Contributions towards the development of the Technical ...
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University of Wollongong Thesis Collections
University of Wollongong Thesis Collection
University of Wollongong Year
Contributions towards the development
of the Technical Report IEC/TR
61000-3-13 on voltage unbalance emission
allocation
Prabodha ParanavithanaUniversity of Wollongong
Paranavithana, Prabodha, Contributions towards the development of the Technical ReportIEC/TR 61000-3-13 on voltage unbalance emission allocation, PhD thesis, School of Elec-trical, Computer and Telecommunications Engineering, University of Wollongong, 2009.http://ro.uow.edu.au/theses/834
This paper is posted at Research Online.
http://ro.uow.edu.au/theses/834
Contributions Towards the Development of the
Technical Report IEC/TR 61000-3-13 on Voltage
Unbalance Emission Allocation
A thesis submitted in fulfilment of the
requirements for the award of the degree
Doctor of Philosophy
from
University of Wollongong
by
Prabodha Paranavithana, BSc(Eng)
School of Electrical, Computer and Telecommunications
Engineering
March 2009
Acknowledgements
It is a pleasure to be able to thank many people to whom I am indebted for the
development of this thesis.
First and foremost, I wish to express my utmost gratitude to my principal super-
visor, Associate Professor Sarath Perera of the University of Wollongong (UoW), for
enabling me to pursue postgraduate studies at the University of Wollongong and the
support given throughout the study period in many ways. Your dedication, patience,
knowledge and experience could not have been surpassed. I admire your guidance
towards growing me up academically and personally over last few years.
Thanks to my co-supervisor, Professor Danny Sutanto of the UoW, for the assis-
tance provided. I would also like to offer many appreciations to Dr. Duane Robinson
of Beca, Australia for proofreading this thesis. To Mr. Robert Koch of Eskom
Holdings Limited, South Africa and Dr. Zia Emin of National Grid Electricity Trans-
mission, United Kingdom go many thanks for their insightful technical contributions
and helpful attitude. LATEX assistance received from Dr. Timothy Browne, previ-
ously with the Integral Energy Power Quality and Reliability Centre (IEPQRC) at
the UoW, is much appreciated.
Funding for this project was provided by SP AusNet, Victoria and the IEPQRC.
I am grateful to Mr. Dhammika Adihetti, Mr. Shiva Bellur and Mr. Sanath Peiris of
SP AusNet for arraigning this. Many thanks to Mr. Jeff Sultana, Mr. Shem Cardosa
and Mr. Mahinda Wickramasuriya of SP AusNet for the support given in collecting
the required data for Chapter 7 of this thesis.
Thanks to Dr. Vic Smith and Sean Elphick of the IEPQRC who have graciously
responded to many administrative and software related requests. My thanks also go to
Roslyn Causer-Temby of the School of Electrical, Computer and Telecommunications
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Engineering (SECTE) at the UoW, Tracey O’Keefe and Maree Burnett who are
former members of the SECTE staff, and Esperanza Riley of the IEPQRC for solving
many administrative problems and providing perspective. The SECTE workshop
staff have cheerfully provided the technical assistance.
Very special thanks go to my friend Dr. Sankika Tennakoon, previously with the
IEPQRC, for being generously supportive especially during hard times along the way.
Your contribution to my PhD experience is also appreciated.
My heartiest gratitude goes to my parents Mithrananda and Manike for all encour-
agements, guidance and sacrifices made on behalf of me to come this far. Finally, my
thanks go to the rest of my family and friends particularly Pinky, Dimuthu, Radley,
Matthew and Nishad for being supportive in many ways.
Certification
I, Prabodha Paranavithana, declare that this thesis, submitted in fulfilment of the
requirements for the award of Doctor of Philosophy, in the School of Electrical, Com-
puter and Telecommunications Engineering, University of Wollongong, is entirely my
own work unless otherwise referenced or acknowledged. This manuscript has not been
submitted for qualifications at any other academic institute.
Prabodha Paranavithana
Date: 31 March 2009
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Abstract
Although voltage unbalance is a well understood concept, its presence as a power
quality problem in electricity transmission and distribution networks has continued
to be an issue of concerns primarily due to difficulties found by some network service
providers in maintaining acceptable levels. This emphasises the lack of recommenda-
tions on engineering practices governing voltage unbalance that would facilitate the
provision of adequate supply quality to connected customers.
The International Electrotechnical Commission (IEC) has recently released the
Technical Report IEC/TR 61000-3-13 which provides guiding principles for coordi-
nating voltage unbalance between various voltage levels of a power system through
the allocation of emission limits to installations. Although the IEC report is based
on widely accepted basic concepts and principles, it requires refinements and original
developments in relation to some of the key aspects. This thesis primarily focuses on
making contributions for further improvements to the IEC report so as to present a
more comprehensive voltage unbalance allocation procedure.
Similar to the counterpart IEC guidelines for harmonics (IEC 61000-3-6) and
flicker (IEC 61000-3-7) allocation, IEC/TR 61000-3-13 also apportions the global
emission allowance to an installation in proportion to the ratio between the agreed
apparent power, and the total available apparent power of the system seen at the
busbar where it is connected. However, noting that voltage unbalance at a busbar
can arise as a result of both load and system (essentially lines) asymmetries, IEC/TR
61000-3-13 applies an additional factor which is referred to as ‘Kue’ to the appor-
tioned allowance. This factor Kue represents the fraction of the global emission
allowance that can be allocated to customers, whereas the factor K ′ue (= 1−Kue)
accounts for voltage unbalance which arises as a result of line asymmetries. Although
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IEC/TR 61000-3-13 recommends system operators to assess the factors Kue and
K ′ue for prevailing system conditions, a systematic method for its evaluation is not
provided other than a rudimentary direction. This thesis initially examines, employ-
ing radial systems, the influence of line asymmetries on the global emission levels
in medium voltage (MV) and high voltage (HV) power systems in the presence of
various load types/bases including three-phase induction motors. It is shown that
the factor K ′ue is seen to be dependant not only on line parameters as evident from
IEC/TR 61000-3-13, but also on the downstream load composition. In essence, the
global emission levels in HV power systems is seen to arise as a result of both the local
HV lines and the downstream MV lines in the presence of considerable proportions of
induction motor loads. Eventually, generalised methodologies, covering both radial
and interconnected networks, for the assessment of the global emission in MV and
HV power systems which arises due to line asymmetries are proposed.
In allocating voltage unbalance based on the IEC/TR 61000-3-13 recommenda-
tions, quantitative measures of its propagation from higher voltage to lower voltage
levels in terms of transfer coefficients, and from one busbar to other neighbouring bus-
bar of a sub-system in terms of influence coefficients are required. IEC/TR 61000-3-13
gives a method for evaluating the MV to LV transfer coefficient suggesting a value
less than unity for industrial load bases containing large proportions of mains con-
nected three-phase induction motors, and a value of unity for passive loads in general.
Upon detailed examination, it is noted that a transfer coefficient > 1 can arise in the
presence of commonly prevailing constant power loads. Incorporating these different
influences exhibited by various load types under unbalanced supply conditions on the
propagation, comprehensive methods for assessing the MV to LV and HV to MV
transfer coefficients are proposed. A systematic approach for estimating influence
coefficients for interconnected network environments taking their dependency on the
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downstream load composition into account is developed.
The IEC allocation policy with regard to harmonics and flicker has been found not
to guarantee that the emission limits allocated to customers ensure non-exceedance
of the set planning levels. This thesis reports that the above is an issue with voltage
unbalance as well. Overcoming this problem, an alternative allocation technique
referred to as ‘constraint bus voltage’ (CBV) method which closely aligns with the
IEC approach has been suggested for harmonics and flicker. The work presented in
this thesis extends the suggested CBV method to voltage unbalance allocation adding
appropriate revisions to address the additional aspect of the emission which arises as
a result of line asymmetries.
In the application of the IEC/TR 61000-3-13 principles to better manage existing
networks already experiencing excessive voltage unbalance levels, the initial develop-
ment of insights into the influences made by various sources of unbalance is required.
Employing an existing 66kV interconnected sub-transmission system as the study
case, deterministic studies are carried out in a systematic manner considering each of
the asymmetrical elements. Approaches for studying the voltage unbalance behaviour
exhibited by various sources which exist in interconnected network environments are
established. These are employed to identify the most favourable line transposition
options for the study system. Further, this knowledge that facilitates the identifi-
cation of contributions made by individual unbalanced sources forms a platform for
developing techniques to assess the compliance with emission limits, which is another
subject of relevance to future editions of IEC/TR 61000-3-13.
As an essential tool for carrying out the studies, an unbalanced load flow program
based on the phase coordinate reference frame incorporating the component level load
flow constraints and the three-phase modelling of system components is developed.
List of Principal Symbols and Abbreviations
a, b, c refer to the three phases
α summation law exponent
CBV constraint bus voltage
CIGRE International Council on Large Electric Systems
CIRED International Conference on Electricity Distribution
Es:x emission limit of any busbar x of any sub-system S [VUF]
Es:x−j emission limit of any installation j to be connected at any
busbar x of any sub-system S [VUF]
EHV extra high voltage
hm refers to a HV-MV coupling transformer
HV high voltage
I refers to a constant current load
[I] matrix of nodal currents
Iλ:t λ (= 0, +,−) sequence current in any line t [A]
Iλ:x λ (= 0, +,−) sequence component of Ix [A]
Ix nodal current at any busbar x [A]
I−:c/e negative sequence current in any system element e (e = t, tf, busbar x)
caused by any source of unbalance c (c = t, td, lines, Ux) [A]
IEC International Electrotechnical Commission
IEEE Institute of Electrical and Electronics Engineers
IM refers to a three-phase induction motor load
ka allocation constant
ki−x influence coefficient from any busbar i to any other busbar x
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x
klv fraction of LV loads supplied by any higher voltage (MV, HV) busbar
km ratio between the rated motor load (in MVA) and the total
load (in MVA) supplied by an LV system
kmmv ratio between the rated motor load (in MVA) and the total
load (in MVA) supplied by an MV system
kpq ratio between the constant power load (in MVA) and the total
load (in MVA) supplied by an LV system
kpqmv ratio between the constant power load (in MVA) and the total
load (in MVA) supplied by an MV system
ks ratio between the positive and negative sequence impedances of the
aggregated motor load supplied by an LV system
ksc−s ratio between the short-circuit capacity (in MVA) at any busbar S
and the total load (in MVA) supplied by the busbar S
kz ratio between the constant impedance load (in MVA) and the total
load (in MVA) supplied by an LV system
kzmv ratio between the constant impedance load (in MVA) and the total
load (in MVA) supplied by an MV system
Kues:x fraction of the busbar emission allowance at any busbar x of any
sub-system S that can be allocated to installations
K ′ues:x fraction of the busbar emission allowance at any busbar x of any
sub-system S that accounts for the emission arising as a result of
system inherent asymmetries
LF load flow
LV low voltage
ml refers to a MV-LV coupling transformer
MV medium voltage
NECA National Electricity Code Australia
NEMA National Equipment Manufacturer’s Association
PCC point of common coupling
PQ refers to a constant power load
PS refers to a passive load
xi
rec receiving end busbar of any line t
S represents any sub-system (S = HV, MV, LV)
Ssc:s short-circuit capacity at any busbar S [MVA]
Ss:x total apparent power to be supplied by any busbar x of any
sub-system S [MVA]
Ss:x−ds part of Ss:x supplied at the downstream (DS) [MVA]
Ss:x−j agreed apparent power of any installation j to be connected
at any busbar x of any sub-system S [MVA]
Ss:x−local part of Ss:x supplied locally [MVA]
Ss:x−total total apparent power, as seen at any busbar x of any
sub-system S, to be supplied by the sub-system S [MVA]
send sending end busbar of any line t
t any radial local line of any sub-system under evaluation
td any radial downstream line of any sub-system under evaluation
tij any line between busbars i and j of any sub-system
under evaluation
tf refers to a coupling transformer
Tus−s US to S transfer coefficient
θpf :x power factor angle at any busbar x [deg.]
θpf :z, θpf :pq power factor angle of the constant impedance and constant
power loads respectively supplied by an LV system [deg.]
θpf :zmv , θpf :pqmv power factor angle of the constant impedance and constant
power loads respectively supplied by an MV system [deg.]
θY−+:x phase angle of the admittance Y−+:x [deg.]
θZ−+:tdphase angle of the impedance Z−+:td [deg.]
θZλ∆:tphase angle of the impedance Zλ∆:t [deg.]
θIλ:tphase angle of the current Iλ:t [deg.]
Ug/s global emission allowance of any sub-system S [VUF]
Ug/s:x emission allowance of any busbar x of any sub-system S [VUF]
U loadsg/s:x global emission arising as a result of unbalanced installations
at any busbar x of any sub-system S [VUF]
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U linesg/s:x global emission arising as a result of system inherent asymmetries
at any busbar x of any sub-system S [VUF]
Uj/s:x emission level caused by any source of unbalance j
at any busbar x of any sub-system S [VUF]
U results:x resultant emission level at any busbar x of any sub-system S [VUF]
Ux voltage unbalance at any busbar x [VUF]
UIE International Union for Electricity Applications
US represents any upstream system of any sub-system S
(US = EHV, HV, MV)
[V ] matrix of nodal voltages
Vλ:x λ (= 0, +,−) sequence component of Vx [V]
Vλ:s−us λ (= 0, +,−) sequence voltage, referred to US, at any busbar S [V]
Vn−s nominal line-line voltage of any sub-system S [V]
Vx voltage at any busbar x [V]
V lines−:g/s:x global negative sequence voltage arising as a result of line
asymmetries at any busbar x of any sub-system S [V]
V−:Ui/x negative sequence voltage at any busbar x caused by
the voltage unbalance Ui that exists at any other busbar i
V Rt voltage regulation of any line t
V Rtd voltage regulation of any line td
VUF voltage unbalance factor [%]
[Y ] matrix of nodal admittances
Yλ∆:xy λ−∆ (λ, ∆ = 0, +,−) sequence coupling admittance
component of Yxy [S]
Yxy nodal admittance between any busbar x and any
other busbar y [S]
Y−−:x−im downstream negative sequence admittance seen at any
busbar x taking only induction motors into account [S]
Y−+:x downstream negative-positive sequence coupling
admittance seen at any busbar x [S]
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Z refers to a constant impedance load
Zλ∆:t λ−∆ (λ, ∆ = 0, +,−) sequence coupling impedance
of any line t [Ω]
Zλλ:x downstream λ (λ = 0, +,−) sequence impedance seen
at any busbar x [Ω]
Zλλ:tf−s λ (λ = 0, +,−) sequence impedance, referred to S, of any
coupling transformer [Ω]
Z−−:x−im downstream negative sequence impedance seen at any
busbar x taking only induction motors into account [Ω]
Z−+:td negative-positive sequence coupling impedance
of any line td [Ω]
Z−+:td−us negative-positive sequence coupling impedance, referred
to US, of any line td [Ω]
0, +,− refer to zero, positive and negative sequences respectively
Publications Arising from the Thesis
1. Prabodha Paranavithana, Sarath Perera, and Danny Sutanto. Impact of Un-
transposed 66kV Sub-transmission Lines on Voltage Unbalance. In Proc. Aus-
tralasian Universities Power Engineering Conference (AUPEC 2006), paper 28,
Melbourne, Australia, December 2006.
2. P. Paranavithana, S. Perera, and D. Sutanto. Analysis of System Asymmetry
of Interconnected 66kV Sub-transmission Systems in relation to Voltage Unbal-
ance. In Proc. IEEE Power Engineering Society Conference and Exposition in
Africa (PowerAfrica ’07), Johannesburg, South Africa, July 2007.
3. Prabodha Paranavithana, Sarath Perera, Danny Sutanto, and Robert Koch.
A Systematic Approach Towards Evaluating Voltage Unbalance Problem in In-
terconnected Sub-transmission Networks: Separation of Contribution by Lines,
Loads And Mitigation. In Proc. 13th IEEE International Conference on Har-
monics and Quality of Power (ICHQP 2008), Wollongong, Australia, September-
October 2008.
4. Prabodha Paranavithana, Sarath Perera, and Robert Koch. An Improved
Methodology for Determining MV to LV Voltage Unbalance Transfer Coeffi-
cient. In Proc. 13th IEEE International Conference on Harmonics and Quality
of Power (ICHQP 2008), Wollongong, Australia, September-October 2008.
5. Robert Koch, Alex Baith, Sarath Perera, and Prabodha Paranavithana. Volt-
age Unbalance Emission Limits for Installations - General Guidelines and Sys-
tem Specific Considerations. In Proc. 13th IEEE International Conference
on Harmonics and Quality of Power (ICHQP 2008), Wollongong, Australia,
September-October 2008.
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6. Prabodha Paranavithana, Sarath Perera, and Danny Sutanto. Management of
Voltage Unbalance Through Allocation of Emission Limits to Installations. In
Proc. Australasian Universities Power Engineering Conference (AUPEC 2008),
paper 017, Sydney, Australia, December 2008.
7. Prabodha Paranavithana, Sarath Perera, and Robert Koch. Propagation of
Voltage Unbalance from HV to MV Power Systems. In Proc. 21st International
Conference on Electricity Distribution (CIRED 2009), paper 0497, Prague, June
2009.
8. Prabodha Paranavithana, Sarath Perera, and Robert Koch. A Generalised
Methodology for Evaluating Voltage Unbalance Influence Coefficients. In Proc.
21st International Conference on Electricity Distribution (CIRED 2009), paper
0500, Prague, June 2009.
9. Prabodha Paranavithana and Sarath Perera. Location of Sources of Voltage Un-
balance in an Interconnected Network. In Proc. IEEE Power Engineering So-
ciety General Meeting (panel session on “Developments in Determining Power
Quality Disturbance Sources and Harmonic Source Contributions”) , Calgary,
Alberta, Canada, July 2009.
10. Prabodha Paranavithana and Sarath Perera. A Robust Voltage Unbalance
Allocation Methodology Based on the IEC/TR 61000-3-13 Guidelines. In Proc.
IEEE Power Engineering Society General Meeting , Calgary, Alberta, Canada,
July 2009.
11. P. Paranavithana, S. Perera, R. Koch, and Z. Emin. Global Voltage Unbalance
in MV Power Systems due to Line Asymmetries. Accepted for publication in
IEEE Trans. on Power Delivery.
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12. P. Paranavithana, S. Perera, R. Koch, and Z. Emin. Global Voltage Unbalance
in HV Power Systems due to Line Asymmetries: Dependency on Loads And an
Evaluation Methodology. Accepted for publication in IEEE Trans. on Power
Delivery.
13. Prabodha Paranavithana, Sarath Perera, and Danny Sutanto. Management
of Voltage Unbalance Through Allocation of Emission Limits to Installations.
Accepted for publication in Australian Journal of Electrical and Electronics
Engineering (reproduction of Proc. AUPEC 2008 ).
Table of Contents
1 Introduction 11.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Research Objectives and Methodologies . . . . . . . . . . . . . . . . . 41.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Literature Review 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Definition of Voltage Unbalance . . . . . . . . . . . . . . . . . . . . . 112.3 Sources of Voltage Unbalance . . . . . . . . . . . . . . . . . . . . . . 132.4 Effects of Voltage Unbalance . . . . . . . . . . . . . . . . . . . . . . . 142.5 Mitigation Techniques of Voltage Unbalance . . . . . . . . . . . . . . 172.6 Measurement and Indices of Voltage Unbalance . . . . . . . . . . . . 182.7 Limits of Voltage Unbalance . . . . . . . . . . . . . . . . . . . . . . . 21
2.7.1 Compatibility Levels . . . . . . . . . . . . . . . . . . . . . . . 212.7.2 Voltage Characteristics . . . . . . . . . . . . . . . . . . . . . . 222.7.3 Planning Levels . . . . . . . . . . . . . . . . . . . . . . . . . . 252.7.4 Customer Emission Limits . . . . . . . . . . . . . . . . . . . . 26
2.8 Guiding Principles of IEC/TR 61000-3-13 [1] for Voltage UnbalanceEmission Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.8.1 Basic Concepts Used in IEC/TR 61000-3-13 . . . . . . . . . . 282.8.2 Emission Limits: Stages 1, 2 and 3 . . . . . . . . . . . . . . . 302.8.3 Development of Stage 2 Emission Limits . . . . . . . . . . . . 312.8.4 Voltage Unbalance Transfer Coefficients . . . . . . . . . . . . 392.8.5 Factor K ′ue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.9 A Revised Harmonics/Flicker Allocation Technique Based on the IECGuidelines - A Preamble to Voltage Unbalance Allocation . . . . . . . 43
2.10 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Global Voltage Unbalance in MV Power Systems due to System InherentAsymmetries 493.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Influence of Line Asymmetries on the Global Emission and its Depen-
dency on Load Types/Bases . . . . . . . . . . . . . . . . . . . . . . . 523.2.1 Constant Impedance (Z) Loads . . . . . . . . . . . . . . . . . 543.2.2 Constant Current (I) Loads . . . . . . . . . . . . . . . . . . . 553.2.3 Constant Power (PQ) Loads . . . . . . . . . . . . . . . . . . 553.2.4 Induction Motor (IM) Loads . . . . . . . . . . . . . . . . . . 563.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.6 Mixes of Passive and Induction Motor Loads . . . . . . . . . . 58
3.3 Methodology for Evaluating the Global Emission Arising Due to LineAsymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
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3.4 Verification of the Methodology . . . . . . . . . . . . . . . . . . . . . 663.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Global Voltage Unbalance in HV Power Systems due to System Inherent Asym-metries 704.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2 Influence of Line Asymmetries on the Global Emission in the Presence
of Induction Motor Loads . . . . . . . . . . . . . . . . . . . . . . . . 744.3 Methodology for Evaluating the Global Emission Arising Due to Line
Asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.4 Verification of the Methodology Using a Three-bus Test System . . . 854.5 Verification of the Methodology Using the IEEE 14-bus Test System . 894.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 Propagation of Voltage Unbalance 945.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.2 Voltage Unbalance Transfer Coefficients . . . . . . . . . . . . . . . . . 97
5.2.1 MV to LV Transfer Coefficient, Tmv−lv . . . . . . . . . . . . . 1035.2.2 HV to MV Transfer Coefficient, Thv−mv . . . . . . . . . . . . . 110
5.3 Voltage Unbalance Influence Coefficients . . . . . . . . . . . . . . . . 1175.3.1 Preliminary Investigations - Dependency of Influence Coeffi-
cients on Load Types/Bases . . . . . . . . . . . . . . . . . . . 1175.3.2 Methodology for Evaluating Influence Coefficients . . . . . . . 1215.3.3 Verification of the Methodology Using a Three-bus MV Test
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.3.4 Verification of the Methodology Using the IEEE 14-bus Test
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 A Revised Voltage Unbalance Allocation Technique Based on the IEC/TR61000-3-13 Guidelines 1316.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.2 Examination of the IEC/TR 61000-3-13 Approach . . . . . . . . . . 132
6.2.1 Calculation of Individual Emission Limits . . . . . . . . . . . 1346.2.2 Resulting Busbar Emission Levels and Examination Remarks . 138
6.3 A Revised Voltage Unbalance Allocation Technique Based on the CBVAllocation Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.4 Examination of the Revised Voltage Unbalance Allocation Technique 1426.4.1 Calculation of Individual Emission Limits . . . . . . . . . . . 1426.4.2 Resulting Busbar Emission Levels and Examination Remarks . 144
6.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
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7 Analysis of the Problem of Voltage Unbalance in Interconnected Power Sys-tems 1477.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477.2 Voltage Unbalance Behaviour of Line Asymmetries . . . . . . . . . . 150
7.2.1 Impact of the Line Asymmetries of the Study System on theVoltage Unbalance Problem . . . . . . . . . . . . . . . . . . . 150
7.2.2 Voltage Unbalance Behaviour of the Individual Lines of theStudy System - as Standalone Lines . . . . . . . . . . . . . . . 152
7.2.3 Voltage Unbalance Behaviour of the Individual Lines of theStudy System - as Elements in the Interconnected Network . . 155
7.2.4 General Outcomes - Representation of the Voltage UnbalanceBehaviour of an Asymmetrical Line as an Element in an Inter-connected Network . . . . . . . . . . . . . . . . . . . . . . . . 160
7.2.5 General Outcomes - Representation of the Interaction of AllAsymmetrical Lines . . . . . . . . . . . . . . . . . . . . . . . . 160
7.3 Voltage Unbalance Behaviour of Load Asymmetries . . . . . . . . . . 1677.3.1 Impact of the Load Asymmetries of the Study System on the
Voltage Unbalance Problem . . . . . . . . . . . . . . . . . . . 1677.3.2 Voltage Unbalance Behaviour of the Individual Loads of the
Study System - as Elements in the Interconnected Network . . 1697.3.3 General Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 174
7.4 Combined Voltage Unbalance Behaviour of Line and Load Asymmetries1767.4.1 Combined Impact of the Line and Load Asymmetries of the
Study System on the Voltage Unbalance Problem . . . . . . . 1767.4.2 Representation of the Voltage Unbalance Behaviour of the En-
tire System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1767.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8 Conclusions and Recommendations for Future Work 1848.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1848.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . 191
Appendices
A Derivation of (3.5) 204
B Radial MV-LV Test System(Fig. 3.2) 207
C Derivation of (3.14) 209
D Y−−:x−im for an MV Network 212
E Application of the Methodology Given by (3.25) to the Three-bus MV TestSystem (Fig. 3.7) 214
xx
F Derivation of (4.7) 218
G Derivation of (4.9) 221
H Test Case Description of the Radial HV-MV-LV System (Fig. 4.2) 224
I Y−+:x for an HV Network 227
J Application of the Methodology Given by (3.22) to the Three-bus HV TestSystem (Fig. 4.6) 229
K Data of the IEEE 14-bus Test System (Fig. 4.9) 233
L Derivation of (5.18) 237
M Application of the Methodology Given by (5.37) to the Three-bus MV TestSystem (Fig. 5.16) 240
N 66kV Sub-transmission Interconnected Study System (Fig. 7.1) - AdditionalData/Information 243N.1 Operating Conditions at the Considered Time Stamp . . . . . . . . . 243N.2 Line Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246N.3 An Explanation on the Influence of the Location of an Asymmetri-
cal Line of an Interconnected Network on the Voltage Unbalance Be-haviour of the Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
N.4 A Demonstration of the Linearity of Negative Sequence Voltages . . . 247
O Development of a Method for Unbalanced Load Flow Analysis 249O.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249O.2 Symmetrical Component Versus Phase Coordinate Reference Frames
for Unbalanced Load Flow Analysis . . . . . . . . . . . . . . . . . . . 250O.3 Special Considerations in Developing an Unbalanced Load Flow Program250O.4 Representation of System Components . . . . . . . . . . . . . . . . . 251
O.4.1 Synchronous Generators . . . . . . . . . . . . . . . . . . . . . 251O.4.2 Passive Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 254O.4.3 Overhead Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 255O.4.4 Capacitor Banks . . . . . . . . . . . . . . . . . . . . . . . . . 256O.4.5 Three-phase Voltage Regulators/Transformers . . . . . . . . . 256O.4.6 Three-phase Induction Motors . . . . . . . . . . . . . . . . . . 256O.4.7 Network Interactions . . . . . . . . . . . . . . . . . . . . . . . 280
O.5 Load Flow Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280O.6 Related References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
List of Figures
2.1 Derating of three-phase induction motors (UIE) . . . . . . . . . . . . 152.2 Statistical interpretation of the compatibility level (IEC 61000-2-2,
IEC 61000-2-12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 Statistical interpretation of the planning level (IEC 61000-2-2, IEC 61000-
2-12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Interpretation of the emission level (IEC/TR 61000-3-13) . . . . . . . 302.5 Illustration of the global emission allowance (IEC/TR 61000-3-13) . . 352.6 Interconnected sub-system S . . . . . . . . . . . . . . . . . . . . . . . 372.7 System representation of any busbar x of the system S shown in Fig. 2.6 372.8 Variation of Tmv−lv with km established using (2.17) for various com-
binations of ks and ksc−lv values . . . . . . . . . . . . . . . . . . . . . 40
3.1 Simple MV network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Radial MV-LV system . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3 Variation of |V t
−:g/mv:rec| with |I+:t| (V Rt values corresponding to vari-
ous |I+:t| are also indicated) for the four basic load types . . . . . . . 573.4 Variation of U t
g/mv:rec with km for the cases where klv = 1, klv = 0.5and klv = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5 Interconnected MV sub-system . . . . . . . . . . . . . . . . . . . . . 613.6 System representation of any busbar x of the MV system shown in
Fig. 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.7 Three-bus MV test system considered for applying the proposed method-
ology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.8 Emissions U lines
g/mv:x for the three-bus MV test system for the two caseswhere km:2 = 0 and km:2 = 1 . . . . . . . . . . . . . . . . . . . . . . . 68
4.1 Simple HV network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.2 Radial HV-MV-LV system . . . . . . . . . . . . . . . . . . . . . . . . 754.3 Variation of U t+td
g/hv:rec with klvr for the two cases where kmr = 0 andkmr = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4 Interconnected HV sub-system . . . . . . . . . . . . . . . . . . . . . . 804.5 System representation of any busbar x of the HV system shown in
Fig. 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.6 Three-bus HV test system considered for applying the proposed method-
ology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.7 Emissions U lines
g/hv:x for the three-bus HV test system for the cases wherekm:2 = 0 and km:2 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.8 Emissions U linesg/hv:x for the three-bus HV test system for the case where
km:2 = 1 in relation to the Phase arrangements I and II of the MV lines 894.9 IEEE 14-bus test system . . . . . . . . . . . . . . . . . . . . . . . . . 914.10 Emissions U lines
g/hv:x for the IEEE 14-bus test system . . . . . . . . . . . 91
xxi
xxii
5.1 Variation of Tmv−lv with ksc−lv obtained for constant power loads usingunbalanced load flow analysis . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Radial system considered for the illustration of transfer coefficients . 975.3 Variation of Tmv−lv with ksc−lv for constant current loads: I - 0.99
lagging pf, II - 0.9 lagging pf . . . . . . . . . . . . . . . . . . . . . . . 1045.4 Variation of Tmv−lv with ksc−lv for constant power loads: I - 0.99 lagging
pf, II - 0.9 lagging pf . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.5 Variation of Tmv−lv with ksc−lv for induction motor loads with ks = 6.7
and pf = 0.9 lagging . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.6 Variation of Tmv−lv with ksc−lv: I - for a load base dominated by in-
duction motors, II - for a load base dominated by passive elements . . 1085.7 Variation of Tmv−lv with km for ksc−lv ≈ 25 and ksc−lv ≈ 10: I - for
load mixes of Z and IM loads, II - for load mixes of PQ and IM loads 1095.8 Variation of Tmv−lv with km established using the IEC method, (5.19),
(5.20) and unbalanced load flow analysis . . . . . . . . . . . . . . . . 1105.9 Variation of Thv−mv with klv for ksc−mv = 12 (loads are supplied directly
at the MV busbar): I - for load mixes of Z and IM loads, II - for loadmixes of PQ and IM loads . . . . . . . . . . . . . . . . . . . . . . . . 115
5.10 Variation of Thv−mv with klv for ksc−mv = 4 (loads are supplied directlyat the MV busbar): I - for load mixes of Z and IM loads, II - for loadmixes of PQ and IM loads . . . . . . . . . . . . . . . . . . . . . . . . 116
5.11 Variation of Thv−mv with klv (LV loads are supplied through MV lines):I - for ksc−mv = 12, II - for ksc−mv = 4 . . . . . . . . . . . . . . . . . . 116
5.12 Radial MV-LV system (reproduction of Fig. 3.2) . . . . . . . . . . . . 1175.13 Variation of ksend−rec with km for the cases where klv = 1, klv = 0.5
and klv = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.14 Interconnected sub-system S (reproduction of Fig. 2.6) . . . . . . . . 1225.15 System representation of any busbar x of the MV system shown in
Fig. 5.14 (reproduction of Fig. 3.6) . . . . . . . . . . . . . . . . . . . 1245.16 Three-bus MV test system considered for applying the proposed method-
ology (reproduction of Fig. 3.7) . . . . . . . . . . . . . . . . . . . . . 1275.17 Variations of k1−2 and k1−3 with km:2 for the three-bus MV test system 1275.18 IEEE 14-bus test system (reproduction of Fig. 4.9) . . . . . . . . . . 1285.19 Influence coefficients k4−x (x = 1 − 14, x 6= 4) for the IEEE 14-bus
test system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.1 Three-bus HV test system considered for examining the IEC/TR 61000-3-13 approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2 A comparison of the influence coefficients for the test system derivedusing the proposed method: (5.37), and unbalanced load flow analysis 135
6.3 A comparison of the K ′uex factors for the test system derived usingthe proposed method: (4.16), and unbalanced load flow analysis . . . 138
xxiii
6.4 Comparison of the busbar emission limits Ehv:x derived according toIEC/TR 61000-3-13 and the revised method for the test system: I -for Case 1, II - for Case 2 . . . . . . . . . . . . . . . . . . . . . . . . 144
6.5 Comparison of the resulting emission levels U reultg/hv:x derived according
to IEC/TR 61000-3-13 and the revised method for the test system: I- for Case 1, II - for Case 2 . . . . . . . . . . . . . . . . . . . . . . . . 145
7.1 66kV sub-transmission interconnected system under study . . . . . . 1487.2 Measured nodal VUF values for the study system . . . . . . . . . . . 1497.3 Nodal VUF values (load flow results) which arise as a result of the line
asymmetries, in comparison to the measured values . . . . . . . . . . 1517.4 Variation of |V t
−:rec| with |I+:t| for the individual lines . . . . . . . . . 1537.5 Variation of θV t
−:recwith |I+:t| for the individual lines . . . . . . . . . . 154
7.6 Nodal VUF values arising as a result of the individual lines . . . . . . 1577.7 Phase angles of the nodal negative sequence voltages introduced by the
individual lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.8 Global emission vectors of the individual lines (drawn approximately
to a scale) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.9 Resultant influence of the interaction of all asymmetrical lines (drawn
approximately to a scale) . . . . . . . . . . . . . . . . . . . . . . . . . 1627.10 Nodal contributions made by the individual lines to the resultant volt-
age unbalance levels . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.11 (I) Deduced from Fig. 7.8 (II) Effect of the transposition of line F
only (III) Effect of the transposition of lines A and F together (drawnapproximately to a scale) . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.12 Effects, obtained using unbalanced load flow analysis, of the transpo-sition of line F only, and lines A and F together . . . . . . . . . . . . 166
7.13 Nodal VUF values which arise as a result of the load asymmetries, incomparison to that of the line asymmetries . . . . . . . . . . . . . . . 168
7.14 Nodal VUF values which arise as a result of the individual loads . . . 1707.15 Phase angles of the nodal negative sequence voltages introduced by the
individual loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1717.16 Global emission vectors of the individual loads (drawn approximately
to a scale) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1737.17 Resultant influence of the interaction of all unbalanced loads (drawn
approximately to a scale) . . . . . . . . . . . . . . . . . . . . . . . . . 1757.18 Nodal VUF values which arise as a result of both the line and load
asymmetries, in comparison to that of the line asymmetries alone, andthe load asymmetries alone, and also to the measured values . . . . . 177
7.19 Resultant influence of the interaction of all lines and loads (drawnapproximately to a scale) . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.20 Nodal contributions made by the line and load asymmetries to theoverall voltage unbalance levels . . . . . . . . . . . . . . . . . . . . . 179
xxiv
7.21 Nodal contributions made by the individual sources of unbalance tothe overall voltage unbalance levels . . . . . . . . . . . . . . . . . . . 181
O.1 Synchronous generator model . . . . . . . . . . . . . . . . . . . . . . 252O.2 Load model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255O.3 Equivalent circuit of a voltage regulator/transformer . . . . . . . . . 257O.4 Three-phase induction motor model proposed in [4, 5] . . . . . . . . . 257O.5 Variation of the real (P) and reactive (Q) power with the supply voltage
level for a typical three-phase induction motor . . . . . . . . . . . . . 259O.6 Variation of the real (P) and reactive (Q) power with kp (motor loading
levels corresponding to various kp is also given as a percentage to therated output power) for a 2250hp induction motor . . . . . . . . . . . 260
O.7 Variation of the speed with kp (motor loading levels corresponding tovarious kp is also given as a percentage to the rated output power) fora 2250hp induction motor . . . . . . . . . . . . . . . . . . . . . . . . 261
O.8 Impedance type induction motor model . . . . . . . . . . . . . . . . . 261O.9 PQ type induction motor model . . . . . . . . . . . . . . . . . . . . . 262O.10 Sequence equivalent circuits of a three-phase induction motor: I - pos-
itive sequence, II - negative sequence . . . . . . . . . . . . . . . . . . 263O.11 Variation of |Yim:s| cos(−θim:s)
|Y nim:s| cos(−θn
im:s)of P
′x−xx with ωrt
ωnrt
for the 3hp, 220V motor 270
O.12 Variation of |Yim:m2| sin(−θim:m2−1200)|Y n
im:m2| sin(−θnim:m2−1200)
of Q′x−xz with ωt
ωnrt
for the 3hp, 220V
motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271O.13 Variation of ηim with ωrt for the 3hp, 220V motor . . . . . . . . . . . 275O.14 Variation of the per phase input active and reactive power with the
motor loading level for the 3hp, 220V motor excited at the rated voltage(balanced) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
O.15 Variation of the per phase input active and reactive power componentswith the motor loading level for the 3hp, 220V motor excited at reducedand unbalanced voltages . . . . . . . . . . . . . . . . . . . . . . . . . 277
O.16 Variation of the per phase input active and reactive power componentswith the motor loading level for a 2250hp, 2.3kV motor excited atreduced and unbalanced voltages . . . . . . . . . . . . . . . . . . . . 278
O.17 Variation of Pim:a with kp for the existing and proposed induction motormodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
O.18 Variation of Qim:a with kp for the existing and proposed inductionmotor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
List of Tables
2.1 Requirements of background disturbances in assessing the uncertaintyof Class A instruments for the measurement of voltage unbalance (IEC61000-4-30) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Indicative planning levels given in IEC/TR 61000-3-13 . . . . . . . . 292.3 Indicative values for the factor K ′ue given in IEC/TR 61000-3-13 . . 42
6.1 Influence coefficients for the test system shown in Fig. 6.1 . . . . . . 1356.2 Shv:x, Shv:x−total and Ug/hv:x for the test system shown in Fig. 6.1 . . . 1356.3 U lines
g/hv:x, K ′uex and Kuex for Case 2 of the test system shown in Fig. 6.1137
6.4 Ehv:x according to IEC/TR 61000-3-13 for the test system shown inFig. 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.5 U reultg/hv:x arising as a result of the IEC/TR 61000-3-13 allocation proce-
dure for the test system shown in Fig. 6.1 . . . . . . . . . . . . . . . 1396.6 Values of the RHS of (6.8) in relation to the test system shown in Fig.
6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.7 ka for the test system shown in Fig. 6.1 . . . . . . . . . . . . . . . . . 1436.8 Kuex and Ehv:x according to the revised allocation method for the test
system shown in Fig. 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . 1436.9 U reult
g/hv:x arising as a result of the revised allocation procedure for thetest system shown in Fig. 6.1 . . . . . . . . . . . . . . . . . . . . . . 144
7.1 Ranking of the sub-transmission lines based on the associated degreeof asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.2 Parameters, operating features and emission levels of the individuallines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.3 Distribution of the active and reactive power across the three phasesat each of the load busbars of the study system . . . . . . . . . . . . 167
7.4 Operating features and emission levels of the individual loads of thestudy system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
H.1 Values of ksc−lvragg and σ for various klvr . . . . . . . . . . . . . . . . 226
K.1 Voltage controlled bus data . . . . . . . . . . . . . . . . . . . . . . . 233K.2 Static capacitor data: susceptances . . . . . . . . . . . . . . . . . . . 233K.3 Generator and load bus data: three-phase MW and MVAr values . . 234K.4 Transformer data: impedances and secondary tap settings (1st and 2nd
bus numbers refer to the primary and the secondary respectively) . . 234K.5 Nodal positive sequence voltages . . . . . . . . . . . . . . . . . . . . . 235K.6 Transmission line data: lengths and impedances . . . . . . . . . . . . 236
L.1 Replacement factors for a mix of various load types . . . . . . . . . . 239
N.1 System details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
xxv
xxvi
N.2 Voltage controlled bus data . . . . . . . . . . . . . . . . . . . . . . . 244N.3 Generator and load bus data: three-phase MW and MVAr values . . 244N.4 Voltage regulator data: impedances and secondary tap settings . . . . 245N.5 Static capacitor data: susceptances . . . . . . . . . . . . . . . . . . . 245N.6 Generator impedance data . . . . . . . . . . . . . . . . . . . . . . . . 245N.7 Lengths and impedances (Z−+ and Z−+) of the sub-transmission lines 246N.8 Negative sequence voltages V t
−:S2 caused by the individual lines A - Nat the busbar S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
N.9 Resultant negative sequence voltage V lines−:S2 at the busbar S2 . . . . . . 248
O.1 Parameters of a 60Hz, 3hp, 220V induction motor . . . . . . . . . . . 270O.2 Power components P n
x−xx - Qnx−xz for the 3hp, 220V motor . . . . . . 270
O.3 Speed coefficients corresponding to the power components Px−xx -Qx−xz for a range of induction motors . . . . . . . . . . . . . . . . . 272
O.4 Efficiency coefficients for a range of induction motors . . . . . . . . . 275
Chapter 1
Introduction
1.1 Statement of the Problem
Excessive voltage unbalance1 levels in electrical power systems arising as a result of
unbalanced installations and system inherent asymmetries can cause damage to, and
degradation and maloperation of, customer and utility equipment. Despite the exis-
tence of voltage unbalance regulatory codes, some network service providers are facing
difficulties in complying with stipulated levels. This emphasises the need for recom-
mendations based on well researched engineering practices governing the management
of the problem of voltage unbalance, which this thesis aims to fulfil.
The IEC, one of the world’s leading organisation for standardisation on power
quality, has recently released the Technical Report IEC/TR 61000-3-13 [1] which
provides guiding principles for coordinating voltage unbalance between various voltage
levels of a power system through the allocation of emission limits to installations.
The philosophy of this voltage unbalance allocation process is similar to that of the
counterpart IEC approaches to harmonics (IEC 61000-3-6 [2]) and flicker (IEC 61000-
3-7 [3]) allocation. The absorption capacity or the allowed global emission of a sub-
1In the context of the thesis, this is limited to negative sequence unbalance.
1
2
system of a power system is established such that the total emission level derived using
the general summation law, taking the upstream contribution in terms of a transfer
coefficient into account, at any point is maintained at or below the set planning
level. The global emission allowance of the sub-system is allocated to its busbars in
proportion to the ratio between the total apparent power to be supplied by the busbar
under evaluation, and the total available apparent power of the sub-system as seen
at the busbar. Voltage unbalance contributions from neighbouring busbars are taken
into account using influence coefficients in determining the total available apparent
power of the sub-system as seen at the busbar. This busbar emission allowance is then
apportioned to individual customers in proportion to the ratio between the agreed
apparent power, and the total apparent power supplied by the busbar.
In the case of voltage unbalance, the global emission at a busbar generally arises
not only as a result of unbalanced installations but also as a result of system inherent
asymmetries (essentially lines). Thus, the apportioning of the total headroom to
installations as in the case of harmonics and flicker can lead to exceedances of the
set planning levels. Hence, IEC/TR-61000-3-13 applies an additional factor which is
referred to as ‘Kue’ to the apportioned allowance. This factor Kue represents the
fraction of the emission allowance that can be allocated to customers, whereas the
factor K ′ue (= 1−Kue) accounts for the emission which arises as a result of system
inherent asymmetries. It is recommended that system operators assess the factors
Kue and K ′ue for prevailing system conditions in their specific networks. However,
a systematic method for its evaluation is not provided other than a rudimentary
direction together with a set of indicative values.
The Technical Report IEC/TR 61000-3-13 gives a method for estimating the MV
to LV transfer coefficient considering the system and load characteristics and the
downstream load composition. This suggests a value less than unity for the trans-
3
fer coefficient in the presence of industrial load bases containing large proportions
of mains connected three-phase induction motors, and a unity transfer coefficient in
relation to passive loads in general. Although a transfer coefficient of unity is math-
ematically trivial for constant impedance loads, its validity has not been cautiously
examined in relation to constant current and constant power loads which may exhibit
different behaviours under unbalanced supply conditions. Further, systematic meth-
ods for assessing the HV to MV and EHV to HV transfer coefficients and influence
coefficients are yet to be developed.
The IEC allocation policy with regard to harmonics and flicker has been found not
to guarantee that the emission limits allocated to individual customer installations
ensure non-exceedance of the set planning levels [4, 5] 2. Overcoming this problem, an
alternative allocation technique that is referred to as ‘constraint bus voltage’ (CBV)
method which closely aligns with the IEC approach has been suggested for harmon-
ics and flicker [4, 5]. Being based on a common philosophy, the above problem is
anticipated to be experienced also by the recently introduced voltage unbalance allo-
cation approach of IEC/TR 61000-3-13. Thus, it is vital to examine the application
of IEC/TR 61000-3-13 which also involves an additional aspect, i.e. the emission aris-
ing due to system inherent asymmetries. Extension of the CBV method to voltage
unbalance allocation requires revisions addressing this new aspect.
In the application of the IEC/TR 61000-3-13 principles to better manage existing
networks already experiencing excessive voltage unbalance levels, the initial develop-
ment of insights into the influences made by various sources of unbalance is required.
In some circumstances, especially in sub-transmission networks where line transposi-
tion is not a usual practice, the emission which arises as a result of system inherent
asymmetries would not allow an equitable share of busbar emission allowances to in-
2References [4, 5] are the only sources which provide evidence in support of this statement.
4
stallations. The present knowledge which describes the voltage unbalance behaviour
in interconnected network environments is seen to be limited. Fundamental theories
on a standalone asymmetrical line/load would not provide a comprehensive basis for
understanding the interactive behaviour of various sources of unbalance that exist
in interconnected networks. As an example, in cases where voltage unbalance lev-
els arising as a result of line asymmetries themselves are excessive, making decisions
on line transposition options which effectively correct these networks is seen to be a
challenge.
1.2 Research Objectives and Methodologies
The aim of the work presented in this thesis is to make contributions for further
improvements to the current Technical Report IEC/TR 61000-3-13 through:
• The development of novel/refined methodologies for the assessment of the global
emission arising as a result of system inherent asymmetries, and the propagation
of voltage unbalance.
• The examination of the IEC/TR 61000-3-13 principles in relation to the antic-
ipated problem of exceedance of the set planning levels, and proposing appro-
priate revisions.
