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A FRAMEWORK FOR

POWER SYSTEM RESTORATION

by

DANIEL LINDENMEYER

Diplom-Ingenieur, Universit�at Karlsruhe (TH), Germany, 1996

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in

THE FACULTY OF GRADUATE STUDIES

(Department of Electrical and Computer Engineering)

We accept this thesis as conformingto the required standard

THE UNIVERSITY OF BRITISH COLUMBIA

September 2000

c Daniel Lindenmeyer, 2000

Abstract

The problem of restoring power systems after a complete or partial blackout is as old as thepower industry itself. In recent years, due to economic competition and deregulation, powersystems are operated closer and closer to their limits. At the same time, power systems haveincreased in size and complexity. Both factors increase the risk of major power outages. Aftera blackout, power needs to be restored as quickly and reliably as possible, and consequently,detailed restoration plans are necessary.

In recent years, there has also been an increasing demand in the power industry forthe automation and integration of tools for power system planning and operation. This isparticularly true for studies in power system restoration where a great number of simulations,taking into account di�erent system con�gurations, have to be carried out. In the past, thesesimulations were mostly performed using power- ow analysis, in order to �nd a suitablerestoration sequence. However, several problems encountered during practical restorationprocedures were found to be related to dynamic e�ects. On the one hand, simple rules thathelp to quickly assess these problems are needed. On the other hand, accurate modelingtechniques are necessary in order to carry out time-domain simulations of restoration studies.

In this work, the concept of a new framework for black start and power system restora-tion is presented. Its purpose is to quickly evaluate the feasibility of restoration steps,and if necessary, to suggest remedial actions. This limits the number of time-consumingtime-domain simulations, based on the trial{and{error principle, and helps to eÆciently �ndfeasible restoration paths. The framework's principle is to subdivide the problem of assess-ing the multitude of di�erent phenomena encountered during a restoration procedure intosubproblems. These can be assessed by simple rules formulated in the frequency and Laplacedomain.

This thesis concentrates on the initial stages of restoration where three major problemareas are identi�ed. The system frequency behavior after the energization of loads is assessedusing analysis in the Laplace domain and simpli�ed generator control system models. Theoccurrence of overvoltages is assessed in the frequency domain. Sensitivity analysis is usedin order to �nd the most eÆcient network change that can be applied to limit overvoltages.In case time-domain simulations need to be carried out, a method based on Prony analysisand fuzzy logic helps to limit the overall calculation time. Problems related to motor startsare evaluated by rules formulated in the frequency and Laplace domain. For these studies,a new induction motor parameter estimation method is developed that helps to build moreaccurate motor models. All the proposed rules are validated using time-domain simulationsbased on actual system data.

Of crucial importance in the restoration process is the black and emergency start oflarge thermal power plants with small hydro or gas turbines. These cases represent islanded

ii

ABSTRACT iii

system conditions with large frequency and voltage excursions that need careful investiga-tion. It is shown how they can be simulated using the Electromagnetic Transients Program(Emtp) by means of an emergency motor start case whose simulation results are comparedto measurements.

Contents

Abstract ii

List of Tables viii

List of Figures ix

Acknowledgement xiii

Quote xiv

1 Introduction 1

1.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Goals and Steps in Restoration . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Problems in Restoration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Thesis Motivation and Objective . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Literature Overview 8

2.1 Overview of Restoration Process . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Active Power System Characteristics and Frequency Control . . . . . . . . . 9

2.3 Reactive Power System Characteristics and Voltage Control . . . . . . . . . 10

2.4 Switching Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Protective Systems and Local Control . . . . . . . . . . . . . . . . . . . . . . 11

2.6 Power System Restoration Planning . . . . . . . . . . . . . . . . . . . . . . . 11

2.7 Power System Restoration Training . . . . . . . . . . . . . . . . . . . . . . . 12

2.8 Power System Restoration Case Studies . . . . . . . . . . . . . . . . . . . . . 12

2.9 Analytical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.10 Expert Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

iv

CONTENTS v

3 Structure of Framework 14

3.1 On-line and O�-Line Restoration Planning . . . . . . . . . . . . . . . . . . . 15

3.2 Framework Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Framework Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Framework Analytical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4.1 Power Flow Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4.2 Steady-State and Harmonic Analysis . . . . . . . . . . . . . . . . . . 19

3.4.3 Electromagnetic Transient Analysis . . . . . . . . . . . . . . . . . . . 20

3.4.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Frequency-Response Analysis 22

4.1 Overview of Frequency-Response Estimation Methods . . . . . . . . . . . . . 23

4.1.1 Simple Frequency-Response Estimation Methods . . . . . . . . . . . . 23

4.1.2 Other Frequency-Response Estimation Methods . . . . . . . . . . . . 24

4.2 Frequency Behavior of Hydro Units for Static Load Pick-up . . . . . . . . . 26

4.2.1 Assumptions for Model Development . . . . . . . . . . . . . . . . . . 26

4.2.2 Derivation and Veri�cation of Equations . . . . . . . . . . . . . . . . 28

4.2.3 Example for Load Pick-up . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.4 Example for Load Rejection . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.5 Remedial Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Voltage Response Analysis 35

5.1 Frequency-dependent Thevenin Impedance . . . . . . . . . . . . . . . . . . . 37

5.1.1 Impedance Change between Bus and Neutral . . . . . . . . . . . . . . 39

5.1.2 Impedance Change between two Buses . . . . . . . . . . . . . . . . . 40

5.1.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Sustained Overvoltages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Harmonic Overvoltages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3.1 Analysis and Assessment of Harmonic Overvoltages . . . . . . . . . . 47

5.3.2 Harmonic Characteristic of Transformers . . . . . . . . . . . . . . . . 51

5.3.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

CONTENTS vi

5.4 EMTP Time-Domain Result Evaluation . . . . . . . . . . . . . . . . . . . . 60

5.4.1 Prony Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4.2 Filter for Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4.3 Prony Analysis Results Evaluation . . . . . . . . . . . . . . . . . . . 65

5.4.4 Reasoning Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.4.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.5 Switching Transient Overvoltages . . . . . . . . . . . . . . . . . . . . . . . . 75

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6 Auxiliary System Analysis 77

6.1 Motor Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.1.1 Overview of Parameter Estimation Methods . . . . . . . . . . . . . . 77

6.1.2 Input Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.1.3 Nonlinear Optimization Procedure . . . . . . . . . . . . . . . . . . . 79

6.1.4 Motor Model without Saturation . . . . . . . . . . . . . . . . . . . . 80

6.1.5 Motor Model with Saturation . . . . . . . . . . . . . . . . . . . . . . 86

6.1.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2 Rules for Motor Start-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2.1 Thevenin Equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2.2 Estimation of Voltage Drop . . . . . . . . . . . . . . . . . . . . . . . 94

6.2.3 Estimation of Inrush Current . . . . . . . . . . . . . . . . . . . . . . 95

6.2.4 Estimation of Start-up Time . . . . . . . . . . . . . . . . . . . . . . . 95

6.2.5 Estimation of Thermal Behavior . . . . . . . . . . . . . . . . . . . . . 98

6.2.6 Estimation of Frequency Drops . . . . . . . . . . . . . . . . . . . . . 100

6.2.7 Remedial Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7 Emergency Start Case Study 105

7.1 Modeling of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.1.1 Electrical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.1.2 Synchronous Generator . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.1.3 Excitation System Model . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.1.4 Governor System Model . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.1.5 Turbine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

CONTENTS vii

7.1.6 Initialization of Excitation System, Governor System, and TurbineModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.1.7 Induction Motor Model . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8 Conclusions and Recommendations for Future Work 116

Bibliography 118

A Determination of Motor Load Characteristics from Measurements 137

B Soft Start of Motors 139

List of Tables

1.1 Frequency thresholds for Cyprus power system . . . . . . . . . . . . . . . . . 5

1.2 Voltage limits during restoration for Hydro-Qu�ebec power system . . . . . . 5

4.1 Governor data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Comparison of time constants . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1 Total sensitivity vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 Index for harmonic overvoltage assessment . . . . . . . . . . . . . . . . . . . 51

5.3 Sensitivity vectors for f=120 Hz . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.4 Total sensitivity vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.1 Induction motor nameplate data . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.2 Induction motor characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.3 Induction motor parameters in per unit . . . . . . . . . . . . . . . . . . . . . 87

6.4 Deviation of induction motor parameters . . . . . . . . . . . . . . . . . . . . 88

6.5 Induction motor parameters in per unit . . . . . . . . . . . . . . . . . . . . . 90

viii

List of Figures

1.1 Power System Operating States . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Power system restoration goals . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.1 On-line database for system restoration framework . . . . . . . . . . . . . . . 16

3.2 O�-line database for system restoration framework . . . . . . . . . . . . . . 17

3.3 Assessment of feasibility of restoration steps . . . . . . . . . . . . . . . . . . 18

3.4 Overview of analytical tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1 Generator supplying isolated load . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Governor block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Frequency deviation for 14 MW load pick-up . . . . . . . . . . . . . . . . . . 25

4.4 Overview of algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.5 Control block diagram of hydro unit . . . . . . . . . . . . . . . . . . . . . . 28

4.6 Simpli�ed control block diagram of hydro unit . . . . . . . . . . . . . . . . . 29

4.7 Comparison between full and simpli�ed model . . . . . . . . . . . . . . . . . 30

4.8 Frequency deviation for 14 MW load pick-up . . . . . . . . . . . . . . . . . . 32

4.9 Frequency deviation for 14 MW load rejection . . . . . . . . . . . . . . . . . 33

4.10 Minimum frequency as a function of load . . . . . . . . . . . . . . . . . . . . 33

4.11 Time at which minimum frequency occurs as a function of load . . . . . . . . 34


5.2 Overvoltage-versus-time capabilities of typical 230 kV system equipment . . 37

5.3 Thevenin equivalent circuit with variable parameters . . . . . . . . . . . . . 38

5.4 System at the beginning of a restoration procedure . . . . . . . . . . . . . . 41

5.5 60 Hz impedance at bus B3 as function of number of generators . . . . . . . 41

5.6 540 Hz impedance at bus B3 as function of number of generators . . . . . . . 42

5.7 Assessment whether power ow is needed . . . . . . . . . . . . . . . . . . . . 43

5.8 System during restoration procedure . . . . . . . . . . . . . . . . . . . . . . 45

ix

LIST OF FIGURES x

5.9 Voltage at bus B7 as function of �Yk= ~Y where ~Y = (2� j � 0:884) mS . . . . 46

5.10 Impedance at bus B3 for Cases 1 to 4 . . . . . . . . . . . . . . . . . . . . . . 47

5.11 Harmonic content of transformer inrush current at bus B3 for Cases 1 to 3 . 48

5.12 Voltage at bus B3 for Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.13 Voltages at bus B3 for Case 1 and Case 2 . . . . . . . . . . . . . . . . . . . . 49

5.14 Voltages at bus B3 for Case 3 and Case 4 . . . . . . . . . . . . . . . . . . . . 49

5.15 Harmonic content of transformer inrush current at bus B3 for Case 4 . . . . 50

5.16 Ideal saturation characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.17 Transformer saturation characteristic . . . . . . . . . . . . . . . . . . . . . . 53

5.18 Harmonics of transformer inrush current . . . . . . . . . . . . . . . . . . . . 54

5.19 System during restoration procedure . . . . . . . . . . . . . . . . . . . . . . 57

5.20 Impedance ZB1(120Hz) as a function of resistive changes . . . . . . . . . . . 58

5.21 Impedance ZB1(120Hz) as a function of inductive changes . . . . . . . . . . 59

5.22 Harmonic characteristic of transformer . . . . . . . . . . . . . . . . . . . . . 59

5.23 Weighting function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.24 Impedance ZB1 for resistive changes as a function of frequency . . . . . . . 60

5.25 Impedance ZB1 for inductive changes as a function of frequency . . . . . . . 61


5.27 Reasoning function for �lter . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.28 Membership function for relative appearance . . . . . . . . . . . . . . . . . . 65

5.29 Membership function for signal-to-noise ratio (SNR) . . . . . . . . . . . . . . 65

5.30 Monotonic reasoning in order to determine total relevance of mode . . . . . . 66

5.31 Amplitude membership function . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.32 Damping membership function . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.33 Membership function for tendency of damping . . . . . . . . . . . . . . . . . 68

5.34 Monotonic reasoning in order to determine total \closeness to steady state"of mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.35 Amplitude membership function . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.36 Damping membership function . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.37 Tendency of damping membership function . . . . . . . . . . . . . . . . . . . 71

5.38 Membership function for variance of damping . . . . . . . . . . . . . . . . . 72

5.39 Final reasoning process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.40 Emtp signal and predicted signal . . . . . . . . . . . . . . . . . . . . . . . . 73

5.41 Emtp signal and predicted signal for m>15 . . . . . . . . . . . . . . . . . . 73

LIST OF FIGURES xi

5.42 Amplitude of 180 Hz mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.43 Damping of 180 Hz mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.1 Induction motor model without saturation . . . . . . . . . . . . . . . . . . . 80

6.2 Induction motor model with stator part replaced by Thevenin equivalent circuit 83

6.3 Values for objective function . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.4 Number of iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.5 Induction motor model with saturation . . . . . . . . . . . . . . . . . . . . . 88

6.6 Calculation of currents and describing functions . . . . . . . . . . . . . . . . 89

6.7 Current characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.8 Powerfactor characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.9 Torque characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


6.11 Sample system for motor start . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.12 Excitation system of sample system . . . . . . . . . . . . . . . . . . . . . . . 94

6.13 Thevenin equivalent circuit of sample system . . . . . . . . . . . . . . . . . . 94

6.14 Voltage drop during motor start-up . . . . . . . . . . . . . . . . . . . . . . . 95

6.15 Inrush current during motor start-up . . . . . . . . . . . . . . . . . . . . . . 96

6.16 Mechanical and electrical torque for f=1:0 p:u: and V =0:5 p:u: . . . . . . . 98

6.17 Current for f=1:0 p:u: and V =0:5 p:u: . . . . . . . . . . . . . . . . . . . . . 98

6.18 Rotor speed during motor start-up . . . . . . . . . . . . . . . . . . . . . . . 99

6.19 Motor relay characteristic compared to averaged starting current . . . . . . . 99

6.20 Active power during motor start-up compared to approximations . . . . . . . 102

6.21 Frequency deviation for motor load pick-up . . . . . . . . . . . . . . . . . . . 103

7.1 One-line diagram of the Oconee emergency electrical power system (ESF) . . 106

7.2 Block diagram of the Keowee generator excitation system . . . . . . . . . . . 107

7.3 Block diagram of the Keowee governor system . . . . . . . . . . . . . . . . . 108

7.4 Block diagram of the Keowee hydro turbine . . . . . . . . . . . . . . . . . . 109

7.5 Transient current characteristic, HPI motor . . . . . . . . . . . . . . . . . . 112

7.6 Transient voltage characteristic, HPI motor . . . . . . . . . . . . . . . . . . . 112

7.7 ESF test: Comparison between measured and simulated frequency . . . . . . 113

7.8 ESF test: Comparison between measured and simulated voltage . . . . . . . 113

7.9 ESF test: Comparison between measured and simulated current . . . . . . . 114

LIST OF FIGURES xii

7.10 ESF test: Comparison between measured and simulated �eld voltage . . . . 114

7.11 ESF test: Comparison between measured and simulated �eld current . . . . 115

7.12 ESF test: Comparison between measured and simulated power . . . . . . . . 115

A.1 Thevenin equivalent for locked-rotor operation (t=0+) . . . . . . . . . . . . 138

A.2 Thevenin equivalent for full-load operation (t!1) . . . . . . . . . . . . . . 138

B.1 Start-up time at voltage V = 1:0 p:u: . . . . . . . . . . . . . . . . . . . . . . 140

B.2 Start-up time at frequency f = 1:0 p:u: . . . . . . . . . . . . . . . . . . . . . 140

B.3 Start-up current at voltage V =1:0 p:u: . . . . . . . . . . . . . . . . . . . . . 141

B.4 Start-up current at frequency f=1:0 p:u: . . . . . . . . . . . . . . . . . . . . 141

Acknowledgement

I wish to express my sincere gratitude to all the people who have assisted me in the accom-plishment of this thesis work:

To my extraordinary, beautiful, and magni�cent wife Lise. Thank you very much foryour love, understanding, never-ending patience, and constant encouragement. Thank youvery much for the great time and memories we share.

To my parents for the in�nite love, support, and encouragement through all my dreams,adventures, and life. To my parents in-law, and all my relatives and friends who supportedme with their love, kindness, and happiness.

I would like to express my deepest gratitude to my thesis supervisors, Dr. HermannW. Dommel and Dr. Prabha Kundur, whom I respect and admire tremendously. In partic-ular, I would like to thank Dr. Prabha Kundur for the great opportunity to carry out a partof my work at Powertech Labs. Thank you both very much for your encouragement andsupport.

Special thanks to Dr. Ali Moshref for helping me to �nd the topic for this thesis, for hisfriendship, encouragement, support, and technical guidance. His expertise and backgroundin the area of power systems not only made this work possible but also tremendously enrichedmy professional life.

Many thanks to Dr. Jos�e Mart�� for introducing me to the principle of the Emtp in hisoutstanding course at UBC. To Dr. Tak Niimura whose expertise in arti�cial intelligence andfuzzy logic helped me enormously during the development of the results evaluation algorithm.

To all the other former and current members of the UBC Power Group for their friendship,support, encouragement, and cooperation.

I am much indebted to Dr. Amir M. Miri, whom I admire and respect tremendously, forhis support and encouragement to go to Canada and pursue a Ph.D. program.

The �nancial assistance of the Natural Science and Engineering Council of Canada, andof B.C. Hydro & Power Authority, through funding provided for the NSERC-B.C. HydroIndustrial Chair in Advanced Techniques for Electric Power System Analysis, Simulationand Control, are gratefully acknowledged.

Many thanks to the engineers at Powertech Labs for providing me with the system dataand technical expertise needed for my studies, and to Chris Schae�er and Aldean Benge atDuke Energy for their cooperation and for letting me use the results of our emergency startcase study in this thesis.

Vancouver, BC, Canada Daniel LindenmeyerSeptember 8, 2000

xiii

\Everything should be made as simple as possible, but not simpler."

Albert Einstein

xiv

Chapter 1

Introduction

1.1 General Background

The operating conditions of power systems can be classi�ed as �ve di�erent states: normal,alert, emergency, in extremis, and restorative (see Figure 1.1) [73, 143]. In the normal

Normal E,I

Restorative E, I Alert E,I

In extremis E,I Emergency E,I

"E": Equality constraints"I": Inequality constraints"-": Negation

Figure 1.1: Power System Operating States

operating state, all system variables are within the normal range and no equipment is beingoverloaded. The system can withstand a contingency without its security being threatened,and both equality and inequality constraints are satis�ed. In the alert operating state,the system's security level is reduced, but the equality and inequality constraints are still

1

1.2. Goals and Steps in Restoration 2

satis�ed. Even though the system is still operated within allowable limits, a contingencymight lead to an emergency state, or, in case of a severe disturbance, to an in extremis state.In the emergency state, the inequality constraints are violated when the system operates at alower frequency with abnormal voltages, or with some portion of the equipment overloaded.After the system has entered an emergency state as a consequence of a severe disturbance, itmay be led back to an alert state by applying emergency control actions. If these measuresare not applied successfully, the system may enter an in extremis state where both inequalityand equality constraints are violated. Cascading outages might lead to partial or completeshut-downs of the system [143].

The system condition where control action is being taken to reconnect all the facilities,to restore system load, and to eventually bring the system back to its normal state, is calledthe restorative state. In this state the inequality constraints are satis�ed and the equalityconstraints are violated.

Due to the deregulation of the power industry worldwide, and the almost revolutionarychanges in the industry structure, power systems are operated closer and closer to theirlimits. Furthermore, in recent years, they have grown considerably in size and complexity.This has led to an increasing number of major blackouts, such as the large power outageson the West Coast of North America in 1996 or the Brazilian blackout in 1999.

Disturbances that can cause such power blackouts are natural disasters, line overloads,system instabilities, etc. Furthermore, temporary faults such as lightning, even if clearedimmediately, can initiate a \domino e�ect" that might lead to a partial or complete outage,involving network separation into several subsystems, and load shedding. After power black-outs, the system has to be restored as quickly and eÆciently as possible. In this restoration,the initial cause of the outage is of secondary importance and it might be futile to investigateit [2].

Although power outages di�er in cause and scale, virtually every utility has experiencedblackouts and gone through restoration procedures. As a consequence, there is an increasinginterest in systematic restoration procedures, tools, and models for on-line restoration, aswell as for restoration planning. A speedy, e�ective, and orderly restoration process reducesthe impact of a power outage on the public and the economy, while reducing the probabilityof equipment damage [2].

1.2 Goals and Steps in Restoration

Even though each power blackout and restoration scenario is a unique event, there are certaingoals and steps that are common in all restoration procedures. The goals in restoration, asgenerally de�ned in [75, 153, 155], are shown in Figure 1.2. They involve almost all aspectsof power system operation and planning.

Each restoration procedure that follows a complete or partial blackout of a power systemcan be subdivided into the following steps [2, 36]:

1. Determination of System Status. In this stage, the boundaries of energized areas are

1.2. Goals and Steps in Restoration 3

Tie LineUtilization

LoadManagement

MWManagement

MVARManagement

StabilityInspection

PathManagement

System Restoration

Black-StartUnit Operation

Non-Black-StartUnit Operation

MechanicalInspection

FuelInspection

Man-PowerDispatch

Static MVARInspection

SwitchingSequenceInspection

TransientVoltage

Inspection

FaultInspection

CapacityInspection

Figure 1.2: Power system restoration goals

identi�ed, and frequencies and voltages within these areas are assessed. Furthermore,in cases where no connections to neighboring systems exist, black start (or cranking)sources are identi�ed in each subsystem and critical loads are located.

2. Black Start of Large Thermal Power Plants. Large thermal power plants have to berestarted within a certain period of time. For example, hot restart of drum type boilersis only possible within thirty minutes. If it cannot be accomplished and the boiler isnot available for four to six hours, a cold restart has then to be performed. Thermalpower plants can be restarted by means of smaller units with black start capability,i. e. power plants that can be started and brought online without external help andwithin a short period of time. Power plants with black start capability are hydro, gas,or diesel power plants. After such a power plant has been brought to full operation,a high voltage path to a large thermal power station is built and the thermal unit'sauxiliaries which are driven by large induction motors are started. Along the path,\ballast" loads have to be supplied to maintain the voltage pro�le within acceptablelimits and to prepare a load base for the thermal units. Additional smaller units canalso be brought online through the path to improve system stability [46].

3. Energization of Subsystems. In case of a large power blackout, it is advantageousin most cases to section the power system into subsystems in order to allow parallelrestoration of islands, and to reduce the overall restoration time. Within each sub-system, starting from a large thermal power station, the skeleton of the bulk powersystem is energized. Paths to other power plants and to the major load centers arebuilt, and loads are energized to �rm up the transmission system. At the end of thisstep, the network has suÆcient power and stability to withstand transients as a resultof further load pick-up and addition of large generating units.

1.3. Problems in Restoration 4

4. Interconnection of Subsystems. In this stage of the power system restoration process,the subsystems are interconnected. Eventually, remaining loads are picked up and thesystem performs its transition to the alarm or normal state.

