# Line Loss Analysis and Calculation of Electric Power Systems

# Line Loss Analysis and Calculation of Electric Power Systems

ISBN: 978-1-118-86723-5 October 2015 384 Pages

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## Description

**Presents the fundamentals and calculation of transmission line losses, their reduction, and economic implications**• Written by a very experienced expert in this field

• Introduces various technical measures for loss reduction, and appended with a large number of examples

• Offers a progressive and systematic approach to various aspects of the problems

• A timely and original book to meet the challenges of power and grid industry development

## Table of contents

Foreword xiii

Preface xvii

Introduction xix

**1 Overview 1**

1.1 Active Power Loss and Electric Energy Loss 1

1.1.1 Main Types of Active Power Loss 1

1.1.2 Calculation of Electric Energy Loss 2

1.1.3 Electricity Line Loss and Line Loss Rate 3

1.1.4 Calculation and Analysis of Line Loss 5

1.2 Calculation of AC Resistance 7

1.3 Influence of Temperature and Voltage Changes on Line Loss in the Measuring Period 7

1.3.1 Influence of Temperature Change on Line Loss in the Measuring Period 7

1.3.2 Influence of Voltage Change on Line Loss in the Measuring Period 9

1.4 Influence of Load Curve Shape on Line Loss 10

1.4.1 Load Curve and Load Duration Curve 10

1.4.2 Parameters of Characterization Load Curve 12

1.4.3 Relationship Between Loss Factor and Load Factor 15

1.5 Influence of Load Power Factor and Load Distribution on Line Loss 16

1.5.1 Influence of Load Power Factor 16

1.5.2 Influence of Load Distribution of Multi-Branch Line 17

1.6 Influence of Measuring Instrument Error on Line Loss 18

1.6.1 Composition of Electric Energy Metering System and Constitution of Metering Error 18

1.6.2 Composition of Electronic Watt-Hour Meter Error 18

1.6.3 Influence of Metering System Error on the Calculation of Line Loss Rate 19

**2 Calculation of Line Loss by Current Load Curve 21**

2.1 RMS Current Method and Loss Factor Method 21

2.1.1 RMS Current Method 21

2.1.2 Loss Factor Method 22

2.1.3 Other Calculation Methods 22

2.2 Derivation of Functional Relationship *F*( *f *) by Ideal Load Curve 23

2.2.1 Derivation of *F*( *f *) Formula by Ideal Load Curve with Two Variables 23

2.2.2 Derivation of *F*( *f *) Curve by Ideal Load Curve with Four Variables 26

2.3 Derivation of Approximate Formula of *F*( *f *) by Statistical Mathematical Method 28

2.3.1 Binomial Approximate Formula of *F*( *f*) 29

2.3.2 Trinomial Approximate Formula of *F*( *f*) 30

2.3.3 Approximate Formula of Family of *F*( *f *) Curves with Four Variables 30

2.4 Derivation of *F*( *f *) Formula by Mathematical Analysis Method 31

2.4.1 Direct Integration Method 31

2.4.2 Subsection Integration Method 32

**3 Probability Theory Analysis of Current Load Curve 33**

3.1 Probability Meanings of Load Curve and Its Parameters 33

3.1.1 Probability Meaning of Load Duration Curve 33

3.1.2 Probability Meanings of Minimum Load Rate and Load Rate 34

3.1.3 Barth Formula of Loss Factor 35

3.2 Analysis of Rossander Formula as Distribution Function 35

3.2.1 Rossander Formula of Load Duration Curve 35

3.2.2 Exponential Distribution Function 36

3.2.3 Derivation of Loss Factor Formula 37

3.2.4 Comparison of Direct Integration Method and Distribution Function Analysis Method 40

3.3 Comparison of Various Loss Factor Formulas 40

3.3.1 Loss Factor Formula Comparison Procedures Prepared by Monte Carlo Method 41

3.3.2 Comparison Results of Various Loss Factor Formulas 41

3.4 Three-Mode Division of Active Load Duration Curve 42

3.4.1 Three Modes of Load Management in the Electric Power System 42

3.4.2 Differences and Relations of the Three Operation Modes 43

3.4.3 Probability Division of Three Operation Modes 43

**4 Calculation of Line Loss by Power Load Curve 49**

4.1 Line Loss Calculation Considering Power Factor 49

4.1.1 The Maximum Apparent Power is Caused by the Maximum Active Power 49

4.1.2 The Maximum Apparent Power is Caused by the Maximum Reactive Power 50

4.2 Maximum Load Power Factor Method of Tröger 51

4.3 Annual Average Power Factor Method of Glazynov 51

4.4 Equivalent Load Curve Method 53

4.4.1 Equivalent Load Curve Method of Cweink 53

4.4.2 Improvement and Extension of the Cweink Method 54

4.4.3 Equal Time Equivalent Load Curve Method 55

4.4.4 Unequal Time Equivalent Load Curve Method 57

4.5 Analysis of Errors of Various Line Loss Calculation Methods 64

4.5.1 Analysis of Relative Error in Line Loss Calculated by rms Current Method 64

4.5.2 Analysis of Relative Error in Line Loss Calculated by Loss Factor Method 66

**5 Line Loss Calculation after Reactive Compensation 69**

5.1 Calculation of Load Curve Parameters after Reactive Compensation 69

5.1.1 Calculation of Reactive Load Curve Parameters at Under-Compensation 69

5.1.2 Calculation of Reactive Load Curve Parameters at Weak Over-Compensation 70

5.1.3 Calculation of Reactive Load Curve Parameters at Strong Over-Compensation 72

5.2 Calculation of Loss Reduction Effect of Reactive Compensation 72

5.2.1 Calculation of Compensation Effect at High Natural Power Factor 73

5.2.2 Calculation of Reactive Compensation Effect at Low Natural Power Factor 75

5.3 Calculation Curves of Annual Electric Energy Losses for Power Grid Planning and Design 79

5.3.1 Calculation Curves of Annual Electric Energy Losses of 35–110 kV Transmission Lines 79

5.3.2 Calculation Curves of Annual Electric Energy Losses of 220 kV Transmission Lines 81

5.3.3 Calculation Curves of Annual Electric Energy Losses of Agricultural Electric Lines Consuming Electricity on a Quarterly Basis 82

