MODELING, SYSTEM ANALYSIS AND CONTROL OF A PROTON EXCHANGE MEMBRANE FUEL CELL

Transcription

1 MODELING, SYSTEM ANALYSIS AND CONTROL OF A PROTON EXCHANGE MEMBRANE FUEL CELL by SUDARSHAN KOLAR A thesis submitted to the Graduate School-New Brunswick Rutgers, The State University of New Jersey In partial fulfillment of the requirements For the degree of Master of Science Graduate Program in Electrical and Computer Engineering Written under the direction of Professor Zoran Gajić And approved by New Brunswick, New Jersey October, 2016

2 ABSTRACT OF THE THESIS Modeling, System Analysis and Control of A Proton Exchange Membrane Fuel Cell by Sudarshan Kolar Thesis Director: Professor Zoran Gajić This thesis presents a control technique for a 9 th -order linearized Proton Exchange Membrane fuel cell model. This work starts with giving a brief introduction about the construction and working of PEM fuel cell. Then, various fuel cell subsystems and their corresponding non-linear dynamical equations are presented. These equations are simulated to obtain steady state operating points of the model which is further used in Jacobian linearization. The linearized model consists of nine states as opposed to the eight states of the linearized model available in the literature. A pole placement controller is designed for the linearized model to obtain desired transient performance. This work concludes with inspiring the readers about some future works that can be carried out on this model. ii

3 Acknowledgements I would like to express heartfelt gratitude to Prof. Zoran Gajić. I learnt a lot under his tutelage and more importantly the learning was insightful. I would like to dedicate this thesis to him, because without his supervision the thesis might never have reached it s culmination. My sincerest thanks to my undergraduate advisor Dr. Prasheel Suryawanshi, who offered all his guidance during my pursuit of Master s degree all the way from India. I feel blessed to have been a student of some of the elite faculties of our university, Prof. Zoran Gajic, Prof. Lawrence Rabiner, Prof. Kristin Dana, Prof. Jerrold Tunnell and Prof. Abraham Borno. I learnt a great deal from all your courses and I will miss your class. Thanks to the committee members Prof. Hana Godrich, Prof. Prasheel Suryawanshi and Prof. Zoran Gajić for taking time out of your busy schedule and for your encouragement. A big thank you to my seniors Heojong Yoo and Kliti Kodra and friends Gayatri Powar, Remya Ramakrishnan and Saie Saraf for all the inputs and for being great friends during my journey. Finally, I would like to bow down to my parents and the divine for all the love and support. iii

4 Table of Contents Abstract ii Acknowledgements iii List of Figures vii 1. Introduction Fuel Cells: Principle, Construction and Working Principle of Operation Construction Basic Operation of PEM Fuel Cells V-I Characteristics Literature Review Modeling of Fuel Cells Fuel Cell System State Space Model Fuel Cell Subsystem Compressor Supply Manifold Cathode Return Manifold Anode Membrane Hydration Non Linear Equations iv

5 3. Simulation of the Fuel Cell Model SIMULINK implementation of Subsystems Compressor Supply Manifold Cathode Return Manifold Anode Membrane Hydration PEM Fuel Cell model Steady State Operating Point Determination Control of Linearized Fuel Cell System Jacobian Linearization Linearized Model System Analysis Stability Analysis of Linearized PEM Fuel Cell Model Controllability and Observability test Popov Belevitch Eigenvalue Test Controllability of Linearized PEM Fuel Cell Observability of Linearized PEM Fuel Cell Popov Belevitch Eigenvector Test Controllability of Linearized PEM Fuel Cell Observability of Linearized PEM Fuel Cell Model Reduction Truncated Model Modal Transformation Controller Design Eigenvalue Assignment Controller v

6 7. Conclusions and Future Work Conclusions Future Work References vi

7 List of Figures 1.1. Operating Principle of PEM Fuel Cell V-I Characteristics of PEM Fuel Cell Compressor Block Supply Manifold Cathode Model Return Manifold Anode Model Membrane Hydration Model Compressor Subsystem Inside the Compressor Subsystem τ cm Subsystem τ cp Subsystem Supply Manifold Subsystem Inside the Supply Manifold Subsystem m sm Generating Subsystem P sm Generating Subsystem t cp,out Generating Subsystem Cathode Subsystem Inside the Cathode Subsystem O 2 and N 2 Inlet Flow Rate Generating Subsystem Inlet Water Flow Rate Generating Subsystem O 2 and N 2 Outlet Flow Rate Generating Subsystem Outlet Water Flow Rate Generating Subsystem Return Manifold Subsystem vii

8 3.17. Inside the Return Manifold Subsystem Anode Inside the Anode Subsystem Inlet Flow Rate Generating Subsystem Partial Pressure Generating Subsystem Membrane Hydration Model Inside the Membrane Hydration Model PEM Fuel Cell Steady State Operating Points Pole Placement Control Design viii

9 1 Chapter 1 Introduction Fuel cells were developed around mid 19 th century by Sir William Grove. The principle of operation, however, is believed to be discovered by Friedrich Schonbein [1, 2]. Over the last few decades, fuel cell research is in boom, owning to the environmental impacts of the fossil fuels and their fast depletion. The advantage with fuel cell is obvious. It feeds on oxygen and hydrogen gases and generates clean electrical energy with water as by-product of the reaction. Since fuel cells convert chemical energy directly into electrical energy, it has higher efficiency in comparison to conventional heat engines [3]. The fuel cells were first employed by NASA in their Gemini Program in early 1960s. Fuel cells were also employed in Apollo Program to support guidance and communication [1]. Today, fuel cells are used in many applications from automobiles, power generation, heating to various space programs. Undeniably, fuel cells are the future of renewable energy. 1.1 Fuel Cells: Principle, Construction and Working Principle of Operation Fuel cells are electrochemical devices that convert chemical energy into electricity, without generating carbon-dioxide. The operation principle of fuel cells is exactly opposite to that of water electrolysis. In water electrolysis electric energy is supplied to dissociate water into constituent hydrogen and oxygen. In fuel cells, oxygen and hydrogen are made to react to form water, hence releasing electrical energy during the process.