Among the above four general steps of the restoration process, the second and third step arethe most critical ones. Mistakes in these stages can lead to unwanted tripping of generatorsand load shedding due to extensive frequency and voltage deviations, and consequently toa recurrence of the system outage. Because of time-critical boiler-turbine start-up char-acteristics and possible further equipment damage, extensively prolonged restoration timesmay occur, resulting in a much higher impact on the public and industry, and an increaseddamage to the economy.

1.3 Problems in Restoration

During power system restoration, a multitude of di�erent phenomena and abnormal condi-tions may occur. The problems encountered during restoration can be subdivided into threegeneral areas [2, 36, 90, 91, 96]:

1. Active Power Balance and Frequency Response. During the restoration process, twodi�erent aspects of this type of problem can be identi�ed. The �rst one is the blackstart of large thermal power plants, where large auxiliary motor loads are picked up,using relatively small hydro generators, diesel, or gas turbines. This can result in largefrequency excursions and consequently in an activation of underfrequency load sheddingrelays and, in the worst case, in the loss of already restored load and a recurrence ofthe blackout [36]. Due to the importance of this problem, it is de�ned as a problemarea in its own, as discussed further below.

The second aspect is the pick-up of cold loads. When the network is extended, powerplants are added to the generation, and loads are picked up, it is necessary to preservea balance between active load and generation. In the case this balance is disturbed,frequency deviations result. If these are extensive, an unwanted activation of load-shedding schemes can occur, and newly connected loads can be lost again. In theworst case, the frequency decline may reach levels that can lead to the tripping ofsteam turbine generating units as a consequence of the operation of underfrequencyprotective relays. This is due to the fact that the operation of steam turbines belowa frequency of 58.8 Hz is severely restricted as a result of vibratory stress on the longlow-pressure turbine blades [143].

Thus, in order keep the frequency deviations within allowable limits, the load incre-ments should not exceed a certain level. However, if the load increments are too small,the overall restoration time will be unnecessarily prolonged [5, 46, 90].

As an example for an islanded system, Table 1.1 shows typical load shedding frequencylimits for the (50 Hz) Cyprus power system [39]. The table indicates that at a frequencydecline of 3 Hz almost all of the load will be disconnected.

1.3. Problems in Restoration 5

Table 1.1: Frequency thresholds for Cyprus power system

Frequency / [Hz] 49.0 48.8 48.4 47.8 47.0Load shedding / [ % ] 15 20 25 10 10

Delay / [s] 0.2 0.2 0.35 0.35 0.35Cumulative shed / [ % ] 15 35 60 70 80

2. Reactive Power Balance and Voltage Response. Analogous to the active power balanceit is necessary to maintain a balance in reactive power. High charging currents, orig-inating from lightly loaded transmission lines, can lead to the violation of generatorreactive capability limits and to the occurrence of sustained (power frequency) over-voltages. These may cause underexciation, selfexcitation, and instability. Sustainedovervoltages can also cause the overexcitation of transformers and the generation ofharmonic distortions.

Transient overvoltages are a consequence of switching operations on long transmissionlines, or of switching of capacitive devices, and may result in arrester failures.

Harmonic resonance overvoltages are a result of system resonance frequencies close tomultiples of the fundamental frequency in combination with the injection of harmonics,mainly caused by transformer switching. They may lead to long-lasting overvoltages,resulting in arrester failures and system faults. Due to the small amount of loadconnected to the system, especially at the beginning of a restoration process, thevoltage oscillations are lightly damped and can last for a long time, reaching very highamplitudes. This e�ect can be aggravated by transformer overexcitation as a result ofsustained overvoltages, and power electronics [90].

As an example, the tolerable voltage deviations during restoration for the HydroQu�ebec power system are shown in Table 1.2 [172]. It shows that in the case of tem-porary overvoltages the overvoltage duration has to be taken into account in additionto the amplitude.

Table 1.2: Voltage limits during restoration for Hydro-Qu�ebec power system

Steady-state voltages / [p.u.] 0:9 � V � 1:05Switching overvoltages / [p.u.] V � 1:8Temporary overvoltages / [p.u.] V � 1:5 (8 cycles)

3. Auxiliary Systems. The auxiliaries of large thermal power plants are essentially largeinduction motors driving pumps and fans. When they are energized during black oremergency starts, several problems can occur. The high reactive currents that aredrawn during start-up may lead to voltage depressions which can result in overheatingand permanent damage of machine windings. Furthermore, the high reactive inrush

1.4. Thesis Motivation and Objective 6

currents can exceed the reactive capability limits of generators which may trip due tothe operation of underexcitation limiters [8, 95, 102].

In severe cases, the decrease in accelerating torque as a result of reduced voltage orfrequency may lead to a failure in bringing the motor up to its rated speed. The start-up of large auxiliary motors may also lead to frequency dips, causing unwanted loadshedding, or generator tripping [36, 102].

1.4 Thesis Motivation and Objective

We can conclude from the previous sections, that a number of complex and serious problemsneed to be resolved during power system restoration. Detailed analysis during both thesystem planning stage and during on-line restoration is therefore necessary. As will beshown in the literature survey in Chapter 2, most of the research work that has been done sofar in the area of power system restoration focuses on the application of arti�cial intelligencetechniques in restoration path development. Expert systems or similar approaches, however,are not always easily generalized since circumstances and philosophies are di�erent for eachutility.

The evaluation of the feasibility of restoration steps has been mainly assessed by meansof (quasi-)steady-state analysis while ignoring dynamic e�ects. However, there is a trend inindustry to include dynamic e�ects in power system restoration planning. In order to takethese e�ects into consideration, a number of additional analytical tools, such as transientstability and electromagnetic transient programs have to be utilized, and new models andmodeling techniques need to be developed to allow for the simulation of extreme o�-nominalfrequency and voltage conditions [35, 36].

The multitude of di�erent phenomena and abnormal conditions during restoration makesit impractical to solve all problems at once, and the combinatorial nature of restorationproblems makes it diÆcult or even impossible to investigate the feasibility of every restorationstep combination. Applying the above mentioned simulation tools results in a large amountof time necessary to build suitable sequences during restoration planning, and makes theirapplication in on-line restoration almost infeasible due to time limitations.

The objective of this thesis is to give simple and approximate rules to assess the feasibilityof restoration steps, and to provide modeling techniques for the simulation of abnormalvoltage and frequency conditions. The rules provide operators with a simple and speedymethodology during on-line restoration, and limit the number of time-consuming simulationsbased on the trial{and{error principle during restoration planning.

In Chapter 2, a comprehensive overview of the literature published in the area of restora-tion during the last two decades is given. Chapter 3 presents the general structure of theproposed framework. In Chapter 4 rules that deal with the estimation of frequency responsesof prime movers are introduced. Chapter 5 describes methods that help to estimate and con-trol overvoltages, and to shorten calculation times during time-domain simulations. Chapter6 introduces rules for the start-up of induction motors and a technique that helps to buildmore accurate induction motor models from manufacturer data. All rules and methods are

1.4. Thesis Motivation and Objective 7

validated using examples based on practical system data 1. Chapter 7 explains the modeldevelopment for Emtp time-domain simulations of abnormal system conditions, by means ofa practical emergency start case study whose results have been veri�ed by measurements 2.Chapter 8 �nally summarizes the contributions of this thesis and gives an outlook on futurework to be done.

1System data has been kindly provided by Powertech Labs Inc., Surrey, BC, Canada.2System data and measurements have been kindly provided by Duke Energy, North Carolina, USA.

Chapter 2

Literature Overview

This chapter gives a literature survey that provides an overview of the relevant areas inrestoration. It is subdivided into ten di�erent topics. The �rst topic provides a generaloverview of the restoration process. It is followed by a discussion of active and reactive powersystem characteristics, and control of frequency and voltages during restoration. Then, topicscovering the basic restoration switching strategies are explained, and protective system andlocal control problems are discussed. After overviews of power system restoration planningand of case studies, the training of operators in restoration is reviewed. The �nal two topicscover the applications of analytical tools and expert systems in power system restoration.The restoration of distribution systems is not speci�cally addressed in this survey.

2.1 Overview of Restoration Process

This section covers publications that give a general introduction to the restoration process,as well as reports by technical committees dealing with system restoration.

In a typical restoration procedure, the stages of the restoration process are summarizedas follows [2, 3, 15, 36, 73, 110, 119, 189]: in the �rst stage, the system status is assessed,initial cranking sources are identi�ed, and critical loads are located. In the following stage,restoration paths are identi�ed and subsystems are energized. These subsystems are theninterconnected to provide a more stable system. In the �nal stage, the bulk of unservedloads is restored.

In 1986 the Power System Restoration Task Force was established by the IEEE PES Sys-tem Operation Subcommittee in order to review current operating practices, and to promoteinformation exchange. Its �rst two reports [90, 91] give a general overview and a comprehen-sive introduction to power system restoration. Restoration plans, active and reactive powersystem characteristics, and various restoration strategies are reviewed. Furthermore, a sur-vey on selected power disturbances and their restoration issues are given. These restorationissues are further discussed by the same authors in [3].

An international survey by CIGRE Study Committee 38.02.02 (modeling of abnormalconditions) on black start and power system restoration in 1990 [35], and a paper based

8

2.2. Active Power System Characteristics and Frequency Control 9

on this survey in 1993 [36], identi�ed the needs of the power industry for system restora-tion planning. These papers give an overview of black start and restoration methods. Inaddition, a number of examples based on actual operating experiences are given, and require-ments for modeling and simulation are recommended. Another CIGRE Task Force 38.06.04investigated expert system applications for power system restoration [37].

Other publications that provide a general overview of power system restoration can befound in [2, 15, 92, 94, 96].

2.2 Active Power System Characteristics and Frequency

Control

An overview of di�erent types of generating units, and their characteristics relevant torestoration is provided in [90]. Special emphasis is placed on the treatment of steam unitsin [45], whereas [100] focuses on nuclear power plants and provision for o�-site power duringrestoration.

Depending on the available cranking power, thermal units are either restarted hot orcold. Hot restarts allow for start-up with hot turbine metal temperature, whereas coldrestarts require a slow start-up in order to keep the turbine metal temperature changeswithin given limits [96]. These characteristics result in di�erent start-up times for di�erenttypes of generating units that have to be estimated and taken into account during restoration[2, 7, 49, 90, 96, 116].

The black start of generating units and subsequent cranking of other large thermal unitsis an important topic in restoration [35, 36], since mistakes in this early stage can lead toprolonged start-up times of thermal power plants, and consequently to a signi�cant delayin the restoration process. A number of publications deal with this topic: a method for theidenti�cation of black start sources is described in [155], and the proper start-up sequenceof power plants is discussed in [3, 96, 151, 155]. Several case studies show how gas turbines[102, 160, 203] or hydro units [46, 80, 149, 150, 160] were utilized as black start sources.In addition, a number of papers deal with the treatment of nuclear power plants after ablackout, using on-site diesel engines [74, 76, 232] or combustion turbines [102, 177] asemergency supply sources.

The pick-up of cold loads is discussed in a recently published doctoral thesis [12], andin a number of other papers [28, 31, 96, 103, 205, 233]. Picking up heavy loads during theinitial stages of restoration can lead to large frequency dips beyond a point of no return. Theprediction of frequency dips and the optimal distribution of generator reserves are subject of[5, 49, 133]. Load shedding schemes that can be utilized during restoration are introducedin [4, 39, 152, 151, 198, 226].

2.3. Reactive Power System Characteristics and Voltage Control 10

2.3 Reactive Power System Characteristics and Volt-

age Control

The overvoltages of concern during power system restoration are classi�ed as: sustainedpower frequency overvoltages, switching transients, and harmonic resonance overvoltages.They may lead to failure of equipment, such as transformers, breakers, arresters, etc. [95],and thus need to be analyzed thorougly. Systematic procedures dealing with the control ofsustained overvoltages by means of optimal power ow programs can be found in [87, 88, 114,172], and methods that deal with harmonic overvoltages in [147, 148, 172, 176]. Simple andapproximate methods that deal with the evaluation of transient and sustained overvoltages,and asymmetry issues during transmission line energization are discussed in [1, 9].

The preparation of the network for reenergization and the energization of high voltagetransmission lines during restoration are discussed in [2, 3, 80], and the special treatmentof cables in underground transmission systems in [96]. The application of shunt reactorsfor overvoltage control, particularly in the cases of lightly loaded transmission lines in thebeginning of the restoration procedures, is treated in [11, 12, 13, 80, 148, 172].

Another issue of importance, especially in the early stages of the restoration process,is the lead and lag reactive power capability limits of synchronous machines. These limitsare important for high charging current requirements of lightly loaded transmission lines,or for the high reactive currents drawn by the start-up of power plant auxiliary motors[6, 8, 49, 102]. The generator reactive power resources must therefore be optimized in suchcases [10, 155].

2.4 Switching Strategies

Two di�erent switching strategies can be applied during restoration. For the \all open"strategy, all breakers are opened immediately after the loss of voltage, whereas for the\controlled operation" strategy only selected breakers are opened [3, 96].

A great amount of research work that has been done in power system restoration concen-trates on �nding suitable restoration paths. Most of these approaches are based on arti�cialintelligence techniques that try to capture an operator's knowledge [68, 77, 86, 106, 123, 130,131, 159, 181, 195]. They are mostly restricted to speci�c systems and restoration philoso-phies. A general approach to this problem that subdivides each restoration process intogeneric restoration actions common to all utilities is proposed in [66, 151, 228].

Restoration paths have to be validated with respect to di�erent phenomena. An overviewof steady-state and time-domain based tools that can be utilized for this purpose can be foundin [2, 15, 98, 227]. Cases where simulation tools were actually applied to the validation ofrestoration paths can be found in [82, 169, 207].

2.5. Protective Systems and Local Control 11

2.5 Protective Systems and Local Control

The continual change in power system con�gurations and their operating conditions duringrestoration might lead to undesired operation of relays, since their settings are optimizedfor normal operating conditions. An overview of resulting delays, possible modi�cations ofrelays, and other protective system issues is provided in [101].

Relay schemes that allow for low frequency isolation, whereby local generation and localload are matched in order to avoid extensive delays due to the otherwise necessary completegenerator shutdowns, and controlled islanding schemes are treated in [3, 90]. A networkprotection scheme that focuses on stability issues is introduced in [194].

In cases where a dramatic decline in frequency occurs during the restoration process, itis necessary to reduce the amount of load that is connected, which can be accomplished bythe application of underfrequency load shedding schemes [39, 134]. In closing network loops,e.g., reconnecting two subsystems, sometimes a signi�cant standing phase angle di�erence(SPA) appears across the circuit breakers. This di�erence has to be reduced in order to closethe circuit breaker without causing instability of the system [81, 98, 229].

2.6 Power System Restoration Planning

The organization and implementation of a restoration plan will substantially determine itssuccess. Reports that deal with the actual deployment of restoration plans can be foundin [73, 184]. An overview of how restoration tasks can be shared most eÆciently betweenoperator and supporting computer systems is given in [94, 234]. Telecommunication issuesduring restoration are discussed in [65, 96, 184], and alarm issues in [92, 96].

One can distinguish between two di�erent restoration approaches: the \bottom-up",and the \top-down" restoration strategies [2, 90]. For the \top-down" approach, the bulkpower transmission system is established �rst, using interconnection assistance or hydroplants with large reactive absorbing capability. Subsequently, transmission stations and therequired substations are energized, generators are resynchronized, and loads are picked up[2, 68, 87, 88, 90, 119, 172, 204]. For the \bottom-up" strategy, the system is �rst dividedinto subsystems, each with black-start capability. Then, each subsystem is stabilized, andeventually the subsystems are interconnected [2, 73, 90, 110, 111, 160].

The veri�cation of steady-state models used for power system restoration planning isdiscussed in [111]. A number of publications describe the modeling of boilers, turbines,generators, controls, motors, etc. for black start studies in the time-domain, using stabilityprograms [46, 74, 177, 203], or the Electromagnetic Transients Program (Emtp) [76, 232].Special emphasis on the modeling of transmission lines for black start studies is given in [26],and the modeling of steam plants is treated in [45]. More information on analytical tools forrestoration planning is given in Section 2.9.

2.7. Power System Restoration Training 12

2.7 Power System Restoration Training

In order to provide operators with the necessary experience to con�dently deal with time-critical restoration problems, thorough training is essential. A general overview of operatortraining techniques for power system restoration is given in [92, 97], and methods showinghow restoration drills can be conducted and evaluated in practice are described in [224, 225].

Interactive and realistic operator training can be accomplished by means of an operatortraining simulator (OTS) [3, 55, 94, 167, 192, 208, 209, 210, 219, 236]. OTS can also beused to verify restoration plans [124], or in combination with knowledge-based systems forpower system restoration planning [75, 132, 189]. Information on simulator-expert-systemcombinations that are speci�cally designed to train operators at utilities and to replacehuman instructors, can be found in [32, 33, 34, 93, 113, 126, 138, 193].

2.8 Power System Restoration Case Studies

Most of the published case studies come from North America. A number of black startstudies can be found in [74, 76, 150, 203, 232]. The restoration of large power systemsin North America for Paci�c Northwest, Ontario-Hydro and Hydro-Qu�ebec is covered in[18, 87, 88, 172, 196, 204], and a report dealing with the restoration of a metropolitanelectrical system in [110]. A Mexican study of restoration policies and their application istreated in [73].

European case studies describe restoration experiences in French, Greek, and SwedishSystems [12, 40, 69, 119], Italy [46, 160], Slovenia [177], and Germany [218].

2.9 Analytical Tools

An overview of analytical tools and their application for solutions of power system restorationproblems is given in [36, 98]. The analytical tools can be subdivided into di�erent categoriesdepending on the frequency range of their application. The �rst type of tool is based onsteady-state analysis: (optimum) power ow programs [87, 88, 114, 125] are the most basictools used in system restoration. An overview of how they can be applied to di�erent typesof restoration problems is provided in [227], and applications to the control of sustainedovervoltages can be found in [95, 172]. Other tools, based on steady-state models, allowfrequency scans that can be used to investigate and control resonance conditions duringpower system restoration [95, 148, 172].

Operator training simulators (OTS) can be regarded as quasi steady-state tools, since inaddition to analyzing the power system's electrical behavior using power ow methods, theyallow taking into account long-term dynamics, i.e. the electromechanical and mechanicalaspects of power systems. Analytical tools that are applied to study frequency transientsof short-, mid-, and long-term range are introduced in [69, 207]. Electromagnetic transientsduring restoration are investigated using the Emtp [26, 95] or similar programs [169].

2.10. Expert Systems 13

Methods that apply stability and security methods to restoration can be found in [65, 185,186]. Other analytical tools that are of importance in restoration are short-circuit programs[98], tools for the reduction of standing phase angles (SPA) [81, 229], or tools that help toidentify system islands [145].

There is an increasing demand in the power industry to integrate existing tools and todevelop combination tools that allow studying di�erent phenomena over a broad range offrequency [36, 98]. Attempts in integration or combination of analytical tools, and theirapplications to practical problems, are described in [69, 75, 82, 207].

2.10 Expert Systems

Power system restoration problems are of a combinatorial nature, and their solution is oftenbased on the operator's knowledge and experience. Consequently, it is not surprising thatmost of the research that has been done in the area of system restoration has concentratedon arti�cial intelligence applications.

A general bibliographical survey on the application of expert systems to electric powersystems can be found in [235]. References [197, 237] give an overview of expert systemapplications in power system operation. An international survey among utilities that inves-tigated restoration expert systems and their application is presented in [37]. A number ofpublications address the requirements for knowledge-based systems, and give an overviewof how expert systems can be applied during di�erent stages of power system restoration[94, 99, 164].

The development of expert systems for restoration requires the transfer of operator knowl-edge into heuristic rules. That process has been the topic of a great number of publications:[25, 32, 43, 47, 51, 66, 70, 71, 72, 104, 106, 108, 109, 112, 113, 114, 115, 116, 117, 118, 120, 121,122, 123, 124, 126, 127, 128, 129, 130, 131, 137, 138, 139, 140, 141, 152, 154, 155, 158, 159,165, 166, 173, 174, 180, 181, 188, 195, 198, 199, 212, 214, 215, 217, 220, 221, 226, 227, 237].Practical implementations of prototype knowledge-based systems in energy management sys-tems can be found in [37, 48, 56, 58, 75, 86, 120, 122, 123, 128, 132, 146, 156, 159, 168, 170,189, 202].

To reduce the number of rules of an expert system, mathematical programming [131, 174,200] or other analytical optimization methods [173, 175, 180, 181, 200, 201] can be appliedin combination with knowledge-based systems. The veri�cation of restoration solutions pro-vided by expert systems is accomplished using an integration with time-domain simulators[47, 75, 124, 132, 189].

Other arti�cial intelligence methods that are applied to black start and power systemrestoration problems are the use of petri nets [68, 77, 228], the application of fuzzy logic torestoration automation [147], the utilization of genetic algorithms for load restoration [133],the generation of switching sequences [135, 179], or as a hybrid approach of an expert system,the application of neural networks [108, 237].

Chapter 3

Structure of Framework

Each restoration case is highly complex and unique, which makes it diÆcult to developsystematic restoration procedures, and to �nd methods that can be generalized in order tobe applied to other systems and scenarios. However, restoration cases can be considered asa succession of simple restoration steps that are common in all restoration procedures, asdescribed in more detail in [228].

The same principle is true when the feasibility of restoration steps needs to be assessed:the investigation of an aggregation of complex phenomena can be simpli�ed by de�ningdi�erent problem areas that are common in all restoration scenarios and by developing rulesthat address speci�c problems during restoration. In this thesis project we focus mainlyon problems that are related to the initial phases of a restoration process. However, thesame principles can be extended to include other problem areas, such as the analysis of thereintegration of subsystems.

Three di�erent areas are identi�ed that are considered to be important in the initialstages of a restoration process:

1. Voltage-response analysis

� Steady-state overvoltages after changes in network

� Harmonic overvoltages during transformer / line switching

� Transient overvoltages during line switching

2. Frequency-response analysis

� Frequency drops due to static load pick-up

3. Auxiliary system analysis

� Voltage drops during motor starts

� Inrush currents during motor starts

� Thermal overload behavior during motor starts

� Frequency drops during motor starts

14

3.1. On-line and O�-Line Restoration Planning 15

In the following, we explain the assumptions that underly the proposed framework, describethe general idea behind the rules that deal with the above problems, and give an overviewof the analytical tools that are used within the framework.

3.1 On-line and O�-Line Restoration Planning

We deal with two di�erent aspects of restoration: on-line restoration, and o�-line restorationor restoration planning. In the former the pressure on operators is very high and decisionshave to be made very quickly. These are mainly based on prede�ned restoration plans, oper-ator experience, data given by state estimation or other data sources. Prede�ned restorationplans have the advantage to allow for a detailed study of the system behavior. However,since each blackout is of a di�erent nature, and consequently, each restoration procedure willstart from di�erent initial conditions, these prede�ned plans can only give the operator ageneral idea on how to behave during restoration. This means that prede�ned restorationplans will give a strategy rather than a detailed set of instructions on how to proceed, sincesetting up restoration plans for each possible combination of restoration actions and for eachsystem condition would be infeasible. Consequently, the operator has still to decide at eachstep whether his action is feasible or not, and to plan future actions. The rules developedin this thesis can help operators to assess the feasibility of restoration steps and to planrestoration sequences, using no or only a small number of costly time-domain simulations.

