generalised power components definitions …khalsarc.com/khalsarc/measurements_files/hk thesis rev...

GENERALISED POWER COMPONENTS DEFINITIONS

FOR SINGLE AND THREE-PHASE ELECTRICAL

POWER SYSTEMS UNDER NON-SINUSOIDAL AND

NONLINEAR CONDITIONS

by

Harnaak Singh Khalsa B. E. (Hons), University of Malaya

M. Eng Sc., Monash University

Thesis

Submitted by Harnaak Singh Khalsa

for the fulfilment of the Requirements for the Degree of

Doctor of Philosophy

Supervisor: Dr Jingxin Zhang

Associate Supervisor: Dr Grahame Holmes

Electrical and Computer Systems Engineering

Monash University

December, 2007

© Copyright

by

Harnaak Singh Khalsa

2007

Dedication

To my INVISIBLE FRIEND who was there for me through thick and thin, whether rain or shine,

guiding me, in the most difficult of times, never letting me down no matter what.

GOD

grant me the boon to see through the thorny bush, at the beautiful flowers beyond,

and the ability to detect

what is right and what is wrong, and

the strength, to stay committed on what I see as right.

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electrical power?

definitions?

mushrooms?

the answer lies within!

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Errata

Errata

Errata-1

ERRATA The errata list included in the hard bound copy has be incorporated in the text of this soft copy.

I

GENERALISED POWER COMPONENTS DEFINITIONS FOR SINGLE AND THREE-PHASE ELECTRICAL

POWER SYSTEMS UNDER NON-SINUSOIDAL AND NONLINEAR CONDITIONS

TABLE OF CONTENTS Abstract i

Declaration iv

Acknowledgement v

List of Publications vi

1 Introduction…………………………………………... 11.1 Background to the research - Problems with current definitions

of power ……………………………………………………….…

1

1.2 Need for Generalised Power Definitions…………….…………... 3

1.3 Research of the thesis …….………..…………….……………… 4

1.4 Justification for the research………..…………….……………… 8

1.4 Methodology in writing this thesis .…………….………………. 9

1.5 Outline of the thesis…………….……………………….……….. 10

1.6 Definitions and Nomenclature…………….……………………... 10

1.7 Delimitation of scope and key assumptions …………….………. 11

1.8 Conclusion …………….……………………….………………... 12

2 Analysis of Current Power Theories………..……….... 132.1 History………….…………………….…………………….……. 13

2.2 Background Technical Information………….………………….. 16

2.2.1 Powers in the time domain………….…………………….……. 16

2.2.1.1 Single-Phase………….…………………….…………………. 16

2.2.1.2 Three-Phase………….…………………….………………….. 17

2.2.2 Discussion on cross-harmonic powers………….……………… 18

II

2.2.3 Does the instantaneous active current always have the same

scaled waveshape as the voltage waveform? ………….……… 21

2.2.4 Does the average active power in a three-phase system remains

the same irrespective of the choice of reference conductor?…… 23

2.2.5 Is the three-phase instantaneous active power p3Ph(t) (sum of

instantaneous active power of each of the phases) waveform the

same as that obtained from the sum of the three-phase

instantaneous total power s3PH(t) of each of the phases……….. 25

2.2.6 Does the current vector projected onto the voltage vector in

presence of harmonics always give the active current?………… 30

2.2.7 Apparent power and line loss………….…………………….….. 32

2.3 Power theories/definitions………….…………………….…….. 33

2.3.1 RMS voltage and current based definitions………….…………. 34

2.3.1.2 RMS based powers and relationship between “average” and

time quantities ………………………………………………… 35

2.3.2 Definition proposed by C. Budeanu (1927) ………….………… 45

2.3.3 Definition proposed by S. Fryze (1932) ………….……………. 48

2.3.4 Definition proposed by W. Shepherd and P Zakikhani (1972)…. 50

2.3.5 Definition proposed by Sharon (1973) ………….……………... 51

2.3.6 Definition proposed by Kusters and Moore (1980) ………….… 53

2.3.7 Definition proposed by Czarnecki (1985/1988) ………….……. 54

2.3.8 The theory of instantaneous power in three-phase four wire

systems proposed by Akagi et al (1983/1994) ………….……… 59

2.3.9 The FBD-Method by Depenbrock (1993) ………….………….. 62

2.3.10 Definition proposed by Ferrero and Superti-Furga (1991)……... 66

2.3.11 Definitions proposed by Willems (1992, 1993) ………….……. 67

2.3.12 Generalised instantaneous reactive power theory for three-phase

systems proposed by Peng and Lai (1996) ………….…………. 69

2.3.13 Definitions in IEEE Standard 1459 (2000) ………….…………. 71

2.4 Conclusion………….…………………….……………………… 75

III

3 Requirements On A Power Definitions And

Benchmark Case Studies For Evaluation……….…... 773.1 Requirements on a power theory or definitions…………….…… 77

3.1.1 What other researchers say …………….………………………. 77

3.1.2 The Requirements…………….……………………….………... 79

3.2 Background Technical Information…………….………………... 80

3.2.1 Source voltage and currents in a resistive single-phase circuit…. 81

3.2.2 Source voltage and currents in an inductive single-phase circuit. 81

3.2.3 Source voltage and currents in a capacitive single-phase circuit.. 82

3.2.4 Source voltage and currents in a linear parallel resistive-

inductive single-phase circuit…..…………….………………… 83

3.2.5 Source voltage and currents in a linear series resistive-inductive

single-phase circuit…………….……………………….……… 84

3.2.6 Determination of active current in R-C single-phase circuit…… 84

3.2.7 Powers in the single-phase circuit…………….………………… 84

3.2.8 Discussion of source voltage and driving voltage……………… 85

3.2.9 Does a diode-R load consume non-active power? …………….. 87

3.3 Evaluation benchmarks…………….………………...…………... 92

3.3.1 Single-phase case…………….…………………….…………… 93

3.3.2 Three-phase case…………….…………………….……………. 94

3.3.3 Evaluation Criteria…………….…………………….………….. 97

3.4 Computation of Waveforms and Energy Transfer…………….…. 98

3.4.1 Single-Phase Cases…………….…………………….…………. 98

3.4.2 Three-Phase Cases…………….…………………….…………. 105

3.5 Conclusion…………….…………………….…………………… 118

4 Single-Phase Power Component Definitions For

Instantaneous And Average Powers…….……….……. 1194.1 Introduction……….……………….……………….……………. 119

4.2 Background Technical Information……….……………….……. 121

4.2.1 Discussion on non-linear Diode RL parallel and series load…… 122

4.3 The Proposed Single Phase Instantaneous Power Definitions…… 123

IV

4.3.1 Load Model……….……………….………………………….… 123

4.3.2 Sinusoidal System……….……………….……………….…….. 123

4.3.3 Non-sinusoidal System……….……………….………………... 124

4.3.3.1 Current Decomposition……….……………….………………. 126

4.3.3.2 Powers……….……………….……………….………………. 127

4.3.3.3 Discussion of the components and application of definitions…. 132

4.3.4 Average active and non-active power……….……………….…. 133

4.3.4.1 Instantaneous Power and its active and non-active components 133

4.3.4.2 Energy Transfer……….……………….……………….……… 134

4.3.4.3 Average power……….……………….……………….………. 136

4.4 Evaluation of the Proposed Single Phase Instantaneous Power’

Definitions……….……………….……………….……………... 142

4.4.1 Computation……….……………….……………….………….. 142

4.4.2 Results of Computation……….……………….……………….. 143

4.4.2.1 Waveforms .……………….……… .……………….………… 143

4.4.2.2 Energy transfer and average power.……………….…………. 145

4.4.3 Evaluation based on Requirements of the Definitions………….. 146

4.5 Analysis and discussion of results……….……………….……… 146

4.5.1 Additional example……….……………….……………….…… 147

4.6 Experimental verification of the viability of proposed definition

algorithm……….……………….……………….………………..

148

4.6.1 Introduction……….……………….……………….…………… 148

4.6.2 Algorithm Implementation……….……………….…………….. 148

4.6.3 Experimental setup……….……………….……………….……. 149

4.6.4 Results and discussion ………...……….……………….………. 150

4.7 Conclusion……….……………….……………….……………... 152

5 Choice Of Reference Conductor In Three Phase

Systems.…...…...…..…...…...…...…...…...….....….....…

153

5.1 Introduction…….………………….………………….………….. 153

5.2 New approach And Formulae…….…………….……………….. 155

5.2.1 Instantaneous Active Current …….………………….………… 155

V

5.2.2 Conductor loss in the 3-phase system …….………………….… 156

5.3 Case Study and Computation …….………………….………….. 156

5.3.1 Cases 1 to 4: 3Ph 3W with 2-phase Load …….………………... 158

5.3.2 Cases 5 to 8: 3Ph 4W star load …….………………….……….. 159

5.3.3 Cases 9 to 11: 3Ph 4W star/delta mixed load …….……………. 160

5.3.4 Cases 12 to 16: 3Ph 3W delta load …….………………….…… 160

5.4 Result of Computation …….………………….…………………. 161

5.5 Results and analysis of results …….………………….…………. 163

5.6 Conclusion …….………………….………………….………….. 166

6 Three Phase Power Component Definitions For Instantaneous

And Average Powers …………………………………………… 169

6.1 Introduction …………….………….……………………….……. 169

6.2 Background Technical Information …………….……………….. 170

6.3 Proposed Three Phase Power Component Definitions ………….. 170

6.3.1 Three-phase powers based on single-phase …………….……… 170

6.3.2 Three-phase powers on collective three-phase basis …………... 170

6.3.3 Unbalance …………….………….……………………….…… 174

6.3.4 Three-phase component powers on collective three-phase basis. 176

6.3.5 Three-Phase Powers as applicable to space-vector transform …. 176

6.3.6 Discussion of the components and application of definitions …. 177

6.4 Evaluation of proposed three-phase instantaneous powers’

definitions ………….……………….……………….…………… 177

6.4.1 Computation…………….……………………….……………… 177

6.4.2 Results of Computation …………….……………………….….. 178

6.4.2.1 Waveforms …………….………………………………….…. 178

6.4.2.2 Average power and energy transfer …………….…………….. 183

6.4.2.3 Additional Examples …………….……………………….…… 185

6.5 Analysis and discussion of results …………….………………… 190

6.6 Experimental Work - Digital Power Meter …………….……….. 190

6.6.1 Introduction …………….……………………….……………… 191

6.6.2 Input Stage ……………………….………………….…….…… 191

6.6.3 Signal Conditioning Stage …………….………………….……. 191

6.6.4 ADC and processing Stage …………….………………………. 192

VI

6.6.5 Experimental setup …………….………………….…………… 193

6.6.6 Results and discussion ……………………...…………………. 193

6.7 Conclusion .….……………………….……………….…….……. 195

7 Application of Definitions ………….………….……. 1977.1 Introduction ……………….………………….………………….. 197

7.2 Background Technical Information ……………….…………….. 199

7.2.1 Compensation Concepts in relation to the needs of this thesis…. 199

7.2.1.1 Compensation of source generated harmonics ……………….. 200

7.2.1.2 Compensation of load generated harmonics ……………….…. 202

7.2.1.3 Compensation of load generated DC ……………….………… 202

7.2.2 Summary….….….….….….….….….….….….….….….….….. 202

7.3 Measurement ……………….………………….………………… 202

7.4 Compensation ……………….………………….……………….. 202

7.4.1 Application Example 1 ……………….………………….…….. 203


7.4.3 Summary – some rules on compensation .……………….…….. 222

7.5 Detection of source of distortion ……………….……………….. 223



7.6 Power quality ……………….………………….……………….. 237



7.6.3 Summary – some comments on power quality ………………… 241

7.7 General ……………….………………….………………….…… 241


7.8 Conclusion ……………….………………….………………….. 246

8 Relationship of the Proposed Definitions with some

existing Definitions …………………………………... 2478.1 DC System ………………………………………………………. 247

8.2 Sinusoidal Systems ……………………………………………… 247

VII

8.3 RMS based powers ……………………………………………… 247

8.4 Budeanu’s Definitions ………………………………………….. 248

8.5 Relationship of the proposed definitions with that of Fryze …… 249

8.6 IEEE Standard 1459-2000 ………………………………………. 251

8.7 Conclusions and future research .….……………………………. 251

9 Conclusions and Future Research …………………… 253

10 References ……………………………………………... 255

Appendix A – Parallel equivalent of series RL load

Appendix B - Comparison of the Proposed Defintion and Fryze’s Definition

Appendix C - Determination of phase angle nγ

Appendix D – Report of company CO

Abstract i


POWER SYSTEM UNDER NON-SINUSOIDAL AND NONLINEAR CONDITIONS

Harnaak Singh Khalsa B.E.(Hons) , M. Eng Sc. Monsah University 2007 Supervisor: Dr Jingxin Zhang [email protected] Associate supervisor: Associate Professor Grahame Holmes [email protected] ABSTRACT There is a need for generalised definitions of electrical powers to provide a

simultaneous common base for measurement, compensation, power quality and

identification of source of distortion.

The major problem area today is definitions of powers in the presence of harmonics and

nonlinear loads in the electrical power system. In such a scenario, there is a problem to

accurately measure especially reactive (nonactive) power. This is important for

accurate energy billing. Another important area is the mitigation equipment used to

remove unwanted polluting quantities from the power system. Definitions of powers

have an important role to play in providing the correct information for the optimal

design and performance of such equipment. Evaluation of the quality of the power

system to enable appropriate allocation costs to those causing deterioration in the power

quality also cannot be discounted. To enable this cost allocation, there is a need to

identify the polluters and the definitions should indicate degradation in power quality as

well as identify the source of this degradation. Finally, it would be very useful if the

definitions could also be used to perform a general analysis of the power system.

Abstract

Abstract ii

This thesis commenced with investigation of the problem with an in-depth study of the

existing definitions, and what other researchers have indicated about this problem, from

the definitions perspective. The issues identified with current definitions are that some

definitions do not possess the attributes that are related to source-load properties, and

others are based on mathematical consideration and lack physical meaning. One issue

in measurement of nonactive power is its nature of having zero average value. Another

contributing factor is that the presence of source impedance is neglected in definitions.

The use of RMS quantities to determine powers, especially instantaneous powers, in the

presence of multi-frequency voltages and currents also contributes to the problem.

Additionally, RMS based definitions are based on heating effect while not all source-

load relationships are totally of a heating nature. The RMS based definitions also do

not satisfy the energy conservation principle. Another issue is that though harmonic

currents are used, current definitions still utilise the RMS value of the voltage wave thus

losing harmonic information.

The solution is to decompose, as accurately as possible, the total instantaneous power

into active and nonactive components utilising DC, fundamental and harmonics of

voltage and current as well as being based on the power system properties. To enable

this, the load model must closely represent the reality. This thesis presents the new

instantaneous power definitions to achieve this. In addition to the fundamental, five

sub-components for each of the active and nonactive parts are defined. The definitions

are based on both the voltage and current DC, fundamental and harmonic components

thus retaining harmonic information. Thus these definitions are not only mathematically

based but also have a direct relationship with the load. The definitions do not make the

assumption of zero source impedance. With good knowledge of the time profile of

active and nonactive power components, an accurate time-domain measurement of the

active and nonactive power is achieved. The components of powers introduced in the

proposed definitions can be utilised to gauge power quality, to identify the source of

distortion and to achieve optimal compensation. Based on the new instantaneous power

definitions, the definitions for average values of the powers are also proposed. The

recognition of positive going and negative going parts of the nonactive power waveform

in defining the average nonactive power alleviates the problem of the “zero average

Abstract iii

nature” of nonactive power. It also retains energy information and satisfies the principle

of energy conservation.

The new definitions are evaluated for linear and non-linear loads in the presence of

harmonics using benchmark case studies. Evaluation results demonstrate good

performance of the proposed definitions. The practical applications of the definitions

are explored with a number of examples from the areas of measurement of power and

energy, compensation, detection of source of distortion and power quality. An

application example showing the capability of the definitions in general analysis of a

system is also presented. Good and useful results are obtained for all these examples.

The proposed definitions are implemented on prototype systems with digital signal

processors to demonstrate their practical usability. The proposed definitions are shown

to be consistent with the traditional definitions under the conventional sinusoidal

conditions, and their relationships to the commonly used existing definitions are also

revealed.

Declaration

Declaration

iv


POWER SYSTEM UNDER NON-SINUSOIDAL AND NONLINEAR CONDITIONS

Declaration I declare that this thesis is my own work and has not been submitted in any form for another degree or diploma at any university or other institute of tertiary education. Information derived from the published and unpublished work of others has been acknowledged in the text and references given. ______________________________________ Harnaak Singh Khalsa December 18, 2007 This manuscript shall not be reproduced in any form except with the expressed written permission of the author.

Acknowledgement

Acknowledgement v

Acknowledgement I would like to thank my present supervisors Dr Jingxin Zhang and Associate Prof Grahame Holmes. My deepest gratitude goes to Dr Jingxin Zhang for his invaluable assistance and guidance throughout the research. I would also like to thank my past supervisors Professor Bob Morrison and Dr Wlad Mielczarcski. I would like to thank Monash University who gave me an oppoturnity to undertake this degree. Additionally, I would like to thank ABB, with whom I am under full time employment, for their support and provision of test equipment and facilities through the years in this research. I am sincerely grateful to them for the study leave extended to me, enabling me to execute the research activity. My sincere thanks also to the organization referred to as CO (this is done to maintain anonymity) in this thesis who have kindly permitted the inclusion of their report (Appendix D) that investigated the source of fifth harmonic between the organization and the supply authority. My thanks also goes to Visahan Gunaratnam whose work on the project with LabVIEW for the first time practically validated the viability of the algorithm implementing the proposed definitions. I must also tender my thanks to Sebastian Rafael Castro, San Yau Foo and Ping Jia Ong for their work on the project that showed possibility of the implementation of the definition algorithm in a DSP. I am indeed grateful to Sivajith Selvarajah who gave his full-hearted support with his unreserved and “always willing” assistance to both projects as well as the ongoing project. My wife Ajit and daughter Kiranjeet have been a constant source of inspiration and strength in me completing this degree. I sincerely thank them for their kind support, encouragement and understanding for the many late night work and noise generated by me in the course of doing this degree. I must not forget my work colleagues, many relatives and friends who have given me encouragement throughout and sincerely thank them for their support. Special thanks goes to my God-brother Narinderpal Singh for proof reading this thesis. I must not forget the thank GOD for the strength HE gave me to finalise this thesis under the stress and pressures of full time employment as well as the research activities while doing this degree. Last but not least, I would like to thank the reader of this thesis without whom any writing may be left in the bookshelf gathering dust.

Harnaak Singh Khalsa

September, 2007

List of Publications

List of Publications vi

List of Publications The following research papers support the thesis: International Journals: 1. Harnaak Khalsa and Jingxin Zhang (2006) "A New Definition of Non-Active

Power," International Journal of Emerging Electric Power Systems: Vol. 7 : Iss. 4, Article 3, 2006, Available at: http://www.bepress.com/ijeeps/vol7/iss4/art3

International Conference Proceedings: 1. H Khalsa, J Zhang, Three Phase Generalised Power Theory for the Measurement

of Powers in an Electric Power System, IEEE Tencon 2005, Melbourne, November 2005

2. H Khalsa, J Zhang, Choice of Reference Conductor in Three Phase Systems – A

Paradigm Shift, IEEE Tencon 2005, Melbourne, November 2005 3. H Khalsa, J Zhang, A New Single-Phase Power Component Theory for Powers in

an Electric Power System, 7th International Power Engineering Conference - IPEC2005, Singapore, November 2005.

4. H. Khalsa, J. Zhang, Performance of the IEEE Standard Definitions for the

Measurement of Electrical Power Under Nonsinusoidal Conditions, AUPEC Conference, Brisbane, Australia, September 2004.

5. H. Khalsa, J. Zhang, Investigation Of A Uni-And Bi-Directional Definition and the

IEEE Standard Definition for The Measurement of Powers, AUPEC Conference, Christchurch, New Zealand, September 2003

6. H. Khalsa, R E Morrisson, A new technique for the measurement of powers in the

presence of harmonics in an electric power system, AUPEC Conference, Melbourne, September 2002.

7. H. Khalsa, W Mielczarski, A concept of Unidirectional and Bi-directional

Components to Define Power Flow in Non-Sinusoidal Circuits, 8th International Conference on Harmonics and Quality of Power, Vol 2, pp 672-677, Greece, 14-16th October, 1998.

Introduction

Chapter 1 1

1. INTRODUCTION

Definitions of power have an impact on the measurement, control and efficient use of

electrical energy. Their development dates back to Edison’s days in the electrical era.

During Edison’s days, the measurement of electricity consumption was as important as

it is today. Edison started by charging his customers a lump sum for every lamp

installed [1]. The power used was defined in terms of the number of connected loads.

As this approach was found to be inequitable, the method of charging was changed to

using an electrolytic meter [2]. Even here the measurement was cumbersome and

difficult for user to verify. It is noted that the definition changed from connected load

to utilizing a chemical measure. This eventually led to the introduction of an electro-

mechanical meter developed by Elihu Thompson, who was working with Edison, that is

a precursor to the watt-hour meter of today. The watt-hour meter utilised a definition

based on the motor principle. It is thus apparent that the definition was improved over

time to meet the changing needs of the day.

1.1 Background to the research – problems with current definitions of power Today, as in the days of Edison, definition of powers is still a problem though the cause

is different. The major problem area today is harmonics and nonlinear loads in the

electrical system. The definitions of powers also contribute to the problem as has been

been highlighted by many a researcher as outlined below.

The accuracy of electricity meters [3,4] is affected by nonsinusoidal waveforms and

unbalance. Due to the measuring principle of the induction disk meter, definitions had

less of an impact, the measurement being based on motor principle. Digital and

numerically based instruments, being widely used today, are much more versatile than

the induction disk types in the computation of powers and thus become more dependent

on the definitions used in the algorithm. Electronically controlled devices, such as

adjustable speed drives and static compensators, commonly encountered in today’s

Introduction

Chapter 1 2

power system are also dependent on the measurement of power for their performance

[5].

Many researchers have pointed out the issues with current definitions and techniques for

measuring electrical power, especially ‘reactive power’, under nonsinusoidal conditions.

Some definitions do not posses attributes that are related to the power phenomena in

electrical circuits and the load properties [6-8]. Others originate from mathematical

consideration and lack physical meaning [9-14]. Yet others eliminate inconsistencies in

one area but introduce problems in other areas [15]. Though active power has an

accepted definition, there is ambiguity in defining reactive power [16-18] mainly

because its average over time is zero [19]. The control philosophy of compensation

systems (compensators) is still a major unsolved problem. Proper control philosophy

can only be derived if the definitions of all the components of electric power, under

nonsinusoidal conditions, prove to be accurate and have an interpretation in terms of the

load connected [18, 20-22]. The presence of source impedance also causes inconsistent

results as it is neglected in definitions [23,24]. Some definitions are introduced for a

specific need, e.g., line loss evaluation, identification of sources of pollution [25,26].

There are issues with geometric and arithmetic powers in three phase definitions under

unbalanced conditions [27,28]. The assumption that active and reactive powers are

orthogonal is not necessarily true for nonlinear situations [29]. Further confusion is

added as the different definitions diverge and emphasize different qualities suited to

different applications [18,26]. Definitions based on time-domain and frequency domain

analysis do not fully agree [30]. Definitions for single-phase systems become

ambiguous and controversial for multi-phase, non-sinusoidal and nonlinear systems [31-

33].

Based on the comments of the researchers outlined above, the causes of the problem are

briefly discussed and will be detailed in later chapters.

• For sinusoidal systems, powers are defined using RMS quantities. Applying this

directly to non-sinusoidal systems using RMS values of voltage and current is not

satisfactory. This is because RMS is a derived quantity that is a representation of

the current or voltage based on heating effect. Use of RMS quantity to represent a

single frequency sinusoidal signal is acceptable because the frequency information

Introduction

Chapter 1 3

is not lost, the single frequency being known. However, when a multi-frequency

non-sinusoidal signal is represented with a single RMS quantity, the frequency

information is lost. Additionally the characteristics of the load are not wholly of

“heating” nature.

• Other definitions are made for a specific purpose. For example most nonactive

power theories have a particular compensation in mind and hence are not generally

applicable. Not only that, they may not satisfy the load properties or phenomena in

electrical systems.

• Yet others adapt techniques from other application. Adapting definition used for

generators, which has as basis a single rotating field, to measurement in non-

sinusoidal systems is one example of this.

• Many use mathematical methods and adapt these for non-sinusoidal systems.

• Others decompose current into components then calculate powers but however use

RMS equivalent of the distorted voltage without any corresponding decomposition.

• Another issue is the assumption that the decomposed current components are

orthogonal. This may not be true for nonlinear conditions.

• Some definitions are only applicable for three-phase systems and are not applicable

for single phase. In fact one researcher showed mathematically that single-phase

system has no instantaneous reactive power [11]. It is noted that three-phase system

evolved from the single-phase and the electrical phenomena are similar. Any

definition should be applicable to both systems.

• Another point of view is to take the phases or conductors in a system as all being

equal. Voltage measure requires a reference phase or conductor. As voltage is an

important factor in determining power, the need for a reference conductor should

not be ignored.

1.2 Need for generalised power definitions Many publications, some of which are highlighted below, have stated a need for a

power theory that should explain the phenomena, especially due to nonlinear loads,

occurring in the power systems today. It should also cater for measurement as well as

compensation. The author of [34] states that there is not a power theory that explains

the power properties of three phase-asymmetrical systems under non-sinusoidal

Introduction

Chapter 1 4

conditions. Reference [35] points out a need for a consistent approach to the definition

of apparent power. The IEEE Working group [15] indicates that present definitions are

not adequate for economic studies in nonsinusoidal and/or unbalanced and/or nonlinear

systems. Strong practical reasons are pointed out in [36] for reviewing power

definitions. Reference [37] states for the need of a generalised power theory that can

provide simultaneous common base for energy billing, evaluation of electric energy

quality, detection of major sources of waveform distortion and provide information for

design of mitigation equipment. Likewise references [38,39] also point to the need for a

new set of definitions of power quantities. This need is additionally evidenced by

substantial research in this area over the last twenty or so years [11-17,20-23,28-31,33-

35,37-38,40–69].

1.3 Research of the thesis The main focus of this thesis is the proposal of new generalised power definitions, with

active and nonactive powers’ components, to meet essentially the requirement stated in

[37] that is a generalised power theory that can provide simultaneous common base for

energy billing, evaluation of electric energy quality, detection of major sources of

waveform distortion and provide information for design of mitigation equipment. The

author’s working experience, for the last thirty one years, in the electrical power

industry and his particular interest in the measuring algorithms both in the metering as

well as protective relaying equipment, coupled with problems experienced in the field in

relation to these devices, has had a profound influence on the thought presented in this

thesis. The definitions are proposed for both single and three-phase systems. The

proposal does not attempt to include definitions for efficiency of utilization of supply,

this being left to the existing methods and definitions for example in [37]. The main

intent is to present the time characteristics or profile of the active and nonactive

currents and powers, as close as possible to what they truly are, in a real system, be

it single-phase or three-phase. Hence, the prevailing thinking, for example, that the in

phase component of the fundamental will contribute to useful power while harmonics

do not and thus so should not be classed as active, is not agreed with. Similarly also, is

the case with harmonic cross-product components. The direction taken is to identify as

faithfully as possible a component’s contribution, be it harmonic or cross-harmonic

component, to active (or nonactive) power even if that component is not providing any

Introduction

Chapter 1 5

useful power. A useful layman analogy here is that a poisonous mushroom is not

classed as something else just because it is poisonous and cannot be consumed. It is

classed as a mushroom but identified as poisonous so that it is made known that it is not

for consumption. In a like manner, different components with specific properties in

relation to source/load are defined for the total, active and nonactive powers. This

enables judgment of the usefulness of a particular power component, be it active or

nonactive and the required action for example to remove it or in another case recover

costs for the existence of that component.

The single-phase system uses two conductors while three-phase systems may have three

or four conductors. First, discuss the single-phase system where the source voltage is

usually the electromotive force or driving voltage behind the current flow. The

instantaneous power, the product of voltage and current (it shall be termed

instantaneous total power in this thesis), is decomposed into instantaneous active and

nonactive powers. The idea of decomposition, some recent publications being

[14,29,40,43-44,54], is not new but the manner in which the decomposition is and how

active and nonactive parts are determined are different. These active and nonactive

parts arise from the presence of energy consuming elements (resistors), storing elements

(inductors and capacitors) and generating elements (sources) in the load. The active part

contributes to the energy consumed by the load and the nonactive part to the energy that

is oscillating bidirectionally between source and load. The active part is also oscillating

but usually unidirectionally. The active part represents the power or energy that is

transformed into another form while the latter represents the energy that is stored in one

part of the cycle and then released in another part of the cycle. The flow of energy of

nonactive power in one direction is always equal to that flow in the opposite direction

and this is a characteristic of nonactive energy. This stored energy cyclically changing

hands between source and load causes increased current requirement from the source

and is generally termed “useless” energy. The load system, beyond the metering point,

is represented by an equivalent parallel time variant conductance and susceptance

(Figures 1.1 and 1.2). The voltage at the metering point represents the supply source.

Introduction

Chapter 1 6

i(t)

~v(t)

~ Load B(t)G(t)~ v(t)~

i(t)

Fig 1.1: Load Model

B(t)G(t)

meteringpoint

~v(t) ~

i(t)

loadsource

Fig 1.2: Metering point

The source voltage connected to the load, gives rise to active and nonactive currents that

manifest as the measurable total current at the metering point. For any particular

source-load arrangement, instantaneous total power can be decomposed to active and

nonactive power components. This is dependent on the source/load characteristics. In

the proposed definitions, the universally known and understood sinusoidal system

concept is extended to the nonsinusoidal system. The determination of the active and

nonactive power is based on the harmonic components of the measurable voltage

and current at the metering point. The current is decomposed into active and

nonactive current based on the property that phase angle between active current

and voltage is zero (in-phase) and nonactive current and voltage is ±90 degrees (in-

quadrature). The inspiration behind the decomposition into active and nonactive

power also lay in study of the power terms of instantaneous total power. The concept

behind the decomposition is depicted in Figure 1.3.

v(t)

i(t)

Harmoniccomponents

Powercomponents

Activepower

p(t)

Non-activepower

q(t)

sum

sum

X

X

+Totalpower

s(t)

voltagecomponents

currentcomponents

DC or in-phasecomponent

quadraturecomponent

Decomposedcurrent

components

active powercomponents

non-activepower

components

Figure 1.3: Concept of the proposed definitions

Introduction

Chapter 1 7

The current for each harmonic is separated into components, in-phase and in quadrature

with the corresponding voltage harmonic. Active power is that contribution by the

current harmonic component in phase with the corresponding voltage harmonic and

includes cross-harmonic products of all voltages and in-phase current harmonic

components. Nonactive, likewise, is the contribution from the quadrature current

component. For non-sinusoidal systems, however, because of the presence of DC and

harmonics in addition to the fundamental, five sub-components for each of the active

and nonactive parts are defined. The active and nonactive components proposed exhibit

a meaning in the sense that they have a direct relationship with the source and load.

This is because the voltage and current harmonic components that are used to define the

powers are a function of the source and load which is dependent on the properties of the

source and load. Therein lies the basic concept of the proposed definitions.

In three-phase system there could be three or four conductors. To quantify the source

voltage, there is a need for a reference conductor. There is an option of a number of

possibilities (A-phase, B-phase, C-phase, neutral or virtual neutral for three-phase

systems) for this choice. Thus, for a particular choice of reference conductor, the

driving voltage (which is the EMF that is driving the current) behind the current flowing

into the load may not necessarily be the source voltage. Hence a load type connected to

the system may appear different when considered from the source voltage standpoint.

For example a resistor connected between two phases may not appear as a resistor when

viewed from phase to neutral viewpoint. Here the driving voltage is the phase-to-phase

voltage while the source voltage from the metering viewpoint may be the phase-to-

neutral voltage. The main intent of pointing this out is to highlight that there is a

difference between source voltage and driving voltage in a system with more than two

conductors or even with two conductors but where there is a generating element in the

load. A comprehensive treatment of this is given in Chapter 2 Section 2.2.8. However

the basic idea of the definition is the same as for the single-phase, that is, the use of an

equivalent load and decomposition of total current in relation to the source voltage still

applies in the three-phase system. Additional definitions, both time domain and

average, describing the three-phase system as a single system are also proposed. It is

interesting, in fact exciting, to highlight that the average powers obtained from the

Introduction

Chapter 1 8

proposed three-phase definitions match those obtained from the arithmetic powers using

the RMS method (see Chapter 6 for more information).

1.3 Justification for the research The IEEE Standard [37] in the introduction explicitly states the justifications for this

research. In the words of the standard “there is not yet available a generalised power

theory that can provide simultaneous common base for energy billing, evaluation of

electric energy quality, detection of major sources of waveform distortion as well as

provide information for design of mitigation equipment”. The problem investigated is

important because deficiencies in power definitions means

• inaccurate billing,

• incomplete compensation which means higher losses on transmission lines,

• non-optimal use of resources.

Additionally, aptly defined power definitions can

• aid in the detection of source of distortions thus making it possible to penalise

the polluters,

• introduce a direct power quality measure.

The proposed definitions are significant because the load model at the measuring point

is an attempt to closely represent the behaviour of the actual. This leads to good

knowledge of the time profile of active and nonactive components. With this

knowledge of the time profile of the powers accurate measurement of the powers,

especially nonactive, can be made. The instantaneous nonactive power defined can be

used for compensation. Accurate knowledge of the time profile of the instantaneous

nonactive power facilitates the reduction of the source current to the minimum possible.

Additionally the power components can be utilised to gauge power quality. These

components also provide possibility of identifying the direction of flow of active

components, enabling detection of source from which the polluting components (that

can cause distortion) arise. An insight into the source-load relationship can also be

derived from a study of the proposed powers’ components.

Introduction

Chapter 1 9

1.4 Methodology in writing this thesis The author of this thesis has taken the approach of the principle of “KISS”, which is to

keep the presentation of concepts as simple as possible at the expense of compactness.

Liberal use of examples with figures, both analytical and numerical, is relied upon to

clarify, explain or illustrate the concepts. It is known that reading and evaluating a

thesis is a challenging task, hence this approach is easy for the reader.

The process is to first present and discuss some current power definitions. This is

followed by an outline of the requirements on new definitions based mainly on the

recommendations from research conducted thus far. Bearing in mind the stipulation in

the requirements, the benchmark case studies as well as the criteria to evaluate a

definition are then outlined. The study cases are designed so that performance of the

definitions is tested for nonlinear, nonsinusoidal as well unbalanced conditions. Next

the proposed single-phase definitions will be outlined and the evaluated using the case

studies and criteria outlined. The concept of reference conductor is then addressed

followed by three-phase definitions. The three-phase definitions are then evaluated

using the stated case studies and criteria. The applications of the definitions are then

addressed with liberal use of examples. Next, the relationship of the proposed

definitions with existing definitions is outlined followed by concluding remarks.

Experimental work performed in testing the viability of the definitions is also included

in the related chapters. A section termed “background technical information” is

included in many chapters. This is a very important section where current

thought and understanding as well as main issues are discussed. Additional

examples, besides the case studies for evaluation, as necessary are included to aid

explanation and understanding of the important concepts. Software used in the analysis

is mainly Mathcad, ATP/ATP Draw and Mathematica. Mathcad (see

www.mathsoft.com) is a commercial mathematical software and ATP/ATP Draw (see

www.emtp.org) is royalty free licensed software that is regarded as one of the leading

time domain simulation software for power systems. Henceforth in this thesis the

ATP/ATP Draw simulation package will be referred to as ATP. Mathematica (see

www.wolfram.com) is a commercial mathematical software that has a powerful

symbolic computation engine. It is used in this thesis to perform symbolic

computations. Labview 7.1 (see www.ni.com/labview) as well as Texas Instruments

Introduction

Chapter 1 10

Code Composer (see http://focus.ti.com) were used to implement the definitions’

algorithm in an experimental setup.

1.5 Outline of the thesis Following this introduction, a brief overview of current definitions is presented

beginning with historical background to the electrical power system in Chapter 2.

Presentation and critical discussion of some current power definitions (RMS powers in

substantial depth and others briefly) are included in this chapter. In Chapter 3,

requirements on power definitions plus the benchmark case studies and criteria to gauge

the performance of the definitions are outlined. Chapter 4 presents and evaluates the

single-phase definitions. The benchmark case studies and criteria outlined in Chapter 2

are used to evaluate the proposed definitions. Experimental results to evaluate the

viability of the algorithm implementing the definitions are also presented in this chapter.

The concept of reference conductor is then presented in Chapter 5. This is followed by

the three-phase definitions in Chapter 6 which are also evaluated using the case studies

and criteria stated in Chapter 2. In Chapter 7 the applications of the proposed

definitions to measurement, compensation, detection of source of distortion and power

quality are shown with application examples presented. Chapter 8 reviews the

relationship of the proposed definitions with some current definitions and conclusions

follows in Chapter 9.

1.6 Definitions and nomenclature Nomenclature adopted by researchers is not uniform, so key and controversial terms are

now defined to establish the position taken in this thesis. In the sequel where the terms

“the proposed definition” or “the proposed definitions” are referred to, these usually

mean the new definitions proposed in this thesis.

Unless otherwise stated, only the terms “total” (product of voltage and current), “active”

and “nonactive”, will be used for powers when discussing the proposed definitions.

There is a preference to use nonactive as against reactive because reactive as used by

other researchers could define a different quantity. For example reactive in Budeanu’s

definitions is the product of quadrature current with the corresponding harmonic voltage

but excludes cross products. Going along the lines of sinusoidal system, active power is

Introduction

Chapter 1 11

taken as that resulting from the current in phase with its corresponding voltage

harmonic (including DC) and nonactive is that in quadrature with the corresponding

voltage harmonic. Both active and nonactive include cross-harmonic components. The

total, active and nonactive powers are further sub classified into components that

provide some information about the characteristics of the source load relationship. The

“driving voltage” is the EMF that is driving the current flowing into the load. The

“source voltage” is the voltage measured at the metering point for a particular choice of

reference conductor. The source voltage is measurable and easily known but the driving

voltage is usually difficult to determine unless complete details of the source side as

well as the load side are known.

In the thesis, voltages, currents and powers are in the time domain and will be referred

to as “instantaneous”. The letters “v” and “i” will be used to designate voltage and

current instantaneous values while capitals V and I are used for the corresponding RMS

values. Subscripts “p” and “q” (unless otherwise stated) will designate active or

nonactive current. The letters “s”, “p” and “q”, with subscripts as necessary, are used to

designate the total, instantaneous active and nonactive powers respectively. Capitals S

(apparent), P (active), N (nonactive) will be used to represent the corresponding

“average” values. Note that the IEEE standard 1459 [37] uses the terms p, pa and pq for

the s, p and q instantaneous quantities. In fact most researchers use p for instantaneous

total power. In this thesis, s, p and q are chosen for two reasons. Firstly, subscripts will

be used to define the various components and multiple subscripts could make

readability cumbersome. Secondly, the lower case letters selected for the time

quantities were picked in view of the universally used S, P and Q for “average”

apparent, active and reactive (nonactive) powers. Now to discuss briefly the use of

terms “average” and “instantaneous”. The term “average” power is used to indicate the

equivalent numerical value of a power waveform that gives an equivalent energy

transfer for an integral number of periods and applies to active, nonactive and total

powers.

1.7 Delimitation of scope and key assumptions The scope of this thesis is limited to the definitions of active and nonactive power, both

instantaneous as well as average, plus outlining the application possibility of these

Introduction

Chapter 1 12

definitions in the area of measurement, compensation, identification of source of

pollution and power quality. This application possibility is limited to identifying the

quantity or property that can be used for these purposes. No attempt is made to present

the actual practically utilised methods of compensation and/or definition of the indices

for power quality. However the application examples clearly illustrate that the

information provided by the definitions is suitable for use in such practical applications.

The definition of these indices can be a subject of future research.

The definitions are based on the following key assumptions.

1. The voltages and currents are periodic.

2. The DC factor as well as the phase angle between harmonic voltage (if it exists) and

the respective harmonic current is a measure of the load property (how

resistive/inductive or capacitive the load is) for that harmonic.

3. The phase angle is an important property and is an indication of not only the vector

location of the current with respect to its corresponding harmonic voltage vector but

also with respect to other harmonic voltage vectors.

4. If the harmonic voltage is zero, the phase angle of that harmonic current is

determined from the phase angle of the fundamental. In this manner this angle is

linked to the load property.

5. There is zero radiation of energy in the system.

1.8 Conclusion Now that there is a broad picture of the thesis, the details are delved into commencing

with a review of some current power theories.

Current power theories

Chapter 2 13

2. ANALYSIS OF CURRENT POWER THEORIES

Many researchers have exhaustively studied practically every definition of powers.

Hence it is not the intent of this thesis to make an exhaustive study into these

definitions. The approach will be limited to briefly mentioning the salient points of a

few with the RMS definition being analysed in substantial depth. This will be

reviewed commencing with some milestone events in history relating to measurement in

electricity, followed by the overview of some most important definitions.

2.1 History Thomas Edison (USA) developed the practical incandescent lamp by 1879. To

commercialise this he went on to invent many system elements (e.g. the parallel circuit,

a durable light bulb, an improved dynamo, the underground conductor network, the

devices for maintaining constant voltage, safety fuses and insulating materials, and light

sockets with on-off switches). He also had to develop a means by which to measure the

usage of electric lights [1]. He was a proponent of DC distribution system. In 1882 he

established the first commercial power station providing electricity power to customers

one square mile in area [70].

Charles Proteus Steinmetz (USA) in 1893 gave a lecture, hailed as a great

contribution to electrical engineering, describing the mathematics of alternating current

phenomena, not previously explained or grasped by earlier engineers. This enabled

engineers to move from designing by trial and error to designing using mathematics

[71]. Earlier in 1889 he had published his research in magnetic hysteresis. The work in

relation to definitions of voltage, currents and powers thus far was basically for DC and

sinusoidal conditions; AC power definitions being mainly formalised by Steinmetz [72,

73].

Lyon (1920) is said to be the first to state that power factor is the ratio of the actual

active power to the greatest possible power that can be absorbed by a load with the same

rms voltage and current. Lyon later also criticised Budeanu’s theory [74].


Chapter 2 14

Buchholz (1922) suggested expression for effective voltage and current and hence

equivalent effective apparent power for a three-phase system [74],

C Budeanu (1927)* major contribution concerned deformed or distortion power via his

monograph entitled “Puissances réactives et fictives” in 1927.

S Fryze (1932)* introduced the most general definition of the reactive power, based on

the concept of the load current splitting. The load current is decomposed into two

components, that is, active and reactive currents. The active current, defined by Fryze in

1932, is the smallest load current that is necessary [13] if the load at the supply voltage

has the active power and has the same waveform as the supply voltage.

M A Iliovici Goodhue (1933) explained the effective definitions by Buchholz. Other notable researches in the area of definitions of power with the major contribution

year stated in brackets are

• W Shepherd and P Zakikhani – Definition of reactive power (1972)*

• D Sharon – Reactive power definitions (1973)*

• N L Kuster’s and M J M Moore – Definition of reactive power (1980)*

• C H Page – Reactive power definition (1980)*

• G Nomoweisjki – Generalised theory of electrical power (1981)

• Akagi and Nabae - Original (1983) and modified (1994) p-q theory*

• L S Czarnecki – CPC Theory (1985/1988)*

• M D Slonim and J D Van Wyck – Definition of active, reactive and apparent powers

with clear physical interpretation (1988)

• J H Enslyn and J D Van Wyck – Load related time domain generalised definition

(1988)

• P S Fillipski– Elucidation of apparent power and power factor (1988, 1991)

• I Takahashi – Instantaneous Vectors (1988)

• M J Robinson and P H G Allen – Power factor and quadergy definitions (1989)

• T Furuhasi – Theory of instantaneous reactive power (1990)

• A Ferrero and G Superti-Furga – Powers using Parks transform (1991)*

• Willems – Instantaneous voltage and current vectors (1992, 1993)*


Chapter 2 15

• E H Watanabe – Generalised theory of instantaneous powers α-β-0 transformation (

(1993)

• M Depenbrock – FBD Method (1993, 2003)*

• A E Emanuel – Definitions of apparent power (1993, 1998)

• IEEE Working Group - Practical power definitions (1995)

• F Z Peng and J S Lai – Generalised instantaneous reactive power theory (1996)*

• D Sharon – Power factor definitions (1996)

• A Nabae and T Tanaka – Powers based on instantaneous space vector (1996)

• L M Dalgerti – Concepts based on instantaneous complex power approach (1996)

• H Akagi and K Hyosung – Instantaneous power theory based on mapping matrices

(1996)

• Nils and Marja – Vector space decomposition of reactive power (1997)

• F Ghassemi – Definition of apparent power based on modified voltage (1999, 2000)

• K Hyusong and H Akagi – Instantaneous p-q-r power theory (1999)

• J Cohen, F de Keon and K M Hernandez – Time domain representation of powers

(1999)

• F Z Peng and L M Tolbert – Definitions of nonactive power from compensation

standpoint (2000)

• Shin-Kuan Chen and G W Chang – Instantaneous power theory based on active

filter (2000)

• Shun Li Lu et al (2000)

• Zhang– Universal instantaneous power theory (single phase) (2000)

• IEEE Std 1459 – Definitions for the measurement of electric power quantities

(2000)*

• M T Haque – Single phases p-q theory (2002)

• H Lev-Ari and A M Stankovic – Reactive power definition via local Fourier

transform (2002).

• D Xianzhong, L Guohai and G Ralf – Generalised theory of instantaneous reactive

power for multiphase system (2004)

• J Seong-Jeub – Generalised power theory for transmission lines with unequal

resistances (2005)


Chapter 2 16

The most important works (indicated with *) in the above list will be analysed in the

sequel. Before embarking on the analysis useful technical background is introduced.

This background information is a summary of some basic concepts commonly used in

electrical engineering and lays the foundation for the analysis of the current theories

presented later in the chapter.

2.2 Background technical information 2.2.1 Powers in the time domain

2.2.1.1 Single-phase In the time domain, power is the product of voltage and current [72]. For single phase,

the voltage v(t) and current i(t) is represented by cosine Fourier components.

v(t) = V0 + v1 + vh + vg :=

0 1 1 h h g gh g

V 2 V cos( t ) 2 V cos(h t ) 2 V cos(g t )+ ω −α + ω −α + ω −α∑ ∑ (2.1)

i(t) = Io + i1 + ih + ig :=

0 1 1I 2 I cos( t )+ ω −β h h g gh g

2 I cos(h t ) 2 I cos(g t )+ ω −β + ω −β∑ ∑ (2.2)

where V0 and I0 is DC voltage and current, V1, Vh, Vg, I1, Ih and Ig are RMS values of

harmonic components v1, vh, vg, i1, ih and ig , ω is the angular frequency 2 fπ , f is

fundamental frequency, t is the time, xα and xβ (x = 1, h, g) the voltage and current

phase angle. "1" represents the fundamental and "h" represents the harmonics that are

source driven and "g" the harmonics that are not driven from the source [34, 37].

Total power is given by s(t) = v(t) i(t) [72]. Note that in [72] the definition of

instantaneous power was p ei= . In fact most researches use p(t) for instantaneous total

power. In this thesis “s” is used instead of “p” as explained in the introduction.

It is known that s(t) = v(t) i(t) (2.3) s(t) = 0 0 0 1 1V I 2 V I cos( t )+ ω −β + 0 m m

m h,g2 V I cos(m t )

=

ω −β∑

+ 1 0 12 V I cos( t )ω −α + m 0 mm h,g

2 V I cos(m t )=

ω −α∑


Chapter 2 17

+ 1 1 1 12 V I cos( t ) cos( t )ω −α ω −β + h h h hh

2 V I cos(h t ) cos(h t )ω −α ω −β∑

+ g g g gg

2 V I cos(g t ) cos(g t )ω −α ω −β∑

+ m n m nm nm 1,h,gn 1,h

2 V I cos(m t )cos(n t )≠==

ω −α ω −β∑

+ m n m nm gm 1,h,gn g


ω −α ω −β∑ . (2.4)

The above different terms are separately written as follows. This will aid in the

analyses that follow.

(a) s0 (t) := 0 0 0 1 1V I 2 V I cos( t )+ ω −β + 0 m mm h,g

2 V I cos(m t )=

ω −β∑

+ 1 0 12 V I cos( t )ω −α + m 0 mm h,g

2 V I cos(m t )=

ω −α∑ (2.5)

(b) s1 (t) := 1 1 1 12 V I cos( t ) cos( t )ω −α ω −β (2.6)

(c) sh (t) := h h h hh

2 V I cos(h t ) cos(h t )ω −α ω −β∑ (2.7)

(d) sg (t) := g g g gg

2 V I cos(g t ) cos(g t )ω −α ω −β∑ (2.8)

(e) sXh(t) := m n m nm nm 1,h,gn 1,h


ω −α ω −β∑ (2.9)

(f) sXg(t) := m n m nm gm 1,h,gn g


ω −α ω −β∑ (2.10)

2.2.1.2 Three-phase For three phase systems the equation (2.1) to (2.10) apply individually to each phase

with respect to the reference conductor. Reference [37] uses the neutral as reference

conductor for three-phase four-wire systems and virtual neutral for three-phase three-

wire systems (see Chapter 5 for a discussion on this). The sum of time domain powers

in each of the phases is useful in providing information about the balance or unbalance

nature of the three-phase system. Further discussion of this is taken up in Chapter 6.


Chapter 2 18

2.2.2 Discussion on cross-harmonic powers The general consensus, for example [37], is that cross harmonic powers, given in (e)

and (f), that is sXh(t) sXg(t), wholly contribute to nonactive power. It is shown below

that this is not necessarily the case.

Resistive load with nonsinusoidal source

Consider a resistive load, R, being supplied by a voltage source with 2 frequencies as

follows.

v(t) = 1 1 2 22 V cos( t) 2 V cos( t)ω + ω . (2.11)

The current flowing into the resistor is given by

i(t) = 1 21 2

V V2 cos( t) 2 cos( t)R R

ω + ω . (2.12)

Total power is then given by s(t) = v(t) i(t) that is

s(t) = 2 2

2 21 2 1 21 2 1 2

V V V V2 cos ( t) 2 cos ( t) 4 cos( t) cos( t)R R R

ω + ω + ω ω (2.13)

Since the current is active it can be rewritten using hha

V IR

= (where h = 1,2) as

s(t) = 2 21 1a 1 2 2a 22V I cos ( t) 2V I cos ( t)ω + ω

1 2a 1 2 2 1a 2 12V I cos( t)cos( t) 2V I cos( t)cos( t)+ ω ω + ω ω (2.14)

The current flowing in the circuit is completely absorbed by the resistor. It can thus be

concluded that p(t) = s(t), that is, the power flow is wholly active and nonactive power

q(t) is zero. The first two terms are the active power due to frequencies 1ω and 2ω , and

the last two terms are the cross product of terms of frequency 1ω and 2ω . Thus all the

terms in (2.13) including the cross harmonic power term 1 21 2

V V4 cos( t) cos( t)R

ω ω

contribute to active power. Note that terms where the oscillating part is of the nature

“cos(..)cos(..)” contribute to active power. This third term though having a net zero

value when integrated over a period, plays an important role in ensuring correct shape

of the active power waveform p(t) to match s(t). This scenario is illustrated in Figure

2.1.


Chapter 2 19

pω1 t( )

t

s t( )

t

The sum of the powerdue to ω1, pω1(t) anddue to ω2, pω2(t) andcross harmic power

pXω(t) gives p(t)which is equal to s(t)

+

pω2 t( )

t

pωX t( )

t

p t( )

t

+ =

Figure 2.1: Components of Active power However, based on the consensus that cross harmonic power contributes to nonactive

power, then the resulting waveform does not match that of s(t). This is reflected in

Figure 2.2.

pωX t( )

t

s t( )

t

The sum of the powerdue to ω1, pω1(t) and

due to ω2, pω2 (t) but notcross harmic power

pXω(t) gives p(t) which isnot equal to s(t)

pω1 t( )

t

pω2 t( )

t

p t( )

t

+ =

Figure 2.2: Components of Nonactive power The “average” value of the active power is however the same whether the cross

harmonic power is taken into consideration or not because its integral over a period is

zero. The “average” power is given by

P = 2 2

1 2V VR R

+ . (2.15)


Chapter 2 20

Inductive load with nonsinusoidal source

Next examine the behaviour of the cross-harmonic power in a purely inductive load.

Consider an inductive load, L, being supplied by a voltage source with 2 frequencies as

follows.

v(t) = 1 1 2 22 V cos( t) 2 V cos( t)ω + ω . (2.16)

The current flowing into the inductor is given by

i(t) = 1 21 2

1 2

V V2 cos( t ) 2 cos( t )L 2 L 2

π πω − + ω −

ω ω. (2.17)

Total power is given by s(t) = v(t) i(t), that is,

s(t) = 2 2

1 21 1 2 2

V V2 cos( t)sin( t) 2 cos( t) cos( t)R R

ω ω + ω ω

1 2 1 21 2 1 2

V V V V2 cos( t)sin( t) 2 sin( t) cos( t)R R

+ ω ω + ω ω (2.18)

The current flowing in the circuit is inductive. It can thus be said that p(t) = 0 and q(t) =

s(t) that is the power flow is wholly nonactive and active power is zero. The first term is

the nonactive power due to frequency 1ω , the second term 2ω and the third and fourth

terms the cross product of 1ω and 2ω . All the terms in (2.18) including the cross

harmonic power terms 1 21 2

V V2 cos( t)sin( t)R

ω ω and 1 21 2

V V2 sin( t) cos( t)R

ω ω

contribute to nonactive power. Note that terms with the oscillating part “sin(..)cos(..)

or cos(..)sin(..)” contribute to nonactive power. This scenario is illustrated in Figure

2.3.


Chapter 2 21

qω1 t( )

t

s t( )

t

The sum of the powerdue to ω1, qω1(t) anddue to ω2, qω2(t) andcross harmic power

qXω(t) gives q(t)which is equal to s(t)

+

qω2 t( )

t

qωX t( )

t

q t( )

t

+ =

Figure 2.3: Components of Nonactive power Thus it is shown that cross-harmonic power can contribute to both active and

nonactive power and is dependent on the load characteristics.

From study into cross harmonic terms it can be generally said that oscillating terms

(including cross terms) of the form “cos(..)cos(..)” or “sin(..)sin(..)” contribute to active

power and of the form “cos(..)sin(..)” or “sin(..)cos(..)” to nonactive power.

2.2.3 Does the instantaneous active current always have the same scaled waveshape as the voltage waveform? The current flowing in the resistive-inductive circuit, with resistance R and inductance

L, is given by the equation

i(t) = 1 h1 1 h h

1 h

V V2 cos( t ) 2 cos(h t )Z Z

ω −α −δ + ω −α −δ∑ . (2.19)

where 2 21 1Z R ( L)= + ω , 1 1

1Ltan

R− ω⎛ ⎞δ = ⎜ ⎟⎝ ⎠

, i = 1, h.

Using resolution of the fundamental and harmonic currents on harmonic basis and using

the property that for each harmonic the oscillating part of active current is in phase with

the voltage oscillating part and the nonactive in quadrature. The active current ip(t) and

nonactive current iq(t) are given respectively by

ip(t) = 1 h1 1 h h

1 hh

V V2 cos( ) cos( t ) 2 cos( )cos(h t )Z Z

δ ω −α + δ ω −α∑ , (2.20)


Chapter 2 22

iq(t) = 1 h1 1 h h

1 hh

V V2 sin( )sin( t ) 2 sin( )sin(h t )Z Z

δ ω −α + δ ω −α∑ . (2.21)

Apparently, the nonactive current iq(t), which is the oscillating part of the terms, is in

lagging quadrature relationship with the corresponding voltage term.

The observation is that the amplitude for the fundamental and harmonic active current

does not have the same ratio with respect to the amplitude of the corresponding terms in

the voltage v(t) waveform, that is,

1 2 h

1 1 1 2 2 2 h h h

2V 2V 2V...2V cos( ) Z 2V cos( ) Z 2V cos( ) Z

≠ ≠ ≠δ δ δ

. Thus the active current

waveform is not a scaled version of the voltage

waveform. It follows that the equivalent parallel

conductance G(t) pi (t)v(t)

= of the load is not

constant (see Figure 2.4).

The above discussion reveals an important fact

that for a series non-resistive load subject to non-sinusoidal source voltage, its

active current cannot be obtained by assuming that the parallel equivalent

conductance is constant. It is easy to show that for a non-resistive load Z = Rs + jXs

can be written equivalently in the parallel form Zeq = 1

p p

1 1R X

−⎛ ⎞

+⎜ ⎟⎜ ⎟⎝ ⎠

, where Rp =

s s

s

R XR+ and Xp = s s

s

R XX+ (see Appendix A for details). Because Xs= j Lω or 1

j Cω is

frequency dependent, so is Rp. Therefore, Rp is not a constant for all harmonic terms of

voltage. Note that if the assumption of constant equivalent parallel conductance based

on average power transmitted is made, the resulting nonactive current (difference

between load current and active current) information may give rise to non-optimal

compensation.

v t( )

ip t( )

iq t( )

G t( )

t Figure 2.4: Series R-L load


Chapter 2 23

2.2.4 Does the average active power in a three-phase system remain the same irrespective of the choice of reference conductor? This discussion is taken up by looking at the three-wire system and four-wire system

separately.

Three-wire system

With B as reference determine average powers as follows.

Phase A-B T

ab ab a0

1P v i dtT

= ∫

Phase C-B T

cb cb c0

1P v i dtT

= ∫

3-Phase T T

3PhBref ab a cb c0 0

1 1P v i dt v i dtT T

= +∫ ∫

With C as reference obtain average powers as follows.

Phase A-C T

ac ac a0

1P v i dtT

= ∫

Phase B-C T

bc bc b0

1P v i dtT

= ∫

3-Phase T T

3PhCref ac a bc b0 0

1 1P v i dt v i dtT T

= +∫ ∫

Knowing that c a bi (i i )= − + and bc ac bav v v= + , the following can be written

T T T T

3PhBref ab a cb c ab a cb a b0 0 0 0

1 1 1P v i dt v i dt v i dt v (i i )dtT T T

= + = − +∫ ∫ ∫ ∫

T T T T T T

ab a cb a bc b ab a ac ba a bc b0 0 0 0 0 0

1 1 1 1v i dt v i dt v i dt v i dt (v v ) i dt v i dtT T T T

= − + = + + +∫ ∫ ∫ ∫ ∫ ∫

T T T T

ab a ac a ba a bc b0 0 0 0

1 1 1v i dt v i dt v i dt v i dtT T T

= + + +∫ ∫ ∫ ∫

T T T T

ab a ac a ab a bc b0 0 0 0

1 1 1v i dt v i dt v i dt v i dtT T T

= + − +∫ ∫ ∫ ∫

T T

ac a bc b 3PhCref0 0

1 1v i dt v i dt PT T

= + =∫ ∫

Hence the average power with B as reference is identical to the average power with C as

reference. A similar result is obtained if A is chosen as the reference. So generally, it


Chapter 2 24

can be said that the 3-phase average active power remains the same irrespective of the

reference for 3-wire systems.