10 Construction The construction of a fuel cell is very similar to a triode. It consists of an anode, membrane and cathode. Based on the material of the membrane, there are 6 different classifications of fuel cells presented below [1]. 1. Alkaline fuel cells AFCs; 2. Proton Exchange Membrane or Polymer Electrolyte Membrane fuel cells PEM- FCs; 3. Phosphoric Acid fuel cells PAFCs; 4. Molten Carbonate fuel cells MCFCs; 5. Solid Oxide fuel cells SOFCs; 6. Direct Methanol fuel cells DMFCs. Out of the above, following fuel cells are predominantly used in practice: 1. PEM fuel cells. PEM stands for proton exchange membrane. It is also called polymer electrolyte membrane. The membrane is solid, teflon like material and is an excellent conductor of protons and isolates electrons. 2. SOFC. SOFC stands for solid oxide fuel Cells. The membrane is made up of ceramic type metal oxide. These membranes are excellent conductors of negatively charged ions electrons. PEM fuel cells are the most developed and the best understood types of fuel cells and are gaining popularity in the automobile applications. SOFCs are pre-dominantly being researched for distributed electric power generations. Some of the significant features of the PEM fuel cells are 1. Long cell life; 2. Low corrosion;

11 3 3. High power density of around 2000Wh/Kg; 4. Low operating temperatures; 5. Higher energy conversion efficiency of around 50% against 25% of internal combustion engines Basic Operation of PEM Fuel Cells A PEM fuel cell is presented below in Figure 1.1 The chemical reactions taking place Figure 1.1: Operating Principle of PEM Fuel Cell in a PEM fuel cell are represented by 2 H 2 4 H e Anode 4 H e + O 2 2 H 2 O Cathode The membrane, being a electron isolator, forces electrons to flow through the load, generating electric current. Water is the by-product of a fuel cell chemical reaction V-I Characteristics The V-I characteristics of a fuel cell is illustrated in Figure 1.2. The fuel cell characteristics deviates from the ideal one because of the activation, ohmic and concentration losses. One fuel cell on an average produces 0.7 V and has a current density of 0.8 A cm 2. A fuel cell with area of 100 cm 2 hence, can produce around 56 W of power, sufficient

12 4 Figure 1.2: V-I Characteristics of PEM Fuel Cell to power up a single bulb.since fuel cells are very thin, of the order of 1 mm, they are stacked together in series to get higher voltage levels. For example, 100 fuel cells stacked up together can produce approximately 6 kw of power, but will still be only 1 cm thick. 1.2 Literature Review A lot of research has been done on fuel cell modeling. A simple third order linear model has been proposed in [4, 5]. Third order bi-linear model, US-DoE US Department of Energy, has been presented in [6]. Then, a third order non-linear model has been shown in [7]. The models of [5, 6, 4, 7] now ignored membrane humidity and pressure of nitrogen in cathode. The model in [8] incorporates these states in a fifth order nonlinear model. Finally, an extensive work has been carried out in [9, 10, 11] to develop a fuel cell model for an automobile application. This model is comprehensive and gives a deeper insight into the fuel cell sub-system. A number of different control strategies have been proposed for the fuel cell. To start with, static and dynamic feedforward controls are developed in [3], followed by LQR feedback control for the 8 th order linearized PEM fuel cell model. An H control features in [12] to achieve robust voltage tracking. Sliding mode control for PEM fuel cell is proposed in [13, 14, 15]. A more comprehensive listing of various control strategies that can be applied to fuel cell can be found in [12].

13 5 Chapter 2 Modeling of Fuel Cells 2.1 Fuel Cell System This work is inspired from the 9 th order model developed in [3]. The outline of this chapter is as follows: various fuel cell subsystems are presented first followed by their non-linear equations. This chapter concludes with a summary of all the non-linear equations and corresponding constants State Space Model State space approach is used for modeling the fuel cell. In state space model, each state x represents a physical parameter of the system. In electrical circuits for example, current through inductor or voltage across capacitor represents a state. In mechanical system, displacement,velocity or acceleration of the body represents a state. In the fuel cell model, mass and pressure of various gases are considered to be the states. A non-linear state space model is represented as: ẋ = fx, u 2.1 Where, x is the state of the system and u is the control input. Following section discusses all the 9 non-linear state equations with derivation Fuel Cell Subsystem Compressor A static compressor map is used to determine the air flow rate through the compressor. The compressor speed, one of the state variables in the model, is defined using the

14 6 concepts of mechanics. The model for the compressor and corresponding non-linear equations are presented below. The dynamics of the compressor speed ω cp is given by dω cp dt = τ cm τ cp 2.2 Where, τ cp [N-m] is the torque required for driving the compressor; τ cm [N-m] is the compressor motor torque. Further, the torques are given by τ cp = C p ω cp T atm η cp [ psm p atm γ 1 γ 1 ] W cp 2.3 k t τ cm = η cm v cm k v ω cp 2.4 R cm C p is the specific heat capacity of air; γ is ratio of specific heats of air; p sm and p atm are the supply manifold and atmospheric pressures respectively in atm; k t, R cm and k v are motor constants given in Table 2.1.2; η cm is the mechanical efficiency of the motor. The air temperature at the compressor outlet T cp,out is calculated through [ T cp,out = T atm + T γ 1 ] atm psm γ 1 η cp p atm 2.5 η cp is the maximum efficiency of the compressor. Compressor air mass flow rate W cp is given by δ W cp = W cr θ 2.6 Where, W cr is the corrected mass mass flow rate, which takes into account variations in the inlet flow pressure and temperature of the compressor. W cr = φρ a π 4 d2 cu c 2.7 ρ a is the air density [kg/m 3 ]; d c is the compressor diameter [m]; U c is the compressor blade tip speed [m/s]; φ is the normalized compressor flow rate. U c is determined as follows U c = π 60 d cn cr 2.8

15 7 N cr is the corrected compressor speed in rpm given by N cr = N cp / θ and corrected temperature, θ = T cp,in 288. φ is given by the following set of equations ] ψ φ = φ max [1 exp β 1 ψ max Here, dimensionless head parameter ψ is a given as [ pcp,out γ 1 ] γ U 2 ψ = C p T cp,in / c 2 p cp,in 2.11 and, φ max, β and ψ max are polynomial function of the Mach number, M given by M = U c γra T cp,in 2.12 φ max = a 4 M 4 + a 3 M 3 + a 2 M 2 + a 1 M + a β = b 2 M 2 + b 1 M + b ψ max = c 5 M 5 + c 4 M 4 + c 3 M 3 + c 2 M 2 + c 1 M + c The regression coefficients a i, b i and c i are given in Table The following table lists all the constants required for compressor modeling: P arameter V alue Units p cp,in Pa T cp,in K R a J/kg.K ρ a 1.23 kg/m 3 d c m δ 1 θ 298/288 contd...

