Current Interruption Transients Calculation E-Book

David F. Peelo

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley - IEEE
• Sprache: Englisch

Seitenzahl: 291

Veröffentlichungsjahr: 2019

Beschreibung

Provides an original, detailed, and practical description of current interruption transients, origins, and the circuits involved, and shows how they can be calculated Based on a course that has been presented by the author worldwide, this book teaches readers all about interruption transients calculation--showing how they can be calculated using only a hand calculator and Excel. It covers all the current interruption cases that occur on a power system and relates oscillatory circuit (transients) and symmetrical component theory to the practical calculation of current interruption transients as applied to circuit breaker application. The book explains all cases first in theory, and then illustrates them with practical examples. Topics featured in Current Interruption Transients Calculation, Second Edition include: RLC Circuits; Pole Factor Calculation; Terminal Faults; Short Line Faults; Inductive Load Switching; and Capacitive Load Switching. The book also features numerous appendices that cover: Differential Equations; Principle of Duality; Useful Formulae; Euler's Formula; Asymmetrical Current-Calculating Areas Under Curves; Shunt Reactor Switching; and Generator Circuit Breaker TRVs. * Offers a clear explanation of how to calculate transients without the use of specialist software, showing how four basic circuits can represent all transients * Describes every possible current interruption case that can arise on a power system, explaining them through theory and practical examples * Analyses oscillatory circuit (transients) and symmetrical component theory in detail * Takes a practical approach to the subject so engineers can use the knowledge in circuit breaker applications Current Interruption Transients Calculation, Second Edition is an ideal book for power electrical engineers, as well as transmission and distribution staff in the areas of planning and system studies, switchgear application, specification and testing, and commissioning and system operation.