During system restoration, planning engineers are faced with a large number of theoreticalpossibilities of how a system can be restored. These have to be explored based on theengineers' experience and on computer simulations carried out with power ow software,stability tools, electromagnetic transient programs, and harmonic analysis software. Simplerules that help to assess restoration steps with respect to their feasibility help to limitthe number of trial{and{error simulations, and consequently, to shorten the overall timenecessary to develop restoration strategies and restoration plans. Furthermore, the rules canbe integrated into a tool that aids automated restoration planning, or they can be added toalready available tools, such as the one described in [75].

The proposed framework has therefore two modes: an on-line restoration mode and arestoration planning mode. In the on-line mode, the rules support the operator in his decisionmaking process. In the o�-line mode, they help system planning engineers to �nd restorationplans in a faster and more eÆcient way.

3.2 Framework Database

As a basis of our analysis we assume that all necessary data is readily available in a database.This is schematically shown in Figure 3.1 for on-line restoration and in Figure 3.2 for restora-tion planning. In the case of on-line restoration, we assume that we have state-estimatordata available, which give us bus voltages, generator output powers, switching status, etc. atthe beginning of the restoration process. We further assume that we have suÆcient informa-tion about the power system devices, such as generator data, data on the generator controls,

3.3. Framework Rules 16

line- and cable data, etc. Since the start-up behavior of motors is particularly important inthe early stages of a restoration procedure, the estimation of motor parameters is depictedin an extra block and explained in more detail in a subsequent section.

The data sources for system restoration planning are usually power ow, short-circuit,and dynamic data. In the case of single-phase calculations, the data for the electrical networkfor dynamic simulations can be extracted from power ow data. Saturation characteristics,overvoltage-versus-time capability characteristics, etc. have to be provided separately in formof tables, given by the manufacturer. In the case of three-phase simulations, zero-sequencedata can be extracted from short-circuit data.

Dynamic dataDatabase

Equipment data

Motor data

Motorparameterestimation

Rules

Simulationtools

State estimator data

Figure 3.1: On-line database for system restoration framework

3.3 Framework Rules

Time-domain simulations give the most accurate assessment of whether a restoration stepis feasible or not. However, they require a great amount of time, particularly when di�er-ent combinations of restoration actions need to be assessed. Therefore, the rules that aredescribed in more detail in the following chapters are based on analysis in the frequencyand Laplace domain, and time-domain simulations are only performed for veri�cation. Thegeneral principle that is the basis for the rules is schematically shown in Figure 3.3. Theobjective is to decrease the overall calculation time while still keeping a reasonable accuracy.

As a basis for the analysis in the frequency domain, a software tool with harmonic analysisfeatures is used. After an initial frequency scan is performed, new network conditions canbe assessed by applying matrix calculations.

Transfer functions for control systems, formulated in the Laplace domain, can be calcu-lated and if necessary, simpli�ed, for each control system type of interest, leaving only theinverse Laplace transform to be performed during the restoration process.

3.4. Framework Analytical Tools 17

Power flow data

Database

Equipment data

Motor dataMotor

parameterestimation

Rules

Simulationtools

Short-circuit data

Dynamic data

Figure 3.2: O�-line database for system restoration framework

Two di�erent ways to avoid problems during restoration actions can be distinguished.In case of e. g. a load pick-up, only a single parameter, namely the amount of load canbe changed. Depending on the operating guidelines of each utility, a maximum allowablefrequency drop is given. Based on this parameter, the corresponding maximum allowableload that can be safely picked up can be estimated.

In cases where a number of di�erent parameters can be changed in order to remedy agiven problem, e. g. in the case of sustained overvoltages, the most e�ective network changecan be found by means of sensitivity analysis. If sensitivity analysis cannot be applied, themost e�ective network change has to be determined based on trial{and{error.

3.4 Framework Analytical Tools

This section gives an overview of the analytical tools that are used within the framework.In the following, only a brief introduction into the theoretical aspects of the analyticaltools is given since the underlying theory cannot be covered in detail in this thesis. Thesoftware packages that are used for the framework and that are listed subsequently are allcommercially available products and have been in use in industry for a number of years. Themethods that are described subsequently are not limited to speci�c software products butcan be used with any software package that can perform the tasks described in the followingsections. Examples for other products that can ful�ll the same tasks are Powerfactory[50] and Netomac [142].

A schematic overview of the tools and software packages used for this project is providedin Figure 3.4. Additional tools that can be used for restoration studies are operator training


Start

Frequency scan

Thevenin equivalent

New restoration step

Laplace transform

Remedial action /Network changes

No

End

Time-domain simulation

Problem Yes

No

Yes

more steps ?

Problem

No

Yes

Figure 3.3: Assessment of feasibility of restoration steps

simulators (OTS), optimal power ow (OPF) programs, or short-circuit programs. Literaturethat deals more thoroughly with analytical tools in power system restoration can be foundin Section 2.9.

3.4.1 Power Flow Analysis

Power ow analysis is the most common analysis method for power systems. The system isassumed to be balanced, which allows for single-phase positive-sequence calculation. Whenwriting the nodal equations of the network we obtain the matrix equation

I = Y �V (3.1)

where I stands for the phasor currents owing into the network, V for the phasor voltages toground, and Y for the node admittance matrix. The e�ects of generators, nonlinear loads,and other devices are re ected in the node current, and constant impedance loads appear in


Power flow

IPFLOW

Steady state(Frequency scan)

EMTP

ElectromechanicalTransients

ETMSP

ElectromagneticTransients

EMTP

Database

Rules

Figure 3.4: Overview of analytical tools

the node admittance matrix. The current at node k can be written as

Ik =Pk � jQk

V �k

(3.2)

where Pk and Qk stand for the active and reactive power, respectively, fed into the networkat node k. Substituting Equation (3.2) into Equation (3.1) gives

Pk � jQk

V �k

= YkkVk +nXi=1i6=k

YkiVi (3.3)

where k = 1; : : : ; n and n stands for the number of nodes in the system. Equation (3.3) isnonlinear and has to be solved iteratively using numerical algorithms such as the Newton-Raphson Method. More details on the theory that underlies power ow analysis can befound in [64, 143]. In this work, the software Interactive Power Flow (Ipflow) [61] is usedin order to solve power ow problems.

3.4.2 Steady-State and Harmonic Analysis

The basis for this type of analysis is Equation (3.1). When the currents I that ow intothe network are readily given, this equation can be solved non-iteratively to obtain the nodevoltages V. The calculations can be either carried out for a three-phase system or|in casethe system is balanced|as a positive sequence calculation.


A harmonic analysis is a steady-state analysis that covers a whole frequency range withfrequency steps �f . Since the network elements are frequency dependent, the node admit-tance matrix changes for each frequency and we obtain [20, 54]:

I(f) = Y(f) �V(f) (3.4)

where

f = fmin; fmin +�f; fmin + 2 ��f; � � � ; fmax

In order to �nd the frequency-dependent impedance of a network seen from a particularlocation, all voltage sources are short-circuited, and all current sources are removed. Then,a current of value 1 A is injected into the node where the impedance is to be determined. Thebranch voltage will be equal to the impedance [54, 161]. In order to carry out harmonic (orsteady-state) analysis, the frequency scan feature of the Electromagnetic Transients Program(Emtp) [54, 63] is utilized.

3.4.3 Electromagnetic Transient Analysis

When analysis in the time domain is performed, the di�erential equations that describe thedynamic behavior of the system have to be integrated with respect to the time.

For the analysis of electromagnetic transients, the trapezoidal is usually used, because ofits very good numerical stability and accuracy [161]. In case of e. g. a simple inductance L,the relationship between voltage and current is expressed by the di�erential equation

v(t) = L � didt

(3.5)

It can be integrated using the trapezoidal rule and we obtain [54, 161]

v(t) + v(t��t)

2= L � i(t)� i(t��t)

�t(3.6)

Transmission lines can be described by the equations

i12(t) =1

Z� v1(t) + hist12(t� �) (3.7)

hist12 = � 1

Z� v2(t� �)� i21(t� �) (3.8)

where � stands for the transmission line traveling time and the subscripts \1" and \2" forthe nodes the line is connected to. Similar operations can be carried out for all networkelements and the system matrix equation is obtained as

Gv(t) = i(t)� hist (3.9)

where hist stands for known \history" terms. For this project, the Emtp is used in orderto simulate the electromagnetic transients. It allows for single- or three-phase calculations[63].


3.4.4 Stability Analysis

The generating units and other dynamic devices of the system can be expressed in thestate-space as

_x = f(x;V) (3.10)

where x stands for the state vectors of devices such as generators, motors, exciters, orgovernors. For stability studies the network and generator stator transients can be neglected.The electrical network with the currents being a function of the state variables x and thenode voltages V can be expressed by the relationship

I(x;V) = Y �V (3.11)

The di�erential Equation (3.10) can be solved using explicit integration rules such as Runge-Kutta, or implicit integration rules using the trapezoidal rule together with the Newtonmethod [143]. Stability programs are based on balanced network conditions and thereforeEquation (3.11) represents the electrical network single-phase with positive-sequence param-eters. In this thesis we use the Extended Transient Midterm Stability Program (Etmsp)[60, 62] to perform stability analysis.

Chapter 4

Frequency-Response Analysis

During power system restoration, it is desirable to reconnect loads as quickly and reliablyas possible. At the very beginning of a restoration process, the loads that help to maintaina reasonable voltage pro�le in the transmission system, and help to stabilize the generatingunits, and the induction motors that drive the auxiliary systems of thermal power plants,are energized. Power plants that are in operation at this stage are small generating unitswith black start capability, such as hydro power plants.

Frequency deviations that occur as a consequence of load pick-ups are of major concernin early stages of a restoration procedure since the ratio between available generating powerand loads to be energized is the lowest. This ratio increases in later stages when thermalpower plants are dominant, and the system is more stabilized. Consequently, frequencydeviations as a result of load pick-ups are of less and less concern as the restoration of thesystem progresses [5].

When frequency deviations exceed certain limits, relay operations may be triggered and|in the worst case|a recurrence of the outage may result. On the one hand, the load that ispicked up should not be too large in order to not exceed these limits. On the other hand,if the amount of load is too small, the restoration duration will be unnecessarily prolonged.Hence, it is desirable to �nd a way to estimate the frequency drops that are the result ofload energization. An assessment of the maximum load that can be picked up safely at eachtime shortens the time during on-line restoration as well as during restoration planning [5].

This chapter introduces methods that deal with the prediction of frequency variationsdue to load changes. The considerations are limited to the pick-up of static loads that can beapproximated by simple steps in the electric energy output. In a later chapter that focuseson the analysis of auxiliary systems, the algorithm is extended to allow for the assessmentof motor pick-ups.

Since a load rejection can be considered as a negative load pick-up, the rules that aredemonstrated in the following can also be used in order to estimate frequency swings duringload shedding. The latter can play an important role during restoration, when systemislanding situations occur as a result of a separation of the system into subsystems. Thesystem frequencies in this case can be below the allowable limits and consequently, the loadneeds to be reduced in an eÆcient and reliable way [4].

22

4.1. Overview of Frequency-Response Estimation Methods 23

4.1 Overview of Frequency-Response Estimation Meth-

ods

4.1.1 Simple Frequency-Response Estimation Methods

Operators often rely on their intuition and experience acquired during normal system op-eration or on fairly crude rules to predict the system frequency behavior when loads arereconnected. The frequency behavior of an islanded system, such as the one shown in Fig-ure 4.1, is controlled by the governors. In case of a load pick-up they detect the decline offrequency and as a consequence, increase the gate or valve position in order to bring up themechanical output power of the turbine, until a new equilibrium with an acceptable systemfrequency is reached.

Turbine G

Governor

Valve / gateSteam / water

SpeedLoad

Tm

Te

Generator

Figure 4.1: Generator supplying isolated load

The two most common rules used by operators in order to estimate the frequency responseof prime movers ignore the action of the governing system and give a reasonable predictiononly for the initial frequency behavior. Both methods are based on the fact that shortlyafter a load change the frequency behavior is solely determined by the change in load, theturbine's inertia constant, and the damping constant, and is not in uenced by any governorcontrol action.

The equation that governs the behavior of an electro-mechanical system is the equationof motion. For small changes in power it can be written in per unit as [143]

2H � d�!dt

= �Pm ��Pe �D ��! (4.1)

where H stands for the inertia constant, D for the damping constant, and �! for thedeviation from the nominal frequency. �Pe represents changes in the electrical and �Pmchanges in the mechanical power.

According to Equation (4.1), a linear function, that de�nes the initial rate of frequencydecay, following a load change of ��Pe, can be written as:

f(t) = (1:0� �Pe2H

� t) � 60 Hz (4.2)


A more elaborate form for the estimation of the frequency behavior can be developed whenEquation (4.1) is transformed into the Laplace domain:

�!(s) = ��Pe2H

� 1

s � �s+ D2H

� (4.3)

where ��Pe represents a load change. This equation can be transformed back into the timedomain, giving the exponential function [143]

f(t) =

�1:0� �Pe

D� [1� exp (�t=TPe)]

�� 60 Hz (4.4)

where

TPe =2 �HD

(4.5)

Example

A 14 MW static load pick-up at the terminals of a generator, whose governor block dia-gram is displayed in Figure 4.2 and whose data are given in Table 4.1, is simulated using thestability program Etmsp, and compared to above rules. The results are shown in Figure 4.3.The linear method allows one to assess the initial rate of decay only, whereas the exponen-tial method allows for a reliable frequency prediction up to around 3 s. Although Equations(4.2) and (4.4) allow for a fast assessment of the frequency behavior, their usefulness is lim-ited when the minimum frequency that is reached is of interest as well. Consequently, bothmethods should be only applied if solely the initial rate of decay is of interest.

Table 4.1: Governor data

Sgen [MVA] Vgen [kV] Tw [s] TD [s] Tp [s] Rp Rt At vo vc87.5 13.8 1.0 s 15.0 s 0.035 s 0.02 0.7 1.1 0.0559 -0.04781

4.1.2 Other Frequency-Response Estimation Methods

In the past, the minimum frequency during load pick-up has been mostly determined bymeans of time-domain simulations using stability programs. This, however, can lead tolong overall simulation times, particularly in cases where a multitude of scenarios has to bestudied. Hence, there is a need to �nd the minimum frequency by simpler rules that are notbased on time-domain simulations.

A general overview of relevant publications in the area of frequency response analysisin power system restoration was given in Chapter 2. Simple guidelines for the estimationof the frequency response of prime movers during restoration are listed in [5]. They are


1

1+sTp

11

s

vo

vu

1-sTw

1+0.5sTw

At

Rp

Rt

sTD

1+sTD

ω

ωref

Pm

1.0

0.0

-

-+

+

+

Figure 4.2: Governor block diagram

0 2 4 6 8 10 12 14 16 18 2053

54

55

56

57

58

59

60

61

Time t [s]

Frequencyf[Hz]

ETMSP simulationLinear method

Exponential method

Figure 4.3: Frequency deviation for 14 MW load pick-up

based on look-up tables that relate frequency dips to sudden load increases and that areproduced based on a number of time-domain simulations. Other methods, outlined in [52,133], are based on neural networks that are trained by running a large number of time-domainsimulations.

An alternative to above methods is the application of the Laplace transform. All methods,based on Laplace-domain calculations, are average frequency models, i. e. oscillations betweengenerators are �ltered out and only an average system frequency is retained. This means that

4.2. Frequency Behavior of Hydro Units for Static Load Pick-up 26

in case more than one generator is in operation, the generators are lumped into one dynamicequivalent. An approach that belongs to this group of methods and that is based on stronglysimpli�ed governor and turbine models is presented in [22, 23]. It gives a simple function forthe system's frequency behavior after a load pick-up. A low-order system frequency responsemodel and the analysis of the frequency-response behavior of steam reheat turbines in anislanded condition is described in [16].

Other similar methods that are based on dynamic equivalents, and extended applicationof the Laplace transform are outlined in [30, 42]. An extended analytical analysis of frequencydecay rates can be found in [21].

4.2 Frequency Behavior of Hydro Units for Static Load

Pick-up

This section introduces a new Laplace transform method, based on [16], to determine thefrequency drops caused by load pick-up during the early stages of system restoration whenhydro units are in operation. An overview of the method is given in Figure 4.4.

At the beginning of power system restoration we mainly deal with hydro units that pro-vide cranking power for larger thermal units. Since [16] provides a method that can bereadily applied to estimate frequency drops in the case of steam turbine units, thermal tur-bines are not considered further. Furthermore, the aggregation of generators is not discussedin this thesis. An overview of dynamic aggregation methods can e. g. be found in [144]. Incase only a small number of hydro units with similar parameters is considered, these can beaggregated according to the method outlined in [16].

4.2.1 Assumptions for Model Development

The development of the equations in this section is based on the following assumptions:

� The operation of the governor-turbine system is unconstrained, i. e. limiters are note�ective. This restriction is necessary since nonlinearities cannot be represented usingthe Laplace transform [16, 22, 23, 67].

� If more than one hydro generator is in operation, the generators are lumped into asingle machine. Topologically, this replacement can be regarded as a single generatorthat is connected to the individual generator buses by ideal phase shifters [16].

� Oscillations between hydro generators are neglected and only an average frequency istaken into consideration [16]. That means that all generators remain in synchronismand do not fall out of step [30].

� Only the mechanical part of the system is taken into account|voltage e�ects areignored [30].


Determine P limit = F (flimit)

PL < Plimit ?

Remedial ActionDetermine load P L to be picked up

Yes

No

Start

ETMSP simulation

f < flimit ?Yes

No

Next restoration step

Load pick-up ?

No

End

Yes

Figure 4.4: Overview of algorithm

� The transformer and generator impedances place a limitation on the amount of loadthe generating units can absorb. It is assumed that the disturbance is small enoughand that the equivalent machine model is able to absorb this change [16].

� Governor and turbine models are chosen in a way that allows for a compromise betweene�ort and eÆciency. The governor model represents a typical proportional control cir-cuit with transient droop [62, 143, 178]. The turbine is represented by its classicaltransfer function [143, 178] which implies an inelastic water column, which is of suÆ-cient accuracy for our purpose [5].


4.2.2 Derivation and Veri�cation of Equations

Simpli�ed Governor and Turbine Control Block Diagram

With the assumptions made in the previous section, we obtain the system in Figure 4.5.Based on this diagram, the function �!(s) can be determined and then transformed back

∆ ω1-sTw

1+0.5sTw

At

1

2Hs

D

1

1+sTp

1

s

Rp

Rt

sTD

1+sTD

-

-

-

-

Pe

gate speedgate

Turbine Generator

Governor

Pm

X2

X1

Figure 4.5: Control block diagram of hydro unit

into the time domain. However, the attempt to develop such a function, taking into accountthe full block diagram in Figure 4.5, leads to an unnecessarily complicated equation. In thefollowing, the block diagram is therefore simpli�ed.

Additional Simpli�cations

The �rst simpli�cation can be accomplished by comparing the time constants listed in Table4.2: the time constant Tp is considerably smaller than the constants Tw; TD; TL, and it can


therefore be set to zero. This reduces the respective block to a constant of value 1 whichcan be omitted from the block diagram.

Table 4.2: Comparison of time constants

Time constant Tw TD Tp TL[s] 1.0 15.0 0.035 9.8

In the next step, we consider the transfer function of the feedback loop which can beformulated in the Laplace domain as:

X2 = (Rp +Rt) �1Tpt

+ s1TD

+ s�X1 (4.6)

where

Tpt =Rp +Rt

Rp

� TD = 178:6 s (4.7)

Since Tpt � TD, we can simplify Equation (4.6) to

X2 = (Rp +Rt) � s1TD

+ s�X1 (4.8)

The simpli�cations result in the new simpli�ed control block diagram displayed in Figure4.6.

1-sTw

1+0.5sTw

At

-

Pe

1/(2H)

D/(2H)+s

1/TD+s

(Rp+Rt+1/TD)+s

1

s

∆ ω

Figure 4.6: Simpli�ed control block diagram of hydro unit


Example

In order to prove the validity of above simpli�cations, a comparison between Simulink [163]simulations of the full and reduced model are carried out. The same 14 MW load pick-upthat was described earlier is simulated. Figure 4.7 shows that noticable deviations betweenthe results for the full and simpli�ed model only occur from around 30 s onwards. Since weare only interested in the initial behavior of the frequency until around the time when theminimum is reached, our simpli�cations are justi�ed.

0 5 10 15 20 25 30 35 40 45 5056

56.5

57

57.5

58

58.5

59

59.5

60

Time t [s]

Frequencyf[Hz]

Full modelSimpli�ed governor

Figure 4.7: Comparison between full and simpli�ed model

Development of Transfer Function

In the Laplace domain, a sudden load change of �Pe can be represented as

Pe(s) = ��Pe � 1s

(4.9)

where

�Pe > 0 for load pick-ups�Pe < 0 for load rejections

After some algebraic operations, the transfer function for the model displayed in Figure4.6 can be formulated as:

�!(s) = K � N(s)

D(s)(4.10)


where

K = ��Pe2H

(4.11)

N(s) =

�2

Tw+ s

��

1

TN+ s

�(4.12)

D(s) =At

H � Tw � TD +

�D

H � Tw � TN +At � (TD � Tw)

H � TD � Tw

�� s+

+

�D

2H � TN +D

H � Tw �At

H+

2

Tw � TN

�s2 + (4.13)

+

�D

2H+

1

TN+

2

Tw

�s3 + s4

TN =1

Rp +Rt + 1=TD(4.14)

Whereas the roots of the numerator of Equation (4.10) are readily available, the roots of thedenominator are not readily given. They are determined using the numerical routine roots[162]. Once the roots are available, Equations (4.13) can be formulated in a factored formas

D(s) =4Y

j=1

(s� zj) (4.15)

where zj are the roots of Equation (4.13). Equation (4.10) can then be expanded in partialfractions and we obtain [67, 143]

�!(s) = K �4X

j=1

Rj

s� zj(4.16)

where Rj represents the residues at the poles zj which can be computed by

Rj = [�!(s) � (s� zj)]s=zj (4.17)

Eventually, following [67], Equation (4.16) can be transformed back into the time-domain,and we �nally obtain the system frequency:

f(t) =

"1�

4Xj=1

(Rj � exp (zj � t))#� 60 Hz (4.18)

4.2.3 Example for Load Pick-up

The result is veri�ed using the same 14 MW static-load pick-up example as before. Figure4.8 shows the results obtained with the new method as compared to results obtained with


the stability program Etmsp. The results are very close and prove the validity of the aboveapproach. It can be further seen that for a load pick-up of 14 MW (0.16 in per unit of thegenerator rating) the limits given for the gate position and speed have no e�ect on the result.For this case, the calculation time when using the new method is approximately 1% of thesimulation time using Etmsp.

The minimum frequency of 56.15 Hz at 7.8 s is obtained from Equation (4.18) using thenumerical routine fmin [162].

0 2 4 6 8 10 12 14 16 18 2056

56.5

57

57.5

58

58.5

59

59.5

60

60.5

61

Time t [s]

Frequencyf[Hz]

ETMSP simulationNew method

Figure 4.8: Frequency deviation for 14 MW load pick-up

4.2.4 Example for Load Rejection

Analogous to the previous example we perform a 14 MW load rejection. The result of theanalytical calculation, as compared to an Etmsp simulation, is depicted in Figure 4.9. Usingthe same numerical routines as above we obtain the maximum frequency 63.86 Hz at 7.8 s.The analytical results agree well with the Etmsp time-domain simulation results and provethat the method can be applied for both load pick-up and rejection.