Four-wire systems

With B as reference average powers are

Phase A-B T

ab ab a0

1P v i dtT

= ∫

Phase C-B T

cb cb c0

1P v i dtT

= ∫

Phase N-B T

nb nb n0

1P v i dtT

= ∫

3-Phase T T T

3PhBref ab a cb c nb n0 0 0

1 1 1P v i dt v i dt v i dtT T T

= + +∫ ∫ ∫

With C as reference average powers are

Phase A-C T

ac ac a0

1P v i dtT

= ∫

Phase B-C T

bc bc b0

1P v i dtT

= ∫

Phase N-C T

nc nc n0

1P v i dtT

= ∫

3-Phase T T T

3PhCref ac a bc b nc n0 0 0


= + +∫ ∫ ∫

Using the relationship c a b ni (i i i )= − + + and bc nc nbv v v= − the equation above can be

written as T T T

3PhBref ab a cb c nb n0 0 0


= + +∫ ∫ ∫

= T T T

ab a bc a b n nb n0 0 0

1 1 1v i dt v (i i i )dt v i dtT T T

+ + + +∫ ∫ ∫

= T T T T T

ab a bc a bc b bc n nb n0 0 0 0 0

1 1 1 1 1v i dt v i dt v i dt v i dt v i dtT T T T T

+ + + +∫ ∫ ∫ ∫ ∫

= T T T T

ab a bc a bc b nc n0 0 0 0

1 1 1 1v i dt v i dt v i dt v i dtT T T T

+ + +∫ ∫ ∫ ∫

Since ab ac cbv v v= − ,


Chapter 2 25

T T T

ac a bc b nc n0 0 0

1 1 1v i dt v i dt v i dtT T T

+ +∫ ∫ ∫ = 3PhCrefP .

Again the average power with B as reference is identical to the average power with C as

reference. A similar result is obtained for other chosen references A or N. So

generally, it can be said that the 3-phase average active power for 3-wire systems

remains the same irrespective of the reference.

Hence it can be generally concluded that for three-phase systems the three-phase

average active power remains the same irrespective of the reference conductor

chosen. This means that the choice of reference is arbitrary for the computation of

average active power.

2.2.5 Is the three-phase instantaneous active power p3PH(t) obtained from the sum of instantaneous active power of each phase the same as the three-phase instantaneous total power s3PH(t) obtained from the sum of the instantaneous total power each phase? It is noted from the above analysis in 2.2.4 that the average three phase active power is

obtained from sum of the integral of the instantaneous total powers sx = vxREFix (x is the

phase and REF the reference) of each of the phases. Then, can it be said that the

instantaneous active power waveform p3PH(t) is obtained by the sum of the

instantaneous total power of the phases that is x xREF xx a,b,c x a,b,c

s v i= =

=∑ ∑ ? This question is

now explored.

A simple parallel R-L linear star connected load and a two-frequency source (for this

example the source voltage is the driving voltage) voltage is used in the analysis.

Consider the parallel load Ra and La in phase A, Rb and Lb in phase B and Rc and Lc in

phase C with voltages Van, Vbn and Vcn, respectively, and with n as the reference. The

voltages, with fundamental plus harmonics, are given below.

van(t) = 1 1 2 22 V cos( t) 2 V cos( t)ω + ω . (2.22)

vbn(t) = 1 1 2 22 22 V cos( t ) 2 V cos( t )3 3π π

ω − + ω − . (2.23)

vcn(t) = 1 1 2 22 22 V cos( t ) 2 V cos( t )3 3π π

ω + + ω + . (2.24)


Chapter 2 26

From the knowledge of the loads in each phase, the instantaneous active current is

iaACTIVE(t) = 1 21 2

a a

V V2 cos( t) 2 cos( t)R R

ω + ω , (2.25)

ibACTIVE(t) = 1 21 2

b b

V 2 V 22 cos( t ) 2 cos( t )R 3 R 3

π πω − + ω − , (2.26)

icACTIVE(t) = 1 21 2

c c

V 2 V 22 cos( t ) 2 cos( t )R 3 R 3

π πω + + ω + , (2.27)

and instantaneous nonactive current is

iaNONACTIVE(t) = 1 21 2

1 a 2 a

V V2 cos( t ) 2 cos( t )L 2 L 2

π πω − + ω −

ω ω, (2.28)

ibNONACTIVE(t) = 1 21 2

1 b 2 b

V 2 V 22 cos( t ) 2 cos( t )L 3 2 L 3 2

π π π πω − − + ω − −

ω ω, (2.29)

icNONACTIVE(t) = 1 21 2

1 c 2 c

V 2 V 22 cos( t ) 2 cos( t )L 3 2 L 3 2

π π π πω + − + ω + −

ω ω. (2.30)

The instantaneous phase current is then given by

ia(t) = iaACTIVE(t) + iaNONACTIVE(t), (2.31)

ib(t) = ibACTIVE(t) + ibNONACTIVE(t), (2.32)

ic(t) = icACTIVE(t) + icNONACTIVE(t). (2.33)

The three phase instantaneous total power is given by

s3PH(t) = van(t)ia(t) + vbn(t)ib(t) + vcn(t)ic(t)

= van(t)iaACTIVE(t) + vbn(t)ibACTIVE(t) + vcn(t)icACTIVE(t)

+ van(t)iaNONACTIVE(t) + vbn(t)ibNONACTIVE(t) + vcn(t)icNONACTIVE(t) (2.34)

and the three phase instantaneous active power is given by

p3PH(t) = van(t)iaACTIVE(t) + vbn(t)ibACTIVE (t) + vcn(t)icACTIVE (t). (2.35)

Is the waveform of (2.34) and (2.35) always the same? It will be the same on the

condition that

van(t)iaNONACTIVE(t) + vbn(t)ibNONACTIVE(t) + vcn(t)icNONACTIVE(t) = 0 (2.36)

and this will be the case if

• Nonactive current is zero, that is, the load current is in phase relationship with

the source voltage i.e. load is time invariant and purely resistive. This is valid

for both sinusoidal and non-sinusoidal source voltages.


Chapter 2 27

• The sum of the nonactive power in the three phases is zero that is condition in

(2.36). This occurs when the voltage is sinusoidal or symmetrical nonsinusoidal

(meaning that the nonsinusoidal waveform of each of the three phases is

identical but shifted in phase by 120 degrees) and the equivalent parallel

inductance (or capacitance) is time invariant and of equal in value in all the

phases.

Thus it can be concluded generally that

the waveform of three-phase

instantaneous total power s3PH(t) and

the waveform of three-phase

instantaneous active power p3PH(t) are

not necessarily identical except under

certain conditions. Note however that

the time average of s3PH(t) over a period

is equal to the period time average of the active power p3PH(t). This is because the

nonactive power is zero average over a period.

This is illustrated with a numerical example of six cases. A parallel R-L load (Figure

2.5) is used with a source voltage 115 Volts RMS fundamental for Cases 1 and 2. For

Cases 3 to 6 third harmonic of magnitude one-fifth of fundamental is added to the

fundamental. A parallel R-L load is used because the active part of the current that is

the current flowing in the resistor can be easily determined. Hence the expected active

power is easily calculated. The R-L load is varied as shown in the Table 2.1 below.

The computation is performed using Mathcad.

~

~

~~in

A

B

C

N

ai

ib

ic

L cL

cR

bL

bR

La

Ra

Figure 2.5: Three phase parallel R-L load


Chapter 2 28

Table 2.1: Data for cases Load R (Ohm) Load L (H) Case Source Voltage Ra Rb Rc La Lb Lc

Case 1 115 V Sinusoidal 10 10 10 ∝ ∝ ∝ Case 2 115 V Sinusoidal 10 8 10 0.1 0.1 0.1 Case 3 115 V Sinusoidal + 20% third 10 8 10 ∝ ∝ ∝ Case 4 115 V Sinusoidal + 20% third 10 10 10 0.1 0.1 0.1 Case 5 115 V Sinusoidal + 20% third 10 8 10 0.1 0.1 0.1 Case 6 115 V Sinusoidal + 20% third 10 10 10 0.1 0.05 0.1 The formulae are as follows.

Active current

iaR t( ) 2 V1⋅1

Ra⋅ sin ω1 t⋅( )⋅ 2 V2⋅

1Ra⋅ sin ω2 t⋅( )⋅+:= A phase

ibR t( ) 2 V1⋅1

Rb⋅ sin ω1 t⋅

2 π⋅

3−⎛⎜

⎝⎞⎟⎠

⋅ 2 V2⋅1

Rb⋅ sin ω2 t⋅

2 π⋅

3−⎛⎜

⎝⎞⎟⎠

⋅+:= B phase

icR t( ) 2 V1⋅1

Rc⋅ sin ω1 t⋅

2 π⋅

3+⎛⎜

⎝⎞⎟⎠

⋅ 2 V2⋅1

Rc⋅ sin ω2 t⋅

2 π⋅

3+⎛⎜

⎝⎞⎟⎠

⋅+:= C phase

Nonactive current

iaL t( ) 2 V1⋅1

ω1 La⋅⋅ sin ω1 t⋅

π

2−⎛⎜

⎝⎞⎟⎠

⋅ 2 V2⋅1

ω2 La⋅⋅ sin ω2 t⋅

π

2−⎛⎜

⎝⎞⎟⎠

⋅+:= A phase

ibL t( ) 2 V1⋅1

ω1 Lb⋅⋅ sin ω1 t⋅

5π6

+⎛⎜⎝

⎞⎟⎠

⋅ 2 V2⋅1

ω2 Lb⋅⋅ sin ω2 t⋅

5π6

+⎛⎜⎝

⎞⎟⎠

⋅+:= B phase

icL t( ) 2 V1⋅1

ω1 Lc⋅⋅ sin ω1 t⋅

π

6+⎛⎜

⎝⎞⎟⎠

⋅ 2 V2⋅1

ω2 Lc⋅⋅ sin ω2 t⋅

π

6+⎛⎜

⎝⎞⎟⎠

⋅+:= C phase

Total current

ia t( ) iaR t( ) iaL t( )+:= A phase

ib t( ) ibR t( ) ibL t( )+:= B phase

ic t( ) icR t( ) icL t( )+:= C phase

Three phase power

p3PH t( ) van t( ) iaR t( )⋅ vbn t( ) ibR t( )⋅+ vcn t( ) icR t( )⋅+:= Based on active current

Based on sum of phase apparerntpowerss3PH t( ) van t( ) ia t( )⋅ vbn t( ) ib t( )⋅+ vcn t( ) ic t( )⋅+:=

The results are presented in the graphs in Figure 2.6. Active powers and the voltages

and currents are shown in the graphs.


Chapter 2 29

0 0.005 0.01 0.015 0.02

2000

2000

4000

6000

8000

200

200

400

600

800

p3PH t( )

s3PH t( )

van t( )

vbn t( )

vcn t( )

5 ia t( )

5 ib t( )

5 ic t( )

t

Case 1

0 0.005 0.01 0.015 0.02

2000

2000

4000

6000

8000

200

200

400

600

800

p3PH t( )

s3PH t( )

van t( )

vbn t( )

vcn t( )

5 ia t( )

5 ib t( )

5 ic t( )

t Case 2

0 0.005 0.01 0.015 0.02

2000

2000

4000

6000

8000

200

200

400

600

800

p3PH t( )

s3PH t( )

van t( )

vbn t( )

vcn t( )

5 ia t( )

5 ib t( )

5 ic t( )

t

Case 3

0 0.005 0.01 0.015 0.02

2000

2000

4000

6000

8000

200

200

400

600

800

p3PH t( )

s3PH t( )

van t( )

vbn t( )

vcn t( )

5 ia t( )

5 ib t( )

5 ic t( )

t

Case 4

0 0.005 0.01 0.015 0.02

2000

2000

4000

6000

8000

200

200

400

600

800

p3PH t( )

s3PH t( )

van t( )

vbn t( )

vcn t( )

5 ia t( )

5 ib t( )

5 ic t( )

t

Case 5

0 0.005 0.01 0.015 0.02

2000

2000

4000

6000

8000

200

200

400

600

800

p3PH t( )

s3PH t( )

van t( )

vbn t( )

vcn t( )

5 ia t( )

5 ib t( )

5 ic t( )

t Case 6

Figure 2.6: Cases comparing p3PH(t) to s3PH(t) The deviation of active power waveform s3PH(t) calculated using the sum of phase

apparent powers from the expected active power p3PH(t) is apparent in Cases 4 to 6

when harmonic voltage is introduced and the load is unbalanced. A similar conclusion

can be drawn when the voltage is unbalanced.


Chapter 2 30

2.2.6 Does the current vector projected onto the voltage vector (as done in definitions based on instantaneous space vectors) in the presence of harmonics always give the active current? The discussion below is important reference for Subsection 2.3.11. This is easiest

illustrated with a numerical example. A parallel R-L load is used with a source voltage

26.87 Volts RMS fundamental plus second harmonic of magnitude one-third of

fundamental. A parallel R-L load is used because the active part of the current that is

the current flowing in the resistor can be easily determined. The R-L load is varied such

that fundamental S1 = V1RMSI1RMS is constant and the fundamental phase angle is varied

between 0 and 90 degrees. This simulates a load that changes from fully resistive to

fully inductive. The computation is performed using Mathcad. Formulae and

computation from Mathcad are presented below. The R and L values are as follows

R

5

5.3926737

7.0710678

13.3473358

8.1658894 1016×

⎛⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎠

= L

9.1189065 1014×

0.0424859

0.0225079

0.0171654

0.0159155

⎛⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎠

=

(2.37) Note large value of R or L is used to simulate open circuit. The voltage v(t), active iR(t)

and nonactive iL(t) currents are given by

v t( ) 2 V1 eω1 t⋅( ) j⋅

⋅ V2 eω2 t⋅ α2−( ) j⋅

⋅+⎡⎢⎣

⎤⎥⎦⋅:=

iR t( ) 2V1R

eω1 t⋅( ) j⋅

⋅V2R

eω2 t⋅ α2−( ) j⋅

⋅+⎡⎢⎣

⎤⎥⎦

⋅:=

iL t( ) 2V1

ω1 L⋅ j⋅eω1 t⋅( ) j⋅

⋅V2

ω2 L⋅ j⋅eω2 t⋅ α2−( ) j⋅

⋅+⎡⎢⎣

⎤⎥⎦

⋅:=

(2.38) and the current vector projected onto the voltage to give the active current is given by

i t( ) iR t( ) iL t( )+:= total current

α t( ) arg v t( )( ):= voltage angle

β t( ) arg i t( )( ):= current angle

ip t( ) i t( ) cos α t( ) β t( )−( )⋅ e α t( )( ) j⋅⋅⎡⎣ ⎤⎦

→⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

:= projected current. (2.39)


Chapter 2 31

The real part of the vectors v(t), iR(t) and ip(t) gives the time domain waveform of the

respective vector. This is plot for the different values of R and L in Figure 2.7.

L~

V(t)

LR~

i (t) i (t)i(t) LR

The circuit for the analysis

n 0:= Rn 5= Ln 9.1189065 1014×=

0 0.005 0.01 0.015 0.02

20

10

10

20

Re iR t( )n

⎛⎝

⎞⎠

Re ip t( )n

⎛⎝

⎞⎠

0.1 Re v t( )( )⋅

t

.

n 1:= Rn 5.3926737= Ln 0.0424859=

0 0.005 0.01 0.015 0.02

10

10

Re iR t( )n

⎛⎝

⎞⎠

Re ip t( )n

⎛⎝

⎞⎠

0.1 Re v t( )( )⋅

t

n 2:= Rn 7.0710678= Ln 0.0225079=

0 0.005 0.01 0.015 0.02

10

10

Re iR t( )n

⎛⎝

⎞⎠

Re ip t( )n

⎛⎝

⎞⎠

0.1 Re v t( )( )⋅

t

n 3:= Rn 13.3473358= Ln 0.0171654=

0 0.005 0.01 0.015 0.02

10

5

5

10

Re iR t( )n

⎛⎝

⎞⎠

Re ip t( )n

⎛⎝

⎞⎠

0.1 Re v t( )( )⋅

t

n 4:= Rn 8.1658894 1016×= Ln 0.0159155=

0 0.005 0.01 0.015 0.02

10

5

5

10

Re iR t( )n

⎛⎝

⎞⎠

Re ip t( )n

⎛⎝

⎞⎠

0.1 Re v t( )( )⋅

t

Figure 2.7: Comparing current vector projected onto the voltage with active current From the above analysis and the waveforms, it is observed that the projected current

vector ip(t) matches the active current iR(t) when inductance L is not included in the

load. The deviation becomes more apparent with the increase in the parallel inductance.


Chapter 2 32

Hence it can said that generally, that current vector projected onto the voltage

vector in presence of harmonics does not necessarily give the active current

flowing in the circuit. Therefore the instantaneous active power calculated using

this projected vector may not give the correct instantaneous active power.

2.2.7 Apparent power and line loss Apparent power S = VRMS IRMS obtained from RMS quantities, though reflecting

magnitude, is not an algebraic quantity and does not generally satisfy the energy

conservation principle as pointed out by many authors for example [13, 14, 54]. Some

researchers state that apparent power cannot be assigned any physical significance [14].

To give it a physical meaning it is linked to line loss via the equation

2LineLoss 2

RMS

rP SV

= (where r is the line per unit length resistance) [28]. Line loss is

dependent on the current flowing through the line. Taking “r” as resistance per unit

length and “vr(t)” the voltage drop across “r”, the average line loss per unit length is

given by

PLineLoss = t T

rt

1 v (t) i(t)dtT

+

∫ = t T

2

t

1 r i(t) dtT

+

∫ = r IRMS2 =

2

RMS

SrV

⎛ ⎞⎜ ⎟⎝ ⎠

which is equation stated above. Thus, it is by definition of RMS quantities, that is VRMS

IRMS = S, that the relationship given above applies. To the best knowledge of the author

of this thesis, it has not been pointed out by any researcher that apparent power SRMS is

not directly linked to line loss but through IRMS and the definition SRMS = VRMS IRMS. It

is noted, though, that it has been indirectly implied in [38] that the product VRMS IRMS is

not preferable to be taken as a definition for SRMS (it, SRMS, being the maximal active

power that can be transferred for a given voltage and current) from the point of view of

physical meaning. Additionally in [39] the author states that apparent power is not a

physical quantity but characterizes physical phenomena. Should the power loss

relationship then be used as an indicator of SRMS being a physical quantity especially

since it is not a direct relation but through a mathematical definition of apparent power?


Chapter 2 33

2.3 Power theories/definitions This section presents the commonly recognised power theories. A detailed analysis is

not performed except for the most commonly used RMS based definitions. Most of

these have been well discussed in the literature. The main intent is to present the

essential formulae with a brief critical analysis. These fall into

• frequency domain,

• time domain or

• both.

The main theories are based on

• orthogonal current decomposition

• instantaneous space vector

Note: Frequency domain is a term used to describe the analysis of mathematical

functions or signals with respect to frequency. Time domain is a term used to describe

the analysis of mathematical functions, or physical signals, with respect to time. A time

domain graph shows how a signal changes over time, whereas a frequency domain

graph shows how much of the signal lies within each given frequency band over a range

of frequencies. A frequency domain representation can also include information on the

phase shift that must be applied to each sinusoid in order to be able to recombine the

frequency components to recover the original time signal. The frequency domain

relates to the Fourier transform or Fourier series by decomposing a function into an

infinite or finite number of frequency components. This is based on the concept of

Fourier series that any periodic waveform can be expressed as a sum of sinusoids.

The mathematical notations used in the following are not necessarily those used by the

authors in the publications referred to. Subscripts may be used to identify the source of

the definition, for example for quantities related to RMS subscript “RMS” will be used

and for definitions by Budeanu subscript “B” will be used. This enables unique

referencing. Generally, instantaneous or time values are denoted by lower case, while

RMS or “average” values by upper case and this is maintained in this thesis.

It is humbly stated that any analysis is done in good faith. It is intended as a

means to better understand the definitions and not at all to discredit any

definition. Also, the analysis is performed with the view that all definitions


Chapter 2 34

produce results that are consistent for the purposes of that definition and the needs

at the point in time the definition was introduced, and are by no means superior or

inferior to other definitions.

2.3.1 RMS voltage and current based definitions An almost universally accepted definition of powers is the definition that is based on

instantaneous as well as RMS values of current and voltage as outlined in (2.40), (2.41)

and (2.42). This definition is also stated in Section 2.1.2.15 of [37] as well as [79]. In

the ensuing analysis the terms, both “average” and time domain, based on to RMS

values will have subscript “RMS, for example average total or apparent power is

denoted by SRMS. The subscript does not mean that the quantity is an RMS quantity but

that it is based on RMS voltage or current. Because of its universality, a detailed

analysis is performed to have an in depth understanding. This will aid to improving

power definitions. For ease of reference these will be termed RMS powers in this

theisis.

Active power is given by

PRMS = T

0

1 v(t) i(t)dtT ∫ , (2.40)

and apparent power by SRMS = VI, (2.41) where V and I are RMS values of the voltage and current. Using (2.40) and (2.41) the nonactive power is defined as

2 2RMS RMS RMSN S P= − (2.42)

This definition can be “loosely” considered to be based on time (2.40) and frequency

(2.41) and (2.42) domain quantities. equations (2.41) and (2.42) can be considered

frequency domain because RMS voltage and current can be obtained from the square

root of the sum of squares of the magnitudes of harmonic components that make up the

voltage or current wave.


Chapter 2 35

It however does not directly use orthogonal current decomposition in the computation

of power, but it can, considering equation (2.42), sort of be classed to be based on

orthogonal decomposition. Average values of powers are provided by the definitions.

2.3.1.2 RMS based powers and relationship between “average” and time quantities RMS powers are an “average” value of instantaneous powers. In the electrical system

voltages and currents are time varying. Hence powers are also time varying (mainly

periodic). The RMS value is a single numerical representation of this varying quantity.

It is thus useful, to relate the RMS based value to its time varying value. This will

enable obtaining energy information from this quantity since power is the rate of energy

transfer [37], be it unidirectional or zero average. It is noted that in defining powers

using RMS quantities some information about the waveform, for example harmonic

content, is lost at the expense of a simple single valued representation of the power

waveform. However this “average” value is a useful measure of the time varying

quantity and renders easy comprehension and reference to this quantity. The analysis to

relate RMS based value to an equivalent time domain wave and the relationship with

energy transfer follows.

First elucidate what RMS based power means starting from a waveform. This is

graphically shown in Figure 2.8 for an arbitrary non-sinusoidal voltage v(t) stating in

Figure 2.8(a) to the RMS value in Figure 2.8(d).


Chapter 2 36

(a) voltage waveform

v t( )

t

v t( )

(b) SQUARE of voltage waveform

vsq t( )

t

vsq t( ) v t( )2:=

vsqAV1T

0

Ttvsq t( )( )⌠

⎮⌡

d⋅:= VRMS1T

0

Ttvsq t( )( )⌠

⎮⌡

d⋅:=

(c) MEAN of SQUARE of voltage waveform

vsqAV

vsq t( )

t

(d) ROOT MEAN SQUARE of voltage waveform

VRMS

v t( )

t

Figure 2.8: Determination of the RMS value of the voltage waveform (blue line) Figure 2.8(d) shows the RMS voltage (blue) and the voltage (pink dashed) on the same

graph. The harmonic content of the waveform v(t) is lost in the RMS quantity.

Next the relationship between apparent power, equivalent sinusoidal wave of the

apparent power, instantaneous total power, voltage and current is shown in Figure 2.9.

The product of the RMS voltage VRMS Figure 2.9(a) and RMS current IRMS Figure

2.9(b) gives the apparent power SRMS Figure 2.9(c). In the electrical power system,

power flow is a varying quantity so it is useful to represent apparent power SRMS with

sinusoidal wave (because originally RMS definition was defined for sinusoidal

waveforms). This can be achieved by an equivalent sinusoidal wave sRMS(t) that has the

same energy transfer as SRMS. The sinusoidal equivalent wave will then have its

amplitude equal to SRMS. Note that SRMS and its equivalent sinusoidal wave sRMS(t) have

lost the harmonic information as well as the information about the negative going part

of the instantaneous total power s(t) as reflected in Figure 2.9(d)


Chapter 2 37

(a) voltage v(t) and RMS voltage

v t( )

VRMS

t

(b) current i(t) and RMS current

i t( )

IRMS

t

X

(c) Apparent Power S = VI

SRMS

10 v t( )⋅

10 i t( )⋅

t

(d) Apparent power and equivalent sinusoidal power

SRMS

sRMS t( )

s t( )

t

= =>

Figure 2.9: Product of VRMS and IRMS gives apparent power SRMS (blue line) with equivalent sinusoidal of apparent power sRMS(t) and instantaneous total power s(t) Proceed with further analysis, now for active and nonactive powers. Consider a load,

supplied by a voltage source with 2 frequencies as follows.

v(t) = 1 1 2 22 V cos( t) 2 V cos( t)ω + ω . (2.43)

and the current by

i(t) = 1 1 1 2 2 22I cos( t ) 2 I cos( t )ω −θ + ω −θ . (2.44)

First consider sinusoidal case whence V2 and I2 are both zero.

Write total power s(t) as follows.

s(t) = 21 1 1 12V I cos( )cos ( t)θ ω 1 1 1 1 12 V I sin( ) cos( t) sin( t)+ θ ω ω . (2.45)

The first term gives the active power, hence write

pRMS(t) = 2RMS 12P cos ( t)ω (2.46)

where PRMS = 1 1 1V I cos( )θ , the amplitude of pRMS(t) is the “average” power

and the second term in (2.45) is nonactive power,


Chapter 2 38

qRMS(t) = s(t) - pRMS(t) = RMS 1 12 N cos( t)sin( t)ω ω (2.47)

with NRMS = 1 1 1V I sin( )θ and NRMS is also the amplitude of qRMS(t) and is taken as the

“average” power.

Note that according to [37] PRMS is the “average” value of the first term

2RMS 12P cos ( t)ω . Further it states that NRMS is the “amplitude of the oscillating power”

RMS 1 12 N cos( t)sin( t)ω ω . In this thesis, PRMS is taken as being the amplitude of the

waveform mainly so that both PRMS and NRMS are the amplitude of their corresponding

waves. This is done to keep the same relationship of the quantity to its waveform for

both P and N. However, note that the “average” value for P as per [37] and the

amplitude as used in this thesis give identical results.

From (2.46) and (2.47), the instantaneous total power sRMS(t) can be writen as

sRMS(t) = pRMS(t) + qRMS(t). (2.48)

For the sinusoidal system, the

amplitude of sRMS(t) is equal to

SRMS. This is reflected in

Figure 2.10 (Note that in the

figures the origin is shifted to

ease showing the amplitudes)

which shows how the

“average” value is related to the

time instantaneous waveform.

It can then be stated that for

the sinusoidal system the “average” powers are the amplitudes of the respective

instantaneous powers.

Having established the relationship of RMS based power and it’s equivalent waveform,

the non-sinusoidal case is explored.

In nonsinusoidal case, the voltages and currents are nonsinusoidal and hence the

powers’ waveforms are also nonsinusoidal. However the RMS value of the voltages

and current is single valued and does not retain the harmonic information. A solution is

sRMS t( )

pRMS t( )

qRMS t( )

t Figure 2.10: Average and Instantaneous Powers

T 0.5T

2PRMS

2NRMS


Chapter 2 39

to “mimic” the sinusoidal

system. Represent the

nonsinusoidal active and

nonactive power by an

“equivalent” sinusoidal

waveform with the amplitude

that is equal to the RMS based

power as has been shown in the

above analysis. This

“equivalent” sinusoidal

waveform has the same absolute (by absolute it is meant that the sign of the negative

going area is ignored) area under the waveform as that for the non-sinusoidal waveform.

For example, as shown in Figure 2.11, the waveform of active power p(t) (blue

continuous line) is represented by the sinusoidal waveform of active power pRMS(t)

(blue dashed line) which has an amplitude PRMS. The same can be said for NRMS. This

is reflected in Figure 2.11 (green and green dashed lines). In this manner the

nonsinusoidal power waveform is related to the “average” value, as in the sinusoidal

case, that is, the “average” value is the amplitude of the corresponding equivalent

sinusoidal waveform. SRMS can be determined using equation (2.42).

The relationship between the “average” powers based on RMS quantities and the

equivalent instantaneous waveforms using an equivalent sinusoidal waveform has been

shown. Hence generally, the nonsinusoidal wave is represented by an “average”

value that is equal to the amplitude of an equivalent sinusoidal. This applies to

active and nonactive power.

A simple example as in Figure 2.12 is used to test these relationships. A source voltage

v(t), of 100 Volts rms (50 Hz) with 33.33% 3rd harmonic was used in the computation

of this example. The pure load resistance was 15 ohm and the inductance 25 mH. The

average values are calculated as follows.

s t( )

p t( )

q t( )

sRMS t( )

pRMS t( )

qRMS t( )

t

Figure 2.11: Average and Instantaneous Powers (nonsinusoidal case)

T0.5T

2PRMS

2NRMS


Chapter 2 40

~v(t)

R~

i(t)

Lmetering

point

Figure 2.12: Parallel R-L load

Average Values

Computing “average” values using (2.40), (2.41) and

(2.42) obtain

PRMS = 23.02 W,

SRMS = 24.04 VA,

NRMS = 6.93 Var.

Results for energy transfer

Active energy EpRMS and absolute nonactive energy EqRMS for the RMS based active and

nonactive power calculated using the equivalent sinusoidal waveform of the RMS

average value are as follows.

EpRMS0

TtpRMS t( )

⌠⎮⌡

d:= EpRMS 14.814815= active energy

and nonactive energy is

π

EqRMS0

TtqRMS t( )

⌠⎮⌡

d⎛⎜⎜⎝

⎞⎟⎟⎠

:= EqRMS 17.1934646= nonactive energy

The calculation using the actual active and nonactive power waveform gives the

following.

Ep0

Ttp t( )

⌠⎮⌡

d:= Ep 14.8148148= active energy

and nonactive energy is

Eq0

Ttq t( )

⌠⎮⌡

d⎛⎜⎜⎝

⎞⎟⎟⎠

:= Eq 20.014061= nonactive energy

Discussion of results

It is apparent from the results above that for active power the RMS based method gave

the correct energy information. For nonactive power, however, the RMS based

definition does not give the correct nonactive energy information. This is true for non-

sinusoidal systems irrespective of whether the load is linear or nonlinear. Thus the

RMS based active power definition correctly captures the energy information


Chapter 2 41

while for nonactive power this is not the case. Reference [75] corroborates this by

stating the following: “The apparent power is the product of the root-mean-square (rms)

magnitudes of voltage and current. When these quantities are used in calculation with

non-sinusoidal voltage and/or current, the results may not be correct unless careful

attention is paid to the fundamental definitions of the quantities”.

Explanation of the differences

Now explore why this discrepancy exists and commence with identifying a few

important points as follows:

2.1 RMS or root mean square as stated in [70] is “The effective value, or the value

associated with joule heating, of a periodic electromagnetic wave. The rms

value is obtained by taking the square root of the mean of the squared value of a

function”. Thus this definition is based on resistive load and this effective value

is the value of the equivalent direct current that would produce the same power

dissipation in a given resistor.

2.2 The characteristic of energy change is not the same for active and nonactive

power (this is explored below).

2.3 Definition of active power is

based directly on instantaneous

voltage v(t) and current i(t) as in

(2.40), while that for nonactive

power equation (2.42) is not.

Point 2.1 gives the cue to an explanation

of the discrepancy. Both the RMS

current and voltage are effective values

that are related to heating effects where

the energy is wholly absorbed by the

load (resistor). SRMS is determined from RMS values and hence it implies that its

energy is fully absorbed by the load. Based on that SRMS is fully absorbed, the

waveform of S(t) (use capital letter S(t) to differentiate it from sRMS(t)) would be given

by

S(t) = 2RMS 12S cos ( t )ω −θ (2.49)

sRMS t( )

vs t( )

10 is t( )

t Figure 2.13: Instantaneous Powers based on effective value associated with heating


Chapter 2 42

and reflected in Figure 2.13.

Similar to (2.49), the “average” values PRMS and NRMS can be related to the

corresponding oscillating time domain waveforms P(t) and N(t) as follows.

S(t) = 2RMS 12S cos ( t )ω −θ (2.50)

P(t) = 2RMS 12P cos ( t)ω (2.51)

N(t) = 2RMS 12 N sin ( t)ω (2.52)

and the corresponding waveforms are given in Figure 2.14.

S t( )

v t( )

i t( )

t

P t( )

v t( )

iR t( )

t

N t( )

v t( )

iL t( )

t Figure 2.14: Wavefroms of S(t), P(t) and N(t) based on fully absorbed power Since SRMS, PRMS, NRMS are also valid for nonsinusoidal situations, equations (2.50),

(2.51) and (2.52) apply to both sinusoidal and nonsinusoidal conditions. For the

purposes of discussion the method of representing the RMS power with equivalent

waveforms is termed “the shifted power waveform” method. Note that for nonactive

average power, this is the status quo where the “reactive” power is determined by

shifting the voltage waveform by a quarter of the fundamental period and then

computing the integral of the product with the current waveform, which as stated in [76]

was a recommendation in the IEC standard 145 (1963). Reference [17] also has a

similar approach in the definition of powers.

The average value of the waveform is the amplitude which is SRMS, PRMS and NRMS

respectively for S(t), P(t) and N(t). The relationship between energy transfer and

amplitude is given by the relationship

EXRMS = XRMST, where X is S, P or N. (2.53)

Thus, “the shifted power waveform” method satisfies equation (2.42) as well as energy

relationship with equation (2.53).


Chapter 2 43

Having related the RMS powers to the corresponding powers waveforms now observe

what meaning can be derived from these results?

• The RMS powers are equivalent to the average of the fundamental sinusoidal

equivalent of the waveform (voltage and current) considering the RMS voltage is in

phase relationship with the RMS current. This applies equally to sinusoidal as well

as non-sinusoidal conditions.

• The VA power SRMS is the maximum (VA) power/energy that can be consumed

by the particular voltage and current. This happens if that current flows in a

resistive load (since voltage is taken in phase with the current in the RMS

computation).

• The reactive power QRMS (termed N in the IEEE Standard) is the maximum

(nonactive) power/energy that can be consumed by the particular voltage and

current. Again this happens if that current flows in a resistive load (since voltage is

taken in phase with the current in the RMS computation).

• The third and this dot-point indicate that the RMS powers are a measure of

capacity requirement from the source for the particular voltage/current/load case.

It need not necessarily represent the actual energy/power taken by the load as seen

below in the expected waveforms.

It is noted that, though this (shifted power waveform) satisfies the relationship (2.42), it

does not truly represent the status of instantaneous total power sRMS(t) or s(t) which is

the product of voltage and current as reflected in Figures 2.10 (red waveform) and 2.11

(red dashed waveform). This is because there should be negative going portion of the

wave in the presence of nonactive power. This will be illustrated with the example of

Figure 2.12. Figure 2.15 shows the power waveforms with zero third harmonic while

Figure 2.16 the waveform with third harmonic included.


Chapter 2 44

0 0.005 0.01 0.015 0.02

2000

1000

1000

2000

3000

s t( )

p t( )

q t( )

t

Figure 2.15: Sinusoidal source voltage showing powers s(t), p(t) and q(t)

0 0.005 0.01 0.015 0.02

2000

1000

1000

2000

3000

s t( )

p t( )

q t( )

sRMS t( )

pRMS t( )

qRMS t( )

t Figure 2.16: Non-sinusoidal source voltage showing powers s(t), p(t) and q(t) plus equivalent RMS based powers sRMS(t), pRMS(t) and qRMS(t)

Comparing Figures 2.14, 2.15 and 2.16, it is observed that the actual average

power/energy information in relation to the load is not reflected in RMS powers based

on “shifted power waveform” with the exception of active power. The negative going

part of the total and nonactive power is lost in the “shifted power waveform”.

Point 2.1 also points to another important fact. Active power PRMS is determined

directly from time quantities while nonactive power NRMS is determined from a

processed quantity (apparent power) which has lost some information of the waveform.

The determination of PRMS is related to energy consumed by the actual load but

nonactive power NRMS is related to energy consumed by fully energy absorbing load

which may not be representative of actual energy transfer in the load. The definition of

“average” nonactive power NRMS should be based on energy transfer, as is the definition

of PRMS. The property of the instantaneous nonactive power that is the positive going

part is equal to the negative going part can be used as a measure of energy transfer.

This can be used to define “average” nonactive power and thus maintain energy

information related to the actual load.

The above discussion explains why the apparent SRMS, reactive QRMS and nonactive

NRMS powers obtained from RMS quantities, though reflecting magnitude, do not

generally satisfy the energy conservation principle as pointed out by many authors, for

example [13, 14, 54].


Chapter 2 45

The conclusion is that the amplitude of the nonactive power wave does not have a direct

constant relationship with the energy transfer if RMS currents are used to determine the

powers. This is the reason for the difference shown in Tables 2.1 and 2.2. Thus, as far

as correct energy information is concerned, NRMS does not truly represent the nonactive

power taken by the load. Thus whether N should be defined based on correct energy

information or the present definition as per equation (2.42) needs serious consideration.

It is pointed out that the latter means that energy information will be based on the

“shifted power waveform” which as stated does not truly reflect the correct energy

information.

The author of this thesis is inclined toward the former, that is, to determine the

nonactive power N directly from the time quantities of voltage and current. It should be

related to the correct energy transfer between the source and load like that done for

active power P.

This above discussion however does not discount the importance of SRMS as is outlined

in the following section.

Importance of RMS based total “average” (apparent) power

The RMS based “average” total power SRMS is a measure of the capacity requirement

[28, 31, 37, 39, 77] for a particular source voltage and current supply combination, to

the load from the source of supply. Additionally it, together with the active power by

means of the power factor, is a measure of utilization of the source capacity.

Now proceed to look at some other power theories or definitions. As stated earlier only

a simple treatment is attempted with presentation of the main formulae or concept

behind the theories/definitions.

2.3.2 Definition proposed by C. Budeanu (1927) Budeanu’s definitions, widely discussed in literature for example [7, 14, 26, 40, 72, 78],

are based on the sinusoidal definitions of power given by the following.

Active power is

1 1P V I cos( )= φ , (2.54)


Chapter 2 46

where φ is the angle between the voltage and current (also termed phase angle) and

subscript 1 represents the fundamental component.

Reactive power is

1 1Q V I sin( )= φ . (2.55)

The apparent power is given by

2 2S P Q= − . (2.56)

Budeanu extended the sinusoidal definitions, above using Fourier series decomposition

for the voltage and current, for nonsinusoidal situations. In words Budeanu’s

definitions are that active power is given by the sum of the product of each harmonic

voltage including the fundamental and the corresponding in phase current. Reactive

power is, in a similar manner, the product of the voltage and quadrature current. Finally

distortion power satisfies the orthogonal relationship with active and reactive power to

give apparent power. The definitions for nonsinusoidal situations are as follows.

Active power is

m m mm

P V I cos( )= φ∑ , (2.57)

Reactive power

B m m mm

Q V I sin( )= φ∑ , (2.58)

where m represents the fundamental and harmonics for which both voltage and current

exist. In fact these are “average” powers.

Since 22BS P Q≠ − , Budeanu further defined an additional power

22 2B BD S P Q= − − , (2.59)

which was termed distortion power because it is caused by distorted voltages and

currents.

Discussion about Budeanu’s definitions

Next analyse the above definitions in light of the powers in the time domain given in

Subsection 2.2.1.


Chapter 2 47

As shown below, equations (2.57) and (2.58) are the “average” active and nonactive

powers of the time domain fundamental and harmonic powers given in (2.6) and (2.7).

s1(t) + sh (t) = 1 1 1 12 V I cos( t ) cos( t )ω −α ω −β + h h h h

h 1

2 V I cos(h t ) cos(h t )>

ω −α ω −β∑

With A1 = 1tω −α , Ah = hh tω −α , 1δ = 1 1α −β (fundamental phase angle) and hδ =

h hα −β (harmonic phase angle) the above becomes

s1(t) + sh (t) = 1 1 1 1 12 V I cos(A ) cos(A )− δ + h h h h hh 1

2 V I cos(A ) cos(A )>

− δ∑

s1(t) + sh (t) = [ ] [ ]1 1 1 1 h h h hh 1

V I cos( ) 1 cos(2A ) V I cos( ) 1 cos(2A )>

δ + + δ +∑

+ [ ] [ ]1 1 1 1 h h h hh 1

V I sin( ) sin(2A ) V I sin( ) sin(2A )>

δ + δ∑ (2.60)

The first two terms of (2.60) [ ] [ ]1 1 1 1 h h h h

h 1

V I cos( ) 1 cos(2A ) V I cos( ) 1 cos(2A )>

δ + + δ +∑

make up the active power and the second two terms

[ ] [ ]1 1 1 1 h h h hh 1

V I sin( ) sin(2A ) V I sin( ) sin(2A )>

δ + δ∑

the nonactive (reactive) power. The “average” value, 1 1 1V I cos( )δ + h h hh 1

V I cos( )>

δ∑ of

the first two terms is Budeanu’s definition for active power as given in (2.54). The sum

of the amplitudes of sinusoidal fundamental and harmonic nonactive power waves,

1 1 1 h h hh 1

V I sin( ) V I sin( )>

δ + δ∑ , is described in Budeanu’s equation (2.58) for reactive

power.

The distortion power DB hence represents the contribution from the remaining terms

given in (2.5), (2.9) and (2.10) which represent DC based and cross harmonic powers.

The distortion power DB represents the “average” value of the DC based and cross

harmonic powers. Note that in the presence of source impedance, the load generated

currents in Budeanu’s definition may be included in the active and quadrature reactive

power instead of distortion power.

Other researchers’ comments on Budeanu’s definitions


Chapter 2 48

Budeanu’s definitions do not possess the attributes related to power phenomena in

circuits [7, 10, 14, 40] and also the distortion power does not give information about

waveform distortion [7]. Reference [14] indicates that confusion over reactive power

dates back to the definitions by Budeanu. Despite these comments by the researchers,

Budeanu’s ideas are important and have been included in the IEEE Standard 1459-2000

[37].

2.3.3 Definition proposed by S. Fryze (1932) Fryze’s definitions are also widely discussed in the literature for example [6, 18, 20, 31,

78, 80, 81]. The definitions are time domain based and split the current into orthogonal

active and nonactive components. In words, Fryze decomposes the current into a

component ia(t) which is a scaled value of v(t) such that the active power given by ia(t)

is maintained with the remaining current ib(t) termed “reactive” current. This is outlined

in the following equations.

Active current

a 2rms

Pi (t) v(t)V

= , (2.61)

where P = T

1 v(t) i(t)dtT ∫ and T is equal to one period

Nonactive current

b ai (t) i(t) i (t)= − . (2.62)

Based on the above, the reactive power is given by

F rms bQ V I rms= , (2.63)

where Ib is the rms value of ib(t).

The apparent power is given by

22FS P Q= − (2.64)

Discussion about Fryze’s definitions

Next analyse Fryze’s definitions in light of the powers in the time domain given in the

Section 2.2.1 above.


Chapter 2 49

equation (2.61) is the active current and is given by extracting the currents in phase with

the voltage from equation (2.2) as follows.

Define G = rms

PV

(2.65)

and use the subscript F to flag Fryze’s definitions. Then (2.61) can be written as

iaF(t) = I0F + i1aF + ihaF + igaF := 0 1 1V G 2 V G cos( t )+ ω −α

h h g gh g

2 V G cos(h t ) 2 V G cos(g t )+ ω −α + ω −α∑ ∑ . (2.66)

This implies that the active current is

the current driven by all the voltage

components through a constant

conductance G in the load, based on the

assumption that the load has an

equivalent constant conductance within

a period. The balance of the current is

driven through a susceptance B(t) which

may be time varying. The equivalent circuit is as shown in Figure 2.17. Of particular

interest is the vg(t) and ig(t). The voltage vg(t) will result from voltage drop in the

source impedance. Since vg(t) is not driven by the source, the component in ia(t) as a

result of vg(t), that is g g2 V G cos(g t )ω −α∑ , is a portion of the load generated

current depending on the magnitude of vg(t). However, since ig(t) is load generated it

should be removed from the source current during compensation. Based on Fryze’s

definition, however, not the whole portion of the load-generated current will be

removed since the part given by g g2 V G cos(g t )ω −α∑ is not separately identified

according to the definition. It is included in the active current. Additionally, the

equivalent load conductance for series circuits, as outlined in Subsection 2.2.3 above,

may not be a constant within a period. Thus the active current obtained using this

definition may not truly represent the actual active current flowing in the load. A

detailed analysis on the Fryze’s definition in comparison with the new proposed

definitions using compensation is included in Appendix B. The information in Chapter

4 is useful as a precursor to reading this appendix.

~V(t)

B(t)G~

i(t) i (t)a i (t)b

Figure 2.17: Fryze’s load model


Chapter 2 50

Other researchers’ comments on Fryze’s definitions

Reference [6] states that QF can be easily measured but it is not related directly to the

load properties and parameters. Fryze definitions are stated to originate from “mere

mathematical consideration” [13], implying they are not related to the load properties.

Reference [14] mentions that Fryze’s definition does not obey conservation. However,

despite the above stated issues Reference [57] states “most of the existing nonactive

power theories and definitions based on time-domain can be extended and deduced”

from the definition by Fryze.

2.3.4 Definition proposed by W. Shepherd and P Zakikhani (1972) Shepherd’s and Zakhikhani’s definitions [79] are also well discussed in the literature,

for example [6, 78, 80, 81, 82]. The definitions are based on frequency domain analysis

and separate the apparent power into three components, that is, active apparent power

SR, reactive apparent power SX and distortion apparent power SD powers. These three

quantities result from corresponding active IR, reactive IX and distortion ID currents and

satisfy the orthogonal relationship to give total apparent power S. The currents are

RMS quantities. This is outlined in the following equations.

S2 = SR

2 + SX2 + SD

2 (2.67) V = VN + VP (2.68)

I = IN + IM (2.69) The apparent powers SR

2, SX2 and SD

2 are give by SR

2 = VN2 INR

2 = [ ]22n n n

n nV I cos( )θ∑ ∑ (2.70)

SX2 = VP

2 INX2 = [ ]22

n n nn n

V I sin( )θ∑ ∑ (2.71)

SD2 = VN

2 IM2 + VP

2 (IN2+ IM

2) = 2 2 2 2 2n m p n m

n m p n m

V I V I I⎛ ⎞+ +⎜ ⎟⎝ ⎠

∑ ∑ ∑ ∑ ∑ (2.72)

where VN = 2

nn

V∑ , VP = 2p

p

V∑ , IN = 2n

n

I∑ , IM = 2m

m

I∑ , (2.73)

and “n” means common harmonics (includes fundamental) for which both voltage and

current exist, “p” means harmonics for which corresponding current is zero, “m” means


Chapter 2 51

harmonics for which voltage is zero. The current IN can be further decomposed to in-

phase INR and quadrature INX components as follows

INR = [ ]2n n

n

I cos( )θ∑ , (2.74)

INX = [ ]2n n

n

I sin( )θ∑ . (2.75)

INX = [ ]2n n

n

I sin( )θ∑ . (2.76)

The apparent powers SR

2, SX2 and SD

2 are give by SR

2 = VN2 INR

2 = [ ]22n n n

n nV I cos( )θ∑ ∑ (2.77)

SX2 = VP

2 INX2 = [ ]22

n n nn n

V I sin( )θ∑ ∑ (2.78)

SD2 = VN

2 IM2 + VP

2 (IN2+ IM

2) = 2 2 2 2 2n m p n m

n m p n m

V I V I I⎛ ⎞+ +⎜ ⎟⎝ ⎠

∑ ∑ ∑ ∑ ∑ (2.79)

Note that all V and I are RMS quantities. Discussion about Shepherd and Zakhikhani’s Definitions

Next analyse the Shepherd and Zakhikhani’s definitions in light of the powers in the

time domain given in the Subsection 2.2.1 above. The separation of powers according

to equations (2.77), (2.78) and (2.79) is along the lines as follows,

• equation (2.77) represents the active part V0I0 of (2.5) and active part of (2.6), (2.7),

(2.8). Note that (2.8) may not be included here if the vg(t) is zero.

• equation (2.78) represents the nonactive part of (2.6), (2.7), (2.8). Note that (2.8)

may not be included here if the vg(t) is zero.

• equation (2.79) then represents the remaining parts which are part of (2.5), (2.9),

(2.10), possibly (2.8).

The difference is that RMS quantities are used by Shepherd and Zakikhani.

Other researchers’ comments on Shepherd and Zakikhani’s definitions

Reference [6] states that the concepts are useful for power factor improvement but the

“nature of the quantity SR is vague and does not provide any information about the

possibilities of minimization”. Reference [78] states that in the presence of source

impedance, there will be no uncommon harmonics in which instance SD will always be

zero.


Chapter 2 52

2.3.5 Definition proposed by Sharon (1973) Sharon’s definitions [77] are also well discussed in literature, for example [6, 78, 81,

83]. The definitions are based on the frequency domain analysis. The active power is

given by the sum of the product of each harmonic voltage including the fundamental

and the corresponding in phase current. Quadrature reactive power is the product of the

RMS voltage and RMS value of quadrature currents of the fundamental and harmonics.

Finally distortion power satisfies the orthogonal relationship with active and reactive

power to give apparent power. The definitions for nonsinusoidal situations are

presented below

Active power is defined as

n n nn

P V I cos( )= φ∑ , (2.80)

quadrature reactive power as

2 2Q rms n n

n

S V I sin ( )= φ∑ , (2.81)

and complimentary reactive power as

2 2 2 22C m n n rms p

m n pS V I cos ( ) V I

⎧= φ +⎨⎩∑ ∑ ∑

12

n n

1 V I cos( ) V I cos( )2 β γ γ γ β β

β= γ=

⎫⎡ ⎤+ φ − φ ⎬⎣ ⎦

⎭∑∑ , (2.82)

where n represents the fundamental and harmonics for which both voltage and currents

exist, m for which only voltage harmonic exist and p where only current harmonics

exist.

Discussion on Sharon’s definitions

equations (2.81) represents the “average” active power of the time domain fundamental

and harmonic powers given in (2.6) and (2.7). The analysis in Subsection 2.3.2 on the

active power of Budeanu’s definition also applies here. The quadrature reactive power,

equation (2.81), represents the nonactive part of (2.6) and (2.7) plus source generated

cross harmonic power given in equation (2.9). The residual reactive power hence

represents the remaining parts in equations (2.8) and (2.10). Note that in the presence of


Chapter 2 53

source impedance, the load generated currents in Sharon’s definitions may become

included in the active power and quadrature reactive power.

Other researchers’ comments on Sharon’s definitions

Reference [6] states that Sharon has not explained the physical meaning of the

definitions. Reference [78] indicates the compensation recommendation may be

affected in the presence of source impedance.

2.3.6 Definition proposed by Kusters and Moore (1980) Kusters and Moore’s definitions [84] are also well discussed in literature, for example

[6, 18, 78, 83, 80]. The definitions are based on time domain method. The active

current ip(t) is defined in the same manner as defined by Fryze. The remaining current

i(t) – ip(t) is further decomposed into capacitive reactive current iqc(t) and iqcr(t) or into

inductive reactive current iql(t) and iqlr(t). This is outlined in the following equations.

Active current is

p 2rms

Pi (t) v(t)V

= . (2.83)

Capacitive reactive current is

[ ]DERT

qc DER2rmsDER

1 v (t) i dtT

i (t) v (t)V

=∫

, (2.84)

qcr p qci (t) i(t) i (t) i (t)= − − . (2.85) Inductive reactive current is

[ ]INTT

ql INT2rmsINT

1 v (t) i dtT

i (t) v (t)V

=∫

, (2.86)

qlr p qli (t) i(t) i (t) i (t)= − − , (2.87) where VrmsDER is the RMS value of DERv (t) = d

dt v(t) and VrmsINT is the RMS value of

INTv (t) = v(t)dt∫ .

The powers are then given as follows:


Chapter 2 54

Apparent power

S = Vrms Irms . (2.88)

Active power

P = Vrms Ip . (2.89)

Inductive reactive power

Ql = Vrms Iql . (2.90)

Capacitive reactive power

Qc = Vrms Iqc . (2.91)

Power related to residual inductive reactive current

Qlr = 22 2lS P Q− − (2.92)

Power related to residual capacitive reactive current

Qcr = 22 2cS P Q− − . (2.93)

Discussion on Kusters’ and Moore’s definitions

Kusters’ and Moore’s definitions, as in the case of Fryze’s definitions (Section 2.3.3),

also define the active current to be a scaled value of v(t). Hence the definitions assume

a constant conductance within a period and the comments in Section 2.3.3 are also

applicable here. The comment about vg(t) also applies.

Other researchers’ comments on Kusters’ and Moore’s definitions

Reference [6] states that the “capacitive reactive power was formulated for ideal voltage

sources and loses some properties for real sources”. Reference [78] also indicates this

by stating that the definitions “are valid if source impedance is negligible”.

2.3.7 Definition proposed by Czarnecki (1985/1988) Czarnecki’s proposed both a single [6] and three-phase power theory [34]. The

definitions are based on currents’ physical components (CPC) and are essentially in the

frequency domain.


Chapter 2 55

Single-phase definitions

The single-phase theory in [6] was defined

for linear non-sinusoidal systems as shown

in Figure 2.18. The active current ia(t) is

defined in the same manner as Fryze. The

remaining current i(t) – ia(t) is further

decomposed into scattered current is(t) and reactive current ir(t). This is outlined in the

following equations.

Active current is

a ei (t) G v(t)= , (2.94)

where e 2rms

PGV

= is the resistive equivalent load.

Scattered current is given b

s 0 e 0 n e n nn

i (t) (G G )V 2(G G ) V cos(n t )= − + − ω −α∑ ,

where 00

0

IGV

= ,

n nn

n

I cos( )GV

θ= , I0 and In are RMS currents, θn is the phase angle

and n represents the fundamental and harmonics.

Reactive current is given by

r n n nn

i (t) 2 B V sin(n t )= ω −α∑ ,

where

n nn

n

I sin( )BVθ

= .

The RMS currents are given as follows.

Active current is

a e rmsrms

PI G VV

= = , (2.95)

where Ia is the RMS active current.

Scattered current is given by

~i v

Linear load

Source ofnonsinusoidal

supply

Figure 2.18: 1-phase circuit structure


Chapter 2 56

2 22 2s 0 e 0 n e n

nI (G G ) V (G G ) V= − + −∑

where Is is RMS scattered current.


2 2r n n

nI B V= ∑

where

Ir is RMS reactive current.

The average powers are then given as follows:

Apparent power

S = Vrms Irms . (2.96)

Active power

P = Vrms Ia . (2.97)

Scattered power

Ds = Vrms Is . (2.98)

Reactive power

Q = Vrms Ir . (2.99)

Currents and powers relationships

Irms2 = Ia

2 + Is2 + Ir

2 , (2.100)

S2 = P2 + Ds2 + Q2 . (2.101)

Three-phase definitions

The definitions [34] are for three phase three wire

systems with symmetrical source of non-

sinusoidal voltage and nonlinear or periodically

variant load. In these definitions, the neutral of

the source is taken as the reference for

determination of the voltage as shown in Figure

2.19. The three-phase voltage and currents are represented by a single generalised RMS

value that is used for the subsequent decomposition of currents and determination of

powers.

~

~

~~i =0N

A

B

C

N

Ai

iB

iC

Av

vB

vC

Non-linearor periodicallyvariant load

Symmetricalsource of

nonsinusoidalsupply

Figure 2.19: 3-phase circuit structure


Chapter 2 57

The active power is defined by

P = [ ]T

A A B B C C0

1 v (t) i (t) v (t) i (t) v (t) i (t) dtT

+ +∫ (2.102)

Generalised RMS value of three phase voltage and current

V = ( ) ( ) ( )T

2 2 2A B C

0

1 v (t) v (t) v (t) dtT

⎡ ⎤+ +⎣ ⎦∫ and

I = ( ) ( ) ( )T

2 2 2A B C

0

1 i (t) i (t) i (t) dtT

⎡ ⎤+ +⎣ ⎦∫ (2.103)

and on harmonic basis

nV = ( ) ( ) ( )T

2 2 2nA nB nC

0

1 v (t) v (t) v (t) dtT

⎡ ⎤+ +⎣ ⎦∫ or

nV = ( ) ( ) ( )2 2 2nA nB nCV V V+ + (2.104)

where n = 1, h and VnA,VnB, VnC are RMS values for each harmonic.

Then equivalent conductance Ge is defined as

e 2

PGV

= , (2.105)

and harmonic equivalent conductance Gne is

nne 2

n

PGV

= , (2.106)

while equivalent harmonic susceptance Bne is

nne 2

n

QBV

= , (2.107)

where Pn and Qn are determined from each individual harmonic (n = 1,h) voltage,

current and phase angle.

Active current is

a eI G V= . (2.108) Scattered current is given by


Chapter 2 58

2s ne e n

n 1,h

I (G G ) V=

= −∑

. (2.109)


2r ne n

n 1,hI B V

=

= ∑ . (2.110)

Unbalance current is given by

2 2 2 2u n ne ne n

n 1,hI I (G B ) V

=

⎡ ⎤= − +⎣ ⎦∑ . (2.111)

Generated current is given by

2

g gm g

I I=

= ∑ . (2.112)

The average powers are then given as follows:

Apparent power

S = V I . (2.113) Active power

P = aV I . (2.114) Scattered power

Ds = sV I . (2.115) Reactive power

Q = rV I . (2.116) Unbalance power

Q = uV I . (2.117) Generated power

Q = gV I . (2.118)


Chapter 2 59

Currents and powers relationships 22 2 2 2 2

a s r u gI I I I I I= + + + + , (2.119)

S2 = P2 + Ds2 + Q2 + Du

2 + Dg2 . (2.120)

Discussion on Czarnecki’s definitions

Czarnecki’s definitions, as in the case of Fryze’s definitions (Section 2.3.3), also define

the active current to be a scaled value of v(t). Please refer discussion on Fryze’s

definition in Section 2.3.3 that has relevance in this discussion. However, as the

conductance, generally, could be a varying quantity (see Subsection 2.2.3 for discussion

on this) Czarnecki has included the scatter of conductance about the constant equivalent

conductance by defining the scatter current. The single-phase definitions are defined

for linear load and hence do not cater for load generated harmonic currents (Ig or ig(t)).