Leseprobe

Table of Contents

Cover

Preface to the Second Edition

Preface to First Edition

1 Introduction

1.1 Background

1.2 Short‐Circuit Rating Basis for High‐Voltage Circuit Breakers

1.3 Current Interruption Terminology

Further Reading

2 RLC Circuits

2.1 General

2.2 Series RLC Circuit with Step Voltage Injection

2.3 Source‐Free Series RLC Circuit with Precharged Capacitor

2.4 Source‐Free Parallel RLC Circuit with Precharged Capacitor

2.5 Parallel RLC Circuit with Ramp Current Injection

2.6 Alternative Equations

2.7 Traveling Wave Basics

2.8 Summary

References

Further Reading

3 Pole Factor Calculation

3.1 General

3.2 Pole Factors: Effectively Earthed Systems

3.3 Pole Factors: Non‐Effectively Earthed Systems

3.4 Alternative Pole Factor Calculation Method

3.5 Three‐Phase Test Circuit Arrangement

3.6 Summary

Further Reading

4 Terminal Faults

4.1 General Considerations

4.2 Standard TRV Derivation

4.3 Effect of Added Capacitance

4.4 Effect of Added Resistance

4.5 Effect of Series Reactors

4.6 Out‐of‐Phase Switching

4.7 Asymmetrical Currents

4.8 Double Earth Faults

4.9 Summary

Further Reading

5 Short Line Faults

5.1 General

5.2 Line Side Voltage Calculation

5.3 Effect of Added Capacitance

5.4 Discussion

Further Reading

6 Inductive Load Switching

6.1 General

6.2 General Shunt Reactor Switching Case

6.3 Shunt Reactors with Isolated Neutrals

6.4 Shunt Reactors with Neutral Reactor Earthed Neutrals

6.5 Shunt Reactors with Earthed Neutrals

6.6 Reignitions

6.7 Unloaded Transformer Switching

6.8 Discussion

6.9 Summary

Further Reading

7 Capacitive Load Switching

7.1 General

7.2 Shunt Capacitor Banks

7.3 Transmission Lines

7.4 Cables

7.5 Special Case: Interrupting Small Capacitance Currents

7.6 Summary

References

Further Reading

8 Circuit Breaker Type Testing

8.1 Introduction

8.2 Circuit Breaker Interrupting Time

8.3 Inherent Transient Recovery Voltages

8.4 Inductive Load Switching

8.5 Capacitive Current Switching

Further Reading

Appendix A: Differential EquationsDifferential Equations

Further Reading

Appendix B: Principle of DualityPrinciple of Duality

Appendix C: Useful FormulaeUseful Formulae

Appendix D: Euler's FormulaEuler's Formula

Further Reading

Appendix E: Asymmetrical Current‐Calculating Areas Under CurvesAsymmetrical Current‐Calculating Areas Under Curves

Appendix F: Shunt Reactor Switching – First‐Pole‐to‐Clear Circuit RepresentationShunt Reactor Switching – First‐Pole‐to‐Clear Circuit Representation

Appendix G: Special Case: Generator Circuit Breakers TRVsSpecial Case

G.1 Introduction

G.2 System‐Source Faults

G.3 Generator‐Source Faults

G.4 Out‐of‐phase Current Switching

G.5 Step‐up Transformer High Side Fault

G.6 Load Current Switching

G.7 Discussion

References

Appendix H: Evolution of Transient Recovery VoltagesEvolution of Transient Recovery Voltages

H.1 Introduction

H.2 TRVs: Terminal Faults

H.3 Terminal Fault TRV Standardization

H.4 Short Line Fault

H.5 Inductive and Capacitive Load Current Switching

H.6 Terminal Fault TRV Calculation

References

Further Reading

Appendix I: Equation Plotting Using ExcelEquation Plotting Using Excel

I.1 General

I.2 Underdamped Series RLC Circuit

I.3 T100 TRV Calculation for 245 kV Circuit Breaker

I.4 T30 TRV Calculation for 245 kV Circuit Breaker

I.5 Series Reactor Application Case

I.6 Generator Circuit Breaker Asymmetrical Current

Index

End User License Agreement

List of Tables

Chapter 1

Table 1.1 Standard TRV values for 245 kV rated circuit breaker on an effective...

Table 1.2 Standard TRV values for 72.5 kV rated circuit breaker on a non‐effe...

Chapter 2

Table 2.1 Series RLC circuit equations.

Table 2.2 Parallel RLC circuit equations.

Chapter 3

Table 3.1 Operator

a

summary.

Table 3.2 Summary of pole‐factor calculations.

Chapter 4

Table 4.1 IEC TRVs requirements for 245 kV circuit breakers.

Table 4.2 IEC TRV requirements for 72.5 kV circuit breakers.

Table 4.3 Equations for T100, T60, T30, and T10 test duties with 1 μF added c...

Table 4.4 IEC TRV requirements for 1100 kV circuit breakers.

Table 4.5 1100 kV circuit breaker test circuit parameters for T100, T60, T30,...

Table 4.6 Base TRV equations for 1100 kV circuit breaker for T100, T60, T30, ...

Table 4.7 1100 kV circuit breaker test circuit parameters for T100, T60, T30,...

Table 4.8 TRV equations for 1100 kV circuit breaker for T100, T60, T30, and T...

Table 4.9 Ratio of

I

pk

/

I

for IEC 62271‐100 time constants.

Table 4.10 Relationships between the IEC time constants and

X

/

R

ratios.

Chapter 5

Table 5.1 L90, L75, and L60 line side transient characteristics for 245 kV, 4...

Table 5.2 L90 line side transient characteristics for 245 kV, 50 Hz circuit b...

Table 5.3 L90 line side transient characteristics for 245 kV, 40 kA, 50 Hz ci...

Chapter 6

Table 6.1 Absolute and per unit values.

Table 6.2 Circuit breaker chopping numbers.

Table 6.3 Shunt reactor switching equation summary.

Chapter 7

Table 7.1 Inrush current mitigation measures.

Table 7.2 Voltage test factors for single‐phase capacitive current switching ...