4.2.5 Remedial Action

In the following, it is investigated whether a simple rule for hydro turbine systems, indicatedin Figure 4.4, can be developed that relates the minimum frequency and the time where thisminimum occurs to the amount of load that is energized. Figure 4.10 shows the minimumfrequency and Figure 4.11 the corresponding time as a function of the amount of load beingpicked up (in the range of 0.02 to 0.2 per unit).


0 2 4 6 8 10 12 14 16 18 2060

60.5

61

61.5

62

62.5

63

63.5

64

64.5

65

Time t [s]

Frequencyf[Hz]


Figure 4.9: Frequency deviation for 14 MW load rejection

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.255

55.5

56

56.5

57

57.5

58

58.5

59

59.5

60

Load �Pe [p.u.]

Frequencyf[Hz]


Figure 4.10: Minimum frequency as a function of load

The time where the minimum occurs is not a function of the load. The minimum fre-quency is a linear function of the amount of load. This result can be used in order todetermine the maximum amount of load that can be picked up safely at once, when a min-imum allowable frequency is given. Therefore, the minimum frequencies fL1; fL2 for twodi�erent arbitrary loads PL1 and PL2 are calculated. The maximum load that can be picked

4.3. Summary 34

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.27.8

7.8005

7.801

7.8015

7.802

Load �Pe [p.u.]

Timeforminimumt min

[s]


Figure 4.11: Time at which minimum frequency occurs as a function of load

up safely can then be calculated by

Plimit = PL1 � (PL1 � PL2)

(fL1 � fL2)(flimit � fL2) (4.19)

where flimit stands for the minimum allowable frequency.

4.3 Summary

In this chapter a method was introduced that predicts the frequency behavior of hydro primemovers after the rejection or pick-up of static loads. The method is based on calculations inthe Laplace domain and simpli�ed governor and turbine models. Comparisons between thestability program Etmsp and the new method show good agreement and prove the validityof the rule.

Chapter 5

Voltage Response Analysis

This chapter introduces a concept for overvoltage control during restoration. Overvoltagescan be classi�ed as switching transient, sustained, and harmonic resonance overvoltages.Steady-state overvoltages occur at the receiving end of lightly loaded transmission lines asa consequence of reactive power imbalance. Excessive sustained overvoltages may lead todamage of transformers and other power system equipment. Transient overvoltages are aconsequence of switching operations on long transmission lines, or switching of capacitivedevices, and may result in arrester failures. Harmonic resonance overvoltages are a resultof system resonance frequencies close to multiples of the fundamental frequency, which areexcited by switching of transformers, or switching of transformer-terminated transmissionlines. They may last for a very long time and result in arrester failures and system faults[95].

The overvoltages of major concern during power system restoration are sustained andharmonic overvoltages [95, 172]. Therefore, this chapter focuses mainly on their analysisand control. The analysis and control of switching transient overvoltages is described onlybrie y for the sake of completeness.

When energizing the transmission system, two con icting issues arise. It is desirableto energize line sections as large as possible. This, however, involves the risk of higherovervoltages, and consequently of equipment damage. Bringing more generators online orenergizing only small sections of transmission lines might reduce these overvoltages, but leadsto an increase in restoration time [1].

Traditionally, sustained overvoltages have been studied using power ow programs. Switch-ing transient and harmonic overvoltages have been mainly investigated using the Emtp. Theapplication of Emtp, however, requires long simulation times. This is especially true duringsystem restoration when a large number of simulations need to be carried out in order toassess all possible restoration scenarios.

In this chapter a method for fast and approximate assessment of the feasibility of switch-ing restoration steps, using simple rules and the frequency scan feature of Emtp, is intro-duced. For the purpose of an approximate assessment, a single-phase representation of thesystem is chosen. The frequency scan has to be performed only once. Then, using matrixmanipulation, changes to the system can be accommodated in a fast and eÆcient way.

35

36

Estimate bus voltage

Sustained overvoltage?

Remedial ActionDetermine network matrix [ Z ]

Yes

No

Harmonic overvoltage?

Transient overvoltage?

No

Yes

Yes

EMTP simulation

Overvoltage ?

No

Yes

No


Line / transformerswitching ?

No

Start

End


A schematic overview of the proposed method is given in Figure 5.1: Once a suitablerestoration path is found by approximate rules, it can be veri�ed by an Emtp time-domainsimulation, together with overvoltage-versus-time-capability-curves to assess whether a de-

5.1. Frequency-dependent Thevenin Impedance 37

vice can withstand the applied voltage. Typical characteristics of transformers, shunt reac-tors, and MOV arresters are displayed in Figure 5.2.

When time-domain simulations are carried out, signal processing techniques and rulesbased on the application of arti�cial intelligence, are applied to terminate the simulation aftera minimum amount of time. If a switching situation is found to be infeasible, sensitivityanalysis is applied in order to �nd the most eÆcient network changes that remedy theproblem.

101

102

103

260

280

300

320

340

360

380

400

420

Time t [s]

VoltageV[kV]

180 kV MOV arrester253 kV shunt reactor230 kV transformer

Figure 5.2: Overvoltage-versus-time capabilities of typical 230 kV system equipment

5.1 Frequency-dependent Thevenin Impedance

The combinatorial nature of a restoration procedure and the many resulting system con�g-urations could require a large number of frequency scans using the Emtp. Furthermore, ineach case the admittance matrix Y would need to be build, factored, and solved, resultingin large calculation times. A method is therefore derived which requires only one Emtpfrequency scan. The results are then used as a basis for quickly calculating the impedancefor any arbitrary set of parameters using simple matrix operations.

In the �rst step of this procedure the multi-port Thevenin equivalent circuit is deter-mined. The terminals of this circuit include the bus where the next switching operation, orrestoration step in general, is to be performed, and buses where network element changesare possible. Examples are buses where the number of generators can be varied, where loadscan be changed, or shunt reactances can be added.

Such a multi-port Thevenin equivalent circuit is schematically shown in Figure 5.3. Inorder to �nd the matrix Z0 = (Y0)

�1representing the Thevenin equivalent, currents of


magnitude 1 A are successively injected into each of its terminals. For this, voltage sourcesin the system are short circuited, and current sources are open circuited. The voltages atthe nodes of the Thevenin equivalent then give|column by column|the matrix elements ofZ0. More details about multi-port Thevenin equivalents can be found in [54, 161].

Z Zi

Reactances

Generators

Loads

Variable elements

Switching bus

][

Figure 5.3: Thevenin equivalent circuit with variable parameters

If an element is added at a certain node, one possibility for calculating the new resultingmatrix Z1 (besides carrying out another Emtp frequency scan) is to invert the matrix Z0,then to add the new network element �Yk and to invert the resulting matrix Y1 again,obtaining the new matrix Z1. Although this method is much faster than using the Emtpfrequency scan for each di�erent system con�guration, it is still too time consuming if agreat number of di�erent system con�gurations for a large system needs to be examined.

Therefore, a new method is derived in the following, based on the application of theSherman-Morrison [187] or Woodbury Formula [57], or the inverse matrix inversion lemma[14]. It allows the direct calculation of a modi�ed matrix Z1 from the matrix Z0 and agiven network change �Yk, without using time-consuming matrix inversions, by providingan explicit expression

Z1 = f�Z0;�Yk

�(5.1)

where �Yk stands for a network change at node k.

In the following sections, this method is described for elements added from one of thenodes to neutral and for elements added between two nodes.


5.1.1 Impedance Change between Bus and Neutral

Examples of impedance changes from bus to ground are the addition (or omission) of gener-ators, loads, and shunt reactors. If we add an element from bus k to ground we obtain thenew admittance matrix

Y1 =

0BBBBBB@

Y 011 : : : : : : Y 0

1n...

. . ....

... Y 0kk +�Yk

......

. . ....

Y 0n1 : : : : : : Y 0

nn

1CCCCCCA

= Y0 +

0BBBBBB@

0 : : : : : : 0...

. . ....

... �Yk...

.... . .

...0 : : : : : : 0

1CCCCCCA

(5.2)

This matrix can be written in a more convenient way as

Y1 =�Y0 +�Yk � ek � eTk

�(5.3)

where ek stands for the k-th unit vector. Inverting the matrixY1 and applying the Sherman-

Morrison Formula [187] we get

Z1 = (Y1)�1 =�Y0 +�Yk � ek � eTk

��1(5.4)

= Z0 � �Yk � (Z0 � ek) � (eTk � Z0)1 + �Yk � eTk � Z0 � ek

(5.5)

This equation can be simpli�ed to

Z1 = Z0 �Z0(1:::n)k � Z0k(1:::n) ��Yk

1 + Z0kk ��Yk

(5.6)

where Z0(1:::n)k describes the kth column and Z0k(1:::n) the kth row of Z0.

The impedance at bus j can then be obtained as

Zj = Z1jj = Z0

jj �Z0jk � Z0

kj ��Yk1 + Z0

kk ��Yk(5.7)

This equation can be extended to the case where an arbitrary number of elements fromdi�erent nodes to ground are changed:

Zj = Z1jj = Z0

jj � Xi=k;l;m;:::

�Yi � Z0ji

!� (5.8)

� Xi=k;l;m;:::

Z0ij

!� 1 +

Xr=k;l;m;:::

Xi=k;l;m;:::

Z0ri ��Yi

!�1

(5.9)

where k; l;m; : : : represents the nodes where the changes are possible.


5.1.2 Impedance Change between two Buses

The above method can be extended to impedance changes between arbitrary buses. Exam-ples are the addition of parallel transformers, series capacitances, etc. In the case elementsare added between two nodes k and l, the impedance matrix can be formulated as

Y1 = Y0 +

0BBBBBBBBBB@

0 : : : : : : : : : 0...

. . ....

: : : �Ykl : : : ��Ykl : : :...

......

...: : : ��Ykl : : : �Ykl : : :

.... . .

0 : : : : : : : : : 0

1CCCCCCCCCCA

(5.10)

(5.11)

This equation can be again written in a more convenient way as

Y1 =�Y0 +�Ykl ��ekl ��eTkl

�(5.12)

where �ekl = ek � el. Following the same steps as in the previous section, we obtain

Z1 = Z0 � �Z0(1:::n)kl ��Z0kl(1:::n) ��Ykl1 + (Z0

kk + Z0ll � (Z0

kl + Z0lk)) ��Ykl

(5.13)

where

�Z0(1:::n)kl = Z0(1:::n)k � Z0(1:::n)l (5.14)

�Z0kl(1:::n) = Z0

k(1:::n) � Z0l(1:::n)

From Equation (5.13) the impedance at node j is obtained as

Zj = Z1jj = Z0

jj ��Z0jk � Z0

jl

� � �Z0kj � Z0

lj

� ��Ykl1 + (Z0

kk + Z0ll � (Z0

kl + Z0lk)) ��Ykl

(5.15)

5.1.3 Example

The above method is tested using a simple two-port Thevenin equivalent, created fromthe system displayed in Figure 5.4. It represents a power system at the early stages of arestoration procedure in which, starting from the generators connected to bus B4, a path toa large power station is built. The impedance seen from bus B3 is calculated as a function ofthe impedance connected to the left of bus B4, which could represent the number of parallelgenerators if Zload is constant. The above equations as well as Emtp frequency scans areused for this calculation. The results for the frequencies 60 Hz and 540 Hz are shown inFigures 5.5 and 5.6. They are so close that they are hardly distinguishable, which provesthe validity of our approach.

5.2. Sustained Overvoltages 41

Xr

G1

G3

G2

Zload

B4 B3

t = 0s

Figure 5.4: System at the beginning of a restoration procedure

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6300

350

400

450

500

550

600

Number of generators at bus B4

ImpedanceZB

3(60Hz)[]

EMTP frequency scanMatrix method

Figure 5.5: 60 Hz impedance at bus B3 as function of number of generators

5.2 Sustained Overvoltages

Sustained overvoltages are caused by the charging currents of lightly loaded transmissionlines. If not controlled, they may cause serious reactive power imbalances resulting in phe-nomena such as generator self-excitation and runaway voltage rise. They may further leadto overexcitation of transformers and generation of harmonic distortions which can lead tothe excitation of harmonic resonant overvoltages. Furthermore, sustained overvoltages canlead to high transient overvoltages when line switching procedures are performed [95].

The occurrence of sustained overvoltages in the system can be assessed using the results


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6760

780

800

820

840

860

880

900

920

Number of generators at bus B4

ImpedanceZB

3(540Hz)[]

EMTP frequency scanMatrix method

Figure 5.6: 540 Hz impedance at bus B3 as function of number of generators

of the frequency scan at the base frequency 60 Hz. As outlined in the previous sections, thesystem can be extended step by step by basic matrix operations, and therefore an assessmentof the voltage level at the switching bus can be performed without the necessity for additionalEmtp frequency scans, provided that the loads are constant impedances and the generatorsdo not exceed their reactive power output limits [1]. Therefore, additional calculationsare added at the end of the block \sustained overvoltage" in Figure 5.1: In case the busvoltage limit or the generator reactive power are exceeded, a more time-consuming load owcalculation is performed, as indicated in the schematic shown in Figure 5.7.

5.2.1 Sensitivity Analysis

When a steady-state overvoltage occurs, it is desirable to �nd the bus where network changesfor reducing the voltage level are most e�ective. This can be accomplished by means ofsensitivity analysis. This produces the sensitivity of the voltage magnitude at the bus wherean overvoltage occurs with respect to network changes.

The sensitivity at the switching bus j with respect to shunt admittance changes at busk is de�ned as

Sjk =@jVj(�Yk)j@j�Ykj j�Yk=@(�Yk) (5.16)

where Vj stands for the voltage at bus j, �Yk for a change in admittance at bus k, and@ (�Yk) for a very small number representing an incremental perturbation.


Steady state generatorvoltage Vgen and current I gen

Bus voltage V bus

Compute generator reactive powerQgen=Imag [ V . I* ]

Start

Qgen > Qmax , Q gen< Q min

Vbus > Vmax , Vbus < Vmin

or ?

NoRun power flow calculation

Yes

End

Figure 5.7: Assessment whether power ow is needed

The voltage magnitude at bus j can be formulated as:

jVjj = jnXr=1

(Z1jr � Ir)j (5.17)

= jVjR + jVjIj (5.18)

Substituting Equation (5.17) into Equation (5.16) then yields

Sjk =1

jVjj ��VjR � @VrR

@j�Ykj + VjI � @VrI@j�Ykj

�(5.19)

where

@VjR@j�Ykj =

nXr=1

�@Z1

jrR

@j�Ykj � IrR �@Z1

jrI

@j�Ykj � IrI�

(5.20)

@VjI@j�Ykj =

nXr=1

�@Z1

jrR

@j�Ykj � IrI +@Z1

jrI

@j�Ykj � IrR�

(5.21)

The impedances are separated into real and imaginary parts according to

Z1jr = Z1

jrR + j � Z1jrI jZ1

jrj =q�

Z1jrR

�2+�Z1jrI

�2(5.22)


The real and imaginary parts are de�ned as

Z1jrR = Z0

jrR �A1 � j�Ykj+ A2 � j�Ykj2

1 +B1 � j�Ykj+B2 � j�Ykj2 (5.23)

Z1jrI = Z0

jrI �A3 � j�Ykj+ A3 � j�Ykj2

1 +B3 � j�Ykj+B4 � j�Ykj2 (5.24)

and their partial derivatives are given by

@Z1jrR

@j�Ykj =�A1 � 2 � A2 � j�Ykj+ (A1 �B2 � B1 � A2) � j�Ykj2

(1 +B1 � j�Ykj+B2 � j�Ykj2)2(5.25)

@Z1jrI

@j�Ykj =�A3 � 2 � A4 � j�Ykj+ (A3 �B4 � B3 � A4) � j�Ykj2

(1 +B3 � j�Ykj+B4 � j�Ykj2)2(5.26)

where

A1 = jZ0jkj � jZ0

krj � cos ('jk + 'kr + '�Yk)

A2 = jZ0jkj � jZ0

krj � jZ0kkj � [cos ('jk + 'kr + '�Yk) � cos ('kk + '�Yk) +

+ sin ('kk + '�Yk) � sin ('kk + '�Yk)]

A3 = jZ0jkj � jZ0

krj � sin ('jk + 'kr + '�Yk) (5.27)

A4 = jZ0jkj � jZ0

krj � jZ0kkj � [sin ('jk + 'kr + '�Yk) � cos ('kk + '�Yk) +

� cos ('kk + '�Yk) � sin ('kk + '�Yk)]

B1 = B3 = 2 � jZ0kkj � cos ('kk + '�Yk)

B2 = B4 = jZ0kkj2

with 'jk, 'kr, and '�Yk being the angles of Z0jk, Z

0kr, and �Yk, respectively.

The sensitivities de�ned in Equation (5.16) are calculated for all nodes of the Theveninequivalent at which a remedial action can be performed. They are then arranged in asensitivity vector

S =

2666664

Sj1...Sjk...Sjn

3777775 (5.28)

This vector can be normalized by referring its elements to the maximum element:

S =1

maxi (Sji)�

2666664

Sj1...Sjk...Sjn

3777775 (5.29)


Since we deal with either resistive, inductive or resistive-inductive changes depending on theavailable devices, two di�erent sensitivity vectors can be de�ned|one for resistive (SR) andone for inductive (SI) changes. This is also re ected in the incremental change @ (�Yk),which in the case of resistive changes can be de�ned as a real number and in the case ofinductive changes as an imaginary number. Capacitive changes can be considered as negativeinductive changes. In case only loads are to be connected to the buses, the resistive andinductive sensitivity vectors can be combined to:

Stotal = jSR + j � tan (') � SIj (5.30)

where tan' takes into account the average powerfactor pf for the loads to be added whichis de�ned as:

pf = cos (') (5.31)

5.2.2 Example

As an example for the application of the above method, the system in Figure 5.8 is used.It represents the same system as the one in Figure 5.4, in a later stage of the restorationprocedure. The most e�ective load changes for reducing an overvoltage at bus B7 are inves-

G1

G3

G2 B4B3

B7G4

B2

G5

Figure 5.8: System during restoration procedure

tigated. Figure 5.9 shows the voltage behavior at bus B7 with respect to load changes atbuses B2, B3, B4, and B7, obtained from full solutions. The load changes �Yk are referred toa typical bus load ~Y during normal operation. The results from sensitivity analysis with thesensitivity vector of Equation (5.30), for an average power factor of pf = 0:9, are displayedin Table 5.1. It can be seen that the higher the values of elements of the sensitivity vector,the more eÆcient the changes are at the corresponding buses for reducing the voltage levelat bus B7.

Table 5.1: Total sensitivity vector

Bus k 2 3 4 7

Stotal(7k) 0.2908 1.0000 0.1553 0.4578

5.3. Harmonic Overvoltages 46

0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Admittance change �Yk= ~Y

BusvoltageVB

7

[p:u:]

4

2

7

3

Figure 5.9: Voltage at bus B7 as function of �Yk= ~Y where ~Y = (2� j � 0:884) mS

5.3 Harmonic Overvoltages

During the restoration of power systems after a complete or partial blackout, resonance con-ditions di�erent from the ones during normal operation are encountered. If the frequencycharacteristic shows resonance peaks around multiples of the fundamental frequency, highovervoltages of long duration may occur when the system is excited by a harmonic distur-bance. Such a disturbance can originate from the saturation of transformers, from powerelectronics, etc.

The major cause of harmonic resonance overvoltage problems during restoration is theswitching of lightly loaded transformers at the end of transmission lines or the switching oftransmission lines, to which transformers are connected. When a transformer is switched,inrush currents with signi�cant harmonic content up to frequencies around 600 Hz are cre-ated. They can be represented by a harmonic current source connected to the transformerbus [172]. The relation between nodal voltages, network matrix, and current injections canbe represented by

V(h � 60 Hz) = Z(h � 60 Hz) � I(h � 60 Hz) (5.32)

where h stands for the order of the harmonic frequencies f=120; 180; : : : Hz.

The harmonic components of the same frequency as a system resonance frequency areampli�ed in case of parallel resonance, thereby creating higher voltages at the transformerterminals. This leads to a higher level of saturation resulting in higher harmonic componentsof the inrush current which again results in increased voltages. This can happen particularlyin lightly damped systems, common at the beginning of a restoration procedure when a pathfrom a black start source to a large power plant is being established and only a few loads


are restored yet [90, 172].

5.3.1 Analysis and Assessment of Harmonic Overvoltages

In the following, four di�erent system con�gurations of the sample system shown in Figure 5.4are used in order to study conditions leading to harmonic resonance overvoltages. In Case1 the impedance at bus B4 is omitted and an Emtp frequency scan is carried out. Itsresult is displayed in Figure 5.10. It shows a resonance peak at the third harmonic. Thesteady-state voltage at bus B3 at the moment of switching is 197.2 kV (1.05 p.u.). Thegeneration of harmonics from the transformer is shown in Figure 5.11. It has signi�cantharmonic content up to a frequency of around 600 Hz. When the transformer is energized,

0 100 200 300 400 500 6000

20

40

60

80

100

120

Case 1 & Case 4Case 2Case 3

Frequency f [Hz]

ImpedanceZB

3

[]

Figure 5.10: Impedance at bus B3 for Cases 1 to 4

the resonance condition results in the overvoltage VB3 (t) shown in Figure 5.12. If there wereno overvoltage limiting equipment, the voltage would reach a level of 279.5 kV (1.49 p.u.).Figure 5.12 also shows the Fast Fourier Transform (FFT) of the signal for the period wherethe voltage reaches its maximum. It re ects the amplifying e�ect of the system resonancefrequency: the magnitude of the third harmonic is almost half of the fundamental.

There are several methods for preventing harmonic overvoltages. In Case 2, a highlyresistive load can be added to bus B4, leading to a decrease in the magnitude of the impedanceZB3, and consequently to a reduced ampli�cation. The frequency-dependent impedance andthe voltage at bus B3 are shown in Figures 5.10 and 5.13, respectively.

Another method that can be used in order to prevent harmonic resonant overvoltages isto bring additional generators online: a higher number of generators results in a lower overall


120 180 240 300 360 420 480 540 6000

5

10

15

Frequency f [Hz]

I inrush

B

3(f)[A]

Figure 5.11: Harmonic content of transformer inrush current at bus B3 for Cases 1 to 3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−300

−200

−100

0

100

200

300

0 60 120 180 240 300 360 420 480 540 6000

50

100

150

200

Time t [s]

VB

3(t)[kV]

VB3(0�) = 1:05 p.u.

Frequency f [Hz]

VB

3(f)[kV]

Figure 5.12: Voltage at bus B3 for Case 1

inductance, and consequently in a higher resonance frequency, according to the equation

fresonance =1

2�pL � C

This means that if generators are added to bus B4, the resonance peak is shifted to higherfrequencies|if generators are omitted, it is shifted to lower frequencies. The frequency-


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−300

−200

−100

0

100

200

300

Time t [s]

VoltageVB

3

[kV]

| Case 1 - - - Case 2

Figure 5.13: Voltages at bus B3 for Case 1 and Case 2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2−300

−200

−100

0

100

200

300

Time t [s]

VoltageVB

3

[kV]

| Case 3 - - - Case 4

Figure 5.14: Voltages at bus B3 for Case 3 and Case 4

dependent impedance for Case 3, where another generator is added to the system, is shownin Figure 5.10, and the resulting voltage in Figure 5.14.

