Dynamic Modeling and Predictive Control in Solid Oxide Fuel Cells

Preface xi

Acknowledgments xiii

List of Figures xv

List of Tables xxi

1 Introduction 1

1.1 Overview of Fuel Cell Technology 1

1.1.1 Types of Fuel Cells 2

1.1.2 Planar and Tubular Designs 3

1.1.3 Fuel Cell Systems 4

1.1.4 Pros and Cons of Fuel Cells 5

1.2 Modelling, State Estimation and Control 5

1.3 Book Coverage 6

1.4 Book Outline 6

Part I Fundamentals

2 First Principle Modelling for Chemical Processes 11

2.1 Thermodynamics 11

2.1.1 Forms of Energy 11

2.1.2 First Law 12

2.1.3 Second Law 13

2.2 Heat Transfer 13

2.2.1 Conduction 14

2.2.2 Convection 15

2.2.3 Radiation 17

2.3 Mass Transfer 18

2.4 Fluid Mechanics 20

2.4.1 Viscous Flow 21

2.4.2 Velocity Distribution 21

2.4.3 Bernoulli Equation 21

2.5 Equations of Change 22

2.5.1 The Equation of Continuity 23

2.5.2 The Equation of Motion 23

2.5.3 The Equation of Energy 24

2.5.4 The Equations of Continuity of Species 26

2.6 Chemical Reaction 26

2.6.1 Reaction Rate 26

2.6.2 Reversible Reaction 28

2.6.3 Heat of Reaction 29

2.7 Notes and References 29

3 System Identification I 31

3.1 Discrete-time Systems 31

3.2 Signals 36

3.2.1 Input Signals 36

3.2.2 Spectral Characteristics of Signals 41

3.2.3 Persistent Excitation in Input Signals 44

3.2.4 Input Design 49

3.3 Models 50

3.3.1 Linear Models 50

3.3.2 Nonlinear Models 54

3.4 Notes and References 56

4 System Identification II 57

4.1 Regression Analysis 57

4.1.1 Autoregressive Moving Average with Exogenous Input Models 57

4.1.2 Linear Regression 59

4.1.3 Analysis of Linear Regression 60

4.1.4 Weighted Least Squares Method 61

4.2 Prediction Error Method 64

4.2.1 Optimal Prediction 65

4.2.2 Prediction Error Method 70

4.2.3 Prediction Error Method with Independent Parameterisation 74

4.2.4 Asymptotic Variance Property of PEM 75

4.2.5 Nonlinear Identification 76

4.3 Model Validation 79

4.3.1 Model Structure Selection 79

4.3.2 The Parsimony Principle 80

4.3.3 Comparison of Model Structures 81

4.4 Practical Consideration 82

4.4.1 Treating Non-zero Means 82

4.4.2 Treating Drifts in Disturbances 83

4.4.3 Robustness 83

4.4.4 Additional Model Validation 83

4.5 Closed-loop Identification 84

4.5.1 Direct Closed-loop Identification 85

4.5.2 Indirect Closed-loop Identification 87

4.6 Subspace Identification 92

4.6.1 Notations 92

4.6.2 Subspace Identification via Regression Analysis Approach 97

4.6.3 Example 100

4.7 Notes and References 102

5 State Estimation 103

5.1 Recent Developments in Filtering Techniques for Stochastic Dynamic Systems 103

5.2 Problem Formulation 105

5.3 Sequential Bayesian Inference for State Estimation 107

5.3.1 Kalman Filter and Extended Kalman Filter 110

5.3.2 Unscented Kalman Filter 112

5.4 Examples 116

5.5 Notes and References 120

6 Model Predictive Control 121

6.1 Model Predictive Control: State-of-the-Art 121

6.2 General Principle 122

6.2.1 Models for MPC 122

6.2.2 Free and Forced Response 125

6.2.3 Objective Function 125

6.2.4 Constraints 126

6.2.5 MPC Law 126

6.3 Dynamic Matrix Control 127

6.3.1 Prediction 127

6.3.2 DMC without Penalising Control Moves 129

6.3.3 DMC with Penalising Control Moves 130

6.3.4 Feedback in DMC 130

6.4 Nonlinear MPC 134

6.5 General Tuning Guideline of Nonlinear MPC 136

6.6 Discretisation of Models: Orthogonal Collocation Method 137

6.6.1 Orthogonal Collocation Method with Prediction Horizon 1 137

6.6.2 Orthogonal Collocation Method with Prediction Horizon N 140

6.7 Pros and Cons of MPC 142

6.8 Optimisation 142

6.9 Example: Chaotic System 144

6.10 Notes and References 145

Part II Tubular SOFC

7 Dynamic Modelling of Tubular SOFC: First-Principle Approach 149