• The establishment of theoretical bases to broaden the understanding of the
voltage unbalance behaviour exhibited by various sources that exist in inter-
connected network environments, which would provide additional assistance in
the application of the voltage unbalance allocation methodologies.
To develop a comprehensive understanding of the influence of line asymmetries
on the global voltage unbalance levels in MV and HV power systems and the de-
pendency of this influence on various load types/bases, preliminary investigations are
5
carried out in relation to radial power systems. A basis towards the development
of methodologies for evaluating the global emission in MV and HV power systems
which arises as a result of line asymmetries is established through the extension of the
nodal equations [I] = [Y ][V ] to the sequence domain. This basis is integrated with the
outcomes obtained from the preliminary studies for ascertaining the methodologies.
Development of a systematic approach for the assessment of influence coefficients is
also facilitated by an approach similar to above. Verification of the methodologies is
accomplished using unbalanced load flow analysis3.
Dependency of the propagation of voltage unbalance from MV to LV and HV to
MV levels on specific load types is initially examined through the development of
theoretical bases which describe the behaviour of these load types under unbalanced
supply conditions. Employing these, the impact of a load base which consists of
various load types on the propagation is established in terms of transfer coefficients.
Examination of the IEC/TR 61000-3-13 principles is achieved through two steps
employing a simple three-bus test system. Consideration to cases both with and with-
out the inclusion of the influence of system inherent asymmetries is given. Firstly, the
emission limits to installations are calculated using the prescribed approach together
with some of the methodologies proposed in this thesis. Secondly, the resulting bus-
bar voltage unbalance levels are established using the general summation law when
all installations inject their allocated limits, and examined against the set planning
level. Extension of the suggested CBV allocation technique to voltage unbalance is
accomplished by introducing its principles while addressing the emission which arises
as a result of system inherent asymmetries according to IEC/TR 61000-3-13.
3This is a program developed in MATLABR. This, which is described in Appendix O, is basedon the phase coordinate reference frame and incorporates the component level load flow constraintsand the three-phase modelling of power system components.
6
To develop theoretical bases which provide an insight into the problem of voltage
unbalance in interconnected network environments, deterministic studies supported
by unbalanced load flow analysis are carried out employing a 66kV sub-transmission
system that is known to experience excessive voltage unbalance levels. Using a new
concept termed ‘voltage unbalance emission vector’ which is derived based on IEC/TR
61000-3-13, the behaviour of each of the lines treating as standalone lines and also as
elements in the interconnected system, and of each of the loads operating in the inter-
connected environment is observed. Through an extensive analysis of these results,
approaches for ascertaining the influence of an unbalanced source, in a global sense,
in terms of a single emission vector (which is referred to as ‘global emission vector’)
are established. Employing the linearity of negative sequence variables, these global
emission vectors of individual unbalanced sources are added forming a basis which
provides a comprehensive understanding of the voltage unbalance behaviour of the
entire system.
1.3 Outline of the Thesis
A brief description of the contents of the remaining chapters is given below:
Chapter 2, a literature review, provides an overview on various general aspects
of voltage unbalance, and a critical discussion on IEC/TR 61000-3-13 on which the
thesis is primarily based. A basic introduction, followed by a review on sources, ef-
fects and mitigation techniques of voltage unbalance is given. Various standards and
documents governing measurement and evaluation procedures, indices and limits of
voltage unbalance are reviewed. The key section of this chapter describes concepts,
principles and related aspects prescribed in IEC/TR 61000-3-13 establishing the back-
grounds for Chapters 3 - 6. The last section discusses fundamental deficiencies of,
and suggested revisions to, the IEC allocation policy with regard to harmonics and
7
flicker forming the background for Chapter 6.
Chapter 3 addresses the aspect of the global voltage unbalance emission which
arises as a result of line asymmetries in relation to MV power systems. Investigations
carried out employing a simple radial network on the influence of line asymmetries on
the global emission levels and its dependency on various load types/bases is presented,
emphasising limitations associated with the respective direction given in IEC/TR
61000-3-13. A systematic approach, covering radial and interconnected networks, for
the evaluation of this global emission at nodal level is proposed. Results established
using this method for a three-bus test system are compared with those obtained using
unbalanced load flow analysis.
As a continuation of the work presented in Chapter 3, Chapter 4 addresses the
same, however in relation to HV power systems which require further investigations
on the influence of the downstream MV line asymmetries on the global emission levels.
Dependency of the global voltage unbalance levels on the local HV line asymmetries
as well as on the untransposed downstream MV lines in the presence of passive and
induction motor loads is presented employing a radial network. Additional limita-
tions, applicable to HV networks, associated with the approach given in IEC/TR
61000-3-13 are emphasised. A methodology for evaluating the global emission which
arises as a result of both the local and downstream line asymmetries is proposed.
Results established using this method for a three-bus test system and also for the
IEEE 14-bus test system are compared with those obtained using unbalanced load
flow analysis.
Chapter 5 focuses on the aspect of the propagation of voltage unbalance. Firstly,
the propagation from upstream higher voltage to downstream lower voltage levels
in terms of transfer coefficients is addressed. Following initial studies on the depen-
8
dency of the MV to LV and HV to MV transfer coefficients on specific load types
which include different passive components and three-phase induction motors, gen-
eralised expressions for their estimation are proposed. Ranges of variation of these
transfer coefficients are demonstrated. The accuracy of this new method for esti-
mating the MV to LV transfer coefficient is compared with the respective method
given in IEC/TR 61000-3-13. Secondly, the propagation from one busbar to other
neighbouring busbars of a sub-system in terms of influence coefficients is addressed.
Preliminary studies carried out employing a radial network on the dependency of
these influence coefficients on various load types is presented. A systematic approach
for the evaluation of influence coefficients for interconnected network environments is
proposed. Results established using this method for a three-bus test system and also
for the IEEE 14-bus test system are compared with those obtained using unbalanced
load flow analysis.
Chapter 6 firstly examines the IEC/TR 61000-3-13 voltage unbalance allocation
principles employing a three-bus test system. The calculation procedure of the emis-
sion limits to installations using the prescribed formulae together with some of the
above proposed methodologies is described. The resulting busbar voltage unbalance
levels when all installations inject their allocated limits are derived, and examined.
Secondly, the principles of the suggested CBV allocation policy are introduced to
voltage unbalance ensuring a robust allocation. These new allocation principles are
examined employing the above three-bus test system.
Chapter 7 establishes theoretical bases for studying the problem of voltage un-
balance in interconnected network environments. Deterministic studies carried out
employing a 66kV interconnected sub-transmission system in relation to its line and
load asymmetries are separately described. Outcomes from these studies are presented
in a generalised form such that a systematic approach which allows a comprehensive
9
understanding of the voltage unbalance behaviour of the system is developed. The
proposed approach is applied for assessing the study system on major contributors
to voltage unbalance levels, and line transposition options which effectively correct
the asymmetrical network. These assessments are validated employing the results
obtained using unbalanced load flow analysis.
Finally, Chapter 8 summarises the major outcomes of the work presented in the
thesis, and makes recommendations and suggestions for future work.
Chapter 2
Literature Review
2.1 Introduction
This chapter provides an overview on various general aspects of voltage unbalance,
and a critical discussion on IEC/TR 61000-3-13 on which the thesis is primarily based.
A brief introduction to voltage unbalance, followed by a review on various methods
used in different standards and documents for its quantification is given in Section 2.2.
Sections 2.3 - 2.5 cover sources, effects and mitigation techniques of voltage unbalance
respectively as reported in the literature. The widely used IEC 61000-4-30 and other
standards/documents governing measurement and evaluation procedures and indices
of voltage unbalance are examined in Section 2.6. Various categories of voltage unbal-
ance limits: compatibility levels, voltage characteristics, planning levels and customer
emission limits are discussed, and a review on limiting values is given in Section 2.7.
The key section of this chapter, Section 2.8, describes concepts, principles and related
aspects prescribed in IEC/TR 61000-3-13 establishing the backgrounds for Chapters
3 - 6. Section 2.9 discusses fundamental deficiencies of, and suggested revisions to, the
IEC allocation policy with regard to harmonics and flicker forming the background
for Chapter 6. The chapter is summarised in Section 2.10.
10
11
2.2 Definition of Voltage Unbalance
The Technical Report IEC/TR 61000-3-13 [1] defines voltage unbalance as a condition
in poly-phase electric power systems in which the magnitudes of the fundamental
phase voltages and/or the associated phase angles of separation are not equal. This
is a steady-state condition, and hence short-term unbalance which can occur during
events such as unsymmetrical faults does not fall under the definition [1]. Voltage
unbalance can exist in two forms in three-phase power systems: zero and negative
sequence unbalance. Where there exists a path for the flow of zero sequence currents
such as in grounded-neutral systems, the presence of zero sequence voltage can become
an issue [6, 7] especially when the coupling transformer allows zero sequence currents
to flow from higher voltage to lower voltage systems and vice-versa. Zero sequence
unbalance is generally a parameter of little concern as it does not affect ungrounded-
neutral systems and dual-phase installations, and also as it can be controlled through
system design and maintenance [1, 8]. As the negative sequence voltage propagates
through all power system components similar to the positive sequence voltage, it is
the quantity of significant concern. Thus, it is common in practice to associate voltage
unbalance with the negative sequence.
The modulus of the ratio of the fundamental negative sequence (V−) to posi-
tive sequence (V+) voltage components, which is known as ‘voltage unbalance factor’
(VUF), as given by (2.1), is used in IEC/TR 61000-3-13 to quantify the degree of
negative sequence voltage unbalance. This seems to be in agreement with most other
standards/codes1 and international working groups2. This is also referred to as ‘true
value’ of negative sequence voltage unbalance [17, 18].
1e.g. European EN 50160 [9], South African NRS 048-2 [10] and National Electricity Code Aus-tralia (NECA) [11].
2e.g. CIGRE/CIRED [12, 13, 14], International Union for Electricity Applications (UIE) [15, 16].
12
V UF =
∣∣∣∣V−
V+
∣∣∣∣ (2.1)
Reproducing from IEC 61000-4-30 [19], IEC/TR 61000-3-13 gives a practical method
for establishing the VUF using the three fundamental line-line rms voltage magni-
tudes as:
V UF =
√1−
√3− 6ε
1 +√
3− 6ε(2.2)
where,
ε = |Vab|4+|Vbc|4+|Vca|4(|Vab|2+|Vbc|2+|Vca|2)2
Vab, Vbc and Vca - fundamental line-line rms voltages
Alternative methods for the quantification of voltage unbalance are given by the
National Electricity Manufacturer’s Association (NEMA)3 and the Institute of Elec-
trical and Electronics Engineries (IEEE)4. The NEMA definition which is known
as ‘line voltage unbalance rate’ (LVUR), and the IEEE definition which is known as
‘phase voltage unbalance rate’ (PVUR) that exists in two different forms (PV UR1 and
PV UR2) are given by (2.3), (2.4) and (2.5) respectively. However, the recent IEEE
power quality monitoring standard IEEE 1159 [23] lists both the PV UR1 and the VUF.
LV UR =Maximum voltage deviation from the average line-line voltage
Average line-line voltage(2.3)
PV UR1 =Maximum voltage deviation from the average phase voltage
Average phase voltage(2.4)
PV UR2 =Difference between the maximum and the minimum phase voltages
Average phase voltage
(2.5)
3NEMA MG1 [20].4IEEE 112 [21] and IEEE 100 [22].
13
Although angle unbalance is excluded, the LVUR which does not take the presence
of zero sequence voltage into account is similar to the VUF or the true value for
more realistic levels of voltage unbalance [22, 24]. However, the PVUR which is
influenced by the presence of zero sequence voltage deviates significantly away from
the true value in the presence of zero sequence voltage even at lower levels of voltage
unbalance [24]. Among the two IEEE definitions, the PV UR1 is reasonably close to
the true unbalance in the absence of zero sequence voltage [24].
Although the absolute value of the ratio V−V+
or the VUF is the parameter in general
use, it is worthwhile noting that voltage unbalance is also associated with a phase
angle. One may, in the same way, define this phase angle as the angle between the
fundamental negative and positive sequence voltage components [25]. This concept
of voltage unbalance as a vector is also applied in IEC/TR 61000-3-13 in defining the
emission level5 introduced by an unbalanced installation at a particular point.
2.3 Sources of Voltage Unbalance
Voltage unbalance is caused mainly by the uneven distribution and/or the uneven
connection of single-phase and dual-phase loads6 across the three phases and the op-
eration of unbalanced three-phase loads7 through the injection of unbalanced phase
currents or negative sequence currents into the system. Unequal mutual impedances
which arise as a result of the asymmetrical electromagnetic coupling between the
conductors of untransposed/partially transposed single circuit [29]/multi circuit [30,
31, 32] transmission and distribution [33, 34] overhead lines, which lead to unbal-
anced voltage drops across the three phases, is also a well known source of voltage
unbalance. Although limited, electrostatic unbalance of untransposed/partially trans-
5See Section 2.8.1.6e.g. LV appliances, electric traction motors [26, 27], induction furnaces.7e.g. arc furnaces [28, 16].
14
posed overhead transmission lines [30, 35] and asymmetrical transformer banks [36]
in particular open-wye open-delta transformer banks [37] have also been reported as
additional sources of voltage unbalance.
2.4 Effects of Voltage Unbalance
The influence of voltage unbalance on the adverse performance of three-phase induc-
tion motors is well documented [38, 39, 40]. When an induction motor is exposed to
unbalanced voltages, the negative sequence voltage component produces an air gap
flux that rotates against the rotor which is forced by the positive sequence torque,
thus generating an unwanted reverse torque. This results in a reduction of the net
motor torque and speed, in addition to torque and speed pulsations and increased
motor vibration and noise. Further, due to the relatively small negative sequence mo-
tor impedance, unbalance in phase currents drawn by a motor can be 6 to 10 times
the supply voltage unbalance [20] causing increased motor losses and heating. On the
whole, the motor efficiency and lifetime (primarily as a consequence of the prolonged
overheating) will be reduced. To be able to deal with this extra heating, the motor
must be derated, or a motor of a large power rating may be required. According to
the International Union for Electricity Applications (UIE) [16]8, an induction motor
has to be derated depending on the prevailing degree of voltage unbalance as depicted
by Fig. 2.1.
Power electronic converters having uncontrolled diode rectifier front-ends9 [42,
43] and arc furnaces [42] produce uncharacteristic triplen harmonics in addition to
the characteristic harmonics in the input current in the presence of supply voltage
8The derating curve given in [16] is preferred, as it uses the VUF in quantifying voltage unbalance,in comparison to other recommendations such as given in the standards NEMA MG1 [20] (whichuses the LVUR) and AS 1359.31 [41]/IEC Report 892 (which uses the PV UR1).
9e.g. adjustable speed drives.
15
Figure 2.1: Derating of three-phase induction motors (UIE)
unbalance. Significant third harmonic currents can increase harmonics and resonance
problems in power systems, and require large filter ratings. As the degree of voltage
unbalance increases, the input current drawn by a converter becomes significantly
unbalanced and changes from a double pulse waveform to a single pulse waveform as
a result of the asymmetric conduction of the diodes. This results in excessive currents
in one or two of the phases10, which can lead to the tripping of overload protection
circuits, under voltage and increased ripple on the dc-link, and decreased lifetime of
the diodes and the dc-link capacitor.
Modern ac drive systems comprising synchronous pulse width modulated (PWM)
rectifier front ends generate a second order harmonic component on the dc-link when
they are exposed to supply voltage unbalance [45]. This results in increased ripple
on the dc-link affecting the life and size of the dc-link capacitor. Further, this second
order harmonic component reflects in the input current and also in the inverter output
10Measurements taken on an adjustable speed drive system has shown 50% over-current for asupply voltage unbalance where the highest voltage magnitude was 3.6% higher than the lowestvoltage magnitude [44].
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16
voltage by generating a third harmonic component and sub-harmonic components11
respectively.
The impact of some fault conditions (other than the traditionally studied three-
phase fault) on the transient stability of synchronous generators has been seen to
be more severe in the presence of voltage unbalance [46, 47]. This indicates the
requirement of advanced algorithms and computer programs for power system stabil-
ity studies.
Power system components such as synchronous generators, transmission and dis-
tribution overhead lines and cables and transformers can also be affected by voltage
unbalance, which is intensified by the fact that a small degree of unbalance in phase
voltages can cause a disproportionately large unbalance in phase currents as discussed
earlier. Synchronous generators exhibit a phenomenon similar to that in induction
motors in the presence of negative sequence current resulting in excess machine losses
and heating and possible hazards to structural components [48]. According to the
Australian standard AS 1359.10112 [49], synchronous machines shall be capable of
operating continuously in unbalanced systems if none of the phase currents exceeds
the rated current and the ratio of the negative sequence current component and the
rated current does not exceed a value between 5% and 10% depending on the type
of construction, the method of cooling and the machine capacity. Flow of negative
sequence currents in overhead lines, cables and transformers increases power losses
lowering their capacity [50, 51]. From a more theoretical point of view, current un-
balance affects the definitions and the measurement techniques of apparent power
and power factor [52, 53] influencing the aspects of the power system economics. In
addition, current unbalance has been seen to result in a degraded power factor [53].
11These sub-harmonics will be replaced by a dc component when the inverter output frequency isequal to twice the system frequency.
12This is based on IEC 34-1: Rotating electrical machines - part 1 - rating and performance.
17
2.5 Mitigation Techniques of Voltage Unbalance
Methods of voltage unbalance mitigation primarily concern the distribution of power
which includes single-phase, dual-phase and unbalanced three-phase loads evenly
across the three phases. This has been facilitated through the development of tech-
niques for the manual or automatic reconfiguration of distribution systems [54, 55], the
phase rearrangement between distribution transformers and primary feeders [55, 56]
and the switching of connected customers between different phases of a particular
feeder [55] such that unbalance in phase currents is minimised.
Theoretically, the complete transposition of overhead lines or the use of tower
arrangements which provide an equilateral triangular spacing between the three phase
conductors (for single-circuit lines) nullify the emission which arises as a result of
line asymmetries [57]. However, as these ideal conditions can rarely be achieved in
practice due to economic constraints and practical difficulties, the implementation of
more appropriate design options in terms of the tower configuration [30, 32], and the
phase positioning/swapping at transposition points of multi-circuit lines [30, 57, 58]
has been recommended.
Furthermore, the increase of the fault level at the point of common coupling (PCC)
of inherently unbalanced large loads (e.g. traction loads and arc furnaces) can make
some contribution towards reducing their impact on voltage unbalance [16].
In cases where excessive voltage unbalance levels are unavoidable, special balanc-
ing equipment can be installed at the utility and/or plant level. Power electronic
based shunt connected static compensators13 where the compensation is achieved by
the injection or the absorbtion of reactive power to or from the system have been pro-
posed with the development of suitable control algorithms for dynamically correcting
13e.g. passive static var compensators (SVC) [59], active static synchronous compensators (STAT-COM) [60] and distribution STATCOMs (DSTATCOM) [61].
18
voltage unbalance. Series connected static compensators14 which provide an active
correction through the injection of a compensating voltage signal in series with the
supply have also been reported as a means for mitigating voltage unbalance. Com-
prehensive techniques such as unified power quality conditioners (UPQC) [66, 67]
and hybrid active and passive filters [68] which are capable of compensating various
power quality disturbances simultaneously have been further advanced also to handle
voltage unbalance.
The principle of the representation of an unbalanced three-phase load (three-wire)
using an equivalent balanced section and a two-phase section has been employed in
reducing the influence of large unbalanced loads (e.g. traction loads) by the use of
special transformer connection topologies such as Scott, V and Le-Blance at their
supply sub-stations [16, 26, 69]. Installation of Steinments compensators consisting
of inductive and capacitive elements at supply sub-stations of large unbalanced loads
is also a well known technique of load unbalance reduction [69].
2.6 Measurement and Indices of Voltage Unbalance
Purpose of the measurement of a power quality disturbance which is stochastic in na-
ture is to obtain statistical information on the performance of the supply or connected
equipment. Site indices are used to provide a statistical description of the disturbance
at a particular site. System indices which are derived using site indices of various
sites15 based on a certain statistical criteria are representatives of the disturbance
over a part of the power system. This section discusses measurement procedures and
indices of voltage unbalance described in various standards and documents16.
14e.g. static synchronous series compensators (SSSC) [62, 63, 64], dynamic voltage restor-ers (DVR) [65].
15Typically of a particular voltage level, or a group of similar voltage levels.16IEC/TR 61000-3-13 [1] is excluded here. This will be reviewed in Section 2.8 covering all related
aspects.
19
The widely accepted standard IEC 61000-4-30 [19] for the measurement of power
quality disturbances prescribes also the voltage unbalance measurement and evalu-
ation procedure for instruments with Class A17 performance. Measurement of the
fundamental component of the three line-line rms voltages over 10-cycle and 12-cycle
intervals for 50Hz and 60Hz systems respectively is specified. A minimum mea-
surement period of one week is recommended. Aggregated values are obtained over
standard time intervals of 3-second, 10-minute and 2-hour18. The method of quan-
tification is as per (2.2). For instruments with Class B19 performance, the above
specifications are to be provided by manufacturers.
Due to the concurrent existence of various power quality disturbances in typi-
cal power systems, the measurement of a particular disturbance can be affected by
the presence of other background disturbances in the input electrical signal to the
measuring instrument. Thus, IEC 61000-4-30 defines limits for the uncertainty of
instruments with Class A performance when each background disturbance is within
a specified range of variation. For the measurement of voltage unbalance, when other
disturbances exist in the input signal fulfil the requirements given in Table 2.1, except
for voltage unbalance levels in the range of 1% to 5% of the declared input voltage
(Udin), an instrument shall present an uncertainty less than ±0.15%. For instruments
with Class B performance, the uncertainty is specified by manufacturers.
Derivation of site indices using a high percentile (e.g. 95%, 99%) of the aggre-
gated values (3-second, 10-minute, 2-hour) is preferred in general in most standards
and documents [9, 10, 12, 19]. The 95% percentile of the 10-minute aggregated val-
ues over a measurement period of one week is seen to be strongly recommended
for most power quality disturbances including voltage unbalance. This is the only
17That is, precise measurements such as for the verification of compliance with standards.18A 2-hour value is obtained by combining twelve number of 10-minute values.19That is, less precise measurements such as for statistical surveys.
20
Table 2.1: Requirements of background disturbances in assessing the uncertainty ofClass A instruments for the measurement of voltage unbalance (IEC 61000-4-30)
Disturbance Requirement
Power frequency fn ± 0.5Hz (fn - nominal frequency)
Voltage magnitude Udin ± 1%
Flicker Pst < 1 (Pst - short-term flicker severity index)
Harmonics 0% to 3% of Udin
Inter-harmonics 0% to 0.5% of Udin
index used in the European standard EN 50160 [9]. IEC 61000-4-30 proposes a num-
ber of voltage unbalance site indices for contractual applications including the 95%
percentile of the 10-minute and 2-hour aggregated values over a week. The issue
of voltage unbalance indices has also been addressed by the CIGRE/CIRED Joint
Working Group C4.07 [12], and the above index together with the 95% percentile of
the 3-second values over a day has been recently recommended. The South African
standard NRS 048-2 [10] uses the highest of the 10-minute values over a week in
addition to the above index as preliminary site indices. Site indices over long mea-
surement periods are typically calculated as the highest of the daily or weekly indices
(e.g. NRS 048-2).
Among various methods of calculating system indices [25], the choice of a high
percentile of site indices is seen to be popular [12, 70, 71]. The IEC electromagnetic
compatibility standards IEC 61000-2-2 [70] and IEC 61000-2-12 [71] use the 95%
percentile of the 95% site indices as the system index. In addition to the above system
index, [25] recommends the highest of the 95% or 99% site indices as a system index
for voltage unbalance. An alternative approach is given by the CIGRE/CIRED Joint
Working Group C4.07 [12] for a system index in assessing a set voltage unbalance
limit as a low percentage of sites (e.g. 1% and 5%) that exceeds the limit.
21
2.7 Limits of Voltage Unbalance
2.7.1 Compatibility Levels
Connection of equipment to a power system requires that it be able to withstand
any disturbance to which it is subjected by itself and other equipment. Alternatively,
the emission of the disturbance must be limited to a level which is tolerable by the
connected equipment. The primary mechanism defined by the IEC20 to achieve a
balance between the emission and the immunity is the compatibility level. Equip-
ment must be designed to ensure the immunity to the disturbance at least up to
the compatibility level, and utilities are required to maintain the disturbance at or
below the compatibility level. Due to the stochastic nature of the power quality phe-
nomenon, an absolute limit or an expectation of 100% compliance at all times and
locations with a set limit is not sensible. Thus, the compatibility levels are generally
set allowing a small exceeding probability (e.g. 5%) as illustrated in Fig. 2.2 where
the probability density function21 of the disturbance level which represent both time
and space variations and the probability density function of the equipment immunity
level are shown.
The IEC compatibility standards IEC 61000-2-2 [70] and IEC 61000-2-12 [71] give
a value for the voltage unbalance compatibility level in LV and MV power systems
respectively of 2% allowing an excursion up to 3% in some areas where predominantly
single-phase loads are connected. These are based on the 95% non-exceeding proba-
bility level of statistical distributions which represent both time and space variations
of the disturbance. The IEC does not define compatibility levels for HV and EHV
systems. One of the CIGRE papers (Working Group 36.05) [72] proposes a com-
patibility level of 1% for HV networks together with the 2% level for LV and MV
20IEC 61000-2-2 [70] and IEC 61000-2-12 [71].21Inherently normal distributions.
22
Figure 2.2: Statistical interpretation of the compatibility level (IEC 61000-2-2,IEC 61000-2-12)
systems22. Use of the compatibility levels is seen to be the usual practice in limiting
voltage unbalance at LV, and the value of 2% is commonly applied (e.g. Belgium,
Italy, United Kingdom, Germany) [16].
2.7.2 Voltage Characteristics
Voltage characteristics are limits within which any user of a public power system
can expect the voltage to remain at the point of utilistion under normal operating
conditions [12]. That is, these limits should not be exceeded for 100% of locations for
a high percentile of time [25].
The European standard EN 50160 [9] defines main voltage characteristics for LV
and MV distribution systems. According to this, the 95% of the 10-minute VUF
values during each measurement period of one week shall be limited at 2%. An
excursion up to 3% is allowed in some areas with partly single-phase or dual-phase
22However, a clearly defined system index is not given other than stating the 95% of the daily3-second values to retain to compare against the compatibility levels.
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23
installations. The CIGRE/CIRED Joint Working Group C4.07 [12] recommends the
EN 50160 limit for MV networks, and further defines limits of 2% and 1.5% for HV
and EHV systems respectively based on the 95% percentile of the weekly 10-minute
values.
The South African standard NRS 048-2 [10] uses the EN 50160 limit for LV net-
works, and extends this limit to MV and HV levels together with the 3% excursion as
long as the 2% limit is not exceeded for more than 80% of time over the assessment
period. This also gives a limit of 1.5% for EHV networks. The compliance with these
limits is assessed based on the highest of the 95% weekly 10-minute values over the
full measurement period.
The National Electricity Code Australia (NECA) [11] specifies various voltage
unbalance limits as given below:
• Except as a consequence of a contingency event, the average voltage unbalance
measured at a connection point should not vary by more than 0.5%, 1.3% and
2% for Vn > 100kV , 10kV < Vn ≤ 100kV and Vn ≤ 10kV (Vn - nominal supply
voltage) respectively when determined over a 30-minute averaging period.
• As a consequence of a credible contingency event, the average voltage unbalance
measured at a connection point should not vary by more than 0.7%, 1.3% and
2% for Vn > 100kV , 10kV < Vn ≤ 100kV and Vn ≤ 10kV respectively when
determined over a 30-minute averaging period.
• Average voltage unbalance measured at a connection point should not vary
by more than 1%, 2% and 2.5% for Vn > 100kV , 10kV < Vn ≤ 100kV and
Vn ≤ 10kV respectively when determined over a 10-minute averaging period.
• Average voltage unbalance measured at a connection point should not vary
more often than once per hour by more than 2%, 2.5% and 3% for Vn > 100kV ,
24
10kV < Vn ≤ 100kV and Vn ≤ 10kV respectively when determined over a
1-minute averaging period.
Beyond the NECA, different states in Australia have adopted their own electricity
distribution codes. As an example, Victorian electricity distribution code [73] specifies
that a distributor must maintain voltage unbalance at a PCC to a customer’s three-
phase electrical installation at or below 1% with an excursion up to 2% only for a
total of 5 minutes in every 30-minute period.
The American National Standards Institute (ANSI) standard C84.1-1995 devel-
oped by the NEMA states that electrical supply systems should be designed and oper-
ated to limit the maximum voltage unbalance23 to 3% when measured at the electric-
utility revenue meter under no-load conditions [43]24. Concurrently, the NEMA stan-
dard MG1-1993 [20] recommends that three-phase induction motors should be derated
for voltage unbalance levels25 greater than 1%. Some other ANSI/IEEE standards26
indicate that some electronic equipment, such as computers, may experience problems
for voltage unbalance levels more than 2% or 2.5%.
As given in [12] and [48], voltage unbalance limits used in some other coun-
tries/provinces are listed below:
• France - 2% and 1% limits for MV and HV-EHV networks respectively based
on the 10-minute values over a minimum measurement period of one week [12].
• Quebec (Canada) - 2%, 1.5% and 1% limits for MV, HV and EHV networks
respectively based on the 95% percentile of the 2-hour values over a week [12].
23Note that this is given in terms of the LVUR.24This 3% value stated in ANSI C84.1-1995 is based on the minimum combined cost to utilities
and manufacturers for voltage unbalance related issues.25In terms of the LVUR.26e.g. ANSI/IEEE standard 141-1993 and ANSI/IEEE standard 241-1990.
25
• Russia - 2% and 4% based on the 95% percentile and the maximum value
respectively [48].
2.7.3 Planning Levels
Planning level is a concept adopted by the IEC27, and is a limit set for a particular
voltage level by the body responsible for the planning and operation of the supply
system. This can be considered as an internal quality objective of the system opera-
tor. Setting of the planning levels is aimed at the coordination of voltage unbalance
between various voltage levels such that the MV/LV compatibility level is not ex-
ceeded. Planning level is usually equal to or lower than the compatibility level, and
may differ from case to case depending on the network structure and circumstances.
In practice, planning levels are reflected in MV, HV and EHV systems. A statistical
interpretation of the planning level is given in Fig. 2.3.
Figure 2.3: Statistical interpretation of the planning level (IEC 61000-2-2, IEC 61000-2-12)
27IEC 61000-2-2 [70] and IEC 61000-2-12 [71].
Please see print copy for image
26
As given in [12] and [16], planning levels used in some countries are listed below:
• United Kingdom - 2% for MV, HV and EHV systems.
• Belgium, Italy - 2% and 1% for MV and HV systems respectively.
Considering practices in various countries and measurement results28, the CIGRE/CI-
RED Joint Working Group C4.07 [12] recommends planning levels of 2%, 1.5% and
1% for MV, HV and EHV systems respectively based on the 95% daily 3-second
values and/or the 99% weekly 10-minute values. Indicative values for planning levels
provided in IEC/TR 61000-3-13 [1] will be given in Section 2.8.
2.7.4 Customer Emission Limits
Beyond the three basic categories of voltage unbalance limits: compatibility levels,
voltage characteristics and planning levels, some countries impose emission limits on
major customers such as high speed railway systems as a means for managing the
global limits. The limits applied in various countries are stated below.
• Denmark - for connecting a single-phase railway supply station to an HV net-
work, the mean VUF value during a 1-minute period is not allowed to exceed
2% at the supply point [16].
• France - an emission limit of 0.7% for 10-minute periods applies to single-phase
loads connected to MV or HV networks. Permissible emission levels for high
speed trains are 1%, and 1.5% for periods equal to or greater than 15 minutes
and less than 15 minutes respectively [16].
• Germany - it is indicated that the specified 2% LV compatibility level can be
normally met if any connected equipment does not cause emission levels higher
than 0.7% for time ranges of minutes and 1% for time ranges of seconds [16].
28Reported on in [12].
27
• Italy - if the subscribed power Sj of an individual user does not meet the
requirements: Sj ≤ 30kV A and Sj ≤ 1% of Ssc, Sj ≤ 500kV A and Sj ≤
1% of Ssc, and Sj ≤ 1% of Ssc for LV, MV and HV systems respectively (Ssc -
fault level at the PCC), emission limits of 0.7% for time ranges of minutes and
1% for time ranges of seconds are applied on the customer [16].
• Spain - for high speed train systems, a voltage unbalance limit of 1% is applied
at the PCC. An average value of 2% for 10-minute periods is allowed [16].
• Australia - according to the NECA [11], a customer connected at a 30kV or
higher voltage level must ensure that the current in any phase of a three-phase
electrical installation does not deviate away from the average current by more
than 2%. This limit is 5% for a customer connected at a voltage level lower
than 30kV. The Victorian distribution code [73] defines this limit as 5% and
2% for voltage levels up to 1kV and above 1kV respectively29.
Furthermore, IEC/TR 61000-3-13 [1] prescribes guiding principles for the assess-
ment of emission limits to three-phase unbalanced installations in an equitable manner
rather than simply specifying limiting values. This will be critically reviewed in the
next section.
2.8 Guiding Principles of IEC/TR 61000-3-13 [1] for Voltage
Unbalance Emission Allocation
The recently released IEC Technical Report IEC/TR 61000-3-13, which is based on
the work that has been undertaken by the CIGRE/CIRED Joint Working Group
C4.103 [13, 14], is the most comprehensive document available in international con-
sensus governing voltage unbalance (negative sequence). The Scope of this report
29Excursions up to 10% and 4% respectively are permitted for periods less than 2 minutes.
28
covers the provision of guiding principles to system operators, which can be used as a
basis for determining the requirements for the connection of unbalanced installations
to MV, HV and EHV public power systems such that adequate service quality to
all connected customers is ensured. The report addresses the coordination of voltage
unbalance between various voltage levels of a power system through the allocation of
the system capacity to absorb voltage unbalance to individual customers.
This report refers to an unbalanced installation as a complete three-phase instal-
lation, i.e. including both balanced and unbalanced parts, causing voltage unbalance.
Connection of single-phase and dual-phase customer equipment is not specifically ad-
dressed, and the distribution of these loads evenly across the three phases is considered
as a responsibility of the system operator.
2.8.1 Basic Concepts Used in IEC/TR 61000-3-13
Development of emission limits to individual customer installations is based on the
effects of these emissions will have on the quality of the voltage. The concepts of
compatibility level30, planning level and emission level are used to evaluate the voltage
quality.
Based on the 2% compatibility level at LV, existing practices in MV, HV and
EHV systems and measurement results31, IEC/TR 61000-3-13 gives indicative plan-
ning levels for MV, HV and EHV systems as reproduced in Table 2.2. These levels
consider the need to provide margins between LV, MV and HV-EHV for the purpose
of the overall electromagnetic compatibility coordination. IEC/TR 61000-3-13 also
provides the general guidance for adopting planning levels for specific systems32. The
30Compatibility levels given in IEC/TR 61000-3-13 are reproductions of the IEC compatibilitystandards IEC 61000-2-2 [70] and IEC 61000-2-12 [71] which were discussed above in Section 2.7.1.
31Reported on in [74] by the GIGRE/CIRED Joint Working Group C4.103.32Refer to Annex A of IEC/TR 61000-3-13.
29
measurement and evaluation procedure for the assessment of planning levels against
actual voltage unbalance levels is as per IEC 61000-4-30 [19] for instruments with
Class A performance33. The minimum measurement period is a week with normal
business activities. Use of one or more of the following indices is recommended34.
• 95% of the weekly 10-minute values should not exceed the planning level.
• The highest of the 99% daily 3-second values should not exceed the planning
level times a multiplying factor35 (e.g. 1.25 - 2) which is to be specified by the
system operator depending on the system characteristics and the short-term
capability of equipment along with their protection devices.
Table 2.2: Indicative planning levels given in IEC/TR 61000-3-13
Voltage level MV HV EHV
Planning level (VUF%) 1.8 1.4 0.8
Emission level introduced by an unbalanced installation into the connected power
system is defined as the magnitude of the unbalanced voltage vector which the con-
sidered installation gives rise to at the point of evaluation36. This, of an installation j,
is the magnitude of the vector Uj illustrated in Fig. 2.437. When this vector results
in an increased38 voltage unbalance level (i.e. |Upost-connection| > |Upre-connection|), the
emission level as defined above (i.e. |Uj| in terms of the VUF) is required to be at
or below the emission limit. As the recommended voltage unbalance coordination
33Refer to Section 2.6.34Use of the second index is only needed for installations having a significant impact on the system.35Customers are allowed to cause higher emission levels for short periods of time such as during
bursts or start-up conditions.36This can be a point of connection or a PCC.37Upre-connection, Upost-connection are unbalanced voltage vectors under pre-connection and post-
connection conditions respectively.38In some cases, the interaction between the installation and the remaining part of the supply
system may result in a decreased voltage unbalance level (i.e. |Upost-connection| < |Upre-connection|).
30
approach relies on individual emission limits which are being derived39 from plan-
ning levels, the compliance against these emission limits should be assessed based
on the same measurement and evaluation procedure and indices applied for assessing
planning levels against actual voltage unbalance levels.
Figure 2.4: Interpretation of the emission level (IEC/TR 61000-3-13)
2.8.2 Emission Limits: Stages 1, 2 and 3
IEC/TR 61000-3-13 specifies three stages governing the approval for the connection
of unbalanced installations into MV, HV and EHV networks.
• Stage 1 - connection of small installations or installations with a small degree
of unbalance, which fulfil the criteria given by (2.6), can be accepted without a
detailed evaluation of their emission characteristics. That is, no emission limit
will be imposed on these installations.
Suj
Ssc
≤ 0.2% (2.6)
39The procedure of the derivation of individual emission limits will be discussed in Section 2.8.3.
Please see print copy for image
31
where,
Suj - single-phase power equivalent (line-line or line-neutral equivalent) of the
load unbalance of an installation j
Ssc - three-phase short-circuit capacity at the point of evaluation
• Stage 2 - if an installation does not meet Stage 1 requirement, it should com-
ply with an emission limit imposed based on a certain criteria by the system
operator. This approach of setting individual emission limits will be discussed
in Section 2.8.3.
• Stage 3 - connection of an installation which would fail to comply with Stage 2
emission limit can be conditionally accepted under some circumstances.
2.8.3 Development of Stage 2 Emission Limits
This stage of the IEC/TR 61000-3-13 voltage unbalance management procedure ap-
portions the system absorption capacity to individual customers. The philosophy
of this allocation approach is similar to that of the counterpart IEC harmonics [2]
and flicker [3] allocation methods. However, an additional aspect is involved in the
case of voltage unbalance, i.e. the emission which arises as a result of system inher-
ent asymmetries40. The general principle of the proposed allocation approach is such
that, when a system is utilised to its designed capacity and all connected installations
inject their individual limits, taking also the emission arising due to system inherent
asymmetries into account, the resultant emission level which arises at any point of
the system should be restricted at or below the set planning level.
Determination of the emission limits to individual customers in an equitable man-
ner is achieved through the following sequence steps:
40Essentially untransposed or partially transposed overhead transmission/distribution lines.
32
• Adoption of a general summation law for combining emissions arising due to
numerous sources of unbalance.
• Sharing of the system capacity to absorb voltage unbalance between various sub-
systems or voltage levels. This share of a sub-system is referred to as ‘global
emission allowance’ (Ug/s for a sub-system S).
• Apportioning of the global emission allowance of the considered sub-system to
its busbars (Ug/s:x for a busbar x of the sub-system S).
• Allocation of the busbar allowance Ug/s:x to connected customers in an equitable
manner taking into account the emission which arises as a result of system
inherent asymmetries.
General Summation Law
Theoretically, a resultant unbalanced voltage at a point of a system which arises as
a result of the interaction of various sources of unbalance41 is the vector summation
of unbalanced voltage components caused by individual sources at the considered
point. These vectors (magnitude and/or phase) of numerous unbalanced installations
are inherently random and independent. Although voltage unbalance arising as a
result of system inherent asymmetries which primarily depends on line construction
practices is not random, it also varies with the load. Thus, representing these vectors
as stochastic quantities, the following general summation law which avoids the need
for phase angle information is adopted on the basis of experience:
U results:x = α
√∑(Uj/s:x)α (2.7)
41i.e. numerous unbalanced installations and system inherent asymmetries.
33
where,
U results:x - resultant emission level (a probabilistic quantity) at a busbar x of a system S
Uj/s:x - emission level (a probabilistic quantity) caused by any source of unbalance j
to be combined at the busbar x
α - summation law exponent
The exponent α depends upon the chosen value of probability for the actual volt-
age unbalance level not to exceed the calculated value, and the degree to which the
combined individual unbalanced voltages vary randomly in magnitude and phase.
Considering the 95% non-exceeding probability and the fact that the operation of
most unbalanced installations are unlikely to produce simultaneous or in-phase emis-
sions in practice, IEC/TR 61000-3-13 gives an indicative value for α in the absence of
specific information42 as 1.4. This indicative value would be more applicable in cases
of about eight or more number of dominating unbalanced installations.
Global Emission Allowance
Determination of the global emission allowance of a sub-system is illustrated using the
simple radial system shown in Fig. 2.5. Each of the busbars (US, S and DS) represents
a sub-system43, where S is the sub-system under evaluation and US and DS are the
upstream and the downstream sub-systems respectively. Adopted planning levels of
the sub-systems US and S are Lus and Ls respectively.
According to IEC/TR 61000-3-13, the global emission in a particular sub-system is
the emission arising due to unbalanced sources that exist in the considered sub-system
and its downstream. Then, two sources which contribute to voltage unbalance in the
42e.g. when it is known that unbalances are likely to be in-phase and coincident in time, asummation exponent closer to unity should be used.
43e.g. US - HV, S - MV and DS - LV.
34
system S can be identified: the global emission (Ug/s), and the voltage unbalance
(UUus/s) which propagates from US. By combining these two emissions using the
general summation law, the resultant voltage unbalance level U results at S can be
established as:
(U results )α = (Ug/s)
α + (UUus/s)α (2.8)
Voltage unbalance UUus/s is expressed as a product of a transfer factor and the voltage
unbalance Uus prevailing at US, i.e. UUus/s = Tus−sUus. This factor Tus−s referred
to as ‘US to S voltage unbalance transfer coefficient’ will be further discussed in
Section 2.8.4. Rearranging (2.8) while restricting the voltage unbalance levels Uus
and Us to the respective planning levels Lus and Ls, the global emission allowance of
the sub-system S can be established as:
Ug/s = α√
(Ls)α − (Tus−sLus)α (2.9)
This is the level of voltage unbalance that unbalanced installations supplied by the
sub-system S44 including customers supplied at its downstream, and asymmetrical
lines which exist in the sub-system and its downstream are allowed to cause at any
busbar of the sub-system.
Apportioning of the Global Emission Allowance to Busbars
Avoiding this intermediate step, IEC/TR 61000-3-13 directly gives the formula for
allocating the global emission allowance to individual customers. However, the inclu-
sion of this sequence of steps would assist in providing a comprehensive description
of the allocation process.
44This can be a single-bus or a multi-bus system.
35
Figure 2.5: Illustration of the global emission allowance (IEC/TR 61000-3-13)
The underlying technique employed for apportioning the global emission allowance
of a sub-system45 to its busbars uses two power definitions, which are described below:
• Total apparent power of installations which are to be supplied by the busbar
under evaluation in foreseeable future. This can be estimated as the sum of
all power flows leaving the busbar while ignoring all power flows between the
considered busbar and other busbars. Considering the sub-system S (Figs. 2.6 -
2.7), this power Ss:x for the busbar x includes customers to be supplied directly
at the busbar (Ss:x−local) and also at the downstream system (Ss:x−ds), i.e. Ss:x =
Ss:x−local + Ss:x−ds.
• Total apparent power, as seen at the busbar under evaluation, of installations
which are to be supplied by the entire sub-system. A preliminary approximation
for this power is given as the sum of power flows leaving the busbar while in-
cluding all power flows between the considered busbar and other busbars. This
approximation assumes that the emissions of installations supplied by neighbor-
45Consider the interconnected sub-system S shown in Fig. 2.6, where Fig. 2.7 shows the systemrepresentation of any busbar x of the sub-system.
Please see print copy for image
36
ing busbars make a direct impact on the considered busbar, which is intended
to be conservative particularly for highly meshed HV and EHV networks. Thus,
the following method given by (2.10) which takes the influences of neighbouring
busbars using a transfer factor into account is recommended for assessing this
power.
Ss:x−total = k1−xSs:1 + k2−xSs:2 + ... + Ss:x + ... + ki−xSs:i + ... + kn−xSs:n (2.10)
where,
Ss:x−total - total available power of the entire sub-system as seen at the busbar x
Ss:i - total power46 to be supplied by a busbar i, where i = 1, 2, 3..., n, i 6= x
ki−x - voltage unbalance influence coefficient between the busbar i and the
busbar x, that is defined as the voltage unbalance which arises at the busbar x
when 1pu of negative sequence voltage source is applied at the busbar i
The global emission allowance Ug/s is then apportioned to the busbar x in proportion
to the ratio between Ss:x and Ss:x−total as:
Ug/s:x = Ug/sα
√Ss:x
Ss:x−total
(2.11)
This is the level of voltage unbalance that unbalanced installations supplied by the
busbar x including customers supplied at its downstream, and asymmetrical lines
existing in the sub-system and the downstream system supplied by the busbar x are
allowed to cause at the busbar.
46The first power definition.
37
Busbar 2
… … Busbar 1
Busbar 3
Busbar x
Sub-system S
Busbar n Upstream
system
Figure 2.6: Interconnected sub-system S
Downstream system supplied by the busbar x
Busbar x
Ss:x-local
Ss:x-ds
Figure 2.7: System representation of any busbar x of the system S shown in Fig. 2.6
38
Individual Emission Limits
As in the case of the IEC harmonics and flicker allocation, IEC/TR 61000-3-13 also
considers that the allocation of the busbar allowance Ug/s:x to a customer to be con-
nected at the busbar x based on the ratio between the agreed apparent power and
the total power to be supplied by the busbar as an equitable criteria. However, not-
ing that the global voltage unbalance at the busbar is generally caused not only by
unbalanced installations (U loadsg/s:x ) but also by system inherent asymmetries (U lines
g/s:x) as
expressed by (2.12), the allocation of the total Ug/s:x to installations may result in
exceedance of the set planning levels.
