Dynamic Modeling and Predictive Control in Solid Oxide Fuel Cells First Principle and Data-based Approaches

Biao Huang University of Alberta, Canada

Yutong Qi Corporate Electronics, Canada

AKM Monjur Murshed Shell Canada, Canada

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 Design 3

1.1.3 Fuel Cell Systems 5

1.1.4 Pros and Cons of Fuel Cells 5

1.2 Modelling, Filtering and Control 6

1.3 Book Coverage 7

1.4 Book Outline 7

Part One Fundamentals 9

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 14

2.2.1 Conduction 14

2.2.2 Convection 16

2.2.3 Radiation 17

2.3 Mass Transfer 19

2.4 Fluid Mechanics 21

2.4.1 Viscous Flow 21

2.4.2 Velocity Distributions 22

2.4.3 Bernoulli Equation 22

2.5 Equations of Change 23

2.5.1 The Equation of Continuity 24

2.5.2 The Equation of Motion 24

2.5.3 The Equation of Energy 25

2.5.4 Equations of Continuity of Species 27

2.6 Chemical Reaction 28

2.6.1 Reaction Rate 28

2.6.2 Reversible Reaction 29

2.6.3 Heat of Reaction 30

2.7 Notes and References 31

3 Data Driven Modelling: Preliminaries for System Identification 33

3.1 Discrete-time Systems 33

3.2 Signals 39

3.2.1 Input signals 39

3.2.2 Spectral Characteristics of Signals 44

3.2.3 Persistent Excitation in Input Signal 48

3.2.4 Input Design 53

3.3 Models 53

3.3.1 Linear Models 53

3.3.2 Nonlinear Models 58

3.4 Notes and References 60

4 Data Driven Modelling: System Identification 61

4.1 Regression Analysis 61

4.1.1 Autoregressive Moving Average with Exogenous Input Models 61

4.1.2 Linear Regression 63

4.1.3 Analysis of Linear Regression 64

4.1.4 Weighted Least Squares Method 65

4.2 Prediction Error Method 68

4.2.1 Optimal Prediction 70

4.2.2 Prediction Error Method 75

4.2.3 Prediction Error Method with Independent Parameterization 79

4.2.4 Asymptotic Variance Property of PEM 80

4.2.5 Nonlinear Identification 82

4.3 Model Validation 84

4.3.1 Model Structure Selection 84

4.3.2 The parsimony principle 86

4.3.3 Comparison of model structures 86

4.4 Practical Consideration 87

4.4.1 Treating nonzero means 88

4.4.2 Treating drifts in disturbances 88

4.4.3 Robustness 89

4.4.4 Model Validation 89

4.5 Closed-loop Identification 90

4.5.1 Direct Closed-loop Identification 91

4.5.2 Indirect Closed-loop Identification 93

4.6 Subspace Identification 98

4.6.1 Notations 98

4.6.2 Subspace Identification via Regression Analysis Approach 104

4.6.3 Example 106

4.7 Notes and References 109

5 State Estimation 111

5.1 Recent Development of Filtering Techniques for Stochastic Dynamic Systems 111

5.2 Problem Formulation 113

5.3 Sequential Bayesian Inference for State Estimation 115

5.3.1 Kalman Filter and Extended Kalman Filter 119

5.3.2 Unscented Kalman Filter 121

5.4 Examples 126

5.5 Notes and References 130

6 Model Predictive Control 131

6.1 Model Predictive Control: State-of-the-Art 131

6.2 General Principle 132

6.2.1 Models for MPC 132

6.2.2 Free and forced response 135

6.2.3 Objective Function 136

6.2.4 Constraints 136

6.2.5 MPC control law 137

6.3 Dynamic Matrix Control 137

6.3.1 The Prediction Model 137

6.3.2 Unconstrained DMC Design 140

6.3.3 Penalizing the Control Action 140

6.3.4 Handling Disturbances in DMC 141

6.4 Nonlinear MPC 144

6.5 General Tuning Guideline of nonlinear MPC 147

6.6 Discretization of Models: Orthogonal Collocation Method 148

6.6.1 Orthogonal Collocation Method with Prediction Horizon 1 149

6.6.2 Orthogonal Collocation Method with Prediction Horizon N 151

6.7 Pros and Cons of MPC 153

6.8 Optimization 154

6.9 Example: Chaotic System 155

6.10 Notes and References 157

Part Two Tubular SOFC 159

7 Dynamic Modelling: First-Principle Approach 161

