Electrical Load Forecasting: Modeling and Model Construction Book

Acknowledgments xiii

Introduction xv

1 Mathematical Background and State of the Art 1

1.1 Objectives 1

1.2 Matrices and Vectors 1

1.3 Matrix Algebra 3

1.3.1 Addition of Matrices 3

1.3.2 Matrix Subtraction (Difference) 4

1.3.3 Matrix Multiplication 4

1.3.4 Inverse of a Matrix (Matrix Division) 6

1.4 Rank of a Matrix 8

1.5 Singular Matrix 8

1.6 Characteristic Vectors of a Matrix 9

1.7 Diagonalization 9

1.8 Partitioned Matrices 12

1.9 Partitioned Matrix Inversion 13

1.10 Quadratic Forms 15

1.11 State Space Representation 17

1.12 Difference Equations 19

1.13 Some Optimization Techniques 20

1.13.1 Unconstrained Optimization 21

1.13.2 Constrained Optimization 25

1.14 State of the Art 29

References 40

2 Static State Estimation 45

2.1 Objectives 45

2.2 The Static Estimation Problem Formulation 45

2.2.1 Linear Least Error Squares Estimation 46

2.2.2 Weighted Linear Least Error Squares (WLES) Estimation 47

2.2.3 Constrained Least Error Squares (CLES) Estimation 50

2.2.4 Recursive Least Error Squares (RLES) Estimation 52

2.2.5 Nonlinear Least Error Squares (NLLES) Estimation 53

2.3 Properties of Least Error Squares Estimation 57

2.4 Least Absolute Value Static State Estimation 58

2.4.1 Historical Perspective 58

2.4.2 Least Absolute Value of Error Estimation 59

2.4.3 Least Absolute Value Based on Linear Programming 60

2.4.4 Schlossmacher Iterative Algorithm 62

2.4.5 Sposito and Hand Algorithm 63

2.4.6 Soliman and Christensen Algorithm 63

2.5 Constrained LAV Estimation 70

2.6 Nonlinear Estimation Using the Soliman and Christensen Algorithm 72

2.7 Leverage Points 75

2.8 Comparison between LES Estimation and LAV Estimation Algorithms 77

References 78

3 Load Modeling for Short-Term Forecasting 79

3.1 Objectives 79

3.2 Introduction 79

3.3 Base Load 79

3.4 Weather-Dependent Load 80

3.4.1 Temperature 80

3.4.2 Wind Speed 81

3.4.3 Humidity 81

3.4.4 Illumination 81

3.5 Residual Load 82

3.6 Short-Term Load Models 82

3.6.1 Multiple Linear Regression 82

3.6.2 General Exponential Smoothing 83

3.6.3 Stochastic Time Series 84

3.6.4 Qualities of Forecasting Models 85

3.7 Special Load-Forecasting Models 86

3.7.1 Model A: Multiple Linear Regression Model 87

3.7.2 Model B: Harmonics Model 90

3.7.3 Model C: Hybrid Model 92

References 93

4 Fuzzy Regression Systems and Fuzzy Linear Models 99

4.1 Objectives 99

4.2 Fuzzy Fundamentals 99

4.3 Fuzzy Sets and Membership 102

4.3.1 Membership Functions 103

4.3.2 Basic Terminology and Definitions 103

4.3.3 Support of a Fuzzy Set 104

4.3.4 Normality 104

4.3.5 Convexity and Concavity 104

4.3.6 Basic Operation 105

4.4 Fuzzy Linear Estimation 109

4.4.1 Nonfuzzy Output (Y_j = m_j) 109

4.4.2 Fuzzy Output Systems 112

4.5 Fuzzy Short-Term Load Modeling 120

4.5.1 Multiple Fuzzy Linear Regression Model: Crisp Data 121

4.5.2 Multiple Fuzzy Linear Regression Model: Fuzzy Data 129

4.5.3 Fuzzy Load Model B 133

4.5.4 Fuzzy Load Model C 134

4.6 Conclusion 136

References 136

5 Dynamic State Estimation 139

5.1 Objectives 139

5.2 Discrete Time Systems 139

5.3 Discrete Time-Optimal Filtering 141

5.3.1 Kalman Filter 143

5.3.2 Initialization of the Kalman Filter 150

5.3.3 Divergence Problems in Kalman Filter 150

5.3.4 Soliman and Christensen Filter: Weighted Least Absolute Value Filter (WLAVF) 151

5.4 Recursive Least Error Squares 157

References 158

6 Load-Forecasting Results Using Static State Estimation 159

6.1 Objectives 159

6.2 Description of the Data 159

6.3 Offline Simulation (Static Load Forecasting Estimation) 159

6.4 Model A Results 160

6.4.1 Model Parameters Estimation for Every Hour in a Summer Weekday (24 Sets) 161

6.4.2 Estimation of Constant Model Parameters for Weekday (One Set) 161

6.4.3 Model Parameter Estimation for Every Hour in a Summer Weekend Day (24 Sets) 164

6.4.4 Estimation of Constant Model Parameters for a Summer Weekend Day (One Set) 166

6.4.5 General Remarks for Summer Model A 168

6.4.6 Winter Predictions 169

6.5 Model B 169

6.5.1 Summer Weekday 170

6.5.2 Summer Weekend Day 173

6.5.3 General Remarks for Summer Model B 175

6.5.4 Winter Predictions 175

6.6 Model C Results 175

6.6.1 General Remarks for Summer Model C 179

6.6.2 Winter Predictions 179

6.7 Conclusion 198

Appendix 6.1 Winter Static Load Results for Model A 199