**6 Change Law for the Electric Energy Losses of Power Grids 87**

6.1 Basis of Analysis of Line Loss Changes 87

6.1.1 Line Loss Binomial 87

6.1.2 Condition of Minimum Line Loss Rate 88

6.2 Calculation and Analysis of No-load Loss 89

6.2.1 Higher-Order Expression 89

6.2.2 Square Expression 91

6.2.3 Quasi Square Expression 91

6.3 Calculation and Analysis of Load Loss Coefficient *C *92

6.3.1 Calculation of Load Loss Coefficient 92

6.3.2 Inclusion of Load Loss Coefficient 94

6.4 Determination of Voltage Level by Loss Reduction Requirement 105

6.4.1 Voltage Characteristics of Various Loads and Comprehensive Loads of Distribution Lines 105

6.4.2 Control of Voltage Level and Reduction of Electric Energy Loss 106

**7 Analysis and Control of Line Loss Rate Indicators of Power Grids 109**

7.1 Analysis of Line Loss Rate Composition 109

7.1.1 Line Loss Rates and Total Line Loss Rate of Different Voltage Grids 109

7.1.2 No-load Line Loss Rate and Load Line Loss Rate 110

7.2 Analysis of Influence of Grid Electric Supply Structure on Line Loss Rate 113

7.2.1 Repeated Electric Supply Rate 113

7.2.2 Calculation of Loss Reduction Effect of Reducing Repeated Electric Supply Rate 115

7.3 Analysis of Power Sales Quantity Composition 117

7.3.1 Influence of Power Sales Quantity without Loss or Power Sales Quantity with Loss on Line Loss Rate 117

7.3.2 Calculation of Influence of Transit Electric Supply on Line Loss Rate 119

7.4 Multiple-factor Analysis of Changes in Electricity Line Losses 122

7.4.1 Loss Structure Coefficient and Electricity Line Loss Increase Rate Function 122

7.4.2 Loss Structure Function and Calculation of Increase in Electricity Line Losses 125

7.5 Marginal Line Loss Rate and Optimal Distribution of Increase in Electric Supply 126

7.5.1 Marginal Line Loss Rate 126

7.5.2 Optimal Distribution of Increase in Electric Supply 126

**8 Theoretical Calculation of Electric Energy Losses of Power Grid Units 131**

8.1 Classification of Electric Energy Losses 131

8.1.1 Classification of Electric Energy Losses by Whether Theoretical Calculation is Feasible 131

8.1.2 Classification of Calculable Technical Losses by Change Law 131

8.1.3 Classification of Electric Energy Losses by Different Power Grid Units 131

8.2 Calculation of Electric Energy Losses of Overhead Lines 132

8.2.1 Calculation of Corona Losses of Power Transmission Lines 132

8.2.2 Calculation of Resistance Heat Losses of Overhead Lines 136

8.2.3 Calculation of Electric Energy Losses of Low-voltage Lines 137

8.3 Calculation of Electric Energy Losses of Cable Lines 141

8.3.1 Calculation of No-load Losses (Dielectric Losses in Insulating Layers) of Cable Lines 141

8.3.2 Calculation of Load Losses of Cable Lines 141

8.4 Calculation of Electric Energy Losses of Main Transformers 145

8.4.1 Active Power Losses of Main Transformers 145

8.4.2 Calculation of Electric Energy Losses of Main Transformers 146

8.4.3 Calculation of Electric Energy Losses of Main Transformers in Parallel Operation 147

8.5 Calculation of Electric Energy Losses of Other Electrical Equipment 148

8.5.1 Shunt Capacitors 148

8.5.2 Shunt Reactors and Series Current-limiting Reactors 149

8.5.3 Synchronous Compensator 149

8.5.4 Watt-hour Meter and Other Instruments 150

**9 Calculation of Electric Energy Losses of Multi-branch Lines 151**

9.1 Basic Method for Calculating Electric Energy Losses of Multi-branch Lines 151

9.1.1 Weighted Average Method 151

9.1.2 Point by Point Section Simplification Method 153

9.2 Equivalent Resistance Method and Calculation of Electric Energy Losses of Distribution Transformers 157

9.2.1 Equivalent Resistance of a Line 157

9.2.2 Calculation of Electric Energy Losses of Distribution Transformers 159

9.2.3 Equivalent Resistance and Equal Resistance of Common Distribution Transformers 159

9.2.4 Calculation of Electric Energy Losses by Equivalent Resistance Method 161

9.3 Double Component Balance Method 162

9.4 Dispersion Coefficient Method 168

9.4.1 Calculation of Power Losses of Typically Distributed Loads 168

9.4.2 Dispersion Coefficient 169

9.4.3 Conversion of Length Under Different Sectional Areas of Conductors 170