16 8 P arameter V alue Units a a a a a b b b c c c c c c J cp kg.m 2 k v V/rad/sec k t N-m/Amp R cm 0.82 Ω η cm 98% Table 2.1: Compressor Constants Following figure depicts the Compressor block diagram with all the inputs, outputs and state. Compressor speed, ω cp, is one of the nine state variables of the fuel cell system and is designated x 4

17 9 Figure 2.1: Compressor Block Supply Manifold The supply manifold is fed by the compressor which in turn feeds air into the cathode. Usually, the air in the supply manifold is not sufficiently humidified and hence needs humidification before being fed into the cathode. The supply manifold pressure and mass are the two state variables of this sub-system. The mass conservation principle is used to determine the mass flow rate through supply manifold and ideal gas equation is used to determine the manifold filling dynamics. The dynamics of the supply manifold are given by dm sm dt dp sm dt = W cp W sm,out 2.16 = γr a V sm W cp T cp,out W sm,out T sm 2.17 From the supply manifold perspective, W cp is the inlet mass flow rate from the compressor and W sm,out is the outlet mass flow rate. Further, V sm is the supply manifold volume and T sm is the supply manifold air temperature. The outlet mass flow rate is calculated using a linearized nozzle flow equation W sm,out = k sm,out p sm p ca 2.18 k sm,out is the supply manifold outlet flow constant and p ca is the cathode pressure. P arameter V alue Units V sm 0.02 m 3 k sm,out kg/s.pa Table 2.2: Supply Manifold Constants

18 10 The two states associated with supply manifold, the pressure, p sm and mass of gas m sm are designated as states x 5 and x 6 respectively. Figure 2.2: Supply Manifold Cathode This model computes the mass flow rates of various gases in the cathode. The input for the cathode is humidified air from the compressor. Ideal gas equations and massconservation principles are used to obtain differential equations for the states: mass of oxygen, nitrogen, water vapor inside the cathode. The three state dynamics corresponding to the cathode subsystem are dm O2,ca dt dm N2,ca = W O2,ca,in W O2,ca,out W O2,reacted 2.19 = W N2,ca,in W O2,reacted 2.20 dt dm w,ca = W v,ca,in W v,ca,out + W v,ca,gen + W v,membr 2.21 dt W O2,ca,in is the oxygen inlet mass flow rate; W O2,ca,out is the oxygen outlet mass flow rate; W O2,reacted is the rate of oxygen reacted; W v,membr is the water flow rate across the fuel cell membrane. The same notation has been extended for N 2 and H 2 O ca. The partial pressures of oxygen, nitrogen, and the water vapor inside the cathode are calculated using ideal gas law Oxygen gas partial pressure: p O2,ca = m O 2,caR O2 T st V ca 2.22 Nitrogen gas partial pressure: p N2,ca = m N 2,caR N2 T st V ca 2.23 Vapor partial pressure: p v,ca = m v,car v T st V ca 2.24

19 11 Here, T st is the stack temperature assumed to be constant at 353 K; V ca is the cathode volume; R O2, R N2 and R v are gas constants of oxygen, nitrogen and water vapor respectively. The partial pressure of dry air is p a,ca = p O2,ca + p N2,ca 2.25 Total cathode pressure is given by p ca = p a,ca + p v,ca 2.26 The oxygen mole fraction is defined as and, relative humidity is given by y O2,ca = p O 2,ca p a,ca 2.27 φ ca = p v,ca p sat T st 2.28 where, p sat T st is the vapor saturation pressure as a function of the stack temperature T st. Inlet gas vapor partial pressure can be determined as p v,ca,in = φ ca,in p sat T ca,in 2.29 Partial pressure of inlet dry air can be obtained p a,ca,in = p ca,in p v,ca,in 2.30 where, p ca,in is the total inlet pressure at cathode The humidity ratio is The, air molar mass is given by ω ca,in = M v M a,ca,in p v,ca,in p a,ca,in 2.31 M a,ca,in = y O2,ca,inM O2 + 1 y O2,ca,inM N2 2.32

20 12 where, M O2 and M N2 are the molar masses of oxygen and nitrogen respectively. y O2,ca,in is assumed a constant = With all the above data, we can now calculate various inlet flows as follows Mass flow rate of dry air, W a,ca,in = Mass flow rate of vapor entering cathode, ω ca,in W ca,in 2.33 W v,ca,in = W ca,in W a,ca,in 2.34 Mass flow rate of oxygen, W O2,ca,in = x O2,ca,inW a,ca,in 2.35 Mass flow rate of nitrogen, W N2,ca,in = 1 x O2,ca,inW a,ca,in 2.36 x O2,ca,in is the oxygen mass fraction defined by x O2,ca,in = y O2,ca,inM O2 y O2,ca,inM O2 + 1 y O2,ca,in M N The total mass flow rate at the cathode exit is given by a linearized nozzle equation W ca,out = k ca,out p ca p rm 2.38 where, P ca is the cathode pressure, p rm is the return manifold pressure, and k ca,out is the orifice constant. From the knowledge of W ca,out we can determine outlet flow rates of oxygen, nitrogen and vapor following the exact same steps as was done in equations

21 to 2.37 M a,ca = y O2,caM O2 + 1 y O2,caM N ω ca,out = W a,ca,out = M v p v,ca M a,ca p a,ca W ca,out 1 + ω ca,out 2.41 W v,ca,out = W ca,out W a,ca,out 2.42 x O2,ca = y O2,caM O2 y O2 M O2 + 1 y O2,caM N W O2,ca,out = x O2,caW a,ca,out 2.44 W N2,ca,out = 1 x O2,caW a,ca,out 2.45 Using the principles of electrochemistry mass flow rates of oxygen reacted and vapor generated can be calculated as Here, I st is the stack current. W O2,reacted = M O2 ni st 4F W v,ca,gen = M v ni st 2F Cathode inlet flow W ca,in is the composition of the dry air from the compressor and water vapor from the humidifier. W ca,in and cathode inlet pressure p ca,in are calculated using a static humidifier model as follows W v,inj is the rate of vapor injected and is given by W ca,in = W sm,out + W v,inj 2.48 p ca,in = p a,cl + φ des p sat T cl 2.49 W v,inj = M v M a φ des p sat T cl p a,cl W a,cl W v,cl 2.50 M v and M a are the molar mass of vapor and dry air respectively, φ des is the desired inlet humidity, p sat T cl is the saturation pressure at T cl = 353 K. The dry air mass flow rate W a,cl and vapor mass flow rate W v,cl is computed as W a,cl = ω cl W sm,out 2.51 W v,cl = W sm,out W a,cl 2.52

