiments with a thermal management system. The cell temperature was effectively regulated 
by the coolant pump and cooling fan based on feedback control.  
 
21 
 
As previously discussed, thermal, air, and water management strongly affects the 
performance and reliability of cells, and further research and development of novel designs and 
operating strategies is required. 
 
1.4. Objectives of this dissertation  
The PEM fuel cell system is a candidate to replace the internal combustion engine. To ensure 
high performance, reliability, and durability, humidified air should be properly supplied and the 
heat produced should be rejected. Therefore, the objectives of this dissertation are to research 
and develop control strategies for the air, water, and temperature. To accomplish these objectives, 
dynamic models of components were developed and associated controls were designed in 
combination with new system configurations that work more effectively and efficiently than 
current systems. Simulations were conducted to analyze the static and dynamic performance of 
the system configurations with controls, and the subsequent effects on behavior. Finally, 
experiments were performed to validate and evaluate the performance of the controls. 
 
The objectives of this dissertation are summarized as follows: 
1. Improvement of dynamic response at an abrupt change of loads. 
2. Maintenance of the optimal oxygen excess ratio. 
3. Prevention of membrane dehydration and water flooding. 
4. Optimal heat transfer by increased cooling effectiveness. 
5. Maximizing the net power by reducing parasitic power sources. 
 
 
22 
 
Chapter 2 Model Development of PEM Fuel Cell 
 
Most of the internal variables in an operating stack are hard to measure because the cells, 
composed of thin layers, are fully sealed and inaccessible. Moreover, the relationship between 
variables in operating cells (as well as in stacks) makes it difficult for design engineers to 
understand the interrelated mechanisms of the operating variables, and to design the stack. 
Therefore, dynamic models were developed and simulations were conducted to study the 
phenomena and behavior of PEM fuel cells.  
Many researchers have proposed models for a single cell and stack. Amphlett et al. [39] 
proposed empirical equations representing the current-voltage (I-V) characteristics of a cell at 
different operating temperatures. However, their model is based on empirical equations and is 
not able to represent the dynamics of gases and water inside the fuel cell, or the varying load. 
Pukrushpan et al. [40] proposed a fuel cell stack model that considers the dependence of proton 
conductivity on water concentration and temperature. However, the water concentration of the 
membrane was obtained from the relative humidity (RH) of the membrane that was assumed to 
be the average of the anode and the cathode RH values. The water content in the membrane was 
greater than that of the cathode and the anode sides. Their developed model assumed no 
temperature gradient in the stack, and the behavior of the stack was obtained from the product of 
the number of cells multiplied by the single cell outputs. A one-dimensional (1-D) thermal model 
developed by Khandelwal et al. [41] was used to predict the temperature distribution for a cold 
23 
 
start. However, the model did not provide a transient analysis of the temperature for a varying 
load. The effects of a varying load on the dynamics of temperature were investigated by Shan 
and Choe [42]. However, all of the aforementioned models assumed that the water in a stack 
exists as vapor, or ignored phase changes. McKay et al. [43] developed a simple two-phase 
model for a gas diffusion layer, and used it to represent the effects of liquid water and vapor on 
the performance of the cells. The proposed model did not consider the variation of temperature, 
and its effects on the performance of a stack and the water balance in the membrane.  
 
In this chapter, several developed fuel cell stack models are described for conditions such as 
isothermal, non-isothermal, and two-phase effects.  
 
2.1. Dynamic fuel cell model  
2.1.1. Model setup 
A schematic diagram of the stack model is shown in Figure 2-1. The stack model consists of 
sub-models that include a stack terminal voltage, an anode, a cathode, and a membrane model. 
Pressure, temperature, reactant gas partial pressures, and membrane humidity are the main 
variables that affect stack voltage. 
24 
 
 
Figure 2-1. Schematic diagram of a fuel cell. 
 
The main assumptions made for the fuel cell stack model are as follows: 
1. All gases obey the ideal gas law. 
2. The temperature of the flow inside the cathode flow channel is equal to the stack 
temperature of 80 ?C. 
3. When the relative humidity of the gas exceeds 100%, vapor condenses into liquid form. 
This liquid water does not leave the stack, and either evaporates in the cathode gas if the 
humidity drops below 100%, or accumulates in the cathode. 
4. The flow channel and cathode backing layer are lumped into one volume. 
 
Based on these assumptions, four sub-models were developed as shown in Figure 2-2. The 
stack voltage, cathode flow, anode flow, and membrane model are described in the following 
sections. 
25 
 
 
Figure 2-2. Block diagram for a fuel cell stack at constant temperature. 
 
2.1.2. Governing equations 
a. Stack voltage 
The current-voltage (I-V) characteristic is the difference between the open circuit voltage and 
over-potentials that include the ohmic over-potential in the membrane, the activation over-
potential in the catalyst on the cathode side, and the concentration over-potential. The 
relationship for a single cell may be written as a function of physical parameters such as the 
reactant partial pressure, temperature, current, and membrane water content [39], as shown in 
Figure 2-2. The output terminal characteristic of a stack is assumed to be a multiple of a single 
cell characteristic, and is given as follows:  
26 
 
? ? ? ? ? ? ? ?
c e l ls t a c k
c o n cm e m bo h ma c tc e l l vnv iTpviTviTpvTpEv ?? ???? ,,,,,,, ?   (2- 1) 
where E is the open circuit voltage (V) given by converting chemical energy into electrical 
energy; vact is the activation over-potential (V) necessary for electron transfer as well as 
dissolving and forming chemical bonds of electrons at the catalysts; vohm is the ohmic over-
potential (V) that occurs because of the resistance of the membrane to the transfer of protons and 
the resistance of the electrode and collector plate to the transfer of electrons; and vconc presents a 
concentration over-potential (V) resulting from the changes in concentration of the reactants as 
they are consumed in the reaction. 
 
The open circuit voltage E is expressed using an energy balance between chemical energy 
and electrical energy [39]: 
? ? ? ? ? ??????? ??????????? ?? 22 ln21ln1031.42981085.023.1 33 OHfcfc ppTTE    (2- 2) 
The activation over-potential vact is obtained using the Tafel equation [44]: 
? ?tcaa c t evvv ?????? 110          (2- 3) 
where v0 and va are functions of cell temperature, cathode pressure, and oxygen partial pressure, 
and c1 is equal to 10. 
The ohmic over-potential vohm [45] is calculated from Ohm?s law: 
ohmohm Riv ??            (2- 4) 
where the ohmic resistance Rohm (?) is a function of membrane conductivity ?memb (S m) and 
membrane thickness tmemb (m) as follows:  
memb
membohm tR ??            (2- 5) 
27 
 
The membrane conductivity at different hydration levels of the membrane can be given in the 
following form [44]: 
???
?
???
? ??
?
???
?
? ????
fcm e m b T
130311268e x p30??         (2- 6) 
where ?30 is a function of the membrane water content. The value of the membrane water content 
varies between 0 and 14, which can be equivalently related to relative humidity values of 0 and 
100% [44]. 
The concentration over-potential vconc is given by  
3
m a x2
c
c onc i iciv ???????? ???
          (2- 7) 
where c2 is a fitting constant described in [46], and imax and c3 are 2.2 and 2.0, respectively.  
 
b. Cathode and anode flows 
The dynamics of the air flow on the cathode side is described using mass conservation 
equations in the control volume, separately considering the mass rates of oxygen, nitrogen, and 
water on the cathode side: 
m e m bcavg e nvoutcavincavcaw
outcaNincaNcaN
r e a c t e dOoutcaOincaOcaO
WmWWm
WWm
mWWm
?????
??
???
,,,,,,,
,,,,,
,,,,,,
222
2222
??
?
??
       (2- 8) 
where Wv,ca-memb is the mass flow rate of water across the membrane.  
The mass rates of oxygen reacted and water generated by the chemical reaction are functions 
of the stack current: 
28 
 
F
InMW
F
InMW
st
vg e nv
st
Ore a c te dO
?
???
?
???
2
4
,
, 22          (2- 9) 
where M denotes the molar mass (kg mol-1). 
Similarly, the mass rate on the anode side can be described as  
m e m banvoutanvinanvanw
r e a c te dHoutanHinanHanH WWWm mWWm
????
???
,,,,,,
,,,,,, 2222? ??        (2- 10) 
where the mass rates of hydrogen reacted by the chemical reaction is a function of the stack 
current: 
FInMW stHre a c te dH ???? 222 ,          (2- 11) 
 
c. Membrane 
The membrane water content is the factor that determines proton conductivity. The dynamics 
of the water content are described by two forces: the electro-osmotic driving force exerted by the 
different electrochemical potentials at the anode and cathode, and diffusion caused by the 
gradient of the water concentration at the two boundaries. Considering the water mass that flows 
at the boundaries of the membrane layer, the dynamics of the water concentration in the 
membrane can be improved compared to the membrane model in [40] as follows [42]:  
OHOH
m e m b
m e m bd r y
OHOH
m e m b MCb
M
MC
22
22
, ??
? ??         (2- 12) 
? ? cam e m bd i f f canm e m bd i f fcam e m be l e canm e m be l e cm e m bc e l lOHm e m bw WWWWdt tACdm ???? ??????? ,,,,, 2?   (2- 13) 
29 
 
where C is the mass concentration (kg m-3), M is the mole mass (kg mol-1), b is the parameter 
given in [40], ? is the membrane dry density (kg m-3),  and Acell is the fuel cell area (m2). 
 
The electro-osmotic driving force determines the water mass flows of Welec,memb-an and 
Welec,memb-ca at the boundaries of the membrane layer. In addition, diffusion produces the mass 
flows of Wdiff,memb-an and Wdiff,memb-ca. These relationships are described as follows, as proposed in 
[39]:  
192 104.305.00029.0 ??????? iidn ??        (2- 14) 
FinAMW idc e llOHim e m be le c ????? ,, 2        (2- 15) 
? ?FCCDAMW m e m biwc e l lOHim e m bd i f f ??????
2,
       (2- 16) 
where the diffusion coefficient Dw is calculated using the following empirical equation: 
???
?
???
? ??
?
???
?
? ????
fcw T
DD 130312416e x p?         (2- 17) 
? ?? ?
? ?? ?
?
?
?
???
?
??
??????
??????
?
?
?
?
?
?
5.4,1025.1
5.43,367.1310
32,22110
2,10
6
6
6
6
i
ii
ii
i
D
?
??
??
?
?       (2- 18) 
Boundary water content ?i is a function of water activity ai, which is calculated from the 
water vapor partial pressure as follows:  
? ?
??
???
?
?????
????????
?
i
ii
iii
i
a
aa
aaaa
3,8.16
13,14.114
01,3685.398.17043.0 321
?      (2- 19) 
30 
 
? ?iisat ivi Tp pa , ,?            (2- 20) 
 
2.2. Non-isothermal dynamic fuel cell model  
The model described in Section 2.1 ignores the effects of the temperature gradient on the 
behavior of a real fuel cell. The developed non-isothermal model [42] includes the temperature 
distribution in the through-plane of a cell, its effects on the characteristics of each component in 
the fuel cell, and the dynamic water balance in the membrane.  
 
2.2.1. Model setup 
The proposed fuel cell model was developed on the basis of layers in a single cell stack that 
consist of a membrane-electrode-assembly (MEA), two gas diffusion layers, and two gas flow 
channels sandwiched between coolant channels, as shown in Figure 2-3. The input variables for 
the model are the load current, mass flow rates, temperature, pressure, relative humidity of the 
reactants, and the temperature and flow rate of coolants at the inlets.  
31 
 
 
Figure 2-3. Schematic diagram of a single cell stack. 
 
The main assumptions made in the development of the model are as follows: 
1. The reactants are ideal gases. 
2. No pressure gradient exists between the cathode and the anode (gas is transported only by 
diffusion). 
3. No pressure drop exists along the gas flow channels. 
4. The temperature gradient is linear across the layers in the fuel cell. 
5. Anodic over-potential is negligible. 
6. There is no current density gradient across the cathode catalyst layer. 
7. Latent heat is not considered. 
 
32 
 
Models for the layers in a fuel cell were developed based on these assumptions, and are 
described in following sections. 
 
2.2.2. Governing equations 
a. Stack voltage 
The I-V characteristic is given by the difference between the open circuit voltage E and the 
over-potentials that include the activation over-potential in the catalyst on the cathode side vact; 
the ohmic over-potential in the membrane vohm; and the concentration over-potential vconc. Thus, 
we have 
co n co h ma ctcel l vvvEv ????          (2- 21) 
The equations for the open circuit voltage and the ohmic over-potential are the same as those 
presented in Section 2.1.2. However, the equations for the activation and concentration over-
potentials should consider temperature and therefore need to be modified. 
The activation over-potential vact can be expressed as a function of working temperature, 
oxygen concentration, and current [39]: 
? ? ? ?stOa c t ITcTTv lnln 4321 2 ????????? ????       (2- 22) 
where ? is the constant parametric coefficient, I is the load current (A), and c is the concentration 
(kg m-3). 
The concentration over-potential vconc is given by the following approximation: 
???????? ???? m a x1ln i ibv c o n c          (2- 23) 
33 
 
where b is a parametric coefficient (V) that was derived from the single cell characteristic and is 
assumed to be identical throughout all of the cells in the stack, and i and imax denote the actual 
and the maximum current density (A cm-2), respectively. 
Finally, the stack voltage is the product of the number of cells and the terminal voltage of the 
single cell voltage: 
cellst vnv ??            (2- 24) 
Electrochemical parameters used for the stack model were obtained by curve-fitting method 
based on experimental data [47].  
 
b. Energy balance 
Consider a cell that is assembled with cubical layers in which the thermo-physical properties 
are isotropic and constant. Then, according to the energy conservation equation, the total energy 
changes in a controlled volume are equal to the sum of the energy exchange at the boundaries 
and the internal energy resources. The energy exchanges at boundaries occur due to two factors:  
conduction across the cell, and convection occurring between bipolar plates with the coolant, 
reactants, and water. The thermal behavior can thus be described by the following energy 
conservation equation [42]:  
? ? ? ? ?
S o u r c e sf e rh e a t  t r a n s C o n d u c t i o nf e rh e a t  t r a n sC o n v e t i n  inf l o wM a s s s o uc o n dc o n vi cvinjin
cvii QQQTTCpWdtdTCpm ?????? ???? ?? ?????????? ?  (2- 25) 
The internal energy source is composed of the entropy losses and the chemical energy 
required for protons to overcome the barrier of the over-potentials in both catalyst layers. In 
addition, there are ohmic losses caused by transport of electrons and protons in the cell [48]. 
Thus, 
34 
 
?????? ??????????? m e m bc e l la c tc e l ls o u RAivF sTAiQ 4?
      (2- 26) 
where the entropy change ?s is equal to -163.0 (J mol-1 K-1) [49], vact is given in [39], and Rmemb 
is the membrane resistance ?. We note that the anodic entropy change becomes zero under the 
assumption that the anode is a standard electrode. 
 