Another possibility for reducing harmonic overvoltages is represented by Case 4: a de-crease of the generators' scheduled voltage leads to a proportional decrease of the pre-switching steady-state voltages. This e�ect results in a change of the transformer inrushcurrent. Figure 5.15 gives the harmonic content of the transformer inrush current for a pre-


switching voltage at bus B3 of 178.4 kV (0.95 p.u.). The comparison to Figure 5.11 showsthat the harmonic content of the inrush current is signi�cantly reduced.

120 180 240 300 360 420 480 540 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Frequency f [Hz]

I inrush

B

3(f)[A]

Figure 5.15: Harmonic content of transformer inrush current at bus B3 for Case 4

From the previous results it can be concluded that three di�erent criteria for the oc-currence of harmonic resonance overvoltages have to be considered: the impedance at thebus where the switching operation is performed, and the harmonic content of the trans-former inrush current which is a function of the pre-switching steady-state voltage level atthe switching bus, and the transformer saturation characteristic.

Based on the above observations we can de�ne a rule for the assessment of harmonicovervoltages for switching operations at bus j that is similar to the total harmonic distortion(THD) [20] known from power system harmonic analysis:

H =

vuut 10Xh=2

[Zjj(h � 60 Hz) � Ij (h � 60 Hz; Vj (0�) ; Lj (�))]2 > Hlimit (5.33)

Ij stands for the transformer inrush current which is a function of the frequency h � 60 Hz,the pre-switching voltage level Vj(0�), and the transformer saturation characteristic Lj. InSection 5.3.2, it is shown how the harmonic characteristic of transformers can be estimated.The constant Hlimit stands for the maximum allowable limit. It is de�ned by the systemoperator or restoration planner. The results for H for Cases 1 to 4 in Table 5.2 re ect thevoltage behavior at bus B3 displayed in Figures 5.13 and 5.14.

The above rule can only provide an approximate assessment of whether a restoration stepis feasible or not with respect to the occurrence of harmonic overvoltages. It is thereforerecommended to always carry out time-domain simulations when a certain level of H is


Table 5.2: Index for harmonic overvoltage assessment

Case 1 2 3 4H 6.33 0.22 0.15 0.35

exceeded. In order to keep the simulation time as short as possible in that case, a methodfor the reduction of the simulation time of Emtp simulations is introduced in Section 5.4.

5.3.2 Harmonic Characteristic of Transformers

For an assessment of harmonic overvoltages during transformer switching, the harmoniccharacteristic, i. e. the harmonics of the inrush current as a function of the voltage of thetransformer that is energized, is needed. In case the harmonic characteristic is not givenby the manufacturer, it can be determined from the saturation characteristic and the trans-former terminal voltage. In the following, analytical equations are introduced to determinethe harmonic content of the transformer inrush current.

Analytical Calculation of Transformer Inrush Current

The development of the equations for the calculation of transformer inrush currents is de-scribed in detail in [27, 211] and, is therefore omitted from this section.

When applying a sinusoidal voltage

v(t) = Vmax � cos(!t+ ')

directly to a saturated inductance, the harmonics of the transformer inrush current can becalculated as

Iinrush(0 Hz) =1

2�� [2 cos(�) + (2�� )] � Vmax

!Ls

(5.34)

Iinrush(60 Hz) =1

2�� [� � 2�� sin(2�)] � Vmax

!Ls

(5.35)

Iinrush(h � 60 Hz) =1

h�

�sin (h� 1) (� + �=2)

h� 1� sin (h + 1) (� + �=2)

h+ 1

�� Vmax

!Ls

(5.36)

(h = 2; 3; 4; : : :)

where

� = arcsin

�(1� 0

s) � !Ls

Vmax

�(5.37)

0 = r � Vmax

!sin'


and where s stands for the saturation ux linkage, r for the remanent ux, and Ls for thesaturated inductance.

The simpli�ed saturation characteristic, on which Equations (5.34) to (5.36) are based,is shown in Figure 5.16. It consists of two slopes|the �rst one with an in�nitely high valueand the second one with value Ls.

i

ψ0

Ls

Current

ψF

lux

Figure 5.16: Ideal saturation characteristic

Saturation characteristics are usually given as n pairs of numbers [ (1); i(1)] , [ (2); i(2)], : : : , [ (n); i(n)]. The magnetizing inductance and the saturation ux linkage can then becalculated from the equations

�Ls = (n)� (n� 1)

i(n)� i(n� 1)(5.38)

s = (n)� �Ls � i(n) (5.39)

In addition to the nonlinear magnetizing inductance, transformers have a leakage inductanceLleakage, resistances R1 and R2 representing the I2R-losses, and resistance Rc representingalso the core losses. For our approximation, the resistances R1 and R2 can be neglected sinceR1; R2 � !Lleakage. The core loss resistance Rc can be neglected as well since Rc � !Ls.We can then include the inductance Lleakage in the magnetizing inductance Ls and obtain:

Ls = �Ls + Lleakage (5.40)

Using this value in Equations (5.34) to (5.36), we obtain the harmonic characteristic of thetransformer inrush current.


Example

As an example for the above method, the harmonic characteristic of the transformer that isswitched at bus B3 in Figure 5.4 is determined. Its saturation characteristic is shown in Fig-ure 5.17. The harmonics for a transformer terminal voltage of 1.05 per unit calculated withabove equations are compared to an Emtp time-domain simulation of the inrush current,followed by FFT, in Figure 5.18. The results show good agreement and therefore prove thevalidity of the approach.

0 20 40 60 80 100 1200

100

200

300

400

500

600

700

Current i [A]

Flux [Vs]

Figure 5.17: Transformer saturation characteristic

5.3.3 Sensitivity Analysis

In cases where the power system during restoration has already grown to a considerable size,and a harmonic overvoltage condition has been detected, it is not always clear at which nodesadmittance changes should be made in order to obtain an optimum impedance-frequencycharacteristic at the switching node. Therefore, a method is developed which determinesthe nodes where changes are most e�ective. The method is based on sensitivity analysisfor the harmonic frequencies [147]. Only harmonic frequencies are of interest since themajor components of the frequency spectrum of the voltages and currents during transformersaturation are multiples of the fundamental frequency 60 Hz.

The algorithm consists of four steps. Steps 2 to 4 are described in the following, whilethe �rst step has already been dealt with in Chapter 5.1.

1. Calculate the Thevenin equivalent circuit for the system.


120 180 240 300 420 480 540 6000

5

10

15

120 180 240 300 420 480 540 6000

5

10

15

FFT of EMTP Simulation

f [Hz]

I inrush(f)[A]

Analytical Calculation

f [Hz]

I inrush(f)[A]

Figure 5.18: Harmonics of transformer inrush current

2. Determine the sensitivity for each harmonic frequency.

3. Determine the weighting function for the inrush current.

4. Calculate the total sensitivity.

Individual Sensitivity

We de�ne the sensitivity of an impedance as its magnitude's derivative with respect to anadmittance change at each node of the Thevenin equivalent circuit. For a harmonic frequencyof order h, the sensitivity with respect to a change at node k can be written as:

Sjk (h) =@jZjj(h � 60 Hz;�Yk)j

@j�Ykj j�Yk=@(�Yk) (5.41)

where Zjj stands for the self impedance at bus j, �Yk for a change in admittance at bus k,and @ (�Yk) for a very small number representing an incremental perturbation. Equation(5.41) can be formulated as

Sjk =1

jZ1jjj��Z1jjR �

@Z1jjR

@j�Ykj + Z1jjI �

@Z1jjI

@j�Ykj�

(5.42)

where

Z1jj = Z1

jjR + j � Z1jjI ; jZ1

jjj =qZ2jjR + Z2

jjI

The values for Z1R, Z

1I ,

@Z1R

@j�Ykj, and

@Z1I

@j�Ykjcan be calculated with the equations developed in

Section 5.2.1.


Calculating the sensitivities for all the nodes of the Thevenin equivalent circuit gives thesensitivity vector for the harmonic frequencies:

S(h) =

2666664

Sj1(h)...

Sjk(h)...

Sjn(h)

3777775 (5.43)

If only sensitivities for single frequencies are considered, the vectors can be normalized byreferring the vector S(h) to the value of its maximum element, i. e. we get

S(h) =1

maxi (Sji(h))�

2666664

Sj1(h)...

Sjk(h)...

Sjn(h)

3777775 (5.44)

Since we deal with either resistive, inductive or resistive-inductive changes depending on theavailable devices, two di�erent sensitivity vectors can be de�ned|one for resistive (SR) andone for inductive (SI) changes. This is also re ected in the incremental change @ (�Yk),which in the case of resistive changes can be de�ned as a real number and in the case ofinductive changes as an imaginary number. Capacitive changes can be considered as negativeinductive changes.

Total Sensitivity

Calculating the sensitivities for each of the harmonics results in a sensitivity matrix, de�nedas

S =

26664S11;120 S11;180 : : : S11;600S12;120 S12;180 : : : S12;600

......

S1n;120 S1n;180 : : : S1n;600

37775 (5.45)

An overall sensitivity taking into account the sensitivities for all the harmonics up to afrequency of 600 Hz can then be calculated using a weighting vector

Iw =

26664Iw;120Iw;180...

Iw;600

37775 (5.46)


which is described in more detail in the following section. Multiplying the weighting vectorand the sensitivity matrix (5.45) results in

Stotal = S � Iw =

26664P10

h=2 (S11;h�60 � Iw;h�60)P10h=2 (S12;h�60 � Iw;h�60)

...P10h=2 (S1n;h�60 � Iw;h�60)

37775 (5.47)

where n stands for the number of buses where network changes are possible.

Weighting Vector

The lower harmonics of transformer inrush currents are of higher magnitude than the higherones. Therefore a resonance peak at a lower harmonic is worse with respect to resonant over-voltage conditions than one at a higher harmonic. Consequently, changes at lower harmonicsare considered more important than changes at higher harmonics. This is taken into accountby a weighting vector Iw which is based on the harmonic characteristic of the transformerinrush current. As pointed out in Section 5.3.2, this characteristic is either given by themanufacturer or can be obtained by simulating the energization of an unloaded transformerusing the Emtp and taking the FFT of the inrush current, or by the application of themethod outlined in Section 5.3.2.

In order to �nd a suitable weighting vector Iw, a normalized current for each transformerterminal voltage VT is calculated as

I (h � 60; VT ) = I (h � 60; VT )maxh (I (h � 60; VT )) (5.48)

where h stands for the harmonic frequency and VT for the pre-switching steady-state trans-former voltage. We then obtain the normalized average by

Iw (h � 60) =PVTmax

VT=VTminI (h � 60)

maxh

�PVTmax

VT=VTminI (h � 60)

� (5.49)

Example

As an example, the system in Figure 5.19 is examined. It represents the same system as theone in Figure 5.4 and Figure 5.8, but a few restoration steps later. The impedance sensi-tivity at bus B1 is investigated with respect to changes at buses B2, B3, B4, B5, B6 and B7.

1. Thevenin Equivalent Circuit

As mentioned earlier, the Emtp frequency scan in combination with matrix calculations canbe utilized to calculate the Thevenin equivalent circuit for the desired range of harmonics.In this, only the frequencies from 60 to 600 Hz are of interest.


B2

B5

B6

B7

B1

G1 G3G2

B4

B3

G4 G5

Figure 5.19: System during restoration procedure

2. Individual Sensitivity

Representative for all harmonics up to a frequency of 600 Hz, the determination of thesensitivity for 120 Hz is presented in the following.

The elements of the resistive sensitivity vector SR and of the inductive sensitivity vectorSI can be found in Table 5.3. In order to investigate the validity of these results, theimpedance at bus B1 as function of admittance changes at the other nodes is calculated.They are varied in the range �Yk = (0:2 : : : 2) ~Y . For the inductive case, the value ~Y =�j � 4:421 � 10�4 S is chosen, which corresponds to an inductance L = 3000mH at thefrequency f =120Hz. For the resistive case ~Y =2 � 10�3 S is chosen, which corresponds to aresistance of R = 500. The results are shown in Figure 5.20 and Figure 5.21. The numbersin the graphs refer to the number k in Tables 5.3.

The results show good agreement with the corresponding sensitivities in Table 5.3. Themost e�ective impedance changes for the frequency f = 120Hz at bus B1 are achieved bychanging the admittance values at either bus B1 or B3.


Table 5.3: Sensitivity vectors for f=120 Hz

Bus k 1 2 3 4 5 6 7

SR(1k) -1.0000 0.1174 -0.3297 0.0140 -0.0034 -0.0033 -0.0022

SI(1k) 0.6388 0.4081 1.0000 0.1966 0.0022 0.0022 0.0006

160

180

200

220

240

260

280

300

0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8


ImpedanceZB

1

(�Yk)[k]

2

2

4,5,6,7

31

1

Figure 5.20: Impedance ZB1(120Hz) as a function of resistive changes

3. Weighting Function

Figure 5.22 shows the frequency components of the transformer inrush current, for terminalvoltages between 0.95 per unit and 1.05 per unit, and for frequencies from 120 Hz to 600 Hz.The result for the weighting function is depicted in Figure 5.23.

4. Total Sensitivity

The results for the total resistive and inductive sensitivities can be found in Table 5.4. In or-

Table 5.4: Total sensitivity vectors

Bus k 1 2 3 4 5 6 7

SR(1k) -1.0000 -0.0306 -0.1406 -0.2630 -0.0034 -0.0034 -0.0001

SI(1k) 1.0000 0.0455 0.2051 0.2636 0.0035 0.0034 0.0002

der to verify their validity, the impedance at the switching bus B2 as a function of frequencyis calculated. Figure 5.24 shows the results for resistive changes at the buses of the Thevenin


3

4

1

5,6,7


ImpedanceZB

1

(�Yk)[k]

260

280

300

320

340

360

380

400

420

440

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2

2

Figure 5.21: Impedance ZB1(120Hz) as a function of inductive changes

0.940.96

0.981

1.021.04

1.06100150

200250

300350

400450

500550

600

0

50

100

150

VT [p.u.]

I [kA]

f [Hz]

Figure 5.22: Harmonic characteristic of transformer

equivalent circuit. The index 0 stands for the case where no change is made and 1; 3 and4 stand for resistive changes of R = 100 at the respective nodes listed in Table 5.4. Onlythe sensitivities with a magnitude of relevance are shown. The same procedure was carriedout for inductive changes of L = 2000mH and its outcome is depicted in Figure 5.25. Theresults correlate well with their total sensitivities. The most e�ective changes for reachingan optimum resonance avoidance are changes of the resistances at bus B1. Inductive changesshould be avoided at all of the buses.

5.4. EMTP Time-Domain Result Evaluation 60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

120 180 240 300 360 420 480 540 600Frequency f [Hz]

WeightingFunctionI w(f)[p:u:]

Figure 5.23: Weighting function

Frequency f [Hz]

ImpedanceZB

1(f)[k]

0

0

4

3

1

0

0.5

1

1.5

2

2.5

3

3.5

100 200 300 400 500 600 700

Figure 5.24: Impedance ZB1 for resistive changes as a function of frequency

5.4 EMTP Time-Domain Result Evaluation

Although the methods introduced in the previous section allow for a reduction of the numberof Emtp time-domain simulations, it will be still necessary to perform a number of suchsimulations. This chapter investigates the possibilities of �nding criteria in order to terminate


0431

10

1

2

3

4

5

6

7

8

100 200 300 400 500 600 700Frequency f [Hz]

ImpedanceZB

1(f)[k]

Figure 5.25: Impedance ZB1 for inductive changes as a function of frequency

Emtp transformer energization studies after a minimum amount of time, to limit the overallsimulation time [148]. The method that is described in the following sections is based onProny analysis combined with fuzzy logic rules. An overview of the method is shown inFigure 5.26.

5.4.1 Prony Analysis

Prony analysis is a fast and e�ective method for obtaining modal information from time-domain signals. It has been mainly used for the analysis of time-domain results of stabilitysimulations [61, 78], mostly as a complement to small signal stability analysis [143]. However,only a few attempts have been made to use this method for the processing of results fromelectromagnetic studies (see e. g. [85]).

Prony analysis extends the Fourier Transform, which gives the frequency, amplitude andphase of modal components, by providing explicit damping information. In the following,the theory behind Prony analysis is brie y described. More details can be found in [61, 62,78, 79, 85, 143, 183, 216].


Run EMTP simulation for 2 cycles

Start

Voltage limits exceeded ?

Prony Analysis on the last 2cycles of EMTP signal

Mode filter

Relevance of mode Closeness to steady state

Fuzzy Reasoning

Continue EMTP simulation for 1 cylce

No

YesContinue EMTP simulation ?

End

Yes

No


Theoretical Background

If a signal y(t) is given as a record ofN evenly spaced samples y(tk)=y(k); k=0; 1; : : : ; N�1,the Prony algorithm �ts a function of the form [78, 61]:

y(t) =MXi=1

Ai � exp (�it) � cos(!it+ �i) (5.50)

This is accomplished with the following procedure [79]:

� Fit the function y(t) with the discrete linear prediction model

y(tn) = �n�1 � y(tn�1) + : : :+ �0 � y(t0) (5.51)

� Find the roots of the characteristic polynomial associated with the �tting.

� Using the roots as complex modal frequencies for the signal, determine the amplitudeand initial phase of the signal's modal components.

As a result of this calculation we get the modal frequency, amplitude, phase angle, anddamping.


Practical Issues

It is not possible to identify the modes of a system using Prony analysis when it is subjectedto a large disturbance such as network switching or system faults. After such a disturbance,the nonlinearities in the system{e. g. transformer saturation characteristics{are re ected inthe current and voltage signals obtained by time-domain simulation. However, Prony anal-ysis assumes linear system behavior. As a consequence, when Prony analysis is appliedsuccessively using a sliding window, the analysis results will change as a function of time[59, 79]. The purpose of the application of Prony analysis in our case is not to identify systemmodes and investigate system properties. Instead, it is utilized to �nd the frequency com-ponents of the signal in each window, in a way similar to the Short-Time Fourier Transform(STFT) (see e. g. [190]) and the case presented in [27].

A single sample window is chosen to cover two 60 Hz cycles, which allows the capture ofthe fundamental frequency component and leads to a suÆcient resolution of the harmoniccomponents of the signal. The �rst analysis is performed after two 60 Hz cycles of an Emtpsimulation have passed. Then the sample window is moved by one cycle, allowing for aone-cycle overlap of two successive windows. This leads to more reliable results since eachcycle is covered twice by an analysis window.

5.4.2 Filter for Modes

The results obtained by Prony analysis are sometimes ambiguous and therefore not easilyinterpreted. Modes, needed for correct �tting of the signal in one window may disappearin the next one. Furthermore, the signal-to-noise ratio (SNR), which is a measure for thequality of �tting [79], may be too small to make a correct statement about the spectralcontent of the signal analyzed. A number of modes can therefore be discarded from furtherevaluation using approximate reasoning [213].

All of the fuzzy logic techniques described in the following are based on Monotonic(Proportional) Reasoning [41]. This method allows for a fast and eÆcient decision makingprocess, since the rules described in the following can be implemented with a few simpleequations. It is well suited for problems using fuzzy rules with the same consequent fuzzyset, such as ours, since it avoids the saturation of the solution fuzzy set. Details aboutfuzzy logic in general and Monotonic (Proportional) Reasoning in particular can be foundin [41, 213]. In addition, an application similar to ours, where the same method is used forthe step-size control of a time-domain power system simulator can be found in [82, 83].

Approximate Reasoning for Filter

The underlying rule for this reasoning process can be expressed as:

\If the mode has recently appeared and if the signal-to-noise ratio (SNR)of the analysis is suÆciently high, then the mode is kept for further eval-uation, else it is discarded."


This process is depicted in Figure 5.27 and can be expressed by the following equation:

�filter(m) =

��TA(m) + �SNR(m)

2

�'

(5.52)

where m stands for the number of the current Prony analysis and ' allows for the adjustmentof the �lter sensitivity.

total appearance µTA

signal-to-noise ratio µSNR

filter function µfilter

>0.75

yes no

keep mode discard mode

Figure 5.27: Reasoning function for �lter

Membership Function for Total Appearance

The total appearance assigns to each mode a membership function which depends on howlong ago and how often it has appeared in the past. This is shown in Figure 5.28 where therelative appearance membership function �RA is displayed for the last N =5 analyses. Themodes are weighed the higher the more recent their appearance has been. From the relativeappearance we can then determine the total appearance using the equation

�TA(m; h) = �

"NXi=1

(m� i; h) � �RA(i)#

(5.53)

where

(m� i; h) =n

1 if mode appears at (m� i)th analysis

0 else

and h stands for harmonic order, and � normalizes the maximum of the function to one.

Membership Function for Signal-to-Noise Ratio

The Signal-to-Noise Ratio (SNR) is a measure of how well the original signal is �tted by themodes as obtained by Prony analysis, i.e. a high SNR indicates an accurate analysis result.The SNR is de�ned as

SNR = 20 � log ky(tk)� y(tk)kky(tk)k dB (5.54)

where k : : : k stands for the root-mean-square norm [79]. The membership function for theSNR, �SNR, is shown in Figure 5.29.


1

m-5 m-1

1/5

µ RA

Figure 5.28: Membership function for relative appearance

µ SN

R

1

SNRmax SNR

Figure 5.29: Membership function for signal-to-noise ratio (SNR)

5.4.3 Prony Analysis Results Evaluation

After a mode passes the �lter it is evaluated with respect to two di�erent criteria: itsimportance and its \closeness to steady state". The number of Prony analyses N , on whichthe statement whether an Emtp simulation should be �nished or not is based, results in thefollowing con icting matters: if N is smaller, the Emtp calculation time will be smaller andthe statement will be less reliable, and vice versa. An analysis depth N between 5 and 8 hasshown to be a good compromise between reliability and e�ectiveness of the determinationof the termination criteria.

In the following, the various membership functions used for this process are explained,and the reasoning process leading to a conclusion whether the Emtp simulation can beterminated, is described.


Relevance of Modes

The underlying rules for the evaluation of the relevance of a particular mode can be sum-marized as:

1. \If the amplitude of the mode is large compared to the fundamental frequency mode,then the relevance of the mode is high"

2. \If the damping is small, then the relevance of the mode is high"

3. \If the average damping changes quickly, then the relevance of the mode is high"

This reasoning process, which is based on monotonic chaining and leads to the membershipfunction �TR, is depicted schematically in Figure 5.30.

1

µ tend

R

tendency β

1

µ ampl

itude

R

amplitude A

1

µ dam

ping

R

damping d relevance level

relevance level

relevance level

total relevance level

relevance of mode

1µ

TR

Figure 5.30: Monotonic reasoning in order to determine total relevance of mode

1. Membership Function for Amplitude

This function, shown in Figure 5.31, can be obtained by referring the amplitudes of eachmode to the 60 Hz steady-state value as obtained with the initialEmtp steady-state solution.It means that the higher the amplitude of a mode, the higher its relevance:

�amplitudeR(m; h) =n A(m;h)

A(60Hz)for A(m; h) � A(60 Hz)

1 for A(m; h) > A(60 Hz)(5.55)


1

µ ampl

itude

R (m

,h)

A(60 Hz) A(m,h)

Figure 5.31: Amplitude membership function

2. Membership Function for Damping

The smaller the damping of a mode, the more important it is. The membership function fordamping

�dampingR(m; h) =

(1 for d(m; h) < dmin

1� 1dmax�dmin

� (d(m; h)� dmin) for dmin � d(m; h) � dmax

0 for d(m; h) > dmax

(5.56)

is depicted in Figure 5.32.