22 14 Where ω cl is the humidity ratio given by The dry air pressure p a,cl is given as p sm, the supply manifold pressure was defined in 2.17 ω cl = M v M a p v,cl p a,cl 2.53 p v,cl = φ cl p sat T cl 2.54 p a,cl = p sm p v,cl 2.55 P arameter V alue Units V ca 0.01 m 3 k ca,out kg/s.pa y O2,ca,in 0.21 T ca,in 353 K Table 2.3: Cathode Constants The cathode states, x 1,x 3 and x 8 correspond to mass of oxygen, nitrogen and water vapor respectively. Figure 2.3: Cathode Model Return Manifold The return manifold releases un-reacted or partially reacted gases to the atmosphere. The return manifold pressure is the only state variable of this model. Ideal gas law at isothermic conditions is used to determine filling dynamics as follows dp rm dt = R at rm V rm W ca,out W rm,out 2.56

23 15 Here, it is assumed that changes in air temperature inside return manifold are negligibly small. V rm is the return manifold volume and T rm is the temperature of the gas inside the manifold. W ca,out is cathode outlet flow, as discussed in equation The outlet mass flow of return manifold W rm,out is determined using nozzle equations. W rm,out = C D,rmA T,rm p rm RTrm patm p rm 1 { [ γ 2γ 1 γ 1 patm p rm γ 1 ]} 1 2 γ 2.57 Where, A T,rm is the throttle opening area in m 2 ; C D,rm is the discharge coefficient of the nozzle; R is the universal gas constant. P arameter V alue Units V rm m 3 T rm 353 K C D,rm A T,rm m 2 Table 2.4: Return Manifold Constants The return manifold with it s state x 9 = return manifold pressure is shown below. Figure 2.4: Return Manifold Anode Anode is fed with hydrogen from a tank. The states for this model are the hydrogen mass inside anode m H2 and water vapor mass m w,an. The dynamic equations are dm H2,an = W H2,an,in W H2,reacted 2.58 dt dm w,an = W v,an,in W v,an,out W v,membr 2.59 dt

24 16 where, W H2,an,in is the inlet hydrogen mass flow rate; W H2,reacted is the rate of hydrogen reacted inside anode; W v,membr is mass flow rate of water transfer across the membrane. In this work, it is assumed that all outlet flow rates from anode are zero. Similar principles discussed for cathode can be applied to anode to determine various supporting values and parameters. The partial pressures are, Hydrogen Gas Partial Pressure: Water Vapor Partial Pressure: Total Anode Pressure: p H2,an = m H 2,anR H2 T st V an 2.60 p v,an = m v,anr v T st V an 2.61 p an = p H2,an + p v,an 2.62 Relative humidity of the gas inside anode is φ an = p v,an p sat T st 2.63 p sat T st is the saturation pressure as a function of stack temperature. by Inlet vapor pressure p v,an,in and inlet hydrogen partial pressure p H2,an,in are given Now, we find the anode humidity ratio as p v,an,in = φ an,in p sat T an,in 2.64 p H2,an,in = p an,in p v,an,in 2.65 ω an,in = M v M H2 p v,an,in p a,an,in 2.66 Here, M v is the molar mass of vapor and M H2 is the molar mass of the hydrogen gas. Finally, with all the above data, we can find various inlet flow rates as W H2,an,in = ω an,in W an,in 2.67 W v,an,in = W an,in W H2,an,in 2.68

25 17 Where, the anode inlet flow W an,in is given by W an,in = K 1 K 2 p sm p an 2.69 W an,in is assumed to be controlled by a simple proportional control that minimizes the pressure difference across the membrane. Since cathode pressure cannot be measured directly, it s approximate value equal to K 2 p sm is used in the equation. Here, K 2 takes into account the pressure drop between the supply manifold and cathode and K 1 is the gain of the proportional controller. The rate of hydrogen reacted or consumed during the electrochemical reaction is given by W H2,reacted = M H2 ni st 2F 2.70 P arameter V alue Units V an kg/m 3 T an,in 353 K φ an,in 1 Table 2.5: Anode Constants The anode block diagram and its corresponding designated states are presented below Figure 2.5: Anode Model Membrane Hydration This model captures the dynamics associated with water mass flow rate across the membrane. The mass flow rate obtained in this model is used in both the cathode and anode model.

26 18 The water flow across the membrane is given by i N v,membr = n d F D c v,ca c v,an w 2.71 t m Where, t m is the membrane thickness [cm]; n d is the electro-osmotic drag coefficient; D w is diffusion coefficient [cm 2 /sec]; c v is the water concentration [mol/cm 3 ]. Each of the above constants are presented below. Water concentration at the membrane surface is given by c v,an = ρ m,dry M m,dry λ an 2.72 c v,ca = ρ m,dry M m,dry λ ca 2.73 Where, ρ m,dry is the membrane dry density in kg/cm 3 and M m,dry is the membrane dry equivalent weight in kg/mol. The water content in the membrane, λ i is defined as a i 39.85a 2 i λ i = a3 i, 0 < a i < a i 1, 1 < a i 3. Where, a i is the water activity and the subscript i denotes either anode an, cathode ca or membrane m. These activities are defined below a i = p v,i p sat,i 2.75 a m = a an + a ca Electro-osmotic drag coefficient can be determined from the membrane water content λ m as n d = λ 2 m λ m The diffusion coefficient is given by 1 D w = D λ exp T fc 2.78 Where D λ is assumed to be a constant equal to and T fc is the fuel cell temperature, assumed to be equal to the stack temperature. N v,membr [mol/sec.cm 2 ]