7.1 SOFC Stack Design 149

7.2 Conversion Process 150

7.2.1 Electrochemical Reactions 150

7.2.2 Electrical Dynamics 153

7.3 Diffusion Dynamics 155

7.3.1 Transfer Function of Diffusion 156

7.3.2 Simplified Transfer Function of Diffusion 157

7.3.3 Dynamic Model of Diffusion 158

7.3.4 Diffusion Coefficient 159

7.4 Fuel Feeding Process 160

7.4.1 Reforming/Shift Reaction 160

7.4.2 Mass Transport 162

7.4.3 Momentum Transfer 164

7.4.4 Energy Transfer and Heat Exchange 165

7.5 Air Feeding Process 166

7.5.1 Mass Transport in the Cathode Channel 166

7.5.2 Cathode Channel Momentum Transfer 167

7.5.3 Energy Transfer in the Cathode Channel 168

7.5.4 Air in Injection Channel 168

7.6 SOFC Temperature 169

7.6.1 Dynamic Energy Exchange Process 169

7.6.2 Conduction 170

7.6.3 Convection 171

7.6.4 Radiation 172

7.6.5 Cell Temperature Model 174

7.6.6 Injection Tube Temperature Model 174

7.7 Final Dynamic Model 175

7.7.1 I/O Variables 175

7.7.2 State Space Model 176

7.7.3 Model Validation 180

7.8 Investigation of Dynamic Properties through Simulations 181

7.8.1 Dynamics of Diffusion 182

7.8.2 Dynamics of Fuel Feeding Process 184

7.8.3 Dynamics of Air Feeding Process 186

7.8.4 Dynamics due to External Load 188

7.9 Notes and References 190

8 Dynamic Modelling of Tubular SOFC: Simplified First-Principle Approach 193

8.1 Preliminary 193

8.1.1 Relation of Process Variables 194

8.1.2 Limits to Power Output 194

8.2 Low-order State Space Modelling of SOFC Stack 195

8.2.1 Physical Processes 195

8.2.2 Modelling Assumptions 197

8.2.3 I/O Variables 197

8.2.4 Voltage 198

8.2.5 Partial Pressures 199

8.2.6 Flow Rates 200

8.2.7 Temperatures 203

8.3 Nonlinear State Space Model 204

8.4 Simulation 205

8.4.1 Validation 205

8.4.2 Step Response to the Inputs 207

8.4.3 Step Responses to the Disturbances 209

8.5 Notes and References 211

9 Dynamic Modelling and Control of Tubular SOFC: System Identification Approach 213

9.1 Introduction 213

9.2 System Identification 213

9.2.1 Selection of Variables 213

9.2.2 Step Response Test 214

9.2.3 Non-typical Step Response 217

9.2.4 Input Design 218

9.2.5 Linear System Identification 220

9.2.6 Nonlinear System Identification 234

9.3 PID Control 241

9.3.1 Set Point Tracking 243

9.3.2 Disturbance Rejection 243

9.3.3 Internal Model Control for Discrete-time Processes 243

9.3.4 Application of Discrete-time IMC to Multi-loop Control of SOFC 254

9.4 Closed-loop Identification 257

9.5 Notes and References 263

Part III Planar SOFC

10 Dynamic Modelling of Planar SOFC: First-Principle Approach 267

10.1 Introduction 267

10.2 Geometry 268

10.3 Stack Voltage 268

10.4 Mass Balance 270

10.5 Energy Balance 271

10.5.1 Lumped Model 272

10.5.2 Detail Model 273

10.6 Simulation 277

10.6.1 Steady-state Response 277

10.6.2 Dynamic Response 278

10.7 Notes and References 280

11 Dynamic Modelling of Planar SOFC System 283

11.1 Introduction 283

11.2 Fuel Cell System 283

11.2.1 Fuel and Air Heat Exchangers 284

11.2.2 Reformer 286

11.2.3 Burner 287

11.3 SOFC along with a Capacitor 287

11.4 Simulation Result 289

11.4.1 Fuel Cell System Simulation 290

11.4.2 SOFC Stack with Ultra-capacitor 292

11.5 Notes and References 292

12 Model Predictive Control of Planar SOFC System 295

12.1 Introduction 295

12.2 Control Objective 296

12.3 State Estimation: UKF 297

12.4 Steady-state Economic Optimisation 298

12.5 Control and Simulation 301

12.5.1 Linear MPC 301

12.5.2 Nonlinear MPC 303

12.5.3 Optimisation 305

12.6 Results and Discussions 306

12.7 Notes and References 307

Appendix A Properties and Parameters 309

A.1 Parameters 309

A.2 Gas Properties 309

References 315

Index 321