Hence they are not applicable when load is nonlinear. The three-phase definition

however caters for non linear loads via the definition of gI . However the limitation of

the three phase supply voltages to be symmetrical must be noted. Such may not be the

case in real systems.

Other researchers’ comments on Budeanu’s definitions

Reference [13] states that “Czarnecki’s theory attempts to give physical meaning” to the

definitions. However, since the decomposed currents are multiplied with the RMS

voltage to obtain the different power components, the definitions are “intrinsically

apparent powers”. “A major flaw of the decomposition” pointed out in [45] is that the

definitions do not “allow easy handling of the interaction of harmonic and sequence

components”. In [24] it is pointed out that Czarnecki’s theory does not give consistent

results in the presence of source impedance and asymmetric supply voltages, which in

fact are conditions that arise in a real system.


Chapter 2 60

2.3.8 The theory of instantaneous power in three-phase four wire systems proposed by Akagi et al (1993/1994) [41, 50, 85] Akagi et al proposed the original p-q

theory by applying Park’s transform to a

three-phase four wire system in 1983. In

1994 the modified theory was formulated

also for three-phase four wire systems.

The original theory

The three-phase A-B-C system as shown

in Figure 2.20 is changed to an 0−α −β system using Park’s vector as given in

equations below.

0 A

B

C

1 1 12 2 2v v

2 1 1v 1 v3 2 2

v v3 30

2 2

α

β

⎡ ⎤⎢ ⎥

⎡ ⎤ ⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎢ ⎥−⎢ ⎥⎣ ⎦

and (2.121)

0 A

B

C

1 1 12 2 2i i

2 1 1i 1 i3 2 2

i i3 30

2 2

α

β

⎡ ⎤⎢ ⎥

⎡ ⎤ ⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎢ ⎥−⎢ ⎥⎣ ⎦

(2.122)

The 0−α −β system is then used to determine the powers as follows.

0 0 0p v 0 0 ip 0 v v iq 0 v v iαβ α β α

αβ β α β

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(2.123)

where p0 is the instantaneous power in zero sequence circuit, pαβ is the instantaneous real power, qαβ is the instantaneous imaginary power. As shown in (2.124) below, the inverse matrix of (2.123) gives the 0−α −β currents

that provide information for compensation.

~

~

~~i =0N

A

B

C

N

Ai

iB

iC

Av

vB

vC

AZ

BZ

CZ

Figure 2.20: 3-phase system


Chapter 2 61

20 0

0 020

0 0

i (v ) 0 0 p1i 0 v v v v p

v (v )i 0 v v v v q

αβ

α α β αβαβ

β β α αβ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

(2.124)

where 2 2 2(v ) (v ) (v )αβ α β= + . Modified theory

The 0−α −β voltages and current are the same as those defined in the original theory.

However the power are defined differently and are given by

0

00

0

0

p v v vi

q 0 v vi

q v 0 vi

q v v 0

α β

β αα

α ββ

β α

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥− ⎢ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥− ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦−⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

(2.125)

where p is the instantaneous real power, 0q , qα and qβ the instantaneous imaginary

powers. The zero sequence power has been included in the real power and the single

imaginary power of the original theory has been split into the three imaginary powers

0q , qα and qβ . One of the imaginary powers q0 is zero sequence imaginary power. In

the original theory imaginary power could not flow in the zero sequence circuit. As for

the active power p it can flow in the 0, α and β phases while in the original theory it

flowed only in the α and β phases.

The inverse matrix of (2.125) gives the 0−α −β currents that provide information for

compensation.

0 00

020

0

pi v 0 v v

q1i v v 0 vq(v )

i v v v 0q

β α

α α βααβ

β β αβ

⎡ ⎤⎡ ⎤ ⎡ ⎤− ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎣ ⎦

(2.126)

where 2 2 2 2

0 0(v ) (v ) (v ) (v )αβ α β= + + . Discussion on the original and modified theory

Since this is an instantaneous theory, the discussion will be on the instantaneous

quantity. The theory gives a single instantaneous power for the three-phase system. In

the three-phase system, the sum of the instantaneous active power for each phase gives


Chapter 2 62

the instantaneous total active power for the three-phase system (see Subsection 2.2.5 for

related discussion). For a selected reference, this should remain the same irrespective of

the definition. For instantaneous nonactive power, this cannot be conclusively stated at

this point in this thesis. Hence it is not used as comparison here. The discussion is thus

limited to the three-phase instantaneous active power. Note that for this discussion, the

zero power p0 and the αβ-phase real power pαβ are summed to give the total real power

and this is equal to the real power p of the modified theory. Study reveals that the

Akagi et al’s instantaneous active power deviates from the expected if the three-phase

voltage (voltage may be nonsinusoidal) is not symmetrical for any load types except for

parallel equivalent load being time invariant and purely resistive. This is because the

theory is more suited for thee-phase three-wire systems. With symmetrical voltage,

deviation is observed when the load is not symmetrical, that is, the currents at the

metering point, are not in phase relationship with the voltage and of the identical time

profile in each phase. The phase relationship occurs, for instance, with sinusoidal

symmetrical voltage and the equivalent parallel nonactive load being equal and time

invariant in all the phases when the theory results do not deviate. Essentially, source

voltage symmetry and the load symmetry is key for the definitions of the theory unless

the equivalent load is purely resistive and time invariant.

Other researchers’ comments on Akagai et al’s definitions

Tolbert and Habetler [18] state for the original theory that “for asymmetrical voltage

sources, q is not equal to the three phase reactive power”. Note that q is the imaginary

power. Depenbrock [98] indicates that both theories can be successfully used to control

compensators without energy storage elements if the zero sequence voltage content is

not too high. It is also indicated that the imaginary quantities do not generally have a

physical relevance. Simulation results by Chang [86] show that the original theory

“failed due to lack of capabilities to compensate zero sequence components”.

Czarnecki [8] states from the power theory point of view that the p-q theory has “major

deficiencies”, for example “it does not identify power properties in the three-phase

system even in sinusoidal conditions” for unbalanced loads. He goes on to add that

though the theory is termed instantaneous, it “has not advantage over frequency domain

approach” because “additional analysis is required to identify active, reactive,

unbalanced and apparent powers after p and q are recorded over the entire period of


Chapter 2 63

their variability”. Also “instantaneous reactive current can occur when reactive power

is zero”.

2.3.9 The FBD-method by Depenbrock (1993) [42] The FBD method describes electrical conditions in an m-wire system.

Electrical conditions are the transferring of energy from source to load characterised as

follows.

• Currents xi (x = 1 to m) flow from source to load in the m conductors. All the m

conductors are considered equal.

• There are “m-1” voltages xyv (x, y = 1 to m) between the conductors at the junction

between source and load. These measured or calculated voltages are ideal voltage

sources in the source system.

For simple cases (Figure 2.21) current xi in

each conductor is split into power component

xpi and zero power component xzi

Instantaneous total power is given by

x x xpp = v i . (2.127)

Collective magnitude of sets of currents and

voltages are defined as follows.

Collective current and voltage for instantaneous

case

m2x

x 1xi i

=Σ = ∑ and

m2x

x 1xv v

=Σ = ∑ (2.128)

and for RMS case

m

2xRMS

x 1xRMSI I

=Σ = ∑ and

m2xRMS

x 1xRMSV V

=Σ = ∑ (2.129)

where branch “x” voltage m

x xyy 1

1v vm =

= ∑ and m

xRMS xyRMSy 1

1V Vm =

= ∑ .

TOTAL CURRENT IX

POWER CURRENT IXp

ZEROCURRENT IXz

TOTAL CURRENT IX

SOU

RCE

PART

LOAD

PAR

T

SOURCEVOLTAGE VX

REF0

0

Figure 2.21: Current decomposition for simple circuits


Chapter 2 64

The relationships m

xx 1

i 0=

=∑ and m

xx 1

v 0=

=∑ are applicable.

Total apparent power is given by

RMS RMS RMSS v iΣ Σ Σ= ⋅ (2.130)

Relationship between voltage and power current is

p xxpi G (t) v= ⋅ and pp xi G (t) vΣ Σ= ⋅ (2.131)

where p 2p (t)

G (t)vΣΣ= is equivalent instantaneous conductance and

m

x 1xp (t) p (t)

=Σ =∑

m

x 1xxREFv i

=

= ⋅∑ (“REF” denotes that vx is measured to a common reference REF).

Zero-power currents are given by xz x xpi i i= − (2.132)

The zero current xzi can be compensated with a compensating device connected in

parallel with the load without any time delay. Since m

x 1x xzzp (t) v i 0

=Σ = ⋅ =∑ , there is no

demand on the storage capability of the compensator. By compensation, zero currents xzi are reduced and RMS value of source current can

be reduced to 2 2xRMS zRMSpRMSI I IΣ ΣΣ = − .

The active currents xai are source currents that are decreased to the smallest possible

collective active current aRMSiΣ and are proportional to mean value p (t)Σ of the

instantaneous collective power called collective active power aPΣ or shortened to PΣ G can be determined for the active power as follows.

For single branch instantaneous values are

xa xi G v= ⋅ , xaRMS xRMSI G V= ⋅ and 2xxap (t) G v= ⋅ , (2.133)

and collective values

ai G vΣ Σ= ⋅ , aRMS RMSI G VΣ Σ= ⋅ and 2ap (t) G vΣΣ = ⋅ , (2.134)


Chapter 2 65

where the equivalent active conductance

22RMS

p (t) pG

vv ΣΣ

Σ Σ= = is not possible to obtain

without time delay as mean values of p (t)Σ

and 2vΣ cannot be determined before averaging

time interval. If source system only delivers

instantaneous active current xai then power

factor is unity.

For a not simple case (Figure 2.22) at a given

load current xi , the complete nonactive

instantaneous current xni has to be compensated. The relationship is

xn x xa xz xvi i i i i= − = + . (2.135)

If the power factor is unity then instantaneous collective power is 2

2a2

RMS RMS

2aa

v vp (t) G v P P f fv v

(t) and (t)Σ ΣΣ

Σ ΣΣ Σ Σ= ⋅ = ⋅ = ⋅ = . (2.136)

If there is any deviation from the function completely characterised by 2

af (t) ,

depending only on voltages, the difference shall be denoted by instantaneous collective

“variation” power vp (t)Σ . Thus any instantaneous power function can be split into its

active and variation component.

2pa vp (t) p (t) p (t) G G (t) vΣΣ Σ Σ ⎡ ⎤= + = + Δ ⋅⎣ ⎦

Compensation of total nonactive current is xn x xa xz xvi i i i i= − = + . Instantaneous

collective power for a compensator is the negative of instantaneous collective variation

power vp (t)Σ produced by xvi , that is, vp (t)Σ− .

2pvp (t) G (t) vΣΣ = Δ ⋅ p pG (t) G (t) GΔ = −

xxv pi G (t) v= Δ ⋅ pvi G (t) vΣ Σ= Δ ⋅

TOTAL CURRENTIX = IXR + IXL

POWER CURRENT IXp

ZEROCURRENT IXz

ACTIVECURRENTIXa

NON-ACTIVE CURRENT IXn

VARIATION

CURRENT Ixv

TOTAL CURRENTIX

SOU

RCE

PART

LOAD

PAR

T

SOURCEVOLTAGE VX

REF0

0 Figure 2.22: Current decomposition for complex circuits


Chapter 2 66

Also according to FBD theory xai and xzi are orthogonal.

Discussion on the FBD method

Determination of the equivalent instantaneous conductance Gp(t) requires instantaneous

collective power. This implies that instantaneous collective power ppΣ is active in

nature. Though this is true for the average active three-phase power as shown in

Subsection 2.2.4, it may not necessarily be so for the instantaneous power as per

discussion in Subsection 2.2.5 above.

Other researchers’ comments on FBD method

Depenbrock and Staudt [87] stated that some problems with compensating when

conductance G becomes negative. Moreno and Pigazo [88] state that the instantaneous

load conductance g(t) “does not allow proper compensation of classical reactive power”.

2.3.10 Definition proposed by Ferrero and Superti-Furga (1991) [13, 89] Ferrero and Superti-Furga proposed definitions based on Park’s vector. The three-phase

A-B-C system is changed to d-q plane using Park’s vector as given in equations below.

d A

q B

0 C

2 1 13 6 6

v v1 1v 0 v2 2

v v1 1 13 3 3

⎡ ⎤− −⎢ ⎥

⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

and (2.137)

d A

q B

0 C

2 1 13 6 6

i i1 1i 0 i2 2

i i1 1 13 3 3

⎡ ⎤− −⎢ ⎥

⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(2.138)

where vA, vB, vC and iA, iB, iC are the phase instantaneous voltages and currents respectively.


Chapter 2 67

Park’s powers

The powers are defined using the d-q plane vectors as follows.

Park’s real power is

pp(t) = vdid + vqiq . (2.139)

Park’s imaginary power is

qp(t) = vqid – vdiq . (2.140)

And the zero-sequence power is

p0(t) = v0i0 . (2.141)

Instantaneous power is given by

p(t) = pp(t) + p0(t). (2.142)

Note that Park’s powers, though defining zero sequence powers, are applicable “more

so” to three-phase three wire systems.

Currents from Park’s powers

The currents defined are applicable to three-phase three-wire system.

Aa Ap

Ba B2

Ca C

i vP

i vV

i vΣ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

and Ax A Aa

Bx B Bb

Cx C Cc

i i ii i ii i i

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(2.143)

where iAa, iBa and iCa are the active phase currents, iAx, iBx and iCx the residual phase

currents, p pT

P p (t)dt= ∫ and 2 2 2 2p q 0V V V VΣ = + + is the RMS value of the Park voltage

vector.

The current iAx , iBx ,and iCx can be utilised for compensation purposes.

Discussion on definitions by Ferrero and Superti-Furga

The power definitions by Ferrero and Superti-Furga just as Akagi et al’s are based on

Park’s vector. Hence the discussion in Section 2.3.8 are applicable here. As for the

determination of current from Park’s vector the discussion in Section 2.3.3 applies


Chapter 2 68

because the definition is, in the words of the authors, a “straightforward extension of

Fryze and Kusters and Moore’s theories to three-phase systems”.

2.3.11 Definitions proposed by Willems (1992) [90] Willems proposed definitions based on instantaneous voltage and current vectors. The

definitions are valid for single-phase as well as a system with m phases. The proposed

definitions are outlined as follows.

For an m phase system, the instantaneous voltages and currents related to the m phases

are represented by m element voltage v(t)→

and current i(t)→

vectors. The instantaneous

power transmitted to the load is given by the internal product of the instantaneous

voltage and current vectors.

p(t)→

= Tv(t) t(t)→ →

(2.144)

The instantaneous active current vector pi (t)→

is given by the orthogonal projection of

the instantaneous current vector i(t)→

onto the voltage vector v(t)→

. This is given by

pi (t)→

=T

2v(t) t(t) v(t)

v(t)

→ →

, (2.145)

where x(t)→

is the length of the vector x(t) and is given by Tx(t) x(t)→ →

.

The instantaneous nonactive current is then given by

qi (t)→

=i(t)→

- pi (t)→

, (2.146)

with qi (t)→

being orthogonal to v(t)→

.

The following is valid for single phase

2i(t)→

=2

pi (t)→

+2

qi (t)→

, (2.147)

and for three-phase


Chapter 2 69

2i(t)→

= 2Ai (t)→

+ 2Bi (t)→

+ 2Ci (t)→

. (2.148)

The instantaneous real and imaginary powers are given by

pp(t) v(t) i (t)= (2.149)

and

qq(t) v(t) i (t)= i . (2.150)

Additionally, the relationships

2

pi (t)→

=2

2p(t)v(t)

(2.151)

and

2

qi (t)→

=2

2q(t)v(t)

(2.152)

also apply.

Discussion on definitions by Willems

The essence in the definition lies in the resolution of the current vector onto the voltage

vector to determine active current. The discussion in Section 2.2.6 above on this matter

should be considered here.

2.3.12 Generalised instantaneous reactive power theory for three-phase systems proposed by Peng and Lai (1996) [46] Peng and Lai proposed definitions

based on instantaneous voltage and

current vectors along the lines of Akagi

et al and Willems. The definitions are

valid for sinusoidal or nonsinusoidal,

balanced or unbalanced three phase

systems with or without zero sequence

currents/voltages. Figure 2.23 shows

the load model.

~

~

~~i =0

A

B

C

N

Ai

iB

iC

Av

vB

vC

LOAD

Figure 2.23: Source Load Model


Chapter 2 70

For the system, voltage v(t)→

and current i(t)→

vectors are as follows

v(t)→

= A

B

C

vvv

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

and i(t)→

= A

B

C

iii

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(2.153)

The instantaneous power transmitted to the load is given by the inner product or dot

product of the instantaneous voltage and current vectors.

p(t) = v(t) i(t)→ →

⋅ or p(t) = vAiA + vBiB + vCiC (2.154) The instantaneous nonactive power is defined as vector q(t) which is the cross product

of the voltage and current vectors

q(t)→

= v(t) i(t)→ →

× (2.155)

where x(t)→

is the length of the vector x(t) which gives the following

q(t)→

= A

B

C

qqq

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

=

B C

B C

C A

C A

A B

A B

v vi i

v vi i

v vi i

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= B C C B

C A A C

A B B A

v i v iv i v iv i v i

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥−⎣ ⎦

. (2.156)

and

q(t) = 2 2 2A B Cq q q+ + . (2.157)

The instantaneous active and nonactive current can be obtained from

pi (t)→

= Ap

Bp

Cp

iii

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= p(t) v(t)v(t) v(t)

→

→ →

i (2.158)

and

qi (t)→

= Aq

Bq

Cq

iii

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

= q(t) v(t)

v(t) v(t)

→ →

→ →

×

i. (2.159)

The instantaneous apparent power is given by

s(t) = v3Ph(t) i3Ph(t), (2.160)


Chapter 2 71

where v3Ph(t) = 2 2 2A B Cv v v+ + , i3Ph(t) = 2 2 2

A B Ci i i+ + ,

and instantaneous power factor is given by

p(t)(t)s(t)

λ = . (2.161)

Discussion on Peng and Lai’s definitions

The discussion for Akagi el al (Section 2.3.8) and Willems definitions (Section 2.3.11)

are also applicable here. The discussion in Section 2.3.10 also have some relevance

here.

2.3.13 Definitions in IEEE Standard 1459 (2000) [37, 74] The standard proposes definitions that are as stated in the introduction “meant to serve

the user who wants to measure and design instrumentation for energy and power

quantification”.

Single-phase definitions

Only some of the definitions for nonsinusoidal systems are stated herewith.

Instantaneous power is given by

p(t) = v(t) i(t) = pa(t) + pq(t) (2.162)

where

pa(t) = [ ]h h hh

V I cos 1 cos(2h t)θ − ω∑ (2.163)

is the instantaneous non-zero average power and

pq(t) = h h hh

V I sin sin(2h t)θ ω∑ + m n m nm n

2V I sin(m t )sin(n t )≠

ω + α ω + β∑ (2.164)

is the instantaneous zero average power.

Average active power is given by

P = T

1 v(t) i(t)dtT ∫ (2.165)

where and P = P1 + PH and P1 = 1 1 1V I cos( )θ is fundamental active power and PH =

h h hh 1

V I cos( )>

θ∑ the harmonic active power.


Chapter 2 72

Average reactive power is given by

Q1 = 11 1

T

i (t) v (t)dt dtTω ⎡ ⎤

⎣ ⎦∫ ∫ = 1 1 1V I cos( )θ (2.166)

where Q1 is the fundamental reactive power and harmonic reactive power is defined

using Budeanu’s definition (2.58) with m > 1.

Apparent power is defined as

S = VI. (2.167)

Apparent power can be subdivided into fundamental apparent power

S1 = V1I1 (2.168)

and non fundamental apparent power

SN2 = S2 – S1

2 = (V1IH)2 + (V1IH)2 + (VHI1)2 + (VHIH)2 . (2.169)

The terms of the non-fundamental power in equation (2.169) are further classified as

current distortion and voltage distortion powers

DI = V1IH and DV = VHI1 (2.170)

respectively and harmonic distortion apparent power

DH = VHIH. (2.171)

Nonactive power, which lumps together both fundamental and non-fundamental

nonactive components, is

N = 2 2S P− . (2.172)

Power factor is defined as

PF = PS

, (2.173)

and fundamental power factor as

PF1 = 1

1

PS

. (2.174)

Three-phase definitions


Chapter 2 73

Only some of the most general definitions for unbalanced nonsinusoidal systems are

stated herewith, except for instantaneous power which, in the standard, is defined only

for sinusoidal balanced and unbalanced systems.

Note that in the standard the recommendation is to use neutral as the reference for three-

phase four wire systems and artificial neutral for three-phase three wire systems.

Further discussion on this matter is given in Reference [91].

Instantaneous Power (sinusoidal balanced and unbalanced systems) is for three-phase

four-wire system

p = va(t)ia(t) + vb(t)ib(t) + vc(t)ic(t), (2.175)

and for three-phase three-wire system

p = vab(t)ia(t) + vcb(t)ic(t) = vac(t)ia(t) + vbc(t)ib(t) = vba(t)ib(t) + vca(t)ic(t). (2.176)

The RMS effective voltage Ve and current Ie are defined as follows.

Ve = 2 2e1 eHV V+ and Ie = 2 2

e1 eHI I+ , (2.177)

where for a four-wire system the equivalent voltage is

Ve = ( ) ( )2 2 2 2 2 2a b c ab bc ca

1 3 V V V V V V18

⎡ ⎤+ + + + +⎣ ⎦ ,

Ve1 = ( ) ( )2 2 2 2 2 2a1 b1 c1 ab1 bc1 ca1

1 3 V V V V V V18

⎡ ⎤+ + + + +⎣ ⎦ ,

VeH = ( ) ( )2 2 2 2 2 2aH bH cH abH bcH caH

1 3 V V V V V V18

⎡ ⎤+ + + + +⎣ ⎦ = 2 2e e1V V− ,

and equivalent current is

Ie = 2 2 2

a b cI I I3

+ + ,

Ie1 = 2 2 2

a1 b1 c1I I I3

+ + ,

IeH = 2 2 2

aH bH cHI I I3

+ + = 2 2e e1I I− ,

whilst for a three-wire system the equivalent voltage is

Ve = 2 2 2

ab bc caV V V9

+ + ,


Chapter 2 74

Ve1 = 2 2 2

ab1 bc1 ca1V V V9

+ + ,

VeH = 2 2 2

abH bcH caHV V V9

+ + = 2 2e e1V V− ,

and equivalent current is

Ie = 2 2 2

a b cI I I3

+ + ,

Ie1 = 2 2 2

a1 b1 c1I I I3

+ + ,

IeH = 2 2 2

aH bH cHI I I3

+ + = 2 2e e1I I− .

Effective apparent power is given by

Se = 3VeIe (2.178)

and

Se2 = Se1

2 + SeN2 (2.179)

where Se1 = 3Ve1Ie1 is the fundamental effective apparent power and SeN is a non-

fundamental effective power and is given by SeN2 = Se

2 + Se12 = DeI

2 + DeV2 + SeH

2 (DeI

and DeV are the equivalent current and voltage distortion powers and SeH is the

equivalent harmonic distortion apparernt power), DeI = 3Ve1IeH, DeV = 3VeHIe1, SeH =

3VeHIeH and DeH = 2 2eH eHS P− .

Load unbalance is evaluated using the fundamental. It is defined as

SU1 = 2 2e1 1S (S )+− (2.180)

where 1S+ = 2 21 1(P ) (Q )+ ++ is the fundamental positive sequence apparent power,

1 1 1 1P 3V I cos( )+ + + += θ the fundamental positive sequence active power and

1 1 1 1Q 3V I sin( )+ + + += θ the fundamental positive sequence reactive power.

The fundamental positive sequence power factor is

1F1

1

PPS

++

+= , (2.181)


Chapter 2 75

and power factor is

Fe

PPS

= . (2.182)

Discussion on the IEEE Standard 1459 definitions

Discussions under Section 2.3.1 are relevant to average nonactive power definition

given in equation (2.172). Comments in Section 2.2.2 on cross-harmonic components

apply to zero average power defined in equation (2.164). On this point it is noted that

the standard does not explicitly state that zero power is nonactive power but there is

indirect implication by the use of pq(t) in this equation. The discussion under Sections

2.2.4 and 2.2.5 has relevance to three phase instantaneous power.

2.4 Conclusion A brief historical account of electricity has been given. Notable researchers in the area

of power definitions have been listed. Discussion on some common power

theories/definitions has been presented. Substantial analysis has been performed on the

RMS based definitions while the others have been briefly discussed.


Chapter 2 76

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Requirements on a power definition and benchmark case studies for evaluation

Chapter 3 77

3. REQUIREMENTS ON POWER DEFINITIONS AND BENCHMARK CASE STUDIES FOR EVALUATION

The first step to realising a goal is to identify the objectives of the goal. Additionally, as

the goal is being achieved, there must be means to evaluate its success in meeting the

objectives. Likewise, before a theory or definition can be stated, it is important to

identify what the requirements are of it and additionally, the benchmarks by which these

requirements will be met must be identified. This chapter addresses these aspects.

3.1 Requirements on power definitions 3.1.1 What other researchers say The inspiration behind the requirements on power definitions is derived from the work

of many researchers who have spent substantial time studying the problem. Many share

common views on this matter. Some common and essential ones are reviewed.

The goal in power systems is to reduce the currents flowing in conductors to the

minimum possible and/or to reduce losses to minimum as stated in [27, 54]. The power

theory should also answer the simple question: why the source current RMS value in

such systems is higher than that necessary for the active power transmission [34]. The

new definitions must “be accurate and have interpretation in terms of the load

connected” [22]. Almost universally researchers, [9, 11, 29, 50, 52, 67, 92-97] to name

some, state the need to attribute a physical meaning to the powers.

A number of requirements are listed in [42] and some essential ones quoted below.

• To define powers and for active compensation of nonactive currents full

information representing their time functions is needed.

• Definitions and standards should state rules on how to determine these time

functions. The rules should be applicable in any case without contradictions and

be as simple as possible.

• New generally applicable definitions and standards should include the ones

approved in the past as more special cases.

Requirements on a power definition and case studies for evaluation

Chapter 3 78

• Nonactive powers are quantities of only secondary importance, normally they

have to be derived from the nonactive currents, not vice versa.

• The definitions of quantities should lead to rules for their determination from

measured voltages and currents.

Reference [23] points out that any definition must consider source equivalent

impedance since this affects measurements.

A systematic approach to study of the electrical power system and what it means is

espoused in [25]. The essential points are summarised as follows

• a set of equations relating current, voltage and physical properties of the system,

• study system behaviour from the energy point of view,

• lay correct specifications for constraints,

• need of a number of equations to describe the power system under nonsinusoidal

conditions,

• physical quantities (voltage, current, energy, active, reactive and apparent

power) only must be used and are sufficient to describe the system i.e. do not

come up with new quantities,

• most likely use both time and frequency domain,

• must be practically realisable.

Reference [36] lists the objectives of a new theory as

• consistent for all system topologies,

• cover single and polyphase system,

• valid for balance and unbalance polyphase system,

• valid for pure or distorted waveforms,

• does not violate any electrical engineering principle,

• readily implemented in practical equipment.

The IEEE Standard [37] lists

• a common base for energy billing,

• evaluation of electric energy quality,


Chapter 3 79

• detection of major sources of waveform distortion

• provision of information for design of mitigation equipment

as the essential requirements.

In [39] the author states that the new set of definitions of power quantities should

• be relevant for the new situations,

• but remain valid in the classical situations.

3.1.2 The requirements Having reviewed the researchers comments in Section 3.1.1, based on these the

requirements are listed and explained.

1. The definitions should give time functions and be generally applicable. This

means they should apply equally to any source/load combination, single or

polyphase system, sinusoidal or distorted as well as balanced or unbalanced

conditions without additional consideration. Neither should they avail to

contradictions for different source/load conditions or power system

configuration. Average values are defined from the time functions.

2. The definitions should use only voltages and current as the base and should

utilise both frequency and time domains as well as maintain energy transfer

between source and load.

3. The definitions should be such that the powers are defined from currents not

vice-versa.

4. The definitions should be independent of source impedance.

5. Any assumptions or constraints should be clearly identified.

6. The definitions should be either coherent with existing definitions or be clearly

explicable of any discrepancies should these exist.

7. The definition should provide sufficient information to achieve the goal of

minimising currents flowing in conductors and hence reduce losses to a

minimum. Essentially the power system supply side conductors should not carry

any unnecessary current that is not usefully utilised. The theory should thus lend

itself to compensation of these “useless” quantities.

8. The theory should preferably attribute a physical meaning to the powers. In this

thesis, this is taken to mean that the power must bear a direct relationship with


Chapter 3 80

the source/load characteristics based on known electrical phenomena and not be

obtained by some mathematical means. Every power component defined thus

must preferably base its formula directly on (not some derivation of) the voltage

and current. There will be a likelihood of a number of equations in the

definitions.

9. The definitions must be practically realizable.

10. Currents calculated from the defined powers should match the currents used to

calculate the powers.

Having identified the requirements, how the definition will be verified or evaluated is

next addressed, but first some background technical information is provided.

3.2 Background technical Information Following the trend in Chapter 2 some basic concepts are first presented. These

concepts will then be used, to determine the instantaneous active and nonactive power

waveforms as well as their energy content, in the benchmark cases developed later in

the chapter. The powers and energy information computed for the benchmark cases are

termed the expected powers and expected energy transfer.

Measurement in the power system in the mainstream uses largely average values. In

this thesis the main focus is on timed values. Thus voltages and currents in the time

domain are used in the analysis. The analysis of simple linear circuits under non-

sinusoidal conditions and the observation of the nature of voltages and currents in the

circuit is first performed. The nature of current in a diode-resistor circuit is also

elucidated.

For the following analysis the source or driving voltage is as given in equation (3.1).

v(t) = 1 1 h h2 V cos( t ) 2 V cos(h t )ω −α + ω −α∑ (3.1)

where V1, Vh, are RMS values of fundamental (1) and harmonic components (h), ω is

the fundamental angular frequency 2 fπ (f is fundamental frequency in Hz), t is the time,


Chapter 3 81

xα (x = 1, h) the voltage phase angle. For the ensuing analysis the following

terminology will be used.

11

V2 cos( t )R

ω −α

amplitude oscillating part 3.2.1 Source voltage and currents in a resistive single-phase circuit The current flowing in the resistive circuit with resistance R is given by the equation

i(t) = 1 h1 h

V V2 cos( t ) 2 cos(h t )R R

ω −α + ω −α∑ . (3.2)

The current i(t) is the active current ip(t) while nonactive current iq(t) is zero. Hence

ip(t) = 1 h1 h

V V2 cos( t ) 2 cos(h t )R R

ω −α + ω −α∑ . (3.3)

It is observed that the amplitudes for the

fundamental and harmonics have the same ratio

with respect to the amplitudes of the

corresponding terms in the voltage waveform v(t)

that is 1 2 h

1 2 h

2V 2V 2V... R2V R 2V R 2V R

= = = = .

Thus the current waveform is a scaled version of the voltage waveform (as shown

above). This indicates that the conductance G(t) i(t)v(t)

= , red curve in Figure 3.1, of the

load is has a constant value 1R

⎛ ⎞=⎜ ⎟⎝ ⎠

during the period. The oscillating parts are the

same. Thus for active currents, the oscillating part of each of the terms is identical

to the corresponding voltage oscillating part. This enables us to observe the current

oscillating terms and determine if they are active or not

3.2.2 Source voltage and currents in an inductive single-phase circuit

v t( )

ip t( )

G t( )

t Figure 3.1: Resistive load


Chapter 3 82

The current flowing in the inductive circuit, with inductance L, would then be given by

the equation

i(t) = 1 h1 h

h

V V2 cos( t ) 2 cos(h t )L 2 h L 2

π πω −α − + ω −α −

ω ω∑

= 1 h1 h

h

V V2 sin( t ) 2 sin(h t )L h L

ω −α + ω −αω ω∑ . (3.4)

The current i(t) in this case is nonactive current iq(t), active current ip(t) being zero. iq(t)

= 1 h1 h

h

V V2 cos( t ) 2 cos(h t )L 2 h L 2

π πω −α − + ω −α −

ω ω∑

= 1 h1 h

h

V V2 sin( t ) 2 sin(h t )L h L

ω −α + ω −αω ω∑ . (3.5)

The observation is that oscillating parts of the

terms are shifted by 2π

− , that is they are in

quadrature and lagging. in relation to the

corresponding voltage terms. Thus for

inductive load, the inductive nonactive

current, the oscillating part of each of the

terms is in quadrature (lagging) to the

corresponding voltage oscillating part. This enables us to observe the current terms

and determine if they are nonactive. The amplitudes for the fundamental and harmonics

have the same ratio with respect to the amplitudes of the corresponding terms in the

voltage v(t) waveform when divided by the harmonic number that is

1 2 h

1 2 h

2V 1 2V 1 2V...2 h2V L 2V 2 L 2V h L

= = =ω ω ω

. The equivalent susceptance B(t)

i(t)v(t)

= , red curve in Figure 3.2, of the load is time varying.

3.2.3 Source voltage and currents in a capacitive single-phase circuit The current flowing in the capacitive circuit, with capacitance C, would then be given

by the equation

v t( )

iq t( )

B t( )

t Figure 3.2: Inductive Load


Chapter 3 83

i(t) = 1 1 h hh

2 CV cos( t ) 2h CV cos(h t )2 2π π

ω ω −α + + ω ω −α +∑

= 1 1 h hh

2 CV sin( t ) 2h CV sin(h t )⎡ ⎤ ⎡ ⎤− ω ω −α + − ω ω −α⎣ ⎦ ⎣ ⎦∑ . (3.6)

The current i(t) in this case is nonactive current

iq(t), active current ip(t) being zero. Again it is

observed that the oscillating parts are shifted by

2π but in leading quadrature relationship with

the corresponding voltage term. For the

capacitive nonactive currents, the oscillating

part of each of the terms is in quadrature (leading) to the corresponding voltage

oscillating part. The amplitudes for the fundamental and harmonics have the same

ratio with respect to the amplitudes of the corresponding terms in the voltage v(t)

waveform when multiplied by the harmonic number that is

1 2 h

1 2 h

2V 2V 2V2 ... h2V C 2V 2 C 2V h C

= = =ω ω ω

. The equivalent susceptance B(t) i(t)v(t)

= ,

red curve in Figure 3.3, of the load is time varying. However its form is sort of inverted

about the time axis as compared to the inductance case above.

3.2.4 Source voltage and currents in a linear parallel resistive-inductive single-phase circuit For a parallel R-L load, the active current ip(t) is given by (3.3) and the nonactive

current iq(t) by equation (3.5). The total current flowing is then equal to i(t) = ip(t) +

iq(t).

For active currents, the oscillating part of each of the terms is identical (in phase)

to the corresponding voltage oscillating term and for nonactive currents, the

oscillating part of each of the terms is in quadrature with the corresponding

voltage oscillating part.

v t( )

iq t( )

B t( )

t Figure 3.3: Capacitive load


Chapter 3 84

Thus the amplitudes for the fundamental and harmonic active currents will have the

same ratio with respect to the amplitudes of the corresponding terms in the voltage v(t)

waveform as is the case in Section 3.2.1. The active current waveform being a constant

scaled version of the voltage waveform indicates a constant equivalent conductance of

the load.

3.2.5 Source voltage and currents in a linear series resistive-inductive single-phase circuit The analysis in Subsection 2.2.3 in Chapter 2 also

applies for this subsection. In summary it can be

stated that for a series non-resistive load subject

to non-sinusoidal source voltage, its active

current cannot be obtained by assuming that the

parallel equivalent conductance is constant.

This is reflected in Figure 3.4 (graph is reproduced

from Chapter 2) where G(t) is not constant.

v t( )

ip t( )

iq t( )

G t( )

t Figure 3.4: Series R-L load

3.2.6 Determination of active current in R-C single-phase circuit Parallel and series circuits can be analysed in a similar manner to the R-L and similar

results are obtained.

3.2.7 Discussion of source voltage and driving voltage In the single phase system the source voltage is usually (except when there is generating

source in the load) the electromotive force (EMF) or the driving voltage behind the

current flow.

For three-phase systems, on the other hand, for a particular choice of reference

conductor, the driving voltage (which is the EMF that is driving the current) behind the

current flowing through the load may not necessarily be the source voltage. This is

illustrated using a resistive load connected between two phases of a three-phase three-

wire system as shown below.


Chapter 3 85

Though the load is purely resistive there is

presence of nonactive power in phase B

when using reference N. The reason for this

is as follows. The flow of current is through

conductor B into RA via RB, then through

conductor A and back to neutral N. This is

shown with a red line in Figure 3.4. Note the

generating element (blue arrow) in the loop

(red line) for source voltage Vbn (red arrow).

Vbn, which is the source voltage with N as

reference, is not the driving voltage for the

current flowing in the loop (or the resistive load) because the generating element Van

(blue) appears as a part of the load. This is because the load is viewed from B (with

reference N) as shown in the equivalent circuit. The driving voltage is thus the vector

sum of voltage Vbn and the voltage Van. So the current flowing out of B into the load

does not have in-phase relationship with the voltage Vbn resulting in presence of

nonactive power.

If Vab voltage is used as reference, then

only the resistive load is in the loop (see

Figure 3.6), hence nonactive power is

zero for this reference. Also the current

out of the source at B towards the load

would be in phase relationship with Vba.

Vba is the source as well as the driving voltage.

3.2.8 Powers in the single-phase and three phase circuit From Sections 3.2.1 to 3.2.6 some simple rules can be summarised that are used to

study the voltage and currents (represented as Fourier components) in the circuit. These

rules enable decomposing the total current terms into active and nonactive currents that

will facilitate computation of powers.

~~~~

A

B

C

RA

BR

RA

BRN

generating element

Vbn

Van

~~

AB

C

RARA

N

Vbn

VanRRB

Thevenin's equivalent

Figure 3.4: Resistive load in phases A and B and source with N as reference

~~~~

A

B

C

RA

BR

RA

BRN

Vba

Figure 3.6: Resistive load in phases A and B with B as reference


Chapter 3 86

• For active currents the oscillating part of fundamental or harmonic terms is the same

as that of the corresponding fundamental or harmonic terms of the voltage

waveform.

• For nonactive currents the oscillating part of fundamental or harmonic terms is in

lagging quadrature (inductive) or leading quadrature (capacitive) relationship to that

of the corresponding fundamental or harmonic terms of the voltage waveform.

• For parallel R-L or R-C or R-L-C circuits, the active current waveform is a scaled

version of the voltage waveform and the equivalent conductance of the load is a

constant.

• For series R-L or R-C or R-L-C circuits, the active current waveform is not a scaled

version of the voltage waveform and the equivalent conductance of the load is a not

a constant.

It can be concluded that knowledge of the voltage and current in Fourier components

form is sufficient to determine active and nonactive current at the metering point. Any

knowledge of the load is thus not necessary in determination of the active as well as

nonactive current at the metering point.

Powers in single-phase circuits

In the above Sections 3.2.1 to 3.2.6, the active power p(t) is given by the product of the

voltage v(t) and active current ip(t) ,

p(t) = v(t) ip(t) (3.7)

while nonactive power is given by the product of the voltage v(t) and nonactive current

iq(t),

q(t) = v(t) iq(t). (3.8)

equations (3.7) and (3.8) are used to calculate active and nonactive powers in the case

studies in Section 3.3.

Powers in three-phase circuits


Chapter 3 87

For three-phase circuits, equations (3.7) and (3.8) can be applied on a phase by phase

basis to determine the powers in each phase. The concept of driving point voltage has

to be taken into consideration when doing this.

3.2.9 Does a diode-R load consume nonactive power? A nonlinear load using a diode is used to create benchmark cases of a nonlinear load.

As such, it is necessary to be explicit as to the powers that are flowing in such a circuit.

This is addressed in the analysis that follows.

There is a consensus that a diode-R load with a sinusoidal source and negligible source

impedance, being nonlinear, gives rise to presence of nonactive power. The following

example shows to the contrary that the ideal diode-R load draws only active power.

The diode-R load with sinusoidal source voltage has been termed the “resistive load

paradox” in [9] in 1988 and recently in 2006 this circuit was still discussed at the 7th

International Workshop "Angelo Barbagelata" on Power Definitions and Measurements

under Nonsinusoidal Conditions held at Cagliari, Italy in July 10-12, 2006.

Consider the circuit shown in Figure 3.7. Source resistance is used so that the phase

between the generated harmonic currents and its corresponding voltage harmonics

exhibit at the metering point can be determined. For a source voltage given by

vs(t) = 2 V1 cos(ω1 t ) (3.9) the current and voltage at the metering point is given by

i(t) = 12V(R r)π +

+ ( )11

2V cos t2(R r)

ω+

+ n 11 1

n

2 2V cos(2n t)( 1)(R r) (2n 1)(2n 1)

+⎡ ⎤ω−⎢ ⎥π + − +⎣ ⎦

∑ (3.10)

and

vm(t) = 1r 2V(R r)

−π +

+ ( )1 1r2V 1 cos t

2(R r)⎛ ⎞− ω⎜ ⎟+⎝ ⎠

n 11 1

n

2r 2V cos(2n t)( 1)(R r) (2n 1)(2n 1)

+⎡ ⎤ω− −⎢ ⎥π + − +⎣ ⎦

∑ . (3.11)

To ease comparison, the terms in equations (3.10) and (3.11) are presented in Table 3.1

below.


Chapter 3 88

Table 3.1 Type Voltage at metering point Current in circuit DC

1r 2V(R r)

−π +

12V(R r)π +

Funda- mental ( )1 1

r2V 1 cos t2(R r)

⎛ ⎞− ω⎜ ⎟+⎝ ⎠

( )11

2V cos t2(R r)

ω+

Harmonic n = 1,2…

n 11 12r 2V cos(2n t)( 1)(R r) (2n 1)(2n 1)

+⎡ ⎤ω− −⎢ ⎥π + − +⎣ ⎦

n 11 12 2V cos(2n t)( 1)(R r) (2n 1)(2n 1)

+⎡ ⎤ω−⎢ ⎥π + − +⎣ ⎦

It is observed from Table 3.1.

• that fundamental current is in-phase relationship with the fundamental voltage with

zero phase angle,

• the harmonic current is in-phase relationship with the corresponding harmonic

voltage with 180 degree phase angle.

Hence both the fundamental and harmonic currents are active in nature, meaning there

are no storage elements in the load. The DC current, hence, is also active. Therefore all

the current components are active. The sum of these current will be active and thus

only active current flows in the circuit. Therefore, only active power will be consumed

by the load; nonactive power being zero.

This is illustrated with a numerical example data as per Figure 3.7. The voltage and

current at the metering point is determined from simulation using ATP and is shown in

Figure 3.8(a). The cosine Fourier components is calculated from the waveform data

from ATP and presented in Table 3.2. The metering point voltage and current

waveform determined from the Fourier components is in Figure 3.8(b). This confirms

correctness of the Fourier components as the ATP graph 3.8(a) and the recreated 3.8(b)

are same.

Source voltage = 10 Volts RMS and source resistance r = 0.1 ohm. The diode is ideal with zero internal resistance. The resistor R is 5 ohms.

meteringpoint~

v(t)R~

i(t)

r

Figure 3.7: Diode-R circuit and system data


Chapter 3 89

Table 3.2: Fourier components of voltage and current at metering point

Subscript Voltage Vmx Vm angle αx Current Imx Im angle βx Phase angle k volts RMS deg amps RMS deg θx deg

DC 0 -0.0882373 - 0.8823726 - - Fundamental 1 9.9019608 90 0.9803922 90 0

2nd harm 2 0.0416502 0 0.4165024 180 -180 4th harm 4 0.0083630 0 0.0836303 180 -180 6th harm 6 0.0036079 0 0.0360786 180 -180 8th harm 8 0.0020230 0 0.0202298 180 -180

10th harm 10 0.0013028 0 0.0130278 180 -180 12th harm 12 0.0009152 0 0.0091518 180 -180

(a) Waveform from ATP

0 0.005 0.01 0.015 0.02

15

10

5

5

10

15

vm t( )

2 im t( )

t

vm t( ) V01

12

k

2 Vmk⋅ cos k ω⋅ t⋅ αk−( )⋅⎛

⎝⎞⎠∑

=

+:=

im t( ) I01

12

k

2 Imk⋅ cos k ω⋅ t⋅ βk−( )⋅⎛

⎝⎞⎠∑

=

+:=

(b) Waveform determined from the Fourier components in Table 3.2 Figure 3.8: Voltage and current waveforms at the metering point The intent is to remove all active current components from im(t). By removing all the

active current components, the remaining current if any, will be nonactive current

indicating existence of nonactive power. First remove the fundamental active current

using the property that the phase angle between active current and voltage is zero.

This is followed by the load generated harmonic active currents and then the load


Chapter 3 90

generated DC current. The current that is finally left is the nonactive current and it will

be contributing to nonactive power.

First, remove the fundamental active current. The equation is follows

im1 t( ) im t( ) 2 Im1⋅ cos θ1( )⋅ cos ω t⋅ α1−( )⋅−:=

with θ1 = (α1 - β1) is the fundamental phase angle. The resulting waveforms are shown

in Figure 3.9.

0 0.005 0.01 0.015 0.02

15

10

5

5

10

15

vm t( )

4 im1 t( )

t Figure 3.9: Metering point current after removal of fundamental active current Next remove the load generated active current and the equation follows

im2 t( ) im t( ) 2 Im1⋅ cos θ1( )⋅ cos ω t⋅ α1−( )⋅−

2

12

k

2 Imk⋅ cos θk( )⋅ cos k ω⋅ t⋅ αk−( )⋅⎛

⎝⎞⎠∑

=

−:=.

where θk = (αk - βk) is the harmonic phase angle. The resulting waveforms are shown in

Figure 3.10.

0 0.005 0.01 0.015 0.02

15

10

5

5

10

15

vm t( )

5 im2 t( )

t Figure 3.10: Metering point current after removal of fundamental and load generated harmonic active currents


Chapter 3 91

As evidenced from Figure 3.10, only DC current is remaining. The fundamental phase-

angle is zero. Hence the DC is not generated by storage elements in the load. It thus

contributes to active power.

Finally remove the load generated DC current and the equation follows

im3 t( ) im t( ) 2 Im1⋅ cos θ1( )⋅ cos ω t⋅ α1−( )⋅−

2

12

k

2 Imk⋅ cos θk( )⋅ cos k ω⋅ t⋅ αk−( )⋅⎛

⎝⎞⎠∑

=

− I0−:=

with the resulting waveform is shown in Figure 3.11.

0 0.005 0.01 0.015 0.02

15

10

5

5

10

15

vm t( )

100 im3 t( )

t Figure 3.11: Metering point current after removal of fundamental and load generated active harmonic and load generated DC currents Thus, after removing all the active current components, practically zero current is left as

shown in Figure 3.11. This indicates absence of nonactive power. Thus all the current

flowing in the load contributes to active power.

After removal of the active fundamental, active load generated harmonics and DC

current the resulting current is zero. Thus the load current did not have any nonactive

current content that could contribute to nonactive power. It has been shown that a diode

resistor load draws only active power.

It must be pointed out, though, that this is not an efficient utilisation of the supply. The

diode has created a condition where the full capability of the source is not utilised

resulting in inefficient utilisation. However, recalling the layman’s analogy of the

poisonous mushroom used in Chapter 1, this should not be taken to attribute existence


Chapter 3 92

of nonactive power for this case. Other means must be used to address this inefficient

utilisation. This is taken up in a later chapter of this thesis.

3.3 Evaluation benchmarks Useful definitions of powers must represent the physical realities of electrical systems

listed in the requirements 1 to 10 in Section 3.1.2. To judge whether or not a definition

satisfies these requirements, some benchmarks are needed to evaluate the validity of the

definitions. For this a number of benchmark case studies are created and used in the

thesis to evaluate/validate different definitions of powers. The details are given in the

sequel.

The load, in the case studies, is chosen so that the expected instantaneous active and

nonactive powers can be calculated without any ambiguity for a chosen reference

conductor. For the nonsinusoidal source, besides the fundamental both even and odd

harmonics are used. This is done, though even harmonics are uncommon in the power

system, to ensure the definitions’ response to both odd and even harmonics is evaluated.

Also different load arrangements are used to enable evaluation of the definitions for

different conditions. Similar cases have been used by many researchers for example

References [7, 98, 99] for evaluation or discussion of power definitions. The source-

load arrangement used in the case studies have

• sinusoidal and nonsinusoidal source voltage,

• symmetrical and unsymmetrical source voltage,

• different linear and nonlinear load combinations,

• balanced and unbalanced load,

and will provide a faithful evaluation of the definitions.

This will test the requirement 1 under section “requirements” above. To comply with

requirement 2, only the computed voltage and current at the measuring point will be

used for the definitions. In the case study the source impedance is taken to be zero.

This eases the computation and comparison. Likewise, other researches e.g. [7,50,98]

also neglect source impedance in their examples/simulations when comparing or

explaining definitions and their meanings.


Chapter 3 93

3.3.1 Single-phase case This section presents the benchmark case studies for the single-phase case. The six case

studies S1 to S6 shown in Figures 3.12 to 3.16 are used in the evaluation.

A source voltage v(t), of 26.87 Volts rms (50 Hz for S1, S4, S5 and 60 Hz for S2 , S3)

with 33.33% 2nd and 20% 5th harmonic was used in the computation for Cases S1 to S3.

For Cases S4 and S5, only the fundamental voltage was used. The pure load resistance

for Cases S1 and S4 was 5 ohm. For Case S2 the pure load resistance was 15 ohm and

the inductance 100 mH. Case S3 is the same as Case S2 but the inductance is replaced

with a capacitor of value 100 μF. An ideal diode was used in series with the pure

resistor to create a nonlinear load for Case S4. Case S5 is the same as Case S4 but with

a 100 mH inductor instead of the resistor. Case S6 is a series R-L-C load with R being

5 ohms, inductor 20 mH and capacitor 1000 microfarad. Source voltage for S6 is 100

volts at 50 Hz with 30% 4th harmonic. Ideal loads are used to ease computation of

actual powers. This however still enables faithful evaluation.

~v(t)

~

i(t)

Rmeteringpoint

Figure 3.12: 1-Phase Load Case S1

~v(t)

R~

i(t)

Lmetering

point

Figure 3.13: 1-Phase Load Case S2 and S3 For Case S3, L is replaced by C

~v(t)

R~

i(t)

meteringpoint


~v(t)

L~

i(t)

meteringpoint


C

~v(t) R

~

i(t)

Lmeteringpoint

Figure 3.16: 1-Phase series R-L-C Load Case S6


Chapter 3 94

Case S1 tests the performance of the given definitions for nonsinusoidal source supply

with a fully energy-absorbing load. There is zero nonactive or energy storing load and

hence there should be zero nonactive power. Case S2 and S3 have both active and

nonactive power and due to the parallel R-L or R-C load the instantaneous active and

nonactive power can be easily calculated. This evaluates the definitions for active and

nonactive power. Case S4 to S7 will test the performance of definitions for nonlinear

load, S4 for a purely resistive or fully absorbing load (see Section 3.3.9 for a discussion

on this), S5 for and inductive load fully storing load and S6 for a parallel RL load where

active and nonactive power is known. Case S7 is used to test the performance of the

definitions for a series load. For Case S7 active and nonactive currents are determined

using the method used in Section 3.3.5.

3.3.2 Three-phase case Similar to the single-phase case, five simple benchmark case studies are created. The

cases are for 3Ph 3W and 3Ph 4W systems having balanced or unbalanced star (wye)

connected source voltage that is made up of fundamental (50 Hz) plus two harmonics.

The choice of cases tests the definitions for different conditions similar to the singe-

phase case.

Case T1 is 3Ph 3W with 2-phase load which is an example taken from [6] shown in

Figure 3.17. Case T2 is 3Ph 4W star load shown in Figure 3.18. Case T3 is similar to

Case T2 but the load is capacitive. Symmetrical source voltage is used for Cases T1, T2

and T3. Symmetrical here is taken to mean that the magnitudes of the fundamental and

harmonic voltage components are the same in all phases. Case T4 is similar to Case T2

but the source voltage is un-symmetrical. Case T5 (Figure 3.19) is case with a diode

before the resistive star connected load. For Case T1 the computation is done with B-

phase as well as virtual neutral as the reference conductor, while the neutral conductor

is used for the other cases. Source voltage v(t) and load data for these case studies are

given in Table 3.3.


