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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
7.4.2 Application Example 2 ……………….………………….…….. 210
7.4.3 Summary – some rules on compensation .……………….…….. 222
7.5 Detection of source of distortion ……………….……………….. 223
7.5.1 Application Example 1 ……………….………………….…….. 223
7.5.2 Application Example 2 ……………….………………….…….. 231
7.6 Power quality ……………….………………….……………….. 237
7.6.1 Application Example 1 ……………….………………….…….. 238
7.6.2 Application Example 2 ……………….………………….…….. 240
7.6.3 Summary – some comments on power quality ………………… 241
7.7 General ……………….………………….………………….…… 241
7.7.1 Application Example 1 ……………….………………….…….. 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
GENERALISED POWER COMPONENTS DEFINITIONS FOR SINGLE AND THREE-PHASE ELECTRICAL
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
GENERALISED POWER COMPONENTS DEFINITIONS FOR SINGLE AND THREE-PHASE ELECTRICAL
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].
Current power theories
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)*
Current power theories
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)
Current power theories
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∑
Current power theories
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
2 V I cos(m t )cos(n t )≠==
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
2 V I cos(m t )cos(n t )≠==
ω −α ω −β∑ (2.9)
(f) sXg(t) := m n m nm gm 1,h,gn g
2 V I cos(m t )cos(n t )≠==
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.
Current power theories
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.
Current power theories
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)
Current power theories
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.
Current power theories
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)
Current power theories
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
Current power theories
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
Current power theories
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
1 1 1P v i dt v i dt v i dtT T T
= + +∫ ∫ ∫
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
1 1 1P v i dt v i dt v i dtT T T
= + +∫ ∫ ∫
= 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= − ,
Current power theories
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)
Current power theories
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.
Current power theories
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
Current power theories
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.
Current power theories
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.
Current power theories
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)
Current power theories
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.
Current power theories
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?
Current power theories
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
Current power theories
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.
Current power theories
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).
Current power theories
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)
Current power theories
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,
Current power theories
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
Current power theories
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
Current power theories
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
Current power theories
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
Current power theories
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).
Current power theories
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.
Current power theories
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].
Current power theories
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)
Current power theories
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.
Current power theories
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
Current power theories
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.
Current power theories
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
Current power theories
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
Current power theories
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 non- 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 (2.79) 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.
Current power theories
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
Current power theories
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:
Current power theories
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.
Current power theories
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
Current power theories
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.
Reactive current is given by
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
Current power theories
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
Current power theories
Chapter 2 58
2s ne e n
n 1,h
I (G G ) V=
= −∑
. (2.109)
Reactive current is given by
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)
Current power theories
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.
Current power theories
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
Current power theories
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
Current power theories
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
Current power theories
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
Current power theories
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)
Current power theories
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
Current power theories
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.
Current power theories
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
Current power theories
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
Current power theories
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
Current power theories
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)
Current power theories
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.
Current power theories
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
Current power theories
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
+ + ,
Current power theories
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)
Current power theories
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.
Current power theories
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,
Requirements on a power definition and benchmark case studies for evaluation
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
Requirements on a power definition and case studies for evaluation
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,
Requirements on a power definition and benchmark case studies for evaluation
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
Requirements on a power definition and case studies for evaluation
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
Requirements on a power definition and benchmark case studies for evaluation
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
Requirements on a power definition and case studies for evaluation
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.
Requirements on a power definition and benchmark case studies for evaluation
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
Requirements on a power definition and case studies for evaluation
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
Requirements on a power definition and benchmark case studies for evaluation
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.
Requirements on a power definition and case studies for evaluation
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
Requirements on a power definition and benchmark case studies for evaluation
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
Requirements on a power definition and case studies for evaluation
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
Requirements on a power definition and benchmark case studies for evaluation
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
Requirements on a power definition and case studies for evaluation
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.
Requirements on a power definition and benchmark case studies for evaluation
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
Figure 3.14: 1-Phase Load Case S4
~v(t)
L~
i(t)
meteringpoint
Figure 3.15: 1-Phase Load Case S5
C
~v(t) R
~
i(t)
Lmeteringpoint
Figure 3.16: 1-Phase series R-L-C Load Case S6
Requirements on a power definition and case studies for evaluation
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.
Requirements on a power definition and benchmark case studies for evaluation
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
Requirements on a power definition and case studies for evaluation
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
Requirements on a power definition and benchmark case studies for evaluation
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)
Requirements on a power definition and case studies for evaluation
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.
