Mathematical Models of Electrical Network Systems (eBook)

Preface 6
Contents 8
Mathematical Model of Network System 11
1 Introduction 12
1.1 Network System 12
1.2 Modeling Network System 13
1.3 Solving Network System 14
1.4 Network Laws 15
2 Basic Notions 17
2.1 Network Graph Notions 17
2.2 Electrical Network Notions 17
3 Algebraic Model of Network Graph 20
3.1 Cut-Set and Loop-Set Vectors 20
3.2 Topology of Network 21
3.3 Topological Model of Loop-Sets 22
3.4 Topological Model of Cut-Sets 23
3.5 Orthogonality of Cut-Sets and Loop-Sets 25
3.6 Linear Space Model of Network Topology 26
3.7 Topological Transformation 27
3.8 Topological Matrix T 29
4 Algebraic Model of Network Currents 31
4.1 Matrix Model of Network Currents 31
4.2 Linear Space Model of Network Currents 32
4.3 Topological Transformation of Current Space 34
4.4 Current Relations Using Topological Matrix T 35
5 Algebraic Model of Network Voltages 37
5.1 Matrix Model of Network Voltages 37
5.2 Linear Space Model of Network Voltages 39
5.3 Topological Transformation of Voltage Space 40
5.4 Voltage Relations Using Topological Matrix T 41
6 Algebraic Model of Current–Voltage Vectors 43
6.1 Orthogonality of Current and Voltage Vectors 43
6.2 Current–Voltage Vector 44
6.3 Linear Space Model of Current–Voltage Vectors 45
6.4 Current–Voltage Vectors’ Relations Using Matrix T 46
6.5 Comment 47
7 Kirchhoff’s Laws Using Matrix T 49
7.1 Basic b-Dimensional Formulation 49
7.2 Pseudo-Unit Formulation 51
7.3 Current–Voltage Formulation 52
8 Current–Voltage Functional Relations 53
8.1 Branch Parameters 53
8.2 Ohm’s Law of Network System—Algebraic Model 56
8.3 Linear Space Model of Current–Voltage State Vectors in Network System 58
8.4 Formulation of Ohm’s Law Using Matrix T 61
8.5 Matrices of System Admittance and System Impedance 63
9 General Comments to Part I 65
9.1 Part II Application Examples 66
Application Examples 67
10 Applications of b-Dimensional Formulation of Kirchhoff’s Laws 68
10.1 Basic b-Dimensional Formulation 68
10.2 Pseudo-Unit Formulation 70
11 Network Solution Method Using Algebraic Network Model 74
11.1 Derivation of Method Using Matrix T 74
11.2 Classical Network Solution Methods 76
12 Method of Current and Voltage Sensitivity Analysis 78
12.1 Derivation of Method 78
12.2 Numerical Example 80
13 Method of Arbitrary Input Data 82
13.1 Description of Arbitrary Input Data 82
13.2 Derivation of Solution Equations 83
13.3 Solvability Analysis 85
13.4 Numerical Example 86
14 Logical Optimization Method of Network System 91
14.1 Example of Network System Matrices 91
14.2 Logical Analysis of Network System Equations 93
14.3 Application Possibilities 98
15 Current-Based Method of Load Flow Solution 100
15.1 Current State-Oriented Topology of Network 100
15.2 Current State Vector and Current Flow Method 101
15.3 Current-Based Load Flow Method 104
15.4 Verification of Method 105
15.5 Conclusions 107