(Ug/s:x)α = (U loads
g/s:x )α + (U linesg/s:x)
α (2.12)
IEC/TR 61000-3-13 addresses this issue by introducing a new factor Kues:x. This
factor which is defined by (2.13) represents the fraction of Ug/s:x that can be allocated
to installations. Conversely, the factor K ′ues:x (= 1−Kues:x) which is given by (2.14)
represents the fraction of Ug/s:x that accounts for the emission arising as a result of
system inherent asymmetries47.
Kues:x =
(U loads
g/s:x
Ug/s:x
)α
(2.13)
K ′ues:x =
(U lines
g/s:x
Ug/s:x
)α
(2.14)
Then, the busbar allowance Ug/s:x is allocated to any customer installation j to be
connected at the busbar x as:
Es:x−j = Ug/s:xα
√Kues:x
(Ss:x−j
Ss:x
)(2.15)
47Section 2.8.5 gives a further discussion on this factor.
39
where,
Es:x−j - emission limit of the customer installation j to be connected at the busbar x
Ss:x−j - agreed apparent power of the installation j to be connected at the busbar x
2.8.4 Voltage Unbalance Transfer Coefficients
As indicated in Section 2.8.3 by (2.9), the prescribed voltage unbalance allocation
procedure requires quantitative measures of the propagation of voltage unbalance
from upstream higher voltage (Uus) to downstream lower voltage (UUus/s) systems
in terms of transfer coefficients. In general terms, the transfer coefficient Tus−s is
defined as:
Tus−s =UUus/s
Uus
(2.16)
Precise estimation of transfer coefficients facilitates an equitable allocation as the
underestimation of these results in emissions above the set planning levels, whereas
the overestimation causes unnecessary limitations on individual customers.
In addition to the indicative values of 0.9 and 0.95 given for the MV to LV and
HV to MV transfer coefficients respectively, IEC/TR 61000-3-13 prescribes a formula
for estimating the MV to LV transfer coefficient (Tmv−lv) using the system and load
characteristics and the downstream load composition as:
Tmv−lv =1
1 + km
(ks−1
ksc−lv+1
) (2.17)
where,
km - ratio between the rated motor load (in MVA) and the total load (in MVA) sup-
plied by the downstream LV system
40
ks - ratio between the positive and negative (which is inductive) sequence impedances
of the aggregated motor load supplied by the LV system48
ksc−lv - ratio between the short-circuit capacity (in MVA) at the LV busbar and the
total load (in MVA) supplied by the LV system
Equation (2.17) indicates a value less than unity for Tmv−lv in the presence of indus-
trial load bases containing large proportions of mains connected three-phase induction
motors, and a unity transfer coefficient in relation to passive loads49 in general. That
is, motor loads help attenuating voltage unbalance as it propagates from higher volt-
age to lower voltage systems, whereas there is no attenuation or amplification in the
presence of passive loads. Fig. 2.8 illustrates the variation of Tmv−lv with km de-
rived using (2.17) for various combinations of ks and ksc−lv values demonstrating that
Tmv−lv can be as small as 0.6. This attenuation has also been seen from measure-
ments taken by the CIGRE/CIRED Joint Working Group C4.103 [74] at a remote
mine with a large proportion of induction motor loads.
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.2 0.4 0.6 0.8 1.0
km
T mv-
lv
ks = 5, ksc-lv = 20ks = 5, ksc-lv = 10ks = 5, ksc-lv = 5ks = 7, ksc-lv = 20ks = 7, ksc-lv = 10
Figure 2.8: Variation of Tmv−lv with km established using (2.17) for various combina-tions of ks and ksc−lv values
48Typically, ks can be in the range of 5 to 7.49e.g. constant current, constant impedance, constant power loads.
41
2.8.5 Factor K ′ue
As defined in Section 2.8.3 by (2.14), the factor K ′ue represents the fraction of a
busbar emission allowance that accounts for the emission arising as a result of sys-
tem inherent asymmetries. IEC/TR 61000-3-13 recommends system operators to
determine this factor based on the line construction practices and the system char-
acteristics in their specific networks. In any case, system operators are responsible
to maintain their networks such that K ′ue allows an equitable share of the busbar
allowance between unbalanced installations and system inherent asymmetries.
The technical report gives a rudimentary direction towards evaluating this factor
together with a set of indicative values which are reproduced in Table 2.8.5. Consid-
ering a simple radial network having an asymmetrical line50, the direction mentioned
above indicates that the global emission caused by the line at its receiving end busbar
can be derived using51:
U tg/s:rec ≈
|Z−+:tI+:t|Vnp
(2.18)
where,
t - represents the asymmetrical line
U tg/s:rec - global voltage unbalance (in terms of the VUF) caused by the line t at its
receiving end busbar
Z−+:t - negative-positive sequence coupling impedance of the line t
I+:t - positive sequence current in the line t
Vnp - nominal phase voltage of the system
50An overhead, single-circuit line.51The nominal system voltage is taken as the prevailing positive sequence voltage.
42
Table 2.3: Indicative values for the factor K ′ue given in IEC/TR 61000-3-13
System characteristics K ′ue
- Highly meshed systems with generation locally connected near load
centers
- Transmission lines are fully transposed, otherwise lines are very
short (few km) 0.1-0.2
- Distribution systems supplying high density load areas with short
lines or cables
- Meshed systems with some radial lines which are either fully or
partly transposed
- Mix of local and remote generation with some long lines 0.2-0.4
- Distribution systems supplying a mix of high density and suburban
areas with relatively short lines (< 10km)
- Long transmission lines which are generally transposed, generation
mostly remote
- Radial sub-transmission lines which are partly transposed or
untransposed
- Distribution systems supplying a mix of medium and low density 0.5-0.4
load areas with relatively long lines (> 20km)
- three-phase motors account for only a small part of the peak load
(e.g. 10%)
43
2.9 A Revised Harmonics/Flicker Allocation Technique Based
on the IEC Guidelines - A Preamble to Voltage Unbal-
ance Allocation
As stated earlier, the IEC approach for managing continuous power quality distur-
bances (IEC/TR 61000-3-13 [1] for voltage unbalance, IEC 61000-3-6 [2] for harmon-
ics, IEC 61000-3-7 [3] for flicker) in power systems through the allocation of emission
limits to individual customer installations is based on a common philosophy. This
section reviews the work that has been undertaken on fundamental deficiencies of, and
suggested revisions to, the counterpart harmonics and flicker allocation approaches.
Being based on a common philosophy, these deficiencies and revisions may appli-
cable to the recently introduced voltage unbalance allocation approach of IEC/TR
61000-3-13, which will be addressed in Chapter 6.
Four key objectives are assumed in the IEC allocation policy:
• Firstly, the level of disturbance at any point of any part of the power system
should not lead to the LV compatibility level is being exceeded. Planning levels
in higher voltage parts should be set accordingly.
• Secondly, the allocation should not distinguish between different types of cus-
tomer installations, i.e. installations of equal MVA demand connected at a
common busbar should receive equal emission limits.
• Thirdly, the allocation should be equitable in some sense, i.e. larger installations
should be entitled to larger emission levels.
• Finally, the emission levels should be as large as possible utilising as much of
the network absorption capacity as possible.
44
Australian/New Zealand standards AS/NZS 61000-3-6 [75] and AS/NZS 61000-3-
7 [76] are essentially based on the respective IEC technical reports on harmonics and
flicker allocation. As a result of difficulties found by Australian utilities in applying
these standards, rigorous studies addressing associated deficiencies have been carried
out by the Integral Energy Power Quality and Reliability Centre at the University of
Wollongong [4, 5]. Consequently, Standards Australia commissioned the writing of
the handbook HB-264-2003 [4] which gives more prescriptive procedures for the use of
these standards. Arising as a result of these studies, it has been revealed that the ap-
plication of the standards to even a simple radial network, let alone relatively complex
meshed systems, is a highly non-trivial exercise. The prescribed allocation procedure
has been seen to lead to situations where the set planning levels are being exceeded
even when no customer exceeds the allocated emission limit. This behaviour has been
identified when using a uniform planning level across the entire network (at a partic-
ular voltage level) as per the IEC approach. Reference [5] demonstrates, employing
a simple example, that it is not possible to derive a practical set of emission limits
such that all busbars in a network reach an uniform planning level when all emission
limits are met. Evidently, a requirement exists for either non-uniform planning levels
for various busbars, or a more suitable allocation policy.
A revised allocation technique for both harmonics and flicker, which closely aligns
with the IEC policy, whereby emission levels at network busbars are explicitly forced
to be at or below the set planning level when all loads inject their limits derived
under the new approach, has been introduced [4, 5]. The principles of this technique
which is referred to as ‘constraint bus voltage (CBV) method’ will be reviewed in the
remaining part of this section.
Consider the requirement that two customer installations (say, m and n) of agreed
apparent power Ss:x−m and Ss:x−n supplied at a busbar x shall receive the same
45
combined emission limit Es:x−mn as a load of Ss:x−mn = Ss:x−m + Ss:x−n connected at
the same busbar. According to the general summation law, the combined emission
limit Es:x−mn can be written in terms of the individual emission limits Es:x−m and
Es:x−m as:
(Es:x−mn)α = (Es:x−m)α + (Es:x−n)α (2.19)
Now, suppose that the emission limit of an installation p is some function f of its
MVA demand Ss:x−p:
Es:x−p = f(Ss:x−p) (2.20)
Then, (2.19) can hold true if and only if:
(Es:x−m)α + (Es:x−n)α = fα(Ss:x−m) + fα(Ss:x−n) = fα(Ss:x−m + Ss:x−n) (2.21)
That is, fα is associative. The simplest way to satisfy (2.21) is to make the function
f proportional to α√
Ss:x−p:
f(Ss:x−p) = Es:x−p = kaα√
Ss:x−p (2.22)
where ka is a proportionality coefficient which is referred to as ‘allocation constant’.
This constant is yet to be determined.
The allocation policy as suggested by the IEC gives the emission limit Es:x−p as52:
Es:x−p =Ug/s
α√
Ss:x−total
α√
Ss:x−p (2.23)
By comparing (2.22) and (2.23), it can be observed that the allocation constant under
the IEC allocation method can be given by:
52For the cases of harmonics (IEC 61000-3-6) and flicker (IEC 61000-3-7).
46
ka =Ug/s
α√
Ss:x−total
(2.24)
Note that, under the IEC allocation policy, ka is a busbar dependant parameter as
the power Ss:x−total53 is busbar dependant.
The simplest way to ensure that busbar emission levels do not exceed the set
planning level is the relaxation of the constraint imposed by (2.24). Instead, the
allocation constant ka, making it a global constant, can be chosen simply to be the
largest value such that:
U resultg/s:x ≤ Ug/s for every busbar x (2.25)
The resulting global emission level U resultg/s:x at any busbar x can be derived using the
general summation law in terms of busbar emission limits Es:i54 for i = 1, 2, ..., x, ..., n
and influence coefficients ki−x between busbars i and the busbar x as:
U resultg/s:x = α
√(k1−xEs:1)α + (k2−xEs:2)α + ... + (Es:x)α + ... + (kn−xEs:n)α (2.26)
Employing (2.22), (2.26) can be written as:
U resultg/s:x = ka
α√
kα1−xSs:1 + kα
2−xSs:2 + ... + Ss:x + ... + kαn−xSs:n (2.27)
53See Section 2.8.3.54i.e. the combined emission limit of a load of which the agreed apparent power is equal to the
total apparent power Ss:i supplied by the busbar.
47
Then, ka can be established in order to satisfy (2.25) as:
ka =Ug/s
max
Ss:x + α
√√√√ n∑i=1,i6=x
(kα
i−xSs:i
) (2.28)
In summary, the suggested allocation policy is given by (2.22) with ka determined
using (2.28). This new policy meets the four key allocation objectives stated above.
Firstly, (2.25) ensures that the set planning levels are not exceeded when all con-
sumers inject their allocated emission limits. Secondly, based on (2.22), customer
installations of equal MVA demand, whether connected at the same busbar or dif-
ferent busbars, receive identical emission limits. Thirdly, (2.22) ensures that larger
customer installations (in MVA demand terms) to receive larger emission levels than
smaller installations. Finally, the absorption capacity of the network is fully utilised
in the sense that at least one busbar will reach the network planning level.
2.10 Chapter Summary
This chapter has provided general information in relation to voltage unbalance, which
include methods of quantification, sources, effects, mitigation techniques, measure-
ment and evaluation procedures, indices and limits.
The key section of the chapter has given a critical discussion on the IEC/TR
61000-3-13 guidelines for voltage unbalance allocation on which the thesis is primarily
based. Step by step procedure of the development of Stage 2 emission limits together
with the related aspects, i.e. the propagation from higher voltage to lower voltage
levels, the propagation from one busbar to other neighboring busbars of a particular
sub-system, and the emission arising as a result of system inherent asymmetries, has
been clearly described establishing the backgrounds for the remaining chapters.
48
The last section of the chapter has covered the counterpart IEC approaches to
harmonics and flicker allocation in relation to their fundamental deficiencies and
suggested revisions. The principles of this revised allocation technique for harmon-
ics/flicker have been explained forming the background for Chapter 6.
Chapter 3
Global Voltage Unbalance in MV
Power Systems due to System
Inherent Asymmetries
3.1 Introduction
As indicated in Chapter 2 by (2.15), the recommended voltage unbalance allocation
approach requires the determination of the factor K ′ue (= 1−Kue) which is system
dependant. For completeness, the definition of K ′ue is replicated here by (3.1):
K ′uex =
(U lines
g/s:x
Ug/s:x
)α
(3.1)
where,
K ′uex - factor K ′ue at any busbar x of any sub-system S (see Figs. 2.6 and 2.7)
Ug/s:x - emission allowance of the busbar x
U linesg/s:x - voltage unbalance at the busbar x arising as a result of line asymmetries that
exist in the sub-system S and the downstream system supplied by the busbar x
49
50
The emission which arises as a result of the asymmetries associated with LV feeders
is not of significant concern owing to their shorter lengths and smaller loading levels.
Thus, when aiming at MV power systems (i.e. where sub-system S is an MV system)
on which this chapter is focused, it is only the local MV lines that are responsible for
the emission U linesg/mv:x.
Considering a simple radial MV network (refer to Fig. 3.1) with an untransposed
overhead single-circuit line (labelled ‘t’), voltage unbalance U tg/mv:rec caused by the
line at its receiving end busbar (labelled ‘rec’) can be expressed as:
U tg/mv:rec =
∣∣∣∣∣Vt−:g/mv:rec
V+:rec
∣∣∣∣∣ (3.2)
where,
|V t−:g/mv:rec| = |V t
−:g/mv:send − (Z−+:tI+:t + Z−−:tI−:t/t + Z−0:tI0:t)| (3.3)
V t−:g/mv:rec - negative sequence voltage caused by the line at its receiving end busbar
V+:rec - positive sequence voltage at the receiving end busbar
Z−+:t, Z−0:t - negative-positive, negative-zero sequence coupling impedances respec-
tively of the line
Z−−:t - negative sequence impedance of the line, which is inherently equal to its pos-
itive sequence impedance Z++:t1
I+:t, I0:t - positive and zero sequence currents respectively in the line
I−:t/t - negative sequence current in the line arising as a result of the asymmetry
associated with the line itself
V t−:g/mv:send - negative sequence voltage caused by the line at its sending end busbar
(labelled ‘send’), which arises as a result of the flow of negative sequence current
1Impedance Z−−:t will be replaced with Z++:t hereafter.
51
(that is caused by the line) through the transformer coupling the upstream system
and the considered system
Upstream HV system
MV line (t)
send
MV system under consideration
rec
Figure 3.1: Simple MV network
Assuming that zero sequence unbalance in the network is controlled through sys-
tem design, maintenance and operation or an ungrounded-neutral system, the term
Z−0:tI0:t can be set to zero. Then, by comparing (3.3) with the IEC approach stated
in Chapter 2 by (2.18), the IEC approach is seen to assume that the negative se-
quence current I−:t/t is not significant enough to introduce considerable influences by
the terms Z++:tI−:t/t and V t−:g/mv:send on the negative sequence voltage |V t
−:g/mv:rec|.
This seems equitable when the line supplies primarily passive loads (e.g. constant
impedance loads) due to the high negative sequence impedance2 associated with such
loads. However, the validity of the above simplification is questionable when the line
supplies an industrial load base containing a large proportion of mains connected
three-phase induction motors, considering the fact that the current unbalance can
2Which is equal to the positive sequence impedance.
52
be 6 to 10 times the supply voltage unbalance [20] for an induction motor due to
its relatively low negative sequence impedance. Referring to Table 2.8.5 (last entry),
IEC/TR 61000-3-13 also indicates that K ′ue is related to the proportion of motor
loads. These emphasise that although by definition K ′ue is a parameter which ac-
counts for system inherent asymmetries it has some degree of load dependency, an
aspect which requires detailed examination.
Objectives of the work presented in this chapter are:
• To investigate the influence of line asymmetries on the global voltage unbalance
in MV power systems, and its dependency on various load types/bases including
three-phase induction motors. The work carried out in this regard in relation
to a radial line is presented in Section 3.2.
• To develop a generalised methodology, covering radial and interconnected net-
works, for the evaluation of the global voltage unbalance in MV systems that
is caused by line asymmetries at nodal level3. This facilitates the assessment of
the nodal K ′ue factors. This is given in Section 3.3.
The proposed methodology is verified in relation to a three-bus test system using un-
balanced load flow analysis in Section 3.4. Section 3.5 summarises the work presented
in the chapter emphasising major conclusions.
3.2 Influence of Line Asymmetries on the Global Emission
and its Dependency on Load Types/Bases
Consider the radial MV-LV system shown in Fig. 3.24 where the MV line is untrans-
posed. The purpose is to assess the voltage unbalance U tg/mv:rec caused by the MV line
3i.e. emissions U linesg/mv:x referring to the system shown in Fig. 3.5.
4Loads, MV-LV coupling transformers and downstream LV busbars supplied by the MV line arerepresented as aggregated elements.
53
at its receiving end busbar while the voltage at the sending end busbar is balanced.
For this, the loads Smv:rec−local and Smv:rec−ds are considered as balanced.
The emission U tg/mv:rec can be expressed by (3.2) where |V t
−:g/mv:rec| given by (3.3)
can be simplified for the considered scenario (i.e. V t−:g/mv:send = 0) as:
|V t−:g/mv:rec| = |Z−+:tI+:t + Z++:tI−:t/t| (3.4)
The behaviour of the negative sequence current I−:t/t or the influence of the term
Z++:tI−:t/t on |V t−:g/mv:rec| may depend on the load type/base supplied by the line.
Thus, the aim of this section is to develop theoretical bases which describe the be-
haviour exhibited by four basic load types5 and also various load bases6 in this regard.
send
Downstream LV system
MV line (t)
rec
Smv:rec-local (MVA)
Smv:rec-ds (MVA)
Figure 3.2: Radial MV-LV system
5i.e. constant impedance, constant current, constant power and three-phase induction motorloads.
6i.e. mixes of various load types.
54
3.2.1 Constant Impedance (Z) Loads
When the loads Smv:rec−local and Smv:rec−ds (Fig. 3.2) represent constant impedance
loads (balanced: decoupled and equal positive, negative and zero sequence impedances)7,
(3.4) can be re-expressed8 using the decoupled nature of sequence impedances as:
|V t−:g/mv:rec| ≈
∣∣∣∣Z−+:tI+:t
(Z−−:rec
Z−−:send
)∣∣∣∣ (3.5)
where, Z−−:rec, Z−−:send - downstream9 negative sequence impedances seen at the
receiving end and the sending end busbars respectively of the MV line. Employing
the fact that the negative sequence impedances associated with all system components
are equal to their positive sequence impedances, (3.5) can be re-expressed as:
|V t−:g/mv:rec| ≈ |Z−+:tI+:t| (1− V Rt) (3.6)
where, V Rt - voltage regulation of the line, which is defined as the ratio between
the positive sequence voltage drop across the line (i.e. Z++:tI+:t) and the sending
end positive sequence voltage. That is, for constant impedance loads, the negative
sequence current I−:t/t behaves in such a manner that the term Z++:tI−:t/t causes
the negative sequence voltage |V t−:g/mv:rec| to be smaller than the term |Z−+:tI+:t|
considered in the IEC approach: (2.18) by the factor (1− V Rt).
7i.e. the three phase supply is connected to equal impedances where each of the impedances ismodelled as per O.5 - O.6 (Appendix O, Section O.4.2) with λp = λq = 2.
8Derivation of (3.5) is given in Appendix A.9This term, which will be using frequently in the thesis, is employed to indicate an impedance
that is seen at a particular point into the direction where the power flows.
55
3.2.2 Constant Current (I) Loads
The negative sequence current I−:t/t can be considered to be negligible when the line
supplies constant current loads (balanced)10, as such loads draw equal magnitudes of
three phase currents regardless of the prevailing voltage condition. Hence, (3.4) can
be simplified for constant current loads as:
|V t−:g/mv:rec| ≈ |Z−+:tI+:t| (3.7)
That is, in contrary to the case of constant impedance loads, the IEC approach: (2.18)
requires no modifications for constant current loads.
3.2.3 Constant Power (PQ) Loads
As the linearisation of system equations and simplifying assumptions (as in the case
of constant current loads) are not supported by constant power loads11, through
careful examination of the results obtained from unbalanced load flow analysis, (3.8)
is established considering operating scenarios most likely to occur in practice as a
close approximation to the negative sequence voltage |V t−:g/mv:rec|:
|V t−:g/mv:rec| ≈ |Z−+:tI+:t| (1− V Rt)
β (3.8)
where, β ≈ −1 and −2 for low (∼ 0.9) and high (∼ 1) lagging power factor (pf)
conditions respectively. That is, for constant power loads, the term Z++:tI−:t/t causes
10i.e. the three phase supply is connected to equal loads where each of the loads is modelled asper O.5 - O.6 (Appendix O, Section O.4.2) with λp = λq = 1. In this case, the three-phase loadbank draws equal magnitudes of three phase currents regardless of the voltage condition (includingunbalance) which prevails at the terminals of the load bank.
11i.e. when the three phase supply is connected to equal loads where each of the loads is modelledas per O.5 - O.6 (Appendix O, Section O.4.2) with λp = λq = 0. In this case, the power drawn byeach of the three phases of the load bank is equal and does not depend on the voltage condition(including unbalance) which prevails at the terminals of the load bank.
56
the negative sequence voltage |V t−:g/mv:rec| to be greater than the term |Z−+:tI+:t|
considered in the IEC approach: (2.18) by the factor (1− V Rt)β.
3.2.4 Induction Motor (IM) Loads
Three-phase induction motors12 can be represented using decoupled, unequal and
constant13 sequence impedances. Hence, (3.4) can be re-expressed14 in the form given
by (3.5). This can be expressed15 in terms of the system, line and load characteristics
and the system operating conditions as:
|V t−:g/mv:rec| ≈ |Z−+:tI+:t|
1
1 +(
V Rt
1−V Rt
)(1
1ks
+ 1ksc−lvagg
) (3.9)
where,
ks - ratio between the positive and negative sequence impedances of the aggregated
motor load supplied by the aggregated LV busbar16
ksc−lvagg =Ssc−lvagg
Smv:rec−ds
Smv:rec−ds - total load (in MVA) supplied by the aggregated LV busbar (see Fig. 3.2)
Ssc−lvagg = (Vn−lv)2
|Z++:sys−lv |, the short-circuit capacity (in MVA) at the aggregated LV
busbar, that is derived using the positive sequence system impedance Z++:sys−lv which
exists between the busbar under evaluation and the downstream LV busbar17
Vn−lv - nominal line-line voltage of the LV system
12Induction motors are considered to be supplied at the LV level, which is generally the case.Hence, for this case of motor loads, Smv:rec−local = 0.
13For a given motor speed.14Employing the decoupled nature of sequence impedances.15This specific case with klv = 1 and km = 1 can be deduced using the generalised expression (3.14)
of which the derivation is given in Appendix C.16Typically, ks can be in the range of 5 to 7.17This definition for Ssc−lvagg
, which is given in general, will be applied in Chapters 4 and 5 aswell. For the particular case of the network shown in Fig. 3.2, Z++:sys−lv is equal to the impedanceZ++:ml−lv of the MV-LV coupling transformer, and the MV busbar rec is the busbar under evalua-tion.
57
Equation (3.9) implies that the negative sequence current I−:t/t in the presence of
induction motor loads behaves in such a manner that the term Z++:tI−:t/t causes
the negative sequence voltage |V t−:g/mv:rec| to be smaller than the term |Z−+:tI+:t|
considered in the IEC approach: (2.18) by the factor 1
1+“
V Rt1−V Rt
”0@ 11
ks+ 1
ksc−lvagg
1A .
0
40
80
50 150 250 350 450 550 650
|Vt -:g
/mv:
rec|
(V)
Z loads: LFZ loads: Eq. (3.6)I loads: LFI loads: Eq. (3.7)PQ loads: LFPQ loads: Eq. (3.8)IM loads: LFIM loads: Eq. (3.9)
IEC approach: |Z-+:t I+:t|
121086531
|I+:t| (A)
VRt (%)
Figure 3.3: Variation of |V t−:g/mv:rec| with |I+:t| (V Rt values corresponding to various
|I+:t| are also indicated) for the four basic load types
3.2.5 Discussion
Taking a generalised view, the additional factors associated with the above proposed
expressions with respect to the IEC approach can be expressed in a form (1− V Rt)γ
for the three passive load types (where γ = 1 for constant impedance loads, γ = 0
for constant current loads, and −2 ≤ γ ≤ −1 for constant power loads). Consider-
ing most practical circumstances where V Rt < 10%, the factor (1 − V Rt)γ can be
approximated to unity (in other words I−:t/t ≈ 0) supporting the IEC approach for
passive loads in general.
58
The additional factor associated with the proposed expression for induction motor
loads is considerably smaller18 than that for passive loads, implying that the IEC
approach is conservative for induction motor loads as expected.
Fig. 3.3 illustrates the variation of the negative sequence voltage |V t−:g/mv:rec| with
the current |I+:t| for the four load types19 obtained employing the test system20 de-
scribed in Appendix B. This shows the variations established using the proposed
expressions (3.6) - (3.9) in comparison to the results obtained using unbalanced load
flow (LF) analysis21, justifying the new formulation and also the above discussion.
3.2.6 Mixes of Passive and Induction Motor Loads
When the MV line supplies a mix of passive loads (at MV and/or LV) and induction
motors (at LV), the negative sequence current I−:t/t can be decomposed as:
I−:t/t = I−:t/ps + I−:t/im (3.10)
where, I−:t/ps, I−:t/im - negative sequence currents (referred to MV side) in passive
(PS) and induction motor (IM) branches respectively arising as a result of the asym-
metry of the MV line. Employing (3.10), (3.4) can be expanded as:
|V t−:g/mv:rec| = |Z−+:tI+:t + Z++:tI−:t/ps + Z++:tI−:t/im| (3.11)
As discussed in Section 3.2.5, the current component I−:t/ps associated with passive
loads is not significant enough to introduce considerable influence on the negative
18e.g. 0.6 and 0.9 for induction motor loads and constant impedance loads respectively withksc−lvagg = 20, ks = 6.7 and V Rt = 10%.
19Supplied at the LV level.20Where, |Z−+:t| = 0.112Ω, ksc−lvagg
≈ 19 and ks = 6.7.21Refer to Appendix O. Out of the two models proposed for three-phase induction motors (Section
O.4.6), the impedance type model is used in the results presented in this thesis.
59
sequence voltage |V t−:g/mv:rec|. Thus, (3.11) can be simplified as:
|V t−:g/mv:rec| ≈ |Z−+:tI+:t + Z++:tI−:t/im| (3.12)
Equation (3.12) can be rearranged in a form similar to that of (3.5) as:
|V t−:g/mv:rec| ≈
∣∣∣∣Z−+:tI+:t
(Z−−:rec−im
Z−−:send−im
)∣∣∣∣ (3.13)
where, Z−−:rec−im, Z−−:send−im - downstream negative sequence impedances seen at
the receiving end and the sending end busbars respectively of the MV line taking
into account only induction motors supplied by the downstream LV busbar. Re-
expressing22 (3.13):
|V t−:g/mv:rec| ≈ |Z−+:tI+:t|
1
1 +(
V Rt
1−V Rt
)(klv
1kskm
+ 1ksc−lvagg
) (3.14)
where,
km - ratio between the rated motor load (in MVA) and the total load (in MVA) sup-
plied by the aggregated LV busbar
klv = Smv:rec−ds
Smv:rec−local+Smv:rec−ds, the fraction of the LV loads supplied by the MV busbar
rec under evaluation
The emission U tg/mv:rec can be established in a generalised form by substituting (3.14)
in (3.2) as:
U tg/mv:rec ≈
|Z−+:tI+:t|
1 +(
V Rt
1−V Rt
)(klv
1kskm
+ 1ksc−lvagg
) × 1
|V+:rec|(3.15)
22Derivation of (3.14) is given in Appendix C.
60
Fig. 3.4 illustrates the variation of U tg/mv:rec with km established using (3.15) in
comparison to the results obtained using unbalanced load flow analysis for the test
system23 in relation to three cases where:
• klv = 1 (i.e. Smv:rec−local = 0)
• klv = 0.5
• klv = 0 (i.e. Smv:rec−ds = 0, which implies that no motor loads are supplied by
the MV line)
The results presented in Fig. 3.4 correspond to a selected operating scenario where
the MV line supplies a total of 10MVA load at 0.9 lagging pf resulting in |I+:t| ≈ 470A,
V Rt ≈ 8.5% and |V+:rec| ≈ 7.2kV (1pu). These results confirm the above basis given
by (3.15) which describes the behaviour of various load bases in relation to the global
emission arising as a result of line asymmetries, also demonstrating the dependency
of the global emission on the motor proportion.
3.3 Methodology for Evaluating the Global Emission Arising
Due to Line Asymmetries
Considering the interconnected network shown in Fig. 3.524 as an MV system with
untransposed lines (t12, t13, t23, ..., tij, ...), the system representation of any busbar x
of the system can be taken as per Fig. 3.625 where the downstream system represents
an aggregated LV system. For the purpose of assessing the global emission arising
as a result of the line asymmetries, the voltage at the upstream system and all loads
supplied by the network are considered to be balanced.
23Appendix B: |Z−+:t| = 0.112Ω, ksc−lvagg≈ 19, ks = 6.7.
24Reproduction of Fig. 2.6.25Reproduction of Fig. 2.7.
61
0.4
0.5
0.6
0.7
0.8
0.0 0.2 0.4 0.6 0.8 1.0km
Ut g/
mv:
rec (%
)
Load flow resultsEq. (3.15)
klv = 1 (Smv:rec-local = 0)
klv = 0 (Smv:rec-ds = 0, i.e. no motor loads are supplied)
klv = 0.5
Figure 3.4: Variation of U tg/mv:rec with km for the cases where klv = 1, klv = 0.5 and
klv = 0
… … Busbar 1
Busbar 3
Busbar x
MV sub-system
Busbar n
Busbar 2
Upstream HV system
Figure 3.5: Interconnected MV sub-system
62
Downstream system supplied by the busbar x
Busbar x
Smv:x-local
Smv:x-ds
Figure 3.6: System representation of any busbar x of the MV system shown in Fig. 3.5
Employing the linearity of negative sequence variables [1], the resultant negative
sequence voltage V lines−:g/mv:x arising as a result of the interaction of the untransposed
lines t12, t13, t23, ..., tij, ... at any busbar x can be written as:
V lines−:g/mv:x = V t12
−:g/mv:x + V t13−:g/mv:x + V t23
−:g/mv:x + ... + Vtij−:g/mv:x + ... (3.16)
where, Vtij−:g/mv:x - negative sequence voltage caused by any line tij on its own at the
busbar x. Then, the emission U linesg/mv:x can be expressed as:
U linesg/mv:x =
∣∣∣∣∣Vlines−:g/mv−x
V+:x
∣∣∣∣∣ (3.17)
where, V+:x - positive sequence voltage at the busbar x.
Extending the nodal equations [I] = [Y ][V ]26 to the sequence domain, xth element
of [I], (x, y)th element of [Y ] and xth element of [V ] can be written respectively for
the considered MV network as:
26Where, [I], [Y ] and [V ] are the nodal current, admittance and voltage matrices respectively.
63
Ix =
I0:x
I+:x
I−:x
(3.18)
Yxy =
Y00:xy Y0+:xy Y0−:xy
Y+0:xy Y++:xy Y+−:xy
Y−0:xy Y−+:xy Y−−:xy
(3.19)
Vx =
V0:x
V+:x
V−:x
(3.20)
where,
Ix - xth element of [I] or the nodal current vector at the busbar x, which is considered
to be negative when leaving the node
Iλ:x - λ (= 0, +,−) sequence component of Ix
Yxy - (x, y)th element of [Y ], where:
• for x = y, Yxy is equal to the summation of all network admittances connected
to the busbar x (= y)
• for x 6= y, Yxy is equal to the negative value of the admittance of the network
element which connects the busbar x and any other busbar y
Yλ∆:xy - λ−∆ sequence coupling admittance element of Yxy for λ 6= ∆, and λ sequence
impedance element of Yxy for λ = ∆, where λ = 0, +,− and ∆ = 0, +,−
Vx - xth element of [V ] or the voltage at the busbar x
Vλ:x - λ (= 0, +,−) sequence component of Vx
64
The nodal equations [I] = [Y ][V ] of which the elements are given by (3.18) - (3.20)
can be expanded while setting zero sequence voltages and currents to zero to establish
negative sequence nodal currents as:
−I−:1
−I−:2
...
−I−:n
=
Y−+:11 Y−+:12 . . . Y−+:1n
Y−+:21 Y−+:22 . . . Y−+:2n
......
......
Y−+:n1 Y−+:n2 . . . Y−+:nn
V+:1
V+:2
...
V+:n
+
+
Y−−:11 Y−−:12 . . . Y−−:1n
Y−−:21 Y−−:22 . . . Y−−:2n
......
......
Y−−:n1 Y−−:n2 . . . Y−−:nn
V−:1
V−:2
...
V−:n
(3.21)
Equation (3.21) can be rewritten in a concise form also employing the inherent rela-
tionship: Y−−:xy = Y++:xy (for x = y and x 6= y) as:
−[I−] = [Y−+][V+] + [Y++][V−] (3.22)
Emphasising that these negative sequence nodal currents and voltages arise as a re-
sult of the untransposed lines of the considered network, the matrices [I−] and [V−]
are relabelled as [I−:lines] and [V lines−:g/mv] respectively for consistency. As shown in Sec-
tion 3.2 above, the influence of the negative sequence currents [I−:lines] on the negative
sequence voltages [V lines−:g/mv] should be taken into account in the presence of consider-
able proportions of induction motor loads. Thus, the negative sequence nodal current
I−:lines/x at any busbar x can be generally written when the busbar supplies a mix of
passive (at MV and/or LV) and motor (at LV) loads as:
65
I−:lines/x ≈ Y−−:x−im V lines−:g/mv:x (3.23)
where, Y−−:x−im - downstream negative sequence admittance seen at the busbar x
taking into account only induction motors supplied by the downstream LV busbar.
This admittance Y−−:x−im, which is inherently inductive owing to the inductive nature
of the associated induction motor negative sequence impedances and MV-LV trans-
former impedances, can be expressed27 in terms of the system and load characteristics,
system operating conditions and downstream load composition as:
Y−−:x−im ≈ −j
(klv:x
1ksc−lvagg :x
+ 1ks:xkm:x
)(√3 |I+:x|Vn−mv
)(3.24)
where,
ks:x, ksc−lv:x, km:x, klv:x - as defined for (3.9) and (3.14)28
Vn−mv - nominal line-line voltage of the MV system
Substitution of (3.23) in (3.22) and rearrangement gives:
[V lines−:g/mv]n×1 ≈ −[Y ′
++]−1n×n[Y−+]n×n[V+]n×1 (3.25)
where,
Y ′++:xy ≈ Y++:xy + Y−−:x−im for x = y
Y ′++:xy = Y++:xy for x 6= y
27Derivation of (3.24) is given in Appendix D.28Additional subscript ‘x’ indicates the quantities corresponding to the busbar x.
66
That is, taking the nodal positive sequence voltages as known quantities as they
can be easily obtained from conventional balanced load flow analysis, the negative
sequence voltage V lines−:g/mv:x at any busbar x can be established using (3.25).
3.4 Verification of the Methodology
The proposed methodology is applied to the three-bus MV system (60Hz, 12.47kV,
three-wire) shown in Fig. 3.7. Considered operating scenario and resulting positive
sequence system conditions29 are also indicated in Fig. 3.7. Lengths of the lines
which are taken as identical in construction (including the phase positioning) and
untransposed are shown alongside the lines. Relevant admittance data of the lines
are30:
• positive sequence admittance = (1.0098− j2.0630)Skm
• negative-positive sequence coupling admittance = (0.0258 + j0.1821)Skm
Busbars 1 and 3 supply MV loads with equal compositions of constant impedance
and constant power elements. That is, km:x = 0 implying that Y−−:x−im ≈ 0 for
x = 1 and 3. Busbar 2 supplies loads at the LV level which account for 40% of the
total load supplied by the system. Two cases based on the type of LV loads are
considered:
• Case 1 - LV loads represent passive elements31. That is, km:2 = 0 implying that
Y−−:2−im ≈ 0.
• Case 2 - LV loads represent three-phase induction motors. That is, km:2 = 1.
Aggregation of 50hp motors of which the details are given in Appendix B is
considered. This results in an admittance Y−−:2−im ≈ −j0.1316S.
29Nodal voltages and line currents, which are obtained using load flow analysis.30See Appendix B for further details.31With equal compositions of constant impedance and constant power loads.
67
Fig. 3.8 illustrates the emissions U linesg/mv:x at the MV busbars for the two cases
listed above. This shows the results established using the proposed methodology32 in
comparison to the results obtained using unbalanced load flow analysis validating the
proposed technique. Further, these results reveal that the presence of considerable
proportions of induction motor loads increases the emission U linesg/mv:x when x represents
the busbar that is directly connected to the upstream system (e.g. busbar 1 of the
system shown in Fig. 3.7), compared to the case where only passive loads are supplied
by the network. In addition, induction motor loads tend to reduce the emission U linesg/mv:x
at all other busbars (e.g. busbars 2 and 3 of the system shown in Fig. 3.7), compared
to the case where only passive loads exist.
204A
MV busbar 2 LV (460V)
MV busbar 1 (1.05pu, -4.070)
MV busbar 3 (1.01pu, -5.570)
10km 135A
10km 59A
5km 152A
(0.98pu, -6.740) 4MVA 0.9 lagging pf
Upstream HV (66kV) system (1.05pu, 00)
4MVA 0.9 lagging pf 2MVA
0.9 lagging pf
HV-MV coupling transformer – 12MVA, winding resistance = 1%, leakage reactance = 10%, secondary tap setting = 1.05pu MV-LV coupling transformer – aggregated representation of fully loaded 1MVA transformers with winding resistance = 1%, leakage reactance = 5% and secondary tap setting = 1.05pu
Figure 3.7: Three-bus MV test system considered for applying the proposed method-ology
32Application of the methodology to the test system is described in Appendix E.
68
0.0
0.2
0.4
0.6
0.8
1 2 3Busbar
Load flow: Case 1Methodology: Case 1Load flow: Case 2Methodology: Case 2
(%
)/lines
xm
vg
U−
Figure 3.8: Emissions U linesg/mv:x for the three-bus MV test system for the two cases
where km:2 = 0 and km:2 = 1
3.5 Chapter Summary
This chapter has addressed the global voltage unbalance in MV power systems which
arises as a result of line asymmetries. This is a key aspect in assessing emission limits
to individual installations connected to MV power systems essentially based on the
IEC/TR 61000-3-13 recommendations.
The dependency of the global emission on various load types/bases including three-
phase induction motors has been examined in relation to an untransposed radial MV
line. The following major conclusions can be drawn from the study:
• The approach given in IEC/TR 61000-3-13 to assess the influence of an asym-
metrical radial line on the global emission can be applied to an MV network
only when the system supplies primarily passive loads. In this case, the impact
of the negative sequence currents arising as a result of line asymmetries on the
global emission is insignificant.
69
• The IEC approach has been seen to be conservative when the network supplies
a large proportion of induction motor loads. In this case, the negative sequence
currents arising as a result of line asymmetries make a significant influence
on the global emission. The degree of this influence is dependant primarily
on the proportion of motor loads, and secondarily on the system and motor
characteristics.
A systematic approach, covering radial and interconnected networks, for evalu-
ating the global emission caused by line asymmetries at nodal level taking the line,
system and load characteristics, system operating conditions and downstream load
composition into account has been proposed. The results established using the pro-
posed methodology in relation to a three-bus test system has been seen to be in close
agreement with the results obtained using unbalanced load flow analysis. Further,
these results clearly demonstrated the following:
• The presence of considerable proportions of induction motor loads at down-
stream LV systems increases the global emission at the MV busbar which is
directly connected to the upstream system, compared to the case where only
passive loads exist.
• Induction motor loads tend to reduce the global emission levels at all other
busbars of the network, compared to the case where only passive loads exist.
Chapter 4
Global Voltage Unbalance in HV
Power Systems due to System
Inherent Asymmetries
4.1 Introduction
The global voltage unbalance in MV power systems which arises as a result of line
asymmetries and the dependency of this emission on the proportion of induction
motor loads has been investigated, emphasising the limitations associated with the
IEC approach1 in Chapter 3. As a continuation of the work presented in Chapter 3,
this chapter focuses on HV systems with regard to the same subject.
When aiming at an HV network (i.e. where sub-system S is an HV system),
the emission U linesg/hv:x at any busbar x can arise as a result of the local HV lines and
also the MV lines that is present in the downstream system supplied by the busbar.
Considering a simple radial HV network (refer to Fig. 4.1) with an asymmetrical
HV line (labelled ‘t’) which supplies an untransposed MV line (labelled ‘td’) at the
1Stated in Chapter 2 by (2.18).
70
71
downstream, the emission U t+tdg/hv:rec caused by the lines2 at the receiving end busbar
(labelled ‘rec’) of the HV line can be expressed as:
U t+tdg/hv:rec =
∣∣∣∣∣Vt+td−:g/hv:rec
V+:rec
∣∣∣∣∣ (4.1)
where,
|V t+td−:g/hv:rec| = |V t
−:g/hv:rec + V td−:g/hv:rec| (4.2)
V t+td−:g/hv:rec - negative sequence voltage caused both by the HV and MV lines at the
receiving end busbar of the HV line
V t−:g/hv:rec - negative sequence voltage caused only by the HV line3 at its receiving end
busbar
V td−:g/hv:rec - negative sequence voltage caused only by the MV line4 at the receiving
end busbar of the HV line
V+:rec - positive sequence voltage at the receiving end busbar of the HV line
As in the case of the radial MV network considered in Chapter 35, the negative
sequence voltage V t−:g/hv:rec that is caused by the local HV line can be expressed,
ignoring zero sequence unbalance, as:
V t−:g/hv:rec = V t
−:g/hv:send − (Z−+:tI+:t + Z−−:tI−:t/t) (4.3)
The symbols in (4.3) are as defined in Chapter 3 for (3.3)6. The second term V td−:g/hv:rec
in (4.2), which is caused by the downstream MV line can be expressed as7:
2i.e. both HV and MV lines.3i.e. taking the MV line as balanced.4i.e. taking the HV line as balanced.5See Section 3.1.6Note that t represents the local HV line here.7The HV line is treated as balanced in this case, i.e. Z−+:t = 0.
72
V td−:g/hv:rec = V td
−:g/hv:send − Z−−:tI−:td/t (4.4)
where,
I−:td/t - negative sequence current in the HV line, which arises as a result of the
asymmetry associated with the MV line
V td−:g/hv:send - negative sequence voltage caused by the MV line at the sending end
busbar (labelled ‘send’) of the HV line, which arises as a result of the flow of the
negative sequence current (that is caused by the MV line) through the transformer
coupling the upstream system and the considered HV system
Upstream system
HV line (t)
HV system under consideration
send
rec
MV line (td)
Figure 4.1: Simple HV network
73
By comparing (4.2) of which the two terms are given by (4.3) and (4.4) respectively,
with the IEC approach, it can be identified that the IEC approach infers the following
when applied to HV networks:
• Negligible influence from the negative sequence currents that are caused by the
local HV lines, which is similar to the associated simplification in assessing MV
systems as discussed in Chapter 3. As evident from Chapter 3, this simplifica-
tion can lead to a significant degree of error when the load supplied by an HV
network accounts for a large proportion of induction motors.
• Negligible influence from the negative sequence currents that are caused by the
downstream MV lines. In other words, the IEC approach leaves out the presence
of the downstream line asymmetries in evaluating the global emission in HV
networks. However, the asymmetry associated with MV lines, which was seen
in Chapter 3 to introduce negative sequence currents that are significant enough
to influence the local emission in the presence of large motor proportions, can
also make a considerable impact on the global emission in HV networks. This
demands additional investigations on the subject applicable to HV systems.
Objectives of the work presented in this chapter are:
• To investigate the influence of line asymmetries, which include the local HV lines
and downstream MV lines, on the global emission in HV systems in the presence
of induction motor loads. The work carried out in this regard in relation to a
simple radial network is presented in Section 4.2.
• To develop a generalised methodology, covering interconnected network envi-
ronments, for the evaluation of the global voltage unbalance in HV systems
that is caused by the local and downstream line asymmetries at nodal level.
This is covered in Section 4.3.
74
The proposed methodology is applied to a three-bus test system and the results are
compared with those obtained using unbalanced load flow analysis in Section 4.4.
Section 4.5 further verifies the proposed methodology employing the IEEE 14-bus
test system which supplies passive loads locally8. Section 4.6 summarises the work
presented in the chapter emphasising major conclusions.
4.2 Influence of Line Asymmetries on the Global Emission
in the Presence of Induction Motor Loads
Consider the radial HV-MV-LV network shown in Fig. 4.29 where the HV and MV
lines of interest are untransposed. The purpose is to assess the emission U t+tdg/hv:rec that
is caused by the line asymmetries at the receiving end busbar of the HV line while
the voltage at the sending end busbar is balanced. For this, all loads supplied by the
system are considered to be balanced.
The emission U t+tdg/hv:rec can be expressed by (4.1) where the two components of
the resultant negative sequence voltage |V t+td−:g/hv:rec| which are given by (4.3) and (4.4)
respectively can be simplified for the considered scenario (i.e. V t−:g/hv:send = 0 and
V td−:g/hv:send = 0) as10:
V t−:g/hv:rec = −(Z−+:tI+:t + Z++:tI−:t/t) (4.5)
V td−:g/hv:rec = −Z++:tI−:td/t (4.6)
8i.e. at the HV level itself.9Fig. 4.2 gives a generalised representation of the downstream system supplied by the HV line.