Chapter 3 95

~~~~

A

B

C

ai

ib

ic

Ra

bR

R

R

metering point

Figure 3.17: Load in 2 phases – Cases T1

~

~

~~in

A

B

C

N

ai

ib

ic

LcL

cR

bL

bR

L a

Ra

metering point

Figure 3.18: Star Load – Cases T2, T3 and T4 For Case T3, La, Lb and Lc are replaced by Ca, Cb and Cc

~

~

~~in

A

B

C

N

ai

ib

ic cR

bR

Ra

metering point

Figure 3.19: Star Load with diode - Case T5 Table 3.3: Source Voltage and Load Data Case Load Case Data T1 Source: Symmetrical 3 Phase (Vph = 15 V RMS fundamental + 33.33% 2nd

+ 20% 3rd harmonic) voltage Load: Ra = 0.7 ohm, Rb = 0.3 ohm, Rc = open circuit.

T2 Source: Symmetrical 3 Phase (Vph = 115 V RMS fundamental + 33.33% 2nd + 20% 5th harmonic) voltage Load: Ra = 10.6 ohm, Rb = 8.2 ohm, Rc = 13.2 ohm, La = 0.036 H, Lb = 0.062 H, Lc = 0.042 H


Chapter 3 96

T3 Source: Symmetrical 3 Phase (Vph = 115 V RMS fundamental + 33.33% 2nd + 20% 5th harmonic) voltage Load: Ra = 10.6 ohm, Rb = 8.2 ohm, Rc = 13.2 ohm, Ca = 280 μF, Cb = 160 μF, Lc = 340 μF

T4

Source: Un-Symmetrical 3 Phase (VphA = VphB =115 V RMS fundamental + 33.33% 2nd + 20% 5th harmonic, VphC = 0.8x115V RMS fundamental + 33.33% 2nd + 20% 5th harmonic) voltage Load: Ra = 10.6 ohm, Rb = 8.2 ohm, Rc = 13.2 ohm, La = 0.036 H, Lb = 0.062 H, Lc = 0.042 H

T5 Source: Symmetrical Sinusoidal 3 Phase (Vph = 115 V fundamental) voltage (ideal diode in each phase) Load: Ra = 10.6 ohm, Rb = 8.2 ohm, Rc = 13.2 ohm

For Case T1, k1 = k2 = 1 and the voltages are va0 t( ) 2 V1⋅ sin ω1 t⋅( )⋅ 2 V2⋅ sin ω2 t⋅( )⋅+ 2 V3⋅ sin ω3 t⋅( )⋅+:=

vb0 t( ) k1 2⋅ V1⋅ sin ω1 t⋅2 π⋅

3−⎛⎜

⎝⎞⎟⎠

⋅ k1 2⋅ V2⋅ sin ω2 t⋅2 π⋅

3−⎛⎜

⎝⎞⎟⎠

⋅+ k1 2⋅ V3⋅ sin ω3 t⋅2 π⋅

3−⎛⎜

⎝⎞⎟⎠

⋅+:=

vc0 t( ) k2 2⋅ V1⋅ sin ω1 t⋅2 π⋅

3+⎛⎜

⎝⎞⎟⎠

⋅ k2 2⋅ V2⋅ sin ω2 t⋅2 π⋅

3+⎛⎜

⎝⎞⎟⎠

⋅+ k2 2⋅ V3⋅ sin ω3 t⋅2 π⋅

3+⎛⎜

⎝⎞⎟⎠

⋅+:=

vab t( ) va0 t( ) vb0 t( )−:=

vcb t( ) vc0 t( ) vb0 t( )−:=

For Case T2, k1 = k2 = 1. For Case T4 k1=1, k2 = 0.8, For Case T5 only fundamental voltage is used i.e. V2 = V3 = 0. The voltages are

van t( ) 2 V1⋅ sin ω1 t⋅( )⋅ 2 V2⋅ sin ω2 t⋅( )⋅+ 2 V3⋅ sin ω3 t⋅( )⋅+:=

vbn t( ) k1 2⋅ V1⋅ sin ω1 t⋅2 π⋅

3−⎛⎜

⎝⎞⎟⎠

⋅ k1 2⋅ V2⋅ sin ω2 t⋅2 π⋅

3−⎛⎜

⎝⎞⎟⎠

⋅+ k1 2⋅ V3⋅ sin ω3 t⋅2 π⋅

3−⎛⎜

⎝⎞⎟⎠

⋅+:=

vcn t( ) k2 2⋅ V1⋅ sin ω1 t⋅2 π⋅

3+⎛⎜

⎝⎞⎟⎠

⋅ k2 2⋅ V2⋅ sin ω2 t⋅2 π⋅

3+⎛⎜

⎝⎞⎟⎠

⋅+ k2 2⋅ V3⋅ sin ω3 t⋅2 π⋅

3+⎛⎜

⎝⎞⎟⎠

⋅+:= .

For Case T3, k1 = k2 = 1 and the voltages are van t( ) 2 V1⋅ sin ω1 t⋅( )⋅ 2 V2⋅ sin ω2 t⋅( )⋅+ 2 V3⋅ sin ω3 t⋅( )⋅+:=

vbn t( ) k1 2⋅ V1⋅ sin ω1 t⋅2 π⋅

3−⎛⎜

⎝⎞⎟⎠

⋅ k1 2⋅ V2⋅ sin ω2 t⋅ω2ω1

2 π⋅

3⋅−

⎛⎜⎜⎝

⎞⎟⎟⎠

⋅+ k1 2⋅ V3⋅ sin ω3 t⋅ω3ω1

2 π⋅

3⋅−

⎛⎜⎜⎝

⎞⎟⎟⎠

⋅+:=

vcn t( ) k2 2⋅ V1⋅ sin ω1 t⋅2 π⋅

3+⎛⎜

⎝⎞⎟⎠

⋅ k2 2⋅ V2⋅ sin ω2 t⋅ω2ω1

2 π⋅

3⋅+

⎛⎜⎜⎝

⎞⎟⎟⎠

⋅+ k2 2⋅ V3⋅ sin ω3 t⋅ω3ω1

2 π⋅

3⋅+

⎛⎜⎜⎝

⎞⎟⎟⎠

⋅+:= .

Case T1 tests the performance of the definitions for nonsinusoidal source supply with a

fully energy-absorbing load. Using phase B as reference only active power is

consumed. It shows how nonactive power exists when source voltage (using the virtual


Chapter 3 97

neutral as reference) is not the driving voltage (refer Section 3.3.7 for more

information). Case T2, T3 and T4 have both active and nonactive power and due to the

parallel R-L or C load the instantaneous active and nonactive power are both known.

This evaluates the definitions for active and nonactive power and symmetrical/un-

symmetrical source voltage. Case T5 will test the performance of definitions for

nonlinear load purely resistive or fully absorbing load.

3.3.3 Evaluation criteria The two measures used to perform the evaluation are

• waveforms of the powers of the circuit and

• energy transfer between source and load.

Since the source load arrangement is known the expected instantaneous active and

nonactive powers can be calculated without, any ambiguity for a selected reference

conductor. The instantaneous active and nonactive power waveforms obtained using

the definitions are then compared with the corresponding expected powers.

It is well known that power is the rate of flow of the energy [37], and that there should

always be a unique relationship between power and its energy transfer. Therefore, a

correct power definition should carry the correct information about energy transfer. For

this reason, the energy transfer between the source and load is used as a quantitative

measure to evaluate the definitions. The energy transfer can be one directional (active

energy) or bi-directional (nonactive energy) within one period. Active energy transfer as

defined in the IEEE dictionary [100] is the area under the active power waveform. The

same does not apply to the nonactive case because nonactive power has zero average

value. The solution to this is given below.

Active energy transfer per period T

As defined in IEEE dictionary [100] the active energy transfer, due to flow of

instantaneous active power p(t), is given by T

p0

E p(t)dt= ∫ (3.12)


Chapter 3 98

Nonactive energy transfer per period T

The integral of nonactive power over a period has a zero value. To get around this

problem the energy transfer of the nonactive power wave is taken as the absolute (by

absolute it is meant that the sign of the negative going area is ignored) area under the

waveform. This is reflected in Figure 3.20.

q t( )

t

q t( )

t

==>

Figure 3.20: Instantaneous nonactive power waveform and its absolute value waveform

The energy transfer due to flow of instantaneous nonactive power q(t) is then

T

q0

E q(t) dt= ∫ , (3.13)

where “ q(t) ” is the absolute value of p(t).

3.4 Computation of waveforms and energy transfer 3.4.1 Single-phase cases Given the source voltage and the load, the resistive and inductive current is determined

for each case using the formula given in Subsection 2.2. The expected powers are then

determined using the voltages and the calculated active and nonactive currents. The

voltages, currents and instantaneous powers are presented in graphical format while the

energy transfer is tabulated. The vertical scale on the graphs is in the measured quantity

units (volts, amps, Watts and Vars) e.g. for voltage it is volts. The horizontal scale is in

seconds. Note that for voltage and current the value may be magnified so that it can be

viewed on the common scale. The magnification is shown in the graph. The

computation is outlined below.


Chapter 3 99

Waveforms

Voltages and currents

The resistive current iR(t), active ip(t), inductive current iL(t), capacitive current iC(t) and

nonactive current iq(t) for the applied source voltage, v(t) are first calculated. Note that

for Cases S1 and S4, the inductive current is zero.

Case S1: Resistive load

The voltage and current are given by

[ ]1

13

m mm 1

v(t) 2 V cos m t=

= ω −α∑ (3.14)

13

m mR 1m 1

i (t) 2 I cos m t=

= ω −α⎡ ⎤⎣ ⎦∑ (3.15)

iL(t) = 0 (3.16) where

[V1..Vm..V13] = [26.87, 8.957, 0, 0, 5.374, 0, 0, 0, 0, 0, 0, 0, 0], mm m

VI ,R 2

π= α = , R = 5

ohm. Case S2: Parallel resistive/inductive load

The equation for v(t) is same as equation (3.14). The resistive current iR(t) and

inductive current iL(t) is given by

[ ]m

13

mR Rm 1 mi (t) 2 I cos m t=

= ω −α∑ (3.17)

13

mL L 1m 1 mi (t) 2 I cos m t

2=

π⎡ ⎤= ω −α −⎢ ⎥⎣ ⎦∑ (3.18)

where

[V1..Vm..V13] = [26.87, 8.957, 0, 0, 5.374, 0, 0, 0, 0, 0, 0, 0, 0],

mmR

VIR

= , m

mmL

VIL

=ω ⋅

, R = 15 ohm, L = 100 mH.

Case S3: Parallel resistive/capacitive load

The equation for v(t) is same as equation (3.14). The resistive current iR(t) and

capacitive current iC(t) is given by

[ ]m m

13

R Rm 1 mi (t) 2 I cos h t=

= ω −α∑ (3.19)


Chapter 3 100

C m

13

C 1m 1

mi (t) 2 I cos m t2=

π⎡ ⎤= ω −α +⎢ ⎥⎣ ⎦∑ (3.20)

where

[V1..Vm..V13] = [26.87, 8.957, 0, 0, 5.374, 0, 0, 0, 0, 0, 0, 0, 0],

mmR

VIR

= , m mmCI V C= ω , R = 15 ohm, C = 100 μF.

Case S4: Diode resistive nonlinear load

The equation for v(t) is same as equation (3.14) but only fundamental voltage is used.

The current calculated up to 13th harmonic is as follows 13

R R RDC 1m 1 m mi (t) I 2 I cos m t=

⎡ ⎤= + ω −β⎣ ⎦∑ (3.21)

iL(t) = 0 (3.22) where

[V1..Vm..V13] = [26.87, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],

IDC = 3.419,

[I1..Im..I13] = [3.687, 1.141, 0, 0.228, 0, 0.098,0, 0.055, 0, 0.035, 0, 0.024, 0],

R Rm1,

2π

β = β = π for m = 2 to 13.

R = 5 ohm, L= 0 mH. Case S5: Diode inductive nonlinear load

The equation for v(t) is same as equation (3.14) but only fundamental voltage is used.

The current calculated up to 13th harmonic is as follows

L

13

DC 1m 1

m mi (t) I 2 I cos m t=

⎡ ⎤= + ω −β⎣ ⎦∑ (3.23)

where

[V1..Vh..V13] = [26.87, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],

IDC = 1.2096

[I1..Ih..I13] = [0.8553, 0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0],

R Rm1, 0β = π β = for m = 2 to 13.

R = 0 ohm, L = 100 mH.

Case S6: Series R-L-C load


Chapter 3 101

The determination of active/nonactive current for this case is not so straightforward.

However the rules given in the summary of Section 3.2 can be used to determine the

active and nonactive currents.

The equation for v(t) is same as equation (3.14). The active current ip(t) and nonactive

current iq(t) is given by

p m

13

a m1m 1

mi (t) 2 I cos( )cos m t=

= δ ω −α⎡ ⎤⎣ ⎦∑ (3.24)

m

13

q q m1m 1

mi (t) 2 I sin( )sin m t=

= δ ω −α⎡ ⎤⎣ ⎦∑ (3.25)

where

[V1..Vm..V13] = [100, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0],

mp

mm

VIZ

= , mq

mm

VIZ

= , 2 2m 1Z R (m L)= + ω , 1 1

mm Ltan

R− ω⎛ ⎞δ = ⎜ ⎟⎝ ⎠

, R = 5 ohm, L = 20

mH and C = 1000 μF.

Instantaneous powers

The instantaneous total power is not used in the comparison, as there is no problem with

the definition.

Expected instantaneous active power

Cases S1 , S2, S3, S4, S5 and S6 : Instantaneous active power for S1 to S5 and S6 is

given respectively by

EXP Rp (t) v(t) i (t)= , EXP pp (t) v(t) i (t)= . (3.26) Expected instantaneous nonactive power

Cases S1, S2, S3, S4 and S5: Nonactive power for S1 to S5 and S6 is given respectively

by

EXP Lq (t) v(t) i (t)= , EXP qq (t) v(t) i (t)= . (3.27) The current, voltage and powers waveforms are given in Figures 3.21 to 3.33.


Chapter 3 102

Case S1: Resistive load

0 0.005 0.01 0.015 0.02

50

25

25

50

v t( )

3 i t( )⋅

t

0 0.005 0.01 0.015 0.02

50

25

25

50

v t( )

3 ip t( )⋅

3 iq t( )⋅

t. Figure 3.21: Case S1 Voltage and currents, total i(t) active ip(t) and nonactive iq(t)

0 0.005 0.01 0.015 0.02

100

200

300

400

500

Active power

pEXP t( )

t

0 0.005 0.01 0.015 0.02

10

5

5

10

Non-active power

qEXP t( )

t

Figure 3.22: Case S1 – Expected active and nonactive powers Case S2: Resistive/Inductive Load

0 0.005 0.01 0.015 0.02

50

25

25

50

v t( )

10 i t( )⋅

t

0 0.005 0.01 0.015 0.02

50

25

25

50

v t( )

10 ip t( )⋅

10 iq t( )⋅


0 0.005 0.01 0.015 0.02

30

60

90

120

150

Active power

pEXP t( )

t

.

0 0.005 0.01 0.015 0.02

50

25

25

50

Non-active power

qEXP t( )

t

Figure 3.24: Case S2 – Expected active and nonactive powers


Chapter 3 103

Case S3: Resistive/Capacitive Load

0 0.005 0.01 0.015 0.02

50

25

25

50

v t( )

5 i t( )⋅

t

0 0.005 0.01 0.015 0.02

50

25

25

50

v t( )

5 ip t( )⋅

5 iq t( )⋅

t Figure 3.25: Case S3 Voltage and currents, total i(t) active ip(t) and nonactive iq(t)

0 0.005 0.01 0.015 0.02

30

60

90

120

150

Active power

pEXP t( )

t

0 0.005 0.01 0.015 0.02

100

50

50

100

Non-active power

qEXP t( )

t..

Figure 3.26: Case S3 – Expected active and nonactive powers Case S4: Diode Resistive Nonlinear Load

0 0.005 0.01 0.015 0.02

40

20

20

40

v t( )

3 i t( )⋅

t

0 0.005 0.01 0.015 0.02

40

20

20

40

v t( )

3 ip t( )⋅

3 iq t( )⋅


0 0.005 0.01 0.015 0.02

50

100

150

200

250

300

Active power

pEXP t( )

t

0 0.005 0.01 0.015 0.02

10

5

5

10

Non-active power

qEXP t( )

t

.

Figure 3.28: Case S4 – Expected active and nonactive powers


Chapter 3 104

Case S5: Diode Inductive Nonlinear Load

0 0.005 0.01 0.015 0.02

40

20

20

40

v t( )

5 i t( )⋅

t

0 0.005 0.01 0.015 0.02

40

20

20

40

v t( )

5 ip t( )⋅

5 iq t( )⋅


0 0.005 0.01 0.015 0.02

10

5

5

10

Active power

pEXP t( )

t

0 0.005 0.01 0.015 0.02

75

50

25

25

50

75

Non-active power

qEXP t( )

t

.

Figure 3.30: Case S5 – Expected active and nonactive powers Case S6: Series R-L Load

0 0.005 0.01 0.015 0.02

200

100

100

200

v t( )

5 i t( )⋅

t

0 0.005 0.01 0.015 0.02

200

100

100

200

v t( )

5 ip t( )⋅

5 iq t( )⋅



Chapter 3 105

0 0.005 0.01 0.015 0.02

2000

800

400

1600

2800

4000

Active power

pEXP t( )

t

0 0.005 0.01 0.015 0.02

2000

1000

1000

2000

Non-active power

qEXP t( )

t

Figure 3.32: Case S6 – Expected active and nonactive powers Energy transfer

The energy transfer per period is computed using equations (3.12) and (3.13). The

results are presented in the Table 3.4 below.

Table 3.4: Energy Transfer per period

Expected Energy Transfer Case S1

Active (EP) 3.324395 W sec Nonactive (EN) 0 Var sec

Case S2 Active (EP) 0.923443 W sec

Nonactive (EN) 0.225434 Var Case S3

Active (EP) 0.923443 W sec Nonactive (EN) 0.460962 Var sec


Nonactive (EN) 0 Var sec Case S5

Active (EP) 0 W sec Nonactive (EN) 0.585229 Var sec


Nonactive (EN) 13.202998 Var sec 3.4.2 Three-phase cases Similar to the single-phase case, using the “driving voltage” and the known load, the

conductor current is determined for each case. This is then resolved into in-phase

(active) and quadrature (nonactive) current using the conductor to reference source

voltage. The expected powers are then determined using the source voltages and the


Chapter 3 106

calculated active and nonactive currents. The voltages, currents and instantaneous

powers are presented in graphical format while the energy transfer is tabulated. The

vertical scale on the graphs is in the measured quantity units (volts, amps, Watts and

Vars) e.g. for voltage it is volts. The horizontal scale is in seconds. Note that for

voltage and current the value may be magnified so that it can be viewed on the common

scale. The magnification is shown in the graph. The computation is outlined below.

Voltages and vurrents

The current flow as driven by the driving voltage is first determined for each conductor.

The conductor current is then decomposed into resistive (active) current iR(t) and

inductive/capacitive (nonactive) current iL(t)/iC(t) (note that resistive and

inductive/capacitive current is used here instead of active and nonactive to ease

readability) based on the applied source voltage. In the formulae the subscripts a, b, c

and n are used to represent the phases A, B, C and N while subscripts 1, 2 …, 13

represent the fundamental and harmonics.

Note that the current in the reference conductor is not determined as it is not required

for the powers calculation.

Case T1: 3Ph 3W unsymmetrical source voltage with 2-phase unbalanced resistive load

The voltages are then given by 13

an 1hh 1

v (t) 2 V sin h t=

= ω⎡ ⎤⎣ ⎦∑ (3.28)

13

1bn hh 1

2v (t) 2 V sin h t3=

π⎡ ⎤= ω −⎢ ⎥⎣ ⎦∑ (3.29)

13

cn 1hh 1

2v (t) 2 V sin h t3=

π⎡ ⎤= ω +⎢ ⎥⎣ ⎦∑ (3.30)

(note that for Case T1 “0” replaces “n” in (3.28), (3.29) and (3.30))

where

[V1..Vh..V13] = [15, 5, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] volt for Case T1,

[V1..Vh..V13] = [115, 38.33, 0, 0, 23, 0, 0, 0, 0, 0, 0, 0, 0] volt for Case T2 to T4 and

[V1..Vh..V13] = [115, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] volt for Case T5.

Using B-phase as reference

The vector diagram for the fundamental is shown in Figure 3.33.


Chapter 3 107

~~~~

A

B

C

ai

ib

ic

Ra

bR

R

R

A

BC

Va

Vb

Ia

Ib

30 deg

30 deg

Vab

Figure 3.33: Vectors Case T1 (B-phase as reference)

The driving voltage behind the current flow in Ra and Rb is Vab.

Vab = Va0 - Vb0 13

1ab hh 1

v (t) 6 V sin h t6=

π⎡ ⎤= ω +⎢ ⎥⎣ ⎦∑ (3.31)

Vcb = Vc0 - Vb0

13

1cb hh 1

v (t) 6 V sin h t2=

π⎡ ⎤= ω +⎢ ⎥⎣ ⎦∑ (3.32)

The conductor currents are given by

a b

13h

aR 1h 1

Vi (t) 6 sin h t

R R 6=

π⎡ ⎤= ω +⎢ ⎥+ ⎣ ⎦∑ (3.33)

aLi (t) 0= (3.34)

cRi (t) 0= (3.35)

cLi (t) 0= (3.36) Using virtual neutral as reference


~~~~

A

B

C

ai

ib

ic

Ra

bR

R

R

A

BC

Van

Vbn

Ia

30 deg

30 deg

Vcn

Figure 3.34: Vectors Case T1 (Virtual neutral as reference)


Chapter 3 108

The voltages are as given in equations (3.28), (3.29) and (3.30). The conductor currents are given by

a b

13h

a 1h 1

Vi (t) 6 sin h t

R R 6=

π⎡ ⎤= ω +⎢ ⎥+ ⎣ ⎦∑ (3.37)

a b

13h

1bh 1

V 5i (t) 6 sin h tR R 6=

π⎡ ⎤= ω −⎢ ⎥+ ⎣ ⎦∑ (3.38)

ci (t) 0= . (3.39) Decomposing into to active and nonactive currents give

a b

13h

aR 1h 1

Vi (t) 6 sin h t cos

R R 6=

π⎡ ⎤= ω⎡ ⎤⎣ ⎦ ⎢ ⎥+ ⎣ ⎦∑ (3.40)

a b

13h

aL 1h 1

Vi (t) 6 sin h t sin

R R 2 6=

π π⎡ ⎤ ⎡ ⎤= ω +⎢ ⎥ ⎢ ⎥+ ⎣ ⎦ ⎣ ⎦∑ (3.41)

a b

13h

1bRh 1

V 2i (t) 6 sin h t cosR R 3 6=

π π⎡ ⎤ ⎡ ⎤= ω −⎢ ⎥ ⎢ ⎥+ ⎣ ⎦ ⎣ ⎦∑ (3.42)

a b

13h

1bLh 1

V 5i (t) 6 sin h t sinR R 6 6=

π π⎡ ⎤ ⎡ ⎤= ω +⎢ ⎥ ⎢ ⎥+ ⎣ ⎦ ⎣ ⎦∑ (3.43)

cRi (t) 0= (3.44)

cLi (t) 0= . (3.45)

Case T2: 3Ph 4W with unsymmetrical source voltage and unbalanced star load


~

~

~~in

A

B

C

N

ai

ib

ic

L cL

cR

bL

bR

La

Ra

A

BC

Van

Vbn

Ia

Ib

Vcn

Ic

Figure 3.35: Vectors Case T2 (Neutral as reference) The voltages are as given in equations (3.28), (3.29) and (3.30).


Chapter 3 109

The active and nonactive currents are

a

13h

aR 1h 1

Vi (t) 2 sin h t

R== ω⎡ ⎤⎣ ⎦∑ (3.46)

a

13h

aL 11h 1

Vi (t) 2 sin h t

h L 2=

π⎡ ⎤= ω −⎢ ⎥ω ⎣ ⎦∑ (3.47)

b

13h

1bRh 1

V 2i (t) 2 sin h tR 3=

π⎡ ⎤= ω −⎢ ⎥⎣ ⎦∑ (3.48)

b

13h

1bL1h 1

V 5i (t) 2 sin h th L 6=

π⎡ ⎤= ω +⎢ ⎥ω ⎣ ⎦∑ (3.49)

c

13h

cR 1h 1


π⎡ ⎤= ω +⎢ ⎥⎣ ⎦∑ (3.50)

c

13h

11h 1

ViL(t) 2 sin h t

h L 6=

π⎡ ⎤= ω +⎢ ⎥ω ⎣ ⎦∑ . (3.51)

Case T3: 3Ph 4W with symmetrical source voltage and unbalanced star load


~

~

~~in

A

B

C

N

ai

ib

ic

Cc

cR

bC

bR

Ca

Ra

A

BC

Van

Vbn

Ia

Ib

Vcn Ic Figure 3.36: Vectors Case T3 (Neutral as reference)

The voltages are as given in equations (3.28), (3.29) and (3.30). The active and nonactive currents are

a

13h

aR 1h 1

Vi (t) 2 sin h t

R== ω⎡ ⎤⎣ ⎦∑ (3.52)

a

13aL 1 1h

h 1i (t) 2V h C sin h t

2=

π⎡ ⎤= ω ω +⎢ ⎥⎣ ⎦∑ (3.53)

b

13h

1bRh 1


π⎡ ⎤= ω −⎢ ⎥⎣ ⎦∑ (3.54)


Chapter 3 110

b

131 1bL

h 1i (t) 2 h C sin h t

6=

π⎡ ⎤= ω ω −⎢ ⎥⎣ ⎦∑ (3.55)

c

13h

cR 1h 1


π⎡ ⎤= ω +⎢ ⎥⎣ ⎦∑ (3.56)

cL c

131 1

h 1

2i (t) 2 h C sin h t3=

π⎡ ⎤= ω ω +⎢ ⎥⎣ ⎦∑ . (3.57)



~

~

~~in

A

B

C

N

ai

ib

ic

L cL

cR

bL

bR

L a

Ra

A

BC

Van

Vbn

Ia

IbVcn

Ic

Figure 3.37: Vectors Case T4 (Neutral as reference)


a

13h

aR 1h 1

Vi (t) 2 sin h t

R== ω⎡ ⎤⎣ ⎦∑ (3.58)

a

13h

aL 11h 1

Vi (t) 2 sin h t

h L 2=

π⎡ ⎤= ω −⎢ ⎥ω ⎣ ⎦∑ (3.59)

b

13h

1bRh 1


π⎡ ⎤= ω −⎢ ⎥⎣ ⎦∑ (3.60)

b

13h

1bL1h 1

V 5i (t) 2 sin h th L 6=

π⎡ ⎤= ω +⎢ ⎥ω ⎣ ⎦∑ (3.61)

c

13h

cR 1h 1

0.8V 2i (t) 2 sin h tR 3=

π⎡ ⎤= ω +⎢ ⎥⎣ ⎦∑ (3.62)

c

13h

11h 1

0.8ViL(t) 2 sin h t

h L 6=

π⎡ ⎤= ω +⎢ ⎥ω ⎣ ⎦∑ (3.63)


Chapter 3 111

Case T5: 3Ph 4W with symmetrical source voltage and nonlinear unbalanced star load


~

~

~~in

A

B

C

N

ai

ib

ic cR

bR

Ra

A

BC

Van

Vbn

Ia

Ib

Vcn

Ic

Figure 3.38: Vectors Case T5 (Neutral as reference)


a

13h

aR 1h 1

Vi (t) 2 sin h t

R== ω⎡ ⎤⎣ ⎦∑ if Van>0, 0 otherwis (3.64)

aLi (t) 0= (3.65)

b

13h

1bh 1


π⎡ ⎤= ω −⎢ ⎥⎣ ⎦∑ if Vbn >0, 0 otherwise (3.66)

bLi (t) 0= (3.67)

c

13h

c 1h 1


π⎡ ⎤= ω +⎢ ⎥⎣ ⎦∑ if Vcn >0, 0 otherwise (3.68)

cLi (t) 0= . (3.69)

Instantaneous powers

The instantaneous total power is not used in the comparison, as there is no problem with

the definition.

Expected instantaneous active power

Case T1 (B-phase as reference):

Instantaneous active power is given for phase AB and CB respectively by

abEXP aRp (t) v(t) i (t)= (3.70)

cbEXP cRp (t) v(t) i (t)= . (3.71)

Cases T1(virtual neutral as reference), T2, T3, T4 and T5:


Chapter 3 112

Instantaneous active power for phases AN, BN and CN is given respectively by

aEXP aRp (t) v(t) i (t)= (3.72)

bEXP bRp (t) v(t) i (t)= . (3.73)

cEXP cRp (t) v(t) i (t)= . (3.74)

Expected instantaneous nonactive power

Case T1 with B-phase as reference:

Instantaneous nonactive power is given for phase AB and CB respectively by

abEXP aLq (t) v(t) i (t)= (3.75)

cbEXP cLq (t) v(t) i (t)= . (3.76)

Cases T1(virtual neutral as reference), T2, T3, T4 and T5:

Instantaneous nonactive power for phases AN, BN and CN is given respectively by

aEXP aLq (t) v(t) i (t)= (3.77)

bEXP bLq (t) v(t) i (t)= (3.78)

cEXP cLq (t) v(t) i (t)= . (3.79)

The voltage, current and powers waveforms are given in Figures 3.39 to 3.50

Case T1: 3Ph 3W unsymmetrical source voltage with 2-phase unbalanced resistive load

B-phase as reference

0.02 0.025 0.03 0.035 0.04

50

25

25

50

75

vab t( )

vcb t( )

0.5 ia t( )⋅

ic t( )

t.

Figure 3.39: Case T1 (B-phase as reference) Voltage and current


Chapter 3 113

0.02 0.025 0.03 0.035 0.04

1000

2000

3000

Active powers

pabEXP t( )

pcbEXP t( )

t

0.02 0.025 0.03 0.035 0.04

1

0.5

0.5

1

Non-active powers

qabEXP t( )

qcbEXP t( )

t

Figure 3.40: Case T1 (B-phase as reference) Expected active and nonactive powers Virtual Neutral as reference

0.02 0.025 0.03 0.035 0.04

50

25

25

50

75

A-phase

van t( )

ia t( )

iaR t( )

iaL t( )

t

0.02 0.025 0.03 0.035 0.04

75

50

25

25

50

B-Phase

vbn t( )

ib t( )

ibR t( )

ibL t( )

t

0.02 0.025 0.03 0.035 0.04

20

10

10

20

30

40

C-phase

vcn t( )

ic t( )

icR t( )

icL t( )

t

Figure 3.41: Case T1 Voltage and current

0.02 0.025 0.03 0.035 0.04

500

1000

1500

2000

Active powers

paEXP t( )

pbEXP t( )

pcEXP t( )

t

0.02 0.025 0.03 0.035 0.04

500

250

250

500

Non-active powers

qaEXP t( )

qbEXP t( )

qcEXP t( )

t

Figure 3.42: Case T1 (Virtual neutral as reference) Expected active and nonactive powers


Chapter 3 114


0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

A-phase

van t( )

5 ia t( )

5 iaR t( )

5 iaL t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

B-Phase

vbn t( )

5 ib t( )

5 ibR t( )

5 ibL t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

C-phase

vcn t( )

5 ic t( )

5 icR t( )

5 icL t( )

t

.


0.02 0.025 0.03 0.035 0.04

2000

4000

6000

8000

Active powers

paEXP t( )

pbEXP t( )

pcEXP t( )

t

0.02 0.025 0.03 0.035 0.04

2000

1000

1000

2000

Non-active powers

qaEXP t( )

qbEXP t( )

qcEXP t( )

t

Figure 3.44: Case T2 Expected active and nonactive powers


Chapter 3 115


0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

A-phase

van t( )

5 ia t( )

5 iaR t( )

5 iaL t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

B-Phase

vbn t( )

5 ib t( )

5 ibR t( )

5 ibL t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

C-phase

vcn t( )

5 ic t( )

5 icR t( )

5 icL t( )

t

.


0.02 0.025 0.03 0.035 0.04

2000

4000

6000

Active powers

paEXP t( )

pbEXP t( )

pcEXP t( )

t

0.02 0.025 0.03 0.035 0.04

5000

2500

2500

5000

Non-active powers

qaEXP t( )

qbEXP t( )

qcEXP t( )

t



Chapter 3 116


0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

A-phase

van t( )

5 ia t( )

5 iaR t( )

5 iaL t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

B-Phase

vbn t( )

5 ib t( )

5 ibR t( )

5 ibL t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

C-phase

vcn t( )

5 ic t( )

5 icR t( )

5 icL t( )

t

.


0.02 0.025 0.03 0.035 0.04

2000

4000

6000

8000

Active powers

paEXP t( )

pbEXP t( )

pcEXP t( )

t

0.02 0.025 0.03 0.035 0.04

2000

1000

1000

2000

Non-active powers

qaEXP t( )

qbEXP t( )

qcEXP t( )

t



Chapter 3 117


0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

A-phase

van t( )

5 ia t( )

5 iaR t( )

5 iaL t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

B-Phase

vbn t( )

5 ib t( )

5 ibR t( )

5 ibL t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

C-phase

vcn t( )

5 ic t( )

5 icR t( )

5 icL t( )

t

.


0.02 0.025 0.03 0.035 0.04

1000

2000

3000

4000

Active powers

paEXP t( )

pbEXP t( )

pcEXP t( )

t

0.02 0.025 0.03 0.035 0.04

1

0.5

0.5

1

Non-active powers

qaEXP t( )

qbEXP t( )

qcEXP t( )

t

Figure 3.50: Case T5 Expected active and nonactive powers Energy transfer and average power

The energy transfer per period is computed using equations (3.12) and (3.13) for each

phase. The results for energy transfer are presented in the Table 3.5.


Chapter 3 118

Table 3.5: Energy Transfer per period Expected Energy Transfer

Phase AB Phase CB Units Case T1 (Ref - B)

Active ET (EP) 15.54 0 W sec Nonactive ET (EN) 0 0 Var sec

Phase A Phase B Phase C Units Case T1 (Ref-Neutral)

Active ET (EP) 7.77 7.77 0 W sec Nonactive ET (EN) 3.891 3.837 0 Var sec

Case T2 Active ET (EP) 28.723480 37.130352 23.065825 W sec

Nonactive ET (EN) 16.517157 9.360886 13.818451 Var sec Case T3

Active ET (EP) 28.72348 37.130352 23.065825 W sec Nonactive ET (EN) 23.666532 13.523660 28.737659 Var sec

Case T4 Active ET (EP) 28.72348 37.130352 14.762128 W sec

Nonactive ET (EN) 16.517157 9.360886 8.843808 Var sec Case T5

Active ET (EP) 12.476415 16.128019 10.018921 W sec Nonactive ET (EN) 0 0 0 Var sec

3.5 Conclusion In this chapter the essential requirements of definitions have been highlighted and the

benchmark case studies to be used in later chapters to test the definitions and the

expected results have been outlined.

Single phase power component definitions for instantaneous and average power

Chapter 4 119

4. SINGLE-PHASE POWER COMPONENT DEFINITIONS FOR INSTANTANEOUS AND AVERAGE POWERS

A brief overview and analysis of some current theories and definitions was presented in

Chapter 2 followed, in Chapter 3, by listing the requirements on power definitions and

the creation of benchmark case studies designed for evaluating the definitions of power.

This Chapter presents the proposed single-phase definitions. One important aspect to

bear in mind is the background technical information outlined in Chapter 2 Section 2.2

and Chapter 3 Section 3.2 as well as the analysis of RMS powers in Chapter 2

Subsection 2.3.1. Therein lie the reasons adopted for the approach taken in the

proposed definitions.

4.1 Introduction The prime objective in the quest for the definitions was to ensure that the source-load

properties are faithfully subscribed to. Some important conclusions in the analysis

performed in the background technical information to attain this are listed below.

L4.1. Analysis in Subsection 2.2.3 indicates that generally the current waveform is

not a scaled version of the voltage waveform. The conductance of a load may

not be linear within a period. Hence the equivalent conductance of the load is

taken to be time variant.

L4.2. Cross-harmonic power can contribute to both active and non-active powers as

outlined in Subsection 2.2.2.

L4.3. Information about the voltage and current waveform is lost when determining

its RMS value (Subsection 3.3.1) unless the waveform is sinusoidal (that is of

single harmonic). Thus only RMS values of a single harmonic will be utilised

in the definitions.

L4.4. The approach using space vectors with current projection was not used as it

has been shown in subection 2.2.6 to have difficulties in the presence of

harmonics with reactance in the load.

In addition to L4.1 to L4.3, the requirements listed in Chapter 3 Subsection 3.1.2 are

also considered. With the above consideration the approach was to use both frequency


Chapter 4 120

and time domains. The frequency domain was necessary to enable analysis using each

harmonic and to use orthogonal decomposition of the harmonic current. The

contribution DC current component, which exists for example in nonlinear situation,

must to be addressed carefully in determination of active and nonactive powers because

they may not be orthogonal. This is pointed out by Cohen [29] who questioned the

assumption of orthogonality in defining active and “reactive” (nonactive) powers for

nonlinear situations. DC current is apportioned between active and non-active based on

the fundamental phase angle. This apportioning of DC current into active and non-

active is necessary since as shown in Subsection 2.2.9 and examples S4 and S5 in

Subsection 2.4.1 that DC current component can contribute to both active and non-

active power. The time domain, which lends itself to easy algebraic manipulation, is

used to “collate” the harmonic terms to obtain the instantaneous powers’ waveforms.

Instantaneous total power is composed of instantaneous active and non-active power.

These active and non-active parts arise from the presence of energy consuming elements

(resistors), storing elements (inductors and capacitors) and generating elements

(sources) in the load. The source voltage connected to the load, gives rise to active and

non-active currents that manifest as the measurable total current at the metering point

(Figure 4.1).

meteringpoint

~v(t) ~

i(t)

loadsource

Figure 4.1: Metering point

For any particular load the total instantaneous power can be decomposed into active and

non-active power components based on source/load characteristics.

For sinusoidal system, the active current is determined from the total current and the

phase angle on the basis of the known fact that active current will be in phase with the

voltage and the non-active current in quadrature with the voltage. The active current

gives rise to active power and non-active current to non-active power. Only the

fundamental component exists in sinusoidal systems


Chapter 4 121

For the nonsinusoidal system, the concept of the sinusoidal system, as outlined in the

above paragraph, is extended in the proposed definitions. The determination of the

active and non-active power is based on the harmonic components of the measurable

voltage and current at the metering point. The current for each harmonic is separated

into components, in-phase and in quadrature with the corresponding voltage. Active

power is the contribution by the current harmonic component in phase with the

corresponding voltage harmonic including cross-harmonic products of all voltages and

in-phase current harmonic components (L4.2 above refers). Non-active power,

likewise, is the contribution from the quadrature current component. For non-sinusoidal

systems, however, because of the presence of DC and harmonics in addition to the

fundamental, five sub-components for each of the active and non-active parts are

defined. The active and non-active components proposed exhibit a meaning in the sense

that they have a direct relationship with the source and load and are indicative of some

characteristic of the source-load relationship. This is because the voltage and current

harmonic components, that are used to define the powers, are a function of the source

and load. They depend on the properties of the source and load. Therein lies the basic

concept of the proposed definitions.

The proposed definitions are significant because the load model at the measuring point

is an attempt to closely represent the actual. This leads to good knowledge of the time

profile of active and non-active components and allows accurate measurement and

optimal compensation.

The definitions are evaluated using the methodology outlined in Chapter 2 plus more

cases that highlight additional points. These will be outlined in the ensuing analysis.

4.2 Background technical Information Following the trend of Chapters 2 and 3 some technical information directly relevant to

this chapter is outlined.


Chapter 4 122

4.2.1 Discussion on nonlinear Diode RL parallel and series load

meteringpoint

~v(t)

R~

i(t)

L meteringpoint

~v(t) R

~

i(t)

L

i (t) i (t)R L

(a) (b) Figure 4.2: Diode RL parallel and series load Consider the parallel and series diode RL circuit as shown in Figure 4.2(a) and 4.2(b)

respectively. For the parallel RL case the voltage and current waveforms are shown in

Figure 4.3(a) and for the series case in Figure 4.3(b).

v t( )

i t( )

iR t( )

iL t( )

t (a)

v t( )

i t( )

t (b)

Figure 4.3: Voltage and current waveforms for the diode RL series and parallel load From Figure 4.3(a) it is apparent that for the parallel case the DC component of current,

which is a part of iL(t), flows through the inductor only (green waveform). The current

flowing through the resistor iR(t) has no DC component. Thus the DC component of the

current contributes only to the non-active power. For the series case, the DC component

is a part of the current i(t). This current i(t) flows through both the resistor and inductor.

Thus the DC component for the series case contributes to both active and non-active

powers. This simple example illustrates that generally DC component of the current

can contribute to both active and non-active power. It can however, contribute

wholly to non-active part (for a diode parallel RL load) even if the load appears

resistive-reactive as viewed from the metering point.


Chapter 4 123

4.3 The proposed single phase instantaneous power definitions The new definitions are based on the concept of the universally accepted sinusoidal

system. In the sequel, voltages, currents and powers are in the time domain and will be

referred to as “instantaneous”. The letters “s”, “p” and “q”, with subscripts as

necessary, are used to designate the total, active and non-active instantaneous powers

respectively. These will be explained as they are used.

4.3.1 Load model

i(t)

~v(t)

~ Load B(t)G(t)~ v(t)~

i(t)

Figure 4.4: Load Model The load model used is shown in Figure 4.4. The electrical system is represented by an

equivalent parallel time variant conductance G(t) (L4.1 above refers) and susceptance

B(t). Active power is consumed by G(t) and non-active power results from B(t).

4.3.2 Sinusoidal system Since the new definitions are based on the generally accepted concept of the sinusoidal

system, the sinusoidal system is first discussed. In sinusoidal systems the total power,

s1, is composed of active and non-active power. The total fundamental current, i1, is

decomposed into active current, i1a , and non-active current, i1q , using the phase

angle 1θ .

The active current is in-phase with the voltage v1, and the non-active current is in-

quadrature with the voltage. Note that “1” is used to represent the fundamental, “p” the

active and “q” the non-active parts. Therefore

1 1p 1qi i i= + . (4.1)

The product of active current and the voltage gives the active power, p1, and the

quadrature current and voltage the non-active power, q1.

1 1 1pp v i= , 1 1 1qq v i= and s1 = v1 i1 = 1 1p 1qv ( i i )+ (4.2)


Chapter 4 124

Figure 4.5 shows the concept and Table 4.1 tabulates the components for the sinusoidal

case.

v(t)

i(t)

Powercomponents

Activepower

p(t)

Non-activepower

q(t)

X

X

+Totalpower

s(t)i 1

1v

q1

p1

i 1p

i 1q

Figure 4.5 Concept of powers (sinusoidal system) Table 4.1: Voltage and current components for sinusoidal case

Harmonic Voltage Phase angle

In-phase current or current contributing to active power

Quadrature current or current contributing to non-active power

Fund. 1 12 V cos( t )ω −α 1θ 1p 1 1 1i 2 I cos( t )cos= ω −α θ 1q 1 1 1i 2 I sin( t )sin= ω −α θ

Note: V1, I1 is the fundamental voltage and current RMS value, 1α the voltage phase angle, 1θ the phase angle between voltage and current, ω is the angular frequency 2 fπ , f is fundamental frequency. 4.3.3 Non-sinusoidal system The proposed definitions extend the concept of sinusoidal case to the non-sinusoidal

case. The difference as compared to the sinusoidal case is that harmonic voltage and

current components are also present. Figure 4.6 shows the concept of the proposed

definitions and Table 4.2 tabulates the components for the non-sinusoidal case. The

voltage v(t) and current i(t) are represented by cosine Fourier components.

v(t) = V0 + v1 + vh+ vg := 0 1 1V 2 V cos( t )+ ω −α

h h g gh g

2 V cos(h t ) 2 V cos(g t )+ ω −α + ω −α∑ ∑ (4.3)

i(t) = Io + i1 + ih + ig := 0 1 1I 2 I cos( t )+ ω −β

h h g gh g

2 I cos(h t ) 2 I cos(g t )+ ω −β + ω −β∑ ∑ (4.4)

where V0 and I0 are respectively DC voltage and current, V1, Vh, Vg, I1, Ih and Ig are

RMS values of harmonic components v1, vh, i1, ih and ig ; ω is the angular frequency


Chapter 4 125

2 fπ ; f is fundamental frequency; t is the time, xα and xβ (x = 1, h, g) the voltage and

current phase angle. Subscript "1"represents the fundamental, "h" represents the source

generated voltage/current harmonics and "g" (g ≠ h) the load generated voltage/current

harmonics. Note that, as per L4.3 in Section 4.1, only the RMS values for a particular

harmonic are used in the definitions. It is pointed out that in [37] “h” and “g” are

combined in the analysis, while in [16] “h” and “g” are separately treated as is done in

this thesis. Load generated current harmonics flow from the load toward the source

[16]. The metering point voltage is used to determine the direction of the currents. The

direction of fundamental

current is taken as the

reference for positive direction

that defines source to load

feed. Note that for harmonics

that arise in both the source

and load only the nett will

exist as one of the elements of

“h” or “g”.

v(t)

i(t)

Harmoniccomponents

iiii

01hg

Power components

Activepower

p(t)

Non-activepower

q(t)

sum

sum

iiii

0p1phpgp

iiii

0q1qhqgq

Totalpower

s(t)

X

X

+

01hg

vvvv

pp

pp

0D0X

X1Xh

01hgXHXg

pppppp

01hgXHXg

qqqqqq

pppppp

01hgXHXg

q

qq

0X

X1Xh

Figure 4.6: Concept of powers (nonsinusoidal system)

Table 4.2: Voltage and current components for non-sinusoidal case

Harm- onic Source Voltage Phase

angle

In-phase current or current contributing to active

power

Quadrature current or current contributing to

non-active power

DC

V0 , zero or load generated DC voltage

drop that will be negative

0 0p 0 xI I k0= ( x = DC,1,h,g) 0q 0 xI I (1 k0 )= − ( x = DC,1,h,g)

Fund. 1 1 1v 2 V cos( t )= ω −α 1θ 1p 1 1 1i 2 I cos( t )cos= ω −α θ 1q 1 1 1i 2 I sin( t )sin= ω −α θ

hth h h hv 2 V cos(h t )= ω −α hθ hp h h hi 2 I cos(h t )cos= ω −α θ hq h h hi 2 I sin(h t )sin= ω −α θ

gth vg , zero or load generated voltage drop gγ gp g g gi 2 I cos(g t )cos= ω −α γ gq g g gi 2 I sin(g t )sin= ω −α γ

Note: xk0 is DC apportioning factor, 1 handθ θ the fundamental and harmonic phase angle between

corresponding voltage and source generated current harmonic, gγ is the phase angle defined for load generated current harmonics The current for each harmonic including DC, is decomposed into active and non-active

components based on harmonic phase angle. Thus

ip(t) = 0p 1p hp gpI i i i+ + + (4.5)


Chapter 4 126

iq(t) = 0q 1q hq gqI i i i+ + + . (4.6)

4.3.3.1 Current decomposition The decomposition of the current is based on the following.

DC current apportioning factor k0x

The presence of DC could be caused by the presence of DC voltage in the source as

well as load that is nonlinear. The nonlinear load can be made up of energy consuming

as well as energy storing elements. Hence I0 can contribute to both active and non-

active power. This had been discussed in Subsection 4.2.1. The DC apportioning factor

“k0x'' gives the active part and (1 – k0x) the contribution to the non-active part. “k0x” is

determined from the fundamental and harmonic phase-angle xcosθ (x = DC, 1, h, g). It

is decided that the DC current apportioning is proportional to the load characteristics at

the fundamental or particular harmonic. The proposal is that relationship is k0x =

( )2xcosθ defines this factor. Note that k0x changes with harmonic (for example for DC

k0DC = 1 and for fundamental k01 = ( )21cosθ ). When viewed from the metering point,

there is no knowledge of the load elements. The characteristic of load is expected to be

related to the fundamental phase-angle. This is the reason for basing the k0x factor on

the fundamental phase-angle. However it must be highlighted as has been shown in

Subsection 4.2.1 that the DC part of current may wholly contribute to non-active power

which can be a source of error. In this thesis a simple rule is set up to detect this

condition. This condition is detected if the DC current exceeds the sum of the

magnitudes of the fundamental and all the harmonics. Hence the rule for defining the

DC factor is as follows.

k0x = ( )2 0 n

n 1,h,gx if I I

0 Otherwise

cosθ=

< ∑ (4.7)

This is a “first pass” simple rule and further study into improving the factor k0 is an

indicated research area. The research should involve determination of the series or

parallel nature of the load. Knowledge of this can be used to improve k0 to reflect more

closely a nonlinear load.


Chapter 4 127

Determination of fundamental/ source generated harmonic phase angle mθ

Angle “ mθ ” (m=1,h) is the phase angle between the fundamental/harmonic voltage and

corresponding fundamental /harmonic current.

Determination of load generated harmonic phase angle gγ

If the source impedance is not negligible γg is obtained from the Fourier components as

for the case of source-generated harmonic, that is, the phase angle between the voltage

and corresponding current harmonic. When the source impedance is negligible, the

determination of γg is based on the fundamental phase angle 1θ . The phase angle γg for

harmonic “g” is defined as follows

1 1g

tanθγ : tang

− ⎛ ⎞= + π⎜ ⎟

⎝ ⎠. (4.8)

Refer Appendix C for the derivation. This is a “first pass” definition and further

research may be useful. The research should involve determination of the series or

parallel nature of the load, knowledge of which can be used to improve γg to reflect a

nonlinear load closely. However, likelihood of using equation (4.8) in real systems is

very small, in fact negligible, given that real systems have non-negligible source

impedance.

Presently both definitions k0x and γg seem to give good results. However, note that the

power definitions given below are independent of the method of determination of k0x

and γg. This permits changes to k0x and γg should it be necessary since these factors are

new definitions.

4.3.3.2 Powers The powers are determined with the aid of Table 4.2 and Figure 4.6. Active power is

contributed by the product of voltage and in-phase current (active current components)

and includes cross-harmonic products of voltages with the in-phase current components.

0 1 h g 0p 1p hp gpp(t) (V v v v ) (I i i i )= + + + + + + (4.9)

Non-active power, likewise, is the contribution from the voltage and “quadrature

current” component.

0 1 h g 0q 1q hq gqq(t) (V v v v ) (I i i i )= + + + + + + (4.10)


Chapter 4 128

Compared to the existing definitions for example [37], the major difference in the

proposed definitions is that voltage and current product of non-identical harmonics,

(cross-harmonic components) contribute to both active and non-active power. In

Reference [37] these components are taken to have zero contribution to active power.

Similarly, the DC components also contribute to both active and non-active power.

The power is separated into components as outlined below. The components are

designated with letters “sx(t)”, “px (t)” and “qx (t)” where subscript “x” designates

component. Subscript “0D” designates the DC power component and “0X” the DC

cross-harmonic component related to the DC voltage and current. "1" represents the

fundamental, "h" the source generated voltage and current harmonic, "g" the load

generated voltage and current harmonic, “X1” and “Xh” the cross-harmonic for source

generated current harmonics and "Xg" the cross-harmonic for load generated current

harmonics. Using “s(t) = v(t)i(t)”, equations (4.9) and (4.10), and Table 4.2, the total,

active and non-active instantaneous powers are defined below.

Instantaneous power

The total instantaneous power s(t) which is equal to the product of v(t) and i(t) is given

by the following components

s(t) = v(t) i(t)

= s0D(t) + s0X(t) + s1(t) + sh(t) + sg(t) + + sX1(t) + sXh(t) + sXg(t) (4.11)

The components of total instantaneous power are detailed in (a) to (f):

(a) DC based power s0(t)

DC based power s0(t) is made up of two sub-components that is DC power s0D(t) and

DC cross harmonic power s0X(t) with

s0(t) = s0D(t) + s0X(t). (4.12)

(a1) DC power

s0D(t) := V0 I0 ; (4.13)

(a2) DC cross-harmonic power s0X(t) resulting from the presence of DC voltage and

current components.

s0X (t) := 0 1 1 0 m mm h,g

2 V I cos( t ) 2 V I cos(h t )=

ω −β + ω −β∑


Chapter 4 129

1 0 1 m 0 mm h,g

2 V I cos( t ) 2 V I cos(m t )=

+ ω −α + ω −α∑ (4.14)

(b) Fundamental power s1(t) resulting from the presence of fundamental voltage and

current components

s1 (t) := 1 1 1 12 V I cos( t ) cos( t )ω −α ω −β (4.15) (c) Source generated harmonic power sh(t)

resulting from the presence of harmonic voltage and current components.

sh (t) := h h h hh

2 V I cos(h t )cos(h t )ω −α ω −β∑ (4.16)

(d) Load generated harmonic power sg(t)

resulting from the presence of harmonic voltage and load generated current components

that do not have a corresponding harmonic driving voltage from the source.

sg (t) := g g g gg

2 V I cos(g t )cos(g t )ω −α ω −β∑ (4.17)

(e) Source generated cross power sXH(t) resulting from the cross products of non-

identical fundamental/harmonic voltage and source generated current components. This

can be further subdivided into fundamental based and harmonic based.

(e1) Cross-fundamental powers

sX1 (t) := m n m nm nm 1,h,gn 1


ω −α ω −β∑ (4.18)

(e2) Cross-harmonic powers

sXh (t) := m n m nm nm 1,h,gn h


ω −α ω −β∑ (4.19)

These can be combined or kept separate depending on the needs of the application.

Further discussions on this are taken up in Chapter 7.

(f) Load generated cross-harmonic power sXg(t) resulting from the cross products of

non-identical fundamental/harmonic voltage and load generated current components

that do not have a corresponding harmonic driving voltage from the source.

sXg (t) := m n m nm nm 1,h,gn g


ω −α ω −β∑ (4.20)


Chapter 4 130

Active instantaneous power

The active instantaneous power components are given as follows.

(a) Active DC based power

The active DC based power p0(t) is made of two sub-components that is DC power

pDC0(t) and DC cross harmonic power p0X(t) with

p0(t) = p0D(t) + p0X(t). (4.21)

(a1) Active DC Power

p0D(t) := V0 I0 ; (4.22)

(a2) Active DC cross-harmonic power

p0X(t) := 0 1 1 12V I cos( t ) cosω −α θ + 0 h h h2 V I cos( t ) cosω −α θ∑ +

0 g g g2V I cos(g t ) cosω −α γ∑ + 1 0 1 12 V I cos( t ) k0ω −α +

m 0 m mm h,g

2 V I cos(m t ) k0=

ω −α∑ ; (4.23)

(b) Fundamental active power

p1(t) := 21 1 1 12V I cos ( t ) cosω −α θ ; (4.24)

(c) Source generated harmonic active power

ph(t) := 2h h h h

h2 V I cos (h t )cosθω −α∑ ; (4.25)

(d) Load generated harmonic active power

pg(t) := 2g g g g

g

2 V I cos (g t )cosω −α γ∑ ; (4.26)

(e) Source generated cross active power

(e1) Cross-fundamental active powers

pX1(t) := m n m n nm nm 1,h,gn 1

2 V I cos(m t ) cos(n t ) cos≠==

ω −α ω −α θ∑ ; (4.27)

(e2) Cross-harmonic active powers

pXh(t) := m n m n nm nm 1,h,gn h

2 V I cos(m t ) cos(n t ) cos≠==

ω −α ω −α θ∑ ; (4.28)


Chapter 4 131

(f) Load generated cross-harmonic active power

pXg (t) := m n m n nm nm 1,h,gn g

2 V I cos(m t )cos(n t ) cos(γ )≠==

ω −α ω −α∑ . (4.29)

Total active instantaneous power

p(t) = p0D(t) + p0X(t) + p1(t) + ph(t) + pg(t) + pXh(t) + pXg(t) (4.30)

Non-active instantaneous power

The non-active instantaneous power components are given by

(a) Non-active DC cross-harmonic power

q0X(t) := 0 1 1 12V I sin( t )sinω −α θ + 0 h h h2 V I sin( t )sinω −α θ∑

+ 0 g g g2V I sin(g t )sinω −α γ∑ + 1 0 1 12 V I cos( t ) (1 k0 )ω −α −

+ m 0 m mm h,g

2 V I cos(m t ) (1 k0 )=

ω −α −∑ ; (4.31)

(b) Fundamental non-active power

q1(t) := 1 1 1 1 12 V I cos( t ) sin( t )sinω −α ω −α θ ; (4.32)

(c) Source generated harmonic non-active power

qh (t) := h h h h hh

2 V I cos(h t )sin(h t )sinω −α ω −α θ∑ ; (4.33)

(d) Load generated harmonic non-active power

qg (t) := g g g g gg

2 V I cos(g t )sin(g t )sinω −α ω −α γ∑ ; (4.34)

(e) Source generated cross-harmonic non-active power

(e1) Cross-fundamental non-active powers

qX1 (t) := m n m n nm nm 1, h,gn 1

2 V I cos(m t )sin(n t )sin≠==

ω −α ω −α θ∑ ; (4.35)

(e2) Cross-harmonic non-active powers

qXh (t) := m n m n nm nm 1, h,gn h

2 V I cos(m t )sin(n t )sin≠==

ω −α ω −α θ∑ ; (4.36)


Chapter 4 132

(f) Load generated cross-harmonic non-active power

qXg (t) := m n m n nm nm 1,h,gn g

2 V I cos(m t )sin(n t )sin( )≠==

ω −α ω −α γ∑ (4.37)

Total non-active instantaneous power

q(t) = q0X(t) + q1(t) + qh(t) + qg(t) + qXh(t) + qXg(t) (4.38)

The above is based on the following assumptions.

• The voltages and currents are periodic.

• The DC factor as well as the phase angle between harmonic voltage (if it not

negligible) and the respective harmonic current is a measure of the load property

(how resistive/inductive or capacitive the load is) for that harmonic.

• If the harmonic voltage is negligible, the phase angle of that harmonic current with

respect to the voltage is determined from the phase angle of the fundamental. Thus

this angle is also linked to the load property.

The definitions are valid in the presence of source impedance since they are hinged on

the voltage and current at the metering point, and the voltage and current at the metering

point are a function of the source (including source impedance) and the load.

4.3.3.3 Discussion of the components and application of definitions For sinusoidal sources with linear load only component (b) exists. Components (a), (d)

and (f) will arise when the load, as viewed from the measuring point, is nonlinear and

the source does not have DC content. Component (a1) active DC part only will exist if

the source is DC and the load is linear. Component (c) and (e) will be present when

voltage harmonics exist in the source voltage supplying a linear load. If (c) and (e) are

present then it is possible to fully compensate the non-active power with passive

elements. On the other hand if components (a), (d) and (f) are present, then passive

components alone may not be sufficient to provide complete non-active power

compensation. It is possible to remove (d) and (f) by filtering out Ig. Active

compensation is required to compensate (a).


Chapter 4 133

With knowledge of the time profile of the powers accurate measurement of the powers,

especially non-active, can be made. The proposed definitions of instantaneous non-

active power can be used as information for compensation. Accurate knowledge of the

time profile of the instantaneous non-active power facilitates the reduction of the source

current. Additionally the components (a to f) can be utilised to gauge power quality as

well as to detect the source of distortion at the metering point. The average powers can

be used for metering purposes. A discussion of these applications is included in

Chapter 8.

4.3.4 Average active and non-active power There is a need to represent the active p(t) and non-active q(t) instantaneous powers by

a numerical value to maintain consistence with existing practice where commonly

active and non-active powers are referred to by a value (for example “…. active power

of 27 watts”). The numerical value will be termed “average” and represented by letters

S, P and N for total, active and non-active powers respectively, with the subscript “AV”

as necessary. The key to defining the average power is energy transfer, discussion of

which follows. A brief introduction to this has been given in Subsection 2.3.3.

This section expands on Subsection 2.3.3 on the definition of average active and non-

active powers as well as energy transfer applicable to both sinusoidal and non-

sinusoidal conditions. Similar to the existing definition of active power, the new

definition for non-active power is based on the energy transfer. It is well known that

power is the rate of flow of the energy [37], and that there should always be a unique

relationship between power and its energy transfer. This is the reason for definition

being based on energy transfer.

Power has active and/or non-active parts. It can be inferred likewise for energy since

power is rate of flow of energy.

4.3.4.1 Instantaneous power and its active and non-active components Consider a source voltage v(t) supplying a current i(t) to a load. The total instantaneous

power is given by s(t) = v(t)i(t) and is depicted for illustration by the waveform shown

in Figure 4.7.


Chapter 4 134

s t( )

t

0 T

Figure 4.7: Instantaneous total power s(t)

p t( )

q t( )

t

0 T

Figure 4.8: Instantaneous active p(t) and non-active power q(t)

The instantaneous total power s(t) consists of instantaneous active power p(t) and non-

active power q(t) as shown in Figure 4.8. The relationship is given by

p(t) + q(t) = s(t). (4.39)

The area under the waveform is the energy ‘changing hands’ between the source and

load. Active energy is absorbed by the load while non-active energy vibrates between

the source and load.

4.3.4.2 Energy transfer Power is the rate of flow of energy. There should thus be a unique relationship between

power (active and non-active) and its energy transfer. Therefore, a correct power

definition should carry the correct information about energy transfer. For this reason,

the energy transfer between the source and load is used as basis for the definition.

Two types of energy transfer, active due to p(t) and non-active due to q(t) are defined.

Active energy transfer per fundamental period T

The active energy transfer, defined as “electric energy” in [100], EP (subscript P implies

active) due to flow of instantaneous active power p(t) is defined by t T

Pt

E p(t) dt+

= ∫ . (4.40)


Chapter 4 135

T 0

Figure 4.9: Active Energy The area under the graph of p(t) is the energy transfer and is shown shaded in Figure

4.9. EP is a measure of the unidirectional (from the source to load) energy transfer per

fundamental period T (henceforth period shall imply fundamental period).

Due to nature of the system it could be possible that within a period, the active power

does become negative. Integral over one period could show zero power. An example

of this is cross-product based active powers. If there is a need to know the magnitude of

the positive going and negative going parts (this is necessary for example in detection of

the source of unwanted powers – refer Chapter 7 for more details) of the active energy

the following definitions can be used. t T

Ppos post

E p (t)dt+

= ∫ , t T

Pneg negt

E p (t)dt+

= ∫ (4.41)

where ppos(t) = p(t) if p(t) > 0, qpos(t) = 0 otherwise (this is the positive going part of the

waveform) and pneg(t) = p(t) if p(t) < 0, qneg(t) = 0 otherwise (this is the negative going

part of the waveform).

Non-active energy transfer per period T The energy transfer due to flow of instantaneous non-active power q(t) is bidirectional

since the energy vibrates between the source and load. In the IEEE standard [37] or

IEEE Dictionary [100], there is no definition for the measurement of such energy

transfer. Because of its bi-directional nature, the energy transfer due to q(t) cannot be

defined in a similar manner to that of p(t) using (4.40), because its value will be zero.

To overcome this difficulty a measure of bi-directional energy transfer is introduced.

An explanation of this follows.


Chapter 4 136

T 0

Figure 4.10: Non-active Energy Consider the instantaneous non-active power q(t) as shown in Figure 4.10. The shaded

areas are under the positive portions of q(t) above the x-axis, and the hatched areas

under the negative portions of q(t) below the x-axis. These areas represent respectively

the energy transfer from the source to the load (shaded) and load to source (hatched).

Because q(t) oscillates between the source and load, it incurs no net energy consumption

by the load, the positive (shaded part) and negative (hatched part) areas are equal

irrespective of the waveform of q(t). The definition must be able to quantify this energy

transfer that is taking place. With this consideration, the non-active energy transfer EN

(subscript N implies non-active) of q(t) is defined as twice the area of the shaded parts

as follows t T

N post

E 2 q (t)dt+

= ∫ , (4.42)

where qpos(t) = q(t) if q(t) > 0, qpos(t) = 0 otherwise (this is the positive going part of the

waveform).

4.3.4.3 Average power The average power of instantaneous active or non-active power is defined as the

amplitude of an equivalent sinusoidal power waveform that has the same energy

transfer as the corresponding instantaneous active or non-active power. Note that

the definition applies to both active and non-active powers.

This is explained below first for sinusoidal and then for non-sinusoidal case.


Chapter 4 137

Sinusoidal power

The relationship between the sinusoidal active and non-active power and the

corresponding energy transfer is outlined. This result is then used to generalise to the

non-sinusoidal case.

t

0 T

NAV

q(t)

-NAV

NAV

Figure 4.11: Non-active Power

t

0 T

P

2P

Pp(t)

Figure 4.12: Active Power

For sinusoidal case, average non-active power NAV is equal to the amplitude NAV of the

sinusoidal waveform of the non-active instantaneous (reactive) power q(t) = NAV

sin(2ωt) [37, 10] as shown in Figure 4.11. The amplitude P of the sinusoidal waveform

of active instantaneous power p(t) = P + P cos(2ωt) is likewise defined as the average

active power P (see Figure 4.12). Thus the relationship of the average active power P

and non-active power NAV to the corresponding active and non-active energy transfer is

easily determined. This is defined as follows.

Active Energy Transfer per period

EP = P T (4.43)

Non-active Energy Transfer per period

EN = 2π

NAVT (4.44)

The explanation of equations (4.43) and (4.44) follows.

The average value of the area under the active power sinusoidal curve p(t) in Figure

4.12 is P. The hatched part shows the energy transfer as given by equation (4.43). The

energy transfer is given by the product “active power x time” and is stated in equation

(4.43).

P

2π

NAV


Chapter 4 138

Consider non-active power normalised to amplitude “1”. The area under the shaded

part (positive going energy transfer) in Figure 4.13 is given by 2ω

. Using equation

(4.42) the non-active energy transfer is thus 4ω

. Thus the energy transfer of a

normalised sinusoidal wave is equal to 4ω

. Averaging this value ( 4ω

) to per period (T =

2πω

) gives 2π

. Thus a normalised (NAV =1) sinusoidal non-active power wave has an

energy transfer of 2π

. Generalising this to normalised case, it can be stated that this

factor 2π

multiplied by the amplitude NAV and the period T gives the non-active energy

transfer per period for non-active power which is given by equation (4.44). This is

reflected by hatched part in Figure 4.11.

1.5

1

0.5

0.5

1

1.5

t

Nor

mal

ised

non

-act

ive

pow

er

0 T

amplitude

Figure 4.13 : Normalised non-active power


Chapter 4 139

Non-sinusoidal power

p t( )

psin t( )

t0 T

2P

Figure 4.14 : Active Power This leads to the definition of average powers for non-sinusoidal waveforms. Figure

4.14 graphically represents the non-sinusoidal instantaneous active power p(t) and its

equivalent sinusoidal instantaneous power psin(t) of amplitude P which has the same

energy transfer as p(t). P is the average active power. The relationship to energy

transfer is given by equation (4.43). Note that EP is equal to the area under p(t) as per

equation (4.40). This is as follows.

P = PE

T, (4.45)

where EP is the total energy taken by the load.

Using equations (4.40), (4.45) and (4.41) gives the general definition of P as follows

P = t T

t

1 p(t) dtT

+

∫ , (4.46)

Ppos = t T

post

1 p (t)dtT

+

∫ , Pneg = t T

negt

1 p (t)dtT

+

∫ . (4.47)

Similarly, using the definition in Subsection 4.3.4.3, the nonsinusoidal instantaneous

non-active power waveform q(t) is represented by an equivalent sinusoidal waveform

qsin(t) with the same area under the graphs, as shown in the Figure 4.15. The amplitude

of the equivalent sinusoidal waveform is the ‘average non-active power’ NAV. The

rationale behind this is that the average power gives the same energy transfer as the

non-sinusoidal non-active instantaneous power.


Chapter 4 140

q t( )

qsin t( )

t

0 T

NAV

Figure 4.15: Non-active power The relationship between average non-active power NAV and energy transfer EN is then

given by

NAV = NE2Tπ , (4.48)

where EN is a measure of the total energy vibrating between the source and load.

Using equations (4.42) and (4.48) gives the general definition of NAV

NAV = t T

post

q (t)dtT

+π∫ . (4.49)

The total average power SAV can be defined using P and NAV as follows.

22AV AVS P N= − . (4.50)

The definitions given by equations (4.40), (4.42), (4.46), (4.49) and (4.50) can be used

for the proposed instantaneous active and non-active component powers defined in

Subsection 4.3.3.2 to obtain average values for utilisation in various applications.

However arithmetic or vector sum of the components average values may not give the

average values of the instantaneous total powers of equations (4.11), (4.30) and (4.38).

The intent of using average power value for the components is mainly for the purpose of

quantifying the component waveform to ease the analysis during application of the

proposed definitions.

equation (4.50) is proposed based on the assumption that P and NAV are orthogonal. It

must be borne in mind, as pointed out in Subsection 4.4.1, that the orthogonal

relationship may not strictly apply in the presence of DC component.


Chapter 4 141

The total average power SAV is load related. It is a measure of the total average power

taken by the load. The author of this thesis views RMS based total average power (or

apparent power) SRMS (=VRMSIRMS) as the source capacity (this includes ratings of all

the equipment or devices) required to supply the load (that is SAV). As stated in

Subsection 2.3.1 SRMS is a very important quantity. It is a measure of the source

capacity for a particular voltage and current at the metering point and, together with the

average active power (power factor), a measure of utilization of the source.

Both SAV and SRMS have their respective uses. In light of the discussion above, SRMS is

the VA that defines the capacity required from the source to faithfully supply the

load, and SAV gives the information of the actual VA taken by the load. SRMS is

related to the demand power that is used to recover installed capacity. SVA is the load

related VA power, which is an indication of the actual VA taken and has relationship

with the running costs to provide electricity.

The average non-active power NAV as defined above does not satisfy the universally

used non-active power definition [37, 100] equation (2.42) rewritten below

2 2RMS RMS RMSN S P= − . (4.51)

Discussion in Subsection 3.3.1 has shown that the amplitude of the non-active power

wave does not have a direct relationship with the energy transfer if RMS currents are

used to determine the powers. Thus as far as full energy information is concerned,

NRMS does not truly represent the non-active power taken by the load. Whether N

should be defined by

• using energy information as discussed above

• or the present definition as per equation (4.51) with energy transfer based on the

“shifted power waveform”

is an important consideration. The “shifted power waveform” method was discussed in

Subsection 2.3.1.

The “crossroads” has sort of been reached at this point. If equation (4.51) is to be

respected, then energy relationship is depicted using the “shifted powers waveform”

method and the issue with energy conservation remains. On the other hand if the actual


Chapter 4 142

energy transfer is to be adhered to, there is an issue in defining average powers using

the RMS method since equation (4.51) will not be satisfied. There is a need for a

consensus on this.

The author of this thesis is inclined toward the latter that is to determine the non-active

power N directly from the time quantities of voltage and current and be related to the

energy transfer of the actual load as outlined above. These quantities NAV and SAV

provide load related information and are a candidate for determining running cost of

electricity. This is further corroborated by findings in Chapter 6 that the three-phase

average powers obtained using the proposed definitions match the three-phase

arithmetic powers obtained by using RMS quantities for sinusoidal conditions

irrespective of source voltage and/or load unbalance.

4.4 Evaluation of the proposed single phase instantaneous power definitions This section evaluates the performance of the powers defined above using the single-

phase case studies S1 to S6 outlined in Subsections 3.3.1 and 3.4.1.

4.4.1 Computation The voltage and current at the metering point are known. The powers based on the new

definitions are determined using the voltage and the current (measurable) at the

metering point. This is outlined below. The proposed total active and non-active

powers are is identified using subscript “hk”.

Total instantaneous power

The total instantaneous power is not used in the comparison, as there is no problem with

the existing definition [37].

Proposed instantaneous active and non-active powers

Active instantaneous power phk(t) is determined using v(t), i(t) and equation (4.30) and

non-active power qhk(t) is determined using v(t), i(t) and equation (4.38).

Average active and non-active powers and energy transfer

These are determined using equations (4.40), (4.42), (4.46) and (4.49).


Chapter 4 143

4.4.2 Results of computation The voltages, currents and instantaneous powers are presented in graphical format. The



voltage and current the value may be magnified in order that it can be viewed on the

common scale. The magnification is shown in the graph. The powers obtained by the

proposed definition are shown on the same graphs with the expected powers to enable

easy comparison.

4.4.2.1 Waveforms The current, voltage and powers waveforms are given in Figures 4.16 to 4.21.

Case 1: Resistive Load

0.02 0.025 0.03 0.035 0.04

100

100

200

300

400

500

Active power

phk t( )

pEXP t( )

t

0.02 0.025 0.03 0.035 0.04

10

5

5

10

Non-active power

qhk t( )

qEXP t( )

t

Figure 4.16: Case S1 – Proposed and expected active and non-active powers Case S2: Resistive/Inductive Load

0.0167 0.0208 0.025 0.0292 0.0333

50

100

150

Active power

phk t( )

pEXP t( )

t

0.0167 0.0208 0.025 0.0292 0.0333

50

50

Non-active power

qhk t( )

qEXP t( )

t

Figure 4.17: Case S2 – Proposed and expected active and non-active powers


Chapter 4 144

Case S3: Resistive/Capacitive Load

0.0167 0.0208 0.025 0.0292 0.0333

50

50

100

150

Active power

phk t( )

pEXP t( )

t

0.0167 0.0208 0.025 0.0292 0.0333

100

50

50

100

Non-active power

qhk t( )

qEXP t( )

t

Figure 4.18: Case S3 – Proposed and expected active and non-active powers Case S4: Diode Resistive Nonlinear Load

0.02 0.025 0.03 0.035 0.04

100

100

200

300

Active power

phk t( )

pEXP t( )

t

0.02 0.025 0.03 0.035 0.04

10

5

5

10

Non-active power

qhk t( )

qEXP t( )

t

Figure 4.19: Case S4 – Proposed and expected active and non-active powers Case S5: Diode Inductive Nonlinear Load

0.02 0.025 0.03 0.035 0.04

10

5

5

10

Active power

phk t( )

pEXP t( )

t

0.02 0.025 0.03 0.035 0.04

50

50

Non-active power

qhk t( )

qEXP t( )

t

Figure 4.20: Case S5 – Proposed and expected active and non-active powers


Chapter 4 145

Case S6: Series R-L Load

0.02 0.025 0.03 0.035 0.04

2000

2000

4000

Active power

phk t( )

pEXP t( )

t

0.02 0.025 0.03 0.035 0.04

2000

1000

1000

2000

Non-active power

qhk t( )

qEXP t( )

t

Figure 4.21: Case S6 – Proposed and expected active and non-active powers 4.4.2.2 Energy transfer and average power The energy transfer per period is computed using equations (4.40) and (4.42) and

average powers using (4.46) and (4.49). The results are presented in Table 4.3 and Table

4.4 below. Note that only energy transfer is compared with the benchmark cases. The

average power, which is representative of the waveform since it is defined in terms of

the energy transfer, is given to show that the waveform can be represented by a

numerical value, this being useful for measurement of powers.

Table 4.3: Energy Transfer per period Expected Energy Transfer Proposed Expected Case S1

% Difference

Active (EP) 3.324395 W sec 3.324395 W sec 0.00Non-active (EN) 0 Var sec 0 Var sec 0.00Case S2 Active (EP) 0.923443 W sec 0.923443 W sec 0Non-active (EN) 0.225107 Var sec 0.225434 Var se -0.15Case S3 Active (EP /) 0.923443 W sec 0.923443 W sec 0.00Non-active (EN) 0.461544 Var sec 0.460962 Var sec 0.13Case S4 Active (EP) 1.443994 W sec 1.443994 W sec 0.00Non-active (EN) 0.000252 Var sec 0 Var sec smallCase S5 Active (EP) 0 W sec 0 W 0.00Non-active (EN) 0.585313 Var sec 0.585229 Var 0.01Case S6 Active (EP) 29.03874 W sec 29.03874 W 0.00Non-active (EN) 12.203153 Var sec 12.202998 Var 0.001


Chapter 4 146

Table 4.4: Average Powers Average Power Proposed Case S1 Active (P) 166.220 W Non-active (N) 0 Var Case S2 Active (P) 55.407 W Non-active (N) 21.216 Var Case S3 Active (P) 55.407 W Non-active (N) 45.499 Var Case S4 Active (P) 72.200 W Non-active (N) 0.019816 Var Case S5 Active (EP / P) 0 W Non-active (N) 45.970 Var Case S6 Active (EP / P) 1451.93675 W Non-active (N) 958.43342 Var 4.4.3 Evaluation based on requirements of the definitions A number of requirements were identified in Subsection 2.1.2. The proposed

definitions will be reviewed based on these requirements. At this stage the definitions

can be said to satisfy all the requirements 1 to 10 except 6, 7 and 9. Requirement 6 is

shown to be complied with in Chapter 8. Compliance to requirement 7 is shown in

Chapter 7 and requirement 9 is shown to be satisfied via the experimental work included

in Section 4.6 below and Section 6.6.

4.5 Analysis and discussion of results To evaluate the proposed definitions, the waveforms and energy transfer, of the active

and non-active powers obtained for the cases, using definitions of the proposed

definitions, are compared with the expected obtained in Chapter 3.

Figures 4.14 to 4.19 show that the active and non-active power waveforms, obtained by

the definitions, match the expected. Tables 4.3 corroborate the results of the

waveforms comparison because the energy transfers using the proposed definitions

match the expected with negligible difference.


Chapter 4 147

The definitions give the same waveform for the active and non-active powers as

expected. These results, especially for Cases 4 and 5 where the instantaneous powers in

a nonlinear load are correctly determined, are encouraging support for the definitions.

However, as pointed out in Subsection 4.2.1 the definitions may have issues with

nonlinear load. This is illustrated with an example of a diode parallel RL load below.

4.5.1 Additional example Consider the diode with parallel RL load as shown in Figure 4.22.

meteringpoint

~v(t)

R~

i(t)

L

i (t) i (t)R L

Data Source voltage = 28.86 Volt RMS Sinusoidal Resistor = 5 ohms Inductor = 100 mH Diode is ideal.

Figure 4.22: Diode in series with parallel RL load ATP is used to perform the simulation which produces the resistor iR(t) and inductor

iL(t) current. The currents obtained are given in Figure 4.23. Knowing the resistive and

inductive current flowing through the resistor/inductor, the instantaneous expected

powers pEXP(t) = v(t) iR(t) and qEXP(t) = v(t) iL(t) are easily determined. The proposed

phk(t), qhk(t) are also determined from the v(t), i(t) and equations (4.30) and (4.38). The

results of both are reflected in Figure 4.24.

0.02 0.025 0.03 0.035 0.04

50

40

30

20

10

10

20

30

40

50

v t( )

i t( )

iR t( )

iL t( )

t Figure 4.23: Voltage and currents

0.02 0.025 0.03 0.035 0.04

400

300

200

100

100

200

300

400

pEXP t( )

phk t( )

qEXP t( )

qhk t( )

t Figure 4.24: Expected and proposed powers


Chapter 4 148

0.02 0.025 0.03 0.035 0.04

50

30

10

10

30

50

70

90

110

130

150

v t( )

i t( )

iR t( )

iL t( )

t Figure 4.25: Voltage and currents with 1 volt DC in the source

0.02 0.025 0.03 0.035 0.04

5000

3750

2500

1250

1250

2500

3750

5000

pEXP t( )

phk t( )

qEXP t( )

qhk t( )

t Figure 4.26: Expected and proposed powers with DC in source

It observed from Figure 4.24 that the proposed definitions (purple and dark-green

dashed line) are able to predict the expected powers (blue and green continuous line).

When DC is present in the source, however, a discrepancy is observed as shown in

Figure 4.26 (a DC voltage of 1 volt and source resistance of 0.0095 ohm was included

in the source). This is because the proposed definitions decipher the DC voltage in the

source as also supplying current to the resistor, which is not the case in the simulation

result shown in Figure 4.25. Note that such a condition is not a norm in a power

system. The simple rule for determining k0 has been found to be quite satisfactory for

simple circuits. This is evidenced by the results obtained for cases S4, S5, application

examples in Sections 7.4.2, 7.5.1, 7.5.2 and 7.7.1, where k0 correctly identifies the

existence or non-existence of DC based power. However, further research into this

factor is indicted.

4.6 Experimental verification of the viability of proposed definition algorithm 4.6.1 Introduction A project was set up to test the viability of the algorithm. This project was the subject

of the thesis [101] of a final year undergraduate student who worked under the guidance

of the author of this thesis. The goal was to validate the viability of the algorithm

implementing the definitions experimentally. A brief overview and some results are

presented below.


Chapter 4 149

4.6.2 Algorithm Implementation The algorithm was implemented with LabVIEW (see www.ni.com/labview). The block

diagram is shown in Figure 4.27.

DAQ

Assistant Input

signals Sliding window

FFT

Converting to polar

coordinates

Output to screen Implementing

power equations (C code) Output to

file

Figure 4.27: Block diagram of algorithm implementation with LabVIEW

4.6.3 Experimental setup The block diagram of the test setup is shown in Figure 4.28 and a picture identifying the

main components in the setup is given in Figure 4.39.

Secondary Injection Test set FREJA

NI Multi -function

DAQ USB 6008

PC with Labview VI

implementing the

Measuring algorithm

Figure 4.28: Block diagram of test setup

Figure 4.29: Picture of the test setup

Lab view running in computer

DAQ USB6800

FREJA secondary injection test set


Chapter 4 150

A secondary injection test set (brand name FREJA) was used to provide the voltage and

current signal. This signal was sampled with the National Instruments DAQ USB6800

data acquisition device and processed by LabVIEW running the algorithm.

4.6.4 Results and discussion Some results obtained are presented in Figures 4.30 to 4.32.

Result 1 The data for the test is as follows: Voltage – 5 volts RMS Current – 3 amps RMS Phase angle – 0 deg Frequency – 50 Hz Data acquisition sampling rate 32 samples per cycle

0.02 0.025 0.03 0.035 0.04

20

10

10

20

30

Sk

Pk

Qk

vk

ik

k dt⋅ . Calculated results (Mathcad)

-5

0

5

10

15

20

25

30

35

245 250 255 260 265

s(t) p(t) q(t)

Labview Output file presented in Excel

Labview screen shot

Figure 4.30: Results for sinusoidal waveform with phase angle 0 deg Result 2 The data for the test is as follows: Voltage – 5 volts RMS Current – 3 amps RMS Phase angle – 60 deg lag Frequency – 50 Hz Data acquisition sampling rate 32 samples per cycle

0.02 0.025 0.03 0.035 0.04

20

10

10

20

30

Sk

Pk

Qk

vk

ik

k dt⋅ . Calculated results (Mathcad)


Chapter 4 151

-15

-10

-5

0

5

10

15

20

25

220 225 230 235 240

s(t) p(t) q(t)

Labview Output file presented in Excel Labview screen shot

Figure 4.31: Results for sinusoidal waveform with phase angle 60 deg lagging Result 3: Fund + 3rd The data for the test is as follows: V1=5 volt RMS at 0 deg V3=1.67 volt RMS at 0 deg I1=2.183 amp RMS at -43.30 deg I3=0.334 amp RMS at -70.50 deg Fundamental frequency – 50 Hz Data acquisition sampling rate 32 samples per cycle

0.02 0.025 0.03 0.035 0.04

15

10

5

5

10

15

20

Sk

Pk

Qk

vk

ik

k dt⋅

.

Calculated results (Mathcad)

-15

-10

-5

0

5

10

15

20

25

160 165 170 175 180

s(t) p(t) q(t)

Labview Output file presented in Excel

Not available Labview screen shot

Figure 4.32: Results for sinusoidal harmonic waveforms The results show that the algorithm is realisable. However, it was found that the

LabVIEW program could not executed continuously in real time. It executed for some

tens of cycles and then stopped with error message indicating that some samples were


Chapter 4 152

lost. The reason for this issue was that the computation time of the algorithm was quite

long. The input data for processing was stored in memory. When the memory was full

the execution terminated. The processing overhead of the algorithm is quite high. It

must be noted that LabVIEW was run on a standard Windows computer that is not

really suitable for real time high-speed measurement and processing. A dedicated DSP

is required to meet the requirements of the algorithm. See Chapter 6 for such a system.

4.7 Conclusion The new definitions define instantaneous powers based on the properties of the power

system. Instantaneous active and non-active powers and corresponding components are

defined in the definitions. A meaning has been attributed to each of the components

defined. Average powers and energy transfer definitions have also been stated and the

link to running cost of electricity identified. A comparison with the average powers

presently used and related issues have been highlighted. The need for a consensus on

the direction of definition for average non-active and total power with energy

consideration has been stated.

The application of the definitions in the areas of measurement, compensation, detection

of distortion and gauging power quality have been briefly mentioned.

The waveforms of the powers based on the proposed definitions are identical to the

expected. The average powers and energy transfer results obtained are matching the

expected with negligible difference. This indicates viability of the definitions. An

example showing deviation of the result, in the presence of DC voltage in the source,

has been highlighted. The need for further research on the k0 factor has been stated.

Experimental results evaluating the viability of the algorithm have been presented. The

definitions can be practically implemented but are quite computation intensive.

However, with modern high-speed digital signal processors, the required computations

can be implemented without any problem. The implementation of these definitions in a

digital signal processor based instrument is presented in Chapter 6.

Choice of reference conductor in three phase systems

Chapter 5 153

5. CHOICE OF REFERENCE CONDUCTOR IN THREE PHASE SYSTEMS

The values of non-active (reactive) and total (apparent) power for three-phase systems

change with choice of reference conductor [27, 30, 36, 102]. The value of average

active power, on the other hand, is independent of the reference conductor. Since the

active power consumed for a particular load case is fixed irrespective of the reference

conductor, the choice of reference conductor should be made on the basis of active

currents flowing in the conductors. The objective should be to choose the reference to

minimise the active currents. Presently, the neutral in a four-wire system and virtual

neutral in three wire systems is the commonly used reference in power definitions and is

recommended in IEEE Standard [37]. This chapter presents a new approach to

determine the optimal reference conductor and compares it with the IEEE recommended

choices. The investigation is done with case studies. The case studies use three-phase

three and four wire systems with different resistive, inductive and capacitive load

combinations.

5.1 Introduction For single-phase systems, the choice of the reference is explicit. There is no ambiguity

in applying the definitions of power and a unique solution is obtained for every case.

However, for three phase systems, especially unbalanced systems (voltage unbalance

and load unbalance), this is not the case. Different choices of references lead to

different results for the powers (non-active N and total S).

Fig 5.1: Unbalance resistive load

Table 5.1: Powers’ with different reference Reference B

Phase C

Phase A

Phase V0

Phase P (W) 675 675 675 675N (Var) 0 1169 0 389.7

Note that “V0 Phase” is the artificial neutral. A, B, C are the phases.

This problem is illustrated with the circuit in Figure 5.1 (from Figure 2 in Reference

[27]) for values of P and N. The values are given in Table 5.1 where it can be seen that

the value of non-active power N is different and dependent on reference used. For


Chapter 5 154

example using “B” as reference gives “N = 0 Var” while using virtual neutral as

reference it is “N = 389.7 Var”.

This ambiguity has been highlighted in many publications for example [27, 30, 36,

102]. Recently there has been increased interest in this topic for example [39, 123.

103]. The “neutral” or “artificial neutral node” also termed “virtual neutral” is common

amongst researchers [16, 20, 34, 50, 80, 91, 102, 103] as the reference in the definitions

for determination of powers in three phase systems. IEEE Standard 1459-2000 [37]

likewise also proposes the neutral or “artificial” neutral for the reference. Based on this

practice, artificial neutral will be the reference for Figure 5.1. Thus the results of P and

N in column “V0 Phase” would be obtained. The existence of non-active power P =

389.7 Var implies the need for non-active power compensation. However, if phase A or

B is chosen as the reference conductor then compensation is not indicated. It is

apparent from above and as shown in the sequel that the there is need for an optimal

choice for reference conductor.

The choice of reference is important.

Consider a three-phase load, which is

generally considered as one entity that

may have three or four terminals. There

is voltage on and current flowing into

each of the terminals. The voltage on

each terminal is measured with respect to

T1i

iT2

iT3

iT4

T1

T2

T3

T4

reference

T1-T3v

T2-T3v

T4-T3v

SOURCE LOAD

Fig 5.2: Voltages, currents and reference

a reference terminal (which may be artificial). This is reflected in Figure 5.2. The

“voltages on” and “currents flowing into” the terminals are used in the determination of

the powers (P, N, S) to the load as well as the determination of compensation of non-

active power. As seen from the simple circuit of Figure 5.1 and Table 5.1, there is

ambiguity in the choice of reference and which would be the best choice. It is important

to select an optimal reference so as to obtain unique, precise and useful information

about the load to enable correct measurement and optimal compensation.

To investigate the impact of the choice of reference conductor on the system, a number

of cases with different load combinations are studied. A method to determine the most

appropriate reference is presented.


Chapter 5 155

5.2. New approach and formulae The goal in power systems is to reduce the currents flowing in conductors to a minimum

as implied in [27] and/or to “minimise power losses” as stated in [54]. The minimum

current in conductors is realised when only active current is flowing (all non-active

current that gives rise to useless power, having been compensated) and this minimum

losses in conductors can be used to identify the optimal reference conductor. This basis

is underpinned by the fact that for any particular load condition the average active

power is constant irrespective of the choice of reference. Refer Subsection 2.2.4 for a

discussion on this point.

Thus the key in determining the reference is to utilise the conductor active current and

to compute the total conductor loss in all the conductors. Only the fundamental

quantities are used in the determination of the optimal reference conductor. The optimal

reference conductor is obtained when the total conductor loss computed is minimum as

compared to the choice of other reference conductors. The active current is calculated

by resolving the conductor current in phase with the conductor voltage with respect to

the chosen reference conductor. Mathematically, for a chosen reference conductor, this

is as follows.

5.2.1. Instantaneous active current Consider the voltage v(t) on a conductor with respect to a chosen reference conductor

and the current i(t) flowing in the conductor

[ ]rmsv(t) : 2 V cos t= ω −α , (5.1)

[ ]rmsi(t) : 2 I cos t= ω −β . (5.2) The active current is given by

[ ] [ ]active rmsi : 2 I cos t cos= ω −α α −β . (5.3) The above general formulae are used to compute the active current flowing in each

conductor. This is repeated for the remaining conductors including the reference

conductor. The current in the reference conductor is the sum of the currents in each of

the other conductors.


Chapter 5 156

5.2.2. Conductor loss in the 3-phase system The loss in a conductor is proportional to Icond

2 (Icond is RMS value). The assumption is

that all conductors have the same resistance, R, per unit length. References [37, 123]

also have made a similar assumption in the computation of line losses. The unit length

conductor loss per conductor is thus given by 2

condLoss condP : I R= (5.4) The total 3-phase system conductor loss per unit length is given by the arithmetic sum

of the unit length conductor loss for each conductor.

totalLoss condLoss

AllConductorsP : P= ∑ (5.5)

5.3. Case study and computation To evaluate the approach presented in Section 5.2, this section considers application of

the approach to most representative load types in three-phase systems.

The study is conducted on sixteen

cases using the circuits shown in

Figures 5.3 to 5.5. The cases are

for 3Ph 3W and 3Ph 4W systems

with balanced or unbalanced

fundamental (50 Hz) source

voltages with star (wye) connected,

delta connected, or mixed star/delta connected loads to give a good mix of load types.

The cases are divided into four groups. Cases 1 to 4 are for 3Ph 3W with 2-phase load

[27] shown in Figure 5.3. Cases 5 to 8 are for 3Ph 4W star load shown in Figure 5.4.