27 19 gives the water flow rate per unit area in one fuel cell. The total mass flow rate across the entire fuel cell stack is given by W v,membr = N v,membr M v A fc 10 4 n 2.79 M v is the molar mass of vapor, A fc is the fuel cell area in cm 2 and n is the number of fuel cells in the stack. P arameter V alue Units ρ m,dry kg/cm 3 M m,dry 1.1 kg/mol t m m A fc m 2 D λ λ m 14 Table 2.6: Membrane Hydration Constants The membrane hydration model generates constants for the rest of the subsystem and hence has no states associated with it. The following block diagram shows the input and output of the membrane hydration model. Figure 2.6: Membrane Hydration Model

28 Non Linear Equations In this section, all the nine non-linear equations of the PEM fuel cell are presented. These equations are derived from the discussion of various subsystems, presented previously. They are represented using state space form with x 1 = Mass of oxygen in cathode x 2 = Mass of hydrogen in anode x 3 = Mass of nitrogen in cathode x 4 = Compressor angular speed x 5 = Supply manifold pressure x 6 = Mass of gas in supply manifold x 7 = Mass of water in anode x 8 = Mass of water in cathode x 9 = Return manifold pressure All masses are expressed in grams, pressures are in bar and the compressor speed is in rad/s.

29 21 x1 = MO2 y O2,ca,in MO2 y O2,ca,in MN2 y O2,ca,in 1 ksm,out x5 t st R O2 x 1 +R N2 x 3 +Rv x 8 Mv p sat T atm φ atm x 5 Ma P atm x 5 p sat T atm φ atm x +1 5 P atm + Ma Mv ksm,out psattcl φ des x5 p sat T atm φ atm x 5 P atm x5 t st R O2 x 1 +R N2 x 3 +Rv x 8 Mv p sat T atm φ atm x 5 Ma P atm x 5 p sat T atm φ atm x +1 5 P atm Mv psattatm psattca,in x5 2 Patm psattcl MO 2 yo 2,ca,in MN 2 yo 2,ca,in 1 x5+psattcl φ des p sat T atm φ atm x 5 p sat T atm p sat T ca,in x 5 P atm 2 P atm p sat T cl + 1 M i O2 st n 4 F + MO2 R O2 k ca,out tst x1 Mv Rv tst x8 RO2 t st x 1 + R N t 2 st x 3 M N2 R O2 t st x 1 RO2 t st x 1 + R N t 2 st x 3 x9 tst R O 2 x1+rn 2 x3+rv x8 1 M O2 R O2 t st x 1 RO2 t st x 1 + R N t 2 st x 3 1 R O 2 tst x1 + R N 2 tst x3 MN2 RO 2 tst x1 RO2 t st x 1 + R N t 2 st x MO 2 RO 2 tst x1 RO2 t st x 1 + R N t 2 st x

30 22 x2 = K1 K2 x5 tst R H 2 x2+rv x7 Van Mv psattan,in φan,in MH 2 pan,in + 1 M i H2 st n 2 F x3 = MO2 y O2,ca,in MO2 y O2,ca,in MN2 y O2,ca,in 1 1 ksm,out x5 t st R O2 x 1 +R N2 x 3 +Rv x 8 Mv p sat T atm φ atm x 5 Ma P atm x 5 p sat T atm φ atm x +1 5 P atm + Ma Mv ksm,out psattcl φ des x5 p sat T atm φ atm x 5 P atm x5 t st R O2 x 1 +R N2 x 3 +Rv x 8 Mv p sat T atm φ atm x 5 Ma P atm x 5 p sat T atm φ atm x +1 5 P atm Mv psattatm psattca,in x5 2 Patm psattcl MO 2 yo 2,ca,in MN 2 yo 2,ca,in 1 x5+psattcl φ des p sat T atm φ atm x 5 p sat T atm p sat T ca,in x 5 P atm 2 P atm p sat T cl x9 t st RO2 x 1 + RN2 x 3 + Rv x8 + 1 kca,out MO 2 RO 2 tst x1 RO2 t st x 1 RO2 t st x 1 + R N t 2 st x 3 + R N t 2 st x 3 M N2 M N2 R O2 t st x 1 RO2 t st x 1 + R N t 2 st x 3 Mv Rv tst x8 R O2 t st x 1 RO2 t st x 1 + R N t 2 st x M O2 R O2 t st x 1 RO2 t st x 1 + R N t 2 st x 3 M O2 R O2 t st x 1 RO2 t st x 1 + R N t 2 st x

31 23 x4 = ηcm kt u kv x4 e ηcp Jcp + π Cp dc 3 ρa δ Tatm x 5 Patm 8 ηcm θjcp γ 1 8 cp T atm θ x5 dc 2 x 2 c 4 c dc 3 x 3 4 c 8 θ 3 2 Ra γ T atm dc 5 x θ 5 2 Ra γ T atm γ 1 P atm γ 1 γ 1 c 1 dc x c 2 dc2 x 2 4 θ Ra γ T atm 4 Ra γ T atm θ + c 4 dc 4 x Ra 2 γ 2 T atm 2 θ 2 a0 + a3 dc 3 x4 3 8 θ 3 2 Ra γ Tatm a1 dc x4 2 θ Ra γ Tatm + a 2 dc 2 x4 2 4 Ra γ Tatm θ + a4 dc 4 x Ra 2 γ 2 Tatm 2 θ 2 1 b0+ b 1 dc x b 2 dc2 x 2 4 θ Ra γ T atm 4 Ra γ T atm θ

32 24 x5 = R a γ Vsm ksm,out Vsm x5 x5 tst R O 2 x1+rn 2 x3+rv x8 Ra x6 R a γ Vsm e π dc 3 ρa δ x4 8 θ 8 cp T atm θ x5 dc 2 x 2 c 4 c dc 3 x 3 4 c 8 θ 3 2 Ra γ T atm dc 5 x θ 5 2 Ra γ T atm P atm γ 1 γ 1 c 1 dc x c 2 dc2 x 2 4 θ Ra γ T atm 4 Ra γ T atm θ + c 4 dc 4 x Ra 2 γ 2 T atm 2 θ 2 1 b0+ b 1 dc x b 2 dc2 x 2 4 θ Ra γ T atm 4 Ra γ T atm θ Tatm + 5 Tatm x 5 Patm 4 γ 1 γ 1 a0 + a3 dc 3 x4 3 8 θ 3 2 Ra γ Tatm a1 dc x4 2 θ Ra γ Tatm + a 2 dc 2 x4 2 4 Ra γ Tatm θ + a4 dc 4 x Ra 2 γ 2 Tatm 2 θ