9.4.4 Calculation of Power Losses of Complexly Distributed Loads 170

9.4.5 Calculation of Electric Energy Losses by Dispersion Coefficient Method 171

9.5 Calculation of Electric Energy Losses of Multi-branch Lines by Voltage Drop Method 173

9.5.1 Calculation of Line Loss Rate by Proportionality Coefficient Method 173

9.5.2 Calculation of Line Losses by Voltage Drop Measurements 175

9.6 Comparison and Selection of Calculation Methods of Electric Energy Losses of Multi-branch Lines 178

9.7 Calculation of Loss Reduction Benefits after Connection of Distributed Resources to System 178

9.7.1 Calculation of Loss Reduction Benefits During Generation Period of Distributed Resources 179

9.7.2 Calculation of Line Loss Change During Consumption Period of Distributed Resources 181

9.7.3 Calculation of Loss Reduction Benefits During the Full Period of Distributed Resources 181

9.7.4 Benefit Evaluation of Distributed Energy System 181

**10 Calculation of High-voltage Power Grid Losses 183**

10.1 Characteristics and Requirements of Loss Calculation 183

10.1.1 Classification of High-voltage Power Grids 183

10.1.2 Characteristics of Regional Power Grids and Requirements of Loss Calculation 184

10.1.3 Characteristics of Prefectural Power Grids and Requirements of Loss Calculation 184

10.2 Real-time Loss Measuring Method for High-voltage Power Grids 184

10.2.1 Function and Method of State Estimation 185

10.2.2 Real-time Calculation of Losses by State Estimation Combined with Excel 186

10.2.3 Typical Day Method Based on Actual Load Measurement and State Estimation 187

10.2.4 Comprehensive Analysis Method of Losses Based on Real-time System Data 189

10.3 Equivalent Node Power Method for Calculation of High-voltage Power Grid Losses 190

10.3.1 Equivalent Node Power and its Distribution 190

10.3.2 Relationship between Power Losses and Electric Energy Losses Under Distribution of Equivalent Node Power 191

10.3.3 Analysis of Equivalent Node Power Method 195

10.4 Calculation of Losses of High-voltage Power Grids Based on Power Losses under Three Modes 196

10.5 Calculation and Analysis of Samples 197

10.5.1 Verification of Loss Calculation of Standard Power Grid with 39 Nodes 197

10.5.2 Three-mode Calculation Based on Total Loads and Measured Loss Power over 24 h in one Province During 2004 204

**11 Analysis and Calculation of Loss Allocation 209**

11.1 Occurrence of Loss Allocation Problem and Possible Solutions 209

11.1.1 Analysis of Double Load Power Supply Model 210

11.1.2 Analysis of Triple Load Power Supply Model 211

11.1.3 Possible Solutions to Loss Allocation 214

11.2 Theoretical Preparation for Loss Allocation 214

11.2.1 Three-mode Section Division of Active Load Duration Curve 214

11.2.2 Calculation of Influence of Transit Electric Supply on Electricity Line Losses 214

11.2.3 Calculation of Marginal Line Loss Rate 215

11.2.4 Calculation of Optimal Distribution of Increased Electric Supply 215

11.3 Analysis and Calculation of Allocation of Increased Losses in Regional Power Grids 216

11.3.1 Allocation of Losses in the Main Part of Regional Power Grids to Provincial Power Grids 216

11.3.2 Allocation of Increased Losses Caused by Power Transmission and Reception in Inter-Provincial Power Grids 216

11.4 Calculation of Loss Allocation Under Complex Trading Setup 223

11.4.1 Loss Allocation for Pilot Project of Direct Electricity Purchase by Large Customers Under “One to Many” Model 223

11.4.2 Shapley Method of “Many to Many” Loss Allocation 224

11.4.3 Marginal Loss Coefficient-Based GMM Method 228

**12 Technical Measures for the Reduction of Line Losses 231**

12.1 Selection of Reasonable Connection Mode and Operation Mode 231

12.1.1 Introduction of High-voltage Grids to Large Cities or Load Centers 231

12.1.2 Stepping up of Power Grid Voltage, Simplification of Voltage Class, and Reduction of Repeated Substation Capacity 232

12.1.3 Reasonable Determination of Closed Loop Operation or Open Loop Operation of Loop Net, or Change of Break Points of Loop Net 232

12.1.4 Realization of Economic Power Distribution by Longitudinal and Transverse Voltage Regulating Transformer or Series Capacitor 235