7.1 SOFC Stack Design 161

7.2 Conversion Process 162

7.2.1 Electrochemical Reactions 162

7.2.2 Electrical Dynamics 165

7.3 Diffusion Dynamics 168

7.3.1 Transfer Function of Diffusion 169

7.3.2 Simplified Transfer Function of Diffusion 170

7.3.3 Dynamic Model of Diffusions 171

7.3.4 Diffusion Coefficient 172

7.4 Fuel Feeding Process 173

7.4.1 Reforming/Shift Reaction 173

7.4.2 Mass Transport 175

7.4.3 Momentum Transfer 177

7.4.4 Energy Transfer and Heat Exchange 178

7.5 Air Feeding Process 179

7.5.1 Mass Transportation in the Cathode Channel 179

7.5.2 Cathode Channel Momentum Transfer 181

7.5.3 Energy Transfer in the Cathode Channel 181

7.5.4 Air in Injection Channel 182

7.6 SOFC Temperature 183

7.6.1 Dynamic Energy Exchange Process 183

7.6.2 Conduction 184

7.6.3 Convection 185

7.6.4 Radiation 186

7.6.5 Cell Temperature Model 188

7.6.6 Injection Tube Temperature Model 188

7.7 Final Dynamic Model 189

7.7.1 I/O Variables 190

7.7.2 State Space Model 191

7.7.3 Model Validation 194

7.8 Investigation of Dynamic Properties through Simulations 197

7.8.1 Dynamics of Diffusion 197

7.8.2 Dynamics of Fuel Feeding Process 201

7.8.3 Dynamics of Air Feeding Process 203

7.8.4 Dynamics due to External Load 205

7.9 Notes and References 207

8 Dynamic Modelling: Simplified First-principle Approach 209

8.1 Preliminary 209

8.1.1 Relation of Property Variables 210

8.1.2 Limits to Power Output 210

8.2 Low-order State Space Modelling of SOFC Stack 211

8.2.1 Physical Processes 211

8.2.2 Modelling Assumptions 213

8.2.3 I/O Variables 213

8.2.4 Voltage 215

8.2.5 Partial Pressures 215

8.2.6 Flow Rates 218

8.2.7 Temperatures 219

8.3 Nonlinear State-space Model 221

8.4 Simulation 222

8.4.1 Validation 222

8.4.2 Step Response to the Inputs 225

8.4.3 Step Responses to Disturbance 226

8.5 Notes and References 227

9 Dynamic Modelling and Control: Data-driven Approach 229

9.1 Introduction 229

9.2 System Identification 229

9.2.1 Selection of Variables 229

9.2.2 Step Response Test 230

9.2.3 Nontypical Step Response 233

9.2.4 Input Design 234

9.2.5 Linear System Identification 237

9.2.6 Nonlinear System Identification 255

9.3 PI Control 258

9.3.1 Setpoint Tracking 261

9.3.2 Disturbance Rejection 262

9.3.3 Internal Model Control for Discrete-time Processes 262

9.3.4 Application of Discrete-time IMC to Multiloop Control of SOFC 273

9.4 Closed-loop Identification 274

9.5 Notes and References 283

Part Three Planar SOFC 285

10 Dynamic Modelling: First-Principle Approach 287

10.1 Introduction 287

10.2 Geometry 288

10.3 Stack Voltage 289

10.4 Material Balance 290

10.5 Energy Balance 292

10.5.1 Lumped Model 292

10.5.2 Detail Model 293

10.6 Simulation 297

10.6.1 Steady state response 298

10.6.2 Dynamic Response 299

10.7 Notes and references 301

11 Dynamic Modelling: a Fuel Cell System 303

11.1 Introduction 303

11.2 Fuel Cell System 303

11.2.1 Fuel and air heat exchangers 305

11.2.2 Reformer 306

11.2.3 Burner 308

11.3 SOFC along with a capacitor 308

11.4 Simulation Result 310

11.4.1 Fuel Cell System Simulation 310

11.4.2 SOFC stack with ultra-capacitor 313

11.5 Notes and references 313

12 Control of Planar SOFC System 315

12.1 Introduction 315

12.2 Control Objective 317

12.3 State Estimation: UKF 318

12.4 Steady State Economic Optimization 321

12.5 Control and Simulation 322

12.5.1 Linear MPC 323

12.5.2 Nonlinear MPC 325

12.5.3 Optimization 326

12.6 Results and Discussions 327

12.7 Notes and References 329

A Properties and Parameters 331

A-I. Parameters 331

A-II. Gas properties 331

References 335

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