33 25 x6 = ksm,out x5 t st RO2 x 1 + RN2 x 3 + Rv x8 π d c 3 ρa δ x4 8 θ e 8 cp T atm θ x5 dc 2 x 2 c 4 c dc 3 x 3 4 c 8 θ 3 2 Ra γ T atm dc 5 x θ 5 2 Ra γ T atm P atm γ 1 γ 1 c 1 dc x c 2 dc2 x 2 4 θ Ra γ T atm 4 Ra γ T atm θ + c 4 dc 4 x Ra 2 γ 2 T atm 2 θ 2 1 a0 + a3 dc 3 x4 3 8 θ 3 2 Ra γ Tatm a1 dc x4 2 θ Ra γ Tatm + a 2 dc 2 x4 2 4 Ra γ Tatm θ + a4 dc 4 x Ra 2 γ 2 Tatm 2 θ 2 x7 = K Afc Mv n K2 x5 t st RH2 x 2 + Rv x7 Van Dλ e t ρm,dry st K1 Mm,dry ρ m,dry K2 x5 tst R H 2 x2+rv x7 Van Mv psattan,in φan,in 10 2 tm MH 2 pan,in γ + 1 R t st x Mv p sat T st Van 1 Mm,dry + i st nd 10 8 Afc F b0+ b 1 dc x b 2 dc2 x 2 4 θ Ra γ T atm 4 Ra γ T atm θ

34 26 x8 = kca,out + + ksm,out x9 t st RO2 x 1 + RN2 x 3 + Rv x8 x5 t st R O2 x 1 +R N2 x 3 +Rv x 8 Mv p sat T atm φ atm x 5 Ma P atm x 5 p sat T atm φ atm x +1 5 P atm + Ma Mv ksm,out psattcl φ des x5 p sat T atm φ atm x 5 P atm x5 t st R O2 x 1 +R N2 x 3 +Rv x 8 Mv p sat T atm φ atm x 5 Ma P atm x 5 p sat T atm φ atm x +1 5 P atm Mv psattatm psattca,in x5 2 Patm psattcl MO 2 yo 2,ca,in MN 2 yo 2,ca,in 1 x5+psattcl φ des p sat T atm φ atm x 5 p sat T atm p sat T ca,in x 5 P atm 2 P atm p sat T cl ksm,out Ma Patm RO2 t st x 1 x5 tst R O 2 x1+rn 2 x3+rv x8 Mv psattatm φatm x5 + 1 x5 p sat T atm φ atm x 5 P atm + R N t 2 st x 3 kca,out M N2 x9 tst R O 2 x1+rn 2 x3+rv x8 Mv Rv tst x8 R O2 t st x 1 RO2 t st x 1 + R N t 2 st x 3 1 M O2 R O2 t st x 1 RO2 t st x 1 + R N t 2 st x M v ist n 2 F Afc Mv n Dλ e t ρm,dry st Mm,dry ρ m,dry 10 2 tm γ R t st x Mv p sat T st Van 1 Mm,dry + i st nd 10 8 Afc F + Ma Mv ksm,out psattcl φ des x5 psattatm φatm x 5 Patm x5 tst R O 2 x1+rn 2 x3+rv x8 Mv psattatm φatm x5 + 1 Ma Patm x5 p sat T atm φ atm x 5 P atm

35 27 x9 = Ra Trm k ca,out x9 tst R O 2 x1+rn 2 x3+rv x8 Vrm + 7 A T,rm CD,rm x9 γ 1 Patm γ 1 x9 R 1 Patm γ x9 π dc 3 ρa δ x4 e 8 cp T atm θ dc 2 x 2 c 4 c dc 3 x 3 4 c 8 θ 3 2 Ra γ T atm dc 5 x θ 5 2 Ra γ T atm 5 2 x 5 patm γ 1 γ 1 + c 1 dc x c 2 dc2 x 2 4 θ Ra γ T atm 4 Ra γ T atm θ + c 4 dc 4 x Ra 2 γ 2 T atm 2 θ 2 y1 = 8 θ a0 + a3 dc 3 x4 3 8 θ 3 2 Ra γ Tatm a1 dc x4 2 θ Ra γ Tatm + a 2 dc 2 x4 2 4 Ra γ Tatm θ + a4 dc 4 x Ra 2 γ 2 Tatm 2 θ 2 y2 = x5 1 b b 1 dc x b 2 dc2 x 2 4 θ Ra γ T atm 4 Ra γ T atm θ

36 28 y3 = n e c 1 i st 10 4 A fc 29 t fc t fc n tfc ln 9 t fc p sat R t O2 st x vca 1173 p sat t st R O2 x 1 +R N2 x 3 +Rv x vca p atm p sat R t O2 st x vca ln psat t st R O2 x 1 +R N2 x 3 +Rv x 8 vca patm n n tfc ln RO2 t st x 1 p atm vca 2 + ln R h2 tst x2 patm van ist ist tfc psat R O t 2 st x vca 10 5 Afc imax 10 5 Afc + n + ist tm e b2 1 1 t fc Afc b t fc c3 2.91

37 29 Chapter 3 Simulation of the Fuel Cell Model 3.1 SIMULINK implementation of Subsystems In this section, the SIMULINK models of the fuel cell subsystems are presented. Various subsystem blocks are individually discussed and all the subsystem blocks are finally inter-connected to form an entire PEM fuel cell. The blocks are implementation of the non-linear equations discussed in Chapter Compressor Figure 3.1: Compressor Subsystem Figure 3.2: Inside the Compressor Subsystem The masked subsystems generating the compressor motor torque τ cm and torque required to drive the compressor τ cp are presented next.