12.1.5 Prevention of Remote Supply by Nearby Power or Round about Power Supply 236

12.1.6 Reasonable Arrangement of Equipment Overhaul and Practice of Live-Line Overhaul 237

12.1.7 Replacement of Conductors, Installation of Composite Conductors, or Construction of Secondary Loop Lines 238

12.2 Reasonable Determination of Voltage Level of Power Grids 238

12.3 Utilization of Reactive Power Compensation Equipment and Increase in Power Factor 239

12.3.1 Calculation of Loss Reduction Effect of Reactive Compensation 240

12.3.2 Optimal Configuration of Reactive Compensation Equipment in Power Grids 244

12.3.3 Exploitation of Reactive Potential and Reduction of Reactive Consumption 245

12.4 Economical Operation of Transformers 245

12.4.1 Economical Operation of Two-Winding Transformers of the Same Model 245

12.4.2 Economical Operation of Two-Winding Transformers of Different Models 247

12.4.3 Economical Operation of Three-Winding Transformers of Different Models 251

12.5 Adjustment and Balancing of Loads 255

12.5.1 Adjustment of Load Curves 255

12.5.2 Balancing the Loads of Lines or Transformers, and Adjusting the Power Sources of Dual Power Customers 256

12.5.3 Balancing Three-Phase Loads 258

12.6 Strengthen Power Grid Maintenance 258

12.7 Strengthen Power Consumption Management and Measuring Management 259

12.8 Application of New Designs, New Materials, and New Technologies 259

12.8.1 New Design for Loss Reduction in Ground Wires of High-voltage Transmission Line 260

12.8.2 Application of Energy-saving Hardware and Energy-saving Conductors 260

12.8.3 Application of Harmonic Control Technology and High-Temperature Superconducting Technology 261

**13 Line Loss Prediction and Loss Reduction Plan for Power Grids 263**

13.1 Univariate Prediction of Electricity Line Losses and Line Loss Rate 263

13.1.1 Basis for Predicting the Indicator of Line Loss Rate 263

13.1.2 Univariate Prediction of Electricity Line Losses 266

13.1.3 Univariate Prediction of Line Loss Rate 266

13.2 Multivariable Prediction of Electricity Line Losses and Line Loss Rate 267

13.2.1 Multivariable Prediction of Electricity Line Losses 268

13.2.2 Multivariable Prediction of Line Loss Rate 268

13.2.3 Rolling Prediction Method 272

13.3 Main Content and Preparation Process of Loss Reduction Plan 276

13.3.1 Content and Preparation Basis of the Loss Reduction Plan 276

13.3.2 Preparation of the Loss Reduction Plan 277

13.3.3 Implementation and Monitoring of the Loss Reduction Plan 279

13.3.4 Introduction of an Example of the Loss Reduction Plan 280

**14 Analysis of the Influence of Power Grid Line Losses on Power Grid Enterprises 281**

14.1 Influence of Line Losses on the Profits of Power Grid Enterprises 281

14.1.1 Calculating the Profits of Power Grid Enterprises 281

14.1.2 Break-Even Point Power Sales Quantity 282

14.1.3 Profit and Tax Amount per Unit Power Sales Quantity 282

14.1.4 Analysis of Factors Affecting Profits 282

14.2 Link Cost and Link Electricity Price 283

14.2.1 Significance of Division of Internal Links of Power Grid Enterprises 283

14.2.2 Calculation Model of Link Electricity Price Under a Simple Electric Supply Structure 283

14.2.3 Calculation Model of Link Electricity Price Under a Complicated Electric Supply Structure 284

14.2.4 Equivalent Merging of Parallel Electric Supply Structure 285

14.3 Influence of Line Losses on the Composition of Multi-section Electricity Prices 289

14.3.1 Type of Electricity Price and Comparison of Calculation Methods 289