A block diagram between two control volumes for the energy balance is shown in Figure 2-4.  
 
Figure 2-4. Energy balance model. 
 
2.3. Non-isothermal and two-phase dynamic fuel cell model  
The proposed model has the following major improvements over models in section 2.2: a) 
the addition of the temperature distribution though the cell and its effects on components of the 
fuel cell; b) inclusion of two-phase effects in the GDL and GFC; and c) a coupled 20-cell stack. 
35 
 
The developed model was used to study two-phase effects, the start-up behavior of a 20-cell 
stack, and transient behavior at a step load.  
 
2.3.1. Two-phase model for GDL   
The phase of water in the gas diffusion layer plays a significant role in the transport of water 
and reactants. The proposed model for the GDL considers mass transport in two-phase flow, and 
the phase transition between liquid water and water vapor in which the diffusion reflects by 
introducing an effective diffusivity that describes the diffusion behavior of vapor and liquid 
water in a capillary [43], [50]. The effective diffusivity <Dm> (m2 s-1) is a function of the 
diffusion coefficient at a single phase Dm (m2 s-1), the porosity of the diffusion layer ?, and the 
liquid water saturation ratio s between the volume of liquid water and the pores (m3). Thus, 
? ?? ?2785.0 111.01 11.0 ksDD mm ???????? ??????? ??         (2- 27) 
where k is number of each domain in the GDL.  
 
A setup of one-half of a cell for the GDL on the cathode side is shown in Figure 2-5. To 
calculate gradients of the species, the GDL was divided into three sub-domains. Points 0 and 4 
are the boundary conditions, and points 1, 2, and 3 are the middle points of the sub-domains. 
36 
 
 
Figure 2-5 Schematic diagram for two-phase GDL.  
 
The water vapor produced Nv,0 and the oxygen consumed NO2,0 are expressed as a function of 
current. The total flux of water vapor at the boundary is given as follows: 
F
IN
F
IN
O
v
??
??
4
;2
0,
0,
2
           (2- 28) 
m e m bvvg d lc a tlv NNN ,0,, ???          (2- 29) 
where Nv,memb is the molar flux of water vapor from the membrane to the cathode catalyst. 
The molar flux is expressed as a function of the effective diffusivity, the gas concentration 
gradient, and the ratio of the species molar flux zca(k): 
? ? ? ?? ? ? ?? ? ? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
g d l
vv
ca
v
v
v
g d l
OO
caO
O
O
ty
kckc
kzkx
kD
kN
ty
kckc
kzkx
kD
kN
?
???
?????? ???
?
?
?
??
?
???
?
?
1
111
;
1
11
22
2
2
2
      (2- 30) 
37 
 
where x is the molar ratio of the species and y is a coefficient that depends on the number of the 
sub-domain. The ratio of species molar flux zca (k) is given by 
? ? ? ?
? ????
???
?
???
?
?
3a n d2f o r,11
1f o r,
2
2 0,
0,
kkN kN
kNN
kz
O
v
O
v
ca         (2- 31) 
The time derivatives of the gas concentrations are a function of the ratio of species molar flux 
[43]: 
? ? ? ? ? ?
? ? ? ? ? ? ? ?kRty kNkNkdtdc
ty
kNkNk
dt
dc
e v a p
gdl
vvO
gdl
OOO
?? ????
?
????
1
1
2
222
       (2- 32) 
where Revap is the evaporation rate determined by the difference between the saturation and vapor 
pressure [43], [50], and is given by 
? ? ? ?? ? ? ?? ?kTR kpkTpkR vsate v a p ? ??? ?         (2- 33) 
where ? is the volumetric condensation coefficient (s-1), p is pressure (Pa), and R is the ideal gas 
constant (J mol-1 K-1). 
 
Using a mass balance, the rate of the liquid water volume at each of the points was obtained 
as follows: 
? ?
? ? ? ?
? ? ? ? ? ?
???
???
?
????????
??????
?
3a n d2f o r1
1f o r
kkWkWkRVM
kkWkRVM
dt
kdV
l i q u i d
l i q u i dl i q u i de v a pG D Lv
l i q u i d
l i q u i de v a pG D Lv
l i q u i d
?
?
?
?
   (2- 34) 
where ? is the density of the liquid water (kg m-3). 
38 
 
The mass flow of the liquid water, Wliquid, was derived from the pressure gradient-driven flow 
in a capillary [43], [50]: 
? ? ? ? ? ?G D Lcl i q u i d rll i q u i dl i q u i d ty kSkSsdSdpKKAkW ? ????????? 1??      (2- 35) 
where K is the absolute permeability (m2), Krl is the relative permeability of liquid water that is 
assumed to be equal to S3, ? is the viscosity of water (kg m-1 s-1), |dpc/dS| is the slope of the 
capillary pressure (Pa), S is the reduced liquid water saturation, and sim is the immobile saturation. 
 
A summary of the parameters for the GDL model is listed in [47]. 
 
2.4. Model for a stack and experimental validation  
Multi-cell stack models were developed to analyze cooling effects on the temperature and 
gas dynamics inside the fuel cell stack, as shown in Figure 2-7.  
 
Figure 2-6. Schematic diagrams of (a) 2-cell stack, and (b) 20-cell stack. 
 
39 
 
An experiment using a 2-cell stack and a 20-cell stack was conducted to validate the 
developed multi-cell stack model [47]. The experimental apparatus used for validation of the 
stack model is shown in Figure 2-7. The test setup consists of a two-cell stack and the BOP, 
which includes an air supply, a humidifier, a hydrogen delivery system, a coolant circuit, 
controls, and an electronic load. The stack was designed such that two cells were separated by a 
thermally conductive plate to minimize the potential influence of the coolants on the working 
temperature. LabVIEW software (ver. 7.1.1., National Instruments Corporation, TX) was used to 
control valves and actuators for flows and coolants, and to simultaneously collect the process 
data.  
 
Figure 2-7. Experimental apparatus. 
 
A two-cell stack model was simulated using the developed model. The I-V characteristics 
calculated for the stack and the individual cells were compared to experimental data collected at 
the installed test station. The operating conditions were as follows. The stoichiometric ratio was 
40 
 
1.2 for the anode and 3.0 for the cathode. The gas pressure and supply gas temperature on both 
sides were 1.0 bar and 333.15 K, respectively. The relative humidity for the anode gas was 0% 
and 100% for the cathode gas. The average temperature of the coolants at the exits of cells 1 and 
2 was maintained at 333.15 K by controlling the coolant temperature. 
The simulated and experimental stack voltage results were in good agreement, but the 
deviation of the two cell voltages obtained from the simulation is not as large as that measured 
experimentally. The reference [47] compared a single-phase and a two-phase model for a 2-cell 
stack and represented that the differences in the two cells could have been caused by non-
uniform characteristics of the individual cells such as reactant distributions, and by the properties 
of the GDL and the membrane.  
 
2.5. Summary  
Several stack models were developed to describe the phenomenon or behavior in the stack. 
An isothermal and single-phase stack model is used to develop the control for the air supply 
system. However, Isothermal model does not represent temperature peak in the catalysts and 
membranes that is usually several degree higher than the average temperature. Therefore, the 
heat generated cannot optimally rejected by control strategies. Single-phase does not represent 
losses of potentials caused by mass transports of reactants and water. Thus, static and dynamic 
characteristics are different than that of two-phase. The model cannot be used for analyzing 
effects of humidity on the cell performances and associated controls designed in this dissertation. 
Therefore 1D PEM fuel cell stack model was proposed, used to design the controls for air, water, 
and thermal management, and validated by experiments.  
41 
 
Chapter 3 Air and Water Management 
 
When a fuel cell operates, oxygen contained in the air is supplied to the fuel cell. Insufficient 
oxygen supply at abrupt change of loads causes oxygen starvation, which results in cell reversal 
in the cathode. Reduced hydrogen in the cathode generates heat on the catalysts and results in 
local hot spots in the membrane electrode assembly (MEA), which could give permanent damage 
to the fuel cells. On the other hand, an excessive oxygen supply increases the parasitic power 
dissipated by a blower, which leads to low efficiency of the overall system. Therefore, an oxygen 
supply should be optimized.   
During the operation of the fuel cell, water is produced in the cathode catalyst. Insufficient 
water in the membrane causes dehydration that decreases proton conductivity and increases 
voltage loss, which leads to a reduction in output power. However, the water produced by 
chemical reactions causes flooding in the cathode, which blocks oxygen transport to the catalysts 
and reduces the catalyst activation areas. As a result, the degradation of components is 
accelerated. Therefore, water management is required to remove excessive water in the cell, and 
at the same time to fully humidify the membrane.  
 
3.1. System configurations 
The air supply system mainly supplies oxygen by air to the fuel cell stack. A typical 
configuration of the air supply system [40] consists of a compressor and a blower to produce 
42 
 
pressurized air, inlet and outlet manifolds, and a humidifier that humidifies the entering air, as 
shown in Figure 3-1.  
 
Figure 3-1. Air supply system with humidifier [51]. 
 
Since the exhaust gas exiting from the fuel cell stack contains water, a gas-to-gas humidifier 
is preferred (which does not need an additional water supply system), as shown in Figure 3-2(a). 
One of the drawbacks of the system is that water flooding may occur in the high current density 
region because all exhaust gas is fully humidified and flows though the humidifier. In addition, 
the humidity of the inlet gases cannot be actively manipulated. Thus, a new configuration is 
proposed that employs an extra three-way bypass valve. This valve diverts the flow of the 
exhaust gas in the humidifier or free air, as shown in Figure 3-2(b). 
 
43 
 
 
Figure 3-2. Air supply system with gas-to-gas humidifier and three-way valves. 
 
3.2. BOP models 
3.2.1. Model for air supplier 
A compressor or blower can be used as an air supplier. A blower is generally preferred 
because of its higher air flow rate compared to that of a compressor. In addition, the parasitic 
power of a blower is significantly less than that of a compressor in the high net power region 
[52]. Different types of blowers can be used for this application. One such blower is the axial 
flow blower, which has blades that force air to move parallel to a shaft with rotating blades  
A centrifugal blower uses the centrifugal force generated by a rotating disk with blades 
mounted at right angles to the disk, as shown in Figure 3-3. It consists of an impeller wheel 
contained within a blower housing. A centrifugal blower is capable of delivering highly 
pressurized air in a very small size with good overall efficiency. 
44 
 
                     
Figure 3-3 Structures of (a) centrifugal blower and (b) regenerative blower [53].  
 
A regenerative blower allows some air passing over a blade of the impeller to slip past it. The 
air is moved forward by another blade. The structure of the blower is relatively simple, 
consisting of an impeller with blades. Regenerative blowers operate with high efficiency and are 
usually relatively compact in size for the large amount of air they move. They produce lower 
amounts of acoustical noise and vibration than many other types of blowers. 
 
3.2.1.1. Modeling a blower 
A blower is operated by a direct current (DC) brushless motor. The dynamic characteristics 
of the blower system can be described by the sum of all moments of inertia of the motor and the 
impeller, and the torque produced by the motor. Hence, the torque produced by the motor ?bl,m (J), 
is a function of stator resistance Rs,bl,m (?), flux linkage ?bl,m (V s rad-1), the number of poles 
nbl,m,pl, and the stator voltage Vbl,m (V):  
45 
 
?
?
?
?
?
? ?
???
?
???
??
???
?
???
? ?
???
?
???
??
???
?
???
? ???
blmbl
plmbl
mbl
mbls
mblplmbl
mblmbl
bla m bbl
mblblbl
mbl
bl
bl
NV
R
n
PW
Jdt
d
???
???
???
,
,,
,
,,
,,,
,,
,
,
222
3
1
     (3- 1) 
where ? is the angular velocity (rad?s-1), J is the rotational inertia (kg?m2), ? is the efficiency, p is 
the pressure (Pa), and ? is the air density (kg?m-3). The air blower flow rate is a function of 
angular velocity and pressure, and the efficiency is a function of flow rate and angular velocity 
[54]:  
? ?? ?? ??
?
??? ??????? ???????????
??
???
4*3*
4*5*32*
109106409.17973.1 109101333.4104415.1581.20 pf o rp pf o rppW blblbl ??
  (3- 2) 
17945.0103087.1105115.9108831.2 428313 ???????????????????????????????????
bl
bl
bl
bl
bl
blbl WWW ????   (3- 3) 
where p* (Pa s2 rad-2) is ??
?
???
?
? ?
2bl ambtube
pp ? . 
The parameters for the blower were derived using characteristic data and specifications 
provided by Phoenix Analysis & Design Technologies (PADT) [53], and include both the flow 
parameter and the overall efficiency versus head parameter. A block diagram of the blower 
model is shown in Figure 3-4. 
 
46 
 
 
Figure 3-4. Block diagram of the blower model. 
 