1

µ dam

ping

R (m

,h)

d(m,h)dmin dmax

Figure 5.32: Damping membership function

3. Membership Function for Tendency of Damping

Besides the value for damping it is also necessary to take into account its tendency. Thisis accomplished by using a statistical method called linear regression [182]. With this, the


slope of a linear interpolation through the last N values of d is de�ned as

�(m; h) =12PN

i=1

�i� 1�N

2

�d (m�(N + 1) + i)

�N +N3(5.57)

The higher the value of the tendency, the more important its change. Its membershipfunction, which is displayed in Figure 5.33 is de�ned as

�tendR(m; h) =

(1 for �(m; h) � �min2; �(m; h) � �max2

0 for �min1 � �(m; h) � �max1

1� 1�min1��min2

(�(m; h)� �min2) for �min2 � �(m; h) � �min1

1�max2��max1

(�(m; h)� �max1) for �max1 � �(m; h) � �max2

(5.58)

1

µ tend

R (m

,h)

β (m,h)βmin2 βmin1 βmax1 βmax2

Figure 5.33: Membership function for tendency of damping

Closeness to Steady State

In this section each mode is assigned a membership function for \closeness to steady state".The rules can be summarized as:

1. \If the amplitude of the mode is small, then steady state is close"

2. \If the damping is large and positive, then steady state is close"

3. \If the tendency of damping is moderate and negative, then steady state is close"

4. \If the variance of damping is small, then steady state is close"

The result of these rules is the \closeness to steady state" membership function �SST . Theunderlying reasoning process is displayed in Figure 5.34 and the membership functions are


1

µ tend

S

tendency β1

damping d steady state level

steady state level

total steady state level

1

1

µ ampl

itude

S

amplitude A

1

µ dam

ping

S

1

µ var

ianc

eS

variance Vdamping

steady state level

steady state level

closeness tosteady stateµ

SS

T

Figure 5.34: Monotonic reasoning in order to determine total \closeness to steady state" ofmode

described subsequently.

1. Membership Function for Amplitude

This function, displayed in Figure 5.35, is obtained by inverting the amplitude membershipfunction in Equation (5.55):

�amplitudeS = 1� �amplitudeR (5.59)

It means that the mode is the closer to steady state the smaller its amplitude.

2. Membership Function for Damping

Here, the inverse of the membership function (5.56) is used, since the higher the damping,the closer we are to steady state:

�dampingS = 1� �dampingR (5.60)

The membership function is displayed in Figure 5.36.


1

µ ampl

itude

S (m

,h)

A(60 Hz) A(m,h)

Figure 5.35: Amplitude membership function

1

µ dam

ping

S (m

,h)

d(m,h)dmaxdmin

Figure 5.36: Damping membership function

3. Membership Function for Tendency of Damping

This function is shown in Figure 5.37 and is de�ned as:

�tendS(m; h) =

(0 for �(m; h) � �min3; �(m; h) � �max3

1�medium3��min3

(�(m; h)� �min3) for �min3 � �(m; h) � �medium3

1� 1�max3��medium3

(�(m; h)� �medium3) for �medium3 � �(m; h) � �max3

(5.61)

4. Membership for Variance of Damping

It is not enough to only consider the damping by its tendency, since the latter may besmall although there is still a high uctuation. As a measure for this uctuation we use the


β (m,h)

µ tend

S (m

,h)

βmin3 βmedium3 βmax3

1

Figure 5.37: Tendency of damping membership function

variance of damping [182]:

Vdamping(m; h)=

vuut 1

N

m�1Xi=m�N

(d(i; h)�dlinear(i))2 (5.62)

where dlinear(i) stands for the linear regression through the last N samples of damping d.The variance is normalized by dividing by dlinear(m):

�Vdamping(i) =Vdamping

dlinear(m)(5.63)

The membership function, displayed if Figure 5.38 can then be de�ned by:

�varianceS(m; h) =

(1 for Vdamping(m; h) � Vdmin

1� 1Vdmax�Vdmin

(Vdamping(m; h)� Vdmin)

for Vdmin < Vdamping(m; h) < Vdmax

0 for Vdamping(m; h) � Vdmax

(5.64)

5.4.4 Reasoning Process

Based on above, we can conduct a �nal decision making process, giving us a statementwhether we are close to steady state and whether the Emtp simulation in progress can beterminated or not. The underlying rule can be formulated as:

\If each important mode is close to steady state, we are overall close tosteady state and the Emtp simulation can be terminated."


1

µ va

rianc

eS (m

,h)

Vdmin Vdmax Vdamping (m,h)

Figure 5.38: Membership function for variance of damping

Mathematically this can be de�ned by:

�total(m) =

Phmax

h=0 (�SST (m; h) � �TR(m; h))Phmax

h=0 �TR(m; h)(5.65)

where h=0 stands for the fundamental frequency, h=1 for the �rst harmonic, etc.

This process is depicted in Figure 5.39. A signal consisting purely of the fundamentalfrequency would lead to a total membership function value of �total = 1. The value of0.95 in Figure 5.39 can be increased, depending on the speci�c requirements concerning the\closeness to steady state".

60 Hz total steady stateµ SST (m,0)

60 Hz total relevanceµ TR (m,0)

hmax . 60Hz total steady state

µ SST (m,hmax)

hmax . 60Hz total relevanceµ TR (m,hmax)

120 Hz total steady stateµ SST (m,1)

120 Hz total relevanceµ TR (m,1)

yes

total membership functionµtotal (m)

>0.95

no

stop EMTPsimulation

continue EMTPsimulation

Figure 5.39: Final reasoning process


0 0.1 0.2 0.3 0.4 0.5−300

−200

−100

0

100

200

300|{ EMTP signal � � � predicted signal

Time t [s]

VoltageVB

2(t)[kV]

Figure 5.40: Emtp signal and predicted signal

0.25 0.3 0.35 0.4 0.45 0.5−300

−200

−100

0

100

200

300|{ EMTP signal � � � predicted signal

Time t [s]

VoltageVB

2(t)[kV]

Figure 5.41: Emtp signal and predicted signal for m>15

5.4.5 Example

The system under study is shown in Figure 5.4. It has a resonance frequency at 180 Hz.The restoration step under consideration is the switching of the transformer at bus B3.

For comparison a full Emtp simulation for 35 cycles has been carried out. Both Figures5.40 and 5.41 show the Emtp simulation result. Since the resonance frequency is 180 Hz, the


0 5 10 15 20 25 30 350

10

20

30

40

50

60

70

80

90

Number of Prony analysis m

Amplitudea(m;3)[kV]

Figure 5.42: Amplitude of 180 Hz mode

0 5 10 15 20 25 30 35−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Number of Prony analysis m

Dampingd(m;3)[s�

1]

Figure 5.43: Damping of 180 Hz mode

only harmonic of relevance is the third. Its amplitude and damping are shown in Figures 5.42and 5.43, respectively. The fuzzy reasoning algorithm terminates the Emtp simulation afterm = 15 Prony analyses. Subsequently, the signal can be predicted using Equation (5.50).The predicted signal is also displayed in Figures 5.40 and 5.41. It is so close to the Emtpsimulation result that the di�erences are barely noticable.

5.5. Switching Transient Overvoltages 75

The application of the routine for this example shortens the overall calculation timeby approximately 37 %. This percentage increases for larger networks, since the Emtpsimulation time increases with increasing network size, whereas the total overhead time forthe Prony analysis and fuzzy logic algorithms stays constant.

5.5 Switching Transient Overvoltages

Switching surges are fast transients with fundamental frequencies between 100 Hz and 1000Hz. They occur when transmission lines are energized, and consist of a transient componentthat is superimposed on a power frequency component.

In the worst case, switching surges can lead to ashovers, and to serious damage toequipment. However, they are usually not of particular concern during the reenergizationof the transmission system during restoration. As long as the steady-state voltages remainbelow 1.2 per unit they can be easily limited by surge arresters [95]. In cases where nosurge arresters are available, line ashovers do not impose a problem on the restorationprocess as well, since line insulators are usually designed conservatively to withstand steady-state overvoltages during heavy fog conditions, resulting in very high impulse strength. Anexception is the case where transformer terminated lines are energized. That case may lead tothe occurrence of harmonic resonance overvoltages [95] and has been investigated in Section5.3.

As a consequence of the above, the control of transient switching voltages is not in-vestigated in this work, and only general observations are made. The parameters of mostin uence on the line switching overvoltages are [38]:

� Line parameters:

{ line length

{ degree of parallel compensation

{ line termination (open or transformer terminated)

{ trapped charges

� Circuit breaker parameters:

{ Closing resistors

{ Phase angle at moment of switching

� Supply side parameters:

{ Total short-circuit power

{ Inductive / complex network

In the case where underground cables are energized, the transients are dominated by thecable capacitances. When switching cables, traveling wave e�ects are not prevalent but there

5.6. Summary 76

is a higher potential for sustained overvoltage problems. The latter is dealt with in Section5.2.

The in uence of di�erent parameters on switching overvoltages is currently investigatedin detail as part of a Ph.D. thesis research project of Awad Ibrahim at the University ofBritish Columbia.

5.6 Summary

In this chapter, a methodology has been outlined that assesses the feasibility of restorationsteps with respect to overvoltages. It is based on analysis in the frequency domain using thefrequency scan feature of the Emtp, and analytical matrix manipulations. Using approxi-mate rules allows one to estimate whether an overvoltage occurs and whether time-domainsimulations or power- ow calculations need to be carried out. In the case of time-domainsimulations, a method is introduced that allows one to shorten the simulation time using amethod based on Prony analysis and fuzzy logic. When overvoltage problems occur, the mosteÆcient remedial action is determined by sensitivity analysis. The validity of the methodshas been veri�ed using several examples based on actual system data.

Chapter 6

Auxiliary System Analysis

After a widespread blackout, the power of large thermal generating units has to be restoredquickly and reliably to bring major loads back into service as soon as possible. If no assistancefrom other utilities is available, a restoration path from a relatively small hydro generatoror gas turbine has to be built to the large thermal power station to start up the thermalpower plant's auxiliaries. These auxiliaries are mostly driven by large induction motors,drawing high currents and requiring long starting times. To ensure a successful start of athermal power plant, it is crucial to investigate the feasibility of such auxiliary motor start-ups [36]. This is usually accomplished using time-domain simulation programs such as theElectromagnetic Transients Program (Emtp), or stability programs (see Chapter 3.4).

The implementation of the induction motor di�erential equations in most simulationprograms can be considered as mature, and in most cases no further improvement is required[36]. However, currently available methods for determining induction motor parametersoften do not take into account all available motor data. This may lead to inaccuracies inthe motor model. Therefore, a new parameter estimation method for induction motors isintroduced. It creates induction motor models from manufacturer nameplate data and frommotor performance characteristics. The induction motor models can then be used for blackand emergency start studies with the Electromagnetic Transients Program (Emtp), or withstability programs, and for the rules outlined in this chapter.

Time-domain simulations of motor start-ups require a vast amount of calculation time.Therefore, rules are developed in the following that estimate motor start-up times, currents,voltage drops, and frequency drops during motor starts.

6.1 Motor Parameter Estimation

6.1.1 Overview of Parameter Estimation Methods

Generally, induction motor parameter estimation methods can be classi�ed into �ve di�erentcategories, depending on what data is available, and what the data is used for.

77

6.1. Motor Parameter Estimation 78

1. Parameter calculation from motor construction data. This method requires a detailedknowledge of the machine's construction, such as geometry and material parameters.On the one hand, it is the most accurate procedure, since it is most closely related tothe physical reality. On the other hand, it is the most costly one since it is based on�eld calculation methods, such as the �nite element method [24, 53]. This method ismainly applied in induction motor design.

2. Parameter estimation based on steady-state motor models. The methods in this cate-gory use iterative solutions based on induction motor steady-state network equationsand given manufacturer data [89, 105, 191, 222]. This is the most common type of pa-rameter estimation for system studies since the data needed for it is usually available.

3. Frequency-domain parameter estimation. The stand-still frequency response (SSFR)method is based on measurements that are performed at standstill. The motor param-eters are estimated from the resulting transfer function [223]. The major advantageof this method is its accuracy. However, stand-still tests are not common industrypractice, and this method can therefore not be used very often.

4. Time-domain parameter estimation. For this type of method, time-domain motormeasurements are performed and model parameters are adjusted to match the mea-surements [44, 107, 171]. Since not all parameters can be observed using measurablequantities, the motor models need to be simpli�ed [107]. The method is costly, andthe required data is usually not available.

5. Real-time parameter estimation. This type of parameter estimation is used to tunethe controllers of induction motor drive systems. This requires real-time parameterestimation techniques, using simpli�ed induction motor models, that are fast enoughto continuously update the motor parameters and therefore prevent the \detuning" ofinduction machine controllers [84, 206].

The motor parameter estimation method proposed in this section belongs to the secondgroup of methods. It is suitable for system studies since suÆcient data is usually availableto determine a motor model of suÆcient accuracy. Most methods in this category have thedisadvantage that they only accept nameplate data [89, 191], even when more data, such asperformance characteristics, are available. Furthermore, they often ignore constraints on themachine parameters or saturation e�ects [105, 222].

This paper proposes a new approach to overcome these drawbacks and to make it possibleto exibly determine motor models for any given combination of manufacturer data. The newmotor parameter estimation routine can be considered as a generalization and combinationof the methods introduced in [105, 191].

In the following, the input data to the motor parameter estimation routine is described.Then, a parameter estimation algorithm neglecting saturation is developed, tested and ex-tended to allow for saturation e�ects. Finally, the results for a typical 600 HP pump motor,such as commonly used in large thermal power plants, are given.


6.1.2 Input Data

The proposed parameter estimation routine allows the input of di�erent types of data, de-pending on their availability. The �rst type of data is the one given on the motor nameplate,where subscript \FL" refers to full load, and \LR" to locked rotor test.

� Rated power PFL

� Rated voltage Vrated

� EÆciency �FL

� Power factor pfFL

� Rated slip sFL

� Starting current ILR

� Starting torque TLR

� Breakdown torque Tmax

This set of data is not always suÆcient to obtain a motor model which behaves accuratelyover the entire speed range. The new parameter estimation routine can therefore use extramotor performance data such as:

� Current-slip characteristic Im:f:(s)

� Torque-slip characteristic Tm:f:(s)

� Power factor-slip characteristic pfm:f:(s)

where \m:f:" stands for \manufacturer data".

Additional data, such as rotational losses Prot FL, can also be taken into account, whichcreate constraints and boundaries for the nonlinear optimization method of the followingsection.

6.1.3 Nonlinear Optimization Procedure

Our algorithm is based on the nonlinear optimization routine Solnp [230, 231], which solvesnonlinear programming problems of the general form:

minimize F (X) (6.1)

subject to: G(X) = 0 (6.2)

Lh � H(X) � Uh (6.3)

Lx � X � Ux (6.4)


where X�Rn, F (X) : Rn �! R, G(X) : Rn �! Rm1 , H(X) : Rn �! Rm2 , Lh;Uh�Rm2 ,

Lh < Uh, Lx;Ux�Rn and Lx < Ux, Lx < X0 < Ux, X�R

n.

The function F (X) represents the objective function. Equation (6.2) stands for theequality constraints, and Equation (6.3) for the inequality constraints, and Equation (6.4)represents the boundaries on the variables. X0 is the initial estimate for the solution. Theunderlying mathematical algorithm uses major iterations, in which a linearly constrainedoptimization problem is solved based on a Lagrangian objective function. Within each majoriteration, minor iterations are carried out, based on linear and quadratic programming. Thefunctions (6.1) to (6.4), as used for our parameter estimation routine, will be de�ned in thefollowing sections.

6.1.4 Motor Model without Saturation

Objective Function

The objective function F (X) is developed starting from the steady-state equations for adouble-cage induction motor. These equations can also be used for induction motors withdeep-bar rotors and with a single-cage [143].

For the derivation of the steady-state equations the motor equivalent circuit of Figure6.1 is used [54, 143]. All parameters and equations are given in per unit. The impedances

R1 X1

XM

X2

X21 X22

R21/s R22/s

I1 I2

I21 I22V1

Figure 6.1: Induction motor model without saturation

for the two rotor circuits in Figure 6.1 are

Zr1 =R21

s+ jX21 (6.5)

Zr2 =R22

s+ jX22 (6.6)

This results in the total rotor impedance

Zr = Rr + jXr = jX2 +Zr1 � Zr2

Zr1 + Zr2(6.7)


and the total motor impedance as seen from the motor terminals

Zmot = R1 + jX1 +jXm � Zr

jXm + Zr

(6.8)

The motor current as a function of the slip follows as

I1 =V1Zmot

(6.9)

where V1 represents the motor terminal voltage. Using current divider equations we then get

I2 =jXm

jXm + Zr

� I1 (6.10)

I21 =Zr2

Zr1 + Zr2

� I2 (6.11)

I22 =Zr1

Zr1 + Zr2

� I2 (6.12)

From the above equations we �nally obtain the equations for current magnitude, power factorand torque:

Ic(s) = jI1j (6.13)

pfc(s) = cos\(Zmot) (6.14)

Tc(s) =R21

s� jI21j2 + R22

s� jI22j2 (6.15)

where the index \c" stands for \calculated". Equations (6.13) to (6.15) form the basis for theobjective function F (X) in Equation (6.1), which is de�ned as a quadratic error function:

F (X) = WI �nIXi=1

�2I(si) +WT �

nTXi=1

�2T (si) +Wpf �

npfXi=1

�2pf(si) (6.16)

(6.17)

where

X = [R1; X1; XM ; X2; R21; X21; R22; K;m] (6.18)

with K and m de�ned in Section 6.1.4, and

�I(si) =Ic(si)� Im:f:(si)

Im:f:(si)i = 1; : : : ; nI (6.19)

�T (si) =T (si)� Tm:f:(si)

Tm:f:

i = 1; : : : ; nT (6.20)

�pf(si) =pf(si)� pfm:f:(si)

pfm:f:(si)i = 1; : : : ; npf (6.21)

The quantity si represents discrete values for the induction motor slip, and nI , nT , npf arethe total number of data points available for current, torque, and power factor, respectively.The factors WI , WT , and Wpf are weighting factors described next.


Weighting Factors

The choice of the weighting factors for the objective function F (X) is of great importancefor the optimization process. It is desirable to give each of the torque-slip, powerfactor-slip, or current-slip characteristics equal weight. Otherwise, if e. g. a large number of datapoints for the torque-slip characteristic and only a small number of points for the current-slipcharacteristic is provided, the latter would only have a negligible in uence on the value of theobjective function if each point had equal weight. This would lead to signi�cant di�erencesbetween manufacturer and calculated values in the current-slip characteristic.

Generally, nameplate parameters are more reliable than motor performance character-istics. The latter do not always give exact numbers, but rather indication of the genericperformance behavior [17]. Based on these observations we de�ne the following weightingfactors:

WI = nT + npf (6.22)

WT = nI + npf (6.23)

Wpf = nI + nT (6.24)

The importance of nameplate parameters is re ected by multiplying its weighting factors byan additional factor

WNP =nI + nT + npf

3(6.25)

Constraints and Boundaries

If neither constraints nor boundaries are used for the optimization, the result vector X maycontain \non-physical" values, such as negative values for resistances. Consequently, wede�ne the following boundaries:

R1; X1; XM ; X2; R21; X21; R22 � 0 (6.26)

We further de�ne a boundary condition for the design factor m [191]:

m =R21 +R22

X21 +X22

(6.27)

0:4 � m � 1:1 (6.28)

when neither the rotor type nor m are known. In cases where the rotor type is known, theboundaries in Equation (6.28) are adjusted. The following boundary conditions are chosen,allowing for a �10% variation of the values given in [191]:

0:45 � m � 0:65 for deep bar rotors (6.29)

0:90 � m � 1:10 for double cage rotors (6.30)

If the rotor is of type single cage, the constraint for m is removed since a design factor inthis case is not de�ned.


Based on nameplate data, four constraints G1(X), : : : , G4(X) are de�ned:

X2 �K �X1 = 0 (6.31)

Xr �X1 = 0 (6.32)

Tmax m:f: � Tmax(X) = 0 (6.33)

PFL � (R1 � I21 FL +R21 � I221 FL +R22 � I222 FL + Prot FL)

PFL� �FL = 0 (6.34)

The factor K in Equation (6.31) is subject to another boundary, whose value is chosenaccording to [17, 29, 191]:

0 � K � 0:4 (6.35)

Equation (6.32) comes from the assumption that the rotor and stator reactance are equal[136, 143].

The maximum or breakdown torque Tmax is reached when the power ow from stator torotor is a maximum, i. e. when the following condition is ful�lled:

Rr(smax)

smax

=q(R2

stat + (Xstat +Xr(smax))2) (6.36)

The values for Rstat and Xstat are obtained from the Thevenin equivalent impedance thatreplaces the stator part of the induction motor (see Figure 6.2):

Zstat = Rstat + jXstat =jXM � (R1 + jX1)

R1 + j(X1 +XM)(6.37)

From Equation (6.7) we can further calculate the resistance Rr and reactance Xr of the

Rstat Xstat X2

X21X22

R21/s R22/s

I1 I2

I21 I22Vstat

Stator Rotor

Zr

Figure 6.2: Induction motor model with stator part replaced by Thevenin equivalent circuit

rotor as:

Rr =1

s� R21 �R22 � (R21 +R22) + s2 � (R21 �X2

22 +R22 �X221)

(R21 +R22)2 + s2 � (X21 +X22)2(6.38)

Xr =1

s� R

221 �X22 +R2

22 �X21 + s2 �X21 �X22 � (X21 +X22)

(R21 +R22)2 + s2 � (X21 +X22)2+X2 (6.39)


Equation (6.36) can be solved analytically for smax and the maximum torque Tmax is thendetermined from Equation (6.15).

Initialization

It is important to start the optimization process with initial estimatesX0 as close as possibleto the values that lead to a minimum for the objective function F (X). This is accomplishedusing approximate equations based on [191]:

�0FL = 0:25 + 0:75 � �FL (6.40)

R1 = pfFL ��1� �0FL

1� sFL

�(6.41)

RFL = sFL � �0FL1� sFL

� pfFL (6.42)

XM =�0FL

(1� sFL) � sin�FL (6.43)

RLR = TFL � �0FL � pfFL=I2LR

1� sFL(6.44)

m = 0:7 (6.45)

R21 = RLR � (1 +m2)�RFL �m2 (6.46)

R22 = R21 � RFL

R1 �RFL

(6.47)

X21 = 0 (6.48)

X22 =R21 +R22

m�X21 (6.49)

Xl =

s�1:0

ILR

�2

� (R1 +RLR)2 (6.50)

X1 =Xl

2(6.51)

X2 = X1 �RFL � R21

R22� m

m2 + 1(6.52)

K =X2

X1(6.53)

Veri�cation of Algorithm

In case a set of adverse initial estimates is chosen, the optimization routine may converge toa local minimum for the objective function that does not represent the best possible solution.In order to ensure with a high probability that the parameter estimation algorithm leads toan absolute minimum, the initial estimates are varied randomly between 0% and 200% ofthe values calculated according to Equations (6.40) to (6.53), except for the parameters mand K (Equations (6.29) and (6.35)), which are randomly varied between their boundaries.If the initial estimates are chosen close enough to the set of parameters that leads to the


absolute minimum of the objective function, the objective function value should always beequal or smaller than the values we obtain for random initial estimates.