14.3.2 Analysis of Composition of Two-Section Electricity Prices Under the Single Electricity Purchaser Model 290

14.3.3 Recursive Calculation of Multi-Section Electricity Prices 292

14.3.4 Controlling the Aggregate Level of Electricity Price 294

14.3.5 Analysis and Discussion 296

14.4 Analysis of Coal–Electricity Price Linkage 297

14.4.1 Interpretation of the Existing Policy of Coal–Electricity Price Linkage 297

14.4.2 Analysis of the Linkage between On-Grid Price and Coal Price 300

14.4.3 Linkage between Sales Price and On-Grid Price 302

14.5 Analysis of Electricity Price Factor in Post Project Evaluation 303

14.5.1 Reverse Calculation of Mark-Up in Link Output End 304

14.5.2 Calculation of Mark-Up Allocation Coefficient of Simplified Electric Supply Network 304

14.5.3 Mark-Up Calculation of Complicated Electric Supply Structure 305

14.5.4 Calculation of Annual Power Sales Mark-Up Revenue of a Single Transmission and Transformation Project 307

**15 Management and Utilization of Line Loss Mass Information for an Electric Power System 309**

15.1 Evaluation and Functions of Two Management Information Systems Under the Guidelines 309

15.1.1 Functional Design Requirements for Two Types of Software 309

15.1.2 Functions of Line Loss Calculation and Management Information Systems Developed by Provincial Power Grid Enterprises 310

15.1.3 Integrated Management System for Theoretical Calculation of Line Losses Developed by Regional Power Grid Enterprise 311

15.1.4 New Management Requirements 312

15.2 Value Creation and Support Processes for Power Grid Enterprises 313

15.2.1 Information-Oriented Development of Large Enterprises and Application of Enterprise Resource Planning 313

15.2.2 Value Creation and Support Processes of Power Grid Enterprises 314

15.3 Composition of Model Driven Decision Support System 315

15.3.1 Structure and Functions of Decision Support System 315

15.3.2 Intelligent Decision Support System and Group Decision Support System 318

15.3.3 Conceptual Model of Power Grid Enterprise 319

15.3.4 Business Conceptual Model of Power Grid Enterprise 320

15.4 Utilization of Line Loss Mass Information 322

15.4.1 Basic Concept of Data Warehouse 322

15.4.2 Basic Concepts of Data Mining and Online Analysis 323

15.4.3 Application of Data Warehouse Technology in Electric Power Dispatching and Marketing Systems 324

15.4.4 In-Depth Utilization of Line Loss Mass Information – Integration of Data in Dispatching and Marketing Systems 328

**Appendix A Calculation Curve of Corona Loss Power Δ P_{cor }335**

**Appendix B Calculation of Electrical Parameters of Power Grid Units 341**

B.1 Parameters of Overhead Lines 341

B.1.1 Parameters of Overhead Transmission Lines 341

B.1.2 Parameters of Steel Conductor Overhead Lines 343

B.1.3 Parameters of Two-Wire One-Ground Overhead Lines 343

B.2 Parameters of Transformer 343

B.2.1 Parameters of Two-Winding Transformer 343

B.2.2 Parameters of Three-Winding Transformer 344

**Appendix C Derivation of Loss Factor Formula by Subsection Integration Method 347**

**Appendix D Actual Measurement Analysis of No-load Power Losses and Relationship between No-load Current and Voltage of Distribution Transformers 351**

D.1 Actual Measurement Analysis of Δ*P*_{0}(*U*) of General Transformers 351

D.2 Actual Measurement Analysis of Δ*P*_{0}0(*U*) of Low Loss Transformers 351

D.3 Actual Measurement Analysis of *I*_{0}(*U*) of General Transformers 352

References 353

Index 357