3.2.2. Model for humidifier 
Different types of humidifiers can be used for this application. One is the bubble-type 
humidifier [55]. A bubble-type humidifier consists of air that is bubbled into a container of water 
and becomes saturated when it leaves the container. Although a very high dew point can be 
achieved at low flow rates, the effectiveness of humidification rapidly decreases at higher flow 
rates. In addition, the humidity cannot be controlled, and a water reservoir is needed.  
The enthalpy wheel is another choice. An enthalpy wheel seeks to absorb moisture from the 
air exiting the fuel cell and transfer it to the relatively dry air entering the fuel cell [56]. As 
shown in Figure 3-5(a), the device employs an electric motor to turn the housing of a desiccant 
capable of absorbing and distributing humidity in the presence of vapor gradients. This device 
has good operation and a compact design. The reactant gases should be preheated, and a motor is 
required to turn the wheel to exchange vapor and heat.   
47 
 
A membrane humidifier can be classified into three types: a plate-type humidifier, a liquid-
to-gas membrane humidifier, and a gas-to-gas humidifier. The plate-type humidifier consists of 
sandwiched membranes that contain flow channels for humid and dry air streams. This type of 
humidifier contains sliding plates that can be used to control the heat and humidity flowing into 
the fuel cell. The sliding plates block one of the flow channels and transfers heat and water from 
one channel to another channel, as shown in Figure 3-5(b). This device has the advantage of no 
moving parts (except the sliding plates), and no associated energy. 
 
 
Figure 3-5. Diagram of (a) an enthalpy wheel [56], and (b) a plate-type humidifier [57]. 
 
A liquid-to-gas humidifier contains a membrane that separates the flow of liquid and gas. 
The liquid water humidifies the dry gas that enters the fuel cell. This type of humidifier is 
capable of humidifying dry gas to near saturation at high flow rates. However, the humidifier 
needs a reservoir and a pump to circulate water on one side of the membrane; therefore, the 
device is heavy. In addition, it is difficult to prevent leakage in this type of humidifier. 
A gas-to-gas humidifier does not have the drawbacks of the liquid-to-gas humidifier. The 
advantages of the gas-to-gas humidifier are no moving parts and no additional energy input. 
48 
 
When the reaction starts in the fuel cell, the generated heat is contained in the exhaust gases 
entering the shell side of the humidifier. Then, the dry air flowing into the fuel cell is heated and 
humidified though the tube side of the humidifier. This gas-to-gas humidifier is considered to be 
the most efficient way to humidify a fuel cell. In addition, this type of humidifier can provide 
heat for the fuel cell stack at subzero conditions.  
 
Figure 3-6. Structure of a gas-to-gas humidifier [58]. 
 
3.2.2.1. Humidifier model  
The principle of a gas-to-gas humidifier is shown in Figure 3-7.  
 
Figure 3-7. Principle of a gas-to-gas humidifier  
49 
 
 
Since a shell-and-tube humidifier is used to transport water from exhaust gases on the shell 
side to dry gases on the tube side, there is no mass change of oxygen or nitrogen through the tube 
and the shell. Therefore, the dynamic equations representing the humidifier are  
? ?
? ? ? ?t r a n swiniwiniwi
iOH
is a t
iw
o u tia i rinia i r
i
ia i ra i ri
WWWV TRTpdtda
WWV TRdtdp
,,,,,
,
,,,,
,
21 ???
???
????
      (3- 4) 
where i denotes a tube or a shell. 
Vapor transfer within the humidifier is a result of concentration gradients within the 
membrane material. Therefore, the amount of vapor transfer exhibited under a specific flow 
condition is dependent on the humidity difference between the tube side and the relatively humid 
shell side. The amount of vapor mass transferred during the operation from the annulus to the 
tube is given by 
Amt CCDW v
m e m
tu b es h e llwtr a n sw ?????????? ???,
  
      (3- 5) 
where C denotes the water mass concentrations in each channel, and t is the thickness of the 
membrane under the proposed single tube representation.  
 
Heat transfer behavior is governed by an energy equation with coefficients representing 
tubular geometry. The heat transfer rate across the membrane is given as 
TAUdtdQ i ????           (3- 6) 
where i denotes a tube or a shell, U is the overall heat transfer coefficient (W m-2 K-1), which is a 
function of the convective heat transfer coefficients of the air, h. We have 
50 
 
? ?
o u t e rs h e l lm e m b
i n n e ro u t e r
i n n e rt u b e AhkL
DDAhAU ???????? 12ln11 ?      (3- 7) 
where h is a function of thermal conductivity k and the Nusselt number Nu, and Nu is an 
empirical function depending on the direction of the flow:  
s h e lls id e )t u b e s id e ,(, ?? iDkNuh iiii        (3- 8) 
 
3.2.3. Model for inlet and outlet manifolds 
Since no heat changes in the manifolds are assumed, the dynamic characteristics of the inlet 
and outlet manifold pressures can be described using the ideal gas law and the mass conservation 
equation: 
? ?outiinii iai WWV TRdtdp ,, ????          (3- 9) 
where i denotes the inlet or outlet manifolds. 
 
3.2.4. Model for a bypass valve 
The bypass valve allows exhaust gas to flow from the stack to the bypass, or to flow into a 
humidifier. When the opening of a bypass valve can be assumed to be linear with a factor k, the 
exhaust gas flow entering the shell side of the humidifier is the product of the exhaust gas flow at 
stack outlet Wst,ex,out (kg s-1) and the opening factor of a bypass valve k:   
o u texstinexs h e l l
b y p a s sexinexs h e l lo u texst WkW WWW
,,,,
,,,,, ?? ??         (3- 10) 
 
51 
 
3.3. Control strategies 
3.3.1. Air  
When a load current is suddenly applied, the oxygen consumption increases. Thus, the 
oxygen partial pressure will drop, which causes a depletion of the oxygen on the cathode side 
and potentially damages the catalyst on the cathode side [7]. Therefore, the air supply system 
must continuously replenish oxygen by supplying air to the cathode, which allows the stack to 
follow the current command as quickly as possible. For better judgment of the dynamic oxygen 
supply, an oxygen excessive ratio is introduced as a new variable that defines the ratio of the 
oxygen supplied and consumed. The air supplied by the blower is controlled to keep the oxygen 
excess ratio ?O2 at the desired level. 
 
A block diagram of an air supply system using a blower is shown in Figure 3-8.  
 
Figure 3-8. Air supply system using a blower. 
 
The control objectives of the air supply system are to maintain the desired oxygen ratio and 
to prevent oxygen starvation that might occur during abrupt changes in the load current. To 
52 
 
design the controls, we first assumed that operating temperature is 343 K and the air supplied to 
the fuel cell is fully humidified. 
 
Figure 3-9 shows two control configurations for the air supply system: a static feed-forward 
controller (sFF), and a feedback controller with a state feedback controller and a static feed-
forward controller (sFB). The current is the reference input for the system.  
 
Figure 3-9. Control configurations of (a) a sFF, and (b) a sFB with a sFF. 
 
The sFF control uses the reference current to calculate the flow rate of the fuels, which 
represents open loop control. An optimal blower motor voltage is calculated based on a given 
reference current, which maintains the desired oxygen excess ratio at 2.0. The set point of 2.0 
53 
 
was determined to avoid oxygen starvation and minimize the parasitic power of the blower 
during operation. This relationship is shown in Figure 3-10. 
 
Figure 3-10. Optimal blower motor voltage for static feed-forward controller. 
 
Even though the sFF shows an excellent dynamic response in rejection of the disturbance, the 
sFF has a steady state error. Conversely, state feedback (sFB) control with an integral control 
employs a closed loop control that removes the error completely. Figure 3-11 shows a block 
diagram of sFB. When the stack current as the disturbance w applies to the fuel cell system, the 
controlled input u is determined by the sFF that is a function of the stack current Ist. In addition, 
the errors of the reference values for the air flow rate and the state variables are amplified by a 
feedback controller, and then added to the blow voltage given by the sFF and the steady state 
error is removed by the integral controller. The controllers regulate the performance variables z2: 
the oxygen excess ratio ?O2. 
54 
 
 
 
Figure 3-11. Block diagram for feedback control with a feed-forward control.  
 
The design of the sFB is based on the state variable equation. Since the fuel cell system is 
nonlinear, the system is linearized to obtain the state equation. The operating point is set as 
follows: a maximal net power, pnetmax = 68 kW; an oxygen excess ratio, ?O2opt = 2; the stack 
current, w0 = Ist = 258 A; and the blower motor voltage, u0 = Vblopt = 118 V.  
 
A general form of the linearized state variable equations can be expressed as 
wDuDxCy
wDuDxCz
wBuBxAx
ywyuy
zwzuz
wu
????
????
????
??????
??????
???????
       (3- 11) 
55 
 
where ? denotes the derivative operator at an operating point, and the variables of the fuel cell 
system can be defined as follows: 
 
? ?omanwSMimblNHO pmmpmmmx ,,,,,,, ,222 ??
   (States) 
 blVu?         (Control variable)  
 stIw?         (Disturbance) 
 ? ?stimbl VPWy ,,?       (Output) 
 ? ?2, OnetPz ??       (Performance variables) 
 
The matrix values of the linearized system, including A, Bu, Bw, Cz, Dzu, Dzw, Cy, Dyu, and Dyw, 
are listed in Appendix A. The units of states and outputs were rescaled for a comparison: mass in 
grams, pressure in bar, rotational speed in kRPM, mass flow rate in grams per second, power in 
kilowatts, voltage in volts, and current in amps. 
 
The linear quadratic method (LQR) presented in Appendix B was used to obtain optimal 
gains for the state feedback controller with an integral controller. The weighting factor QI only 
affects the integrator gain. When QI increases, its influence on the cost function increases, which 
leads to a large value of ?z2. Consequently, the dynamic response of the recovery behavior of the 
oxygen excess ratio increases and reaches steady state with an overshoot. Figure 3-12 shows the 
dynamic response of ?z2 for different weighting factors QI. In contrast, the weighting factor Qz 
hardly improves the transition behavior, which can be seen in Figure 3-13.  
56 
 
19 20 21 22
1 .6
1 .8
2 .0
2 .2
2 .4
 0 . 1
 0 . 0 1
 0 . 0 0 1
 0 . 0 0 0 1
 
 
O
2
 
e
xce
ss 
ra
t
i
o
T i m e ,  se c
 
Figure 3-12. Effect of weighting factor QI on the recovery behavior of oxygen excess. 
9 .9 1 0 .0 1 0 .1 1 0 .2 1 0 .3 1 0 .4 1 0 .5 1 0 .6 1 0 .7
1 .5
1 .6
1 .7
1 .8
1 .9
2 .0
2 .1
2 .2
2 .3
 1
 0 . 1
 0 . 0 1
 0 . 0 0 1
 
 
O
2
 
e
xcess ratio
T i m e ,  se c
 
Figure 3-13. Effect of weighting factor Qz on the oxygen excess ratio. 
 
57 
 
The optimum values of the determined weighting factors are Qz = 1 and QI = 0.01. The gains 
for the PI controller were obtained accordingly, and the second and the seventh state variables 
are zeros. Both variables are related to the mass of the hydrogen and water on the anode side, 
which were assumed to have been proportionally delivered.   
? ?1 6 2 3.3 5 5 8 7.3300 0 1 9.0107 6 8 1.2717.20102 6 3 6.10101 0 6 0.1 355?? ???????
I
pKK  (3- 12) 
 
3.3.2. Air and water management (AWM) 
As a matter of fact, the air supply system is coupled with the water management. Thus, the 
control objectives for a blower with a humidifier are to maintain the desired level of membrane 
water content to prevent membrane dehydration and water flooding in the cathode, in addition to 
the objectives of the blower given in Section 3.3.1. Three assumptions were made in the design 
of the controls: the operating temperature is 343 K, dry hydrogen is supplied to the anode, and 
the exhaust gas temperature is the same as that of the working temperature in the cell. 
A block diagram of the air and water management system, including the gas-to-gas 
humidifier with a state feedback controller, is shown in Figure 3-14. When the load current 
applies to the fuel cell stack, the blower and the bypass valve operates to control the flow rate 
and the relative humidity of the air supplied to the stack, respectively. The controlled inputs such 
as the blower motor voltage Vbl and the bypass valve opening factor kbp are determined by the 
static feed-forward controller that is a function of the load current Iref, and the state feedback 
controller. The controller regulates the performance variables z: the oxygen excess ratio ?O2 and 
the membrane water content ?memb. 
58 
 
B l owe r
Fu el  
cell
s t ack
B y pa s s  
v a l v e
r e fIw ?
F ( I
r e f
)
*u ?
*x
- K
u?
bpku ?2
blVu ?1
H u m i d if ier
O u tl e t 
m a n if o l d
I n l e t 
m a n if o l d
Pl an t
x?
x? ?
Co n trollers
blW
e x h a u s tW
w e tW
a m bW
e x h a u s tbp Wk ?
incaW ,
outcaW ,
?
z
 
Figure 3-14. Block diagram for control of the air flow rate with a humidifier. 
 
The terminal voltage drops due to the presence of liquid water, particularly when the current 
increases, and the effects of concentration over-potential increase. Therefore, the design of the 
controls was approached using two divided regions: a single-phase region and a two-phase 
region. The design of the controls is shown in Figure 3-15. The operating conditions for the 
single- and two-phase regions are dependent on current density ranges of 0 to 0.3 (A cm-2) and 
greater than 0.3 (A cm-2), respectively.  
59 
 
 
Figure 3-15. Control configuration of air and water management system. 
 