LV busbars, where motor loads are connected, which can be supplied through MV lines (labelled‘LVr’ where subscript ‘r’ indicates the receiving ends of the MV lines) and also directly by HV-MVcoupling transformers (i.e. not through MV lines, labelled as ‘LVs’ where subscript ‘s’ indicatesthe sending ends of the MV lines) are separately considered. Loads, HV-MV and MV-LV couplingtransformers, and downstream MV lines and MV and LV busbars supplied by the HV line arerepresented as aggregated elements.
10The impedance Z−−:t is replaced with Z++:t.
75
HV line (t)
MV line (td)
LVr
LVs
rec
Downstream system supplied by the HV line
send
Figure 4.2: Radial HV-MV-LV system
The resultant negative sequence voltage |V t+td−:g/hv:rec|, which is the absolute value of
the sum of (4.5) and (4.6), can be generally expressed11 when the network supplies a
mix of passive loads (at HV, MV and/or LV) and induction motors (at LV) as:
|V t+td−:g/hv:rec| ≈ |Z−+:tI+:t (1− µ− ζ)| (4.7)
where,
µ =∣∣∣ Z++:t
Z−−:send−im
∣∣∣11Derivation of (4.7) is given in Appendix F.
76
Z−−:send−im - downstream negative sequence impedance seen at the sending end bus-
bar of the HV line taking into account only induction motors supplied by the LV
busbars (i.e. LVs and LVr)
ζ = µktdknσ
σ =Z−+:td−hv
Z−+:t(a complex quantity)
Z−+:td−hv - negative-positive sequence coupling impedance, referred to HV side, of
the MV line
ktd - ratio between the total load supplied by the MV line, and the total load supplied
by the HV busbar rec
kn - ratio between the downstream negative sequence impedance12 seen at the sending
end busbar of the MV line taking into account the total motor load13 supplied by the
HV line, and the downstream negative sequence impedance seen at the sending end
busbar of the MV line taking into account only the motor load14 supplied the MV line
The emission U t+tdg/hv:rec can be established in a generalised form by substituting (4.7) in
(4.1) as:
U t+tdg/hv:rec ≈
∣∣∣∣Z−+:tI+:t (1− µ− ζ)
V+:rec
∣∣∣∣ (4.8)
The factors µ and ζ (= µktdknσ) represent the impact of the term Z++:tI−:t/t in
(4.5) and of the term given by (4.6) or of the downstream MV line respectively on the
emission U t+tdg/hv:rec in the presence of induction motor loads. On the whole, the term
µ + ζ = µ (1 + ktdknσ) represents the overall influence introduced by motor loads,
of which the magnitude can be greater or smaller than µ and ζ noting the phase
angle involved with σ. For the purpose of demonstrating the level of this influence,
12Ignoring the presence of any passive loads.13i.e. the motor load supplied both by the busbars LVs and LVr.14i.e. the motor load supplied only by the busbar LVr.
77
the factor µ is expressed15 in terms of the system and load characteristics, system
operating conditions and downstream load composition for a simplified case where
motor loads are supplied only by the MV line at the busbar LVr16 as:
µr ≈1
1 +(1−V Rt)(1−V Rtd
)2„
1ksrkmr
+ 1ksc−lvragg
«V Rtklvr
(4.9)
where,
µr - factor µ corresponding to the simplified case where motor loads are supplied only
by the busbar LVr
V Rt, V Rtd - voltage regulations of the HV and MV lines respectively17
ksr - ratio between the positive and negative sequence impedances of the aggregated
motor load supplied by the busbar LVr18
kmr - ratio between the rated motor load (in MVA) and the total load (in MVA)
supplied by the busbar LVr
klvr - ratio between the total load (in MVA) supplied by the busbar LVr, and the
total load (in MVA) supplied by the HV busbar rec under evaluation
ksc−lvragg - ratio between the short-circuit capacity19 (in MVA) at the aggregated
busbar LVr, and the the total load (in MVA) supplied by the busbar LVr
15Derivation of (4.9) is given in Appendix G.16i.e. the busbar LVs, which is directly supplied by the HV-MV coupling transformer, supplies
primarily passive loads resulting in a kn = 1.17Voltage regulation is defined as the ratio between the positive sequence voltage drop across the
line (e.g. Z++:tI+:t for the HV line), and the sending end positive sequence voltage.18Typically, ksr can be in the range of 5 to 7.19Which is derived using the positive sequence system impedance Z++:sys−lv = Z++:hm−lv +
Z++:td−lv +Z++:mlr−lv that exists between the HV busbar rec under evaluation and the downstreambusbar LVr, where Z++:hm−lv, Z++:td−lv, Z++:mlr−lv are the positive sequence impedances, referredto LV, of the HV-MV coupling transformer, MV line and MV-LV coupling transformer supplyingthe busbar LVr respectively.
78
Fig. 4.3 illustrates the variation of U t+tdg/hv:rec with klvr established using (4.8) in
comparison to the results obtained using unbalanced load flow analysis for the sim-
plified test case20 described in Appendix H in relation to two cases where:
• kmr = 0 (i.e. busbar LVr supplies only passive loads)
• kmr = 1 (i.e. busbar LVr supplies only induction motors)
The operating scenario considered corresponds to |I+:t| ≈ 490A, V Rt ≈ 7%, V Rtd ≈
9% and |V+:rec| ≈ 39.6kV (1.04pu). Values of ksc−lvragg and σ corresponding to various
klvr are given in Appendix H. Further, the levels of the influence of the factors µ21
(i.e. of the term Z++:tI−:t/t) and ζ22 (i.e. of the downstream MV line) on the emission
in each case are also indicated in Fig. 4.323. Arising from these results, the following
can be concluded:
• For passive loads, the influence of the factors µ and ζ on the emission U t+tdg/hv:rec
is insignificant implying that the IEC approach can be accepted only when the
line supplies primarily passive loads also in assessing HV systems. That is, in
the case of passive loads, the local HV lines are totally responsible for the global
emission and the contribution made by the downstream MV lines is negligible.
• The presence of induction motors affects the level of emission U t+tdg/hv:rec given by
the IEC approach noticeably24. This dependency of the global emission on the
20Where, |Z−+:t| = 0.5226Ω, ksr = 6.7, kn = 1, ktd= klvr.
21For the case of kmr = 0, µ is the difference between the results established using (4.8) forkmr = 0, and those obtained using unbalanced load flow analysis for kmr = 0 without introducingthe effects of the downstream MV line asymmetry. For the case of kmr = 1, µ is the differencebetween the results established using (4.8) for kmr = 0, and those obtained using unbalanced loadflow analysis for kmr = 1 without introducing the effects of the downstream MV line asymmetry.
22For the both cases of kmr = 0 and kmr = 1, ζ is the difference between the results establishedusing unbalanced load flow analysis with and without introducing the effects of the downstream MVline asymmetry.
23The overall influence introduced by motor loads on the emission for the test case is given by thesummation µ + |ζ| noting that the phase angle (see Table H.1) involved with σ is zero (i.e. phaseangles of the impedances Z−+:t and Z−+:td
of the HV and MV lines respectively are equal).24e.g. motor loads weighted only 20% cause 10% reduction in the emission compared to that when
only passive loads exist.
79
motor proportion can be seen in two forms which are explained by the factors µ
and ζ respectively. Prominently, in the case of motor loads, not only the local
HV line but also the downstream MV line are equally responsible in determining
the global emission.
0.4
0.5
0.6
0.7
0.0 0.2 0.4 0.6 0.8 1.0
Load flow resultsEq. (4.8)
for kmr = 0
for kmr = 1
|| :: dtt IZ ++−ζ
(%)
:/dt
tre
chv
gU
+
lvrk
kmr = 1caused by both t and td
kmr = 1caused only by t
kmr = 0 caused only by t
caused by both t and td
for kmr = 0
for kmr = 1|| :: dtt IZ ++−ζ
|| :: dtt IZ ++−μ
|| :: dtt IZ ++−μ
Figure 4.3: Variation of U t+tdg/hv:rec with klvr for the two cases where kmr = 0 and
kmr = 1
4.3 Methodology for Evaluating the Global Emission Arising
Due to Line Asymmetries
Considering the network shown in Fig. 4.425 as an HV system with asymmetrical lines
(t12, t13, t23, ...), the system representation of any busbar x of the system can be taken
as per Fig. 4.5. This system supplied by the busbar x consists of a radial26 MV-LV
network with an untransposed MV line td:x. For assessing the global emission arising
25Reproduction of Fig. 2.6.26Which is the usual practice.
80
as a result of the line asymmetries, the voltage at the upstream system and all loads
supplied by the network are considered to be balanced.
… … Busbar 1
Busbar 3
Busbar x
HV sub-system
Busbar n
Busbar 2
Upstream EHV system
Figure 4.4: Interconnected HV sub-system
As seen in Section 4.2 above, the global voltage unbalance in HV networks is
determined not only by the local line asymmetries but also by the downstream MV
lines asymmetries when the network supplies considerable proportions of motor loads.
Thus, the resultant negative sequence voltage V lines−:g/hv:x which arises as a result of the
line asymmetries at the busbar x can be generally written as [1]:
V lines−:g/hv:x = (V t12
−:g/hv:x + V t13−:g/hv:x + V t23
−:g/hv:x + ... + Vtij−:g/hv:x + ...)
+ (V td:1
−:g/hv:x + V td:2
−:g/hv:x + ... + V td:x
−:g/hv:x + ... + V td:i
−:g/hv:x + ...)
(4.10)
where,
Vtij−:g/hv:x - negative sequence voltage caused by any local HV line tij on its own at
the busbar x
81
V td:i
−:g/hv:x - negative sequence voltage caused by any downstream MV line td:i on its
own at the busbar x
Then, the emission U linesg/hv:x and the factor K ′uex can be expressed in the forms which
are similar to (3.17) and (3.1) respectively given in Chapter 3.
Busbar x
MV line (td:x)
LVr:x
LVs:x
Downstream system supplied by the HV busbar x
Figure 4.5: System representation of any busbar x of the HV system shown in Fig. 4.4
As described in Chapter 3 (Section 3.3), the nodal negative sequence currents
([I−:lines]) which arise as a result of the line asymmetries can be written in terms
of the nodal negative-positive sequence coupling admittances ([Y−+]), nodal positive
sequence admittances ([Y++]) and nodal positive ([V+]) and negative([V lines
−:g/hv])
se-
quence voltages as:
82
−[I−:lines] = [Y−+][V+] + [Y++][V lines−:g/hv] (4.11)
The current I−:lines/x which is the xth element of [I−:lines] or the negative sequence
nodal current at the busbar x can be generally written when the busbar supplies a
mix of passive (at HV, MV and/or LV) and motor (at LV) loads as:
I−:lines/x ≈ Y−−:x−im V lines−:g/hv:x + Y−+:x V+:x (4.12)
The admittance Y−−:x−im is the downstream negative sequence admittance seen at
the busbar x taking into account only induction motors that are normally supplied
at the LV level, which is inductive in nature. As an example, this can be expressed27
for the simplified case where motor loads are supplied only by the MV line td:x at the
busbar LVr:x as:
Y−−:x−imr ≈ −jklvr:x
(1− V Rtd:x)2(
1ksr:xkmr:x
+ 1ksc−lvragg :x
) (√3 |I+:x|Vn−hv
)(4.13)
where,
Y−−:x−imr - admittance Y−−:x−im corresponding to the simplified case stated above
V Rtd:x, ksr:x, kmr:x, klvragg :x, ksc−lvr:x - as defined for (4.9)28
Vn−hv - nominal line-line voltage of the HV system
The admittance Y−+:x is the downstream negative-positive sequence coupling ad-
mittance seen at the busbar x, which arises as a result of the asymmetry of the MV
line td:x. This admittance can be generally expressed29 by (4.14) and (4.15):
27Derivation of (4.13) can be described using (D.1), (D.3) and (G.6).28Additional subscript ‘x’ indicates the quantities corresponding to the busbar x.29Refer to Appendix I for the derivation.
83
|Y−+:x| ≈ ktd:xkn:x
(√3 |I+:x|Vn−hv
)|Y−−:x−im Z−+:td:x−hv| (4.14)
θY−+:x ≈ 900 + θZ−+:td:x+ θpf :x (4.15)
where,
ktd:x, kn:x - as defined for (4.7)
Z−+:td:x−hv - negative-positive sequence coupling admittance, referred to HV, of the
MV line td:x
θY−+:x , θZ−+:td:x, - phase angles of the admittance Y−+:x and the negative-positive se-
quence coupling impedance Z−+:td:xof the MV line td:x respectively
θpf :x - pf angle30 at the busbar x
Substitution of (4.12) in (4.11) and rearrangement gives:
[V lines−:g/hv]n×1 ≈ −[Y ′
++]−1n×n[Y−+]n×n[V+]n×1 (4.16)
where,
Y ′++:xy ≈ Y++:xy + Y−−:x−im for x = y
Y ′−+:xy ≈ Y−+:xy + Y−+:x for x = y
Y ′++:xy = Y++:xy for x 6= y
Y ′−+:xy = Y−+:xy for x 6= y
That is, taking the nodal positive sequence voltages as known quantities, the negative
sequence voltage V lines−:g/hv:x at any busbar x can be established using (4.16).
30− and + for lagging and leading conditions respectively.
84
The presence of positive sequence voltage controlled components such as PV gen-
erators and synchronous condensers in a system force the negative sequence voltage
at the connected busbars to be zero, disregarding the existence of sources of unbal-
ance. Equation (4.16) which gives the nodal negative sequence voltages arising as a
result of line asymmetries does not consider the presence of such components, and
thus requires suitable adjustments such that the influence of zero voltage unbalance
(or of voltage controlled components) at given busbars on the emission levels at other
busbars is accommodated.
Consider that a voltage controlled component is connected at any busbar i of
the considered HV network (see Fig. 4.4). Hence, the negative sequence voltage V−:i
at the busbar i is zero, and there are only (n − 1) number of busbars at which the
negative sequence voltage to be determined. Then, by expanding the basic equation
(3.21) given in Chapter 3, the negative sequence current (labelled I−ci :x31) at any
other busbar x can be written as:
I−ci :x= (Y−+:x1V+:1 + ... + Y−+:xiV+:i + ... + Y−+:xnV+:n) + (Y++:x1V−:1 + ...
... + Y++:xhV−:h + Y++:xjV−:j + ... + Y++:xnV−:n) (4.17)
Noting that the term associating the admittance Y++:xi is absent in (4.17), the ma-
trix equation (3.22) can be modified accordingly to incorporate the influence of the
constraint V−:i = 0 on the emission arising as a result of line asymmetries at other
busbars by:
• Reducing the dimension of the matrix [Y ′++] down to (n − 1) × (n − 1) by
removing both the ith row and column.
31Subscript ci indicates the additional constraint of the controlled voltage unbalance at the bus-bar i.
85
• Reducing the dimension of the matrix [Y ′−+] down to (n − 1) × n by removing
the ith row.
4.4 Verification of the Methodology Using a Three-bus Test
System
The proposed methodology is applied to the three-bus HV network (66kV, 60Hz,
three-wire) shown in Fig. 4.6. Considered operating scenario and resulting positive
sequence system conditions32 are also indicated in Fig. 3.7. Lengths of the HV
lines which are taken as identical in construction (including the phase positioning33:
©a ©b ©c) and untransposed are shown alongside the lines. Relevant admittance
data of the lines are34:
• positive sequence admittance = (0.6265− j2.3517)Skm
• negative-positive sequence coupling admittance = (0.1040 + j0.1779)Skm
(0.2061∠600Skm)
Busbars 1 and 3 supply MV loads with equal compositions of constant impedance
and constant power elements directly at the HV-MV coupling transformers. This
implies that Y−−:x−im ≈ 0 and Y−+:x ≈ 0 for x = 1 and 3. Busbar 2 supplies loads at
the LV level through 3.2187km of untransposed MV lines35, which account for 40%
of the total load supplied by the system. Two cases based on the type of LV loads
are considered:
32Nodal voltages and line currents, which are obtained using load flow analysis.33This shows the considered arrangement of the three phase conductors (a, b and c) of the hori-
zontal tower.34Refer to Appendix H for further details.35Refer to Appendix B for the tower construction and conductor data.
86
HV busbar 1 (1.07pu, -8.30)
HV busbar 3 (1.04pu, -9.60)
50km 130A
40km 78A 20km
167A
10MVA 0.9 lagging pf
EHV-HV coupling transformer – 60MVA, winding resistance = 1%, leakage reactance = 20%, secondary tap setting = 1.1pu HV-MV coupling transformers – 12MVA, winding resistance = 1%, leakage reactance = 10% MV-LV coupling transformers – aggregated representation of fully loaded 1MVA transformers with winding resistance = 1% , leakage reactance = 5%, and secondary tap setting = 1.08pu
10MVA 0.9 lagging pf
10MVA 0.9 lagging pf
10MVA 0.9 lagging pf
tap
= 1p
u
tap = 1.03pu
tap = 1.1pu
MV lines – 12.47kV 3.2187km 150A VR = 9%
Upstream EHV (230kV) system
(1.1pu, 00)
HV busbar 2 (1.01pu, -10.70) LV (460V)
10MVA 0.9 lagging pf
208A
Figure 4.6: Three-bus HV test system considered for applying the proposed method-ology
87
• Case 1 - LV loads represent passive elements36. That is, km:2 = 0, implying that
Y−−:2−im ≈ 0 and Y−+:2 ≈ 0.
• Case 2 - LV loads represent three-phase induction motors. That is, km:2 = 1.
Aggregation of 50hp motors of which the details are given in Appendix B is
considered. This results in an admittance Y−−:2−im ≈ −j0.0154S.
Fig. 4.7 illustrates the emissions U linesg/hv:x at the HV busbars established using the
proposed methodology37 in comparison to the results obtained using unbalanced load
flow analysis for the two cases listed above, clearly indicating the influence of the
downstream line asymmetries38 in each case. The considered phase positioning of
the MV lines39 is similar to that of the HV lines resulting in a Y−+:2 = (−0.0077 +
j0.1093)× 10−3S.
Fig. 4.8 illustrates the emission U linesg/hv:x at each of the HV busbars established
using the proposed methodology and unbalanced load flow analysis for the case where
km:2 = 1 (i.e. when busbar 2 supplies motor loads) in relation to two different phase
arrangements of the MV lines, indicating their influence40 in each case on the resultant
emission levels.
• Phase positioning I - ©a © b © c (case considered in Fig. 4.7), of which
the negative-positive sequence coupling impedance and admittance values are
0.0349∠300Ω/km and 0.1839∠820Skm respectively as given in Appendix B.
Note that this phase positioning is as same as that of the local HV lines.
36With equal compositions of constant impedance and constant power loads.37Application of the methodology to the test system is described in Appendix J.38Which is the difference between the results established using unbalanced load flow analysis with
and without introducing the effects of the downstream MV line asymmetries.39See Appendix B.40Which is the difference between the results established using unbalanced load flow analysis with
and without introducing the effects of the downstream MV line asymmetries.
88
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1 2 3
Busbar
Caused both by the HV andMV lines for Case 1: load flow
Case 1: methodology
Caused both by the HV andMV lines for Case 1: load flow
Caused only by the HV linesfor Case 2: load flow
Caused only by the HV linesfor Case 2: methodology
Caused both by the HV andMV lines for Case 2: load flow
Caused both by the HV andMV lines for Case 2:methodology
(%
):
/lines
xhv
gU
Figure 4.7: Emissions U linesg/hv:x for the three-bus HV test system for the cases where
km:2 = 0 and km:2 = 1
• Phase positioning II - ©a © c © b, of which the negative-positive se-
quence coupling impedance and admittance values are 0.0349∠1500Ω/km and
0.1839∠−1580Skm respectively. This gives rise to an admittance Y−+:2 ≈
(−0.0908− j0.0613)× 10−3S.
Figs. 4.7 and 4.8 demonstrate that there is a good agreement between the results
obtained using the proposed technique and unbalanced load flow analysis. Further,
based on the results presented in Figs. 4.7 and 4.8, the following can be revealed:
• As in the case of MV networks discussed in Chapter 3, the emission/s arising as
a result of the local HV lines in the presence of considerable motor proportions
is higher at the busbar which is directly connected to the upstream system
(e.g. busbar 1 of the system shown in Fig. 4.7) and lower at all other busbars
(e.g. busbars 2 and 3 of the system shown in Fig. 4.7), compared to the case
where only passive loads exist.
89
0.0
0.1
0.2
0.3
0.4
0.5
0.6
1 2 3
Busbar
Caused only by the HV lines:load flow
Caused only by the HV lines:methodology
Caused both by the HV and MVlines for Phase positioning I:load flow
Caused both by the HV and MVlines for Phase positioning I:methodology
Caused both by the HV and MVlines for Phase positioning II:load flow
Caused both by the HV and MVlines for Phase positioning II:methodology
(%
):
/lines
xhv
gU
Figure 4.8: Emissions U linesg/hv:x for the three-bus HV test system for the case where
km:2 = 1 in relation to the Phase arrangements I and II of the MV lines
• In addition, the global emission in HV networks is dependant on the downstream
line asymmetries in the presence of considerable motor proportions as seen
in Section 4.2 above. This influence of the downstream MV lines can either
decrease (e.g. for Phase positioning I in Fig. 4.8) or increase (e.g. for Phase
positioning II in Fig. 4.8) the resultant emission U linesg/hv:x compared to the local
emission levels depending on the impedance/admittance characteristics of the
downstream lines relative to the local lines.
4.5 Verification of the Methodology Using the IEEE 14-bus
Test System
The proposed methodology is further applied to the IEEE 14-bus test system shown
in Fig. 4.9 which consists of positive sequence voltage controlled busbars (busbars
1, 2, 3, 6 and 8), taking it as a 66kV, 60Hz and three-wire network supplying constant
90
power loads at the HV level. The system data are as per [77] with appropriate
and minor modifications, which are given in Appendix K together with the nodal
positive sequence voltages41. Lines are taken as identical in construction (including
phase positioning) and untransposed, of which the relevant admittances per km as
pu quantities (on a 100MV A base) are42:
• Positive sequence admittance = (0.2729 + j1.0244)× 102pu
• Negative-positive sequence coupling admittance = (0.4530 + j0.7749)× 10pu
Lengths43 of the lines are established such that their positive sequence impedance
magnitudes (in pu) are approximately as per [77].
Fig. 4.10 illustrates the emission U linesg/hv:x at each of the HV busbars established
using the methodology in comparison to the results obtained using unbalanced load
flow analysis, further validating the proposed technique.
4.6 Chapter Summary
As a continuation of the work presented in Chapter 3, this chapter has addressed
the global voltage unbalance in HV power systems which arises as a result of line
asymmetries. This is a key aspect in assessing emission limits to individual installa-
tions connected to HV power systems essentially based on the IEC/TR 61000-3-13
recommendations.
The dependency of the global emission on the local HV lines as well as on the
downstream MV lines in the presence of passive and induction motor loads has been
examined in relation to a simple radial network. The following major conclusions can
be drawn from the study:
41Which are obtained using load flow analysis.42See Appendix H for further details.43Given in Appendix K.
91
Figure 4.9: IEEE 14-bus test system
0.0
0.1
0.2
0.3
0.4
0.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Busbar
Load flow resultsMethodology
(%
):
/lines
xhv
gU
Figure 4.10: Emissions U linesg/hv:x for the IEEE 14-bus test system
Please see print copy for image
92
• The direction given in IEC/TR 61000-3-13 to assess the influence of an asym-
metrical radial line on the global emission can be applied also to an HV system
however only when the network supplies primarily passive loads. In this case,
the local HV lines are totally responsible for the global emission, and the con-
tribution made by the downstream MV lines is negligible.
• The presence of induction motors has been seen to affect the level of emission
given by the IEC approach noticeably. This dependency of the global emission
on the motor proportion can be seen in two forms:
– As in the case of MV networks, the local emission or the emission arising
as a result of HV lines is influenced by motor loads.
– In addition, the presence of motor loads makes the downstream emission
or the emission arising as a result of MV lines accountable for the global
emission in HV networks.
A systematic approach, covering interconnected environments, for evaluating the
global emission caused both by the local and downstream line asymmetries at nodal
level has been proposed. The results established using the proposed methodology
in relation to a three-bus test system has been seen to be in close agreement with
the results obtained using unbalanced load flow analysis. Furthermore, these results
clearly demonstrated the following:
• As in the case of MV networks, the emission/s arising as a result of the local
HV lines in the presence of considerable motor proportions is higher at the HV
busbar which is directly connected to the upstream system and lower at all other
busbars of the network, compared to the case where only passive loads exist.
• The influence of the downstream MV lines can either decrease or increase the
resultant emission levels with respect to the local emission levels depending on
93
the impedance/admittance characteristics of the downstream lines relative to
the local lines.
The proposed methodology is further validated in relation to the IEEE 14-bus test
system.
Chapter 5
Propagation of Voltage Unbalance
5.1 Introduction
As discussed in Chapter 2 (Section 2.8.3), the propagation of voltage unbalance is a
key aspect considered in the IEC/TR 61000-3-13 allocation procedure. Quantitative
measures of this propagation exist in two forms:
• Transfer coefficients1 which give a measure of the propagation from upstream
higher voltage to downstream lower voltage systems through coupling trans-
formers. This is employed in the allocation procedure in determining the global
emission allowance Ug/s of any sub-system S to quantify the level of voltage
unbalance which propagates from the upstream sub-system.
• Influence coefficients which give a measure of the propagation from one busbar
to another busbar of a sub-system at a particular voltage level through lines.
The influence coefficient ki−x between busbars i and x is defined as the voltage
unbalance which arises at the busbar x when 1pu of negative sequence voltage
source is applied at the busbar i. This is employed in the allocation procedure in
1See Section 2.8.4.
94
95
determining the total available apparent power Stotal−x of the entire sub-system
as seen at the busbar x under evaluation to take into account the contributions
from neighbouring busbars.
0.90
0.95
1.00
1.05
1.10
1.15
10 13 16 19 22 25ksc-lv
T mv-
lv
Figure 5.1: Variation of Tmv−lv with ksc−lv obtained for constant power loads usingunbalanced load flow analysis
The method given in IEC/TR 61000-3-13 for the evaluation of the MV to LV
transfer coefficient Tmv−lv, which is reproduced in Section 2.8.4 by (2.17), assumes
a unity transfer coefficient in relation to passive loads in general. A transfer coeffi-
cient = 1 is mathematically trivial for constant impedance loads. However, it may not
be valid for other load types such as constant power and constant current loads, owing
to the different behaviours exhibited by these load types under unbalanced supply
conditions as noted in Chapter 3 (Section 3.2). As an example, Fig. 5.1 illustrates
the variation of the transfer coefficient Tmv−lv with ksc−lv2) established when an LV
system supplies a load base primarily having constant power loads with 0.9 lagging pf
using unbalanced load flow analysis, compared against unity. Noting that the transfer
coefficient Tmv−lv is considerably greater than unity at relatively lower ksc−lv values
2As defined in Chapter 2 for (2.17), ksc−lv is the ratio between the short-circuit capacity (inMVA) at an LV busbar and the total load (in MVA) supplied by the LV busbar.
96
(i.e. heavily loaded LV systems), it is evident that the assumption of a unity transfer
coefficient for passive loads used in the IEC method cannot be generally applied with
a high degree of accuracy. That is, a requirement exists for cautious examination
of the dependency of transfer coefficients on various load types/bases. Moreover,
systematic approaches for assessing other transfer coefficients (e.g. HV to MV) and
influence coefficients are not covered in IEC/TR 61000-3-13.
Objectives of the work presented in this chapter are:
• To develop theoretical bases which describe the behaviour exhibited by various
load types with regard to the propagation of voltage unbalance from higher
voltage to lower voltage systems, and to propose improved/novel approaches for
evaluating the MV to LV and HV to MV transfer coefficients. This is covered
in Section 5.2.
• To carry out preliminary studies in order to investigate the dependency of in-
fluence coefficients on various load types/bases, and to develop a generalised
methodology for their evaluation covering interconnected network environments.
Section 5.3 presents this, together with a comparison of the results obtained us-
ing the proposed technique and unbalanced load flow analysis in relation to an
MV three-bus test system and the IEEE 14-bus test system.
97
5.2 Voltage Unbalance Transfer Coefficients
Considering the system shown in Fig. 5.23 where the US (upstream higher voltage
system) to S (lower voltage system under evaluation) transfer coefficient Tus−s which
is defined in Section 2.8.4 by (2.16) can be written in an expanded form as:
Tus−s =
∣∣∣∣V−:Uus/s
V+:s
∣∣∣∣× 1∣∣∣V−:us
V+:us
∣∣∣ =
∣∣∣V−:Uus/s−us
V−:us
∣∣∣∣∣∣V+:s−us
V+:us
∣∣∣ (5.1)
where,
V+:s, V+:us - positive sequence voltages at the busbars S and US respectively
V−:us - negative sequence voltage at the busbar US
V−:Uus/s - negative sequence voltage at the busbar S which is transferred from US
V+:s−us, V−:Uus/s−us - V+:s and V−:Uus/s respectively referred to US
US
DS
S
Uus/s
Ss-local
Ss-ds
Figure 5.2: Radial system considered for the illustration of transfer coefficients
3Reproduction of Fig. 2.5.
98
The positive sequence voltage ratio∣∣∣V+:s−us
V+:us
∣∣∣ can be expressed in a general form,
disregarding the load type, as:
∣∣∣∣V+:s−us
V+:us
∣∣∣∣ ≈ 1∣∣∣1 + j∣∣∣Z++:tf−s
Z++:s
∣∣∣∠θpf :s
∣∣∣ (5.2)
where,
Z++:tf−s - positive sequence impedance (assumed as inductive), referred to S, of the
US-S coupling transformer
Z++:s - downstream positive sequence impedance (or equivalent) seen at the busbar S
θpf :s - pf angle4 at the busbar S
The impedance ratio∣∣∣Z++:tf−s
Z++:s
∣∣∣ can be written as:
∣∣∣∣Z++:tf−s
Z++:s
∣∣∣∣ =1
ksc−s
(5.3)
where,
ksc−s = Ssc−s
Ss
Ssc−s = (Vn−s)2
|Z++:tf−s|, the short-circuit capacity (in MVA) at the busbar S
Ss = (Vn−s)2
|Z++:s| , the total load (in MVA) supplied by the system S
Vn−s - nominal line-line voltage of the system S
Then, the positive sequence voltage ratio∣∣∣V+:s−us
V+:us
∣∣∣ can be rewritten as:
∣∣∣∣V+:s−us
V+:us
∣∣∣∣ ≈ 1∣∣∣1 + j 1ksc−s
∠θpf :s
∣∣∣ (5.4)
4− and + for lagging and leading conditions respectively.
99
The negative sequence voltage |V−:Uus/s−us| can be generally written, ignoring zero
sequence unbalance, as:
|V−:Uus/s−us| = |V−:us − (Z−−:tf−usI−:Uus/tf )| (5.5)
where,
Z−−:tf−us - negative sequence impedance5, referred to US, of the US-S coupling trans-
former
I−:Uus/tf - negative sequence current (referred to US) in the US-S coupling trans-
former, which arises as a result of the voltage unbalance at the busbar US
Based on the evidence from Chapter 3 (Section 3.2), the behaviour of the negative
sequence current I−:Uus/tf seem to be influenced by the load type/base supplied by
the system S making the transfer coefficient Tus−s dependant on the load type/base.
The IEC approach6 for assessing the MV to LV transfer coefficient accounts for this
influence, however it distinguishes only motor loads from passive loads, or in other
words it does not take the differences7 that exist between various passive load types
into account. This section addresses four basic load types8 and various load bases
developing theoretical bases which describe their behaviours in this regard.
Constant Impedance (Z) loads
When the system S supplies constant impedance loads (balanced), (5.5) can be re-
arranged such that the negative sequence voltage ratio∣∣∣V−:Uus/s−us
V−:us
∣∣∣ is equal to the
5Which is inherently equal to the positive sequence impedance Z++:tf−us of the transformer.6Given in Chapter 2 by (2.17).7As seen from Fig. 5.1.8i.e. constant impedance, constant current, constant power and three-phase induction motor
loads.
100
positive sequence voltage ratio given by (5.4), as the positive and negative sequence
impedances of the loads and the US-S coupling transformer are equal:
∣∣∣∣V−:Uus/s−us
V−:us
∣∣∣∣ =1∣∣∣1 + j 1
ksc−s∠θpf :s
∣∣∣ (5.6)
Substitution of (5.4) and (5.6) in (5.1) gives:
Tus−s = 1 (5.7)
That is, for constant impedance loads, the term Z++:tf−usI−:Uus/tf causes the down-
stream negative sequence voltage |V−:Uus/s−us| to be smaller than the upstream neg-
ative sequence voltage |V−:us| by the factor 1˛1+j 1
ksc−s∠θpf :s
˛ , leading to a unity Tus−s.
Constant Current (I) loads
As in Section 3.2.2, the negative sequence current I−:Uus/tf can be considered to be
negligible when the system supplies constant current loads (balanced), as such loads
draw equal magnitudes of three phase currents regardless of the prevailing voltage
condition. Hence, (5.5) can be simplified for constant current loads as:
∣∣∣∣V−:Uus/s−us
V−:us
∣∣∣∣ ≈ 1 (5.8)
Substitution of (5.4) and (5.8) in (5.1) gives:
Tus−s ≈∣∣∣∣1 + j
1
ksc−s
∠θpf :s
∣∣∣∣ (5.9)
That is, in contrary to the case of constant impedance loads, the downstream negative
sequence voltage |V−:Uus/s−us| is equal to the upstream negative sequence voltage
101
|V−:us| in the presence of constant current loads, resulting in an increase in Tus−s by
the factor∣∣∣1 + j 1
ksc−s∠θpf :s
∣∣∣ relative to unity.
Constant Power (PQ) Loads
As in Section 3.2.3, through careful examination of the results obtained using unbal-
anced load flow analysis, (5.10) is established as a close approximation to the negative
sequence voltage ratio∣∣∣V−:Uus/s−us
V−:us
∣∣∣:∣∣∣∣V−:Uus/s−us
V−:us
∣∣∣∣ ≈ 1∣∣∣1 + j 1ksc−s
∠θpf :s
∣∣∣β (5.10)
where, β ≈ −1 and −2 for low (∼ 0.9) and high (∼ 1) lagging pf conditions respec-
tively. Substitution of (5.4) and (5.10) in (5.1) gives:
Tus−s ≈∣∣∣∣1 + j
1
ksc−s
∠θpf :s
∣∣∣∣1−β
(5.11)
That is, for constant power loads, the term Z++:tf−usI−:Uus/tf causes the downstream
negative sequence voltage |V−:Uus/s−us| to be greater than the upstream negative se-
quence voltage |V−:us| by the factor 1˛1+j 1
ksc−s∠θpf :s
˛β , resulting in an increase in Tus−s
by the factor∣∣∣1 + j 1
ksc−s∠θpf :s
∣∣∣1−β
relative to unity.
Discussion
Taking a generalised view, the scaling factors applied to the downstream negative
sequence voltage |V−:Uus/s−us| relative to the upstream negative sequence voltage
|V−:us| can be expressed in a form∣∣∣1 + j 1
ksc−s∠θpf :s
∣∣∣γ for the three passive load types,
where γ = 1 for constant impedance loads, γ = 0 for constant current loads, and
−2 ≤ γ ≤ −1 for constant power loads. Note that this exponent γ, which distin-
102
guishes various load behaviours, is similar to that established in Section 3.2 in relation
to the emission arising as a result of an asymmetrical radial line. Alternatively, the
transfer coefficient Tus−s can be expressed in a form∣∣∣1 + j 1
ksc−s∠θpf :s
∣∣∣τ , where τ = 0
for constant impedance loads, τ = 1 for constant current loads, and 2 ≤ τ ≤ 3 for
constant power loads. Noting that the factor ksc−s can be in the range of 59 to 2510
for various systems and the exponent τ can vary in the range of 0 to 3 for different
load types, a uniform behaviour or the behaviour of constant impedance loads as
assumed in the IEC approach cannot be used to represent all load types with a high
degree of accuracy. The accuracy of the formulation Tus−s ≈∣∣∣1 + j 1
ksc−s∠θpf :s
∣∣∣τ will
be demonstrated in Sections 5.2.1 and 5.2.2 as applicable to MV to LV and HV to
MV transfer coefficients respectively.
Induction Motor (IM) Loads
When the system S supplies three-phase induction motors11 which can be repre-
sented using decoupled, unequal and constant12 sequence impedances, (5.5) can be
written as: ∣∣∣∣V−:Uus/s−us
V−:us
∣∣∣∣ =1∣∣∣1 +
Z++:tf−s
Z−−:s
∣∣∣ (5.12)
where, Z−−:s - downstream negative sequence impedance seen at the busbar S. When
US and S represent MV and LV (i.e. s = lv) systems respectively (i.e. the case of
the MV to LV propagation), the impedance Z−−:lv is equal to the negative sequence
impedance Z−−:im of the aggregated motor load supplied by the aggregated LV sys-
tem13. In the case of the HV to MV (i.e. s = mv) propagation, the impedance Z−−:mv
9e.g. fully loaded 60MVA HV-MV transformer with 20% impedance.10e.g. fully loaded 400kVA MV-LV transformer with 4% impedance.11Which are usually supplied at the LV level.12For a given motor speed.13Note that the busbar DS of Fig. 2.5, which represents the downstream system of the system S,
does not exist in this case.
103
is not simply equal to Z−−:im due to the additional system impedance Z++:sys−lv14
that exists between the MV busbar under evaluation and the downstream LV systems
at which motor loads are supplied. That is, transfer coefficients, in the presence of
motor loads, seem to depend on the systems in which the propagation is being con-
sidered. This will be further discussed separately in relation to the MV to LV and
HV to MV propagation in Sections 5.2.1 and 5.2.2 respectively.
5.2.1 MV to LV Transfer Coefficient, Tmv−lv
This section considers that the busbars US and S (Fig. 5.2) represent MV and LV
systems respectively (i.e. the subscripts ‘us’ and ‘s’ used in the above formulae (5.1)
- (5.12) are to be replaced with ‘mv’ and ‘lv’ respectively).
Passive Loads
As derived above, Tmv−lv ≈∣∣∣1 + j 1
ksc−lv∠θpf :lv
∣∣∣τ for passive loads. Representing most
practical circumstances, the factor ksc−lv for LV systems can take a value in the
range of 1015 to 2516. Figs. 5.3: I − II and 5.4: I − II illustrate the variation of
Tmv−lv with ksc−lv established using this formulation in comparison to the results
obtained using unbalanced load flow analysis for constant current and constant power
loads respectively, where sub-figures I and II correspond to 0.99 and 0.9 lagging pf
conditions respectively. These illustrate the accuracy of the new formulation, while
demonstrating the deviation of the actual transfer coefficient from the unity value
assumed in the IEC approach.
14e.g. MV-LV coupling transformer and MV line impedances. The subscript ‘lv’ indicates theimpedance referred to the LV level.
15e.g. fully loaded 10MVA transformer with 10% impedance.16e.g. fully loaded 400kVA transformer with 4% impedance.
104
I - 0.99 lagging pf
0.90
0.95
1.00
1.05
1.10
10 13 16 19 22 25ksc-lv
T mv-
lv
Load flow results
Proposed formulation
II - 0.9 lagging pf
0.90
0.95
1.00
1.05
1.10
10 13 16 19 22 25ksc-lv
T mv-
lv
Load flow results
Proposed formulation
Figure 5.3: Variation of Tmv−lv with ksc−lv for constant current loads: I - 0.99 laggingpf, II - 0.9 lagging pf
I - 0.99 lagging pf
0.90
0.95
1.00
1.05
1.10
1.15
10 13 16 19 22 25ksc-lv
T mv-
lv
Load flow results
Proposed formulation
II - 0.9 lagging pf
0.90
0.95
1.00
1.05
1.10
1.15
10 13 16 19 22 25ksc-lv
T mv-
lv
Load flow results
Proposed formulation
Figure 5.4: Variation of Tmv−lv with ksc−lv for constant power loads: I - 0.99 laggingpf, II - 0.9 lagging pf
105
Induction Motor (IM) Loads
Equation (5.12) is reproduced here for the propagation of the negative sequence volt-
age from MV to LV as:
∣∣∣∣V−:Umv/lv−mv
V−:mv
∣∣∣∣ =1∣∣∣1 + Z++:ml−lv
Z−−:im
∣∣∣ (5.13)
where, the subscript ‘ml’ in Z++:ml−lv is a replacement of the subscript ‘tf ’ used
in Z++:tf−s to specifically indicate the MV-LV coupling transformer. Noting the
inductive nature of the impedances Z++:ml−lv and Z−−:im, (5.13) can be rewritten as:
∣∣∣∣V−:Umv/lv−mv
V−:mv
∣∣∣∣ =1(
1 + ks
ksc−lv
) (5.14)
where, ks - ratio between the positive and negative (which is inductive) sequence
impedances of the aggregated motor load supplied by the LV system17. Substitution
of (5.4) and (5.14) in (5.1) gives:
Tmv−lv ≈
∣∣∣1 + j 1ksc−lv
∠θpf :lv
∣∣∣(1 + ks
ksc−lv
) (5.15)
That is, for induction motor loads, the term Z++:ml−mvI−:Umv/ml causes the down-
stream negative sequence voltage |V−:Umv/lv−mv| to be smaller than the upstream neg-
ative sequence voltage |V−:mv| by the factor 1„1+ ks
ksc−lv
« . Noting that 5 < ks < 7, this
results in a Tmv−lv < 1. This reduction in Tmv−lv relative to unity is significant com-
pared to the increment in Tmv−lv caused by passive loads18. Fig. 5.5 illustrates the
variation of Tmv−lv with ksc−lv, for motor loads with ks = 6.719 and pf = 0.9 lagging,
17Typically, ks can be in the range of 5 to 7.18e.g. Tmv−lv = 0.81 for motor loads with ks = 6.7, ksc−lv = 25 and pf = 0.9 lagging, whereas
Tmv−lv = 1.04 for constant power loads with ksc = 25 and pf = 0.9 lagging.19e.g. 50hp motors described in Appendix B.
106
established using the IEC method20 and (5.15) in comparison to the results obtained
using unbalanced load flow analysis. This demonstrates that both the IEC method
and the proposed new formulation are satisfactory for estimating Tmv−lv for mo-
tor loads.
0.50
0.60
0.70
0.80
0.90
1.00
1.10
10 13 16 19 22 25ksc-lv
T mv-
lv
Load flow results
Prposed formulation
IEC method
Figure 5.5: Variation of Tmv−lv with ksc−lv for induction motor loads with ks = 6.7and pf = 0.9 lagging
Generalisation for Mixes of Various Load Types
Consider a mix of constant impedance, constant current, constant power and induc-
tion motor loads is supplied by the LV system. Then, the negative sequence current
I−:Umv/ml can be decomposed as:
I−:Umv/ml = I−:z + I−:i + I−:pq + I−:im (5.16)
where, I−:z, I−:i, I−:pq, I−:im - negative sequence currents (referred to MV side) in Z,
I, PQ and IM loads respectively arising as a result of the MV unbalance. Employing
20Given by (2.17).
107
(5.16), (5.5) can be written in an expanded form as:
|V−:Umv/lv−mv| =
∣∣∣∣∣V−:mv −∑
L=z,i,pq,im
(Z++:ml−mv I−:L)
∣∣∣∣∣ (5.17)
The impact of the components Z++:ml−mvI−:L for the four load elements can be com-
bined, forming the resultant influence on the propagation, as21:
∣∣∣∣V−:Umv/lv−mv
V−:mv
∣∣∣∣ ≈ 1∣∣∣1 + j kz
ksc−lv∠θpf :z
∣∣∣ ∣∣∣1 + j kpq
ksc−lv∠θpf :pq
∣∣∣β (1 + kmks
ksc−lv
) (5.18)
where,
km - ratio between the rated motor load (in MVA) and the total load (in MVA) sup-
plied by the LV system
kz, kpq - ratios of the constant impedance and constant power loads (in MVA) respec-
tively to the total load (in MVA) supplied by the LV system
θpf :z, θpf :pq - power factor angles of the constant impedance and constant power loads
respectively supplied by the LV system
The terms∣∣∣1 + j kz
ksc−lv∠θpf :z
∣∣∣, ∣∣∣1 + j kpq
ksc−lv∠θpf :pq
∣∣∣β and(1 + kmks
ksc−lv
)in the numera-
tor of (5.18) account for the influence of the constant impedance, constant power and
induction motor elements respectively in the mix on the propagation of the negative
sequence voltage. Note that these terms are modified versions of the respective terms
for the individual load types22, where the modifications are introduced by multiply-
ing(
1ksc−lv
)of each of the terms for the individual load types by the respective load
proportion kL (L = z, pq, im). Substituting (5.4) and (5.18) in (5.1), the transfer
21See Appendix L for details.22i.e.
∣∣∣1 + j 1ksc−lv
∠θpf :lv
∣∣∣γ with different γ values for the various passive loads types, and(1 + ks
ksc−lv
)for motor loads.
108
coefficient Tmv−lv can be expressed in a generalised form as:
Tmv−lv ≈
∣∣∣1 + j 1ksc−lv
∠θpf :lv
∣∣∣∣∣∣1 + j kz
ksc−lv∠θpf :z
∣∣∣ ∣∣∣1 + j kpq
ksc−lv∠θpf :pq
∣∣∣β (1 + kmks
ksc−lv
) (5.19)
Figs. 5.6: I − II illustrate the variation of Tmv−lv with ksc−lv for two load bases
which are dominated by motor loads (Z − 10%, I − 5%, PQ− 15% and IM − 70%)
and passive loads (Z − 25%, I − 5%, PQ− 60%, IM − 10%) respectively established
using the IEC method, the proposed formulation (5.19) and unbalanced load flow
analysis. A lagging pf of 0.9 for all load components and a ks = 6.7 for motor loads
are assumed. These results demonstrate that although the IEC method provides an
accurate estimation to Tmv−lv for load bases containing large proportions of induc-
tion motors, it associates a considerable degree of error for load bases dominated by
passive elements. Furthermore, the proposed new formulation gives a more accurate
estimation to Tmv−lv for both of the loads bases.
I - Dominated by induction motors
0.70
0.75
0.80
0.85
0.90
10 13 16 19 22 25ksc-lv
T mv-
lv
Load flow results
Proposed formulation
IEC method
II - Dominated by passive elements
0.90
0.95
1.00
1.05
10 13 16 19 22 25
ksc-lv
T mv-
lv
Load flow results
Proposed formulation
IEC method
Figure 5.6: Variation of Tmv−lv with ksc−lv: I - for a load base dominated by inductionmotors, II - for a load base dominated by passive elements
109
Figs. 5.7: I−II illustrate the variation23 of Tmv−lv with km, considering load mixes
of constant impedance and motor loads (i.e. km = 1 − kz) and constant power and
motor loads (i.e. km = 1− kpq) respectively, for two extreme cases where ksc−lv ≈ 10
and ksc−lv ≈ 25. A lagging pf of 0.9 for all load components and ks = 6.7 for motor
loads are assumed. These demonstrate that the transfer coefficient Tmv−lv can be in
the range of 0.6 to 1.1.