Chapter 8

Table 8.1 Short‐circuit making and breaking mandatory type testing.

Table 8.2 Application‐related current making and breaking type testing.

Appendix A

Table A.1 Solutions for

.

Table A.2 Perspective solutions for non‐homogenous second‐order differential ...

Appendix B

Table B.1 Electrical dualities.

Table B.2 Comparison of series and parallel RLC circuits with respect to degr...

Appendix G

Table G.1 Circuit breaker TRV requirements for 400 MVA generator and step‐up ...

Table G.2 System voltage and short‐circuit current values.

Table G.3 Generator and step‐up transformer rated values.

Table G.4 Typical generator reactance and associated time constant values.

Table G.5 Generator rated values.

Appendix H

Table H.1 Historical development of terminal fault TRVs.

Table H.2 Development of current interruption transients calculation.

List of Illustrations

Chapter 1

Figure 1.1 Current interruption terminology: timing related quantities.

Figure 1.2 Current interruption terminology: current related quantities.

Figure 1.3 Current interruption terminology: voltage related quantities.

Chapter 2

Figure 2.1 Basic RLC circuits: (a) Series RLC circuit with step voltage inje...

Figure 2.2 Generic damping curves for a series RLC circuit with step voltage...

Figure 2.3 Source‐free series RLC circuit with precharged capacitor.

Figure 2.4 Generic damping curves for series RLC circuit with precharged cap...

Figure 2.5 Source‐free parallel RLC circuit with precharged capacitor.

Figure 2.6 Generic damping curves for parallel RLC circuit with precharged c...

Figure 2.7 Parallel RLC circuit with ramp current injection.

Figure 2.8 Generic damping curves for parallel RLC circuit with ramp current...

Figure 2.9 Amplitude factor for oscillation starting at 1 pu (see Figure 2.4...

Figure 2.10 Amplitude factor for oscillation starting at zero (see Figure 2....

Figure 2.11 Amplitude factor

k

af

as a function of degree of damping for the ...

Figure 2.12 Traveling waves on a terminated transmission line.

Figure 2.13 Steady‐state wave representation.

Figure 2.14 Steady‐state wave travel progression.

Chapter 3

Figure 3.1 Symmetrical component set.

Figure 3.2 Positive‐, negative‐ and zero‐sequence networks.

Figure 3.3 Symmetrical component calculation approach.

Figure 3.4 Equivalent circuit for three‐phase to earth fault.

Figure 3.5 Sequence network for three‐phase to earth fault.

Figure 3.6 Three‐phase to earth fault clearing in an effectively earthed sys...

Figure 3.7 Three‐phase unearthed fault clearing in a non‐effectively earthed...

Figure 3.8 Equivalent circuit for double‐line to earth fault.

Figure 3.9 Sequence network for double‐line to ground fault.

Figure 3.10 Hammarlund's equivalent circuit for

k

pp

1

calculation.

Figure 3.11 Hammarlund's sequence impedance network.

Figure 3.12 Equivalent circuit for single‐line to earth fault.

Figure 3.13 Sequence network for single‐line to earth fault.

Figure 3.14 Equivalent circuit for line‐to‐line fault.

Figure 3.15 Sequence network for line‐to‐line fault.

Figure 3.16 Equivalent circuit for line‐to‐line second and third‐pole factor...

Figure 3.17 Sequence network for line‐to‐line fault second‐ and third‐pole f...

Figure 3.18 General circuit for pole factor calculation.

Figure 3.19 Sequence network for general‐pole factor calculation.

Figure 3.20 Three‐phase sequence network for first‐pole clearing in an effec...

Figure 3.21 Three‐phase test circuit for first‐pole clearing in an effective...

Figure 3.22 TRV axis of oscillation and peak‐value calculation.

Chapter 4

Figure 4.1 Distribution of T10 TRV between the source impedance and transfor...

Figure 4.2 Single line diagram for first‐pole‐to‐clear TRV calculation for e...