Similar to the design of the air flow rate described in Section 3.3.1, the fuel cell system was 
linearized using a Taylor expansion at the operating points, ist = 0.1 A cm-2 for the single-phase 
region and 1.0 A cm-2 for the two-phase region. For the single-phase region, the integrated 
system of the AWM with the fuel cell stack can be written as Eq. (3-11). The variables are 
defined as follows: 
? ?s h e l lws h e l lomanwcawcaimt u b ewt u b ebl appaappapx ,,,,??   (States) 
? ?bpbl kVu?          (Control variable)  
stIw?          (Disturbance) 
? ?s h e l lomcaimt u b ebl pppppy ??     (Output) 
60 
 
? ?membOz ?? 2?        (Performance variables) 
 
From the simulation results at the operating point, 1.0 A cm-2, the water activities in the 
cathode and the shell side of the humidifier are 1.0. Thus, the dynamics of water activities in two 
control volumes can be neglected in the controller design for the two-phase region. The state 
vector becomes:  
? ?s h e l lomanwcaimt u b ewt u b ebl ppappapx ,,??     (States) 
 
Similar to the air supply system, optimization of the feedback gains was performed using the 
LQR method. Since an integrator was used to suppress any steady state errors, a new state 
variable was considered in the cost function: 
blobl WWq ???             (3- 13) 
After several iterations with different weighting factors, the optimal control matrices Kp and 
KI for the single-phase region were found to be  
 
?????? ?????? ????????? 3.14104.51036.814.0643.11028.11052.88.21038.664.2 010573.51097.100101854.71045.9010736.125.16 23333 33333, spK
 (3- 14) 
???????? 0418.2,sIK           (3- 15) 
And the optimal control matrices Kp and KI for the two-phase region were found to be  
?????? ?????? ????????? ? 33333 33343, 10326.610865.231.010252.510167.9234.910876.2284.6 10671.71067.4010152.81085.2010872.448.25tpK  (3- 16) 
???????? 0067.1,tIK           (3- 17) 
 
61 
 
3.4. Simulation and analysis 
An integrated system with all components and the fuel cell are shown in Figure 3-16. The 
whole system has ten state variables. There are two input variables connected to the controller: 
the voltage of the blower motor Vbl, and the bypass opening factor k.  
 
Figure 3-16. Block diagram of the proposed integrated system model.  
 
Simulations were performed to analyze the static and dynamic behavior of the stack along 
with the air and water management system and their associated controllers. The parameters and 
reference data for the models chosen are given in Table 3-1. 
62 
 
n 381 b 11 0 . 5 1 3 9
A fc 0 . 0 2 5 m
2
b 12 0 . 3 2 6
b 2 350
P 0 1 . 0 b ar n d f ( C w a t e r )
T r e f 3 5 3 . 1 5 K D w f ( T ,  C w a t e r )
E r e f 1 . 2 2 9 V
A c a t l , e f f /A c e l l f ( I ,  T ,  P o2 )
T h ick n ess Den s ity Hea t c o n d u ctiv ity
m W  m
-1
K
-1
J  k g
-1
K
-1
C o o lan t Ch an n el 0 . 0 0 1 1400 30
P late 0 . 0 0 1 1400 52
Gas Ch an n el 0 . 0 0 1 1400 52
C ataly s t la y er 0 . 0 0 0 0 6 5 387 0 . 2
Me m b r an e lay er 0 . 0 0 0 1 8 3 1967 0 . 2 1 1100
k g  m
-3
935
935
935
P r o to n  C o n d u ctin g  Mo d el
E lectr o ch em ical R ea tio n  Mo d el
Fu el C ell
Geo m etr ical d ata f o r  lay er s
Sp ec if ic h ea t
770
 
Table 3-1. Parameters of fuel cell model. 
 
3.4.1. Oxygen excess ratio  
Two analyses were performed to assess the performance of the designed controllers. The 
response of the oxygen access ratio at a multi-step current for two different control strategies is 
shown in Figure 3-17. The sFBI controller can reject the disturbance better than sFF, for which 
the recovery time for the oxygen excess ratio is short. The recovery time reached 0.37 sec. In 
addition, the sFBI decreased the oxygen excess ratio drop compared to that of sFF control. The 
sFBI controller performed more effectively and maintained the optimal oxygen excess ratio more 
efficiently than the sFF. 
63 
 
0 20 40 60 80 100 120
0
50
100
150
200
250
300
350
 
 
S
t
a
c
k
 
c
u
r
r
e
n
t
,
 
A
Tim e ,  se c
0 10 20 30 40 50 60 70 80 90 100 110 120
1. 4
1. 5
1. 6
1. 7
1. 8
1. 9
2. 0
2. 1
2. 2
2. 3
 sF F
 sF B I
 
 
O
x
y
g
e
n
 
e
x
c
e
s
s
 
r
a
t
i
o
Tim e ,  se c
38 39 40 41 42 43 44
1. 5
1. 6
1. 7
1. 8
1. 9
2. 0
2. 1
2. 2
2. 3
2. 4
2. 5
 sF F
 S F B I
 
 
 
 
Figure 3-17. Simulated oxygen excess ratio for two different controllers. 
 
64 
 
3.4.2. Membrane water content and cell voltage 
When a constant current of 0.2 A cm-2 was applied for 120 sec, membrane water content 
without humidification kept decreasing as shown in Figure 3-18. This indicates that the amount 
of water generated in a cell was not sufficient to humidify the membrane without the use of 
external humidification. On the other hand, controlled humidification maintained the membrane 
water content at 9.21, which suggests that the membrane was fully humidified at 70 oC. 
Decreasing membrane water content leads ohmic over-potential to increase, which results in 
decreasing cell voltage as shown in Figure 3-18 (b).  
0 20 40 60 80 100 120
6
7
8
9
10
11
 
 
M
e
m
b
r
a
n
e
 
w
a
t
e
r
 
c
o
n
t
e
n
t
Tim e ,  se c
 No humi dif ic a ti on
 C ontroll e d humi dif ic a ti on
(a)
 
65 
 
0 20 40 60 80 100 120
0. 740
0. 745
0. 750
0. 755
0. 760
0. 765
0. 770
 
 
C
e
l
l
 
v
o
l
t
a
g
e
,
 
V
Tim e ,  se c
 No hum if ic a ti on
 C ont r oll e d hum idi f ic a ti on
(b)
 
Figure 3-18. Comparison of (a) membrane water content and (b) cell voltages at 0.2 A cm-2. 
 
At a high current density of 1.0 A cm-2, the fuel cell requires external humidification. 
Therefore, fuel cell systems using uncontrolled and controlled humidification were simulated and 
compared. While uncontrolled humidification keeps the relative humidity of the stack inlet air at 
1.0, controlled humidification regulated the relative humidity of stack inlet air to maintain full 
humidification of the membrane. As a result, Controlled humidification maintained the level of 
the membrane water content at 9.21. On the other hand, membrane water content with 
uncontrolled humidification was higher than the desired level, which indicates that water 
flooding occurred. As time goes on, water flooding becomes severe, which results in decreasing 
cell voltage as shown in Figure 3-19 (b). 
66 
 
0 20 40 60 80 100 120
6
8
10
12
14
16
 
 
M
e
m
b
r
a
n
e
 
w
a
t
e
r
 
c
o
n
t
e
n
t
Tim e ,  se c
 I de a l humidi f ic a ti on
 C ontroll e d humi dif ic a ti on
(a)
 
0 20 40 60 80 100 120
0. 30
0. 35
0. 40
0. 45
0. 50
0. 55
0. 60
 
 
C
e
l
l
 
v
o
l
t
a
g
e
,
 
V
Tim e ,  se c
 I de a l humi f ic a ti on
 C ont r oll e d hum idi f ic a ti on
(b)
 
Figure 3-19. Comparison of (a) membrane water content and (b) cell voltages at 1.0 A cm-2. 
 
67 
 
When the step current profile shown in Figure 3-20 was applied, the uncontrolled 
humidification did not maintain the membrane water content at 9.21. For 60 sec, the membrane 
water content increased until it reached at 15.5. The membrane water content at 1.0 A cm-1 was 
less than that at 0.2 A cm-1 because the chemical reaction rate at the high current density was 
reduced due to severe water flooding. This resulted in less water generation. Unlike uncontrolled 
humidification, controlled humidification regulated the membrane water content at 9.21. 
0 20 40 60 80 100 120 140 160 180
0. 0
0. 2
0. 4
0. 6
0. 8
1. 0
1. 2
 
 
C
u
r
r
e
n
t
 
d
e
n
s
i
t
y
,
 
A
/
c
m
2
Tim e ,  se c
 
68 
 
0 20 40 60 80 100 120 140 160 180
8
9
10
11
12
13
14
15
16
17
18
 
 
M
e
m
b
r
a
n
e
 
w
a
t
e
r
 
c
o
n
t
e
n
t
Tim e ,  se c
 I de a l humidi f ic a ti on
 C ontroll e d humi dif ic a ti on
 
Figure 3-20. Membrane water content change with respect to the step current profile. 
 
3.5. Experimental validation 
We designed two test setups: a humidifier, and an air and water management system. 
Experiments were conducted to validate the model for the gas-to-gas humidifier and to validate 
the effects of the controls.  
 
3.5.1. Model validation for a gas-to-gas humidifier 
3.5.1.1. Experimental setup 
As shown in Figure 3-21, the experimental setup for static and dynamic experiments with a 
humidifier designed by Dunlavy [58] consisted of the humidifier (FC-200-780-10, Perma Pure 
LLC, NJ [59]), an air flow system, a heating and humidification system, sensors, and DAQ 
computer.  
69 
 
 
Figure 3-21. Layout of the test station for humidifier experiments [58]. 
 
Three blowers and associated power supplies used for blower speed control were located on 
the right side of the test station. One blower supplied dry air flow to the tube side of the 
humidifier, and two were used to produce wet air flow entering the shell side in combination 
with a liquid-to-gas humidifier to overcome the back pressure generated in the liquid-to-gas 
humidifier. This approach gave the humidifier a counter-flow configuration as a result of the 
shell-side flow entering from the top left. In-line flow meters were used directly after the blower 
so that the temperature at each flow meter would be equal, thereby ensuring that the same 
volumetric flow rate was read by each flow meter. The dry air flow passed through piping that 
was heated by a heater coil. The wet air flow was heated by heated water passing through the 
liquid-to-gas humidifier. Thermocouples and humidity sensors were located at the inlets and 
outlets of the humidifier where the temperature and humidity directly before entering the device 
could be known, and so that measurement error due to heat loss could be minimized. A humidity 
sensor was not placed at the exhaust outlet because the humidity condition at this point is not 
important. Finally, the thermocouples were connected to the thermocouple reader that was 
connected to the data recording PC.  
70 
 
 
3.5.1.2. Uncertainty analysis 
Since the objective of this experiment is to determine how accurate the humidifier model is, 
it is essential that the uncertainty of the physical measurements also be known with confidence. 
In order to do this, an uncertainty analysis of measurements is presented so that the total 
uncertainty of all components in the system can be known. Typically, the total uncertainty of the 
measurement devices is the root mean square of the uncertainties of all components given by: 
??? ki itotal FF 1 2          (3- 18) 
where F is uncertainty and k is number of components in hypothetical case. In this experiment, 
there are two measurements that are presented as results: the temperature and relative humidity at 
the humidifier outlet. The uncertainty of the humidifier outlet temperature can be estimated by 
considering the accuracies of the thermocouples. According to data sheets of thermocouples, 
each thermocouple has an uncertainty of ? 0.5 ?C for the temperature range used in the 
experiments. Thus the total uncertainty of all four thermocouples used in the experiments is 
given by 
? ? ? ? ? ? ? ? C15.05.05.05.0 o2222, ?????t o t a lTCF       (3- 19) 
whereby the most accurate measurement of the humidifier outlet temperature can be known with 
confidence to be less than or equal to 1 ?C above or below the measurement. 
Similarly, three relative humidity sensors have accuracies of ? 3 %, ? 1.5 %, and ? 2 %. 
Therefore, the total uncertainty of the relative humidity at humidifier outlet can be given by 
? ? ? ? ? ? %9.325.13 222, ????to ta lHSF        (3- 20) 
71 
 
whereby the best measurement of the relative humidity is less than or equal to approximately 3.9 
% above or below the measured value.  
With this in mind, it should be noted that the two-phase humidifier model developed [58] is 
within the uncertainty of the measured values. However the two-phase model for the relative 
humidity is more often within the uncertainty of the measured values than that for the 
temperature. 
 
3.5.1.3. Experimental procedure 
Experiments were conducted to evaluate the active control strategy that regulated the 
humidity of the air flow entering the fuel cell stack though the tube side of the humidifier by 
adjusting the shell side inlet exhaust gas flow rate. This process involved setting flow rates equal 
to each other, and then keeping the tube-side flow rate constant while incrementally decreasing 
the exhaust flow rate. The theory is that as the exhaust gas flow rate is decreased, the 
temperature and the humidity of the tube-side outlet air would decrease as well.  
Since the PEM fuel cell is likely to experience a situation similar to a step increase power 
requirement whenever the vehicle demands extra speed or power, the step input has a practical 
basis. Thus, a test was performed to examine the system?s response to step changes in exhaust 
gas flow rate. In the test, a pulse pattern was used so that flow rates were stepped up and down at 
specific time intervals, thereby repeating the process several times. 
 
3.5.1.4. Experimental results and analysis 
The static and dynamic behavior of the humidifier was used to validate the model. As shown 
in Figure 3-22, the simulated results were initially in relatively good agreement with the 
72 
 
experimental results. When the flow rate exceeded 100 slpm, the error became approximately 10% 
because the exhaust air temperature varied at the different exhaust flow rates.  
0 20 40 60 80 100 120
28
32
36
40
44
48
52
56
 S im ulation
 Expe r im e nt
 
 
T
a
i
r
,
h
u
,
o
u
t
, 
o
C
W
s t, ou t
,  sl pm
 
Figure 3-22. Temperatures for variable exhaust flow rates (dry air flow rate = 140 slpm). 
 
As shown in Figure 3-23, both results for the wet outlet humidity match fairly well. The 
simulated value reached only 99.7% because of the drop in the high exhaust flow rate. When the 
flow rate dropped below 40 slpm, the relative humidity of the wet air decreased, and the exhaust 
flow rate increased from 80 to 100 slpm. 
73 
 
0 20 40 60 80 100 120
50
60
70
80
90
100
110
120
 S im ulation
 Expe r im e nt
 
 
RH
w
e
t
, %
W
e x h a u s t
,  sl pm
 
Figure 3-23. Humidity for variable exhaust flow rates (dry air flow rate = 140 slpm). 
 