~~~~

Ai

iB

iC

RA

BR

CR

L A

BLBL

L CL

A

B

C

V0 Virtual Neutral Figure 5.5: Delta load – Cases 12 to 16

~~~~

RA

BR

L A

BLBL

Ai

iB

iC

A

B

C

Note: for Case 2A, LA and LB are replaced by CA and CB Figure 5.3: Load in 2 phases – Cases 1 to 4

~~~~

RA

BR

L A

BLBL

ABR ABLL

CR L CL

Ai

iB

iC

iN

A

B

C

N0

Note: for Case 7A, LB is replaced by CB Figure 5.4: Star and Mixed Load – Cases 5 to 11


Chapter 5 157

Cases 9 to 11 are for 3Ph 4W star/delta mixed load also shown in Figure 5.4. Cases 12

to 16 are for 3Ph 3W delta load according to Figure 5.5. It is essential to appreciate the

actual reference of a circuit. The actual reference is the reference consistent with a

single driving voltage (a comprehensive treatment of driving voltage has been taken up

in Subsection 3.2.7) for the current flowing through the load. The actual reference used

is identified in the analysis.

The source voltages and loads are as follows. There are different ways to perform the

required computations. In the ensuing analysis the time domain approach is taken to

perform the computations.

Voltages

Source phase voltages are

viN(t) := RMSi i2 V cos( t )ω −α , (5.6)

vNi(t) = -viN(t), (5.7)

where i = A, B, C

The phase to phase voltages are derived from (5.6) and are represented as follows

vij(t) = viN(t) – vjN(t) , (5.8)

vij(t) := RMSi j i j2 V cos( t )ω −α , (5.9)

where αi j = αi - αj and j = A, B, C.

The phase to virtual neutral voltages derived from (5.9) are represented as follows

vA0(t) = [ ]AB AC1 v (t) v (t)3

+ (5.10)

= A0rms A02 V cos( t )ω −α , (5.11)

vB0(t) = [ ]BA BC1 V (t) V (t)3

+ (5.12)

= B0rms B02 V cos( t )ω −α , (5.13)

vC0(t) = [ ]CA CB1 V (t) V (t)3

+ , (5.14)

= C0rms C02 V cos( t )ω −α . (5.15) Loads

The loads are given by


Chapter 5 158

2 2i i iZ : R +( L ) ⎡ ⎤= ω⎣ ⎦ ,

i

1 iZ

i

L: tan R

− ⎛ ⎞ωδ = ⎜ ⎟

⎝ ⎠, (5.16)

2 2i1 i

i

1Z : R +( )C

⎡ ⎤= ⎢ ⎥ω⎣ ⎦

,i1

1Z

i i

1: tanC R

− ⎛ ⎞δ = −⎜ ⎟ω⎝ ⎠

, (5.17)

i = A, B, C and j = A, B, C,

2 2i j i j i jZ : R +( L ) ⎡ ⎤= ω⎣ ⎦ ,

i j

i j1Z

i j

L: tan

R−⎛ ⎞ω

δ = ⎜ ⎟⎜ ⎟⎝ ⎠

. (5.18)