Di�erent rotor parameter sets may lead to the same value for the objective function. Tocompare between the results for di�erent initial estimates, one of the rotor parameters is�xed for this test. Otherwise the objective function may lead to the same value althoughthe motor parameters are di�erent. As suggested in [143, 191], the rotor reactance X21 istherefore limited to a small value:

0 < X21 < 10�3 (6.54)

For the test, we use the motor data of a 600 HP pump motor, which is currently in use ina large thermal power plant. Its nameplate data is listed in Table 6.1 and the torque-slip,current-slip, and powerfactor-slip characteristics are displayed in Table 6.2.

Table 6.1: Induction motor nameplate data

PFL [HP] 600Vrated [kV] 4

Synchr: speed [RPM] 900

TFL [Nm] 4812IFL [A] 75.5pfFL 0.914�FL 0.936

sFL [%] 1.333

TLR [Nm] 4861ILR [A] 480.8pfLR 0.29

Tmax [Nm] 13810

For 100 di�erent sets of initial estimates, the algorithm converged in 81 cases to the sameobjective function values and delivered the same motor parameters. The values for the othercases were signi�cantly higher and therefore belong to local minima that do not representthe best possible solution. The objective function values with F (X)< 0:2 are displayed inFigure 6.3 and the number of major iterations in Figure 6.4. The �rst set of initial estimatesis the one calculated with Equations (6.40) to (6.53). In Table 6.3 the mean of the motorparameters X and the percentage deviation �X from the mean are given. The deviationsare negligible, despite the variation in the number of major iterations between 3 and 10.

As a second test, the algorithm's robustness is investigated. \Correct" current-, torque-,and powerfactor-slip characteristics are created from the motor parameters in Table 6.3.These are then fed back into the algorithm. The di�erence �Xm:f :�corr: between the param-eters obtained from the \correct" characteristics and from the manufacturer data, given inTables 6.1 and 6.2, is shown in Table 6.4. The only signi�cant di�erence can be observed forvariable X21. However, since its maximum value is limited to the small value of 10�3, it isof no relevance.


Table 6.2: Induction motor characteristics

Speed [RPM] I [A] pf T [Nm]

0.0 480.8 0.290 4861.0150 { { 5288.0180 { 0.306 {400 { { 6454.0405 { 0.339 {600 { { 8243.0630 { 0.416 {750 { { 12202.0765 { 0.570 {850 { 0.813 12609.0888 75.5 0.914 4812.0

10 20 30 40 50 60 70 800.0604

0.0606

0.0608

0.061

0.0612

0.0614

0.0616

0.0618

Set of initial estimates

ObjectivefunctionF(X)

Figure 6.3: Values for objective function

6.1.5 Motor Model with Saturation

In this section, we discuss how saturation e�ects can be taken into account. Only thesaturation of the stator leakage reactance X1 and the rotor mutual leakage reactance X2

are considered, which is a good approximation for most motor start-up studies [191]. Theequivalent circuit for a motor including saturation e�ects is depicted in Figure 6.5 [143]. Inthe following, only the changes needed for the saturation e�ects are outlined.


10 20 30 40 50 60 70 803

4

5

6

7

8

9

10

Set of initial estimates

Numberofiterations

Figure 6.4: Number of iterations

Table 6.3: Induction motor parameters in per unit

X X / [p.u.] �X / [%]

R1 0.024111 0.0005X1 0.076779 0.0006R21 0.027005 0.0257X21 0.00099999 0.0416R22 0.027389 0.0271X2 0.070127 0.0415XM 3.2316 0.0293m 0.60206 0.0996K 0.91336 0.5426

Objective Function

To represent saturation of the leakage reactances, we use the \describing function" DFde�ned in [191]:

DF = 1 for > 1 (6.55)

DF =2

�� (� + 0:5 � sin (2�)) for � 1 (6.56)

=IsatI

(6.57)

� = arcsin ( ) (6.58)


Table 6.4: Deviation of induction motor parameters

X �Xm:f :�corr: / [%]

R1 0.0459X1 0.0263R21 0.9137X21 25.606R22 0.8115X2 0.4028XM 0.2592m 0.5889K 0.4292

R1 X1u

XM

DF . X2sX21 X22

R21/s R22/s

I1 I2

I21 I22V1

DF . X1sX2u

Figure 6.5: Induction motor model with saturation

where Isat stands for the saturation threshold current and I for the current owing throughthe reactance.

It is assumed that both stator and rotor leakage reactance have the same percentage partsaturated, and de�ne the modi�ed reactances as:

X1 = X1u +DF1 �X1s = (1� SAT ) �X10 +DF1 � SAT �X10 (6.59)

X2 = X2u +DF2 �X2s = (1� SAT ) �X20 +DF2 � SAT �X20 (6.60)

The factor SAT represents the per unit value of the saturable parts of the leakage reactancesX1s and X2s with respect to the unsaturated leakage reactances X10 and X20:

SAT =X1s

X10=X2s

X20(6.61)

The values of the currents and the describing functions are now determined iteratively,since the currents depend on the values of the reactances, and vice versa. This is shown inthe diagram in Figure 6.6, where the superscript \n" stands for the nth iteration.

The objective function F (X) for the saturated induction motor model has the same formas the one without saturation, with two more variables added:

X = [R1; X1; XM ; X2; R21; X21; R22; K;m; Isat; SAT ] (6.62)


X1n = X10 ; X2

n = X20

I 1n ; I2

n

DF1 ; DF2

X1 ; X2

I 1n+1 ; I2

n+1

| I1n+1 - I1

n | < ε

| I2n+1 - I2

n | < ε?

I1n = I1

n+1 ; I2n = I2

n+1

No

Yes

I c ; pfc ; Tc

Figure 6.6: Calculation of currents and describing functions

If detailed saturation data is available, the saturation threshold current can be �xed, intro-ducing an additional boundary.

Constraints, Boundaries, Weighting Factors, and Initialization

In addition to the boundaries described earlier, we add [191]:

0:2 � SAT � 0:8 (6.63)

1:5 � Isat � 3:0 (6.64)

This allows a deviation of �60% from the value SAT =0:5 suggested in [191].


The breakdown torque for the saturated case can no longer be calculated analyticallysince, as outlined earlier, the currents have to be determined iteratively. Instead, it is foundwith an additional maximum �nding routine. Since the torque function may have another(local) maximum, the range where the maximum torque is located is limited to:

0:0 � smax � 0:2 (6.65)

For the initialization procedure, the values calculated in Equations (6.40) to (6.53) remainthe same, and two initial values for the new variables are added [191]:

Isat = 2:0 (6.66)

SAT = 0:5 (6.67)

Veri�cation of Algorithm

For the veri�cation of the algorithm that includes saturation of leakage reactances, thesame motor is investigated and the same procedures are carried out as for the case withoutsaturation. For 100 di�erent initial estimates, the algorithm converged in 78 cases to thesame objective function values. The objective function values and the number of iterationsas a function of the set of initial estimates look similar to Figure 6.3 and 6.4. Table 6.5shows the mean of the motor parameters and the percentage deviation from the mean.

Table 6.5: Induction motor parameters in per unit

X X / [p.u.] �X / [%]

R1 0.02369 0.00009X1 0.11078 0.00060R21 0.02965 0.00190X21 0.00002 0.00030R22 0.02372 0.00120X2 0.10257 0.00080XM 0.20314 0.00690m 1.10000 0.03970K 0.92592 0.00950SAT 0.49999 0.00020Isat 1.50000 0.00080

6.1.6 Results

Figures 6.7 to 6.9 show the results obtained from the algorithm for the 600 HP motor. Goodagreement between the manufacturer's and model characteristics can be observed. Thematch for the model with saturation is slightly better than the one for the model without


saturation. This di�erence is not signi�cant for this particular motor. It can be explainedby the additional degree of freedom that is introduced by the inclusion of saturation, whichleads to a better overall �tting. The results also show that close to the rated voltage, themodel without saturation gives a reasonably good �t, and can therefore be used for caseswhere voltage variations are small.

0 100 200 300 400 500 600 700 800 9000

50

100

150

200

250

300

350

400

450

500

Speed [RPM]

Current[A]

No saturationSaturation

Manufacturer data

Figure 6.7: Current characteristic

0 100 200 300 400 500 600 700 800 9000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Speed [RPM]

Powerfactor


Manufacturer data

Figure 6.8: Powerfactor characteristic

6.2. Rules for Motor Start-up 92

0 100 200 300 400 500 600 700 800 9000

2000

4000

6000

8000

10000

12000

14000

Speed [RPM]

Torque[Nm]


Manufacturer data

Figure 6.9: Torque characteristic

6.2 Rules for Motor Start-up

A general overview of the methodology described in this section is given in Figure 6.10. Forthe formulation of the rules, it is assumed that only one motor is started at a time. In caseseveral motors are started at once, they are lumped into a single machine. Methods thatdeal with the aggregation of static and motor loads were investigated in a Ph.D. thesis thatwas recently published at the University of British Columbia [157]. Thus, motor aggregationmethods are not discussed in this report.

In order to evaluate the following rules [149], the sample system shown in Figure 6.11 isused. It represents the worst-case scenario in a power system where a number of inductionmotors are emergency-started by a hydro generator. The turbine, governor system, andgenerator of the system have already been used in the example in Section 4.1.1. The blockdiagram that represents the excitation system is displayed in Figure 6.12. All inductionmotors are lumped into a single machine of power 14.4 MW. Data for this aggregated loadmodel has been provided by the utility.

6.2.1 Thevenin Equivalent

A multi-port Thevenin equivalent impedance Z with terminals for the motor and generatoris determined by Emtp frequency scans. This is done only once, since changing networkconditions during the extension of the system can be calculated using the matrix calcula-tions outlined in Section 5.1. All variable elements, such as loads, reactances, capacitances,and motors are assumed to be included in the impedance matrix Z. To one of the termi-nals we connect the generator subtransient reactance X 00

d and to the other the locked rotormotor impedance ZLR, calculated according to Equation (6.8) (with s=1). The Thevenin


Build equivalent motor model

Yes

No

Yes

No

Determine voltage drop ∆V

Determine I start , Qstart

R I2 > Wmax(tstart,I)Yes

No

Remedial actionDetermine network matrix [Z]

Yes

fsystem < fmin


No

End

EMTP Simulation

Problem ?Yes

Yes

No

Start

No

No. of motors > 1

∆V > ∆Vmin ?

Qstart > Qmax ?Yes

Motor Start ?



TransformerCable /

circuit breaker Induction motorGenerator Cable

Figure 6.11: Sample system for motor start

Vt Efd

1+sTC

1+sTD

VIMAX

VIMIN

Ka

1+sTa

Ifd

Vref

+

+-

Kf

1+sTf

Vt.VRMAX - Kc

.Ifd

Vt.VRMIN - Kc

.Ifd

Figure 6.12: Excitation system of sample system

equivalent circuit for the sample system is displayed in Figure 6.13.

[Z(f=60Hz) ] Zmot(s=1)

∆V

V0

2Generator X''d 1 Motor

Figure 6.13: Thevenin equivalent circuit of sample system

6.2.2 Estimation of Voltage Drop

The start-up of large induction motors or a group of induction motors can lead to signi�cantvoltage drops. These can be determined using the voltage divider equation

�V [p:u:] = 1� j ZLR

ZLR + Z2(60 Hz) + j �X 00d

j (6.68)

where Z2 represents the Thevenin equivalent impedance between terminals 1 and 2. It isimportant to notice that the initial voltage drop is always independent of the generator exci-


tation system since every exciter has a delayed response to disturbances due to its inherenttime constants.

For the validation of Equation (6.68) we use our sample system and assume that initiallyboth frequency and voltage are equal to their rated values. This results in a voltage drop of0.38 per unit. The voltage behavior obtained by an Emtp simulation of the system is shownin Figure 6.14. Its value of 0.4 per unit is very close to the estimated one.

0 1 2 3 4 5 6 7 8 9 100

500

1000

1500

2000

2500

3000

3500

4000

4500

MotorterminalvoltageVm

ot

[V]

Time t [s]

Figure 6.14: Voltage drop during motor start-up

6.2.3 Estimation of Inrush Current

The inrush current is calculated according to

Imot = j V0ZLR + Z2(60 Hz) + j �X 00

d

j (6.69)

In case of our sample system it gives a value of 6147.2 A. It agrees well with 6200 A obtainedfrom the Emtp time-domain simulation shown in Figure 6.15. As for the voltage dropcalculation, the initial value of the current is independent of the action of the generatorexcitation system. In cases where the motor is started at frequencies di�erent from thenominal frequency, the Thevenin equivalent impedance can be determined by an Emtp

steady-state solution at that particular frequency.

6.2.4 Estimation of Start-up Time

The start-up time of induction motors can be predicted, using a quasi-steady-state calcu-lation. For this, we use the equation of motion that governs the mechanical behavior of


0 1 2 3 4 5 6 7 8 9 100

1000

2000

3000

4000

5000

6000

7000

8000

9000

MotorcurrentI mot

[A]

Time t [s]

Figure 6.15: Inrush current during motor start-up

the induction motor, together with its electrical steady-state equations. Following the basicphysical law that relates induction motor speed ! and acceleration � we get

�(!) =d!

dt) tstart =

Z !0

0

1

�(!)� d! (6.70)

The equation of motion is

J � d!dt

= Te(!)� Tm(!) (6.71)

) �(!) =Te(!)� Tm(!)

J(6.72)

Tm represents the mechanical torque which can be generally formulated as [19]

Tm = �0 + �1 � s+ �3 � s2 (6.73)

where �0; �1; �2 are constants that are either given by the manufacturer or, can be determinedas outlined in Appendix A. Te represents the electrical torque of the motor calculatedwith Equation (6.15). Combining Equation (6.70) and Equation (6.71), and considering therelationships

! = (1� s) � !m base

d! = �!m base � dsgives us the integral

tstart = J � !m base �Z s0

1

ds

Te(s)� Tm(s)(6.74)

= J � !m base �Z s0

1

(s)ds (6.75)


The operating slip s0 is reached when the condition

Te(s0)� Tm(s0) = 0 (6.76)

is ful�lled. It is determined numerically by the zero-�nding routine fzero [162]. Equation(6.74) can be solved by writing the sum

tstart = J � !m base ��s �n�sXi=1

i (6.77)

where

�s =s0n�s

; i = i + i�1

2(6.78)

s0 = (�final � (1� s0) + s0) (6.79)

�final = 0:002 (6.80)

Since the actual �nal motor speed is an asymptote, the numerical calculation of the start-up time may result in di�erent values for di�erent slip increments �s. Therefore, we de�nethe operating slip as Equation (6.79) and the associated speed as

!0 = (1� s0) � !m base (6.81)

When calculating the start-up time, the question arises whether a feasible operating conditionexists at all. We de�ne an infeasible operating condition when the current in the operatingpoint is larger than the limit provided by the manufacturer. The torque behavior for sucha condition is shown in Figure 6.16. It may occur e. g. during the soft start of motors. Asshown in Figure 6.17, the motor draws very high currents continuously, leading to possibleoverheating and machine damage. The soft start of motors is discussed in more detail inAppendix B.

For the validation of the above equations we again use our sample system. Due to thecontrol action of the excitation system, the duration of the initial voltage drop is muchsmaller than the motor start-up time. Therefore, the generator terminal voltage can beapproximated to be constant during start-up. By adding the Thevenin impedance of thesystem to Equation (6.13) we determine the current for the calculation of the electricaltorque Te, as

Imot = j V0ZLR + Z2(60 Hz)

j (6.82)

If a more conservative start-up time estimation is desired, Equation (6.69) instead of Equa-tion (6.82) can be used for the electrical torque calculation.

The rotor speed characteristic as calculated with the Emtp is given in Figure 6.18. Itsstart-up time of 3.7 s agrees well with the estimated time of 3.82 s.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2000

4000

6000

8000

10000

12000

14000

16000

18000

TorqueTe

andTm

[N�

m�

s]

Slip

Operating PointEl. torque Te

Mech. torque Tm

Figure 6.16: Mechanical and electrical torque for f=1:0 p:u: and V =0:5 p:u:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

2000

2500

3000

3500

CurrentI mot[A]

Slip

Operating PointCurrent

Figure 6.17: Current for f=1:0 p:u: and V =0:5 p:u:

6.2.5 Estimation of Thermal Behavior

During emergency or black start, any unwanted trip of motors should be avoided. To examinethe motor protection coordination, the motor current is integrated in the time range whereits value is above the relay current pickup. This time range can be approximated by themotor start-up time calculated according to Equation (6.74). A conservative estimate of theaverage motor inrush current is given by Equation (6.69). As an example for this principle,


0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

250

300

350

Motorspeed!

� rad�

s�1

�

Time t [s]

Figure 6.18: Rotor speed during motor start-up

Figure 6.19 shows the comparison of the relay characteristic of the 600 HP induction motorof Table 6.1 to the averaged and estimated currents. The averaged current has been obtainedby integration of the current given by an Emtp motor-start simulation.

0 200 400 600 800 1000 12000

1

2

3

4

5

6

7

8

9

10

Time[s]

Relay characteristicAveraged starting currentEstimated starting current

Current [A]

Figure 6.19: Motor relay characteristic compared to averaged starting current


6.2.6 Estimation of Frequency Drops

In this section the behavior of prime movers under static load pick-up, discussed in Chapter4, is extended to allow for the estimation of frequency drops that occur as a consequence ofthe pick-up of motor loads. For this, only a minimum amount of e�ort needs to be added tothe method for static load pick-up, as will be shown subsequently.

Approximation of Motor Power

For the determination of prime mover frequency deviations it is only necessary to considerthe active power drawn by a load. As compared to static loads which can be approximatedby single step functions, motor loads need to be represented by two successive steps:

Pe(t) = � [Pe1 � �(t) + (Pe2 � Pe1) � �(t� tstart)] (6.83)

where �(t) and �(t � tstart) stand for unity step functions, starting at times 0 and tstart,respectively. The parameter Pe2 is the steady-state active power drawn by the motor (t!1),and Pe1 the average inrush active power during the motor start (t!0+). In this section, inorder to avoid confusion with the Laplace operator \s", we use the variable \slip" when werefer to the motor slip.

The power Pe1 is obtained by averaging the motor active power that is derived fromEquation (6.69)

Pe(slip) = real

�p3 � V 2

0

Zslip + Z2 + j �X 00d

�(6.84)

over the slip range from 1 to the steady-state slip0:

Pe1 =

Z slip0

1

Pe(slip) � dslip (6.85)

When Equation (6.83) is transformed into the Laplace-domain we obtain:

Pe(s) = ��Pe1s

+Pe2 � Pe1

s� exp(�tstart � s)

�(6.86)

If this equation (as a replacement of Equation (4.9)) is used for the development of thetransfer function, a complicated expression is obtained that cannot be easily transferred backinto the time-domain. Therefore, we replace Equation (6.83) by an exponential function

�Pe(t) =��Pe1 � �Pe2

� � exp�� t

Tmot

�+ �Pe2 (6.87)

When this function is transferred into the Laplace domain we obtain the rational function

�Pe(s) =�Pe1 � �Pe2s+ 1

Tmot

+�Pe2s

(6.88)

=1

Tmot� �Pe2 + �Pe1 � s

s ��s+ 1

Tmot

�


that can be transformed back into the time-domain more easily.

The parameter �Pe2 simply equals Pe2. However, the values of the variables �Pe1; �Pe2; tx;and Tmot have still to be determined. This can be accomplished by an optimization routinethat determines the parameters in a way to satisfy the conditionZ tx

0

�Pe(t) � dt =Z tx

0

Pe(t) � dt (6.89)

Integrating Equations (6.83) and (6.87) results in the objective function:

F ( �Pe1; tx; Tmot) = j tstart � (Pe1 � Pe2) + (6.90)

�Tmot � ( �Pe1 � Pe2) ��1� exp

�� txTmot

��j = 0

Furthermore, the power �Pe(tx) needs to be equal to Pe2. Since we are dealing with anexponential function, Pe2 can only be an asymptote for �Pe(t), and we therefore add thetolerance " to obtain an additional constraint for the optimization:

G( �Pe1; tx; Tmot) = Pe(tx)� "� Pe2 = 0 (6.91)

where " = 0:1 � Pe2. The optimization problem can be solved in a few iterations with themathematical optimization routine Solnp described in Section 6.1.3.

The exponential function, the step function calculated according to Equation (6.83),and the active power obtained by an Emtp simulation, are displayed in Figure 6.20. Sincethe calculation of the steady-state power Pe2 is accomplished by assuming rated voltageand frequency, it is higher then the actual power calculated with Emtp, which adds someconservatism to the frequency drop estimation.

Frequency Drop Estimation

For the estimation of frequency drops during motor starts we follow the same steps outlinedin Chapter 4 and obtain the transfer function

�!(s) = K � N(s)

D(s)(6.92)


0 5 10 15 20 25 30 35 40 45 500.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

Time t [s]

PowerP[p.u.]

EMTP simulationStep functionE-function

Figure 6.20: Active power during motor start-up compared to approximations

where

K = �Pe12H

(6.93)

N(s) =

�2

Tw+ s

��

1

TN+ s

��

1

TPe+ s

�(6.94)

D(s) = [At

H � Tw � TD +

�D

H � Tw � TN +At � (TD � Tw)

H � TD � Tw

�� s+

+

�D

2H � TN +D

H � Tw �At

H+

2

Tw � TN

�s2 + (6.95)

+

�D

2H+

1

TN+

2

Tw

�s3 + s4] �

�s+

1

Tmot

�

TN =1

Rp +Rt + 1=TD(6.96)

TPe = Tmot � Pe1Pe2

(6.97)

The only di�erence to Equations (4.12) and (4.13) is the addition of one root to both thenumerator and denominator. This root, however, is easily found. As in the case of static loadpick-up, we only have to solve a fourth order polynomial numerically to obtain the roots,and obtain the transform back into the time-domain the same way as described in Section4.2.2.

6.3. Summary 103

Example

The Emtp result for the frequency obtained for our sample system as compared to the esti-mated result is displayed in Figure 6.21. Although the frequency obtained by the analyticalequation gives only an approximation of the value obtained by the time-domain simulation,it is still accurate enough to allow for an assessment of the frequency drop.

−10 0 10 20 30 40 5057.5

58

58.5

59

59.5

60

Time t [s]

Frequencyf[Hz]

EMTP simulationNew method

Figure 6.21: Frequency deviation for motor load pick-up

6.2.7 Remedial Actions

In case the reactive power capability limits are exceeded due to the high reactive power of themotors during start-up, additional generating units can be brought on-line. Furthermore, theauxiliary transformer tap positions can be changed [102]. In this case and in the case the busvoltage needs to be increased as a consequence of excessive voltage drops, the voltages can becontrolled following the method outlined in Section 5.2. In order to decrease the frequencydrops during motor load pick-up, the number of generators online can be increased.

6.3 Summary

This chapter deals with the analysis of auxiliary systems during restoration. A new in-duction motor parameter estimation technique is introduced that produces motor modelsby exibly taking into account di�erent types of data, such as nameplate data and motorperformance characteristics. Rules for the feasibility of motor start-ups taking into account

6.3. Summary 104

inrush currents, voltage drops, thermal behavior, and frequency drops during the energiza-tion of motors are introduced. The induction motor parameter estimation as well as therules are validated using actual system data.