II - Load mixes of PQ and IM loads
0.6
0.7
0.8
0.9
1.0
1.1
0.0 0.2 0.4 0.6 0.8 1.0km = 1-kpq
T mv-
lv
I - Load mixes of Z and IM loads
0.6
0.7
0.8
0.9
1.0
1.1
0.0 0.2 0.4 0.6 0.8 1.0km = 1-kz
T mv-
lv
ksc-lv ≈ 25
ksc-lv ≈ 10
ksc-lv ≈ 25
ksc-lv ≈ 10
Figure 5.7: Variation of Tmv−lv with km for ksc−lv ≈ 25 and ksc−lv ≈ 10: I - for loadmixes of Z and IM loads, II - for load mixes of PQ and IM loads
The generalised expression (5.19) can be simplified, easing the computation, for
industrial load bases which contain large proportions of motor loads by neglecting
the negative sequence current components I−:z, I−:i, and I−:pq compared to I−:im in
(5.16) as:
Tmv−lv ≈
∣∣∣1 + j 1ksc−lv
∠θpf :lv
∣∣∣(1 + kmks
ksc−lv
) (5.20)
Fig. 5.8 provides a comparison of the variations of Tmv−lv with km established using
the IEC method, the generalised expression (5.19), the simplified expression (5.20)
for industrial load bases, and unbalanced load flow analysis. Load mixes of constant
23Derived using the proposed formulation.
110
power and motor loads, a lagging pf of 0.9, ks = 6.7 and ksc−lv ≈ 10 are considered.
This shows that the estimation from the various methods are in close agreement for
km values above 0.5. For lower km values, (5.19) gives the closest estimation to the
actual24 Tmv−lv, and (5.20) is seen to provide a better estimation compared to the
IEC method.
0.6
0.7
0.8
0.9
1.0
1.1
1.2
0.0 0.2 0.4 0.6 0.8 1.0km
T mv-
lv
Load flow resultsEq. (5.19)Eq. (5.20)IEC method
Figure 5.8: Variation of Tmv−lv with km established using the IEC method, (5.19),(5.20) and unbalanced load flow analysis
5.2.2 HV to MV Transfer Coefficient, Thv−mv
This section considers that the busbars US, S and DS of Fig. 5.2 represent HV, MV
and LV25 systems respectively (i.e. the subscripts ‘us’ and ‘s’ in the above formulae
(5.1) - (5.12) are to be replaced with ‘hv’ and ‘mv’ respectively).
24i.e. the values obtained using unbalanced load flow analysis.25This is an aggregated representation of all LV systems supplied by the upstream MV system.
111
Passive Loads
As derived above, Thv−mv ≈∣∣∣1 + j 1
ksc−mv∠θpf :mv
∣∣∣τ for passive loads. Representing
most practical circumstances, the factor ksc−mv for MV systems can take a value in
the range of 526 to 1527. Noting that the values of ksc−mv for MV systems are usually
smaller than the ksc−lv values for LV systems, the amplification which takes place in
the presence of passive loads in the HV to MV propagation is greater than that in
the case of the MV to LV propagation.
Induction Motor (IM) Loads
Considering that the MV system under evaluation supplies only induction motors at
the aggregated LV busbar, (5.12) is reproduced here for the HV to MV propagation as:
∣∣∣∣V−:Uhv/mv−hv
V−:hv
∣∣∣∣ =1∣∣∣1 + Z++:hm−mv
Z−−:mv
∣∣∣ (5.21)
where, the subscript ‘hm’ in Z++:hm−mv is a replacement of the subscript ‘tf ’ used
in Z++:tf−s to specifically indicate the HV-MV coupling transformer. Noting that
the impedance Z−−:mv is equal to the sum of the impedances Z−−:im and Z++:sys−lv
(referred to MV), the rearrangement of (5.21) in terms of the load and system char-
acteristics gives :
∣∣∣∣V−:Uhv/mv−hv
V−:hv
∣∣∣∣ =1[
1 +(
1ksc−mv
)(ks
1+ ksksc−lvagg
)] (5.22)
26e.g. fully loaded 60MVA transformer with 20% impedance.27e.g. fully loaded 10MVA transformer with 6.5% impedance.
112
where, ksc−lvagg - ratio between the short-circuit capacity28 (in MVA) at the aggregated
LV busbar, and the total load (in MVA) supplied by the LV system. Substitution of
(5.4) and (5.22) in (5.1) gives:
Thv−mv ≈
∣∣∣1 + j 1ksc−mv
∠θpf :mv
∣∣∣[1 +
(1
ksc−mv
)(ks
1+ ksksc−lvagg
)] (5.23)
That is, for induction motor loads, the term Z++:hm−hvI−:Uhv/hm causes the down-
stream negative sequence voltage |V−:Uhv/mv−hv| to be smaller than the upstream neg-
ative sequence voltage |V−:hv| by the factor 1241+“
1ksc−mv
”0@ ks
1+ ksksc−lvagg
1A35 . Noting that
(ks
1+ ksksc−lvagg
)> 1, a value less than unity can be expected for the transfer coefficient
Thv−mv. Similar to the case of the MV to LV propagation, this reduction in Thv−mv
relative to unity is significant compared to the increment in Thv−mv introduced by
passive loads29. Note that the degree of this reduction in the HV to MV transfer
coefficient Thv−mv is lower than that in the HV to MV transfer coefficient Tmv−lv for
similar system and load characteristics30. However, usually ksc−mv < ksc−lv, and thus
a higher degree of this reduction can be expected in the HV to MV propagation than
that in the MV to LV propagation31.
28Which is derived using the positive sequence system impedance Z++:sys−lv that exists betweenthe MV busbar under evaluation and the downstream LV busbar.
29e.g. Thv−mv = 0.79 for motor loads with ks = 6.7, ksc−mv = ksc−lvagg= 15 and pf = 0.9 lagging,
whereas Tmv−lv = 1.06 for constant power loads with ksc−mv = 15 and pf = 0.9 lagging.30e.g. Tmv−lv = 0.71 for ks = 6.7, ksc−lv = 15 and pf = 0.9 lagging, and Thv−mv = 0.79 for
ks = 6.7, ksc−mv = ksc−lv = 15 (i.e. the HV-MV transformer supplies a similar MV-LV transformer.)and pf = 0.9 lagging.
31e.g. Thv−mv = 0.76 and Tmv−lv = 0.81 for ks = 6.7, ksc−mv = 15, ksc−lv = ksc−lvagg= 25 and
pf = 0.9 lagging.
113
Generalisation for Mixes of Various Load Types
Consider that the MV system under evaluation supplies a mix of constant impedance,
constant current, constant power loads (at MV and/or LV) and induction motors (at
LV). Based on the same approach32 used to establish (5.18) in the case of the MV to
LV propagation, the various load behaviours in the above load mix can be combined33,
forming the resultant impact of the load mix on the HV to MV propagation, as:
∣∣∣∣V−:Uhv/mv−hv
V−:hv
∣∣∣∣ ≈1˛
1+jkzmv
ksc−mv∠θpf :zmv
˛˛1+j
kpqmvksc−mv
∠θpf :pqmv
˛β241+“
kmmvksc−mv
”0@ ks
1+ kmksksc−lvagg
1A35(5.24)
where,
kzmv , kpqmv , kmmv - ratios of the constant impedance, constant power and motor loads
(in MVA) respectively to the total load (in MVA) supplied by the MV system under
evaluation
θpf :zmv , θpf :pqmv - power factor angles of the constant impedance and constant power
loads respectively supplied by the MV system
Substituting (5.4) and (5.24) in (5.1), and replacing kmmv = kmklv (where klv is
the fraction of LV loads supplied by the MV system under evaluation), the transfer
coefficient Thv−mv can be expressed in a generalised form as:
32Described in Appendix L.33Replacing 1
ksc−mvand/or 1
ksc−lvaggof the respective terms for the individual load types by kLmv
ksc−mv
and kL
ksc−lvaggrespectively for L = z, pq, im.
114
Thv−mv ≈
∣∣∣1 + j 1ksc−mv
∠θpf :mv
∣∣∣∣∣∣1 + j kzmv
ksc−mv∠θpf :zmv
∣∣∣ ∣∣∣1 + j kpqmv
ksc−mv∠θpf :pqmv
∣∣∣β [1 +(
kmklv
ksc−mv
)(ks
1+ kmksksc−lvagg
)](5.25)
Figs. 5.9: I - II and 5.10: I - II illustrate the variation of Thv−mv with klv for two
extreme cases where ksc−mv = 12 and ksc−mv = 4 respectively considering systems
where the loads are supplied directly at the MV busbar34. Sub-figures I and II cor-
respond to load mixes of constant impedance and motor loads, and constant power
and motor loads respectively. Each figure shows the variation established using (5.25)
in comparison to the result obtained using unbalanced load flow analysis for three
sub-cases where:
• km = 1 (i.e. LV system supplies only motor loads)
• km = 0.5 (i.e. LV system supplies equal proportions of passive and motor loads)
• km = 0 (i.e. LV system supplies only passive loads)
A case where ksc−lvagg = 2035, pf = 0.9 lagging for all load components and ks = 6.7
for induction motors is considered.
Figs. 5.11: I - II illustrate the variation of Thv−mv with klv for the two cases where
ksc−mv = 12 and ksc−mv = 4 respectively considering systems where the LV loads are
supplied through MV lines36. Each figure shows the variation for the three sub-cases
km = 1, km = 0.5 and km = 0 established using (5.25) in comparison to the result
obtained using unbalanced load flow analysis. Passive loads are represented as a mix
of constant impedance and constant power elements with equal compositions. A case
34LV loads are supplied through MV-LV transformers, i.e. Z++:sys−lv accounts for the impedancesof the MV-LV coupling transformers.
35e.g. aggregation of fully loaded 1MVA transformers with 5% impedance.36i.e. Z++:sys−lv accounts for the impedances of the MV lines and the MV-LV coupling transform-
ers.
115
where ksc−lvagg = 637, pf = 0.9 lagging for all load components and ks = 6.7 for
induction motors is considered.
According to Figs. 5.9 - 5.11, the proposed formulation (5.25) provides an estima-
tion to the transfer coefficient Thv−mv, which is seen to be in close agreement with the
values given by unbalanced load flow analysis except for the minor discrepancies38
arise when a system supplies primarily constant power loads under heavy loading
or lower ksc−mv conditions. Further, these demonstrate that the transfer coefficient
Thv−mv can be a value in the range of 0.5 to 1.4, which is a wider range than that of
Tmv−lv (0.6 - 1.1), depending on the prevailing system and load characteristics and
downstream load composition.
I - Load mixes of Z and IM loads
0.6
0.7
0.8
0.9
1.0
1.1
1.2
0.0 0.2 0.4 0.6 0.8 1.0
klv
T hv-
mv
Load flow resultsProposed formulation km = 1
km = 0.5
km = 0
II - Load mixes of PQ and IM loads
0.6
0.7
0.8
0.9
1.0
1.1
1.2
0.0 0.2 0.4 0.6 0.8 1.0klv
T hv-
mv
Load flow resultsProposed formulation
km = 1
km = 0.5km = 0
Figure 5.9: Variation of Thv−mv with klv for ksc−mv = 12 (loads are supplied directlyat the MV busbar): I - for load mixes of Z and IM loads, II - for load mixes of PQand IM loads
37e.g. the MV lines described in Appendix B supplying 10MVA loads per line with 10% voltageregulation through fully loaded 1MVA transformers with 5% impedance and 1.1pu secondary tapsetting.
3810% error in the case considered in Fig. 5.10: II.
116
I - Load mixes of Z and IM loads
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
0.0 0.2 0.4 0.6 0.8 1.0klv
T hv-
lv
Load flow resultsProposed formulation
km = 1
km = 0.5
km = 0
II - Load mixes of PQ and IM loads
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
0.0 0.2 0.4 0.6 0.8 1.0klv
T hv-
mv
Load flow resultsProposed formulation
km = 1
km = 0.5 km = 0
Figure 5.10: Variation of Thv−mv with klv for ksc−mv = 4 (loads are supplied directlyat the MV busbar): I - for load mixes of Z and IM loads, II - for load mixes of PQand IM loads
I - ksc-mv = 12
0.6
0.7
0.8
0.9
1.0
1.1
1.2
0.0 0.2 0.4 0.6 0.8 1.0klv
T hv-
mv
Load flow resultsProposed formulation
II - ksc-mv = 4
0.50.60.70.80.91.01.11.21.31.4
0.0 0.2 0.4 0.6 0.8 1.0klv
T hv-
mv
Load flow resultsProposed formulation
km = 1
km = 0.5
km = 0
km = 1
km = 0.5
km = 0
Figure 5.11: Variation of Thv−mv with klv (LV loads are supplied through MV lines):I - for ksc−mv = 12, II - for ksc−mv = 4
117
5.3 Voltage Unbalance Influence Coefficients
5.3.1 Preliminary Investigations - Dependency of Influence Coefficients
on Load Types/Bases
Consider the radial MV-LV system shown in Fig. 5.1239 where the voltage at the
sending end busbar (labelled ‘send’) of the line is taken as unbalanced. The purpose
is to assess the voltage unbalance that propagates from the sending end busbar to
the receiving end busbar (labelled ‘rec’) of the MV sub-system, or the influence
coefficient ksend−rec between the busbars send and rec. For this, the loads Smv:rec−local
and Smv:rec−ds, and the MV line (t) are considered as balanced.
send
Downstream LV system
MV line (t)
rec
Smv:rec-local (MVA)
Smv:rec-ds (MVA)
Figure 5.12: Radial MV-LV system (reproduction of Fig. 3.2)
39Reproduction of Fig. 3.2.
118
Based on the definition40, the influence coefficient ksend−rec can be expressed as:
ksend−rec =
∣∣∣∣V−:Usend/rec
V−:send
∣∣∣∣ (5.26)
where,
V−:Usend/rec - negative sequence voltage at the receiving end busbar that propagates
from the sending end busbar, which can be expressed as41:
V−:Usend/rec =∣∣V−:send − Z++:tI−:Usend/t
∣∣ (5.27)
V−:send - negative sequence voltage that exists at the sending end busbar
I−:Usend/t - negative sequence current in the line, which arises as a result of the voltage
unbalance existing at the sending end busbar
Equation (5.27) is a similar version of (5.5) considered above in relation to trans-
fer coefficients, where the impedance Z−−:tf−us (= Z++:tf−us) of the US-S coupling
transformer is replaced here with Z++:t of the line. Thus, as in the case of transfer co-
efficients, the negative sequence voltage ratio∣∣∣V−:Usend/rec
V−:send
∣∣∣, i.e. the influence coefficient
ksend−rec, can be written for passive loads as:
∣∣∣∣V−:Usend/rec
V−:send
∣∣∣∣ = ksend−rec ≈1∣∣∣1 + Z++:t
Z++:rec
∣∣∣γ (5.28)
where,
Z++:rec - downstream positive sequence impedance (or equivalent) seen at the MV
busbar rec
40See Section 5.1.41Ignoring zero sequence unbalance, and replacing the negative sequence impedance Z−−:t of the
line with the positive sequence impedance Z++:t.
119
γ =
1 for constant impedance loads
0 for constant current loads
β for constant power loads, where β = −1 and− 2 for low (∼ 0.9) and
high (∼ 1) lagging pf conditions respectively
Equation (5.28) can be rearranged as:
ksend−rec ≈ (1− V Rt)γ (5.29)
where, V Rt - voltage regulation of the line42. That is, the influence coefficient ksend−rec
is equal to unity, smaller than unity by the factor (1−V Rt), and greater than unity by
the factor (1− V Rt)β for constant current, constant impedance, and constant power
loads respectively. In other words, the term Z++:tI−:Usend/t causes the receiving end
negative sequence voltage∣∣V−:Usend/rec
∣∣ to deviate away from the sending end negative
sequence voltage |V−:send| by the factor (1 − V Rt)γ for passive loads. Note that an
identical behaviour was observed in Chapter 3 (Section 3.2) for the negative sequence
current or the term Z++:tI−:t/t which arises as a result of an asymmetrical line, where
the term Z++:tI−:t/t was seen to cause the negative sequence voltage |V t−:g/mv:rec| to
deviate away from the principal term |Z−+:tI+:t| by the factor (1− V Rt)γ for passive
loads. Employing this similarity, for three-phase induction motor loads supplied at
the LV level43, the influence of the term Z++:tI−:send/t on ksend−rec can be written44 as:
ksend−rec ≈1
1 +(
V Rt
1−V Rt
)(1
1ks
+ 1ksc−lvagg
) (5.30)
42Voltage regulation is defined as the ratio between the positive sequence voltage drop across theline (i.e. Z++:tI+:t), and the sending end positive sequence voltage.
43i.e. Smv:rec−local = 0.44Refer to (3.9).
120
As in Chapter 3 (Section 3.2), the factor (1 − V Rt)γ can be approximated to
unity (in other words I−:Usend/t ≈ 0), resulting in a unity ksend−rec for passive loads
in general. However, the influence coefficient ksend−rec for induction motor loads is
considerably smaller than unity45, implying that the impact of the negative sequence
current I−:Usend/t in the presence of motor loads on ksend−rec cannot be ignored as in
the case of passive loads. Based on this, ksend−rec for a mix of passive (at MV and/or
LV) and motor (at LV) loads can be expressed46 in a generalised form as:
ksend−rec ≈1
1 +(
V Rt
1−V Rt
)(klv
1kskm
+ 1ksc−lvagg
) (5.31)
Fig. 5.13 illustrates the variation of ksend−rec with km established using (5.31) in
comparison to the results obtained using unbalanced load flow analysis for the test
system47 described in Appendix B in relation to three cases where:
• klv = 1 (i.e. Smv:rec−local = 0)
• klv = 0.5
• klv = 0 (i.e. Smv:rec−ds = 0, which implies that no motor loads are supplied by
the MV line)
The results presented in Fig. 5.13 correspond to a selected operating scenario where
the MV line supplies a total of 10MVA load at 0.9 lagging pf resulting in a V Rt ≈
8.5%. Fig. 5.13 confirms the above basis given by (5.31) which describes the be-
haviour of different load bases in relation to the propagation of voltage unbalance in
a sub-system, also demonstrating the dependency of this propagation on the motor
proportion.
45e.g. 0.6 for ksc−lvagg= 20, ks = 6.7 and V Rt = 10%.
46Refer to (3.14).47Where ksc−lvagg
≈ 19, ks = 6.7, passive load composition - equal proportions of constantimpedance and constant power elements, and the MV line is taken as ideally transposed.
121
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.0 0.2 0.4 0.6 0.8 1.0
km
k sen
d-re
c
Load flow resultsEq. (5.31)
klv = 1 (Smv:rec:ds
= 0)
klv = 0 (Smv:rec-local = 0, i.e. no motor loads are supplied)
klv=0.5
Figure 5.13: Variation of ksend−rec with km for the cases where klv = 1, klv = 0.5 andklv = 0
5.3.2 Methodology for Evaluating Influence Coefficients
Consider the network shown in Fig. 5.1448. For the purpose of assessing the voltage
unbalance that propagates from any busbar i to other busbars 1, 2, ..., n, i.e. influence
coefficients ki−1, ki−2, ..., ki−n, the voltage at the upstream system, all loads supplied
by the system, and all lines in the network49 are considered to be balanced.
Based on a similar approach used to establish (3.21) in Chapter 3 (Section 3.3), the
nodal negative sequence currents50 I−:Ui/x at the busbars x (x = 1, 2, ..., n and x 6= i)
can be written in terms of the nodal negative sequence admittances Y−−:xy51 and
nodal negative sequence voltages52 V−:Ui/x at the busbars x, and the nodal negative
sequence voltage V−:i which exists at the busbars i as53:
48Reproduction of Fig. 3.5.49Including lines that exist at lower voltage levels, e.g. MV lines when assessing influence coeffi-
cients in an HV network.50Which arise as a result of the voltage unbalance that exists at the busbar i.51Y−−:xy = Y++:xy, where x, y = 1, 2, ..., n and x 6= i.52Which are caused by the voltage unbalance that exists at the busbar i.53Note that Y−+:xy = 0 for ideally transposed lines.
122
−I−:Ui/1
−I−:Ui/2
...
−I−:Ui/n
=
Y++:11 Y++:12 . . . Y++:1i . . . Y++:1n
Y++:21 Y++:22 . . . Y++:2i . . . Y++:2n
......
......
......
Y++:n1 Y++:n2 . . . Y++:ni . . . Y++:nn
V−:Ui/1
V−:Ui/2
...
V−:i
...
V−:Ui/n
(5.32)
Busbar 2
… … Busbar 1
Busbar 3
Busbar x
Sub-system S
Busbar n Upstream
system
Figure 5.14: Interconnected sub-system S (reproduction of Fig. 2.6)
123
Equation (5.32) can be decomposed as:
−I−:Ui/1
−I−:Ui/2
...
−I−:Ui/n
=
Y++:11 Y++:12 . . . Y++:1n
Y++:21 Y++:22 . . . Y++:2n
......
......
Y++:n1 Y++:n2 . . . Y++:nn
V−:Ui/1
V−:Ui/2
...
V−:Ui/n
+ V−:i
Y++:1i
Y++:2i
...
Y++:ni
(5.33)
Equation (5.33) can be written in a concise form as:
−[I−:Ui/x] = [Yxz:++][V−:Ui/x
] + V−:i[Yxi:++] (5.34)
where, x, z = 1, 2, ..., n and x, z 6= i. As shown in Section 5.3.1 above, the influence of
the negative sequence currents [I−:Ui] on the negative sequence voltages [V−:Ui
] should
be taken into account in the presence of considerable proportions of induction motor
loads. Thus, the current I−:Ui/x at any busbar x can be generally written when the
busbar supplies a mix of passive (at HV, MV and/or LV) and motor (at LV) loads as:
I−:Ui/x ≈ Y−−:x−im V−:Ui/x (5.35)
where, Y−−:x−im - downstream negative sequence admittance seen at the busbar x
taking only induction motors into account. As discussed in Chapters 3 (Section 3.3)
and 4 (Section 4.3) with regard to MV and HV systems respectively, this admittance
Y−−:x−im is inductive in nature, and dependant on the system and load characteris-
tics, system operating conditions and downstream load composition. As an example,
taking the network shown in Fig. 5.14 as an MV network with Fig. 5.15 representing
the system54 supplied by any busbar x, the admittance Y−−:x−im can be expressed as
54Where the downstream system represents an aggregated LV system.
124
given by (3.24) which is reproduced here by (5.36) for completeness55:
Y−−:x−im ≈ −j
(klv:x
1ksc−lvagg :x
+ 1ks:xkm:x
)(√3 |I+:x|Vn−mv
)(5.36)
Substitution of (5.35) in (5.34) and rearrangement gives:
[ki−x](n−1)×1 ≈∣∣∣[Y ′
++:xz]−1(n−1)×(n−1)[Y++:xi](n−1)×1
∣∣∣ (5.37)
where,
Y ′++:xz ≈ Y++:xz + Y−−:x−im for x = z
Y ′++:xz = Y++:xz for x 6= z
Downstream system supplied by the busbar x
Busbar x
Smv:x-local
Smv:x-ds
Figure 5.15: System representation of any busbar x of the MV system shown inFig. 5.14 (reproduction of Fig. 3.6)
As discussed in Chapter 4 (Section 4.3) in relation to the emission arising as
a result of system inherent asymmetries, the presence of positive sequence voltage
controlled components such as PV generators and synchronous condensers in a system
force the negative sequence voltage at the connected busbars to be zero disregarding
the existence of sources of unbalance. Equation (5.37) which gives the influence
55Refer to Chapter 3 for the definitions of the symbols.
125
coefficients ki−x between busbar i and other neighbouring busbars x does not consider
the presence of such components, and thus requires suitable adjustments such that
the impact of zero voltage unbalance (or of voltage controlled components) at given
busbars on these influence coefficients is accommodated. Similar to the approach
taken in Chapter 4, the matrix equation (5.37) can be modified to incorporate the
influence of a constraint V−:j = 0 at any busbar j on the influence coefficients [ki−x]
by56:
• Reducing the dimension of the matrix [Y ′++:xz] down to (n − 2) × (n − 2) by
removing both the jth row and column.
• Reducing the dimension of the matrix [Y++:ix] down to (n− 2)×n by removing
the jth row.
5.3.3 Verification of the Methodology Using a Three-bus MV Test
System
The proposed methodology is applied to the three-bus MV network (60Hz, 12.47kV,
three-wire) shown in Fig. 5.1657 for evaluating the voltage unbalance which propa-
gates from busbar 1 to busbars 2 and 3, i.e. influence coefficients k1−x for x = 2, 3.
Lengths of the lines which are taken as identical in construction58 and ideally trans-
posed are shown alongside the lines. The positive sequence admittance per km of the
lines is (1.0098−j2.0630)Skm. Busbar 3 supplies MV loads (2MVA at 0.9 lagging pf)
with equal compositions of constant impedance and constant power elements. That
is, km:x = 0 implying that Y−−:x−im ≈ 0 for x = 3. Busbar 2 supplies a mix (4MVA
56Note that there is only (n − 2) number of influence coefficients are to be determined as theinfluence coefficient ki−j is known to be zero.
57Reproduction of Fig. 3.7.58See Appendix B for further details.
126
at 0.9 lagging pf) of passive loads59 and induction motors (ks = 6.760) at the LV
level, which accounts for 40% of the total load supplied by the system. Note that
Y−−:x−im 6= 0 when km:x > 0 for x = 2.
Fig. 5.17 illustrates the variations of k1−2 and k1−3 with km:2 established using the
proposed methodology61 in comparison to the results obtained using unbalanced load
flow analysis, demonstrating the accuracy of the proposed technique. Further, these
results reveal that motor loads help reducing the voltage unbalance that propagates
between neighboring busbars compared to the case where only passive loads exist.
5.3.4 Verification of the Methodology Using the IEEE 14-bus Test
System
The proposed methodology is further applied to the IEEE 14-bus test system shown
in Fig. 5.18 which consists of positive sequence voltage controlled busbars (busbars
1, 2, 3, 6 and 8), taking it as a 66kV, 60Hz and three-wire network supplying constant
power loads at the HV level. System and line62 data are given in Appendix K.
Fig. 5.19 illustrates the influence coefficients ki−x where i = 4, x = 1−14 and x 6=
463 established using the proposed methodology in comparison to the results obtained
using unbalanced load flow analysis, further validating the proposed technique.
59Which are represented using a mix of constant impedance and constant power elements withequal compositions.
60See Appendix B for further details.61Application of the methodology to the test system is described in Appendix M.62Which are taken as identical in construction as described in Appendix H, and ideally transposed.
The positive sequence admittance per km of the lines = (0.2729 + j1.0244) × 102pu (based on a100MV A base).
63i.e. the propagation of voltage unbalance from busbar 4 to other busbars.
127
204A
MV busbar 2 LV (460V)
MV busbar 1 (1.05pu, -4.070)
MV busbar 3 (1.01pu, -5.570)
10km 135A
10km 59A
5km 152A
(0.98pu, -6.740) 4MVA 0.9 lagging pf
Upstream HV (66kV) system (1.05pu, 00)
4MVA 0.9 lagging pf 2MVA
0.9 lagging pf
HV-MV coupling transformer – 12MVA, winding resistance = 1%, leakage reactance = 10%, secondary tap setting = 1.05pu MV-LV coupling transformer – aggregated representation of fully loaded 1MVA transformers with winding resistance = 1%, leakage reactance = 5% and secondary tap setting = 1.05pu
Figure 5.16: Three-bus MV test system considered for applying the proposed method-ology (reproduction of Fig. 3.7)
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.0 0.3 0.5 0.8 1.0km:2
k 1-2
, k1-
3
Load flow resultsMethodology
k1-3
k1-2
Figure 5.17: Variations of k1−2 and k1−3 with km:2 for the three-bus MV test system
128
Figure 5.18: IEEE 14-bus test system (reproduction of Fig. 4.9)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1 2 3 5 6 7 8 9 10 11 12 13 14
Busbar x
k 4-x
Load flow resultsMethodology
Figure 5.19: Influence coefficients k4−x (x = 1− 14, x 6= 4) for the IEEE 14-bus testsystem
Please see print copy for image
129
5.4 Chapter Summary
This chapter has addressed the propagation of voltage unbalance. This is a key aspect
in assessing emission limits to individual installations connected to EHV, HV, MV
and LV public power systems essentially based on the IEC/TR 61000-3-13 recommen-
dations. This propagation from higher voltage to lower voltage systems in terms of
transfer coefficients (Section 5.2), and from one busbar to other neighbouring busbars
of a sub-system in terms of influence coefficients (Section 5.2) has been addressed.
Theoretical bases which describe the behaviour exhibited by four basic load types
with regard to transfer coefficients and influence coefficients have been developed.
The following can be drawn as a summary of this work:
• Transfer coefficients, for passive loads, have to be scaled up by the factor∣∣∣1 + j 1ksc−s
∠θpf :s
∣∣∣τ relative to the value of unity assumed in the IEC method,
where τ = 0, 1, 2 ∼ 3 for constant impedance, constant current and constant
power loads respectively. Noting that the factor ksc−s can be a value in the
range of 5 to 25 for various systems and the exponent τ varies in the range of 0
to 3 for different load types, a uniform behaviour or the behaviour of constant
impedance loads as assumed in the IEC approach cannot be used to represent
all load types with a high degree of accuracy.
• Transfer coefficients, for induction motor loads, have to be scaled down by the
factors 1„1+ ks
ksc−lv
« and 1241+“
1ksc−mv
”0@ ks
1+ ksksc−lvagg
1A35 relative to unity in the cases of
the MV to LV and HV to MV propagation respectively. Noting that 5 < ks < 7,
the degree of this reduction is significant compared to the increment introduced
by passive loads.
130
• Considering a simple two-bus radial sub-system, the influence coefficients can
be approximated to unity for passive loads in general. However, these influence
coefficients can be considerably smaller than unity when the network supplies
a large proportion of induction motor loads. The conclusion of this observation
is that the negative sequence currents which arise as a result of the voltage
unbalance that exists at a particular busbar introduce a considerable impact
on influence coefficients in the presence of large proportions of motor loads,
although it is insignificant for passive loads.
Systematic methods for evaluating the MV to LV and HV to MV transfer coeffi-
cients, and influence coefficients for interconnected network environments have been
developed. These have been verified using unbalanced load flow analysis. In summary:
• It has been demonstrated that the proposed new formulation for assessing the
MV to LV transfer coefficient gives a more accurate estimation, particularly
for load bases which are dominated by passive elements, compared to the IEC
method.
• The MV to LV and HV to MV transfer coefficients can vary in the ranges
of 0.6 to 1.1 and 0.5 to 1.4 respectively depending on the system and load
characteristics and downstream load composition.
• Verification of the proposed method for estimating influence coefficients has
been undertaken employing a three-bus MV test system and also the IEEE
14-bus test system.
Chapter 6
A Revised Voltage Unbalance
Allocation Technique Based on the
IEC/TR 61000-3-13 Guidelines
6.1 Introduction
The IEC approach of managing continuous power quality disturbances (e.g. har-
monics, flicker and voltage unbalance) through the allocation of emission limits to
installations is based on a common philosophy. Thus, the problem of violating the
set planning levels, which has been identified with regard to harmonics and flicker1,
is an anticipated problem with the new IEC/TR 61000-3-13 voltage unbalance al-
location aproach as well. This chapter examines the IEC/TR 61000-3-13 procedure
employing a simple three-bus HV test system in Section 6.2. The principles of the
constraint bus voltage (CBV) method, that was discussed in Section 2.9 of Chap-
ter 2 as an alternative approach for harmonics and flicker allocation, are introduced
to voltage unbalance so that a robust allocation technique which closely aligns with
1Refer to Section 2.9 of Chapter 2.
131
132
IEC/TR 61000-3-13 is developed in Section 6.3 of this chapter. Section 6.4 examines
this revised allocation technique using the above mentioned test system. A summary
of the chapter is given in Section 6.5.
6.2 Examination of the IEC/TR 61000-3-13 Approach
Consider the three-bus HV system (60Hz, 66kV, three-wire) shown in Fig. 6.1. The
system supplies constant power loads at the HV level. Considered operating scenario
and resulting positive sequence system conditions2 are also indicated in Fig. 6.1.
Lengths of the lines which are taken as identical in construction (including the phase
positioning) are shown alongside the lines. The relevant admittance data (per km)
of the lines are3:
• positive sequence admittance = (0.6265− j2.3517)Skm
• negative-positive sequence coupling admittance when untransposed =
(0.1040+ j0.1779)Skm
The examination procedure involves the following two steps:
• Calculation of the emission limits to individual installations using the IEC/TR
61000-3-13 prescribed formulae which have been reproduced in Section 2.8.3
of Chapter 2, together with the methodologies that have been proposed in
Chapters 3 - 5.
• Derivation of the busbar emission levels, which result in when all individual
installations are injecting at their allocated limits, using the general summa-
tion law.
2Nodal voltages and line currents, which are obtained using load flow analysis.3Refer to Appendix H for further details.
133
HV busbar 2 (0.992pu, -10.840)
HV busbar 1 (1.026pu, -7.980)
HV busbar 3 (1.007pu, -9.510)
50km 112A
40km 64A
20km 151A
20MVA 0.95 lagging pf 10MVA
0.95 lagging pf
20MVA 0.95 lagging pf
EHV-HV coupling transformer – 60MVA, winding resistance = 1%, leakage reactance = 20%, secondary tap setting = 1pu
Upstream EHV (230kV) system (1.100pu, 00)
Figure 6.1: Three-bus HV test system considered for examining the IEC/TR 61000-3-13 approach
134
6.2.1 Calculation of Individual Emission Limits
The procedure of the calculation of the emission limits to the three aggregated loads
supplied by the test system is described in the following steps:
Global Emission Allowance
The voltage at the upstream system of the HV network under evaluation is taken as
balanced. That is, the upstream contribution to voltage unbalance in the considered
HV system is zero, resulting in a global emission allowance Ug/hv which is equal to the
network planning level. As indicated in IEC/TR 61000-3-13, a uniform HV planning
level of 1.4% is assumed, i.e. Ug/hv = 1.4%.
Apportioning of the Global Emission Allowance to Busbars
Referring to (2.10) and (2.11), the derivation of the busbar emission allowances Ug/hv:x
requires the initial evaluation of influence coefficients. A method for estimating influ-
ence coefficients was proposed in Section 5.3 of Chapter 5 as given by (5.37). Table 6.1
gives these influence coefficients for the considered test system, which are derived us-
ing the proposed method4. Fig. 6.2 gives a comparison of these values with those
obtained using unbalanced load flow analysis. Arising from these values, an inter-
esting fact to note is that influence coefficients do not essentially hold the reciprocal
relationship ki−x = kx−i implying that the propagation from busbar i to busbar x is
not necessarily equal to that from busbar x to busbar i.
The total apparent power Shv:x supplied by any busbar x, the total available
apparent power Shv:x−total of the entire sub-system as seen at the busbar x which is
4As the test system supplies constant power loads at all three busbars (i.e. km:x = 0 implyingthat Y−−:x−im ≈ 0 for x = 1, 2, 3), the admittances Y ′
++:xz ≈ Y++:xz not only for x 6= z but also forx = z.
135
Table 6.1: Influence coefficients for the test system shown in Fig. 6.1
k1−2 k1−3 k2−1 k2−3 k3−1 k3−2
1 1 0.57 0.71 0.69 0.86
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Influence coefficient
Val
ue
Load flow resultsMethodology
k1-2 k2-3k2-1k1-3 k3-1 k3-2
Figure 6.2: A comparison of the influence coefficients for the test system derived usingthe proposed method: (5.37), and unbalanced load flow analysis
Table 6.2: Shv:x, Shv:x−total and Ug/hv:x for the test system shown in Fig. 6.1
Busbar (x) Shv:x (MVA) Shv:x−total (MVA) Ug/hv:x (%)
1 20 38.2 0.88
2 20 48.6 0.74
3 10 44.2 0.48
136
derived using (2.10), and the busbar emission allowance Ug/hv:x at the busbar x that is
calculated using (2.11) with a summation law exponent of 1.45 are given in Table 6.2
for each of the three busbars. Note that, as the contributions from the neighboring
busbars 1 and 3 to voltage unbalance at busbar 2 (k1−2 = 1 and k3−2 = 0.86) are seen
to be greater than that at busbar 1 (k2−1 = 0.57 and k3−1 = 0.69), the allocation
approach allows a lower level of emission for busbar 2 than that for busbar 1 although
both busbars 1 and 2 supply loads of equal MVA demand.
Individual Emission Limits
Two cases are considered for examining the IEC/TR 61000-3-13 approach:
• Case 1 - all HV lines are ideally transposed. That is, the contribution from
system inherent asymmetries to the global emission levels is zero, resulting in a
K ′uex = 06 or Kuex = 17 for all busbars. This leads to voltage unbalance allo-
cation formulae which are identical to that of the harmonics/flicker allocation
[2, 3], where the total busbar allowance Ug/hv:x can be allocated to installations.
• Case 2 - the line between busbars 1 and 3 is ideally transposed, and the other
two lines are untransposed. That is, system inherent asymmetries make some
contribution to the global emission levels resulting in a K ′uex > 0 or Kuex < 1
for some or all busbars, implying that only a fraction of the busbar emission
allowance Ug/hv:x can be allocated to installations. A method for evaluating the
global emissions U linesg/hv:x which arise as a result of line asymmetries was proposed
in Section 4.3 of Chapter 4 as given by (4.16). These emissions for the test
5Which is the indicative value given in IEC/TR 61000-3-13.6K ′uex is the factor which accounts for the emission arising as a result of system inherent asym-
metries.7Kuex is the factor which represents the fraction of the busbar emission allowance that can be
allocated to installations.
137
system which are derived using the proposed method8, the K ′uex factors which
are calculated using (2.14) with the above busbar allowances Ug/hv:x (Table 6.2),
and the Kuex factors (= 1 − K ′uex) are given in Table 6.3. Fig. 6.3 gives a
comparison of these K ′uex values with those obtained using unbalanced load
flow analysis.
The busbar emission limits Ehv:x9 or the emission limits to the three aggregated
loads, which are derived using (2.15) with the above busbar allowances Ug/hv:x (Table
6.2) and Kuex factors (unity for Case 1, and as given in Table 6.3 for Case 2), are
given in Table 6.4 for Cases 1 and 2. Note that, in Case 2, only 70% of the busbar 2
allowance can be allocated to the connected load as the emission arising as a result
of line asymmetries at the busbar is considerably high.
Table 6.3: U linesg/hv:x, K ′uex and Kuex for Case 2 of the test system shown in Fig. 6.1
Busbar (x) U linesg/hv:x (%) K ′uex Kuex
1 0 0 1
2 0.39 0.41 0.59
3 0.05 0.04 0.96
Table 6.4: Ehv:x according to IEC/TR 61000-3-13 for the test system shown in Fig. 6.1
Busbar (x) Ehv:x for Case 1 (%) Ehv:x for Case 2 (%)
1 0.88 0.88
2 0.74 0.51
3 0.48 0.47
8As the test system supplies constant power loads at all three busbars (i.e. km:x = 0 implyingthat Y−−:x−im ≈ 0 for x = 1, 2, 3), the admittances Y ′
++:xy ≈ Y++:xy and Y ′−+:xy ≈ Y−+:xy not only
for x 6= y but also for x = y.9i.e. the combined emission limit of a load of which the agreed apparent power is equal to the
total apparent power Shv:x supplied by the busbar.
138
0.0
0.1
0.2
0.3
0.4
0.5
1 2 3
Busbar (x)
K'u
e x
Load flow resultsMethodology
Figure 6.3: A comparison of the K ′uex factors for the test system derived using theproposed method: (4.16), and unbalanced load flow analysis
6.2.2 Resulting Busbar Emission Levels and Examination Remarks
When an installation supplied by a sub-system is injecting at its allocated limit, the
installation introduces an emission level at the connected busbar which is equal to
the imposed limit while making some contributions to voltage unbalance at other
neighboring busbars as well. According to IEC/TR 61000-3-13, this emission level
introduced at a neighboring busbar can be quantified10 as a product of the emission
limit and the influence coefficient between the two busbars. Thus, when all instal-
lations supplied by the system are injecting at their individual limits, the resulting
global emission level U reultg/s:x at any busbar x of the sub-system can be expressed using
the general summation law, also taking the global emission which arises as a result
of system inherent asymmetries into account, as:
U reultg/s:x =
[(k1−xEs:1)
α + (k2−xEs:2)α + ... + (Es:x)
α + ... + (kn−xEs:n)α
+ (U linesg/s:x)
α]1/α
(6.1)
10Refer to the definition of the influence coefficient given in Section 2.8.3 of Chapter 2.
139
The busbar emission levels for the test system, which result in when the three
loads are injecting at the above allocated limits (Table 6.4), are derived using (6.1)
for the two cases and given in Table 6.5.
Table 6.5: U reultg/hv:x arising as a result of the IEC/TR 61000-3-13 allocation procedure
for the test system shown in Fig. 6.1
Busbar (x) U reultg/hv:x for Case 1 (%) U reult
g/hv:x for Case 2 (%)
1 1.24 1.15
2 1.52 1.51
3 1.40 1.30
Note that although the resulting emission levels at busbars 1 and 3 are below the set
planning level of 1.4%, this at busbar 2 exceeds the planning level by approximately
8% in each of the cases. This indicates, as anticipated, that the IEC allocation policy
is unlikely to be robust enough to comply with one of the key allocation objectives not
only in the cases of harmonics and flicker but also with regard to voltage unbalance.
6.3 A Revised Voltage Unbalance Allocation Technique Based
on the CBV Allocation Principles
The CBV allocation method which has been suggested for harmonics and flicker11
cannot be applied in its present form to voltage unbalance, as the case of voltage
unbalance involves an additional aspect which is the emission arising as a result of
system inherent asymmetries. Due to the presence of this additional emission, the
total busbar emission allowance Ug/s:x cannot be allocated to installations. That is,
in the case of voltage unbalance, the busbar emission limit Es:x12 is usually smaller
11Refer to Section 2.9.12Es:x of any busbar x is the emission limit of the load of power Ss:x.
140
than the busbar emission allowance Ug/s:x13, whereas Es:x = Ug/s:x for the cases of
harmonics and flicker. Thus, appropriate revisions addressing this issue are required
in extending this CBV method to voltage unbalance allocation.
Closely following the IEC/TR 61000-3-13 approach, in which the busbar emis-
sion allowance is derived14 by apportioning the global emission allowance Ug/s in
proportion to the term α√
Ss:x, the principle given by (2.22) for determining the bus-
bar emission limit under the CBV harmonics/flicker allocation policy is extended to
voltage unbalance for deriving the busbar allowance Ug/s:x as:
Ug/s:x = kaα√
Ss:x (6.2)
The allocation constant ka is yet to be determined. As addressed in IEC/TR 61000-
3-1315, the emission which arises as a result of system inherent asymmetries can be
accounted for by using the factor Kuex (= 1 − K ′uex)16 in determining the busbar
emission limit Es:x as:
Es:x = α√
KuexUg/s:x = kaα√
KuexSs:x (6.3)
The factor Kuex can be expressed17 for the new allocation policy as a function of the
allocation constant as:
Kuex = 1−
(U lines
g/s:x
Ug/s:x
)α
= 1−(U lines
g/s:x)α
kαa Ss:x
(6.4)
13Ug/s:x for any busbar x accounts for the emissions which arise at the busbar as a result of boththe load of power Ss:x and line asymmetries.
14See (2.11).15See (2.15).16The factors Kuex and K ′uex are defined by (2.13) and (2.14) respectively.17i.e. substituting Ug/s:x = ka
α√
Ss:x.
141
Substitution of (6.4) in (6.3) gives:
Es:x == α
√kα
a Ss:x − (U linesg/s:x)
α (6.5)
As suggested in relation to harmonics and flicker, ensuring voltage unbalance levels
at network busbars do not exceed the set planning level, the allocation constant ka
can be chosen to be the largest value such that:
U resultg/s:x ≤ Ug/s for every busbar x (6.6)
where, the resulting emission level U resultg/s:x at any busbar x when all installations
are injecting at their individual limits can be written as given by (6.1) which can
rearranged for the new allocation policy given by (6.5) as:
U resultg/s:x =
(kα
a
(kα
1−xS1 + kα2−xS2 + ... + Sx + ... + kα
n−xSn
)−
(U linesg/s:1)
α + (U linesg/s:2)
α + ... + (U linesg/s:n)α
)1/α
(6.7)
Then, ka can be established in order to satisfy (6.6) as:
ka = min
α
√√√√√√√√√(Ug/s)α +
n∑i=1,i6=x
(U lines
g/s:x
)αSs:x +
n∑i=1,i6=x
(kα
i−xSs:i
) (6.8)
In summary, under the suggested alternative voltage unbalance allocation policy,
the emission limit to any customer installation j of which the MVA demand is Ss:x−j
to be connected at any busbar x can be derived using:
142
Es:x−j = kaα√
KuexSs:x−j (6.9)
where the allocation constant ka is determined using (6.8), and the Kuex factor is
derived from (6.4).
6.4 Examination of the Revised Voltage Unbalance Alloca-
tion Technique
The same three-bus HV test system shown in Fig. 6.1 is employed for examining the
alternative voltage unbalance allocation approach suggested in Section 6.3 above.
6.4.1 Calculation of Individual Emission Limits
The calculation procedure, under the revised allocation policy, of the emission limits
to the three aggregated loads supplied by the test system is described in the follow-
ing steps:
Allocation Constant ka
Table 6.6 gives the values18, corresponding to each of the three busbars, of the right
hand side (RHS) of (6.8) in relation to the two cases listed above. Choosing the
smallest value of the RHS of (6.8), which corresponds to busbar 2 in each of the
cases, the allocation constant ka for the two cases can be identified as given in Table
6.7. That is, the allocation constant is chosen such that the resulting emission level
at busbar 219 is constrained at the allowed global emission level of 1.4%.
18α = 1.4, Ug/hv = 1.4%, apparent power values Shv:x as given in Table 6.2, and influencecoefficients as given in Table 6.1, and emissions U lines
g/hv:x = 0 for Case 1 and as given in Table 6.3 forCase 2 are used in the calculation.
19Note that this is the busbar at which the resulting emission level was observed to be above theset planning level under the IEC/TR 61000-3-13 allocation procedure.