Appendix 6.2 Winter Static Load Results for Model B 215

Appendix 6.3 Winter Static Load Results for Model C 223

7 Load-Forecasting Results Using Fuzzy Systems 229

7.1 Objectives 229

7.2 Fuzzy Load Model A 229

7.2.1 Load Parameters for a Summer Weekday 229

7.2.2 Load Estimation for a Summer Weekday 230

7.2.3 Load Prediction for a Summer Weekday 230

7.2.4 Load Parameters for a Summer Weekend Day 231

7.2.5 Load Prediction for a Summer Weekend Day 232

7.2.6 Load Estimation and Prediction for a Winter Weekday and a Winter Weekend Day 233

7.3 Fuzzy Load Model B 234

7.3.1 Load Parameters for Model B 234

7.3.2 Load Estimation and Prediction 237

7.4 Fuzzy Load Model C 237

7.4.1 Load Parameters for Model C 237

7.4.2 Load Estimation and Prediction for a Summer Day 238

7.4.3 Load Estimation and Prediction for a Winter Day 239

7.5 Conclusion 260

Appendix 7.1 Winter Load Forecasting: Fuzzy Case Model A 261

Appendix 7.2 Winter Load Forecasting: Fuzzy Case Model B 274

Appendix 7.3 Winter Load Forecasting: Fuzzy Case Model C 275

8 Dynamic Electric Load Forecasting 291

8.1 Objectives 291

8.2 Introduction 291

8.3 Load Regression Models 292

8.4 Estimating the Next Year's Load Contour 295

8.5 Annual Load Growth 299

8.6 Examples 300

8.6.1 Multiple Regression Models Results 300

8.6.2 Estimating 1995 Year Load Contour 301

8.6.3 Annual Load Growth Results 302

8.6.4 Remarks 303

8.7 Kalman Filtering Algorithm with Moving Window Weather 304

8.7.1 Load-Forecasting Model 306

8.7.2 Winter Model 308

8.8 Kalman Filter Parameter Estimation 309

8.8.1 Basic Kalman Filter 309

8.8.2 Prediction of the Kalman Filter Model 311

8.8.3 Examples and Results 311

8.8.4 Order of the Load Model 312

8.8.5 One-Hour Prediction 312

8.8.6 Twenty-Four-Hour Prediction 314

8.8.7 Weekdays and Weekends Profiles 315

8.8.8 Conclusions 326

8.9 Fuzzy Load Forecasting Using the Kalman Filter 326

8.9.1 Fuzzy Linear Model 328

8.9.2 Fuzzy Parameter Estimation Using Kalman Filtering 330

8.9.3 Kalman Filter Prediction Model 330

8.9.4 Fuzzy Rule-Based Inference 331

8.10 Model Validation and Results 335

8.10.1 One-Day Parameter Estimation and Load Prediction 335

8.10.2 Up to 60 Days of Load Prediction 338

8.10.3 Conclusions 340

8.11 Recursive Least Error Squares 342

8.11.1 Testing the Algorithm 343

8.11.2 Conclusions 351

References 351

9 Electric Load Modeling for Long-Term Forecasting 353

9.1 Introduction 353

9.2 Peak-Load-Demand Model 354

9.2.1 Example 355

9.2.2 A More Detailed Model 356

9.2.3 A Time-Dependent Model 358

9.3 Time-Series Analysis 359

9.3.1 Example for the Time-Series Model 360

9.3.2 Remarks 360

9.4 Kalman Filtering Algorithm 361

9.4.1 Estimating Multiple Regression Models 362

9.4.2 Estimating the Next Year's Load Contour 365

9.5 Annual Load Growth 366

9.5.1 Load Modeling for the Kalman Filtering Algorithm 369

9.5.2 Kalman Filter Parameter Estimation Algorithm 369

9.6 Computer Exercises 370

9.6.1 Multiple: Regression Models Results 370

9.6.2 Estimating the 1995 Load Contour 371

9.6.3 Kalman filter Prediction Results 372

9.6.4 Remarks 373

9.7 Long-Term/Mid term Forecasting (Short-Term Correlation and Annual Growth) 377

9.7.1 Load Regression Models 377

9.7.2 Estimating the Next Year's Load Contour 380

9.7.3 Annual Load Growth 383

9.8 Examples of Long-Term/Mid Term Forecasting 384

9.8.1 Multiple Regression Model Results 384

9.8.2 Estimating the 1995 Load Contour 385

9.8.3 Annual Load Growth Results 385

9.8.4 Remarks 389

9.9 Fuzzy Regression for Peak-Load Forecasting 389

9.9.1 Modeling of Electric Annual Peak Load 390

9.9.2 A Nonfuzzy Peak Load with Fuzzy Parameters 390

9.9.3 A Fuzzy Peak-Load Demand 391

9.10 Testing the Algorithm 392

9.10.1 Nonfuzzy Annual Peak Load 392

9.10.2 Fuzzy Annual Peak Load 393

9.10.3 Remarks 394

9.11 Time-Series Models 394

9.11.1 Time Series 395

9.11.2 Forecasting Methods 395

9.11.3 Forecasting Errors 395

9.12 Power System Load Forecasting 396

9.12.1 A Simple Example of Power System Load Forecasting 396

9.13 Linear Regression Method 398

9.14 Autoregressive (AR) Model 399

9.15 Moving Average (MA) Model 400

9.16 Autoregressive Moving Average (ARMA, or Box-Jenkins) Model 401

9.17 Autoregressive Integrated Moving Average (ARIMA) Model 402

9.18 ARMAX and ARIMAX Models 403

9.18.1 Remarks 403

References 403

Index 407