Requirements on a power definition and benchmark case studies for evaluation
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)
Requirements on a power definition and case studies for evaluation
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
Requirements on a power definition and benchmark case studies for evaluation
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.
Requirements on a power definition and case studies for evaluation
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( )⋅
t. Figure 3.23: Case S2 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
50
25
25
50
Non-active power
qEXP t( )
t
Figure 3.24: Case S2 – Expected active and nonactive powers
Requirements on a power definition and benchmark case studies for evaluation
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( )⋅
t. Figure 3.27: Case S4 Voltage and currents, total i(t) active ip(t) and nonactive 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
Requirements on a power definition and case studies for evaluation
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( )⋅
t Figure 3.29: Case S5 Voltage and currents, total i(t) active ip(t) and nonactive 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( )⋅
t Figure 3.31: Case S6 Voltage and currents, total i(t) active ip(t) and nonactive iq(t)
Requirements on a power definition and benchmark case studies for evaluation
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
Case S4 Active (EP) 1.443994 W sec
Nonactive (EN) 0 Var sec Case S5
Active (EP) 0 W sec Nonactive (EN) 0.585229 Var sec
Case S6 Active (EP) 29.038735 W 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
Requirements on a power definition and case studies for evaluation
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.
Requirements on a power definition and benchmark case studies for evaluation
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
The vector diagram for the fundamental is shown in Figure 3.34.
~~~~
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)
Requirements on a power definition and case studies for evaluation
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
The vector diagram for the fundamental is shown in Figure 3.35.
~
~
~~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).
Requirements on a power definition and benchmark case studies for evaluation
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
V 2i (t) 2 sin h tR 3=
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
The vector diagram for the fundamental is shown in Figure 3.36.
~
~
~~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
V 2i (t) 2 sin h tR 3=
π⎡ ⎤= ω −⎢ ⎥⎣ ⎦∑ (3.54)
Requirements on a power definition and case studies for evaluation
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
V 2i (t) 2 sin h tR 3=
π⎡ ⎤= ω +⎢ ⎥⎣ ⎦∑ (3.56)
cL c
131 1
h 1
2i (t) 2 h C sin h t3=
π⎡ ⎤= ω ω +⎢ ⎥⎣ ⎦∑ . (3.57)
Case T4: 3Ph 4W with unsymmetrical source voltage and unbalanced star load
The vector diagram for the fundamental is shown in Figure 3.37.
~
~
~~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)
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⎡ ⎤⎣ ⎦∑ (3.58)
a
13h
aL 11h 1
Vi (t) 2 sin h t
h L 2=
π⎡ ⎤= ω −⎢ ⎥ω ⎣ ⎦∑ (3.59)
b
13h
1bRh 1
V 2i (t) 2 sin h tR 3=
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)
Requirements on a power definition and benchmark case studies for evaluation
Chapter 3 111
Case T5: 3Ph 4W with symmetrical source voltage and nonlinear unbalanced star load
The vector diagram for the fundamental is shown in Figure 3.38.
~
~
~~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)
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⎡ ⎤⎣ ⎦∑ if Van>0, 0 otherwis (3.64)
aLi (t) 0= (3.65)
b
13h
1bh 1
V 2i (t) 2 sin h tR 3=
π⎡ ⎤= ω −⎢ ⎥⎣ ⎦∑ if Vbn >0, 0 otherwise (3.66)
bLi (t) 0= (3.67)
c
13h
c 1h 1
V 2i (t) 2 sin h tR 3=
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:
Requirements on a power definition and case studies for evaluation
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
Requirements on a power definition and benchmark case studies for evaluation
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
Requirements on a power definition and case studies for evaluation
Chapter 3 114
Case T2: 3Ph 4W with unsymmetrical source voltage and unbalanced star load
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
.
Figure 3.43: Case T2 Voltage and current
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
Requirements on a power definition and benchmark case studies for evaluation
Chapter 3 115
Case T3: 3Ph 4W with symmetrical source voltage and unbalanced star load
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
.
Figure 3.45: Case T3 Voltage and current
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
Figure 3.46: Case T3 Expected active and nonactive powers
Requirements on a power definition and case studies for evaluation
Chapter 3 116
Case T4: 3Ph 4W with unsymmetrical source voltage and unbalanced star load
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
.
Figure 3.47: Case T4 Voltage and current
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 3.48: Case T4 Expected active and nonactive powers
Requirements on a power definition and benchmark case studies for evaluation
Chapter 3 117
Case T5: 3Ph 4W with symmetrical source voltage and nonlinear unbalanced star load
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
.