Figure 4.3 Sequence network for first‐pole‐to‐clear calculation for effectiv...

Figure 4.4 Current injection circuit for first‐pole‐to‐clear TRV calculation...

Figure 4.5 Traveling wave representation for first‐pole‐to‐clear TRV calcula...

Figure 4.6 Double exponential TRV due to initial circuit breaker bus transie...

Figure 4.7 Overdamped TRV characterization.

Figure 4.8 Four parameter TRV representation.

Figure 4.9 Two parameter TRV representation.

Figure 4.10 Single‐phase test circuit.

Figure 4.11 T100 TRV for 245 kV, 50 kA, 60 Hz circuit breaker.

Figure 4.12 T100 TRV hyperbolic versus exponential representation for 245 kV...

Figure 4.13 T100 TRV time delay region hyperbolic versus exponential represe...

Figure 4.14 T60 TRV for 245 kV, 50 kA, 60 Hz circuit breaker.

Figure 4.15 Ratio

t

3

/

T

for underdamped oscillations.

Figure 4.16 T30 TRV for 245 kV, 50 kA, 60 Hz circuit breaker.

Figure 4.17 T10 TRV for 245 kV, 50 kA, 60 Hz circuit breaker.

Figure 4.18 T100, T60, T30, and T10 TRVs for 245 kV, 50 kA, 60 Hz circuit br...

Figure 4.19 Single‐phase test circuit with added capacitance.

Figure 4.20 T100 TRV for 245 kV, 50 kA, 60 Hz circuit breaker without and wi...

Figure 4.21 T60 TRV for 245 kV, 50 kA, 60 Hz circuit breaker without and wit...

Figure 4.22 T30 TRV for 245 kV, 50 kA, 60 Hz circuit breaker without and wit...

Figure 4.23 T10 TRV for 245 kV, 50 kA, 60 Hz circuit breaker without and wit...

Figure 4.24 Overall effect of 1 μF added capacitance on TRVs for a 245 kV, 5...

Figure 4.25 Test circuit with added opening resistor.

Figure 4.26 T100, T60, T30, and T10 TRVs for 1100 kV circuit breaker at 50 H...

Figure 4.27 First‐pole‐to‐clear factor k

pp

as a function of the ratio X

R

/X

1

....

Figure 4.28 Representative circuits for series reactor effect TRV calculatio...

Figure 4.29 Circuits for TRV calculation for 225 kV case with series reactor...

Figure 4.30 Calculated TRVs for the circuits shown in Figure 4.29: (a) Serie...

Figure 4.31 Circuit for TRV calculation with the series reactor on the fault...

Figure 4.32 Calculated TRVs for the circuit shown in Figure 4.31.

Figure 4.33 Circuit for TRV calculation for the 24 kV case.

Figure 4.34 Calculated TRV for the circuit shown in Figure 4.33.

Figure 4.35 Johnson's conversion for coupled to de‐coupled circuits.

Figure 4.36 Basic circuit for out‐of‐phase switching condition.

Figure 4.37 Out‐of‐phase switching TRV in comparison to other terminal fault...

Figure 4.38 Illustration of out‐of‐phase switching individual system TRVs.

Figure 4.39 Symmetrical and asymmetrical currents in relation to the recover...

Figure 4.40 Circuit for asymmetrical current calculation.

Figure 4.41 Asymmetrical current derivation.

Figure 4.42 Asymmetrical current for a 63 kA, 50 Hz circuit breaker with a t...

Figure 4.43 Asymmetrical currents for currents of 63 and 50kA with time cons...

Figure 4.44 Expanded 20–40 ms range for the currents shown in Figure 4.43.

Figure 4.45 Double earth fault circuit representation.

Figure 4.46 Double earth fault

k

pp

calculation.

Figure 4.47 Double earth fault current calculation.

Chapter 5

Figure 5.1 Short line fault circuit representation.

Figure 5.2 Short line fault traveling wave analysis approach.