143
Table 6.6: Values of the RHS of (6.8) in relation to the test system shown in Fig. 6.1
Busbar (x) RHS of (6.8) RHS of (6.8)
for Case 1 for Case 2
1 0.110 0.124
2 0.088 0.089
3 0.096 0.108
Table 6.7: ka for the test system shown in Fig. 6.1
Case 1 Case 2
ka 0.088 0.089
Individual Emission Limits
The Kuex factors calculated20 using (6.4), and the busbar emission limits Ehv:x or
the emission limits to the three aggregated loads derived21 from (6.9) for the two
different cases are given in Table 6.8. Figs. 6.4: I - II provides a comparison of the
individual emission limits derived according to IEC/TR 61000-3-13 and the revised
allocation method for Cases 1 and 2 respectively. This illustrates that the allocated
limits under IEC/TR 61000-3-13 are greater, specially at busbar 1, than that under
the revised method.
Table 6.8: Kuex and Ehv:x according to the revised allocation method for the testsystem shown in Fig. 6.1
Busbar Kuex for Kuex for Ehv:x for Ehv:x for
(x) Case 1 Case 2 Case 1 (%) Case 2 (%)
1 1 1 0.75 0.76
2 1 0.60 0.75 0.53
3 1 0.96 0.46 0.45
20α = 1.4, allocation constant ka as given in Table 6.7, apparent power values Shv:x as given inTable 6.2, and emissions U lines
g/hv:x = 0 for Case 1 and as given in Table 6.3 for Case 2 are used in thecalculation.
21α = 1.4, allocation constant ka as given in Table 6.7, apparent power values Shv:x as given inTable 6.2, and the Kuex factors as given in column 1 of this table are used in the calculation.
144
II - Case 2
0.0
0.2
0.4
0.6
0.8
1.0
1 2 3
Busbar (x)
Ehv
:x (%
)
Revised method
IEC/TR 61000-3-13
I - Case 1
0.0
0.2
0.4
0.6
0.8
1.0
1 2 3Busbar (x)
Ehv
:x (%
)
Figure 6.4: Comparison of the busbar emission limits Ehv:x derived according toIEC/TR 61000-3-13 and the revised method for the test system: I - for Case 1,II - for Case 2
6.4.2 Resulting Busbar Emission Levels and Examination Remarks
The busbar emission levels which result in when the three loads are injecting at the
above allocated limits (Table 6.8) are derived22 using (6.1) for the two cases, and given
in Table 6.9. Figs. 6.5: I - II provides a comparison of the resulting busbar emission
levels under IEC/TR 61000-3-13 and the revised allocation procedure for Cases 1
and 2 respectively. This illustrates that the resulting busbar emission levels under
IEC/TR 61000-3-13 are greater, at all three busbars, than that under the revised
method.
Table 6.9: U reultg/hv:x arising as a result of the revised allocation procedure for the test
system shown in Fig. 6.1
Busbar (x) U reultg/hv:x for Case 1 (%) U reult
g/hv:x for Case 2 (%)
1 1.12 1.04
2 1.40 1.40
3 1.28 1.18
22α = 1.4, influence coefficients as given in Table 6.1, and emissions U linesg/hv:x = 0 for Case 1 and
as given in Table 6.3 for Case 2 are used in the calculation.
145
II - Case 2
0.0
0.4
0.8
1.2
1.6
2.0
1 2 3
Busbar (x)
Revised methodIEC/TR 61000-3-13
(%
):
/lines
xhv
gU
I - Case 1
0.0
0.4
0.8
1.2
1.6
2.0
1 2 3Busbar (x)
(%
):
/lines
xhv
gU
Planning level = 1.4%
Figure 6.5: Comparison of the resulting emission levels U reultg/hv:x derived according to
IEC/TR 61000-3-13 and the revised method for the test system: I - for Case 1,II - for Case 2
As expected, the revised allocation technique restricts the resulting emission level
at the constrained busbar (e.g. busbar 2 of the test system) to the allowed global
emission level (= 1.4% for the test case) while maintaining the emission levels at other
busbars (e.g. busbars 1 and 3 of the test system) below the set limit. That is, this
proposed alternative technique allows a robust allocation in the sense that it satisfies
all four key allocation objectives23:
• The allocation constant ka determined using (6.8) ensures that the resulting
emission levels at all network busbars are maintained at or below the set plan-
ning level.
• The revised allocation policy given by (6.9) ensures that customer installations
of equal MVA demand (whether connected at the same busbar or a different
busbars) receive identical emission limits, and also that larger customer instal-
lations (in MVA demand terms) to receive larger emission levels than smaller
installations.
23See Section 2.9.
146
• In determining the allocation constant ka using (6.8), the resulting emission level
at a busbar is constrained at the allowed global emission level. This implies that
the network absorption capacity is fully utilised.
6.5 Chapter Summary
This chapter has examined the recently introduced IEC/TR 61000-3-13 voltage unbal-
ance allocation approach. It was seen, as in the case of the counterpart IEC harmonics
and flicker allocation methods, the prescribed voltage unbalance allocation procedure
also leads to planning levels are being exceeded even when no customer exceeds the
allocated limit.
Closely aligning with the IEC/TR 61000-3-13 guidelines, a revised voltage un-
balance allocation policy based on the CBV harmonics/flicker allocation principles,
whereby the emission levels at network busbars are explicitly forced to be at or below
the set planning levels when all loads inject their limits derived under the new ap-
proach, has been proposed. This revised allocation technique was seen to restrict the
resulting emission level at the constrained busbar to the allowed global emission level
while maintaining the emission levels at other busbars below the set limit. Further,
this proposed alternative technique allows a robust allocation in the sense that it
satisfies all four key allocation objectives.
Chapter 7
Analysis of the Problem of Voltage
Unbalance in Interconnected
Power Systems
7.1 Introduction
The presence of voltage unbalance in electricity transmission and distribution net-
works has continued to be a power quality issue of concern primarily due to difficul-
ties found by some network service providers in maintaining acceptable levels. As
an example, consider the 66kV sub-transmission interconnected system (this will be
referred to as ‘study system’ hereinafter) shown in Fig. 7.1, which is owned and
operated by an Australian (Victoria) utility. This is connected to the EHV trans-
mission system at S1 (bulk supply point: BSP) where the level of voltage unbalance
has been measured to be negligible. The system is divided into three sub-parts: up-
stream (US), central part (CP) and downstream (DS) for convenience in presenting.
Some of the sub-transmission lines are longer than 50km, and are not systematically
147
148
ZS1: bulk supply point
Voltage regulators
PV generator: operates continuously
PV generator: operates only in limited time periods
Loads
Capacitor bank s
ABC
G
H
F
E
D
K
L
J
I
N
M
ZS4
ZS2ZS3
ZS6
ZS7
ZS8
ZS9
ZS5
Local generator:operates continuously
Local generator:operates only in limited time periods
Figure 7.1: 66kV sub-transmission interconnected system under study
149
0.0
0.4
0.8
1.2
1.6
2.0
S2 S3 S4 S5 S6 S7 S8 S9
Busbar
VU
F (%
)
Figure 7.2: Measured nodal VUF values for the study system
transposed1. Despite the applicable voltage unbalance limit of 1% according to the
Victorian electricity distribution code2 [73], the measured voltage unbalance levels3
during the peak demand periods at S6, S7 and S8 have been seen to exceed the limit
in addition to significant levels (0.6% - 0.8%) at the upstream busbars S2 and S4.
These measured values corresponding to a selected time stamp4 that lies within the
system peak are illustrated in Fig. 7.2. Although effort has been put to address
this issue by balancing loads at some of the busbars, no significant improvements
have been noted indicating that asymmetries associated with the sub-transmission
lines can play a vital role in leading up to these excessive voltage unbalance levels.
However, due to insufficient knowledge at present in systematically evaluating such a
problem, especially in meshed network environments, the system operator/owner has
found challenges in better managing the network.
1Line transposition is not a common practice at this voltage level due to economic reasons.2See Section 2.7.3These measurements, provided by the system operator/owner, are not based on a well defined
measurement and evaluation procedure such as described in the standard IEC 61000-4-30. Therefore,the accuracy of these VUF values are uncertain, however they can be considered to provide a generalsense of the problem. Measurements are not available for the busbars S3, S5 and S9.
4Operating conditions at this time stamp are given in Appendix N.
150
Objectives of the work presented in this chapter are:
• To carry out deterministic studies5 which are required to develop an insight into
the problem experienced by the study system.
• To develop systematic approaches which would facilitate the evaluation of such
a problem including the identification of major contributors to voltage unbal-
ance levels, and transposition options that can effectively reduce the emission
arising as a result of line asymmetries. These approaches would provide addi-
tional assistance in the application of the IEC/TR 61000-3-13 voltage unbalance
allocation methodologies.
Sections 7.2 and 7.3 present deterministic studies and outcomes, together with
the verification of theoretical findings, in relation to line and load asymmetries re-
spectively. The overall system behaviour or the combined influence of line and load
asymmetries is addressed in Section 7.4. The chapter is summarised in Section 7.5.
7.2 Voltage Unbalance Behaviour of Line Asymmetries
7.2.1 Impact of the Line Asymmetries of the Study System on the
Voltage Unbalance Problem
The impact of the interaction of all asymmetrical lines of the study system on the
problem of voltage unbalance is established in terms of the VUF values at the various
busbars using unbalanced load flow analysis. This is accomplished by synthesising the
system operation6, however under balanced loading7 conditions. Fig. 7.3 illustrates
5These studies are supported by the developed unbalanced load flow program which is describedin Appendix O.
6As revealed by measurements, voltage at the BSP is taken as balanced. This eliminates thecontribution made by the upstream EHV system to voltage unbalance in the study system.
7Constant power loads are assumed.
151
these results in comparison to the measured voltage unbalance levels8 (Fig. 7.2)
with respect to the selected time stamp. These results demonstrate that the line
asymmetries themselves introduce excessive voltage unbalance levels at the busbars
located in the central part (S6 and S7 - 1%) and in the downstream part (S8 and S9 -
1.4%) of the system, in addition to the considerable levels at the upstream busbars (S2
and S4 - 0.5%). This indicates the importance of proper line transposition practices
even at sub-transmission voltage levels, which is currently seen not to be a common
practice. However, the transposition of each single line is not economically viable
and/or practically feasible, and thus further knowledge is required in order to assist
a careful selection of lines for the transposition.
0.0
0.4
0.8
1.2
1.6
2.0
S2 S3 S4 S5 S6 S7 S8 S9
Busbar
VU
F (%
)
Line asymmetriesMeasured values
Figure 7.3: Nodal VUF values (load flow results) which arise as a result of the lineasymmetries, in comparison to the measured values
8Note that these measured levels arise as a result of the interaction of both the line and loadasymmetries which exist in the actual system.
152
7.2.2 Voltage Unbalance Behaviour of the Individual Lines of the
Study System - as Standalone Lines
When an asymmetrical line supplying balanced passive loads operates as a standalone
line9 which is energised by balanced supply voltages, as evident from Chapters 3 - 410,
the negative sequence voltage V t−:rec at the receiving end busbar of the line (t) can be
written for operating scenarios most likely to occur in practice as11:
V t−:rec ≈ −Z−+:tI+:t (7.1)
where,
Z−+:t - negative-positive sequence coupling impedance of the line
I+:t - positive sequence current in the line
That is, the magnitude of V t−:rec is given by the term |Z−+:tI+:t|. Thus, for a given
line, as illustrated in Fig. 7.4 which shows the results obtained using unbalanced
load flow analysis for each of the lines12 of the study system, the variation of |V t−:rec|
with |I+:t| is approximately linear with a gradient that is being equal to |Z−+:t|. In
addition, the phase angle θV t−:rec
of the negative sequence voltage V t−:rec: θV t
−:rec≈
1800 + θZ−+:t + θI+:t13, where θZ−+:t , θI+:t are phase angles of the impedance Z−+:t
and the current I+:t respectively. Fig. 7.5 illustrates the variation of θV t−:rec
with |I+:t|
obtained using unbalanced load flow analysis for each of the lines, justifying this re-
9i.e. without introducing any effect of interactions which exist in interconnected network envi-ronments, and of other sources of unbalance.
10The impact of the negative sequence current I−:t/t in the line arising as a result of the lineitself (or of the term Z++:tI−:t/t where Z++:t is the positive sequence impedance of the line) on thenegative sequence voltage V t
−:rec is negligible when the line supplies passive loads.11The study system is three-wired. Thus, the impact of zero sequence variables on the negative
sequence voltage V t−:rec can be ignored.
12Descriptive data (lengths and impedances) of the lines is given in Appendix N (Table N.7).13Under high load power factor conditions, this can be further simplified as: θV t
−:rec≈ 1800+θZ−+:t .
153
lationship of the phase angle θV t−:rec
. For a given line, the angle θV t−:rec
remains almost
constant for various |I+:t| values, where this distinct angle is determined by the angle
θZ−+:t as the angle θI+:t is governed by the load power factor14.
0
100
200
300
400
500
600
0 50 100 150 200 250 300
|I+:t| (A)
|Vt -:r
ec| (
V)
ABCDEFGHIJKLMN
Figure 7.4: Variation of |V t−:rec| with |I+:t| for the individual lines
In summary, the term −Z−+:tI+:t (which is a vector) governs the voltage unbal-
ance behaviour exhibited by a standalone asymmetrical line. The impedance element
|Z−+:t| which is the principal intrinsic parameter behind the voltage unbalance emis-
sion can be used as a measure for assessing the degree of asymmetry associated with
a line. Employing this, a rank for the lines of the study system can be assigned as
given in Table 7.115.
14The behaviour associated with line L (in both Figs. 7.4 and 7.5) is slightly out of ordinary athigher |I+:t| values. This arises as a result of its relatively very large positive sequence impedance(Table N.7) which leads to a considerable level of impact from the neglected term Z++:tI−:t/t onthe negative sequence voltage V t
−:rec. However, these |I+:t| values at which the line exhibits an outof ordinary behaviour are unlikely to arise in practice (e.g. voltage regulation of the line > 20% for|I+:t| > 200A).
15The purpose of this ranking is to give an assessment of the degree of asymmetry associated withthe lines. The overall influence of a line on voltage unbalance levels, which is determined by other
154
-200
-150
-100
-50
0
50
100
150
200
25 75 125 175 225 275|I+:t| (A)
ABCDEFGHIJKLMN
.)
(deg
:t rec
V −θ
Figure 7.5: Variation of θV t−:rec
with |I+:t| for the individual lines
Table 7.1: Ranking of the sub-transmission lines based on the associated degree ofasymmetry
Line/s |Z−+:t| (Ω) Rank
M ∼2.0 Extraordinarily High
F, I ∼1.3 Very high
A, D ∼0.65 High
B, N ∼0.5 Moderate
J, L, E, C ∼0.3 Low
G, K, H ∼0.1 Very low
155
7.2.3 Voltage Unbalance Behaviour of the Individual Lines of the
Study System - as Elements in the Interconnected Network
The voltage unbalance behaviour exhibited by each of the lines, when operating in the
interconnected network, is established employing the concept of voltage unbalance as a
vector16 (referred to as ‘voltage unbalance emission vector’ [1]) using unbalanced load
flow analysis. This is accomplished by synthesising the system operation, however
parameterising such that:
• The line of which the behaviour is to be observed (referred to as ‘line under
observation’, and labelled as t) to represent its actual construction.
• All other lines are ideally transposed.
• All loads are balanced.
This leaves the line under observation as the only source of unbalance that exists in
the system.
A line under observation (t) introduces voltage unbalance at various busbars where:
• The negative sequence voltage V t−:rec−t at its receiving end busbar has to satisfy
the following relationship:
V t−:rec−t = V t
−:send−t − Z−+:tI+:t − Z++:tI−:t/t (7.2)
where, V t−:send−t is the negative sequence voltage introduced by the line under
observation at its sending end busbar.
additional parameters such as the line loading level (|I+:t|), will be discussed in Section 7.2.3 whenthe line operates in the interconnected network environment.
16See Section 2.2.
156
• The negative sequence voltage V t−:rec−tany
at the receiving end busbar of any
other ideally transposed line (tany) has to satisfy the following relationship17:
V t−:rec−tany
= V t−:send−tany
− Z++:tanyI−:t/tany (7.3)
where,
V t−:send−tany
- negative sequence voltage introduced by the line under observation
at the sending end busbar of any other ideally transposed line tany
Z++:tany - positive sequence impedance of any other ideally transposed line tany
I−:t/tany - negative sequence current in any other ideally transposed line tany,
which arises as a result of the line under observation t
As evident from Chapters 3 - 4 (see footnote 10 above), the influence of the terms
Z++:tI−:t/t in (7.2) and Z++:tanyI−:t/tany in (7.3) on the negative sequence voltages
is insignificant compared to that of the term Z−+:tI+:t. That is, as in the case of a
standalone line, the voltage unbalance behaviour exhibited by an asymmetrical line at
various busbars of an interconnected network is also governed by the term −Z−+:tI+:t,
however dependant on the location of the line in the network18.
Fig. 7.6 illustrates the VUF values at the various busbars (S2 - S9 of Fig. 7.1)
corresponding to the selected time stamp obtained by applying each of the lines
(one at a time) as a line under observation. Table 7.2 gives further details of the
lines including commentaries on the values of |Z−+:t| (columns 2), |I+:t| (columns 3)
and |Z−+:tI+:t| (columns 4), and also on the location in the network (column 5) by
assigning into the three sub-parts: US, CP and DS (Fig. 7.1). Commentary on the
location also indicates whether or not a line is in the direct path connecting the BSP
17Note that the negative-positive coupling impedance Z−+:tany of any other line is zero, as it istaken as ideally transposed.
18Further explanation on this influence of the location of a line is given in Appendix N.
157
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
A B C D E F G H I J K L M NLine under observation
VU
F (%
)
S2S3S4S5S6S7S8S9
Figure 7.6: Nodal VUF values arising as a result of the individual lines
and CP and/or DS of the network. Depending on the emission levels which arise as a
result of the individual lines at the various busbars, a rank19 is assigned to the lines as
given in column 6. Referring to the entries in Table 7.2, the following can be derived:
• Line M which carries a relatively low level of positive sequence current is not
seen to be a major source of emission, although it is the most asymmetrical
line of the network. Line J which carries the highest level of positive sequence
current is seen to be a major source of emission, although it has a relatively
low level of asymmetry. This indicates, as implied by (7.2) and (7.3), that the
product |Z−+:tI+:t| is the term of concern when assessing the level of emission
introduced by an asymmetrical line as an element in an interconnected network.
• In contrary to the above, lines B and N introduce only a low level of emission
although their |Z−+:tI+:t| is equal to that of line J, a behaviour which can be
attributed to the location of these lines in the network20. Lines B and N are not
19Taking the highest emission level, i.e. the tallest bar of the group of bars of a line underobservation (Fig. 7.6), as the reference.
20See Appendix N.
158
in the path connecting the BSP, CP and/or DS, and thus their contributions to
voltage unbalance levels at the remaining busbars is reduced compared to that
of line J.
-200
-100
0
100
200
S2 S4 S6 S7 S8 S9
Busbar
Pha
se a
ngle
(deg
.)
ABCDFIJLMN
Figure 7.7: Phase angles of the nodal negative sequence voltages introduced by theindividual lines
Fig. 7.7 illustrates the phase angles of the negative sequence voltages at S2, S4 and
S6 - S921 caused by lines A - D, F, I, J and L - N22 individually. This demonstrates
that each of the above individual lines leads to a unique and nearly constant phase
angle across all busbars. Comparison of Figs. 7.5 (behaviour as standalone lines)
and 7.7 (behaviour as elements in the interconnected network) clearly demonstrates
the similarity of the voltage unbalance behaviour of a line as a standalone line and
as an element in an interconnected network. This further justifies the governance of
the behaviour exhibited by a line as an element in an interconnected network by the
term −Z−+:tI+:t.
21Which are affected significantly by the asymmetrical network as noted in Fig. 7.3.22Which introduce considerable levels of emission as noted in Table 7.2.
159
Table 7.2: Parameters, operating features and emission levels of the individual lines
Line/s |Z−+:t| |I+:t| |Z−+:tI+:t| Location Emission
level
M Extra large Very low Very low DS: an element Low
(∼2Ω) (∼15A) (.30V) in the BSP-CP-DS
connection
F Very large High Very high US: tie line Very high
(∼1.3Ω) (∼150A) (∼230V)
I Very large High Very high US: an element Very high
in the BSP-CP-DS
connection
A, D Large Very High High US: elements in High
(∼0.65Ω) (∼200A) (∼150V) the BSP-CP-DS
connection
B, N Moderate High Moderate US: not elements Low
(∼0.5Ω) (∼65V) in the BSP-CP-DS
connection
J Small Extra High Moderate CP: an element High
(∼0.3Ω) (∼250A) in the BSP-CP-DS
connection
L Small Low Very low CP: an element Low
(∼75A) in the BSP-CP-DS
connection
E Small Low Very low US: tie line Very low
C Small Very high Low US: not an element Low
(∼40V) in the BSP-CP-DS
connection
G, H Very Small High Very low US: elements in Very Low
(∼0.1Ω) the BSP-CP-DS
connection
K Very Small High Very low CP: element Very Low
in the BSP-DS
connection
160
7.2.4 General Outcomes - Representation of the Voltage Unbalance
Behaviour of an Asymmetrical Line as an Element in an Inter-
connected Network
An asymmetrical line of an interconnected network exhibits a voltage unbalance be-
haviour which is vectorial in nature. This behaviour, in a global sense, can be ascer-
tained by a single vector (referred to as ‘global emission vector’) of which:
• The magnitude, as summarised in Table 7.2 - column 6 for the lines of the study
system, can be approximately assessed by referring to the product |Z−+:tI+:t| of
the line and its location23 in the network.
• The phase angle, as illustrated in Fig. 7.7 for the lines of the study system, can
be approximately derived using the term −Z−+:tI+:t24.
The global emission vectors of the individual lines25 A - D, F, I, J and L - N of
the study system are illustrated in Fig. 7.8.
7.2.5 General Outcomes - Representation of the Interaction of All
Asymmetrical Lines
The behaviour of negative sequence variables is known to be linear [1]. That is, a
resultant negative sequence voltage at a busbar, which arises as a result of the inter-
action of various sources of unbalance, is equal to the vector summation of negative
sequence voltage components caused by individual sources at the considered busbar26.
23The work presented in this thesis does not cover an absolute measure of the impact of thelocation of the line on the magnitude of the vector rather than suggesting a relative assessment.This is possibly a subject of interest for further research.
24Further simplified term −Z−+:t can be used under high load power factor conditions, which isgenerally the case in higher voltage systems.
25Which introduce considerable emission levels.26This is further demonstrated in Appendix N employing results obtained using unbalanced load
flow analysis in relation to the study system.
161
J
N C
B
M
A
L
I
D
F
Figure 7.8: Global emission vectors of the individual lines (drawn approximately toa scale)
Thus, the integration of global emission vectors of individual lines, as illustrated in
Fig. 7.8 for the lines of study system, establishes a basis that provides a comprehensive
understanding of the manner in which various asymmetrical lines interact with each
other to form the resultant influence. This basis can be used to derive the following:
• The resultant influence of the interaction of all lines in terms of a single vector
which can be established by the summation of individual global emission vectors.
This, for the study system, is illustrated by the vector Rlines in Fig. 7.9.
• Major contributing lines to the resultant influence.
– Assessments on the study system using the proposed technique:
Referring to Fig. 7.9, lines A and F can be identified as dominant contrib-
utors as the respective global emission vectors lie in close proximity to the
resultant vector. Although the global emission vector of line I is displaced
slightly away from the resultant vector, it, being the line which introduces
the highest level of emission on its own, can also make a significant contri-
162
J
N C
L
A
M
F
B
I
D
Rlines
Figure 7.9: Resultant influence of the interaction of all asymmetrical lines (drawnapproximately to a scale)
163
bution. The phase deviation close to 900 of the vector of line J with respect
to the resultant vector can make it a less of a contributor. The positioning
of the vector of line D with respect to the resultant vector suggests that it
can make a negative contribution, or in other words it assists in counter
balancing some of the emissions caused by the other lines.
– Validation of the above assessments using unbalanced load flow analysis:
Fig. 7.10 illustrates the contributions, quantified using (7.4)27 while em-
ploying the results (presented in Figs. 7.6 and 7.7) obtained using unbal-
anced load flow analysis, made by each of the lines to the resultant voltage
unbalance levels (presented in Fig. 7.3) at the various busbars (S2, S4 and
S6 - S928).
Ct/Si =|V t
−:Si|cos(θV t−:Si
− θV lines−:Si
)
|V lines−:Si |
× 100% (7.4)
where,
Ct/Si - contribution made by any line t to the resultant voltage unbalance
level at any busbar Si
V t−:Si - negative sequence voltage caused by the line t at the busbar Si
V lines−:Si - resultant negative sequence voltage at the busbar Si
θV t−:Si
, θV lines−:Si
- phase angles of the negative sequence voltages V t−:Si and
V lines−:Si respectively
Fig. 7.10 clearly demonstrates the dominant contributions made by lines
A, F and I as identified above using the proposed technique. Further, the
assessments made on the role of lines J and D above as a less of a con-
27This quantifies the fraction of the negative sequence voltage caused by any line t which is in-phase with the resultant negative sequence voltage, as a ratio to the magnitude of the resultantnegative sequence voltage.
28Significantly affected busbars.
164
tributor and as a negative contributor respectively are also seen to be in
agreement with the results presented in Fig. 7.10.
A B C D E F G H I J K L M NS2
S6S8-10
0
10
20
30
40
50
Line
Busbar
S2S4S6S7S8S9
Con
tribu
tion
(%)
Figure 7.10: Nodal contributions made by the individual lines to the resultant voltageunbalance levels
• Transposition options for better managing the emission arising as a result of
line asymmetries.
– Assessments on the study system using the proposed technique:
Referring to Fig. 7.11 (I), which is deduced using Fig. 7.8 by representing
the smaller global emission vectors of lines B, C, L, M and N using a single
vector (labelled ‘B+C+L+M+N’), the most effective option to correct the
network through the transposition of a single line is seen to be the selection
165
DB+C+L+M+N DD
Rlines
B+C+L+M+N B+C+L+M+N
JAF
I
J
I
JA
Resultant vector after transposing line F
I
Resultant vector after transposing lines F and A
(I) (II) (III)
Figure 7.11: (I) Deduced from Fig. 7.8 (II) Effect of the transposition of line F only(III) Effect of the transposition of lines A and F together (drawn approximately to ascale)
of line F as it is being represented by the largest vector among the group
of vectors (i.e. B+C+L+M+N, A and F) clustered together. Further
correction can be introduced to the network effectively by transposing lines
A and F. This results in the vectors B+C+L+M+N, D, I and J to remain,
out of which the vectors B+C+L+M+N and I lie in close proximity. This
is the case with the vectors D and J as well. The phase difference close
to 900 between these two groups (i.e. B+C+L+M+N and I, and D and
J) suggests that the emissions arising as a result of lines B, C, I and L
- N assist in counter balancing some of the emissions of lines D and J.
The effects of the transposition of line F only and lines A and F together
are illustrated in Figs. 7.11 (II) and (III) respectively. These demonstrate
that the transposition of line F can introduce approximately 30% reduction
in the resultant influence, whereas approximately 50% reduction in the
resultant influence can be expected by transposing both lines A and F.
– Validation of the above assessments using unbalanced load flow analysis:
Fig. 7.12 illustrates the effects, in terms of the residual VUF values at the
166
various busbars obtained using unbalanced load flow analysis29, of the two
line transposition options identified above. This also shows the voltage
unbalance levels which arise as a result of the existing line asymmetries
(reproduction of Fig. 7.3). Fig. 7.12 demonstrates, as identified above
using the proposed technique, that the transposition of line F introduces
approximately 30% reduction in the voltage unbalance levels caused by the
existing line asymmetries at S2, S4 and S6 - S9 (busbars which are signifi-
cantly affected by the line asymmetries as noted in Fig. 7.3), whereas this
reduction is approximately 50% when both lines A and F are transposed.
0.0
0.4
0.8
1.2
1.6
S2 S3 S4 S5 S6 S7 S8 S9
Busbar
VU
F (%
)
Exisitng line asymmetries
After transposing line F
After transposing lines A and F
Figure 7.12: Effects, obtained using unbalanced load flow analysis, of the transposi-tion of line F only, and lines A and F together
29Operating scenario corresponds to the selected time stamp.
167
7.3 Voltage Unbalance Behaviour of Load Asymmetries
7.3.1 Impact of the Load Asymmetries of the Study System on the
Voltage Unbalance Problem
It has been noted, based on measurements, that the loads supplied by the study
system exhibit unbalance with regard to their active power components, whereas the
reactive power components have been seen to be reasonably balanced30. Table 7.3
gives the distribution of the active (Pa, Pb and Pc) and reactive (Qa, Qb and Qc)
power across the three phases at each of the load busbars, which corresponds to the
selected time stamp. It also gives the degree of asymmetry associated with the three
active power components, in terms of the standard deviation (µP ), at each of the
busbars.
Table 7.3: Distribution of the active and reactive power across the three phases ateach of the load busbars of the study system
Load busbar S2 S3 S4 S7 S8 S9
Pa (MW) 6.32 6.24 3.87 11.3 2.04 0.57
Pb (MW) 5.87 6.08 3.43 11.3 1.89 0.55
Pc (MW) 5.87 5.96 3.37 11.38 2.04 0.54
Qa ≈ Qb ≈ Qc 0.12 2.37 1.705 4.84 0.63 0.08
(MVAr)
µP 0.26 0.14 0.27 0.05 0.09 0.02
Similar to the case of the line asymmetries presented in Section 7.2.1, the impact
of the interaction of all unbalanced loads of the study system on the problem of volt-
30This can be considered as a general case for higher voltage systems which are usually associatedwith loads with a near unity power factor. That is, in comparison to the level of active power,systems supply only a low level of reactive power at a load point, of which the level of unbalancecan be considered to be insignificant compared to that of the reactive power.
168
age unbalance is established in terms of the VUF values at the various busbars using
unbalanced load flow analysis. This is accomplished by synthesising the system oper-
ation31 while parameterising the loads as given in Table 7.3, and the lines assuming
they are ideally transposed. Fig. 7.13 illustrates these results, in comparison to the
emission levels arising as a result of the line asymmetries (reproduction of Fig. 7.3).
This reveals that the emission which arises due to the loads themselves is also consid-
erable (highest VUF: 1% at S9), although the line asymmetries are seen to dominate
the problem.
0.0
0.3
0.6
0.9
1.2
1.5
S2 S3 S4 S5 S6 S7 S8 S9
Busbar
VU
F (%
)
Load asymmetriesLine asymmetries
Figure 7.13: Nodal VUF values which arise as a result of the load asymmetries, incomparison to that of the line asymmetries
31Which corresponds to the selected time stamp.
169
7.3.2 Voltage Unbalance Behaviour of the Individual Loads of the
Study System - as Elements in the Interconnected Network
Based on a similar approach taken to observe the voltage unbalance behaviour of
the individual lines as elements in the interconnected network32, this for the indi-
vidual loads is obtained using unbalanced load flow analysis by considering a single
load (referred to as ‘load under observation’) at a time, where it is applied as de-
fined in Table 7.3 while parameterising all other loads and all lines to represent a
balanced behaviour.
Fig. 7.14 illustrates the VUF values which arise as a result of each of the loads
as a load under observation at the various busbars. These results are summarised
in Table 7.4 by giving a rank to each of the loads based on its emission level in
a global sense33 (column 5). For brevity, the information given in Table 7.3 are
repeated together with commentaries on the degree of asymmetry associated with
the real power components (column 2), and the three-phase loading level (column 3).
A commentary on the location (i.e. US, CP, DS) of each of the loads in the system
is also given in column 4. Referring to the entries in Table 7.4, the following can
be derived:
• Three-phase loading level is not a factor which governs the level of emission
introduced by an unbalanced load34.
• As it is known, the degree of asymmetry associated with a load is the primary
factor that needs consideration when assessing the level of emission introduced
by the load35.
32See Section 7.2.3.33Taking the highest emission level as the reference, as in the case of the line asymmetries.34e.g. compare the emission levels introduced by the loads at S3 and S8 referring to their three-
phase loading levels (column 3).35e.g. compare the emission levels introduced by the loads at S9 and S8 referring to their µP .
170
• In contrary to the above, the load supplied by S8 (µP = 0.09) located in the
downstream part of the system is seen to introduce the highest level of emission,
although the associated degree of asymmetry is considerably lower than that of
the loads supplied by S2 and S4 (µP ≈ 0.3) located in the upstream part. It
is therefore evident that, for a similar degree of asymmetry, the loads supplied
by the busbars located in the downstream part of the system tend to introduce
increased emission levels compared to those located in the upstream.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
S2 S3 S4 S7 S8 S9Load under observation
VU
F (%
)
S2S3S4S5S6S7S8S9
Figure 7.14: Nodal VUF values which arise as a result of the individual loads
Fig. 7.15 illustrates the phase angles of the negative sequence voltages which arise
as a result of the individual loads supplied by S2, S4, S7 - S936 at the various busbars
(S2 - S9). This demonstrates that, as in the case of the line asymmetries (Fig. 7.7),
the above individual loads yield a unique and nearly constant phase angle across all
the busbars.
36Which introduce considerable emission levels as noted in Table 7.4.
171
Table 7.4: Operating features and emission levels of the individual loads of the studysystem
Load µP Three-phase loading Location in Emission
busbar level (MW/MVAr) the network level
S4 High High US High
(0.27) (11/5.115)
S2 High Very high US High
(0.26) (18/0.36)
S3 Low Very high US: a remote Negligible
(0.14) (18/7.11) busbar
S8 Low Low DS Very high
(0.09) (6/1.89)
S7 Very low Extra high CP Low
(0.05) (34/14.52)
S9 Very low Very low DS Low
(0.02) (2/0.08)
0
50
100
150
200
250
Busbar
Pha
se a
ngle
(deg
.)
Load at S2Load at S4Load at S7Load at S8Load at S9
S3 S4 S5 S6 S7 S8 S9S2
Figure 7.15: Phase angles of the nodal negative sequence voltages introduced by theindividual loads
172
Arising as a result of a load under observation (LSi), the negative sequence volt-
age V LSi−:rec−tany
at the receiving end busbar of any line tany37 (in other words, at any
busbar) has to satisfy:
V LSi−:rec−tany
= V LSi−:send−tany
− Z++:tanyI−:LSi/tany (7.5)
where,
V LSi−:send−tany
- negative sequence voltage introduced by the load under observation at
the sending end busbar of any line tany
I−:LSi/tany - negative sequence current in the line tany, which arises as a result of the
load under observation
Equation (7.5) provides an explanation to the results presented in Figs. 7.15 and 7.16:
• The negative sequence currents I−:LSi/tany flow through the lines (i.e. through
the impedances Z++:tany) are drawn by the load under observation. The level
of the negative sequence current drawn by this load is governed by the degree
of its asymmetry regardless of the three-phase loading level. That is, as seen
in Fig. 7.15, the principal intrinsic parameter that determines the negative se-
quence voltage V LSi−:rec−tany
is the degree of asymmetry associated with the load
under observation. The location of this load in the system determines the part
of the network in which the negative sequence currents I−:LSi/tany flow38, making
the negative sequence voltage V LSi−:rec−tany
dependant secondarily on the location
of the load under observation as indicated by the results presented in Fig. 7.15.
37Note that lines are taken as transposed for this case of observing the voltage unbalance behaviourof individual loads. Thus, the impedance Z−+:tany
= 0.38e.g. when the load is supplied by a upstream busbar, the negative sequence currents I−:LSi/tany
mainly flow through the lines located in the upstream part of the network.
173
• The phase angles of the negative sequence voltages which arise as a result of the
load under observation at the various busbars should be associated with the term
−Z++:tanyI−:LSi/tany . The phase angle of the impedance Z++:tany depends only
on the X/R ratio of the line which has been noted to be nearly identical39 for
all lines of the study network. The phase angle of the negative sequence current
I−:LSi/tany is governed by the order of the distribution of the three active power
components of the load under observation across the three phases. Thus, the
phase angles of the negative sequence voltages at the various busbars remain
similar and unique for a particular load under observation as seen in Fig. 7.15.
Load at S8
Load at S9
Load at S7
Load at S2
Load at S4
Figure 7.16: Global emission vectors of the individual loads (drawn approximately toa scale)
39This can be considered as a general case.
174
7.3.3 General Outcomes
As for an asymmetrical line40, the voltage unbalance behaviour exhibited by an unbal-
anced load of an interconnected network can be represented using a global emission
vector of which:
• The magnitude, as summarised in Table 7.4 - column 5 for the loads of the study
system, can be approximately assessed by referring to the degree of asymmetry
associated with the load, and the location of the load in the network.
• The phase angle, as illustrated in Fig. 7.15 for the loads of the study system,
can be approximately derived by referring to the order of the distribution of
power of the load under observation across the three phases, and the X/R ratio
associated with lines of the network.
The global emission vectors of the individual loads41 S2, S4 and S7 - S9 of the
study system are illustrated in Fig. 7.16.
Employing the linearity of negative sequence variables, as for line asymmetries42,
the global emission vectors of individual loads can be integrated to establish a basis
that provides a comprehensive understanding of the manner in which various un-
balanced loads interact with each other to form the resultant influence. Fig. 7.17
illustrates this resultant influence for the study system using a single vector (Rloads)
which is obtained by the summation of the individual global emission vectors shown
in Fig. 7.16.
40See Section 7.2.4.41Which introduce considerable emission levels.42See Section 7.2.5.
175
Rloads Load at S8
Load at S2
Load at S4
Load at S9
Load at S7
Figure 7.17: Resultant influence of the interaction of all unbalanced loads (drawnapproximately to a scale)
176
7.4 Combined Voltage Unbalance Behaviour of Line and Load
Asymmetries
7.4.1 Combined Impact of the Line and Load Asymmetries of the
Study System on the Voltage Unbalance Problem
When both the line and load asymmetries exist concurrently in the study system43,
their impact on the problem of voltage unbalance is established in terms of the VUF
values at the various busbars using unbalanced load flow analysis. This is accom-
plished by synthesising the system operation44 while parameterising the loads as given
in Table 7.3, and the lines based on their actual construction. Fig. 7.18 illustrates
these results, in comparison to the emission levels which arise as a result of the line
asymmetries alone (reproduction of Fig. 7.3) and the load asymmetries alone (re-
production of Fig. 7.13). The measured voltage unbalance levels (reproduction of
Fig. 7.2) are also shown in Fig. 7.18. This reveals that the interaction of the various
line and load asymmetries causes voltage unbalance levels up to 1.8% at S8 and S9
(downstream), 1.2% at S6 and S7 (central part) and 0.6% at S2 and S4 (upstream),
which are seen to be in close agreement with the measured values.
7.4.2 Representation of the Voltage Unbalance Behaviour of the Entire
System
Employing the linearity of negative sequence variables, the global emission vectors
of individual sources of unbalance (i.e. individual lines and loads) can be integrated
to establish a basis that provides a comprehensive understanding of the manner in
43Which represents the real operating scenario, where voltage unbalance levels arise as a result ofthe interaction of both the line and load asymmetries.
44Which corresponds to the selected time stamp.
177
0.0
0.4
0.8
1.2
1.6
2.0
S2 S3 S4 S5 S6 S7 S8 S9
Busbar
VU
F (%
)
Load asymmetries Line asymmetriesBoth line and load asymmetriesMeasured values
Figure 7.18: Nodal VUF values which arise as a result of both the line and load asym-metries, in comparison to that of the line asymmetries alone, and the load asymmetriesalone, and also to the measured values
which various untransposed lines and unbalanced loads interact with each other to
form the overall influence. Fig. 7.19 illustrates this overall influence for the study
system using a single vector (Rsystem) which is obtained by the summation of the
individual global emission vectors shown in Figs. 7.9 (for lines) and 7.16 (for loads).
Fig. 7.19 also shows the vectors Rlines (Fig. 7.10) and Rloads (Fig. 7.17) where
Rsystem ≈ Rlines + Rloads.
This can be used to make the following assessments on the problem of voltage
unbalance, which are also confirmed using unbalanced load flow analysis:
• Contributions made by the line and load asymmetries to the overall voltage
unbalance levels.
– Using the proposed technique:
Referring to the vectors Rsystem, Rlines and Rloads, the component of Rlines
which is in-phase with Rsystem accounts approximately for 60% of the mag-
178
Rload
Rlines
Rsystem
Figure 7.19: Resultant influence of the interaction of all lines and loads (drawn ap-proximately to a scale)
179
nitude of Rsystem, whereas that of Rloads is approximately 30%. That is,
the asymmetry associated with the lines is the dominant contributor to
the problem, whereas the load asymmetries play only a secondary role.
– Using unbalanced load flow analysis:
Fig. 7.20 illustrates the contributions, quantified45 employing the results
obtained using unbalanced load flow analysis, made by the line and load
asymmetries to the overall voltage unbalance levels (presented in Fig. 7.18)
at the various busbars (S2, S4 and S6 - S946). Fig. 7.20 clearly demon-
strates that, as identified above using the proposed technique, the line
asymmetries contribute approximately by 60% - 70% to the overall volt-
age unbalance levels at S2, S4 and S6 - S9, whereas it is approximately
25% - 30% by the load asymmetries.
0
20
40
60
80
S2 S4 S6 S7 S8 S9
Busbar
Con
tribu
tion
(%)
Line asymmetriesLoad asymmetries
Figure 7.20: Nodal contributions made by the line and load asymmetries to the overallvoltage unbalance levels
45Based on the approach given by (7.4).46Significantly affected busbars.
180
• Contributions made by the individual sources of unbalance to the overall voltage
unbalance levels.
– Using the proposed technique:
Observation of the global emission vectors of the individual lines and loads
illustrated in Figs. 7.9 and 7.16 respectively together with the vector
Rsystem (Fig. 7.19) suggests that, among all sources of unbalance, lines
F and I, these being represented by the largest and the closest vectors to
Rsystem, can be identified as the major contributors to the overall voltage
unbalance levels. In addition, the vectors of line A and the loads sup-
plied by S2 and S4, having relatively large magnitudes and being closer
to Rsystem, can contribute significantly to the problem supporting the two
major contributors (i.e. lines F and I). The phase deviation close to 900
of the vector of the load supplied by S8 with respect to Rsystem can make
it a less of a contributor. The positioning of the vectors of lines D and J
and the load supplied by S7 with respect to Rsystem suggests that they can
make negative contributions.
– Using unbalanced load flow analysis:
Fig. 7.21 illustrates the contributions, quantified employing the results
(presented in Figs. 7.6, 7.7, 7.14 and 7.15) obtained using unbalanced load
flow analysis, made by each of the sources of unbalance to the overall
voltage unbalance levels (presented in Fig. 7.18) at the various busbars
(S2, S4 and S6 - S9). This clearly demonstrates the dominant contributions
made by lines F and I as identified above using the proposed technique.
Further, the assessments made on the role of line A and the loads supplied
by S2 and S4 as significant contributors, the load supplied by S8 as a less
of a contributor, and lines D and J and the load supplied by S7 as negative
181
contributors are also seen to be in agreement with the results presented in
Fig. 7.21.
Load at S2
Load at S3
Load at S4
Load at S7
Load at S8
Load at S9
A B C D E F G H I J K L M N S2
S7
-20
-10
0
10
20
30
40
50
Con
tribu
tion
(%)
Source of unbalance
Busbar
S2S4S6S7S8S9
Figure 7.21: Nodal contributions made by the individual sources of unbalance to theoverall voltage unbalance levels
7.5 Chapter Summary
This chapter has established theoretical bases to broaden the understanding of the
voltage unbalance behaviour exhibited by various sources that exist in interconnected
network environments. These would provide additional assistance in the application
of the IEC/TR 61000-3-13 voltage unbalance allocation guidelines.
182
Employing a 66kV interconnected sub-transmission system as the study case, de-
terministic studies were carried out with the view to develop insights into the in-
fluences made by line and load asymmetries on the problem in a systematic man-
ner considering each of the asymmetrical elements. The following can be drawn
from the study:
• The voltage unbalance behaviour (in terms of the level of voltage unbalance
and its phase angle) exhibited by an asymmetrical line at various busbars of
an interconnected network is primarily governed by the term −Z−+:tI+:t, and
secondarily dependant on the location of the line in the network. This be-
haviour, in a global sense, can be ascertained by a single vector referred to as
‘global emission vector’ of which the magnitude can be approximately assessed
by referring to the product |Z−+:tI+:t| of the line and the location of the line in
the network, and the phase angle can be approximately derived using the term
−Z−+:tI+:t.
• As for an asymmetrical line, the voltage unbalance behaviour exhibited by an
unbalanced load of an interconnected network can also be represented using
a global emission vector. Of this global emission vector, the magnitude can
be approximately assessed by referring to the degree of asymmetry associated
with the load and the location of the load in the network. The phase angle
can be approximately derived by referring to the order of the distribution of
power of the load under observation across the three phases, and the X/R ratio
associated with lines of the network.
• The global emission vectors of individual sources of unbalance (i.e. of individual
lines and loads) can be integrated to establish a basis that provides a compre-
hensive understanding of the manner in which various untransposed lines and
183
unbalanced loads interact with each other to form the overall system behaviour.
That is, the overall influence of various sources of unbalance that exist in an in-
terconnected network can be represented, in a global sense, using a single vector.
This basis can be used to assess the interconnected networks on the problem of
voltage unbalance in different ways, e.g. principal contributors, most favorable
corrective options. This proposed technique was applied to the study system
to identify the major contributors to voltage unbalance levels and the effective
line transposition options, which were also confirmed using unbalanced load
flow analysis.
Chapter 8
Conclusions and Recommendations
for Future Work
8.1 Conclusions
This thesis has focused on making contributions for further development of the re-
cently released Technical Report IEC/TR 61000-3-13 which provides guiding prin-
ciples for coordinating voltage unbalance between various voltage levels of a power
system through the allocation of emission limits to installations. A number of related
aspects of this voltage unbalance allocation procedure such as the global emission
arising as a result of system inherent asymmetries, the propagation of voltage unbal-
ance and associated deficiencies have been addressed. Furthermore, theoretical bases
to broaden the understanding of the influence made by various sources of unbalance
that exist in interconnected networks on voltage unbalance, which would provide ad-
ditional assistance in the application of these voltage unbalance allocation guidelines
have been established.
184
185
As an essential tool required for specific applications of the work presented in this
thesis, an unbalanced load flow algorithm based on the phase coordinate reference
frame was developed. The component level load flow constraints and the three-phase
modelling of system components were incorporated. The major task of this work
was to develop models for the representation of three-phase induction motors, which
overcome the limitations associated with existing models.