38 30 Figure 3.3: τ cm Subsystem Figure 3.4: τ cp Subsystem Supply Manifold Figure 3.5: Supply Manifold Subsystem Masked subsystems generating the mass of air in supply manifold m sm and supply manifold pressure p sm are presented below

39 31 Figure 3.6: Inside the Supply Manifold Subsystem Figure 3.7: m sm Generating Subsystem

40 32 The t cp,out is generated as follows Figure 3.8: P sm Generating Subsystem Figure 3.9: t cp,out Generating Subsystem

41 Cathode Figure 3.10: Cathode Subsystem Figure 3.11: Inside the Cathode Subsystem Here, the SIMULINK block Subsystem1,Subsystem4 and Subsystem3 generate the inlet flow, outlet flow and other cathode parameters respectively. They have been expanded below:

42 34 Figure 3.12: O 2 and N 2 Inlet Flow Rate Generating Subsystem Figure 3.13: Inlet Water Flow Rate Generating Subsystem Figure 3.14: O 2 and N 2 Outlet Flow Rate Generating Subsystem

43 Figure 3.15: Outlet Water Flow Rate Generating Subsystem 35

44 Return Manifold Figure 3.16: Return Manifold Subsystem below: The return manifold implementation is simple and straightforward, as presented Figure 3.17: Inside the Return Manifold Subsystem Anode Similar to cathode, Subsystem5 and Subsystem4 are used to compute inlet flow properties and Anode internal properties respectively. Their in-depth implementation is given below:

45 37 Figure 3.18: Anode Figure 3.19: Inside the Anode Subsystem Figure 3.20: Inlet Flow Rate Generating Subsystem Figure 3.21: Partial Pressure Generating Subsystem

46 Membrane Hydration Figure 3.22: Membrane Hydration Model The membrane hydration is implemented as follows: Figure 3.23: Inside the Membrane Hydration Model PEM Fuel Cell model All the different SIMULINK blocks presented above can be inter-connected as shown in Figure 3.24 to form the entire PEM fuel cell. Various constants used in the model are presented below

47 39 P arameter V alue Units R O R N R v R H R a J/kg.K R J/mol.K M O kg/mol M N kg/mol M v kg/mol M H kg/mol M a kg/mol Table 3.1: Gas Constants and Molar Masses P arameter V alue Units p atm Pa T atm K T cl 353 K T st 353 K T fc 353 K ρ a 1.23 kg/m 3 γ 1.4 C p 1004 J/kg/K F Coulombs φ des 1 φ atm 0.5 n 381 Table 3.2: Simulation Constants

48 40 Figure 3.24: PEM Fuel Cell 3.2 Steady State Operating Point Determination The PEM fuel cell model in Figure 3.24 was used to determine the system steady state or equilibrium operating point. The overall system has only one input, the compressor motor input voltage, v cm. A steady state value of 164 V was applied to the system and resulting state values were recorded in MATLAB. Figure 3.25 shows steady state operating points for different states. Figure 3.25: Steady State Operating Points

49 41 These steady state operating points are given by P arameter V alue Units m ss O Kg m ss H Kg m ss N Kg ω ss cp rad/sec p ss sm Pa m ss sm Kg m ss w,an Kg m ss w,ca Kg p ss rm Pa Table 3.3: Steady State Operating Points These values are further used in Chapter 4 for Jacobian linearization.

50 42 Chapter 4 Control of Linearized Fuel Cell System 4.1 Jacobian Linearization Jacobian linearization is one of the several techniques used to linearize a non-linear system. It is based on the Taylor series expansion of a non-linear differential equation around a nominal operating point[16]. Consider the following time invariant non-linear dynamical system given by ẋt = fxt, ut 4.1 ft, xt R n, ut R m Let u o t be the nominal or steady state input and x o t be the resulting nominal state trajectory. We can approximate the state trajectory around the nominal operating points using the Taylor series expansion as xt = x o t + δxt 4.2 ut = u o t + δut 4.3 ẋ o t = fx o t, u o t 4.4 Expanding equation 4.1 using Taylor series we get ẋ o t + δẋt = fx o t + δxt, u o t + δut 4.5 f x f x = fx o t, u o t + δxt + δut + h.o.t x o t,u o t u o t,u o t 4.6

51 43 Where, h.o.t stands for high order terms. Since δx and δu are small, we can neglect the higher order terms. The matrices of partial derivatives are given by f 1 f x x n f x f 2 f = A x n n x = x n o t,u o t f n x f n x n f 1 f u u m f x f 2 f = B u n m u = u m o t,u o t f n u f n u m x o t,u o t x o t,u o t The matrices A and B are called Jacobian matrices and are evaluated at nominal/operating points. We can extend the same concept to the output equation as follows. yt = gxt, ut 4.9 where yt, gt R p. Applying the Taylor series expansion around y o t in equation 4.9 we have δy = g x g x δxt + δut + h.o.t 4.10 x o t,u o t u o t,u o t Here, g x = C x p n = o t,u o t g x = D u p m = o t,u o t g 1 g x x n g 2 g x x n g p x g p x n g 1 g u u m g 2 g u u m g p u g p u m x o t,u o t x o t,u o t

52 44 Combining the results from equation 4.7,4.8,4.11,4.12 we can write δẋt = Aδxt + Bδut 4.13 δyt = Cδxt + Dδut 4.14 The above equations are linearized versions of the non-linear equations 4.1 and Linearized Model MATLAB symbolic toolbox is used for Jacobian linearization. All the non-linear equations from 2.80 to 2.88 are expressed as symbolic equations in MATLAB and the command jacobian does the linearization. The linearized symbolic model is then evaluated at the steady state operating point presented in Table 3.3. MATLAB code for achieving the linearization is presented below. % Define the symbolic variables syms x1 x2 x3 x4 x5 x6 x7 x8 x9 u i_st % Write the non-linear equations in terms of symbolic variables %state equations f1=y_o2_ca_in*m_o2/y_o2_ca_in*m_o2+1-y_o2_ca_in*m_n2*... f2=1/1+m_v/m_h2*phi_an_in*10^-1.69*10^-10*t_an_in^ f9=r_a*t_rm/v_rm*k_ca_out*x1*r_o2+x3*r_n2+x8*r_v*t_st/v_ca-x9-... %output equations g1=a_4*pi/60*d_c*60*x4/2*pi/sqrttheta/sqrt1.4*298*2.869*100^4+... g2=x5 g3=n* *10^-4*t_fc *10^-5*t_fc*...