Figure 5.3 Short line fault right‐going traveling wave.

Figure 5.4 Short line fault left‐going traveling wave.

Figure 5.5 Voltage profile on the line versus travel time.

Figure 5.6 Voltage at circuit breaker as function of travel time.

Figure 5.7 Sequence network for a single line to ground fault.

Figure 5.8 L90, L75, and L60 line side transients for 245 kV, 40 kA, 50 Hz c...

Figure 5.9 L90 line side transients for 245 kV, 50 Hz circuit breakers rated...

Figure 5.10 L90 line side transients for 245 kV, 40 kA, 50 Hz circuit breake...

Figure 5.11 Circuit representations for first and third pole short line faul...

Chapter 6

Figure 6.1 Shunt reactor load circuit variations.

Figure 6.2 General case for shunt reactor switching analysis.

Figure 6.3 First‐pole‐to‐clear three‐phase representation.

Figure 6.4 First‐pole‐to‐clear circuit.

Figure 6.5 Shunt reactor switching phenomena.

Figure 6.6 Load side circuit generic oscillations.

Figure 6.7 Current interruption voltages in absolute values.

Figure 6.8 Current interruption voltages in per unit values.

Figure 6.9 Circuit breaker capacitance circuit.

Figure 6.10 TRV for switching out a 12 kV 35 Mvar shunt reactor with isolate...

Figure 6.11 TRV for switching out a 12 kV 35 Mvar shunt reactor with isolate...

Figure 6.12 TRV for switching out a 170 kV 35 Mvar shunt reactor with isolat...

Figure 6.13 TRVs for switching out a 516 kV 135 Mvar shunt reactor with a 10...

Figure 6.14 Second‐parallel reignition oscillatory circuit.

Figure 6.15 Reignition overvoltage schematic representation.

Figure 6.16 Chopping number as function of arcing time for case discussed in...

Figure 6.17 Suppression peak overvoltage ka as function of arcing time for c...

Figure 6.18 TRV peak as function of arcing time for case discussed in Sectio...

Chapter 7

Figure 7.1 Capacitive current interruption recovery voltage.

Figure 7.2 Circuit for capacitive inrush current calculation.

Figure 7.3 Circuit for back‐to‐back capacitive inrush current calculation.

Figure 7.4 Basic circuit for single and back‐to‐back capacitor switching exa...

Figure 7.5 Inrush current for single capacitor bank switching example.

Figure 7.6 Derivation of effective circuit for back‐to‐back switching exampl...

Figure 7.7 Inrush current for back‐to‐back capacitor bank switching example....

Figure 7.8 Circuit for inrush current calculation for a single capacitor ban...

Figure 7.9 Back‐to‐back switching inrush current example with 1 mH series re...

Figure 7.10 Back‐to‐back switching inrush current with circuit resistance eq...

Figure 7.11 Single‐phase circuit for single capacitor bank with earthed neut...

Figure 7.12 Current interruption for shunt capacitor bank with unearthed neu...

Figure 7.13 Applied and trapped charge voltage distributions for recovery vo...

Figure 7.14 Recovery voltages for shunt capacitor bank switching on a‐, b‐ a...

Figure 7.15 Current interruption for shunt capacitor bank with unearthed neu...

Figure 7.16 Applied and trapped charge voltage distributions for recovery vo...

Figure 7.17 Recovery voltage for a‐phase with delayed interruption in the se...

Figure 7.18 Definition of reignitions and restrikes.

Figure 7.19 Load side voltage escalation due to multiple restriking.

Figure 7.20 Circuit for restriking voltage calculation.

Figure 7.21 Circuit for shunt capacitor bank outrush current calculation.

Figure 7.22 Representative circuit for unloaded transmission lines.

Figure 7.23 Circuit element network and sequence network for unloaded transm...

Figure 7.24 Circuit for transmission line first‐pole‐to‐clear factor (voltag...

Figure 7.25 Transmission line recovery voltage peak versus

C

1

/

C

0

ratio.