To validate the dynamic responses, the dry flow rate was maintained at approximately 140 
slpm while the exhaust flow rate was stepped up and down between 15 and 30 slpm. The goal 
was to allow the system enough time to approach a steady-state value before changing the flow 
rate again. Thus, 15 min (except for the first 5) were allowed for each flow rate, and the total 
length of the run was one hr. Because the exhaust inlet temperature changed as the flow rate was 
changed, a record was kept of all of the data that could be recorded; therefore, when running the 
simulation, the record could be mapped into the inputs of the model and run.  
Figure 3-24 shows the response of the wet outlet temperature with time corresponding to the 
simulation and experiment. The results appear to follow the same trend, although the 
experimental data appears to be slightly delayed after the flow rate increased or decreased, and is 
slightly lower (or higher) in amplitude depending on the time. In contrast, the simulation appears 
74 
 
to have reacted almost simultaneously to the step increase or decrease in flow rate, and reached 
higher amplitudes. The temperature for the simulation at 15 slpm is higher than that of 
experiment.   
0 600 1200 1800 2400 3000 3600
294
296
298
300
302
304
306
308
310
312
314
10
20
30
40
50
60
 
 
T
w
e
t
, 
o
C
Tim e ,  se c
 S im ula ti on
 Ex pe r im e nt
 W
e
x
h
a
u
s
t
,
 
s
l
p
m
 
Figure 3-24. Temperatures for step exhaust flow rate. 
Figure 3-25 shows humidity changes for the step exhaust flow rate. The relative humidity of 
the wet air flow from simulation was lower than that from experiment because of the temperature 
difference between simulation and experiment, as shown in Figure 3-24.   
75 
 
0 600 1200 1800 2400 3000 3600
0. 5
0. 6
0. 7
0. 8
0. 9
1. 0
1. 1
 
 
RH
w
e
t
, %
Tim e ,  se c
 S im ula ti on
 Ex pe r im e nt
 
Figure 3-25. Humidity for step exhaust flow rate. 
 
3.5.2. Experiments for controls of BOP  
3.5.2.1. Experimental setup 
The experimental setup consisted of a two-cell stack and the BOP, which includes an air 
supply and humidification system, a hydrogen delivery system, a thermal circuit, an electronic 
load, sensors, and processing units, as shown in Figure 3-26. 
76 
 
.  
Figure 3-26. Fuel cell test station. 
 
a. Fuel cell and stack 
The cell has an active area of 140 cm2. The membrane and the GDL have thicknesses of 
0.035 mm and 0.4 mm, respectively. The plate for the gas flow channel is 1.5 mm thick and is 
made of graphite. The plates for the coolant channels and the separator are made of graphite with 
thicknesses of 3 mm and 1.5 mm, respectively. The thickness of the endplate and the bus plate 
are made of garolite with thicknesses of 28.5 mm and gold-plated stainless steel with thicknesses 
of 1.5 mm, respectively. 
The stack was constructed with two cells separated by a thermal conductive plate in order to 
minimize the potential influence of the coolant on the working temperature. Except for the 
77 
 
separator, the other components of the stack are the same as those of a typical two-cell stack with 
the coolant channel between the cells, as shown in Figure 2-6 (a).  
 
b. Design of BOP 
Three subsystems were designed to validate the proposed control strategies for the BOP. 
These include an air and water management system, a hydrogen supply system, and a coolant 
system. 
The air and water management system is shown in Figure 3-27. This management system 
was designed to supply humidified air to the fuel cell stack. The air flow was controlled by an 
800 W WindjammerTM 5.7 inch (145 mm) brushless DC blower (AMETEK, PA) capable of 
producing up to approximately 67 ft3 min-1 (> 1800 slpm) of flow at a no-load condition. The 
blower was controlled either electronically or manually with a speed control of 0-10 DC V. A 
power supply with a range of 0-30 V was used for the manual speed control and the power for 
the blower came from an AC outlet.  
The humidifier used in the experiment was a gas-to-gas humidifier (FC-200-780-10, Perma 
Pure LLC, NJ) with an operating range from 50 to 300 slpm and up to 80 oC. The humidifier 
contains 780 individual tubes though which the dry air travels, and around which the exhaust gas 
from the fuel cell stack travels. 
 The conditions of the air at the inlet and outlet of the stack were monitored using LABVIEW 
software. Two mass flow meters (TSI 4226-01-01, TSI Incorporated, MN) were used to monitor 
the flow rate of the dry air and exhaust gas flows entering the humidifier. The relative humidity 
and temperature of the air flow at the fuel cell stack inlet were measured by a 
humidity/temperature transmitter. The humidity of the supplied air was mainly determined by the 
78 
 
flow rate, the relative humidity, and the temperature of the exhaust gas flow entering the 
humidifier. Thus, the conditions of the exhaust gas flow were measured by a humidity sensor and 
thermocouple, and by a flow meter.   
 
Figure 3-27. Structure of air supply and humidification system. 
 
The hydrogen supply system is shown in Figure 3-28. The hydrogen system supplied pure 
hydrogen gas from a high pressure cylinder to the fuel cell, thereby reducing the hydrogen 
pressure to an appropriate level. From the manual valve, the hydrogen flowed through an in-line 
15 micron filter to remove debris from the gas stream to protect downstream components and the 
fuel cell. The mass flow meter sensed flow over a 0-20 slpm range. The high pressure hydrogen 
was reduced to 2.7-5.5 psig by a low pressure regulator. After passing though the regulator, the 
flow into the fuel cell stack was controlled by a supply solenoid valve. A pressure gauge 
indicated the local hydrogen pressure at the inlet of the fuel cell stack. During the operation of 
the fuel cell stack, a purge solenoid valve located at the outlet of the fuel cell stack removed the 
water accumulated on the anode side.  
79 
 
 
Figure 3-28. Structure of hydrogen supply system. 
 
The thermal circuit shown in Figure 3-29 circulated deionized (DI) water to control the 
temperature of the water exiting the fuel cell stack during operation. When DI water leaked from 
the thermal circuit, DI water addition to the water reservoir was accomplished using an 
automatic solenoid valve triggered by a float switch. The water flow rate could be manually 
throttled with a valve located at the pump outlet. Flow indication was provided by a rotameter 
downstream of the throttle valve (HW-07187-04, Cole-Parmer, IL). If the water pump cannot 
supply enough water to cool the temperature of the water exiting the stack while the stack is running, 
the stack temperature will rise and the cooling fans contacting the radiator will run in order to reject 
the heat contained in the water.  
80 
 
 
Figure 3-29. Structure of thermal circuit. 
 
The fuel cell stack was connected to an electronic load module operating at a maximum of 1 
kW. The operating modes of the E-load include constant current, an IV curve, and a load profile 
that can be input by the user.  
 
Two T-type thermocouple probes (TMTSS-062U-6, Omega Engineering, INC., CT) with a 
working range of -200 to 350 oC and an accuracy of ? 5 ?C were used to measure the 
temperatures of the dry air and the exhaust gas flows at the humidifier inlets. A humidity sensor 
and probe were used at the inlet of the shell side, and the humidity sensor and transmitter 
measured the humidity of the dry air at the inlet of the tube side. This sensor was located at the 
inlet of the tube-side because it had the lowest accuracy compared to other humidity sensors, and 
no software was installed to record data. However, the disadvantages of this sensor were not a 
81 
 
problem in the experiments because the humidity at the dry inlet rarely varied by more than 1% 
RH when the temperature and flow rate were set. 
A handheld thermocouple reader used in the experiments (HH147, Omega Engineering, INC., 
CT) recorded the temperatures measured by four thermocouples. This device can monitor the 
temperatures at all four ports simultaneously, Furthermore, the device is capable of recording up 
to 10,000 data points via an RS-232 cable connected to its own recording software installed in a 
computer. 
A computer equipped with National Instruments Data Acquisition (DAQ) hardware and 
LabVIEW software was used to monitor the data sets measured by the flow meters, 
thermocouples, and humidity sensors, and to control the test station. The data acquisition and 
signal conditioning system incorporated analog and digital inputs and outputs. 
 
3.5.2.2. Uncertainty analysis 
In this experiment, there are three measurements that are strongly related with the stack 
voltage as experimental result: the flow rate, temperature, and relative humidity at the humidifier 
outlet. Similar to the uncertainty analysis for the humidifier experiments, two flow meters have 
accuracies of ? 2 %, and ? 2 % from the data sheets. Thus the total uncertainty of two flow 
meters used in the experiments is given by: 
? ? ? ? %8.222 22, ???to ta lFMF         (3- 21) 
whereby the most accurate measurement of the humidifier outlet temperature can be known with 
confidence to be less than or equal to 2.8 % above or below the measurement. 
82 
 
Each thermocouple has an uncertainty of ? 0.5 ?C for the temperature range used in the 
experiments. Thus the total uncertainty of all four thermocouples used in the experiments is 
given by: 
? ? ? ? ? ? C86.05.05.05.0 o222, ????t o t a lTCF        (3- 22) 
whereby the most accurate measurement of the humidifier outlet temperature can be known with 
confidence to be less than or equal to 0.86 ?C above or below the measurement. 
Similarly, two relative humidity sensors have accuracies of ? 3 %, and ? 2 % from the data 
sheets. Therefore, the total uncertainty of the relative humidity at humidifier outlet can be given 
by 
? ? ? ? %6.323 22, ???to ta lHSF         (3- 23) 
whereby the best measurement of the relative humidity is less than or equal to approximately 3.6 
% above or below the measured value.  
 
3.5.2.3. Experimental procedure 
Experiments on the static behavior of the fuel cell stack were conducted to evaluate the 
performance of the active control strategy using a three-way valve that adjusted the exhaust gas 
flow rate entering the shell side of the humidifier. The process included keeping the dry air flow 
rate constant while incrementally decreasing the exhaust flow rate from the stack at 10 and 60 A, 
and keeping the stoichiometric number at 2 while increasing the current density from 0 to 60 A. 
The theory is that as the three-way valve opening factor is increased at low current, the stack 
voltages would increase as well. As the three-way valve opening factor is decreased at high load 
current, the cell voltages increases. 
83 
 
Experiments on the dynamic behavior of the fuel cell stack were conducted to determine the 
response of the fuel cell stack to dynamic inputs of the three-way valve opening factor at 10 A.   
 
3.5.2.4. Experimental results and analysis  
The first experiment involved varying three-way valve opening factor by 25 % incrementally 
at 10 A where the flow rate, relative humidity and temperature of the dry air were maintained at 
8 slpm, 46 % and 57 oC, respectively. The stack voltage was measured every 15 min at different 
opening factors. As the valve opening factor increases, the stack voltage increases because the 
membrane becomes humidified and proton conductivity is increased.  
0 25 50 75 100
1. 0
1. 2
1. 4
1. 6
1. 8
 I  =  10 A
 
 
S
t
a
c
k
 
v
o
l
t
a
g
e
,
 
V
Thr e e - wa y  va lve  ope nin g  f a c tor,  %
 
Figure 3-30. Stack voltages with different three-way valve opening factors at 10 A. 
 
The second experiment was conducted by the same process as the first experiment, but at 60 
A, where the fuel cell operation is safe. The flow rate, relative humidity and temperature of the 
84 
 
dry air were maintained at 25 slpm, 44 % and 56 oC, respectively. It has to be noted that the 
relative humidity of the exhaust gas at stack outlet was almost 1.0. The peak of the stack voltage 
was obtained at 25 % of the valve opening factor. When all the exhaust gas flows to the ambient, 
the stack voltage is a little lower than that of 25 %. On the other hand, the stack voltage 
decreased as the valve opening factor was increased. It indicates that the water flooding was 
occurred and results in degradation of the stack voltage. 
0 25 50 75 100
1. 0
1. 1
1. 2
1. 3
1. 4
1. 5
 I  =  60 A
 
 
S
t
a
c
k
 
v
o
l
t
a
g
e
,
 
V
Thr e e - va lve  ope nin g  f a c tor,  %
 
Figure 3-31. Stack voltages with different three-way valve opening factors at 60 A. 
 
Experimental result is illustrated in Figure 3-32. Since the relative humidity of the exhaust 
gas at stack outlet is less than 100% at low current region, the three-way valves for both 
humidification systems were fully opened for all the exhaust gas to flow into the shell side of the 
humidifier. Therefore there was little difference of the stack voltages between controlled and 
uncontrolled systems at I = 10A and 20A. However, it was observed that the difference of the 
85 
 
stack voltages became greater as the current increased. It indicates that uncontrolled 
humidification supplied more amount of the exhaust flow than the required one, which resulted 
in occurrence of water flooding. 
0 10 20 30 40 50 60
1. 0
1. 2
1. 4
1. 6
1. 8
2. 0
 
 
S
t
a
c
k
 
v
o
l
t
a
g
e
,
 
V
C ur r e nt, A
 C ont r oll e d hum idi f ic a ti on
 Unc ont r oll e d hum idi f ic a ti on
 
Figure 3-32. I-V characteristics of stack between different humidification. 
 
For the experiments on the dynamic behavior of the stack depending upon the three-way 
valve opening factor, the flow rate of the dry air at 10 A was maintained at 8 slpm. The three-
way valve didn?t allow the exhaust gas to flow into the humidifier for 20 min. After then, the 
opening factor of the three-way valve became 100 % and all the exhaust gas flowed into the 
humidifier. Finally, the three-way valve opening factor came back to zero at 30 min. The 
experimental result showed that the stack voltage gradually changed even though the three-way 
valve was suddenly opened or closed as shown in Figure 3-33. In particular, when the valve was 
fully opened, the relative humidity of the stack inlet air slowly increased, which resulted in 
86 
 
increase of the stack voltage as well as the relative humidity of the exhaust gas at stack outlet. 
Consequently, more water was transferred from the exhaust gas to the dry air though the 
humidifier.  
10 15 20 25 30 35 40
1. 40
1. 45
1. 50
1. 55
1. 60
 
 
S
t
a
c
k
 
v
o
l
t
a
g
e
,
 
V
Tim e ,  mi n
 
Figure 3-33. Dynamic behavior of the stack voltage for step valve opening factor at 10 A. 
 