53 45 % Linearize the equations A=jacobian[f1; f2; f3; f4; f5; f6; f7; f8; f9],[x1 x2 x3 x4 x5 x6 x7 x8 x9]; B=jacobian[f1; f2; f3; f4; f5; f6; f7; f8; f9],u; C=jacobian[g1; g2; g3],[x1 x2 x3 x4 x5 x6 x7 x8 x9]; D=jacobian[g1; g2; g3],u; % Define the steady state values x1= ; x2= e-004; x3= ; x4= e+003; x5= e+005; x6= ; x7= ; x8= ; x9= e+005; %Evaluate the matrices A=evalA; B=evalB; C=evalC; D=evalD;

54 46 The linearization results are as follows: A = B = [ C = D = [ 00 0 ] ] Note that the linearized fuel cell model is a 9 th order model. The next chapter discusses Controllability, Observability and Stability issues of this model.

55 47 Chapter 5 System Analysis This chapter discusses stability, controllability, and observability of the linearized PEM fuel cell model presented in Chapter Stability Analysis of Linearized PEM Fuel Cell Model The eigenvalues of the system matrix A are λa = i i Though one of the eigenvalue is very small and is very close to the origin, the system is still asymptotically stable. Next section examines controllability and observability of the fuel cell system. 5.2 Controllability and Observability test For the eigenvalue placement feedback control technique, the system has to be controllable. For designing observers the system has to be observable. There are several tests to determine whether the given linear dynamical system is controllable observable or

56 48 not, like the grammiam test, rank of controllability observability matrix etc. [16]. But the most elegant tests are the Popov Belevitch eigenvalue and eigenvector tests. These tests are very accurate over other tests and are highly recommended in environments like MATLAB, where the possibility of numerical precision errors are not in the programmer s control Popov Belevitch Eigenvalue Test This theorem states that a system is controllable if the matrix [A λi B] has full rank for every λ where λ = eiga, i.e. rank[a λ i I B] = n λ i A, i = 1, 2..., n The test, as applied to the fuel cell model 4.15, is presented below. Controllability of Linearized PEM Fuel Cell rank[a λ 1 I B] = rank[a λ 2 I B] = rank[a λ 3 I B] = rank[a λ 4 I B] = rank[a λ 5 I B] = rank[a λ 6 I B] = rank[a λ 7 I B] = rank[a λ 8 I B] = rank[a λ 9 I B] = From , it is clear that the linearized model 4.15 is completely controllable.

57 49 Observability of Linearized PEM Fuel Cell Popov Belevitch test for observability states that: A system is observable if, rank λ ii A = n λ i A, i = 1, 2..., n C This test, as applied to the linearized fuel cell model, is presented below. rank λ 1I A = C rank λ 2I A = C rank λ 3I A = C rank λ 4I A = C rank λ 5I A = C rank λ 6I A = C rank λ 7I A = C rank λ 8I A = C rank λ 9I A = C Obviously, from , 4.15 is completely observable Popov Belevitch Eigenvector Test This is another elegant test for determining whether a system is controllable observable or not. The controllability test says that, a system is controllable if, no left

58 50 eigenvector of A is orthogonal to B [17]. i.e. if v is the left eigenvector of A, corresponding to any λ then v A = v λ v B 0 Controllability of Linearized PEM Fuel Cell The controllability of the linearized 9 th order PEM fuel cell model in 4.15 is determined using the Popov Belevitch eigenvector test. The resulting vectors v B for controllability v1b = e v2b = v3b = v4b = v5b = v6b = i = v6b = v7b = i = v7b = v8b = v9b = Where, v i is the left eigenvector corresponding to λ i. The above results confirms the conclusion obtained in the previous section. Note: The Popov Belevitch eigenvector test can not give the controllability measure of a particular state variable. Observability of Linearized PEM Fuel Cell The results presented above can be extended to observability. For a system to be observable [17], Cx i 0 for Ax i = λx i i = 1, 2,..., n

59 51 Cx i for the observability test are presented below. Cx 1 = [ ] T = Cx 1 = Cx 2 = [ ] T = Cx 2 = Cx 3 = [ ] T = Cx 3 = Cx 4 = [ ] T = Cx 4 = Cx 5 = [ ] T = Cx 5 = Cx 6 = [ i i i] T 5.34 = Cx 6 = Cx 7 = [ i i i] T 5.36 = Cx 7 = Cx 8 = [ ] T = Cx 8 = Cx 9 = [ ] T = Cx 9 = Where, x i is the eigenvector corresponding to λ i It can be seen from that the fuel cell model is completely observable. Note: The Popov Belevitch eigenvector test can not give the observability measure of a particular state variable. 5.3 Model Reduction If the state of a system is weakly controllable and weakly observable, it can be discarded from the model without affecting the system dynamics to a reasonable extent. This technique is called system order reduction [16]. One of the methods to achieve model reduction is through balanced transformation or simply balancing. Balancing is a similarity transformation that puts the system such that its controllability and observability grammiams are identical and diagonal, with Hankel singular values on the diagonal of the grammiam matrix [18]. Hankel singular values give the controllability and observability measure of each state. The Hankel

60 52 singular values of the fuel cell model presented in 4.15 is presented below σ = The last singular value being extremely small, indicates the presence of a weakly controllable and weakly observable mode, and can be removed from the system dynamics Truncated Model The original 9 th order model has been reduced to an 8 th order model using balanced truncation. The model is presented below: A r = B r = C r = [ ] D r =

61 53 The eigen values of the truncated system are: λa r = i i This system is controllable and observable with respect to all the eigenvalues. The claim has been verified using Popov Belevitch test. Comments: State x 9 is truncated from the balanced system. The next step is to determine the relation between the eliminated x 9 in the balanced coordinate and the states in the original coordinate system. The similarity transformation matrix T in the transformation, x b = T x is used to determine the relation. The subscript b indicates the balanced system. x 9,b = x x x x x x x x x From equation 5.44, one things that stands out is x 9,b is predominantly dependent on mass of hydrogen in anode, x 2 ; pressure of gas in supply manifold, x 5, and mass of water in anode, x 7. x 9,b x x x It seems like one of these three states is weakly controllable and weakly observable. To get a better picture of the scenario, the system needs to be analysed in a new coordinate system, which is presented in the next section. 5.4 Modal Transformation The intention of this section is to get a more clear picture, if possible, to determine which state in original state space coordinates is weakly controllable and weakly observable