Figure 7.26 Circuit for unloaded cable inrush current calculation.

Figure 7.27 Circuit for back‐to‐back cable switching.

Figure 7.28 Three‐phase screened and belted cables and equivalent circuits....

Figure 7.29 Capacitive current switching circuit.

Figure 7.30 Capacitive current interruption with 300 kV air‐break disconnect...

Figure 7.31 Capacitive current interruption with 300 kV air‐break disconnect...

Chapter 8

Figure 8.1 Arcing windows for circuit breakers applied on non‐effectively ea...

Figure 8.2 Arcing windows with respect to current zero crossings for circuit...

Figure 8.3 Arcing windows and minimum arcing times for circuit breakers appl...

Figure 8.4 Arcing windows and maximum arcing times for circuit breakers appl...

Figure 8.5 Arcing windows with respect to current zero crossings for circuit...

Figure 8.6 Suppression peak overvoltages versus chopping numbers for varying...

Appendix A

Figure A.1 RL circuit.

Figure A.2 RC circuit.

Appendix B

Figure B.1 Current injection series RLC circuit.

Figure B.2 Series reactor location relative to source.

Appendix D

Figure D.1 Euler formula plot.

Figure D.2 Euler formula right side plot.

Appendix E

Figure E.1 Application of Trapezoidal Rule.

Figure E.2 Application of Simpson's Rule.

Appendix F

Figure F.1 General circuit for shunt reactor switching calculation.

Figure F.2 Sequence network for first‐pole‐to‐clear calculation.

Figure F.3 AC load circuit representation.

Figure F.4 Transient load circuit representation.

Figure F.5 Exceptional case transient load circuit representation.

Appendix G

Figure G.1 Basic generator plant single‐line diagram.

Figure G.2 Standard TRVs for 20 kV rated circuit breaker and 400 MVA rated g...

Figure G.3 Circuits for system‐source fault analysis.

Figure G.4 First‐pole‐to‐clear circuit for 13.8 kV rated circuit breaker for...

Figure G.5 TRVs for 13.8 kV rated circuit breaker application for system‐sou...

Figure G.6 Asymmetrical current for 72.3 kA system‐source fault case.

Figure G.7 Degree of asymmetry at contact parting.

Figure G.8 Symmetrical rms current for generator‐source fault.

Figure G.9 TRVs for generator‐source fault.

Figure G.10 Asymmetrical current for generator‐source fault.

Figure G.11 Asymmetrical current at contact parting for generator‐source fau...

Figure G.12 Out‐of‐phase current interruption TRVs with and without added su...

Figure G.13 Out‐of‐phase current interruption generator‐side and system‐side...

Figure G.14 Step‐up transformer high‐side fault TRV.

Figure G.15 Load current switching circuits.

Figure G.16 Load current switching TRV.

Figure G.17 Load current switching TRV based on separate generator and trans...

Figure G.18 Load current switching TRV as for Figure G.17 except with added ...

Figure G.19 Comparison overview of TRVs for the 13.8 kV generator case.

Appendix H

Figure H.1 IEEE exponential‐cosine TRV representation.

Figure H.2 IEEE exponential‐cosine TRV for 245 kV rated circuit breaker.

Figure H.3 Circuit for first pole to clear pole factor calculation [85].

Figure H.4 Circuit for second pole to clear pole factor calculation [85].

Figure H.5 Impedance circuits for first, second and third pole clearing pole...

Appendix I

Figure I.1 Series RLC circuit: (a) Exponential function calculation, (b) Cos...

Figure I.2 245 kV circuit breaker T100 TRV: (a) Exponential function calcula...

Figure I.3 245 kV circuit breaker T30 TRV: (a) Exponential function calculat...

Figure I.4 225 kV circuit breaker series reactor case: (a) Source side per u...

Figure I.5 Generator‐source asymmetrical fault current: (a) Conversion secon...