3.6. Summary 
In this chapter, new configuration of AWM using a gas-to-gas humidifier and a bypass valve 
was proposed and the component models for the proposed configuration were developed. While 
the blower regulated the dry air flow rate supplying to the fuel cell, the bypass valve adjusted the 
exhaust gas flow rate entering he humidifier to maintain the membrane water content at desired 
level. The integrated model including an isothermal and two-phase stack model and BOP models 
were simulated. The developed feedback controllers for AWM were shown to have successfully 
maintained the oxygen excess ratio and membrane water content at the desired level. As a result, 
87 
 
the net power of the stack increased due to high chemical reaction rate and low ohmic over-
potential. 
Experiments were performed to characterize the static and dynamic behavior of the 
humidifier installed in the fuel cell station. The results showed that the relative humidity of the 
air flow exiting the tube side of the humidifier could be controlled by the wet air flow rate 
entering the shell side. Furthermore, the temperature of the air flow at the outlet of the tube side 
was increased though the humidifier. After the humidifier experiment, a fuel cell test station was 
modified to verify the performance of the proposed configuration, including the humidifier and 
the bypass valve. It was confirmed that the proposed control strategy was very effective in 
improving the degradation of stack power output caused by membrane dehydration and/or water 
flooding. 
88 
 
Chapter 4 Thermal Management 
 
When a fuel cell generates electricity, heat is generated at the same time. The generated heat 
increases the temperature in the fuel cell. The elevated temperature can improve gas 
transportation, ease the removal of water in the catalyst, and increase the mobility of water vapor 
in the membrane. However, excessive heat could create local hotspots that damage thin layers 
and result in degradation of the components. Conversely, a low temperature decreases efficiency 
of the fuel cell because of low reaction rates and increased concentration over-potential due to 
increased condensation of water. Therefore, a thermal management system should be designed to 
optimally control the temperature. In this section, different configurations of thermal 
management system (TMS) and their associated control strategies are described. 
 
4.1. System configurations 
A typical thermal management system consists of a radiator, a fan, a reservoir, and a coolant 
pump, as shown in Figure 4-1. The system has only one path for coolant flow where heat 
exchange takes place. Thus, there is little freedom to manipulate the heat exchange rates. As a 
result, power consumption of the coolant circuit cannot be optimally controlled for a given heat 
rejection rate, which decreases the overall system efficiency.  
A potential solution to resolve this limitation is to use an additional three-way valve as 
shown in Figure 4-1. The three-way valve allows coolant exiting from the stack outlet to bypass 
to the reservoir, or to flow into the radiator to exchange heat with the ambient air. In addition
89 
 
the reaction time of the valve is faster than that of the coolant pump. Therefore, the coolant 
temperature entering the fuel cell stack can be manipulated more quickly than by using the 
coolant pump, and the coolant pump can supply less coolant than is needed to reject the same 
amount of heat from the stack compared to a typical system.  
  
Figure 4-1. Configuration of thermal management using a bypass valve. 
 
When the installed power increases, the physical size of thermal management system also 
increases. Because of strict spatial limitations in vehicle applications, a second thermal circuit 
can be added for better cooling and packaging effectiveness, as shown in Figure 4-2. The inner 
thermal circuit consists of a bypass valve, heat exchanger, water reservoir and water pump. The 
bypass valve allows the stack outlet water to a reservoir directly, or to flow into the heat 
exchanger. The heat exchanger transfers heat energy contained in water exiting the stack to the 
coolant. The water pump provides the cooled water from the water reservoir to the stack. The 
outlet thermal circuit consists of a heat exchanger, radiator, fan and coolant reservoir and coolant 
pump. The radiator exchanges the heat transferred from the hot water with the ambient and the 
90 
 
fan enhances the heat convection. The coolant pump supplies coolant from coolant reservoir to 
the heat exchanger. 
 
Figure 4-2. Configurations of thermal management with two thermal circuits. 
 
4.2. Models for system components 
4.2.1. Bypass valve 
Similar to the bypass valve used for air and water management, the bypass valve in a thermal 
circuit allows the coolant flow at the stack outlet to bypass or to flow into a radiator. Coolant 
flow entering a radiator to reject heat contained in the coolant to the ambient is the product of 
coolant flow at the stack outlet Wst,c,out (kg s-1) and the opening factor of the bypass valve, k: 
? ? outcstb y p a s sc outcstincr a d
b y p a s scincr a doutcst
WkW
WkW
WWW
,,,
,,,,
,,,,,
1 ???
??
??          (4- 1) 
 
91 
 
4.2.2. Radiator and fan 
A radiator in the thermal circuit was used to maintain the temperature of the reservoir by 
rejecting the excess heat contained in stack outlet coolant to the ambient. Prediction of the heat 
transfer capability is the most challenging issue in modeling a radiator. Kroger [60] proposed 
empirical equations for the heat transfer coefficient hrad (kW m-2 C-1) and pressure drop pr (kPa) 
as functions of the air flow rate though the radiator, Wair (kg s-1): 
1 1 5 7.09 0 4 5.54 4 9 5.1 2 ?????? a i ra i rr a d WWh        (4- 2) 
? ? 325.101396.7532612 ???? a i rr Wp        (4- 3) 
 
If the heat of the coolant is fully transferred to the radiator without loss, the heat capacity of 
the coolant is identical to that of the radiator. Thus, the radiator outlet coolant temperature can be 
expressed as a function of radiator geometry and heat convection caused by the temperature 
difference between the ambient and the radiator?s outgoing air temperature [60]: 
? ?
cc
r a da m bincr a da r e aincr a doutcr a d CpW hTTFrTT ? ?????? ,,,,,, 5.0      (4- 4) 
where a constant (0.5) is the fitting factor (the author of [60] used a 0.5 m2 radiator in the 
derivation of the heat transfer coefficient). Hence, Frarea denotes the frontal area (m2) of the 
radiator and Trad,c,in denotes the radiator inlet coolant temperature (K). Then, the electric power 
for the fan, Pfan (W), can be calculated using the thermal-dynamic relationship between pressure 
drop and air flow rate [48] as follows: 
???
?
???
? ????
??
?????? ?? 111 kk
ra m ba i ra i rf a ne l e cf a n pTCpWP ??
       (4- 5) 
 
92 
 
 
4.2.3. Reservoir 
The reservoir was assumed to be thermally insulated such that heat exchange takes place only 
at the heat exchanger or radiator by convection. Then, a variation of heat in the reservoir can be 
expressed as the sum of the heat that the water or coolant carries and the heat being exchanged 
with the ambient though plumbing. Thus, the variation of the reservoir coolant temperature Tres,c 
(K) is given by [48] 
? ? ? ?a m binir e splplinir e sir e siinir e sir e sir e s TTAhTTCpWdtdTm C p ????????? ,,,,,,,,,    (4- 6) 
where i denotes water or coolant, mCpres is the heat capacity of the fluid contained in the 
reservoir (J), Wres,in is the reservoir inlet flow rate (kg s-1) that equals the flow rate at the stack 
outlet Wst,,out, hpl is the heat transfer coefficient of plumbing (J m-2 K-1) and Apl is surface area of 
plumbing (m2).  
 
The water temperature at the reservoir inlet, Tres,w,in (K), is expressed as a function of bypass 
valve opening factor k, the water temperature at stack outlet Tst,w,out, and the heat exchanger 
outlet The,w,out. Thus, 
? ? o u twheo u twstinwr e s TkTkT ,,,,,, 1 ?????         (4- 7) 
where the water flow rate at the reservoir outlet is equal to the water flow rate supplied to the 
fuel cell stack by the water pump. 
 
4.2.4. Pump 
A coolant pump circulates the coolant though the thermal circuit to remove heat from the fuel 
cell stack. Under the assumptions that no heat exchange occurs between the coolant and pump, 
93 
 
and no heat resistance exists between the heat generated in the stack and the coolant, the flow 
rate of the coolant is expressed by the following relationship with the heat source in the stack: 
TCpQW c stsouincst ??? ,,,
?           (4- 8) 
 
The heat generation rate in the stack Q? sou,st is the product of the number of cells and the heat 
generated in a cell (Eq. 2-26):  
cellsoustsou QnQ ,, ?? ??           (4- 9) 
It is noted that the coolant pump power has the same trend as the coolant flow rate that is 
required to maintain a constant stack temperature during operation. 
 
4.2.5. Heat exchanger 
A heat exchanger is a device that enables heat energy from a gas or liquid to transfer to 
another gas or liquid, and has many applications in chemical plants, refrigerators, and air 
conditioners. Heat exchangers can be classified as a shell-and-tube type, and a plate-type. The 
shell-and-tube type is preferred for industrial applications because of high reliability and wide 
operating ranges, and where a large volume for gases at high temperature and pressure is needed. 
In contrast, the plate-type is preferred for a small heat exchange capacity where high heat 
transfer efficiency is required. A shell-and-tube type heat exchanger consists of a shell 
surrounding a bundle of tubes. Water runs through the tubes and coolant flows over the tubes 
(through the shell) to transfer heat between the two fluids. Conversely, the plate-type heat 
exchanger is uses metal plates to transfer heat between two fluids. Heat transfer in the plate-type 
takes place more effectively than in the shell-and-tube type because fluids spread out over the 
94 
 
plates and are exposed to large surface areas. This exchanger consists of a series of thin 
corrugated plates that are welded or brazed together, depending on applications of the heat 
exchanger. The plates are compressed together in a rigid frame to form an arrangement of 
parallel flow channels with alternating hot and cold fluids. 
 
4.2.5.1. Model for a plate heat exchanger 
To develop a model for the heat exchanger, the following assumptions were made: 
1. The heat exchanger is regarded as a control volume. 
2. No heat convection between the fluids and the ambient takes place. 
3. The plate wall resistance is negligible.  
4. The fluid flow rate is constant in the heat exchanger. 
5. No phase change occurs in the coolant. 
 
The Nusselt number, which represents a correlation for heat transfer coefficient, is a function 
of the thermal conductivity of fluid k and channel equivalent diameter De, and can be expressed 
by the heat transfer factor and the Prandtl number as follows: 
3/1Pr???? he JkDhNu          (4- 10) 
where the heat transfer factor Jh is a function of Ch and y:  
yhh CJ Re??            (4- 11) 
 
The effectiveness of the heat exchanger ? is defined as the actual heat transfer divided by the 
maximum possible heat transfer: 
95 
 
maxQ
Q????            (4- 12) 
The energy balance of the system is 
? ? ? ?o u tcinccinwo u tww TTm C pTTm C pQ ,,,, ???????       (4- 13) 
 
The maximum heat transfer rate is the product of the low heat capacity and the temperature 
difference between the water and coolant: 
? ?incinwc TTm C pQ ,,m a x ????          (4- 14) 
 
The effectiveness is obtained using Eq. (4-13) and (4-14): 
? ?? ?
incinwc
o u tcincc TTm C p TTm C p
,,
,, ?? ????          (4- 15) 
 
The temperature deviations of the water and coolant depend on the heat transfer between the 
water and coolant, and the enthalpy change between the inlet and outlet. Therefore, the 
temperature changes of the water and the coolant can be expressed by 
? ?outwinwwwheww TTCpWQdtdTm C p ,, ?????? ?       (4- 16) 
? ?outcinccchecc TTCpWQdtdTm C p ,, ?????? ?        (4- 17) 
The heat transfer rate is the product of the overall heat transfer coefficient Uoverall, the surface 
area A, and the log mean temperature difference LMTD: 
L M T DAUQ o v e ra llhe ????          (4- 18) 
96 
 
where U is the overall heat transfer coefficient (W m-2 K-1), A is the surface area (m2), and LMTD 
is the log mean temperature difference (K). 
The overall heat transfer coefficient is calculated from 
cw
overall
hh
U 11 1??           (4- 19) 
 
4.3. Control strategies 
A control structure called bang-bang control is a well-known classical on-off control method 
used to keep the temperature of the coolant constant. However, bang-bang control does not 
consider optimization of the energy dissipated at the actuators, and thus the parasitic power in 
bang-bang control is greater than that for the state feedback control that is optimized in terms of 
the dynamic response.  
 
Figure 4-3. Block diagram of the coolant pump. 
 
4.3.1. Temperature controls  
As described in Section 4.1, new control strategies for non-isothermal fuel cell stack should 
be designed for the coolant circuit with a bypass valve shown in Figure 4-1. Variation of the 
97 
 
temperature of the stack can be obtained from the sum of a heat source and the heat exchanged 
with the water as follows: 
? ?incststwws o uststst TTCpWQdtdTCpm ,,??????? ?       (4- 20) 
where Cpst is the specific heat of the stack (J kg-1 K-1), Ww is the water flow rate (kg s-1) as the 
control variable, and Q? sou is the internal energy source (J s-1) and is a function of the load current. 
Since Eq. (4-20) and the reservoir model (Eq. 4-6) are nonlinear, a Taylor expansion was 
used to linearize the equations at an operating point where the reservoir temperature and coolant 
flow rate were set at 64 oC and 0.93 kg/s, respectively, and the stack current and voltage are 140 
A and 198 V, respectively. The state equations and variables are defined as follows: 
xCy wBuBxAx wu?? ???? ?? ???????
         (4- 21) 
? ?resst TTx?         (States) 
? ?kWu c?       (Controlled input)  
stIw?           (Disturbance) 
? ?resst TTy ?         (Output) 
where the matrices of the linearized system (A, Bu, Bw , and C for a single cell and a two-cell 
stack) are listed in Appendix C.  
 
Based on Eq. (4-21), state feedback control with an integral control was designed as shown 
in Figure 4-4.  
98 
 
 
Figure 4-4. Block diagram of a state feedback controller with an integral controller.  
 