Figure 3.49: Case T5 Voltage and current
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.
Requirements on a power definition and case studies for evaluation
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
Single phase power component definitions for instantaneous and average power
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
Single phase power component definitions for instantaneous and average power
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.
Single phase power component definitions for instantaneous and average power
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.
Single phase power component definitions for instantaneous and average power
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)
Single phase power component definitions for instantaneous and average power
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
Single phase power component definitions for instantaneous and average power
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
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)
Single phase power component definitions for instantaneous and average power
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.
Single phase power component definitions for instantaneous and average power
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)
Single phase power component definitions for instantaneous and average power
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∑
Single phase power component definitions for instantaneous and average power
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
2 V I cos(m t )cos(n t )≠==
ω −α ω −β∑ (4.18)
(e2) Cross-harmonic powers
sXh (t) := m n m nm nm 1,h,gn h
2 V I cos(m t )cos(n t )≠==
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
2 V I cos(m t )cos(n t )≠==
ω −α ω −β∑ (4.20)
Single phase power component definitions for instantaneous and average power
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)
Single phase power component definitions for instantaneous and average power
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)
Single phase power component definitions for instantaneous and average power
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).
Single phase power component definitions for instantaneous and average power
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.
Single phase power component definitions for instantaneous and average power
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)
Single phase power component definitions for instantaneous and average power
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.
Single phase power component definitions for instantaneous and average power
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.
Single phase power component definitions for instantaneous and average power
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
Single phase power component definitions for instantaneous and average power
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
Single phase power component definitions for instantaneous and average 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.
Single phase power component definitions for instantaneous and average 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.
Single phase power component definitions for instantaneous and average power
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
Single phase power component definitions for instantaneous and average power
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).
Single phase power component definitions for instantaneous and average power
Chapter 4 143
4.4.2 Results of computation The voltages, currents and instantaneous powers are presented in graphical format. 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 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
Single phase power component definitions for instantaneous and average power
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
Single phase power component definitions for instantaneous and average power
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
Single phase power component definitions for instantaneous and average power
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.
Single phase power component definitions for instantaneous and average power
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
Single phase power component definitions for instantaneous and average power
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.
Single phase power component definitions for instantaneous and average power
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
Single phase power component definitions for instantaneous and average power
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)
Single phase power component definitions for instantaneous and average power
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
Single phase power component definitions for instantaneous and average power
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
Choice of reference conductor in three phase systems
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.
Choice of reference conductor in three phase systems
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.
Choice of reference conductor in three phase systems
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
Choice of reference conductor in three phase systems
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
Choice of reference conductor in three phase systems
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)
Choice of reference conductor in three phase systems
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)
Choice of reference conductor in three phase systems
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
(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”.
5.3.4 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
Choice of reference conductor in three phase systems
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
(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 “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.
Choice of reference conductor in three phase systems
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
Choice of reference conductor in three phase systems
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
Choice of reference conductor in three phase systems
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
Choice of reference conductor in three phase systems
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
Choice of reference conductor in three phase systems
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.
Choice of reference conductor in 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.
Choice of reference conductor in three phase systems
Chapter 5 168
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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.
Three phase power component definitions for instantaneous and average powers
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
Three phase power component definitions for instantaneous and average powers
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
Three phase power component definitions for instantaneous and average 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
Three phase power component definitions for instantaneous and average powers
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)
Three phase power component definitions for instantaneous and average powers
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
⎧ − − >⎪= ⎨⎪⎩
Three phase power component definitions for instantaneous and average powers
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
Three phase power component definitions for instantaneous and average powers
Chapter 6
176
3PhComp PhCompAll Phases FOR EACH POWER COMPONENT
p (t) : p (t)= ∑ (6.22)
3PhComp PhCompAll Phases FOR EACH POWER COMPONENT
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.
Three phase power component definitions for instantaneous and average powers
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
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 powers obtained by the proposed
definition are shown on the same graphs with the expected to enable easy comparison.
6.4.2.1 Waveforms
Three phase power component definitions for instantaneous and average powers
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
Three phase power component definitions for instantaneous and average 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
Three phase power component definitions for instantaneous and average powers
Chapter 6
180
Case T2: 3Ph 4W with unsymmetrical source voltage and unbalanced star load
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
Figure 6.5: Case T2 (Virtual neutral as reference) Voltages, currents, proposed and expected active and non-active powers
Three phase power component definitions for instantaneous and average powers
Chapter 6 181
Case T3: 3Ph 4W with symmetrical source voltage and unbalanced star load
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
Three phase power component definitions for instantaneous and average powers
Chapter 6
182
Case T4: 3Ph 4W with unsymmetrical source voltage and unbalanced star load
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
Figure 6.7: Case T4 (Virtual neutral as reference) Voltages, currents, proposed and expected active and non-active powers
Three phase power component definitions for instantaneous and average powers
Chapter 6 183
Case T5: 3Ph 4W with symmetrical source voltage and nonlinear unbalanced star load
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.