Chapter 7

Emergency Start Case Study

This chapter describes how the Emtp can be used for time-domain simulations of emer-gency or black start studies. The simulation of a hydro generator start and the subsequentenergization of a power system, are described. The power system represents the emergencysafeguard functions (ESF) of a nuclear power plant under o�-nominal frequency and voltageconditions. The chapter describes the modeling of the generator, turbine, generator controls,and induction motors that drive the power plant's auxiliaries, and compares the results withmeasurements.

Among the di�erent types of power plants, nuclear stations require special treatment,since, following a system blackout, there always has to be suÆcient energy available for theirsafe shutdown. Emergency supply sources for nuclear plants, in case of a loss of o�-sitepower (LOOP), are black start combustion turbines, conventional or pumped hydro units,or diesel generators [100]. LOOP cases and the subsequent use of emergency power sourceslead to an islanded system operation that can result in abnormal conditions with respectto frequency and voltage behavior. This situation requires careful analysis, using accuratesystem models and simulation tools.

Studies similar to ours can be found in [74, 76, 102, 177, 232]. All the cases have incommon that they investigate the energization of power plant auxiliary motors with gen-erators running at nominal speed. In the following, we describe the individual models andgive indications of how they were veri�ed by measurements. We focus on modeling issuesand refer to nuclear technical aspects only when necessary. An overall model validation isperformed, using a system test and comparing simulations to measurements.

7.1 Modeling of the System

7.1.1 Electrical System

A single-line diagram of the emergency electrical system for the Oconee nuclear power plantthat is simulated in our study is shown in Figure 7.1. Since the cables in the system areelectrically short and traveling wave e�ects can be neglected, they are modeled using lumped

105

7.1. Modeling of the System 106

elements.

RBS

RBCF

LPSW HPI LPI

RBCF

RBS LPSW HPI LPI EFW

RBCF

LPI HPI EFW

Keowee Hydro Plant

Standby Bus

Figure 7.1: One-line diagram of the Oconee emergency electrical power system (ESF)

7.1.2 Synchronous Generator

The synchronous generator to be modeled is a hydro generator of 87.5 MVA and 13.8 kV. Itis implemented using the Emtp synchronous machine model SM59 [63]. This model includesthe e�ects of the amortisseur windings as well as saliency and saturation e�ects [54, 143].

The generator parameters are veri�ed by an open circuit magnetization test for thevalidation of saturation parameters and D-axis and Q-axis parameter tests in order to identifysub-transient, transient and steady state parameters.

7.1.3 Excitation System Model

The excitation system, and the other controllers described subsequently, are implemented inthe Transient Analysis of Control Systems (Tacs) module [63] of the Emtp. A control blockdiagram of the excitation system model used for our study is shown in Figure 7.2. It is astatic exciter that is represented by the IEEE ST1 exciter model, modi�ed with the additionof a V/Hz limiter, IEEE droop compensator, and the replacement of the �eld voltage by a�eld current feedback.

In reality, the static exciter is a recti�er bridge, whose �ring angle is controlled by a �ringcircuit logic. Since we are dealing with a generator start-up from zero initial voltage andspeed, �eld ashing is applied until the generator terminal voltage is suÆciently high. This


is accomplished by modeling a battery in parallel to the recti�er bridge, which delivers the�eld current in the beginning.

| Vt + It . ( Rc + j Xc ) |

1

1+sTrv

ItVt

Vt

FT

1

1+sTrf

ω -Kvl

1+sTvl

Efd

FT

1+sTC

1+sTD

VIMAX

VIMIN

Ka

1+sTa

MIN

Vt.VRMAX - Kc

.Ifd

Vt.VRMIN - Kc

.Ifd

Ifd

Vref+Vs+

+-

Kf

1+sTf

Kv

-

VLLR+

VB

RB-

S1

+Field Flashing

Ifd

Figure 7.2: Block diagram of the Keowee generator excitation system

Modeling the whole recti�er bridge would be too detailed for the purpose of our study,since the accuracy gained with it is negligible as compared to the added complexity. However,modeling of the recti�er bridge might be justi�ed if the objective of the study is to examinethe behavior of the recti�er and its control circuits.

For our study, we model the �eld ashing separately, using a simple equation in Tacs

that gives the �eld voltage, as provided by the battery:

Efd (FieldF lash) = VB � RB � Ifd (7.1)

where VB stands for the battery voltage, RB for the battery's internal resistance, and Ifd forthe �eld current. After a certain V/Hz level is reached, the �eld voltage output is switchedfrom the battery voltage to the exciter voltage output, as indicated in Figure 7.2. Thisresults in a small discontinuity in the �eld voltage when switching from battery to recti�erdc output. The excitation system model is veri�ed using a reactive droop compensation testand step tests to identify the individual elements of the block diagram in Figure 7.2.

The reactive droop compensation is implemented as an algebraic function. The outputvoltage of a droop compensator as a function of the terminal voltage Vt, the current It, andthe compensator's resistance Rc and reactance Xc are given as

jVcj = jVt + (Rc + j �Xc) � Itj (7.2)

Writing the generator terminal voltage and current as complex equations, we get

Vt = Vd + j � Vq (7.3)

It = Id + j � Iq (7.4)

Substituting Equations (7.3) and (7.4) into (7.2) then leads to

jVcj =q(Vd +RcId �XcIq)2 + (Vq +RcIq +XcId)2 (7.5)


The quantities Vd, Vq, Id, and Iq are directly obtained from the Emtp generator modeloutput.

7.1.4 Governor System Model

The hydro governor system model used for this study is shown in Figure 7.3. It is a pro-portional control with transient droop and is described in more detail in [143, 178]. For thegenerator start-up, the model has to be modi�ed since it is only valid for a certain speedrange. Outside that range the upper and lower speed limits are used as integrator inputs inFigure 7.3, and in that case the feedback loop is inoperative. Furthermore, the gate closingspeed is a function of the gate position, as indicated schematically in Figure 7.3.

1

1+sTp

Q

Vo if RPM <= RPM 1

if RPM 2 < RPM < RPM 3

1

s

Gmax

Vc Gmin

Gc

1+sTg

Vo

Vc GATE

Rp

s.Rt.Tr

1+sTr

ω ref

ω

∆ω+

-

+

+

-

-

if GATE >= G buff then VC=VC1

if GATE < G buff then VC=VC2

Figure 7.3: Block diagram of the Keowee governor system

7.1.5 Turbine Model

For the turbine a non-linear model including water column traveling wave e�ects [143, 178]is chosen. Its block diagram is depicted in Figure 7.4.

A crucial factor in the modeling of hydro turbines for generator start-up studies andother abnormal system conditions was found to be the damping term

DAMP = D ��! �GATE (7.6)

Although Equation (7.6) led to a good agreement between simulation and measurement forthe cases investigated in our study, it cannot be considered valid for all possible operatingconditions.


Π

ABS

Zt

ft

fpΠ

Zp

2 e-2Tts

2 e-2Tts

Π

Π

AtΠ

Href

GATE

∆ωD.

PM

FLOW

QNL

HEAD

FLOWfrom other unit

-

-

-

-

-

-

+

+ +

+

+

+

+

+

+ +

+

+

+

-

Figure 7.4: Block diagram of the Keowee hydro turbine

The turbine model time constants are calculated from manufacturer data. They areveri�ed by applying step changes to the governor set point, and measuring gate position andactive power output.

7.1.6 Initialization of Excitation System, Governor System, and

Turbine Models

A very important aspect, discussed subsequently, is the initialization of control system modelsin Tacs. Although Tacs initializes some of the control blocks automatically, others have tobe initialized manually by the user. We recommend to initialize all state variables manuallyin order to avoid faulty initializations and to allow for a better veri�cation of the model.

This section gives hints on how a proper control system initialization can be accomplishedfor practical case studies. One way is to run a time-domain simulation without applying adisturbance at the control's inputs and without prior initialization of the state variables.The simulation is continued until the state variables settle down to their steady-state values.


These can then be used to initialize the control and to perform another non-disturbancesimulation. If the model implementation and initialization is done correctly, the control'sstate variables should remain constant at all times.

One disadvantage of this method is that it doesn't give any indication of whether thecontrol itself has been set up properly in Tacs. Even when this is not the case, the statevariables may settle down to steady-state values. Furthermore, state variables may not settledown to �nal values at all. Instead, the state variables' values may increase continuously tovery high numbers, resulting in numerical problems and an unwanted early termination ofthe simulation. Also, control systems with large time constants might need a long time inorder to settle down, resulting in overly long simulation times.

Therefore, it is better to calculate the initial values prior to the simulation and performa non-disturbance simulation only in order to ensure that the control is initialized correctly,and that all control parameters are entered properly. In that case, all state variables remainconstant.

As an example for such a procedure, we focus on the initialization of the synchronousmachine and the most important state variables of the turbine model as shown in Figure 7.4.Since a start from exactly zero frequency and voltage cannot be simulated with the Emtp,due to the implementation of its generator model, we approximate such a zero conditionby choosing a small generator base voltage (Vbase = 10�2 � Vnominal) and frequency (fbase =10�2 � fnominal). The Emtp generator parameters must then be modi�ed as well since theyhave to be entered in per unit. As a consequence of the change of the base voltage, the baseimpedance decreases by a factor of 10�4. Therefore, the resistances entered into Emtp are104 times higher. The reactances increase by a factor 104=102 = 100, since they have to becorrected as well with respect to the modi�ed base frequency.

Furthermore, Emtp requires to load the generator with a small initial load, which is ac-complished by a high resistance R0 at the generator terminals. This is removed immediatelyafter the simulation is started, since with growing voltage, it would represent a signi�cantload, giving erroneous results.

Since in steady state, the mechanical power PM equals the electrical power PEL, we cancalculate the initial value for PM as:

PM = PEL = V 20 =R0 (7.7)

We can now go \backwards" in the block diagram in Figure 7.4. The initial value for thehead can be calculated as

HEAD = Href ; (7.8)

The other inputs to the respective summer are zero since the factors ft and fp are negligible,and the initial output of the summers, representing the traveling wave e�ects, are zero. Thisresults in the following values for ow and gate:

FLOW =PM

At �Href

+QNL (7.9)

GATE =FLOWpHEAD

(7.10)

7.2. Simulation Results 111

Starting from these equations, the initialization of the other variables is then straightforwardsince the values only have to be multiplied by constant factors, e. g. the output of the blockwith proportionality factor Zp is FLOW � Zp, etc.

The exciter model is initialized as well, even though initially, its output is not connectedto the generator, since the �eld voltage is given by the �eld ashing battery. An initializationas if the controls were already in full operation is performed since initial transients in thecontrol systems, caused by non- or insuÆcient initialization, may not settle down before the�eld voltage is switched from �eld ashing mode to exciter control, and may therefore leadto erroneous results.

7.1.7 Induction Motor Model

For our study we modeled six di�erent motor loads:

� Reactor building cooling fans (RBCF)

� Emergency feedwater pumps (EFW)

� High pressure injection pumps (HPI)

� Low pressure service water pumps (LPS)

� Reactor building spray pumps (RBS)

The induction motor model selected for this study is the universal machine model UM3of the Emtp [63]. Although the UM3 model allows for an arbitrary number of rotor circuits,experience has shown that a model with two rotor circuits is suÆcient over the whole rangeof operation for motor start-up studies (see e. g. [105, 191]).

The model parameters are derived from manufacturer nameplate data and current-speed,torque-speed, and powerfactor-speed performance characteristics of the motor, following theprinciples outlined in Section 6.1. The load characteristics are modeled by quadratic func-tions [19] according to Appendix A, based on measurements. Representatively, Figures 7.5and 7.6 show the Emtp simulation results for a motor start-up (signature test) of a highpressure injection pump (HPI) motor as compared to measurements. Measurement andsimulation results agree well.

7.2 Simulation Results

The test carried out for validating theEmtpmodel of the Oconee emergency electrical systemis an engineered safeguards functions (ESF) test during a simulated loss of coolant accident(LOCA) concurrent with a loss of o�-site power (LOOP). The Keowee hydro generator isused for this to provide emergency power.

The nuclear plant's auxiliaries are energized by closing the switch shown in the single linediagram in Figure 7.1, when the voltage at the main feeder reaches 0.43 per unit, provided


0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12

Current[p.u.]

Time [s]

SimulationMeasurements

Figure 7.5: Transient current characteristic, HPI motor

0 0.5 1 1.5 2 2.5 31.02

1.025

1.03

1.035

1.04

1.045

1.05

1.055

1.06

Voltage[p.u.]

Time [s]

SimulationMeasurements

Figure 7.6: Transient voltage characteristic, HPI motor

11 seconds have elapsed since the loss of o�-site power to the standby bus. This delay givesenough time for the generator to build voltage and frequency up to around 0.6 per unit Thiscondition represents a motor soft-start which is analyzed in more detail in Appendix B.

The most relevant simulation results are shown in Figures 7.7 to 7.12. The shape of thegenerator current suggests that some of the motors (with lower inertia) have locked to thesystem frequency (drop-o� of the start-up current) sooner than others. The zero �eld current


0 5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

Frequency[Hz]

Time [s]

MeasurementSimulation

Figure 7.7: ESF test: Comparison between measured and simulated frequency

0 5 10 15 20 25 30 35 400

5

10

15

Voltage[kV]

Time [s]


Figure 7.8: ESF test: Comparison between measured and simulated voltage

of the measurement result as compared to a continuously rising current in the simulation isa result of the measurement not taking into account the initial current delivered by the �eld ashing circuit. Overall, the simulation results agree well with the measurements.

7.3. Summary 114

0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Current[kA]

Time [s]


Figure 7.9: ESF test: Comparison between measured and simulated current

0 5 10 15 20 25 30 35 400

50

100

150

200

250

300

FieldVoltage[V]

Time [s]


Figure 7.10: ESF test: Comparison between measured and simulated �eld voltage

7.3 Summary

In this chapter we describe the modeling of a hydro generator and its controls in Emtp

and Tacs. The successive development and test of each of the models is outlined. As animportant aspect of building control system models for system studies in Emtp, we identi�edthe need for proper initialization, and the correct modeling of induction motors and of the

7.3. Summary 115

0 5 10 15 20 25 30 35 400

100

200

300

400

500

600

700

800

900

1000

FieldCurrent[A]

Time [s]


Figure 7.11: ESF test: Comparison between measured and simulated �eld current

0 5 10 15 20 25 30 35 400

5

10

15

20

25

Power[MW]

Time [s]


Figure 7.12: ESF test: Comparison between measured and simulated power

generator damping.

Field tests and simulations of an engineered safeguards test show good correlation. Ourcase study has demonstrated the potential of Emtp to be utilized as a system stability toolfor the simulation of abnormal conditions, such as islanded operation during black start andpower system restoration procedures.

Chapter 8

Conclusions and Recommendations

for Future Work

This dissertation presents a new framework for power system restoration. The results of anextensive bibliographical research showed that there is a need for a methodology that allowsone to assess restoration steps quickly and eÆciently, and that supports operators duringon-line restoration, and system planners during o�-line restoration planning. Moreover, itrevealed that models and modeling techniques are needed that help to take into account theabnormal voltage and frequency conditions during restoration.

The new method is based on a subdivision of the aggregation of complex phenomenaencountered during restoration into simpler problems that can be assessed using simple andeÆcient rules formulated in the frequency and Laplace domain. The rules allow for fastscreening of the large number of possible combinations of restoration steps, and support theselection of the most promising restoration paths.

The time-domain modeling techniques that have been developed for this thesis allowEmtp simulations of the large deviations in frequency and voltage that are of particularconcern during the black or emergency start phase of a system restoration procedure. Specialemphasis is thereby given to the modeling of large induction motor loads.

The major conclusions and contributions of this work are:

� A new method for the assessment of the frequency response behavior of hydro powerplants has been developed. By using calculations in the Laplace domain and simplifyinggovernor control circuits, the initial rate of frequency decline, the minimum frequency,and the time when this minimum occurs can be calculated with a minimum numberof iterations.

� Matrix manipulation techniques have been developed for determining the matrix changesof the network impedance matrix directly from the network changes of single networkelements, without the need for additional frequency scans or matrix inversions. Inthe case of overvoltages, it permits a quick assessment of di�erent network conditionswith respect to overvoltages. A sensitivity analysis method for sustained and harmonicovervoltages helps to �nd the most eÆcient network changes. When time-domain sim-

116

117

ulations need to be carried out, an algorithm based on Prony analysis in combinationwith fuzzy logic shortens the overall simulation time.

� A new induction motor parameter estimation algorithm helps to build accurate induc-tion motor models for restoration studies. It exibly takes into account the di�erentdata sets. Frequency-domain based rules help to assess the feasibility of motor startswith respect to overcurrents, voltage drops, and thermal behavior.

� Using a practical emergency-start case study, it is shown how the Emtp can be utilizedas a stability tool. It is demonstrated how generator controls, turbines, etc. can bemodeled. An overall system test is carried out and compared to measurements thatcon�rm the validity of the simulation results.

In summary, the new methodology gives an approximate assessment of the feasibilityof restoration steps. It can be used in conjunction with commercially available analyticaltools, such as electromagnetic transient programs, stability programs, harmonic analysistools, power ows, and operator training simulators, or combination tools. By eliminatinginfeasible restoration sequences, the overall restoration time can be reduced signi�cantly.

The following studies are suggested for future research:

� The frequency-response analysis procedure could be extended to other types of powerplants, such as gas turbines, and the feasibility of approximations taking into accountconstrained generator operation could be investigated.

� For cases where more than one generator is in operation, a simpli�ed method forthe aggregation of generators should be developed, to obtain a single-generator modelsuitable for frequency-response analysis.

� Laplace-domain based frequency-response methods could be investigated for the designof load shedding schemes that are in use, e. g. when the system disintegrates into severalislands with the system frequencies below the allowable limits.

� Rules that help to assess transient switching overvoltages could be investigated.

� The possibility of replacing Prony analysis by a method such as wavelets that is moresuited for time-frequency analysis should be explored.

� A simpli�ed motor aggregation method could be developed, and the interaction of indi-vidual motors when several large motors are started in sequence should be investigated.

� Rules that cover other parts of the restoration process, such as the integration ofsubsystems or the protection system, could be developed.

� A more detailed representation for the generator damping for Emtp time-domain sim-ulations should be developed.

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Appendix A

Determination of Motor Load

Characteristics from Measurements

If a motor load characteristic is not known, it can be determined from motor start-up mea-surements if available. This is described in the following.

A load characteristic that is commonly used is [19]

Tm = Cm � !kmm (A.1)

where Cm is an unknown proportionality constant. The exponent km depends on the loadtype, and should therefore be known. Typical factors km and load types are:

km = 1 for fan-type loads (A.2)

km = 2 for centrifugal pumps (A.3)

Often, during motor start-ups, the rotor speed is not recorded due to higher technical re-quirements and costs. Therefore we determine Cm using the measurements of motor currentand voltage. The network that drives the motor can be approximated by a simple circuit,as outlined in the following.

Starting from the measurements, we determine the initial voltage drop V m(t = 0+)and the steady-state voltage V m(t!1). From Equation (6.8) we obtain the locked-rotorimpedance Zmot(s=1) and the motor impedance at full load Zmot(s=s0). Applying voltagedivider equations to the circuits shown in Figure A.1 and A.2, for locked-rotor and full-loadoperation, we then obtain:

V 2th

Rmot(s=1)2 + (Xth +Xmot(s=1))2=

[V m(t=0+)]2

jZmot(s=1)j2 (A.4)

V 2th

Rmot(s=s0)2 + (Xth +Xmot(s=s0))2=

[V m(t!1)]2

jZmot(s=s0)j2 (A.5)

where Zmot = Rmot + j � Xmot, and where the system driving the motor is assumed to bepurely inductive. Using Equations (A.4) and (A.5) we get the following parameters for the

137

138

Xth

Zmot (s=1) V(t→0+)Vth

Figure A.1: Thevenin equivalent for locked-rotor operation (t=0+)

Xth

Zmot (s=s0) V(t→∞)Vth

Figure A.2: Thevenin equivalent for full-loadoperation (t!1)

Thevenin equivalent circuit:

Xth = � b2+

rb2

4� c (A.6)

Vth =V m(t=0+)

Zmot(s=1)

qjZ2

mot(s=1)j2 +X2Thev + 2XThevXmot(s=1) (A.7)

where

a =

�V m(t=0+)

Zmot(s=1)� V m(t!1)

Zmot(s=s0)

�2(A.8)

b =2 � (a �Xmot(s=1)�Xmot(s=s0))

a� 1(A.9)

Thus, the motor current at full load follows as:

Imot(s = s0) =VThev

j �Xthev + Zmot(s = s0)(A.10)

The operating slip s0 is not known a priori and can be determined by a simple iterationusing the zero-�nding routine fzero [162] to solve:

Imot(s = s0)� Im(t!1) = 0 (A.11)

This permits to determine the associated torque Te(s0) using Equation (6.15). In steadystate, we have

2H � d!dt

= Te(s0)� Tm(s0) = 0 (A.12)

and therefore we eventually obtain Cm as

Cm =Te(s0)

[(1� s0) � !m base]2 (A.13)

Appendix B

Soft Start of Motors

When auxiliaries are started on reduced voltage, this is referred to as "soft start" condi-tion. In that case, one expects to observe lower starting currents and longer start-up times.However, during emergency starts of power plants (see Chapter 7), the motors are pickedup under lower frequency as well. It is expected that the motors will lock to the systemfrequency sooner, and hence draw start-up currents for a shorter time. Since in this casewe deal with both an under-voltage and under-frequency condition, the analysis is morecomplex and less intuitive.

In order to analyze the starting current and time as a function of frequency and voltage,both are calculated using the equations developed in Section 6.2. Values of current and start-up time are normalized to their values at f =V =1:0 per unit. The start-up time at ratedvoltage as a function of frequency calculated according to Equation (6.74) and comparedto the start-up time obtained with Emtp are shown in Figure B.1. The start-up time as afunction of voltage at rated frequency is displayed in Figure B.2. The currents as functionof voltage and frequency as compared to the Emtp results are shown in Figures B.3 andB.4. Only the range where a feasible operating condition can be reached is displayed. Theestimated start-up times and currents agree well with the Emtp results.

The results show that the motor start-up time increases exponentially with increasingfrequency, and decreasing voltage. The relationship between motor start-up current and volt-age is linear, and the relationship between start-up current and frequency is approximatelylinear.

139

140

0.5 0.6 0.7 0.8 0.9 1 1.10

0.5

1

1.5

2

2.5

Estimated timeEmtp time

Motorstart-uptimet start

[p:u:]

Frequency f [p:u:]

Figure B.1: Start-up time at voltage V = 1:0 p:u:

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

0.5

1

1.5

2

2.5

3

3.5

Estimated timeEmtp time

Motorstart-uptimet start

[p:u:]

Voltage V [p:u:]

Figure B.2: Start-up time at frequency f = 1:0 p:u:.

141

0.5 0.6 0.7 0.8 0.9 1 1.10.5

1

1.5

2

Estimated currentEmtp current

Motorstart-upcurrentI mot

[p:u:]

Frequency f [p:u:]

Figure B.3: Start-up current at voltage V =1:0 p:u:

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

Estimated currentEmtp current

Motorstart-upcurrentI mot

[p:u:]

Voltage V [p:u:]

Figure B.4: Start-up current at frequency f=1:0 p:u:.

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