The state feedback control handles the parasitic power dissipated in the coolant pump as one 
of its control objectives. Since the parasitic power of the coolant pump is directly proportional to 
the coolant flow rate, the coolant flow rate is included as a variable in the cost function as 
follows: 
?????? ??? resores stost TT TTq?           (4- 22) 
? ?? ? ?????????? 0 dtuRuqQqxQxJ TITzT ????       (4- 23) 
where the superscript o denotes the value at the operating point and QI is the weighting matrix for 
the integrator. 
 
99 
 
When the weighing matrix R is larger than the weighting matrix Q, the role of the coolant 
flow rate in the cost function is increased; subsequently, the gains of the controller are chosen 
such that the parasitic power is minimized. After several iterations with different values of 
weighting factor Q, the optimal control matrices Kp and KI were obtained:  
?????? ?? ???????? ???? 1.00027.0 0027.01.0,1479.30521.0 01.02015.1 Ip KK      (4- 24) 
 
4.3.2. Temperature controls based on a reduced-order observer  
The aforementioned designed controller was based on the assumption that the stack is 
isothermal and that the temperature profile inside the stack is constant. The controller controls 
coolant flows based on the measured temperature of the coolant outlet. Under non-isothermal 
conditions, heat generation is location-dependent and the temperature gradient is formed through 
the stack that cannot be simply rejected from the proposed control. The heat source terms in the 
stack are expressed as a function of the current drawn, and are regarded as disturbances in the 
control loop. As a result, the heat rejected is less than the heat produced. A countermeasure is to 
estimate the temperature rise in layers that are directly related to the magnitude of the current 
load and compensate it as a feed-forward component to the temperature control loop.  
The relationship between the current and the stack temperature was derived using Eqs. (4-20) 
and (2-26), which yield the following transfer functions in which vact represents the activation 
over-potential: 
? ?
? ?
? ?
? ? AhWCpsCpm
R
sI
sT
AhWCpsCpm
vF sT
sI
sT
o
ccstst
m e m bst
o
ccstst
a c t
st
???????
??????
?????
?
2
4
       (4- 25) 
100 
 
 
The source terms for the energy conservation equation yield the heat produced in the stack, 
and are dependent on the current. Since the load current as an input variable is known prior to 
determining the coolant flow rate, the resulting heat can be calculated based on the current. In 
[61], the feed-forward block is based on the relationship between the current and the stack 
temperature, and was introduced to suppress the surge in temperature. Due to the ignored source 
terms of layers and effect of coolant, the estimation in [61] was inaccurate. One of the drawbacks 
of their approach was insufficient rejection of heat by the coolant control. We propose a reduced-
order observer that should continuously estimate the temperature in the catalyst. A block diagram 
for coolant control with a state feedback controller and a reduced-order observer is shown in 
Figure 4-5. 
Fu el  ce l l
S t ack
C oo l an t
pu m p
R ad i at or
&  f an
W a t er
r ese r voi r
cW
re fI
R ed u ced - order
ob server
St a t e  f e e d b a ck 
& i n t e g ra l  
co n t ro l l e r
outcT ,
r e sT
B y pa ss
val ve
cWk ?
? ? cWk ??1
k
cW
?
?
?
?
?
?
o
r e s
o
outc
T
T ,
outcT ,
c a t l
o
c a t l TT
??
 
Figure 4-5. Block diagram for coolant control with a sFB and a reduced-order observer. 
101 
 
The estimated temperature obtained from the observer was used as an input variable for the 
coolant flow control loop. The coolant temperature was easily measured, whereas the 
temperature inside the cell was difficult to measure. 
 
A two-cell stack model consists of 21 layers including 3 coolant channels, 4 bipolar plates, 4 
gas flow channels, 4 GDL, 4 catalyst layers and 2 membranes as shown in Figure 2-6. Using 
thermal resistance and heat transfer equation, a bipolar plate, gas flow channel and GDL in the 
cathode or anode are combined to the cathode (ca) or the anode (can), and two catalyst layers and 
a membrane are combined to MEA as shown in Figure 4-6. 
 
Figure 4-6. Example of the order reduction: cathode. 
 
After a state space form for the temperature of individual layers was obtained using the 
energy conservation equation, the coolant temperature was defined as a measurable variable, and 
the temperatures inside cells were defined as an immeasurable state variable (for which an 
observer was designed). In addition, the load current was regarded as a disturbance, and the input 
variables were the coolant flow rate and the opening variable for the three-way bypass valve. The 
state equation was reformulated for a reduced-order observer, and the temperature of coolant 
channels and other layers including cathode (ca), anode (an) and MEA were defined as variable 
vectors xb and xa, respectively.  
102 
 
? ?? ?
3,2,1,
2,2,2,1,1,1, ,
cccb
anM E AcaanM E Acaa TTTx TTTTTTx ??        (4- 26) 
where 1 and 2 denotes cell number. 
 
The state matrix was partitioned into two parts: 
? ? ?
?
?
??
??
??
?
??
??
??
?
??
??
??
?
??
?
??
?
??
??
??
?
??
?
b
a
bw
aw
bu
au
b
a
bbba
abaa
b
a
x
xy
wBBuBBxxAA AAxx
?
??
??????
01
,
,
,
,
?
?
      (4- 27) 
The state variable, ?x?b, in Eq. (4-27) includes the measurable state ?xa, which was replaced 
by the following equation:  
wBuBxAyAyx awaubabaaa ?????? ????????? ,,??       (4- 28) 
The new equation includes a rule for the reduced-order observer:  
? ?babawauaabwbubabbbb xAwBuByAyLwBuByAxAx ?? ,,,, ?????????? ?????????????????? ??  (4- 29) 
The gain of the observer loop was optimized by applying the linear quadratic Gaussian 
method. However, the last term in Eq. (4-29) includes variable y? that is very sensitive to high 
frequency noise. Therefore, a new state xc was introduced, and (Eq. 4-27) was rewritten as 
? ? ? ? ? ? ? ? wBLBuBLByALAxALAx yLxx awbwaubuaabababbbc bc ????? ??? ???????????????? ??? ,,,,? ???  (4- 30) 
After several iterations with different weighting factors, the reduced-order observer gain L 
is given by  
??
?
?
?
??
?
?
?
???
???
??
?
379.9312.3749.20764.1719.0033.0
81.4773.7297.144086.90784.0425.0
748.19383.379.11018.2322.19174.21
TL
    (4- 31) 
 
103 
 
4.3.3. Temperature controls for two thermal circuits 
Two coolants circuits are preferred for high power applications of more than 100 kW, where 
reduction of the physical size of the thermal management system and reduction of parasitic 
power are required. The objectives of the state feedback controls are to keep the temperature at a 
desired level, and to reduce the parasitic power dissipated in the water and coolant pumps.  
 
A block diagram of TMS for inner and outer thermal circuits with a state feedback controller 
and an integral controller is shown in Figure 4-7. 
F ue l c e ll
sta c k
H e at  
e xcha ng e r
Ra dia tor
&  f a n
C oolant
pum p
Coola nt
r e se r v oir
W a ter  
r e se r v oir
st
I
S t a c k  c urre nt
W a t e r fl o w
Co o l a nt  fl o w
Me a s ure d t e m pe ra t ure s
Co nt ro l l e d i nput s
sF B I
W at e r  
P um p
k
w
W
B y pas s
v al v e
R e f .  
T e m pe r at ur e s
st
T
wr e s
T
,
cr e s
T
,
whe
T
,
che
T
,
c
W
air
W
O ut e r t he rm a l  c i rc ui t
I nne r t he rm a l  c i rc ui t
?
?
 
Figure 4-7. Block diagram for temperature control with a state feedback controller 
 
Since the parasitic power of the pumps is proportional to the water or coolant flow rates, the 
water or coolant flow rates are included as variables in the cost function. The optimization of the 
gains was performed using the LQR method. 
104 
 
 
Integrator was used to suppress potential steady state errors. Thus, the errors of both closed 
loops were defined as a new state variable that was considered in the cost function as follows: 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
che
o
che
whe
o
whe
cres
o
cres
wres
o
wres
st
o
st
TT
TT
TT
TT
TT
q
,,
,,
,,
,,
?
          (4- 32) 
where the superscript o denotes the value at the operating point. 
Then, the rules for the optimal controlled inputs are obtained: 
? ? qKxKqxKu IpT ???????? ???         (4- 33) 
 
When weighting matrix R in the cost function J (Eq. 4-23) is larger than weighting matrix Q, 
the role of the coolant flow rate in the cost function increases. Subsequently, the gains of the 
controller were chosen to minimize the parasitic power of the pump. After several iterations with 
different weighting factors, the optimal control matrices Kp and KI are given by  
?
?
?
?
?
?
?
?
?
?
?
?
??
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
??
??
??
?
1.0003.0000
003.01.0000
00026.0094.0022.0
00062.0034.0071.0
,
164.3027.0000
031.0875.0000
00192.1921.05.0
0092.2372.0729.1
,
I
wp
K
K
      (4- 34) 
 
105 
 
4.4. Simulation and analysis 
An integrated system with the fuel cell stack and thermal circuits is shown in Figure 4-8. The 
system has 5 state variables, 4 input variables connected to the controller: the water flow rate Ww, 
the bypass valve opening factor kbp, the coolant flow rate Wc, and the air flow rate provided to 
the radiator by the fan Wair. 
wW
W a t e r pum p
W a t e r re s e rv o i r
( T
w , re s
)
o u tr e swW ,,
instwW ,,
instwT ,,
Ra di a t o r &  fa n
F ue l  c e l l  s t a c k
( T
st
)
By pa s s
bpk
o u thewW ,,
stI
outstwW ,,
ca t lT
Dis tu r b an c e
C o n tr o lle d  in p u t
Ou tp u t
Fl o w
outr e swT ,,
o u thewT ,,
outradcT ,,
o u tstwT ,,
o u tbpwW ,,
o u tbpwT ,,
Co o l a nt  re s e rv o i r
( T
c , re s
)
H e a t e xc ha ng e r
( T
w , h e
, T
c , h e
)
outstwW ,, o u tstwT ,,
Co o l a nt  pum p
inhecW ,,
inhecT ,,
outhecW ,, o u thecT ,,
o u tr a dcW ,,
o u tr e scW ,, o u tr e scT ,,
airW
cW
 
Figure 4-8. Block diagram of the proposed integrated system model for TMS. 
 
Simulations were performed to analyze the dynamic behavior of the non-isothermal stack 
along with the thermal system and the associated control strategies. Analyses of temperature 
variations, the dynamics of water content in the membranes, and the oxygen excess ratio at the 
load currents are described in the following chapters. The parameters and reference data for the 
106 
 
models chosen are shown in Table 3-1. All models were coded using blocks provided in 
MATLAB? /Simulink?  (R2010b, MathWorks, Natick, MA).  
 
4.4.1. Temperature distribution in two cells 
The temperature distribution through the plane of two cells with controllers is shown in 
Figure 4-9. As the current density increases stepwise from 0.5 to 0.7 A/cm2, the temperatures of 
both cells decrease accordingly because more coolant is supplied to maintain the temperatures of 
the catalyst layers at the set value. In addition, the temperature difference between cells increased 
as the current density increased. 
344
346
348
350
352
354
356
358
0 . 5  A / c m 2
0 . 6  A / c m 2
0 . 7  A / c m 2
No. 1 cell No . 2 c ell
T
e
m
pe
r
a
ture
, 
K
T
c a t l , c a
T
c a t l , a n
T
c , mi d d l e
T
c , r i g h t
T
ca t l ,c a
T
c a t l , a n
T
c, l e f t
 
Figure 4-9. Temperature variation in the two cells depending on current density. 
 
107 
 
4.4.2. Effect of temperature on membrane water content 
The membrane water contents between two cells at a step load current were compared, as 
shown in Figure 4-10. In general, the membrane water content is strongly influenced by the 
relative humidity, which is determined by the saturated vapor pressure depending on the 
temperature and vapor pressures of the cathode and anode sides. Even though the reactants on 
the anode and cathode sides are supplied with full humidification, the relative humidity in the gas 
channels on the anode side of the cells drops due to the effect of temperature on the gas flow 
channels. In addition, we note that the elevated temperature of the stack near a high load current 
leads to high saturated vapor pressure and low relative humidity on both sides of the cell. As a 
result, the membrane water content decreases.  
108 
 
0 300 600 900 1200 1500 1800 2100
0. 40
0. 45
0. 50
0. 55
0. 60
0. 65
0. 70
0. 75
0. 80
0. 85
0. 90
 
 
L
o
a
d
 
c
u
r
r
e
n
t
 
d
e
n
s
i
t
y
,
 
A
/
c
m
2
Tim e ,  se c
0 300 600 900 1200 1500 1800 2100
8 .0
8 .5
9 .0
9 .5
1 0 .0
1 0 .5
1 1 .0
1 1 .5
 
 
Me
mb
ra
n
e
 
w
a
t
e
r 
co
n
t
e
n
t
T i m e ,  se c
 N o . 1  ce l l
 N o . 2  ce l l
 
Figure 4-10. Water content in the membrane. 
 
4.4.3. Effect of temperature on oxygen excess ratio  
Figure 4-11 illustrates the oxygen excess ratio at a constant operating temperature with 
dynamically varying temperature and coolant control. At a constant operating temperature, the 
109 
 
oxygen excess ratio was maintained at 2.0. However, the oxygen excess ratio with variation of 
stack temperature was not kept at 2. The coolant supplied to the stack for reduction of heat 
generated in the stack decreased the temperature of the gas flow channel next to the coolant 
channel, which resulted in a lower cathode pressure than that at 80 oC. Since the air flow rate 
entering the stack depends on the difference in pressure between the cathode and inlet manifold, 
and the blower regulated by the controller developed for non-isothermal stack model supplied 
the same amount of air to the stack, the oxygen excess ratio was higher than 2.0.  
0 300 600 900 1200 1500 1800 2100
1 .8
1 .9
2 .0
2 .1
2 .2
2 .3
2 .4
 
 
O
xyg
e
n
 
e
xce
ss 
ra
t
i
o
T i m e ,  se c
 C onst a nt  st a c k  tem pe r a tu r e
 Va r in g  st a c k  tem pe r a tu r e
 
Figure 4-11. Comparison of oxygen excess ratio at constant and varying temperatures. 
 