Three phase power component definitions for instantaneous and average powers
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
Three phase power component definitions for instantaneous and average powers
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)
Three phase power component definitions for instantaneous and average powers
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
Three phase power component definitions for instantaneous and average 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
Three phase power component definitions for instantaneous and average 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
Three phase power component definitions for instantaneous and average 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
Three phase power component definitions for instantaneous and average powers
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
Three phase power component definitions for instantaneous and average powers
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
Three phase power component definitions for instantaneous and average powers
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
Three phase power component definitions for instantaneous and average powers
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.
Three phase power component definitions for instantaneous and average powers
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.
Three phase power component definitions for instantaneous and average powers
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.
Three phase power component definitions for instantaneous and average powers
Chapter 6
196
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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
Applications of the definitions
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.
Applications of the definitions
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
Applications of the definitions
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
Applications of the definitions
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
xrms FOR EACH HARMONIC x
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)
Applications of the definitions
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.
Applications of the definitions
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.
Applications of the definitions
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)
Applications of the definitions
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.
Applications of the definitions
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.
Applications of the definitions
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
Applications of the definitions
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
Applications of the definitions
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
Applications of the definitions
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.
7.4.2 Application example 7.4.2 Introduction
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
Applications of the definitions
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)
Applications of the definitions
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
Applications of the definitions
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.
Applications of the definitions
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
Applications of the definitions
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.
Applications of the definitions
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.
Applications of the definitions
Chapter 7 217
(f ile ExDRL3.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 7.18: 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
Applications of the definitions
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
Applications of the definitions
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
(f ile ExDRL4.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 7.22: Voltage and current waveforms after compensation of fundamental nonactive power plus load-generated Dc and harmonic currents
Applications of the definitions
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
Applications of the definitions
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
Applications of the definitions
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
Applications of the definitions
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.
7.5.1 Application example 7.5.1 Introduction
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.
Applications of the definitions
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.
Applications of the definitions
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.
Applications of the definitions
Chapter 7 226
0.08 0.085 0.09 0.095 0.1
400
200
200
400
vload
5iResistor
t Metering point Voltage and current
0.08 0.085 0.09 0.095 0.1
40
20
20
40
s0Dk
p0Dk
q0Dk
0.1 vk⋅
0.5 ik⋅
k dt⋅ DC power (a1)
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⋅
k dt⋅ Fundamental powers (b)
0.08 0.085 0.09 0.095 0.1
50
50
100
shk
phk
qhk
0.1 vk⋅
0.5 ik⋅
k dt⋅ Source generated harmonic powers (c)
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)
Applications of the definitions
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⋅
k dt⋅ Load generated cross-harmonic powers (f)
0.08 0.085 0.09 0.095 0.1
2000
2000
4000
6000
8000
s k
p k
qk
5 vk
50 ik⋅
k dt⋅ Total powers
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.
Applications of the definitions
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
t Metering point Voltage and current
0.08 0.085 0.09 0.095 0.1
40
20
20
40
s0Dk
p0Dk
q0Dk
0.1 vk⋅
0.5 ik⋅
k dt⋅ DC power (a1)
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⋅
k dt⋅ Fundamental powers (b)
0.08 0.085 0.09 0.095 0.1
50
50
shk
phk
qhk
0.1 vk⋅
0.5 ik⋅
k dt⋅ Source generated harmonic powers (c)
0.08 0.085 0.09 0.095 0.1
200
100
100
200
sgk
pgk
qgk
0.001 vk⋅
0.005 ik⋅
k dt⋅ Load generated harmonic powers (d)
Applications of the definitions
Chapter 7 229
0.08 0.085 0.09 0.095 0.1
4000
2000
2000
4000
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
40
20
20
40
sXhk
pXhk
qXhk
0.1 vk
0.5 ik
k dt⋅ Source generated cross-harmonic powers (e2)
0.08 0.085 0.09 0.095 0.1
2000
500
1000
2500
4000
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
1 .104
1 .104
2 .104
sk
p k
qk
5 vk
50 ik⋅
k dt⋅ Total powers
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
Applications of the definitions
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.