With the voltages and loads

defined, the load currents in each

conductor can be calculated. The

active current in each conductor is

then determined. For a particular

reference, the active current is the

source current after non-active

current has been “compensated

~~~~

Asourcei

iBsource

iCsource

iNsource

A

B

C

N

Aloadi

iBload

iCload

iNload

Compensator

iAcomp

Load

iBcomp iCcomp iNcomp

Aactive=i

=iBactive

=iCactive

=iNactive

Figure 5.6: Active conductor current

for” (Figure 5.6), which is the same as the load current resolved in phase with the

voltage vector. The compensator is taken to be ideal such that it can provide complete

compensation of non-active currents. Knowing the active conductor current, the

total unit length conductor loss for a particular reference conductor can be obtained.

Note that “R” is taken as the conductor resistance per unit length and used to compute

the conductor loss per unit length. The procedure is repeated for each reference

conductor. This is outlined below for each group of the cases.

5.3.1 Cases 1 to 4 (Figure 5.3): 3Ph 3W with 2-phase load For this circuit the actual reference can be either conductor A or B (the same results are

obtained for both).

Using conductor B as reference with voltage vAB(t) given by (5.9), the conductor

currents are calculated as follows.

Use (5.18) to determine the load where RAB = RA+ RB and LAB = LA + LB and calculate

the conductor currents.

iA(t) = AB

ABrmsAB Z

AB

V2 cos( t )Z

ω −α −δ (5.19)


Chapter 5 159

iB(t) = AB

ABrmsAB Z

AB

V2 cos( t )Z

− ω −α −δ (5.20)

iC(t) = 0 (5.21) With B as reference, the conductor active currents are then calculated as follows

iAactive(t) = AB

ABrmsAB Z

AB

V2 cos( t ) cos( )Z

ω −α δ (5.22)

iCactive(t) = 0 (5.23)

iBactive(t) = -[iAactive(t) + iCactive(t)] (5.24)

The conductor loss for each conductor and the total conductor loss are calculated with

(5.4) and (5.5) using the "RMS" value of the conductor active currents as follows.

AB

2

ABrmsAcondLoss Z

AB

VP : cos( ) RZ

⎛ ⎞= δ⎜ ⎟⎝ ⎠

(5.25)

CcondLossP : 0= (5.26)

AB

2

ABrmsBcondLoss Z

AB

VP : cos( ) RZ

⎛ ⎞= δ⎜ ⎟⎝ ⎠

(5.27)

totalLoss AcondLoss CcondLoss BcondLossP : P P P= + + (5.28) The computation is repeated for the other references - “C” and “virtual neutral”.

5.3.2 Cases 5 to 8 (Figure 5.4): 3Ph 4W star load The neutral conductor is the actual reference for this circuit.

Using voltage given in equations (5.6) and loads as in (5.16), the conductor currents are

given by

ii(t) := RMS

i

ii Z

i

V2 cos( t )

Zω −α −δ (5.29)

i = A, B, C

iN(t) = -[iA(t) + iB(t) + iC(t) ] := N N2 I cos( t )ω −δ (5.30)

With conductor B as reference, the conductor active currents are calculated as follows.

iAactive(t) = A

ArmsAB AB A Z

A

V2 cos( t ) cos( )Z

ω −α α −α + δ (5.31)

iCactive (t) = C

CrmsCB CB C Z

C

V2 cos( t ) cos( )Z

ω −α α −α + δ (5.32)


Chapter 5 160

iNactive (t) = N B B N2 I cos( t ) cos( )ω −α + π α + π+ δ (5.33)

iBactive(t) = -[iAactive(t) + iCactive(t) + iNactive(t)] (5.34)

The conductor loss for each conductor and the total conductor loss is calculated with

(5.4) and (5.5) using the "RMS" value of the conductor active currents.

The computation is repeated for the other references “C”, “A” and “N”. 5.3.3 Case 9 to 11 (Figure 5.4): 3Ph 4W star/delta mixed load For this circuit there are two reference conductors - neutral for the star connected load

and A or B for the phase-phase connected load. Essentially this is a case of mixed

reference from the load point of view. However, a single reference conductor has to be

determined. The conductor currents are determined as follows.

Star load currents are determined as in Subsection 5.3.2 and phase AB load currents

determined as in 5.3.1. The sum gives the conductor currents as follows.

iA(t) := A1 A1 A12 I cos( t )ω +α −δ (5.35) iB(t) := B1 B1 B12 I cos( t )ω +α −δ (5.36)

iC(t) := C

CrmsC Z

C

V2 cos( t )Z

ω +α −δ (5.37)

iN(t) = -[iA(t) + iB(t) + iB(t) ] := N1 N12 I cos( t )ω −δ (5.38) With conductor B as reference, the conductor active currents are calculated as follows.

iAactive(t) = A1 AB AB A1 A12 I cos( t ) cos( )ω +α α −α + δ (5.39)

iCactive (t) = C

CrmsCB CB C Z

C

V2 cos( t ) cos( )Z

ω +α α −α + δ (5.40)

iNactive (t) = N1 B B N12 I cos( t ) cos( )ω +α + π α + π+ δ (5.41)

iBactive(t) = -[iAactive(t) + iCactive(t)+ iNactive(t)] (5.42)

The conductor loss for each conductor and the total conductor loss is calculated with


The computation is repeated for the other references “C”, “A” and “N”.

5.3.4 Cases 12 to 16 (Figure 5.5): 3Ph 3W delta load This is also a mixed reference case. There is no unique reference conductor - conductor

A is the reference for load BA and CA. Similarly, B is for load AB and CB and C is for


Chapter 5 161

load AC and BC as reference. Again, a single reference conductor should be

determined. Conductor currents are determined as follows.

The current in conductor A is made of two components as follows

iA(t) =AB

ABrmsAB Z

AB

V2 cos( t )Z

ω +α −δ + CA

ACrmsAC Z

CA

V2 cos( t )Z

ω +α −δ . (5.43)

This can be simplified to

IA(t) := A2 A2 A12 I cos( t )ω +α −δ . (5.44)

Similarly the current in the other conductors is given by

iB(t) = B2 B2 B22 I cos( t )ω +α −δ , (5.45)

iC(t) = C2 C2 C22 I cos( t )ω +α −δ , (5.46)

iN(t) = -[iA(t) + iA(t) + iA(t) ] = N2 N22 I cos( t )ω −δ . (5.47) With conductor B as reference, the conductor active currents are calculated as follows.

iAactive(t) = A2 AB AB A2 A22 I cos( t ) cos( )ω +α α −α + δ , (5.48)

iCactive (t) = C2 CB CB C2 C22 I cos( t ) cos( )ω +α α −α + δ , (5.49)

iNactive (t) = N2 B B N22 I cos( t ) cos( )ω +α + π α + π+ δ , (5.50)

iBactive(t) = -[iAactive(t) + iCactive(t)+ iNactive(t)]. (5.51) The conductor loss for each conductor and the total conductor loss is calculated with


The computation is repeated for the other references - “C”, “A” and “0” (virtual

neutral). The results of the computation are presented in the next section.

5.4. Computation results This section presents the computation results of currents and conductor losses for

sixteen cases using the data in Table 5.2. The RMS conductor currents and total

conductor loss, for each of the references, calculated as outlined in Section 5.3 are

tabulated in Table 5.3. Note that the source currents IXsource (X = A, B, C) are the active

source currents after the load currents have been compensated and Pcondloss is the total

conductor loss resulting from the flow of these active source currents.


Chapter 5 162

Table 5.2: Case Data

Note: Voltages are RMS values, frequency is 50 Hz, resistance in Ohms, inductance in

Henry and capacitance in μF.

Case Load Case Data Case Load Case Data 1 Ra = 0.7, Rb = 0.3, Rc = open

circuit. Symmetrical 3 Phase (Vph = 15 V) voltage

9 Ra = 1.06, Rb = 1.32, Rc = 1.32, Rab = 1. Symmetrical 3 Phase (Vph = 115 V) voltage

2, 2A

Ra = 0.7, Rb = 0.3, Rc = open ckt, La = 0.01, Lb = 0.01, Lc = open ckt. For Case 2A Ca = Cb = 1013 replaces La, Lb. Symmetrical 3 Phase (Vph = 15 V) voltage

10 Ra = 1.06, Rb = 0, Rc = 1.32, Rab = 0, La = 0.0036, Lb = 0.0042, Lc = 0.0042, Lab = 0.005. Symmetrical 3 Phase (Vph = 115 V) voltage

3 Ra = 0, Rb = 0, Rc = open ckt, La = 0.01, Lb = 0.01, Lc = open ckt. Symmetrical 3 Phase (Vph = 15 V) voltage

11 Ra = 1.06, Rb = 0, Rc = 1.32, Rab = 1.0, La = 0.0036, Lb = 0.0042, Lc = 0.0042, Lab = 0. Symmetrical 3 Phase (Vph = 115 V) voltage

4 Ra = 0.6, Rb = 0.9, Rc = open ckt, La = 0.005, Lb = 0.004, Lc = open ckt. Symmetrical 3 Phase (Vph = 15 V) voltage

12 Rab = 1.06, Rbc = 1.32, Rca = 1.32, Lab = 0.0, Lbc = 0.0, Lca = 0.02. Symmetrical 3 Phase (Vph = 115 V) voltage

5 Ra = 1.06, Rb = 1.32, Rc = 1.32. Symmetrical 3 Phase (Vph = 115 V) voltage

13 Rab = 1.06, Rbc = 1.32, Rca = 1.32. Un-Symmetrical 3 Phase (VphA = 199.1858 V, VphB = 199.1858 V, VphC = 179.6357 V ) voltage

6 Ra = 1.06, Rb = 1.32, Rc = 1.32. Un-Symmetrical 3 Phase (VphA = 115 V, VphB = 115V, VphC = 0.8x115V ) voltage

14 Rab = 1.06, Rbc = 1.32, Rca = 1.32. Un-Symmetrical 3 Phase (VphAB = 189.3139 V, VphBC = 169.4056 V, VphCA = 179.6357 V ) voltage

7,7A Ra = 1.06, Rb = 1.32, Rc = 1.32, La = 0.0036, Lb = 0.0042, Lc = 0.0042. For Case 7A Cb = 2412 replaces Lb. Symmetrical 3 Phase (Vph = 115 V) voltage

15 Rab = 1.06, Rbc = 1.32, Rca = 1.32, Lab = 0.0036, Lbc = 0.0042, Lca = 0.0042. Symmetrical 3 Phase (Vph = 115 V) voltage

8 Ra = 1.06, Rb = 1.32, Rc = 1.32, La = 0.0036, Lb = 0.0042, Lc = 0.0042. Un-Symmetrical 3 Phase (VphA = 115 V, VphB = 115V, VphC = 0.8x115V ) voltage

16 Rab = 1.06, Rbc = 1.32, Rca = 1.32, Lab = 0.0036, Lbc = 0.0042, Lca = 0.0042. Un-Symmetrical 3 Phase (VphA = 199.1858 V, VphB = 199.1858 V, VphC = 179.6357 V ) voltage


Chapter 5 163

5.5. Results and analysis The result of the computation is presented in Table 5.3.

Table 5.3: Conductor Currents and Total Unit Length Conductor Loss Note: Minimum loss is shown in bold italics Reference B Phase C Phase A Phase N Phase 0

Phase Case 1- Source currents / unit length loss after compensation IAsource (A) 25.98 12.99 22.56 IBsource (A) 25.98 12.99 22.5 ICsource (A) 0 22.50 22.38 INsource (A) Pcondloss (W) 1350R 843.75R 1516R Case 2 -Source currents / unit length loss after compensation IAsource (A) 0.642 3.813 2.607 IBsource (A) 0.642 3.172 1.461 ICsource (A) 0 3.536 3.542 INsource (A) Pcondloss (W) 0.824R 37.11R 21.47R Case 2A -Source currents / unit length loss after compensation IAsource (A) 0.640 3.171 1.537 IBsource (A) 0.640 3.812 2.57 ICsource (A) 0 3.537 3.531 INsource (A) Pcondloss (W) 0.819R 37.09R 21.44R Case 3 - Source currents / unit length loss after compensation IAsource (A) 0 3.581 2.122 IBsource (A) 0 3.581 2.067 ICsource (A) 0 3.581 3.581 INsource (A) Pcondloss (W) 0R 34.20R 21.60R Case 4 - Source currents / unit length loss after compensation IAsource (A) 3.804 8.112 6.901 IBsource (A) 3.804 4.308 0.291 ICsource (A) 0 7.030 7.048 INsource (A) Pcondloss (W) 28.94R 133.8R 97.384


Chapter 5 164

Reference B Phase C Phase A Phase N Phase 0

Phase Case 5 - Source currents / unit length loss after compensation IAsource (A) 93.466 93.466 152.05 108.49 IBsource (A) 136.34 75.449 75.449 87.121 ICsource (A) 75.499 136.34 75.449 87.121 INsource (A) 10.685 10.942 21.369 25.369 Pcondloss (kW) 32.23R 33.23R 34.96R 27.41R Case 6 - Source currents / unit length loss after compensation IAsource (A) 93.956 97.236 147.89 108.49 IBsource (A) 133.09 78.083 75.449 87.121 ICsource (A) 58.005 118.73 58.005 69.697 INsource (A) 1.973 28.109 30.138 33.654 Pcondloss (kW) 29.91R 30.33R 31.84R 25.35R Case 7 - Source currents / unit length loss after compensation IAsource (A) 16.872 71.003 75.687 51.187 IBsource (A) 74.836 15.959 59.52 43.578 ICsource (A) 59.52 63.364 15.959 43.578 INsource (A) 5.576 12.733 7.278 9.867 Pcondloss (kW) 9.459R 10.13R 9.579R 6.516R Case 7A - Source currents / unit length loss after compensation IAsource (A) 16.872 71.003 110.46 51.187 IBsource (A) 74.836 59.52 15.959 43.578 ICsource (A) 59.52 88.182 15.959 43.578 INsource (A) 5.576 25.509 82.884 9.867 Pcondloss (kW) 9.459R 10.70R 19.58R 6.516R Case 8 - Source currents / unit length loss after compensation IAsource (A) 16.872 69.48 68.994 51.187 IBsource (A) 76.523 19.737 59.52 43.578 ICsource (A) 48.335 59.57 9.693 34.863 INsource (A) 17.479 21.449 4.869 15.766 Pcondloss (kW) 8.872R 9.226R 8.842R 5.983R Case 9 - Source currents / unit length loss after compensation IAsource (A) 275.03 184.50 321.87 264.55 IBsource (A) 309.38 165.99 256.53 243.94 ICsource (A) 75.499 292.99 75.449 87.121 INsource (A) 10.685 10.685 21.369 167.75 Pcondloss (kW) 177.2R 147.5R 175.6R 165.8R


Chapter 5 165

Reference B Phase C Phase A Phase N Phase 0

Phase Case 10 - Source currents / unit length loss after compensation IAsource (A) 16.872 180.82 45.579 351.00 IBsource (A) 35.319 153.40 43.578 63.406 ICsource (A) 59.52 181.39 15.959 43.578 INsource (A) 38.002 46.811 10.741 355.57 Pcondloss (kW) 6.519R 91.32R 4.347R 255.5R Case 11 - Source currents / unit length loss after compensation IAsource (A) 216.05 170.60 243.73 223.31 IBsource (A) 215.38 50.015 242.76 172.5 ICsource (A) 59.52 249.72 15.959 43.578 INsource (A) 38.002 46.811 10.741 160.03 Pcondloss (kW) 98.06R 96.79R 118.7R 107.1R Case 12 - Source currents / unit length loss after compensation IAsource (A) 263.36 244.85 424.50 - 293.43 IBsource (A) 424.50 244.85 263.36 - 293.42 ICsource (A) 226.35 424.10 226.35 - 261.36 INsource (A) - - - - 31.898 Pcondloss (kW) 300.8R 299.8R 300.8R - 241.5R Case 13 - Source currents / unit length loss after compensation IAsource (A) 263.36 240.27 399.93 - 285.92 IBsource (A) 399.93 240.27 263.36 - 285.92 ICsource (A) 188.52 399.92 188.52 - 226.52 INsource (A) - - - - 37.147 Pcondloss (kW) 264.8R 275.4R 264.8R - 216.2R Case 14 - -Source currents / unit length loss after compensation IAsource (A) 257.45 239.57 384.91 - 279.76 IBsource (A) 384.88 218.19 243.16 - 266.45 ICsource (A) 184.52 384.92 189.07 - 222.28 INsource (A) - - - - 32.279 Pcondloss (kW) 248.5R 253.2R 243.0R - 199.8R Case 15 - Source currents / unit length loss after compensation IAsource (A) 60.274 200.61 218.86 - 151.58 IBsource (A) 215.13 38.22 190.96 - 132.31 ICsource (A) 178.56 222.20 47.878 - 130.73 INsource (A) - - - - 25.686 Pcondloss (kW) 81.8R 91.08R 86.66R - 58.23R Case 16 - Source currents / unit length loss after compensation IAsource (A) 68.986 194.82 201.42 - 147.96 IBsource (A) 203.60 38.761 182.24 - 127.21 ICsource (A) 157.09 212.78 31.504 - 113.30 INsource (A) - - - - 22.875 Pcondloss (kW) 70.89R 84.73R 74.78R - 51.44R


Chapter 5 166

The following is observed from Table 5.3. For Cases 1 to 4 we know the reference

conductor to be B-phase (or A-Phase). The proposed method however identifies

conductor C for Case 1 with minimum loss 843.75R. For Cases 2 to 4 it identifies

conductor B to be the reference with the minimum loss (shown in bold italics) 0.824R,

0.819R and 28.94R respectively for Cases 2 to 4. However note that this type of load is

not common in a practical system.

The actual reference conductor for Cases 5 to 8 is the neutral conductor and the

proposed method identifies this conductor (with minimum loss 27.14R, 25.35R, 6.516R,

6.516R and 5.938R for Cases 5 to 8 respectively) as reference. This type of load is

common in the real system.

For Cases 9 to 11, where there is no unique reference, the proposed approach does

identify a reference conductor which gives minimum conductor loss for a non-active

power compensated system. Loads connected across two phases are not common in a

star system, the majority of the loads being star-connected. Thus in a practical system

where only a minority loads would be phase-phase connected, the reference would be

biased towards the neutral.

For Cases 12 to 16 the method identifies the virtual neutral as the choice for reference

conductor. This is also a common load arrangement practically.

The presently used or recommended choice of virtual neutral is satisfactory for three-

phase three-wire system. For four-wire system the neutral, as presently used, is the

choice for reference conductor. However, for special cases where there is a need, the

optimal reference conductor can be identified using the above method.

5.6. Conclusion A new approach to determine the optimal reference conductor for a three-phase system

has been presented.

The currents and total power loss for a number of cases with unbalanced conditions are

investigated. The proposed method identifies a reference conductor that would give

minimum conductor loss for a compensated system. The proposed method applies to

both balanced and unbalanced three-phase systems.


Chapter 5 167

The case studies corroborate the present practice of utilising the neutral for four-wire

systems and virtual neutral for three-wire system. The case studies also show that the

proposed new approach is consistent with the present practice under the normal

conditions and is able to identify the optimal reference conductor under abnormal

condition when the present practice may not be applicable.

Using the reference as per present practice or determined by the proposed method,

single-phase power definitions can be extended for three phase systems.


Chapter 5 168


Three phase power component definitions for instantaneous and average powers

Chapter 6 169

6. THREE PHASE POWER COMPONENT DEFINITIONS FOR INSTANTANEOUS AND AVERAGE POWERS

The concepts of the single-phase definitions presented in Chapter 4 are extended to

encompass the three-phase system in this chapter. The importance of the background

technical information outlined in Section 2.2 and Section 3.2 as well as the analysis of

RMS powers in Subsection 2.3.1 cannot be discounted for this chapter. List L4.1 to

L4.4 in Chapter 4 is also an important precursor to this chapter.

6.1. Introduction The three-phase electrical power system is made up of three single-phase systems with

their fundamental voltage 120 electrical degrees apart. Many researchers, especially

those using space vectors [11, 13, 41, 42, 46, 47, 50, 65], analyse the three-phase system

as a single system. These definitions are not applicable to a single-phase system. The

author of this thesis has taken the approach as [6, 30, 34] where the three-phase system

has the same basis as the single-phase system. The proposed definitions can analyse the

power system as three single-phase systems or a single equivalent system. The

definitions/formulae developed for a single-phase system are used to describe the three-

phase system from an each phase perspective. The single-phase definitions were

presented in Chapter 4. In three-phase systems there are generally three conductors -

phases A, B, C or four conductors - phases A, B, C and neutral (N). One of these

conductors is used as the reference conductor, the analysis of which was outlined in

Chapter 5. To avert confusion between phase, neutral and reference, the term conductor

is used in preference over phase. As pointed out above the definitions can also be used

if the three phase system is analysed as a single system where equivalent single

instantaneous voltage and current quantities are first determined. The discussion on this

is also briefly taken up in this chapter.

The theory is evaluated using the methodology outlined in Chapter 3.

6.2 Background technical information Sections 2.2 and 3.2 as well as the analysis of RMS powers in Subsection 2.3.1 refer.

List L4.1 to L4.4 in Chapter 4 is also useful reference for this chapter.


Chapter 6

170

6.3 Proposed three phase power component definitions 6.3.1 Three-phase powers based on single-phase basis Substantial insight about the three-phase system can be obtained by considering each

phase separately. This requires the selection of a reference conductor. The choice for

reference is presented in Chapter 5. In summary, the virtual neutral is recommended for

three-phase three wire systems, the neutral for three-phase four wire systems and if

necessary for special cases the reference conductor can be determined using the method

given in Chapter 5. Once this is done all the definitions for the single-phase system of

Chapter 4 can be used for the analysis of the three-phase system.

Thus the three-phase powers are defined by considering each phase individually using

the single-phase power definitions given in Chapter 4 and the reference conductor given

in Chapter 5.

6.3.2 Three-phase powers on collective three-phase basis Presently three-phase instantaneous or time domain based power definitions in the

majority use space vectors [40, 46, 47, 85], voltage and current vectors decomposition

[14, 44], orthogonal decomposition [42] or transform (example Park’s transform) of the

three-phase voltages and currents [13, 41, 47, 85].

The approach presented in this thesis is different. It is based on the energy exchange

occurring between the source and load, the definitions developed for the single-phase in

Chapter 4, and the reference conductor presented in Chapter 5.

The concept behind the definitions is first outlined. Consider phase to reference

voltages

vaREF(t), vbREF(t), vcREF(t), (6.1) phase active currents

iaACTIVE(t), ibACTIVE(t) and icACTIVE(t) (6.2) and phase non-active currents

iaNONACTIVE(t), ibNONACTIVE(t) and icNONACTIVE(t). (6.3) The instantaneous total phase currents is then given by


Chapter 6 171

ia(t) = iaACTIVE(t) + iaNONACTIVE(t), (6.4)

ib(t) = ibACTIVE(t) + ibNONACTIVE(t), (6.5)

ic(t) = icACTIVE(t) + icNONACTIVE(t). (6.6)

The next equation equation (6.7) is essentially the definition of instantaneous power in

[37] under Subsection 3.2.2.1. But in this thesis “s” is used instead of “p” to represent

the instantaneous total power.

The three phase instantaneous total power is given by

s3ph(t) = vaREF(t)ia(t) + vbREF(t)ib(t) + vcREF(t)ic(t) = vaREF(t)iaACTIVE(t) + vbREF(t)ibACTIVE(t) + vcREF(t)icACTIVE(t)

+ vaREF(t)iaNONACTIVE(t) + vbREF(t)ibNONACTIVE(t) + vcREF(t)icNONACTIVE(t). (6.7) Likewise define three-phase instantaneous active power p3ph(t) = vaREF(t)iaACTIVE(t) + vbREF(t)ibACTIVE (t) + vcREF(t)icACTIVE (t) (6.8) and the three phase instantaneous nonactive power q3ph(t) = vaREF(t)iaNONACTIVE(t) + vbREF(t)ibNONACTIVE (t) + vcREF(t)icNONOACTIVE (t). (6.9)

s3ph t( )

p3ph t( )

q3ph t( )

t

0 T

(a) Balanced voltage and load

s3ph t( )

p3ph t( )

q3ph t( )

t

0 T

(b) Balanced voltage and unbalanced load

s3ph t( )

p3ph t( )

q3ph t( )

t

0 T

(c) Unbalanced voltage and balanced load

s3ph t( )

p3ph t( )

q3ph t( )

t

0 T

(d) Unbalanced voltage and load

Figure 6.1: Instantaneous three-phase powers


Chapter 6

172

The graphs of equations (6.7), (6.8) and (6.9) for some sinusoidal balance and

unbalance conditions are shown in Figure 6.1.

It is observed the s3ph(t) has the same wavefrom as p3ph(t) and q3ph(t) is zero under

balanced conditions. However under unbalanced conditions (voltage and/or load) s3ph(t)

does not have the same wavefrom as p3ph(t) and q3ph(t) has a value oscillating about

zero. The instantaneous three-phase powers are an indication of unbalance.

For the balance condition, the equations give the magnitude for p3ph(t). Three-phase

power s3ph(t) has the same magnitude as p3ph(t) while q3ph(t) has zero magnitude. It is

known that balanced non-active current is flowing in the system and non-active power

should be indicated. Hence equations (6.7), (6.8) and (6.9) are not useful as indication

of magnitude of the three phase instantaneous total s3ph(t) and non-active power q3ph(t).

The reason for this is that the phase components of both s3ph(t) and q3ph(t) have positive

and negative going parts which cancel out each other in the summation and impair the

magnitude information. This is apparent from the waveforms in Figure 6.2.

Total power phase components

sA t( )

sB t( )

sC t( )

t

Active power phase components

pA t( )

pB t( )

pC t( )

t

Non-active power phase components

qA t( )

qB t( )

qC t( )

t

0 T

0 T

0 T

Figure 6.2: Phase components of instantaneous powers To retain the magnitude information, the three phase instantaneous powers are defined

with two equations, one giving positive part and the other the negative part, for each of

the total, active and non-active power as follows.

Total instantaneous power s3phPOS t( ) sA t( ) sA t( ) 0>if

0 otherwise

sB t( ) sB t( ) 0>if

0 otherwise

+ sC t( ) sC t( ) 0>if

0 otherwise

+:=

s3phNEG t( ) sA t( ) sA t( ) 0<if

0 otherwise

sB t( ) sB t( ) 0<if

0 otherwise

+ sC t( ) sC t( ) 0<if

0 otherwise

+:=

, (6.10) active instantaneous power


Chapter 6 173

p3phPOS t( ) pA t( ) pA t( ) 0>if

0 otherwise

pB t( ) pB t( ) 0>if

0 otherwise

+ pC t( ) pC t( ) 0>if

0 otherwise

+:=

p3phNEG t( ) pA t( ) pA t( ) 0<if

0 otherwise

pB t( ) pB t( ) 0<if

0 otherwise

+ pC t( ) pC t( ) 0<if

0 otherwise

+:=

(6.11) and non-active instantaneous power

q3phPOS t( ) qA t( ) qA t( ) 0>if

0 otherwise

qB t( ) qB t( ) 0>if

0 otherwise

+ qC t( ) qC t( ) 0>if

0 otherwise

+:=

q3phNEG t( ) qA t( ) qA t( ) 0<if

0 otherwise

qB t( ) qB t( ) 0<if

0 otherwise

+ qC t( ) qC t( ) 0<if

0 otherwise

+:=

(6.12) These definitions give the desired result in measuring the powers as shown in Figure

6.3.

Total

s3phPOS t( )

s3phNEG t( )

t

Active

p3phPOS t( )

p3phNEG t( )

t

Non-active

q3phPOS t( )

q3phNEG t( )

t

0 T 0 T 0 T

Figure 6.2: Positive and negative instantaneous powers Each of the total, active and non-active powers is composed of two components, the

positive and negative component. Note that though the active is generally positive

going, the definition caters for both positive and negative going active power. Negative

going active power is very unlikely except in the case when one phase active power is

moving in the opposite direction. In such a case, the negative going active power will

also be reflected in the waveform. Note that the above definitions are such that energy

content is retained. This is very important, as there is a unique relationship between

power and energy. The powers’ waveform must retain the energy information.

Average powers can be defined from the positive/negative powers using the basis

developed in Subsection 4.3.4.2. This is given by the following formulae.

PΣ3phAV = t T

3PhPOSt

1 p (t)dtT

+

∫ or PΣ3phAVREV = t T

3PhNEGt

1 p (t)dtT

+

∫ (6.13)


Chapter 6

174

where PΣ3phAVREV is for the power flowing in the reverse direction.

NΣ3phAV = [ ]t T

3PhPOSt

q (t) dtT

+π∫ (6.14)

2 23PhAV 3PhAV 3PhAVS P NΣ Σ Σ= − (6.15)

The energy transfer definitions developed in Subsection 4.3.4.2 also apply.

t T

3PhP 3PhPOSt

E p (t)dt+

Σ = ∫ or t T

3PhPREV 3PhNEGt

E p (t)dt+

Σ = ∫ (6.16)

t T

3PhN 3PhPOSt

E 2 q (t)dt+

Σ = ∫ (6.17)

It is interesting to highlight here that equations (6.13), (6.14) and (6.15) give practically

identical results to the RMS arithmetic powers under sinusoidal conditions and linear

load irrespective of source voltage and/or load unbalance. The results start to deviate

when harmonics are present in the source voltage.

The above definitions (6.7) through (6.17) are applicable for sinusoidal or non-

sinusoidal and balanced or unbalanced three phase systems.

6.3.3 Unbalance In Figure 6.1 it is shown that unbalance is reflected in the waveform of s3ph(t), p3ph(t)

and q3ph(t). Under unbalanced conditions, both s3ph(t) and p3ph(t) deviate from a straight

line while q3ph(t) deviates from zero value. This characteristic under unbalance

conditions can be used to gauge the degree of unbalance. The straight line in the case of

s3ph(t) and p3ph(t) is the average value of this waveform. The degree of unbalance is

defined for total, active and non-active power as follows.

Total power unbalance in per unit

3phUNBAL3phUNBALpu

3phAV

SS

SΣ

= (6.18)

where

t T

3phUNBAL 3phPOSt

S s dtT

+π= ∫ , ( ) ( )3phPOS 3phAV 3phPOS 3phAV

3phPOS

s (t) S if s (t) S 0s (t)

0 otherwise

⎧ − − >⎪= ⎨⎪⎩


Chapter 6 175

and t T

3phAV 3pht

1S s (t)dtT

+

= ∫ .

Active power unbalance in per unit

3phUNBAL3phUNBALpu

3phAV

PP

PΣ

= (6.19)

where

t T

3phUNBAL 3phPOSt

P p dtT

+π= ∫ , ( ) ( )3phPOS 3phAV 3phPOS 3phAV

3phPOS

p (t) P if p (t) P 0p (t)

0 otherwise

⎧ − − >⎪= ⎨⎪⎩

and t T

3phAV 3pht

1P p (t)dtT

+

= ∫ .

Non-active power unbalance in per unit

3phUNBAL3phUNBALpu

3phAV

NN

NΣ

= (6.20)

where

t T

3phUNBAL 3phPOSt

N q dtT

+π= ∫ , 3phPOS 3phPOS

3phPOS

q (t) if q (t) 0q (t)

0 otherwise

>⎧= ⎨⎩

Based on the above definitions 1 per unit (or 100%) unbalance occurs when only one

phase is loaded for a sinusoidal three phase system. In the presence of harmonics,

unbalance can exceed 1 per unit. A balanced sinusoidal system will give 0 per unit

unbalance.

6.3.4 Three-phase component powers on collective three-phase basis The instantaneous three-phase power components are given by sum of the

corresponding components (DC, fundamental and harmonic, cross-harmonic) for all

conductors

3PhComp PhCompAll Phases FOR EACH POWER COMPONENT

s (t) : s (t)= ∑ (6.21)

Similarly for the active and non-active power components


Chapter 6

176


p (t) : p (t)= ∑ (6.22)


q (t) : q (t)= ∑ (6.23)

The sum of the components gives the total instantaneous powers. The total three-phase

instantaneous total power is thus given by

3Ph 3PhCompAll components

s (t) : s (t)= ∑ (6.24)

Similarly for the total three-phase active and non-active powers

3P 3PhCompAll components

p (t) : p (t)= ∑ (6.25)

3Ph 3PhCompAll components

q (t) : q (t)= ∑ (6.26)

The definitions in equations (6.10) to (6.17) also apply to each of the component powers

as defined in equations (6.21) to (6.23).

6.3.5 Three-phase powers as applicable to space-vector transform The definitions above also apply to space-vector transform where a single equivalent

voltage and current is defined for the three-phase system [14,104].

In this thesis it is not the intention to perform an analysis on these methods but to point

out the possibilities of the proposed definition in application to any method where the

voltages and currents are represented in the time domain.

6.3.6 Discussion on the components and application of definitions The discussion of the different components in Subsection 4.3.3.3 also applies to the

three-phase system.

The comments made in Subsection 4.3.3.3 about the applications in measurement,

compensation, detection of source of distortion and power quality are also valid for the

three-phase case. Additionally, the three-phase total, active and non-active powers

provide the possibility of detecting unbalance. An application example is given in

Chapter 7.


Chapter 6 177

6.4 Evaluation of proposed three-phase instantaneous powers’ definitions This section evaluates the performance of proposed three-phase powers defined above

using the three-phase benchmark case studies given in Subsections 2.3.2 and 2.4.2.

6.4.1 Computation The voltages and currents at the metering point are known. The powers based on the

new definitions are determined using the voltages and the currents (measurable) at the

metering point. This is outlined below. The proposed total active and non-active power

is identified using subscript “HK”.

Total Instantaneous Power

The total instantaneous power is not used in the comparison, as there is no problem with

the existing definition [37].

Proposed instantaneous active and non-active powers for each phase to reference

The voltages and currents at the metering point are known. Active and non-active

instantaneous power for each phase, using the selected reference conductor, is

determined using the single-phase equation (4.28) for active power and equation (4.35)

for non-active power.

Average active and non-active powers and energy transfer for each phase to

reference

These are determined using equations (4.37), (4.38), (4.42) and (4.44).

6.4.2 Results of computation The voltages, currents and instantaneous powers are presented in graphical format. The



voltage and current the value may be magnified so that it can be viewed on the common

scale. The magnification is shown in the graph. The powers obtained by the proposed

definition are shown on the same graphs with the expected to enable easy comparison.

6.4.2.1 Waveforms


Chapter 6

178

The current, voltage and powers waveforms are given in Figures 6.3 to 6.8.

Case T1: 3Ph 2W with unsymmetrical source voltage with 2-phase unbalanced resistive

load (reference – B phase)

0.02 0.025 0.03 0.035 0.04

50

25

25

50

75

vab t( )

vcb t( )

0.5 ia t( )⋅

ic t( )

t.

0.02 0.025 0.03 0.035 0.04

1000

2000

3000

Active powers

pabEXP t( )

pcbEXP t( )

pabHK t( )

pcbHK t( )

t

0.02 0.025 0.03 0.035 0.04

1

0.5

0.5

1

Non-active powers

qabEXP t( )

qcbEXP t( )

qabHK t( )

qcbHK t( )

t

Figure 6.3: Case T1 (B-phase as reference) Voltages, currents, proposed and expected active and non-active powers


Chapter 6 179

Case T1: 3Ph 2W with unsymmetrical source voltage with 2-phase unbalanced resistive

load (reference - virtual neutral)

0.02 0.025 0.03 0.035 0.04

60

40

20

20

40

60

A-phase

van t( )

ia t( )

t

0.02 0.025 0.03 0.035 0.04

60

40

20

20

40

60

B-Phase

vbn t( )

ib t( )

t

0.02 0.025 0.03 0.035 0.04

60

40

20

20

40

60

C-phase

vcn t( )

ic t( )

t

0.02 0.025 0.03 0.035 0.04

500

500

1000

1500

2000

Active powers

paEXP t( )

pbEXP t( )

pcEXP t( )

paHK t( )

pbHK t( )

pcHK t( )

t

0.02 0.025 0.03 0.035 0.04

600

400

200

200

400

600

Non-active powers

qaEXP t( )

qbEXP t( )

qcEXP t( )

qaHK t( )

qbHK t( )

qcHK t( )

t

Figure 6.4: Case T1 (Virtual neutral as reference) Voltages, currents, proposed and expected active and non-active powers


Chapter 6

180


0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

A-phase

van t( )

ia t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

B-Phase

vbn t( )

ib t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

C-phase

vcn t( )

ic t( )

t

0.02 0.025 0.03 0.035 0.04

2000

4000

6000

8000

Active powers

paEXP t( )

pbEXP t( )

pcEXP t( )

paHK t( )

pbHK t( )

pcHK t( )

t

0.02 0.025 0.03 0.035 0.04

2000

1000

1000

2000

Non-active powers

qaEXP t( )

qbEXP t( )

qcEXP t( )

qaHK t( )

qbHK t( )

qcHK t( )

t



Chapter 6 181


0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

A-phase

van t( )

5 ia t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

B-Phase

vbn t( )

5 ib t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

C-phase

vcn t( )

5 ic t( )

t

0.02 0.025 0.03 0.035 0.04

1000

2000

3000

4000

5000

Active powers

paEXP t( )

pbEXP t( )

pcEXP t( )

paHK t( )

pbHK t( )

pcHK t( )

t

0.02 0.025 0.03 0.035 0.04

6000

4000

2000

2000

4000

6000

Non-active powers

qaEXP t( )

qbEXP t( )

qcEXP t( )

qaHK t( )

qbHK t( )

qcHK t( )

t

Figure 6.6: Case T3 (neutral as reference) Voltages, currents, proposed and expected active and non-active powers


Chapter 6

182


0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

A-phase

van t( )

5 ia t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

B-Phase

vbn t( )

5 ib t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

C-phase

vcn t( )

5 ic t( )

t

0.02 0.025 0.03 0.035 0.04

2000

4000

6000

8000

Active powers

paEXP t( )

pbEXP t( )

pcEXP t( )

paHK t( )

pbHK t( )

pcHK t( )

t

0.02 0.025 0.03 0.035 0.04

2000

1000

1000

2000

Non-active powers

qaEXP t( )

qbEXP t( )

qcEXP t( )

qaHK t( )

qbHK t( )

qcHK t( )

t



Chapter 6 183


0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

A-phase

van t( )

5 ia t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

B-Phase

vbn t( )

5 ib t( )

t

0.02 0.025 0.03 0.035 0.04

250

150

50

50

150

250

C-phase

vcn t( )

5 ic t( )

t

0.02 0.025 0.03 0.035 0.04

1000

1000

2000

3000

4000

Active powers

paEXP t( )

pbEXP t( )

pcEXP t( )

paHK t( )

pbHK t( )

pcHK t( )

t

0.02 0.025 0.03 0.035 0.04

500

300

100

100

300

500

Non-active powers

qaEXP t( )

qbEXP t( )

qcEXP t( )

qaHK t( )

qbHK t( )

qcHK t( )

t

Figure 6.8: Case T5 (Virtual neutral as reference) Voltages, currents, proposed and expected active and non-active powers 6.4.2.2 Average power and energy transfer The energy transfer per period is computed using equations (4.37) and 4.38) and

average powers using (4.42) and (4.44). The results are presented in the Tables 6.1 to

6.3.


Chapter 6

184

Table 6.1: Energy transfer per and average power per period for Phase A/AB Expected Energy Transfer Average Power Phase AB Proposed Expected Units Proposed Units Case T1 (Ref - B) Active ET (EP) 15.54 15.54 W sec 777 W Non-active ET (EN) 0 0 Var sec 0 Var Phase A Proposed Expected Units Proposed Units Case T1 (Ref-Neutral) Active ET (EP) 7.77 7.77 W sec 388.5 W Non-active ET (EN) 2.890 2.891 Var sec 226.962 Var Case T2 Active ET (EP) 28.723480 28.723480 W sec 1436.174 W Non-active ET (EN) 16.512032 16.517157 Var sec 1296.852 Var Case T3 Active ET (EP) 28.72348 28.72348 W sec 1436.174 W Non-active ET (EN) 23.651217 23.666532 Var sec 1857.562 Var Case T4 Active ET (EP) 28.72348 28.72348 W sec 1436.174 W Non-active ET (EN) 16.512032 16.517157 Var sec 1296.852 Var Case T5 Active ET (EP) 12.476415 12.476415 W sec 623.821 W Non-active ET (EN) 0.001330 0 Var sec 0.014 Var Table 6.2: Energy transfer per and average power per period for Phase B Expected Energy Transfer Average Power Phase B Proposed Expected Units Proposed Units Case T1 (Ref-Neutral) Active ET (EP) 7.77 7.77 W sec 388.5 W Non-active ET (EN) 2.837 2.837 Var sec 222.800 Var Case T2 Active ET (EP) 37.130352 37.130352 W sec 1856.518 W Non-active ET (EN) 9.363273 9.360886 Var sec 735.390 Var Case T3 Active ET (EP) 37.130352 37.130352 W sec 1856.518 W Non-active ET (EN) 13.528130 13.523660 Var sec 1062.497 Var Case T4 Active ET (EP) 37.130352 37.130352 W sec 1856.518 W Non-active ET (EN) 9.363273 9.360886 Var sec 735.390 Var Case T5 Active ET (EP) 16.128021 16.128019 W sec 623.821 W Non-active ET (EN) 0.000573 0 Var sec 0.045 Var


Chapter 6 185

Table 6.3: Energy transfer per and average power per period for Phase C/CB Expected Energy Transfer Average Power Phase CB Proposed Expected Units Proposed Units Case T1 (Ref - B) Active ET (EP) 0 0 W sec 0 W Non-active ET (EN) 0 0 Var sec 0 Var Phase C Proposed Expected Units Proposed Units Case T1 (Ref-Neutral)

Active ET (EP) 0 0 W sec 0 W Non-active ET (EN) 0 0 Var sec 0 Var Case T2 Active ET (EP) 23.065825 23.065825 W sec 1153.291 W Non-active ET (EN) 13.821974 13.818451 Var sec 1085.575 Var Case T3 Active ET (EP) 23.065825 23.065825 W sec 1153.291 W Non-active ET (EN) 28.747276 28.737659 Var sec 2257.806 Var Case T4 Active ET (EP) 14.762128 14.762128 W sec 738.106 W Non-active ET (EN) 8.846063 8.843808 Var sec 694.768 Var Case T5 Active ET (EP) 10.018956 10.018921 W sec 500.948 W Non-active ET (EN) 0.000356 0 Var sec 0.027956 Var 6.4.2.3 Additional examples Additional examples are included to show the implementation of the collective three-

phase powers as well as the relationship with RMS based arithmetic powers. The

waveforms and average values for three-phase powers for the Cases T1 to T5 according

to equations (6.7) to (6.9) and (6.11) to (6.15) were determined. The waveforms and

average values obtained are given in Figures 6.9 to 6.15. The average values from the

proposed definitions are compared with RMS values for sinusoidal voltage also for

Cases T1 to T5 (Note for Case T1 only the case with virtual reference is included). The

RMS powers were calculated using the following equations.

XRMS XRMS XRMSS v i= (6.27) t T

XRMS X Xt

P v (t) i (t)+

= ∫ (6.28)

2 2XRMS XRMS XRMSN S P= − (6.29)

3RMS XRMS

X A,B,CP P

=

= ∑ (6.30)


Chapter 6

186

3RMS XRMSX A,B,C

Q Q=

= ∑ (6.31)

2 23RMS 3RMS 3RMSS P Q= + (6.32)

where X represents the phase. These results are in Table 6.4.

Case T1 B as reference

0.02 0.025 0.03 0.035 0.04

1000

2000

3000

s3Ph t( )

p3Ph t( )

q3Ph t( )

t

0.02 0.025 0.03 0.035 0.04

1000

2000

3000

p3PhPOS t( )

q3PhPOS t( )

q3PhNEG t( )

t (a) (b)

P3phAV = 777 W N3phAV = 0 Var S3PhAV = 777 VA Figure 6.9: Three phase instantaneous and average total, active and non-active powers Case T1 Neutral as reference

0.02 0.025 0.03 0.035 0.04

1000

1000

2000

3000

s3Ph t( )

p3Ph t( )

q3Ph t( )

t

0.02 0.025 0.03 0.035 0.04

1000

1000

2000

p3PhPOS t( )

q3PhPOS t( )

q3PhNEG t( )

t (a) (b)

P3phAV = 777 W N3phAV = 449.762 Var S3PhAV = 897.783 VA Figure 6.10: Three phase instantaneous and average total, active and non-active powers


Chapter 6 187

Case T2

0.02 0.025 0.03 0.035 0.04

5000

5000

1 .104

s3Ph t( )

p3Ph t( )

q3Ph t( )

t

0.02 0.025 0.03 0.035 0.04

5000

5000

1 .104

p3PhPOS t( )

q3PhPOS t( )

q3PhNEG t( )

t (a) (b)

P3phAV = 4445.983 W N3phAV = 3117.817 Var S3PhAV = 5430.244VA Figure 6.11: Three phase instantaneous and average total, active and non-active powers Case T3

0.02 0.025 0.03 0.035 0.04

1 .104

5000

5000

1 .104

1.5 .104

s3Ph t( )

p3Ph t( )

q3Ph t( )

t

0.02 0.025 0.03 0.035 0.04

1 .104

5000

5000

1 .104

p3PhPOS t( )

q3PhPOS t( )

q3PhNEG t( )

t (a) (b)

P3phAV = 4445.983 W N3phAV = 5177.865 Var S3PhAV = 6824.738 VA Figure 6.12: Three phase instantaneous and average total, active and non-active powers


Chapter 6

188

Case T4

0.02 0.025 0.03 0.035 0.04

5000

5000

1 .104

s3Ph t( )

p3Ph t( )

q3Ph t( )

t

0.02 0.025 0.03 0.035 0.04

5000

5000

1 .104

p3PhPOS t( )

q3PhPOS t( )

q3PhNEG t( )

t (a) (b)

P3phAV = 4030.798 W N3phAV = 2727.01 Var S3PhAV = 4866.612 VA Figure 6.13: Three phase instantaneous and average total, active and non-active powers Case T5

0.02 0.025 0.03 0.035 0.04

1000

1000

2000

3000

4000

s3Ph t( )

p3Ph t( )

q3Ph t( )

t

0.02 0.025 0.03 0.035 0.04

1000

1000

2000

3000

4000

p3PhPOS t( )

q3PhPOS t( )

q3PhNEG t( )

t (a) (b)

P3phAV = 1931.17 W N3phAV = 0.177 Var S3PhAV = 1931.17 VA Figure 6.14: Three phase instantaneous and average total, active and non-active powers


Chapter 6 189

Case T4 Sinusoidal Balanced Load R= 10.6 ohm and L = 0.42 H

0.02 0.025 0.03 0.035 0.04

1000

1000

2000

3000

4000

s3Ph t( )

p3Ph t( )

q3Ph t( )

t

0.02 0.025 0.03 0.035 0.04

2000

1000

1000

2000

3000

4000

p3PhPOS t( )

q3PhPOS t( )

q3PhNEG t( )

t (a) (b)

P3phAV = 3742.925 W N3phAV = 3508.023 Var S3PhAV = 5129.884 VA Figure 6.15: Three phase instantaneous and average total, active and non-active powers Table 6.4: Compare Proposed and RMS powers for Cases T1 to T5 for sinusoidal

voltage

Case T1 (Ref: Virtual Neutral) Proposed Sinusoidal Units Total S3PhAV / S3RMS 779.418 779.423 Watt Active P3phAV / P3RMS 675 675 Var Non-active N3phAV / Q3RMS 389.702 389.711 VA Case T2 Proposed Sinusoidal Units Total S3PhAV / S3RMS 4800.37 4800.386 Watt Active P3phAV / P3RMS 3862.34 3862.34 Var Non-active N3phAV / Q3RMS 2850.593 2850.619 VA Case T3 Proposed Sinusoidal Units Total S3PhAV / S3RMS 5041.799 5041.813 Watt Active P3phAV / P3RMS 3862.34 3862.34 Var Non-active N3phAV / Q3RMS 3240.689 3240.71 VA Case T4 Proposed Sinusoidal Units Total S3PhAV / S3RMS 4296.571 4296.589 Watt Active P3phAV / P3RMS 3501.659 3501.659 Var Non-active N3phAV / Q3RMS 2489.761 2489.792 VA Case T5 Proposed Sinusoidal Units Total S3PhAV / S3RMS 1931.17 2731.087 Watt Active P3phAV / P3RMS 1931.17 1931.17 Var Non-active N3phAV / Q3RMS 0.177 1931.17 VA


Chapter 6

190

6.5 Analysis and discussion of results To evaluate the proposed definitions, the waveforms and energy transfer of the active

and non-active powers obtained for the cases using definitions of the proposed

definitions are compared with the expected.

Figures 6.3 to 6.7 show that the active and non-active power waveforms, obtained by

the definitions, match the expected. Tables 6.1 and 6.3 corroborate the results obtained

from the waveform comparison because the energy transfers matching with negligible

difference.

The definitions give the same waveform for the active and non-active powers as those

expected.

Figures 6.9 to 6.14 show the graphs implementing equations (6.7) to (6.9) in graph (a)

and equations (6.11) and (6.12) in graph (b). The average powers according to

equations (6.13) to (6.15) are also shown in the figures. The powers’ waveforms and

average values are easily determined without any restrictions for different source

voltage and load combinations.

Table 6.4 shows that under sinusoidal and linear load conditions, the values give by the

proposed definitions are practically the same (the differences being attributed to

completely different methods of computation) as that given by the RMS based

arithmetic powers. Note that Case T5 is a nonlinear load, hence the difference. This is

a very exciting finding as it lays credence to the proposed method. Additionally, since

the proposed method is based on energy transfer, this also shows that the RMS based

arithmetic powers maintain energy information for sinusoidal linear load conditions.

6.6 Experimental work - digital power meter Subsequent to the work done with LabVIEW (see Section 4.6 for details) a project was

undertaken to fabricate a prototype digital meter coded with algorithms implementing

the proposed definitions. The main intent of the project was to investigate the

feasibility of the algorithm implementation in a processor based environment but not the

fabrication of an accurate digital power meter. This project was the subject of the thesis

[105,106, 107, 108, 109] of two groups final year undergraduate students who worked


Chapter 6 191

under the guidance of the author of the thesis. A brief overview and some results are

presented below.

6.6.1 Introduction The block diagram of the digital meter is shown in Figure 6.16.

Figure 6.16: Digital power meter block diagram

There are three main stages to the prototype digital power meter as shown in the block

diagram. The input stage, the anti-aliasing stage (signal conditioning stage) and the

processor board (processing stage).

6.6.2 Input stage The input voltage and current transformers scale down the secondary voltage (110 volts

three-phase and neutral – phase to neutral voltage 63.5 voltage nominal) and secondary

current (1 amp three-phase), from the power system voltage and current measurement

transformer, to values suitable for the processor. There are three voltage input

transformers (one each for phases A, B and C) and three current input transformers (one

each for phases A, B and C).

6.6.3 Signal conditioning stage The anti-aliasing filters receive the signal from the voltage and current input

transformers and process the signal. There are three voltage and three current input

transformers. The anti-aliasing filter is a low pass filter that removes harmonics higher

than 13th and conditions the signal suitable for input to the ADC on the processor board.

One anti-aliasing input filter is used for each of the analog inputs from the input

transformers. The six anti-aliasing filters were identical with the exception that the

output from the voltage and current input transformer were conditioned to give an

identical peak voltage into the anti-aliasing filter.

Voltage / current input

transformer s

Anti-aliasing filter with

gain

Processor board


Chapter 6

192

The input and signal conditioning was first implemented on a stripboard and later

implemented on a printed circuit board as shown in Figure 6.17.

(a)

(b)

Figure 6.17: Input and signal conditioning on (a) stripboard and (b) printed circuit board

6.6.4 ADC and processing stage

Figure 6.18: Block diagram of the process

The output from the anti-aliasing filter is passed on to the ADC, which samples the

signal. The power computation algorithm was in the processor. The block diagram of

the process is given in Figure 6.18. FFT is performed on the ADC signal and then

passed on to the powers computation algorithm. The output was written to disk as a text

file for offline use. Due to time constraint the output stage in the form of an LCD

display is not yet implemented. This will be the subject of future project. Figure 6.19

shows the ADC and processor hardware.

Power Calculation performed on the

FFT’s data

FFT performed on the ADC’s data

ADC on FPGA samples the analog signal and

converts it into digital data

Anti-Aliasing Filters

Result output/display


Chapter 6 193

(a)

(b)

Figure 6.19: (a) The processor board and (b) the ADC mounted on the processor board

6.6.5 Experimental setup The experimental setup identifying the main components is shown in Figure 6.20.

Harmonics Oscilloscope Digital Meter synthesizing PC

Figure 6.20: Experimental setup The voltage and current signals were generated in the PC and output through the PC’s audio card as continuous time signals that are fed into the signal conditioning stage of the digital power meter. The digital power meter performs all the computations and writes the computation results to a file in the PC. 6.6.6 Result and discussion Result

The powers calculation was successfully implemented and tested. The result for a

single phase is given in Figure 6.21. As a comparison, the corresponding result

obtained using MathCad is shown in Figure 6.22.


Chapter 6

194

-150.00

-100.00

-50.00

0.00

50.00

100.00

150.00

200 220 240 260 280 300 320

Sample Number

Vol

tage

(V)/P

ower

s (W

)

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

Curr

ent (

A)

VP DCP fundP HarmP cross AP Cross BP TotalI

Figure 6.21: Voltage, current, component power and total power output from meter

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

150

100

50

50

100

150

2

1

1

2

seconds

Vol

ts, W

atts

Am

ps

vk

P0k

P1k

Phk

PXA k

PXBk

Pk

ik

k dt⋅

Figure 6.22: Voltage, current, component power and total power using Mathcad Note that nonactive power is zero since voltage and current are in phase relationship.

Discussion of result

The result obtained from digital power meter shown in Figure 6.21 matches that from

MathCad computation in Figure 6.22, showing that the algorithm was successfully

implemented in the meter.


Chapter 6 195

6.7 Conclusion The proposed three-phase definitions of instantaneous powers can be utilised on a phase

basis and are defined using the definitions proposed for single-phase. Definitions for

three-phase instantaneous powers, total active and non-active on a collective basis are

also proposed. Three-phase collective powers for the different power components, as

defined for the single-phase, have also been proposed. Three-phase positive and

negative instantaneous total, active and non-active powers are also proposed. Average

three-phase powers have been defined based on energy transfer and the positive and

negative instantaneous powers. These average powers have been shown to be

consistent with the RMS arithmetic powers for sinusoidal linear circuits. A measure of

unbalance in three-phase circuits has been defined.

The applications of the three-phase have been stated to be the same as the single-phase

for measurement, compensation, detection of source of distortion and power quality.

The test cases show that the waveforms of the powers on a phase basis are identical to

the expected and likewise for the average powers and energy transfer. Additional

examples have been included to show the determination of the average powers using the

proposed collective powers definitions. This also includes a comparison with the RMS

arithmetic powers.

The experimental work done on a prototype digital power meter has been briefly

outlined with presentation of some results. The algorithm was successfully

implemented in the processor.


Chapter 6

196


Applications of the definitions

Chapter 7 197

7. APPLICATIONS OF THE DEFINITIONS

The ultimate objective and test of any definition is application. This chapter explores

the application possibilities of the new definitions. The objective is to identify and

show that the information obtained from the definitions is useful for application and is

faithful in achieving its aim. No attempt is made to show a comprehensive analysis

with actual devices, for example using a compensator to achieve compensation. It is left

to experts in the respective fields to utilise the information provided by the proposed

definitions to achieve their objectives. The examples used in this chapter are mainly

drawn from the literature as used by other researchers as well as developed by the

author specifically to illustrate the usefulness of the powers components.

7.1 Introduction This section identifies the application areas. Before considering this, the information

provided by the components, which has been outlined in Chapters 4 and 6, is reviewed.

This recapitulation is useful since the defined power components provide information

that aids in the application area.

The components have certain characteristics that are useful in providing information

about the system from the metering point perspective. The components are listed below

(a) DC based power comprising

(a1) DC power s0D(t), p0D(t), q0D(t) and

(a2) DC based s0X(t), p0X(t), q0X(t).

(b) Fundamental power s1(t), p1(t), q1(t).

(c) Source generated harmonic powers sh(t), ph(t), qh(t).

(d) Load generated harmonic powers sg(t), p g(t), q g(t).

(e) Source generated cross powers sXH(t), pXH(t), qXH(t).

(e1) Cross-fundamental powers sX1(t), pX1(t), qX1(t) and

(e2) Cross-harmonic powers sXh(t), pXh(t), qXh(t).

(f) Load generated cross-harmonic powers sXg(t), pXg(t), qXg(t).

For sinusoidal sources with linear load only component (b) exists. Components (a), (d)

and (f) will arise when the load, as viewed from the measuring point, is nonlinear and


Chapter 7 198

the source does not have DC content. Component (a1) DC power only will exist if the

source is DC and load is linear. Components (c) and (e) will be present when voltage

harmonics exist in the source voltage supplying a linear load.

The average powers can be used for metering and billing purposes.

If (c) and (e) are present then it is possible to fully compensate the nonactive power with

passive elements. On the other hand if components (a), (d) and (f) are present, then

passive components alone may not be sufficient to provide complete nonactive power

compensation. It is possible to remove (d) and (f) by filtering out Ig. Active

compensation is required to compensate (a).

Components (a) to (f) can be utilised to gauge power quality. Components (a1), (b), (c) and (d) are directional. The direction can be used to detect

the source of that component in relation to the metering point. This can be used to

detect the source of the unwanted component at the metering point. Component (b),

fundamental power, is used to define the correct or positive direction of power flow

(source to load). Components (a1), (c) and (d) are the main components that can be

used in detection of source of pollution. Components (a2), (e2) and (f) are supporting

components for (a1), (c) and (d) respectively in decision-making. The reason for this is

that components (a1), (c) and (d) could be quite small in value to make a definitive

decision.

The application areas identified are listed below. For each area a brief statement of the

aspect of the definition that is useful to that area, is made.

Measurement: Knowledge of the time profile of the powers allows accurate

measurement of the powers (especially nonactive)

Compensation: Knowledge of the time profile of the instantaneous active and

nonactive powers as well as the different components facilitates the selective reduction

of the source current. The power components defined give information of action to take

in terms of removing unwanted currents/powers, reducing source current as well as

improving the source voltage and current waveform.


Chapter 7 199

Source of distortion: The power components (a) to (f) can be used to identify the

source of the distortion in power systems because the existence of the cross-harmonic

components supports the directional information available from the corresponding

active components.

Power quality: The presence of the power components (a) DC based power, (c) source

generated harmonic power, (d) load generated harmonic power, (e2) source generated

cross-harmonic, (f) load generated cross-harmonic power, can be used as an indication

of power quality.

General: The definitions in general have been found to be quite useful in proving

substantial information about the system from the perspective of the measuring point.

Each of the above areas will be investigated separately. ATP is used to simulate the

electrical system close to reality and to provide voltages and currents at points of

interest. Mathcad, where the powers algorithms are implemented, is then used to

process and analyse the data obtained from the ATP.

7.2 Background technical information Following the trend of Chapters 2 and 3 some additional technical information relevant

to this chapter is outlined.

7.2.1 Compensation concepts used in this thesis To obtain optimal results in compensation, accurate knowledge of the components to be

removed is required. Therefore, if the definition correctly gives the active and/or

nonactive power, this knowledge provided by the definition can be used to completely

compensate (using shunt elements and DC sources) the unwanted components, leaving

the source supplying only the wanted power. The concept behind this is explained as

follows.

Compensation will be considered from the concepts point of view. Conceptually

compensation is

• removal of unwanted currents (or powers) with the intent of reducing source

currents


Chapter 7 200

• improving both source voltage and source current waveform from distortion caused

by load generated currents.

Passive compensation uses static components (inductors and capacitors) to achieve

these goals and active compensation uses active means (e.g static var compensator) to

achieve this.

Passive compensation removes the current that can be compensated from the source to

the passive element providing compensation. There are two types. In one type the

passive element temporarily stores the compensating current during some parts of the

cycle and then returns it to the load during other parts of the cycle. Source generated

nonactive powers are candidates for this type of passive compensation. The other type

is when it is used as a filter to filter-out load generated currents. Load generated powers

(currents), especially when source impedance is not negligible, are candidates for this

type of passive compensation. Passive compensation can reduce losses in source

resistance.

7.2.1.1 Compensation of source generated harmonics Essentially the nonactive power can be compensated with passive elements. This is

achieved by removing the nonactive current that contributes to the nonactive power

harmonic by harmonic. The sketch in Figure 7.1 illustrates this pictorially.

~R

fundamentalharmonicpass filter

fundamentalvar supply

sourcewithharmonics

harmonicpass filterand var

supply forother

harmonics

.....

.....

L

C

iS iL

i1C ihC

Figure 7.1: Compensation (using shunt elements) of nonactive current The passive “harmonic pass filter”, tuned to a particular frequency, is used to ensure

that only the selected harmonic flows in the leg. The “var supply” is used to supply the

nonactive current for that particular harmonic. This is done for each harmonic. In this

manner it is possible to compensate for the entire nonactive current and hence power,

harmonic by harmonic. This method will show if a definition faithfully defines the


Chapter 7 201

active/nonactive power (or current). The method is based on the C-filter. The treatment

here is focused on showing complete compensation of predicted nonactive power using

the new definition, regardless of commercial economics. Hence the like components of

the harmonic pass tuned filter and the Var supply are not combined. The choice of the

values of the components of the tuned filter is important for minimising the leakage of

other harmonic currents. This will minimise error and give faithful results for the

simulation. Hence the chosen values for these components may not be of a practical

magnitude.

The instantaneous nonactive power or nonactive current is used to determine the

compensating current. Generally this is

xcx

x

q (t)i (t)v (t)

= − or cx naxi (t) i (t)= − (7.1)

where “x” is the harmonic index; icx(t) is the compensation current, qx(t) the nonactive

power, inax(t) is the nonactive current and vx(t) is the source voltage for harmonic x.

For each harmonic the compensating capacitance/inductance, depending on whether

inductive or capacitive vars are required, is given by

Cc = cxrms

xrms FOR EACH HARMONIC x

iVω

or Lc= xrms


VIω

. (7.2)

Each harmonic compensating current, i1c(t) and/or ihc(t), is realised by a series tuned LC

filter in series connection with a compensating capacitor Cc or inductor Lc, which is the

“var supply”. The compensating capacitors/inductors provide the compensating

currents that supply the nonactive current required by the load.

The harmonic pass filter for each frequency is determined from

2FOR EACH HARMONIC

1LC=ω

(7.3)

Thus the compensating part of the circuit shown in Figure 7.1 is known. Since all the

components of the circuit are known, the source current after compensation is(t) for the

circuit, can be determined using circuit analysis. The compensated source current will

comprise harmonic components as follows

is(t) = i1s(t) + ihs(t) (7.4)


Chapter 7 202

For a fully compensated system the fundamental i1s(t) will be in phase with v1(t) and

each harmonic ihs(t) in phase with vh(t). Also the source current will have the minimum

magnitude for a fully compensated system. The magnitude (RMS value) of is(t) is

obtained using

Isrms = ( )T

2s

0

1 i (t) dtT ∫ (7.5)

7.2.1.2 Compensation of load generated harmonics Load generated harmonics are compensated by filtering them. The idea behind this is

similar to that for source generated harmonics but only the harmonic pass filter (see

Figure 7.1) is used. This tuned filter will remove the load-generated harmonic currents.

7.2.1.3 Compensation of load generated DC This will be compensated with DC current injection using a DC current source.

7.2.2 Summary The steps for compensation have been detailed in Subsections 7.2.1. The detailed

mechanics of compensation have been listed. This is also a good educational tool in

better understanding compensation concepts in a power system. It is evident that

compensation can be performed solely using DC injection and passive elements. This

has not been shown before and could lead to new and unique compensation methods.

In the application example, see Section 7.4, the compensation will be performed using

ATP and Mathcad.

7.3 Measurement The powers measurement capability of the definitions have been illustrated with the

case studies in Subsection 4.4.2 of Chapter 4 and 6.4.2 of Chapter 6.

7.4 Compensation Application examples using the concepts outlined in Subsection 7.2.1 are given below.


Chapter 7 203

7.4.1 Application example 7.4.1 Introduction

The example is based on that used in [6]. It shows the application of the definitions in

compensating a linear load. Complete removal of nonactive power is possible with

passive components. Detailed calculations outlining the process are shown in the

ensuing analysis.

The source has harmonics and RLC load with data as given in Figure 7.2.

~

L

R

Cmetering

pointv =v + v1 2

V1 100 volt⋅:= ω1 100 π⋅1

sec⋅:=

V2 0.3 V1⋅:= ω2 4 ω1⋅:=

R 5 ohm⋅:= L 0.02 H⋅:= CL 1 10 3−⋅ F⋅:=

Source impedance is neglected. The voltages above are RMS values with phase relationship as in equation (7.6).

Figure 7.2: System data The voltage and current are given by

v t( ) 2 V1⋅ cos ω1 t⋅( )⋅ 2 V2⋅ cos ω2 t⋅( )⋅+:= (7.6)

and

i t( ) 2V1Z1⋅ cos ω1 t⋅ δ1−( )⋅ 2

V2Z2⋅ cos ω2 t⋅ δ2−( )⋅+:=

(7.7) where Z1 and Z3 are respectively the magnitude of the fundamental and harmonic

impedances with δ1 and δ3 the corresponding impedance angle for the load and

i1 t( ) 2V1Z1⋅ cos ω1 t⋅ δ1−( )⋅:= i2 t( ) 2

V2Z2⋅ cos ω2 t⋅ δ2−( )⋅:= . (7.8)

Calculation, results and discussion

The voltage and current as given in (7.6) and (7.7) are used to calculate the component

active and nonactive powers, using equations (4.20) to (4.35) from Chapter 4, and the

waveforms plots are shown in Figure 7.3. Computations are performed with Mathcad.


Chapter 7 204

0 0.005 0.01 0.015 0.02

200

150

100

50

50

100

150

200

v t( )

4 i t( )⋅

t Metering point Voltage and current

0.02 0.025 0.03 0.035 0.04

1000

500

500

1000

1500

2000

2500

3000

3500

4000

s1 t( )

p1 t( )

q1 t( )

t Fundamental powers (b)

0.02 0.025 0.03 0.035 0.04

40

30

20

10

10

20

30

40

50

60

sh t( )

ph t( )

qh t( )

t Source generated harmonic powers (c)

0.02 0.025 0.03 0.035 0.04

1500

1200

900

600

300

300

600

900

1200

1500

sxh t( )

pxh t( )

qxh t( )

t Source generated cross-harmonic powers (e)

0.02 0.025 0.03 0.035 0.04

4000

3000

2000

1000

1000

2000

3000

4000

5000

6000

s t( )

p t( )

q t( )

25 v t( )⋅

t

Total powers

• DC power p0D= 0 (a1)

• DC based cross powers p0X, q0X = 0 (a2)

• Load generated harmonic powers

pg, qg = 0 (d)

• Load generated cross-harmonic powers

pXg, qXg = 0 (f)

Figure 7.3: Powers based on proposed definition The total RMS (current without compensation) and source RMS current (the current left

after compensation i.e. active current) are determined. These are calculated as follows

irms1T T

2Tti t( )( )2⌠

⎮⌡

d⋅:= irms 17.040756 A=

(7.9)

and active RMS current

iactive t( )p t( )v t( )

:= IActiverms1T

T

2T

tiactive t( )2⌠⎮⌡

d⋅:= IActiverms 14.448512A= . (7.10)


Chapter 7 205

For this example, nonactive power is compensated by passive means. The fundamental

(Figure 7.3 (b)) and harmonic (Figure 7.3 (c)) nonactive power is to be removed. It is

used to determine the compensating current. Removal of the fundamental and harmonic

nonactive current will also remove the cross-harmonic nonactive power (Figure 7.3

component (e)). Using this compensating current, the concepts of Subsection 7.2.1 and

equations (7.1) to (7.3), the passive components of the compensating filter part of the

circuit are determined.

Fundamental pass filter L1, C1 and compensating capacitor Cc1

L1 506.606 mH= C1 20 μF= Cc1 285.112 μF=

(7.11)

Harmonic pass filter L3, C3 and compensating capacitor Cc3 for 4th harmonic

L2 316.629 mH= C2 2 μF= Cc2 31.374 μF=

(7.12)

The circuit with compensation components is shown in Figure 7.4. Note that C1 and

Cc1 (C2 and Cc2) are shown in series to illustrate the concept. Practically they can be

combined to a single capacitor.

L2

C2

Cc2

Harmfilter

L1

C1

Cc1

Fundfilter

~L

R

CL

Figure 7.4: Source, load and shunt compensating elements The source current for the fundamental is1(t) after compensation is determined by circuit

analysis using only fundamental voltage as the source. The fundamental source current

after compensation is shown in Figure 7.5. The first graph shows one cycle of the

waveform and the second shows the first crossing of the voltage and current on an

expanded scale to show where it occurs. Theoretically it should occur at v1(t) = 0, but

in the Figures it does not. This is attributed to the leakage currents flowing through the

compensating legs.


Chapter 7 206

0.02 0.03 0.04

250

150

50

50

150

250

v1 t( )

10 is1 t( )⋅

t

0.0249 0.025 0.0251

5

5

v1 t( )

10 is1 t( )⋅

t Figure 7.5: Fundamental current after compensation Similarly the harmonic source current is2(t) is determined for the harmonic. The result

likewise is shown in Figure 7.6.

0.02 0.025 0.03

50

50

v2 t( )

20 is2 t( )⋅

t

0.0237 0.02375 0.0238

5

5

v2 t( )

20 is2 t( )⋅

t

Figure 7.6: 4th harmonic current after compensation The resulting source current is the sum of the fundamental and harmonic and is given by