Guide

Cover

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Current Interruption Transients Calculation

Second Edition

David F. Peelo

Consultant, former Specialist Engineer at BC Hydro, Vancouver, Canada

This edition first published 2020© 2020 John Wiley & Sons Ltd

Edition HistoryJohn Wiley & Sons Ltd (1e, 2014)

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HB ISBN: 9781119547211

Cover Design: WileyCover Images: Courtesy of David F. Peelo; Background© chanchai howharn/Shutterstock

Preface to the Second Edition

The intent of this second edition is to improve on the first edition content by way of changes to and rearranging the original text as well as the addition of new material. The changes include: a common first‐pole‐to‐clear calculation for both effectively and non‐effectively earthed systems; series reactor application is covered in much more depth; short‐line fault transient recovery voltage (TRV) derivation and capacitive current recovery voltage calculation have been expanded to provide better understanding; and the derivation and use of a generic TRV equation for shunt reactor switching. There are two major additions: the calculation of generator circuit breaker TRVs in Appendix G replacing the former small capacitive current switching content which has now moved to Chapter 7; and, as requested by many, key calculation examples on the use of Excel for plotting TRV equations in Appendix I.

I am grateful to all those who provided comments, criticism, ideas, and suggestions, most of which have been adopted. Once again, I am greatly indebted to Sandra Giasson for her patient and diligent word processing of the text.

David PeeloVancouver, BC, CanadaJune 2019

Preface to First Edition

After a fortunate and rewarding career that started at ASEA in Ludvika, Sweden and was followed by 28 years at BC Hydro in Vancouver, Canada, I took early retirement in May 2001. Not long afterwards, I was asked by the Association of Professional Engineers and Geoscientists of BC if I would be interested in presenting continuing professional development (CPD) courses on circuit breaker application and this started a second career in teaching.

The first course was four hours long and eventually grew into far more detailed courses, some up to five days’ duration. Experience with the courses showed that the part that generated the most questions from participants related to all types of current interruption transients and I started to consider developing a course on transients alone. Around about the same time, the engineering manager at one of my consulting clients lamented the fact that engineers today, particularly the younger generation of engineers, are much too dependent on software and have lost sight of theory and practical reality. He asked if a course could be developed to provide a fundamental understanding of transients and enable estimations using only a hand calculator and a spreadsheet program.

The approach taken (after a number of false starts) was to draw the circuit diagrams for all possible making, breaking, reignition, and restriking cases. Comparison showed that practically all cases are covered by four basic circuits (Tables 2.1 and 2.2). Some exceptions do, of course, occur but are variations on a common theme. Three of the circuits involve second‐order linear homogenous differential equations which, instead of individually resorting to Laplace transformation‐based solutions, have a common solution of the form:

where the roots r1 and r2 are derived from the circuit RLC components and the constants A and B from the initial or boundary conditions. The equation, in turn, has three possible variations: the roots are real, equal, or complex corresponding to overdamping, critical damping, and underdamping, respectively. Once derived, the three equations enable a generic approach to RLC oscillatory circuit calculations (Table A.1).

The fourth case involves a second‐order non‐homogenous differential equation which is more difficult to solve than the homogenous case. However, mathematicians have long resolved the difficulty by providing look‐up tables, basically making a guess at the solution, and then using the method of undetermined coefficients to solve the equation (Appendix A).

At this stage, we now have three equations for each of the four circuits incorporating the r1 and r2 roots. The next step is to apply the boundary (initial) conditions and the equations for current or voltage in real time are derived. The final step is to convert the equations to a generic format by expressing the circuit damping and time in relative terms, i.e. damping relative to critical damping and time relative to the period of the frequency of the transient oscillation (Tables 2.1 and 2.2). General curves can then be drawn and are easily convertible to current or voltage in real time for any switching case.