Applications of the definitions
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.
Application example conclusion
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.
7.5.2 Application example 7.5.2 Introduction
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
Applications of the definitions
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⋅
t Metering point Voltage and current
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⋅
k dt⋅ DC power (a1)
Applications of the definitions
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⋅
k dt⋅ Source generated harmonic powers (c)
0.08 0.085 0.09 0.095 0.1
500
500sgk
pgk
qgk
0.001 vk⋅
0.005 ik⋅
k dt⋅ Load generated harmonic powers (d)
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
k dt⋅ Source generated cross-fundamental powers (e1)
0.08 0.085 0.09 0.095 0.1
5 .105
5 .105
sXhk
pXhk
qXhk
2 vk
10 ik
k dt⋅ Source generated cross-harmonic powers (e2)
Applications of the definitions
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⋅
k dt⋅ Total powers
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
Applications of the definitions
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.
Applications of the definitions
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.
Application example conclusion
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.
Applications of the definitions
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.
Applications of the definitions
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.
Applications of the definitions
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
Applications of the definitions
Chapter 7 240
and nonactive powers. The active and nonactive powers could then be considered to
find out which is the worst contributor.
Application example conclusion
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.
7.6.2 Application example 7.6.2 Introduction
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.
Applications of the definitions
Chapter 7 241
Application example conclusion
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
Applications of the definitions
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).
Applications of the definitions
Chapter 7 243
0 0.01 0.02 0.03 0.04
2
4
6
8
10
vs t( )
is t( )
t Metering point Voltage and current
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
Applications of the definitions
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
Applications of the definitions
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
Applications of the definitions
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
Relationship of the proposed definitions with existing some definitions
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).
Relationship of the proposed definitions with existing some definitions
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
Relationship of the proposed definitions with existing some definitions
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)
Relationship of the proposed definitions with existing some definitions
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.
Relationship of the proposed definitions with existing some definitions
Chapter 8 252
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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.
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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
Comparison of the proposed definition and Fryze’s definition
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.
Comparison of the proposed definition and Fryze’s definition
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
Comparison of the proposed definition and Fryze’s definition
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)
Comparison of the proposed definition and Fryze’s definition
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.
Comparison of the proposed definition and Fryze’s definition
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”.
Comparison of the proposed definition and Fryze’s definition
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
xrms FOR EACH HARMONIC x
iVω
or Lc= xrms
xrms FOR EACH HARMONIC x
VIω
. (B.11)
Comparison of the proposed definition and Fryze’s definition
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.
Comparison of the proposed definition and Fryze’s definition
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
Comparison of the proposed definition and Fryze’s definition
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
Comparison of the proposed definition and Fryze’s definition
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
Comparison of the proposed definition and Fryze’s definition
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
Comparison of the proposed definition and Fryze’s definition
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.
Comparison of the proposed definition and Fryze’s definition
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.
Comparison of the proposed definition and Fryze’s definition
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.
Comparison of the proposed definition and Fryze’s definition
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
Comparison of the proposed definition and Fryze’s definition
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.
Comparison of the proposed definition and Fryze’s definition
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
Comparison of the proposed definition and Fryze’s definition
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.
Comparison of the proposed definition and Fryze’s 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.
Report of company CO
Appendix D
Appendix D
PCC
0.02
0.02
50.
030.
035
0.04
1. 1
04
5000
5000
1. 1
04
v s 10i s⋅
t
App
licat
ion
Cas
e: S
ourc
e of
dis
tort
ion
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
Unipower PQSecure (C)
THD
Volta
ge[%
]
Time3 Mon
Oct 20054 Tue 5 Wed 6 Thu 7 Fri 8 Sat 9 Sun 10 Mon
1.0
1.5
2.0
2.5
3.0THDF_U1 THDF_U2 THDF_U3
Figure 1. Total Harmonic Voltage Distortion @ PCC
Between 3/10/2005 and 10/10/2005
Unipower PQSecure (C)
[%]
Time3 Mon
Oct 20054 Tue 5 Wed 6 Thu 7 Fri 8 Sat 9 Sun 10 Mon
0.5
1.0
1.5
2.0
2.5
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
00.20.40.60.8
11.21.4
230 235 240 245 250 255 260
Frequency (Hz)
Impe
danc
e (O
hms)
HF, 5th, Red Impedance Phase
-100
-50
0
50
100
230 235 240 245 250 255 260
Frequency (Hz)
Phas
e (D
egre
es)
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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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