is t( ) is1 t( ) is2 t( )+:=

(7.13)

The source voltage v(t), the source current after compensation is(t) and the load current

iL(t) are shown in Figure 7.7. Note the phase relationship between the source voltage

and source current. Both exhibit symmetry about the same half-cycle of the

fundamental. This is akin to the in-phase relationship of voltage and current in

sinusoidal systems and is an indication of voltage and current “in-phase” relationship

for non-sinusoidal systems.


Chapter 7 207

0.02 0.03 0.04 0.05 0.06

300

200

100

100

200

300

v t( )

10 is t( )⋅

10 iL t( )⋅

t Figure 7.7: Voltage, load and source current after compensation using Mathcad The RMS source current after compensation is

Isrms1T

0.sec

T

tis t( )2⌠⎮⌡

d⋅:= Isrms 14.448736 A=

(7.14) This value is practically the same as the value of active current 14.448512 amp

determined earlier from the active power proposed by the definition. The difference is

attributed to leakage current through the compensating legs. There will some analysis

about a similar leakage in application Example 7.4.2.

The next step is to determine the RMS source current for varying values of Cc1 and Cc2

to check the value of compensated minimum source current. This is done by repeating

the above computation with various values of Cc1 and Cc3. Table 7.1 and the graph in

Figure 7.8 show the results.

Table 7.1: Source current (RMS value) for varying compensating capacitors

Cc1 279.1 281.1 283.1 285.1 287.1Cc227.5 14.449857 14.449442 14.449301 14.449434 14.44983929.5 14.449262 14.448849 14.448710 14.448843 14.44925031.5 14.449062 14.448650 14.448512 14.448646 14.44905433.5 14.449255 14.448844 14.448707 14.448843 14.44925235.5 14.449842 14.449432 14.449296 14.449432 14.449842


Chapter 7 208

279.1 281.1 283.1 285.1 287.127.5

31.5

35.5

14.44800014.44820014.44840014.44860014.44880014.44900014.44920014.44940014.44960014.44980014.450000

14.449800-14.45000014.449600-14.44980014.449400-14.44960014.449200-14.44940014.449000-14.44920014.448800-14.44900014.448600-14.44880014.448400-14.44860014.448200-14.44840014.448000-14.448200

Figure 7.8: Plot of RMS compensated source current for varying compensating capacitors The result shows that the minimum current 14.448512 amp is very close to the

compensated source current 14.448736 amp obtained by the method above. The

difference between the compensated source current and the expected minimum current,

as stated above, is due to the leakage current. This value, 14.448512 amp, is however

equal to active current in equation (7.10) obtained from the definition. This means that

the proposed definitions accurately predict the active current and thus the nonactive

current.

Finally ATP is used to simulate the system with the compensating capacitance values as

obtained using the proposed definitions. The system modelled in “ATP Draw” is as

shown in Figure 7.9. The simulation is run for a time period of 20 secs with a sampling

time of 0.1 msec.

Source VI

Comp

LoadI

I

20 mH

1000 microF

5 ohm

20 microF

505.606

0.01 ohm

285.112 31.374

2 microF

0.05 ohm

316.629

100 V,

50 Hz

100 V200 Hz

microF

mH mH

microF

0.05 ohm

Figure 7.9: ATP Draw system used for analysis with ATP


Chapter 7 209

The waveforms for the source voltage, load voltage, source current and load current

obtained are as shown in Figure 7.10. Note that source resistor and those in the

compensating legs were added to minimise oscillations. Hence the circuit is slightly

different from that analysed using Mathcad.

(f ile ExRLC01.pl4; x-v ar t) factors:offsets:

10

v :SOURCE 10

c:SOURCE-XX0002 100

c:XX0002-LOAD 100

19.960 19.965 19.970 19.975 19.980 19.985 19.990 19.995 20.000*10-300

-200

-100

0

100

200

300

The legend for the plots: Red = Metering point voltage, Green = source current, magnified by 10, after compensation, Blue = load current magnified by 10. Figure 7.10: Voltage, load and source current after compensation using ATP The waveforms from ATP and Mathcad are similar corroborating the Mathcad analysis.

However the RMS value of the currents, source current after compensation 14.280904

and load current 16.797322, are slightly different and this can be attributed to the

additional components in the ATP circuit.

Note that after complete compensation of the nonactive power, the source current

waveform is not a scaled version of the voltage waveform. The reason for this is

because the equivalent conductance of the load as perceived from the metering point is

not constant. Subsection 2.2.5 of Chapter 2 has an analysis of this.

Application example conclusion

The information from the proposed definitions has been used to perform passive

compensation of nonactive power using ATP simulation and Mathcad analysis. The

results show that practically complete compensation has been achieved.

Note that after compensation, the current waveform, for example in Figures 7.7 and

7.10, is not the same shape as the voltage waveform. This current would be a scaled

version of voltage for definitions where constant conductance of the load is assumed


Chapter 7 210

(e.g. Fryze’s definition any others based on this premise). Though the minimum current

is obtained theoretically by this method, but when the nonactive current obtained by this

method is compensated by passive means, optimal compensation may not be achieved.

Further if an active compensator is used to obtain the scaled waveform active current,

additional current will have to be provided by the compensator, which adds to the

source current to give a net total current that will be higher than that obtained above

using the proposed definitions.


This example shows the application of the definitions in compensating a nonlinear load

supplied by a source with impedance. Detailed calculations are shown in the ensuing

analysis.

As shown in Figure 7.11, the voltage is sinusoidal with resistive-inductive source

impedance. The load is a rectified RL load. The data is given in Figure 7.11. ATP is

used to simulate currents while MathCad is used to analyse and determine

compensating information for the system. The output from ATP is obtained as time

stamped data. The computations were performed using this time stamped data where

subscript “k” represents the “kth” data value. The time “dt” is the time interval between

data values.

Load

Comp

XX0003

UXX0007

Source

Amplitude

Freq = 50 Hz

5 mH

0.5 Ohm

33 Ohm +

10 microF

15 mH

10 Ohm

= 26.87 V

Generator voltage amplitude = 26.87 Volts Other data given in the figure. The diode is ideal with zero internal resistance. XX0003 is the metering point

Figure 7.11: Example 7.4.2 System data ATP is used to determine the voltage and current for the circuit of Figure 7.11. The

simulation is run for a time period of 10 seconds with a sampling time of 0.2 msec. The


Chapter 7 211

voltage and current obtained is given in Figure 7.12. The component powers were

computed using the proposed definitions and are graphed in Figure 7.12.

0.02 0.025 0.03 0.035 0.04

40

20

20

40

vk

5 ik

k dt⋅ Metering point Voltage and current

0.02 0.025 0.03 0.035 0.04

4

2

2

4

s0Dk

p0Dk

q0Dk

0.1 vk⋅

0.5 ik⋅

k dt⋅ DC power (a1)

0.02 0.025 0.03 0.035 0.04

40

20

20

40

s0Xk

p0Xk

q0Xk

0.5 vk

5 ik

k dt⋅ DC based cross powers (a2)

0.02 0.025 0.03 0.035 0.04

40

20

20

40

60

s1k

p1k

q1k

vk

5 ik⋅

k dt⋅ Fundamental powers (b)

0.02 0.025 0.03 0.035 0.04

0.4

0.2

0.2

0.4

shk

phk

qhk

0.01 vk⋅

0.05 ik⋅

k dt⋅ Source generated harmonic powers (c)

0.02 0.025 0.03 0.035 0.04

1

0.67

0.33

0.33

0.67

1

sgk

pgk

qgk

0.01 vk⋅

0.05 ik⋅

k dt⋅ Load generated harmonic powers (d)


Chapter 7 212

0.02 0.025 0.03 0.035 0.04

20

10

10

20

30

sXhk

pXhk

qXhk

0.5 vk⋅

2.5 ik⋅

k dt⋅ Source generated cross-harmonic powers (e)

0.02 0.025 0.03 0.035 0.04

20

10

10

20

30

sXgk

pXgk

qXgk

0.5 vk⋅

2.5 ik⋅

k dt⋅ Load generated cross-harmonic powers (f)

0.02 0.025 0.03 0.035 0.04

50

50

100

150

sk

p k

qk

vk

3 ik⋅

k dt⋅ Total powers

Figure 7.12: Powers based on proposed definition

The waveforms in Figure 7.12 are analysed in the following. The active DC power is

negative (a1) indicating that this power is flowing from the load to the metering point,

i.e., toward source. Thus the origin of the current I0 contributing to this power is the

load. The nonlinear load is the cause of this. For compensation the current contributing

to s0D and s0X must be removed if it originates from the load. This removal is by active

compensation. The fundamental power (b) flows from the source to the load. For

compensation the current contributing to q1 can be removed as it originates from the

source (fundamental almost always originates from the load). This removal is by

passive compensation. Source generated harmonic powers (c) are negligible, indicating

that the source does not generate harmonic currents. Should these exist then the

compensation method would be similar to that for the fundamental (similar to

application Example 7.4.1 Subsection 7.4.1). The load generated active harmonic

power (d) is negative. This is an indication of flow from the load to the metering point,

i.e., from load to source. The currents contributing to this power are generated in the


Chapter 7 213

load. The load-generated powers are small in magnitude because of dependence on the

small voltage drop across the source impedance as a result of flow of these load-

generated currents. However, load generated cross-harmonic powers (f) corroborate

their existence. For compensation the current contributing to sg must be removed since

it originates from the load. These currents will be filtered out with tuned filters as

outlined in Subsection 7.2.1 above.

Progressing with the compensation in steps. First the load generated DC currents will

be removed. This will be followed by the removal of load generated harmonic currents.

The reason for removing the load-generated harmonics before the fundamental is

because the filtering components will also draw fundamental leakage current (akin to a

real system when simulated with ATP) and this tends to compensate the fundamental

nonactive power. Finally the fundamental nonactive power will be removed. Both

Mathcad and ATP will be used in the ensuing analysis. This enables comparison of

results. In Mathcad the compensating current will be mathematically removed from the

source current while in ATP the compensating components will be used in the circuit to

determine the compensated source current by simulation.

Compensation of load-generated DC current

Mathcad computation

The first step is to remove the load generated DC currents. The resulting waveforms are

as shown below. The load-generated DC current is given by

icompDCGENkI0k−:=

(7.15) The source current is then given by

isource kik icompDCGENk+:= .

(7.16)

After removal of each component, the metering point voltage is calculated using

vsource kvgk

isource kRs⋅− Ls

isource k 1+isource k 1−

−

2 dt⋅⋅−:=

(7.17)

where vg is the source voltage before the source impedance (Rs + jLs) at the generator

terminals, and vsource the voltage at the metering point.

The resulting waveforms after compensation of DC currents is given in Figure 7.13.


Chapter 7 214

0.04 0.05 0.06 0.07 0.08

10

7.5

5

2.5

2.5

5

7.5

10

ik

icomp k

isourcek

0.25 vsk⋅

0.25 vsourcek⋅

k dt⋅ Figure 7.13: Voltage and current waveforms after compensation of DC current Legend: vs (equal to v) - source voltage at the metering point before compensation, vsource the same voltage after compensation, i is the load current, icomp the compensation current, isource source current after compensation ATP simulation

Compensate for the load generated DC current of magnitude 1.0773261 amps with a

DC current source. This is reflected in the ATP circuit in Figure 7.14. Note that in the

ATP circuit, the DC current source has a RC shunt which filters off high frequency

voltage harmonics. The resulting waveforms are graphed in Figure 7.15.

Load

Comp

U

Source

Amplitude

Freq = 50 Hz

5 mH

0.5 Ohm

33 Ohm +

10 microF

15 mH

10 Ohm

= 26.87 V

10 Ohm

33 Ohm +

10 microFDC current

source

Highfrequency

filter

Figure 7.14: Circuit for compensation of load-generated DC current


Chapter 7 215

(f ile ExDRL2.pl4; x-v ar t) factors:offsets:

10

v :SOURCE 0.250

c:SOURCE-XX0002 10

c:XX0002-COMP 10

c:XX0002-LOAD 10

9.960 9.965 9.970 9.975 9.980 9.985 9.990 9.995 10.000-10.0

-7.5

-5.0

-2.5

0.0

2.5

5.0

7.5

10.0

Figure 7.15: Voltage and current waveforms after compensation of fundamental nonactive power plus load-generated harmonic and DC currents The waveforms in Figure 7.15 closely match that obtained using Mathcad in Figure

7.13.

Compensation of load generated harmonic currents

Mathcad computation

The load generated compensating current is

icompGk1

nmax 1−

n

2 Igk n,⋅ cos Ck n,( )⋅⎛

⎝⎞⎠∑

=

−:=

(7.18)

and the resulting source current is given by

isource kik icompDCGENk+ icompGk

+:=

(7.19)

where ik is load current and isource is the source current after compensation. The voltage after compensation is calculated with equation (7.17). Compensation of load generated harmonic currents results in the waveforms shown in

Figure 7.16.


Chapter 7 216

0.04 0.05 0.06 0.07 0.08

10

7.5

5

2.5

2.5

5

7.5

10

ik

icompG k

isourcek

0.25 vsk⋅

0.25 vsourcek⋅

k dt⋅ Figure 7.16: Voltage and current waveforms after compensation of load generated DC and harmonic currents ATP simulation

Using the technique outlined in Subsection 7.2.1 the load generated harmonic currents

are filtered off by tuned filters. The components for this example are shown in the ATP

circuit in Figure 7.17.

Load

Comp

U

Source

Amplitude

Freq = 50 Hz

5 mH

0.5 Ohm

33 Ohm +

10 microF

15 mH

10 Ohm

= 26.87 V

253.30 mH

0.001 ohm

10 microF

112.58 mH

0.001 ohm

10 microF

63.33 mH

0.001 ohm

10 microF

40.53 mH

0.001 ohm

10 microF

28.14 mH

0.001 ohm

10 microF

20.68 mH

0.001 ohm

10 microF

15.83 mH

0.001 ohm

10 microF

12.51 mH

0.001 ohm

10 microF

10.13 mH

0.001 ohm

10 microF

8.37 mH

0.001 ohm

10 microF

7.03 mH

0.001 ohm

10 microF

6.00 mH

0.001 ohm

10 microF

10 Ohm

33 Ohm +

10 microFDC current

source

Highfrequency

filter

Figure 7.17: Circuit for compensation of load generated DC and harmonic currents The waveforms resulting from the simulation are graphed in Figure 7.18.


Chapter 7 217


10

v :SOURCE 0.250

c:SOURCE-XX0002 10

c:XX0002-COMP 10

c:XX0002-LOAD 10

9.960 9.965 9.970 9.975 9.980 9.985 9.990 9.995 10.000-10.0

-7.5

-5.0

-2.5

0.0

2.5

5.0

7.5

10.0

Figure 7.18: Voltage and current waveforms after compensation of load generated DC and harmonic currents

Comparing Figure 7.18 to Figure 7.16 it is noted that the ATP source current has a

similar shape as that from Mathcad, except the phase is different. The Mathcad source

current is lagging while that from ATP is leading. The reason for this is the

fundamental leakage current through the harmonic tuned filters that remove the load-

generated harmonic currents. This leakage current is not reflected in the Mathcad

computation because the currents are mathematically compensated. This leakage

current has overcompensated the fundamental nonactive power/current. This source

current from ATP when analysed using the proposed definitions gives fundamental

powers as shown in Figure 7.19.

0.02 0.025 0.03 0.035 0.04

40

20

20

40

60

80

S1k

P1k

Q1k

vk

5 ik⋅

k dt⋅ Figure 7.19: Fundamental powers in the ATP source current after compensation of load generated currents

The existence of capacitive nonactive power is quite apparent in Figure 7.19.

Comparing the nonactive power in this figure with fundamental powers (b) in Figure


Chapter 7 218

7.12 shows the overcompensation. This will be further analysed in the next step where

compensation for the fundamental nonactive current is considered.

Compensation of fundamental nonactive power

Mathcad computation

The compensating current is obtained from q1 (fundamental powers (b) in Figure 7.12)

and is given by

icompFUNDk2 Ik 0,⋅ sin Ak 0,( ) sin Bk 0,( )⋅( )⋅⎡⎣ ⎤⎦−:=

(7.20)

The total compensation current, including the fundamental nonactive current, is

icompkicompDCGENk

icompGk+ icompFUNDk

+:=

(7.21)

and the source current after compensation is given by

isource kik icompDCGENk+ icompGk

+ icompFUNDk+:=

(7.22)

where ik is load current and isource is the source current after compensation of load-

generated harmonic currents and fundamental nonactive power.

The voltage after compensation is calculated with equation (7.17). The waveforms

after compensating fundamental, load-generated harmonic and DC currents are given in

Figure 7.20.

0.04 0.05 0.06 0.07 0.08

10

7.5

5

2.5

2.5

5

7.5

10

ik

icomp k

isourcek

0.25 vsk⋅

0.25 vsourcek⋅

k dt⋅ Figure 7.20: Voltage and current waveforms after compensation of fundamental nonactive power plus load generated DC and harmonic currents


Chapter 7 219

ATP simulation

Analysis of the nonactive power in the source current after load-generated harmonic

current compensation (Figure 7.19) gives the fundamental compensating components

L1, C1 and compensating inductor Lc1 as

L1 = 506.61 mH C1 = 20 microF Lc1 = 115.9 mH. This is reflected in the ATP circuit for fundamental nonactive power compensation is

given in Figure 7.21. Note that in the ATP circuit, the components compensating the

harmonic load-generated current have been grouped (red box). The resulting

waveforms are graphed in Figure 7.22.

Load

Comp

XX0002

UXX0006

Source

Comp

GR

OU

Ptf-2-13

Amplitude

Freq = 50 Hz

5 mH

0.5 Ohm

33 Ohm +

10 microF

15 mH

10 Ohm

= 26.87 V

506.61mH

0.2 ohm

115.9 mH

20 microF

33 Ohm +

10 microFDC current

source

Highfrequency

filter

Figure 7.21: Circuit for compensation of fundamental nonactive power plus load-generated harmonic currents


10

v :SOURCE 0.250

c:SOURCE-XX0002 10

c:XX0002-COMP 10

c:XX0002-LOAD 10

9.960 9.965 9.970 9.975 9.980 9.985 9.990 9.995 10.000-10.0

-7.5

-5.0

-2.5

0.0

2.5

5.0

7.5

10.0

Figure 7.22: Voltage and current waveforms after compensation of fundamental nonactive power plus load-generated Dc and harmonic currents


Chapter 7 220

The waveform in Figure 7.22 closely matches the results by Mathcad in Figure 7.20. Next consider the content of the compensated waveforms. The Fourier components of

the voltages and currents for this application example are summarized in Table 7.2 and

graphed in Figure 7.23. The phase angle of the fundamental is listed in the footnote of

the table. Additionally, Figures 7.24 and 7.25 give the Mathcad computed and ATP

simulated waveforms of fundamental and total powers after compensation using the

proposed theory. It is apparent that good compensation has been achieved.

Table 7.2: Voltages and current magnitude frequency spectrum before and after compensation

Voltage and Load Current

After Complete Comp Mathcad

After Complete Comp ATP Harmonic

V (volt) I (amp) V (volt) I (amp) V (volt) I (amp) DC -0.53903 1.078062 0.000028 0 -0.0245 0.048959 1 25.6779 1.13083 26.30101 1.059124 26.24202 1.142069 2 1.34701 0.423301 0.001352 2.68E-09 0.177776 0.055813 3 0.311676 0.065722 0.000724 1.03E-09 0.034169 0.007189 4 0.444273 0.070393 0.001785 1.70E-09 0.068649 0.010834 5 0.3361 0.04262 0.002101 1.92E-09 0.045175 0.005693 6 0.221406 0.02339 0.00199 4.65E-09 0.043262 0.00453 7 0.261288 0.023643 0.003183 4.85E-09 0.050707 0.004533 8 0.142673 0.011285 0.00227 4.10E-09 0.032317 0.002515 9 0.141582 0.009942 0.002844 1.95E-09 0.04664 0.003209 10 0.111939 0.007064 0.002776 1.10E-09 0.03493 0.002149 11 0.066941 0.003834 0.002009 2.54E-10 0.045221 0.002511 12 0.082564 0.004327 0.002937 1.34E-09 0.06847 0.003458 13 0.041304 1.078062 0.001731 6.84E-10 0.004572 0.000211

Note: The phase angle of the fundamental after compensation is for Mathcad 0.43164 deg leading and for ATP 0.0994897 deg leading


Chapter 7 221

Voltage after Comp - MCAD

0

5

10

15

20

25

30

DC 1 2 3 4 5 6 7 8 9 10 11 12 13

Current after Comp - MCAD

0

0.2

0.4

0.6

0.8

1

1.2

DC 1 2 3 4 5 6 7 8 9 10 11 12 13

Voltage before Comp

-50

5101520

2530

DC 1 2 3 4 5 6 7 8 9 10 11 12 13

Load Current

0

0.2

0.40.6

0.8

1

1.2

DC 1 2 3 4 5 6 7 8 9 10 11 12 13

Voltage after Comp - ATP

-5

0

5

1015

20

25

30

DC 1 2 3 4 5 6 7 8 9 10 11 12 13

Current after Comp - ATP

0

0.2

0.4

0.6

0.8

1

1.2

DC 1 2 3 4 5 6 7 8 9 10 11 12 13

Figure 7.23: Voltage and current frequency magnitude spectrum before and after compensation


Chapter 7 222

0.08 0.085 0.09 0.095 0.1

40

20

20

40

60

s1k

p1k

100 q1k⋅

0.1 vk

0.5 ik⋅

k dt⋅ (a)

0.06 0.065 0.07 0.075 0.08

40

20

20

40

60

80

s1k

p1k

100 q1k

vk

5 ik⋅

k dt⋅.

(b) Figure 7.24: Fundamental powers after complete compensation. (a) Mathcad and (b) ATP. Note nonactive power is magnified 100 times.

0.08 0.085 0.09 0.095 0.1

50

50

100

sk

p k

qk

vk

3 ik⋅

k dt⋅ (a)

0.06 0.065 0.07 0.075 0.08

50

50

100

sk

p k

qk

vk

3 ik⋅

k dt⋅ (b)

Figure 7.25: Total powers after complete compensation. (a) Mathcad and (b) ATP Application example conclusion The information from the proposed definitions has been used to perform compensation

of unwanted quantities. The result shows that practically complete compensation has

been achieved. This result obtained by Mathcad computation and verification is

corroborated by simulation using ATP. This shows that the proposed definitions

provide accurate information to enable complete compensation.

7.4.3 Summary - some rules on compensation The component powers are first computed. If only the fundamental power (b) is present

the compensation will most likely be for nonactive power and this can be achieved with

passive components. The passive components are determined using equations (7.1) and

(7.2). If additionally power component powers (c) and (e) are also present then it is


Chapter 7 223

possible to fully compensate the nonactive power with passive elements using the

technique described in Subsection 7.2.1.1. On the other hand, if DC component (a) and

load generated power components (d) and (f) are present, then passive components

alone may not be sufficient to provide complete nonactive power compensation.

Components (d) and (f) can be removed using tuned filters that remove Ig while active

compensation will be required to compensate for DC component (a). However it is

possible to use active compensation to compensate for all the components. The final

choice of the method, components or devices to use for the compensation rests on

economic factors related to the various compensation options, that is, passive, active or

mixed active and passive.

7.5 Detection of source of distortion As stated before components (a1), (b), (c) and (d) are directional. Their directions can

be used to detect the source of the component. This property of the component can be

used to determine the direction from which that component originated as observed at the

metering point. Components (e2) and (f) are supporting components for (c) and (d)

respectively in decision-making. The reason for this is that components (c) and (d),

especially (d), could be quite small in magnitude to make a definitive decision. This

capability of the proposed definitions is illustrated with the following application

example.


The example system is taken from reference [24]. It is selected because, as indicated by

the author, it is a model of a practical network.

This example illustrates the application of the definitions to detect the source of

polluting harmonics. The analysis is done for the resistive load and the rectifier load

separately and the results are then compared. The analysis is performed for phase A of

the example system only. Similar analysis is applicable for the other phases.


Chapter 7 224

The system

The ATP model and data of the example system are given in Figure 7.26.

Generator U voltage: Symmetrical sinusoidal 3 phase 338.85 Volts amplitude at –30 degrees angle, frequency 50 Hz. Note that all other generators are set to zero voltage. Source impedance (RLC X005 – X0040): RA = 0.17185 Ohm, LA = 1.72441 mH, RB = 0.8577 Ohm, LB =3.43226 mH, RC = 1.46381 Ohm, LC = 1.62251 mH. Line impedance (RLC X042 – X0044): RA = 0.16434 Ohm, LA = 0.16394 mH, RB = 0.85943 Ohm, LB =0.17211 mH, RC = 1.54908 Ohm, LC = 0.17291 mH. Resistive Load: RA = 13.778 ohm, RB = 10.3335 ohm, RC = 17.2225 ohm. Rectifier DC load = L = 1 mH, C = 10 μF, R = 7 ohm. The diodes are ideal. Metering point: XX0044 Figure 7.26: System and system data The simulation provides voltage and current at a number of points as shown in Figure

7.26. The results from the simulation are then imported into MathCad for analysis. The

voltage and current waveforms of phase A are shown in Figure 7.27. The ensuing

analysis is done for phase A only. Similar results are also obtained for phases B and C.


Chapter 7 225

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

400

200

200

400

vgen

vload

4 iResistor⋅

4iRectifier

t.

0.08 0.085 0.09 0.095 0.1

400

200

200

400

vgen

vload

4 iResistor⋅

4iRectifier

t

Legend waveforms as follows: Variable Colour Remarks vgen Red dashed Voltage at generator before generator impedance – X0005 vload Red Voltage at metering point – X0044 iResistor Blue Resistive load current – X0044 ⇒ X0013 iRectifier Magenta Rectifier load current – X0044 ⇒ X0001

Figure 7.27: Metering point voltage and current waveform for phase A Calculation, results and discussion

Resistive load

The component power waveforms are determined using the proposed definitions and the

resulting waveforms are presented in Figure 7.28.


Chapter 7 226

0.08 0.085 0.09 0.095 0.1

400

200

200

400

vload

5iResistor


0.08 0.085 0.09 0.095 0.1

40

20

20

40

s0Dk

p0Dk

q0Dk

0.1 vk⋅

0.5 ik⋅


0.08 0.085 0.09 0.095 0.1

40

20

20

40

s0Xk

p0Xk

q0Xk

0.1 vk

0.5 ik

k dt⋅ DC based active and nonactive cross powers (a2)

0.08 0.085 0.09 0.095 0.1

4000

1666.67

666.67

3000

5333.33

7666.67

1 .104

s1k

p1k

q1k

10 vk⋅

50 ik⋅


0.08 0.085 0.09 0.095 0.1

50

50

100

shk

phk

qhk

0.1 vk⋅

0.5 ik⋅


0.08 0.085 0.09 0.095 0.1

0.4

0.2

0.2

0.4

sgk

pgk

qgk

0.001 vk⋅

0.005 ik⋅

k dt⋅

.

Load generated harmonic powers (d)


Chapter 7 227

0.08 0.085 0.09 0.095 0.1

1000

500

500

1000

sX1k

pX1k

qX1k

2 vk

10 ik

k dt⋅ Source generated cross-fundamental powers (e1)

0.08 0.085 0.09 0.095 0.1

1000

500

500

1000

sXhk

pXhk

qXhk

2 vk

10 ik

k dt⋅ Source generated cross-harmonic powers (e2)

0.08 0.085 0.09 0.095 0.1

0.4

0.2

0.2

0.4

sXgk

pXgk

qXgk

0.001 vk

0.005 ik⋅


0.08 0.085 0.09 0.095 0.1

2000

2000

4000

6000

8000

s k

p k

qk

5 vk

50 ik⋅


Figure 7.28: Powers based on proposed definition for resistive load

The DC and DC cross power (a1 and a2) is negligible. The fundamental active power p1 flows from the source to the load. Nonactive power q1

is negligible.

Source generated active harmonic power ph is positive. As in the case of fundamental

active power this indicates that flow from source to the load. The source generated

nonactive power qh is negligible.

Load generated active and nonactive harmonic powers pg and qg are negligible

indicating that there is no load generated harmonics.

Source based cross-harmonic powers indicate that the origin of the harmonic power is

the source.


Chapter 7 228

Load based cross harmonic powers are negligible.

Rectifier load The component power waveforms are determined using the proposed definitions and the

waveforms presented in Figure 7.29.

0.08 0.085 0.09 0.095 0.1

400

200

200

400

vload

5iRectifier


0.08 0.085 0.09 0.095 0.1

40

20

20

40

s0Dk

p0Dk

q0Dk

0.1 vk⋅

0.5 ik⋅


0.08 0.085 0.09 0.095 0.1

40

20

20

40

s0Xk

p0Xk

q0Xk

0.1 vk

0.5 ik

k dt⋅ DC based active and nonactive cross powers (a2)

0.08 0.085 0.09 0.095 0.1

4000

4000

8000

1.2 .104

1.6 .104

2 .104

s1k

p1k

q1k

10 vk⋅

50 ik⋅


0.08 0.085 0.09 0.095 0.1

50

50

shk

phk

qhk

0.1 vk⋅

0.5 ik⋅


0.08 0.085 0.09 0.095 0.1

200

100

100

200

sgk

pgk

qgk

0.001 vk⋅

0.005 ik⋅



Chapter 7 229

0.08 0.085 0.09 0.095 0.1

4000

2000

2000

4000

sX1k

pX1k

qX1k

2 vk

10 ik


0.08 0.085 0.09 0.095 0.1

40

20

20

40

sXhk

pXhk

qXhk

0.1 vk

0.5 ik


0.08 0.085 0.09 0.095 0.1

2000

500

1000

2500

4000

sXgk

pXgk

qXgk

0.5 vk

1 ik⋅


0.08 0.085 0.09 0.095 0.1

1 .104

1 .104

2 .104

sk

p k

qk

5 vk

50 ik⋅


Figure 7.29: Powers based on proposed definitions for rectifier load

The DC and DC cross harmonic power is negligible.

The fundamental active power p1 flows from the source to the load. There is a small

magnitude of fundamental nonactive power q1.

Source generated active and nonactive harmonic power ph and qh is negligible,

indicating that there is no source-generated harmonics.

Load generated active harmonic power pg is negative. This indicates that it is opposite

to the direction of flow of fundamental active power. Thus, it is flowing from load to

source. The nonactive power is due to the filter inductor in the DC part of the rectifier.

The presence of active load generated cross harmonic power is also an indicator of load-

generated harmonics. It is especially useful when the source impedance is negligible

and load generated harmonic power is negligible. The absence of this power can thus


Chapter 7 230

be taken as an indicator of non-existence of load generated harmonic power as in the

case above for resistive load.

Average powers resistor and rectifier load

The average values of the components are presented in Table 7.3.

Table 7.3: Average values of the power components of waveforms in Figures 7.28 and 7.29 Power component Resistive load Rectifier load Units DC P0DAV a1 0 0 Watt DC cross P0XAVPOS a2 0.00000002 0.00000004 Watt DC cross N0XAVPOS a2 0 0.00000001 Var Fundamental active P1AV b 3359.2092629 9852.9112286 Watt Fundamental nonactive N1AV b 0.0000031 410.2083456 Var Source harmonic active PhAV c 45.4863356 0 Watt Source harmonic nonactive NhAV c 0.0000016 0 Var Load harmonic active PgAV d 0 -59.2859516 Watt Load harmonic nonactive NgAV d 0 65.8211836 Var Source cross fund. active PX1AVPOS e1 103.5546722 303.7366572 Watt Source cross fund nonactive NX1AV e1 0.0000005 64.9157508 Var Source cross harm. active PXhAVPOS e2 98.9250388 0 Watt Source cross harm nonactive NXhAV e2 0.0000820 0 Var Load cross harm. active PXgAVPOS f 0 163.3946762 Watt Load cross harm nonactive NXgAV f 0 1599.3733286 Var Note: For cross-based active powers the average of positive going part is used as a measure of the existence of that power. This is because these have a zero average value over one period. Discussion of results

From the waveforms and/or the average values the following can be concluded.

1. Traces of DC (a1) and DC based cross (a2) powers mean that the voltage and

current are symmetric about the x-axis and that there is negligible DC component.

So for this example the DC does not assist in identifying the source of pollution.

2. The fundamental active power (b) flows from the source to the load for both

resistive and rectifier load. This gives the direction of desired power flow.

3. For the resistive load there exists source generated harmonic active power (c) in the

same direction as the fundamental power while for the rectifier load it is zero. This

means that the source side is the cause of the harmonics flowing into the resistive

load. This is supported by source generated cross-harmonic power (e2) see point 5

below.


Chapter 7 231

4. For the rectifier there exists load generated harmonic active power (d), while for the

resistive load this is zero. The load generated harmonic active power flows in the

direction opposite to that of the fundamental. This means that the load is generating

the harmonics flowing into the system. This is corroborated by load generated

cross-harmonic power (f) see point 6 below.

5. Source generated cross-harmonic active power (e2) exits at the metering point for

the resistive load. This supports the source generated harmonic active power (c),

see point 3, indicating that the cause of harmonics flowing in the resistive load is the

source side.

6. Load generated cross-harmonic power (f) exists in the rectifier load but is negligible

in the resistive load. This supports the load generated harmonic active power (d),

see point 4, indicating the rectifier load is producing the harmonics. The rectifier

load causes metering point voltage distortion that results in the flow of source

generated harmonic power in the resistive load.


It has been illustrated by the example that the defined components can be used to

identify the source of harmonics at the metering point. In the simulated case study the

source of pollution is determined to be the rectifier load. The key in this determination

is the harmonic active power supported by the corresponding cross power. This, that is

source of pollution, together with power quality measure can be incorporated into future

billing practice.


This is a real life example provided by an organisation referred to as CO in the sequel

for confidentiality. CO operates harmonic filters installed in 1994 and 1999 throughout

their 11kV network. For the early part of their lives these filters performed well,

absorbing harmonic currents and maintaining the site’s VAR demand within the limits

prescribed in the power supply agreement. However, subsequent to 2000 several

problems occurred. Firstly the number of filter protection trips steadily increased. Such

trips are always associated with the 5th harmonic leg(s). Secondly, a number of reactors

and capacitors have failed; once again all associated with the 5th harmonic elements. In


Chapter 7 232

2004 the organization commissioned an investigation into this problem. This

investigation revealed that the problem was due to the flow of the 5th harmonic currents

from the supply authority into the filter that caused the trips and also failure of the

reactors and capacitors. The portion of the related system and report of the

invesstigation is included in Appendix D. The author of this thesis was able to obtain

one voltage and current recording from the organization in relation to the problem for

feeder FZ9. This recording was analysed to detect the direction of the source of

distortion using the proposed definitions. Presently the author of this thesis is involved

in further investigation on a similar problem on another feeder FZ6. This investigation

is ongoing and some preliminary results are presented here. As a first step, three phase

voltage and current measurements were made every minute for 15 hours. This data is

being used for the analysis.

The system

The related part of the system is given in Appendix E. The voltages and currents used

in this analysis were obtained from feeder FZ9 point PCC shown in the system diagram

in Appendix D. The system for Feeder FZ6 is similar to FZ9

Calculation, results and discussion

Feeder FZ9

The component power waveforms are determined using the proposed definitions and the

waveforms presented in Figures 7.30.

0.02 0.025 0.03 0.035 0.04

1 .104

5000

5000

1 .104

vc

vs

10 is⋅


0.08 0.085 0.09 0.095 0.1

10

5

5

10

s0D k

p0D k

q0Dk

0.001 vk⋅

0.005 ik⋅



Chapter 7 233

0.08 0.085 0.09 0.095 0.1

1 .104

5000

5000

1 .104

s0Xk

p0Xk

q0Xk

0.5 vk

5 ik

k dt⋅

.

DC based active and nonactive cross powers (a2)

0.08 0.085 0.09 0.095 0.1

2 .106

3.33 .105

1.33 .106

3 .106

4.67 .106

6.33 .106

8 .106

s1k

p1k

50 q1k

10 vk⋅

50 ik⋅

k dt⋅

Fundamental powers (b)

0.08 0.085 0.09 0.095 0.1

1000

500

500

1000

1500

2000

2500

shk

phk

qhk

0.1 vk⋅

0.5 ik⋅


0.08 0.085 0.09 0.095 0.1

500

500sgk

pgk

qgk

0.001 vk⋅

0.005 ik⋅


0.08 0.085 0.09 0.095 0.1

2 .105

1 .105

1 .105

2 .105

sX1k

pX1k

qX1k

2 vk

10 ik


0.08 0.085 0.09 0.095 0.1

5 .105

5 .105

sXhk

pXhk

qXhk

2 vk

10 ik



Chapter 7 234

0.08 0.085 0.09 0.095 0.1

1 .105

2.5 .104

5 .104

1.25 .105

2 .105

sXgk

pXgk

qXgk

0.5 vk

1 ik⋅

k dt⋅

Load generated cross-harmonic powers (f)

0.08 0.085 0.09 0.095 0.1

2 .106

2 .106

4 .106

6 .106

8 .106

s k

p k

qk

5 vk

50 ik⋅


Figure 7.30: Powers based on proposed definition for Feeder FZ9

The DC and DC cross harmonic power is very small but indicates flow of a small DC

current towards the source.

The fundamental active power p1 flows from the source to the load. There is a small

magnitude of fundamental nonactive power q1.

Source generated active and nonactive harmonic power ph and qh is not negligible,

indicating that there are source-generated harmonics.

Load generated active harmonic power pg is not negligible but smaller than the source

generated currents. This indicates that it is opposite to the direction of flow of

fundamental active power. Thus it is flowing from load to source.

The presence of active source and load generated cross harmonic power is also an

indicator of source and load-generated harmonics. It is especially useful when the

source impedance is negligible and load generated harmonic power is negligible. The

absence of this power can thus be taken as an indicator of non-existence of the

corresponding source or load generated harmonic power. This is also reflected in the

average powers in Table 7.4. Further analysis of the harmonic current in the source

generated power up to the 13th harmonic are listed in Table 7.5.

Average powers

The average values of the components are presented in Table 7.4


Chapter 7 235

Table 7.4: Average values of the power components of waveforms in Figures 7.30 Power component Feeder FZ9 Units DC P0DAV a1 -1.9287109 Watt DC cross P0XAVPOS a2 3112.4830698 Watt DC cross N0XAVPOS a2 135.0133419 Var Fundamental active P1AV b 3727699.1615237 Watt Fundamental nonactive N1AV b 23561.4067414 Var Source harmonic active PhAV c 961.4506207 Watt Source harmonic nonactive NhAV c 785.5747614 Var Load harmonic active PgAV d -81.1516683 Watt Load harmonic nonactive NgAV d 452.0005581 Var Source cross fund. active PX1AVPOS e1 22185.4074803 Watt Source cross fund nonactive NX1AV e1 375.4897326 Var Source cross harm. active PXhAVPOS e2 22185.4074803 Watt Source cross harm nonactive NXhAV e2 184111.7036119 Var Load cross harm. active PXgAVPOS f 3306.0537271 Watt Load cross harm nonactive NXgAV f 68097.0659825 Var Note: For cross-based active powers the average of positive going part is used as a measure of the existence of that power. This is because these have a zero average value over one period. Table 7.5: Harmonic currents Feeder FZ9 Harmonic Voltage Current Phase angle Active Power DC Fund 6344.226009 587.5852282 -0.3618111 3727699.162 2nd 11.9045933 1.0018953 278.5185854 1.7667721 3rd 4th 3.609462 1.2766011 -0.0904539 4.6078374 5th 34.7863936 32.0407649 44.8088186 790.75435 6th 4.4191135 1.1838319 -0.0603026 5.2314846 7th 4.7001661 4.3750358 -0.051688 20.5633868 8th 11.3351182 0.9874021 11.1468918 10.981174 9th 6.7547267 1.7434636 274.093109 0.8405855 10th 7.1593478 0.7568903 -46.3660589 3.7392625 11th 6.6114244 15.0546843 3.2433132 99.3734834 12th 5.8577454 0.3331151 -0.0301513 1.951303 13th

Feeder FZ6

The long term measurements on feeder FZ6 have been performed. The measurements

were made every minute for 15 hours. The three phase power for each harmonic was

computed for each minute and the hourly average was calculated which are plot in

Figure 7.31.


Chapter 7 236

3 PhaseOne Hour AveragePower

-600

-400

-200

0

200

400

600

800

1000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-7200

-5200

-3200

-1200

800

2800

4800

6800

8800

10800

Thou

sand

DC AV2nd AV3rd AV4th AV5th AV6th AV7th AV8th AV9th AV10th AVFund AV

Figure 7.31: Three Phase harmonic powers Feeder FZ6 Note: In the figure the fundamental value is on the right axis

Discussion of results

Feeder FZ9

Analysis of the waveform for feeder FZ9 shows that the 5th harmonic is flowing from

the source towards the load at the metering point (PCC) as shown in Table 7.5. This is

in agreement with the conclusion of the report prepared by the organisation.

Feeder FZ6

The results in Figure 7.31 also show that the 5th harmonic power (brown triangle ),

which has the same direction as fundamental power (blue diamond ♦), is flowing from

the source towards the load up to the hour number 14 at the metering point.


Preliminary practical study shows that the proposed definition can be used as an aid to

detect source of distortion. Further study into this possibility of the definitions is

currently being carried out.


Chapter 7 237

7.5.3 Summary - some rules on detection on source of pollution The component powers at the point of interest, where it is necessary to determine the

direction of distortion, are computed. The point of interest is the metering point and the

supply side and load side are identified at this point using the fact that fundamental

flows from the source side to the supply side. The direction of fundamental is taken to

be the positive direction. The direction of DC power component (a1) if positive

indicates that it is originating in the source. If it is negative, then it is generated in the

load side and it means, most likely, that the load is nonlinear. Presence of source

generated power (c) and load generated power (d), as the name implies, means

respectively that the source or the load is producing the harmonics. If these values are

very small, then supporting components source generated cross-harmonic (e2) and load

generated cross-harmonic (f) can be used to assist the detection of the origin. It is

important to take note, when determining the source of harmonics, that care must be

exercised when there are multiple active nonlinear loads, for example thyristor

controlled rectifiers, on one bus. The direction of flow, when measured for one of the

active nonlinear loads connected to the bus, can change very fast in the sense that at one

instant the flow could be in one direction and at the next instant the flow could be in the

other direction. This is the case even when the active nonlinear loads are known to be a

source of harmonics. When active nonlinear loads are connected together, one could

manifest as a generator (as a supply side) and the other as a sink (as a load side) with the

status quo changing very fast, depending on the firing angle. Care must be exercised

when detecting the direction of distortion.

7.6 Power quality Introduction

The ideal condition of an electrical system is that the voltages and currents are

sinusoidal and the current is balanced in a three-phase system. Deviation from this ideal

condition is called distortion or (harmonic) pollution and the quantities causing this

distortion are generally called “useless” or “unwanted” quantities.

In the proposed definitions the following have been defined.


Chapter 7 238

(a) DC based power comprising

(a1) DC power s0D(t), p0D(t), q0D(t) and

(a2) DC based s0X(t), p0X(t), q0X(t).

(b) Fundamental power s1(t), p1(t), q1(t).

(c) Source generated harmonic powers sh(t), ph(t), qh(t).

(d) Load generated harmonic powers sg(t), p g(t), q g(t).

(e) Source generated cross powers sXH(t), pXH(t), qXH(t).

(e1) Cross-fundamental powers sX1(t), pX1(t), qX1(t) and

(e2) Cross-harmonic powers sXh(t), pXh(t), qXh(t).

(f) Load generated cross-harmonic powers sXg(t), pXg(t), qXg(t).

Ideally, component (b) only should exist at the metering point. The presence of the

other components is an indication of degradation of the power quality and these

components can be used to gauge power quality.

Additionally, definitions have been made to detect unbalance in the system in Chapter

6. These are definitions of unbalance (in per unit values) for total power unbalance

S3PhUNBALpu, active power unbalance P3PhUNBALpu and nonactive power unbalance

S3PhUNBALpu. equations (6.18) to (6.20) in Chapter 6 refer.

7.6.1 Application example 7.6.1 The rectifier load of application Example 7.5.1 above is used to illustrate power quality.

The components that cause distortion are presented as a percentage of the fundamental

for the active average power in Table 7.6, nonactive average power in Table 7.7 and

total average power in Table 7.8.


Chapter 7 239

Results Table 7.6: Measure of distortion active power Power component Units Rectifier load % fund DC P0DAV a1 Watt 0 0.00%DC cross P0XAVPOS a2 Watt 0.00000004 0.00%Fundamental active P1AV b Watt 9852.911229 100.00%Source harmonic active PhAV c Watt 0 0.00%Load harmonic active PgAV d Watt -59.2859516 -0.60%Source cross fund. active PX1AVPOS e1 Watt 303.7366572 3.08%Source cross harm. active PXhAVPOS e2 Watt 0 0.00%Load cross harm. active PXgAVPOS f Watt 163.3946762 1.66% Table 7.7: Measure of distortion nonactive power Power component Units Rectifier load % fund DC N0DAV a1 Watt 0 0.00%DC cross N0XAV a2 Var 0.00000001 0.00%Fundamental nonactive N1AV b Var 410.2083456 100.00%Source harmonic nonactive NhAV c Var 0 0.00%Load harmonic nonactive NgAV d Var 65.8211836 16.05%Source cross fund nonactive NX1AV e1 Var 64.9157508 15.83%Source cross harm nonactive NXhAV e2 Var 0 0.00%Load cross harm nonactive NXgAV f Var 1599.373329 389.89% Table 7.8: Measure of distortion total power Power component Units Rectifier load % fund DC S0DAV a1 VA 0 0.00%DC cross S0XAV a2 VA 4.12311E-08 0.00%Fundamental total S1AV b VA 9861.446677 100.00%Source harmonic total ShAV c VA 0 0.00%Load harmonic total SgAV d VA 88.58471803 0.90%Source cross fund total SX1AV e1 VA 310.5962196 3.15%Source cross harm total SXhAV e2 VA 0 0.00%Load cross harm total SXgAV f VA 1607.698002 16.30% Discussion of results

From Table 7.6 it is apparent that largest contribution to the loss of quality is nonactive

power. The total power would be preferable for purposes related to a measure of power

quality. This is because the total power encompasses power quality of both the active


Chapter 7 240

and nonactive powers. The active and nonactive powers could then be considered to

find out which is the worst contributor.


A power quality measure of the distortion power has been illustrated. The practical use

of this is in quantifying distortion level, which is useful for future billing practice that

could incorporate power quality together with source of distortion in billing. This is a

future research area.


Cases T1 to T5 as well as the application example 7.5.1 are used to illustrate the

unbalance. Also included is variation of Case T4 using sinusoidal balanced voltage and

balanced load (R= 10.6 ohm and L = 0.036 H).

Results The results of the computation using equations (6.18) to (6.20) in Chapter 6 are

presented in Table 7.9.

Table 7.9: Measure of unbalance Unbalance per unit S3PhUNBALpu P3PhUNBALpu N3PhUNBALpu Case T1 (Ref - B) 1.087 1.087 0 Case T1 (Ref-Virtual Neutral) 0.941 0.776 0.869 Case T2 0.515 0.642 0.249 Case T3 0.735 0.559 0.841 Case T4 0.546 0.691 0.263 Case T4 (sinusoidal balanced) 0.000 0.000 0.000 Case T5 0.398 0.398 0.000 Application example 7.5.1 Resistive load 0.363 0.363

0.000

Rectifier load 0.388 0.313 0.363 Discussion of results

The unbalance is quantified on power type, that is, total, active and nonactive, basis. In

Case T1, there is zero unbalance in the nonactive power but unbalance exists in active

and total powers. The same can be said for application Example 7.5.1. With both

balanced sinusoidal source voltage and balanced load Case T4 (sinusoidal balanced)

shows zero unbalance. All other cases show the degree of unbalance with the worse

being Case T1.


Chapter 7 241


A measure of unbalance has been demonstrated. This information can be used for

billing purposes or possible impetus to improvement of balance in the system. This is

also a future research area.

7.6.3 Summary - some comments on power quality The distortion level and unbalance are useful in quantifying power quality. These measures can be usefully built into future billing systems. 7.7 General 7.7.1 Application example 7.7.1 Introduction

This example illustrates the capability of the proposed definitions, in providing

substantial insight into the nature of the system, from the voltages and currents at the

measuring point. A waveform similar to the waveform in Figure 7.32 was put forward

during the discussion between Prof P Tenti and Prof M Slonim during the 7th

International Workshop "Angelo Barbagelata" on Power Definitions and Measurements

under Nonsinusoidal Conditions in July 2006. The question was “how come apparent

power (S = VRMS IRMS) is not zero while instantaneous power s(t) = v(t)i(t) is zero”?

This is a very simple and rather academic example, but it shows how the proposed

definitions can be utilised to gain insight into the system using the data available at the

measuring point.

0 0.01 0.02 0.03 0.04 0.05 0.06

2

4

6

8

10

vs t( )

is t( )

t Figure 7.32: Voltage and current waveforms at the measuring point The waveform of amplitude of 10 units both for the voltage (magenta) and current (black), frequency 50 Hz, is used in the following analysis. The wavefrom is given by the following


Chapter 7 242

s

10sin(100 t) for sin(100 t) 0v (t)

0 otherwiseπ π >⎧

= ⎨⎩

(7.23)

s

10sin(100 t ) for sin(100 t ) 0i (t)

0 otherwiseπ + π π + π >⎧

= ⎨⎩

(7.24)

Analysis of the question

Conventional definition of apparent Power

The voltage andd current waveforms give RMS values as follows:

RMS voltage ( )T

2RMS s

0

1V v (t) dtT

= ∫ = 5 volts , (7.25)

RMS current ( )T

2RMS s

0

1V i (t) dtT

= ∫ = 5 amps, (7.26)

Where T is time for one period.

Apparent power = VRMSIRMS = 25 VA.

Analysis using the proposed definitions

As a starting point, the source-load model, with the voltage current waveforms as shown

in Figure 7.32, is assumed as shown in Figure 7.33.

source load

is(t)

vs(t)meteringpoint

Figure 7.33: Assumed system Component power waveforms

The component power waveforms and their average values are shown in Figure 7.34.

The component instantaneous powers are determined using equations (4.11) to (4.38).

The average powers for the components are computed using equations (4.45) and

(4.47).


Chapter 7 243

0 0.01 0.02 0.03 0.04

2

4

6

8

10

vs t( )

is t( )


0 0.005 0.01 0.015 0.02

5

5

10

15

p0D t( )

q0D. t( )

vs t( )

is t( )

t P0DAV = 10.1304517 (a1) DC powers

0 0.005 0.01 0.015 0.02

20

10

10

20

p0X t( )

q0X. t( )

vs t( )

is t( )

t P0XAVpos = 4.27255, P0XAVneg = - 4.27255, P0XAV = P0XAVpos + P0XAVneg = 0, Q0XAV = 0 (a2) DC based cross-harmonic powers

0.02 0.025 0.03 0.035

30

20

10

10

p1 t( )

q1. t( )

vs t( )

is t( )

t P1AV = -12.5, Q1AV = 0 (b) Fundamental powers

0 0.005 0.01 0.015 0.02

5

5

10

ph t( )

qh. t( )

vs t( )

is t( )

t PhAV = 0, Q0XAV = 0 (c) Source generated harmonic powers

0 0.005 0.01 0.015 0.02

5

5

10

pg t( )

qg. t( )

vs t( )

is t( )

t PgAV = 2.3678919, QgAV = 0 (d) Load generated harmonic power


Chapter 7 244

0 0.005 0.01 0.015 0.02

10

5

5

10

pX1 t( )

qX1. t( )

vs t( )

is t( )

t PX1AVpos = 1.92149, PX1AVneg = - 1.92149, P0XAV = PX1AVpos + PX1AVneg = 0, QX1AV = 0 (e1) Source generated cross-fundamental powers

0 0.005 0.01 0.015 0.02

5

5

10

pXh t( )

qXh. t( )

vs t( )

is t( )

t PXhAV = 0, QXhAV = 0 (e2) Source generated cross-harmonic powers

0 0.005 0.01 0.015 0.02

15

10

5

5

10

pXg t( )

qXg. t( )

vs t( )

is t( )

t PXgAVpos = 2.03175, PXgAVneg = - 2.03175, PXgAV = PXgAVpos + PXgAVneg = 0, QXgAV = 0 (f) Load generated cross-harmonic powers

0 0.005 0.01 0.015 0.02

5

5

10

s t( )

p t( )

q t( )

vs t( )

is t( )

t P0XAV = -0.0016564, Q0XAV = 0 (g) Total powers

Figure 7.34: Powers based on proposed definition Discussion of the waveforms and average powers

The unidirectional energy transfer is contributed by the active power components where

both the voltage and current are of the same “harmonic” (DC, fundamental and

harmonic) i.e. active power components (a1), (b), (c) and (d); cross-active components

(a2), (f2) and (g) being zero average (there is bidirectional flow of these within a

period). The nonactive components are all zero for this example. The nature of the load

(resistive only or has reactance) can be gauged mainly from components (b) and (c).

Presence of nonactive power in (b) and (c) indicates the presence of susceptance in the

load. Since the nonactive power is zero, the load is resistive in nature.

In this example, DC power (a1) and harmonic power (c) is positive and the fundamental

(b) is negative. The average power after summing these components (= -0.00166) is


Chapter 7 245

essentially zero. However the sum of the absolute values of the unidirectional

components POD, P1, Ph and Pg

P0DAV P1AV+ PhAV+ PgAV+ 24.9983436=

is practically 25 watts and this is equal to the apparent power.

This means that, in essence, the

“load” assumed in Figure 7.32

does not absorb any energy since

the sum of the components

powers across the load is zero.

However since the absolute value

is non-zero, it appears that the

energy is passing through this

“load”, which indicates that the

load of Figure 32 is possibly a

“switch” that is supplying a load,

see sketch Figure 7.35. The

source switch

is(t)

vs(t)

load

vsource(t)

Figure 7.35: Predicted system

switch turns on every second half of the cycle to feed the load. Also based on the

knowledge of the voltage at and current through the switch it is predicted that the source

voltage is a rectified voltage as per graph in Figure 7.36. The current is also shown in

the waveform in Figure 7.36. The load is resistive.

0 0.01 0.02 0.03 0.04

2.5

5

7.5

10Source Voltage

vsource t( )

t

0 0.01 0.02 0.03 0.04

2.5

5

7.5

10Load Current

is t( )

t Source voltage and load current Figure 7.36: Predicted system Application example conclusion

Though instantaneous total power s(t) is zero, energy is passing through the switch.

The apparent power “rates” the switch i.e. gives the capacity of the switch to handle this


Chapter 7 246

particular situation. This means that switch must have insulation level to prevent

breakdown for the peak voltage of 10 volts when it is open and be able to withstand the

current passing through it when it is closed. Thus, the apparent power is important in

that it quantifies the rating of a device or capacity required for a particular load.

However the apparent power does not give more information about the device or load.

For this purpose, the proposed definitions used to make the above analysis are very

useful in that they are able to provide substantial information about the system using

only the voltage and current information at the metering point, as illustrated in the

above example.

7.8 Conclusions The use of the proposed definition in the areas of measurement, compensation, detection

of source of distortion, power quality and general application in system analysis has

been illustrated with examples. Good and useful results have been obtained for all the

examples. Analysis with Mathcad is corroborated with ATP simulation, showing

practical applicability of the definitions. The information provided by the proposed

definitions can be used for static or dynamic compensation of nonactive power and

removal of the distorting components. The knowledge of direction of source of

distortion together with power quality measure can be incorporated into future possible

real time billing.

Relationship of the proposed definitions with existing some definitions

Chapter 8 247

8. RELATIONSHIP OF THE PROPOSED DEFINITIONS WITH SOME EXISTING DEFINITIONS

In this chapter the relationship between the proposed definitions and some commonly

used definitions is revealed. These commonly definitions are the DC system

definitions, sinusoidal system definitions, RMS based definition, Budeanu’s definition,

Fryze’s definition and the definitions in IEEE Standard 1459. These definitions have

been presented and critically discussed in Chapter 2.

8.1 DC system In a DC system V1, Vh, Vg, I1, Ih and Ig are equal to zero. Hence equations (4.30) and

(4.38) in Chapter 4 reduce to

P = V0I0 and Q = 0 (8.1)

which is the power equation in DC systems. The proposed definitions are consistent

with the equation for DC systems.

8.2 Sinusoidal systems For sinusoidal circuits, since only component (b) is present, equations (4.30) and (4.38)

in Chapter 4 reduce to

p1(t) = 1P(1 cos 2( t ))+ ω −α (8.2)

and

q1(t) = 1Q sin 2( t )ω −α (8.3) where 1 1 1P V I cos= θ and 1 1 1Q V I sin= θ . The proposed definition is consistent with the traditional definition for p(t) and q(t) as

well as the definitions in IEEE Standard [37] for sinusoidal systems.

8.3 RMS based powers Substantial discussion on this has been given in Subsection 2.3.1 in Chapter 2. The

average value of instantaneous active power equation (4.30), of the proposed definition

is equal to the RMS based power defined in equation (2.40). Using the “equivalent


Chapter 8 248

sinusoidal waveform” plus the “shifted power waveform” method (see Subsection 2.3.1

in Chapter 2) for equation (4.38) to calculate average nonactive power will give the

same value as the RMS based nonactive average power.

8.4 Budeanu’s definitions For a detailed discussion on Budeanu’s definition refer Subsection 2.3.2 in Chapter 3.

The key formulae, for which the relationship is shown, of Budeanu’s definitions of

active and reactive average powers are reproduced below.

Average active power m m mm 1,h

P V I cos( )=

= θ∑ (8.4)

and nonactive B m m mm 1,h

Q V I sin( )=

= θ∑ (8.5)

The sum of the proposed instantaneous fundamental active power (b) and source

generated harmonic active power (c) is

p1(t)+ph(t) = 21 1 1 12V I cos ( t ) cosω −α θ + 2

h h h hh

2 V I cos (h t ) cosθω −α∑ . (8.6)

The average value of this active power is

P1hAV = 1 1 1V I cosθ + h h hh

V I cosθ∑ . (8.7)

Next, similarly, consider the nonactive fundamental and harmonic powers. The sum is

given by

q1(t)+qh(t) = 1 1 1 1 12 V I cos( t )sin( t ) sinω −α ω −α θ + h h h h h

h2 V I cos(h t )sin(h t )sinω −α ω −α θ∑ . (8.8)

Using the “shifted power waveform” method (see equation (3.52) in Subsection 3.3.1 in

Chapter 3) equation (8.8) can be written as

q1SPM(t)+qhSPM(t) = 2

1 1 1 12V I sin ( t )sinω −α θ + 2h h h h

h2 V I sin (h t )sinω −α θ∑ , (8.9)

where subscript “SPM” represents shifted power method. The average value of

equation (8.9) is

Q1hAV = 1 1 1V I sin θ + h h hh

V I sin θ∑ , (8.10)

which is the same as Budeanu’s reactive power in equation (8.5).


Chapter 8 249

This shows that the proposed definitions are related to Budeanu’s active and reactive

powers using the shifted power method.

8.5 Relationship of the proposed definitions with that of Fryze’s For a detailed discussion on Fryze’s definitions refer to Subsection 2.3.3 in Chapter 2.

Fryze’s definitions are given below.

Active current

aF 2rms

Pi (t) v(t)V

= , (8.11)

where P = T

1 v(t) i(t)dtT ∫ and T is equal to one period

Nonactive current

bF aFi (t) i(t) i (t)= − . (8.12)

The equations for voltage v(t), active current ip(t) and nonactive current iq(t) are given

by

v(t) = V0 + v1(t) + vh(t),

ip(t) = I0p + i1p(t) + ihp(t) + igp(t) and

iq(t) = I0q + i1q(t) + ihq(t) + igq(t). (8.13)

With the assumption of constant equivalent parallel resistance R of the load the

following can be obtained

I0p = 0VR

, 11p = 1vR

, I0p = hvR

and igp = 0. (8.14)

Hence ia(t) is given by

ia(t) = 0 1 hV v v+R R R

+ . (8.15)

The active p(t) and nonactive powers q(t) are given by

p(t) = v(t) ip(t) and q(t) = v(t) iq(t). (8.16)

Thus p(t) and q(t) are given by


Chapter 8 250

p(t) = ( )0 1 hV + v (t) + v (t) 0 1 hV v (t) v (t)+R R R

⎛ ⎞+⎜ ⎟⎝ ⎠

and

q(t) = ( )0 1 hV + v (t) + v (t) ( )0q 1q hq gqI + i (t) + i (t) + i (t) . (8.17)

Expanding p(t) gives, 2 2 2

0 0 1 0 h1 1 h hV 2V v (t) 2V v (t)v (t) 2v (t)v (t) v (t)+ + + +R R R R R R

⎛ ⎞+⎜ ⎟

⎝ ⎠. (8.18)

Simplifying the above expanded p(t) gives

( )20 1 hV v (t) v (t)

R+ +

. (8.19)

Thus equation (4.30) in Chapter 4 can be written as

p(t) = 21 v(t)R

. (8.20)

Therefore ip(t) is given by

ip(t) = v(t)R

(8.21)

This is equivalent to Fryze definition of active current iaF(t) in equation (8.11) where

iaF(t) = 2rms

P v(t)V

= v(t)R

(8.22)

with R = 2

rmsVP

.

Using this information in the equation of q(t) we proceed as follows.

Since igp(t) = 0, igq(t) = ig(t), q(t) is given by

q(t) = ( )0 1 hV + v (t) + v (t) ( )0q 1q hq gqI + i (t) + i (t) + i (t)

= ( )0 1 hV + v (t) + v (t) ( ) ( ) ( ) ( )( )0 0p 1 1p h hp gI I + i (t) i (t) + i (t) i (t) + i (t)− − −

= ( )0 1 hV + v (t) + v (t) ( ) 0 1 h0 1 h g

V v (t) v (t)I + i (t) + i (t) + i (t) +R R R

⎛ ⎞⎛ ⎞− +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(8.23)


Chapter 8 251

Since 0 1 h gI + i (t) + i (t) + i (t) = i(t) , 0 1 hV v (t) v (t)+R R R

⎛ ⎞+⎜ ⎟⎝ ⎠

= ip(t) and

( )0 1 hV + v (t) + v (t) = v(t) , then

q(t) = v(t) ( )pi(t) i (t)−

iq(t) = q(t)v(t)

= ( )pi(t) i (t)− (8.24)

Hence equation (4.38) in Chapter 4 is simplified to

q pi (t) i(t) i (t)= − . (8.25)

This result is the same as Fryze’s definition of nonactive current bF aFi (t) i(t) i (t)= −

given in equation (8.12).

Thus the definitions of proposed theory result in Fryze definition if the load is

considered to have a constant parallel resistance.

This analysis shows clearly, that the proposed definitions coincide with the definitions

by Fryze (see Subsection 2.3.3 Chapter 3) as well as with the definitions in Reference

[57] with integrating time of one period. It is noted that Reference [57] states that “most

of the existing nonactive power theories and definitions based on time-domain can be

extended and deduced from the definition by Fryze”. This means that to some extent,

the proposed definitions encompass these nonactive power theories.

8.6 IEEE standard 1459-2000 [37] Instantaneous powers

The cross component active powers (a2), (e) and (f) as defined in the proposed

definitions are included as zero average powers (nonactive) in the IEEE standard. The

IEEE standard defines the fundamental and harmonic power components while the

proposed definitions define additional components besides these two.

8.7 Conclusion The relationship of the proposed definitions with some commonly used definitions has

been revealed.


Chapter 8 252


Conclusions and future research

Chapter 9 253

9. CONCLUSIONS AND FUTURE RESEARCH

New single-phase definitions, defining instantaneous active and non-active powers and

corresponding components, based on the properties of the power system have been

introduced. A meaning, by virtue of the definitions’ relationship to power system

properties, has been attributed to each of the components defined. New average powers

and energy transfer definitions, linked to the running cost of electricity, have also been

introduced. These average power and energy transfer definitions are based on the

energy content of the waveform and therefore satisfy the principle of energy

conservation which many present definitions do not comply with. The definitions,

being generalised, form a (much quested for) common base for the measurement of

powers, compensation and mitigation of unwanted quantities in the power system,

detection of source of distortion as well determination of power quality. The new

definitions also encompass many of existing commonly used definitions.

The new definitions are applicable in the presence of nonlinear load and harmonics

which current power definitions have problems with.

A new method to identify the optimal reference conductor for a three-phase system has

been presented. This approach corroborates the present practice of utilising the neutral

for four-wire systems and virtual neutral for three-wire system while at the same time

provides a new method to identify the optimal reference conductor under abnormal

conditions when the present practice may not be applicable.

New three-phase definitions for instantaneous powers have been proposed. These can

be considered on a single-phase basis and are defined using the definitions proposed for

single-phase or on a collective three phase basis. The collective three phase

instantaneous power definitions have been presented. These are novel in the sense that

they use the collective energy content and are defined as being made up of positive and

negative going parts. Average three-phase powers, complying with the energy

conservation principle, based on energy transfer have also been defined from the

collective instantaneous three-phase powers. A measure of unbalance in three-phase

Conclusions and future research

Chapter 9 254

systems, which can be used as an indicator of power quality, has been introduced.

These three-phase definitions, under sinusoidal linear load irrespective of balance or

unbalance conditions, corroborate the RMS based arithmetic powers, indicating that the

RMS based arithmetic powers meet energy conservation under certain conditions.

Numerous examples simulating real cases using ATP have shown both the viability as

well as the practical applicability of the new definitions. Experimental work has shown

that the definition algorithms can be realised in a digital signal processor. This enables

the use of the definition algorithms in power meters for billing purposes, power

analysers or mitigation and compensation equipment.

Further research into the definition of DC factor k0 and generated harmonic phase angle

γg is indicated. Additionally, there is a need to define indices that can be used as power

quality measures and aid detection of source of pollution. The use of the newly defined

quantities in billing practice, especially real time billing, is another area of investigation.

Research into the use of unbalance measure defined to correct for unbalance in three-

phase systems would also be useful to improve utilisation of the supply.

References

References 255

10. REFERENCES

[1] R. W Clark, Edison The Man Who Made The Future, G P Putnam’s Sons, New

York, 1977.

[2] G. S. Bryan, Edison The Man and His Work, Alfred A Knopf, London, year Not

Known.

[3] D.P. Hartman, An Electrical Utility’s Experience with Nonsinusoidal Waveforms

and Electrical Metering, IEEE Tutorial Course, pp 23 – 24, 1990.

[4] R. H. Stevens, Harmonics and Related Factors Affecting Metering Accuracy,

IEEE Tutorial Course, pp 61 – 66, 1990.

[5] D Czarkowski & A Domijan, Harmonic Content of PWM Adjustable Speed

Drive Waveform - Analysis and Metering Implications, Proceeding 7th

International Conference on Harmonics and Quality of Power, Las Vegas, NV,

pp 48 – 53, October 1996.

[6] L.S. Czarnecki, Considerations on the Reactive Power in Nonsinusoidal

Situations, IEEE Transactions on Instrumentation and Measurement, Vol. IM-34,

pp 399-404, September 1985.

[7] L.S. Czarnecki, What is Wrong with Budeanu Concept of Reactive and Distortion