Preliminary studies carried out on the global emission which arises as a result
of system inherent asymmetries revealed that this emission has some degree of load
dependency. The approach given in IEC/TR 61000-3-13 to evaluate the influence
of an asymmetrical radial line on the global emission does not consider this load
dependency, and thus it can be applied only when the line supplies passive loads. This
IEC approach was noted to be conservative when the line supplies large proportions
of three-phase induction motors. In this case of motor loads, the negative sequence
currents arising as a result of line asymmetries made a significant influence on the
global emission, whereas it was seen to be insignificant for passive loads. The degree
of this influence was seen to be dependant primarily on the motor proportion, and
secondarily on the system and motor characteristics. In essence, the global emission
levels in HV power systems, in the presence of considerable proportions of induction
motors, were seen to arise not only as a result of the local HV lines but also as a
result of the downstream MV line asymmetries.
Systematic approaches, covering radial and interconnected networks, for evaluat-
ing the global emission which arises due to line asymmetries at nodal level, taking
the load dependency of this emission into account were proposed. These were verified
using unbalanced load flow analysis in relation to simple test systems. These test
results revealed that the presence of considerable proportions of induction motors
increases the emission, compared to emission levels when only passive loads exist, at
186
the busbar which is directly connected to the upstream system. At all other network
busbars, induction motor loads were seen to reduce the emission levels which arise due
to the local line asymmetries, compared to that when only passive loads exist. In HV
power system, in the presence of motor loads, the influence of the downstream MV
line asymmetries was noted to either decrease or increase the resultant emission lev-
els with respect to the local emission levels depending on the impedance/admittance
characteristics of the downstream lines relative to the local HV lines. The proposed
approach was further verified using the IEEE 14-bus test system taking it as an HV
system supplying passive loads at the HV level itself.
The propagation of voltage unbalance from higher voltage to lower voltage systems
in terms of transfer coefficients was initially examined in the presence of four basic
load types (i.e. constant impedance, constant current, constant power and three-
phase induction motor loads).
• It was seen that the value of unity which has been assumed in IEC/TR 61000-
3-13 for the MV to LV transfer coefficient in the presence of passive loads in
general is conservative in relation to some passive load types such as commonly
prevailing constant power loads. Transfer coefficients for passive loads were
seen to be scaled up by the factor∣∣∣1 + j 1
ksc−s∠θpf :s
∣∣∣τ relative to unity, where
τ = 0, 1, 2 ∼ 3 for constant impedance, constant current and constant power
loads respectively1. Noting that the factor ksc−s can be a value in the range
of 5 to 25 for various systems and the exponent τ varies in the range of 0 to
3 for different load types, a uniform behaviour or the behaviour of constant
impedance loads as assumed in the IEC approach cannot be used to represent
all load types with a high degree of accuracy. That is, voltage unbalance can get
1ksc−s - ratio between the short-circuit capacity (in MVA) at the busbar S under evaluation andthe total load (in MVA) supplied by the busbar S, and θpf :s - pf angle (− and + for lagging andleading conditions respectively) at the busbar S.
187
significantly amplified, especially for constant power loads, as it propagates from
higher voltage to lower voltage levels. As the values of ksc−s for MV systems
are usually smaller than those for LV systems, this amplification in the case of
the HV to MV propagation is greater than that in the MV to LV propagation.
• As suggested in IEC/TR 61000-3-13 in relation to the MV to LV transfer coeffi-
cient, voltage unbalance was seen to get significantly attenuated as it propagates
from higher voltage to lower voltage levels in the presence of induction motor
loads. This attenuation, relative to unity, can be given by the scaling factors
1„1+ ks
ksc−lv
« and 1241+“
1ksc−mv
”0@ ks
1+ ksksc−lvagg
1A35 in the cases of the MV to LV and HV
to MV propagation respectively2. As ksc−mv for MV systems < ksc−lv for LV
systems in practice, a higher degree of this attenuation can be expected in the
HV to MV propagation than that in the MV to LV propagation. Noting that
5 < ks < 7, the degree of this reduction of the transfer coefficients relative to
unity is significant compared to the increment introduced by passive loads.
Incorporating the above, systematic methods for evaluating the MV to LV and HV to
MV transfer coefficients were developed. These were verified using unbalanced load
flow analysis for various scenarios. It was demonstrated that, compared to the IEC
method, the proposed new formulation for assessing the MV to LV transfer coefficient
gives a more accurate estimation, particularly for load bases which are dominated by
passive elements. The MV to LV and HV to MV transfer coefficients were noted to
vary in the ranges of 0.6 to 1.1 and 0.5 to 1.4 respectively depending on the system
and load characteristics and the downstream load composition.
2Motor loads are considered to be supplied at the LV level. ks - ratio between the positive andnegative sequence impedances of the aggregated motor load supplied by the LV system, ksc−lvagg -ratio between the short-circuit capacity (in MVA) at, and the total load (in MVA) supplied by, theaggregated LV busbar.
188
Preliminary studies carried out on the propagation of voltage unbalance from one
busbar to other neighbouring busbars of a sub-system in terms of influence coefficients
revealed that, for a simple two-bus radial sub-system, the influence coefficients can
be approximated to unity in the presence of passive loads in general. However, these
influence coefficients were seen to be considerably smaller than unity (in other words,
than that in the case of passive loads) for induction motor loads. The conclusion of
this observation is that the negative sequence currents which arise as a result of the
voltage unbalance that exists at a particular busbar introduce a considerable impact
on influence coefficients for motor loads, whereas it is insignificant for passive loads.
Employing this, a systematic method for evaluating influence coefficients for inter-
connected network environments was developed. This was verified using unbalanced
load flow analysis in relation to a simple test system, and also to the IEEE 14-bus
test system supplying local passive loads.
The IEC/TR 61000-3-13 voltage unbalance allocation approach was examined
employing a simple test system with and without the inclusion of the influence of
system inherent asymmetries. For both of the cases stated above, the IEC/TR 61000-
3-13 procedure was noted to lead to situations where the set planning level are being
exceeded even when no customer exceeds the allocated limit. Closely aligning with the
IEC/TR 61000-3-13 guidelines, a revised voltage unbalance allocation policy based
on the constrained bus voltage (CBV) harmonics/flicker allocation principles was
proposed. In this CBV allocation technique, emission levels at network busbars are
explicitly forced, introducing a new factor referred to as ‘allocation constant’, to be at
or below the set planning levels when all loads inject their limits derived under the new
approach. The issue of the emission arising as a result of system inherent asymmetries
involved with the case of voltage unbalance was taken into account according to
IEC/TR 61000-3-13 using the factor Kue. This revised allocation technique was
189
examined employing the above mentioned test system, and it was seen to restrict the
resulting emission level at all network busbars at or below the set limit. Further, this
proposed alternative technique was noted to be able to satisfy all four key allocation
objectives.
With the view of developing theoretical bases which describe the voltage unbal-
ance behaviour exhibited by various sources that exist in interconnected network
environments, deterministic studies supported by unbalanced load flow analysis were
carried employing a 66kV interconnected sub-transmission system. This study net-
work has been monitored to experience voltage unbalance levels above the applicable
regulatory limit of 1%.
• The influence made by each of the untransposed lines, as a standalone line
and also as an element in the interconnected network, on voltage unbalance
was established employing a new concept termed ‘voltage unbalance emission
vector’, and extensively analysed. The voltage unbalance behaviour (in terms of
the magnitude and the phase angle) exhibited by an asymmetrical line at various
busbars of an interconnected network was seen to be primarily governed by the
term −Z−+:tI+:t as in the case of a standalone line, and secondarily dependant
on the location of this line in the network. This behaviour, in a global sense, was
ascertained by a single vector referred to as ‘global emission vector’ of which the
magnitude can be approximately assessed by referring to the product |Z−+:tI+:t|
of the line and the location of this line in the network. The phase angle of this
vector can be approximately derived using the term −Z−+:tI+:t.
• Following an analysis of the voltage unbalance emission vectors introduced by
each of the unbalanced loads as an element in the interconnected system, it was
seen, as for an asymmetrical line, that the voltage unbalance behaviour exhib-
190
ited by an unbalanced load of an interconnected network can also be represented
using a global emission vector. Of this global emission vector, the magnitude
can be approximately assessed by referring to the degree of asymmetry associ-
ated with the load and the location of this load in the network, and the phase
angle can be approximately derived by referring to the order of the distribution
of power of the load under observation across the three phases and the X/R
ratio associated with lines of the network.
• It was seen that the voltage unbalance behaviour exhibited by individual lines
and loads can be ascertained, in a global sense, in terms of the global emission
vectors. Based on the linearity of negative sequence variables, these individual
global emission vectors were integrated to establish a basis that provides a com-
prehensive understanding of the manner in which various untransposed lines and
unbalanced loads interact with each other to establish the overall system be-
haviour. That is, the overall influence of various sources of unbalance that exist
in an interconnected network can be represented, in a global sense, using a single
vector. This basis can be used to assess interconnected networks in relation to
the problem of voltage unbalance in different ways, e.g. principal contributors,
most favourable corrective options. Further, this knowledge which facilitates
the identification of contributions made by individual unbalanced sources forms
a platform for developing techniques to assess the compliance against emission
limits, which is another subject of relevance to future editions of IEC/TR 61000-
3-13. This proposed technique was applied to the study system to identify the
major contributors to voltage unbalance levels and the effective line transposi-
tion options, which were also confirmed using unbalanced load flow analysis.
191
• In the study system, the line asymmetries were seen to contribute approxi-
mately by 60% - 70% to the overall voltage unbalance levels, whereas the con-
tribution made by the load asymmetries was only 25% - 30%. This indicates
the requirement of proper line transposition practices even at relatively low
sub-transmission voltage levels, which is currently not seen to be a common
practice.
8.2 Recommendations for Future Work
The Technical Report IEC/TR 61000-3-13, which will possibly be adopted as an
Australian standard in the future, provides valuable techniques for managing voltage
unbalance in public power systems. In the cases of the counterpart Australian/New
Zealand harmonics (AS/NZS 61000-3-6 [75]) and flicker (AS/NZS 61000-3-6 [76])
standards, which are essentially based on the respective IEC technical reports, amend-
ments has been shown to require in order to allow accurate and straightforward
implementation by utilities and customers. Consequently, Standards Australia com-
missioned the writing of the handbook HB-264-2003 [4] which gives more prescriptive
procedures for the use of these harmonics and flicker standards. Similarly, in adopting
IEC/TR 61000-3-13 as an Australian/New Zealand standard, further work is seen to
require in order to assist the application of such a standard to complex systems such
as MV distribution systems with relatively long feeders.
From a utility point of view, a significant improvement to the voltage allocation
process would be the development of software modules which are able to produce
various factors (e.g. Kue factor) and coefficients (e.g. transfer and influence coeffi-
cients) directly from network databases. Such software modules would also benefit
customers, as the emission limits can be evaluated without substantial involvement
or effort on the part of the responsible utility.
192
Although the Technical Report IEC/TR 61000-3-13 provides the guidance for eval-
uating customer emission limits, it does not discuss the assessment of the compliance
against these limits. Development of systematic techniques to carry out this would sig-
nificantly assist in further developing IEC/TR 61000313 such that a complete voltage
unbalance management procedure is prescribed. In fact, the compliance assessment
has been noted to be a subject of interest to a currently active CIGRE/CIRED work-
ing group which is responsible for updating IEC/TR 61000-3-13. The work presented
in Chapter 7 of this thesis provides a fundamental basis towards the development of
such techniques, yet a substantial amount of further work is required for completing
the task.
Variation in network topologies and loads makes the setting of prescriptive generic
planning levels and voltage unbalance allocation procedures impractical. At present,
the planning levels proposed in IEC/TR 61000-3-13 are given as indicative values. The
significant differences between networks mean that there are likely to be situations
in which it may suit utilities to choose alternative planning levels. Guidance on how
this might best be accomplished would be a useful advancement.
The Technical Report IEC/TR 61000-3-13, as well as this thesis, has focused
mainly on Stage 2 (i.e. allocation of emission limits to individual customers) of the
voltage unbalance management process. Prescriptive Methodologies governing the
third and final stage, or as to how a customer who would fail to comply with the
Stage 2 emission limit to be treated are yet to be covered.
Verification of the methodologies proposed in Chapters 3 - 5 through laboratory
level experiments or field measurements is encouraged. To confirm the outcomes pre-
sented in Chapter 7, extended work such as the carrying out suitable field measure-
ments and the examination of the proposed techniques in relation to other networks
193
with different topologies are recommended. Chapter 7, which can be considered as an
initial step towards a broad research area, requires more deterministic, analytical and
theoretical work such that the techniques are further developed to be more definitive.
As an example, Chapter 7 only covers a relative assessment of the influence and the
location of a source of unbalance that exists in an interconnected network, for which
absolute assessment techniques can be developed.
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public low-voltage power supply systems, Ed. 2. Technical report, International
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age quality with relation to harmonics, flicker and unbalance. In Proc. CIGRE
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CIGRE/CIRED JWG C4.103 results. In Proc. 19th International Conference
on Electricity Distribution (CIRED 2007), paper 0892, Vienna, May 2007.
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of emission limits for distorting loads in MV and HV power systems. Technical
report, Standards Australia/New Zealand, 2001.
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of emission limits for fluctuating loads in MV and HV power systems. Technical
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Appendix A
Derivation of (3.5)
Equation (3.4) in relation to the considered radial MV-LV network shown in Fig. 3.2
is replicated here by (A.1):
|V t−:g/mv:rec| = |Z−+:tI+:t + Z++:tI−:t/t| (A.1)
When the loads Smv:rec−local and Smv:rec−ds represent constant impedance loads, the
currents I+:t and I−:t/t can be written1 respectively as:
I+:t =V+:send
Z++:send
(A.2)
I−:t/t = Y−+:send V+:send (A.3)
where,
V+:send - positive sequence voltage at the sending end busbar (labelled ‘send’)
Z++:send, Y−+:send - as defined below when xr = send
1Note the voltage at the sending end busbar is balanced.
204
205
The downstream negative-positive sequence coupling admittance Y−+:xr seen at a
busbar xr of a radial network can be generally expressed2 using various impedance
elements seen at the busbar as:
Y−+:xr ≈−Z−+:xr
Z++:xr Z−−:xr
(A.4)
Alternatively, the downstream negative-positive sequence coupling impedance Z−+:xr
seen at the busbar xr can be expressed using various admittance elements seen at the
busbar as:
Z−+:xr ≈−Y−+:xr
Y++:xr Y−−:xr
(A.5)
where,
Z++:xr , Z−−:xr - downstream positive and negative sequence impedances respectively
seen at the busbar xr
Y++:xr , Y−−:xr - downstream positive and negative sequence admittances respectively
seen at the busbar xr
For the considered system where the loads are balanced (i.e. decoupled sequence
impedances) and the MV line is untransposed: Z−+:send = Z−+:t. Thus, the admit-
tance Y−+:send can be given by:
Y−+:send ≈−Z−+:t
Z++:send Z−−:send
(A.6)
Using (A.2), (A.3) and (A.6), (A.1) can be developed as:
|V t−:g/mv:rec| ≈
∣∣∣∣Z−+:tI+:t
(1− Z++:t
Z−−:send
)∣∣∣∣ (A.7)
2Through the inversion of the impedance matrix and the simplification.
206
Noting that Z−−:send = Z−−:rec + Z++:t, (A.8) can be rewritten as:
|V t−:g/mv:rec| ≈
∣∣∣∣Z−+:tI+:t
(Z−−:rec
Z−−:send
)∣∣∣∣ (A.8)
Appendix B
Radial MV-LV Test System
(Fig. 3.2)
This test system is designed mainly based on the data given in [78].
• System details - 60Hz, three-wire, MV - 12.47kV , LV - 460V
• MV line -
– Line details: 3.2187km, untransposed
– Tower construction details: 1.143m flat and horizontal,
phase positioning1: ©a © b © c
– Conductor data:
geometric mean radius = 7.7724mm
resistance = 0.19014Ω/km
earth resistivity = 100Ωm
1This shows the considered arrangement of the three phase conductors (a, b and c) of the hori-zontal tower.
207
208
– Relevant impedance/admittance data:
positive sequence impedance (Ω/km) = 0.1901 + j0.3937 (0.4372∠640)
negative-positive sequence coupling impedance (Ω/km) = 0.0302+j0.0174
(0.0349∠300)
positive sequence admittance (Skm) = 1.0098− j2.0630 (2.2969∠−640)
negative-positive sequence coupling admittance (Skm) = 0.0258+ j0.1821
(0.1839∠820)
• MV-LV coupling transformer - aggregated representation of fully loaded 1MVA,
12.47kV/460V transformers with winding resistance = 1%, leakage reactance
= 5%, and secondary tap setting = 1.03pu. This gives a value of 19 for the
factor ksc−lvagg .
• Induction motor load - aggregated representation of 50hp, 460V, 1705rpm mo-
tors [15] supplying the rated mechanical load.
– Parameters per motor:
stator resistance = 0.087Ω
rotor resistance (referred to stator side) = 0.228Ω
stator, rotor (referred to stator side) leakage reactance = 0.302Ω
magnetising reactance = 13.08Ω
– Other relevant details: input power = 50kVA at 0.9 lagging pf and ks = 6.7
under rated operating conditions
• Passive loads - aggregated representation of passive components with a rated
power of 50kVA at 0.9 lagging pf which consist of equal shares constant power
and constant impedance elements.
• Sending end positive sequence voltage = 7.78∠00kV (1.08pu)
Appendix C
Derivation of (3.14)
Equation (3.13) in relation to the considered radial MV-LV network shown in Fig. 3.2
is replicated here by (C.1):
|V t−:g/mv:rec| ≈
∣∣∣∣Z−+:tI+:t
(Z−−:rec−im
Z−−:send−im
)∣∣∣∣ (C.1)
Noting Z−−:send−im = Z−−:rec−im + Z++:t and the inductive nature of the associated
impedance elements1, (C.1) can be rewritten as:
|V t−:g/mv:rec| ≈ |Z−+:tI+:t|
1
1 +∣∣∣ Z++:t
Z++:rec
∣∣∣ ∣∣∣ Z++:rec
Z−−:rec−im
∣∣∣ (C.2)
where, Z++:rec - downstream equivalent positive sequence impedance seen at the re-
ceiving end busbar of the MV line. The ratio∣∣∣ Z++:t
Z++:rec
∣∣∣ can be expressed as:
∣∣∣∣ Z++:t
Z++:rec
∣∣∣∣ ≈ V Rt
1− V Rt
(C.3)
1e.g. the impedance Z−−:rec−im is made up of the negative sequence impedances of inductionmotors and the MV-LV transformer.
209
210
Substituting Z−−:rec−im = (nml)2(Z−−:im + Z++:ml−lv) and rearranging, the ratio∣∣∣ Z++:rec
Z−−:rec−im
∣∣∣ can be written as:
∣∣∣∣ Z++:rec
Z−−:rec−im
∣∣∣∣ =
(1
(nml)2|Z−−:im+Z++:ml−lv ||V+:rec|2
)(1
|V+:rec|2|Z++:rec|
)(C.4)
where,
nml - operating turns ratio of the MV-LV coupling transformer
Z−−:im - negative sequence impedance of the aggregated motor load
The term |V+:rec|2|Z++:rec| represents the total MVA load (i.e. Smv:rec−local + Smv:rec−ds) sup-
plied by the MV busbar. Using the relationships |V+:rec| = nml|V+:lv| and |Z−−:im +
Z++:ml−lv| ≈ |Z−−:im|+ |Z++:ml−lv|, (C.4) can be further rearranged as:
∣∣∣∣ Z++:rec
Z−−:rec−im
∣∣∣∣ ≈ 1
(Smv:rec−local + Smv:rec−ds)
1∣∣∣Z−−:im
Z++:im
∣∣∣ ∣∣∣Z++:im
Z++:lv
∣∣∣ |Z++:lv ||V+:lv |2
+ |Z++:ml−lv ||V+:lv |2
(C.5)
where,
V+:lv - positive sequence voltage at the LV busbar
Z++:im - positive sequence impedance of the aggregated motor load
Z++:lv - equivalent downstream positive sequence impedance seen at the LV busbar
Noting that:∣∣∣Z−−:im
Z++:im
∣∣∣ = 1ks
,∣∣∣Z++:im
Z++:lv
∣∣∣ = 1km
, |Z++:lv ||V+:lv |2
= 1Smv:rec−ds
|Z++:ml−lv ||V+:lv |2
= 1Ssc−lvagg
, Smv:rec−ds
Smv:rec−local+Smv:rec−ds= klv,
Ssc−lvagg
Smv:rec−ds= ksc−lvagg
211
(C.5) can be written as:
∣∣∣∣ Z++:rec
Z−−:rec−im
∣∣∣∣ ≈ klv
1kskm
+ 1ksc−lvagg
(C.6)
Substitution of (C.3) and (C.6) in (C.2) gives:
|V t−:g/mv:rec| ≈ |Z−+:tI+:t|
1
1 +(
V Rt
1−V Rt
)(klv
1kskm
+ 1ksc−lvagg
) (C.7)
Appendix D
Y−−:x−im for an MV Network
The admittance Y−−:x−im can be generally written noting the inductive nature of the
respective impedance Z−−:x−im as:
Y−−:x−im ≈ −j |Y++:x|∣∣∣∣ Z++:x
Z−−:x−im
∣∣∣∣ (D.1)
where, Y++:x, Z++:x - downstream equivalent positive sequence admittance and imped-
ance respectively seen at the busbar x. Noting that the system representation of the
MV busbar x shown in Fig. 3.6 and that of the receiving end busbar of the MV line
considered in Fig. 3.2 are similar, the ratio∣∣∣Z++:x
Z−−:x
∣∣∣ can be given according to (C.6) as:
∣∣∣∣Z++:x
Z−−:x
∣∣∣∣ ≈ klv:x
1ks:xkm:x
+ 1ksc−lvagg :x
(D.2)
The absolute value of the admittance Y++:x can be expressed in terms of the system
voltage and the nodal current as:
|Y++:x| ≈√
3 |I+:x|Vn−mv
(D.3)
212
213
Substitution of (D.2) and (D.3) in (D.1) gives:
Y−−:x−im ≈ −j
(klv:x
1ksc−lvagg :x
+ 1ks:xkm:x
)(√3 |I+:x|Vn−mv
)(D.4)
Appendix E
Application of the Methodology
Given by (3.25) to the Three-bus
MV Test System (Fig. 3.7)
The matrix equation (3.25) which gives the proposed methodology for MV networks
is replicated here by (E.1):
[V lines−:g/mv]n×1 ≈ −[Y ′
++]−1n×n[Y−+]n×n[V+]n×1 (E.1)
where,
Y ′++:xy ≈ Y++:xy + Y−−:x−im for x = y
Y ′++:xy = Y++:xy for x 6= y
214
215
The matrix [V+]1 for the test system is:
[V+] =
1.05∠−4.070
0.98∠−6.740
1.01∠−5.570
×(
12.47√3
)kV (E.2)
Noting that the positive sequence admittance per km of the lines = 1.0098−j2.0630S
and also that the impedance Y++:11 has to be incorporated with the HV-MV trans-
former impedance, the matrix [Y++] is:
[Y++] =
0.3722− j1.3119 −0.1010 + j0.2063 −0.2020 + j0.4126
−0.1010 + j0.2063 0.2020− j0.4126 −0.1010 + j0.2063
−0.2020 + j0.4126 −0.1010 + j0.2063 0.3029− j0.6189
S (E.3)
Noting that the negative-positive sequence coupling admittance per km of the lines
= 0.0258 + j0.1821S, the matrix [Y−+] is:
[Y−+] =
0.0078 + j0.0546 −0.0026− j0.0182 −0.0052− j0.0364
−0.0026− j0.0182 0.0052 + j0.0364 −0.0026− j0.0182
−0.0052− j0.0364 −0.0026− j0.0182 0.0078 + j0.0546
S (E.4)
As busbars 1 and 3 supply passive loads (i.e. km:1 = 0, km:3 = 0), the admittances
Y−−:1−im ≈ 0 and Y−−:3−im ≈ 0.
• Case 1 - busbar 2 supplies passive loads or km:2 = 0. That is, the admittance
Y−−:2−im ≈ 0 implying that [Y ′++] ≈ [Y++]. Then, the substitution of (E.2),
(E.3) and (E.4) in (E.1) gives:
1This is obtained using load flow analysis.
216
[V lines−:g/mv] =
0
47.38
26.64
V (E.5)
Expressing (E.5) as VUFs:
[U linesg/mv] =
0
0.67
0.37
% (E.6)
• Case 2 - busbar 2 supplies motor loads at the LV level or km:2 = 1. Equation
(3.24) gives the admittance Y−−:2−im for klv:2 = 1, ksc−lvagg :2 = 18, ks:2 = 6.7,
I+:2 = 194A and Vn−mv = 12.47kV as:
Y−−:2−im ≈ −j0.1316S (E.7)
Then, the matrix [Y ′++] can be obtained as:
[Y ′++] ≈ [Y++] +
0 0 0
0 −j0.1316 0
0 0 0
S (E.8)
Substitution of (E.2), (E.3) and (E.8) in (E.1) gives:
[V lines−:g/mv] =
5.94
31.46
17.38
V (E.9)
Appendix F
Derivation of (4.7)
Equation (4.5) which gives the negative sequence voltage V t−:g/hv:rec caused by the
local HV line t of the considered radial HV-MV-LV network shown in Fig. 4.2 is
replicated here by (F.1):
V t−:g/hv:rec = −(Z−+:tI+:t + Z++:tI−:t/t) (F.1)
Similar to (A.8) in the case of the radial MV-LV network considered in Fig. 3.2, (F.1)
can be written noting that the local HV line is the element which gives rise to the
negative-positive sequence coupling impedance seen at the sending end busbar of the
HV line as1:
V t−:g/hv:rec = −Z−+:tI+:t
(1− Z++:t
Z−−:send−im
)(F.2)
Due to the inductive nature of the impedances Z++:t and Z−−:send−IM , Z++:t
Z−−:send−im≈∣∣∣ Z++:t
Z−−:send−im
∣∣∣ = µ. Thus, (F.2) can be rewritten as:
V t−:g/hv:rec = −Z−+:tI+:t (1− µ) (F.3)
1The downstream MV line is treated as balanced in this case.
218
219
Equation (4.6) which gives the negative sequence voltage V td−:g/hv:rec caused by the
downstream MV line td is replicated here by (F.4):
V td−:g/hv:rec = −Z++:tI−:td/t (F.4)
The negative sequence current I−:td/t = Y−+:sendV+:send where, according to (A.4),
Y−+:send = −Z−+:send
Z++:sendZ−−:send−im
2. Noting that the impedance Z−+:send arises as a result
of the downstream MV line in this case3, it is not simply equal to the impedance
Z−+:td . Employing (A.4) and (A.5), Z−+:send can be established as4:
Z−+:send ≈ (nhm)2ktdknZ−+:td (F.5)
where, nhm - operating turns ratio of the HV-MV coupling transformer. Then, (F.4)
can be written as:
V td−:g/hv:rec ≈ (nhm)2ktdknZ−+:td
(Z++:t
Z−−:send−im
)(V+:send
Z++:send
)(F.6)
Noting that:
Z++:t
Z−−:send−im= µ, V+:send
Z++:send= I+:t,
(nhm)2Z−+:td
Z−+:t= σ
(F.6) can be rearranged as:
V td−:g/hv:rec ≈ (µktdknσ)Z−+:tI+:t = ζZ−+:tI+:t (F.7)
2New notations which have been used here are as defined in Appendix A while using the subscript‘send’ with reference to the sending end busbar of the local HV line.
3The local HV line is treated as balanced in this case.4A uniform pf across the network is assumed.
220
Summation of (F.3) and (F.7) gives the negative sequence voltage V t+td−:g/hv:rec which
arises as a result of both the HV and MV lines as:
|V t+td−:g/hv:rec| ≈ |Z−+:tI+:t (1− µ− ζ)| (F.8)
Appendix G
Derivation of (4.9)
New notations used in this appendix, unless defined here or in Chapter 4 (Section 4.2),
are as per Appendix C while using the subscripts ‘send’ and ‘rec’ with reference to the
sending and receiving end busbars respectively of the local HV line of the considered
HV-MV-LV radial network shown in Fig. 4.2.
The factor µ is defined as:
µ =
∣∣∣∣ Z++:t
Z−−:send−im
∣∣∣∣ (G.1)
As Z−−:send−im = Z−−:rec−im + Z++:t, and the associated impedances are inductive in
nature, (G.1) can be written as:
µ ≈ 1
1 +∣∣∣Z−−:rec−im
Z++:rec
∣∣∣ ∣∣∣Z++:rec
Z++:t
∣∣∣ (G.2)
The ratio∣∣∣Z++:rec
Z++:t
∣∣∣ can be expressed in terms of the voltage regulation of the line t as:
∣∣∣∣Z++:rec
Z++:t
∣∣∣∣ ≈ 1− V Rt
V Rt
(G.3)
221
222
Noting that Z−−:rec−im = (nmlnhm)2(Z−−:im+Z++:mlr−lv+Z++:td−lv+Z++:hm−lv) when
motor loads are supplied only by the MV line at the busbar LVr, and the inductive
nature of the associated impedances, the ratio∣∣∣Z−−:rec−im
Z++:rec
∣∣∣ can be given as:
∣∣∣∣Z−−:rec−im
Z++:rec
∣∣∣∣ ≈ ( |V+:rec|2
|Z++:rec|
)(|Z−−:im||V+:rec|2
(nmlnhm)2
+|Z++:mlr−lv + Z++:td−lv + Z++:hm−lv|
|V+:rec|2(nmlnhm)2
)(G.4)
The term |V+:rec|2|Z++:rec| represents the total load Srec (in MVA) supplied by the HV busbar
rec. Further, |V+:rec| ≈ nmlnhm|V+:lv |(1−V Rtd
). Substitution of these and rearrangement of (G.4)
gives:
∣∣∣∣Z−−:rec−im
Z++:rec
∣∣∣∣ ≈ Srec(1− V Rtd)2
(|Z−−:im||V+:lv|2
+|Z++:mlr−lv + Z++:td−lv + Z++:hm−lv|
|V+:lv|2
)(G.5)
Noting that:
|Z−−:im||V+:lv |2
=(
1ksrkmr
)(1
Slvragg
)(see (C.5) - (C.6))
|Z++:mlr−lv+Z++:td−lv+Z++:hm−lv ||V+:lv |2
= 1Ssc−lvragg
Srec
Slvragg= 1
klvr
Slvragg
Ssc−lvragg= 1
ksc−lvragg
(G.5) can be given as:
∣∣∣∣Z−−:rec−im
Z++:rec
∣∣∣∣ ≈ (1− V Rtd)2
klvr
(1
ksrkmr
+1
ksc−lvragg
)(G.6)
223
Substitution of (G.3) and (G.6) in (G.2) gives:
µr ≈1
1 +(1−V Rt)(1−V Rtd
)2„
1ksrkmr
+ 1ksc−lvragg
«V Rtklvr
(G.7)
Appendix H
Test Case Description of the Radial
HV-MV-LV System (Fig. 4.2)
• System details - 60Hz, three-wire, HV - 66kV , MV - 12.47kV , LV - 460V
• System loads -
– Shv = 0
– Smvsagg - represents passive loads
– Smvragg = 0 (i.e. ktd = klvr)
– Slvsagg = 0 (i.e. kn = 1 for the case where kmr = 1)
– Slvragg - represents passive loads and induction motors for the cases where
kmr = 0 and kmr = 1 respectively
• HV line -
– Line details: 15km, untransposed
– Tower construction details: 1.44m flat and horizontal,
phase positioning1: ©a © b © c
1This shows the considered arrangement of the phases a, b and c of the horizontal tower.
224
225
– Conductor data:
geometric mean radius = 9mm
resistance = 0.105Ω/km
earth resistivity = 100Ωm
– Relevant impedance/admittance data:
positive sequence impedance (Ω/km) = 0.1050 + j0.4001 (0.4136∠750)
negative-positive sequence coupling impedance (Ω/km) = 0.0302+j0.0174
(0.0349∠300)
positive sequence admittance (Skm) = 0.6265− j2.3517 (2.4337∠−750)
negative-positive sequence coupling admittance (Skm) = 0.1040+ j0.1779
(0.2061∠600)
• HV-MV coupling transformer - aggregated representation of equally loaded
(10MVA per transformer under the considered operating scenario) 12MVA,
66kV/12.47kV transformers with winding resistance and leakage reactance of
1% and 10% respectively [78].
• MV line - aggregated representation of equally loaded (10MVA per line under
the considered operating scenario) 12.47kV, 3.2187km lines of which the details
are given in Appendix B.
• MV-LV coupling transformer - aggregated representation of fully loaded 1MVA,
12.47kV/460V transformers with winding resistance and leakage reactance of
1% and 5% respectively [78].
• Induction motor and passive loads - as described in Appendix B.
226
• Considered operating scenario -
– Positive sequence voltage at the sending end busbar of the HV line =
41.916kV ∠00 (1.1pu)
– Total load supplied by the HV line = 50MVA at 0.9 lagging pf
– Secondary tap setting of the HV-MV coupling transformer = 1.1pu
– Secondary tap setting of the MV-LV coupling transformer = 1.05pu
– Resulting system conditions2:
I+:t ≈ 490A , V Rt ≈ 7% , V Rtd ≈ 9% , |V+:rec| ≈ 39.6kV (1.04pu)
• Number of 12.47kV, 3.2187km lines and the values of the factors ksc−lvragg and
σ corresponding to various klvr are given in Table H.1 below:
Table H.1: Values of ksc−lvragg and σ for various klvr
klvr No. of 12.47kV , ksc−lvragg σ
3.2187km lines
0.0 0 N/A N/A
0.2 1 5.7 5.0 ∠00
0.4 2 5.0 2.5 ∠00
0.6 3 4.5 1.7 ∠00
0.8 4 4.1 1.3 ∠00
1.0 5 3.8 1.0 ∠00
2Which are obtained using load flow analysis.
Appendix I
Y−+:x for an HV Network
According to (A.4), the admittance Y−+:x which arises as a result of line asymmetries
that exist in the downstream system supplied by the busbar x can be written as1:
Y−+:x ≈−Z−+:x
Z++:x Z−−:x−im
≈ j Z−+:x Y++:x |Y−−:x−im| (I.1)
Noting that:
• the impedance Z−+:send given by (F.5) in the case where the untransposed MV
line gives rise to unbalance in the considered radial HV-MV-LV network shown
in Fig. 4.2, is equal to the respective impedance Z−+:rec that is seen at the
receiving end busbar of the HV line, as the local HV line is treated as balanced
in this case, and
• the system representation of any HV busbar x shown in Fig. 4.5 and that of the
receiving end busbar of the radial HV-MV-LV network considered in Fig. 4.2
are similar,
1Note that the impedance Z−−:x−im is inductive.
227
228
the impedance Z−+:x can be expressed as:
Z−+:x ≈ (nhm:x)2ktd:x
kn:xZ−+:td:x(I.2)
where, nhm:x - operating turns ratio of the HV-MV coupling transformer supplied by
the busbar x. Similar to (D.3), the admittance Y++:x can be expressed as:
Y++:x ≈
(√3 |I+:x|Vn−hv
)∠θpf :x (I.3)
Substitution of (I.2) and (I.3) in (I.1) gives:
|Y−+:x| ≈ ktd:xkn:x
(√3 |I+:x|Vn−hv
)|Y−−:x−im Z−+:td:x−hv| (I.4)
θY−+:x ≈ 900 + θZ−+:td:x+ θpf :x (I.5)
Appendix J
Application of the Methodology
Given by (3.22) to the Three-bus
HV Test System (Fig. 4.6)
The matrix equation (3.22) which gives the proposed methodology for HV networks
is replicated here by (J.1):
[V lines−:g/hv]n×1 ≈ −[Y ′
++]−1n×n[Y−+]n×n[V+]n×1 (J.1)
where,
Y ′++:xy ≈ Y++:xy + Y−−:x−im for x = y
Y ′−+:xy ≈ Y−+:xy + Y−+:x for x = y
Y ′++:xy = Y xy
++ for x 6= y
Y ′−+:xy = Y xy
−+ for x 6= y
229
230
The matrices [V+]1, [Y++] and [Y−+] for the three-bus HV test system shown in
Fig. 4.6 are:
[V+] =
1.07∠−8.30
1.01∠−10.70
1.04∠−9.60
×(
66√3
)kV (J.2)
[Y++] =
0.0467− j0.2214 −0.0125 + j0.0470 −0.0313 + j0.1176
−0.0125 + j0.0470 0.0282− j0.1058 −0.0157 + j0.0588
−0.0313 + j0.1176 −0.0157 + j0.0588 0.0470− j0.1764
S (J.3)
[Y−+] =
0.0073 + j0.0124 −0.0021− j0.0036 −0.0052− j0.0089
−0.0021− j0.0036 0.0047 + j0.0080 −0.0026− j0.0044
−0.0052− j0.0089 −0.0026− j0.0044 0.0078 + j0.0133
S (J.4)
As busbars 1 and 3 supply passive loads (i.e. km:1 = 0, km:3 = 0), the admittances
Y−−:x−im ≈ 0 and Y−+:x ≈ 0 for x = 1, 3.
• Case 1 - busbar 2 supplies passive loads or km:2 = 0. That is, the admittances
Y−−:2−im ≈ 0 and Y−+:2 ≈ 0 implying that [Y ′++] ≈ [Y++] and [Y ′
−+] ≈ [Y−+]
respectively. Then, the substitution of (J.2), (J.3) and (J.4) in (J.1) gives:
[V lines−:g/hv] =
0
226.89
119.02
V (J.5)
1This is obtained using load flow analysis.
231
Expressing (J.5) as VUFs:
[U linesg/hv ] =
0
0.59
0.30
% (J.6)
• Case 2 - busbar 2 supplies motor loads at the LV level or km:2 = 1. Equation
(4.13) gives the admittance Y−−:2−im for klvr:2 = 1, ksc−lvragg :2 = 3.6, ksr:2 = 6.7,
I+:2 = 208A, V Rtd:2= 9% and Vn−hv = 66kV as:
Y−−:2−im ≈ −j0.0154S (J.7)
Equation (4.14) gives the admittance Y−+:2 for ktd:2= 1, kn:x = 1, I+:2 = 208A,
Vn−hv = 66kV , Y−−:2−im ≈ −j0.0154S, Z−+:td:x−hv = 1.3∠300Ω, and θpf :x =
−260 as:
Y−+:2 ≈ (−0.0077 + j0.1093)× 10−3S (J.8)
Then, the matrices [Y ′++] and [Y ′
−+] can be obtained as:
[Y ′++] ≈ [Y++] +
0 0 0
0 −j0.0154 0
0 0 0
S (J.9)
[Y ′−+] ≈ [Y−+] +
0 0 0
0 −0.0077 + j0.1093 0
0 0 0
× 10−3S (J.10)
232
Substitution of (J.2), (J.9) and (J.10) in (J.1) gives:
[V lines−:g/hv] =
94.19
73.52
5.51
V (J.11)
Expressing (J.11) as VUFs:
[U linesg/hv ] =
0.23
0.19
0.01
% (J.12)
Appendix K
Data of the IEEE 14-bus Test
System (Fig. 4.9)
Note that the impedance/admittance values given in pu are based on a 100MV A
base.
Table K.1: Voltage controlled bus data
Bus Voltage Minimum Maximum
number magnitude (pu) MVAr capability MVAr capability
2 1.045 40 50
3 1.010 0 40
6 1.070 6 24
8 1.090 6 24
Table K.2: Static capacitor data: susceptances
Bus Susceptance
number (pu)
9 0.19
233
234
Table K.3: Generator and load bus data: three-phase MW and MVAr values
Bus Generation Load
number MW MVAr MW MVAr
1 (reference bus) 0 0 0 0
2 40 0 21.7 12.7
3 0 0 94.2 19.0
4 0 0 47.8 3.9
5 0 0 7.6 1.6
6 0 0 11.2 7.5
7 0 0 0 0
8 0 0 0 0
9 0 0 29.5 16.6
10 0 0 9.0 5.8
11 0 0 3.5 1.8
12 0 0 6.1 1.6
13 0 0 13.5 5.8
14 0 0 14.9 5.0
Table K.4: Transformer data: impedances and secondary tap settings (1st and 2nd
bus numbers refer to the primary and the secondary respectively)
Transformer Impedance Secondary tap
number (pu) setting
4-7 j0.20912 1.022
4-9 j0.55618 1.032
5-6 j.25202 1.073
7-8 j0.17615 1
7-9 j0.11001 1
235
Table K.5: Nodal positive sequence voltages
Bus number Magnitude (pu) Phase angle (deg.)
1 1.0600 0
2 1.0450 −5.00
3 1.0100 −12.60
4 1.0137 −10.19
5 1.0158 −8.64
6 1.0700 −14.67
7 1.0605 −13.55
8 1.0900 −13.55
9 1.0558 −15.21
10 1.0517 −15.45
11 1.0579 −15.23
12 1.0579 −15.61
13 1.0541 −15.76
14 1.0409 −16.61
236
Table K.6: Transmission line data: lengths and impedances
Line Length Positive sequence
(km) impedance (pu)
1-2 6.56 0.0158 + j0.0602
1-5 24.17 0.0583 + j0.2220
2-3 21.43 0.0517 + j0.1968
2-4 19.55 0.0471 + j0.1796
2-5 19.27 0.0464 + j0.1770
3-4 19.34 0.0466 + j0.1777
4-5 4.65 0.0112 + j0.0427
6-11 23.21 0.0560 + j0.2132
6-12 29.89 0.0720 + j0.2745
6-13 15.39 0.0371 + j0.1413
9-10 9.51 0.0229 + j0.0873
9-14 31.46 0.0758 + j0.2890
10-11 22.00 0.0530 + j0.2020
12-13 31.37 0.0756 + j0.2882
13-14 40.83 0.0984 + j0.3750
Appendix L
Derivation of (5.18)
Consider a series of constant impedance loads L1, L2, ..., Ln (balanced: decoupled and
equal positive, negative and zero sequence impedances) is supplied by the LV sys-
tem shown in Fig. 5.2, where the busbars US and S represent MV and LV systems
respectively. For this, (5.5) can be written as:
∣∣∣∣V−:Umv/lv−mv
V−:mv
∣∣∣∣ =1∣∣∣1 + Z++:ml−lv
(1
Z++:L1+ 1
Z++:L2+ ... + 1
Z++:Ln
)∣∣∣ (L.1)
where, Z++:Li- positive sequence impedance of any load Li (i = 1, 2, ..., n). Employing
(5.3), (L.1) can be expressed in terms of the system and load characteristics and the
load composition as:
∣∣∣∣V−:Umv/lv−mv
V−:mv
∣∣∣∣ =1∣∣∣1 + j
kL1
ksc−lv∠θpf :L1 + j
kL2
ksc−lv∠θpf :L2 + ... + j
kLn
ksc−lv∠θpf :Ln
∣∣∣ (L.2)
where,
kLi- ratio between the load Li (in MVA) and the total load (in MVA) supplied by
the LV system
237
238
θpf :Li- power factor angle1 of the load Li
Noting thatkLi
kLj
(ksc−lv)2 1,
kLi
ksc−lv
2, (L.2) can be written, neglectingkLi
kLj
(ksc−lv)2and higher
order terms3, as:
∣∣∣∣V−:Umv/lv−mv
V−:mv
∣∣∣∣ ≈ 1∣∣∣1 + jkL1
ksc−lv∠θpf :L1
∣∣∣ ∣∣∣1 + jkL2
ksc−lv∠θpf :L2
∣∣∣ ... ∣∣∣1 + jkLn
ksc−lv∠θpf :Ln
∣∣∣(L.3)
Alternatively, the negative sequence voltage |V−:Umv/lv−mv| can be written in an ex-
panded form by decomposing the negative sequence current I−:Uus/tf (I−:Uus/tf =
I−:L1 + I−:L2 + ... + I−:Ln) for the above considered case as:
|V−:Umv/lv−mv| =
∣∣∣∣∣V−:mv −n∑
i=1
(Z++:ml−mvI−:Li)
∣∣∣∣∣ (L.4)
where, I−:Li- negative sequence current (referred to MV) in the load Li, which
arises as a result of the MV unbalance. Noting that the influence of the term
Z++:tf−mvI−:Umv/tf on the ratio∣∣∣V−:Umv/lv−mv
V−:mv
∣∣∣ has been replaced by the factor (referred
to as ‘replacement factor’) 1˛1+j 1
ksc−lv∠θpf :lv
˛ for an aggregated constant impedance
load4, the comparison of (L.3) and (L.4) indicates, for the series of loads, that the in-
fluence of each of the Z++:ml−mvI−:Licomponents on
∣∣∣V−:Umv/lv−mv
V−:mv
∣∣∣ has been replaced
by the factor 1˛1+j
kLiksc−lv
∠θpf :Li
˛ which involves an additional factor kLi. This observa-
tion suggests that the impact of a share I−:Liof the total negative sequence current
I−:Uus/tf (or of a share Z++:ml−mvI−:Liof the total Z++:ml−mvI−:Uus/tf ) with a unique
behaviour, which is determined by the load type and its characteristics, on the prop-
agation of the negative sequence voltage from higher voltage to lower voltage systems
1− and + for lagging and leading conditions respectively.2As kLi , kLj < 1 and 10 < ksc−lv < 25.3i.e.
kLikLj
(ksc−lv)3 ,kLi
kLj
(ksc−lv)4 etc.4Refer to (5.5) and (5.6).
239
can be represented in the form of (L.3). Equation (L.3) is a product of a number of
terms where an individual term accounts for an unique I−:Lior a load element Li.
Generalising the above outcome, the influence of a series of load elements L1,
L2,..., Ln5 on the propagation of the negative sequence voltage from higher voltage
to lower voltage systems can be represented as a product of a number of terms in the
form given by (L.3). An individual term of this product, which corresponds to any
load element Li, can be obtained by modifying the replacement factor for its load
type6 where(
1ksc−lv
)is multiplied by the load proportion kLi
. Applying this to a mix
of constant impedance (Z), constant current (I), constant power (PQ) and induction
motor (IM) loads (i.e. Li = z, i, pq, im), the individual terms to form the product
can be obtained as given in Table L.1.
Table L.1: Replacement factors for a mix of various load types
Load type Replacement factor
Z 1˛1+j kz
ksc−lv∠θpf :z
˛
I 1
PQ 1˛1+j
kpqksc−lv
∠θpf :pq
˛β
IM 1„1+ kmks
ksc−lv
«
5Li can represent any type (i.e. constant impedance, constant current, constant power andinduction motor loads) of load element.