For multi‐phase faults, sequential interruption of the fault current in the individual circuit breaker poles leads to AC recovery voltages higher than rated voltage. The AC recovery voltages are related to rated voltage − actually pre‐fault voltage at the point of the fault − by pole factors calculated using the method of symmetrical components. A number of approaches are considered including a generic approach to first‐pole‐to‐clear pole factor calculation.

As the reader will learn, there is a certain symmetry to current interruption transients. For any switching event, first taking the status before the switching operation and then the status after the operation, the transient is the transition from “before” to “after.” On this basis, all transients have a starting point, an aiming point or axis of oscillation and a maximum point. Take for example, the transient recovery voltage (TRV) for a terminal fault on an effectively earthed system: the starting point is zero, the axis of oscillation is around the AC recovery voltage and the maximum value is dependent on the damping and nature of the involved circuit. Understanding this overall concept enables a graphical approach to transients’ calculation in many cases (see Figures 6.7 and 6.8).

This is not a book about circuit breaker application and readers are referred in this regard to the references following Chapter 1. Also, it is not a book about how to use Excel for equation-based calculations: guidance is readily available in instruction manuals and online. Using the generic approach to transient calculation is well suited to Excel because generic time is always in radians, a prerequisite for plotting sinusoidal and hyperbolic functions. A note of caution with respect to plotting in Excel is that, in contrast to software which permits the plotting of functions, Excel plots points. This means, for example, if no point is calculated at a maximum value, then the maximum value will not appear in the plot. A further note is in combining plots with different frequencies, such as the case of adding series reactors, all plots have to be referred to common real‐time coordinates before attempting addition or subtraction.

The book is intended to be inclusive. The switching cases are covered in detail in the main text and supporting calculations and information can be found in Appendices A–G. The evolution of TRVs and their understanding is interesting and is reviewed in Appendix H. The first circuit breakers became commercially available around 1910 and technical papers started to appear within a few years in AIEE publications. The notion of a TRV was first recognized in 1927 by J.D. Hilliard of GE, who used the descriptive term “voltage kick” for the concept. The first standards for fault current TRVs were developed in the 1950s and evolved further into the standards of today.

I would not have been able to write this book without the support that made my career possible. I am grateful to BC Hydro for supporting my participation in learned societies, principally CIGRE and the IEEE, and in the development of circuit breaker standards in IEC; to my colleagues past and present at BC Hydro and in the IEEE Switchgear Committee, CIGRE Study Committee A3 and IEC Technical Committee 17A; to those who have attended the course and asked the great questions that contributed to the book content; and, most of all, to Sandra Giasson for her patient and diligent word‐processing of the text through several drafts to the final version.

Writing the book took 10 months but really it has been 30 years in the making. Now it's done and I hope that you will find it to be useful and of value.

David PeeloVancouver, BC, CanadaAugust 2013

1Introduction

1.1 Background

The intent of this textbook is to explain the origin and nature of the transients associated with fault and inductive and capacitive load current interruption. The transients in general have a power frequency and an oscillatory component. The oscillatory components have a RLC circuit basis with such a degree of commonality between the above current interruption cases that a generic calculation approach is possible. The power frequency component is either a balanced or momentarily unbalanced quantity and, in some cases, is the axis of oscillation for the oscillatory component. In overview, the following transients will be analyzed and the resulting equations applied to real current interruption cases:

Fault current interruption: The transient of interest is the

transient recovery voltage

(

TRV

) that appears across the circuit breaker after current interruption. For terminal faults, i.e. a fault at the circuit breaker, the power frequency component is dependent on the system earthing and the type of fault. The oscillatory component can be either overdamped or underdamped with traveling waves contributing to the former oscillation. The TRV may be on one side of the circuit breaker only, for example, a three‐phase‐to‐earth fault on an effectively earthed system, or on both sides of the circuit breaker as for the out‐of‐phase switching and short‐line fault cases.

Inductive current interruption: The transients for consideration in this case are the TRV, which is the difference between the source power frequency and the load circuit oscillation, and also the transients due to reignitions. The load circuit and reignition transient oscillations are underdamped.