Power and Why It Should be Abandoned, IEEE Transactions on Instrumentation

and Measurement, Vol. IM-36, pp 834-837, September 1987.

[8] L.S. Czarnecki, On Some Misinterpretations of the Instantaneous Reactive Power

p-q Theory, IEEE Transactions on Power Electronics, pp 828-836, Vol 19, No 3,

May 2004.

[9] M D Slonim, J D Van Wyk, Power components in a system with sinusoidal and

nonsinusoidal voltages and/or currents, IEE Proceedings – Electric Power

Applications, Vol. 135 Issue 2, pp 76-84, March 1988.

[10] A. E. Emanuel, Powers in nonsinusoidal situations-a review of definitions and

physical meaning, IEEE Transactions on Power Delivery, Volume: 5 Issue: 3, pp

1377 –1389, July 1990.

[11] T. Furuhashi et al., A study on the theory of instantaneous reactive power, IEEE

Transactions on Industrial Electronics, Volume: 37 Issue: 1, Page(s): 86 –90,

Feb. 1990.

References

References 256

[12] E. B. Makram, S. Varadan, Definition of power components in the presence of

distorted waveforms using time domain technique, Twenty-Third Proceedings

Southeastern Symposium on System Theory, 1991, pp 536 –539, March 1991.

[13] A. Ferrero, G. Superti-Furga, A new approach to the definition of power

components in three-phase systems under nonsinusoidal conditions, IEEE

Transactions on Instrumentation and Measurement, Volume: 40 Issue: 3 , pp 568

-577, June 1991.

[14] Niels La White, Marija D. Illic, Vector Space Decomposition of Reactive Power

for Periodic Nonsinusoidal Signal, IEEE Transactions on Circuits and Systems 1:

Fundamental Theory and Applications, Vol 44 Issue 4, pp 338-346, April 1997.

[15] IEEE Working Group on Nonsinusoidal Situations: Effects on Meter Performance

and Definitions of Power, Practical Definitions for Powers in Systems with

Unbalanced Loads: A Discussion, IEEE Transactions on Power Delivery, Vol. 11

Issue 1, pp 79-101, January 1996.

[16] Czarnecki, L.S., Orthogonal decomposition of the currents in a 3-phase nonlinear

asymmetrical circuit with a nonsinusoidal voltage source, IEEE Transactions on

Instrumentation and Measurement, Volume: 37 Issue: 1, pp 30 -34, March 1988.

[17] F. Ghassemi, New Apparent Power and Power Factor with Non-sinusoidal

Waveforms, IEEE Power Engineering Society Winter Meeting, 2000, Vol 4, pp

2852-2857, January 2000.

[18] L.M. Tolbert, T.G. Habetler, Comparison of Time Based Nonactive Power

Definitions for Active Filtering, Power Electronics Congress CIEP 2000, pp 73-

79, October 2000.

[19] R. Fetea, A. Petroianu, Can the Reactive Power be Used?, Proceedings Power

Systems Technology, PowerCon 2000, Vol 3, pp 1251 -1255, December 2000.

[20] P.S. Filipski, R. Arseneau, Definition and Measurement of Apparent Power Under

Distorted Waveform Conditions, IEEE Tutorial Course on Nonsinusoidal

Situations: Effecton on Performance of Meters. Course Text 90EH0327-7-PWR,

pp 35-42, 1990.

[21] J H Enslin, J D Van Wyk, A New Control Philosophy for Power Electronic

Converters as Fictitious Power Compensators, 19th Annual IEEE Conference

Power Electronics Specialist Conference, PESC ’88, Vol. 2, pp 1188-1196, 11-14

April 1988.

References

References 257

[22] J H Enslin, J D Van Wyk, Measurement and Compensation of Fictitious Power

Under Nonsinusoidal Voltage and Current Conditions, IEEE Transactions on

Instrumentation and Measurement, Vol. 37 Issue 3, pp 403-408, September 1988.

[23] R. Sasdelli, Compensable Power for Electrical Systems in Nonsinusoidal

Conditions, IEEE Transactions on Instrumentation and Measurement, Vol. 43

No. 4, pp 592-598, August 1994.

[24] A P J Rens, P H Swart, Investigating the validity of the Czarnecki three phase

power definitions, 6th Africon Conference in Africa, 2002, IEEE AFRICON, Vol

2, pp 815 – 821, October 2002.

[25] A.M. Ferrero, Some Considerations about the Different Possible Approaches to

the Study of Electrical Power Systems under Nonsinusoidal Conditions, 8th

International Conference on Harmonics and Quality of Power, vol 1, pp 14 – 19,

October 1998.

[26] J H C Pretorius et al, An Evaluation of Some Alternative Methods of Power

Resolution in a Large Industrial Plant, IEEE Transactions on Power Delivery,

Vol. 15 Issue 2, pp 1052-1059,July 2000.

[27] L.S. Czarnecki, Power related phenomena in three-phase unbalanced systems,

IEEE Transactions on Power Delivery, Vol. 10 No. 3, pp 1168-1176, July 1995.

[28] A E Emanuel, Apparent Power Definitions for Three-Phase Systems, IEEE

Transactions on Power Delivery, Vol 14, No 3, pp 762-772, July 1999.

[29] J Cohen et al, Physical Time Domain Representation of Powers in Linear and

Nonlinear Electrical Circuits, IEEE Transactions on Power Delivery, Vol 14, No

4, pp 1240-1249, October 1999.

[30] F. Ghassemi, New Concept in AC PowerTheory, IEE Proceedings Generation,

Transmission and Distribution, Volume: 147 Issue: 6, pp 417-424, November

2000.

[31] P.S. Filipski, Polyphase Apparent Power and Power Factor Under Distorted

Waveform Conditions, IEEE Transactions on Power Delivery, Vol 6 No. 3, pp

1161 - 1165, July 1991.

[32] L.S. Czarnecki, Power related phenomena in three-phase unbalanced systems,

IEEE Transactions on Power Delivery, pp 1168-1176, Vol. 10, July 1995.

[33] Hyusong Kim et al, Spectral Analysis of Instantaneous Powers in Single-Phase

and Three-Phase Systems with the use of p-q-r Theory, IEEE 32th Annual Power

References

References 258

Electronics Specialist Conference, PESC 2001, Vol. 1, 17-21, pp 54-61, June

2001.

[34] L.S. Czarnecki, New Power Theory of the 3-Phase Non-linear Assymmetrical

Circuits Supplied from Nonsinusoidal Voltage Sources, IEEE International

Symposium on Circuits and Systems, Vol. 2, pp 1627-1630, June 1988.

[35] P.S. Filipski, R. Arseneau, Definition and Measurement of Apparent Power Under

Distorted Waveform Conditions, Third International Conference on Harmonics in

Power Systems, Indiana, USA, pp 127 – 131, October 1988.

[36] F. Ghassemi, What is Wrong with Electric Power Theory and How It Should be

Modified, Metering and Tariffs for Energy Supply, Conference Publication No

462, pp 109-114, May 1999.

[37] IEEE Standard Definitions for the Measurement of Electric Power Quantities

Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions, IEEE

Standard 1459-2000, New York, 2000.

[38] J L Willems, J A Ghijselen, Apparent Power and Power Factor Concepts in

Unbalanced and Nonsinusoidal Situations, 2003 IEEE Bologna Power Conference

Bologna, Italy, Vol 3, pp , 23-26 June 2003.

[39] J L Willems, Reflection on Apparent Power and Power Factor Nonsinusoidal and

Polyphase Situations, IEEE Transactions on Power Delivery, pp 7, 835-840, Vol

19 No 2, April 2004.

[40] I. Takahashi, Analysis of instantaneous current and power using space switching

functions, 19th Annual IEEE Power Electronics Specialists Conference, 1988.

PESC '88 Record, 11-14, vol.1, pp 42 -49, April 1988.

[41] E. H. Watanabe et al, New Concepts of Instantaneous Active and Reactive Powers

in Electrical Systems with Generic Loads, IEEE Transactions on Power Delivery,

Volume: 8, No: 2, pp 697 –703, April 1993.

[42] M. Depenbrock, The FBD-method, a generally applicable tool for analyzing

power relations, IEEE Transactions on Power Systems, Volume: 8, Issue: 2, pp

381-387, May 1993.

[43] A. E. Emanuel, On the definition of power factor and apparent power in

unbalanced polyphase circuits with sinusoidal voltage and currents, IEEE

Transactions on Power Delivery, Volume: 8 Issue: 3 , pp 841–848, July 1993.

References

References 259

[44] J. L. Willems, D. Aeyeis, New decomposition for 3-phase current in power

systems, IEE Proceedings on Generation, Transmission and Distribution, Volume

140 Issue 4, pp 307-310, July 1993.

[45] L Cristaldi et al, Current Decomposition in Assymmetrical, Unbalanaced Three-

Phase Systetms under Nonsinusoidal Conditions, IEEE Transactions on

Instrumentation and Measurement, Volume: 43, No: 1, pp 63 –68, February 1994.

[46] F. Z. Peng, J. Lai, Generalized Instantaneous Reactive Power Theory for Three

Phase Power Systems, IEEE Transactions on Power Instrumentation and

Measurements, Volume: 45, No: 1 , pp 293 –297, February 1996.

[47] A. Nabae, T. Tanaka, A New Definition of Instantaneous Active-Reactive Current

and Power Based on Instantaneous Space Vectors on Polar Coordinates in Three-

Phase Circuits, IEEE Transactions on Power Delivery, Volume: 11, No: 3, pp

1238 –1243, July 1996.

[48] Milanez, D.L., New concepts of the power received by ideal energy storage

elements: the instantaneous complex power approach, IEEE 39th Midwest

symposium on Circuits and Systems, 1996, Volume: 3, pp 1038 -1041, 18-21 Aug.

1996.

[49] Emanuel, A.E., The Buchholz-Goodhue apparent power definition: the practical

approach for nonsinusoidal and unbalanced systems, IEEE Transactions on Power

Delivery, Volume: 13 Issue: 2, pp 344 -350, April 1998.

[50] H. Akagi, et al, The Theory of Instantaneous Power in Three-Phase Four-Wire

Systems: A Comprehensive Approach, Conference Record of the 1999 Thirty-

Fourth IAS Annual Meeting Industry Applications Conference, Volume: 1, pp 431

–439, October 1999.

[51] Shin-Kuan Chen; G. W. Chang, A new instantaneous power theory-based three-

phase active power filter, IEEE Power Engineering Society Winter Meeting,

2000., Volume: 4, pp 2687 - 2692, 23-27 Jan. 2000.

[52] Shun-Li Lu et al, Suggested power definition and measurement due to harmonic

load, Elsevier Journals, Electric Power System Research, Vol 53, Issue 2, pp 73-

81, February 2000.

[53] Wilczynski, E, Total apparent power of the electrical system for periodic,

deformed waveforms, IEE Proceedings - Electric Power Applications, Volume:

147 Issue: 4 , pp 281 –285, July 2000.

References

References 260

[54] F. Z. Peng, L. M. Tolbert , Compensation of Non-Active Current in Power

Systems – Defintions from Compensation Standpiont, Power Engineering Society

Summer Meeting 2000, Volume: 2 , pp 983-987, 16-20 July 2000.

[55] D. Zhang, N. Huang, Universal instantaneous power theory for DC, single-phase

AC sinusoidal and nonsinusoidal circuits, Proceedings Power Electronics and

Motion Control Conference PIEMC 2000, Beijing, pp 1142-1147, August 2000.

[56] Lev-Ari, H.; Stankovic, A.M, Defining reactive power in circuit transients via

local Fourier coefficients, IEEE International Symposium on Circuits and

Systems, 2002. ISCAS 2002, Volume: 5, pp V-689 -V-692, 26-29 May 2002.

[57] F. Z. Peng et al, Definitions and compensation of non-active current in power

systems, IEEE 33rd Annual Power Electronics Specialists Conference, 2002.

Volume: 4 , pp 1779 – 1784, June 2002.

[58] Haque, M.T., Single-phase PQ theory, IEEE 33rd Annual Power Electronics

Specialists Conference, 2002. PESC 02. 2002, Volume: 4, pp 1815 -1820, 23-27

June 2002.

[59] Milanez, D.L.; Emanuel, A.E., The instantaneous-space-phasor: a powerful

diagnosis tool, IEEE Transactions on Instrumentation and Measurement, Volume:

52 Issue: 1, pp 143 –148, Feb. 2003.

[60] Lev-Ari, H.; Stankovic, A.M, Hilbert space techniques for modeling and

compensation of reactive power in energy processing systems, IEEE Transactions

on Circuits and Systems I: Fundamental Theory and Applications, Volume: 50

Issue: 4, pp 540 -556 April 2003.

[61] Yan Xu et al, Compensation-based nonactive power definition, IEEE Power

Electronics Letters, , Volume: 1 , Issue: 2, pp 45 – 50, June 2003.

[62] Driesen, J.L.J.; Belmans, R.J.M., Wavelet-based power quantification approaches,

IEEE Transactions on Instrumentation and Measurement, Volume: 52 Issue: 4, pp

1232 –1238, August 2003.

[63] Hongfa Ding et al, Instantaneous power components definition for three-phase

circuits based on the four- dimension complex number, IEEE PES Transmission

and Distribution Conference and Exposition 2003, Volume 1, Vol.1, 7-12, pp132

– 137, Sept. 2003.

[64] Zhibin Ling et al, The definition of power factor in single phase system with

arbitrary waveform according to similarity, The Fifth International Conference on

References

References 261

Power Electronics and Drive Systems, 2003. PEDS 2003. Volume 2, pp 1323 -

1328, 17-20 November 2003.

[65] Dai, X.; Liu, G.; Gretsch, R.; Generalized Theory of Instantaneous Reactive

Quantity for Multiphase Power System, IEEE Transactions on Power Delivery,

Volume: 19 , Issue: 3, pp 965 – 972, July 2004.

[66] Seong-Jeub Jeon, Definitions of apparent power and power factor in a power

system having transmission lines with unequal resistances, IEEE Transactions on

Power Delivery, Volume 20, Issue 3, pp 1806 – 1811, July 2005.

[67] Pajic, S. Emanuel, A.E., A comparison among apparent power definitions, IEEE

Power Engineering Society General Meeting, pp 8, 18-22 June 2006.

[68] Pajic, S.; Emanuel, A.E., Modern apparent power definitions: theoretical versus

practical Approach-the general case, IEEE Transactions on Power

Delivery,Volume 21, Issue 4, pp1787 – 1792, October 2006.

[69] J. L. Wyatt, M. Ilic, Time-domain reactive power concepts for nonlinear,

nonsinusoidal or nonperiodic networks, IEEE International Symposium on

Circuits and Systems, 1990, pp 387 -390 vol.1, May 1990.

[70] Author Unknown, Thomas Edison (1847-1931), Electricity and light bulb -

history, http://inventors.about.com/library/inventors/bledison.htm, Date:

Unknown.

[71] Tom Flynn, Charles Proteus Steinmetz, Inventor, http://www.yonkershistory.org/

stein.html, Date: Unknown.

[72] Steinmetz C.P., Theory and Calculation of Alternating Current Phenomena,

McGraw-Hill Book Company, New York, 1900.

[73] Steinmetz C.P., Theoretical Elements of Electrical Engineering, Electrical World

and Engineer, New York, 1900, Part 1 Section 4 and 8

[74] Emanuel A.E., Summary of IEEE Standard 1459: Definitions for the

Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal,

Balanced, or Unbalanced Conditions, IEEE Transactions on Industry

Applications, Volume: 40 , Issue: 3 , pp 869 – 876, May-June 2004

[75] Sutherland P.E., On The Definition Of Power In An Electrical System, IEEE

Transactions on Power Delivery, Volume 22, pp 1100 – 1107, April 2007

References

References 262

[76] Branislav D, Petar B, Miroslav B, New Method for Reactive Power and Energy

Measurement, IEEE Transactions on Instrumentation and Measurement, Volume:

41 , No. 2, pp 280 – 285, April 1992.

[77] Sharon D., Reactive power definition and power factor improvement in non-linear

systems, IEE Proceedings, Vol 120, No 6, pp 704-706, June 1973.

[78] Svensson S., “Power measurement techniques for non-sinusoidal conditions”,

Doctoral thesis for the degree of Doctor of Philosophy, Department of Electric

Power Engineering, Chalmers University of Technology, Goteborg, Sweden 1999.

[79] Shepherd W, Zakikhani P., Suggested definitions of reactive power for

nonsinusoidal situations, Proceedings of Institute of Electrical Engineers, Vol

119, pp 1361-1362, Sept 1972.

[80] Czarnecki, L.S., Comparison of Power Definitions for Circuits with

Nonsinusoidal Waveforms, IEEE Tutorial Course, pp 43 – 50, 1990.

[81] Balci, M.E., Hocaoglu, M.H., Comparison of power definitions for reactive power

compensation in nonsinusoidal conditions, 11th International Conference on

Harmonics and Quality of Power 2004, pp 519 – 524, 12-15 Sept. 2004.

[82] de Leon, F.; Cohen, J., A practical approach to power factor definitions:

transmission losses, reactive power compensation, and machine utilization, IEEE

Power Engineering Society General Meeting, 2006, , pp 1 – 7, 18-22 June 2006.

[83] P.S.Filipski, P.S. Labaj, Evaluation of reactive power meters in the presence of

high harmonic distortion, IEEE Transactions on Power Delivery, Volume 7,

Issue 4, pp 1793 – 1799, Oct. 1992.

[84] Kuster’s N.L., Moore W.J.M., On the definition of reactive power under non-

sinusoidal conditions, IEEE Transactions on Power Apparatus and Systems, Vol

PAS-99, No. 5,pp 1845 – 1850, Sept/Oct 1980.

[85] Kim H., Akagi H., The instantaneous power theory based on mapping matrices in

three-phase four-wire systems, PCC’97 Nagaoka, Pages 361 – 366, 1997.

[86] Chang, G.W., Tai-Chang Shee, A comparative study of active power filter

reference compensation approaches, IEEE Power Engineering Society Summer

Meeting, 2002, Volume: 2, pp 1017 –1021, 21-25 July 2002.

[87] Depenbrock, M. Staudt V., The FBD-method as tool for compensating total non-

active currents, 8th International Conference on Harmonics and Quality of Power

ICHQP ‘98, Athens, Greece, pp 320 –324, October 1998.

References

References 263

[88] Victor M. Moreno, Alberto Pigazo, Modified FBD Method in Active Power

Filters to Minimize the Line Current Harmonics, IEEE Transactions on Power

Delivery, Volume 22, Issue 1, pp 735 – 736, Jan. 2007.

[89] Ferrero, A.; Leva, S.; Morando, A.P., A park-vector approach to displacement and

distortion in three-phase systems: the role of the power factor and the imaginary

power, 11th International Conference on Harmonics and Quality of Power, 2004,

pp 745 - 75212-15 Sept. 2004.

[90] Willems, J.L., A new interpretation of the Akagi-Nabae power components for

nonsinusoidal three-phase situations, IEEE Transactions on Instrumentation and

Measurement, Volume: 41 , Issue: 4 , pp 523 – 527, Aug. 1992.

[91] J L Willems, J A Ghijselen, The Apparent Power Concept and the IEEE Standard

1459-2000, IEEE Transactions on Power Delivery, Volume 20, Issue 2, pp 876 –

884, April 2005.

[92] Czarnecki, L.S.; Swietlicki, T., Powers in nonsinusoidal networks: their

interpretation, analysis, and measurement, IEEE Transactions on Instrumentation

and Measurement, Volume: 39, Issue: 2, pp 340 – 345, April 1990.

[93] Chen Xiangxun, Physical nature and exact definition of electric power,

Conference on Precision Electromagnetic Measurements, 1990. CPEM '90

Digest., pp 145 – 146, 11-14 June 1990.

[94] A. E. Emanuel, Powers in nonsinusoidal situations-a review of definitions and

physical meaning, IEEE Transactions on Power Delivery, Volume: 5 Issue: 3 , pp

1377 –1389, July 1990.

[95] S. H. Hosseini, G. Shannon, New definition of reactive power and implementation

of wide bandwidth Varmeters, Elsevier Journals, Electric Power System

Research, Vol 22, Issue 3, pp 195-201, December 1991.

[96] E. B. Makram, S Varadan, Analysis of reactive power and power factor correction

in the presence of harmonic distortion, Elsevier Journals, Electric Power System

Research, Vol 26, Issue 3, pp 211-218, April 1993.

[97] Czarnecki, L.S., Misinterpretations of some power properties of electric circuits,

IEEE Transactions on Power Delivery, Volume: 9 Issue: 4, pp 1760 -1769, Oct.

1994.

[98] M. Depenbrock et al, A Concise Assessment of Original and Modified

Instantaneous Power Theory Applied to Four-Wire Systems, Proceedings of the

References

References 264

Power Conversion Conference, Osaka, Japan, vol.1 Volume: 1, pp 60-67, April

2002.

[99] L.S. Czarnecki, On Some Misinterpretations of the Instantaneous Reactive Power

p-q Theory, IEEE Transactions on Power Electronics, Vol 19, No 3, pp 828-836,

May 2004.

[100] The New IEEE Standard Dictionary of Electrical and Electronics Terms, IEEE

Std 100-1996, Sixth Edition, IEEE, New York, 1996.

[101] “Visahan G., Digital Power Meter for Nonsinusoidal Systems”, Bachelors thesis

for the degree of Bachelor of Engineering, Department of Electric and Computer

Systems Engineering, Monash Unviersity, Victoria, Australia, 2005.

[102] P.S. Filipski et al, Discussion of Power Definitions contained in the IEEE

Dictionary, IEEE Transactions on Power Delivery, Vol 9 No. 3, pp 1237 – 1244,

July 1994.

[103] J L Willems, J A Ghijselen, A E Emannuel, The Relation between Generalised

Apparent Power Concept and the Voltage Reference, L’Energia Elettrica, Volume

81, 2004 Page(s): 37 – 45.

[104] J M Aller, B Alexander, T Pagi, Power System analysis using Space-Vector

Transformation, IEEE Transactions on Power System, Vol 17 No. 4, pp 937 –

965, November 2002.

[105] S. Y. Foo, “Power Meter for Nonsinusoidal Systems”, Bachelors thesis for the

degree of Bachelor of Engineering, Department of Electric and Computer Systems

Engineering, Monash University, Victoria, Australia, 2006.

[106] P J Ong., “Digital Power Meter for Nonsinusoidal Systems”, Bachelors thesis for

the degree of Bachelor of Engineering, Department of Electric and Computer

Systems Engineering, Monash University, Victoria, Australia, 2006.

[107] S R Castro., “Digital Power Meter for Nonsinusoidal Systems”, Bachelors thesis

for the degree of Bachelor of Engineering, Department of Electric and Computer


[108] Md Asifuzzaman., “Digital Power Meter for Nonsinusoidal Systems”, Bachelors

thesis for the degree of Bachelor of Engineering, Department of Electric and

Computer Systems Engineering, Monash University, Victoria, Australia, 2007.

References

References 265

[109] C. Y Tan., “Digital Power Meter for Nonsinusoidal Systems”, Bachelors thesis for

the degree of Bachelor of Engineering, Department of Electric and Computer


Parallel Equivalent of Series RL load

Appendix A A-1

APPENDIX A PARALLEL EQUIVALNT OF A SERIES RL LOAD

s s sZ R jX= +

1

p1 1ZR jX

−⎛ ⎞

= +⎜ ⎟⎝ ⎠

= 1

jX RjRX

−⎛ ⎞+⎜ ⎟⎝ ⎠

= jRXjX R+

= 2 2

jRX(R jX)R X

−+

= 2 2

RX(X jR)R X

++

2

s p s 2 2

RXZ Z RR X

= ↔ =+

(A.1)

2

s 2 2

R XXR X

↔ =+

(A.2)

s

s

X RtanR X

↔ = θ = (A.3)

s s

s s

X RR X, X RR X

↔ = = (A.4)

Substituting X from (A.4) in (A.1) gives

2 2s s

s

1R (R X )R

= + (A.5)

Substituting R from (A.4) in (A.2) gives

2 2s s

s

1X (R X )X

= + (A.6)

Rs

LsZs

R XZs

Comparison of the proposed definition and Fryze’s definition

Appendix B B-1

APPENDIX B Comparison of the proposed defintion and Fryze’s definition Introduction (Linear series RLC circuit) The main intent of this appendix is to fully compensate for non-active power using the

proposed definitions in part 1 and then repeat the same using Fryze’s definition in part

2. The two results are compared. Software used for analysis is Mathcad and ATP.

The System

The source and load data are given in Figure 1.

V1 100 volt⋅:= ω1 100 π⋅1

sec⋅:=

V3 0.3 V1⋅:= ω3 4 ω1⋅:=

R 5 ohm⋅:= L 0.02 H⋅:= CL 1 10 3−⋅ F⋅:=

Source impedance is neglected.

The voltages above are RMS values with

phase relationship as in equation (B.1).

Figure 1: System data

The voltage and current is given by

v t( ) 2 V1⋅ cos ω1 t⋅( )⋅ 2 V3⋅ cos ω3 t⋅( )⋅+:= voltage (B.1)

and

i t( ) 2V1Z1⋅ cos ω1 t⋅ δ1−( )⋅ 2

V3Z3⋅ cos ω3 t⋅ δ3−( )⋅+:= current

(B.2)

where Z1 and Z3 are the fundamental and harmonic impedance magnitude and δ1 and δ3

are the corresponding impedance angle for the load.

Part1: Using the proposed definition Compensation of non-active power using the proposed definitions. Powers waveforms using proposed definitions

The total power is given by


Appendix B B-2

s(t) = v(t) i(t) (B.3)

The graph for voltage, current and total power is given in Figure 2.

0 0.005 0.01 0.015 0.02

4000

3000

2000

1000

1000

2000

3000

4000

5000

6000

10 v t( )⋅

100 i t( )⋅

s t( )

t . Figure 2: Voltage, current and total power waveforms

In the analysis, sx(t) represents total “instantaneous” power, px(t) represents active

“instantaneous” power and sx(t) represents non-active “instantaneous” power. In this

example the use of the term “power” implies “instantaneous power”. Also

compensation implies shunt compensation.

Using the proposed definitions the concept of which is shown in Figure 3, the

component powers are determined.

v(t)

i(t)

Harmoniccomponents

ppppp

01hIhIg

01h

vvv

iiii

01hg

qqqqq

01hIhIg

Powercomponents

Activepower

p(t)

Non-activepower

q(t)

sum

sum

iiii

0a1ahaga

iiii

0q1qhqgq

X

X

+Totalpower

s(t)

Figure 3: Powers in nonsinusoidal system

The waveforms for component powers follow.


Appendix B B-3

DC based Powers s0(t), p0(t), q0(t)

These are all zero since the load is linear.

s0(t) = 0

p0(t) = 0

q0(t) = 0 (B.4)

Fundamental Powers s1(t), p1(t), q1(t)

Fundamental powers are shown in Figure 4.

0.02 0.025 0.03 0.035 0.04

1000

500

500

1000

1500

2000

2500

3000

3500

4000

s1 t( )

p1 t( )

q1 t( )

t .

Figure 4: Fundamental powers waveforms

Source Generated Harmonic Powers sh(t), ph(t), qh(t)

Source generated harmonic powers are shown in the Figure 5.

0.02 0.025 0.03 0.035 0.04

40

30

20

10

10

20

30

40

50

60

sh t( )

ph t( )

qh t( )

t . .

Figure 5: Harmonic powers waveforms


Appendix B B-4

Load Generated Harmonic Powers sg(t), pg(t), qg(t)

Load generated harmonic powers are zero since the load is linear.

sg(t) = 0

pg(t) = 0

qg(t) = 0 (B.5)

Source Generated Cross Harmonic Powers sxh(t), pxh(t), qxh(t)

Cross harmonic type A powers are shown in the Figure 6.

0.02 0.03 0.04 0.05 0.06

1500

1200

900

600

300

300

600

900

1200

1500

sxh t( )

pxh t( )

qxh t( )

t Figure 6: Source generated harmonic powers waveforms

Load Generated Cross Harmonic Powers sxg(t), pxg(t), qxg(t)

Load generated cross harmonic powers are zero since load is linear.

sxg(t) = 0

pxg(t) = 0

qxg(t) = 0 (B.6)

Total Powers s(t), p(t), q(t)

The sum of all the above gives total powers

Sk S0DkS0Xk

+ S1k+ Shk

+ Sgk+ SXhk

+ SXgk+:=

Pk P0DkP0Xk

+ P1k+ Phk

+ Pgk+ PXhk

+ PXgk+:=

Qk Q0DkQ0Xk

+ Q1k+ Qhk

+ Qgk+ QXhk

+ QXgk+:=

(B.7)


Appendix B B-5

and the waveforms are

0.02 0.025 0.03 0.035 0.04

4000

3000

2000

1000

1000

2000

3000

4000

5000

6000

s t( )

p t( )

q t( )

25 v t( )⋅

t.

Figure 7: Total powers waveforms

Note that v(t) is magnified by 25 to enable plotting on the same graph. This

(magnification of quantities) is used henceforth as necessary for this purpose.

Is decomposition faithful? Now the question is “how do we know that the above is correct decomposition into

active and non-active powers”, i.e., is the definition is able to faithfully predict active

and non-active powers? It is a known that if non-active power is removed, only active

power will remain. This will give the lowest current (for that particular source-load

case) being drawn from the source. Thus lowest current can be used as an indicator.

Hence in the above example, if the non-active power q(t) is removed, only p(t) will

remain. The resulting active current, which is now the current is(t) supplied by the

source, will be the lowest possible current in the circuit. The waveforms of the voltage

v(t), load i(t) and source is(t) currents (source current is determined from active power)

are shown in Figure 8.


Appendix B B-6

0.02 0.03 0.04 0.05 0.06

300

200

100

100

200

300

v t( )

10 is t( )⋅

10 i t( )⋅

t .

Figure 8: Voltage, load and source current after removal of non-active power

Let us determine the total RMS current without compensation and source RMS current

(current left after compensation = active current). These are calculated as follows.

Total RMS current

irms1T T

2Tti t( )( )2⌠

⎮⌡

d⋅:= irms 17.040756 A=

(B.8)

and active RMS current

iactive t( )p t( )v t( )

:= IActiverms1T

T

2T

tiactive t( )2⌠⎮⌡

d⋅:= IActiverms 14.448512A= . (B.9)

Check if decomposition is faithful using compensation with shunt elements As stated above, we have to show that the definition faithfully predicts the active and

non-active decomposition. The method to be used to test the definition is essentially “if

the definition correctly gives the active and/or non-active power, then the knowledge of

the non-active power provided by the definition, can be used to completely compensate

(using shunt elements) the non-active power, resulting in the source supplying only

active power”.


Appendix B B-7

Figure 9: Compensation (using shunt elements) of non-active current

The sketch above illustrates the idea. The passive "harmonic pass filter" is used to

ensure that only the selected harmonic flows in the leg. The “var supply” is then used

to supply the non-active power for that particular harmonic. This is done for each

harmonic. In this manner it is possible to compensate for all the non-active power

harmonic by harmonic and thus show if a definition faithfully defines the active/non-

active power (or current). This is based on the ideas of the commercially used C-filter.

The treatment here is more so to show complete compensation of predicted non-active

power using the new definition from the concepts viewpoint, without regard to

commercial economics, thus the “like” components of the harmonic pass tuned filer and

the var supply (e.g. C and Cc as below) are not combined.

The instantaneous non-active power or non-active current is used to determine the

compensating current. Generally this is

xcx

x

q (t)i (t)v (t)

= − or cx naxi (t) i (t)= − (B.10)

where “x” is the harmonic; icx(t) is the compensation current, qx(t) the non-active

power, inax(t) is the non-active current and vx(t) is the source voltage for harmonic x.

For each harmonic the compensating capacitance/inductance depending on whether

inductive or capacitive vars are required is given by

Cc = cxrms


iVω

or Lc= xrms


VIω

. (B.11)


Appendix B B-8

Each harmonic compensating current (i1c(t), ihc(t)) is realised by a series tuned LC filter

in series with a compensating capacitor Cc or inductor Lc, which is the “var supply”.

The compensating capacitors/indutors provide the compensating currents that supply the

non-active current required by the load.

The harmonic pass filter, for each frequency is determined from

2FOR EACH HARMONIC

1LC=ω

(B.12)

The compensating part of the circuit shown in Figure 10 is then known. Since all the

components of the circuit are known, the source current is(t) for the circuit, can be

determined using circuit analysis,. The source current will comprise harmonic

components as follows

is(t) = i1s(t) + ihs(t) (B.13)

For a fully compensated system i1s will in phase with v1 and each harmonic ih in phase

with vh. Also the source current will have the minimum magnitude for a fully

compensated system. The magnitude (RMS value) of is(t) is obtained using

Isrms = ( )T

2s

0

1 i (t) dtT ∫ (B.14)

For the above example the fundamental and 4th harmonic is present. The pass filter and

compensating equipment data is as follows.

Fundamental pass filter L1, C1 and compensating capacitor Cc1

L1 506.606mH= C1 20μF= Cc1 285.112μF= (B.15)

Harmonic pass filter L3, C3 and compensating capacitor Cc3 for 4th harmonic

L3 316.629mH= C3 2μF= Cc3 31.374μF= (B.16)

The circuit with compensation shown is as follows.


Appendix B B-9

L3

C3

Cc3

Harmfilter

L1

C1

Cc1

Fundfilter

~L

R

CL

Note: C and Cc1 (C3 and Cc3) are shown in series as this aided the computation when determining the

minimum current by trial and error (see below). They can be combined to a single capacitor.

Figure 10: Source, load and shunt compensating elements

The source current for the fundamental is1(t) is first determined by circuit analysis with

only fundamental voltage as the source. The fundamental source current after

compensation is as shown in the graph below. The first graph shows one cycle of the

waveform and the second shows the first crossing, on an expanded scale, to show where

it occurs. Theoretically it should occur at v1(t) = 0.

0.02 0.03 0.04

250

150

50

50

150

250

v1 t( )

10 is1 t( )⋅

t

0.0249 0.025 0.0251

5

5

v1 t( )

10 is1 t( )⋅

t Figure 11: Fundamental current after compensation

Similarly the harmonic source current is3(t) is determined for the harmonic. The result

likewise is shown below.

0.02 0.025 0.03

50

50

v3 t( )

100 is3 t( )⋅

t

0.0237 0.02375 0.0238

1

1

v3 t( )

10 is3 t( )⋅

t Figure 12: 4th harmonic current after compensation


Appendix B B-10

The resulting source current is the sum of the fundamental and harmonic and is

is t( ) is1 t( ) is3 t( )+:= . (B.17)

The RMS source current after compensation is then

Isrms1T

0

Ttis t( )( )2⌠

⎮⌡

d⋅:= Isrms 14.448649 A=

(B.18)

This value is very close to value of “14.448512 A” determined earlier from the active

power.

As stated above the crossing should occur at amplitude = 0. The compensating

capacitor values to give this were determined by “trial and error”. The values obtained

were

Cc1 283.1μF= Cc3 31.5μF= . (B.19)

The resulting fundamental and harmonic is shown in fig 12 where is1M(t) and is3M(t) are

the source current using these compensating capacitor values. Note the zero crossing.

0.02 0.03 0.04

250

150

50

50

150

250

v1 t( )

10 is1M t( )⋅

t

0.0249 0.025 0.0251

5

5

v1 t( )

10 is1M t( )⋅

t

0 0.005 0.01

50

50

v3 t( )

100 is3M t( )⋅

t

0.0037 0.00375 0.0038

1

1

v3 t( )

10 is3M t( )⋅

t Figure 13: Fundamental, harmonic current after compensation with elements giving

minimum source current


Appendix B B-11

0.02 0.03 0.04 0.05 0.06

300

200

100

100

200

300

v t( )

10 isM t( )⋅

10 i t( )⋅

t .

Figure 14: Voltage, load and source current after compensation using compensating

capacitors that give fundamental/harmonic voltage/current crossing at zero

The source current after compensation is IsourceRMS = 14.448512 which is identical to that

obtained by compensating for the non-active power given by the proposed definitions.

Hence it can be said that the proposed definitions faithfully predict non-active power

that gives rise, when compensated, to minimum source current.

Next step was to determine the RMS source current for varying values of Cc1 and Cc3

to confirm that the compensated current minimum source current does occur at Cc1 =

283.1 μF and Cc3 = 31.5 μF. Table 1 and the graph in Figure 15 shows this to be the

case.

Table 1: Source current for varying compensating capacitors

Cc1 279.1 281.1 283.1 285.1 287.1Cc327.5 14.449857 14.449442 14.449301 14.449434 14.44983929.5 14.449262 14.448849 14.448710 14.448843 14.44925031.5 14.449062 14.448650 14.448512 14.448646 14.44905433.5 14.449255 14.448844 14.448707 14.448843 14.44925235.5 14.449842 14.449432 14.449296 14.449432 14.449842


Appendix B B-12

279.1 281.1 283.1 285.1 287.127.5

31.5

35.5

14.44800014.44820014.44840014.44860014.44880014.44900014.44920014.44940014.44960014.44980014.450000

14.449800-14.45000014.449600-14.44980014.449400-14.44960014.449200-14.44940014.449000-14.44920014.448800-14.44900014.448600-14.44880014.448400-14.44860014.448200-14.44840014.448000-14.448200

Figure 15: Plot of RMS compensated source current for varying compensating

capacitors

Perform the same process using ATP

Finally ATP was used to simulate the system with the compensating capacitance values

as obtained using the proposed definitions. The system modelled in “ATP Draw” is as

shown below. The simulation was run for a time period of 20 secs.

U

U

XX0003

XX0013

XX0015

XX0017

XX0023

XX0024

XX0025

XX0008

Switch close at5.6100405 ms for comp

Ex01sRLC.adp

Figure 16: ATP Draw system used for analysis with ATP

The waveforms for the source voltage, load voltage, source current and load current

obtained are as shown in Figure 17. Note that resistors XX00 (0.1 ohm), XX0017 (0.01


Appendix B B-13

ohm) and XX0025 (0.01 ohm) were added to minimise oscillations. Hence the circuit is

slightly different from that analysed using Mathcad.

(file Ex01sRLC.pl4; x-var t) factors:offsets:

10

v:XX0001 10

v:XX0008 10

c:XX0008-XX0003 100

c:XX0001-XX0015 100

19.960 19.965 19.970 19.975 19.980 19.985 19.990 19.995 20.000-300

-200

-100

0

100

200

300

Figure 17: Voltage, load and source current after compensation using ATP

The legend for the plots are Red = Generator voltage (location XX0001 on system circuit), Green = voltage across load (location XX0008 on system circuit), Pink = source current, magnified by 10, after compensation (location XX0008-XX0003 on system circuit), Blue = load current magnified by 10 (location XX0001-XX0015 on system circuit). The RMS value of the currents as follows:

Source current after compensation = 14.341308

Load current = 16.79639.


Appendix B B-14

Comprison of source current from the proposed theory and ATP

0.02 0.025 0.03 0.035 0.04

300

200

100

100

200

300

v t( )

10 isM t( )⋅

10 is t( )⋅

10 isATPl⋅

t t, t, l dta⋅, .

Figure 18: Comparison of compensated source currents

The source current waveforms obtained from ATP are practically the same as those

obtained above from the proposed definitions (Figure 8) and from compensation using

passive elements (Figure 14). This comparison is shown in Figure 18. There is a slight

difference in the waveform and RMS values because of inclusion of resistors to control

oscillation and also due to the presence of slight oscillation at the end of the 20-second

simulation.


Appendix B B-15

From definition From Mathcad From ATP

Active current 14.448512 - -

Source current

after

compensation

14.448512 14.448649 14.341308

Load current 17.040756 17.040756 16.79639

Conclusion From the analysis above, we can say the proposed definition does faithfully decompose

the total power into active and non-active powers for this example. The resultant active

power gives active current that will be the minimum source current.


Appendix B B-16

Part 2: Using Fryze’s definition Compensation of non-active power using the Fryze’s definitions. The definitions by Fryze’s are

PF1T 0.sec

Ttp t( )

⌠⎮⌡

d⋅:= ia t( )PF

Vrms( )2v t( )⋅:= ib t( ) i t( ) ia t( )−:=

The factor ( )

FeqF 2

rms

PrV

= is a constant. This means that the active current time profile

is a constant scaled version of the voltage profile i.e iA(t) = k v(t) where k is constant

and equal to eqF

1r

which is the equivalent conductance geqF of the load. Thus Fryze

model assumes a constant conductance within a period as seen from the metering point.

In the proposed definitions, the conductance is not assumed to be constant but a

function of time within the period.

For this the equivalent conductance and susceptance of the load using Fryze’s definition

are shown in Figure 1.1.

geqF t( )ia t( )

v t( ):= Equivalent conductance

of the loadbeqF t( )

ib t( )

v t( ):= Equivalent susceptance

of the load

0 0.005 0.01 0.015 0.02

0.5

0.25

0.25

0.5

geqF t( )

0.003 v t( )⋅

t

0 0.005 0.01 0.015 0.02

0.5

0.25

0.25

0.5

beqF t( )

0.003 v t( )⋅

t Figure 1.1: Conductance and susceptance based on Fryze’s defintions

Using the proposed definition the equivalent conductance and susceptance for the

example is as per Figure 1.2


Appendix B B-17

geq t( )p t( )

v t( )( )2:= Equivalent conductance

of the loadbeq t( )

q t( )

v t( )( )2:= Equivalent susceptance

of the load

0 0.005 0.01 0.015 0.02

0.5

0.25

0.25

0.5

geq t( )

0.003 v t( )⋅

t

0 0.005 0.01 0.015 0.02

0.5

0.25

0.25

0.5

beq t( )

0.003 v t( )⋅

t Figure 1.2: Conductance and susceptance based on proposed definitions

ATP was also used the to obtain the minimum source current for Example 1 by “trial

and error”. The minimum source current is the active current. The active and non-active

current is shown in Figure 1.3.

iactiveATPlisourceATP l

:= active current = minimum source current

iNonactiveATP liloadATPl

isourceATP l−:= non active current = load current - source current

0.02 0.025 0.03 0.035 0.04

300

200

100

100

200

300

v t( )

10 isourceATPl⋅

10 iloadATPl⋅

t l dta⋅,

0.02 0.025 0.03 0.035 0.04

300

200

100

100

200

300

v t( )

10 iactiveATPl⋅

10 iNonactiveATPl⋅

t l dta⋅, Figure 1.3: Active and non-active currents obtained from on ATP computation

The conductance and susceptance is determined from the active and non-active current.


Appendix B B-18

geqATPl

iactiveATPl

vsourcel1 10 15−⋅ V⋅+

:= Equivalent conductanceof the load

beqATPl

iNonactiveATPl

vsourcel1 10 15−⋅ V⋅+

:= Equivalent susceptanceof the load

.

0 0.005 0.01 0.015 0.02

0.5

0.25

0.25

0.5

geqATPl

0.003 vsourcel⋅

l dta⋅

0 0.005 0.01 0.015 0.02

0.5

0.25

0.25

0.5

beqATPl

0.003 vsourcel⋅

l dta⋅ Figure 1.4: Conductance and susceptance obtained from on ATP computation

The Fryze model works very well with purely parallel circuits but the assumption of

constant parallel conductance in a series circuit gives discrepancy in active current and

subsequently of non-active current for the example. This is shown in Figure 1.5. Note

that iact(t) is the current after compensation based on the proposed definition.

0 0.005 0.01 0.015 0.02

25

15

5

5

15

25

ia t( )

iact t( )

iactiveATPl

t t, l dta⋅,

0 0.005 0.01 0.015 0.02

20

10

10

20

ib t( )

inonact t( )

iNonactiveATPl

t t, l dta⋅,

ia(t) – Fryze’s active current ib(t) – Fryze’s nonactive current iact(t) – proposed active current inonact(t) –proposed nonactive current iactiveATP(t) – ATP active current iNonactiveATP(t) – ATP nonactive current

Figure 1.5: Active and non-active current by Fryze, the proposed definition and ATP simulation

Figure 1.5 compares the active and non active currents given by the Fryze definition, the

proposed definition and that obtained from ATP simulation. Using the method of part 1

above, that is, using the non-active current ib(t) = ib1(t) + ib3(t) provided by the

definition to determine passive compensation. The result of computation taking each

harmonic separately is presented as follows


Appendix B B-19

Determine capacitance to compensate fundamental current

ib1rms1T

0 sec⋅

T

tib1c t( )2⌠⎮⌡

d⋅:=ib1rms 9.027551A=

Xb1cV1

ib1rms:= Xb1c 11.077201Ω= C1F

1ω1 Xb1c⋅

:=

C1F 2.873559 10 4−× F=

Determine capacitance to compensate 3rd harmonic current

ib3rms1T

0 sec⋅

T

tib3c t( )2⌠⎮⌡

d⋅:= ib3rms 3.935116A=

Xb3cV3

ib3rms:= Xb3c 7.623664Ω= C3F

1ω3 Xb3c⋅

:=

C3F 1.043822 10 4−× F=

Replacing Cc1 with C1F and Cc3 with C3F in Figure 10 of part 1, the source current isF(t)

obtained is plot together with the active current iaF(t) in the following graph.

0 0.005 0.01 0.015 0.02

500

333.33

166.67

166.67

333.33

500

v t( )

20 isF t( )⋅

20 iaF t( )⋅

t Figure 1.6: Fryze’s compensated source and active currents

The blue waveform is the source current and the pink the Fryze’s active current. The

compensated source current does not match the active current given by the definition.

This indicates that for the series case, the non-active current does no provide correct

information to completely remove non-active power or current. The RMS value of the

compensated source current is 14.708018 amp which is greater than the minimum value

14.448512 obtained by the proposed definition.


Appendix B B-20

The source current after compensation by the proposed defintion isM(t) and by Fryze’s

definition isF(t) is given in the Figure 1.7.

0 0.005 0.01 0.015 0.02

500

333.33

166.67

166.67

333.33

500

v t( )

20 isF t( )⋅

20 isM t( )⋅

t Figure 1.7: Fryze’s (blue) and the proposed (pink)

definition compensated source current

Thus the Fryze’s definition does not accurately predict the active/non-active current for

series load cases. This is because of the assumption that the equivalent conductance of

the circuit is linear during the period. This that is constant parallel equivalent

conductance is not necessarily true of a series circuit in the presence of harmonics.

Conclusion It is concluded from parts 1 and 2 that Fryze’s definition does not provide non-active

power/current information, in the case of series R-L load, to enable optimal

compensation. This is generally true for series loads that have resistance and reactance.

Determination of phase angle nγ

Appendix C C-1

Appendix C Determination of phase angle nγ The phase angle nγ is determined from the fundamental phase angle. To understand

how this angle is derived, consider a simple series and parallel RL circuit as in Figures

C1 and C2.

~v(t) R

~

i(t)

Lmeteringpoint

Figure C1: Series equivalent of power system

~v(t)

R~

i(t)

Lmetering

point

Figure C2: Parallel equivalent of power system

Series Circuit

At any frequency for the series R-L equivalent circuit of a power system,

s sZ R j L= + ω , that is ( )22s sZ R L= + ω and load angle 1 s

sLtan

R− ω⎛ ⎞θ = ⎜ ⎟⎝ ⎠

. For the

fundamental, 1 11

LtanR

− ω⎛ ⎞θ = ⎜ ⎟⎝ ⎠

giving 11

LtanRω

θ = . For the nth

harmonic, 1 1n

n LtanR

− ω⎛ ⎞θ = ⎜ ⎟⎝ ⎠

giving 1 1n 1

n L Ltan n n tanR Rω ω⎛ ⎞θ = = = θ⎜ ⎟

⎝ ⎠. Hence for

harmonic n in the case of series circuit

( )1n 1tan n tan−θ = θ . (C.1)

Parallel Circuit

At any frequency for the parallel R-L equivalent circuit of a power system

pj L RZ

R j Lω

=+ ω

, hence 2 2 2

p 2 2 2

L R R LZ

R Lω +ω

=+ω

and load angle 1p

RtanL

− ⎛ ⎞θ = ⎜ ⎟ω⎝ ⎠

. For

the fundamental 11

1

RtanL

− ⎛ ⎞θ = ⎜ ⎟ω⎝ ⎠

giving 11

RtanL

θ =ω

. For the nth harmonic

Determination of phase angle nγ

Appendix C

C-2

1n

1

Rtann L

− ⎛ ⎞θ = ⎜ ⎟ω⎝ ⎠

giving 1n

1 1

R 1 R tantann L n L n

⎛ ⎞ θθ = = =⎜ ⎟ω ω⎝ ⎠

. Hence for the

harmonic n in the case of a parallel circuit

1 1n

tantann

− θ⎛ ⎞θ = ⎜ ⎟⎝ ⎠

. (C.2)

This angle nθ represents the expected angle between the nth harmonic current and its

non-existent harmonic voltage i.e the phase angle nγ for the harmonic g where vg is

non-existent. It is seen that the phase angle nγ is different for a parallel as against

series circuit. Generally, in an electrical power system, the loads are mixed and exhibit

behaviour that is somewhere between parallel and series when observed from the

measuring point. Since most loads, in a power system, are shunt connected, it is

expected that they behave closer to parallel circuits. Hence the proposal is to use

1 1n

tanθγ tann

− ⎛ ⎞= ⎜ ⎟⎝ ⎠

. (C.3)

Note that the power definitions are independent of the method of determination of nγ .

Hence if knowledge of the parallel/series character of the load is available an

appropriate value of nγ can be used.

Note that in the presence of source impedance there will be a corresponding “non-zero

voltage” (Vg) for Ig. Then the above does not apply, as the phase angle nγ will be

obtained from the Fourier components. This is generally the case in the practical

system. The need for the above relationship is mainly in theoretical study.

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Appendix D

Appendix D

PCC

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Rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr November 2005

Harmonic Filter Overload Investigation

Interim Report

Zzzzzzz Hhhhhh Smelter

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NOTES: The company, supply authority and substation names have been blacked out and replaced as follows: HHHHHHH – The Company HHHHHHH – The Supply Autority HHHHHHH – The Supply Autority Substation No 1 HHHHHHH – The Supply Autority Substation No 2 HHHHHHH – The Supply Autority Substation No 3 HHHHHHH – One of the consultants involved in the investigation

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November 2005

Harmonic Filter Overload Investigation - Interim Report.

1 Introduction. The Zzzzzzz Hhhhhh Smelter operates three Harmonic Filters throughout its 11kV network. Two filters were installed in 1994, along with four new phase-controlled rectifiers. In 1999 a third filter was added. While the initial purpose of these filters to provide a sink for harmonic currents produced by the site’s rectifier fleet, in 1997 they became critical to meeting the reactive power limitations, after the Power Supply Agreement (PSA) was renewed. For the early part of their lives these filters performed well, absorbing harmonic currents and maintaining the site’s VAr demand within the limits prescribed in the new PSA. However in recent years several problems have occurred. Firstly the number of filter protection trips has steadily increased. Such trips are always associated with the 5th leg(s). Secondly, within the last five years four reactors and numerous capacitors have failed; once again all associated with the 5th harmonic elements. Apart from the expense of repair, this damage to the site’s harmonic filters has restricted Zzzzzzz’s ability to adhere to the reactive MD conditions of its PSA. With increased production output the power consumption on site has increased however this has been achieved with existing electrical plant. As no major harmonic generating equipment has been added to the network, Zzzzzzz has for some time been of the opinion that harmonic filter overloads are the result increased in the levels of imported harmonic current due to a steady increase in the background levels of the 5th harmonic at the PCC. Until recently the above postulation has been difficult to confirm, and in an effort to do so, Zzzzzzz has recently purchased a Power Quality Meter data-logger and, has installed a permanent device to monitor the level of 5th harmonic current flowing into one of its three filters. In addition, the latter has been used to partially control the level of the 5th harmonic. This has been achieved by reducing the DC output current of the only 6-pulse rectifier on site, in response to elevated levels of harmonic current flowing into the associated filter. In this way the integrity of at least one filter has been increased, albeit at the expense of zinc production. With a body of supporting evidence, Zzzzzzz recently approached Hhhhhhhh Networks for help. Hhhhhhhh’s engineers immediately appreciated the problem and agreed to undertake a study designed to determine the background levels of the harmonics, (particularly the 5th) at the PCC, as well as to determine the associated system impedances. This interim report has been written as a result of harmonic information supplied to Zzzzzzz by Hhhhhhhh, and research work undertaken within the Zzzzzzz site. It aims to update all participants and to suggest possible project methodology for the Hhhhhhhh study.

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2 Background Harmonic Levels Harmonic distortion information at the PCC supplied by Hhhhhhhh reveals several interesting features, as shown in the figures below. Figure 1 shows THVD at PCC and Figure 2 the 3rd, 5th and 7th harmonic distortion components at the PCC for the period 3 - 10 October ‘05. Comparison of these waveforms shows that the THD is dominated by the 5th harmonic. Further as seen in Figure 2, the level of the 5th is frequently in excess of 2% of the fundamental. Figures 3 and 4 showing THD at Rrrrrr from the 26th Oct to 2nd Nov support the above observations. It was also noted that during this period THVD levels of close to 3.5% were recorded.

Between 3/10/2005 and 10/10/2005

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Figure 1. Total Harmonic Voltage Distortion @ PCC

Between 3/10/2005 and 10/10/2005

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[%]

Time3 Mon

Oct 20054 Tue 5 Wed 6 Thu 7 Fri 8 Sat 9 Sun 10 Mon

0.5

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HU1_3[%] -Temporary meter-XXXXxx HU1_5[%] -Temporary meter-

HU1_7[%] -Temporary meter-

Figure 2. 3rd, 5th & 7th Harmonic Voltage Distortion @ PCC

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Figure 3. 3rd, 5th and 7th Phase Voltage Harmonic Distortion at the PCC

Figure 4. Total Harmonic Phase Voltage Distortion at PCC

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Figure 5 shows the 5th harmonic current inflow to Harmonic Filter #2 at Zzzzzzz over the period from 4 – 10 Oct. Except for a period during the 6th, there is a very strong correlation between the 5th harmonic distortion at the PCC (see Fig 2) and the current flowing into HF#2. This information suggests there is a very strong cause and effect linkage between the harmonics seen at the PCC and those measured at the Zzzzzzz filter.

Figure 5. 5th Harmonic Current Flowing into ZHS Harmonic Filter #2

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3 Zzzzzzz Filter Design and Harmonic Levels The Zzzzzzz harmonic filters were designed in 1993, at a time when AS2279 was the authorative document on harmonic levels. It suggested limits for individual HV system harmonics of 1.5% THD and 1% for any odd harmonic, (Table 1, p8). It was on this basis that the filters were designed; that is a maximum level of 1% at the 5th was assumed to exist at the PCC, and filter capacity was provided accordingly for an inflow of 5th current from the PCC, limited only by the impedance of the Rrrrrr transformers of the day. This assumption proved to be valid for several years. Since then several things in the network have changed. Firstly new transformers were provided as part of with the Rrrrrr substation upgrade. However since the maintenance of the 11kV fault level at Zzzzzzz was paramount in the design, there has been no significant change in the effective transformer impedance. (ZTX ≅11% on a 45MVA base, 3 windings.) Secondly, anecdotal evidence would suggest that there has been a steady increase in the background levels of 5th harmonic distortion since that time. This may be due to the rapid influx of heat pumps into the LV network, (most of which include sizeable AC-DC inverters), or the proliferation of nonlinear electronic devices with switch mode power supplies, or perhaps simply an increase in nonlinear industrial load. In 2001 AS2279 was superseded by AS61000, which saw an upward revision in the recommended maximum harmonic levels, probably in order to reflect reality. This document suggests that the THD on an HV system should be 3% or less while the maximum levels of individual non-triplen odd harmonics should be less than 2%, (Table 2, p3). The snapshot data provided by Hhhhhhhh suggest that at times THD levels exceeds the threshold and the level of the 5th frequently exceeds the specified 2%. Further, the 5th seems now to be the dominant harmonic within the network. When the Zzzzzzz filters were designed they had the ability to control the THD on the 11kV bus by absorbing 5th, 7th, 11th and 13th harmonics, with four proportionally dimensioned tuned LC circuits. Today not only have harmonic levels risen, but also their distribution apparently has changed in favour of the 5th, this makes it very difficult to maintain the THD within limits without a substantial increase in the capacity of the 5th harmonic elements, which Zzzzzzz has done. In 1999 a 3rd filter was added and in 2003 the capacity of the 5th leg of No 1 Harmonic filter was increased by 50%.

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4 Harmonic Current Direction Although the original filter designers considered that Zzzzzzz would be a net importer of harmonic current from the wider network and dimensioned their filters accordingly, thus far importation of harmonics has been difficult to demonstrate. This has led to suggestions that Zzzzzzz may in fact be contributing to the THD at the PCC rather that reducing it. However with the information that has recently become available and a little elementary analysis, Zzzzzzz believes that they can now demonstrate that their filters import harmonic current from the PCC, and therefore assist in reducing the THD there. There are several pieces of information that support this assertion. Firstly as the upper graph in Figure 6 shows, the voltage THD on one Zzzzzzz 11kV bus is close to 1.8%, (whilst that at the PCC is typically around 2.5%). This level remains substantially constant until the harmonic filter on the bus in question trips (due to an overload of the 5th element), late on Tuesday Nov 1st. At this point a sudden increase in THVD is observed since there is now no local sink available to absorb the harmonics produced by the two rectifiers on that bus section. As will be demonstrated, the impedance of the Rrrrrr transformers is too large to allow much harmonic current from Zzzzzzz to flow back to the PCC and therefore the voltage distortion on the Zzzzzzz bus rises to around 5.5%. Secondly, the lower portion of Figure 6 shows the THD of the incoming current in feeder FZ12, which supplies the filter and rectifiers concerned. At the instant that the filter trips the harmonic current distortion falls substantially, since in the absence of the filter, the Zzzzzzz network presents a high impedance to harmonic current inflow from the PCC. The events associated with Figure 6 suggest a method of estimating the system impedance at the PCC to the 5th harmonic. By removing the Zzzzzzz filters one at a time and observing the resulting increase in the level of the 5th at the PCC, an estimate of the system impedance there can be made, provided that the current inflow into Zzzzzzz is known in advance. As will be shown in section 4.1, the transformer impedance alone effectively determines this and thus the Zzzzzzz inflow can be determined by measuring the level of 5th at the PCC a priori. This technique could be incorporated into the project methodology.

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Figure 6. Upper graph: 11kV Voltage THD; Lower Graph: Feeder FZ12 Current THD

j0.12 j1.48

Figure 7. Equivalent Circuit of the Zzzzzzz /Rrrrrr Interface as seen at the Zzzzzzz 11kV bus

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4.1 Fifth Harmonic Equivalent Circuit The impedance of the Rrrrrr transformers effectively prevents the export of much harmonic current from Zzzzzzz and yet permits the import of a portion of harmonic current from the PCC as can be seen by considering the equivalent circuit in Figure 7. This represents the interface between Zzzzzzz and Rrrrrr as seen at the Zzzzzzz 11kV bus. It is essentially the same circuit as presented by Dr Gggggii in his paper on the project methodology. The impedances shown are calculated at the 5th harmonic. The dominant impedance is that of the Rrrrrr transformers (j1.48Ω), and is based on three parallel windings supplying the Zzzzzzz bus, presenting an impedance of 55% on a 45MVA base at the 5th harmonic. The Zzzzzzz 5th harmonic filter's resonant impedance has been recently measured at about 0.2 Ohms, as shown in Figure 8.

Figure 8. Zzzzzzz 5th Harmonic Element Impedance (Magnitude and Phase)

The current source on the right hand side of Figure 7 corresponds to the Zzzzzzz harmonic contribution, which under normal operation is about 60A. The voltage source on the left-hand side represents the background voltage distortion at the PCC, and the associated inductance represents the 5th harmonic system impedance. The principle of superposition can be used to determine how current is imported to or exported from the Zzzzzzz site. Consider the voltage source acting alone, driving harmonic current from the PCC into the Zzzzzzz filter. Because the filter presents such a low impedance, it is the transformer’s impedance that largely determines the magnitude of the current flowing. Consider for example a 1% THD at the PCC. This corresponds to 635 volts at the PCC, or 63.5 volts at Zzzzzzz. The resulting current flowing into the filter is therefore 63.5/1.49 = 43 amps. On the other hand consider the Zzzzzzz current source acting alone in order to determine the current exported from Zzzzzzz to Rrrrrr. The low resonant impedance will result in most of the current flowing into the filter, therefore of the 60 amps generated within Zzzzzzz, only 60*0.2/1.49 = 8 amps will flow back to the PCC. Therefore the net current flowing into the Zzzzzzz filter will be the superposition of these two current components, the magnitude of which will depend on the phase

HF3, 5th, Red Impedance Magnitude

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relationship between the two. If for example they are exactly in phase, the inflow will be about 35 amps, (43-8) and if they are out of phase it will be about 51 amps, (43+8). Because the phase relationship between the two sites will vary significantly with time, the best that can be said is that the magnitude of the imported 5th component will lie somewhere in the range of 30 to 50 amps for every 1% distortion at the PCC. Further, the total current flowing into the Zzzzzzz filters will be the vectorial sum of the imported and the locally generated components. The overall magnitude of which will also be a function of the phase relationship between the two. 5 Zzzzzzz Filter Upgrades It has been apparent to Zzzzzzz for some time that an upgrade of its harmonic filters is necessary. Until recently the difficulty has been in knowing how much capacity should be provided, especially at the 5th harmonic. With all the information now at hand Zzzzzzz believes that this question has been answered and intends to upgrade the capacity of its 5th harmonic elements to a level that will cope with both the site's contribution as well as that imported from Rrrrrr, provided that the level of PCC voltage distortion at the 5th does not exceed 3%. This effectively assumes that the limiting THD at the PCC is exclusively 5th harmonic. Zzzzzzz has chosen to undertake this upgrade in the interest of maintaining low levels of both the THD within their 11kV network and their reactive energy demands. Zzzzzzz therefore requests Hhhhhhhh to recognise that should transient levels of the 5th rise beyond 3% at the PCC the Zzzzzzz protection will remove the filters from service and thus it will not be possible to maintain its VAr MD within the PSA limits in the short term. Zzzzzzz is also concerned that the system 5th harmonic impedance at the PCC may be found to be quite low. This assertion is based on the high levels of 5th voltage distortion already seen at the PCC despite the (occasional?) presence of the Rrrrrr, Nnnnnnnnnn or Cccccc St capacitors, each of which is likely to present a reasonably low shunt impedance at the 5th. Should this turn out to be the case, then the increased capacity of the Zzzzzzz filters will not significantly reduce the THD at the PCC. Conversely however, future increases in average harmonic levels at the PCC will generate greater harmonic inflows to Zzzzzzz resulting in chronic filter overloads. In order to avoid this situation Zzzzzzz seeks from Hhhhhhhh an understanding that future harmonic distortion levels at the PCC will not be permitted to exceed the limits prescribed in AS61000. This being the case, then the proposed filter upgrade will provide sufficient capacity to enable Zzzzzzz to maintain low levels of THD across its site while simultaneously adhering to the reactive demand limits imposed by the PSA.

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6 Estimation of the System Impedance at the 5th Harmonic. As mentioned earlier, Figure 3 shows the typical daily variations in the 5th harmonic as seen on the Rrrrrr 110kV bus. This waveform is mirrored closely by that observed on the Zzzzzzz site. The random variations in background level are punctuated with sudden step changes in amplitude, which the authors have suspected may be due to capacitor bank switching on the 110kV network. Recently it has been possible to compare the switching of some of the Southern Region Capacitors with the level of 5th voltage distortion at the PCC, as shown below in Figure 9. Here the upper graph shows the VAr loading of several of the southern capacitor banks, while the lower one shows the 5th harmonic voltage distortion at the PCC, over the same time interval.

Figure 9: Upper Graph: Southern Region Cap bank Switching Lower Graph: 5th Harmonic Voltage Distortion at Rrrrrr 110kV Bus

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As expected, the step changes in the level of the 5th do correspond with the switching of local capacitors. For example a drop in the level of the 5th of about 0.7% occurs when one of the Cccccc St capacitor is switched on, (Figure 9, left hand side). The level of the 5th is briefly restored when this capacitor is removed; however it falls again when the other Cccccc St capacitor is energised instead. This effect, together with a knowledge of the design of the Cccccc St capacitor installation, can be used to obtain an estimate of the 5th harmonic system impedance as seen from the 11kV bus, (see Figure 7). By noting the drop in the 5th as a result of bringing a known impedance onto the bus, one can estimate the system impedance at this frequency. Calculations show that this is about j0.12 Ohms, when referred to the 11kV bus. This value is reasonably low, as had been suspected. 6 Capacitor Bank Tuning During this work a partial reason for the frequently excessive levels of the 5th became apparent. Clearly the presence of either of the Cccccc St capacitors assists in reducing the background level of the 5th; however the same cannot be said for the Rrrrrr capacitors. As shown by the blue trace (upper graph) when a Rrrrrr capacitor is energised the 5th harmonic voltage distortion actually increases by about 0.4%. This is quite significant, since with both Rrrrrr capacitors energised, the level of the 5th will be about 0.8% higher than it might otherwise have been. Further, with the background levels already frequently in excess of the planning level (2%), such an elevation is unacceptable. Why should one set of capacitors behave so differently to another? The answer to this can be found by examining the design of the individual capacitor installations. Each installation has detuning reactors fitted. These are provided for two reasons; firstly to limit the inrush current at switch-on and secondly to ensure that series resonance does not occur at a frequency associated with a system harmonic. In the case of the Cccccc St capacitors, series resonance occurs at 203Hz while the Rrrrrr capacitors are resonant at 2580Hz. The difference between these frequencies is very significant and it explains the difference in behaviour when these capacitors are connected to the 110kV bus. The Cccccc St resonance lies below the 5th harmonic (ie 250Hz), therefore this circuit presents an inductive impedance at the 5th. Because the system impedance at the 5th is also inductive, the Cccccc St capacitor bank will cause an attenuation of the 5th component at the PCC, (as observed). On the other hand the Rrrrrr capacitor installation is resonant well above 250Hz. As a result it presents a capacitive impedance at this frequency, which generates an increase in the 5th voltage seen at the PCC, in exactly the same way as capacitive support increases the voltage at the fundamental. In summary, the tuning of the Rrrrrr capacitors is not ideal from the point of view of reducing the levels of the 5th harmonic within the 110kV network, while the Cccccc St installations are well suited in this respect.

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In order to rectify this situation there are two things that can be done. Firstly, in the short term the use of the Rrrrrr capacitors should be avoided, especially both at once. Secondly in the longer term, consideration should be given to retuning the installation at or below the 5th harmonic. This would involve replacing the reactors with devices whose inductance is considerably higher so that the resonant frequency can be lowered. A resonant frequency of 250Hz would be the most appropriate as this would significantly assist in reducing the level of the 5th harmonic and therefore the THD at the PCC as well.

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generalised power components definitions …khalsarc.com/khalsarc/measurements_files/hk thesis rev...

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