6i.e. 1˛1+j 1
ksc−lv∠θpf:lv
˛γ where γ = 1, 0,−2 ∼ −1 for constant impedance, constant current and
constant power loads respectively, and 1“1+ ks
ksc−lv
” for induction motor loads.
Appendix M
Application of the Methodology
Given by (5.37) to the Three-bus
MV Test System (Fig. 5.16)
The matrix equation (5.37) which gives the proposed methodology for evaluating influ-
ence coefficients [ki−x] between any busbar i and other busbars x (x = 1, 2, ..., n x 6= i)
is reproduced here by (M.1):
[ki−x](n−1)×1 ≈∣∣∣[Y ′
++:xz]−1(n−1)×(n−1)[Y++:xi](n−1)×1
∣∣∣ (M.1)
where,
Y ′++:xz ≈ Y++:xz + Y−−:x−im for x = z
Y ′++:xz = Y++:xz for x 6= z
For the purpose of evaluating the influence coefficients k1−2 and k1−3 between
busbar 1 (i.e. i = 2) and busbars 2 and 3 (i.e. x, z = 2, 3) of the three-bus MV test
240
241
system shown in Fig. 5.16, the matrices [Y++:xz] and [Y++:xi] are:
[Y++:xz] =
0.2020− j0.4126 −0.1010 + j0.2063
−0.1010 + j0.2063 0.3029− j0.6189
S (M.2)
[Y++:xi] =
−0.1010 + j0.2063
−0.2020 + j0.4126
S (M.3)
As busbar 3 supplies passive loads (i.e. km:3 = 0), the admittance Y−−:3−im ≈ 0.
• Case 1 - busbar 2 supplies passive loads or km:2 = 0. That is, the admittance
Y−−:2−im ≈ 0 implying that [Y ′++:xz] ≈ [Y++:xz]. Then, the substitution of (M.2)
and (M.3) in (M.1) gives: k1−2
k1−3
=
1
1
(M.4)
• Case 2 - busbar 2 supplies motor loads at the LV level or km:2 = 1. Equation
(5.36) gives the admittance Y−−:2−im for klv:2 = 1, ksc−lvagg :2 = 18, ks:2 = 6.7,
I+:2 = 194A and Vn−mv = 12.47kV as:
Y−−:2−im ≈ −j0.1316S (M.5)
Then, the matrix [Y ′++:xz] can be obtained as:
[Y ′++:xz] ≈ [Y++:xz] +
0 0
0 −j0.1316
S (M.6)
Appendix N
66kV Sub-transmission
Interconnected Study System
(Fig. 7.1) - Additional
Data/Information
N.1 Operating Conditions at the Considered Time Stamp
Note that the impedance values given in pu are based on a 100MV A base.
Table N.1: System details
Nominal voltage 66kV (line-line)
Nominal frequency 60Hz
Connection type three-wire
243
244
Table N.2: Voltage controlled bus data
Busbar Voltage
magnitude (pu)
S1 1.023
S5 0.975
Table N.3: Generator and load bus data: three-phase MW and MVAr values
Busbar Generation Load
MW MVAr MW MVAr
S1 (reference bus) 0 0 0 0
S2 0 0 18.06 0.36
S3 0 0 18.27 7.11
S4 0 0 10.68 5.13
S5 13.50 0 0 0
S61 0 0 0.09 3.99
S7 0 0 33.99 14.52
S8 0 0 5.97 1.89
S9 0 0 1.65 0.241This represent the operation (balanced) of a static Var compensator.
245
Table N.4: Voltage regulator data: impedances and secondary tap settings
Busbar Impedance Secondary tap
(pu) setting
S2 0.0014 + j0.0192 1.058
S7 0.0014 + j0.0192 1.111
Table N.5: Static capacitor data: susceptances
Busbar Susceptance
(pu)
S2 0.0218
S3 0.0597
S4 0.0409
S7 0.1277
S8 0.0130
Table N.6: Generator impedance data
Busbar Sequence impedances (pu)
Zero Positive Negative
S1 0 0 0
S5 j0.5 j0.6 j0.6
246
N.2 Line Data
Notations:
Z++- positive sequence impedance of a line
Z−+ - negative-positive sequence coupling impedance of a line
Table N.7: Lengths and impedances (Z−+ and Z−+) of the sub-transmission lines
Line Length Z−+ Z++
(km) (Ω) (Ω)
A 67.65 0.62∠840 24.09∠730
B 19.16 0.52∠300 8.36∠530
C 17.83 0.25∠3360 6.37∠720
D 71.49 0.74∠1780 24.81∠710
E 19.59 0.27∠650 9.63∠480
F 45.37 1.25∠300 22.81∠450
G 66.29 0.14∠340 32.50∠450
H 56.46 0.03∠470 28.80∠450
I 55.32 1.40∠300 18.98∠720
J 11.40 0.31∠1480 4.01∠720
K 15.57 0.08∠1210 5.47∠730
L 80.65 0.30∠1310 41.00∠450
M 83.20 1.91∠550 29.90∠640
N 21.16 0.45∠230 9.23∠590
N.3 An Explanation on the Influence of the Location of an
Asymmetrical Line of an Interconnected Network on the
Voltage Unbalance Behaviour of the Line
• When a line (e.g. line J of the study system) is located in the downstream part
of its network, it introduces voltage unbalance primarily at the downstream
247
busbars as the negative sequence voltage V t−:rec−tany
at any other upstream bus-
bar (e.g. consider S2 and S6 in the case of line J) has to satisfy the relationship
given by (7.3) where V t−:send−tany
is approximately zero1.
• Alternatively, a line (e.g. line A of the study system) which is located in the
upstream part of its network can introduce voltage unbalance at the upstream
busbars as well as the downstream busbars (e.g. consider the upstream busbar
S2 and the downstream busbar S7 in the case of line J) as the line is in a position
to take part of V t−:send−tany
for all other lines.
• When a line (e.g. line B of the study system) is isolated from the rest of
the network, its influence at other main busbars, compared to that of a line
(e.g. line J of the study system) which is located in the main part of the network,
is somewhat reduced. Because, the negative sequence voltage V t−:rec−tany
at any
other busbar (e.g. consider S2 in the case of line B) has to satisfy the relationship
(7.3) where there can exist several connections (e.g. lines E and A for S2) to
the busbar resulting in several V t−:send−tany
(e.g. V t−:send−E ≈ Z−+:BI+:B, and
V t−:send−A = 0 for S2) out of which at least one or more can be approximately
zero.
N.4 A Demonstration of the Linearity of Negative Sequence
Voltages
Table N.8 gives the negative sequence voltages2 (V t−:S2) introduced by the individual
lines A - N of the study system at the busbar S2. Table N.9 gives a comparison of
1Because the voltage at the bulk supply is balanced, and the term Z++:tanyI−:t/tany
in (7.3) isinconsiderable.
2These, which are obtained using unbalanced load flow analysis, correspond to the selected timestamp.
248
the resultant negative sequence voltage3 (V lines−:S2 ) at S2 obtained using the summation
of the individual vectors given in Table N.8, and unbalance load flow analysis.
Table N.8: Negative sequence voltages V t−:S2 caused by the individual lines A - N
at the busbar S2Line (t) V t
−:S2 (V )
A 58.560∠−127.1760
B 14.691∠−120.6320
C 12.124∠151.7370
D 45.203∠−24.2310
E 4.401∠−139.6360
F 95.222∠−137.7970
G 0.950∠−134.2140
H 0.257∠−119.4290
I 54.234∠176.4530
J 14.402∠−47.7590
K 1.850∠−75.2630
L 1.409∠−68.1720
M 0.361∠−170.8340
N 13.425∠176.6590
Table N.9: Resultant negative sequence voltage V lines−:S2 at the busbar S2
V lines−:S2 (V )
Vector summation 207.735∠−133.9680
Unbalanced load flow analysis 205.703∠−134.3440
3i.e. as a result of the interaction of all lines.
Appendix O
Development of a Method for
Unbalanced Load Flow Analysis
O.1 Introduction
Unbalanced load flow is an essential tool for analysing steady-state unbalanced power
system problems (e.g. voltage unbalance). Due to the need for careful representation
of power system components and the lack of widespread availability of comprehensive
commercial packages, a generaliased unbalanced load flow program which is based on
MATLABR is developed. This appendix gives a detailed description on the program
including the background information (Sections O.2 and O.3), and the representation
of power system components (Section O.4) employed for the formulation of load flow
equations. The key section, Section O.4.5, discusses existing models of three-phase
induction motors that have been used for unbalanced load flow studies, and proposes
two types of models which overcome the limitations associated with the existing
models.
249
250
O.2 Symmetrical Component Versus Phase Coordinate Ref-
erence Frames for Unbalanced Load Flow Analysis
References [1, 2, 3, 4, 5] propose two basic approaches for unbalanced load flow analy-
sis, which are based on the symmetrical component and the phase coordinate reference
frames respectively. Analysis of unbalanced power system problems has been tradi-
tionally based on the symmetrical component reference frame due to the advantages
of the availability of sequence impedances for power system components, and the
decoupled nature of most power system components in the symmetrical component
reference frame [1]. However, the use of the phase coordinate reference frame has been
identified as the best way to represent three-phase power system components, as it
facilitates the maintenance of the initial physical identity of the system with regard
to line parameters and variables such as nodal voltages and line currents [2, 4, 5].
The only drawback of this approach is that the size of the problem is significantly
large compared to that based on the symmetrical component reference frame [3].
Disregarding the computational advantages associated with the symmetrical com-
ponent approach, an unbalanced load flow program that is based on the phase coor-
dinate reference frame is developed for specific applications of the work presented in
this thesis.
O.3 Special Considerations in Developing an Unbalanced Load
Flow Program
Compared to balanced load flow algorithms, a number of additional issues are to be
addressed in developing an unbalanced load flow method:
• First is the question on how to formulate unbalanced load flow equations in
251
a generalised manner so that the incorporation of numerous component con-
nections (e.g. single-phase and dual-phase loads) can be easily achieved. The
concept of specifying load flow constraints for each bus or each phase of a bus,
as used in the traditional balanced or unbalanced load flow methods, cannot
take various component connections into account. In view of the fact that the
power constraints such as the specified power generation and consumption are
properties of system components instead of buses, [4, 5] proposes a new concept
that is referred to as ‘component level power flow constraints’ which allows the
incorporation of numerous component connections.
• Second is the question on how to represent the behaviour of system components
when subjected to unbalanced voltage/current excitations. As an example, a
three-phase induction motor responds differently to the positive and negative se-
quence applied voltages, whereas a typical residential load would exhibit nearly
identical behaviours with both the positive and negative sequence voltages.
The developed unbalanced load flow program incorporates the suggested concept
of the component level power flow constraints, and takes into account the second
requirement in the modelling of system components.
O.4 Representation of System Components
O.4.1 Synchronous Generators
The synchronous generator model used in the developed program is based on [4,
5], which takes the different machine responses for the positive, negative and zero
sequence current injections into account. As illustrated in Fig. O.1 and given by
(O.1), this is a positive sequence voltage source behind the generator admittance
matrix [Yg] with no pre-determined connection form.
252
a2Eg
[Yg]
Eg a
k m b
aEg c
Figure O.1: Synchronous generator model
[Ikm:g] = [Yg]([Vk]− [Vm]− [Eg]) (O.1)
where,
[Vk] = [Va:k Vb:k Vc:k]t, matrix of phase voltages on side k
[Vm] = [Va:m Vb:m Vc:m]t, matrix of phase voltages on side m
[Eg] = [Eg a¯
2Eg a¯Eg]
t, matrix of the generator internal phase voltages (balanced)
[Ikm:g] = [Ikm:g−a Ikm:g−b Ikm:g−c]t, matrix of the generator currents flow from side
k to side m
[Yg] = T [Yg:seq]T−1
[Yg:seq] =
Yg:0 0 0
0 Yg:+ 0
0 0 Yg:−
T =
1 1 1
1 a¯
2 a¯
1 a¯
a¯
2
Yg:0, Yg:+, Yg:− - zero, positive and negative sequence admittances respectively of
the generator
253
superscript t - represents the transpose of a matrix
a¯
= 1∠1200
If the generator operates as a slack machine, the magnitude and the phase angle
of the positive sequence voltage at the generator terminals are constrained:
[T+]([Vk]− [Vm]) = Vg:spec∠θg:spec (O.2)
where,
[T+] = 13[1 a
¯a¯
2]
|Vg:spec|, θg:spec - specified magnitude and phase angle respectively of the positive se-
quence voltage at the generator terminals
If the generator operates as a PV machine1, the three-phase active power and the
magnitude of the positive sequence voltage at the generator terminals are constrained:
Real(− [Ikm:g]
c([Vk]− [Vm]))
= Pg:spec (O.3)
|[T+]([Vk]− [Vm])| = |Vg:spec| (O.4)
where,
Pspec - specified generation of the three-phase active power
superscript c - represents the conjugate of a vector
1This refers to a power and voltage controlled generator.
254
O.4.2 Passive Loads
The exponential load model [6] which takes the voltage dependency of the active
and reactive power into account is employed. As illustrated in Fig. O.2, this is
represented as a branch between two nodes allowing the incorporation of different
load configurations.
P = P0
(|Vk − Vm|
V0
)λp
(O.5)
Q = Q0
(|Vk − Vm|
V0
)λq
(O.6)
where,
P , Q - active power and reactive power respectively consumed by the load
P0, Q0 - P and Q respectively under the rated voltage
V0 - rated voltage
λp, λq - voltage coefficients of active power and reactive power respectively, where
λp = λq = 2 for constant impedance loads , λp = λq = 1 for constant current loads,
λp = λq = 0 for constant power loads, and different values can be chosen to represent
the aggregated effects of various load mixes
The active and reactive power consumed by this load are constrained:
Ickm:l(Vk − Vm) = Pl:spec + jQl:spec (O.7)
where,
Ikm:l - load current flows from side k to side m
Pl:spec, Ql:spec - specified consumption of the active and reactive power respectively of
the load
255
P + jQ
k m
Figure O.2: Load model
O.4.3 Overhead Lines
Overhead lines are modelled as electromagnetically coupled2 impedance matrices in
the phase coordinate reference frame. The phase impedance matrix([Zt:xy](3×3)
)for
a three-wire line3 with the earth return is derived using [8]:
Zt:xy = Rearth+Rconductor+j
(k ln
(De
Dxy
))in Ω/m, when x = y, i.e. self impedance
(O.8)
Zt:xy = Rearth + j
(k ln
(De
Dxy
))in Ω/m, when x 6= y, i.e. mutual impedance
(O.9)
where,
Rearth = 9.869× 10−7f in Ω/m, earth resistance
De = 658.376×√
ro
fin m
k = 2× 10−7 in H/m
Dpq−
conductor geometric mean radius (m), when x = y
geometric mean distance between the conductor x and the conductor y (m),
when x 6= y
Rconductor - ac resistance of the conductor x (Ω/m)
f - operating frequency (Hz)
ro - earth resistivity (Ωm)
x, y - represents the three phases a, b and c
2Capacitive effects are ignored.3Single-circuit.
256
O.4.4 Capacitor Banks
Capacitor banks are represented as constant impedance elements which allow their
reactive power injection to be determined according to the prevailing voltage condi-
tion.
O.4.5 Three-phase Voltage Regulators/Transformers
Voltage regulators/transformers are represented using three single-phase star con-
nected units. Each unit is modelled as an impedance in series with an ideal trans-
former having taps on the secondary. The equivalent circuit is shown in Fig. O.3
where [Avr:x], [Bvr:x] and [Cvr:x] are 3 × 3 diagonal matrices of which the elements
Avr:x, Bvr:x and Cvr:x respectively are given by [7]:
Avr:x = tapx yvr:x (O.10)
Bvr:x = tapx(tapx − 1) yvr:x (O.11)
Cvr:x = (1− tapx) yvr:x (O.12)
i, j - represent the busbars at which the primary and the secondary respectively are
connected
tapxy - tap position, as a ratio of the primary voltage and the secondary voltage, of
any phase x (= a, b, c)
yvr:x - series impedance, referred to the secondary, of the phase x
O.4.6 Three-phase Induction Motors
A number of different models to represent the behaviour of an induction motor in
unbalanced load flow studies has been proposed [4, 5, 11, 13]. A simple model based
on the sequence equivalent circuits is given in [4, 5], where the positive sequence
257
[Avr:x]
[Cvr:x] [Bvr:x]
Busbar i Busbar j
Figure O.3: Equivalent circuit of a voltage regulator/transformer
impedance is considered as an unknown variable which is to be determined so that
the total real and reactive power drawn by the motor are equal to specified values. In
this model, the unknown positive sequence impedance is represented by an unknown
voltage source behind the known negative sequence impedance, and the voltage source
is solved to meet the power flow constraints stated above. The phase domain model
of the above, which is established by transforming the sequence elements into the
phase domain, is illustrated in Fig. O.4 and expressed by (O.13). The considered
power flow constraints are given by (O.14).
a2Eim
[Yim]
Eim
a
k m b
aEim c
Figure O.4: Three-phase induction motor model proposed in [4, 5]
258
[Ikm:im] = [Yim]([Vk]− [Vm]− [Eim]) (O.13)
−[Ikm:im]c([Vk]− [Vm]) = Pim:spec + jQim:spec (O.14)
where,
[Yim] = T [Yim:seq]T−1
[Ysequence] =
Yim:0 0 0
0 Yim:− 0
0 0 Yim:−
Yim:0, Yim:− - zero and negative sequence admittances respectively of the induction mo-
tor
[Ikm:im] = [Ikm:im−a Ikm:im−b Ikm:im−c]t, matrix of the induction motor currents flow
from side k to side m
Eim - unknown voltage
Pim:spec, Qim:spec - specified motor input active and reactive power respectively
Reference [10] reports that although the active power drawn by a three-phase
induction motor is nearly independent of the supply voltage level until the point of
stalling, the reactive power is more sensitive to the voltage level. Fig. O.5, which
is reproduced using [10], illustrates the variation of the real (P) and reactive (Q)
power drawn by a typical induction motor with the voltage level justifying the above
statement4. Hence, the power flow constraints used in the model proposed in [4, 5],
which controls the reactive power drawn by the motor at a specified value, does not
represent the actual motor behaviour resulting erroneous outcomes.
The subject of the modelling of three-phase induction motors for unbalanced load
flow studies has received increased attention recently by the IEEE Distribution System
4The rated mechanical load and balanced supply voltages are assumed.
259
0.20.40.60.81.01.21.41.61.82.02.22.42.6
0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Voltage (pu)
P, Q
(pu)
PQ
Figure O.5: Variation of the real (P) and reactive (Q) power with the supply voltagelevel for a typical three-phase induction motor
Analysis Sub-committee with the view to provide reference test cases for developers
to validate the induction machine models under unbalanced conditions [11, 12, 13].
The model discussed in [11, 12] is also based on the sequence equivalent circuits,
and considers both the positive and negative sequence impedances as functions of
the motor slip which is taken as a new state variable to be determined such that the
input real power is equal to a specified value. The reactive power is then adjusted by
the load flow algorithm according to the existing supply voltage condition. This is a
valid representation only if the specified power corresponds to the actual operating
speed which is determined by the characteristics of both the electromechanical torque
developed by the motor and the mechanical torque demanded by the load. However,
the nominal input real power corresponding to the rated motor speed has been used
as the specified power in [11, 12]. In other words, the rated mechanical load which
lead the motor to operate at the rated speed has been assumed. Although the real
power drawn by an induction motor exhibits a relatively low dependency on supply
voltage conditions until the point of stalling (see Fig. O.5), it is highly sensitive to
260
the changes in mechanical loading conditions. As an example, Figs. O.6 and O.7
illustrate the variation5 of the real (P) and reactive (Q) power and the motor speed
respectively of a 2250hp motor with various characteristics6 of pump systems, when
the motor is excited at the rated voltage (balanced). These demonstrate that the
changes in mechanical loading conditions have a high degree of influence on the real
and reactive power, whereas such changes have only a minor influence on the motor
speed. The rated mechanical loading condition assumed in the model discussed in
[11, 12] may not always arise in practice, and hence this model also has limitations
in generalised applications.
0.0
0.5
1.0
1.5
2.0
2.5
0.3 0.5 0.7 0.9 1.1kp
P (M
W),
Q (M
VA
r)
PQ
kp:rated = 0.93
Loading level (%) 33 54 76 97 118
Figure O.6: Variation of the real (P) and reactive (Q) power with kp (motor loadinglevels corresponding to various kp is also given as a percentage to the rated outputpower) for a 2250hp induction motor
In the remaining part of this section, two types of induction motor models which
overcome the above stated limitations are proposed:
5Obtained using PSCAD/EMTDC.6Various characteristics are represented using kp, where Torque = kp×speed2. kp:rated represents
the rated load.
261
• Impedance type - the operation of an induction motor is represented using
impedance/admittance elements ([Yim]) as shown in Fig. O.8, together with a
power flow constraint.
• PQ type - the operation of an induction motor is represented using active and
reactive power components as shown in Fig. O.97, together with a power flow
constraint.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.3 0.5 0.7 0.9 1.1kp
Mot
or s
peed
(pu)
Loading level (%) 33 54 76 97 118
kp:rated = 0.93
Figure O.7: Variation of the speed with kp (motor loading levels corresponding tovarious kp is also given as a percentage to the rated output power) for a 2250hpinduction motor
Busbar k
c
b
a
[Yim]
Figure O.8: Impedance type induction motor model
7Where, Pim:a, Pim:b, Pim:c - active power drawn by the phases a, b and c respectively of themotor, Qim:a, Qim:b, Qim:c - reactive power drawn by the phases a, b and c respectively of the motor.
262
Busbar k a Pim:a + jQim:a
b Pim:b + jQim:b
c Pim:c + jQim:c
Figure O.9: PQ type induction motor model
Impedance Type Induction Motor Model
This is an extension to the model given in [11, 12], which is based on the sequence
equivalent circuits. The positive and negative sequence equivalent circuits of an in-
duction motor are shown in Fig. O.10. The positive (Zim:+) and negative (Zim:−) se-
quence impedances, as functions of the motor speed, are given by (O.15) and (O.16)
respectively. The zero sequence impedance is irrelevant because of the three-wire
connection associated with motors.
Zim:+ = rst + jxst +1
1jxmag
+ 1rrtωsyn
ωsyn−ωrt+jxrt
(O.15)
Zim:− = rst + jxst +1
1jxmag
+ 1rrtωsyn
ωsyn+ωrt+jxrt
(O.16)
where,
rst, xst - stator resistance and leakage reactance respectively
rrt, xrt - rotor resistance and leakage reactance respectively (referred to the stator
side)
xmag - magnetizing reactance
ωsyn - synchronous speed
ωrt - rotor angular speed
263
rtsyn
synrtrωω
ω−rst jxst jxrt
jxmag
I
rtsyn
synrtrωω
ω+rst jxst jxrt
jxmag
II
Figure O.10: Sequence equivalent circuits of a three-phase induction motor: I - posi-tive sequence, II - negative sequence
264
In the sequence domain, a three-phase induction motor can be represented as:
f1 = V0:kYim:0 − I0:im = 0 (O.17)
f2 = V+:kYim:+ − I+:im = 0 (O.18)
f3 = V−:kYim:− − I−:im = 0 (O.19)
where,
Yim:0 = 0 (noting the three-wire connection)
Yim:+ = 1Zim:+
Yim:− = 1Zim:−
V0:k, V+:k, V−:k - zero, positive and negative sequence voltages respectively at the
motor terminals (busbar k)
I0:im, I+:im, I−:im - zero, positive and negative sequence currents respectively drawn
by the motor
As the load flow method is based on the phase coordinate reference frame, the se-
quence domain equations (O.17) - (O.19) are transformed into the phase domain. Thus:
f1 = f(Iim:a, Iim:b, Iim:c, Va:k, Vb:k, Vc:k)
f2 = f(Iim:a, Iim:b, Iim:c, Va:k, Vb:k, Vc:k, ωrt)
f3 = f(Iim:a, Iim:b, Iim:c, Va:k, Vb:k, Vc:k, ωrt)
where,
Va:k, Vb:k, Vc:k - three phase voltages at the supply terminals of the motor
Ia:im, Ib:im, Ic:im - three phase currents drawn by the motor
265
The motor speed ωrt associated with (O.17) - (O.19) is a new variable to be de-
termined by the load flow algorithm. Thus, an additional equation is required to
complete the load flow formulation, which essentially has to be the power flow con-
straint under which the induction motor operates. The general situation where the
electromechanical torque/power developed by the motor is equal to the mechanical
torque/power demanded by the rotating load is applied as the power flow constraint
instead of constraining the power drawn by the motor at the nominal values as in the
case of the existing models.
The resultant electromechanical torque developed by an induction motor is equal
to the sum of the two torque components corresponding to the positive and negative
sequence voltage/current inputs. The positive (Tim:+) and negative (Tim:−) sequence
torque components can be expressed in terms of the three phase voltages, motor
parameters and motor speed as:
Tim:+ =1
3
∣∣∣∣∣(Va:k + a¯Vb:k + a
¯2Vc:k)
(Zmag
Zmag+Zst
)ZmagZst
Zmag+Zst+ jxrt + rrtωsyn
ωsyn−ωrt
∣∣∣∣∣2( rrt
ωsyn − ωrt
)(O.20)
Tim:− = −1
3
∣∣∣∣∣(Va:k + a¯
2Vb:k + a¯Vc:k)
(Zmag
Zmag+Zst
)ZmagZst
Zmag+Zst+ jxrt + rrtωsyn
ωsyn+ωrt
∣∣∣∣∣2( rrt
ωsyn + ωrt
)(O.21)
where,
Zst = rst + jxst
Zmag = jxmag
Considering pump systems, torque characteristics of the mechanical load (Tload) can
be expressed as:
Tload = kl−1 + kl−2 ω2rt (O.22)
266
where, kl−1, kl−2 are constants for a given pump system. Then, the suggested power
flow constraint can be given by:
f4 = Tim:+ + Tim:− − Tload = 0 (O.23)
where,
f4 = f(Va:k, Vb:k, Vc:k, ωrt)
The functions f1, f2, f3 and f4 which give the complete load flow formulation for
a three-phase induction motor allow the load flow algorithm to determine the motor
speed and three stator phase voltages and currents thus naturally adjusting the input
real and reactive power.
PQ Type Induction Motor Model
The three-phase induction motor model proposed in this section is in line with the
exponential load model that has been used to represent induction motors in balanced
load flow studies [10, 14].
The real and reactive power drawn by each of the three phases of an induction
motor can be expressed using the stator phase voltages and the phase admittance
matrix as:Pim:a + jQim:a
Pim:b + jQim:b
Pim:c + jQim:c
=
Va:k 0 0
0 Vb:k 0
0 0 Vc:k
[Yim]c
Va:k
Vb:k
Vc:k
c
(O.24)
267
where,
[Yim] =
Yim:s Yim:m1 Yim:m2
Yim:m2 Yim:s Yim:m1
Yim:m1 Yim:m2 Yim:s
Yim:s = Yim:0 + Yim:+ + Yim:−, self admittance
Yim:m1 = Yim:0 + a¯Yim:+ + a
¯2Yim:−, a mutual admittance
Yim:m2 = Yim:0 + a¯
2Yim:+ + a¯Yim:−, a mutual admittance
Referring to (O.24), the complex power per phase can be generally written as:
Pim:x + jQim:x = Y cim:s|Vx:k|2 + Y c
im:m1Vx:kVcy:k + Y c
im:m2Vx:kVcz:k (O.25)
where, x, y, z - represent the three phases a, b and c. The real and reactive power
components in (O.25) can be separated as:
Pim:x = Pim:x−xx + Pim:x−xy + Pim:x−xz (O.26)
Qim:x = Qim:x−xx + Qim:x−xy + Qim:x−xz (O.27)
where,
Pim:x−xx = |Yim:s|cos(−θim:s)|Vx:k|2
Pim:x−xy = |Yim:m1|cos(−θim:m1 + θx:k − θy:k)|Vx:k||Vy:k|
Pim:x−xz = |Yim:m2|cos(−θim:m2 + θx:k − θz:k)|Vx:k||Vz:k|
Qim:x−xx = |Yim:s|sin(−θim:s)|Vx:k|2
Qim:x−xy = |Yim:m1|sin(−θim:m1 + θx:k − θy:k)|Vx:k||Vy:k|
Qim:x−xz = |Yim:m2|sin(−θim:m2 + θx:k − θz:k)|Vx:k||Vz:k|
θim:s, θim:m1, θim:m2 - phase angles of Yim:s, Yim:m1 and Yim:m2 respectively
θx:k, θy:k, θz:k - phase angles of Vx:k, Vy:k and Vz:k respectively
268
For the case of balanced voltage angles8 (i.e. θx:k−θy:k = 1200 and θx:k−θz:k = −1200),
the power components Pim:x−xx - Qim:x−xz can be rearranged as:
P′
im:x−xx = P nim:x−xx
( |Yim:s|cos(−θim:s)
|Y nim:s| cos(−θn
im:s)
)( |Vx:k||V n
x:k|
)2
(O.28)
P′
im:x−xy = P nim:x−xy
( |Yim:m1|cos(−θim:m1 + 1200)
|Y nim:m1|cos(−θn
im:m1 + 1200)
)( |Vx:k||Vy:k||V n
x:k||V ny:k|
)(O.29)
P′
im:x−xz = P nim:x−xz
( |Yim:m2|cos(−θim:m2 − 1200)
|Y nim:m2|cos(−θn
im:m2 − 1200)
)( |Vx:k||Vz:k||V n
x:k||V nz:k|
)(O.30)
Q′
im:x−xx = Qnim:x−xx
( |Yim:s|sin(−θim:s)
|Y nim:s|sin(−θn
im:s)
)( |Vx:k||V n
x:k|
)2
(O.31)
Q′
im:x−xy = Qnim:x−xy
( |Yim:m1|sin(−θim:m1 + 1200)
|Y nim:m1|sin(−θn
im:m1 + 1200)
)( |Vx:k||Vy:k||V n
x:k||V ny:k|
)(O.32)
Q′
im:x−xz = Qnim:x−xz
( |Yim:m2|sin(−θim:m2 − 1200)
|Y nim:m2|sin(−θn
im:m2 − 1200)
)( |Vx:k||Vz:k||V n
x:k||V nz:k|
)(O.33)
where,
′- refers to the condition of balanced voltage angles
superscript n - refers to the parameters corresponding to the nominal conditions
(rated voltage and rated motor speed)
P nim:x−xx = |V n
x:k|2|Y nim:s|cos(−θn
im:s)
P nim:x−xy = |V n
x:k||V ny:k||Y n
im:m1|cos(−θnim:m1 + 1200)
P nim:x−xz = |V n
x:k||V nz:k||Y n
im:m2|cos(−θnim:m2 − 1200)
Qnim:x−xx = |V n
x:k|2 |Y nim:s|sin(−θn
im:s)
8In most practical circumstances unbalance in phase voltages arise mainly due to the unbalancein their magnitudes. Thus, this condition of balanced voltage angles can be considered as a generalcase.
269
Qnim:x−xy = |V n
x:k||V ny:k||Y n
im:m1|sin(−θnim:m1 + 1200)
Qnim:x−xz = |V n
x:k||V nz:k||Y n
im:m2|sin(−θnim:m2 − 1200)
Referring to (O.28) - (O.33), the power components P′im:x−xx - Q
′im:x−xz can be written
in a generalised form as:
S′
im:x−ij = Snim:x−ij
( ∣∣∣∣ Y
Y n
∣∣∣∣ )( |Vi:k||Vj:k||V n
i:k||V nj:k|
)(O.34)
where,
S′im:x−ij - represents any P
′im:x−ij or Q
′im:x−ij where i, j = x, y, z
Snim:x−ij - nominal value of S
′im:x−ij (i.e. at the rated voltage and the rated motor
speed)
|Y | - represents the magnitude of various admittance components which are functions
of the motor speed
|Y n| - nominal value of |Y | (i.e. at the rated motor speed)
For the purpose of illustrating the characteristics of the various power and admit-
tance elements in (O.34), a 60Hz, 3hp, 220V induction motor with the parameters
given in Table O.1 [15] is used. Table O.2 gives the power components P nx−xx - Qn
x−xz
under the nominal conditions for the above motor.
The admittance ratio∣∣ YY n
∣∣ can be shown to be approximately equal to an exponen-
tial form of the speed ratio ωrt
ωnrt
as given by (O.35). As examples, Figs. O.11 and O.12 il-
lustrate the variation of |Yim:s| cos(−θim:s)|Y n
im:s| cos(−θnim:s)
of P′x−xx and |Yim:m2| sin(−θim:m2−1200)
|Y ni m:m2| sin(−θn
im:m2−1200)of Q
′x−xz
respectively with ωrt
ωnrt
over a range of ωrt corresponding to 125% - 25% of the rated
motor loading level.
∣∣∣∣ Y
Y n
∣∣∣∣ =(ωrt
ωnrt
)λsp:s−ij
(O.35)
270
Table O.1: Parameters of a 60Hz, 3hp, 220V induction motor
Parameter Value
ωsync 1800rpm
ωnrt 1710rpm
rst 0.435Ω
xst 0.754Ω
rrt 0.816Ω
xrt 0.754Ω
xmag 26.13Ω
Table O.2: Power components P nx−xx - Qn
x−xz for the 3hp, 220V motor
P n (W) Qn (VAr)
x− xx 1834.94 2968.26
x− xy 1923.00 -2483.74
x− xz -2842.67 166.41
0.85
0.90
0.95
1.00
1.05
1.10
0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05
)co
s()
cos(
::
::
ns
imn
sim
sim
sim YY
θθ −−
Approximation: y = x-3.3828
Actual variation
nrt
rt
ωω
Figure O.11: Variation of |Yim:s| cos(−θim:s)|Y n
im:s| cos(−θnim:s)
of P′x−xx with ωrt
ωnrt
for the 3hp, 220V motor
271
where, λsp:s−ij - speed coefficient corresponding to the power component Sim:x−ij,
which is a constant for a given motor. Hence, (O.34) can be written as:
S′
im:x−ij = Snim:x−ij
(ωrt
ωnrt
)λsp:s:ij
(|Vi:k||Vj:k||V n
i:k||V nj:k|
)(O.36)
A speed range corresponding to 125% - 25% of the rated motor loading level is
usually sufficient to cover most practical circumstances, within which the accuracy
of the approximation (O.35) is seen to be acceptable. A higher degree of accuracy
can be achieved if the speed range under consideration can be further narrowed. The
speed coefficients corresponding to the power components Px−xx - Qx−xz for a range
of induction motors [15] are given in Table O.3.
0.7
0.8
0.9
1.0
1.1
1.2
0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05
Approximation: y=x-5.5278
Actual variation
)12
0co
s()
120
cos(
02
:2
:
02
:2
:
−−
−−
nm
imn
mim
mim
mim YY
θθ
nrt
rt
ωω
Figure O.12: Variation of |Yim:m2| sin(−θim:m2−1200)|Y n
im:m2| sin(−θnim:m2−1200)
of Q′x−xz with ωt
ωnrt
for the 3hp, 220Vmotor
The power components Px−xx - Qx−xz which represent the behaviour of an induc-
tion motor under both unbalanced voltage magnitudes and unbalanced voltage angles
272
Table O.3: Speed coefficients corresponding to the power components Px−xx - Qx−xz
for a range of induction motors
Motor specification
λsp:s−ij 3hp, 220V 50hp, 460V 500hp, 2.3kV 2250hp, 2.3kV
λsp:p−xx -3.3828 -5.9151 -40.658 -94.6090
λsp:p−xy -2.9136 -3.0044 -12.089 -19.8360
λsp:p−xz 2.0121 2.7340 13.9710 23.2230
λsp:q−xx -0.0848 -0.4081 -2.9381 -4.2029
λsp:q−xy 0.2243 0.6279 5.6215 8.3997
λsp:q−xz -5.5278 3.8814 10.469 11.9090
can be expressed in terms of the power components P′x−xx - Q
′im:x−xz (which represent
the behaviour under the condition of balanced voltage angles) as:
Pim:x−xx = P′
im:x−xx (O.37)
Pim:x−xy = P′
im:x−xy cos(θxy:k)−Q′
im:x−xy sin(θxy:k) (O.38)
Pim:x−xz = P′
im:x−xz cos(θxz:k)−Q′
im:x−xy sin(θxz:k) (O.39)
Qim:x−xx = Q′
im:x−xx (O.40)
Qim:x−xy = Q′
im:x−xy cos(θxy:k) + P′
im:x−xy sin(θxy:k) (O.41)
Qim:x−xz = Q′
im:x−xz cos(θxz:k) + P′
im:x−xy sin(θxz:k) (O.42)
where,
θxy:k = (θx:k − θy:k)− 1200
θxz:k = (θx:k − θz:k) + 1200
The power flow constraint considered in the impedance type model, which is given
by (O.23), can be rewritten in terms of power (= torque× ωrt) as:
Pim:+ + Pim:− = Pload (O.43)
273
where,
Pim:+, Pim:− - gross electromechanical power developed by the motor corresponding
to the positive and negative sequence voltage/current inputs respectively
Pload - mechanical power demanded by the load
The total gross electromechanical power developed by the motor (i.e. Pim:+ + Pim:−)
can be expressed in terms of the input active power components (given by (O.26)) in
the phase domain and the motor efficiency (ηim) as:
Pim:+ + Pim:− = ηim(Pim:a + Pim:b + Pim:c) (O.44)
Based on the observation that the real power drawn by an induction motor is nearly
independent of the supply voltage condition until the point of stalling (Fig. O.5),
the motor efficiency ηim can be expressed in terms of the motor parameters and the
speed as:
ηim =Pgross at V n
Pin at V n
(O.45)
where,
Pgross at V n - total gross electromechanical power developed by the motor at the rated
voltage V n (balanced)
Pgross at V n = 3
∣∣∣∣∣ V n
(Zmag
Zmag+Zst
)ZmagZst
Zmag+Zst+jxrt+
rrtωsyncωsync−ωrt
∣∣∣∣∣2(
ωrtrrt
ωsync−ωrt
)Pin at V n - motor input real power at the rated voltage V n (balanced)
Pin at V n = 3 real(
|V n|2Zc
im:+
)
The motor efficiency ηim, as a function of the motor speed ωrt, can be shown to
be approximated using (O.46) with a reasonable degree of accuracy. As an exam-
ple, Fig. O.13 illustrates the variation of ηim with ωrt obtained using (O.45) and the
274
approximation (O.46) for the 3hp, 220V motor, where the speed range considered
corresponds to 125% - 25% of the rated motor loading level.
ηim = λeff−1 (ωrt)λeff−2 (O.46)
where, λeff−1, λeff−2 - efficiency coefficients, which are constants for a given mo-
tor. Table O.4 gives these efficiency coefficients for the range of induction motors
considered earlier. Thus, the power flow constraint given by (O.43) can be modified
employing (O.44) and (O.46) as:
λeff−1 (ωrt)λeff−2(Pim:a + Pim:b + Pim:c) = Pload (O.47)
Collecting all related equations together, the complete load flow formulation rep-
resenting the behaviour of a three-phase induction motor can be given by:
f1(Ia:im, Ib:im, Ic:im, Va:k, Vb:k, Vc:k, ωrt) = VaIca:im − (Pim:a + jQim:a) = 0
f2(Ia:im, Ib:im, Ic:im, Va:k, Vb:k, Vc:k, ωrt) = VbIcb:im − (Pim:b + jQim:b) = 0
f3(Ia:im, Ib:im, Ic:im, Va:k, Vb:k, Vc:k, ωrt) = VcIcc:im − (Pim:c + jQim:c) = 0
f4(Ia:im, Ib:im, Ic:im, Va:k, Vb:k, Vc:k, ωrt) =
λeff−1 (ωrt)λeff−2(Pim:a + Pim:b + Pim:c)− Pload = 0
The model requires the nominal power components P nx−xx - Qn
x−xz, speed coefficients
αsp:p−xx - αsp:q−xz, efficiency coefficients λeff−1, λeff−2 and mechanical loading char-
acteristics as inputs. Estimation of these input parameters is an additional task
involved with this model in comparison to the impedance type model. However, this
model makes the load flow formulation and the derivation of the jacobian matrix
much simpler, especially in the case where the condition of balanced voltage angles
can be applied.
275
Table O.4: Efficiency coefficients for a range of induction motors
Motor specification
3hp, 220V 50hp, 460V 500hp, 2.3kV 2250hp, 2.3kV
λeff−1 95.776 99.135 99.565 99.824
λeff−2 0.9019 1.2673 2.2260 2.1890
85
87
89
91
93
95
97
0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00ωrt (pu)
η im
(%)
Approximation: y = 95.776x0.9019
Actual variation
Figure O.13: Variation of ηim with ωrt for the 3hp, 220V motor
276
Comparison of the Proposed Impedance Type and PQ Type
Induction Motor Models
Cases 1 - 3 listed below demonstrate that the PQ type model behaves almost similar
to the impedance type model, although it incorporates some approximations9 and
assumptions10.
• Case 1 - demonstration of the impact of the approximations (O.35) and (O.46)
incorporated with the PQ type model
Fig. O.14 illustrates the variation of the per phase input active and reactive
power with the motor loading level (given as a % to the rated motor load) for
the 3hp, 220V motor excited at the rated voltage (balanced).
• Case 2 - demonstration of the impact of the assumption, incorporated with
the PQ type model, of the motor efficiency is being independent of the supply
voltage condition
Fig. O.15 illustrates the variation of the per phase input active and reactive
power components with the motor loading level (given as a % to the rated
motor load) for the 3hp, 220V motor excited at reduced (|V+:k| = 0.9pu) and
unbalanced (V UF = 3%) voltages11.
• Case 3 - demonstration of the validity of the PQ type model for large motors
Fig. O.16 illustrates the variation of the per phase input active and reactive
power components with the motor loading level (given as a % to the rated motor
load) for a 2250hp, 2.3kV motor [15] excited at reduced (|V+:k| = 0.9pu) and
unbalanced (V UF = 3%) voltages12.
9See (O.35) and (O.46).10The motor efficiency is being independent of the supply voltage condition.11Where,Va:k = 0.84∠00pu, Vb:k = 0.92∠− 1200pu, Vc:k = 0.92∠1200pu.12Where, Va:k = 0.84∠20pu, Vb:k = 0.92∠ − 1200pu, Vc:k = 0.92∠1200pu. Note that the supply
voltage associates also with an angle unbalance.
277
0
200
400
600
800
1000
1200
20 40 60 80 100 120
Loading level ( %)
P im
:a (W
), Q
im:a (V
Ar)
Pim:a - impedance type
Pim:a - PQ type
Qim:a - impedance type
Qim:a - PQ type
Figure O.14: Variation of the per phase input active and reactive power with themotor loading level for the 3hp, 220V motor excited at the rated voltage (balanced)
0
200
400
600
800
1000
1200
1400
20 40 60 80 100 120
Loading level (%)
P im
:x (W
), Q
im:x (V
Ar)
Pim:a - impedance typePim:a - PQ typePim:b - impedance typePim:b - PQ typePim:c - impedance typePim:c - PQ typeQim:a - impedance typeQim:a - PQ typeQim:b - impedance typeQim:b - PQ typeQim:c - impedance typeQim:c - PQ type
Figure O.15: Variation of the per phase input active and reactive power componentswith the motor loading level for the 3hp, 220V motor excited at reduced and unbal-anced voltages
278
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
20 40 60 80 100 120
Loading level (%)
P im
:x (M
W),
Qim
:x (M
VAr)
Pim:a - impedance typePim:a - PQ typePim:b - impedance typePim:b - PQ typePim:c - impedance typePim:c - PQ typeQim:a - impedance typeQim:a - PQ typeQim:b - impedance typeQim:b - PQ typeQim:c - impedance typeQim:c - PQ type
Figure O.16: Variation of the per phase input active and reactive power componentswith the motor loading level for a 2250hp, 2.3kV motor excited at reduced and un-balanced voltages
0
200
400
600
800
1000
0.15 0.4 0.65 0.9 1.15
kp
P im
:a (W
)
Proposed impedance type model
Proposed PQ type model
Mode given in [4,5]
Mode given in [11,12]
kp:rated = 0.87
Figure O.17: Variation of Pim:a with kp for the existing and proposed induction motormodels
279
Comparison of the existing and proposed models
Figs. O.17 and O.18 illustrate the variation of the input real and reactive power
respectively for the phase a with kp of pump systems obtained employing the ex-
isting and proposed models in relation to the 3hp, 220V motor excited at reduced
(|V+:k| = 0.9pu) and unbalanced (V UF = 3%) voltages13. These demonstrate that
the model discussed in [11, 12]14 behaves similar to the proposed models only when
the motor supplies the rated mechanical load15. The model given in [4, 5]16 does not
represent the motor behaviour accurately even when the motor is loaded with the
rated mechanical load17.
150
250
350
450
0.15 0.4 0.65 0.9 1.15
kp
Qim
:a (V
Ar)
Proposed impedance type model
Proposed PQ type model
Model given in [4,5]
Model given in [11,12]
kp:rated = 0.87
Figure O.18: Variation of Qim:a with kp for the existing and proposed induction motormodels
13Where, Va:k = 0.84∠00pu, Vb:k = 0.92∠− 1200pu, Vc:k = 0.92∠1200pu.14This constraints the real power drawn by the motor at the rated value, and allows the reactive
power to be adjusted by the load flow algorithm according to the existing supply voltage condition.15Note the deviation in both Pim:a and Qim:a associated with the model given in [11, 12] compared
to the proposed models when kp 6= kp:rated.16This constraints both the real and reactive power drawn by the motor at the rated values.17Note the deviation in Qa at kp:rated associated with the model given in [4, 5] compared to the
other models.
280
O.4.7 Network Interactions
With the representation of individual system components as described above, the
interaction between them are obtained using the component branch currents by:
[Ykm][Vk] + [Ik] = 0 (O.48)
where,
[Ykm] - matrix of nodal admittances
[Vk] - matrix of nodal voltages
[Ik] - matrix of nodal currents
Ykm - represents any element (a 3× 3 matrix) of the matrix [Ykm], where:
• for k = m, Ykm is equal to the summation of all phase admittance matrices
(3× 3) connected to the busbar k (= m),
• for x 6= y, Yxy is equal to the negative value of the phase admittance matrix
(3× 3) that exists between the busbars k and m
Vk = [Va:k Vb:k Vc:k]t, any element of the matrix [Vk]
Ik = [Ia:k Ib:k Ic:k]t, any element of the matrix [Ik], which is equal to the summation
of the phase current matrices (3 × 1) correspond to individual system components
(e.g. generator and load currents) connected at the busbar k
O.5 Load Flow Solution
The load flow equations are solved employing the well established Newton-Raphson
iterative technique (full version).
281
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