4.4.4. Response of temperatures with a step current profile  
When a multi-step current was applied to the stack, the temperature in the stack rapidly 
increased, particularly in the catalyst on the cathode side. The catalyst temperature (the dotted 
red line in Figure 4-12) increased 3 to 7 K higher than the average temperature in the stack, 
110 
 
where the coolant temperature was fully controlled for the reference temperature of 349 K (see 
the dotted blue line in Figure 4-12). We note that the catalyst and membrane layers could be 
overheated and might be damaged. Therefore, a reduced-order observer was used to estimate the 
catalyst temperature, which was set as the reference for the control system. As a result, the 
control with the reduced-order observer maintained the catalyst temperature at 353 K, and was 
capable of quickly responding to a change in the current and rejecting the excess heat in the 
catalyst layer effectively. 
0 300 600 900 1200 1500 1800 2100
346
348
350
352
354
356
358
360
362
 
 
T
e
m
p
e
r
a
t
u
r
e
,
 
K
Tim e ,  se c
 T
c a tl ,  c on tol  on l y
 T
c ool a n t,  c on tol  on l y
 T
c a tl ,  c on tol  wi th  obs e r v e r
 T
c ool a n t,  c on tol  wi th  obs e r v e r
 
Figure 4-12. Comparison of temperatures of the catalyst layer and coolant channel. 
 
111 
 
4.4.5. Response of the accumulated total parasitic power with a step current profile 
Figure 4-13 shows simulation results for the responses of accumulated total parasitic power 
for bang-bang and state feedback controls with a step current profile. In a thermal circuit, the 
blower and the fan consume the power to run. While the fan provides the constant air flow to the 
radiator, the blower supplies the varying coolant according to the load current. Therefore the 
difference of the accumulated total parasitic power between bang-bang and state feedback 
controls depends on the parasitic power of the blower. Since the blower controlled by the state 
feedback controller operates more accurately than bang-bang controller, the total parasitic power 
for the state feedback controls during 2100 sec is 8% less than that for the bang-bang control. 
0 300 600 900 1200 1500 1800 2100
0
50
100
150
200
250
300
350
T
o
t
a
l
 
p
a
r
a
s
i
t
i
c
 
p
o
w
e
r
,
 
W
h
 B a ng - ba ng  c ontrol
 S ta te  f e e dba c k c ontrols
 
 
Tim e ,  se c
 
Figure 4-13. Comparison of accumulated total parasitic power between bang-bang and state 
feedback controller. 
 
112 
 
4.4.6. Responses of controllers and temperatures for the coolant leak  
When a coolant leaks, the coolant pump cannot provide the sufficient coolant to the heat 
exchanger at the given load current. Thus, the bypass valve is gradually opened to decrease the 
temperature of the coolant supplied to the heat exchanger as shown in Figure 4-14 (b). The 
controller regulates the coolant temperature by 240 sec, and then it breaks down as shown in 
Figure 4-14 (c), which results in rapid increase of the temperature inside a cell and permanent 
damage of the cell. 
0 100 200 300 400
0. 0
0. 5
1. 0
1. 5
2. 0
2. 5
3. 0
3. 5
4. 0
4. 5
 
 
C
o
o
l
a
n
t
 
f
l
o
w
 
r
a
t
e
,
 
k
g
 
s
-1
Tim e ,  se c
( a )
 
113 
 
0 100 200 300 400
0. 0
0. 2
0. 4
0. 6
0. 8
1. 0
 
 
B
y
p
a
s
s
 
v
a
l
v
e
 
o
p
e
n
i
n
g
 
f
a
c
t
o
r
Tim e ,  se c
( b)
 
 
0 50 100 150 200 250 300 350 400
332. 0
332. 4
332. 8
333. 2
333. 6
334. 0
334. 4
 
 
T
e
m
p
e
r
a
t
u
r
e
,
 
K
Tim e ,  se c
 T
c a tl , de s i r e d
 T
c a tl
 T
c ool a n t, de s i r e d
 T
c ool a n t
( c)
 
Figure 4-14. (a) Coolant flow rate, (b) bypass valve opening factor, and (c) temperatures of the 
catalyst layer and coolant channel for the coolant leak. 
114 
 
 
4.4.7. Responses of temperatures with FUDS  
Figure 4-15 shows simulation results for the responses of state feedback control using the 
reduced-order observer and a current profile experimentally obtained from a vehicle tested at 
Federal Urban Driving Schedule (FUDS). The peak temperature in the catalyst layer was 6 K 
higher than the working temperature using the control system without the observer, even though 
the coolants were fairly well controlled around the set reference temperature as shown in Figure 
4-15(b). Figure 4-15(c) shows the temperature of the catalyst and coolants with control using the 
observer. The peak temperature of the control system with the observer was similar to that of the 
control system without the observer for the first 200 sec, but was substantially suppressed in the 
following intervals compared to that of the control system without the observer as shown in 
Figure 4-15(b). The excursion duration of the catalyst temperature decreased, and finally the heat 
energy imposed to the thin layers could be reduced. This significantly reduced the heat stress on 
the layers. Correspondingly, the oxygen excess ratio becomes higher than the desired value of 
2.0 shown in Figure 4-15(d). 
115 
 
0 200 400 600 800 1000 1200 1400
1 .9 5
2 .0 0
2 .0 5
2 .1 0
2 .1 5
2 .2 0
 
 
O
xyge
n
 
e
xcess ratio
T i m e ,  se c
0 200 400 600 800 1000 1200 1400
344
346
348
350
352
354
356
358
360
362
 
 
T
e
mp
e
ra
t
u
re
,
 
K
T i m e ,  se c
 T
c a t a l y s t
 T
c o o l a n t
 T
d e s i r e d
0 200 400 600 800 1000 1200 1400
346
348
350
352
354
356
358
360
362
 
T
e
mp
e
ra
t
u
re
,
 
K
 T
ca t a l y st
 T
d e si re d
 T
co o l a n t
 
 
T i m e ,  se c
0 200 400 600 800 1000 1200 1400
120
140
160
180
200
220
240
260
 
C
u
rre
n
t
,
 
A
 I
loa d
 
 
T i m e ,  se c
( a)
( c)
( b )
( d )
 
Figure 4-15. (a) FUDS and a current profile with a base load of 120 A, (b) temperature of the 
catalysts and coolants for control only, (c) temperature of the catalysts and coolants for control 
with the reduced-order observer, and (d) the oxygen excess ratio. 
 
4.5. Summary 
In this chapter, new configuration with two thermal circuits to increase the cooling 
effectiveness was proposed. For the proposed thermal system, a state feedback controller 
optimized using LQR method regulated the stack temperature at the desired level and reduced 
the parasitic power of the system. In fact, the temperature of the catalyst where the chemical 
reactions occur is 3-7 K higher than the stack temperature. To maintain the catalyst temperature 
at 80 ?C, the reduced-order observer was developed to estimate the catalyst temperature that was 
116 
 
used to feed-forward to reset the reference temperatures. The results calculated in the FUDS 
showed that the use of a state feedback controller based on a reduced-order observer improved the 
controllability of the stack temperature, and that the peak of the catalyst temperature could be 
dynamically suppressed during a varying load. 
 
117 
 
Chapter 5 Conclusion 
 
In this study, dynamic models of PEM fuel cell components were developed and integrated 
with the fuel cell stack model. The integrated system model was used to help determine control 
strategies and to aid in the design of optimal controllers based on an observer that estimates 
inaccessible variables.  
For air and water management, a bypass valve was added to control the membrane water 
content by adjusting the exhaust gas flow rate entering the humidifier. The developed feedback 
controllers for AWM were shown to have successfully maintained the oxygen excess ratio and 
membrane water content at the desired level. As a result, the net power of the stack increased due 
to high chemical reaction rate and low ohmic over-potential.  
For thermal management, the proposed configuration reduced the size of the weight of the 
overall system. Moreover, cooling effectiveness was increased using the proposed thermal 
circuits with a state feedback controller based on input from a reduced-order observer. The 
designed reduced-order observer was able to follow the estimate state of the catalyst temperature 
that was used as input to the feed-forward block. The results calculated in the FUDS showed that 
the use of a reduced-order observer improved the controllability of the stack temperature, and 
that the peak of the catalyst temperature can be dynamically suppressed and the duration of the 
above desired catalyst temperature can be reduced during a varying load. Moreover, the 
efficiency of the total system was increased by reducing parasitic power.  
118 
 
Experiments were performed to characterize the static and dynamic behavior of the 
humidifier installed in the fuel cell station. The results show that the relative humidity of the air 
flow exiting the tube side of the humidifier could be controlled by the wet air flow rate entering 
the shell side. Furthermore, the temperature of the air flow at the outlet of the tube side was 
increased though the humidifier, which means that there is no need for an additional heater to 
heat the air flow entering the fuel cell stack. After the humidifier experiment, a fuel cell test 
station was modified to verify the performance of the proposed configuration, including the 
humidifier and the bypass valve. The experiments were performed for constant dry air and 
exhaust gas flow rates entering the humidifier. The stack voltages measured at different valve 
opening factor under the constant dry air flow were compared, and we confirmed that the 
proposed control strategy was very effective in improving the degradation of stack power output 
caused by membrane dehydration and/or water flooding. 
 
In future work, the proposed fuel cell station will be optimized and, in particular, the thermal 
circuit will be modified to improve cooling effectiveness and reduce the parasitic power of the 
coolant pump. Additional work involving the AWM and the thermal management system will 
include development of an integrated control system that accounts for interaction between the 
reactant gas flow, water, and temperature, and the optimization of controllers using advanced 
control algorithms.  
 
 
 
119 
 
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126 
 
Appendix A Matrices for air supply system 
 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
??
??
??
?
22.510000621.40045.4
083.8001.142005.4490
0004.4790622.503.38028.33
00106.07.383973.0968.20598.2
0002.20335.17000
3.158007.275031.46057.37
012.18064.51009.1610
05.240074.83095.10062.12
A
, 
? ?00009 4 7.3000?TuB , 
? ?005244.0000000398.003159.0 ????TwB , 
?????? ?? ?? 00096.130462.10280.1 00001966.0089.1091.294.4zC , 
???????? 0169.0zuD , 
???????? 01055.0 1802.0zwD  
 
127 
 
Appendix B Linear Quadratic Regulator method 
 
A linear quadratic regulator (LQR) is used for the optimization of control gains by summing 
the squares of the errors. The gains become optimal if the cost function of the LQR, J, obtains a 
minimum value. Thus, 
dtuRuqQqzQzJ TITzT? ? ??? 0 22 )( ????        (B-1) 
where Qz and QI  represent weighting matrices amplifying the errors of the control objects, and 
weighting matrix R is used to suppress the effect of the manipulating variable.  
The variable ?z2 includes the disturbance ?w (Eq. (3-11)), and consequently in the cost 
function. If the disturbance term in Eq. (3-11) is set to zero, the variable ?z2 yields a new variable 
with xCz z ?? 2'2 ? that is reflected in cost function J:  
dtuRuqQqxCQCxJ TITzzTzT? ? ??? 0 )( 22 ????       (B-2) 
Consequently, the cost function includes a control input as well as the objects. If the control 
input is substituted by the function of a controller, the cost function will include control objects 
that facilitate a determination of both weighting matrices that are only dependent on the control 
objects. Eq. (B-3) describes the control input as a function of the control objects:  
? ? qKxKqxKu IpT ????? ~~ ???
        (B-3) 
where 
xxx d e s i r e d??? ??~
 and K represents the gain for the proportional and integral controller. 
?xdesired is the reference value of the state variables and is obtained at ?z/ ?w=0.  
128 
 
According to optimal control theory, the variation of the cost function at a fixed end time is 
given as 
dtxwuxfuRuqQqxCQCxJ TTITzzTzT? ? ????? 0 ))),,((( 22 ??????????    (B-4) 
where ? is defined as a co-state vector that can be an arbitrary value set because the variation 
of the cost function vanishes along an optimal trajectory. The last term in Eq. (B-4), 
dtxT???0 ???
, 
can be resolved into three parts. The first and the second terms are zeros because of the initial 
and terminal conditions that are assumed to be fixed:  
?? ?? ?????? 00 )0()( txxxdtx TTTT ????????? ????      (B-5) 
Finally, Eq. (B-5) is resolved into three parts: the state, output, and control input variables. 
Thus,  
dtufRuqfQqxfCQCxJ uTTwTITTxTzzTzT? ? ??????? 0 ))()()( 22 ????????? ?  (B-6) 
where fx= Atxxxdtx TTTT ??????? ??
??
00
)0()( ????????? ???? , fu= Bu, and fw = Bw.  
If the cost function is minimized, each of the three terms in the variation of the cost function 
(Eq. B-6) should be zeros. Thus, 
??? TTzzz AxCQC ??? 22?          (B-7) 
0?? ?? TwI BwQ         
  (B-8) 
0?? ?? TuBuR           (B-9) 
If ?=P?x, then Eqs. B-7, B-8, and B-9 can be expanded using Eq. 3-11, which results in the 
following equations:  
129 
 
01122 ????? ?? PBQPBPBRPBCQCPAPA TwIwTuuzzzT      (B-10) 
Finally, an algebraic Riccati equation (Eq. (B-10)) is obtained. Its solution P can be solved 
by the MATLAB function ?lqr?. As a result, the gain for the PI controller is obtained from Eqs. 
B-8 and B-9 with P: 
qKxKPqBQxPBRu IpTwITu ?????? ?? ~11 ???    
  (B-11) 
130 
 
Appendix C Matrices for thermal management 
 
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10
01
,
0
1034.6
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4 7 5 6.20 0 8 9.0
03 1 7 9.0
,
0 4 7 7.00 3 7 3.0
0 2 4 5.00 2 4 5.0
6
C
B
B
A
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u